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--- abstract: | We study lattices in free abelian groups of infinite rank that are invariant under the action of the infinite symmetric group, with emphasis on finiteness of their equivariant bases. Our framework provides a new method for proving finiteness results in algebraic statistics. As an illustration, we show that every invariant lattice in $\mathbb{Z}^{(\mathbb{N}\times[c])}$, where $c\in\mathbb{N}$, has a finite equivariant Graver basis. This result generalizes and strengthens several finiteness results about Markov bases in the literature. address: - | Universität Osnabrück\ Osnabrück, Germany - | Universität Osnabrück\ Osnabrück, Germany author: - Dinh Van Le - Tim Römer title: Equivariant lattice bases --- # Introduction There are several notions of bases of a lattice in $\mathbb{Z}^n$ that are related in a hierarchy as follows (see, e.g. [@DSS Chapter 1]): $$\begin{aligned} \text{lattice basis } &\subset \text{ Markov basis } \subset \text{ Gr\"{o}bner basis} \\ &\subset \text{universal Gr\"{o}bner basis } \subset \text{ Graver basis.} \end{aligned}$$ These bases have different origins and applications. Markov bases were introduced in the seminal paper by Diaconis and Sturmfels [@DS] as a key tool for sampling algorithms that are used in Fisher's exact test (see [@Su Chapter 9]). Graver bases [@Gr] play an important role in the theory of integer programming as they allow solving linear and various nonlinear integer programming problems in polynomial time (see [@DHK Part II]). As a powerful algebraic tool, Gröbner bases can be used to computed Markov bases (via the so-called fundamental theorem of Markov bases [@DS Theorem 3.1]) and Graver bases (via the Lawrence lifting technique [@Stu Chapter 7]). The primary goal of this paper is to extend the above notions of bases to lattices in the free abelian group $\mathbb{Z}^{(I)}$, where the basis $I$ need not be finite, and to study them systematically. With motivations from algebraic statistics, we will mainly focus on the case $I=\mathbb{N}^d\times[c]$, where $\mathbb{N}$ denotes the set of positive integers, $c,d\in\mathbb{N}$, and $[c]=\{1,\dots,c\}.$ In this situation, there is a natural action of the infinite symmetric group $\mathop{\mathrm{Sym}}$ on $\mathbb{Z}^{(I)}$ that acts diagonally on the unbounded indices and keeps the bounded index unchanged. We are interested in lattices in $\mathbb{Z}^{(I)}$ that are invariant under this action and their equivariant bases. More specifically, the main problem that we study in this paper is the following: Given a $\mathop{\mathrm{Sym}}$-invariant lattice $L$ in $\mathbb{Z}^{(I)}$, determine whether $L$ has a finite equivariant generating set (respectively, Markov basis, Gröbner basis, universal Gröbner basis, Graver basis). This problem is inspired by finiteness results and questions in algebraic statistics involving chains of increasing lattices of finite rank; see, e.g. [@AT; @AH07; @HS12; @HoS; @SSt]. In fact, any lattice $L$ in $\mathbb{Z}^{(I)}$ corresponds to a chain of increasing lattices $\mathfrak{L}=(L_n)_{n\ge1}$, where $L_n=L\cap\mathbb{Z}^{(I_n)}$ is the truncation of $L$ in the free abelian group $\mathbb{Z}^{(I_n)}$ with finite basis $I_n=[n]^d\times [c]$ for $n\ge1$. One may view $L$ as a *global* lattice and its truncations as *local* ones. It is also useful to view $L=\bigcup_{n\ge1}L_n$ as the *limit* of the chain $\mathfrak{L}$. One of the major questions in algebraic statistics concerns finiteness up to symmetry of Markov bases of certain chains of local lattices $\mathfrak{L}=(L_n)_{n\ge1}$. A well-established method for studying this question combines algebraic and combinatorial tools. First, each lattice $L_n$ is assigned a lattice ideal $\mathfrak{I}_{L_n}$ and the finiteness up to symmetry of Markov bases is translated into the stabilization up to symmetry of the chain of ideals $(\mathfrak{I}_{L_n})_{n\ge1}$ (via the fundamental theorem of Markov bases mentioned above). Next, the stabilization up to symmetry of the chain $(\mathfrak{I}_{L_n})_{n\ge1}$ is characterized by the finite generation up to symmetry of the limit ideal $\bigcup_{n\ge1}\mathfrak{I}_{L_n}$ in an infinite-dimensional polynomial ring. Finally, the latter is usually proved using a combinatorial result called Higman's lemma [@Hig]. See, e.g. [@AH07; @HS12] for details. See also [@HM] for a similar approach that is used to prove finiteness up to symmetry of generating sets of chains of local lattices via ideals in Laurent polynomial rings. It should be noted that the technique of passing to infinite-dimensional limits has also been applied to study chains of varieties (see [@Dr10; @Dr14; @DK14]), cones and monoids (see [@KLR; @LR21]), and convex sets (see [@LC]). This idea is closely related to a general phenomenon known as *representation stability*, which is formalized in [@CF] and has appeared in diverse areas, including linear programming [@GHL] and machine learning [@LD]. In this paper, we provide a more direct approach to proving finiteness up to symmetry of lattice bases. Namely, instead of using ring theoretic tools, we characterize finiteness up to symmetry of various bases (generating sets, Markov bases, Gröbner bases, Graver bases) of a chain of local lattices in terms of the corresponding finiteness property of the global lattice (see Theorems [Theorem 16](#stabilization){reference-type="ref" reference="stabilization"}, [Theorem 21](#stabilization-others){reference-type="ref" reference="stabilization-others"} and ). These *local-global* results allow us to study lattices without translating them to ideals, which has advantages in certain situations. The main result of this paper, which generalizes and strengthens the finiteness results about Markov bases of hierarchical models in [@AT; @HoS; @SSt], illustrates the advantages of our approach. Suppose that $I=\mathbb{N}\times[c]$. Then every $\mathop{\mathrm{Sym}}$-invariant lattice in $\mathbb{Z}^{(I)}$ has a finite equivariant Graver basis. It seems difficult and inconvenient to prove the previous result using the ring theoretic tools described above. Our proof employs a close relationship between the Graver basis of a lattice and the Hilbert basis of a related monoid (see ). It then uses Higman's lemma to show that this monoid has a finite equivariant Hilbert basis (see ). Notably, there is a somewhat simpler proof in the case $c=1$ that does not make use of Higman's lemma (see ). As another contribution to the study of , we show that every $\mathop{\mathrm{Sym}}$-invariant lattice in $\mathbb{Z}^{(I)}$ has a finite equivariant generating set (see ), extending a result of Hillar and Martín del Campo [@HM]. We also provide a global version of the independent set theorem of Hillar and Sullivant [@HS12], which is one of the major results on finiteness of equivariant Markov bases (see ). The paper is organized as follows. In we define various bases for any lattice in $\mathbb{Z}^{(I)}$, discuss their relationship, and prove some general local-global results. deals with $\mathop{\mathrm{Sym}}$-invariant lattices and their equivariant bases, focusing on local-global results, which have stronger forms in this setting. The proof and consequences of are given in . Finally, we discuss finiteness of equivariant generating sets and Markov bases in . # Bases of lattices {#sec-bases} In this section we extend several notions of lattice bases to lattices of infinite rank. After briefly reviewing their algebraic interpretations, we discuss implications among these bases, as well as a close relationship between the Graver basis of a lattice and the Hilbert basis of its nonnegative monoid. The section closes with local-global results for lattice bases and Hilbert bases. To begin, let $I$ be an arbitrary set. Denote by $\mathbb{Z}^{(I)}$ the free abelian group with basis $I$. Write each element ${\boldsymbol u}\in\mathbb{Z}^{(I)}$ as ${\boldsymbol u}=(u_{\boldsymbol i})_{{\boldsymbol i}\in I}$, where $u_{\boldsymbol i}\in\mathbb{Z}$ and all but finitely many of them are zero. We call $$\mathop{\mathrm{supp}}({\boldsymbol u})=\{{\boldsymbol i}\in I\mid u_{\boldsymbol i}\ne0\}$$ the *support* of ${\boldsymbol u}$ and $$\|{\boldsymbol u}\|=\sum_{{\boldsymbol i}\in I}|u_{\boldsymbol i}|$$ the *norm* of ${\boldsymbol u}$. Let $\mathbb{Z}_{\ge0}^{(I)}$ be the submonoid of $\mathbb{Z}^{(I)}$ consisting of elements ${\boldsymbol u}=(u_{\boldsymbol i})_{{\boldsymbol i}\in I}$ with $u_{\boldsymbol i}\ge0$ for all ${\boldsymbol i}\in I$. So $\mathbb{Z}_{\ge0}^{(I)}$ is nothing but the free commutative monoid generated by $I$. Any ${\boldsymbol u}\in\mathbb{Z}^{(I)}$ can be decomposed as ${\boldsymbol u}={\boldsymbol u}^{+}-{\boldsymbol u}^-$ with ${\boldsymbol u}^{+},{\boldsymbol u}^-\in\mathbb{Z}_{\ge0}^{(I)}$ given by $$u_{\boldsymbol i}^+=\max\{u_{\boldsymbol i},0\} \ \text{ and }\ u_{\boldsymbol i}^-=\max\{-u_{\boldsymbol i},0\} \ \text{ for all }\ {\boldsymbol i}\in I.$$ Note that ${\boldsymbol u}^{+}$ and ${\boldsymbol u}^{-}$ have disjoint supports. A *term order* $\prec$ on $\mathbb{Z}_{\ge0}^{(I)}$ is a well-ordering that is additive, i.e. if ${\boldsymbol u}\prec{\boldsymbol v}$, then ${\boldsymbol u}+{\boldsymbol w}\prec{\boldsymbol v}+{\boldsymbol w}$ for all ${\boldsymbol w}\in\mathbb{Z}_{\ge0}^{(I)}.$ **Example 1**. Later, we will be mostly concerned with the case $I=\mathbb{N}^d\times[c]$, where $c,d$ are positive integers and $[c]=\{1,\dots,c\}.$ In this case, familiar term orders on $\mathbb{Z}_{\ge0}^{n}$ can be extended to $\mathbb{Z}_{\ge0}^{(I)}$. Let us consider first the subcase $I=\mathbb{N}^d$ (i.e. $c=1$) for the sake of clarity. Denote by $B=\{{\boldsymbol e}_{\boldsymbol i}\}_{{\boldsymbol i}\in\mathbb{N}^d}$ the standard basis of $\mathbb{Z}^{(I)}$. The definition below works for any well-ordering on $B$. However, for our purposes, we always employ a particular well-ordering $\prec$ on $B$ that is a modification of the lexicographic order on $\mathbb{N}^d$. Let $\max({\boldsymbol i})$ denote the maximal component of ${\boldsymbol i}\in\mathbb{N}^d$. Then for ${\boldsymbol e}_{\boldsymbol i},{\boldsymbol e}_{{\boldsymbol i}'}\in B$ we define ${\boldsymbol e}_{\boldsymbol i}\prec{\boldsymbol e}_{{\boldsymbol i}'}$ if either $\max({\boldsymbol i})< \max({\boldsymbol i}')$, or $\max({\boldsymbol i})=\max({\boldsymbol i}')$ and the leftmost nonzero component of the vector ${\boldsymbol i}-{\boldsymbol i}'$ is negative. For instance, when $d=2$, this order reads $$e_{(1,1)}<e_{(1,2)}<e_{(2,1)}<e_{(2,2)}<e_{(1,3)}<e_{(2,3)}<e_{(3,1)}<\cdots.$$ Now list the terms of each element ${\boldsymbol u}=\sum_{{\boldsymbol i}\in I}u_{\boldsymbol i}{\boldsymbol e}_{\boldsymbol i}\in\mathbb{Z}^{(I)}$ according to the order $\prec$. Let $\mathfrak{f}({\boldsymbol u})$ and $\mathfrak{l}({\boldsymbol u})$ be respectively the coefficients of the first and last nonzero terms of ${\boldsymbol u}$. So if $${\boldsymbol u}=2e_{(1,2)} + e_{(1,3)}+3e_{(3,1)}-4e_{(2,4)} ,$$ then $\mathfrak{f}({\boldsymbol u})=2$ and $\mathfrak{l}({\boldsymbol u})=-4$. Let ${\boldsymbol v},{\boldsymbol w}\in\mathbb{Z}_{\ge0}^{(I)}$. One can check that the following orders are term orders on $\mathbb{Z}_{\ge0}^{(I)}$: 1. *Lexicographic order*: ${\boldsymbol v}\prec_{\mathop{\mathrm{lex}}} {\boldsymbol w}$ if $\mathfrak{l}({\boldsymbol v}-{\boldsymbol w})<0.$ 2. *Degree lexicographic order*: ${\boldsymbol v}\prec_{\mathop{\mathrm{dlex}}} {\boldsymbol w}$ if either $\|{\boldsymbol v}\|<\|{\boldsymbol w}\|$, or $\|{\boldsymbol v}\|=\|{\boldsymbol w}\|$ and $\mathfrak{l}({\boldsymbol v}-{\boldsymbol w})<0.$ 3. *Reverse lexicographic order*: ${\boldsymbol v}\prec_{\mathop{\mathrm{rev}}} {\boldsymbol w}$ if either $\|{\boldsymbol v}\|<\|{\boldsymbol w}\|$, or $\|{\boldsymbol v}\|=\|{\boldsymbol w}\|$ and $\mathfrak{f}({\boldsymbol v}-{\boldsymbol w})>0.$ These orders are defined in the same manner for the general case when $c\ge1$, provided that a well-ordering on the standard basis of $\mathbb{N}^d\times[c]$ is given. Such an order can be obtained by extending the order $\prec$ above as follows: for standard basis elements ${\boldsymbol e}_{{\boldsymbol i},j}$ and ${\boldsymbol e}_{{\boldsymbol i}',j'}$ of $\mathbb{Z}^{(I)}$ with ${\boldsymbol i},{\boldsymbol i}'\in\mathbb{N}^d$ and $j, j'\in [c]$, we define $${\boldsymbol e}_{{\boldsymbol i},j}\prec{\boldsymbol e}_{{\boldsymbol i}',j'} \ \text{ if either }\ j<j', \text{ or } j= j' \text{ and } {\boldsymbol e}_{\boldsymbol i}\prec{\boldsymbol e}_{{\boldsymbol i}'}.$$ In what follows, by a *lattice* $L$ in $\mathbb{Z}^{(I)}$ we mean a subgroup of $\mathbb{Z}^{(I)}$. Thus, in particular, $L$ is itself a free abelian group. For ${\boldsymbol u}\in\mathbb{Z}_{\ge0}^{(I)}$ the *$L$-fiber* of ${\boldsymbol u}$ is defined as $$F_L({\boldsymbol u})=\{{\boldsymbol v}\in\mathbb{Z}_{\ge0}^{(I)}\mid {\boldsymbol u}-{\boldsymbol v}\in L\}.$$ We will simply write this fiber as $F({\boldsymbol u})$ if $L$ is clear from the context. For a subset $\mathcal{B}\subseteq L$ let $G({\boldsymbol u},\mathcal{B})$ be the (undirected) graph with vertices $F({\boldsymbol u})$ and edges $({\boldsymbol v},{\boldsymbol w})$ if ${\boldsymbol v}-{\boldsymbol w}\in\pm \mathcal{B}.$ In general, the fiber $F({\boldsymbol u})$ might be infinite, and hence $G({\boldsymbol u},\mathcal{B})$ need not be a finite graph. Let $\prec$ be a term order on $\mathbb{Z}_{\ge0}^{(I)}$. We denote by $G_\prec({\boldsymbol u},\mathcal{B})$ the directed graph with underlying undirected graph $G({\boldsymbol u},\mathcal{B})$ and an edge $({\boldsymbol v},{\boldsymbol w})$ being directed from ${\boldsymbol v}$ to ${\boldsymbol w}$ if ${\boldsymbol w}\prec{\boldsymbol v}$. Since $\prec$ is a well-ordering, $F({\boldsymbol u})$ always has a unique $\prec$-minimal element for any ${\boldsymbol u}\in\mathbb{Z}_{\ge0}^{(I)}$. We need another partial order on $\mathbb{Z}^{(I)}$ defined as follows: $${\boldsymbol u}\sqsubseteq{\boldsymbol v} \ \text{ if }\ u_{\boldsymbol i}v_{\boldsymbol i}\ge0 \ \text{ and }\ |u_{\boldsymbol i}|\le|v_{\boldsymbol i}| \ \text{ for all }\ {\boldsymbol i}\in I.$$ Obviously, if ${\boldsymbol u}\sqsubseteq{\boldsymbol v}$, then $\|{\boldsymbol u}\|\le\|{\boldsymbol v}\|$. This implies that any non-empty subset of $\mathbb{Z}^{(I)}$ has a minimal element with respect to $\sqsubseteq$. In other words, $\sqsubseteq$ is a well-founded ordering. Note that the restriction of $\sqsubseteq$ on $\mathbb{Z}_{\ge0}^{(I)}$ is a partial order that is coarser than any term order $\prec$ on $\mathbb{Z}_{\ge0}^{(I)}$, i.e. if ${\boldsymbol u}\sqsubseteq{\boldsymbol v}$, then ${\boldsymbol u}\prec{\boldsymbol v}$ for all ${\boldsymbol u},{\boldsymbol v}\in\mathbb{Z}_{\ge0}^{(I)}$. Let us now extend several notions of lattice bases to our setting. In the next definition, the classical notions correspond to the case where $I$ is finite. **Definition 2**. Let $L\subseteq\mathbb{Z}^{(I)}$ be a lattice and $\mathcal{B}\subseteq L$. Let $\prec$ be a term order on $\mathbb{Z}_{\ge0}^{(I)}$. We say that $\mathcal{B}$ is 1. a *generating set* of $L$ if $\mathcal{B}$ generates the abelian group $L$, i.e. $L=\mathbb{Z}\mathcal{B}$; 2. a *Markov basis* of $L$ if the graph $G({\boldsymbol u},\mathcal{B})$ is connected for every ${\boldsymbol u}\in\mathbb{Z}_{\ge0}^{(I)}$; 3. a *Gröbner basis* of $L$ with respect to $\prec$ if for every ${\boldsymbol u}\in\mathbb{Z}_{\ge0}^{(I)}$ there exists a directed path in $G_\prec({\boldsymbol u},\mathcal{B})$ from ${\boldsymbol u}$ to the unique $\prec$-minimal element of $F({\boldsymbol u})$; 4. a *universal Gröbner basis* of $L$ if $\mathcal{B}$ is a Gröbner basis of $L$ with respect to every term order on $\mathbb{Z}_{\ge0}^{(I)}$; 5. the *Graver basis* of $L$ if $\mathcal{B}$ is the set of all $\sqsubseteq$-minimal elements in $L\setminus\{{\boldsymbol 0}\}$. **Remark 3**. The following reformulations of Markov basis and Graver basis are well-known in the classical case, and continue to hold in the general setting of . 1. $\mathcal{B}$ is a Markov basis of $L$ if and only if for any ${\boldsymbol u}\in\mathbb{Z}_{\ge0}^{(I)}$ and any ${\boldsymbol v},{\boldsymbol w}\in F({\boldsymbol u})$ there exist ${\boldsymbol b}_1,\dots,{\boldsymbol b}_s\in\pm \mathcal{B}$ such that $${\boldsymbol v}={\boldsymbol w}+\sum_{i=1}^s{\boldsymbol b}_i \quad\text{and}\quad {\boldsymbol w}+\sum_{i=1}^t{\boldsymbol b}_i\in F({\boldsymbol u}) \ \text{ for all }\ t\in [s].$$ This is immediate from the definition: even if the graph $G({\boldsymbol u},\mathcal{B})$ is infinite, the connectivity of the graph still means that any two vertices are connected by a finite path; see, e.g. [@Di Chapter 8]. 2. The Graver basis of $L$ is precisely the set of all elements ${\boldsymbol u}\in L\setminus\{{\boldsymbol 0}\}$ that do not have a conformal decomposition. Here, a *conformal decomposition* of ${\boldsymbol u}$ is an expression of the form ${\boldsymbol u}={\boldsymbol v}+{\boldsymbol w}$ with ${\boldsymbol v},{\boldsymbol w}\in L\setminus\{{\boldsymbol 0}\}$ and $|u_{\boldsymbol i}|=|v_{\boldsymbol i}|+|w_{\boldsymbol i}|$ for all ${\boldsymbol i}\in I$. Indeed, if ${\boldsymbol u}={\boldsymbol v}+{\boldsymbol w}$ is a conformal decomposition, then ${\boldsymbol v},{\boldsymbol w}\sqsubset{\boldsymbol u}$, and hence ${\boldsymbol u}$ is not $\sqsubseteq$-minimal in $L\setminus\{{\boldsymbol 0}\}$. Conversely, if ${\boldsymbol u}$ is not $\sqsubseteq$-minimal in $L\setminus\{{\boldsymbol 0}\}$, then there exists ${\boldsymbol v}\in L\setminus\{{\boldsymbol 0}\}$ with ${\boldsymbol v}\sqsubset{\boldsymbol u}$, leading to a conformal decomposition ${\boldsymbol u}={\boldsymbol v}+({\boldsymbol u}-{\boldsymbol v})$. As in the classical case, each of the above notion of basis has an algebraic interpretation, which we now briefly discuss. Let $R=K[x_{\boldsymbol i}\mid {\boldsymbol i}\in I]$ be the polynomial ring over a field $K$ with variables indexed by $I$. Also, let $R^\pm=R[x_{\boldsymbol i}^{-1}\mid {\boldsymbol i}\in I]$ be the ring of Laurent polynomials in the variables of $R$. For any ideal $\mathfrak{I}\subseteq R$ let $\mathfrak{I}^\pm=\mathfrak{I}R^\pm$ denote the extension of $\mathfrak{I}$ in $R^\pm$. So each element of $\mathfrak{I}^\pm$ is of the form $fg^{-1}$, where $f\in \mathfrak{I}$ and $g$ is a monomial in $R$. Any element ${\boldsymbol u}=(u_{\boldsymbol i})_{{\boldsymbol i}\in I}\in\mathbb{Z}_{\ge0}^{(I)}$ corresponds to a monomial ${\boldsymbol x}^{{\boldsymbol u}}=\prod_{{\boldsymbol i}\in I}x_{\boldsymbol i}^{u_{\boldsymbol i}}\in R.$ Likewise, any term order $\prec$ on $\mathbb{Z}_{\ge0}^{(I)}$ corresponds to a term order on $R$, which is also denoted by $\prec$. Let $L$ be a lattice in $\mathbb{Z}^{(I)}$. The binomial ideal $$\mathfrak{I}_L=\langle {\boldsymbol x}^{{\boldsymbol u}}-{\boldsymbol x}^{\boldsymbol v} \mid{\boldsymbol u},{\boldsymbol v}\in\mathbb{Z}_{\ge0}^{(I)},{\boldsymbol u}-{\boldsymbol v}\in L\rangle \subseteq R$$ is called the *lattice ideal* associated to $L$. It is easy to see that $$\mathfrak{I}_L=\langle {\boldsymbol x}^{{\boldsymbol u}^+}-{\boldsymbol x}^{{\boldsymbol u}^-} \mid{\boldsymbol u}\in L\rangle.$$ A binomial ${\boldsymbol x}^{{\boldsymbol u}^+}-{\boldsymbol x}^{{\boldsymbol u}^-}\in \mathfrak{I}_L$ is said to be *primitive* if there is no other binomial ${\boldsymbol x}^{{\boldsymbol v}^+}-{\boldsymbol x}^{{\boldsymbol v}^-}$ in $\mathfrak{I}_L$ such that ${\boldsymbol x}^{{\boldsymbol v}^+}$ divides ${\boldsymbol x}^{{\boldsymbol u}^+}$ and ${\boldsymbol x}^{{\boldsymbol v}^-}$ divides ${\boldsymbol x}^{{\boldsymbol u}^-}$. One can check that ${\boldsymbol x}^{{\boldsymbol u}^+}-{\boldsymbol x}^{{\boldsymbol u}^-}$ is primitive if and only if ${\boldsymbol u}$ is a $\sqsubseteq$-minimal element of $L\setminus\{{\boldsymbol 0}\}$. For a subset $\mathcal{B}\subseteq L$ denote $\mathfrak{b}(\mathcal{B})=\{{\boldsymbol x}^{{\boldsymbol b}^+}-{\boldsymbol x}^{{\boldsymbol b}^-} \mid{\boldsymbol b}\in \mathcal{B}\}$ and $$\mathfrak{I}_\mathcal{B}=\langle\mathfrak{b}(\mathcal{B})\rangle =\langle {\boldsymbol x}^{{\boldsymbol b}^+}-{\boldsymbol x}^{{\boldsymbol b}^-} \mid{\boldsymbol b}\in \mathcal{B}\rangle.$$ The next result gives algebraic formulations of lattice bases and can be proved in a similar manner to the classical case; see [@HM Lemma 23], [@Stu Chapter 5] and [@DHK Chapter 11]. The details are left to the reader. **Proposition 4**. Let $L\subseteq\mathbb{Z}^{(I)}$ be a lattice and $\mathcal{B}\subseteq L$. The following statements hold: 1. $\mathcal{B}$ is a generating set of $L$ if and only if $\mathfrak{I}_L^{\pm}=\mathfrak{I}_\mathcal{B}^{\pm}$. 2. $\mathcal{B}$ is a Markov basis of $L$ if and only if $\mathfrak{I}_L=\mathfrak{I}_\mathcal{B}$. 3. $\mathcal{B}$ is a Gröbner basis of $L$ with respect to a term order $\prec$ if and only if $\mathfrak{b}(\mathcal{B})$ is a Gröbner basis of the lattice ideal $\mathfrak{I}_L$ with respect to $\prec$. 4. $\mathcal{B}$ is a universal Gröbner basis of $L$ if and only if $\mathfrak{b}(\mathcal{B})$ is a universal Gröbner basis of $\mathfrak{I}_L$. 5. $\mathcal{B}$ is the Graver basis of $L$ if and only if $\mathfrak{b}(\mathcal{B})$ is the set of primitive binomials of $\mathfrak{I}_L$. *Proof.* (i) If $B$ is a generating set of $L$, then $I_L^{\pm}=I_B^{\pm}$ by [@HM Lemma 23]. Conversely, assume that $I_L^{\pm}=I_B^{\pm}$. Let ${\boldsymbol u}\in L$. Then ${\boldsymbol x}^{{\boldsymbol u}^+}-{\boldsymbol x}^{{\boldsymbol u}^-}\in I_L^{\pm}=I_B^{\pm}$. So there exists ${\boldsymbol a}\in\mathbb{Z}^{\infty}_{\ge0}$ such that ${\boldsymbol x}^{{\boldsymbol a}}({\boldsymbol x}^{{\boldsymbol u}^+}-{\boldsymbol x}^{{\boldsymbol u}^-})\in I_B$. From the proof of (ii) below we deduce that there are ${\boldsymbol b}_1,\dots,{\boldsymbol b}_s\in\pm B$ such that $${\boldsymbol u}=({\boldsymbol a}+{\boldsymbol u}^+)-({\boldsymbol a}+{\boldsymbol u}^-)=\sum_{i=1}^s{\boldsymbol b}_i.$$ Thus, ${\boldsymbol u}$ belongs to the lattice generated by $B$, and we are done. \(ii\) Assume $B$ is a Markov basis of $L$ and ${\boldsymbol u}\in L$. Since ${\boldsymbol u}^+,{\boldsymbol u}^-\in F({\boldsymbol u}^+)$, there exist ${\boldsymbol v}_1,\dots,{\boldsymbol v}_s\in F({\boldsymbol u}^+)$ such that $${\boldsymbol u}^-={\boldsymbol v}_1,\ {\boldsymbol u}^+={\boldsymbol v}_s,\ \text{ and }\ {\boldsymbol v}_{i+1}-{\boldsymbol v}_{i}\in\pm B \ \text{ for }\ i=1,\dots,s-1.$$ This yields ${\boldsymbol x}^{{\boldsymbol v}_{i+1}}-{\boldsymbol x}^{{\boldsymbol v}_i}\in I_B$ for $i=1,\dots,s-1$, and hence, $${\boldsymbol x}^{{\boldsymbol u}^+}-{\boldsymbol x}^{{\boldsymbol u}^-} =\sum_{i=1}^{s-1}({\boldsymbol x}^{{\boldsymbol v}_{i+1}}-{\boldsymbol x}^{{\boldsymbol v}_i})\in I_B.$$ Conversely, assume that $I_L=I_B$. Let ${\boldsymbol v},{\boldsymbol w}\in F({\boldsymbol u})$ for some ${\boldsymbol u}\in\mathbb{Z}^{\infty}$. Then we can write $${\boldsymbol x}^{{\boldsymbol v}}-{\boldsymbol x}^{{\boldsymbol w}}= \sum_{i=1}^{s}\epsilon_i{\boldsymbol x}^{{\boldsymbol a}_i}({\boldsymbol x}^{{\boldsymbol b}_{i}^+}-{\boldsymbol x}^{{\boldsymbol b}_i^-}),$$ where $\epsilon_i\in\{\pm 1\},{\boldsymbol a}_i\in\mathbb{Z}^{\infty}_{\ge0}$ and ${\boldsymbol b}_i={\boldsymbol b}_{i}^+-{\boldsymbol b}_{i}^-\in B$ for $i=1,\dots,s$. We show that ${\boldsymbol v}$ is connected to ${\boldsymbol w}$ in $G({\boldsymbol u},B)$ by induction on $s$. If $s=1$, then ${\boldsymbol v}-{\boldsymbol w}\in\pm B$ and we are done. For $s>1$ we may assume $\epsilon_1=1$ and ${\boldsymbol v}={\boldsymbol a}_1+{\boldsymbol b}_1^+$. Then ${\boldsymbol v}-({\boldsymbol a}_1+{\boldsymbol b}_1^-)={\boldsymbol b}_1\in B$ and $${\boldsymbol x}^{{\boldsymbol a}_1+{\boldsymbol b}_1^-}-{\boldsymbol x}^{{\boldsymbol w}}= \sum_{i=2}^{s}\epsilon_i{\boldsymbol x}^{{\boldsymbol a}_i}({\boldsymbol x}^{{\boldsymbol b}_{i}^+}-{\boldsymbol b}^{{\boldsymbol b}_i^-}).$$ So the desired conclusion follows from the induction hypothesis. \(iii\) If $({\boldsymbol v},{\boldsymbol w})$ is a directed edge in $G_\prec({\boldsymbol u},B)$, then ${\boldsymbol x}^{\boldsymbol w}$ is a reduction of ${\boldsymbol x}^{\boldsymbol v}$ with respect to $\mathfrak{b}(B)$. First assume that $\mathfrak{b}(B)$ is a Gröbner basis of $I_L$. Let ${\boldsymbol u}\in\mathbb{Z}^{\infty}_{\ge0}$. If ${\boldsymbol r}$ is the $\prec$-minimal element of $F({\boldsymbol u})$, then ${\boldsymbol x}^{\boldsymbol r}$ must be the normal form of ${\boldsymbol x}^{\boldsymbol u}$ modulo $\mathfrak{b}(B)$, since ${\boldsymbol x}^{\boldsymbol u}-{\boldsymbol x}^{\boldsymbol r}\in I_L$. Thus, there is a directed path in $G_\prec({\boldsymbol u},B)$ connecting ${\boldsymbol u}$ and ${\boldsymbol r}$. For the converse take ${\boldsymbol u}\in L_\succ$. Since $F({\boldsymbol u}^+)=F({\boldsymbol u}^-)$, both ${\boldsymbol x}^{{\boldsymbol u}^+}$ and ${\boldsymbol x}^{{\boldsymbol u}^-}$ have the normal form ${\boldsymbol x}^{\boldsymbol r}$ modulo $\mathfrak{b}(B)$, where ${\boldsymbol r}$ is the $\prec$-minimal element of $F({\boldsymbol u}^+)$. Hence, ${\boldsymbol x}^{{\boldsymbol u}^+}-{\boldsymbol x}^{{\boldsymbol u}^-}$ has normal form $0$ modulo $\mathfrak{b}(B)$. This implies that ${\boldsymbol x}^{{\boldsymbol u}^+}$ is divisible by $\mathop{\mathrm{in}}_\prec({\boldsymbol x}^{{\boldsymbol b}^+}-{\boldsymbol x}^{{\boldsymbol b}^-})$ for some ${\boldsymbol b}\in B$, and we are done. \(iv\) This is immediate from (iii). \(v\) This follows from the fact that a binomial ${\boldsymbol x}^{{\boldsymbol u}^+}-{\boldsymbol x}^{{\boldsymbol u}^-}$ is primitive precisely when ${\boldsymbol u}$ is a $\sqsubseteq$-minimal element of $L$. ◻ Among different bases of a lattice there is the following relationship that is well-known in the classical case; see, e.g. [@DSS Section 1.3]. **Proposition 5**. For any lattice $L\subseteq\mathbb{Z}^{(I)}$, the following implications hold for its bases: $$\begin{aligned} \text{Graver basis } &\Rightarrow \text{ universal Gr\"{o}bner basis } \Rightarrow \text{ Gr\"{o}bner basis} \\ &\Rightarrow \text{ Markov basis } \Rightarrow \text{ generating set.} \end{aligned}$$ *Proof.* We only prove the first implication, since the last three ones are immediate from (or ). The argument is similar to the proof of [@Stu Lemma 4.6], but we include it here for the convenience of the reader. Let $\mathcal{G}$ be the Graver basis of $L$ and $\prec$ a term order on $\mathbb{Z}_{\ge0}^{(I)}$. Denote by $\mathop{\mathrm{in}}_\prec(\mathfrak{I}_\mathcal{G})$ the initial ideal of $\mathfrak{I}_\mathcal{G}$ with respect to $\prec$. Also, let $L_{\succ}=\{{\boldsymbol u}\in L:{\boldsymbol u}\succ{\boldsymbol 0}\} =\{{\boldsymbol u}\in L:{\boldsymbol u}^+\succ{\boldsymbol u}^-\}.$ It suffices to show that ${\boldsymbol x}^{{\boldsymbol u}^+}\in \mathop{\mathrm{in}}_\prec(\mathfrak{I}_\mathcal{G})$ for every ${\boldsymbol u}\in L_{\succ}$. Suppose on the contrary that ${\boldsymbol x}^{{\boldsymbol u}^+}\not\in \mathop{\mathrm{in}}_\prec(\mathfrak{I}_\mathcal{G})$ for some ${\boldsymbol u}\in L_{\succ}$. Since $\prec$ is a well-ordering, we may choose such ${\boldsymbol u}$ with ${\boldsymbol u}^-$ being $\prec$-minimal. We show that ${\boldsymbol x}^{{\boldsymbol u}^-}\not\in \mathop{\mathrm{in}}_\prec(\mathfrak{I}_\mathcal{G}).$ Indeed, if ${\boldsymbol x}^{{\boldsymbol u}^-}\in \mathop{\mathrm{in}}_\prec(\mathfrak{I}_\mathcal{G})$, then there exists ${\boldsymbol b}\in \mathcal{G}\cap L_{\succ}$ such that ${\boldsymbol x}^{{\boldsymbol b}^+}$ divides ${\boldsymbol x}^{{\boldsymbol u}^-}$, say, ${\boldsymbol x}^{{\boldsymbol u}^-}={\boldsymbol x}^{{\boldsymbol a}}{\boldsymbol x}^{{\boldsymbol b}^+}$ for some ${\boldsymbol a}\in\mathbb{Z}_{\ge0}^{(I)}$. From ${\boldsymbol u},{\boldsymbol b}\succ{\boldsymbol 0}$ it follows that ${\boldsymbol u}+{\boldsymbol b}\succ{\boldsymbol 0}$, i.e. ${\boldsymbol u}+{\boldsymbol b}\in L_{\succ}$. Moreover, since $$({\boldsymbol u}+{\boldsymbol b})^+-({\boldsymbol u}+{\boldsymbol b})^- = {\boldsymbol u}+{\boldsymbol b} ={\boldsymbol u}^+-{\boldsymbol u}^-+ {\boldsymbol b}^+-{\boldsymbol b}^- ={\boldsymbol u}^+-({\boldsymbol a}+{\boldsymbol b}^-)$$ and $({\boldsymbol u}+{\boldsymbol b})^+,({\boldsymbol u}+{\boldsymbol b})^-$ have disjoint supports, we get $({\boldsymbol u}+{\boldsymbol b})^+\sqsubseteq{\boldsymbol u}^+$ and $({\boldsymbol u}+{\boldsymbol b})^-\sqsubseteq {\boldsymbol a}+{\boldsymbol b}^-$. Consequently, ${\boldsymbol x}^{({\boldsymbol u}+{\boldsymbol b})^+}\not\in \mathop{\mathrm{in}}_\prec(\mathfrak{I}_\mathcal{G})$ and $$({\boldsymbol u}+{\boldsymbol b})^-\prec {\boldsymbol a}+{\boldsymbol b}^- \prec {\boldsymbol a}+{\boldsymbol b}^+ = {\boldsymbol u}^-.$$ But this contradicts the minimality of ${\boldsymbol u}^-$. Hence, we must have ${\boldsymbol x}^{{\boldsymbol u}^-}\not\in \mathop{\mathrm{in}}_\prec(\mathfrak{I}_\mathcal{G}).$ Now let ${\boldsymbol v}\in \mathcal{G}$ be a $\sqsubseteq$-minimal element of $L$ with ${\boldsymbol v}\sqsubseteq{\boldsymbol u}$. Then ${\boldsymbol x}^{{\boldsymbol v}^+}$ divides ${\boldsymbol x}^{{\boldsymbol u}^+}$ and ${\boldsymbol x}^{{\boldsymbol v}^-}$ divides ${\boldsymbol x}^{{\boldsymbol u}^-}$. Since either ${\boldsymbol x}^{{\boldsymbol v}^+}$ or ${\boldsymbol x}^{{\boldsymbol v}^-}$ belongs to $\mathop{\mathrm{in}}_\prec(\mathfrak{I}_\mathcal{G})$, so does either ${\boldsymbol x}^{{\boldsymbol u}^+}$ or ${\boldsymbol x}^{{\boldsymbol u}^-}$. This contradiction completes the proof. ◻ Graver bases of lattices are also closely related to Hilbert bases of monoids. To present this relationship, let us first recall some notions. Given a monoid $M\subseteq\mathbb{Z}^{(I)}$, we say that $M$ is *generated* by a subset $A\subseteq M$ if any element of $M$ is a *$\mathbb{Z}_{\geq 0}$-linear combination* of elements of $A$, i.e. $$M=\Big\{\sum_{i=1}^lm_i{\boldsymbol a}_i\mid l\in\mathbb{N},\ {\boldsymbol a}_i\in A,\ m_i\in\mathbb{Z}_{\geq 0}\Big\}.$$ Any minimal generating set of $M$ is called a *Hilbert basis*. It is known that if $M$ is contained in $\mathbb{Z}_{\ge0}^{(I)}$ and finitely generated, then it has a unique Hilbert basis that consists of all *irreducible* elements, i.e. those elements ${\boldsymbol u}\in M\setminus\{{\boldsymbol 0}\}$ with the property that if ${\boldsymbol u}={\boldsymbol v}+{\boldsymbol w}$ for ${\boldsymbol v},{\boldsymbol w}\in M$, then either ${\boldsymbol v}={\boldsymbol 0}$ or ${\boldsymbol w}={\boldsymbol 0}$; see, e.g. [@BG Definition 2.15]. This can be extended to our general setting as follows. **Lemma 6**. Let $M\subseteq \mathbb{Z}_{\ge0}^{(I)}$ be a monoid. Then the following statements hold: 1. $M$ has a unique Hilbert basis $\mathcal{H}$ that consists of all irreducible elements of $M$. 2. If there is a lattice $L\subseteq \mathbb{Z}^{(I)}$ such that $M=L\cap \mathbb{Z}_{\ge0}^{(I)}$, then $\mathcal{H}$ is exactly the set of all $\sqsubseteq$-minimal elements of $M\setminus\{{\boldsymbol 0}\}$. In particular, $\mathcal{H}=\mathcal{G}\cap\mathbb{Z}_{\ge0}^{(I)}$, where $\mathcal{G}$ denotes the Graver basis of $L$. *Proof.* (i) Let $\mathcal{H}$ denote the set of all irreducible elements of $M$. Obviously, $\mathcal{H}$ is contained in any generating set of $M$. So to prove $\mathcal{H}$ is the unique Hilbert basis of $M$, it suffices to show that $\mathcal{H}$ generates $M$, i.e. every element ${\boldsymbol u}\in M$ can be written as a $\mathbb{Z}_{\geq 0}$-linear combination of elements of $\mathcal{H}$. We proceed by induction on $\|{\boldsymbol u}\|$. The case $\|{\boldsymbol u}\|=0$ is trivial. Suppose $\|{\boldsymbol u}\|>0$. If ${\boldsymbol u}$ is irreducible, then we are done. Otherwise, there exist ${\boldsymbol v},{\boldsymbol w}\in M\setminus\{{\boldsymbol 0}\}$ such that ${\boldsymbol u}={\boldsymbol v}+{\boldsymbol w}$. This implies $\|{\boldsymbol w}\|,\|{\boldsymbol v}\|<\|{\boldsymbol u}\|$ since ${\boldsymbol v},{\boldsymbol w}\in\mathbb{Z}_{\ge0}^{(I)}$. So the induction hypothesis yields the desired conclusion. \(ii\) Evidently, every $\sqsubseteq$-minimal element of $M\setminus\{{\boldsymbol 0}\}$ is irreducible. Therefore, it remains to show that any irreducible element ${\boldsymbol u}\in M$ is also $\sqsubseteq$-minimal in $M\setminus\{{\boldsymbol 0}\}$. Suppose not. Then there would exist ${\boldsymbol v}\in M\setminus\{{\boldsymbol 0}\}$ with ${\boldsymbol v}\sqsubset {\boldsymbol u}$. From ${\boldsymbol u}-{\boldsymbol v}\in L\cap\mathbb{Z}_{\ge0}^{(I)}=M$ and ${\boldsymbol u}-{\boldsymbol v}\ne{\boldsymbol 0},$ it follows that ${\boldsymbol u}={\boldsymbol v}+({\boldsymbol u}-{\boldsymbol v})$ is reducible, a contradiction. ◻ We conclude this section with results of the *local-global* type that was developed in [@KLR; @LR21]. From now on we restrict our attention to the case $I=\mathbb{N}^d\times[c]$, where $c$ and $d$ are positive integers. For any $n\in\mathbb{N}$ let $I_n=[n]^d\times[c]\subset I$. Then each group $\mathbb{Z}^{(I_n)}$ can be regarded as a subgroup of $\mathbb{Z}^{(I)}$ via the natural inclusion. More specifically, each element $(u_{\boldsymbol i})_{{\boldsymbol i}\in I_n}\in\mathbb{Z}^{(I_n)}$ is identified with $(u_{\boldsymbol i})_{{\boldsymbol i}\in I}\in\mathbb{Z}^{(I)}$, where $u_{\boldsymbol i}=0$ for all ${\boldsymbol i}\in I\setminus I_n.$ In this manner, we obtain an increasing chain of subgroups $$\mathbb{Z}^{(I_1)}\subset \mathbb{Z}^{(I_2)}\subset \cdots\subset \mathbb{Z}^{(I_n)}\subset \cdots,$$ whose limit is $\mathbb{Z}^{(I)}=\bigcup_{n\ge1}\mathbb{Z}^{(I_n)}.$ By intersecting with this chain, any lattice $L\subseteq\mathbb{Z}^{(I)}$ can be approximated through a chain of sublattices of finite rank $$L_1\subset L_2\subset \cdots\subset L_n\subset \cdots.$$ Here, each $L_n=L\cap\mathbb{Z}^{(I_n)}$ is a truncation of $L$, and conversely, $L=\bigcup_{n\ge1}L_n$ is the limit of the chain $(L_n)_{n\ge1}$. So one can investigate $L$ (viewed as a *global* lattice) using its truncations $L_n$ (viewed as *local* lattices), and vice versa. It should be mentioned that similar ideas have been applied to study ideals, varieties, monoids, and cones in infinite dimensional ambient spaces; see, e.g. [@AH07; @Dr14; @HS12; @IY; @KLR; @LR21]. The next result shows how to construct bases of a global lattice from its local bases. For Graver bases the other way around is also possible. **Proposition 7**. Let $L\subseteq\mathbb{Z}^{(I)}$ be a lattice with truncations $L_n=L\cap\mathbb{Z}^{(I_n)}$ for $n\ge1$. 1. If $\mathcal{B}_n\subseteq L_n$ is a generating set (respectively, Markov basis, universal Gröbner basis, Graver basis) of $L_n$ for all $n\ge 1$, then $\mathcal{B}=\bigcup_{n\ge1}\mathcal{B}_n$ is so of $L$. This also holds for Gröbner bases provided that the term orders considered are compatible: if $\prec$ is a term order on $\mathbb{Z}_{\ge0}^{(I)}$ and each $\mathcal{B}_n$ is a Gröbner basis of $L_n$ with respect to the restriction of $\prec$ on $\mathbb{Z}_{\ge0}^{(I_n)}$, then $\mathcal{B}$ is a Gröbner basis of $L$ with respect to $\prec$. 2. If $\mathcal{G}$ is the Graver basis of $L$, then $\mathcal{G}_n=\mathcal{G}\cap\mathbb{Z}^{(I_n)}$ is the Graver basis of $L_n$ for all $n\ge1$. *Proof.* (i) We prove the statement for Gröbner bases. The other cases can be argued similarly and are left to the reader. For a term order $\prec$ on $\mathbb{Z}_{\ge0}^{(I)}$ we denote its restriction on each $\mathbb{Z}_{\ge0}^{(I_n)}$ also by $\prec$. Assume that $\mathcal{B}_n$ is a Gröbner basis of $L_n$ with respect to $\prec$ for all $n\ge1$. Let ${\boldsymbol u}\in\mathbb{Z}_{\ge0}^{(I)}$ and let ${\boldsymbol v}$ be the unique $\prec$-minimal element of the fiber $F_L({\boldsymbol u})$. Choose $n$ large enough such that ${\boldsymbol u},{\boldsymbol v}\in \mathbb{Z}_{\ge0}^{(I_n)}$. Then ${\boldsymbol v}\in F_{L_n}({\boldsymbol u})$, and moreover, ${\boldsymbol v}$ is the unique $\prec$-minimal element of $F_{L_n}({\boldsymbol u})$ since $F_{L_n}({\boldsymbol u})\subseteq F_L({\boldsymbol u})$. So there exists a directed path in $G_\prec({\boldsymbol u},\mathcal{B}_n)$ from ${\boldsymbol u}$ to ${\boldsymbol v}$. As $\mathcal{B}_n\subseteq \mathcal{B}$, this is also a directed path in $G_\prec({\boldsymbol u},\mathcal{B})$. \(ii\) If ${\boldsymbol u}\sqsubseteq{\boldsymbol v}$ and ${\boldsymbol v}\in \mathbb{Z}^{(I_n)}$, then ${\boldsymbol u}\in \mathbb{Z}^{(I_n)}$. So if $\mathcal{G}$ is the set of all $\sqsubseteq$-minimal elements in $L\setminus\{{\boldsymbol 0}\}$, then each $\mathcal{G}_n=\mathcal{G}\cap L_n$ is the set of all $\sqsubseteq$-minimal elements in $L_n\setminus\{{\boldsymbol 0}\}$. ◻ One might ask whether (ii) also holds for other bases of a lattice. The following example shows that this is not the case for generating sets and Markov bases. In the equivariant setting, however, these bases behave better (see ) **Example 8**. Consider the case $I=\mathbb{N}$ (i.e. $c=d=1$). Then as usual, we write $\mathbb{Z}^n$ instead of $\mathbb{Z}^{(I_n)}$ for $n\ge1$. Denote by $\{{\boldsymbol e}_n\}_{n\in\mathbb{N}}$ the standard basis of $\mathbb{Z}^{(\mathbb{N})}$. Let $$\mathcal{B}_1=\emptyset \quad\text{and}\quad \mathcal{B}_{n}=\mathcal{B}_{n-1}\cup\{{\boldsymbol e}_{n-1}-2{\boldsymbol e}_{n},2{\boldsymbol e}_{n}\} \ \text{ for }\ n\ge 2.$$ Then $\mathcal{B}=\bigcup_{n\ge 1}\mathcal{B}_n$ generates $\mathbb{Z}^{(\mathbb{N})}$ since $${\boldsymbol e}_n=({\boldsymbol e}_n-2{\boldsymbol e}_{n+1})+2{\boldsymbol e}_{n+1} \ \text{ for }\ n\ge 1.$$ Moreover, $\mathcal{B}$ is also a Markov basis of $\mathbb{Z}^{(\mathbb{N})}$ because $$\mathfrak{I}_\mathcal{B}=\langle x_{n-1}-x_{n}^2,\ x_{n}^2-1\mid n\ge 2\rangle = \langle x_{n}-1\mid n\ge 1\rangle = \mathfrak{I}_{\mathbb{Z}^{(\mathbb{N})}}.$$ On the other hand, it is clear that $\mathcal{B}_n=\mathcal{B}\cap\mathbb{Z}^n$ and ${\boldsymbol e}_n$ does not belong to the lattice generated by $\mathcal{B}_n$ for all $n\ge 1$. Hence, $\mathcal{B}_n$ is not a generating set of $\mathbb{Z}^n=\mathbb{Z}^{(\mathbb{N})}\cap \mathbb{Z}^{n}$ for all $n\ge1$. The next result shows that (ii) can be extended to Gröbner bases with respect to the lexicographic order $\prec_{\mathop{\mathrm{lex}}}$ defined in . It would be interesting to know whether this also holds for other term orders. **Proposition 9**. Let $L\subseteq\mathbb{Z}^{(I)}$ be a lattice with truncations $L_n=L\cap\mathbb{Z}^{(I_n)}$ for $n\ge1$ and let $\mathcal{B}\subseteq L$. Then the following statements are equivalent: 1. $\mathcal{B}$ is a Gröbner basis of $L$ with respect to $\prec_{\mathop{\mathrm{lex}}}$; 2. $\mathcal{B}_n=\mathcal{B}\cap\mathbb{Z}^{(I_n)}$ is a Gröbner basis of $L_n$ with respect to $\prec_{\mathop{\mathrm{lex}}}$ for all $n\ge1$. *Proof.* By , it suffices to prove (i) $\Rightarrow$ (ii). For ${\boldsymbol u}\in\mathbb{Z}_{\ge0}^{(I_n)}$ let ${\boldsymbol v}$ be the unique $\prec_{\mathop{\mathrm{lex}}}$-minimal element of $F_L({\boldsymbol u})$. Then there is a directed path $P$ in $G_{\prec_{\mathop{\mathrm{lex}}}}({\boldsymbol u},\mathcal{B})$ from ${\boldsymbol u}$ to ${\boldsymbol v}$. For any vertex ${\boldsymbol w}$ of $P$ it holds that ${\boldsymbol w}\prec_{\mathop{\mathrm{lex}}} {\boldsymbol u}$, which yields ${\boldsymbol w}\in\mathbb{Z}^{(I_n)}$ by the definition of $\prec_{\mathop{\mathrm{lex}}}$. Thus, ${\boldsymbol v}$ is the unique $\prec_{\mathop{\mathrm{lex}}}$-minimal element of $F_{L_n}({\boldsymbol u})$ and $P$ is a path in $G_{\prec_{\mathop{\mathrm{lex}}}}({\boldsymbol u},\mathcal{B}_n)$. Hence, $\mathcal{B}_n$ is a Gröbner basis of $L_n$ with respect to $\prec_{\mathop{\mathrm{lex}}}$. ◻ Finally, we record the following local-global result for Hilbert bases of monoids. **Proposition 10**. Consider a monoid $M\subseteq \mathbb{Z}_{\ge0}^{(I)}$ and its truncated submonoids $M_n=M\cap \mathbb{Z}^{(I_n)}$ for $n\ge1$. Let $\mathcal{H}$ be a subset of $M$ and denote $\mathcal{H}_n=\mathcal{H}\cap M_n$ for $n\ge1$. Then the following statements are equivalent: 1. $\mathcal{H}$ is the Hilbert basis $M$; 2. $\mathcal{H}_n$ is the Hilbert basis $M_n$ for all $n\ge1$. *Proof.* For any $n\ge1$, it is clear that ${\boldsymbol u}$ is an irreducible of $M_n$ if and only if ${\boldsymbol u}\in\mathbb{Z}^{(I_n)}$ and ${\boldsymbol u}$ is an irreducible of $M$. Therefore, the result follows easily from (i). ◻ # Equivariant lattice bases {#sec-equi-bases} We now restrict our attention to the main objects of study in this work, namely, lattices in $\mathbb{Z}^{(I)}$ that are invariant under the infinite symmetric group and their equivariant bases. For those lattices and bases, the local-global results presented in the previous section can be substantially improved. Let us begin with symmetric groups and their actions. ## Symmetric groups {#sec:Sym} For $n\in \mathbb{N}$ let $\mathop{\mathrm{Sym}}(n)$ be the symmetric group on $[n]$. Identifying $\mathop{\mathrm{Sym}}(n)$ with the stabilizer subgroup of $n+1$ in $\mathop{\mathrm{Sym}}(n+1)$, we obtain an increasing chain of finite symmetric groups $$\mathop{\mathrm{Sym}}(1)\subset \mathop{\mathrm{Sym}}(2)\subset\cdots\subset \mathop{\mathrm{Sym}}(n)\subset\cdots.$$ The limit of this chain is the infinite symmetric group $$\mathop{\mathrm{Sym}}(\infty)=\bigcup_{n\geq 1} \mathop{\mathrm{Sym}}(n)$$ that consists of all finite permutations of $\mathbb{N}$, i.e. permutations that fix all but finitely many elements of $\mathbb{N}$. In what follows, for brevity, we will write $\mathop{\mathrm{Sym}}(\infty)$ simply as $\mathop{\mathrm{Sym}}$. ## Sym action {#sec:Sym-action} As before, let $I=\mathbb{N}^d\times[c]$ and $I_n=[n]^d\times[c]$ for $n\in\mathbb{N}$, where $c$ and $d$ are given positive integers. There is a natural action of $\mathop{\mathrm{Sym}}$ on $\mathbb{Z}^{(I)}$, which we now describe. For the sake of clarity, let us first consider the case $c=1$, i.e. $I=\mathbb{N}^d$. Let $\{{\boldsymbol e}_{\boldsymbol i}\}_{{\boldsymbol i}\in\mathbb{N}^d}$ denote the standard basis of $\mathbb{Z}^{(I)}$. For any ${\boldsymbol i}=(i_1,\dots,i_d)\in\mathbb{N}^d$ and $\sigma\in\mathop{\mathrm{Sym}}$ we define $$\sigma({\boldsymbol i})=(\sigma(i_1),\dots,\sigma(i_d)).$$ Now the action of $\mathop{\mathrm{Sym}}$ on $\mathbb{Z}^{(I)}$ is the linear extension of the following action on the basis elements: $$\sigma({\boldsymbol e}_{\boldsymbol i})={\boldsymbol e}_{\sigma({\boldsymbol i})} \ \text{ for any } \sigma\in\mathop{\mathrm{Sym}} \text{ and } {\boldsymbol i}\in\mathbb{N}^d.$$ So if ${\boldsymbol u}=(u_{\boldsymbol i})_{{\boldsymbol i}\in\mathbb{N}^d}=\sum_{{\boldsymbol i}\in\mathbb{N}^d}u_{\boldsymbol i}{\boldsymbol e}_{\boldsymbol i}\in\mathbb{Z}^{(I)}$ and $\sigma\in\mathop{\mathrm{Sym}}$, then $$\sigma({\boldsymbol u})=\sigma\Big(\sum_{{\boldsymbol i}\in\mathbb{N}^d}u_{\boldsymbol i}{\boldsymbol e}_{\boldsymbol i}\Big) =\sum_{{\boldsymbol i}\in\mathbb{N}^d}u_{\boldsymbol i}\sigma({\boldsymbol e}_{\boldsymbol i}) =\sum_{{\boldsymbol i}\in\mathbb{N}^d}u_{\boldsymbol i}{\boldsymbol e}_{\sigma({\boldsymbol i})}.$$ When $c>1$, $\mathop{\mathrm{Sym}}$ acts on $\mathbb{Z}^{(I)}$ by permuting the *unbounded indices* while keeping the *bounded index* unchanged. More precisely, if ${\boldsymbol e}_{{\boldsymbol i},j}$ is a standard basis element of $\mathbb{Z}^{(I)}$ with ${\boldsymbol i}\in\mathbb{N}^d$ and $j\in [c]$, then the action of $\mathop{\mathrm{Sym}}$ on $\mathbb{Z}^{(I)}$ is defined via $$\sigma({\boldsymbol e}_{{\boldsymbol i},j})={\boldsymbol e}_{\sigma({\boldsymbol i}),j} \ \text{ for any } \sigma\in\mathop{\mathrm{Sym}}.$$ Observe that this action induces an action of $\mathop{\mathrm{Sym}}(n)$ on $\mathbb{Z}^{(I_n)}$ for $n\ge1$. A subset $A\subseteq\mathbb{Z}^{(I)}$ is called *$\mathop{\mathrm{Sym}}$-invariant* if $A$ is stable under the action of $\mathop{\mathrm{Sym}}$, i.e. $$\mathop{\mathrm{Sym}}(A)\coloneqq\{\sigma({\boldsymbol u})\mid \sigma\in \mathop{\mathrm{Sym}},\ {\boldsymbol u}\in A\} \subseteq A.$$ Similarly, for any $n\ge1$ a subset $A_n\subseteq\mathbb{Z}^{(I_n)}$ is *$\mathop{\mathrm{Sym}}(n)$-invariant* if $\mathop{\mathrm{Sym}}(n)(A_n)\subseteq A_n$. Note that if $A\subseteq\mathbb{Z}^{(I)}$ is $\mathop{\mathrm{Sym}}$-invariant, then each truncation $A_n=A\cap \mathbb{Z}^{(I_n)}$ is $\mathop{\mathrm{Sym}}(n)$-invariant since $$\mathop{\mathrm{Sym}}(n)(A_n) = \mathop{\mathrm{Sym}}(n)(A\cap \mathbb{Z}^{(I_n)}) \subseteq \mathop{\mathrm{Sym}}(n)(A)\cap \mathop{\mathrm{Sym}}(n)(\mathbb{Z}^{(I_n)}) \subseteq A\cap \mathbb{Z}^{(I_n)} = A_n.$$ ## Equivariant bases {#sec:Equivariant-bases} With motivations from algebraic statistics, one is interested in lattices in $\mathbb{Z}^{(I)}$ or $\mathbb{Z}^{(I_n)}$ (with $n\ge1$) that are $\mathop{\mathrm{Sym}}$-invariant or $\mathop{\mathrm{Sym}}(n)$-invariant, respectively; see, e.g. [@AH07; @HM; @HS12; @KKL]. Such lattices require appropriate notions of bases that reflect the actions of symmetric groups. In [@KKL], equivariant generating sets and equivariant Markov bases for those lattices have been defined. The next definition extends these notions to other bases. **Definition 11**. Let $L\subseteq\mathbb{Z}^{(I)}$ (respectively, $L\subseteq\mathbb{Z}^{(I_n)}$ with $n\ge1$) be a $\mathop{\mathrm{Sym}}$- (respectively, $\mathop{\mathrm{Sym}}(n)$-)invariant lattice and let $\mathcal{B}\subseteq L$. Then $\mathcal{B}$ is called an *equivariant generating set* (respectively, *equivariant Markov basis*, *equivariant Gröbner basis*, *equivariant universal Gröbner basis*, *equivariant Graver basis*) of $L$ if $\mathop{\mathrm{Sym}}(\mathcal{B})$ (respectively, $\mathop{\mathrm{Sym}}(n)(\mathcal{B})$) is a generating set (respectively, Markov basis, Gröbner basis, universal Gröbner basis, Graver basis) of $L$. **Remark 12**. As pointed out in [@KKL Remark 1.1], there is no well-defined notion of an equivariant lattice basis. The reason is that the independence of basis elements is generally not preserved when taking orbits. For instance, among the elements in the $\mathop{\mathrm{Sym}}(2)$-orbit of $(1,-1)\in\mathbb{Z}^2$ there is the following nontrivial linear relation: $$(1,-1)+(-1,1)=(0,0).$$ Similarly to , we define: **Definition 13**. Let $M\subseteq\mathbb{Z}^{(I)}$ (respectively, $M\subseteq\mathbb{Z}^{(I_n)}$ with $n\ge1$) be a $\mathop{\mathrm{Sym}}$- (respectively, $\mathop{\mathrm{Sym}}(n)$-)invariant monoid and let $\mathcal{H}\subseteq M$. Then $\mathcal{H}$ is called an *equivariant Hilbert basis* of $M$ if $\mathop{\mathrm{Sym}}(\mathcal{H})$ (respectively, $\mathop{\mathrm{Sym}}(n)(\mathcal{H})$) is a Hilbert basis of $M$. **Example 14**. Consider the case $I=\mathbb{N}^2\times[c]$ with $r\ge1$ (i.e. $d=2$). Let ${\boldsymbol e}_{i_1,i_2,j}$ with $i_1,i_2\in\mathbb{N}$, $j\in[c]$ denote the standard basis elements of $\mathbb{Z}^{(I)}$. Then $\mathcal{G}=\{\pm {\boldsymbol e}_{i_1,i_2,j}\mid i_1,i_2\in\mathbb{N}, j\in[c]\}$ is the Graver basis of $\mathbb{Z}^{(I)}$ and $\mathcal{H}=\{{\boldsymbol e}_{i_1,i_2,j}\mid i_1,i_2\in\mathbb{N}, j\in[c]\}$ is the Hilbert basis of $\mathbb{Z}_{\ge0}^{(I)}$. It follows that $\mathcal{G}_1=\{\pm {\boldsymbol e}_{1,1,j},\ \pm {\boldsymbol e}_{1,2,j}\mid j\in[c]\}$ is an equivariant Graver basis of $\mathbb{Z}^{(I)}$ and $\mathcal{H}_1=\{{\boldsymbol e}_{1,1,j},\ {\boldsymbol e}_{1,2,j}\mid j\in[c]\}$ is an equivariant Hilbert basis of $\mathbb{Z}_{\ge0}^{(I)}$, since $\mathcal{G}=\mathop{\mathrm{Sym}}(\mathcal{G}_1)$ and $\mathcal{H}=\mathop{\mathrm{Sym}}(\mathcal{H}_1)$. Note that $\mathcal{G}_2=\mathcal{G}_1\setminus\{{\boldsymbol u}\}\cup\{\sigma({\boldsymbol u})\}$ and $\mathcal{H}_2=\mathcal{H}_1\setminus\{{\boldsymbol v}\}\cup\{\sigma({\boldsymbol v})\}$ for any ${\boldsymbol u}\in\mathcal{G}_1,{\boldsymbol v}\in\mathcal{H}_1$ and $\sigma\in\mathop{\mathrm{Sym}}$ are also an equivariant Graver basis of $\mathbb{Z}^{(I)}$ and an equivariant Hilbert basis of $\mathbb{Z}_{\ge0}^{(I)}$, respectively. So equivariant Graver bases and equivariant Hilbert bases are not uniquely determined. Concerning equivariant lattice bases, the following problem is of central importance. **Problem 15**. For a $\mathop{\mathrm{Sym}}$-invariant lattice $L\subseteq\mathbb{Z}^{(I)}$, decide whether $L$ has a finite equivariant generating set (respectively, Markov basis, Gröbner basis, universal Gröbner basis, Graver basis). The remaining part of the paper is devoted to studying this problem. First, let us establish connections between local and global equivariant lattice bases. As seen in the previous section, any lattice in $\mathbb{Z}^{(I)}$ is the limit of the chain of its truncations and can therefore be examined using this chain. In what follows, it is convenient to allow more general chains that do not necessarily consist of truncations of a global lattice. By a *chain of lattices* we mean an arbitrary chain $\mathfrak{L}=(L_n)_{n\ge1}$ with $L_n$ a lattice in $\mathbb{Z}^{(I_n)}$ for all $n\ge1$. This chain is called *$\mathop{\mathrm{Sym}}$-invariant* if the following conditions are satisfied: 1. $\mathfrak{L}$ is an increasing chain, i.e. $L_m\subseteq L_n$ for all $1\le m\le n$; 2. $L_n$ is $\mathop{\mathrm{Sym}}(n)$-invariant for all $n\ge1.$ It is easy to verify that for any $\mathop{\mathrm{Sym}}$-invariant chain $\mathfrak{L}=(L_n)_{n\ge1}$, its limit $L=\bigcup_{n\ge1}L_n$ is a $\mathop{\mathrm{Sym}}$-invariant lattice in $\mathbb{Z}^{(I)}$. Conversely, if $L\subseteq\mathbb{Z}^{(I)}$ is a $\mathop{\mathrm{Sym}}$-invariant lattice, then its truncations $L_n=L\cap\mathbb{Z}^{(I_n)}$ form a $\mathop{\mathrm{Sym}}$-invariant chain that is also called the *saturated* chain of $L$. Note that among $\mathop{\mathrm{Sym}}$-invariant chains having the same limit, the saturated chain is the largest one. Given a $\mathop{\mathrm{Sym}}$-invariant chain of lattices $\mathfrak{L}=(L_n)_{n\ge1}$, it holds that $$\mathop{\mathrm{Sym}}(n)(L_m)\subseteq \mathop{\mathrm{Sym}}(n)(L_n)\subseteq L_n,$$ hence $L_n$ contains the lattice generated by $\mathop{\mathrm{Sym}}(n)(L_m)$ for all $n\ge m\ge 1$. We say that the chain $\mathfrak{L}$ *stabilizes* if this containment becomes an equality when $n\ge m\gg0$, i.e. there exists $p\in\mathbb{N}$ such that $$L_n=\mathbb{Z}\mathop{\mathrm{Sym}}(n)(L_m) \ \text{ for all } n\ge m\ge p.$$ This property is interesting because it characterizes the equivariant finite generation of the global lattice, as shown in the next result; see [@KLR; @LR21] for related results on cones, monoids, and ideals. For ${\boldsymbol u}\in\mathbb{Z}^{(I)}$ we call the cardinality of $\mathop{\mathrm{supp}}({\boldsymbol u})$ the *support size* of ${\boldsymbol u}$ and denote it by $|\mathop{\mathrm{supp}}({\boldsymbol u})|$. **Theorem 16**. Let $\mathfrak{L}=(L_n)_{n\ge1}$ be a $\mathop{\mathrm{Sym}}$-invariant chain of lattices with limit $L$. Then the following are equivalent: 1. $\mathfrak{L}$ stabilizes; 2. There exist $p\in\mathbb{N}$ and an equivariant generating set $\mathcal{B}_p$ of $L_p$ such that $\mathcal{B}_p$ is also an equivariant generating set of $L_n$ for all $n\ge p$; 3. There exist $q, q'\in\mathbb{N}$ such that for all $n\ge q$ the following hold: 1. $L_{n} = L \cap \mathbb{Z}^{(I_n)}$, 2. $L_{n}$ is generated by elements of support size at most $q'$; 4. $L$ has a finite equivariant generating set. The proof of this result requires some preparations. The following lemma is used to reduce the *width* of the support of an equivariant generating set. **Lemma 17**. Let $S\subseteq I_n=[n]^d\times[c]$ with $|S|=m$. Suppose that $n\ge p=dm+1$. Then there exists $\sigma\in\mathop{\mathrm{Sym}}(n)$ such that $\sigma(S)\subseteq I_p=[p]^d\times[c].$ *Proof.* Denote $$T_S=\bigcup\{\{i_1,\dots,i_d\}\mid (i_1,\dots,i_d,j)\in S \ \text{ for some } j\in [c]\}\subseteq [n].$$ We argue by induction on the cardinality $t$ of the set $T_S\setminus[p].$ If $t=0$, then $T_S\subseteq [p]$, and we can simply take $\sigma=\mathop{\mathrm{id}}_{[n]}$, the identity map of $[n]$. Now consider the case $t>0$. Choose an element $k\in T_S\setminus[p].$ Since $$|T_S| \le d|S|\le dm<p,$$ there must exist an $l\in [p]\setminus T_S.$ Let $\pi\in\mathop{\mathrm{Sym}}(n)$ be the transposition that swaps $k$ and $l$. It is easy to see that $$T_{\pi(S)}\setminus[p] =T_S\setminus[p]-\{k\}.$$ So by the induction hypothesis, there exists $\tau\in\mathop{\mathrm{Sym}}(n)$ such that $\tau(\pi(S))\subseteq I_p.$ The proof is completed by taking $\sigma=\tau\circ\pi$. ◻ We also need the following technical but useful observation. **Lemma 18**. Let $\sigma_1,\dots,\sigma_h\in \mathop{\mathrm{Sym}}$ and let $m,n\in\mathbb{N}$. Denote $$D=\bigcup_{j=1}^h\sigma_j([m])\cap [n].$$ Suppose $n\ge hm+1$. Then there exist $\sigma\in\mathop{\mathrm{Sym}}$ and $\tau_1,\dots,\tau_h\in \mathop{\mathrm{Sym}}(n)$ such that $$\sigma|_{D}=\mathop{\mathrm{id}}_D \ \text{ and }\ \sigma\circ\sigma_j|_{[m]}=\tau_j|_{[m]} \ \text{ for all } j\in[h],$$ where $\sigma|_{D}$ denotes the restriction of $\sigma$ on $D$. *Proof.* The argument is similar to that of the previous lemma. Let $$T(\sigma_1,\dots,\sigma_h)=\bigcup_{j=1}^h\sigma_j([m]).$$ We proceed by induction on $t=|T(\sigma_1,\dots,\sigma_h)\setminus[n]|.$ If $t=0$, then $\sigma_j([m])\subseteq [n]$ for all $j\in[h].$ Consequently, there exist $\tau_1,\dots,\tau_h\in \mathop{\mathrm{Sym}}(n)$ such that $\tau_j|_{[m]}=\sigma_j|_{[m]}$ for all $j\in[h]$, and we are done by choosing $\sigma=\mathop{\mathrm{id}}_{\mathbb{N}}.$ Now assume $t>0$. Choose an element $k\in T(\sigma_1,\dots,\sigma_h)\setminus[n].$ Since $$|T(\sigma_1,\dots,\sigma_h)|\le \sum_{j=1}^h|\sigma_j([m])| =hm< n,$$ there exists $l\in [n]\setminus T(\sigma_1,\dots,\sigma_h).$ Let $\pi\in\mathop{\mathrm{Sym}}$ be the transposition that swaps $k$ and $l$. Put $\sigma_j'=\pi\circ\sigma_j$ for $j\in[h]$. Then for any $i\in [m]$ and $j\in[h]$ one has $$\sigma_j'(i)=\pi(\sigma_j(i)) =\begin{cases} l&\text{if } \sigma_j(i) =k,\\ \sigma_j(i)&\text{otherwise}. \end{cases}$$ It follows that $D':=\bigcup_{j=1}^h\sigma_j'([m])\cap [n]=D\cup\{l\}.$ Moreover, $$T(\sigma_1',\dots,\sigma_h')\setminus[n] =T(\sigma_1,\dots,\sigma_h)\setminus[n]-\{k\}.$$ So by the induction hypothesis, there exist $\rho\in\mathop{\mathrm{Sym}}$ and $\tau_1,\dots,\tau_h\in \mathop{\mathrm{Sym}}(n)$ such that $$\rho|_{D'}=\mathop{\mathrm{id}}_{D'} \ \text{ and }\ \rho\circ\sigma_j'|_{[m]}=\tau_j|_{[m]} \ \text{ for all } j\in[h].$$ Letting $\sigma=\rho\circ\pi$ it is easy to check that $\sigma$ and $\tau_1,\dots,\tau_h$ fulfill the required condition. ◻ We are now ready to prove . *Proof of .* We will show (i) $\Rightarrow$ (ii) $\Rightarrow$ (iv) $\Rightarrow$ (iii) $\Rightarrow$ (i). \(i\) $\Rightarrow$ (ii): When $\mathfrak{L}$ stabilizes, we can choose $p\in\mathbb{N}$ so that $L_n$ is generated by $\mathop{\mathrm{Sym}}(n)(L_p)$ for all $n\ge p$. Thus if $\mathcal{B}_p$ is any equivariant generating set of $L_p$, then $\mathop{\mathrm{Sym}}(n)(\mathop{\mathrm{Sym}}(p)(\mathcal{B}_p))$ generates $L_n$ for all $n\ge p$. Since $\mathop{\mathrm{Sym}}(n)(\mathop{\mathrm{Sym}}(p)(\mathcal{B}_p))=\mathop{\mathrm{Sym}}(n)(\mathcal{B}_p)$, we are done. \(ii\) $\Rightarrow$ (iv): Choose $p$ and $\mathcal{B}_p$ as in (ii). We may assume that $\mathcal{B}_p$ is finite because $L_p$ is finitely generated. Since $L=\bigcup_{n\ge1}L_n$, it is easily seen that $\mathcal{B}_p$ is an equivariant generating set of $L$. \(iv\) $\Rightarrow$ (iii): Suppose $\mathcal{B}=\{{\boldsymbol b}_1,\dots,{\boldsymbol b}_h\}$ is a finite equivariant generating set of $L$. Since $L=\bigcup_{n\ge1}L_n$, there exists $m\in\mathbb{N}$ such that $\mathcal{B}\subseteq L_m\subseteq\mathbb{Z}^{(I_m)}$. Thus, each element of $\mathcal{B}$ has support size at most $|I_m|=m^dc$. The $\mathop{\mathrm{Sym}}(n)$-invariance of $L_n$ yields $$\mathbb{Z}\mathop{\mathrm{Sym}}(n)(\mathcal{B})\subseteq L_n \subseteq L \cap \mathbb{Z}^{(I_n)} \ \text{ for all } n\ge m.$$ Observe that the action of $\mathop{\mathrm{Sym}}(n)$ does not alter the support size of any element of $\mathbb{Z}^{(I_n)}$. So it suffices to prove the existence of some $q\ge m$ such that $$L \cap \mathbb{Z}^{(I_n)}=\mathbb{Z}\mathop{\mathrm{Sym}}(n)(\mathcal{B}) \ \text{ for all } n\ge q.$$ We show that $q=hm+1$ is such a number. Indeed, let $n\ge q$ and take any ${\boldsymbol u}\in L \cap \mathbb{Z}^{(I_n)}$. Since $L =\mathbb{Z}\mathop{\mathrm{Sym}}(\mathcal{B})$, there exist $z_1,\dots,z_h\in \mathbb{Z}$ and $\sigma_1,\dots,\sigma_h\in \mathop{\mathrm{Sym}}$ such that $${\boldsymbol u}=\sum_{k=1}^hz_k\sigma_k({\boldsymbol b}_k).$$ Using we find $\sigma\in\mathop{\mathrm{Sym}}$ and $\tau_1,\dots,\tau_h\in \mathop{\mathrm{Sym}}(n)$ satisfying $$\sigma|_{D}=\mathop{\mathrm{id}}_D \ \text{ and }\ \sigma\circ\sigma_k|_{[m]}=\tau_k|_{[m]} \ \text{ for all } k\in[h],$$ where $D=\bigcup_{k=1}^h\sigma_k([m])\cap [n]$. Evidently, if ${\boldsymbol i}=(i_1,\dots,i_d,j)\in \mathop{\mathrm{supp}}({\boldsymbol u})$, then $i_l\in D$ for all $l\in[d].$ So from $\sigma|_{D}=\mathop{\mathrm{id}}_D$ it follows that $\sigma({\boldsymbol u})={\boldsymbol u}$. Similarly, from $\sigma\circ\sigma_k|_{[m]}=\tau_k|_{[m]}$ we get $\sigma\circ\sigma_k({\boldsymbol b}_k)=\tau_k({\boldsymbol b}_k)$ for all $k\in[h].$ Hence $${\boldsymbol u}=\sigma({\boldsymbol u})=\sum_{k=1}^hz_k\sigma\circ\sigma_k({\boldsymbol b}_k) =\sum_{k=1}^hz_k\tau_k({\boldsymbol b}_k),$$ and therefore, ${\boldsymbol u}\in \mathbb{Z}\mathop{\mathrm{Sym}}(n)(\mathcal{B})$. \(iii\) $\Rightarrow$ (i): Denote $p=\max\{q,dq'+1\}$ and let $n\ge p$. Then $L_n$ has a generating set $\mathcal{B}_n$ that consists of elements of support size at most $q'$. Take any ${\boldsymbol b}\in\mathcal{B}_n$ and set $S=\mathop{\mathrm{supp}}({\boldsymbol b})$. Since $|S|\le q'$, there exists $\sigma\in\mathop{\mathrm{Sym}}(n)$ such that $\sigma(S)\subseteq I_p$ by virtue of . This implies $\sigma({\boldsymbol b})\in \mathbb{Z}^{(I_p)}\cap L=L_p.$ In other words, ${\boldsymbol b}=\sigma^{-1}({\boldsymbol b}')$ for some ${\boldsymbol b}'\in L_p.$ Hence, we can find a subset $\mathcal{B}_n'\subseteq L_p$ such that $\mathcal{B}_n\subseteq \mathop{\mathrm{Sym}}(n)(\mathcal{B}_n')$. From $$\mathbb{Z}\mathcal{B}_n\subseteq \mathbb{Z}\mathop{\mathrm{Sym}}(n)(\mathcal{B}_n') \subseteq \mathbb{Z}\mathop{\mathrm{Sym}}(n)(L_p) \subseteq L_n$$ and $L_n=\mathbb{Z}\mathcal{B}_n$, it follows that $L_n=\mathbb{Z}\mathop{\mathrm{Sym}}(n)(L_p)$ for all $n\ge p$. That is, $\mathfrak{L}$ stabilizes. ◻ As we will see in , every $\mathop{\mathrm{Sym}}$-invariant lattice in $\mathbb{Z}^{(I)}$ is equivariantly finitely generated. Hence, the statements (i), (ii), (iii) in always hold true. Let us now discuss the stabilization of other equivariant lattice bases. The next definition is motivated by . **Definition 19**. Let $\mathfrak{L}=(L_n)_{n\ge1}$ be a $\mathop{\mathrm{Sym}}$-invariant chain of lattices. We say that $\mathfrak{L}$ *Markov-* (respectively, *Gröbner-*, *universal Gröbner-*, *Graver-*)*stabilizes* if there exist $p\in\mathbb{N}$ and an equivariant Markov (respectively, Gröbner, universal Gröbner, Graver) basis $\mathcal{B}_p$ of $L_p$ such that $\mathcal{B}_p$ is also an equivariant Markov (respectively, Gröbner, universal Gröbner, Graver) basis of $L_n$ for all $n\ge p$. **Remark 20**. yields the following relationship among different kinds of stabilization of a $\mathop{\mathrm{Sym}}$-invariant chain of lattices: $$\begin{aligned} \text{Graver-stabilization } &\Rightarrow \text{ universal Gr\"{o}bner-stabilization } \Rightarrow \text{ Gr\"{o}bner-stabilization} \\ &\Rightarrow \text{ Markov-stabilization } \Rightarrow \text{ stabilization.} \end{aligned}$$ For equivariant Markov and Graver bases, can be extended as follows. **Theorem 21**. Let $\mathfrak{L}=(L_n)_{n\ge1}$ be a $\mathop{\mathrm{Sym}}$-invariant chain of lattices with limit $L$. Then the following are equivalent: 1. $\mathfrak{L}$ Markov- (respectively, Graver-)stabilizes; 2. There exist $q, q'\in\mathbb{N}$ such that for all $n\ge q$ the following hold: 1. $L_{n} = L \cap \mathbb{Z}^{(I_n)}$, 2. $L_{n}$ has a Markov (respectively, Graver) basis consisting of elements of support size at most $q'$; 3. $L$ has a finite equivariant Markov (respectively, Graver) basis. *Proof.* (i) $\Rightarrow$ (iii): This follows analogously to the implication (ii) $\Rightarrow$ (iv) in the proof of . Note that if $\mathfrak{L}$ Markov-stabilizes and $\mathcal{M}_p\subseteq L_p$ is an equivariant Markov basis of $L_n$ for $n\ge p$, then we may assume $\mathcal{M}_p$ is finite by using (ii) and Hilbert's basis theorem. Similarly, if $\mathfrak{L}$ Graver-stabilizes and $\mathcal{G}_p\subseteq L_p$ is an equivariant Graver basis of $L_n$ for $n\ge p$, then $\mathcal{G}_p$ must be finite by e.g. Gordan--Dickson lemma (see [@DHK Lemma 2.5.6]). \(iii\) $\Rightarrow$ (ii): We obtain (ii)(a) by using . So it remains to prove (ii)(b). Assume first that $L$ has a finite equivariant Markov basis $\mathcal{M}$. Choose $m\in\mathbb{N}$ with $\mathcal{M}\subseteq L_m$. It suffices to show that $\mathop{\mathrm{Sym}}(n)(\mathcal{M})$ is a Markov basis of $L_n$ for all $n\ge p=dm+1.$ To this end, take any ${\boldsymbol u}\in\mathbb{Z}_{\ge0}^{(I_n)}$ and let ${\boldsymbol v},{\boldsymbol w}\in F_{L_n}({\boldsymbol u}).$ Writing ${\boldsymbol v}-{\boldsymbol w}=({\boldsymbol v}-{\boldsymbol w})^+-({\boldsymbol v}-{\boldsymbol w})^-$, we may assume that ${\boldsymbol v}$ and ${\boldsymbol w}$ have disjoint supports. In particular, $\mathop{\mathrm{supp}}({\boldsymbol w})\subseteq\mathop{\mathrm{supp}}({\boldsymbol v}-{\boldsymbol w})$. Since ${\boldsymbol v},{\boldsymbol w}\in F_{L}({\boldsymbol u})$ and $\mathop{\mathrm{Sym}}(\mathcal{M})$ is a Markov basis of $L$, there exist ${\boldsymbol m}_1,\dots,{\boldsymbol m}_s\in\pm\mathcal{M}$ and $\sigma_1,\dots,\sigma_s\in\mathop{\mathrm{Sym}}$ such that $${\boldsymbol v}-{\boldsymbol w}=\sum_{i=1}^s\sigma_i({\boldsymbol m}_i) \quad\text{and}\quad {\boldsymbol w}+\sum_{i=1}^t\sigma_i({\boldsymbol m}_i)\in F_L({\boldsymbol u}) \ \text{ for all }\ t\in [s].$$ Applying to the first equation (as in the proof of ), we find $\sigma\in\mathop{\mathrm{Sym}}$ and $\tau_1,\dots,\tau_s\in \mathop{\mathrm{Sym}}(n)$ such that $$\sigma({\boldsymbol v}-{\boldsymbol w}) = {\boldsymbol v}-{\boldsymbol w} \quad\text{and}\quad \sigma\circ\sigma_i({\boldsymbol m}_i)=\tau_i({\boldsymbol m}_i) \ \text{ for all }\ i\in [s].$$ It holds furthermore that $\sigma({\boldsymbol w}) = {\boldsymbol w}$ since $\mathop{\mathrm{supp}}({\boldsymbol w})\subseteq\mathop{\mathrm{supp}}({\boldsymbol v}-{\boldsymbol w})$. Therefore, $${\boldsymbol v}-{\boldsymbol w}=\sigma({\boldsymbol v}-{\boldsymbol w}) =\sum_{i=1}^s\sigma\circ\sigma_i({\boldsymbol m}_i) =\sum_{i=1}^s\tau_i({\boldsymbol m}_i)$$ and $${\boldsymbol w}+\sum_{i=1}^t\tau_i({\boldsymbol m}_i) =\sigma\Big({\boldsymbol w}+\sum_{i=1}^t\sigma_i({\boldsymbol m}_i)\Big) \in \sigma(F_L({\boldsymbol u}))\cap\mathbb{Z}^{(I_n)} \subseteq\mathbb{Z}_{\ge0}^{(I_n)} \ \text{ for all }\ t\in [s].$$ This confirms that $\mathop{\mathrm{Sym}}(n)(\mathcal{M})$ is indeed a Markov basis of $L_n$ for all $n\ge p.$ Next, suppose that $L$ has a finite equivariant Graver basis $\mathcal{G}$. Then we can find a $q\in\mathbb{N}$ such that $\mathcal{G}\subseteq L_q$ and $L_{n} = L \cap \mathbb{Z}^{(I_n)}$ for all $n\ge q$. By (ii), $\mathop{\mathrm{Sym}}(\mathcal{G})\cap \mathbb{Z}^{(I_n)}$ is the Graver basis of $L_n$ for $n\ge q$. Since $\mathop{\mathrm{Sym}}(\mathcal{G})\cap \mathbb{Z}^{(I_n)}=\mathop{\mathrm{Sym}}(n)(\mathcal{G})$ for $n\ge q$, we are done. \(ii\) $\Rightarrow$ (i): Let $p=\max\{q,dq'+1\}$. First suppose that $L_{n}$ has a Markov basis $\mathcal{M}_n$ that consists of elements of support size at most $q'$ for $n\ge p$. Then arguing as in the proof of we find a subset $\mathcal{M}_n'\subseteq L_p$ such that $\mathcal{M}_n\subseteq \mathop{\mathrm{Sym}}(n)(\mathcal{M}_n')$ for $n\ge p$. In particular, $\mathcal{M}_n'$ is an equivariant Markov basis of $L_n$ for $n\ge p$. It follows that $$\mathcal{M}=\bigcup_{n\ge p}\mathcal{M}_n' \subseteq L_p$$ is an equivariant Markov basis of $L_n$ for all $n\ge p$. The argument for Graver bases is similar. Note that if ${\boldsymbol u}\in\mathbb{Z}^{(I)}$ is $\sqsubseteq$-minimal, then $\sigma({\boldsymbol u})$ is also $\sqsubseteq$-minimal for any $\sigma\in \mathop{\mathrm{Sym}}.$ For $n\ge p$, let $\mathcal{G}_n$ be the Graver basis of $L_n$ and let $\mathcal{G}_n'\subseteq L_p$ be the set constructed as in the proof of such that $\mathcal{G}_n\subseteq \mathop{\mathrm{Sym}}(n)(\mathcal{G}_n')$. By construction, all the elements of $\mathcal{G}_n'$ are $\sqsubseteq$-minimal because they are of the form $\sigma({\boldsymbol u})$ with $\sigma\in\mathop{\mathrm{Sym}}(n)$ and ${\boldsymbol u}\in\mathcal{G}_n$. So if we set $$\mathcal{G}=\bigcup_{n\ge p}\mathcal{G}_n' \subseteq L_p,$$ then for $n\ge p$, $\mathop{\mathrm{Sym}}(n)(\mathcal{G})$ is a subset of $L_n$ that consists of $\sqsubseteq$-minimal elements. This means that $\mathop{\mathrm{Sym}}(n)(\mathcal{G})\subseteq\mathcal{G}_n$ for $n\ge p$. On the other hand, $\mathcal{G}_n\subseteq \mathop{\mathrm{Sym}}(n)(\mathcal{G})$ since $\mathcal{G}_n'\subseteq \mathcal{G}$. Hence, $\mathop{\mathrm{Sym}}(n)(\mathcal{G})=\mathcal{G}_n$ and $\mathcal{G}$ is an equivariant Graver basis of $L_n$ for all $n\ge p$. ◻ We do not know whether can also be fully extended to equivariant (universal) Gröbner bases. Nevertheless, at least a partial extension is possible. **Proposition 22**. Consider the following conditions for a $\mathop{\mathrm{Sym}}$-invariant chain of lattices $\mathfrak{L}=(L_n)_{n\ge1}$ with limit $L$: 1. $\mathfrak{L}$ Gröbner- (respectively, universal Gröbner-)stabilizes. 2. There exist $q, q'\in\mathbb{N}$ such that for all $n\ge q$ the following hold: 1. $L_{n} = L \cap \mathbb{Z}^{(I_n)}$, 2. $L_{n}$ has a Gröbner (respectively, universal Gröbner) basis consisting of elements of support size at most $q'$. 3. $L$ has a finite equivariant Gröbner (respectively, universal Gröbner) basis. Then it holds that $$\text{(i) } \Leftrightarrow \text{(ii) } \Rightarrow \text{(iii)}.$$ Moreover, for (equivariant) Gröbner bases with respect to the lexicographic order $\prec_{\mathop{\mathrm{lex}}}$, one has $\text{(i) }\Leftrightarrow \text{(ii) }\Leftrightarrow \text{(iii)}.$ *Proof.* The implications (i) $\Rightarrow$ (iii) and (ii) $\Rightarrow$ (i) follow analogously to the corresponding implications in the proof of . For the implication (i) $\Rightarrow$ (ii), it suffices to prove (i) $\Rightarrow$ (ii)(b) by . But this is clear because if $\mathcal{B}_p\subseteq L_p$ is an equivariant (universal) Gröbner basis of $L_n$ for $n\ge p$, then for such $n$, $\mathop{\mathrm{Sym}}(n)(\mathcal{B}_p)$ is a (universal) Gröbner basis of $L_n$. Finally, the implication (iii) $\Rightarrow$ (ii) for equivariant Gröbner bases with respect to $\prec_{\mathop{\mathrm{lex}}}$ follows from , using an argument similar to that for Graver bases in the proof of the implication (iii) $\Rightarrow$ (ii) of . ◻ **Remark 23**. The action of $\mathop{\mathrm{Sym}}$ on $I$ induces actions of this group on the polynomial ring $R=K[x_{{\boldsymbol i}}\mid {\boldsymbol i}\in I]$ and Laurent polynomial ring $R^\pm=R[x_{{\boldsymbol i}}^{-1}\mid {\boldsymbol i}\in I]$ that are determined by $$\sigma(x_{{\boldsymbol i}})=x_{\sigma({\boldsymbol i})} \ \text{ and }\ \sigma(x_{{\boldsymbol i}}^{-1})=x_{\sigma({\boldsymbol i})}^{-1} \ \text{ for any } \sigma\in\mathop{\mathrm{Sym}} \text{ and } {\boldsymbol i}\in I.$$ Analogously to lattices, one defines $\mathop{\mathrm{Sym}}$-invariant chains of (Laurent) ideals and their stabilization. Then for any $\mathop{\mathrm{Sym}}$-invariant chain of lattices $\mathfrak{L}=(L_n)_{n\ge1}$, its stabilization and Markov-stabilization can be characterized in algebraic terms as follows: 1. $\mathfrak{L}$ stabilizes if and only if the chain of Laurent ideals $(\mathfrak{I}^{\pm}_{L_n})_{n\ge1}$ stabilizes; 2. $\mathfrak{L}$ Markov-stabilizes if and only if the chain of lattice ideals $(\mathfrak{I}_{L_n})_{n\ge1}$ stabilizes. See [@HM] and [@HS12] for details in the cases $c=1$ (i.e. $I=\mathbb{N}^d$) or $d=1$ (i.e. $I=\mathbb{N}\times[c]$), which can be easily extended to our setting. One might ask whether there is a similar interpretation for the Gröbner-stabilization. Here, one obstruction is that for a given term order $\prec$, the chain of initial ideals $(\mathop{\mathrm{in}}_\prec(\mathfrak{I}_{L_n}))_{n\ge1}$ might be not $\mathop{\mathrm{Sym}}$-invariant. As a remedy, one replaces the action of the symmetric group $\mathop{\mathrm{Sym}}$ by that of the monoid $\mathop{\mathrm{Inc}}$ of increasing functions. Then for a term order $\prec$ such that the chain $(\mathop{\mathrm{in}}_\prec(\mathfrak{I}_{L_n}))_{n\ge1}$ is $\mathop{\mathrm{Inc}}$-invariant, the Gröbner-stabilization with respect to $\prec$ of $\mathfrak{L}$ can be characterized in terms of the $\mathop{\mathrm{Inc}}$-stabilization of the chain $(\mathop{\mathrm{in}}_\prec(\mathfrak{I}_{L_n}))_{n\ge1}$; see [@HS12; @NR17]. To conclude this section, let us derive a local-global result for equivariant Hilbert bases of monoids in the flavor of . As for lattices, a chain of monoids $\mathfrak{M}=(M_n)_{n\ge1}$ with $M_n\subseteq \mathbb{Z}^{(I_n)}$ for $n\ge 1$ is called *$\mathop{\mathrm{Sym}}$-invariant* if it is increasing and each monoid $M_n$ is $\mathop{\mathrm{Sym}}(n)$-invariant. In this case, we say that $\mathfrak{M}$ *stabilizes* if there exists $p\in\mathbb{N}$ such that $$M_n=\mathbb{Z}_{\ge0}\mathop{\mathrm{Sym}}(n)(M_m) \ \text{ for all } n\ge m\ge p.$$ The following result generalizes [@KLR Corollary 5.13] and [@LR21 Lemma 5.1]. **Theorem 24**. Let $\mathfrak{M}=(M_n)_{n\ge1}$ be a $\mathop{\mathrm{Sym}}$-invariant chain of nonnegative monoids, i.e. $M_n\subseteq \mathbb{Z}_{\ge0}^{(I_n)}$ for all $n\ge 1$. Denote $M=\bigcup_{n\ge1}M_n$. Then the following are equivalent: 1. $\mathfrak{M}$ stabilizes and $M_n$ is finitely generated for $n\gg0$; 2. There exist $p\in\mathbb{N}$ and a finite equivariant Hilbert basis $\mathcal{H}_p$ of $M_p$ such that $\mathcal{H}_p$ is also an equivariant Hilbert basis of $M_n$ for all $n\ge p$; 3. There exist $q, q'\in\mathbb{N}$ such that for all $n\ge q$ the following hold: 1. $M_{n} = M \cap \mathbb{Z}^{(I_n)}$, 2. $M_{n}$ has a finite Hilbert basis consisting of elements of support size at most $q'$; 4. $M$ has a finite equivariant Hilbert basis. *Proof.* Observe that if ${\boldsymbol u}\in M$ is an irreducible element, then $\sigma({\boldsymbol u})$ is also an irreducible element of $M$ for any $\sigma\in\mathop{\mathrm{Sym}}.$ \(i\) $\Rightarrow$ (ii): If $\mathfrak{M}$ stabilizes, there exists $p\in\mathbb{N}$ such that $M_n$ is generated by $\mathop{\mathrm{Sym}}(n)(M_p)$ for all $n\ge p$. Choosing $p$ large enough, we may assume that $M_p$ is finitely generated. So if $\mathcal{H}_p$ is any equivariant Hilbert basis of $M_p$, then $\mathcal{H}_p$ is finite and the set $$\mathop{\mathrm{Sym}}(n)(\mathop{\mathrm{Sym}}(p)(\mathcal{H}_p))=\mathop{\mathrm{Sym}}(n)(\mathcal{H}_p)$$ generates $M_n$ for all $n\ge p$. Since $\mathop{\mathrm{Sym}}(n)(\mathcal{H}_p)$ consists of irreducible elements, it must be the Hilbert basis of $M_n$ by . \(iv\) $\Rightarrow$ (iii): To show that $M_{n} = M \cap \mathbb{Z}^{(I_n)}$ for $n\gg0$ one can argue similarly to the proof of . However, employing the nonnegativity of $\mathfrak{M}$, this can be proven without using . Let $\mathcal{H}$ be a finite equivariant Hilbert basis of $M$ and choose $q\in\mathbb{N}$ such that $\mathcal{H}\subseteq M_q$. We show that $M_{n} = M \cap \mathbb{Z}^{(I_n)}$ for all $n\ge q$. Indeed, take any ${\boldsymbol u}\in M \cap \mathbb{Z}^{(I_n)}$ with $n\ge q$. Since $M=\mathbb{Z}_{\ge0}\mathop{\mathrm{Sym}}(\mathcal{H})$, there exist ${\boldsymbol h}_1,\dots,{\boldsymbol h}_s\in\mathcal{H}$, $m_1,\dots,m_s\in\mathbb{Z}_{\ge0}$ and $\sigma_1,\dots,\sigma_s\in\mathop{\mathrm{Sym}}$ such that ${\boldsymbol u}=\sum_{j=1}^sm_j\sigma_j({\boldsymbol h}_j).$ We may assume that all the coefficients $m_j$ are positive. Then the nonnegativity of the elements ${\boldsymbol h}_j$ yields $$\mathop{\mathrm{supp}}(\sigma_j({\boldsymbol h}_j))\subseteq\mathop{\mathrm{supp}}({\boldsymbol u})\subseteq I_n \ \text{ for all }\ j\in [s].$$ It follows that $\sigma_j({\boldsymbol h}_j)=\tau_j({\boldsymbol h}_j)$ with $\tau_j\in\mathop{\mathrm{Sym}}(n)$ for all $j\in [s]$. Hence, $${\boldsymbol u}=\sum_{j=1}^sm_j\tau_j({\boldsymbol h}_j)\in \mathbb{Z}_{\ge0}\mathop{\mathrm{Sym}}(n)(M_q)\subseteq M_n,$$ and therefore $M_{n} = M \cap \mathbb{Z}^{(I_n)}$, as desired. Now using we see that $\mathop{\mathrm{Sym}}(\mathcal{H})\cap M_n=\mathop{\mathrm{Sym}}(n)(\mathcal{H})$ is the Hilbert basis of $M_n$ for $n\ge q.$ Finally, the implications (ii) $\Rightarrow$ (iv) and (iii) $\Rightarrow$ (i) follow similarly to the corresponding implications in the proof of . ◻ # Finiteness of equivariant Graver bases {#sec-equi-Graver} This section deals with the case $d=1$. Our goal is to prove the following finiteness result for equivariant Graver bases and discuss its consequences. Throughout the section, let $c$ be a positive integer. **Theorem 25**. Suppose that $I=\mathbb{N}\times[c]$. Then every $\mathop{\mathrm{Sym}}$-invariant lattice in $\mathbb{Z}^{(I)}$ has a finite equivariant Graver basis. Before proving this result, let us derive some corollaries. First, using we get the following algebraic version of . **Corollary 26**. Let $I=\mathbb{N}\times[c]$. Then for any $\mathop{\mathrm{Sym}}$-invariant lattice $L\subseteq\mathbb{Z}^{(I)}$, the lattice ideal $\mathfrak{I}_L$ has a finite equivariant Graver basis. The next consequence is immediate from . **Corollary 27**. Let $I=\mathbb{N}\times[c]$. Then every $\mathop{\mathrm{Sym}}$-invariant lattice in $\mathbb{Z}^{(I)}$ has 1. a finite equivariant universal Gröbner basis, 2. a finite equivariant Gröbner basis with respect to an arbitrary term order on $\mathbb{Z}_{\ge0}^{(I)}$, 3. a finite equivariant Markov basis, 4. a finite equivariant generating set. As a local version of , the result below is obtained by combining Theorems [Theorem 25](#Finiteness-Graver){reference-type="ref" reference="Finiteness-Graver"}, [Theorem 21](#stabilization-others){reference-type="ref" reference="stabilization-others"} and . **Corollary 28**. Let $I=\mathbb{N}\times[c]$. Then every $\mathop{\mathrm{Sym}}$-invariant chain of lattices in $\mathbb{Z}^{(I)}$ Graver-stabilizes. In particular, such a chain 1. universal Gröbner-stabilizes, 2. Gröbner-stabilizes with respect to an arbitrary term order on $\mathbb{Z}_{\ge0}^{(I)}$, 3. Markov-stabilizes, 4. stabilizes. This corollary generalizes and strengthens a finiteness result for Markov bases of hierarchical models due to Hoşten and Sullivant [@HoS Theorem 1.1]. To present this result, let us recall some necessary notions; see, e.g. [@Su Chapter 9] for further details. Hierarchical models are used in statistics to represent interactions among random variables. Such a model is determined by a simplicial complex $\Delta\subseteq 2^{[m]}$ and a vector ${\boldsymbol r}=(r_1,\dots,r_m)\in\mathbb{N}^m$, where $m\in\mathbb{N}$ is a given number. While the simplicial complex $\Delta$ describes the interactions among the random variables, the vector ${\boldsymbol r}$ encodes their numbers of states. Let $F_1,\dots,F_s$ be the facets of $\Delta$. Denote $\mathcal{R}=\prod_{i=1}^m[r_i]$ and $\mathcal{R}_{F_k}=\prod_{j\in F_k}[r_j]$ for $k\in[s]$. For ${\boldsymbol i}=(i_1,\dots,i_m)\in \mathcal{R}$ let ${\boldsymbol i}_{F_k}=(i_j)_{j\in F_k}\in \mathcal{R}_{F_k}$. The *$F_k$-marginal* of each element of $\mathbb{Z}^{(\mathcal{R})}$ can be computed using the linear map $\mu_{F_k}:\mathbb{Z}^{(\mathcal{R})}\to \mathbb{Z}^{(\mathcal{R}_{F_k})}$, which is defined by $\mu_{F_k}({\boldsymbol e}_{\boldsymbol i})={\boldsymbol e}_{{\boldsymbol i}_{F_k}}$, where ${\boldsymbol e}_{\boldsymbol i}$ and ${\boldsymbol e}_{{\boldsymbol i}_{F_k}}$ are standard basis elements of $\mathbb{Z}^{(\mathcal{R})}$ and $\mathbb{Z}^{(\mathcal{R}_{F_k})}$, respectively. The minimal sufficient statistics of the model are then given by the *$\Delta$-marginal map* that puts all facet marginals together: $$\mu_{\Delta,{\boldsymbol r}}:\mathbb{Z}^{(\mathcal{R})}\to \bigoplus_{k=1}^s\mathbb{Z}^{(\mathcal{R}_{F_k})}, \ \ {\boldsymbol u}\mapsto (\mu_{F_1}({\boldsymbol u}),\dots,\mu_{F_s}({\boldsymbol u})).$$ One is interested in Markov bases of the lattice $\ker \mu_{\Delta,{\boldsymbol r}}$ as they are useful for performing Fisher's exact test. In particular, it is desirable to know how the structure of these Markov bases depends on $\Delta$ and ${\boldsymbol r}$. Hoşten and Sullivant [@HoS] studied this problem when $\Delta$ and the numbers $r_2,\dots,r_m$ are fixed, while $r_1$ is allowed to vary. Their remarkable finiteness result [@HoS Theorem 1.1] says that, in such case, there is always a finite Markov basis up to symmetry. To interpret this result in our language, we identify $\mathcal{R}$ with $[r_1]\times [c]$, where $c=\prod_{i=2}^mr_i$. In this way, $\ker \mu_{\Delta,{\boldsymbol r}}$ is identified with a lattice $L_{\Delta,{\boldsymbol r}}$ in $\mathbb{Z}^{([r_1]\times[c])}$. Consider the action of $\mathop{\mathrm{Sym}}(r_1)$ on $\mathbb{Z}^{([r_1]\times[c])}$ by permuting the first index as before. Then it is easy to see that the chain of lattices $\mathfrak{L}_{\Delta,{\boldsymbol r}}=(L_{\Delta,{\boldsymbol r}})_{r_1\ge1}$ is $\mathop{\mathrm{Sym}}$-invariant. The result of Hoşten and Sullivant can now be stated as a special case of (iii) as follows. **Corollary 29**. Keep the notation as above. If $\Delta$ and $r_2,\dots,r_m$ are fixed, then the chain of lattices $\mathfrak{L}_{\Delta,{\boldsymbol r}}=(L_{\Delta,{\boldsymbol r}})_{r_1\ge1}$ Markov-stabilizes. It should be mentioned that this result generalizes finiteness results for the no three-way interaction model due to Aoki and Takemura [@AT] and for logit models due to Santos and Sturmfels [@SSt]. A far-reaching generalization of this corollary, the so-called independent set theorem, will be discussed in the next section (see ). Let us now turn to the proof of . We first discuss the case $c=1$ before giving a proof for the general case. ## The case $c=1$ We consider the case $c=1$ separately for two reasons. First, in this case, we obtain more specific information about the Graver basis. Second, the proof of this case is more elementary, in the sense that it does not make use of Higman's lemma. It should be mentioned that Higman's lemma is essential to all known proofs of the equivariant Noetherianity of the polynomial ring $K[x_i\mid i\in\mathbb{N}]$; see [@HM Remark 15]. As before, we denote the standard basis of $\mathbb{Z}^{(\mathbb{N})}$ by $\{{\boldsymbol e}_i\}_{i\in\mathbb{N}}$. For ${\boldsymbol u}=(u_i)_{i\in\mathbb{N}}\in\mathbb{Z}^{(\mathbb{N})}$ let $$g({\boldsymbol u})=\gcd(u_i\mid i\in\mathbb{N})$$ be the greatest common divisor of the components of ${\boldsymbol u}$. Moreover, for a lattice $L\subseteq\mathbb{Z}^{(\mathbb{N})}$ consider the element $$\label{eq-g} {\boldsymbol g}_L=(g_L,-g_L)=g_L({\boldsymbol e}_1-{\boldsymbol e}_2)\in \mathbb{Z}^{(\mathbb{N})}, \quad\text{where }\ g_L=\gcd(g({\boldsymbol u})\mid {\boldsymbol u}\in L).$$ The role of this element will be made clear in . At first, we observe: **Lemma 30**. Let $L\subseteq\mathbb{Z}^{(\mathbb{N})}$ be a $\mathop{\mathrm{Sym}}$-invariant lattice and let ${\boldsymbol u}=(u_i)_{i\in\mathbb{N}}\in L.$ Then the following statements hold: 1. $(u_i,-u_i)\in L$ for all $i\in\mathbb{N}$. 2. ${\boldsymbol g}_L\in L$. *Proof.* (i) Note that $u_n=0$ for $n\gg0$. So for any $i\in\mathbb{N}$, there exist $\sigma_1,\sigma_2\in\mathop{\mathrm{Sym}}$ such that $$\sigma_1({\boldsymbol u})=(u_i,0,u_1,u_2,\dots,u_{i-1},u_{i+1},\dots),\ \sigma_2({\boldsymbol u})=(0,u_i,u_1,u_2,\dots,u_{i-1},u_{i+1},\dots).$$ Since $L$ is $\mathop{\mathrm{Sym}}$-invariant, we get $$(u_i,-u_i)=\sigma_1({\boldsymbol u})-\sigma_2({\boldsymbol u})\in L.$$ \(ii\) By definition of $g({\boldsymbol u})$, there exist $i_1,\dots,i_k\in\mathbb{N}$ and $z_{i_1},\dots,z_{i_k}\in\mathbb{Z}$ such that $$g({\boldsymbol u})=\sum_{j=1}^kz_{i_j}u_{i_j}.$$ Thus from (i) it follows that $$(g({\boldsymbol u}),-g({\boldsymbol u}))=\sum_{j=1}^kz_{i_j}(u_{i_j},-u_{i_j})\in L.$$ Similarly, by definition of $g_L$, there exist ${\boldsymbol v}_1,\dots,{\boldsymbol v}_l\in L$ and $y_{1},\dots,y_{l}\in\mathbb{Z}$ such that $$g_L=\sum_{j=1}^ly_{j}g({\boldsymbol v}_{j}).$$ Hence, $${\boldsymbol g}_L=(g_L,-g_L)=\sum_{j=1}^ly_{j}(g({\boldsymbol v}_{j}),-g({\boldsymbol v}_{j}))\in L,$$ as desired. ◻ As a consequence of the preceding lemma, the next result shows an interesting property of *generic* $\mathop{\mathrm{Sym}}$-invariant lattices in $\mathbb{Z}^{(\mathbb{N})}$: they always contain certain multiples of standard basis elements. **Corollary 31**. Let $L\subseteq\mathbb{Z}^{(\mathbb{N})}$ be a $\mathop{\mathrm{Sym}}$-invariant lattice. For any ${\boldsymbol u}=(u_i)_{i\in\mathbb{N}}\in \mathbb{Z}^{(\mathbb{N})}$ set $s({\boldsymbol u})=\sum_{i\in\mathbb{N}}u_i$. Denote $s_L=\gcd(s({\boldsymbol u})\mid {\boldsymbol u}\in L)$. Then the following hold: 1. $s_L{\boldsymbol e}_1\in L.$ 2. If there exists ${\boldsymbol u}\in L$ such that $s({\boldsymbol u})\ne0$, then $s_L{\boldsymbol e}_1$ belongs to the Graver basis of $L$. *Proof.* (i) Take any ${\boldsymbol u}\in L$ and choose $n$ such that $u_i=0$ for all $i\ge n$, i.e. ${\boldsymbol u}=(u_1,\dots,u_n)$. implies ${\boldsymbol f}=u_n({\boldsymbol e}_{n-1}-{\boldsymbol e}_n)=(0,\dots,0,u_{n},-u_{n})\in L.$ Hence, $${\boldsymbol u}'={\boldsymbol u}+{\boldsymbol f} =(u_1,\dots,u_{n-2},u_{n-1}+u_{n}) \in L.$$ Repeating the above argument, we see that $s({\boldsymbol u}){\boldsymbol e}_1\in L.$ Since $s_L$ is a $\mathbb{Z}$-linear combination of finitely many $s({\boldsymbol u})$ with ${\boldsymbol u}\in L$, the desired conclusion follows. \(ii\) If $s({\boldsymbol u})\ne0$ for some ${\boldsymbol u}\in L$, then $s_L\ne 0$. In this case, $s_L{\boldsymbol e}_1$ is a $\sqsubseteq$-minimal element of $L\setminus\{{\boldsymbol 0}\}$. Indeed, let ${\boldsymbol v}\in L\setminus\{{\boldsymbol 0}\}$ with ${\boldsymbol v}\sqsubseteq s_L{\boldsymbol e}_1$. Then ${\boldsymbol v}$ must be of the form $z{\boldsymbol e}_1$ for some $z\in\mathbb{Z}$ with $0<z\le s_L$. Since $s({\boldsymbol v})=z$ is divisible by $s_L$, we deduce that $z=s_L$, and thus ${\boldsymbol v}=s_L{\boldsymbol e}_1$, as desired. ◻ In order to determine the Graver basis of a $\mathop{\mathrm{Sym}}$-invariant lattice $L\subseteq\mathbb{Z}^{(\mathbb{N})}$, we relate it to the Hilbert basis of the monoid $M=L\cap\mathbb{Z}^{(\mathbb{N})}_{\ge0}$; see for analogous ideas. The following key lemma is reminiscent of the equivariant Gordan's lemma established in [@LR21 Theorem 6.1]. **Lemma 32**. For every $\mathop{\mathrm{Sym}}$-invariant lattice $L\subseteq\mathbb{Z}^{(\mathbb{N})}$, the monoid $M=L\cap\mathbb{Z}^{(\mathbb{N})}_{\ge0}$ has a finite equivariant Hilbert basis. *Proof.* Set $L_n=L\cap \mathbb{Z}^n$ and $M_n=L_n\cap\mathbb{Z}^{n}_{\ge0}$ for $n\ge1$. Moreover, let $\mathcal{H}_n$ denote the Hilbert basis of $M_n$. Obviously, $\mathfrak{M}=(M_n)_{n\ge1}$ is a $\mathop{\mathrm{Sym}}$-invariant chain of nonnegative monoids with limit $\bigcup_{n\ge1}M_n=M$. So by , it suffices to show that each $\mathcal{H}_n$ is finite and the support sizes of its elements do not exceed a fixed number $q$. First of all, $L_n$ is a finitely generated monoid as it is a finitely generated abelian group. Hence, $M_n=L_n\cap\mathbb{Z}^{n}_{\ge0}$ is also a finitely generated monoid; see, e.g. [@BG Corollary 2.11(a)]. This implies that $\mathcal{H}_n$ is finite for all $n\ge 1$. Set $$\|\mathcal{H}_{n}\|=\max\{\|{\boldsymbol u}\|\mid {\boldsymbol u}\in \mathcal{H}_{n}\}.$$ Evidently, $|\mathop{\mathrm{supp}}({\boldsymbol u})|\le \|{\boldsymbol u}\|$ for all ${\boldsymbol u}\in\mathbb{Z}^{(\mathbb{N})}$. Thus, in order to verify that the support sizes of elements of $\mathcal{H}_n$ are uniformly bounded, we only need to show that $\|\mathcal{H}_{n+1}\|\le \|\mathcal{H}_{n}\|$ for all $n\ge 1.$ To this end, take any ${\boldsymbol u}=(u_1,\dots,u_{n+1})\in \mathcal{H}_{n+1}$. From it follows that $${\boldsymbol f}=u_{n+1}({\boldsymbol e}_n-{\boldsymbol e}_{n+1})=(0,\dots,0,u_{n+1},-u_{n+1})\in L.$$ Hence, $${\boldsymbol u}'={\boldsymbol u}+{\boldsymbol f} =(u_1,\dots,u_{n-1},u_{n}+u_{n+1}) \in L\cap\mathbb{Z}^{n}_{\ge0} = M_{n}.$$ Since $\|{\boldsymbol u}\|=\|{\boldsymbol u}'\|$, the desired inequality will follow if we show that ${\boldsymbol u}'\in\mathcal{H}_{n}.$ Assume the contrary that ${\boldsymbol u}'\not\in\mathcal{H}_{n}$. Then there exist ${\boldsymbol v}'=(v_1,\dots,v_{n}),{\boldsymbol w}'=(w_1,\dots,w_{n})\in M_{n}\setminus\{{\boldsymbol 0}\}$ such that ${\boldsymbol u}'={\boldsymbol v}'+{\boldsymbol w}'.$ This gives $u_{n}+u_{n+1}=v_{n}+w_{n}.$ We may suppose $u_{n}\le v_{n}.$ Since ${\boldsymbol u}-{\boldsymbol v}'\in L$, yields $${\boldsymbol f}'=(u_n-v_{n})({\boldsymbol e}_n-{\boldsymbol e}_{n+1}) =(0,\dots,0,u_n-v_{n},v_{n}-u_{n}) \in L.$$ It follows that $${\boldsymbol v}={\boldsymbol v}'+{\boldsymbol f}' =(v_1,\dots,v_{n-1},u_{n},v_{n}-u_{n}) \in L\cap\mathbb{Z}^{n+1}_{\ge0} = M_{n+1}.$$ On the other hand, ${\boldsymbol w}=(w_1,\dots,w_{n-1},0,w_{n})\in M_{n+1}$ since ${\boldsymbol w}'=(w_1,\dots,w_{n},0)\in M_{n+1}.$ Now from ${\boldsymbol u}'={\boldsymbol v}'+{\boldsymbol w}'$ we get ${\boldsymbol u}={\boldsymbol v}+{\boldsymbol w}$, contradicting the assumption that ${\boldsymbol u}\in \mathcal{H}_{n+1}$. ◻ We are now ready to prove the following more specific version of in the case $c=1$. **Theorem 33**. Let $L\subseteq\mathbb{Z}^{(\mathbb{N})}$ be a $\mathop{\mathrm{Sym}}$-invariant lattice. Suppose that $\mathcal{H}$ is a finite equivariant Hilbert basis of the monoid $M=L\cap\mathbb{Z}^{(\mathbb{N})}_{\ge0}$. Then $L$ has a finite equivariant Graver basis $\mathcal{G}$ satisfying $$\pm\mathcal{H}\subseteq\mathcal{G}\subseteq\pm\mathcal{H}\cup\{\pm{\boldsymbol g}_L\},$$ where ${\boldsymbol g}_L$ is defined as in [\[eq-g\]](#eq-g){reference-type="eqref" reference="eq-g"}. *Proof.* First, note that $\mathcal{H}$ exists by . Moreover, it follows from that every element of $\pm\mathcal{H}$ is $\sqsubseteq$-minimal in $L\setminus\{{\boldsymbol 0}\}$. So to prove the theorem, it suffices to show that $\mathop{\mathrm{Sym}}(\mathcal{H}')$ contains all $\sqsubseteq$-minimal elements of $L\setminus\{{\boldsymbol 0}\}$, where $\mathcal{H}'=\pm\mathcal{H}\cup\{\pm{\boldsymbol g}_L\}$. Take any ${\boldsymbol u}\in L\setminus\{{\boldsymbol 0}\}$. We need to prove that $\sigma({\boldsymbol h})\sqsubseteq {\boldsymbol u}$ for some $\sigma\in\mathop{\mathrm{Sym}}$ and ${\boldsymbol h}\in\mathcal{H}'.$ If ${\boldsymbol u}\in M$ or $-{\boldsymbol u}\in M$, then we are done since $\mathcal{H}$ is an equivariant Hilbert basis of $M$. Now suppose ${\boldsymbol u}=(u_i)_{i\in\mathbb{N}}\not\in\pm M$. Then $\mathop{\mathrm{supp}}({\boldsymbol u}^+),\mathop{\mathrm{supp}}({\boldsymbol u}^-)\ne\emptyset$. Replacing ${\boldsymbol u}$ with $\pm\tau({\boldsymbol u})$ for a suitable $\tau\in\mathop{\mathrm{Sym}}$ (if necessary), we may assume that $1\in\mathop{\mathrm{supp}}({\boldsymbol u}^+)$, $2\in\mathop{\mathrm{supp}}({\boldsymbol u}^-)$ and $|u_1|\le |u_2|$. Then ${\boldsymbol u}={\boldsymbol v}+{\boldsymbol w}$ is a conformal decomposition of ${\boldsymbol u}$, where $${\boldsymbol v}=(u_1,-u_1) \quad\text{and}\quad {\boldsymbol w}=(0,u_1+u_2,u_3,\dots).$$ From the definition of ${\boldsymbol g}_L$ it is clear that ${\boldsymbol v}=z{\boldsymbol g}_L$ for some $z\in\mathbb{Z}\setminus\{0\}$. Hence, we have either ${\boldsymbol g}_L\sqsubseteq{\boldsymbol v}\sqsubseteq{\boldsymbol u}$ or $-{\boldsymbol g}_L\sqsubseteq{\boldsymbol v}\sqsubseteq {\boldsymbol u}.$ This completes the proof. ◻ **Example 34**. Consider the lattice $L=\mathbb{Z}\mathop{\mathrm{Sym}}(\{{\boldsymbol u},{\boldsymbol v}\})\subseteq\mathbb{Z}^{(\mathbb{N})}$ generated by the $\mathop{\mathrm{Sym}}$-orbits of the elements $${\boldsymbol u}=(1,3,5)\ \text{ and }\ {\boldsymbol v}= (2,4,6).$$ Let $M=L\cap\mathbb{Z}^{(\mathbb{N})}_{\ge0}$. Denote by $\mathcal{G}$ an equivariant Graver basis of $L$. Then it follows from that $\mathcal{H}=\mathcal{G}\cap\mathbb{Z}^{(\mathbb{N})}_{\ge0}$ is an equivariant Hilbert basis of $M$. By , we may assume that $3{\boldsymbol e}_1\in \mathcal{G}$ since $s_L=\gcd(s({\boldsymbol u}),s({\boldsymbol v}))=3.$ Hence, $3{\boldsymbol e}_1\in \mathcal{H}$. So the proof of implies that we may choose $\mathcal{H}\subseteq\{3{\boldsymbol e}_1,{\boldsymbol e}_1+2{\boldsymbol e}_2\}$. On the other hand, by , ${\boldsymbol g}_L={\boldsymbol e}_1-{\boldsymbol e}_2$ could be an element of $\mathcal{G}$. Computations with 4ti2 [@4ti2] and Macaulay2 [@GS] confirm that we can take $\mathcal{H}=\{3{\boldsymbol e}_1,{\boldsymbol e}_1+2{\boldsymbol e}_2\}$ and $\mathcal{G}=\pm\mathcal{H}\cup\{\pm{\boldsymbol g}_L\}$. Note that, in general, ${\boldsymbol g}_L$ is not necessarily an element of the Graver basis of $L$. An obvious example is $L=\mathbb{Z}^{(\mathbb{N})}$, of which $\mathcal{G}=\{\pm{\boldsymbol e}_1\}$ is an equivariant Graver basis. ## The general case {#sec-Graver} A key step in the proof of , as in the case $c=1$, is to relate the Graver basis of a lattice to the Hilbert basis of a certain monoid. We present this relationship in the general setting as it might be of independent interest. Let $I$ be an arbitrary set. Denote by $2I$ the disjoint union $I\sqcup I$. Then there is a natural isomorphism of abelian groups $$\psi:\mathbb{Z}^{(2I)}\cong \mathbb{Z}^{(I)}\oplus\mathbb{Z}^{(I)}.$$ Identifying $\mathbb{Z}^{(2I)}$ with $\mathbb{Z}^{(I)}\oplus\mathbb{Z}^{(I)}$ using this isomorphism, we write each element of $\mathbb{Z}^{(2I)}$ in the form $({\boldsymbol u},{\boldsymbol v})$ with ${\boldsymbol u},{\boldsymbol v}\in\mathbb{Z}^{(I)}$. Consider the map $$\varphi: \mathbb{Z}^{(2I)}\to \mathbb{Z}^{(I)},\ ({\boldsymbol u},{\boldsymbol v})\mapsto {\boldsymbol u}-{\boldsymbol v}.$$ Via this map, the Graver basis of any lattice in $\mathbb{Z}^{(I)}$ is intimately related to the Hilbert basis of a nonnegative monoid in $\mathbb{Z}^{(2I)}$. As usual, we denote standard basis of $\mathbb{Z}^{(I)}$ by $\{{\boldsymbol e}_{\boldsymbol i}\}_{{\boldsymbol i}\in I}$. **Proposition 35**. Let $L\subseteq\mathbb{Z}^{(I)}$ be a lattice and set $M=\varphi^{-1}(L)\cap \mathbb{Z}^{(2I)}_{\ge0}$. Denote by $\mathcal{G}$ the Graver basis of $L$ and $\mathcal{H}$ the Hilbert basis of $M$. Then $L=\varphi(M)$. Moreover, the following hold: 1. $\mathcal{G}=\varphi(\mathcal{H})\setminus\{{\boldsymbol 0}\}$. 2. $\mathcal{H}=\{({\boldsymbol u}^+,{\boldsymbol u}^-)\mid {\boldsymbol u}={\boldsymbol u}^+-{\boldsymbol u}^-\in \mathcal{G}\}\cup\{({\boldsymbol e}_{\boldsymbol i},{\boldsymbol e}_{\boldsymbol i})\mid {\boldsymbol i}\in I\}$. *Proof.* Since every element ${\boldsymbol u}\in L$ can be written in the form ${\boldsymbol u}={\boldsymbol u}^+-{\boldsymbol u}^-$, we see that ${\boldsymbol u}=\varphi(({\boldsymbol u}^+,{\boldsymbol u}^-))\in \varphi(M)$. Thus $\varphi(M)=L.$ \(i\) Let ${\boldsymbol u}\in \mathcal{G}$. Then $({\boldsymbol u}^+,{\boldsymbol u}^-)\in M\setminus\{{\boldsymbol 0}\}$. So by , there exists $({\boldsymbol v},{\boldsymbol w})\in\mathcal{H}$ such that $({\boldsymbol v},{\boldsymbol w})\sqsubseteq ({\boldsymbol u}^+,{\boldsymbol u}^-)$. This means that ${\boldsymbol v}\sqsubseteq {\boldsymbol u}^+$ and ${\boldsymbol w}\sqsubseteq {\boldsymbol u}^-$. In particular, ${\boldsymbol v}\ne{\boldsymbol w}$ as ${\boldsymbol u}^+$ and ${\boldsymbol u}^-$ have disjoint supports. Hence, $${\boldsymbol 0}\ne\varphi(({\boldsymbol v},{\boldsymbol w}))= {\boldsymbol v}-{\boldsymbol w} \sqsubseteq {\boldsymbol u}^+-{\boldsymbol u}^-={\boldsymbol u}.$$ Since ${\boldsymbol u}\in \mathcal{G}$, we must have ${\boldsymbol u}=\varphi(({\boldsymbol v},{\boldsymbol w}))\in\varphi(\mathcal{H})\setminus\{{\boldsymbol 0}\}$. Conversely, take ${\boldsymbol v}-{\boldsymbol w}\in \varphi(\mathcal{H})\setminus\{{\boldsymbol 0}\}$ with $({\boldsymbol v},{\boldsymbol w})\in\mathcal{H}$. Then there exists ${\boldsymbol u}\in\mathcal{G}$ such that ${\boldsymbol u}={\boldsymbol u}^+-{\boldsymbol u}^-\sqsubseteq {\boldsymbol v}-{\boldsymbol w}$. This implies $({\boldsymbol u}^+,{\boldsymbol u}^-)\sqsubseteq ({\boldsymbol v},{\boldsymbol w})$ since ${\boldsymbol u}^+$ and ${\boldsymbol u}^-$ have disjoint supports. The fact that $({\boldsymbol v},{\boldsymbol w})\in\mathcal{H}$ forces $({\boldsymbol u}^+,{\boldsymbol u}^-)= ({\boldsymbol v},{\boldsymbol w})$. Hence, ${\boldsymbol v}-{\boldsymbol w}={\boldsymbol u}\in\mathcal{G}.$ \(ii\) Evidently, $({\boldsymbol e}_{\boldsymbol i},{\boldsymbol e}_{\boldsymbol i})\in\mathcal{H}$ for all ${\boldsymbol i}\in I$. Moreover, arguing as in the first part of the proof of (i), we see that $({\boldsymbol u}^+,{\boldsymbol u}^-)\in\mathcal{H}$ for all ${\boldsymbol u}\in\mathcal{G}$. Conversely, let $({\boldsymbol v},{\boldsymbol w})\in\mathcal{H}$. If ${\boldsymbol v}={\boldsymbol w}$, then $({\boldsymbol e}_{\boldsymbol i},{\boldsymbol e}_{\boldsymbol i})\sqsubseteq ({\boldsymbol v},{\boldsymbol w})$ for some ${\boldsymbol i}\in I$, which implies $({\boldsymbol v},{\boldsymbol w})=({\boldsymbol e}_{\boldsymbol i},{\boldsymbol e}_{\boldsymbol i})$. Otherwise, if ${\boldsymbol v}\ne{\boldsymbol w}$, then arguing as in the second part of the proof of (i), we get $({\boldsymbol v},{\boldsymbol w})=({\boldsymbol u}^+,{\boldsymbol u}^-)$ for some ${\boldsymbol u}\in\mathcal{G}$. ◻ When $I=\mathbb{N}^d\times[c]$, one can identify $2I$ with $\mathbb{N}^d\times[2c]$. In this case, it is apparent that the maps $\psi$ and $\varphi$ are compatible with the action of the group $\mathop{\mathrm{Sym}}$, i.e. $$\psi(\sigma({\boldsymbol u}))=\sigma(\psi({\boldsymbol u})),\quad \varphi(\sigma({\boldsymbol u}))=\sigma(\varphi({\boldsymbol u})) \quad\text{for all }\ {\boldsymbol u}\in \mathbb{Z}^{(2I)}, \sigma\in\mathop{\mathrm{Sym}}.$$ Thus, in particular, $\varphi^{-1}(L)$ is a $\mathop{\mathrm{Sym}}$-invariant lattice in $\mathbb{Z}^{(2I)}$ for any $\mathop{\mathrm{Sym}}$-invariant lattice $L\subseteq\mathbb{Z}^{(I)}$. **Corollary 36**. Suppose $I=\mathbb{N}^d\times[c]$, where $c,d\in\mathbb{N}$. Let $L\subseteq\mathbb{Z}^{(I)}$ be a $\mathop{\mathrm{Sym}}$-invariant lattice and let $M=\varphi^{-1}(L)\cap \mathbb{Z}^{(2I)}_{\ge0}$. Then the following statements are equivalent: 1. $M$ has a finite equivariant Hilbert basis; 2. $L$ has a finite equivariant Graver basis. *Proof.* (i)$\Rightarrow$(ii): Suppose $\mathcal{H}$ is an equivariant Hilbert basis of $M$. Since $\varphi$ is compatible with the $\mathop{\mathrm{Sym}}$-action, we have $$\mathop{\mathrm{Sym}}(\varphi(\mathcal{H})\setminus\{{\boldsymbol 0}\}) = \varphi(\mathop{\mathrm{Sym}}(\mathcal{H}))\setminus\{{\boldsymbol 0}\}.$$ By , $\varphi(\mathop{\mathrm{Sym}}(\mathcal{H}))\setminus\{{\boldsymbol 0}\}$ is the Graver basis of $L$. Hence, $\varphi(\mathcal{H})\setminus\{{\boldsymbol 0}\}$ is an equivariant Graver basis of $L$. If $\mathcal{H}$ is finite, then of course $\varphi(\mathcal{H})\setminus\{{\boldsymbol 0}\}$ is also finite. (ii)$\Rightarrow$(i): Let $\mathcal{G}$ be an equivariant Graver basis of $L$. Denote $$\mathcal{H}=\{({\boldsymbol u}^+,{\boldsymbol u}^-)\mid {\boldsymbol u}={\boldsymbol u}^+-{\boldsymbol u}^-\in \mathcal{G}\} \cup\{({\boldsymbol e}_{\boldsymbol i},{\boldsymbol e}_{\boldsymbol i})\mid {\boldsymbol i}\in [d]^d\times[c]\}.$$ Then it is easy to check that $$\mathop{\mathrm{Sym}}(\mathcal{H})=\{({\boldsymbol u}^+,{\boldsymbol u}^-)\mid {\boldsymbol u}={\boldsymbol u}^+-{\boldsymbol u}^-\in \mathop{\mathrm{Sym}}(\mathcal{G})\} \cup\{({\boldsymbol e}_{\boldsymbol i},{\boldsymbol e}_{\boldsymbol i})\mid {\boldsymbol i}\in I\}.$$ Hence, $\mathcal{H}$ is an equivariant Hilbert basis of $M$ by . Clearly, $\mathcal{H}$ is finite if $\mathcal{G}$ is. ◻ In view of the previous corollary, will follow from the next lemma that is a generalization of . **Lemma 37**. Let $I=\mathbb{N}\times[c]$, where $c\in\mathbb{N}$. Then for any $\mathop{\mathrm{Sym}}$-invariant lattice $L\subseteq\mathbb{Z}^{(I)}$, the monoid $M=L\cap\mathbb{Z}^{(I)}_{\ge0}$ has a finite equivariant Hilbert basis. We do not know whether the "elementary\" proof of can be adapted to prove this result. Our proof here uses a different argument that requires Higman's lemma. For the proof, let us first recall a few notions. Let $\le$ be a partial order on a set $S$. Then $\le$ is called a *well-partial-order* if any infinite sequence $s_1,s_2,\dots$ in $S$ is *good*, i.e. $s_i\le s_j$ for some indices $i<j$. For example, $\sqsubseteq$ is a well-partial-order on $\mathbb{Z}^n$ by Gordan--Dickson lemma (see [@DHK Lemma 2.5.6]). A *final segment* is a subset $\mathcal{F}\subseteq S$ with the property that if $s\in \mathcal{F}$ and $s\le t\in S$, then $t\in \mathcal{F}$. The final segment *generated* by a subset $T\subseteq S$ is defined as $$\mathcal{F}(T)\coloneqq\{s\in S\mid s\le t \text{ for some } t\in T\}.$$ A classical characterization of well-partial-orders (see, e.g. [@Hig Theorem 2.1]) is that every final segment is finitely generated. Let $S^*=\bigcup_{n\in\mathbb{N}}S^n$ denote the set of finite sequences of elements of $S$. Define the *Higman order* $\le_H$ on $S^*$ as follows: $(s_1,\dots,s_p)\le_H (s_1',\dots,s_q')$ if there exists a strictly increasing map $\pi:[p]\to [q]$ such that $s_i\le s_{\pi(i)}'$ for all $i\in[p]$. Then Higman's lemma [@Hig Theorem 4.3] states that $\le_H$ is a well-partial-order on $S^*$. In order to apply Higman's lemma to the situation of , it is convenient to consider the monoid of strictly increasing functions $$\mathop{\mathrm{Inc}}= \{ \pi \colon \mathbb{N}\to \mathbb{N}\mid \pi(i)<\pi(i+1) \text{ for all } i\in \mathbb{N}\}.$$ This monoid acts on $\mathbb{Z}^{(I)}$ when $I=\mathbb{N}^d\times[c]$ analogously to $\mathop{\mathrm{Sym}}$. For simplicity, we only describe the action in the case $I=\mathbb{N}\times[c]$, which is enough for the proof of . Let $\{{\boldsymbol e}_{i,j}\}_{(i,j)\in\mathbb{N}\times[c]}$ be the standard basis of $\mathbb{Z}^{(I)}$. Then the action of $\mathop{\mathrm{Inc}}$ on $\mathbb{Z}^{(I)}$ is determined by $$\pi({\boldsymbol e}_{i,j})= {\boldsymbol e}_{\pi(i),j} \ \text{ for any } \pi\in\mathop{\mathrm{Inc}} \text{ and } (i,j)\in\mathbb{N}\times[c].$$ This action is closely related to that of $\mathop{\mathrm{Sym}}$; see [@KLR] for details in the case $c=1$. Note that for any $\pi\in\mathop{\mathrm{Inc}}$ and ${\boldsymbol u}\in\mathbb{Z}^{(I)}$, there exists $\sigma\in\mathop{\mathrm{Sym}}$ such that $\pi({\boldsymbol u})=\sigma({\boldsymbol u}).$ Indeed, $\mathop{\mathrm{supp}}({\boldsymbol u})$ is contained in $[n]\times[c]$ for some $n\in\mathbb{N}$. Since the restriction $\pi|_{[n]}$ is injective, there exists $\sigma\in\mathop{\mathrm{Sym}}$ such that $\pi|_{[n]}=\sigma|_{[n]}$, which yields $\pi({\boldsymbol u})=\sigma({\boldsymbol u}).$ We are now ready to prove . *Proof of .* Define a partial order $\sqsubseteq_{\mathop{\mathrm{Inc}}}$ on $\mathbb{Z}_{\ge0}^{(I)}$ as follows: ${\boldsymbol u}\sqsubseteq_{\mathop{\mathrm{Inc}}}{\boldsymbol v}$ if there exists $\pi\in \mathop{\mathrm{Inc}}$ such that $\pi({\boldsymbol u})\sqsubseteq{\boldsymbol v}.$ We show that $\sqsubseteq_{\mathop{\mathrm{Inc}}}$ is a well-partial-order. Indeed, recall that $\sqsubseteq$ is a well-partial-order on $\mathbb{Z}_{\ge0}^{c}$ by Gordan--Dickson lemma. So by Higman's lemma, the Higman order $\sqsubseteq_H$ is a well-partial-order on $(\mathbb{Z}_{\ge0}^{c})^*$. Via the natural bijection $$\mathbb{Z}_{\ge0}^{(I)}=\mathbb{Z}_{\ge0}^{(\mathbb{N}\times [c])} =\bigcup_{n\in\mathbb{N}}\mathbb{Z}_{\ge0}^{([n]\times [c])} \cong \bigcup_{n\in\mathbb{N}}(\mathbb{Z}_{\ge0}^{c})^n =(\mathbb{Z}_{\ge0}^{c})^*,$$ we may identify the partially ordered set $(\mathbb{Z}_{\ge0}^{(I)},\sqsubseteq_{\mathop{\mathrm{Inc}}})$ with $((\mathbb{Z}_{\ge0}^{c})^*,\sqsubseteq_H)$. Therefore, $\sqsubseteq_{\mathop{\mathrm{Inc}}}$ is a well-partial-order on $\mathbb{Z}_{\ge0}^{(I)}$. Now consider the final segment generated by $M\setminus\{{\boldsymbol 0}\}$ with respect to $\sqsubseteq_{\mathop{\mathrm{Inc}}}$: $$\mathcal{F}=\mathcal{F}(M\setminus\{{\boldsymbol 0}\})=\{{\boldsymbol v}\in \mathbb{Z}_{\ge0}^{(I)} \mid {\boldsymbol u}\sqsubseteq_{\mathop{\mathrm{Inc}}}{\boldsymbol v}\text{ for some } {\boldsymbol u}\in M\setminus\{{\boldsymbol 0}\}\}.$$ Since $\sqsubseteq_{\mathop{\mathrm{Inc}}}$ is a well-partial-order, $\mathcal{F}$ is finitely generated. That is, $\mathcal{F}=\mathcal{F}(\mathcal{H})$ for some finite subset $\mathcal{H}\subseteq M\setminus\{{\boldsymbol 0}\}$. We prove that $\mathcal{H}$ contains an equivariant Hilbert basis of $M$. By , it suffices to show that $\mathop{\mathrm{Sym}}(\mathcal{H})$ contains all $\sqsubseteq$-minimal elements of $M\setminus\{{\boldsymbol 0}\}$. Indeed, for any ${\boldsymbol v}\in M\setminus\{{\boldsymbol 0}\}$, there exist ${\boldsymbol u}\in\mathcal{H}$ and $\pi\in\mathop{\mathrm{Inc}}$ such that $\pi({\boldsymbol u})\sqsubseteq {\boldsymbol v}$. Since $\pi({\boldsymbol u})=\sigma({\boldsymbol u})$ for some $\sigma\in\mathop{\mathrm{Sym}}$, the desired assertion follows. ◻ Let us conclude this section with the proof of . *Proof of .* The result follows easily from . ◻ # Finiteness of equivariant generating sets and Markov bases {#sec-Markov} Throughout this section, let $I=\mathbb{N}^d\times[c]$ with $c,d\in\mathbb{N}$. We study here for equivariant generating sets and Markov bases of $\mathop{\mathrm{Sym}}$-invariant lattices in $\mathbb{Z}^{(I)}$. Based on a result of Hillar and Martín del Campo [@HM Theorem 19], we show that every $\mathop{\mathrm{Sym}}$-invariant lattice in $\mathbb{Z}^{(I)}$ has a finite equivariant generating set. Concerning finiteness of equivariant Markov bases, we provide a global version of the independent set theorem of Hillar and Sullivant [@HS12 Theorem 4.7]. ## Equivariant Noetherianity We say that $\mathbb{Z}^{(I)}$ is an *equivariantly Noetherian $\mathbb{Z}$-module* if every $\mathop{\mathrm{Sym}}$-invariant lattice in $\mathbb{Z}^{(I)}$ has a finite equivariant generating set. Our goal here is to prove the following result and discuss its consequences. **Theorem 38**. $\mathbb{Z}^{(I)}$ is an equivariantly Noetherian $\mathbb{Z}$-module. When $c=1$, this result follows from [@HM Theorem 19]. It is therefore enough to prove the following lemma. **Lemma 39**. Let $I^{(1)}=\mathbb{N}^d\times[c_1]$ and $I^{(2)}=\mathbb{N}^d\times[c_2]$, where $c_1,c_2\in\mathbb{Z}_{\ge0}$ with $c_1+c_2=c$. Then $\mathbb{Z}^{(I)}$ is an equivariantly Noetherian $\mathbb{Z}$-module if $\mathbb{Z}^{(I^{(1)})}$ and $\mathbb{Z}^{(I^{(2)})}$ are. *Proof.* The natural isomorphism $\mathbb{Z}^{(I)}\cong\mathbb{Z}^{(I^{(1)})}\oplus \mathbb{Z}^{(I^{(2)})}$ yields the short exact sequence $$\xymatrix{ 0 \ar[r] & \mathbb{Z}^{(I^{(1)})} \ar[r]^-{\iota} & \mathbb{Z}^{(I)} \ar[r]^-{\eta} & \mathbb{Z}^{(I^{(2)})}\ar[r] & 0, }$$ where $\iota$ and $\eta$ are compatible with the $\mathop{\mathrm{Sym}}$-action. For any $\mathop{\mathrm{Sym}}$-invariant lattice $L\subseteq\mathbb{Z}^{(I)}$, it is then routine to construct a finite equivariant generating set of $L$ from finite equivariant generating sets of the lattices $\eta(L)\subseteq\mathbb{Z}^{(I^{(2)})}$ and $\iota^{-1}(L)\subseteq\mathbb{Z}^{(I^{(1)})}$. ◻ Let us now derive a few consequences of . First, combining yields the next corollary, which can be seen as a local version of . **Corollary 40**. For any $\mathop{\mathrm{Sym}}$-invariant chain of lattices $\mathfrak{L}=(L_n)_{n\ge1}$ in $\mathbb{Z}^{(I)}$, the statements (i), (ii), (iii) in always hold true. The following algebraic version of is obtained by using . **Corollary 41**. For any $\mathop{\mathrm{Sym}}$-invariant lattice $L\subseteq\mathbb{Z}^{(I)}$, the Laurent ideal $\mathfrak{I}_L^\pm$ is equivariantly finitely generated. Now together with yields the following generalization of [@HM Theorem 3]. **Corollary 42**. For any $\mathop{\mathrm{Sym}}$-invariant chain of lattices $\mathfrak{L}=(L_n)_{n\ge1}$ in $\mathbb{Z}^{(I)}$, the chain of Laurent ideals $(\mathfrak{I}_{L_n}^\pm)_{n\ge1}$ stabilizes. It should be noted that is no longer true when $d\ge2$ if generating set is replaced with Markov basis. **Example 43**. Consider the *no $3$-way interaction model*, i.e. the hierarchical model defined by $\Delta=\{\{1,2\},\{1,3\},\{2,3\}\}$ and ${\boldsymbol r}=(r_1,r_2,r_3)\in\mathbb{N}^3$. For such a model, the $\Delta$-marginal map $$\mu_{\Delta,{\boldsymbol r}}:\mathbb{Z}^{([r_1]\times[r_2]\times[r_3])}\to \mathbb{Z}^{(([r_1]\times[r_2])}\oplus\mathbb{Z}^{(([r_1]\times[r_3])} \oplus \mathbb{Z}^{(([r_2]\times[r_3])}$$ computes all $2$-way marginals of each $3$-way table in $\mathbb{Z}^{([r_1]\times[r_2]\times[r_3])}$. It is known that (see [@DS] and also [@HS12 Example 4.3]) when $r_3=c\ge2$ is fixed and $r_1=r_2=n$ vary, any Markov basis of the $\mathop{\mathrm{Sym}}$-invariant lattice $L_n=\ker\mu_{\Delta,{\boldsymbol r}}\subseteq\mathbb{Z}^{([n]^2\times[c])}$ contains the following element $${\boldsymbol u}=\sum_{i=1}^n({\boldsymbol e}_{i,i,1}-{\boldsymbol e}_{i,i,2}) + \sum_{i=1}^{n-1}({\boldsymbol e}_{i,i+1,2}-{\boldsymbol e}_{i,i+1,1}) +({\boldsymbol e}_{n,1,2}-{\boldsymbol e}_{n,1,1}),$$ where ${\boldsymbol e}_{i,j,k}$ denotes a standard basis element of $\mathbb{Z}^{(\mathbb{N}^2\times[c])}$. Obviously, the support size of ${\boldsymbol u}$ is unbounded when $n$ tends to $\infty$. So by [Theorem 21](#stabilization-others){reference-type="ref" reference="stabilization-others"}, the lattice $L=\bigcup_{n\ge1}L_n\subseteq\mathbb{Z}^{(\mathbb{N}^2\times[c])}$ has no finite equivariant Markov bases. When ${\boldsymbol r}=(r_1,r_2,3)$, where $r_1,r_2$ vary, De Loera and Onn [@DO Theorem 1.2] even showed that Markov bases of $\ker\mu_{\Delta,{\boldsymbol r}}$ can be arbitrarily complicated: for any ${\boldsymbol v}\in\mathbb{N}^k$, there exist $r_1,r_2$ such that any Markov basis of $\ker\mu_{\Delta,{\boldsymbol r}}$ contains an elements whose restriction to some $k$ entries is precisely ${\boldsymbol v}$. ## The independent set theorem In view of , it is of great interest to know which $\mathop{\mathrm{Sym}}$-invariant lattices in $\mathbb{Z}^{(I)}$ have finite equivariant Markov bases when $d\ge2$. We discuss here one of the major results in this direction, the independent set theorem of Hillar and Sullivant [@HS12 Theorem 4.7], which is a generalization of the finiteness result of Hoşten and Sullivant [@HoS] mentioned in the previous section. For related results see [@Do; @DoS; @DEKL; @KKL]. Consider the hierarchical model determined by a simplicial complex $\Delta\subseteq 2^{[m]}$ and a vector ${\boldsymbol r}=(r_1,\dots,r_m)\in\mathbb{N}^m$. In the independent set theorem, several entries of ${\boldsymbol r}$, indexed by an independent set, can vary, while the others are fixed. Here, a subset $T\subseteq[m]$ is called an *independent set* of $\Delta$ if $|T\cap F_k|\le 1$ for all $k\in [s]$, where $F_1,\dots,F_k$ are the facets of $\Delta$. Fix $r_j$ for $j\in[m]\setminus T$ and let $r_i=n$ for all $i\in T$. Denote $d=|T|$, $c=\prod_{j\in[m]\setminus T} r_j$ and $d_k=|T\cap F_k|$, $c_k=\prod_{j\in F_k\setminus T} r_j$ for $k\in [s]$. Then $\mathcal{R}=\prod_{i=1}^m[r_i]$ and $\mathcal{R}_{F_k}=\prod_{j\in F_k}[r_j]$ can be identified with $[n]^d\times[c]$ and $[n]^{d_k}\times[c_k]$, respectively. The $\Delta$-marginal map now becomes a map $$\mu_{\Delta,{\boldsymbol r}}:\mathbb{Z}^{([n]^d\times[c])}\to \bigoplus_{k=1}^s\mathbb{Z}^{([n]^{d_k}\times[c_k])}.$$ We will write $\mu_{\Delta,{\boldsymbol r}}$ as $\mu_{\Delta,T,n}$ to emphasize its dependence on $T$ and $n$. Consider the action of $\mathop{\mathrm{Sym}}(n)$ on $\mathbb{Z}^{([n]^d\times[c])}$ and $\mathbb{Z}^{([n]^{d_k}\times[c_k])}$ as before. It is easy to see that $\mu_{\Delta,T,n}$ is compatible with the $\mathop{\mathrm{Sym}}(n)$-action. Thus, the lattice $\ker\mu_{\Delta,T,n}$ is $\mathop{\mathrm{Sym}}(n)$-invariant, and moreover, the chain $\mathfrak{L}_{\Delta,T}=(\ker\mu_{\Delta,T,n})_{n\ge1}$ is $\mathop{\mathrm{Sym}}$-invariant. With the above setup, the independent set theorem of Hillar and Sullivant can be stated as follows. **Theorem 44**. Consider the hierarchical model defined by $\Delta\subseteq 2^{[m]}$ and ${\boldsymbol r}=(r_1,\dots,r_m)$. Suppose $T\subseteq[m]$ is an independent set of $\Delta$. Fix $r_j$ for $j\in[m]\setminus T$ and let $r_i=n$ vary for all $i\in T$. Then the $\mathop{\mathrm{Sym}}$-invariant chain of lattices $\mathfrak{L}_{\Delta,T}=(\ker\mu_{\Delta,T,n})_{n\ge1}$ Markov-stabilizes. Extending the maps $\mu_{\Delta,T,n}$, we obtain the *global $\Delta$-marginal map* $$\mu_{\Delta,T}:\mathbb{Z}^{(\mathbb{N}^d\times[c])}\to \bigoplus_{k=1}^s\mathbb{Z}^{(\mathbb{N}^{d_k}\times[c_k])}$$ with $\mu_{\Delta,T}|_{\mathbb{Z}^{([n]^d\times[c])}}=\mu_{\Delta,T,n}$ for all $n\ge 1$. Evidently, $\ker\mu_{\Delta,T}=\bigcup_{n\ge1}\ker\mu_{\Delta,T,n}$. thus yields the following global version of . **Corollary 45**. Keep the assumption of . Then the lattice $\ker\mu_{\Delta,T}$ has a finite equivariant Markov basis. Note that the assumption that $T$ is an independent set cannot be omitted in , as shows. **Remark 46**. Hillar and Sullivant actually proved the algebraic version of , i.e. the chain of lattice ideals $(\mathfrak{I}_{L_n})_{n\ge1}$ stabilizes, where $L_n=\ker\mu_{\Delta,T,n}$ for $n\ge1$. It would be interesting to have *purely combinatorial* proofs of . The framework developed in this paper might be helpful in finding such proofs. It is still open whether the lattice $\ker\mu_{\Delta,T}$ always has a finite equivariant Gröbner basis. On the other hand, this lattice does not have a finite equivariant Graver basis in general. **Example 47**. Consider the *independence model* for 2-way tables, i.e. the hierarchical model determined by $\Delta=\{\{1\},\{2\}\}$ and ${\boldsymbol r}=(r_1,r_2)\in\mathbb{N}^2$. In this model, $T=\{1,2\}$ is an independent set of $\Delta$. Let $r_1=r_2=n$ vary. Then it is known that the Graver basis of $\ker\mu_{\Delta,T,n}$ consists of the elements $${\boldsymbol u}={\boldsymbol e}_{i_1,j_1}-{\boldsymbol e}_{i_1,j_2}+{\boldsymbol e}_{i_2,j_2}-{\boldsymbol e}_{i_2,j_3} +\cdots+ {\boldsymbol e}_{i_k,j_k}-{\boldsymbol e}_{i_k,j_1},$$ where $2\le k\le n$ and each index vector $(i_1,\dots,i_k),(j_1,\dots,j_k)\in[n]^k$ has pairwise distinct entries; see [@AHT Proposition 4.2]. Hence, $\ker\mu_{\Delta,T}$ has no finite equivariant Graver bases by . We close this section with the following cofiniteness property of the lattice $\ker\mu_{\Delta,T}$. **Proposition 48**. Keep the assumption of . Then the quotient abelian group $\mathbb{Z}^{(\mathbb{N}^d\times[c])}/\ker\mu_{\Delta,T}$ is a lattice that has a finite equivariant Graver basis. The proof is based on the next result that slightly generalizes . **Lemma 49**. Let $c,m\in\mathbb{Z}_{\ge0}$. Consider the action of $\mathop{\mathrm{Sym}}$ on $\mathbb{Z}^{(\mathbb{N}\times[c])}\oplus\mathbb{Z}^m$ that extends its action on $\mathbb{Z}^{(\mathbb{N}\times[c])}$ with a trivial action on $\mathbb{Z}^m$, i.e. $$\sigma({\boldsymbol u},{\boldsymbol v})=(\sigma({\boldsymbol u}),{\boldsymbol v}) \ \text{ for any } {\boldsymbol u}\in \mathbb{Z}^{(\mathbb{N}\times[c])}, {\boldsymbol v}\in \mathbb{Z}^m, \sigma\in\mathop{\mathrm{Sym}}.$$ Then every $\mathop{\mathrm{Sym}}$-invariant lattice in $\mathbb{Z}^{(\mathbb{N}\times[c])}\oplus\mathbb{Z}^m$ has a finite equivariant Graver basis. *Proof.* Let $L\subseteq \mathbb{Z}^{(\mathbb{N}\times[c])}\oplus\mathbb{Z}^m$ be an arbitrary $\mathop{\mathrm{Sym}}$-invariant lattice. Arguing similarly to the proofs of , one can show that $L$ has a finite equivariant Graver basis if and only if the monoid $\varphi^{-1}(L)\cap (\mathbb{Z}_{\ge0}^{(\mathbb{N}\times[2c])}\oplus\mathbb{Z}_{\ge0}^{[2m]})$ has a finite equivariant Hilbert basis, where $$\varphi: \mathbb{Z}^{(\mathbb{N}\times[2c])}\oplus\mathbb{Z}^{[2m]}\cong (\mathbb{Z}^{(\mathbb{N}\times[c])}\oplus\mathbb{Z}^m)\oplus(\mathbb{Z}^{(\mathbb{N}\times[c])}\oplus\mathbb{Z}^m) \to \mathbb{Z}^{(\mathbb{N}\times[c])}\oplus\mathbb{Z}^m$$ is defined as in . So by renaming $c$ and $m$, it suffices to show that the monoid $M=L'\cap (\mathbb{Z}_{\ge0}^{(\mathbb{N}\times[c])}\oplus\mathbb{Z}_{\ge0}^{[m]})$ has a finite equivariant Hilbert basis for any $\mathop{\mathrm{Sym}}$-invariant lattice $L'\subseteq \mathbb{Z}^{(\mathbb{N}\times[c])}\oplus\mathbb{Z}^m$. Consider the following partial order on $\mathbb{Z}_{\ge0}^{(\mathbb{N}\times[c])}\oplus\mathbb{Z}_{\ge0}^{[m]}$: $$({\boldsymbol u},{\boldsymbol v})\preceq ({\boldsymbol u}',{\boldsymbol v}') \ \text{ if }\ {\boldsymbol u}\sqsubseteq_{\mathop{\mathrm{Inc}}} {\boldsymbol u}' \ \text{ and } {\boldsymbol v}\sqsubseteq {\boldsymbol v}',$$ where $\sqsubseteq_{\mathop{\mathrm{Inc}}}$ is defined in the proof of . Since $\sqsubseteq_{\mathop{\mathrm{Inc}}}$ and $\sqsubseteq$ are both well-partial-orders, it is easy to check that $\preceq$ is also a well-partial-order (see, e.g. [@Dr14]). Now an argument similar to the proof of shows that $M$ has a finite equivariant Hilbert basis, as desired. ◻ Let us now prove . *Proof of .* First, $\mathbb{Z}^{(\mathbb{N}^d\times[c])}/\ker\mu_{\Delta,T}$ is a lattice because it is isomorphic to a subgroup of $\bigoplus_{k=1}^s\mathbb{Z}^{(\mathbb{N}^{d_k}\times[c_k])}$. 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--- abstract: | In this note we explore how standard statistical distances are equivalent for discrete log-concave distributions. Distances include total variation distance, Wasserstein distance, and $f$-divergences. author: - Arnaud Marsiglietti and Puja Pandey title: A Note on Statistical Distances for Discrete Log-Concave Measures --- **Keywords:** log-concave, total variation distance, Wasserstein distance, $f$-divergence. # Introduction The study of convergence of probability measures is central in probability and statistics, and may be performed via statistical distances for which the choice has its importance (see, e.g., [@GS02], [@Rac]). The space of probability measures, say over the real numbers, is infinite dimensional, therefore there is a priori no canonical distance, and distances may not be equivalent. Nonetheless, an essential contribution made by Meckes and Meckes in [@MM] demonstrates that certain statistical distances between continuous log-concave distributions turn out to be equivalent up to constants that may depend on the dimension of the ambient space (see also [@CG] for improved bounds, and [@MP] for the extension to the broader class of so-called $s$-concave distributions). The goal of this note is to develop quantitative comparisons between distances for discrete log-concave distributions. Let us denote by $\mathbb{N}= \{0, 1, 2, \dots\}$ the set of natural numbers and by $\mathbb{Z}$ the set of integers. Recall that the probability mass function (p.m.f.) associated with an integer valued random variable $X$ is $$p(k) = \mathbb{P}(X = k), \quad k \in \mathbb{Z}.$$ An integer-valued random variable $X$ is said to be log-concave if its probability mass function $p$ satisfies $$p(k)^{2} \geq p(k-1)p(k+1)$$ for all $k \in \mathbb{Z}$ and the support of $X$ is an integer interval. Discrete log-concave distributions form an important class. Examples include Bernoulli, discrete uniform, binomial, geometric and Poisson distributions. We refer to [@Sta], [@Bre94], [@SW], [@Bra15] for further background on log-concavity. Let us introduce the main distances we will work with (we refer to [@LV], [@Dud], [@GS02], [@Rac] for further background on statistical distances). Our setting is the real line $\mathbb{R}$ equipped with its usual Euclidean structure $d(x,y) = |x-y|$, $x,y \in \mathbb{R}$. 1. The bounded Lipschitz distance between two probability measures $\mu$ and $\nu$ is defined as $$d_{BL}(\mu,\nu) = \sup_{\|g\|_{BL} \leq 1} \left| \int g \, d\mu - \int g \, d\nu \right|,$$ where for a function $g \colon \mathbb{R}\to \mathbb{R}$, $$\|g\|_{BL} = \max \left\{ \|g\|_{\infty}, \, \sup_{x \neq y} \frac{|g(x)-g(y)|}{|x-y|} \right\}.$$ 2. The Lévy-Prokhorov distance between two probability measures $\mu$ and $\nu$ is defined as $$d_{LP}(\mu,\nu) = \inf \left\{ \epsilon > 0 : \mu(A) \leq \nu(A^{\varepsilon}) + \epsilon \mbox{ for all Borel set } A \subset \mathbb{R}\right\},$$ where $A^{\varepsilon} = \{x \in \mathbb{R}: d(x,A) < \varepsilon\}$. Using the Ky-Fan distance, which is defined for two random variables $X$ and $Y$ as $$K(X,Y) = \inf \{ \varepsilon> 0 : \mathbb{P}(|X-Y| > \varepsilon) < \varepsilon\},$$ the Lévy-Prokhorov distance admits the following coupling representation, $$\label{LV-coupl} d_{LP}(\mu,\nu)= \inf K(X,Y),$$ where the infimum runs over all random variables $X$ with distribution $\mu$ and random variables $Y$ with distribution $\nu$ (see, e.g., [@Rac]). 3. The total variation distance between two probability measures $\mu$ and $\nu$ is defined as $$\begin{aligned} d_{TV}(\mu,\nu) = 2 \sup_{A \subset \mathbb{R}} |\mu(A) - \nu(A)|.\end{aligned}$$ The total variation distance admits the following coupling representation, $$\label{d-tv-coupl} d_{TV}(\mu,\nu) = \inf \mathbb{P}(X \neq Y),$$ where the infimum runs over all random variables $X$ with distribution $\mu$ and random variables $Y$ with distribution $\nu$ (see, e.g., [@GS02]). Moreover, for integer valued measures, one has the following identity, $$\label{d-tv-dis} d_{TV}(\mu,\nu) = \sum_{k \in \mathbb{Z}} |\mu(\{k\}) - \nu(\{k\})|.$$ 4. The $p$-th Wasserstein distance, $p \geq 1$, between two probability measures $\mu$ and $\nu$ is defined as $$\begin{aligned} W_{p}(\mu,\nu) = \inf \mathbb{E}[|X-Y|^{p}]^{\frac{1}{p}}, \end{aligned}$$ where the infimum runs over all random variables $X$ with distribution $\mu$ and random variables $Y$ with distribution $\nu$. 5. Let $f \colon [0, +\infty) \to \mathbb{R}$ be a convex function such that $f(1) = 0$. The $f$-divergence between two probability measures $\mu$ and $\nu$ on $\mathbb{Z}$ is defined as $$d_f(\mu || \nu) = \sum_{k \in \mathbb{Z}} \nu(\{k\}) f\left( \frac{\mu(\{k\})}{\nu(\{k\})} \right).$$ Note that the choice of convex function $f(x) = x\log(x)$, $x \geq 0$, leads to the Kullback-Leibler divergence $$D(\mu || \nu) = \sum_{k \in \mathbb{Z}} \mu(\{k\}) \log \left( \frac{\mu(\{k\})}{\nu(\{k\})} \right),$$ the function $f(x) = (x-1)^2$ yields the so-called $\chi^2$-divergence, while $f(x) = |x-1|$ returns us to the total variation distance. Let us review the known relationships between the above distances. It is known [@Dud68 Corollaries 2 and 3] that bounded Lipschitz and Lévy-Prokhorov distances are equivalent, $$\frac{1}{2} d_{BL}(\mu,\nu) \leq d_{LP}(\mu,\nu) \leq \sqrt{\frac{3}{2} d_{BL}(\mu,\nu)}.$$ One also has $$d_{LP}(\mu,\nu) \leq d_{TV}(\mu,\nu),$$ and, for $\mu, \nu$ integer valued probability measures, $$d_{TV}(\mu,\nu) \leq W_1(\mu,\nu),$$ see [@GS02]. By Hölder's inequality, if $p \leq q$, then $$W_{p}(\mu,\nu) \leq W_{q}(\mu,\nu).$$ As for divergences, the Pinsker-Csiszár inequality ([@Pin], [@C]) states that $$d_{TV}(\mu,\nu) \leq \sqrt{2 D(\mu || \nu)}.$$ Also, one has $$D(\mu || \nu) \leq \log(1 + \chi^2(\mu || \nu)).$$ The article is organized as follows. In Section [2](#2){reference-type="ref" reference="2"}, we establish properties for log-concave distributions on $\mathbb{Z}$ that are of independent interests. In section [3](#3){reference-type="ref" reference="3"}, we present our results and proofs. # Preliminaries {#2} In this section we gather the main tools used throughout the proofs. First, recall that a real-valued random variable $X$ is said to be isotropic if $$\mathbb{E}[X] = 0, \quad \mathbb{E}[X^2] = 1.$$ We start with an elementary lemma that will allow us to pass results for log-concave distributions on $\mathbb{N}$ to log-concave distributions on $\mathbb{Z}$, however with sub-optimal constants. **Lemma 1**. *If $X$ is symmetric log-concave on $\mathbb{Z}$, then $|X|$ is log-concave on $\mathbb{N}$.* *Proof.* Denote by $p$ (resp. $q$) the p.m.f. of $X$ (resp. $|X|$). Then, $q(0) = p(0)$ and $q(k) = 2p(k)$ for $k \geq 1$. Therefore, $$q^2(1) = 4p^2(1) \geq 4 p(0) p(2) = 2q(0) q(2),$$ and for all $k \geq 2$, $$q^2(k) = 4p^2(k) \geq 4 p(k+1) p(k-1) = q(k+1) q(k-1).$$ Hence, $q$ is log-concave. ◻ We note that Lemma [Lemma 1](#sym){reference-type="ref" reference="sym"} no longer holds for non-symmetric log-concave random variables, as can be seen by taking $X$ supported on $\{-1,0,1,2,3\}$ with distribution $\mathbb{P}(X=-1) = \mathbb{P}(X=3) = 0.1$, $\mathbb{P}(X=0) = \mathbb{P}(X=2) = 0.2$, and $\mathbb{P}(X=1) = 0.4$. In this case, $\mathbb{P}(|X| = 2)^2 < \mathbb{P}(|X|=1)\mathbb{P}(|X|=3)$. The next lemma provides moments bounds for log-concave distributions on $\mathbb{Z}$. **Lemma 2**. *If $X$ is log-concave on $\mathbb{Z}$, then for all $\beta \geq 1$, $$\mathbb{E}[|X - \mathbb{E}[X]|^{\beta}]^{\frac{1}{\beta}} \leq \Gamma(\beta + 1)^{\frac{1}{\beta}} (2\mathbb{E}[|X - \mathbb{E}[X]|] + 1).$$* *Proof.* It has been shown in [@MM22 Corollary 4.5] that for all log-concave random variable $X$ on $\mathbb{N}$, for all $\beta \geq 1$, $$\label{moment} \mathbb{E}[X^{\beta}]^{\frac{1}{\beta}} \leq \Gamma(\beta + 1)^{\frac{1}{\beta}} (\mathbb{E}[X] + 1).$$ Let $X$ be a symmetric log-concave random variable on $\mathbb{Z}$, then by Lemma [Lemma 1](#sym){reference-type="ref" reference="sym"}, $|X|$ is log-concave on $\mathbb{N}$ so one may apply inequality [\[moment\]](#moment){reference-type="eqref" reference="moment"} to obtain $$\label{moment-sym} \mathbb{E}[|X|^{\beta}]^{\frac{1}{\beta}} \leq \Gamma(\beta + 1)^{\frac{1}{\beta}} (\mathbb{E}[|X|] + 1).$$ Now, let $X$ be a log-concave random variable on $\mathbb{Z}$. Let $Y$ be an independent copy of $X$, so that $X-Y$ is symmetric log-concave. Applying inequality [\[moment-sym\]](#moment-sym){reference-type="eqref" reference="moment-sym"}, we deduce that $$\mathbb{E}[|X - \mathbb{E}[X]|^{\beta}]^{\frac{1}{\beta}} \leq \mathbb{E}[|X-Y|^{\beta}]^{\frac{1}{\beta}} \leq \Gamma(\beta + 1)^{\frac{1}{\beta}} (\mathbb{E}[|X-Y|] + 1) \leq \Gamma(\beta + 1)^{\frac{1}{\beta}} (2\mathbb{E}[|X - \mathbb{E}[X]|] + 1),$$ where the first inequality follows from Hölder's inequality and the last inequality from triangle inequality. ◻ Let us derive concentration inequalities for log-concave distributions on $\mathbb{Z}$. **Lemma 3**. *For each log-concave random variable $X$ on $\mathbb{Z}$, one has for all $t \geq 0$, $$\mathbb{P}(|X - \mathbb{E}[X]| \geq t) \leq 2e^{- \frac{t}{2(2\mathbb{E}[|X - \mathbb{E}[X]|] + 1)}}.$$* *Proof.* The proof is a standard application of the moments bounds obtained in Lemma [Lemma 2](#moment-lem){reference-type="ref" reference="moment-lem"} (see, e.g., [@V22]). For $\lambda > 0$, $$\begin{aligned} \mathbb{E}[e^{\lambda |X - \mathbb{E}[X]|}] = 1 + \sum_{\beta \geq 1} \frac{\lambda^{\beta}}{\beta !} \mathbb{E}[|X - \mathbb{E}[X]|^{\beta}] & \leq & 1 + \sum_{\beta \geq 1} \frac{\lambda^{\beta}}{\beta !} \beta ! (2\mathbb{E}[|X - \mathbb{E}[X]|] + 1)^{\beta} \\ & = & \sum_{\beta \geq 0} \left[ \lambda (2\mathbb{E}[|X - \mathbb{E}[X]|] + 1) \right]^{\beta} \\ & = & \frac{1}{1-\lambda (2\mathbb{E}[|X - \mathbb{E}[X]|] + 1)}, \end{aligned}$$ where the last identity holds for all $0 < \lambda < \frac{1}{2\mathbb{E}[|X - \mathbb{E}[X]|] + 1}$. Choosing $\lambda = \frac{1}{2(2\mathbb{E}[|X - \mathbb{E}[X]|] + 1)}$ yields $$\mathbb{E}[e^{\lambda |X - \mathbb{E}[X]|}] \leq 2.$$ Therefore, by Markov's inequality, $$\mathbb{P}(|X - \mathbb{E}[X]| \geq t) = \mathbb{P}(e^{\lambda |X - \mathbb{E}[X]|} \geq e^{\lambda t}) \leq \mathbb{E}[e^{\lambda |X - \mathbb{E}[X]|}] e^{-\lambda t} \leq 2e^{- \frac{t}{2(2\mathbb{E}[|X - \mathbb{E}[X]|] + 1)}}.$$ ◻ The following lemma, which provides a bound on the variance and maximum of the probability mass function of log-concave distributions on $\mathbb{Z}$, was established in [@BMM] and [@Ara] (see, also, [@BC15], [@JMNS]). **Lemma 4** ([@BMM], [@Ara]). *Let $X$ be a log-concave distribution on $\mathbb{Z}$ with probability mass function $p$, then $$\sqrt{1 + \mathop{\mathrm{Var}}(X)} \leq \frac{1}{\|p\|_{\infty}} \leq \sqrt{1+ 12 \mathop{\mathrm{Var}}(X)}.$$* The following lemma is standard in information theory and provides an upper bound on the entropy of an integer valued random variable (see [@Ma88]). Recall that the Shannon entropy of an integer valued random variable $X$ with p.m.f. $p$ is defined as $$H(X) = \mathbb{E}[-\log(p(X))] = - \sum_{k \in \mathbb{Z}} p(k) \log(p(k)).$$ **Lemma 5** ([@Ma88]). *For any integer valued random variable $X$ with finite second moment, $$H(X) \leq \frac{1}{2} \log \left( 2 \pi e \left( \mathop{\mathrm{Var}}(X) + \frac{1}{12} \right) \right).$$* The last lemma of this section provides a bound on the second moment of the information content of a log-concave distribution on $\mathbb{Z}$. **Lemma 6**. *Let $X$ be a discrete log-concave random variable on $\mathbb{Z}$ with probability mass function $p$. Then, $$\mathbb{E}[\log^2(p(X))] \leq 4 \left( 4 e^{-2} + 1 + \frac{H^2(X)}{\|p\|_{\infty}} \right).$$* *Proof.* Let $X$ be a log-concave random variable with p.m.f. $p$. Then $p$ is unimodal, that is, there exists $m \in \mathbb{Z}$ such that for all $k \leq m$, $p(k-1) \leq p(k)$ and for all $k \geq m$, $p(k) \geq p(k+1)$. Note that $p(m) = \|p\|_{\infty}$. Define, for $k \in \mathbb{Z}$, $$p^{\nearrow}(k) = \frac{p(k)}{\sum_{l \leq m} p(l)} 1_{\{k \leq m\}},$$ and $$p^{\searrow}(k) = \frac{p(k)}{\sum_{l \geq m} p(l)} 1_{\{k \geq m\}}.$$ Note that both $p^{\nearrow}$ and $p^{\searrow}$ are monotone log-concave probability mass functions. Denote by $X^{\nearrow}$ (resp. $X^{\searrow}$) a random variable with p.m.f. $p^{\nearrow}$ (resp. $p^{\searrow}$). Denote also $a = \sum_{l \leq m} p(l)$ and $b=\sum_{l \geq m} p(l)$. On one hand, by a result of Melbourne and Palafox-Castillo [@MP-C Theorem 2.5], $$\mathop{\mathrm{Var}}(\log(p^{\nearrow}(X^{\nearrow}))) \leq 1, \qquad \mathop{\mathrm{Var}}(\log(p^{\searrow}(X^{\searrow}))) \leq 1.$$ On the other hand, $$H(X^{\nearrow}) = \sum_{k \leq m} \frac{p(k)}{a} \log \left( \frac{a}{p(k)} \right) \leq \frac{1}{a} \sum_{k \leq m} p(k) \log \left( \frac{1}{p(k)} \right) \leq \frac{1}{a} H(X),$$ and similarly, $$H(X^{\searrow}) \leq \frac{H(X)}{b}.$$ Therefore, $$\mathbb{E}[\log^2(p^{\nearrow}(X^{\nearrow}))] = \mathop{\mathrm{Var}}(\log(p^{\nearrow}(X^{\nearrow}))) + H^2(X^{\nearrow}) \leq 1 + \frac{H^2(X)}{a^2},$$ and similarly, $$\mathbb{E}[\log^2(p^{\searrow}(X^{\searrow}))] \leq 1 + \frac{H^2(X)}{b^2}.$$ We deduce that $$\begin{aligned} \mathbb{E}[\log^2(p(X))] & = & \sum_{k \in \mathbb{Z}} p(k) \log^2(p(k)) \\ & \leq & \sum_{k \in \mathbb{Z}} ap^{\nearrow}(k) \log^2(ap^{\nearrow}(k)) + \sum_{k \in \mathbb{Z}} bp^{\searrow}(k) \log^2(bp^{\searrow}(k)) \\ & \leq & 2 \left( a \log^2(a) + a\mathbb{E}[\log^2(p^{\nearrow}(X^{\nearrow}))] + b \log^2(b) + b\mathbb{E}[\log^2(p^{\searrow}(X^{\searrow}))] \right) \\ & \leq & 2 \left(4 e^{-2} + a + \frac{H^2(X)}{a} + 4 e^{-2} + b + \frac{H^2(X)}{b} \right) \\ & \leq & 4 \left( 4 e^{-2} + 1 + \frac{H^2(X)}{\|p\|_{\infty}} \right),\end{aligned}$$ where we used the fact that $a,b \in [\|p\|_{\infty}, 1]$. ◻ **Remark 7**. *For an isotropic log-concave random variable $X$ on $\mathbb{Z}$ with probability mass function $p$, the above bounds reduce to $$\begin{aligned} \label{moment-iso} \mathbb{E}[|X|^{\beta}]^{\frac{1}{\beta}} & \leq & 3 \Gamma(\beta + 1)^{\frac{1}{\beta}}, \quad \beta \geq 1, \\ \label{concentration-iso} \mathbb{P}(|X| \geq t) & \leq & 2 e^{- \frac{t}{6}}, \quad t \geq 0, \\ \label{infinity-iso} \sqrt{2} & \leq & \frac{1}{\|p\|_{\infty}} \leq \sqrt{13}, \\ \label{entropy-iso} H(X) & \leq & \frac{1}{2} \log \left( 2 \pi e \left( 1 + \frac{1}{12} \right) \right) \leq 3,\end{aligned}$$ in particular, we also deduce $$\label{varent-iso} \mathbb{E}[\log^2(p(X))] \leq 4(4e^{-2} + 1 + 9 \sqrt{13}) \leq 136.$$* # Main results and proofs {#3} This section contains our main results together with the proofs. The first theorem establishes quantitative reversal bounds between 1-Wasserstein distance and Lévy-Prokhorov distance. **Theorem 8**. *Let $\mu$ and $\nu$ be isotropic log-concave probability measures on $\mathbb{Z}$, then $$W_1(\mu, \nu) \leq 12 d_{LP}(\mu, \nu) \log \left( \frac{4 e}{ d_{LP}(\mu, \nu)} \right).$$* *Proof.* Let $R > 0$. Let $X$ (resp. $Y$) be distributed according to $\mu$ (resp. $\nu$). Note that for all $t \geq 0$, $$\mathbb{P}(|X-Y| > t) = \mathbb{P}(|X-Y| \geq \lfloor t \rfloor + 1) \leq \mathbb{P}(|X-Y| \geq 1) \leq K(X,Y),$$ therefore, $$\begin{aligned} \mathbb{E}[|X-Y|] & = & \int_0^{R} \mathbb{P}(|X-Y| > t) dt + \int_R^{\infty} \mathbb{P}(|X-Y| > t) dt \\ & \leq & R K(X,Y) + \int_R^{\infty} \mathbb{P}\left(|X| > \frac{t}{2} \right) dt + \int_R^{\infty} \mathbb{P}\left(|Y| > \frac{t}{2} \right) dt. \end{aligned}$$ Applying inequality [\[concentration-iso\]](#concentration-iso){reference-type="eqref" reference="concentration-iso"}, we obtain $$W_1(\mu, \nu) \leq \mathbb{E}[|X-Y|] \leq R K(X,Y) + 2 \int_R^{\infty} 2e^{- \frac{t}{12}} dt = R K(X,Y) + 48 e^{-\frac{R}{12}}.$$ The above inequality being true for any random variable $X$ with distribution $\mu$ and any random variable $Y$ with distribution $\nu$, we deduce by taking infimum over all couplings that $$W_1(\mu, \nu) \leq R d_{LP}(\mu, \nu) + 48 e^{-\frac{R}{12}}.$$ Choosing $R = 12 \log(4 / d_{LP}(\mu, \nu))$, which is nonnegative, yields the desired result. ◻ The next theorem demonstrates that Wasserstein distances are equivalent for discrete log-concave distributions. **Theorem 9**. *Let $\mu$ and $\nu$ be isotropic log-concave probability measures on $\mathbb{Z}$, then for all $1 \leq p \leq q$, $$W_q^q(\mu, \nu) \leq 24^{q-p} W_p^p(\mu, \nu) \log^{q-p} \left( \frac{6^{q} \sqrt{\Gamma(2q + 1)}}{W_p^p(\mu, \nu)} \right) + 2 W_p^p(\mu, \nu),$$* *Proof.* Let $X$ (resp. $Y$) be distributed according to $\mu$ (resp. $\nu$). Let $R > 0$. One has $$\begin{aligned} \mathbb{E}[|X-Y|^q] & = & \mathbb{E}[|X-Y|^{q-p+p} 1_{\{ |X-Y| < R \}}] + \mathbb{E}[|X-Y|^q 1_{\{ |X-Y| \geq R\}}] \\ & \leq & R^{q-p} \mathbb{E}[|X-Y|^{p}] + \sqrt{\mathbb{P}(|X-Y| \geq R) \mathbb{E}[|X-Y|^{2q}] }, \end{aligned}$$ where we used the Cauchy-Schwarz inequality. Note that by inequality [\[moment-iso\]](#moment-iso){reference-type="eqref" reference="moment-iso"}, $$\mathbb{E}[|X-Y|^{2q}]^{\frac{1}{2q}} \leq \mathbb{E}[|X|^{2q}]^{\frac{1}{2q}} + \mathbb{E}[|Y|^{2q}]^{\frac{1}{2q}} \leq 6 \Gamma(2q + 1)^{\frac{1}{2q}}.$$ Moreover, by inequality [\[concentration-iso\]](#concentration-iso){reference-type="eqref" reference="concentration-iso"}, $$\mathbb{P}(|X-Y| \geq R) \leq \mathbb{P}(|X| \geq \frac{R}{2}) + \mathbb{P}(|Y| \geq \frac{R}{2}) \leq 4e^{-\frac{R}{12}}.$$ Combining the above and taking infimum over all couplings yield $$W_q^q(\mu, \nu) \leq R^{q-p} W_p^p(\mu, \nu) + 6^q 2 \sqrt{\Gamma(2q + 1)} e^{-\frac{R}{24}}.$$ The result follows by choosing $R = 24 \log \left( \frac{6^q \sqrt{\Gamma(2q + 1)}}{W_p^p(\mu, \nu)} \right)$, which is nonnegative since by inequality [\[moment-iso\]](#moment-iso){reference-type="eqref" reference="moment-iso"} and log-convexity of the Gamma function, $$W_p^p(\mu,\nu) \leq \left( \mathbb{E}[|X|^p]^{\frac{1}{p}} + \mathbb{E}[|Y|^p]^{\frac{1}{p}} \right)^p \leq 6^p \Gamma(p+1) \leq 6^q \sqrt{\Gamma(2q+1)}.$$ ◻ Let us now turn to $f$-divergences. Considering $f$-divergences, such as the Kullback-Leibler divergence, the main question lies in figuring out the distribution of the reference measure. In general, if the support of a measure $\mu$ is not included in the support of a measure $\nu$, then $D(\mu || \nu) = + \infty$. Our choice of reference measure will therefore be a measure fully supported on $\mathbb{Z}$, but it turns out that it needs not be log-concave. Given $a > 0$ and $c \geq 1$, let us introduce the following class of functions: $$\mathcal{Q}(a,c) = \{q \colon \mathbb{Z}\to [0,1], \, \forall k \in \mathbb{Z}, q(k) > 0 \mbox{ and } \log \left( \frac{1}{q(k)} \right) \leq ak^2 + \log(c) \}.$$ Before stating our next result, let us note that important distributions belong to such a class. **Remark 10**. *The isotropic symmetric Poisson distribution, whose probability mass function is $$q(k) = C \frac{\lambda^{|k|}}{|k|!}, \quad k \in \mathbb{Z},$$ with $\lambda > 0$ such that $\sum_{k \in \mathbb{Z}} k^2 q(k) = 1$ and $C = (2 e^{\lambda} - 1)^{-1}$ being the normalizing constant, belongs to $\mathcal{Q}(1+\log(4), 2e -1)$. Indeed, since $$1 = \sum_{k \in \mathbb{Z}} k^2 q(k) = \frac{2e^{\lambda}}{2 e^{\lambda} - 1}\lambda(1 + \lambda),$$ then one may choose $\lambda \in [1/4,1]$. Therefore, using $|k|! \leq |k|^{|k|}$, $$0 \leq \log \left( \frac{1}{q(k)} \right) = \log(|k|!) + |k| \log \left( \frac{1}{\lambda} \right) + \log(2 e^{\lambda} - 1) \leq (1+\log(4)) k^2 + \log(2e - 1).$$* *One may also note that the isotropic symmetric geometric distribution and isotropic discretized Gaussian distribution (whose p.m.f. is of the form $q(k) = Ce^{-\lambda k^2}$) belong to $\mathcal{Q}(a,c)$ for some numerical constants $a,c>0$. The above three measures are natural candidates as a reference measure for Kullback-Leibler divergence.* *As for examples of non-log-concave distributions, consider p.m.f. of the form $C e^{-\lambda k^{\alpha}}$, for $\alpha \in (0,1)$.* The next result provides a comparison between total variation distance and $f$-divergences. The result is general as it holds for arbitrary convex function $f$, however the statement is not in a closed form formula. We state it as a lemma, and then apply it to two specific convex functions, yielding a comparison with Kullback-Leibler divergence and $\chi^2$-divergence. **Lemma 11**. *Let $f \colon [0, +\infty) \to \mathbb{R}$ be a convex function such that $f(1) = 0$. Let $a > 0$ and $c \geq 1$. Let $\nu$ be a measure on $\mathbb{Z}$ whose p.m.f. $q$ belongs to the class $\mathcal{Q}(a,c)$. Let $\mu$ be an isotropic log-concave measure on $\mathbb{Z}$ with p.m.f. $p$. Then, denoting by $Y$ a random variable with distribution $\mu$ and $W=p(Y)/q(Y)$, $$\begin{gathered} d_f(\mu||\nu) \leq \inf_{R \geq c} \left[ \left(\max\{f(0),0\} + \frac{f(R)}{R-1} \right) d_{TV}(\mu, \nu) \, + \right. \\ \left. \sqrt{\mathbb{E}\left[ \left(\frac{f(W)}{W} \right)^2 1_{\{W > 1\}} \right]} \sqrt{2} e^{-\frac{1}{12} \sqrt{\frac{1}{a} \log \left( \frac{R}{c} \right)} } \right].\end{gathered}$$* *Proof.* The idea of proof comes from [@MM] (see also [@MP]). Denote by $p$ (resp. $q$) the p.m.f. of $\mu$ (resp. $\nu$). Denote by $Y$ a random variable with p.m.f. $p$, by $Z$ a random variable with p.m.f. $q$, and denote $$X = \frac{p(Z)}{q(Z)}, \qquad W = \frac{p(Y)}{q(Y)}.$$ Using identity [\[d-tv-dis\]](#d-tv-dis){reference-type="eqref" reference="d-tv-dis"}, one has $$\label{tot-var} \mathbb{E}[|X-1|] = d_{TV}(\mu, \nu).$$ Let $R \geq 1$ and write $$d_f(\mu||\nu) = \mathbb{E}[f(X)] = \mathbb{E}[f(X) 1_{\{X < 1\}}] + \mathbb{E}[f(X) 1_{\{1 \leq X \leq R\}}] + \mathbb{E}[f(X) 1_{\{X > R\}}].$$ Let us bound all three parts. For the first part, since $f$ is convex and $f(1) = 0$, it holds that for all $x \in [0,1]$, $$f(x) \leq f(0)|x-1| \leq \max\{f(0),0\} |x-1|.$$ Therefore, using [\[tot-var\]](#tot-var){reference-type="eqref" reference="tot-var"}, $$\label{ineq-1} \mathbb{E}[f(X) 1_{\{X < 1\}}] \leq \max\{f(0),0\} \mathbb{E}[|X-1| 1_{\{X < 1\}}] \leq \max\{f(0),0\} d_{TV}(\mu, \nu).$$ For the second part, since $f$ is convex and $f(1) = 0$, it holds that for all $x \in [1,R]$, $$f(x) \leq \frac{f(R)}{R-1} (x-1).$$ Hence, using [\[tot-var\]](#tot-var){reference-type="eqref" reference="tot-var"}, $$\label{ineq-2} \mathbb{E}[f(X) 1_{\{1 \leq X \leq R\}}] \leq \frac{f(R)}{R-1} \mathbb{E}[(X-1) 1_{\{1 \leq X \leq R\}}] \leq \frac{f(R)}{R-1} d_{TV}(\mu, \nu).$$ For the last part, note that $$\label{ineq-33} \mathbb{E}[f(X) 1_{\{X > R\}}] = \mathbb{E}\left[ \frac{f(W)}{W} 1_{\{W > R\}} \right] \leq \sqrt{\mathbb{E}\left[ \left(\frac{f(W)}{W} \right)^2 1_{\{W > 1\}} \right]} \sqrt{\mathbb{P}(W > R)},$$ where we used the Cauchy-Schwarz inequality. It remains to upper bound $\mathbb{P}(W > R)$. Using that $q \in \mathcal{Q}(a,c)$ and $\|p\|_{\infty} \leq 1$, we have $$\mathbb{P}(W > R) = \mathbb{P}\left( \frac{p(Y)}{q(Y)} > R \right) \leq \mathbb{P}\left( \log(c) + a Y^2 > \log(R) \right).$$ Using [\[concentration-iso\]](#concentration-iso){reference-type="eqref" reference="concentration-iso"}, we deduce that for all $R \geq c$, $$\begin{aligned} \label{ineq-3} \mathbb{P}(W > R) \leq \mathbb{P}\left( |Y| > \sqrt{\frac{1}{a} \log \left( \frac{R}{c} \right)} \right) \leq 2 e^{-\frac{1}{6} \sqrt{\frac{1}{a} \log \left( \frac{R}{c} \right)} }.\end{aligned}$$ The result follows by combining [\[ineq-1\]](#ineq-1){reference-type="eqref" reference="ineq-1"}, [\[ineq-2\]](#ineq-2){reference-type="eqref" reference="ineq-2"}, [\[ineq-33\]](#ineq-33){reference-type="eqref" reference="ineq-33"}, and [\[ineq-3\]](#ineq-3){reference-type="eqref" reference="ineq-3"}, and by taking infimum over all $R \geq c$. ◻ Applying Lemma [Lemma 11](#f-div){reference-type="ref" reference="f-div"} to the convex function $f(x)=x\log(x)$, $x \geq 0$, yields a comparison between total variation distance and Kullback-Leibler divergence. **Theorem 12**. *Let $a > 0$ and $c \geq 2$. Let $\nu$ be a measure on $\mathbb{Z}$ whose p.m.f. belongs to the class $\mathcal{Q}(a,c)$. Let $\mu$ be an isotropic log-concave measure on $\mathbb{Z}$. Then, $$D(\mu || \nu) \leq d_{TV}(\mu, \nu) \left(288 a \log^2 \left( \frac{ \left( \sqrt{136} + 46a + \log(c) \right) \sqrt{2}}{24 \sqrt{a} d_{TV}(\mu, \nu)} \right) + 2 \log(c) + 24\sqrt{a} \right).$$* *Proof.* Recall the notation $W=p(Y)/q(Y)$ from Lemma [Lemma 11](#f-div){reference-type="ref" reference="f-div"}. With the choice of convex function $f(x) = x\log(x)$, $x \geq 0$, Lemma [Lemma 11](#f-div){reference-type="ref" reference="f-div"} tells us that for all $R \geq c \geq 2$, $$D(\mu||\nu) \leq 2\log(R) d_{TV}(\mu, \nu) + \sqrt{\mathbb{E}[|\log(W)|^{2}]} \sqrt{2}e^{-\frac{1}{12} \sqrt{\frac{1}{a} \log \left( \frac{R}{c} \right)} }.$$ Next, let us upper bound the term $\mathbb{E}[|\log(W)|^{2}]^{1/2}$. On one hand, since $q \in \mathcal{Q}(a,c)$, $$\label{first} \mathbb{E}[|\log(q(Y))|^{2}] \leq \mathbb{E}[\left( \log(c) + a Y^2 \right)^2] \leq \log^2(c) + 2a \log(c) + 1944 a^2 :=C(a,c),$$ where we used [\[moment-iso\]](#moment-iso){reference-type="eqref" reference="moment-iso"}. On the other hand, by inequality [\[varent-iso\]](#varent-iso){reference-type="eqref" reference="varent-iso"}, $$\label{second} \mathbb{E}[|\log(p(Y))|^2] \leq 136.$$ Therefore, combining [\[first\]](#first){reference-type="eqref" reference="first"} and [\[second\]](#second){reference-type="eqref" reference="second"}, $$\begin{aligned} \label{1st} \mathbb{E}[|\log(W)|^{2}]^{\frac{1}{2}} \leq \mathbb{E}[|\log(p(Y))|^2]^{\frac{1}{2}} + \mathbb{E}[|\log(q(Y))|^{2}]^{\frac{1}{2}} \leq \sqrt{136} + \sqrt{C(a,c)},\end{aligned}$$ implying $$D(\mu || \nu) \leq 2\log(R) d_{TV}(\mu, \nu) + (\sqrt{136} + \sqrt{C(a,c)}) \sqrt{2} e^{-\frac{1}{12} \sqrt{\frac{1}{a} \log \left( \frac{R}{c} \right)} }.$$ Putting $t = \sqrt{ \log(\frac{R}{c})}$, the above inequality reads $$D(\mu || \nu) \leq 2\log(c) d_{TV}(\mu, \nu) + 2 d_{TV}(\mu, \nu) t^2 + (\sqrt{136} + \sqrt{C(a,c)}) \sqrt{2} e^{-\frac{t}{12 \sqrt{a}}}.$$ Choosing $t = 12 \sqrt{a} \log \left( \frac{ \left( \sqrt{136} + \sqrt{C(a,c)} \right) \sqrt{2} }{24 \sqrt{a} d_{TV}(\mu, \nu)} \right)$, which is nonnegative, and using the bound $$\sqrt{C(a,c)} = \sqrt{(a+\log(c))^2 + 1943 a^2} \leq 46a + \log(c)$$ yields the desired result. ◻ As a last illustration, the next result provides a comparison between total variation distance and $\chi^2$-divergence, under an extra moment assumption. **Theorem 13**. *Let $a > 0$ and $c \geq 1$. Let $\nu$ be a measure on $\mathbb{Z}$ whose p.m.f. belongs to the class $\mathcal{Q}(a,c)$. Let $\mu$ be an isotropic log-concave measure on $\mathbb{Z}$. Under the moment assumption $$\mathbb{E}[e^{2aY^2}] < +\infty,$$ where $Y$ denotes a random variable with distribution $\mu$, one has $$d_{\chi^2}(\mu || \nu) \leq c \left( d_{TV}(\mu, \nu) + \sqrt{d_{TV}(\mu, \nu)} \right) + c \sqrt{\mathbb{E}[e^{2a Y^2}]} \sqrt{2 }e^{-\frac{1}{12} \sqrt{\frac{1}{a} \log \left( 1 + \frac{1}{\sqrt{d_{TV}(\mu, \nu)}} \right)} }.$$* *Proof.* Recall the notation $W=p(Y)/q(Y)$ from Lemma [Lemma 11](#f-div){reference-type="ref" reference="f-div"}. With the choice of convex function $f(x) = (x-1)^2$, $x \geq 0$, Lemma [Lemma 11](#f-div){reference-type="ref" reference="f-div"} tells us that for all $R \geq c$, $$d_{\chi^2}(\mu || \nu) \leq R d_{TV}(\mu, \nu) + \sqrt{\mathbb{E}\left[ \frac{(W-1)^4}{W^2} 1_{\{W > 1\}} \right]} \sqrt{2} e^{-\frac{1}{12} \sqrt{\frac{1}{a} \log \left( \frac{R}{c} \right)} }.$$ Note that $$\mathbb{E}\left[ \frac{(W-1)^4}{W^2} 1_{\{W > 1\}} \right] \leq \mathbb{E}\left[ W^2 \right] \leq \mathbb{E}\left[ \frac{1}{q^2(Y)} \right] \leq c^2 \mathbb{E}[e^{2a Y^2}],$$ where the last inequality comes from $q \in \mathcal{Q}(a,c)$. Therefore, $$d_{\chi^2}(\mu || \nu) \leq R d_{TV}(\mu, \nu) + c \sqrt{\mathbb{E}[e^{2a Y^2}]} \sqrt{2 }e^{-\frac{1}{12} \sqrt{\frac{1}{a} \log \left( \frac{R}{c} \right)} }.$$ Choosing $R = c \left(1 + \frac{1}{\sqrt{d_{TV}(\mu, \nu)}} \right)$ yields the desired result. ◻ 50 H. Aravinda. 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On the Poincaré constant of log-concave measures. Geometric Aspects of Functional Analysis: Israel Seminar (GAFA). 2017-2019. Volume I. LNM 2256, Springer Verlag, 171--217, 2020. R. M. Dudley. Distances of probability measures and random variables. Ann. Math. Statist. 39 (1968), 1563-1572. R. M. Dudley. Real analysis and probability.(English summary) Revised reprint of the 1989 original Cambridge Stud. Adv. Math., 74 Cambridge University Press, Cambridge, 2002. x+555 pp. A. L. Gibbs, F. E. Su. On choosing and bounding probability metrics. Preprint, arXiv:math/0209021. J. Jakimiuk, D. Murawski, P. Nayar, S. Słobodianiuk. Log-concavity and discrete degrees of freedom. Preprint, arXiv:2205.04069. F. Liese, I. Vajda. Convex statistical distances. With German, French and Russian summaries Teubner-Texte Math., 95\[Teubner Texts in Mathematics\] BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1987. 224 pp. A. Marsiglietti, J. Melbourne. Moments, concentration, and entropy of log-concave distributions. Preprint, arXiv:2205.08293. A. Marsiglietti, P. Pandey. On the equivalence of statistical distances for isotropic convex measures, Mathematical Inequalities & Applications, Volume 25, Issue 3, 881-901, 2022. J. L. Massey. On the entropy of integer-valued random variables. In: Proc. 1988 Beijing Int. Workshop on Information Theory, pages C1.1-C1.4, July 1988. E. Meckes, M. Meckes. On the Equivalence of Modes of Convergence for Log-Concave Measures. In Geometric aspects of functional analysis (2011/2013), volume 2116 of Lecture Notes in Math., pages 385-394. Springer, Berlin, 2014. J. Melbourne, G. Palafox-Castillo. A discrete complement of Lyapunov's inequality and its information theoretic consequences. Preprint, arXiv:2111.06997. M. S. Pinsker. Information and information stability of random variables and processes. Translated and edited by Amiel Feinstein Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964, xii+243 pp. S. T. Rachev, L. B. Klebanov, S. V. Stoyanov, and F. J. Fabozzi. The methods of distances in the theory of probability and statistics. Springer-Verlag, New York, 2013. A. Saumard, J. A. Wellner. Log-concavity and strong log-concavity: a review. Stat. Surv. 8, 45-114, 2014. R. P. Stanley. Log-concave and unimodal sequences in algebra, combinatorics, and geometry. In Graph theory and its applications: East and West (Jinan, 1986), volume 576 of Ann. New York Acad. Sci., pages 500-535. New York Acad. Sci., New York, 1989. R. Vershynin. High-dimensional probability. An introduction with applications in data science. With a foreword by Sara van de Geer. Cambridge Series in Statistical and Probabilistic Mathematics, 47. Cambridge University Press, Cambridge, 2018. xiv+284 pp. Arnaud Marsiglietti\ Department of Mathematics\ University of Florida\ Gainesville, FL 32611, USA\ a.marsiglietti\@ufl.edu Puja Pandey\ Department of Mathematics\ University of Florida\ Gainesville, FL 32611, USA\ puja.pandey\@ufl.edu

arxiv_math

--- author: - | Gurpreet Jagdev$^a$, Na Yu$^a$\ $^a$Department of Mathematics, Toronto Metropolitan University, Toronto, Canada\ bibliography: - bib.bib title: "**Noise-induced synchronization and regularity in feed-forward-loop motifs**" --- 0 0 **Title** # Introduction Network motifs, a fundamental concept in network analysis, are recurring interaction patterns observed in various systems [@Mangan2003]. These motifs often retain their specific dynamical functions as well as integrate into the collective behaviors of the broader network structures, thus they are often considered as the building blocks of complex networks [@Shen2002; @reigl2004search]. Among the common three-node network motifs, coherent and incoherent feedforward loops (FFL) have proven to be notably more prevalent than their counterparts [@reigl2004search; @Song2005]. Here we call them type-1 motif (T1) and type-2 motif (T2) for simplicity and they are demonstrated in Fig. 1. In motif T1, three unidirectional connections are positive (i.e., excitatory coupling). In motif T2, the connections from node 1 to node 2 and from node 1 to node 3 are positive but the connection from node 2 to node 3 is negative (i.e., inhibitory coupling). FFL network motifs have been found in many actual biological networks, such as gene expression in bacteria and yeast [@Milo2002; @Lee2002], human and mouse genomes [@Boyer2005; @tsang2007], cat cortex [@sporns2004], and the nervous system of roundworm [@milo2002network]. Moreover, both motifs are constructed such that node 1 may be considered an input layer, and node 3 an output layer of the network. There are two parallel signalling pathways: one direct pathway from the input layer to the output layer, and one indirect pathway from input to output by means of node 2 (the layer of internodes or interneurons). Such parallel information transmission structure has been discovered in the auditory cortex [@Eggermont1998] and electrosensory system [@Middleton2006]. The significance of FFL network motifs leads to the question of how they interact with one another and perform specific information-processing roles in the presence of noise, a ubiquitous factor in real life. Under the conditions of equal coupling and symmetric noise, FFL motifs have been shown to produce coherence [@Gui2016] or resonance [@Krauss2019]. However, to our best knowledge, no prior have clearly demonstrated their performance in a heterogeneous setting: unequal coupling and asymmetric noise. To address this unexplored aspect, our research will focus on noise-induced synchrony and regularity of T1 and T2 motifs in the heterogeneous setting. The remainder of this paper is structured as follows. Sec. 2 introduces the mathematical model and methods. Sec. 3.1 discusses the noise-induced dynamics of our networks. Sec. 3.2 studies the effects of noise on network synchronization. Sec. 3.3 studies the effects of noise on output regularity. Sec. 3.4 studies the effects of network connectivity on network synchrony and output regularity. Sec. 3.5 studies the effects of the bifurcation parameter $\lambda_0$ on network synchronization and output regularity. And a discussion is given in Sec. 4. {#fig:motif} # Mathematical model and methods ## Model We consider a system comprised of three coupled oscillators arranged in FFL patterns (see Fig. 1), where the dynamics of each oscillator are described by the canonical model for the normal form of a Hopf bifurcation (HB), the $\lambda-\omega$ system, with additive noise and diffusive coupling terms. The $i$th oscillator is modelled by the system of stochastic differential equations (SDEs) $$\begin{aligned} dx_i &= \Big[\lambda(r_i)x_i - \omega(r_i)y_i + \sum_{j \neq i}d_{j,i}(x_j-x_i)\Big]dt+\delta_i d\eta_i(t), \label{x_i}\\ dy_i & = \Big[\omega(r_i)x_i + \lambda(r_i)y_i + \sum_{j \neq i} d_{j,i}(y_j-y_i)\Big]dt, \label{y_i} %r_i^2 &= x_i^2 + y_i^2, \label{r_i}\end{aligned}$$ where $i,j=1,2,3$. The amplitude and phase of $i$th oscillator can be obtained by $r_i = \sqrt{x_i^2 + y_i^2}$ and $\phi_i = \tan^{-1} \frac{y_i}{x_i}$, respectively. The function $\lambda(r_i) = \lambda_0 + \alpha r_i^2 + \gamma r_i^4$ governs the modulation of the amplitude of $i$th oscillator. $\lambda_0$ serves as the control parameter, and the Hopf bifurcation (HB) occurs when $\lambda_0=0$. The parameters $\alpha$ and $\gamma$ influence the system's behavior away from this bifurcation point. Additionally, the function $\omega(r_i) = \omega_0 + \omega_1 r_i^2$ controls the modulation of the $i$th oscillator's frequency, with $\omega_1$ governing how the frequency evolves concerning the amplitude $r_i$. Notably, when $\omega_1=0$, changes in amplitude do not directly impact the phase. We consider a supercritical Hopf bifurcation and the corresponding parameter values are $\alpha=-0.2$, $\gamma=-0.2$, $\omega_0=2$, and $\omega_1=0$. The term $\delta_i d\eta_i(t)$ represents intrinsic white noise applied to $x_i$, and $\eta_i(t)$ characterizes a Wiener process with zero mean and unit variance. The parameter $\delta_i$ is the noise intensity. Additionally, the terms $\sum_{j \neq i}d_{i,j}(x_j-x_i)$ and $\sum_{j \neq i} d_{i,j}(y_j-y_i)$ denote the diffusive coupling that one neuronal oscillator receives, where $d_{i,j}$ signifies the connection strength between oscillators $i$ and $j$, indicating the signal transmitted from oscillator $i$ to oscillator $j$. To create the Feed-Forward Loop (FFL) motifs, as illustrated in Figure 1, we configure $d_{j,1}=0$ and $d_{3,i}=0$ for all $i$ and $j$. This setup ensures that oscillator 1 does not receive input from other oscillators within the FFL, and oscillator 3 does not transmit output to other oscillators within the FFL. In motif T1, all three connections are excitatory, meaning that $d_{1,2}>0$, $d_{1,3}>0$, and $d_{2,3}>0$. Conversely, in motif T2, there are two excitatory connections ($d_{1,2}>0$ and $d_{1,3}>0$) along with one inhibitory connection ($d_{2,3}<0$). ## Methods All computational tasks, including simulation, numerical analysis, and figure generation, are carried out using MATLAB. We employ the Euler-Maruyama method to calculate the numerical solutions to SDEs and use the time interval $[50, 200]$ and time step of $dt=0.01$. We initiate the simulations with arbitrary, small random initial conditions $x_i(0),y_i(0) \sim N(0,0.008^2)$ for $i=1,2,3$. To address the challenges posed by the high-frequency fluctuations inherent in noise-induced oscillations during numerical analysis, we implement a low-pass filter. This filter is constructed using a Gaussian-weighted moving average spanning a window of $100$ data points, ensuring enhanced result consistency. Finally, we compute the coherence measures $\sigma$, $\gamma$, and $R$ outlined in Equations 4 through 6, averaging these values over a total of $N=200$ simulations. # Results ## Noise-induced dynamics {#sec: noise-induced dynamics} {#fig:fig2} We consider the dynamics of our network motifs in the excitable regime but in the vicinity of a supercritical HB at $\lambda_0=0$ (e.g., $\lambda_0=-0.3$ in Fig. 2). The deterministic systems associated with motifs T1 and T2 ($\delta_i=0$ for $i=1,2,3$) converge to the stable fixed point $(0,0)$, as shown in Fig. 2a and 2d, respectively. We also observed that all three oscillators of the T1 motif reach in-phase during the transient time, despite their different initial values (Fig. 2a). The deterministic T2 oscillators 1 and 2 exhibit in-phase, whereas oscillator 3 show anti-phase with the others during the transient time (Fig. 2d). The presence of intrinsic noise induces sustained limit-cycle oscillations in all oscillators within the T1 and T2 motifs (Fig. 2b and 2e where $\delta_i$ equals 0.01 for $i=1,2,3$). This kind of oscillation is commonly referred to as \"noise-induced oscillation\". Examining the dynamics of the two input oscillators (specifically, oscillator 1) in both T1 and T2 motifs, we observe similarities in their periods and time-varying amplitudes, as indicated by the blue dashed lines in Fig. 2b and 2e. A similar resemblance is found for oscillator 2, as depicted by the gray dotted lines. However, a notable distinction emerges when we consider the output oscillators (oscillator 3) of the T1 and T2 motifs. The output oscillator of the T2 motif (red curve in Fig. 2e) exhibits more regular oscillations compared to its counterpart in the T1 motif (red curve in Fig. 2b). For example, the amplitude of the output oscillator in motif T2 remains relatively stable, characterized by more pronounced peaks and fewer small-amplitude perturbations. To determine whether this is the case for other noise intensities, we compute the time-averaged amplitude of $x_i$, $A_i = \frac{1}{T}\int^T_0|x_i(t)|\ dt,$ $i=1,2,3$, as a function of $\delta_i$, and average over $N=200$ trials. Our results are displayed in Fig. 2c for motif T1 and Fig. 2f for motif T2. The dashed blue lines represent $A_1$, the dotted black lines represent $A_2$, and the solid red lines represent $A_3$. We find that $A_1$ and $A_2$ are overlapped for both motifs T1 and T2, implying that oscillators 1 and 2 However, $A_3$ differs between the T1 and T2 motifs. In particular, for motif T2, $A_3$ is greater than its counterpart in motif T1. For instance, when $\delta_1,\delta_2,\delta_3=0.1$, $A_3 \approx 0.12$ for the T1 motif and $A_3 \approx 0.19$ for the T2 motif (see the dashed grey lines in Figs. 2c and 2f). Furthermore, not only does $A_3$ itself differ across the two FFL motifs, but the relative size of $A_3$ compared to $A_1$ and $A_2$ differs across the two FFL motifs as well. In particular, for motif T2 $A_3 \geq A_1$, $A_2$, whereas $A_3 \leq A_1$, $A_2$ for motif T1. ## The effects of noise on network synchronization {#sec:network sync} Next we consider the impact of noise on the synchronization of noise-induced dynamics. To excite oscillations from rest, we fix the noise intensity applied to $x_2$ and $x_3$ (e.g., $\delta_2,\delta_3=0.01$ in Fig. 3) and vary the noise intensity applied to $x_1$ (e.g. $\delta_3 \in [0.001,10^{1/2}]$ in Fig. 3). Then, we consider network synchronization from two perspectives. We begin first from a broad perspective, where we analyze the synchronization between every oscillator in each FFL motif. In order to quantify the degree of network synchrony, we use the root mean square deviation [@wang2009; @gao2001] $$\label{sigma} \sigma = \frac{1}{T}\int^T_{0}\sigma_t\ dt,$$ where $$\label{sigma(t)} \sigma_t = \sqrt{\frac{1}{M}\sum_{i=1}^M \left( \frac{x_i(t)}{A_i}\right)^2 - \left(\frac{1}{M}\sum_{i=1}^M \frac{x_i (t)}{A_i}\right)^2}$$ and $M$ is the total number of oscillators. Note that $\sigma_t$ is computed using the amplitude-normalized time series, $x_i(t)/A_i$, rather than $x_i(t)$. This normalization ensures that $\sigma$ is a measure of temporal synchronization, or phase synchronization, rather than complete synchronization, which depends on alignments in both amplitude and phase [@rosenblum2001phase]. Furthermore, since $\sigma$ quantifies the degree of variability between $x_1$, $x_2$, and $x_3$, it follows that smaller values of $\sigma$ indicate greater levels of synchrony. {#fig:network_sync} The results of our analysis are presented in Fig. 3a which displays the root mean square deviation, $\sigma$, as a function of the driving noise intensity, $\delta_1$, for the T1 (blue curve) and T2 (black curve) motifs. We observe that both of the T1 and T2 motifs display a resonant response to $\delta_1$, which indicates the occurrence of coherence resonance (CR) [@gang1993stochastic; @pikovsky1997coherence]. That is, when $\delta_1$ is weak (e.g. $\delta_1<0.12$ for motif T2 in Fig. 3a), network synchrony increases with the increment of $\delta_1$, or equivalently, the value of $\sigma$ decreases. Then, at an intermediate intensity of $\delta_1$, for example, $\delta_1 \approx 0.12$ for motif T2 in Fig. 3a, network synchronization reaches an optimal state as $\sigma$ attains its absolute minimum. Then, as the noise intensity increases further, network synchrony exhibits a sharp decline which suggests that the noise has begun to dominate network dynamics. Indeed, our results in Fig. 3 indicate that the network synchronization (i.e. $\sigma$) is the same for the T1 and T2 motifs across the strong intensity range of $\delta_1$ (e.g. $\delta_1>0.5$ in Fig. 3). We observe that both motifs display a resonant character and behave similarly over the strong intensity range of $\delta_1$, however, they differ considerably over the weak and intermediate intensity ranges (e.g. $\delta_1 \leq 0.1$ in Fig. 3a). Over the weak intensity range of $\delta_1$, the $\sigma$ curve which corresponds to the T1 motif (blue curve in Fig. 3a) is always less than the black curve in Fig. 3a which corresponds to the T2 motif. This suggests that the T1 motif displays comparatively greater network synchrony. Moreover, we see that the optimal driving noise intensity, denoted by $\delta_1^*$, and the absolute minimum which it induces, denoted $\sigma^*$, are both smaller for the T1 motif than the T2 motif. For example, Fig. 3a shows that the optimal driving noise intensity and its corresponding minimum are located at $\delta_1^*=0.07$ and $\sigma^*=0.215$ for motif T1, and $\delta_1^*=0.12$ and $\sigma^*=0.218$ for motif T2 (see solid squares in Fig. 3a). A second perspective from which to study network synchronization comes by examining the synchrony solely between the input and output oscillators of the networks. To quantify the synchronization between the input-output pair $x_1$ and $x_3$ we analyze the distribution of their phase differences, $\Delta\phi=\phi_1-\phi_3$, using a measure known as the mean phase coherence [@rosenblum2001phase; @mormann2000mean]: $$\label{gamma} \gamma= \sqrt{\left(\frac{1}{T} \int^{T}_{t_0} \sin{\Delta \phi} \ dt\right)^2 + \left(\frac{1}{T} \int^{T}_{t_0} \cos{\Delta \phi} \ dt\right)^2}.$$ The range of $\gamma$ is $0 \leq \gamma \leq 1$, such that larger values of $\gamma$ indicate a greater degree of phase synchronization; in particular, $\gamma=1$ indicates a state of perfect synchronization and $\gamma=0$ indicates a completely chaotic state. The results of our analysis are presented in Fig. 3b, where the blue and black curves correspond to the T1 and T2 motifs, respectively. As in Fig. 2a, $\gamma$ is computed over the range $0.001 \leq \delta_1 \leq 10^{1/2}$ with $\delta_2$ and $\delta_3$ fixed at 0.01. The results of Fig. 3b corroborate those in Fig. 3a, and reveal a resonant behaviour in the $\gamma$ curves which mirrors that in Fig. 3a. That is, for weak noise intensities (e.g. $\delta_1<0.12$ for motif T1), increasing $\delta_1$ leads to an increase in network synchronization as indicated by an increase in $\gamma$. Then, at an intermediate noise level (e.g. $\delta_1= 0.12$ for motif T1), synchronization reaches an optimal state, where $\gamma$ reaches its absolute maximum. For stronger noise intensities, network synchronization declines which is indicated by a falling $\gamma$. This suggests that optimal phase synchrony between $x_1$ and $x_3$ is achieved at an intermediate level of $\delta_1$. Moreover, we find that the T1 motif requires a lesser intensity of noise to attain an optimal level of synchrony than the T2 motif. For example, in Fig. 3b, the optimal noise intensity for the T1 motif, $\delta_1^*=0.12$, is smaller than that for the T2 motif which is $\delta_1^*=0.188$, with a larger corresponding maximum in $\gamma$, $\gamma^*=0.728$, relative to the T2 motif, for which $\gamma^*=0.713$. Furthermore, we note that $\gamma$ for the T1 motif is consistently higher than that of the T2 motif, and that this difference diminishes as $\delta_1$ increases (as in Fig. 3a). Although the results shown in Figs. 3a and 3b have the same qualitative characteristics, there are quantitative differences such as the optimal driving noise intensities. These differences emerge naturally as a result of the inherent differences in the measures $\gamma$ and $\sigma$ defined in Eqs. [\[gamma\]](#gamma){reference-type="ref" reference="gamma"} and [\[sigma\]](#sigma){reference-type="ref" reference="sigma"}, respectively. Nevertheless, both measures indicate that the T1 motif exhibits a greater propensity for network synchronization than the T2 motif over the weak to intermediate noise intensity ranges and requires less noise to achieve an optimal level of network synchronization. ## The effects of noise on output regularity {#sec: regularity} In addition to studying the effects of noise on network synchronization, we also consider the effects of noise on output regularity by analyzing the regularity of the time series $x_3$. By regularity, we refer to the degree to which the dynamics of $x_3$ are periodic and quantify this using the coefficient of variation of the inter-spike intervals (ISIs). The coefficient of variation, $R$, is defined as the standard deviation of the ISIs divided by the mean of the ISIs [@bonsel2022control; @lu2019phase], that is, $$\label{coeff variation} R = \frac{\sqrt{\frac{1}{K-1}\sum_{k=1}^{K-1} \left(\text{ISI}_k\right)^2 - \left(\frac{1}{K-1}\sum_{k=1}^{K-1}\text{ISI}_k \right)^2}}{\frac{1}{K-1}\sum_{k=1}^{K-1} \left(\text{ISI}_k\right)},$$ where $\text{ISI}_k$ denotes the $k$th ISI. In order to compute an ISI we mark the occurrence of a spike by a peak in the time series of $x_3$ (e.g., inset of Fig. 4a) and compute the difference between consecutive spike times. Furthermore, because $R$ is a measure of central variability, larger values of $R$ indicate a more irregular firing pattern and consequently lower regularity, and vice versa. {#fig:regularity} We present the results of our analysis in Fig. 4a, where $R$ is plotted against the driving noise intensity, $\delta_1$, for the T1 motif (black curve) and T2 motif (blue curve). We find the that both motifs display a resonant character, which indicates the occurrence of CR. That is, $R$ is relatively low at weak intensities of $\delta_1$ (e.g. $\delta_1=0.0023$ in Fig. 4a), and with an increase in $\delta_1$, $R$ decreases, indicating that output regularity is increasing, or, the firing pattern is becoming more regular. Then, as $\delta_1$ surpasses an optimal intensity, $R$ starts to increase again, and the output of our networks becomes more chaotic. Thus, the addition of intrinsic noise can optimize the regularity of the output oscillator, $x_3$, of both the T1 and T2 motifs at an intermediate intensity. We find however, that motif T2 exhibits greater regularity than motif T1 over the range $0.001 \leq \delta < 0.16$; before the optimal point B in Fig. 4a which occurs at $\delta_1=0.16$. That is, when $0.001 \leq \delta < 0.16$, the $R$ of the T2 motif is less than the $\gamma$ of the T1 motif (Fig. 4a). As $\delta_1$ approaches the optimal point $\delta_1=0.16$, the difference between the two curves becomes increasingly small. Interestingly, the optimal noise intensity $\delta_1^*$ and the corresponding absolute minimum in $R$, denoted $R^*$, are both the same for motifs T1 and T2. For example, in Fig. 4a, $\delta_1^*=0.162$ and $R^*=0.116$. Moreover, as $\delta_2$ increases further, the blue and black curves in Fig. 4a change at the same rate, and therefore the output regularity of motifs T1 and T2 is the same over this region. The density functions of the ISIs presented in panels b and c of Fig. 4 corroborate our findings in Fig. 4a. We consider the density functions of the ISIs for three disparate intensities of the driving noise which correspond to points A, B, and C in Fig. 4a. Point A is a low level of noise with intensity $\delta_1=0.0023$ and corresponds to the orange curve, point B is a moderate level of noise with intensity $\delta_1=0.16$ and corresponds to the purple curve, and point C is a high level of noise with intensity $\delta_1=1.35$ and corresponds to the green curve. First, we find that the peak of the purple density function, which corresponds to the optimal noise intensity, is more pronounced in both panels b and c than those of the orange and green density functions which correspond to noise intensities that are either too weak or too strong. This implies that the ISIs at point B are more tightly grouped, and agrees with our results in Fig. 4a. Furthermore, the orange density function in panel c displays a much more pronounced peak (i.e. larger peak and smaller half-width) than its counterpart, the orange density function in panel b, which has relatively broad bimodal peaks. This suggests that the output of the T2 motif is more regular than that of the T1 motif over the low intensity range of $\delta_1$. Conversely, the purple and green density functions appear to be equivalent across panel b and c of Fig. 4. This is consistent with Fig. 4a, which suggests that sufficiently strong intensities of the driving noise can eliminate the differences in the output regularity of the T1 and T2 motifs. Overall, we observe that the regularity of the output of both motifs T1 and T2 increases as the noise intensity increases from low levels until it reaches an optimal point. Before this optimal point, the T2 motif shows greater output reliability (as measured by a lower value of $R$). And Beyond this optimal point, the regularity of the T1 and T2 motifs is identical, and with the increment of $\delta_1$, the regularity begins to decline as the noise intensity becomes overpowering. ## The effects of network connectivity on network synchrony and output regularity {#fig:coupling} In Secs. 3.2 and 3.3, we investigated the impact of the driving noise intensity, $\delta_1$, on network synchronization and output regularity with fixed coupling strengths $d_{3,1}$, $d_{3,2}$, and $d_{2,1}$. Next we will consider the effects of the network connectivity on network synchronization and output regularity by considering a variable coupling strength $d$, where $d=d_{3,1},d_{3,2},d_{2,1}$ for the T1 motif, and $d=d_{3,1},-d_{3,2},d_{2,1}$ for the T2 motif. To quantify network synchronization we use the root mean square deviation, $\sigma$, in Eq. [\[sigma\]](#sigma){reference-type="ref" reference="sigma"} and to quantify output regularity we use the coefficient of variation, $R$, in Eq. [\[coeff variation\]](#coeff variation){reference-type="ref" reference="coeff variation"}. Our findings are presented in Fig. 5, in the form of contour maps of $\sigma$ and $R$ as functions of $\delta_1$ and $d$ for the T1 motif (panels a and b, respectively) and T2 motif (panels c and d, respectively). As indicated by the colour bars, warmer colours correspond to larger values of both $\sigma$ and $R$. First, we see that there exist threshold values of the common coupling strength, $d$, denoted $d_T$, such that when $d \leq d_T$, changes in $\delta_1$ have no significant effect on network synchronization ($\sigma$) and output regularity ($R$). In the case of network synchronization, we find that $d_T \approx 0.02$ for both T1 and T2 motifs (Fig. 5a and 5c) and for output regularity, $d_T \approx 0.03$ for both the T1 and T2 motifs (Fig. 5b and 5d). Furthermore, we see that the contour maps in panels c and d, which correspond to motif T2, are notably different than the contour maps in panels a and b, which correspond to motif T1, when the noise intensity is weak (e.g. $\delta_1<0.01$ for $\sigma$ and $\delta_1<0.03$ for $R$ in Fig. 5) and the coupling strength is strong (e.g. $d>0.01$ for $\sigma$ and $d>0.02$ for $R$); or simply, the bottom right corners of the contour maps in Fig. 5. We find that the T1 motif has a smaller $\sigma$ in this region relative to the T2 motif, but a larger $R$. In other words, the T1 motif shows a greater propensity for network synchrony but a lesser propensity for output regularity than the T2 motif in this region---clearly, the converse holds as well. Moreover, we see that the differences in $\sigma$ and $R$ disappear as $\delta_1$ increases, which is consistent with our findings in the sections (upper right region of Fig. 5). {#fig:optimal} The contours in Fig. 5 suggest that the noise intensity required for both motifs to optimize both network synchrony ($\sigma$) and/or output regularity ($R$) is dependent on $d$. To consider the effects the network connectivity, $d$, more comprehensively, we compute $\sigma^*$ and $R^*$ and their corresponding $\delta_1^*$ values as functions of $d$. Our findings are presented in Fig. 6, where: panel (a) displays $\sigma^*$ vs. $d$; (b) displays $R^*$ vs. $d$; and (c) and (d) display their corresponding optimal noise intensities, $\delta_1^*$, vs. $d$, respectively. For motifs T1 and T2, we find that $\sigma^*$ is negatively correlated with $d$. This suggests that the optimal level of network synchrony increases as the network connectivity, $d$, increases. Additionally, in Fig. 6a, the $\sigma^*$ curve for the T1 motif is less than or equal to the $\sigma^*$ curve for the T2 motif. This is consistent with the results presented in Sec. 3.2 which highlight the T1 motif's propensity for greater network synchrony. Furthermore, Fig. 6c reveals that the optimal noise intensity, $\delta_1^*$ is negatively correlated with $d$. That is, as network connectivity increases, the intensity of noise required to optimize network synchrony decreases. In addition, the $\delta_1^*$ curve for motif T1 in Fig. 6c (blue curve) is less than or equal to the $\delta_1^*$ curve for the motif T2 (black curve). This suggests that the T1 motif requires weaker noise to maximize network synchronization than to the T2 motif (as in Fig. 3a). From Fig. 6b we see the $R^*$ is a non-monotone function of $d$. For both motifs T1 (black curve) and T2 (blue curve), $R^*$ is decreasing when $d<0.05$, reaches a minimum at $d=0.05$, and then increases for $d>0.05$. When $d=0.05$, both motifs have the same optimal output regularity: $R^*=0.108$. Indeed, there is no clear difference between the $R^*$ curves for the T1 and T2 motifs in Fig. 6b. This corroborates the findings in Fig. 4, which show that the T1 and T2 motifs have similar levels of optimal output regularity. Fig. 6d, which displays the optimal noise intensities associated with the curves in Fig. 6b suggests $\delta_1^*$ decreases as $d$ increases for both types of motifs, and further, that there is no consistent difference between the $\delta_1$ required to minimize $R$ for the T1 motif and T2 motif. This is consistent with our results in Fig. 4. ## The effects of $\lambda_0$ on network synchrony and output regularity {#fig:lambda} In the preceding sections, we have only considered $\lambda_0=-0.1$. However, many studies (e.g., [@yu2009constructive; @yu2021noise]) show network synchronization in excitable systems is dependent on the distance of the critical parameter from the excitation threshold. Furthermore, we consider the effects of $\lambda_0$ on network synchronization and output regularity. To quantify network synchronization and output regularity we use the measures $\sigma$ in Eq. [\[sigma\]](#sigma){reference-type="ref" reference="sigma"} and $R$ in Eq. [\[coeff variation\]](#coeff variation){reference-type="ref" reference="coeff variation"}. We calculate $\sigma$ and $R$ within the range of $-1 \leq \lambda_0 \leq -0.001$ and $0.001\leq \delta_1 \leq 5$, and then compute the noise-averaged values of $\sigma$ and $R$, denoted by $\langle \sigma \rangle$ and $\langle R \rangle$, respectively, which allows us to consider $\langle \sigma \rangle$ and $\langle R \rangle$ as functions of $\lambda_0$. Our findings are presented in Figs. 7a and 7b, where the blue curves in both figures represent motif T1 and the black curves represent motif T2. Surprisingly, we find that there is no correlation between $\langle \sigma \rangle$ and $\lambda_0$ (Fig. 7a), both $\langle \sigma \rangle$ curves in Fig. 7a are constant with $\langle \sigma \rangle \approx 0.308$ for motif T1 and $\langle \sigma \rangle \approx 0.372$ for motif T2. That is, network synchronization is not correlated with the distance of $\lambda_0$ from the excitation threshold. Our results further reveal that $\langle \sigma \rangle$ is lower for the T1 motif relative to the T2 motif across the range $-1 \leq \lambda_0 \leq -0.08$. This finding is in agreement with our previous findings that suggest that motif T1 exhibits a greater degree of network synchronization than motif T2. On the other hand, from Fig. 7b we see that there is a correlation between $\langle R \rangle$ and $\lambda_0$. For both motifs we find that $\langle R \rangle$ decreases linearly as $\lambda_0$ approaches the excitation threshold and then relaxes toward a minimum when $-0.01 \leq \lambda_0 < 0$, after which moving the networks closer to the excitation threshold does not affect $\langle R \rangle$. Our results further indicate that the black curve in Fig. 7b which corresponds to the T2 motif is always below the blue curve in Fig. 7b which corresponds to the T1 motif. This is in agreement with our previous findings, which suggest that the T2 motif exhibits greater output regularity than the T1 motif over the weak intensity range (e.g. Fig. 4a). Next we consider how $\lambda_0$ affects the optimal driving noise intensity, $\delta_1^*$, needed to minimize $\sigma$ and $R$. Figs. 2c and 2d display the optimal driving noise intensities which correspond to $\sigma^*$ and $R^*$, respectively, as functions of $\lambda_0$. The T1 motifs are represented by blue curves and the T2 motifs are represented by black curves. We find that the $\delta_1^*$ corresponding $\sigma$ is a constant function of $\lambda_0$ for both the T1 and T2 motifs. Namely, $\delta_1^* \approx 0.07$ for the T1 motif and $\approx 0.12$ for the T2 motif (Fig. 7c). These results are consistent with our earlier findings in Fig. 3a, which suggest that the T1 motif requires a lesser intensity of $\delta_1$ to minimize $\sigma$ relative to the T2 motif. Conversely, in Fig. 7d we find that the $\delta_1$ needed to minimize $R$ is a decreasing function of $\lambda_0$, and as in Fig. 7b, the rate of change of the $\delta_1^*$ decreases as $\lambda_0$ approaches the excitation threshold. Finally, the noise intensities required to optimize output regularity are approximately the same for both the T1 and T2 type motifs. # Discussion We explore the impact of noise on network synchronization and output regularity within the context of three-neuron FFL motifs. Our investigation focuses exclusively on two distinct motif types: T1, which has purely excitatory connections (as depicted in Fig. 1a); and T2, which has a combination of both excitatory and inhibitory connections (as illustrated in Fig. 1b). We choose to analyze these specific motifs due to their prevalence in neuronal networks [@reigl2004search; @Song2005]. Nevertheless, there are other three-neuron motifs that are also prevalent within the brain. More generally, neural motifs (i.e., frequently recurring wiring patterns) are thought to serve important functions within the brain, and may be viewed as the fundamental building blocks of larger, more complex neuronal networks [@Song2005; @alon2007network; @sporns2004motifs]. We model the dynamics of each oscillator using the $\lambda-\omega$ system, the canonical model for the normal form of a HB, since is a well-established framework that captures the dynamics of the transition from quiescence to oscillations observed in real neurons [@izhikevich2000neural]. We consider the excitable regime, which is quiescent in the absence of noise but can be excited by the addition of an intrinsic noise. Our result show that the addition of a pure noise stimulus can maximize both network synchronization and output reliability at an intermediate intensity. Our findings agree with existing studies which find CR and other noise-induced effects in similar network motifs [@lu2019; @guo2009; @lou2014stochastic]. Furthermore, our results indicate that the T1 motif shows a greater propensity for noise-induced synchronization; it exhibits a greater degree of synchronicity than the T2 motif and requires a comparatively lower noise intensity to optimize network synchrony. Conversely, we found that the T2 motif displays greater output reliability over the weak to intermediate range than the T1 motif but does not require a lesser intensity of noise to reach an optimal point. Importantly, our observations indicate that these differences between the two motifs are robust in that they persist over a wide range of noise intensities and parameter regimes, and only vanish when the intrinsic noise stimulus becomes excessively strong or the network connectivity becomes excessively weak. These results may in part be understood through our results in Fig. 2, which indicate: (i) the T2 motif shows both in-phase and anti-phase synchronization, whereas the T1 motif shows only in-phase synchronization (Fig. 2a, 2d); and (ii) the amplitude of the T2 oscillator is larger and appears to be more stable than the amplitude of the output oscillator of the T1 motif (Fig. 2c, 2f). Since we consider in-phase synchronization it is clear why (i) may lead motif T1 to exhibit a greater propensity for network synchrony than the T2 motif. On the other hand, from (ii) we find that the oscillations of $x_3$ of motif T2 have more pronounced and larger peaks than those which correspond to motif T1. This makes it easier to distinguish between spurious subthreshold oscillations and actual peaks and may therefore promote greater output regularity (or reliability) as we consider regularity in the context of the ISI variability. Our results further suggest that network connectivity is positively correlated with synchronization and negatively correlated with the intensity of noise required to maximize synchronization, which is in agreement with existing studies (e.g., [@lu2019; @guo2009; @ge2020propagation]). Conversely, we find that the output regularity reaches an optimum point and an intermediate level of network connectivity ($d=0.05$; Fig. 6b). Finally, we find that moving our model closer to the excitation threshold has no effect on network synchronization or the amount of noise needed to maximize network synchrony, whereas it can indeed enhance output regularity and decrease the level of noise needed to optimize output regularity. Overall, we find that the T1 (excitatory FFL) and T2 (excitatory and inhibitory FFL) motifs, which are common in biological networks, differ in terms of their respective propensities for network synchronization and output regularity. Our results emphasize the functional importance of neural motifs and the diverse roles that they may hold within the brain. Possible extensions of our work could include: (a) an investigation into how uneven interactions (like coupling and noise) impact emergent dynamics in neural motifs; and (b) an analysis of the functional differences between neural motifs embedded within larger networks; in addition to an examination of the stochastic dynamics networks composed of such motifs.

arxiv_math

ALMOST-REDUCTIVE AND ALMOST-ALGEBRAIC LEIBNIZ ALGEBRAS DAVID A. TOWERS Department of Mathematics and Statistics Lancaster University Lancaster LA1 4YF England d.towers\@lancaster.ac.uk **Abstract** This paper examines whether the concept of an almost-algebraic Lie algebra developed by Auslander and Brezin in [@ab] can be introduced for Leibniz algebras. Two possible analogues are considered: almost-reductive and almost-algebraic Leibniz algebras. For Lie algebras these two concepts are the same, but that is not the case for Leibniz algebras, the class of almost-algebraic Leibniz algebras strictly containing that of the almost-reductive ones. Various properties of these two classes of algebras are obtained, together with some relationships to $\phi$-free, elementary, $E$-algebras and $A$-algebras. : 17B05, 17B20, 17B30, 17B50. Leibniz algebra, Frattini ideal, $\phi$-free, elementary, $E$-algebra, $A$-algebra, almost-algebraic, almost-reductive. # Introduction An algebra $L$ over a field $F$ is called a *Leibniz algebra* if, for every $x,y,z \in L$, we have $$[x,[y,z]]=[[x,y],z]-[[x,z],y]$$ In other words the right multiplication operator $R_x : L \rightarrow L : y\mapsto [y,x]$ is a derivation of $L$. As a result such algebras are sometimes called *right* Leibniz algebras, and there is a corresponding notion of *left* Leibniz algebras, which satisfy $$[x,[y,z]]=[[x,y],z]+[y,[x,z]].$$ Clearly the opposite of a right (left) Leibniz algebra is a left (right) Leibniz algebra, so, in most situations, it does not matter which definition we use. A *symmetric* Leibniz algebra $L$ is one which is both a right and left Leibniz algebra and in which $[[x,y],[x,y]]=0$ for all $x,y\in L$. This last identity is only needed in characteristic two, as it follows from the right and left Leibniz identities otherwise (see [@jp Lemma 1]). Symmetric Leibniz algebras $L$ are flexible, power associative and have $x^3=0$ for all $x\in L$ (see [@feld Proposition 2.37]), and so, in a sense, are not far removed from Lie algebras. Put $I=\langle\{x^2:x\in L\}\rangle$. Then $I$ is an ideal of $L$ and $L/I$ is a Lie algebra called the *liesation* of $L$. We define the following series: $$L^1=L,L^{k+1}=[L^k,L] \hbox{ and } L^{(1)}=L,L^{(k+1)}=[L^{(k)},L^{(k)}] \hbox{ for all } k=2,3, \ldots$$ Then $L$ is *nilpotent* (resp. *solvable*) if $L^n=0$ (resp.$L^{(n)}=0$) for some $n \in {\mathbb N}$. The *nilradical*, $N(L)$, (resp. *radical*, $\Gamma(L)$) is the largest nilpotent (resp. solvable) ideal of $L$. Throughout, $L$ will denote a (right) Leibniz algebra over a field $F$ of characteristic zero unless otherwise specified. The Frattini ideal of $L$, $\phi(L)$, is the largest ideal of $L$ contained in all maximal subalgebras of $L$. The Lie/Leibniz algebra $L$ is called *$\phi$-free* if $\phi(L) = 0$, and *elementary* if $\phi(B)=0$ for every subalgebra $B$ of $L$. Lie/Leibniz algebras all of whose nilpotent subalgebras are abelian are called *$A$-algebras*; Lie/Leibniz algebras $L$ such that $\phi(B)\leq \phi(L)$ for all subalgebras $B$ of $L$ are called *$E$-algebras*. The *abelian socle*, $Asoc(L)$, of a Lie/Leibniz algebra $L$ is the sum of its minimal abelian ideals. A linear Lie algebra $L\leq {\rm gl}(V)$ is *almost algebraic* if $L$ contains the nilpotent and semisimple Jordan components of its elements; an abstract Lie algebra $L$ is then called almost algebraic if ${\rm ad}L\leq {\rm gl}(L)$ is almost algebraic. Here we are exploring whether an analogous concept to this last one can be developed for Leibniz algebras and then to determine properties of, and inter-relationships between, these five classes of algebras analogous to those obtained by Towers and Varea in [@further]. In section 2 the concepts of an almost-reductive Leibniz algebra and of an almost-algebraic Leibniz algebra are introduced and various basic properties of them are produced. Descriptions of symmetric Leibniz algebras wich are almost algebraic, and of those with an almost-reductive radical are obtained. In additionh, some analogues of the results in [@further] are found for symmetric Leibniz algebras. In section 3 the inner derivation algebra, $R(L)$, of $L$ is defined. This is a Lie algebra and some properties of almost-reductive and almost-algebraic Leibniz algebras $L$ are related to corresponding properties of $R(L)$. We also introduce the concept of an $L$-split element of a Leibniz algebra $L$ and show that $L$ is almost algebraic if every element of $L$ is $L$-split. It is also shown that if every element of a subalgebra $B$ of $L$ is $L$-split then the idealiser of $B$ in $L$ is almost algebraic. In the final section we determine some consequences of these results for symmetric Leibniz $A$-algebras. # Definitions and Preliminary results ****Definition** 1**. *We call $L$ **almost reductive** if $L=N(L)\dot{+}\Sigma$, where $\Sigma$ is a Lie algebra and $N(L)$ is a completely reducible $\Sigma$-bimodule.* ****Lemma** 1**. *Let $L$ be an almost-reductive Leibniz algebra. Then $\Sigma=C\oplus S$, where $S$ is a semisimple Lie algebra, $C$ is an abelian Lie algebra and $R_c|_{N(L)}$ is semisimple for all $c\in C$.* **Proof.** If $A$ is an irreducible $\Sigma$-bimodule of $L$ we have that $[\Sigma,A]=0$ or $[\sigma,a]=-[a,\sigma]$ for all $a\in A, \sigma \in \Sigma$, by [@barnes Lemma 1.9]. It follows that $A$ is an irreducible right $\Sigma$-module of $L$ and the result now follows from [@jac Theorem 11, p.47]. $\Box$ ****Definition** 2**. *We call $L$ **almost algebraic** if $L/I$ is an almost-algebraic Lie algebra.* ** **Theorem** 2**. *Let $L$ be an almost-reductive Leibniz algebra. Then $L$ is almost algebraic.* **Proof.** Let $L=N(L)\dot{+} \Sigma$ be an almost-reductive Leibniz algebra, where $\Sigma=C\oplus S$, and let $N/I$ be the nilradical of $L/I$. Then $N(L)\subseteq N$ and $N=N(L)+N\cap \Sigma$. Since $N\cap \Sigma$ is a solvable ideal of $\Sigma$, $N\cap \Sigma \subseteq C$. Let $C=N\cap \Sigma \oplus D$. Then $L/I=N/I\dot{+} (\Sigma'+I)/I$ where $\Sigma'=D\oplus S$ is reductive. Hence $L/I$ is an almost-algebraic Lie algebra. $\Box$ The converse of the above result is false, as the following example shows. **Example 1**. *Let $L$ be the four-dimensional solvable cyclic Leibniz algebra with basis $a,a^2,a^3,a^4$ and $[a^4,a]=a^4$. Then $I=L^2$ and $L/I$ is trivially almost algebraic. But $N(L)=I$ and $L$ is not completely reducible: for example, there is no ideal $A$ of $L$ such that $I=A\oplus (Fa^3+Fa^4)$.* In fact, it even fails for symmetric Leibniz algebras, as shown in the next example. **Example 2**. *Let $L$ be the three-dimensional symmetric Leibniz algebra with basis $e_1,e_2,e_3$ and non-zero products $[e_1,e_2]=e_1$, $[e_2,e_1]=-e_1$, $e_2^2=e_3$. Then $I=Fe_3$ and $L/I\cong Fe_1+Fe_2$, which has nilradical $Fe_1$ and is clearly almost algebraic. However, $N(L)=Fe_1+Fe_3$ and $L$ does not split over this ideal, so $L$ is not almost reductive.* ** **Theorem** 3**. *Let $L$ be a Leibniz algebra.* - *If $L$ is $\phi$-free, then $L$ is almost reductive (and so, almost algebraic).* - *Let $L$ be almost reductive. Then $L$ is $\phi$-free if and only if its nilradical is abelian.* **Proof.** (i) Let $L$ be $\phi$-free. By [@stit Theorem 2.4 and Corollary 2.9] we have that $L = N(L) \dot{+} V$ where $V$ is a Lie subalgebra of $L$ acting completely reducibly on $N(L)$ and $N(L)=$ Asoc $L$. It follows that $L$ is almost reductive. \(ii\) Suppose that $L$ is almost reductive and that $N(L)$ is abelian. Then $N(L) =$ Asoc $L$, so $L$ is $\phi$-free by the same argument as in [@frat]. The converse follows from [@stit Theorem 2.4 and Corollary 2.9]. $\Box$ ****Corollary** 4**. *Let $L$ be an almost reductive Leibniz algebra. Then $L$ is $\phi$-free if and only if $L=\Lambda \dot{+} I$, where $\Lambda=\Sigma\dot{+}A$ is a $\phi$-free almost-algebraic Lie algebra with nilradical $A$, $N(L)=A\oplus I$, $[I,\Sigma]=I$ and $I$ is a completely reducible $\Sigma$-bimodule.* **Proof.** First, let $L$ be $\phi$-free. Then $L= (A_1\oplus \ldots\oplus A_n)\dot{+}\Sigma$ where $\Sigma$ is as described in Lemma [**Lemma** 1](#red){reference-type="ref" reference="red"} and $A_i$ is an abelian irreducible $\Sigma$-bimodule for each $1\leq i\leq n$. Suppose that $[a,\sigma]=-[\sigma,a]$ for all $a\in A_1\oplus \ldots\oplus A_r$, $\sigma\in \Sigma$, but that there is an $a_i\in A_i$ and a $\sigma_i \in \Sigma$ such that $[a_i,\sigma_i]\neq -[\sigma_i,a_i]$ for each $r+1\leq i\leq n$. Put $\Lambda=A_1\oplus \ldots\oplus A_r\dot{+}\Sigma$. Then $[A_i,\Sigma]$ is a $\Sigma$-bimodule, and so $[A_i,\Sigma]=A_i$ or $0$ for each $i$. Now $[\Sigma,A_i]=0$ for $r+1\leq i\leq n$, by [@barnes Lemma 1.9], so $[A_i,\Sigma]=A_i$ for $r+1\leq i\leq n$, by the choice of $r$. If $a\in A_i$ and $\sigma\in \Sigma$, then $[a,\sigma]=[a,\sigma+[\sigma,a]\in I$ for $r+1\leq i\leq n$. It follows that $A_i=[A_i,\Sigma] \subseteq I$ for each $r+1\leq i\leq n$. Clearly, $A_{r+1}\oplus\ldots\oplus A_n\subseteq I$ since $\Lambda$ is a Lie algebra. Now let $L=\Lambda \dot{+} I$, where $\Lambda=\Sigma\dot{+}A$ is a $\phi$-free almost-algebraic Lie algebra with nilradical $A$, $N(L)=A\oplus I$, $[I,\Sigma]=I$ and $I$ is a completely reducible $\Sigma$-bimodule. Then $L$ is $\phi$-free since its nilradical is abelian.. $\Box$ The following result is a generalisation of [@ao Theorem 6 and its Corollary]. ****Corollary** 5**. *Every $\phi$-free Leibniz algebra in which $I\subseteq Z(L)$ is a Lie algebra; in particular, every $\phi$-free symmetric Leibniz algebra is a Lie algebra.* **Proof.** Let $L$ be a $\phi$-free Leibniz algebra in which $I\subseteq Z(L)$. It follows from Corollary [**Corollary** 4](#arphifree){reference-type="ref" reference="arphifree"} that $I=0$, and so $L$ is a Lie algebra. $\Box$ ****Definition** 3**. *The **right centre** of a Leibniz algebra is the set $Z_r(L)=\{z\in L \mid [x,z]=0 \hbox{ for all } x\in L\}$.* For any (right) Leibniz algebra $L$, $Z_r(L)$ is an abelian ideal of $L$ (see [@feld Proposition 2.9]). A special case of the above result is the following. ****Proposition** 6**. *(cf. [@ao Theorem 6 and its Corollary]) If $L/Z_r(L)$ is semisimple and $\dim Z_r(L)=1$, then $L$ is a Lie algebra.* **Proof.** By Levi's Theorem for Leibniz algebras, $L=Z_r(L)\dot{+} S$ where $S$ is a semisimple Lie algebra. Let $Z_r(L)=Fz$ and let $s_1,s_2\in S$. Then $[z,s_i]=\lambda_i z$ for $i=1,2$ and $[z,[s_1,s_2]]=[[z,s_1],s_2]-[[z,s_2],s_1]=\lambda_1\lambda_2 z - \lambda_2\lambda_1 z=0$. Thus $[Z_r,L]=[Z_r,S]=[Z_r,S^2]=0$, whence $Z_r(L)=Z(L)$. It is now clear that $L$ is a Lie algebra. Note that $L$ is $\phi$-free and $I=0$, so this is a special case of Corollary [**Corollary** 5](#lie1){reference-type="ref" reference="lie1"}. $\Box$ ****Lemma** 7**. *Let $B$ be a subalgebra of a Leibniz algebra $L$. If $B$ is almost algebraic, then so the Lie algebra $(B+I)/I$.* **Proof.** Let $J$ be the Leibniz kernel of $B$. Then $B/J$ is almost algebraic. But now $$\frac{B+I}{I} \cong \frac{B}{B\cap I} \cong \frac{B/J}{(B\cap I)/J},$$ and $(B\cap I)/J$ is abelian and so is almost algebraic. The result then follows from [@ab Lemma 4.1]. $\Box$ ** **Theorem** 8**. *Let $L$ be a Leibniz algebra with radical $\Gamma$. Then,* - *if $\Gamma$ is almost algebraic, then so is $L$; and* - *if $L$ is almost reductive, then so is $\Gamma$.* **Proof.** (i) If $\Gamma$ is the radical of $L$, $\Gamma/I$ is the radical of $L/I$. Let $\Gamma$ be almost algebraic. Then $\Gamma/I$ is almost algebraic, by Lemma [**Lemma** 7](#frac){reference-type="ref" reference="frac"}. It follows from [@ab Corollary 3.1] that $L/I$, and hence $L$, is almost algebraic. \(ii\) This is clear from Lemma [**Lemma** 1](#red){reference-type="ref" reference="red"}. $\Box$ For Lie algebras the converse is true. However, it appears that this may not be the case even for symmetric Leibniz algebras, though examples are not easy to construct. The best that we can achieve at the moment is given by the following two results. ** **Theorem** 9**. *Let $L$ be a almost-algebraic symmetric Leibniz algebra with radical $\Gamma$ and nilradical $N$. Then $L=N+\Sigma$, where $N\cap \Sigma=I$ and $\Sigma=\Gamma\cap\Sigma\oplus S$ with $(\Gamma\cap\Sigma)^3=0$, $S$ semisimple and $R_{c+I}$ acting semisimply on $N/I$ for all $c\in \Gamma\cap \Sigma$.* **Proof.** We have that $L=\Gamma\dot{+}S$, where $S$ is a semisimple Lie algebra, by Levi's Theorem for Leibniz algebras. Also, $L/I$ is almost reductive. Thus, $L/I=N/I\dot{+}\Sigma/I$ where $\Sigma$ is a subalgebra of $L$, $(\Gamma\cap\Sigma)/I$ is abelian, $\Sigma/I=(\Gamma\cap\Sigma)/I\oplus (S\dot{+}I)/I$ with $S$ a semismple Lie algebra and $R_{c+I}$ acting semisimply on $N/I$ for all $c\in \Gamma\cap \Sigma$. Hence $L=N+\Sigma$ where $N\cap\Sigma=I$ and $(\Gamma\cap\Sigma)^2\subseteq I$, so $(\Gamma\cap\Sigma)^3=0$. Moreover, $$\begin{aligned} & = [\Gamma\cap\Sigma,S^2]\subseteq [[\Gamma\cap\Sigma,S],S]\subseteq [I,S]=0, \hbox{ and } \\ [S,\Gamma\cap\Sigma]& = [S^2,\Gamma\cap\Sigma]\subseteq [S,[S,\Gamma\cap\Sigma]]+[[S,\Gamma\cap\Sigma],S] \\ & \subseteq [S,I]+[I,S]=0.\end{aligned}$$ $\Box$ ** **Theorem** 10**. *Let $L$ be a symmetric Leibniz algebra with an almost-reductive radical $\Gamma$. Then $L$ is as described in Theorem [ **Theorem** 9](#symm1){reference-type="ref" reference="symm1"} above and $\Gamma=N\dot{+}C$ where $C$ is an abelian subalgebra and $R_c\mid_N$ is semisimple for all $c\in C$.* **Proof.** Suppose that $\Gamma$ is almost reductive, so that $\Gamma=N\dot{+}C$ where $C$ is an abelian subalgebra and $R_c\mid_N$ is semisimple for all $c\in C$. Moreover, $\Gamma$ is almost-algebraic, by Theorem [ **Theorem** 2](#t:aa){reference-type="ref" reference="t:aa"}, and hence so is $L$, by Theorem [ **Theorem** 8](#rad1){reference-type="ref" reference="rad1"}(i). The result follows. $\Box$ ****Proposition** 11**. *Let $L$ be a Leibniz algebra $L$.* - *If $L$ is almost algebraic and $J$ is an almost-algebraic ideal of $L$, then $L/J$ is almost algebraic.* - *If $L$ is almost reductive and $J$ is an ideal of $L$ with $J\subseteq \phi(L)$, then $L/J$ is almost reductive.* **Proof.** (i) It follows from Lemma [**Lemma** 7](#frac){reference-type="ref" reference="frac"} and [@ab Lemma 4.1] that $(J+I)/I$ and hence $L/(J+I)\cong (L/I)/((J+I)/I)$ is an almost-algebraic Lie algebra. But $(J+I)/J$ is the Leibniz kernel of $L/J$ and $(L/J)/((J+I)/J)\cong L/(J+I)$. Thus $L/J$ is almost algebraic. \(ii\) We have that $N(L/J)=N(L)/J$ as in [@nil Lemma 2.3]. Let $N(L)=A_1\oplus \ldots \oplus A_n$ where $A_i$ is an irreducible $\Sigma$-bimodule of $L$. Then $A_i\cap J= 0$ or $A_i$ for each $i=1,\ldots,n$. Let $A_1\cap J=\ldots=A_r\cap J=0$, $J=A_{r+1}\oplus \ldots \oplus A_n$, so $L/J\cong (A_1\oplus \ldots \oplus A_r)\dot{+} \Sigma$, which is almost reductive. $\Box$ ****Corollary** 12**. *Let $L$ be an almost-reductive Leibniz algebra. Then $\phi(L) = N^2$, where $N$ is the nilradical of $L$.* **Proof.** First, $N^2=\phi(N)\subseteq \phi(L)$, as in [@frat Theorem 6.5]. Hence $N(L/N^2)=N/N^2$, by [@nil Lemma 2.3], giving that $N(L/N^2)$ is abelian. Moreover, $L/N^2$ is almost reductive, by Proposition [**Proposition** 11](#p:fac){reference-type="ref" reference="p:fac"} (ii), and so $L/N^2$ is $\phi$-free, by Theorem [ **Theorem** 3](#phifree){reference-type="ref" reference="phifree"} (ii). It follows that $\phi(L) \subseteq N^2$. $\Box$ Note that the above Corollary is false if 'almost-reductive' is replaced by 'almost-algebraic', as the following example shows. **Example 3**. *Let $L$ be as in Example [Example 1](#ex1){reference-type="ref" reference="ex1"}. Then the only maximal subalgebras are $I$ and $F(a-a^2)+F(a^2-a^3)+F(a^3-a^4)$ (see [@st proof of Proposition 6.1]. Hence $\phi(L)= F(a^2-a^3)+F(a^3-a^4)\neq 0=N^2$.* Once again, it is not even true if $L$ is a symmetric Leibniz algebra. **Example 4**. *Let $L$ be as in Example [Example 2](#ex2){reference-type="ref" reference="ex2"}. Then $\phi(L)=Fe_3\neq 0 = N^2$.* ****Proposition** 13**. *Let $L$ be an almost-reductive symmetric Leibniz algebra. If every almost-algebraic subalgebra of $L$ is $\phi$-free, then $L$ is an elementary Lie algebra.* **Proof.** Since $\phi(L)=0$, we have that $L$ is a Lie algebra, by Corollary [**Corollary** 5](#lie1){reference-type="ref" reference="lie1"}. The result now follows from [@further Proposition 2.3]. $\Box$ ****Proposition** 14**. *Let $L$ be an almost-reductive symmetric Leibniz algebra. If every almost-algebraic subalgebra of $L/I$ is $\phi$-free, then $\phi(L)=N^2=I$ and $L$ is an $E$-algebra.* **Proof.** We have that $L/I$ is almost algebraic, by Theorem [ **Theorem** 2](#t:aa){reference-type="ref" reference="t:aa"}, so $N^2=\phi(L)\subseteq I$, by Corollary [**Corollary** 12](#c:aalg){reference-type="ref" reference="c:aalg"}. Now, if $M$ is a maximal subalgebra of $L$ with $Z(L)\not \subseteq M$, $L=M+Z(L)$ which gives that $L^2\subseteq M$. Hence $Z(L)\cap L^2\subseteq \phi(L)$. Thus $$N^2\subseteq I\subseteq Z(L)\cap L^2\subseteq \phi(L)=N^2.$$ Let $B$ be a subalgebra of $L$. Then $I_{L/I}((B+I)/I)$ is almost algebraic, by [@ab Theorem 2.3], and so is $\phi$-free, by assumption. It follows that $(B+I)/I$ is $\phi$-free, by [@frat Lemma 4.1], whence, $\phi(B)\subseteq I=\phi(L)$ and $L$ is an $E$-algebra. $\Box$ # The inner derivation algebra of a Leibniz algebra ****Definition** 4**. *The **inner derivation algebra** of $L$ is the set $R(L)=\{R_x\mid x\in L\}$.* Note that $R(L)$ is a Lie algebra under bracket product. For every subset $U$ of $L$ we will write $R_U=\{R_x\mid x\in U\}$. It is easy to check that $R_{[y,x]}=[R_x,R_y]$. To simplify notation, put $[y,_nx]=R_x^n(y)$. Then we have ****Lemma** 15**. *$R_{[y,_nx]}=(-1)^n [R_y,_{n-1}R_x]$.* **Proof.** We use induction on n. If $n=1$ we have $$R_{[y,x]}=[R_x,R_y]=-[R_y,R_x].$$ So suppose it holds for $n=k$. Then $$\begin{aligned} R_{[y,_{k+1}x]}=R_{[[y,_k x],x]}=-[R_{[y,_k x]},R_x]& =(-1)^{k+1}[[R_y,_{k-1} R_x],R_x] \\ & =(-1)^{k+1}[R_y,_k R_x].\end{aligned}$$ $\Box$ ****Definition** 5**. *For any algebra $A$, the **opposite algebra**, $A^{\circ}$, has the same underlying vector space and the opposite multiplication, $(x,y)\mapsto x\star y =yx$, where juxtaposition denotes the multiplication in $A$.* The following is easy to check (see, for example, [@feld Proposition 2.26]). ****Proposition** 16**. *For any (right) Leibniz algebra $L$, the map $\theta : L \rightarrow R(L)^{\circ} : x\mapsto R_x$ is a homomorphism with kernel $Z_r(L)$, so the Lie algebra $L/Z_r(L)$ is isomorphic to $R(L)^{\circ}$.* The following lemma is easy to see. ****Lemma** 17**. *For any Lie algebra $L$,* - *$U$ is a subalgebra of $L$ if and only if $U^{\circ}$ is a subalgebra of $L^{\circ}$;* - *$U$ is an ideal of $L$ if and only if $U^{\circ}$ is an ideal of $L^{\circ}$;* - *$U$ is solvable if and only if $U^{\circ}$ is solvable;* - *$U$ is nilpotent if and only if $U^{\circ}$ is nilpotent; and* ****Lemma** 18**. *For every Leibniz algebra $L$ we have* - *If $U$ is a subalgebra of $L$ then $R_U$ is a subalgebra of $R(L)$;* - *Every subalgebra of $R(L)$ is of the fom $R_U$ where $U$ is a subalgebra of $L$.* - *If $U$ is an ideal of $L$ then $R_U$ is an ideal of $R(L)$.* - *Every ideal of $R(L)$ is of the fom $R_U$ where $U$ is an ideal of $L$.* **Proof.** $$\begin{aligned} \hbox{(i) } U \hbox{ is a subalgebra of } L & \Rightarrow \lambda x + \mu y, [x,y]\in U \hbox{ for all } x,y\in U, \lambda,\mu\in F \\ & \Rightarrow \lambda R_x + \mu R_y = R_{\lambda x + \mu y}, [R_x,R_y]=R_{[y,x]}\in R_U.\end{aligned}$$ (ii) Let $K$ be a subalgebra of $R(L)$. Put $U=\{x\in L \mid R_x\in K\}$. Then $U$ is a subalgebra of $L$ as in (i). $$\begin{aligned} \hbox{(iii) } U \hbox{ is an ideal of } L & \Rightarrow [x,y], [y,x]\in U \hbox{ for all } x\in U, y\in L \hspace{2.4cm} \\ & \Rightarrow [R_y,R_x], [R_x,R_y]=\pm R_{[x,y]}\in R_U.\end{aligned}$$ (iv) This is similar to (ii). $\Box$ ****Lemma** 19**. *Let $L$ be a Leibniz algebra. Then* - *$\Gamma$ is the radical of $L \Leftrightarrow R_{\Gamma}$ is the radical of $R(L)$;* - *If $Z_r(L)\subseteq \phi(L)$, then $N$ is the nilradical of $L\Leftrightarrow R_N$ is the nilradical of $R(L)$.* **Proof.** (i) Clearly, $\Gamma(L/Z_r(L))=\Gamma/Z_r(L)$ and so $\Gamma/Z_r(L)\cong\Gamma(R(L)^{\circ})=\Gamma(R(L))$. Moreover, $\theta\mid_{\Gamma}$ is a homomorphism from $\Gamma$ onto $R_{\Gamma}$, whence the result. \(ii\) Clearly, $Z_r(L)\subseteq N$ and so $N/Z_r(L)\subseteq N(L/Z_r(L))=K/Z_r(L)$, say. But $K$ is nilpotent, by [@barnes Theorem 5.5], so $N/Z_r(L)=N(L/Z_r(L))$. The proof now follows in similar manner to (i). $\Box$ ****Proposition** 20**. *If the Leibniz algebra $L$ is almost algebraic then so is the Lie algebra $R(L)$.* **Proof.** Let $L$ be almost algebraic. Then $L/I$ is almost algebraic. Now $L/Z_r(L)\cong (L/I)/(Z_r(L)/I)$ and $Z_r(L)/I$ is abelian and so is almost algebraic. It follows that $L/Z_r(L)$ is almost algebraic, by [@ab Lemma 4.1]. The result follows from Proposition [**Proposition** 16](#factor){reference-type="ref" reference="factor"}. $\Box$ ****Definition** 6**. *If $B$ is a subalgebra of $L$, the **idealiser of $B$ in $L$**, $I_L(B)=\{ x\in L \mid [x,b], [b,x]\in B \hbox{ for all } b\in B\}$.* ****Corollary** 21**. *Let $B$ be a subalgebra of an almost-algebraic Leibniz algebra $L$. Then the idealiser, $I_{R(L)}(R_B)$, of $R_B$ in $R(L)$ is an almost-algebraic Lie algebra.* **Proof.** This follows from Proposition [**Proposition** 20](#main){reference-type="ref" reference="main"} and [@ab Theorem 2.3]. $\Box$ ****Definition** 7**. *The element $x\in L$ is called **$L$-split** if there exist elements $s,n\in L$ such that $R_x=R_s+R_n$ is the decomposition of $R_x$ into its semisimple and nilpotent parts.* ****Proposition** 22**. *If every element of the Leibniz algebra $L$ is $L$-split, then $L$ is almost algebraic.* **Proof.** Let $x\in L$. Then $R_{x+I}=R_{s+I}+R_{n+I}$ if $R_x=R_s+R_n$, and $R_{s+I}, R_{n+I}$ are the semisimple and nilpotent parts of $R_{x+I}$, so the result follows from [@ab Theorem 2]. $\Box$ The following result is now proved as in [@ab Theorem 2.3]. ****Proposition** 23**. *Let $B$ be a subalgebra of a Leibniz algebra $L$ in which every element is $L$-split. Then the idealiser, $I_L(B)$, of $B$ in $L$ is almost algebraic.* **Proof.** Let $J=I_L(B)$. Since $R_L(B)$ leaves $B$ invariant, so does its algebraic hull. In particular, if $x\in J$, both the semisimple and nilpotent parts of $R_L(x)$ leave $B$ invariant. Hence,every element of $J$ is $J$-split, and so, by Proposition [**Proposition** 22](#split){reference-type="ref" reference="split"}, $J$ is almost algebraic. $\Box$ # Leibniz $A$-algebras ****Proposition** 24**. *Let $L$ be a Lie $A$-algebra and let $K$ be an ideal of $L$ with $K\subseteq Z(L)$. If $L/K$ is almost algebraic then so is $L$.* **Proof.** Let $L/K$ be almost algebraic and let $R$ be the radical of $L$. Then $R/K$ is almost algebraic, by [@ab Corollary 3.1], and so $\phi(R/K)=0$, by [@further Lemma 2.1 (ii)]. Hence $\phi(R)\subseteq K\subseteq Z(R)$, by [@frat Corollary 4.4]. It follows that $\phi(R)\subseteq Z(R)\cap R^2=0$, by [@lie Theorem 3.3], since $R$ is an $A$-algebra. Thus $R$ is almost algebraic, by [@further Proposition 2.1], whence so is $L$, by [@ab Corollary 3.1] again. $\Box$ ****Corollary** 25**. *Let $L$ be a Leibniz $A$-algebra and let $K$ be an ideal of $L$ with $K\subseteq Z(L)$. If $L/K$ is almost algebraic then so is $L$.* **Proof.** Let $L/K$ be almost algebraic. Then $$\frac{L/I}{(I+K)/I}\cong \frac{L/K}{(I+K)/K},$$ which is almost algebraic, by Proposition [**Proposition** 11](#p:fac){reference-type="ref" reference="p:fac"}. Moreover, $(I+K)/I\subseteq Z(L/I)$ and $L/I$ is a Lie $A$-algebra, by [@lie Lemma 2], so $L/I$ is almost algebraic, by Proposition [**Proposition** 24](#lie){reference-type="ref" reference="lie"}. Hence $L$ is almost algebraic. $\Box$ ****Lemma** 26**. *Let $L$ be an almost-reductive symmetric Leibniz $A$-algebra. Then $L$ is a Lie algebra.* **Proof.** Since $L$ is almost reductive, $\phi(L)=N^2=0$, since $L$ is an $A$-algebra. The result now follows from Corollary [**Corollary** 5](#lie1){reference-type="ref" reference="lie1"}. $\Box$ ****Lemma** 27**. *If $L$ is a symmetric Leibniz $A$-algebra, then $L/I$ is a Lie $A$-algebra.* **Proof.** If $K/I$ is a nilpotent subalgebra of $L/I$, $K^r\subseteq I$ for some $r>0$, whence $K^{r+1}=0$. It follows that $K$ is nilpotent and thus abelian. $\Box$ ** **Theorem** 28**. *Let $L$ be a symmetric Leibniz $A$-algebra. Then $L$ is an almost-reductive algebra if and only if it is an elementary Lie algebra.* **Proof.** $(\Rightarrow)$ Let $L$ be an almost-reductive symmetric Leibniz $A$-algebra. Then $L$ is a Lie algebra, by Lemma [**Lemma** 26](#ars){reference-type="ref" reference="ars"}.It now follows that it is elementary, by [@further Theorem 2.4]. $(\Leftarrow)$ The converse follows from [@further Theorem 2.4]. $\Box$ ****Corollary** 29**. *Let $L$ be a symmetric Leibniz $A$-algebra with radical $\Gamma$. If $\Gamma$ is $\phi$-free, then $L$ is an elementary Lie algebra.* **Proof.** Assume that $\Gamma$ is $\phi$-free. Then $\Gamma$ is almost reductive, by Corollary [ **Theorem** 3](#phifree){reference-type="ref" reference="phifree"} (i). It follows that $L$ is as described in Theorem [ **Theorem** 10](#symm2){reference-type="ref" reference="symm2"}. Moreover, $(\Gamma\cap \Sigma)^2=0$, since $L$ is an $A$-algebra, and, if $\sigma\in \Gamma\cap \Sigma$, $\sigma=n+c$ for some $n\in N$, $c\in C$. Hence $[n',\sigma]=[n',c]$ for all $n'\in N$, and so $R_{\sigma}\mid_N$ is semisimple. It follows that $L$ is almost reductive and hence that $L$ is an elementary Lie algebra, by Theorem [ **Theorem** 28](#A){reference-type="ref" reference="A"}. $\Box$ ****Corollary** 30**. *Let $L$ be an almost-reductive symmetric Leibniz $A$-algebra. Then $L$ splits over each of its ideals.* **Proof.** This follows from Lemma [**Lemma** 26](#ars){reference-type="ref" reference="ars"} and [@further Corollary 2.6]. $\Box$ ****Proposition** 31**. *Let $L$ be a Leibniz algebra over any field. Then $L$ is an $E$-algebra if and only if $L/\phi(L)$ is elementary.* **Proof.** The proof is the same as for the Lie case in [@ernie Proposition 2]. $\Box$ ****Proposition** 32**. *Let $L$ be a symmetric Leibniz $A$-algebra. Then $L$ is an $E$-algebra.* **Proof.** Let $L$ be a Leibniz $A$-algebra. Then $L/\phi(L)$ is an $A$-algebra , by [@Aalg Lemma 2]. But $L/\phi(L)$ is $\phi$-free and so is almost-reductive, by Corollary [**Corollary** 12](#c:aalg){reference-type="ref" reference="c:aalg"} (i). Hence $L/\phi(L)$ is elementary, by Theorem [ **Theorem** 28](#A){reference-type="ref" reference="A"}, and so $L$ is an $E$-algebra, by Proposition [**Proposition** 31](#E){reference-type="ref" reference="E"}. $\Box$ 1 L. Auslander and J. Brevin, 'Almost algebraic Lie algebras', *J. Algebra* **8** (1968), 295--313. Sh.A. Ayupov ,B.A. Omirov, 'On Leibniz algebras', *Algebra and Operators Theory, Proceeding of the Colloquium in Tashkent*, (1997), Kluwer Academic Publishers, (1998), 1--13. D.W. Barnes, 'Some theorems on Leibniz algebras', *Comm. Algebra* **39(7)** (2011), 2463--2472. C. Batten Ray, L. Bosko-Dunbar, A. Hedges, J.T. Hird, K. Stagg and E. Stitzinger, 'A Frattini theory for Leibniz algebras', *Comm. Alg.* **41(4)** (2013), 1547--1557. J. Feldvoss, 'Leibniz algebras as non-associative algebras.' *Nonassociative mathematics and its applications*, 115--149, Contemp. Math., **721**, *Amer. Math. Soc., Providence, RI*, 2019. A. Fialowski and E.Z. Mihálka, 'Representations of Leibniz algebras', *Algebras and Representation Theory*, **18(2)** (2015), 477--490. N. Jacobson, *Lie Algebras* (Interscience, New York-London, 1962). , 'Lie theory for symmetric Leibniz algebras', *J. Homotopy and Related Structures* **15** (2020), 167-183. E.L. Stitzinger, 'Frattini subalgebras of a class of solvable Lie algebras', *Pacific J. Math.* **34 (1)** (1970), 177-182. D.A. Towers, 'A Frattini theory for algebras', *Proc. London Math. Soc.* (3) **27** (1973), 440--462. D.A. Towers, 'Solvable Lie $A$-algebras', *J. Algebra* **340 (1)** (2011), 1-12. D.A. Towers and V.R. Varea, 'Further results on elementary Lie algebras and Lie $A$-algebras', *Comm. Alg.* **41 (4)** (2013), 1432-1441. D.A. Towers, 'Leibniz $A$-algebras', *Comm. Math.* **28** (2020), 103-121. D.A. Towers, 'On the nilradical of a Leibniz algebra', *Comm. Alg.* **49 (10)** (2021), 4345-4347.

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--- abstract: | The nonlinear Poisson-Boltzmann equation (NPBE) is an elliptic partial differential equation used in applications such as protein interactions and biophysical chemistry (among many others). It describes the nonlinear electrostatic potential of charged bodies submerged in an ionic solution. The kinetic presence of the solvent molecules introduces randomness to the shape of a protein, and thus a more accurate model that incorporates these random perturbations of the domain is analyzed to compute the statistics of quantities of interest of the solution. When the parameterization of the random perturbations is high-dimensional, this calculation is intractable as it is subject to the curse of dimensionality. However, if the solution of the NPBE varies analytically with respect to the random parameters, the problem becomes amenable to techniques such as sparse grids and deep neural networks. In this paper, we show analyticity of the solution of the NPBE with respect to analytic perturbations of the domain by using the analytic implicit function theorem and the domain mapping method. Previous works have shown analyticity of solutions to linear elliptic equations but not for nonlinear problems. We further show how to derive *a priori* bounds on the size of the region of analyticity. This method is applied to the trypsin molecule to demonstrate that the convergence rates of the quantity of interest are consistent with the analyticity result. Furthermore, the approach developed here is sufficiently general enough to be applied to other nonlinear problems in uncertainty quantification. author: - Trevor Norton, Jie Xu, Brian Choi, Mark Kon, Julio Enrique Castrillón-Candás bibliography: - ref_nPBE.bib - citations.bib - nPBEapplicationsALL.bib title: Uncertainty quantification and complex analyticity of the nonlinear Poisson-Boltzmann equation for the interface problem with random domains --- **Keywords.** Non-linear PDEs, Uncertainty Quantification, Sparse Grids, non-linear solvers, Interface problem **MSC 2020** 65N35, 65N12, 65N15, 65C20, 35G20, 35J57, 35J60 # Introduction Nonlinear elliptic partial differential equations (PDEs) are frequently used as models for applications in electrostatics. In particular, a salient problem in the field is the modeling of potential fields generated by molecules in solvents. The nonlinear Poisson-Boltzmann Equation (NPBE) serves as an accurate representation of the molecule-solvent interactions and is employed in molecular dynamics simulations and chemical applications [@Gray2018; @Stein2019]. It has been used in the modeling of electrode-electrolyte interfaces [@Sundararaman2017; @Sundararaman2018; @Nattino2019] and in solvers such as the Adaptive Poisson--Boltzmann Solver (APBS) in determining the electrostatic potential for biomolecular processes [@Baker2001; @jurrus2018improvements]. The NPBE also finds applications in other various scientific disciplines. It has been studied in many fields such as applied mathematics [@Cai2013; @Rubinstein1990; @Li2009], biophysical chemistry [@Ohshima2010; @Edsall1958], biochemistry [@Bergethon1998], chemical physics [@Frenkel1946; @Kirkwood1961], colloids [@Butt2013; @Israelachvili1991; @Verwey1948; @Evans1999; @Hunter2001; @Lyklema1991], condensed matter physics [@Chaikin1995], electrochemistry [@Schmickler2010; @Bockris1970; @Sparnaay1972], electrolyte solutions [@Barthel1998; @Friedman1962], liquid state theory [@Hansen2013; @Fawcett2004; @March1984], many-body theory [@Brout1963; @Giuliani2003], materials science [@Chavazaviel1999], medical physics [@Hobbie1988], molecular biology [@Sneppen2005], physiology [@Bayliss1959; @Hober1947], physical chemistry [@Berry2000; @Atkins1986], plasma physics [@Morrison1967; @Ichimaru1984], polymer physics [@Muthukumar2011], soft matter [@Doi2016; @Dean2014; @Holm2001; @Poon2006], solid-state physics [@Ashcroft1976; @Kittel1996; @McKelvey1966], statistical mechanics [@Blum1992; @Landau1958; @McQuarrie1976], surface science [@Israelachvili1991; @Butt2013], thermodynamics [@Glasstone1947; @Lewis1961], among others. The NPBE can be written as $$\label{npbe} \begin{aligned} -\nabla \cdot (\epsilon(\mathbf{x}) \nabla u(\mathbf{x})) + \overline{\kappa}^2(\mathbf{x}) \sinh(u(\mathbf{x})) &= f(\mathbf{x}), &\text{for }\mathbf{x}&\in \mathcal D, \\ u(\mathbf{x}) &= g(\mathbf{x}), &\text{for } \mathbf{x}&\in \partial \mathcal D, \end{aligned}$$ where $\mathcal D\subset \mathbb{R}^3$ is the domain, $\epsilon(\mathbf{x})>0$ is a dimensionless dielectric function, $\overline \kappa(\mathbf{x})\geq 0$ is the modified Debye-Hückel parameter, and $f(\mathbf{x})$ gives the charge of the particles in the region. The desired solution $u$ represents the (dimensionless) potential function. One typically assumes the domain is separated into three parts: the solvent, the molecular region, and the ion-exclusion layer. From this assumption, $\epsilon(\mathbf{x})$ becomes discontinuous at the interfaces between these regions, and so this PDE is sometimes referred to as an *elliptic interface problem*. Variational methods can be used to show that a unique weak solution (i.e. a function $u\in H^1$) exists under certain conditions [@holst1994poisson]. When computing molecular dynamics (MD), the presence of thermal fluctuations and solvent interactions (among other factors) can lead to random conformations of the molecules, and more accurate models incorporate this stochasticity. For instance, stochastic initial velocities are used in [@Allen2004; @Neumaier1997] when computing MD. Other approaches to MD which factor in stochasticity include Langevin dynamics [@Ceriotti2009; @Hanggi1995; @Jung1987] and Markov random models [@Xia2013]. In this paper we assume that the random domain conformations are represented using a finite dimensional model with $N$ random variables such as Karhunen-Loève expansions or similar random field stochastic representations. Quadrature methods can compute stochastic measures for the Quantity of Interest (QoI) of the potential field under random configurations of the molecules. However, for each quadrature point we must compute the solution of the NPBE. As the number of dimensions, $N$, increases the calculation quickly becomes intractable. One strategy to reducing the cost of the calculations is to show that the QoI varies analytically with respect to the stochastic parameters. In this case, one can compute the $N$-dimensional quadrature using a sparse grid [@nobile2008a], which gives sub-exponential or algebraic decay in the error as a function of the number of interpolation points. Thus the "curse of dimensionality" from the $N$ can be ameliorated, and the problem becomes tractable. If the QoI depends analytically on the solution $u$, then it is sufficient to prove that the solution varies analytically with respect to the stochastic parameters. Previous studies in uncertainty quantification (UQ) have explored the analyticity of solutions to linear partial differential equations with random domains [@castrillon2016; @castrillon2021stochastic]. The authors in [@Heitzinger2018] explore the utilization of stochastic collocation and Galerkin methods for the NPBE. The NPBE is treated as a semi-linear stochastic boundary valued problem and the existence of a unique solution is proved. This approach extends the existence and uniqueness result found in the deterministic case [@holst1994poisson] and follows a similar proof strategy. However, due to the absence of analytic regularity results, convergence rates for implementing the stochastic collocation method are not derived. For the case of elliptic interface problems, the regularity of point evaluations of solutions and how to approximate them with Deep Neural Networks (DNNs) have been studied [@scarabosio2022deep]. Furthermore, in [@Opschoor2021] the authors show that given a holomorphic map, such as an analytic extension of the solution of a PDE, there exists a DNN with exponential accuracy with respect to the dimensionality of the DNN. In the case of the NPBE, our objective is to demonstrate that analytical deformations of the domain result in the analytic variation of the solution $u$. This particular investigation introduces two challenges that were not addressed in previous research: 1. *Nonlinearity*. Previous results have been proved by showing that the solutions satisfy the Cauchy equations and are thus analytic. This becomes difficult to do with the nonlinearity introduced by $\sinh(u)$. A more subtle complication due to the nonlinearity is that outputs of the function might not end up in the desired space. That is, for potential weak solutions $u \in H^1$, it is not guaranteed that $\sinh(u)\in L^2$. 2. *Interfaces.* In [@choi2021existence] the analyticity properties of the NPBE are studied where $\epsilon$ exhibits some degree of regularity. However in our case, the assumption that $\epsilon$ is Lipschitz continuous on the entire domain is relaxed to account for the interface problem. The main strategy of this paper is to use the implicit function theorem and the domain mapping method (introduced in [@castrillon2016analytic]) to show that $u$ is analytic with respect to the stochastic parameters. This avoids trying to show the Cauchy equations hold, and it is a general strategy that can be applied to other nonlinear PDEs. In order to apply the implicit function theorem, we have to specify a function domain for our solution. As noted above, we cannot take $u\in H^1$ since this does not imply $\sinh(u)$ is in $L^2$. If $u$ were in $H^2$ then the fact that the Sobolev space is a Banach algebra (see [@adams2003sobolev Thm. 4.39] for a proof) would let us conclude the nonlinearity is in $L^2$, but the discontinuity of $\epsilon$ in the problem means that $u$ is not (weakly) differentiable across the interfaces. However, we can instead define a "piecewise $H^2$" space, in which $u$ naturally lies. Thus we can then apply the implicit function theorem to get analyticity. From there, estimates of the rate of convergence of the sparse grid method can be obtained by getting *a priori* bounds on the region of analyticity of the solution. The results here are also notable in that they can easily be generalized to other UQ problems that come from nonlinear PDEs with interfaces. The paper will be structured as follows. In [2](#linear-pde-section){reference-type="ref" reference="linear-pde-section"}, we introduce the problem of a linear elliptic PDE with interfaces. We introduce a suitable Banach space in which a strong solution naturally exists: a "piecewise $H^2$" space denote by $\mathcal{H}(U)$. It is then shown that linear problem has unique strong solutions that induces an isomorphism between $\mathcal{H}(U)$ and the Banach space of the forcing functions. In [3](#npbe-reference-domain-section){reference-type="ref" reference="npbe-reference-domain-section"}, the NPBE is reformulated onto a reference domain with solutions in $\mathcal{H}(U)$. From there, the implicit function theorem is used to get an analytic mapping from the parameter space to the solutions of the NPBE. Furthermore, we give details on how to get *a priori* bounds on the size of the region of analyticity after applying the implicit function theorem. gives an overview of applying sparse grids to the efficient computation of integrals of analytic functions. Finally, numerical experiments are performed in [5](#numerics){reference-type="ref" reference="numerics"} to demonstrate the convergence results. # The Linear Elliptic PDE with Interfaces {#linear-pde-section} ## Definitions and notations We first consider a linear elliptic PDE with possibly discontinuous coefficients at interfaces. For our problem, the domain is split up into three subdomains. The coefficients of the PDE are sufficiently regular on each subdomain, and the subdomains are nested within each other. The boundary between the subdomains form the interfaces in our problem. It is straightforward to generalize this to an arbitrary number of nested subdomains. **Definition 1**. *We say that a connected, bounded open set $U\subset \mathbb{R}^3$ is **properly decomposed** into $l$ subdomains $U_1,U_2,\ldots, U_l$ if the following holds: There is a sequence of compactly embedded subsets $$U^{(l-1)} \subset\subset U^{(l-2)} \subset\subset \cdots \subset \subset U^{(1)} \subset\subset U$$ where $$\begin{aligned} U_l &= U \setminus \overline{U^{(1)}} \\ U_{l-1} &= U^{(1)} \setminus \overline{U^{(2)}} \\ &\hspace{0.5em}\vdots \\ U_2 &= U^{(l-2)} \setminus \overline{U^{(l-1)}} \\ U_1 &= U^{(l-1)}.\end{aligned}$$ We define the interfaces $I_1, \ldots, I_{l-1}$ to be $I_i = \partial U_i \cap \partial U_{i+1}$ for $i = 1, \ldots, l-1$.* *Remark 1*. The results for this section could also be generalized to the case where subdomains are no longer strictly nested within each other. However, we will use the above definition since it is sufficient for our application and it keeps the notation simple. For convenience, we shall refer to $\partial U$ by $I_l$. We now assume that $U\subset \mathbb{R}^3$ is properly decomposed into $l$ subdomains $U_1,\ldots, U_l$ where the interfaces and the boundary of $U$ are all of class $C^{1,1}$. We choose our domain to be in $\mathbb{R}^3$ for our application; other dimensions are possible, but the choices of Sobolev spaces will be affected. Denote by $\nu_k$ the outward facing normal for the surface $I_k$. On each of these surfaces we can define trace operators. For $1\leq k \leq l-1$, there are two trace operators depending on if we take the domain to be $H^1(U_k)$ or $H^1(U_{k+1})$. Let $$\gamma^+_k : H^1(U_k) \to H^{1/2}(I_k)$$ be the trace operator from the domain $U_k$ to its outer boundary (for $1\leq k \leq l$), and similarly let $$\gamma^-_k : H^1(U_{k+1}) \to H^{1/2}(I_k)$$ be the trace operator from the domain $U_{k+1}$ to its inner boundary (for $1\leq k\leq l-1$). Define a second-order elliptic operator ${\mathcal P}_k$ on each $U_k$ by $$\label{second-order-operator-k} {\mathcal P}_k u := - \sum_{i=1}^3 \sum_{j=1}^3 \partial_i(a_{ij} \partial_j u) + c u$$ where $a_{ij} \in C^{0,1}(\overline{U_k})$ for each $i,j=1,2,3$ and $k=1,2,\ldots, l$ and $c \in L^\infty(U)$. We assume that $a_{ij} = a_{ji}$ for all $i$ and $j$. We further assume that each ${\mathcal P}_k$ satisfies a uniform ellipticity condition on $U_k$, i.e., there exists a constant $\theta > 0$ such that $$\sum_{i,j=1}^3 a_{ij}(x) \xi_i\xi_j \geq \theta |\xi|^2$$ for a.e. $x\in U_k$ and all $\xi \in \mathbb{R}^3$. We can choose $\theta$ independently of $k$. The operator ${\mathcal P}_k$ is naturally associated with the bilinear map $\Phi_k$ given by $$\Phi_k(u,v) = \int_{U_k} \left( \sum_{i=1}^3 \sum_{j=1}^3 a_{ij} \partial_j u \partial_i v + c u v \right)\, \mathrm d x$$ The operators ${\mathcal P}_k$ define co-normal derivatives on the interfaces and boundary. We define ${\mathcal B}^{\pm}_k$ by $${\mathcal B}_k^\pm u := \sum_{i=1}^3 (\nu_k)_i \gamma_k^{\pm} \left( \sum_{j=1}^3 a_{ij} \partial_j u \right),$$ where $(\nu_k)_i$ denotes the $i$th component of the normal vector $\nu_k$. This gives maps ${\mathcal B}_k^+ : H^2(U_k)\to H^{1/2}(I_k)$ and ${\mathcal B}_k^- :H^2(U_{k+1}) \to H^{1/2}(I_k)$. By specifying the value $f\in H^{-1}(U_k)$ of ${\mathcal P}_k u$, the conormal derivative can be extended to $H^1(U_k)$ functions. If the choice of $f\in H^{-1}(U_k)$ is clear, then we will simply say the distribution ${\mathcal B}^\pm_k u \in H^{-1/2}(I_k)$ is the conormal derivative of $u$. These definitions allow us to use the following Green's identity for $2\leq k \leq l$: $$\label{greens-1} \Phi_k(u,v) = ({\mathcal P}_k u,v)_{U_k} + ({\mathcal B}^+_k u, \gamma_k^+ v)_{I_k} - ({\mathcal B}^-_{k-1}u, \gamma_{k-1}^- v)_{I_{k-1}}, \quad \text{for all } u\in H^2(U_k), v\in H^1(U_k).$$ And in the case of $k=1$ we have $$\label{greens-2} \Phi_1(u,v) = ({\mathcal P}_1 u,v)_{U_1} + ({\mathcal B}^+_1 u, \gamma_1^+ v)_{I_1}, \quad \text{for all } u\in H^2(U_1), v\in H^1(U_1).$$ ## Weak and strong forms of elliptic problem with interfaces {#subsection-linear-elliptic-problems} Similar to standard elliptic PDE theory, the existence and uniqueness of the weak solution for the discontinuous interface problem will first be established. Consequently, this result will be used to show existence and uniqueness of the strong solution. However, to motivate the weak formulation, we first start off by defining the strong form of the problem. Normally, the strong solution of an elliptic PDE would lie in the space $H^2(U)$, but this cannot be the case for our problem since we can lose regularity at the interfaces. The next best option is to require a "piecewise $H^2$ regularity" for the strong solution, where the function is $H^2$ when restricting to the subdomains. **Definition 2**. *Let $$\mathcal{H}(U;U_1,U_2,\ldots, U_l) := \{u \in H^1(U) \mid u|_{U_k} \in H^2(U_k) \text{ for } k=1,2,\ldots, l\}.$$ This is a Banach space with a norm given by $$\| u \|_{\mathcal{H}} = \|u\|_{H^1(U)} + \sum_{k=1}^l \left\| u|_{U_k} \right\|_{H^2(U_k)}.$$* This Banach space depends on our decomposition of $U$, but when this decomposition is clear we will simply write $\mathcal{H}(U)$ instead of $\mathcal{H}(U;U_1,U_2,\ldots, U_l)$. Throughout the paper, we will denote $u|_{U_k}$ or $f|_{U_k}$ by $u_k$ or $f_k$, respectively, to cut down on notation. Requiring the strong solution $u$ to lie in $\mathcal{H}(U)$ is insufficient to define a unique strong solution for the elliptic problem. If the strong solution were only required to satisfy ${\mathcal P}_k u_k = f_k$ on each $U_k$ along with a Dirichlet boundary condition, then infinitely many solutions would be possible; for instance, in the case where $c \equiv 0$ adding a constant value to $u_1$ on $U_1$ would give another solution to the problem. Unique solutions exist if certain jump conditions are satisfied at the interfaces. For $u\in \mathcal{H}(U;U_1,U_2,\ldots,U_l)$, define $$_{I_k} = {\mathcal B}_k^+u_k - {\mathcal B}^-_k u_{k+1} \in H^{1/2}(I_k).$$ Then the strong form of the elliptic PDE with interfaces is stated as follows: **Problem 1** (Strong form of elliptic PDE with interfaces). *Suppose $U$ is properly decomposed into subdomains $U_1,U_2,\ldots,U_l$, where the interfaces and boundary are of class $C^{1,1}$. Let ${\mathcal P}_k$ be defined as in [\[second-order-operator-k\]](#second-order-operator-k){reference-type="ref" reference="second-order-operator-k"}. Fix $f \in L^2(U)$, $g_k\in H^{1/2}(I_k)$ for $k=1,2,\ldots, l-1$, and $g_l \in H^{3/2}(\partial U)$. A function $u \in \mathcal{H}(U;U_1,U_2,\ldots, U_l)$ is a strong solution of the elliptic PDE with interfaces if $$\begin{aligned} {\mathcal P}_k u_k &= f_k & &\text{for } k=1,2,\ldots, l, \\ \left[{\mathcal B}_k u\right]_{I_k} &= g_k & &\text{for } k = 1,2,\ldots, l-1, \\ \gamma^+_l u_l &= g_l. \end{aligned}$$* The boundary condition can be set to zero by setting $w\in H^2(U)$ to be such that $\gamma_l^+ w_l = g_l$. Then we can break up $u$ into $u = \tilde u + w$ where $\tilde u \in H^1_0(U)\cap \mathcal{H}(U)$. The weak formulation of the problem is derived by taking $${\mathcal P}\tilde u = f - {\mathcal P}w$$ (where ${\mathcal P}$ is the differential operator that is locally ${\mathcal P}_k$ on each $U_k$), multiplying each side of the equation by $v\in H^1_0(U)$, and integrating over $U$. Applying [\[greens-1,greens-2\]](#greens-1,greens-2){reference-type="ref" reference="greens-1,greens-2"} and summing up the terms gives us $$\label{formal-integration} \sum_{k=1}^l \Phi_k(\tilde u_k , v_k) = \left(\sum_{k=1}^l ( f_k, v_k)_{U_k} - ({\mathcal P}_k w_k, v_k)_{U_k} \right)+ \left(\sum_{k=1}^{l-1} (g_k, \gamma_k^+ v_k)_{I_k} - ([{\mathcal B}_k w]_{I_k} , \gamma_k^+ v_k)_{I_k}\right).$$ Note that since $v\in H^1_0(U)$, the functions $\gamma^+_k v_k$ and $\gamma^-_{k+1} v_{k+1}$ will be equal, which allowed us to combine terms in the equation above. makes sense even in the case where $\tilde u$ is not in $\mathcal{H}(U)$, so we use this equation to define a weak solution in $H^1_0(U)$. **Problem 2** (Weak form of elliptic PDE with interfaces). *Suppose $U$ is properly decomposed into subdomains $U_1,U_2,\ldots,U_l$, where all interfaces and the boundary are of class $C^{1,1}$. Let ${\mathcal P}_k$ be defined as in [\[second-order-operator-k\]](#second-order-operator-k){reference-type="ref" reference="second-order-operator-k"}, and let $\Phi_k$ be the bilinear maps associated with ${\mathcal P}_k$. Fix $f \in L^2(U)$, $g_k\in H^{1/2}(I_k)$ for $k=1,2,\ldots, l-1$, and $g_l \in H^{3/2}(\partial U)$. Take $w \in H^2(U)$ so that $\gamma_l^+w = g_l$. A function $u = \tilde u + w$ with $\tilde u \in H^1_0(U)$ is a weak solution to the elliptic PDE with interfaces if $$\label{weak-eqn} \sum_{k=1}^l \Phi_k(\tilde u_k , v_k) = \left(\sum_{k=1}^l ( f_k, v_k)_{U_k} - ({\mathcal P}_k w_k, v_k)_{U_k} \right)+ \left(\sum_{k=1}^{l-1} (g_k, \gamma_k^+ v_k)_{I_k} - ([{\mathcal B}_k w]_{I_k} , \gamma_k^+ v_k)_{I_k}\right), \quad \forall v \in H^1_0(U).$$* *Remark 2*. The formulation of [Problem 2](#weak-linear-interface-problem){reference-type="ref" reference="weak-linear-interface-problem"} agrees with the weak form of the linear Poisson Bolztmann equation in the case where $g_1,g_2,\ldots, g_{l-1}$ are set to zero (c.f. [@holst1994poisson]), which suggests that this is the appropriate formulation of the weak problem for our application. Although in practice there will be no forcing terms on the interfaces, allowing the possibility of non-zero $g_k$'s is of theoretical importance when we later apply the implicit function theorem. Showing that [Problem 2](#weak-linear-interface-problem){reference-type="ref" reference="weak-linear-interface-problem"} has unique solutions follows from applying the Lax-Milgram theorem in a similar way to how it is applied in standard linear elliptic theory. **Proposition 1**. *Suppose $c\in L^\infty(U)$ is non-negative. Then we have a unique solution to [Problem 2](#weak-linear-interface-problem){reference-type="ref" reference="weak-linear-interface-problem"}.* *Remark 3*. The regularity of the data can be loosened in [Proposition 1](#existence-uniqueness-linear-weak-solns){reference-type="ref" reference="existence-uniqueness-linear-weak-solns"}; for instance, we can let $f\in H^{-1}(U)$ and still have unique weak solutions. However, for our purposes we do not need these low regularity results. The uniqueness of weak solutions can now be used to show the existence and uniqueness of strong solutions. **Theorem 1**. *Suppose that $c\in L^\infty(U)$ is non-negative. Then there exists a unique solution $u\in \mathcal{H}(U)$ to [Problem 1](#strong-linear-interface-problem){reference-type="ref" reference="strong-linear-interface-problem"}. Moreover, [Problem 1](#strong-linear-interface-problem){reference-type="ref" reference="strong-linear-interface-problem"} defines an isomorphism between $\mathcal{H}(U)$ and $L^2(U) \times \prod_{k=1}^{l-1} H^{1/2} (I_k) \times H^{3/2}(I_l)$ by associating solutions $u$ with data $(f,g_1,g_2,\ldots, g_{l-1}, g_l)$.* *Proof.* From [Proposition 1](#existence-uniqueness-linear-weak-solns){reference-type="ref" reference="existence-uniqueness-linear-weak-solns"}, it follows that the weak solution $u \in H^1(U)$. To show that $u \in \mathcal{H}(U)$, we can appeal to [@mclean2000strongly Thm. 4.20]. Namely we have that $$\begin{aligned} u_k &\in H^1(U_k), & &\text{for } k=1,2,\ldots, l, \\ [{\mathcal B}_k u]_{I_k} &\in H^{1/2}(I_k), & &\text{for } k=1,2,\ldots,l-1, \text{ and } \\ \gamma_l^+ u_l &\in H^{3/2}(I_l), \end{aligned}$$ which implies that $u_k \in H^2(U_k)$ for each $k=1,2, \ldots, l$ and thus $u \in \mathcal{H}(U)$. Applying the Green's identities in [\[greens-1,greens-2\]](#greens-1,greens-2){reference-type="ref" reference="greens-1,greens-2"} demonstrates that $u$ is a solution of [Problem 1](#strong-linear-interface-problem){reference-type="ref" reference="strong-linear-interface-problem"}. Since solutions of [Problem 1](#strong-linear-interface-problem){reference-type="ref" reference="strong-linear-interface-problem"} are also solutions of [Problem 2](#weak-linear-interface-problem){reference-type="ref" reference="weak-linear-interface-problem"}, the strong solution $u$ is unique. The isomorphism result follows immediately from existence and uniqueness of the strong solution and the continuity of the solution map with respect to the boundary and forcing data. ◻ # The Nonlinear Poisson Bolztmann Equation on a Reference Domain {#npbe-reference-domain-section} ## Existence of Region of Analyticity We now return to our main focus of the paper: solutions for the NPBE. The NPBE given in [\[npbe\]](#npbe){reference-type="ref" reference="npbe"} is a nonlinear elliptic PDE. For our applications, we will have the main domain properly decomposed into three subdomains. We also allow for the possibility of random perturbations of the boundary and interfaces. Let $\Omega$ be the sample space. Each outcome $\omega \in \Omega$ designates a random domain $\mathcal D(\omega)$ on which the NPBE will evaluated. The domain $\mathcal D(\omega)$ is properly decomposed into three subdomains $\mathcal D_1(\omega)$, $\mathcal D_2(\omega)$, and $\mathcal D_3(\omega)$ with interfaces ${\mathcal I}_1(\omega)$ and ${\mathcal I}_2(\omega)$. The parameters $\epsilon$, $\overline{\kappa}^2$, $g$, and $f$ will also depend on $\omega$. From here one can define strong and weak solutions of the NPBE on the stochastic domain in a similar way to that in [@castrillon2016analytic]. In practice, one usually assumes that the value of each parameter is given as a function of the random vector $\mathbf Y(\omega) = (Y_1(\omega), Y_2(\omega),\ldots, Y_N(\omega))$ taking values on the compact set $\Gamma \subset \mathbb{R}^N$ and with known density $\rho :\Gamma \to \mathbb{R}_{\geq0}$. Typically $\Gamma = [-1,1]^N$ with $\rho$ a truncated normal distribution, although the distribution can be more general. Often the parameters will vary analytically with respect to the value of $\mathbf Y$ and are usually polynomials in $\mathbf Y$. Thus the NPBE can be stated as a problem with parameters $\mathbf{y}\in \Gamma \subset \mathbb{R}^N$. To parameterize the random domain, we assume that the random domain has a pullback onto some fixed open set for each $\omega$. In particular, take $U\subset \mathbb{R}^3$ to be a bounded, open set that is properly decomposed into three subdomains $U_1$, $U_2$, and $U_3$ with interfaces $I_1 = \partial U_1 \cap \partial U_2$ and $I_2 = \partial U_2 \cap \partial U_3$. The interfaces and boundary are taken to have $C^{1,1}$ regularity. We assume for each $\mathbf{y}\in \Gamma$ that there is $F(\cdot ; \mathbf{y}) \in C^2_{\mathrm{diff}}(\mathbb{R}^3, \mathbb{R}^3)$ such that $$F(U_k;\mathbf{y}) = \mathcal D_k(\mathbf{y}) \quad \text{for } k = 1,2,3$$ and $$F(I_k;\mathbf{y}) = {\mathcal I}_k(\mathbf{y}) \quad \text{for } k = 1,2.$$ The Jacobian matrix of $F(\cdot; \mathbf{y})$ will be denoted by $J(\cdot;\mathbf{y})$. Note that since $F$ is a $C^2$ diffeomorphism, the regularity of the interfaces and the boundary are preserved under the mapping. To distinguish between the coordinates in each domain, we will denote by $\boldsymbol{r}$ elements of $U$ and by $\mathbf{x}$ elements of $\mathcal D(\mathbf{y})$. Similarly $\nabla_{\boldsymbol{r}}$ and $\nabla_{\mathbf{x}}$ will be used to distinguish between the derivatives in $U$ and $\mathcal D(\mathbf{y})$, respectively. Hence, we can define the strong form of the NPBE on the random domain. Again, we will use subscripts to denote restrictions to the subdomains (e.g. $u_k = u |_{\mathcal D_k}$). The trace operators $\gamma_k^{\pm}$ are defined similar to those in [2.2](#subsection-linear-elliptic-problems){reference-type="ref" reference="subsection-linear-elliptic-problems"}. Also, we have the conormal derivative ${\mathcal B}_k$ using the elliptic operator defined by $a_{ij} (\mathbf{x};\mathbf{y}) = \epsilon(\mathbf{x};\mathbf{y}) \delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta. **Problem 3**. *The function $\Gamma \ni \mathbf{y}\mapsto u(\cdot; \mathbf{y}) \in \mathcal{H}(\mathcal D(\mathbf{y});\mathcal D_1(\mathbf{y}), \mathcal D_2(\mathbf{y}), \mathcal D_3(\mathbf{y}))$ is a strong solution for the NPBE on the random domain if for each $\mathbf{y}\in \Gamma$ we have $u(\cdot;\mathbf{y})$ satisfies $$\begin{aligned} -\nabla_\mathbf{x}\cdot (\epsilon_k(\mathbf{x}; \mathbf{y}) \nabla_\mathbf{x}u_k(\mathbf{x}; \mathbf{y})) + \overline{\kappa}^2_k(\mathbf{x};\mathbf{y}) \sinh(u_k(\mathbf{x}; \mathbf{y})) &= f_k(\mathbf{x};\mathbf{y}), & \text{for } k &= 1,2,3 \\ [{\mathcal B}_k u(\cdot;\mathbf{y})]_{{\mathcal I}_k(\mathbf{y})} &= 0 & \text{for } k&=1,2 \\ \gamma_3^+(u_3(\cdot;\mathbf{y})) &= g(\cdot;\mathbf{y}). \end{aligned}$$* From the strong form, we can derive a weak version of the NPBE by integrating against a test function and using integration by parts. Here we define $w(\cdot;\mathbf{y}) \in H^2(\mathcal D(\mathbf{y}))$ to be the inverse trace of $g(\cdot;\mathbf{y})$ so that $\gamma_3^+(w) = g$. **Problem 4**. *The function $\Gamma \ni \mathbf{y}\mapsto u(\cdot ;\mathbf{y}) = \tilde u(\cdot;\mathbf{y}) + w(\cdot;\mathbf{y})$ is a weak solution for the NPBE on the random domain if for each $\mathbf{y}\in \Gamma$ we have $\tilde u (\cdot;\mathbf{y}) \in H^1_0(\mathcal D(\mathbf{y}))$ and $$\begin{gathered} \int_{\mathcal D(\mathbf{y})} \epsilon(\mathbf{x};\mathbf{y}) \nabla_\mathbf{x}\tilde u(\mathbf{x};\mathbf{y}) \cdot \nabla_\mathbf{x}v(\mathbf{x}) + \overline{\kappa}^2(\mathbf{x};\mathbf{y}) \sinh(\tilde u(\mathbf{x};\mathbf{y}) + w(\mathbf{x};\mathbf{y})) \, d\mathbf{x}\\ = \int_U f(\mathbf{x};\mathbf{y}) v(\mathbf{x}) \, d\mathbf{x}- \int_{\mathcal D(\mathbf{y})} \epsilon(\mathbf{x};\mathbf{y}) \nabla_\mathbf{x}w(\mathbf{x};\mathbf{y}) \cdot \nabla_\mathbf{x}v(\mathbf{x})\, d\mathbf{x}, \quad \forall v \in H^1_0(\mathcal D(\mathbf{y})). \end{gathered}$$* The weak form of the NPBE given above agrees with the standard definition of the weak form for this equation (c.f. [@holst1994poisson §2.1.5]). To pull back onto the reference domain, we construct an equivalent weak form of the NPBE for the pullback of $u$ onto $U$ given by $u^*(\cdot;\mathbf{y}) = u(F(\cdot;\mathbf{y});\mathbf{y})$. Following results given in [@castrillon2016analytic; @castrillon2021hybrid; @castrillon2021stochastic], we can write the weak form of the pullback onto the reference domain. **Problem 5**. *The function $\Gamma \ni \mathbf{y}\mapsto u^*(\cdot ;\mathbf{y}) = \tilde u^*(\cdot;\mathbf{y}) + w^*(\cdot;\mathbf{y})$ is a weak solution for the NPBE on the reference domain if for each $\mathbf{y}\in \Gamma$ we have $\tilde u^* (\cdot;\mathbf{y}) \in H^1_0(U)$ and $$\begin{gathered} \label{weak-eqn-ref-dom} \int_{U} \epsilon^*(\boldsymbol{r};\mathbf{y})\Big(J^{-1}(\boldsymbol{r};\mathbf{y}) J^{-\mathrm{T}}(\boldsymbol{r};\mathbf{y}) \det J(\boldsymbol{r};\mathbf{y}) \nabla_{\boldsymbol{r}} \tilde u^*(\boldsymbol{r};\mathbf{y}) \Big)\cdot \nabla_{\boldsymbol{r}} v(\boldsymbol{r}) \\ + (\overline{\kappa}^2)^*(\boldsymbol{r};\mathbf{y}) \sinh(\tilde u^*(\boldsymbol{r};\mathbf{y}) + w^*(\boldsymbol{r};\mathbf{y})) v(\boldsymbol{r}) \det J(\boldsymbol{r};\mathbf{y})\, d\boldsymbol{r}\\ = \int_{U} f^* (\boldsymbol{r};\mathbf{y}) v(\boldsymbol{r})\det J(\boldsymbol{r};\mathbf{y})\, d\boldsymbol{r}\\ - \int_U \epsilon^*(\boldsymbol{r};\mathbf{y}) \Big(J^{-1}(\boldsymbol{r};\mathbf{y}) J^{-\mathrm{T}}(\boldsymbol{r};\mathbf{y}) \det J(\boldsymbol{r};\mathbf{y}) \nabla_{\boldsymbol{r}} w^*(\boldsymbol{r};\mathbf{y}) \Big)\cdot\nabla_{\boldsymbol{r}} v(\boldsymbol{r})\, d\boldsymbol{r}, \quad \forall v\in H^1_0(U). \end{gathered}$$* The strong form of the NPBE on the reference domain is defined in an analogous way to [Problem 1](#strong-linear-interface-problem){reference-type="ref" reference="strong-linear-interface-problem"}. This also corresponds to the weak problem given in [Problem 5](#weak-npbe-ref-domain){reference-type="ref" reference="weak-npbe-ref-domain"} in that assuming sufficient regularity of the weak solution and integrating by parts gives the strong formulation. **Problem 6**. *The function $\Gamma \ni \mathbf{y}\mapsto u^*(\cdot; \mathbf{y}) \in \mathcal{H}(U; U_1, U_2, U_3)$ is a strong solution for the NPBE on the random domain if for each $\mathbf{y}\in \Gamma$ we have $u^*(\cdot;\mathbf{y})$ satisfies $$\begin{aligned}[] [{\mathcal L}_k(\mathbf{y}))] \big(u^*_k(\cdot;\mathbf{y})\big) + (\overline \kappa^2)^*_k(\cdot;\mathbf{y}) \sinh(u^*_k(\cdot;\mathbf{y})) \det J_k(\cdot;\mathbf{y}) &= f_k^*(\cdot;\mathbf{y}), & &\text{for } k=1,2,3, \\ [{\mathcal B}_k u^* ]_{I_k} &= 0, & &\text{for } k =1,2, \\ \gamma_3^+ u_3^* &= g^*, \end{aligned}$$ where $$\label{linear-operator} [{\mathcal L}_k(\mathbf{y})] \big(v\big) := -\nabla_{\boldsymbol{r}} \cdot \left( \epsilon_k^*(\cdot;\mathbf{y}) J_k^{-1}(\cdot;\mathbf{y})J_k^{-\mathrm T}(\cdot;\mathbf{y}) \det J_k(\cdot;\mathbf{y}) \nabla_{\boldsymbol{r}} v \right), \quad \text{for } v \in H^2(U_k).$$* To summarize, we have four problems that we can consider for the NPBE depending on whether we want the weak or strong solution and whether the domain is random or fixed. We can move from the the random domain problems to the reference domain problems by using the pullback $F^*$. Transitioning between weak and strong forms is done by integrating-by-parts or having increased regularity of the solution. illustrates the relationship between these problems. We will be working with the reference domain moving forward: first showing that a weak solution exists, and then applying that result for the strong solution. We can recapture results on the random domain by composing solutions on the reference domain with the diffeomorphism $F$. To get solutions for the NPBE, we must make some assumptions on the parameters. The assumptions that $\epsilon^*$ and $(\overline{\kappa}^2)^*$ are positive and non-negative, respectively, come from the physics of the simulation and are generally satisfied. The function $f^*$ is used to model the point charges, and ideally would be made to be a sum of Dirac deltas. However, there are few results for the nonlinear version of the Poisson-Boltzmann equation with forcing functions in $H^{-2}(U)$. Thus in practice, one approximates the point charges with $L^2$ functions (e.g. Gaussian functions centered at the location of the charge), and so we take $f^*\in L^2$. The Dirichlet boundary condition is typically taken to be the long-distance approximation of the potential from the point charges and is smooth on $\partial U=I_3$, but simply requiring $g^*\in H^{3/2}(I_3)$ gives sufficient regularity. Finally, we assume all the parameters vary analytically with respect to $\mathbf{y}\in \Gamma$, which is reasonable to assume when computing numerical solutions. Thus we make the following assumptions: **Assumption 1**. *For $\mathbf{y}\in \Gamma$, we have that* - *$\epsilon^*_k(\cdot;\mathbf{y}) \in C^{0,1}(\overline{U_k})$ for $k=1,2,3$* - *$(\overline\kappa^2)^*(\cdot;\mathbf{y})\in L^\infty(U)$* - *$f^*(\cdot;\mathbf{y}) \in L^2(U)$* - *$g^*(\cdot;\mathbf{y})\in H^{3/2}(I_3)$* - *$F(\cdot;\mathbf{y}) \in C^2_{\mathrm{diff}}(\mathbb{R}^3,\mathbb{R}^3) \subset C^2(\mathbb{R}^3,\mathbb{R}^3)$* *and the maps from $\Gamma$ into the respective Banach spaces are analytic. Since the inverse trace operator is linear, we also have $\mathbf{y}\mapsto w^*(\cdot;\mathbf{y}) \in H^2(U)$ is analytic.* **Assumption 2**. *There exists $c_1>0$ such that for any $\mathbf{y}\in \Gamma$ $$\epsilon^*(\boldsymbol{r};\mathbf{y})\geq c_1, \quad \forall \boldsymbol{r}\in U.$$* **Assumption 3**. *For any $\mathbf{y}\in \Gamma$, we have that $$(\overline\kappa ^2)^*(\boldsymbol{r};\mathbf{y})\geq 0, \quad \text{for a.e.\ } \boldsymbol{r}\in U.$$* We will also need to assume that $\det J(\boldsymbol{r};\mathbf{y})$ is bounded away from $0$ for $\mathbf{y}\in \Gamma$ and $\boldsymbol{r}\in U$. This is essentially assuming that the map $F$ is non-singular and preserves the orientation of the domain, both of which are reasonable when considering small perturbations of the interface. **Assumption 4**. *There exists $c_2>0$ such that for any $\mathbf{y}\in \Gamma$ we have $$\det J(\boldsymbol{r};\mathbf{y}) \geq c_2, \quad \forall \boldsymbol{r}\in U.$$* Hence, we can prove the existence and uniqueness of weak solutions. **Proposition 2**. *If [\[asum1,asum2,asum3,asum4\]](#asum1,asum2,asum3,asum4){reference-type="ref" reference="asum1,asum2,asum3,asum4"} hold, then [Problem 5](#weak-npbe-ref-domain){reference-type="ref" reference="weak-npbe-ref-domain"} has a unique solution $\mathbf{y}\mapsto u^*(\cdot, \mathbf{y})$. Furthermore, $\sinh(u^*(\cdot;\mathbf{y}))$ is in $L^2(U)$ for each $\mathbf{y}\in \Gamma$.* *Proof.* By [Assumption 4](#asum4){reference-type="ref" reference="asum4"}, we have that $\det J \geq c_2 > 0$ and so the matrix $$J^{-1}(\boldsymbol{r};\mathbf{y}) J^{-\mathrm{T}}(\boldsymbol{r};\mathbf{y}) \det J(\boldsymbol{r};\mathbf{y})$$ is symmetric positive definite for each $\boldsymbol{r}\in U$ and $\mathbf{y}\in [-1,1]^N$. The proof of weak solutions to the NPBE given in [@holst1994poisson Thm. 2.14] can easily be adjusted to the case where the matrix $\overline{\mathbf a}:=J^{-1}(\boldsymbol{r};\mathbf{y}) J^{-\mathrm{T}}(\boldsymbol{r};\mathbf{y}) \det J(\boldsymbol{r};\mathbf{y})$ is symmetric positive definite. That argument gives unique solutions to [Problem 5](#weak-npbe-ref-domain){reference-type="ref" reference="weak-npbe-ref-domain"}. The fact that $\sinh(u^*(\cdot;\mathbf{y}))$ is an $L^2$ function also follows from the proof in [@holst1994poisson]. ◻ *Remark 4*. The same assumptions also imply that there is a unique solution for [Problem 4](#weak-npbe-rand-domain){reference-type="ref" reference="weak-npbe-rand-domain"} which also satisfies $\sinh(u(\cdot;\mathbf{y})) \in L^2(\mathcal D(\mathbf{y}))$ for each $\mathbf{y}\in \Gamma$. *Remark 5*. The next result follows from the analytic version of the Implicit Function Theorem on Banach spaces. The statement of this theorem is analogous to the finite-dimensional version, and an exact statement of it can be found in [@whittlesey1965analytic]. One key change from the finite-dimensional version to the infinite-dimensional version is that we now use Fréchet derivatives and require that the derivative is an isomorphism between Banach spaces. **Theorem 2**. *Suppose [\[asum1,asum2,asum3,asum4\]](#asum1,asum2,asum3,asum4){reference-type="ref" reference="asum1,asum2,asum3,asum4"} hold. Then there exists a unique solution to [Problem 6](#strong-npbe-ref-domain){reference-type="ref" reference="strong-npbe-ref-domain"}. Furthermore, there exists a complex neighborhood of $\Gamma$ given by $\mathcal N \subset \mathbb{C}^N$ such that there is a function $$\mathbf{y}\mapsto u^*(\cdot, \mathbf{y}) \in \mathcal{H}(U), \quad \text{for } \mathbf{y}\in \mathcal N$$ where* 1. *the map is analytic from $\mathcal N$ into $\mathcal{H}(U)$, and* 2. *the map agrees with the strong solution on $\Gamma$.* *Proof.* From [Proposition 2](#unique-weak-solns){reference-type="ref" reference="unique-weak-solns"}, there are unique weak solutions $u^*(\cdot,\mathbf{y}) \in H^1(U)$ for each $\mathbf{y}\in [-1,1]^N$ and that $\sinh(u^*(\cdot;\mathbf{y}))\in L^2(U)$. Setting $$\tilde f(\boldsymbol{r};\mathbf{y}) = f^*(\boldsymbol{r};\mathbf{y}) \det J(\boldsymbol{r};\mathbf{y})- (\overline{\kappa}^2)^*(\boldsymbol{r};\mathbf{y}) \sinh(\tilde u^*(\boldsymbol{r};\mathbf{y}) + w^*(\boldsymbol{r};\mathbf{y})) \det J(\boldsymbol{r};\mathbf{y}),$$ we get that $\tilde f(\cdot;\mathbf{y}) \in L^2(U)$ and [\[weak-eqn-ref-dom\]](#weak-eqn-ref-dom){reference-type="ref" reference="weak-eqn-ref-dom"} can be written as $$\begin{gathered} \int_{U} \epsilon^*(\boldsymbol{r};\mathbf{y})\Big(J^{-1}(\boldsymbol{r};\mathbf{y}) J^{-\mathrm{T}}(\boldsymbol{r};\mathbf{y}) \det J(\boldsymbol{r};\mathbf{y}) \nabla_{\boldsymbol{r}} \tilde u^*(\boldsymbol{r};\mathbf{y}) \Big)\cdot \nabla_{\boldsymbol{r}} v(\boldsymbol{r}) \\ = \int_{U} \tilde f(\boldsymbol{r};\mathbf{y}) v(\boldsymbol{r}), d\boldsymbol{r}- \int_U \epsilon^*(\boldsymbol{r};\mathbf{y}) \Big(J^{-1}(\boldsymbol{r};\mathbf{y}) J^{-\mathrm{T}}(\boldsymbol{r};\mathbf{y}) \det J(\boldsymbol{r};\mathbf{y}) \nabla_{\boldsymbol{r}} w^*(\boldsymbol{r};\mathbf{y}) \Big)\cdot\nabla_{\boldsymbol{r}} v(\boldsymbol{r})\, d\boldsymbol{r}, \quad \forall v\in H^1_0(U), \end{gathered}$$ which is in the same form as [\[weak-eqn\]](#weak-eqn){reference-type="ref" reference="weak-eqn"}. Then applying [Theorem 1](#linear-regularity-weak-soln){reference-type="ref" reference="linear-regularity-weak-soln"} gives that $u^*(\cdot;\mathbf{y}) \in \mathcal{H}(U)$. To show analyticity, we apply the analytic version of the implicit function theorem. We define a mapping $$\label{mcF-function} \mathcal{F}: \Gamma \times \mathcal{H}(U;U_1,U_2) \to L^2(U) \times H^{1/2}(I_1) \times H^{1/2}(I_2) \times H^{3/2}(I_3) =: Z$$ such that $\mathcal{F}(\mathbf{y}, u^*) = (0,0,0,0)$ if and only if $u^*$ is a strong solution on the reference domain for that fixed $\mathbf{y}$. The first component of $\mathcal{F}$ is defined on each $U_k$ by $$(u^*_k) + (\overline \kappa^2)^*_k(\cdot;\mathbf{y}) \sinh(u^*_k) \det J_k(\cdot;\mathbf{y}) - f_k^*(\cdot;\mathbf{y}) \det J_k(\cdot;\mathbf{y})$$ for $k=1,2,3$. This defines $L^2$ function on each $U_k$ (since $\sinh(u^*_k) \in H^2(U_k)$) and thus can be used to define an $L^2$ function on all of $U$. The second and third component of $\mathcal{F}(\mathbf{y}, u^*)$ is defined to be $[{\mathcal B}_k u^* ]_{I_k}$ for $k=1$ and $k=2$, respectively. The final component of $\mathcal{F}(\mathbf{y}, u^*)$ is given by $\gamma_3^+ u_3^* - g^*$. The second and third components define linear maps, and the first and fourth component are easily verified to be analytic. Thus $\mathcal{F}$ is an analytic map between Banach spaces. Fix $\mathbf{y}_0\in \Gamma$. Then since $u^*(\cdot;\mathbf{y})$ is a strong solution we have $$\label{implicit-equation} \mathcal{F}(\mathbf{y}_0, u^*(\cdot;\mathbf{y}_0)) = (0, 0, 0, 0).$$ We want to apply the implicit function theorem to [\[implicit-equation\]](#implicit-equation){reference-type="ref" reference="implicit-equation"} to get $u^*$ as an analytic function of $\mathbf{y}$ in a neighborhood around $\mathbf{y}_0$. To do this, we must check that the derivative of $\mathcal{F}$ with respect to $u^*$ at $(\mathbf{y}_0, u^*(\cdot;\mathbf{y}_0))$ is an isomorphism between $\mathcal{H}(U)$ and $Z$. ([To apply the Implicit Function Theorem to [\[implicit-equation\]](#implicit-equation){reference-type="ref" reference="implicit-equation"} to get $u^*$ as an analytic function of $\mathbf{y}$ in a neighborhood around $\mathbf{y}_0$, it must checked that the derivative of $\mathcal{F}$ with respect to $u^*$ at $(\mathbf{y}_0, u^*(\cdot;\mathbf{y}_0))$ is an isomorphism between $\mathcal{H}(U)$ and $Z$.]{style="color: black"}) One can compute that $$(v) = ([{\mathcal L}(\mathbf{y}_0)](v) + (\overline \kappa^2)^*(\cdot;\mathbf{y}_0) \cosh(u^*(\cdot;\mathbf{y}_0)) \det J(\cdot;\mathbf{y}_0) v, [{\mathcal B}_1 v ]_{I_1} , [{\mathcal B}_2 v ]_{I_2}, \gamma_3^+ v_3),$$ where ${\mathcal L}$ is defined locally in [\[linear-operator\]](#linear-operator){reference-type="ref" reference="linear-operator"}. This linear operator is of the same form of [Problem 1](#strong-linear-interface-problem){reference-type="ref" reference="strong-linear-interface-problem"}, and so [Proposition 1](#existence-uniqueness-linear-weak-solns){reference-type="ref" reference="existence-uniqueness-linear-weak-solns"} implies that $D_{u^*} \mathcal{F}(\mathbf{y}_0, u^*(\cdot;\mathbf{y}_0))$ is in fact an isomorphism. Therefore the map $\mathbf{y}\mapsto u^*(\cdot;\mathbf{y})$ is analytic in a neighborhood of $\mathbf{y}_0\in\mathbb{R}^N$. This map can be extended to a complex neighborhood of $\mathbf{y}_0\in \mathbb{C}^N$. Applying this argument to every point in $\Gamma$ gives the complex neighborhood $\mathcal N$ for which $\mathcal N \ni \mathbf{y}\mapsto u^*(\cdot;\mathbf{y})$ is analytic. ◻ ## Estimates on the Region of Analyticity shows that there exists a region of analyticity for the strong solution of the NPBE. This will be used to prove convergence results relating to the quantity of interest by using sparse grids [@nobile2008a]. [In this section, a quantitative bound on the size of the region of analyticity is derived, which is applied to obtain the aforementioned convergence rates; see and the discussion that follows.]{style="color: black"} However, the rate of convergence we depend on the size of the region of analyticity. The typical application of the implicit function theorem does not give *a priori* bounds on the size of the region of analyticity. For the finite-dimensional case, an application of Rouché's theorem can give estimates of this region. The results in [@chang2003analytic] give simple bounds on the radius of the region of analyticity. We will show a similar result holds for Banach spaces. **Theorem 3**. *Let $\mathcal{F}: \mathbb{C}^n \times X \to Y$ be an analytic function with $X$ and $Y$ Banach spaces. Suppose that $\mathcal{F}(0,0) = 0$ and $D_x \mathcal{F}(0,0): X \to Y$ is an isomorphism. Let $\|D_x\mathcal{F}(0,0)^{-1} \|_{{\mathcal L}(Y,X)} \leq a$ and suppose that $\|\mathcal{F}(z,x)\|_Y \leq M$ on B where $B = \{(z,x): |z|, \|x\|_X \leq R\}$. Also, suppose that $D_x \mathcal{F}(z,x)^{-1}$ exists and is a bounded operator for each $(z,x)\in B$. Then the analytic function $z \mapsto x(z)$ is defined in a region containing the ball $$|z| <\Theta(M,a,R;\mathcal{F}) := \frac{\Big(aMR - \sqrt{aMR^2(aM+R)}\Big)\Big(aMR + R^2 - \sqrt{aMR^2(aM+R)}\Big)}{2a^2M^2R - R\sqrt{aMR^2(aM+R)} + aM \Big(2R^2 - 3 \sqrt{aMR^2(aM+R)}\Big)}$$ where $$\| x(z) \|_X < \Xi (M, a, R;\mathcal{F}) := \frac{aMR+ R^2 - \sqrt{a^2M^2R^2 + aMR^3}}{aM + R} \quad \text{ for } |z|<\Theta.$$* *Proof.* The proof follows in the spirit of arguments from degree theory. We find $r > 0$ such that $\mathcal F(z,x) \neq 0$ for $z\neq 0$ sufficiently small and $\|x\|_X = r$. This -- along with the assumption on the inverse of $D_x\mathcal{F}$ -- guarantees the zero from the implicit function theorem does not leave the ball, bifurcate, or vanish. Obviously, $z \mapsto x(z)$ is continuous and so if $\|x(z_1)\|_X > r$ for some $z_1$ then there must be some point $z_0$ where $\|x(z_0)\|_X = r$ and $\mathcal{F}(z_0, x(z_0)) = 0$, which contradicts our assumed bound. By repeatedly applying the implicit function theorem, one can show that we cannot have the function lose analyticity if the function $x(z)$ does not leave the ball of radius $r$. Thus we can increase $|z|$ up to the point where the zero can leave the ball of radius $r$ and this becomes our estimate for the region of analyticity. We first find an $r > 0$ such that $\|\mathcal{F}(0, x)\|_{Y} > 0$ for all $0< \|x\|_X \leq r$. Because $\mathcal{F}$ is analytic, we can write $\mathcal{F}(0,x)$ as a power series centered at $(0,0)$: $$\label{eq:power-series} \mathcal{F}(0, x) = D_x \mathcal{F}(0,0) x + \sum_{k=2}^\infty a_k(\underbrace{x, \ldots, x}_{k \text{ times}})$$ where $a_k$ are $k$-linear maps. Using a Cauchy estimate we have that $$\|a_k(x,\ldots, x)\|_Y = M \left(\frac{\|x\|_X}{R}\right)^k.$$ Then rearranging [\[eq:power-series\]](#eq:power-series){reference-type="ref" reference="eq:power-series"}, applying $D_x\mathcal{F}(0,0)^{-1}$ to both sides, and taking norms gives $$\begin{aligned} \| x\|_X &= \| D_x\mathcal{F}(0,0)^{-1} [\mathcal{F}(0,x) - \sum_{k=2}^\infty a_k(x,\cdots, x)] \|_X \\ &\leq a \left(\|\mathcal{F}(0,x) \|_Y + \sum_{k=2}^\infty M \left(\frac{\|x\|_X}{R} \right)^k \right) \\ &= a \left(\|\mathcal{F}(0,x) \|_Y + \frac{M\|x\|_X^2}{R^2 - \|x\|_X R}\right). \end{aligned}$$ Thus we have that $$\|F(0, x) \|_Y \geq \frac{\|x\|_X} a - \frac{M\|x\|_X^2}{R^2 - \|x\|_X R}.$$ To guarantee the right-hand side is strictly greater than zero we need that $$0 < \|x\|_{X} < \frac{R^2}{R+ aM}.$$ Thus we can choose any $r$ such that $$0 < r < \frac{R^2}{R+ aM}.$$ Now we want to find $\theta> 0$ such that if $|z| < \theta$ then $\mathcal{F}(z, x) \neq 0$ when $\|x\|_X = r$. It will be sufficient to find a $\theta$ where for any $|z| < \theta$ $$\label{rouche-ineq} \| \mathcal{F}(0,x) - \mathcal{F}(z, x)\|_{Y} < \|\mathcal{F}(0,x)\|_Y, \quad \text{ for } x \text{ where } \|x \| = r.$$ By using a power series expansion around $(0,x)$ with respect to $z$ and the Cauchy estimate we get that $$\| \mathcal{F}(0,x) - \mathcal{F}(z, x)\|_{Y} \leq \frac{M|z|}{R - |z|}.$$ Then [\[rouche-ineq\]](#rouche-ineq){reference-type="ref" reference="rouche-ineq"} holds if $$\frac{M|z|}{R - |z|} < \frac r a - \frac{Mr^2}{R^2 - rR},$$ which holds if $$|z| < \frac{r R^3 - r^2R^2 - Mr^2 a R}{aMR^2+rR^2 - aMR - r^2R - Mr^2 a}.$$ Setting $\theta$ equal to the right-hand side gives the desired result. Furthermore, the value on the right-hand side is maximized for fixed values of $a$, $M$, and $R$ when $$r = \Xi (M, a, R;\mathcal{F}) := \frac{aMR+ R^2 - \sqrt{a^2M^2R^2 + aMR^3}}{aM + R}.$$ Thus the optimal radius can be given by plugging in this value for $r$, from which we get $$\Theta(M,a,R;\mathcal{F}) := \frac{\Big(aMR - \sqrt{aMR^2(aM+R)}\Big)\Big(aMR + R^2 - \sqrt{aMR^2(aM+R)}\Big)}{2a^2M^2R - R\sqrt{aMR^2(aM+R)} + aM \Big(2R^2 - 3 \sqrt{aMR^2(aM+R)}\Big)}.$$ ◻ For the purposes of showing sub-exponential convergence of the sparse grid, we want to find the largest polyellipse in the region of analyticity. For one dimension, a Bernstein ellipse ${\mathcal E}_{\sigma}$ is given by $${\mathcal E}_\sigma = \left\{z\in \mathbb{C}: \mathop{\text{\rm Re}}{z} = \frac{e^{\hat\sigma} + e^{-{\hat\sigma}}}{2} \cos(\theta) , \mathop{\text{\rm Im}}{z} = \frac{e^{\hat\sigma} - e^{-{\hat\sigma}}}{2} \sin(\theta), \theta\in[0,2\pi), 0\leq \hat{\sigma} \leq \sigma \right\}.$$ For multiple dimensions, we define a polyellipse to be a direct product of Bernstein ellipses: $${\mathcal E}_{\sigma_1 ,\sigma_2, \ldots, \sigma_n} := \prod_{k=1}^n {\mathcal E}_{\sigma_k} \subset \mathbb{C}^n.$$ Applying [Theorem 3](#theorem-analytic-ift-region){reference-type="ref" reference="theorem-analytic-ift-region"} directly using uniform estimates for each point in $\Gamma$ gives the analytic domain $$\mathcal{G}_\Theta := \bigcup_{\mathbf{y}\in\Gamma} B_\Theta(\mathbf{y}),$$ where $B_\Theta(\mathbf{y})$ is a ball of radius $\Theta = \Theta(M, a, R; \mathcal{F})$ centered at $\mathbf{y}$. Thus we want to fit the largest Bernstein ellipse into $\mathcal G_\Theta$, as shown in [\[errorestimates:figure1\]](#errorestimates:figure1){reference-type="ref" reference="errorestimates:figure1"}. The following is a simple result following from [Theorem 3](#theorem-analytic-ift-region){reference-type="ref" reference="theorem-analytic-ift-region"}. **Corollary 1**. *Take $\mathcal{F}$ to be the function designated in [\[mcF-function\]](#mcF-function){reference-type="ref" reference="mcF-function"}. Set the positive constants $R$, $M$, and $a$ such that the following hold:* 1. *$R> 0$ is small enough so that $[D_{u^*} \mathcal{F}(\mathbf{y}, u^*)]^{-1}$ exists whenever $\operatorname{dist}(\mathbf{y}, \Gamma) \leq R$ and $\|\mathop{\text{\rm Im}}u^* \|_{\mathcal{H}(U)} \leq R$.* 2. *$M>0$ is large enough so that $\| \mathcal{F}(\mathbf{y}^0 + \mathbf{y}, u^*(\cdot,\mathbf{y}^0) + u^*) \|_Z \leq M$ whenever $\mathbf{y}^0 \in \Gamma$ and $|\mathbf{y}|, \|u^*\|_{\mathcal{H}(U)} \leq R$.* 3. *$a>0$ is large enough so that $\|[D_{u^*} \mathcal{F}(\mathbf{y}^0, u^*(\cdot;\mathbf{y}^0))]^{-1}\|_{{\mathcal L}(Z, \mathcal{H}(U))} \leq a$ for all $\mathbf{y}^0\in \Gamma$.* *Then defining $$\label{sigma-star} \sigma_* := \log \left(\sqrt{\Theta^2 + 1} + \Theta \right)$$ where $\Theta = \Theta(M,a,R;\mathcal{F})$, we have that the polyellipse ${\mathcal E}_{\sigma_1, \sigma_2,\ldots,\sigma_N}$ is inside the region of analyticity for the solution $\mathbf{y}\mapsto u^*(\cdot;\mathbf{y})$ if $\sigma_1 = \sigma_2 =\cdots = \sigma_N = \sigma_*$.* *Proof.* By applying [Theorem 3](#theorem-analytic-ift-region){reference-type="ref" reference="theorem-analytic-ift-region"} to each point $\mathbf{y}^0 \in \Gamma$, we get that there is a region of analyticity for the solution $\mathbf{y}\mapsto u^*(\cdot;\mathbf{y})$, where a ball of radius $\Theta$ centered at any $\mathbf{y}^0\in \Gamma$ is contained in the region. The largest polyellipse ${\mathcal E}_{\sigma_1, \sigma_2,\ldots, \sigma_N}$ with $\sigma_1 = \sigma_2 = \cdots = \sigma_N$ in the region of analyticity can be computed. From [@castrillon2016analytic; @castrillon2021stochastic], we know the largest polyellipse occurs when $\sigma^*$ is defined as in [\[sigma-star\]](#sigma-star){reference-type="ref" reference="sigma-star"}. ◻ *Remark 6*. Conditions (ii) and (iii) of [Corollary 1](#cor:implicit){reference-type="ref" reference="cor:implicit"} are straightforwardly made to satisfy the conditions of [Theorem 3](#theorem-analytic-ift-region){reference-type="ref" reference="theorem-analytic-ift-region"}. The condition (i) follows from the specific form of the PDE in question. A sufficient condition for the inverse $[D_{u^*} \mathcal{F}(\mathbf{y}, u^*)]^{-1}$ to exist is $\epsilon^*(\cdot;\mathbf{y}) > 0$, $\det J(\cdot;\mathbf{y}) > 0$, $\overline{\kappa}^2(\cdot;\mathbf{y}) \geq 0$, and $\mathop{\text{\rm Re}}\cosh(u^*)\geq 0$. The first three inequalities can be satisified by choosing $\mathbf{y}$ to be sufficiently close to $\Gamma$. The term $\mathop{\text{\rm Re}}\cosh(u^*)$ will be strictly positive when $u^*$ is real-valued, and only becomes negative when $\mathop{\text{\rm Im}}u^*$ is sufficiently large. *Remark 7*. The estimate for the size of the polyellipse takes each $\sigma_k$ for $k=1,2,\ldots, N$ to be equal to $\sigma^*$. This will only give the optimal estimate of the decay rate when using an isotropic sparse grid. For anisotropic sparse grids, we would need to choose different values for each $\sigma_k$. show how we can obtain *a priori* bounds on the region of analyticity after applying the implicit function theorem. To apply these bounds, one needs to find the values for the constants $a$ and $M$ (for a fixed $R$). For our problem, there must be some choice of constants that work, but computing them is tricky. The constant $a$ can be difficult to estimate because it involves getting bounds on the solution of a *backward* problem. That is, we ultimately want to find a bound on the norm of $[D_x \mathcal{F}(0, 0)]^{-1}$, which typically means solving a linear PDE. For simple linear operators and domains -- for example, a Helmholtz operator $-\Delta + k^2$ on the sphere-- this norm is possible to calculate explicitly. However, our domain has interfaces, which makes estimation difficult. For the moment, we set aside the problem of bounding the constant $a$ and leave the task of optimizing the bounds for that value to future work. The constant $M$ can be more easily estimated since we are now solving a *forward* problem. That is, given some inputs to our (known) function $\mathcal{F}$ we want to determine the size of the outputs. The remainder of this section is devoted to showing how the estimate for $M$ can be obtained. To get the explicit bounds needed to apply [Corollary 1](#cor:implicit){reference-type="ref" reference="cor:implicit"}, we will need to make assumptions on the parameters in the NPBE. Suppose that $\Gamma = [-1,1]^N$. To simplify the arguments, we will assume that $\epsilon^*_k$ and $(\overline{\kappa}^2)^*$ are piecewise constant and that $g^* = 0$. Suppose that $F$ has the form $$F(\boldsymbol{r};\mathbf{y}) = \boldsymbol{r}+ \sum_{k=1}^N \sqrt{\mu_k} b_k(\boldsymbol{r}) y_k$$ where $b_k \in C^2$. Assume that the $b_k$'s are normalized so that $$\| b_k \|_{L^\infty(U)} = 1 \quad \text{ for } k =1,2,\ldots N,$$ and that the $\mu_k$'s are decreasing in value. Then the Jacobian, $J(\boldsymbol{r};\mathbf{y})$ has the form $$J(\boldsymbol{r};\mathbf{y}) = I + \sum_{k=1}^N \sqrt{\mu_k} B_k(\boldsymbol{r}) y_k$$ where $B_k(\boldsymbol{r}) = \partial b_k(\boldsymbol{r})$. Denote $$\label{mcB-defn} {\mathcal B}\mathbf{y}:= \sum_{k=1}^N \sqrt{\mu_k} B_k(\cdot) y_k$$ so that $J(\cdot;\mathbf{y}) = I + {\mathcal B}\mathbf{y}$. Note that we can treat ${\mathcal B}$ as a linear map from $\mathbb{C}^N$ into $C^1(U,\mathbb{C}^{3\times 3})$. We will also assume that $f^*$ has the form $$f^*(\boldsymbol{r};\mathbf{y}) = \sum_{k=1}^{N_f} \xi(F(\boldsymbol{r};\mathbf{y}) - F(\eta_k;\mathbf{y}))$$ where $\xi$ is a Gaussian function and each $\eta_k \in U$ is a fixed point. Suppose that $\mathcal{F}(\mathbf{y}^0,u^0) = 0$ for some $\mathbf{y}^0 \in \Gamma$ and $u^0 \in \mathcal{H}(U)$. To apply [Corollary 1](#cor:implicit){reference-type="ref" reference="cor:implicit"}, we want an estimate on $$\label{difference-estimate} \|\mathcal{F}(\mathbf{y}^0 + \mathbf{y}, u^0 + u) - \mathcal{F}(\mathbf{y}^0, u^0)\|_{Z}.$$ Note that the norm on the last three coordinates of $\mathcal{F}$ (see [\[mcF-function\]](#mcF-function){reference-type="ref" reference="mcF-function"}) can be estimated by finding bounds for the trace operators and co-normal derivatives. Let us focus on the first coordinate of [\[difference-estimate\]](#difference-estimate){reference-type="ref" reference="difference-estimate"}, which is given by $$\begin{aligned} &[{\mathcal L}_k(\mathbf{y}^0 + \mathbf{y})] (u^0_k + u_k) + (\overline \kappa^2)^*_k \sinh(u^0_k + u_k) \det J_k(\cdot;\mathbf{y}^0 + \mathbf{y}) - f_k^*(\cdot;\mathbf{y}^0 + \mathbf{y}) \det J_k(\cdot;\mathbf{y}^0 + \mathbf{y}) \\ &\quad - [{\mathcal L}_k(\mathbf{y}^0)] (u^0_k) + (\overline \kappa^2)^*_k \sinh(u^0_k) \det J_k(\cdot;\mathbf{y}^0) - f_k^*(\cdot;\mathbf{y}^0) \det J_k(\cdot;\mathbf{y}^0 ) \end{aligned}$$ for $k = 1,2,3$. Estimating the above term in the $L^2(U)$ requires us to define the norms of several other spaces in order for the calculation to be tractable. Norms in finite-dimensional vector spaces (e.g. $\mathbb{R}^n$) will be denoted with single bars, $|\cdot|$, while norms in infinite-dimensional function spaces will be denoted with double bars, $\| \cdot \|$. Similar notation will be used for the norms induced on linear operators on normed spaces. In particular, $|\cdot|_p$ for $1\leq p \leq \infty$ will be used to denote the typical $\ell^p$ norms in $\mathbb{R}^n$ or $\mathbb{C}^n$ as well as the associated matrix norms. So if $\mathbf{v}\in \mathbb{C}^n$ and $A\in \mathbb{C}^{n\times n}$, then $|\mathbf v|_2$ will be the standard Euclidean norm of $\mathbf{v}$ and $$|A|_2 = \sup_{\mathbf{x}\in \mathbb{C}^n\setminus\{0\}} \frac{|A\mathbf{x}|_2}{|\mathbf{x}|_2}.$$ We will assume $L^2(U;\mathbb{C}^n)$ to have the norm $$\| \mathbf{v}(\cdot) \|_{L^2(U;\mathbb{C}^n)} = \| \,|\mathbf{v}(\cdot)|_2 \,\|_{L^2(U;\mathbb{R})} = \left( \int_U |\mathbf{v}(\boldsymbol{r}) |_2^2\, d\boldsymbol{r}\right)^{1/2}.$$ We will also assume that $H^1(U;\mathbb{C}^n)$ has the norm $$\| \mathbf{v}(\cdot) \|_{H^1(U;\mathbb{C}^n)} = \|\mathbf{v}(\cdot)\|_{L^2(U;\mathbb{C}^n)} + \sum_{i=1}^n \left\| \frac{\partial \mathbf{v}}{\partial r_i} (\cdot) \right \|_{L^2(U;\mathbb{C}^n)}.$$ For the space $C^1(U;\mathbb{C}^{3\times 3})$, we introduce the norm $$\| B(\cdot) \|_{C^1(U;\mathbb{C}^{3\times 3})} := \max_{\substack{k=0,1 \\ i = 1,2,3}} \sup_{\boldsymbol{r}\in U} \left|\frac{\partial^k}{\partial r_i^k}B(\boldsymbol{r})\right|_2.$$ Note that for any $B\in C^1(U;\mathbb{C}^{3\times3})$ and $\mathbf{v}\in H^1(U;\mathbb{C}^3)$, we have $$\| B \mathbf{v}\|_{H^1(U;\mathbb{C}^3)} \leq 4 \| B \|_{C^1(U;\mathbb{C}^{3\times 3})} \| \mathbf{v}\|_{H^1(U;\mathbb{C}^3)},$$ which implies the continuous imbedding $C^1(U;\mathbb{C}^{3\times 3}) \hookrightarrow {\mathcal L}(H^1(U;\mathbb{C}^3))$. Recall that ${\mathcal B}$ as defined in [\[mcB-defn\]](#mcB-defn){reference-type="ref" reference="mcB-defn"} is a linear map from $\mathbb{C}^N$ into $C^1(U;\mathbb{C}^{3\times3})$, and so inherits a natural norm from being a bounded linear operator between two Banach spaces. We introduce a different norm for these maps that will be easier to estimate and be an upper bound for the linear operator norm. Let $$\| {\mathcal B}\|_p := \left(\sum_{k=1}^N |\mu_k|^{p/2} \| B_k \|_{C^1(U,\mathbb{C}^{3\times 3})}^p \right)^{1/p}$$ for $1\leq p < \infty$ and $$\| {\mathcal B}\|_\infty := \max_{k=1,2,\ldots,N} \sqrt{\mu_k} \| B_k \|_{C^1(U, \mathbb{C}^{3\times 3})}.$$ Then for $1\leq p , q \leq \infty$ with $\frac 1 p + \frac 1 q = 1$, we have that $$\label{holder} \| {\mathcal B}\mathbf{y}\|_{C^1(U,\mathbb{C}^{3\times 3})} \leq \|{\mathcal B}\|_{p} |\mathbf{y}|_q.$$ For the following estimates on various parameters, the hypothesis $$\label{small-b} \| {\mathcal B}\|_1 < \frac 1 4.$$ is assumed, which defines $J^{-1}(\mathbf{y}^0+\mathbf{y})$ in ${\mathcal L}(H^1(U;\mathbb{C}^3))$. **Proposition 3**. *Let ${\mathcal L}= {\mathcal L}(H^1(U;\mathbb{C}^3))$ denote the space of bounded linear operators from $H^1(U;\mathbb{C}^3)$ to itself. Suppose that $\mathbf{y}^0\in\Gamma$ and $$|\mathbf{y}|_\infty < \frac 1 {4\|{\mathcal B}\|_1} - |\mathbf{y}^0|_\infty.$$ Then we have the following bounds: $$\begin{aligned} \| J^{-1}(\mathbf{y}^0) \|_{\mathcal L}&\leq \frac{1}{1 - 4 \| {\mathcal B}\|_1 |\mathbf{y}^0|_\infty}, \label{Jinf-L} \\ \| J^{-1}(\mathbf{y}^0 + \mathbf{y}) \|_{\mathcal L}&\leq \frac{1}{1 - 4 \| {\mathcal B}\|_1 (|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)}, \label{Jinf-L2} \\ \| (I + J^{-1}(\mathbf{y}^0){\mathcal B}\mathbf{y})^{-1} - I\|_{{\mathcal L}} &\leq \frac{4 \| {\mathcal B}\|_1 |\mathbf{y}|_\infty}{1 - 4 \| {\mathcal B}\|_1 (|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)}, \label{I-J-B} \\ |\det J(\mathbf{y}^0)| &\leq \frac 1 {(1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty)^3},\label{absdet} \\ |\det J(\mathbf{y}^0 + \mathbf{y})| &\leq \frac 1 {(1 - \|{\mathcal B}\|_1 (|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty))^3}, \label{absdet2}\\ \| \det J(\mathbf{y}^0) \|_{{\mathcal L}} & \leq \frac 4 {(1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty)^3} \label{det-L} \\ \| \det J(\mathbf{y}^0 + \mathbf{y}) \|_{{\mathcal L}} & \leq \frac 4 {(1 - \|{\mathcal B}\|_1 (|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty))^3} \label{det-L2}\\ \| \det(I + J^{-1}(\mathbf{y}^0){\mathcal B}\mathbf{y}) -1 \|_{\mathcal L}&\leq 4 \left[ \left(\frac{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty }{1 - \|{\mathcal B}\|_1(|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)} \right)^3 - 1\right] \label{det-I-J-B}. \end{aligned}$$* *Proof.* See [7](#app:computations){reference-type="ref" reference="app:computations"} for the proof. ◻ Hence we have that $$\label{estimate-linear} \begin{aligned} \|&[{\mathcal L}_k(\mathbf{y}^0 + \mathbf{y})] (u^0_k + u_k) - [{\mathcal L}_k(\mathbf{y}^0)] (u^0_k)\|_{L^2(U_k)} \\ &\leq |\epsilon_k^*|\, \| J^{-1}(\mathbf{y}^0 + \mathbf{y})J(\mathbf{y}^0+\mathbf{y})^{-\mathrm{T}} \det J(\mathbf{y}^0+\mathbf{y}) \nabla(u^0_k + u_k) \\ &\hspace{20em} - J^{-1}(\mathbf{y}^0)J(\mathbf{y}^0)^{-\mathrm{T}} \det J(\mathbf{y}^0) \nabla u^0_k\|_{H^1(U_k;\mathbb{C}^3)} \\ &\leq |\epsilon_k^*| \Big(a(\mathbf{y}^0, \mathbf{y}) \| u_k \|_{H^2(U_k)} + b(\mathbf{y}^0, \mathbf{y}) \|u^0_k\|_{H^2(U_k)} \Big) \end{aligned}$$ where $$a(\mathbf{y}^0, \mathbf{y}) := \left(\frac{1}{1 - 4 \| {\mathcal B}\|_1 (|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)}\right)^2 \times \frac 4 {(1 - \|{\mathcal B}\|_1 (|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty))^3}$$ and $$\begin{aligned} b(\mathbf{y}^0,\mathbf{y}) := &\Bigg[ \left(\frac{4 \| {\mathcal B}\|_1 |\mathbf{y}|_\infty}{1 - 4 \| {\mathcal B}\|_1 (|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)}\right)^2 \times 4 \left( \left(\frac{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty }{1 - \|{\mathcal B}\|_1(|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)} \right)^3 - 1\right) \\ &\quad+2 \times \frac{4 \| {\mathcal B}\|_1 |\mathbf{y}|_\infty}{1 - 4 \| {\mathcal B}\|_1 (|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)} \times 4 \left( \left(\frac{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty }{1 - \|{\mathcal B}\|_1(|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)} \right)^3 - 1\right) \\ &\quad+ \left(\frac{4 \| {\mathcal B}\|_1 |\mathbf{y}|_\infty}{1 - 4 \| {\mathcal B}\|_1 (|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)}\right)^2 \\ &\quad + 2 \times \frac{4 \| {\mathcal B}\|_1 |\mathbf{y}|_\infty}{1 - 4 \| {\mathcal B}\|_1 (|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)} \\ &\quad + 4 \left( \left(\frac{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty }{1 - \|{\mathcal B}\|_1(|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)} \right)^3 - 1\right) \Bigg] \times \left( \frac{1}{1 - 4 \| {\mathcal B}\|_1 |\mathbf{y}^0|_\infty} \right)^2 \times \frac 4 {(1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty)^3}. \end{aligned}$$ It can then be shown that the norm in $L^2(U)$ is bounded by $$\sqrt 3 \, \epsilon_{\max}\left( a(\mathbf{y}^0,\mathbf{y}) \| u \|_{\mathcal{H}} + b(\mathbf{y}^0, \mathbf{y}) \|u^0\|_\mathcal{H}\right),$$ where $\epsilon_{\max} := \max\{\epsilon_1, \epsilon_2,\epsilon_3\}$. Let $C_k>0$ denote the constant associated with the Banach algebra $H^2(U_k)$. That is, $C_k>0$ is a constant such that $$\|uv\|_{H^2(U_k)}\leq C_k \| u \|_{H^2(U_k)} \times \| v \|_{H^2(U_k)}\quad \forall u,v\in H^2(U_k).$$ Then we have $$\begin{aligned} &\|\overline{\kappa}_k^2 \sinh(u^0_k + u_k) \det J(\mathbf{y}^0 + \mathbf{y}) - \overline{\kappa}_k^2 \sinh(u^0_k) \det J(\mathbf{y}^0) \|_{L^2(U_k)} \\ &\quad\leq \frac {2 |\overline{\kappa}^2_k| \cosh(C_k(\|u^0_k\|_{H^2(U_k)} + \frac 1 2 \|u_k\|_{H^2(U_k)} )) \sinh(C_k\frac 1 2 \|u_k\|_{H^2(U_k)})} {(1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty)^3} \left(\frac{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty }{1 - \|{\mathcal B}\|_1(|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)} \right)^3 \\ &\qquad + \frac { |\overline{\kappa}^2_k| C_k^{-1} \sinh(C_k \|u_k\|_{H^2(U_k)})} {(1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty)^3} \left[ \left(\frac{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty }{1 - \|{\mathcal B}\|_1(|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)} \right)^3 - 1\right]. \end{aligned}$$ Thus the $L^2(U)$ norm is bounded by $$\begin{gathered} \label{estimate-nonlinear} \frac{\sqrt 3\, \overline{\kappa}^2_{\max}}{(1 - \| {\mathcal B}\|_1 |\mathbf{y}^0|_\infty)^3} \Bigg[ 2 \cosh(C_{\max}(\|u^0_k\|_{\mathcal{H}} + \frac 1 2 \|u_k\|_{\mathcal{H}} )) \sinh(C_{\max}\frac 1 2 \|u_k\|_{\mathcal{H}}) \left(\frac{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty }{1 - \|{\mathcal B}\|_1(|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)} \right)^3 \\ + C_{\max}^{-1} \sinh(C_{\max} \|u_k\|_{\mathcal{H}}) \left[ \left(\frac{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty }{1 - \|{\mathcal B}\|_1(|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)} \right)^3 - 1\right] \Bigg]\end{gathered}$$ where $C_{\max} :=\max\{C_1, C_2, C_3\}$. Finally the forcing term can be dealt with by finding the derivatives with respect to $y_k$ for $k=1,2,\ldots, N$. We have that $$\begin{aligned} &\frac{\partial}{\partial y_i} \xi(F(r;\mathbf{y}) - F(\eta_k;\mathbf{y})) \det J(\boldsymbol{r};\mathbf{y}) \\ &\quad = \sum_{j=1}^3 \frac{\partial \xi}{\partial x_j} (F(\boldsymbol{r}; \mathbf{y}) - F(\eta_k; \mathbf{y})) [\sqrt{\mu_i} (b^j_i(\boldsymbol{r}) - b^j_i(\mathbf{\eta}_k))] \det J(\boldsymbol{r};\mathbf{y}) + \xi(F(r;\mathbf{y}) - F(\eta_k;\mathbf{y})) \\ &\qquad + \xi(F(r;\mathbf{y}) - F(\eta_k;\mathbf{y})) \det J(\boldsymbol{r};\mathbf{y}) \mathrm{tr}\left(J^{-1}(\boldsymbol{r};\mathbf{y}) \sqrt{\mu_i} B_i(\boldsymbol{r})\right) \end{aligned}$$ and so $$\left \| \frac{\partial}{\partial y_i} \xi(F(\cdot;\mathbf{y}) - F(\eta_k;\mathbf{y})) \det J(\cdot;\mathbf{y})\right\|_{L^2(U)} \leq 6 \sqrt{\mu_i} \max_{j=1,2,3} \left \| \frac{\partial \xi}{\partial x_j} \right\|_{L^2(\mathbb{R}^3)} + \|\xi\|_{L^2(\mathbb{R}^3)} \times \frac{3}{1-\|{\mathcal B}\|_1 |\mathbf{y}|} \times \|{\mathcal B}\|_\infty$$ Thus we get that $$\label{estimate-f} \begin{aligned} &\| f^*(\cdot;\mathbf{y}^0 + \mathbf{y}) \det J(\cdot;\mathbf{y}^0 + \mathbf{y}) - f^*(\cdot;\mathbf{y}^0 ) \det J(\cdot;\mathbf{y}^0 ) \|_{L^2(U)} \\ &\quad\leq N_f \cdot |\mathbf{y}| \cdot \left( 6 \sqrt{\mu_i} \max_{j=1,2,3} \left \| \frac{\partial \xi}{\partial x_j} \right\|_{L^2(\mathbb{R}^3)} + \frac{3 \|\xi\|_{L^2(\mathbb{R}^3)} \|{\mathcal B}\|_\infty}{1-\|{\mathcal B}\|_1 |\mathbf{y}|} \right). \end{aligned}$$ Combining the estimates in [\[estimate-linear,estimate-nonlinear,estimate-f\]](#estimate-linear,estimate-nonlinear,estimate-f){reference-type="ref" reference="estimate-linear,estimate-nonlinear,estimate-f"} gives a bound on the $L^2(U)$ part of $\mathcal{F}(\mathbf{y}^0 + \mathbf{y}, u^0 + u) - \mathcal{F}(\mathbf{y}^0, u^0)$. We still have constants that have not been explicitly given (such as $C_{\max}$ and the norm of the solution $u^0$), but estimating these would be difficult to do in the space of this paper. # Sparse Grids {#sparsegrids} Sparse grids are a mathematical technique used to efficiently approximate functions and solve problems in high-dimensional spaces. They provide a way to reduce the computational cost associated with high-dimensional problems by exploiting the sparsity of the underlying function. In many real-world applications, such as optimization, machine learning, and scientific simulations, the dimensionality of the problem can be quite large. Traditional numerical methods often struggle to handle these high-dimensional scenarios due to the exponential growth of computational requirements with increasing dimensions. Sparse grids offer a solution to this problem by selectively evaluating the function only at a subset of points in the high-dimensional space. The idea is to concentrate computational effort on regions that contribute the most to the overall approximation accuracy, while ignoring or approximating the function in less significant regions. Sparse grids are constructed from tensor products of Lagrange iterpolation. Given a set of data points $(\zeta_0, z_0), (\zeta_1, z_1), \ldots, (\zeta_p, z_p) \in \tilde \Gamma \times \mathbb{R}$, where we define $\tilde \Gamma := [-1,1]$ and the $\zeta_i$ values are distinct, Lagrange interpolation constructs a polynomial $P(\zeta)$ of degree at most $n$ that satisfies: $$P(\zeta_i) = z_i, \quad \text{for } i = 0, 1, \ldots, p$$ The polynomial $P(\zeta)$ is defined as the linear combination of Lagrange basis polynomials $l_i(\zeta)$, which are constructed to ensure that $P(\zeta_i) = z_i$ for each data point: $$P(\zeta) = \sum_{i=0}^{p} z_i l_i(\zeta)$$ The Lagrange basis polynomials are defined as: $$l_i(\zeta) = \prod_{\substack{j=0 \\ j \neq i}}^{n} \frac{\zeta - \zeta_j}{\zeta_i - \zeta_j}$$ These basis polynomials have the property that $l_i(\zeta_i) = 1$ and $l_i(\zeta_j) = 0$ for $j \neq i$, ensuring that the polynomial $P(\zeta)$ passes through the corresponding data point $(\zeta_i, z_i)$. It is clear that $P(\zeta) \in {\mathcal P}_{p}(\tilde \Gamma) := \text{\rm span}\{\zeta^m : \,m=0,\dots,p\}$. We can now extend Lagrange interpolation to using tensor products of 1D interpolants. Let $$C^{0}(\Gamma) : = \{ v: \Gamma \rightarrow V\,\, \mbox{is continuous on $\Gamma$ and } \max_{y\in \Gamma} |v(y)| < \infty \},$$ $\mathbf{p}= (p_1,\ldots,$ $p_{N})$, and ${\mathcal P}_{ p_n}(\tilde \Gamma):=\text{\rm span}(y_n^m,\,m=0,\dots,p_n),$ for each dimension $n = 1,\dots,N$. Let ${\mathcal I}^{m(i)}:C^{0}(\tilde \Gamma) \rightarrow {\mathcal P}_{m(i)-1}(\tilde \Gamma)$ be the Lagrange interpolation operator where $i \in \mathbb{N}_0$, $m(0) = 0$, $m(1) = 1$ and in general $m(i) \in \mathbb{N}_0$ is the number of evaluation points at level $i$. Note that if $m(i) = 0$ then let ${\mathcal P}_{-1}(\Gamma) := \emptyset$. Consider the vector of approximations $\mathbf{i}= (i_1,i_2, \dots,i_N) \in \mathbb{N}^{N}_{0}$, and form the space ${\mathcal P}_{\mathbf{p}}(\Gamma) = \bigotimes_{n=1}^{N}\;{\mathcal P}_{p_n}(\tilde \Gamma)$ then the Lagrange interpolation for $N$ dimensions operator ${\mathcal I}^{N}_{\mathbf{i}} :C^{0}(\Gamma) \rightarrow {\mathcal P}_{\mathbf{p}}(\Gamma)$ can now be built as $${\mathcal I}^{N}_{\mathbf{i}} = {\mathcal I}^{m(i_1)}_{1} \otimes {\mathcal I}^{m(i_2)}_{2} \otimes \dots \otimes {\mathcal I}^{m(i_N)}_{N}.$$ More explicitly for each dimension $n = 1, \dots, N$ let $\{y^{n}_{1},\dots,y^{n}_{m(i)}\} \subset \tilde \Gamma$ be a sequence of abcissas for the Lagrange interpolation operator ${\mathcal I}^{m(i_n)}_{n}$. Thus for any $\nu \in C^{0}(\Gamma)$ $${\mathcal I}^{N}_{\mathbf{i}} \nu(\mathbf{y}) = \sum_{k_1 = 1}^{m(i_1)} \sum_{k_2 = 1}^{m(i_2)} \dots \sum_{k_N = 1}^{m(i_N)} \nu(y^{i_{1}}_{k_1},y^{i_{2}}_{k_2},\dots,y^{i_{N}}_{k_N}) l^{i_1}_{k_1} \otimes l^{i_2}_{k_2} \otimes \dots \otimes l^{i_N}_{k_N},$$ where $$l^i_j(y) = \prod_{\substack{k=0 \\ j \neq k}}^{N} \frac{y - y^i_k}{y^i_k - y^j_k}.$$ However, the dimensionality of ${\mathcal P}_{\mathbf{p}}$ explodes as $\prod_{n=1}^N$ $(p_n+1)$ making Lagrange interpolation intractable for even a number of moderate dimensions. In contrast, if there exists a complex analytic extension of $\nu(\mathbf{y})$ with respect to $\mathbf{y}$ then sparse grids are a better choice [@Smolyak63; @Novak_Ritter_00; @Back2011; @nobile2008a]. They provide almost the same convergence accuracy of full tensor product grids, but with significant reductions in dimensionality. This is achieved by judiciously selecting a reduced set of monomials from the full tensor product. Let $\mathbf{m}(\mathbf{i}) = (m(i_1),\ldots,m(i_{N})) \in \mathbb{Z}^{N}$ be the vector of the number of evaluations points for each dimension. For a given non-negative integer $w$, we define the index set $\Lambda^{m,g}(w)$ as follows: $$\Lambda^{m,g}(w) = \{\mathbf{p}\in\mathbb{N}_0^{N}, \;\; g(\mathbf{m}^{-1}(\mathbf{p}+\boldsymbol{1}))\leq w\}.$$ In this context, the function $g:\mathbb{Z}^{N} \rightarrow \mathbb{Z}$ acts as a restriction function along each dimension of the complete tensor grid. The indices in $\Lambda^{m,g}(w)$ constitute the set of permissible polynomial moments $\mathbb{P}_{\Lambda^{m,g}(w)}(\Gamma)$, subject to the restrictions imposed by $(\mathbf{m},g,w)$. Specifically, this polynomial set is defined as: $$\mathbb{P}_{\Lambda^{m,g}(w)}(\Gamma) := \mathrm{span}\left\{\prod_{n=1}^{N} y_n^{p_n}, \text{ with } \mathbf{p}\in\Lambda^{m,g}(w)\right\}.$$ Let's consider the difference operator along the $n^{th}$ dimension of $\Gamma$, denoted as $${\Delta_n^{m(i)} :=} {\mathcal I}_n^{m(i)}-{\mathcal I}_n^{m(i-1)}.$$ By taking the tensor product of these difference operators across all dimensions, we can construct a sparse grid. In this context, $w \in \mathbb{N}_0$ represents the desired approximation level. The sparse grid approximation of $\nu$ is then obtained as follows: $$\mathcal{S}^{m,g}_w[\nu] = \sum_{\mathbf{i}\in\mathbb{Z}^{N}: g(\mathbf{i})\leq w} \;\; \bigotimes_{n=1}^{N} {\Delta_n^{m(i_n)}}(\nu(\mathbf{y})).$$ We have the flexibility to choose different values for the parameters $m$ and $g$. Our main objective is to achieve accurate results while controlling the increase in dimensionality of the space $\mathbb{P}_{\Lambda^{m,g}(w)}(\Gamma)$. To address this, we can utilize the well-known Smolyak sparse grid method introduced by Nobile et al. (2008), which can be constructed using the following formulas: $$m(i) = \begin{cases} 1, & \text{for } i=1 \\ 2^{i-1}+1, & \text{for } i>1 \end{cases}\quad \text{ and } \quad g(\mathbf{i}) = \sum_{n=1}^N (i_n-1).$$ For this particular choice, the index set $\Lambda^{m,g}(w)$ is defined as follows: $\Lambda^{m,g}(w):=\{\mathbf{p}\in\mathbb{N}_0^{N}: \;\; \sum_n f(p_n) \leq w\}$ where $$f(p) = \begin{cases} 0, \; p=0 \\ 1, \; p=1 \\ \lceil \log_2(p) \rceil, \; p\geq 2 \end{cases}.$$ Alternative choices, such as the Total Degree (TD) and Hyperbolic Cross (HC) grids, are described in [@castrillon2016analytic]. The last component of the sparse grid is the selection of the abcissas $\{y^{n}_{1},\dots,y^{n}_{m(i)}\} \subset [-1,1]$ along each dimension. One option is the extrema of Chebyshev polynomials: $$y^n_j = -\cos \left( \frac{\pi(j-1)}{m(i) - 1} \right).$$ This popular choice are denoted as Clenshaw-Curtis abscissas. It is worth noting that not all stochastic dimensions need to be treated equally. Some dimensions may contribute more to the sparse grid approximation than others. By customizing the restriction function $g$ according to the input random variables $y_n$ for $n = 1, \dots, N$, a more accurate *anisotropic* sparse grid can be obtained [@Schillings2013; @nobile2008b]. In this paper, for the sake of simplicity, we focus on isotropic sparse grids. However, extending the approach to an anisotropic setting is a straightforward. Our current focus is on establishing error bounds for the sparse grid, specifically the norm $\|\nu - \mathcal{S}^{m,g}[\nu]$ $\|_{{L^{\infty}(\Gamma)}}$. This bound can be controlled by three key factors. Firstly, the number of dimensions, denoted as $N$, influences the bound. Secondly, the number of knots, denoted as $\eta$, in the sparse grid plays a role. However, the most crucial factor is the size of the complex region in the analytic extension of $\nu(\mathbf{y})$ onto $\mathbb{C}^{N}$ and the following bound on the polyellipse $$\tilde{M}(\nu) := \sup_{\mathbf{z}\in {\mathcal E}_{\sigma_1, \dots, \sigma_{N}}} |\nu(\mathbf{z})|.$$ With the parameters analytic extension parameters $(\sigma^{*}, \tilde M(\nu))$, the number of dimensions $N$ and the level of the sparse grid $w$ the error of the sparse grid $\|\nu - \mathcal{S}^{m,g}[\nu]$ $\|_{{L^{\infty}(\Gamma)}}$ can be bounded. Define the following constants $$\begin{aligned} \sigma &= \sigma^*/2, & \tilde{C}_{2}(\sigma) &= 1 + \frac{1}{\log{2}}\sqrt{\frac{\pi}{2\sigma}}, & \delta^{*}(\sigma) &= \frac{e\log{(2)} - 1}{\tilde{C}_2 (\sigma)}, \\ \mu_1 &= \frac{\sigma}{1 + \log (2N)}, & \mu_2(N) &= \frac{\log(2)}{N(1 + \log(2N))}, & \mu_3 &= \frac{\sigma \delta^{*} \tilde C_2(\sigma)}{1 + 2\log(2N)},\end{aligned}$$ $$\begin{aligned} a(\delta,\sigma) &=\exp{\left(\delta \sigma \left\{\frac{1}{\sigma \log^{2}{(2)}}+ \frac{1}{\log{(2)}\sqrt{2 \sigma}}+ 2\left( 1 + \frac{1}{\log{(2)}} \sqrt{ \frac{\pi}{2\sigma} }\right)\right\}\right)}, \\ C_1(\sigma,\delta,\tilde M(\nu)) &= \frac{4\tilde{M}(\nu) C(\sigma)a(\delta,\sigma)}{ e\delta\sigma}, \\ {\mathcal Q}(\sigma,\delta^{*}(\sigma),N, \tilde M(\nu)) &= \frac{ C_1(\sigma,\delta^{*}(\sigma),\tilde M(\nu))}{\exp(\sigma\delta^{*}(\sigma) \tilde{C}_2(\sigma) )}\frac{\max\{1,C_1(\sigma,\delta^{*}(\sigma),\tilde M(\nu))\}^{N}}{|1- C_1(\sigma,\delta^{*}(\sigma),\tilde M(\nu))|}.\end{aligned}$$ **Theorem 4**. *Suppose that $\nu \in C^{0}(\Gamma;\mathbb{R})$ admits an analytic extension on ${\mathcal E}_{\sigma_1, \dots, \sigma_{N}}$. Let $\mathcal{S}^{m,g}_{w}[\nu]$ be the sparse grid approximation of the function $\nu$ with Clenshaw-Curtis abcissas. If $w > N / \log{2}$ then $$\|\nu - \mathcal{S}^{m,g}_{w}[\nu] \|_{L^{\infty}(\Gamma)} \leq {\mathcal Q}(\sigma,\delta^{*}(\sigma),N, \tilde M(\nu)) \eta^{\mu_3(\sigma,\delta^{*}(\sigma),N)}\exp \left(-\frac{N \sigma}{2^{1/N}} \eta^{\mu_2(N)} \right) , \\ \label{erroranalysis:sparsegrid:estimate}$$ Furthermore, if $w \leq N / \log{2}$ then the following algebraic convergence bound holds: $$\begin{split} \| \nu - \mathcal{S}^{m,g}_{w}[\nu] \|_{L^{\infty}(\Gamma)} &\leq \frac{C_1(\sigma,\delta^{*}(\sigma),\tilde M(\nu)) \max{\{1,C_{1}(\sigma,\delta^{*}(\sigma),\tilde{M}(\nu)) \}}^N }{ |1 - C_1(\sigma,\delta^{*}(\sigma),\tilde M(\nu))| }\eta^{-\mu_1}. \end{split} \label{erroranalysis:sparsegrid:estimate2}$$ [\[erroranalysis:theorem1\]]{#erroranalysis:theorem1 label="erroranalysis:theorem1"}* *Proof.* Theorem 3.10 and 3.11 in [@nobile2008a]. ◻ # Numerical Results {#numerics} We now test the complex analyticity result for the NPBE by computing the potential field of the Trypsin protein (PDB:1ppe [@Berman2000], $n = 1{,}852$ atoms) submerged in a solvent. In Figure [\[NumericalResults:Fig1\]](#NumericalResults:Fig1){reference-type="ref" reference="NumericalResults:Fig1"} (a) the secondary structure of the Trypsin molecule is rendered with a meshed surface of the molecular boundary. This corresponds to the molecular boundary obtained by rolling a solvent atom around the molecule. This boundary corresponds to the interface $I_1$. Inside the molecule the dielectric is set to $\epsilon = 70$ and outside the boundary it is set to $\epsilon = 1$ (e.g. solvent dielectric). Note that these dielectric values are unit-less. The second boundary $I_2$ (not rendered) corresponds to the ion-accessible surface. The Debye-Hückel parameter, $\overline{\kappa}$, is set to zero inside the surface and non-zero outside. The entire protein is contained in a cubic domain $\mathcal D$ measuring $70 \times 70 \times 70$ Å  and the Dirichlet boundary conditions are set to zero, i.e., $u \equiv 0$ on $\partial \mathcal D$. The temperature of the solvent is set to $T = 310$ Kelvin. Let ${\mathcal C}:= \{ \mathbf{x}_1, \dots, \mathbf{x}_n\}$ correspond to set of the location of the molecular atoms. This information is contained in the PDB file. In theory and in practice these locations represent point charges that are replaced with functions $L^{2}(\Omega)$ as this guarantee the existence of a unique solution for the NPBE [@holst1994poisson]. The potential field, which corresponds to the solution of the NPBE, is then solved using APBS. ------- ------- \(a\) \(b\) ------- ------- The protein will be now shifted using a stochastic model inside the domain $\mathcal D$. Since boundary conditions are set to zero the solution will not be a simple translation, but will depend nonlinearly on the shift of the atom locations given by ${\mathcal C}$. For each shift, the domain of the Trypsin molecule is discretized and the potential field is solved using APBS. Let ${\mathcal C}(\omega)$ correspond to the set of atom locations shifted by the event outcome $\omega \in \Omega$, i.e. $${\mathcal C}(\omega) = \{\mathbf{x}= x_0 + \sum_{k = 1}^{N} \alpha_{k} \mathbf{e}_{k}Y_{k}(\omega) \,|\, x_0 \in {\mathcal C}\},$$ where $$\mathbf{e}_1 = [1,0,0]^{T}, \mathbf{e}_2 = [0,1,0]^{T}, \mathbf{e}_3 = [0,0,1]^{T}, \alpha_1 = \alpha_2 = \alpha_3 = 10,$$ $N \leq 3$, and $Y_1(\omega)$, $Y_2(\omega)$, $Y_3(\omega)$ are uniformly distributed over the domain $[-\sqrt{3},\sqrt{3}]$ and independent of each other. Thus for each event $\omega \in \Omega$ each element of ${\mathcal C}$ are shifted as follows: $$\mathbf{x}_k \rightarrow \mathbf{x}_k + \sum_{k = 1}^{N} \alpha_{k} \mathbf{e}_{k}Y_{k}(\omega),$$ for $k = 1,\dots, N$. Suppose $\hat u(x,Y_1,\dots,Y_N)$ is the solution of the APBS with respect to $Y_1,\dots,Y_N$ and let $$\label{QoI} Q(\hat u) := \int_{\mathcal D} \hat u(x,Y_1,\dots,Y_N)\,\mbox{d}x$$ with respect to stochastic domain shifts $Y_1,\dots,Y_N$. Using the sparse grid the error $|{\mathbb E} \left[ Q(\hat u) \right] - {\mathbb E} \left[ \mathcal{S}^{m,g}_{w}[Q(\hat u)] \right] |$ is computed. In Figure [\[NumericalResults:Fig1\]](#NumericalResults:Fig1){reference-type="ref" reference="NumericalResults:Fig1"} (b) the convergence graphs are plotted for $w = 2,3,4,5$ for $N = 2$. We assume that $w = 7$ is the actual value for ${\mathbb E} \left[ Q(\hat u) \right]$. Notice that the convergence rate decays algebraically. This is consistent with [\[erroranalysis:theorem1\]](#erroranalysis:theorem1){reference-type="ref" reference="erroranalysis:theorem1"}. We observe the similar algebraic decay for $N = 3$. # Conclusion In this paper we show the existence of complex analytic extension of the solution of the NPBE with respect to the random perturbations of the domain, and its application to UQ is explored. This is a difficult problem due to the exponentially-growing nonlinear term and the discontinuity of the interface. The analyticity of solutions holds significant practical implications for efficiently computing quantities of interest. Any bounded linear map $Q: \mathcal{H}(U) \rightarrow \mathbb{R}$ can be computed at an algebraic or sub-exponential rate, since they are analytic with respect to solutions $\tilde{u} \in \mathcal{H}(U)$. Notably, the linearity of surface integrals allows for the direct application of these results in such cases. We have also given estimates on the region of analyticity for the solution of the NPBE, which would allow for explicit calculations of the convergence rate of the sparse grid quadrature. However the estimate of the region relies on constants, whose exact values are difficult to determine, a topic of interest that are left for future work. More generally, the framework developed here is applicable to other problems in UQ. The strategy to applying these results to other problems is roughly as follows: 1. Rewrite the problem as a functional equation, $\mathcal{F}(\mathbf{y}, u) = 0$, where $\mathbf{y}$ are the stochastic parameters in $\mathbb{R}^N$ or $\mathbb{C}^N$ and $u$ represents the solution to the nonlinear PDE or equation in an appropriate Banach space. If the problem involves an interface, then a space similar to $\mathcal{H}(U)$ may be useful. 2. Applying the implicit function theorem (similar to how it was done for [Theorem 2](#existence-of-region){reference-type="ref" reference="existence-of-region"}) gives a region of analyticity for the solution $\mathbf{y}\mapsto u(\mathbf{y})$. It is important that the Banach space for $u$ is appropriately chosen so that the Fréchet derivative $D_u \mathcal{F}$ is an isomorphism. 3. can be applied to estimate the region of analyticity. This gives useful convergence results for numerical methods (e.g. sparse grid quadratures). To get explicit bounds, one needs to estimate the norm of the linear operator $[D_u F]^{-1}$, which for simple domains and operators can be computed. Thus the results in this paper can be more broadly implemented in other nonlinear UQ problems in order to demonstrate the analyticity of solutions and observables as well as determine estimates on the region of analyticity. # Computations for estimates {#app:computations} *Proof of [Proposition 3](#prop:computations){reference-type="ref" reference="prop:computations"}.* The space ${\mathcal L}$ is nice to work with since it is a Banach algebra, i.e., for $A_1, A_2 \in {\mathcal L}$ we have $$\|A_1A_2\|_{{\mathcal L}} \leq \|A_1\|_{\mathcal L}\|A_2\|_{\mathcal L}.$$ We will use this fact frequently in the proof. From [\[holder\]](#holder){reference-type="ref" reference="holder"}, we have that $$\sigma_{\mathrm{max}} ({\mathcal B}\mathbf{y}(\boldsymbol{r})) = |{\mathcal B}\mathbf{y}(\boldsymbol{r})|_2 \leq \|{\mathcal B}\|_1 |\mathbf{y}|_\infty \quad \text{for any } \boldsymbol{r}\in U$$ where $\sigma_{\mathrm{max}}(\cdot)$ denotes the largest singular value of a matrix. Thus (suppressing the $\boldsymbol{r}\in U$ term) $$\begin{aligned} \sigma_{\mathrm{max}}(J(\mathbf{y})) &= |J(\mathbf{y})|_2 \leq 1 + \| {\mathcal B}\|_1 |\mathbf{y}|_\infty \\ \sigma_{\mathrm{min}} (J(\mathbf{y})) &\geq 1 - \sigma_{\mathrm{max}}({\mathcal B}\mathbf{y}) \geq 1 - \|{\mathcal B}\|_1 |\mathbf{y}|_\infty, \end{aligned}$$ where $\sigma_{\mathrm{min}}(\cdot)$ represents the smallest singular value of a matrix. We will need estimates on the norms $J^{-1}(\mathbf{y}^0 + \mathbf{y})$ and $\det J(\mathbf{y}^0 + \mathbf{y})$, and so the following identity will be useful: $$J(\mathbf{y}^0 + \mathbf{y}) = J(\mathbf{y}^0) [ I + J^{-1}(\mathbf{y}^0) {\mathcal B}\mathbf{y}].$$ By the assumption [\[small-b\]](#small-b){reference-type="ref" reference="small-b"}, we have that $J(\mathbf{y}^0)$ is invertible, and the inverse is given by the Neumann series $$J^{-1}(\mathbf{y}^0) = I + \sum_{k=1}^\infty (-1)^k ({\mathcal B}\mathbf{y}^0)^k.$$ Thus $$\begin{aligned} \| J^{-1}(\mathbf{y}^0) \|_{{\mathcal L}} &\leq 1 + \sum_{k=1}^\infty (\|{\mathcal B}\mathbf{y}^0\|_{\mathcal L})^k \\ &\leq 1 + \sum_{k=1}^\infty (4 \| {\mathcal B}\mathbf{y}^0 \|_{C^1})^k \\ &= \frac{1}{1 - 4 \| {\mathcal B}\|_1 |\mathbf{y}^0|_\infty}, \end{aligned}$$ which give [\[Jinf-L\]](#Jinf-L){reference-type="ref" reference="Jinf-L"}. is shown similarly. Also, $$\begin{aligned} \|(I + J(\mathbf{y}^0){\mathcal B}\mathbf{y})^{-1} - I \|_{\mathcal L}&\leq \sum_{k=1}^\infty \| J^{-1}(\mathbf{y}^0) {\mathcal B}\mathbf{y}\|_{\mathcal L}^k \\ &\leq \sum_{k=1}^\infty \left( \frac{1}{1 - 4 \| {\mathcal B}\|_1 |\mathbf{y}^0|_\infty} \right)^k (4 \|{\mathcal B}\|_1 |\mathbf{y}|_\infty)^k \\ &= \frac{4 \| {\mathcal B}\|_1 |\mathbf{y}|_\infty}{1 - 4 \| {\mathcal B}\|_1 (|\mathbf{y}|_\infty + |\mathbf{y}^0|_\infty)}, \end{aligned}$$ which gives [\[I-J-B\]](#I-J-B){reference-type="ref" reference="I-J-B"}. To estimate $\|\det J(\mathbf{y}) \|_{\mathcal L}$, we will need to bound both $|\det J(\mathbf{y})|$ and $|\frac{\partial}{\partial r_i} \det J (\mathbf{y}) |$ over $U$. For a matrix $A$ with $\sigma_{\mathrm{max}}(A) < 1$, we have $$\det(I+A) = 1 + \sum_{k=1}^\infty \frac 1 {k!} \left( - \sum_{j=1}^\infty \frac{(-1)^j}{j}\mathrm{tr}(A^j)\right)^k.$$ Therefore, $$\begin{aligned} |\det J(\mathbf{y}^0) | &= |\det (I + {\mathcal B}\mathbf{y}^0) | \\ &\leq 1 + \sum_{k=1}^\infty \frac 1 {k!} \left( \sum_{j=1}^\infty \frac{1}{j}\mathrm{tr}( ({\mathcal B}\mathbf{y}^0)^j)\right)^k \\ &\leq 1 + \sum_{k=1}^\infty \frac 1 {k!} \left( \sum_{j=1}^\infty \frac{3}{j}\sigma_{\mathrm{max}}( {\mathcal B}\mathbf{y}^0)^j\right)^k \\ &\leq 1 + \sum_{k=1}^\infty \frac 1 {k!} \left( \sum_{j=1}^\infty \frac{3}{j} \|{\mathcal B}\|_1^j |\mathbf{y}^0|_\infty^j\right)^k \\ &= 1 + \sum_{k=1}^\infty \frac 1 {k!}(-3 \ln(1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty))^k \\ &= \frac{1}{(1- \|{\mathcal B}\|_1 |\mathbf{y}^0|)^3}, \end{aligned}$$ which gives [\[absdet\]](#absdet){reference-type="ref" reference="absdet"}. The inequality in [\[absdet2\]](#absdet2){reference-type="ref" reference="absdet2"} is similarly shown. Using Jacobi's formula, we have that $$\begin{aligned} \left|\frac{\partial}{\partial r_i} \det J(\mathbf{y}^0) \right| &\leq |\det J(\mathbf{y}^0)| \times \left|\mathrm{tr}(J^{-1}(\mathbf{y}^0) \frac{\partial J}{\partial r_i}(\mathbf{y}^0))\right| \\ &\leq \frac{1}{(1- \|{\mathcal B}\|_1 |\mathbf{y}^0|)^3} \times 3 \sigma_{\mathrm{min}}(J(\mathbf{y}^0))^{-1} \times \sigma_{\mathrm{max}} \left( \frac \partial {\partial r_i} {\mathcal B}\mathbf{y}^0\right) \\ &\leq \frac{1}{(1- \|{\mathcal B}\|_1 |\mathbf{y}^0|)^3} \times \frac{3}{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|} \times \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty \\ &= \frac{3 \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty}{(1- \|{\mathcal B}\|_1 |\mathbf{y}^0|)^4} \end{aligned}$$ Therefore, $$\begin{aligned} \|\det J(\mathbf{y}^0) \|_{\mathcal L}&\leq 4 \| \det J(\mathbf{y}^0) \|_{C^1} \\ &\leq 4 \max\left\{ \frac{1}{(1-\|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty)^3}, \frac{3 \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty}{(1- \|{\mathcal B}\|_1 |\mathbf{y}^0|)^4}\right\} \\ &= \frac{4}{(1-\|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty)^3} \end{aligned}$$ where the last line follows from our assumptions [\[small-b\]](#small-b){reference-type="ref" reference="small-b"} and $\mathbf{y}^0 \in \Gamma = [-1,1]^N$. This gives [\[det-L\]](#det-L){reference-type="ref" reference="det-L"}, and [\[det-L2\]](#det-L2){reference-type="ref" reference="det-L2"} is proved similarly. Lastly, we want to get the estimate [\[det-I-J-B\]](#det-I-J-B){reference-type="ref" reference="det-I-J-B"}. We have that $$\begin{aligned} |\det(I + J^{-1}(\mathbf{y}^0){\mathcal B}\mathbf{y}) -1 | &\leq \sum_{k=1}^\infty \frac 1 {k!} \left( \frac 1 j |\mathrm{tr}\left((J^{-1}(\mathbf{y}^0){\mathcal B}\mathbf{y}\right)^j)|\right)^k \\ &\leq \sum_{k=1}^\infty \frac 1 {k!} \left( \sum_{j=1}^\infty \frac 3 j \left( \frac{\|{\mathcal B}\|_1 |\mathbf{y}|_\infty}{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty} \right)^j \right)^k \\ &= \left(\frac{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|}{1 - \|{\mathcal B}\|_1(|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)}\right)^3 - 1. \end{aligned}$$ Using Jacobi's formula, we have that the corresponding derivative can be bounded as follows: $$\begin{aligned} &\left|\frac \partial {\partial r_i} \det (I + J^{-1}(\mathbf{y}^0){\mathcal B}\mathbf{y}) \right| \\ &\quad \leq |\det(I + J^{-1}(\mathbf{y}^0){\mathcal B}\mathbf{y})| \times \left| \mathrm{tr} \left( (I + J^{-1}(\mathbf{y}^0){\mathcal B}\mathbf{y})^{-1} \left(\frac{\partial J^{-1}(\mathbf{y}^0)}{\partial r_i} {\mathcal B}\mathbf{y}+ J^{-1}(\mathbf{y}^0) \frac{\partial {\mathcal B}\mathbf{y}}{\partial r_i}\right) \right) \right|. \end{aligned}$$ Note that $$\begin{aligned} \sigma_{\mathrm{min}}(I + J^{-1}(\mathbf{y}^0) {\mathcal B}\mathbf{y}) &= \sigma_{\mathrm{min}}(J^{-1}(\mathbf{y}^0) (I + {\mathcal B}\mathbf{y}^0 + {\mathcal B}\mathbf{y})) \\ &\geq \sigma_{\mathrm{min}}(J^{-1}(\mathbf{y}^0)) \sigma_{\mathrm{min}}(I + {\mathcal B}\mathbf{y}^0 + {\mathcal B}\mathbf{y}) \\ &\geq \frac{1 - \|{\mathcal B}\|_1(|\mathbf{y}^0| + |\mathbf{y}|_\infty)}{1 + \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty}. \end{aligned}$$ We also have that $$\frac{\partial J^{-1}(\mathbf{y}^0)}{\partial r_i} = - J^{-1}(\mathbf{y}^0) \frac{\partial {\mathcal B}\mathbf{y}^0}{\partial r_i}J^{-1}(\mathbf{y}^0)$$ so that $$\sigma_{\mathrm{max}} \left( \frac{\partial J^{-1}(\mathbf{y}^0)}{\partial r_i} \right) \leq \frac{\|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty}{(1-\|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty)^2}.$$ Therefore, $$\begin{aligned} &\left|\frac{\partial}{\partial r_i} \det (I + J^{-1}(\mathbf{y}^0) {\mathcal B}\mathbf{y})\right| \\ &\quad\leq \left[ \left(\frac{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|}{1 - \|{\mathcal B}\|_1(|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)}\right)^3 - 1 \right] \times 3 \times \frac{1 + \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty}{1 - \|{\mathcal B}\|_1(|\mathbf{y}^0| + |\mathbf{y}|_\infty)} \\ &\qquad\times \left[\frac{\|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty}{(1-\|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty)^2} \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty + \frac{\|{\mathcal B}\|_1|\mathbf{y}^0|_\infty}{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|_\infty}\right] \end{aligned}$$ It is possible to show that $$\left|\frac{\partial}{\partial r_i} \det (I + J^{-1}(\mathbf{y}^0) {\mathcal B}\mathbf{y})\right| \leq \left(\frac{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|}{1 - \|{\mathcal B}\|_1(|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)}\right)^3 - 1$$ using the assumptions of the norms of $\mathbf{y}$, $\mathbf{y}^0$, and ${\mathcal B}$. Thus $$\begin{aligned} \|\det (I + J^{-1}(\mathbf{y}^0) {\mathcal B}\mathbf{y}) \|_{\mathcal L}&\leq \|\det (I + J^{-1}(\mathbf{y}^0) {\mathcal B}\mathbf{y})\|_{C^1} \\ &\leq 4 \left[ \left(\frac{1 - \|{\mathcal B}\|_1 |\mathbf{y}^0|}{1 - \|{\mathcal B}\|_1(|\mathbf{y}^0|_\infty + |\mathbf{y}|_\infty)}\right)^3 - 1 \right]. \end{aligned}$$ ◻

arxiv_math

--- abstract: | A *clique transversal* in a graph is a set of vertices intersecting all maximal cliques. The problem of determining the minimum size of a clique transversal has received considerable attention in the literature. In this paper, we initiate the study of the "upper" variant of this parameter, the *upper clique transversal number*, defined as the maximum size of a minimal clique transversal. We investigate this parameter from the algorithmic and complexity points of view, with a focus on various graph classes. We show that the corresponding decision problem is NP-complete in the classes of chordal graphs, chordal bipartite graphs, and line graphs of bipartite graphs, but solvable in linear time in the classes of split graphs and proper interval graphs. clique transversal, upper clique transversal number, vertex cover 05C69, 05C85, 05C75, 05C76, 68R10 author: - | Martin Milanič\ FAMNIT and IAM, University of Primorska, Koper, Slovenia\ `martin.milanic@upr.si` - | Yushi Uno\ Graduate School of Informatics, Osaka Metropolitan University, Sakai, Osaka, Japan\ `yushi.uno@omu.ac.jp` title: "Upper Clique Transversals in Graphs[^1]" --- # Introduction A set of vertices of a graph $G$ that meets all maximal cliques of $G$ is called a *clique transversal* in $G$. Clique transversals in graphs have been studied by Payan in 1979 [@MR539710], by Andreae, Schughart, and Tuza in 1991 [@MR1099264], by Erdős, Gallai, and Tuza in 1992 [@MR1189850], and also extensively researched in the more recent literature (see, e.g., [@MR1201987; @MR1375117; @MR1413638; @MR1423977; @MR1737764; @MR4213405; @MR4264990; @MR3875141; @MR3350239; @MR3325542; @MR3131902; @MR2203202]). What most of these papers have in common is that they are interested in questions regarding the *clique transversal number* of a graph, that is, the minimum size of a clique transversal of the graph. For example, Chang, Farber, and Tuza showed in [@MR1201987] that computing the clique transversal number for split graphs is NP-hard, and Guruswami and Pandu Rangan showed in [@MR1737764] that the problem is NP-hard for cocomparability, planar, line, and total graphs, and solvable in polynomial time for Helly circular-arc graphs, strongly chordal graphs, chordal graphs of bounded clique size, and cographs. In this paper, we initiate the study of the "upper" version of this graph invariant, the *upper clique transversal number*, denoted by $\tau_c^+(G)$ and defined as the maximum size of a minimal clique transversal, where a clique transversal in a graph $G$ is said to be *minimal* if it does not contain any other clique transversal. The corresponding decision problem is defined as follows. Our study contributes to the literature on upper variants of graph minimization problems, which already includes the upper vertex cover (also known as maximum minimal vertex cover; see [@MR2772557; @MR3717814; @MR3399960]), upper feedback vertex set (also known as maximum minimal feedback vertex set; see [@MR4322274; @lampis2023parameterized]), upper edge cover (see [@MR4075059]), upper domination (see [@MR1088560; @MR3807977; @MR3767516]), and upper edge domination (see [@MR4266848]). We provide a first set of results on the algorithmic complexity of [Upper Clique Transversal]{.smallcaps}. Since clique transversals have been mostly studied in the class of chordal graphs and related classes, we also find it natural to first focus on this interesting graph class and its subclasses. In this respect, we provide an NP-completeness result as well as two very different linear-time algorithms. We show that UCT is NP-complete in the class of chordal graphs, but solvable in linear time in the classes of split graphs and proper interval graphs. Note that the result for split graphs is in contrast with the aforementioned NP-hardness result for computing the clique transversal number in the same class of graphs [@MR1201987]. In addition, we provide NP-completeness proofs for two more subclasses of the class of perfect graphs, namely for chordal bipartite graphs, and for line graphs of bipartite graphs. The diagram in summarizes the relationships between various graph classes studied in this paper and indicates some boundaries of tractability of the UCT problem. We define those graph classes in the corresponding later sections in the paper. For further background and references on graph classes, we refer to [@MR1686154]. {#fig:graphclass width="75%"} Our approach is based on connections with a number of graph parameters. For example, the NP-completeness proofs for the classes of chordal bipartite graphs and of line graphs of bipartite graphs are based on the fact that for triangle-free graphs without isolated vertices, minimal clique transversals are exactly the minimal vertex covers, and they are closely related with minimal edge covers via the line graph operator. In particular, if $G$ is a triangle-free graph without isolated vertices, then the upper clique transversal number of $G$ equals the upper vertex cover number of $G$, that is, the maximum size of a minimal vertex cover. Since the upper vertex cover number of a graph $G$ plus the independent domination number of $G$ equals the order of $G$, there is also a connection with the independent dominating set problem. Let us note that, along with a linear-time algorithm for computing a minimum independent set in a tree [@MR0485473], the above observations suffice to justify the polynomial-time solvability of the upper clique transversal problem on trees, as indicated in . The NP-completeness proof for the class of chordal graphs is based on a reduction from Spanning Star Forest, the problem of computing a spanning subgraph with as many edges as possible that consists of disjoint stars; this problem, in turn, is known to be closely related to the dominating set problem. The linear-time algorithm for computing the upper clique transversal number of proper interval graphs relies on a linear-time algorithm for the maximum induced matching problem in bipartite permutation graphs due to Chang [@MR2024264]. More precisely, we prove that the upper clique transversal number of a given graph cannot exceed the maximum size of an induced matching of a derived bipartite graph, the *vertex-clique incidence graph*, and show, using new insights on the properties of the matching computed by Chang's algorithm, that for proper interval graphs, the two quantities are the same. The linear-time algorithm for computing the upper clique transversal number of a split graph is based on a characterization of minimal clique transversals of split graphs. A clique transversal that is an independent set is also called a *strong independent set* (or *strong stable set*; see [@MR4273625] for a survey). It is not difficult to see that every strong independent set is a minimal clique transversal. We show that every split graph has a maximum minimal clique transversal that is independent (and hence, a strong independent set). In we introduce the relevant graph theoretic background. Hardness results are presented in . Linear-time algorithms for UCT in the classes of split graphs and proper interval graphs are developed in , respectively. We conclude the paper in . # Preliminaries {#sec:prelim} Throughout the paper, graphs are assumed to be finite, simple, and undirected. We use standard graph theory terminology, following West [@MR1367739]. A graph $G$ with vertex set $V$ and edge set $E$ is often denoted by $G=(V, E)$; we write $V(G)$ and $E(G)$ for $V$ and $E$, respectively. The set of vertices adjacent to a vertex $v\in V$ is the *neighborhood* of $v$, denoted $N(v)$; its cardinality is the *degree* of $v$, denoted $\deg(v)$. The *closed neighborhood* is the set $N[v]$, defined as $N(v)\cup \{v\}$. An *independent set* in a graph is a set of pairwise non-adjacent vertices; a *clique* is a set of pairwise adjacent vertices. An independent set (resp., clique) in a graph $G$ is *maximal* if it is not contained in any other independent set (resp., clique). A *clique transversal* in a graph is a subset of vertices that intersects all the maximal cliques of the graph. A *dominating set* in a graph $G = (V,E)$ is a set $S$ of vertices such that every vertex not in $S$ has a neighbor in $S$. An *independent dominating set* is a dominating set that is also an independent set. The *(independent) domination number* of a graph $G$ is the minimum size of an (independent) dominating set in $G$. Note that a set $S$ of vertices in a graph $G$ is an independent dominating set if and only if $S$ is a maximal independent set. In particular, the independent domination number of a graph is a well-defined invariant leading to a decision problem called Independent Dominating Set. The *clique number* of $G$ is denoted by $\omega(G)$ and defined as the maximum size of a clique in $G$. An *upper clique transversal* of a graph $G$ is a minimal clique transversal of maximum size. The *upper clique transversal number* of a graph $G$ is denoted by $\tau_c^+(G)$ and defined as the maximum size of a minimal clique transversal in $G$. A *vertex cover* in $G$ is a set $S\subseteq V(G)$ such that every edge $e\in E(G)$ has at least one endpoint in $S$. A vertex cover in $G$ is *minimal* if it does not contain any other vertex cover. These notions are illustrated in . Note that if $G$ is a triangle-free graph without isolated vertices, then the maximal cliques of $G$ are exactly its edges, and hence the clique transversals of $G$ coincide with its vertex covers. {#fig:uct width=".90\\textwidth"} # Intractability of UCT for some graph classes {#sec:hardness} In this section we prove that Upper Clique Transversal is NP-complete in the classes of chordal graphs, chordal bipartite graphs, and line graphs of bipartite graphs. First, let us note that for the class of all graphs, we do not know whether the problem is in NP. If $S$ is a minimal clique transversal in $G$ such that $|S|\ge k$, then a natural way to verify this fact would be to certify separately that $S$ is a clique transversal and that it is a minimal one. Assuming that $S$ is a clique transversal, one can certify minimality simply by exhibiting for each vertex $u\in S$ a maximal clique $C$ in $G$ such that $C\cap S = \{u\}$. However, unless P = NP, we cannot verify the fact that $S$ is a clique transversal in polynomial time. This follows from a result of Zang [@MR1344757], showing that it is co-NP-complete to check, given a weakly chordal graph $G$ and an independent set $S$, whether $S$ is a clique transversal in $G$. A graph $G$ is *weakly chordal* if neither $G$ nor its complement contain an induced cycle of length at least five. We do not know whether Upper Clique Transversal is in NP when restricted to the class of weakly chordal graphs. However, for their subclasses chordal graphs and chordal bipartite graphs, membership of UCT in NP is a consequence of the following proposition. **Proposition 1**. *Let $\mathcal{G}$ be a graph class such that every graph $G\in \mathcal{G}$ has at most polynomially many maximal cliques. Then, [Upper Clique Transversal]{.smallcaps} is in NP for graphs in $\mathcal{G}$.* *Proof.* Given a graph $G\in \mathcal{G}$ and an integer $k$, a polynomially verifiable certificate of the existence of a minimal clique transversal $S$ in $G$ such that $|S|\ge k$ is any such set $S$. Indeed, in this case we can enumerate all maximal cliques of $G$ in polynomial time by using any of the output-polynomial algorithms for this task (e.g., [@MR2159537]). In particular, we can verify that $S$ is a clique transversal of $G$ in polynomial time. We can also verify minimality in polynomial time, by determining whether for each vertex $u\in S$ there exists a maximal clique $C$ in $G$ such that $C\cap S = \{u\}$. ◻ A *star* is a graph that has a vertex that is adjacent to all other vertices, and there are no other edges. A *spanning star forest* in a graph $G = (V,E)$ is a spanning subgraph $(V,F)$ consisting of vertex-disjoint stars. Some of our hardness results will make use of a reduction from Spanning Star Forest, the problem that takes as input a graph $G$ and an integer $\ell$, and the task is to determine whether $G$ contains a spanning star forest $(V,F)$ such that $|F|\ge \ell$. **Theorem 2**. *[Spanning Star Forest]{.smallcaps} is NP-complete in the class of bipartite graphs with minimum degree at least $2$.* *Proof.* Membership in NP is clear. [Spanning Star Forest]{.smallcaps} is NP-complete due to its close relationship with Dominating Set, the problem that takes as input a graph $G$ and an integer $k$, and the task is to determine whether $G$ contains a dominating set $S$ such that $|S|\le k$. The connection between the spanning star forests and dominating sets is as follows: a graph $G$ has a spanning star forest with at least $\ell$ edges if and only if $G$ has a dominating set with at most $|V|-\ell$ vertices (see [@MR1887943; @MR2421073]). Dominating Set is known to be NP-complete in the class of bipartite graphs (see, e.g., [@MR761623]) and even in the class of chordal bipartite graphs, as shown by Müller and Brandstädt [@MR918093]. The graphs constructed in the NP-hardness reduction from [@MR918093] do not contain any vertices of degree zero or one, hence the claimed result follows. ◻ We present the hardness results in increasing order of difficulty of the proofs, starting with the class of chordal bipartite graphs. A *chordal bipartite* graph is a bipartite graph in which all induced cycles are of length four. **Theorem 3**. *[Upper Clique Transversal]{.smallcaps} is NP-complete in the class of chordal bipartite graphs.* *Proof.* Proposition [Proposition 1](#prop:poly-many-maximal-cliques){reference-type="ref" reference="prop:poly-many-maximal-cliques"} implies that UCT is in NP when restricted to any class of bipartite graphs. To prove NP-hardness, we make a reduction from Independent Dominating Set in chordal bipartite graphs, the problem that takes as input a chordal bipartite graph $G$ and an integer $k$, and the task is to determine whether $G$ contains a maximal independent set $I$ such that $|I|\le \ell$. As proved in [@MR1081460], this problem is NP-complete. We may assume without loss of generality that the input graph does not have any isolated vertices. Then, given a set $I\subseteq V(G)$, the following statements are equivalent: (i) $I$ is a (maximal) independent set in $G$, (ii) $V(G)\setminus I$ is a (minimal) vertex cover in $G$, and (iii) $V(G)\setminus I$ is a (minimal) clique transversal in $G$. Statements (i) and (ii) are equivalent for any graph, while the equivalence between statements (ii) and (iii) follows from the fact that the maximal cliques in $G$ are precisely its edges. It follows that $G$ has a maximal independent set $I$ such that $|I|\le \ell$ if and only if $G$ has a minimal clique transversal $S$ such that $|S|\ge k$ where $k = |V(G)|-\ell$. This completes the proof. ◻ We next consider the class of line graphs of bipartite graphs. The *line graph* of a graph $G$ is the graph $H$ with $V(H) = E(G)$ in which two distinct vertices are adjacent if and only if they share an endpoint as edges in $G$. **Lemma 4**. *Let $G$ be a triangle-free graph with minimum degree at least $2$ and let $H$ be the line graph of $G$. Then, the maximal cliques in $H$ are exactly the sets $E_v$ for $v\in V(G)$, where $E_v$ is the set of edges in $G$ that are incident with $v$.* *Proof.* Since $G$ is triangle-free, any clique in $H$ corresponds to a set of edges in $G$ having a common endpoint. Furthermore, since $G$ is of minimum degree at least $2$, any two sets $E_u$ and $E_v$ for $u\neq v$ are incomparable with respect to inclusion. ◻ An *edge cover* of a graph $G$ is a set $F$ of edges such that every vertex of $G$ is incident with some edge of $F$. **Lemma 5**. *Let $G$ be a triangle-free graph with minimum degree at least $2$ and let $H$ be the line graph of $G$. Then, a set $F\subseteq E(G)$ is a clique transversal in $H$ if and only if $F$ is an edge cover in $G$. Consequently, a set $F\subseteq E(G)$ is a minimal clique transversal in $H$ if and only if $F$ is a minimal edge cover in $G$.* *Proof.* Immediate from the definitions and . ◻ Using and , we can now prove the following. **Theorem 6**. *[Upper Clique Transversal]{.smallcaps} is NP-complete in the class of line graphs of bipartite graphs.* *Proof.* To argue that the problem is in NP, we show that every line graph of a bipartite graph has at most polynomially many maximal cliques. Let $G$ be a line graph of a bipartite graph. Fix a bipartite graph $H$ such that $G = L(H)$. Clearly, we may assume that $H$ has no isolated vertices. Since $H$ is triangle-free, any clique in $G$ corresponds to a set of edges in $H$ having a common endpoint, and consequently any maximal clique in $G$ corresponds to an inclusion-maximal set of edges in $H$ having a common endpoint. The number of such sets is bounded by the number of vertices in $H$. Since $$|V(H)| = \sum_{v\in V(H)} 1 \le \sum_{v\in V(H)}\deg(v) = 2|E(H)| = 2|V(G)|\,,$$ it follows that the number of maximal cliques in $G$ is at most $2|V(G)|$. By , the problem is in NP. To prove NP-hardness, we make a reduction from Spanning Star Forest in the class of bipartite graphs with minimum degree at least $2$. By , this problem is NP-complete. Let $H$ be the line graph of $G$. By , a set $F\subseteq E(G)$ is a minimal clique transversal in $H$ if and only if $F$ is a minimal edge cover in $G$. Therefore, the graph $G$ contains a minimal edge cover with at least $\ell$ edges if and only if its line graph, $H$, contains a minimal clique transversal with at least $\ell$ vertices. As observed by Hedetniemi [@hedetniemi1983max], the maximum size of a minimal edge cover equals the maximum number of edges in a spanning star forest (in fact, a set of edges in a graph without isolated vertices is a minimal edge cover if and only if it is a spanning star forest, see Manlove [@MR1670163]). Therefore, the graph $G$ contains a minimal edge cover with at least $\ell$ edges if and only if $G$ contains a spanning star forest with at least $\ell$ edges. The claimed NP-hardness result follows from . ◻ We now prove intractability of UCT in the class of chordal graphs. A graph is *chordal* if it does not contain any induced cycles on at least four vertices. We first recall a known result on maximal cliques in chordal graphs. **Theorem 7** (Berry and Pogorelcnik [@MR2816655]). *A chordal graph $G = (V,E)$ has at most $|V|$ maximal cliques, which can be computed in time $\mathcal{O}(|V|+|E|)$.* **Theorem 8**. *[Upper Clique Transversal]{.smallcaps} is NP-complete in the class of chordal graphs.* *Proof.* Membership in NP follows from and . To prove NP-hardness, we reduce from Spanning Star Forest. Let $G = (V,E)$ and $\ell$ be an input instance of Spanning Star Forest. We may assume without loss of generality that $G$ has an edge and that $\ell\ge 2$, since if any of these assumptions is violated, then it is trivial to verify if $G$ has a spanning star forest with at least $\ell$ edges. We construct a chordal graph $G'$ as follows. We start with a complete graph with vertex set $V$. For each edge $e=\{u,v\}\in E$, we introduce two new vertices $x^e$ and $y^e$, and make $x^e$ adjacent to $u$, to $v$, and to $y^e$. The obtained graph is $G'$. We thus have $V(G') = V\cup X\cup Y$, where $X = \{x^e:e\in E\}$ and $Y = \{y^e:e\in E\}$. See Fig. [3](#fig:reduced-1){reference-type="ref" reference="fig:reduced-1"} for an example. Clearly, $G'$ is chordal. Furthermore, let $k = \ell+|E|$. {#fig:reduced-1 width="75%"} To complete the proof, we show that $G$ has a spanning star forest of size at least $\ell$ if and only if $G'$ has a minimal clique transversal of size at least $k$. First, assume that $G$ has a spanning star forest $(V,F)$ such that $|F|\ge \ell$. Since $(V,F)$ is a spanning forest in which each component is a star, each edge of $F$ is incident with a vertex of degree one in $(V,F)$. Let $S$ be a set obtained by selecting from each edge in $F$ one vertex of degree one in $(V,F)$. Then every edge of $F$ has one endpoint in $S$ and the other one in $V\setminus S$. In particular, $|S| = |F|\ge \ell$. Let $S' = S\cup\{x^e:e\in E\setminus F\}\cup \{y^f:f\in F\}$. See Fig. [4](#fig:reduced-2){reference-type="ref" reference="fig:reduced-2"} for an example. {#fig:reduced-2 width="75%"} Clearly, the size of $S'$ is at least $\ell+|E| = k$. We claim that $S'$ is a minimal clique transversal of $G'$. There are three kinds of maximal cliques in $G'$: the set $V$, sets of the form $\{u,v,x^e\}$ for all $e = \{u,v\}\in E$, and sets of the form $\{x^e,y^e\}$ for all $e\in E$. Since $|S| = |F|\ge \ell\ge 2$, the set $S$ is non-empty, and thus the set $S'$ intersects $V$. Furthermore, since $S$ contains one endpoint of each edge in $F$, set $S'$ intersects all cliques of the form $\{u,v,x^f\}$ for all $f = \{u,v\}\in F$. For all $e = \{u,v\}\in E\setminus F$, set $S'$ contains vertex $x^e$ and thus also intersects the clique $\{u,v,x^e\}$, as well as the clique $\{x^e,y^e\}$. Finally, for each $f\in F$, we have that $y^f\in S'$ and hence $S'$ intersects $\{x^f,y^f\}$. Thus, $S'$ is a clique transversal of $G'$. To argue minimality, we need to show that for every $u\in S'$ there exists a maximal clique in $G'$ missed by $S'\setminus \{u\}$. Suppose first that $u\in V$. Then $u\in S$ and there is an edge $f\in F$ such that $u$ is an endpoint of $f$. Let $v$ be the other endpoint of $f$. Then $v\not\in S$ and thus also $v\not\in S'$. Note also that $x^f\not\in S'$. In particular, this implies that the set $\{u,v,x^f\}$ is a maximal clique of $G'$ missed by $S'\setminus \{u\}$. Next, suppose that $u\in X$. Then $u = x^e$ for some edge $e\in E\setminus F$ and $y^e\not\in S'$, hence the set $\{x^e,y^e\}$ is a maximal clique of $G'$ missed by $S'\setminus \{u\}$. Finally, suppose that $u\in Y$. Then $u = x^f$ for some edge $f\in F$. Then $x^f\not\in S'$, therefore the set $\{x^f,y^f\}$ is a maximal clique of $G'$ missed by $S'\setminus \{u\}$. This shows that $S'$ is a minimal clique transversal of $G'$, as claimed. For the converse direction, let $S'$ be a minimal clique transversal of $G'$ such that $|S'|\ge k$. First we show that $S'\cap Y\neq\emptyset$. Suppose for a contradiction that $S'\cap Y= \emptyset$. Then $X\subseteq S'$, since otherwise the maximal clique $\{x^e,y^e\}$ of $G'$ would be missed by $S'$ for every $x^e\in X\setminus S'$. Furthermore, since $V$ is a maximal clique in $G'$, there is a vertex $u\in S'$ such that $u\in V$. Since the set $X\cup\{u\}$ is a clique transversal in $G'$, the minimality of $S'$ implies that $S' = X\cup\{u\}$. Using the fact that $k = \ell+|E|$ and $k\le |S'| = |X|+1 = |E|+1$, we then obtain that $\ell\le 1$. This contradicts our assumption that $\ell\ge 2$ and shows that $S'\cap Y\neq\emptyset$. Let $S = S'\cap V$. Recall that for every edge $e\in E$ we denote by $x^e$ the unique vertex in $X$ that is adjacent in $G'$ to both endpoints of $e$. We claim that for each vertex $u\in S$ there exists a vertex $v\in V$ such that $e = \{u,v\}\in E$ and $S'\cap \{u,v,x^e\} = \{u\}$. Let $u\in S$ and suppose for a contradiction that for all vertices $v$ such that $e = \{u,v\}\in E$ we have $S'\cap \{u,v,x^e\} \neq \{u\}$. This implies that the set $S'\setminus\{u\}$ intersects all maximal cliques in $G'$ of the form $\{u,v,x^e\}$ for some $e = \{u,v\}\in E$. Since $S'$ is a minimal clique transversal of $G'$, we infer that the maximal clique of $G'$ missed by $S'\setminus\{u\}$ is $V$. In particular, we have $S = S'\cap V = \{u\}$, which in turn implies that for all vertices $v\in V$ such that $e = \{u,v\}\in E$ we have $S'\cap \{u,v,x^e\} = \{u,x^e\}$. Since $S'\cap Y\neq\emptyset$, there exists an edge $e=\{w,z\}$ of $G$ such that $y^e\in S'$. Then $x^e\not\in S'$, and therefore $u$ is not an endpoint of $e$. However, since, $x^e\not\in S'$ but $S'$ intersects the maximal clique $\{w,z,x^e\}$, it follows that an endpoint of $e$ belongs to $S$. This contradicts the fact that $S = \{u\}$ and $u$ is not an endpoint of $e$. By the above claim, we can associate to each vertex $u\in S$ a vertex $v(u)\in V$ such that $e = \{u,v(u)\}\in E$ and $S'\cap \{u,v(u),x^e\} = \{u\}$. For each $u\in S$, let us denote by $e(u)$ the corresponding edge $\{u,v(u)\}$, and let $F = \{e(u):u\in S\}$ (see Fig. [5](#fig:reduced-3){reference-type="ref" reference="fig:reduced-3"}). We next claim that the mapping $u\mapsto e(u)$ is one-to-one, that is, for all $u_1,u_2\in S$, if $e(u_1) = e(u_2)$ then $u_1 = u_2$. Suppose that $e(u_1) = e(u_2)$ for some $u_1\neq u_2$. Then $e(u_1) = e(u_2) = \{u_1,u_2\}$, $v(u_1) = u_2$, and $v(u_2) = u_1$. Furthermore, $\{u_1\} = S'\cap \{u_1,v(u_1),x^{e(u_1)}\} = S'\cap \{u_2,v(u_2),x^{e(u_2)}\} = \{u_2\}$, which is in contradiction with $u_1\neq u_2$. Since the mapping $u\mapsto e(u)$ is one-to-one, we have $|F| = |S|$. Furthermore, every vertex in $S$ has degree one in $(V,F)$. Therefore, the graph $(V,F)$ is a spanning star forest of $G$. {#fig:reduced-3 width="83%"} Since $S'$ is a minimal clique transversal of $G'$, for each edge $e\in E$ exactly one of $x^e$ and $y^e$ belongs to $S'$. Therefore, $|F| = |S| = |S'|-|E|\ge k-|E| = \ell$. Thus, $G$ has a spanning star forest of size at least $\ell$. ◻ # A linear-time algorithm for UCT in split graphs {#sec:split-graphs} A *split graph* is a graph that has a *split partition*, that is, a partition of its vertex set into a clique and an independent set. We denote a split partition of a split graph $G$ as $(K,I)$ where $K$ is a clique, $I$ is an independent set, $K\cap I = \emptyset$, and $K\cup I = V(G)$. We may assume without loss of generality that $I$ is a maximal independent set. Indeed, if this is not the case, then $K$ contains a vertex $v$ that has no neighbors in $I$, and $(K\setminus \{v\},I\cup \{v\})$ is a split partition of $G$ such that $I\cup\{v\}$ is a maximal independent set. In what follows, we repeatedly use the structure of maximal cliques of split graphs. If $G$ is a split graph with a split partition $(K,I)$, then the maximal cliques of $G$ are as follows: the closed neighborhoods $N[v]$, for all $v\in I$, and the clique $K$, provided that it is a maximal clique, that is, every vertex in $I$ has a non-neighbor in $K$. Given a graph $G$ and a set of vertices $S\subseteq V(G)$, we denote by $N(S)$ the set of all vertices in $V(G)\setminus S$ that have a neighbor in $S$. Moreover, given a vertex $v\in S$, an *$S$-private neighbor* of $v$ is any vertex $w\in N(S)$ such that $N(w)\cap S = \{v\}$. The following proposition characterizes minimal clique transversals of split graphs. **Proposition 9**. *Let $G$ be a split graph with a split partition $(K,I)$ such that $I$ is a maximal independent set and let $S\subseteq V(G)$. Let $K'= K\cap S$ and $I'= I\cap S$. Then, $S$ is a minimal clique transversal of $G$ if and only if the following conditions hold:* (i) *[\[condition:domination-2\]]{#condition:domination-2 label="condition:domination-2"} $K'\neq \emptyset$ if $K$ is a maximal clique.* (ii) *[\[condition:domination-and-minimality\]]{#condition:domination-and-minimality label="condition:domination-and-minimality"} $I'= I\setminus N(K')$.* (iii) *[\[condition:minimality-2\]]{#condition:minimality-2 label="condition:minimality-2"} Every vertex in $K'$ has a $K'$-private neighbor in $I$.* *Proof.* Assume first that $S$ is a minimal clique transversal of $G$. We prove that $S$ satisfies each of the three conditions. Condition [\[condition:domination-2\]](#condition:domination-2){reference-type="eqref" reference="condition:domination-2"} follows from the fact that $S$ is a clique transversal. To show condition [\[condition:domination-and-minimality\]](#condition:domination-and-minimality){reference-type="eqref" reference="condition:domination-and-minimality"}, we first show the inclusion $I\setminus S\subseteq N(K')$, which is equivalent to $I\setminus N(K') \subseteq I\setminus (I\setminus S) = I\cap S = I'$. Consider an arbitrary vertex $v\in I\setminus S$. Since $N[v]$ is a maximal clique in $G$ and $S$ is a clique transversal not containing $v$, set $S$ must contain a neighbor $w$ of $v$. As $N(v)\subseteq K$, we conclude that $w$ belongs to $K'$. The converse inclusion, $I'\subseteq I\setminus N(K')$, is equivalent to the condition that there are no edges between $I'$ and $K'$. Suppose for a contradiction that $G$ contains an edge $uv$ with $u\in I'$ and $v\in K'$. Since $N[u]$ is the only maximal clique of $G$ containing $u$ and $\{u,v\}\subseteq S\cap N[u]$, it follows that $S\setminus \{u\}$ is a clique transversal of $G$, contradicting the minimality of $S$. This establishes [\[condition:domination-and-minimality\]](#condition:domination-and-minimality){reference-type="eqref" reference="condition:domination-and-minimality"}. To show condition [\[condition:minimality-2\]](#condition:minimality-2){reference-type="eqref" reference="condition:minimality-2"}, consider an arbitrary vertex $v\in K'$. If $K'= \{v\}$, then any neighbor of $v$ in $I$ is a $K'$-private neighbor of $v$, and $v$ has a neighbor in $I$ since $I$ is a maximal independent set. Thus we may assume that $|K'|\ge 2$. Suppose for a contradiction that $v$ does not contain any $K'$-private neighbor in $I$. The maximal cliques of $G$ containing $v$ are $N[w]$ for $w\in N(v)\cap I$ and possibly $K$ (if $K$ is a maximal clique). For every $w\in N(v)\cap I$, the assumption on $v$ implies that there exists a vertex $v'\in K'\setminus\{v\}$ adjacent to $w$; hence $\{v,v'\}\subseteq S\cap N[w]$. Moreover, we had already justified that $|K'|\ge 2$. It follows that the set $S\setminus \{v\}$ intersects all maximal cliques in $G$; this contradicts the minimality of $S$ and shows condition [\[condition:minimality-2\]](#condition:minimality-2){reference-type="eqref" reference="condition:minimality-2"}. Assume now that $S$ is a set of vertices satisfying conditions [\[condition:domination-2\]](#condition:domination-2){reference-type="eqref" reference="condition:domination-2"}--[\[condition:minimality-2\]](#condition:minimality-2){reference-type="eqref" reference="condition:minimality-2"}. We prove that $S$ is a minimal clique transversal by verifying both conditions in the definition. Consider an arbitrary maximal clique $C$ of $G$. If $C = N[v]$ for some $v\in I$, then either $v\in S$, in which case $v\in S\cap C$, or $v\in I\setminus S$, in which case condition [\[condition:domination-and-minimality\]](#condition:domination-and-minimality){reference-type="eqref" reference="condition:domination-and-minimality"} guarantees that $v$ has a neighbor $w\in K'$; hence $w\in S\cap C$ and $S$ intersects $C$. If $C = K$, then $S\cap C \neq \emptyset$ by condition [\[condition:domination-2\]](#condition:domination-2){reference-type="eqref" reference="condition:domination-2"}. Hence $S$ is a clique transversal. To show minimality, suppose for a contradiction that $S$ contains a vertex $v$ such that $S\setminus\{v\}$ is also a clique transversal of $G$. Suppose that $v\in I$. Since the set $S\setminus\{v\}$ intersects the maximal clique $N[v]$, there is a vertex $w\in (S\setminus\{v\})\cap N[v]$. Since $w\neq v$, we have $w\in N(v)$ and hence $w\in K$. In particular, $w\in K'$ and thus $v\in N(K')\cap I'$; this contradicts condition [\[condition:domination-and-minimality\]](#condition:domination-and-minimality){reference-type="eqref" reference="condition:domination-and-minimality"}. It follows that $v\not\in I$ and hence $v\in K'$. Condition [\[condition:minimality-2\]](#condition:minimality-2){reference-type="eqref" reference="condition:minimality-2"} implies that $v$ has a $K'$-private neighbor $w\in I$. Since $N(w)\subseteq K$ and $w$ is a $K'$-private neighbor of $v$, we have $S \cap N(w) = N(w)\cap S = N(w)\cap K'= \{v\}$, which implies $(S\setminus \{v\})\cap N(w) = \emptyset$. Moreover, condition [\[condition:domination-and-minimality\]](#condition:domination-and-minimality){reference-type="eqref" reference="condition:domination-and-minimality"} implies that $w\not\in S$; hence $(S\setminus \{v\})\cap \{w\} = \emptyset$. It follows that $(S\setminus \{v\})\cap N[w] = ((S\setminus \{v\})\cap N(w))\cup ((S\setminus \{v\})\cap \{w\}) = \emptyset$. Since the set $S\setminus\{v\}$ misses the maximal clique $N[w]$, it is not a clique transversal, a contradiction. ◻ leads to the following result about maximum minimal clique transversals in split graphs. We denote by $\alpha(G)$ the *independence number* of a graph $G$, that is, the maximum size of an independent set in $G$. **Theorem 10**. *Let $G$ be a split graph with a split partition $(K,I)$ such that $I$ is a maximal independent set. Then:* 1. *If $K$ is not a maximal clique in $G$, then $I$ is a maximum minimal clique transversal in $G$; in particular, we have $\tau_c^+(G) = \alpha(G)$ in this case.* 2. *If $K$ is a maximal clique in $G$, then for every vertex $v\in K$ with the smallest number of neighbors in $I$, the set $\{v\}\cup (I\setminus N(v))$ is a maximum minimal clique transversal in $G$; in particular, we have $\tau_c^+(G) = \alpha(G)-\delta_G(I,K)+1$ in this case, where $\delta_G(I,K) = \min\{|N(v)\cap I|: v\in K\}$.* *Consequently, every split graph $G$ satisfies $\tau_c^+(G)\le \alpha(G)$.* *Proof.* Let $S$ be a minimal clique transversal of $G$ that is of maximum possible size and, subject to this condition, contains as few vertices from $K$ as possible. Let $K'= K\cap S$ and $I'= I\cap S$. If $K'= \emptyset$, then $K$ is not a maximal clique in $G$, and we have $S = I$, implying $\tau_c^+(G) = |S| = \alpha(G)$. Suppose now that $K'\neq\emptyset$. We first show that $|K'|= 1$. Suppose for a contradiction that $|K'|\ge 2$ and let $v\in K'$. Let $I_v$ denote the set of $K'$-private neighbors of $v$ in $I$ and let $S' = (S\setminus \{v\})\cup I_v$. Let $I_v$ denote the set of $K'$-private neighbors of $v$ in $I$ and let $S' = (S\setminus \{v\})\cup I_v$. By , conditions [\[condition:domination-2\]](#condition:domination-2){reference-type="eqref" reference="condition:domination-2"}--[\[condition:minimality-2\]](#condition:minimality-2){reference-type="eqref" reference="condition:minimality-2"} hold for $S$. We claim that set $S'$ also satisfies conditions [\[condition:domination-2\]](#condition:domination-2){reference-type="eqref" reference="condition:domination-2"}--[\[condition:minimality-2\]](#condition:minimality-2){reference-type="eqref" reference="condition:minimality-2"} from . Since $S'\cap K = K'\setminus\{v\}$, the assumption $|K'|\ge 2$ implies that $S'\cap K\neq\emptyset$, thus condition [\[condition:domination-2\]](#condition:domination-2){reference-type="eqref" reference="condition:domination-2"} holds for $S'$. Since condition [\[condition:domination-and-minimality\]](#condition:domination-and-minimality){reference-type="eqref" reference="condition:domination-and-minimality"} holds for $S$, we have $$I\cap S' = I'\cup I_v = (I\setminus N(K'))\cup I_v = I\setminus N(K'\setminus\{v\}) = I\setminus N(S'\cap K)\,,$$ that is, condition [\[condition:domination-and-minimality\]](#condition:domination-and-minimality){reference-type="eqref" reference="condition:domination-and-minimality"} holds for $S'$. Finally, since $S'\cap K \subseteq K'$, condition [\[condition:minimality-2\]](#condition:minimality-2){reference-type="eqref" reference="condition:minimality-2"} for $S$ immediately implies condition [\[condition:minimality-2\]](#condition:minimality-2){reference-type="eqref" reference="condition:minimality-2"} for $S'$. It follows that $S'$ is a minimal clique transversal in $G$. Furthermore, since $v\in K'$, vertex $v$ has an $K'$-private neighbor in $I$, that is, the set $I_v$ is nonempty. This implies that $|S'|\ge |S|$; in particular, $S'$ is a maximum minimal clique transversal in $G$. However, $S'$ contains strictly fewer vertices from $K$ than $S$, contradicting the choice of $S$. This shows that $|K'|= 1$, as claimed. Let $w$ be the unique vertex in $K'$. Since Condition [\[condition:domination-and-minimality\]](#condition:domination-and-minimality){reference-type="eqref" reference="condition:domination-and-minimality"} from holds for $S$, we have $I'= I \setminus N(w)$. Hence $S = \{w\}\cup (I \setminus N(w))$ and $|S| = 1+ |I|-|N(w)\cap I|$. Since $w\in K$, we have $|N(w)\cap I|\ge \delta_G(I,K)$ and hence $\tau_c^+(G)= |S| \le \alpha(G)-\delta_G(I,K)+1$. On the other hand, for every vertex $z\in K$ the set $X_z:=\{z\}\cup (I \setminus N(z))$ satisfies conditions [\[condition:domination-2\]](#condition:domination-2){reference-type="eqref" reference="condition:domination-2"}--[\[condition:minimality-2\]](#condition:minimality-2){reference-type="eqref" reference="condition:minimality-2"} from . Conditions [\[condition:domination-2\]](#condition:domination-2){reference-type="eqref" reference="condition:domination-2"} and [\[condition:domination-and-minimality\]](#condition:domination-and-minimality){reference-type="eqref" reference="condition:domination-and-minimality"} hold by the definition of $X_z$. Since $I$ is a maximal independent set in $G$, vertex $z\not\in I$ has a neighbor in $I$, and any neighbor of $z$ in $I$ is trivially an $(X_z\cap K)$-private neighbor of $z$. Thus Condition [\[condition:minimality-2\]](#condition:minimality-2){reference-type="eqref" reference="condition:minimality-2"} holds, too. It follows that $X_z$ is a minimal clique transversal in $G$. Choosing $z$ to be a vertex in $K$ with the smallest number of neighbors in $I$, we obtain a set $X_z$ of size $\alpha(G)-\delta_G(I,K)+1$. Thus $\tau_c^+(G)\ge |X_z| = \alpha(G)-\delta_G(I,K)+1$ and since we already proved that $\tau_c^+(G)\le \alpha(G)-\delta_G(I,K)+1$, any such $X_z$ is optimal. Since $I$ is a maximal independent set and $K$ is nonempty, we have $\delta_G(I,K)\ge 1$. Thus, $\tau_c^+(G) \le \alpha(G)$. Suppose that $K$ is not a maximal clique in $G$. Then $I$ is a minimal clique transversal in $G$ and therefore $\tau_c^+(G)\ge |I| = \alpha(G)\ge \tau_c^+(G)$. Hence equalities must hold throughout and $I$ is a maximum minimal clique transversal. Finally, suppose that $K$ is a maximal clique in $G$. Then every minimal clique transversal $S$ in $G$ satisfies $S\cap K\neq\emptyset$. In this case, the above analysis shows that for every vertex $v\in K$ with the smallest number of neighbors in $I$, the set $\{v\}\cup (I\setminus N(v))$ is a maximum minimal clique transversal in $G$. ◻ **Corollary 11**. *Upper Clique Transversal can be solved in linear time in the class of split graphs.* *Proof.* Let $G = (V,E)$ be a given split graph. Hammer and Simeone showed that split graphs can be characterized by their degree sequences; furthermore, that characterization yields a linear-time algorithm to compute a split partition $(K,I)$ of $G$ (see [@MR637832]). If there exists a vertex in $K$ that is not adjacent to $I$, then we move it to $I$. Thus, in linear time we can compute a split partition $(K,I)$ of $G$ such that $I$ is a maximal independent set. Clearly, $K$ is a maximal clique if and only if no vertex in $I$ is adjacent to all vertices of $G$. If $K$ is not a maximal clique, then the algorithm simply returns $I$. If $K$ is a maximal clique, then the algorithm first computes, for each vertex $v\in K$, the number of neighbors of $v$ in $I$. For a vertex $v\in K$ with the smallest number of neighbors in $I$, the set $\{v\}\cup (I\setminus N(v))$ is returned. ◻ **Remark 12**. Recall that a *strong independent set* in a graph $G$ is an independent clique transversal. If $I$ is a strong independent set in $G$, then for every vertex $v\in I$, every maximal clique $K$ containing $v$ satisfies $K\cap I = \{v\}$; it follows that every strong independent set is a minimal clique transversal. implies that every split graph has a maximum minimal clique transversal that is independent, that is, it is a strong independent set. Consequently, the problem of computing a maximum minimal clique transversal of a split graph $G$ reduces to the problem of computing a maximum strong independent set in $G$. A linear-time algorithm for a more general problem, that of computing a maximum weight strong independent set in a vertex-weighted chordal graph, was developed by Wu [@MR1325490]. This gives an alternative proof of . # A linear-time algorithm for UCT in proper interval graphs {#sec:PIGs} A graph $G = (V,E)$ is an *interval graph* if it has an *interval representation*, that is, if its vertices can be put in a one-to-one correspondence with a family $(I_v:v\in V)$ of closed intervals on the real line such that two distinct vertices $u$ and $v$ are adjacent if and only if the corresponding intervals $I_u$ and $I_v$ intersect. If $G$ has a *proper interval representation*, that is, an interval representation in which no interval contains another, then $G$ is said to be a *proper interval graph*. Our approach towards a linear-time algorithm for Upper Clique Transversal in the class of proper interval graphs is based on a relation between clique transversals in $G$ and induced matchings in the so-called vertex-clique incidence graph of $G$. This relation is valid for arbitrary graphs. ## UCT via induced matchings in the vertex-clique incidence graph Given a graph $G=(V,E)$, we denote by $B_G$ the *vertex-clique incidence graph* of $G$, a bipartite graph defined as follows. The vertex set of $B_G$ consists of two disjoint sets $X$ and $Y$ such that $X = V$ and $Y = {\mathcal C}_G$, where ${\mathcal C}_G$ is the set of maximal cliques in $G$. The edge set of $B_G$ consists of all pairs $x\in X$ and $C\in {\mathcal C}_G$ that satisfy $x\in C$. An *induced matching* in a graph $G$ is a set $M$ of pairwise disjoint edges such that the set of endpoints of edges in $M$ induces no edges other than those in $M$. Given two disjoint sets of vertices $A$ and $B$ in a graph $G$, we say that $A$ *dominates $B$ in $G$* if every vertex in $B$ has a neighbor in $A$. Given a matching $M$ in a graph $G$ and a vertex $v\in V(G)$, we say that $v$ is *$M\!$-saturated* if it is an endpoint of an edge in $M$. Clique transversals and minimal clique transversals of a graph $G$ can be expressed in terms of the vertex-clique incidence graph as follows. **Lemma 13**. *Let $G$ be a graph, let $B_G = (X,Y;E)$ be its vertex-clique incidence graph, and let $S\subseteq V(G)$. Then:* 1. *$S$ is a clique transversal in $G$ if and only if $S$ dominates $Y$ in $B_G$.* 2. *$S$ is a minimal clique transversal in $G$ if and only if $S$ dominates $Y$ in $B_G$ and there exists an induced matching $M$ in $B_G$ such that $S$ is exactly the set of $M\!$-saturated vertices in $X$.* *Proof.* The first statement follows immediately from the definitions. For the second statement, we prove each of the two implications separately. Assume first that $S$ is a minimal clique transversal in $G$. Since $S$ is a clique transversal in $G$, it dominates $Y$ in $B_G$. Furthermore, the minimality of $S$ implies that for every vertex $s\in S$ there exists a maximal clique $y_s\in Y (= \mathcal{C}_G)$ such that $y_s\cap S = \{s\}$. Let $M = \{\{s,y_s\}\mid s\in S\}$. We claim that $M$ is an induced matching $M$ in $B_G$ such that $S$ is exactly the set of $M\!$-saturated vertices in $X$. First, note that each $s\in S$ is adjacent in $B_G$ to $y_s$, since $s$ belongs to the maximal clique $y_s$. Second, $M$ is a matching in $B_G$ since every $s\in S$ is by construction incident with only one edge in $M$, and if $y_{s_1} = y_{s_2}$ for two vertices $s_1,s_2\in S$, then $\{s_1\} = y_{s_1}\cap S = y_{s_2}\cap S = \{s_2\}$ and thus $s_1 = s_2$. Third, $M$ is an induced matching in $B_G$, since otherwise $B_G$ would contain an edge of the form $\{s_1,y_{s_2}\}$ for two distinct vertices $s_1,s_2\in S$, which would imply that $s_1$ belongs to the maximal clique $y_{s_2}$, contradicting the fact that $y_{s_2}\cap S = \{s_2\}$. Finally, the fact that $S$ is exactly the set of $M\!$-saturated vertices in $X$ follows directly from the definition of $M$. For the converse direction, assume that $S$ dominates $Y$ in $B_G$ and there exists an induced matching $M$ in $B_G$ such that $S$ is exactly the set of $M\!$-saturated vertices in $X$. The fact that $S$ dominates $Y$ in $B_G$ implies that $S$ is a clique transversal in $G$. To see that $S$ is a minimal clique transversal, we will show that for every $s\in S$, the set $S\setminus \{s\}$ misses a maximal clique in $G$. Let $s\in S$. By the assumptions on $M$, vertex $s$ has a unique neighbor $y_s$ in $B_G$ such that $\{s,y_s\}$ is an edge of $M$. Furthermore, since $M$ is an induced matching in $B_G$, vertex $y_s$ is not adjacent in $B_G$ to any vertex in $S\setminus \{s\}$. Thus, the set $S\setminus \{s\}$ misses the maximal clique $y_s$. We conclude that $S$ is a minimal clique transversal. ◻ The *induced matching number* of a graph $G$ is the maximum size of an induced matching in $G$. immediately implies the following. **Corollary 14**. *For every graph $G$, the upper clique transversal number of $G$ is at most the induced matching number of $B_G$.* As another consequence of , we obtain a sufficient condition for a set of vertices in a graph to be a minimal clique transversal of maximum size. **Corollary 15**. *Let $G$ be a graph, let $B_G = (X,Y;E)$ be its vertex-clique incidence graph, and let $S\subseteq V(G)$. Suppose that $S$ dominates $Y$ in $B_G$ and there exists a maximum induced matching $M$ in $B_G$ such that $S$ is exactly the set of $M\!$-saturated vertices in $X$. Then, $S$ is a minimal clique transversal in $G$ of maximum size.* To apply to proper interval graphs, we first state several characterizations of proper interval graphs in terms of their vertex-clique incidence graphs, establishing in particular a connection with bipartite permutation graphs. ## Characterizing proper interval graphs via their vertex-clique incidence graphs We first recall some concepts and results from the literature. A bipartite graph $G = (X,Y;E)$ is said to be *biconvex* if there exists a *biconvex ordering* of (the vertex set of) $G$, that is, a pair $(<_X,<_Y)$ where $<_X$ is a linear ordering of $X$ and $<_Y$ is a linear ordering of $Y$ such that for every $x\in X$, the vertices in $Y$ adjacent to $x$ appear consecutively with respect to the ordering $<_Y$, and, similarly, for every $y\in Y$, the vertices in $X$ adjacent to $y$ appear consecutively with respect to the ordering $<_X$. We will need the following property of biconvex graphs. Let $(<_X,<_Y)$ be a biconvex ordering of a biconvex graph $G = (X,Y;E)$. Two edges $e$ and $f$ of $G$ are said to *cross* (each other) if there exist vertices $x_1,x_2\in X$ and $y_1,y_2\in Y$ such that $\{e,f\} = \{\{x_1,y_2\},\{x_2,y_1\}\}$, $x_1<_X x_2$, and $y_1<_Y y_2$. A biconvex ordering $(<_X,<_Y)$ of a biconvex graph $G = (X,Y;E)$ is said to be *induced-crossing-free* if for any two crossing edges $e = \{x_1,y_2\}$ and $f = \{x_2,y_1\}$, either $x_1$ is adjacent to $y_1$ or $x_2$ is adjacent to $y_2$. **Theorem 16** (Abbas and Stewart [@MR1762199]). *Every biconvex graph has an induced-crossing-free biconvex ordering.* Given a bipartite graph $G = (X,Y;E)$, a *strongly induced-crossing-free ordering* (or simply a *strong ordering*) of $G$ is a pair $(<_X,<_Y)$ of linear orderings of $X$ and $Y$ such that for any two crossing edges $e = \{x_1,y_2\}$ and $f = \{x_2,y_1\}$, vertex $x_1$ is adjacent to $y_1$ and vertex $x_2$ is adjacent to $y_2$. A *permutation graph* is a graph $G=(V,E)$ that admits a permutation model, that is, vertices of $G$ can be ordered $v_1,\ldots, v_n$ such that there exists a permutation $(a_1,\ldots, a_n)$ of the set $\{1,\ldots, n\}$ such that for all $1\le i< j\le n$, vertices $v_i$ and $v_j$ are adjacent in $G$ if and only if $a_i>a_j$. A *bipartite permutation graph* is a graph that is both a bipartite graph and a permutation graph. The following characterization of bipartite permutation graphs follows from Theorem 1 in [@MR917130] and its proof. **Theorem 17** (Spinrad, Brandstädt, and Stewart [@MR917130]). *The following statements are equivalent for a bipartite graph $G = (X,Y;E)$:* 1. *$G$ is a bipartite permutation graph.* 2. *$G$ has a strong ordering.* 3. *$G$ has a strong biconvex ordering.* implies the following property of bipartite permutation graphs equipped with a strong ordering. **Corollary 18**. *Let $G = (X,Y;E)$ be a bipartite permutation graph, let $(<_X,<_Y)$ be a strong ordering of $G$, and let $M$ be an induced matching in $G$. Then, no two edges in $M$ cross.* We will also use the following well-known characterization of proper interval graphs (see, e.g., Gardi [@MR2364171]). **Theorem 19**. *A graph $G$ is a proper interval graph if and only if there exists an ordering $\sigma= (v_1,\ldots, v_n)$ of the vertices of $G$ and an ordering $\tau = (C_1,\ldots, C_k)$ of the maximal cliques of $G$ such that for each $i\in \{1,\ldots, n\}$ the maximal cliques containing vertex $v_i$ appear consecutively in the ordering $\tau$, and for each $j\in \{1,\ldots, k\}$ clique $C_j$ consists of consecutive vertices with respect to ordering $\sigma$.* The following theorem gives several characterizations of proper interval graphs in terms of their vertex-clique incidence graphs. **Theorem 20**. *Let $G$ be a graph. Then, the following statements are equivalent:* 1. *[\[statement-1\]]{#statement-1 label="statement-1"} $G$ is a proper interval graph.* 2. *[\[statement-2\]]{#statement-2 label="statement-2"} $B_G$ is a biconvex graph.* 3. *[\[statement-4\]]{#statement-4 label="statement-4"} $B_G$ is a bipartite permutation graph.* 4. *[\[statement-5\]]{#statement-5 label="statement-5"} $B_G$ has a strong ordering.* 5. *[\[statement-6\]]{#statement-6 label="statement-6"} $B_G$ has a strong biconvex ordering.* 6. *[\[statement-3\]]{#statement-3 label="statement-3"} $B_G$ has an induced-crossing-free biconvex ordering.* *Proof.* Theorem [Theorem 19](#thm:proper-interval){reference-type="ref" reference="thm:proper-interval"} implies the equivalence between statements [\[statement-1\]](#statement-1){reference-type="ref" reference="statement-1"} and [\[statement-2\]](#statement-2){reference-type="ref" reference="statement-2"}. Equivalence between statements [\[statement-2\]](#statement-2){reference-type="ref" reference="statement-2"} and [\[statement-3\]](#statement-3){reference-type="ref" reference="statement-3"} follows from Theorem [Theorem 16](#thm:induced-crossing-free){reference-type="ref" reference="thm:induced-crossing-free"}. Equivalence among statements [\[statement-4\]](#statement-4){reference-type="ref" reference="statement-4"}, [\[statement-5\]](#statement-5){reference-type="ref" reference="statement-5"}, and [\[statement-6\]](#statement-6){reference-type="ref" reference="statement-6"} follows from Theorem [Theorem 17](#thm:strongly-induced-crossing-free){reference-type="ref" reference="thm:strongly-induced-crossing-free"}. Clearly, statement [\[statement-6\]](#statement-6){reference-type="ref" reference="statement-6"} implies statement [\[statement-2\]](#statement-2){reference-type="ref" reference="statement-2"}. Finally, we show that statement [\[statement-1\]](#statement-1){reference-type="ref" reference="statement-1"} implies statement [\[statement-5\]](#statement-5){reference-type="ref" reference="statement-5"}. Fix a proper interval representation $(I_v:v\in V(G))$ of $G$. Let $B_G = (X,Y;E)$ where $X = V(G)$ and $Y = \mathcal{C}_G$. Let $<_X$ be the ordering of $X$ corresponding to the left-endpoint order of the intervals. (Note that since no interval properly contains another, the left-endpoint order and the right-endpoint order are the same.) As shown in [@MR2364171], every maximal clique $C\in \mathcal{C}_G (= Y)$ consists of consecutive vertices with respect to $<_X$. Since the cliques are maximal, no two cliques in $Y$ have the same first vertex with respect to $<_X$, hence there is a unique and well defined ordering $<_Y$ of $Y$ that orders the cliques in increasing order of their first vertices in the vertex order. We claim that the pair $(<_X, <_Y)$ is a strong ordering of $B_G$. Consider any two crossing edges $e = \{x_1,y_2\}$ and $f = \{x_2,y_1\}$. We may assume that $x_1 <_X x_2$ and $y_1<_Y y_2$. Since $y_1<_Y y_2$, we have $s_1<_X s_2$, where $s_i$ is the first vertex of $y_i$ for $i\in \{1,2\}$. Furthermore, since $x_1$ and $y_2$ are adjacent in $B_G$, vertex $x_1$ belongs to $y_2$, and thus $s_2\le_X x_1$. Consequently, $s_1<_X s_2 \le_X x_1 <_X x_2$. Thus, since $x_2$ belongs to $y_1$, also $x_1$ belongs to $y_1$. This implies that $x_1$ and $y_1$ are adjacent in $B_G$. Finally, since $y_1<_Y y_2$, clique $y_2$ ends strictly after clique $y_1$, and since $x_2$ belongs to $y_1$, we conclude that $x_2$ also belongs to $y_2$. Thus, $x_2$ and $y_2$ are adjacent in $B_G$. It follows that the pair $(<_X, <_Y)$ is a strong ordering of $B_G$, as claimed. ◻ ## Maximum induced matchings in bipartite permutation graphs, revisited Our goal is to show that if $G$ is a proper interval graph, then the sufficient condition given by is satisfied, namely, there exists a maximum induced matching $M$ in $B_G$ such that the set $S$ of $M\!$-saturated vertices in $X$ dominates $Y$ in $B_G$. By , this will imply $\tau_c^+(G) = |S| = |M|$. We show the claimed property of $B_G$ as follows. First, by applying , we infer that the graph $B_G$ is a bipartite permutation graph. Second, by construction, $B_G$ does not have any isolated vertices and no two distinct vertices in $Y$ have comparable neighborhoods in $X$. It turns out that these properties are already enough to guarantee the desired conclusion. We show this by a careful analysis of the linear-time algorithm due to Chang from [@MR2024264] for computing a maximum induced matching in bipartite permutation graphs. The linear time complexity also relies on the following result. **Theorem 21** (Sprague [@MR1369371] and Spinrad, Brandstädt, and Stewart [@MR917130]). *A strong biconvex ordering of a given bipartite permutation graph can be computed in linear time.* **Theorem 22**. *Given a bipartite permutation graph $G = (X,Y;E)$, there is a linear-time algorithm that computes a maximum induced matching $M$ in $G$ such that, if $G$ has no isolated vertices and no two vertices in $Y$ have comparable neighborhoods in $G$, then the set of $M\!$-saturated vertices in $X$ dominates $Y$.* *Proof.* Let $G = (X,Y;E)$ be a bipartite permutation graph. We consider two cases. First, assume first that $G$ either contains an isolated vertex or two vertices in $Y$ with comparable neighborhoods in $G$. In this case, it suffices to show that there is a linear-time algorithm that computes a maximum induced matching in $G$. We may assume without loss of generality that $G$ is connected; otherwise, we compute in linear time the connected components of $G$ using breadth-first search, solve the problem on each component, and combine the solutions. Assuming $G$ is connected, we compute a maximum induced matching $M$ in $G$ in linear time using Chang's algorithm [@MR2024264]. Assume now that $G$ has no isolated vertices and no two vertices $y,y'\in Y$ have comparable neighborhoods in $G$, that is, $N(y)\subseteq N(y')$ or $N(y')\subseteq N(y)$, if and only if $y = y'$. Again, we first argue that it suffices to consider the case of connected graphs. In the general case, we proceed as follows. First, the connected components of $G$ can be computed in linear time using breadth-first search. Second, since no two vertices in $Y$ have comparable neighborhoods in $G$, the same is also true for each connected component. Third, assume that each connected component $C = (X_C,Y_C;E_C)$ has a maximum induced matching $M_C$ such that, if no two vertices in $Y_C$ have comparable neighborhoods in $G$, then the set of $M_C$-saturated vertices in $X_C$ dominates $Y_C$. Thus, the union of all such maximum induced matchings $M_C$ yields a maximum induced matching $M$ in $G$ such that the set of $M\!$-saturated vertices in $X$ dominates $Y$. Assume now that $G$ is connected. As shown by Chang [@MR2024264], a maximum induced matching $M$ of $G$ can be computed in linear time. We show that the set of $M\!$-saturated vertices in $X$ dominates $Y$. To do that, we first explain Chang's algorithm. The algorithm is based on a strong biconvex ordering $(<_X,<_Y)$ of $G$, which can be computed in linear time (see ). Let $x_1,\ldots,x_s$ be the ordering of $X$ such that for all $i,j\in \{1,\ldots, s\}$, we have $i<j$ if and only if $x_i<_X x_j$. Similarly, let $y_1,\ldots,y_t$ be the ordering of $Y$ such that for all $i,j\in \{1,\ldots, t\}$, we have $i<j$ if and only if $y_i<_Y y_j$. For each vertex $v\in X$, let $\min(v)$ and $\max(v)$ denote the smallest and the largest $i$ such that $y_i$ is adjacent to $v$, respectively; for vertices in $Y$, $\min(v)$ and $\max(v)$ are defined similarly. The pseudocode is given as Algorithm [\[alg:BPG\]](#alg:BPG){reference-type="ref" reference="alg:BPG"}. compute a strong biconvex ordering $(<_X,<_Y)$ of $B_G$ compute the values $\min(v)$ and $\max(v)$ for all $v\in V(G)$ $M \leftarrow \{\{x_s,y_t\}\}$ let $i = s$ and $j = t$ let $p = \min(y_j)$ and $q = \min(x_i)$ $M\leftarrow M\cup \{\{x_{p-1},y_{q-1}\}\}$ $i \leftarrow p-1$ $j \leftarrow q-1$ $M\leftarrow M\cup \{\{x_{\max(y_{q-1})},y_{q-1}\}\}$ $i \leftarrow \max(y_{q-1})$ $j \leftarrow q-1$ $M\leftarrow M\cup \{\{x_{p-1},y_{\max(x_{p-1})}\}\}$ $i \leftarrow p-1$ $j \leftarrow \max(x_{p-1})$ Let $M$ be the matching computed by the above algorithm and suppose for a contradiction that there exists a vertex $y\in Y$ that is not adjacent to any $M$-saturated vertex in $X$. Clearly, $y$ is not an endpoint of a matching edge. By construction, no two edges of $M$ cross. Thus, we may order the edges of $M$ linearly as $M = \{\{x_{i_1},y_{j_1}\},\ldots, \{x_{i_r},y_{j_r}\}\}$ so that $i_1<\dots<i_r = s$ and $j_1<\dots<j_r = t$. Note that the algorithm added the edges to $M$ in the order $\{x_{i_r},y_{j_r}\}, \{x_{i_{r-1}},y_{j_{r-1}}\}, \ldots, \{x_{i_1}y_{j_1}\}$. Since $i_r = s$ and $j_r = t$, there exists a smallest integer $k\in \{1,\ldots, r\}$ such that $y<_Y y_{j_k}$. Furthermore, since no two vertices in $Y$ have comparable neighborhoods, there exists a vertex $x\in X$ adjacent to $y$ but not to $y_{j_k}$. The edge $\{x_{i_k},y_{j_k}\}$ belongs to the matching $M$, and hence the vertex $x_{i_k}$ is adjacent to $y_{j_k}$ but not to $y$, since no neighbor of $y$ is $M$-saturated. Next, observe that $x<_X x_{i_k}$, since otherwise the presence of the edges $\{x_{i_k},y_{j_k}\}$ and $\{x,y\}$ would imply, using the fact that $(<_X,<_Y)$ is a strong ordering of $G$, that $x_{i_k}$ is adjacent to $y$. Consider the iteration of the **while** loop of the algorithm right after the edge $\{x_{i_k},y_{j_k}\}$ was added to $M$. Then $i = i_k$ and $j = j_k$ at the beginning of that loop. Since $x<_Xx_{i}$ and $y<_Y y_i$, the facts that $x_i$ and $y_j$ are non-adjacent to $y$ and $x$, respectively, and that $(<_X,<_Y)$ is a strong ordering of $G$, imply that the condition $\min(x_i)\neq 1$ and $\min(y_j)\neq 1$ of the **while** loop is satisfied. Hence, the algorithm enters the **while** loop. Let $p = \min(y_j)$ and $q = \min(x_i)$. Using the fact that $(<_X,<_Y)$ is a strong ordering of $G$, we infer that $x<_Xx_p$ and $y<_Yy_q$. Since the graph $G$ is connected, exactly one of the conditions of the three **if** statements within the **while** loop will be satisfied and the algorithm adds at least one more edge $e = \{x_{i_{k-1}},y_{j_{k-1}}\}$ to $M$. In particular, $(i_{k-1},j_{k-1})\in \{(p-1,q-1),(\max(y_{q-1}),q-1),(p-1,\max(x_{p-1}))\}$. By the definition of $k$, we have $y_{j_{k-1}}<_Yy$. Since we also have $y<_Yy_q$, we infer that $j_{k-1}<q-1$ and therefore $j_{k-1}= \max(x_{p-1})$ and consequently $i_{k-1} = p-1$. The vertex $x_{p-1} = x_{i_{k-1}}$ is an endpoint of an edge in $M$ and therefore not adjacent to $y$, since no neighbor of $y$ is $M$-saturated. In particular, $x_{p-1}\neq x$ and thus $x<_Xx_p$ implies that $x<_Xx_{p-1}$. But now, the presence of the edges $\{x,y\}$ and $\{x_{p-1},y_{j_{k-1}}\}$ together with $x<_Xx_{p-1}$, $y_{j_{k-1}}<_Yy$, and the fact that $(<_X,<_Y)$ is a strong ordering of $G$, implies that $x_{p-1}$ is adjacent to $y$, a contradiction. ◻ ## Solving UCT in proper interval graphs in linear time The following result is a consequence of and the fact that every proper interval graph is a chordal graph. **Corollary 23**. *The vertex-clique incidence graph of a proper interval graph $G$ can be computed in linear time.* We now have everything ready to prove the announced result. **Theorem 24**. *Upper Clique Transversal can be solved in linear time in the class of proper interval graphs.* *Proof.* The algorithm proceeds in three steps. In the first step, we compute from the input graph $G=(V,E)$ its vertex-clique incidence graph $B_G$, with parts $X = V$ and $Y = {\mathcal C}_G$. By Theorem [Theorem 20](#thm:pig-characterizations-via-BG){reference-type="ref" reference="thm:pig-characterizations-via-BG"}, the graph $B_G$ is a bipartite permutation graph. In the second step of the algorithm, we compute a maximum induced matching $M$ of $B_G$, using . Finally, the algorithm returns the set of $M$-saturated vertices in $X$. The pseudocode is given as Algorithm [\[alg:PIG\]](#alg:PIG){reference-type="ref" reference="alg:PIG"}. compute the vertex-clique incidence graph $B_G$, with parts $X = V$ and $Y = {\mathcal C}_G$ compute a maximum induced matching $M$ of $B_G$; compute the set $M_X$ of $M$-saturated vertices in $X$ *Correctness.* By construction, the set $M_X$ returned by the algorithm is a subset of $X$, and thus a set of vertices of $G$. Since every vertex of $G$ belongs to a maximal clique, and every maximal clique contains a vertex, $B_G$ does not have any isolated vertices. Furthermore, since the vertices of $Y$ are precisely the maximal cliques of $G$, no two vertices in $Y$ have comparable neighborhoods in $B_G$. Therefore, by , the set $M_X$ dominates $Y$. By Corollary [Corollary 15](#cor:optimality){reference-type="ref" reference="cor:optimality"}, $M_X$ is a maximum minimal clique transversal in $G$. *Time complexity.* Computing the vertex-clique incidence graph $B_G$ can be done in linear time by Corollary [Corollary 23](#cor:BG){reference-type="ref" reference="cor:BG"}. Since $B_G$ is a bipartite permutation graph, a maximum induced matching of $B_G$ can be computed in linear time, see Theorem [Theorem 22](#thm:MIM-bip-perm){reference-type="ref" reference="thm:MIM-bip-perm"}. The set of $M$-saturated vertices in $X$ can also be computed in linear time. Thus, the overall time complexity of the algorithm is $\mathcal{O}(|V|+|E|)$. ◻ The above proof also shows the following. **Theorem 25**. *For every proper interval graph $G$, the upper clique transversal number of $G$ is equal to the induced matching number of $B_G$.* We conclude the section by showing that the result of does not generalize to the class of interval graphs. **Observation 26**. *There exist interval graphs such that the difference between the induced matching number of their vertex-clique incidence graph and the upper clique transversal number of the graph is arbitrarily large.* *Proof.* Let $q\ge 2$ and let $G$ be the graph obtained from two disjoint copies of the star graph $K_{1,q}$ by adding an edge between the two vertices of degree $q$. It is easy to see that $G$ is an interval graph. We claim that the upper clique transversal number of $G$ is at most $q+1$, while the induced matching number of $B_G$ is at least $2q$. To see that the upper clique transversal number of $G$ is at most $q+1$, consider an arbitrary minimal clique transversal $S$ of $G$. Then $S$ must contain at least one of the vertices of degree $q+1$; let $u$ be such a vertex. Then, since $S$ is minimal, it cannot contain any of the $q$ neighbors of $u$ that are of degree $1$ in $G$. Thus, $S$ either consists of the two vertices of degree $q+1$ in $G$, or contains $u$ and all its non-neighbors in $G$. In either case, $S$ is of size at most $q+1$. It remains to show that the induced matching number of the vertex-clique incidence graph of $G$ is at least $2q$. As usual, let $B_G = (X,Y;E)$, with $X = V(G)$ and $Y = \mathcal{C}_G$. Since $G$ is triangle-free and has no isolated vertices, the maximal cliques of $G$ are exactly the edges of $G$, and the edges of $B_G$ are the pairs $\{x,e\}$ where $x\in V(G)$, $e\in E(G)$, and $x$ is an endpoint of $e$. Thus, $B_G$ is isomorphic to the graph obtained from $G$ by subdividing each edge. Let $M$ be the set of edges of $B_G$ of the form $\{x,e\}$ where $x$ is a vertex in $B_G$ of degree $1$ and $e$ is the unique edge incident with it. Then $M$ is an induced matching in $B_G$ of size $2q$ and hence the induced matching number of $B_G$ is at least $2q$. ◻ # Conclusion {#sec:conclusion} We performed a systematic study of the complexity of [Upper Clique Transversal]{.smallcaps} in various graph classes, showing, on the one hand, NP-completeness of the problem in the classes of chordal graphs, chordal bipartite graphs, and line graphs of bipartite graphs, and, on the other hand, linear-time solvability in the classes of split graphs and proper interval graphs. Our work leaves open several questions. **Question 1**. *What is the complexity of computing a minimal clique transversal in a given graph?* **Question 2**. *What is the complexity of [Upper Clique Transversal]{.smallcaps} in the class of interval graphs?* **Question 3**. *For what graphs $G$ does the upper clique transversal number equal to the induced matching number of the vertex-clique incidence graph?* While not all interval graphs have the stated property, shows that the property is satisfied by every proper interval graph. But there is more; for example, all cycles have the property. The upper clique transversal number is a trivial upper bound for the clique transversal number; however, the ratio between these two parameters can be arbitrarily large in general. For instance, in the complete bipartite graph $K_{1,q}$ the former one has value $q$ while the latter one has value $1$. This motivates the following. **Question 4**. *For which graph classes is the ratio (or even the difference) between the clique transversal number and the upper clique transversal number bounded?* **Question 5**. *What is the parameterized complexity of [Upper Clique Transversal]{.smallcaps} (with respect to its natural parameterization)?* Using hypergraph techniques, it can be shown that the problem is in XP, see [@boros2023dually]. ## Acknowledgements {#acknowledgements .unnumbered} We are grateful to Nikolaos Melissinos and Haiko Müller for their helpful comments. The work of the first named author is supported in part by the Slovenian Research Agency (I0-0035, research program P1-0285 and research projects N1-0102, N1-0160, J1-3001, J1-3002, J1-3003, J1-4008, and J1-4084). 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--- abstract: | This paper concerns asymptotic stability, instability, and bifurcation of constant steady state solutions of the parabolic-parabolic and parabolic-elliptic chemotaxis models on metric graphs. We determine a threshold value $\chi^*>0$ of the chemotaxis sensitivity parameter that separates the regimes of local asymptotic stability and instability, and, in addition, determine the parameter intervals that facilitate global asymptotic convergence of solutions with positive initial data to constant steady states. Moreover, we provide a sequence of bifurcation points for the chemotaxis sensitivity parameter that yields non-constant steady state solutions. In particular, we show that the first bifurcation point coincides with threshold value $\chi^*$ for a generic compact metric graph. Finally, we supply numerical computation of bifurcation points for several graphs. address: - Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA - Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA - Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA author: - Hewan Shemtaga - Wenxian Shen - Selim Sukhtaiev title: Stability and bifurcation for logistic Keller--Segel models on compact graphs --- [^1] # Introduction This paper is centered around the Keller--Segel model given by the following initial value problem for a system of reaction-advection-diffusion equations on a metric graph $\Gamma=(\mathcal{V},\mathcal{E})$ $$\label{parabolic-parabolic-eq} \begin{cases} u_t=\partial_x\big(\partial_{x} u-\chi u\partial_xv\big)+u(a-bu), \quad { x\in\mathcal{E}}, \cr \tau v_t=\partial_{xx}^2 v-v+u, \quad { x\in\mathcal{E}},\cr u(0, x)=u_0(x),\ v(0,x)=v_0(x),\quad {x\in \mathcal{E}}, \end{cases}$$ where $\chi, a,b>0, \tau\ge 0$, ${\mathcal V}$ is the set of vertices and ${\mathcal E}$ is the set of edges of the graph. This pair of PDEs describes population dynamics in presence of attracting substances. In the context of the directed movement of microorganisms in response to a chemical attractant this model is often referred to as the chemotaxis model, see [@KJP] for illuminating discussion of chemotaxis phenomena in biomedical and social sciences. The vast mathematical literature on this model includes [@MR3351175; @MR3698165; @MR2409228; @MR2448428; @MR2013508; @IsSh1; @IsSh2; @MR3925816; @KS1; @KoWeXu; @MR3294344; @MR3620027; @MR3397319; @MR2334836; @MR1654389; @MR3147229; @WaYaGa; @MR2445771; @MR2644137; @MR2754053; @MR2825180; @MR3115832; @MR3210023; @MR3462549; @MR3335922; @MR3286576]. The two quantities central to chemotaxis models are the population density $u=u(t,x)$ and the concentration of the chemical substance $v=v(t,x)$. The classical Fick's law of diffusion combined with population drifts along the chemical gradient yield the first equation in [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"}, where $\chi>0$ is the chemotaxis sensitivity parameter and $\chi u\partial_x v$ is the taxis-flux term. The second equation in [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} is the usual reaction-diffusion equation, with $\tau>0$ corresponding to moderate diffusion rate of the chemical substances and $\tau=0$ corresponding to rapid diffusion thereof. In this paper, we consider [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} in two regimes: 1. $\tau=0$, in which case [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} is referred to as the parabolic-elliptic system, 2. $\tau>0$, in which case [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} is referred to as the parabolic-parabolic system. Treating both regimes by different methods, we focus on local asymptotic stability of the constant steady state $(u_0, v_0)=(a/b, a/b)$ and existence of non-trivial steady states bifurcating from $(u_0, v_0)$ in response to small variation of the chemotaxis sensitivity parameter $\chi$. We investigate [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} posed on arbitrary connected compact metric graphs $\Gamma={(\mathcal{V},\mathcal{E})}$ and consider solutions $u=u(t,x)$, $v=v(t,x)$ satisfying natural Neumann--Kirchhoff vertex conditions describing continuity and preservation of flux at all vertices $\vartheta\in{\mathcal V}$, that is, $$\label{NKsol} \begin{cases} u_e(\vartheta)=u_{e'}(\vartheta), v_e(\vartheta)=v_{e'}(\vartheta), \,\, e\sim \vartheta, e'\sim\vartheta, \,\, { \vartheta\in\mathcal{V}}\text{ (continuity at vertices)},\cr \sum\limits_{\vartheta \sim e} \partial_{\nu}u_e(\vartheta)=0\sum\limits_{\vartheta \sim e} \partial_{\nu}v_e(\vartheta)=0,\,\, { \vartheta\in\mathcal{V}}. \text{ (conservation of current),} \end{cases}$$ where $\partial_{\nu}u_e(\vartheta)$ denotes the inward normal derivative of $u$ along the edge $e$ at the vertex $\vartheta$. In [@HWS], we have established well-posedness for general chemotaxis systems on arbitrary compact metric graphs including [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"}, [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"} as a special case, see Theorem [\[Lp\]](#Lp){reference-type="ref" reference="Lp"}. In this paper, we investigate the stability, instability, and bifurcation of the constant solution $(\frac{a}{b},\frac{a}{b})$ of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"}, [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"}. Our first result provides a threshold value $\chi^*>0$ of the chemotaxis sensitivity parameter that separates the regimes of local asymptotic stability and instability of the constant solution $(u_0, v_0)$. **Theorem 1** (Local asymptotic stability and instability). *Let $\Gamma$ be a connected compact metric graph and let $\chi(\lambda)$, $\chi^*\in(0,\infty)$ be defined by $$\label{chi-lambda-eq} {\chi(\lambda):=\frac{b(\lambda-a)(1-\lambda)}{a\lambda}}, \lambda<0,$$ and $$\begin{aligned} &\chi^*:=\min\left \{{\chi(\lambda)}: \lambda\in\mathop{\mathrm{Spec}}(\Delta)\setminus\{0\} \right\}, \end{aligned}$$ where $\mathop{\mathrm{Spec}}(\Delta)$ is the spectrum of the Neumann--Kirchhoff Laplacian acting in $L^2(\Gamma)$. Then the following assertions hold for $\tau\geq 0$.* 1. *If $0<\chi<\chi^*$ then the constant solution $(\frac{a}{b},\frac{a}{b})$ of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"}, [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"} is locally asymptotically stable.* 2. *If $\chi>\chi^*$ then the constant solution $(\frac{a}{b},\frac{a}{b})$ of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"}, [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"} is unstable.* To prove Theorem [Theorem 1](#local-stability-thm){reference-type="ref" reference="local-stability-thm"} we compute the spectrum, via finding zeros of the perturbation determinant, of the linearization of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} or, equivalently, of $$\begin{bmatrix} 1& 0\\ 0&\tau \end{bmatrix}\begin{bmatrix} \partial_tu\\ \partial_t v \end{bmatrix}={ \mathcal{H}(u,v, \chi)}:=\begin{bmatrix} \partial_x\big(\partial_{x} u-\chi u\partial_xv\big)+u(a-bu) \\ \partial_{xx}^2 v-v+u \end{bmatrix},$$ about the constant steady state, see Lemma [Lemma 1](#eigenvalue-lm2){reference-type="ref" reference="eigenvalue-lm2"}. Our next result stems from a simple observation that $\mathcal{H}(a/b,a/b, \chi)=0$ for all $\chi\geq 0$. That is, for both parabolic-parabolic and parabolic-elliptic systems $(a/b, a/b, \chi)$ is the line of constant solutions in the space $(u,v,\chi)\in \widehat W^{2,2}({\Gamma})\times \widehat W^{2,2}({\Gamma})\times (0,\infty)$ (see [\[L-p-eq\]](#L-p-eq){reference-type="eqref" reference="L-p-eq"} in Appendix A for the definition of $\widehat W^{2,2}(\Gamma)$ and other functional spaces on graphs). We show that the eigenvalues of the Neumann--Kirchhoff Laplacian give rise to a sequence $\{\chi_n\}_{n\geq 1}$ of bifurcation points that is bounded from below and that accumulates only at $+\infty$. Importantly, the first bifurcation point is precisely the threshold value $\chi^*$ where stability of the constant steady state ceases to take place. **Theorem 2**. *Let $\Gamma$ be a connected compact metric graph. Let $\lambda\in\mathop{\mathrm{Spec}}(\Delta)$ be a simple eigenvalue of the Neumann--Kirchhoff Laplacian on $\Gamma$, let $\varphi$ be the corresponding eigenfunction and define $$\begin{aligned} &\label{D-eq} {\mathcal D}:=\left\{u\in \widehat W^{2,2}(\Gamma): \sum\limits_{\vartheta \sim e} \partial_{\nu}u_e(\vartheta)=0,\ u_e(\vartheta)=u_{e'}(\vartheta), \vartheta\sim e, e'\right\},\\ & \hspace{5cm} \chi_{\lambda}:={\chi(\lambda)}. \label{chieq} \end{aligned}$$ Assume, in addition, that $\chi_{\lambda}\not= \chi_{\mu}$ for $\mu\in\mathop{\mathrm{Spec}}(\Delta)\setminus\{\lambda\}$ then $\chi_{\lambda}$ is a bifurcation point of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"}, [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"}, that is, there exist $\varepsilon>0$ and $\chi\in C^2((-\varepsilon, \varepsilon), {\mathbb{R}})$, $\Phi\in C^2((-\varepsilon, \varepsilon), {\mathcal D}\times {\mathcal D})$ such that ${\mathcal{H}}\left(\Phi(s), \chi(s)\right)=0$ for $s\in (-\varepsilon, \varepsilon)$ and $$\begin{aligned} &\chi(0)=\chi_{\lambda}, \Phi(s)=\begin{bmatrix} a/b\\ a/b \end{bmatrix}+ s\begin{bmatrix} u\\ v \end{bmatrix}\varphi+ o(s) \text{\ in $\widehat W^{2,2}(\Gamma)\times \widehat W^{2,2}(\Gamma)\ $as\ }s\rightarrow 0, \end{aligned}$$ where $$\label{auxmatnew} \begin{bmatrix} u\\ v \end{bmatrix}\in \operatorname{ker}M,\ \ M:=\begin{bmatrix} \lambda-a-\frac{\chi a}{b}& \frac{\chi a}{b} \\ 1&\lambda-1 \end{bmatrix}.$$ Moreover, there exists an open set $U\subset {\mathcal D}\times {\mathcal D}\times{\mathbb{R}}$ containing $\left(\frac ab, \frac ab, \chi_{\lambda}\right)$ such that $$\begin{aligned} &\left\{ (u,v, \chi)\in U: { \mathcal{H}} (u,v,\chi)=0, (u,v)\not=\left(\frac ab, \frac ab\right)\right\}=\left\{(\Phi(s), \chi(s)): |s|<\varepsilon\right\}. \end{aligned}$$* We stress that both assumptions of Theorem [Theorem 2](#prop4.4){reference-type="ref" reference="prop4.4"} hold automatically for a generic connected graph $\Gamma$ that has no vertices of degree $2$ and that is not a circle. That is, for a given combinatorial graph $({\mathcal V}, {\mathcal E})$ that has no loops there exists a dense $G_{\delta}$ set ${\mathcal S}\subset {\mathbb{R}}^{|{\mathcal E}|}_+$ of edge lengths such that the corresponding metric graph $\Gamma$ with edge lengths $\{\ell_e, e\in{\mathcal E}\}\in {\mathcal S}$ satisfies the assumptions of Theorem [Theorem 2](#prop4.4){reference-type="ref" reference="prop4.4"}. Indeed, the eigenvalues of the Neumann--Kirchhoff Laplacian are simple for generic graph $\Gamma$, see [@MR2151598]. In addition, since the eigenvalues of the Neumann--Kirchhoff Laplacian depend continuously on edge lengths, the function $\mathop{\mathrm{Spec}}(\Delta)\ni\lambda\mapsto \chi_{\lambda}\in(0,\infty)$ is injective up to small variation of $\{\ell_e, e\in{\mathcal E}\}\in {\mathcal S}$. Figure [2](#dumbbell){reference-type="ref" reference="dumbbell"} illustrates numerically the bifurcation points for the dumbell graph, see also Section [4](#numerics){reference-type="ref" reference="numerics"} for more numerical examples. We note that in the absence of chemotaxis, that is, when $\chi=0$, the constant solution $(\frac{a}{b},\frac{a}{b})$ of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"}, [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"} is globally stable. In particular, no non-constant steady states of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"}, [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"} exists for $\chi=0$. In this context, Theorem [Theorem 2](#prop4.4){reference-type="ref" reference="prop4.4"} shows that chemotaxis induces non-constant steady states via the bifurcation of the constant steady state. In the special case of $\Gamma$ being a single interval the bifurcation of the constant steady state and existence of spiky solutions have been investigated, for example, in [@CaXuLi; @KoWeXu; @MR2334836; @WaYaGa; @WaXu]. {#dumbbell width="1.3\\linewidth"} {#dumbbell} The next two theorems concern global asymptotic convergence of solutions with non-trivial non-negative initial data to the constant steady state $(a/b, a/b)$ in the following regimes: - for $\chi$ satisfying $$\begin{aligned} \label{kap} &\frac {b^2}{\chi^2} > \frac{\left(a+\chi \kappa(a,b,\chi) \right)^2}{a-\chi \kappa(a,b,\chi) } \text{\ and\ } a-\chi \kappa(a,b,\chi) >0,\end{aligned}$$ where $$\begin{aligned} &\kappa(a,b,\chi):={ C \left(\frac{2a^2}{b^2}+\frac{a^2\chi}{b^2}+\frac{a^2\chi^2}{b^3}+\frac{a^3\chi^3}{b^4}+\frac{a^4\chi^4}{b^5} \right)}, \ C=C(\Gamma)>0,\end{aligned}$$ in the parabolic-parabolic model, see Theorem [Theorem 3](#global-stability-thm){reference-type="ref" reference="global-stability-thm"}, - for $\chi\in(0,b/2)$ in the parabolic-elliptic model, see Theorem [Theorem 4](#4.4){reference-type="ref" reference="4.4"}. **Theorem 3** (Global stability for parabolic-parabolic model). *Let $\Gamma$ be a connected compact metric graph. Let $u_0\in\widehat C(\overline{\Gamma})$, $v_0\in \widehat C^1(\overline{\Gamma})$ be non-negative initial data $u_0\not \equiv 0$ and let $u=u(x, t; u_0,v_0)$, $v=v(x, t; u_0,v_0)$ be a global unique positive solution of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} with $\tau>0$ satisfying the Neumann--Kirchhoff vertex conditions [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"} and the initial condition $(u(x,0;u_0,v_0), v(x,0;u_0,v_0))=(u_0(x),v_0(x))$ [^2]. Then there exists $C=C(\Gamma)>0$ such that for $\chi>0$ satisfying [\[kap\]](#kap){reference-type="eqref" reference="kap"} one has $$\label{asympstab1} \lim_{t \rightarrow \infty}\left(\left\|u(t, \cdot;u_0, v_0)-\frac{a}{b}\right\|_{L^{\infty}(\Gamma)} + \left\|v(t, \cdot;u_0, v_0)-\frac{a}{b}\right\|_{L^{\infty}(\Gamma)} \right)=0.$$* **Theorem 4** (Global stability for parabolic-elliptic model). *Let $\Gamma$ be a connected compact metric graph. Let $u_0\in\widehat C(\overline{\Gamma})$ be non-negative initial data $u_0\not\equiv 0$ and let $u=u(x, t; u_0)$, $v=v(x, t; u_0)$ be a global unique positive solution of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} with $\tau=0$ satisfying the Neumann--Kirchhoff vertex conditions [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"} and the initial condition $u(x,0;u_0)=u_0(x)$ [^3]. Then for $\chi\in (0, b/2)$ one has $$\lim_{t \rightarrow \infty}\left(\left\|u(t, \cdot;u_0)-\frac{a}{b}\right\|_{L^{\infty}(\Gamma)} + \left\|v(t, \cdot;u_0)-\frac{a}{b}\right\|_{L^{\infty}(\Gamma)} \right)=0.$$* The global stability of the positive constant solution for [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} with $\tau>0$ on regular convex domains $\Omega$ with Neumann boundary condition is studied in [@LiMu; @MR3210023; @MR3335922]. These works heavily rely on the following inequality $$\frac{\partial |\nabla v|^2}{\partial \nu}\le 0,\quad x\in\partial\Omega,$$ where $\frac{\partial}{\partial\nu}$ denotes the outward normal derivative. Such an inequality is not available in the setting of metric graphs. For parabolic-parabolic models, i.e. $\tau>0$, we offer a new alternative approach which does apply to regular domains, see Section [3](#sec3){reference-type="ref" reference="sec3"}. We also note that the global stability of the positive constant solution for [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} with $\tau=0$ on regular domains $\Omega$ with Neumann boundary condition have been studied in [@IsSh1; @MR3620027; @MR2334836]. We adopt the approach established in [@MR3620027] to prove global stability of the positive constant solution for [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} with $\tau=0$. The rest of the paper is organized as follows. In Section [2](#sec2){reference-type="ref" reference="sec2"}, we study the local stability, instability, and bifurcation of the constant solution $(\frac{a}{b},\frac{a}{b})$ of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"}, [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"} and prove Theorems [Theorem 1](#local-stability-thm){reference-type="ref" reference="local-stability-thm"} and [Theorem 2](#prop4.4){reference-type="ref" reference="prop4.4"}. In Section [3](#sec3){reference-type="ref" reference="sec3"}, we investigate the global stability of the constant solution $(\frac{a}{b},\frac{a}{b})$ of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"}, [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"} and prove Theorems [Theorem 3](#global-stability-thm){reference-type="ref" reference="global-stability-thm"} and [Theorem 4](#4.4){reference-type="ref" reference="4.4"}. We supply numerical computation of bifurcation points for several graphs in Section [4](#numerics){reference-type="ref" reference="numerics"}. Finally, in Appendix [5](#functionalspaces){reference-type="ref" reference="functionalspaces"}, we record several facts about fractional power spaces generated by the Neumann--Kirchhoff Laplacian on compact metric graphs. # Local stability, instability, and bifurcation of constant steady states {#sec2} In this section, we study the local stability, instability, and bifurcation of the constant solution $(\frac{a}{b},\frac{a}{b})$ of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"}, [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"} via spectral analysis of the linearizations of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"}, [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"} about the steady state solution $(\frac{a}{b},\frac{a}{b})$. We first recall a global well-posedness result from [@HWS] in Section 2.1. We then prove Theorems [Theorem 1](#local-stability-thm){reference-type="ref" reference="local-stability-thm"} and [Theorem 2](#prop4.4){reference-type="ref" reference="prop4.4"} in Sections 2.2 and 2.3, respectively. ## Well-posedness of Keller--Segel model on graphs First, let us record a result concerning well-posedness of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} subject to vertex conditions [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"} in $L^p(\Gamma)$ (see [\[L-p-eq\]](#L-p-eq){reference-type="eqref" reference="L-p-eq"} in Appendix A for the definition of $L^p(\Gamma)$ and other functional spaces on metric graphs) and regularity of solutions, in particular, their membership to the fractional power spaces ${\mathcal X}^{\beta}_p$ generated by the Neumann--Kirchhoff Laplacian $\Delta$ and to the space of Hölder continuous functions $\widehat C^{\nu}(\overline{\Gamma})$ (see Appendix [5](#functionalspaces){reference-type="ref" reference="functionalspaces"} for definition of ${\mathcal X}_p^\beta$, $\widehat C^{\nu}(\overline{\Gamma})$). **Theorem 5**. *[@HWS]. [\[Lp\]]{#Lp label="Lp"} Let $\Gamma$ be a connected compact metric graph. Then there exists $p_0\geq 1$ such that the following assertions hold for $p\geq p_0$.* *(1) Assume that $\tau=0$. Then for arbitrary $u_0\in L^p(\Gamma)$, [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} has a unique global classical solution $u=u(t, x; u_0)$, $v=v(t, x; u_0)$, $t\geq 0$ satisfying Neumann--Kirchhoff vertex conditions [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"}. For such a solution one has $$\begin{aligned} \label{rubp} u\in C((0, \infty), \widehat C^{\nu}(\overline{\Gamma}))\cap C([0, \infty), L^p(\Gamma))\cap C^{0,\beta}((0, \infty), {\mathcal X}^{\beta}_r), \end{aligned}$$ for arbitrary $r\geq1$, $\beta\in(0,1/8)$, $\nu<\beta$. Moreover, if $u_0\in\widehat C(\overline{\Gamma})$ is non-negative and not identically zero, then $u=u(t, x; u_0)>0$, $v=v(t, x; u_0)>0$ for all $t> 0$, $x\in\Gamma$.* *(2) Assume that $\tau>0$ and let $(u_0, v_0)\in L^p(\Gamma)\times \widehat W^{1, p}(\Gamma)$. Then [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} has a unique global classical solution $u=u(x, t; u_0, v_0)$, $v=v(x, t; u_0, v_0)$, $t\in [0, \infty)$ satisfying Neumann--Kirchhoff vertex conditions [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"}. For such a solution one has $$\begin{aligned} \begin{split}\label{rmilduvsolsmax} &u\in C([0, \infty), L^p(\Gamma))\cap C((0, \infty), L^r(\Gamma))\cap C^{0,\beta}((0, \infty), {\mathcal X}^{\beta}_r)\cap C^{0,\beta}((0, \infty), \widehat C^{\nu}(\overline{\Gamma})), \\ &v\in C([0, \infty), \widehat W^{1, p}(\Gamma))\cap C^{0,\beta}((0, \infty), \widehat W^{2,r}(\Gamma))\cap C^{0,\beta}((0, \infty), {\mathcal X}^{\beta}_r)\cap C^{0,\beta}((0, \infty), \widehat C^{\nu}(\overline{\Gamma})), \end{split} \end{aligned}$$ for arbitrary $r\geq1$, $\beta\in(0,1/8)$, $\nu<\beta$. Moreover, if $u_0\in\widehat C(\overline{\Gamma})$, $v_0\in \widehat C^1(\overline{\Gamma})$ are non-negative with $u_0\not\equiv 0$ the $u=u(t, x; u_0, v_0)>0$, $v=v(t, x; u_0, v_0)> 0$ for all $t> 0$, $x\in\Gamma$.* We note that well-posedness of Keller--Segel model on subset of ${\mathbb{R}}^n$, $n\geq 1$ has been investigated by numerous authors, see, for example, [@MR4188348; @MR3698165; @MR3620027; @MR2334836; @MR2445771; @MR2825180] and references therein. ## Local asymptotic stability In this subsection we first discuss spectral properties of linearizations of parabolic-parabolic and parabolic-elliptic equations about the steady state solution $(a/b, a/b)$ and then prove Theorem [Theorem 1](#local-stability-thm){reference-type="ref" reference="local-stability-thm"}. In particular, we show that the non-selfadjoint linearized operators have compact resolvents, hence, their spectra is discrete and compute (in general, complex) eigenvalues in terms of the eigenvalues of Neumann-Kirchhoff Laplacian, see Lemmas [Lemma 1](#eigenvalue-lm2){reference-type="ref" reference="eigenvalue-lm2"} and [Lemma 2](#eigenvalue-lm1){reference-type="ref" reference="eigenvalue-lm1"}. Then we prove the following: - if $\chi\in (0,\chi^*)$ then all eigenvalues of the linearized operators have negative real part, - if $\chi\in (\chi^*, \infty)$ then the linearized operators exhibit eigenvalues with positive real part. Let us introduce the following semi-linear mappings corresponding to parabolic-parabolic and parabolic-elliptic equations respectively $$\begin{aligned} \begin{split}\label{nonlinopnew} &{{\mathcal F}}(u,v,\tau, \chi): {\mathcal D}\times {\mathcal D}{\times [0,\infty)}\times \mathbb{R} \rightarrow L^2(\Gamma),\ \\ &{{\mathcal F}}(u,v, \tau, \chi):=\begin{bmatrix} \partial_x\big(\partial_{x} u-\chi u\partial_xv\big)+u(a-bu)\\ \tau^{-1}(\partial_{xx}^2 v-v+u) \end{bmatrix},\\ \end{split}\end{aligned}$$ and $$\begin{aligned} \begin{split}\label{nonlinopnew0} &F(u,\chi): {\mathcal D}\times \mathbb{R} \rightarrow L^2(\Gamma)\times L^2(\Gamma),\ \\ &F(u, \chi)u:=\partial_{xx}^2 u+\chi\partial_x\left(u \partial_x (\Delta-I)^{-1}u)\right)+u(a-bu), \\ \end{split}\end{aligned}$$ where $\mathcal{D}$ is as in [\[D-eq\]](#D-eq){reference-type="eqref" reference="D-eq"} and $\mathbb{R}^+=(0,\infty)$. Let us recall, from [@BK Section 3.1.1, Theorem 1.4.19], see also [@MR3748521], that the spectrum of the Neumann--Kirchhoff Laplacian $\Delta$ on a compact graph is discrete[^4] and bounded from above. **Lemma 1**. *The linerization of ${\mathcal F}(u,v,\tau, \chi)$ about $(a/b, a/b)$ is given by $$D_{(u,v)}{\mathcal F}\left(a/b, a/b,\tau, \chi\right)=D(a,b,\tau, \chi),$$ where $D(a,b,\tau, \chi): L^{2}(\Gamma)\times L^{2}(\Gamma)\rightarrow L^{2}(\Gamma)\times L^{2}(\Gamma)$ is a non-selfadjoint block operator matrix given by $$\begin{aligned} &\operatorname{dom}\left(D(a,b,\tau, \chi)\right):= \operatorname{dom}(\Delta)\times\operatorname{dom}(\Delta),\\ &D(a,b,\tau, \chi):=\begin{bmatrix} \Delta -aI_{L^{2}(\Gamma)}& {-} \frac{\chi a}{b}\Delta\\ \tau^{-1} I_{L^{2}(\Gamma)}& \tau^{-1}{(}\Delta -I_{L^{2}(\Gamma)}{)} \end{bmatrix}, \end{aligned}$$ where $\Delta$ denotes the Neumann--Kirchhoff Laplacian on a compact graph $\Gamma$, $a,b, \tau, \chi$ are positive constants.* *Then the spectrum of $D(a,b,\tau, \chi)$ is discrete, that is, it consist of isolated eigenvalues of finite multiplicity and it is given by $$\mathop{\mathrm{Spec}}\left(D(a,b,\tau, \chi)\right)=\left\{\mu\in{\mathbb{C}}: \operatorname{det}(\lambda A+B-\mu T)=0, \lambda\in \mathop{\mathrm{Spec}}(\Delta)\right\},$$ where $$\begin{aligned} A:=\begin{bmatrix} 1 & -\frac{\chi a}{b}\\ 0& 1 \end{bmatrix}, B:= \begin{bmatrix} -a & 0\\ 1&-1 \end{bmatrix}, T:= \begin{bmatrix} 1& 0\\ 0&\tau \end{bmatrix}.\end{aligned}$$* *Concretely, $\mu\in \mathop{\mathrm{Spec}}\left(D(a,b,\tau, \chi)\right)$ if and only if $$\label{muplus} \mu=\frac{-\Big(1-(1+\tau)\lambda+a\tau\Big)+ \sqrt { \Big(1-(1+\tau)\lambda+a\tau\Big)^2-4 \tau \Big((a-\lambda)(1-\lambda)+\chi\frac{a}{b}\lambda\Big)}}{2\tau},$$ or $$\label{muminus} \mu=\frac{-\Big(1-(1+\tau)\lambda+a\tau\Big)- \sqrt { \Big(1-(1+\tau)\lambda+a\tau\Big)^2-4 \tau \Big((a-\lambda)(1-\lambda)+\chi\frac{a}{b}\lambda\Big)}}{2\tau}.$$ for some eigenvalue $\lambda$ of the Neumann--Kirchhoff Laplacian.* *Proof.* In the first step we find the eigenvalues of $D=D(a,b,\tau, \chi)$, in the second step we will prove that $D-\mu$ is boundedly invertible, that is, $(D-\mu)^{-1}\in{\mathcal B}(L^{2}(\Gamma)\times L^{2}(\Gamma))$ whenever $\mu\in{\mathbb{C}}$ is not an eigenvalue. *Step one.* Let $\Delta_2:=\Delta\oplus \Delta$ and $$\begin{aligned} &{\mathcal A}:=A\otimes I_{L^{2}(\Gamma)} ={\begin{bmatrix} I_{L^{2}(\Gamma)} & -\frac{\chi a}{b} I_{L^{2}(\Gamma)}\\ 0& I_{L^{2}(\Gamma)} \end{bmatrix}},\\ &{\mathcal B}:=A\otimes I_{L^{2}(\Gamma)}=\begin{bmatrix} -aI_{L^{2}(\Gamma)}& 0_{L^{2}(\Gamma)}\\ I_{L^{2}(\Gamma)}& -I_{L^{2}(\Gamma)}\\ \end{bmatrix}, \label{ab}\\ &{\mathcal T}=T\otimes I_{L^{2}(\Gamma)}=\begin{bmatrix} I_{L^{2}(\Gamma)}& 0_{L^{2}(\Gamma)}\\ 0_{L^{2}(\Gamma)}& \tau I_{L^{2}(\Gamma)}\\ \end{bmatrix}.\end{aligned}$$ Then one has $$\begin{aligned} D={\mathcal T}^{-1}({\mathcal A}\Delta_2 +{\mathcal B}),\end{aligned}$$ and $\mu$ is an eigenvalue of $D$ if and only if $$\label{eq:kernelcond} \operatorname{ker}\left({{\mathcal A}\Delta _2 }+({\mathcal B}-{\mathcal T}\mu) \right)\not=\{0\}.$$ Since $1\not\in\mathop{\mathrm{Spec}}(\Delta)$, $0\not\in\mathop{\mathrm{Spec}}({\mathcal A})$ one has $${\mathcal A}\Delta_2+({\mathcal B}-{\mathcal T}\mu)={\mathcal A}(\Delta_2-I)(I+(\Delta_2-I)^{-1}({\mathcal A}^{-1}({\mathcal B}-{\mathcal T}\mu)+I)),$$ and [\[eq:kernelcond\]](#eq:kernelcond){reference-type="eqref" reference="eq:kernelcond"} is equivalent to $$\label{kercond2} \operatorname{ker}(I+(\Delta_2-I)^{-1}({\mathcal A}^{-1}({\mathcal B}-{\mathcal T}\mu)+I))\not=\{0\}.$$ Since $(\Delta_2-I)^{-1}$ is a Hilbert--Schmidt operator and $({\mathcal A}^{-1}({\mathcal B}-{\mathcal T}\mu)+I)$ is bounded, the operator $$V_{\mu}:=(\Delta_2-I)^{-1}({\mathcal A}^{-1}({\mathcal B}-{\mathcal T}\mu)+I)$$ is trace class. Hence, $I+V_{\mu}$ is boundedly invertible if and only if $\operatorname{det}(I+V_{\mu})\not=0$, cf.,e.g, [@MR1130394 Theorem VII. 7.1]. Next, we compute this perturbation determinant explicitly. Let $P_{t}:=\chi_{(t,\infty)}(\Delta_2)$ be the spectral projection of $\Delta_2$ corresponding to the interval $(t,\infty)$. Since the spectrum of $\Delta_2$ is discrete and bounded from above, one has $\dim\text{\rm{ran}}(P_{t})<\infty$, $t\in{\mathbb{R}}$ and $\lim\limits_{t\rightarrow -\infty} P_t=I_{L^{2}(\Gamma)\times L^{2}(\Gamma)}$. Then one has $$\begin{aligned} \begin{split}\label{detcomp} \operatorname{det}(I+V_{\mu})&=\lim\limits_{t\rightarrow-\infty}\operatorname{det}(I_{\text{\rm{ran}}P_{t}}+P_{t}(\Delta_2-I)^{-1}({\mathcal A}^{-1}({\mathcal B}-\mu{\mathcal T})+I)P_{t})\\ &=\lim\limits_{t\rightarrow-\infty}\operatorname{det}(I_{\text{\rm{ran}}P_{t}}+P_{t}(\Delta_2-I)^{-1}P_{t}({\mathcal A}^{-1}({\mathcal B}-\mu{\mathcal T})+I)P_{t})\\ &=\lim\limits_{t\rightarrow-\infty}\prod_{\substack{\lambda\in\mathop{\mathrm{Spec}}(\Delta_2)\\ \lambda>t} }\operatorname{det}(I_2+(\lambda-1)^{-1}(A^{-1}(B-\mu T)+I))\\ &=\prod_{\substack{\lambda\in\mathop{\mathrm{Spec}}(\Delta_2)} }\frac{\operatorname{det}(\lambda A +B-\mu T)}{\operatorname{det}(A) (\lambda-1)^{2}}, \end{split}\end{aligned}$$ where we used that fact that $({\mathcal A}^{-1}({\mathcal B}-\mu{\mathcal T})+I)$ and $P_t$ commute. The latter is inferred, for example, from the matrix representation of these operators with respect to spectral the decomposition $$L^{2}(\Gamma)\oplus L^{2}(\Gamma)=\bigoplus_{\lambda\in\mathop{\mathrm{Spec}}(\Delta_2)} \text{\rm{ran}}\chi_{\{\lambda\}}(\Delta_2),$$ $$\begin{aligned} ({\mathcal A}^{-1}({\mathcal B}-\mu{\mathcal T})+I)&= \begin{bmatrix} (A^{-1}(B-\mu T )+I_2)& 0_2&...\\ 0_2& (A^{-1}(B-\mu T )+I_2)& ...\\ \vdots & \vdots &\ddots\\ \end{bmatrix},\\ P_t&= \begin{bmatrix} \lambda_1 I_2& 0_2&...& 0_2&0_2&...\\ 0_2& \lambda_2 I_2& ...& 0_2&0_2&...\\ \vdots & \vdots &\ddots& \vdots&\vdots&\vdots\\ 0_2&0_2& ...&\lambda_k I_2&0_2&0_2\\ 0_2&0_2& ...&0_2&0_2&\ddots\\ \end{bmatrix}, \end{aligned}$$ where $\lambda_1\geq ... \geq \lambda_k$ are eigenvalues of $\Delta_2$ and $\lambda_k$ is the smallest eigenvalue satisfying $\lambda_k>t$. Then [\[detcomp\]](#detcomp){reference-type="eqref" reference="detcomp"} yields [\[kercond2\]](#kercond2){reference-type="eqref" reference="kercond2"} which in turn shows that $\mu$ is an eigenvalue of $D$ if and only if $\operatorname{det}(\lambda A+B-\mu T)=0$ for some $\lambda\in \mathop{\mathrm{Spec}}(\Delta)$, that is $\mu$ is as in [\[muplus\]](#muplus){reference-type="eqref" reference="muplus"}, [\[muminus\]](#muminus){reference-type="eqref" reference="muminus"}. *Step two.* For $\mu\in {\mathbb{C}}$ one has $$\begin{aligned} \label{dminusmu} D-\mu={\mathcal T}^{-1}{\mathcal A}(\Delta_2-I)(I+(\Delta_2-I)^{-1}({\mathcal A}^{-1}({\mathcal B}-{\mathcal T}\mu)+I)).\end{aligned}$$ Let us pick $\mu\in{\mathbb{C}}$ such that $\operatorname{det}(\lambda A +B-\mu T)\not=0$ for all $\lambda\in\mathop{\mathrm{Spec}}(\Delta_2)$[^5]. Then by step one the operator in the right-hand side of [\[dminusmu\]](#dminusmu){reference-type="eqref" reference="dminusmu"} is boundedly invertible, hence, $$\begin{aligned} (D-\mu)^{-1}=(I+(\Delta_2-I)^{-1}({\mathcal A}^{-1}({\mathcal B}-{\mathcal T}\mu)+I))^{-1}(\Delta_2-I)^{-1}{\mathcal A}^{-1}{\mathcal T}.\end{aligned}$$ Since $(\Delta_2-I)^{-1}$ is compact and all other factors are bounded we infer that $(D-\mu)^{-1}$ is also compact. Therefore, the spectrum of $D$ purely discrete and consists of eigenvalues given by [\[muplus\]](#muplus){reference-type="eqref" reference="muplus"}, [\[muminus\]](#muminus){reference-type="eqref" reference="muminus"}. ◻ **Lemma 2**. *The linerization of $F(u, \chi)$ about $a/b$ is given by $$D_{u}F \left(a/b, \chi\right)=D(a,b,\chi),$$ where $D(a,b, \chi): L^{2}(\Gamma)\rightarrow L^{2}(\Gamma)$ is a self-adjoint operator given by $$\begin{aligned} &\operatorname{dom}\big(D(a,b, \chi)\big):= \operatorname{dom}(\Delta),\\ &D(a,b,\chi):=\Delta { - }\chi\frac{a}{b} \left(\Delta-I\right)^{-1} -\left(a-\chi\frac{a}{b}\right),\ \end{aligned}$$ where $\Delta$ denotes the Neumann--Kirchhoff Laplacian on a compact graph $\Gamma$, $a,b, \chi$ are positive constants. Then the spectrum of $D(a,b, \chi)$ is discrete, that is, it consists of isolated eigenvalues and $\mu\in\mathop{\mathrm{Spec}}\big(D(a,b, \chi)\big)$ if and only if $$\mu=\lambda-\frac{\chi a}{b(1-\lambda)}-\left(a-\chi\frac{a}{b}\right)$$ for some $\lambda\in \mathop{\mathrm{Spec}}(\Delta)$.* *Proof.* Let $f(t):=t{ -} \chi a b^{-1}(1-t)^{-1}-(a-\chi ab^{-1})$, $t\leq0$. Then $D(a,b,\chi)=f(\Delta)$ and the assertions follow form the spectral theorem combined with the fact that $\Delta$ has compact resolvent. ◻ We now ready to prove Theorem [Theorem 1](#local-stability-thm){reference-type="ref" reference="local-stability-thm"}. *Proof of Theorem [Theorem 1](#local-stability-thm){reference-type="ref" reference="local-stability-thm"}.* To prove local asymptotic stability of $(\frac{a}{b},\frac{a}{b})$ in Part (1) and the instability of $(\frac{a}{b},\frac{a}{b})$ in Part (2), it suffices to show that the spectrum of the linearized operator is a subset of $\{z\in{\mathbb{C}}: \operatorname{Re }z <0\}$ if $0<\chi<\chi^*$, and that it intersects the set $\{z\in {\mathbb{C}}:\operatorname{Re }z>0\}$ if $\chi>\chi^*$, cf., e.g., [@Henry Theorem 5.1.1]. We prove this for the cases $\tau=0$ and $\tau>0$ separately. First, consider the case that $\tau=0$. Recall [\[nonlinopnew0\]](#nonlinopnew0){reference-type="eqref" reference="nonlinopnew0"} and its linearization $D(a,b, \chi)$ from Lemma [Lemma 2](#eigenvalue-lm1){reference-type="ref" reference="eigenvalue-lm1"}. Then $\mu\in\mathop{\mathrm{Spec}}\big(D(a,b, \chi)\big)$ if and only if $$\mu=\lambda{ -a}+\chi\frac{a}{b}\left(1-\frac{1}{1-\lambda}\right).$$ for some $\lambda\in\mathop{\mathrm{Spec}}(\Delta)$. Since $\lambda\leq 0$, one has $\mu=\mu(\chi)$ is a non-decreasing and vanishes at $$\chi(\lambda):=\frac{ b(\lambda-a)(1-\lambda)}{a\lambda}, \quad \lambda\in\mathop{\mathrm{Spec}}(\Delta).$$ Hence, $\mu<0$ whenever $0<\chi<\min\{\chi(\lambda): \lambda\in\mathop{\mathrm{Spec}}(\Delta)\}=\chi^*$. That is $$\mathop{\mathrm{Spec}}\left(D(a,b, \chi)\right)\subset \{z\in{\mathbb{C}}: \operatorname{Re }z <0\}, \chi\in(0,\chi^*).$$ Moreover, if $\chi>\chi^*$, then for some $\lambda\in \mathop{\mathrm{Spec}}(\Delta)$ one has $\mu=\lambda{-a}+\chi\frac{a}{b}\left(1-\frac{1}{1-\lambda}\right)>0$, which concludes the proof of the case $\tau=0$. Next, we consider the case$\tau>0$. Recall [\[nonlinopnew0\]](#nonlinopnew0){reference-type="eqref" reference="nonlinopnew0"} and its linearization $D(a,b, \tau, \chi)$ from Lemma [Lemma 2](#eigenvalue-lm1){reference-type="ref" reference="eigenvalue-lm1"}. Then the spectrum of $D(a,b, \tau, \chi)$ is given by the following eigenvalues $$\mu_{\pm}(\lambda,\chi)=\frac{-\Big(1-(1+\tau)\lambda+a\tau\Big)\pm\sqrt { \Big(1-(1+\tau)\lambda+a\tau\Big)^2-4 \tau \Big((a-\lambda)(1-\lambda)+\chi\frac{a}{b}\lambda\Big)}}{2\tau}$$ for $\lambda\in\mathop{\mathrm{Spec}}(\Delta)$. First, let us observe that $$1-(1+\tau)\lambda+a\tau>0.$$ Hence $\operatorname{Re }(\mu_-(\lambda,\chi))<0$ for all $\lambda\in\mathop{\mathrm{Spec}}(\Delta)$, $\chi>0$. To show $\operatorname{Re }(\mu_+(\lambda,\chi))<0$ for all $\lambda\in\mathop{\mathrm{Spec}}(\Delta)$, $\chi\in(0, \chi^*)$, we first note that $$\mu_+(0,\chi)=\frac{-(1+a\tau)+\sqrt { \Big(1+a\tau\Big)^2-4 \tau a}}{2\tau}<0.$$ If $\lambda\in\mathop{\mathrm{Spec}}(\Delta)\setminus\{0\}$ then either $$\Big(1-(1+\tau)\lambda+a\tau\Big)^2\geq 4 \tau \Big((a-\lambda)(1-\lambda)+\chi\frac{a}{b}\lambda\Big),$$ in which case $\operatorname{Re }(\mu_+(\lambda,\chi))<0$, or $$\Big(1-(1+\tau)\lambda+a\tau\Big)^2< 4 \tau \Big((a-\lambda)(1-\lambda)+\chi\frac{a}{b}\lambda\Big),$$ in which case $\chi\mapsto \mu_+(\lambda,\chi)$ is a real-valued, non-decreasing function of $\chi$. In the latter case, the equation $\mu_+(\lambda,\chi)=0$ reads $$\Big(1-(1+\tau)\lambda+a\tau\Big)=\sqrt { \Big(1-(1+\tau)\lambda+a\tau\Big)^2-4 \tau \Big((a-\lambda)(1-\lambda)+\chi\frac{a}{b}\lambda\Big)}$$ and yields $$\chi=\frac{ b(\lambda-a)(1-\lambda)}{a\lambda}, \lambda\in\mathop{\mathrm{Spec}}(\Delta).$$ Hence, $\operatorname{Re }(\mu_+(\lambda,\chi))=\mu_+(\lambda,\chi)<0$ whenever $\chi\in(0,\chi^*)$ as required. To finish the proof, we note that if $\chi>\chi^*$ then there exists $\lambda\in \mathop{\mathrm{Spec}}(\Delta)$ such that $\mu_+(\lambda,\chi) >0$.  ◻ ## Local bifurcation In this subsection, we prove Theorem [Theorem 2](#prop4.4){reference-type="ref" reference="prop4.4"} via Crandall--Rabinowitz's Theorem, cf. [@MR1956130 Theorem 8.3.1]. *Proof of Theorem [Theorem 2](#prop4.4){reference-type="ref" reference="prop4.4"}.* Our goal is to verify conditions of Crandall--Rabinowitz's Theorem as stated in [@MR1956130 Theorem 8.3.1]. To that end, we first note that steady state solutions of both parabolic-parabolic and parabolic-elliptic equations stem for the same system $$\begin{aligned} \begin{cases} \partial_x\big(\partial_{x} u-\chi u\partial_xv\big)+u(a-bu)=0,\\ \partial_{xx}^2 v-v+u=0, \end{cases} \end{aligned}$$ or, equivalently, $\mathcal{F}(u,v,1, \chi)=0$, cf. [\[nonlinopnew\]](#nonlinopnew){reference-type="eqref" reference="nonlinopnew"}. The partial derivative with respect to $(u,v)$ of this nonlinear mapping is given by $$\begin{aligned} &L:=D_{(u,v)}{\mathcal F}\left(u, v, 1, \chi\right)\in{\mathcal B}({\mathcal D}\times {\mathcal D}\times{\mathbb{R}}, L^2(\Gamma)\times L^2(\Gamma)), \\ &D_{(u,v)}{\mathcal F}(u,v, \chi)=L_2+L_1\text{ where},\\ &L_2\begin{bmatrix} f\\g \end{bmatrix} := \begin{bmatrix} I_{L^2(\Gamma)}& -\chi u I_{L^2(\Gamma)}\\ 0_{L^2(\Gamma)}&I_{L^2(\Gamma)} \end{bmatrix} \begin{bmatrix} \Delta & 0_{L^2(\Gamma)}\\ 0_{L^2(\Gamma)}&\Delta \end{bmatrix},\label{el2}\\ &L_1\begin{bmatrix} f\\g \end{bmatrix} := \begin{bmatrix} -\chi v'f'-\chi v''f-\chi u'g'+af-2buf \\ -g +f \end{bmatrix},\label{el1} \end{aligned}$$ here $f,g\in{\mathcal D}$ and ${\mathcal D}$ is considered as a Banach space with $\widehat W^{2,2}(\Gamma)-$norm. This shows that $F\in C^2\left( {\mathcal D}\times {\mathcal D}\times{\mathbb{R}}, L^2(\Gamma)\times L^2(\Gamma)\right)$. Next, we show that $D_{(u,v)}{\mathcal F}(u,v,1, \chi)$ is Fredholm with index zero as an operator from ${\mathcal D}\times {\mathcal D}\times{\mathbb{R}}$ to $L^2(\Gamma)\times L^2(\Gamma)$. Let us recall that the Neumann--Kirchhoof Laplacian $\Delta\in {\mathcal B}(\widehat W^{2,2}(\Gamma), L^2(\Gamma))$ is Fredholm with index zero and the first term in the right-hand side of [\[el2\]](#el2){reference-type="eqref" reference="el2"} is Fredhlom in $L^2(\Gamma)\times L^2(\Gamma)$ with index zero. Therefore by [@MR929030 Theorem 3.16], $L_2$ is Fredholm with index zero as a mapping from $\widehat W^{2,2}(\Gamma)$ to $L^2(\Gamma)$. Next, $L_1\in{\mathcal B}(\widehat W^{2,2}(\Gamma), \widehat W^{1,2}(\Gamma))$ and the embedding $\widehat W^{1,2}(\Gamma)\hookrightarrow L^2(\Gamma)$ is compact, therefore $L_1$ is compact as a mapping from $\widehat W^{2,2}(\Gamma)$ to $L^2(\Gamma)$. Thus by [@MR929030 Theorem 3.17] the operator $L=L_1+L_2$ is a Fredholm with index zero. Recalling ${\mathcal A}, {\mathcal B}, \Delta_2$ from [\[ab\]](#ab){reference-type="eqref" reference="ab"} we obtain $$\begin{aligned} L&=D_{(u,v)}{\mathcal F}\left(a/b, a/b, 1, \chi\right)={\mathcal A}\Delta_2+{\mathcal B}. \end{aligned}$$ Hence, one has $$\begin{aligned} \operatorname{ker}L=\operatorname{ker}\left( \begin{bmatrix} \Delta & 0_{L^2(\Gamma)}\\ 0_{L^2(\Gamma)}&\Delta \end{bmatrix}+\begin{bmatrix} \left(-a{+\frac{\chi a}{b}}\right)I_{L^2(\Gamma)}& { -\frac{\chi a}{b}I_{L^2(\Gamma)}}\\ I_{L^2(\Gamma)}&-I_{L^2(\Gamma)} \end{bmatrix} \right). \end{aligned}$$ Then $\operatorname{ker}L\not=\{0\}$ if and only if for some $\lambda\in\mathop{\mathrm{Spec}}(\Delta)$ one has $$\operatorname{det}\left( \begin{bmatrix} \lambda& 0_{L^2(\Gamma)}\\ 0_{L^2(\Gamma)}&\lambda \end{bmatrix}+\begin{bmatrix} \left(-a+{\frac{\chi a}{b}}\right)I_{L^2(\Gamma)}& {-\frac{\chi a}{b}I_{L^2(\Gamma)}}\\ I_{L^2(\Gamma)}&-I_{L^2(\Gamma)} \end{bmatrix} \right)=0,$$ that is, if and only if one has $$\left(a-\frac{\chi a}{b}-\lambda\right)(1-\lambda)+\frac{\chi a}{b}=0,$$ or, equivalently, the identity [\[chieq\]](#chieq){reference-type="eqref" reference="chieq"} holds. By assumptions, we then obtain $\dim \operatorname{ker}(L)=1$ and $L\xi_0=0,\ \xi_0:=[ u, v]^{\top}\varphi$. Let us now show that the transversality condition in [@MR1956130 Theorem 8.3.1] is also satisfied. That is, for $K:=D^2_{(u,v),\chi}{\mathcal F}(a/b, a/b,1, \chi )$ we show that $K [\xi_0, 1]^{\top}\not \in \text{\rm{ran}}(L)$. It suffices to show $\operatorname{ker}(L^*)=0$. Since $L^*=({\mathcal A}^{-1})^{*}(\Delta_2+({\mathcal B}{\mathcal A}^{-1})^*)$, we note that $\operatorname{ker}(L^{*})\not =\{0\}$ yields a $\lambda\in\mathop{\mathrm{Spec}}(\Delta)$ such that $$\operatorname{det}( \lambda+(BA^{-1})^*)=0,$$ that is $$(\lambda-a)(\lambda-1)=0,$$ which contradicts $\lambda\leq 0$, $a>0$. ◻ # Global stability of constant steady states {#sec3} In this section, we study the global stability of the constant solution $(\frac{a}{b},\frac{a}{b})$ of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"}, [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"} and prove Theorems [Theorem 3](#global-stability-thm){reference-type="ref" reference="global-stability-thm"} and [Theorem 4](#4.4){reference-type="ref" reference="4.4"}. Throughout this section, $C$ denotes a positive constant independent of $a,b, \chi$ and the solutions of [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"}, [\[NKsol\]](#NKsol){reference-type="eqref" reference="NKsol"}. We first establish some lemmas and then prove Theorems [Theorem 3](#global-stability-thm){reference-type="ref" reference="global-stability-thm"} and [Theorem 4](#4.4){reference-type="ref" reference="4.4"}. **Lemma 3**. *Assume the setting of Theorem [Theorem 3](#global-stability-thm){reference-type="ref" reference="global-stability-thm"}. Then for $T>0$ one has $$\label{bounds-eq1} \limsup_{t\to\infty} \int_\Gamma u(t;u_0,v_0)dx\le \frac{a|\Gamma|}{b}.$$* *Proof.* Integrating both sides of the first equation in [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} over $\Gamma$ we obtain $$\frac{d}{dt}\int_\Gamma u(t;u_0,v_0)dx=a\int_\Gamma u(t;u_0,v_0)dx-b\int_\Gamma u^2(t;u_0,v_0)dx.$$ Combining this with $$\int_\Gamma u^2(t;u_0,v_0)dx\ge \frac{1}{|\Gamma|}\left(\int_\Gamma u(t;u_0,v_0)dx\right)^2,$$ we arrive at $$\label{L1-estimate-eq1} \frac{d}{dt}\int_\Gamma u(t;u_0,v_0)dx\le a\int_\Gamma u(t;u_0,v_0)-\frac{b}{|\Gamma|}\left(\int_\Gamma u(t;u_0,v_0)\right)^2dx.$$ Then, since $f(t)=|\Gamma|ab^{-1}$ solves the differential equation $f'=af-b(|\Gamma|)^{-1}f^2$, the comparison principle yields $$\limsup_{t\to\infty}\int_\Gamma u(t;u_0,v_0)dx\le \frac{a|\Gamma|}{b},$$ as asserted. ◻ **Lemma 4**. *Assume the setting of Theorem [Theorem 3](#global-stability-thm){reference-type="ref" reference="global-stability-thm"}. Then there exists a constant $C=C(\Gamma)>0$ such that $$\label{bounds-eq2} \limsup_{t\to\infty} \| v(t)\|_{ \widehat C^1(\overline\Gamma)} {\le C \frac{a}{b}}.$$* *Proof.* Let us fix $\beta\in \left(\frac{1}{2},1\right)$ and $q>1$ satisfying $2\beta -q^{-1}>1$. The by Theorem [Theorem 6](#analytic-semigroup-thm2){reference-type="ref" reference="analytic-semigroup-thm2"} one has $$\label{emb1} {\mathcal X}_q^\beta \hookrightarrow \widehat C^1(\overline\Gamma).$$ By the Duhamel principle, the second equation in [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} yields $$v(t)=e^{(\Delta-I)\frac{(t-t_0)}{\tau}}v(t_0)+\frac{1}{\tau}\int_{t_0}^{t}e^{(\Delta-I)\frac{t-s}{\tau}}u(s)ds,$$ where we abbreviated $v(t)=v(t;u_0,v_0)$. Then one obtains $$\begin{aligned} & { \| v(t)\|_{\widehat C^1(\overline \Gamma)}} \underset{\eqref{emb1}}{\leq} C \|v(t)\|_{{\mathcal X}_q^\beta}\le C \| e^{(\Delta-I)\frac{t-t_0}{\tau}}(I-\Delta)^\beta v(t_0)\|_{L^q(\Gamma)} \\ &\hspace{5cm}+C\int_{t_0}^{t}\|(I-\Delta)^\beta e^{(\Delta-I)\frac{t-s}{2\tau}}e^{(\Delta-I)\frac{t-s}{2\tau}}u(s)\|_{L^q(\Gamma)}ds \nonumber\\ &\le C \| e^{(\Delta-I)\frac{t-t_0}{\tau}}(I-\Delta)^\beta v(t_0)\|_{L^q(\Gamma)}\\ &\hspace{4cm}+C \int_{t_0}^{t} \Big(\frac{t-s}{2\tau}\Big)^{-\beta}e^{-\frac{t-s}{4\tau}} \| e^{(\Delta-I)\frac{t-s}{2\tau}}u(s)\|_{L^q(\Gamma)} ds\nonumber\\ &\le C \| e^{(\Delta-I)\frac{t-t_0}{\tau}}(I-\Delta)^\beta v(t_0)\|_{L^q(\Gamma)}\\ &\hspace{4cm}+C\int_{t_0}^{t}\left(\frac{t-s}{2\tau}\right)^{-\beta-\frac{1}{2}(1-\frac{1}{q})}e^{-\frac{t-s}{2\tau}}\|u(s)\|_{L^{1}(\Gamma)}ds\\ &\leq Ce^{\frac{t_0-t}{2\tau}}\| (I-\Delta)^\beta v(t_0)\|_{L^q(\Gamma)} +C\sup_{t\in [t_0,\infty)} \|u(t)\|_{L^{1}(\Gamma)} \int_{t_0}^{t}\left(\frac{t-s}{2\tau}\right)^{-\beta-\frac{1}{2}(1-\frac{1}{q})}e^{-\frac{t-s}{2\tau}}ds.\end{aligned}$$ This implies [\[bounds-eq2\]](#bounds-eq2){reference-type="eqref" reference="bounds-eq2"} by choosing sufficiently large $t_0$ such that $\sup_{t\in[t_0,\infty)}\|u(t)\|_{L^1(\Gamma)}\le 2|\Gamma|\frac{a}{b}$ and then letting $t\to\infty$.  ◻ **Lemma 5**. *Assume the setting of Theorem [Theorem 3](#global-stability-thm){reference-type="ref" reference="global-stability-thm"}. Then one has $$\label{bounds-eq3} \limsup_{t \to \infty} \|u(t)\|_{L^{4}(\Gamma)}\leq C\left({\frac{a}{b}}+\frac{a^2\chi^2}{b^3}\right).$$* *Proof.* Multiplying both sides of the first equation in [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} by $u^3$ and integrating over $\Gamma$ we obtain $$\begin{aligned} \label{firstequmultiplied} \int_{\Gamma}u_tu^3dx=\int_{\Gamma}u^3\partial_x\big(\partial_{x} u-\chi u\partial_xv\big)dx+\int_{\Gamma}u^4(a-bu)dx. \end{aligned}$$ We note that $$\begin{aligned} \label{ibp1} \int_{\Gamma} u^3u_{xx}dx&= -\int_{\Gamma} (u^3)_xu_{x}dx+\sum_{\theta\in {\mathcal V}}\sum_{e\sim \theta}u_e^3(\theta)\partial_{\nu}u_e(\theta)\\ & = -\int_{\Gamma} (u^3)_xu_{x}dx+\sum_{\theta\in {\mathcal V}}u^{3}(\theta)\sum_{e\sim \theta}\partial_{\nu}u_e(\theta)=-\int_{\Gamma} (u^3)_xu_{x}dx=-3\int_{\Gamma} (uu_{x})^2dx,\end{aligned}$$ where we used the fact that $u$ satisfies the Neumann-Kirchhoff vertex conditions. Similarly, one has $$\begin{aligned} \begin{split} \label{ibp2} &\int_{\Gamma} u^3\partial_x(uv_{x})dx= -\int_{\Gamma}(u^3)_xuv_{x}dx+\sum_{\theta\in {\mathcal V}}\sum_{e\sim \theta}u^4_e(\theta)\partial_{\nu}v_e(\theta)=-\int_{\Gamma}(u^3)_xuv_{x}dx.\\ \end{split}\end{aligned}$$ Therefore, we have $$\label{firsteqm2} \frac{1}{4} \frac{d}{dt}\int_{\Gamma} u^{4}dx= -3 \int_{\Gamma} (uu_{x})^2dx +3\chi \int_{\Gamma}u^{3}u_x v_x dx + \int_{\Gamma}u^4(a-bu)dx.$$ We note that Young's inequality with exponents $2,2$ yields $$\begin{aligned} \begin{split} \chi \int_{\Gamma}u^{3}u_x v_x dx&= \int_{\Gamma}(uu_x)(\chi u^2v_{x})dx\leq \frac{\chi^2}{4}\int_{\Gamma}u^{4} |v_x|^2dx+\int_{\Gamma} (uu_x)^2dx\label{young1}. \end{split}\end{aligned}$$ By Hölder's inequality gives $$\label{holder1} \int_\Gamma u^5 dx\geq \frac{1}{|\Gamma|^{1/5}}\left(\int_{\Gamma} u^4dx\right)^{5/4}.$$ Combining these inequalities with [\[firsteqm2\]](#firsteqm2){reference-type="eqref" reference="firsteqm2"} we obtain $$\begin{aligned} \frac{1}{4}\frac{d}{dt}\int_{\Gamma}u^{4}dx&=-3 \int_{\Gamma} (uu_{x})^2dx +3\chi \int_{\Gamma}u^{3}u_x v_x dx + \int_{\Gamma}u^4(a-bu)dx\\ % &= -\int_{\Gamma} |u_x|^2dx+\chi\int_{\Gamma}u u_x v_xdx +\int_{\Gamma}u^{2} (a- b u)dx\\ &\underset{\eqref{young1}}{\leq } \frac{3\chi^2}{4}\int_{\Gamma}u^{4} |v_x|^2dx +a \int_{\Gamma}u^{4}dx -b\int_\Gamma u^{5}dx\\ &{ \le \frac{3\chi^2}{4}\|v\|_{\widehat C^1(\overline \Gamma)}^2 \int_{\Gamma}u^{4} dx +a \int_{\Gamma}u^{4}dx -\frac{b}{|\Gamma|^{\frac{1}{5}}}\Big(\int_\Gamma u^{4}dx\Big)^{\frac{5}{4}}}\\ &{=\Big ( \frac{3\chi^2}{4}\|v\|_{\widehat C^1(\overline \Gamma)}^2 +a - \frac{b}{|\Gamma|^{\frac{1}{5}}} \big(\int_{\Gamma}u^{4}dx\big)^{\frac{1}{4}}\Big) \int_\Gamma u^{4}dx.}\end{aligned}$$ This together with [\[bounds-eq2\]](#bounds-eq2){reference-type="eqref" reference="bounds-eq2"} imply [\[bounds-eq3\]](#bounds-eq3){reference-type="eqref" reference="bounds-eq3"}.  ◻ **Lemma 6**. *Assume the setting of Theorem [Theorem 3](#global-stability-thm){reference-type="ref" reference="global-stability-thm"}. Then for arbitrary $\gamma\in(0,1/2)$ one has $$\label{bounds-eq4} \limsup_{t \to \infty}\| (I-\Delta)^{\gamma }u(t)\|_{L^{2}(\Gamma)}\leq C{ \left(\frac{2a^2}{b^2}+\frac{a^2\chi}{b^2}+\frac{a^2\chi^2}{b^3}+\frac{a^3\chi^3}{b^4}+\frac{a^4\chi^4}{b^5} \right).}$$* *Proof.* By Duhamel's principle, the first equation in [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} yields $$\begin{aligned} \begin{split}\label{lem4.4e1} u(t)=e^{(\Delta-I)(t-t_0)}u(t_0)&-\chi\int_{t_0}^{t}e^{(\Delta-I)(t-s)}\partial_x\left(u(s)\partial_x v(s)\right)ds\\ &+\int_{t_0}^{t}e^{(\Delta-I)(t-s)}u(s)(a-bu(s))ds. \end{split}\end{aligned}$$ Let us note the following auxiliary inequalities $$\begin{aligned} \begin{split}\label{lem4.4e2} &\|(I-\Delta)^{\gamma} e^{(\Delta-I)\frac{t-s}{2}} e^{(\Delta-I)\frac{t-s}{2}} \partial_x(u(s)\partial_x v(s))\|_{L^{2}(\Gamma)}\\ &\underset{\eqref{fracpow}}{\leq} C (t-s)^{-\gamma}e^{\frac{t-s}{4}}\|e^{(\Delta-I)\frac{t-s}{2}} \partial_x(u(s)\partial_x v(s))\|_{L^{2}(\Gamma)}\\ &\underset{\eqref{tgradlplq}}{\leq} C (t-s)^{-\gamma-1/2}e^{\frac{t-s}{4}}\|u(s)\partial_x v(s)\|_{L^{2}(\Gamma)}\\ &\leq C (t-s)^{-\gamma-1/2}e^{\frac{t-s}{2}}{ \sup\limits_{r\geq t_0}\|\partial_x v(r)\|_{L^{\infty}(\Gamma)}}\|u(s)\|_{L^{2}(\Gamma)}, \end{split}\end{aligned}$$ $$\begin{aligned} \begin{split}\label{lem4.4e3} &\|(I-\Delta)^{\gamma } e^{(\Delta-I)(t-s)}u(s)(a-bu(s))\|_{L^{2}(\Gamma)}\\ &\underset{\eqref{fracpow}}{\leq} C (t-s)^{-\gamma}e^{\frac{t-s}{2}}\|au(s)-bu^2(s)\|_{L^{2}(\Gamma)}\\ &\leq C (t-s)^{-\gamma}e^{\frac{t-s}{2}}\left(a\|u(s)\|_{L^{2}(\Gamma)}+b{\|u(s)\|_{L^{4}(\Gamma)}^{2}}\right), \end{split}\end{aligned}$$ and $$\label{lem4.4e00} { \|u(s)\|_{L^2(\Gamma)}\le |\Gamma|^{\frac{1}{4}}\|u(s)\|_{L^4(\Gamma)}.}$$ Combining [\[lem4.4e1\]](#lem4.4e1){reference-type="eqref" reference="lem4.4e1"}, [\[lem4.4e2\]](#lem4.4e2){reference-type="eqref" reference="lem4.4e2"}, [\[lem4.4e3\]](#lem4.4e3){reference-type="eqref" reference="lem4.4e3"}, [\[lem4.4e00\]](#lem4.4e00){reference-type="eqref" reference="lem4.4e00"} we obtain $$\begin{aligned} \| (I-\Delta)^{\gamma}u(t)\|_{L^{2}(\Gamma)}&\le Ce^{\frac{t-t_0}2}\|(I-\Delta)^{\gamma}u(t_0)\|_{L^{2}(\Gamma)}\\ &\qquad { +} C \chi \int_{t_0}^{t}(t-s)^{-\gamma-1/2}e^{\frac{t-s}{2}}\sup\limits_{ r\geq t_0}\|\partial_x v(r)\|_{L^{\infty}(\Gamma)}\|u(s)\|_{L^{2}(\Gamma)}ds\\ & \qquad +C\int_{t_0}^{t} (t-s)^{-\gamma}e^{\frac{t-s}{2}}\left(a\|u(s)\|_{L^{2}(\Gamma)}+b\|u(s)\|_{L^{4}(\Gamma)}^{{ 2}}\right)ds\\ &{\le Ce^{\frac{t-t_0}2}\|(I-\Delta)^{\gamma}u(t_0)\|_{L^{2}(\Gamma)}}\\ &\qquad { + C \chi \sup\limits_{r\geq t_0}\|\partial_x v(r)\|_{L^{\infty}(\Gamma)} \sup\limits_{t\ge t_0}\|u(r)\|_{L^{4}(\Gamma)} \int_{t_0}^{t}(t-s)^{-\gamma-1/2}e^{\frac{t-s}{2}}ds}\\ & \qquad{+C \left(a\sup\limits_{r\ge t_0}\|u(r)\|_{L^{4}(\Gamma)}+b\sup\limits_{r\ge t_0}\|u(r)\|_{L^{4}(\Gamma)}^{ 2}\right) \int_{t_0}^{t} (t-s)^{-\gamma}e^{\frac{t-s}{2}}ds}\end{aligned}$$ This together with [\[bounds-eq2\]](#bounds-eq2){reference-type="eqref" reference="bounds-eq2"} and [\[bounds-eq3\]](#bounds-eq3){reference-type="eqref" reference="bounds-eq3"} implies [\[bounds-eq4\]](#bounds-eq4){reference-type="eqref" reference="bounds-eq4"} (choosing $t_0$ sufficiently large). ◻ **Lemma 7**. *Assume the setting of Theorem [Theorem 3](#global-stability-thm){reference-type="ref" reference="global-stability-thm"}. Then one has $$\label{bounds-eq6} \limsup_{t \to \infty} \|\Delta v(t)\|_{L^{\infty}(\Gamma)}\leq \kappa(a,b,\chi),$$ where $\kappa(a,b,\chi)$ is as in [\[kap\]](#kap){reference-type="eqref" reference="kap"}.* *Proof.* The second equation in [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} together with Duhamel's principle yields $$v(t)=e^{(\Delta-I)\frac{t-t_0}{\tau}}v(t_0)+\frac{1}{\tau}\int_{t_0}^{t}e^{(\Delta-I)\frac{t-s}{\tau}}u(s)ds,$$ hence, $$\label{duhamel} (I-\Delta)v(t)=e^{(\Delta-I)\frac{t-t_0}{\tau}}(I-\Delta)v(t_0)+\frac{1}{\tau}\int_{t_0}^{t}(I-\Delta)e^{(\Delta-I)\frac{t-s}{\tau}}u(s)ds.$$ Let us estimate $L^{\infty}(\Gamma)$ norm of each term above. To that end, we first note the following embedding $$\label{emb2-0} { {\mathcal X}^{\alpha }_2\hookrightarrow \widehat C^{\nu }(\overline{\Gamma}), \,\, \frac{1}{4}<\alpha<1,\,\, 0<\nu <2\alpha-\frac{1}{2}}.$$ Choose $\frac{1}{4}<\alpha<\frac{1}{2}<\gamma<1$. One has $$\begin{aligned} &\int_{t_0}^ t \|(I-\Delta) e^{(\Delta-I)\frac{t-s}{\tau}}u(s)\|_{L^{\infty}(\Gamma)} ds\\ % &\hspace{1.9cm}{\color{green}=\int_{t_0}^ t \|(I-\Delta)^{1/2}(I-\Delta)^{1/2} e^{(\Delta-I)\frac{t-s}{\tau}}u(s)\|_{\el{\infty}} ds}\\ &\hspace{1.9cm}\le \int_{t_0}^ t \|(I-\Delta) e^{(\Delta-I)\frac{t-s}{\tau}}u(s)\|_{C^\nu(\overline{\Gamma})} ds\\ &\hspace{1.9cm}\le C \int_{t_0}^ t \|(I-\Delta) e^{(\Delta-I)\frac{t-s}{\tau}}u(s)\|_{ {\mathcal X}^{\alpha }_2 } ds\\ &\hspace{1.9cm}\le C \int_{t_0}^ t \|(I-\Delta)^{1+\alpha} e^{(\Delta-I)\frac{t-s}{\tau}}u(s)\|_{ L^2(\Gamma) } ds\\ &\hspace{1.9cm}\le C \int_{t_0}^ t \|(I-\Delta)^{1+\alpha-\gamma} e^{(\Delta-I)\frac{t-s}{\tau}}(I-\Delta)^{\gamma}u(s)\|_{ L^2(\Gamma) } ds\\ &\hspace{1.9cm}\le C \int_{t_0}^ t (t-s)^{-(1+\alpha-\gamma)} e^{-\frac{t-s}{2\tau}}\|(I-\Delta)^{\gamma}u(s)\|_{ L^2(\Gamma) } ds\end{aligned}$$ Combining this with $$\begin{aligned} \| (I-\Delta)e^{(\Delta-I) \frac{t-t_0}{\tau}}v(t_0)\|_{L^{\infty}(\Gamma)} &\le \| (I-\Delta)e^{(\Delta-I) \frac{t-t_0}{\tau}}v(t_0)\|_{C^\nu(\overline{\Gamma})} \\ &\le \| (I-\Delta)e^{(\Delta-I) \frac{t-t_0}{\tau}}v(t_0)\|_{{\mathcal X}^{\alpha }_2}\\ &\le C e^{\frac{t_0-t}{2\tau}}\| (I-\Delta)^{1+\alpha } v(t_0)\|_{L^2(\Gamma)}\\\end{aligned}$$ we obtain $$\begin{aligned} \|(I-\Delta) v(t)\|_{L^{\infty}(\Gamma)} &\le C e^{\frac{t_0-t}{2\tau}}\| (I-\Delta)^{1+\alpha } v(t_0)\|_{L^2(\Gamma)}\\ &\qquad+ C \int_{t_0}^ t \|(t-s)^{-(1+\alpha-\gamma)} e^{-\frac{t-s}{2\tau}}\|(I-\Delta)^{\gamma}u(s)\|_{ L^2(\Gamma) } ds. %&\le C e^{-\lambda_1\frac{t-t_0}{\tau}}\|A v(t_0)\|_\infty %+C \int_{t_0}^t \Big(1+(t-s)^{-\frac{7}{8}}\Big) e^{-\lambda_1(t-s)} \|A^{\frac{3}{8}}u(s)\|_{L^2}ds.\end{aligned}$$ This inequality together with [\[bounds-eq2\]](#bounds-eq2){reference-type="eqref" reference="bounds-eq2"} and [\[bounds-eq4\]](#bounds-eq4){reference-type="eqref" reference="bounds-eq4"} (which is applicable since $\frac{1}{4}<\alpha<\frac{1}{2}<\gamma<1$) yield [\[bounds-eq6\]](#bounds-eq6){reference-type="eqref" reference="bounds-eq6"}. ◻ We now prove Theorem [Theorem 3](#global-stability-thm){reference-type="ref" reference="global-stability-thm"}. *Proof of Theorem [Theorem 3](#global-stability-thm){reference-type="ref" reference="global-stability-thm"}.* Let us observe that for arbitrary $\varepsilon>0$ Lemma [Lemma 7](#bounds-lm5){reference-type="ref" reference="bounds-lm5"} yields a $\chi-$independent constant such that for arbitrary $x\in\overline{\Gamma}$, $t>0$ and $\chi>0$ one has $$\label{deltavbound} -(\kappa(a,b,\chi)+\varepsilon)\leq v_{xx}(t,x)\leq \kappa(a,b,\chi)+\varepsilon.$$ Employing [\[deltavbound\]](#deltavbound){reference-type="eqref" reference="deltavbound"} we obtain the following inequalities $$\begin{aligned} u_t&=u_{xx}-\chi(u_x v)_x+u(a-bu)\\ &=u_{xx}-\chi u_x v_x-\chi u_{xx} v+u(a-bu)\\ &\ge u_{xx}-\chi u_x v_x- (\kappa(a,b,\chi)+\varepsilon) u+ u(a-bu), \end{aligned}$$ and $$\begin{aligned} u_t&=u_{xx}-\chi(u_x v)_x+u(a-bu)\\ &=u_{xx}-\chi u_x v_x-\chi u_{xx} v+u(a-bu)\\ &\le u_{xx}-\chi u_x v_x+(\kappa(a,b,\chi)+\varepsilon) u+ u(a-bu). \end{aligned}$$ Therefore the partial differential equation $$\varphi_t=\varphi_{xx}-\chi \varphi_x v_x + (\kappa(a,b,\chi)+\varepsilon) \varphi+ \varphi(a-b\varphi),$$ exhibits the subsolution $u$ and a constant solution $$\begin{aligned} &\overline{u}_\varepsilon :=\frac{a+\chi(\kappa(a,b,\chi)+\varepsilon) }{b}, %&\underline{u}_\varepsilon :=\frac{a-C\chi(1+\chi^2+\chi^3+\varepsilon)}{b},\end{aligned}$$ and, similarly $u$ is a supersolution of $$\varphi_t=\varphi_{xx}-\chi \varphi_x v_x - \chi(\kappa(a,b,\chi)+\varepsilon) \varphi+ \varphi(a-b\varphi),$$ while a constant solution is given by $$\begin{aligned} &\underline{u}_\varepsilon :=\frac{a-\chi(\kappa(a,b,\chi)+\varepsilon) }{b}. %&\underline{u}_\varepsilon :=\frac{a-C\chi(1+\chi^2+\chi^3+\varepsilon)}{b},\end{aligned}$$ Therefore, one obtains $$\label{solbound} \underline{u}_\varepsilon \le u(t,x)\le\overline{u}_\varepsilon,\ x\in\Gamma, t>0.$$ Next, let us introduce $$U(t,x):=u(t,x)-\frac{a}{b},\quad V(t,x):=v(t,x)-\frac{a}{b}.$$ Then we have $$\label{Uequation} U_t=U_{xx}-\chi (u_x V)_x-b u U,$$ and $$\label{Vequation} \tau V_t=V_{xx}-V+U.$$ Since $U$ satisfies the Neumann--Kirchhoff vertex conditions, multiplying [\[Uequation\]](#Uequation){reference-type="eqref" reference="Uequation"} and integrating by parts as in the proof of Lemma [Lemma 3](#bounds-lm1){reference-type="ref" reference="bounds-lm1"} we obtain $$\begin{aligned} \begin{split}\label{vspteq1} \frac{1}{2}\frac{d}{dt}\int_\Gamma U^2dx&=-\int_\Gamma |U_x|^2dx+\chi\int_\Gamma u U_xV_xdx-b\int_\Gamma u U^2dx\\ &\le \frac{\chi^2}{4}\int_\Gamma u^2 |V_x|^2dx -b \int_\Gamma u U^2dx\\ &\le \frac{\chi^2}{4}\bar u_\varepsilon ^2 \int_\Gamma |V_x|^2dx-b\underline u_\varepsilon \int_\Gamma U^2dx \end{split} \end{aligned}$$ we used [\[solbound\]](#solbound){reference-type="eqref" reference="solbound"} and $$\chi\int_\Gamma u U_xV_xdx=\int_\Gamma \chi V_xu U_x dx\leq \int_\Gamma |U_x|^2dx+\frac{\chi^2}{4}\int_\Gamma u^2 |V_x|^2dx$$ which, in turn, follows from Young's inequality. Similarly, multiplying [\[Vequation\]](#Vequation){reference-type="eqref" reference="Vequation"} by $V$ and integrating by parts yields $$\begin{aligned} \begin{split}\label{vspteq2} \frac{\tau}{2}\frac{d}{dt}\int_\Gamma V^2dx&=-\int_\Gamma |V_x|^2-\int_\Gamma V^2dx+\int_\Gamma U Vdx\\ &\le -\int_\Gamma |V_x|^2dx-\frac{1}{2}\int_\Gamma V^2dx+\frac{1}{2}\int_\Gamma U^2dx, \end{split} \end{aligned}$$ where in the last step we used Young's inequality. Hence, combining [\[vspteq1\]](#vspteq1){reference-type="eqref" reference="vspteq1"}, [\[vspteq2\]](#vspteq2){reference-type="eqref" reference="vspteq2"} we arrive at $$\begin{aligned} \begin{split}\label{vsp3} &\frac{1}{2}\frac{d}{dt}\int_\Gamma U^2dx+\frac{\tau}{2}\frac{\chi^2}{4}\overline{u}_\varepsilon^2 \int_\Gamma V^2dx\\ &\le -\frac{1}{2}\frac{\chi^2}{4}\overline{u}_\varepsilon^2 \int_\Gamma V^2dx -\Big(b\underline u_\varepsilon -\frac{\chi^2}{4}\overline{u}_\varepsilon^2 \Big)\int_\Gamma U^2dx. \end{split} \end{aligned}$$ Provided [\[kap\]](#kap){reference-type="eqref" reference="kap"} we have $$b \underline u_\epsilon-\frac{\chi^2}{4}\overline{u}_\epsilon^2>0,$$ which together with [\[vsp3\]](#vsp3){reference-type="eqref" reference="vsp3"} yield $$\label{vsp4} \lim_{t\to\infty}\int_\Gamma (U^2+V^2)dx=0.$$ Let us now switch to the prove of [\[asympstab1\]](#asympstab1){reference-type="eqref" reference="asympstab1"}. Assume that $$\limsup_{t\to\infty}\left(\left\|u-\frac{a}{b}\right\|_{L^{\infty}(\Gamma)}+\left\|v-\frac{a}{v}\right\|_{L^{\infty}(\Gamma)}\right)>0,$$ Then for some $\epsilon_0>0$, $t_n\to\infty$ and $x_n\in\Gamma$, $n\in{\mathbb{N}}$ one has $$\left|u(t_n,x_n)-\frac{a}{b}\right|+\left|v(t_n,x_n)-\frac{a}{b}\right|\ge \epsilon_0.$$ Combining this inequality with the uniform continuity of $u$ and $v$ yields a $\delta_0>0$ such that $$\left|u(t_n,x)-\frac{a}{b}\right|+\left|v(t_n,x)-\frac{a}{b}\right|\ge \frac{\epsilon_0}{2},\ n\in{\mathbb{N}}, x\in\Gamma, |x-x_n|\le\delta_0.$$ This implies that $$\liminf_{n\to\infty}\int_\Gamma (U^2(t_n,x)+V^2(t_n,x))>0,$$ which contradicts [\[vsp4\]](#vsp4){reference-type="eqref" reference="vsp4"}. ◻ Finally, we prove Theorem [Theorem 4](#4.4){reference-type="ref" reference="4.4"}. *Proof of Theorem [Theorem 4](#4.4){reference-type="ref" reference="4.4"}.* Let us define $$\overline{u}= \limsup_{t \to \infty} \left( \sup_{x \in \Gamma} u(x,t)\right) \quad \text{and} \quad \underline{u} =\liminf_{t \to \infty} \left( \inf_{x \in \Gamma} u(x,t)\right),$$ then for $\varepsilon>0$ there exists $t_{\varepsilon}>0$ such that $$\underline{u} - \varepsilon \leq \inf_{x \in \Gamma} u(x,t) \leq u(x,t) \leq \sup_{x \in \Gamma} u(x,t) \leq \overline{u}+ \varepsilon,\quad \ t\geq t_{\varepsilon}.$$ Let us note that $v_{xx}(x,t)-v(x,t)+u(x,t)=0$ together with the comparison principle for elliptic equations yield $$\label{vpeq1} \underline {u}_\epsilon:=\underline{u} - \epsilon \leq v(x,t) \leq \overline{u}+ \epsilon:=\overline u_\epsilon,\quad x \in \Gamma, t\geq t_{\epsilon}.$$ Hence, using the first equation in [\[parabolic-parabolic-eq\]](#parabolic-parabolic-eq){reference-type="eqref" reference="parabolic-parabolic-eq"} we obtain $$\label{parab1} u_t \leq u_{xx}-\chi u_x v_x -\chi u(\underline{u}^\epsilon-u)+u(a-bu).$$ Consider the initial value problem for the following logistic equation $$\label{parab2} \begin{cases} \overline{w}_t&=\chi\overline{w}(\overline{w}- \underline{u}^\epsilon) + \overline{w}(a-b\overline{w})\ \\ &=-(b-\chi)\overline{w}^2 +(a -\chi \underline{u}^\epsilon)\overline{w},\ t \geq t_{\epsilon},\\ \overline{w}(t_\epsilon)&= \underset{x \in \Gamma} \max\ u(x, t_{\varepsilon}), \end{cases}$$ Combining [\[parab1\]](#parab1){reference-type="eqref" reference="parab1"}, [\[parab2\]](#parab2){reference-type="eqref" reference="parab2"} and the comparison principle for parabolic equations one obtains $$\label{upbnd} u(x,t )\leq \overline{w}(t), x \in \Gamma, t \geq t_\varepsilon.$$ Moreover, the logistic equation [\[parab2\]](#parab2){reference-type="eqref" reference="parab2"} yields $$\overline{w}(t) \rightarrow \frac{(a- \chi\underline{u}^\epsilon)_+}{b-\chi} \quad \text{as} \quad t \rightarrow \infty.$$ Then employing [\[upbnd\]](#upbnd){reference-type="eqref" reference="upbnd"} one infers $$\overline{u}\leq \frac{{ (a- \chi(\underline{u}-\varepsilon))_+}}{b-\chi}, \quad \forall\, \varepsilon>0.$$ Hence, one has $$\label{centin2} \overline{u}\leq \frac{{ (a- \chi\underline{u})_+}}{b-\chi}$$ By a similar argument, using $$\label{parab1new} u_t \geq u_{xx}-\chi u_x v_x -\chi u(\overline{u}^\epsilon-u)+u(a-bu).$$ Consider the initial value problem for the following logistic equation $$\label{parab2new} \begin{cases} \underline{w}_t&=\chi\underline{w}(\underline{w}- \overline{u}^\varepsilon) + \underline{w}(a-b\underline{w})\ \\ &=-(b-\chi)\underline{w}^2 +(a -\chi \overline{u}^\varepsilon)\underline{w},\ t \geq t_{\epsilon},\\ \overline{w}(t_\epsilon)&= \underset{x \in \Gamma} \min\ u(x, t_{\varepsilon}), \end{cases}$$ one obtains $$\label{centin} \underline{u}\geq \frac{a- \chi\overline{u}}{b-\chi}.$$ Note that $a-\chi\underline u>0$, for otherwise, by [\[centin2\]](#centin2){reference-type="eqref" reference="centin2"}, we have $\overline u=0$ and then $\underline u=0$. But by [\[centin\]](#centin){reference-type="eqref" reference="centin"}, we have $\underline u\ge \frac{a}{b-\chi}>0$, which yields a contradiction. Hence, one has $a-\chi\underline u>0$. Then, employing [\[centin\]](#centin){reference-type="eqref" reference="centin"}, [\[centin2\]](#centin2){reference-type="eqref" reference="centin2"} we obtain $$\begin{aligned} \begin{split}\label{inaga} a-\chi\overline{u}+ (b-\chi)\overline{u} &\leq (b-\chi)\underline{u}+ (a-\chi\underline{u}),\\ (b-2\chi)\overline{u} &\leq (b-2\chi)\underline{u}. \end{split} \end{aligned}$$ Recalling $b > 2\chi$ and $\overline{u} \geq \underline{u}$ we obtain $\overline{u} =\underline{u}$. This identity together with [\[centin2\]](#centin2){reference-type="eqref" reference="centin2"} (resp. [\[centin\]](#centin){reference-type="eqref" reference="centin"}) gives $\overline u\le \frac{a}{b}$ (resp. $\overline u\ge \frac{a}{b}$). Hence $\overline{u}=\underline{u}= \frac{a}{b}$, which in turn shows $$\lim_{t \rightarrow \infty}\left\|u(t, \cdot;u_0)-\frac{a}{b}\right\|_{L^{\infty}(\Gamma)} =0.$$ Similar inequality for $v$ follows from [\[vpeq1\]](#vpeq1){reference-type="eqref" reference="vpeq1"}. ◻ # Numerical illustrations {#numerics} In this section we provide numerical computation of bifurcation points for the following graphs: - the dumbbell graph, see Figure [2](#dumbbell){reference-type="ref" reference="dumbbell"}, whose eigenvalues $\lambda=k^2$ can be determined from the secular equation $$\sin\frac {\ell_1 k}2\sin\frac {\ell_3 k}2\left[\left(4\sin\frac {\ell_1 k}2\sin\frac {\ell_3 k}2-\cos\frac {\ell_1 k}2\cos\frac {\ell_3 k}2\right)\sin\frac {\ell_2 k}2-2\cos\frac {\ell_2 k}2\sin\frac{(\ell_1+\ell_3)k}{2}\right]=0,$$ - the tadpole graph, see Figure [4](#Tadpole){reference-type="ref" reference="Tadpole"}, whose eigenvalues $\lambda=k^2$ can be determined from the secular equation $$2\cos(\ell_1 k)\cos(\ell_2k)-\sin(\ell_1 k)\sin(\ell_2k)=0,$$ - the figure $8$ graph, see Figure [6](#fig8){reference-type="ref" reference="fig8"}, whose eigenvalues $\lambda=k^2$ can be determined from the secular equation $$\sin\left(\frac {\ell_1 k}2\right)\sin\left(\frac {\ell_2 k}2\right)\sin\left(\frac {(\ell_1+\ell_2) k}2\right)=0,$$ - the 3-star graph, see Figure [8](#3star){reference-type="ref" reference="3star"}, whose eigenvalues $\lambda=k^2$ can be determined from the secular equation $$\begin{aligned} \sin(\ell_1 k) \cos(\ell_2 k)\cos(\ell_3 k)+\cos(\ell_1 k) &\sin(\ell_2 k)\sin(\ell_3 k)+\\ &+ \sin(\ell_1 k) \sin(\ell_2 k)\cos(\ell_3 k)=0. \end{aligned}$$ {#Tadpole width="1.3\\linewidth"} {#Tadpole} {#fig8 width="1.3\\linewidth"} {#fig8} {#3star width="1.3\\linewidth"} {#3star} # Auxiliary results {#functionalspaces} In this section we record several facts about fractional power spaces ${\mathcal X}^{\alpha}_p$, $\alpha\in(0,1)$, $1\leq p< \infty$ generated by the Neumann--Kirchhoff Laplacian on compact metric graphs on $\Gamma=(\mathcal{V},\mathcal{E})$. Let $$\label{L-p-eq} {L^p(\Gamma):=\bigoplus_{e\in{\mathcal E}}L^p(e)\quad {\rm and}\quad \ \widehat{W}^{s,p}(\Gamma):=\bigoplus_{e\in{\mathcal E}} W^{s,p}(e).}$$ By definition, see, e.g., [@Henry Section 1.3], ${\mathcal X}^{\alpha}_p:=\operatorname{dom}((I_{L^p(\Gamma)}-\Delta)^{\alpha})$ is equipped with the graph norm of $(I_{L^p(\Gamma)}-\Delta)^{\alpha}$. Throughout this paper we used bounded embeddings of ${\mathcal X}^{\alpha}_p$ into various function spaces. Such embeddings are well known in the case of classical domains $\Omega\subset{\mathbb{R}}^n$, cf. [@Henry Chapter 1]. To the best of our knowledge, these type of embedding for metric graphs, although expected, have not appeared in print. For completeness of exposition we present them in Theorem [Theorem 6](#analytic-semigroup-thm2){reference-type="ref" reference="analytic-semigroup-thm2"} below. Let us first introduce some notation. The edge-wise direct sum of Banach spaces of functions will be denoted by $\widehat\ $, in particular, we write $$\begin{aligned} \label{hats} \widehat C_0^{\infty}(\Gamma):= \bigoplus_{e\in{\mathcal E}} C_0^{\infty}(e), &\widehat W^{k,p}({\Gamma}):= \bigoplus_{e\in{\mathcal E}}W^{k,p}(e), k\in{\mathbb{N}}_0,\widehat C^{\nu}(\overline{{\Gamma}}) := \bigoplus_{e\in{\mathcal E}}\widehat C^{\nu}(\overline{e}),\end{aligned}$$ where $\nu>0$ and $C^{\nu}(\overline{e})$ denotes the usual space of Hölder continuous functions defined on the closed interval $\overline{e}$ endowed with the norm $$\|u\|_{C^\nu(\bar e)}=\sum_{\alpha\in{\mathbb{N}}_0,\alpha\le [\nu]}\sup_{x\in\bar e} |u^{(\alpha)}(x)|+\sup_{x,y\in\bar e,x\not =y} \frac{|u^{([\nu])}(x)-u^{([\nu])}(y)|}{|x-y|^{\nu-[\nu]}}.$$ Let us note that the edge-wise direct sums introduced in [\[hats\]](#hats){reference-type="eqref" reference="hats"} induce no vertex conditions as oppose to the space of continuous functions on the closure $\overline{{\Gamma}}$ of the graph $$C(\overline \Gamma)=\{u\in \widehat C(\overline \Gamma)\, :\, u\,\, {\rm is \,\, continuous\,\, at\,\, the\,\, vertices\,\, of}\,\, \Gamma\}.$$ In the following theorem $(L^p(\gamma),\widehat{W}^{2,p}(\Gamma))_{\theta,q}$ denotes the interoplation space between $L^p(\Gamma)$ and $\widehat{W}^{2,p}(\Gamma)$ via the $K$-method, where $0<\theta<1$ and $1\le q<\infty$ (see [@Tri Section 1.3.2] for definition). **Theorem 6**. *Suppose that $1\leq p<\infty$. Then one has* - *For any $q\ge p$ and $s-\frac{1}{p}> t-\frac{1}{q}$, there holds $$\label{sobolev-embedding-eq3-0} \widehat W^{s,p} (\Gamma)\hookrightarrow \widehat W^{t,q}(\Gamma),$$ $$\label{sobolev-embedding-eq2} \widehat{W}^{s,p}(\Gamma)\hookrightarrow \widehat{C}^r(\overline \Gamma) \quad r<s-\frac{1}{p},$$ and for $s\in (0,1)\setminus\{\frac{1}{2}\}$ one has $$\label{sobolev-embedding-eq3} (L^p(\Gamma), \widehat{W}^{2,p}(\Gamma))_{s,p}=\widehat{W}^{2s,p}(\Gamma).$$ * - *One has $$\label{sobolev-embedding-eq4} (\widehat{L}^p(\Gamma), X_p^\alpha)_{\theta,p}=(\widehat{L}^p(\Gamma),\mathcal{D}(A_p))_{{\alpha}\theta,p},\quad \, 0<\theta<1,$$* *$$\label{sobolev-embedding-eq5-0} X_p^\alpha \hookrightarrow \widehat{W}^{2\alpha\theta,p}(\Gamma),\quad \, 0<\theta<1,$$* *and $$\label{sobolev-embedding-eq5} X_p^\alpha \hookrightarrow \widehat C^\nu(\overline \Gamma),\quad \, 0<\nu <2\alpha -\frac{1}{p}.$$* *Proof.* (1) [\[sobolev-embedding-eq3-0\]](#sobolev-embedding-eq3-0){reference-type="eqref" reference="sobolev-embedding-eq3-0"} and [\[sobolev-embedding-eq2\]](#sobolev-embedding-eq2){reference-type="eqref" reference="sobolev-embedding-eq2"} follow from [@Ama Theorem 11.5], and [\[sobolev-embedding-eq3\]](#sobolev-embedding-eq3){reference-type="eqref" reference="sobolev-embedding-eq3"} follows from [@Ama Theorem 11.6]. (2) First, [\[sobolev-embedding-eq4\]](#sobolev-embedding-eq4){reference-type="eqref" reference="sobolev-embedding-eq4"} follows from [@Tri (unnumbered) Theorem on page 101]. To prove [\[sobolev-embedding-eq5-0\]](#sobolev-embedding-eq5-0){reference-type="eqref" reference="sobolev-embedding-eq5-0"}, for a given $0<\nu<2\alpha-\frac{1}{p}$, let us choose $\theta\in (0,1)$ such that $\frac{\alpha}{2}\theta\not =1$ and $2 \alpha \theta-\frac{1}{p}>\nu$. Then one has $$\begin{aligned} %\label{embedding-eq5} {\mathcal X}_p^\alpha &\subset (L^p(\Gamma), {\mathcal X}_p^\alpha)_{\theta,p}=(L^p(\Gamma),\mathcal{D}(A))_{{\alpha}\theta,p}\underset{\eqref{sobolev-embedding-eq4}}{\subset} ((L^p(\Gamma),\widehat{W}^{2,p}(\Gamma))_{{\alpha \theta},p}\underset{ \eqref{sobolev-embedding-eq3}}{=}\widehat{W}^{2\alpha\theta,p}(\Gamma). \end{aligned}$$ The embedding [\[sobolev-embedding-eq5\]](#sobolev-embedding-eq5){reference-type="eqref" reference="sobolev-embedding-eq5"} follows from [\[sobolev-embedding-eq2\]](#sobolev-embedding-eq2){reference-type="eqref" reference="sobolev-embedding-eq2"} and [\[sobolev-embedding-eq5-0\]](#sobolev-embedding-eq5-0){reference-type="eqref" reference="sobolev-embedding-eq5-0"}. ◻ **Proposition 1**. *[@MR610244 Theorem 1.4.3]. Let $\sigma>0$, $\alpha\in[0,1)$, $p\in[1, \infty)$. Then there exists $C>0$ such that for arbitrary $t>0$ and $u\in L^p(\Gamma)$, $v\in{\mathcal X}^{\alpha}$ one has $$\begin{aligned} &\|({\sigma I}-\Delta)^{\alpha}e^{(\Delta-{\sigma I})t}u\|_{L^p(\Gamma)}\leq C t^{-\alpha}e^{-\frac{ \sigma}{2} t} \|u\|_{L^p(\Gamma)},\label{fracpow}\\ & \|(e^{(\Delta-\sigma I)t}-I)v\|_{L^p(\Gamma)}\leq C t^{\alpha} \|v\|_{{\mathcal X}^{\alpha}_p}. \label{fracpow2} \end{aligned}$$* **Theorem 7**. *[@HWS].[\[lplq estimate1\]]{#lplq estimate1 label="lplq estimate1"} Let $\{e^{\Delta t}\}_{t\geq 0}$ be analytic semigroup generated by the Neumann--Kirchhoff Laplacian in $L^p(\Gamma)$, $1\leq p<\infty$. 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arxiv_math

--- abstract: | Let $T$ be a complete, model complete o-minimal theory extending the theory of real closed ordered fields and assume that $T$ is power bounded. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring $\mathcal O$ and a $T$-derivation $\der$ such that $\der$ is monotone, i.e., weakly contractive with respect to the valuation induced by $\mathcal O$. We show that the theory of monotone $T$-convex $T$-differential fields, i.e., the common theory of such $K$, has a model completion, which is complete and distal. Among the axioms of this model completion, we isolate an analogue of henselianity that we call $T^\der$-henselianity. We establish an Ax--Kochen/Ershov theorem and further results for monotone $T$-convex $T$-differential fields that are $T^\der$-henselian. address: - Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada - Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Austria author: - Elliot Kaplan - Nigel Pynn-Coates title: Monotone $T$-convex $T$-differential fields --- # Introduction Let $\mathbb{R}_{\mathop{\mathrm{an}}}$ denote the expansion of the real field $\mathbb{R}$ by all globally subanalytic sets. Explicitly, $\mathbb{R}_{\mathop{\mathrm{an}}}$ is the structure obtained by adding a new function symbol for each $n$-ary function $F$ which is real analytic on a neighborhood of $[-1,1]^n$, and by interpreting this function symbol as the restriction of $F$ to $[-1,1]^n$. Let $T_{\mathop{\mathrm{an}}}$ be the elementary theory of $\mathbb{R}_{\mathop{\mathrm{an}}}$ in the language extending the language of ordered rings by the function symbols described above. This theory is model complete and o-minimal [@Ga68; @vdD86]. Let $K$ be a model of $T_{\mathop{\mathrm{an}}}$. A map $\der\colon K\to K$ is said to be a *$T_{\mathop{\mathrm{an}}}$-derivation* if it is a field derivation on $K$ which satisfies the identity $$\der F(u)\ =\ \frac{\partial F}{\partial Y_1}(u)\der u_1+\cdots + \frac{\partial F}{\partial Y_n}(u)\der u_n$$ for each restricted analytic function $F$ and each $u = (u_1,\ldots,u_n) \in K^n$ with $|u_i|< 1$ for each $i$. The only $T_{\mathop{\mathrm{an}}}$-derivation on $\mathbb{R}_{\mathop{\mathrm{an}}}$ itself is the trivial derivation, which takes constant value zero. However, there are a number of interesting examples of nonstandard models of $T_{\mathop{\mathrm{an}}}$ with nontrivial $T_{\mathop{\mathrm{an}}}$-derivations. 1. [\[introex1\]]{#introex1 label="introex1"} Consider $\mathbb{R}(\!(t^{1/\infty})\!)\coloneqq \bigcup_n\mathbb{R}(\!(t^{1/n})\!)$, the field of Puiseux series over $\mathbb{R}$. We totally order $\mathbb{R}(\!(t^{1/\infty})\!)$ by taking $t$ to be a positive infinitesimal element. Then $\mathbb{R}(\!(t^{1/\infty})\!)$ admits an expansion to a model of $T_{\mathop{\mathrm{an}}}$, by extending each restricted analytic function on $\mathbb{R}$ to the corresponding box in $\mathbb{R}(\!(t^{1/\infty})\!)$ via Taylor expansion. We put $x \coloneqq 1/t$, and we let $\der\colon \mathbb{R}(\!(t^{1/\infty})\!)\to \mathbb{R}(\!(t^{1/\infty})\!)$ be the derivation with respect to $x$, so $$\der \sum_{q\in \mathbb{Q}}r_qx^q\ \coloneqq\ \sum_{q\in \mathbb{Q}}r_qqx^{q-1}.$$ As $\der$ commutes with infinite sums, it is routine to verify that $\der$ is indeed a $T_{\mathop{\mathrm{an}}}$-derivation. 2. [\[introex2\]]{#introex2 label="introex2"} The field $\mathbb{T}$ of logarithmic-exponential transseries also admits a canonical expansion to a model of $T_{\mathop{\mathrm{an}}}$ using Taylor expansion [@DMM97 Corollary 2.8]. The usual derivation on $\mathbb{T}$ is a $T_{\mathop{\mathrm{an}}}$-derivation. 3. [\[introex3\]]{#introex3 label="introex3"} Let $\boldsymbol{k}\models T_{\mathop{\mathrm{an}}}$ and let $\der_{\boldsymbol{k}}$ be a $T_{\mathop{\mathrm{an}}}$-derivation on $\boldsymbol{k}$. Let $\Gamma$ be a divisible ordered abelian group, and consider the Hahn field $\boldsymbol{k}(\!(t^\Gamma )\!)$, ordered so that $0<t<\boldsymbol{k}^>$. We expand $\boldsymbol{k}(\!(t^\Gamma )\!)$ to a model of $T_{\mathop{\mathrm{an}}}$ using Taylor expansion (see [@Ka21 Proposition 2.13] for details). Let $c\colon \Gamma\to \boldsymbol{k}$ be an additive map. We use $c$ to define a derivation $\der$ on $\boldsymbol{k}(\!(t^\Gamma )\!)$ as follows: $$\der\sum_\gamma f_\gamma t^\gamma\ \coloneqq\ \sum_\gamma\big(\der_{\boldsymbol{k}}f_\gamma + f_\gamma c(\gamma)\big)t^\gamma.$$ The map $\der$ is the unique derivation which extends $\der_{\boldsymbol{k}}$, commutes with infinite sums, and satisfies the identity $\der t^\gamma = c(\gamma)t^\gamma$ for each $\gamma\in \Gamma$. By [@Ka21 Proposition 3.14], this derivation is even a $T_{\mathop{\mathrm{an}}}$-derivation. We denote this expansion of $\boldsymbol{k}(\!(t^\Gamma)\!)$ by $\boldsymbol{k}(\!(t^\Gamma)\!)_{\mathop{\mathrm{an}},c}$. In each of the above examples, there is a natural convex valuation ring $\mathcal O$ (the convex hull of $\mathbb{R}$ in the first two examples, the convex hull of $\boldsymbol{k}$ in the third). These convex valuation rings are even *$T_{\mathop{\mathrm{an}}}$-convex*, as defined by van den Dries and Lewenberg [@DL95], and the derivation in each example is continuous. In this paper, we work not just with the theory $T_{\mathop{\mathrm{an}}}$, but with any complete, model complete, power bounded o-minimal $\mathcal L$-theory $T$ (where *power boundedness* is the assumption that every definable function is eventually dominated by a power function $x \mapsto x^\lambda$). Let $K$ be a model of $T$. A *$T$-derivation* is a map $\der\colon K\to K$ which satisfies the identity $$\der F(u)\ =\ \frac{\partial F}{\partial Y_1}(u)\der u_1+\cdots + \frac{\partial F}{\partial Y_n}(u)\der u_n$$ for all $\mathcal L(\emptyset)$-definable functions $F$ which are $\mathcal C^1$ at $u$, and a *$T$-convex valuation ring* is a nonempty convex subset $\mathcal O\subseteq K$ which is closed under all $\mathcal L(\emptyset)$-definable continuous functions. A **$T$-convex $T$-differential field** is the expansion of $K$ by a $T$-convex valuation ring and a continuous $T$-derivation. In examples [\[introex1\]](#introex1){reference-type="ref" reference="introex1"} and [\[introex3\]](#introex3){reference-type="ref" reference="introex3"} above, the derivation $\der$ is **monotone**: the logarithmic derivative $\der a/a$ is in the valuation ring $\mathcal O$ for each nonzero $a$. The derivation in example [\[introex2\]](#introex2){reference-type="ref" reference="introex2"} is not monotone. In this paper, our focus is primarily on monotone $T$-convex $T$-differential fields, and in this setting, our assumption that $T$ is power bounded comes almost for free; see Remark [Remark 15](#rem:powerbounded){reference-type="ref" reference="rem:powerbounded"}. We prove the following: **Theorem 1**. *The theory of monotone $T$-convex $T$-differential fields has a model completion. This model completion is complete and distal (in particular, it has NIP).* Our model completion is quite similar to Scanlon's model completion for the theory of monotone valued differential fields [@Sc00]. In the case $T = T_{\mathop{\mathrm{an}}}$, a model of this model completion can be constructed as follows: consider the $T_{\mathop{\mathrm{an}}}$-convex $T_{\mathop{\mathrm{an}}}$-differential field $\boldsymbol{k}(\!(t^\Gamma)\!)_{\mathop{\mathrm{an}},c}$ in example [\[introex3\]](#introex3){reference-type="ref" reference="introex3"}, where the derivation $\der_{\boldsymbol{k}}$ on $\boldsymbol{k}$ is *generic*, as defined in [@FK21], and where $c$ is taken to be the zero map. In axiomatizing this model completion, we introduce an analogue of differential-henselianity for $T$-convex $T$-differential fields, which we call *$T^\der$-henselianity*. This condition states that $\der$ maps the maximal ideal $\smallo$ of $\mathcal O$ into itself (small derivation); that for any $a_0,\ldots,a_r \in \mathcal O$, not all in $\smallo$, there is $y \in K$ such that $a_0y + a_1\der y + \ldots+a_r\der^r y-1\in \smallo$ (linearly surjective differential residue field); and that for every $\mathcal L(K)$-definable function $F\colon K^r\to K$, if $a\in K$ is an approximate zero of the function $y\mapsto y^{(r)} - F(y,\ldots,y^{(r-1)})$, and if this function is well-approximated by a linear differential operator on a neighborhood of $a$, then it has an actual zero in this neighborhood. This last condition is inspired by Rideau-Kikuchi's definition of *$\sigma$-henselianity* for analytic difference valued fields [@Ri17]. We prove an Ax--Kochen/Ershov theorem for monotone $T^\der$-henselian fields, from which we derive the following: **Theorem 2**. *Any monotone $T^\der_{\mathop{\mathrm{an}}}$-henselian field $(K,\mathcal O,\der)$ is elementarily equivalent to some $\boldsymbol{k}(\!(t^\Gamma)\!)_{\mathop{\mathrm{an}},c}$.* Hakobyan [@Ha18] previously established a similar statement for monotone differential-henselian fields. Let $K$ be a $T$-convex $T$-differential field with small derivation and linearly surjective differential residue field. With an eye toward future applications, we develop the theory of $T^\der$-henselian fields as much as possible without the assumption that $\der$ is monotone. We show that if $K$ is spherically complete, then $K$ is $T^\der$-henselian, and that any $T^\der$-henselian $K$ admits a *lift of the differential residue field*: an elementary $\mathcal L$-substructure $\boldsymbol{k}\preccurlyeq_{\mathcal L} K$ which is closed under $\der$ and which maps isomorphically onto the differential residue field $\mathop{\mathrm{res}}(K)$. An essential ingredient in this proof is the fact, due to García Ramírez, that $\mathcal L$-definable functions enjoy the *Jacobian property* [@Ga20]. We also show that if each immediate extension of $K$ has a property we call the *$T^\der$-henselian configuration property*, then $K$ has a unique spherically complete immediate $T$-convex $T$-differential field extension with small derivation. The existence of such a spherical completion was previously established by the first author [@Ka22 Corollary 6.4], and the $T^\der$-henselian configuration property is similar to the differential-henselian configuration property of van den Dries and the second author [@DPC19]. Again making use of the Jacobian property, we show that monotone fields enjoy the $T^\der$-henselian configuration property, thereby establishing the following: **Theorem 3**. *Let $K$ be a monotone $T$-convex $T$-differential field with linearly surjective differential residue field. Then $K$ has a unique spherically complete immediate monotone $T$-convex $T$-differential field extension, up to isomorphism over $K$.* This uniqueness plays an essential role in our Ax--Kochen/Ershov theorem. One may be able to adapt our definition of $T^\der$-henselianity to study compatible derivations on other tame expansions of equicharacteristic zero valued fields. One broad class where our methods may generalize is the class of *Hensel minimal* expansions of valued fields, introduced by Cluckers, Halupczok, and Rideau-Kikuchi [@CHRK22]. The setting of Hensel minimal fields (1-h-minimal fields to be precise), includes our present o-minimal setting, as well as the fields with analytic structure studied by Cluckers and Lipshitz [@CL11]. As an indicator that generalizing to this class may be possible, we note that the Jacobian property, which proves so useful in this paper, holds in the setting of 1-h-minimal fields [@CHRK22 Theorem 5.4.10]. ## Organization of the paper Section [2](#sec:background){reference-type="ref" reference="sec:background"} contains background information on $T$-convex valuation rings and $T$-convex differential fields and results that we need, drawn from [@DL95; @vdD97; @Yi17; @Ga20; @FK21; @ADH17; @Ka22]. In Section [3](#sec:Tdh){reference-type="ref" reference="sec:Tdh"} we introduce $T^\der$-henselianity and record basic consequences in Section [3.1](#sec:Tdhbasic){reference-type="ref" reference="sec:Tdhbasic"}. In Section [3.2](#sec:lift){reference-type="ref" reference="sec:lift"}, we show that $T^\der$-henselianity yields a lift the differential residue field (Theorem [Theorem 20](#thm:reslift){reference-type="ref" reference="thm:reslift"}). Section [3.3](#sec:sphcompTdh){reference-type="ref" reference="sec:sphcompTdh"} relates $T^\der$-henselianity to immediate extensions and shows that every spherically complete $T$-convex $T$-differential field with small derivation and linearly surjective differential residue field is $T^\der$-henselian (Corollary [Corollary 23](#cor:sphcompTdh){reference-type="ref" reference="cor:sphcompTdh"}). The next subsection develops further technical material for use in Section [3.5](#sec:sphcompunique){reference-type="ref" reference="sec:sphcompunique"}, which establishes the uniqueness of spherically complete immediate extensions and related results, conditional on the $T^\der$-hensel configuration property introduced there. Section [4](#sec:monotone){reference-type="ref" reference="sec:monotone"} focuses on monotone fields. In Section [4.1](#sec:monotoneTdhc){reference-type="ref" reference="sec:monotoneTdhc"}, we show that monotone $T$-convex $T$-differential fields have the $T^\der$-hensel configuration property and summarize the consequences. The next subsection records variants of the results in Section [3.5](#sec:sphcompunique){reference-type="ref" reference="sec:sphcompunique"}, of which only Corollary [Corollary 46](#cor:sphcompembed){reference-type="ref" reference="cor:sphcompembed"} is needed later, in the proof of the Ax--Kochen/Ershov theorem. The short Section [5](#sec:RCF){reference-type="ref" reference="sec:RCF"} explains what $T^\der$-henselianity means when $T$ is the theory of real closed ordered fields. Section [6](#sec:AKE){reference-type="ref" reference="sec:AKE"} contains the Ax--Kochen/Ershov theorem and related three-sorted results, while Section [7](#sec:modelcompletion){reference-type="ref" reference="sec:modelcompletion"} contains the model completion result and related one-sorted results. # Background {#sec:background} ## Notation and conventions Throughout, $m,n,q,r$ range over $\mathbb{N}\coloneqq \{0,1,2,\dots\}$. For a unital ring $R$, we let $R^\times$ denote the multiplicative group of units of $R$, and for an ordered set $S$ equipped with a distinguished element $0$, we set $S^> \coloneqq \{ s \in S : s>0 \}$. In this paper, we fix a complete, model complete o-minimal theory $T$ extending the theory of real closed ordered fields in an appropriate language $\mathcal L\supseteq\{0,1,+,\cdot,<\}$. Throughout, $K$ is a model of $T$. Given a subset $A \subseteq K$, we let $\mathop{\mathrm{dcl}}_\mathcal L(A)$ denote the $\mathcal L$-definable closure of $A$. Since $T$ has definable Skolem functions, $\mathop{\mathrm{dcl}}_\mathcal L(A)$ is (the underlying set of) an elementary substructure of $K$. It is well-known that $\mathop{\mathrm{dcl}}_\mathcal L$ is a pregeometry, and we denote the corresponding rank by $\mathop{\mathrm{rk}}_\mathcal L$. Let $M$ be a $T$-extension of $K$ (that is, a model of $T$ which extends $K$) and let $A\subseteq M$. We denote by $K\langle A\rangle$ the intermediate extension of $K$ with underlying set $\mathop{\mathrm{dcl}}_\mathcal L(K\cup A)$. We say that $A$ is **$\mathcal L(K)$-independent** if $A$ is independent over $K$ with respect to the pregeometry $\mathop{\mathrm{dcl}}_\mathcal L$. Otherwise $A$ is said to be **$\mathcal L(K)$-dependent**. If $A$ is $\mathcal L(K)$-independent and $M = K\langle A\rangle$, then $A$ is said to be a **$\mathop{\mathrm{dcl}}_\mathcal L$-basis for $M$ over $K$**. Given $a = (a_1,\ldots,a_n) \in M^n$, we write $K\langle a \rangle$ in place of $K\langle \{a_1,\ldots,a_n\}\rangle$, and we say that $a$ is **$\mathcal L(K)$-independent** if the set $\{a_1,\ldots,a_n\}$ is $\mathcal L(K)$-independent and no components are repeated. Equivalently, $a$ is **$\mathcal L(K)$-independent** if $\mathop{\mathrm{rk}}_\mathcal L(a|K) = n$. For tuples $a,b \in K^n$, we let $a\cdot b\coloneqq a_1b_1+\cdots+a_nb_n$ denote the inner product of $a$ and $b$. A **power function** is an $\mathcal L(K)$-definable endomorphism of the ordered multiplicative group $K^>$. Each power function $f$ is uniquely determined by its derivative at $1$, and if $f'(1) = \lambda$, then we suggestively write $a^\lambda$ instead of $f(a)$ for $a \in K^>$. The collection $\Lambda \coloneqq \{f'(1): f\text{ is a power function}\}$ forms a subfield of $K$, called the **field of exponents of $K$**. If every $\mathcal L(K)$-definable function is eventually bounded by a power function, then $K$ is said to be **power bounded**. Suppose $K$ is power bounded. Then by Miller's dichotomy [@Mi96], every model of $T$ is power bounded and every power function is $\mathcal L(\emptyset)$-definable, so $\Lambda$ does not depend on $K$. We just say that , and we call $\Lambda$ the **field of exponents of $T$**. For the remainder of the paper, we assume that $T$ is power bounded with field of exponents $\Lambda$. ## Background on $T$-convex valuation rings In this subsection, let $\mathcal O$ be a **$T$-convex valuation ring** of $K$; that is, $\mathcal O\subseteq K$ is convex and nonempty and $F(\mathcal O)\subseteq \mathcal O$ for every $\mathcal L(\emptyset)$-definable continuous $F \colon K \to K$. We call $(K, \mathcal O)$ a **$T$-convex valued field**. These structures were introduced and studied by van den Dries and Lewenberg [@DL95] and additionally by van den Dries [@vdD97]. We briefly review notation, following [@Ka22 Section 1], and general facts we use; additional relevant facts on $T$-convex valuation rings can be found there, or in the original papers. For valuation-theoretic notation we follow [@ADH17 Chapter 3]. Let $\mathcal L^{\mathcal O}\coloneqq \mathcal L\cup \{\mathcal O\}$ be the extension of $\mathcal L$ by a unary predicate $\mathcal O$ and $T^{\mathcal O}$ be the theory extending $T$ by axioms stating that $\mathcal O$ is a *proper* $T$-convex valuation ring. This $T^{\mathcal O}$ is complete and model complete, since $T$ is [@DL95 Corollary 3.13]; if $T$ has quantifier elimination and a universal axiomatization, then $T^{\mathcal O}$ has quantifier elimination [@DL95 Theorem 3.10]. Let $\Gamma$ be the value group of the valuation $v \colon K^{\times} \to \Gamma$ induced by $\mathcal O$, which moreover is an ordered $\Lambda$-vector space with scalar multiplication $\lambda va \coloneqq v(a^{\lambda})$ for $a \in K^>$ and $\lambda \in \Lambda$. We extend $v$ to $v \colon K \to \Gamma \cup \{\infty\}$ by $v(0)\coloneqq \infty$, where $\infty$ is a symbol not in $\Gamma$, and extend the ordering and addition of $\Gamma$ to $\Gamma \cup \{\infty\}$ in the natural way. We also extend $v$ to $K^n$ by $va \coloneqq \min\{va_1,\ldots,va_n\}$ for $a \in K^n$. For $a,b \in K$, we set: $$\begin{array}{lc} a \preccurlyeq b\ \Leftrightarrow\ va\geqslant vb,\qquad a \prec b\ \Leftrightarrow\ va>vb,\\ a \asymp b\ \Leftrightarrow\ va=vb,\qquad a\sim b\ \Leftrightarrow\ a-b \prec b. \end{array}$$ The relations $\asymp$ and $\sim$ are equivalence relations on $K$ and $K^{\times}$, respectively. If $a \sim b$, then $a \asymp b$ and also $a>0$ if and only if $b>0$. The unique maximal ideal of $\mathcal O$ is $\smallo$ and $\mathop{\mathrm{res}}(K) \coloneqq \mathcal O/\smallo$ is the **residue field** of $K$. We usually denote $\mathop{\mathrm{res}}(K)$ by $\boldsymbol{k}$ (unlike in [@Ka22]) and let $\bar{a}$ or $\mathop{\mathrm{res}}(a)$ denote the image of $a \in \mathcal O$ under the residue map to $\boldsymbol{k}$. In fact, $\boldsymbol{k}$ can be expanded to a model of $T$ [@DL95 Remark 2.16], and we always construe $\boldsymbol{k}$ this way. Related is the fact that the convex hull of an elementary $\mathcal L$-substructure of $K$ is a $T$-convex valuation ring of $K$; cf. Section [3.2](#sec:lift){reference-type="ref" reference="sec:lift"} on lifting the residue field. If we need to indicate the dependence of these objects on $K$, we do so using a subscript, as in $\Gamma_K$ and $\boldsymbol{k}_K$. Given a $T^{\mathcal O}$-extension $M$ of $K$, we identify $\boldsymbol{k}$ with a $T$-submodel of $\boldsymbol{k}_M$ and $\Gamma$ with an ordered $\Lambda$-subspace of $\Gamma_M$ in the natural way. If $\Gamma_M = \Gamma$ and $\boldsymbol{k}_M = \boldsymbol{k}$, then the $T^{\mathcal O}$-extension $M$ is said to be **immediate**. As a consequence of our power boundedness assumption, we have the following analogue of the Abhyankar--Zariski inequality, established by van den Dries and referred to in the literature as the **Wilkie inequality**. **Fact 1** ([@vdD97 Section 5]). *Let $M$ be a $T^{\mathcal O}$-extension of $K$ and suppose $\mathop{\mathrm{rk}}_\mathcal L(M|K)$ is finite. Then $$\mathop{\mathrm{rk}}_\mathcal L(M|K)\ \geqslant\ \mathop{\mathrm{rk}}_\mathcal L(\boldsymbol{k}_M|\boldsymbol{k})+\dim_\Lambda(\Gamma_M/\Gamma).$$* An **open $v$-ball** is a set of the form $B(a, \gamma) \coloneqq \{ b \in K : v(b-a) > \gamma \}$ for $a \in K$ and $\gamma \in \Gamma$. These open $v$-balls form a basis for the *valuation topology* on $K$, which coincides with its order topology since $\mathcal O$ is assumed to be a proper subring. Similarly, a **closed $v$-ball** is a set of the form $\{ b \in K : v(b-a) \geqslant\gamma \}$; in this latter definition we allow $\gamma \in \Gamma \cup \{\infty\}$ so that singletons are closed $v$-balls. Later we only use open $v$-balls, but closed $v$-balls are needed for the next definitions. A collection of closed $v$-balls is **nested** if any two balls in the collection have nonempty intersection (recall that any two $v$-balls are either disjoint or one is contained in the other). We call $K$ **spherically complete** if every nonempty nested collection of closed $v$-balls has nonempty intersection. For the relationship between spherical completeness and immediate extensions, see the next subsection and the beginning of Section [3.3](#sec:sphcompTdh){reference-type="ref" reference="sec:sphcompTdh"}. In places, we expand $K$ by two quotients. We denote the quotient $K^\times/(1+\smallo)$ by $\mathop{\mathrm{RV}}_{\! K}^\times$, with corresponding quotient map $\mathop{\mathrm{rv}}\colon K^\times \to \mathop{\mathrm{RV}}_{\! K}^\times$. We set $\mathop{\mathrm{RV}}_{\! K}\coloneqq \mathop{\mathrm{RV}}_{\! K}^\times \cup \{0\}$, and we extend $\mathop{\mathrm{rv}}$ to all of $K$ by setting $\mathop{\mathrm{rv}}(0)\coloneqq 0$. The residue map $\mathcal O^\times \to \boldsymbol{k}^\times$ induces a bijection $\mathop{\mathrm{rv}}(\mathcal O^\times) \to \boldsymbol{k}^\times$, which we also call $\mathop{\mathrm{res}}$, by $\mathop{\mathrm{res}}\mathop{\mathrm{rv}}(a) = \bar{a}$ for $a \in \mathcal O^\times$; conversely, in the same way we can recover $\mathop{\mathrm{res}}\colon \mathcal O^{\times} \to \boldsymbol{k}^{\times}$ from $\mathop{\mathrm{res}}\colon \mathop{\mathrm{rv}}(\mathcal O^\times) \to \boldsymbol{k}^\times$. We extend $\mathop{\mathrm{res}}$ to a map $\mathop{\mathrm{rv}}(\mathcal O^\times)\cup \{0\}\to \boldsymbol{k}$ by setting $\mathop{\mathrm{res}}(0)\coloneqq 0$, and we view $\mathop{\mathrm{res}}$ as a partial map from $\mathop{\mathrm{RV}}_{\! K}$ to $\boldsymbol{k}$. Let $\mathcal L^{\mathop{\mathrm{RV}}}$ be the language extending $\mathcal L^{\mathcal O}$ by a sort for $\mathop{\mathrm{RV}}_{\! K}$ (in the language $(\cdot, \mbox{}^{-1}, 1, 0, <)$), a sort for $\boldsymbol{k}$ (in the language $\mathcal L$), and the maps $\mathop{\mathrm{rv}}$ and $\mathop{\mathrm{res}}$. Let $T^{\mathop{\mathrm{RV}}}$ be the following $\mathcal L^{\mathop{\mathrm{RV}}}$-theory. 1. $(K, \mathcal O) \models T^{\mathcal O}$; 2. $\mathop{\mathrm{RV}}_{\! K}^{\times}$ is an abelian (multiplicative) group in the language $(\cdot,\mbox{}^{-1}, 1)$; 3. $\mathop{\mathrm{rv}}\colon K^{\times} \to \mathop{\mathrm{RV}}_{\! K}^{\times}$ is a surjective group homomorphism with $\ker\mathop{\mathrm{rv}}= 1+\smallo$, extended to $K$ by $\mathop{\mathrm{rv}}(0)=0$; 4. $<$ is interpreted in $\mathop{\mathrm{RV}}_{\! K}$ by $\mathop{\mathrm{rv}}(a)<\mathop{\mathrm{rv}}(b)$ if $a<b$ and $a \not\sim b$, for $a, b \in K$; 5. $\mathop{\mathrm{res}}\colon \mathop{\mathrm{rv}}(\mathcal O^{\times}) \to \boldsymbol{k}^{\times}$ is a group isomorphism extended to $\mathop{\mathrm{RV}}_{\! K}$ by $\mathop{\mathrm{res}}(\mathop{\mathrm{RV}}_{\! K}\setminus\mathop{\mathrm{rv}}(\mathcal O^{\times}))=\{0\}$; 6. [\[TRV-k\]]{#TRV-k label="TRV-k"} $\boldsymbol{k}\models T$ and if $F \colon K^n \to K$ is an $\mathcal L(\emptyset)$-definable continuous function (with the corresponding function $\boldsymbol{k}^n \to \boldsymbol{k}$ also denoted by $F$), then $\mathop{\mathrm{res}}(F(a))=F(\mathop{\mathrm{res}}(a))$ for all $a \in \mathcal O^n$. We denote such a structure simply by $(K, \mathop{\mathrm{RV}}_{\! K})$. Yin [@Yi17] introduced the language $\mathcal L^{\mathop{\mathrm{RV}}}$ and the theory $T^{\mathop{\mathrm{RV}}}$ (he denoted them by $\mathcal L_{T\!\mathop{\mathrm{RV}}}$ and $\mathop{\mathrm{TCVF}}$, respectively), although for Yin, the residue field $\boldsymbol{k}$ is a predicate on the sort for $\mathop{\mathrm{RV}}_{\! K}$, not a sort in its own right, and $\mathcal O$ is not in the language $\mathcal L^{\mathop{\mathrm{RV}}}$ (but it is definable). Here are some initial observations about a model $(K, \mathop{\mathrm{RV}}_{\! K})$ of $T^{\mathop{\mathrm{RV}}}$. The relation $<$ in $\mathop{\mathrm{RV}}_{\! K}$ is a linear order induced by $K$ via $\mathop{\mathrm{rv}}$, which makes $\mathop{\mathrm{RV}}_{\! K}^> \coloneqq \{ d \in \mathop{\mathrm{RV}}_{\! K} : d>0 \}$ an ordered abelian group. The map $\mathop{\mathrm{res}}\colon \mathcal O\to \boldsymbol{k}$ has kernel $\smallo$, so it induces an isomorphism $\mathop{\mathrm{res}}(K) \to \boldsymbol{k}$ of (ordered) fields. Moreover, it follows from [\[TRV-k\]](#TRV-k){reference-type="ref" reference="TRV-k"} that the map $\mathop{\mathrm{res}}(K) \to \boldsymbol{k}$ is an isomorphism of $\mathcal L$-structures by [@DL95 Lemma 1.13] and [@Yi17 Remark 2.3] (the latter is a syntactic manoeuvre replacing all primitives of $\mathcal L$ except $<$ by function symbols interpreted as continuous functions). Observe that, again using [@Yi17 Remark 2.3], $\mathop{\mathrm{RV}}_{\! K}$ and $\boldsymbol{k}$ are interpretable in $(K, \mathcal O)$, so $(K, \mathop{\mathrm{RV}}_{\! K})$ is a reduct of $(K, \mathcal O)^{\mathop{\mathrm{eq}}}$, the expansion of $(K, \mathcal O)$ by all imaginaries; more precisely, $(K, \mathop{\mathrm{RV}}_{\! K})$ is an expansion by definitions of such a reduct. It follows on general model-theoretic grounds that $(K, \mathop{\mathrm{RV}}_{\! K})$ is the unique expansion of $(K, \mathcal O)$ to a model of $T^{\mathop{\mathrm{RV}}}$ and that if $(M,\mathcal O_M)$ is an elementary $T^{\mathcal O}$-extension of $(K,\mathcal O)$, then $(M,\mathop{\mathrm{RV}}_{\! M})$ is an elementary $T^{\mathop{\mathrm{RV}}}$-extension of $(K,\mathop{\mathrm{RV}}_{\! K})$; these facts were first observed by Yin in [@Yi17 Proposition 2.13] and [@Yi17 Corollary 2.17], respectively. Next we consider the further extension of $\mathcal L^{\mathop{\mathrm{RV}}}$ by all the imaginary sorts coming from $\mathop{\mathrm{RV}}_{\! K}$. The corresponding language is denoted by $\mathcal L^{\mathop{\mathrm{RV}}^{\mathop{\mathrm{eq}}}}$ and the corresponding theory is denoted by $T^{\mathop{\mathrm{RV}}^{\mathop{\mathrm{eq}}}}$. We let $\mathop{\mathrm{RV}}_{\! K}^{\mathop{\mathrm{eq}}}$ be the structure consisting of the sort $\mathop{\mathrm{RV}}_{\! K}$, the sort $\boldsymbol{k}$, and all of these imaginary sorts. As before, $(K,\mathcal O)$ admits a unique expansion to a model $(K,\mathop{\mathrm{RV}}_{\! K}^{\mathop{\mathrm{eq}}})$ of $T^{\mathop{\mathrm{RV}}^{\mathop{\mathrm{eq}}}}$, and if $(M,\mathcal O_M)$ is an elementary $T^{\mathcal O}$-extension of $(K,\mathcal O)$, then $(M,\mathop{\mathrm{RV}}_{\! M}^{\mathop{\mathrm{eq}}})$ is an elementary $T^{\mathop{\mathrm{RV}}^{\mathop{\mathrm{eq}}}}$-extension of $(K,\mathop{\mathrm{RV}}_{\! K}^{\mathop{\mathrm{eq}}})$. Also note that if $(M,\mathcal O_M)$ is an *immediate* $T^{\mathcal O}$-extension of $(K,\mathcal O)$, then $\mathop{\mathrm{RV}}_{\! M}^{\mathop{\mathrm{eq}}} = \mathop{\mathrm{RV}}_{\! K}^{\mathop{\mathrm{eq}}}$. In addition, any $\mathcal L^{\mathop{\mathrm{RV}}^{\mathop{\mathrm{eq}}}}(K \cup \mathop{\mathrm{RV}}_{\! K}^{\mathop{\mathrm{eq}}})$-definable subset of $K^n$ is already $\mathcal L^{\mathcal O}(K)$-definable. We use these facts in combination with the following key result, the *Jacobian property* for definable functions in models of $T^{\mathcal O}$: **Fact 2** ([@Ga20 Theorem 3.18]). *Let $A \subseteq K\cup \mathop{\mathrm{RV}}_{\! K}^{\mathop{\mathrm{eq}}}$ and let $F\colon K^n \to K$ be an $\mathcal L^{\mathop{\mathrm{RV}}^{\mathop{\mathrm{eq}}}}(A)$-definable function. Then there is an $\mathcal L^{\mathop{\mathrm{RV}}^{\mathop{\mathrm{eq}}}}(A)$-definable map $\chi\colon K^n \to \mathop{\mathrm{RV}}_{\! K}^{\mathop{\mathrm{eq}}}$ such that for each $s \in \chi(K^n)$, if $\chi^{-1}(s)$ contains an open $v$-ball, then either $F$ is constant on $\chi^{-1}(s)$ or there is $d \in K^n$ such that $$v\big(F(x)- F(y) - d\cdot (x-y)\big) \ >\ vd+ v(x-y)$$ for all $x, y \in \chi^{-1}(s)$ with $x \neq y$.* ## Background on $T$-derivations {#sec:Td} Let $\der\colon K \to K$ be a map. For $a \in K$, we use $a'$ or $\der a$ in place of $\der(a)$ and if $a\neq 0$, then we set $a^\dagger\coloneqq a'/a$. Given $r \in \mathbb{N}$, we write $a^{(r)}$ in place of $\der^r(a)$, and we let $\mbox{\small$\mathscr{J}$}_\der^r(a)$ denote the tuple $(a,a',\ldots,a^{(r)})$. We use $\mbox{\small$\mathscr{J}$}_\der^\infty(a)$ for the infinite tuple $(a,a',a'',\ldots)$. Let $F\colon U \to K$ be an $\mathcal L(\emptyset)$-definable $\mathcal C^1$-function with $U \subseteq K^n$ open. We say that $\der$ is **compatible with $F$** if $$F(u)'\ =\ \frac{\partial F}{\partial Y_1}(u)u_1'+\cdots+\frac{\partial F}{\partial Y_n}(u)u_n'$$ for each $u = (u_1,\ldots,u_n)\in U$. We say that $\der$ is a **$T$-derivation on $K$** if $\der$ is compatible with every $\mathcal L(\emptyset)$-definable $\mathcal C^1$-function with open domain. Let $T^\der$ be the $\mathcal L^\der\coloneqq \mathcal L\cup\{\der\}$-theory which extends $T$ by axioms stating that $\der$ is a $T$-derivation. The study of $T$-derivations was initiated in [@FK21]. Any $T$-derivation on $K$ is a *derivation* on $K$, that is, a map satisfying the identities $(a+b)' = a' +b'$ and $(ab)' = a'b+ab'$ for $a,b \in K$ (this follows from compatibility with the functions $(x,y) \mapsto x+y$ and $(x,y) \mapsto xy$). For the remainder of this subsection, $\der$ is a $T$-derivation on $K$. We also call $(K, \der)$ a **$T$-differential field**. It is straightforward to verify the following fact. **Fact 3**. *If $x \mapsto x^\lambda\colon K^>\to K$ is an $\mathcal L(\emptyset)$-definable power function on $K$, then $(y^\lambda)' = \lambda y ^{\lambda-1}y'$ for all $y \in K^>$.* Given an element $a$ in a $T^\der$-extension of $K$, we write $K\langle\!\langle a \rangle\!\rangle$ in place of $K\langle \mbox{\small$\mathscr{J}$}_\der^\infty(a)\rangle$. Then $K\langle\!\langle a \rangle\!\rangle$ is itself a $T^\der$-extension of $K$. We say that $a$ is **$T^\der$-algebraic over $K$** if $\mbox{\small$\mathscr{J}$}_\der^r(a)$ is $\mathcal L(K)$-dependent for some $r$; otherwise, $a$ is said to be **$T^\der$-transcendental**. Equivalently, $a$ is $T^\der$-algebraic over $K$ if $a \in \mathop{\mathrm{cl}}^\der(K)$, where $\mathop{\mathrm{cl}}^\der$ is the $\der$-closure pregeometry considered in [@FK21 Section 3]. More generally, a $T^\der$-extension $L$ of $K$ is said to be **$T^\der$-algebraic over $K$** if each $a \in L$ is $T^\der$-algebraic over $K$ or, equivalently, if $L \subseteq \mathop{\mathrm{cl}}^\der(K)$. As a consequence of [@FK21 Lemma 3.2 (4)], we have the following: **Fact 4**. *Let $a$ be an element in a $T^\der$-extension of $K$. If the tuple $\mbox{\small$\mathscr{J}$}_\der^r(a)$ is $\mathcal L(K)$-dependent, then $K\langle\!\langle a \rangle\!\rangle= K \langle \mbox{\small$\mathscr{J}$}_\der^{r-1} a\rangle$.* Any $T$-extension of $K$ can be expanded to a $T^\der$-extension of $K$, and this expansion depends entirely on how the derivation is extended to a $\mathop{\mathrm{dcl}}_\mathcal L$-basis: **Fact 5** ([@FK21 Lemma 2.13]). *Let $M$ be a $T$-extension of $K$, let $A$ be a $\mathop{\mathrm{dcl}}_\mathcal L$-basis for $M$ over $K$, and let $s\colon A\to M$ be a map. Then there is a unique extension of $\der$ to a $T$-derivation on $M$ such that $a'= s(a)$ for all $a \in A$.* The $T$-derivation $\der$ is said to be **generic** if for each $\mathcal L(K)$-definable function $F\colon U\to K$ with $U \subseteq K^r$ open, there is $a \in K$ with $\mbox{\small$\mathscr{J}$}_\der^{r-1}(a) \in U$ such that $a^{(r)} = F(\mbox{\small$\mathscr{J}$}_\der^{r-1}(a))$. Let $T^\der_{\mathcal G}$ be the $\mathcal L^\der$-theory which extends $T^\der$ by axioms stating that $\der$ is generic. In [@FK21], it was shown that the theory $T^\der_{\mathcal G}$ is the model completion of $T^\der$. We record here the main extension and embedding results which go into this proof, for use in Section [7](#sec:modelcompletion){reference-type="ref" reference="sec:modelcompletion"}. **Fact 6** ([@FK21 Proposition 4.3 and Lemma 4.7]). *There is a model $M \models T^\der_{\mathcal G}$ which extends $K$ with $|M| = |K|$. Given any such extension $M$ and any $|K|^+$-saturated model $M^* \models T^\der_{\mathcal G}$ extending $K$, there is an $\mathcal L^\der(K)$-embedding $M\to M^*$.* Next we recall some terminology and notation about linear differential operators from [@ADH17 Chapter 5]. Let $K\{Y\} \coloneqq K[Y, Y', \dots]$ be the differential ring of differential polynomials over $K$ and let $K[\der]$ be the ring of linear differential operators over $K$. An element in $K[\der]$ corresponds to a $C$-linear operator $y \mapsto A(y)$ on $K$, where $A(Y) = a_0Y+a_1Y' + \dots + a_rY^{(r)} \in K\{Y\}$ is a homogeneous linear differential polynomial. Viewed as an operator in $K[\der]$, we write $A = a_0+a_1\der+\dots+a_r\der^r$, and we freely pass between viewing $A$ as an element of $K[\der]$ and of $K\{Y\}$. Henceforth, $A = a_0+a_1\der+\dots+a_r\der^r \in K[\der]$. If $a_r \neq 0$, we say that $A$ has **order** $r$. Multiplication in the ring $K[\der]$ is given by composition. In particular, for $g \in K^{\times}$ the linear differential operator $Ag \in K[\der]$ corresponds to the differential polynomial $A_{\times g}(Y) = A(gY) \in K\{Y\}$. We call $K$ **linearly surjective** if every $A \in K[\der]^{\neq} \coloneqq K[\der]\setminus\{0\}$ is surjective. ## Background on $T$-convex $T$-differential fields Let $\mathcal L^{\mathcal O,\der}\coloneqq \mathcal L^{\mathcal O}\cup\mathcal L^\der= \mathcal L\cup\{\mathcal O,\der\}$, and let $T^{\mathcal O,\der}$ be the $\mathcal L^{\mathcal O,\der}$-theory which extends $T^\der$ and $T^{\mathcal O}$ by the additional axiom "$\der \smallo \subseteq \smallo$". **Assumption 7**. *For the remainder of this paper, let $K = (K,\mathcal O,\der) \models T^{\mathcal O,\der}$.* This additional axiom, called **small derivation**, ensures that $\der$ is continuous with respect to the valuation topology (equivalently, order topology) on $K$ [@ADH17 Lemma 4.4.6], so every model of $T^{\mathcal O,\der}$ is a $T$-convex $T$-differential field, as defined in the introduction. Also, small derivation gives $\der\mathcal O\subseteq\mathcal O$ too [@ADH17 Lemma 4.4.2], so $\der$ induces a derivation on $\boldsymbol{k}$. Moreover, $\boldsymbol{k}$ is a $T$-differential field (see [@Ka22 p. 280]). In this paper we are interested in the case that the derivation induced on $\boldsymbol{k}$ is nontrivial; indeed, often it will be linearly surjective or even generic. A consequence of the main results of [@Ka22] provides spherically complete extensions in this case: **Fact 8** ([@Ka22 Corollary 6.4]). *If the derivation induced on $\boldsymbol{k}$ is nontrivial, then $K$ has a spherically complete immediate $T^{\mathcal O,\der}$-extension.* It follows that for $K$ such that the derivation induced on $\boldsymbol{k}$ is nontrivial, $K$ is spherically complete if and only if $K$ has no proper immediate $T^{\mathcal O,\der}$-extension. The notion of an $\mathcal L(K)$-definable function being in *implicit form*, defined in the next section, plays an important role in the result above and our work here, as does the notion of *vanishing*, which we define in Section [3.3](#sec:sphcompTdh){reference-type="ref" reference="sec:sphcompTdh"} when it is needed. Now we define an important function on $\Gamma$ defined by a linear differential operator $A \in K[\der]^{\neq}$; for details see [@ADH17 Section 4.5]. The valuation of $K$ induces a valuation $v$ on $K[\der]^{\neq}$ given by $vA \coloneqq \min\{ va_i : 0 \leqslant i \leqslant r\}$. Combining this with multiplicative conjugation yields a strictly increasing function $v_A \colon \Gamma \to \Gamma$ defined by $v_A(\gamma) \coloneqq v(A_{\times g})$ for any $g \in K^{\times}$ with $vg=\gamma$. As a consequence of the Equalizer Theorem [@ADH17 Theorem 6.0.1], $v_A$ is bijective. We extend $v_A$ to $\Gamma \cup \{\infty\}$ by $v_A(\infty)=v(A_{\times 0})\coloneqq \infty$. # $T^\der$-henselianity {#sec:Tdh} Given an $\mathcal L(K)$-definable function $F\colon K^{1+r}\to K$, a linear differential operator $A \in K[\der]^{\neq}$, and an open $v$-ball $B\subseteq K$, we say that **$A$ linearly approximates $F$ on $B$** if $$v\big(F(\mbox{\small$\mathscr{J}$}_\der^r b) - F(\mbox{\small$\mathscr{J}$}_\der^r a) - A(b-a)\big) \ > \ vA_{\times (b-a)}$$ for all $a,b \in B$ with $a \neq b$. Note that if $A$ linearly approximates $F$ on $B$, then $A$ linearly approximates $F$ on any open $v$-ball contained in $B$. In this section, $F\colon K^{1+r}\to K$ will always be an $\mathcal L(K)$-definable function in **implicit form**, that is, $$F \ = \ \mathfrak{m}_F\big(Y_{r}-I_F(Y_0,\dots,Y_{r-1})\big)$$ for some $\mathfrak{m}_F \in K^{\times}$ and $\mathcal L(K)$-definable function $I_F \colon K^{r} \to K$. Additionally, let $A$ range over $K[\der]$, $a$ range over $K$, and $\gamma$ range over $\Gamma$. We say that **$(F,A,a,\gamma)$ is in $T^\der$-hensel configuration** if $A$ linearly approximates $F$ on $B(a,\gamma)$ and $vF(\mbox{\small$\mathscr{J}$}_\der^r a) > v_A(\gamma)$. We say that $K$ is **$T^\der$-henselian** if: 1. its differential residue field $\boldsymbol{k}$ is linearly surjective; 2. for every $(F,A,a,\gamma)$ in $T^\der$-hensel configuration, there exists $b \in B(a,\gamma)$ with $F(\mbox{\small$\mathscr{J}$}_\der^r b) = 0$ and $vA_{\times(b-a)} \geqslant vF(\mbox{\small$\mathscr{J}$}_\der^r a)$. We allow all valuations to be infinite, so if $F(\mbox{\small$\mathscr{J}$}_\der^r a) = 0$, then we may take $b = a$. The definition of $T^\der$-henselianity is inspired by that of *$\sigma$-henselianity* for analytic difference valued fields from [@Ri17]. Implicit form of definable functions was introduced and exploited in [@Ka22]. ## Basic consequences {#sec:Tdhbasic} Here are some consequences of $T^\der$-henselianity needed later. The proofs are mostly routine adaptations of results from [@ADH17 Sections 7.1 and 7.5], using $T^\der$-henselianity instead of $\operatorname{d}$-henselianity, so we give a couple as an illustration and omit most of the others. For the next lemma, we call $A \in K[\der]^{\neq}$ **neatly surjective** if for every $b \in K^{\times}$ there is $a \in K^{\times}$ with $A(a)=b$ and $v_A(va)=vb$. **Lemma 9**. *Suppose that $K$ is $T^\der$-henselian. Then every $A \in K[\der]^{\neq}$ is neatly surjective.* *Proof.* Let $A \in K[\der]^{\neq}$ have order $r$. To see that $A$ is neatly surjective, let $b \in K^{\times}$ and take $\alpha \in \Gamma$ with $v_A(\alpha) = \beta \coloneqq vb$. We need to find $a \in K^{\times}$ with $va=\alpha$ and $A(a)=b$. Take $\phi \in K^{\times}$ with $v\phi = \alpha$ and let $D \coloneqq b^{-1}A\phi \in K[\der]^{\neq}$, so $v(D)=0$. Take an $\mathcal L(K)$-definable function $F \colon K^{1+r} \to K$ in implicit form such that $F(\mbox{\small$\mathscr{J}$}_\der^{r} a)=D(a)-1$ for all $a \in K$. Since $\boldsymbol{k}$ is linearly surjective, we have $u \in \mathcal O^{\times}$ with $D(u)-1 \prec 1$. As $D$ linearly approximates $F$ on $K$, $(F, D, u, 0)$ is in $T^\der$-hensel configuration, and thus we have $y \sim u$ with $F(\mbox{\small$\mathscr{J}$}_\der^{r}y)=0$. Then $a \coloneqq \phi y$ works. ◻ The next corollary follows from Lemma [Lemma 9](#lem:Tdhneatsurj){reference-type="ref" reference="lem:Tdhneatsurj"} as [@ADH17 Corollary 7.1.9] follows from [@ADH17 Lemma 7.1.8]. **Corollary 10**. *If $K$ is $T^\der$-henselian, then $\smallo = (1+\smallo)^{\dagger}$.* Under the assumption of monotonicity this yields additional information as in [@ADH17 Corollary 7.1.11]. We say that $K$ has **many constants** if $v(C^{\times})=\Gamma$. Note that if $K$ has many constants (and small derivation), then $K$ is monotone. **Corollary 11**. *Suppose that $K$ is $T^\der$-henselian and monotone, and $(\boldsymbol{k}^{\times})^{\dagger} = \boldsymbol{k}$. Then $K$ has many constants and $(K^{\times})^{\dagger} = (\mathcal O^{\times})^{\dagger} = \mathcal O$.* Rather opposite to monotonicity, we say that $K$ has **few constants** if $C \subseteq \mathcal O$. Now we record several lemmas about $K$ with few constants adapted from [@ADH17 Section 7.5]. By [@ADH17 Lemma 9.1.1] and Lemma [Lemma 9](#lem:Tdhneatsurj){reference-type="ref" reference="lem:Tdhneatsurj"}, any $T^\der$-henselian $K$ with few constants is **asymptotic** in the sense of [@ADH17 Chapter 9]; that is, for all nonzero $a, b \in \smallo$, we have $a \prec b \iff a' \prec b'$. Conversely, any asymptotic $K$ obviously satisfies $C \subseteq \mathcal O$. The next lemma is analogous to [@ADH17 Lemma 7.5.1] but has a simpler proof. **Lemma 12**. *Suppose that $K$ is $T^\der$-henselian and let $A \in K[\der]^{\neq}$ have order $r$. If $A(1) \prec A$, then $A(y)=0$ for some $y \in K$ with $y \sim 1$ and $vA_{\times (y-1)} \geqslant vA(1)$.* *Proof.* Take an $\mathcal L(K)$-definable function $F \colon K^{1+r} \to K$ in implicit form such that $F(\mbox{\small$\mathscr{J}$}_\der^r a) = A(a)$ for all $a \in K$. If $A(1)\prec A$, then $(F, A, 1, 0)$ is in $T^\der$-hensel configuration, so $T^\der$-henselianity yields the desired $y$. ◻ The next lemma follows from Lemma [Lemma 12](#lem:7.5.1){reference-type="ref" reference="lem:7.5.1"} as [@ADH17 Lemma 7.5.2] follows from [@ADH17 Lemma 7.5.1]. **Lemma 13**. *Suppose that $K$ is $T^\der$-henselian and $C \subseteq \mathcal O$. Let $A \in K[\der]^{\neq}$ have order $r$. There do not exist $b_0, \dots, b_r \in K^{\times}$ such that $b_0 \succ b_1 \succ \dots \succ b_r$ and $A(b_i) \prec Ab_i$ for $i=0, \dots, r$.* The proof of Lemma [Lemma 14](#lem:7.5.5){reference-type="ref" reference="lem:7.5.5"} is adapted from that of [@ADH17 Lemma 7.5.5]. **Lemma 14**. *Suppose that $K$ is $T^\der$-henselian and $C \subseteq \mathcal O$. Let $F \colon K^{1+r} \to K$ be an $\mathcal L(K)$-definable function in implicit form and $A \in K[\der]^{\neq}$ have order $q$. There do not exist $y_0, \dots, y_{q+1} \in K$ such that* 1. *[\[lem:7.5.5i\]]{#lem:7.5.5i label="lem:7.5.5i"} $y_{i-1}-y_i \succ y_i-y_{i+1}$ for $i = 1, \dots, q$ and $y_q \neq y_{q+1}$;* 2. *[\[lem:7.5.5ii\]]{#lem:7.5.5ii label="lem:7.5.5ii"} $F(\mbox{\small$\mathscr{J}$}_\der^ry_i) = 0$ for $i=0, \dots, q+1$;* 3. *[\[lem:7.5.5iii\]]{#lem:7.5.5iii label="lem:7.5.5iii"} $(F, A, y_{q+1}, \gamma)$ is in $T^\der$-hensel configuration and $v(y_0-y_{q+1})>\gamma$ for some $\gamma \in \Gamma$.* *Proof.* Suppose towards a contradiction that we have $y_0, \dots, y_{q+1} \in K$ and $\gamma \in \Gamma$ satisfying [\[lem:7.5.5i\]](#lem:7.5.5i){reference-type="ref" reference="lem:7.5.5i"}--[\[lem:7.5.5iii\]](#lem:7.5.5iii){reference-type="ref" reference="lem:7.5.5iii"}. Below, $i$ ranges over $\{0, \dots, q\}$. Set $G \coloneqq F_{+y_{q+1}}$ and $b_i \coloneqq y_i-y_{q+1}$. Then $b_i \sim y_i-y_{i+1}$, so $b_0 \succ b_1 \succ \dots \succ b_q \neq 0$. Also, $G(\mbox{\small$\mathscr{J}$}_\der^r b_i) = F(\mbox{\small$\mathscr{J}$}_\der^r y_i) = 0$ and $G(0) = F(\mbox{\small$\mathscr{J}$}_\der^r y_{q+1}) = 0$. Since $vb_0>\gamma$, by [\[lem:7.5.5iii\]](#lem:7.5.5iii){reference-type="ref" reference="lem:7.5.5iii"} we have $\varepsilon_i$ such that $v\varepsilon_i > vA_{\times b_i}$ and $0=G(\mbox{\small$\mathscr{J}$}_\der^r b_i) = A(b_i)+\varepsilon_i$. In particular, $vA(b_i)=v(\varepsilon_i)> vA_{\times b_i}$, contradicting Lemma [Lemma 13](#lem:7.5.2){reference-type="ref" reference="lem:7.5.2"} with $q$ in the role of $r$. ◻ **Remark 15**. What of our assumption that $T$ is power bounded? The definitions of $T$-convex valuation ring, $T$-derivation, and $T^\der$-henselianity do not use it, nor do the above consequences of $T^\der$-henselianity. Suppose temporarily that $T$ is not power bounded. Then by Miller's dichotomy [@Mi96], we have an $\mathcal L(\emptyset)$-definable exponential function $E \colon K \to K^>$ (i.e., $E$ is an isomorphism from the ordered additive group of $K$ to the ordered multiplicative group $K^>$ which is equal to its own derivative). As we now explain, this $E$ is not compatible with monotonicity, at least if we suppose in addition that the derivation of $\boldsymbol{k}$ is nontrivial and that $K$ is nontrivially valued. Let $a \in K$ with $a \succ 1$. If $a' \succ 1$, then $E(a)' = a'E(a) \succ E(a)$, so $K$ is not monotone. If $a'\preccurlyeq 1$, taking $u \in K$ with $u \asymp u' \asymp 1$ yields $(au)'=a'u+au' \sim au' \asymp a$, so $b\coloneqq au$ satisfies $b' \succ 1$ and $E(b)' \succ E(b)$. ## Lifting the residue field {#sec:lift} In this subsection we show that every $T^\der$-henselian $T^{\mathcal O,\der}$-model admits a lift of its differential residue field as an $\mathcal L^\der$-structure in the sense defined before Corollary [Corollary 19](#cor:reslift){reference-type="ref" reference="cor:reslift"} (cf. [@ADH17 Proposition 7.1.3]). A **partial $T$-lift of the residue field $\boldsymbol{k}$** is a $T$-submodel $E\subseteq K$ which is contained in $\mathcal O$. If $E$ is a partial $T$-lift of $\boldsymbol{k}$, then the residue map induces an $\mathcal L$-embedding $E \to \boldsymbol{k}$. If this embedding is surjective onto $\boldsymbol{k}$, then $E$ is called a **$T$-lift of $\boldsymbol{k}$**. By [@DL95 Theorem 2.12], $\boldsymbol{k}$ always admits a $T$-lift. **Lemma 16**. *Let $E\subseteq K$ be a partial $T$-lift of $\boldsymbol{k}$ and let $a,b$ be tuples in $\mathcal O^n$ with $a-b \in \smallo^n$. Suppose that $a$ is $\mathcal L(E)$-independent and that $E\langle a \rangle$ is a partial $T$-lift of $\boldsymbol{k}$. Then $a$ and $b$ have the same $\mathcal L^{\mathop{\mathrm{RV}}}$-type over $E \cup \mathop{\mathrm{rv}}\!\big(E\langle a \rangle\big)$. In particular, $E\langle b\rangle$ is also a partial $T$-lift of $\boldsymbol{k}$.* *Proof.* We proceed by induction on $n$, with the case $n = 0$ holding trivially. Assume that $(a_1,\ldots,a_{n-1})$ and $(b_1,\ldots,b_{n-1})$ have the same $\mathcal L^{\mathop{\mathrm{RV}}}$-type over $E \cup \mathop{\mathrm{rv}}\!\big(E\langle a_1,\ldots,a_{n-1} \rangle\big)$. This assumption yields a partial elementary $\mathcal L^{\mathop{\mathrm{RV}}}$-embedding $\imath\colon E\langle a_1,\ldots,a_{n-1} \rangle \to E\langle b_1,\ldots,b_{n-1}\rangle$ which fixes $E$ and $\mathop{\mathrm{rv}}\!\big(E\langle a_1,\ldots,a_{n-1} \rangle\big)$ and which sends $a_i$ to $b_i$ for each $i<n$. Let $g \in E\langle a_1,\ldots,a_{n-1} \rangle$. We claim that $g < a_n \Longleftrightarrow \imath(g)<b_n$. Take some $\mathcal L(E)$-definable function $G$ with $g = G(a_1,\ldots,a_{n-1})$. Since $a$ is $\mathcal L(E)$-independent, the function $G$ is continuous on some $\mathcal L(E)$-definable neighborhood $U$ of $(a_1,\ldots,a_{n-1})$. Note that $(b_1,\ldots,b_{n-1})\in U$ as well, so $G(a_1,\ldots,a_{n-1}) - G(b_1,\ldots,b_{n-1}) \prec 1$ by [@DL95 Lemma 1.13]. Since $E\langle a\rangle$ is a partial $T$-lift of $\boldsymbol{k}$, we have $a_n - G(a_1,\ldots,a_{n-1}) \asymp 1$. By assumption, $a_n - b_n \prec 1$, so $$a_n - g\ =\ a_n - G(a_1,\ldots,a_{n-1})\ \sim\ b_n - G(b_1,\ldots,b_{n-1})\ =\ b_n - \imath(g).$$ In particular, $a_n - g$ and $b_n - \imath(g)$ have the same sign. This allows us to extend $\imath$ to an $\mathcal L(E)$-embedding $\jmath\colon E \langle a \rangle \to E\langle b\rangle$ by mapping $a_n$ to $b_n$. To see that $\jmath$ is even an elementary $\mathcal L^{\mathop{\mathrm{RV}}}$-embedding over $E \cup \mathop{\mathrm{rv}}\!\big(E\langle a \rangle\big)$, it suffices to show that $\mathop{\mathrm{rv}}h = \mathop{\mathrm{rv}}\jmath(h)$ for each $h \in E \langle a \rangle$. We may assume that $h \neq 0$, so $h \asymp 1$. Take some $\mathcal L(E)$-definable function $H$ with $h = H(a)$. Again, $H$ is continuous on an open set containing $a$ and $b$, so we may use [@DL95 Lemma 1.13] to get that $H(a) -H(b) \prec 1$. Thus, $H(a) \sim H(b)$, so $$\mathop{\mathrm{rv}}h\ =\ \mathop{\mathrm{rv}}H(a) \ =\ \mathop{\mathrm{rv}}H(b)\ =\ \mathop{\mathrm{rv}}\jmath(h).\qedhere$$ ◻ **Lemma 17**. *Let $n$ be given, let $E$ be a partial $T$-lift of $\boldsymbol{k}$, let $a \in \mathcal O^\times$ with $\bar{a} \not\in \mathop{\mathrm{res}}(E)$, and suppose that $\mbox{\small$\mathscr{J}$}_\der^{n-1}(a)$ is $\mathcal L(E)$-independent and that $E\langle\mbox{\small$\mathscr{J}$}_\der^{n-1}(a) \rangle$ is a partial $T$-lift of $\boldsymbol{k}$. Let $G\colon K^{1+n}\to K$ be an $\mathcal L(E)$-definable function in implicit form with $\mathfrak{m}_G = 1$. Then there is $A \in K[\der]$ with $vA = 0$ which linearly approximates $G$ on $a+\smallo$.* *Proof.* By applying Fact [Fact 2](#fact:rvapprox){reference-type="ref" reference="fact:rvapprox"} to the function $I_G$, we find an $\mathcal L^{\mathop{\mathrm{RV}}^{\mathop{\mathrm{eq}}}}(E)$-definable map $\chi\colon K^n \to \mathop{\mathrm{RV}}_{\! K}^{\mathop{\mathrm{eq}}}$ such that for each $s \in \chi(K^n)$, if $\chi^{-1}(s)$ contains an open $v$-ball, then either $I_G$ is constant on $\chi^{-1}(s)$ or there is $d \in K^n$ such that $$\label{eq:dependentlift} v\big(I_G(x)- I_G(y) - d\cdot (x-y)\big) \ >\ vd+ v(x-y)$$ for all $x, y \in \chi^{-1}(s)$ with $x \neq y$. Let $s_0\coloneqq \chi\big(\mbox{\small$\mathscr{J}$}_\der^{n-1}(a)\big)$ and let $U\coloneqq \chi^{-1}(s_0)$. Note that if $x\in \mbox{\small$\mathscr{J}$}_\der^{n-1}(a)+\smallo^n$, then $x$ and $\mbox{\small$\mathscr{J}$}_\der^{n-1}(a)$ have the same $\mathcal L^{\mathop{\mathrm{RV}}}$-type over $E \cup \mathop{\mathrm{rv}}E\langle \mbox{\small$\mathscr{J}$}_\der^{n-1}(a)\rangle$ by Lemma [Lemma 16](#lem:closesamechi){reference-type="ref" reference="lem:closesamechi"}, so $x \in U$. In particular, $\mbox{\small$\mathscr{J}$}_\der^{n-1}(u) \in U$ for all $u \in a+\smallo$, since $K$ has small derivation. We choose $A$ as follows: if $I_G$ is constant on $U$, then we let $A$ be the linear differential operator $\der^n$. If $I_G$ is not constant on $U$, then we let $A$ be $$\der^n - d_n \der^{n-1} - \cdots-d_1\ \in \ K[\der],$$ where $d = (d_1,\ldots,d_n) \in K^n$ is chosen such that ([\[eq:dependentlift\]](#eq:dependentlift){reference-type="ref" reference="eq:dependentlift"}) holds for $x,y \in U$ with $x \neq y$. We claim that $vA = 0$. This is clear if $A = \der^n$, so we may assume that $I_G$ is not constant on $U$, and we need to show that $d_i \preccurlyeq 1$ for each $i =1,\ldots,n$. Take $i \leqslant n$ with $vd_i=vd$ and suppose towards contradiction that $d_i \succ 1$. Take tuples $x,y \in \mbox{\small$\mathscr{J}$}_\der^{n-1}(a)+\smallo^n$ such that $x_i - y_i \succ d_i^{-1}$ and $x_j = y_j$ for $j \leqslant n$ with $j\neq i$. Then $d \cdot (x-y) = d_i(x_i-y_i) \succ 1$. Since $E\langle x\rangle$ is a partial $T$-lift of $K$ by Lemma [Lemma 16](#lem:closesamechi){reference-type="ref" reference="lem:closesamechi"}, we have $I_G(x) \preccurlyeq 1$. Likewise, $I_G(y) \preccurlyeq 1$, so $$v\big(I_G(x)- I_G(y) - d\cdot (x-y)\big) \ = \ vd_i+v(x_i-y_i)\ =\ vd+ v(x-y).$$ This is a contradiction, since both $x$ and $y$ are in $U$. Now, we will show that $A$ linearly approximates $G$ on $a+\smallo$. Again, this is clear if $I_G$ is constant on $U$, so we may assume that $I_G$ is not constant on $U$. Let $t,u \in a+\smallo$ with $t\neq u$. We have $$v\big(G(\mbox{\small$\mathscr{J}$}_\der^n u)- G(\mbox{\small$\mathscr{J}$}_\der^n t) - A(u-t)\big) \ = \ v\big(I_G(\mbox{\small$\mathscr{J}$}_\der^{n-1}u)- I_G(\mbox{\small$\mathscr{J}$}_\der^{n-1}t)- d\cdot \mbox{\small$\mathscr{J}$}_\der^{n-1}(u-t)\big) \ >\ vd + v\big(\mbox{\small$\mathscr{J}$}_\der^{n-1}(u-t)\big),$$ and we need to show that $vd+ v\big(\mbox{\small$\mathscr{J}$}_\der^{n-1}(u-t)\big)\geqslant vA_{\times(u-t)}$. We consider two cases. First, if $(u-t)' \preccurlyeq u-t$, then $(u-t)^{(m)} \preccurlyeq(u-t)$ for all $m$ and $vA_{\times (u-t)} = vA + v(u-t)$ by [@ADH17 Lemma 4.5.3]. Using also that $vA \leqslant vd$, this yields $$vd + v\big(\mbox{\small$\mathscr{J}$}_\der^{n-1}(u-t)\big)\ =\ vd+ v(u-t)\ \geqslant\ vA + v(u-t)\ =\ vA_{\times (u-t)},$$ as desired. Next, if $(u-t)' \succ (u-t)$, then $(u-t)^{(m)} \asymp (u-t)\big((u-t)^\dagger\big)^m$ for each $m$ by [@ADH17 Lemma 6.4.1]. In particular, $(u-t)^{(i)}\prec(u-t)^{(n)}$ whenever $i<n$, since $(u-t)^\dagger \succ 1$. This gives $$A_{\times (u-t)}\ \succcurlyeq\ A(u-t) \ =\ (u-t)^{(n)} - d_n (u-t)^{(n-1)} - \cdots-d_1(u-t)\ \asymp (u-t)^{(n)},$$ so we have $$vA_{\times(u-t)}\ \leqslant\ v\big((u-t)^{(n)}\big) \ = \ v\big(\mbox{\small$\mathscr{J}$}_\der^{n-1}(u-t)\big)\ \leqslant\ vd+v\big(\mbox{\small$\mathscr{J}$}_\der^{n-1}(u-t)\big).\qedhere$$ ◻ **Proposition 18**. *Suppose that $K$ is $T^\der$-henselian, let $E$ be a $T$-convex $T$-differential subfield of $K$, and let $a \in \boldsymbol{k}\setminus \mathop{\mathrm{res}}(E)$. Then there is $b \in \mathcal O$ with $\mathop{\mathrm{res}}(b) = a$ and the following properties:* 1. *[\[prop:resexti\]]{#prop:resexti label="prop:resexti"} $\mathop{\mathrm{rk}}_{\mathcal L}\!\big(E\langle\!\langle b\rangle\!\rangle\big| E\big)=\mathop{\mathrm{rk}}_{\mathcal L}\!\big(\mathop{\mathrm{res}}(E)\langle\!\langle a\rangle\!\rangle\big| \mathop{\mathrm{res}}(E)\big)$;* 2. *[\[prop:resextii\]]{#prop:resextii label="prop:resextii"} $v\big(E\langle\!\langle b\rangle\!\rangle^\times\big) = v(E^\times)$;* 3. *[\[prop:resextiii\]]{#prop:resextiii label="prop:resextiii"} If $K^*$ is any $T^\der$-henselian $T^{\mathcal O,\der}$-extension of $E$, then any $\mathcal L^\der(\mathop{\mathrm{res}}(E))$-embedding $\imath\colon \mathop{\mathrm{res}}(E)\langle\!\langle a\rangle\!\rangle\to \mathop{\mathrm{res}}(K^*)$ lifts to an $\mathcal L^{\mathcal O,\der}(E)$-embedding $\jmath\colon E\langle\!\langle b\rangle\!\rangle\to K^*$;* *Proof.* First, consider the case that $a$ is $T^\der$-transcendental over $\mathop{\mathrm{res}}(E)$. Let $b\in K$ be a lift of $a$. We first show that [\[prop:resexti\]](#prop:resexti){reference-type="ref" reference="prop:resexti"} holds. We claim that $b$ is $T^\der$-transcendental over $E$. Suppose not, and take $n$ and an $\mathcal L(E)$-definable function $G\colon K^n\to K$ with $b^{(n)} = G(\mbox{\small$\mathscr{J}$}_\der^{n-1}b)$. Then $\mbox{\small$\mathscr{J}$}_\der^n(b)$ belongs to $\mathop{\mathrm{Gr}}(G)$, the graph of $G$, so $\mbox{\small$\mathscr{J}$}_\der^n(a)$ belongs to $\mathop{\mathrm{res}}(\mathop{\mathrm{Gr}}(G))$. By [@vdD97 Corollaries 1.13 and 1.14], the set $\mathop{\mathrm{res}}(\mathop{\mathrm{Gr}}(G))$ is $\mathop{\mathrm{res}}(E)$-definable of $\mathcal L$-dimension at most $n$, contradicting that $a$ is $T^\der$-transcendental over $\mathop{\mathrm{res}}(E)$. For each $n$, let $E_n\coloneqq E\langle \mbox{\small$\mathscr{J}$}_\der^{n-1} b\rangle$, so $E\langle\!\langle b\rangle\!\rangle= \bigcup_n E_n$ and $$\mathop{\mathrm{rk}}_{\mathcal L}(E_n | E)\ =\ \mathop{\mathrm{rk}}_{\mathcal L}\!\big(\mathop{\mathrm{res}}(E_n)| \mathop{\mathrm{res}}(E)\big)\ =\ n.$$ The Wilkie inequality gives that $v(E_n^\times) = v(E)$, so $v\big(E\langle\!\langle b\rangle\!\rangle^\times\big) = v(E^\times)$ as well, proving [\[prop:resextii\]](#prop:resextii){reference-type="ref" reference="prop:resextii"}. Now, let $K^*$ and $\imath$ be as in [\[prop:resextiii\]](#prop:resextiii){reference-type="ref" reference="prop:resextiii"} and let $b^* \in K^*$ be a lift of $\imath(a)$. Let $\jmath\colon E\langle\!\langle b\rangle\!\rangle\to K^*$ be the map which fixes each element of $E$ and which sends $\mbox{\small$\mathscr{J}$}_\der^\infty(b)$ to $\mbox{\small$\mathscr{J}$}_\der^\infty(b^*)$. By [\[prop:resexti\]](#prop:resexti){reference-type="ref" reference="prop:resexti"}, [\[prop:resextii\]](#prop:resextii){reference-type="ref" reference="prop:resextii"}, and a straightforward induction on $n$, we see that the restriction of $\jmath$ to $E_n$ is an $\mathcal L^{\mathcal O}(E)$-embedding, so $\jmath$ itself is an $\mathcal L^{\mathcal O,\der}(E)$-embedding lifting $\imath$. Now assume that $a$ is $T^\der$-algebraic over $\mathop{\mathrm{res}}(E)$, and let $n$ be minimal such that $a^{(n)} \in \mathop{\mathrm{res}}(E)\langle\mbox{\small$\mathscr{J}$}_\der^{n-1} a\rangle$. Let $b \in K$ be a lift of $a$. By the argument in the previous case, we get that $\mbox{\small$\mathscr{J}$}_\der^{n-1}(b)$ is $\mathcal L(E)$-independent. Let $E_0\subseteq E$ be a $T$-lift of $\mathop{\mathrm{res}}(E)$, so $E_0\langle \mbox{\small$\mathscr{J}$}_\der^{n-1}b\rangle$ is a $T$-lift of $\mathop{\mathrm{res}}(E)\langle\mbox{\small$\mathscr{J}$}_\der^{n-1} a\rangle$. Since $a^{(n)} \in \mathop{\mathrm{res}}(E)\langle\mbox{\small$\mathscr{J}$}_\der^{n-1} a\rangle$, there is some $\mathcal L(E_0)$-definable function $G\colon K^n\to K$ with $b^{(n)} - G(\mbox{\small$\mathscr{J}$}_\der^{n-1}b)\prec 1$. Let $H\colon K^{n+1}\to K$ be the function $$H(Y_0,\ldots,Y_n)\ \coloneqq\ Y_n - G(Y_0,\ldots,Y_{n-1}),$$ so $H$ is in implicit form with $\mathfrak{m}_H = 1$ and $H(\mbox{\small$\mathscr{J}$}_\der^nb) \prec 1$. Applying Lemma [Lemma 17](#lem:dependentlift){reference-type="ref" reference="lem:dependentlift"}, we get a linear differential operator $A$ with $vA = 0$ which linearly approximates $H$ on $b+\smallo$. Then $(H,A,b,0)$ is in $T^\der$-hensel configuration, and since $K$ is assumed to be $T^\der$-henselian, we may replace $b$ with some element in $b+\smallo$ and arrange that $H(\mbox{\small$\mathscr{J}$}_\der^nb) = 0$. Note that then $E\langle \mbox{\small$\mathscr{J}$}_\der^{n-1} b \rangle = E\langle\!\langle b \rangle\!\rangle$, so [\[prop:resexti\]](#prop:resexti){reference-type="ref" reference="prop:resexti"} holds. As above, [\[prop:resextii\]](#prop:resextii){reference-type="ref" reference="prop:resextii"} follows from [\[prop:resexti\]](#prop:resexti){reference-type="ref" reference="prop:resexti"} and the Wilkie inequality. For [\[prop:resextiii\]](#prop:resextiii){reference-type="ref" reference="prop:resextiii"}, let $K^*$ and $\imath$ be given. Arguing as we did with $K$, we can take a lift $b^*$ of $\imath(a)$ in $K^*$ such that $H(\mbox{\small$\mathscr{J}$}_\der^nb^*) = 0$. Let $\jmath\colon E\langle\!\langle b\rangle\!\rangle\to K^*$ be the map that fixes each element of $E$ and sends $\mbox{\small$\mathscr{J}$}_\der^{n-1}(b)$ to $\mbox{\small$\mathscr{J}$}_\der^{n-1}(b^*)$. As above, this map is easily seen to be an $\mathcal L^{\mathcal O,\der}(E)$-embedding lifting $\imath$. ◻ A **partial $T^\der$-lift of the differential residue field $\boldsymbol{k}$** is a $T^\der$-submodel $E\subseteq K$ which is contained in $\mathcal O$. Equivalently, a partial $T^\der$-lift is a partial $T$-lift which is closed under $\der$. If $E$ is a partial $T^\der$-lift of $\boldsymbol{k}$, then the residue map induces an $\mathcal L^\der$-embedding $E \to \boldsymbol{k}$. If $\mathop{\mathrm{res}}(E) = \boldsymbol{k}$, then $E$ is called a **$T^\der$-lift of $\boldsymbol{k}$**. **Corollary 19**. *Suppose that $K$ is $T^\der$-henselian. Then any partial $T^\der$-lift of $\boldsymbol{k}$ can be extended to a $T^\der$-lift of $\boldsymbol{k}$.* *Proof.* Let $E$ be a partial $T^\der$-lift of $\boldsymbol{k}$ and let $a \in \boldsymbol{k}\setminus \mathop{\mathrm{res}}(E)$. Proposition [Proposition 18](#prop:resext){reference-type="ref" reference="prop:resext"} gives us $b \in \mathcal O$ with $\mathop{\mathrm{res}}(b) = a$ such that $E\langle\!\langle b\rangle\!\rangle$ is a partial $T^\der$-lift of $\boldsymbol{k}$. The corollary follows by Zorn's lemma. ◻ Note that the prime model of $T$ (identified with $\mathop{\mathrm{dcl}}_\mathcal L(\emptyset) \subseteq K$ and equipped with the trivial derivation) is a partial $T^\der$-lift of $\boldsymbol{k}$, so Corollary [Corollary 19](#cor:reslift){reference-type="ref" reference="cor:reslift"} has the following consequence: **Theorem 20**. *Suppose that $K$ is $T^\der$-henselian. Then $\boldsymbol{k}$ admits a $T^\der$-lift.* ## Spherically complete implies $T^\der$-henselian {#sec:sphcompTdh} In this subsection we assume that the derivation on $\boldsymbol{k}$ is nontrivial. To establish the result claimed in the subsection heading, under the assumption that $\boldsymbol{k}$ is linearly surjective, we work as usual with pseudocauchy sequences, which we abbreviate as *pc-sequences*; see [@ADH17 §2.2 and §3.2] for definitions and basic facts about them. In particular, recall that $K$ is spherically complete if and only if every pc-sequence in $K$ has a pseudolimit in $K$, and that if $L$ is an immediate $T^{\mathcal O,\der}$-extension of $K$, then every element of $L$ is the pseudolimit of a divergent pc-sequence in $K$ (i.e., a pc-sequence in $K$ that has no pseudolimit in $K$). We are interested in the behavior of $\mathcal L(K)$-definable functions along pc-sequences, which will allow us to study immediate $T^{\mathcal O,\der}$-extensions of $K$. **Lemma 21**. *Suppose that $\boldsymbol{k}$ is linearly surjective, $(F,A,a,\gamma)$ is in $T^\der$-hensel configuration, and $F(\mbox{\small$\mathscr{J}$}_\der^r a) \neq 0$. Then there is $b \in B(a,\gamma)$ such that $(F,A,b,\gamma)$ is in $T^\der$-hensel configuration, $vA_{\times(b-a)} = vF(\mbox{\small$\mathscr{J}$}_\der^r a)$, and $F(\mbox{\small$\mathscr{J}$}_\der^r b) \prec F(\mbox{\small$\mathscr{J}$}_\der^r a)$. Moreover, if $b^*$ is any element of $K$ with $b^* - a \sim b- a$, then $(F,A,b^*,\gamma)$ is in $T^\der$-hensel configuration, $vA_{\times(b^*-a)} = vF(\mbox{\small$\mathscr{J}$}_\der^r a)$, and $F(\mbox{\small$\mathscr{J}$}_\der^r b^*) \prec F(\mbox{\small$\mathscr{J}$}_\der^r a)$.* *Proof.* Let $\alpha \coloneqq vF(\mbox{\small$\mathscr{J}$}_\der^r a)> v_A(\gamma)$ and take $g\in K$ with $v_A(vg)= \alpha$, so $vg>\gamma$. Then for $y \in \mathcal O^{\times}$, we have $a+gy \in B(a,\gamma)$ and $$F\big(\mbox{\small$\mathscr{J}$}_\der^r(a+gy)\big)\ =\ F(\mbox{\small$\mathscr{J}$}_\der^r a) + A(gy) + \varepsilon,$$ where $v\varepsilon>vA_{\times gy} = v_A(vg) = \alpha$. Let $D$ be the linear differential operator $F(\mbox{\small$\mathscr{J}$}_\der^r a)^{-1}A_{\times g}$, so $D \asymp 1$. Since $\boldsymbol{k}$ is linearly surjective, we can find $u \in \mathcal O^\times$ with $1+D(u) \prec 1$. Let $b\coloneqq a+gu$, so $$F(\mbox{\small$\mathscr{J}$}_\der^r b) \ =\ F(\mbox{\small$\mathscr{J}$}_\der^r a) + A(gu)+\varepsilon\ =\ F(\mbox{\small$\mathscr{J}$}_\der^r a)\big(1+D(u)\big)+\varepsilon\ \prec\ F(\mbox{\small$\mathscr{J}$}_\der^r a).$$ Since $b-a \asymp g$, we have $vA_{\times(b-a)} = v_A(vg)=\alpha$. Since $B(b,\gamma) = B(a,\gamma)$ and $vF(\mbox{\small$\mathscr{J}$}_\der^r b)>vF(\mbox{\small$\mathscr{J}$}_\der^r a) > v_A(\gamma)$, the tuple $(F,A,b,\gamma)$ is in $T^\der$-hensel configuration. Now, let $b^* \in K$ with $b^* - a \sim b- a = gu$. Then we have $u^* \in K$ with $b^* =a+ gu^*$ and $u^* \sim u$. In particular, $1+D(u^*) \prec 1$, so the same argument as above gives that $(F,A,b^*,\gamma)$ is in $T^\der$-hensel configuration, $vA_{\times(b^*-a)} = vF(\mbox{\small$\mathscr{J}$}_\der^r a)$, and $F(\mbox{\small$\mathscr{J}$}_\der^r b^*) \prec F(\mbox{\small$\mathscr{J}$}_\der^r a)$. ◻ **Lemma 22**. *Suppose that $\boldsymbol{k}$ is linearly surjective, $(F,A,a,\gamma)$ is in $T^\der$-hensel configuration, and $F(\mbox{\small$\mathscr{J}$}_\der^r b) \neq 0$ for all $b \in B(a,\gamma)$ with $vA_{\times(b-a)} \geqslant vF(\mbox{\small$\mathscr{J}$}_\der^r a)$. Then there is a divergent pc-sequence $(a_\rho)$ in $K$ such that $F(\mbox{\small$\mathscr{J}$}_\der^r a_\rho) \leadsto 0$.* *Proof.* Suppose we have a nonzero ordinal $\lambda$ and a sequence $(a_\rho)_{\rho<\lambda}$ in $B(a,\gamma)$ satisfying 1. [\[divpcseq1\]]{#divpcseq1 label="divpcseq1"} $a_0 = a$ and $(F,A,a_\rho,\gamma)$ is in $T^\der$-hensel configuration for each $\rho<\lambda$; 2. [\[divpcseq2\]]{#divpcseq2 label="divpcseq2"} $v_A(\gamma_\rho)= vF(\mbox{\small$\mathscr{J}$}_\der^r a_\rho)$ whenever $\rho+1<\lambda$, where $\gamma_\rho\coloneqq v(a_{\rho+1}-a_\rho)$; 3. [\[divpcseq3\]]{#divpcseq3 label="divpcseq3"} $vF(\mbox{\small$\mathscr{J}$}_\der^r a_\rho)$ is strictly increasing as a function of $\rho$, for $\rho<\lambda$. Such a sequence exists when $\lambda=1$, so it suffices to extend $(a_\rho)_{\rho<\lambda}$ to a sequence $(a_\rho)_{\rho<\lambda+1}$ in $B(a, \gamma)$ satisfying [\[divpcseq1\]](#divpcseq1){reference-type="ref" reference="divpcseq1"}--[\[divpcseq3\]](#divpcseq3){reference-type="ref" reference="divpcseq3"} with $\lambda+1$ in place of $\lambda$. If $\lambda = \mu+1$ is a successor ordinal, then we use Lemma [Lemma 21](#lem:successorapprox){reference-type="ref" reference="lem:successorapprox"} to find $a_\lambda \in B(a,\gamma)$ such that $(F,A,a_\lambda,\gamma)$ is in $T^\der$-hensel configuration, $vA_{\times(a_\lambda-a_\mu)} = vF(\mbox{\small$\mathscr{J}$}_\der^r a_\mu)$, and $F(\mbox{\small$\mathscr{J}$}_\der^r a_\lambda) \prec F(\mbox{\small$\mathscr{J}$}_\der^r a_\mu)$. This extended sequence $(a_\rho)_{\rho<\lambda+1}$ satisfies conditions [\[divpcseq1\]](#divpcseq1){reference-type="ref" reference="divpcseq1"}--[\[divpcseq3\]](#divpcseq3){reference-type="ref" reference="divpcseq3"}. Suppose that $\lambda$ is a limit ordinal. Then $v_A(\gamma_\rho)$ is strictly increasing as a function of $\rho$, so $\gamma_\rho$ is also strictly increasing by [@ADH17 Lemma 4.5.1(iii)]. Hence, $(a_\rho)_{\rho<\lambda}$ is a pc-sequence with $F(\mbox{\small$\mathscr{J}$}_\der^r a_\rho) \leadsto 0$. If $(a_\rho)_{\rho<\lambda}$ is divergent, then we are done. Otherwise, let $a_\lambda$ be any pseudolimit of $(a_\rho)_{\rho<\lambda}$ in $K$. Since $a_\lambda- a_\rho\sim a_{\rho+1} - a_\rho$ for each $\rho< \lambda$, Lemma [Lemma 21](#lem:successorapprox){reference-type="ref" reference="lem:successorapprox"} gives that $(F,A,a_\lambda,\gamma)$ is in $T^\der$-hensel configuration and that $F(\mbox{\small$\mathscr{J}$}_\der^r a_\lambda) \prec F(\mbox{\small$\mathscr{J}$}_\der^r a_\rho)$ for each $\rho<\lambda$. Thus, the extended sequence $(a_\rho)_{\rho<\lambda+1}$ satisfies [\[divpcseq1\]](#divpcseq1){reference-type="ref" reference="divpcseq1"}--[\[divpcseq3\]](#divpcseq3){reference-type="ref" reference="divpcseq3"}. ◻ **Corollary 23**. *If $\boldsymbol{k}$ is linearly surjective and $K$ is spherically complete, then $K$ is $T^\der$-henselian.* In fact, the divergent pc-sequence we build in Lemma [Lemma 22](#lem:Tdhcdivpc){reference-type="ref" reference="lem:Tdhcdivpc"} is of a specific form analogous to "differential-algebraic type" in valued differential fields with small derivation (see [@ADH17 Sections 4.4 and 6.9]). To introduce this notion and refine Corollary [Corollary 23](#cor:sphcompTdh){reference-type="ref" reference="cor:sphcompTdh"}, let $(a_\rho)$ be a divergent pc-sequence in $K$, and let $\ell$ be a pseudolimit of $(a_\rho)$ in some $T^{\mathcal O,\der}$-extension $L$ of $K$. Then $v(\ell-K) \subseteq \Gamma$ has no greatest element, and we say that a property holds **for all $y \in K$ sufficiently close to $\ell$** if there exists $\gamma \in v(\ell-K)$ such that the property holds for all $y \in K$ with $v(\ell-y)>\gamma$. Under the assumptions that in this paper $K$ has small derivation and in this section $\boldsymbol{k}$ has nontrivial derivation, the definition of vanishing from [@Ka22] simplifies as follows. We say that $F$ **vanishes at $(K, \ell)$** if whenever $a \in K$ and $d \in K^{\times}$ satisfy $\ell-a \prec d$, we have $I_{F_{+a, \times d}}(\mbox{\small$\mathscr{J}$}_\der^{r-1}y) \prec 1$ for all $y \in K$ sufficiently close to $d^{-1}(\ell-a)$. Let $Z_r(K, \ell)$ be the set of all $F$ of arity $1+r$ that vanish at $(K, \ell)$ and $Z(K, \ell) \coloneqq \bigcup_{r} Z_r(K, \ell)$; we have $Z_0(K, \ell)=\emptyset$ by [@Ka22 Lemma 5.2]. We say that $(a_\rho)$ is of **$T^\der$-algebraic type over $K$** if $Z(K,\ell)\neq \emptyset$. Otherwise, $(a_\rho)$ is said to be of **$T^\der$-transcendental type over $K$**. Note that $Z(K,\ell)$ does not depend on $\ell$ or even $(a_\rho)$, only on $v(\ell-K) \subseteq \Gamma$. Thus, if $(a_\rho)$ is of $T^\der$-algebraic type over $K$, then so is $(b_\sigma)$ for any pc-sequence $(b_\sigma)$ in $K$ equivalent to $(a_\rho)$ (recall the various characterizations of equivalence of pc-sequences in [@ADH17 Lemma 2.2.17]). If $(a_\rho)$ is of $T^\der$-algebraic type over $K$ and $r$ is minimal with $Z_r(K,\ell) \neq \emptyset$, then any member of $Z_r(K,\ell)$ is called a **minimal $\mathcal L^\der$-function** of $(a_\rho)$ over $K$. We use the following two propositions to construct immediate extensions. **Fact 24**. *[@Ka22 Proposition 6.1][\[prop:Ka226.1\]]{#prop:Ka226.1 label="prop:Ka226.1"} Suppose that $(a_\rho)$ is of $T^\der$-transcendental type over $K$. Then $\ell$ is $T^\der$-transcendental over $K$ and $K\langle\!\langle\ell\rangle\!\rangle$ is an immediate $T^{\mathcal O,\der}$-extension of $K$. If $b$ is a pseudolimit of $(a_\rho)$ in a $T^{\mathcal O,\der}$-extension $M$ of $K$, then there is a unique $\mathcal L^{\mathcal O,\der}(K)$-embedding $K\langle\!\langle\ell\rangle\!\rangle\to M$ sending $\ell$ to $b$.* **Fact 25**. *[@Ka22 Proposition 6.2][\[prop:Ka226.2\]]{#prop:Ka226.2 label="prop:Ka226.2"} Suppose that $(a_\rho)$ is of $T^\der$-algebraic type over $K$, and let $F$ be a minimal $\mathcal L^\der$-function of $(a_\rho)$ over $K$. Then $K$ has an immediate $T^{\mathcal O,\der}$-extension $K\langle\!\langle a\rangle\!\rangle$ with $F(\mbox{\small$\mathscr{J}$}_\der^{r}a)=0$ and $a_\rho\leadsto a$. If $b$ is a pseudolimit of $(a_\rho)$ in a $T^{\mathcal O,\der}$-extension $M$ of $K$ with $F(\mbox{\small$\mathscr{J}$}_\der^{r}b)=0$, then there is a unique $\mathcal L^{\mathcal O,\der}(K)$-embedding $K\langle\!\langle a\rangle\!\rangle\to M$ sending $a$ to $b$.* Let us connect pc-sequences of $T^\der$-algebraic type to $T^\der$-algebraic $T^{\mathcal O,\der}$-extensions. We say that $K$ is **$T^\der$-algebraically maximal** if $K$ has no proper immediate $T^{\mathcal O,\der}$-extension that is $T^\der$-algebraic over $K$. **Lemma 26**. *The following are equivalent:* 1. *$K$ is $T^\der$-algebraically maximal;* 2. *every pc-sequence of $T^\der$-algebraic type over $K$ has a pseudolimit in $K$.* *Proof.* If $(a_\rho)$ is of $T^\der$-algebraic type over $K$, then Fact [\[prop:Ka226.2\]](#prop:Ka226.2){reference-type="ref" reference="prop:Ka226.2"} provides a proper immediate $T^{\mathcal O,\der}$-extension of $K$ that is $T^\der$-algebraic over $K$. Conversely, if $a$ is an element in a proper immediate $T^{\mathcal O,\der}$-extension that is $T^\der$-algebraic over $K$, then any divergent pc-sequence in $K$ with pseudolimit $a$ is necessarily of $T^\der$-algebraic type over $K$ by Fact [\[prop:Ka226.1\]](#prop:Ka226.1){reference-type="ref" reference="prop:Ka226.1"}. ◻ Now we show that the divergent pc-sequence constructed in Lemma [Lemma 22](#lem:Tdhcdivpc){reference-type="ref" reference="lem:Tdhcdivpc"} is of $T^\der$-algebraic type over $K$. **Lemma 27**. *If $Z_q(K, \ell) = \emptyset$ for all $q<r$ and $F(\mbox{\small$\mathscr{J}$}_\der^r a_\rho) \leadsto 0$, then $F \in Z_r(K, \ell)$. In particular, if $F(\mbox{\small$\mathscr{J}$}_\der^r a_\rho) \leadsto 0$, then $Z(K, \ell)\neq\emptyset$.* *Proof.* Suppose that $Z_q(K, \ell) = \emptyset$ for all $q<r$ and $F \not\in Z_r(K, \ell)$. By [@Ka22 Proposition 5.4], $F(\mbox{\small$\mathscr{J}$}_\der^r \ell) \neq 0$ and $F(\mbox{\small$\mathscr{J}$}_\der^r a_\rho) \sim F(\mbox{\small$\mathscr{J}$}_\der^r \ell)$ for all sufficiently large $\rho$, and thus $F(\mbox{\small$\mathscr{J}$}_\der^r a_\rho) \not\leadsto 0$. ◻ In light of Lemmas [Lemma 26](#lem:TdalgmaxTdalgtype){reference-type="ref" reference="lem:TdalgmaxTdalgtype"} and [Lemma 27](#lem:pconvvanish){reference-type="ref" reference="lem:pconvvanish"}, Lemma [Lemma 22](#lem:Tdhcdivpc){reference-type="ref" reference="lem:Tdhcdivpc"} yields the following refinement of Corollary [Corollary 23](#cor:sphcompTdh){reference-type="ref" reference="cor:sphcompTdh"}. **Corollary 28**. *If $\boldsymbol{k}$ is linearly surjective and $K$ is $T^\der$-algebraically maximal, then $K$ is $T^\der$-henselian.* ## More on behavior along pc-sequences {#sec:Tdhc} In this subsection, we prove some technical lemmas regarding behavior along pc-sequences. For the remainder of this subsection, let $(a_\rho)$ be a divergent pc-sequence in $K$, let $L$ be a $T^{\mathcal O,\der}$-extension of $K$, and let $\ell \in L$ be a pseudolimit of $(a_\rho)$. Set $\gamma_{\rho} \coloneqq v(a_{\rho+1}-a_{\rho})$, and set $B_\rho \coloneqq B(a_{\rho+1},\gamma_\rho)$. In this context, *eventually* means for all sufficiently large $\rho$. The next lemma is an analogue of [@ADH17 Lemma 6.8.1]. **Lemma 29**. *Suppose that $A$ linearly approximates $F$ on $B_{\rho}^L$, eventually. Then there is a pc-sequence $(b_\rho)$ in $K$ equivalent to $(a_\rho)$ such that $F(\mbox{\small$\mathscr{J}$}_\der^r b_\rho) \leadsto F(\mbox{\small$\mathscr{J}$}_\der^r \ell)$.* *Proof.* By removing some initial terms of the sequence, we can assume that for all $\rho$, $A$ linearly approximates $F$ on $B_{\rho}^L$ and $\gamma_\rho = v(\ell-a_\rho)$, and also that $\gamma_\rho$ is strictly increasing as a function of $\rho$. Take $g_\rho \in K$ and $y_\rho \in \mathcal O_L^{\times}$ as in the proof of [@ADH17 Lemma 6.8.1] (with $\ell$, $F$, and $A$ replacing $a$, $G$, and $P$, respectively) so that $vg_\rho = \gamma_\rho$, $b_\rho \coloneqq \ell + g_{\rho}y_{\rho} \in K$, and $vA(g_\rho y_\rho) = v_A(\gamma_\rho)$. Then $(b_\rho)$ is a pc-sequence equivalent to $(a_\rho)$ and $$F(\mbox{\small$\mathscr{J}$}_\der^r b_\rho) - F(\mbox{\small$\mathscr{J}$}_\der^r \ell)\ =\ A(g_\rho y_\rho) + \varepsilon_\rho,$$ where $\varepsilon_\rho \in L$ with $v\varepsilon_\rho > vA_{\times g_\rho y_\rho} = v_A(\gamma_\rho)$. Since $\gamma_\rho$ is strictly increasing and $vA(g_\rho y_\rho) = v_A(\gamma_\rho)$, we have $F(\mbox{\small$\mathscr{J}$}_\der^r b_\rho) \leadsto F(\mbox{\small$\mathscr{J}$}_\der^r \ell)$, as desired. ◻ We say that $(F, A, (a_\rho))$ is in **$T^\der$-hensel configuration** if there is an index $\rho_0$ such that $(F, A, a_{\rho'}, \gamma_\rho)$ is in $T^\der$-hensel configuration for all $\rho'>\rho\geqslant\rho_0$. Tacitly, this is relative to $K$. We say that $(F, A, (a_\rho))$ is in **$T^\der$-hensel configuration in $L$** if when we identify $F$ with $F^L \colon L^{1+r} \to L$ and $A$ with the obvious element of $L[\der]$, $(F, A, (a_\rho))$ is in $T^\der$-hensel configuration relative to $L$. Modulo passing to cofinal subsequences, being in $T^\der$-hensel configuration is preserved by equivalence of pc-sequences: **Lemma 30**. *Suppose that $(F, A, (a_\rho))$ is in $T^\der$-hensel configuration in $L$ and let $(b_{\sigma})$ be a pc-sequence in $K$ equivalent to $(a_\rho)$. There exists a cofinal subsequence $(b_{\lambda})$ of $(b_{\sigma})$ such that $(F, A, (b_{\lambda}))$ is in $T^\der$-hensel configuration in $L$.* *Proof.* By discarding some initial terms, we can assume that $\gamma_\rho$ and $\delta_\sigma \coloneqq v(b_{\sigma+1}-b_{\sigma})$ are strictly increasing. First, take $\rho_0$ and $\sigma_0$ large enough that $(F, A, a_{\rho'}, \gamma_{\rho})$ is in $T^\der$-hensel configuration in $L$ for all $\rho'>\rho \geqslant\rho_0$ and $v(a_\rho-b_\sigma)>\gamma_{\rho_0}$ for all $\rho>\rho_0$ and $\sigma>\sigma_0$. By increasing $\sigma_0$, we can arrange that $\delta_{\sigma_0} \geqslant\gamma_{\rho_0}$. Then for any $\rho>\rho_0$ and $\sigma>\sigma_0$, $A$ linearly approximates $F$ on $B^L_{\rho_0}\supseteq B^L_{\sigma_0}$. Second, take $\sigma_1 > \sigma_0$ and $\rho_1 > \rho_0$ sufficiently large that $v(b_{\sigma}-a_{\rho})>\delta_{\sigma_1}$ for all $\sigma>\sigma_1$ and $\rho>\rho_1$. Let $\sigma>\sigma_1$. To show that $(F, A, b_{\sigma}, \delta_{\sigma_1})$ is in $T^\der$-hensel configuration in $L$, suppose towards a contradiction that $vF(\mbox{\small$\mathscr{J}$}_\der^{r}b_{\sigma}) \leqslant v_A(\delta_{\sigma_1})$. By increasing $\rho_1$, we can assume that $\gamma_{\rho_1}>\delta_{\sigma_1}$. Then for $\rho>\rho_1$ we have $vF(\mbox{\small$\mathscr{J}$}_\der^{r}a_{\rho})>v_A(\gamma_{\rho_1})>v_A(\delta_{\sigma_1})$ and $vA(b_\sigma-a_\rho) \geqslant vA_{\times(b_\sigma-a_\rho)}>v_A(\delta_{\sigma_1})$, and thus $$v\big( F(\mbox{\small$\mathscr{J}$}_\der^{r}b_{\sigma})-F(\mbox{\small$\mathscr{J}$}_\der^{r}a_{\rho}) - A(b_{\sigma}-a_{\rho}) \big)\ =\ vF(\mbox{\small$\mathscr{J}$}_\der^{r}b_{\sigma})\ \leqslant\ v_A(\delta_{\sigma_1})\ <\ vA_{\times (b_{\sigma}-a_{\rho})},$$ contradicting that $A$ linearly approximates $F$ on $B^L_{\rho_0}$. Iterating this second step yields a cofinal subsequence of $(b_{\sigma})$ with the desired property. ◻ If $(F, A, (a_\rho))$ is in $T^\der$-hensel configuration in $L$, then by definition, $A$ linearly approximates $F$ on $B_{\rho}^L$, eventually. The converse holds, under the assumption that $F(\mbox{\small$\mathscr{J}$}_\der^ra_\rho)\leadsto 0$. **Lemma 31**. *Suppose that $F(\mbox{\small$\mathscr{J}$}_\der^ra_\rho)\leadsto 0$ and that $A$ linearly approximates $F$ on $B_{\rho}^{L}$, eventually. Then $(F, A, (a_\rho))$ is in $T^\der$-hensel configuration in $L$.* *Proof.* Take $\rho_0$ such that $A$ linearly approximates $F$ on $B_{\rho_0}^L$. By increasing $\rho_0$, we may assume that $vF(\mbox{\small$\mathscr{J}$}_\der^ra_\rho)$ is strictly increasing for $\rho>\rho_0$. Let $\rho' > \rho> \rho_0$ be given. We need to show that $(F,A,a_{\rho'},\gamma_\rho)$ is in $T^\der$-hensel configuration in $L$. Since $A$ linearly approximates $F$ on $B_\rho^L$, it suffices to show that $vF(\mbox{\small$\mathscr{J}$}_\der^ra_{\rho'}) > v_A(\gamma_\rho)$. Since $a_{\rho'}$ and $a_\rho$ are in $B_{\rho_0}$, we have $$v\big(F(\mbox{\small$\mathscr{J}$}_\der^ra_{\rho'}) - F(\mbox{\small$\mathscr{J}$}_\der^ra_\rho) - A(a_{\rho'}-a_\rho)\big) \ > \ vA_{\times (a_{\rho'}-a_\rho)}\ =\ v_A(\gamma_\rho).$$ Since $vA(a_{\rho'}- a_\rho) \geqslant v_A(\gamma_\rho)$, we must have $v\big(F(\mbox{\small$\mathscr{J}$}_\der^ra_{\rho'}) - F(\mbox{\small$\mathscr{J}$}_\der^ra_\rho)\big) \geqslant v_A(\gamma_\rho)$ as well. This gives $vF(\mbox{\small$\mathscr{J}$}_\der^ra_\rho)\geqslant v_A(\gamma_\rho)$, since $F(\mbox{\small$\mathscr{J}$}_\der^ra_{\rho'}) \prec F(\mbox{\small$\mathscr{J}$}_\der^ra_\rho)$, so $vF(\mbox{\small$\mathscr{J}$}_\der^ra_{\rho'}) > v_A(\gamma_\rho)$, as desired. ◻ The next lemma shows that $T^\der$-hensel configuration in the presence of $T^\der$-henselianity allows us to find a pseudolimit that is also a zero of the definable function, a key step towards our results. **Lemma 32**. *Suppose that $L$ is $T^\der$-henselian and that $(F, A, (a_\rho))$ is in $T^\der$-hensel configuration in $L$. Then there exists $b \in L$ such that $a_\rho \leadsto b$ and $F(\mbox{\small$\mathscr{J}$}_\der^rb)=0$.* *Proof.* Suppose that $(F, A, a_{\rho'}, \gamma_\rho)$ is in $T^\der$-hensel configuration in $L$ for all $\rho'>\rho\geqslant\rho_0$. By increasing $\rho_0$, we can assume that $v(\ell-a_\rho) = \gamma_\rho$ for all $\rho>\rho_0$ and $\gamma_\rho$ is strictly increasing as a function of $\rho>\rho_0$. For $\rho>\rho_0$, an argument similar to one in the previous proof shows that $(F, A, \ell, \gamma_\rho)$ is in $T^\der$-hensel configuration in $L$. Hence $T^\der$-henselianity yields $b \in L$ with $F(\mbox{\small$\mathscr{J}$}_\der^r b)=0$ and $vA_{\times (b-\ell)} \geqslant vF(\mbox{\small$\mathscr{J}$}_\der^r \ell)$. From $vF(\mbox{\small$\mathscr{J}$}_\der^r \ell) > v_A(\gamma_\rho)$ for all $\rho>\rho_0$, we get $v(b-\ell)>\gamma_\rho$ for all $\rho>\rho_0$, and thus $a_\rho \leadsto b$. ◻ ## Uniqueness and the $T^\der$-hensel configuration property {#sec:sphcompunique} In this subsection, we establish the key property needed for our theorem on the uniqueness of spherically complete immediate $T^{\mathcal O,\der}$-extensions of monotone $T^{\mathcal O,\der}$-models. We say that $K$ has the **$T^\der$-hensel configuration property** if whenever we have 1. a divergent pc-sequence $(a_\rho)$ in $K$ of $T^\der$-algebraic type over $K$, 2. a minimal $\mathcal L^\der$-function $F$ of $(a_\rho)$ over $K$, and 3. an immediate $T^{\mathcal O,\der}$-extension $L$ of $K$ containing a pseudolimit of $(a_\rho)$, there is an $A \in K[\der]^{\neq}$ that linearly approximates $F$ on $B(a_{\rho+1},\gamma_\rho)^{L}$ for all sufficiently large $\rho$, where $\gamma_\rho\coloneqq v(a_{\rho+1}-a_\rho)$. The $T^\der$-hensel configuration property is an analogue of the differential-henselian configuration property for valued differential fields with small derivation that was implicitly used in [@ADH17 Chapter 7] and explicitly introduced in [@DPC19]. In this subsection, we will show how uniqueness follows from the $T^\der$-hensel configuration property, without assuming monotonicity. Then we show in Section [4.1](#sec:monotoneTdhc){reference-type="ref" reference="sec:monotoneTdhc"} that monotone $T^{\mathcal O,\der}$-models have the $T^\der$-hensel configuration property. First, we use the $T^\der$-hensel configuration property to give an alternative description of minimal $\mathcal L^\der$-functions. **Lemma 33**. *Suppose that $K$ has the $T^\der$-hensel configuration property, and let $(a_\rho)$ be a divergent pc-sequence in $K$. Then the following are equivalent:* 1. *$(a_\rho)$ is of $T^\der$-algebraic type over $K$ and $F$ is a minimal $\mathcal L^\der$-function for $(a_\rho)$ over $K$.* 2. *$F(\mbox{\small$\mathscr{J}$}_\der^r b_\sigma) \leadsto 0$ for some pc-sequence $(b_\sigma)$ in $K$ equivalent to $(a_\rho)$, and $G(\mbox{\small$\mathscr{J}$}_\der^q b_\sigma) \not\leadsto 0$ for $q<r$, every $\mathcal L(K)$-definable function $G \colon K^{1+q} \to K$ in implicit form, and every pc-sequence $(b_\sigma)$ in $K$ equivalent to $(a_\rho)$.* *Proof.* Suppose that $F$ is a minimal $\mathcal L^\der$-function for $(a_\rho)$ over $K$. Fact [\[prop:Ka226.2\]](#prop:Ka226.2){reference-type="ref" reference="prop:Ka226.2"} yields an immediate $T^{\mathcal O,\der}$-extension $K\langle\!\langle a \rangle\!\rangle$ of $K$ such that $a_\rho \leadsto a$ and $F(\mbox{\small$\mathscr{J}$}_\der^r a)=0$. As $K$ has the $T^\der$-hensel configuration property, Lemma [Lemma 29](#lem:6.8.1){reference-type="ref" reference="lem:6.8.1"} provides a pc-sequence $(b_\rho)$ in $K$ equivalent to $(a_\rho)$ such that $F(\mbox{\small$\mathscr{J}$}_\der^r b_\rho) \leadsto 0$. For each $q<r$, we have $Z_q(K,a) = \emptyset$, so $G(\mbox{\small$\mathscr{J}$}_\der^q b_\sigma) \not\leadsto 0$ for any $\mathcal L(K)$-definable function $G \colon K^{1+q} \to K$ in implicit form and any pc-sequence $(b_\sigma)$ in $K$ equivalent to $(a_\rho)$ by Lemma [Lemma 27](#lem:pconvvanish){reference-type="ref" reference="lem:pconvvanish"}. Now suppose that $F$ is not a minimal $\mathcal L^\der$-function for $(a_\rho)$ over $K$, and fix a pseudolimit $\ell$ of $(a_\rho)$ in some $T^{\mathcal O,\der}$-extension of $K$. If $G \in Z_q(K,\ell)$ is a minimal $\mathcal L^\der$-function for $(a_\rho)$ over $K$ for some $q<r$, then we have $G(\mbox{\small$\mathscr{J}$}_\der^q b_\sigma) \leadsto 0$ for some pc-sequence $(b_\sigma)$ in $K$ equivalent to $(a_\rho)$ by the first part of this proof. If $Z_q(K,\ell) =\emptyset$ for all $q<r$, then $F \not\in Z_r(K,\ell)$, so $F(\mbox{\small$\mathscr{J}$}_\der^r b_\sigma) \not\leadsto 0$ for any pc-sequence $(b_\sigma)$ in $K$ equivalent to $(a_\rho)$ by Lemma [Lemma 27](#lem:pconvvanish){reference-type="ref" reference="lem:pconvvanish"}. ◻ **Assumption 34**. *For the rest of this section, suppose that every immediate $T^{\mathcal O,\der}$-extension of $K$ has the $T^\der$-hensel configuration property.* **Theorem 35**. *Suppose that $\boldsymbol{k}$ is linearly surjective. Any two spherically complete immediate $T^{\mathcal O,\der}$-extensions of $K$ are $\mathcal L^{\mathcal O,\der}$-isomorphic over $K$. Any two $T^\der$-algebraically maximal immediate $T^{\mathcal O,\der}$-extensions of $K$ that are $T^\der$-algebraic over $K$ are $\mathcal L^{\mathcal O,\der}$-isomorphic over $K$.* *Proof.* Let $L_0$ and $L_1$ be spherically complete immediate $T^{\mathcal O,\der}$-extensions of $K$. Let $\mu \colon K_0 \to K_1$ be a maximal $\mathcal L^{\mathcal O,\der}$-isomorphism between $T^{\mathcal O,\der}$-extensions $K_0 \subseteq L_0$ and $K_1 \subseteq L_1$ of $K$. For convenience, we identify $K_0$ and $K_1$ via $\mu$, so $\mu$ becomes the identity, and assume moreover that $K_0=K_1=K$. Suppose that $K \neq L_0$ (equivalently, $K \neq L_1$), so we have $\ell \in L_0 \setminus K$ and a divergent pc-sequence $(a_\rho)$ in $K$ with $a_\rho \leadsto \ell$. If $(a_\rho)$ is $T^\der$-transcendental, then we can take $b \in L_1$ with $a_\rho \leadsto b$ and extend $\mu$ to an $\mathcal L^{\mathcal O,\der}$-isomorphism $K\langle\!\langle\ell \rangle\!\rangle\to K\langle\!\langle b \rangle\!\rangle$ sending $\ell$ to $b$ by Fact [\[prop:Ka226.1\]](#prop:Ka226.1){reference-type="ref" reference="prop:Ka226.1"}, contradicting the maximality of $\mu$. Now suppose that $(a_\rho)$ is $T^\der$-algebraic, and let $F$ be a minimal $\mathcal L^\der$-function of $(a_\rho)$ over $K$. Using Lemma [Lemma 33](#lem:vanishpconv){reference-type="ref" reference="lem:vanishpconv"}, we replace $(a_\rho)$ by an equivalent pc-sequence in $K$ to arrange that $F(\mbox{\small$\mathscr{J}$}_\der^r a_\rho) \leadsto 0$. By assumption $K$ has the $T^\der$-hensel configuration property, so by Lemma [Lemma 31](#lem:Tdhc){reference-type="ref" reference="lem:Tdhc"}, we have $A_0, A_1 \in K[\der]$ such that $(F, A_0, (a_\rho))$ is in $T^\der$-hensel configuration in $L_0$ and $(F, A_1, (a_\rho))$ is in $T^\der$-hensel configuration in $L_1$. By Corollary [Corollary 23](#cor:sphcompTdh){reference-type="ref" reference="cor:sphcompTdh"} and Lemma [Lemma 32](#lem:mindiffzero){reference-type="ref" reference="lem:mindiffzero"}, we have $b_0 \in L_0 \setminus K$ and $b_1 \in L_1 \setminus K$ such that $a_\rho \leadsto b_0$, $a_\rho \leadsto b_1$, $F(\mbox{\small$\mathscr{J}$}_\der^r b_0)=0$, and $F(\mbox{\small$\mathscr{J}$}_\der^r b_1)=0$. Then Fact [\[prop:Ka226.2\]](#prop:Ka226.2){reference-type="ref" reference="prop:Ka226.2"} yields an extension of $\mu$ to an $\mathcal L^{\mathcal O,\der}$-isomorphism $K\langle\!\langle b_0\rangle\!\rangle\to K\langle\!\langle b_1\rangle\!\rangle$, contradicting the maximality of $\mu$. The proof of the second statement is similar but it uses Lemma [Lemma 26](#lem:TdalgmaxTdalgtype){reference-type="ref" reference="lem:TdalgmaxTdalgtype"} and also Corollary [Corollary 28](#cor:TdalgmaxTdh){reference-type="ref" reference="cor:TdalgmaxTdh"} replaces Corollary [Corollary 23](#cor:sphcompTdh){reference-type="ref" reference="cor:sphcompTdh"}. ◻ In the case of few constants, we have two additional results and an easy corollary. The first is a converse to Corollary [Corollary 28](#cor:TdalgmaxTdh){reference-type="ref" reference="cor:TdalgmaxTdh"}, and it does not need the assumption that proper immediate $T^{\mathcal O,\der}$-extensions of $K$ have the $T^\der$-hensel configuration property, only that $K$ itself does. **Theorem 36**. *Suppose that $K$ is $T^\der$-henselian and $C \subseteq \mathcal O$. Then $K$ is $T^\der$-algebraically maximal.* *Proof.* Let $(a_\rho)$ be a pc-sequence in $K$ of $T^\der$-algebraic type over $K$ with minimal $\mathcal L^\der$-function $F$ over $K$. By Lemma [Lemma 26](#lem:TdalgmaxTdalgtype){reference-type="ref" reference="lem:TdalgmaxTdalgtype"}, it suffices to show that $(a_\rho)$ has a pseudolimit in $K$. Assume towards a contradiction that $(a_\rho)$ is divergent. Then we may replace $(a_\rho)$ by an equivalent pc-sequence in $K$ to arrange that $F(\mbox{\small$\mathscr{J}$}_\der^r a_\rho) \leadsto 0$ by Lemma [Lemma 33](#lem:vanishpconv){reference-type="ref" reference="lem:vanishpconv"}. By assumption, $K$ has the $T^\der$-hensel configuration property, so by Lemma [Lemma 31](#lem:Tdhc){reference-type="ref" reference="lem:Tdhc"} we have $A \in K[\der]^{\neq}$ of order $q$ such that $(F,A,(a_\rho))$ is in $T^\der$-hensel configuration in $K$. By removing some initial terms of the sequence, we arrange that $\gamma_\rho \coloneqq v(a_{\rho+1}-a_\rho)$ is strictly increasing as a function of $\rho$, $\gamma_\rho = v(a-a_\rho)$, and $(F,A,a_{\rho'},\gamma_\rho)$ is in $T^\der$-hensel configuration for all $\rho'>\rho$. By $T^\der$-henselianity take for each $\rho$ a $z_\rho \in B(a_{\rho+1},\gamma_\rho)$ with $F(\mbox{\small$\mathscr{J}$}_\der^r z_\rho)=0$. Then $v(a-z_{\rho})>\gamma_\rho$ and $v(z_{\rho}-a_{\rho})=\gamma_\rho$. Since $(\gamma_\rho)$ is cofinal in $v(a-K)$, we can take indices $\rho_0<\dots<\rho_{q+2}$ such that $a-z_{\rho_j} \prec a-z_{\rho_i}$ whenever $0 \leqslant i<j \leqslant q+2$. Set $y_i \coloneqq z_{\rho_{i+1}}$ for $i=0, \dots, q+1$. If $1 \leqslant i \leqslant q$, then $$y_i-y_{i-1}\ \sim\ a-y_{i-1}\ \succ\ a-y_{i}\ \sim\ y_{i+1}-y_{i},$$ so conditions [\[lem:7.5.5i\]](#lem:7.5.5i){reference-type="ref" reference="lem:7.5.5i"} and [\[lem:7.5.5ii\]](#lem:7.5.5ii){reference-type="ref" reference="lem:7.5.5ii"} from Lemma [Lemma 14](#lem:7.5.5){reference-type="ref" reference="lem:7.5.5"} are satisfied. Letting $\gamma \coloneqq \gamma_{\rho_0}$, we will reach a contradiction with that lemma by showing that $(F, A, y_{q+1}, \gamma)$ is in $T^\der$-hensel configuration and $v(y_0-y_{q+1})>\gamma$. The latter holds because $$v(a-y_{q+1})\ >\ v(a-y_0)\ >\ \gamma_{\rho_1}\ >\ \gamma.$$ We also have $B(y_{q+1}, \gamma) = B(y_0, \gamma) = B(a_{\rho_1}, \gamma)$, so since $(F, A, a_{\rho_1}, \gamma)$ is in $T^\der$-hensel configuration, $A$ linearly approximates $F$ on $B(y_{q+1}, \gamma)$. It remains to note that $vF(\mbox{\small$\mathscr{J}$}_\der^r y_{q+1})=\infty>v_A(\gamma)$. ◻ The previous result fails without the assumption $C \subseteq \mathcal O$, as the next example shows. **Example 37**. We build an increasing sequence $(\Gamma_n)$ of divisible ordered abelian groups as follows: set $\Gamma_0 \coloneqq \{0\}$ and given $\Gamma_n$, set $\Gamma_{n+1} \coloneqq \Gamma_n \oplus \mathbb{Q}\gamma_{n}$, where $\gamma_n$ is a new element greater than $\Gamma_n$. Now take a linearly surjective $\boldsymbol{k}\models T^\der_{\mathop{\mathrm{an}}}$. From $\boldsymbol{k}$ and $(\Gamma_n)$ we obtain the ordered Hahn field $\boldsymbol{k}(\!(t^{\Gamma_n})\!)$ with $0<t<\boldsymbol{k}^>$. As in the introduction, we expand $\boldsymbol{k}(\!(t^{\Gamma_n})\!)$ to a model $E_n \coloneqq \boldsymbol{k}(\!(t^{\Gamma_n})\!)_{\mathop{\mathrm{an}},c} \models T^{\mathcal O,\der}_{\mathop{\mathrm{an}}}$ with $c$ the zero map. That is, we expand $\boldsymbol{k}(\!(t^{\Gamma_n})\!)$ to a model of $T_{\mathop{\mathrm{an}}}$ by Taylor expansion; equip it with the convex hull of $\boldsymbol{k}$, which is a $T_{\mathop{\mathrm{an}}}$-convex valuation ring; and equip it with the derivation given by $\der(\sum_{\gamma} f_{\gamma}t^{\gamma}) = \sum_{\gamma} \der(f_{\gamma})t^{\gamma}$, which is a $T^\der_{\mathop{\mathrm{an}}}$-derivation by [@Ka21 Proposition 3.14]. Identifying $E_n$ with an $\mathcal L^{\mathcal O,\der}$-substructure of $E_{n+1}$ in the obvious way, we get an increasing sequence $(E_n)$ of $T^{\mathcal O,\der}_{\mathop{\mathrm{an}}}$-models. Then $E \coloneqq \bigcup_n E_n \models T^{\mathcal O,\der}_{\mathop{\mathrm{an}}}$ by [@Ka21 Corollary 3.16]. Note that $E$ is $T^\der$-henselian since each $E_n$ is by Corollary [Corollary 23](#cor:sphcompTdh){reference-type="ref" reference="cor:sphcompTdh"}. Now set $\Gamma \coloneqq \bigcup_n \Gamma_n$ and $L \coloneqq \boldsymbol{k}(\!(t^{\Gamma} )\!)_{\mathop{\mathrm{an}},c}$ as before, so $E\subseteq L$. We have $\sum_n t^{\gamma_n} \in L \setminus E$ with $\der(\sum_n t^{\gamma_n})=0$, so $E$ is not $T^\der_{\mathop{\mathrm{an}}}$-algebraically maximal. Recall that any $T^\der$-henselian $K$ with $C \subseteq \mathcal O$ is asymptotic (see Section [3.1](#sec:Tdhbasic){reference-type="ref" reference="sec:Tdhbasic"}), so Theorem [Theorem 36](#thm:TdhTdalgmax){reference-type="ref" reference="thm:TdhTdalgmax"} is really about asymptotic $T^{\mathcal O,\der}$-models, as are the next results about minimal $T^\der$-henselian extensions. An immediate $T^{\mathcal O,\der}$-extension $L$ of $K$ is a **$T^\der$-henselization of $K$** if $L$ is $T^\der$-henselian and for every immediate $T^\der$-henselian $T^{\mathcal O,\der}$-extension $M$ of $K$, there is an $\mathcal L^{\mathcal O,\der}$-embedding $L \to M$ that is the identity on $K$. **Theorem 38**. *Suppose that $\boldsymbol{k}$ is linearly surjective and $K$ is asymptotic. Then $K$ has a $T^\der$-henselization that is $T^\der$-algebraic over $K$ and has no proper $T^\der$-henselian $\mathcal L^{\mathcal O,\der}$-substructure containing $K$. In particular, any two $T^\der$-henselizations of $K$ are isomorphic over $K$.* *Proof.* Let $L$ be a $T^\der$-algebraically maximal immediate $T^{\mathcal O,\der}$-extension of $K$ that is $T^\der$-algebraic over $K$. Then $L$ is $T^\der$-henselian by Corollary [Corollary 28](#cor:TdalgmaxTdh){reference-type="ref" reference="cor:TdalgmaxTdh"} and asymptotic by [@ADH18 Lemma 1.12] (or [@ADH17 Lemmas 9.4.2 and 9.4.5]). In particular, $C_L \subseteq \mathcal O_L$, and thus by Theorem [Theorem 36](#thm:TdhTdalgmax){reference-type="ref" reference="thm:TdhTdalgmax"} no proper $\mathcal L^{\mathcal O,\der}$-substructure of $L$ containing $K$ is $T^\der$-henselian. Now let $M$ be an immediate $T^\der$-henselian $T^{\mathcal O,\der}$-extension of $K$, so $M$ is asymptotic and hence $T^\der$-algebraically maximal by Theorem [Theorem 36](#thm:TdhTdalgmax){reference-type="ref" reference="thm:TdhTdalgmax"}, making Lemma [Lemma 32](#lem:mindiffzero){reference-type="ref" reference="lem:mindiffzero"} available. To see that there is an $\mathcal L^{\mathcal O,\der}$-embedding of $L$ into $M$ over $K$, argue as in the proof of Theorem [Theorem 35](#thm:sphcompunique){reference-type="ref" reference="thm:sphcompunique"}. ◻ **Corollary 39**. *Suppose that $\boldsymbol{k}$ is linearly surjective and $K$ is asymptotic. Any immediate $T^\der$-henselian $T^{\mathcal O,\der}$-extension of $K$ that is $T^\der$-algebraic over $K$ is a $T^\der$-henselization of $K$.* *Proof.* Let $M$ be an immediate $T^\der$-henselian $T^{\mathcal O,\der}$-extension of $K$ that is $T^\der$-algebraic over $K$ and let $L$ be a $T^\der$-henselization of $K$. Then the $\mathcal L^{\mathcal O,\der}$-embedding $L \to M$ over $K$ is surjective by Theorem [Theorem 36](#thm:TdhTdalgmax){reference-type="ref" reference="thm:TdhTdalgmax"}. ◻ # Monotone fields {#sec:monotone} ## $T^\der$-hensel configuration property for monotone fields {#sec:monotoneTdhc} In this subsection, we assume that $K$ is monotone and the derivation of $\boldsymbol{k}$ is nontrivial. Let $(a_\rho)$ be a divergent pc-sequence in $K$, and let $\ell$ be a pseudolimit of $(a_\rho)$ in a monotone $T^{\mathcal O,\der}$-extension $M$ of $K$. Let $F\colon K^{1+r} \to K$ be an $\mathcal L(K)$-definable function in implicit form, and assume that $Z_q(K,\ell) = \emptyset$ for all $q < r$. For each $\rho$, set $\gamma_\rho\coloneqq v(a_{\rho+1}-a_\rho)$ and set $B_\rho\coloneqq B(a_{\rho+1},\gamma_\rho)$. We assume that $\gamma_\rho$ is strictly increasing as a function of $\rho$, so $B_{\rho'} \subsetneq B_\rho$ for $\rho'> \rho$. **Proposition 40**. *There is an $A \in K[\der]^{\neq}$ and an index $\rho_0$ such that $A$ linearly approximates $F$ on $B_{\rho_0}^M$.* *Proof.* Note that if $A$ linearly approximates $\mathfrak{m}_F^{-1}F$ on an open $v$-ball $B$, then $\mathfrak{m}_FA$ linearly approximates $F$ on $B$, so we may assume that $\mathfrak{m}_F = 1$. By applying Fact [Fact 2](#fact:rvapprox){reference-type="ref" reference="fact:rvapprox"} to the function $I_F$, we find an $\mathcal L^{\mathop{\mathrm{RV}}^{\mathop{\mathrm{eq}}}}(K)$-definable map $\chi\colon K^r \to \mathop{\mathrm{RV}}_{\! K}^{\mathop{\mathrm{eq}}}$ such that for each $s \in \chi(K^r)$, if $\chi^{-1}(s)$ contains an open $v$-ball, then either $I_F$ is constant on $\chi^{-1}(s)$ or there is $d \in K^r$ such that $$\label{eq:monotoneapprox} v\big(I_F(x)- I_F(y) - d\cdot (x-y)\big) \ >\ vd+ v(x-y)$$ for all $x, y \in \chi^{-1}(s)$ with $x \neq y$. By [@Ka22 Lemma 5.5], $L \coloneqq K\langle\mbox{\small$\mathscr{J}$}_\der^{r-1}\ell\rangle \subseteq M$ is an immediate $T^{\mathcal O}$-extension of $K$. Let $\chi^L$ and $\chi^M$ denote the natural extensions of $\chi$ to $L^r$ and $M^r$, respectively, and let $$s_0\ \coloneqq\ \chi^L(\mbox{\small$\mathscr{J}$}_\der^{r-1} \ell)\ \in\ \mathop{\mathrm{RV}}_{\! L}^{\mathop{\mathrm{eq}}}\ =\ \mathop{\mathrm{RV}}_{\! K}^{\mathop{\mathrm{eq}}}.$$ Let $U\coloneqq \chi^{-1}(s_0) \subseteq K^r$. Then $U$ is $\mathcal L^{\mathcal O}(K)$-definable by [@Yi17 Corollary 2.18]. Since $\mbox{\small$\mathscr{J}$}_\der^{r-1}(\ell) \in U^M$, we can apply [@Ka22 Lemma 5.6] to get that $U$ has nonempty interior and that $\mbox{\small$\mathscr{J}$}_\der^{r-1}(y) \in U$ for all $y \in M$ sufficiently close to $\ell$. Take $\rho_0$ such that $\mbox{\small$\mathscr{J}$}_\der^{r-1}(y) \in U^M$ whenever $v(\ell-y) > \gamma_{\rho_0}$, so $B_{\rho_0}^M \subseteq U^M$. We choose the linear differential operator $A$ as follows: If $I_F$ is constant on $U$, then we let $A$ be $\der^r \in K[\der]$. If $I_F$ is not constant on $U$, then we let $A$ be $$\der^r - d_r \der^{r-1} - \cdots-d_1\ \in \ K[\der],$$ where $d = (d_1,\ldots,d_r) \in K^r$ is chosen such that ([\[eq:monotoneapprox\]](#eq:monotoneapprox){reference-type="ref" reference="eq:monotoneapprox"}) holds for $x,y \in U$ with $x \neq y$. We claim that $A$ linearly approximates $F$ on $B_{\rho_0}^M$. This is clear if $I_F$ is constant on $U$, for then $I_F$ is constant on $U^M$ as well. Suppose that $I_F$ is not constant on $U$, and let $a,b \in B_{\rho_0}^M$ with $a \neq b$. Then $$v\big(F(\mbox{\small$\mathscr{J}$}_\der^rb)- F(\mbox{\small$\mathscr{J}$}_\der^r a) - A(b-a)\big) \ = \ v\big(I_F(\mbox{\small$\mathscr{J}$}_\der^{r-1}b)- I_F(\mbox{\small$\mathscr{J}$}_\der^{r-1}a)- d\cdot \mbox{\small$\mathscr{J}$}_\der^{r-1}(b-a)\big) \ >\ vd + v\big(\mbox{\small$\mathscr{J}$}_\der^{r-1}(b-a)\big),$$ where the inequality holds since $M$ is an elementary $T^{\mathcal O}$-extension of $K$. We have $v\big(\mbox{\small$\mathscr{J}$}_\der^{r-1}(b-a)\big) = v(b-a)$ since $M$ is monotone. By [@ADH17 Corollary 4.5.4], $vA_{\times (b-a)}= vA +v(b-a)$. Using that $vA \leqslant vd$, we get $$v\big(F(\mbox{\small$\mathscr{J}$}_\der^rb)- F(\mbox{\small$\mathscr{J}$}_\der^r a) - A(b-a)\big) \ >\ vd + v\big(\mbox{\small$\mathscr{J}$}_\der^{r-1}(b-a)\big)\ \geqslant\ vA + v(b-a)\ =\ vA_{\times (b-a)}. \qedhere$$ ◻ Note that every immediate extension of $K$ is monotone by [@ADH17 Corollary 6.3.6]. Requiring $M$ to be an immediate extension of $K$ in the previous result thus shows that every monotone $T^{\mathcal O,\der}$-model with nontrivial induced derivation on its differential residue field has the $T^\der$-hensel configuration property. Combining this with the previous section yields the following results (for monotone $K$). **Theorem 41**. *Suppose that $\boldsymbol{k}$ is linearly surjective. Any two spherically complete immediate $T^{\mathcal O,\der}$-extensions of $K$ are $\mathcal L^{\mathcal O,\der}$-isomorphic over $K$. Any two $T^\der$-algebraically maximal immediate $T^{\mathcal O,\der}$-extensions of $K$ that are $T^\der$-algebraic over $K$ are $\mathcal L^{\mathcal O,\der}$-isomorphic over $K$.* **Theorem 42**. *Suppose $\boldsymbol{k}$ is linearly surjective. If $K$ is $T^\der$-algebraically maximal, then $K$ is $T^\der$-henselian. If $K$ is $T^\der$-henselian and $C\subseteq\mathcal O$, then $K$ is $T^\der$-algebraically maximal.* **Theorem 43**. *Suppose that $K$ is asymptotic, and $\boldsymbol{k}$ is linearly surjective. Then $K$ has a $T^\der$-henselization that is $T^\der$-algebraic over $K$ and has no proper $T^\der$-henselian $\mathcal L^{\mathcal O,\der}$-substructure containing $K$. In particular, any two $T^\der$-henselizations of $K$ are isomorphic over $K$.* ## The $T^\der$-hensel configuration property in a class of models {#sec:Tdhcclass} In Section [3.5](#sec:sphcompunique){reference-type="ref" reference="sec:sphcompunique"}, we introduced the $T^\der$-hensel configuration property, and used it to establish results about spherically complete immediate extensions. This property referred to immediate $T^{\mathcal O,\der}$-extensions of $K$, and now we generalize it by considering other kinds of extensions. Let $\mathcal C$ be a class of $T^{\mathcal O,\der}$-models. We say that $(K, \mathcal C)$ has the **$T^\der$-hensel configuration property** if whenever we have: 1. a divergent pc-sequence $(a_\rho)$ in $K$ of $T^\der$-algebraic type over $K$, 2. a minimal $\mathcal L^\der$-function $F$ of $(a_\rho)$ over $K$, and 3. a $T^{\mathcal O,\der}$-extension $L\in \mathcal C$ of $K$ containing a pseudolimit of $(a_\rho)$, there is an $A \in K[\der]^{\neq}$ that linearly approximates $F$ on $B(a_{\rho+1},\gamma_\rho)^{L}$ for all sufficiently large $\rho$, where, again, $\gamma_\rho\coloneqq v(a_{\rho+1}-a_\rho)$. We say that $\mathcal C$ has the **$T^\der$-hensel configuration property** if for every $K \in \mathcal C$ with nontrivial derivation on $\boldsymbol{k}$, $(K, \mathcal C)$ has the $T^\der$-hensel configuration property. In these terms, the previous subsection established: **Corollary 44**. *The class of monotone $T^{\mathcal O,\der}$-models has the $T^\der$-hensel configuration property.* Theorem [Theorem 35](#thm:sphcompunique){reference-type="ref" reference="thm:sphcompunique"} only required the assumption that the class $\mathcal I(K)$ of immediate extensions of $K$ has the $T^\der$-hensel configuration property, but its proof yields a variant involving non-immediate extensions whose Corollary [Corollary 46](#cor:sphcompembed){reference-type="ref" reference="cor:sphcompembed"} is needed later in the proof of the Ax--Kochen/Ershov theorem. **Theorem 45**. *Suppose that $\boldsymbol{k}$ is linearly surjective, $\mathcal C$ has the $T^\der$-hensel configuration property, and $\mathcal I(K) \subseteq \mathcal C$. Let $L \in \mathcal C$ be a $T^\der$-henselian $T^{\mathcal O,\der}$-extension of $K$. If $L$ is $|\Gamma|^+$-saturated, then any immediate $T^{\mathcal O,\der}$-extension of $K$ can be embedded in $L$ over $K$.* *Proof.* Argue as in the proof of Theorem [Theorem 35](#thm:sphcompunique){reference-type="ref" reference="thm:sphcompunique"}, but use saturation instead of spherical completeness to obtain pseudolimits in $L$. ◻ **Corollary 46**. *Suppose that $K$ is monotone and $\boldsymbol{k}$ is linearly surjective, and let $L$ be a monotone $T^\der$-henselian $T^{\mathcal O,\der}$-extension of $K$. If $L$ is $|\Gamma|^+$-saturated, then any immediate $T^{\mathcal O,\der}$-extension of $K$ can be embedded in $L$ over $K$.* *Proof.* Note that every immediate extension of $K$ is monotone by [@ADH17 Corollary 6.3.6]. ◻ These definitions also lead to improvements of Theorem [Theorem 38](#thm:Tdhenselization){reference-type="ref" reference="thm:Tdhenselization"} and Theorem [Theorem 43](#thm:Tdhenselizationmono){reference-type="ref" reference="thm:Tdhenselizationmono"}. We call $L$ a **$\mathcal C$-$T^\der$-henselization of $K$** if $L \in \mathcal C$ is an immediate $T^{\mathcal O,\der}$-extension of $K$ that is $T^\der$-henselian and for any $T^\der$-henselian $T^{\mathcal O,\der}$-extension $M \in \mathcal C$ of $K$, there is an $\mathcal L^{\mathcal O,\der}$-embedding $L \to M$ that is the identity on $K$. The proof of Theorem [Theorem 38](#thm:Tdhenselization){reference-type="ref" reference="thm:Tdhenselization"} shows: **Theorem 47**. *Suppose that $\boldsymbol{k}$ is linearly surjective, $\mathcal C$ has the $T^\der$-hensel configuration property, $\mathcal I(K) \subseteq \mathcal C$, and every $M \in \mathcal C$ is asymptotic. Then $K$ has a $\mathcal C$-$T^\der$-henselization that is $T^\der$-algebraic over $K$ and has no proper $T^\der$-henselian $\mathcal L^{\mathcal O,\der}$-substructure containing $K$. In particular, any two $\mathcal C$-$T^\der$-henselizations of $K$ are isomorphic over $K$.* **Corollary 48**. *Suppose that $K$ is monotone and asymptotic, and $\boldsymbol{k}$ is linearly surjective. Then the $T^\der$-henselization of $K$ embeds into every monotone, asymptotic, $T^\der$-henselian $T^{\mathcal O,\der}$-extension of $K$.* ## The $c$-map {#sec:cmap} In this subsection, assume that $K$ is monotone. A **section for $K$** is a $\Lambda$-vector space embedding $s\colon \Gamma\to K^>$ such that $v(s\gamma) = \gamma$ for all $\gamma \in \Gamma$. An **angular component for $K$** is a $\Lambda$-linear map $\mathop{\mathrm{ac}}\colon K^>\to \boldsymbol{k}^>$ such that $\mathop{\mathrm{ac}}(a) = \mathop{\mathrm{res}}(a)$ whenever $a \asymp 1$. We extend any angular component map $\mathop{\mathrm{ac}}$ to a map $K\to \boldsymbol{k}$ (also denoted by $\mathop{\mathrm{ac}}$) by setting $\mathop{\mathrm{ac}}(0)\coloneqq 0$ and $\mathop{\mathrm{ac}}(-a)\coloneqq -\mathop{\mathrm{ac}}(a)$ for $a \in K^>$. Given any section $s\colon\Gamma\to K^>$, the map $a \mapsto \mathop{\mathrm{res}}\!\big(a/s(va)\big)\colon K^>\to \boldsymbol{k}^>$ is an angular component for $K$, which we say is **induced by $s$**. **Lemma 49**. *Let $A \subseteq K^>$ be a $\Lambda$-subspace such that $\{va:a \in A\} = \Gamma$. Then there is a section $s\colon \Gamma\to K^>$ with image contained in $A$.* *Proof.* The inclusion $\mathcal O^{\times}\cap A\to A$ and the restricted valuation map $A\to \Gamma$ yield an exact sequence $$1\to \mathcal O^{\times}\cap A\to A\to \Gamma\to 0$$ of $\Lambda$-vector spaces. This exact sequence splits, yielding a section $s\colon \Gamma\to A$ as claimed. ◻ **Corollary 50**. *Let $\mathop{\mathrm{ac}}$ be an angular component for $K$. Then $\mathop{\mathrm{ac}}$ is induced by a section for $K$.* *Proof.* Apply the previous lemma to the set $A \coloneqq \{a \in K^>:\mathop{\mathrm{ac}}(a) = 1\}$. ◻ Let $s\colon \Gamma \to K^>$ be a section for $K$. We define a map $c\colon \Gamma\to \boldsymbol{k}$ by setting $c(\gamma) \coloneqq \mathop{\mathrm{res}}\!\big(s(\gamma)^\dagger\big)$ for $\gamma \in \Gamma$. Since $(a^\lambda)^\dagger = \lambda a^\dagger$ for $a \in K^>$ by Fact [Fact 3](#fact:powerderivative){reference-type="ref" reference="fact:powerderivative"}, the map $c$ is $\Lambda$-linear. If $K$ has many constants, then by Lemma [Lemma 49](#lem:sectionexists){reference-type="ref" reference="lem:sectionexists"} with $C^>$ in place of $A$, we can choose $s$ so that its image is contained in the constant field. In this case, $c$ is the zero map. The argument above tells us that we can associate to any monotone $T^{\mathcal O,\der}$-model $K$ a structure $(\boldsymbol{k},\Gamma,c)$ where $\boldsymbol{k}\models T^\der$, $\Gamma$ is an ordered $\Lambda$-vector space, and $c\colon \Gamma\to \boldsymbol{k}$ is $\Lambda$-linear. As a converse, we show below that any such structure $(\boldsymbol{k},\Gamma,c)$ comes from a monotone $T^{\mathcal O,\der}$-model. We need the following fact. **Fact 51** ([@Ka23 Proposition 2.6]). *Let $E$ be a $T$-convex valued field, let $M$ be a $T^{\mathcal O}$-extension of $E$, let $a \in M$ with $a\not\sim f$ for all $f \in E$, and let $F\colon E\to E$ be an $\mathcal L(E)$-definable function. Then $\frac{\partial F}{\partial Y}(a) \preccurlyeq a^{-1}F(a)$.* **Proposition 52**. *Let $\boldsymbol{k}\models T^\der$, let $\Gamma$ be an ordered $\Lambda$-vector space, and let $c\colon \Gamma \to \boldsymbol{k}$ be a $\Lambda$-linear map. Then there is a monotone $T^{\mathcal O,\der}$-model $K$ with differential residue field $\boldsymbol{k}$ and value group $\Gamma$ and a section $s\colon \Gamma \to K^>$ such that $\mathop{\mathrm{res}}\!\big(s(\gamma)^\dagger \big)=c(\gamma)$ for all $\gamma \in \Gamma$.* *Proof.* Let $(\gamma_\alpha)_{\alpha<\beta}$ be a $\Lambda$-basis for $\Gamma$, so we may write each $\gamma \in \Gamma$ uniquely as a sum $\gamma = \sum_{\alpha<\beta}\lambda_\alpha\gamma_\alpha$ where each $\lambda_\alpha$ is in $\Lambda$ and only finitely many $\lambda_\alpha$ are nonzero. Let $K\coloneqq \boldsymbol{k}\langle(t_\alpha)_{\alpha<\beta}\rangle$ be a $T^{\mathcal O}$-extension of $\boldsymbol{k}$, equipped with the convex hull of $\boldsymbol{k}$ as its $T$-convex valuation ring and ordered so that $t_\alpha >0$ and $v(t_\alpha) = \gamma_\alpha$ for each $\alpha$ (such an extension can be built by a transfinite construction, using repeated applications of [@Ka23 Lemma 2.3]). Then $K$ has residue field $\boldsymbol{k}$ and value group $\Gamma$, and the sequence $(t_\alpha)$ is necessarily $\mathcal L(\boldsymbol{k})$-independent. Let $s\colon\Gamma\to K^>$ be the section mapping $\gamma = \sum_{\alpha<\beta}\lambda_\alpha\gamma_\alpha\in \Gamma$ to $\prod_{\alpha<\beta}t_\alpha^{\lambda_\alpha}\in K^>$. Using Fact [Fact 5](#fact:transext){reference-type="ref" reference="fact:transext"}, we extend the $T$-derivation on $\boldsymbol{k}$ to a $T$-derivation on $K$ by setting $t_\alpha' = c(\gamma_{\alpha})t_\alpha$. Since $(t_\alpha^\lambda)^\dagger = \lambda t_\alpha^\dagger$ by Fact [Fact 3](#fact:powerderivative){reference-type="ref" reference="fact:powerderivative"}, we have $s(\gamma)^\dagger =c(\gamma) \in \boldsymbol{k}$ for all $\gamma \in \Gamma$. Thus, we need only check that $K$ is monotone. Each element of $K$ is of the form $G(a,t)$ where $G\colon K^{m+n}\to K$ is $\mathcal L(\emptyset)$-definable, $a = (a_1,\ldots,a_m)$ is an $\mathcal L(\emptyset)$-independent tuple from $\boldsymbol{k}$, and $t = (t_{\alpha_1},\ldots,t_{\alpha_n})$ for some distinct $\alpha_1,\ldots,\alpha_n<\beta$. We fix such an element $G(a,t)$, and we need to show that $G(a,t)' \preccurlyeq G(a,t)$. Viewing $G$ as a function of the variables $X_1,\ldots,X_m,Y_1,\ldots,Y_n$, we have $$G(a,t)'\ =\ \frac{\partial G}{\partial X_1}(a,t)a_1'+\cdots+\frac{\partial G}{\partial X_m}(a,t)a_m'+\frac{\partial G}{\partial Y_1}(a,t)t_{\alpha_1}'+\cdots+\frac{\partial G}{\partial Y_n}(a,t)t_{\alpha_n}'.$$ We will show that $\frac{\partial G}{\partial X_i}(a,t)a_i',\frac{\partial G}{\partial Y_j}(a,t)t_{\alpha_j}' \preccurlyeq G(a,t)$ for all $i \in \{1,\ldots,m\}$ and $j \in \{1,\ldots,n\}$. By symmetry, it suffices to handle the case $i = j = 1$. We start with $\frac{\partial G}{\partial X_1}(a,t)a_1'$. Since $a_1' \preccurlyeq 1$, it suffices to show that $\frac{\partial G}{\partial X_1}(a,t)\preccurlyeq G(a,t)$. Set $E \coloneqq \mathop{\mathrm{dcl}}_\mathcal L(a_2,\ldots,a_m,t)$, so $E$ is an $\mathcal L^{\mathcal O}$-substructure of $K$, and set $E_0\coloneqq \mathop{\mathrm{dcl}}_\mathcal L(a_2,\ldots,a_m) \subseteq E$. Since $E_0$ is trivially valued and $a_1 \not\in E_0$, we have $a_1 \not\sim f$ for any $f \in E_0$. The Wilkie inequality gives $\mathop{\mathrm{res}}(E)= \mathop{\mathrm{res}}(E_0)$, so $a_1 \not\sim f$ for any $f \in E$. Applying Fact [Fact 51](#fact:smallderivsim){reference-type="ref" reference="fact:smallderivsim"} with $a_1$ in place of $a$ and the function $x\mapsto G(x,a_2,\ldots,a_m,t)$ in place of $F$ gives $$\frac{\partial G}{\partial X_1}(a,t)\ \preccurlyeq\ a_1^{-1}G(a,t)\ \asymp\ G(a,t),$$ as desired. Next, we show that $\frac{\partial G}{\partial Y_1}(a,t)t_{\alpha_1}' \preccurlyeq G(a,t)$. This time, we set $E\coloneqq \mathop{\mathrm{dcl}}_\mathcal L(a,t_{\alpha_2},\ldots,t_{\alpha_n})$, so $t_{\alpha_1}\not\asymp f$ for any $f \in E$. In particular, $t_{\alpha_1}\not\sim f$ for any $f \in E$, so Fact [Fact 51](#fact:smallderivsim){reference-type="ref" reference="fact:smallderivsim"} (this time with $x\mapsto G(a,x,t_{\alpha_2},\ldots,t_{\alpha_n})$ in place of $F$) gives $$\frac{\partial G}{\partial Y_1}(a,t)\ \preccurlyeq\ t_{\alpha_1}^{-1}G(a,t).$$ Thus, $\frac{\partial G}{\partial Y_1}(a,t)t_{\alpha_1}'\preccurlyeq\ t_{\alpha_1}^\dagger G(a,t) \preccurlyeq G(a,t)$, since $t_{\alpha_1}^\dagger = c(\gamma_{\alpha_1})\preccurlyeq 1$. ◻ **Lemma 53**. *Let $K$ be a monotone $T^{\mathcal O,\der}$-model, let $c\colon \Gamma\to \boldsymbol{k}$ be a $\Lambda$-linear map, and suppose that for each $\gamma \in \Gamma$, there is $a \in K^>$ with $va= \gamma$ and $\mathop{\mathrm{res}}(a^\dagger) = c(\gamma)$. Then there is a section $s\colon \Gamma\to K^>$ such that $\mathop{\mathrm{res}}\!\big(s(\gamma)^\dagger\big) = c(\gamma)$ for all $\gamma \in \Gamma$. The corresponding angular component map $\mathop{\mathrm{ac}}\colon K\to \boldsymbol{k}$ induced by $s$ satisfies the equality $\mathop{\mathrm{ac}}(a)^\dagger =\mathop{\mathrm{res}}(a^\dagger)- c(va)$ for all $a \in K^\times$.* *Proof.* Let $A \subseteq K^>$ be the set of all $a \in K^>$ with $\mathop{\mathrm{res}}(a^\dagger) = c(va)$. Then $A$ is a multiplicative $\Lambda$-subspace of $K^>$ and $\{va:a \in A\} = \Gamma$, so there is a section $s\colon \Gamma\to K^>$ with image contained in $A$ by Lemma [Lemma 49](#lem:sectionexists){reference-type="ref" reference="lem:sectionexists"}. Let $\mathop{\mathrm{ac}}\colon K\to \boldsymbol{k}$ be the angular component map induced by $s$. Then for $a \in K^\times$, we have $$\mathop{\mathrm{ac}}(a)^\dagger\ =\ \mathop{\mathrm{res}}\!\big(a/s(va)\big)^\dagger\ =\ \mathop{\mathrm{res}}(a^\dagger - s(va)^\dagger)\ =\ \mathop{\mathrm{res}}(a^\dagger) - c(va).\qedhere$$ ◻ # When $T=\mathop{\mathrm{RCF}}$ {#sec:RCF} Let $T=\mathop{\mathrm{RCF}}$ in the language of ordered rings. In this setting we show that the notions and results in the previous sections specialize to the analogous notions and results of [@ADH17 Chapter 7]. By Fact [Fact 8](#fact:sphcomp){reference-type="ref" reference="fact:sphcomp"} and [@ADH17 Corollary 6.9.4], if $\boldsymbol{k}$ has nontrivial derivation, then $K$ has no proper immediate $T^{\mathcal O,\der}$-extension if and only if it has no proper immediate valued differential field extension with small derivation. Additionally: **Lemma 54**. *The $T^{\mathcal O,\der}$-model $K$ is $T^\der$-algebraically maximal if and only if the valued differential field $K$ is $\operatorname{d}$-algebraically maximal in the sense of [@ADH17 Chapter 7].* *Proof.* Since $\mathcal L(K)$-definable functions are semialgebraic, the notions $T^\der$-algebraic and $\operatorname{d}$-algebraic coincide. Thus the right-to-left direction is trivial. Conversely, let $L$ be a $\operatorname{d}$-algebraic immediate valued differential field extension of $K$ with small derivation. Since $\Gamma$ is divisible and $\boldsymbol{k}$ is real closed, the henselization $L^{\mathop{\mathrm{h}}}$ of $L$ is real closed, and it has small derivation by [@ADH17 Proposition 6.2.1]. Additionally, its valuation ring is convex, and hence $T$-convex by (see [@DL95 Proposition 4.2]). Thus $L^{\mathop{\mathrm{h}}}$ is a $T^\der$-algebraic immediate $T^{\mathcal O,\der}$-extension of $K$. ◻ Combining this lemma with the observation preceding it shows that in this case Theorem [Theorem 41](#thm:sphcompuniquemono){reference-type="ref" reference="thm:sphcompuniquemono"} becomes [@ADH17 Theorem 7.4.3], Theorem [Theorem 42](#thm:TdhTdalgmaxmono){reference-type="ref" reference="thm:TdhTdalgmaxmono"} becomes [@ADH17 Theorem 7.0.3], and Theorem [Theorem 43](#thm:Tdhenselizationmono){reference-type="ref" reference="thm:Tdhenselizationmono"} becomes a special case of [@Pyn20 Theorem 3.7]. Lemma [Lemma 54](#lem:Tdalgmaxiffdalgmax){reference-type="ref" reference="lem:Tdalgmaxiffdalgmax"} also yields: **Corollary 55**. *Suppose that $K$ is monotone and asymptotic. Then $K$ is $T^\der$-henselian if and only if $K$ is $\operatorname{d}$-henselian in the sense of [@ADH17 Chapter 7].* *Proof.* Suppose that $\boldsymbol{k}$ is linearly surjective. Then by [@ADH17 Theorems 7.0.1, 7.0.3], $K$ is $\operatorname{d}$-algebraically maximal if and only if $K$ is $\operatorname{d}$-henselian. Likewise, by Theorem [Theorem 42](#thm:TdhTdalgmaxmono){reference-type="ref" reference="thm:TdhTdalgmaxmono"}, $K$ is $T^\der$-algebraically maximal if and only if $K$ is $T^\der$-henselian. The result follows. ◻ This raises the question of whether $T^\der$-henselianity always implies $\operatorname{d}$-henselianity. Moreover, at present the only proof we have of this implication in the case that $K$ is monotone and asymptotic is the roundabout proof given above. # An AKE theorem for monotone $T^\der$-henselian fields {#sec:AKE} To establish our Ax--Kochen/Ershov theorem for monotone $T^\der$-henselian fields, we construe a monotone $T^{\mathcal O,\der}$-model $K$ as a three-sorted structure $\mathcal K= (K, \boldsymbol{k}, \Gamma; \pi, v, c, \mathop{\mathrm{ac}})$ with a sort $\operatorname{f}$ for $K$ as a structure in the language $\mathcal L_{\operatorname{f}} = \mathcal L^\der$, a sort for $\boldsymbol{k}$ as a structure in the language $\mathcal L_{\operatorname{r}} = \mathcal L^\der$, and a sort for $\Gamma$ as a structure in the (one-sorted) language $\mathcal L_{\operatorname{v}}$ ordered $\Lambda$-vector spaces, together with symbols for maps $\pi, v, c, \mathop{\mathrm{ac}}$ connecting the sorts as follows. Suppose that: 1. $K \models T^\der$; 2. $\boldsymbol{k}\models T^\der$; 3. $\Gamma$ is an ordered $\Lambda$-vector space; 4. $v \colon K^{\times} \to \Gamma$ is a (surjective) valuation making $K$ a monotone $T^{\mathcal O,\der}$-model such that $v(a^{\lambda})=\lambda va$ for all $a \in K^>$ and $\lambda \in \Lambda$; 5. $\pi \colon \mathcal O\to \boldsymbol{k}$ is a map such that the map $\mathop{\mathrm{res}}(K) \to \boldsymbol{k}$ induced by $\pi$ is an $\mathcal L^\der$-isomorphism; 6. $c \colon \Gamma \to \boldsymbol{k}$ is $\Lambda$-linear and for every $\gamma \in \Gamma$, there is $a \in K^>$ with $va=\gamma$ and $\pi(a^{\dagger})=c(\gamma)$; 7. $\mathop{\mathrm{ac}}\colon K \to \boldsymbol{k}$ is an angular component map such that $\mathop{\mathrm{ac}}(a)^{\dagger}=\pi(a^{\dagger})-c(va)$ for all $a \in K^{\times}$. Let $\mathcal L_3$ be this three-sorted language of $\mathcal K$, where we extend $v$ and $\pi$ to $K$ by $v(0)=0$ and $\pi(K\setminus\mathcal O)=\{0\}$, respectively. Note that in $\mathcal L_3$ we have two distinct copies of the language $\mathcal L^\der$, one for the sort $\operatorname{f}$ and one for the sort $\operatorname{r}$. Let $T_{\operatorname{mon}}^{\mathop{\mathrm{ac}}}$ be the theory whose models are such $\mathcal K$. Note that by Section [4.3](#sec:cmap){reference-type="ref" reference="sec:cmap"}, any monotone $T^{\mathcal O,\der}$-model can be expanded to a $T_{\operatorname{mon}}^{\mathop{\mathrm{ac}}}$-model. ## Back-and-forth Let $\mathcal K= (K, \boldsymbol{k}, \Gamma; \pi, v, c, \mathop{\mathrm{ac}})$ and $\mathcal K^* = (K^*, \boldsymbol{k}^*, \Gamma^*; \pi^*, v^*, c^*, \mathop{\mathrm{ac}}^*)$ be $T_{\operatorname{mon}}^{\mathop{\mathrm{ac}}}$-models. Our goal is to construct a back-and-forth system between $\mathcal K$ and $\mathcal K^*$, when they are $T^\der$-henselian and appropriately saturated. A **good substructure** of $\mathcal K$ is a triple $\mathcal E= (E, \boldsymbol{k}_{\mathcal E}, \Gamma_{\mathcal E})$ such that: 1. $E$ is a $T^\der$-submodel of $K$; 2. $\boldsymbol{k}_{\mathcal E}$ is a $T^\der$-submodel of $\boldsymbol{k}$ with $\mathop{\mathrm{ac}}(E) \subseteq \boldsymbol{k}_{\mathcal E}$ (so $\pi(E) \subseteq \boldsymbol{k}_{\mathcal E}$); 3. $\Gamma_{\mathcal E}$ is an ordered $\Lambda$-subspace of $\Gamma$ with $v(E^{\times}) \subseteq \Gamma_{\mathcal E}$ and $c(\Gamma_{\mathcal E}) \subseteq \boldsymbol{k}_{\mathcal E}$. Note that in this definition we neither require $\mathop{\mathrm{ac}}(E) = \boldsymbol{k}_E$, let alone $\pi(E) = \boldsymbol{k}_E$, nor $v(E^{\times})=\Gamma_E$. When needed we construe $E$ as a $T^{\mathcal O,\der}$-model $(E, \mathcal O_E)$ with the induced valuation ring $\mathcal O_E \coloneqq \mathcal O\cap E$. If $\mathcal E_1 = (E, \boldsymbol{k}_{\mathcal E_1}, \Gamma_{\mathcal E_1})$ and $\mathcal E_2 = (E_2, \boldsymbol{k}_{\mathcal E_2}, \Gamma_{\mathcal E_2})$ are good substructures of $\mathcal K$, then $\mathcal E_1 \subseteq \mathcal E_2$ means $E_1 \subseteq E_2$, $\boldsymbol{k}_{\mathcal E_1} \subseteq \boldsymbol{k}_{\mathcal E_2}$, and $\Gamma_{\mathcal E_1} \subseteq \Gamma_{\mathcal E_2}$. Now let $\mathcal E= (E, \boldsymbol{k}_{\mathcal E}, \Gamma_{\mathcal E})$ and $\mathcal E^* = (E^*, \boldsymbol{k}_{\mathcal E^*}, \Gamma_{\mathcal E^*})$ be good substructures of $\mathcal K$ and $\mathcal K^*$, respectively. A **good map** $\boldsymbol{f}\colon \mathcal E\to \mathcal E^*$ is a triple $\boldsymbol{f}= (f, f_{\operatorname{r}}, f_{\operatorname{v}})$ consisting of $\mathcal L^\der$-isomorphisms $f \colon E \to E^*$ and $f_{\operatorname{r}} \colon \boldsymbol{k}_{\mathcal E} \to \boldsymbol{k}_{\mathcal E^*}$ and an isomorphism $f_{\operatorname{v}} \colon \Gamma_{\mathcal E} \to \Gamma_{\mathcal E^*}$ of ordered $\Lambda$-vector spaces such that: 1. $f_{\operatorname{r}}\big( \mathop{\mathrm{ac}}(a) \big) = \mathop{\mathrm{ac}}^*\big( f(a) \big)$ for all $a \in E$; 2. $f_{\operatorname{v}}\big( v(a) \big) = v^*\big( f(a) \big)$ for all $a \in E^{\times}$; 3. $(f_{\operatorname{r}}, f_{\operatorname{v}})$ is a partial elementary map $(\boldsymbol{k}, \Gamma; c) \to (\boldsymbol{k}^*, \Gamma^*; c^*)$ (so $f_{\operatorname{r}}\big(c(\gamma)\big) = c^*\big(f_{\operatorname{v}}(\gamma)\big)$ for all $\gamma \in \Gamma_{\mathcal E}$). This lemma handles residue field extensions. **Lemma 56**. *Suppose that $\mathcal K$ and $\mathcal K^*$ are $T^\der$-henselian and let $\boldsymbol{f}\colon \mathcal E\to \mathcal E^*$ be a good map. Let $d \in \boldsymbol{k}_{\mathcal E} \setminus \pi(E)$. Then there are $b \in \mathcal O$ with $\pi(b) = d$ and a good map $\boldsymbol{g}\colon (E\langle\!\langle b \rangle\!\rangle, \boldsymbol{k}_{\mathcal E}, \Gamma_{\mathcal E}) \to \mathcal K^*$ extending $\boldsymbol{f}$.* *Proof.* Proposition [Proposition 18](#prop:resext){reference-type="ref" reference="prop:resext"} gives $b \in \mathcal O$ with $\pi(b) = d$ and $v\big( E\langle\!\langle b \rangle\!\rangle^\times \big) = v(E^\times)$, as well as an $\mathcal L^{\mathcal O,\der}$-embedding $g\colon E\langle\!\langle b \rangle\!\rangle\to K^*$ extending $f$ such that $\pi^*\big(g(a)\big) =f_{\operatorname{r}}\big(\pi(a))$ for all $a \in E\langle\!\langle b \rangle\!\rangle$. Let $\boldsymbol{g}= (g,f_{\operatorname{r}},f_{\operatorname{v}})$. To see that $(E\langle\!\langle b \rangle\!\rangle, \boldsymbol{k}_{\mathcal E}, \Gamma_{\mathcal E})$ is a good substructure and that $\boldsymbol{g}$ is a good map, we only need to show that $\mathop{\mathrm{ac}}(a) \in \boldsymbol{k}_{\mathcal E}$ and that $\mathop{\mathrm{ac}}^*\!\big(g(a)\big) =f_{\operatorname{r}}\big(\mathop{\mathrm{ac}}(a))$ for all $a \in E\langle\!\langle b \rangle\!\rangle$. This holds by [@Ha18B Lemma 4.2], but we repeat the short argument here. Take $y \in E$ with $va = vy$ and take $u \in E\langle\!\langle b \rangle\!\rangle$ with $a = uy$. Then $u \asymp 1$, so $\mathop{\mathrm{ac}}(a) = \mathop{\mathrm{ac}}(u)\mathop{\mathrm{ac}}(y) = \pi(u)\mathop{\mathrm{ac}}(y) \in \boldsymbol{k}_{\mathcal E}$ and $$f_{\operatorname{r}}\big(\mathop{\mathrm{ac}}(a))\ =\ f_{\operatorname{r}}\big(\pi(u)\big)f_{\operatorname{r}}\big(\mathop{\mathrm{ac}}(y)\big)\ =\ \pi^*\big(g(u)\big)\mathop{\mathrm{ac}}^*\!\big(f(y)\big) \ =\ \mathop{\mathrm{ac}}^*\!\big(g(a)\big).\qedhere$$ ◻ The next lemma handles value group extensions. **Lemma 57**. *Suppose that $\mathcal K$ and $\mathcal K^*$ are $T^\der$-henselian and let $\boldsymbol{f}\colon \mathcal E\to \mathcal E^*$ be a good map. Suppose that $\pi(E)=\boldsymbol{k}_{\mathcal E}$ and $E$ is equipped with a $T^\der$-lift of $\boldsymbol{k}_{\mathcal E}$. Let $\gamma \in \Gamma_{\mathcal E} \setminus v(E^{\times})$. Then there are $b \in K^{\times}$ with $vb=\gamma$ and a good map $\boldsymbol{g}\colon (E\langle\!\langle b \rangle\!\rangle, \boldsymbol{k}_{\mathcal E}, \Gamma_{\mathcal E}) \to \mathcal K^*$ extending $\boldsymbol{f}$.* *Proof.* Let $E_0 \subseteq E$ be a $T^\der$-lift of $\boldsymbol{k}_{\mathcal E}$. We will find $b \in K^>$ with $b^{\dagger} \in E_0$, $\pi(b^{\dagger} ) = c(\gamma)$, $vb=\gamma$, and $\mathop{\mathrm{ac}}(b)=1$. By assumption, we have $a \in K^>$ with $\pi(a^{\dagger})=c(\gamma)$ and $va=\gamma$. Take $u \in E_0$ with $\pi(u) = \pi(a^\dagger) = c(\gamma)$, and let $\varepsilon \coloneqq a^{\dagger}-u \in \smallo$. By Corollary [Corollary 10](#cor:7.1.9){reference-type="ref" reference="cor:7.1.9"}, we have $\delta \in \smallo$ with $\varepsilon=(1+\delta)^{\dagger}$, so by replacing $a$ with $a/(1+\delta)$, we arrange that $a^\dagger \in E_0$. By Corollary [Corollary 19](#cor:reslift){reference-type="ref" reference="cor:reslift"}, extend $E_0$ to a $T^\der$-lift $E_1 \subseteq K$ of $\boldsymbol{k}$. Now, take $e \in E_1$ with $\pi(e) = \mathop{\mathrm{ac}}(a)$. We have $$\pi(e^\dagger)\ =\ \pi(e)^\dagger\ =\ \mathop{\mathrm{ac}}(a)^\dagger\ =\ \pi(a^{\dagger})-c(\gamma)\ =\ 0.$$ Since $e^\dagger \in E_1$, it follows that $e^\dagger = 0$. Let $b\coloneqq a/e$, so $b^\dagger = a^\dagger \in E_0$, $vb = va = \gamma$, and $\mathop{\mathrm{ac}}(b) = \mathop{\mathrm{ac}}(a)/\pi(e) = 1$. Now $f(E_0)\subseteq E^*$ is a $T^\der$-lift of $\boldsymbol{k}_{\mathcal E^*}$, so as above take $b^* \in (K^*)^>$ with $(b^*)^{\dagger} \in f(E_0)$, $\pi^*((b^*)^{\dagger} ) = c^*(f_{\operatorname{v}}\gamma)$, $vb^*=f_{\operatorname{v}}\gamma$, and $\mathop{\mathrm{ac}}^*(b^*)=1$. Since $b^\dagger \in E_0 \subseteq E$, we have $E\langle\!\langle b \rangle\!\rangle= E\langle b \rangle$. Then $\mathop{\mathrm{res}}(E\langle b \rangle)=\mathop{\mathrm{res}}(E)$ and $v(E\langle b \rangle^{\times}) = v(E^{\times}) \oplus \Lambda \gamma \subseteq \Gamma_{\mathcal E}$ by the Wilkie inequality. Since $b$ and $b^*$ have the same sign and realize the same cut over $v(E^\times)$, we may use [@Ka23 Lemma 2.3] to get an $\mathcal L^{\mathcal O}$-embedding $g \colon E\langle b \rangle \to K^*$ extending $f$ and satisfying $gb=b^*$. Note that $(b^*)^{\dagger} = f(b^\dagger)$, so $g$ is even an $\mathcal L^{\mathcal O,\der}$-embedding by Fact [Fact 5](#fact:transext){reference-type="ref" reference="fact:transext"}. To verify that $\boldsymbol{g}\coloneqq (g, f_{\operatorname{r}}, f_{\operatorname{v}})$ is a good map, it remains to check that $\boldsymbol{g}$ preserves $\mathop{\mathrm{ac}}$. For this, use that for $\mathcal L(E)$-definable $F \colon K \to K$ with $F(b)\neq 0$, we have $\lambda \in \Lambda$ and $d \in E^{\times}$ with $F(b) \sim b^{\lambda}d$, so $\mathop{\mathrm{ac}}(F(b)) = \mathop{\mathrm{ac}}(b^{\lambda}d) = \mathop{\mathrm{ac}}(b)^{\lambda}\mathop{\mathrm{ac}}(d) = \mathop{\mathrm{ac}}(d)$. ◻ **Theorem 58**. *Suppose that $\mathcal K$ and $\mathcal K^*$ are $T^\der$-henselian. Then any good map $\mathcal E\to \mathcal E^*$ is a partial elementary map $\mathcal K\to \mathcal K^*$.* *Proof.* Let $\kappa$ be an uncountable cardinal with $\max\{ |\boldsymbol{k}_{\mathcal E}|, |\Gamma_{\mathcal E}| \}<\kappa$. By passing to elementary extensions we arrange that $\mathcal K$ and $\mathcal K^*$ are $\kappa^+$-saturated. We call a good substructure $(E_1, \boldsymbol{k}_{1}, \Gamma_{1})$ of $\mathcal K$ **small** if $\max\{ |\boldsymbol{k}_1|, |\Gamma_1| \}<\kappa$. It suffices to show that the set of good maps with small domain is a back-and-forth system from $\mathcal K$ to $\mathcal K^*$. First, we describe several extension procedures. 1. [\[akeext:ressort\]]{#akeext:ressort label="akeext:ressort"} Given $d \in \boldsymbol{k}$, arranging that $d \in \boldsymbol{k}_{\mathcal E}$: By the saturation assumption, we can extend $f_{\operatorname{r}}$ to a map $g_{\operatorname{r}} \colon \boldsymbol{k}_{\mathcal E}\langle\!\langle d \rangle\!\rangle\to \boldsymbol{k}^*$ so that $(g_{\operatorname{r}}, f_{\operatorname{v}})$ is a partial elementary map. Then $(f, g_{\operatorname{r}}, f_{\operatorname{v}})$ is the desired extension of $\boldsymbol{f}$. 2. [\[akeext:valsort\]]{#akeext:valsort label="akeext:valsort"} Given $\gamma \in \Gamma$, arranging that $\gamma \in \Gamma_{\mathcal E}$: First use [\[akeext:ressort\]](#akeext:ressort){reference-type="ref" reference="akeext:ressort"} to arrange $c(\gamma) \in \boldsymbol{k}_{\mathcal E}$, then use saturation as before to extend $f_{\operatorname{v}}$ to $g_{\operatorname{v}} \colon \Gamma_{\mathcal E} \oplus \Lambda\gamma \to \Gamma^*$ with the desired properties. 3. [\[akeext:res\]]{#akeext:res label="akeext:res"} Arranging $\pi(E)=\boldsymbol{k}_{\mathcal E}$: If $d \in \boldsymbol{k}_{\mathcal E} \setminus \pi(E)$, then Lemma [Lemma 56](#lem:resext){reference-type="ref" reference="lem:resext"} yields $b \in K$ and an extension of $\boldsymbol{f}$ to a good map $(g, f_{\operatorname{r}}, f_{\operatorname{v}})$ with small domain $(E\langle\!\langle b\rangle\!\rangle, \boldsymbol{k}_{\mathcal E}, \Gamma_{\mathcal E})$. Iterate this procedure to arrange $\pi(E)=\boldsymbol{k}_{\mathcal E}$. 4. [\[akeext:Tdh\]]{#akeext:Tdh label="akeext:Tdh"} Arranging that $(E, \mathcal O_E)$ is $T^\der$-henselian: By [\[akeext:ressort\]](#akeext:ressort){reference-type="ref" reference="akeext:ressort"} and [\[akeext:res\]](#akeext:res){reference-type="ref" reference="akeext:res"} we can assume that $\boldsymbol{k}_{\mathcal E}$ is linearly surjective and $\pi(E)=\boldsymbol{k}_{\mathcal E}$. Now use Fact [Fact 8](#fact:sphcomp){reference-type="ref" reference="fact:sphcomp"} to take a spherically complete immediate $T^{\mathcal O,\der}$-extension $L$ of $E$. Then $L$ is $T^\der$-henselian by Corollary [Corollary 23](#cor:sphcompTdh){reference-type="ref" reference="cor:sphcompTdh"}, and $L$ embeds over $E$ into both $K$ and $K^*$ by Corollary [Corollary 46](#cor:sphcompembed){reference-type="ref" reference="cor:sphcompembed"}. Let $g$ be the extension of $f$ to an $\mathcal L^{\mathcal O,\der}$-isomorphism between these images of $L$ in $K$ and $K^*$, respectively. Then $(g,f_{\operatorname{r}},f_{\operatorname{v}})$ is a good map by [@Ha18B Corollary 4.4]. 5. [\[akeext:val\]]{#akeext:val label="akeext:val"} Arranging $v(E^{\times})=\Gamma_{\mathcal E}$: We can assume that $\pi(E)=\boldsymbol{k}_{\mathcal E}$ and that $\mathcal E$ is $T^\der$-henselian and is equipped with a $T^\der$-lift of $\boldsymbol{k}_{\mathcal E}$ by Theorem [Theorem 20](#thm:reslift){reference-type="ref" reference="thm:reslift"}. If $\gamma \in \Gamma_{\mathcal E} \setminus v(E^{\times})$, then Lemma [Lemma 57](#lem:valext){reference-type="ref" reference="lem:valext"} yields $b \in K$ and an extension of $\boldsymbol{f}$ to a good map $(g, f_{\operatorname{r}}, f_{\operatorname{v}})$ with small domain $(E\langle\!\langle b \rangle\!\rangle, \boldsymbol{k}_{\mathcal E}, \Gamma_{\mathcal E})$. Iterate this procedure to arrange $v(E^{\times})=\Gamma_{\mathcal E}$. Given $a \in K$, we need to extend $\boldsymbol{f}$ to a good map with small domain containing $a$. By the above, we can assume that $\pi(E)=\boldsymbol{k}_{\mathcal E}$ and $v(E^{\times})=\Gamma_{\mathcal E}$. From $$\mathop{\mathrm{rk}}_\mathcal L\big(\pi(E\langle\!\langle a \rangle\!\rangle) | \pi(E)\big)\ \leqslant\ \mathop{\mathrm{rk}}_\mathcal L(E\langle\!\langle a \rangle\!\rangle| E)\ \leqslant\ \aleph_0,$$ we get $|\pi(E\langle\!\langle a \rangle\!\rangle)|<\kappa$, and from $$\dim_{\Lambda}\big( v(E\langle\!\langle a \rangle\!\rangle^{\times}) | v(E^{\times}) \big)\ \leqslant\ \mathop{\mathrm{rk}}_\mathcal L(E\langle\!\langle a \rangle\!\rangle| E)\ \leqslant\ \aleph_0,$$ we get $|v(E\langle\!\langle a \rangle\!\rangle^{\times})| < \kappa$. Hence by [\[akeext:ressort\]](#akeext:ressort){reference-type="ref" reference="akeext:ressort"}--[\[akeext:val\]](#akeext:val){reference-type="ref" reference="akeext:val"} we extend $\boldsymbol{f}$ to a good map $\boldsymbol{f}_1 = (f_1, f_{1, \operatorname{r}}, f_{1, \operatorname{v}})$ with small domain $\mathcal E_1 = (E_1, \boldsymbol{k}_1, \Gamma_1) \supseteq \mathcal E$ such that $\boldsymbol{k}_1$ is linearly surjective and $$\pi\big(E\langle\!\langle a \rangle\!\rangle\big) \subseteq \boldsymbol{k}_1 = \pi(E_1) \qquad \text{and} \qquad v\big(E\langle\!\langle a \rangle\!\rangle^{\times}\big) \subseteq \Gamma_1 = v(E_1^{\times}).$$ In the same way, we extend $\boldsymbol{f}_1$ to a good map $\boldsymbol{f}_2$ with small domain $\mathcal E_2 = (E_2, \boldsymbol{k}_2, \Gamma_2) \supseteq \mathcal E_1$ such that $\boldsymbol{k}_2$ is linearly surjective and $$\pi\big(E_1\langle\!\langle a \rangle\!\rangle\big) \subseteq \boldsymbol{k}_2 = \pi(E_2) \qquad \text{and} \qquad v\big(E_1\langle\!\langle a \rangle\!\rangle^{\times}\big) \subseteq \Gamma_2 = v(E_2^{\times}).$$ Iterating this procedure and taking unions yields an extension of $\boldsymbol{f}$ to a good map $\boldsymbol{f}_{\omega} = (f_{\omega}, f_{\omega, \operatorname{r}}, f_{\omega, \operatorname{v}})$ with small domain $\mathcal E_{\omega} = (E_{\omega}, \boldsymbol{k}_{\omega}, \Gamma_{\omega}) \supseteq \mathcal E$ such that $\boldsymbol{k}_{\omega}$ is linearly surjective and $$\pi\big(E_{\omega}\langle\!\langle a \rangle\!\rangle\big) = \boldsymbol{k}_{\omega} = \pi(E_{\omega}) \qquad \text{and} \qquad v\big(E_{\omega}\langle\!\langle a \rangle\!\rangle^{\times}\big) = \Gamma_{\omega} = v(E_{\omega}^{\times}).$$ This makes $\big(E_{\omega}\langle\!\langle a \rangle\!\rangle, \mathcal O_{E_{\omega}\langle\!\langle a \rangle\!\rangle}\big)$ an immediate $T^{\mathcal O,\der}$-extension of $(E_{\omega}, \mathcal O_{E_{\omega}})$, so by Fact [Fact 8](#fact:sphcomp){reference-type="ref" reference="fact:sphcomp"} and Corollary [Corollary 46](#cor:sphcompembed){reference-type="ref" reference="cor:sphcompembed"} we can take a spherically complete immediate $T^{\mathcal O,\der}$-extension $(E_{\omega+1}, \mathcal O_{E_{\omega+1}})$ of $\big(E_{\omega}\langle\!\langle a \rangle\!\rangle, \mathcal O_{E_{\omega}\langle\!\langle a \rangle\!\rangle}\big)$ inside $\mathcal K$, which is also an immediate $T^{\mathcal O,\der}$-extension of $(E_{\omega}, \mathcal O_{E_{\omega}})$. Then $\mathcal E_{\omega+1} = (E_{\omega+1}, \boldsymbol{k}_{\omega}, \Gamma_{\omega})$ is a good substructure of $\mathcal K$. Likewise taking a spherically complete immediate $T^{\mathcal O,\der}$-extension of $\big(f_{\omega}(E_{\omega}), f_{\omega}(\mathcal O_{E_{\omega}})\big)$ inside $\mathcal K^*$, Theorem [Theorem 41](#thm:sphcompuniquemono){reference-type="ref" reference="thm:sphcompuniquemono"} and [@Ha18B Corollary 4.4] yield an extension of $\boldsymbol{f}_{\omega}$ to a good map with small domain $\mathcal E_{\omega+1}$ containing $a$. ◻ For the next result, we construe $(\boldsymbol{k}, \Gamma; c)$ as a structure in the two-sorted language $\mathcal L_{\operatorname{r}\!\operatorname{v}, c} = \mathcal L_{\operatorname{r}} \cup \mathcal L_{\operatorname{v}} \cup \{c\}$. **Corollary 59**. *Suppose that $\mathcal K$ and $\mathcal K^*$ are $T^\der$-henselian. Then $\mathcal K\equiv \mathcal K^*$ if and only if $(\boldsymbol{k}, \Gamma; c) \equiv (\boldsymbol{k}^*, \Gamma^*; c^*)$.* *Proof.* The left-to-right direction is obvious. For the converse, suppose that $(\boldsymbol{k}, \Gamma; c) \equiv (\boldsymbol{k}^*, \Gamma^*; c^*)$. Let $\mathbb{P}$ be the prime model of $T$. Then $\mathcal E= (\mathbb{P}, \mathbb{P}, \{0\})$ is a good substructure of $\mathcal K$ and $\mathcal E^* = (\mathbb{P}, \mathbb{P}, \{0\})$ is a good substructure of $\mathcal K^*$, and we have a good map $\mathcal E\to \mathcal E^*$, which is partial elementary by Theorem [Theorem 58](#thm:equiv){reference-type="ref" reference="thm:equiv"}. ◻ For $T=T_{\mathop{\mathrm{an}}}$, this yields a theorem claimed in the introduction. **Corollary 60**. *Any $T^\der_{\mathop{\mathrm{an}}}$-henselian monotone $T^{\mathcal O,\der}_{\mathop{\mathrm{an}}}$-model is elementarily equivalent to $\boldsymbol{k}(\!(t^{\Gamma})\!)_{\mathop{\mathrm{an}},c}$ for some $\boldsymbol{k}\models T^\der_{\mathop{\mathrm{an}}}$, divisible ordered abelian group $\Gamma$, and additive map $c \colon \Gamma \to \boldsymbol{k}$.* **Corollary 61**. *Suppose that $\mathcal K$ is $T^\der$-henselian and let $\mathcal E= (E, \boldsymbol{k}_{\mathcal E}, \Gamma_{\mathcal E}; \pi, v, c, ac) \subseteq \mathcal K$ be a $T^\der$-henselian $T_{\operatorname{mon}}^{\mathop{\mathrm{ac}}}$-model such that $(\boldsymbol{k}_{\mathcal E}, \Gamma_{\mathcal E}; c) \preccurlyeq(\boldsymbol{k}, \Gamma; c)$. Then $\mathcal E\preccurlyeq\mathcal K$.* *Proof.* The identity map on $(E, \boldsymbol{k}_{\mathcal E}, \Gamma_{\mathcal E})$ is a good map from $\mathcal E$ to $\mathcal K$, so $\mathcal E\preccurlyeq\mathcal K$ by Theorem [Theorem 58](#thm:equiv){reference-type="ref" reference="thm:equiv"}. ◻ We can eliminate angular components from the previous corollary. **Corollary 62**. *Suppose that $(K, \boldsymbol{k}, \Gamma; \pi, v, c)$ is $T^\der$-henselian and let $(E, \boldsymbol{k}_{\mathcal E}, \Gamma_{\mathcal E}; \pi, v, c) \subseteq (K, \boldsymbol{k}, \Gamma; \pi, v, c)$ be $T^\der$-henselian such that $(\boldsymbol{k}_{\mathcal E}, \Gamma_{\mathcal E}; c) \preccurlyeq(\boldsymbol{k}, \Gamma; c)$. Then $(E, \boldsymbol{k}_{\mathcal E}, \Gamma_{\mathcal E}; \pi, v, c) \preccurlyeq(K, \boldsymbol{k}, \Gamma; \pi, v, c)$.* *Proof.* Let $\Delta$ be a $\Lambda$-subspace of $\Gamma$ such that $\Gamma = \Gamma_{\mathcal E} \oplus \Delta$ and let $B$ be a $\Lambda$-subspace of $K^>$ such that $K^>=E^>\cdot B$ is the direct sum of $E^>$ and $B$. By Lemma [Lemma 49](#lem:sectionexists){reference-type="ref" reference="lem:sectionexists"}, take a section $s_E \colon \Gamma_{\mathcal E} \to E^>$. By the proof of the same lemma, take a $\Lambda$-vector space embedding $s_B \colon \Delta \to B$ such that $v(s_B(\delta))=\delta$ for all $\delta \in \Delta$. Then $s\colon \Gamma \to K^>$ defined by $s(\gamma+\delta)=s_E(\gamma)s_B(\delta)$ for $\gamma \in \Gamma_{\mathcal E}$ and $\delta \in \Delta$ is a section for $K$. Letting $\mathop{\mathrm{ac}}$ be the angular component induced by $s$, the result now follows from Corollary [Corollary 61](#cor:elementarysubstructure){reference-type="ref" reference="cor:elementarysubstructure"}. ◻ ## Relative quantifier elimination and preservation of NIP In this subsection we eliminate quantifiers relative to the two-sorted structure $(\boldsymbol{k}, \Gamma; c)$ in the language $\mathcal L_{\operatorname{r}\!\operatorname{v}, c} \coloneqq \mathcal L_{\operatorname{r}} \cup \mathcal L_{\operatorname{v}} \cup \{c\}$. Let $x$ be an $l$-tuple of variables of sort $\operatorname{f}$, $y$ be an $m$-tuple of variables of sort $\operatorname{r}$, and $z$ be an $n$-tuple of variables of sort $\operatorname{v}$. A formula in $(x,y,z)$ is **special** if it is of the form $$\psi\big(\mathop{\mathrm{ac}}(F_1(\mbox{\small$\mathscr{J}$}_\der^{r}x)), \dots, \mathop{\mathrm{ac}}(F_{s}(\mbox{\small$\mathscr{J}$}_\der^{r}x)), y, v(G_1(\mbox{\small$\mathscr{J}$}_\der^{r}x)), \dots, v(G_{t}(\mbox{\small$\mathscr{J}$}_\der^{r}x)), z\big)$$ where $F_1, \dots, F_s, G_1, \dots, G_t \colon K^{(1+r)l} \to K$ are $\mathcal L(\emptyset)$-definable functions and $\psi(u_1, \dots, u_s, y, v_1, \dots, v_t, z)$ is an $\mathcal L_{\operatorname{r}\!\operatorname{v}, c}$-formula with $u_1, \dots, u_s$ of sort $\operatorname{r}$ and $v_1, \dots, v_t$ of sort $\operatorname{v}$. **Theorem 63**. *Every $\mathcal L_3$-formula is equivalent to a special formula.* *Proof.* For a $T_{\operatorname{mon}}^{\mathop{\mathrm{ac}}}$-model $\mathcal K$ and $a \in K^{l}$, $d \in \boldsymbol{k}^m$, and $\gamma \in \Gamma^n$, define the **special type** of $(a, d, \gamma)$ to be $$\textrm{sptp}(a, d, \gamma)\ \coloneqq\ \{ \theta(x,y,z) : \mathcal K\models \theta(a, d, \gamma)\ \text{and}\ \theta\ \text{is special}\}.$$ Now let $\mathcal K$ and $\mathcal K^*$ be $T^\der$-henselian $T_{\operatorname{mon}}^{\mathop{\mathrm{ac}}}$-models and $a \in K^{l}$, $a^* \in (K^*)^{l}$, $d \in \boldsymbol{k}^m$, $d^* \in (\boldsymbol{k}^*)^m$, $\gamma \in \Gamma^n$, $\gamma^* \in (\Gamma^*)^n$, and suppose that $(a, d, \gamma)$ and $(a^*, d^*, \gamma^*)$ have the same special type. It suffices to show that $(a, d, \gamma)$ and $(a^*, d^*, \gamma^*)$ have the same type. Let $E$ be the $\mathcal L^\der$-substructure of $K$ generated by $a$, $\Gamma_{\mathcal E}$ be the ordered $\Lambda$-subspace of $\Gamma$ generated by $\gamma$ and $v(E^{\times})$, and $\boldsymbol{k}_{\mathcal E}$ be the $\mathcal L^\der$-substructure of $\boldsymbol{k}$ generated by $\mathop{\mathrm{ac}}(E)$, $c(\Gamma_{\mathcal E})$, and $d$. Then $\mathcal E= (E, \boldsymbol{k}_{\mathcal E}, \Gamma_{\mathcal E})$ is a good substructure of $\mathcal K$. Likewise define a good substructure $\mathcal E^* = (E^*, \boldsymbol{k}_{\mathcal E^*}, \Gamma_{\mathcal E^*})$ of $\mathcal K^*$. Note that for an $\mathcal L(\emptyset)$-definable $F \colon K^{(1+r)l} \to K$, we have $F(\mbox{\small$\mathscr{J}$}_\der^{r}a)=0$ if and only if $\mathop{\mathrm{ac}}(F(\mbox{\small$\mathscr{J}$}_\der^{r}a))=0$ , and likewise in $\mathcal K^*$. From this and the assumption that $(a, d, \gamma)$ and $(a^*, d^*, \gamma^*)$ have the same special type we obtain an $\mathcal L^\der$-isomorphism $f \colon E \to E^*$ with $f(a)=a^*$. We next get an isomorphism $f_{\operatorname{v}} \colon \Gamma_{\mathcal E} \to \Gamma_{\mathcal E^*}$ of ordered $\Lambda$-vector spaces such that $f_{\operatorname{v}}(\gamma)=\gamma^*$ and $f_{\operatorname{v}}(vb)=v^*(f(b))$ for all $b \in E^{\times}$. Finally, we get an $\mathcal L^\der$-isomorphism $f_{\operatorname{r}} \colon \boldsymbol{k}_{\mathcal E} \to \boldsymbol{k}_{\mathcal E^*}$ such that $f_{\operatorname{r}}(d)=d^*$, $f_{\operatorname{r}}(\mathop{\mathrm{ac}}(b))=\mathop{\mathrm{ac}}^*(f(b))$ for all $b \in E$, and $f_{\operatorname{r}}(c(\delta))=c^*(f_{\operatorname{v}}(\delta))$ for all $\delta \in \Gamma_{\mathcal E}$. By the assumption on special types, $(f_{\operatorname{r}}, f_{\operatorname{v}})$ is a partial elementary map $(\boldsymbol{k}, \Gamma; c) \to (\boldsymbol{k}^*, \Gamma^*; c^*)$, so $(f, f_{\operatorname{r}}, f_{\operatorname{v}})$ is a good map $\mathcal E\to \mathcal E^*$. The result now follows from Theorem [Theorem 58](#thm:equiv){reference-type="ref" reference="thm:equiv"}. ◻ It follows that $(\boldsymbol{k}, \Gamma; c)$ is stably embedded in $\mathcal K$: **Corollary 64**. *Any subset of $\boldsymbol{k}^m \times \Gamma^n$ definable in $\mathcal K$ is definable in $(\boldsymbol{k}, \Gamma; c)$.* Now we turn to preservation of NIP. Consider the following languages, each of which has the same three sorts as $\mathcal L_3$: 1. The language $\mathcal L'$, where we drop the derivation on the field sort $\operatorname{f}$. Explicitly, the field sort is an $\mathcal L$-structure, the residue field sort is still an $\mathcal L^\der$-structure, the value group sort is still an ordered $\Lambda$-vector space, and we keep the maps $\pi$, $v$, $c$, and $\mathop{\mathrm{ac}}$. We let $T'$ be the restriction of $T_{\operatorname{mon}}^{\mathop{\mathrm{ac}}}$ to the language $\mathcal L'$; that is, $T'$ consists of all $\mathcal L'$-sentences which hold in all models of $T_{\operatorname{mon}}^{\mathop{\mathrm{ac}}}$. 2. The language $\mathcal L^{\mathop{\mathrm{ac}}}$, where we drop the derivation on both the field sort $\operatorname{f}$ and the residue field sort $\operatorname{r}$, as well as $c$. Explicitly, the field sort and the residue field sort are both $\mathcal L$-structures, the value group sort is still an ordered $\Lambda$-vector space, and we keep the maps $\pi$, $v$, and $\mathop{\mathrm{ac}}$. We let $T^{\mathop{\mathrm{ac}}}$ be the restriction of $T_{\operatorname{mon}}^{\mathop{\mathrm{ac}}}$ to the language $\mathcal L^{\mathop{\mathrm{ac}}}$. **Lemma 65**. *The theory $T^{\mathop{\mathrm{ac}}}$ is complete and has NIP. The residue field and value group are stably embedded in any model of $T^{\mathop{\mathrm{ac}}}$.* *Proof.* Let $\mathcal L^s$ extend $\mathcal L^{\mathop{\mathrm{ac}}}$ by an additional map $s$ from the value group sort to the field sort, and let $T^s$ be the $\mathcal L^s$-theory which extends $T^{\mathop{\mathrm{ac}}}$ by axioms stating that $s$ is a section and that $\mathop{\mathrm{ac}}$ is induced by $s$. By Corollary [Corollary 50](#cor:acinduced){reference-type="ref" reference="cor:acinduced"}, any model of $T^{\mathop{\mathrm{ac}}}$ admits an expansion to a model of $T^s$. The theory $T^s$ is complete and has NIP [@KK23 Corollary 2.2 and Proposition 4.2], and the residue field and value group are stably embedded in any model of $T^s$ [@KK23 Corollary 2.4], so the lemma follows. Note that in [@KK23], the language $\mathcal L^s$ does not contain a function symbol for the angular component, but this is nonetheless definable by $\mathop{\mathrm{ac}}(y) = \pi(y/s(vy))$ for nonzero $y$. ◻ **Theorem 66**. *Let $\mathcal K= (K,\boldsymbol{k},\Gamma;\pi,v,c,\mathop{\mathrm{ac}}) \models T_{\operatorname{mon}}^{\mathop{\mathrm{ac}}}$ and suppose that $\mathop{\mathrm{Th}}(\boldsymbol{k},\Gamma;c)$ has NIP. Then $\mathop{\mathrm{Th}}(\mathcal K)$ has NIP.* *Proof.* By Lemma [Lemma 65](#lem:acNIP){reference-type="ref" reference="lem:acNIP"}, the theory of the reduct $\mathcal K|_{\mathcal L^{\mathop{\mathrm{ac}}}}$ has NIP, with stably embedded residue field and value group. By [@JS20 Proposition 2.5], we may expand the residue field and value group by any additional NIP structure, and the theory of the resulting expansion of $\mathcal K|_{\mathcal L^{\mathop{\mathrm{ac}}}}$ still has NIP. In particular, the theory of the intermediate reduct $\mathcal K|_{\mathcal L'}$ has NIP. Suppose now that $\mathop{\mathrm{Th}}(\mathcal K)$ has IP. After replacing $\mathcal K$ with an elementary extension, we find an $\mathcal L_3$-indiscernible sequence $(a_i)_{i<\omega}$, a tuple of parameters $b$, and an $\mathcal L_3$-formula $\phi(x,y)$ such that $\mathcal K\models \phi(a_i,b)$ if and only if $i$ is even. Note that $x$ and $y$ may span multiple sorts, and we let $x_{\operatorname{f}}$, $x_{\operatorname{r}}$, and $x_{\operatorname{v}}$ denote the parts of $x$ coming from the field sort, residue field sort, and value group sort; likewise for $y_{\operatorname{f}}$, $y_{\operatorname{r}}$, and $y_{\operatorname{v}}$. By Theorem [Theorem 63](#thm:special){reference-type="ref" reference="thm:special"}, we may assume that $\phi(x,y)$ is of the form $$\psi\big(\mathop{\mathrm{ac}}(F_1(\mbox{\small$\mathscr{J}$}_\der^{r}x_{\operatorname{f}},\mbox{\small$\mathscr{J}$}_\der^{r}y_{\operatorname{f}})), \dots, \mathop{\mathrm{ac}}(F_{s}(\mbox{\small$\mathscr{J}$}_\der^{r}x_{\operatorname{f}},\mbox{\small$\mathscr{J}$}_\der^{r}y_{\operatorname{f}})), x_{\operatorname{r}},y_{\operatorname{r}}, v(G_1(\mbox{\small$\mathscr{J}$}_\der^{r}x_{\operatorname{f}},\mbox{\small$\mathscr{J}$}_\der^{r}y_{\operatorname{f}})), \dots, v(G_t(\mbox{\small$\mathscr{J}$}_\der^{r}x_{\operatorname{f}},\mbox{\small$\mathscr{J}$}_\der^{r}y_{\operatorname{f}})), x_{\operatorname{v}},y_{\operatorname{v}}\big),$$ where $\psi$ is an $\mathcal L_{\operatorname{r}\!\operatorname{v},c}$-formula. Now, we augment each $a_i$ to a tuple $a_i^*$ as follows: write $a_i = a_{i,\operatorname{f}}a_{i,\operatorname{r}}a_{i,\operatorname{v}}$ where $a_{i,\operatorname{f}}$, $a_{i,\operatorname{r}}$, and $a_{i,\operatorname{v}}$ denote the parts of $a_i$ coming from the relevant sorts, and put $a_i^*\coloneqq \mbox{\small$\mathscr{J}$}_\der^r(a_{i,\operatorname{f}})a_{i,\operatorname{r}}a_{i,\operatorname{v}}$. We augment $b$ to $b^*$ similarly. Let $x^*_{\operatorname{f}}$ be a field sort variable whose length is $1+r$ times that of $x_{\operatorname{f}}$, likewise for $y^*_{\operatorname{f}}$, and let $\phi^*$ be the $\mathcal L'$-formula $$\phi^*(x^*,y^*)\ \coloneqq\ \psi\big(\mathop{\mathrm{ac}}(F_1(x^*_{\operatorname{f}},y^*_{\operatorname{f}})), \dots, \mathop{\mathrm{ac}}(F_{s}(x^*_{\operatorname{f}},y^*_{\operatorname{f}})), x_{\operatorname{r}},y_{\operatorname{r}}, v(G_1(x^*_{\operatorname{f}},y^*_{\operatorname{f}})), \dots, v(G_t(x^*_{\operatorname{f}},y^*_{\operatorname{f}})), x_{\operatorname{v}},y_{\operatorname{v}}\big).$$ Then $\mathcal K\models \phi^*(a_i^*,b^*)$ if and only if $\mathcal K\models \phi(a_i,b)$ (so if and only if $i$ is even). Since the sequence $(a_i)_{i<\omega}$ is $\mathcal L_3$-indiscernible, the augmented sequence $(a_i^*)_{i<\omega}$ is $\mathcal L'$-indiscernible, so this contradicts the fact that the theory of $\mathcal K|_{\mathcal L'}$ has NIP. ◻ **Remark 67**. Instead of appealing to the theory $T^s$ to show that $T^{\mathop{\mathrm{ac}}}$ has NIP, and then using [@JS20 Proposition 2.5] to deduce that the $\mathcal L'$-reduct of $\mathcal K$ has NIP, one could likely show that $\mathop{\mathrm{Th}}(\mathcal K|_{\mathcal L'})$ has NIP directly, by building an appropriate back-and-forth system between appropriate substructures of models of $T'$ and applying [@JS20 Theorem 2.3]. From there, the rest of the proof would proceed as above. It seems likely that distality and NTP$_2$ also transfer from $\mathop{\mathrm{Th}}(\boldsymbol{k},\Gamma;c)$ to $\mathop{\mathrm{Th}}(\mathcal K)$, but in both cases, certain aspects of the above proof do not work. # The model completion of monotone $T$-convex $T$-differential fields {#sec:modelcompletion} In this section, we return to the one-sorted setting. Let $T_{\operatorname{mon}}$ be the theory of monotone $T$-convex $T$-differential fields, in the language $\mathcal L^{\mathcal O,\der}$; note that unlike in $T^{\mathcal O,\der}$, we do not require the $T$-convex valuation ring to be proper. Let $T^{*}_{\operatorname{mon}}$ be the theory of $T^\der$-henselian $T$-convex $T$-differential fields with many constants, nontrivial valuation, and generic $T$-differential residue field (that is, with differential residue field a model of $T^\der_{\mathcal G}$; see Section [2.3](#sec:Td){reference-type="ref" reference="sec:Td"}). Note that any model of $T^{*}_{\operatorname{mon}}$ is monotone, as it has many constants and small derivation. The purpose of this section is to show that $T^{*}_{\operatorname{mon}}$ is the model completion of $T_{\operatorname{mon}}$. For this, we need the following general proposition on residue field extensions. In this section, $K=(K,\mathcal O,\der)$ is a $T$-convex $T$-differential field with small derivation. The only difference with the earlier standing assumption is that we may have $\mathcal O=K$ (equivalently, $\Gamma=\{0\}$). **Proposition 68**. *Let $E$ be a $T^\der$-extension of $\boldsymbol{k}$. Then there is a $T$-convex $T$-differential field extension $L$ of $K$ with small derivation and the following properties:* 1. *[\[prop:residueexti\]]{#prop:residueexti label="prop:residueexti"} $\boldsymbol{k}_L$ is $\mathcal L^\der(\boldsymbol{k})$-isomorphic to $E$;* 2. *[\[prop:residueextii\]]{#prop:residueextii label="prop:residueextii"} $\Gamma_L= \Gamma$;* 3. *[\[prop:residueextiii\]]{#prop:residueextiii label="prop:residueextiii"} If $K^*$ is any $T^\der$-henselian $T^{\mathcal O,\der}$-extension of $K$, then any $\mathcal L^\der(\boldsymbol{k})$-embedding $\imath\colon \boldsymbol{k}_L \to \mathop{\mathrm{res}}(K^*)$ lifts to an $\mathcal L^{\mathcal O,\der}(K)$-embedding $\jmath\colon L \to K^*$.* *Proof.* If $\Gamma = \{0\}$, then we may take $L = E$, so we will assume that $\Gamma \neq \{0\}$. We may further assume that $E = \boldsymbol{k}\langle\!\langle a\rangle\!\rangle$, where $a \not\in \boldsymbol{k}$. First, we build the extension $L$, and then we verify that it is indeed a $T^{\mathcal O,\der}$-extension $L$ of $K$ satisfying [\[prop:residueexti\]](#prop:residueexti){reference-type="ref" reference="prop:residueexti"}--[\[prop:residueextiii\]](#prop:residueextiii){reference-type="ref" reference="prop:residueextiii"}. Consider the case that $a$ is $T^\der$-transcendental over $\boldsymbol{k}$, so $E = \boldsymbol{k}\langle a,a',\ldots\rangle$. We define $T^{\mathcal O}$-extensions $K = L_0 \subseteq L_1\subseteq \cdots$ as follows: for each $i$, let $L_{i+1}\coloneqq L_i\langle b_i\rangle$, where $b_i$ realizes the cut $$\{y\in L_i: y <\mathcal O_{L_i} \text{ or } y \in \mathcal O_{L_i}\text{ and } \bar{y}<a^{(i)}\},$$ and equip $L_{i+1}$ with the $T$-convex valuation ring $$\mathcal O_{L_{i+1}}\ \coloneqq \ \big\{y\in L_{i+1}:|y|<d\text{ for all }d \in L_i\text{ with } d >\mathcal O_{L_i}\big\}.$$ By [@DL95 Main Lemma 3.6], each $L_{i+1}$ is indeed a $T^{\mathcal O}$-extension of $L_i$. Let $L\coloneqq \bigcup_{i}L_i$, and using Fact [Fact 5](#fact:transext){reference-type="ref" reference="fact:transext"}, extend $\der$ to a $T$-derivation on $L$ such that $\der b_n = b_{n+1}$ for each $n$. Now, consider the case that $a$ is $T^\der$-algebraic over $\boldsymbol{k}$. Let $r>0$ be minimal with $E = \boldsymbol{k}\langle \mbox{\small$\mathscr{J}$}_\der^{r-1}a \rangle$ and build $T^{\mathcal O}$-extensions $K = L_0 \subseteq L_1\subseteq \cdots\subseteq L_r$ as in the $T^\der$-transcendental case, so $L_{i+1}\coloneqq L_i\langle b_i\rangle$ for each $i<r$, where $b_i$ is a lift of $a^{(i)}$. This time, we put $L\coloneqq L_r = K\langle b_0,\ldots,b_{r-1}\rangle$. Let $K_0 \subseteq K$ be a $T$-lift of $\boldsymbol{k}$, so there is an $\mathcal L$-isomorphism $K_0\langle b_0,\ldots,b_{r-1}\rangle\to E$ which agrees with the residue map on $K_0$ and sends $b_i$ to $a^{(i)}$ for each $i<r$. Let $b_{r} \in K_0\langle b_0,\ldots,b_{r-1}\rangle$ be the unique element which maps to $a^{(r)} \in E$, and again using Fact [Fact 5](#fact:transext){reference-type="ref" reference="fact:transext"}, extend $\der$ to a $T$-derivation on $L$ such that $\der b_i = \der b_{i+1}$ for $i<r$. We claim that $L$ has small derivation. Put $b\coloneqq b_0$, so $L = K\langle\!\langle b \rangle\!\rangle$. If $a$ is $T^\der$-algebraic over $\boldsymbol{k}$, then let $r$ be as above, and if $a$ is $T^\der$-transcendental over $\boldsymbol{k}$, then let $r$ be arbitrary. Let $F\colon K^r\to K$ be an $\mathcal L(K)$-definable function with $F(\mbox{\small$\mathscr{J}$}_\der^{r-1}b)\prec 1$. We need to verify that $F(\mbox{\small$\mathscr{J}$}_\der^{r-1}b)' \prec 1$. By [@FK21 Lemma 2.12], there is some $\mathcal L(K)$-definable function $F^{[\der]}\colon K^r\to K$ such that $$F(\mbox{\small$\mathscr{J}$}_\der^{r-1}b)'\ =\ F^{[\der]}(\mbox{\small$\mathscr{J}$}_\der^{r-1}b) + \frac{\partial F}{\partial Y_0}(\mbox{\small$\mathscr{J}$}_\der^{r-1}b)b'+\cdots+\frac{\partial F}{\partial Y_{r-1}}(\mbox{\small$\mathscr{J}$}_\der^{r-1}b)b^{(r)}.$$ For each $i<r$, we note that $b^{(i)} \asymp 1$ and that $b^{(i)} \not\sim g$ for any $g \in K\langle (b^{(j)})_{j<r,j\neq i}\rangle$, so Fact [Fact 51](#fact:smallderivsim){reference-type="ref" reference="fact:smallderivsim"} gives us that $\frac{\partial F}{\partial Y_i}(\mbox{\small$\mathscr{J}$}_\der^{r-1}b)\preccurlyeq F(\mbox{\small$\mathscr{J}$}_\der^{r-1}b)\prec 1$. Since $b^{(i+1)} \preccurlyeq 1$, we conclude that $\frac{\partial F}{\partial Y_i}(\mbox{\small$\mathscr{J}$}_\der^{r-1}b)b^{(i+1)}\prec 1$, so it remains to show that $F^{[\der]}(\mbox{\small$\mathscr{J}$}_\der^{r-1}b)\prec 1$. Suppose that this is not the case, and let $U \subseteq K^r$ be the $\mathcal L^{\mathcal O}(K)$-definable set $$U\ \coloneqq\ \Big\{u \in \mathcal O^r: F^{[\der]}(u) \succcurlyeq 1\text{ and }F(u),\frac{\partial F}{\partial Y_0}(u),\ldots,\frac{\partial F}{\partial Y_{r-1}}(u)\prec 1\Big\}.$$ Note that $U$ is nonempty, since $L$ is an elementary $T^{\mathcal O}$-extension of $K$ and $\mbox{\small$\mathscr{J}$}_\der^{r-1}(b) \in U^L$. Take a tuple $u = (u_0,\ldots,u_{r-1}) \in U$. Then $u_0',\ldots,u_{r-1}' \preccurlyeq 1$ since $K$ has small derivation. But then $$F(u)' \ =\ F^{[\der]}(u) + \frac{\partial F}{\partial Y_0}(u)u_0'+\cdots+\frac{\partial F}{\partial Y_{r-1}}(u)u_{r-1}'\ \asymp\ F^{[\der]}(u)\ \succcurlyeq\ 1,$$ contradicting that $K$ has small derivation. This concludes the proof of the claim. With the claim taken care of, we see that $L$ is a $T^{\mathcal O,\der}$-extension of $K$ and that the differential residue field $\boldsymbol{k}_L$ is $\mathcal L^\der(\boldsymbol{k})$-isomorphic to $E$. By the Wilkie inequality, we have that $\Gamma_L = \Gamma$. Let $K^*$ be a $T^\der$-henselian $T^{\mathcal O,\der}$-extension of $K$ and let $\imath\colon \boldsymbol{k}_L\to \mathop{\mathrm{res}}(K^*)$ be an $\mathcal L^\der(\boldsymbol{k})$-embedding. The argument that $\imath$ lifts to an $\mathcal L^{\mathcal O,\der}(K)$-embedding $L \to K^*$ is very similar to the proof of part [\[prop:resextiii\]](#prop:resextiii){reference-type="ref" reference="prop:resextiii"} of Proposition [Proposition 18](#prop:resext){reference-type="ref" reference="prop:resext"}. If $a$ is $T^\der$-transcendental over $\boldsymbol{k}$, then for any lift $b^*$ of $\imath(a)$, the unique $\mathcal L^{\mathcal O,\der}(K)$-embedding $L \to K^*$ which sends $b$ to $b^*$ is a lift of $\imath$. Suppose that $a$ is $T^\der$-algebraic over $\boldsymbol{k}$. Then by construction $b^{(r)} = F(\mbox{\small$\mathscr{J}$}_\der^{r-1}b)$ for some minimal $r$ and some $\mathcal L(K_0)$-definable function $F$, where $K_0\subseteq K$ is a lift of $\boldsymbol{k}$. As $K^*$ is $T^\der$-henselian, we find $b^* \in K^*$ which lifts $\imath(a)$ and satisfies the identity $(b^*)^{(r)} = F(\mbox{\small$\mathscr{J}$}_\der^{r-1}b^*)$. Then the unique $\mathcal L^{\mathcal O,\der}(K)$-embedding $L \to K^*$ which sends $b$ to $b^*$ is a lift of $\imath$. ◻ **Corollary 69**. *Let $K\models T_{\operatorname{mon}}$. Then $K$ has a $T^{\mathcal O,\der}$-extension $L\models T^{*}_{\operatorname{mon}}$ which embeds over $K$ into any $|K|^+$-saturated $K^*\models T^{*}_{\operatorname{mon}}$ extending $K$.* *Proof.* We construct $L$ in three steps. First, we build a nontrivially valued $T_{\operatorname{mon}}$-model $K_1$ extending $K$. If $K$ itself is nontrivially valued, then we take $K_1\coloneqq K$. If $K$ is trivially valued, then let $K_1\coloneqq K\langle a\rangle$ where $a>K$. In this case, we equip $K_1$ with the convex hull of $K$ as its $T$-convex valuation ring and, using Fact [Fact 5](#fact:transext){reference-type="ref" reference="fact:transext"}, we uniquely extend the $T$-derivation on $K$ to a $T$-derivation on $K_1$ such that $\der a = 0$. Then $\mathop{\mathrm{res}}(K_1) = \boldsymbol{k}$, and it is fairly easy to check that $K_1$ is monotone, arguing as in Proposition [Proposition 52](#prop:fieldbuilding){reference-type="ref" reference="prop:fieldbuilding"}. In fact, one can build $K_1$ explicitly using Proposition [Proposition 52](#prop:fieldbuilding){reference-type="ref" reference="prop:fieldbuilding"} by taking $K$ in place of $\boldsymbol{k}$, taking $\Lambda$ in place of $\Gamma$, and taking $c\colon \Lambda \to K$ to be the zero map. Next we use Fact [Fact 6](#fact:tdextend){reference-type="ref" reference="fact:tdextend"} to take a $T^\der_{\mathcal G}$-model $E$ extending $\boldsymbol{k}$ with $|E| = |\boldsymbol{k}|$. Take a $T^{\mathcal O,\der}$-extension $K_2$ of $K_1$ as in Proposition [Proposition 68](#prop:residueext){reference-type="ref" reference="prop:residueext"}, so $\Gamma_{K_2}= \Gamma_{K_1}$ and $\mathop{\mathrm{res}}(K_2)$ is $\mathcal L^\der(\boldsymbol{k})$-isomorphic to $E$. Finally, we use Fact [Fact 8](#fact:sphcomp){reference-type="ref" reference="fact:sphcomp"} to take a spherically complete immediate $T^{\mathcal O,\der}$-extension $L$ of $K_2$. Then $\Gamma_L= \Gamma_{K_1}$, so $L$ is monotone by [@ADH17 Corollary 6.3.6]. Moreover, $L$ is $T^\der$-henselian by Corollary [Corollary 23](#cor:sphcompTdh){reference-type="ref" reference="cor:sphcompTdh"}, so it has many constants by Corollary [Corollary 11](#cor:7.1.11){reference-type="ref" reference="cor:7.1.11"} (it follows easily from the axioms of $T^\der_{\mathcal G}$ that $(\boldsymbol{k}_L^\times)^\dagger = \boldsymbol{k}_L$). Now let $K^*\models T^{*}_{\operatorname{mon}}$ be a $|K|^+$-saturated $T^{\mathcal O,\der}$-extension of $K$. In the case that $K_1 = K\langle a \rangle$ where $a>K$ and $a' = 0$, we take $a^* \in C_{K^*}$ with $a^*> \mathcal O_{K^*}$ and we let $\imath_1\colon K_1\to K^*$ be the $\mathcal L(K)$-embedding which sends $a$ to $a^*$. By Fact [Fact 5](#fact:transext){reference-type="ref" reference="fact:transext"} and [@DL95 Corollary 3.7], $\imath_1$ is even an $\mathcal L^{\mathcal O,\der}$-embedding. If $K_1 = K$, then we just take $\imath_1$ to be the identity on $K$. Next, as $\mathop{\mathrm{res}}(K^*)$ is a $|\boldsymbol{k}|^+$-saturated model of $T^\der_{\mathcal G}$, the inclusion $\boldsymbol{k}\subseteq \mathop{\mathrm{res}}(K^*)$ extends to an $\mathcal L^\der$-embedding $\mathop{\mathrm{res}}(K_2) \to \mathop{\mathrm{res}}(K^*)$ by Fact [Fact 6](#fact:tdextend){reference-type="ref" reference="fact:tdextend"}. By Proposition [Proposition 68](#prop:residueext){reference-type="ref" reference="prop:residueext"}, we may lift this to an $\mathcal L^{\mathcal O,\der}(K)$-embedding $\imath_2\colon K_2\to K^*$ which extends $\imath_1$. Finally, note that $K^*$ is $|\Gamma_{K_2}|^+$-saturated, since $|\Gamma_{K_2}| = \max\{|\Gamma_K|,|\Lambda|\}\leqslant|K|$, so $\imath_2$ extends to an $\mathcal L^{\mathcal O,\der}(K)$-embedding $\jmath\colon L\to K^*$ by Corollary [Corollary 46](#cor:sphcompembed){reference-type="ref" reference="cor:sphcompembed"}. ◻ **Theorem 70**. *$T^{*}_{\operatorname{mon}}$ is the model completion of $T_{\operatorname{mon}}$. Consequently, $T^{*}_{\operatorname{mon}}$ is complete, and if $T$ has quantifier elimination and a universal axiomatization, then $T^{*}_{\operatorname{mon}}$ has quantifier elimination.* *Proof.* Let $E\models T_{\operatorname{mon}}$, and let $K$ and $K^*$ be models of $T^{*}_{\operatorname{mon}}$ extending $E$. In light of Corollary [Corollary 69](#cor:genericextension){reference-type="ref" reference="cor:genericextension"}, it is enough to show that $K$ and $K^*$ are $\mathcal L^{\mathcal O,\der}(E)$-elementarily equivalent. We may assume that both $K$ and $K^*$ are $|E|^+$-saturated. Let $L\models T^{*}_{\operatorname{mon}}$ be the $T^{\mathcal O,\der}$-extension of $E$ constructed in Corollary [Corollary 69](#cor:genericextension){reference-type="ref" reference="cor:genericextension"}. Then $L$ admits an $\mathcal L^{\mathcal O,\der}(E)$-embedding into both $K$ and $K^*$, so by replacing $E$ with $L$, we may assume that $E \models T^{*}_{\operatorname{mon}}$. It remains to note that $T^{*}_{\operatorname{mon}}$ is model complete by Corollary [Corollary 62](#cor:elementarysubstructure2){reference-type="ref" reference="cor:elementarysubstructure2"}, since $T^\der_{\mathcal G}$ and the theory of nontrivial ordered $\Lambda$-vector spaces are both model complete (in invoking Corollary [Corollary 62](#cor:elementarysubstructure2){reference-type="ref" reference="cor:elementarysubstructure2"}, we take $c$ to be the zero map). Let $\mathbb{P}$ be the prime model of $T$. Then $\mathbb{P}$, equipped with the $T$-convex valuation ring $\mathcal O_\mathbb{P}= \mathbb{P}$ and the trivial $T$-derivation, is a model of $T_{\operatorname{mon}}$ which embeds into every model of $T^{*}_{\operatorname{mon}}$, so $T^{*}_{\operatorname{mon}}$ is complete. Suppose that $T$ has quantifier elimination and a universal axiomatization. Then $T_{\operatorname{mon}}$ has a universal axiomatization, so $T^{*}_{\operatorname{mon}}$ has quantifier elimination. ◻ **Corollary 71**. *For every $\mathcal L^{\mathcal O,\der}$-formula $\varphi$ there is some $r$ and some $\mathcal L^{\mathcal O}$-formula $\tilde{\varphi}$ such that $$T^{*}_{\operatorname{mon}}\ \vdash\ \forall x\big( \varphi(x) \leftrightarrow \tilde{\varphi}\big(\mbox{\small$\mathscr{J}$}_\der^r(x)\big)\big).$$* *Proof.* Argue as in [@FK21 Lemma 4.11], using that every $\mathcal L^{\mathcal O}$-term is an $\mathcal L$-term. ◻ **Corollary 72**. *$T^{*}_{\operatorname{mon}}$ is distal.* *Proof.* Argue as in [@FK21 Theorem 4.15], using Corollary [Corollary 71](#cor:formulaform){reference-type="ref" reference="cor:formulaform"} and the fact that $T^{\mathcal O}$ is distal. ◻ **Corollary 73**. *Let $K\models T^{*}_{\operatorname{mon}}$ and let $C$ be the constant field of $K$. For every $\mathcal L^{\mathcal O,\der}(K)$-definable set $A \subseteq C^n$, there is an $\mathcal L^{\mathcal O}(K)$-definable set $B\subseteq K^n$ such that $A = B\cap C^n$. In particular, the induced structure on $C$ is weakly o-minimal.* *Proof.* Argue as in [@FK21 Lemma 4.23], using Corollary [Corollary 71](#cor:formulaform){reference-type="ref" reference="cor:formulaform"}. ◻ # Acknowledgements {#acknowledgements .unnumbered} Research for this paper was conducted in part at the Fields Institute for Research in Mathematical Sciences. This material is based upon work supported by the National Science Foundation under Grants DMS-2103240 (Kaplan) and DMS-2154086 (Pynn-Coates). Pynn-Coates was also supported by the Austrian Science Fund (FWF) under project ESP 450. 10 M. Aschenbrenner, L. van den Dries, and J. van der Hoeven. . Number 195 in Annals of Mathematics Studies. Princeton University Press, 2017. M. Aschenbrenner, L. van den Dries, and J. van der Hoeven. 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--- abstract: | We give a unified and self-contained proof of the Nielsen--Thurston classification theorem from the theory of mapping class groups and Thurston's characterization of rational maps from the theory of complex dynamics (plus various extensions of these). Our proof follows Bers' proof of the Nielsen--Thurston classification. address: - | James Belk\ School of Mathematics & Statistics\ 15 University Gardens\ University of Glasgow\ G12 8QW\ james.belk\@glasgow.ac.uk - | Dan Margalit\ Department of Mathematics\ Vanderbilt University\ 1326 Stevenson Center Ln\ Nashville, TN 37240\ dan.margalit\@vanderbilt.edu - | Rebecca R. Winarski\ Department of Mathematics and Computer Science\ College of the Holy Cross\ 1 College Street Worcester, MA 01610\ rebecca.winarski\@gmail.com author: - James Belk - Dan Margalit - Rebecca R. Winarski bibliography: - thurston.bib title: Thurston's theorem and the Nielsen--Thurston classification via Teichmüller's theorems --- # Introduction The main theorem of this paper is what we call the Nielsen--Thurston Übertheorem. This is a unification, and extension, of the Nielsen--Thurston classification theorem from the theory of mapping class groups and Thurston's characterization of rational maps from the theory complex dynamics. The unified statement we give here is new, although the content is almost entirely due to Thurston. We give a unified proof of the Übertheorem by extending the Bers proof of the Nielsen--Thurston classification [@bers; @primer] to the case of nontrivial (branched) covers, possibly with marked points that are not post-critical. In Appendix [9](#sec:ext){reference-type="ref" reference="sec:ext"}, we also extend the theorem to treat the cases of non-orientable surfaces, orientation-reversing maps, and equivariant maps. Thurston proved his characterization of rational maps in 1982 and gave several lectures on the proof. The first published proof was given by Douady and Hubbard in 1993 [@DH]. Our proof of the Übertheorem is not only an extension of the Bers proof of the Nielsen--Thurston classification, but it also tracks the Douady--Hubbard paper closely. One aim of this paper is to clarify the connection between these two proofs, which have long been recognized to be similar in spirit but have not heretofore been put into a single framework. The Nielsen--Thurston Übertheorem classifies dynamical branched covers, which we presently define. Let $\Sigma$ be a marked surface, that is, a pair $(S,P)$ where $S$ is a closed surface, and $P$ is a finite set of marked points in $S$. By a *dynamical branched cover* of $\Sigma$, we mean a branched covering map $f\colon \Sigma \to \Sigma$ where $f(P) \subseteq P$ and $P$ contains all of the critical values of $f$. Dynamical branched covers of the sphere with degree at least 2 are traditionally called Thurston maps (according to Douady--Hubbard, this terminology was suggested by Milnor). A dynamical branched cover can be a homeomorphism, a nontrivial covering map, or a nontrivial branched covering map. The last two cases only arise when $S$ is $T^2$ or $S^2$, respectively. A motivation for studying dynamical branched covers is that they make topological operations accessible in the context of rational maps. For instance the mating of two polynomials of degree $d$ is a dynamical branched cover of $S^2$ (the maps on the hemispheres being given by the two polynomials) with no complex structure attached. The Nielsen--Thurston Übertheorem classifies dynamical branched covers up to homotopy. Here, two dynamical branched covers $f$ and $g$ of $\Sigma$ are homotopic if there is a homeomorphism $h$ of $\Sigma$ that is homotopic to the identity (rel $P$) and satisfies $f \circ h = g$ (this relation is finer than the usual notion of Thurston equivalence; see below). Before stating the Übertheorem, we recall the statements of the Nielsen--Thurston classification and Thurston's characterization of rational maps. ## The Nielsen--Thurston classification The Nielsen--Thurston classification theorem for surface homeomorphisms [@primer Theorem 13.2] is a theorem of Thurston from 1974, although the first complete, published proof was given in 1979 by Fathi--Laudenbach--Poénaru [@flp_original] (many other proofs have appeared since then). In the statement we say that a homeomorphism is *periodic* if some nontrivial power is the identity. Every periodic homeomorphism is geometric in the sense that it is an isometry in some metric of constant curvature. Next, we say that a homeomorphism is *reducible* if it preserves a multicurve, that is, a collection of pairwise disjoint simple closed curves in $\Sigma$. Finally, a surface homeomorphism $f$ of $\Sigma = (S,P)$ is *pseudo-Anosov* if there is a pair of transverse measured foliations $(\mathcal F^+,\mathcal F^-)$ on $\Sigma$ that is preserved by $f$ and satisfies $$f^{-1}(\mathcal F^+,\mathcal F^-)=(\lambda\, \mathcal F^+,\tfrac{1}{\lambda}\,\mathcal F^-)$$ for some $\lambda > 1$. The foliations may have 1-pronged singularities and $k$-pronged singularities with $k \geq 3$. Each 1-pronged singularity must be at a point of $P$. As with periodic maps, pseudo-Anosov maps are geometric in that they preserve the affine structure on $\Sigma$ induced by the pair of measured foliations. **Theorem 1** (Nielsen--Thurston classification). *Let $f\colon \Sigma \to \Sigma$ be a homeomorphism, where $\Sigma$ is a closed surface with finitely many marked points. Then $f$ is isotopic to a homeomorphism of one of the following types:* 1. *periodic,* 2. *reducible, or* 3. *pseudo-Anosov.* *Type (3) is exclusive from the other two. If $f$ is of type (3) the pseudo-Anosov structure is unique up to isotopy.* We can rephrase this classification as: every homeomorphism decomposes along reducing curves into homeomorphisms that are geometric, that is, periodic or pseudo-Anosov. Thurston proved the exclusivity by showing that pseudo-Anosov maps increase the length of every simple closed curve exponentially under iteration (see Section [5](#sec:proof){reference-type="ref" reference="sec:proof"} for more details). This clearly fails for periodic and reducible maps (in both cases, some power of the map fixes a curve). So in this sense the Nielsen--Thurston classification says that the only obstructions to pseudo-Anosovity are the "obvious" ones. ## Thurston's characterization of rational maps Our next goal is to state Thurston's characterization of rational maps from the theory complex dynamics. (Within the field of complex dynamics, this theorem is often referred to as simply "Thurston's theorem"; we prefer to avoid this terminology due to the ubiquity of Thurston's work in the fields of mapping class groups, complex dynamics, and beyond.) Our phrasing of the theorem requires the notion of an unmarked map and the notion of a strong reduction system. *Marked and unmarked maps.* We say that a dynamical branched cover $f : (S,P) \to (S,P)$ is unmarked if $P$ is the post-critical set for $f$, that is, the set of $f^k(c)$ where $c$ is a critical point for $f$ and $k \geq 1$. If $P$ strictly contains the post-critical set, then we say that $f$ is marked. We can define isotopy for dynamical branched covers in the same way that we defined homotopy; these notions are equivalent since homotopic homeomorphisms of a marked closed surface are isotopic. *Exceptional maps.* We now define exceptional maps of the torus and the sphere (exceptional maps of $S^2$ are examples of Lattès-type maps; see below). We focus here on the unmarked exceptional maps, the marked exceptional maps being obtained from the unmarked ones by adding additional marked points (the latter being not post-critical). While the notion of exceptional maps allows us to give a sharper and more general theorem, the Übertheorem and its proof make sense without the exceptional cases. First, an (unmarked) dynamical branched cover of $T^2$ is exceptional if it has degree greater than 1. All such maps are (unbranched) covering maps. The exceptional maps of the sphere will be defined in terms of hyperelliptic involutions of $T^2$, which we now discuss. A *hyperelliptic involution* $\iota : T^2 \to T^2$ is a homeomorphism of order 2 that acts by $-I$ on $H_1(T^2)$. Every hyperelliptic involution has exactly four fixed points (this follows, for instance, from the Riemann--Hurwitz formula). One way to obtain a hyperelliptic involution is to choose an affine structure and base point on $T^2$ and take the linear map given by $-I$. All other hyperelliptic involutions of $T^2$ are topologically conjugate to this one. Given a hyperelliptic involution $\iota$, we may regard the quotient $T^2/\iota$ as the sphere $S^2$ with a set $P_0$ of four marked points, the images of the fixed points of $\iota$. Any dynamical branched cover $f: T^2 \to T^2$ that commutes with $\iota$ descends to an unmarked dynamical branched cover $\bar f$ of the quotient $(S^2,P_0)$. We refer to any such $f$ as *symmetric* (note that $f$ may permute the fixed points of $\iota$). Any $\bar f$ constructed in this way is what we call an unmarked exceptional dynamical branched cover of $S^2$. If we regard $\iota$ as the linear map given by $-I$, then every linear map of $T^2$ is symmetric, and thus descends to an unmarked exceptional dynamical branched cover of $S^2$. Further, every dynamical branched cover of $T^2$ is homotopic to a linear one, and so every such cover has a corresponding exceptional map of $S^2$. This correspondence between homotopy classes is not a bijection; for instance the identity map of $T^2$ and translation by $1/2$ in one (or both) factors are homotopic maps of $T^2$, but the corresponding maps of $S^2$ are not homotopic (they act differently on the set of marked points). *Strong reduction systems and stable multicurves.* A labeling of a multicurve is a choice of positive real number for each component of the multicurve. If two components of a multicurve bound an annulus disjoint from $P$, then we may obtain a related multicurve by replacing these components with a single component whose label is the sum of the two labels. We may also obtain a related multicurve by deleting any inessential components. We consider labeled multicurves up to the equivalence relation generated by these two relations and homotopy (where homotopies are not allowed to pass through a marked point). We say that a representative of an equivalence class is *standard* if it has the minimal number of connected components. We may say that a labeled multicurve $\Gamma_1$ *contains* a labeled multicurve $\Gamma_2$ if for each component of the standard representative of $\Gamma_2$ there is a component of the standard representative of $\Gamma_1$ that is homotopic and has a label that is at least as large. Given a dynamical branched cover $f : \Sigma \to \Sigma$ and a labeled multicurve $\Gamma$ we obtain a labeled multicurve $f^*(\Gamma)$ whose components are the components of $f^{-1}(\Gamma)$ and whose label at a component $\alpha$ is $1/\deg(f|\alpha)$ times the label of $f(\alpha)$. Finally, we say that a labeled multicurve $\Gamma$ is a strong reduction system for $f$ if the labeled multicurve $f^*(\Gamma)$ contains the labeled multicurve $\Gamma$. If $\Gamma$ is an unlabeled multicurve (or the unlabeled multicurve underlying a labeled one) and $f^*(\Gamma)$ contains $\Gamma$ as unlabeled multicurves, then we say that $\Gamma$ is *stable*. Similarly, if $f^*(\Gamma)$ equals $\Gamma$ as unlabeled multicurves, we say $\Gamma$ is *invariant*. *Statement of Thurston's characterization of rational maps.* We say that a self-map of $S^2$ is *rational* if, under some homeomorphic identification of $S^2$ with $\hat \mathbb C$, the map is equal to a rational map. It is a fact that the rational maps of $\hat \mathbb C$ are exactly the holomorphic maps. Thurston observed that a strong reduction system is an obstruction to holomorphicity for a non-exceptional dynamical branched cover. We will return to this point after the statement of Thurston's characterization of rational maps. Because of Thurston's observation, strong reduction systems for non-exceptional dynamical branched covers are called Thurston obstructions in the literature. Since strong reduction systems are not obstructions to holomorphicity in the exceptional cases, we avoid this terminology. **Theorem 2** (Thurston's characterization of rational maps). *Let $f\colon \Sigma \to \Sigma$ be an unmarked dynamical branched cover where $\Sigma = (S^2,P)$. If $f$ is not exceptional, then $f$ is isotopic to a dynamical branched cover of one of the following two types:* 1. *rational, or* 2. *strongly reducible.* *The two types are exclusive. If $f$ is of type (1), the complex structure is unique up to isotopy.* Our statement of Thurston's characterization is different from, but equivalent to, the usual statement. One difference is that our statement involves a stable multicurve instead of an invariant multicurve. So in terms of finding an obstruction to rationality, our statement is stronger. Another difference is that our statement makes no reference to a matrix or an eigenvalue (the labels on the strong reduction system play the role of the eigenvector). Pilgrim [@pilgrim03] showed that we can use Thurston's characterization of rational maps to say that every unmarked dynamical branched cover of $(S^2,P)$ reduces into pieces that are geometric, that is, rational. This is analogous to the story for surface homeomorphisms, as above. The uniqueness statement in Theorem [Theorem 2](#thm:thu){reference-type="ref" reference="thm:thu"} is often referred to as Thurston rigidity. Hence the common parlance: Thurston's theorem states that a Thurston map has a Thurston obstruction---meaning that the Thurston matrix has a Thurston eigenvalue greater than or equal to 1---or it is Thurston equivalent to a rational map, which moreover satisfies Thurston rigidity. *Topological polynomials, Levy cycles, and Levy--Berstein.* We say that a dynamical branched cover $f : (S^2,P) \to (S^2,P)$ is a topological polynomial if $P$ contains a fixed point $p$ for $f$ and the local degree of $f$ at $p$ is equal to $\deg f$. We may regard the topological polynomial $f$ as a dynamical branched cover of $(\mathbb R^2,P \setminus p)$ (so $p$ plays the role that $\infty$ plays for a polynomial). Examples of topological polynomials include polynomials acting on $\hat \mathbb C$ (equivalently, acting on $\mathbb C$). A multicurve $\{ \gamma_1, \dots, \gamma_k \}$ for a dynamical branched cover $f$ is a Levy cycle if there is a cyclic permutation $\sigma$ of $\{1,\dots,k\}$ so that for every $i$ there is a component $\tilde \gamma_i$ of $f^{-1}(\gamma_i)$ that is homotopic to $\gamma_{\sigma(i)}$ and that maps with degree 1 onto $\gamma_i$. A Levy cycle is degenerate if each $\gamma_i$ bounds an embedded disk $\Delta_i$ so that for every $i$ some component of $f^{-1}(\Delta_i)$ that is homotopic to $\Delta_{\sigma(i)}$ and maps with degree 1 onto $\Delta_i$. By work of Berstein, Hubbard, Levy, Rees, Tan, and Shishikura [@hubbard Theorem 10.3.7] we have the following refinement of Thurston's characterization of rational maps: *a topological polynomial is either rational or it has a degenerate Levy cycle*. In Appendix [8](#sec:levy){reference-type="ref" reference="sec:levy"} we state and prove a strengthening of Levy's theorem, Proposition [Proposition 15](#prop:levy){reference-type="ref" reference="prop:levy"}. Levy cycles are strong reduction systems. However, they are not always Thurston obstructions since they are not always invariant multicurves. It is a feature of our statement of Thurston's characterization of rational maps that Levy cycles suffice to obstruct rationality. Levy and Berstein give a sufficient criterion for a topological polynomial to be rational: each point of $P$ contains a critical point in its forward orbit. This result is known as the Levy--Berstein theorem. In Appendix [8](#sec:levy){reference-type="ref" reference="sec:levy"} we explain how to derive this statement from Proposition [Proposition 15](#prop:levy){reference-type="ref" reference="prop:levy"}. *Portraits, homotopy, and Thurston equivalence.* Above, we defined two dynamical branched covers $f$ and $g$ of $\Sigma$ to be homotopic if there is a homeomorphism $h$ of $\Sigma$ that is homotopic to the identity (rel $P$) and satisfies $f \circ h = g$. We give here an alternate description of homotopic maps and also compare the notion of homotopy to the more commonly used notion of Thurston equivalence. For the former we require the notion of an extended portrait. The portrait of a dynamical branched cover $f$ is the directed, labeled graph whose vertices are the post-critical points of $f$ and where there is an edge labeled $k$ from $p_1$ to $p_2$ if $f$ maps $p_1$ to $p_2$ with local degree $k$. The extended portrait of $f$ is defined in the same way, except that the vertex set consists of the critical points and the post-critical points of $f$. We may say that two dynamical branched covers of $\Sigma$ are homotopic if they are connected by a homotopy of maps $f_t : \Sigma \to \Sigma$ rel $P$ where each $f_t$ is a dynamical branched cover and all of the $f_t$ have the same extended portraits up to labeled, directed graph isomorphism. This notion agrees with the notion of homotopy given earlier. Let $\Sigma = (S,P)$ and $T = (S,Q)$ be two marked surfaces. In the literature, dynamical branched covers $f : \Sigma \to \Sigma$ and $g : T \to T$ are said to be Thurston equivalent (or combinatorially equivalent) if there are homeomorphisms $h_0,h_1 : \Sigma \to T$ that are homotopic (rel $P$) and satisfy $f \circ h_0 = h_1 \circ g$. If, for example, $f$ and $g$ are polynomials with different post-critical sets, then it does not make sense for $f$ and $g$ to be homotopic, but it does make sense for them to be Thurston equivalent. Because of this, Thurston's characterization of rational maps is usually stated in terms of Thurston equivalence. We will not discuss Thurston equivalence in what follows. *Orbifolds and Thurston obstructions.* Let $\hat{\mathbb{N}}$ denote $\mathbb{N} \cup \{\infty\}$. For our purposes, a (2-dimensional) orbifold is a marked surface $(S,P)$ endowed with a function $\nu : P \to \hat{\mathbb{N}}$. We think of the function $\nu$ as a labeling of the points of $P$ by elements of $\hat{\mathbb{N}}$. To a dynamical branched cover $f : (S,P) \to (S,P)$ there is an associated orbifold structure on $(S,P)$---that is, an associated function $\nu$---defined as follows. For each $k$ and each critical point $c$ of $f^k$ with $f^k(c)=p$, we compute the local degree of $f^k$ at $c$. The label $\nu_p$ is the least common multiple of these local degrees over all such choices of $k$ and $c$ (we take the least common multiple of the empty set to be 1, so the label on a non-postcritical point is 1). We provide geometric meaning to this notion in Appendix [7](#sec:orb){reference-type="ref" reference="sec:orb"}. Briefly, the orbifold for $f$ is the minimal orbifold structure for which $f$ is a partial self-cover (in the orbifold sense). Every orbifold falls into one of three categories---spherical, Euclidean, or hyperbolic---according to whether its Euler characteristic is positive, zero, or negative; see the appendix. Thurston's characterization of rational maps can equivalently be stated in terms of orbifolds instead of exceptional maps. There is a particular orbifold $(S^2,P)$, called the $(2,2,2,2)$-orbifold, where $|P|=4$ and $\nu(p)$ is equal to 2 for all $p \in P$. In the appendix, we show that a dynamical branched cover $f$ of $(S^2,P)$ is exceptional if and only if the orbifold for $f$ is the $(2,2,2,2)$-orbifold. As such, we obtain an alternate statement of Thurston's characterization, namely, that if the orbifold for a dynamical branched cover $f$ of $(S^2,P)$ is not the $(2,2,2,2)$-orbifold, then (up to homotopy) $f$ is either holomorphic or it has a strong reduction system. With this in mind, we may think of Thurston's characterization of rational maps as a statement about maps with hyperbolic orbifold, as opposed to a statement about non-exceptional maps. Indeed, a slight weakening of Theorem [Theorem 2](#thm:thu){reference-type="ref" reference="thm:thu"} is that if $f$ has hyperbolic orbifold, then $f$ is rational if and only if it is not strongly reducible (the only weakening is that this version leaves out non-exceptional Euclidean maps). The $(2,2,2,2)$-orbifold is the only Euclidean orbifold with four cone points. Since there are no essential curves on an orbifold with three marked points, there are no strong reduction systems and so by Thurston's characterization all such dynamical branched covers are rational. To summarize, the reasons why Thurston's dichotomy holds for maps with hyperbolic orbifold and non-exceptional maps with Euclidean orbifold are different: in the former case strong reduction systems are obstructions to holomorphicity, and in the latter case there are no strong reduction systems. In the Appendix [7](#sec:orb){reference-type="ref" reference="sec:orb"}, we use orbifolds to explain why strong reduction systems are obstructions to holomorphicity for maps with hyperbolic orbifold. Unlike previous proofs in the literature, our argument makes no reference to Teichmüller space or the pullback map. Instead, it relies on the geometric characterization of the orbifold for a dynamical branched cover that seems to not appear in the literature but was surely known to Thurston. As with the Nielsen--Thurston classification, we can therefore think of Thurston's characterization of rational maps as saying that the only obstruction to holomorphicity is the "obvious" one. ## The Nielsen--Thurston Übertheorem {#subsec:uber} Before stating the Übertheorem, we introduce affine exceptional maps, which will appear in the statement. We think of these as being geometric representatives of homotopy classes of maps, in the same way that pseudo-Anosov and holomorphic maps are. *Affine exceptional maps.* An unmarked affine exceptional map of $T^2$ is simply that: an exceptional map of $T^2$ (in other words, a map of degree greater than 1) that is unmarked and preserves some affine structure on $T^2$. Again, all unmarked exceptional maps of $T^2$ are homotopic to affine exceptional maps. To translate this notion to the sphere case, we again need to go through the hyperelliptic involution. Fix an affine structure on $T^2$ and choose a base point. As above, there is an associated hyperelliptic involution $\iota$, one of whose fixed points is the base point. All linear maps of $T^2$ are symmetric with respect to $\iota$ and hence descend to unmarked dynamical branched covers of $(S^2,P_0)$, the sphere with four marked points. These are examples of unmarked affine exceptional maps of $(S^2,P_0)$ (there are four marked points but, as per the definition of an unmarked map, they are all post-critical). More generally, if we take a linear map of $T^2$ and compose it with a rotation of $T^2$ by $\pi$ in either or both factors, we obtain an affine map of $T^2$ that descends to a map of $(S^2,P_0)$. Any such map is an unmarked affine exceptional map of $S^2$. While $S^2$ carries no affine structure, it does carry many singular affine structures: those arising from affine structures on $T^2$. Affine maps of $S^2$ preserve these singular affine structures. A dynamical branched cover of a torus or a sphere with four marked points is an unmarked affine exceptional map if it is affine with respect to some choice of (singular) affine structure. A marked exceptional dynamical branched cover is affine if the corresponding unmarked map (obtained by forgetting the extra marked points) is affine. In the case of the torus this means forgetting all the marked points, and in the case of the sphere this means forgetting all but four (all of which being post-critical). We emphasize that a marked map is affine if the corresponding unmarked map is actually an affine map, not just homotopic to an affine map. *Statement of the Übertheorem.* After stating two definitions, we will give the statement of the Übertheorem and explain how to derive the previous two theorems as special cases. A dynamical branched cover $f : \Sigma \to \Sigma$ of degree $d$ is *holomorphic* if it is holomorphic with respect to some complex structure on $\Sigma$. And $f$ is *pseudo-Anosov* if there is a pair of transverse measured foliations $(\mathcal F^+,\mathcal F^-)$ on $\Sigma$ that is preserved by $f$ and satisfies $$f^{-1}(\mathcal F^+,\mathcal F^-)=(\lambda\sqrt{d}\, \mathcal F^+,\tfrac{\sqrt{d}}{\lambda}\,\mathcal F^-)$$ for some $\lambda > 1$. The singularities have the same restrictions as in the case of a pseudo-Anosov homeomorphism. **Nielsen--Thurston Übertheorem 1**. *Let $f\colon \Sigma \to \Sigma$ be a dynamical branched cover. Then $f$ is isotopic to a map $\phi$ of one of the following types:* 1. *holomorphic,* 2. *strongly reducible, or* 3. *pseudo-Anosov.* *If $f$ is of type (1) and of type (2), then either $\deg f = 1$ or $f$ is affine exceptional. If $f$ is of type (2) and of type (3) then $f$ is affine exceptional. If $f$ is of type (3) then either $\deg f = 1$ or $f$ is affine exceptional.* *If $f$ is of type (1) and $f$ is a non-exceptional map with $\deg f > 1$, then the associated complex structure is unique up to isotopy. If $f$ of type (3) then the associated pair of measured foliations is unique up to isotopy.* As mentioned, the Übertheorem has the Nielsen--Thurston classification and Thurston's characterization of rational maps as special cases. To see that the Nielsen--Thurston classification is the $\deg f=1$ case, we must use the following three facts about homeomorphisms of surfaces: (1) a holomorphic homeomorphism of a surface of negative Euler characteristic has finite order (and a holomorphic homeomorphism of the torus is homotopic to a map of finite order), (2) a strong reduction system is nothing other than a reduction system, and (3) a pseudo-Anosov dynamical branched cover of degree 1 is a pseudo-Anosov homeomorphism. To obtain Thurston's characterization of rational maps from the Übertheorem, we use the fact that holomorphic maps of $S^2$ are rational. Since we do not require the marked points of $\Sigma$ to be post-critical, the Übertheorem also implies the generalization of Thurston's characterization due to Buff--Cui--Tan, which extends the theorem to the case of marked dynamical branched covers [@BCT Theorem 2.1]. While we are not aware of any theorems in the literature that combine the exceptional cases of Thurston's characterization of rational maps into the classical statement, a result of Bartholdi--Dudko does give an analogue of the Übertheorem for the exceptional cases themselves [@BD Theorem A]. *Extensions of the Übertheorem: non-orientable surfaces, orientation reversing maps, equivariant maps.* In Appendix [9](#sec:ext){reference-type="ref" reference="sec:ext"}, we explain how our argument for the Nielsen--Thurston Übertheorem applies in even further generality. Specifically, we give extensions to the cases of non-orientable surfaces and the cases of orientation-reversing dynamical branched covers. We also give a version of the Übertheorem for equivariant dynamical branched covers. *The Bers strategy.* As in the Bers proof of the Nielsen--Thurston classification, we prove the Übertheorem by appealing to the action of a dynamical branched cover $f : \Sigma \to \Sigma$ on the Teichmüller space $\mathop{\mathrm{Teich}}(\Sigma)$. A point in $\mathop{\mathrm{Teich}}(\Sigma)$ is an equivalence class of complex structures on $\Sigma$. By pulling back complex structures through $f$, we obtain Thurston's pullback map $\sigma_f : \mathop{\mathrm{Teich}}(\Sigma) \to \mathop{\mathrm{Teich}}(\Sigma)$. (In the original Bers proof, it makes sense to consider either pullback or push-forward, but for covers of higher degree only pullback makes sense in general.) Following Bers, we consider the translation length $\tau$ of $\sigma_f$, that is, the infimum of the distances $d(X,\sigma_f(X))$ over all $X$ in $\mathop{\mathrm{Teich}}(\Sigma)$. There are three cases for $\tau$: it can be 0 and realized, not realized, or nonzero and realized. In the first case, $\sigma_f$ has a fixed point, which means that $f$ is holomorphic. In the second case, we show that $f$ has a reduction system. As in the original Bers proof, this is derived as a consequence of the Mumford compactness criterion. When $\deg f > 1$ we augment the original Bers proof to show that there is an orbit for $\sigma_f$ that goes to infinity (towards the reduction system); this is the content of Proposition [Proposition 6](#prop:metricorbit){reference-type="ref" reference="prop:metricorbit"}. Then, assuming the reduction system is not strong, we show that this orbit is also repelled from infinity, a contradiction. Finally, in the third case, we show that $\sigma_f$ preserves a geodesic ray in $\mathop{\mathrm{Teich}}(\Sigma)$. This phenonenon, which does not seem to have been observed before for $\deg f > 1$, is elucidated in Proposition [Proposition 7](#prop:metricray){reference-type="ref" reference="prop:metricray"}. We show that this only occurs in the exceptional cases and the cases where $\deg f = 1$. Perhaps unexpectedly, the usual discussion for the $\deg f =1$ applies in this more general case. As in the original Bers proof, we then show that a geodesic ray corresponds to a pair of transverse measured foliations, and the translation distance along the ray corresponds to a stretch factor $\lambda$, thus implying that $f$ is pseudo-Anosov. *Examples of non-exclusivity.* Figure [\[fig:venn\]](#fig:venn){reference-type="ref" reference="fig:venn"} gives examples of dynamical branched covers of $T^2$ of all the different types allowed by the Übertheorem when the cover is exceptional and the degree is greater than 1: holomorphic, holomorphic and strongly reducible, strongly reducible, strongly reducible and Anosov, and Anosov. (Here we say "Anosov" instead of "pseudo-Anosov" since the underlying surface is a torus, and hence the corresponding foliations have no singularities.) For the first three examples, we require that $d$ be a perfect square. As demanded by the Übertheorem, the strongly reducible and Anosov example fails to be Anosov when $d=1$. *Comparison to Douady--Hubbard.* The original proof of Thurston's characterization of rational maps is detailed in Douady--Hubbard's paper [@DH] and Hubbard's book [@hubbard]. Our approach is the same in spirit, but differs in the following ways: 1. we appeal to Teichmüller's theorems instead of working with the derivative of the pullback map (our application of Teichmüller's uniqueness theorem is morally equivalent to Lemma 1 of Douady--Hubbard), 2. we avoid explicit mention of hyperbolic surfaces, staying entirely in the category of Riemann surfaces, 3. we give a simplified treatment of the combinatorial topological step (Proposition [Proposition 3](#prop:stable){reference-type="ref" reference="prop:stable"}) and, like Buff--Cui--Tan, we directly address the case where there are marked points that are not post-critical (the cost of our simplification is the loss of sharpness), 4. we isolate in Section [4](#sec:synthetic){reference-type="ref" reference="sec:synthetic"} the basic properties of metric spaces we use, and 5. we clarify the role that orbifolds play in the proof that strong reduction systems are obstructions to holomorphicity in the case of a non-exceptional map. Another feature of our exposition is that we treat many cases of Thurston's characterization of rational maps that were not addressed before, namely, the cases where $S=T^2$, where $S$ is non-orientable, where $f$ reverses orientation, and where $f$ is equivariant with respect to a finite group action. The arguments of Douady--Hubbard could similarly be extended to prove these additional cases. One other philosophical difference between our approach and the prevailing literature is that we make no mention of Thurston equivalence. To wit, instead of considering maps up to homotopy and conjugacy, we only consider maps up to homotopy. This point of view has long been championed by Kevin Pilgrim. We emphasize that there is a general translation between the Douady--Hubbard proof and our proof; and in the text that follows we have indicated the points of similarity. We hope that our exposition will appeal to those already familiar with the Bers proof of the Nielsen--Thurston classification theorem, and will also clarify the relationship between that theorem and Thurston's characterization of rational maps. Work in progress by Drach--Reinke--Schleicher [@DS] gives a new approach to the four theorems of Thurston involving the pullback map (two of which are the ones discussed in this paper). Their approach also uses Teichmüller's theorems instead of the derivative of the pullback map. *Lattès maps and Euclidean maps.* The exceptional maps that we consider overlap with several other notions in the literature, and the terminology is used differently by different authors. A Lattès map is a holomorphic branched cover $S^2\rightarrow S^2$ that is the finite quotient of a holomorphic affine map of $T^2$. Milnor gives a thorough survey and further characterization of Lattès maps [@milnor]. A Lattès-type map is a (not-necessarily-holomorphic) quotient of an affine map of $T^2$ (this is not typically given as the definition of Lattès-type, but Bonk--Meyer prove that it is equivalent [@BM Theorem 1.2]). The exceptional maps we consider are Lattès-type maps where the finite quotient is by the hyperelliptic involution. (Milnor also defines finite quotients of affine maps, which have a similar definition as a Lattès map, except with the torus possibly replaced by a cylinder; these types of maps do not arise in this paper.) Cannon--Floyd--Parry--Pilgrim consider Euclidean maps, which they define as dynamical branched covers of $S^2$ with at most four post-critical points, none of which are critical, such that every critical point is simple (local degree two) [@CFPP]. These are precisely the dynamical branched covers with Euclidean orbifold and at least four (hence exactly four) post-critical points. Our exceptional maps of $S^2$ are the Euclidean maps of Cannon--Floyd--Parry--Pilgrim. Cannon--Floyd--Parry--Pilgrim also introduce and study nearly Euclidean maps, which are branched covers of $S^2$ with exactly four post-critical points and where each critical point is simple (such as the rabbit polynomial). ## Overview of the paper We divide the proof of the Übertheorem into five parts, each with their own section. The first three of these sections isolate three different aspects of the proof, namely, combinatorial topology, Teichmüller theory, and metric space theory. Sections [5](#sec:proof){reference-type="ref" reference="sec:proof"} and [6](#sec:torus){reference-type="ref" reference="sec:torus"} tie these together to prove the theorem for the non-exceptional and exceptional cases, respectively. While the exceptional cases are handled separately, we emphasize that the proof is essentially the same; the main content is already contained in the non-exceptional case, while the exceptional case requires a few extra technical details. In Section [2](#sec:rh){reference-type="ref" reference="sec:rh"}, we give a combinatorial topological statement, Proposition [Proposition 3](#prop:stable){reference-type="ref" reference="prop:stable"}. It says that, under certain hypotheses on $f$, at most 3 marked points have the property that all of their iterated preimages under $f$ are critical or marked. In Section [3](#sec:pullback){reference-type="ref" reference="sec:pullback"} we prove Proposition [Proposition 4](#prop:pullback){reference-type="ref" reference="prop:pullback"}, which is about the pullback map on Teichmüller space $\sigma_f$. The proposition states that if $\deg f > 1$ and if $f$ is not exceptional, then some iterate of $\sigma_f$ is weakly contracting, meaning that it decreases the distance between all pairs of points. The proof uses Proposition [Proposition 3](#prop:stable){reference-type="ref" reference="prop:stable"}. In Section [4](#sec:synthetic){reference-type="ref" reference="sec:synthetic"} we prove three statements about metric spaces, namely, Propositions [Proposition 5](#prop:metricmumford){reference-type="ref" reference="prop:metricmumford"}, [Proposition 6](#prop:metricorbit){reference-type="ref" reference="prop:metricorbit"}, and [Proposition 7](#prop:metricray){reference-type="ref" reference="prop:metricray"}. The purpose is to isolate the parts of the proof of the Übertheorem that only use the theory of metric spaces and not the theory of Teichmüller space. In Section [5](#sec:proof){reference-type="ref" reference="sec:proof"} we follow the Bers proof of the Nielsen--Thurston classification in order to prove the Übertheorem in the non-exceptional cases. Our argument follows the Bers strategy described above. Again, the key idea is to consider the translation length $\tau$ of the pullback map on Teichmüller space and separately investigate the three cases where $\tau$ is 0 and realized, nonzero and realized, and not realized. These three cases exactly correspond to the three cases in the conclusion of the Übertheorem. Finally in Section [6](#sec:torus){reference-type="ref" reference="sec:torus"}, we prove the Übertheorem in the exceptional cases. We prove that in these cases the associated Teichmüller space decomposes as a product in a natural way, and apply the ideas of Section [5](#sec:proof){reference-type="ref" reference="sec:proof"} to the action of a dynamical branched cover on the product structure. Among maps of degree greater than 1, the exceptional $f$ that preserve a horizontal slice are exactly the ones whose associated pullback maps fail to have weakly contracting orbits. This is the reason why exceptional maps require separate consideration. There are three appendices. Appendix [7](#sec:orb){reference-type="ref" reference="sec:orb"} gives a direct proof of the fact that strong reduction systems are obstructions to holomorphicity for dynamical branched covers with hyperbolic orbifold. Along the way, we clarify the geometric meaning of the orbifold structure for a dynamical branched cover. In Appendix [8](#sec:levy){reference-type="ref" reference="sec:levy"} we explain how Thurston's characterization of rational maps specializes in the case of topological polynomials. Appendix [9](#sec:ext){reference-type="ref" reference="sec:ext"} describes how our arguments apply to give generalizations of the Übertheorem to the cases of equivariant dynamical branched covers, dynamical branched covers of non-orientable surfaces, and orientation-reversing dynamical branched covers. *Acknowledgments.* We would like to thank Wolf Jung, Jeremy Kahn, Sanghoon Kwak, Yair Minsky, Insung Park, Kevin Pilgrim, Dierk Schleicher, Roberta Shapiro, and Sam Taylor for helpful comments and conversations. The second author is grateful to the Georgia Institute of Technology for supporting this work. The third author is grateful to the Mathematical Sciences Research Institute for a stimulating work environment. # Stable marked points {#sec:rh} The goal of this section is to prove Proposition [Proposition 3](#prop:stable){reference-type="ref" reference="prop:stable"} below. This is the main ingredient in the proof of Proposition [Proposition 4](#prop:pullback){reference-type="ref" reference="prop:pullback"} in Section [3](#sec:pullback){reference-type="ref" reference="sec:pullback"}. A refined version of this proposition is given by Lemma 2 of Douady--Hubbard. To state our proposition, we require the notion of stability. *Stability of marked points.* Let $\Sigma = (S,P)$ and let $f : \Sigma \to \Sigma$ be a dynamical branched cover. We say that $p \in P$ is stable if $f^{-1}(p) \subseteq P \cup \mathop{\mathrm{Crit}}(f)$. We say that $p$ is infinitely stable if $f^{-k}(p) \subseteq P \cup \mathop{\mathrm{Crit}}(f^k)$ for all $k \geq 0$. If $f$ is exceptional, then each post-critical point is infinitely stable. The following proposition is a sort of converse to this statement. **Proposition 3**. *Let $\Sigma = (S^2,P)$, and let $f : \Sigma \to \Sigma$ be a dynamical branched cover of degree $d > 1$. If $f$ is not exceptional, then $f$ has fewer than 4 infinitely stable marked points.* *Proof.* Let $Q \subseteq P$ be the set of infinitely stable points for $f$, and suppose that $|Q|\geq 4$. We will show that $f$ is exceptional. Let $\tilde{Q} = f^{-1}(Q)$, and let $C = \mathop{\mathrm{Crit}}(f) \cap \tilde Q$. If a non-critical marked point maps to an infinitely stable marked point, then it itself is infinitely stable, that is, $\tilde{Q}\subseteq Q\cup C$. In particular, $$|\tilde{Q}| \leq |C| + |Q|.$$ Since (counting with multiplicity) a critical point of degree $k$ accounts for $k$ pre-images of a point in $Q$, we also have $$|\tilde Q| = |Q|d - \sum_{c\in C} \bigl( \deg_f(c) - 1\bigr).$$ By the Riemann--Hurwitz formula and the preceding equality and inequality we have $$\begin{aligned} 2d-2\geq \sum_{c\in C}\bigl(\deg_f(c)-1\bigr) = |Q|d-|\tilde Q| \geq |Q|d-|Q|-|C| = |Q|(d-1)-|C|.\end{aligned}$$ We conclude that $|C| \geq (|Q|-2)(d-1)$. Since $d > 1$ and a branched cover $S^2 \to S^2$ of degree $d$ has at most $2d-2$ critical points, it follows that $|Q|\leq 4$. By our earlier assumption that $|Q| \geq 4$, we conclude that $|Q|=4$. Replacing $|Q|$ with 4 in the above, we conclude that $|C|=2d-2$, so $C$ is equal to all of $\mathop{\mathrm{Crit}}(f)$ and each critical point is simple. Moreover, the inequality must be an equality, so in particular $|\tilde{Q}|=|C|+|Q|$, which means that $C$ is disjoint from $Q$ and $Q\subseteq \tilde{Q}$. This means that $f(Q)\subseteq Q$. Since $Q$ contains all the critical values of $f$, it follows that $Q$ contains the post-critical set. Because the preimage of each point of $Q$ is either in $Q$ or in $C$ it follows that every point in $Q$ must be post-critical. Since the critical points are all simple, the ramification index at each point of $Q$ is 2. In other words, the orbifold for $f$ is the $(2,2,2,2)$-orbifold. As in the introduction, this is equivalent to the statement that $f$ is exceptional, as desired. ◻ Similar arguments can be used to derive a stronger conclusion if $P$ is the post-critical set: the second iterate $f^2$ must have fewer than $4$ stable marked points, and if $f$ is a topological polynomial then $f$ itself must have fewer than $4$ stable marked points. Combining this with the proof of Proposition [Proposition 4](#prop:pullback){reference-type="ref" reference="prop:pullback"} below, it follows that $\sigma_f^2$ is weakly contracting whenever $P$ is the post-critical set, and $\sigma_f$ is weakly contracting in this case if $f$ is a topological polynomial. # Pullback is a weak contraction {#sec:pullback} The goal of this section is to prove Proposition [Proposition 4](#prop:pullback){reference-type="ref" reference="prop:pullback"}, which states that the pullback map is non-expanding, and in many cases weakly contracting. A refinement of this statement is given in Proposition 3.3 of Douady--Hubbard. Both of these results are in concert with a theorem of Royden, which says that analytic maps of Teichmüller space are weak contractions [@royden]. As in the work of Douady--Hubbard, we will neither use the analyticity of the pullback map nor the Royden result. We begin with the requisite definitions; see [@primer Chapter 11] for more details. *Teichmüller space and the pullback map.* Let $\Sigma = (S,P)$ and let $f : \Sigma \to \Sigma$ be a dynamical branched cover. The Teichmüller space $\mathop{\mathrm{Teich}}(\Sigma)$ is the set of complex structures on $\Sigma$ up to isotopy. More specifically, a complex structure on $\Sigma$ is a complex structure on $S$ and two complex structures $X$ and $Y$ on $\Sigma$ are equivalent if there is a isomorphism $h : X \to Y$ that is isotopic to the identity (here we insist that $h(P)=P$ and that isotopies fix $P$). The pullback map associated to $f$ is the map $$\sigma_f : \mathop{\mathrm{Teich}}(\Sigma) \to \mathop{\mathrm{Teich}}(\Sigma)$$ defined by pulling back complex structures through $f$. *The Teichmüller metric and Teichmüller's theorems.* The Teichmüller metric on $\mathop{\mathrm{Teich}}(\Sigma)$ is defined as follows. For a map $h$ between Riemann surfaces, let $K(h)$ denote the quasi-conformal dilatation. Given $X,Y \in \mathop{\mathrm{Teich}}(\Sigma)$ we set $$K(X,Y) = \inf \{K(h) \mid h : X \to Y \text{ and } h \sim \textrm{id} \}$$ and $$d(X,Y) = \frac{1}{2} \log K(X,Y).$$ Teichmüller's existence theorem gives that the infimum is a minimum, that is, there is a map $h$, called the Teichmüller map, that realizes the infimum [@primer Theorem 11.8]. Teichmuller's uniqueness theorem states that the minimizing map $h$ is unique [@primer Theorem 11.9]. *Teichmüller maps and foliations.* Teichmüller's existence theorem further gives an explicit description of the Teichmüller map $h$. Usually, this description is phrased in terms of quadratic differentials. We avoid this terminology here. For the description of $h$, we need the fact that a pair of transverse measured foliations $(\mathcal F^+,\mathcal F^-)$ on $\Sigma$ induces a complex structure on $\Sigma$; in other words, $(\mathcal F^+,\mathcal F^-)$ represents a point in $\mathop{\mathrm{Teich}}(\Sigma)$. Indeed, $(\mathcal F^+,\mathcal F^-)$ gives a Euclidean structure on $\Sigma$ away from the singularities, and hence (orientation-preserving) charts to the complex plane, well-defined up to rotation. If the charts identify segments of the leaves of $\mathcal F^+$ and $\mathcal F^-$ with horizontal and vertical line segments, then they are called natural coordinates for $(\mathcal F^+,\mathcal F^-)$. These are well defined up to translation in $\mathbb C$. Now, Teichmüller's description of the Teichmüller map $h$ is that there is a pair of measured foliations $(\mathcal F^+,\mathcal F^-)$ on $\Sigma$ so that, setting $\lambda=\sqrt{K(h)}$, we have - $(\mathcal F^+,\mathcal F^-)$ induces $X$, - $(\lambda \,\mathcal F^+,\tfrac{1}{\lambda}\,\mathcal F^-)$ induces $Y$, and - in natural coordinates with respect to these two pairs of foliations, $h$ is given by $$\left(\begin{array}{cc} \lambda & 0 \\ 0 & 1/\lambda \end{array}\right)$$ One way to rephrase Teichmüller's theorems is that every geodesic ray in $\mathop{\mathrm{Teich}}(\Sigma)$ is determined by a pair of transverse measured foliations $(\mathcal F^+,\mathcal F^-)$ on $\Sigma$, and the ray is obtained by multiplying $\mathcal F^+$ by $\lambda \geq 1$ and $\mathcal F^-$ by $1/\lambda$. The measured foliations $\mathcal F^+$ and $\mathcal F^-$ must have singularities if $\chi(S) \neq 0$. If there are any 1-pronged singularities, they must be at points of $P$, for otherwise $K(h)$ is not minimal. *The pullback map is non-expanding or weakly contracting.* Let $(T,d)$ be a metric space and let $\sigma : T \to T$. We say that $\sigma$ is non-expanding if $$d(\sigma(x),\sigma(y)) \leq d(x,y)$$ for all $x,y \in T$. We say that $\sigma$ is weakly contracting if $$d(\sigma(x),\sigma(y)) < d(x,y)$$ for all distinct $x,y \in T$. **Proposition 4**. *Let $\Sigma = (S,P)$, and let $f: \Sigma \to \Sigma$ be a dynamical branched cover.* 1. *[\[nonexp\]]{#nonexp label="nonexp"} The pullback map $\sigma_f$ is non-expanding.* 2. *If $f$ is not exceptional and $\deg(f) > 1$, then $\sigma_f^k$ is weakly contracting for some $k \geq 1$.* *Idea of the proof and pseudo-Teichmüller maps.* Before proving Proposition [Proposition 4](#prop:pullback){reference-type="ref" reference="prop:pullback"}, we explain the main observation used in the proof. Let $f : \Sigma \to \Sigma$ be a dynamical branched cover, let $X,Y \in \mathop{\mathrm{Teich}}(\Sigma)$, and let $h : X \to Y$ be a Teichmüller mapping. Since $h$ is isotopic to the identity, there is a unique map $h^f$, which we call the lifted map, that is isotopic to the identity and so that the following diagram commutes: $$\begin{tikzcd} \sigma_f(X) \arrow[r,"h^f"] \arrow[d,"f"] & \sigma_f(Y) \arrow[d,"f"] \\ X \arrow[r,"h"] & Y \end{tikzcd}$$ We can incorporate the pair of foliations $(\mathcal F^+,\mathcal F^-)$ into the diagram: $$\begin{tikzcd} (\sigma_f(X),f^*(\mathcal F^+,\mathcal F^-)) \arrow[r,"h^f"] \arrow[d,"f"] & (\sigma_f(Y),f^*(\lambda\,\mathcal F^+,\tfrac{1}{\lambda}\,\mathcal F^-)) \arrow[d,"f"] \\ (X,(\mathcal F^+,\mathcal F^-)) \arrow[r,"h"] & (Y,(\lambda\,\mathcal F^+,\tfrac{1}{\lambda}\,\mathcal F^-)) \end{tikzcd}$$ As the pullback $f^*(\lambda\,\mathcal F^+,\tfrac{1}{\lambda}\,\mathcal F^-)$ on the top right of the diagram is equal to $(\lambda\, f^*(\mathcal F^+),\tfrac{1}{\lambda}\,f^*(\mathcal F^-))$ the map $h^f$ has the same quasiconformal dilatation as $h$. It locally behaves like a Teichmüller map whose associated foliations are the pullbacks of the foliations for $h$. However, $h^f$ need not be a Teichmüller map, because it is possible that these foliations have 1-pronged singularities at unmarked preimages of points of $P$. In general, if a map is obtained from a Teichmüller map by forgetting a marked point at one of the associated 1-pronged singularities, we call that map a pseudo-Teichmüller mapping. The key point is that pseudo-Teichmüller mappings are not themselves Teichmüller mappings. *Proof of Proposition [Proposition 4](#prop:pullback){reference-type="ref" reference="prop:pullback"}.* Let $X,Y \in \mathop{\mathrm{Teich}}(\Sigma)$. Let $h : X \to Y$ be the Teichmüller map, which exists by Teichmüller's existence theorem. As above, the lifted map $$h^f : \sigma_f(X) \to \sigma_f(Y)$$ is a Teichmüller map or pseudo-Teichmüller map with the same quasi-conformal dilatation as $h$. The first statement follows now from the definition of the Teichmüller metric. Suppose now that $f$ is not exceptional and $\deg(f) > 1$. In this case $S=S^2$ and $f$ is not the quotient of an affine map by the hyperelliptic involution. Since $S=S^2$, the foliations associated to $h$ must have at least four 1-pronged singularities at points of $P$. By Proposition [Proposition 3](#prop:stable){reference-type="ref" reference="prop:stable"}, there is a $k$ so that at least one of these four points of $P$ fails to be stable for $f^k$. Therefore the pulled back map $$h^{f^k} : \sigma_f^k(X) \to \sigma_f^k(Y)$$ is a pseudo-Teichmüller map and not a Teichmüller map. The second statement follows from Teichmüller's uniqueness theorem and the definition of the Teichmüller metric. ◻ As mentioned, the analogue of Proposition [Proposition 4](#prop:pullback){reference-type="ref" reference="prop:pullback"} in Douady--Hubbard is their Proposition 3.3. The key to that proof is their Lemma 1, which is the analogue of our observation that the pullback of a Teichmüller map is a pseudo-Teichmüller map. There they observe that the pullback of a Beltrami differential $q$ has norm greater than or equal to that of $q$, and that we have equality if and only if the preimages of the images of the poles of $p$ are critical or post-critical. Through the duality between equivalence classes of Beltrami differentials (tangent vectors for Teichmüller space) and holomorphic quadratic differentials (cotangent vectors for Teichmüller space), we see that the two arguments are essentially the same. Indeed, a Beltrami differential can be thought of as an ellipse field, and there is a natural ellipse field associated to a Teichmüller map. In this way, our argument using Teichmüller's theorems recovers the Douady--Hubbard statement that the derivative of (an iterate of) the pullback map is contracting [@DH Proposition 3.3]. # Synthetic Nielsen--Thurston theory {#sec:synthetic} By a synthetic Nielsen--Thurston package, we mean a collection $(T,P,\phi,\sigma)$, where 1. $T$ is a uniquely geodesic metric space where all maximal geodesics are bi-infinite, 2. $P$ is a group acting properly discontinuously on $T$, 3. $\phi : P \dasharrow P$ is a virtual endomorphism, and 4. $\sigma : T \to T$ is a function that is intertwined with $\phi$ and is non-expanding. Here a *virtual endomorphism* $\phi : P \dasharrow P$ is a homomorphism $L \to P$ where $L$ is a finite-index subgroup of $P$. We say $\sigma$ is *intertwined* with $\phi$ if $\sigma(g\cdot x) = \phi(g)\cdot \sigma(x)$ for all $x \in T$ and $g \in L$. In this paper, the only synthetic Nielsen--Thurston packages we will consider are ones where the space $T$ is $\mathop{\mathrm{Teich}}(\Sigma)$ for some marked surface $\Sigma$, where $P$ is the pure mapping class group $\mathop{\mathrm{PMod}}(\Sigma)$, where $\phi$ is the lifting homomorphism associated to a given dynamical branched cover $f : \Sigma \to \Sigma$ (see Section [5](#sec:proof){reference-type="ref" reference="sec:proof"}), and where $\sigma$ is the pullback map $\sigma_f$. Our axiomatic approach is meant to clarify which properties of these objects are essential for the argument. We will write $\tau_\sigma(X)$ for $d(X,\sigma(X))$ and $\tau_\sigma$ for the translation distance, which is the infimum of $\tau_\sigma(X)$ over $X \in T$: $$\tau_\sigma = \inf_{X \in T} \tau_\sigma(X).$$ In this section we prove three propositions about synthetic Nielsen--Thurston packages, Propositions [Proposition 5](#prop:metricmumford){reference-type="ref" reference="prop:metricmumford"}, [Proposition 6](#prop:metricorbit){reference-type="ref" reference="prop:metricorbit"}, and [Proposition 7](#prop:metricray){reference-type="ref" reference="prop:metricray"}. These will be used in the proof of the Nielsen--Thurston Übertheorem to address the cases where 1. $\tau_\sigma$ is not realized and $\sigma_f$ is non-expanding, 2. $\tau_\sigma$ is not realized and $\sigma_f$ is weakly contracting, and 3. $\tau_\sigma$ is realized and $\sigma_f$ is non-expanding. In the proof of the Übertheorem in Section [5](#sec:proof){reference-type="ref" reference="sec:proof"}, these appear in Case 2 ($\deg f=1$ subcase), Case 2 ($\deg f>1$ subcase), and Case 3, respectively. *Translation distances not realized.* The following proposition is a slight generalization of one of the steps in the Bers proof of the Nielsen--Thurston classification [@primer Section 13.6.1, Step 1]. In that classical setting, the map $\phi$ is simply the inner automorphism of the mapping class group corresponding to $f^{-1}$ (this makes sense because the lift of a homeomorphism $g$ under a homeomorphism $f$ is $f^{-1}gf$). **Proposition 5**. *Let $(T,P,\phi,\sigma)$ be a synthetic Nielsen--Thurston package where $\tau_\sigma$ is not realized. If $\{X_n\}$ is a sequence in $T$ with $$\tau_\sigma(X_n) \to \tau_\sigma,$$ then the image of $\{X_n\}$ in $T/P$ is not contained in any compact set.* *Proof.* Suppose to the contrary that the image of $\{X_n\}$ has compact closure. We will find a point $Z$ so that $\tau_\sigma(Z) \leq \tau_\sigma$, contrary to the assumption that $\tau_\sigma$ is not realized. Let $L$ be the domain of $\phi$, and let $\pi : T \to T/L$ be the quotient map. Since $L$ has finite index in $P$, the map $T/L \to T/P$ is finite-to-one. Thus $\{\pi(X_n)\}$ has a limit point, which is $\pi(Y)$ for some $Y \in T$. The desired $Z$ will be in the $L$-orbit of $Y$. To find this $Z$, we define $F \colon T/L \to[0,\infty)$ by $$F(\pi(X)) = \min_{g \in L} \tau_\sigma(g \cdot X).$$ We will prove below that $F$ is well defined, which implies two further statements: 1. $F$ is continuous, and 2. there exists $g \in L$ with $\tau_\sigma(g \cdot Y) \leq \tau_\sigma$ $\Longleftrightarrow$ $F(\pi(Y)) \leq \tau_\sigma$. Moreover, the last inequality follows from the continuity of $F$ and the definition of $Y$. It remains to prove that $F$ is well defined. To this end, we give another description of $F$. Using the definition of $\tau_\sigma(g \cdot X)$, the assumption that $\sigma$ is intertwined with $\phi$, and the fact that elements of $L$ act by isometries on $\mathop{\mathrm{Teich}}(\Sigma)$, we have $$\tau_\sigma(g\cdot X) = d\bigl(g\cdot X,\sigma(g\cdot X)\bigr) = d\bigl(g\cdot X,\phi(g)\cdot \sigma(X)\bigr) = d\bigl(X,g^{-1}\phi(g)\cdot \sigma(X)\bigr).$$ From this we obtain the following description of $F$: $$F(\pi(X)) = \min_{g \in L} \, d(X,g^{-1}\phi(g) \cdot \sigma(X)).$$ The set of points $g^{-1}\phi(g)\cdot \sigma(X)$ is a subset of the $P$-orbit of $\sigma(X)$. Since $P$ acts properly discontinuously, the given minimum exists, which is to say $F$ is well defined. ◻ We remark that the proof of Proposition [Proposition 5](#prop:metricmumford){reference-type="ref" reference="prop:metricmumford"} does not use the non-expanding property of $\sigma$. *Weakly contracting orbits.* The next proposition is essentially the same as Proposition 5.1 of Douady--Hubbard. We begin by giving the definition of a weakly contracting orbit. Given a self-map $\sigma$ of a metric space $T$ and an orbit $\mathcal{O} = (X_i)_{i=1}^\infty$ where $X_i=\sigma^i(X)$, we say that $\mathcal{O}$ is weakly contracting if the sequence $d(X_i,X_{i+1})$ is strictly decreasing (in particular, no two $X_i$ are equal). Since $d(X_{i+1},X_{i+2})$ is equal to $d(\sigma(X_i),\sigma(X_{i+1}))$, it follows that all orbits of a weakly contracting map are weakly contracting. It also follows from the definitions that if all orbits of a map are weakly contracting, then the map has no fixed points. **Proposition 6**. *Let $(T,P,\phi,\sigma)$ be a synthetic Nielsen--Thurston package. If every orbit for $\sigma$ is weakly contracting, then every orbit leaves every compact subset of $T/P$.* Note that Proposition [Proposition 6](#prop:metricorbit){reference-type="ref" reference="prop:metricorbit"} applies whenever $\sigma$ is weakly contracting and $\tau_\sigma$ is not realized, since having a fixed point implies that $\tau_\sigma$ is realized (and is equal to 0). *Proof of Proposition [Proposition 6](#prop:metricorbit){reference-type="ref" reference="prop:metricorbit"}.* Let $\mathcal{O}=(X_i)$ be a $\sigma$-orbit. Suppose for the sake of contradiction that the image of $\mathcal{O}$ in $T/P$ has compact closure. In order to obtain a contradiction, we will find another $\sigma$-orbit $(Y_i)$ whose first three terms satisfy $$d(Y_0,Y_1) = d(Y_1,Y_2).$$ Here is why this is a contradiction. Since $Y_i$ is a $\sigma$-orbit, the above equality is equivalent to $$d(Y,\sigma(Y))=d(\sigma(Y),\sigma^2(Y))$$ where $Y=Y_0$; equivalently, $\tau_\sigma(Y) = \tau_\sigma(\sigma(Y))$. By the weakly contracting property of $\sigma$, this implies that $d(Y,\sigma(Y))=0$, which is to say that $\sigma$ has a fixed point, contrary to the assumption that all orbits of $\sigma$ are weakly contracting. To find such a $Y=Y_0$, our strategy is similar to the one used in the proof of Proposition [Proposition 5](#prop:metricmumford){reference-type="ref" reference="prop:metricmumford"}. Because we need to analyze three consecutive points in an orbit, instead of just two, we need to replace $T/L$ with a further finite cover of $T/P$. To this end, let $$L_2 = \{g \in P \mid \phi^2(g)\text{ is defined}\}.$$ The subgroup $L_2$ has finite index in $P$. Let $\pi$ be the quotient map $$\pi : T \to T/L_2.$$ Since $L_2$ has finite index, the sequence $\{\pi(X_i)\}$ has a limit point, which is $\pi(Y)$ for some $Y \in T$. We will show that, up to replacing $Y$ with another point in its $L_2$-orbit, $\tau_\sigma(Y) = \tau_\sigma(\sigma(Y))$. First we define a function $F$, analogous to the one in the proof of Proposition [Proposition 5](#prop:metricmumford){reference-type="ref" reference="prop:metricmumford"}. Since the sequence $\tau_\sigma(X_i) = d(X_i,X_{i+1})$ is non-negative and strictly decreasing (by the weakly contracting assumption), it converges to some $\delta\geq 0$. We define $F\colon T/L_2 \to [0,\infty)$ by $$F\bigl(\pi(X)\bigr) = \min_{g\in L_2} \left\{ \bigl|\tau_\sigma(g \cdot X) - \delta\bigr| \ + \ \bigl|\tau_\sigma(\sigma(g \cdot X)) - \delta\bigr| \right\}.$$ Assuming $F$ is well defined we have $$F\bigl(\pi(X_i)\bigr) \leq \left|\tau_\sigma(X_i) -\delta\right| \ + \ \left|\tau_\sigma(X_{i+1}) -\delta \right|$$ for each $i$. It follows that $F\bigl(\pi(X_i)\bigr) \to 0$. We now use $F$ to analyze $Y$. Again assuming $F$ is well defined, it is continuous. Therefore, the statement $F\bigl(\pi(X_i)\bigr) \to 0$ implies that $F\bigl(\pi( Y) \bigr)=0$. Thus, after possibly replacing $Y$ with a different point in its $L_2$-orbit, we have $$\bigl|\tau_\sigma(Y)-\delta\bigr|\ +\ \bigl|\tau_\sigma(\sigma(Y))-\delta\bigr| = 0.$$ It follows that $\tau_\sigma(Y)$ and $\tau_\sigma(\sigma(Y))$ are both equal to $\delta$, and in particular are equal to each other, as desired. It remains to prove that $F$ is well defined. Similar to the proof of Proposition [Proposition 5](#prop:metricmumford){reference-type="ref" reference="prop:metricmumford"}, the intertwining with $\phi$ gives that $$F\bigl(\pi(X)\bigr) = \min_{g\in L_2}\left\{ \bigl|d\bigl(X,g^{-1}\phi(g)\cdot\sigma(X)\bigr) - \delta\bigr| + \bigl|d\bigl(\sigma(X),\phi(g)^{-1}\phi^2(g)\cdot\sigma^2(X)\bigr) - \delta\bigr| \right\}.$$ Again, since the action of $P$ on $T$ is properly discontinuous, the same is true for $L_2$. Thus, the minimum exists and $F$ is well defined. ◻ *Forward translations along rays.* The next proposition is a version of one of the steps of the Bers proof of the Nielsen--Thurston classification [@primer Section 13.6.4, Step 1]. Here we generalize to the case where $\tau_\sigma$ is non-expanding. The proof is almost unchanged. We begin by defining forward translation along a ray. Let $\gamma$ be a ray in a metric space $T$, and say that $\gamma$ has a unit speed parameterization as $\gamma : [0,\infty) \to T$. For any interval $J \subset [0,\infty)$ we have a (possibly infinite) segment $\gamma|J$ of $\gamma$. The forward translation of $\gamma|J$ along $\gamma$ by $d$ is the segment $\gamma : J \to \gamma$ given by $$\gamma(t) \mapsto \gamma(t+d).$$ This map is an isometric embedding of $\gamma$ into itself. **Proposition 7**. *Let $(T,P,\phi,\sigma)$ be a synthetic Nielsen--Thurston package. Suppose $\tau_\sigma$ is positive and that $X \in T$ realizes $\tau_\sigma$. Let $\gamma$ be the geodesic ray from $X$ through $\sigma(X)$. Then $\sigma|\gamma$ is the forward translation of $\gamma$ by $\tau_\sigma$. In particular, $\sigma$ is not weakly contracting.* *Proof.* Let $Y$ be a point on $\gamma$ between $X$ and $\sigma(X)$. Using the triangle inequality twice and the assumption that $\sigma$ is non-expanding, we have $$\begin{aligned} d(Y,\sigma(Y)) &\leq d(Y,\sigma(X)) + d(\sigma(X),\sigma(Y)) \\ &\leq d(Y,\sigma(X)) + d(X,Y) \\ & = d(X,\sigma(X)) \\ &= \tau_\sigma.\end{aligned}$$ By the definition of $\tau_\sigma$ as an infimum, each of the above inequalities is an equality. By the first (in)equality and the assumption that $T$ is uniquely geodesic, it must be that $\sigma(Y)$ lies on $\gamma$. By the second (in)equality, $\sigma$ preserves the distance between $X$ and $Y$. Combining the last two statements and the fact that $Y$ was arbitrary, we find that the restriction of $\sigma$ to the initial segment of $\gamma$ from $X$ to $\sigma(X)$ is forward translation along $\gamma$ by $\tau_\sigma$. Inductively, we see that the restriction of $\sigma$ to the segment of $\gamma$ from $\sigma^k(X)$ to $\sigma^{k+1}(X)$ is forward translation along $\gamma$ by $\tau_\sigma$, whence the proposition. ◻ # Proof of the Übertheorem: Non-exceptional cases {#sec:proof} In this section we combine the results of the previous three sections to prove the Nielsen--Thurston Übertheorem in the non-exceptional cases. In preparation, we present some of the requisite terminology and state and prove a series of three lemmas. *Modulus.* For $r > 1$ the modulus of the standard annulus $1 < |z| < r$ is $\ln r/2\pi$. The modulus of an an arbitrary annulus (annular domain) is the modulus of the unique standard annulus to which it is biholomorphic. We note that the standard annulus is conformally equivalent to a Euclidean cylinder of height $\ln r$ and circumference $2\pi$. For $X \in \mathop{\mathrm{Teich}}(\Sigma)$ and $A \subseteq \Sigma$ and embedded annulus we denote by $\mu_x(A)$ the modulus of $A$. Similarly, for $\gamma$ a simple closed curve in $\Sigma$ we denote by $\mu_X(\gamma)$ the supremum of $\mu_X(A)$ over all embedded annuli $A$ in $\Sigma$ homotopic to $\gamma$. We denote by $\mu(X)$ the supremum of $\mu_X(\gamma)$ as $\gamma$ ranges over all simple closed curves in $\Sigma$. *Covering modulus.* We require another version of modulus. Let $\gamma$ be an essential closed curve in a Riemann surface $X$. There is an annular cover $\tilde X_\gamma \to X$ corresponding to $\gamma$, which is unique up to biholomorphism. We define the covering modulus of $\gamma$ to be $$\tilde \mu_X(\gamma) = \mu(\tilde X_\gamma).$$ It is a fact that $\tilde \mu_X(\gamma)$ is $\pi/\ell_X(\gamma)$, where $\ell_X(\gamma)$ is the length of the geodesic in the free homotopy class of $\gamma$, with respect to the hyperbolic metric associated to $X$. *The Margulis number.* The Margulis number $\epsilon$ is a real number with the properties that (1) any closed curve $\gamma$ with covering modulus $\tilde \mu_X(\gamma) > \epsilon$ is a multiple of a simple closed curve, and (2) if $\gamma_1$ and $\gamma_2$ are simple closed curves with $\mu_X(\gamma_i) \geq \epsilon$, then there are disjoint annuli homotopic to $\gamma_1$ and $\gamma_2$, respectively, each of modulus $\epsilon' = \mu_X(\gamma_i) -1$; see [@primer Lemma 13.6]. The second fact, sometimes called the collar lemma, implies that if two simple closed curves in $\Sigma$ have modulus greater than or equal to $\epsilon$ then they are homotopic to disjoint curves. Let $\xi(\Sigma)$ denote the maximum number of pairwise disjoint, pairwise non-homotopic, simple closed curves in $\Sigma$. This is an upper bound for the number of homotopy classes of simple closed curves $\gamma$ with $\mu_X(\gamma) > \epsilon$. *Modulus-degree inequality.* Let $f \colon X' \to X$ be a (holomorphic) covering map of Riemann surfaces, and let $\gamma'$ be a component of $f^{-1}(\gamma)$. We denote by $\deg f|\gamma'$ the degree of the restriction of $f$ to $\gamma'$. Then $$\mu_{X'}(\gamma') \leq \frac{\mu_X(\gamma)+1}{\deg f|\gamma'}.$$ This fact, which we refer to as the modulus-degree inequality, follows from two other facts: (1) the covering modulus multiplies by exactly $\deg f|\gamma'$ under the cover, and (2) the fact that $$\tilde \mu_X(\gamma)-1 \leq \mu_X(\gamma) \leq \tilde \mu_X(\gamma).$$ The right-hand inequality here is immediate, since an annulus in $X$ lifts to an annulus in $\tilde X_\gamma$. The left-hand inequality follows from the quantitative version of the collar lemma given above. The left-hand inequality also follows from Maskit's comparisons between extremal length and modulus [@maskit Propositions 1 and 2]. *The Grötzch inequality.* The next ingredient is a version of the classical Grötzch inequality, adapted from the case of rectangles to the case of annuli; see [@primer Theorem 11.10]. It states that given $X,Y \in \mathop{\mathrm{Teich}}(\Sigma)$, a $K$-quasiconformal map $h : X \to Y$, and a simple closed curve $\gamma$ in $\Sigma$ we have $$\frac{1}{K}\mu_X(\gamma) \leq \mu_Y(h(\gamma)) \leq K \mu_X(\gamma).$$ Applying this fact to the Teichmüller map $h :X \to Y$ we obtain $$\frac{1}{e^{2d(X,Y)}} \mu_X(\gamma) \leq \mu_Y(\gamma) \leq e^{2d(X,Y)} \mu_X(\gamma).$$ *Finding stable multicurves.* If $X\in\mathop{\mathrm{Teich}}(\Sigma)$ and $\Gamma$ is a multicurve in $\Sigma$, let $\mu_X(\Gamma)$ denote the vector of moduli of the components of $\Gamma$ (we emphasize that each component is the modulus of a single curve). Also, for a dynamical branched cover $f : \Sigma \to \Sigma$ and $\Gamma$ a multicurve in $\Sigma$, we define the *full preimage* of $\Gamma$ to be the set of all homotopy classes of simple closed curves in $\Sigma$ that map to components of $\Gamma$ under a power of $f$. The following lemma is essentially the same as Proposition 8.1(a) in Douady--Hubbard. **Lemma 8**. *Let $f \colon \Sigma\to \Sigma$ be a dynamical branched cover, and let $D > 0$. There exists an $N>0$, depending only on $\Sigma$, $\deg f$, and $D$ with the following property: for any multicurve $\Gamma$ in $\Sigma$ and any $X \in \mathop{\mathrm{Teich}}(\Sigma)$ with $$\mu_X(\Gamma) > (N,\ldots,N) \quad \text{and} \quad \tau_{\sigma_f}(X) \leq D,$$ the full preimage of $\Gamma$ is an $f$-stable multicurve.* *Proof.* Let $K=e^{2D}$, and let $N=(Kd)^{\xi(\Sigma)}\epsilon$, where $d$ is the degree of $f$. For each $j \geq 0$ let $\Gamma_j$ be the collection of all homotopy classes of essential curves in $f^{-i}(\Gamma)$ for $0\leq i\leq j$. We claim that for $0 \leq j \leq \xi(\Sigma)$ the collection $\Gamma_j$ is a multicurve. By the properties of the Margulis constant $\epsilon$, it suffices to show that each component of $\Gamma_j$ has modulus bounded below by $\epsilon$. We now prove this. Since $\sigma_f$ is non-expanding and $\tau_{\sigma_f}(X)\leq D$, we have $\tau_{\sigma_f^i}(X) \leq iD$ for all $i\geq 0$, so each of the associated Teichmüller maps $X\to \sigma_f^i(X)$ is $K^i$-quasiconformal. Let $\gamma'$ be a component of $\Gamma_j$; say $\gamma'$ is a component of $f^{-i}(\Gamma)$. By the Grötzch inequality, we have $$\mu_{X}(\gamma') \geq \frac{\mu_{\sigma_f^i(X)}(\gamma')}{K^i} \geq \frac{\mu_X(\gamma)}{K^id^i} \geq \frac{N}{K^id^i} \geq \frac{N}{K^{\xi(\Sigma)}d^{\xi(\Sigma)}} = \epsilon$$ (for the second inequality, we use the fact that if we restrict a degree $d^i$ cover to a cover of annuli, then the latter has degree at most $d^i$). Since $\gamma'$ was arbitrary, the claim follows. We next claim that some $\Gamma_j$ is $f$-stable. Indeed, we have inclusions $\Gamma_0 \subseteq \Gamma_1 \subseteq \cdots \subseteq \Gamma_{\xi(S)}$. Since $\Gamma_{\xi(S)}$ is a multicurve, we know that $|\Gamma_{\xi(S)}|\leq \xi(S)$, so there exists a $j< \xi(S)$ such that $\Gamma_j=\Gamma_{j+1}$, which implies that $\Gamma_j$ is an $f$-stable multicurve, as desired. ◻ *Uniform contraction.* The following lemma is a basic linear algebra fact. We will use it in the proof of the Übertheorem to show that if a stable multicurve is not a strong reduction system, then under pullback (by a suitable power) the moduli of the curves fails to increase. For a matrix $A$, let $\|A\|$ denote the operator norm of a matrix $A$ with respect to the sup norm on $\mathbb R^n$. We also denote by $\| \vec v \|$ the sup norm of $\vec v \in \mathbb R^n$. We denote by $\rho(A)$ the spectral radius of $A$. **Lemma 9**. *There exists a number $p = p(\Sigma,d)$ with the following property. If $f : \Sigma \to \Sigma$ is a dynamical branched cover of degree $d$ with $f$-stable multicurve $\Gamma$ and associated transition matrix $A$ then $$\rho(A) < 1 \quad\Rightarrow\quad \|A^p\| < \frac{1}{2}.$$* Before giving the proof of Lemma [Lemma 9](#lem:m){reference-type="ref" reference="lem:m"}, we remark that for a matrix $A$, the condition that $\rho(A)<1$ does not in general put any upper bound on $\|A\|$. *Proof of Lemma [Lemma 9](#lem:m){reference-type="ref" reference="lem:m"}.* It follows from Jordan canonical form that if $\rho(A)<1$ then $\|A^n\|\to 0$ as $n\to \infty$. In particular, there exists an $N_A$ such that $\|A^n\|<1/2$ for all $n\geq N_A$. For a given degree $d$ and a given $\Sigma$ there are only finitely many possible transition matrices, and in particular finitely many for which $\rho(A)<1$. Taking the maximum of all corresponding $N_A$ yields the desired exponent $p$. ◻ *The transition matrix versus the pullback map.* Let $f : \Sigma \to \Sigma$ be a dynamical branched cover. If $\Gamma$ is an $f$-stable multicurve, there is an associated transition matrix $M$. The $ij$-th entry is $$m_{ij} = \sum_\delta \frac{1}{\deg f|\delta}$$ where $\delta$ is a component of $f^{-1}(\gamma_j)$ homotopic in $\Sigma$ to $\gamma_i$. Here, $\deg f|\delta$ is the degree of the map $f|\delta : \delta \to \gamma_j$, thought of as a map $S^1 \to S^1$. For an $f$-stable multicurve $\Gamma$, the next lemma bounds (under certain conditions) the effect of $\sigma_f$ on $\mu_X(\Gamma)$ in terms of the associated transition matrix. This statement incorporates Theorem 7.1, Proposition 8.1(b), and Proposition 8.2 in Douady--Hubbard as well as part of their proof of Proposition 8.2. For the proof we use the notion of a latitude in an annulus. By definition, an annulus $A$ in a Riemann surface is a subset that is biholomorphic to a standard annulus $A_r$ given by $1 < |z| < r$. A latitude in $A_r$ is any circle centered at 0, and a latitude in $A$ is any corresponding circle in $A$ (under a biholomorphism). A biholomorphism of $A_r$ preserves latitudes, and so the latitudes in $A$ form a well-defined foliation of $A$. **Lemma 10**. *Fix $d \geq 2$ and $\Sigma$ a marked surface. Let $b = (d|P|+1)(\epsilon+2)$. If $f \colon \Sigma \to \Sigma$ is a dynamical branched cover of degree $d$ with stable multicurve $\Gamma$ and associated transition matrix $M$, and for some $X \in \mathop{\mathrm{Teich}}(\Sigma)$ the multicurve $\Gamma$ includes all simple closed curves $\gamma$ with $\mu_X(\gamma) > \epsilon$, then $$\mu_{\sigma_f(X)}(\Gamma) \leq M \mu_X(\Gamma) + (b,\dots,b).$$* *Proof.* The given inequality is a vector inequality, which must hold separately for each component. Specifically, for each curve $\gamma$ of $\Gamma$, we must prove that $$\mu_{\sigma_f(X)}(\gamma)\leq \sum_{\delta \in \Delta_\gamma} \frac{\mu_X(\gamma)}{\deg f|\delta}+ b$$ where $\Delta_\gamma$ is the set of all components of $\Delta = f^{-1}(\Gamma)$ that are homotopic to $\gamma$ in $\Sigma$. Let $A$ be an annulus in $\sigma_f(X)$ homotopic to $\gamma$. It suffices to prove that $$\mu_{\sigma_f(X)}(A) \leq \sum_{\delta\in \Delta_\gamma} \frac{\mu_X(\gamma)}{\deg f|\delta}+ b.$$ We now set about proving this inequality. Let $\tilde X$ be the marked Riemann surface obtained from $\sigma_f(X)$ by adding additional marked points: the set of marked points $\tilde P$ is the full $f$-preimage of the marked points in $X$. We have $|\tilde P| \leq d|P|$, and hence the maximal number of parallel, disjoint curves in $\tilde X$ is bounded above by $d|P|+1$. In particular, $|\Delta_\gamma| \leq d|P|+1$. Decompose $A$ into sub-annuli $A_1,\ldots,A_n$ by cutting it along all latitudes that pass through marked points of $\tilde X$. For each $i$, let $\alpha_i$ be a latitude of $A_i$; we have $\mu_{\tilde X}(\alpha_i)\geq \mu_{\tilde X}(A_i)$. The curves $\alpha_1,\ldots,\alpha_n$ are pairwise non-isotopic in $\tilde{X}$---this is obvious except for the bottom curve $\alpha_1$ and the top curve $\alpha_n$, but if $n\geq 2$ then $\tilde{P}$ and hence $P$ must be nonempty, in which case any point of $P$ separates $\alpha_1$ from $\alpha_n$. As in the last paragraph, it follows that $n \leq d|P|+1$. Since we decomposed $A$ along latitudes, we have $$\mu_{\sigma_f(X)}(A) = \sum_{i=1}^n \mu_{\tilde X}(A_i).$$ Set $$\mathcal A^\leq = \{A_i \mid \mu_{\tilde X}(A_i) \leq \epsilon+1 \} \qquad \text{and} \qquad \mathcal A^> = \{A_i \mid \mu_{\tilde X}(A_i) > \epsilon+1 \}$$ where $\epsilon$ is the Margulis constant. We will prove two claims that provide upper bounds on the sum of moduli in $\mathcal A^\leq$ and $\mathcal A^>$ in turn, beginning with $\mathcal A^\leq$. We first claim that $$\sum_{\mathcal A^\leq} \mu_{\tilde X}(A_i) \leq (d|P|+1)(\epsilon+1).$$ This follows from the fact that $n \leq (d|P|+1)$, and the definition of $\mathcal A^\leq$. We next claim that $$\sum_{\mathcal A^>} \mu_{\tilde X}(A_i) \leq \sum_{\delta\in \Delta_\gamma} \frac{\mu_X(\gamma)}{\deg f|\delta} + (d|P|+1).$$ Consider an $A_i\in\mathcal A^>$. Since each point of $f^{-1}(P)\subseteq \tilde X$ is marked, the image of $\alpha_i$ under $f$ is a curve $\gamma_i$ in $X$. This curve satisfies $$\tilde{\mu}_X(\gamma_i) = \tilde{\mu}_{\tilde X}(\alpha_i) \geq \mu_{\tilde X}(\alpha_i) \geq \mu_{\tilde X}(A_i) > \epsilon+1,$$ Here the first step uses the fact that the annular cover for $\gamma_i$ is the same as the annular cover for $\alpha_i$, the second step uses the fact that any annulus homotopic to $\alpha_i$ lifts to the annular cover, the third step uses the fact that $A_i$ is an annulus homotopic to $\alpha_i$, and the last step uses the definition of $\mathcal A^>$. Since $\tilde{\mu}_X(\gamma_i) > \epsilon+1> \epsilon$, we have that $\gamma_i$ is homotopic to a multiple of a simple closed curve for each $A_i\in\mathcal A^>$, and $\mu_X(\gamma_i)> \epsilon$ by the collar lemma. By hypothesis, it follows that $\gamma_i$ is homotopic to a multiple of a component of $\Gamma$. Then $\alpha_i$ must be homotopic to a multiple of some curve $\delta_i\in\Delta_\gamma$, and since $\alpha_i$ is simple it must be homotopic to $\delta_i$. By the modulus-degree inequality, we have $$\mu_{\tilde X}(A_i) \leq \mu_{\tilde X}(\alpha_i) = \mu_{\tilde X}(\delta_i) \leq \frac{\mu_X(\gamma)+1}{\deg f|\delta_i}.$$ Since the curves $\alpha_i$ are pairwise non-isotopic in $\tilde{X}$, the $\delta_i$'s are all distinct, so $$\sum_{\mathcal A^>} \mu_{\tilde X}(A_i) \leq \sum_{\mathcal A^>} \frac{\mu_X(\gamma)+1}{\deg f|\delta_i} \leq \sum_{\delta\in\Delta_\gamma} \frac{\mu_X(\gamma)+1}{\deg f|\delta} \leq \sum_{\delta\in\Delta_\gamma} \frac{\mu_X(\gamma)}{\deg f|\delta} + (d|P|+1).$$ The first inequality is as above, the second inequality comes from the fact that the $\delta_i$'s are all distinct, and the third comes from two facts, namely, that $1/(\deg f|\delta) \leq 1$ and that $|\Delta_\gamma| \leq d|P|+1$. This completes the proof of the claim. We may now complete the proof of the lemma. We have $$\begin{aligned} \mu_{\sigma_f(X)}(A) = \sum_{i=1}^n \mu_{\tilde X}(A_i) = \sum_{\mathcal A^\leq} \mu_{\tilde X}(A_i) + \sum_{\mathcal A^>} \mu_{\tilde X}(A_i) \leq \sum_{\delta\in \Delta_\gamma} \frac{\mu_X(\gamma)}{\deg f|\delta} + b\end{aligned}$$ The first equality was explained above. The second equality is true since $\{A_i\}$ is equal to the disjoint union $\mathcal A^\leq \cup \mathcal A^>$. The last inequality is the combination of the two claims and the definition of $b$. ◻ *Mapping class groups and virtual endomorphisms.* Let $\Sigma = (S,P)$. The pure mapping class group $\mathop{\mathrm{PMod}}(\Sigma)$ is the group of homotopy classes of homeomorphisms of $\Sigma$, where homeomorphisms and homotopies are required to fix $P$ pointwise. Let $f : \Sigma \to \Sigma$ be a dynamical branched cover. There is an associated virtual endomorphism $$\phi : \mathop{\mathrm{PMod}}(\Sigma) \dasharrow \mathop{\mathrm{PMod}}(\Sigma)$$ defined by lifting (homotopy classes of) homeomorphisms through $f$. It follows from the usual lifting criterion in algebraic topology and the fact that the degree of $f$ is finite that the domain of $\phi$ has finite index in $\mathop{\mathrm{PMod}}(\Sigma)$. Since isotopies always lift through $f$, the map $\phi$ is well defined. There is a natural action of $\mathop{\mathrm{PMod}}(\Sigma)$ on $\mathop{\mathrm{Teich}}(\Sigma)$ by pullback: given $h \in \mathop{\mathrm{PMod}}(\Sigma)$ and $X \in \mathop{\mathrm{Teich}}(\Sigma)$ we obtain $h \cdot X$ by pulling back the complex structure given by a representative of $X$ through a representative of $h$. It follows from the definitions that the pullback map $\sigma_f$ is intertwined with $\phi$. *Mumford's compactness criterion.* We refer to the quotient of $\mathop{\mathrm{Teich}}(\Sigma)$ by $\mathop{\mathrm{PMod}}(\Sigma)$ as moduli space (often moduli space refers to the quotient by a larger group, the full mapping class group). Mumford's compactness criterion states that if $X_i$ is a sequence in $\mathop{\mathrm{Teich}}(\Sigma)$ and if the images of the $X_i$ leave every compact set in moduli space then $\limsup \mu(X_i) \to \infty$. *Proof of the Übertheorem: Non-exceptional cases.* As in the statement of the theorem, $f\colon \Sigma \to \Sigma$ is a dynamical branched cover where $\Sigma = (S,P)$. Assume that $f$ is not exceptional. Let $\phi\colon \mathop{\mathrm{PMod}}(\Sigma) \dasharrow \mathop{\mathrm{PMod}}(\Sigma)$ be the virtual endomorphism associated to $f$, and let $\sigma\colon \mathop{\mathrm{Teich}}(\Sigma) \to \mathop{\mathrm{Teich}}(\Sigma)$ denote the pullback map. It follows from Teichmüller's theorems that the space $\mathop{\mathrm{Teich}}(\Sigma)$ is uniquely geodesic and that all maximal geodesics are bi-infinite. It is also known that the action of $\mathop{\mathrm{PMod}}(\Sigma)$ on $\mathop{\mathrm{Teich}}(\Sigma)$ is properly discontinuous [@primer Theorem 12.2]. We already stated that $\sigma$ is intertwined with $\phi$. By Proposition [Proposition 4](#prop:pullback){reference-type="ref" reference="prop:pullback"}, the map $\sigma$ is non-expanding. In other words, the collection $$(\mathop{\mathrm{Teich}}(\Sigma), \mathop{\mathrm{PMod}}(\Sigma), \phi, \sigma)$$ is a (not-at-all synthetic) synthetic Nielsen--Thurston package. Following the Bers proof of the Nielsen--Thurston classification, we treat three cases in turn: 1. $\tau_\sigma=0$ and is realized 2. $\tau_\sigma$ is not realized 3. $\tau_\sigma > 0$ and is realized We will show in the three cases that $f$ is holomorphic, strongly reducible, and pseudo-Anosov, respectively. *Case 1.* In this case it follows from the definitions that $f$ preserves a complex structure on $\Sigma$, which implies that $f$ has a holomorphic representative. *Case 2, $d=1$.* Let $D = \tau_\sigma + 1$, and let $N$ be the resulting constant from Lemma [Lemma 8](#lem:InvariantMulticurves){reference-type="ref" reference="lem:InvariantMulticurves"}. Let $X_i$ be a sequence of points in $\mathop{\mathrm{Teich}}(\Sigma)$ with $\tau_\sigma(X_i) \to \tau_\sigma$. By Proposition [Proposition 5](#prop:metricmumford){reference-type="ref" reference="prop:metricmumford"}, the (images of the) $X_i$ leave every compact subset of moduli space. By Mumford's compactness criterion, we may choose a $k$ so that $\mu(X_k) > N$. In particular there is a simple closed curve $\gamma$ in $\Sigma$ with $\mu_{X_k}(\gamma) > N$. By Lemma [Lemma 8](#lem:InvariantMulticurves){reference-type="ref" reference="lem:InvariantMulticurves"}, the full preimage of $\gamma$ is a stable multicurve $\Gamma$. This $\Gamma$ is a reduction system and hence a strong reduction system. *Case 2, $d>1$.* By Proposition [Proposition 4](#prop:pullback){reference-type="ref" reference="prop:pullback"}, some iterate of $\sigma$ is weakly contracting. Applying Proposition [Proposition 6](#prop:metricorbit){reference-type="ref" reference="prop:metricorbit"} to this iterate, we conclude that (the image of) every orbit leaves every compact subset of moduli space. Fix one such orbit $Y_i$. Again by Mumford's compactness criterion the $\mu(Y_i)$ tend to infinity. We now introduce several constants. Let $p = p(\Sigma,d)$ the the constant obtained from Lemma [Lemma 9](#lem:m){reference-type="ref" reference="lem:m"}. Since $\sigma$ is non-expanding, there exists a $D>0$ so that $\tau_\sigma(Y_i) \leq D$ for all $i$, namely, $D=\tau_\sigma(Y_0)$. For this $D$, let $N=N(\Sigma,d,D)$ be the constant from Lemma [Lemma 8](#lem:InvariantMulticurves){reference-type="ref" reference="lem:InvariantMulticurves"}. Next, let $b=b(\Sigma,d)$ be the constant from Lemma [Lemma 10](#lemma:fudge){reference-type="ref" reference="lemma:fudge"}, and let $$r = \max_M \big\| M^{p-1} + \cdots + M\big\| \bigl\|(b,\ldots,b)\big\|$$ where the maximum is taken over all transition matrices $M$ for dynamical branched covers of degree $d$ over $\Sigma$ (there are finitely many such matrices). Finally, let $$C = \max \{ N, 2r , \epsilon \}.$$ Since $\limsup \mu(Y_i) = \infty$, there exists a smallest $n$ with $\mu(Y_n)>C$. Increasing $C$ if necessary, we may assume that $n\geq p$. Let $\gamma$ be a simple closed curve in $\Sigma$ so that $\mu_{Y_n}(\gamma) > C$. Then $\mu_{Y_n}(\gamma)> N$, so Lemma [Lemma 8](#lem:InvariantMulticurves){reference-type="ref" reference="lem:InvariantMulticurves"} tells us that the full $f$-preimage $\Gamma$ of $\gamma$ is an $f$-stable multicurve. Suppose for the sake of contradiction that $\Gamma$ is not the multicurve underlying some strong reduction system for $f$, that is, the transition matrix $M$ for $\Gamma$ has $\rho(M)<1$. By Lemma [Lemma 9](#lem:m){reference-type="ref" reference="lem:m"} we have $\|M^p\| \leq 1/2$. We thus have $$\begin{aligned} \mu_{Y_n}(\gamma) \leq \| \mu_{Y_n}(\Gamma) \| &\leq \bigl\| M^p\, \mu_{Y_{n-m}}(\Gamma) + (M^{p-1} + \cdots + M)(b,\ldots,b) \bigr\| \\ &\leq \|M^p\|\,\|\mu_{Y_{n-p}}(\Gamma) \| + \big\| M^{p-1} + \cdots + M\big\| \bigl\|(b,\ldots,b) \bigr\| \\ &< \frac{1}{2}C + r \leq \frac{1}{2}C + \frac{1}{2}C = C.\end{aligned}$$ In order, we used the definition of the sup norm, Lemma [Lemma 10](#lemma:fudge){reference-type="ref" reference="lemma:fudge"} (iteratively), the triangle inequality and the definition of the operator norm, Lemma [Lemma 9](#lem:m){reference-type="ref" reference="lem:m"} and the choices of $n$ and $r$, the choice of $C$, and basic algebra. The resulting inequality $\mu_{Y_n}(\gamma) \leq C$ contradicts the earlier assumption that $\mu_{Y_n}(\gamma)> C$, and we are done. *Case 3.* Let $X \in \mathop{\mathrm{Teich}}(\Sigma)$ be a point with $\tau_\sigma(X)=\tau_\sigma$. Let $\gamma$ be the unique geodesic ray passing through $X$ and $\sigma(X)$. Since $\tau_\sigma>0$ by assumption, Proposition [Proposition 7](#prop:metricray){reference-type="ref" reference="prop:metricray"} implies the restriction of $\sigma$ to $\gamma$ is forward translation by $\tau_\sigma$. In particular, $\sigma^2(X)$ lies on $\gamma$ and $d(X,\sigma^2(X))$ is twice $d(X,\sigma(X))$. The ray $\gamma$ is determined by an ordered pair of measured foliations $(\mathcal F^+,\mathcal F^-)$ on $\Sigma$, each well defined up to scaling and isotopy. The Teichmüller map $h : X \to \sigma(X)$ has $(\mathcal F^+,\mathcal F^-)$ as its associated foliations. As in Section [3](#sec:pullback){reference-type="ref" reference="sec:pullback"} there is a commutative diagram $$\begin{tikzcd} (\sigma(X),f^*(\mathcal F^+,\mathcal F^-)) \arrow[r,"h^f"] \arrow[d,"f"] & (\sigma^2(Y),f^*(\lambda\,\mathcal F^+,\tfrac{1}{\lambda}\,\mathcal F^-)) \arrow[d,"f"] \\ (X,(\mathcal F^+,\mathcal F^-)) \arrow[r,"h"] & (\sigma(X),(\lambda\,\mathcal F^+,\tfrac{1}{\lambda}\,\mathcal F^-)), \end{tikzcd}$$ where $h^f$ is a pseudo-Teichmüller map with the same dilatation as $h$ and where $\lambda = e^{\tau_\sigma}$. Since $d(X,\sigma(X))=d(\sigma(X),\sigma^2(X))$, it follows that $h^f$ is in fact a Teichmüller map. We claim that the top-left and bottom-right corners of the diagram are scalar multiples. More precisely, we claim $$f^*(\mathcal F^+,\mathcal F^-) = ((\sqrt{d}\lambda)\,\mathcal F^+,(\sqrt{d}/\lambda)\,\mathcal F^-),$$ where $d=\deg(f)$. This claim gives that $f$ is pseudo-Anosov, and so it remains to prove the claim. (One is tempted to worry about the fact that $X \neq \sigma(X)$, but if we forget the complex structures, we can replace both $X$ and $\sigma(X)$ in the claim with $\Sigma$, making it clear how the claim implies that $f$ is pseudo-Anosov.) Firstly, the underlying (unmeasured) foliations must be equal, for if not, the composition $h^f \circ h$ would have dilatation less than $\lambda^2$ and hence $d(X,\sigma^2(X))$ would be strictly less than $2d(X,\sigma(X))$, a contradiction. As for the measures, the Euclidean areas of the pairs of foliations on the bottom row are equal, and pulling back by $f$ multiplies area by $d$, and so the claim follows. *Exclusivity.* We now prove the exclusivity statement in the non-exceptional case. As discussed in the introduction---and proved in the appendix---a strong reduction system is an obstruction to holomorphicity when $d > 1$. This implies that cases 1 and 2 are exclusive when $d>1$. We would now like to show that cases 2 and 3 are exclusive. To this end, we first point out that in the above argument for Case 3, Proposition [Proposition 7](#prop:metricray){reference-type="ref" reference="prop:metricray"} further implies that $\sigma$ is not weakly contracting. Since we are in the non-exceptional case, Proposition [Proposition 4](#prop:pullback){reference-type="ref" reference="prop:pullback"} then implies $\deg f = 1$, that is, $f$ is an element of the mapping class group of $\Sigma$. Therefore, the exclusivity of Cases 2 and 3 follows as in the Nielsen--Thurston classification theorem (a pseudo-Anosov mapping class stretches the lengths of all curves exponentially, but a reducible mapping class does not [@primer Theorem 14.23]). *Uniqueness.* Finally, we prove the uniqueness statements of the theorem. If $f$ is non-exceptional with $\deg f > 1$ then it follows from Proposition [Proposition 4](#prop:pullback){reference-type="ref" reference="prop:pullback"} that $\sigma_f$ has an iterate that is weakly contracting. In particular, $\sigma_f$ has at most one fixed point, and so there is at most one complex structure for which $f$ is holomorphic. The other uniqueness statement is the same as in the case of mapping class groups, since (as above) all non-exceptional pseudo-Anosov maps have degree 1. See [@FLP Corollary 12.4] for the argument. The idea is that, under iteration, a pseudo-Anosov map acts with source-sink dynamics on the space of projective measured foliations. ◻ We record here two statements that were established in the course of the proof of the Übertheorem in the non-exceptional cases. These statements will be applied in the proof for the exceptional cases. **Proposition 11**. *Let $f : \Sigma \to \Sigma$ be a dynamical branched cover. Suppose that the pullback map $\sigma_f$ has an orbit whose image in moduli space leaves every compact set. Then $f$ is strongly reducible.* **Proposition 12**. *Let $f : \Sigma \to \Sigma$ be a dynamical branched cover. Suppose that the pullback map $\sigma_f$ preserves a geodesic ray in $\mathop{\mathrm{Teich}}(\Sigma)$ and acts by forward translation on that ray. Then $f$ is pseudo-Anosov.* Even though our proof of Case 3 in the non-exceptional case reduces to the case of $\deg f = 1$, we gave the argument for arbitrary degree precisely so that we could give Proposition [Proposition 12](#case2){reference-type="ref" reference="case2"}. # Proof of the Übertheorem: Exceptional cases {#sec:torus} In this section we prove the Übertheorem in the remaining cases, the exceptional cases. As above, these are the cases where $\deg f > 1$ and $f$ either a torus map or a sphere map obtained from a torus map through the hyperelliptic involution. The proof uses many of the tools developed in Section [5](#sec:proof){reference-type="ref" reference="sec:proof"}. The main obstacle is that Proposition [Proposition 3](#prop:stable){reference-type="ref" reference="prop:stable"} gives no information in the exceptional cases, and hence Proposition [Proposition 4](#prop:pullback){reference-type="ref" reference="prop:pullback"} does not hold (as we will see, there are indeed cases where the pullback map has no iterate which is a weak contraction, namely, the cases of affine exceptional maps). We will instead take advantage of a product structure on Teichmüller space that is special to the exceptional cases. (In the case of an unmarked exceptional map, the product structure is trivial, and so these cases could be equally well have been addressed in Section [5](#sec:proof){reference-type="ref" reference="sec:proof"}.) The paper by Douady--Hubbard gives a detailed account of the dynamical branched covers with Euclidean orbifold, including a catalogue of all such maps [@DH Section 9]. *Exceptional surfaces and maps.* In order to give proofs that work simultaneously for the torus and the sphere, we will slightly alter our notation for a marked surface. Specifically, in this section, a marked surface $\Sigma$ is a pair $(S,P)$ where $S = (S_0,P_0)$ itself is a surface with marked points in the usual sense (so $S_0$ is a closed surface) and $P \subseteq S_0 \setminus P_0$. The relevant marked surfaces $\Sigma$ for this section are $((T^2,\emptyset),P)$ and $((S^2,P_0),P)$ with $|P_0|=4$. When we say that a dynamical branched cover $f : \Sigma \to \Sigma$ is exceptional, we will take $\Sigma$ to be $(S,P)$ where $S = (S_0,P_0)$ as above and $P_0$ is the post-critical set of $f$. So in all cases $P$ is the set of marked points that are not post-critical. *Teichmüller maps in the exceptional cases.* For the torus $T^2$, a Teichmüller map is the same thing as an orientation-preserving affine homeomorphism. This follows from the same reasoning as in the resolution of Grötzsch's problem about extremal maps between rectangles [@primer Theorem 11.10]. As a consequence, we see that Teichmüller maps on $T^2$ are closed under composition. We can identify $\mathop{\mathrm{Teich}}(T^2)$ with $\mathop{\mathrm{Teich}}(S_{1,1})$, the Teichmüller space of the torus with one marked point (this is the space of complex structures on the torus, modulo pullback by diffeomorphisms that fix the marked point and are homotopic to the identity). For the latter, the Teichmüller maps are exactly the orientation-preserving linear homeomorphisms and they are thus unique. In what follows, when we refer to *the* Teichmüller map between two points of $\mathop{\mathrm{Teich}}(T^2)$, we mean the linear one (here we are abusing the identification of $\mathop{\mathrm{Teich}}(T^2)$ with $\mathop{\mathrm{Teich}}(S_{1,1})$). Every point in $\mathop{\mathrm{Teich}}(T^2)$ comes equipped with a holomorphic hyperelliptic involution. The quotient is a Riemann surface that may be regarded as a sphere with four marked points. Each marked point corresponds to a fixed point of the hyperelliptic involution, also called a Weierstrass point. This correspondence gives a homeomorphic identification of $\mathop{\mathrm{Teich}}(T^2)$ with $\mathop{\mathrm{Teich}}(S_{0,4})$, the Teichmüller space of a sphere with four marked points. The Teichmüller maps for $S_{0,4}$ are exactly the quotients under the hyperelliptic involution of the affine maps of $T^2$ preserving the set of four Weierstrass points. By the same token, the above correspondence of $\mathop{\mathrm{Teich}}(T^2)$ with $\mathop{\mathrm{Teich}}(S_{0,4})$ is an isometry. *A product decomposition on Teichmüller space.* Let $\Sigma = (S,P)$ be an exceptional marked surface. Again, either $S$ is $(T^2,\emptyset)$, or it is $S=(S^2,P_0)$ with $|P_0|=4$, and in either case $P \cap P_0 = \emptyset$. There is a forgetful map $$\pi_v : \mathop{\mathrm{Teich}}(\Sigma) \to \mathop{\mathrm{Teich}}(S)$$ obtained by forgetting the set of marked points $P$. Let $X_\square \in \mathop{\mathrm{Teich}}(S)$ be some basepoint for $\mathop{\mathrm{Teich}}(S)$ (for instance when $S = T^2$, we may take $X_\square$ to be the unit square torus where the generators for $\pi_1(T^2)$ have length 1). We denote $\pi_v^{-1}(X_\square)$ by $\mathop{\mathrm{Teich}}(X_\square,P)$. Having defined $\mathop{\mathrm{Teich}}(X_\square,P)$ we may define a map $$\nu : \mathop{\mathrm{Teich}}(S)\times \mathop{\mathrm{Teich}}(X_\square,|P|) \to \mathop{\mathrm{Teich}}(\Sigma).$$ The formula for $\nu$ is $$\nu(X,Y) = (h_X)_*(Y)$$ where $h_X : X_\square \to X$ is the Teichmüller map and $(h_X)_*$ is the push forward of the complex structure $Y$. The marked points in $\nu(X,Y)$ are defined to be the $h_X$-images of the marked points in $Y$. In what follows we will refer to a subset $\mathop{\mathrm{Teich}}(S)\times \{Y\}$ of $\mathop{\mathrm{Teich}}(S)\times \mathop{\mathrm{Teich}}(X_\square,|P|) \to \mathop{\mathrm{Teich}}(\Sigma)$ as a horizontal slice, and we will write it as $\mathop{\mathrm{Teich}}(S)\times Y$ for simplicity. We have a similar definition and notation for vertical slices. **Proposition 13**. *Let $\Sigma = (S,P)$ be an exceptional marked surface and fix some $X_\square \in \mathop{\mathrm{Teich}}(S)$.* 1. *[\[torus:product\]]{#torus:product label="torus:product"} The map $$\nu : \mathop{\mathrm{Teich}}(S)\times \mathop{\mathrm{Teich}}(X_\square,|P|) \to \mathop{\mathrm{Teich}}(\Sigma)$$ is a homeomorphism.* 2. *[\[horisom\]]{#horisom label="horisom"} The map $\nu$ restricts to an isometry on each horizontal slice $\mathop{\mathrm{Teich}}(S)\times Y$.* 3. *[\[torus:hor\]]{#torus:hor label="torus:hor"} Two points $Z_1,Z_2 \in \mathop{\mathrm{Teich}}(\Sigma)$ lie in the $\nu$-image of a slice $\mathop{\mathrm{Teich}}(S)\times Y$ if and only if the Teichmüller map between them has no 1-pronged singularities at points of $P$.* 4. *[\[verticalproj\]]{#verticalproj label="verticalproj"} The projection $\pi_v : \mathop{\mathrm{Teich}}(\Sigma) \to \mathop{\mathrm{Teich}}(S)$ is non-expanding. Further $d(\pi_v(Z_1),\pi_v(Z_2)) = d(Z_1,Z_2)$ if and only if $Z_1$ and $Z_2$ lie in the same horizontal slice $\mathop{\mathrm{Teich}}(S) \times Y$.* *Proof.* We begin with the first statement. To prove it, we define an inverse map to $\nu$. The inverse has two coordinate functions. The first is the projection map $\pi_v$. The second coordinate function is: $$\rho(Z) = h_{X}^*(Z)$$ where $X=\pi_v(Z)$ and $h_{X}^*$ is pullback by the Teichmüller map $h_{X} : X_\square \to X$. The maps $\nu$, $\pi_v$, and $\rho$ are well defined and continuous by Teichmüller's theorems. The maps $\nu$ and $\pi_v \times \rho$ are inverses of each other by definition, and so both are homeomorphisms, proving the first statement. We proceed to the second statement. Let $(X_1,Y)$ and $(X_2,Y)$ be two points of $\mathop{\mathrm{Teich}}(S)\times \mathop{\mathrm{Teich}}(X_\square,|P|)$, and let $Z_1$ and $Z_2$ be their $\nu$-images. Let $h : X_1 \to X_2$ be the Teichmüller map. Since $\nu$ is defined in terms of Teichmüller maps from $X_\square$ and since Teichmüller maps of exceptional surfaces are closed under composition, it follows that $h$ may be regarded as the Teichmüller map $Z_1 \to Z_2$. Since we have Teichmüller maps $X_1 \to X_2$ and $Z_1 \to Z_2$ with the same stretch factor (in fact it is the same underlying map), the second statement follows. The third statement follows from the previous paragraph. Indeed, if two points lie in the $\nu$-image of a horizontal slice, then we have from the previous paragraph a Teichmüller map with the desired properties. For the other direction, suppose $h: Z_1 \to Z_2$ is a Teichmüller map where $Z_i = \nu(X_i,Y_i)$ and suppose $h$ has no singularities at the points of $P$. We would like to show $Y_1 = Y_2$. We may regard $h$ as a Teichmüller map $X_1 \to X_2$. If $h_i : X_\square \to X_i$ is the Teichmüller map for each $i$ then $h \circ h_1 = h_2$. Since $Y_i = h_i^*(Z_i)$, we have $$Y_2 = h_2^*(Z_2) = (h \circ h_1)^*(Z_2) = h_1^* h^*(Z_2) = h_1^*(Z_1) = Y_1,$$ We now prove the fourth statement. The projection $\pi_v$ is non-expanding because a Teichmüller map $h : Z_1 \to Z_2$ induces a pseudo-Teichmüller map $\bar h : \pi_v(Z_1) \to \pi_v(Z_2)$, as in Section [3](#sec:pullback){reference-type="ref" reference="sec:pullback"}. The pseudo-Teichmüller map $\bar h$ is a Teichmüller map if and only if $h$ has no singularities at a point of $P$. The fourth statement now follows from the third. ◻ Since Teichmüller maps between points in a horizontal slice are affine, the space $\mathop{\mathrm{Teich}}(X_\square,P)$---or indeed any of the vertical slices in the product decomposition in Proposition [Proposition 13](#prop:torus){reference-type="ref" reference="prop:torus"}---can be identified with the space of affine structures on $\Sigma$. *Pullback and the product decomposition.* Given the product decomposition from Proposition [Proposition 13](#prop:torus){reference-type="ref" reference="prop:torus"}, our next goal is to elaborate on the interaction between the product structure and the pullback map. The statement of the following proposition uses the following observation: an exceptional dynamical branched cover $f : (S,P) \to (S,P)$ induces a dynamical branched cover $\bar f : S \to S$. In particular, there is an induced pullback map on $\mathop{\mathrm{Teich}}(S)$. **Proposition 14**. *Let $\Sigma = (S,P)$ and let $f: \Sigma \to \Sigma$ be an exceptional dynamical branched cover of degree $d$.* 1. *[\[preserveproduct\]]{#preserveproduct label="preserveproduct"} The pullback map $\sigma_f$ preserves the product structure on $\mathop{\mathrm{Teich}}(\Sigma)$.* 2. *[\[horisom\]]{#horisom label="horisom"} If $\sigma_f$ preserves a horizontal slice $H$ of $\mathop{\mathrm{Teich}}(\Sigma)$ then $f$ is affine, $\sigma_f|H$ is an isometry, and $\sigma_f|H$ is conjugate under $\pi_v|H$ to the induced pullback map $\sigma_f^{hor}$ on $\mathop{\mathrm{Teich}}(S)$.* 3. *[\[noslice\]]{#noslice label="noslice"} If $\sigma_f$ preserves no horizontal slice of $\mathop{\mathrm{Teich}}(\Sigma)$, then all $\sigma_f$-orbits are weakly contracting.* For an exceptional $\Sigma = (S,P)$ we have that $\mathop{\mathrm{Teich}}(S)$ is isometric to $\mathbb H^2$ (up to scale). And by Proposition [\[verticalproj\]](#verticalproj){reference-type="ref" reference="verticalproj"} the restriction of $\pi_v$ to each horizontal slice of $\mathop{\mathrm{Teich}}(\Sigma)$ is an isometry to $\mathop{\mathrm{Teich}}(S)$. Thus, Proposition [\[horisom\]](#horisom){reference-type="ref" reference="horisom"}, implies that $\sigma_f$ is isometrically conjugate, through $\pi_v$, to an isometry of $\mathbb H^2$. *Proof of Proposition [Proposition 14](#prop:toruspullback){reference-type="ref" reference="prop:toruspullback"}.* We begin with the first statement. It follows from the definitions that $\sigma_{\bar f} \circ \pi_v = \pi_v \circ \sigma_f$, and hence that $\sigma_f$ preserves the set of vertical slices of the product. Now suppose that $Z_1$ and $Z_2$ lie in the same horizontal slice. By Proposition [\[torus:hor\]](#torus:hor){reference-type="ref" reference="torus:hor"} the Teichmüller map $h : Z_1 \to Z_2$ has no 1-pronged singularities at $P$. Since the map $h$ is homotopic to the identity, it has a lift through $f$. We denote this lift by $\tilde h$. By the definition of the pullback, we have that $\tilde h$ maps $\sigma_f(Z_1)$ to $\sigma_f(Z_2)$, in the sense that $\tilde h^*(\sigma_f(Z_2))=\sigma_f(Z_1)$. By Proposition [\[torus:hor\]](#torus:hor){reference-type="ref" reference="torus:hor"}, the first statement is a consequence of the following claim: the map $\tilde h$ is the Teichmüller map $\sigma_f(Z_1) \to \sigma_f(Z_2)$ and the singularities for the associated foliations all lie at $P_0$. Since $\tilde h$ is the lift of $h$ through $f$, it is a pseudo-Teichmüller map whose foliations are the preimages of the foliations for $h$. Since the 1-pronged singularities for the latter all lie at points of $P_0$, and since in both exceptional cases the preimage of $P_0$ is the union of $P_0$ with the set of critical points for $f$, it follows that the foliations for $\tilde h$ have 1-pronged singularities only at $P_0$ and that $\tilde h$ is a Teichmüller map, as desired. Suppose now that $\sigma_f$ preserves a horizontal slice $H$ of $\mathop{\mathrm{Teich}}(\Sigma)$. From the equality $\sigma_{\bar f} \circ \pi_v = \pi_v \circ \sigma_f$ used above, we conclude that $\sigma_f|H$ is conjugate under $\pi_v|H$ to the induced pullback map $\sigma_f^{hor} \colon \mathop{\mathrm{Teich}}(S) \to \mathop{\mathrm{Teich}}(S)$, as in the second statement. We next prove that if $\sigma_f$ preserves a horizontal slice, then $f$ is affine (as in the second statement). By the definition of the product structure on $\mathop{\mathrm{Teich}}(\Sigma)$, its horizontal slices correspond exactly to the (singular) affine structures on $\Sigma$. Therefore, if $f$ preserves a horizontal slice, it preserves an affine structure, and hence is affine. The remaining two statements (really the third statement and the second conclusion of the second statement) will be consequences of the following claim: if $X$ and $Y$ are points of $\mathop{\mathrm{Teich}}(\Sigma)$, then $d(\sigma_f(X),\sigma_f(Y))$ is strictly less than $d(X,Y)$ if and only if $X$ and $Y$ lies in different horizontal slices. Indeed, by Proposition [\[torus:hor\]](#torus:hor){reference-type="ref" reference="torus:hor"}, $X$ and $Y$ lie in different horizontal slices if and only if the foliations for the Teichmüller map $h : X \to Y$ have a 1-pronged singularity at a point of $P$. Since the points of $P$ are not post-critical (by definition), the latter is true if and only if the foliations for the lifted map $\tilde h : \sigma_f(X) \to \sigma_f(Y)$ have a 1-pronged singularity at a point of $f^{-1}(P)$. Since $f^{-1}(P)$ is disjoint from $P_0$ (again using the fact that the points of $P$ are not post-critical), the claim now follows from a second application of Proposition [\[torus:hor\]](#torus:hor){reference-type="ref" reference="torus:hor"}. Suppose that $\sigma_f$ preserves a horizontal slice $H$. By the claim and the fact that $\sigma_f$ is non-expanding (Proposition [\[nonexp\]](#nonexp){reference-type="ref" reference="nonexp"}), it follows that $\sigma_f|H$ is an isometry. Finally, if $\sigma_f$ preserves no horizontal slice then by the first statement it follows that for any $Z \in \mathop{\mathrm{Teich}}(\Sigma)$, the image $\sigma_f(Z)$ lies in a different horizontal slice of $\mathop{\mathrm{Teich}}(\Sigma)$. Combining this with the claim completes the proof. ◻ *Proof of the Übertheorem: Exceptional cases.* As in the statement of the theorem, $f\colon \Sigma \to \Sigma$ is an exceptional dynamical branched cover with degree $d > 1$. In particular, we have that $\Sigma = (S,P)$ with either $S=(T^2,\emptyset)$ or $S=(S^2,P_0)$ with $|P_0|=4$. In either case, the marked points of $S$ are the post-critical points for $f$. By Lemma [\[preserveproduct\]](#preserveproduct){reference-type="ref" reference="preserveproduct"}, $\sigma_f$ preserves the product structure on $\mathop{\mathrm{Teich}}(\Sigma)$. We treat two cases, according to whether or not $\sigma_f$ preserves a horizontal slice of $\mathop{\mathrm{Teich}}(\Sigma)$. If $\sigma_f$ preserves no horizontal slice then by Lemma [\[noslice\]](#noslice){reference-type="ref" reference="noslice"}, each $\sigma_f$-orbit is weakly contracting. By Proposition [Proposition 6](#prop:metricorbit){reference-type="ref" reference="prop:metricorbit"}, each $\sigma_f$-orbit leaves every compact subset of moduli space. Then by Proposition [Proposition 11](#case3){reference-type="ref" reference="case3"}, the map $f$ strongly reducible. Now suppose $\sigma_f$ does preserve a horizontal slice $H$. By parts (2) and (3) of Proposition [Proposition 14](#prop:toruspullback){reference-type="ref" reference="prop:toruspullback"}, the restriction $\sigma_f|H$ is isometrically conjugate to an isometry $\varphi$ of $\mathop{\mathrm{Teich}}(S) \cong \mathbb H^2$. There are three possibilities for $\varphi$: it can be elliptic, loxodromic, or parabolic. If $\varphi$ is elliptic then $\sigma_f|H$, hence $\sigma_f$, has a fixed point and $f$ is holomorphic. And if $\varphi$ is loxodromic, then by Proposition [\[horisom\]](#horisom){reference-type="ref" reference="horisom"} and Proposition [Proposition 12](#case2){reference-type="ref" reference="case2"}, the map $f$ is pseudo-Anosov. In the remainder of the proof we deal with the case where $\varphi$ is parabolic. In this case, the translation length of $\varphi$ is 0. It then follows from Proposition [\[horisom\]](#horisom){reference-type="ref" reference="horisom"} that the translation length $\tau_f$ is 0. It also follows from Proposition [\[horisom\]](#horisom){reference-type="ref" reference="horisom"} and Proposition [\[verticalproj\]](#verticalproj){reference-type="ref" reference="verticalproj"} that this translation length is not realized by $f$ (translation distances in $\mathop{\mathrm{Teich}}(\Sigma)$ are no smaller than the corresponding translation distances in $H$). By Proposition [\[horisom\]](#horisom){reference-type="ref" reference="horisom"}, the map $f$ is an affine torus map or a hyperelliptic quotient of an affine torus map. We first treat the case where $\Sigma$ is a torus and $f$ is affine. Since $\varphi$ is parabolic, the linear map homotopic to $f$ must have a single repeated eigenvalue, namely $\sqrt{d}$. We can change coordinates so that $f$ is of the form $$\left(\begin{array}{cc} \sqrt{d} & \ast \\ 0 & \sqrt{d} \end{array}\right)$$ where $d = \deg(f)$. It must be that $\sqrt{d}$ is a natural number. The preimage under $f$ of any horizontal curve in $T^2$ is a collection of horizontal curves. We will construct a strong reduction system consisting of horizontal curves. Let $\Gamma = \{\gamma_1,\dots,\gamma_k\}$ be a maximal multicurve in $\Sigma$ consisting of horizontal curves. The number of components $k$ is the same as the number of horizontal curves in $T^2$ that pass through a marked point of $\Sigma$ (although such curves are not permitted to be components of $\Gamma$, exactly because they pass through marked points). We label each component $\gamma_i$ by its modulus (equivalently, the supremum of Euclidean widths of annuli in $\Sigma$ that have horizontal boundary curves and that contain the given $\gamma_i$). These numbers are the vertical distances between marked points with distinct, but consecutive, coordinates in the vertical direction. We claim that the resulting labeled multicurve, which we still call $\Gamma$, is a strong reduction system for $f$. For each $i$, we may choose a closed annulus $A_i$ that has horizontal boundary, that has Euclidean width $\ell_i$, and that is homotopic in $\Sigma$ to $\gamma_i$. If $\Sigma$ has marked points, then each $A_i$ has at least one marked point on each boundary component, and the union of all of the $A_i$ is $\Sigma$. (If $\Sigma$ has no marked points, then $k=1$ and $A_1$ should be taken to be all of $T^2$.) Each $f^{-1}(A_i)$ is a collection of $\sqrt{d}$ annuli, each with width $\ell_i/\sqrt{d}$ (the above matrix for $f$ stretches in the vertical direction by $\sqrt{d}$). Since $f$ is a covering map, the union over $i$ of the $f^{-1}(A_i)$ is all of $\Sigma$, from which it follows that $\Gamma$ is a strong reduction system and so $f$ is strongly reducible, as desired. Suppose now that $\Sigma = ((S^2,P_0),P)$. Since $f$ is exceptional and $\sigma_f$ preserves a horizontal slice of $\mathop{\mathrm{Teich}}(\Sigma)$, it follows from Proposition [\[horisom\]](#horisom){reference-type="ref" reference="horisom"} that $f$ is affine. Thus, $f$ lifts to an affine map $\tilde f$ of $T^2$. What is more, $\tilde f$ can be regarded as an affine map of $\tilde \Sigma = (T^2,\tilde P)$, where $\tilde P$ is the preimage of $P$ under the hyperelliptic involution. As above we obtain a strong reduction system $\tilde \Gamma$ in $\tilde \Sigma$, which we may assume is horizontal. By construction, $\tilde \Gamma$ is invariant under the hyperelliptic involution. Hence it gives rise to a labeled multicurve $\Gamma$ in $\Sigma$. Let $\mathcal {HM}(\Sigma)$ denote the set of labeled horizontal multicurves on $\Sigma$ and let $\mathcal {SHM}(\tilde \Sigma)$ denote the set of symmetric labeled horizontal multicurves on $\tilde \Sigma$ (we concentrate on horizontal curves to avoid curves in $\Sigma$ with connected preimage). There is a commutative diagram $$\begin{tikzcd} \mathcal {SHM}(\tilde \Sigma) \arrow[r,"\tilde f^\ast"] \arrow[d,"\cong"] & \mathcal {SHM}(\tilde \Sigma) \arrow[d,"\cong"] \\ \mathcal {HM}(\Sigma) \arrow[r,"f^\ast"] & \mathcal {HM}(\Sigma) \end{tikzcd}$$ (where the horizontal maps are the natural pullback maps). The symmetric, horizontal strong reduction system for $\tilde f$ thus gives a (horizontal) strong reduction system for $f$, as desired. For exceptional maps, the only exclusivity statement is that types 1 and 3 are exclusive. This follows by the same reasoning as in the non-exceptional case. (In Appendix [7](#sec:orb){reference-type="ref" reference="sec:orb"} we explain why the argument for exclusivity of types 1 and 2 only applies in the non-exceptional cases.) The uniqueness statement for type 3 (pseudo-Anosov) maps follows from the same argument as in the non-exceptional case. ◻ We end by pointing out one consequence of the proof that is heretofore unmentioned: an exceptional dynamical branched cover of $\Sigma=(S,P)$ is affine if and only if it has no strong reduction system that is inessential in $S$. # Strong reduction systems and Thurston obstructions {#sec:orb} Our main goal in this appendix is to give a geometric characterization of the orbifold for a dynamical branched cover. With this characterization, we accomplish two goals: 1. we give a direct proof that strong reduction systems are obstructions to holomorphicity for dynamical branched covers with hyperbolic orbifold, 2. we show that a dynamical branched cover of the sphere is exceptional if and only if its orbifold is the $(2,2,2,2)$-orbifold, and The first item explains why strong reduction systems are the "obvious" obstructions to holomorphicity for a non-exceptional dynamical branched cover. The second justifies our characterization of exceptional maps in the introduction. All of the material in this section was surely known to Thurston, although the authors are unable to find the arguments in the existing literature. The argument in Theorem 4.1 of Douady--Hubbard is very similar to our argument for the first item. Their proof concludes by considering the derivative of the pullback map on Teichmüller space, which in turn relies on their analogue of our Proposition [Proposition 3](#prop:stable){reference-type="ref" reference="prop:stable"}. Our argument ends by simply considering the lifted map of the hyperbolic plane. *Orbifolds for dynamical branched covers.* For our purposes, a (2-dimensional) orbifold is a marked surface $(S,P)$ endowed with a labeling of $P$ by $\mathbb{N} \cup \{\infty\}$, that is, a function $\nu_P : P \to \mathbb{N} \cup \{\infty\}$. If $\nu_P(p) > 1$ then we refer to $p$ as a cone point. We will explain below the geometric meaning of an orbifold, which will allow us to use geometry to study dynamical branched covers. A map $f : (S,P) \to (T,Q)$ is an *orbifold cover* if it induces a branched covering map $S \to T$ and whenever we have $p \in P$, $q \in Q$, and $f(p)=q$, then $$(\deg f_p) \cdot \nu_p = \nu_q.$$ Here, $\deg f_p$ is the local degree of $f$ at $p$. For two orbifolds $(S,P)$ and $(S',P')$ we write $(S',P') \sqsubseteq (S,P)$ if - $S' \subseteq S$, - $P' \subseteq P$, and - for each $p \in P'$ we have $\nu_{P}(p) \mid \nu_{P'}(p)$. A *partial orbifold cover* from $(S,P)$ to $(T,Q)$ is an orbifold cover $$(S',P') \to (T,Q)$$ with $(S',P') \sqsubseteq (S,P)$. And a *partial self-orbifold cover* of an orbifold $(S,P)$ is a partial orbifold cover from $(S,P)$ to itself. To our knowledge this definition has not appeared in the literature, although we strongly suspect it was known to Thurston. A partial self cover of surfaces is a covering map $S' \to S$ where $S' \subseteq S$ (we sometimes require $S'$ to be open in $S$). We can think of this as a special case of a partial self-orbifold cover, since a deleted point can be regarded as an orbifold point with label $\infty$. For a given dynamical branched cover $f : (S,P) \to (S,P)$, a basic problem is to understand all orbifold structures on $(S,P)$ so that $f$ induces a partial self-orbifold cover of $(S,P)$. Specifically, this means that there is some $(S',P') \sqsubseteq (S,P)$ so that the induced map $f : (S',P') \to (S,P)$ is an orbifold cover. Once we explain the geometric meaning of orbifolds below, we will be able to use the geometry of the orbifold to study $f$. Given $f : (S,P) \to (S,P)$, there is a minimal labeling of $P$ so that $f$ is a partial self-orbifold covering map. The label at $p \in P$ is determined as follows. For each $k$ and each critical point $c$ with $f^k(c)=p$, we compute the local degree of $f^k$ at $c$. The label $\nu_p$ is the least common multiple of these local degrees over all such choices of $k$ and $c$. For each $q \in f^{-1}(P) \setminus P$, the label $\nu_q$ is defined to be $\nu_p$, where $p=f(q)$. So, for example, if $c \in P$ is critical and $f^k(c)=c$ for some $k$ (that is, the portrait for $f$ has a loop based at $c$) then $\nu_c = \infty$. It is a fact that every orbifold structure on $(S,P)$ for which $f$ is a partial self-orbifold covering map is a multiple of the one constructed above. As such, this orbifold structure is often referred to as *the* orbifold for $f$. *Euler characteristic and hyperbolic orbifolds.* The Euler characteristic of an orbifold $(S,P)$ is given by the Riemann--Hurwitz formula $$\chi(S,P) = \chi(S) + \sum_P \left(\frac{1}{\nu_p}-1\right)$$ (here $\chi(S)$ is the usual Euler characteristic for surfaces). We can think of an orbifold topologically as the surface obtained from $S$ by deleting a disk around each $p \in P$ and gluing in a fraction of a disk, namely, one $\nu_p$th of a disk; hence the formula. We say that $(S,P)$ is hyperbolic, Euclidean, or spherical if $\chi(S,P)$ is negative, zero, or positive, respectively. Under an orbifold covering map $f : \Sigma \to T$ of degree $d$ we have the usual multiplicative property $$\chi(\Sigma) = d \cdot \chi(T).$$ It follows that in an orbifold covering, both orbifolds are of the same type: hyperbolic, Euclidean, or spherical. *Geometric orbifolds.* There is an entirely geometric approach to orbifolds. Let $X$ be $\mathbb R^2$, $\mathbb H^2$, or $S^2$, and let $G$ be a discrete group of isometries of $X$ (unlike a covering space action, the action of $G$ might not be free). The quotient $\Sigma = X/G$ is naturally described as an orbifold: the label of a point in $\Sigma$ is the cardinality of the stabilizer in $G$ of any preimage in $X$. We think of a point labeled $\nu$ as a cone point of order $\nu$. We refer to any orbifold constructed in this way as a geometric orbifold. The space $X$ is the orbifold universal cover of $\Sigma$ and $G$ its orbifold fundamental group. Thurston determined exactly which orbifolds are geometric [@Thurston Theorem 13.3.6]. In particular, he proved that all hyperbolic and Euclidean orbifolds are geometric: they arise as quotients of $\mathbb H^2$ and $\mathbb R^2$ by discrete groups of isometries as above. He also proved that all orbifolds with three or more cone points are geometric. It follows from the Gauss--Bonnet theorem that the space $X \in \{\mathbb R^2,\mathbb H^2,S^2\}$ is determined uniquely by the orbifold $X/G$. *Lifting to the universal cover.* Now that we have given geometric meaning to the notion of an orbifold, we can do the same for the notion of an orbifold covering map. Specifically, it is a fact that any orbifold covering map lifts to a map of their orbifold universal covers. In other words, if $f : \Sigma \to T$ is a partial orbifold covering map and $\pi_\Sigma : X \to \Sigma$ and $\pi_T : X \to T$ are the universal covering maps, then there is a map $\tilde f$ so that the following diagram commutes $$\begin{tikzcd} X \arrow[r,"\tilde f"] \arrow[d,"\pi_\Sigma",right] & X \arrow[d,"\pi_T"] \\ \Sigma \arrow[r,"f"] & T \end{tikzcd}$$ Indeed, the definition of a partial orbifold covering map implies that $f$ induces a well-defined homomorphism of orbifold fundamental groups. As such, the natural analogue of the usual lifting criterion from algebraic topology applies, implying the existence of $\tilde f$. If $f$ is holomorphic then, since $\pi_\Sigma$ and $\pi_T$ are holomorphic by definition, the induced map $\tilde f$ is holomorphic. *Compatible measured foliations.* Let $\Sigma = (S,P)$ be a marked surface endowed with a complex structure, and let $(\mathcal F^+,\mathcal F^-)$ be a pair of transverse measured foliations on $\Sigma$ (as usual, any 1-pronged singularities of the singularities must lie at points of $P$). Let $Q$ be the set of singular points of the pair of foliations. The pair $(\mathcal F^+,\mathcal F^-)$ induces a pair of transverse, nonsingular foliations on $\Sigma \setminus Q$. Further, these foliations induce a complex structure on $\Sigma \setminus Q$, hence on $\Sigma$ (by the removable singularity theorem). The charts for this complex structure map open sets in $\Sigma \setminus Q$ to $\mathbb C$ in such a way that $(\mathcal F^+,\mathcal F^-)$ map to the measured foliations on $\mathbb C$ given by horizontal and vertical lines, the measures for the latter being $|dy|$ and $|dx|$, respectively. We say that the pair $(\mathcal F^+,\mathcal F^-)$ is compatible with the complex structure on $\Sigma$ if the complex structures agree. The reader familiar with quadratic differentials will recognize that a compatible pair of foliations on $\Sigma$ is the same as an integrable meromorphic quadratic differential on $\Sigma$ with all (simple) poles at points of $P$. Since every marked surface with a complex structure admits a nontrivial quadratic differential (on $S_g$ there is a $(6g-6)$-dimensional vector space of these), every complex structure has a compatible pair of measured foliations. A pair of measured foliations on $\Sigma$ gives more information than a complex structure: it gives a Euclidean structure on $\Sigma \setminus Q$, and a singular Euclidean structure on $\Sigma$. In particular, we have an area form as well as a total area. *The Jenkins extremal problem.* Let $\Sigma = (S,P)$ be a marked surface endowed with a complex structure, and let $\Gamma = \{\gamma_1,\dots,\gamma_k\}$ be a labeled multicurve in $\Sigma$. We denote the weight on $\gamma_i$ by $w(\gamma_i)$. A multi-annulus in $\Sigma$ is a disjoint union of domains, each biholomorphic to an open annulus in $\mathbb C$, and each disjoint from $P$. We consider the following extremal problem: given the labeled multicurve $\Gamma$ as above, find a multi-annulus $A = \{A_1,\dots,A_k\}$ with the following properties: 1. each $A_i$ is homotopic to $\gamma_i$, 2. $(\mu(A_1),\dots,\mu(A_k))$ is a multiple of $(w(\gamma_1),\dots,w(\gamma_k))$, and 3. $(\mu(A_1),\dots,\mu(A_k))$ is maximal with respect to the first two properties. Jenkins proved that when $S$ is not the torus, this extremal problem has a unique solution [@jenkins Theorem 1]. This solution corresponds to a pair of measured foliations $(\mathcal F^+,\mathcal F^-)$ that is compatible with the complex structure. The singular leaves of $\mathcal F^+$ form a finite graph in $S$ (with singular points as vertices) whose complement is a disjoint union of open annuli, each foliated by smooth closed leaves of $\mathcal F^-$. The modulus of each annulus with respect to the complex structure is the modulus of the corresponding Euclidean annulus (the modulus of a Euclidean annulus with circumference $C$ and heights $H$ is $2{\pi}H/C$). By the uniqueness of the extremal problem, it follows that the pair $(\mathcal F^+,\mathcal F^-)$ is unique up to scale. *Strong reduction systems as Thurston obstructions.* Let $f : \Sigma \to \Sigma$ be a dynamical branched cover. Suppose that - $f$ is holomorphic and - $f$ has a strong reduction system $\Gamma$. We will show that either $\deg f = 1$ or $f$ has Euclidean orbifold. This means that for $f$ with hyperbolic orbifold and degree greater than 1, strong reduction systems are obstructions to holomorphicity (and vice versa). Fix a complex structure on $\Sigma$ with respect to which $f$ is holomorphic (we may have to replace $f$ with a homotopic map). Let $(A_1,\dots,A_k)$ be the multi-annulus that gives the solution to the Jenkins extremal problem associated to $\Gamma$, and let $(\mathcal F^+,\mathcal F^-)$ be a corresponding pair of measured foliations. By the definition of a strong reduction system, the preimage is an equal or larger solution to the extremal problem. Indeed, the preimage of the collection $(A_1,\dots,A_k)$ is, after consolidating parallel annuli, a multi-annulus where the moduli are given by the weights on $f^*(\Gamma)$ (this uses three basic facts: (1) an $m$-fold cover of annuli multiplies modulus by $m$, (2) the modulus of a union of the closures of two adjacent annuli is the sum of the moduli, and (3) modulus is monotone under inclusion). By the previous paragraph, and the fact the compatible foliations for a solution to the Jenkins problem is unique up to scale, it must be that $(f^*\mathcal F^+,f^*\mathcal F^-)$ is a positive multiple of $(\mathcal F^+,\mathcal F^-)$. Moreover, since a cover of degree $d$ reduces Euclidean area by a factor of $d$, we have $$(f^*\mathcal F^+,f^*\mathcal F^-) = \sqrt{d} \cdot (\mathcal F^+,\mathcal F^-)$$ Therefore, if we lift the map $f$ to the universal cover, we obtain a biholomorphic homothety where the scaling factor is $\sqrt{d}$. Biholomorphic maps of the hyperbolic plane are isometries, and so it must be that $d=1$ or that the orbifold for $f$ is Euclidean, as desired. *Orbifolds and exceptional maps.* We have one more loose end to tie up with respect to orbifolds and Thurston's characterization of rational maps. As promised in the introduction, we explain here why an (unmarked) dynamical branched cover of the sphere has orbifold the $(2,2,2,2)$-orbifold if and only if is a hyperelliptic quotient of a torus map. This statement is originally due to Cannon--Floyd--Parry--Pilgrim [@CFPP Theorem 1.4]. We explained one direction in the introduction: hyperelliptic quotients of torus maps have the $(2,2,2,2)$-orbifold as their orbifold. Now suppose that $f : (S^2,P) \to (S^2,P)$ is a dynamical branched cover with $(2,2,2,2)$-orbifold. We would like to show that $f$ lifts---through the hyperelliptic involution---to a map of the torus. In other words, we would like to show that there is a map $\tilde f$ as in the following diagram: $$\begin{tikzcd} T^2 \arrow[r,dashed,"\tilde f"] \arrow[d,"p "] & T^2 \arrow[d,"p "] \\ (S^2,P) \arrow[r,"f"] & (S^2,P) \end{tikzcd}$$ where $p$ is the quotient map $T^2 \to T^2/\langle \iota \rangle = (S^2,P)$. The orbifold fundamental group of $(S^2,P)$ has the presentation $$\pi_1^{orb}(S^2,P) \cong \langle a_1,a_2,a_3,a_4 \mid a_1^2=a_2^2=a_3^2=a_4^2=abcd=1 \rangle$$ and the image of the induced map $$p_* : \pi_1(T^2) \to \pi_1^{orb}(S^2,P)$$ is the even subgroup of $\pi_1^{orb}(S^2,P)$, that is, the kernel of the map $$\begin{aligned} \pi_1^{orb}(S^2,P) &\to \mathbb Z/2 \\ a_i &\mapsto 1.\end{aligned}$$ Since all four points of $P$ carry the label 2, it follows that the local degree of $f$ at each point of $P$ is 1. Thus, the induced map $f_*$ maps the even subgroup of $\pi_1^{orb}(S^2,P)$ to itself. Finally, by the lifting criterion for orbifold covering maps implies the existence of $\tilde f$, as desired. # Topological polynomials, Levy cycles, and Levy--Berstein {#sec:levy} In this appendix we prove a strong form of the theorem which says that if a topological polynomial is not rational then it has a degenerate Levy cycle. Again, this theorem is due to the work of Berstein, Hubbard, Levy, Rees, Tan, and Shishikura. Our strengthening is Proposition [Proposition 15](#prop:levy){reference-type="ref" reference="prop:levy"} below. In the statement, we say that a strong reduction system is *minimal* if all multicurves with fewer components fail to underlie a strong reduction system. If a dynamical branched cover has a strong reduction system, then it has a minimal one. **Proposition 15**. *Let $f \colon (\mathbb R^2,P) \to (\mathbb R^2,P)$ be a topological polynomial. Every minimal strong reduction system for $f$ is a degenerate Levy cycle. In particular, if $f$ has a strong reduction system then it has a degenerate Levy cycle.* As in the introduction, the Levy--Berstein theorem says that if $f$ is a topological polynomial and each point of $P$ has a critical point in its forward $f$-orbit then $f$ is rational. This is immediate from Proposition [Proposition 15](#prop:levy){reference-type="ref" reference="prop:levy"}, since the union of the disks for a degenerate Levy cycle contains no critical points. Our argument for Proposition [Proposition 15](#prop:levy){reference-type="ref" reference="prop:levy"} is a modification of the argument in Hubbard's book for an analogous statement about Thurston obstructions [@hubbard Theorem 10.3.7]. We use two tools, innermost curves and lifting graphs. *Innermost curves.* The main feature that makes topological polynomials different from topological rational maps---and what allows us to prove Proposition [Proposition 15](#prop:levy){reference-type="ref" reference="prop:levy"}---is that every curve in $(\mathbb R^2,P)$ has a well-defined interior: the compact region of $\mathbb R^2$ bounded by the curve. Moreover, if $\delta$ is a component of $f^{-1}(\gamma)$ then $f$ maps the interior of $\delta$ onto the interior of $\gamma$. Given a multicurve $\Gamma$ we will denote by $\Gamma^\circ$ the multicurve given by its innermost components. *Lifting graphs.* If $\Gamma$ is an $f$-stable, labeled multicurve for a dynamical rational map $f$, we define a corresponding a directed graph, the lifting graph, as follows: the vertices are the components of $\Gamma$ and there is a directed edge from $\gamma$ to $\delta$ if $\delta$ is homotopic to a component of $f^{-1}(\gamma)$ (note that $f^{-1}(\gamma)$ may have components that are inessential or are essential and not homotopic to a component of $\Gamma$). We label each vertex by the corresponding labels on the curves of $\Gamma$ and we label each edge by a natural number, the degree of $f|\delta : \delta \to \gamma$. We can interpret the action of $f^*$ on $\Gamma$ in terms of the lifting graph. Under $f^*$, the labels on the vertices change as follows: the new label on a vertex $v$ is the sum of $w_i/d_i$ where $w_i$ is the weight on the $i$th vertex with a directed edge pointing to $v$ and $d_i$ is the label on that edge. *Proof of Proposition [Proposition 15](#prop:levy){reference-type="ref" reference="prop:levy"}.* Let $\Gamma$ be a labeled multicurve in $(\mathbb R^2,P)$ giving a minimal strong reduction system for $f$. Let $G$ be the corresponding lifting graph. We first claim that each vertex of $G$ has at least one incoming edge, that is, $G$ has no initial vertices. This follows from the stability of $\Gamma$, since an initial vertex would be a component of $\Gamma$ not parallel to a component of $f^{-1}(\Gamma)$. We next claim that each vertex of $G$ has at least one outgoing edge, that is, $G$ has no terminal vertices. Indeed, suppose that a vertex $\gamma$ is terminal. It cannot be that $\gamma$ is the only vertex of $G$, for then $G$ would have no edges, and it would be impossible for $\Gamma = \gamma$ to underly a strong reduction system. Now, if we delete $\gamma$ from $\Gamma$, then the multicurve that remains---which is nonempty by the previous sentence---still underlies a strong reduction system for $f$, violating the minimality of $\Gamma$. We now claim that the set of vertices of $G$ corresponding to innermost curves of $\Gamma$ determines a closed subgraph $G^\circ$ of $G$, that is, a directed edge starting at an innermost curve ends at an innermost curve. Suppose there is a directed edge from some curve $\gamma$ to a curve $\delta$ that is not innermost. We will show that $\gamma$ is not innermost. Let $\epsilon$ be a curve of $\Gamma$ in the interior of $\delta$ (and not parallel to $\delta$). Since $G$ has no initial vertices, $\epsilon$ lies in the $f$-preimage of a curve $\phi$ of $\Gamma$. And because $f$ maps interiors to interiors, this $\phi$ would have to lie in the interior of $\gamma$. Also, since the components of $\Gamma$ are not parallel pairwise and since $f$ is a function, $\phi$ is not parallel to $\gamma$, and the claim is proved. We next claim that $G^\circ$ is equal to $G$. Suppose not. Then the subgraph $G'$ of $G$ spanned by the vertices not in $G^\circ$ is nonempty. We will show that $G'$ represents a strong reduction system for $f$, which will violate the minimality of $\Gamma$. We first show that $G'$ represents a stable multicurve, and then check the condition on labels. Since $G$ has no terminal vertices, each vertex of $G'$ is the end point of an edge of $G$. As $G^\circ$ is closed, it must be that the edges terminating in $G'$ have origins in $G'$. This is to say that $G'$ represents a stable multicurve for $f$. The action of $f^*$ on the labels of $G'$ agrees with the restriction of its action on the labels of $G$, and so $G'$ does indeed represent a strong reduction system for $f$, the desired contradiction. We now claim that no two directed edges of $G$ have the same endpoint. Indeed, by the previous claim all vertices of $G$ are innermost curves of $\Gamma$. Any two innermost curves are un-nested, that is, neither lies in the interior of the other. It follows that the components of their preimages un-nested. In particular, the preimages cannot be parallel, whence the claim. At this point, we have shown that $G$ has no initial or terminal vertices and that no two edges has the same endpoints. It follows that $G$ is a union of directed cycles. By minimality, $G$ is a single directed cycle. If $G$ has an edge label greater than 1, then there are no positive labels of the vertices of $G$ that satisfy the condition for a strong reduction system. Therefore all of the edges are labeled 1. This is to say that $G$ represents a Levy cycle. Since each curve of $\Gamma$ maps to the next with degree 1, the disks interior to these curves also map to the next with degree 1, meaning that $\Gamma$ is a degenerate Levy cycle, as desired. ◻ # Further extensions of the Übertheorem {#sec:ext} In this third and final appendix, we explain several generalizations of the Nielsen--Thurston Übertheorem. There are three versions: for equivariant maps, for non-orientable surfaces, and for orientation-reversing maps. All of these are straightforward extensions of the Übertheorem. In theory, we could combine all of the extensions into one Superübertheorem, but for clarity we prefer to state them separately. We also state the extensions informally, because some of the details are left to the reader. *Equivariant maps.* Let $\Sigma = (S,P)$, let $f : \Sigma \to \Sigma$ be a dynamical branched cover. Let $G$ be a finite group that acts on $\Sigma$. As usual, we say that $f$ is $G$-equivariant is $f(g \cdot x) = g \cdot f(x)$ for all $x \in S$. For example, we say that $f$ is an odd map of $(S^2,P)$ if it is $\mathbb Z/2$-equivariant, where $\mathbb Z/2$ acts by the antipodal map. If we assume that the map $f$ in the statement of the Übertheorem is $G$-equivariant, then the Übertheorem (of course) still holds, but with the added conclusion that the resulting homotopic map $\phi$ is also $G$-equivariant. We have the following consequences: 1. if $\phi$ is holomorphic then $G$ preserves the complex structure, 2. if $\phi$ is strongly reducible, then $G$ preserves the strong reduction system, and 3. if $\phi$ is pseudo-Anosov, then $G$ preserves the measured foliations. The key observation required to prove this enhancement of the Übertheorem is that the pullback of any geometric object (complex structure, strong reduction system, measured foliation, etc.) under a $G$-equivariant map is $G$-invariant. So, for example, the image of the pullback map $\sigma_f$ is contained in the subspace of $\mathop{\mathrm{Teich}}(\Sigma)$ fixed by the action of $G$. *Non-orientable surfaces.* For a non-orientable, closed surface $S$, we can define a marked surface $\Sigma = (S,P)$ and a dynamical branched cover $f : \Sigma \to \Sigma$ as in the orientable case. Such maps arise naturally even when studying dynamical branched covers of orientable surfaces. For instance, any odd map of $\Sigma = (S^2,P)$ descends to a dynamical branched cover of $({\mathbb R}{\mathrm P}^2,\bar P)$ where $\bar P$ is the image of $P$ under the quotient of $S^2$ by the antipodal map. The natural analogue of a complex structure in this setting is a conformal structure, by which we mean a map that preserves angles, up to sign, in the tangent space. This is equivalent to the existence of an atlas where the charts map to the complex plane and transition maps are holomorphic or anti-holomorphic. For orientable surfaces, complex structures and conformal structures are the same thing. Given $f : \Sigma \to \Sigma$ for non-orientable $\Sigma$, we obtain a dynamical branched cover $\tilde f : \tilde \Sigma \to \tilde \Sigma$ of the orientation double cover $\tilde \Sigma$. The deck group for this (characteristic) cover is $G \cong \mathbb Z/2$ and the map $\tilde f$ is $G$-equivariant. As above the map $\tilde f$ is (up to homotopy) either holomorphic, strongly reducible, or pseudo-Anosov. And moreover these corresponding geometric structures are $G$-invariant. These means that $f$ is either conformal, strongly reducible, or pseudo-Anosov, giving our second extension of the Übertheorem. There is an important subtlety in the above argument. When we modify $\tilde f$ by isotopy, we need to know that we can modify $f$ accordingly. In other words, we need to know that the isotopy of $\tilde f$ can be pushed down to an isotopy of $f$. In the theory of mapping class groups, it is true that homotopic $G$-equivariant homeomorphisms are $G$-equivariantly homotopic; this fact is known as the Birman--Hilden theorem (see the expository paper by the second- and third-named authors [@MW]). The analogue of the Birman--Hilden theorem does indeed hold for $G$-equivariant maps of degree greater than 1 (which is what we need here). In fact, the Maclachlan--Harvey proof of the Birman--Hilden theorem, which is based on Teichmüller theory, applies almost directly to this more general case (see page 13 of the aformentioned survey for a discussion). The only change needed is to replace all of the groups in the proof with monoids, since maps of degree greater than 1 do not have inverses. *Orientation-reversing maps.* Let $\Sigma = (S,P)$ be a marked surface, and suppose that $\Sigma$ is oriented. We say that an orientation-reversing map $f : \Sigma \to \Sigma$ is a dynamical branched cover if $f$ restricts to an (unbranched) covering space over $S \setminus P$. One way to construct such an $f$ is to take an (orientation-preserving) dynamical branched cover $(S^2,P) \to (S^2,P)$ where $P$ is preserved by the antipodal map and post-compose with the antipodal map. Let $f : \Sigma \to \Sigma$ is an orientation-reversing dynamical branched cover. We claim that $f$ is homotopic to a map that is either anti-holomorphic, strongly reducible, or pseudo-Anosov, and moreover this follows from our proof of the Übertheorem. The only required observation is that if an orientation-reversing map fixes a point in Teichmüller space then it is anti-holomorphic with respect to the corresponding complex structure. For the non-exceptional cases, this statement was already stated and proved by Geyer [@geyer Theorem 3.9].

arxiv_math

--- abstract: | Let $\Sigma$ be a compact orientable surface with nonempty boundary, let $\varphi: \Sigma \to \Sigma$ be an orientation-preserving pseudo-Anosov homeomorphism, and let $M = \Sigma \times I / \stackrel{\varphi}{\sim}$ be the mapping torus of $\Sigma$ over $\varphi$. Let $\mathcal{F}^{s}$ denote the stable foliation of $\varphi$ in $\Sigma$. Let $T_1, \ldots, T_k$ denote the boundary components of $M$. With respect to a canonical choice of meridian and longitude on each $T_i$, the degeneracy locus of the suspension flow of $\varphi$ on $T_i$ can be identified with a pair of integers $(p_i; q_i)$ such that $p_i > 0$ and $-\frac{1}{2}p_i < q_i \leqslant \frac{1}{2}p_i$. Let $c_i$ denote the number of components of $T_i \cap (\Sigma \times \{0\})$. Assume that $\mathcal{F}^{s}$ is co-orientable and $\varphi$ reverses the co-orientation on $\mathcal{F}^{s}$. We show that the Dehn filling of $M$ along $\partial M$ with any multislope in $J_1 \times \ldots \times J_k$ admits a co-orientable taut foliation, where $J_i$ is one of the two open intervals in $\mathbb{R} \cup \{\infty\} \cong \mathbb{R}P^{1}$ between $\frac{p_i}{q_i + c_i}, \frac{p_i}{q_i - c_i}$ which doesn't contain $\frac{p_i}{q_i}$. For some hyperbolic fibered knot manifolds, the slopes given above contain all slopes that yield non-L-space Dehn filllings. The examples include (1) the exterior of the $(-2,3,2q+1)$-pretzel knot in $S^{3}$ for each $q \in \mathbb{Z}_{\geqslant 3}$ (see [\[Kri\]](#Kri) for a previous proof), (2) the exteriors of many L-space knots in lens spaces. address: Department of Mathematics, University at Buffalo, Buffalo, NY 14260, USA author: - Bojun Zhao title: co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy --- # Introduction Throughout this paper, all $3$-manifolds are connected, orientable and irreducible. The L-space conjecture ([\[BGW\]](#BGW), [\[J\]](#J)) predicts that the following statements are equivalent for a closed orientable irreducible $3$-manifold $M$: \(1\) $M$ is a non-L-space. \(2\) $\pi_1(M)$ is left orderable. \(3\) $M$ admits a co-orientable taut foliation. The implication (3) $\Longrightarrow$ (1) is confirmed in [\[OS\]](#OS) (see also [\[Bo\]](#Bo), [\[KazR2\]](#KazR2)). In the case that $M$ has positive first Betti number, (2) is proved in [\[BRW\]](#BRW), and (3) is proved in [\[Ga1\]](#Ga1). The L-space conjecture has been verified when $M$ is a graph manifold ([\[BC\]](#BC), [\[Ra\]](#Ra), [\[HRRW\]](#HRRW)). There are two useful ways to construct closed orientable $3$-manifolds: Dehn surgeries on knots or links in $S^{3}$, and Dehn fillings of mapping tori of compact orientable surfaces. Both of these two operations can produce all closed orientable $3$-manifolds ([\[A\]](#A), [\[Lic\]](#Lic), [\[W\]](#W)). One practical approach to the L-space conjecture is to find and identify the (multi)slopes of those Dehn surgeries or fillings that yield manifolds satisfying (1), (2), (3) listed above. A slope on a knot or a knot manifold is called an *NLS* or *LO* or *CTF surgery/filling slope* if it yields a manifold satisfying (1) or (2) or (3), respectively. Also a slope on a knot or a knot manifold is called an *L-space surgery/filling slope* if it yields an L-space. And we use the similar terminologies *NLS, LO, CTF, L-space surgery/filling multislopes* for links and link manifolds (see Subection [2.1](#subsection 2.1){reference-type="ref" reference="subsection 2.1"} for the convention of multislopes). In this paper we focus on the problem of finding CTF filling (multi)slopes on pseudo-Anosov mapping tori of compact orientable surfaces. Let $\Sigma$ be a compact orientable surface with nonempty boundary and let $\varphi: \Sigma \to \Sigma$ be an orientation-preserving pseudo-Anosov homeomorphism. Let $M = \Sigma \times I / \stackrel{\varphi}{\sim}$ be the mapping torus of $\Sigma$ over $\varphi$. We fix an orientation on $M$. We choose a canonical meridian/longitude coordinate system on each component $T$ of $\partial M$, following [\[Ro2\]](#Ro2) (see Convention [\[slope\]](#slope){reference-type="ref" reference="slope"} for details). If $M$ is the exterior of a fibered knot $K$ in $S^{3}$, this coordinate system coincides with the standard meridian/longitude coordinate system except a special case (see Remark [\[slope consistent\]](#slope consistent){reference-type="ref" reference="slope consistent"} for details). With respect to the canonical coordinate system, a slope on each component $T$ of $\partial M$ will be identified with an element of $\mathbb{Q} \cup \{\infty\}$ as usual. We introduce some notations related to $M$ and $\varphi$, as explained in [\[GO\]](#GO), [\[Ga3\]](#Ga3), [\[HKM\]](#HKM), [\[KazR1\]](#KazR1): **Notation 1**. *(a) Let $\Psi$ denote the suspension flow of $\varphi$ in $M$. Then all closed orbits of $\Psi$ contained in the same boundary component of $M$ are parallel essential simple closed curves. Let $T$ be a component of $\partial M$. The *degeneracy slope* of $T$ (denoted $\delta_T$) is the slope of the closed orbits of $\Psi$ contained in $T$. Note that there are $2n$ closed orbits of $\Psi$ on $T$ for some $n \in \mathbb{N}_+$, and $n$ of them contain $(t,0)$ for some $t \in \partial \Sigma$ which is a singular point of the stable foliation of $\varphi$. The *degeneracy locus* of $\Psi$ on $T$ ([\[Ga3\]](#Ga3), see also [\[Ro2\]](#Ro2)) is the union of these $n$ closed orbits, and we denote it by $d(T) = (nu; nv)$ for $u, v \in \mathbb{Z}$ such that $\frac{u}{v} = \delta_{T}$, $u > 0$, and $\gcd(u, v) = 1$ (where $u = 1, v = 0$ if $\delta_T = \infty$). And we call $n$ the *multiplicity* of $d(T)$.* *(b) If $\varphi(C) = C$ for some component $C$ of $\partial \Sigma$ and $T$ is the component $C \times I$ of $\partial M$, the *fractional Dehn twist coefficient* of $C$ (denoted $f_C(\varphi)$) is defined to be $\frac{1}{\delta_T}$ (where $f_C(\varphi) = 0$ if $\delta_T = \infty$). In the case that $\varphi$ takes each component of $\partial \Sigma$ to itself, $\varphi$ is called *right-veering* (resp. *left-veering*) if $f_C(\varphi) > 0$ (resp. $f_C(\varphi) < 0$) for each component $C$ of $\partial \Sigma$.* We note that under the canonical coordinate system on $\partial M$, for each component $T$ of $\partial M$, $\delta_T \in (\mathbb{Q} \cup \{\infty\}) - [-2,2)$. Moreover, if $d(T) = (p;q)$, then $-\frac{1}{2}p < q \leqslant \frac{1}{2}p$, $\frac{p}{q} = \delta_{T_i}$, and $\gcd(p,q)$ is the multiplicity of $d(T)$. And we adopt the following conventions for Dehn fillings of $M$ and slopes on $\partial M$: **Convention 2**. *[\[filling convention\]]{#filling convention label="filling convention"} (a) Let $k$ denote the number of boundary components of $M$. For any multislope $\textbf{s} \in (\mathbb{Q} \cup \{\infty\})^{k}$ of $\partial M$, $M(\textbf{s})$ denotes the Dehn filling of $M$ along $\partial M$ with the multislope $\textbf{s}$.* *(b) Throughout this paper, for a slope $\frac{p}{q}$ on a component $T$ of $\partial M$, we allow $\gcd(p,q) > 1$. In this case, we consider $\frac{p}{q}$ as the corresponding fraction in reduced form, i.e. $\frac{p}{q}$ is identified with $\frac{u}{v}$ for which $u = \frac{p}{\gcd(p,q)}$, $v = \frac{q}{\gcd(p,q)}$.* Now we review some known results on CTF filling slopes of $M$. To be convenient, we state the following two theorems in our setting, although they do not need to restrict $\varphi$ to be pseudo-Anosov. Roberts ([\[Ro1\]](#Ro1), [\[Ro2\]](#Ro2)) proves the following theorem: **Theorem 3** (Roberts). *Suppose that $\Sigma$ has exactly one boundary component.* *(a) If $\varphi$ is right-veering, then $M(s)$ admits a co-orientable taut foliation for any rational slope $s \in (-\infty, 1)$.* *(b) If $\varphi$ is left-veering, then $M(s)$ admits a co-orientable taut foliation for any rational slope $s \in (-1, +\infty)$.* *(c) If $\varphi$ is neither right-veering nor left-veering, then $M(s)$ admits a co-orientable taut foliation for any rational slope $s \in (-\infty, +\infty)$.* *Moreover, the core curve of the filling solid torus is transverse to the foliation in each case.* For the case that $\Sigma$ has multiple boundary components, Kalelkar and Roberts ([\[KalR\]](#KalR)) prove **Theorem 4** (Kalelkar-Roberts). *Let $k$ denote the number of boundary components of $M$. There is a neighborhood $J \subseteq \mathbb{R}^{k}$ of $(0,\ldots,0)$ such that for any rational multislope $(s_1,\ldots,s_k) \in J$, $M(s_1,\ldots,s_k)$ admits a co-orientable taut foliation. Moreover, the core curves of the filling solid tori are transverse to the foliation.* Let $\mathcal{F}^{s}, \mathcal{F}^{u}$ denote the stable and unstable foliations of $\varphi$. **Definition 5**. *[\[co-orientable\]]{#co-orientable label="co-orientable"} We call $\varphi$ *co-orientable* if $\mathcal{F}^{s}$ is co-orientable, and we call $\varphi$ *co-orientation-preserving* (resp. *co-orientation-reversing*) if $\varphi$ is co-orientable and preserves (resp. reverses) the co-orientation on $\mathcal{F}^{s}$.* **Remark 6**. *In Definition [\[co-orientable\]](#co-orientable){reference-type="ref" reference="co-orientable"}, $\mathcal{F}^{s}$ can be replaced by $\mathcal{F}^{u}$. If $\mathcal{F}^{s}$ is co-oriented, then the co-orientation on $\mathcal{F}^{s}$ defines continuously varying orientations on the leaves of $\mathcal{F}^{u}$, which implies that $\mathcal{F}^{u}$ is orientable and thus is co-orientable. It follows that $\mathcal{F}^{u}$ is co-orientable if and only if $\mathcal{F}^{s}$ is co-orientable, and $\varphi$ preserves (resp. reverses) the co-orientation on $\mathcal{F}^{u}$ if and only if $\varphi$ is co-orientation-preserving (resp. co-orientation-reversing).* We refer to [\[T\]](#T), [\[M\]](#M) for some information on the co-orientability of $\varphi$ and refer to [\[BB, Lemma 4.3\]](#BB), [\[DGT\]](#DGT) for criterions for $\varphi$ being co-orientation-preserving/reversing. $\varphi$ is co-orientable implies that each singularity of $\mathcal{F}^{s}$ in $Int(\Sigma)$ has an even number of prongs and each component of $\partial \Sigma$ contains an even number of singularities of $\mathcal{F}^{s}$. Note that if $\varphi$ is non-co-orientable, then $\Sigma$ has a double cover branched over $\{\text{singularities of } \mathcal{F}^{s} \text{ in } Int(\Sigma) \text{ with an odd number of prongs}\}$, denoted $\widetilde{\Sigma}$, such that the pull-back of $\mathcal{F}^{s}$ to $\widetilde{\Sigma}$ is co-orientable, and $\varphi$ lifts to two co-orientable pseudo-Anosov homeomorphisms $\varphi_1, \varphi_2$ on $\widetilde{\Sigma}$ which are co-orientation-preserving, co-orientation-reversing respectively, see for example [\[LT, page 14\]](#LT). For general pseudo-Anosov mapping tori of compact surfaces with more than one boundary components, co-orientation-preserving is the only known case to have an explicit multi-interval such that all rational multislopes in it are CTF filling multislopes. The following theorem is implicitly contained in [\[Ga2\]](#Ga2): **Theorem 7** (Gabai). *Assume that $\varphi$ is co-orientable and co-orientation-preserving. Let $k$ denote the number of boundary components of $M$. For any multislope $(s_1, \ldots, s_k) \in (\mathbb{Q} \cup \{\infty\})^{k}$ such that each $s_i$ is not the degeneracy slope of the corresponding boundary component, $M(s_1, \ldots, s_k)$ admits a co-orientable taut foliation. Moreover, the core curves of the filling solid tori are transverse to the foliation.* Unlike the co-orientation-preserving case, many $M(s_1,\ldots,s_k)$ (with each $s_i$ being not the degeneracy slope) don't admit co-orientable taut foliation when $\varphi$ is co-orientation-reversing, since there are many hyperbolic fibered L-space knots in $S^{3}$ and lens spaces that have co-orientation-reversing monodromy (see Proposition [Proposition 13](#pretzel){reference-type="ref" reference="pretzel"}, Examples [\[v0751\]](#v0751){reference-type="ref" reference="v0751"}$\sim$[\[nonsharp\]](#nonsharp){reference-type="ref" reference="nonsharp"}). We concentrate on the co-orientation-reversing case in this paper. **Remark 8**. *Foliations in the above theorems are very useful in showing that many of these Dehn fillings have left orderable fundamental group. See [\[BH\]](#BH) and [\[H\]](#H) for results on LO filling slopes derived from foliations in Theorem [Theorem 3](#Roberts){reference-type="ref" reference="Roberts"}. And see [\[Z\]](#Z) for a result on LO filling multislopes derived from foliations in Theorem [Theorem 7](#co-orientation-preserving){reference-type="ref" reference="co-orientation-preserving"}.* ## The main results {#subsection 1.1} Let $T_1, \ldots, T_k$ denote the boundary components of $M$. For each $i \in \{1, \ldots, k\}$, we choose a boundary component $C_i$ of $\Sigma$ such that $C_i \times \{0\} \subseteq T_i$, and let $c_i$ denote the order of $C_i$ under $\varphi$ (i.e. $c_i = \min \{k \in \mathbb{N}_+ \mid \varphi^{k}(C_i) = C_i\}$). Let $(p_i; q_i)$ denote the degeneracy locus of the suspension flow of $\varphi$ on each $T_i$. We note that if $\varphi$ is co-orientable, then $2 \mid p_i$, and moreover, if $\varphi$ is co-orientation-reversing, then $q_i \equiv c_i \text{ } (\text{mod } 2)$ (this is because $2 \mid q_i$ if and only if $\varphi^{c_i}$ is co-orientation-preserving, see Remark [\[remark\]](#remark){reference-type="ref" reference="remark"} (a)). **Theorem 9**. *Assume that $\varphi$ is co-orientable and co-orientation-reversing (then $2 \mid p_i$ and $q_i \equiv c_i \text{ } (\emph{mod } 2)$ for each $i$). Then $M(s_1,\ldots,s_k)$ admits a co-orientable taut foliation for any multislope $(s_1,\ldots,s_k) \in (J_1 \times \ldots \times J_k) \cap (\mathbb{Q} \cup \{\infty\})^{k}$, where each $J_i$ is a set of slopes on $T_i$ such that* *$$J_i = \begin{cases} (-\infty, \frac{p_i}{q_i + c_i}) \cup (\frac{p_i}{q_i - c_i}, +\infty) \cup \{\infty\} & \emph{if } q_i > c_i > 0 \\ (-\infty, \frac{p_i}{2q_i}) & \emph{if } q_i = c_i > 0 \\ (-\frac{p_i}{c_i - q_i}, \frac{p_i}{q_i + c_i}) & \emph{if } c_i > q_i \geqslant 0 \\ (-\frac{p_i}{|q_i| + c_i}, \frac{p_i}{c_i - |q_i|}) & \emph{if } - c_i < q_i < 0 \\ (-\frac{p_i}{2|q_i|}, +\infty) & \emph{if } q_i = -c_i < 0 \\ (-\infty, -\frac{p_i}{|q_i| - c_i}) \cup (-\frac{p_i}{|q_i| + c_i}, +\infty) \cup \{\infty\} & \emph{if } q_i < -c_i < 0. \end{cases}$$ Moreover, the core curves of the filling solid tori are transverse to the leaves of these foliations.* In the case that $\varphi$ preserves each component of $\partial \Sigma$, $c_i = 1$ for each $1 \leqslant i \leqslant k$. **Corollary 10**. *Assume that $\varphi$ is co-orientable, co-orientation-reversing and $\varphi(C) = C$ for each component $C$ of $\partial \Sigma$. Then $M(s_1,\ldots,s_k)$ admits a co-orientable taut foliation for any multislope $(s_1,\ldots,s_k) \in (J_1 \times \ldots \times J_k) \cap (\mathbb{Q} \cup \{\infty\})^{k}$, where each $J_i$ is a set of slopes on $T_i$ such that* *$$J_i = \begin{cases} (-\infty, \frac{p_i}{q_i + 1}) \cup (\frac{p_i}{q_i - 1}, +\infty) \cup \{\infty\} & \emph{if } q_i > 1 \\ (-\infty, \frac{p_i}{2}) & \emph{if } q_i = 1 \\ (-\frac{p_i}{2}, +\infty) & \emph{if } q_i = -1 \\ (-\infty, -\frac{p_i}{|q_i| - 1}) \cup (-\frac{p_i}{|q_i| + 1}, +\infty) \cup \{\infty\} & \emph{if } q_i < -1. \end{cases}$$* **Remark 11**. *[\[remark\]]{#remark label="remark"} (a) Note that $p_i$ is the number of singular points of $\mathcal{F}^{s}$ contained in $C_i$. We assign each $C_i$ an orientation consistent with the orientation on the longitude of $T_i$, and let $v_1,\ldots,v_{p_i}$ denote the $p_i$ singular points of $\mathcal{F}^{s}$ in $C_i$ (consecutive along the orientation on $C_i$). Then $\varphi^{c_i}(v_j) = v_{j+q_i}$ (mod $p_i$) for each $1 \leqslant j \leqslant p_i$.* *(b) Assume that $\varphi$ is co-orientable, co-orientation-reversing and $\partial \Sigma$ is connected. Then the degeneracy slope of $\partial M$ is not $\infty$, and so $\partial \Sigma$ has nonzero fractional Dehn twist coefficient. Thus, one of Theorem [Theorem 3](#Roberts){reference-type="ref" reference="Roberts"} (a), (b) applies to $M$ and Theorem [Theorem 3](#Roberts){reference-type="ref" reference="Roberts"} (c) doesn't apply. In this case, Theorem [Theorem 9](#main){reference-type="ref" reference="main"} expands the known range of CTF filling slopes when $(p_1;q_1) \ne (2;1)$.* *(c) In Theorem [Theorem 9](#main){reference-type="ref" reference="main"}, if we regard the set of slopes $\mathbb{R} \cup \{\infty\}$ on each $T_i$ as $\mathbb{R}P^{1} \cong S^{1}$, then $J_i$ is one of the two open intervals in $\mathbb{R} \cup \{\infty\}$ between $\frac{p_i}{q_i + c_i}, \frac{p_i}{q_i - c_i}$ that doesn't contain the degeneracy slope $\frac{p_i}{q_i}$ on $T_i$. To be convenient, we will call $J_i$ an interval and call $J_1 \times \ldots \times J_k$ a multi-interval.* ## The application to Dehn surgeries on knots In this subsection, we apply Corollary [Corollary 10](#open book){reference-type="ref" reference="open book"} to Dehn surgeries on knots in $S^{3}$ or spherical manifolds. At first, we describe some properties of L-space filling slopes on knot manifolds. A knot manifold is *Floer simple* if it has at least two distinct L-space filling slopes (compare with [\[RR, Proposition 1.3\]](#RR)), and a knot in a closed $3$-manifold is called an *L-space knot* if its exterior is Floer simple. We note from [\[RR\]](#RR) that for a Floer simple knot manifold $N$, there is an interval $\mathcal{L}(N)$ of slopes such that $$\{\text{L-space filling slopes of } N\} = \mathcal{L}(N) \cap (\mathbb{Q} \cup \{\infty\}),$$ and in particular, $\mathcal{L}(N)$ either consists of all slopes except the homological longitude, or is a closed interval ([\[RR, Theorem 1.6\]](#RR)). We call $\mathcal{L}(N)$ the *maximal L-space filling interval* of $N$. If $\mathcal{L}(N)$ is a closed interval, then we call its complement the *maximal NLS filling interval* of $N$. A knot in a closed $3$-manifold is called a *fibered knot* if its exterior is an once-punctured surface bundle over $S^{1}$. Note that all L-space knots in $S^{3}$ are fibered ([\[Gh\]](#Gh), [\[Ni\]](#Ni)). Let $K$ be a hyperbolic fibered L-space knot in a closed $3$-manifold $W$ which is either $S^{3}$ or a spherical manifold, and we fix an orientation on $W$. Let $N$ denote the exterior of $K$ in $W$, let $T = \partial N$, and let $\phi$ denote the monodromy of $N$. We assign $N$ an orientation induced from $W$. Combining Theorem [Theorem 3](#Roberts){reference-type="ref" reference="Roberts"} with [\[OS\]](#OS), $\phi$ is either left-veering or right-veering. Fix the canonical coordinate system on $\partial N$. Let $\delta_T, d(T)$ denote the degeneracy slope on $T$ and the degeneracy locus of the suspension flow of $\phi$ on $T$, respectively, and we denote $d(T)$ by $(p;q)$ (then $\delta_T = \frac{p}{q}$). Let $m_K$ denote the slope on $\partial N$ with $N(m_K) = W$. Then $$\Delta(m_K, d(T)) = \gcd(p, q) \Delta(m_K, \delta_T) < 2$$ since $W$ admits no essential lamination ([\[GO\]](#GO)), where $\Delta(m_K, d(T)), \Delta(m_K, \delta_T)$ denote the minimal geometric intersection numbers of $m_K, d(T)$ and of $m_K, \delta_T$. Thus $m_K$ belongs to one of Cases [\[case1\]](#case1){reference-type="ref" reference="case1"}$\sim$[\[case3\]](#case3){reference-type="ref" reference="case3"}: **Case 1**. *[\[case1\]]{#case1 label="case1"} $m_K = \delta_T$.* **Case 2**. *[\[case2\]]{#case2 label="case2"} $\Delta(m_K, \lambda_T) = 1$. In this case, either (1) $m_K = \infty$ or (2) $m_K = 1$ and $\delta_T = 2$. In the case (2), we can choose the opposite orientation on $W$ to make $m_K$ become $\infty$ under the coordinate system induced from this orientation (see Conventions [\[slope\]](#slope){reference-type="ref" reference="slope"}, [\[orientation\]](#orientation){reference-type="ref" reference="orientation"}). In the case of $m_K = \infty$, $|q| = 1$ and $2 \leqslant p \leqslant 4g-2$ ([\[Ga3\]](#Ga3)). Thus $d(T)$ has multiplicity $1$.* **Case 3**. *[\[case3\]]{#case3 label="case3"} $m_K \ne \delta_T$ and $\Delta(m_K, \lambda_T) > 1$. Equivalently, $m_K \notin \{\delta_T, \infty, 1\}$. In this case, $\gcd(p,q) = \Delta(m_K, \delta_T) = 1$, and thus $d(T)$ has multiplicity $1$.* We call $K$ a *type-I* (resp. *type-II*, *type-III*) knot in $W$ if Case [\[case1\]](#case1){reference-type="ref" reference="case1"} (resp. Case [\[case2\]](#case2){reference-type="ref" reference="case2"}, Case [\[case3\]](#case3){reference-type="ref" reference="case3"}) holds. As explained in Case [\[case2\]](#case2){reference-type="ref" reference="case2"}, we may always assume $m_K = \infty$ when $K$ is a type-II knot in $W$. Let $g$ denote the fibered genus of $N$. When $W = S^{3}$, $K$ can only be a type-II knot, and the maximal NLS filling interval of $N$ is $(-2g+1, +\infty)$ if $\phi$ is left-veering and is $(-\infty, 2g-1)$ if $\phi$ is right-veering ([\[KMOS\]](#KMOS)). However, when $W$ is a lens space and $K$ is a type-II knot in $W$, the maximal NLS filling interval of $N$ doesn't have to be $(-2g+1, +\infty)$ or $(-\infty, 2g-1)$ (for instance, Examples [\[v0751\]](#v0751){reference-type="ref" reference="v0751"}, [\[example1\]](#example1){reference-type="ref" reference="example1"}). Also, $K$ is possible to be a type-I knot or a type-III knot when $W$ is a lens space (Examples [\[v0751\]](#v0751){reference-type="ref" reference="v0751"}, [\[general\]](#general){reference-type="ref" reference="general"}). **Remark 12**. *[\[veering\]]{#veering label="veering"} We note that if $\phi$ is left-veering (resp. right-veering), then we can choose an opposite orientation on $W$ to get a right-veering (resp. left-veering) monodromy of $N$. In the case of $\delta_T \ne 2$, the meridian on $T$ doesn't change, and any other slope $s \in \mathbb{Q}$ becomes $-s$ under the canonical coordinate system induced from the opposite orientation on $N$.* We give some examples in the remainder of this subsection. For $q \in \mathbb{N}$ with $q \geqslant 3$, the $(-2,3,2q+1)$-pretzel knot $K$ in $S^{3}$ is a hyperbolic L-space knot with Seifert genus $g(K) = q+2$ ([\[LM\]](#LM), [\[O1\]](#O1)). We have **Proposition 13**. *Let $K$ denote the $(-2,3,2q+1)$-pretzel knot in $S^{3}$ with $q \geqslant 3$. We fix an orientation on $S^{3}$ so that $K$ has right-veering monodromy.* *(a) $K$ has co-orientable and co-orientation-reversing monodromy.* *(b) The degeneracy slope on $K$ is $4g(K)-2$.* *(c) All rational slopes in $(-\infty, 2g(K) - 1)$ are CTF surgery slopes of $K$.* *(d) For each $n \in \mathbb{N}$ with $n \geqslant 2$, the $n$-fold cyclic branched cover of $S^{3}$ over $K$, denoted $\Sigma_n(K)$, admits a co-orientable taut foliation.* **Remark 14**. *(1) Proposition [Proposition 13](#pretzel){reference-type="ref" reference="pretzel"} (c) has been proved in [\[Kri\]](#Kri).* *(2) In Proposition [Proposition 13](#pretzel){reference-type="ref" reference="pretzel"} (d), the foliations are obtained from Theorem [Theorem 7](#co-orientation-preserving){reference-type="ref" reference="co-orientation-preserving"} when $n$ is even and from Corollary [Corollary 10](#open book){reference-type="ref" reference="open book"} when $n$ is odd. When $n \geqslant 4g(K) - 2 = 4(q+2)-2 = 4q+6$, it can be deduced from Theorem [Theorem 3](#Roberts){reference-type="ref" reference="Roberts"} that $\Sigma_n(K)$ admits a co-orientable taut foliation (see [\[BH\]](#BH) for an explanation). It follows from (d) and [\[OS\]](#OS) that $\Sigma_n(K)$ is a non-L-space. In fact, it's already known from [\[BBG, Theorem 1.2\]](#BBG) that $\Sigma_n(K)$ is a non-L-space (see [\[Nie, Subsection 3.2\]](#Nie) for an explanation that [\[BBG, Theorem 1.2\]](#BBG) can be applied to $\Sigma_n(K)$). By [\[BGH, Theorem 1.2\]](#BGH), $\Sigma_n(K)$ also has left orderable fundamental group.* It follows that **Corollary 15**. *For every $(-2,3,2q+1)$-pretzel knot $K$ in $S^{3}$ with $q \geqslant 3$, and for each $n \geqslant 2$, the $n$-fold cyclic branched cover of $K$ satisfies each of (1), (2), (3) of the L-space conjecture.* In the census [\[D3\]](#D3), there are $3242$ $1$-cusped hyperbolic fibered $3$-manifolds with monodromies satisfying that all singularities of the stable foliations have even number of prongs. $805$ of them have co-orientation-preserving monodromy, $2214$ of them have co-orientation-reversing monodromy, and the rest $223$ of them don't have co-orientable monodromy. Let $\mathcal{N}$ denote the set of these $2214$ manifolds with co-orientation-reversing monodromy. For each $N \in \mathcal{N}$, we can obtain a CTF filling interval from our main results. In [\[D1\]](#D1), the manifolds in $\mathcal{N}$ are tested for being Floer simple, and some Dehn fillings of them are tested for being L-spaces. We can find some L-space filling slopes and NLS filling slopes of the manifolds in $\mathcal{N}$ from [\[D2\]](#D2) (the data associated to [\[D1\]](#D1)). We list some examples in $\mathcal{N}$ below with explicit CTF filling intervals obtained from Corollary [Corollary 10](#open book){reference-type="ref" reference="open book"}, as well as other data, such as fibered genera, degeneracy slopes, and maximal NLS filling intervals. We first explain how this data is obtained. For each $1$-cusped manifold $N$ in Examples [\[v0751\]](#v0751){reference-type="ref" reference="v0751"}$\sim$[\[nonsharp\]](#nonsharp){reference-type="ref" reference="nonsharp"}, $\bullet$ The fibered genus and the degeneracy slope of $N$ are known from [\[D3\]](#D3), and $N$ is the complement of a hyperbolic fibered L-space knot in some spherical manifold of type-II or type-III ($N$ is also possible to be the complement of a type-I knot in another spherical manifold). By the degeneracy slope of $N$, we can get a CTF filling interval obtained from Corollary [Corollary 10](#open book){reference-type="ref" reference="open book"} directly. $\bullet$ If $N$ is contained in Examples [\[v0751\]](#v0751){reference-type="ref" reference="v0751"}$\sim$[\[nonspherical\]](#nonspherical){reference-type="ref" reference="nonspherical"}, then the two endpoints of the obtained CTF filling interval of $N$ are verified to be L-space filling slopes in [\[D2\]](#D2). Combined with [\[OS\]](#OS) and [\[RR\]](#RR), the obtained CTF filling interval of $N$ is exactly the maximal NLS filling interval of $N$. $\bullet$ Spherical Dehn fillings of $N$ are known from comparing [\[D2\]](#D2) with snapPy [\[CDGW\]](#CDGW). Let $L(p,q)$ denote the lens space obtained from Dehn surgery along the unknot in $S^{3}$ with the slope $-\frac{p}{q}$. If we write $L(p,q)$ such that $p$ is specified but $q$ is not specified, that means it's valid for the given $p$ and some $q$, for example, $L(5,q)$ denotes an unspecified element of $\{L(5,1), L(5,2)\}$. Also note that all slopes in the following examples are consistent with Convention [\[slope\]](#slope){reference-type="ref" reference="slope"} (instead of the slopes in snapPy). Similar to Convention [\[filling convention\]](#filling convention){reference-type="ref" reference="filling convention"}, for any $1$-cusped $3$-manifold $N$, we denote by $N(s)$ the Dehn filling of $N$ with slope $s$ for any $s \in \mathbb{Q} \cup \{\infty\}$. **Example 16**. *[\[v0751\]]{#v0751 label="v0751"} We list some examples in $\mathcal{N}$, each of which has three distinct lens space Dehn fillings, and it can be identified with the complement of a type-I$\sim$III knot in these three lens spaces respectively. The obtained CTF filling intervals of these examples are equal to their maximal NLS filling intervals.* *\| l \| l \| l \| l \| l \| manifold & genus & degeneracy slope & obtained CTF filling interval & maximal NLS filling interval\ $m122$ & $g = 2$ & $\delta = 4$ & $(-\infty, 2)$ & $(-\infty, 2)$\ &\ $m280$ & $g = 2$ & $\delta = -4$ & $(-2, +\infty)$ & $(-2, +\infty)$\ &\ $v0751$ & $g = 3$ & $\delta = -6$ & $(-3, +\infty)$ & $(-3, +\infty)$\ &\ * **Example 17**. *[\[general\]]{#general label="general"} We give some more examples in $\mathcal{N}$ which are complements of type-III knots in lens spaces, and the obtained CTF filling intervals of them are equal to their maximal NLS filling intervals.* *\| l \| l \| l \| l \| l \| manifold & genus & degeneracy slope & obtained CTF filling interval & maximal NLS filling interval\ $s297$ & $g = 2$ & $\delta = -6$ & $(-3,+\infty)$ & $(-3,+\infty)$\ &\ $s408$ & $g = 3$ & $\delta = -8$ & $(-4,+\infty)$ & $(-4,+\infty)$\ &\ $o9_{26541}$ & $g = 3$ & $\delta = -\frac{8}{3}$ & $(-\infty, -4) \cup (2, +\infty) \cup \{\infty\}$ & $(-\infty, -4) \cup (2, +\infty) \cup \{\infty\}$\ &\ * **Example 18**. *[\[example1\]]{#example1 label="example1"} We list some examples in $\mathcal{N}$, each of which is the complement of a type-II knot in some $L(p,q)$ with relatively small $p$, and its obtained CTF filling interval is exactly its maximal NLS filling interval.* ------------ ----------- ------------------ ------------------------ ----------------------- ------------------------------------- *manifold* *genus* *degeneracy* *obtained CTF filling* *maximal NLS filling* *lens space filling* *slope* *interval* *interval* *$m146$* *$g = 3$* *$\delta = -10$* *$(-5, +\infty)$* *$(-5, +\infty)$* *$m146(\infty) = \mathbb{R}P^{3}$* *$v2585$* *$g = 4$* *$\delta = 14$* *$(-\infty,7)$* *$(-\infty, 7)$* *$v2585(\infty) = \mathbb{R}P^{3}$* *$m036$* *$g = 2$* *$\delta = -6$* *$(-3, +\infty)$* *$(-3, +\infty)$* *$m036(\infty) = L(3,1)$* *$s313$* *$g = 3$* *$\delta = -10$* *$(-5, +\infty)$* *$(-5, +\infty)$* *$s313(\infty) = L(3,1)$* *$v3327$* *$g = 3$* *$\delta = 10$* *$(-\infty, 5)$* *$(-\infty, 5)$* *$v3327(\infty) = L(3,1)$* ------------ ----------- ------------------ ------------------------ ----------------------- ------------------------------------- ---------------- ----------- ------------------ ------------------------ ----------------------- --------------------------------- *manifold* *genus* *degeneracy* *obtained CTF filling* *maximal NLS filling* *lens space filling* *slope* *interval* *interval* *$v3248$* *$g = 4$* *$\delta = -14$* *$(-7, +\infty)$* *$(-7, +\infty)$* *$v3248(\infty) = L(3,1)$* *$o9_{36980}$* *$g = 4$* *$\delta = 14$* *$(-\infty, 7)$* *$(-\infty, 7)$* *$o9_{36980}(\infty) = L(3,1)$* *$s479$* *$g = 2$* *$\delta = 6$* *$(-\infty, 3)$* *$(-\infty, 3)$* *$s479(\infty) = L(4,1)$* *$v2296$* *$g = 3$* *$\delta = -10$* *$(-5, +\infty)$* *$(-5, +\infty)$* *$v2296(\infty) = L(4,1)$* *$t11938$* *$g = 3$* *$\delta = -10$* *$(-5, +\infty)$* *$(-5, +\infty)$* *$t11938(\infty) = L(4,1)$* *$t11829$* *$g = 4$* *$\delta = -14$* *$(-7, +\infty)$* *$(-7, +\infty)$* *$t11829(\infty) = L(4,1)$* *$o9_{39567}$* *$g = 5$* *$\delta = -18$* *$(-9, +\infty)$* *$(-9, +\infty)$* *$o9_{39567}(\infty) = L(4,1)$* *$v1682$* *$g = 2$* *$\delta = -6$* *$(-3, +\infty)$* *$(-3, +\infty)$* *$v1682(\infty) = L(5,q)$* *$t07148$* *$g = 3$* *$\delta = -10$* *$(-5, +\infty)$* *$(-5, +\infty)$* *$t07148(\infty) = L(5,q)$* *$t09415$* *$g = 3$* *$\delta = 8$* *$(-\infty, 4)$* *$(-\infty, 4)$* *$t09417(\infty) = L(5,q)$* *$o9_{35581}$* *$g = 3$* *$\delta = -10$* *$(-5, +\infty)$* *$(-5, +\infty)$* *$o9_{35581}(\infty) = L(5,q)$* *$t06948$* *$g = 4$* *$\delta = 12$* *$(-\infty, 6)$* *$(-\infty, 6)$* *$t06948(\infty) = L(5,q)$* *$o9_{35298}$* *$g = 4$* *$\delta = 14$* *$(-\infty, 7)$* *$(-\infty, 7)$* *$o9_{35298}(\infty) = L(5,q)$* *$t04406$* *$g = 2$* *$\delta = -6$* *$(-3, +\infty)$* *$(-3, +\infty)$* *$t04406(\infty) = L(6,1)$* *$o9_{18836}$* *$g = 3$* *$\delta = -10$* *$(-5, +\infty)$* *$(-5, +\infty)$* *$o9_{18836}(\infty) = L(6,1)$* *$v2678$* *$g = 2$* *$\delta = 6$* *$(-\infty, 3)$* *$(-\infty, 3)$* *$v2678(\infty) = L(7,q)$* *$o9_{10415}$* *$g = 2$* *$\delta = -6$* *$(-3, +\infty)$* *$(-3, +\infty)$* *$o9_{10415}(\infty) = L(7,q)$* *$t05634$* *$g = 3$* *$\delta = 8$* *$(-\infty, 4)$* *$(-\infty, 4)$* *$t05634(\infty) = L(7,q)$* *$o9_{14062}$* *$g = 3$* *$\delta = -6$* *$(-3, +\infty)$* *$(-3, +\infty)$* *$o9_{14062}(\infty) = L(7,q)$* *$o9_{34972}$* *$g = 4$* *$\delta = -12$* *$(-6, +\infty)$* *$(-6, +\infty)$* *$o9_{34972}(\infty) = L(7,q)$* *$v1076$* *$g = 2$* *$\delta = 4$* *$(-\infty, 2)$* *$(-\infty, 2)$* *$v1076(\infty) = L(9,q)$* *$o9_{21619}$* *$g = 3$* *$\delta = -8$* *$(-4, +\infty)$* *$(-4, +\infty)$* *$o9_{21619}(\infty) = L(9,q)$* ---------------- ----------- ------------------ ------------------------ ----------------------- --------------------------------- **Example 19**. *[\[nonspherical\]]{#nonspherical label="nonspherical"} Here are some examples in $\mathcal{N}$ which are complements of type-II knots in some spherical manifolds other than lens spaces, and their obtained CTF filling intervals are equal to their maximal NLS filling intervals.* *\| l \| l \| l \| l \| l \| manifold & genus & degeneracy slope & obtained CTF filling interval & maximal NLS filling interval\ $t08752$ & $g = 2$ & $\delta = -6$ & $(-3,+\infty)$ & $(-3,+\infty)$\ &\ $o9_{23699}$ & $g = 2$ & $\delta = -6$ & $(-3,+\infty)$ & $(-3,+\infty)$\ &\ * **Example 20**. *[\[nonsharp\]]{#nonsharp label="nonsharp"} The CTF filling interval obtained from Corollary [Corollary 10](#open book){reference-type="ref" reference="open book"} may not be the maximal NLS filling interval for every Floer simple $1$-cusped manifold satisfying the assumptions of Corollary [Corollary 10](#open book){reference-type="ref" reference="open book"}. For instance, $o9_{19364}$ is an example in $\mathcal{N}$ which is the complement of an L-space knot in $S^{3}$ with Seifert genus $14$ and degeneracy slope $48$. Corollary [Corollary 10](#open book){reference-type="ref" reference="open book"} gives a CTF filling interval $(-\infty,24)$, but the maximal NLS filling interval of $o9_{19364}$ is $(-\infty, 27)$.* ## Organization In Section [2](#section 2){reference-type="ref" reference="section 2"}, we set up some conventions and review some background material on branched surfaces. We prove Theorem [Theorem 9](#main){reference-type="ref" reference="main"} in Section [3](#section 3){reference-type="ref" reference="section 3"}. In Subsection [3.1](#subsection 3.1){reference-type="ref" reference="subsection 3.1"}, we construct a branched surface $B(\alpha)$ in $M$. We prove that $B(\alpha)$ is a laminar branched surface in Subsection [3.2](#subsection 3.2){reference-type="ref" reference="subsection 3.2"}. In Subsection [3.3](#subsection 3.3){reference-type="ref" reference="subsection 3.3"}, we describe the boundary train tracks of $B(\alpha)$ in $\partial M$ and choose some simple closed curves carried by it. In Subsection [3.4](#subsection 3.4){reference-type="ref" reference="subsection 3.4"}, we prove that the boundary train tracks of $B(\alpha)$ realize all rational multislopes contained in the multi-interval given in Theorem [Theorem 9](#main){reference-type="ref" reference="main"}. And we complete the proof of Theorem [Theorem 9](#main){reference-type="ref" reference="main"} in Subsection [3.5](#proof){reference-type="ref" reference="proof"}. In Section [4](#section 4){reference-type="ref" reference="section 4"}, we prove Proposition [Proposition 13](#pretzel){reference-type="ref" reference="pretzel"}. ## Acknowledgements The author wishes to thank Xingru Zhang for his guidance, patience, encouragement, and for many helpful discussions and comments on this work. He is grateful to Nathan Dunfield for providing him the census of examples [\[D3\]](#D3) and answering him several questions about the examples. He thanks Cameron Gordon and Rachel Roberts for telling him the fractional Dehn twist coefficients of $(-2,3,2q+1)$ pretzel knots and some relevant works. He thanks Chi Cheuk Tsang for telling him that the monodromies of $(-2,3,2q+1)$-pretzel knots have co-orientable stable foliations, and for some helpful comments. He thanks Cagatay Kutluhan, Johanna Mangahas, William Menasco, Tao Li and Diego Santoro for some helpful conversations. He thanks Tech Topology Summer School 2023 and its organizers for providing him a great chance to communicate and discuss, and for their travel support. # Preliminaries {#section 2} ## Conventions {#subsection 2.1} For a set $X$, let $|X|$ denote the cardinality of $X$. For two metric spaces $A$ and $B$, let $A \setminus \setminus B$ denote the closure of $A - B$ under the path metric. For a link manifold $N$ such that $\partial N$ is a union of tori $\bigcup_{i=1}^{n}S_n$, a *multislope* on $\partial N$ is an $n$-tuple of slopes on $S_1, \ldots, S_n$ respectively. For a link in some closed $3$-manifold, the multislopes on it are defined as the multislopes on its exterior. Now we illustrate our conventions of slopes and orientations for pseudo-Anosov mapping tori of compact orientable surfaces. Let $\Sigma$ be a compact orientable surface with nonempty boundary and let $\varphi: \Sigma \to \Sigma$ be an orientation-preserving pseudo-Anosov homeomorphism. Let $M = \Sigma \times I / \stackrel{\varphi}{\sim}$ be the mapping torus of $\Sigma$ over $\varphi$, and we fix an orientation on $M$. **Notation 21**. *(a) For two slopes $\alpha, \beta$ in the same boundary component of $M$, let $\Delta(\alpha, \beta)$ denote the minimal geometric intersection number of $\alpha, \beta$.* *(b) For two closed oriented curves $\gamma, \eta$ in the same boundary component of $M$, let $\langle \gamma, \eta \rangle$ denote the algebraic intersection number of $\gamma, \eta$ (see Convention [\[orientation\]](#orientation){reference-type="ref" reference="orientation"} (d) for more details on this setting).* We adopt the following conventions for slopes on the boundary components of $M$, as described in [\[Ro2\]](#Ro2): **Convention 22** (Slope conventions). *[\[slope\]]{#slope label="slope"} Let $T$ be a boundary component of $M$. Let $\delta_T$ denote the degeneracy slope of $T$.* *(a) We call the slope of $T \cap (\Sigma \times \{0\})$ on $T$ the *longitude* of $T$ and denote it by $\lambda_T$. We choose a slope $\mu_T$ on $T$ such that $\Delta(\lambda_T, \mu_T) = 1$ and* *$\Delta(\mu_T,\delta_T) \leqslant \Delta(s,\delta_T)$ for any slope $s$ of $T$ with $\Delta(\lambda_T, s) = 1$,* *and we call $\mu_T$ the *meridian* of $T$. We fix a canonical orientation on $\lambda_T$ and on $\mu_T$, see Convention [\[orientation\]](#orientation){reference-type="ref" reference="orientation"} (c) for details. Note that $\mu_T$ has a unique choice if $\Delta(\lambda_T, \delta_T) \ne 2$. See (c) for the choice of $\mu_T$ in the case of $\Delta(\lambda_T, \delta_T) = 2$.* *(b) For an essential simple closed curve $\gamma$ on $T$, we identify the slope of $\gamma$ with the number $$\frac{\langle \gamma, \lambda_T \rangle}{\langle \mu_T, \gamma \rangle} \in \mathbb{Q} \cup \{\infty\}.$$* *(c) If $\Delta(\lambda_T, \delta_T) = 2$, then there are two choices of $\mu_T$ satisfying (a). $\delta_T$ is equal to $-2$, $2$ in these two choices respectively. We choose $\mu_T$ so that $\delta_T = 2$.* Next, we assign orientations to $\Sigma \times \{0\}$ and the meridians and longitudes on $\partial M$. **Convention 23** (Orientation conventions). *[\[orientation\]]{#orientation label="orientation"} (a) Note that each orientation on $\Sigma \times \{0\}$ determines a normal vector field with respect to the orientation on $M$. We assign $\Sigma \times \{0\}$ an orientation so that the induced normal vector field is consistent with the increasing orientation on the second coordinates in $\Sigma \times I$.* *(b) The orientation on $\Sigma \times \{0\}$ induces an orientation on $\Sigma$. For any oriented curve in $\Sigma$, it has well-defined left and right sides with respect to the orientation on $\Sigma$. Throughout this paper, the left and right side of an oriented curve on $\Sigma$ will always be with respect to the orientation on $\Sigma$. Each boundary component of $\Sigma$ has an orientation induced from the orientation on $\Sigma$ such that, for any properly embedded arc $\gamma: I \to \Sigma$, choose a normal vector field $\{v(x) \mid x \in \gamma(I)\}$ pointing to the right side of $\gamma$, then $v(\gamma(0))$ is consistent with the positive orientation on $\partial \Sigma$ and $v(\gamma(1))$ is consistent with the negative orientation on $\partial \Sigma$. For each component $C$ of $\partial \Sigma$, we also assign $C \times \{0\} \subseteq \partial M$ an orientation consistent with the orientation on $C$.* *(c) Let $T$ be a component of $\partial M$. We assign the longitude of $T$ an orientation consistent with the positive orientation on each component of $T \cap (\partial \Sigma \times \{0\})$. We choose a curve $\gamma$ on $T$ such that $\gamma$ represents the meridian on $T$ and $\gamma$ is transverse to the fibered surfaces $\{\Sigma \times \{t\} \mid t \in I\}$, and we assign $\gamma$ an orientation consistent with the increasing orientation on the second coordinates. We assign an orientation to the meridian on $T$ consistent with the orientation on $\gamma$.* *(d) For a boundary component $T$ of $M$, we set $\langle \mu_T, \lambda_T \rangle = - \langle \lambda_T, \mu_T \rangle = 1$, where $\mu_T, \lambda_T$ denotes the meridian and longitude on $T$ respectively.* At last, we describe the relation between our canonical coordinate system on $\partial M$ and the standard meridian/longitude coordinate system for knots in $S^{3}$, when $M$ is the exterior of a fibered knot in $S^{3}$. **Remark 24**. *[\[slope consistent\]]{#slope consistent label="slope consistent"} Let $M$ be the exterior of a fibered knot $K$ in $S^{3}$. Let $T = \partial M$, and let $\mu_K, \lambda_K$ denote the standard meridian and longitude of $K$ in $S^{3}$. As explained in [\[Ro2, Corollary 7.4\]](#Ro2), $\lambda_K = \lambda_T, \mu_K = \mu_T$ when $\Delta(\lambda_T, \delta_T) \ne 2$. In the case of $\Delta(\lambda_T, \delta_T) = 2$, the two choices of orientations on $M$ give two distinct canonical coordinate systems on $T$, denoted $\mathcal{C}_1, \mathcal{C}_2$. By Convention [\[slope\]](#slope){reference-type="ref" reference="slope"} (c), $\delta_T$ has slope $2$ with respect to both of $\mathcal{C}_1, \mathcal{C}_2$, which implies that the slopes $\infty, 1$ with respect to $\mathcal{C}_1$ are the slopes $1, \infty$ with respect to $\mathcal{C}_2$, respectively. It also follows from [\[Ro2, Corollary 7.4\]](#Ro2) that $\mu_K$ has slope $\infty$ with respect to one of $\mathcal{C}_1, \mathcal{C}_2$ and has slope $1$ with respect to the other one. So the standard coordinate system on $T$ coincides with exactly one of $\mathcal{C}_1, \mathcal{C}_2$.* ## Branched surfaces {#subsection 2.2} [\[standard spine picture\]]{#standard spine picture label="standard spine picture"} {#standard spine picture width="25%"} {#standard spine picture width="25%"} {#standard spine picture width="25%"} [\[branched surface\]]{#branched surface label="branched surface"} {#branched surface width="25%"} {#branched surface width="25%"} {#branched surface width="25%"} The branched surface is an important tool to describe foliations and laminations. We review some backgrounds on branched surfaces in this subsection. Our notations follow from [\[FO\]](#FO), [\[O2\]](#O2). **Definition 25**. *[\[standard spine\]]{#standard spine label="standard spine"} A *standard spine* is a $2$-complex such that every point has a neighborhood modeled as Figure [3](#standard spine picture){reference-type="ref" reference="standard spine picture"}.* **Definition 26**. *[\[branched surface definition\]]{#branched surface definition label="branched surface definition"} A *branched surface* $B$ is a standard spine with well-defined cusp structure (at the set of points without Euclidean neighborhoods), such that every point has a neighborhood modeled as Figure [6](#branched surface){reference-type="ref" reference="branched surface"}.* Let $B$ be a branched surface in a compact orientable $3$-manifold $M$, and let $$L(B) = \{t \in B \mid t \text{ has no Euclidean neighborhood in } B\}.$$ Call $L(B)$ the *branch locus* of $B$, call each component of $B \setminus \setminus L(B)$ a *branch sector*, call each point in $L(B)$ without an $\mathbb{R}$-neighborhood in $L(B)$ a *double point* of $L(B)$, and call each component of $L(B) \setminus \setminus \{\text{double points in } L(B)\}$ a *segment* in $L(B)$. For each segment $s$ in $L(B)$, we assign $s$ a normal vector in $B$ with orientation consistent with the direction of the cusp, and we call it the *cusp direction* at $s$. $B$ is *co-orientable* if there exists a continuous normal vector field on $B$, or equivalently, all branch sectors of $B$ have compatible co-orientations. A *fibered neighborhood* $N(B)$ of $B$ is a regular neighborhood of $B$ locally modeled as Figrue [7](#fibered neighborhood){reference-type="ref" reference="fibered neighborhood"}. We regard $N(B)$ as an interval bundle over $B$ and call its fibers *interval fibers*. $\partial N(B)$ can be decomposed to two (possibly non-connected) compact subsurfaces $\partial_h N(B)$ (called the *horizontal boundary* of $N(B)$) and $\partial_v N(B)$ (called the *vertical boundary* of $N(B)$) such that $\partial_h N(B)$ is transverse to the interval fibers and $\partial_v N(B)$ is tangent to the interval fibers. Let $\pi: N(B) \to B$ denote the canonical projection that sends every interval fiber to a single point. We call $\pi$ the *collapsing map* for $N(B)$. [\[fibered neighborhood\]]{#fibered neighborhood label="fibered neighborhood"} {#fibered neighborhood width="70%"} **Definition 27**. *A lamination $\mathcal{L}$ is *carried* by $B$ if we can choose a fibered neighborhood $N(B)$ of $B$ such that $\mathcal{L} \subseteq N(B)$ and every leaf of $\mathcal{L}$ is transverse to the interval fibers of $N(B)$. And $\mathcal{L}$ is *fully carried* by $B$ if $\mathcal{L}$ is carried by $B$ and intersects all interval fibers of $N(B)$.* In [\[GO\]](#GO), Gabai and Oertel introduce essential branched surfaces to describe essential laminations. **Definition 28** (Essential branched surface). *[\[essential\]]{#essential label="essential"} A branched surface $B$ in a compact orientable $3$-manifold $M$ is *essential* if all of the following conditions hold:* *(a) There is no disk of contact, where a disk of contact is an embedded disk $D \subseteq N(B)$ transverse to the interval fibers of $N(B)$ and $\partial D \subseteq Int(\partial_v N(B))$. And there is no half disk of contact, where a half disk of contact is an embedded disk $D \subseteq N(B)$ transverse to the interval fibers such that there is a connected segment $\alpha \subseteq \partial D$ with $\alpha \subseteq \partial M \cap \partial N(B)$ and $\partial D - Int(\alpha) \subseteq Int(\partial_v N(B)) - \partial M$.* *(b) $\partial_h N(B)$ is incompressible and $\partial$-incompressible in $M -Int(N(B))$, no component of $M - Int(N(B))$ is a monogon, and no component of $\partial_h N(B)$ is a sphere or a disk properly embedded in $M$.* *(c) $M - Int(N(B))$ is irreducible, and $\partial M - Int(N(B))$ is incompressible in $M - Int(N(B))$.* *(d) $B$ contains no Reeb branched surface (see [\[GO\]](#GO) for the definition of Reeb branched surface).* *(e) $B$ fully carries a lamination.* Gabai and Oertel ([\[GO\]](#GO)) prove that **Theorem 29** (Gabai-Oertel). *(a) Any essential lamination in a compact orientable $3$-manifold is fully carried by an essential branched surface.* *(b) Any lamination in a compact orientable $3$-manifold fully carried by an essential branched surface is an essential lamination.* Now we describe the laminar branched surface introduced by Li in [\[Li1\]](#Li1), [\[Li2\]](#Li2). **Definition 30**. *Let $B$ be a branched surface. A *sink disk* of $B$ is a disk component $D$ of $B \setminus \setminus L(B)$ such that $D \cap \partial M = \emptyset$ and the cusp directions at all segments in $\partial D$ point in $D$.* *(b) A *half sink disk* of $B$ is a disk component $D$ of $B \setminus \setminus L(B)$ such that $D \cap \partial M \ne \emptyset$ and the cusp directions at all segments in $\partial D \cap L(B)$ point in $D$.* For a branched surface $B$ in a $3$-manifold $M$, A *trivial bubble* in $M - Int(N(B))$ is a $3$-ball component $Q$ of $M - Int(N(B))$ such that $\partial Q \cap \partial_h N(B)$ has $2$ components and each of them is a disk, and moreover, $\pi \mid_{\partial Q \cap N_h(B)}$ is injective. And we call $\pi(\partial Q)$ a *trivial bubble* of $B$. **Definition 31** (Laminar branched surface). *[\[laminar\]]{#laminar label="laminar"} Let $B$ be a branched surface in a compact orientable $3$-manifold $M$. $B$ is a *laminar branched surface* if $B$ satisfies Conditions (b)$\sim$(d) of Definition [\[essential\]](#essential){reference-type="ref" reference="essential"}, $B$ contains no trivial bubble, and $B$ contains no sink disk or half sink disk.* In [\[Li1\]](#Li1), [\[Li2\]](#Li2), Li proves that **Theorem 32** (Li). *Let $M$ be a compact orientable $3$-manifold.* *(a) Every laminar branched surface in $M$ fully carries an essential lamination.* *(b) For every essential lamination in $M$ which is not a lamination by $2$-planes, it is fully carried by a laminar branched surface.* The laminar branched surface is a useful tool to construct taut foliations in Dehn fillings ([\[Li2, Theorem 2.2\]](#Li2)): **Theorem 33** (Li). *Let $M$ be a compact orientable, irreducible $3$-manifold with tori boundary $\partial M = \bigcup^{n}_{i=1}T_i$. Let $B$ be a laminar branched surface in $M$ such that $\partial M \setminus \setminus B$ is a union of bigons. Suppose that $$\textbf{s} = (s_1,\ldots,s_n) \in (\mathbb{Q} \cup \{\infty\})^{n}$$ is a rational multislope in $\partial M$ such that each $s_i$ is realized by the boundary train track $B \cap T_i$ and $B$ does not carry a torus that bounds a solid torus in $M(\textbf{s})$, then $B$ fully carries an essential lamination $\mathcal{L}_{\textbf{s}}$ that meets each $T_i$ transversely in a collection of simple closed curves of slope $s_i$. Moreover, $\mathcal{L}_{\textbf{s}}$ can be extended to an essential lamination in the Dehn filling of $M$ along $\partial M$ with the multislope $\textbf{s}$.* **Remark 34**. *In [\[Li2, Theorem 2.2\]](#Li2), Li only states this theorem in the case that $M$ has connected boundary. As noted in [\[KalR, Subsection 2.4\]](#KalR), the argument holds for the case that $M$ has multiple boundary components.* # Proof of the main theorem {#section 3} We prove Theorem [Theorem 9](#main){reference-type="ref" reference="main"} in this section. Let $\Sigma$ be a compact orientable surface with nonempty boundary and let be $\varphi: \Sigma \to \Sigma$ be an orientation-preserving pseudo-Anosov homeomorphism. Let $M = \Sigma \times I / \stackrel{\varphi}{\sim}$ be the mapping torus of $\Sigma$ over $\varphi$. We fix an orientation on $M$, and we adopt Conventions [\[slope\]](#slope){reference-type="ref" reference="slope"}, [\[orientation\]](#orientation){reference-type="ref" reference="orientation"} for $M$. We assume **Assumption 35**. *$\varphi$ is co-orientable and co-orientation-reversing.* Let $\mathcal{F}^{s}, \mathcal{F}^{u}$ denote the stable and unstable foliations of $\varphi$. Then $\mathcal{F}^{s}, \mathcal{F}^{u}$ are co-orientable. We fix a co-orientation on $\mathcal{F}^{s}$. ## Construction of the branched surface {#subsection 3.1} In this subsection, we construct a branched surface of $M$. We first choose a union of oriented properly embedded arcs $\alpha \subseteq \Sigma$ and then construct a branched surface $B(\alpha)$ by adding a union of product disks $\alpha \times I$ to the fibered surface $\Sigma \times \{0\}$. Let $C$ be a boundary component of $\Sigma$. Let $p$ denote the number of singularities of $\mathcal{F}^{s}$ contained in $C$, and let $v_1, v_2, \ldots, v_p \in C$ denote these $p$ singularities (consecutive along the positive orientation on $C$). Note that $2 \mid p$ since $\mathcal{F}^{s}$ is co-orientable. $v_1, v_2, \ldots, v_p$ divides $C$ into $p$ segments, denoted $(v_i, v_{i+1})$ for each $i \in \{1,\ldots,p\}$ (by convention, $v_{p+1} = v_1$), and we call each of them a *stable segment* of $C$. For each stable segment $(v_i, v_{i+1})$, we choose a point $w_i$ in its interior, then $w_i$ is the starting endpoint of a transversal of $\mathcal{F}^{s}$, which is either positively oriented or negatively oriented. $(v_i, v_{i+1})$ is said to be *negative* (resp. *positive*) if $w_i$ is the starting endpoint of some positively oriented transversal (resp. negatively oriented transversal) of $\mathcal{F}^{s}$. Note that a stable segment is positive (resp. negative) implies that its two adjacent stable segments are negative (resp. positive). [\[smoothing\]]{#smoothing label="smoothing"} {#smoothing width="30%"} {#smoothing width="30%"} **Construction 36**. *[\[arcs\]]{#arcs label="arcs"} (a) Let $$A_+ = \{\text{positive stable segments in } \partial \Sigma\},$$ $$A_- = \{\text{negative stable segments in } \partial \Sigma\}.$$ Note that $|A_+| = |A_-|$. We choose a bijection $j: A_- \to A_+$. For each $\sigma \in A_-$, we choose a path $r_\sigma$ that starts at some point in $Int(\sigma)$, positively transverse to $\mathcal{F}^{s}$ and disjoint from the singularities of $\mathcal{F}^{s}$, and ends at some point in $Int(j(\sigma))$ as follows:* *$\bullet$ We draw a positively oriented transversal $\tau_1: I \to \Sigma$ that starts at some point in $Int(\sigma)$, and we draw a negatively oriented transversal $\tau_2: I \to \Sigma$ that starts at some point in $Int(j(\sigma))$. And we assume $\tau_1, \tau_2$ are disjoint from the singularities of $\mathcal{F}^{s}$. Recall from [\[FM, Corollary 14.15\]](#FM), every leaf of $\mathcal{F}^{s}$ that is not contained in $\partial \Sigma$ is dense in $\Sigma$. So there exists a non-singular leaf $\lambda$ of $\mathcal{F}^{s}$ and $t_1, t_2 \in I$ such that $\tau_1(t_1), \tau_2(t_2) \in \lambda$. Let $r^{''}_{\sigma}$ denote the immersed path that starts at $\tau_1(0)$ and goes along $\tau_1([0,t_1])$ to $\tau_1(t_1)$, then goes along $\lambda$ to $\tau_2(t_2)$, finally goes along the inverse direction of $\tau_2([0,t_2])$ and ends at $\tau_2(0)$. Then $r^{''}_{\sigma}$ is positively transverse to $\mathcal{F}^{s}$ except a subarc contained in $\lambda$. Let $\eta$ denote this subarc. Since $\lambda$ is a non-singular leaf and $\eta$ is compact, $\eta$ has a product neighborhood $\eta \times I$ in $\mathcal{F}^{s}$. We can isotope $r^{''}_{\sigma}$ in $\eta \times I$ to make it transverse to $\mathcal{F}^{s}$ and keep it still disjoint from the singularities of $\mathcal{F}^{s}$. Then we obtain an immersed path $r^{'}_{\sigma}$ which is positively transverse to $\mathcal{F}^{s}$. At last, we smooth every self-intersection point of $r^{'}_{\sigma}$ with respect to the orientation on the path (Figure [9](#smoothing){reference-type="ref" reference="smoothing"}), and we can obtain a properly embedded arc and a (possibly empty) union of circles. We delete all these circles, and we denote by $r_\sigma$ the remained properly embedded arc.* *We can construct $\{r_\sigma \mid \sigma \in A_-\}$ one-by-one to make them only have double intersection points (this can be guaranteed since each $r_\sigma$ is disjoint from the singularities of $\mathcal{F}^{s}$) and satisfy that $\varphi$ does not take any endpoint in $\{r_\sigma \mid \sigma \in A_-\}$ to another endpoint in it. We smooth every double intersection point of $\bigcup_{\sigma \in A_-} r_\sigma$ (which is a non-singular point of $\mathcal{F}^{s}$) with respect to the orientations on the paths (Figure [9](#smoothing){reference-type="ref" reference="smoothing"}), and we can obtain a finite union of disjoint oriented properly embedded arcs which are positively transverse to $\mathcal{F}^{s}$. Let $\alpha$ denote this union of oriented arcs.* We note that **Fact 37**. *[\[endpoint\]]{#endpoint label="endpoint"} Each negative stable segment contains exactly one starting endpoint of some path in $\alpha$ and no ending endpoint, and each positive stable segment contains exactly one ending endpoint of some path in $\alpha$ and no starting endpoint.* Now we construct a branched surface $B(\alpha)$. **Definition 38**. *[\[B(alpha)\]]{#B(alpha) label="B(alpha)"} (a) Let $$B(\alpha) = (\Sigma \times \{0\}) \cup (\alpha \times I)$$ be a branched surface, where we orient the branch sectors so that for each component $\gamma$ of $\alpha$, the cusp direction at $\gamma \times \{0\}$ points to its left side in $\Sigma \times \{0\}$, and the cusp direction at $\gamma \times \{1\} = \varphi(\gamma) \times \{0\}$ points to its right side in $\Sigma \times \{0\}$.* *(b) For each component $\gamma$ of $\alpha$, we call $\gamma \times I$ a *product disk* and call $\gamma \times \{0\}$ (resp. $\gamma \times \{1\}$) the *lower arc* (resp. *upper arc*) of this product disk.* *(c) Note that $\partial \alpha \cap \varphi(\partial \alpha) = \emptyset$ and both of $\alpha, \varphi(\alpha)$ contain no singularity of $\mathcal{F}^{s}$. There is a union of oriented properly embedded arcs $\beta$ in $\Sigma$ such that, $\beta$ is isotopic to $\varphi(\alpha)$ relative to the endpoints, $\beta$ is transverse to $\mathcal{F}^{s}$, and $\alpha,\beta$ only have double intersection points. We isotope $\alpha \times I$ relative to $(\alpha \times \{0\}) \cup (\partial \alpha \times I)$ so that the upper arcs of $\alpha \times I$ are isotoped to $\beta \times \{0\}$. This makes $B(\alpha)$ locally modeled as in Figure [6](#branched surface){reference-type="ref" reference="branched surface"}.* ## Verifying that $B(\alpha)$ is laminar {#subsection 3.2} In this subsection, we verify that $B(\alpha)$ is a laminar branched surface. **Lemma 39**. *Let $\rho$ be an oriented simple closed curve or an oriented properly embedded arc in $\Sigma$ such that $\rho$ can be divided into finitely many segments which are either positively transverse to $\mathcal{F}^{s}$ or tangent to $\mathcal{F}^{s}$, and at least one of these segments is positively transverse to $\mathcal{F}^{s}$. Then $\rho$ is essential in $\Sigma$.* We offer two proofs of this lemma. *The first proof of Lemma [Lemma 39](#transverse){reference-type="ref" reference="transverse"}.* We first consider the case that $\rho$ is an oriented simple closed curve. We assume that $\rho$ is non-essential. Then there is an embedded disk $D \subseteq \Sigma$ such that $\rho = \partial D$. Let $\lambda$ be a leaf of $\mathcal{F}^{s}$ such that $\rho, \lambda$ have transverse intersections but have no tangent intersection. Then $\rho$ is positively transverse to $\lambda$ at each point of $\rho \cap \lambda$. Note that $\mathcal{F}^{s}$ can be split open along the singular leaves to a geodesic lamination with respect to some hyperbolic metric on $\Sigma$ with geodesic boundary. So $\lambda \cap D$ is a union of closed segments. At each component $s$ of $\lambda \cap D$, $\rho$ is positively transverse to $\lambda$ at one endpoint of $s$ and is negatively transverse to $\lambda$ at the other endpoint of $s$ (Figure [10](#intersection){reference-type="ref" reference="intersection"}). This is a contradiction. [\[intersection\]]{#intersection label="intersection"} {#intersection width="35%"} Now we consider the case that $\rho$ is an oriented properly embedded arc. If $\rho$ is non-essential, then there is an embedded disk $D \subseteq \Sigma$ with $\partial D \subseteq \rho \cup \partial \Sigma$. Similar to the above case, this is a contradiction. ◻ *The second proof of Lemma [Lemma 39](#transverse){reference-type="ref" reference="transverse"}.* We can split open $\mathcal{F}^{s}$ along its singular leaves to obtain a geodesic lamination $\Lambda^{s}$ of $\Sigma$. Let $\widetilde{\Sigma}$ be the universal cover of $\Sigma$. Let $\widetilde{\Lambda^{s}}$ denote the pull-back lamination of $\Lambda^{s}$ in $\widetilde{\Sigma}$, and let $L(\Lambda^{s})$ denote the leaf space of $\widetilde{\Lambda^{s}}$. We note that $L(\Lambda^{s})$ is an order tree, and the co-orientation on $\mathcal{F}^{s}$ induces a co-orientation on $\Lambda^{s}$ and an orientation on $L(\Lambda^{s})$. We may consider $\rho$ as a path $\rho: I \to \Sigma$ such that (1) either $\rho(0) = \rho(1)$ or $\rho(0), \rho(1) \in \partial \Sigma$, (2) $\rho$ can be divided into finitely many segments $\rho_1, \ldots, \rho_n$ which are either positively transverse to $\Lambda^{s}$ or tangent to $\Lambda^{s}$, (3) at least one of $\rho_1, \ldots, \rho_n$ is positively transverse to $\Lambda^{s}$. Let $\widetilde{\rho}: I \to \widetilde{\Sigma}$ be a lift of $\rho$, and let $\widetilde{\rho_i}$ be the lift of $\rho_i$ to $\widetilde{\Sigma}$ contained in $\widetilde{\rho}$. Let $$\Omega = \{i \in \{1,\ldots,n\} \mid \rho_i \text{ is positively transverse to } \Lambda^{s}\}.$$ Then $\Omega \ne \emptyset$. $\bigcup_{i \in \Omega} \widetilde{\rho_i}$ can be cannonically identified with a positively oriented path in $L(\Lambda^{s})$, and the two endpoints of this path are distinct since $L(\Lambda^{s})$ is simply connected. It follows that $\rho$ is essential. ◻ Because each component of $\alpha$ is positively transverse to $\mathcal{F}^{s}$, **Corollary 40**. *Each component of $\alpha$ is essential in $\Sigma$.* To verify that $B(\alpha)$ contains no trivial bubble, it suffices to show that $B(\alpha)$ has no $3$-ball complementary region. Note that $M \setminus \setminus B(\alpha) = (M \setminus \setminus \alpha) \times I$. If there is a $3$-ball complementary region of $B(\alpha)$, then $M \setminus \setminus \alpha$ must have some disk component. Now we exclude this case. **Lemma 41**. *$\Sigma \setminus \setminus \alpha$ contains no disk component.* *Proof.* [\[no disk\]]{#no disk label="no disk"} {reference-type="ref" reference="alpha"}. The dots are singularities of $\mathcal{F}^{s}$ contained in $\partial \Sigma$, the blue lines are leaves of $\mathcal{F}^{s}$, and the segments labeled with $+, -$ are positive stable segments, negative stable segments respectively.](17.png "fig:"){#no disk width="45%"} Assume that $\Sigma \setminus \setminus \alpha$ contains a disk component $D$. In the following discussions, the clockwise and anticlockwise orientations on $\partial D$ will always be with respect to the orientation on $\Sigma$ (then $D$ is in the left side of $\partial D$ if $\partial D$ has anticlockwise orientation). Let $\gamma: I \to \Sigma$ be a component of $\alpha$ contained in $\partial D$, and we may assume that the orientation on $\gamma$ is consistent with the anticlockwise orientation on $\partial D$. Now draw a path $\eta: I \to \partial D$ that starts at $\gamma(1)$, goes along the anticlockwise orientation on $\partial D$, and ends at $\gamma(1)$. Recall that $\gamma(1)$ is contained in a positive stable segment in $\partial \Sigma$, and this positive stable segment does not contain any other endpoint of $\alpha$ (Fact [\[endpoint\]](#endpoint){reference-type="ref" reference="endpoint"}). So $\eta$ first goes to a negative stable segment adjacent to the positive stable segment containing $\gamma(1)$, and then goes through another arc in $\alpha$ along its orientation, and so on (Figure [11](#no disk){reference-type="ref" reference="no disk"}). So the anticlockwise orientation on $\partial D$ is consistent with the orientation on each component of $\alpha$ contained in $\partial D$. Thus, the anticlockwise orientation on $\partial D$ is positively transverse to $\mathcal{F}^{s}$ in $\partial D \cap \alpha$ and tangent to $\mathcal{F}^{s}$ in $\partial D \cap \partial \Sigma$. This contradicts Lemma [Lemma 39](#transverse){reference-type="ref" reference="transverse"}. So $\Sigma \setminus \setminus \alpha$ contains no disk component. ◻ It follows that **Corollary 42**. *$B(\alpha)$ contains no trivial bubble.* We've verified that each component of $\alpha$ is essential and $\Sigma \setminus \setminus \alpha$ contains no disk component. As shown in [\[S, Lemma 3.16\]](#S), $B(\alpha)$ satisfies Conditions (b)$\sim$(d) of Definition [\[essential\]](#essential){reference-type="ref" reference="essential"}. **Remark 43**. *[\[S, Lemma 3.16\]](#S) has an assumption that the upper arcs and the lower arcs of the product disks intersect efficiently in $\Sigma \times \{0\}$, but our $B(\alpha)$ doesn't satisfy this. We explain that [\[S, Lemma 3.16\]](#S) still holds for our $B(\alpha)$ below. Let $B^{'}(\alpha)$ be a branched surface obtained from isotoping the product disks of $B(\alpha)$ relative to $(\alpha \times \{0\}) \cup (\partial \alpha \times I)$ so that the upper arcs and the lower arcs of $\alpha \times I$ intersect efficiently in $\Sigma \times \{0\}$. Then [\[S, Lemma 3.16\]](#S) applies to $B^{'}(\alpha)$, and there exists a homeomorphism between $M - Int(N(B^{'}(\alpha)))$ and $M - Int(N(B(\alpha)))$ that takes $\partial_h N(B^{'}(\alpha)), \partial_v N(B^{'}(\alpha))$ to $\partial_h N(B(\alpha)), \partial_v N(B(\alpha))$ respectively. Note that each of Condition (b), (c) of Definition [\[essential\]](#essential){reference-type="ref" reference="essential"} only depends on the complementary regions of the branched surface, and the proof for Condition (d) of Definition [\[essential\]](#essential){reference-type="ref" reference="essential"} in [\[S, Lemma 3.16\]](#S) only uses a property of the complementary regions. It follows that $B(\alpha)$ satisfies Conditions (b)$\sim$(d) of Definition [\[essential\]](#essential){reference-type="ref" reference="essential"}.* It remains to prove that $B(\alpha)$ contains no sink disk or half sink disk. **Proposition 44**. *$B(\alpha)$ has no sink disk or half sink disk.* *Proof.* [\[sink disk obstruction\]]{#sink disk obstruction label="sink disk obstruction"} We first show that every product disk in $B(\alpha)$ is not a sink disk or a half sink disk. For every product disk $S$ of $B(\alpha)$, the cusp directions at both of its upper arc and lower arc point out of $S$. So $S$ is not a sink disk or a half sink disk. Now assume that $B(\alpha)$ contains a sink disk $D$. Then $D$ is not contained in $\alpha \times I$. So $D \subseteq \Sigma \times \{0\}$, and thus $\partial D \subseteq (\alpha \times \{0\}) \cup (\beta \times \{0\})$. For a segment $\sigma$ of $\partial D$, $\bullet$ Assume $\sigma \subseteq \alpha \times \{0\}$. Then the cusp direction at $\sigma$ points to the left side. Since the cusp direction at $\sigma$ points in $D$, $\sigma$ has anticlockwise orientation (Figure [\[sink disk obstruction\]](#sink disk obstruction){reference-type="ref" reference="sink disk obstruction"} (b), where the solid lines are subarcs of $\alpha \times \{0\}$). As $\sigma \subseteq \alpha \times \{0\}$, $\sigma$ is positively transverse to $\mathcal{F}^{s} \times \{0\} \subseteq \Sigma \times \{0\}$. $\bullet$ Assume $\sigma \subseteq \beta \times \{0\}$. In this case, the cusp direction at $\sigma$ points to the right side and points in $D$, and thus $\sigma$ has clockwise orientation (Figure [\[sink disk obstruction\]](#sink disk obstruction){reference-type="ref" reference="sink disk obstruction"} (b), where the dashed lines are subarcs of $\beta \times \{0\}$). Because $\varphi$ is co-orientation-reversing and $\beta$ is isotopic to $\varphi(\alpha)$ relative to the endpoints, $\beta$ is negatively transverse to $\mathcal{F}^{s}$. So $\sigma$ is negatively transverse to $\mathcal{F}^{s} \times \{0\}$. So every segment of $\partial D$ either has anticlockwise orientation and is positively transverse to $\mathcal{F}^{s} \times \{0\}$, or has clockwise orientation and is negatively transverse to $\mathcal{F}^{s} \times \{0\}$. Therefore, the anticlockwise orientation on $\partial D$ is positively transverse to $\mathcal{F}^{s} \times \{0\}$ (Figure [\[sink disk obstruction\]](#sink disk obstruction){reference-type="ref" reference="sink disk obstruction"} (c)). However, $\partial D$ is non-essential in $\Sigma$ since it bounds the disk $D$. This contradicts Lemma [Lemma 39](#transverse){reference-type="ref" reference="transverse"}. So $B(\alpha)$ contains no sink disk. Similarly, if $B(\alpha)$ contains a half sink disk $D$, then the anticlockwise orientation on $\partial D$ is positively transverse to $\mathcal{F}^{s} \times \{0\}$ at every segment contained in $(\alpha \times \{0\}) \cup (\beta \times \{0\})$ and is tangent to $\mathcal{F}^{s} \times \{0\}$ at $\partial D \cap (\partial \Sigma \times \{0\})$. This also contradicts Lemma [Lemma 39](#transverse){reference-type="ref" reference="transverse"}. So $B(\alpha)$ also contains no half sink disk. It follows that $B(\alpha)$ is laminar. ◻ Thus **Corollary 45**. *$B(\alpha)$ is a laminar branched surface.* ## Simple closed curves carried by boundary train tracks {#subsection 3.3} Let $\tau(\alpha) = B(\alpha) \cap \partial M$. In this subsection, we choose some simple closed curves carried by $\tau(\alpha)$ and compute their slopes. We first give some descriptions for $\tau(\alpha)$. **Definition 46**. *[\[vertical edges\]]{#vertical edges label="vertical edges"} Let $\gamma: I \to \partial \Sigma$ be a component of $\alpha$. Then the product disk $\gamma \times I$ intersects $\partial M$ at $(\{\gamma(0)\} \times I) \cup (\{\gamma(1)\} \times I)$. We call $\{\gamma(0)\} \times I$ (resp. $\{\gamma(1)\} \times I$) a *positive vertical edge* (resp. *negative vertical edge*) of $\tau(\alpha)$.* Under the assumption of Definition [\[vertical edges\]](#vertical edges){reference-type="ref" reference="vertical edges"}, let $C_1, C_2$ denote the boundary components of $\Sigma$ that contain $\gamma(0), \gamma(1)$ respectively. Recall from Convention [\[orientation\]](#orientation){reference-type="ref" reference="orientation"} (b), the positive orientation on $C_1$ goes from the left side of $\gamma$ to its right side at $\gamma(0)$, and the positive orientation on $C_2$ goes from the right side of $\gamma$ to its left side at $\gamma(1)$, see Figure [\[type\]](#type){reference-type="ref" reference="type"} (a) for an example. Recall from Definition [\[B(alpha)\]](#B(alpha)){reference-type="ref" reference="B(alpha)"}, the cusp direction at $\gamma \times \{0\}$ points to the left, and the cusp direction at $\varphi_*(\gamma) \times \{0\}$ points to the right. Thus [\[type\]]{#type label="type"} **Fact 47**. *(a) For a positive vertical edge $\{a\} \times I$ ($a \in \partial \Sigma$) of $\tau(\alpha)$, the cusp direction at the point $(a,0)$ is consistent with the negative orientation on $\partial \Sigma \times \{0\}$, and the cusp direction at $(a,1) = (\varphi(a), 0)$ is consistent with the positive orientation on $\partial \Sigma \times \{0\}$. For example, see the right one of the two vertical edges in Figure [\[type\]](#type){reference-type="ref" reference="type"} (a).* *(b) For a negative vertical edge $\{b\} \times I$ ($b \in \partial \Sigma$) of $\tau(\alpha)$, the cusp direction at $(b,0)$ is consistent with the positive orientation on $\partial \Sigma \times \{0\}$, and the cusp direction at $(b,1) = (\varphi(b), 0)$ is consistent with the negative orientation on $\partial \Sigma \times \{0\}$. For example, see the left one of the two vertical edges in Figure [\[type\]](#type){reference-type="ref" reference="type"} (a).* For a positive or negative vertical edge $\{a\} \times I$ (where $a \in \partial \Sigma$), we call $(a,0)$ its *lower endpoint* and call $(a,1) = (\varphi(a),0)$ its *upper endpoint*. Since $\varphi$ is co-orientation-reversing, $\varphi$ takes all positive stable segments to negative stable segments and takes all negative stable segments to positive stable segments. Thus, for a positive vertical arc (resp. negative vertical arc), its lower endpoint is contained in a negative stable segment (resp. positive stable segment) and its upper endpoint is contained in a positive stable segment (resp. negative stable segment), compare with Figure [\[type\]](#type){reference-type="ref" reference="type"} (b). Let $C$ be a boundary component of $\Sigma$ and let $c$ denote the order of $C$ under $\varphi$ (i.e. $c = \min \{k \in \mathbb{N}_+ \mid \varphi^{k}(C) = C\}$). Let $T$ denote the boundary component of $M$ containing $C \times \{0\}$ and let $\tau = \tau(\alpha) \cap T$. Let $p$ denote the number of singularities of $\mathcal{F}^{s}$ contained in $C$. And let $v_1,\ldots,v_p$ denote the $p$ singularities of $\mathcal{F}^{s}$ contained in $C$ (consecutive along the positive orientation on $C$). Let $q \in \mathbb{Z}$ for which $(p;q)$ is the degeneracy locus of the suspension flow of $\varphi$ on $T$. As explained in Remark [\[remark\]](#remark){reference-type="ref" reference="remark"} (a), $v_{j+q} = \varphi^{c}(v_j)$ (mod $p$) for each $j \in \{1,\ldots,p\}$. We note that $2 \mid p$ since $\varphi$ is co-orientable, and $q \equiv c \text{ } (\text{mod } 2)$ since $2 \mid q$ if and only if $\varphi^{c}$ is co-orientation-preserving. **Definition 48**. *[\[varphi triple\]]{#varphi triple label="varphi triple"} Under the assumption as above, we call $(c,p,q)$ the *$\varphi$-triple* for $C$.* **Proposition 49**. *$\tau$ carries a simple closed curve of slope $\frac{p}{q+c}$ and a simple closed curve of slope $\frac{p}{q-c}$, where $\frac{p}{q + c} = \infty$ if $q + c = 0$ and $\frac{p}{q - c} = \infty$ if $q - c = 0$.* *Proof.* Let $\mu_T, \lambda_T$ denote the meridian and longitude on $T$ (see Convention [\[orientation\]](#orientation){reference-type="ref" reference="orientation"} for the orientations on them). We may assume that every $(v_{2i-1}, v_{2i})$ is a negative stable segment and every $(v_{2i}, v_{2i+1})$ is a positive stable segment. Let $t_j = (v_j,0) \in \Sigma \times \{0\}$ for each $j \in \{1,\ldots,p\}$. Let $\gamma: I \to \tau$ be the path that starts at $t_1$ and do the following steps again and again, (1) once $\gamma$ reaches $T \cap (\Sigma \times \{0\})$, $\gamma$ goes along the positive orientation on $T \cap (\Sigma \times \{0\})$ until it meets the lower endpoint of some positive vertical edge, (2) when $\gamma$ meets the lower endpoint of some positive vertical edge, $\gamma$ goes along this positive vertical edge and reaches $T \cap (\Sigma \times \{0\})$ again, (3) $\gamma$ stops by $t_1$ at its second time to meet $t_1$. Compare with Figure [\[boundary train track\]](#boundary train track){reference-type="ref" reference="boundary train track"} (a) for the picture when $\gamma$ starts. Note that $\gamma$ arrives at the positive stable segment $(t_{q+c}, t_{q+c+1})$ (mod $p$) at it's second time meeting $C \times \{0\}$, and then $\gamma$ goes along the positive orientation on $C \times \{0\}$ to $t_{q+c+1}$. If $t_{q+c+1} = t_1$, then $\gamma$ stops at $t_1$. Otherwise, $\gamma$ repeats the steps given above, passes $t_{2(q+c)+1}, t_{3(q+c)+1}, \ldots$, and arrives at $t_1$ at last. Thus $$\langle \gamma, \lambda_T \rangle = \frac{p}{\gcd (p,q+c)},$$ $$\langle \mu_T, \gamma \rangle = \frac{q+c}{p} \cdot \langle \gamma, \lambda_T \rangle = \frac{q+c}{\gcd (p,q+c)},$$ by convention $\gcd(p,0) = p$. Therefore, $$\text{slope}(\gamma) = \frac{(\frac{p}{\gcd (p,q+c)})}{(\frac{q+c}{\gcd (p,q+c)})} = \frac{p}{q+c}.$$ [\[boundary train track\]]{#boundary train track label="boundary train track"} Next, we choose a simple closed curve $\nu$ carried by $\tau$ with $\text{slope}(\nu) = \frac{p}{q-c}$. Let $\nu: I \to \tau$ be the path starting at $t_1$ such that (1) when $\gamma$ gets to $T \cap (\Sigma \times \{0\})$, $\gamma$ goes along the negative orientation on $T \cap (\Sigma \times \{0\})$ until it meets the lower endpoint of some negative vertical edge, (2) when $\gamma$ meets the lower endpoint of some negative vertical edge, $\gamma$ goes along this negative vertical edge and gets to $T \cap (\Sigma \times \{0\})$ again, (3) $\gamma$ stops by $t_1$ at the second time to meet $t_1$. Compare with Figure [\[boundary train track\]](#boundary train track){reference-type="ref" reference="boundary train track"} (b) for the picture when $\nu$ starts. $\nu$ arrives at the positive stable segment $(t_{q-c+1}, t_{q-c+2})$ (mod $p$) at the second time to meet $C \times \{0\}$. Then $\nu$ goes along the negative orientation on $C \times \{0\}$ to $t_{q-c+1}$. If $t_{q-c+1} = t_1$, then $\nu$ stops at this time. Otherwise, $\nu$ repeats the above steps and passes $t_{2(q-c)+1}, t_{3(q-c)+1}, \ldots,$ and $\nu$ arrives at $t_1$ at last. Then $$\langle \nu, \lambda_T \rangle = \frac{p}{\gcd (p,q-c)},$$ $$\langle \mu_T, \nu \rangle = \frac{q-c}{p} \cdot \langle \nu, \lambda_T \rangle = \frac{q-c}{\gcd (p,q-c)},$$ and thus $$\text{slope}(\nu) = \frac{(\frac{p}{\gcd (p,q-c)})}{(\frac{q-c}{\gcd (p,q-c)})} = \frac{p}{q-c}.$$ ◻ ## Slopes realized by boundary train tracks {#subsection 3.4} We first briefly review some ingredients in measures on train tracks. For a train track $\tau$, let $E(\tau)$ denote the set of edges of $\tau$. $\bullet$ A *measure* $m: E(\tau) \to \mathbb{R}_{\geqslant 0}$ on $\tau$ is an assignment of nonnegative numbers to $E(\tau)$ that satisfies the cusp relation (Figure [\[cusp relation\]](#cusp relation){reference-type="ref" reference="cusp relation"} (a)) at each cusp, i.e. for any three edges $x,y,z \in E(\tau)$ that has a common endpoint $p$, if the cusp direction at $p$ points toward $z$, then $m(x) + m(y) = m(z)$. [\[cusp relation\]]{#cusp relation label="cusp relation"} $\bullet$ Measures on $\tau$ are in one-to-one correspondence with measured laminations carried by $\tau$. For a measure $m$ on $\tau$, call the corresponding measured lamination the *companion lamination* of $m$ (compare with Figure [\[cusp relation\]](#cusp relation){reference-type="ref" reference="cusp relation"} (b)). $\bullet$ $\tau$ is *orientable* if its edges have continuously varying orientations. If $\tau$ is orientable, then every measured lamination carried by $\tau$ is orientable. Moreover, given any measure on an oriented train track, any closed curve has a well-defined algebraic intersection number with the companion lamination of this measure. We refer the reader to [\[FM, Chapter 15\]](#FM) for more details. **Notation 50**. *Let $\tau$ be a train track.* *(a) Let $m_1, m_2$ be two measures on $\tau$. Then $m_1 + m_2$ denotes the measure on $\tau$ with $(m_1 + m_2)(e) = m_1(e) + m_2(e)$ for each $e \in E(\tau)$.* *(b) Let $\rho$ be a simple closed curve carried by $\tau$ and let $t \in \mathbb{R}_+$. Then we can regard $\rho \times [0,t]$ as a measured lamination carried by $\tau$. Let $\rho(t)$ denote the measure on $\tau$ for which $\rho \times [0,t]$ is the companion measured lamination of $\rho(t)$.* **Proposition 51**. *Let $C$ be a component of $\partial \Sigma$ and let $T$ be a boundary component of $M$ containing $C \times \{0\}$. Let $\tau = \tau(\alpha) \cap T$. Let $(c,p,q)$ denote the $\varphi$-triple for $C$ (Definition [\[varphi triple\]](#varphi triple){reference-type="ref" reference="varphi triple"}).* *(a) If $q > c > 0$, then $\tau$ realizes all rational slopes in $(-\infty,\frac{p}{q+c}) \cup (\frac{p}{q-c}, +\infty) \cup \{\infty\}$.* *(b) If $q = c > 0$, then $\tau$ realizes all rational slopes in $(-\infty,\frac{p}{2q})$.* *(c) If $c > q \geqslant 0$, then $\tau$ realizes all rational slopes in $(-\frac{p}{c-q},\frac{p}{q+c})$.* *(d) If $-c < q < 0$, then $\tau$ realizes all rational slopes in $(-\frac{p}{|q|+c},\frac{p}{c-|q|})$.* *(e) If $q = -c < 0$, then $\tau$ realizes all rational slopes in $(-\frac{p}{2|q|},+\infty)$.* *(f) If $q < -c < 0$, then $\tau$ realizes all rational slopes in $(-\infty, -\frac{p}{|q|-c}) \cup (-\frac{p}{|q|+c},+\infty) \cup \{\infty\}$.* *Proof.* We only prove (a). The proofs of (b)$\sim$(f) are similar to (a). Let $\mu_T, \lambda_T$ denote the meridian and longitude on $T$. For every vertical edge $\{a\} \times I$ in $\tau$, its *upward orientation* (resp. *downward orientation*) is the orientation on it consistent with the increasing orientation (resp. decreasing orientation) on the second coordinates. Assume $q > c > 0$. We first orient $\tau$ so that $T \cap (\Sigma \times \{0\})$ has positive orientation, every positive vertical edge in $\tau$ has upward orientation, and every negative vertical edge in $\tau$ has downward orientation. By Proposition [Proposition 49](#maximal slope){reference-type="ref" reference="maximal slope"} (a), $\tau$ carries a simple closed curve $\gamma$ of slope $\frac{p}{q+c}$. Let $v = \frac{p}{\gcd(p, q+c)}$, $u = \frac{q+c}{\gcd(p, q+c)}$. Then $\frac{p}{q+c} = \frac{v}{u}$, $u, v > 0$, and $\gcd(u,v) = 1$. We assign $\gamma$ an orientation consistent with the orientation on $\tau$, then $\langle \gamma, \lambda_T \rangle = v$, $\langle \mu_T, \gamma \rangle = u$. For every edge $e$ of $\tau$, we can choose a simple closed curve $\rho_e$ carried by $\tau$ such that $\rho_e$ contains $e$ and $\text{slope}(\rho_e) = 0$. And we assign each $\rho_e$ an orientation consistent with the orientation on $\tau$. Then $\langle \rho_e, \lambda_T \rangle = 0$, $\langle \mu_T, \rho_e \rangle = 1$. Let $m_0$ denote the measure $\sum_{e \in E(\tau)} \rho_e(1)$, and let $\Lambda_0$ denote the companion lamination of $m(0)$. Then $\langle \Lambda_0, \lambda_T\rangle = 0$, and thus $\tau$ realizes the slope $0$. We define a family of one-parameter measures $m_1(t)$ with $t \in (0,+\infty)$ such that $$m_1(t) = \gamma(1) + \sum_{e \in E(\tau)} \rho_e(t).$$ Let $\Lambda_1(t)$ denote the companion lamination of $m_1(t)$, and we assign $\Lambda_1(t)$ an orientation induced from the orientation on $\tau$. Let $N = |E(\tau)|$. Then $$\frac{\langle \Lambda_1(t), \lambda_T\rangle}{\langle \mu_T, \Lambda_1(t) \rangle} = \frac{v}{u+tN}.$$ For any rational number $x \in (0,\frac{p}{q+c})$, we can choose some $t > 0$ so that $$\frac{v}{u+tN} = x,$$ which implies that $\tau$ realizes the slope $x$. Now we prove that $\tau$ realizes all rational slopes in $(-\infty, 0) \cup \{\infty\} \cup (\frac{p}{q-c}, +\infty)$. We re-orient $\tau$ so that $T \cap (\Sigma \times \{0\})$ has negative orientation, every positive vertical edge in $\tau$ has downward orientation, and every negative vertical edge in $\tau$ has upward orientation. By Proposition [Proposition 49](#maximal slope){reference-type="ref" reference="maximal slope"} (a), $\tau$ carries a simple closed curve $\nu$ of slope $\frac{p}{q-c}$. We assign $\nu$ an orientation induced from the orientation on $\tau$. Let $s= \frac{p}{\gcd(p, q-c)}$, $r = \frac{q-c}{\gcd(p, q-c)}$. Then $\frac{p}{q-c} = \frac{s}{r}$, $r, s > 0$, and $\gcd(r,s) = 1$. And we have $\langle \nu, \lambda_T \rangle = s$, $\langle \mu_T, \nu \rangle = r$. For every edge $e \in E(\tau)$, we can choose a simple closed curve $\eta_e$ carried by $\tau$ that contains $e$ and has slope $0$. We assign each $\eta_e$ an orientation consistent with the orientation on $\tau$. Because $T \cap (\Sigma \times \{0\})$ has negative orientation, $\langle \eta_e, \lambda_T \rangle = 0$, $\langle \mu_T, \eta_e \rangle = -1$. We define a family of one-parameter measures $m_2(t)$ with $t \in (0,+\infty)$ such that $$m_2(t) = \nu(1) + \sum_{e \in E(\tau)} \eta_e(t).$$ Let $\Lambda_2(t)$ denote the companion lamination of $m_2(t)$, and we assign $\Lambda_2(t)$ an orientation induced from the orientation on $\tau$. Recall that $|E(\tau)| = N$, so $$\frac{\langle \Lambda_2(t), \lambda_T\rangle}{\langle \mu_T, \Lambda_2(t) \rangle} = \frac{s}{r-tN}.$$ If we choose $t = \frac{r}{N}$, then $\frac{s}{r-tN} = \infty$. For any rational number $x \in (\frac{p}{q-c}, +\infty)$, we can choose some $0 < t < \frac{r}{N}$ so that $\frac{s}{r-tN} = x$. And for any rational number $x \in (-\infty, 0)$, we can choose some $t > \frac{r}{N}$ so that $\frac{s}{r-tN} = x$. Thus, $\tau$ realizes all rational slopes in $(-\infty, 0) \cup \{\infty\} \cup (\frac{p}{q-c}, +\infty)$. ◻ ## The proof of Theorem [Theorem 9](#main){reference-type="ref" reference="main"} {#proof} We first explain that $B(\alpha)$ carries no torus. Assume that $B(\alpha)$ carries a torus $T$. Since all product disks in $B(\alpha)$ intersect $\partial M$ and $T$ is a closed surface, $\pi(T)$ contains no product disk, and thus $\pi(T) \subseteq \Sigma \times \{0\}$. However, $\Sigma \times \{0\}$ carries no closed surface, which contradicts $\pi(T) \subseteq \Sigma \times \{0\}$. So $B(\alpha)$ carries no torus. Choose a multislope $\textbf{s} = (s_1, \ldots, s_k) \in (\mathbb{Q} \cap \{\infty\})^{k}$ contained in the multi-inverval as given in Theorem [Theorem 9](#main){reference-type="ref" reference="main"}. Combining Corollary [Corollary 45](#laminar branched surface){reference-type="ref" reference="laminar branched surface"}, Proposition [Proposition 51](#realize){reference-type="ref" reference="realize"} with Theorem [Theorem 33](#boundary){reference-type="ref" reference="boundary"}, since $B(\alpha)$ carries no torus, $B(\alpha)$ fully carries an essential lamination $\mathcal{L}_{\textbf{s}}$ in $M$ such that $\mathcal{L}_{\textbf{s}}$ intersects each $T_i$ in a collection of simple closed curves of slope $s_i$. Since $B(\alpha)$ has product complementary regions, each complementary region of $\mathcal{L}_{\textbf{s}}$ is an $I$-bundle over a surface. We can extend $\mathcal{L}_{\textbf{s}}$ to a foliation $\mathcal{F}_{\textbf{s}}$ in $M$ that intersects each $T_i$ in a foliation by simple closed curves of slope $s_i$. We can then extend $\mathcal{F}_{\textbf{s}}$ to a foliation $\widehat{\mathcal{F}_{\textbf{s}}}$ in $M(\textbf{s})$. $\widehat{\mathcal{F}_{\textbf{s}}}$ is taut since the union of core curves of the filling solid tori is transverse to $\widehat{\mathcal{F}_{\textbf{s}}}$ and intersects all leaves of $\widehat{\mathcal{F}_{\textbf{s}}}$. We assign $\Sigma \times \{0\}$ a co-orientation. For each product disk of $B(\alpha)$, we can assign it a co-orientation compatible with the co-orientation on $\Sigma \times \{0\}$. This defines a co-orientation on $B(\alpha)$, which induces a co-orientation on $\mathcal{L}_{\textbf{s}}$ and on $\mathcal{F}_{\textbf{s}}$. It follows that $\widehat{\mathcal{F}_{\textbf{s}}}$ is co-orientable. This completes the proof of Theorem [Theorem 9](#main){reference-type="ref" reference="main"}. # The proof of Proposition [Proposition 13](#pretzel){reference-type="ref" reference="pretzel"} {#section 4} Let $q \in \mathbb{N}$ with $q \geqslant 3$, and let $K$ be the $(-2,3,2q+1)$-pretzel knot in $S^{3}$. Let $g(K)$ denote the Seifert genus of $K$ (then $g(K) = q + 2$). Let $X = S^{3} - Int(N(K))$, let $S$ be a fibered surface of $X$, and let $\phi: S \to S$ denote the pseudo-Anosov monodromy of $X$. Fix an orientation on $S^{3}$ so that $\phi$ is right-veering. In this section, we prove Proposition [Proposition 13](#pretzel){reference-type="ref" reference="pretzel"}: **Proposition [Proposition 13](#pretzel){reference-type="ref" reference="pretzel"} 1**. *(a) $\phi$ is co-orientable and co-orientation-reversing.* *(b) $K$ has degeneracy slope $4g(K)-2$.* *(c) All rational slopes in $(-\infty, 2g(K) - 1)$ are CTF surgery slopes of $K$.* *(d) For each $n \geqslant 2$, the $n$-fold cyclic branched cover of $K$ admits a co-orientable taut foliation.* *The proof of (a).* Let $\mu, \delta$ denote the merdian and degeneracy slope on $K$ respectively. Since $\phi$ is right-veering, $\mu \ne \delta$. Let $\widetilde{X_2}$ denote the double cyclic cover of $X$, and let $\Sigma_2(K)$ denote the double branched cover of $K$. Then $\widetilde{X_2}$ is the mapping torus of $S$ with monodromy $\phi^{2}$. Because $K$ is a hyperbolic L-space knot in $S^{3}$, $\Delta(\mu, \delta) = 1$ (see Case [\[case2\]](#case2){reference-type="ref" reference="case2"}). The suspension flow of $\phi^{2}$ in $\widetilde{X_2}$ induces a pseudo-Anosov flow $\Upsilon$ of $\Sigma_2(K)$ since $2 \Delta(\mu,\delta) = 2$ ([\[F\]](#F)). Let $\mathcal{E}^{s}$ denote the weak stable foliation of $\Upsilon$. As explained in [\[BS, A.4\]](#BS), $\Sigma_2(K)$ is a Seifert fibered $3$-manifold with base orbifold $S^{2}(2,3,2q+1)$ (i.e. $S^{2}$ with three cone points of index $2, 3, 2q+1$ respectively). Note that all pseudo-Anosov flows in Seifert fibered $3$-manifolds are $\mathbb{R}$-covered Anosov flows ([\[Ba\]](#Ba)). Thus $\Upsilon$ is an $\mathbb{R}$-covered Anosov flow and $\mathcal{E}^{s}$ is an $\mathbb{R}$-covered foliation. $\mathcal{E}^{s}$ contains no compact leaf since the stable foliation of $\varphi$ contains no compact leaf. By [\[Br, Corollary 7\]](#Br), $\mathcal{E}^{s}$ can be isotoped to be transverse to the Seifert fibers of $\Sigma_2(K)$. Because the base orbifold of $\Sigma_2(K)$ is orientable, the Seifert fibers of $\Sigma_2(K)$ have continuously varying orientations, and these orientations define a co-orientation on $\mathcal{E}^{s}$. Thus $\phi^{2}$ is co-orientable and co-orientation-preserving, and therefore $\phi$ is co-orientable. Since $\phi$ is right-veering and $K$ is a knot in $S^{3}$, $\phi$ can only be co-orientation-reversing. ◻ *The proof of (b).* It can be deduced from [\[FS\]](#FS) that $\delta = 4g(K) - 2$. We give another proof here using (a). Since $\Upsilon$ is an $\mathbb{R}$-covered Anosov flow, $\Upsilon$ has no singular orbit. So the stable foliation of $\phi$ has no singularity in $Int(S)$. It follows that the stable foliation of $\phi$ has $4g(K) - 2$ singularities in $\partial S$ ([\[FM, Proposition 11.4\]](#FM)). As $\Delta(\mu, \delta) = 1$, we have $\delta = 4g(K) - 2$. ◻ *The proof of (c).* This follows from (b) and Corollary [Corollary 10](#open book){reference-type="ref" reference="open book"} directly. ◻ *The proof of (d).* Let $\widetilde{X_n}$ denote the $n$-fold cyclic cover of $X$, which is also the mapping torus of $S$ with monodromy $\phi^{n}$. If $n$ is even, then $\phi^{n}$ is co-orientation-preserving, so the result can be deduced from Theorem [Theorem 7](#co-orientation-preserving){reference-type="ref" reference="co-orientation-preserving"} directly. Now we assume that $n$ is odd (then $\phi^{n}$ is co-orientation-reversing). We denote $4g(K) - 2$ by $p$. There is a unique $a \in \mathbb{Z}_{\geqslant 0}$ with $-\frac{1}{2}p < n - ap \leqslant \frac{1}{2}p$, and let $b = n - ap$. We fix the canonical coordinate system on $\partial \widetilde{X_n}$. By Remark [\[remark\]](#remark){reference-type="ref" reference="remark"} (a), $(p; b)$ is the degeneracy locus of the suspension flow of $\phi^{n}$ on $\partial \widetilde{X_n}$. Applying Corollary [Corollary 10](#open book){reference-type="ref" reference="open book"} to $\widetilde{X_n}$, a slope $s$ on $\partial \widetilde{X_n}$ is a CTF filling slope if $s \notin [\frac{p}{b+1}, \frac{p}{b-1}]$ (when $b > 1$) or $s \notin [\frac{p}{2}, + \infty) \cup \{\infty\}$ (when $b = 1$). The inverse image of the slope $\infty$ on $\partial X$ is the slope $- \frac{1}{a}$ on $\partial \widetilde{X_n}$. When $b = 1$, we have $a \geqslant 1$ since $n > 1$, and thus $- \frac{1}{a} \notin [\frac{p}{2}, + \infty) \cup \{\infty\}$. When $b > 1$, we have $\frac{p}{b+1}, \frac{p}{b-1} \in (0, +\infty)$ and $- \frac{1}{a} \in (-\infty, 0) \cup \{\infty\}$, which implies $- \frac{1}{a} \notin [\frac{p}{b+1}, \frac{p}{b-1}]$. The result follows directly. ◻ 10 J. Alexander, [\[A\]]{#A label="A"} G. Band and P. Boyland, [\[BB\]]{#BB label="BB"} T. Barbot, [\[Ba\]]{#Ba label="Ba"} M. Boileau, S. Boyer, and C. Gordon, [\[BBG\]]{#BBG label="BBG"} J. Bowden, [\[Bo\]]{#Bo label="Bo"} S. Boyer and A. Clay, [\[BC\]]{#BC label="BC"} S. Boyer, C. Gordon and L. Watson, [\[BGW\]]{#BGW label="BGW"} S. Boyer, C. Gordon, and Y. Hu, [\[BGH\]]{#BGH label="BGH"} S. Boyer and Y. Hu, [\[BH\]]{#BH label="BH"} S. Boyer, D. Rolfsen and B. Wiest, [\[BRW\]]{#BRW label="BRW"} F. Bonahon and L. Siebenmann, [\[BS\]]{#BS label="BS"} M. Brittenham, [\[Br\]]{#Br label="Br"} M. Culler, N. Dunfield, M. Goerner and J. Weeks, [\[CDGW\]]{#CDGW label="CDGW"} S. Dowdall, R. Gupta and S. Taylor, [\[DGT\]]{#DGT label="DGT"} N. Dunfield, [\[D1\]]{#D1 label="D1"} N. Dunfield, [\[D2\]]{#D2 label="D2"} N. Dunfield, [\[D3\]]{#D3 label="D3"} B. Farb and D. Margalit, [\[FM\]]{#FM label="FM"} R. Fintushel and R. Stern, [\[FS\]]{#FS label="FS"} W. Flyod and U. Oertel, [\[FO\]]{#FO label="FO"} D. Fried, [\[F\]]{#F label="F"} D. Gabai, [\[Ga1\]]{#Ga1 label="Ga1"} D. Gabai, [\[Ga2\]]{#Ga2 label="Ga2"} D. Gabai, [\[Ga3\]]{#Ga3 label="Ga3"} D. Gabai and U. Oertel, [\[GO\]]{#GO label="GO"} Paolo Ghiggini, [\[Gh\]]{#Gh label="Gh"} J. Hanselman, J. Rasmussen, S. Rasmussen and L. Watson, [\[HRRW\]]{#HRRW label="HRRW"} K. Honda, W. Kazez and G. Matić, [\[HKM\]]{#HKM label="HKM"} Y. Hu, [\[H\]]{#H label="H"} A. Juhász, [\[J\]]{#J label="J"} T. Kalelkar and R. Roberts, [\[KalR\]]{#KalR label="KalR"} W. Kazez and R. Roberts, [\[KazR1\]]{#KazR1 label="KazR1"} W. Kazez and R. Roberts, [\[KazR2\]]{#KazR2 label="KazR2"} S. Krishna, [\[Kri\]]{#Kri label="Kri"} P. Kronheimer, T. Mrowka, P. Ozsváth and Z. Szabó, [\[KMOS\]]{#KMOS label="KMOS"} E. Lanneau and J.-L. Thiffeault, [\[LT\]]{#LT label="LT"} T. Li, [\[Li1\]]{#Li1 label="Li1"} T. Li, [\[Li2\]]{#Li2 label="Li2"} W. Lickorish, [\[Lic\]]{#Lic label="Lic"} T. Lidman and A. H. Moore, [\[LM\]]{#LM label="LM"} C. McMullen, [\[M\]]{#M label="M"} Y. Ni, [\[Ni\]]{#Ni label="Ni"} Z. Nie, [\[Nie\]]{#Nie label="Nie"} U. Oertel, [\[O1\]]{#O1 label="O1"} U. Oertel, [\[O2\]]{#O2 label="O2"} P. Ozsváth and Z. Szabó, [\[OS\]]{#OS label="OS"} J. Rasmussen and S. Rasmussen, [\[RR\]]{#RR label="RR"} S. Rasmussen, [\[Ra\]]{#Ra label="Ra"} R. Roberts, [\[Ro1\]]{#Ro1 label="Ro1"} R. Roberts, [\[Ro2\]]{#Ro2 label="Ro2"} D. Santoro, [\[S\]]{#S label="S"} W. Thurston, [\[T\]]{#T label="T"} A. Wallace, [\[W\]]{#W label="W"} J. Zung, [\[Z\]]{#Z label="Z"}

arxiv_math

--- abstract: | To each complex reflection group $\Gamma$ one can attach a canonical symplectic singularity $\mathcal{M}_{\Gamma}$ [@Bea00]. Motivated by the 4D/2D duality [@BeeLemLie15; @BeeRas], Bonetti, Meneghelli and Rastelli [@BMR19] conjectured the existence of a supersymmetric vertex operator superalgebra $\mathsf{W}_{\Gamma}$ whose associated variety is isomorphic to $\mathcal{M}_{\Gamma}$. We prove this conjecture when the complex reflection group $\Gamma$ is the symmetric group $S_N$ by constructing a sheaf of $\hbar$-adic vertex operator superalgebras on the Hilbert scheme of $N$ points in the plane. For that case, we also show the free-field realisation of $\mathsf{W}_{\Gamma}$ in terms of $\mathop{\mathrm{rk}}(\Gamma)$ many $\beta\gamma bc$-systems proposed in [@BMR19], and identify the character of $\mathsf{W}_{\Gamma}$ as a certain quasimodular form of mixed weight. In physical terms, the vertex operator superalgebra $\mathsf{W}_{S_N}$ constructed in this article corresponds via the 4D/2D duality [@BeeLemLie15] to the four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory with gauge group $\mathop{\mathrm{SL}}_N$. author: - Tomoyuki Arakawa^a^, Toshiro Kuwabara^b^ and Sven Möller^a,c^ bibliography: - refs.bib title: Hilbert Schemes of Points in the Plane and Quasi-Lisse Vertex Algebras with $\mathcal{N}=4$ Symmetry --- [^1] [^2] [^3] [^4] # Introduction {#sec:intro} Let $\Gamma$ be a complex reflection group of rank $\mathop{\mathrm{rk}}(\Gamma)$. By definition, $\Gamma$ is a subgroup of $\mathop{\mathrm{GL}}(V_\Gamma)$ that is generated by reflections, where $V_\Gamma={\mathbb C}^{\mathop{\mathrm{rk}}(\Gamma)}$. The group $\Gamma$ acts diagonally on the symplectic vector space $T^*V_\Gamma=V_\Gamma\oplus V_\Gamma^*$, preserving its symplectic form, and it is known that the orbit space $$\mathcal{M}_\Gamma\coloneqq(V_\Gamma\oplus V_\Gamma^*)/\Gamma=\mathop{\mathrm{Spec}}{\mathbb C}[V_\Gamma\oplus V_\Gamma^*]^\Gamma$$ has a symplectic singularity [@Bea00]. Motivated by the 4D/2D duality [@BeeLemLie15; @BeeRas], Bonetti, Meneghelli and Rastelli [@BMR19] conjectured the existence of a vertex operator superalgebra $\mathsf{W}_\Gamma$ of central charge $c_\Gamma\coloneqq-3\sum_{i=1}^{\mathop{\mathrm{rk}}(\Gamma)}(2p_i-1)$ equipped with $\mathcal{N}=2$ supersymmetry, such that, among other things, $$\label{eq:ass-variety-with-Higgs} X_{\mathsf{W}_\Gamma}\cong \mathcal{M}_\Gamma,$$ where $p_1,\dots ,p_{\mathop{\mathrm{rk}}(\Gamma)}$ are the degrees of the fundamental invariants of ${\mathbb C}[V_\Gamma]^\Gamma$ and $X_V$ denotes the associated variety of a vertex superalgebra $V$ [@Arakawa12]. When $\Gamma$ is even a Coxeter group, [@BMR19] further conjectured that the $\mathcal{N}=2$ symmetry of $\mathsf{W}_\Gamma$ should be enhanced to the small $\mathcal{N}=4$ supersymmetry. That is, $\mathsf{W}_\Gamma$ should be a conformal extension of the small $\mathcal{N}=4$ superconformal algebra $\operatorname{Vir}_{\mathcal{N}=4}^{c_\Gamma}$ of level $c_\Gamma/6$, a vertex operator superalgebra of central charge $c_\Gamma$. Note that [\[eq:ass-variety-with-Higgs\]](#eq:ass-variety-with-Higgs){reference-type="eqref" reference="eq:ass-variety-with-Higgs"} implies that the vertex operator superalgebra $\mathsf{W}_\Gamma$ is quasi-lisse in the sense of [@AraKaw18] since a normal variety with symplectic singularities has only finitely many symplectic leaves [@Kaledin:2006kq]. We now mention the meaning of $\mathsf{W}_\Gamma$ in the context of the 4D/2D duality $${\mathbb V}\colon\{\text{4D $\mathcal{N}=2$ SCFTs}\}\longrightarrow \{\text{VOAs}\},$$ which associates a vertex operator superalgebra ${\mathbb V}(\mathcal{T})$ with each four-dimensional $\mathcal{N}=2$ superconformal field theory $\mathcal{T}$ [@BeeLemLie15]. It is believed that ${\mathbb V}$ is a finite map and that the associated variety of ${\mathbb V}(\mathcal{T})$ satisfies $$\label{eq:Hiigs-branch} X_{{\mathbb V}(\mathcal{T})}\cong \operatorname{Higgs}(\mathcal{T}),$$ where $\operatorname{Higgs}(\mathcal{T})$ is the Higgs branch of $\mathcal{T}$, as conjectured in [@BeeRas]. Since the Higgs branch of $\mathcal{T}$ is expected to have a symplectic singularity, this Higgs branch conjecture [\[eq:Hiigs-branch\]](#eq:Hiigs-branch){reference-type="eqref" reference="eq:Hiigs-branch"} would imply that the vertex operator superalgebras ${\mathbb V}(\mathcal{T})$ are quasi-lisse, and it is moreover believed that they are of CFT-type, simple and strongly finitely generated. According to [@BMR19], in the case where the Coxeter group $\Gamma$ is the Weyl group $W(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$, a central property of the conjectured vertex operator superalgebra $\mathsf{W}_{W(\mathfrak{g})}$ is that $$\mathsf{W}_{W(\mathfrak{g})}\cong{\mathbb V}(\operatorname{SYM}_{\mathfrak{g}}),$$ where $\operatorname{SYM}_{\mathfrak{g}}$ is the four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory with gauge Lie algebra $\mathfrak{g}$. Since $$\operatorname{Higgs}(\operatorname{SYM}_{\mathfrak{g}})\cong\mathcal{M}_{W(\mathfrak{g})},$$ the isomorphism $X_{\mathsf{W}_{\Gamma}}\cong\mathcal{M}_{\Gamma}$ in [\[eq:ass-variety-with-Higgs\]](#eq:ass-variety-with-Higgs){reference-type="eqref" reference="eq:ass-variety-with-Higgs"} is exactly the Higgs branch conjecture [\[eq:Hiigs-branch\]](#eq:Hiigs-branch){reference-type="eqref" reference="eq:Hiigs-branch"} of Beem and Rastelli for the four-dimensional theory $\mathcal{T}=\operatorname{SYM}_{\mathfrak{g}}$. However, despite of their importance, the vertex operator superalgebras $\mathsf{W}_\Gamma$ have been studied very little since the paper [@BMR19] appeared. The main difficulty is that $\mathsf{W}_\Gamma$ is in general expected to be a W-algebra in the sense that it is not generated by a Lie algebra, and therefore one cannot expect to define $\mathsf{W}_\Gamma$ by generators and relations (operator product expansions) in a closed form. In this article, we prove the conjecture of Bonetti, Meneghelli and Rastelli [@BMR19] in the case where $\Gamma$ is the symmetric group $S_N=W(\mathfrak{sl}_N)$, corresponding to the four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory with gauge group $\mathop{\mathrm{SL}}_N$. That is, we construct a family of vertex operator superalgebras $\mathsf{W}_{S_N}$, $N\ge2$, with the desired properties. In this situation, the symplectic variety $\mathcal{M}_{S_N}$ is essentially[^5] the $N$-th symmetric power ${\mathbb C}^{2N}/S_N$ of the affine plane ${\mathbb C}^2$, and it is well-known that the singular variety $M_0\coloneqq{\mathbb C}^{2N}/S_N$ admits the conical symplectic resolution $$\mathop{\mathrm{Hilb}}^N({\mathbb C}^2)\longrightarrow{\mathbb C}^{2N}/S_N,$$ where $M\coloneqq\mathop{\mathrm{Hilb}}^N({\mathbb C}^2)$ is the Hilbert scheme of $N$ points in the affine plane. In the classical setting, Kashiwara and Rouquier [@KR08] constructed a sheaf $\mathcal{A}_{\hbar}$ of $\hbar$-adic algebras on the Hilbert scheme $M$ such that the algebra $[\mathcal{A}_{\hbar}(M)]^{{\mathbb C}^\times}$ of its global sections (and taking the invariants in some sense under a certain action of the one-dimensional torus ${\mathbb C}^\times$) is isomorphic to the spherical rational Cherednik algebra [@EG02] associated with $S_N$, which is a natural quantisation of the coordinate ring of $M_0={\mathbb C}^{2N}/S_N$. The notion of a sheaf $\mathcal{A}_{\hbar}$ of $\hbar$-adic algebras was upgraded to that of a sheaf of $\hbar$-adic vertex algebras in [@AKM15]. Therefore, it is natural to try to apply the method in [@AKM15] to $M=\mathop{\mathrm{Hilb}}^N({\mathbb C}^2)$ and chiralise the construction of [@KR08]. However, this straightforward attempt does not work for the Hilbert scheme $M$ due to some obstruction in constructing sheaves of vertex algebras (cf. [@GMS04]). In order to overcome this problem, we replace $M$ with structure sheaf $\mathcal{O}_M$ by the supervariety whose underlying topological space is again $M$ but whose structure sheaf is a certain superalgebra analogue $\widetilde{\mathcal{O}}_M$ of $\mathcal{O}_M$. In terms of the Nakajima quiver variety description [@Nakajima] of the Hilbert scheme $M$, this corresponds to replacing the vector space ${\mathbb C}$ on the framing vertex of the Jordan quiver by the superspace ${\mathbb C}^{1|1}$ (see for details): $$\begin{tikzpicture} \node (1) at (0,0) {${\mathbb C}^{1|1}$}; \node (2) at (2,0) {${\mathbb C}^N$}; \begin{scope}[transform canvas={yshift=2pt}] \draw[->] (1) to (2); \end{scope} \begin{scope}[transform canvas={yshift=-2pt}] \draw[->] (2) to (1); \end{scope} \draw[->] ([shift={(24pt,0)}]2.center) +(20:-22pt) arc (20:340:-22pt); \draw[->] ([shift={(24pt,0)}]2.center) +(340:-26pt) arc (340:20:-26pt); \end{tikzpicture}$$ Then, the machinery of [@AKM15] becomes applicable and we are able to construct a sheaf $\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}$ of $\hbar$-adic vertex superalgebras on $M$ quantising the $\infty$-jet bundle $\widetilde{\mathcal{O}}_{J_\infty M} = J_{\infty} \widetilde{\mathcal{O}}_M$ associated with the sheaf $\widetilde{\mathcal{O}}_M$. The space of global sections $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ is naturally equipped with the structure of an $\hbar$-adic vertex operator superalgebra with a certain action of the one-dimensional torus ${\mathbb C}^\times$. To obtain a usual vertex operator algebra, we consider the (in some sense) ${\mathbb C}^\times$-invariant subalgebra $[\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)]^{{\mathbb C}^\times}$ of $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ (for details, see ). Then the vertex operator superalgebra $[\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)]^{{\mathbb C}^\times}$ is essentially the conjectured vertex operator superalgebra $\mathsf{W}_{S_N}$ (see ): **Main Theorem 1**. *For $N\ge 2$, there exists a sheaf $\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}$ of $\hbar$-adic vertex operator superalgebras of central charge $c=-3N^2$ on the Hilbert scheme $M=\mathop{\mathrm{Hilb}}^N({\mathbb C}^2)$ such that $\mathop{\mathrm{Gr}}\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}\cong\pi_*\mathcal{O}_{J_{\infty}M_{\mathcal{V}}}[[\hbar]]$ and whose global sections $\mathsf{V}_{S_N}\coloneqq [\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)]^{{\mathbb C}^\times}$ have the associated variety $$X_{\mathsf{V}_{S_N}}\cong{\mathbb C}^{2N}/S_N.$$ Moreover, the vertex operator superalgebra $\mathsf{V}_{S_N}$ decomposes as a tensor product $$\mathsf{V}_{S_N}\cong\mathsf{W}_{S_N}\otimes\beta\gamma\otimes\mathsf{SF},$$ where $\mathsf{W}_{S_N}$ is a conformal extension of some quotient of the small $\mathcal{N}=4$ superconformal algebra $\operatorname{Vir}_{\mathcal{N}=4}^{c_{S_N}}$ of central charge $c_{S_N}=-3(N^2-1)$, and $$X_{\mathsf{W}_{S_N}}\cong\mathcal{M}_{S_N}.$$ The vertex operator superalgebras $\mathsf{V}_{S_N}$ and $\mathsf{W}_{S_N}$ are of CFT-type and quasi-lisse.* Here, $\beta\gamma$ denotes the $\beta\gamma$-system vertex algebra and $\mathsf{SF}$ the symplectic fermion vertex superalgebra, and in the theorem they are endowed with conformal structures of central charges $c=-1$ and $c=-2$, respectively. We suspect that $\mathsf{V}_{S_N}$, $\mathsf{W}_{S_N}$ and the quotient of the small $\mathcal{N}=4$ superconformal algebra inside $\mathsf{W}_{S_N}$ are simple. This is true for $N=2$, where we identify $\mathsf{W}_{S_N}$ as the simple small $\mathcal{N}=4$ superconformal algebra of level $k=-3/2$ [@GMS05; @Adamovic16]. The global sections containing the tensor factor $\beta\gamma\otimes\mathsf{SF}$ is merely an artefact due to the Nakajima quiver variety construction of $M=\mathop{\mathrm{Hilb}}^N({\mathbb C}^2)$ being associated with $\mathfrak{gl}_N$ rather than $\mathfrak{sl}_N$. This is mirrored by the corresponding decomposition ${\mathbb C}^{2N}/S_N\cong \mathcal{M}_{S_N}\times T^* {\mathbb C}$ of the associated varieties. It is an immediate consequence of our sheaf construction that by considering the local sections over a Zariski open subset $U\subset M$ we obtain the vertex operator superalgebra embedding $$\mathsf{V}_{S_N}=[\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)]^{{\mathbb C}^\times} \hookrightarrow [\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U)]^{{\mathbb C}^\times}.$$ In particular, by choosing an appropriate open set $U$ we obtain the free-field realisation in terms of $\mathop{\mathrm{rk}}(S_N)=\mathop{\mathrm{rk}}(\mathfrak{sl}_N)=N-1$ many tensor copies of the $\beta\gamma b c$-system vertex superalgebra that is conjectured in [@BMR19]: **Theorem 1**. *For $N\ge2$, there is an embedding $\mathsf{V}_{S_N}\hookrightarrow(\beta\gamma b c)^{\otimes(N-1)}\otimes\beta\gamma\otimes\mathsf{SF}\hookrightarrow(\beta\gamma b c)^{\otimes N}$, which restricts to the free-field realisation $$\mathsf{W}_{S_N }\hookrightarrow(\beta\gamma b c)^{\otimes(N-1)}.$$* Note that the conformal structure of $\mathsf{V}_{S_N}$ fixes conformal structures of the $\beta\gamma b c$-systems of central charges $c=-3(2p_i-1)$ with $p_i=N,N-1,\ldots,1$ and with $p_i=N,N-1,\ldots,2$ in the case of $\mathsf{W}_{S_N}$. We recall that in the latter case the $p_i$ are the $\mathop{\mathrm{rk}}(S_N)=N-1$ many degrees of the fundamental invariants of $S_N$. The last $\beta\gamma b c$-system of central charge $c=-3$ is the obvious conformal extension of $\beta\gamma\otimes\mathsf{SF}$, which splits from the chosen local sections of $\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}$ and, in fact, implies the splitting result stated in the main theorem for the global sections $\mathsf{V}_{S_N}$. As a corollary, we also obtain a free-field realisation of (some quotient of) the small $\mathcal{N}=4$ superconformal algebra $\operatorname{Vir}_{\mathcal{N}=4}^{c_{S_N}}$ of central charge $c_{S_N}=-3(N^2-1)$ in terms of $(\beta\gamma b c)^{\otimes(N-1)}$, generalising results for $N=2$ and $3$ in [@GMS05; @Adamovic16] and [@BMR19], respectively. Besides the Higgs branch, another important datum associated with a four-dimensional $\mathcal{N}=2$ superconformal field theory $\mathcal{T}$ is the Schur index $\mathcal{I}_{\mathcal{T}}(q)$. In the context of the 4D/2D duality, it is conjectured that this Schur index $\mathcal{I}_{\mathcal{T}}$, which should coincide with the supercharacter of the corresponding vertex operator superalgebra ${\mathbb V}(\mathcal{T})$, is a quasimodular form [@BeeRas]. This is true for $\mathcal{T}=\operatorname{SYM}_{\mathfrak{sl}_N}$ with ${\mathbb V}(\mathcal{T})\cong\mathsf{W}_{S_N}$: **Theorem 2**. *For $N\ge2$, the supercharacter of $\mathsf{W}_{S_N}$ is $$\operatorname{sch}_{\mathsf{W}_{S_N}}(q)=\frac{1}{\eta(q)^3}{\sum_{n=0}^\infty}(-1)^n\biggl(\binom{N+n}{N}+\binom{N+n-1}{N}\biggr)q^{(N+2n)^2/8}$$ and coincides with the Schur index of the four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory $\operatorname{SYM}_{\mathfrak{sl}_N}$ with gauge group $\mathop{\mathrm{SL}}_N$. This supercharacter is a sum of holomorphic quasimodular forms of weights $N-1,N-3,\dots\in{\mathbb Z}_{\ge0}$ for the full modular group $\mathop{\mathrm{SL}}_2({\mathbb Z})$ if $N$ is odd and for $\Gamma^0(2)$ with some character if $N$ is even.* Here, $\eta(q)$ denotes the Dedekind eta function. The proof relies on the exact integration formulae in [@PP22] and results for the Schur indices for $\mathop{\mathrm{SL}}_N$ in [@BDF15]. For example, for $N=3$, the supercharacter equals $$\operatorname{sch}_{\mathsf{W}_{S_3}}(q)=(1-E_2(q))/24,$$ where $E_2(q)$ denotes the quasimodular Eisenstein series of weight $2$. Finally, we note that our results in this article give the first non-trivial examples of chiral quantisation of Nakajima quiver varieties. We plan to extend our results to other Nakajima quiver varieties in forthcoming work. ## Outline {#outline .unnumbered} In we recall the construction of the $N$-point Hilbert scheme $M=\mathop{\mathrm{Hilb}}^N({\mathbb C}^2)$ of the plane as a Nakajima quiver variety, i.e. by Hamiltonian reduction. Then, by adding fermionic coordinate functions, we construct a sheaf $\widetilde{\mathcal{O}}_{M}$ of commutative superalgebras on $M$. In we review the notions of vertex superalgebras, vertex Poisson superalgebras and $\hbar$-adic vertex superalgebras. Then we introduce $\hbar$-adic versions $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^n)_{\hbar}$ of the $\beta\gamma$-system, $C\ell_{\hbar}(T^* {\mathbb C}^n)$ of the $bc$-system, $V^k(\mathfrak{g})_{\hbar}$ of the affine vertex algebras and $\mathsf{SF}_{\hbar}$ of the symplectic fermion vertex superalgebra. We also construct the microlocalisation $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^n, \hbar}$ of $\mathcal{D}^\mathrm{ch}(T^*{\mathbb C}^n)_{\hbar}$, a certain sheaf of $\hbar$-adic vertex algebras on the affine space $T^* {\mathbb C}^n$ with global sections $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^n)_{\hbar}$. In we construct the central object of this text, a sheaf $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}$ of $\hbar$-adic vertex superalgebras on the Hilbert scheme $M$. The construction is based on a vertex superalgebra analogue of quantum Hamiltonian reduction, which we call semi-infinite BRST cohomology. We then prove a vanishing (or no-ghost) theorem for this BRST cohomology. In we study the $\hbar$-adic vertex superalgebra of global sections $\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}(M)$, which we then reduce to a vertex superalgebra $\mathsf{V}_{S_N}\coloneqq [\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}(M)]^{{\mathbb C}^\times}$ of invariants under a certain torus action. $\mathsf{V}_{S_N}$ is endowed with a natural conformal structure that makes it into a vertex operator superalgebra of CFT-type of central charge $c=-3N^2$. We show that the associated variety of $\mathsf{V}_{S_N}$ is the symplectic quotient variety ${\mathbb C}^{2N}/S_N$ so that $\mathsf{V}_{S_N}$ is quasi-lisse and that $\mathsf{V}_{S_N}$ contains a quotient of the small $\mathcal{N}=4$ superconformal algebra $\operatorname{Vir}_{\mathcal{N}=4}^{c_{S_N}}$ of central charge $c_{S_N}=-3(N^2-1)$. In , by considering the local sections of $\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}$ over an appropriately chosen open subset $U\subset M$, we obtain a free-field realisation of $\mathsf{V}_{S_N}$. This also implies the factorisation $\mathsf{V}_{S_N}=\mathsf{W}_{S_N}\otimes \mathcal{D}^\mathrm{ch}(T^*{\mathbb C}^1)\otimes\mathsf{SF}$. Then, $\mathsf{W}_{S_N}$ is a vertex operator superalgebra of CFT-type of central charge $c_{S_N}=-3(N^2-1)$ and a conformal extension of some quotient of $\operatorname{Vir}_{\mathcal{N}=4}^{c_{S_N}}$. Moreover, $\mathsf{W}_{S_N}$ has the associated variety $\mathcal{M}_{S_N}$, is quasi-lisse and has a free-field realisation in terms of a $\beta\gamma bc$-system of rank $N-1=\mathop{\mathrm{rk}}(S_N)$. $\mathsf{W}_{S_N}$ is the vertex superalgebra for the reflection group $S_N$ conjectured by Bonetti, Meneghelli and Rastelli [@BMR19]. In we determine the supercharacter of the vertex operator superalgebra $\mathsf{W}_{S_N}$ and show that it is a quasimodular form of mixed weight. Finally, in we consider the special case of $N=2$. Then, the sheaf $\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}$ is essentially the $\hbar$-adic version of the twisted chiral de Rham algebra on ${\mathbb P}^1$ with parameter $\alpha=1/2$ introduced in [@GMS05]. Moreover, $\mathsf{W}_{S_2}$ is equal to the simple quotient of $\operatorname{Vir}_{\mathcal{N}=4}^{-9}$ and the free-field realisation in terms of the $\beta\gamma bc$-system coincides with the one given in [@Adamovic16]. ## Acknowledgements {#acknowledgements .unnumbered} We thank Dražen Adamović, Philip Argyres, Christopher Beem, Anirudh Deb, Martin Möller and Leonardo Rastelli for valuable discussions. Tomoyuki Arakawa was supported by Grant-in-Aid KAKENHI JP21H04993 by the *Japan Society for the Promotion of Science*. Toshiro Kuwabara was supported by Grant-in-Aid KAKENHI JP21K03174 by the *Japan Society for the Promotion of Science*. Sven Möller acknowledges support through the Emmy Noether Programme by the *Deutsche Forschungsgemeinschaft* (project number 460925688). Sven Möller was also supported by a Postdoctoral Fellowship for Research in Japan and Grant-in-Aid KAKENHI JP20F40018 by the *Japan Society for the Promotion of Science*. ## Notation {#notation .unnumbered} Let $G$ be a group and $V$ a $G$-module. We denote the subspace of all $G$-invariant elements of $V$ by $V^G$. For a character $\theta\colon G \longrightarrow{\mathbb C}^\times$, we denote the subspace of all $G$-semi-invariants of weight $\theta$ by $V^{G,\theta}=\{v\in V\,|\,g\cdot v=\theta(g)x\}$. All algebras and vector spaces will be over the base field ${\mathbb C}$. By a commutative (super)algebra we mean a commutative, associative and unital (super)algebra. For a commutative algebra $R$ and a vector space $V$, denote by $R\langle V\rangle=\bigoplus_{n=0}^{\infty}R\otimes_{{\mathbb C}} V^{\otimes n}$ the tensor algebra of $V$ over $R$. Let $S_{R}(V)=R\langle V\rangle/(a\otimes b-b\otimes a\,|\,a,b\in V)$ and $\Lambda_{R}(V)=R\langle V\rangle/(a\otimes b+b \otimes a\,|\,a,b\in V)$ be the symmetric and the exterior algebras of $V$ over $R$, respectively. Given a basis $\{a_1,\dots,a_r\}$ of $V$, we also write $R\langle a_1,\dots,a_r\rangle=R\langle V\rangle$, $R[a_1,\dots,a_r]=S_{R}(V)$ and $\Lambda_{R}(a_1,\dots,a_r)=\Lambda_{R}(V)$. For a commutative algebra $A$, let $\mathop{\mathrm{Spec}}A$ be the affine scheme associated with $A$. For a commutative, graded algebra $A=\bigoplus_{n\in {\mathbb Z}_{\ge0}}A_n$, let $\mathop{\mathrm{Proj}}A$ be the projective scheme over $\mathop{\mathrm{Spec}}A_0$ associated with $A$. Throughout the paper, we only consider integral, separated and reduced schemes over ${\mathbb C}$ and call them algebraic varieties. Let $M$ be an algebraic variety. For a sheaf $\mathcal{F}$ on $M$ and an open subset $U\subset M$, we denote the set of sections of $\mathcal{F}$ over $U$ by $\mathcal{F}(U)$. We denote the structure sheaf of $M$ by $\mathcal{O}_M$ and the coordinate ring of $M$ by ${\mathbb C}[M]=\mathcal{O}_M(M)$. # Hilbert Scheme of Points in the Plane {#sec:Hilb} In this section, we recall Nakajima's construction of the Hilbert scheme $M$ of points in the affine plane and construct a certain sheaf $\widetilde{\mathcal{O}}_{M}$ of commutative superalgebras on it. ## Quiver Variety Construction {#sec:quiver-const} We review the construction of the Hilbert scheme $M\coloneqq\mathop{\mathrm{Hilb}}^N({\mathbb C}^2)$ of $N$ points in the affine complex plane ${\mathbb C}^2$ as a Nakajima quiver variety for the Jordan quiver. We refer the reader to [@Nakajima] for details of the construction. Throughout the text, let $N\in{\mathbb Z}_{>0}$. Set $V=\mathop{\mathrm{End}}({\mathbb C}^N)\oplus{\mathbb C}^N$ and consider the corresponding symplectic vector space given by the cotangent bundle $$T^*V=V\oplus V^*=\mathop{\mathrm{End}}({\mathbb C}^N)^{\oplus2}\oplus({\mathbb C}^N)^{\oplus2}.$$ We write the standard coordinate functions of $\mathop{\mathrm{End}}({\mathbb C}^N)\subset V$ by $x_{ij}$ and the ones of $\mathop{\mathrm{End}}({\mathbb C}^N)\subset V^*$ by $y_{ij}$ for $i,j=1,\dots,N$. We further write the coordinate functions of the vector space ${\mathbb C}^N\subset V$ by $\gamma_i$ and the ones of ${\mathbb C}^N\subset V^*$ dual to $\gamma_i$ by $\beta_i$ for $i=1,\dots,N$. Then, the coordinate ring ${\mathbb C}[T^*V]$ of the symplectic vector space $T^*V$ is a Poisson algebra with Poisson bracket $\{y_{ij}, x_{kl}\}=\delta_{il}\delta_{jk}$, $\{\beta_i, \gamma_j\}=\delta_{ij}$ and zero otherwise. Let $G\coloneqq\mathop{\mathrm{GL}}_N({\mathbb C})$ be the general linear group of rank $N$. We consider an action of $G$ on ${\mathbb C}[T^*V]$ given by $$g*X=gXg^{-1},\quad g*Y=gYg^{-1},\quad g*\gamma=g\gamma\quad\text{and}\quad g*\beta=\beta g^{-1},$$ where $g=(g_{ij})_{i,j}$ is an element of $G$ and $X=(x_{ij})_{i,j}$, $Y=(y_{ij})_{i,j}$, $\gamma={}^t(\gamma_i)_{i}$ and $\beta=(\beta_i)_{i}$. The action induces a $G$-action on the symplectic vector space $T^*V$. This action is Hamiltonian, and we let $\mu\colon T^*V\longrightarrow \mathfrak{g}^*$ be a moment map associated with the action, where $\mathfrak{g}=\mathop{\mathrm{Lie}}(G)=\mathfrak{gl}_N=\mathop{\mathrm{End}}({\mathbb C}^N)$ is the Lie algebra of $G$. We define the linear map $$\mu^*\colon\mathfrak{g}\longrightarrow{\mathbb C}[T^*V],\qquad E_{ij}\mapsto\sum_{p=1}^{N} x_{ip}y_{pj}-\sum_{p=1}^{N}x_{pj}y_{ip}+\gamma_i\beta_j,$$ where $E_{ij}\in\mathfrak{g}$ is the matrix whose $(i,j)$-th entry is $1$ and all other entries vanish. The map $\mu^*$ satisfies $$\begin{aligned} \frac{d}{dt}(\exp(tA)*f)|_{t=0}&=\{\mu^*(A),f\},\\ \mu^*([A,B])&=\{\mu^*(A),\mu^*(B)\}\end{aligned}$$ for $A,B\in\mathfrak{g}$, $f\in{\mathbb C}[T^*V]$. The moment map $\mu\colon T^*V\longrightarrow\mathfrak{g}^*$ is the morphism of affine varieties associated with the comoment map $\mu^*$. Let $\mathfrak{X}\subset T^*V$ be the subset defined by the following stability condition: a point $p\in T^*V$ belongs to $\mathfrak{X}$ if and only if the vector $\gamma(p)\in{\mathbb C}^N$ is a cyclic vector of ${\mathbb C}^N$ with respect to the action of the matrices $X(p),Y(p)\in\mathop{\mathrm{End}}({\mathbb C}^N)$. It is known that $\mathfrak{X}$ is a Zariski open subset, and $G$ acts freely on $\mathfrak{X}$. Now, we consider the Hamiltonian reduction of $T^*V$ with respect to the $G$-action. The $G$-action is closed on the inverse image $\mu^{-1}(0)$ under the moment map $\mu$. Set $$M_0=\mu^{-1}(0)/\!\!/G=\mathop{\mathrm{Spec}}{\mathbb C}[\mu^{-1}(0)]^G,$$ the affine variety associated with the invariant subalgebra ${\mathbb C}[\mu^{-1}(0)]^G$, and $$M=(\mu^{-1}(0)\cap\mathfrak{X})/G,$$ the quotient manifold of closed $G$-orbits in $\mu^{-1}(0)\cap\mathfrak{X}$. We denote the projection $\mu^{-1}(0)\cap\mathfrak{X}\longrightarrow M$ by $\rho$. The inclusion map $\mu^{-1}(0)\cap\mathfrak{X}\longrightarrow\mu^{-1}(0)$ induces a map $\pi\colon M\longrightarrow M_0$. Note that the Poisson bracket on ${\mathbb C}[T^*V]$ induces a Poisson bracket on the algebra ${\mathbb C}[\mu^{-1}(0)]^G$ and on the structure sheaf $\mathcal{O}_{M}$ of $M$. Moreover, the manifold $M$ is endowed with the structure of a holomorphic symplectic manifold by the Poisson bracket. The symmetric group $S_N$ acts on ${\mathbb C}^{2N}=({\mathbb C}^2)^N$. We obtain: **Proposition 1**. *$M_0\simeq{\mathbb C}^{2N}/S_N$ and $M\simeq\mathop{\mathrm{Hilb}}^N({\mathbb C}^2)$, and the map $\pi\colon M\longrightarrow M_0$ gives a resolution of singularities. Moreover, the corresponding homomorphism $\pi^*\colon {\mathbb C}[M_0] = {\mathbb C}[{\mathbb C}^{2N}]^{S_N}\longrightarrow\mathcal{O}_{M}(M)$ is a homomorphism of Poisson algebras, i.e. $\pi$ is a symplectic resolution.* The structure sheaf $\mathcal{O}_M$ of $M$ can be realised also by Hamiltonian reduction of the structure sheaf $\mathcal{O}_{\mathfrak{X}}$ of $\mathfrak{X}$ as follows: for an open subset $U$ of $M$, take an open subset $\widetilde{U}$ of $\mathfrak{X}$ such that $\rho(\widetilde{U}\cap \mu^{-1}(0))=U$. Then the sheaf associated with the presheaf given by $$\label{eq:Hamilton-str-sh} U\mapsto\Bigl(\mathcal{O}_{\mathfrak{X}}(\widetilde{U})\bigm/\sum_{i,j=1}^{N}\mathcal{O}_{\mathfrak{X}}(\widetilde{U}) \,\mu^*(E_{ij})\Bigr)^\mathfrak{g}$$ coincides with the structure sheaf $\mathcal{O}_M$ of $M$. ## A Sheaf of Poisson Superalgebras on the Hilbert Scheme {#sec:superHilb} By adding fermionic coordinate functions to $\mathcal{O}_{\mathfrak{X}}$, we construct a sheaf $\widetilde{\mathcal{O}}_{M}$ of commutative superalgebras on the Hilbert scheme $M$ by Hamiltonian reduction. Consider $\widetilde{\mathcal{O}}_{\mathfrak{X}}\coloneqq\mathcal{O}_{\mathfrak{X}}\otimes_{{\mathbb C}}\Lambda_{{\mathbb C}}(\psi_i,\phi_i\,|\,i=1,\dots,N)$, a sheaf of superalgebras on $\mathfrak{X}$, and define a Poisson bracket on $\Lambda_{{\mathbb C}}(\psi_i,\phi_i\,|\,i=1,\dots,N)$ by setting $\{\psi_i,\phi_j\}=\delta_{ij}$ for $i,j=1,\dots,N$ and zero otherwise. Then $\widetilde{\mathcal{O}}_{\mathfrak{X}}$ is a sheaf of Poisson superalgebras. We define a homomorphism of Lie superalgebras $$\label{eq:tldmu} \widetilde{\mu}\colon\mathfrak{g}\longrightarrow\widetilde{\mathcal{O}}_{\mathfrak{X}}(\mathfrak{X}),\qquad E_{ij}\mapsto\sum_{p=1}^{N}x_{ip}y_{pj}-\sum_{p=1}^{N}x_{pj}y_{ip}+\gamma_i\beta_j+\psi_i\phi_j.$$ The comoment map $\widetilde{\mu}$ induces an action of $\mathfrak{g}$ on $\widetilde{\mathcal{O}}_{\mathfrak{X}}$ by $A\mapsto\{\widetilde{\mu}(A),\raisebox{0.5ex}{\rule{2ex}{0.4pt}}\}$, and the corresponding $G$-action. As $\psi_i\phi_j$ is nilpotent for any $i,j=1,\dots,N$, note that the quotient sheaf $\widetilde{\mathcal{O}}_{\mathfrak{X}}\big/\sum_{i,j}\widetilde{\mathcal{O}}_{\mathfrak{X}}\,\widetilde{\mu}(E_{ij})$ is supported on the closed subset $\mu^{-1}(0)\subset\mathfrak{X}$. Define $$\widetilde{\mathcal{O}}_{M}=\Bigl(\rho_{*}\bigl(\widetilde{\mathcal{O}}_{\mathfrak{X}}\bigm/\sum_{i,j=1}^{N} \widetilde{\mathcal{O}}_{\mathfrak{X}}\,\widetilde{\mu}(E_{ij})\bigr)\Bigr)^\mathfrak{g},$$ a sheaf of superalgebras on $M$. The Poisson bracket on $\widetilde{\mathcal{O}}_{\mathfrak{X}}$ induces a Poisson bracket on $\widetilde{\mathcal{O}}_{M}$. ## Local Coordinates over a Certain Open Subset {#sec:big-cell} In the following, we describe the local coordinates of the sheaf $\widetilde{\mathcal{O}}_{M}$ over a certain affine open subset $U_{(N)}\subset M$ of the Hilbert scheme (cf. ). This will be useful in , when we describe a certain free-field realisation. Consider the Zariski open subset $$\widetilde{U}_{(N)}\coloneqq\{p\in T^*V\,|\,\gamma(p),\,X(p)\gamma(p),\,\dots,\,X(p)^{N-1} \gamma(p)\text{ span }{\mathbb C}^N\},$$ of $\mathfrak{X}$, and let $$U_{(N)}\coloneqq\rho(\widetilde{U}_{(N)}\cap\mu^{-1}(0))$$ be the corresponding Zariski open subset of $M$. We describe the local coordinates over the open subset $U_{(N)}$ explicitly. Set $B_{(N)}=(\gamma,X \gamma,\dots,X^{N-1}\gamma)$, an $N$-by-$N$ matrix with entries in ${\mathbb C}[T^* V]$. Note that $B_{(N)}$ is an invertible matrix in $\mathcal{O}_{\mathfrak{X}}(\widetilde{U}_{(N)})$. We then define the functions $[X^N:X^{i}]_{(N)}$ and $[Y:X^{i}]_{(N)}\in\widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U}_{(N)})$ for $i=0,\dots,N-1$ by the equalities $$\begin{aligned} {}^t([X^N:X^{i-1}]_{(N)})_{i=1,\dots,N}&=B_{(N)}^{-1}X^N\gamma,\\ {}^t([Y:X^{i-1}]_{(N)})_{i=1,\dots,N}&=B_{(N)}^{-1}Y\gamma\end{aligned}$$ and the fermionic functions $\{\psi:X^{i}\gamma\}_{(N)}\in\widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U}_{(N)})$ for $i=0,\dots,N-1$ by $${}^t(\{\psi:X^{i-1}\gamma\}_{(N)})_{i=1,\dots,N}=B_{(N)}^{-1}\psi,$$ where we write $\psi=(\psi_i)_{i=1,\dots,N}$. We also consider $\phi X^i\gamma$ for $i=0,\dots,N-1$ with $\phi={}^t(\phi_i)_{i=1,\dots,N}$. Then, all these functions are $G$-invariant and we shall see that $$\begin{aligned} \label{eq:local-coord-X} \widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U}_{(N)})&={\mathbb C}[[X^N:X^i]_{(N)},[Y:X^i]_{(N)}\,|\,i=0, \dots,N-1]\\ &\quad\otimes\Lambda_{{\mathbb C}}(\{\psi:X^i\gamma\}_{(N)},\phi X^{i}\gamma\,|\, i=0,\dots,N-1)\nonumber\\ &\quad\otimes{\mathbb C}[(B_{(N)}^{\pm 1})_{ij}\,|\,i,j=1,\dots,N]\otimes{\mathbb C}[\widetilde{\mu}(E_{ij})\,|\,i,j=1,\dots,N].\nonumber\end{aligned}$$ By [\[eq:Hamilton-str-sh\]](#eq:Hamilton-str-sh){reference-type="eqref" reference="eq:Hamilton-str-sh"}, this implies the following description of the local coordinates over $U_{(N)}$ so that $U_{(N)}\simeq{\mathbb C}^{2N}$ is affine: $$\begin{aligned} \label{eq:local-coord-Hilb} \widetilde{\mathcal{O}}_{M}(U_{(N)})&={\mathbb C}[[X^N:X^i]_{(N)},[Y:X^i]_{(N)}\,|\,i=0,\dots,N-1]\\ &\quad\otimes\Lambda_{{\mathbb C}}(\{\psi:X^i\gamma\}_{(N)},\phi X^{i}\gamma\,|\,i=0,\dots,N-1).\nonumber\end{aligned}$$ We now show [\[eq:local-coord-X\]](#eq:local-coord-X){reference-type="eqref" reference="eq:local-coord-X"}. Since $\mathcal{O}_{\mathfrak{X}}(\widetilde{U}_{(N)})={\mathbb C}[T^* V][(B_{(N)}^{-1})_{ij}\,|\,i,j=1,\dots,N]$, the right-hand side is contained in the left-hand side. To prove the opposite inclusion, it suffices to show that all the generators belong to the right-hand side. By the above definition, $\psi=B_{(N)}{}^t(\{\psi:X^{i-1}\}_{(N)})_{i=1,\dots,N}$ and $\phi=(\phi X^{i-1}\gamma)_{i=1,\dots,N}B_{(N)}^{-1}$ so that $\psi_i$ and $\phi_i$ belong to the right-hand side for any $i=1,\dots,N$. From the identity $X B_{(N)}=(X\gamma,X^2\gamma,\dots,X^N\gamma)$ it follows that $X=(X\gamma,X^2\gamma,\dots,X^N\gamma)B_{(N)}^{-1}$, and this implies that $x_{ij}$ belongs to the right-hand side for any $i,j=1,\dots,N$ since $X^N\gamma=B_{(N)}{}^t ([X^N:X^{i-1}]_{(N)})_{i=1,\dots,N}$. In order to apply the same reasoning to $Y$, we need two lemmata. To this end, set $Z=(\widetilde{\mu}(E_{ij}))_{i,j=1,\dots,N}=XY-YX+\gamma\beta+\psi\phi$. **Lemma 2**. *For $i\in{\mathbb Z}_{\ge0}$, $\beta X^i\gamma$ lies in the right-hand side of [\[eq:local-coord-X\]](#eq:local-coord-X){reference-type="eqref" reference="eq:local-coord-X"}.* *Proof.* Note that $$\begin{aligned} \beta X^i\gamma %=\Tr(\beta X^i\gamma) &=\mathop{\mathrm{Tr}}(X^i\gamma\beta)=-\mathop{\mathrm{Tr}}(X^{i+1}Y)+\mathop{\mathrm{Tr}}(X^iYX)-\mathop{\mathrm{Tr}}(X^i\psi\phi)+\mathop{\mathrm{Tr}}(X^iZ)\\ &=-\mathop{\mathrm{Tr}}(X^i\psi\phi)+\mathop{\mathrm{Tr}}(X^i Z)=\phi X^i\psi+\mathop{\mathrm{Tr}}(X^iZ).\end{aligned}$$ Since $\psi_i$, $\phi_i$, $x_{ij}$ and $Z_{ij}=\widetilde{\mu}(E_{ij})$ belong to the right-hand side for all $i,j$, the same holds for $\beta X^i\gamma$ for $i\in{\mathbb Z}_{\ge0}$. ◻ **Lemma 3**. *For $m,i=0,\dots,N-1$, let $[YX^m:X^i]_{(N)}\in\widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U}_{(N)})$ be defined by $([YX^m:X^{i-1}]_{(N)})_{i=1,\dots,N}=B_{(N)}^{-1}YX^m\gamma$. Then, $[YX^m:X^i]_{(N)}$ belongs to the right-hand side of [\[eq:local-coord-X\]](#eq:local-coord-X){reference-type="eqref" reference="eq:local-coord-X"}.* *Proof.* We proceed by induction over $m$. If $m=0$, $[YX^m:X^{i}]_{(N)}=[Y:X^i]_{(N)}$ belongs to the right-hand side of [\[eq:local-coord-X\]](#eq:local-coord-X){reference-type="eqref" reference="eq:local-coord-X"} for $i=0,\dots,N-1$. For the induction step, assume that $[YX^m:X^i]_{(N)}$ lies in the right-hand side for all $i=0,\dots,N-1$. Then, $YX^{m+1}\gamma=XYX^{m}\gamma+\gamma\beta X^{m}\gamma+\psi\phi X^{m}\gamma-ZX^m\gamma$. By the induction hypothesis, $XYX^m\gamma$ lies in the right-hand side. By , $\gamma(\beta X^m\gamma)$ belongs to the right-hand side, and clearly so does $\psi(\phi X^m\gamma)$. Therefore, $[YX^{m+1}:X^i]_{(N)}$ also lies in the right-hand side of [\[eq:local-coord-X\]](#eq:local-coord-X){reference-type="eqref" reference="eq:local-coord-X"} for all $i=0,\dots,N-1$. ◻ By , all entries of the matrix $YB_{(N)}=(Y\gamma,YX\gamma,\dots,YX^{N-1}\gamma)$ lie in the right-hand side of [\[eq:local-coord-X\]](#eq:local-coord-X){reference-type="eqref" reference="eq:local-coord-X"}. Thus, so does $y_{ij}$ for all $i,j=1,\dots,N$. Similarly, all entries of the row vector $\beta B_{(N)}=(\beta\gamma,\beta X\gamma,\dots,\beta X^{N-1}\gamma)$ lie in the right-hand side, and hence so do all $\beta_i$. This proves [\[eq:local-coord-X\]](#eq:local-coord-X){reference-type="eqref" reference="eq:local-coord-X"}. ## Local Trivialisation of the Hamiltonian Reduction {#sec:local-trivial} The Hilbert scheme $M=\mathop{\mathrm{Hilb}}^N({\mathbb C}^2)$ has a well-known affine open covering $M=\bigcup_{\lambda\vdash N}U_{\lambda}$ indexed by the partitions $\lambda$ of $N$, which includes the set $U_{(N)}$ from the preceding section. We now describe the local trivialisations of the sheaf $\widetilde{\mathcal{O}}_{M}$ over the open sets $U_\lambda$. Let $\lambda=(\lambda_0\ge\lambda_1\ge\dots)$ be a partition of $N$, denoted by $\lambda\vdash N$. For a pair of non-negative integers $(i,j)\in{\mathbb Z}_{\ge 0}^2$, we write $(i,j)\in\lambda$ if $i<\lambda_j$. Generalising the definition in , we let $\widetilde{U}_{\lambda}\subset T^*V$ be the subset of those $p\in T^*V$ satisfying that $\{X(p)^iY(p)^j\gamma(p)\,|\,(i,j)\in\lambda\}$ spans the vector space ${\mathbb C}^N$. With $B_{\lambda}\coloneqq(X^iY^j\gamma)_{(i,j)\in\lambda}$, $\widetilde{U}_{\lambda}$ may be written as $\widetilde{U}_{\lambda}=\{p\in T^*V\,|\,\det B_{\lambda}(p)\ne0\}$. Thus, $\widetilde{U}_{\lambda}$ is a Zariski open subset of $\mathfrak{X}$, and moreover there is an affine open covering $\mathfrak{X}=\bigcup_{\lambda\vdash N}\widetilde{U}_{\lambda}$. It induces an affine open covering $$M=\bigcup_{\lambda\vdash N}U_{\lambda},\quad\text{with}\quad U_{\lambda}=(\widetilde{U}_{\lambda}\cap\mu^{-1}(0))/G$$ the affine open subset of $M$ associated with $\widetilde{U}_{\lambda}$. For $P=P(X,Y)\in{\mathbb C}\langle X,Y\rangle$ and $(i,j)\in\lambda$, define a bosonic section $[P:X^iY^j]_{\lambda}$ and a fermionic section $\{\psi:X^iY^j\gamma\}_{\lambda}$ in $\widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U}_{\lambda})$ by $$\begin{aligned} {}^t([P:X^iY^j]_{\lambda})_{(i,j)\in\lambda}&=B_{\lambda}^{-1}P\gamma,\\ {}^t(\{\psi:X^iY^j\gamma\}_{\lambda})_{(i,j)\in\lambda}&=B_{\lambda}^{-1}\psi,\end{aligned}$$ respectively. Then, $[P:X^iY^j]_{\lambda}$ is $G$-invariant for any $P\in{\mathbb C}\langle X,Y\rangle$, $(i,j)\in\lambda$. Also, $\{\psi:X^iY^j\gamma\}_{\lambda}\in\widetilde{\mathcal{O}}_{\mathfrak{X}}(\mathfrak{X})$ is a $G$-invariant fermionic section for $(i,j)\in\lambda$. Define $A_{\lambda}$ to be the subalgebra of $\mathcal{O}_{\mathfrak{X}}(\widetilde{U}_{\lambda})={\mathbb C}[T^*V][(B_{\lambda}^{-1})_{ij}\,|\,i,j=1,\dots,N]\subset\widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U}_{\lambda})$ generated by $[P:X^iY^j]_{\lambda}$ for all $P\in {\mathbb C}\langle X,Y\rangle$ and $(i,j)\in\lambda$. **Proposition 4**. *For $\lambda\vdash N$, there is the following trivialisation of the Hamiltonian reduction $$\begin{aligned} \widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U}_{\lambda})&=A_{\lambda}\otimes\Lambda(\{\psi:X^iY^j \gamma\}_{\lambda},\phi X^iY^j\gamma\,|\,(i,j)\in\lambda)\\ &\quad\otimes{\mathbb C}[(B_{\lambda}^{\pm 1})_{ij}\,|\,i,j=1,\dots,N]\otimes{\mathbb C}[\widetilde{\mu}(E_{ij})\,|\,i,j=1,\dots,N]\eqqcolon\widetilde{A}_{\lambda}.\end{aligned}$$ This localisation yields the isomorphism $$\widetilde{\mathcal{O}}_M(U_{\lambda})\simeq A_{\lambda}\otimes\Lambda(\{\psi:X^iY^j\gamma\}_{\lambda},\phi X^iY^j\gamma\,|\,(i,j)\in\lambda).$$* **Remark 5**. It is easy to see that $A_{\lambda}=\mathcal{O}_M(U_{\lambda})$. For further details of the combinatorial nature of the affine open covering $M=\bigcup_{\lambda\vdash N}U_{\lambda}$ see, e.g., Chapter 18 of [@MS05]. *Proof of .* We have to show that all generators of $\widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U}_{\lambda})$ belong to $\widetilde{A}_{\lambda}$. Evidently, all entries of the vectors $X^aY^b\gamma=B_{\lambda}{}^t([X^aY^b:X^iY^j]_{\lambda})_{(i,j)\in\lambda}$, $\psi=B_{\lambda}{}^t(\{\psi:X^iY^j\gamma\}_{\lambda})_{(i,j)\in\lambda}$ and $\phi=(\phi X^iY^j\gamma)_{(i,j)\in\lambda}B_{\lambda}^{-1}$ lie in $\widetilde{A}_{\lambda}$. Consider the matrix $X B_{\lambda}=(X^{k+1}Y^l\gamma)_{(k,l)\in\lambda}$. All its entries are of the form $[X^{k+1}Y^l:X^iY^j]_{\lambda}$ and lie in $A_{\lambda}$. Thus, $X=(X^{k+1}Y^l\gamma)_{(k,l)\in\lambda}B^{-1}_{\lambda}$ implies that $x_{ij}$ lies in $\widetilde{A}_{\lambda}$ for $i,j=1,\dots,N$. Similarly, $Y=(YX^kY^l\gamma)_{(k,l)\in\lambda}B_{\lambda}^{-1}$ implies that $y_{ij}$ lies in $\widetilde{A}_{\lambda}$ for $i,j=1,\dots,N$. Therefore, we obtain the identity of . ◻ # Sheaves of $\hbar$-Adic Vertex Superalgebras {#sec:sheaves-SVA} In this section, we review the notion of ($\hbar$-adic) vertex superalgebras. We also recall jet schemes and arc spaces. Then we introduce $\hbar$-adic versions of some well-known examples of vertex superalgebras. ## Vertex Superalgebras and $\hbar$-Adic Vertex Superalgebras {#sec:vertex-algebras} In the following, we define vertex superalgebras, vertex Poisson superalgebras and $\hbar$-adic vertex superalgebras. A vertex superalgebra $V$ (see, e.g., [@Kac]) is a tuple $(V, \mathbf{1}, \partial, Y(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z))$, where $V$ is a ${\mathbb Z}/2{\mathbb Z}$-graded vector space, $\mathbf{1}\in V$ is a distinguished even element called the vacuum vector, $\partial$ is an even, linear map $\partial\colon V \longrightarrow V$ called the translation operator, and $Y(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z)$ is the state-field correspondence, an even, linear map, $$\begin{aligned} Y(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z)\colon V &\longrightarrow \mathop{\mathrm{End}}_{{\mathbb C}}(V)[[z, z^{-1}]],\\ a&\longmapsto Y(a, z) = \sum_{n \in {\mathbb Z}} a_{(-n-1)} z^{n},\end{aligned}$$ subject to the following axioms: 1. $Y(a, z)$ is a field, i.e. $a_{(n)} b = 0$ for any $a,b \in V$ if $n \gg 0$, 2. $Y(\mathbf{1}, z) = \operatorname{Id}_V$, 3. $Y(a, z) \mathbf{1}\in V[[z]]$ and $Y(a, z) \mathbf{1}|_{z=0} = a$ for any $a \in V$, 4. $[\partial, Y(a, z)] = Y(\partial a, z) = \partial_z Y(a, z)$ for any $a \in V$, and $\partial \mathbf{1}= 0$, 5. (Borcherds' identity) for any $a,b \in V$ and $l,m,n \in {\mathbb Z}$, $$\begin{aligned} \label{eq:Borcherds} &\sum_{j=0}^{\infty} \binom{m}{j} (a_{(n+j)} b)_{(l+m-j)}\\ &= \sum_{j=0}^{\infty} (-1)^j \binom{n}{j} \{a_{(m+n-j)} b_{(l+j)} - (-1)^{n+p(a)p(b)} b_{(l+n-j)} a_{(m+j)}\},\nonumber\end{aligned}$$ where $p(a) \in {\mathbb Z}/2{\mathbb Z}$ is the parity of $a$. The $Y(a,z)$, $a\in V$, are called vertex operators and sometimes written as $a(z)$. The vertex superalgebra $V$ is said to be commutative if $a_{(n)} = 0$ for any $a\in V$ and $n \in {\mathbb Z}_{\ge 0}$. Equivalently, all vertex operators (super)commute with each other. In this case, the $(-1)$-product endows $V$ with the structure of a commutative superalgebra (with an even derivation). It is well-known that the commutation relations between the operators $a_{(m)}$ and $b_{(n)}$ for $a,b \in V$ and $m,n \in {\mathbb Z}$ are encoded into the identity $$\label{eq:OPE} Y(a, z) Y(b, w) = \sum_{n \in {\mathbb Z}} \frac{Y(a_{(n)} b, w)}{(z-w)^{n+1}} = \sum_{n \ge 0} \frac{Y(a_{(n)} b, w)}{(z-w)^{n+1}} + {:}Y(a, z) Y(b, w) {:},$$ called the operator product expansion (or OPE for short), where ${:}\raisebox{0.5ex}{\rule{2ex}{0.4pt}}{:}$ is the normally ordered product (see, e.g., [@FBZ]). It is often written in the form $$Y(a, z) Y(b, w) \sim \sum_{n \ge 0} \frac{Y(a_{(n)} b, w)}{(z-w)^{n+1}},$$ omitting the term ${:}Y(a, z) Y(b, w) {:}$, which is regular at $z=w$ and does not affect the commutators $[a_{(m)}, b_{(n)}]$. A conformal vertex superalgebra of central charge $c\in{\mathbb C}$ is a vertex superalgebra $V$ together with an even vector $T\in V$ satisfying 1. $[T_{(m+1)},T_{(n+1)}]=(m-n)T_{(m+n+1)}+\frac{m^3-m}{12}\delta_{m+n,0}\,c\operatorname{Id}_V$ for any $m,n\in{\mathbb Z}$, 2. $T_{(1)}$ acts semisimply on $V$ with eigenvalues in $\frac{1}{2}{\mathbb Z}$, called weights, 3. $\operatorname{wt}(a_{(n)})=\operatorname{wt}(a)-n-1$ for any $a\in V$, $n\in{\mathbb Z}$, 4. $T_{(0)}=\partial$. We write $V=\bigoplus_{n\in\frac{1}{2}{\mathbb Z}}V_n$ for the corresponding eigenspace decomposition of $V$. Note that $\mathbf{1}\in V_0$, $\partial\in V_1$ and $T\in V_2$. We point out that we do not require "correct statistics", i.e. that $\bigoplus_{n\in{\mathbb Z}}V_n$ and $\bigoplus_{n\in\frac{1}{2}+{\mathbb Z}}V_n$ be exactly the vectors of even and odd parity, respectively. A conformal vertex superalgebra $V$ is called vertex operator superalgebra if additionally $V_n=\{0\}$ for $n\ll 0$ and $\mathop{\mathrm{dim}}_{\mathbb C}V_n<\infty$ for all $n\in\frac{1}{2}{\mathbb Z}$. A vertex Poisson superalgebra $V$ (see, e.g., [@FBZ] in the even case) is a tuple $(V, \mathbf{1}, \partial, Y_{-}(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z), Y_{+}(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z))$, where $$Y_{\pm}(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z)\colon V \longrightarrow \mathop{\mathrm{End}}_{{\mathbb C}}(V)[[z, z^{-1}]]$$ are even, linear maps to fields $$Y_{-}(a, z) = \sum_{n \in {\mathbb Z}_{\ge 0}} a_{(-n-1)} z^n, \quad Y_{+}(a, z) = \sum_{n \in {\mathbb Z}_{< 0}} a_{(-n-1)} z^n,$$ $a\in V$, such that $(V, \mathbf{1}, \partial, Y_{-}(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z))$ is a commutative vertex superalgebra and $(V, \partial, Y_{+}(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z))$ is a vertex Lie superalgebra, i.e. the operators $a_{(n)}$ for $n \in {\mathbb Z}_{\ge 0}$ satisfy the following axioms: 1. $a_{(n)} b = (-1)^{n+p(a)p(b)+1} \sum_{j \ge 0} (-1)^j \partial^j (b_{(n+j)} a) / j!$, 2. $[a_{(m)},b_{(n)}] = \sum_{j \ge 0} \binom{m}{j} (a_{(j)} b)_{(m+n-j)}$, 3. $[\partial, Y_{+}(a, z)] = Y_{+}(\partial a, z) = \partial_z Y_{+}(a, z)$ for any $a,b \in V$ and $m,n \in {\mathbb Z}_{\ge 0}$. For a vertex Poisson superalgebra, in addition, we require that $Y_{-}(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z)$ and $Y_{+}(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z)$ be compatible in the sense that the $a_{(n)}$ for $n \in {\mathbb Z}_{\ge 0}$ are derivations with respect to the $(-1)$-product, i.e. that 1. $a_{(n)} (b_{(-1)} c) = (a_{(n)} b)_{(-1)} c + (-1)^{p(a)p(b)} b_{(-1)} (a_{(n)} c)$ for all $a,b,c \in V$ and $n \in {\mathbb Z}_{\ge 0}$. Following [@Li04], we introduce the notion of $\hbar$-adic vertex superalgebras. To this end, let $\hbar$ be an indeterminate that commutes with all other operators. An $\hbar$-adic vertex superalgebra $V$ is a tuple $(V, \mathbf{1}, \partial, Y(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z))$, where $V$ is a flat ${\mathbb C}[[\hbar]]$-module complete in the $\hbar$-adic topology, the even vacuum vector $\mathbf{1}\in V$ and the parity-preserving ${\mathbb C}[[\hbar]]$-linear map $\partial\colon V \longrightarrow V$ satisfy the same axioms as for a vertex superalgebra and $$Y(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z)\colon V \longrightarrow \mathop{\mathrm{End}}_{{\mathbb C}[[\hbar]]}(V)[[z, z^{-1}]]$$ is a ${\mathbb C}[[\hbar]]$-linear map such that the $(n)$-products are continuous with respect to the $\hbar$-adic topology and $(V / \hbar^k V, \mathbf{1}, \partial, Y(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z))$ is a vertex superalgebra for each $k \in {\mathbb Z}_{>0}$. Note that an $\hbar$-adic vertex superalgebra is not necessarily a vertex superalgebra over ${\mathbb C}[[\hbar]]$ since $Y(a, z)$ is not a field on $V$. Indeed, for any $k \in {\mathbb Z}_{>0}$, $Y(a, z) = \sum_{n \in {\mathbb Z}} a_{(n)} z^{-n-1}$ satisfies $a_{(n)} b \equiv 0$ modulo $\hbar^k$ if $n \gg 0$ but not necessarily $a_{(n)} b = 0$. Let $(V, \mathbf{1}, \partial, Y(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z))$ be an $\hbar$-adic vertex superalgebra. Assume that $V / \hbar V$ is commutative. Then, $Y_{+}(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z) \coloneqq \hbar^{-1} Y(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z) \bmod \hbar$ satisfies the axioms of a vertex Lie superalgebra. Thus, $(V / \hbar V, \mathbf{1}, \partial, Y(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z) \bmod \hbar, \hbar^{-1} Y(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z) \bmod \hbar)$ is a vertex Poisson superalgebra. For $\hbar$-adic vertex superalgebras $V$ and $W$, we denote the $\hbar$-adic completion of the tensor product $V \otimes_{{\mathbb C}[[\hbar]]} W$ by $V \mathop{\mathrm{\widehat{\otimes}}}W$. This tensor product $V \mathop{\mathrm{\widehat{\otimes}}}W$ is also an $\hbar$-adic vertex superalgebra and it includes $V$ and $W$ as $\hbar$-adic vertex subalgebras. One can also introduce the notions of conformal $\hbar$-adic vertex superalgebras and $\hbar$-adic vertex operator superalgebras in analogy to the non-$\hbar$-adic versions above. For instance, the main properties of a conformal vector in an $\hbar$-adic vertex superalgebra are stated in . ## Jet Bundles {#sec:jet-bundles} We briefly recall jet bundles, jet schemes and arc spaces. In the following, a differential algebra is a commutative ${\mathbb C}$-superalgebra $A$ with an even derivation $\partial$. Note that a differential algebra $A$ can be regarded as a commutative vertex superalgebra, where $Y(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z)$ is given by $Y(a, z) = \mathrm{e}^{z \partial} a$ for $a \in A$. For a finitely generated, unital, commutative superalgebra $R$, let $J_{\infty}{R}$ be the unique differential algebra satisfying the following universal property: there is an algebra homomorphism $j \colon R \longrightarrow J_{\infty}{R}$ such that for any algebra homomorphism $\varphi \colon R \longrightarrow A$ from $R$ to a differential algebra $A$, there is a unique differential algebra homomorphism $\widetilde{\varphi} \colon J_{\infty}{R} \longrightarrow A$ satisfying $\widetilde{\varphi} \circ j = \varphi$. The differential algebra $J_{\infty}{R}$ can also be characterised as the unique unital, commutative ${\mathbb C}$-superalgebra such that $\mathop{\mathrm{Hom}}_{\mathrm{Alg}}(J_{\infty}{R}, A) \simeq \mathop{\mathrm{Hom}}_{\mathrm{Alg}}(R, A[[t]])$ for any unital, commutative ${\mathbb C}$-superalgebra $A$. For $m \in {\mathbb Z}_{\ge 0}$ and $R$ as above, let $J_{m}{R}$ be the unique unital, commutative ${\mathbb C}$-superalgebra satisfying $\mathop{\mathrm{Hom}}_{\mathrm{Alg}}(J_{m}{R}, A) \simeq \mathop{\mathrm{Hom}}_{\mathrm{Alg}}(R, A[t]/(t^{m+1}))$ for any unital, commutative ${\mathbb C}$-superalgebra $A$. There exists a natural homomorphism $J_{m}{R} \longrightarrow J_{n}{R}$ for $m \le n$, and $J_{\infty}{R}$ is the injective limit associated with the direct system $J_{0}{R} \longrightarrow J_{1}{R} \longrightarrow J_{2}{R} \longrightarrow \dots$ We may assume that the commutative superalgebra $R$ is of the form $$R = ({\mathbb C}[x_1, \dots, x_N] \otimes \Lambda(\psi_1, \dots, \psi_M)) / (f_1, \dots, f_r)$$ with bosonic variables $x_1,\dots,x_N$, fermionic variables $\psi_1,\dots,\psi_M$ and $f_1,\dots,f_r \in {\mathbb C}[x_1, \dots, x_N] \otimes \Lambda(\psi_1, \dots, \psi_M)$. Then, $J_{\infty}{R}$ is the differential algebra given by $$J_{\infty}{R} = ({\mathbb C}[x_{i (-n)} \,|\, \substack{i = 1, \dots, N \\ n = 1, 2, \dots}] \otimes \Lambda(\psi_{i (-n)} \,|\, \substack{i = 1, \dots, M \\ n = 1, 2, \dots})) / ( \partial^n f_i \,|\, \substack{i = 1, \dots, r \\ n = 1, 2, \dots}),$$ where $x_{i (-n)} = (\partial^{n-1} / (n-1)!) x_i$ and $\psi_{i (-n)} = (\partial^{n-1} / (n-1)!) \psi_i$. For $m \in {\mathbb Z}_{\ge 0}$, the superalgebra $J_{m}{R}$ is the subalgebra of $J_{\infty}{R}$ generated by $x_{i (-n)}$ and $\psi_{j (-n)}$ with $i=1,\dots,N$, $j=1,\dots,M$ and $n=0,1,\dots,m$. Let $X$ be a topological space and $\widetilde{\mathcal{O}}_X$ a sheaf of unital, commutative superalgebras on $X$. We define the sheaf of differential algebras $\widetilde{\mathcal{O}}_{J_\infty X} \coloneqq J_{\infty}{\widetilde{\mathcal{O}}_{X}}$ associated with $\widetilde{\mathcal{O}}_X$ as the sheaf $U \mapsto J_{\infty}{(\widetilde{\mathcal{O}}_{X}(U))}$ for open subsets $U \subset X$. Note that there is a natural inclusion $j \colon \widetilde{\mathcal{O}}_X \hookrightarrow J_{\infty}{\widetilde{\mathcal{O}}_{X}}$. Similarly, for $m \in {\mathbb Z}_{\ge 0}$, let $J_{m}{\widetilde{\mathcal{O}}_{X}}$ be the sheaf $U \mapsto J_{m}{(\widetilde{\mathcal{O}}_{X}(U))}$ for open subsets $U \subset X$. We call the sheaf $\widetilde{\mathcal{O}}_{J_\infty X} = J_{\infty}{\widetilde{\mathcal{O}}_{X}}$ the $\infty$-jet bundle associated with $\widetilde{\mathcal{O}}_{X}$. Let $X$ be a scheme of finite type with structure sheaf $\mathcal{O}_X$. The arc space (or $\infty$-jet scheme) over $X$ is the unique scheme $J_{\infty}{X}$ satisfying $$\mathop{\mathrm{Hom}}_{\mathrm{Scheme}}(\mathop{\mathrm{Spec}}A, J_{\infty}{X}) \simeq \mathop{\mathrm{Hom}}_{\mathrm{Scheme}}(\mathop{\mathrm{Spec}}A[[t]], X)$$ for any unital, commutative ${\mathbb C}$-superalgebra $A$. Similarly, for $m \in {\mathbb Z}_{\ge 0}$, the $m$-jet scheme over $X$ is the unique scheme $J_{m}{X}$ such that $$\mathop{\mathrm{Hom}}_{\mathrm{Scheme}}(\mathop{\mathrm{Spec}}A, J_{m}{X}) \simeq \mathop{\mathrm{Hom}}_{\mathrm{Scheme}}(\mathop{\mathrm{Spec}}A[t] / (t^{m+1}), X)$$ for any unital, commutative ${\mathbb C}$-superalgebra $A$. By definition, for $m \in {\mathbb Z}_{\ge 0} \cup \{\infty\}$, there is a natural morphism $\pi_m \colon J_{m}{X} \longrightarrow X$. For $m \in {\mathbb Z}_{\ge 0} \cup \{\infty\}$, the sheaf $J_{m}{\mathcal{O}_X}$ associated with $\mathcal{O}_X$ is related to the structure sheaf $\mathcal{O}_{J_{m}{X}}$ of the $m$-jet scheme $J_{m}{X}$ by $(\pi_m)_{*} \mathcal{O}_{J_{m}{X}} = J_{m}{\mathcal{O}_{X}}$. Now assume that the commutative superalgebra $R$ is a Poisson superalgebra with Poisson bracket $\{\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, \raisebox{0.5ex}{\rule{2ex}{0.4pt}}\}$. The Poisson bracket $\{\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, \raisebox{0.5ex}{\rule{2ex}{0.4pt}}\}$ of $\widetilde{\mathcal{O}}_X$ induces a natural vertex Poisson superalgebra structure on the differential algebra $J_{\infty}{R}$ satisfying $f_{(n)} g = \delta_{n0} \{f, g\}$ for $f,g \in R$ and $n \in {\mathbb Z}_{\ge 0}$. See [@Arakawa12], Section 2.3, for details. ## The $\beta\gamma$-System and Its Microlocalisation {#sec:h-adic-betagamma} In the following, we introduce the $\hbar$-adic analogue $\mathcal{D}^\mathrm{ch}(T^*{\mathbb C}^{N})_{\hbar}$ of the $\beta\gamma$-system $\mathcal{D}^\mathrm{ch}(T^*{\mathbb C}^{N})=(\beta\gamma)^{\otimes N}$ of rank $N$, also called Weyl vertex algebra or bosonic ghost vertex algebra. We then construct the microlocalisation $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}$ of $\mathcal{D}^\mathrm{ch}(T^*{\mathbb C}^{N})_{\hbar}$ as a sheaf of $\hbar$-adic vertex algebras on the affine space $T^* {\mathbb C}^{N}$, as discussed in Section 2.2 of [@AKM15], with global sections $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(T^* {\mathbb C}^N) = \mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^N)_{\hbar}$. Since we only deal with the affine space $T^* {\mathbb C}^N$, the construction looks simpler than the one in [@AKM15], where we considered the cotangent bundle of a flag manifold. Denote by $x_1,\dots,x_N$ and $y_1,\dots,y_N$ be the standard coordinate functions on $T^*{\mathbb C}^N = {\mathbb C}^{N} \oplus ({\mathbb C}^N)^*$. They are Darboux coordinates with respect to the standard symplectic form. Recall that the $\beta\gamma$-system $\mathcal{D}^{\mathrm{ch}}(T^* {\mathbb C}^N)=(\beta\gamma)^{\otimes N}$ is the vertex algebra strongly generated by $x_i$ and $y_i$ for $i=1,\dots,N$ being subject to the operator product expansions $x_i(z) y_j(w) \sim -\delta_{ij}/(z-w)$ and $x_i(z) x_j(w) \sim y_i(z) y_j(w) \sim 0$. The $\hbar$-adic $\beta\gamma$-system on $T^* {\mathbb C}^N$ is the $\hbar$-adic vertex algebra $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^{N})_{\hbar}$ isomorphic as a ${\mathbb C}[[\hbar]]$-module to $$\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^{N})_{\hbar} = {\mathbb C}[[\hbar]][x_{1 (-n)}, \dots, x_{N (-n)}, y_{1 (-n)}, \dots, y_{N (-n)} \,|\, n \in {\mathbb Z}_{>0}]$$ with operator product expansions $$\begin{aligned} x_i(z) y_j(w) &\sim - \delta_{ij} \hbar /(z-w),\\ x_i(z) x_j(w) &\sim 0,\\ y_i(z) y_j(w) &\sim 0\end{aligned}$$ for $i,j=1,\dots,N$, where we write $x_i(z) = Y(x_{i (-1)},z)$ and $y_i(z) = Y(y_{i (-1)},z)$. Evidently, it is an $\hbar$-adic analogue of the $\beta\gamma$-system vertex algebra. The notion of chiral differential operators (CDO), which is the chiral analogue of sheaves of differential operators on complex manifolds, was studied in [@GMS00; @GMS04; @GMS03; @MSV99; @MS99]. Following their construction, we can localise the $\beta\gamma$-system $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^N)$ generated by $x_i$ and $y_i$ for $i=1,\dots,N$ as a sheaf of vertex algebras on the complex vector space ${\mathbb C}^N$. Moreover, as discussed in [@AKM15], we can also localise the above $\hbar$-adic $\beta\gamma$-system $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^{N})_{\hbar}$ as a sheaf $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}$ of $\hbar$-adic vertex algebras on the cotangent bundle $T^* {\mathbb C}^N$. We shall give the details of this construction below, which we can summarise as follows: for the $\hbar$-adic $\beta\gamma$-system, the operator product expansions (and hence the $(n)$-products) between the vertex operators are determined by the Wick formula and thus they turn out to be bidifferential operators in the variables $x_{i (-n)}$, $y_{i (-n)}$. Therefore, even for rational functions in $x_{i (-n)}$, $y_{i (-n)}$, the same bidifferential operators give well-defined operator product expansions (or $(n)$-products) between them. Hence, we obtain a sheaf of $\hbar$-adic vertex algebras $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}$ on $T^* {\mathbb C}^N$. As we discussed in the previous section, the jet bundle $\mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}$ on the symplectic vector space $T^* {\mathbb C}^N$ is equipped with a vertex Poisson algebra structure. The $\hbar$-adic $\beta\gamma$-system $\mathcal{D}^\mathrm{ch}_{T^*{\mathbb C}^N, \hbar}$ is a quantisation of $\mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}$. Indeed, the quotient $\mathcal{D}^\mathrm{ch}_{T^*{\mathbb C}^N, \hbar} / \hbar \mathcal{D}^\mathrm{ch}_{T^*{\mathbb C}^N, \hbar}$ is isomorphic to $\mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}$ as vertex Poisson algebras. We now construct the sheaf of $\hbar$-adic vertex algebras $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}$ over $T^* {\mathbb C}^N$ that localises $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^N)_{\hbar}$. We discuss the localisation in Zariski topology. The key is that the operator product expansion of the ($\hbar$-adic) $\beta\gamma$-system satisfies the Wick formula, which implies that it can be realised by certain bidifferential operators. This fact allows us to construct the localisation of $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}$ similarly to the construction of the ring of microlocal differential operators on a holomorphic symplectic manifold (cf. [@KR08; @Losev10]). While the $(-n)$-products on the $\hbar$-adic $\beta\gamma$-system are not associative, the variables $x_{i (-n)}$ and $y_{i (-n)}$ for $i=1,\dots,N$ and $n \in {\mathbb Z}_{>0}$ mutually commute in $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^N)_{\hbar}$. Thus, any element of $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^N)_{\hbar}$ can be viewed as a polynomial in the variables $\{x_{i (-n)}, y_{i (-n)} \,|\, i=1, \dots, N; n \in {\mathbb Z}_{>0}\}$. Namely, we do not need to care about the order of the variables, but we do care about the placement of brackets. We identify the variables $x_{i (-1)}$ and $y_{i (-1)}$ with the coordinate functions $x_i$ and $y_i$, respectively, in ${\mathbb C}[T^* {\mathbb C}^N] = {\mathbb C}[x_1, \dots, x_N, y_1, \dots, y_N]$ for $i=1,\dots,N$. Recall the $\infty$-jet bundle $\mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}$ on $T^* {\mathbb C}^N$. For an open subset $U \subset T^* {\mathbb C}^N$, $$\mathcal{O}_{J_{\infty}T^* {\mathbb C}^N}(U) = \mathcal{O}_{T^* {\mathbb C}^N}(U) \otimes_{{\mathbb C}[T^* {\mathbb C}^N]} {\mathbb C}[x_{i (-n)}, y_{i (-n)} \,|\, i=1, \dots, N; n \in {\mathbb Z}_{>0}],$$ and we set $$\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U) = \mathcal{O}_{J_{\infty}T^* {\mathbb C}^N}(U)[[\hbar]]$$ as a ${\mathbb C}[[\hbar]]$-module. For example, if $U = U_f = \{ f \ne 0 \} \subset T^* {\mathbb C}^N$ is an affine open subset associated with a polynomial $f \in {\mathbb C}[x_{i (-1)}, y_{i (-1)} \,|\, i=1, \dots, N] = {\mathbb C}[T^* {\mathbb C}^N]$, then $$\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U_f) = {\mathbb C}[[\hbar]][x_{i (-n)}, y_{i (-n)}, f^{-1} \,|\, i=1, \dots, N; n \in {\mathbb Z}_{>0}]$$ as a ${\mathbb C}[[\hbar]]$-module. Moreover, for open subsets $U' \subset U$, the restriction morphism $\mathop{\mathrm{res}}^U_{U'}\colon \mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U) \longrightarrow \mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U')$ is induced by the restriction morphism $\mathcal{O}_{T^* {\mathbb C}^N}(U) \longrightarrow \mathcal{O}_{T^* {\mathbb C}^N}(U')$. Hence, we obtain a sheaf of ${\mathbb C}[[\hbar]]$-modules $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}$ on the affine space $T^* {\mathbb C}^N$. The global sections are $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(T^* {\mathbb C}^N) = \mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^N)_{\hbar}$. Then, in order to define an $\hbar$-adic vertex algebra structure on $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}$, we recall the following observation (see also [@AKM15], Lemma 2.8.1.1). For a polynomial $f \in {\mathbb C}[x_{i (-n)}, y_{i (-n)} \,|\, i=1, \dots, N; n \in {\mathbb Z}_{>0}]$, we write $f(z) = Y(f \mathbf{1}, z)$ for the corresponding vertex operator. Then, the following lemma follows from the Wick formula (see [@Kac], Theorem 3.3, [@FBZ], Lemma 12.2.6). **Lemma 6**. *There exist bidifferential operators $P_{nk}$ on the polynomial algebra ${\mathbb C}[x_{i (-m)}, y_{i (-m)} \,|\, i=1, \dots, N; m \in {\mathbb Z}_{>0}]$ for $k \in {\mathbb Z}_{\ge 0}$, $n \in {\mathbb Z}$ such that the operator product expansion of $f(z) g(w)$ in $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^N)_{\hbar}$ satisfies $$f(z) g(w) = \sum_{k = 0}^{\infty} \sum_{n \in {\mathbb Z}} \frac{\hbar^{k}}{(z-w)^n} P_{nk}(f, g)$$ for $f,g \in \mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}(T^* {\mathbb C}^N) = {\mathbb C}[x_{i (-m)}, y_{i (-m)} \,|\, i=1, \dots, N; m \in {\mathbb Z}_{>0}]$.* For an open subset $U \subset T^* {\mathbb C}^N$, we shall define the operator product expansion of $f(z) g(w)$ for $f,g \in \mathcal{O}_{J_{\infty }T^* {\mathbb C}^N}(U)$. In order to do so by using , we regard the elements $f$ and $g$ as rational functions in the variables $x_{i (-n)}$ and $y_{i (-n)}$ for $i=1,\dots,N$, $n \in {\mathbb Z}_{>0}$. By definition, there is a projection $$\sigma_0\colon \mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U) \longrightarrow \mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}(U), \qquad f \mapsto f \mod \hbar,$$ which we call the symbol map. Let $$\label{eq:lift-map} \iota\colon \mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}(U) \longrightarrow \mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U), \qquad f \mapsto \iota(f) = f \mathbf{1}$$ be a natural section of the symbol map $\sigma_0$. Note that $\iota$ does not send products to $(-1)$-products, i.e. $\iota(f)_{(-1)} \iota(g)$ is in general different from $\iota(f g)$. By abuse of notation, we also denote $\iota(f)$ simply by $f$. Recall that we write $f(z) = Y(\iota(f), z)$ for the vertex operator on $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)$. Note that $f$, $g$ are rational functions in $\{x_{i (-n)}, y_{i (-n)} \,|\, i=1, \dots, N; n \in {\mathbb Z}_{>0}\}$, and thus only finitely many variables may appear in them. This implies that the bidifferential operators $P_{nk}$ also act on $\mathcal{O}_{J_{\infty}T^* {\mathbb C}^N}(U)$ and $P_{nk}(f, g) \in \mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}(U)$ for any $k \in {\mathbb Z}_{\ge 0}$, $n \in {\mathbb Z}$. Hence, we define the operator product expansion by $$\label{eq:12} f(z) g(w) =\sum_{k = 0}^{\infty} \sum_{n \in {\mathbb Z}} \frac{\hbar^{k}}{(z-w)^n} P_{nk}(f, g) \in \mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)[[(z-w)^{-1}]]$$ for $f,g \in \mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}(U)$. We extend the operator product expansion ${\mathbb C}[[\hbar]]$-linearly and continuously in the $\hbar$-adic topology for general $f,g \in \mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)$. The translation operator $\partial$ can be defined on $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)$ in the same way since it is also a differential operator. Note that the above definition of the operator product expansion [\[eq:12\]](#eq:12){reference-type="eqref" reference="eq:12"} induces $(n)$-products on $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)$ through the formula [\[eq:OPE\]](#eq:OPE){reference-type="eqref" reference="eq:OPE"} $f(z) g(w) = \sum_{n \in {\mathbb Z}} Y(\iota(f)_{(n)} \iota(g), w) (z-w)^{-n-1}$, namely $$\label{eq:16} \iota(f)_{(n)} \iota(g) = \sum_{k \ge 0} \hbar^{k} \iota(P_{n k}(f, g)).$$ Then, we obtain an $\hbar$-adic vertex algebra $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)$ with vacuum vector $\mathbf{1}$ by the following lemma: **Lemma 7**. *For any open subset $U \subset T^* {\mathbb C}^N$, Borcherds' identity [\[eq:Borcherds\]](#eq:Borcherds){reference-type="eqref" reference="eq:Borcherds"} holds on $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)$ with respect to the $(n)$-products defined above.* *Proof.* For a point $p \in U$, let $\widehat{\mathcal{O}}_{T^* {\mathbb C}^N, p}$ be the formal completion of the stalk $\mathcal{O}_{T^* {\mathbb C}^N, p}$ with respect to the unique maximal ideal $\mathfrak{m}_p$. Set $$\bigl(\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}\bigr)^{\wedge}_p = \widehat{\mathcal{O}}_{T^* {\mathbb C}^N, p} \otimes_{{\mathbb C}[T^* {\mathbb C}^N]} {\mathbb C}[[\hbar]][x_{i (-n)}, y_{i (-n)} \,|\, i=1, \dots, N; n \in {\mathbb Z}_{>0}].$$ Then we define the $(n)$-products on $(\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar})^{\wedge}_p$ in the same way as the ones on $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)$. By definition, the $(n)$-products are continuous in the $\mathfrak{m}_p$-adic topology on $(\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar})^{\wedge}_p \times (\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar})^{\wedge}_p$. Moreover, the natural embedding $$\label{eq:14} \mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U) \hookrightarrow \bigl(\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}\bigr)^{\wedge}_p$$ given by Taylor expansion commutes with the $(n)$-products. We furthermore set $\tilde{x}_{i (-1)} = x_{i (-1)} - x_i(p)$, $\tilde{y}_{i (-1)} = y_{i (-1)} - y_i(p)$, $\tilde{x}_{i (-n)} = x_{i (-n)}$ and $\tilde{y}_{i (-n)} = y_{i (-n)}$ for $i=1,\dots,N$ and $n \in {\mathbb Z}_{\ge 2}$. For any polynomials in the variables $\{\tilde{x}_{i (-n)}, \tilde{y}_{i (-n)} \,|\, i=1, \dots, N; n \in {\mathbb Z}_{>0}\}$, Borcherds' identity holds on $(\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar})^{\wedge}_p$ since it coincides with the one on the $\hbar$-adic $\beta\gamma$-system $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^N)_{\hbar}$ by the definition of the $(n)$-products. Then, Borcherds' identity must hold on $(\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar})^{\wedge}_p$ for any formal power series in the variables $\{\tilde{x}_{i (-n)}, \tilde{y}_{i (-n)} \,|\, i=1, \dots, N; n \in {\mathbb Z}_{>0}\}$ since the $(n)$-products are continuous in the $\mathfrak{m}_p$-adic topology, and thus it holds on $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)$ by the embedding [\[eq:14\]](#eq:14){reference-type="eqref" reference="eq:14"}. ◻ Clearly, the restriction $\mathop{\mathrm{res}}^{U}_{U'}\colon \mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U) \longrightarrow \mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U')$ is an $\hbar$-adic vertex algebra homomorphism for open subsets $U' \subset U$, and hence we obtain the sheaf of $\hbar$-adic vertex algebras $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}$ over the affine space $T^* {\mathbb C}^N$. Finally, the operator product expansion [\[eq:12\]](#eq:12){reference-type="eqref" reference="eq:12"} yields that $f(z) g(w) \equiv {:}f(z) g(w) {:}\bmod \hbar$ for any symbol $f,g \in \mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}(U)$ since $P_{n0}(f, g) = 0$ for $n \ge 0$, and thus there is natural isomorphism of vertex Poisson algebras $$\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)/\hbar \mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U) \simeq \mathcal{O}_{J_{\infty}T^* {\mathbb C}^N}(U)$$ for any open subset $U$. Therefore, $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}$ is a quantisation of $\mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}$. For an invertible function $f \in \mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}(U)$, the element $\iota(1/f) \in \mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)$ is in general not an inverse of $\iota(f)$ with respect to the $(-1)$-product (the normally ordered product). However, the following proposition ensures the existence of the inverse with respect to the $(-1)$-product: **Proposition 8**. *Assume that $f_0 = \sigma_0(f)$ is invertible in the commutative algebra $\mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}(U)$. We write $1/f_0$ for the inverse element of $f_0$ in $\mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}(U)$ with respect to the usual multiplication. Then, there exists an element $g \in \mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)$ such that $f_{(-1)} g = \mathbf{1}$ in $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)$ and $\sigma_0(g) = 1/f_0$. In particular, $${:}f(z) g(z) {:}= Y(f_{(-1)} g, z) = \operatorname{Id}_{\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)}.$$* *Proof.* Write $f = \sum_{k \ge 0} \hbar^k \iota(f_k)$, $g = \sum_{k \ge 0} \hbar^k \iota(g_k)$, where $f_k$, $g_k \in \mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}(U)$. Then, by [\[eq:16\]](#eq:16){reference-type="eqref" reference="eq:16"}, $$\begin{aligned} \label{eq:15} f_{(-1)} g &= \sum_{k \ge 0} \hbar^{k} \iota(P_{-1 k}(f, g)) = \iota(f_0 g_0) + \hbar \bigl( \iota(f_0 g_1 + f_1 g_0) + \iota(P_{-1, 1}(f_0, g_0)) \bigr) \\ &\quad + \hbar^2 \bigl(\iota(f_0 g_2 + f_1 g_1 + f_2 g_0) \nonumber\\ &\quad + \iota(P_{-1, 1}(f_0, g_1) + P_{-1, 1}(f_1, g_0)) + \iota(P_{-1, 2}(f_0, g_0)) \bigr) + \dots\nonumber\end{aligned}$$ Setting $g_0 = 1/f_0$ and $$g_1 = - \frac{1}{f_0}\bigl( f_1 g_0 + P_{-1, 1}(f_0, g_0) \bigr) \in \mathcal{O}_{J_{\infty} T^* {\mathbb C}^N}(U),$$ it follows that $f_{(-1)} (\iota(g_0) + \hbar \iota(g_1)) - \mathbf{1}\equiv 0 \bmod \hbar^2$. By induction on $k=1,2,\dots$ we can determine $g_k$ such that $$f_{(-1)} (\iota(g_0) + \hbar^1 \iota(g_1) + \dots + \hbar^{k} \iota(g_{k})) - \mathbf{1}\equiv 0 \mod \hbar^{k+1}.$$ Then, since $\mathcal{D}^\mathrm{ch}_{T^* {\mathbb C}^N, \hbar}(U)$ is complete in the $\hbar$-adic topology, we obtain the element $g = \sum_{k \ge 0} \hbar^k \iota(g_k)$ with the desired properties. ◻ ## The $bc$-System {#sec:h-adic-bc} In the following, we also introduce the $\hbar$-adic analogue $C\ell_{\hbar}(T^*{\mathbb C}^N)$ of the $bc$-system $C\ell(T^*{\mathbb C}^N)=(bc)^{\otimes N}$ of rank $N$, also called Clifford vertex superalgebra or $bc$-ghost vertex superalgebra. Again, consider the symplectic vector space $T^* {\mathbb C}^N = {\mathbb C}^N \oplus ({\mathbb C}^N)^*$. We write $\Pi V$ for the odd vector space corresponding to an even vector space $V$. Let $\psi_1,\dots,\psi_N$, $\phi_1,\dots,\phi_N$ be the standard Poisson coordinate functions on the odd vector space $\Pi T^* {\mathbb C}^N$, i.e. $\{\psi_i, \phi_j \} = \delta_{ij}$ and $\{\psi_i, \psi_j\} = \{\phi_i, \phi_j\} = 0$ for $i,j=1,\dots,N$, where $\{\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, \raisebox{0.5ex}{\rule{2ex}{0.4pt}}\}$ denotes the Poisson superbracket of the Poisson superalgebra ${\mathbb C}[\Pi T^* {\mathbb C}^N]$. The $\hbar$-adic $bc$-system $C\ell_{\hbar}(T^* {\mathbb C}^N)$ on $\Pi T^* {\mathbb C}^N$ is the $\hbar$-adic vertex superalgebra generated by the fermions $\psi_1,\dots,\psi_N$, $\phi_1,\dots,\phi_N$ with operator product expansions $$\begin{aligned} \psi_i(z) \phi_j(w) &\sim \delta_{ij} \hbar /(z-w), \\ \psi_i(z) \psi_j(w) &\sim 0, \\ \phi_i(z) \phi_j(w) &\sim 0\end{aligned}$$ for $i,j=1,\dots,N$, where we again write $\psi_i(z) = Y(\psi_i, z)$, $\phi_i(z) = Y(\phi_i, z)$. As a ${\mathbb C}[[\hbar]]$-module, $$C\ell_{\hbar}(T^* {\mathbb C}^N) = \Lambda_{{\mathbb C}[[\hbar]]}(\psi_{1 (-n)}, \dots, \psi_{N (-n)}, \phi_{1 (-n)}, \dots, \phi_{N (-n)} \,|\, n \in {\mathbb Z}_{>0}).$$ Let $\Lambda^\mathrm{vert}(T^* {\mathbb C}^N) = C\ell_{\hbar}(T^* {\mathbb C}^N) / \hbar C\ell_{\hbar}(T^* {\mathbb C}^N)$. It follows from the definition that $\psi_i(z) \phi_j(w) \sim \psi_i(z) \psi_j(w) \sim \phi_i(z) \phi_j(w) \sim 0 \bmod \hbar$ for $i,j=1,\dots,N$, and thus $\Lambda^\mathrm{vert}(T^* {\mathbb C}^N)$ is a commutative vertex superalgebra. Note that $\Lambda^\mathrm{vert}(T^* {\mathbb C}^N)$ becomes a vertex Poisson superalgebra upon defining $$Y_{+}(\psi_i, z) = \hbar^{-1} \sum_{n \ge 0} \psi_{i (n)} z^{-n-1}, \quad Y_{+}(\phi_i, z) = \hbar^{-1} \sum_{n \ge 0} \phi_{i (n)} z^{-n-1}$$ for $i=1,\dots,N$. To construct the BRST cohomology, we shall later also need the (ghost) $\hbar$-adic $bc$-system $C\ell_{\hbar}(T^* \mathfrak{g})$ associated with the symplectic vector space $T^* \mathfrak{g}= \mathfrak{g}\oplus \mathfrak{g}^*$ for $\mathfrak{g}= \mathfrak{gl}_N$. In this case, we consider the standard coordinate functions $\Psi_{ij} \in \mathfrak{g}^* \subset {\mathbb C}[\mathfrak{g}]$ and $\Phi_{ij} \in \mathfrak{g}\subset {\mathbb C}[\mathfrak{g}^*]$ for $i,j=1,\dots,N$, which correspond to the matrices $E_{ij} \in \mathfrak{g}$ (see above). Thus, $$C\ell_{\hbar}(T^* \mathfrak{g}) = \Lambda_{{\mathbb C}[[\hbar]]}(\Psi_{ij (-n)}, \Phi_{ij (-n)} \,|\, \substack{i,j=1, \dots, N \\ n = 1, 2, \dots})$$ as a ${\mathbb C}[[\hbar]]$-module, and the operator product expansions between the generators are given by $$\begin{aligned} \Psi_{ij}(z) \Phi_{kl}(w) &\sim \delta_{ik} \delta_{jl} \hbar / (z-w), \\ \Psi_{ij}(z) \Psi_{kl}(w) &\sim 0, \\ \Phi_{ij}(z) \Phi_{kl}(w) &\sim 0.\end{aligned}$$ Note that the $\hbar$-adic $bc$-system $C\ell_{\hbar}(T^* \mathfrak{g})$ is naturally ${\mathbb Z}$-graded by the charge (or ghost) grading given by $\deg(\Psi_{ij (-n)}) = 1$, $\deg(\Phi_{ij (-n)}) = -1$ for $i,j=1,\dots,N$ and $n \in {\mathbb Z}_{>0}$. Moreover, the homogeneous subspace $C\ell^n_{\hbar}(T^* \mathfrak{g})$ of charge $n \in {\mathbb Z}$ can be decomposed as a ${\mathbb C}[[\hbar]]$-module as $$\label{eq:bc-double-grade} C\ell^n_{\hbar}(T^* \mathfrak{g}) = \prod_{p+q=n} C\ell^{p,q}_{\hbar}(T^* \mathfrak{g})$$ with $$C\ell^{p,q}_{\hbar}(T^* \mathfrak{g}) = \Lambda^{p}_{{\mathbb C}[[\hbar]]}(\Psi_{ij (-n)} \,|\, \substack{i,j=1, \dots, N \\ n = 1, 2, \dots}) \mathop{\mathrm{\widehat{\otimes}}}\Lambda^{-q}_{{\mathbb C}[[\hbar]]}(\Phi_{ij (-n)} \,|\, \substack{i,j=1, \dots, N \\ n = 1, 2, \dots}), \nonumber$$ where $\Lambda^{p}_{{\mathbb C}[[\hbar]]}(\raisebox{0.5ex}{\rule{2ex}{0.4pt}})$ is the space of $p$-th exterior powers. Note that $C\ell^{p,q}_{\hbar}(T^* \mathfrak{g}) = 0$ unless $p \ge 0$ and $q \le 0$. ## Affine Vertex Algebras and Small $\mathcal{N}=4$ Superconformal Algebra {#sec:h-adic-affine-VA} We also introduce the $\hbar$-adic affine vertex algebras $V^k(\mathfrak{g})_{\hbar}$ and the $\hbar$-adic small $\mathcal{N}=4$ superconformal algebra $\operatorname{Vir}_{\mathcal{N}=4,\hbar}^c$. Let $\mathfrak{g}$ be a finite-dimensional Lie algebra with invariant bilinear form $\kappa\colon \mathfrak{g}\times \mathfrak{g}\longrightarrow {\mathbb C}$. Consider the universal affine vertex algebra $V^{\kappa}(\mathfrak{g})$, and define $V^{\kappa}(\mathfrak{g})[\hbar] = V^{\kappa}(\mathfrak{g}) \otimes {\mathbb C}[\hbar]$. For an element $A \in \mathfrak{g}$, we write $\widehat{A} = \hbar A_{(-1)} \mathbf{1}\in V^{\kappa}(\mathfrak{g})[\hbar]$. Then, $$\label{eq:OPE-affine-VA} \widehat{A}_{(0)} \widehat{B} = \hbar \, \widehat{[A, B]}, \quad \widehat{A}_{(1)} \widehat{B} = \hbar^2 \,\kappa(A, B) \mathbf{1}, \quad \widehat{A}_{(n)} \widehat{B} = 0 \text{ if } n > 1$$ for $A$, $B \in \mathfrak{g}$. The universal $\hbar$-adic affine vertex algebra $V^{\kappa}(\mathfrak{g})_{\hbar}$ associated with $\mathfrak{g}$ and $\kappa$ is the $\hbar$-adic completion of the vertex subalgebra generated by the elements in $\{\widehat{A} \,|\, A \in \mathfrak{g}\}$. Note that $V^{\kappa}(\mathfrak{g})_{\hbar} / (\hbar)$ is a vertex Poisson algebra, and it satisfies $V^{\kappa}(\mathfrak{g})_{\hbar} / (\hbar) \simeq S(\mathfrak{g}\otimes {\mathbb C}[t^{-1}] t^{-1})$. By [\[eq:OPE-affine-VA\]](#eq:OPE-affine-VA){reference-type="eqref" reference="eq:OPE-affine-VA"}, the vertex Poisson structure $Y_{-}(\raisebox{0.5ex}{\rule{2ex}{0.4pt}}, z)$ on it does not depend on $\kappa$. In the present article, we consider the case where $\mathfrak{g}$ is the general linear Lie algebra $\mathfrak{gl}_N({\mathbb C})$. In this case, the invariant bilinear form is given explicitly by $\kappa(A, B) = \kappa_k(A, B) = k \mathop{\mathrm{Tr}}(AB) - (k/N) \mathop{\mathrm{Tr}}(A) \mathop{\mathrm{Tr}}(B)$ for some level $k \in {\mathbb C}$. We also use the notation $V^k(\mathfrak{g})_{\hbar}$ for $V^{\kappa_k}(\mathfrak{g})_{\hbar}$. Similarly, we obtain an $\hbar$-adic analogue of the small $\mathcal{N}=4$ superconformal algebra. The universal small $\mathcal{N}=4$ superconformal algebra $\operatorname{Vir}_{\mathcal{N}=4}^c$ of central charge $c$ (or equivalently of level $k=c/6$) is a vertex operator superalgebra strongly generated by bosonic elements $J^0$, $J^{\pm}$ and the conformal vector $T_{\mathcal{N}=4}$ and the fermionic ones $G^{\pm}$ and $\widetilde{G}^{\pm}$ satisfying the operator product expansions $$\begin{aligned} J^0(z) J^{\pm}(w) &\sim \frac{\pm 2}{z-w} J^{\pm}(w), \quad J^{0}(z) J^{0}(w) \sim \frac{c/3}{(z-w)^2}, \\ J^{+}(z) J^{-}(w) &\sim \frac{-c/6}{(z-w)^2} + \frac{-1}{z-w} J^{0}(w) \\ J^{0}(z) B^{\pm}(w) &\sim \frac{\pm 1}{z-w} B^{\pm}(w), \quad J^{\pm}(z) B^{\mp}(w) \sim \frac{\mp 1}{z-w} B^{\pm}(w),\end{aligned}$$ as well as $$\begin{aligned} G^{a}(z) \widetilde{G}^{b}(w) &\sim \frac{(c/3) \epsilon^{ab}}{(z-w)^3} + \frac{2 J^{ab}(w)}{(z-w)^2} + \frac{\epsilon^{ab} T_{\mathcal{N}=4}(w) + \partial J^{ab}(w)}{z-w}, \\ T_{\mathcal{N}=4}(z) T_{\mathcal{N}=4}(w) &\sim \frac{c/2}{(z-w)^4} + \frac{2}{(z-w)^2} T_{\mathcal{N}=4}(w) + \frac{1}{z-w} \partial T_{\mathcal{N}=4}(w), \\ T_{\mathcal{N}=4}(z) A(w) &\sim \frac{1}{(z-w)^2} A(w) + \frac{1}{z-w} \partial A(w),\\ T_{\mathcal{N}=4}(z) B^{\pm}(w) &\sim \frac{3 / 2}{(z-w)^2} B^{\pm}(w) + \frac{1}{z-w} \partial B^{\pm}(w),\end{aligned}$$ with the operator product expansions of other pairs among the elements being trivial, for $A = J^0, J^{\pm}$, $B = G,\widetilde{G}$ and $a,b = \pm$, where $\epsilon^{+-} = 1$, $\epsilon^{-+} = -1$, $\epsilon^{\pm \pm} = 0$ and $J^{+-} = J^{-+} = J^0/2$, $J^{\pm\pm} = J^{\pm}$. Note that the vertex superalgebra $\operatorname{Vir}_{\mathcal{N}=4}^c$ contains the universal affine vertex algebra $V^k(\mathfrak{sl}_2)$ of level $k=c/6$, which is also the level of $\operatorname{Vir}_{\mathcal{N}=4}^c$ by definition. Consider the vertex subalgebra of $\operatorname{Vir}_{\mathcal{N}=4}^c[\hbar]$ generated by the elements $\hbar J^0$, $\hbar J^{\pm}$, $\hbar^2 T_{\mathcal{N}=4}$, $\hbar^{3/2} G^{\pm}$ and $\hbar^{3/2} \widetilde{G}^{\pm}$, and define the universal $\hbar$-adic small $\mathcal{N}=4$ superconformal algebra $\operatorname{Vir}_{\mathcal{N}=4,\hbar}^c$ of central charge $c$ as its $\hbar$-adic completion. ## Symplectic Fermion Vertex Superalgebra {#sec:symplec-ferm} Finally, we introduce the $\hbar$-adic version $\mathsf{SF}_{\hbar}$ of the symplectic fermion vertex superalgebra. Let $V = {\mathbb C}\Lambda_1 \oplus {\mathbb C}\Lambda_2$ be a symplectic vector space with symplectic bilinear form given by $(\Lambda_1, \Lambda_2) = -1$ and $(\Lambda_i, \Lambda_i) = 0$ for $i=1,2$. The vertex superalgebra of symplectic fermions $\mathsf{SF}$ associated with $V$ is the vertex superalgebra generated by fermionic elements $\Lambda_1$ and $\Lambda_2$ with operator product expansion $u(z) v(w) \sim (u, v)/(z-w)^2$ for $u,v \in V$. The $\hbar$-adic analogue of $\mathsf{SF}$ is denoted by $\mathsf{SF}_{\hbar}$. It is an $\hbar$-adic vertex superalgebra generated by $\Lambda_1$ and $\Lambda_2$ with operator product expansion given by $$u(z) v(w) \sim \hbar^2 (u, v)/(z-w)^2$$ for $u,v \in V$. # Semi-Infinite BRST Reduction {#sec:semi-infinite-red} In this section, for $N\ge1$ we construct a sheaf $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}$ of $\hbar$-adic vertex superalgebras on the Hilbert scheme $M=\mathop{\mathrm{Hilb}}^N({\mathbb C}^2)$. This is the central object of this text. The construction is based on a vertex algebra analogue of quantum Hamiltonian reduction, which we call semi-infinite BRST cohomology (see [@AKM15; @Kuwabara21]). We also prove a vanishing (or no-ghost) theorem for this BRST cohomology. ## Construction of the BRST Cohomology {#sec:brst-construction} In the following, we define a BRST cohomology sheaf $\mathcal{H}_{\hbar\mathrm{VA}}^{\infty/2 + \bullet}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar})$ on $M$, which we later call $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}$, after having proved a vanishing theorem for the cohomological spaces. Recall the symplectic vector space $T^* V = \mathop{\mathrm{End}}({\mathbb C}^N)^{\oplus 2} \oplus ({\mathbb C}^N)^{\oplus 2}$ and its standard symplectic coordinates $\{x_{ij}, y_{ij}\}_{i,j=1, \dots, N} \cup \{\gamma_i, \beta_{i}\}_{i=1, \dots, N}$ from . Let $\mathcal{D}^\mathrm{ch}_{T^* V, \hbar}$ be the microlocalisation of the $\hbar$-adic $\beta\gamma$-system, a sheaf of vertex algebras on $T^* V$, and let $C\ell_{\hbar}(T^* {\mathbb C}^N)$ be the $\hbar$-adic $bc$-system. Then, define the sheaf $\widetilde{\mathcal{D}}^\mathrm{ch}_{T^*V, \hbar} = \mathcal{D}^\mathrm{ch}_{T^*V, \hbar} \mathop{\mathrm{\widehat{\otimes}}}C\ell_{\hbar}(T^* {\mathbb C}^N)$ of $\hbar$-adic vertex superalgebras on $T^*V$. Restricting onto $\mathfrak{X}\subset T^*V$, we obtain a sheaf of $\hbar$-adic vertex superalgebras $\widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar} \coloneqq \widetilde{\mathcal{D}}^\mathrm{ch}_{T^*V, \hbar} |_{\mathfrak{X}}$. As a ${\mathbb C}[[\hbar]]$-module, $$\begin{aligned} \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\widetilde{U}) &= \mathcal{O}_{\mathfrak{X}}(\widetilde{U}) \otimes_{{\mathbb C}[\mathfrak{X}]} {\mathbb C}[[\hbar]][x_{ij (-n)}, y_{ij (-n)}, \gamma_{i (-n)}, \beta_{i (-n)} \,|\, \substack{i,j = 1, \dots, N \\ n = 1, 2, \dots}] \\ &\quad \mathop{\mathrm{\widehat{\otimes}}}\Lambda_{{\mathbb C}[[\hbar]]}(\psi_{i (-n)}, \phi_{i (-n)} \,|\, \substack{i = 1, \dots, N \\ n = 1, 2, \dots})\end{aligned}$$ for any open subset $\widetilde{U}\subset \mathfrak{X}$. To construct the BRST reduction of $\widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}$, we introduce a chiralisation $\widetilde{\mu}_\mathrm{ch}$ of the comoment map $\widetilde{\mu}\colon \mathfrak{g}\longrightarrow \widetilde{\mathcal{O}}_{\mathfrak{X}}(\mathfrak{X})$, which we call chiral comoment map. To this end, consider the $\hbar$-adic affine vertex algebra $V^{-2N}(\mathfrak{g})_{\hbar}$ associated with the general linear Lie algebra $\mathfrak{g}= \mathfrak{gl}_N$ of level $-2N$. Note that the corresponding invariant bilinear form $\kappa_{-2N}(A, B) = - 2N \mathop{\mathrm{Tr}}(AB) + 2 \mathop{\mathrm{Tr}}(A) \mathop{\mathrm{Tr}}(B)$ is the negative of the Killing form of $\mathfrak{g}= \mathfrak{gl}_N$. Define the ${\mathbb C}[\partial]$-linear map $\widetilde{\mu}_\mathrm{ch}\colon V^{-2N}(\mathfrak{g})_{\hbar}\longrightarrow\widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\mathfrak{X})$, $$\widetilde{\mu}_\mathrm{ch}(E_{ij}) = \sum_{p=1}^{N} (x_{ip (-1)} y_{pj} - x_{pj (-1)} y_{ip}) + \gamma_{i (-1)} \beta_j + \psi_{i (-1)} \phi_j.$$ By direct verification, we obtain the following lemma. **Lemma 9**. *The chiral comoment map $\widetilde{\mu}_\mathrm{ch}\colon V^{-2N}(\mathfrak{g})_{\hbar}\longrightarrow\widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\mathfrak{X})$ is a homomorphism of $\hbar$-adic vertex superalgebras.* Now, recall the ghost $\hbar$-adic $bc$-system $C\ell_{\hbar}(T^* \mathfrak{g})$ with generators $\Psi_{ij}$, $\Phi_{ij}$ for $i,j=1,\dots,N$. It is ${\mathbb Z}$-graded by the charge (or ghost) grading defined in , i.e. $C\ell_{\hbar}(T^* \mathfrak{g}) = \prod_{\bullet \in {\mathbb Z}} C\ell^{\bullet}_{\hbar}(T^* \mathfrak{g})$. Let $c_{ijkl}^{pq}$ denote the structure constants of the Lie algebra $\mathfrak{g}= \mathfrak{gl}_N$, i.e. $[E_{ij}, E_{kl}] = \sum_{pq} c_{ijkl}^{pq} E_{pq}$ for $i,j,k,l=1,\dots,N$. Then, via $$J(E_{ij})\coloneqq - \sum_{klpq} c_{ijkl}^{pq} \Psi_{kl} \Phi_{pq}$$ we define a ${\mathbb C}[\partial]$-linear map $J\colon V^{2N}(\mathfrak{g})_{\hbar}\longrightarrow C\ell_{\hbar}^0(T^* \mathfrak{g})$. Again, a straightforward computation shows: **Lemma 10**. *The map $J\colon V^{2N}(\mathfrak{g})_{\hbar}\longrightarrow C\ell_{\hbar}^0(T^* \mathfrak{g})$ is a homomorphism of $\hbar$-adic vertex superalgebras.* We then define the sheaf $$\widetilde{C}_{\hbar\mathrm{VA}} = \prod_{\bullet \in {\mathbb Z}} \widetilde{C}_{\hbar\mathrm{VA}}^{\bullet}, \qquad \widetilde{C}_{\hbar\mathrm{VA}}^{\bullet} = \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar} \mathop{\mathrm{\widehat{\otimes}}}C\ell^{\bullet}_{\hbar}(T^* \mathfrak{g}),$$ of ${\mathbb Z}$-graded $\hbar$-adic vertex superalgebras on $\mathfrak{X}$. On this sheaf we consider the odd global section $$\begin{aligned} Q &\coloneqq \sum_{i,j} \bigl(\widetilde{\mu}_\mathrm{ch}(E_{ij})+\frac{1}{2}J(E_{ij})\bigr)_{(-1)}\Psi_{ij}\\ &=\sum_{i,j} \widetilde{\mu}_\mathrm{ch}(E_{ij})_{(-1)} \Psi_{ij} - \frac{1}{2} \sum_{ijklpq} c_{ijkl}^{pq} \Psi_{ij} \Psi_{kl} \Phi_{pq} \in \widetilde{C}_{\hbar\mathrm{VA}}^1(\mathfrak{X})\end{aligned}$$ of charge $+1$. Note that the image of $Q_{(0)}$ lies in $\hbar \widetilde{C}_{\hbar\mathrm{VA}}$, and we define a derivation $d_{\hbar\mathrm{VA}} = (1/\hbar) Q_{(0)}$ on $\widetilde{C}_{\hbar\mathrm{VA}}$, which is homogeneous of charge $+1$. Crucially, a straightforward calculation shows that $Q_{(0)} Q = 0$, which implies $$(d_{\hbar\mathrm{VA}})^2 = (Q_{(0)})^2 = (1/2) (Q_{(0)} Q)_{(0)} = 0.$$ Therefore: **Proposition 11**. *For any $\widetilde{U}\subset \mathfrak{X}$ open, $(\widetilde{C}_{\hbar\mathrm{VA}}(\widetilde{U}), d_{\hbar\mathrm{VA}})$ is a cochain complex.* We define the global sections $$\widetilde{E}_{ij} = Q_{(0)} \Phi_{ij} = \widetilde{\mu}_\mathrm{ch}(E_{ij}) + J(E_{ij}) \in \widetilde{C}_{\hbar\mathrm{VA}}^{0}(\mathfrak{X})$$ for $i,j=1,\dots,N$, which induce a ${\mathbb C}[\partial]$-linear map $V^0(\mathfrak{g})_{\hbar}\longrightarrow\widetilde{C}_{\hbar\mathrm{VA}}^0(\mathfrak{X})$. Then and imply: **Lemma 12**. *The map $E_{ij}\mapsto\widetilde{E}_{ij}$ induces a homomorphism $V^0(\mathfrak{g})_{\hbar}\longrightarrow\widetilde{C}_{\hbar\mathrm{VA}}^0(\mathfrak{X})$ of $\hbar$-adic vertex superalgebras.* In particular, the operators $\{ (1/ \hbar) \widetilde{E}_{ij (0)} \,|\, i, j = 1, \dots, N \}$ form the general linear Lie algebra $\mathfrak{g}= \mathfrak{gl}_N$, and make $\widetilde{C}_{\hbar\mathrm{VA}}(\mathfrak{X})$ into a $\mathfrak{g}$-module. It shall be advantageous to consider the BRST cohomology only on a relative subcomplex. To this end, we consider the subspace of $\widetilde{C}_{\hbar\mathrm{VA}}$ given by $$C_{\hbar\mathrm{VA}} \coloneqq \{ c \in \widetilde{C}_{\hbar\mathrm{VA}} \,|\, \widetilde{E}_{ij (0)} c = \Phi_{ij (0)} c = 0 \text{ for all } i, j = 1, \dots, N \}.$$ As $Q_{(0)}$ is a derivation, $$\begin{aligned} \widetilde{E}_{ij (0)} Q_{(0)} c &= Q_{(0)} \widetilde{E}_{ij (0)} c - (Q_{(0)} \widetilde{E}_{ij})_{(0)} c = (Q_{(0)}^2 \Phi_{ij})_{(0)} c = 0,\\ \Phi_{ij (0)} Q_{(0)} c &= - Q_{(0)} \Phi_{ij (0)} c + (Q_{(0)} \Phi_{ij})_{(0)} c = \widetilde{E}_{ij (0)} c = 0\end{aligned}$$ for any section $c \in C_{\hbar\mathrm{VA}}$ and $i,j=1,\dots,N$. Therefore, $(C_{\hbar\mathrm{VA}}, d_{\hbar\mathrm{VA}})$ is a sheaf of subcomplexes of $(\widetilde{C}_{\hbar\mathrm{VA}}, d_{\hbar\mathrm{VA}})$. For an open subset $\widetilde{U}\subset \mathfrak{X}$, we write the cohomology $$H^{\infty/2+\bullet}_{\hbar\mathrm{VA}}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\widetilde{U})) \coloneqq H^{\bullet}(C_{\hbar\mathrm{VA}}(\widetilde{U}), d_{\hbar\mathrm{VA}})$$ and call it the (relative) BRST cohomology of $\widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\widetilde{U})$ with respect to $\mathfrak{g}$. **Lemma 13**. *The BRST cohomology $H_{\hbar\mathrm{VA}}^{\infty/2+\bullet}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\widetilde{U}))$ is equipped with the structure of a (${\mathbb Z}$-graded) $\hbar$-adic vertex superalgebra.* *Proof.* Again using that the coboundary operator $Q_{(0)}$ is a derivation, we see that $Q_{(0)} (a_{(n)} b) = (-1)^{p(a)} a_{(n)} Q_{(0)} b + (Q_{(0)} a)_{(n)} b$ for $a,b \in C_{\hbar\mathrm{VA}}$ and $n \in {\mathbb Z}$. This implies that $a_{(n)} b \in \mathop{\mathrm{Ker}}d_{\hbar\mathrm{VA}}$ for $a,b\in\mathop{\mathrm{Ker}}d_{\hbar\mathrm{VA}}$. Also, $a_{(n)} (Q_{(0)} b) = Q_{(0)} (a_{(n)} b)$ for $a \in \mathop{\mathrm{Ker}}d_{\hbar\mathrm{VA}}$ and $b \in C_{\hbar\mathrm{VA}}$. Thus, $(\mathop{\mathrm{Ker}}d_{\hbar\mathrm{VA}})_{(n)} (\operatorname{Im}d_{\hbar\mathrm{VA}}) \subset \operatorname{Im}d_{\hbar\mathrm{VA}}$. ◻ Now, using the BRST cohomology, we construct a sheaf of $\hbar$-adic vertex superalgebras on the Hilbert scheme $M$. For an open subset $U \subset M$, take an open subset $\widetilde{U}\subset \mathfrak{X}$ such that $\widetilde{U}$ is closed under the $G$-action and $U = (\widetilde{U}\cap \mu^{-1}(0))/G$. Crucially, by below, the BRST cohomology $H_{\hbar\mathrm{VA}}^{\infty/2 + \bullet}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\widetilde{U}))$ does not depend on the choice of the open subset $\widetilde{U}$ satisfying these conditions. Therefore, it defines a presheaf $U \mapsto H_{\hbar\mathrm{VA}}^{\infty/2 + \bullet}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\widetilde{U}))$ over $M$, which we use to define the BRST cohomology sheaf: **Definition 14**. The sheaf on the Hilbert scheme $M=\mathop{\mathrm{Hilb}}^N({\mathbb C}^2)$ associated with the presheaf $U \mapsto H_{\hbar\mathrm{VA}}^{\infty/2 + \bullet}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\widetilde{U}))$ is denoted by $\mathcal{H}_{\hbar\mathrm{VA}}^{\infty/2 + \bullet}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar})$. We comment that the BRST cochain complex can be decomposed into a double cochain complex. Recall the decomposition [\[eq:bc-double-grade\]](#eq:bc-double-grade){reference-type="eqref" reference="eq:bc-double-grade"} of the $\hbar$-adic $bc$-system $C\ell^{p,q}_{\hbar}(T^* \mathfrak{g})$. Set $\widetilde{C}_{\hbar\mathrm{VA}}^{p,q} = \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar} \mathop{\mathrm{\widehat{\otimes}}}C\ell^{p,q}_{\hbar}(T^* \mathfrak{g})$ for $p,q \in {\mathbb Z}$. Then, $\widetilde{C}_{\hbar\mathrm{VA}}^{n} = \prod_{p + q = n} \widetilde{C}_{\hbar\mathrm{VA}}^{p, q}$ for any $n \in {\mathbb Z}$. Define $$\begin{aligned} \label{eq:d-plus} d^{+}_{\hbar\mathrm{VA}} &\coloneqq \hbar^{-1} \sum_{i,j} \sum_{n \ge 0} \Psi_{ij (-n-1)} \widetilde{E}_{ij (n)} \\ &\quad + \hbar^{-1} \sum_{ijklpq} \sum_{n \ge 0} \sum_{m=0}^{n} c_{ijkl}^{pq} \Psi_{ij (-n+m-1)} \Psi_{kl (-m)} \Phi_{pq (n)} \nonumber \\ &= \sum_{i,j} \sum_{n \ge 0} \Psi_{ij (-n-1)} (\hbar^{-1} \widetilde{E}_{ij (n)}) \nonumber \\ &\quad + \sum_{ijklpq} \sum_{n \ge 0} \sum_{m=0}^{n} c_{ijkl}^{pq} \Psi_{ij (-n+m-1)} \Psi_{kl (-m)} \frac{\partial}{\partial \Psi_{pq (-n-1)}} \nonumber\end{aligned}$$ and $$\begin{aligned} \label{eq:d-minus} d^{-}_{\hbar\mathrm{VA}} &\coloneqq \hbar^{-1} \sum_{i,j} \sum_{n \ge 0} \widetilde{\mu}_\mathrm{ch}(E_{ij})_{(-n-1)} \Psi_{ij (n)} \\ &\quad + \hbar^{-1} \sum_{ijklpq} \sum_{n \ge 0} \sum_{m=0}^{n} c_{ijkl}^{pq} \Phi_{pq (-n-1)} \Psi_{ij (n-m)} \Psi_{kl (m)} \nonumber \\ &= \sum_{i,j} \sum_{n \ge 0} \widetilde{\mu}_\mathrm{ch}(E_{ij})_{(-n-1)} \frac{\partial}{\partial \Phi_{ij (-n-1)}} \nonumber \\ &\quad + \hbar \sum_{ijklpq} \sum_{n \ge 0} \sum_{m=0}^{n} c_{ijkl}^{pq} \Phi_{pq (-n-1)} \frac{\partial}{\partial \Phi_{ij (-n+m-1)}} \frac{\partial}{\partial \Phi_{kl (-m-1)}}. \nonumber\end{aligned}$$ Then, $d^{+}_{\hbar\mathrm{VA}}$ maps from $\widetilde{C}_{\hbar\mathrm{VA}}^{p, q}$ to $\widetilde{C}_{\hbar\mathrm{VA}}^{p+1, q}$, $d^{-}_{\hbar\mathrm{VA}}$ maps from $\widetilde{C}_{\hbar\mathrm{VA}}^{p, q}$ to $\widetilde{C}_{\hbar\mathrm{VA}}^{p, q+1}$, and $d_{\hbar\mathrm{VA}} = d^+_{\hbar\mathrm{VA}} + d^-_{\hbar\mathrm{VA}}$ and $d^+_{\hbar\mathrm{VA}} \circ d^-_{\hbar\mathrm{VA}} = - d^-_{\hbar\mathrm{VA}} \circ d^+_{\hbar\mathrm{VA}}$ hold. Thus, we obtain a double complex $(\widetilde{C}_{\hbar\mathrm{VA}}, d^+_{\hbar\mathrm{VA}}, d^-_{\hbar\mathrm{VA}})$ whose total complex is the cochain complex $(\widetilde{C}_{\hbar\mathrm{VA}}, d_{\hbar\mathrm{VA}})$. Setting $C_{\hbar\mathrm{VA}}^{p,q} = C_{\hbar\mathrm{VA}} \cap \widetilde{C}^{p,q}_{\hbar\mathrm{VA}}$, we obtain a double complex $(C_{\hbar\mathrm{VA}}^{p,q}, d_{\hbar\mathrm{VA}}^{+}, d_{\hbar\mathrm{VA}}^{-})$ whose total complex is the BRST complex $(C_{\hbar\mathrm{VA}}, d_{\hbar\mathrm{VA}})$. ## Poisson BRST Reduction {#sec:Poisson-brst} By considering the associated graded, we make the sheaf $(C_{\hbar\mathrm{VA}}, d_{\hbar\mathrm{VA}})$ of cochain complexes of $\hbar$-adic vertex superalgebras into a sheaf $(C_{\mathrm{VPA}}, d^{+}_{\mathrm{VPA}}, d^{-}_{\mathrm{VPA}})$ of double complexes of vertex Poisson superalgebras. The cochain complex $\widetilde{C}_{\hbar\mathrm{VA}} = \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar} \mathop{\mathrm{\widehat{\otimes}}}C\ell_{\hbar}(T^* \mathfrak{g})$ is naturally equipped with a filtration ${}^{\hbar} \!F_{\bullet} \widetilde{C}_{\hbar\mathrm{VA}}$ by powers of $\hbar$: ${}^{\hbar} \!F_p \widetilde{C}_{\hbar\mathrm{VA}} \coloneqq \hbar^p \widetilde{C}_{\hbar\mathrm{VA}}$ for $p \in {\mathbb Z}_{\ge 0}$. For each $p \in {\mathbb Z}_{\ge 0}$, the associated graded space is $${}^{\hbar} \!\mathop{\mathrm{Gr}}_{p} \widetilde{C}_{\hbar\mathrm{VA}} = {}^{\hbar} \!F_{p} \widetilde{C}_{\hbar\mathrm{VA}} / {}^{\hbar} \!F_{p+1} \widetilde{C}_{\hbar\mathrm{VA}} \simeq \widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}} \otimes \Lambda^\mathrm{vert}(T^* \mathfrak{g})$$ as commutative vertex superalgebra. Recall that there is a natural vertex Poisson superalgebra structure on ${}^{\hbar} \!\mathop{\mathrm{Gr}}_{p} \widetilde{C}_{\hbar\mathrm{VA}}$ with the Poisson structure $Y_{+}(a, z)$ for $a \in {}^{\hbar} \!\mathop{\mathrm{Gr}}_{p} \widetilde{C}_{\hbar\mathrm{VA}}$ given by $Y_{+}(a, z) = \hbar^{-1} Y(a, z) \bmod \hbar$. By abuse of notation, we write $a_{(n)}$ for the modes of $Y_{+}(a, z) = \sum_{n \ge 0} a_{(n)} z^{-n-1}$, i.e. the operator $a_{(n)}$ on ${}^{\hbar} \!\mathop{\mathrm{Gr}}_{p} \widetilde{C}_{\hbar\mathrm{VA}}$ is the one induced from $\hbar^{-1} a_{(n)} \bmod \hbar$ on $\widetilde{C}_{\hbar\mathrm{VA}}$. Then, the isomorphism ${}^{\hbar} \!\mathop{\mathrm{Gr}}_{p} \widetilde{C}_{\hbar\mathrm{VA}} \simeq \widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}} \otimes \Lambda^\mathrm{vert}(T^* \mathfrak{g})$ is one of vertex Poisson superalgebras. By restriction, we obtain the filtered complex ${}^{\hbar} \!F_p C_{\hbar\mathrm{VA}} = C_{\hbar\mathrm{VA}} \cap {}^{\hbar} \!F_p \widetilde{C}_{\hbar\mathrm{VA}}$. For $p \in {\mathbb Z}_{\ge 0}$, the associated graded space is the vertex Poisson superalgebra $${}^{\hbar} \!\mathop{\mathrm{Gr}}_{p} C_{\hbar\mathrm{VA}} = \{ c \in \widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}} \otimes \Lambda^\mathrm{vert}(T^* \mathfrak{g}) \,|\, \widetilde{E}_{ij (0)} c = \Phi_{ij (0)} c = 0 \text{ for } i,j = 1, \dots, N \}.$$ Note that ${}^{\hbar} \!\mathop{\mathrm{Gr}}_{p} \widetilde{C}_{\hbar\mathrm{VA}}$ and ${}^{\hbar} \!\mathop{\mathrm{Gr}}_{p} C_{\hbar\mathrm{VA}}$ are independent of $p \in {\mathbb Z}_{\ge 0}$. Then, the coboundary operators $d_{\hbar\mathrm{VA}}^{+}$ and $d_{\hbar\mathrm{VA}}^{-}$ induce coboundary operators on ${}^{\hbar} \!\mathop{\mathrm{Gr}}_{p} \widetilde{C}_{\hbar\mathrm{VA}}$ and ${}^{\hbar} \!\mathop{\mathrm{Gr}}_p C_{\hbar\mathrm{VA}}$. They can be explicitly described as $$\begin{aligned} \label{eq:d-VPA-plus} d^{+}_{\mathrm{VPA}} &\coloneqq \sum_{i,j} \sum_{n \ge 0} \Psi_{ij (-n-1)} \widetilde{E}_{ij (n)} \\ &\quad + \sum_{ijklpq} \sum_{n \ge 0} \sum_{m=0}^{n} c_{ijkl}^{pq} \Psi_{ij (-n+m-1)} \Psi_{kl (-m)} \Phi_{pq (n)} \nonumber \\ &= \sum_{i,j} \sum_{n \ge 0} \Psi_{ij (-n-1)} \widetilde{E}_{ij (n)} \nonumber \\ &\quad + \sum_{ijklpq} \sum_{n \ge 0} \sum_{m=0}^{n} c_{ijkl}^{pq} \Psi_{ij (-n+m-1)} \Psi_{kl (-m)} \frac{\partial}{\partial \Psi_{pq (-n-1)}} \nonumber\end{aligned}$$ and $$\label{eq:d-VPA-minus} d^{-}_{\mathrm{VPA}} \coloneqq \sum_{i,j} \sum_{n \ge 0} \widetilde{\mu}_{\infty}(E_{ij})_{(-n-1)} \Psi_{ij (n)} = \sum_{i,j} \sum_{n \ge 0} \widetilde{\mu}_{\infty}(E_{ij})_{(-n-1)} \frac{\partial}{\partial \Phi_{ij (-n-1)}},$$ where $\widetilde{\mu}_{\infty} \colon S(\mathfrak{g}\otimes {\mathbb C}[t^{-1}] t^{-1}) \longrightarrow \widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\mathfrak{X})$ is a homomorphism of vertex Poisson superalgebras induced from the Lie algebra homomorphism $\widetilde{\mu}\colon \mathfrak{g}\longrightarrow \widetilde{\mathcal{O}}_{\mathfrak{X}}(\mathfrak{X})$. Note that $$\widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\mathfrak{X}) = {\mathbb C}[x_{ij (-n)}, y_{ij (-n)}, \gamma_{i (-n)}, \beta_{i (-n)}, \psi_{i (-n)}, \phi_{i (-n)} \,|\, \substack{i, j = 1, \dots, N \\ n = 1, 2, \dots}\,]$$ as a commutative superalgebra. Then, the homomorphism $\widetilde{\mu}_{\infty}$ is given by $$\widetilde{\mu}_{\infty}(E_{ij}) = \sum_{p=1}^{N} (x_{ip (-1)} y_{pj} - x_{pj (-1)} y_{ip}) + \gamma_{i (-1)} \beta_{j} + \psi_{i (-1)} \phi_{j}$$ for the matrices $E_{ij} \in \mathfrak{g}$ for $i,j=1,\dots,N$. Setting $\widetilde{C}_{\mathrm{VPA}} = {}^{\hbar} \!\mathop{\mathrm{Gr}}_{p} \widetilde{C}_{\hbar\mathrm{VA}} = \widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}} \otimes \Lambda^\mathrm{vert}(T^* \mathfrak{g})$ and $C_{\mathrm{VPA}} = {}^{\hbar} \!\mathop{\mathrm{Gr}}_p C_{\hbar\mathrm{VA}}$, we obtain double complexes $(\widetilde{C}_{\mathrm{VPA}}, d^{+}_{\mathrm{VPA}}, d^{-}_{\mathrm{VPA}})$ and $(C_{\mathrm{VPA}}, d^{+}_{\mathrm{VPA}}, d^{-}_{\mathrm{VPA}})$, respectively, each defined on a vertex Poisson superalgebra. ## A Koszul Complex Associated with the BRST Complex {#sec:koszul-cpx} In the following, using the vertex Poisson algebra complex of the previous section, we associate a Koszul complex with the BRST complex. For $p \in {\mathbb Z}$, we consider the vertex Poisson algebra complex $(\widetilde{C}^{p, \bullet}_{\mathrm{VPA}}, d^{-}_{\mathrm{VPA}})$ from above. By the explicit description [\[eq:d-VPA-minus\]](#eq:d-VPA-minus){reference-type="eqref" reference="eq:d-VPA-minus"} of the coboundary operator $d^{-}_{\mathrm{VPA}}$, for an open subset $\widetilde{U}\subset \mathfrak{X}$, the complex $(\widetilde{C}^{p, \bullet}_{\mathrm{VPA}}(\widetilde{U}), d^{-}_{\mathrm{VPA}})$ coincides with the Koszul complex $K_{- \bullet}(\{\partial^m \widetilde{\mu}_{\infty}(E_{ij}) \,|\, \substack{i,j = 1, \dots, N \\ m = 1, 2, \dots}\}, \widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\widetilde{U}))$ associated with the sequence of sections $\{\partial^{m} \widetilde{\mu}_{\infty}(E_{ij}) \,|\, \substack{i,j = 1, \dots, N \\ m = 1, 2, \dots}\}$ on $\widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\widetilde{U})$. **Lemma 15**. *For any open subset $\widetilde{U}\subset \mathfrak{X}$, the sequence $\{\partial^m \widetilde{\mu}_{\infty}(E_{ij}) \,|\, \substack{i,j = 1, \dots, N \\ m = 1, 2, \dots}\}$ is a regular sequence on $\widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\widetilde{U})$.* *Proof.* Since the moment map $\mu \colon T^* V \longrightarrow \mathfrak{g}^*$ is a flat morphism, the associated algebra homomorphism $\mu^*$ maps $S(\mathfrak{g})$ injectively into ${\mathbb C}[T^* V] = \mathcal{O}_{\mathfrak{X}}(\mathfrak{X})$. Thus, $\{ \mu^*(E_{ij}) \,|\, i, j = 1, \dots, N \}$ is a regular sequence on ${\mathbb C}[T^* V]$, and $\mu^{-1}(0)$ is a complete intersection. By Proposition 1.4 in [@Mustata01], $J_n \mu^{-1}(0)$ is also a locally complete intersection for any $n \in {\mathbb Z}_{\ge 0}$. By comparing the dimensions of the manifolds $\mathop{\mathrm{dim}}_{{\mathbb C}} J_{n} \mathfrak{X}= \mathop{\mathrm{dim}}_{{\mathbb C}} J_{n} M + 2 \mathop{\mathrm{dim}}_{{\mathbb C}} J_{n} G$, we see that $\mathop{\mathrm{dim}}_{{\mathbb C}} J_n \mathfrak{X}= \mathop{\mathrm{dim}}_{{\mathbb C}} J_n \mu^{-1}(0) + \#\{ \partial^m \mu^*(E_{ij}) \,|\, \substack{i,j = 1, \dots, N \\ m = 0, 1, \dots, n}\}$. This implies that $\{ \partial^m \mu^*(E_{ij}) \,|\, \substack{i,j = 1, \dots, N \\ m = 0, 1, \dots, n}\}$ is again a regular sequence on ${\mathbb C}[J_n T^* V]$ for any $n \in {\mathbb Z}_{\ge 0}$. Therefore, $\{ \partial^m \mu^*(E_{ij}) \,|\, \substack{i,j = 1, \dots, N \\ m = 0, 1, \dots}\}$ is a regular sequence on ${\mathbb C}[J_{\infty} T^* V]$. We arrange the sequence as $r_s \coloneqq \partial^{m_s} \mu^*(E_{i_s j_s})$ for $s = 1, 2, \dots$ and define $n_s = \partial^{m_s}\widetilde{\mu}_{\infty}(E_{i_s j_s}) - r_{s} = \partial^{m_s} (\psi_{i_s (-1)} \phi_{j_s})$. Now, assume for the sake of contradiction that the sequence $\{ r_s + n_s \,|\, s = 1, 2, \dots \}$ is not regular on $\widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\mathfrak{X}) = {\mathbb C}[J_{\infty} T^* V] \otimes \Lambda(\psi_{i (-n)}, \phi_{i (-n)} \,|\, \substack{i = 1, \dots, N \\ n = 1, 2, \dots})$. Then, there is an $s \in {\mathbb Z}_{>0}$ such that $r_s + n_s$ is a zero-divisor on $\widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\mathfrak{X}) / (r_1 + n_1, \dots, r_{s-1} + n_{s-1})$. This implies that there exist $a_1,\dots,a_s \in {\mathbb C}[J_{\infty} T^* V]$ and $b_1,\dots,b_s \in \Lambda(\psi_{i (-n)}, \phi_{i (-n)} \,|\, \substack{i = 1, \dots, N \\ n = 1, 2, \dots})$ such that $\sum_{i=1}^{s} (a_s + b_s)(r_s + n_s) = 0$ in $\widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\mathfrak{X})$. Taking the component ${\mathbb C}[J_{\infty} T^* V] \otimes 1 \subset \widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\mathfrak{X})$, the equation implies that $\sum_{i=1}^{s} a_s r_s = 0$ in ${\mathbb C}[J_{\infty} T^* V]$, contradicting the fact that $\{ r_s \,|\, s = 1, 2, \dots \}$ is a regular sequence on ${\mathbb C}[J_{\infty} T^* V]$. Thus $\{ \partial^{n_s} \widetilde{\mu}_{\infty}(E_{i_s j_s}) = r_s + n_s \,|\, s = 1, 2, \dots \}$ is a regular sequence on $\widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\mathfrak{X})$. ◻ For an open subset $\widetilde{U}\subset \mathfrak{X}$, let $\tau_{\ge \bullet} \widetilde{C}_{\mathrm{VPA}}(\widetilde{U})$ be the column filtration associated with the double complex $(\widetilde{C}_{\mathrm{VPA}}(\widetilde{U}), d^{+}_{\mathrm{VPA}}, d^{-}_{\mathrm{VPA}})$, i.e. $$\tau_{\ge p} \widetilde{C}_{\mathrm{VPA}}(\widetilde{U}) = \prod_{k \ge p, q \le 0} \widetilde{C}^{k,q}_{\mathrm{VPA}}(\widetilde{U})$$ for $p \in {\mathbb Z}$. We consider the spectral sequence ${}^{\tau} \!E_r^{p,q}(\widetilde{U})$ associated with this column filtration. Then the zeroth term of the spectral sequence coincides with the tensor product of the Koszul complex and the exterior algebra $${}^{\tau} \!E_0^{p,q}(\widetilde{U}) = K_{-q}(\{\partial^m \widetilde{\mu}_{\infty}(E_{ij}) \,|\, \substack{i,j = 1, \dots, N \\ m = 1, 2, \dots}\}, \widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\widetilde{U})) \otimes \Lambda^p(\Psi_{ij (-n)} \,|\, \substack{i, j = 1, \dots, N \\ n = 1, 2, \dots})$$ as discussed above. Thus, the first term is the tensor product of the Koszul homology and the exterior algebra $${}^{\tau} \!E^{p,q}_1 = H_{-q}^{\mathrm{Kosz}}(\{\partial^m \widetilde{\mu}_{\infty}(E_{ij}) \,|\, \substack{i,j = 1, \dots, N \\ m = 1, 2, \dots}\}, \widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\widetilde{U})) \otimes \Lambda^p(\Psi_{ij (-n)} \,|\, \substack{i, j = 1, \dots, N \\ n = 1, 2, \dots}).$$ Then, by above, $$\label{eq:10} {}^{\tau} \!E^{p,q}_{1} = \begin{cases} \widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\widetilde{U}) / \sum_{i,j,m} \widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\widetilde{U}) \partial^m \widetilde{\mu}_{\infty}(E_{ij}) \otimes \Lambda^{p}(\Psi_{ij (-n)} \,|\, \substack{i, j = 1, \dots, N \\ n = 1, 2, \dots}), & \!\!\! q = 0, \\ 0, & \!\!\! q \ne 0. \end{cases}$$ The coboundary operator $d_1 = d^{+}_{\mathrm{VPA}}\colon {}^{\tau} \!E^{p,q}_{1} \longrightarrow {}^{\tau} \!E^{p+1, q}_{1}$ is given by [\[eq:d-VPA-plus\]](#eq:d-VPA-plus){reference-type="eqref" reference="eq:d-VPA-plus"}, and thus $({}^{\tau} \!E^{p,q}_{1}, d_1 = d^{+}_{\mathrm{VPA}})$ coincides with the Chevalley complex of the Lie algebra $\mathfrak{g}[t]$ with coefficients in the Koszul homology. **Lemma 16**. *For any open subset $\widetilde{U}\subset \mathfrak{X}$, the spectral sequence ${}^{\tau} \!E_r^{p,q}(\widetilde{U})$ converges to the total cohomology, i.e. ${}^{\tau} \!E_r^{p,q}(\widetilde{U}) \Longrightarrow H^{p+q}(\widetilde{C}_{\mathrm{VPA}}(\widetilde{U}), d_{\mathrm{VPA}})$.* *Proof.* Note that $\widetilde{C}^{p,q}_{\mathrm{VPA}} = 0$ unless $p \ge 0$ and $q \le 0$. To prove the asserted convergence, we construct subcomplexes that are bounded both from above and below. For $m \in {\mathbb Z}_{\ge 0}$, set $(\widetilde{C}_{\mathrm{VPA}})_m(\widetilde{U}) = \partial^m R$, where $R \coloneqq \mathcal{O}_{\mathfrak{X}}(\widetilde{U}) \otimes \Lambda(T^* {\mathbb C}^N) \otimes \Lambda(T^* \mathfrak{g})$ is regarded as a subalgebra of $J_{\infty}{R}$ by the canonical embedding $j \colon R \hookrightarrow J_{\infty}{R}$. By [\[eq:d-VPA-plus\]](#eq:d-VPA-plus){reference-type="eqref" reference="eq:d-VPA-plus"} and [\[eq:d-VPA-minus\]](#eq:d-VPA-minus){reference-type="eqref" reference="eq:d-VPA-minus"}, $d_{\mathrm{VPA}}$ preserves the subspace $(\widetilde{C}_{\mathrm{VPA}})_0(\widetilde{U})$. Since $d_{\mathrm{VPA}}$ commutes with the translation operator $\partial$, $d_{\mathrm{VPA}}$ also preserves $(\widetilde{C}_{\mathrm{VPA}})_m(\widetilde{U})$ for any $m \in {\mathbb Z}_{\ge 0}$. Therefore, $((\widetilde{C}_{\mathrm{VPA}})_m(\widetilde{U}), d^+_{\mathrm{VPA}}, d^-_{\mathrm{VPA}})$ is a double subcomplex of $(\widetilde{C}_{\mathrm{VPA}}(\widetilde{U}), d^+_{\mathrm{VPA}}, d^-_{\mathrm{VPA}})$. Consider the spectral sequence $({}^{\tau} \!E_r^{p,q})_m(\widetilde{U})$ associated with the double complex $((\widetilde{C}_{\mathrm{VPA}})_m(\widetilde{U}), d^+_{\mathrm{VPA}}, d^-_{\mathrm{VPA}})$. Since $(\widetilde{C}_{\mathrm{VPA}})_m(\widetilde{U})$ is bounded, the spectral sequence $({}^{\tau} \!E_r^{p,q})_m(\widetilde{U})$ converges. Hence, so does the spectral sequence ${}^{\tau} \!E_r^{p,q}(\widetilde{U})$. ◻ As a consequence, we obtain the following lemma. **Lemma 17**. *Let $\widetilde{U}\subset \mathfrak{X}$ be an open subset.* 1. *If $\widetilde{U}\cap \mu^{-1}(0) = \emptyset$, then $H^{n}(\widetilde{C}_{\mathrm{VPA}}^{\bullet}(\widetilde{U}), d_{\mathrm{VPA}}) = 0$ for all $n \in {\mathbb Z}$.* 2. *The cohomology $H^{n}(\widetilde{C}_{\mathrm{VPA}}^{\bullet}(\widetilde{U}), d_{\mathrm{VPA}})$ vanishes if $n < 0$.* *Proof.* Since $\widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\widetilde{U}) / \sum_{i,j,m} \widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\widetilde{U}) \partial^m \widetilde{\mu}_{\infty}(E_{ij}) = 0$ if $\widetilde{U}\cap \mu^{-1}(0) = \emptyset$ in [\[eq:10\]](#eq:10){reference-type="eqref" reference="eq:10"}, it follows that ${}^{\tau} \!E^{p,q}_{1} = 0$ for any $p,q \in {\mathbb Z}$. Thus, $H^{p+q}(\widetilde{C}_{\mathrm{VPA}}^{\bullet}(\widetilde{U}), d_{\mathrm{VPA}}) = 0$ for any $p,q \in {\mathbb Z}$ by , proving the first assertion. By [\[eq:10\]](#eq:10){reference-type="eqref" reference="eq:10"}, ${}^{\tau} \!E^{p,q}_{1} = 0$ unless $p \in {\mathbb Z}_{\ge 0}$ and $q = 0$. Thus, by , the cohomology $H^{p+q}(\widetilde{C}_{\mathrm{VPA}}^{\bullet}(\widetilde{U}), d_{\mathrm{VPA}})$ vanishes unless $p+q \ge 0$. ◻ ## Cohomology of the Poisson BRST Reduction {#sec:cohom-Poisson-BRST} In the following, we compute the cohomology of the sheaf $(C_{\mathrm{VPA}}, d_{\mathrm{VPA}})$ of vertex Poisson algebra complexes from above. Recall that the first term of the spectral sequence $({}^{\tau} \!E^{p,q}_{1}, d_1 = d^{+}_{\mathrm{VPA}})$ coincides with the Chevalley complex of the Lie algebra $\mathfrak{g}[t]$ with coefficients in the Koszul homology. Therefore, the second term ${}^{\tau} \!E_{2}^{p,q}$ is isomorphic to the Lie algebra cohomology, and thus $$\begin{aligned} \label{eq:13} {}^{\tau} \!E_2^{p,q}(\widetilde{U}) &\simeq H^{p}(H^{q}(\widetilde{C}_{\mathrm{VPA}}(\widetilde{U}), d^{-}_{\mathrm{VPA}}), d^{+}_{\mathrm{VPA}}) \\ &\simeq H^{p}(\mathfrak{g}, H^{\mathrm{Kosz}}_{-q}(\{\partial^{m} \widetilde{\mu}_{\infty}(E_{ij}) \}_{i,j,m}, \widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U}))) \nonumber \\ &\simeq \begin{cases} H^{p}(\mathfrak{g}, \widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\widetilde{U}) / \sum_{i,j,m} \widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\widetilde{U}) \partial^m \widetilde{\mu}_{\infty}(E_{ij})), & q = 0, \\ 0, & q \ne 0. \nonumber \end{cases}\end{aligned}$$ Recall the affine open covering $\mathfrak{X}= \bigcup_{\lambda \vdash N} \widetilde{U}_{\lambda}$ in terms of partitions $\lambda$ of $N$ introduced in . By , $$\begin{aligned} \label{eq:18} \widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U}_{\lambda}) &= A_{\lambda} \otimes \Lambda(\{\psi : X^i Y^j \gamma \}_{\lambda}, \phi X^i Y^j \gamma \,|\, (i,j) \in \lambda) \\ &\quad \otimes {\mathbb C}[(B_{\lambda}^{\pm})_{ij} \,|\, i, j = 1, \dots, N] \otimes {\mathbb C}[\widetilde{\mu}(E_{ij}) \,|\, i, j = 1, \dots, N], \nonumber\end{aligned}$$ where $A_{\lambda}$ is the subalgebra of $\widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U}_{\lambda})$ generated by the $G$-invariant bosonic sections defined in . Note that the Lie group $G$ acts trivially on the subalgebra $A_{\lambda} \otimes \Lambda(\{\psi : X^i Y^j \gamma \}_{\lambda}, \phi X^i Y^j \gamma \,|\, (i,j) \in \lambda)$, while ${\mathbb C}[(B_{\lambda}^{\pm})_{ij} \,|\, i, j = 1, \dots, N] \otimes {\mathbb C}[\widetilde{\mu}(E_{ij}) \,|\, i, j = 1, \dots, N]$ is as $G$-module isomorphic to the coordinate ring ${\mathbb C}[T^* G]$ of the cotangent bundle $T^* G \simeq G \times \mathfrak{g}^*$. The $\infty$-jet bundle $\widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\widetilde{U}_{\lambda}) = J_{\infty}{\widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U}_{\lambda})}$ is thus the tensor product $\widetilde{\mathcal{O}}_{J_\infty \mathfrak{X}}(\widetilde{U}_{\lambda}) \simeq \widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda}) \otimes {\mathbb C}[J_{\infty}{T^* G}]$ of the $J_{\infty}{G}$-invariant subalgebra $\widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda}) \!=\! J_{\infty}{(A_{\lambda} \otimes \Lambda(\{\psi : X^i Y^j\}_{\lambda}, (\phi X^i Y^j \gamma) \,|\, (i, j) \in \lambda))}$ and the subalgebra ${\mathbb C}[J_{\infty}{T^* G}] \simeq {\mathbb C}[J_{\infty}{G}] \otimes {\mathbb C}[J_{\infty}{\mathfrak{g}^{*}}]$. Therefore, by [\[eq:13\]](#eq:13){reference-type="eqref" reference="eq:13"}, ${}^{\tau} \!E_2^{p,0} \simeq \widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda}) \otimes H^p(\mathfrak{g}[t], {\mathbb C}[J_{\infty}{G}])$. The Lie algebra cohomology $H^p(\mathfrak{g}[t], {\mathbb C}[J_{\infty}{G}])$ is naturally isomorphic to the de Rham cohomology $H^p_{\mathrm{dR}}(G, {\mathbb C})$. Hence, $$\label{eq:17} {}^{\tau} \!E^{p,q}_2(\widetilde{U}_{\lambda}) \simeq \begin{cases} \widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda}) \otimes H^p_{\mathrm{dR}}(G, {\mathbb C}), & q = 0, \\ 0, & q \ne 0. \end{cases}$$ Then, the spectral sequence ${}^{\tau} \!E^{p,q}_r$ collapses at the second term and it converges by , and thus we obtain: **Proposition 18**. *For a partition $\lambda \vdash N$ and $n\in{\mathbb Z}$, the vertex Poisson algebra cohomology satisfies $$H^n(\widetilde{C}_{\mathrm{VPA}}(\widetilde{U}_{\lambda}), d_{\mathrm{VPA}}) \simeq \widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda}) \otimes H^n_{\mathrm{dR}}(G, {\mathbb C}).$$ The superalgebra $\widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda}) = J_{\infty}{(A_{\lambda} \otimes \Lambda(\{\psi : X^i Y^j\}_{\lambda}, (\phi X^i Y^j \gamma) \,|\, (i, j) \in \lambda))}$ is naturally isomorphic to the commutative superalgebra $$J_{\infty}{\mathcal{O}_M(U_{\lambda})} \otimes \Lambda\bigl(\{\psi : X^i Y^j\}_{\lambda, (-n)}, (\phi X^i Y^j \gamma)_{(-n)} \,\bigm|\, \substack{ (i, j) \in \lambda \\ n = 1, 2, \dots }\bigr).$$* Now, we determine the cohomology of the subcomplex $(C_{\mathrm{VPA}}, d_{\mathrm{VPA}})$. By the above discussion, for a partition $\lambda \vdash N$, $${}^{\tau} \!E^{p,q}_{1}(\widetilde{U}_{\lambda}) = \widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda}) \otimes {\mathbb C}[J_{\infty}{G}] \otimes \Lambda^{p}(\Psi_{ij (-n)} \,|\, \substack{i, j = 1, \dots, N \\ n = 1, 2, \dots})$$ if $q=0$, and ${}^{\tau} \!E^{p,q}_1 = 0$ otherwise. It is a Lie algebra cohomology complex associated with the action of $\mathfrak{g}[t]$ on $\widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda}) \otimes {\mathbb C}[J_{\infty}{G}]$, as discussed above. We consider the Hochschild-Serre spectral sequence associated with the Lie subalgebra $\mathfrak{g}\subset \mathfrak{g}[t]$. We refer the reader to [@HS53] for details of the construction of the spectral sequence and its fundamental properties. Define a filtration ${}^{\mathrm{HS}} \!F_{\bullet} {}^{\tau} \!E_1^{n,0}$ by $$\begin{aligned} &{}^{\mathrm{HS}} \!F_p {}^{\tau} \!E_1^{n,0}\\ &= \Bigl\{\sum_{\underline{i}, \underline{j}, \underline{m}} a_{\underline{i}, \underline{j}, \underline{m}} \Psi_{i_1 j_1 (-m_1)} \dots \Psi_{i_n j_n (-m_n)} \,\Bigm|\, a_{\underline{i}, \underline{j}, \underline{m}} = 0 \text{ if } \#\{k \,|\, m_k = 1\} > n - p \Bigr\}\end{aligned}$$ for $p\in{\mathbb Z}$. Let ${}^{\mathrm{HS}} \!E_r^{p,q}$ be the spectral sequence associated with this filtration. The zeroth term ${}^{\mathrm{HS}} \!E_0^{p,q} = {}^{\mathrm{HS}} \!\mathop{\mathrm{Gr}}_p {}^{\tau} \!E_1^{p+q,0}$ is the Lie algebra cohomology complex associated with the action of the Lie subalgebra $\mathfrak{g}$. Indeed, since $G$ acts on $J_{\infty}{G}$ freely, the complex $E_0^{p,q}$ factorises as $$\begin{aligned} {}^{\mathrm{HS}} \!E_0^{p,q} &= \bigl(\widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda}) \otimes {\mathbb C}[J_{\infty}{G}]^G \otimes \Lambda^{p}(\Psi_{ij (-n)} \,|\, \substack{i, j = 1, \dots, N \\ n = 2, \dots}) \bigr) \\ &\quad \otimes \bigl({\mathbb C}[G] \otimes \Lambda^{q}(\Psi_{ij (-1)} \,|\, i, j = 1, \dots, N) \bigr),\end{aligned}$$ where the second factor is the Lie algebra cohomology complex of the Lie subalgebra $\mathfrak{g}$ with coefficients in ${\mathbb C}[G]$. Then, the first term ${}^{\mathrm{HS}} \!E_1^{p,q}$ is obtained by de Rham cohomology $H^q_{\mathrm{dR}}(G, {\mathbb C})$ as $$\label{eq:19} {}^{\mathrm{HS}} \!E_1^{p,q} \simeq \left(\widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda}) \otimes {\mathbb C}[J_{\infty}{G}]^G \otimes \Lambda^{p}(\Psi_{ij (-n)} \,|\, \substack{i, j = 1, \dots, N \\ n = 2, \dots}) \right) \otimes H_{\mathrm{dR}}^{q}(G, {\mathbb C}).$$ On the other hand, it follows from [\[eq:18\]](#eq:18){reference-type="eqref" reference="eq:18"} that $$\begin{aligned} C^n_{\mathrm{VPA}}(\widetilde{U}_{\lambda}) &= \bigoplus_{p+q=n} \widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda}) \otimes {\mathbb C}[J_{\infty}{G}]^G \otimes {\mathbb C}[\partial^m \widetilde{\mu}_{\infty}(E_{ij}) \,|\, \substack{i,j = 1, \dots, N \\ m = 1, 2, \dots}] \\ &\quad \otimes \Lambda^{p}(\Psi_{ij (-m)} \,|\, \substack{i, j = 1, \dots, N \\ m = 2, 3 \dots}) \otimes \Lambda^{q}(\Phi_{ij (-m)} \,|\, \substack{i, j = 1, \dots, N \\ m = 1, 2, \dots}).\end{aligned}$$ Thus, the cohomology $H^{q}(C^{p, \bullet}_{\mathrm{VPA}}(\widetilde{U}_{\lambda}), d^{-}_{\mathrm{VPA}})$ with respect to the coboundary operator $d^{-}_{\mathrm{VPA}}$ coincides with the first factor of [\[eq:19\]](#eq:19){reference-type="eqref" reference="eq:19"} if $q = 0$ and vanishes otherwise. As a consequence, $${}^{\mathrm{HS}} \!E_1^{p,q} \simeq H^0(C^{p,\bullet}_{\mathrm{VPA}}(\widetilde{U}_{\lambda}), d^{-}_{\mathrm{VPA}}) \otimes H^{q}_{\mathrm{dR}}(G, {\mathbb C}).$$ Considering the coboundary operator $d^{+}_1 \colon {}^{\mathrm{HS}} \!E_1^{p,q} \longrightarrow {}^{\mathrm{HS}} \!E_1^{p+1, q}$ induced from $d^{+}_{\mathrm{VPA}}$, we obtain $${}^{\mathrm{HS}} \!E_2^{p,q} \simeq H^p(H^0(C_{\mathrm{VPA}}(\widetilde{U}_{\lambda}), d^{-}_{\mathrm{VPA}}), d^{+}_{\mathrm{VPA}}) \otimes H_{\mathrm{dR}}^{q}(G, {\mathbb C}).$$ By the same arguments as for , we obtain the isomorphism $$H^p(C_{\mathrm{VPA}}(\widetilde{U}_{\lambda}), d_{\mathrm{VPA}}) \simeq H^p(H^0(C_{\mathrm{VPA}}(\widetilde{U}_{\lambda}), d^{-}_{\mathrm{VPA}}).$$ In particular, $H^0(C_{\mathrm{VPA}}(\widetilde{U}_{\lambda}), d_{\mathrm{VPA}}) \simeq \widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda})$ by [\[eq:19\]](#eq:19){reference-type="eqref" reference="eq:19"}. The above spectral sequence collapses at the second term, and it converges. Therefore, $$H^n(\widetilde{C}_{\mathrm{VPA}}(\widetilde{U}_{\lambda}), d_{\mathrm{VPA}}) \simeq \bigoplus_{p+q=n} H^p(C_{\mathrm{VPA}}(\widetilde{U}_{\lambda}), d_{\mathrm{VPA}}) \otimes H^q_{\mathrm{dR}}(G, {\mathbb C}).$$ By the isomorphism $H^0(C_{\mathrm{VPA}}(\widetilde{U}_{\lambda}), d_{\mathrm{VPA}}) \simeq \widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda})$ and , the above isomorphism implies that $H^p(C_{\mathrm{VPA}}(\widetilde{U}_{\lambda}), d_{\mathrm{VPA}}) = 0$ unless $p=0$. **Proposition 19**. *For a partition $\lambda \vdash N$ and $n\in{\mathbb Z}$, the vertex Poisson algebra cohomology satisfies $$H^n(C_{\mathrm{VPA}}(\widetilde{U}_{\lambda}), d_{\mathrm{VPA}}) \simeq \begin{cases} \widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda}), & n=0, \\ 0, & n \ne 0. \end{cases}$$* ## Vanishing Theorem {#sec:vanish-brst} Finally, we use the results of the preceding sections in order to prove a vanishing (or no-ghost) theorem for the BRST cohomology sheaf $\mathcal{H}_{\hbar\mathrm{VA}}^{\infty/2 + \bullet}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar})$, which we then define as the sheaf $\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}$. Recall the $\hbar$-adic filtration ${}^{\hbar} \!F_{\bullet} C_{\hbar\mathrm{VA}}$. Let ${}^{\hbar} \!E_{r}^{p,q}$ be the spectral sequence associated with this filtration. The zeroth term ${}^{\hbar} \!E_0^{p,q}$ is the associated graded complex ${}^{\hbar} \!\mathop{\mathrm{Gr}}_p C_{\hbar\mathrm{VA}}^{p+q} \simeq \hbar^p C_{\mathrm{VPA}}^{p+q}$ with the coboundary operator $d_{\mathrm{VPA}}$. **Lemma 20**. *For an open subset $\widetilde{U}\subset \mathfrak{X}$, the spectral sequence ${}^{\hbar} \!E^{p,q}_{r}(\widetilde{U})$ converges to ${}^{\hbar} \!\mathop{\mathrm{Gr}}_{p} H^{p+q}(C_{\hbar\mathrm{VA}}(\widetilde{U}), d_{\hbar\mathrm{VA}})$.* *Proof.* Because the filtration ${}^{\hbar} \!F_{\bullet} C_{\hbar\mathrm{VA}}(\widetilde{U})$ is bounded from above and complete, the spectral sequence ${}^{\hbar} \!E^{p,q}_{r}(\widetilde{U})$ converges by the complete convergence theorem (see Theorem 5.5.10 in [@Weibel]). ◻ First, consider the case where $\widetilde{U}\cap \mu^{-1}(0) = \emptyset$. By  (1) it follows that ${}^{\hbar} \!E^{p,q}_{1}(\widetilde{U}) \simeq \hbar^p H^{p+q}(C_{\mathrm{VPA}}(\widetilde{U}), d_{\mathrm{VPA}}) = 0$. The spectral sequence ${}^{\hbar} \!E^{p,q}_{r}$ collapses at the first term, and hence ${}^{\hbar} \!E^{p,q}_{r} = 0$ for any $p,q \in {\mathbb Z}$ and $r \in {\mathbb Z}_{>0}$. Then implies: **Lemma 21**. *For an open subset $\widetilde{U}\subset \mathfrak{X}$ such that $\widetilde{U}\cap \mu^{-1}(0) = \emptyset$, the cohomology $H^n(C_{\hbar\mathrm{VA}}(\widetilde{U}), d_{\hbar\mathrm{VA}}) = 0$ for all $n\in{\mathbb Z}$.* As remarked above, this lemma completes the definition of the BRST cohomology sheaf $\mathcal{H}^{\infty/2 + \bullet}_{\hbar\mathrm{VA}}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar})$ (see and the preceding discussion). For a partition $\lambda \vdash N$, yields the isomorphism $$\label{eq:hE1-local-isom} {}^{\hbar} \!E_1^{p,q}(\widetilde{U}_{\lambda}) \simeq \begin{cases} \hbar^p \widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda}), & p + q = 0, \\ 0, & p + q \ne 0. \end{cases}$$ This implies that the coboundary operator $d_1 \colon {}^{\hbar} \!E^{p,q}_{1}(\widetilde{U}_{\lambda}) \longrightarrow {}^{\hbar} \!E^{p+1,q}_{1}(\widetilde{U}_{\lambda})$ on the first term is identically zero. Thus, the spectral sequence ${}^{\hbar} \!E^{p,q}_{r}$ collapses at the first term. By , $$H^{\infty/2+n}_{\hbar\mathrm{VA}}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\widetilde{U}_{\lambda})) = H^n(C_{\hbar\mathrm{VA}}(\widetilde{U}_{\lambda}), d_{\hbar\mathrm{VA}}) = 0$$ for $n \ne 0$. Therefore, we obtain the following vanishing theorem for BRST cohomology sheaf. **Theorem 22**. *The BRST cohomology sheaf $\mathcal{H}^{\infty/2+n}_{\hbar\mathrm{VA}}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar})$ vanishes for $n \neq 0$.* We come to the central definition of this text: **Definition 23**. We set $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar} \coloneqq \mathcal{H}^{\infty/2+0}_{\hbar\mathrm{VA}}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}) = \mathcal{H}^{\infty/2+\bullet}_{\hbar\mathrm{VA}}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar})$. This is a sheaf of $\hbar$-adic vertex superalgebras on the Hilbert scheme $M$. The isomorphism in [\[eq:hE1-local-isom\]](#eq:hE1-local-isom){reference-type="eqref" reference="eq:hE1-local-isom"} together with induces an isomorphism $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{\lambda}) \simeq \widetilde{\mathcal{O}}_{J_\infty M}(U_{\lambda})[[\hbar]]$ for any $\lambda \vdash N$. However, this isomorphism depends on the local trivialisation [\[eq:18\]](#eq:18){reference-type="eqref" reference="eq:18"} and does not induce an isomorphism of sheaves. The isomorphism only implies: **Proposition 24**. *There is an isomorphism of sheaves of vertex Poisson superalgebras over $M$, $$\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar} / \hbar \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar} \simeq \widetilde{\mathcal{O}}_{J_\infty M}.$$ We say that the sheaf of $\hbar$-adic vertex superalgebras $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}$ quantises the sheaf of vertex Poisson superalgebras $\widetilde{\mathcal{O}}_{J_\infty M}$.* # Vertex Superalgebra of Global Sections {#sec:F-action} In the previous section, we introduced the sheaf of $\hbar$-adic vertex superalgebras $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}$ on the Hilbert scheme $M=\mathop{\mathrm{Hilb}}^N({\mathbb C}^2)$. Now, we construct a vertex operator superalgebra $\mathsf{V}_{S_N}$ of central charge $c=-3N^2$ from the $\hbar$-adic vertex superalgebra $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ of global sections and study its conformal structure and associated variety. ## Equivariant Torus Action and Global Sections {#sec:global-VA} In the following, we consider a natural equivariant ${\mathbb C}^\times$-action on the BRST cohomology sheaf $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}$. This allows us to define a vertex superalgebra $\mathsf{V}_{S_N}=[\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)]^{{\mathbb C}^\times}$ from the global sections by taking the invariants under this torus action in some sense. The resolution of singularities $M = \mathop{\mathrm{Hilb}}^N({\mathbb C}^2) \longrightarrow M_0 = {\mathbb C}^{2N}/S_N$ is known as an example of conical symplectic resolution. The conical structure is given by the following action of the one-dimensional torus ${\mathbb C}^\times$: consider the action of ${\mathbb C}^\times$ on $\mathfrak{X}$ that induces an equivariant action on the structure sheaf $\mathcal{O}_{\mathfrak{X}}$ such that the weights of the generators with respect to it are given by $\operatorname{twt}(x_{ij}) = \operatorname{twt}(y_{ij}) = \operatorname{twt}(\gamma_i) = \operatorname{twt}(\beta_i)= 1/2$ for $i,j=1,\dots,N$. Here, an element $x$ has (torus) weight $\operatorname{twt}(x)=m$ if it is semi-invariant for the character ${\mathbb C}^\times\longrightarrow{\mathbb C}^\times$, $t\longmapsto t^m$, i.e. if $t\cdot x=t^m x$ for $t\in{\mathbb C}^\times$. Note that, with respect to this action, the Poisson bracket on $\mathcal{O}_{\mathfrak{X}}$ is homogeneous of weight $-1$. Since the ${\mathbb C}^\times$-action commutes with the $G$-action, we obtain an induced ${\mathbb C}^\times$-action on $M$. Moreover, there is the equivariant ${\mathbb C}^\times$-action on the sheaf $\widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}$ over ${\mathbb C}$ with the torus weights of the generators given by $$\begin{aligned} \operatorname{twt}(x_{ij (-n)}) &= \operatorname{twt}(y_{ij (-n)}) = \operatorname{twt}(\gamma_{i (-n)}) = \operatorname{twt}(\beta_{i (-n)}) = 1/2,\\ \operatorname{twt}(\psi_{i (-n)}) &= \operatorname{twt}(\phi_{i (-n)}) = 1/2,\qquad\operatorname{twt}(\hbar) = 1\end{aligned}$$ for $i,j=1,\dots,N$ and $n \in {\mathbb Z}_{>0}$. Note that the operator product expansions of $\widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}$ are homogeneous of weight $0$ with respect to the ${\mathbb C}^\times$-action. We extend this torus action to the BRST complexes $\widetilde{C}_{\hbar\mathrm{VA}}$ and $C_{\hbar\mathrm{VA}}$ by $$\operatorname{twt}(\Psi_{ij (-n)}) = 0,\quad\operatorname{twt}(\Phi_{ij (-n)}) = 1$$ for $i,j=1,\dots,N$ and $n \in {\mathbb Z}_{>0}$. Then, the element $Q \in \widetilde{C}_{\hbar\mathrm{VA}}$ is homogeneous of weight $\operatorname{twt}(Q) = 1$, and hence the coboundary operator $d_{\hbar\mathrm{VA}} = \hbar^{-1} Q_{(0)}$ is a homogeneous operator of weight $0$ on the complexes $\widetilde{C}_{\hbar\mathrm{VA}}$ and $C_{\hbar\mathrm{VA}}$. This implies that the BRST cohomology sheaf $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar} = \mathcal{H}^{\infty/2+\bullet}_{\hbar\mathrm{VA}}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar})$ is also equipped with the induced equivariant ${\mathbb C}^\times$-action over $M$. In particular, the space of global sections $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ is a ${\mathbb C}[[\hbar]]$-module with a ${\mathbb C}^\times$-action over ${\mathbb C}$. Recall the affine open covering $\mathfrak{X}= \bigcup_{\lambda \vdash N} \widetilde{U}_{\lambda}$. For a partition $\lambda$, the open subset $\widetilde{U}_{\lambda}$ is closed under the ${\mathbb C}^\times$-action. Each column of the matrix $B_{\lambda} = (X^i Y^j \gamma)_{(i,j) \in \lambda}$ is homogeneous of weight $(i+j+1)/2$ and thus its determinant and minors are again homogeneous. This implies that the $(i,j)$-th row of its inverse $B_{\lambda}^{-1}$ is homogeneous of weight $-(i+j+1)/2$. Therefore, the complex $\widetilde{C}_{\hbar\mathrm{VA}}(\widetilde{U}_{\lambda})$ is generated by homogeneous sections and it can be decomposed into a direct product of weight spaces. The coboundary operator $d_{\hbar\mathrm{VA}}$ commutes with the ${\mathbb C}^\times$-action, and hence the BRST cohomology $H^{\bullet}(\widetilde{C}_{\hbar\mathrm{VA}}(\widetilde{U}_{\lambda}), d_{\hbar\mathrm{VA}})$ is also a direct product of weight spaces. This implies, in particular, the weight-space decomposition $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{\lambda}) = \prod_{m} \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{\lambda})^{{\mathbb C}^\times\!, m}$. Since the space of global sections $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ is the intersection of the $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{\lambda})$ for all $\lambda \vdash N$, it can also be decomposed into the direct product of weight spaces $$\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M) = \prod_{m \ge 0} \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)^{{\mathbb C}^\times\!, m}.$$ We note that the weights $m \in \frac{1}{2} {\mathbb Z}_{\ge 0}$ of the global sections are non-negative and $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)^{{\mathbb C}^\times\!, 0} = {\mathbb C}\mathbf{1}$. Consider the subspace $$\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)_\mathrm{fin}\coloneqq \bigoplus_{m \in \frac{1}{2} {\mathbb Z}_{\ge 0}} \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)^{{\mathbb C}^\times\!, m}$$ of the direct sum of weight spaces. This subspace is a ${\mathbb C}[\hbar]$-module since the weights of the global sections are non-negative and $\operatorname{twt}(\hbar)=1$. Moreover, because the operator product expansions preserve the ${\mathbb C}^\times$-weight, they also preserve this subspace. Now we set $$\label{eq:sfW} \mathsf{V}_{S_N} = \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)_\mathrm{fin}\bigm/ (\hbar - 1),$$ the quotient space by the ideal generated by $\hbar - 1$. It is a ${\mathbb C}$-vector space equipped with operator product expansions induced from the ones on $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$. Since all the identities involving the vertex operators of $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ are also satisfied by the ones of $\mathsf{V}_{S_N}$, the ${\mathbb C}$-vector space $\mathsf{V}_{S_N}$ is a vertex superalgebra. **Remark 25**. Alternatively, we can think of $\mathsf{V}_{S_N}$ as $$[\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)]^{{\mathbb C}^\times} \coloneqq \big(\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)\otimes_{{\mathbb C}[[\hbar]]}{\mathbb C}((\sqrt{\hbar}))\big)^{{\mathbb C}^\times}=\bigoplus_{m \in \frac{1}{2} {\mathbb Z}_{\ge 0}} \mathsf{V}_{S_N}^{{\mathbb C}^\times\!, m}\hbar^{-m}$$ (and then forget about $\hbar$), where taking the invariants under the natural extension of the ${\mathbb C}^\times$-action precludes the appearance of infinite sums (cf. [@KR08; @AKM15]). ## Associated Variety {#sec:associated-variety} In the following, we determine the associated variety of the vertex superalgebra $\mathsf{V}_{S_N}$ and, as a consequence, show that $\mathsf{V}_{S_N}$ is quasi-lisse. For a vertex superalgebra $V$ (or an $\hbar$-adic vertex superalgebra), consider the quotient vector space $$R(V) \coloneqq V / C_2(V)\quad\text{with}\quad C_2(V) = \{ a_{(-2)} b \,|\, a, b \in V\}.$$ The $(-1)$-product of $V$ induces a commutative and associative product on $R(V)$ [@Z96]. Moreover, the $(0)$-product of $V$ defines a Poisson bracket on $R(V)$ by $\{a, b\} = a_{(0)} b$ (or $\{a, b\} = \hbar^{-1} a_{(0)} b$, respectively) modulo $V_{(-2)} V$ for $a,b \in V$. Thus, $R(V)$ is a Poisson superalgebra over ${\mathbb C}$ (or ${\mathbb C}[[\hbar]]$, respectively), and is called the $C_2$-Poisson algebra of $V$. The affine algebraic variety $\mathop{\mathrm{Spec}}(R(V)_{\mathrm{red}})$ associated with $R(V)_{\mathrm{red}}$ is called the associated variety of $V$, where $R(V)_{\mathrm{red}} = R(V) / N$ is the quotient algebra by the nilradical $N$ of $R(V)$ [@Arakawa12]. For any open subset $U$ of the Hilbert scheme $M$, the exact sequence $$0 \longrightarrow \hbar \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar} \longrightarrow \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar} \longrightarrow \widetilde{\mathcal{O}}_{J_\infty M} \longrightarrow 0$$ induces an injective homomorphism $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U) / \hbar \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U) \hookrightarrow \widetilde{\mathcal{O}}_{J_\infty M}(U)$. Then, the composition of the above homomorphism $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U) \longrightarrow \widetilde{\mathcal{O}}_{J_\infty M}(U)$ and the canonical projection $\widetilde{\mathcal{O}}_{J_\infty M}(U) \longrightarrow R(\widetilde{\mathcal{O}}_{J_\infty M}(U))$ induces an injective homomorphism $$\label{eq:3} \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U) \bigm/ \bigl(C_2(\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U)) + \hbar \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U)\bigr) \hookrightarrow R(\widetilde{\mathcal{O}}_{J_\infty M}(U)).$$ **Lemma 26**. *For $U\subset M$ open, the homomorphism [\[eq:3\]](#eq:3){reference-type="eqref" reference="eq:3"} induces an injective homomorphism of Poisson superalgebras $R(\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U)) / \hbar R(\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U)) \hookrightarrow \widetilde{\mathcal{O}}_{M}(U)$.* *Proof.* By the isomorphism theorem, $$\begin{aligned} R(\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U)) / \hbar R(\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U)) &= \frac{\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U) / C_2(\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U))}{ \hbar \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U) / (C_2(\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U)) \cap \hbar \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U))} \\ &\simeq \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U) / (C_2(\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U)) + \hbar \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U)).\end{aligned}$$ The asserted homomorphism is then given by the composition of this isomorphism and the one in [\[eq:3\]](#eq:3){reference-type="eqref" reference="eq:3"} mapping to $R(\widetilde{\mathcal{O}}_{J_\infty M}(U)) \simeq \widetilde{\mathcal{O}}_{M}(U)$. ◻ implies the following proposition about the $C_2$-Poisson algebras of $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ and $\mathsf{V}_{S_N}$. **Proposition 27**. *The $C_2$-Poisson algebra $R(\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M))$ of $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ is a subalgebra of $\widetilde{\mathcal{O}}_M(M)[[\hbar]]$, and $R(\mathsf{V}_{S_N})$ is a subalgebra of $\widetilde{\mathcal{O}}_{M}(M)$.* **Lemma 28**. *For any $m \in {\mathbb Z}_{>0}$, $\mathop{\mathrm{Tr}}(X^m)$ and $\mathop{\mathrm{Tr}}(Y^m)$ are cocycles in $C_{\hbar\mathrm{VA}}(\mathfrak{X})$.* *Proof.* We shall see in the proof of that the operator product expansion $\mu_\mathrm{ch}(E_{ij})(z) \mathop{\mathrm{Tr}}(X^m)(w) \sim \mu_\mathrm{ch}(E_{ij})(z) \mathop{\mathrm{Tr}}(Y^m)(w) \sim 0$ holds for all $i,j=1,\dots,N$. Thus, we obtain $d_{\hbar\mathrm{VA}}(\mathop{\mathrm{Tr}}(X^m)) = d_{\hbar\mathrm{VA}}(\mathop{\mathrm{Tr}}(Y^m)) = 0$ and moreover $\widetilde{E}_{ij(0)} \mathop{\mathrm{Tr}}(X^m) = \widetilde{E}_{ij(0)} \mathop{\mathrm{Tr}}(Y^m) = 0$ for any $i,j=1,\dots,N$. ◻ By the above lemma, there are the elements $\mathop{\mathrm{Tr}}(X^m),\mathop{\mathrm{Tr}}(Y^m) \in \mathsf{V}_{S_N}$ and their images in the Poisson algebra $R(\mathsf{V}_{S_N})_{\mathrm{red}} \subset \mathcal{O}_{M}(M)$. Under the isomorphism $\mathcal{O}_{M}(M) \simeq {\mathbb C}[{\mathbb C}^{2N}]^{S_N}$, the elements $\mathop{\mathrm{Tr}}(X^m)$ and $\mathop{\mathrm{Tr}}(Y^m)$ lie in ${\mathbb C}[x_1, \dots, x_N]^{S_N}$ and ${\mathbb C}[y_1, \dots, y_N]^{S_N}$, respectively. In fact, the sets $\{\mathop{\mathrm{Tr}}(X^m) \,|\, m = 1, \dots, N \}$ and $\{\mathop{\mathrm{Tr}}(Y^m) \,|\, m = 1, \dots, N \}$ generate ${\mathbb C}[x_1, \dots, x_N]^{S_N}$ and ${\mathbb C}[y_1, \dots, y_N]^{S_N}$, respectively, by fundamental properties of symmetric polynomials. The following lemma is also a well-known fact for diagonal invariant algebras. **Lemma 29** ([@Wallach93], Theorem 2.1). *As a Poisson algebra, ${\mathbb C}[{\mathbb C}^{2N}]^{S_N}$ is generated by the subalgebras ${\mathbb C}[x_1, \dots, x_N]^{S_N}$ and ${\mathbb C}[y_1, \dots, y_N]^{S_N}$.* With this result, we conclude that the $C_2$-Poisson algebra $R(\mathsf{V}_{S_N})_{\mathrm{red}}$ includes ${\mathbb C}[{\mathbb C}^{2N}]^{S_N} = \mathcal{O}_M(M)$. Then, we obtain the following theorem as a consequence of the above arguments. Recall that the Hilbert scheme $M$ is a resolution of singularities of $M_0 \simeq {\mathbb C}^{2N} / S_N$. **Theorem 30**. *The associated variety of the vertex superalgebra $\mathsf{V}_{S_N}$ coincides with the symplectic quotient variety ${\mathbb C}^{2N} / S_N$. In particular, $\mathsf{V}_{S_N}$ is quasi-lisse.* We also recall that ${\mathbb C}^{2N}/S_N\cong\mathcal{M}_{S_N}\times T^*{\mathbb C}$ as symplectic varieties, where $\mathcal{M}_{S_N}$ is the canonical symplectic singularity associated with the complex reflection group $S_N$ [@Bea00] (cf. ). ## Conformal Structure {#sec:conformal} We now equip the sheaf $\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}$ with a natural conformal structure of central charge $c=-3N^2$ that is inherited via the BRST construction. This also makes $\mathsf{V}_{S_N}$ into a vertex operator superalgebra of CFT-type of that central charge. A comment on notation: for an open subset $\widetilde{U}\subset \mathfrak{X}$, a section $f$ in $\widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U}) = \mathcal{O}_{\mathfrak{X}}(\widetilde{U}) \otimes \Lambda(\psi_i, \phi_i \,|\, i=1, \dots, N)$ determines a section $\iota(f) \in \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\widetilde{U})$ using the natural map $\iota$ defined in [\[eq:lift-map\]](#eq:lift-map){reference-type="eqref" reference="eq:lift-map"}. For the remainder of the text, we shall regard vectors and matrices with entries in $\widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U})$ as ones with entries belonging to the $\hbar$-adic vertex superalgebra $\widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\widetilde{U})$. Moreover, we identify sections in $\widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\widetilde{U})$ with ones in the $\hbar$-adic vertex subalgebra $\widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\widetilde{U}) \otimes \mathbf{1}\subset \widetilde{C}_{\hbar\mathrm{VA}}(\widetilde{U}) = \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\widetilde{U}) \mathop{\mathrm{\widehat{\otimes}}}C\ell_{\hbar}(T^* \mathfrak{g})$ of the BRST cochain complex $\widetilde{C}_{\hbar\mathrm{VA}}(\widetilde{U})$. For example, the matrix product $\phi X \gamma$ means $\sum_{i,j=1}^{N} \iota(\phi_i x_{ij} \gamma_{j}) = \sum_{i,j=1}^{N} x_{ij (-1)} \gamma_{j (-1)} \phi_{i (-1)} \otimes \mathbf{1}\in \widetilde{C}_{\hbar\mathrm{VA}}(\mathfrak{X})$. First, we endow the sheaf $\widetilde{C}_{\hbar\mathrm{VA}} = \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar} \mathop{\mathrm{\widehat{\otimes}}}C\ell_{\hbar}(T^* \mathfrak{g})$ of $\hbar$-adic vertex superalgebras with a conformal structure of central charge $c=-3N^2$ for which the free-field generators have weights $$\begin{aligned} \operatorname{wt}(x_{ij})&=\operatorname{wt}(y_{ij})=\operatorname{wt}(\gamma_i)=\operatorname{wt}(\beta_i)=\operatorname{wt}(\psi_i)=\operatorname{wt}(\phi_i)=1/2,\\ \operatorname{wt}(\Psi_{ij})&=0,\qquad\operatorname{wt}(\Phi_{ij})=1\end{aligned}$$ for $i,j=1,\dots,N$. That is, we consider the global section $$\label{eq:free-field-conformal} T\coloneqq \frac{\hbar}{2}\bigl(\mathop{\mathrm{Tr}}(\partial X Y) - \mathop{\mathrm{Tr}}(X \partial Y) + \beta \partial \gamma - \partial \beta \gamma + \partial \phi \psi - \phi \partial \psi\bigr) + \hbar\mathop{\mathrm{Tr}}(\partial \Psi {}^{t} \Phi)$$ in $\widetilde{C}_{\hbar\mathrm{VA}}(\mathfrak{X})$ satisfying the operator product expansions $$\begin{aligned} T(z) T(w) &\sim \frac{-3N^2 \hbar^4 / 2}{(z-w)^4} + \frac{2 \hbar^2}{(z-w)^2} T(w) + \frac{\hbar^2}{z-w} \partial T(w),\\ T(z) A(w) &\sim \frac{\operatorname{wt}(A)\hbar^2}{(z-w)^2} A(w) + \frac{\hbar^2}{z-w} \partial A(w)\end{aligned}$$ for $A=x_{ij},y_{ij},\gamma_i,\beta_i,\psi_i,\phi_i,\Psi_{ij},\Phi_{ij}$. We observe that $\widetilde{E}_{ij (0)} T = \Phi_{ij (0)} T = 0$ for all $i, j = 1, \dots, N$ so that, by definition, $T\in C_{\hbar\mathrm{VA}}(\mathfrak{X})$. This means that $T$ also defines a conformal structure on the subalgebra $C_{\hbar\mathrm{VA}}(\mathfrak{X})$ of $\widetilde{C}_{\hbar\mathrm{VA}}(\mathfrak{X})$. **Remark 31**. We recall the homomorphisms $\widetilde{\mu}_\mathrm{ch}\colon V^{-2N}(\mathfrak{g})_{\hbar}\longrightarrow\widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar}(\mathfrak{X})$ and $J\colon V^{2N}(\mathfrak{g})_{\hbar}\longrightarrow C\ell_{\hbar}(T^* \mathfrak{g})$ of $\hbar$-adic vertex superalgebras from . Both affine vertex algebras can be equipped with the standard Sugawara conformal vector, which we call $T_{\mathfrak{g}}$ and $\widetilde{T}_{\mathfrak{g}}$, respectively. Then a straightforward computation shows that the conformal structure on $\widetilde{C}_{\hbar\mathrm{VA}} = \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar} \mathop{\mathrm{\widehat{\otimes}}}C\ell_{\hbar}(T^* \mathfrak{g})$ defined by $T$ is compatible with the conformal structures defined by $\widetilde{\mu}_\mathrm{ch}(T_{\mathfrak{g}})$ and $J(\widetilde{T}_{\mathfrak{g}})$ on the images of $\widetilde{\mu}_\mathrm{ch}$ and $J$, respectively, i.e. $\widetilde{\mu}_\mathrm{ch}(T_{\mathfrak{g}})$ and $J(\widetilde{T}_{\mathfrak{g}})$ both have $T_{(1)}$-weights $2$ and satisfy $T_{(2)}\widetilde{\mu}_\mathrm{ch}(T_{\mathfrak{g}})=T_{(2)}J(\widetilde{T}_{\mathfrak{g}})=0$. The following property is crucial in order to endow also the BRST cohomology sheaf $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}$ with a conformal structure inherited from $T$. **Proposition 32**. *The global section $T$ satisfies $T\in\mathop{\mathrm{Ker}}d_{\hbar\mathrm{VA}}$, i.e. it is a cocycle in the cochain complex $(\widetilde{C}_{\hbar\mathrm{VA}}(\mathfrak{X}),d_{\hbar\mathrm{VA}})$, and in the subcomplex $(C_{\hbar\mathrm{VA}}(\mathfrak{X}),d_{\hbar\mathrm{VA}})$.* This will follow from below, where we show that $T$ coincides with a certain vector $T_{\mathcal{N}=4}+ T_{\beta\gamma}+ T_{\sf{SF}}\in \mathop{\mathrm{Ker}}d_{\hbar\mathrm{VA}}$ modulo $\operatorname{Im}d_{\hbar\mathrm{VA}}$. The proposition implies that $T$ induces a conformal structure of central charge $c=-3N^2$ on the cohomology associated with the complex $(\widetilde{C}_{\hbar\mathrm{VA}}(\mathfrak{X}),d_{\hbar\mathrm{VA}})$. The same is true for the (relative) cohomology $\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}(M)$ associated with the subcomplex $(C_{\hbar\mathrm{VA}}(\mathfrak{X}), d_{\hbar\mathrm{VA}})$. **Remark 33**. A direct calculation reveals that any conformal vector of $C_{\hbar\mathrm{VA}}(\mathfrak{X})$ lying in the kernel of $d_{\hbar\mathrm{VA}}$ must be of form $$\begin{aligned} T_{k_1, k_2} &= k_1 \mathrm{Tr}(\partial X Y) + (k_1-1) \mathop{\mathrm{Tr}}(X \partial Y)\\ &\quad+ (k_2-1) (\partial \beta \gamma - \partial \phi \psi) + k_2 (\beta \partial \gamma - \phi \partial \psi) + \mathop{\mathrm{Tr}}(\partial \Psi {}^t \Phi)\end{aligned}$$ for some $k_1,k_2\in{\mathbb C}$. Moreover, $T_{k_1, k_2} - T_{k_1, 1/2} \in \operatorname{Im}d_{\hbar\mathrm{VA}}$ for all $k_1$, $k_2$, and hence we obtain a one-parameter family $T_{k_1, 1/2}$, $k_1 \in {\mathbb C}$, of conformal vectors of $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$. The conformal vector $T = T_{1/2, 1/2}$ is the unique choice that gives the same conformal structure as one discussed in [@BMR19] (see below). By definition of $C_{\hbar\mathrm{VA}}$, the $\hbar$-adic vertex superalgebra $C_{\hbar\mathrm{VA}}(\mathfrak{X})$ is generated by fields whose weights are in $\frac{1}{2}{\mathbb Z}_{>0}$. This implies that $C_{\hbar\mathrm{VA}}(\mathfrak{X})$ is an $\hbar$-adic vertex operator superalgebra of CFT-type. This also holds for $\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}(M)$: **Proposition 34**. *The vector $T\bmod\operatorname{Im}d_{\hbar\mathrm{VA}}$ is a conformal vector of the $\hbar$-adic vertex operator superalgebra $\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}(M)$ of CFT-type of central charge $c=-3N^2$.* In particular, the global sections satisfy a $\frac{1}{2} {\mathbb Z}_{\ge 0}$-graded decomposition into weight spaces for $T_{(1)}$, $$\widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M) = \prod_{n \in \frac{1}{2}{\mathbb Z}_{\ge 0}} \widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M)_n, \quad \widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M)_n = \{ a \in \widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M) \,|\, T_{(1)} a = n a \},$$ with $\widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M)_0 = {\mathbb C}[[\hbar]] \mathbf{1}$. Finally, the reduction of $\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}(M)$ to $\mathsf{V}_{S_N}$ endows the latter with the structure of a vertex operator superalgebra: **Corollary 35**. *$\mathsf{V}_{S_N}=[\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}(M)]^{{\mathbb C}^\times}$ is a vertex operator superalgebra of CFT-type of central charge $c=-3N^2$.* Indeed, the following $\frac{1}{2} {\mathbb Z}_{\ge 0}$-graded decomposition into finite-dimensional weight spaces for $T_{(1)}$ holds: $$\mathsf{V}_{S_N} = \bigoplus_{n \in \frac{1}{2}{\mathbb Z}_{\ge 0}} \mathsf{V}_{S_N, n}, \qquad \mathsf{V}_{S_N, n} = \{ a \in \mathsf{V}_{S_N} \,|\, T_{(1)} a = n a \},$$ with $\mathsf{V}_{S_N, 0} = {\mathbb C}\mathbf{1}$. We point out that the vertex operator superalgebra $\mathsf{V}_{S_N}$ does not have "correct statistics", i.e. it is not the case that states of integral $T_{(1)}$-weight are exactly those with even parity. ## Small $\mathcal{N}=4$ Superconformal Algebra {#sec:N4-SCA} In the following, we assume that $N\ge2$ and show that the global sections $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ contain a quotient of the small $\mathcal{N}=4$ superconformal algebra $\operatorname{Vir}_{\mathcal{N}=4,\hbar}^{c_{S_N}}$ of central charge $c_{S_N}=-3(N^2-1)$. We also show that the conformal structure of the superconformal algebra is essentially the one described in the previous section, inherited from the BRST reduction. We define the global sections $$\begin{aligned} \label{eq:gen-N4-SCA} J^{+} &= \frac{1}{2}\Bigl(\mathop{\mathrm{Tr}}(X^2) - \frac{1}{N} \mathop{\mathrm{Tr}}(X)^2\Bigr), & J^{-} &= \frac{1}{2}\Bigl(\mathop{\mathrm{Tr}}(Y^2) - \frac{1}{N} \mathop{\mathrm{Tr}}(Y)^2\Bigr), \\ J^{0} &= \mathop{\mathrm{Tr}}(X Y) - \frac{1}{N} \mathop{\mathrm{Tr}}(X) \mathop{\mathrm{Tr}}(Y), \nonumber\\ G^{+} &= \phi X \gamma - \frac{1}{N} \mathop{\mathrm{Tr}}(X) \phi \gamma, & G^{-} &= \phi Y \gamma - \frac{1}{N} \mathop{\mathrm{Tr}}(Y) \phi \gamma, \nonumber\\ \widetilde{G}^{+} &= - \beta X \psi + \frac{1}{N} \mathop{\mathrm{Tr}}(X) \beta \psi, & \widetilde{G}^{-} &= - \beta Y \psi + \frac{1}{N} \mathop{\mathrm{Tr}}(Y) \beta \psi, \nonumber \\ T_{\mathcal{N}=4}&= \frac{1}{\hbar} G^{+}_{(0)}\widetilde{G}^{-} - \frac{\hbar}{2} \partial J^0 \nonumber\end{aligned}$$ in $\widetilde{C}_{\hbar\mathrm{VA}}(\mathfrak{X})$. **Remark 36**. By direct computation of the operator product expansions, one can show that $T_{\mathcal{N}=4}$ takes the form $$\begin{aligned} T_{\mathcal{N}=4}&= \phi X Y \psi - \beta Y X \gamma + \beta \gamma \phi \psi - \frac{1}{N} \phi \gamma \beta \psi \\ &\quad + \frac{\hbar}{2}\Bigl(\mathop{\mathrm{Tr}}(\partial X Y) - \mathop{\mathrm{Tr}}(X \partial Y) - \frac{1}{N} \mathop{\mathrm{Tr}}(\partial X) \mathop{\mathrm{Tr}}(Y) + \frac{1}{N} \mathop{\mathrm{Tr}}(X) \mathop{\mathrm{Tr}}(\partial Y) \Bigr) \\ &\quad + \hbar \Bigl(N - \frac{1}{N}\Bigr) (\beta \partial \gamma - \partial \phi \psi).\end{aligned}$$ Once again, we show that the above global sections are closed, i.e. in $\mathop{\mathrm{Ker}}d_{\hbar\mathrm{VA}}$. **Proposition 37**. *The global sections [\[eq:gen-N4-SCA\]](#eq:gen-N4-SCA){reference-type="eqref" reference="eq:gen-N4-SCA"} are cocycles in $\widetilde{C}_{\hbar\mathrm{VA}}(\mathfrak{X})$, and in the subcomplex $C_{\hbar\mathrm{VA}}(\mathfrak{X})$.* *Proof.* It suffices to show that $\mu_\mathrm{ch}(E_{ij})_{(n)}A = 0$ for $i,j=1,\dots,N$ and $n \in {\mathbb Z}_{\ge 0}$ and for any section $A$ defined in [\[eq:gen-N4-SCA\]](#eq:gen-N4-SCA){reference-type="eqref" reference="eq:gen-N4-SCA"}. Then, since these sections do not contain any ghost fields $\Psi_{ij}$ or $\Phi_{ij}$, this proves both assertions. Recall that the Wick formula is realised by bidifferential operators in the variables $x_{ij (-n)},y_{ij (-n)},\dots$, as explained in . For $m \in {\mathbb Z}_{\ge 0}$, we obtain by direct calculation $$\begin{aligned} &\mu_\mathrm{ch}(E_{ij})(z) \mathop{\mathrm{Tr}}(X^m)(w) \\ &\sim \sum_{p=1}^{N} (x_{ip} y_{pj} - x_{pj} y_{ip})(z) \mathop{\mathrm{Tr}}(X^m)(w) \nonumber\\ &\sim \frac{\hbar}{z-w} \sum_{p=1}^{N} \Bigl(\frac{\partial}{\partial y_{pj}} (x_{ip} y_{pj})\Bigr)(z) \Bigl(\frac{\partial}{\partial x_{jp}} \mathop{\mathrm{Tr}}(X^m)\Bigr)(w) \nonumber\\ &\quad - \frac{\hbar}{z-w} \sum_{p=1}^{N} \Bigl(\frac{\partial}{\partial y_{ip}} (x_{pj} y_{ip})\Bigr)(z) \Bigl(\frac{\partial}{\partial x_{pi}} \mathop{\mathrm{Tr}}(X^m)\Bigr)(w) \nonumber\\ & \sim \frac{\hbar}{z-w} \sum_{p=1}^{N} \sum_{k=0}^{m-1} \left\{ \left(x_{ip} \mathop{\mathrm{Tr}}(X^{k} E_{jp} X^{m-k-1})\right)(w) - \left(x_{pj} \mathop{\mathrm{Tr}}(X^{k} E_{pi} X^{m-k-1})\right)(w) \right\} \nonumber\\ &= \frac{\hbar}{z-w} \sum_{p, q=1}^{N} \sum_{k=0}^{m-1} \left\{ \left(x_{ip} (X^{k})_{qj} (X^{m-k-1})_{pq} \right)(w) - \left(x_{pj} (X^{k})_{qp} (X^{m-k-1})_{iq} \right)(w) \right\} \nonumber\\ &= \frac{m \hbar}{z-w} \bigl\{ \left(X^m\right)_{ij}(w) - \left(X^m \right)_{ij}(w) \bigr\} = 0. \nonumber\end{aligned}$$ This implies that $\mu_\mathrm{ch}(E_{ij})_{(n)} J^{+} = 0$ for any $i,j=1,\dots,N$ and $n \in {\mathbb Z}_{\ge 0}$. Similarly to the above calculation, we also obtain $\mu_\mathrm{ch}(E_{ij})_{(n)} \mathop{\mathrm{Tr}}(Y^m) = 0$, and thus $\mu_\mathrm{ch}(E_{ij})_{(n)} J^{-} = 0$ for any $i$, $j$ and $n$. Moreover, for $m \in {\mathbb Z}_{\ge 0}$, $$\begin{aligned} &\mu_\mathrm{ch}(E_{ij})(z) (\phi Y^m \gamma)(w) \\ & \sim \frac{\hbar}{z-w} \biggl\{\sum_{p=1}^{N} \Bigl(- y_{pj} \frac{\partial \phi Y^m \gamma}{\partial y_{pi}} + y_{ip} \frac{\partial \phi Y^m \gamma}{\partial y_{jp}} \Bigr)(w) \\ &\quad + \Bigl(\gamma_i \frac{\partial \phi Y^m \gamma}{\partial \gamma_{j}} \Bigr)(w) - \Bigl(\phi_j \frac{\partial \phi Y^m \gamma}{\partial \phi_{i}} \Bigr)(w) \biggr\} \\ & = \frac{\hbar}{z-w} \biggl\{ \sum_{p=1}^{N} \sum_{k=0}^{m-1} \left(- y_{pj} \phi Y^{k} E_{pi} Y^{m-k-1} \gamma + y_{ip} \phi Y^{k} E_{jp} Y^{m-k-1} \gamma \right)(w) \\ &\quad + \left(\gamma_i \phi Y^m \mathbf{e}_j \right)(w) - \left(\phi_j {}^t \mathbf{e}_i Y^m \gamma \right)(w) \biggr\} \\ & = \frac{\hbar}{z-w} \biggl\{ \sum_{k=0}^{m-1} \left(- (\phi Y^{k+1})_{j} (Y^{m-k-1} \gamma)_i + (\phi Y^{k})_{j} (Y^{m-k} \gamma)_{i} \right)(w) \\ &\quad + \left(\gamma_i (\phi Y^m)_j \right)(w) - \left(\phi_j (Y^m \gamma)_i \right)(w) \biggr\} = 0.\end{aligned}$$ Therefore, $\mu_\mathrm{ch}(E_{ij})_{(n)} G^{-} = 0$ for any $i,j=1,\dots,N$ and $n \in {\mathbb Z}_{\ge 0}$. By similar calculations, one can check that $\mu_\mathrm{ch}(E_{ij})_{(n)} A = 0$ for $A = J^0, G^{+}, \widetilde{G}^{+}, \widetilde{G}^{-}$, as desired. ◻ By , the global sections $J^{+}, J^{0}, J^{-}, \dots \in C_{\hbar\mathrm{VA}}(\mathfrak{X})$ define elements in the cohomology $H^{0}(C_{\hbar\mathrm{VA}}(\mathfrak{X}), d_{\hbar\mathrm{VA}}) = \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ denoted by the same symbols. The operator product expansions between these elements can also be obtained as a direct consequence of the Wick formula. First, note that $\mathop{\mathrm{Tr}}(X)$ and $\mathop{\mathrm{Tr}}(Y) \in \widetilde{C}_{\hbar\mathrm{VA}}(\mathfrak{X})$ are also cocycles and define elements in the cohomology. It is easy to see that $\mathop{\mathrm{Tr}}(X)(z) A(w) \sim \mathop{\mathrm{Tr}}(Y)(z) A(w) \sim 0$ for any element $A = J^{+}, J^{0}, \dots$ defined in [\[eq:gen-N4-SCA\]](#eq:gen-N4-SCA){reference-type="eqref" reference="eq:gen-N4-SCA"}. Indeed, for example, $$\begin{aligned} \label{eq:9} &\mathop{\mathrm{Tr}}(Y)(z) \mathop{\mathrm{Tr}}(X^m)(w) \sim \frac{\hbar}{z-w} \sum_{i=1}^{N} \mathop{\mathrm{Tr}}(E_{ii})(w) \Bigl(\frac{\partial \mathop{\mathrm{Tr}}(X^m)}{\partial x_{ii}}\Bigr)(w) \\ &= \frac{\hbar}{z-w} \sum_{i=1}^{N} \sum_{k=0}^{m-1} \mathop{\mathrm{Tr}}(X^k E_{ii} X^{m-k-1})(w) = \frac{m \hbar}{z-w} \mathop{\mathrm{Tr}}(X^{m-1})(w) \nonumber\end{aligned}$$ for $m \in {\mathbb Z}_{\ge 0}$, which implies $\mathop{\mathrm{Tr}}(Y)(z) J^{+}(w) \sim 0$. We shall later identify $\mathop{\mathrm{Tr}}(X)$ and $\mathop{\mathrm{Tr}}(Y)$ as generators of an $\hbar$-adic $\beta\gamma$-system in $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$. The following six lemmata aim at studying the operator product expansions among the global sections [\[eq:gen-N4-SCA\]](#eq:gen-N4-SCA){reference-type="eqref" reference="eq:gen-N4-SCA"}. We shall see that they define a small $\mathcal{N}=4$ superconformal algebra. **Lemma 38**. *The elements $J^{+}$, $J^{0}$ and $J^{-}$ generate an $\hbar$-adic affine vertex algebra associated with $\mathfrak{sl}_2$ of level $k=-(N^2-1)/2$, i.e. the operator product expansions $$\begin{aligned} J^0(z) J^{\pm}(w) &\sim \frac{\pm 2 \hbar}{z-w} J^{\pm}(w), \quad J^{0}(z) J^{0}(w) \sim \frac{-(N^2-1) \hbar^2}{(z-w)^2}, \\ J^{+}(z) J^{-}(w) &\sim \frac{(N^2-1)\hbar^2 /2}{(z-w)^2} + \frac{- \hbar}{z-w} J^{0}(w)\end{aligned}$$ hold, and other operator product expansions between the generators are trivial.* More precisely, what we mean is that $J^{+}$, $J^{0}$ and $J^{-}$ form some quotient of the universal $\hbar$-adic affine vertex algebra associated with $\mathfrak{sl}_2$ of level $k=-(N^2-1)/2$. *Proof.* It follows from the Wick formula that $$\begin{aligned} &\mathop{\mathrm{Tr}}(XY)(z) \mathop{\mathrm{Tr}}(X^m)(w) \sim \frac{\hbar}{z-w} \sum_{i,j=1}^{N} \Bigl(\frac{\partial \mathop{\mathrm{Tr}}(XY)}{\partial y_{ij}}\Bigr)(w) \Bigl(\frac{\partial \mathop{\mathrm{Tr}}(X^m)}{\partial x_{ji}}\Bigr)(w) \\ &\sim \frac{\hbar}{z-w} \sum_{i,j=1}^{N} \sum_{k=0}^{m-1} \mathop{\mathrm{Tr}}(X E_{ij})(w) \mathop{\mathrm{Tr}}(X^k E_{ji} X^{m-k-1})(w) \\ &\sim \frac{\hbar}{z-w} \sum_{i,j,p=1}^{N} \sum_{k=0}^{m-1} x_{ji}(w) \left( (X^k)_{pj} (X^{m-k-1})_{ip} \right)(w) = \frac{m \hbar}{z-w} \mathop{\mathrm{Tr}}(X^m)(w)\end{aligned}$$ for $m \in {\mathbb Z}_{\ge 0}$. Thus, $J^{0}(z) J^{+}(w) \sim 2 J^{+}(w) \hbar/(z-w)$. Moreover, $$\begin{aligned} &\mathop{\mathrm{Tr}}(X^2)(z) \mathop{\mathrm{Tr}}(Y^2)(w) \\ &\sim \frac{\hbar^2/2}{(z-w)^2} \sum_{i,j,k,l=1}^{N} \Bigl(\frac{\partial^2 \mathop{\mathrm{Tr}}(X^2)}{\partial x_{ij} \partial x_{kl}} \Bigr)(w) \Bigl(\frac{\partial^2 \mathop{\mathrm{Tr}}(Y^2)}{\partial y_{ji} \partial y_{lk}} \Bigr)(w) \\ &\quad - \frac{\hbar}{z-w} \sum_{i,j=1}^{N} \Bigl(\frac{\partial \mathop{\mathrm{Tr}}(X^2)}{\partial x_{ij}} \Bigr)(w) \Bigl(\frac{\partial \mathop{\mathrm{Tr}}(Y^2)}{\partial y_{ji}} \Bigr)(w) \\ &= \frac{\hbar^2/2}{(z-w)^2} \sum_{i,j,k,l=1}^{N} \mathop{\mathrm{Tr}}(E_{ij}E_{kl} + E_{kl} E_{ij})(w) \mathop{\mathrm{Tr}}(E_{ji}E_{lk} + E_{lk} E_{ji})(w) \\ &\quad - \frac{\hbar}{z-w} \sum_{i,j=1}^{N} \mathop{\mathrm{Tr}}(E_{ij} X + X E_{ij})(w) \mathop{\mathrm{Tr}}(E_{ji} Y + Y E_{ji})(w) \\ &= \frac{2N^2 \hbar^2}{(z-w)^2} - \frac{4 \hbar}{z-w} \mathop{\mathrm{Tr}}(XY)(w),\end{aligned}$$ and $\mathop{\mathrm{Tr}}(X^2)(z) \mathop{\mathrm{Tr}}(Y)^2(w) \sim 2 N \hbar^2/(z-w)^2 - 4 \hbar (\mathop{\mathrm{Tr}}(X) \mathop{\mathrm{Tr}}(Y))(w)/(z-w)$ by [\[eq:9\]](#eq:9){reference-type="eqref" reference="eq:9"}. It follows that $J^{+}(z) J^{-}(w) \sim ((N^2-1) \hbar^2 / 2) / (z-w)^2 - \hbar J^0(w) / (z-w)$. The other operator product expansions are obtained analogously. ◻ By , there is an action of the $\hbar$-adic affine Lie algebra $\widehat{\mathfrak{sl}}_2$ of level $k=-(N^2-1)/2$ on $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$. A natural question is how the affine Lie algebra $\widehat{\mathfrak{sl}}_2$ acts on the other elements $G^{+}$, $G^{-}$, $\widetilde{G}^{+}$ and $\widetilde{G}^{-}$. By direct verification, similar to the above lemmata, we obtain: **Lemma 39**. *For $A = G,\widetilde{G}$, the elements $\{A^{+}, A^{-}\}$ generate a certain quotient of the Weyl module of the affine Lie algebra $\widehat{\mathfrak{sl}}_2$ of level $k=-(N^2-1)/2$ associated with the two-dimensional irreducible representation of $\mathfrak{sl}_2$, i.e. the following operator product expansions hold: $$J^{0}(z) A^{\pm}(w) \sim \frac{\pm \hbar}{z-w} A^{\pm}(w), \;\; J^{\pm}(z) A^{\mp}(w) \sim \frac{\mp \hbar}{z-w} A^{\pm}(w), \;\; J^{\pm}(z) A^{\pm}(w) \sim 0.$$* **Lemma 40**. *For $A = G,\widetilde{G}$, we have $A^{+}(z) A^{-}(w) \sim A^{\pm}(z) A^{\pm}(w) \sim 0$.* *Proof.* It is clear that $A^{\pm}(z) A^{\pm}(w) \sim 0$. It remains to show $G^{+}(z) G^{-}(w) \sim 0$ and $\widetilde{G}^{+}(z) \widetilde{G}^{-}(w) \sim 0$. To conclude the former, we observe that $$\begin{aligned} (\phi X \gamma)(z) (\phi Y \gamma)(w) &\sim \frac{-\hbar}{z-w} \sum_{i,j=1}^{N} \Bigl(\frac{\partial \phi X \gamma}{\partial x_{ij}} \Bigr)(w) \Bigl(\frac{\partial \phi Y \gamma}{\partial y_{ji}} \Bigr)(w) \\ &= \frac{-\hbar}{z-w} \sum_{i,j=1}^{N} \left(\phi E_{ij} \gamma \right)(w) \left(\phi E_{ji} \gamma \right)(w) \\ &= \frac{-\hbar}{z-w} \sum_{i,j=1}^{N} (\phi_i \gamma_j \phi_j \gamma_i)(w) = \frac{-\hbar}{z-w} \sum_{i,j=1}^{N} (\phi \gamma)^2(w) = 0\end{aligned}$$ since $\phi \gamma$ is an odd element. Also, $\mathop{\mathrm{Tr}}(X)(z) (\phi Y \gamma)(w) \sim - (\phi \gamma)(w)/(z-w)$, and thus we obtain $G^{+}(z) G^{-}(w) \sim 0$. The operator product expansion $\widetilde{G}^{+}(z) \widetilde{G}^{-}(w) \sim 0$ can be obtained in a similar way. ◻ The remaining operator product expansions between the elements defined in [\[eq:gen-N4-SCA\]](#eq:gen-N4-SCA){reference-type="eqref" reference="eq:gen-N4-SCA"} are of the form $G^{a}(z) \widetilde{G}^{b}(w)$ for $a,b = +,-$. To compute them, we first need the following lemma. **Lemma 41**. *For $m \in {\mathbb Z}_{\ge 0}$, the global sections $\beta X^m \gamma - \phi X^m \psi$ and $\beta Y^m \gamma - \phi Y^m \psi$ in $\widetilde{C}_{\hbar\mathrm{VA}}(\mathfrak{X})$ are coboundaries, i.e. in $\operatorname{Im}d_{\hbar\mathrm{VA}}$.* *Proof.* Consider the element $\mathop{\mathrm{Tr}}(X^m \Phi)$, where $\Phi = (\Phi_{ij})_{i,j=1}^{N}$. Then, using the Wick formula, we see that $d_{\hbar\mathrm{VA}} \mathop{\mathrm{Tr}}(X^m \Phi) = (1/\hbar) Q_{(0)} \mathop{\mathrm{Tr}}(X^m \Phi) = \beta X^m \gamma - \phi X^m \psi$. ◻ **Lemma 42**. *The element $T_{\mathcal{N}=4}$ defines a conformal vector of central charge $c_{S_N}=-3(N^2-1)$ of the subalgebra of $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ generated by the elements $J^{\pm}$, $J^{0}$, $G^{\pm}$, $\widetilde{G}^{\pm}$, $T_{\mathcal{N}=4}$ in [\[eq:gen-N4-SCA\]](#eq:gen-N4-SCA){reference-type="eqref" reference="eq:gen-N4-SCA"}, i.e. the following operator product expansions hold: $$\begin{aligned} T_{\mathcal{N}=4}(z) T_{\mathcal{N}=4}(w) &\sim \frac{-3 \hbar^4 (N^2-1) / 2}{(z-w)^4} + \frac{2 \hbar^2}{(z-w)^2} T_{\mathcal{N}=4}(w) + \frac{\hbar^2}{z-w} \partial T_{\mathcal{N}=4}(w), \\ T_{\mathcal{N}=4}(z) A(w) &\sim \frac{\hbar^2}{(z-w)^2} A(w) + \frac{\hbar^2}{z-w} \partial A(w),\\ T_{\mathcal{N}=4}(z) B(w) &\sim \frac{3 \hbar^2 / 2}{(z-w)^2} B(w) + \frac{\hbar^2}{z-w} \partial B(w)\end{aligned}$$ for $A = J^{\pm},J^{0}$ and $B = G^{\pm},\widetilde{G}^{\pm}$.* *Proof.* By direct calculation, we verify $G^{-}_{(0)} \widetilde{G}^{+} - (\hbar^2/2) \partial J^0 = - \hbar T_{\mathcal{N}=4}$. The operator product expansions for $T_{\mathcal{N}=4}$ then follow from , below and Borcherds' identity $(a_{(0)} b)_{(n)} = a_{(0)} b_{(n)} - (-1)^{p(a) p(b)} b_{(n)} a_{(0)}$. ◻ **Lemma 43**. *For $a,b = +,-$, the following operator product expansions hold: $$G^{a}(z) \widetilde{G}^{b}(w) \sim \frac{-(N^2-1) \epsilon^{ab} \hbar^3}{(z-w)^3} + \frac{2 J^{ab}(w) \hbar^2}{(z-w)^2} + \frac{\epsilon^{ab} T_{\mathcal{N}=4}(w) \hbar + \partial J^{ab}(w) \hbar^2}{z-w},$$ where $\epsilon^{+-} = 1$, $\epsilon^{-+} = -1$, $\epsilon^{\pm \pm} = 0$ and $J^{+-} = J^{-+} = J^0/2$, $J^{\pm \pm} = J^{\pm}$.* *Proof.* Again by the Wick formula, we compute $$\begin{aligned} &(\phi X \gamma)(z) (\beta X \psi)(w) \sim \frac{- \hbar^2}{(z-w)^2} \sum_{i,j=1}^{N} \Bigl(\frac{\partial^2 \phi X \gamma}{\partial \phi_i \partial \gamma_j}\Bigr)(w) \Bigl(\frac{\partial^2 \beta X \psi}{\partial \psi_i \partial \beta_j}\Bigr)(w) \\ &\quad + \frac{- \hbar^2}{z-w} \sum_{i,j=1}^{N} \Bigl(\partial_w \frac{\partial^2 \phi X \gamma}{\partial \phi_i \partial \gamma_j}\Bigr)(w) \Bigl(\frac{\partial^2 \beta X \psi}{\partial \psi_i \partial \beta_j}\Bigr)(w) \\ &\quad + \frac{\hbar}{z-w} \sum_{i=1}^{N} \Bigl(\frac{\partial \phi X \gamma}{\partial \phi_i}\Bigr)(w) \Bigl(\frac{\partial \beta X \psi}{\partial \psi_i}\Bigr)(w) + \frac{-\hbar}{z-w} \sum_{i=1}^{N} \Bigl(\frac{\partial \phi X \gamma}{\partial \gamma_i}\Bigr)(w) \Bigl(\frac{\partial \beta X \psi}{\partial \beta_i}\Bigr)(w) \\ &= \frac{- \hbar^2}{(z-w)^2} \sum_{i,j=1}^{N} (x_{ij} x_{ji})(w) + \frac{- \hbar^2}{z-w} \sum_{i,j=1}^{N} (\partial x_{ij} x_{ji})(w) \\ &\quad + \frac{\hbar}{z-w} \sum_{i=1}^{N} \left\{ (X \gamma)_i(w) (\beta X)_i(w) - (\phi X)_{i}(w) (X \psi)_i(w) \right\} \\ &= \frac{- \hbar^2}{(z-w)^2} \mathop{\mathrm{Tr}}(X^2)(w) + \frac{- \hbar^2}{z-w} \frac{1}{2} \partial_w \mathop{\mathrm{Tr}}(X^2)(w) + \frac{\hbar}{z-w}(\beta X^2 \gamma - \phi X^2 \psi)(w) \\ &\equiv \frac{- \hbar^2}{(z-w)^2} \mathop{\mathrm{Tr}}(X^2)(w) + \frac{- \hbar^2}{z-w} \frac{1}{2} \partial_w \mathop{\mathrm{Tr}}(X^2)(w),\end{aligned}$$ where we use for the equivalence modulo $\operatorname{Im}d_{\hbar\mathrm{VA}}$. Similarly, $$(\phi X \gamma)(z) (\mathop{\mathrm{Tr}}(X) \beta \psi)(w) \sim \frac{- \hbar^2}{(z-w)^2} \mathop{\mathrm{Tr}}(X)^2(w) + \frac{- \hbar^2}{z-w} \frac{1}{2} \partial_w \mathop{\mathrm{Tr}}(X)^2(w)$$ modulo $\operatorname{Im}d_{\hbar\mathrm{VA}}$, and hence we obtain the desired operator product expansion $G^{+}(z) \widetilde{G}^{+}(w)$ of the lemma. The other operator product expansions can be verified in the same way. ◻ As a conclusion of Lemmata [Lemma 38](#lemma:sl2-OPE){reference-type="ref" reference="lemma:sl2-OPE"}--[Lemma 43](#lemma:doublet-OPE){reference-type="ref" reference="lemma:doublet-OPE"}, we obtain: **Proposition 44**. *For $N\ge2$, the elements [\[eq:gen-N4-SCA\]](#eq:gen-N4-SCA){reference-type="eqref" reference="eq:gen-N4-SCA"} define a homomorphism of $\hbar$-adic vertex superalgebras $$\operatorname{Vir}_{\mathcal{N}=4,\hbar}^{c_{S_N}}\longrightarrow \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M).$$* We denote the image of the above homomorphism by $V_{\mathcal{N}=4,\hbar}$. It is some quotient of the universal $\hbar$-adic small $\mathcal{N}=4$ superconformal algebra $\operatorname{Vir}_{\mathcal{N}=4,\hbar}^{c_{S_N}}$ of central charge $c_{S_N}=-3(N^2-1)$. The analogous statement holds for the vertex operator superalgebra $\mathsf{V}_{S_N}$ with a vertex algebra homomorphism $\operatorname{Vir}_{\mathcal{N}=4}^{c_{S_N}}\longrightarrow \mathsf{V}_{S_N}$, whose image we denote $V_{\mathcal{N}=4}$. Finally, we study further cocycles in $C_{\hbar\mathrm{VA}}(\mathfrak{X})$, corresponding to non-zero elements in $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$, and show that together with $V_{\mathcal{N}=4,\hbar}$ they generate a conformal subalgebra of the $\hbar$-adic vertex operator superalgebra $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$. **Lemma 45**. *The elements $\mathop{\mathrm{Tr}}(X)/\sqrt{N}$ and $\mathop{\mathrm{Tr}}(Y)/\sqrt{N} \in \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ form an $\hbar$-adic $\beta\gamma$-system $\mathcal{D}^\mathrm{ch}(T^*{\mathbb C}^1)_{\hbar}$.* *Proof.* This follows directly from $$\begin{aligned} \mathop{\mathrm{Tr}}(X)(z)\mathop{\mathrm{Tr}}(Y)(w) &\sim - N \hbar/ (z-w),\\ \mathop{\mathrm{Tr}}(X)(z)\mathop{\mathrm{Tr}}(X)(w) &\sim \mathop{\mathrm{Tr}}(Y)(z)\mathop{\mathrm{Tr}}(Y)(w) \sim 0.\qedhere\end{aligned}$$ ◻ **Lemma 46**. *The elements $\Lambda_1 \coloneqq \phi \gamma / \sqrt{N}$ and $\Lambda_2 \coloneqq \beta \psi / \sqrt{N} \in \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ generate an $\hbar$-adic symplectic fermion vertex superalgebra $\mathsf{SF}_{\hbar}$.* *Proof.* The assertion follows from the operator product expansions $$\begin{aligned} \Lambda_1(z)\Lambda_2(w) &\sim -\hbar^2/ (z-w)^2,\\ \Lambda_1(z)\Lambda_1(w) &\sim \Lambda_2(z)\Lambda_2(w) \sim 0.\qedhere\end{aligned}$$ ◻ Moreover, one can easily verify that the operator product expansions among these three vertex subalgebras are trivial. Thus, $$V_{\mathcal{N}=4,\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathcal{D}^\mathrm{ch}(T^*{\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar}\subset \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$$ as $\hbar$-adic vertex superalgebras. Now, consider the conformal vectors $$\begin{aligned} T_{\beta\gamma}&= \frac{\hbar}{2N}\bigl(\mathop{\mathrm{Tr}}(\partial X) \mathop{\mathrm{Tr}}(Y) - \mathop{\mathrm{Tr}}(X) \mathop{\mathrm{Tr}}(\partial Y) \bigr) \in \mathcal{D}^\mathrm{ch}(T^*{\mathbb C}^1)_{\hbar},\\ T_{\sf{SF}}&= \Lambda_1 \Lambda_2 = \frac{1}{N}( \phi \gamma \beta \psi + \hbar \beta \partial \gamma - \hbar \partial \phi \psi) \in \mathsf{SF}_{\hbar}\end{aligned}$$ of the $\hbar$-adic vertex subalgebras $\mathcal{D}^{\mathrm{ch}}(T^*{\mathbb C}^1)_{\hbar}$ and $\mathsf{SF}_{\hbar}$. They have central charge $c=-1$ and $-2$, respectively. Overall, the tensor product $V_{\mathcal{N}=4,\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathcal{D}^\mathrm{ch}(T^*{\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar}$ inside $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ has a conformal structure of central charge $c=-3(N^2-1) + (-1) + (-2) = -3 N^2$ defined by $T_{\mathcal{N}=4}+ T_{\beta\gamma}+ T_{\sf{SF}}$. We show that it coincides with the natural conformal structure of $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ defined in : **Proposition 47**. *For $N\ge2$, the $\hbar$-adic vertex operator superalgebra $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ of central charge $c=-3N^2$ is a conformal extension of $V_{\mathcal{N}=4,\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathcal{D}^\mathrm{ch}(T^*{\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar}$.* *Proof.* We need to show that for $T$ from [\[eq:free-field-conformal\]](#eq:free-field-conformal){reference-type="eqref" reference="eq:free-field-conformal"}, $$\label{eq:4} T \equiv T_{\mathcal{N}=4}+ T_{\beta\gamma}+ T_{\sf{SF}}$$ modulo $\operatorname{Im}d_{\hbar\mathrm{VA}}$. Indeed, one can easily verify the identity $$\begin{aligned} &T - (T_{\mathcal{N}=4}+ T_{\beta\gamma}+ T_{\sf{SF}}) \\ &= \beta Y X \gamma - \phi X Y \psi - \beta \gamma \phi \psi - \hbar\Bigl(N - \frac{1}{2}\Bigr)\beta \partial \gamma - \frac{\hbar}{2} \partial \beta \gamma \\ &\quad - \frac{\hbar}{2} \phi \partial \psi + \hbar\Bigl(N + \frac{1}{2}\Bigr) \partial \phi \psi + \hbar\mathop{\mathrm{Tr}}(\partial \Psi {}^t \Phi) \\ &= \frac{1}{2} d_{\hbar\mathrm{VA}} \bigl(\mathop{\mathrm{Tr}}(X Y \Phi) + \mathop{\mathrm{Tr}}(Y X \Phi) - \beta \Phi \gamma + \phi \Phi \psi + (\beta\gamma) \mathop{\mathrm{Tr}}(\Phi) \\ & \phantom{=\frac{1}{2} d_{\hbar\mathrm{VA}} \bigl(} - (\phi \psi) \mathop{\mathrm{Tr}}(\Phi) - N \hbar \mathop{\mathrm{Tr}}(\partial \Phi) \bigr) \in \operatorname{Im}d_{\hbar\mathrm{VA}}.\qedhere\end{aligned}$$ ◻ In the next section, we shall actually see that $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ is isomorphic to a tensor product of $\mathcal{D}^\mathrm{ch}(T^*{\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar}$ and some conformal extension of $V_{\mathcal{N}=4,\hbar}$. That is, we can split off the (unimportant) $\beta\gamma$-system and the symplectic fermion completely. As an immediate consequence of the proposition we obtain: **Corollary 48**. *For $N\ge2$, the vertex operator superalgebra $\mathsf{V}_{S_N}$ of central charge $c=-3N^2$ is a conformal extension of $V_{\mathcal{N}=4} \otimes \mathcal{D}^\mathrm{ch}(T^*{\mathbb C}^1) \otimes \mathsf{SF}$.* # Free-Field Realisation {#sec:Wakimoto} In this section, we consider the local sections of the sheaf $\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}$ over the affine open subset $U_{(N)}$ defined in , and thus obtain a free-field realisation of the global sections, and of $\mathsf{V}_{S_N}$, given by the sheaf restriction morphism. This also shows a factorisation of $\mathsf{V}_{S_N}$ that allows us to split off the vertex operator superalgebra $\mathsf{W}_{S_N}$ of central charge $c_{S_N}=-3(N^2-1)$ from $\mathsf{V}_{S_N}$. The latter is the vertex operator superalgebra for the reflection group $S_N$ conjectured by Bonetti, Meneghelli and Rastelli [@BMR19]. We show that $\mathsf{W}_{S_N}$ is a conformal extension of the small $\mathcal{N}=4$ superconformal algebra of central charge $c_{S_N}$. Moreover, $\mathsf{W}_{S_N}$ has the associated variety $\mathcal{M}_{S_N}$, is quasi-lisse and has a free-field realisation in terms of a $\beta\gamma bc$-system of rank $N-1$ that coincides with the one proposed in [@BMR19]. We recall from the affine open subset $\widetilde{U}_{(N)} \subset \mathfrak{X}$ and the sections $[X^N : X^i]_{(N)}$, $[Y : X^i]_{(N)}$ and $\{\psi : X^i \gamma\}_{(N)} \in \widetilde{\mathcal{O}}_{\mathfrak{X}}(\widetilde{U}_{(N)})$ for $i=0,\dots,N-1$. In this section, we omit the subscript ${(N)}$ from these sections and simply write $[X^N : X^i]$, $[Y : X^i]$ and $\{\psi : X^i \gamma\}$, respectively. Via the embedding $\widetilde{\mathcal{O}}_M \hookrightarrow \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar} \subset \widetilde{C}_{\hbar\mathrm{VA}}$, we regard these sections as elements of $\widetilde{C}_{\hbar\mathrm{VA}}(\widetilde{U}_{(N)})$. **Lemma 49**. *The matrix $B_{(N)} = (\gamma, X \gamma, \dots, X^{N-1} \gamma)$ satisfies the following identities for $p,q=1,\dots,N$:* 1. *$(\partial / \partial x_{pq}) B_{(N)} = \sum_{k=0}^{N-2} (X^k \gamma)_q (0, \dots, 0, \mathbf{e}_p, X \mathbf{e}_p, \dots, X^{N-k-1} \mathbf{e}_p)$,* 2. *$(\partial / \partial x_{pq}) B^{-1}_{(N)} = - \sum_{k=0}^{N-2} (X^k \gamma)_q B^{-1}_{(N)} (0, \dots, 0, \mathbf{e}_p, X \mathbf{e}_p, \dots, X^{N-k-1} \mathbf{e}_p) B^{-1}_{(N)}$.* *Proof.* By the Leibniz rule, we obtain $(\partial / \partial x_{pq}) X^m \gamma = \sum_{k=0}^{m-1} X^{m-k-1} E_{pq} X^{k} \gamma = \sum_{k=0}^{m-1} (X^k \gamma)_q X^{m-k-1} \mathbf{e}_p$, which implies (1). The identity (2) follows from (1) and $B_{(N)} (\partial B_{(N)}^{-1} / \partial x_{pq}) + (\partial B_{(N)} / \partial x_{pq}) B^{-1}_{(N)} = 0$. ◻ **Lemma 50**. *The elements $[X^N : X^i]$, $[Y : X^i]$ and $\{\psi : X^i \gamma\} \in \widetilde{C}_{\hbar\mathrm{VA}}(\widetilde{U}_{(N)})$ for $i=0,\dots,N-1$ are cocycles with respect to the coboundary operator $d_{\hbar\mathrm{VA}}$.* *Proof.* Again, it suffices to show that $\mu_\mathrm{ch}(E_{ij})_{(n)} A = 0$ for $A = [X^N : X^k], {[Y : X^k]},\allowbreak \{\psi : X^k \gamma\}$ and $k=0,\dots,N-1$. We consider the operator product expansion of $\mu_\mathrm{ch}(A_{ij})(z) (B_{(N)}^{-1} Y \gamma)(w)$ for the vector $B_{(N)}^{-1} Y \gamma = {}^t([Y : X^{k-1}])_{k=1, \dots, N}$. By the Wick formula, $$\begin{aligned} &\mu_\mathrm{ch}(E_{ij})(z) (B_{(N)}^{-1} Y \gamma)(w) \\ &\sim \frac{- \hbar^2}{(z-w)^2} \sum_{i,j,p=1}^{N} \biggl(\frac{\partial^2 B_{(N)}^{-1} Y \gamma}{\partial x_{jp} \partial y_{pi}} - \frac{\partial^2 B_{(N)}^{-1} Y \gamma}{\partial x_{pi} \partial y_{jp}}\biggr)(w) + \frac{\hbar}{z-w} ( \dots ).\end{aligned}$$ The second term $\hbar ( \dots ) / (z-w)$ comes from the single contraction of the operator product expansion. Note that $[Y : X^k]$ is $G$-invariant for any $k=0,\dots,N-1$, which implies that the second term vanishes. We then note that $$\begin{aligned} \sum_{p=1}^{N} \frac{\partial^2 B_{(N)}^{-1} Y \gamma}{\partial x_{jp} \partial y_{pi}} &= - \sum_{p=1}^{N} \sum_{k=0}^{N-2} (X^k \gamma)_p B^{-1}_{(N)} (0, \dots, 0, \mathbf{e}_j, \dots, X^{N-k-1} \mathbf{e}_j) B^{-1}_{(N)} \gamma_i \mathbf{e}_p \\ &= - \sum_{k=0}^{N-2} \gamma_i B^{-1}_{(N)} (0, \dots, 0, \mathbf{e}_j, \dots, X^{N-k-1} \mathbf{e}_j) B^{-1}_{(N)} X^k \gamma \\ &= - \sum_{k=0}^{N-2} \gamma_i B^{-1} (0, \dots, 0, \mathbf{e}_j, \dots, X^{N-k-1} \mathbf{e}_j) \mathbf{e}_{k+1}.\end{aligned}$$ The first $k+1$ columns of the matrix $(0, \dots, 0, \mathbf{e}_j, \dots, X^{N-k-1} \mathbf{e}_j)$ vanish so that $(0, \dots, 0, \mathbf{e}_j, \dots, X^{N-k-1} \mathbf{e}_j) \mathbf{e}_{k+1} = 0$. Hence, $\mu_\mathrm{ch}(E_{ij})(z) (B^{-1}_{(N)} Y \gamma)(w) \sim 0$. On the other hand, the elements $[X^N : X^i]$ and $\{\psi : X^i \gamma\}$ are $G$-invariant and the operator product expansions of $\mu_\mathrm{ch}(E_{ij})(z) [X^N : X^i](w)$ and $\mu_\mathrm{ch}(E_{ij})(z) \{\psi : X^i \gamma\}(w)$ do not cause multiple contractions. Thus, these elements are cocycles. ◻ Since the elements $[X^N : X^i]$, $[Y : X^i]$, $\{\psi : X^i \gamma\}$ and $\phi X^i \gamma$ for $i=0,\dots,N-1$ are $G$-invariant, they lie in the subcomplex $C_{\hbar\mathrm{VA}}(\widetilde{U}_{(N)})$. By the above lemma, they define elements in $H^0(C_{\hbar\mathrm{VA}}(\widetilde{U}_{(N)}), d_{\hbar\mathrm{VA}}) = \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{(N)})$. In fact, the $\hbar$-adic vertex superalgebra $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{(N)})$ of sections over $U_{(N)}$ is strongly generated by these sections, i.e. $$\begin{aligned} \label{eq:1} \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{(N)}) &= {\mathbb C}[[\hbar]][[X^N : X^i]_{(-n)}, [Y : X^i]_{(-n)} \,|\, \substack{ i = 0, \dots, N-1 \\ n = 1, 2, \dots}] \\ &\quad \mathop{\mathrm{\widehat{\otimes}}}\Lambda_{{\mathbb C}[[\hbar]]}(\{\psi : X^i \gamma\}_{(-n)}, (\phi X^i \gamma)_{(-n)} \,|\, \substack{ i = 0, \dots, N-1 \\ n = 1, 2, \dots}) \nonumber\end{aligned}$$ as ${\mathbb C}[[\hbar]]$-module. We shall show that the $\hbar$-adic vertex superalgebra $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{(N)})$ is isomorphic to an $\hbar$-adic $\beta\gamma bc$-system of rank $N-1$. First, we determine the operator product expansions between the above generators. **Lemma 51**. *For $i=0,\dots,N-1$, the operator product expansions $$\begin{aligned} \label{eq:8} \mathop{\mathrm{Tr}}(X)(z) [Y : X^i](w) &\sim \begin{cases} \displaystyle \frac{\hbar}{z-w}, & i = 0, \\ 0, & i \ne 0, \end{cases} \\ \label{eq:11} \mathop{\mathrm{Tr}}(Y)(z) [Y : X^i](w) &\sim \begin{cases} 0, & i = N-1, \\ \displaystyle - \frac{(i+1) \hbar}{z-w} [Y : X^{i+1}], & i \ne N-1, \end{cases} \\ \nonumber \mathop{\mathrm{Tr}}(Y)(z) [X^N : X^i](w) &\sim \begin{cases} \displaystyle \frac{N \hbar}{z-w}, & i = N-1, \\ \displaystyle - \frac{(i+1) \hbar}{z-w} [X^N : X^{i+1}](w), & i \ne N-1, \end{cases} \\ \nonumber \mathop{\mathrm{Tr}}(Y)(z) \{\psi : X^i \gamma\}(w) &\sim \begin{cases} 0, & i = N-1, \\ \displaystyle - \frac{(i+1) \hbar}{z-w} \{ \psi : X^{i+1} \gamma \}(w), & i \ne N-1, \end{cases} \\ \nonumber \mathop{\mathrm{Tr}}(Y)(z) (\phi X^i \gamma)(w) &\sim \frac{i \hbar}{z-w} (\phi X^{i-1} \gamma)(w)\end{aligned}$$ hold, and other operator product expansions between $\mathop{\mathrm{Tr}}(X)$, $\mathop{\mathrm{Tr}}(Y)$ and the elements in [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"} are trivial.* *Proof.* Since the elements $[X^N : X^i]$, $\{\psi : X^i \gamma\}$ and $\phi X^i \gamma$ do not contain the variables $y_{pq}$, $p,q=1,\dots,N$, they have trivial operator product expansions with $\mathop{\mathrm{Tr}}(X)$. We note that $$\begin{aligned} &\mathop{\mathrm{Tr}}(X)(z) (B^{-1}_{(N)} Y \gamma)(w) \sim \frac{- \hbar}{z-w} \sum_{p=1}^{N}\biggl(\frac{\partial B^{-1}_{(N)} Y \gamma}{\partial y_{pp}} \biggr)(w) \\ &= \frac{- \hbar}{z-w} \sum_{p=1}^{N}\left(B^{-1}_{(N)} E_{pp} \gamma \right)(w)= \frac{- \hbar}{z-w}\left(B^{-1}_{(N)} \gamma\right)(w) = \frac{- \hbar}{z-w} \mathbf{e}_1,\end{aligned}$$ which implies [\[eq:8\]](#eq:8){reference-type="eqref" reference="eq:8"}. To see [\[eq:11\]](#eq:11){reference-type="eqref" reference="eq:11"}, we use  (2) and $$\begin{aligned} &\mathop{\mathrm{Tr}}(Y)(z) (B^{-1}_{(N)} Y \gamma)(w) \sim \frac{\hbar}{z-w} \sum_{p=1}^{N} \biggl(\frac{\partial B^{-1}_{(N)}}{\partial x_{pp}} Y \gamma \biggr)(w) \\ &\sim \frac{- \hbar}{z-w} B^{-1}_{(N)} (0, \gamma, 2 X \gamma, \dots, (N-1) X^{N-2} \gamma) B^{-1}_{(N)} Y \gamma \\ &= \frac{- \hbar}{z-w} (0, \mathbf{e}_1, 2 \mathbf{e}_2, \dots, (N-1) \mathbf{e}_{N-1})\, {}^t \!\left([Y : X^{i-1}]\right)_{i=1, \dots, N} \\ &= \frac{- \hbar}{z-w} {}^t\!\left(j [Y : X^j]\right)_{j=1, \dots, N}.\end{aligned}$$ The other operator product expansions can be obtained in a similar way. ◻ **Lemma 52**. *For $i,j=0,\dots,N-1$, there are the operator product expansions $$\begin{aligned} (z) [X^N : X^j](w) &\sim \begin{cases} \displaystyle \frac{\hbar}{z-w}, & i=N-j-1, \\ \displaystyle \frac{-\hbar}{z-w} [X^N : X^{i+j+1}](w), & i=0, \dots, N-j-2, \\ 0, & i = N-j, \dots, N-1, \end{cases} \\ [Y : X^i](z) [Y : X^j](w) &\sim 0, \\ [Y : X^i](z) \{\psi : X^j \gamma\}(w) &\sim \begin{cases} \displaystyle \frac{-\hbar}{z-w} \{\psi : X^{i+j+1} \gamma\}(w), & i=0, \dots, N-j-2, \\ 0, & i=N-j-1, \dots, N-1, \end{cases} \\ [Y : X^i](z) (\phi X^{j} \gamma)(w) &\sim \begin{cases} \displaystyle \frac{\hbar}{z-w} (\phi X^{j-i-1} \gamma)(w), & i=0, \dots, j-1, \\ 0, & i=j, \dots, N-1. \end{cases}\end{aligned}$$ * *Proof.* By , for $p=1,\dots,N$, $$\begin{aligned} \sum_{q=1}^{N} \gamma_{q} \frac{\partial}{\partial x_{qp}} B^{-1}_{(N)} &= \sum_{k=0}^{N-2} (X^k \gamma)_{p} B^{-1}_{(N)} (0, \dots, 0, \gamma, X \gamma, \dots, X^{N-k-1} \gamma) B^{-1}_{(N)}, \\ \sum_{q=1}^{N} \gamma_{q} \frac{\partial}{\partial x_{qp}} X^N \gamma &= \sum_{k=0}^{N-1} (X^k \gamma)_p X^{N-k-1} \gamma.\end{aligned}$$ This implies that $$\begin{aligned} &(Y \gamma)_{p}(z) (B^{-1}_{(N)} X^N \gamma)(w) \sim \frac{\hbar}{z-w} \sum_{q=1}^{N} \Bigl(\gamma_q \frac{\partial}{\partial x_{qp}} B^{-1}_{(N)} X^N \gamma\Bigr)(w) \\ &= \frac{\hbar}{z-w} \biggl\{\sum_{k=0}^{N-1} \bigl((X^k \gamma)_p B^{-1}_{(N)} X^{N-k-1} \gamma\bigr)(w) \\ &\quad - \sum_{k=0}^{N-k-2} \bigl((X^k \gamma)_{p} B^{-1}_{(N)} (0, \dots, 0, \gamma, X \gamma, \dots, X^{N-k-1} \gamma) B^{-1}_{(N)} X^{N} \gamma\bigr)(w)\biggr\} \\ &= \frac{\hbar}{z-w} {\Big.}^t\!\Bigl( (X^{N-j} \gamma)_{p}(w) - \sum_{k=0}^{N-2} (X^k \gamma)_p [X^N : X^{k+j}] \Bigr)_{j=1, \dots, N}\end{aligned}$$ since $B^{-1}_{(N)} X^{j-1} \gamma = \mathbf{e}_{j}$ for $j=1,\dots,N$. Multiplying by $\sum_{p=1}^{N} (B^{-1}_{(N)})_{ip}$ on both sides, we obtain the first identity of the lemma. The other operator product expansions can be obtained in a similar fashion. ◻ **Lemma 53**. *For $i,j=0,\dots,N-1$, there is the operator product expansion $(\phi X^i \gamma)(z) \{\psi : X^j \gamma\}(w) \sim \delta_{ij} \hbar / (z-w)$.* *Proof.* By the Wick formula, $$\begin{aligned} (\phi X^i \gamma)(z) \{\psi : X^j \gamma\}(w) &\sim \frac{\hbar}{z-w} \sum_{p,q=1}^{N} \left((X^i \gamma)_p \frac{\partial}{\partial \psi_p} (B^{-1}_{(N)})_{j+1,q} \psi_q \right)(w) \\ &= \frac{\hbar}{z-w} \sum_{p=1}^{N} \left((B^{-1}_{(N)})_{j+1,p} (X^i \gamma)_p \right)(w) = \frac{\hbar}{z-w} \delta_{ij}.\qedhere\end{aligned}$$ ◻ Obviously, the operator product expansions between $[X^N : X^i]$ and $\{\psi : X^j \gamma\}$, $\phi X^j \gamma$ are trivial for any $i,j=0,\dots,N-1$. Note that all operator product expansions in Lemmata [Lemma 51](#lemma:OPE-Tr-local){reference-type="ref" reference="lemma:OPE-Tr-local"}--[Lemma 53](#lemma:10){reference-type="ref" reference="lemma:10"} have simple poles, and they have a certain triangular property, e.g., $[Y : X^{i}](z) [X^N : X^j](w) \sim 0$ and $[Y : X^{i}](z) \{\psi : X^j \gamma\}(w) \sim 0$ unless $i < N-j-1$. This enables us to diagonalise the operator product expansions, as stated in the following proposition. For $m=1,\dots,N-1$, we define $$\begin{aligned} \widetilde{b}_m &= \sum_{k=1}^{m} \left(\frac{-1}{N}\right)^{m-k} \binom{m}{k} \mathop{\mathrm{Tr}}(X)^{m-k} \Bigl( \phi X^k \gamma - \frac{1}{N} \mathop{\mathrm{Tr}}(X^k) \phi \gamma \Bigr), \\ \widetilde{c}_m &= \sum_{k=m}^{N-1} \left(\frac{1}{N}\right)^{m-k} \binom{k}{m} \mathop{\mathrm{Tr}}(X)^{k-m} \{\psi : X^k \gamma\}, \\ \widetilde{\beta}_{m} &= \frac{1}{m+1} \biggl(\, \sum_{k=2}^{m+1} \left(\frac{-1}{N}\right)^{m-k+1} \binom{m+1}{k} \mathop{\mathrm{Tr}}(X)^{m-k+1} \mathop{\mathrm{Tr}}(X^k) \\ &\quad + m \left(\frac{-1}{N}\right)^{m} \mathop{\mathrm{Tr}}(X)^{m+1} \biggr), \\ \widetilde{\gamma}_{m} &= \sum_{k=m}^{N-1} \left(\frac{1}{N}\right)^{k-m} \binom{k}{m} \mathop{\mathrm{Tr}}(X)^{k-m} [Y : X^k] - \sum_{j=1}^{N-m-2} b_j c_{m+j+1}\\ &\quad - \sum_{j=1}^{N-m-3} \frac{j+1}{N} \widetilde{\beta}_j \phi\gamma \widetilde{c}_{m+j+2} - \frac{N-m-1}{N} \phi\gamma \widetilde{c}_{m+1}.\end{aligned}$$ **Proposition 54**. *The $\widetilde{\beta}_i$, $\widetilde{\gamma}_i$ are bosonic and the $\widetilde{b}_i$, $\widetilde{c}_i$ fermionic elements in $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{(N)})$ for $i=1,\dots,N-1$. They satisfy the operator product expansions $$\widetilde{\beta}_i(z) \widetilde{\gamma}_j(w) \sim - \delta_{ij} \hbar / (z-w)\quad\text{and}\quad\widetilde{b}_i(z) \widetilde{c}_j(w) \sim \delta_{ij} \hbar / (z-w),$$ and all other operator product expansions between them vanish, i.e. they form an $\hbar$-adic $\beta\gamma bc$-system of rank $N-1$. Moreover, they have trivial operator product expansions with $\mathop{\mathrm{Tr}}(X)$, $\mathop{\mathrm{Tr}}(Y)$ and with $\phi \gamma$, $\beta \psi$.* By the above proposition, there is the following isomorphism of $\hbar$-adic vertex superalgebras $$\label{eq:factor-h-Wakimoto} \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{(N)}) \simeq \mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^{N-1})_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}C\ell_{\hbar}(T^* {\mathbb C}^{N-1}) \mathop{\mathrm{\widehat{\otimes}}}\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar},$$ where $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^{N-1})_{\hbar}$ is the $\hbar$-adic $\beta\gamma$-system of rank $N-1$ generated by $\widetilde{\beta}_{i}$ and $\widetilde{\gamma}_i$ for $i=1,\dots,N-1$, $C\ell_{\hbar}(T^* {\mathbb C}^{N-1})$ is the $\hbar$-adic $bc$-system of rank $N-1$ generated by $\widetilde{b}_{i}$ and $\widetilde{c}_i$ for $i=1,\dots,N-1$, $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^1)_{\hbar}$ is another $\hbar$-adic $\beta\gamma$-system generated by $\mathop{\mathrm{Tr}}(X)/\sqrt{N}$ and $\mathop{\mathrm{Tr}}(Y)/\sqrt{N}$ and $\mathsf{SF}_{\hbar}$ is the $\hbar$-adic symplectic fermion vertex superalgebra generated by $\phi \gamma/\sqrt{N}$ and $\beta \psi/\sqrt{N}$. That is, the sheaf restriction morphism (see the proof of for the injectivity) $$\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M) \hookrightarrow \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{(N)})$$ defines a free-field realisation. Recall the conformal structure on $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ of central charge $c=-3N^2$. We extend it to the free-field vertex superalgebra $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{(N)})$ via the above restriction morphism. Then, the generators have weights $$\operatorname{wt}(\widetilde{\beta}_m)=\frac{m+1}{2},\;\operatorname{wt}(\widetilde{\gamma}_m)=\frac{-m+1}{2},\; \operatorname{wt}(\widetilde{b}_m)=\frac{m+2}{2},\;\operatorname{wt}(\widetilde{c}_m)=-\frac{m}{2}$$ for $m=1,\dots,N-1$. This shows that the $\beta\gamma$-system generated by $\widetilde{\beta}_m$, $\widetilde{\gamma}_m$, which generally affords a one-parameter family of conformal structures, has a conformal structure of central charge $c=-1+3m^2$, while similarly the $bc$-system generated by $\widetilde{b}_m$, $\widetilde{c}_m$ has a conformal structure of central charge $c=-2-6m-3m^2$ for $m=1,\dots,N-1$. Moreover, we already saw that the $\beta\gamma$-system generated by $\mathop{\mathrm{Tr}}(X)/\sqrt{N}$, $\mathop{\mathrm{Tr}}(Y)/\sqrt{N}$ and the symplectic fermion vertex superalgebra have central charge $c=-1$ and $c=-2$, respectively, extending the above formulae for the central charges to $m=0$. We also obtain an analogous free-field realisation of the vertex operator superalgebra $\mathsf{V}_{S_N}=[\widetilde{\mathcal{D}}^\mathrm{ch}_{M,\hbar}(M)]^{{\mathbb C}^\times}$: **Theorem 55**. *The restriction morphism $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M) \hookrightarrow \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{(N)})$ induces a free-field realisation of $\mathsf{V}_{S_N}$, i.e. an embedding $$\mathsf{V}_{S_N}\hookrightarrow\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^{N-1}) \otimes C\ell(T^* {\mathbb C}^{N-1}) \otimes \mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^1) \otimes \mathsf{SF}$$ of vertex operator superalgebras.* The conformal structure on the right-hand side is the one with central charge $c=\sum_{m=0}^{N-1}((-1+3m^2)+(-2-6m-3m^2))=-3N^2$ described in the preceding paragraph. *Proof.* First, we show that the restriction $\mathop{\mathrm{res}}^{M}_{U_{(N)}}\colon \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M) \longrightarrow \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{(N)})$ is injective. Let $f \in \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ be a global section such that $\mathop{\mathrm{res}}^{M}_{U_{(N)}}(f) = 0$. We assume for the sake of contradiction that $f \not\in \hbar\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$. By the isomorphism $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar} / \hbar \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar} \simeq \widetilde{\mathcal{O}}_{J_\infty M}$ of , its $0$-th symbol $f_0 \coloneqq \sigma_0(f)$ is a non-zero global section of $\widetilde{\mathcal{O}}_{J_\infty M}$. Consider the grading $\deg x_{ij (-n)} = \deg y_{ij (-n)} = \deg \beta_{i (-n)} = \deg \gamma_{i (-n)} = 0$ and $\deg \psi_{i (-n)} = \deg \phi_{i (-n)} = 1$ for all $i$, $j$ and $n$. This grading induces a filtration $\{F_r \widetilde{\mathcal{O}}_{J_\infty M} \}_{r=0}^{\infty}$ of $\widetilde{\mathcal{O}}_{J_\infty M}$ defined by $F_r \widetilde{\mathcal{O}}_{J_\infty M} = \{ f \in \widetilde{\mathcal{O}}_{J_\infty M} \;|\; \deg f \le r \}$. Let $\mathop{\mathrm{Gr}}_r \widetilde{\mathcal{O}}_{J_\infty M}$ be the quotient sheaf $F_{r} \widetilde{\mathcal{O}}_{J_\infty M} / F_{r+1} \widetilde{\mathcal{O}}_{J_\infty M}$ for $r\in{\mathbb Z}_{\ge0}$. Then $\mathop{\mathrm{Gr}}_r \widetilde{\mathcal{O}}_{J_\infty M}$ is a locally free $\mathcal{O}_{M}$-module of infinite rank. The short exact sequence $0 \rightarrow F_{r+1} \widetilde{\mathcal{O}}_{J_\infty M} \rightarrow F_{r} \widetilde{\mathcal{O}}_{J_\infty M} \rightarrow \mathop{\mathrm{Gr}}_r \widetilde{\mathcal{O}}_{J_\infty M} \rightarrow 0$ induces the commutative diagram $$\begin{tikzcd} 0 \arrow[r] & F_{r+1} \widetilde{\mathcal{O}}_{J_\infty M}(M) \arrow[r] \arrow[d, "\mathop{\mathrm{res}}^{M}_{U_{(N)}}" swap] & F_{r} \widetilde{\mathcal{O}}_{J_\infty M}(M) \arrow[r, "\tilde{\sigma}_0"] \arrow[d, "\mathop{\mathrm{res}}^{M}_{U_{(N)}}"] & \mathop{\mathrm{Gr}}_r \widetilde{\mathcal{O}}_{J_\infty M}(M) \arrow[d, "\mathop{\mathrm{res}}^{M}_{U_{(N)}}"]\\ 0 \arrow[r] & F_{r+1} \widetilde{\mathcal{O}}_{J_\infty M}(U_{(N)}) \arrow[r] & F_{r} \widetilde{\mathcal{O}}_{J_\infty M}(U_{(N)}) \arrow[r, "\tilde{\sigma}_0"] & \mathop{\mathrm{Gr}}_r \widetilde{\mathcal{O}}_{J_\infty M}(U_{(N)}) \end{tikzcd}$$ where each row is exact. Let $g \in \mathop{\mathrm{Gr}}_r \widetilde{\mathcal{O}}_{J_\infty M}$ be a global section and assume $\mathop{\mathrm{res}}^M_{U_{(N)}}(g) = 0$. Since $\mathop{\mathrm{Gr}}_r \widetilde{\mathcal{O}}_{J_\infty M}$ is a locally free $\mathcal{O}_{M}$-module, the zero locus $\{p \in M \;|\; g(p) = 0 \}$ is a Zariski closed subset of $M$. By assumption, $g(p) = 0$ for any point $p$ of the Zariski open subset $U_{(N)}$, and hence $g(p) = 0$ for any point $p$ of the closure $\overline{U}_{(N)} = M$. Then, $g = 0$ and thus the restriction $\mathop{\mathrm{res}}^M_{U_{(N)}}\colon\mathop{\mathrm{Gr}}_r \widetilde{\mathcal{O}}_{J_\infty M}(M) \longrightarrow \mathop{\mathrm{Gr}}_r \widetilde{\mathcal{O}}_{J_\infty M}(U_{(N)})$ is injective. Since $f_0 \in \widetilde{\mathcal{O}}_{J_\infty M}(M)$ is a non-zero global section, there exists a unique $r\in{\mathbb Z}_{\ge0}$ such that $f_0 \in F_r \widetilde{\mathcal{O}}_{J_\infty M}(M) \setminus F_{r+1} \widetilde{\mathcal{O}}_{J_\infty M}(M)$. By assumption, $\mathop{\mathrm{res}}^{M}_{U_{(N)}}(\tilde{\sigma}_0(f_0)) = \tilde{\sigma}_0(\mathop{\mathrm{res}}^{M}_{U_{(N)}}(f_0)) = 0$, and thus we obtain $\tilde{\sigma}_0(f_0) = 0$ by the above injectivity. This implies that $f_0 \in F_{r+1} \widetilde{\mathcal{O}}_{J_\infty M}(M)$, contradicting the definition of $r$. By contradiction, we conclude that $\mathop{\mathrm{res}}^{M}_{U_{(N)}}(f) = 0$ implies $f = 0$. The image of the ${\mathbb C}[\hbar]$-submodule $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)_\mathrm{fin}$ with respect to the restriction homomorphism is clearly included in the ${\mathbb C}[\hbar]$-submodule of all superpolynomials in $\widetilde{\beta}_{i (-n)}$, $\widetilde{\gamma}_{i (-n)}$, $\widetilde{b}_{i (-n)}$, $\widetilde{c}_{i (-n)}$, $\mathop{\mathrm{Tr}}(X)_{(-n)}$, $\mathop{\mathrm{Tr}}(Y)_{(-n)}$, $(\phi \gamma)_{(-n)}$, $(\beta \psi)_{(-n)}$ for $i=1,\dots,N-1$ and $n \in {\mathbb Z}_{>0}$. Taking the quotient by the ideal generated by $(\hbar - 1)$, these elements strongly generate the vertex superalgebra $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^{N-1}) \otimes C\ell(T^* {\mathbb C}^{N-1}) \otimes \mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^1) \otimes \mathsf{SF}$, and hence there is a homomorphism of vertex superalgebras from $\mathsf{V}_{S_N}$ to it. ◻ **Remark 56**. One can verify that the free-field realisation of the small $\mathcal{N}=4$ superconformal algebra of level $k=-(N^2-1)/2$ obtained in coincides for $N=2$ with the one given by Adamović in [@Adamovic16] and for $N=3$ with the one obtained by Bonetti, Meneghelli and Rastelli in [@BMR19], Section 4.2.2. **Remark 57**. The free-field realisation of is an analogue of the Wakimoto realisation for affine vertex algebras studied in [@Wakimoto86; @FF90; @Frenkel05]. Finally, as an application of this free-field realisation, we discuss a certain factorisation of the $\hbar$-adic vertex superalgebra $\widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M)$. Recall the $\hbar$-adic vertex operator subalgebra $\mathcal{D}^{\mathrm{ch}}(T^*{\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar}$ of $\widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M)$ generated by $\mathop{\mathrm{Tr}}(X)$, $\mathop{\mathrm{Tr}}(Y)$, $\Lambda_1 = \phi \gamma/\sqrt{N}$ and $\Lambda_2 = \beta\psi/\sqrt{N}$. Let $$\widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M)^{\perp}\coloneqq\mathop{\mathrm{Com}}_{\widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M)}(\mathcal{D}^{\mathrm{ch}}(T^* {\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar})$$ be the commutant vertex operator superalgebra of $\mathcal{D}^{\mathrm{ch}}(T^* {\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar}$ in $\widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M)$. We show that the double commutant is again $\mathcal{D}^{\mathrm{ch}}(T^* {\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar}$, and not some extension of it. In other words: **Proposition 58**. *The following factorisation of the $\hbar$-adic vertex operator superalgebra $\widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M)$ of global sections holds: $$\widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M) \cong \widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M)^{\perp} \mathop{\mathrm{\widehat{\otimes}}}\mathcal{D}^{\mathrm{ch}}(T^* {\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar}.$$* *Proof.* Recall the isomorphism [\[eq:factor-h-Wakimoto\]](#eq:factor-h-Wakimoto){reference-type="eqref" reference="eq:factor-h-Wakimoto"}. Consider any element of the form $\sum_{i} a_i \otimes b_i$ of $\widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M) \subset \widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(U_{(N)})$, where $a_i \in \mathcal{D}^{\mathrm{ch}}(T^* {\mathbb C}^{N-1})_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}C\ell_{\hbar}(T^* {\mathbb C}^{N-1})$ and $b_i \in \mathcal{D}^{\mathrm{ch}}(T^* {\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar}$. By induction on the conformal weight with respect to the conformal vector $T_{\beta\gamma}+ T_{\sf{SF}}\in \mathcal{D}^{\mathrm{ch}}(T^* {\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar}$, the simplicity of $\mathcal{D}^{\mathrm{ch}}(T^* {\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar}$ implies that $a_i \otimes \mathbf{1}$ lies in $\widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M) / (\hbar^m)$ for any $i$ and any positive integer $m$. Since $\widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M)^{\perp} = \widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M) \cap (\mathcal{D}^{\mathrm{ch}}(T^* {\mathbb C}^{N-1}))_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}C\ell_{\hbar}(T^* {\mathbb C}^{N-1}))$, this implies $a_i \in \widetilde{\mathcal{D}}^{\mathrm{ch}}_{M, \hbar}(M)^{\perp}$ for any $i$. ◻ Analogously, let $$\mathsf{W}_{S_N}\coloneqq\mathop{\mathrm{Com}}_{\mathsf{V}_{S_N}}(\mathcal{D}^{\mathrm{ch}}(T^*{\mathbb C}^1) \otimes \mathsf{SF})$$ be the commutant of $\mathcal{D}^{\mathrm{ch}}(T^*{\mathbb C}^1) \otimes \mathsf{SF}$ inside $\mathsf{V}_{S_N}$. Then implies the following factorisation of the vertex operator superalgebra $$\mathsf{V}_{S_N} = \mathsf{W}_{S_N} \otimes \mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^1) \otimes \mathsf{SF},$$ which allows us to split off the uninteresting tensor factor $\mathcal{D}^{\mathrm{ch}}(T^*{\mathbb C}^1) \otimes \mathsf{SF}$. As a consequence of , , and , we conclude the following theorem, which summarises the properties of $\mathsf{W}_{S_N}$. **Theorem 59**. *For $N\ge2$, the vertex subalgebra $\mathsf{W}_{S_N}\subset \mathsf{V}_{S_N}$ is a vertex operator superalgebra of CFT-type of central charge $c_{S_N}=-3(N^2-1)$ satisfying:* 1. *$\mathsf{W}_{S_N}$ is a conformal extension of some quotient of the universal small $\mathcal{N}=4$ superconformal vertex superalgebra $\operatorname{Vir}_{\mathcal{N}=4}^{c_{S_N}}$.* 2. *The associated variety of $\mathsf{W}_{S_N}$ is $\mathcal{M}_{S_N}=(V_{S_N} \oplus V_{S_N}^*) / S_N$, where $V_{S_N}={\mathbb C}^{N-1}$ is the reflection representation of the symmetric group $S_N$. In particular, $\mathsf{W}_{S_N}$ is quasi-lisse.* 3. *There is a free-field realisation $\mathsf{W}_{S_N}\hookrightarrow\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^{N-1}) \otimes C\ell(T^* {\mathbb C}^{N-1})$ in terms of $\mathop{\mathrm{rk}}(S_N)=N-1$ many copies of the $\beta\gamma b c$-system with central charges $c=-3(2p_i-1)$, where $p_i=2,\dots,N$ are the degrees of the fundamental invariants of $S_N$.* $\mathsf{W}_{S_N}$ is the vertex operator superalgebra for the reflection group $S_N$ conjectured by Bonetti, Meneghelli and Rastelli [@BMR19], i.e. the vertex superalgebra corresponding to the four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory $\operatorname{SYM}_{\mathfrak{sl}_N}$ with gauge group $\mathop{\mathrm{SL}}_N$ via the 4D/2D duality [@BeeLemLie15]. We remark that in the case of $N=1$, $\mathsf{W}_{S_N}$ reduces to the trivial vertex operator algebra ${\mathbb C}\mathbf{1}$. **Remark 60**. The small $\mathcal{N}=4$ superconformal vertex superalgebra $\operatorname{Vir}_{\mathcal{N}=4}^{c_{S_N}}$ has the group of outer automorphisms isomorphic to $\operatorname{SL}_2({\mathbb C})$, which acts on the odd generators of conformal weight $3/2$ by $G^+ \mapsto a G^+ - b \widetilde{G}^+$, $\widetilde{G}^+ \mapsto - c G^+ + d \widetilde{G}^+$, $G^- \mapsto a G^- - b \widetilde{G}^-$ and $\widetilde{G}^- \mapsto - c G^- + d \widetilde{G}^-$ for $a,b,c,d \in {\mathbb C}$ with $ad - bc = 1$ and trivially on the other generators $J^{\pm}$, $J^0$ and $T_{\mathcal{N}=4}$ (see [@CLW22], Section 3). It can be hypothesised that this action lifts to an action on the vertex operator superalgebras $\mathsf{W}_{S_N}$ and $\mathsf{V}_{S_N}$ given by $\phi P \gamma \mapsto a (\phi P \gamma) + b (\beta P \psi)$, $\beta P \psi \mapsto c (\phi P \gamma) + d (\beta P \psi)$ and $\mathop{\mathrm{Tr}}P \mapsto \mathop{\mathrm{Tr}}P$ for $a,b,c,d \in {\mathbb C}$ with $ad - bc = 1$ and $P = P(X, Y) \in {\mathbb C}\langle X, Y\rangle$. # Characters and Schur Indices {#sec:chars} In the following, we determine the supercharacters of the vertex operator superalgebras $\mathsf{V}_{S_N}$ and $\mathsf{W}_{S_N}$, which coincide with the Schur indices of the four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theories with gauge groups $\mathop{\mathrm{GL}}_N$ and $\mathop{\mathrm{SL}}_N$, respectively. Using the Euler-Poincaré principle and the exact integration formulae in [@PP22] we show that these characters are quasimodular forms (see, e.g., [@Zag08] for an introduction). Further quasimodularity results for vertex operator algebras in the context of the 4D/2D duality are obtained, e.g., in [@Mil22]. Recall the BRST cohomology sheaf on the Hilbert scheme $M$, whose only non-vanishing degree component is $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}=\mathcal{H}^{\infty/2+0}_{\hbar\mathrm{VA}}(\mathfrak{g}, \widetilde{\mathcal{D}}^\mathrm{ch}_{\mathfrak{X}, \hbar})$ of ghost degree $0$. The vertex superalgebra $\mathsf{V}_{S_N}=[\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)]^{{\mathbb C}^\times}$ is obtained from the global sections of this sheaf by taking the invariants under the torus action. The conformal vector $T$ from makes $\mathsf{V}_{S_N}$ a vertex operator superalgebra of central charge $c=-3N^2$ that is $\frac{1}{2}{\mathbb Z}_{\ge0}$-graded by weights, of CFT-type and whose (super)character is $$\operatorname{(s)ch}_{\mathsf{V}_{S_N}}(q)=\mathop{\mathrm{Tr}}_{\mathsf{V}_{S_N}}(\pm1)^Pq^{T_{(1)}-c/24}=q^{N^2/8}(1+\mathcal{O}(q^{1/2})),$$ where $P$ is the usual parity operator with eigenvalues in ${\mathbb Z}/2{\mathbb Z}$. In the following, we determine this supercharacter and study its modular properties. The global sections $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$ are given by the cohomology of $d_{\hbar\mathrm{VA}} = (1/\hbar) Q_{(0)}$ on the $\hbar$-adic vertex superalgebra $C_{\hbar\mathrm{VA}}(\mathfrak{X})\subset\widetilde{C}_{\hbar\mathrm{VA}}(\mathfrak{X})$. The latter is the $\hbar$-adic free-field vertex superalgebra $\widetilde{C}_{\hbar\mathrm{VA}}(\mathfrak{X}) = \mathcal{D}^\mathrm{ch}(T^* V)_{\hbar} \otimes C\ell_{\hbar}(T^* {\mathbb C}^N) \otimes C\ell_{\hbar}(T^* \mathfrak{g})$. For the torus invariants this entails that $$\mathsf{V}_{S_N}=H^\bullet(C, Q_{(0)})=H^0(C, Q_{(0)})$$ is the cohomology of $Q_{(0)}$ on the vertex superalgebra $$C \coloneqq \{ c \in \widetilde{C}\,|\, \widetilde{E}_{ij (0)} c = \Phi_{ij (0)} c = 0 \text{ for all } i, j = 1, \dots, N \} \subset \widetilde{C}$$ given as the intersection of the kernels of $\widetilde{E}_{ij (0)}$ and $\Phi_{ij (0)}$ for $i,j=1,\dots,N$ on the free-field vertex operator superalgebra $$\widetilde{C}\coloneqq \mathcal{D}^\mathrm{ch}(T^* V) \otimes C\ell(T^* {\mathbb C}^N) \otimes C\ell(T^* \mathfrak{g}),$$ where $\mathcal{D}^\mathrm{ch}(T^* V) = {\mathbb C}[x_{ij (-n)}, y_{ij (-n)}, \gamma_{i (-n)}, \beta_{i (-n)} \,|\, \substack{i,j=1, \dots, N \\ n = 1, 2, \dots}]$ is the $\beta\gamma$-system associated with the symplectic vector space $T^* V$, $V = \mathop{\mathrm{End}}({\mathbb C}^N) \oplus {\mathbb C}^N$, with generators $x_{ij}$, $y_{ij}$, $\gamma_i$ and $\beta_i$, $C\ell(T^* {\mathbb C}^N) = \Lambda_{{\mathbb C}}(\psi_{i (-n)}, \phi_{i (-n)} \,|\, \substack{i=1, \dots, N \\ n = 1, 2, \dots})$ is the $bc$-system with generators $\psi_i$ and $\phi_i$ and $C\ell(T^* \mathfrak{g}) = \Lambda_{\mathbb C}(\Psi_{ij (-n)}, \Phi_{ij (-n)} \,|\, \substack{i,j=1, \dots, N \\ n = 1, 2, \dots})$ is the (ghost) $bc$-system generated by $\Psi_{ij}$ and $\Phi_{ij}$. The ${\mathbb Z}$-grading by ghost degree on $\widetilde{C}$ is defined via $\deg(\Psi_{ij (-n)}) = 1$ and $\deg(\Phi_{ij (-n)}) = -1$. For convenience we also introduce the intermediate kernel $$C \subset \bar{C} \coloneqq \{ c \in \widetilde{C}\,|\, \Phi_{ij (0)} c = 0 \text{ for all } i, j = 1, \dots, N \} \subset \widetilde{C}.$$ Taking the kernels of the $\Phi_{ij (0)}$ in $\widetilde{C}$ simply amounts to removing all underived $\Psi$-fields from the tensor factor $C\ell(T^* \mathfrak{g})$, i.e. $$\bar{C}=\mathcal{D}^\mathrm{ch}(T^* V) \otimes C\ell(T^* {\mathbb C}^N)\otimes\Lambda_{\mathbb C}(\Psi_{ij (-n-1)}, \Phi_{ij (-n)} \,|\, \substack{i,j=1, \dots, N \\ n = 1, 2, \dots}).$$ As an application of the Euler-Poincaré principle, the (super)character of $\mathsf{V}_{S_N}$ can be computed as $$\begin{aligned} \operatorname{(s)ch}_{\mathsf{V}_{S_N}}(q)&=\mathop{\mathrm{Tr}}_{\mathsf{V}_{S_N}}(\pm1)^Pq^{T_{(1)}-c/24}=\mathop{\mathrm{Tr}}_{H^0(C, Q_{(0)})}(\pm1)^{P_\text{m}}q^{T_{(1)}-c/24}\\ &=\sum_{i\in{\mathbb Z}}(-1)^i\mathop{\mathrm{Tr}}_{H^i(C, Q_{(0)})}(\pm1)^{P_\text{m}}q^{T_{(1)}-c/24}\\ &=\sum_{i\in{\mathbb Z}}(-1)^i\mathop{\mathrm{Tr}}_{C^i}(\pm1)^{P_\text{m}}q^{T_{(1)}-c/24}\\ &=\mathop{\mathrm{Tr}}_{C}(\pm1)^{P_\text{m}}(-1)^{P_\text{gh}}q^{T_{(1)}-c/24},\end{aligned}$$ where $P=P_\text{m}+P_\text{gh}$ is the parity operator on $\widetilde{C}$ and $P_\text{m}$, $P_\text{gh}$ the parity operators only acting on matter and ghost fields, respectively (so, $P_\text{gh}$ gives the ghost degree modulo 2). For the fourth equality we used the Euler-Poincaré principle and that $Q_{(0)}$ commutes with $T_{(1)}$ and $P_\text{m}$. For the supercharacter of $\mathsf{V}_{S_N}$ we then obtain the simple formula $$\operatorname{sch}_{\mathsf{V}_{S_N}}(q)=\operatorname{sch}_{C}(q).$$ Hence, we need to compute the supercharacter of the kernel $C$ in the free-field vertex superalgebra $\bar{C}$. To understand this intersection of the kernels of the $\widetilde{E}_{ij (0)} = (Q_{(0)} \Phi_{ij})_{(0)}$, recall from that they form a representation of $\mathfrak{g}= \mathfrak{gl}_N$ on $\widetilde{C}$, which restricts to $\bar{C}$. We shall describe this representation in the following. For any finite-dimensional representation $V$ of $\mathfrak{g}$ we modify the usual symmetric algebra $S(V)=\bigoplus_{k=0}^\infty S^k(V)$ by introducing a formal variable $x$, $$S_x(V)\coloneqq S(Vx)=\bigoplus_{k=0}^\infty S^k(V)x^k\subset S(V)[[x]]$$ and analogously for the exterior algebra $\Lambda(V)$. Let $\bar{C}=\bigoplus_{p,r,n}\bar{C}^{p,r}_n$ be the decomposition of $\bar{C}$ into simultaneous eigenspaces for $F_\text{m}$, $F_\text{gh}$ and $T_{(1)}$ with eigenvalues $p,r\in{\mathbb Z}$ and $n\in\frac{1}{2}{\mathbb Z}_{\ge0}$ respectively, where $F_\text{gh}$ denotes the ghost degree (or ghost fermion number operator) defined above by $F_\text{gh}\Psi_{ij}=\Psi_{ij}$ and $F_\text{gh}\Phi_{ij}=-\Phi_{ij}$ and $F_\text{m}$ is the analogous fermion number operator for the matter fields defined by $F_\text{m}\psi_i=\psi_i$ and $F_\text{m}\phi_i=-\phi_i$. Note that $(-1)^{F_\text{m}}=(-1)^{P_\text{m}}$ and $(-1)^{F_\text{gh}}=(-1)^{P_\text{gh}}$. Then we define $$\bar{C}_{y,z,q}=\bigoplus_{\substack{p,r\in{\mathbb Z}\\n\in\frac{1}{2}{\mathbb Z}_{\ge0}}}\bar{C}^{p,r}_ny^pz^rq^n\subset\bar{C}[[y^{\pm1},z^{\pm1},q^{1/2}]].$$ It is not difficult to see: **Lemma 61**. *As a module for $\mathfrak{gl}_N$, $$\begin{aligned} \bar{C}_{y,z,q}&\cong\bigotimes_{n=0}^\infty\Bigl(S_{q^{n+1/2}}(\mathfrak{gl}_N)\otimes S_{q^{n+1/2}}(\mathfrak{gl}_N)\otimes S_{q^{n+1/2}}({\mathbb C}^N)\otimes S_{q^{n+1/2}}(({\mathbb C}^N)^*)\\ &\quad\otimes\Lambda_{yq^{n+1/2}}({\mathbb C}^N)\otimes\Lambda_{y^{-1}q^{n+1/2}}(({\mathbb C}^N)^*)\otimes\Lambda_{zq^{n+1}}(\mathfrak{gl}_N)\otimes\Lambda_{z^{-1}q^{n+1}}(\mathfrak{gl}_N)\Bigr),\end{aligned}$$ where $\mathfrak{gl}_N$ denotes the (self-dual) adjoint representation and ${\mathbb C}^N$ and $({\mathbb C}^N)^*$ the vector and dual vector representation, respectively.* The lemma also encodes the eigenvalues of the semisimple operator $y^{F_\text{m}}z^{F_\text{gh}}q^{T_{(1)}}$ on $\bar{C}$. For example, for any $n$, the $x_{ij (n)}\mathbf{1}$ for $i,j=1,\dots,N$ span an adjoint representation of $\mathfrak{gl}_N$ supported in $T_{(1)}$-weight $n+1/2$, amounting to the first tensor product in the above expression. Then, for the (super)character we shall take the trace over the specialisation $(\pm1)^{F_\text{m}}(-1)^{F_\text{gh}}q^{T_{(1)}}=(\pm1)^{P_\text{m}}(-1)^{P_\text{gh}}q^{T_{(1)}}$ of this operator. The vertex superalgebra $\widetilde{C}$ is nothing but the sum of all copies of the trivial representation of $\mathfrak{gl}_N$ in this tensor-product representation $\bar{C}$. The trivial representation may be selected by integrating over the Haar measure of the corresponding compact Lie group $\operatorname{U}(N)$ with suitable normalisation. Such integrals are well-studied in the physics literature in the context of computing the Schur index of four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theories. We refer to [@BDF15; @PP22; @PWZ22; @BSR22] in the case of the gauge groups $\mathop{\mathrm{GL}}_N$ and $\mathop{\mathrm{SL}}_N$. Indeed, the above integral coincides precisely with that given for $\mathop{\mathrm{GL}}_N$ in these references (in particular, see [@BDF15], but take note of a slightly different convention here that leads to a factor of $q^{N^2/8}$ compared to [@BDF15]). This is of course to be expected, as the prediction from [@BeeLemLie15] is that the Schur index of the four-dimensional theory is recovered by the supercharacter of the corresponding vertex operator superalgebra. Let $\eta(q)=q^{1/24}\prod_{n=0}^\infty(1-q^n)$ be the well-known Dedekind eta function. As an immediate corollary we obtain [@BDF15]: **Proposition 62**. *The supercharacters of $\mathsf{V}_{S_N}$ and $\mathsf{W}_{S_N}$ are $$\begin{aligned} \operatorname{sch}_{\mathsf{V}_{S_N}}(q)&=\frac{\eta(q)}{\eta(q^{1/2})^2}{\sum_{n=0}^\infty}(-1)^n\biggl(\binom{N+n}{N}+\binom{N+n-1}{N}\biggr)q^{(N+2n)^2/8},\\ \operatorname{sch}_{\mathsf{W}_{S_N}}(q)&=\frac{1}{\eta(q)^3}{\sum_{n=0}^\infty}(-1)^n\biggl(\binom{N+n}{N}+\binom{N+n-1}{N}\biggr)q^{(N+2n)^2/8},\end{aligned}$$ and they coincide with the Schur indices of the four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theories with gauge groups $\mathop{\mathrm{GL}}_N$ and $\mathop{\mathrm{SL}}_N$, respectively.* Ignoring the eta factors and the signs, the coefficients in the $q$-series have a combinatorial interpretation as $N$-dimensional square pyramidal numbers. The two characters (or Schur indices) differ precisely by $$\frac{\eta(q)^2}{\eta(q^{1/2})^2}\eta(q)^2=\operatorname{sch}_{\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^1) \otimes \mathsf{SF}}(q),$$ i.e. the above factorisation $\mathsf{V}_{S_N} = \mathsf{W}_{S_N} \otimes \mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^1) \otimes \mathsf{SF}$ corresponds to considering $\mathfrak{gl}_N$ rather than $\mathfrak{sl}_N$ in the quiver construction (or $\mathop{\mathrm{GL}}_N$ rather than $\mathop{\mathrm{SL}}_N$ on the level of the Schur indices). Finally, using the exact integration formulae in [@PP22], we can study the modular properties of these characters. As usual, we consider $q=\mathrm{e}^{2\pi\mathrm{i}\tau}$, where $\tau$ is from the complex upper half-plane $\mathbb{H}=\{z\in{\mathbb C}\,|\,\operatorname{Im}(z)>0\}$ equipped with an action of $\mathop{\mathrm{SL}}_2({\mathbb Z})=\langle S,T\rangle$. Let $\Gamma^0(2)=\langle T^2,STS\rangle$ denote the congruence subgroup of matrices whose upper right entry vanishes modulo 2. **Proposition 63** ([@PP22]). *The supercharacter $\operatorname{sch}_{\mathsf{W}_{S_N}}(q)$ converges to a sum of quasimodular forms of weights $N-1,N-3,\dots\in{\mathbb Z}_{\ge0}$ for the full modular group $\mathop{\mathrm{SL}}_2({\mathbb Z})$ if $N$ is odd and for $\Gamma^0(2)$ with some character if $N$ is even.* *Proof.* The supercharacter (or the corresponding Schur index) can be written as $\mathop{\mathrm{rk}}(\mathfrak{sl}_N)=N-1$ many iterated contour integrations. After each integration, one obtains an expression in terms of certain generalised Eisenstein series. Applying the precise integration formulae in [@PP22] to the Schur index for $\mathop{\mathrm{SL}}_N$, the statement follows by induction over $N$. ◻ In particular, for odd $N$, the Fourier expansion of $\operatorname{sch}_{\mathsf{W}_{S_N}}(q)=q^{(N^2-1)/8}+\ldots$ contains only integral $q$-powers. For even $N$, on the other hand, the $q$-powers are in $\frac{3}{8}+\frac{1}{2}{\mathbb Z}$ or $\frac{7}{8}+\frac{1}{2}{\mathbb Z}$. By multiplying by $\eta(q)^4/\eta(q^{1/2})^2$ we obtain the corresponding statement for the supercharacter $\operatorname{sch}_{\mathsf{V}_{S_N}}(q)=q^{N^2/8}+\ldots$ of $\mathsf{V}_{S_N}$, which is then a quasimodular form of weights $N$, $N-2$, $\dots$ for $\Gamma^0(2)$ with some character. It is in principle straightforward to write down bases for the relevant (finite-dimensional) spaces of quasimodular forms in terms of Eisenstein series and Jacobi theta functions so that one may express the above characters in these bases for small $N$. This has been done for up to $N\approx 10$ [@PP22; @BSR22]. For instance, for $N=2$, $3$ the characters are given by $$\begin{aligned} \operatorname{sch}_{\mathsf{W}_{S_2}}(q)&=\frac{\eta(q^{1/2})^2}{\eta(q)^4}\big(-E_2(q)/24+\theta_2(q)^4+\theta_3(q)^4\big)\\ &=q^{3/8}(1+3q-4q^{3/2}+9q^2-12q^{5/2}+22q^3+\ldots),\\ \operatorname{sch}_{\mathsf{W}_{S_3}}(q)&=\frac{1}{24}(1-E_2(q))\\ &=q(1 + 3 q + 4 q^2 + 7 q^3 + 6 q^4 + 12 q^5 + 8 q^6+\ldots).\end{aligned}$$ Here $E_2(q)=1-24\sum_{n=0}^\infty\sigma_1(n)q^n$ denotes the quasimodular Eisenstein series of weight 2, and $\theta_2(q)=\sum_{n\in{\mathbb Z}}q^{(n+1/2)^2/2}=2\eta(q^2)^2/\eta(q)$ and $\theta_3(q)=\sum_{n\in{\mathbb Z}}q^{n^2/2}=\eta(q)^5/\eta(q^{1/2})^2/\eta(q^2)^2$ are the standard Jacobi theta functions, which like the eta function $\eta(q)$ itself are modular forms of weight $1/2$ for some suitably defined congruence subgroup (and character). In [@PP22], they even conjecture a general formula for arbitrary $N$. **Remark 64**. We comment on the appearance of quasimodular forms of mixed weight as supercharacters of the vertex superalgebras $\mathsf{W}_{S_N}$ (see, e.g., the discussion in the introduction of [@BSR22]). Let $f\colon\mathbb{H}\longrightarrow{\mathbb C}$ be a (holomorphic) quasimodular form of weight $k$ and depth $p$, say, for the full modular group $\mathop{\mathrm{SL}}_2({\mathbb Z})$, but the following argument works similarly for $\Gamma^0(2)$. Recall that necessarily $p\leq k/2$. Then, by definition, applying the weight-$k$ Petersson slash operator $|_k M$ to $f$ for some $M\in\mathop{\mathrm{SL}}_2({\mathbb Z})$ will result in a function $f|_k M$ that is a polynomial of degree $p$ in $c/(c\tau+d)$ with quasimodular coefficients. If, however, we apply the weight-$w$ Petersson slash operator $|_w$ for any weight $w\leq k-p$, then $f|_w M$ is a polynomial of degree $k-w$ in $\tau$ or equivalently $\log(q)$ with quasimodular coefficients. This argument applies in particular to $w=0$, and hence we can interpret any quasimodular form of mixed weight, say with some highest weight $k$, as sitting inside of a vector-valued function $F\colon\mathbb{H}\longrightarrow{\mathbb C}^d$ (which is not unique, of course) whose components are polynomials in $\tau$ of degree $k$ with quasimodular coefficients that transforms with weight 0 under $\mathop{\mathrm{SL}}_2({\mathbb Z})$. This point of view is sensible when considering the supercharacter of $\mathsf{W}_{S_N}$. Indeed, $\mathsf{W}_{S_N}$ is quasi-lisse and hence satisfies a modular linear differential equation of weight 0 [@AraKaw18; @Li23]. This is compatible with $\mathsf{W}_{S_N}$ being a component of a vector-valued quasimodular form on $\mathbb{H}$ with $\log(q)$-terms. Finally, we point out that the characters $\operatorname{sch}_{\mathsf{W}_{S_N}}(q)$ can be generalised by introducing additional variables so that they probably transform as quasi-Jacobi forms [@KM15]. The modular properties of these characters should reveal information about the representation theory of $\mathsf{W}_{S_N}$ and its free-field realisation in . This is studied for the case $N=2$ in [@PP22; @PWZ22]. # $N=2$ Case: Twisted Chiral de Rham Algebra on ${\mathbb P}^1$ {#sec:n=2-case} In this section, we study the easiest non-trivial case of $N=2$, i.e. the sheaf of $\hbar$-adic vertex superalgebras $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}$ on the Hilbert scheme $M = \mathop{\mathrm{Hilb}}^2({\mathbb C}^2)$. We see that this sheaf essentially coincides with the twisted chiral de Rham algebra on the projective line ${\mathbb P}^1$ with parameter $\alpha = 1/2$, introduced in [@GMS05]. Moreover, the vertex operator superalgebra $\mathsf{W}_{S_2}$ of global sections is equal to the simple quotient of $\operatorname{Vir}_{\mathcal{N}=4}^{-9}$ and the free-field realisation in terms of the $\beta\gamma bc$-system coincides with the one given in [@Adamovic16]. First, we discuss the local coordinates of the Hilbert scheme $M = \mathop{\mathrm{Hilb}}^2({\mathbb C}^2)$. In this section, we write $U_{0} = U_{(2)}$ and $U_{\infty} = U_{(1^2)}$ so that there is an affine open covering $M = U_{0} \cup U_{\infty}$. Recall the local coordinates discussed in . There are the functions $[X^2 : 1]_{0}$, $[X^2 : X]_{0}$, $[Y : 1]_{0}$, $[Y : X]_{0}$ defined over $U_{0}$, and $[Y^2 : 1]_{\infty}$, $[Y^2 : Y]_{\infty}$, $[X : 1]_{\infty}$, $[X : Y]_{\infty}$ defined over $U_{\infty}$. The traces $\mathop{\mathrm{Tr}}(X)$, $\mathop{\mathrm{Tr}}(Y)$ are functions defined over $M$ and are written in the local coordinates as $$\begin{aligned} \mathop{\mathrm{Tr}}(X) &= [X^2 : X]_{0} = 2 [X : 1]_{\infty} + [X : Y]_{\infty} [Y^2 : Y]_{\infty}, \\ \mathop{\mathrm{Tr}}(Y) &= 2 [Y : 1]_{0} + [Y : X]_{0} [X^2 : X]_{0} = [Y^2 : Y]_{\infty}.\end{aligned}$$ We define $$\begin{aligned} \bar{x}&= [Y : X]_{0}, &\bar{\xi}&= - \mathop{\mathrm{Tr}}(X^2) + (1/2) \mathop{\mathrm{Tr}}(X)^2 = - 2 [X^2 : 1]_{0} - (1/2) \mathop{\mathrm{Tr}}(X)^2, \\ \bar{y}&= [X : Y]_{\infty}, &\bar{\eta}&= \mathop{\mathrm{Tr}}(Y^2) - (1/2) \mathop{\mathrm{Tr}}(Y)^2 = 2 [Y^2 : 1]_{\infty} + (1/2) \mathop{\mathrm{Tr}}(Y)^2.\end{aligned}$$ Then, $(\bar{x}, \bar{\xi}, \mathop{\mathrm{Tr}}(X), \mathop{\mathrm{Tr}}(Y))$ and $(\bar{y}, \bar{\eta}, \mathop{\mathrm{Tr}}(X), \mathop{\mathrm{Tr}}(Y))$ give local coordinates for $M$ over $U_{0}$ and $U_{\infty}$, respectively. The transition map between these local coordinates can be seen to be $\bar{y}= {1}/{\bar{x}}$, $\bar{\eta}= - \bar{x}^2 \bar{\xi}$. This implies the well-known fact that the Hilbert scheme $M$ is isomorphic to the direct product $$M = \mathop{\mathrm{Hilb}}^2({\mathbb C}^2) \simeq T^* {\mathbb P}^1 \times T^* {\mathbb C}^1.$$ It is not difficult to see that this isomorphism is an isomorphism of holomorphic symplectic manifolds, not just one of algebraic varieties. Now we discuss the structure of the sheaf $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}$ with respect to the above local coordinates. As discussed in , the following local trivialisations hold: $$\begin{aligned} \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{0}) &= {\mathbb C}[[\hbar]][x_{(-n)}, \xi_{(-n)} \,|\, n = 1, 2, \dots] \mathop{\mathrm{\widehat{\otimes}}}\Lambda_{{\mathbb C}[[\hbar]]}(b^{0}_{(-n)}, c^{0}_{(-n)} \,|\, n = 1, 2, \dots) \\ &\quad \mathop{\mathrm{\widehat{\otimes}}}\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar}, \\ \widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(U_{\infty}) &= {\mathbb C}[[\hbar]][y_{(-n)}, \eta_{(-n)} \,|\, n = 1, 2, \dots] \mathop{\mathrm{\widehat{\otimes}}}\Lambda_{{\mathbb C}[[\hbar]]}(b^{\infty}_{(-n)}, c^{\infty}_{(-n)} \,|\, n = 1, 2, \dots) \\ &\quad \mathop{\mathrm{\widehat{\otimes}}}\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}}\mathsf{SF}_{\hbar}\end{aligned}$$ with $$\begin{aligned} {2} x&= [Y : X]_{0},\qquad & \xi&= - \mathop{\mathrm{Tr}}(X^2) + (1/2) \mathop{\mathrm{Tr}}(X)^2,\\ y&= [X : Y]_{\infty},\qquad & \eta&= \mathop{\mathrm{Tr}}(Y^2) - (1/2) \mathop{\mathrm{Tr}}(Y)^2\end{aligned}$$ and $$\begin{aligned} {2} b^{0}&= \phi X \gamma - (1/2) \mathop{\mathrm{Tr}}(X) \phi \gamma,\qquad & c^{0}&= \{\psi : X \gamma\}_{0},\\ b^{\infty}&= \phi Y \gamma - (1/2) \mathop{\mathrm{Tr}}(Y) \phi \gamma,\qquad & c^{\infty}&= \{\psi : Y \gamma\}_{\infty}\end{aligned}$$ and where $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^1)_{\hbar}$ and $\mathsf{SF}_{\hbar}$ are generated by $\mathop{\mathrm{Tr}}(X)/\sqrt{2}$, $\mathop{\mathrm{Tr}}(Y)/\sqrt{2}$ and $\phi \gamma/\sqrt{2}$, $\beta \psi/\sqrt{2}$, respectively. It is not difficult to verify that $$\begin{gathered} b^{\infty}= x_{(-1)} b^{0}, \qquad c^{\infty}= (1/x)_{(-1)} c^{0},\end{gathered}$$ and thus $b^{\infty}_{(-1)} c^{\infty}= b^{0}_{(-1)} c^{0}$. The global section $J^0 = \mathop{\mathrm{Tr}}(XY) - (1/2) \mathop{\mathrm{Tr}}(X) \mathop{\mathrm{Tr}}(Y)$ can be written in terms of the above elements as $$\label{eq:2} J^0 = - 2 x_{(-1)} \xi+ b^{0}_{(-1)} c^{0}= 2 y_{(-1)} \eta- b^{\infty}_{(-1)} c^{\infty}.$$ Combining this identity with $y= 1/x$ and $b^{\infty}_{(-1)} c^{\infty}= b^{0}_{(-1)} c^{0}$, we obtain that $\eta= - x_{(-1)}^2 \eta+ x_{(-1)} b^{0}_{(-1)} c^{0}- (3/2) \hbar x_{(-2)} \mathbf{1}$. The above description of the transition homomorphism implies that the sections $(x, \xi, b^{0}, c^{0})$ defined over $U_{0}$ and the sections $(y, \eta, b^{\infty}, c^{\infty})$ over $U_{\infty}$ form the $\hbar$-adic version of the twisted chiral de Rham algebra $\Omega^\mathrm{ch}_{{\mathbb P}^1, \alpha}$ on ${\mathbb P}^1$ for $\alpha=1/2$ introduced by Gorbounov, Malikov and Schechtman in [@GMS05]. On the other hand, there are the bosonic global section $\mathop{\mathrm{Tr}}(X)$ and $\mathop{\mathrm{Tr}}(Y)$ and the fermionic ones $\phi \gamma$ and $\beta \psi$ in $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}(M)$. **Proposition 65**. *There is an isomorphism of sheaves of $\hbar$-adic vertex superalgebras over $M = \mathop{\mathrm{Hilb}}^2({\mathbb C}^2) \simeq T^* {\mathbb P}^1 \times T^* {\mathbb C}^1$, $$\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar} \simeq \Omega_{T^* {\mathbb P}^1, 1/2,\hbar}^\mathrm{ch}\mathop{\mathrm{\widehat{\otimes}}}\mathcal{D}^\mathrm{ch}(T^*{\mathbb C}^1)_{\hbar} \mathop{\mathrm{\widehat{\otimes}}} \mathsf{SF}_{\hbar},$$ where $\Omega^\mathrm{ch}_{T^* {\mathbb P}^1, 1/2,\hbar}$ is a certain $\hbar$-adic analogue of the twisted chiral de Rham algebra $\Omega^\mathrm{ch}_{{\mathbb P}^1, 1/2}$ defined as a sheaf of $\hbar$-adic vertex superalgebras on $T^* {\mathbb P}^1$, $\mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^1)_{\hbar}$ is the $\hbar$-adic $\beta\gamma$-system generated by $\mathop{\mathrm{Tr}}(X)/\sqrt{2}$, $\mathop{\mathrm{Tr}}(Y)/\sqrt{2}$ and $\mathsf{SF}_{\hbar}$ is the $\hbar$-adic symplectic fermion vertex superalgebra generated by $\phi \gamma/\sqrt{2}$, $\beta \psi/\sqrt{2}$.* Finally, we describe the global sections of $\widetilde{\mathcal{D}}^\mathrm{ch}_{M, \hbar}$ in the case of $N=2$. The elements $J^+ = \xi$, $J^0$ defined by [\[eq:2\]](#eq:2){reference-type="eqref" reference="eq:2"}, $J^{-} = \eta$, $G^+ = b^{0}$, $G^{-} = b^{\infty},\dots$ form the quotient $V_{\mathcal{N}=4}$ of the universal small $\mathcal{N}= 4$ superconformal algebra. The description of these global sections in the sections $(x, \xi, b^{0}, c^{0})$ defined over $U_{0}$ gives a free-field realisation of the superconformal algebra that coincides with the free-field realisation of the simple quotient introduced by Adamović in [@Adamovic16]. The main consequence of this is, that for $N=2$, $V_{\mathcal{N}=4}$ is the simple quotient of $\operatorname{Vir}_{\mathcal{N}=4}^{-9}$. Moreover, comparing the global sections in [@GMS05] with the free-field realisation in [@Adamovic16] shows that $\mathsf{W}_{S_2}$ is actually equal to $V_{\mathcal{N}=4}$, rather than a conformal extension of it, as would happen for $N>2$. **Proposition 66**. *The vertex operator superalgebra $\mathsf{V}_{S_2}$ of global sections coincides with the tensor product $$\mathsf{V}_{S_2}\cong V_{\mathcal{N}=4} \otimes \mathcal{D}^\mathrm{ch}(T^* {\mathbb C}^1) \otimes \mathsf{SF},$$ where $V_{\mathcal{N}=4}=\mathsf{W}_{S_2}$ is the simple quotient of the small $\mathcal{N}=4$ superconformal algebra $\operatorname{Vir}_{\mathcal{N}=4}^{-9}$ of level $k=-3/2$. In particular, $\mathsf{W}_{S_2}$ and $\mathsf{V}_{S_2}$ are simple.* We are led to believe that the vertex operator algebras of global sections $\mathsf{V}_{S_N}$, or equivalently $\mathsf{W}_{S_N}$, and the quotient $V_{\mathcal{N}=4}$ of the small $\mathcal{N}=4$ superconformal algebra are simple for all $N\ge2$. [^1]: ^a^Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan [^2]: ^b^University of Tsukuba, Tsukuba, Japan [^3]: ^c^Universität Hamburg, Hamburg, Germany [^4]: Email: [`arakawa@kurims.kyoto-u.ac.jp`](mailto:arakawa@kurims.kyoto-u.ac.jp), [`toshiro.kuwa@gmail.com`](mailto:toshiro.kuwa@gmail.com), [`math@moeller-sven.de`](mailto:math@moeller-sven.de) [^5]: To be precise, ${\mathbb C}^{2N}/S_N\cong\mathcal{M}_{S_N}\times T^*{\mathbb C}$ as symplectic varieties.

arxiv_math

--- abstract: | In a recent preprint published on arXiv (see arXiv:2308.02993v2, referred here as [@NXH]), N.X. Hong stated that every plurifinely open set $U\subset {\mathbb C}^n$, $n\geq 1$, is of the form $U=\bigcup \{\varphi_j>-1\}$, where each $\varphi_j$ is a negative plurisubharmonic function defined on an open ball $B_j\subset {\mathbb C}^n$ and used this result to prove an equality result on complex Monge-Ampère measures. Unfortunately, this result is wrong as we will see below. address: | University of Mohammed V\ Department of Mathematics,\ Faculty of Science\ P.B. 1014, Rabat\ Morocco author: - Mohamed El Kadiri title: A note on the structure of plurifinely open sets and the equality of some complex Monge-Ampère measures --- # Introduction The plurifine topology on an open set $\Omega\subset {\mathbb C}^n$, $n\geq 1$, is the coarsest topology on $\Omega$ that makes continuous all plurisubharmonic functions on $\Omega$. For properties of this topology we refer the reader to [@BT]. An open set relative to this topology is called a plurifinely open set. In a recent preprint on arXiv, cf. [@NXH], N.X Hong gave a description of plurifinely open sets in ${\mathbb C}^n$ ($n\geq 2$) and applied it to establish a result on complex Monge-Ampère measures. More precisely, he stated the following results: **Theorem 1** ([@NXH Theorem 1.1]). *Let $\Omega$ be a subset of ${\mathbb C}^n$ and let $\{z_j\}\subset \Omega$ be such that $\overline {\{z_j\}} =\Omega$. Then, $\Omega$ is plurifinely open if and only if there exists a sequence of negative plurisubharmonic functions $\varphi_{j,k})$ in $B(z_j, 2^{-k})$ such that $$\Omega=\bigcup_{j,k=1}\{\varphi_{j,k} >-1\}.$$* **Corollary 2**. *Every plurifinely open set is Borel.* **Proposition 3** ([@NXH Proposition 2.2]). *Let $D$ be a Euclidean open set and let $\mathop{\mathrm{PSH}}(D)$ be the set of all plurisubharmonic functions in $D$. Then, for every plurifinely open set $\Omega$, the function $$u := \sup\{\varphi\in \mathop{\mathrm{PSH}}(D) : \varphi\leq -1 \text{ on } D\setminus \Omega \text{ and } \varphi \leq 0 \text{ in } D\}$$ is plurisubharmonic in $D$ and $u = -1$ on $D\setminus \Omega$.* **Theorem 4**. *Let $D$ be a bounded hyperconvex domain and let $\Omega$ be a plurifinely open subset of $D$. Assume that $u_0, \cdots . . . , u_n \in \mathcal E(D)$ such that $u_0 = u_1$ on $\Omega$. Then $$\label{eq1.2} dd^cu_0 \wedge T = ddcu_1 \wedge T$$ on $\Omega$. Here $T := dd^cu_2 \wedge . . . \wedge dd^cu_n.$* Note that Theorem 1.1 and Theorem 1.2 in [@NXH] are respectively Theorem 1.1 and Theorem 1.2 of the first version arXiv:2308.02993v1 of [@NXH]. # Theorem 1.1 is wrong Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} and its corollary are wrong as we will show by giving an example of a plurifinely open subset of ${\mathbb C}^n$ that is not a Borel set. Recall that for $n=1$ pluri-polar and plurifinely open subsets of ${\mathbb C}^1={\mathbb C}$ are just the classical polar subsets and finely open subsets of ${\mathbb C}\cong {\mathbb R}^2$. Any polar set $E\subset {\mathbb C}$ is finely closed (in ${\mathbb C}^n$, $n\geq 2$, this property is not true, a pluripolar set is not necessarily plurifinely closed) and there are polar subsets of ${\mathbb C}$ that are not countable, see [@D I.V.3, p. 59]. Let $E\subset {\mathbb C}$ be a non countable polar set and denote by $\mathcal B({\mathbb C})$ the $\sigma$-algebra of Borel subsets of ${\mathbb C}$. It is well known that $\# \mathcal B({\mathbb C})=\mathfrak{c} <2^{\mathfrak c}=\#\mathfrak P(E)$ (the latter equality holds because $\#E=\mathfrak c$). It then follows that $\mathfrak P(E)\nsubseteq \mathcal B({\mathbb C})$ and, consequently, there are subsets of $E$ that are not Borel subsets of ${\mathbb C}$. Since every subset of $E$ is a polar, we conclude that there are polar subsets of ${\mathbb C}$ that are not Borel sets. In the same we can show that there are pluripolar subsets of ${\mathbb C}^n$, $n\geq 2$, that are not Borel sets. Let $F$ be a polar subset of ${\mathbb C}$ that is not a Borel set and let $U={\mathbb C}\setminus F$. Then $U$ is a finely open subset of ${\mathbb C}$ which is not a Borel set. Let $n\geq 2$, then $U^n$ is a plurifinely open subset of ${\mathbb C}^n$. The function $f:{\mathbb C}\rightarrow {\mathbb C}^n$ defined by $f(z)=(z,\cdots, z)$ is Borel measurable (because it is continuous) and hence $U^n\subset {\mathbb C}^n$ is not a Borel set because otherwise $U=f^{-1}(U^n)$ would be a Borel subset of ${\mathbb C}$. In the same way $U\times {\mathbb C}^{n-1}$ is not a Borel subset of ${\mathbb C}^n$. We then have proved that Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} is wrong for every $n\geq 1$. # Proposition 1.3 is wrong Proposition 1.3, on which is based the proof of the (wrong) theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}, is also wrong. Here we give a concrete example showing this. Let $U \subset {\mathbb C}$ be an irregular bounded domain such that $U\subset B(0,1)$, and let $z\in \partial U$ be an irregular point. For every (Euclidean) domain $O\subset B(0,1)$ and every $A\subset O$ we denote by ${^OR}_f^A$ and ${^O\widehat R}_f^A$ (or ${R}_f^A$ and ${\widehat R}_f^A$ if there is no risk of confusion) respectively, the reduced and the balayage of a function $f:O\rightarrow [0,+\infty]$ relative to $O$ (see [@AG p. 129]), and by $\mathcal S_+(O)$ the cone of nonnegative superharmonic functions on $O$. Recall that $$^OR_f^A=\inf\{u\in \mathcal S_+(O): u\geq f \text{ on } A\}$$ and that ${^O\widehat R}_f^A$ is the l.s.c. regularized function of $^OR_f^A$, that is, the function defined by $${^O\widehat R}_f^A(z)=\liminf_{\zeta\to z} {^OR}_f^A(\zeta)$$ for every $z\in O$. It is clear $O$ and $O'$ are two open sets such that $O\subset O'\subset B(0,1)$ we have $^OR_f^A\leq {^{O'}R}_f^A|_O$ and $^O{\widehat R}_f^A\leq {^{O'}}{\widehat R}_f^A|_O$. Since $z$ is an irregular point of $U$, it follows by [@AG Theorem 7.3.1 (ii)] that there is an open ball $B(z,r)\subset \overline B(z,r)\subset B(0,1)$ such that $$\widehat R_1^{B(z,r)\setminus U}(z):= {^{B(0,1)}\widehat R}_1^{B(z,r)\setminus U}(z)<1$$ and hence $$\label{eq1} {^{B(z,r)}}\widehat R_1^{B(z,r)\setminus U\cap B(0,r)}(z)= {^{B(z,r)}}\widehat R_1^{B(z,r)\setminus U}(z)<1.$$ Take $D=B(z,r)$ and $\Omega=U\cap B(z,r)$ and let $u$ be the function $$u:=\sup\{\varphi\in \mathop{\mathrm{PSH}}_-(D): u\leq -1 \text{ on } D\setminus U\}.$$ We have $${^{B(0,r)}}\widehat R_1^{B(z,r)\setminus \Omega}=-u^*$$ and by ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"}) that $u(z)<u^*(z).$ This proves that Proposition 2.2 is wrong. # The correct form of Theorem 1.4 As it is stated, Theorem [Theorem 4](#thm1.4){reference-type="ref" reference="thm1.4"} is not sure, because the given proof of this theorem in [@NXH] is based on the wrong theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}. However, that proof is correct if $\Omega$ has the form given in Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}. So, Theorem [Theorem 4](#thm1.4){reference-type="ref" reference="thm1.4"} should be stated correctly as follows: **Theorem 5**. *Let $D\subset {\mathbb C}^n$ ($n\geq 1$) be a bounded hyperconvex domain and let $\Omega$ be a plurifinely open subset of $D$ of the form $\Omega=\bigcup_j\{\varphi_j>0\}$, where each $\varphi_j$ is a negative plurisubharmonic function defined on an open ball $B_j\subset D$. Assume that $u_0,. . . , u_n \in \mathcal E(D)$ such that $u_0 = u_1$ on $\Omega$. Then $$\label{eq1.5} dd^cu_0 \wedge T = dd^cu_1 \wedge T$$ on $\Omega$. Here $T := dd^cu_2 \wedge . . . \wedge dd^cu_n.$* *Proof.* It suffices to prove the theorem in case $\Omega=\{\varphi>0\}$, where $\varphi$ is a nonpositive plurisubharmonic function on an open ball $B_j\subset D$. To do this, we bring from [@NXH] the part of the proof of Theorem [Theorem 4](#thm1.4){reference-type="ref" reference="thm1.4"} corresponding to this case (which is correct). Without loss of generality, we may assume that $u_k \in {\mathcal F}(D)$, $0 \leq k \leq n$. Let $\psi \in {\mathcal E}_0(D)$ and define $u_{j,k}:= \max\{j\psi, u_k\}$, $k = 0, 1, ..., n.$ We easily have $${\mathcal E}_0(\Omega) \ni u_{j,k}\downarrow u_k \text{ as } j\uparrow +\infty, \ k = 0, ...,, n$$ and, by the hypotheses, $$\label{eq1.6} u_{j,0} = u_{j,1} \text{ on } \Omega, \forall j \geq 1.$$ Write $T_j:= dd^cu_{j,2} \wedge ... \wedge dd^cu_{j,n}.$ Since the $u_{j,k}$ are bounded plurisubharmonic functions in $D$, it follows by Corollary 3.4 in [@EKW] that $$dd^cu_{j,0} \wedge T_j = dd^cu_{j,1} \wedge T_j \text{ on } \Omega.$$ and hence $$\label{eq1.7} \varphi dd^cu_{j,0} \wedge T_j = \varphi dd^cu_{j,1} \wedge T_j \text{ on } D$$ because $\varphi=0$ on $D\setminus \Omega$. By Corollary 5.2 in [@Ce] we have $$\varphi dd^c\wedge u_{j,0} \wedge T_j \to \varphi dd^cu_0 \wedge T$$ and $$\varphi dd^c\wedge u_{j,1} \wedge T_j \to \varphi dd^cu_1 \wedge T$$ weakly\* on $D$ as $j \to +\infty$. Hence, by ([\[eq1.7\]](#eq1.7){reference-type="ref" reference="eq1.7"}), $$\varphi dd^cu_0 \wedge T = \varphi dd^cu_1 \wedge T \text{ on } D,$$ and therefore $$dd^cu_0 \wedge T = dd^cu_1 \wedge T \text{ on } \Omega.$$ ◻ Theorem [Theorem 5](#thm1.5){reference-type="ref" reference="thm1.5"} is a partial extension of [@EK Theorem 4.5]. In a forthcoming paper we shall extend Theorem [Theorem 5](#thm1.5){reference-type="ref" reference="thm1.5"} to every plurifnely open subset of a given bounded hyperconvex domain $D\subset {\mathbb C}^n$. 00 D.H Armitage, S.J. Gardiner: Classical potential theory. Springer Monographs in Mathematics. *Springer-Verlag London, Ltd., London*, 2001. E. Bedford, B. A. Taylor: Fine topology, Šilov boundary and $(dd^{c})^{n}$. *J. Funct. Anal.* **72** (1987), no. 2, 225--251. U. Cegrell: The general definition of the Monge-Ampère operator. *Ann. Inst. Fourier*, **54** (2004), no. 1, 159--179. J. L. Doob: Classical potential theory and its probabilistic counterpart. Grundlehren der mathematischen Wissenschaften, 262. *Springer-Verlag, New York*, 1984. M. El Kadiri: An equality on Monge-Ampère measures. *J. Math. Anal. Appl.* 519 (2023), no. 2, Paper No. 126826. Plurifinely psubharmonic functions and the Monge-Ampère operator, *Potential Anal.* **41** (2014) no. 2, 469--485. N.X. Hong: The plurifinely open sets and complex Monge-Ampère measures. arXiv:2308.02993v2.

arxiv_math

--- abstract: | We study a tensor optimal transport (TOT) problem for $d\ge 2$ discrete measures. This is a linear programming problem on $d$-tensors. We introduces an interior point method (ipm) for $d$-TOT with a corresponding barrier function. Using a \"short-step\" ipm following central path within $\varepsilon$ precision we estimate the number of iterations. address: Department of Mathematics and Computer Science, University of Illinois at Chicago, Chicago, Illinois, 60607-7045, USA author: - Shmuel Friedland date: October 29, 2023 title: Interior point method in tensor optimal transport --- # Introduction {#sec:intro} For $i\in\{1,2\}$ let $X_i$ be a random variables on $\Omega_i$ which has a finite number of values: $X_i:\Omega_i\to [n_i]$, where $[n]\stackrel{\mathclap{\normalfont\mbox{def}}}{=}\{1,\ldots,n\}\subset \mathbb{N}$. Assume that $\mathbf{p}_i=(p_{1,i},\ldots,p_{n_i,i})$ is the column probability vector that gives the distribution of $X_i$: $\mathbb{P}(X_i=j)=p_{j,i}$. Then the discrete Kantorovich optimal transport problem (OT) can be states as follows [@Kan]. (See [@Vil03; @Vil09] for modern account of OT.) Let $Z$ be a random variable $Z: \Omega_1\times \Omega_2\to [n_1]\times[n_2]$ with contingency matrix (table) $U\in\mathbb{R}_+^{n_1\times n_2}$ that gives the distribution of $Z$: $\mathbb{P}(Z=(j_1,j_2))=u_{j_1,j_2}$. (Here $\mathbb{R}_+=[0,\infty), \mathbb{R}_{++}=(0,\infty)$.) Let $P=(\mathbf{p}_1,\mathbf{p}_2)$ and $\mathrm{U}(P)$ be the convex set of all probability matrices with marginals $\mathbf{p}_1,\mathbf{p}_2$: $$\label{defUPmat} \mathrm{U}(P)=\{U=[u_{j_1,j_2}]\in\mathbb{R}_+^{n_1\times n_2}, \sum_{j_2=1}^{n_2} u_{j_1,j_2}=p_{j_1,1}, \quad \sum_{j_1=1}^{n_1} u_{j_1,j_2}=p_{j_2,2}\}.$$ Let $C=[c_{i_1,i_2}]\in \mathbb{R}^{n_1\times n_2}$ be the cost matrix of transporting a unit $j_1\in[n_1]$ to $j_2\in[n_2]$. Then the optimal transport problem is the linear programming problem (LP): $$\label{TOTmat} \tau(C,P)=\min\{\langle C,U\rangle, U\in \mathrm{U}(P)\}.$$ (Here $\langle C,U\rangle=\mathop{\mathrm{Tr}}C^\top U$.) For $n_1=n_2=n$ and a nonnegative symmetric cost matrix $C=[c_{ij}]$ with zero diagonal satisfying the triangle inequality $c_{ik}\le c_{ij}+c_{jk}$, the quantitty $\tau(C,P)$ gives rise to a distance between probability vectors $\mathbf{p}_1$ and $\mathbf{p}_2$, which can be viewed as two histograms. It turns out that $\tau(C,P)$ has many recent applications in machine learning [@AWR17; @ACB17; @LG15; @MJ15; @SL11], statistics [@BGKL17; @FCCR18; @PZ16; @SR04] and computer vision [@BPPH11; @RTG00]. A related problem to OT is quantum optimal transport (QOT), see [@FECZ22; @CEFZ23] and references therein. QOT is a semidefinite programming problem [@VB96], which are effectively solved using the interior point methods (ipm) [@NN94; @Ren01; @YFFKNN]. In this paper we don't treat QOT, but we do use ipm for solving OT and d-multi-marginal transport problem that we call $d$-tensor optimal transport abbreviated as $d$-TOT. Assume that $n_1=n_2=n$. Then the complexity of finding $\tau(C,P)$ is $O(n^3 \log n)$, as this problem can be stated in terms of flows [@PW09]. In applications, when $n$ exceeds a few hundreds, the cost is prohibitive. One way to improve the computation of $\tau(C,P)$ is to replace the linear programming with problem of OT with convex optimization by introducing an entropic regularization term as in [@Cut13]. This regularization terms gives an $\varepsilon$-approximation to $\tau(C,P)$, where $\varepsilon>0$ is given. The regularization term gives almost linear time approximation $O(n^2)$, ignoring the logarithmic terms, using a variation of the celebrated Sinkhorn algorithm for matrix diagonal scaling [@AWR17; @LHCJ22; @Fri20]. The aim of this paper is to introduce the interior point method for $d$-TOT, which correspond to the set of $d$-probability measures $\mathbf{p}_i\in\mathbb{R}^{n_i}$ for $i\in[d]$. For the case $d=2$ the set $\mathrm{U}(P)$ is the set of probability matrices $U\in \mathbb{R}_+^{n_1\times n_2}$ satisfying the marginal conditions $U \mathbf{1}_{n_2}=\mathbf{p}_1, U^\top \mathbf{1}_{n_1}=\mathbf{p}_2$. Here $\mathbf{1}_n\in\mathbb{R}^n$ is the vector whose coordinates are all $1$. For $d\ge 3$ we introduce $d$-mode tensors $\otimes_{k=1} ^d \mathbb{R}^{n_k}$. We denote by $\mathcal{U}\in \otimes _{k=1}^d \mathbb{R}^{n_k}$ a tensor whose entries are $u_{i_1,\ldots,i_d}$, i.e., $\mathcal{U}=[u_{i_1,\ldots,i_d}]$. Assume that $\mathcal{C},\mathcal{U}\in \otimes _{k=1}^d \mathbb{R}^{n_k}, \mathcal{X}\in \otimes^{j\in[d]\setminus\{k\}}\mathbb{R}^{n_j}$. Denote by $\langle \mathcal{C},\mathcal{U}\rangle$ the Hilbert-Schmidt inner product $\sum_{i_k\in[n_k],k\in[d]} c_{i_1,\ldots,i_d}u_{i_1,\ldots,i_d}$. For $k\in [d]$ denote by $\mathbf{y}=\mathcal{U}\times_{\bar k} \mathcal{X}\in \mathbb{R}^{n_k}$ the contraction on all but the index $k$: $$y_{i_k}=\sum_{i_j\in[n_j], j\in[d]\setminus\{k\}}u_{i_1,\ldots,i_d} x_{i_1,\ldots,i_{k-1},i_{k+1},\ldots,i_d}.$$ Let $\mathcal{J}_{d-1,k}\in\otimes_{j\in [d]\setminus\{k\}}\mathbb{R}^{n_j}$ be the tensor whose all coordinates are $1$. Define $$\label{defU(P)ten} \mathrm{U}(P)=\{\mathcal{U}\in \otimes_{k=1}^d\mathbb{R}_+^{n_k}, \mathcal{U}\times_{\bar k} \mathcal{J}_{d-1,k}=\mathbf{p}_k, k\in[d]\}, P=(\mathbf{p}_1,\ldots,\mathbf{p}_d).$$ Then the tensor optimal transport problem (TOT) is $$\begin{aligned} \label{TOT} \tau(\mathcal{C},P)\stackrel{\mathclap{\normalfont\mbox{def}}}{=}\min\{\langle \mathcal{C}, \mathcal{U}\rangle, \mathcal{U}\in \mathrm{U}(P)\}.\end{aligned}$$ TOT problem is a LP problem with $\prod_{k=1}^d n_k$ nonnegative variables and $1+\sum_{k=1}^d (n_k-1)$ constraints. The TOT was considered in [@Pi68; @Po94] in the context of multidimensional assignment problem, where the entires of the tensor $\mathcal{U}$ are either $0$ or $1$. There is a vast literature on continuous multidimensional optimal transport problem. See for example [@FV18; @BCN19; @TDGU20; @HRCK; @LHCJ22] and the references therein. The TOT problem can be viewed as a discretization the continuous multidimensional optimal transport problem. We point out that that the ipm approach is easily adopted for variations of TOT. Indeed, for $d>2$ there is another well known variation of the set marginals. Namely, let $Z$ be a random variable $Z: \Omega_1\times\cdots \times\Omega_d\to [n_1]\times\cdots\times [n_d]$ with contingency tensor (table) $\mathcal{U}\in \mathbb{R}_+^{n_1\times\cdots n_d}$ that gives the distribution of $Z$: $\mathbb{P}\big(Z=(j_1,\ldots, j_d)=u_{j_1,\ldots,j_d}\big)$. Denote by $P_k\in \mathbb{R}_+^{n_1\times \cdots n_{k-1}\times n_{k+1}\times \cdots\times n_d}$ the marginal of $Z_k$ obtained from $Z$ with respect to $X_k:\Omega_k\to [n_k]$. Assume that we are given a distribution of $Z_0$ with a positive contingency tensor $\mathcal{V}=(v_{j_1,\ldots,j_d})\in \otimes_{i=1}^d\mathbb{R}^{n_i}$ that have the above marginals. Let $\mathrm{U}(P)$ be a nonempty set of such $Z$. See for example [@LZZ17] and references therein. Then one can use a similar ipm algorithm to find an approximate algorithm to the problem [\[TOT\]](#TOT){reference-type="eqref" reference="TOT"}. In subsection [2.3](#subsec:cest){reference-type="ref" reference="subsec:cest"} we show that if $\mathbf{p}_1,\ldots,\mathbf{p}_d\in\mathbb{R}_+^n$ are weak $K-\ell$ uniform distributions, see Definition [Definition 8](#defwKldist){reference-type="ref" reference="defwKldist"}, then the number iterations of the ipm algorithm has complexity $O(n^{d/2})$, ignoring the logarithmic terms. # The interior point method {#sec:ipm} We first recall some notations and definitions that we will use in this section. $$\label{defellsnrm} \begin{aligned} \|\mathbf{x}\|_s=\bigl(\sum_{i=1}^n |x_i|^s\bigr)^{1/s}, \, s\in[1,\infty],\, \mathbf{x}=(x_1,\ldots,x_n)\in\mathbb{R}^n,\\ \|\mathbf{x}\|\stackrel{\mathclap{\normalfont\mbox{def}}}{=}\|\mathbf{x}\|_2,\\ \mathrm{B}(\mathbf{x},r)=\{\mathbf{y}\in\mathbb{R}^n, \|\mathbf{y}-\mathbf{x}\|\le r\} \textrm{ for } r\ge 0. \end{aligned}$$ Let $f\in\mathrm{C}^3(\mathrm{B}(\mathbf{x},r))$ for $r>0$. Denote $$f_{,i_1\ldots i_d}(\mathbf{x})=\frac{\partial ^d}{\partial x_{i_1}\ldots\partial x_{i_d}}f(\mathbf{x}), \quad i_1,\ldots,i_d\in[n], d\in[3].$$ Recall the Taylor expansion of $f$ at $\mathbf{x}$ of order $3$ for $\mathbf{u}\in\mathbb{R}^n$ with a small norm: $$\begin{aligned} f(\mathbf{x}+\mathbf{u})\approx f(\mathbf{x})+\nabla f(\mathbf{x})^\top\mathbf{u}+\frac{1}{2}\mathbf{u}^\top \partial^2 f(\mathbf{x})\mathbf{u}+\frac{1}{6}\partial ^3 f(\mathbf{x})\otimes \mathbf{u}^{3\otimes},\\ \nabla f(\mathbf{x})=(f_{,1}(\mathbf{x}),\ldots,f_{,n}(\mathbf{x}))^\top, \quad \partial^2 f(\mathbf{x})=[f_{,ij}(\mathbf{x})], i,j\in[n],\\ \partial^3 f(\mathbf{x})=[f_{,ijk}(\mathbf{x})], i,j,k\in[n], \, \partial^3 f(\mathbf{x})\otimes\mathbf{u}^{3\otimes}=\sum_{i,j,k\in[n]}f_{,ijk}(\mathbf{x})u_i u_j u_k, \end{aligned}$$ where $\nabla f, \partial^2 f, \partial ^3 f$ are called the gradient, the Hessian, and the 3-mode symmetric partial derivative tensor of $f$. A set $\mathrm{D}\subset \mathbb{R}^n$ is called a domain if $\mathrm{D}$ is an open connected set. **Definition 1**. *Assume that $f: \mathrm{D}\to\mathbb{R}$ is a convex function in a convex domain $\mathrm{D}\subset \mathbb{R}^n$, and $f\in\mathrm{C}^3(\mathrm{D})$. The function $f$ is called $a (>0)$-self-concordant, or simply self-concordant, if the following inequality hold $$\label{defconcconst1} |\langle \partial^3f(\mathbf{x}),\otimes^3\mathbf{u}\rangle|\le 2a^{-1/2} (\mathbf{u}^\top \partial^2f(\mathbf{x})\mathbf{u})^{3/2}, \textrm{ for all }\mathbf{x}\in \mathrm{D},\mathbf{u}\in\mathbb{R}^n.$$ The function $f$ is called a standard self-concordant if $a=1$, and a strongly $a$-self-concordant if $f(\mathbf{x}_m)\to\infty$ if the sequence $\{\mathbf{x}_m\}$ converges to the boundary of $\mathrm{D}$.* *The complexity value $\theta(f)\in[0,\infty]$ of an a-self-concordant function $f$ in $\mathrm{D}$, called a self-concordant parameter in [@NN94 Definition 2.3.1], is $$\label{defsconc0} \begin{aligned} \theta(f)=sup_{\mathbf{x}\in\mathrm{D}} \inf\{\lambda^2\in[0,\infty], |\nabla f(\mathbf{x})^\top \mathbf{u}|^2\\ \le \lambda^2 a\big(\mathbf{u}^\top\partial^2 f(\mathbf{x})\mathbf{u}\big),\forall \mathbf{u}\in\mathbb{R}^n\}. \end{aligned}$$ A strongly self-concordant function with a finite $\theta(f)$ is called a barrier (function).* The following lemma is probably well known, and we give its short proof for completeness: **Lemma 2**. *Let $\mathrm{D}\subset\mathbb{R}^n$ be a convex domain and assume that $f$ is a self-concordant function in $\mathrm{D}$. Then one of the following conditions hold* (a) *The function $f$ is affine on $\mathrm{D}$.* (b) *The Hessian $\partial^2 f$ is positive definite on $\mathrm{D}$.* (c) *There is an orthogonal change of coordinates $\mathbf{y}=(y_1,\ldots,y_n)^\top=Q\mathbf{x}$, such that $$f(\mathbf{y})=f_1((y_1,\ldots,y_m)^\top)+by_{m+1}$$ for some $m\in[n-1]$, such that $f_1$ has a positive definite Hessian on $\mathrm{D}_1\subset \mathbb{R}^m$, where $\mathrm{D}_1$ is the projection of $\mathrm{D}$ on the first $m$ coordinates.* *Suppose furthermore that $\theta(f)<\infty$. Then either $f$ is constant on $\mathrm{D}$, the Hessian of $f$ is positive definite in $\mathrm{D}$, or the condition (c) holds with $b=0$. In particular, $\nabla f(\mathbf{x})$ is orthogonal to the kernel of $\partial^2 f(\mathbf{x})$ for $\mathbf{x}\in\mathrm{D}$.* *Proof.* Corollary 2.1.1 in [@NN94] states that the nullity subspace $\mathbf{U}(\mathbf{x})\subseteq\mathbb{R}^n$ of $\partial^2 f(\mathbf{x}),\mathbf{x}\in\mathrm{D}$ does not depend on $\mathbf{x}\in\mathrm{D}$. Set $\mathbf{U}=\mathbf{U}(\mathbf{x})$ for $\mathbf{x}\in\mathrm{D}$. If $\mathbf{U}=\mathbb{R}^n$ then (a) holds. If $\mathbf{U}=\{\mathbf{0}\}$ then (b) holds. If $\dim \mathbf{U}=n-m$ it is straightforward to show that (c) holds. Assume that $\theta(f)<\infty$. Suppose that $\mathbf{0}\ne\mathbf{u}\in \ker \partial^2 f(\mathbf{x})$. Then $\nabla f(\mathbf{x})^\top \mathbf{u}=0$. If the condition (a) satisfied then $f$ is a constant function. Suppose that the condition (c) is satisfied. Then $b=0$. In particular, $\nabla f(\mathbf{x})$ is orthogonal to the kernel of $\partial^2 f(\mathbf{x})$ for $\mathbf{x}\in\mathrm{D}$. ◻ Denote by $\mathrm{S}_n\supset\mathrm{S}_{n,+}\supset \mathrm{S}_{n,++}$ the space of $n\times n$ symmetric, the cone of positive semidefinite and the open set of positive definite matrices respecctively. For $A,B\in \mathrm{S}_n$ we denote $A\succ B\,(A\succeq B)$ if $A-B\in\mathrm{S}_{n,++},\, (A-B\in\mathrm{S}_{n,+})$. For a matrix $A\in\mathbb{R}^{n\times n}$ denote by $A^\dagger\in\mathbb{R}^{n\times n}$ the Moore-Penrose inverse of $A$ [@Frib]. Recall that if $A$ is invertible then $A^\dagger=A^{-1}$. In particular for $a\in\mathbb{R}$: $a^\dagger =a^{-1}$ if $a\ne 0$, and $a^\dagger =0$ if $a=0$. Assume that $A\in \mathrm{S}_n$. Then $A=Q\mathop{\mathrm{diag}}(\lambda_1,\ldots,\lambda_n) Q^\top$, where $Q\in\mathbb{R}^{n\times n}$ is an orthogonal matrix and $$\lambda_{\max}=\lambda_1\ge\ldots\ge\lambda_n=\lambda_{\min}$$ are the eigenvalues of $A$. Then $A^\dagger=Q \mathop{\mathrm{diag}}(\lambda_1^\dagger,\ldots,\lambda_n^\dagger) Q^\top$. In particular, $\ker A=\ker A^\dagger$. In what follows we will use the following lemma: **Lemma 3**. *Let $A\in\mathrm{S}_{n,+}, \mathbf{y}\in\mathbb{R}^{n}$. Suppose that $\mathbf{y}^\top \ker A=0$. Then $$\label{maxcharlem1} \begin{aligned} \mathbf{y}^\top A^\dagger\mathbf{y}=\inf\{\lambda>0, |\mathbf{y}^\top\mathbf{u}|^2\le \lambda^2 \mathbf{u}^\top A\mathbf{u}, \\ \forall \mathbf{u}\in\mathbb{R}^n\}= \max_{\mathbf{u}\in\mathbb{R}^n} 2\mathbf{y}^\top \mathbf{u}-\mathbf{u}^\top A\mathbf{u}. \end{aligned}$$ Furthermore, if $A\succeq \mathbf{y}\mathbf{y}^\top$ then $\mathbf{y}^\top A^\dagger\mathbf{y}\le 1$.* *Proof.* Clearly, it is enough to consider the case $\mathbf{y}\ne \mathbf{0}$. Let $$\mu=\inf\{\lambda>0, |\mathbf{y}^\top\mathbf{u}|^2\le \lambda^2 \mathbf{u}^\top A\mathbf{u}, \forall \mathbf{u}\in\mathbb{R}^n\}.$$ Suppose first that $A$ is positive definite. Then $\mu=\max_{\mathbf{u}\ne \mathbf{0}}\big(\mathbf{u}^\top\mathbf{y}\mathbf{y}^\top \mathbf{u}\big)/\big(\mathbf{u}^\top A\mathbf{u}\big)$. Let $B=\sqrt A\succ 0$, and $B\mathbf{u}=\mathbf{v}$. Then $\mu$ is the maximum eigenvalue of the rank-one matrix $B^{-1} \mathbf{y}\mathbf{y}^\top B^{-1}$. Thus $$\mu=\mathop{\mathrm{Tr}}B^{-1} \mathbf{y}\mathbf{y}^\top B^{-1}=\mathbf{y}^\top B^{2}\mathbf{y}=\mathbf{y}^\top A^{-1}\mathbf{y}.$$ This proves the first equality in [\[maxcharlem\]](#maxcharlem){reference-type="eqref" reference="maxcharlem"}. We now show the second equality in [\[maxcharlem\]](#maxcharlem){reference-type="eqref" reference="maxcharlem"}. Fix $\mathbf{w}\ne \mathbf{0}$ and let $\mathbf{u}=t\mathbf{w}$. Set $\phi(t)=2\mathbf{y}^\top(t\mathbf{w})-(t\mathbf{w})^\top A(t\mathbf{w})$. The maximum of $\phi(t)$ is achieved at $t=\frac{\mathbf{y}^\top\mathbf{w}}{\mathbf{w}^\top A\mathbf{w}}$ and is equal to $\frac{|\mathbf{y}^\top \mathbf{w}|^2}{\mathbf{w}^\top A\mathbf{w}}$. Use the first equality of [\[maxcharlem\]](#maxcharlem){reference-type="eqref" reference="maxcharlem"} to deduce the second equality in [\[maxcharlem1\]](#maxcharlem1){reference-type="eqref" reference="maxcharlem1"}. Assume now that $A\in\mathrm{S}_{n,+}$ is singular. Suppose that $\mathbf{u}^\top A\mathbf{u}=0$. Then $\mathbf{u}\in\ker A$. Hence, $\mathbf{y}^\top \mathbf{u}=0$. Therefore, its is enough to consider the case where $\mathbf{u}\in$range$\, A$. As $\mathbf{y}^\top\ker A=0$ it follows that $\mathbf{y}\in$range$\,A$. Let $C$ be the restriction of $A$, viewed as a linear operator $\mathbb{R}^n\to\mathbb{R}^n$, to range $A$. So $C$ is positive definite and we can use the previous case. Observe that $\mathbf{y}^\top C^{-1}\mathbf{y}=\mathbf{y}^\top A^\dagger\mathbf{y}$, and the first equality of [\[maxcharlem\]](#maxcharlem){reference-type="eqref" reference="maxcharlem"} follows. The second equality follows similarly. Suppose that $A\succeq \mathbf{y}\mathbf{y}^\top$. That is, $A=\mathbf{y}\mathbf{y}^\top +B$ for some $B\in\mathrm{S}_{n,+}$. Hence $\ker A=\ker (\mathbf{y}\mathbf{y}^\top)\cap\ker B$. Therefore $\mathbf{y}$ is orthogonal to $\ker A$. We now use the second equality in [\[maxcharlem1\]](#maxcharlem1){reference-type="eqref" reference="maxcharlem1"}. Observe that $\mathbf{u}^\top A\mathbf{u}\ge \mathbf{u}^\top (\mathbf{y}\mathbf{y}^\top)\mathbf{u}$. Hence $$\mathbf{y}^\top A^\dagger \mathbf{y}\le \mathbf{y}^\top (\mathbf{y}\mathbf{y}^\top)^\dagger\mathbf{y}=1.$$ ◻ **Corollary 4**. *Let $\mathrm{D}\subset\mathbb{R}^n$ be a convex domain and assume that $f$ is a nonconstant $a$-self-concordant function in $\mathrm{D}$. Suppose furthermore that $\nabla f(\mathbf{x})^\top \ker \partial^2 f(\mathbf{x})=0$ for each $\mathbf{x}\in\mathrm{D}$. Then $$\label{defconcconst1} \theta(f)=a^{-1}\sup_{\mathbf{x}\in \mathrm{D}}\nabla f(\mathbf{x})^\top (\partial^2f)^\dagger(\mathbf{x})\nabla f(\mathbf{x}).$$* This equality is well known if $\partial^2f$ is invertible [@NN94 top of page 16]. A simple example of a strongly standard self-concordant function for the interior of the cone $\mathbb{R}_+^n$, denoted as $\mathbb{R}_{++}^n$, where $\mathbb{R}_{++}=(0,\infty)$, is $$\label{logxbar} \begin{aligned} \sigma(\mathbf{x})=-\sum_{i=1}^n \log x_i,\\ \theta(\sigma)=n. \end{aligned}$$ The equality $a=1$ follows from the well known fact that the norm $\|\mathbf{x}\|_s$ is decreasing for $s\in[1,\infty]$. Use Corollary [Corollary 4](#charthetf){reference-type="ref" reference="charthetf"} to deduce the second equality of [\[logxbar\]](#logxbar){reference-type="eqref" reference="logxbar"}. Assume that $\mathrm{D}$ is a bounded convex domain, $\mathbf{x}\in\mathrm{D}$ and $\mathrm{L}$ is a line through $\mathbf{x}$. Denote by $d_{\max}(\mathbf{x},\mathrm{L})\ge d_{min}(\mathbf{x},\mathrm{L})$ the two distances from $\mathbf{x}$ to the end points of $\mathrm{L}\cap\partial\mathrm{D}$. Then $\textrm{sym}(\mathbf{x},\mathrm{D})$ is the infimum of $\frac{d_{\min}(\mathbf{x},L)}{ d_{\max}(\mathbf{x},L)}$ for all lines $\mathrm{L}$ through $\mathbf{x}$. Observe that if $\mathrm{B}(\mathbf{x},r)\subset \textrm{Closure}(\mathrm{D})\subset \mathrm{B}(\mathbf{x},R)$ then $\textrm{sym}(\mathbf{x},\mathrm{D})\ge r/R$. Recall that Renegar [@Ren01] deals only with strongly standard self-concordant functions. The complexity value $\theta(f)$, coined in [@Ren01], is called the parameter of barrier $f$ in [@NN94], and is considered only for self-concordant barrier in [@NN94 .1]. We now recall the complexity result to approximate the infimum of a linear functional on a bounded convex domain with whose boundary is given by a barrier function $\beta$. We normalize $\beta$ by assuming that it is strongly self-concordant. A simple implementation of the Newton's method is a \"short-step\" ipm's that follows the central path [@Ren01 .2]. The number of iterations to approximate the minimum of a linear functional within $\varepsilon$ precision starting with an intial point $\mathbf{x}'$ is [@Ren01 Theorem 2.4.1]: $$\label{Renthm} O\big(\sqrt{\theta(\beta)}\log\big(\frac{\theta(\beta)}{\varepsilon \textrm{sym}(\mathbf{x}',\mathrm{D})}\big)\big).$$ ## The number of iterations of ipm for matrix optimal transport {#subsec:mot} Assume that $1<m,n\in\mathbb{N}$. Let $\mathbf{p}=(p_1,\ldots,p_m)^\top,\mathbf{q}=(q_1,\ldots,q_n)^\top$ be two positive probability vectors. Denote by $\mathrm{U}(P)$ the set [\[defUPmat\]](#defUPmat){reference-type="eqref" reference="defUPmat"}, where $n_1=m, n_2=n$ and $\mathbf{p}_1=\mathbf{p}, \mathbf{p}_2=\mathbf{q}$. Set $\mathbf{1}_m=(1,\ldots,1)^\top\in\mathbb{R}^m$, and define $$\mathrm{U}_{0,2}=\{X=[x_{ij}]\in\mathbb{R}^{m\times n}, X\mathbf{1}_n=\mathbf{0}, X^\top \mathbf{1}_m=\mathbf{0}\}.$$ As the sum of all rows of $X$ is equal to the sum of all columns of $X$ it follows that $\dim\mathrm{U}_{0,2}=(m-1)(n-1)$. Let $\mathbf{e}_i=(\delta_{1i},\ldots,\delta_{mi})^\top\in\mathbb{R}^m, i\in[m]$ and $\mathbf{f}_j=(\delta_{1j},\ldots,\delta_{nj})^\top\in\mathbb{R}^n, j\in[n]$ be the standard bases in $\mathbb{R}^m$ and $\mathbb{R}^n$ respectively. Then one has a following simple basis in $\mathrm{U}_{0,2}$: $$\begin{aligned} \mathbf{g}_i\mathbf{h}_j^\top, i\in[m-1], j\in[n-1],\\ \mathbf{g}_i=\mathbf{e}_i-\mathbf{e}_{i+1}, i\in[m-1],\quad \mathbf{h}_j=\mathbf{f}_j-\mathbf{f}_{j+1},j\in[n-1]. \end{aligned}$$ The interior of $\mathrm{U}(P)$, denoted as $\mathrm{U}_o(P)$, is the set of positive matricers in the affine space $$\mathrm{U}_{af,2}(P)=\{X=\mathbf{p}\mathbf{q}^\top +\sum_{i=1}^{m-1}\sum_{j=1}^{n-1} t_{ij}\mathbf{g}_i\mathbf{h}_j^\top, \quad t_{ij}\in\mathbb{R}, i\in[m-1],j\in[n-1]\}.$$ Let $$\sigma(X)=-\sum_{i=1}^m\sum_{j=1}^n \log x_{ij}, \quad X=[x_{ij}]\in\mathbb{R}_{++}^{m\times n}$$ be a barrier function $\mathbb{R}_{++}^{m\times n}$. Recall that $\sigma$ is a standard self-concondant barrier with $\theta(\sigma)=mn$. The restriction of $\tilde \sigma$ to $\mathrm{U}_{af,2}\cap \mathbb{R}_{++}^{m\times n}$ is a standard self-concordant barrier with $$\label{thetsig} \theta(\tilde\sigma)\le mn.$$ **Theorem 5**. *Let $\mathbf{p}=(p_1,\ldots,p_m)^\top\in\mathbb{R}^m,\mathbf{q}=(q_1,\ldots,q_n)^\top\in\mathbb{R}^n$ be positive probability vectors. Consider the minimum problem [\[TOTmat\]](#TOTmat){reference-type="eqref" reference="TOTmat"} on the polytope $\mathrm{U}(P)$ given by [\[defUPmat\]](#defUPmat){reference-type="eqref" reference="defUPmat"}, where $n_1=m, n_2=n$ and $\mathbf{p}_1=\mathbf{p}, \mathbf{p}_2=\mathbf{q}$. The short step interior path algorithm with the barrier $\tilde \sigma$ starting at the point $\mathbf{p}\mathbf{q}^\top$ finds the value $\tau(C,P)$ within precision $\varepsilon>0$ in $$\label{ipmotmat1} O\big(\sqrt{mn}\log\frac{\sqrt{2}mn}{\varepsilon (\min_{i\in[m]}p_i)(\min_{j\in[n]}q_j)}\big)$$ iterations.* *Proof.* In view of [\[Renthm\]](#Renthm){reference-type="eqref" reference="Renthm"} and [\[thetsig\]](#thetsig){reference-type="eqref" reference="thetsig"} it is enough to show that $$\textrm{sym}(\mathbf{p}\mathbf{q}^\top,\mathrm{D})\ge (\min_{i\in[m]}p_i)(\min_{j\in[n]}q_j)\big)/\sqrt{2},\quad \mathrm{D}=\mathrm{U}_{af}(P)\cap \mathbb{R}_{++}^{m\times n}.$$ For $Y=[y_{ij}]\in \mathbb{R}^{m\times n}$ denote: $\|Y\|_2=\sqrt{\sum_{i=1}^m\sum_{j=1}^n y_{ij}^2}$. Assume that $X\in \textrm{Closure}(\mathrm{D})$. Then $X=[x_{ij}]$ is a probability matrix. Note that if $X\in\partial \mathrm{D}$ then $x_{ij}=0$ for some $i\in[m],j\in[n]$. Hence, for $X\in\partial \mathrm{D}$ we have the inequalities: $$\begin{aligned} \sqrt{2}\ge\|\mathbf{p}\mathbf{q}^\top-X\|_2\ge p_iq_j \textrm{ for some } i\in[m],j\in[n]\Rightarrow\\ \sqrt{2}\ge d_{\max}(\mathbf{p}\mathbf{q}^\top,L)\ge d_{\min}(\mathbf{p}\mathbf{q}^\top,L)\ge(\min_{i\in[m]}p_i)(\min_{j\in[n]}q_j)\big). \end{aligned}$$ ◻ ## The number iterations of ipm for tensor optimal transport {#subsec:totipm} Assume that $d>2$. We first consider the TOT of the form [\[TOT\]](#TOT){reference-type="eqref" reference="TOT"}. We now repeat the arguments of the previous subsection. Let $$\begin{aligned} \mathrm{U}_{0,d}=\{\mathcal{U}\in \otimes^d\mathbb{R}^n, \mathcal{U}\times_{\bar k} \mathcal{J}_{d-1}=0, k\in[d]\},,\\ \mathrm{U}_{af,d}(P)=\otimes_{k=1}^d \mathbf{p}_k +\mathrm{U}_{0,d}, \quad P=(\mathbf{p}_1,\ldots,\mathbf{p}_d) \end{aligned}$$ Then $\mathrm{U}(P)=\mathrm{U}_{af,d}(P)\cap \otimes_{k=1}^d \mathbb{R}_+^{n_k}$. The interior of $\mathrm{U}(P)$ is given by $\mathrm{U}_{af,d}\cap \otimes_{k=1}^d \mathbb{R}_{++}^{n_k}$. Let $$\sigma(\mathcal{X})=-\sum_{i_k\in[n_k], k\in[d]}\ \log x_{i_1,\ldots,i_d}, \quad \mathcal{X}\in\otimes_{k=1}^d\mathbb{R}_{++}^{n_k}$$ be a barrier function on $\otimes_{k=1}^d\mathbb{R}_{++}^{n_k}$. Thus, $\sigma$ is a standard self-concondant barrier with $\theta(\sigma)=\prod_{k=1}^d n_k$. The restriction of $\tilde \sigma$ to $\mathrm{U}_{af,2}\cap \mathbb{R}_{++}^{m\times n}$ is a standard self-concordant barrier with $$\label{thetsigd} \theta(\tilde\sigma)\le \prod_{k=1}^d n_k.$$ The arguments of the proof of Theorem [Theorem 5](#ipmotmat){reference-type="ref" reference="ipmotmat"} yield: **Theorem 6**. *Let $\mathbf{p}_k=(p_{1,k},\ldots,p_{n_k,k})^\top\in\mathbb{R}^{n_k}, k\in[d]$ be positive probability vectors. Consider the minimum problem [\[TOT\]](#TOT){reference-type="eqref" reference="TOT"} on the polytope $\mathrm{U}(P)$ given by [\[defU(P)ten\]](#defU(P)ten){reference-type="eqref" reference="defU(P)ten"}. The short step interior path algorithm with the barrier $\tilde \sigma$ starting at the point $\otimes_{k=1}^d\mathbf{p}_k$ finds the value $\tau(C,P)$ within precision $\varepsilon>0$ in $$\label{ipmotmat1} O\big(\sqrt{\prod_{k=1}^d n_k}\log\frac{\sqrt{2}\prod_{k=1}^d n_k}{\varepsilon \prod_{k=1}^d\min_{i_k\in[n_k]}p_{i_k,k}}\big)$$ iterations.* We now consider a variation of the polytope $\mathrm{U}(P)$, which correspond to the problem of $d$-dimensional stochastic tensors [@LZZ17]. For $\mathcal{U}=[u_{i_1,\ldots,i_d}]\in \otimes_{k=1}^d \mathbb{R}^{n_k}$ and a vector $\mathbf{x}=(x_1,\ldots,x_{n_k})^\top\in\mathbb{R}^{n_k}$ denote $$\begin{aligned} \mathcal{U}\times _k \mathbf{x}=\mathcal{W}=[w_{i_1,\ldots,i_{k-1},i_{k+1},\ldots,i_d} ]\in \otimes_{j\in[d]\setminus\{k\}}\mathbb{R}^{n_j},\\ w_{i_1,\ldots,i_{k-1},i_{k+1},\ldots,i_d}=\sum_{i_k=1}^{n_k} u_{i_1,\ldots,i_d} x_{i_k}. \end{aligned}$$ Define $$\label{defV(P)ten} \begin{aligned} \mathrm{V}(P)=\{\mathcal{V}=[v_{i_1,\ldots,i_d}]\in \otimes_{k=1}^d\mathbb{R}_+^{n_k}, \\ \mathcal{V}\times_{k} \mathbf{1}_{n_k}=\otimes_{j\in[d]\setminus{k}}\mathbf{p}_j, k\in[d]\}, P=(\mathbf{p}_1,\ldots,\mathbf{p}_d),\\ \end{aligned}$$ Let $$\label{defV0(P)} \begin{aligned} \mathrm{V}_{0,d}=\{\mathcal{V}=[v_{i_1,\ldots,i_d}]\in \otimes_{k=1}^d\mathbb{R}^{n_k}, \mathcal{V}\times_{k} \mathbf{1}_{n_k}=0\in[d]\},\\ \mathrm{V}_{af,d}(P)=\otimes_{k=1}^d \mathbf{p}_k+\mathrm{V}_{0,d}. \end{aligned}$$ Let $\mathbf{e}_{1,k},\ldots,\mathbf{e}_{n_k,k}\in\mathbb{R}^{n_k}$ be the standard basis in $\mathbb{R}^{n_k}$ for $k\in[d]$. Denote $$\mathbf{g}_{i,k}=\mathbf{e}_{i,k}-\mathbf{e}_{i+1,k} \textrm{ for } i\in[n_k-1].$$ Observe that $\mathbf{1}_{n_k}^\perp=$span$(\mathbf{g}_{1,k},\ldots,\mathbf{g}_{n_k-1,k})$ is the orthogonal complement of $\mathbf{1}_{n_k}$ in $\mathbb{R}^{n_k}$. We claim that $$\label{V0dstr} \mathrm{V}_{0,d}=\otimes_{k=1}^d\mathbf{1}_{n_k}^\perp.$$ Indeed, assume that $\mathcal{V}\in\otimes_{k=1}^d \mathbb{R}^{n_k}$ satisfies $\mathcal{V}\times_1\mathbf{1}_{n_1}=0$. View $\mathcal{V}$ as a matrix in $\mathbb{R}^{n_1}\otimes\big(\otimes_{k=2}^d\mathbb{R}^{n_k}\big)$. The above condition yields that range $\mathcal{V}\subset \mathbf{1}_{n_1}^\perp$, which is equivalent to $\mathcal{V}\in \mathbf{1}_{n_1}^\perp\otimes\big(\otimes_{k=2}^d \mathbb{R}^{n_k}\big)$. Apply this observation to $\mathrm{V}_{0,d}$ to deduce [\[V0dstr\]](#V0dstr){reference-type="eqref" reference="V0dstr"}. Hence, $$\mathrm{V}_{af,d}(P)=\{\mathcal{V}=\otimes_{k=1}^d \mathbf{p}_k +\sum_{i_k\in[n_k-1],k\in[d]} t_{i_1,\ldots,i_d}\otimes_{j=1}^d \mathbf{g}_{i_j,j}, [t_{i_1,\ldots,i_d}]\in\otimes_{l=1}^d \mathbb{R}^{n_l-1}\}.$$ Then $\mathrm{V}(P)=\mathrm{V}_{af,d}(P)\cap \otimes_{k=1}^d \mathbb{R}_+^{n_k}$. The interior of $\mathrm{V}(P)$ is given by $\mathrm{V}_{af,d}\cap \otimes_{k=1}^d \mathbb{R}_{++}^{n_k}$. The arguments of the proof of Theorem [Theorem 5](#ipmotmat){reference-type="ref" reference="ipmotmat"} yield: **Theorem 7**. *Let $\mathbf{p}_k=(p_{1,k},\ldots,p_{n_k,k})^\top\in\mathbb{R}^{n_k}, k\in[d]$ be positive probability vectors. Consider the minimum problem $\tau(\mathcal{C},P)= \min\{\langle \mathcal{C}, \mathcal{U}\rangle, \mathcal{U}\in \mathrm{V}(P)\}$. The short step interior path algorithm with the barrier $\tilde \sigma$ starting at the point $\otimes_{k=1}^d\mathbf{p}_k$ finds the value $\tau(C,P)$ within precision $\varepsilon>0$ in the number of iterations given by [\[ipmotmat1\]](#ipmotmat1){reference-type="eqref" reference="ipmotmat1"}.* ## Iteration estimates for certain probabilities $P$ {#subsec:cest} Our iteration estimate [\[ipmotmat1\]](#ipmotmat1){reference-type="eqref" reference="ipmotmat1"} depends on $P=(\mathbf{p}_1,\ldots,\mathbf{p}_d)$: the product of the minum values of the coordinates of $\mathbf{p}_k$ for $k\in[d]$. **Definition 8**. *A probability vector $\mathbf{p}=(p_1,\ldots,p_n)\in\mathbb{R}_{+}^n$ is called a weak $K-\ell$ uniform distribution if $$\label{defwKldist1} p_j\ge \frac{K}{n^{\ell}}, j\in[n],\quad K>0, \ell\ge 1, K\le n^{\ell-1}.$$* Note that if $K=\ell=1$ the $\mathbf{p}$ is the uniform distribution. Theorem [Theorem 6](#ipmotd){reference-type="ref" reference="ipmotd"} yields: **Corollary 9**. *Let the assumptions Theorem [Theorem 6](#ipmotd){reference-type="ref" reference="ipmotd"} hold. Assume that each $\mathbf{p}_j$ is a weak $K-\ell$ uniform distribution. Then the short step interior path algorithm with the barrier $\tilde \sigma$ starting at the point $\otimes_{k=1}^d\mathbf{p}_k$ finds the value $\tau(C,P)$ within precision $\varepsilon>0$ in $$\label{cipmotmat1} O\big(\sqrt{\prod_{k=1}^d n_k}\log (\sqrt{2}\varepsilon^{-1}K^{-d}\prod_{k=1}^d n_k^{1+\ell})\big)$$ iterations. In particular, if $n_1=\cdots=n_d=n$ then the above estimate is $O\big(n^{d/2}\log(\sqrt{2}\varepsilon^{-1} K^{-d} n^{d(1+\ell)})\big)$.* # Acknowledgment {#acknowledgment .unnumbered} The author is partially supported by the Simons Collaboration Grant for Mathematicians. MMM J. Altschuler, J. Weed and P. 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arxiv_math

--- abstract: | We introduce the notion of *self-concordant smoothing* for minimizing the sum of two convex functions: the first is smooth and the second may be nonsmooth. Our framework results naturally from the smoothing approximation technique referred to as *partial smoothing* in which only a part of the nonsmooth function is smoothed. The key highlight of our approach is in a natural property of the resulting problem's structure which provides us with a variable-metric selection method and a step-length selection rule particularly suitable for proximal Newton-type algorithms. In addition, we efficiently handle specific structures promoted by the nonsmooth function, such as $\ell_1$-regularization and group-lasso penalties. We prove local quadratic convergence rates for two resulting algorithms: `Prox-N-SCORE`, a proximal Newton algorithm and `Prox-GGN-SCORE`, a proximal generalized Gauss-Newton (GGN) algorithm. The `Prox-GGN-SCORE` algorithm highlights an important approximation procedure which helps to significantly reduce most of the computational overhead associated with the inverse Hessian. This approximation is essentially useful for overparameterized machine learning models and in the mini-batch settings. Numerical examples on both synthetic and real datasets demonstrate the efficiency of our approach and its superiority over existing approaches. author: - | Adeyemi D. Adeoye $^{1}$ \>8 [^1] , Alberto Bemporad $^{1}$ [^2] ¶\ \ bibliography: - references.bib title: | Self-concordant Smoothing for Convex Composite Optimization --- # Introduction We consider the composite optimization problem $$\begin{aligned} \min\limits_{x\in{\mathbb R}^n} {\mathcal{L}}(x) \coloneqq f(x) + g(x), \label{eq:prob}\end{aligned}$$ where $f$ is a smooth, convex loss function and $g$ is a closed, proper, convex regularization function which may be nonsmooth. The smoothing approximation framework for solving [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"} involves solving a smooth approximation of ${\mathcal{L}}(x)$ where the nonsmooth function $g$ is sequentially replaced by a smooth function $g_{\hat{s}}$ such that with an efficient algorithm for solving the resulting smooth optimization problem, we may approach the solution of the original problem. In this setting, problem [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"} becomes $$\begin{aligned} \min\limits_{x\in{\mathbb R}^n} {\mathcal{L}}_{\hat{s}}(x) \coloneqq f(x) + g_{\hat{s}}(x). \label{eq:fullsmooth-prob}\end{aligned}$$ However, as noted in [@beck2012smoothing], the nonsmooth function $g$ in [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"} often plays a key role in describing some desirable properties specific to the application in which it appears, such as sparsifying the solution of the problem or enforcing some constraints on $x$, e.g., in sparse signal recovery and image processing [@chambolle2004algorithm; @beck2009gradient], compressed sensing [@donoho2006compressed; @candes2006robust], model predictive control of constrained dynamical systems [@bemporad2002explicit; @richter2009real; @stella2017simple], neural network training [@bemporad2022recurrent], as well as various classification and regression problems in machine learning. In order to retain such properties about the optimization vector $x$ in these applications, [@beck2012smoothing] proposes to keep $g$ unchanged and hence considers a *partial smoothing*. In this case, $g$ is partially approximated, and problem [\[eq:fullsmooth-prob\]](#eq:fullsmooth-prob){reference-type="eqref" reference="eq:fullsmooth-prob"} is replaced by the following optimization problem: $$\begin{aligned} \min\limits_{x\in{\mathbb R}^n} {\mathcal{L}}_s(x) \coloneqq f(x) + \mathop{g_s}\nolimits(x;\mu) + g(x). \label{eq:partialsmooth-prob}\end{aligned}$$ Sometimes, we are interested in more general structures in the solution estimates of [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"}. Prominent cases are found in the lasso and multi-task regression problems with structured sparsity-inducing penalties. In these problems, the regularization function $g(x)$ takes on the form $g(x) = {\mathcal{R}}(x) + \Omega(x)$ where both ${\mathcal{R}}(x)$ and $\Omega(x)$ induce particular structures on the solution estimates. In particular, ${\mathcal{R}}(x)$ is the scaled $\ell_1$-norm penalty $\beta\norm{x}_1$ that encourages sparse estimates of $x$, and $\Omega(x)$ additionally enforces a more specific structure on these estimates, such as groups and fused structures. We do not give a special attention to the particular structure induced by $g(x)$ in our presentation. Yet in , we propose an approach to incorporate certain known structures into our framework, thereby making it amenable to more general structured regularization functions. In particular, for the lasso and multi-task regression problems with structured sparsity-inducing penalties, we consider the equivalence between the Nesterov's smoothing framework and the smoothing framework used in this work, and then synthesize the so-called "prox-decomposition\" property of $g(x)$ with the smoothness property of $g_s(x)$ for easily handling the structures promoted by $g(x)$ on the solution. Two extra points about the partial smoothing formulation in [\[eq:partialsmooth-prob\]](#eq:partialsmooth-prob){reference-type="eqref" reference="eq:partialsmooth-prob"}, vital to the framework proposed in this paper, are highlighted as follows: 1. The first is the generalization of the Moreau-infimal smoothing technique via the infimal convolution smoothing framework developed in [@beck2012smoothing]. We observe that, under certain conditions, this generalization naturally exhibits a structure characterized by the self-concordant regularization (SCORE) framework of [@adeoye2023score]. 2. Secondly, via the notion of *epi-smoothing functions* established in [@burke2013epi] (a weaker notion than the *smoothable functions* of [@beck2012smoothing]), we can combine the partial smoothing technique with the Moreau-infimal-based (proximal) algorithms to handle the nonsmooth function $g$, supposing we can find an efficient method to compute a closed-form solution to the minimization of the sum of $g$ and an auxiliary function $h(x)$. This combination is known to be more efficient in solving [\[eq:partialsmooth-prob\]](#eq:partialsmooth-prob){reference-type="eqref" reference="eq:partialsmooth-prob"} (and hence [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"}) than others typically used in classical algorithms --- such as the subgradient and bundle-type algorithms --- which are employed for directly solving the nonsmooth problem. For instance, with the "fast\" variants of the Moreau-infimal-based algorithms, we can recover an $\mathcal{O}(1/\epsilon)$ complexity rate for any $\epsilon$-optimal solution (in the function value) --- a significant improvement over the $\mathcal{O}(1/\epsilon^2)$ complexity rate for a typical subgradient or bundle-type method [@nesterov2005smooth; @beck2012smoothing]. The fast Moreau-infimal/proximal gradient schemes are essentially characterized by their $\mathcal{O}(1/k^2)$ convergence rates which are achieved by combining one or two proximal (or projected) gradient steps with an extrapolation step (see, e.g., [@nesterov1983method; @nesterov2005smooth; @tseng2008accelerated; @beck2009fast]). While the fast gradient methods prove to be more efficient than the subgradient and bundle-type methods, they are first-order methods which often fall back to weak solution estimates and accuracy. It is evident, from their performance on unconstrained smooth optimization problems, that incorporating second-order information into a gradient scheme often yields superior performance and better solution quality compared to first-order schemes. Efforts have been made (e.g., in [@becker2012quasi; @lee2014proximal; @patrinos2014forward; @tran2015composite; @stella2017forward]) to incorporate (approximate) second-order information into proximal gradient schemes to achieve faster convergence rates with much better solution quality and accuracy, thereby emulating the performance of their relative second-order methods for unconstrained smooth problems. The computational overhead associated with proximal algorithms that use second-order information is often largely mitigated by choosing a special structure for the matrix of the second-order terms of $f$, or by exploiting the specific structure of the function $f$ itself. For example, the authors in [@tran2015composite] assume a self-concordant structure of $f$ which allows to develop efficient step-size and correction techniques for proximal Newton-type and proximal quasi-Newton algorithms. However, because $f$ oftentimes define a loss or data-misfit in real-world applications, the self-concordant assumption is not easy to check for many of these applications. Our self-concordant smoothing framework in this work provides a remedy to this limitation. We propose a new step-size selection technique that is suitable for Newton-type and quasi-Newton-type methods under the infimal convolution smoothing framework. This also exploits a self-concordant structure albeit not imposed on any of functions $f$ and $g$ that define the original problem, but that arise naturally from the infimal convolution smoothing framework under certain conditions, as we shall highlight in the next section. Burke and Hoheisel [@burke2013epi; @burke2017epi] developed the notion of *epi-smoothing* for studying several epigraphical convergence (*epi-convergence*) properties for convex composite functions by combining the infimal convolution smoothing framework due to Beck and Teboulle [@beck2012smoothing] with the idea of *gradient consistency* due to Chen [@chen2012smoothing]. The key variational analysis tool used throughout their development is the *supercoercivity* of the class of regularization kernels studied in [@beck2012smoothing]. In particular, they establish the close connection between epi-convergence of the regularization functions and supercoercivity of the regularization kernel. We synthesize this idea with the notion of *self-concordant regularization* [@adeoye2023score] and the proximal Newton-type algorithms to propose two proximal-type algorithms, viz., `Prox-N-SCORE` (Algorithm [\[alg:NewtonSCOREProx\]](#alg:NewtonSCOREProx){reference-type="ref" reference="alg:NewtonSCOREProx"}) and `Prox-GGN-SCORE` (Algorithm [\[alg:GGNSCOREProx\]](#alg:GGNSCOREProx){reference-type="ref" reference="alg:GGNSCOREProx"}), for convex composite minimization. #### Paper organization. The rest of this paper is organized as follows: In , we present some notations and background on convex analysis. In , we establish our self-concordant smoothing notion with some properties and results. We describe our proximal Newton-type scheme in , and present the `Prox-N-SCORE` and `Prox-GGN-SCORE` algorithms. In , we describe an approach for handling specific structures promoted by the nonsmooth function $g$ in problem [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"}, and propose a practical extension of the so-called *prox-decomposition* property of $g$ for the self-concordant smoothing framework, which has certain in-built smoothness properties. Convergence rates are analyzed for the `Prox-N-SCORE` and `Prox-GGN-SCORE` algorithms in . In , we present some numerical simulation results for our proposed framework with an accompanying Julia package, and compare the results with other state-of-the-art approaches. Finally, we give a concluding remark and discuss prospects for future research in . ## Notation and preliminaries {#sec:notations} We denote by $\bar{{\mathbb R}}\coloneqq {\mathbb R}\cup \{-\infty, +\infty\}$ the set of extended real numbers. ${\mathbb R_{+}}\coloneqq [0, +\infty[$ and ${\mathbb P}\coloneqq {\mathbb R_{+}}\backslash \{0\}$, respectively, denote the set of nonnegative and positive real numbers. Let $\mathop{g\colon{\mathbb R}^n\to {\mathbb R}\cup \{+\infty\}}\nolimits$ be an extended real-valued function. The *(effective) domain* of $g$ is given by $\mathop{\rm dom}\nolimits{g} \coloneqq \{x\in {\mathbb R}^n \mathop{\, | \,}\nolimits g(x) < +\infty \}$ and its *epigraph* (resp., *strict epigraph*) is given by $\mathop{\rm epi}\nolimits{g} \coloneqq \{(x, \gamma)\in {\mathbb R}^n \times {\mathbb R}\mathop{\, | \,}\nolimits g(x) \le \gamma \}$ (resp., $\mathop{\rm epi_s}\nolimits{g} \coloneqq \{(x, \gamma)\in {\mathbb R}^n \times {\mathbb R}\mathop{\, | \,}\nolimits g(x) < \gamma \}$). Given $\gamma \in {\mathbb P}$, the $\gamma$-sublevel set of $g$ is $\Gamma_\gamma(g) \coloneqq \{x\in {\mathbb R}^n \colon g(x) \le \gamma\}$. The standard inner product between two vectors $x,y\in{\mathbb R}^n$ is denoted by $\left<\cdot,\cdot\right>$, that is, $\left<x,y\right> \coloneqq x^\top y$. For an $n\times n$ matrix $H$, we write $H\succ 0$ (resp. $H\succeq 0$) to say $H$ is positive definite (resp., positive semidefinite). ${\mathcal S_{+}^{n}}$ and ${\mathcal S_{++}^{n}}$ respectively denotes the set of $n\times n$ symmetric positive semidefinite and symmetric positive definite matrices. The set $\set{\mathop{\rm diag}\nolimits(v)\mathop{\, | \,}\nolimits v\in{\mathbb R}^n}$, where $\mathop{\rm diag}\nolimits\colon {\mathbb R}^n \to {\mathbb R}^{n\times n}$, defines the set of all diagonal matrices in ${\mathbb R}^{n\times n}$. $I_d$ denotes the $d\times d$ identity matrix. We denote by $\mathop{\rm card}\nolimits({\mathcal{G}})$, the cardinality of a (nonempty) set ${\mathcal{G}}$. For any two functions $f$ and $g$, we define $(f\circ g)(\cdot) \coloneqq f(g(\cdot))$. We denote by ${\mathcal{C}}^k({\mathbb R}^n)$, the class of $k$ times continuously differentiable functions on ${\mathbb R}^n$, $k\ge 0$. A continuous function $g$ belongs to the class ${\mathcal{C}}^0(\mathop{\rm dom}\nolimits g)$. If the $p$-th derivative of a function $g\in {\mathcal C^{k}}({\mathbb R}^n)$ is $L_g$-Lipschitz continuous with $p\le k$, $L_g\ge0$, we write $g\in {\mathcal{C}}_{L_g}^{k,p}({\mathbb R}^n)$. The notation $\norm{\cdot}\equiv\norm{\cdot}_2$ stands for the standard Euclidean (or $2$-) norm. We define the weighted norm induced by $H \in {\mathcal S_{++}^{n}}$ by $\norm{x}_H \coloneqq \left<Hx, x\right>^{\frac{1}{2}}$, for $x\in {\mathbb R}^n$. The associated *dual* norm is $\norm{x}_H^* \coloneqq \left<H^{-1}x, x\right>^{\frac{1}{2}}$. An Euclidean ball of radius $r$ centered at $\bar{x}$ is denoted by $\mathcal{B}_r(\bar{x}) \coloneqq \{x \in {\mathbb R}^n \mathop{\, | \,}\nolimits\norm{x - \bar{x}} \le r \}$. Associated with a given $H \in {\mathcal S_{++}^{n}}$, the (Dikin) ellipsoid of radius $r$ centered at $\bar{x}$ is defined by ${\mathcal{E}}_r(\bar{x})\coloneqq \{x\in {\mathbb R}^n \mathop{\, | \,}\nolimits\norm{x-\bar{x}}_H < r\}$. A convex function $\mathop{g\colon{\mathbb R}^n\to {\mathbb R}\cup \{+\infty\}}\nolimits$ is said to be *proper* if $\mathop{\rm dom}\nolimits{g} \ne \emptyset$. The function $g$ is said to be lower semicontinuous (lsc) at $y$ if $g(y) \le \liminf\limits_{x\to y} g(x)$; if it is lsc at every $y\in \mathop{\rm dom}\nolimits g$, then it is said to be lsc on $\mathop{\rm dom}\nolimits g$. We denote by $\mathop{\Gamma_0(D)}\nolimits$ the set of proper convex lsc functions from $D \subseteq {\mathbb R}^n$ to ${\mathbb R}\cup \{+\infty\}$. Given $g\in {\mathcal C^{3}}(\mathop{\rm dom}\nolimits{g})$, we respectively denote by $g'_t(t)$, $g''_t(t)$ and $g'''_t(t)$ the first, second and third derivatives of $g$, at $t\in {\mathbb R}$, and by $\mathop{\nabla}\nolimits_x g(x)$, $\mathop{\nabla^2}\nolimits_x g(x)$, and $\mathop{\nabla^3}\nolimits_x g(x)$ the gradient, Hessian and third-order derivative tensor of $g$, respectively, at $x\in {\mathbb R}^n$; if the variables with respect to which the derivatives are taken are clear from context, the subscripts are omitted. If $\mathop{\nabla^2}\nolimits g(x) \succ 0$ for a given $x\in {\mathbb R}^n$, then the *local* norm $\norm{\cdot}_x$ with respect to $g$ at $x$ is defined by $\norm{d}_x\coloneqq \left<\mathop{\nabla^2}\nolimits g(x)d,d\right>^{1/2}$, the weighted norm of $d$ induced by $\mathop{\nabla^2}\nolimits g(x)$. The associated dual norm is $\norm{v}_x^*\coloneqq \left<\mathop{\nabla^2}\nolimits g(x)^{-1}v,v\right>^{1/2}$, for $v\in {\mathbb R}^n$. The subdifferential $\partial g \colon {\mathbb R}^n \to 2^{{\mathbb R}^n}$ of a proper function $\mathop{g\colon{\mathbb R}^n\to {\mathbb R}\cup \{+\infty\}}\nolimits$ is defined by $x \mapsto \set{u \in {\mathbb R}^n \mathop{\, | \,}\nolimits(\forall y \in {\mathbb R}^n) \, \left<y-x,u\right> + g(x) \le g(y)}$, where $2^{{\mathbb R}^n}$ denotes the set of all subsets of ${\mathbb R}^n$. The function $g$ is said to be subdifferentiable at $x\in {\mathbb R}^n$ if $\partial g(x) \ne \emptyset$; the subgradients of $g$ at $x$ are the elements of $\partial g(x)$. We say that a function $\mathop{g\colon{\mathbb R}^n\to {\mathbb R}\cup \{+\infty\}}\nolimits$ is coercive if $\liminf\limits_{\norm{x} \to \infty}g(x) = +\infty$, and supercoercive if $\liminf\limits_{\norm{x} \to \infty}\frac{g(x)}{\norm{x}} = +\infty$. The sequence $\{g_k\}$ of functions $g_k\colon {\mathbb R}^n \to \bar{{\mathbb R}}$ is said to epi-converge to the function $g\colon {\mathbb R}^n \to \bar{{\mathbb R}}$ if $\lim\limits_{k\to\infty}\mathop{\rm epi}\nolimits{g_k} = \mathop{\rm epi}\nolimits{g}$; it is said to continuously converge to $g$ if for all $x\in {\mathbb R}^n$ and $\{x_k\}\to x$, we have $\lim\limits_{k\to\infty}g_k = g$; and it converges pointwise to $g$ if for all $x\in{\mathbb R}^n$, $\lim\limits_{k\to\infty}g_k(x) = g(x)$. Epi-convergence, continuous convergence, and pointwise convergence of $\{g_k\}$ to $g$ are respectively denoted by $\mathop{\rm \textrm{e--}\lim}\nolimits g_k=g$ (or $g_k\mathop{\underrightarrow{e}}\nolimits g$), $\mathop{\rm \textrm{c--}\lim}\nolimits g_k=g$ (or $g_k\mathop{\underrightarrow{c}}\nolimits g$), and $\mathop{\rm \textrm{p--}\lim}\nolimits g_k=g$ (or $g_k\mathop{\underrightarrow{p}}\nolimits g$). The conjugate (or Fenchel conjugate, or Legendre transform, or Legendre--Fenchel transform) $\mathop{g^*\colon{\mathbb R}^n\to {\mathbb R}\cup \{+\infty\}}\nolimits$ of a function $\mathop{g\colon{\mathbb R}^n\to {\mathbb R}\cup \{+\infty\}}\nolimits$ is the mapping $y \mapsto \sup\limits_{x\in{\mathbb R}^n}\left\{\left<x, y\right> - g(x)\right\}$, and its biconjugate is $g^{**} = (g^*)^*$. The (scaled) proximal operator associated with a function $g\in \mathop{\Gamma_0({\mathbb R})}\nolimits$ is written as $\mathop{\rm prox}\nolimits_{g}^{Q}\left(\cdot\right)$, and is defined by the unique point in $\mathop{\rm dom}\nolimits g$ that satisfies $$\begin{aligned} \mathop{\fullfunction{\mathop{g \square \frac{1}{2}\norm{\cdot}_{Q}^2}\nolimits}{{\mathbb R}^n}{{\mathbb R}\cup \{+\infty\}}{x}{g(\mathop{\rm prox}\nolimits_{g}^{Q}(x))+\frac{1}{2}\norm{x - \mathop{\rm prox}\nolimits_{g}^{Q}(x)}_Q^2}}\nolimits.\end{aligned}$$ A key property of the $\mathop{\rm prox}\nolimits_{g}^{Q}$ operator is its *nonexpansivenss*: $$\begin{aligned} \label{eq:nonexpansive} \norm{\mathop{\rm prox}\nolimits_{g}^Q(x) - \mathop{\rm prox}\nolimits_{g}^Q(y)}_Q \le \norm{x-y}_Q,\quad \norm{\mathop{\rm prox}\nolimits_{g}^Q(x) - \mathop{\rm prox}\nolimits_{g}^Q(y)}_y \le \norm{x-y}_y^*,\end{aligned}$$ for all $x,y\in{\mathbb R}^n$. # Smoothing via self-concordant regularization {#sec:smoothing} Since the pioneering work of Nesterov and Nemirovskii [@nesterov1994interior] on interior-point methods, the notion of self-concordant functions has helped to better understand the importance of exploiting the problem's structure to improve performance of optimization algorithms. This section introduces our notion of self-concordant smoothing which provides us with structures to exploit in composite optimization problems. We begin by presenting the definition of generalized self-concordant functions given in [@sun2019generalized]. **Definition 1** (Generalized self-concordant function on ${\mathbb R}$). *A univariate convex function $g\in {\mathcal C^{3}}(\mathop{\rm dom}\nolimits g)$, with $\mathop{\rm dom}\nolimits g$ open, is said to be $(M_g,\nu)$-generalized self-concordant, with $M_g\in{\mathbb R_{+}}$ and $\nu\in {\mathbb P}$, if $$\begin{aligned} \abs{\mathop{g'''}\nolimits(t)} \le M_g \mathop{g''}\nolimits(t)^\frac{\nu}{2}, \qquad \forall t \in {\mathbb R}. \end{aligned}$$* **Definition 2** (Generalized self-concordant function on ${\mathbb R}^n$). *A convex function $g\in {\mathcal C^{3}}(\mathop{\rm dom}\nolimits g)$, with $\mathop{\rm dom}\nolimits g$ open, is said to be $(M_g,\nu)$-generalized self-concordant of order $\nu\in {\mathbb P}$, with $M_g\in{\mathbb R_{+}}$, if $\forall x \in \mathop{\rm dom}\nolimits{g}$ $$\begin{aligned} \abs{\left<\mathop{\nabla^3}\nolimits{g}(x)[v]u,u\right>} \le M_g\norm{u}_x^2\norm{v}_x^{\nu-2}\norm{v}^{3-\nu}, \qquad \forall u,v \in {\mathbb R}^n, \end{aligned}$$ where $\nabla^3{g}(x)[v] \coloneqq \lim\limits_{t\to 0} \left\{\left(\nabla^2{g}(x + tv)-\nabla^2{g}(v)\right)/{t}\right\}$.* Note that for an $(M_g,\nu)$-generalized self-concordant function $g$ defined on ${\mathbb R}^n$, the univariate function $\varphi\colon{\mathbb R}\to{\mathbb R}$ defined by $\varphi(t)\coloneqq g(x + tv)$ is $(M_g,\nu)$-generalized self-concordant for every $x,v\in \mathop{\rm dom}\nolimits g$ and $x + tv \in \mathop{\rm dom}\nolimits g$. This provides an alternative definition for the generalized self-concordant function on ${\mathbb R}^n$. A key observation from the above definition is the possibility to extend the theory beyond the case $\nu = 3$ and $u=v$ originally presented in [@nesterov1994interior]. This observation, for instance, allowed the authors in [@bach2010self] to introduce a *pseudo* self-concordant framework, in which $\nu = 2$, for the analysis of logistic regression. In a recent development, the authors in [@ostrovskii2021finite] identified a new class of pseudo self-concordant functions and showed how these functions may be slightly modified to make them *standard* self-concordant (i.e., where $M_g=2, \nu=3, u=v$), while preserving desirable structures. With such generalizations, and stemming from the idea of *Newton decrement* in [@nesterov1994interior], new analytic step-size selection and correction techniques for a number of proximal algorithms were developed in [@tran2015composite]. It is in the same spirit that we propose new step-size selection techniques from the self-concordant smoothing framework developed in this paper. We denote by ${{\mathcal{F}}_{M_{g},\nu}}$ the class of $(M_g,\nu)$-generalized self-concordant functions, with generalized self-concordant parameters $M_g\in{\mathbb R_{+}}$ and $\nu\in {\mathbb P}$. We would like to mention that the possibility to approximate nonsmooth convex functions by generalized self-concordant ones was briefly noted with examples in a subsection of [@sun2019generalized]. Our goal in the following subsections is to concretely establish this notion particularly for algorithmic purposes. ## Self-concordant smoothing {#sec:self-conc} **Definition 3** (Self-concordant smoothing function). *We say that the parameterized function $g_s\colon {\mathbb R}^n \times {\mathbb P}\to {\mathbb R}$ is a self-concordant smoothing function for function $g\in\mathop{\Gamma_0({\mathbb R}^n)}\nolimits$ if the following two conditions are satisfied:* 1. *$\mathop{\rm \textrm{e--}\lim}\nolimits\limits_{\mu\downarrow 0} \mathop{g_s}\nolimits= g$.[\[ass:sc1\]]{#ass:sc1 label="ass:sc1"}* 2. *$\mathop{g_s}\nolimits(\cdot; \mu) \in {{\mathcal{F}}_{M_{g},\nu}}$. [\[ass:sc2\]]{#ass:sc2 label="ass:sc2"}* By construction, the class of functions exhibiting the property in [\[ass:sc1\]](#ass:sc1){reference-type="ref" reference="ass:sc1"} inherits the gradient and/or the Jacobian consistency properties introduced in [@chen2012smoothing] and [@chen1998global], respectively. In [@burke2013epi Lemma 3.4], the authors show the following property for epi-convergent smoothing functions (that is, the ones for which condition [\[ass:sc1\]](#ass:sc1){reference-type="ref" reference="ass:sc1"} holds): $$\begin{aligned} \limsup_{\substack{x\to\bar{x}\\\mu\downarrow 0}} \mathop{\nabla}\nolimits g_s(x) = \partial g(\bar{x}).\label{eq:grad-consistency}\end{aligned}$$ The gradient consistency property holds upon taking the convex hull on both sides of [\[eq:grad-consistency\]](#eq:grad-consistency){reference-type="eqref" reference="eq:grad-consistency"} (see [@chen2012smoothing Equation 4]). However, since $g\in\mathop{\Gamma_0({\mathbb R}^n)}\nolimits$, then $g$ is subdifferentially regular at any point $\bar{x}\in\mathop{\rm dom}\nolimits g$ (see [@rockafellar2009variational Definition 7.25 and Example 7.27]), and hence, as noted in [@burke2013epi], the equivalence between [\[eq:grad-consistency\]](#eq:grad-consistency){reference-type="eqref" reference="eq:grad-consistency"} and gradient/Jacobian consistency holds. Other nice properties of a self-concordant smoothing function are discussed in subsequent subsections. We denote by ${\mathcal{S}_{M_{g},\nu}^{\mu}}$ the set of self-concordant smoothing functions for a function $g\in\mathop{\Gamma_0({\mathbb R}^n)}\nolimits$, that is, ${\mathcal{S}_{M_{g},\nu}^{\mu}}\coloneqq\set{g_s\colon {\mathbb R}^n \times {\mathbb P}\to {\mathbb R}\mathop{\, | \,}\nolimits g_s \mathop{\underrightarrow{e}}\nolimits g,~ g_s \in {{\mathcal{F}}_{M_{g},\nu}}}$. ## Self-concordant smoothing via infimal convolution We next present some important elements of smoothing via infimal convolution which generalizes the Moreau-Yosida proximal regularization framework. **Definition 4** (Infimal convolution). *Let $g$ and $h$ be two functions from ${\mathbb R}^n$ to ${\mathbb R}\cup \{+\infty\}$. The infimal convolution (or "inf-convolution\" or "inf-conv\")[^3] of $g$ and $h$ is the function $\mathop{g \square h}\nolimits\colon {\mathbb R}^n \to \bar{{\mathbb R}}$ defined by $$\begin{aligned} (\mathop{g \square h}\nolimits)(x) = \inf\limits_{w\in{\mathbb R}^n}\left\{g(w)+h(x-w)\right\}. \label{eq:infconv} \end{aligned}$$* The infimal convolution of $g$ with $h$ is said to be *exact at $x \in\mathop{\rm dom}\nolimits g$* if the infimum [\[eq:infconv\]](#eq:infconv){reference-type="eqref" reference="eq:infconv"} is attained. It is *exact* if it is exact at each $x \in\mathop{\rm dom}\nolimits g$, in which case we write $\mathop{g \boxdot h}\nolimits$. Of utmost importance about the inf-conv operation in this paper is its use in the approximation of a function $g\in\mathop{\Gamma_0({\mathbb R}^n)}\nolimits$; that is, the approximation of $g$ by its infimal convolution with a member $h_\mu(\cdot)$ of a parameterized family ${\mathcal{H}}\coloneqq \{h_\mu \, \mathop{\, | \,}\nolimits\, \mu \in {\mathbb P}\}$ of (regularization) kernels. This approximation technique generalizes the Moreau-Yosida regularization process in which $h_\mu(\cdot)=\norm{\cdot}^2/(2\mu)$. In more formal terms, we recall the notion of inf-conv regularization in below. For $h\in \mathop{\Gamma_0({\mathbb R}^n)}\nolimits$ and $\mu \in {\mathbb P}$, we define the function $\mathop{h_\mu\colon{\mathbb R}^n\to {\mathbb R}\cup \{+\infty\}}\nolimits$ by the *epi-multiplication* operation[^4] $$\begin{aligned} h_\mu(\cdot) \coloneqq \mu h\left(\frac{\cdot}{\mu}\right), \quad \mu \in {\mathbb P}. \label{eq:hmu}\end{aligned}$$ **Definition 5** (Inf-conv regularization). *Let $g$ be a function in $\mathop{\Gamma_0({\mathbb R}^n)}\nolimits$. Define $$\begin{aligned} {\mathcal{H}}\coloneqq \left\{(x, w) \mapsto h_\mu(x-w) \mathop{\, | \,}\nolimits x,w\in {\mathbb R}^n, \mu \in {\mathbb P}\right\}, \label{eq:kernelfam} \end{aligned}$$ a parameterized family of regularization kernels. The inf-conv regularization process of $g$ with $h_\mu \in {\mathcal{H}}$ is given by $(\mathop{g \square h}\nolimits_\mu)(x)$, for any $x\in {\mathbb R}^n$.* In the sequel, we assume that the regularization kernel function $h$ is of the form $$\begin{aligned} h(x) = \sum_{i=1}^{n} \phi(x_i), \label{eq:hseparable}\end{aligned}$$ where $\phi$ is a univariate *potential function*. We note that, by convexity, $g$ is lsc and hence $g\in \mathop{\Gamma_0({\mathbb R}^n)}\nolimits$. We are left with the question of what properties must hold for $\phi$ such that $\mathop{g \square h_\mu}\nolimits$ produces $g_s$ satisfying the self-concordant smoothing conditions [\[ass:sc1\]](#ass:sc1){reference-type="ref" reference="ass:sc1"} -- [\[ass:sc2\]](#ass:sc2){reference-type="ref" reference="ass:sc2"}. To this end, we impose the following conditions on $\phi$: 1. $\phi$ is supercoercive. [\[ass:k1\]]{#ass:k1 label="ass:k1"} 2. $\phi \in {{\mathcal{F}}_{M_{\phi},\nu}}$. [\[ass:k2\]]{#ass:k2 label="ass:k2"} Many functions that appear in different settings naturally exhibit the structures in conditions [\[ass:k1\]](#ass:k1){reference-type="ref" reference="ass:k1"} -- [\[ass:k2\]](#ass:k2){reference-type="ref" reference="ass:k2"}. For example, the ones belonging to the class of *Bregman/Legendre functions* introduced by Bauschke and Borwein [@bauschke1997legendre] (see also [@de1986relaxed] for a related characterization of the class of *Bregman functions*). In the context of proximal gradient algorithms for solving [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"}, the recent paper by Bauschke and Borwein [@bauschke2017descent] enlists these functions as satisfying the new descent lemma (a.k.a *descent lemma without Lipschitz gradient continuity*) which the paper introduced. We summarize examples of these regularization kernel functions in **Table** [\[tab:kernel-functions\]](#tab:kernel-functions){reference-type="ref" reference="tab:kernel-functions"}, and extract two practical examples for the smoothing of the $1$-norm below. **Remark 1**. *Suppose that $\mathop{\rm dom}\nolimits h$ is a nonempty bounded subset of ${\mathbb R}^n$, for example, if $\phi \in \mathop{\Gamma_0({\mathbb R})}\nolimits$, then since we have that $g\in\mathop{\Gamma_0({\mathbb R}^n)}\nolimits$ is bounded below as it possesses a continuous affine minorant (in view of [@bauschke2011convex Theorem 9.20]), the less restrictive condition that $\phi$ is coercive sufficiently replaces the condition [\[ass:k1\]](#ass:k1){reference-type="ref" reference="ass:k1"}. In other words, the convergence notion presented below hold similarly for the resulting function $\mathop{g \square h_\mu}\nolimits$ in this case. Our examples in **Table** [\[tab:kernel-functions\]](#tab:kernel-functions){reference-type="ref" reference="tab:kernel-functions"} therefore include both coercive and supercoercive functions, where in either case, we have $\phi \in {{\mathcal{F}}_{M_{\phi},\nu}}$.* #### Examples (Infimal convolution of $\|\cdot\|_1$ with $h_\mu$). For some functions $g$ and $h_\mu$, there exists a closed form solution to $\mathop{g \square h_\mu}\nolimits$. On the other hand, if one gets that $\mathop{g \square h_\mu}\nolimits=\mathop{g \boxdot h_\mu}\nolimits\in \mathop{\Gamma_0({\mathbb R}^n)}\nolimits$, e.g., as a result of [\[thm:exact\]](#thm:exact){reference-type="ref" reference="thm:exact"} below, then knowing in this case that $$\begin{aligned} \mathop{g \square h_\mu}\nolimits = (g^* + h_\mu^*)^*,\end{aligned}$$ we can efficiently estimate $\mathop{g \square h_\mu}\nolimits$ using *fast* numerical schemes (see, e.g., [@lucet1997faster]). Suppose $g(x)= \|x\|_1$, $x\in {\mathbb R}^n$ and $\mu \in {\mathbb P}$. Below, we provide the infimal convolution of the function $g$ with $h_\mu$, where $h$ is given by [\[eq:hseparable\]](#eq:hseparable){reference-type="eqref" reference="eq:hseparable"}, for two of functions $\phi$ given in **Table** [\[tab:kernel-functions\]](#tab:kernel-functions){reference-type="ref" reference="tab:kernel-functions"} such that [\[ass:sc1\]](#ass:sc1){reference-type="ref" reference="ass:sc1"} -- [\[ass:sc2\]](#ass:sc2){reference-type="ref" reference="ass:sc2"} hold. **Example 1**. *Let $p=1$ in $\phi(t) = \frac{1}{p}\sqrt{1+p^2\abs{t}^2}-1$, with $\mathop{\rm dom}\nolimits\phi = {\mathbb R}$. Then, $$\begin{aligned} (\mathop{g \square h_\mu}\nolimits)(x) = \frac{\mu^{2}-\mu \sqrt{\mu^{2}+x^{2}}+x^{2}}{\sqrt{\mu^{2}+x^{2}}}. \end{aligned}$$* **Example 2**. *$\phi(t) = \frac{1}{2}\left[\sqrt{1+4t^2}-1+\log\left(\frac{\sqrt{1+4t^2}-1}{2t^2}\right)\right]$, with $\mathop{\rm dom}\nolimits\phi = {\mathbb R}$: $$\begin{aligned} (\mathop{g \square h_\mu}\nolimits)(x) = \frac{\sqrt{\mu^{2}+4 x^{2}}}{2}-\frac{\mu}{2}\left[1+\log \left(2\right) - \log \left(\frac{2x -\sqrt{\mu^{2}+4 x^{2}}+\mu}{x}\right)-\log \left(\frac{2x + \sqrt{\mu^{2}+4 x^{2}}-\mu}{x}\right)\right]. \end{aligned}$$* {width="\\textwidth"} {width="\\textwidth"} $\phi(t)$ $\mathop{\rm dom}\nolimits{\phi}$ $M_{\phi}$ $\nu$ Remark ----------------------------------------------------------------------------------------- ----------------------------------- ------------- ------- ------------------------------------------- $-\sqrt{1-t^2}$ $[-1,+1]$ $2.25$ $4$ "Hellinger\" $\frac{1}{p}\sqrt{1+p^2\abs{t}^2}-1$, $p\in{\mathbb P}$ ${\mathbb R}$ $2$ $2.6$ $p=1$ $\frac{7}{22\sqrt{t(1-t)}}$ $[0,1]$ $2.02$ $4$ "Arcsine probability density\" $\frac{1}{2}\left[\sqrt{1+4t^2}-1+\log\left(\frac{\sqrt{1+4t^2}-1}{2t^2}\right)\right]$ ${\mathbb R}$ $2\sqrt{2}$ $3$ Ostrovskii & Bach [@ostrovskii2021finite] $\frac{1}{2}t^2$ ${\mathbb R}$ $0$ $3$ "Energy\" $\frac{1}{p}\abs{t}^p$, $p\in (1,2)$ ${\mathbb R_{+}}$ $4$ $6$ $p=1.5$ $\log(1 + \exp(t))$ ${\mathbb R}$ $1$ $2$ "Logistic\" $\exp(-t)$ ${\mathbb R}$ $1$ $2$ "Exponential\" $t\log t - t$ $[0,+\infty]$ $1$ $4$ "Boltzmann-Shannon\" $t\log t + (1-t)\log(1-t)$ $[0,1]$ $1$ $4$ "Fermi-Dirac\" $-\frac{1}{2}\log t$ ${\mathbb P}$ $8$ $3$ "Burg\" $\begin{cases} ${\mathbb R}$ $4$ $3$ De Pierro & Iusem [@de1986relaxed] \frac{1}{2}(t^2 - 4t + 3), \quad \text{if} \,\,\, t\le 1\\ -\log t, \quad \text{otherwise} \end{cases}$ [\[tab:kernel-functions\]]{#tab:kernel-functions label="tab:kernel-functions"} The next two results characterize the functions $h$ and $h_\mu$ defined by supercoercive and generalized self-concordant kernel functions. **Lemma 1**. *Let $\phi \in \mathop{\Gamma_0({\mathbb R})}\nolimits$ be a function from ${\mathbb R}$ to ${\mathbb R}\cup \{+\infty\}$, and let the function $\mathop{h\colon{\mathbb R}^n\to {\mathbb R}\cup \{+\infty\}}\nolimits$ be defined by $h(x) \coloneqq \sum_{i=1}^{n} \lambda_i\phi_i$ with $\phi_i\coloneqq \phi(x_i)$, $x_i\in \mathop{\rm dom}\nolimits{\phi}$, $\lambda_i >0$, $i=1,2,\ldots,n$. Then the following properties hold:* (i) *$h \in \mathop{\Gamma_0({\mathbb R}^n)}\nolimits$.* (ii) *$h$ is supercoercive if and only if $\phi$ is supercoercive on its domain.* (iii) *If $\phi \in {{\mathcal{F}}_{M_{\phi},\nu}}$, where $M_{\phi} \in {\mathbb R_{+}}$ and $\nu \ge 2$, then $h(x)$ is well-defined on $\mathop{\rm dom}\nolimits{h} = \{\mathop{\rm dom}\nolimits{\phi}\}^n$, and $h(x) \in {{\mathcal{F}}_{M_{h},\nu}}$, with $M_h \coloneqq \max\{\lambda_i^{1- \frac{\nu}{2}}M_{\phi} \mathop{\, | \,}\nolimits 1\le i \le n\} \ge 0$.[\[item:hsciii\]]{#item:hsciii label="item:hsciii"}* *Proof.* (i) This statement is a direct consequence of [@bauschke2011convex Corollary 9.4, Lemma 1.27 and Proposition 8.17]. (ii) Follows directly from the definition of supercoercivity. (iii) $h(\cdot) \in {{\mathcal{F}}_{M_{h},\nu}}$ with $M_h \coloneqq \max\{\lambda_i^{1- \frac{\nu}{2}}M_{\phi} \mathop{\, | \,}\nolimits 1\le i \le n\} \ge 0$ follows from [@sun2019generalized Proposition 1].  ◻ **Proposition 1** (Self-concordance of $h_\mu$). *Suppose the conditions of hold such that the function $\mathop{h\colon{\mathbb R}^n\to {\mathbb R}\cup \{+\infty\}}\nolimits$ defined by [\[eq:hseparable\]](#eq:hseparable){reference-type="eqref" reference="eq:hseparable"} is $(M_h,\nu)$-generalized self-concordant. Let $A\in {\mathbb R}^{n\times n}$ be a diagonal matrix defined by $A\coloneqq \mathop{\rm diag}\nolimits(\frac{1}{\mu})$ such that $h(\frac{x}{\mu})\equiv h(Ax)$ is an affine transformation of $h(x)$. Then the following properties hold:* (i) *If $\nu \in (0,3]$, then $h_\mu\in{{\mathcal{F}}_{M_{},\nu}}$ with $M=n^{\frac{3-\nu}{2}}\mu^{\frac{\nu}{2}-2}M_h$.* (ii) *If $\nu >3$, then $h_\mu\in{{\mathcal{F}}_{M_{},\nu}}$ with $M=\mu^{4-\frac{3\nu}{2}}M_h$.* *Proof.* (i) We have $\norm{A} = \frac{\sqrt{n}}{\mu}$. By [@sun2019generalized Proposition 2(a)], $h(\frac{x}{\mu}) \in {{\mathcal{F}}_{M_{},\nu}}$ with $M= \norm{A}^{3-\nu}M_h$. In view of [\[item:hsciii\]](#item:hsciii){reference-type="ref" reference="item:hsciii"}, the scaling $h(\frac{\cdot}{\mu})\mapsto \mu h(\frac{\cdot}{\mu})$ gives $M\mapsto \mu^{1-\frac{\nu}{2}}M$. The result follows. (ii) The value $\mu^2>0$ corresponds to the unique eigenvalues of $A^\top A$. By [@sun2019generalized Proposition 2(b)], $h(\frac{x}{\mu}) \in {{\mathcal{F}}_{M_{},\nu}}$ with $M= \mu^{3-\nu}M_h$. The result follows as in Item (i) above.  ◻ In addition to [\[eq:grad-consistency\]](#eq:grad-consistency){reference-type="eqref" reference="eq:grad-consistency"}, the next result due to [@burke2017epi] concerns the epi-convergence of smoothing via infimal convolution under the condition of supercoercive regularization kernels in $\mathop{\Gamma_0({\mathbb R}^n)}\nolimits$. **Lemma 2**. *[@burke2017epi Theorem 3.8][\[thm:epiconvergence\]]{#thm:epiconvergence label="thm:epiconvergence"} Let $g, h \in \mathop{\Gamma_0({\mathbb R}^n)}\nolimits$ with $h$ supercoercive and $0\in\mathop{\rm dom}\nolimits{h}$. Let $h_\mu$ be defined as in [\[eq:hmu\]](#eq:hmu){reference-type="eqref" reference="eq:hmu"}. Then the following hold:* (i) *$\mathop{\rm \textrm{e--}\lim}\nolimits\limits_{\mu\downarrow 0}\inf\{g^*+\mu h^*\}\ge g^*$.* (ii) *$\mathop{\rm \textrm{e--}\lim}\nolimits\limits_{\mu\downarrow 0}\{g^* + \mu h^*\} = g^*$.* (iii) *$\mathop{\rm \textrm{e--}\lim}\nolimits\limits_{\mu\downarrow 0}\{\mathop{g \square h_\mu}\nolimits\} = g$. [\[thm:lemepi\]]{#thm:lemepi label="thm:lemepi"}* (iv) *If $h(0)\le 0$, we have $$\begin{aligned} \mathop{\rm \textrm{p--}\lim}\nolimits\limits_{\mu\downarrow 0}\{\mathop{g \square h_\mu}\nolimits\} = g. \end{aligned}$$* The main argument for the notion of epi-convergence in optimization problems is that when working with functions that may take infinite values, it is necessary to extend traditional convergence notions by applying the theory of *set convergence* to epigraphs in order to adequately capture local properties of the function (through a resulting calculus of smoothing functions), which on the other hand may be challenging due to the *curse of differentiation* associated with nonsmoothness. We refer the interested reader to [@rockafellar2009variational Chapter 7] for further details on the notion of epi-convergence, and to [@stromberg1994study; @burke2013epi; @burke2017epi] for extended results on epi-convergent smoothing via infimal convolution. The following result highlights key properties of the infimal convolution of $g\in \mathop{\Gamma_0({\mathbb R}^n)}\nolimits$ with $h_\mu$ satisfying $h\in{{\mathcal{F}}_{M_{h},\nu}}$. The proof is given in . **Proposition 2**. *Let $g,h \in \mathop{\Gamma_0({\mathbb R}^n)}\nolimits$. Suppose further that $h$ is $(M_h,\nu)$-generalized self-concordant and supercoercive, and define $g_s \coloneqq \mathop{g \square h_\mu}\nolimits$ for all $\mu > 0$. Then the following hold:* (i) *$\mathop{g \square h_\mu}\nolimits=\mathop{g \boxdot h_\mu}\nolimits\in \mathop{\Gamma_0({\mathbb R}^n)}\nolimits$. [\[thm:exact\]]{#thm:exact label="thm:exact"}* (ii) *$g_s \in {\mathcal{S}_{M_{g},\nu}^{\mu}}$ with $$\begin{aligned} M_g = \begin{cases} n^{\frac{3-\nu}{2}}\mu^{\frac{\nu}{2}-2}M_h, \qquad \text{if } \nu \in (0,3],\\ \mu^{4-\frac{3\nu}{2}}M_h, \qquad \text{if } \nu >3. \end{cases} \end{aligned}$$* (iii) *$g_s$ is locally Lipschitz continuous.[\[thm:g-lip\]]{#thm:g-lip label="thm:g-lip"}* # A proximal Newton-type scheme {#sec:prox-N} Our notion of self-concordant smoothing developed in the previous section is motivated for algorithmic purposes. Specifically, we are interested in composite minimization algorithms that utilize the idea of the *Newton decrement* framework but without imposing the self-concordant structure on the problem's objective functions. In this section, we present proximal Newton-type algorithms that exploit the structure of self-concordant smoothing functions developed in [2](#sec:smoothing){reference-type="ref" reference="sec:smoothing"} for variable-metric selection and computation of their step-lengths. The optimization problem of concern is $$\begin{aligned} \min\limits_{x\in{\mathbb R}^n} {\mathcal{L}}_s(x) \coloneqq f(x) + \mathop{g_s}\nolimits(x;\mu) + g(x), \label{eq:smoothprob}\end{aligned}$$ for which we assume the following[^5]: 1. $f$ is convex and $f\in{\mathcal{C}}_{L_f}^{2,2}({\mathbb R}^n)$.[\[ass:p1\]]{#ass:p1 label="ass:p1"} 2. $\rho_0 I_n \le \mathop{\nabla^2}\nolimits f(x^*) \le LI_n$, $\rho I_n \le \mathop{\nabla^2}\nolimits g_s(x^*) \le L_0I_n$ at a locally optimal solution $x^*$ of [\[eq:smoothprob\]](#eq:smoothprob){reference-type="eqref" reference="eq:smoothprob"} with $L \ge \rho_0 >0$ and $L_0\ge \rho >0$.[\[ass:p2\]]{#ass:p2 label="ass:p2"} 3. $g\in \mathop{\Gamma_0({\mathbb R}^n)}\nolimits$.[\[ass:p3\]]{#ass:p3 label="ass:p3"} 4. $\mathop{g_s}\nolimits\in {\mathcal{S}_{M_{g},\nu}^{\mu}}$.[\[ass:p4\]]{#ass:p4 label="ass:p4"} In particular, we consider $\mathop{g_s}\nolimits(x;\mu)\coloneqq \mathop{g \square h_\mu}\nolimits$ such that $h$ is a suitable regularization kernel for self-concordant smoothing in the sense of . Proximal Newton-type algorithms for solving [\[eq:smoothprob\]](#eq:smoothprob){reference-type="eqref" reference="eq:smoothprob"} consist in minimizing a sequence of *upper approximation* of ${\mathcal{L}}_s$ obtained by summing the nonsmooth part $g(x_k)$ and a local quadratic model of the smooth part $q(x_k)\coloneqq f(x_k) + \mathop{g_s}\nolimits(x_k)$ near $x_k$. That is, for $x\in\mathop{\rm dom}\nolimits{{\mathcal{L}}}\equiv \mathop{\rm dom}\nolimits f \cap \mathop{\rm dom}\nolimits g$, we iteratively define $$\begin{aligned} \hat{q}_k(x) &\coloneqq q(x_k) + \left<\mathop{\nabla}\nolimits q(x_k),x-x_k\right> + \frac{1}{2}\norm{x-x_k}_{Q}^2,\label{eq:proxnewtonstep0}\\ \hat{m}_k(x) &\coloneqq \hat{q}_k(x) + g(x_k),\end{aligned}$$[\[eq:proxnewtonmodel\]]{#eq:proxnewtonmodel label="eq:proxnewtonmodel"} where $Q\in {\mathcal S_{++}^{n}}$, and then solve the subproblem $$\begin{aligned} \delta_k = \arg\min\limits_{d\in{\mathbb R}^n} \hat{m}_k(x_k+d) \coloneqq \hat{q}(x_k+d) + g(x_k+d), \label{eq:subproblem}\end{aligned}$$ for a proximal Newton-type search direction $\delta_k$. With proximal Newton-type algorithms comprising only their special cases, we proceed by recalling the class of *cost approximation (CA) methods* [@patriksson1998cost] which helps us to propose a new method for selecting $\{x_k\}$ from the iterates $\{\delta_k\}$. The necessary optimality conditions for [\[eq:smoothprob\]](#eq:smoothprob){reference-type="eqref" reference="eq:smoothprob"} are defined by $$\begin{aligned} 0\in \mathop{\nabla}\nolimits q(x^*) + \partial g(x^*), \label{eq:optimality}\end{aligned}$$ for $x^*\in {\mathbb R}^n$. To find points $x^*$ satisfying [\[eq:optimality\]](#eq:optimality){reference-type="eqref" reference="eq:optimality"}, CA methods, as the name implies, iteratively approximate $\mathop{\nabla}\nolimits q(x_k)$ by a *cost approximating mapping* $\Phi\colon {\mathbb R}^n \to {\mathbb R}^n$, taking into account the fixed approximation error term $\Phi(x_k) - \mathop{\nabla}\nolimits q (x_k)$. That is, a point $d$ is sought satisfying $$\begin{aligned} 0\in \Phi(d) + \partial g(d) + \mathop{\nabla}\nolimits q(x_k) - \Phi(x_k). \label{eq:ca-optimality}\end{aligned}$$ Let $\Phi$ be the gradient mapping of a continuously differentiable convex function $\psi\colon {\mathbb R}^n\to {\mathbb R}$. A CA method iteratively solves the subproblem $$\begin{aligned} \min\limits_{d\in{\mathbb R}^n} \set{\psi(d) + q(x_k) + g(d) - \psi(x_k) + \left<\mathop{\nabla}\nolimits q(x_k) - \mathop{\nabla}\nolimits\psi(x_k), d-x_k\right>}.\label{eq:ca-prob}\end{aligned}$$ A step is then taken in the direction $\delta_k - x_k$, namely $$\begin{aligned} x_{k+1} = x_k + \alpha_k(\delta_k - x_k),\label{ca-update}\end{aligned}$$ where $\delta_k$ solves [\[eq:ca-prob\]](#eq:ca-prob){reference-type="eqref" reference="eq:ca-prob"} and $\alpha_k>0$ is a step-length typically computed via a line search such that an appropriately selected *merit function* is sufficiently decreased along the direction $\delta_k - x_k$. **Remark 2**. *Evaluating the merit function too many times can be practically intractable. One way to (completely) mitigate this difficulty for large-scale problems is to incorporate "predetermined step-lengths\" [@patriksson1993unified] into the solution scheme of [\[eq:ca-prob\]](#eq:ca-prob){reference-type="eqref" reference="eq:ca-prob"}, so that we may update $x_k$ as $x_{k+1}\equiv\delta_k$. However, methods that use this approach do not, in general, yield monotonically decreasing sequence of objective values, and convergence is instead characterized by a metric that measures the distance from iteration points to the set of optimal solutions [@patriksson1993unified].* In view of , we discuss next a new proximal Newton-type scheme that compromises between minimizing the objective values and decreasing the distance from iteration points to the set of optimal solutions as specified by a curvature-exploiting variable-metric. ## Variable-metric and adaptive step-length selection A very nice feature of the CA framework is that it can help, for instance, through the specific choice of $\Phi$, to efficiently utilize the original problem's structure---a practice which is particularly useful when solving medium- to large-scale problems. This feature fits directly into our self-concordant smoothing framework. We notice that [\[eq:ca-prob\]](#eq:ca-prob){reference-type="eqref" reference="eq:ca-prob"} gives [\[eq:subproblem\]](#eq:subproblem){reference-type="eqref" reference="eq:subproblem"} by setting $\alpha_k=1$ in the following choice of $\psi$: $$\begin{aligned} \psi(\cdot) = \frac{1}{2}\norm{\cdot}_Q^2, \qquad Q\in{\mathcal S_{++}^{n}}.\label{eq:cost-approx-term}\end{aligned}$$ In this case, the optimality conditions and our assumptions give $$\begin{aligned} (Q-\mathop{\nabla}\nolimits q)(x_k) \in (Q + \partial g)(d),\end{aligned}$$ which leads to $$\begin{aligned} \delta_k = \mathop{\rm prox}\nolimits_g^{Q}(x_k - Q^{-1}\mathop{\nabla}\nolimits q(x_k)).\end{aligned}$$ In the proximal Newton-type scheme, $Q$ may be the Hessian of $q(x_k)$ or its approximation[^6]. Although a diagonal structure of $Q$ is often desired due to its ease of implementation in the proximal framework, we most likely throw away relevant curvature information by performing a diagonal or scalar approximation of $\mathop{\nabla^2}\nolimits q(x_k)$, especially when the objective functions are not assumed to be separable. Our consideration in this work entails the following characterization of the optimality conditions: $$\begin{aligned} (H_k-\mathop{\nabla}\nolimits q)(x_k) \in (\mathop{\nabla^2}\nolimits g_s + \partial g)(d),\label{eq:optimality-conditions}\end{aligned}$$ where $H_k$ may be the Hessian, $\mathop{\nabla^2}\nolimits q \equiv H_f + H_g$, of $q$ or its approximation; $H_f\equiv \mathop{\nabla^2}\nolimits f$ and $H_g\equiv \mathop{\nabla^2}\nolimits g_s$. Specifically, we propose the following step update formula: $$\begin{aligned} x_{k+1} = \mathop{\rm prox}\nolimits_{\alpha_k g}^{H_g}(x_k - \bar{\alpha}_k H_k^{-1}\mathop{\nabla}\nolimits q(x_k)),\label{eq:proxnewton-general}\end{aligned}$$ where $\bar{\alpha}_k\in{\mathbb P}$ results from *damping* the Newton-type steps. $x_0\in{\mathbb R}^n$, problem functions $f$, $g$, self-concordant smoothing function $g_s\in {\mathcal{S}_{M_{g},\nu}^{\mu}}$, $\alpha\in(0,1]$ $\mathrm{grad}_k\gets \mathop{\nabla}\nolimits f(x_k) + \mathop{\nabla}\nolimits g_s(x_k)$ $H_g \gets \mathop{\nabla^2}\nolimits g_s(x_k)$; $\eta_k \gets \norm{\mathop{\nabla}\nolimits g_s(x_k)}_{H_g}^*$ $\bar{\alpha}_k = \frac{\alpha}{1 + M_g\eta_k}$ $H_k \gets \mathop{\nabla^2}\nolimits f(x_k)+H_g$; Solve for $\Delta_k$: $H_k \Delta_k = \mathrm{grad}_k$ $x_{k+1} \gets \mathop{\rm prox}\nolimits_{\alpha g}^{H_g}(x_k - \bar{\alpha}_k\Delta_k)$ The validity of this procedure in the present scheme may be seen in the interpretation of the proximal operator $\mathop{\rm prox}\nolimits_{g}^{Q}\left(x^+\right)$ for some $x^+\in\mathop{\rm dom}\nolimits g$ as compromising between minimizing the function $g$ and staying close to $x^+$ (see [@parikh2014proximal Chapter 1]). Here, "closeness\" is quantified in terms of the metric induced by $\mathop{\nabla^2}\nolimits g_s$, and we want the proximal steps to stay close (as much as possible) to the Newton iterates relative to $\norm{\cdot}_{\mathop{\nabla^2}\nolimits g_s}$. To see this, we note that in view of the fixed-point characterization [\[eq:ca-prob\]](#eq:ca-prob){reference-type="eqref" reference="eq:ca-prob"} via CA methods, we may interpret proximal Newton-type algorithms as a fixation of the error term $\mathop{\nabla}\nolimits\psi - \mathop{\nabla}\nolimits q$ at some point in $\mathop{\rm dom}\nolimits q \cap \mathop{\rm dom}\nolimits g$. Let us fix some $\bar{x} \in \mathop{\rm dom}\nolimits q \cap \mathop{\rm dom}\nolimits g$ and introduce the operator $E_{\bar{x}}$ defined by $$\begin{aligned} E_{\bar{x}}(z) \coloneqq \mathop{\nabla^2}\nolimits q(\bar{x})z - \bar{\alpha} \mathop{\nabla}\nolimits q(z),\label{eq:error-term1}\end{aligned}$$ where $\bar{\alpha}\in {\mathbb P}$. Let $Q\equiv Q_k\in{\mathcal S_{++}^{n}}$ be arbitrary in [\[eq:cost-approx-term\]](#eq:cost-approx-term){reference-type="eqref" reference="eq:cost-approx-term"}. We aim to exploit the structure in $g_s$ (and $\mathop{\nabla^2}\nolimits g_s$), so we define an operator $\xi_{\bar{x}}(Q_k,\cdot)$ to quantify the error between $\mathop{\nabla^2}\nolimits g_s$ and $Q_k$ as follows: $$\begin{aligned} \xi_{\bar{x}}(Q_k,z) \coloneqq (\mathop{\nabla^2}\nolimits g_s(\bar{x})-Q_k)(z - x_k).\label{eq:error-term2}\end{aligned}$$ We provide a characterization of the optimality conditions for [\[eq:ca-prob\]](#eq:ca-prob){reference-type="eqref" reference="eq:ca-prob"} in terms of $E_{\bar{x}}$ and $\xi_{\bar{x}}$ in the next result (see for the proof). **Proposition 3**. *Let the operators $E_{\bar{x}}$ and $\xi_{\bar{x}}(Q_k,\cdot)$ be defined by [\[eq:error-term1\]](#eq:error-term1){reference-type="eqref" reference="eq:error-term1"} and [\[eq:error-term2\]](#eq:error-term2){reference-type="eqref" reference="eq:error-term2"}, respectively. Then the optimality conditions for [\[eq:ca-prob\]](#eq:ca-prob){reference-type="eqref" reference="eq:ca-prob"} with $\psi(\cdot) = \frac{1}{2}\norm{\cdot}_{Q_k}^2$ are characterized in terms of $E_{\bar{x}}$ and $\xi_{\bar{x}}(Q_k,\cdot)$ by $$\begin{aligned} E_{\bar{x}}(x_k) + \xi_{\bar{x}}(Q_k, d) \in \mathop{\nabla^2}\nolimits g_s(\bar{x})d + \alpha\partial g(d). \label{eq:new-optimality} \end{aligned}$$ More precisely, [\[eq:optimality-conditions\]](#eq:optimality-conditions){reference-type="eqref" reference="eq:optimality-conditions"} holds with $Q_k=\mathop{\nabla^2}\nolimits g_s(\bar{x})$ whenever $\bar{x}$ is the unique optimizer satisfying [\[eq:new-optimality\]](#eq:new-optimality){reference-type="eqref" reference="eq:new-optimality"}.* Indeed, in view of , $Q=Q_k\equiv\mathop{\nabla^2}\nolimits g_s(x_k)$ is an ideal choice for $Q$ in [\[eq:optimality-conditions\]](#eq:optimality-conditions){reference-type="eqref" reference="eq:optimality-conditions"}. In addition, as we shall see in the GGN approximation discussed below, we may exploit the properties of the function $g_s$ in ensuring stability of the Newton-type steps via the notion of *Newton decrement*. In essence, we consider damping the Newton-type steps such that $$\begin{aligned} \bar{\alpha}_k = \frac{\alpha_k}{1 + M_g\eta_k}, \label{eq:steplength}\end{aligned}$$ where by [\[ass:p4\]](#ass:p4){reference-type="ref" reference="ass:p4"}, $M_g$ is a generalized self-concordant parameter for $g_s$, and $\eta_k \coloneqq \norm{\mathop{\nabla}\nolimits g_s(x_k)}_{x_k}^*$ is the dual norm associated with $g_s$. Note that the above choice for $\bar{\alpha}_k$, in the context of minimizing generalized self-concordant functions, assumes $\nu\ge2$ (see e.g. [@sun2019generalized Equation 12]). Suppose for example $\alpha_k=1$ is fixed and $\nu=3$, then [\[eq:ca-optimality-expanded\]](#eq:ca-optimality-expanded){reference-type="eqref" reference="eq:ca-optimality-expanded"} leads to the standard damped-step proximal Newton-type method (cf. [@tran2015composite; @sun2019generalized]) in the framework of Newton decrement, and we recover the fixed-point characterization [\[eq:ca-optimality\]](#eq:ca-optimality){reference-type="eqref" reference="eq:ca-optimality"}. In view of [\[eq:hseparable\]](#eq:hseparable){reference-type="eqref" reference="eq:hseparable"}, $H_g$ has a desirable diagonal structure and hence can be cheaply updated from iteration to iteration. This structure provides an efficient way to compute the scaled proximal operator $\mathop{\rm prox}\nolimits_{g}^{H_g}$, for example via a special case of the proximal calculus derived in [@becker2019quasi] (see for two practical examples). Overall, by exploiting the structure of the problem, precisely (i) taking adaptive steps that properly capture the curvature of the objective functions, and (ii) scaling the proximal operator of $g$ by a variable-metric $H_g$ which has a simple, diagonal structure, we can adapt to an affine-invariant structure due to the algorithm and ensure we remain close to the Newton-type iterates towards convergence. If we choose $H_k\equiv \mathop{\nabla^2}\nolimits q(x_k)$ in [\[eq:proxnewton-general\]](#eq:proxnewton-general){reference-type="eqref" reference="eq:proxnewton-general"}, we obtain a proximal Newton step (see Algorithm [\[alg:NewtonSCOREProx\]](#alg:NewtonSCOREProx){reference-type="ref" reference="alg:NewtonSCOREProx"}): $$\begin{aligned} x_{k+1} = \mathop{\rm prox}\nolimits_{\alpha_k g}^{H_g}(x_k - \bar{\alpha}_k\mathop{\nabla^2}\nolimits q(x_k)^{-1}\mathop{\nabla}\nolimits q(x_k)). \label{eq:proxnewton}\end{aligned}$$ However, $H_k$ may be any approximation to the Hessian of $q$ at $x_k$. In view of [\[eq:new-optimality\]](#eq:new-optimality){reference-type="eqref" reference="eq:new-optimality"}, this corresponds to replacing the Hessian term $\mathop{\nabla^2}\nolimits q(\bar{x})$ in [\[eq:error-term1\]](#eq:error-term1){reference-type="eqref" reference="eq:error-term1"} by the approximating matrix evaluated at $\bar{x}$. $x_0\in{\mathbb R}^n$, problem functions $f$, $g$, self-concordant smoothing function $g_s\in {\mathcal{S}_{M_{g},\nu}^{\mu}}$, model ${\mathcal{M}}$, input-output pairs $\{u_i,y_i\}_{i=1}^m$ with $y_i \in {\mathbb R}^{n_y}$, $\alpha\in (0,1]$ $H_g \gets \mathop{\nabla^2}\nolimits g_s(x_k)$; $\eta_k \gets \norm{\mathop{\nabla}\nolimits g_s(x_k)}_{H_g}^*$ $\bar{\alpha}_k \gets \frac{\alpha}{1 + M_g\eta_k}$ Compute $\delta_k^{\mathrm{ggn}}$ via [\[eq:ggn-step-approx\]](#eq:ggn-step-approx){reference-type="eqref" reference="eq:ggn-step-approx"} Compute $\delta_k^{\mathrm{ggn}}$ via [\[eq:ggn-step\]](#eq:ggn-step){reference-type="eqref" reference="eq:ggn-step"} $x_{k+1} \gets \mathop{\rm prox}\nolimits_{\alpha g}^{H_g}(x_k + \bar{\alpha}_k\delta_k^{\mathrm{ggn}})$ ## A proximal generalized Gauss-Newton algorithm In describing the proximal GGN algorithm, consider first the simple case $g\equiv0$. Then [\[eq:proxnewton-general\]](#eq:proxnewton-general){reference-type="eqref" reference="eq:proxnewton-general"} with $\bar{\alpha}_k=1$ gives exactly the pure Newton-type direction $$\begin{aligned} \delta_k^{\mathrm{ggn}} = -H_k^{-1}\mathop{\nabla}\nolimits q(x_k).\label{eq:newtonstep}\end{aligned}$$ Now suppose that the function $f$ quantifies a data-misfit or loss between the outputs[^7] $\hat{y}_i$ of a model ${\mathcal{M}}(\cdot;x)$ and the expected outputs $y_i$, for $i=1,2,\ldots,m$, as in a typical machine learning problem, and that $g\ne 0$. Precisely, let $\hat{y}_i \equiv {\mathcal{M}}(u_i;x)$, and suppose that $f$ can be written as $$\begin{aligned} f(x) = \sum_{i=1}^{m}\ell(y_i,\hat{y}_i),\label{eq:f-ggn}\end{aligned}$$ where $\ell\colon{\mathbb R}\times{\mathbb R}\to{\mathbb R}$ is a loss function. Define an "augmented\" Jacobian matrix $J_k\in{\mathbb R}^{(m+1)\times n}$ by [@adeoye2023score] $$\begin{aligned} J_k^T &\coloneqq \begin{bmatrix} \mathop{\hat{y}_1'}\nolimits(x_1) & \mathop{\hat{y}_2'}\nolimits(x_1) & \cdots & \mathop{\hat{y}_m'}\nolimits(x_1) & \mathop{g_1'}\nolimits(x_1) \\ \mathop{\hat{y}_1'}\nolimits(x_2) & \mathop{\hat{y}_2'}\nolimits(x_2) & \cdots & \mathop{\hat{y}_m'}\nolimits(x_2) & \mathop{g_2'}\nolimits(x_2)\\ \vdots&\vdots&&\vdots&\vdots\\ \mathop{\hat{y}_1'}\nolimits(x_n) & \mathop{\hat{y}_2'}\nolimits(x_n) & \cdots & \mathop{\hat{y}_m'}\nolimits(x_n) & \mathop{g_{n}'}\nolimits(x_n) \end{bmatrix}, \label{eq:Jaug}\end{aligned}$$ where $x_1,x_2,\ldots,x_n$ are components of the variable $x$, and $g_1,g_2,\ldots,g_n$ are the respective components of $g_s$. Then GGN approximation of the Newton direction [\[eq:newtonstep\]](#eq:newtonstep){reference-type="eqref" reference="eq:newtonstep"} gives $$\begin{aligned} \delta_k^{\mathrm{ggn}} = -(H_f + H_g )^{-1}\mathop{\nabla}\nolimits q \approx -(J_k^\top V_k J_k + H_g )^{-1}J_k^\top u_k, \label{eq:ggn-step}\end{aligned}$$ where $V_k\equiv\mathop{\rm diag}\nolimits(v_k)$, $v_k\coloneqq[l''_{\hat{y}_1}(y_1,\hat{y}_1;x_k),\ldots,l''_{\hat{y}_m}(y_m,\hat{y}_m;x_k),0]^\top\in{\mathbb R}^{(m+1)}$, and the vector $u_k\coloneqq[l'_{\hat{y}_1}(y_1,\hat{y}_1;x_k),\ldots,l'_{\hat{y}_m}(y_m,\hat{y}_m;x_k),1]^\top\in{\mathbb R}^{m+1}$ defines an augmented "residual\" term. If $m+1<n$ (possibly $m\ll n$), that is, when the model is overparameterized, the following equivalent formulation of [\[eq:ggn-step\]](#eq:ggn-step){reference-type="eqref" reference="eq:ggn-step"} provides a convenient way to compute the GGN search direction [@adeoye2023score]: $$\begin{aligned} \delta_k^{\mathrm{ggn}} = -H_g^{-1}J_k^\top(I_m+V_k J_k H_g^{-1}J_k^\top)^{-1}u_k.\label{eq:ggn-step-approx}\end{aligned}$$ Note that in case the function $g$ (and hence $g_s$) is scaled by some (nonnegative) constant, only the identity matrix $I_m$ may be scaled accordingly. Following [@adeoye2023score Section 4], it suffices to assume stability of the GGN iterates by ensuring the stability of $H_g$. This is achieved, for instance, through the generalized self-concordant structure of $g_s$. Now if we choose $H_k\equiv J_k^\top V_k J_k + H_g$ in the proximal Newton-type scheme of [\[eq:proxnewton-general\]](#eq:proxnewton-general){reference-type="eqref" reference="eq:proxnewton-general"}, we have the proximal GGN update (see Algorithm [\[alg:GGNSCOREProx\]](#alg:GGNSCOREProx){reference-type="ref" reference="alg:GGNSCOREProx"}): $$\begin{aligned} x_{k+1} = \mathop{\rm prox}\nolimits_{\alpha_k g}^{H_g}(x_k + \bar{\alpha}_k\delta_k^{\mathrm{ggn}}), \label{eq:proxggn}\end{aligned}$$ where $\delta_k$ is computed via [\[eq:ggn-step\]](#eq:ggn-step){reference-type="eqref" reference="eq:ggn-step"}, or by [\[eq:ggn-step-approx\]](#eq:ggn-step-approx){reference-type="eqref" reference="eq:ggn-step-approx"} in case $m+1$ is less than $n$, and $\bar{\alpha}_k$ is as defined in [\[eq:steplength\]](#eq:steplength){reference-type="eqref" reference="eq:steplength"}. # Structured penalties {#sec:structured} As we have noted, more general nonsmooth regularized problems impose certain structures on the variables that must be handled explicitly by the algorithm. Such situations can be seen in some lasso and multi-task regression problems in which problem [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"} takes on the form $$\begin{aligned} \min\limits_{x\in{\mathbb R}^n} {\mathcal{L}}(x) \coloneqq f(x) + \underbrace{{\mathcal{R}}(x) + \Omega(Cx)}_{g(x)},\label{eq:structured-prob}\end{aligned}$$ where, in addition to ${\mathcal{R}}(x)$, the function (cf., [@chen2010graph; @chen2012smoothingprox]) $$\begin{aligned} \Omega(Cx) \coloneqq \max_{u\in {\mathcal{Q}}} \langle u, Cx \rangle,\end{aligned}$$ characterizes a specific desired structure of the solution estimates and, for $\mathbb{V}$ a finite-dimensional vector space such that $C \colon {\mathbb R}^n \to \mathbb{V}$ is a linear map, ${\mathcal{Q}}\subseteq \mathbb{V}^*$ is closed and convex, where $\mathbb{V}^*$ is the dual space to $\mathbb{V}$. For example, in the sparse group lasso problem [@friedman2010note; @simon2013sparse], $\Omega(Cx) = \gamma \sum_{j\in {\mathcal{G}}}\omega_j\norm{x_j}$ induces group level sparsity on the solution estimates and ${\mathcal{R}}(x) = \beta\norm{x}_1$ promotes the overall sparsity of the solution, so that the optimization problem is written as $$\begin{aligned} \min\limits_{x\in{\mathbb R}^n} {\mathcal{L}}(x) \coloneqq f(x) + \beta\norm{x}_1 + \beta_{\mathcal{G}}\sum_{j\in {\mathcal{G}}}\omega_j\norm{x_j},\label{eq:glasso}\end{aligned}$$ where $\beta\in{\mathbb P}$, $\beta_{\mathcal{G}}\in {\mathbb P}$, ${\mathcal{G}}= \{j_k, \ldots, j_{n_g}\}$ is the set of variables groups with $n_g=\mathop{\rm card}\nolimits({\mathcal{G}})$, $x_j \in {\mathbb R}^{n_j}$ is the subvector of $x$ corresponding to variables in group $j$ and $\omega_j \in {\mathbb P}$ is the group penalty parameter. Another example is the graph-guided fused lasso for multi-task regression problems [@kim2009multivariate], where the function $\Omega(Cx)=\beta_{\mathcal{G}}\sum_{e=(r,s)\in E,r<s}\tau(\omega_{rs})\abs{x_r - \mathop{\rm sign}\nolimits(\omega_{rs})x_s}$ encourages a fusion effect over variables $x_r$ and $x_s$ shared across tasks through a graph $G\equiv (V,E)$ of relatedness, where $V=\{1,\ldots,n\}$ denotes the set of nodes and $E$ the edges; $\beta_{\mathcal{G}}\in {\mathbb P}$, $\tau(\omega_{rs})$ is a fusion penalty function, and $\omega_{rs}\in {\mathbb R}$ is the weight of the edge $e=(r,s)\in E$. Here, with ${\mathcal{R}}(x) = \beta\norm{x}_1$, $\beta\in{\mathbb P}$, the optimization problem is written as $$\begin{aligned} \min\limits_{x\in{\mathbb R}^n} {\mathcal{L}}(x) \coloneqq f(x) + \beta\norm{x}_1 + \beta_{\mathcal{G}}\sum_{e=(r,s)\in E,r<s}\tau(\omega_{rs})\abs{x_r - \mathop{\rm sign}\nolimits(\omega_{rs})x_s}.\label{eq:gflasso}\end{aligned}$$ ## Structure reformulation for self-concordant smoothing The key observation in problems of the form [\[eq:structured-prob\]](#eq:structured-prob){reference-type="eqref" reference="eq:structured-prob"} is that the function $\Omega(Cx)$ belongs to the class of nonsmooth convex functions that is well-structured for the Nesterov's smoothing [@nesterov2005smooth] in which a smooth approximation $\Omega_s$ of $\Omega$ has the form $$\begin{aligned} \Omega_s(Cx;\mu) = \max_{u\in {\mathcal{Q}}} \left\{\langle u, Cx \rangle - \mu d(u)\right\}, \qquad \mu \in {\mathbb P},\label{eq:nesterovsmooth}\end{aligned}$$ where $d$ is a *prox-function*[^8] of the set ${\mathcal{Q}}$. Note that the Nesterov's smoothing approach assumes the knowledge of the exact structure of $C$. In the sequel, we shall write $\Omega^C(x)\equiv \Omega(Cx)$ or $\Omega_s^C(x;\mu)\equiv \Omega_s(Cx)$, with the superscript "$C$\" to indicate the function is *structure-aware* via $C$. In view of [@beck2012smoothing Theorem 4.1(a), Equation 4.10], the function $\Omega_s^C(x;\mu)$ satisfies the dual formulation of the inf-conv regularization process of $\Omega(x)$ with $h_\mu \in {\mathcal{H}}$ at the point $Cx$, where ${\mathcal{H}}$ is defined by [\[eq:kernelfam\]](#eq:kernelfam){reference-type="eqref" reference="eq:kernelfam"}. In this case, the prox-function $d$ in [\[eq:nesterovsmooth\]](#eq:nesterovsmooth){reference-type="eqref" reference="eq:nesterovsmooth"} is given by $h^*$, the dual of $h$. In the framework of [2](#sec:smoothing){reference-type="ref" reference="sec:smoothing"}, if $h$ is defined by [\[eq:hseparable\]](#eq:hseparable){reference-type="eqref" reference="eq:hseparable"} such that $\phi$ satisfies [\[ass:k1\]](#ass:k1){reference-type="ref" reference="ass:k1"} -- [\[ass:k2\]](#ass:k2){reference-type="ref" reference="ass:k2"}, and if $\Omega_s^C$ satisfies the conditions in Definition [Definition 3](#def:smooth-function){reference-type="ref" reference="def:smooth-function"} with respect to $\Omega$ (e.g., via ), then at any point $x\in {\mathbb R}^n$, $\Omega_s^C(x;\mu)$ provides a self-concordant smooth approximation of $\Omega(x)$. ## Prox-decomposition and smoothness properties {#sec:prox-decomp} An important property of the function $g={\mathcal{R}}+ \Omega^C$ we want to infer here is its prox-decomposition property [@yu2013decomposing] in which the (unscaled) proximal operator of $g$ satisfies $$\begin{aligned} \mathop{\rm prox}\nolimits_{g} = \mathop{\rm prox}\nolimits_{\Omega^C} \circ \mathop{\rm prox}\nolimits_{{\mathcal{R}}}.\end{aligned}$$ Under our assumptions on $g$ and $h$, this property extends for the inf-conv regularization (and hence the self-concordant smoothing framework)[^9]. To see this, let $z \coloneqq (\mathop{{\mathcal{R}} \square h_\mu}\nolimits)(x)$, and note the following equivalent expression for the definition of inf-convolution [\[eq:infconv\]](#eq:infconv){reference-type="eqref" reference="eq:infconv"}: $$\begin{aligned} z \coloneqq (\mathop{{\mathcal{R}} \square h_\mu}\nolimits)(x) = \inf\limits_{\substack{(u,v)\in{\mathbb R}^n\times{\mathbb R}^n\\u+v=x}}\left\{{\mathcal{R}}(u)+h_\mu(v)\right\}.\end{aligned}$$ The next result follows, highlighting what we propose as the *inf-decomposition* property. **Proposition 4**. *Let $g \in \mathop{\Gamma_0({\mathbb R}^n)}\nolimits$ be given as the sum $g(x)={\mathcal{R}}(x) + \Omega^C(x)$. Suppose that the function $h \in \mathop{\Gamma_0({\mathbb R}^n)}\nolimits$ is supercoercive. Then the regularization process $g_s \coloneqq \mathop{g \square h_\mu}\nolimits$, for all $\mu > 0$, is given by the composition $$\begin{aligned} g_s(x) = (\mathop{\Omega^C \square h_\mu}\nolimits)\circ (\mathop{{\mathcal{R}} \square h_\mu}\nolimits)(x). \end{aligned}$$* *Proof.* We have $$\begin{aligned} (\mathop{\Omega^C \square h_\mu}\nolimits)\circ (\mathop{{\mathcal{R}} \square h_\mu}\nolimits)(x) &= \inf\limits_{\substack{(Cu,v)\in\mathbb{V}\times{\mathbb R}^n\\Cu+v=z}}\left\{\Omega^C(u)+h_\mu(v)\right\}\\ &= \inf\limits_{\substack{(u,Cu,v)\in{\mathbb R}^n\times\mathbb{V}\times{\mathbb R}^n\\u+Cu+v=x}}\left\{{\mathcal{R}}(u)+\Omega^C(u)+h_\mu(v)\right\}\\ &= (\mathop{({\mathcal{R}}+ \Omega^C) \square h_\mu}\nolimits)(x) = (\mathop{g \square h_\mu}\nolimits)(x),\end{aligned}$$ where the second equality is inferred from the exactness of the inf-conv regularization process by [\[thm:exact\]](#thm:exact){reference-type="ref" reference="thm:exact"}. ◻ Given the smoothness properties of $\mathop{\Omega^C \square h_\mu}\nolimits$ and $\mathop{{\mathcal{R}} \square h_\mu}\nolimits$, we can apply the chain rule to obtain the derivatives of their composition $\mathop{g \square h_\mu}\nolimits$. Precisely, [@sun2006strong Lemma 2.1] provides sufficient conditions for the validity of the derivatives obtained via the chain rule for composite functions, which are indeed satisfied for $\mathop{g \square h_\mu}\nolimits$ by our assumptions. # Convergence analysis {#sec:convergence} We analyze the convergence of Algorithms [\[alg:NewtonSCOREProx\]](#alg:NewtonSCOREProx){reference-type="ref" reference="alg:NewtonSCOREProx"} and [\[alg:GGNSCOREProx\]](#alg:GGNSCOREProx){reference-type="ref" reference="alg:GGNSCOREProx"} under the smoothing framework of this paper. In view of the numerical examples considered in , we restrict our analyses to the case $2\le\nu\le3$. However, similar convergence properties are expected to hold for the general case $\nu>0$, as the key bounds describing generalized self-concordant functions hold similarly for all of these cases (see, e.g., Section 2 and the concluding remark of [@sun2019generalized]). We define the following metric term associated with the function $g_s$: $$\begin{aligned} \label{eq:d-metric} d_\nu(x,y) \coloneqq \begin{cases} M_g\norm{y-x} &\text{if } \nu = 2,\\ \left(\frac{\nu}{2}-1\right)M_g\norm{y-x}_2^{3-\nu}\norm{y-x}_x^{\nu-2} &\text{if }\nu>2. \end{cases}\end{aligned}$$ We introduce the notations $H_g^*\equiv \mathop{\nabla^2}\nolimits g_s(x^*)$, $H_f^*\equiv \mathop{\nabla^2}\nolimits f(x^*)$ and $H_k^*\equiv\mathop{\nabla^2}\nolimits q(x^*)$. Recall also the notations $H_g\equiv \mathop{\nabla^2}\nolimits g_s(x_k)$, $H_f\equiv \mathop{\nabla^2}\nolimits f(x_k)$ and $H_k\equiv\mathop{\nabla^2}\nolimits q(x_k)$ at $x_k$. Furthermore, we define the following matrices associated with any given twice differentiable function $f$: $$\begin{aligned} \Sigma_f^{x,y} &\coloneqq \int_0^1 \left(\mathop{\nabla^2}\nolimits f(x+\tau(y-x)) - \mathop{\nabla^2}\nolimits f(x)\right)d\tau,\\ \Upsilon_f^{x,y} &\coloneqq \mathop{\nabla^2}\nolimits f(x)^{-1/2} \Sigma_f^{x,y} \mathop{\nabla^2}\nolimits f(x)^{-1/2}.\end{aligned}$$ We begin by stating some useful preliminary results. The following two results respectively provides bounds on the function values of the two smooth terms $f$ and $g_s$ in [\[eq:smoothprob\]](#eq:smoothprob){reference-type="eqref" reference="eq:smoothprob"}. **Lemma 3**. *[@nesterov2018lectures Lemma 1.2.3][\[thm:f-bound\]]{#thm:f-bound label="thm:f-bound"} Let [\[ass:p1\]](#ass:p1){reference-type="ref" reference="ass:p1"}--[\[ass:p2\]](#ass:p2){reference-type="ref" reference="ass:p2"} hold. Then, for any $x,y\in \mathop{\rm dom}\nolimits f$, we have $$\begin{aligned} \abs{f(y)-f(x)-\langle\mathop{\nabla}\nolimits f (x), y-x\rangle} \le \frac{L}{2}\|y-x\|^2. \end{aligned}$$* **Lemma 4**. *[@sun2019generalized Proposition 10][\[thm:g-bound0\]]{#thm:g-bound0 label="thm:g-bound0"} Suppose that [\[ass:p3\]](#ass:p3){reference-type="ref" reference="ass:p3"}--[\[ass:p4\]](#ass:p4){reference-type="ref" reference="ass:p4"} hold. Then, given any $x,y\in \mathop{\rm dom}\nolimits g$, we have $$\begin{aligned} \omega_\nu(-d_\nu(x,y))\|y-x\|_x^2 \le g_s(y) - g_s(x) - \langle\mathop{\nabla}\nolimits g_s(x), y-x\rangle \le \omega_\nu(d_\nu(x,y))\|y-x\|_x^2, \end{aligned}$$ in which, if $\nu>2$, the right-hand side inequality holds if $d_\nu(x,y) < 1$, and $$\begin{aligned} \label{eq:omega-nu} \omega_\nu(\tau) \coloneqq \begin{cases} \frac{\exp(\tau)-\tau-1}{\tau^2} &\text{if } \nu = 2,\\ \frac{-\tau - \ln(1-\tau)}{\tau^2} &\text{if } \nu=3,\\ \frac{(1-\tau)\ln(1-\tau)+\tau}{\tau^2} &\text{if } \nu=4,\\ \left(\frac{\nu-2}{4-\nu}\right)\frac{1}{\tau}\left[\frac{\nu-2}{2(3-\nu)\tau}\left((1-\tau)\frac{2(3-\nu)}{2-\nu}-1\right)-1\right] &\text{otherwise}. \end{cases}\end{aligned}$$* The next result uses a similar argument as in [@sun2019generalized Proposition 9], and provides an estimate on the error between the gradients of $g_s$. We provide the proof in for completeness. **Proposition 5**. *Given any $x,y\in \mathop{\rm dom}\nolimits g$. We have $$\begin{aligned} \label{eq:g-error-bound} \bar{\omega}_\nu(-d_\nu(x,y))\|y-x\|_x \le \mathop{\nabla}\nolimits g_s(y) - \mathop{\nabla}\nolimits g_s(x) \le \bar{\omega}_\nu(d_\nu(x,y))\|y-x\|_x, \end{aligned}$$ in which, if $\nu>2$, the right-hand side inequality holds if $d_\nu(x,y) < 1$, and $$\begin{aligned} \label{eq:baromega-nu} \bar{\omega}_\nu(\tau) \coloneqq \begin{cases} \frac{2\exp\left(\frac{\tau}{2}\right)-2}{\tau} &\text{if } \nu = 2,\\ -\frac{\ln(1-\tau)}{\tau} &\text{if } \nu=3,\\ \left(\frac{\nu-2}{\nu-3}\right)\frac{1-(1-\tau)^{\frac{\nu-3}{\nu-2}}}{\tau} &\text{otherwise}. \end{cases} \end{aligned}$$* The next two lemmas are instrumental in our convergence analysis, and are immediate consequences of the (local) Hessian regularity of the smooth functions $f$ and $g_s$ in [\[eq:smoothprob\]](#eq:smoothprob){reference-type="eqref" reference="eq:smoothprob"}. **Lemma 5**. *[@nesterov2018lectures Lemma 1.2.4][\[thm:gradf-bound\]]{#thm:gradf-bound label="thm:gradf-bound"} For any given $x,y\in \mathop{\rm dom}\nolimits f$, we have $$\begin{aligned} \norm{\mathop{\nabla}\nolimits f(y) - \mathop{\nabla}\nolimits f(x) - \mathop{\nabla^2}\nolimits f(x)(y-x)} \le \frac{L_f}{2}\norm{y-x}^2,\\ \abs{f(y) - f(x) - \langle\mathop{\nabla}\nolimits f(x), y-x\rangle - \frac{1}{2}\langle\mathop{\nabla^2}\nolimits f(x)(y-x),y-x\rangle} \le \frac{L_f}{6}\norm{y-x}^3.\end{aligned}$$* **Lemma 6**. *[@sun2019generalized Lemma 2][\[thm:gradg-bound\]]{#thm:gradg-bound label="thm:gradg-bound"} For any given $x,y\in \mathop{\rm dom}\nolimits g$, $\Upsilon_{g_s}^{x,y}$ satisfies $$\begin{aligned} \|\Upsilon_{g_s}^{x,y}\| \le R_\nu(d_\nu(x,y))d_\nu(x,y), \end{aligned}$$ where, for $\tau\in [0,1)$, $R_\nu(\tau)$ is defined by $$\begin{aligned} \label{eq:r-nu} R_\nu(\tau) \coloneqq \begin{cases} \left(\frac{3}{2}+\frac{\tau}{3}\right)\exp(\tau) & \text{if }\nu=2,\\ \frac{1-\left(1-\tau\right)^{\frac{4-\nu}{\nu-2}}-\left(\frac{4-\nu}{\nu-2}\right)\tau\left(1-\tau\right)^{\frac{4-\nu}{\nu-2}}}{\left(\frac{4-\nu}{\nu-2}\right)\tau^2\left(1-\tau\right)^{\frac{4-\nu}{\nu-2}}} & \text{if } \nu\in(2,3]. \end{cases}\end{aligned}$$* #### Global convergence. We prove a first global result for the proximal Newton-type scheme [\[eq:proxnewton-general\]](#eq:proxnewton-general){reference-type="eqref" reference="eq:proxnewton-general"}. We show that the iterates of this scheme decrease the objective function values with the step-lengths specified by [\[eq:steplength\]](#eq:steplength){reference-type="eqref" reference="eq:steplength"} and $\alpha_k\in(0,1]$, and converge to an optimal solution of [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"}. Let us define the following mapping: $$\begin{aligned} \label{eq:map-G} \mathop{G_{\alpha_k g}(x_k)}\nolimits \coloneqq \frac{1}{\bar{\alpha}_k}H_k\left(x_k - \mathop{\rm prox}\nolimits_{\alpha_k g}(x_k - \bar{\alpha}_kH_k^{-1}\mathop{\nabla}\nolimits q(x_k))\right).\end{aligned}$$ Clearly, [\[eq:proxnewton-general\]](#eq:proxnewton-general){reference-type="eqref" reference="eq:proxnewton-general"} is equivalent to $$\begin{aligned} \label{eq:equiv-proxnewton-general} x_{k+1} = x_k - \bar{\alpha}_kH_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits.\end{aligned}$$ Using [\[eq:optimality-conditions\]](#eq:optimality-conditions){reference-type="eqref" reference="eq:optimality-conditions"} and the definition of the (scaled) proximal operator, $\mathop{G_{\alpha_k g}(x_k)}\nolimits$ satisfies $$\begin{aligned} \label{eq:mapg-optimality} \mathop{G_{\alpha_k g}(x_k)}\nolimits \in \mathop{\nabla}\nolimits q(x_k) + \partial g(x_k - \bar{\alpha}_kH_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits).\end{aligned}$$ Moreover, $\mathop{G_{\alpha_k g}(\bar{x})}\nolimits= 0$ if and only if $\bar{x}$ solves problem [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"}. **Theorem 1**. *Let $\{x_k\}$ be the sequence generated by the scheme [\[eq:proxnewton-general\]](#eq:proxnewton-general){reference-type="eqref" reference="eq:proxnewton-general"} for problem [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"} and let $\bar{\alpha}_k$ be specified by [\[eq:steplength\]](#eq:steplength){reference-type="eqref" reference="eq:steplength"} with $\alpha_k\in(0,1]$. Then $\{x_k\}$ satisfies $$\begin{aligned} {\mathcal{L}}(x_{k+1}) \le {\mathcal{L}}(x_k) - \hat{g}_s(T(x_k)), \end{aligned}$$ where $\hat{g}_s$ is an upper quadratic approximation of the function $g_s$ which is globally minimized at $T(x_k) \coloneqq x_k - \bar{\alpha}_kH_g^{-1}\mathop{\nabla}\nolimits g_s(x_k)$.* *Proof.* See . ◻ **Corollary 1**. *Let $\{x_k\} \subset {\mathbb R}^n$ in . Then every limit point $\bar{x}$ of $\{x_k\}$ at which [\[eq:optim-gs\]](#eq:optim-gs){reference-type="eqref" reference="eq:optim-gs"} holds is a stationary point of the objective function ${\mathcal{L}}$ in problem [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"}.* *Proof.* See . ◻ #### How to choose $\alpha_k$. In previous results, we did not specify a particular way to choose $\alpha_k$. Our algorithms converge for any value of $\alpha_k\in(0,1]$. Compared to the step-length selection rule proposed in [@sun2019generalized], for instance, our approach and analysis do not directly rely on the actual value of $\nu$ in the choice of both $\bar{\alpha}_k$ and $\alpha_k$. Indeed, in the context of minimizing a function $g_s\in{{\mathcal{F}}_{M_{g},\nu}}$, an optimal choice for $\bar{\alpha}_k$, in view of [@sun2019generalized], corresponds to setting $$\begin{aligned} \alpha_k = \begin{cases} \frac{\ln(1+d_k)(1 + M\eta_k)}{d_k} &\text{if } \nu=2,\\ \frac{2(1+M_g\eta_k)}{2+M_g\eta_k} &\text{if } \nu=3, \end{cases}\end{aligned}$$ where $d_k\coloneqq M_g\|\mathop{\nabla^2}\nolimits H_g^{-1}\mathop{\nabla}\nolimits g_s(x_k)\|$ and in each case, it can be shown that $\bar{\alpha}_k\in(0,1)$. However, choosing $\alpha_k$ this way does not guarantee certain theoretical bounds in the context of the framework studied in this work, especially for $\nu=2$. We therefore propose to leave $\alpha_k$ as a hyperparameter that must satisfy $0<\alpha_k\equiv\alpha\le1$. This however gives us the freedom to exploit specific properties about the function $f$, when they are known to hold. One of such properties is the global Lipschitz continuity of $\mathop{\nabla}\nolimits f$, where supposing the Lipschitz constant $L$ is known, one may set $$\begin{aligned} \alpha_k = \min\{1/L,1\}.\end{aligned}$$ #### Local quadratic convergence. We next discuss the local convergence properties of Algorithms [\[alg:NewtonSCOREProx\]](#alg:NewtonSCOREProx){reference-type="ref" reference="alg:NewtonSCOREProx"} and [\[alg:GGNSCOREProx\]](#alg:GGNSCOREProx){reference-type="ref" reference="alg:GGNSCOREProx"}. In our discussion, we take the local norm $\norm{\cdot}_x$ (and its dual) with respect to $g_s$, and the standard Euclidean norm $\norm{\cdot}$ with respect to the (local) Euclidean ball $\mathcal{B}_{r_0}(\cdot) \subset {\mathcal{E}}_r(\cdot)$. **Theorem 2**. *Suppose that [\[ass:p1\]](#ass:p1){reference-type="ref" reference="ass:p1"}--[\[ass:p4\]](#ass:p4){reference-type="ref" reference="ass:p4"} hold, and let $x^*$ be an optimal solution of [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"}. Let $\{x_k\}$ be the sequence of iterates generated by Algorithm [\[alg:NewtonSCOREProx\]](#alg:NewtonSCOREProx){reference-type="ref" reference="alg:NewtonSCOREProx"} and define $\lambda_k\coloneqq 1+M_g\omega_\nu(-d_\nu(x^*,x_k))\norm{x_k-x^*}_{x_k}$, where $\omega_\nu$ is defined by [\[eq:omega-nu\]](#eq:omega-nu){reference-type="eqref" reference="eq:omega-nu"}. Then starting from a point $x_0 \in {\mathcal{E}}_r(x^*)$, if $d_\nu(x^*,x_k)<1$ with $d_\nu$ defined by [\[eq:d-metric\]](#eq:d-metric){reference-type="eqref" reference="eq:d-metric"}, the sequence $\{x_k\}$ converges to $x^*$ according to $$\begin{aligned} \label{eq:conv-rate-xk-alg1} \norm{x_{k+1}-x^*}_{x^*} \le \frac{L(\lambda_k-\alpha_k)}{\lambda_k\sqrt{\rho}} + R_k\norm{x_k-x^*}_{x^*} + \vartheta_k\norm{x_k-x^*}^2, \end{aligned}$$ where $\alpha_k\in(0,1]$, $\vartheta_k \coloneqq \left(\frac{\lambda_k-\alpha_k}{\lambda_k}\right)\bar{\omega}_\nu(d_\nu(x^*,x_k))+\frac{L_f}{2\sqrt{\rho}}$, $R_k\coloneqq R_\nu(d_\nu(x^*,x_k))d_\nu(x^*,x_k)$ with $R_\nu$ defined by [\[eq:r-nu\]](#eq:r-nu){reference-type="eqref" reference="eq:r-nu"}, and $\bar{\omega}_\nu$ is defined by [\[eq:baromega-nu\]](#eq:baromega-nu){reference-type="eqref" reference="eq:baromega-nu"}.* *Proof.* See . ◻ To prove a local convergence rate for Algorithm [\[alg:GGNSCOREProx\]](#alg:GGNSCOREProx){reference-type="ref" reference="alg:GGNSCOREProx"}, we need an additional assumption about the behaviour of the Jacobian matrix $J_k$ near $x^*$. As before, $J_k$ denotes the Jacobian matrix evaluated at $x_k$; likewise, $V_k$ and $u_k$. At $x^*$, we respectively write $J^*$, $V^*$ and $u^*$. We assume the following: 1. $\|J_kv\|\ge \beta_1\|v\|$, $\beta_1 > 0$, for all $x_k$ near $x^*$, and for any $v\in {\mathbb R}^n$.[\[ass:g1\]]{#ass:g1 label="ass:g1"} For $f$ defined by [\[eq:f-ggn\]](#eq:f-ggn){reference-type="eqref" reference="eq:f-ggn"}, condition [\[ass:g1\]](#ass:g1){reference-type="ref" reference="ass:g1"} implies that the singular values of $J_k$ are uniformly bounded away from zero, at least locally. Let the unaugmented version of the residual vector $u_k$ be denoted $\tilde{u}_k$ at $x_k$, that is, $$\begin{aligned} \tilde{u}_k\coloneqq[l'_{\hat{y}_1}(y_1,\hat{y}_1;x_k),\ldots,l'_{\hat{y}_m}(y_m,\hat{y}_m;x_k)]^\top\in{\mathbb R}^m.\end{aligned}$$ Define the following matrix: $$\begin{aligned} W_k^T &\coloneqq \begin{bmatrix} \mathop{\hat{y}_1''}\nolimits(x_1) & \mathop{\hat{y}_2''}\nolimits(x_1) & \cdots & \mathop{\hat{y}_m''}\nolimits(x_1)\\ \mathop{\hat{y}_1''}\nolimits(x_2) & \mathop{\hat{y}_2''}\nolimits(x_2) & \cdots & \mathop{\hat{y}_m''}\nolimits(x_2)\\ \vdots&\vdots&&\vdots\\ \mathop{\hat{y}_1''}\nolimits(x_n) & \mathop{\hat{y}_2''}\nolimits(x_n) & \cdots & \mathop{\hat{y}_m''}\nolimits(x_n) \end{bmatrix} \in {\mathbb R}^{n\times m}. \label{eq:Junaug}\end{aligned}$$ We note that the "full\" Hessian matrix $H_k$ can be expressed as $$\begin{aligned} H_k \equiv J_k^\top V_k J_k + (\mathbf{1}\otimes(W_k^\top\tilde{u}_k))^\top + H_g,\label{eq:hk-hat}\end{aligned}$$ where $\mathbf{1}\in{\mathbb R}^{n\times 1}$ is the $n \times 1$ matrix of ones and $\otimes$ is the outer product notation. By [\[ass:p1\]](#ass:p1){reference-type="ref" reference="ass:p1"}, [\[ass:p2\]](#ass:p2){reference-type="ref" reference="ass:p2"} and the Lipschitz continuity of $g_s$ around $x^*$ in [\[thm:g-lip\]](#thm:g-lip){reference-type="ref" reference="thm:g-lip"}, we have: for $r$ small enough, there exists a constant $\beta_2>0$ such that $\norm{\tilde{u}_k}\le \beta_2$ near $x^*$. Furthermore by our assumptions (see, e.g., [@nocedal1999numerical Theorem 10.1]), we deduce that there exists $\beta_3>0$ such that $\|W_k\|\le \beta_3$ near $x^*$. The next result about the local convergence of Algorithm [\[alg:GGNSCOREProx\]](#alg:GGNSCOREProx){reference-type="ref" reference="alg:GGNSCOREProx"} follows (see for the proof). Note that for Algorithm [\[alg:GGNSCOREProx\]](#alg:GGNSCOREProx){reference-type="ref" reference="alg:GGNSCOREProx"}, we consider the case where $f$ in problem [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"} may, in general, be expressed in the form [\[eq:f-ggn\]](#eq:f-ggn){reference-type="eqref" reference="eq:f-ggn"}. **Theorem 3**. *Suppose that [\[ass:p1\]](#ass:p1){reference-type="ref" reference="ass:p1"}--[\[ass:p4\]](#ass:p4){reference-type="ref" reference="ass:p4"} hold, and let $x^*$ be an optimal solution of [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"} where $f$ is defined by [\[eq:f-ggn\]](#eq:f-ggn){reference-type="eqref" reference="eq:f-ggn"}. Additionally, let [\[ass:g1\]](#ass:g1){reference-type="ref" reference="ass:g1"} hold for the Jacobian matrix $J_k$ defined by [\[eq:Jaug\]](#eq:Jaug){reference-type="eqref" reference="eq:Jaug"}. Let $\{x_k\}$ be the sequence of iterates generated by Algorithm [\[alg:GGNSCOREProx\]](#alg:GGNSCOREProx){reference-type="ref" reference="alg:GGNSCOREProx"}, and define $\lambda_k\coloneqq 1+M_g\omega_\nu(-d_\nu(x^*,x_k))\norm{x_k-x^*}_{x_k}$, where $\omega_\nu$ is defined by [\[eq:omega-nu\]](#eq:omega-nu){reference-type="eqref" reference="eq:omega-nu"}. Then starting from a point $x_0 \in {\mathcal{E}}_r(x^*)$, if $d_\nu(x^*,x_k)<1$ with $d_\nu$ defined by [\[eq:d-metric\]](#eq:d-metric){reference-type="eqref" reference="eq:d-metric"}, the sequence $\{x_k\}$ converges to $x^*$ according to $$\begin{aligned} \label{eq:conv-rate-xk-alg2} \norm{x_{k+1}-x^*}_{x^*} \le \frac{L(\lambda_k-\alpha_k)}{\lambda_k\sqrt{\rho}} + \frac{\tilde{\beta}\norm{x_k - x^*}}{\sqrt{\rho}} + R_k\norm{x_k-x^*}_{x^*} + \vartheta_k\norm{x_k-x^*}^2, \end{aligned}$$ where $\vartheta_k$ and $R_k$ are as defined in , $\alpha_k\in(0,1]$, $\tilde{\beta}\coloneqq\beta_2\beta_3>0$, and $\bar{\omega}_\nu$ is defined by [\[eq:baromega-nu\]](#eq:baromega-nu){reference-type="eqref" reference="eq:baromega-nu"}.* # Numerical experiments {#sec:experiments} In this section, we validate the efficiency of the technique introduced in this paper in numerical examples using both synthetic and real datasets from the LIBSVM repository [@chang2011libsvm]. The approach and algorithms proposed in this paper are implemented in the Julia programming language and are available online as an open-source package[^10]. We test the performance of Algorithms [\[alg:NewtonSCOREProx\]](#alg:NewtonSCOREProx){reference-type="ref" reference="alg:NewtonSCOREProx"} and [\[alg:GGNSCOREProx\]](#alg:GGNSCOREProx){reference-type="ref" reference="alg:GGNSCOREProx"} for various fixed values of $\alpha_k \equiv \alpha \in (0,1]$ (see **Fig.** [13](#fig:alpha-plots){reference-type="ref" reference="fig:alpha-plots"}). Clearly, convergence is fastest with $\alpha_k = 1$, so we fix this value for the two algorithms in the remainder of our experiments. We compare our technique with other algorithms, namely `PANOC` [@stella2017simple], `ZeroFPR` [@themelis2018forward], `OWL-QN` [@andrew2007scalable], proximal gradient [@lions1979splitting], and fast proximal gradient [@beck2009fast] algorithms[^11]. In the sparse group lasso experiments, we compare our approach with the proximal gradient and block coordinate descent (`BCD`)[^12] algorithms. The `BCD` is known to be an efficient algorithm for general regularized problems [@friedman2010regularization], and is used as a standard approach for the sparse group lasso problem [@ida2019fast; @friedman2010note; @simon2013sparse]. Since the problems considered in our experiments use the $\ell_1$ and $\ell_2$ regularizers, we use $\phi(t) = \frac{1}{p}\sqrt{1+p^2\abs{t}^2}-1$ from , with $p=1$ and derive $g_s$ in problem [\[eq:smoothprob\]](#eq:smoothprob){reference-type="eqref" reference="eq:smoothprob"} accordingly (see also **Fig.** [\[fig:g-smooth\]](#fig:g-smooth){reference-type="ref" reference="fig:g-smooth"}). This provides a good (smooth) approximation for the $1$- and $2$-norms with an appropriate value of $\mu$. Its gradient and Hessian are respectively $$\begin{aligned} \mathop{\nabla}\nolimits g_s(x) = x/\sqrt{\mu^{2}+x^{2}}, \qquad \mathop{\nabla^2}\nolimits g_s(x) = \mathop{\rm diag}\nolimits\left(\mu^{2}/(\mu^{2}+x^{2})^{\frac{3}{2}}\right).\end{aligned}$$ For a diagonal matrix $H_g\in{\mathbb R}^{n\times n}$, the scaled proximal operator for the $1$- and $2$-norms are obtained using the proximal calculus derived in [@becker2019quasi]. Let the vector $\hat{d}\in{\mathbb R}^n$ contain the diagonal entries of $H_g$, and let $\beta \in {\mathbb P}$: $\mathop{\rm prox}\nolimits_{\beta \|x\|_1}^{H_g} = \mathop{\rm sign}\nolimits(x)\cdot \max\{|x| - \beta \hat{d},0\}$, and $\mathop{\rm prox}\nolimits_{\beta \|x\|}^{H_g} = x\cdot \max\{1 - \beta\hat{d}/\|x\|,0\}$. All experiments are performed on a laptop with dual (2.30GHz + 2.30GHz) Intel Core i7-11800 H CPU and 32GB RAM. ## Sparse logistic regression {#ss:logexample} We consider the problem of finding a sparse solution $x$ to the following logistic regression problem $$\begin{aligned} \label{eq:logexample} \min\limits_{x\in{\mathbb R}^n} {\mathcal{L}}(x) \coloneqq \underbrace{\sum_{i=1}^{m} \log\left(1 + \exp(-y_i\langle a_i, x\rangle)\right)}_{\eqqcolon f(x)} + \beta \|x\|_1,\end{aligned}$$ where, in view of [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"}, $g(x) \coloneqq \beta \|x\|_1$, $\beta\in{\mathbb P}$, and $a_i \in {\mathbb R}^n, y_i \in \{-1,1\}$ form the data. We perform experiments on both randomly generated data and real datasets summarized in **Table** [\[tab:data-summary\]](#tab:data-summary){reference-type="ref" reference="tab:data-summary"}. For the synthetic data, we set $\beta = 0.2$, while for the real datasets, we set $\beta = 1$. We fix $\mu =1$ in both Algorithms [\[alg:NewtonSCOREProx\]](#alg:NewtonSCOREProx){reference-type="ref" reference="alg:NewtonSCOREProx"} and [\[alg:GGNSCOREProx\]](#alg:GGNSCOREProx){reference-type="ref" reference="alg:GGNSCOREProx"}, and set $\alpha_k = 1/L$ for the proximal gradient algorithm, where $L$ is estimated as $L = \lambda_{max} (A^\top A)$, the columns of $A\in {\mathbb R}^{n\times m}$ are the vectors $a_i$ and $\lambda_{max}$ denotes the largest eigenvalue. For the sake of fairness, we provide this value of $L$ to each of `PANOC`, `ZeroFPR`, and fast proximal gradient algorithms for computing their step-lengths in our comparison. {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} ![Performance profile (CPU time) for the sparse logistic regression problem [\[eq:logexample\]](#eq:logexample){reference-type="eqref" reference="eq:logexample"} using the LIBSVM datasets summarized in **Table** [\[tab:data-summary\]](#tab:data-summary){reference-type="ref" reference="tab:data-summary"}. Here, $\tau$ denotes the performance ratio (CPU times in seconds) averaged over 20 independent runs with different random initializations, and $\rho(\tau)$ is the corresponding frequency. Each algorithm is stopped when $\frac{\|x_k - x_{k-1}\|}{\max\{\|x_{k-1}\|,1\}} < 10^{-6}$ or when the default tolerance is reached.](figures/perf_profile.pdf){#fig:perf-prof} ![Behaviour of `Prox-N-SCORE` and `Prox-GGN-SCORE` for different fixed values of $\alpha_k$ in problem [\[eq:logexample\]](#eq:logexample){reference-type="eqref" reference="eq:logexample"}.](figures/100_4000_alpha_sim_log_prox-newtonscore.pdf){#fig:alpha-plots width="\\textwidth"} ![Behaviour of `Prox-N-SCORE` and `Prox-GGN-SCORE` for different fixed values of $\alpha_k$ in problem [\[eq:logexample\]](#eq:logexample){reference-type="eqref" reference="eq:logexample"}.](figures/100_4000_alpha_sim_log_prox-ggnscore.pdf){#fig:alpha-plots width="\\textwidth"} ![Behaviour of `Prox-N-SCORE` and `Prox-GGN-SCORE` for different fixed values of $\alpha_k$ in problem [\[eq:logexample\]](#eq:logexample){reference-type="eqref" reference="eq:logexample"}.](figures/4000_100_alpha_sim_log_prox-newtonscore.pdf){#fig:alpha-plots width="\\textwidth"} ![Behaviour of `Prox-N-SCORE` and `Prox-GGN-SCORE` for different fixed values of $\alpha_k$ in problem [\[eq:logexample\]](#eq:logexample){reference-type="eqref" reference="eq:logexample"}.](figures/100_4000_alpha_t_sim_log_prox-newtonscore.pdf){#fig:alpha-plots width="\\textwidth"} ![Behaviour of `Prox-N-SCORE` and `Prox-GGN-SCORE` for different fixed values of $\alpha_k$ in problem [\[eq:logexample\]](#eq:logexample){reference-type="eqref" reference="eq:logexample"}.](figures/100_4000_alpha_t_sim_log_prox-ggnscore.pdf){#fig:alpha-plots width="\\textwidth"} ![Behaviour of `Prox-N-SCORE` and `Prox-GGN-SCORE` for different fixed values of $\alpha_k$ in problem [\[eq:logexample\]](#eq:logexample){reference-type="eqref" reference="eq:logexample"}.](figures/4000_100_alpha_t_sim_log_prox-newtonscore.pdf){#fig:alpha-plots width="\\textwidth"} ![Behaviour of `Prox-N-SCORE` and `Prox-GGN-SCORE` for different fixed values of $\alpha_k$ in problem [\[eq:logexample\]](#eq:logexample){reference-type="eqref" reference="eq:logexample"}.](figures/4000_100_alpha_sim_log_prox-ggnscore.pdf){#fig:alpha-plots width="\\textwidth"} ![Behaviour of `Prox-N-SCORE` and `Prox-GGN-SCORE` for different fixed values of $\alpha_k$ in problem [\[eq:logexample\]](#eq:logexample){reference-type="eqref" reference="eq:logexample"}.](figures/alpha_mushrooms_prox-newtonscore.pdf){#fig:alpha-plots width="\\textwidth"} ![Behaviour of `Prox-N-SCORE` and `Prox-GGN-SCORE` for different fixed values of $\alpha_k$ in problem [\[eq:logexample\]](#eq:logexample){reference-type="eqref" reference="eq:logexample"}.](figures/alpha_mushrooms_prox-ggnscore.pdf){#fig:alpha-plots width="\\textwidth"} ![Behaviour of `Prox-N-SCORE` and `Prox-GGN-SCORE` for different fixed values of $\alpha_k$ in problem [\[eq:logexample\]](#eq:logexample){reference-type="eqref" reference="eq:logexample"}.](figures/4000_100_alpha_t_sim_log_prox-ggnscore.pdf){#fig:alpha-plots width="\\textwidth"} ![Behaviour of `Prox-N-SCORE` and `Prox-GGN-SCORE` for different fixed values of $\alpha_k$ in problem [\[eq:logexample\]](#eq:logexample){reference-type="eqref" reference="eq:logexample"}.](figures/alpha_t_mushrooms_prox-newtonscore.pdf){#fig:alpha-plots width="\\textwidth"} ![Behaviour of `Prox-N-SCORE` and `Prox-GGN-SCORE` for different fixed values of $\alpha_k$ in problem [\[eq:logexample\]](#eq:logexample){reference-type="eqref" reference="eq:logexample"}.](figures/alpha_t_mushrooms_prox-ggnscore.pdf){#fig:alpha-plots width="\\textwidth"} Data $m$ $n$ Density ------------- --------- ------- --------- `mushrooms` $8124$ $112$ $0.19$ `phishing` $11055$ $68$ $0.44$ `w1a` $2477$ $300$ $0.04$ `w2a` $3470$ $300$ $0.04$ `w3a` $4912$ $300$ $0.04$ `w4a` $7366$ $300$ $0.04$ `w5a` $9888$ $300$ $0.04$ `w8a` $49749$ $300$ $0.04$ `a1a` $1605$ $123$ $0.11$ `a2a` $2265$ $123$ $0.11$ `a3a` $3185$ $123$ $0.11$ `a4a` $4781$ $123$ $0.11$ `a5a` $6414$ $123$ $0.11$ [\[tab:data-summary\]]{#tab:data-summary label="tab:data-summary"} As shown in **Fig.** [13](#fig:alpha-plots){reference-type="ref" reference="fig:alpha-plots"}, computation time is greatly reduced for `Prox-GGN-SCORE` with $m+n_y<n$. On the other hand (when $m \gg n$), while `Prox-N-SCORE` and `Prox-GGN-SCORE` are seen to terminate after almost the same number of iterations (see also **Fig.** [\[fig:splogl1-loss\]](#fig:splogl1-loss){reference-type="ref" reference="fig:splogl1-loss"}), `Prox-GGN-SCORE` is more computationally demanding than `Prox-N-SCORE`. Hence `Prox-GGN-SCORE` is preferred for problems with $m+n_y<n$ (or $m \ll n$) while `Prox-N-SCORE` is preferred when $n<m+n_y$ (or $n \ll m$). Since the latter is the case for all of the real datasets in **Table** [\[tab:data-summary\]](#tab:data-summary){reference-type="ref" reference="tab:data-summary"}, the performance profile of **Fig.** [1](#fig:perf-prof){reference-type="ref" reference="fig:perf-prof"} evaluates the `Prox-N-SCORE` for three different values of $\alpha$. **Fig.** [\[fig:splogl1-loss\]](#fig:splogl1-loss){reference-type="ref" reference="fig:splogl1-loss"} and **Fig.** [1](#fig:perf-prof){reference-type="ref" reference="fig:perf-prof"} show that our approach outperform other tested algorithms in most cases. ## Sparse group lasso In this example, we consider the sparse group lasso problem: $$\begin{aligned} \label{eq:sgl-example} \min\limits_{x\in{\mathbb R}^n} {\mathcal{L}}(x) \coloneqq \underbrace{\frac{1}{2}\norm{Ax - y}^2}_{\eqqcolon f(x)} + \underbrace{\beta\norm{x}_1 + \beta_{\mathcal{G}}\sum_{j\in {\mathcal{G}}}\omega_j\norm{x_j}}_{\eqqcolon g(x)},\end{aligned}$$ as described in . We use the common example used in the literature [@wang2014two; @tibshirani2012strong], which is based on the model $y = Ax^* + 0.01\epsilon\in {\mathbb R}^{m\times 1}$, $\epsilon \sim {\mathcal{N}}(0,1)$. The entries of the data matrix $A\in {\mathbb R}^{m\times n}$ are drawn from the normal distribution with pairwise correlation $\mathrm{corr}(A_i,A_j) = 0.5^{|i-j|}$, $\forall (i,j) \in [n]^2$. We generate datasets for different values of $m$ and $n$ with $n$ satisfying $(n \mod n_g)=0$. In this problem, we want to further highlight the faster computation time achieved by the approximation in `Prox-GGN-SCORE`, so we consider only overparameterized models (i.e., with $m+1<n$). \|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\| & & & &\ & `alg.A` & `alg.B` & `alg.C` & `alg.A` & `alg.B` & `alg.C` & `alg.A` & `alg.B` & `alg.C` &\ $(500,2000; 19)$ & **1.4106E-01** & & & **19** & **19** & **19** & **2.81** & 13.39 & 3.55 & 1.2\ $(500,4000; 36)$ & **3.3461E-01** & & & **36** & 39 & 39 & **8.44** & 51.91 & 11.60 & 1.6\ $(500,5000; 45)$ & **6.8589E-01** & & & **45** & 52 & 56 & 22.09 & 90.05 & **18.53** & 1.6\ $(1000,5000; 45)$ & **1.3689E+00** & & & **45** & **45** & 46 & **12.69** & 40.71 & 22.14 & 1.6\ $(1000,7000; 65)$ & **2.5171E+00** & & & **65** & 76 & 77 & **30.26** & 139.99 & 70.08 & 1.6\ $(1000,10000; 94)$ & **1.8111E+01** & & & **95** & 141 & 142 & **111.39** & 279.58 & 126.17 & 1.5\ [\[tab:gl-results\]]{#tab:gl-results label="tab:gl-results"} In this problem, the matrix $C$ in the reformulation [\[eq:structured-prob\]](#eq:structured-prob){reference-type="eqref" reference="eq:structured-prob"} is a diagonal matrix with row indices given by all pairs $(i,j) \in \{(i,j)|i\in j, i \in \{1,\ldots,n_g\}\}$, and column indices given by $k\in\{1,\ldots,n_g\}$. That is, $$\begin{aligned} C_{(i,j),k} = \begin{cases} \beta_{\mathcal{G}}\omega_j &\quad\text{if }i=k,\\ 0 &\quad\text{otherwise}. \end{cases}\end{aligned}$$ We construct $x^*$ in a similar way as [@ndiaye2016gap]: We fix $n_g=100$ and break $n$ randomly into groups of equal sizes with $0.1$ percent of the groups selected to be *active*. The entries of the subvectors in the *nonactive* groups are set to zero, while for the active groups, $\lceil\frac{n}{n_g}\rceil\times 0.1$ of the subvector entries are drawn randomly and set to $\mathop{\rm sign}\nolimits(\xi) \times U$ where $\xi$ and $U$ are uniformly distributed in $[0.5,10]$ and $[-1,1]$, respectively; the remaining entries are set to zero. For the sake of fair comparison, each data and the associated initial vector $x_0$ are generated in Julia and exported for the `BCD` implementation in Python. We set $\beta = \tau_1\gamma\|A^Ty\|_\infty$, $\beta_{\mathcal{G}}= (10-\tau_1)\gamma\|A^Ty\|_\infty$ with $\tau_1=0.9$ and $\gamma = 10^{-8}$. For each group $j$, the parameter $\omega_j$ is set to the standard value $\sqrt{n_j}$ [@friedman2010note; @simon2013sparse], where $n_j=\mathop{\rm card}\nolimits(j)$. For fairness, the estimate $\alpha_k = 1/L$ with $L = \lambda_{max} (A^\top A)$ is used in the proximal gradient algorithm. ![Mean squared error (MSE) between the estimates $x_k$ and the true coefficient $x^*$ for `Prox-GGN-SCORE`, `Prox-Grad` and `BCD` on the sparse group lasso problem [\[eq:sgl-example\]](#eq:sgl-example){reference-type="eqref" reference="eq:sgl-example"}.](figures/_500_2000_mseplot_sim_gl.pdf){#fig:sgl-plots width="\\textwidth"} ![Mean squared error (MSE) between the estimates $x_k$ and the true coefficient $x^*$ for `Prox-GGN-SCORE`, `Prox-Grad` and `BCD` on the sparse group lasso problem [\[eq:sgl-example\]](#eq:sgl-example){reference-type="eqref" reference="eq:sgl-example"}.](figures/_500_4000_mseplot_sim_gl.pdf){#fig:sgl-plots width="\\textwidth"} ![Mean squared error (MSE) between the estimates $x_k$ and the true coefficient $x^*$ for `Prox-GGN-SCORE`, `Prox-Grad` and `BCD` on the sparse group lasso problem [\[eq:sgl-example\]](#eq:sgl-example){reference-type="eqref" reference="eq:sgl-example"}.](figures/_500_5000_mseplot_sim_gl.pdf){#fig:sgl-plots width="\\textwidth"} ![Mean squared error (MSE) between the estimates $x_k$ and the true coefficient $x^*$ for `Prox-GGN-SCORE`, `Prox-Grad` and `BCD` on the sparse group lasso problem [\[eq:sgl-example\]](#eq:sgl-example){reference-type="eqref" reference="eq:sgl-example"}.](figures/_1000_5000_mseplot_sim_gl.pdf){#fig:sgl-plots width="\\textwidth"} ![Mean squared error (MSE) between the estimates $x_k$ and the true coefficient $x^*$ for `Prox-GGN-SCORE`, `Prox-Grad` and `BCD` on the sparse group lasso problem [\[eq:sgl-example\]](#eq:sgl-example){reference-type="eqref" reference="eq:sgl-example"}.](figures/_1000_7000_mseplot_sim_gl.pdf){#fig:sgl-plots width="\\textwidth"} ![Mean squared error (MSE) between the estimates $x_k$ and the true coefficient $x^*$ for `Prox-GGN-SCORE`, `Prox-Grad` and `BCD` on the sparse group lasso problem [\[eq:sgl-example\]](#eq:sgl-example){reference-type="eqref" reference="eq:sgl-example"}.](figures/_1000_10000_mseplot_sim_gl.pdf){#fig:sgl-plots width="\\textwidth"} In the experiments, we stop `Prox-GGN-SCORE` and proximal gradient algorithms when $\frac{\|x_k - x_{k-1}\|}{\max\{\|x_{k-1}\|,1\}} < 10^{-9}$. The `BCD` algorithm is stopped when the default tolerance is reached. The simulation results are shown in **Table** [\[tab:gl-results\]](#tab:gl-results){reference-type="ref" reference="tab:gl-results"} and **Fig.** [19](#fig:sgl-plots){reference-type="ref" reference="fig:sgl-plots"}. As shown, `Prox-GGN-SCORE` solves the problem faster with fewer number of iterations in most cases, and gives smaller objective values and much better solution quality than both the proximal gradient and `BCD` algorithms. ## Sparse deconvolution {#ss:lsqexample} In this example, we consider the problem of estimating the unknown sparse input $x$ to a linear system, given a noisy output signal and the system response. That is, $$\begin{aligned} \min\limits_{x\in{\mathbb R}^n} {\mathcal{L}}(x) \coloneqq \underbrace{\frac{1}{2}\norm{Ax - y}^2}_{\eqqcolon f(x)} + \beta \|x\|_p, \label{eq:lsqexample}\end{aligned}$$ where $A\in {\mathbb R}^{n\times n}$ and $y\in{\mathbb R}^{n\times 1}$ are given data about the system which we randomly generate according to [@selesnick2014sparse Example F]. We solve with both $\ell_1$ ($p=1$) and $\ell_2$ ($p=2$) regularizers, and set $\beta = 10^{-3}$. We set $\mu = 5\times 10^{-2}$ in the smooth approximation $g_s$ of $g$. We estimate $L = \lambda_{max} (A^\top A)$ and set $\alpha_k = 1/L$ in the proximal gradient algorithm. Again, for fairness, we provide this value of $L$ to each of `PANOC`, `ZeroFPR`, and fast proximal gradient procedures in our comparison. The results of the simulations are displayed in **Fig.** [20](#fig:deconv-l1){reference-type="ref" reference="fig:deconv-l1"} and **Fig.** [21](#fig:deconv-l2){reference-type="ref" reference="fig:deconv-l2"}. While `Prox-GGN-SCORE` and `Prox-N-SCORE` sometimes have more running time in this problem, they provide better solution quality with smaller reconstruction error than the other tested algorithms, which is often more important for signal reconstruction problems. ![Sparse deconvolution via $\ell_1$-regularized least squares [\[eq:lsqexample\]](#eq:lsqexample){reference-type="eqref" reference="eq:lsqexample"} using `Prox-N-SCORE`, `Prox-GGN-SCORE`, `PANOC`, `ZeroFPR`, proximal gradient, and fast proximal gradient algorithms with $n=1024$. Each algorithm is stopped when $\frac{\|x_k - x_{k-1}\|}{\max\{\|x_{k-1}\|,1\}} < 10^{-6}$ or when the default tolerance is reached.](figures/sol_deconv-l1.pdf){#fig:deconv-l1 width="\\linewidth"} ![Sparse deconvolution via $\ell_2$-regularized least squares [\[eq:lsqexample\]](#eq:lsqexample){reference-type="eqref" reference="eq:lsqexample"} using `Prox-N-SCORE`, `Prox-GGN-SCORE`, `PANOC`, `ZeroFPR`, proximal gradient, and fast proximal gradient algorithms with $n=1024$. Each algorithm is stopped when $\frac{\|x_k - x_{k-1}\|}{\max\{\|x_{k-1}\|,1\}} < 10^{-6}$ or when the default tolerance is reached.](figures/sol_deconv-l2.pdf){#fig:deconv-l2 width="\\linewidth"} # Conclusions {#sec:conclusions} Generalized self-concordant optimization provides very useful tools for implementing and analyzing Newton-type methods for unconstrained problems. This helps to reconcile the geometric properties of Newton-type methods with their implementations, while providing convergence guarantees. In the presence of constraints or nonsmooth terms in the objective functions, it becomes natural to extend these methods via proximal schemes. However, when the (generalized) self-concordant property is uncheckable for the objective functions (from a practical point of view), these convergence guarantees becomes difficult to prove. The self-concordant smoothing framework of this paper addresses this drawback by combining different regularization/smoothing phenomena, namely: partial smoothing, inf-conv smoothing, and self-concordant regularization (SCORE). This approach, leading to two algorithms in this paper (`Prox-N-SCORE` and `Prox-GGN-SCORE`), is able to utilize certain properties of generalized self-concordant functions in the selection of adaptive step-lengths and a simple variable-metric in the proximal Newton-type scheme. We prove global convergence and local convergence rates for the approach. As demonstrated in numerical simulations, in most cases, our approach compares favourably against other state-of-the-art approaches from the literature. In future research, it would be interesting to analyze our framework in the nonconvex setting. In particular, we believe that our notion of self-concordant smoothing could lead to interesting research directions in applications such as deep neural network training, in which the approximation scheme of `Prox-GGN-SCORE` would become very instrumental in scaling our method with respect to the problem size. # Proofs of the results stated in , , {#app:A} #### Proof of . *Proof.* First, as an immediate consequence of [@bauschke2011convex Lemma 1.28, Lemma 1.27 and Proposition 8.17], we have $h_\mu\in\mathop{\Gamma_0({\mathbb R}^n)}\nolimits$. (i) Follows immediately from [@bauschke2011convex Proposition 12.14]. (ii) By Item [\[thm:exact\]](#thm:exact){reference-type="ref" reference="thm:exact"}, $g_s=\mathop{g \boxdot h_\mu}\nolimits\in \mathop{\Gamma_0({\mathbb R}^n)}\nolimits$. As a consequence of [@bauschke2011convex Proposition 12.14], we have $$\begin{aligned} g_s(x,\mu) = \min\limits_{w\in{\mathbb R}^n}\left\{g(w)+h_\mu(x-w)\right\}, \end{aligned}$$ and $g_s \mathop{\underrightarrow{e}}\nolimits g$ (by [@rockafellar2009variational Theorem 11.34]). In view of [@rockafellar2009variational Proposition 7.2], for $x\in \mathop{\rm dom}\nolimits g$ and $$\begin{aligned} w_\mu(x) \in \arg\min\limits_{w\in {\mathbb R}^n}\set{g(w) + h_\mu(x-w)} \ne \emptyset, \end{aligned}$$ $g_s \mathop{\underrightarrow{e}}\nolimits g$ implies that $g_s(x,\mu) \to g(x)$ for at least one sequence $w_\mu(x) \to x$. Hence, we have $$\begin{aligned} (\mathop{g \square h_\mu}\nolimits)(x) = g(w_\mu(x)) + h_\mu(x - w_\mu(x)). \end{aligned}$$ And, given $h\in {{\mathcal{F}}_{M_{h},\nu}}$, we have by that $h_\mu$ is $(M_g,\nu)$-generalized self-concordant, where $M_g$ is defined by $$\begin{aligned} M_g = \begin{cases} n^{\frac{3-\nu}{2}}\mu^{\frac{\nu}{2}-2}M_h, \qquad \text{if } \nu \in (0,3],\\ \mu^{4-\frac{3\nu}{2}}M_h, \qquad \text{if } \nu >3. \end{cases} \end{aligned}$$ Hence, $h_\mu\in{\mathcal C^{3}}(\mathop{\rm dom}\nolimits g)$, and by [@bauschke2011convex Proposition 18.7/Corollary 18.8], we have $$\begin{aligned} \mathop{\nabla^3}\nolimits(\mathop{g \square h_\mu}\nolimits)(x) &= \mathop{\nabla^3}\nolimits h_\mu(x - w_\mu(x)),\\ \abs{\left<\mathop{\nabla^3}\nolimits(\mathop{g \square h_\mu}\nolimits)(x)[v]u,u\right>} &= \abs{\left<\mathop{\nabla^3}\nolimits h_\mu(x - w_\mu(x))[v]u,u\right>}, \qquad \forall u,v \in \mathop{\rm dom}\nolimits g. \end{aligned}$$ By definition, the univariate function $$\begin{aligned} \varphi(t)\coloneqq h_\mu(u_1 + tv_1), \label{eq:univariatesc} \end{aligned}$$ is $(M_g,\nu)$-generalized self-concordant, for every $u_1,v_1 \in \mathop{\rm dom}\nolimits g$. That is, $\forall t\in{\mathbb R}$, $$\begin{aligned} \abs{\mathop{\varphi'''}\nolimits(t)} \le M_g\mathop{\varphi''}\nolimits(t)^{\frac{\nu}{2}}, \end{aligned}$$ which concludes the proof after setting $u_1=x$, $v_1=w(\frac{x}{\mu})$ and $t=-\mu$ in [\[eq:univariatesc\]](#eq:univariatesc){reference-type="eqref" reference="eq:univariatesc"}. (iii) Following the arguments in Items (i) and (ii) above, $w_\mu$ (and hence $g_s$) is finite-valued (see also [@burke2013epi Lemma 4.2]). Then the Lipschitz continuity of $g_s$ near some $\bar{x}\in\mathop{\rm dom}\nolimits g$ follows from the convexity of $g_s$ (see [@rockafellar2009variational Example 9.14]; see also [@burke2017epi Proposition 3.6]).  ◻ #### Proof of . *Proof.* As a result of [\[eq:grad-consistency\]](#eq:grad-consistency){reference-type="eqref" reference="eq:grad-consistency"}, , and by [@rockafellar2009variational Theorem 13.2], there exists $v_g \in {\mathbb R}^n$, in the *extended sense* of differentiability (see [@rockafellar2009variational Definition 13.1]), such that $$\begin{aligned} \limsup_{\substack{x\to\bar{x}\\\mu\downarrow 0}} \mathop{\nabla}\nolimits g_s(x) = \partial g(\bar{x}) = \{v_g\},\\ \emptyset \ne \partial g(d) \subset v_g + \mathop{\nabla^2}\nolimits g_s(\bar{x})(d-\bar{x}) + o(\abs{d-\bar{x}}){\mathcal{E}}_r(\bar{x}). \label{eq:extended-differentiability} \end{aligned}$$ Let $x_k$ be in some neighbourhood of $\bar{x}$ and let $\{x_k\} \to \bar{x}$ be generated by an iterative process. By assumption, the differentiable terms in [\[eq:extended-differentiability\]](#eq:extended-differentiability){reference-type="eqref" reference="eq:extended-differentiability"} are convex and the differential operators are monotone. It then holds that $$\begin{aligned} \partial g(d) \subset v_g + \mathop{\nabla^2}\nolimits g_s(\bar{x})(d-x_k) + o(\abs{d-\bar{x}}){\mathcal{E}}_r(\bar{x}), \label{eq:optimal-differentiability} \end{aligned}$$ for all $x_k$ in the neighbourhood of $\bar{x}$. Since differentiability in the extended sense is necessary and sufficient for differentiability in the *classical sense* (see [@rockafellar2009variational Definition 13.1 and Theorem 13.2]), it holds for some $\mu\in {\mathbb P}$ that $v_g \equiv \mathop{\nabla}\nolimits g_s(\bar{x})$ which is defined through: $$\begin{aligned} \mathop{\nabla}\nolimits g_s(d) = \mathop{\nabla}\nolimits g_s(\bar{x}) + \mathop{\nabla^2}\nolimits g_s(\bar{x})(d-\bar{x}) + o(\abs{d-\bar{x}}). \label{eq:classical-differentiability} \end{aligned}$$ Consequently, using [\[eq:ca-optimality\]](#eq:ca-optimality){reference-type="eqref" reference="eq:ca-optimality"}, [\[eq:extended-differentiability\]](#eq:extended-differentiability){reference-type="eqref" reference="eq:extended-differentiability"}, [\[eq:optimal-differentiability\]](#eq:optimal-differentiability){reference-type="eqref" reference="eq:optimal-differentiability"}, [\[eq:classical-differentiability\]](#eq:classical-differentiability){reference-type="eqref" reference="eq:classical-differentiability"}, and defining the Dikin ellipsoid ${\mathcal{E}}_r(\bar{x})$ in terms of $g_s$ for $r$ small enough, we deduce $$\begin{aligned} Q_k(x_k-d) + \mathop{\nabla^2}\nolimits g_s(\bar{x})(d-x_k) + \mathop{\nabla^2}\nolimits g_s(\bar{x})x_k - \bar{\alpha}\mathop{\nabla}\nolimits q(x_k) \in \mathop{\nabla^2}\nolimits g_s(\bar{x})d + \alpha(\mathop{\nabla}\nolimits g_s + \partial g)(d). \label{eq:ca-optimality-expanded} \end{aligned}$$ Using the definition of $q$ and the convexity of $f$, we assert that when $\bar{x}$ is the unique solution $x^*$ of [\[eq:smoothprob\]](#eq:smoothprob){reference-type="eqref" reference="eq:smoothprob"}, the inclusion $0\in\mathop{\nabla^2}\nolimits f(\bar{x})x_k + \alpha\mathop{\nabla}\nolimits g_s(d)$ holds with equality. Then, in terms of $E_{\bar{x}}$ and $\xi_{\bar{x}}(Q_k,\cdot)$, [\[eq:ca-optimality-expanded\]](#eq:ca-optimality-expanded){reference-type="eqref" reference="eq:ca-optimality-expanded"} may be written as [\[eq:new-optimality\]](#eq:new-optimality){reference-type="eqref" reference="eq:new-optimality"} and $$\begin{aligned} \delta_k = \mathop{\rm prox}\nolimits_{\alpha g}^{\mathop{\nabla^2}\nolimits g_s(\bar{x})}\left(E_{\bar{x}}(x_k) + \xi_{\bar{x}}(Q_k, d)\right). \end{aligned}$$ ◻ #### Proof of . *Proof.* Indeed, for any $x,y\in \mathop{\rm dom}\nolimits g$, we have $$\begin{aligned} \mathop{\nabla}\nolimits g_s(y) - \mathop{\nabla}\nolimits g_s(x) = \int_{0}^{1} \langle\mathop{\nabla^2}\nolimits g_s(x+\tau(y-x)), y-x\rangle d\tau. \end{aligned}$$ Let $y_\tau \coloneqq x+\tau(y-x)$, then we can write $$\begin{aligned} \label{eq:g-grad-mvt} \mathop{\nabla}\nolimits g_s(y) - \mathop{\nabla}\nolimits g_s(x) = \int_{0}^{1} \frac{1}{\tau}\norm{y_\tau - x}_{y_\tau} d\tau. \end{aligned}$$ Now consider the function $\bar{\bar{\omega}}_\nu\colon {\mathbb R}\to {\mathbb R_{+}}$ defined by $$\begin{aligned} \bar{\bar{\omega}}_\nu(\tau) \coloneqq \begin{cases} \exp(\tau) &\text{if } \nu = 2,\\ \frac{1}{(1-\tau)^\frac{2}{\nu-2}} &\text{if } \nu > 2. \end{cases} \end{aligned}$$ with $\mathop{\rm dom}\nolimits\bar{\bar{\omega}}_\nu = (-\infty, 1)$ if $\nu>2$ and $\mathop{\rm dom}\nolimits\bar{\bar{\omega}}_\nu = {\mathbb R}$ if $\nu = 2$. From [@sun2019generalized Proposition 7], we have the estimate $$\begin{aligned} \bar{\bar{\omega}}_\nu(-d_\nu(x,y_\tau))^{\frac{1}{2}}\norm{y_\tau-x}_x \le \norm{y_\tau - x}_{y_\tau} \le \bar{\bar{\omega}}_\nu(d_\nu(x,y_\tau))^{\frac{1}{2}}\norm{y_\tau-x}_x, \end{aligned}$$ which may be rewritten as $$\begin{aligned} \label{eq:w-estimate} \bar{\bar{\omega}}_\nu(-\tau d_\nu(x,y))^{\frac{1}{2}}\norm{y-x}_x \le \frac{1}{\tau}\norm{y_\tau - x}_{y_\tau} \le \bar{\bar{\omega}}_\nu(\tau d_\nu(x,y))^{\frac{1}{2}}\norm{y-x}_x, \end{aligned}$$ where we have used the fact that $d_\nu(x,y_\tau)=\tau d_\nu(x,y)$ and $\norm{y_\tau-x}_x = \tau\norm{y-x}_x$. Substituting [\[eq:w-estimate\]](#eq:w-estimate){reference-type="eqref" reference="eq:w-estimate"} into [\[eq:g-grad-mvt\]](#eq:g-grad-mvt){reference-type="eqref" reference="eq:g-grad-mvt"}, we get $$\begin{aligned} \norm{y-x}_x\int_{0}^{1}\bar{\bar{\omega}}_\nu(-\tau d_\nu(x,y))^{\frac{1}{2}}d\tau \le \mathop{\nabla}\nolimits g_s(y) - \mathop{\nabla}\nolimits g_s(x) \le \norm{y-x}_x \int_{0}^{1}\bar{\bar{\omega}}_\nu(\tau d_\nu(x,y))^{\frac{1}{2}}d\tau. \end{aligned}$$ Evaluating the integrals yields [\[eq:g-error-bound\]](#eq:g-error-bound){reference-type="eqref" reference="eq:g-error-bound"}. ◻ #### Proof of . *Proof.* Letting $y = x_k - \bar{\alpha}_kH_k^{-1}G_{\alpha_kg}(x_k)$ and $x=x_k$ in , where $G_{\alpha_kg}$ is defined by [\[eq:map-G\]](#eq:map-G){reference-type="eqref" reference="eq:map-G"}, we have $$\begin{aligned} \label{eq:f-upper} f(x_{k+1}) &\le f(x_k) - \bar{\alpha}_k(H_k^{-1}\mathop{\nabla}\nolimits f(x_k))^\top\mathop{G_{\alpha_k g}(x_k)}\nolimits + \frac{\bar{\alpha}_k}{2}\norm{H_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits}_{H_f}^2 + \frac{\bar{\alpha}_kL_f}{6}\norm{H_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits}^3. \end{aligned}$$ Using ${\mathcal{L}}(x_{k+1})\coloneqq f(x_{k+1}) + g(x_{k+1})$ and [\[eq:f-upper\]](#eq:f-upper){reference-type="eqref" reference="eq:f-upper"}, we get $$\begin{aligned} \label{eq:l-upper-1} {\mathcal{L}}(x_{k+1}) &\le f(x_k) - \bar{\alpha}_k(H_k^{-1}\mathop{\nabla}\nolimits f(x_k))^\top\mathop{G_{\alpha_k g}(x_k)}\nolimits + \frac{\bar{\alpha}_k}{2}\norm{H_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits}_{H_f}^2 + \frac{\bar{\alpha}_kL_f}{6}\norm{H_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits}^3\nonumber\\ &\quad + g(x_k - \bar{\alpha}_kH_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits)\nonumber\\ &\stackrel{\lemref{thm:gradf-bound}}{\le} f(z) - \langle\mathop{\nabla}\nolimits f(x_k), z-x_k\rangle - \frac{1}{2}\norm{z-x_k}_{H_f}^2 + \frac{L_f}{6}\norm{z-x_k}^3 - \bar{\alpha}_k(H_k^{-1}\mathop{\nabla}\nolimits f(x_k))^\top\mathop{G_{\alpha_k g}(x_k)}\nolimits\nonumber\\ &\quad + \frac{\bar{\alpha}_k}{2}\norm{H_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits}_{H_f}^2 + \frac{\bar{\alpha}_kL_f}{6}\norm{H_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits}^3 + g(x_k - \bar{\alpha}_kH_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits). \end{aligned}$$ In the above, we used the lower bound in on $f(z)$. By the convexity of $g$, we have $g(z) - g(x_{x_{k+1}}) \ge v^\top(z-x_{k+1})$ for all $v\in \partial g(x_{k+1})$. Now since from [\[eq:mapg-optimality\]](#eq:mapg-optimality){reference-type="eqref" reference="eq:mapg-optimality"}, we have $\mathop{G_{\alpha_k g}(x_k)}\nolimits - \mathop{\nabla}\nolimits q(x_k)\in \partial g(x_k - \bar{\alpha}_kH_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits)$, and noting that $\mathop{\nabla}\nolimits q - \mathop{\nabla}\nolimits f = \mathop{\nabla}\nolimits g_s$, [\[eq:l-upper-1\]](#eq:l-upper-1){reference-type="eqref" reference="eq:l-upper-1"} gives $$\begin{aligned} \label{eq:l-upper-2} {\mathcal{L}}(x_{k+1}) &\le f(z) - \langle\mathop{\nabla}\nolimits f(x_k), z-x_k\rangle - \frac{1}{2}\norm{z-x_k}_{H_f}^2 + \frac{L_f}{6}\norm{z-x_k}^3 - \bar{\alpha}_k(H_k^{-1}\mathop{\nabla}\nolimits f(x_k))^\top\mathop{G_{\alpha_k g}(x_k)}\nolimits\nonumber\\ &\quad + \frac{\bar{\alpha}_k}{2}\norm{H_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits}_{H_f}^2 + \frac{\bar{\alpha}_kL_f}{6}\norm{H_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits}^3 + g(z)\nonumber\\ &\quad - (\mathop{G_{\alpha_k g}(x_k)}\nolimits - \mathop{\nabla}\nolimits q(x_k))^\top(z - x_k + \bar{\alpha}_kH_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits)\nonumber\\ &\le {\mathcal{L}}(z) - \langle\mathop{\nabla}\nolimits f(x_k), z-x_k\rangle - \frac{1}{2}\norm{z-x_k}_{H_f}^2 + \frac{L_f}{6}\norm{z-x_k}^3 - \bar{\alpha}_k(H_k^{-1}\mathop{\nabla}\nolimits f(x_k))^\top\mathop{G_{\alpha_k g}(x_k)}\nolimits\nonumber\\ &\quad + \frac{\bar{\alpha}_k}{2}\norm{H_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits}_{H_f}^2 + \frac{\bar{\alpha}_kL_f}{6}\norm{H_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits}^3 - \mathop{G_{\alpha_k g}(x_k)}\nolimits^\top(z - x_k)\nonumber\\ &\quad - \frac{\bar{\alpha}_k}{2}\langle H_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits,\mathop{G_{\alpha_k g}(x_k)}\nolimits\rangle - \mathop{\nabla}\nolimits q(x_k)^\top(z - x_k + \bar{\alpha}_kH_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits) \nonumber\\ &= {\mathcal{L}}(z) + \mathop{G_{\alpha_k g}(x_k)}\nolimits^\top(x_k - z) + \frac{\bar{\alpha}_k}{2} \langle H_k^{-1}(H_fH_k^{-1} - I_n)\mathop{G_{\alpha_k g}(x_k)}\nolimits, \mathop{G_{\alpha_k g}(x_k)}\nolimits\rangle\nonumber\\ &\quad + \mathop{\nabla}\nolimits g_s(x_k)^\top(z-x_k) + \bar{\alpha}_k(H_k^{-1}\mathop{\nabla}\nolimits g_s(x_k))^\top\mathop{G_{\alpha_k g}(x_k)}\nolimits - \frac{1}{2}\norm{z-x_k}_{H_f}^2\nonumber\\ &\quad + \frac{L_f}{6}\norm{z-x_k}^3 + \frac{\bar{\alpha}_k L_f}{6}\norm{H_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits}^3, \end{aligned}$$ where the second inequality results from the fact that $\langle H_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits,\mathop{G_{\alpha_k g}(x_k)}\nolimits\rangle \ge 0$. Now set $z=x_k$ in [\[eq:l-upper-2\]](#eq:l-upper-2){reference-type="eqref" reference="eq:l-upper-2"} and use the following relations from [\[eq:equiv-proxnewton-general\]](#eq:equiv-proxnewton-general){reference-type="eqref" reference="eq:equiv-proxnewton-general"}: $$\begin{aligned} \bar{\alpha}_kH_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits = x_k - x_{k+1},\quad \mathop{G_{\alpha_k g}(x_k)}\nolimits = \frac{1}{\bar{\alpha}_k}H_k(x_k - x_{k+1}). \end{aligned}$$ We get $$\begin{aligned} {\mathcal{L}}(x_{k+1}) &\le {\mathcal{L}}(x_k) + \frac{\bar{\alpha}_k}{2} \langle H_k^{-1}(H_fH_k^{-1} - I_n)\mathop{G_{\alpha_k g}(x_k)}\nolimits, \mathop{G_{\alpha_k g}(x_k)}\nolimits\rangle + \bar{\alpha}_k(H_k^{-1}\mathop{\nabla}\nolimits g_s(x_k))^\top\mathop{G_{\alpha_k g}(x_k)}\nolimits\nonumber\\ &\quad + \frac{\bar{\alpha}_k L_f}{6}\norm{H_k^{-1}\mathop{G_{\alpha_k g}(x_k)}\nolimits}^3\nonumber\\ &= {\mathcal{L}}(x_k) - \left[\mathop{\nabla}\nolimits g_s(x_k)(x_{k+1} - x_k) + \frac{1}{2\bar{\alpha}_k}\langle H_g(x_{k+1} - x_k), x_{k+1} - x_k\rangle + \frac{L_f}{6\bar{\alpha}_k^2}\norm{x_{k+1} - x_k}^3\right], \end{aligned}$$ where the term $$\begin{aligned} \mathop{\nabla}\nolimits g_s(x_k)(x_{k+1} - x_k) + \frac{1}{2\bar{\alpha}_k}\langle H_g(x_{k+1} - x_k), x_{k+1} - x_k\rangle + \frac{L_f}{6\bar{\alpha}_k^2}\norm{x_{k+1} - x_k}^3 \eqqcolon \hat{g}_s(x_{k+1}) \end{aligned}$$ is nothing but a cubic regularization of an upper quadratic approximation of $g_s(x_k)$ (cf. [@nesterov2006cubic]), which is here minimized at the point $x_{k+1}$ such that $g_s$ is sufficiently decreased. Note that in the proximal Newton-type scheme [\[eq:proxnewton-general\]](#eq:proxnewton-general){reference-type="eqref" reference="eq:proxnewton-general"}, the actual minimizer of $g_s$ is the point $T(x_k) \coloneqq x_k - \bar{\alpha}_kH_g^{-1}\mathop{\nabla}\nolimits g_s(x_k)$ satisfying the first-order optimality condition $$\begin{aligned} \label{eq:optim-gs} \mathop{\nabla}\nolimits g_s(x_k) + \frac{1}{\bar{\alpha}_k}H_g(T(x_k)-x_k) = 0. \end{aligned}$$ Since $g_s$ is convex, $\hat{g}_s$ is globally minimized at this point with an appropriate damping parameter $\bar{\alpha}_k$. Since $g_s \in {{\mathcal{F}}_{M_{g},\nu}}$, $\bar{\alpha}_k$ specified by [\[eq:steplength\]](#eq:steplength){reference-type="eqref" reference="eq:steplength"} with some $\alpha_k\in(0,1]$ yields the respective global minimizer (cf. [@nesterov1994interior; @sun2019generalized]). Finally, since $g_s\mathop{\underrightarrow{e}}\nolimits g$, indeed ${\mathcal{L}}(x_{k+1}) \le {\mathcal{L}}(x_k)$. ◻ #### Proof of . *Proof.* We infer existence of the limit point $\bar{x}$ from the discussion in . As a result of , $\{x_k\}$ necessarily converges to an optimal solution $x^*$ satisfying [\[eq:optim-gs\]](#eq:optim-gs){reference-type="eqref" reference="eq:optim-gs"}. The result follows from the gradient consistency property specified by [\[eq:grad-consistency\]](#eq:grad-consistency){reference-type="eqref" reference="eq:grad-consistency"} and our characterization of the optimality conditions in [\[eq:optimality-conditions\]](#eq:optimality-conditions){reference-type="eqref" reference="eq:optimality-conditions"} (cf. [@burke1992optimality Section 3] and [@burke2013epi Section 5]). ◻ #### Proof of . *Proof.* The iterative process of Algorithm [\[alg:NewtonSCOREProx\]](#alg:NewtonSCOREProx){reference-type="ref" reference="alg:NewtonSCOREProx"} is given by $$\begin{aligned} x_{k+1} = \mathop{\rm prox}\nolimits_{\alpha_k g}^{H_g} (x_k - \bar{\alpha}_k\mathop{\nabla^2}\nolimits q(x_k)^{-1}\mathop{\nabla}\nolimits q(x_k)). \end{aligned}$$ In terms of $E_{\bar{x}}$ and $\xi_{\bar{x}}(Q_k,\cdot)$ with $Q_k\equiv H_g$, and using the definition of $q$, we have $$\begin{aligned} \norm{x_{k+1}-x^*}_{x^*} &= \norm{\mathop{\rm prox}\nolimits_{\alpha_k g}^{H_g^*}(E_{x^*}(x_k) + \xi_{x^*}(Q_k,x_{k+1})) - \mathop{\rm prox}\nolimits_{\alpha_k g}^{H_g^*}(E_{x^*}(x^*))}_{x^*}\nonumber\\ &\stackrel{\eqref{eq:nonexpansive}}{\le} \norm{E_{x^*}(x_k) - E_{x^*}(x^*) + \xi_{x^*}(Q_k,x_{k+1})}_{x^*}^*\nonumber\\ &= \norm{H_k^*x_k - \bar{\alpha}_k \mathop{\nabla}\nolimits q(x_k) - H_k^*x^* + \bar{\alpha}_k q(x^*)}_{x^*}^*\nonumber\\ &= \norm{\mathop{\nabla}\nolimits q(x^*) - \mathop{\nabla}\nolimits q(x_k) + (1-\bar{\alpha}_k)(\mathop{\nabla}\nolimits q(x_k) - \mathop{\nabla}\nolimits q(x^*)) + H_k^*(x_k - x^*)}_{x^*}^*\nonumber\\ &\le \norm{\mathop{\nabla}\nolimits q(x_k) - \mathop{\nabla}\nolimits q(x^*) - H_k^*(x_k - x^*)}_{x^*}^* + (1-\bar{\alpha}_k)\norm{\mathop{\nabla}\nolimits q(x_k) - \mathop{\nabla}\nolimits q(x^*)}_{x^*}^*\nonumber\\ &\le \norm{\mathop{\nabla}\nolimits f(x_k) - \mathop{\nabla}\nolimits f(x^*) - H_f^*(x_k - x^*)}_{x^*}^* + \norm{\mathop{\nabla}\nolimits g_s(x_k) - \mathop{\nabla}\nolimits g_s(x^*) - H_g^*(x_k - x^*)}_{x^*}^* \nonumber\\&\quad + (1-\bar{\alpha}_k)\left(\norm{\mathop{\nabla}\nolimits f(x_k) - \mathop{\nabla}\nolimits f(x^*)}_{x^*}^* + \norm{\mathop{\nabla}\nolimits g_s(x_k) - \mathop{\nabla}\nolimits g_s(x^*)}_{x^*}^*\right). \label{eq:xk-bound} \end{aligned}$$ To estimate $\|\mathop{\nabla}\nolimits f(x_k) - \mathop{\nabla}\nolimits f(x^*) - H_f^*(x_k - x^*)\|_{x^*}^*$, we note that for $v\in{\mathbb R}^n$, $\|v\|_{x^*}^* \equiv \|H_g^{*-\frac{1}{2}}v\|$ since we take the dual norm with respect to $g_s$. Now, using [\[ass:p2\]](#ass:p2){reference-type="ref" reference="ass:p2"}, we get that the matrix $H_g^*$ is positive definite and $$\begin{aligned} \label{eq:hg-bound} \|H_g^{*-\frac{1}{2}}\| \le \frac{1}{\sqrt{\rho}}. \end{aligned}$$ Consequently, we have $$\begin{aligned} \norm{\mathop{\nabla}\nolimits f(x_k) - \mathop{\nabla}\nolimits f(x^*) - H_f^*(x_k - x^*)}_{x^*}^* &= \norm{H_f^{*{-\frac{1}{2}}}\left(\mathop{\nabla}\nolimits f(x_k) - \mathop{\nabla}\nolimits f(x^*) - H_f^*(x_k - x^*)\right)}\\ &\le \|H_f^{*{-\frac{1}{2}}}\|\norm{\mathop{\nabla}\nolimits f(x_k) - \mathop{\nabla}\nolimits f(x^*) - H_f^*(x_k - x^*)}\\ &\stackrel{\lemref{thm:gradf-bound}}{\le} \frac{L_f\|x_k - x^*\|^2}{2\sqrt{\rho}}. \end{aligned}$$ To estimate $\|\mathop{\nabla}\nolimits g_s(x_k) - \mathop{\nabla}\nolimits g_s(x^*) - H_g^*(x_k - x^*)\|_{x^*}^*$, we can apply as in the proof of [@sun2019generalized Theorem 5], and get $$\begin{aligned} \norm{\mathop{\nabla}\nolimits g_s(x_k) - \mathop{\nabla}\nolimits g_s(x^*) - H_g^*(x_k - x^*)}_{x^*}^* \le R_\nu(d_\nu(x^*,x_k))d_\nu(x^*,x_k)\norm{x_k-x^*}_{x^*}. \end{aligned}$$ Following [@sun2019generalized p. 195], we can derive the following inequality in a neighbourhood of the sublevel set of ${\mathcal{L}}$ in [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"} using and the convexity of $g_s$: $$\begin{aligned} \|\mathop{\nabla}\nolimits g_s(x_k)\|_{x_k}^* \ge \omega_\nu(-d_\nu(x^*,x_k))\|x_k-x^*\|_{x_k}.\label{eq:hg-positive} \end{aligned}$$ In this regard, [\[eq:steplength\]](#eq:steplength){reference-type="eqref" reference="eq:steplength"} gives $$\begin{aligned} 1-\bar{\alpha}_k \le \frac{\lambda_k - \alpha_k}{\lambda_k}. \end{aligned}$$ Next, using , we can derive the estimate $$\begin{aligned} \norm{\mathop{\nabla}\nolimits g_s(x_k) - \mathop{\nabla}\nolimits g_s(x^*)}_{x^*}^* \le \bar{\omega}_\nu(d_\nu(x^*,x_k))\norm{x_k-x^*}^2, \end{aligned}$$ and using [\[ass:p1\]](#ass:p1){reference-type="ref" reference="ass:p1"}, [\[ass:p2\]](#ass:p2){reference-type="ref" reference="ass:p2"} and [\[eq:hg-bound\]](#eq:hg-bound){reference-type="eqref" reference="eq:hg-bound"}, we get $$\begin{aligned} \norm{\mathop{\nabla}\nolimits f(x_k) - \mathop{\nabla}\nolimits f(x^*)}_{x^*}^* \le \frac{L}{\sqrt{\rho}}. \end{aligned}$$ Finally, putting the above estimates into [\[eq:xk-bound\]](#eq:xk-bound){reference-type="eqref" reference="eq:xk-bound"}, we obtain [\[eq:conv-rate-xk-alg1\]](#eq:conv-rate-xk-alg1){reference-type="eqref" reference="eq:conv-rate-xk-alg1"}. ◻ #### Proof of . *Proof.* Let $\hat{H}_k \coloneqq J_k^\top V_k J_k + H_g$, and consider the iterative process of Algorithm [\[alg:GGNSCOREProx\]](#alg:GGNSCOREProx){reference-type="ref" reference="alg:GGNSCOREProx"} given by $$\begin{aligned} x_{k+1} = \mathop{\rm prox}\nolimits_{\alpha_k g}^{H_g} (x_k - \bar{\alpha}_k \hat{H}_k^{-1}J_k^\top u_k). \end{aligned}$$ We first note that $J_k^\top u_k$ is a compact way of writing $\mathop{\nabla}\nolimits f(x_k) + \mathop{\nabla}\nolimits g_s(x_k) \eqqcolon \mathop{\nabla}\nolimits q(x_k)$, where $f$ is given by [\[eq:f-ggn\]](#eq:f-ggn){reference-type="eqref" reference="eq:f-ggn"}. Following the proof of , we have $$\begin{aligned} \norm{x_{k+1}-x^*}_{x^*} &= \norm{\mathop{\rm prox}\nolimits_{\alpha_k g}^{H_g^*}(E_{x^*}(x_k) + \xi_{x^*}(Q_k,x_{k+1})) - \mathop{\rm prox}\nolimits_{\alpha_k g}^{H_g^*}(E_{x^*}(x^*))}_{x^*}\nonumber\\ &\le \norm{\mathop{\nabla}\nolimits q(x_k) - \mathop{\nabla}\nolimits q(x^*) - \hat{H}_k^*(x_k - x^*)}_{x^*}^* + (1-\bar{\alpha}_k)\norm{\mathop{\nabla}\nolimits q(x_k) - \mathop{\nabla}\nolimits q(x^*)}_{x^*}^*.\label{eq:xk-bound-2} \end{aligned}$$ Substituting [\[eq:hk-hat\]](#eq:hk-hat){reference-type="eqref" reference="eq:hk-hat"} into [\[eq:xk-bound-2\]](#eq:xk-bound-2){reference-type="eqref" reference="eq:xk-bound-2"} and using [\[eq:hg-bound\]](#eq:hg-bound){reference-type="eqref" reference="eq:hg-bound"} in the estimate $$\begin{aligned} \norm{(\mathbf{1}\otimes(W_k^\top\tilde{u}_k))^\top(x_k-x^*)}_{x^*}^*\le \norm{H_g^{*-\frac{1}{2}}(\mathbf{1}\otimes(W_k^\top\tilde{u}_k))^\top}\norm{x_k-x^*}, \end{aligned}$$ where $W_k$ is defined by [\[eq:Junaug\]](#eq:Junaug){reference-type="eqref" reference="eq:Junaug"}, we get $$\begin{aligned} \norm{x_{k+1}-x^*}_{x^*} &\le \norm{\mathop{\nabla}\nolimits q(x_k) - \mathop{\nabla}\nolimits q(x^*) - H_k^*(x_k - x^*)}_{x^*}^* + \norm{(\mathbf{1}\otimes(W_k^\top\tilde{u}_k))^\top(x_k-x^*)}_{x^*}^*\nonumber\\ &\quad + (1-\bar{\alpha}_k)\norm{\mathop{\nabla}\nolimits q(x_k) - \mathop{\nabla}\nolimits q(x^*)}_{x^*}^*\nonumber\\ &\le \norm{\mathop{\nabla}\nolimits f(x_k) - \mathop{\nabla}\nolimits f(x^*) - H_f^*(x_k - x^*)}_{x^*}^* + \norm{\mathop{\nabla}\nolimits g_s(x_k) - \mathop{\nabla}\nolimits g_s(x^*) - H_g^*(x_k - x^*)}_{x^*}^* \nonumber\\&\quad + (1-\bar{\alpha}_k)\left(\norm{\mathop{\nabla}\nolimits f(x_k) - \mathop{\nabla}\nolimits f(x^*)}_{x^*}^* + \norm{\mathop{\nabla}\nolimits g_s(x_k) - \mathop{\nabla}\nolimits g_s(x^*)}_{x^*}^*\right) + \frac{\tilde{\beta}\norm{x_k - x^*}}{\sqrt{\rho}}, \label{eq:xk-bound-3} \end{aligned}$$ where $\tilde{\beta}=\beta_2\beta_3$. Now, using the estimates derived in the proof of in [\[eq:xk-bound-3\]](#eq:xk-bound-3){reference-type="eqref" reference="eq:xk-bound-3"} above, we obtain [\[eq:conv-rate-xk-alg2\]](#eq:conv-rate-xk-alg2){reference-type="eqref" reference="eq:conv-rate-xk-alg2"}. ◻ [^1]: Correspondence: adeyemi.adeoye [^2]: E-mail: {adeyemi.adeoye, alberto.bemporad}\@imtlucca.it [^3]: *Also sometimes called "epigraphic sum\" or "epi-sum\", as its operation yields the (strict) *epigraphic sum* $\mathop{\rm epi}\nolimits f + \mathop{\rm epi}\nolimits g$ [@hiriart2004fundamentals p. 93].* [^4]: It is easy to show that $h_\mu^* = \mu h^*$. [^5]: [\[ass:p1\]](#ass:p1){reference-type="ref" reference="ass:p1"} is a standard assumption on the smooth part of the original problem [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"}, which directly implies that the function $f$ admits a (locally) $L$-Lipschitz gradient. [^6]: If $Q$ is the scaled identity matrix, then we have the *proximal gradient method*, if $Q=\mathop{\nabla^2}\nolimits q$, we have the *proximal Newton method*, and if $Q$ is a quasi-Newton-type, say BFGS, approximation of the Hessian, we have a *proximal quasi-Newton-type method*. [^7]: Note that for the sake of simplicity, we assume here $y_i\in{\mathbb R}$, but it is straightforward to extend the approach that follows to cases where $y_i\in{\mathbb R}^{n_y}$, $n_y>1$. [^8]: A function $d_1$ is called a *prox-function* of a closed and convex set ${\mathcal{Q}}_1$ if ${\mathcal{Q}}_1 \subseteq \mathop{\rm dom}\nolimits d_1$, and $d_1$ is continuous and strongly convex on ${\mathcal{Q}}_1$ with convexity parameter $\rho_1 > 0$. [^9]: Additional assumptions may be required to hold in order to *correctly* define this property in our framework, e.g., nonoverlapping groups in case of the sparse group lasso problem, in which case, $\mathbb{V}$ is the space ${\mathbb R}^n$. [^10]: <https://github.com/adeyemiadeoye/SelfConcordantSmoothOptimization.jl> [^11]: We use the open-source package `ProximalAlgorithms.jl` for the `PANOC`, `ZeroFPR`, and fast proximal gradient algorithms, while we use our own implementation of the `OWL-QN` (modification of <https://gist.github.com/yegortk/ce18975200e7dffd1759125972cd54f4>) and proximal gradient methods which can also be found in our package `SelfConcordantSmoothOptimization.jl`. [^12]: We use the `BCD` method of [@ndiaye2017gap] which is efficiently implemented with a gap safe screening rule. The open-source implementation can be found in <https://github.com/EugeneNdiaye/Gap_Safe_Rules>.

arxiv_math

--- abstract: | We consider disjoint and sliding blocks estimators of cluster indices for multivariate, regularly varying time series in the Peak-over-Threshold framework. We aim to provide a complete description of the limiting behaviour of these estimators. This is achieved by a precise expansion for the difference between the sliding and the disjoint blocks statistics. The rates in the expansion stem from *internal clusters* and *boundary clusters*. To obtain these rates we need to extend the existing results on vague convergence of cluster measures. We reveal dichotomous behaviour between *small blocks* and *large blocks* scenario. author: - "Zaoli Chen[^1]  and Rafał Kulik[^2]" title: "Asymptotic expansions for blocks estimators: PoT framework" --- # Introduction {#sec:intro} Consider a stationary, regularly varying $\mathbb{R}^d$-valued time series $\boldsymbol{X}=\sequence{\boldsymbol{X}}$. We are interested in estimating cluster indices that describe its extremal behaviour. Informally speaking, a cluster is a triangular array $(\boldsymbol{X}_1/\tepseq,\ldots,\boldsymbol{X}_{\dhinterseq}/\tepseq)$ with $\dhinterseq,\tepseq\to\infty$ that converges in distribution in a certain sense. Cluster indices are obtained by applying the appropriate functional $H$ to the cluster. The functionals are defined on $(\mathbb{R}^d)^\mathbb{Z}$ and are such that their values do not depend on coordinates whose entries are small. More precisely, denote $\boldsymbol{X}_{i,j}=(\boldsymbol{X}_i,\ldots, \boldsymbol{X}_j)\in (\mathbb{R}^d)^{(j-i+1)}$. Then, we identify $H(\boldsymbol{X}_{i,j})$ with $H(({\boldsymbol 0},\boldsymbol{X}_{i,j},{\boldsymbol 0}))$, where ${\boldsymbol 0}\in (\mathbb{R}^d)^\mathbb{Z}$ is the zero sequence. Such a functional $H$ will be called a *cluster functional*. Let $w_n:=\mathbb P(\norm{\boldsymbol{X}_0}>\tepseq)$. Given a cluster functional $H$ on $(\mathbb{R}^d)^\mathbb{Z}$, we want to estimate $$\begin{aligned} {\boldsymbol{\nu}}^*(H):= \lim_{n\to\infty} {\boldsymbol{\nu}}^*_{n,\dhinterseq}(H):=\lim_{n\to\infty} \frac{\mathbb E[H(\tepseq^{-1}\boldsymbol{X}_{1,\dhinterseq})]}{\dhinterseq w_n}\;. \label{eq:thelimitwhichisnolongercalledbH}\end{aligned}$$ To guarantee existence of the limit we require additional anticlustering assumptions on $\boldsymbol{X}$. The particular cluster quantity of interest is the (candidate) extremal index obtained with $H(\boldsymbol{x})=\mathbbm{1}{\left\{\boldsymbol{x}^\ast>1\right\}}$, $\boldsymbol{x}=\{\boldsymbol{x}_j,j\in\mathbb{Z}\}\in(\mathbb{R}^d)^\mathbb{Z}$. See Chapter 6 in [@kulik:soulier:2020] and below. Several methods of estimation of the limit ${\boldsymbol{\nu}}^*(H)$ in [\[eq:thelimitwhichisnolongercalledbH\]](#eq:thelimitwhichisnolongercalledbH){reference-type="eqref" reference="eq:thelimitwhichisnolongercalledbH"} may be employed. The natural one is to consider a statistics based on disjoint blocks of size $\dhinterseq$, $$\begin{aligned} \label{eq:blocktype} \tedcluster(H):= \frac{1}{nw_n} \sum_{i=1}^{m_n} H(\tepseq^{-1}\boldsymbol{X}_{(i-1)\dhinterseq+1,i\dhinterseq}) \;,\end{aligned}$$ where $m_n=[n/\dhinterseq]$. Although some special cases were considered (e.g. the extremal index in [@hsing:1991] and [@smith:weissman:1994]), the general theory was developed in the seminal paper [@drees:rootzen:2010]. The appropriately scaled and centered statistics is asymptotically normal with the limiting variance given by ${\boldsymbol{\nu}}^*(H^2)$. A summary of the theory for the disjoint blocks statistics and the corresponding data based estimators (where the threshold $\tepseq$ is replaced with the appropriate intermediate order statistics) can be found in [@kulik:soulier:2020 Chapter 10]. Some recent developments in the context of $\ell^p$-blocks can be found in [@buritica:mikosch:wintenberger:2023]. Alternatively, we can consider the sliding blocks statistics $$\begin{aligned} \label{eq:sliding-block-estimator-nonfeasible-1} \tedclustersl(H):=\frac{1}{n\dhinterseq w_n}\sum_{i=1}^{n-\dhinterseq+1}H\left(\tepseq^{-1}\boldsymbol{X}_{i,i+\dhinterseq-1}\right)\;.\end{aligned}$$ This and the corresponding data based estimators have been studied for some specific functionals $H$, however there was no unified theory available. Recently, [@drees:neblung:2021] used the framework of [@drees:rootzen:2010] and showed that the limiting variance of the sliding blocks estimator never exceeds that of the disjoint blocks one. In case of the extremal index, both variances are equal. In [@cissokho:kulik:2021], building upon [@kulik:soulier:2020 Chapter 10], it has been proven that in case of regularly varying time series the asymptotic behaviour of sliding and disjoint blocks estimators is the same. We note in passing that the same holds for so-called runs estimators, which can be viewed as a special case of the sliding blocks estimators. See [@cissokho:kulik:2022]. We note that all the discussion above is valid in the Peak-over-Threshold (PoT) framework. On the basic technical level, the \"PoT framework\" refers to the assumption $\lim_{n\to\infty}\dhinterseq w_n=0$; see [\[eq:rnbarFun0\]](#eq:rnbarFun0){reference-type="ref" reference="eq:rnbarFun0"} below. To the contrary, in the block maxima framework it has been observed that the asymptotic variance for sliding blocks estimators is strictly smaller as compared to disjoint blocks. We refer to [@bucher:segers:2018sliding] and a review in [@bucher:zhou:2018]. #### The goal of the paper. We aim to provide a mathematical explanation for the aforementioned phenomena observed in the PoT framework. This is achieved by a precise expansion for the difference between the sliding and the disjoint blocks statistics. The rate in the expansion will stem from *internal clusters* and *boundary clusters*. ## Probabilistic tools for asymptotic expansions The techniques we use in the paper stem from Chapters 6 and 10 in [@kulik:soulier:2020]. #### Internal clusters. Intuitively, an internal cluster occurs if there is a large value in a single block, but there are no large values in adjacent blocks. The starting point of our analysis is thus vague convergence of clusters. (see also Theorem 6.2.5 in [@kulik:soulier:2020]) establishes such convergence, where the class of test functions consists of bounded functionals that vanish around zero. The rate of convergence is $\dhinterseq w_n$, which is proportional to the probability of an occurrence of a large value in a single block. In the present context we face a challenge. Starting with a functional $H$, the internal cluster statistics involves another functional $\widetilde{H}_{\mathcal{IC}}$, which is induced from $H$. For example, if $H(\boldsymbol{x})=\mathbbm{1}{\left\{\boldsymbol{x}^*>1\right\}}$, then $\widetilde{H}_{\mathcal{IC}}$ equals a cluster length (subtracted by one), the distance between the location of the last and the first large value. In fact, we shall note that the cluster length functional plays a key role in this paper. Note also that the cluster length is unbounded. Hence, at the first step of our analysis we need to extend vague convergence of clusters to unbounded functionals. Under the appropriate uniform integrability conditions we can recover - cluster functionals still have the rate $\dhinterseq w_n$. The uniform integrability condition holds for tight cluster functionals (such as the cluster length) as long as a new anticlustering condition is valid. The latter in turn is related to *small blocks*. On the other hand, in case of *large blocks* or some unbounded non-tight functionals (such as locations of large values), is no longer valid and cluster functionals grow at a different rate. #### Boundary clusters. When proving central limit theorem (CLT) for (disjoint) blocks estimators, mixing conditions allow to treat the consecutive blocks as if they were independent. In such the case (referred below to as \"piecewise stationarity\") the chance of large values in two consecutive blocks is proportional to $\dhinterseq^2 w_n^2$. It turns out that this CLT-based heuristic fails in the context of the asymptotic expansion. Indeed, in case of *small blocks* the rate for the event \"large values in two consecutive blocks\" is proportional to $w_n$. Intuitively, large values occur at the end of one block and at the beginning of the next block. Bounded cluster functionals grow then at the same rate $w_n$. On the other hand, in case of *large blocks*, the blocks behave as if they were independent and the cluster functionals have a different rate. #### Internal vs. boundary clusters. Both types of clusters have a different asymptotic behaviour. Also, the notion of *small* and *large* blocks is different for both types of clusters. #### Asymptotic expansions. In case of small blocks the rate in the asymptotic expansion stems from both internal and boundary clusters. As a consequence, after the appropriate scaling, the difference between the disjoint and the sliding blocks statistics is of the rate $O_P(1/\dhinterseq)$. On the other hand, in case of large blocks, the difference is of the rate $O_P(\dhinterseq w_n)$. #### Weak dependence assumptions. Most of the results in the paper are valid just under the appropriate anticlustering condition and only some of them involve mixing. This is the situation of the small blocks scenario. The large blocks scenario is presented in case of $\ell$-dependence. We can re-write our results in an expense of cumbersome mixing assumptions. We decided to focus on a very simple dependence structure that is sufficient to understand the difference between the small and the large blocks scenario, as well as between internal and boundary clusters. All phenomena that appear in the paper can be illustrated by a simple, 1-dependent time series! Also, we obtain an expansion for a piecewise stationary time series. In the small blocks scenario, the result is quantitatively different as compared to the weakly dependent case. #### Conclusion for the asymptotic behaviour of blocks estimators. In the PoT framework, disjoint and sliding blocks estimators have the same asymptotic behaviour in either weakly dependent or piecewise stationary situation: the limiting variance in the Central Limit Theorem is the same. This is in contrast to the block maxima framework. ## Structure of the paper contains preliminaries. It introduces the tail process, the relevant class of functions (including the cluster length, which appears to be the most important functional in the context of the paper; see ), cluster measures and cluster indices. introduces different types of anticlustering conditions. recalls vague convergence of cluster measures. is the most relevant result in this context. In we decompose the difference between disjoint and sliding blocks estimators. Internal () and boundary () clusters appear. contains the main results on the asymptotics expansion for disjoint and sliding blocks statistics. deals with the small blocks scenario. We obtain a precise expansion at the rate $O_P(\dhinterseq^{-1})$. Here, both the internal and the boundary clusters contribute at the same rate. In we consider a piecewise stationary case. This is motivated by the following observation. When a central limit theorem for disjoint blocks statistics is considered, a $\beta$-mixing time series has the same asymptotic behaviour as a corresponding piecewise stationary. This idea breaks down when the asymptotic expansion is considered. establishes an expansion in the small blocks scenario. We note that the rate is determined by the internal clusters only. deals with large blocks scenario, which is quantitatively different as compared to the small blocks one. contains the first part of technical details, with some results being of independent interest. includes results related to internal clusters (that is, functionals of a single block). The main goal is to extend vague convergence of Theorem 6.2.5 in [@kulik:soulier:2020] (see below) to unbounded functionals such as the cluster length. Under the small blocks condition (which is roughly equivalent to uniform integrability), the cluster length behaves as predicted by . When the uniform integrability fails, the behaviour of the cluster length is different. The phase transition occurs precisely when small blocks are replaced with large blocks. The results are taken from [@chen:kulik:2023a]. deals with the boundary clusters. The key result is the rate of convergence for the event \"a big jump in two consecutive blocks.\" We have a dichotomous behaviour: in case of small blocks, the rate is $w_n=\mathbb P(\norm{\boldsymbol{X}_0}>\tepseq)$, while for large blocks the rate is $\dhinterseq^2 w_n^2$ See and , respectively. Extensions to unbounded functionals that vanish on the event \"at most one big jump\" follow. The second part of technical details is included in . The theory established in is applied to particular, rather complex, functionals that appear in the context of the asymptotic expansion of blocks statistics. See for small blocks and for large blocks. establishes the growth of both internal and boundary clusters statistics. These statistics are sums of internal and boundary clusters. As such, we use the results from in conjunction with some weak dependence assumptions. The last part of technical details can be found in . There, we provide precise computations for somehow cumbersome expansions. ## What is missing? We provide rather complete theory is the small blocks scenario. The behavior in the large blocks scenario is illustrated under much simpler dependence assumptions. However, from the statistical perspective, large blocks are much less relevant in practice. Also, we do not consider here functionals that do not vanish around zero. For example functionals that lead to the large deviation index (see e.g. [@mikosch:wintenberger:2013]). These functionals may be large due to a cumulation of small values. This is usually prevented by imposing a \"negligibility of small values\" condition, but it is still not suitable for techniques and the framework of the paper. Furthermore, we do not formulate here nor prove the central limit theorems for two types cluster. We refer to [@chen:kulik:2023a] for the appropriate results related to the internal clusters. # Preliminaries {#sec:prel} In this section we fix the notation and introduce the relevant classes of functions. In we recall the notion of the tail and the spectral tail process (cf. [@basrak:segers:2009]). In we define cluster indices; see [@kulik:soulier:2020 Chapter 5] for a detailed introduction. In we introduce anticlustering conditions. The first one, the classical [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"}, is needed, to establish convergence of the cluster measure; see . The results of the latter section are extracted from [@kulik:soulier:2020 Chapter 6]. See also [@planinic:soulier:2018] and [@basrak:planinic:soulier:2018]. Another condition, called here [\[eq:conditionSstronger:gamma\]](#eq:conditionSstronger:gamma){reference-type="ref" reference="eq:conditionSstronger:gamma"}, was introduced in [@chen:kulik:2023a]. In we introduce our dependence assumptions. In we state simple versions of the existing results on disjoint and sliding blocks estimators for stationary time series. In the PoT framework, disjoint blocks estimators are considered in [@drees:rootzen:2010] and [@kulik:soulier:2020 Chapter 10]. Results for sliding blocks are taken from [@cissokho:kulik:2021] (see also [@drees:neblung:2021]). The (somehow surprising) conclusion is that the disjoint and sliding blocks estimator yield the same asymptotic behaviour. ## Notation Let $\norm{\cdot}$ be an arbitrary norm on $\mathbb{R}^d$ and $\{\tepseq\}$, $\{\dhinterseq\}$ be such that $$\begin{aligned} \label{eq:rnbarFun0} \lim_{n\to\infty}\tepseq=\lim_{n\to\infty}\dhinterseq =\lim_{n\to\infty}nw_n = \infty\;, \ \ \lim_{n\to\infty}\dhinterseq/n=\lim_{n\to\infty}\dhinterseq w_n = 0\;, \tag{$\ensuremath{\mathcal{R}}(\dhinterseq,\tepseq)$}\end{aligned}$$ where $w_n=\mathbb P(\norm{\boldsymbol{X}_0}>\tepseq)$. For a sequence $\boldsymbol{x}=\{\boldsymbol{x}_j,j\in\mathbb{Z}\}\in (\mathbb{R}^d)^\mathbb{Z}$ and $i\leqslant j\in\mathbb{Z}\cup\{-\infty,\infty\}$ denote $\boldsymbol{x}_{i,j}=(\boldsymbol{x}_i,\ldots,\boldsymbol{x}_j)\in (\mathbb{R}^d)^{j-i+1}$, $\boldsymbol{x}_{i,j}^\ast=\max_{i\leqslant l\leqslant j}|\boldsymbol{x}_l|$ and $\boldsymbol{x}^\ast=\sup_{j\in\mathbb{Z}}|\boldsymbol{x}_j|$. By ${\boldsymbol 0}$ we denote the zero sequence; its dimension can be different at each of its occurrences. By $\ell_0(\mathbb{R}^d)$ we denote the set of $\mathbb{R}^d$-valued sequences which tend to zero at infinity. ## Tail process {#sec:tail-process} Let $\boldsymbol{X}=\sequence{\boldsymbol{X}}$ be a stationary, regularly varying time series with values in $\mathbb{R}^d$ and tail index $\alpha$. In particular, $$\begin{aligned} \lim_{x\to\infty}\frac{\mathbb P(|\boldsymbol{X}_0|> tx)}{\mathbb P(|\boldsymbol{X}_0|> x)}=t^{-\alpha}\end{aligned}$$ for all $t>0$. Then, there exists a sequence $\boldsymbol{Y}=\sequence{\boldsymbol{Y}}$ such that $$\begin{aligned} %\label{eq:def-numultprob} \mathbb P(x^{-1}(\boldsymbol{X}_i,\dots,\boldsymbol{X}_j) \in \cdot \mid |\boldsymbol{X}_0|>x) \mbox{ converges weakly to } \mathbb P((\boldsymbol{Y}_i,\dots,\boldsymbol{Y}_j) \in \cdot)\end{aligned}$$ as $x\to\infty$ for all $i\leqslant j\in\mathbb{Z}$. We call $\boldsymbol{Y}$ the tail process. See [@basrak:segers:2009]. Equivalently, viewing $\boldsymbol{X}$ and $\boldsymbol{Y}$ as random elements with values in $(\mathbb{R}^d)^\mathbb{Z}$, we have for every bounded or non-negative functional $H$ on $(\mathbb{R}^d)^{\mathbb{Z}}$, continuous with respect to the product topology, $$\begin{aligned} \label{eq:tailprocesstozero} \lim_{x\to\infty} \frac{\mathbb E[H(x^{-1}\boldsymbol{X})\mathbbm{1}{\left\{\norm{\boldsymbol{X}_0}>x\right\}}]} {\mathbb P(\norm{\boldsymbol{X}_0}>x)} & = \mathbb E[H(\boldsymbol{Y})] \; .\end{aligned}$$ The random variable $|\boldsymbol{Y}_0|$ has the Pareto distribution with index $\alpha$ and hence $\norm{\boldsymbol{Y}_0}>1$. The spectral tail process $\boldsymbol{\Theta}=\{\boldsymbol{\Theta}_j,j\in\mathbb{Z}\}$ is defined as $\boldsymbol{\Theta}_j=\boldsymbol{Y}_j/|\boldsymbol{Y}_0|$. ## Classes of functions {#sec:classes} Let $\epsilon>0$. We will need a sequence of exceedance times associated with $\boldsymbol{x}$. Define $$\begin{aligned} T^{(1)}(\boldsymbol{x},\epsilon)=T_{\rm min}(\boldsymbol{x},\epsilon) &= \inf\{ j \in \mathbb{Z}: | \boldsymbol{x}_j | > \epsilon \}\;, \label{eq:exc-times-x-1}\\ T_{\rm max}(\boldsymbol{x},\epsilon)&=\sup\{ j \in \mathbb{Z}: | \boldsymbol{x}_j | > \epsilon \}\;, \label{eq:exc-times-x-2}\\ T^{(i+1)}(\boldsymbol{x},\epsilon)&=\inf\{j>T^{(i)}(\boldsymbol{x},\epsilon): |\boldsymbol{x}_j|>\epsilon\}\;, \ \ i\geqslant 1\;, \label{eq:exc-times-x-3}\\ \Delta T^{(i)}(\boldsymbol{x},\epsilon)&= T^{(i+1)}(\boldsymbol{x},\epsilon)-T^{(i)}(\boldsymbol{x},\epsilon)\;. \label{eq:exc-times-x-4}\end{aligned}$$ We will use the convention $\inf\{\emptyset\}=+\infty$, $\sup\{\emptyset\}=-\infty$. If $\boldsymbol{x}\in \ell_0(\mathbb{R}^d)$, then $\sup\{ j\geqslant 0: | \boldsymbol{x}_j | > \epsilon \}<\infty$ and $\inf\{ j\geqslant 0: | \boldsymbol{x}_j | > \epsilon \}>-\infty$. Hence, when restricted to $\ell_0$, the functionals $T^{(i)}$, $i\geqslant 1$, and $T_{\rm max}$ attain finite values. Formally, the above functionals depend on $\epsilon$. However, without loss of generality we will assume throughout the paper that $\epsilon=1$ and we will use the notation $T^{(i)}(\boldsymbol{x})$ instead. Let again $\boldsymbol{x}\in \ell_0(\mathbb{R}^d)$. We can define the **cluster length functional** as $$\begin{aligned} \label{eq:cluster-length-def} \mathcal L(\boldsymbol{x}) =T_{\rm max}(\boldsymbol{x})-T_{\rm min}(\boldsymbol{x})+1\;\end{aligned}$$ with the convention $\mathcal L(\boldsymbol{x})=0$ whenever $\boldsymbol{x}^\ast=\sup_{j\in\mathbb{Z}}\norm{\boldsymbol{x}_j}\leqslant 1$. We note also that if $\norm{\boldsymbol{x}_0}>1$, while $\sup_{j\in\mathbb{Z}, j\not=0}\norm{\boldsymbol{x}_j}\leqslant 1$, then $\mathcal L(\boldsymbol{x})=1$. Also, neither $\mathcal L$ nor $T^{(i)}$ can be chosen in [\[eq:tailprocesstozero\]](#eq:tailprocesstozero){reference-type="eqref" reference="eq:tailprocesstozero"}, since they are not defined on $(\mathbb{R}^d)^\mathbb{Z}$. We define further the functional $\mathcal E$ by $\mathcal E(\boldsymbol{x})=\sum_{j\in\mathbb{Z}}\mathbbm{1}{\left\{\norm{\boldsymbol{x}_j}>1\right\}}$. It returns the number of exceedances over 1. We shall consider functionals under the following assumptions. **Assumption 1**. *We denote by $\mathcal H$ a collection of functionals $\ell_0(\mathbb{R}^d) \to \mathbb{R}_+$ such that each $H \in \mathcal{H}$ satisfies:* - *$H$ is continuous with respect to the law of the process $\boldsymbol{Y}$;* - *If $\mathcal E(\boldsymbol{x})=0$, then $H(\boldsymbol{x})=0$;* - *If $\mathcal E(\boldsymbol{x})>0$, then $H(\boldsymbol{x}) =H(\boldsymbol{x}_{T_{\rm min}(\boldsymbol{x}),T_{\rm max}(\boldsymbol{x})})$, where $T_{\rm min}(\boldsymbol{x})$ and $T_{\rm max}(\boldsymbol{x})$, are the first and the last exceedance times defined in [\[eq:exc-times-x-1\]](#eq:exc-times-x-1){reference-type="eqref" reference="eq:exc-times-x-1"}-[\[eq:exc-times-x-2\]](#eq:exc-times-x-2){reference-type="eqref" reference="eq:exc-times-x-2"}.* We note that the above assumption allows for unbounded functionals. We will need to control their growth. **Assumption 2**. *We denote by $\mathcal H(\gamma)\subseteq \mathcal H$ a collection of functionals $\ell_0(\mathbb{R}^d) \to \mathbb{R}$ such that each $H \in\mathcal H(\gamma)$ satisfies:* - *There exists a constant $C_H>0$ such that $H(\boldsymbol{x}) \leqslant C_H \big[ \mathcal L(\boldsymbol{x}) \big]^{\gamma}$ for all $\boldsymbol{x}\in \ell_0( \mathbb{R}^d )$.* Some functionals will play a special role: - Obviously, we can take $H$ as the cluster length functional itself: $H=\mathcal L$. Then $\gamma=1$. - Extremal index functional defined as $\mathcal T(\boldsymbol{x})=\mathbbm{1}{\left\{\boldsymbol{x}^\ast>1\right\}}$. It fulfills with $C_{\mathcal T}=1$ and $\gamma=0$. This functional is bounded. In particular, $\mathcal T(\boldsymbol{Y})=1$. We make additional comments: - The class $\mathcal H(\gamma)$ is parametrized by $\gamma$. - If $\gamma=0$, we will denote $\|H\|=\sup_{\boldsymbol{x}\in(\mathbb{R}^d)^\mathbb{Z}}|H(\boldsymbol{x})|$. - If $0<\gamma_1<\gamma_2$, then $\mathcal H(\gamma_1)\subseteq \mathcal H(\gamma_2)$. - If $H\in\mathcal H(\gamma)$, then for $p>0$, $|H|^p\in \mathcal H(p\gamma)$. - Note that the functional $H$ can depend on small values, but only those that occur between large values (\"within a cluster\"). - The property $( \textup{\lowercase{\romannumeral 2}} )$ in will be referred to as \"$H$ vanishes around ${\boldsymbol 0}$\" or \"$H$ has support separated from ${\boldsymbol 0}$\". In particular, $\mathcal L$ and $\mathcal T$ vanish around ${\boldsymbol 0}$. Let $p>0$. Any $H\in \mathcal H(\gamma)$ induces the following three new functionals. $$\begin{aligned} \widetilde{H}_{\mathcal{IC}}(\boldsymbol{x})&:=\sum_{i=1}^{\mathcal E(\boldsymbol{x})-1}\Delta T^{(i)}(\boldsymbol{x})\{H(\boldsymbol{x}_{-\infty,T^{(i)}(\boldsymbol{x})})+H(\boldsymbol{x}_{T^{(i+1)}(\boldsymbol{x}),\infty})-H(\boldsymbol{x})\}\;,\label{eq:new-functional-IC}\\ \widetilde{H}_{\mathcal{BC}}(\boldsymbol{x})&:=\sum_{i=1}^{\mathcal L(\boldsymbol{x})-1} \left\{H(\boldsymbol{x})-H(\boldsymbol{x}_{-\infty,i-1})-H(\boldsymbol{x}_{i,\infty})\right\}\label{eq:new-functional-BC}\;, \\ \widetilde{H}_{\mathcal{BC},p}(\boldsymbol{x})&:=\sum_{i=1}^{\mathcal L(\boldsymbol{x})-1} |H(\boldsymbol{x})-H(\boldsymbol{x}_{-\infty,i-1})-H(\boldsymbol{x}_{i,\infty})|^p\label{eq:new-functional-BC-p} \;.\end{aligned}$$ **Example 3**. 1. *Take $H=\mathbbm{1}{\left\{\boldsymbol{x}^*>1\right\}}$. Then $\widetilde{H}_{\mathcal{IC}}(\boldsymbol{x})=T_{\rm max}(\boldsymbol{x})-T_{\rm min}(\boldsymbol{x})=\mathcal L(\boldsymbol{x})-1$ and $\widetilde{H}_{\mathcal{BC}}(\boldsymbol{x})=-(\mathcal L(\boldsymbol{x})-1)$.* 2. *If $H$ is a linear functional, that is the functional of the form $H(\boldsymbol{x}_{i,j})+H(\boldsymbol{x}_{j+1,k})=H(\boldsymbol{x}_{i,k})$, $i<j<k$, then also $\widetilde{H}_{\mathcal{IC}}=\widetilde{H}_{\mathcal{BC}}=\widetilde{H}_{\mathcal{BC},p}\equiv 0$.* 3. *Take $H=\mathcal L$. Then $\widetilde{H}_{\mathcal{IC}}=-\sum_{i=1}^{\mathcal E(\boldsymbol{x})-1}(\Delta T^{(i)}(\boldsymbol{x}))^2$. We can view $\widetilde{H}_{\mathcal{IC}}$ as a \"measure of sparsity\" in the tail process.* **Remark 1**. - *If $\boldsymbol{X}$ is extremally independent, then $\boldsymbol{Y}_j=0$ for $j\not=0$. Then $\mathcal E(\boldsymbol{Y})=1$ and $\mathcal L(\boldsymbol{Y})=1$. Hence, $\widetilde{H}_{\mathcal{IC}}=\widetilde{H}_{\mathcal{BC}}=\widetilde{H}_{\mathcal{BC},p}\equiv 0$.* - *If $H\in\mathcal H(\gamma)$, then $\widetilde{H}_{\mathcal{IC}}\in \mathcal H(\gamma+1)$. The additional exponent comes from $\sum_{i=1}^{\mathcal E(\boldsymbol{x})-1}\Delta T^{(i)}(\boldsymbol{x})= \mathcal L(\boldsymbol{x})$. Then $|\widetilde{H}_{\mathcal{IC}}|\in \mathcal H(p(\gamma+1))$. Likewise, $H\in\mathcal H(\gamma)$, gives $\widetilde{H}_{\mathcal{BC},p}\in \mathcal H(p\gamma+1)$.* ## Cluster measure and cluster indices {#sec:cluster-index} Consider the infargmax functional $\mathcal A_0$ defined on $(\mathbb{R}^d)^\mathbb{Z}$ by $\mathcal A_0(\boldsymbol{x})=\inf\{j:\boldsymbol{x}_{-\infty,{j}}^\ast=\boldsymbol{x}^\ast\}$, with the convention that $\inf\{\emptyset\}=+\infty$. If $\mathbb P(\mathcal A_0(\boldsymbol{Y})\notin\mathbb{Z})=0$ then we can define $$\begin{aligned} %\label{eq:canditheta-anchor} {\vartheta}= \mathbb P(\mathcal A_0(\boldsymbol{Y})=0) \; .\end{aligned}$$ In fact, $\mathcal A_0$ can be replaced with any anchoring map (see [@planinic:soulier:2018] and [@kulik:soulier:2020 Theorem 5.4.2]). In particular, $$\begin{aligned} \label{eq:canditheta-anchor-conclusion} {\vartheta}=\mathbb P(\mathcal A_0(\boldsymbol{Y})=0)=\mathbb P\left(\boldsymbol{Y}_{-\infty,-1}^\ast\leqslant 1\right)=\mathbb P\left(\boldsymbol{Y}_{1,\infty}^\ast\leqslant 1\right)\;.\end{aligned}$$ Therefore, ${\vartheta}$ can be recognized as the (candidate) extremal index. It becomes the usual extremal index under additional mixing and anticlustering conditions. **Definition 4** (Cluster measure). *Let $\boldsymbol{Y}$ and $\boldsymbol{\Theta}$ be the tail process and the spectral tail process, respectively, such that $\mathbb P(\lim_{|j|\to\infty} |\boldsymbol{Y}_j|=0)=1$. The cluster measure is the measure ${\boldsymbol{\nu}}^*$ on $\ell_0(\mathbb{R}^d)$ defined by $$\begin{aligned} %\label{eq:def-tailmeasurestar-premiere} {\boldsymbol{\nu}}^*= {\vartheta}\int_0^\infty \mathbb E[\delta_{r\boldsymbol{\Theta}}\mathbbm{1}{\left\{\mathcal A_0(\boldsymbol{\Theta})=0\right\}}] \alpha r^{-\alpha-1} \mathrm{d}r \; .\end{aligned}$$* The measure ${\boldsymbol{\nu}}^*$ is boundedly finite on $(\mathbb{R}^d)^\mathbb{Z}\setminus\{{\boldsymbol 0}\}$, puts no mass at ${\boldsymbol 0}$ and is $\alpha$-homogeneous. For every bounded or non-negative functional $H$ such that $H(\boldsymbol{x})=0$ if $\boldsymbol{x}^\ast\leqslant 1$ we have $$\begin{aligned} \label{eq:cluster-measure} {\boldsymbol{\nu}}^*(H) &= \mathbb E[H(\boldsymbol{Y}) \mathbbm{1}{\left\{\mathcal A_0(\boldsymbol{Y})=0\right\}}] =\mathbb E[H(\boldsymbol{Y}) \mathbbm{1}{\left\{\boldsymbol{Y}_{-\infty,-1}^\ast\leqslant 1\right\}}] \; .\end{aligned}$$ **Definition 5** (Cluster index). *We will call ${\boldsymbol{\nu}}^*(H)$ the cluster index associated to the functional $H$.* ## Change of measure {#sec:change-of-measure} It is important to notice the presence of $\mathbbm{1}{\left\{\boldsymbol{Y}_{-\infty,-1}^\ast\leqslant 1\right\}}$ in the definition of ${\boldsymbol{\nu}}^*$. It will be convenient to express some formulas in the language of an auxiliary process $\boldsymbol{Z}$. Define $\boldsymbol{Z}$ as $\boldsymbol{Y}$ conditioned on the first exceedance over 1 to happen at time zero, that is $\boldsymbol{Y}_{-\infty,-1}^*\leqslant 1$. Then, combining [\[eq:canditheta-anchor-conclusion\]](#eq:canditheta-anchor-conclusion){reference-type="eqref" reference="eq:canditheta-anchor-conclusion"} with [\[eq:cluster-measure\]](#eq:cluster-measure){reference-type="eqref" reference="eq:cluster-measure"}, we obtain $$\begin{aligned} %\label{eq:Palm} {\boldsymbol{\nu}}^*(H)=\mathbb E[H(\boldsymbol{Y}) \mathbbm{1}{\left\{\boldsymbol{Y}_{-\infty,-1}^\ast\leqslant 1\right\}}]={\vartheta}\mathbb E[H(\boldsymbol{Z})]\;.\end{aligned}$$ ## Anticlustering conditions {#sec:anticlustering-condition} For each fixed $r\in\mathbb{N}$, the distribution of $\tepseq^{-1}\boldsymbol{X}_{-r,r}$ conditionally on $\norm{\boldsymbol{X}_0}>\tepseq$ converges weakly to the distribution of $\boldsymbol{Y}_{-r,r}$. In order to let $r$ tend to infinity, we must embed all these finite vectors into one space of sequences. By adding zeroes on each side of the vectors $\tepseq^{-1}\boldsymbol{X}_{-r,r}$ and $\boldsymbol{Y}_{-r,r}$ we identify them with elements of the space $\ell_0(\mathbb{R}^d)$. Then $\boldsymbol{Y}_{-r,r}$ converges (as $r\to\infty$) to $\boldsymbol{Y}$ in $\ell_0(\mathbb{R}^d)$ if (and only if) $\boldsymbol{Y}\in\ell_0(\mathbb{R}^d)$ almost surely. However, this is not enough for statistical purposes and we consider the following definition that controls persistence of large values on one block. **Definition 6** ([@davis:hsing:1995], Condition 2.8). *Condition [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} holds if for all $s,t>0$, $$\begin{aligned} \label{eq:conditiondh} \lim_{\ell\to\infty} \limsup_{n\to\infty}\mathbb P\left(\max_{\ell\leqslant |j|\leqslant \dhinterseq}|\boldsymbol{X}_j| > \tepseq s\mid |\boldsymbol{X}_0|> \tepseq t \right)=0 \; . \tag{$\conditiondh[\dhinterseq][\tepseq]$} \end{aligned}$$* Condition [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} is referred to as the (basic) anticlustering condition. It is fulfilled by many models, including geometrically ergodic Markov chains, short-memory linear or max-stable processes. [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} implies that $\boldsymbol{Y}\in \ell_0(\mathbb{R}^d)$ and $\vartheta = \mathbb P\left ( \boldsymbol{Y}^\ast_{-\infty,-1} \leqslant 1 \right)> 0$. Also, [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} holds for sequence of i.i.d. random variables whenever $\lim_{n\to\infty} \dhinterseq w_n=0$. **Definition 7**. *Condition [\[eq:conditionSstronger:gamma\]](#eq:conditionSstronger:gamma){reference-type="ref" reference="eq:conditionSstronger:gamma"}* *holds if for all $s,t>0$ $$\begin{aligned} \label{eq:conditionSstronger:gamma} \lim_{\ell\to\infty} \limsup_{n\to\infty} \frac{1}{ w_n} \sum_{i=\ell}^{\dhinterseq}i^\gamma \mathbb P(\norm{\boldsymbol{X}_0}>\tepseq s,\norm{\boldsymbol{X}_i}>\tepseq t) = 0 \; . \tag{$\conditiondhsumstrongergammapaper{\dhinterseq}{\tepseq}{\gamma}$} \end{aligned}$$* This condition has been introduced in [@chen:kulik:2023a]. **Remark 2**. - *It is obvious that in case of i.i.d. or $\ell$-dependent sequences [\[eq:conditionSstronger:gamma\]](#eq:conditionSstronger:gamma){reference-type="ref" reference="eq:conditionSstronger:gamma"} holds if and only if $\lim_{n\to\infty}\dhinterseq^{\gamma+1}w_n=0$. However, we could not establish that [\[eq:conditionSstronger:gamma\]](#eq:conditionSstronger:gamma){reference-type="ref" reference="eq:conditionSstronger:gamma"} and $\lim_{n\to\infty}\dhinterseq^{\gamma+1}w_n=0$ are equivalent in general.* - *The condition [$\mathcal{S}^{(\gamma_1)}(\dhinterseq, \tepseq)$](#SummabilityAC) implies [$\mathcal{S}^{(\gamma_2)}(\dhinterseq, \tepseq)$](#SummabilityAC) whenever $\gamma_1>\gamma_2\geqslant 0$. The condition [$\mathcal{S}^{(0)}(\dhinterseq, \tepseq)$](#SummabilityAC) implies the condition [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"}.* - *In several places, we will need the new anticlustering condition to hold with $c\dhinterseq$ ($c>1$) instead of $\dhinterseq$. With some abuse of notation, we will not make a distinction.* Next, we note the following bound. Let $c_1>c_1>1$. If [\[eq:conditionSstronger:gamma\]](#eq:conditionSstronger:gamma){reference-type="ref" reference="eq:conditionSstronger:gamma"} holds, then $$\begin{aligned} \label{eq:consequence-condition-s} \sum_{i=c_1\dhinterseq}^{c_2\dhinterseq}\mathbb P(|\boldsymbol{X}_0|>\tepseq,|\boldsymbol{X}_i|>\tepseq)=o(\dhinterseq^{-\gamma}w_n)\;. \end{aligned}$$ ## Conditional convergence of clusters From the definition of the tail process we obtain immediately $$\begin{aligned} \lim_{n\to\infty} \mathbb E[H(\tepseq^{-1} \boldsymbol{X}_{i,j})\mid \norm{\boldsymbol{X}_0}>\tepseq] = \mathbb E[H(\boldsymbol{Y}_{i,j})] \; , %\label{eq:conv-H-cluster} \end{aligned}$$ for $i\leqslant j$ and suitable functionals $H$. Thanks to the anticlustering condition we can replace $i,j$ with a sequence diverging to $\infty$. **Proposition 8** ([@basrak:segers:2009], Proposition 4.2; [@kulik:soulier:2020], Theorem 6.1.4). *Let $\boldsymbol{X}$ be a stationary time series. Assume that [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} holds. Let $H$ be a bounded or non-negative functional defined on $\ell_0(\mathbb{R}^d)$ that is almost surely continuous with respect to the distribution of the tail process $\boldsymbol{Y}$. Then for any constant $C>0$, $$\begin{aligned} \lim_{n\to\infty} \mathbb E[H(\tepseq^{-1} \boldsymbol{X}_{-C\dhinterseq,C\dhinterseq})\mid \norm{\boldsymbol{X}_0}>\tepseq] = \mathbb E[H(\boldsymbol{Y})] \; . %\label{eq:conv-H-cluster} \end{aligned}$$* ## Vague convergence of cluster measure {#sec:cluster-measure-convergence} We now investigate the unconditional convergence of $\tepseq^{-1}\boldsymbol{X}_{1,\dhinterseq}$. Contrary to , where an extreme value was imposed at time 0, a large value in the cluster can happen at any time. Define the measures ${\boldsymbol{\nu}}^*_{n,\dhinterseq}$, $n\geqslant1$, on $\ell_0(\mathbb{R}^d)$ as follows: $$\begin{aligned} % \label{eq:def-tailmeasuresta\dhinterseq} {\boldsymbol{\nu}}^*_{n,\dhinterseq} & = \frac{1} {\dhinterseq w_n} \mathbb E\left[\delta_{\tepseq^{-1}\boldsymbol{X}_{1,\dhinterseq}} \right] \; .\end{aligned}$$ We are interested in convergence of ${\boldsymbol{\nu}}^*_{n,\dhinterseq}$ to ${\boldsymbol{\nu}}^*$. We quote Theorem 6.2.5 in [@kulik:soulier:2020]. **Proposition 9**. *Let condition [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} hold. Then, for all bounded continuous shift-invariant functionals $H$ with support separated from ${\boldsymbol 0}$ we have $$\begin{aligned} %\label{eq:vague-convergence} \lim_{n\to\infty} {\boldsymbol{\nu}}^*_{n,\dhinterseq}(H) = \lim_{n\to\infty}\frac{\mathbb E[ H(\tepseq^{-1}\boldsymbol{X}_{1,\dhinterseq})]} {\dhinterseq w_n} = \boldsymbol{\nu}^\ast(H) \; . \end{aligned}$$* The immediate consequence is the following limit: $$\begin{aligned} \label{eq:extremal-index} \lim_{n\to\infty} \frac{\mathbb P(\boldsymbol{X}_{1,\dhinterseq}^\ast>\tepseq)}{\dhinterseq w_n} ={\vartheta}\; . \end{aligned}$$ ## Dependence assumptions {#sec:dependence-assumptions} Let $\sequence{\boldsymbol{X}}$ be a time series. We shall divide the sample $\{\boldsymbol{X}_1,\ldots, \boldsymbol{X}_n\}$ into $m_n\in\mathbb{N}$ disjoint blocks of same size $\dhinterseq\in\mathbb{N}$. Without loss of generality we will assume that $n=m_n\dhinterseq$. Dependence within each block (\"local dependence\") is controlled by the appropriate anticlustering condition. On the other hand, dependence between the blocks (\"global dependence\") is controlled by the appropriate temporal dependence assumption. We will consider two temporal dependence schemes. - **Stationary, weakly dependent:** $\sequence{\boldsymbol{X}}$ is a stationary, regularly varying $\mathbb{R}^d$-valued time series. Weak dependence will be controlled by $\alpha$-mixing. We will assume that there exists $C,\epsilon>0$ such that $$\begin{aligned} \label{eq:mixing-rates} \alpha_j=O(\exp(-Cj))\;, \ \ \lim_{n\to\infty}\exp\left(-\frac{C\epsilon}{2+\epsilon}\dhinterseq\right)/(\dhinterseq w_n)^{\epsilon/(2+\epsilon)}=0\;. \end{aligned}$$ This assumption will imply all the mixing bounds used in the paper. In most of the results in the paper we can replace the above rate with a polynomial decay. We keep the exponential bound for clarity. - **Piecewise stationary:** For a sample of size $n$, we have observations $$\begin{aligned} (\boldsymbol{X}_1,\ldots,\boldsymbol{X}_n)=((\boldsymbol{X}_{1}^{(1)},\ldots,\boldsymbol{X}_{\dhinterseq}^{(1)}), (\boldsymbol{X}_{1}^{(2)},\ldots,\boldsymbol{X}_{\dhinterseq}^{(2)}),\ldots, (\boldsymbol{X}_{1}^{(m_n)},\ldots,\boldsymbol{X}_{\dhinterseq}^{(m_n)})) \end{aligned}$$ and $\{\boldsymbol{X}_{j}^{(i)},j\in\mathbb{N}\}$, $i=1,\ldots,m_n$, are independent copies of the regularly varying, weakly dependent, time series $\{\boldsymbol{X}_{j},j\in\mathbb{N}\}$. Formally, $(\boldsymbol{X}_1,\ldots,\boldsymbol{X}_n)$ is an array of random elements. The latter assumption will serve for some illustration purposes only. The rationale for this assumption is that, from the point of view of the central limit theorems, both disjoint and sliding blocks estimators behave as if the blocks were independent. We will show that this is not the case when the asymptotic expansion is considered. For a future use, we recall the following mixing inequality. Let $\mathcal{F}_{i,j}$ be the sigma field generated by $\boldsymbol{X}_{i,j}$. Let $p,q,r>0$ be such that $1/r+1/p+1/q$. Then for $U\in \mathcal{F}_{-\infty,\ell}$ and $V\in \mathcal{F}_{\ell+i,\infty}$ we have $$\begin{aligned} \label{eq:mixing-inequality} |\mathrm{Cov}(U,V)|\leqslant 8\alpha_i^{1/r}\|U\|_p\|V\|_q\;. \end{aligned}$$ ## Central Limit Theorem - disjoint blocks estimators {#sec:clt-disjoint} We quote results on blocks estimators in the PoT setting. Let ${\mathbb{G}}$ be the Gaussian process on $L^2({\boldsymbol{\nu}}^*)$ with covariance $$\begin{aligned} \mathrm{Cov}({\mathbb{G}}(H),{\mathbb{G}}(\widetilde{H})) = {\boldsymbol{\nu}}^*(H\widetilde{H}) \; . \end{aligned}$$ We state Theorem 10.2.1 in [@kulik:soulier:2020]. We refer to that theorem for the set of assumptions on functional $H$ and the mixing rates. Theorem is valid in particular for bounded, shift-invariant functionals that vanish around zero. **Proposition 10**. *Let $\boldsymbol{X}=\sequence{\boldsymbol{X}}$ be a stationary, regularly varying $\mathbb{R}^d$-valued time series. Assume that [\[eq:rnbarFun0\]](#eq:rnbarFun0){reference-type="ref" reference="eq:rnbarFun0"}, [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} hold and $\boldsymbol{X}$ is beta-mixing with the appropriate rates. Then $$\begin{aligned} \label{eq:clt-estimator-disj} \sqrt{n\mathbb P(\norm{\boldsymbol{X}_0}>\tepseq)}\left\{\tedcluster(H)-\mathbb E[\tedcluster(H)]\right\}\stackrel{\mbox{\tiny\rm d}}{\longrightarrow} {\mathbb{G}}(H)\;.\end{aligned}$$* For piecewise stationary time series, the central limit theorem follows from Theorem 10.2.1 in [@kulik:soulier:2020], without a need of the mixing assumption. We only need the basic anticlustering condition to control each disjoint block of size $\dhinterseq$. **Proposition 11**. *Let $\boldsymbol{X}=\sequence{\boldsymbol{X}}$ be a piecewise stationary, regularly varying $\mathbb{R}^d$-valued time series. Assume that [\[eq:rnbarFun0\]](#eq:rnbarFun0){reference-type="ref" reference="eq:rnbarFun0"}, [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} hold. Then the central limit theorem [\[eq:clt-estimator-disj\]](#eq:clt-estimator-disj){reference-type="eqref" reference="eq:clt-estimator-disj"} holds.* ## Central Limit Theorem - sliding blocks estimators {#sec:clt-sliding} The next result is from [@cissokho:kulik:2021]. We again refer to that paper for the precise assumptions. **Proposition 12** (Theorem 5.12 in [@cissokho:kulik:2021]). *Let $\boldsymbol{X}=\sequence{\boldsymbol{X}}$ be a stationary, regularly varying $\mathbb{R}^d$-valued time series. Assume that [\[eq:rnbarFun0\]](#eq:rnbarFun0){reference-type="ref" reference="eq:rnbarFun0"}, [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} hold and $\boldsymbol{X}$ is beta-mixing with the appropriate rates. Then $$\begin{aligned} \label{eq:clt-estimator-1} \sqrt{n\mathbb P(\norm{\boldsymbol{X}_0}>\tepseq)}\left\{\tedclustersl(H)-\mathbb E[\tedclustersl(H)]\right\}\stackrel{\mbox{\tiny\rm d}}{\longrightarrow} {\mathbb{G}}(H)\;.\end{aligned}$$* From the asymptotic expansions developed in the paper, we can also infer that (under additional restrictions on the blocks size) the above asymptotics holds for the piecewise stationary case. **Thus, in the PoT setting, the disjoint and sliding blocks estimator for both stationary and piecewise stationary case, lead to the same asymptotic variance.** # Internal and boundary clusters {#sec:two-types-of-clusters} ## Notation {#sec:notation} We consider disjoint blocks of size $\dhinterseq$: $$\begin{aligned} I_{j}=\{(j-1)\dhinterseq+1,\ldots,j\dhinterseq\}\;, \ \ j=1,\ldots,m_n\;.\end{aligned}$$ The subscript $j$ will always indicate the numbering of blocks. We assume without loss of generality that $m_n\dhinterseq=n$. Set $$\begin{aligned} \label{eq:def-DB} {\rm{DB}}_j:={\rm{DB}}_j(H):=\dhinterseq H(\tepseq^{-1}\boldsymbol{X}_{(j-1)\dhinterseq+1,j\dhinterseq})\;, \ \ {\rm{DB}}:={\rm{DB}}(H):=\sum_{j=2}^{m_n-1} {\rm{DB}}_j(H)\;.\end{aligned}$$ Note that to avoid dealing with boundary terms, we consider blocks $j=2,\ldots,m_n-1$ only. Keeping this in mind, the disjoint blocks statistics in [\[eq:blocktype\]](#eq:blocktype){reference-type="eqref" reference="eq:blocktype"} can be written as $$\begin{aligned} \label{eq:disjoint-rep} \tedcluster(H)&= \frac{1}{n\dhinterseq w_n} {\rm{DB}}(H)\;.\end{aligned}$$ Likewise, let $$\begin{aligned} \label{eq:def-SBj} {\rm{SB}}_j:={\rm{SB}}_j(H):=\sum_{i\in I_j} H\left(\tepseq^{-1}\boldsymbol{X}_{i,i+\dhinterseq-1}\right)\;.\end{aligned}$$ That is, ${\rm{SB}}_j(H)$ is the contribution to the sliding blocks statistics coming from the block $I_j$. Note that ${\rm{SB}}_j(H)$ is a function of the random variables $\boldsymbol{X}_i$, $i\in I_j\cup I_{j+1}$ only. Set $$\begin{aligned} \label{eq:def-SB} {\rm{SB}}:={\rm{SB}}(H):=\sum_{j=2}^{m_n-1} {\rm{SB}}_j(H)\;.\end{aligned}$$ Thus, ignoring the boundary effects, the sliding blocks statistics defined in [\[eq:sliding-block-estimator-nonfeasible-1\]](#eq:sliding-block-estimator-nonfeasible-1){reference-type="eqref" reference="eq:sliding-block-estimator-nonfeasible-1"} becomes $$\begin{aligned} \label{eq:sliding-rep} \tedclustersl(H)&=\frac{1}{n \dhinterseq w_n} {\rm{SB}}(H)\;.\end{aligned}$$ The difference between the sliding and disjoint blocks statistics will heavily depend on the number of exceedances in the consecutive blocks. Denote by $$\begin{aligned} N_{j}:=\mathcal E(\tepseq^{-1}\boldsymbol{X}_{(j-1)\dhinterseq+1,j\dhinterseq})=\sum_{i\in I_{j}}\mathbbm{1}{\left\{\norm{\boldsymbol{X}_i}>\tepseq\right\}}\;, \ \ j=1,\ldots,m_n\;,\end{aligned}$$ the number of exceedances in the block $I_{j}$. Set $$\begin{aligned} \label{eq:set-Aj} A_{j} = \{ \exists i : (j-1)\dhinterseq + 1 \leqslant i \leqslant j\dhinterseq , \vert \boldsymbol{X}_i \vert > \tepseq \}=\{\boldsymbol{X}_{(j-1)\dhinterseq + 1,j\dhinterseq}^\ast>\tepseq\}\;.\end{aligned}$$ Recall the notation [\[eq:exc-times-x-1\]](#eq:exc-times-x-1){reference-type="eqref" reference="eq:exc-times-x-1"}-[\[eq:exc-times-x-3\]](#eq:exc-times-x-3){reference-type="eqref" reference="eq:exc-times-x-3"}. If $N_{j}\not=0$, then denote by $$\begin{aligned} t_j{(i)}=T^{(i)}(\tepseq^{-1}\boldsymbol{X}_{(j-1)\dhinterseq+1,j\dhinterseq})\end{aligned}$$ the $i$th exceedance time in the block $j$. Hence, $$\begin{aligned} \label{eq:exceedance} t_j{(1)}<\cdots< t_{j}{(N_{j})}\;, \ \ j=1,\ldots,m_n\;,\end{aligned}$$ are the exceedance times in the $j$th block. We will use the convention $t_{j}(0)\equiv (j-1)\dhinterseq$ and $t_{j}(N_{j}+1)\equiv j\dhinterseq$. Note that it is possible that $t_{j}{(N_{j})}=t_{j}(N_{j}+1)=j\dhinterseq$. This happens when the last jump in the block $j$ occurs at the right-end point, $j\dhinterseq$. On the other hand, since the $j$th block starts at $(j-1)\dhinterseq+1$, $t_{j}(1)$ is strictly larger than $t_{j}(0)$. We will set $t_1(i)=t(i)$ for $i=1,\ldots,N_1$. We will also define $$\begin{aligned} \Delta t_{j}(i) =t_{j}(i+1) -t_{j}(i)\end{aligned}$$ for each $i=0,\ldots, N_j$. With this notation we have $\sum_{i=0}^{N_j}\Delta t_{j}(i) =\dhinterseq$. Note that this is valid regardless whether $N_j=0$ or $N_j\not=0$, thanks to the convention introduced above. Furthermore, using the notation introduced in [\[eq:cluster-length-def\]](#eq:cluster-length-def){reference-type="eqref" reference="eq:cluster-length-def"}, $$\begin{aligned} %\label{eq:def-clusterlength} \sum_{i=1}^{N_j-1}\Delta t_{j}(i) +1= t_{j}{(N_{j})}-t_{j}{(1)}+1=\mathcal L(\tepseq^{-1}\boldsymbol{X}_{(j-1)\dinterseq+1,j\dhinterseq})= :\mathcal L_{j}\end{aligned}$$ is the cluster length in the $j$th block. We use the convention $\sum_{i=\ell}^j=0$ whenever $j<\ell$. Hence, $\mathcal L_{j}=1$ whenever $N_{j}=1$. If there are no exceedances over the threshold $\tepseq$, then we set $\mathcal L_{j}\equiv 0$ (there will be no issue, since $\mathcal L_{j}$ will appear with the appropriate indicator.) We need to keep track of the exceedances in each block. For this, for each $j=1,\ldots,m_n$ and each $1\leqslant k_1 \leqslant k_2 \leqslant N_{j}$, we define $$\begin{aligned} %\label{eq:random-elements-Z} \mathbb{X}_{j}{(k_1:k_2)} = \tepseq^{-1} \big(\boldsymbol{X}_{ t_{j}{(k_1)}}, \ldots, \boldsymbol{X}_{t_{j}{(k_2)} } \big)\;.\end{aligned}$$ Thus, $\mathbb{X}_{j}{(1:N_{j})}$ is a vector of size $\mathcal L_{j}$ that consists of all (scaled) exceedances in the block $j$ and all small values between the first and the last exceedance. We use the convention $\mathbb{X}_{j}{(k_1:k_2)}=0$ whenever $k_1>k_2$. Hence, in particular, $\mathbb{X}_{j}{(1:N_{j})}=0$ if $N_{j}=0$. We will also use the notation $$\begin{aligned} %\label{eq:random-element-entire} \mathbb{X}_{j}=\tepseq^{-1}\boldsymbol{X}_{(j-1)\dhinterseq+1,j\dhinterseq}\;, \ \ j=1,\ldots,m_n\end{aligned}$$ for the scaled block $j$. By the definition, for $H\in \mathcal H(\gamma)$ we have $H(\mathbb{X}_{j})=H(\mathbb{X}_{j}{(1:N_{j})})$. ### Adjacent blocks {#sec:adjacent-blocks} We will also use the notation $$\begin{aligned} \label{eq:DBj,j+1} {\rm{DB}}_{j,j+1}(H)=\dhinterseq H(\tepseq^{-1}\boldsymbol{X}_{(j-1)\dhinterseq+1,(j+1)\dhinterseq})\;\end{aligned}$$ and define $$\begin{aligned} \label{eq:boundary-full-cluster-length} \mathcal{L}_{j,j+1} = t_{j+1}{(N_{j+1})} - t_{j}(1) + 1\;, \end{aligned}$$ the joint cluster length for blocks $j$ and $j+1$. When adjacent blocks with large values are considered, we shall rename $t_{j}(1)< \cdots <t_{j}{(N_j)}< t_{j+1}(1)< \cdots<t_{j+1}{(N_{j+1})}$ as $$\begin{aligned} \label{eq:boundary-full-jump-times} t_{j,j+1}(1)< \cdots <t_{j,j+1}{(N_j)}< t_{j,j+1}{(N_j+1)}< \cdots<t_{j,j+1}{(N_j+N_{j+1})}\;. \end{aligned}$$ By the convention $t_{j,j+1}(0)=(j-1)\dhinterseq$ and $t_{j,j+1}{(N_j+N_{j+1}+1)}=(j+1)\dhinterseq$. We will also use the notation $\Delta t_{j,j+1}(i)=t_{j,j+1}(i+1)-t_{j,j+1}(i)$. ## Two types of blockwise clusters {#subsec:two-types-clusters} We will distinguish between two kinds of clusters, with distinct properties. We shall call the first kind an *internal cluster*: it typically lives inside a block $I_j$, and whose emergence excludes exceedances in neighboring blocks $I_{j-1}$ and $I_{j+1}$. The second kind is referred to as a *boundary cluster*. Its name suggests that it describes simultaneous exceedances in two adjacent disjoint blocks $I_j$ and $I_{j+1}$. The exceedances are typically "near" the boundary points $j\dhinterseq$ and $j\dhinterseq+1$. In this case, the exceedances in $I_{j+1}$ persist from $I_{j}$. ### Internal clusters {#sec:contribution-internal-clusters} Recall the notation [\[eq:set-Aj\]](#eq:set-Aj){reference-type="eqref" reference="eq:set-Aj"}: $A_j=\{\boldsymbol{X}_{(j-1)\dhinterseq+1,j\dhinterseq}^*>\tepseq\}$. Informally speaking, we have an internal cluster in the block $I_j$ if $\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}^c\right\}}=1$. That is, there is a large value in the block $j$, but there are no big values in the adjacent blocks. Formally, the name \"internal cluster\" is due to the following result. We note that the scaled support of exceedance times $\dhinterseq^{-1}\{ t^{(1)}_1,\ldots, t^{(1)}_{N_1} \}$ converges weakly to the random singleton $\{U_1 \}$, which lies in the interior of $(0,1)$. The following result follows from [@chen:kulik:2023a]. **Lemma 13**. *Assume that [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} and [\[eq:rnbarFun0\]](#eq:rnbarFun0){reference-type="ref" reference="eq:rnbarFun0"} hold. Let a random variable $U_1$ be ${\rm Uniform}(0,1)$ and independent of the tail process $\boldsymbol{Y}$. Then $$\begin{aligned} %\label{eq:joint-convergence-interior-cluster} \left( \frac{t_{1}(1)}{\dhinterseq},\frac{t_{1}{(N_1)}}{\dhinterseq}, \mathbbm{1}{\left\{A_0^c\right\}},\mathbbm{1}{\left\{A_2^c\right\}} \right) \xLongrightarrow{\mathbb{P}(\cdot \mid A_1 )} \left( U_1,U_1, 1, 1 \right)\;.\end{aligned}$$* #### Internal clusters statistics. Motivated by , on a general event $A_{j-1}^c \cap A_j \cap A_{j+1}^c$, there will be three blocks estimators capturing the cluster in $I_j$. Namely, the sliding blocks estimators ${\rm{SB}}_{j-1}$ and ${\rm{SB}}_j$, and the disjoint blocks estimator ${\rm{DB}}_j$. To be more specific (see for the detailed explanation): 1. On $A_{j-1}^c\cap A_j$ ($N_{j-1}=0$ and $N_j\not=0$), we have $$\begin{aligned} \label{eq:SBj-1:nojump-jump} {\rm{SB}}_{j-1}(H)&=\sum_{i=(j-2)\dhinterseq+1}^{(j-1)\dhinterseq}H(\tepseq^{-1}\boldsymbol{X}_{i,i+\dhinterseq-1}) =\sum_{i=1}^{N_j}\Delta t_j(i)H(\mathbb{X}_{j}{(1:i)})\;.\end{aligned}$$ 2. On $A_j\cap A_{j+1}^c$ ($N_j\not=0$ and $N_{j+1}=0$), we have $$\begin{aligned} \label{eq:SBj:jump-nojump} {\rm{SB}}_j(H)&=\sum_{i=0}^{N_j-1} \Delta t_j(i)H(\mathbb{X}_{j}{(i+1:N_j)})\;.\end{aligned}$$ 3. On $A_j$, ${\rm{DB}}_j(H)=\dhinterseq H(\mathbb{X}_{j})$. This leads to the following definition, that represents a contribution to the difference between sliding and disjoint blocks, stemming from the internal clusters (see for the detailed calculation): $$\begin{aligned} \label{eq:interior-clusters-def} &\mathcal{IC}:=\mathcal{IC}(H):=\sum_{j=2}^{m_n-1}({\rm{SB}}_{j-1}+{\rm{SB}}_j-{\rm{DB}}_j) \mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}^c\right\}}=:\sum_{j=2}^{m_n-1}\mathcal{IC}_{j}(H)\;,\end{aligned}$$ with $$\begin{aligned} \label{eq:internal-as-functional} \mathcal{IC}_j(H)=\widetilde{H}_{\mathcal{IC}}(\tepseq^{-1}\boldsymbol{X}_{(j-1)\dhinterseq+1,j\dhinterseq})\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}^c\right\}}\;.\end{aligned}$$ We will refer to $\mathcal{IC}_j(H)$ as the *internal cluster* and to $\mathcal{IC}$ as the *internal clusters statistics*. **Example 14**. * If $H(\boldsymbol{x})=\mathbbm{1}{\left\{\boldsymbol{x}^*>1\right\}}$. Here, $H(\mathbb{X}_{j}{(i_1:i_2)})=1$ for any $1\leqslant i_1\leqslant i_2\leqslant N_j$. Hence, $\mathcal{IC}_j(H)$ in [\[eq:internal-as-functional\]](#eq:internal-as-functional){reference-type="eqref" reference="eq:internal-as-functional"} becomes $$\begin{aligned} \mathcal{IC}_j(H)=\left(t_{j}{(N_j)}-t_{j}(1)\right)\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}^c\right\}}=\left(\mathcal L_j-1\right) \mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}^c\right\}}\;.\end{aligned}$$ Obviously, $\mathcal{IC}_j(H)\equiv 0$ if there is only one large jump in the $j$th block. * **Remark 3**. * We note that it is important to combine ${\rm{SB}}_{j-1}$ and ${\rm{SB}}_j$ together. This allows us to reduce the sums $$\begin{aligned} \widetilde{\mathcal{IC}}_{j}(H): &=\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}^c\right\}}\sum_{i=0}^{N_j}\Delta t_{j}{(i)}\left( H(\mathbb{X}_{j}{(1:i)})+ H(\mathbb{X}_{j}{(i+1:N_j)}) -H(\mathbb{X}_{j})\right) \end{aligned}$$ to $\mathcal{IC}_{j}(H)$. Replacing the sum $\sum_{i=0}^{N_j}$ with $\sum_{i=1}^{N_j-1}$ is crucial. Indeed, if $H$ is bounded, then $$|\widetilde\mathcal{IC}_{j}(H)|\leqslant 3\|H\| \dhinterseq \mathbbm{1}{\left\{A_j\right\}}$$ while $$|\mathcal{IC}_{j}(H)|\leqslant 3\|H\|\mathcal L_j \mathbbm{1}{\left\{A_j\right\}}\;.$$ The cluster length $\mathcal L_j$ is tight under the conditional distribution (given $A_j$). See [@chen:kulik:2023a]. Hence, combining ${\rm{SB}}_{j-1}$ with ${\rm{SB}}_j$ allows us to get the optimal convergence rates. * ### Boundary clusters {#sec:contribution-boundary-clusters} Informally, we have a boundary cluster between blocks $I_j$ and $I_{j+1}$ if $\mathbbm{1}{\left\{A_{j}\cap A_{j+1}\right\}}=1$. We have large values in the adjacent blocks. The chance of this event is much smaller than $\mathbb P(A_1)$. If the blocks were independent, then $\mathbb P(A_j\cap A_{j+1})$ would be of the order $\dhinterseq^2 w_n^2$. In case of temporal dependence, the chance of this joint event is different as indicated in the following lemma. The lemma provides the precise asymptotic behaviour in case of small blocks - recall that [$\mathcal{S}^{(1)}(\dhinterseq, \tepseq)$](#SummabilityAC) requires $\dhinterseq^2w_n\to 0$ in e.g. i.i.d. case. **Lemma 15**. *Assume that [$\mathcal{S}^{(1)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. Then $$\begin{aligned} %\label{eq:mdep-P(a1-cap-a2)-smallblocks-at-the-beginning} \lim_{n\to\infty}\frac{\mathbb P(A_1\cap A_2)}{w_n}={\vartheta}\mathbb E[(\mathcal L(\boldsymbol{Z}) - 1)] \;.\end{aligned}$$* Interestingly, in case of large blocks the asymptotic behaviour changes. See . Informally speaking, the above lemma explains the name \"boundary cluster\": a (finite number) of large values occurs at the end of block $j$ and at the beginning of block $j+1$. #### Boundary clusters statistics. We analyse the contribution from the boundary clusters to blocks statistics. See . - On $A_{j-1}^c\cap A_j$, the contribution ${\rm{SB}}_{j-1}(H)$ is the same as in [\[eq:SBj-1:nojump-jump\]](#eq:SBj-1:nojump-jump){reference-type="eqref" reference="eq:SBj-1:nojump-jump"}. - On $A_{j+1}\cap A_{j+2}^c$, the contribution is ${\rm{SB}}_{j+1}(H)$ is the same as in [\[eq:SBj:jump-nojump\]](#eq:SBj:jump-nojump){reference-type="eqref" reference="eq:SBj:jump-nojump"} (with $j$ replaced by $j+1$). - On $A_j$, ${\rm{DB}}_j(H)=\dhinterseq H(\mathbb{X}_{j})$. - On $A_j\cap A_{j+1}$, ${\rm{SB}}_j(H)$ has a cumbersome form (see ). This leads to the following definition, that represents a contribution to the difference between sliding and disjoint blocks, stemming from the boundary clusters: $$\begin{aligned} \label{eq:boundary-clusters-def} \mathcal{BC}:=\mathcal{BC}(H)&=\sum_{j=2}^{m_n-1}({\rm{SB}}_{j-1}+{\rm{SB}}_j-{\rm{DB}}_j+{\rm{SB}}_{j+1}-{\rm{DB}}_{j+1}) \mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}\notag\\ &=:\sum_{j=2}^{m_n-1}\mathcal{BC}_j(H)\;.\end{aligned}$$ We will refer to $\mathcal{BC}_j$ as the *boundary clusters* and to $\mathcal{BC}$ as the *boundary clusters statistics*. We need to expand it further. Recall the notation ${\rm{DB}}_{j,j+1}(H)$ from [\[eq:DBj,j+1\]](#eq:DBj,j+1){reference-type="eqref" reference="eq:DBj,j+1"}. Write [\[eq:boundary-clusters-def\]](#eq:boundary-clusters-def){reference-type="eqref" reference="eq:boundary-clusters-def"} as $$\begin{aligned} \label{eq:boundary-clusters-decomposition} \mathcal{BC}(H):=\mathcal{BC}(H;1)+\mathcal{BC}(H;2)= \sum_{j=2}^{m_n-1}\mathcal{BC}_j(H;1)+\sum_{j=2}^{m_n-1}{\mathcal{BC}}_j(H;2)\end{aligned}$$ with $$\begin{aligned} \mathcal{BC}_j(H;1) = & \left({\rm{DB}}_{j,j+1} - {\rm{DB}}_j - {\rm{DB}}_{j+1} \right)\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}\; \label{eq;difference-SB&DB;boundary-cluster-part;exceedance-times-expansion;main} \end{aligned}$$ and $$\begin{aligned} {\mathcal{BC}}_j(H;2) = & \left( {\rm{SB}}_{j-1} + {\rm{SB}}_j + {\rm{SB}}_{j+1} - {\rm{DB}}_{j,j+1} \right)\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}\;. \label{eq;difference-SB&DB;boundary-cluster-part;exceedance-times-expansion;minor}\end{aligned}$$ Recall [\[eq:DBj,j+1\]](#eq:DBj,j+1){reference-type="eqref" reference="eq:DBj,j+1"}-[\[eq:boundary-full-cluster-length\]](#eq:boundary-full-cluster-length){reference-type="eqref" reference="eq:boundary-full-cluster-length"}. Thus, $\mathcal{BC}_j(H;1)$ has a very simple form: $$\begin{aligned} \label{eq:boundary-1-as-functional} \mathcal{BC}_j(H;1)=\dhinterseq\left(H(\mathbb{X}_{j})+H(\mathbb{X}_{j+1})- H(\mathbb{X}_{j,j+1})\right) \mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}\;.\end{aligned}$$ Next, on the event $\{\mathcal L_{j,j+1}<\dhinterseq\}$, with the definition of $\widetilde{H}_{\mathcal{IC}}(\boldsymbol{x})$ in [\[eq:new-functional-IC\]](#eq:new-functional-IC){reference-type="eqref" reference="eq:new-functional-IC"}, we can write $$\begin{aligned} %\label{eq:boundary-2-as-functional} &{\mathcal{BC}}_j(H;2)=\widetilde{H}_{\mathcal{IC}}(\mathbb{X}_{j,j+1})\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}\;.\end{aligned}$$ with $$\begin{aligned} \label{eq:boundary-2-as-functional} &\widetilde{H}_{\mathcal{IC}}(\mathbb{X}_{j,j+1}):= \sum_{i=1}^{M_j} \Delta t_{j,j+1}(i) \left( H( \mathbb{X}_{j,j+1}{(1:i)} ) + H( \mathbb{X}_{j,j+1}{(i+1:N_{j}+N_{j+1})} ) - H( \mathbb{X}_{j,j+1} )\right)\;,\end{aligned}$$ where $M_j:=N_{j}+N_{j+1}-1$. See again for the details. On the other hand, the event $\{\mathcal L_{j,j+1}\geqslant \dhinterseq\}$ has a small probability; see . **Example 16**. * Let $H(\boldsymbol{x})=\mathbbm{1}{\left\{\boldsymbol{x}^*>1\right\}}$. Then $\mathcal{BC}_j(H;1)=-\dhinterseq\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}$, $$\begin{aligned} \mathcal{BC}_j(H;2)=\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}} \left(\left(t_{j+1}{(N_{j+1})}-t_{j}(1)\right)-\left(t_{j+1}(1)-t_{j}{(N_j)}-\dhinterseq\right)_+\right) \; \end{aligned}$$ and the last part simplifies to $\left(t_{j+1}{(N_{j+1})}-t_{j}(1)\right)$ on the event $\{\mathcal L_{j,j+1}<\dinterseq\}$ * # Asymptotic expansions {#sec:main-results} Throughout this section it is everywhere assumed that [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} and [\[eq:rnbarFun0\]](#eq:rnbarFun0){reference-type="ref" reference="eq:rnbarFun0"} hold. Recall [\[eq:disjoint-rep\]](#eq:disjoint-rep){reference-type="eqref" reference="eq:disjoint-rep"}-[\[eq:sliding-rep\]](#eq:sliding-rep){reference-type="eqref" reference="eq:sliding-rep"}. The difference $\tedcluster(H)-\tedclustersl(H)$ between the sliding blocks and the disjoint blocks statistics is written as $$\begin{aligned} \tedcluster(H)-\tedclustersl(H)= \frac{1}{n \dhinterseq w_n}\mathcal{IC}(H)+\frac{1}{n \dhinterseq w_n}\mathcal{BC}(H) +\frac{1}{n \dhinterseq w_n}\mathcal{R}(H)\;,\end{aligned}$$ where the internal and the boundary clusters statistics are defined in [\[eq:interior-clusters-def\]](#eq:interior-clusters-def){reference-type="eqref" reference="eq:interior-clusters-def"} and [\[eq:boundary-clusters-def\]](#eq:boundary-clusters-def){reference-type="eqref" reference="eq:boundary-clusters-def"}, respectively, while the remainder term will be defined later, cf. . We summarize the results of this section as follows: - In the small blocks scenario, the sliding and the disjoint blocks statistics differ by $O_P(\dhinterseq^{-1})$. In all the examples we consider the rate is in fact $o_P(\dhinterseq^{-1})$, but we are unable to draw a general conclusion. Both the internal and the boundary clusters contribute at the same rate. See . - In the small blocks scenario, if $\boldsymbol{X}$ is piecewise stationary, then the difference is also of the order $O_P(\dhinterseq^{-1})$. This rate is sharp and cannot be improved to $o_P$ in general. Only the internal clusters contribute here. See . - In the large blocks scenario, the expected distance between the disjoint and the sliding blocks is of the order $O(\dhinterseq w_n)$ and this rate cannot be improved in general. Only the internal clusters contribute here. From this perspective, the large blocks scenario resembles the piecewise stationary case. See . For the large blocks scenario we consider a simple case of an MMA(1) process and $H(\boldsymbol{x})=\mathbbm{1}{\left\{\boldsymbol{x}^*>1\right\}}$. To get general results, one needs to establish proper asymptotic results for boundary clusters in the large blocks scenario. We do not pursue this direction. Indeed, one can argue that the large blocks scenario is statistically not relevant: large blocks yield large variance of blocks statistics. ## Small blocks scenario Recall the functionals $\widetilde{H}_{\mathcal{IC}}$ and $\widetilde{H}_{\mathcal{BC}}$; cf. [\[eq:new-functional-IC\]](#eq:new-functional-IC){reference-type="eqref" reference="eq:new-functional-IC"}, [\[eq:new-functional-BC\]](#eq:new-functional-BC){reference-type="eqref" reference="eq:new-functional-BC"}. The result below deals with asymptotic expansion for the disjoint and the sliding blocks statistics. We refer to this result as the \"small blocks scenario\", since the anticlustering condition [$\mathcal{S}^{(\delta)}(\dhinterseq, \tepseq)$](#SummabilityAC) forces the blocks size to fulfill $\dhinterseq^{\delta+1}w_n\to 0$. Note that both internal and boundary clusters contribute at the same rate. **Theorem 17**. *Assume that $\boldsymbol{X}$ is stationary and mixing with the rates [\[eq:mixing-rates\]](#eq:mixing-rates){reference-type="eqref" reference="eq:mixing-rates"}. Assume that [$\mathcal{S}^{(2\gamma+3)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. Let $H\in\mathcal H(\gamma)$. Then $$\begin{aligned} \tedcluster(H)-\tedclustersl(H)=\frac{1}{n \dhinterseq w_n}\mathcal{IC}(H)+\frac{1}{n \dhinterseq w_n}\mathcal{BC}(H)+\textcolor{black}{O_P\left(\frac{\dhinterseq}{n}+\dhinterseq^{-(\gamma+4)}\right)}\;,\end{aligned}$$ where $$\begin{aligned} \label{eq:internal-clusters-convprob} \frac{\mathcal{IC}(H)}{nw_n}\stackrel{\tiny \mathbb{P}}{\longrightarrow}{\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{IC}})\;,\end{aligned}$$ $$\begin{aligned} \label{eq:boundary-clusters-convprob} \frac{\mathcal{BC}(H)}{n w_n} \stackrel{\tiny \mathbb{P}}{\longrightarrow} {\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{BC}})\;.\end{aligned}$$ The latter convergence holds as long as $\dhinterseq/(nw_n)\to 0$.* *Proof.* and give [\[eq:internal-clusters-convprob\]](#eq:internal-clusters-convprob){reference-type="eqref" reference="eq:internal-clusters-convprob"}. and yield [\[eq:boundary-clusters-convprob\]](#eq:boundary-clusters-convprob){reference-type="eqref" reference="eq:boundary-clusters-convprob"}. The rate for the remainder follows from . ◻ We can provide some heuristic for the result above. - For the internal clusters, recall [\[eq:internal-as-functional\]](#eq:internal-as-functional){reference-type="eqref" reference="eq:internal-as-functional"}. Then, $\widetilde{H}_{\mathcal{IC}}$ is a tight cluster functional, the event $A_j$ occurs with the probability proportional to $\dhinterseq w_n$ and we have $m_n$ blocks. Hence, the rate for the internal clusters statistics is $m_n\times \dhinterseq w_n=nw_n$. - For the boundary clusters statistics, recall first [\[eq:boundary-2-as-functional\]](#eq:boundary-2-as-functional){reference-type="eqref" reference="eq:boundary-2-as-functional"}. Again, $\widetilde{H}_{\mathcal{IC}}$ is tight, and the event $A_j\cap A_{j+1}$ occurs with the probability $o(\dhinterseq w_n)$. Hence, the term $\mathcal{BC}(H;2)$ will not contribute. - On the other hand, recall [\[eq:boundary-1-as-functional\]](#eq:boundary-1-as-functional){reference-type="eqref" reference="eq:boundary-1-as-functional"}. Then the functional considered there is or order $\dhinterseq$, the event $A_j\cap A_{j+1}$ occurs with the probability $w_n$. In total, the boundary clusters statistics will contribute at the rate $m_n\times \dhinterseq \times w_n=nw_n$, the same as the internal clusters statistics. **Remark 4**. - *Note that the disjoint and the sliding blocks statistics have the same mean. Thus, if $\sqrt{nw_n}\left\{\frac{1}{\dhinterseq}+\frac{\dhinterseq}{n}\right\}\to 0$, then gives that the disjoint and the sliding blocks statistics yield the same central limit theorem. The restriction on $w_n$ and $\dhinterseq$ is certainly stronger as compared to the results obtained by direct computations.* - *The anticlustering condition [$\mathcal{S}^{(2\gamma+3)}(\dhinterseq, \tepseq)$](#SummabilityAC) may seem too strong, but seems to be optimal to get the general results.* - *Recall $\boldsymbol{Z}$ from . We note that for suitably chosen $g:\mathbb{R}\to\mathbb{R}$ we have $$\begin{aligned} %\label{eq:short-formula-for-Hic} &{\boldsymbol{\nu}}^*(g(\widetilde{H}_{\mathcal{IC}}))={\vartheta}\mathbb E\left[g(\widetilde{H}_{\mathcal{IC}}(\boldsymbol{Z}))\right]\\ &={\vartheta}\mathbb E\left[g\left(\sum_{i=1}^{\mathcal E(\boldsymbol{Z})-1}\Delta T^{(i)}(\boldsymbol{Z})\{ H(\boldsymbol{Z}_{0,T^{(i)}(\boldsymbol{Z})}) + H(\boldsymbol{Z}_{T^{(i+1)}(\boldsymbol{Z}),T_{\rm max}(\boldsymbol{Z})}) - H(\boldsymbol{Z}) \}\right)\right]\;.\notag\end{aligned}$$ and $$\begin{aligned} \label{eq:short-formula-for-Hbc} &{\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{BC},p})={\vartheta}\mathbb E\left[\widetilde{H}_{\mathcal{BC},p}(\boldsymbol{Z})\right] \\ &={\vartheta}\mathbb E\left[\sum_{i=1}^{\mathcal L(\boldsymbol{Z})-1}| H(\boldsymbol{Z}) - H(\boldsymbol{Z}_{0,i-1})- H(\boldsymbol{Z}_{i,T_{\rm max}(\boldsymbol{Z})})|^p\right]\;.\notag\end{aligned}$$ Note also that $T^{(1)}(\boldsymbol{Z})=0$, $T_{\rm max}(\boldsymbol{Z})=\mathcal L(\boldsymbol{Z})-1$.* - *tells us that the difference between the disjoint and the sliding blocks statistics is of the order $O_P(\dhinterseq^{-1})$: $$\begin{aligned} \label{eq:conv-prob-expansion} \dhinterseq \left[\tedcluster(H)-\tedclustersl(H)\right]\stackrel{\tiny \mathbb{P}}{\longrightarrow}{\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{IC}}+\widetilde{H}_{\mathcal{BC}})\;. \end{aligned}$$* - *The expression on the right-hand side of [\[eq:conv-prob-expansion\]](#eq:conv-prob-expansion){reference-type="eqref" reference="eq:conv-prob-expansion"} vanishes in several scenarios. In such the case $$\begin{aligned} \label{eq:conv-prob-expansion-0} \dhinterseq \left[\tedcluster(H)-\tedclustersl(H)\right]\stackrel{\tiny \mathbb{P}}{\longrightarrow}0\;. \end{aligned}$$* - *Take $H(\boldsymbol{x})=\sum_j \phi(x_j)$, where $\phi(x_j)=0$ whenever $|x_j|<1$. Here $\widetilde{H}_{\mathcal{IC}}=\widetilde{H}_{\mathcal{BC},p}\equiv 0$ and hence ${\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{IC}})={\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{BC}})=0$. This is not surprising since the choice of the functional $H$ implies that the sliding and disjoint blocks estimator coincide (up to negligible boundary terms).* - *Take $H=\mathcal L$. Here $\widetilde{H}_{\mathcal{IC}}=\widetilde{H}_{\mathcal{BC},p}\equiv 0$ and hence ${\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{IC}})={\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{BC}})=0$.* - *Take $H=\mathbbm{1}{\left\{\boldsymbol{x}^*>1\right\}}$. We have ${\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{IC}})={\vartheta}\mathbb E[\mathcal L(Z)-1]=-{\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{BC}})$.* - *If $\boldsymbol{X}$ is extremally independent, then ${\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{IC}})={\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{BC}})=0$ for any $H$.* - *Assume that $\boldsymbol{Y}_j\not=0$ for any $j\geqslant 1$. This is for example the case of AR($p$) or ARCH($p$) process. Then ${\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{IC}})\not =0$, ${\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{BC}})\not=0$, but [\[eq:conv-prob-expansion-0\]](#eq:conv-prob-expansion-0){reference-type="eqref" reference="eq:conv-prob-expansion-0"} holds. Indeed, here $T^{(i)}(\boldsymbol{Z})=i-1$ and $\Delta T^{(i)}(\boldsymbol{Z})=1$.* - *However, we are unable to answer the question: Do we always have [\[eq:conv-prob-expansion-0\]](#eq:conv-prob-expansion-0){reference-type="eqref" reference="eq:conv-prob-expansion-0"}?* ## Small blocks scenario: piecewise stationary case {#sec:piecewise} Consider the piecewise stationary time series as defined in . A proof of central limit theorem for disjoint blocks statistics works as follows: we replace the original time series with its piecewise stationary version and prove the limit theorem for the latter. We will show below that in the context of the expansion considered in the paper, this idea breaks down. That is, the blocks in the stationary time series cannot be treated as independent. Recall [\[eq:internal-as-functional\]](#eq:internal-as-functional){reference-type="eqref" reference="eq:internal-as-functional"}. Define $$\begin{aligned} \mathcal{IC}^{\textrm{PS}}(H)=\sum_{j=2}^{m_n-1}\mathcal{IC}_j^{\textrm{PS}}(H)=\sum_{j=2}^{m_n-1}\widetilde{H}_{\mathcal{IC}}(\tepseq^{-1}\boldsymbol{X}_{(j-1)\dhinterseq+1,j\dhinterseq})\mathbbm{1}{\left\{A_j \right\}}\;.\end{aligned}$$ We formulate the following result. We note that only the internal clusters contribute. **Theorem 18**. *Assume that $\boldsymbol{X}$ is piecewise stationary. Assume that [$\mathcal{S}^{(2\gamma+3)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. Then $$\begin{aligned} \tedcluster(H)-\tedclustersl(H)=\frac{1}{n \dhinterseq w_n}\mathcal{IC}^{\textrm{PS}}(H)+\textcolor{black}{O_P\left(\dhinterseq w_n+\frac{\dhinterseq}{n}\right)}\;,\end{aligned}$$ where $$\begin{aligned} %\label{eq:internal-clusters-convprob-piecewise} \frac{\mathcal{IC}^\textrm{PS}(H)}{nw_n}\stackrel{\tiny \mathbb{P}}{\longrightarrow}{\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{IC}})\;.\end{aligned}$$* *Proof.* Since the blocks are independent, and [\[eq:extremal-index\]](#eq:extremal-index){reference-type="eqref" reference="eq:extremal-index"} give $$\begin{aligned} \frac{1}{n\dhinterseq w_n}\mathbb E\left[\left|\mathcal{IC}(H)-\mathcal{IC}^{\textrm{PS}}(H)\right|\right] &\leqslant 2\frac{1}{n\dhinterseq w_n}m_n \mathbb P(A_1)\mathbb E[|\widetilde{H}_{\mathcal{IC}}|(\tepseq^{-1}\boldsymbol{X}_{1,\dhinterseq})\mathbbm{1}{\left\{A_1\right\}}] \\ &=\frac{O(1)}{n\dhinterseq w_n}m_n \dhinterseq w_n \dhinterseq w_n =O(w_n)\;. \end{aligned}$$ Thus, in case of independent blocks, the indicators of small jumps, $\mathbbm{1}{\left\{A_j^c\right\}}$, can be dropped. Furthermore, give $$\begin{aligned} \frac{1}{n\dhinterseq w_n}\mathbb E[|\mathcal{BC}(H)|]=O(m_n)\frac{1}{n\dhinterseq w_n} \dhinterseq^3 w_n^2=O(\dhinterseq w_n)\;. \end{aligned}$$ ◻ ## Large blocks scenario {#sec:large-blocks-MMA(1)} For a simple treatment of the large blocks scenario, we consider the functional $H(\boldsymbol{x})=\mathbbm{1}{\left\{\boldsymbol{x}^*>1\right\}}$ and a very special case of a time series. The result can be extended to a mixing case with a cumbersome mixing rates. Furthermore, unbounded functionals can be considered. However, we decided to keep a very simple formulation of the large blocks scenario to illustrate a quantitative difference as compared to the small blocks scenario. From a statistical point of view, one can argue that large blocks are not relevant. To proceed, consider an MMA($1$) sequence $\boldsymbol{X}=\{X_j,j\in\mathbb{Z}\}$ defined by $$\begin{aligned} X_j=c_0 \xi_{j}\vee c_1 \xi_{j+1}\;,\end{aligned}$$ where $c_0,c_1$ are non-negative numbers and $\xi=\{\xi_j,j\in\mathbb{Z}\}$ is a sequence of i.i.d. non-negative, regularly varying random variables with the tail index $\alpha$. Non-negativity here is assumed for simplicity, to illustrate easily the main messages from the paper. Then, $$\begin{aligned} &\lim_{x\to\infty}\frac{\mathbb P(X_0>x)}{\mathbb P(\xi_0>x)}=c_0^\alpha+c_1^\alpha\;, \ \ \lim_{x\to\infty}\mathbb P(X_1>x\mid X_0>x)=\frac{(c_0\wedge c_{1})^\alpha}{c_0^\alpha+c_1^\alpha}\;. \label{eq:MMA1-tail}\end{aligned}$$ The tail process is $Y_1=\mathrm{b}(c_1/c_0)Y_0$, $Y_{-1}=(1-\mathrm{b})(c_0/c_1)Y_0$, where $\mathrm{b}$ is a Bernoulli random variables with the mean $1/(1+(c_1/c_0)^\alpha)$, and $Y_0$ is a standard Pareto random variable with the tail index $\alpha$. Furthermore, $Y_j=0$ if $|j|\geqslant 2$. Hence $$\begin{aligned} \mathbb P(Y_1>1)=\mathbb P(\mathrm{b}=1)\mathbb P((c_1/c_0)Y_0>1)=\frac{c_0^\alpha}{c_0^\alpha+c_1^\alpha}\left(\left(\frac{c_1}{c_0}\right)^\alpha\wedge 1\right)= \frac{(c_0\wedge c_1)^\alpha}{c_0^\alpha+c_1^\alpha}\;,\end{aligned}$$ (This can be also derived from [\[eq:MMA1-tail\]](#eq:MMA1-tail){reference-type="eqref" reference="eq:MMA1-tail"}, since $\mathbb P(Y_1>1)=\lim_{x\to\infty}\mathbb P(X_1>x\mid X_0>x)$.) The candidate extremal index is then $$\begin{aligned} %\label{eq:MMA1-extremal-index} {\vartheta}=\mathbb P(\boldsymbol{Y}_{1,\infty}^*\leqslant 1)=\mathbb P(Y_1\leqslant 1)=1-\mathbb P(Y_1>1)=1-\frac{(c_0\wedge c_1)^\alpha}{c_0^\alpha+c_1^\alpha}=\frac{(c_0\vee c_1)^\alpha}{c_0^\alpha+c_1^\alpha}\;.\end{aligned}$$ The next result shows that the rates in the asymptotic expansion for blocks statistics are different as compared to the small blocks scenario. Also, similarly to the piecewise stationary case, only the internal clusters contribute. In other words, in the large blocks scenario, the blocks behave as if they were independent. **Theorem 19**. *Assume that $\boldsymbol{X}$ is the MMA(1) process and $H(\boldsymbol{x})=\mathbbm{1}{\left\{\boldsymbol{x}^*>1\right\}}$. Assume $\dhinterseq^{3}w_n\to \infty$. Then $$\begin{aligned} \tedcluster(H)-\tedclustersl(H)=\frac{1}{n \dhinterseq w_n}\mathcal{IC}(H)+o_P(\dhinterseq w_n)+O_P\left(\frac{\dhinterseq^2w_n}{n}\right)\;,\end{aligned}$$ with $$\begin{aligned} \lim_{n\to\infty}\frac{\mathbb E[\mathcal{IC}(H)]}{n\dhinterseq^2w_n^2}=\frac{1}{6}{\vartheta}^2\;. \end{aligned}$$* *Proof.* gives the rate for the internal clusters statistics, yields that the internal clusters statistics is negligible, while the second part of (applied with $\gamma=0$) gives the rate for the remainder. ◻ In particular, $$\begin{aligned} %\label{eq:conv-prob-expansion-large-0} \left[\tedcluster(H)-\tedclustersl(H)\right]=O_P(\dhinterseq w_n)\; \end{aligned}$$ and this rate cannot be improved in general. # Proofs I - Extensions of vague convergence {#sec:technical-details} ## Consequences of the anticlustering conditions ### First and last jump decompositions {#sec:first-and-last-jump-decomposition} Recall the notation [\[eq:exceedance\]](#eq:exceedance){reference-type="eqref" reference="eq:exceedance"} for the exceedance times. Set $t(i)=t_1{(i)}$, $i=1,\ldots,N_1$. Note that for $i=1,\ldots,\dhinterseq$, $$\begin{aligned} %\label{eq:decomposition-first-jump} \{t(1)=i\}=\{\boldsymbol{X}_{1,i-1}^\ast\leqslant \tepseq, \norm{\boldsymbol{X}_{i}}>\tepseq\}\;, \ \ \{t{(N_1)}=i\}=\{\boldsymbol{X}_{i+1,\dhinterseq}^\ast\leqslant \tepseq, \norm{\boldsymbol{X}_{i}}>\tepseq\}\;, \end{aligned}$$ with the convention $\boldsymbol{X}_{1,0}^\ast\equiv 0$ and $\boldsymbol{X}_{\dhinterseq+1,\dhinterseq}^\ast\equiv 0$. These types of decompositions will play a crucial role. ### Conditional convergence of clusters {#sec:conditional-convergence-of-clusters} As a consequence of , for any bounded functional $H$ on $\ell_0(\mathbb{R}^d)$ that is almost surely continuous with respect to the distribution of the tail process $\boldsymbol{Y}$ and any $i_1,i_2,j_1,j_2,s\geqslant 0$, $$\begin{aligned} \lim_{n\to\infty}\mathbb E\left[H(\tepseq^{-1}(\boldsymbol{X}_{-[\dhinterseq s]-j_1,-i_1},\boldsymbol{X}_{i_2,j_2+[\dhinterseq s]})) \mid \norm{\boldsymbol{X}_0}>\tepseq\right]&= \mathbb E\left[H(\boldsymbol{Y}_{-\infty,-i_1},\boldsymbol{Y}_{i_2,\infty})\right]\;.\label{eq:tool-conditional-conv-rn-twosided}\end{aligned}$$ This type of convergence will be referred to as the *conditional convergence of clusters*. ### Vague convergence of clusters {#sec:integral-convergence-of-clusters} In view of the anticlustering condition [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"}, the following argument will be used repeatedly (cf. and the proof of Theorem 6.2.5 in [@kulik:soulier:2020]). Let $G$ be a bounded cluster functional such that $G(\boldsymbol{x})=0$ whenever $\boldsymbol{x}^\ast\leqslant 1$. Then $$\begin{aligned} \label{eq:convergence-series-anticlustering-1} \frac{1}{\dhinterseq}\sum_{i=1}^{\dhinterseq}\mathbb E\left[G(\boldsymbol{X}_{1-i,\dhinterseq-i}/\tepseq) \mathbbm{1}{\left\{\boldsymbol{X}_{1-i,-1}^\ast\leqslant \tepseq\right\}} \mid\norm{\boldsymbol{X}_0}>\tepseq\right] =\int_{0}^1 g_n(s)\mathrm{d}s\end{aligned}$$ with $$g_n(s)=\mathbb E\left[G(\boldsymbol{X}_{1-[\dhinterseq s],\dhinterseq-[\dhinterseq s]}/\tepseq) \mathbbm{1}{\left\{\boldsymbol{X}_{1-[\dhinterseq s],-1}^\ast\leqslant \tepseq\right\}} \mid\norm{\boldsymbol{X}_0}>\tepseq\right]$$ converging to $g(s):= \mathbb E[G(\boldsymbol{Y})\mathbbm{1}{\left\{\boldsymbol{Y}_{-\infty,-1}^\ast\leqslant 1\right\}}]$ for each $s\in (0,1)$. By the dominated convergence the limit of the expression in [\[eq:convergence-series-anticlustering-1\]](#eq:convergence-series-anticlustering-1){reference-type="eqref" reference="eq:convergence-series-anticlustering-1"} becomes $$\begin{aligned} %\label{eq:convergence-series-anticlustering} \mathbb E[G(\boldsymbol{Y})\mathbbm{1}{\left\{\boldsymbol{Y}_{-\infty,-1}^\ast\leqslant 1\right\}}]={\boldsymbol{\nu}}^*(G)\;.\end{aligned}$$ In the similar spirit, since $g_n(s):=\mathbb P(\boldsymbol{X}_{1-[\dhinterseq s],-1}^\ast\leqslant \tepseq)$ converges to $1$ for any $s>0$, we have for $\gamma\geqslant 0$, $$\begin{aligned} \label{eq:P-sum-smallvalues} \lim_{n\to\infty}\frac{1}{\dhinterseq}\sum_{i=1}^{\dhinterseq}\left(\frac{i}{\dhinterseq}\right)^\gamma\mathbb P(\boldsymbol{X}_{1-i,-1}^\ast\leqslant \tepseq)=\lim_{n\to\infty}\int_0^1s^\gamma g_n(s)\mathrm{d}s=\frac{1}{\gamma+1}\;.\end{aligned}$$ We will refer to this type of convergence as the *vague convergence of clusters*. ## Extensions of vague convergence: internal clusters {#sec:internal-clusters} In this section we consider a single block of size $\dhinterseq$. is valid for bounded functionals. The goal is to extend it to unbounded ones. In the small blocks scenario the uniform integrability holds and is still valid. See . It fails in the large blocks scenario. See . In particular, for the $\gamma$ moment of the cluster length, the dichotomous between the small and the large blocks scenario is related to whether $\dhinterseq^{\gamma+1}w_n\to 0$ or $\dhinterseq^{\gamma+1}w_n\to \infty$. All results are stated without a proof. See [@chen:kulik:2023a]. ### Moments of the cluster length - small blocks scenario **Lemma 20**. *Assume that [$\mathcal{S}^{(\gamma+\delta)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. Then for any $G\in\mathcal H(\delta)$ $$\begin{aligned} %\label{eq:cluster-length-gamma-power-UniformIntegrability-interior} \lim_{n\to \infty} \frac{1}{\dhinterseq w_n} \mathbb E\left[\mathcal L^\gamma(\boldsymbol{X}_{1,\dhinterseq}/\tepseq) G(\boldsymbol{X}_{1,\dhinterseq}/\tepseq)\mathbbm{1}{\left\{ A_1 \right\}}\right] = {\boldsymbol{\nu}}^*(G \mathcal L^\gamma)={\vartheta}\mathbb E\left[G(\boldsymbol{Z})\mathcal L^\gamma(\boldsymbol{Z}) \right]\;.\end{aligned}$$* **Remark 5**. * We note that $\mathbbm{1}{\left\{A_1\right\}}$ in the statement of can be replaced with $\mathbbm{1}{\left\{A_0^c\cap A_1\cap A_2^c\right\}}$. In other words, indicators of \"no large jumps\" can be easily added. * Furthermore, with help of , we can obtain the tail bound on the cluster length. **Corollary 21**. *Fix $a>0$. Assume that [$\mathcal{S}^{(\gamma+\delta)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. Then for any $G\in\mathcal H(\delta)$, $$\begin{aligned} \ \mathbb E[\mathbbm{1}{\left\{\mathcal L(\boldsymbol{X}_{1,\dhinterseq}/\tepseq)>a\dhinterseq\right\}}G(\boldsymbol{X}_{1,\dhinterseq}/\tepseq)]=O(w_n\dhinterseq^{1-\gamma})\;. \end{aligned}$$* *Proof.* For any $\gamma>0$ we can write $$\begin{aligned} \mathbb E[\mathbbm{1}{\left\{\mathcal L(\boldsymbol{X}_{1,\dhinterseq}/\tepseq)>a\dhinterseq\right\}}G(\boldsymbol{X}_{1,\dhinterseq}/\tepseq)] \leqslant \frac{1}{a^\gamma\dhinterseq^\gamma}\mathbb E[\mathcal L^{\gamma}(\boldsymbol{X}_{1,\dhinterseq})G(\boldsymbol{X}_{1,\dhinterseq}/\tepseq)\mathbbm{1}{\left\{A_1\right\}}]\;. \end{aligned}$$ Apply . ◻ ### Moments of the cluster length - large blocks scenario We recall that [$\mathcal{S}^{(\gamma)}(\dhinterseq, \tepseq)$](#SummabilityAC) is almost equivalent to $\dhinterseq^{\gamma+1}w_n\to 0$. Thus, we ask what happens if the latter condition is violated. **Lemma 22**. *Assume that $\boldsymbol{X}$ is stationary and $\ell$-dependent. Assume that [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} holds and $\dhinterseq^{\gamma+1}w_n\to \infty$. Then $$\begin{aligned} %\label{eq:moment-clusterlength-largeblocks} &\lim_{n\to\infty}\frac{\mathbb E\left[\mathcal L_1^\gamma \mathbbm{1}{\left\{A_1\right\}}\right]}{\dhinterseq^{\gamma+2} w_n^2}=\frac{1}{(\gamma+1)(\gamma+2)}{\vartheta}^2\;.\end{aligned}$$* We obtain immediately the following counterpart to . **Corollary 23**. *Fix $a>0$. Assume that $\boldsymbol{X}$ is stationary and $\ell$-dependent. Assume that [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} holds and $\dhinterseq^{\gamma+1}w_n\to \infty$. Then $$\begin{aligned} \mathbb P(\mathcal L(\boldsymbol{X}_{1,\dhinterseq})>a\dhinterseq)=O(\dhinterseq^2w_n^2)\;. \end{aligned}$$* ## Extensions of vague convergence: boundary clusters {#sec:technical-boundary} deals with convergence of cluster functionals, conditionally on the event $A_1=\{\boldsymbol{X}_{1,\dhinterseq}^\ast>\tepseq\}$. The present section deals with convergence of cluster functionals, conditionally on $A_1\cap A_2=\{\boldsymbol{X}_{1,\dhinterseq}^\ast>\tepseq,\boldsymbol{X}_{\dhinterseq+1,2\dhinterseq}^\ast>\tepseq\}$. We note the dichotomous behaviour, depending on the blocks size. ### Summary of the results {#sec:summary-boundary} - and give the rates of convergence for $\mathbb P(A_1\cap A_2)$ in the small blocks scenario. The rate is $w_n$, as opposite to the rate $\dhinterseq w_n$ for $\mathbb P(A_1\cap A_2)$. That is, a finite number of $\mathbb P(\norm{\boldsymbol{X}_{i_1}}>\tepseq,\norm{\boldsymbol{X}_{i_2}}>\tepseq)$, $i_1\in I_1,i_2\in I_2$, plays a role (even in the simple case of 1-dependence!). The required assumption on the blocks size is $\dhinterseq^2w_n\to 0$. - extend the previous results to unbounded functionals with a particular focus on cluster length. We note that the small blocks condition for the finite $\gamma$-moment of the cluster length is $\dhinterseq^{\gamma+2}w_n\to 0$. This should be compared to the situation of . There, the small blocks condition for the $\gamma$ moment of the cluster length is $\dhinterseq^{\gamma+1}w_n\to 0$. In other words, small block condition has a different meaning in the case of internal and boundary clusters. - deals with the large blocks scenario and bounded functionals - the blocks behave as if they were independent. The large blocks conditions is again $\dhinterseq^{\gamma+2}w_n\to \infty$ (with $\gamma=0$ for a bounded $H$). - extends to unbounded cluster functionals. To control $\gamma$ moments of the cluster length, the large blocks condition reads $\dhinterseq^{\gamma+2}w_n\to \infty$. The result should be compared to . In both cases of boundary and internal clusters, the jump locations behave as if they were independent. However, note again different large blocks condition: $\dhinterseq^{\gamma+1}w_n\to \infty$ for internal clusters and $\dhinterseq^{\gamma+2}w_n\to\infty$ for boundary clusters. The results of this section are new. All proofs are given in . ### Rates for boundary clusters - small blocks scenario **Proposition 24**. *Assume that [$\mathcal{S}^{(1)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. For any bounded $H\in \mathcal H$ we have $$\begin{aligned} \label{eq:pa1-cap-pa2-bounded} \lim_{n\to\infty}\frac{\mathbb E[H(\tepseq^{-1}\boldsymbol{X}_{1,2\dhinterseq})\mathbbm{1}{\left\{A_1\cap A_2\right\}}]}{w_n}= {\vartheta} \mathbb E\left[ (\mathcal L(\boldsymbol{Z}) - 1) H(\boldsymbol{Z}) \right]\;.\end{aligned}$$* **Corollary 25**. *Assume that [$\mathcal{S}^{(1)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. Then $$\begin{aligned} %\label{eq:mdep-P(a1-cap-a2)-smallblocks} \lim_{n\to\infty}\frac{\mathbb P(A_1\cap A_2)}{w_n}={\vartheta} \mathbb E\left[ (\mathcal L(\boldsymbol{Z}) - 1)\right]\;.\end{aligned}$$* The next result extends to unbounded functionals if the appropriate uniform integrability condition holds. **Proposition 26**. *Assume that [$\mathcal{S}^{(1)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. Let $H\in\mathcal H$. Assume that $$\begin{aligned} \label{eq:pa1-cap-pa2-uniform-integrability} \lim_{\ell\to\infty}\limsup_{n\to\infty}\frac{\mathbb E[ |H|(\tepseq^{-1}\boldsymbol{X}_{1,2\dhinterseq})\mathbbm{1}{\left\{A_1\cap A_2\right\}}\mathbbm{1}{\left\{|H|(\tepseq^{-1}\boldsymbol{X}_{1,2\dhinterseq})>\ell\right\}}]} {w_n}=0\;. \end{aligned}$$ Then [\[eq:pa1-cap-pa2-bounded\]](#eq:pa1-cap-pa2-bounded){reference-type="eqref" reference="eq:pa1-cap-pa2-bounded"} holds.* **Proposition 27**. *Assume that [$\mathcal{S}^{(\gamma+1)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. Let $H\in\mathcal H(\gamma)$. Then the uniform integrability condition [\[eq:pa1-cap-pa2-uniform-integrability\]](#eq:pa1-cap-pa2-uniform-integrability){reference-type="eqref" reference="eq:pa1-cap-pa2-uniform-integrability"} holds.* #### Joint cluster length. Recall that notation [\[eq:boundary-full-cluster-length\]](#eq:boundary-full-cluster-length){reference-type="eqref" reference="eq:boundary-full-cluster-length"} for the joint cluster length $\mathcal{L}_{j,j+1} = t_{j+1}{(N_{j+1})} - t_{j}(1) + 1$ as well as [\[eq:boundary-full-jump-times\]](#eq:boundary-full-jump-times){reference-type="eqref" reference="eq:boundary-full-jump-times"} for the jump times in the adjacent blocks. The following result is a counterpart to and follows directly from . This time we study the behaviour of the cluster length $\mathcal L_{1,2}=\mathcal L(\tepseq^{-1}\boldsymbol{X}_{1,2\dhinterseq})$ defined as a function of two adjacent blocks. Note the rate change as compared to , primarily due to the rates in in the small blocks scenario. **Corollary 28**. *Assume that [$\mathcal{S}^{(\gamma+\delta+1)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. For any $G\in \mathcal H(\delta)$ we have, $$\begin{aligned} %\label{eq:cluster-length-gamma-power-UniformIntegrability-boundary} \lim_{n\to \infty} \frac{\mathbb E\left[\mathcal L^\gamma(\tepseq^{-1}\boldsymbol{X}_{1,2\dhinterseq})G(\tepseq^{-1}\boldsymbol{X}_{1,2\dhinterseq})\mathbbm{1}{\left\{ A_1\cap A_2 \right\}} \right]}{w_n} = {\vartheta}\mathbb E\left[(\mathcal L(\boldsymbol{Z}) - 1 )\mathcal L^\gamma(\boldsymbol{Z})G(\boldsymbol{Z}) \right]\;.\end{aligned}$$* We also state the next corollary to , for a future use. Below, [\[eq;convergence-boundary-cluster-H-full\]](#eq;convergence-boundary-cluster-H-full){reference-type="eqref" reference="eq;convergence-boundary-cluster-H-full"} is obvious. However, the remaining statements require a short argument. **Corollary 29**. *Assume that $H\in\mathcal H(\gamma)$. Under the condition  [$\mathcal{S}^{(\gamma+1)}(\dhinterseq, \tepseq)$](#SummabilityAC), we have* *$$\begin{aligned} &\lim_{n\to\infty}\frac{\mathbb E\left[ H \left( \tepseq^{-1}\boldsymbol{X}_{1,2\dhinterseq}\right) \mathbbm{1}{\left\{ A_1\cap A_2 \right\}} \right] } {w_n}= {\vartheta}\mathbb E\left[ (\mathcal L(\boldsymbol{Z})-1)H(\boldsymbol{Z})\right]\;, \label{eq;convergence-boundary-cluster-H-full} \\ & \lim_{n\to\infty}\frac{ \mathbb E\left[ H \left( \tepseq^{-1}\boldsymbol{X}_{1,\dhinterseq} \right) \mathbbm{1}{\left\{ A_1\cap A_2 \right\}} \right] } {w_n} ={\vartheta}\mathbb E\left[\sum_{j=1}^{\mathcal L(\boldsymbol{Z})-1} H(\boldsymbol{Z}_{0,j-1}) \right]\;, \label{eq;joint-convergence-boundary-cluster;H-front} \\ & \lim_{n\to\infty}\frac{ \mathbb E\left[ H \left( \tepseq^{-1} \boldsymbol{X}_{\dhinterseq+1,2\dhinterseq} \right) \mathbbm{1}{\left\{ A_1\cap A_2 \right\}} \right] } {w_n} = {\vartheta}\mathbb E\left[\sum_{j=1}^{\mathcal L(\boldsymbol{Z})-1} H(\boldsymbol{Z}_{j,\infty}) \right]\;. \label{eq;joint-convergence-boundary-cluster;H-back}\end{aligned}$$* In the spirit of , we obtain the tail asymptotic for the joint cluster length. Note that $\mathcal L_{1,2}\geqslant \dhinterseq$ implies $\mathbbm{1}{\left\{A_1\cap A_2\right\}}=1$. **Corollary 30**. *Assume that [$\mathcal{S}^{(\gamma+\delta+1)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. For any $G\in \mathcal H(\delta)$ we have, $$\begin{aligned} \mathbb E\left[\mathbbm{1}{\left\{\mathcal L(\tepseq^{-1}\boldsymbol{X}_{1,2\dhinterseq})\geqslant \dhinterseq\right\}}G(\tepseq^{-1}\boldsymbol{X}_{1,2\dhinterseq}) \right]=O(w_n\dhinterseq^{-\gamma})\;. \end{aligned}$$* ### Rates for boundary clusters - large blocks scenario gives the rate for $\mathbb P(A_1\cap A_2)$ in the small blocks scenario. The next lemma deals with large blocks - the blocks behave as if they were independent (recall that $\mathbb P(A_1)\sim {\vartheta}\dhinterseq w_n$). Recall also that bounded $H$ corresponds to $\mathcal H(0)$, so the large blocks condition reads $\dhinterseq^{\gamma+2}w_n\to 0$. **Lemma 31**. *Assume that $\boldsymbol{X}$ is stationary and $\ell$-dependent. Assume that [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} holds and $\dhinterseq^{2}w_n\to \infty$. Then for any bounded $H\in \mathcal H$, $$\begin{aligned} \label{eq:mdep-P(a1-cap-a2)-largeblocks} \limsup_{n\to\infty}\frac{\mathbb E[H(\tepseq^{-1}\boldsymbol{X}_{1,2\dhinterseq})\mathbbm{1}{\left\{A_1\cap A_2\right\}}]}{\dhinterseq^2 w_n^2}<\infty\;.\end{aligned}$$* **Remark 6**. * We will show in that the rate in [\[eq:mdep-P(a1-cap-a2)-largeblocks\]](#eq:mdep-P(a1-cap-a2)-largeblocks){reference-type="eqref" reference="eq:mdep-P(a1-cap-a2)-largeblocks"} is sharp. * ### Boundary clusters for the toy example - MMA(1) {#sec:MMA(1)} Boundary clusters seem to be harder to handle with in a large blocks scenario. Hence, we illustrate the precise rates for the toy example MMA(1) introduced in . Proofs are given in . In we obtained a precise asymtptotics for $\mathbb P(A_1\cap A_2)$ in the small block case, while gives a bound in the large block case. Here, we provide additionally the precise asymptotics in the large blocks case, showing that the rate in [\[eq:mdep-P(a1-cap-a2)-largeblocks\]](#eq:mdep-P(a1-cap-a2)-largeblocks){reference-type="eqref" reference="eq:mdep-P(a1-cap-a2)-largeblocks"} is sharp. The proof also shows precisely where does the small blocks rate $w_n$ come from. **Lemma 32**. *Consider the MMA(1) process. Then* - *If $\dhinterseq^2 w_n\to 0$, then $$\begin{aligned} %\label{eq:MMA1-P(a1-cap-a2)-smallblocks} \lim_{n\to\infty}\frac{1}{w_n}\mathbb P(A_1\cap A_2)=\frac{(c_0\wedge c_1)^\alpha}{c_0^\alpha+c_1^\alpha}=\mathbb P(Y_1>1)\;.\end{aligned}$$* - *If $\dhinterseq^2 w_n\to \infty$, then $$\begin{aligned} %\label{eq:MMA1-P(a1-cap-a2)-largeblocks} \lim_{n\to\infty}\frac{1}{\dhinterseq^2w_n^2}\mathbb P(A_1\cap A_2)=\left(\frac{(c_0\vee c_1)^\alpha}{c_0^\alpha+c_1^\alpha}\right)^2={\vartheta}^2\;.\end{aligned}$$* The next result gives the precise asymptotics for the cluster length based on two blocks. In the large blocks scenario, the result should be compared to . In the current situation as well as in the situation discussed in the aforementioned lemma, the jump locations behave as if they were independent. Note also different small and large blocks condition, when comparing to the lemma below. **Lemma 33**. *Consider the MMA(1) process.* - *If $\dhinterseq^{\gamma+2}w_n\to 0$, then $$\begin{aligned} \lim_{n\to\infty}\frac{1}{w_n}\mathbb E\left[\left(t_{2}{(N_2)}-t_1{(1)}\right)^\gamma\mathbbm{1}{\left\{A_1 \cap A_2^c\right\}}\right]=\frac{(c_0\wedge c_1)^\alpha}{c_0^\alpha+c_1^\alpha}=\mathbb P(Y_1>1)\;. \end{aligned}$$* - *If $\dhinterseq^{\gamma+2}w_n\to \infty$, then $$\begin{aligned} \lim_{n\to\infty}\frac{1}{\dhinterseq^{\gamma+2}w_n^2}\mathbb E\left[\left(t_{2}{(N_2)}-t_1{(1)}\right)^\gamma\mathbbm{1}{\left\{A_1 \cap A_2^c\right\}}\right]=\frac{2^{\gamma+2}-1}{(\gamma+1)(\gamma+2)}{\vartheta}^2\;. \end{aligned}$$* Finally, we will need a specific lemma for the boundary clusters. **Lemma 34**. *Consider the MMA(1) process. Then $$\begin{aligned} \lim_{n\to\infty}\frac{1}{\dhinterseq^3 w_n^2}\mathbb E\left[\mathbbm{1}{\left\{A_1\cap A_2\right\}}\left((t_2{(1)}-t_{1}{(N_1)})-\dhinterseq\right)_+\right]=\frac{1}{6}{\vartheta}^2\;.\end{aligned}$$* ### Proofs for boundary clusters {#sec:proof-for-boundary} *Proof of .* Fix an integer $\ell$. For any $n$ such that $\dhinterseq > \ell$, it follows that $$\begin{aligned} \label{eq:boundary-clusters-asymptotics} &\mathbb E\left[ H(\tepseq^{-1}\boldsymbol{X}_{1,2\dhinterseq}) \mathbbm{1}{\left\{ A_1\cap A_2 \right\}} \right] =\sum_{i_1=1}^{\dhinterseq}\sum_{i_2=\dhinterseq+1}^{2\dhinterseq} \mathbb E\left[H(\tepseq^{-1}\boldsymbol{X}_{i_1,i_2})\mathbbm{1}{\left\{t_{1}(1) = i_1, t_{2}{(N_2)} = i_2\right\}}\right]\nonumber\\ & =\sum_{ \substack{ \dhinterseq - \ell + 1 \leqslant i_1 \leqslant \dhinterseq \\ \dhinterseq+1 \leqslant i_2 \leqslant \dhinterseq+\ell} } \mathbb E\left[H(\tepseq^{-1}\boldsymbol{X}_{i_1,i_2})\mathbbm{1}{\left\{t_{1}(1) = i_1, t_{2}{(N_2)} = i_2\right\}}\right] \nonumber\\ &\phantom{=} + \sum_{ \text{other pairs } (i_1,i_2) } \mathbb E\left[H(\tepseq^{-1}\boldsymbol{X}_{i_1,i_2})\mathbbm{1}{\left\{t_{1}(1) = i_1, t_{2}{(N_2)} = i_2\right\}}\right] = : J(H,\dhinterseq;\ell)+ \widetilde{J}(H,\dhinterseq;\ell)\;.\end{aligned}$$ For $J(H,\dhinterseq;\ell)$, we have first by shifting by $i_1$, then by changing the variables $i_1$ into $i_1-\dhinterseq$ and $i_2$ into $i_2-\dhinterseq$: $$\begin{aligned} & J(H,\dhinterseq;\ell) \\ &=\sum_{ \substack{ \dhinterseq - \ell + 1 \leqslant i_1 \leqslant \dhinterseq \\ \dhinterseq+1 \leqslant i_2 \leqslant \dhinterseq+\ell} }\mathbb E\left[H(\tepseq^{-1}\boldsymbol{X}_{i_1,i_2}) \mathbbm{1}{\left\{\boldsymbol{X}^\ast_{ 1,i_1 - 1 } \leqslant \tepseq , |\boldsymbol{X}_{i_1}| > \tepseq,|\boldsymbol{X}_{i_2}| > \tepseq , \boldsymbol{X}^\ast_{i_2+1, 2\dhinterseq} \leqslant \tepseq\right\}}\right] \\ &= \sum_{ \substack{ - \ell + 1 \leqslant i_1 \leqslant 0 \\ 1 \leqslant i_2 \leqslant \ell} } \mathbb E\left[H(\tepseq^{-1}\boldsymbol{X}_{0,i_2-i_1}) \mathbbm{1}{\left\{\boldsymbol{X}^\ast_{ 1-\dhinterseq - i_1, - 1 } \leqslant \tepseq , |\boldsymbol{X}_0|>\tepseq,|\boldsymbol{X}_{i_2 - i_1}| > \tepseq, \boldsymbol{X}^\ast_{i_2 - i_1 + 1, \dhinterseq - i_1} \leqslant \tepseq\right\}} \right] \\ &=w_n\sum_{\substack{- \ell + 1 \leqslant j \leqslant 0 \\1-j\leqslant i\leqslant \ell-j} } \mathbb E\left[H(\tepseq^{-1}\boldsymbol{X}_{0,i})\mathbbm{1}{\left\{\boldsymbol{X}^\ast_{1-\dhinterseq-j,-1}\leqslant \tepseq,\norm{\boldsymbol{X}_{i}}>\tepseq,\boldsymbol{X}^\ast_{i+1,\dhinterseq-j}\leqslant \tepseq\right\}}\mid \norm{\boldsymbol{X}_0}>\tepseq\right]\;.\end{aligned}$$ For each $j$, using the conditional convergence of clusters (cf. [\[eq:tool-conditional-conv-rn-twosided\]](#eq:tool-conditional-conv-rn-twosided){reference-type="eqref" reference="eq:tool-conditional-conv-rn-twosided"}) and the process $\boldsymbol{Z}$, $$\begin{aligned} &\lim_{n\to\infty}\mathbb E\left[H(\tepseq^{-1}\boldsymbol{X}_{0,i})\mathbbm{1}{\left\{\boldsymbol{X}^\ast_{1-\dhinterseq-j,-1}\leqslant \tepseq,\norm{\boldsymbol{X}_{i}}>\tepseq,\boldsymbol{X}^\ast_{i+1,\dhinterseq-j}\leqslant \tepseq\right\}}\mid \norm{\boldsymbol{X}_0}>\tepseq\right]\\ &= \mathbb E\left[H(\boldsymbol{Y}_{0,i})\mathbbm{1}{\left\{\boldsymbol{Y}^\ast_{-\infty,-1}\leqslant 1,\norm{\boldsymbol{Y}_i}>1,\boldsymbol{Y}^\ast_{i+1,\infty}\leqslant 1\right\}}\right]={\vartheta} \mathbb E\left[H(\boldsymbol{Z}_{0,i}) \mathbbm{1}{\left\{\mathcal L(\boldsymbol{Z})= i +1\right\}}\right]\end{aligned}$$ and hence $$\begin{aligned} &\lim_{n\to\infty}\frac{J(H,\dhinterseq;\ell)}{w_n}\\ &= {\vartheta}\sum_{i = 1}^{\ell} i \mathbb E\left[H(\boldsymbol{Z}_{0,i}) \mathbbm{1}{\left\{\mathcal L(\boldsymbol{Z})= i +1\right\}}\right] + {\vartheta}\sum_{i = \ell+1}^{2\ell-1}(2\ell-i)\mathbb E\left[H(\boldsymbol{Z}_{0,i}) \mathbbm{1}{\left\{\mathcal L(\boldsymbol{Z})= i +1\right\}}\right]\\ & =:J_1(\ell)+J_2(\ell) \; .\end{aligned}$$ We note that $$\begin{aligned} \lim_{\ell\to\infty} J_1(\ell)={\vartheta} \mathbb E\left[ (\mathcal L(\boldsymbol{Z}) - 1) H(\boldsymbol{Z}_{0,\mathcal L(\boldsymbol{Z})-1}) \right]= {\vartheta} \mathbb E\left[ (\mathcal L(\boldsymbol{Z}) - 1) H(\boldsymbol{Z}) \right]\end{aligned}$$ and since $H$ is bounded we have $$\begin{aligned} \lim_{\ell\to\infty}J_2(\ell)&\leqslant {\vartheta}\|H\|\lim_{\ell\to\infty} \sum_{i=\ell}^\infty i\mathbb P\left(\mathcal L(\boldsymbol{Z})= i \right) =\|H\|\lim_{\ell\to\infty} \sum_{i=\ell}^\infty i\mathbb P\left(\mathcal L(\boldsymbol{Y})= i,\boldsymbol{Y}_{-\infty,-1}^\ast\leqslant 1 \right)\\ &\leqslant \|H\|\lim_{\ell\to\infty} \sum_{i=\ell}^\infty i\mathbb P\left(\norm{\boldsymbol{Y}_i}>1 \right) =0\end{aligned}$$ on account of [$\mathcal{S}^{(1)}(\dhinterseq, \tepseq)$](#SummabilityAC); cf. the first part of . For $\widetilde{J}(H,\dhinterseq;\ell)$, we note that $$\begin{aligned} \widetilde{J}(H,\dhinterseq;\ell) \leqslant & \|H\|\sum_{ i=\ell } ^{2\dhinterseq-1} i \mathbb P\left( |\boldsymbol{X}_0 | > \tepseq, |\boldsymbol{X}_i| > \tepseq \right)\end{aligned}$$ and hence due to the condition [$\mathcal{S}^{(1)}(\dhinterseq, \tepseq)$](#SummabilityAC), $$\begin{aligned} \lim_{\ell\to \infty} \lim_{n\to \infty} \frac{\widetilde{J}(H,\dhinterseq;\ell)}{w_n} = 0\;.\end{aligned}$$ ◻ *Proof of .* We proceed in a slightly different way as compared to . We use the last jump decomposition in the first block, then the first jump decomposition in the second block, and combine this with stationarity to obtain $$\begin{aligned} \mathbb P\left( A_1 \cap A_2 \right) &= \sum_{j_1=1}^{\dhinterseq} \sum_{j_2=\dhinterseq+1} ^ {2\dhinterseq} \mathbb P\left( \norm{\boldsymbol{X}_0} > \tepseq, \boldsymbol{X}_{1,j_2 - j_1 -1}^* \leqslant \tepseq, \norm{\boldsymbol{X}_{j_2 - j_1}} > \tepseq \right)\;. \label{eq:pr-a1-a2} \\ &\leqslant w_n\sum_{i=1}^{2\dhinterseq-1} \left(i \wedge (2\dhinterseq-i)\right) \mathbb P\left( \tau_n = i \mid |\boldsymbol{X}_0| > \tepseq \right)\;, \nonumber\end{aligned}$$ where $\tau_n$ is the random variable $$\begin{aligned} \tau_n := \inf \{ t \geqslant 1: |\boldsymbol{X}_t| > \tepseq \}\;.\end{aligned}$$ Fix an integer $\ell\geqslant 1$ (we can assume that $\ell\leqslant \dhinterseq$) and split $$\begin{aligned} \mathbb P(A_1\cap A_2)&\leqslant w_n\sum_{i=1}^{\ell} i\cdot \mathbb P\left( \tau_n = i \mid |\boldsymbol{X}_0| > \tepseq \right)+ w_n\sum_{i=\ell+1}^{\dhinterseq} i\cdot \mathbb P\left( \tau_n = i \mid |\boldsymbol{X}_0| > \tepseq \right)\\ &\phantom{=}+w_n\sum_{i=\dhinterseq+1}^{2\dhinterseq} (2\dhinterseq-i) \mathbb P\left( \tau_n = i \mid |\boldsymbol{X}_0| > \tepseq \right)=:J_1(\ell)+J_2(\ell,\dhinterseq)+J_3(\dhinterseq)\;.\end{aligned}$$ For each finite $i\ge 1$ we have $$\begin{aligned} \lim_{n\to\infty} \mathbb P\left( \tau_n = i \mid |\boldsymbol{X}_0| > \tepseq \right)= \mathbb P(\boldsymbol{Y}_{1,i-1}^*\leqslant 1,|\boldsymbol{Y}_i|>1)\;\end{aligned}$$ and hence $J_1(\ell)=O(w_n)$. In case of $\ell$-dependence we have immediately $$\begin{aligned} J_2(\ell,\dhinterseq)+J_3(\dhinterseq)\leqslant w_n\left\{ \sum_{i=\ell+1}^{\dhinterseq} i+\sum_{i=\dhinterseq+1}^{2\dhinterseq}(2\dhinterseq-i) \right\}\mathbb P(\norm{\boldsymbol{X}_{i}}>\tepseq\mid \norm{\boldsymbol{X}_0}>\tepseq)=O(\dhinterseq^2 w_n^2)\;\end{aligned}$$ and the latter rate dominates $w_n$ in the large blocks scenario. ◻ *Proof of .* We mimic the proof of . Recall the term $J(H,\dhinterseq;\ell)$ in [\[eq:boundary-clusters-asymptotics\]](#eq:boundary-clusters-asymptotics){reference-type="eqref" reference="eq:boundary-clusters-asymptotics"}. On the event $\{t_{2}{(N_2)}-t_1{(1)}>\ell/2\}\subseteq\{\mathcal L_{1,2}>\ell/2\}$, we have $J(\dhinterseq;\ell/2)\equiv0$. Since $H\in \mathcal H(\gamma)$, we bound $|H|(\tepseq^{-1}\boldsymbol{X}_{1,\dhinterseq})\leqslant C_H(t_{N_1}-t_1)^\gamma$. Assume without loss of generality that $C_H=1$. Recall again $\widetilde{J}(H,\dhinterseq;\ell)$ defined in [\[eq:boundary-clusters-asymptotics\]](#eq:boundary-clusters-asymptotics){reference-type="eqref" reference="eq:boundary-clusters-asymptotics"}. It follows that $$\begin{aligned} &\mathbb E\left[ |H|(\tepseq^{-1}\boldsymbol{X}_{1,2\dhinterseq})\mathbbm{1}{\left\{|H|(\tepseq^{-1}\boldsymbol{X}_{1,2\dhinterseq})>\ell/2\right\}} \mathbbm{1}{\left\{ A_1\cap A_2 \right\}} \right]\\ &\leqslant \mathbb E\left[ \mathcal L_{1,2}^\gamma\mathbbm{1}{\left\{\mathcal L_{1,2}>(\ell/2)^{\gamma/2}\right\}} \mathbbm{1}{\left\{ A_1\cap A_2 \right\}} \right] \leqslant\widetilde{J}(\mathcal L^\gamma,\dhinterseq;(\ell/2)^{\gamma/2})\;.\end{aligned}$$ We have $$\begin{aligned} \widetilde{J}(H,\dhinterseq;(\ell/2)^{\gamma/2}) \leqslant & \sum_{ i=\ell } ^{2\dhinterseq-1} i^{\gamma+1} \mathbb P\left( |\boldsymbol{X}_0 | > \tepseq, |\boldsymbol{X}_i| > \tepseq \right)\end{aligned}$$ and hence due to the condition [$\mathcal{S}^{(\gamma+1)}(\dhinterseq, \tepseq)$](#SummabilityAC), $$\begin{aligned} \lim_{\ell\to \infty} \lim_{n\to \infty} \frac{\widetilde{J}(H,\dhinterseq;(\ell/2)^{\gamma/2})}{w_n} = 0\;.\end{aligned}$$ The proof of the uniform integrability is finished. ◻ *Proof of .* We will only show [\[eq;joint-convergence-boundary-cluster;H-front\]](#eq;joint-convergence-boundary-cluster;H-front){reference-type="eqref" reference="eq;joint-convergence-boundary-cluster;H-front"}. We assume that $H$ is bounded. The unbounded case follows from . For a bounded case, we proceed in a slightly different manner as compared to . Fix an integer $\ell>0$. For any $n$ such that $\dhinterseq > \ell$, it follows that $$\begin{aligned} &\mathbb E\left[ H \left(\tepseq^{-1}\boldsymbol{X}_{1,\dhinterseq} \right) \mathbbm{1}{\left\{ A_1\cap A_2 \right\}} \right] = \sum_{ i_1 = \dhinterseq +1 - \ell} ^ {\dhinterseq} \mathbb E\bigg[ \mathbbm{1}{\left\{t_{1}(1) = i_1, \mathcal L_{1,2} \geqslant \dhinterseq + 2 - i_1\right\}} H( \tepseq^{-1} \boldsymbol{X}_{i_1,\dhinterseq} ) \bigg] \\ & + \sum_{ i_1 = 1 } ^ {\dhinterseq-\ell} \mathbb E\bigg[ \mathbbm{1}{\left\{t_{1}(1) = i_1, \mathcal L_{1,2} \geqslant \dhinterseq + 2 - i_1\right\}} H( \tepseq^{-1} \boldsymbol{X}_{i_1,\dhinterseq} ) \bigg] = : J(\dhinterseq;\ell)+\widetilde{J}(\dhinterseq;\ell)\;. \end{aligned}$$ We have $$\begin{aligned} J(\dhinterseq;\ell)&=w_n\sum_{i_1=\dhinterseq+1-\ell}^{\dhinterseq}\mathbb E[\mathbbm{1}{\left\{\boldsymbol{X}_{1-i_1,-1}^\ast\leqslant \tepseq,\mathcal L_{1,2} \geqslant \dhinterseq + 2 - i_1\right\}}H(\tepseq^{-1}\boldsymbol{X}_{0,\dhinterseq-i_1})\mid \norm{\boldsymbol{X}_0}>\tepseq]\\ &=w_n\sum_{i=1}^\ell \mathbb E[\mathbbm{1}{\left\{\boldsymbol{X}^\ast_{1-(\dhinterseq+i-\ell),-1}\leqslant\tepseq,\mathcal L_{1,2}\geqslant 2-i+\ell\right\}} H(\tepseq^{-1}\boldsymbol{X}_{0,\ell-i})\mid \norm{\boldsymbol{X}_0}>\tepseq]\;. \end{aligned}$$ Since $H$ is bounded, the conditional convergence of clusters argument yields $$\begin{aligned} &\lim_{\ell\to\infty}\lim_{n\to\infty}\frac{J(\dhinterseq;\ell)}{w_n}=\lim_{\ell\to\infty}\sum_{i=1}^\ell \mathbb E[\mathbbm{1}{\left\{\boldsymbol{Y}_{-\infty,-1}^\ast\leqslant 1,\mathcal L(\boldsymbol{Y})\geqslant 2-i+\ell\right\}}H(\boldsymbol{Y}_{0,\ell-i})]\\ &=\lim_{\ell\to\infty}\sum_{j=2}^{\ell+1}\mathbb E[\mathbbm{1}{\left\{\boldsymbol{Y}_{-\infty,-1}^\ast\leqslant 1,\mathcal L(\boldsymbol{Y})\geqslant j\right\}}H(\boldsymbol{Y}_{0,j-2})] \\&=\sum_{j=2}^\infty \mathbb E[\mathbbm{1}{\left\{\boldsymbol{Y}_{-\infty,-1}^\ast\leqslant 1,\mathcal L(\boldsymbol{Y})\geqslant j\right\}}H(\boldsymbol{Y}_{0,j-2})]\\ &= \mathbb E\left[\mathbbm{1}{\left\{\boldsymbol{Y}_{-\infty,-1}^\ast\leqslant 1\right\}}\sum_{j=2}^{\mathcal L(\boldsymbol{Y})}H(\boldsymbol{Y}_{0,j-2})\right]={\vartheta} \mathbb E\left[\sum_{j=2}^{\mathcal L(\boldsymbol{Z})}H(\boldsymbol{Z}_{0,j-2})\right] \;.\end{aligned}$$ Next, $$\begin{aligned} \widetilde{J}(\dinterseq;\ell)\leqslant \|H\| \sum_{i = \ell}^{2\dhinterseq} i \cdot \mathbb P( | \boldsymbol{X}_0 | > \tepseq , | \boldsymbol{X}_i | > \tepseq ) \end{aligned}$$ and the term vanishes on account of the anticlustering condition. ◻ *Proof of .* As in the proof of , we proceed with the last jump decomposition in the first block, followed by the first jump decomposition in the second block. We have $$\begin{aligned} \mathbb P(A_1\cap A_2)&= \sum_{j_1=1}^{\dhinterseq}\sum_{j_2=\dhinterseq+1}^{2\dhinterseq}\mathbb P(t_1(1)=j_1,t_2(N_2)=j_2)\\ &= \sum_{j_1=1}^{\dhinterseq}\sum_{j_2=\dhinterseq+1}^{2\dhinterseq}\mathbb P(\boldsymbol{X}_{1,j_1-1}^*\leq \tepseq,X_{j_1}>\tepseq,X_{j_2}>\tepseq,\boldsymbol{X}_{j_2+1,2\dhinterseq}^*\leq \tepseq)\;. \end{aligned}$$ By 1-dependence we have $$\begin{aligned} \label{eq:proof-for-mma1-1} \mathbb P(A_1\cap A_2) &= \sum_{j_1=1}^{\dhinterseq}\sum_{j_2=\dhinterseq+1}^{2\dhinterseq}\mathbb P(\boldsymbol{X}_{1,j_1-1}^*\leq \tepseq,X_{j_1}>\tepseq)\mathbb P(X_{j_2}>\tepseq,\boldsymbol{X}_{j_2+1,2\dhinterseq}^*\leq \tepseq)\nonumber\\ &\phantom{=}-w_n^2\mathbb P(\boldsymbol{X}_{1-\dhinterseq,-1}^*\leq \tepseq \mid X_{0}>\tepseq)\mathbb P(\boldsymbol{X}_{1,\dhinterseq-1}^*\leq \tepseq\mid X_{0}>\tepseq)\nonumber\\ &\phantom{=}+\mathbb P(\boldsymbol{X}_{1,\dhinterseq-1}^*\leq \tepseq,X_{\dhinterseq}>\tepseq,X_{\dhinterseq+1}>\tepseq,\boldsymbol{X}_{\dhinterseq+2,2\dhinterseq}^*\leq \tepseq)\nonumber\\ &=:J_1(\dhinterseq)-J_2(\dhinterseq)+J_3(\dhinterseq)\;.\end{aligned}$$ The vague convergence of clusters gives $J_1(\dhinterseq)\sim {\vartheta}^2 \dhinterseq^2 w_n^2$. Next, the conditional convergence of clusters gives $J_2(\dhinterseq)\sim {\vartheta}^2 w_n^2$ as $n\to\infty$. Finally, $$\begin{aligned} &J_3(\dhinterseq)\sim \mathbb P(c_1\xi_{\dhinterseq}\leqslant \tepseq,c_0\xi_{\dhinterseq}\vee c_1\xi_{\dhinterseq+1}>\tepseq,c_0\xi_{\dhinterseq+1}\vee c_1\xi_{\dhinterseq+2}>\tepseq,c_0\xi_{\dhinterseq+2}\leqslant \tepseq)\nonumber\\ &\sim \mathbb P((c_0\wedge c_1)\xi_{\dhinterseq+1}>\tepseq) \sim w_n \mathbb P(Y_1>1)\;.\label{eq:j3-mma1}\end{aligned}$$ In summary, the term $J_2(\dhinterseq)$ does not contribute, while the terms $J_1(\dhinterseq)$ and $J_3(\dhinterseq)$ dominate in the large and the small blocks scenario, respectively. ◻ *Proof of .* We use the notation from [\[eq:proof-for-mma1-1\]](#eq:proof-for-mma1-1){reference-type="eqref" reference="eq:proof-for-mma1-1"} to get $$\begin{aligned} & \mathbb E\left[\left(t_2{(N_2)}-t_1{(1)}\right)^\gamma\mathbbm{1}{\left\{A_1^c \cap A_2^c\right\}}\right]\\ & = \sum_{j_1=1}^{\dhinterseq} \sum_{j_2=\dhinterseq+1} ^ {2\dhinterseq} (j_2-j_1)^\gamma\mathbb P(\boldsymbol{X}_{1,j_1-1}^*\leqslant \tepseq, X_{j_1}> \tepseq, X_{j_2} > \tepseq,\boldsymbol{X}_{j_2+1,2\dhinterseq}^*\leqslant \tepseq)\\ &= w_n^2\sum_{j_1=1}^{\dhinterseq} \sum_{j_2=\dhinterseq+2} ^ {2\dhinterseq} (j_2-j_1)^\gamma\mathbb P(\boldsymbol{X}_{1-j_1,-1}^*\leqslant \tepseq\mid X_{0}> \tepseq) \mathbb P(\boldsymbol{X}_{1,2\dhinterseq-j_1}^*\leqslant \tepseq\mid X_0>\tepseq)\\ &\phantom{=}-J_2(\dhinterseq)+J_3(\dhinterseq)=:w_n^2J_0(\dhinterseq)-J_2(\dhinterseq)+J_3(\dhinterseq)\;. \end{aligned}$$ For $J_0(\dhinterseq)$ we have as $n\to\infty$, $$\begin{aligned} &w_n^2 J_0(\dhinterseq)\sim {\vartheta}^2 w_n^2 \sum_{j=1}^{2\dhinterseq}j^\gamma[ (2\dhinterseq-j)\wedge j]\sim {\vartheta}^2 w_n^2 \dhinterseq^{\gamma+2} \left[\int_0^1 s^{\gamma+1}\mathrm{d}s+\int_1^2 s^\gamma(2-s)\mathrm{d}s\right]\\ & \sim\frac{2^{\gamma+2}-1}{(\gamma+1)(\gamma+2)} {\vartheta}^2w_n^2 \dhinterseq^{\gamma+2}\;. \end{aligned}$$ Combining this with [\[eq:j3-mma1\]](#eq:j3-mma1){reference-type="eqref" reference="eq:j3-mma1"} and negligibility of $J_2(\dhinterseq)$, we conclude the proof. ◻ *Proof of .* The expectation of interest is $$\begin{aligned} &\sum_{j_1=1}^{\dhinterseq}\sum_{j_2=\dhinterseq+j_1}^{2\dhinterseq}(j_2-j_1-\dhinterseq)\mathbb P(X_{j_1}>\tepseq,\boldsymbol{X}_{j_1+1,\ldots,j_2-1}^*\leqslant\tepseq,X_{j_2}>\tepseq)\\ &=\mathbb P(c_0\xi_0\vee c_1\xi_1>\tepseq, c_0\xi_1\leqslant \tepseq)\\ &\phantom{=}\times \mathbb P(c_1\xi_{0}\leqslant \tepseq,c_0\xi_{0}\vee c_1\xi_{1}>\tepseq)\sum_{i=\dhinterseq+1}^{2\dhinterseq-i}(i-\dhinterseq)(2\dhinterseq-i)\mathbb P(D_{2,i-1})\\ &\sim \left(\frac{(c_0\vee c_1)^\alpha}{c_0^\alpha+c_1^\alpha}\right)^2\int_1^2 (s-1)(2-s)\mathrm{d}s=\frac{1}{6}{\vartheta}^2\;.\end{aligned}$$ ◻ # Proofs II - Limit theorems for internal and boundary clusters statistics {#sec:technical-details-block-statistics} In this section we apply the theory established in to particular functionals that appear in the context of the asymptotic expansion of blocks statistics. As a result, we obtain convergence in probability for $\mathcal{IC}(H)$, $\mathcal{BC}(H;1)$ and $\mathcal{BC}(H;2)$. ## Moments of clusters - small blocks {#sec:moments-of-clusters} ### Internal Clusters {#subsubsec:mean-interior-clusters} Recall the notation [\[eq:internal-as-functional\]](#eq:internal-as-functional){reference-type="eqref" reference="eq:internal-as-functional"} for $\mathcal{IC}_j(H)$. We analyse the moments of $\mathcal{IC}_1$. Since one block is involved and $\widetilde{H}_{\mathcal{IC}}$ is tight under the conditional law, the scaling is $\dhinterseq w_n$. Let $p\ge 0$. **Corollary 35**. *Assume that [$\mathcal{S}^{(p(\gamma+1))}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. For any $H\in\mathcal H(\gamma)$ we have $$\begin{aligned} &\lim_{n\to\infty} \frac{ \mathbb E[|\mathcal{IC}_1(H)|^p] } { \dhinterseq w_n } ={\boldsymbol{\nu}}^*(|\widetilde{H}_{\mathcal{IC}}|^p)\;. %\label{eq;mean-interior-cluster-IC1}\end{aligned}$$* **Corollary 36**. *Assume that [$\mathcal{S}^{(2(\gamma+1))}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. For any $H\in H(\gamma)$ we have $$\begin{aligned} \lim_{n\to\infty}\frac{ \mathrm{Var}(\mathcal{IC}_1(H)) } { \dhinterseq w_n } ={\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{IC}}^2)\;. %\label{eq;variance-interior-cluster-IC1}\end{aligned}$$* *Proof of .* The functional $\widetilde{H}_{\mathcal{IC}}$ is clearly continuous with respect to the law of $\boldsymbol{Y}$. We also note that $| \widetilde{H}_{\mathcal{IC}}( \mathbb{X}_{1} ) | \leqslant 3C_H \mathcal L^{\gamma+1}(\mathbb{X}_{1})$ yielding $$\begin{aligned} %\label{eq:internal-simple bound} |\mathcal{IC}_{1}(H)|\leqslant \mathbbm{1}{\left\{A_1\right\}} |\widetilde{H}_{\mathcal{IC}}(\mathbb{X}_{1})|\leqslant 3C_H\mathbbm{1}{\left\{A_1\right\}} \mathcal L^{\gamma+1}(\mathbb{X}_{1})\;. \end{aligned}$$ Thus, the first conclusion follows from along with . With $p=1$, gives $(\mathbb E[\mathcal{IC}_1(H)])^2= O(\dhinterseq^2 w_n^2)=o(\dhinterseq w_n)$. Hence, $$\begin{aligned} \lim_{n\to\infty}\frac{1}{\dhinterseq w_n}\mathrm{Var}(\mathcal{IC}_1(H))= \lim_{n\to\infty}\frac{1}{\dhinterseq w_n}\mathbb E[\mathcal{IC}_1^2(H)]\end{aligned}$$ and the term converges to the desired limit. ◻ ### Boundary clusters {#boundary-clusters} Recall the functionals $\widetilde{H}_{\mathcal{BC}}$ and $\widetilde{H}_{\mathcal{BC},p}$ defined in [\[eq:new-functional-BC\]](#eq:new-functional-BC){reference-type="eqref" reference="eq:new-functional-BC"}-[\[eq:new-functional-BC-p\]](#eq:new-functional-BC-p){reference-type="eqref" reference="eq:new-functional-BC-p"}. Recall [\[eq;difference-SB&DB;boundary-cluster-part;exceedance-times-expansion;main\]](#eq;difference-SB&DB;boundary-cluster-part;exceedance-times-expansion;main){reference-type="eqref" reference="eq;difference-SB&DB;boundary-cluster-part;exceedance-times-expansion;main"}- [\[eq:boundary-2-as-functional\]](#eq:boundary-2-as-functional){reference-type="eqref" reference="eq:boundary-2-as-functional"}: $$\begin{aligned} \mathcal{BC}_j(H;1) = & \dhinterseq\left(H(\mathbb{X}_{j})+H(\mathbb{X}_{j+1})- H(\mathbb{X}_{j,j+1})\right) \mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}\; \end{aligned}$$ and $$\begin{aligned} &\mathcal{BC}_j(H;2) = \big[ {\rm{SB}}_{j-1} + {\rm{SB}}_j + {\rm{SB}}_{j+1} - {\rm{DB}}_{j,j+1} \big]\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}\label{eq:BCj-tilde-0}\\ &= \mathbbm{1}{\left\{\mathcal L_{j,j+1}<\dhinterseq\right\}}\mathcal{BC}_j(H;2) +\mathbbm{1}{\left\{\mathcal L_{j,j+1}\geqslant \dhinterseq\right\}}\mathcal{BC}_j(H;2)\nonumber \\ &= \mathbbm{1}{\left\{\mathcal L_{j,j+1}<\dhinterseq\right\}}\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}\widetilde{H}_{\mathcal{IC}}(\mathbb{X}_{j,j+1})+\mathbbm{1}{\left\{\mathcal L_{j,j+1}\geqslant \dhinterseq\right\}}\mathcal{BC}_j(H;2)\nonumber\\ &\leqslant \mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}\widetilde{H}_{\mathcal{IC}}(\mathbb{X}_{j,j+1})+\mathbbm{1}{\left\{\mathcal L_{j,j+1}\geqslant \dhinterseq\right\}}\mathcal{BC}_j(H;2)\nonumber\\ &=: \widetilde{\mathcal{BC}}_j(H;2)+ \overline{\mathcal{BC}}_j(H;2)\;.\label{eq:BCj-tilde}\end{aligned}$$ The asymptotic behaviour of these terms is significantly different, as illustrated by the results below. For the $p$th moment of $\mathcal{BC}_j(H;1)$ the scaling is $\dhinterseq^p w_n$, due to its specific, simple, structure, with $\dhinterseq$ as a multiplier. For $\mathcal{BC}_j(H;2)$, we note first that the representation [\[eq:BCj-tilde-0\]](#eq:BCj-tilde-0){reference-type="eqref" reference="eq:BCj-tilde-0"} yields a rough bound $$\begin{aligned} \label{eq:Bcj-second-rough} |\mathcal{BC}_j(H;2)|\leqslant \dhinterseq H(\mathbb{X}_{j,j+1})\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}\;.\end{aligned}$$ This bound is not sharp enough - the right hand side of [\[eq:Bcj-second-rough\]](#eq:Bcj-second-rough){reference-type="eqref" reference="eq:Bcj-second-rough"} is of the order $\dhinterseq w_n$, with $w_n$ due to the presence of $A_j\cap A_{j+1}$. The trick is to split $\mathcal{BC}_j(H;2)$ according to small and large values of the cluster length. A small cluster length allows to replace $\dhinterseq H(\mathbb{X}_{j,j+1})$ with $\widetilde{H}_{\mathcal{IC}}(\mathbb{X}_{j,j+1})$. The latter is tight under the conditional law (given $A_j\cap A_{j+1}$). This yields the scaling $w_n$ for $\widetilde{\mathcal{BC}}_j(H;2)$. Finally, for $\overline{\mathcal{BC}}_j(H;2)$ we use the bound [\[eq:Bcj-second-rough\]](#eq:Bcj-second-rough){reference-type="eqref" reference="eq:Bcj-second-rough"} in conjunction with , that controls the tail of the cluster length. **Corollary 37**. *Assume that [$\mathcal{S}^{(p\gamma+1)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. For any $H\in H(\gamma)$ we have $$\begin{aligned} \lim_{n\to\infty} \frac{\mathbb E[|\mathcal{BC}_{1}(H;1)|^p]}{\dhinterseq^p w_n} = {\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{BC},p}) \;. \label{eq;mean-boundary-cluster-BC1,1;1} \end{aligned}$$* *Proof.* Recalling that ${\rm{DB}}_j(H)=\dhinterseq H(\boldsymbol{X}_{(j-1)\dhinterseq+1,j\dhinterseq}/\tepseq)=\dhinterseq H(\mathbb{X}_{j})$ and ${\rm{DB}}_{1,2}(H)=\dhinterseq H(\mathbb{X}_{1,2})$, the limit [\[eq;mean-boundary-cluster-BC1,1;1\]](#eq;mean-boundary-cluster-BC1,1;1){reference-type="eqref" reference="eq;mean-boundary-cluster-BC1,1;1"} follows easily from $( \textup{\lowercase{\romannumeral 1}} )$, along with (we can add the indicators of the small values for free), in conjunction with [\[eq:short-formula-for-Hbc\]](#eq:short-formula-for-Hbc){reference-type="eqref" reference="eq:short-formula-for-Hbc"}. ◻ **Corollary 38**. *Assume that [$\mathcal{S}^{(2\gamma+1)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. For any $H\in H(\gamma)$ we have $$\begin{aligned} & \lim_{n\to\infty} \frac{\mathrm{Var}(\mathcal{BC}_{1}(H;1))}{\dhinterseq^2 w_n} = % \canditheta\esp\left[ %\sum_{j=2}^{\clusterlength(\bsZ)} \left\{H(\bsZ_{0,\infty}) - H (\bsZ_{0,j-2}) - H (\bsZ_{j-1,\infty}) %\right\} ^ 2 %\right] {\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{BC},2})\;. \label{eq:boundary-cluster;First-Term;Variance} \end{aligned}$$* *Proof.* It suffices to calculate $\mathbb E[\mathcal{BC}_{1}^2(H;1)]$. Then [\[eq:boundary-cluster;First-Term;Variance\]](#eq:boundary-cluster;First-Term;Variance){reference-type="eqref" reference="eq:boundary-cluster;First-Term;Variance"} follows from by noting that $H^2 \in \mathcal H({2\gamma})$. ◻ **Corollary 39**. *Assume that [$\mathcal{S}^{(p(\gamma+1)+1)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. For any $H\in H(\gamma)$ we have $$\begin{aligned} & \lim_{n\to\infty} \frac{\mathbb E[|\widetilde{\mathcal{BC}}_{1}(H;2)|^p]}{w_n} = {\vartheta}\mathbb E\left[ (\mathcal L(\boldsymbol{Z}) - 1)|\widetilde{H}_{\mathcal{IC}}(\boldsymbol{Z})|^p \right] %\label{eq;mean-boundary-cluster-BC1,2} \;. \end{aligned}$$* gives immediately: **Corollary 40**. *Assume that [$\mathcal{S}^{(2\gamma+3)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. For any $H\in\mathcal H(\gamma)$ we have $$\begin{aligned} & \lim_{n\to\infty} \frac{\mathrm{Var}(\widetilde{\mathcal{BC}}_1(H;2))}{w_n} = {\vartheta}\mathbb E\left[ (\mathcal L({\boldsymbol{Z}})-1)\widetilde{H}_{\mathcal{IC}}^2(\boldsymbol{Z}) \right]\;. %\label{eq:boundary-cluster;Second-Term;Variance} \end{aligned}$$* *Proof of .* We use the representation [\[eq:BCj-tilde\]](#eq:BCj-tilde){reference-type="eqref" reference="eq:BCj-tilde"}. Since $H\in \mathcal H(\gamma)$, we have $\widetilde{H}_{\mathcal{IC}}\in \mathcal H(\gamma+1)$ and $\widetilde{H}_{\mathcal{IC}}^p\in \mathcal H(p(\gamma+1))$. We apply [\[eq;convergence-boundary-cluster-H-full\]](#eq;convergence-boundary-cluster-H-full){reference-type="eqref" reference="eq;convergence-boundary-cluster-H-full"} to conclude the proof. For this, we need [$\mathcal{S}^{(p(\gamma+1)+1)}(\dhinterseq, \tepseq)$](#SummabilityAC). ◻ **Corollary 41**. *Let $\eta>0$. Assume that [$\mathcal{S}^{(p\gamma+\eta+1)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. For any $H\in H(\gamma)$, $$\begin{aligned} \mathbb E[|\overline{\mathcal{BC}}_{1}(H;2)|^p]=O(w_n\dhinterseq^{p-\eta})\;. \end{aligned}$$* *Proof.* We have $$\begin{aligned} \mathbb E[|\overline{\mathcal{BC}}_{1}(H;2)|^p]\leqslant \dhinterseq^p \mathbb E[|H|^p(\mathbb{X}^{(1,2)})\mathbbm{1}{\left\{\mathcal L_{1,2}\geqslant \dhinterseq\right\}}]\;.\end{aligned}$$ Apply with $\delta=p\gamma$ and $\gamma=\eta$. ◻ ### Boundary clusters - piecewise stationary case **Corollary 42**. *Assume that $\boldsymbol{X}$ is piecewise stationary. Assume that [$\mathcal{S}^{(\gamma)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. For any $H\in\mathcal H(\gamma)$, $$\begin{aligned} \mathbb E[|\mathcal{BC}_1(H;1)|]=O(\dhinterseq^3 w_n^2)\;.\end{aligned}$$* *Proof.* Independence between blocks, give $$\begin{aligned} &\mathbb E[|\mathcal{BC}_1(H;1)|] \\ & \leqslant\dhinterseq \mathbb E[|H|(\tepseq^{-1}\boldsymbol{X}_{1,2\dhinterseq})\mathbbm{1}{\left\{A_1\cap A_2\right\}}]\mathbb P^2(A_1^c) + 2\dhinterseq \mathbb E[|H|(\tepseq^{-1}\boldsymbol{X}_{1,\dhinterseq})\mathbbm{1}{\left\{A_1\right\}}]\mathbb P^2(A_1^c)\mathbb P(A_1)\\ &= O(\dhinterseq \dhinterseq^2 w_n^2)+O(\dhinterseq \dhinterseq w_n \dhinterseq w_n)\;. \end{aligned}$$ ◻ **Corollary 43**. *Assume that $\boldsymbol{X}$ is piecewise stationary. Assume that [$\mathcal{S}^{(\gamma)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. For any $H\in\mathcal H(\gamma)$, $$\begin{aligned} \mathbb E[|{\mathcal{BC}}_1(H;2)|]=O(\dhinterseq^3 w_n^2)\;.\end{aligned}$$* *Proof.* Recall the definition of ${\rm{SB}}_j$ in [\[eq:def-SBj\]](#eq:def-SBj){reference-type="eqref" reference="eq:def-SBj"}. On $A_{j-1}\cap A_j$, ${\rm{SB}}_{j-1}$ is a function of the block $I_j$ only. Hence, the independence between blocks gives $$\begin{aligned} \mathbb E[{\rm{SB}}_{j-1}\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}]&= \mathbb E[{\rm{SB}}_{j-1} \mathbbm{1}{\left\{A_j\right\}}]\mathbb P(A_{j-1})\mathbb P(A_{j+1}^c)\mathbb P(A_{j+2})\\ &\leqslant \dhinterseq \mathbb E[|H|(\tepseq^{-1}\boldsymbol{X}_{1,\dhinterseq})\mathbbm{1}{\left\{A_1\right\}}]\mathbb P^2(A_1^c)\mathbb P(A_1)\;.\end{aligned}$$ Hence, gives $$\begin{aligned} \mathbb E[{\rm{SB}}_{j-1}\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}]&=O(\dhinterseq \dhinterseq w_n \dhinterseq w_n)\;. \end{aligned}$$ Likewise, $$\begin{aligned} \mathbb E[{\rm{SB}}_{j+1}\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}]&= \mathbb E[{\rm{SB}}_{j+1} \mathbbm{1}{\left\{A_{j+1}\right\}}]\mathbb P(A_{j-1}^c)\mathbb P(A_{j})\mathbb P(A_{j+2}^c)=O(\dhinterseq^3 w_n^2)\;. \end{aligned}$$ The term $\mathbb E[{\rm{DB}}_{j,j+1}\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}]$ was bounded in the preceding corollary. ◻ ## Moments of clusters - large blocks {#sec:moments-of-clusters-large} We continue with large blocks. Here, we are going to focus on the special case of $H(\boldsymbol{x})=\mathbbm{1}{\left\{\boldsymbol{x}^*>1\right\}}$ and the toy example MMA(1); see . Then (cf. ) $$\begin{aligned} \mathcal{IC}_j(H)=\left(\mathcal L(\tepseq^{-1}\boldsymbol{X}_{(j-1)\dhinterseq+1,j\dhinterseq})-1\right)\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}^c\right\}}\;.\end{aligned}$$ gives the following counterpart to : **Corollary 44**. *Assume that $\boldsymbol{X}$ is stationary and $\ell$-dependent. Assume that [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} holds and $\dhinterseq^2 w_n\to \infty$. Then $$\begin{aligned} \lim_{n\to\infty}\frac{\mathbb E[\mathcal{IC}_1(H)]}{\dhinterseq^3 w_n^2}= \frac{1}{6}{\vartheta}^2\;.\end{aligned}$$* Next (cf. ), $\mathcal{BC}_j(H;1)=-\dhinterseq\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}$, and $$\begin{aligned} \mathcal{BC}_j(H;2)=\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}} \left[\left(t_{j+1}{(N_{j+1})}-t_{j}(1)\right)-\left(t_{j+1}(1)-t_{j}{(N_j)}-\dhinterseq\right)_+\right] \;. \end{aligned}$$ Thus, for the MMA(1) process we obtain via $$\begin{aligned} \lim_{n\to\infty}\frac{\mathbb E[\mathcal{BC}_1(H;1)]}{\dhinterseq^3 w_n^2}=-{\vartheta}^2\;\end{aligned}$$ as long as $\dhinterseq^2 w_n\to \infty$. Then, give $$\begin{aligned} \lim_{n\to\infty}\frac{\mathbb E[\mathcal{BC}_1(H;2)]}{\dhinterseq^3 w_n^2}=\frac{7}{6}{\vartheta}^2-\frac{1}{6}{\vartheta}^2={\vartheta}^2\;,\end{aligned}$$ whenever $\dhinterseq^3w_n\to \infty$. In summary: **Corollary 45**. *Assume that $\boldsymbol{X}$ is the MMA(1) process. Assume that [\[eq:conditiondh\]](#eq:conditiondh){reference-type="ref" reference="eq:conditiondh"} holds and $\dhinterseq^3 w_n\to \infty$. Then $$\begin{aligned} \lim_{n\to\infty}\frac{\mathbb E[\mathcal{BC}_1(H)]}{\dhinterseq^3 w_n^2}=0\;. \end{aligned}$$* ## Moments of clusters statistics - small blocks scenario {#sec:dependence-clusters} In this section we calculate mean and variance of cluster statistics $\mathcal{IC}(H)$ (cf. [\[eq:interior-clusters-def\]](#eq:interior-clusters-def){reference-type="eqref" reference="eq:interior-clusters-def"} and [\[eq:internal-as-functional\]](#eq:internal-as-functional){reference-type="eqref" reference="eq:internal-as-functional"}) and $\mathcal{BC}(H)$ (cf. [\[eq:boundary-clusters-def\]](#eq:boundary-clusters-def){reference-type="eqref" reference="eq:boundary-clusters-def"} and [\[eq:boundary-clusters-decomposition\]](#eq:boundary-clusters-decomposition){reference-type="eqref" reference="eq:boundary-clusters-decomposition"}). To break dependence between blocks, we need some mixing assumptions. The most important conclusion is that while the expectations of $\mathcal{IC}$ and $\mathcal{BC}$ growth at the same rate $nw_n$, the variances growth at different rates: $\mathrm{Var}(\mathcal{IC}(H))$ at the rate $nw_n$, while the variance of $\mathcal{BC}(H)$ at the rate $n\dhinterseq w_n$. ### Internal Clusters Statistics Recall that the statistics is defined as $\mathcal{IC}(H)=\sum_{j=2}^{m_n-1}\mathcal{IC}_j(H)$; cf. [\[eq:interior-clusters-def\]](#eq:interior-clusters-def){reference-type="eqref" reference="eq:interior-clusters-def"} and [\[eq:internal-as-functional\]](#eq:internal-as-functional){reference-type="eqref" reference="eq:internal-as-functional"}. Recall also the functional $\widetilde{H}_{\mathcal{IC}}$ defined in [\[eq:new-functional-IC\]](#eq:new-functional-IC){reference-type="eqref" reference="eq:new-functional-IC"}. The first result is an immediate consequence of . **Corollary 46**. *Assume that [$\mathcal{S}^{(\gamma+1)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. For any $H\in\mathcal H(\gamma)$ we have $$\begin{aligned} \lim_{n\to\infty}\frac{\mathbb E[\mathcal{IC}(H)]}{nw_n}={\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{IC}})\;. \end{aligned}$$* The next result shows that the internal clusters statistics behaves as if the summands were independent. **Proposition 47**. *Assume that $\boldsymbol{X}$ is stationary and mixing with the rates [\[eq:mixing-rates\]](#eq:mixing-rates){reference-type="eqref" reference="eq:mixing-rates"}. Assume that [$\mathcal{S}^{(2\gamma+2+\delta)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds for some $\delta>0$. For any $H\in\mathcal H(\gamma)$ we have $$\begin{aligned} \lim_{n\to\infty}\frac{\mathrm{Var}(\mathcal{IC}(H))}{nw_n} = {\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{IC}}^2)\;. %\label{eq:interior-cluster;total-variances} \end{aligned}$$* *Proof.* It suffices to show that $$\begin{aligned} %\label{eq:interior-cluster;negligible-sum-covariances} \left| \sum_{j=2}^{m_n-1} \mathrm{Cov}(\mathcal{IC}_1, \mathcal{IC}_{j}) \right| = o\left( \mathrm{Var}(\mathcal{IC}_1) \right) = o (\dhinterseq w_n)\;. \end{aligned}$$ We divide the proof into the following four steps. $( \textup{\lowercase{\romannumeral 1}} )$: Since $\mathcal{IC}_{j}$ is based on the indicator $\mathbbm{1}{\left\{A_{j-1}^c\cap A_j \cap A_{j+1}^c\right\}}$, for $j=2$ we have by $$\left| \mathrm{Cov}(\mathcal{IC}_1, \mathcal{IC}_{2}) \right| = (\mathbb E[\mathcal{IC}_1] ) ^2 = O(\dhinterseq^2 w_n^2)=o(\dhinterseq w_n)\;.$$ For this we need [$\mathcal{S}^{(\gamma+1)}(\dhinterseq, \tepseq)$](#SummabilityAC). $( \textup{\lowercase{\romannumeral 2}} )$: Let $j=3,4$. Take $p = 2 + \epsilon$ for any $\epsilon>0$ such that $$(\gamma + 1 ) (2 + \epsilon ) \leqslant 2\gamma + 2+\delta\;.$$ It follows from and the mixing inequality [\[eq:mixing-inequality\]](#eq:mixing-inequality){reference-type="eqref" reference="eq:mixing-inequality"}, $$\begin{aligned} | \mathrm{Cov}(\mathbbm{1}{\left\{A_1\right\}} \widetilde{H}_{\mathcal{IC}}(\mathbb{X}_{1}), \mathbbm{1}{\left\{A_j\right\}} \widetilde{H}_{\mathcal{IC}}(\mathbb{X}_{j}) |\leqslant \alpha^{1/r}_{(j-2)\dhinterseq} \Vert \mathbbm{1}{\left\{A_1\right\}} \mathcal L_{1}^{\gamma + 1} \Vert_p^2 \leqslant & ( \dhinterseq w_n )^{ 2/(2+\epsilon) } \alpha^{ \epsilon/(2+\epsilon) }_{ (j-2)\dhinterseq }\;. \end{aligned}$$ It is $o(\dhinterseq w_n)$ by applying [\[eq:mixing-rates\]](#eq:mixing-rates){reference-type="eqref" reference="eq:mixing-rates"}. For this we need [$\mathcal{S}^{(2\gamma+2+\delta)}(\dhinterseq, \tepseq)$](#SummabilityAC). Next, $$\begin{aligned} |\mathbb E[\mathcal{IC}_1(H)-\mathbbm{1}{\left\{A_1\right\}} \widetilde{H}_{\mathcal{IC}}(\mathbb{X}_{1})]|\leqslant \mathbb E[\mathbbm{1}{\left\{A_1\right\}}\mathbbm{1}{\left\{A_2^c\right\}} |\widetilde{H}_{\mathcal{IC}}|(\mathbb{X}_{1})]=O(w_n)=o(\dhinterseq w_n)\end{aligned}$$ by [\[eq;joint-convergence-boundary-cluster;H-front\]](#eq;joint-convergence-boundary-cluster;H-front){reference-type="eqref" reference="eq;joint-convergence-boundary-cluster;H-front"}. Hence, together, $\mathrm{Cov}(\mathcal{IC}_1(H),\mathcal{IC}_j(H))=o(\dhinterseq w_n)$. $( \textup{\lowercase{\romannumeral 3}} )$: In this step, we will show that for $M>4$, $$\lim_{M \to \infty} \limsup_{n\to \infty} \; \sum_{j=M} ^{m_n-1} \mathrm{Cov}(\mathcal{IC}_1, \mathcal{IC}_{j}) = o(\dhinterseq w_n)\;.$$ We apply the covariance inequality under mixing to get $$\begin{aligned} | \mathrm{Cov}(\mathcal{IC}_1, \mathcal{IC}_{j}) | \leqslant \mathrm{cst}\; \alpha^{1/r}_{(j-4)\dhinterseq} \Vert \mathcal{IC}_1 \Vert_p ^2 \leqslant\mathrm{cst}\; \; \alpha^{1/r}_{(j-4)\dhinterseq} \Vert \mathbbm{1}{\left\{A_1\right\}} \mathcal L_1^{\gamma + 1} \Vert_p^2 \;. \end{aligned}$$ Take $p = 2 + \epsilon$ for any $\epsilon>0$ such that $$(\gamma + 1 ) (2 + \epsilon ) \leqslant 2\gamma + 2+\delta\;.$$ It follows from $$\begin{aligned} | \mathrm{Cov}(\mathcal{IC}_1, \mathcal{IC}_{j}) |\leqslant \alpha^{1/r}_{(j-4)\dhinterseq} \Vert \mathbbm{1}{\left\{A_1\right\}} \mathcal L_{1}^{\gamma + 1} \Vert_p^2 \leqslant & ( \dhinterseq w_n )^{ 2/(2+\epsilon) } \alpha^{ \epsilon/(2+\epsilon) }_{ (j-4)\dhinterseq }\;. \end{aligned}$$ Thus, $$\begin{aligned} \sum_{j=M} ^{m_n-1}| \mathrm{Cov}(\mathcal{IC}_1, \mathcal{IC}_{j})|\leqslant ( \dhinterseq w_n )^{ 2/(2+\epsilon) }\sum_{j=M} ^{m_n-1} \alpha^{ \epsilon/(2+\epsilon) }_{ (j-4)\dhinterseq }\end{aligned}$$ and the result follows by applying [\[eq:mixing-rates\]](#eq:mixing-rates){reference-type="eqref" reference="eq:mixing-rates"}. ◻ ### Boundary Clusters Statistics Recall that the boundary clusters statistics is defined as $$\begin{aligned} \mathcal{BC}(H):=\mathcal{BC}(H;1)+\mathcal{BC}(H;2)= \sum_{j=2}^{m_n-1}\mathcal{BC}_j(H;1)+\sum_{j=2}^{m_n-1}\mathcal{BC}_j(H;2)\;, \end{aligned}$$ cf. [\[eq:boundary-clusters-def\]](#eq:boundary-clusters-def){reference-type="eqref" reference="eq:boundary-clusters-def"} and [\[eq:boundary-clusters-decomposition\]](#eq:boundary-clusters-decomposition){reference-type="eqref" reference="eq:boundary-clusters-decomposition"}. Recall also the decomposition [\[eq:BCj-tilde\]](#eq:BCj-tilde){reference-type="eqref" reference="eq:BCj-tilde"}. Set $$\begin{aligned} \widetilde\mathcal{BC}(H;2)=\sum_{j=2}^{m_n-1}\widetilde{\mathcal{BC}}_j(H;2)\;, \ \ \overline\mathcal{BC}(H;2)=\sum_{j=2}^{m_n-1}\overline{\mathcal{BC}}_j(H;2)\;. \end{aligned}$$ In view of , it is enough to consider $\mathcal{BC}(H;1)$ only. The first result is an immediate consequence of . **Corollary 48**. *Assume that [$\mathcal{S}^{(\gamma+2+\delta)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds for some $\delta>0$. For any $H\in H(\gamma)$ we have $$\begin{aligned} \lim_{n\to\infty} \frac{\mathbb E[\mathcal{BC}(H)]}{n w_n} = {\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{BC}}) \;. \end{aligned}$$* **Proposition 49**. *Assume that $\boldsymbol{X}$ is stationary and mixing with the rates [\[eq:mixing-rates\]](#eq:mixing-rates){reference-type="eqref" reference="eq:mixing-rates"}. Assume that [$\mathcal{S}^{(2\gamma+2+\delta)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds for some $\delta>0$. For any $H\in\mathcal H(\gamma)$ we have $$\begin{aligned} &\lim_{n\to \infty} \frac{\mathrm{Var}(\mathcal{BC}(H;1))}{n\dhinterseq w_n} = \lim_{n\to \infty} \frac{\mathrm{Var}(\mathcal{BC}_{1}(H;1))}{\dhinterseq^2 w_n} ={\boldsymbol{\nu}}^*(\widetilde{H}_{\mathcal{BC},2}) %\label{eq:boundary-clusters-full-variance} \end{aligned}$$* *Proof.* We will show that, as $n\to\infty$, $$\begin{aligned} \frac{1}{\dhinterseq^2 w_n} \left| \sum_{j=2}^{m_n-1} \mathrm{Cov}(\mathcal{BC}_1(H;1),\mathcal{BC}_j(H;1)) \right| \to 0\;. \label{eq:boundary-clusters-small-full-covariance}\end{aligned}$$ We shall divide the proof of [\[eq:boundary-clusters-small-full-covariance\]](#eq:boundary-clusters-small-full-covariance){reference-type="eqref" reference="eq:boundary-clusters-small-full-covariance"} into the following three steps. $( \textup{\lowercase{\romannumeral 1}} )$: For $j=2,3$, we have by $$\begin{aligned} \mathrm{Cov}(\mathcal{BC}_1(H;1),\mathcal{BC}_j(H;1)) = - (\mathbb E[\mathcal{BC}_1(H;1)]) ^ 2 = (\dhinterseq w_n) ^ 2 = o(\dhinterseq^2 w_n)\;.\end{aligned}$$ $( \textup{\lowercase{\romannumeral 2}} )$: For $j=4,5,\ldots$, we note first the trivial bound $|\mathcal{BC}_j(H;1)|\leqslant \mathrm{cst}\;\dhinterseq^{\gamma+1}$. Indeed, $|H|\leqslant \mathrm{cst}\;\mathcal L^{\gamma}\leqslant \mathrm{cst}\; \dhinterseq^{\gamma}$. We have $$\begin{aligned} & \big| \mathbb E[\mathcal{BC}_1(H;1) \mathcal{BC}_j(H;1) ] \big|\leqslant \mathrm{cst}\; \dhinterseq^{2\gamma+2} \mathbb P( A_1 \cap A_2\cap A_{j}\cap A_{j+1} ) \leqslant \mathrm{cst}\;\dhinterseq^{2\gamma + 2 } \mathbb P( A_1\cap A_{j} ) \\ &\leqslant \dhinterseq^{2\gamma + 2 } \cdot \dhinterseq \cdot \sum_{i = (j-2)\dhinterseq }^{j\dhinterseq} \mathbb P( | \boldsymbol{X}_0 | > \tepseq , | \boldsymbol{X}_i | > \tepseq) \\ & =\dhinterseq^{2\gamma + 2 } \cdot \dhinterseq \cdot o( \dhinterseq^{- 2\gamma - 1} w_n ) = o( \dhinterseq^2 w_n )\end{aligned}$$ whenever [$\mathcal{S}^{(2\gamma+1)}(\dhinterseq, \tepseq)$](#SummabilityAC) is satisfied; cf. [\[eq:consequence-condition-s\]](#eq:consequence-condition-s){reference-type="eqref" reference="eq:consequence-condition-s"}. $( \textup{\lowercase{\romannumeral 3}} )$: We have $$\begin{aligned} \sum_{i=\ell}^{m_n-1} \big| \mathrm{Cov}[\mathcal{BC}_1(H;1), \mathcal{BC}_j(H;1)] \big| \leqslant\mathrm{cst}\;. \sum_{i=\ell}^{m_n-1}\alpha_{(i-4)\dhinterseq}^{1/r}\|\mathcal{BC}_{1}(H;1)\|_p^2\end{aligned}$$ and we conclude in the same way as in the case of . The assumption [$\mathcal{S}^{(2\gamma+2+\delta)}(\dhinterseq, \tepseq)$](#SummabilityAC) is needed here. ◻ ## Remainder terms {#sec:remainder} The difference between the sliding and the disjoint blocks statistics can be decomposed as follows: $$\begin{aligned} \label{eq;difference-SB&DB;three-parts} {\rm{SB}}(H) - {\rm{DB}}(H) = \mathcal{IC}(H) + \mathcal{BC}(H) + \mathcal{R}(H)\;,\end{aligned}$$ where ${\rm{SB}}(H)$, ${\rm{DB}}(H)$, $\mathcal{IC}(H)$, $\mathcal{BC}(H)$ are defined in [\[eq:def-SB\]](#eq:def-SB){reference-type="eqref" reference="eq:def-SB"}, [\[eq:def-DB\]](#eq:def-DB){reference-type="eqref" reference="eq:def-DB"}, [\[eq:internal-as-functional\]](#eq:internal-as-functional){reference-type="eqref" reference="eq:internal-as-functional"} and [\[eq:boundary-clusters-def\]](#eq:boundary-clusters-def){reference-type="eqref" reference="eq:boundary-clusters-def"}, respectively, and the remainder term $\mathcal{R}(H)$ is given by the sum of three type of terms $\mathcal{R}^{\rm IC}(H)$, $\mathcal{R}^{\rm BC}(H)$, $\mathcal{R} ^ {\rm NC} (H)$ to be defined below. The main result of this section is follows. Note that we are concerned in weakening the assumptions of the result - the assumptions are determined by the considerations on the internal and the boundary clusters. **Lemma 50**. - *Assume that $H\in \mathcal H(\gamma)$ and [$\mathcal{S}^{(2\gamma+3)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. Then $$\begin{aligned} \frac{1}{n\dhinterseq w_n}\mathcal{R}=O_P\left(\frac{\dhinterseq}{n}+\dhinterseq^{-(\gamma+4)}\right)\;.\end{aligned}$$* - *Assume that $\boldsymbol{X}$ is $\ell$-dependent and $H$ is bounded. Assume that $\dhinterseq w_n\to \infty$. Then $$\begin{aligned} \frac{1}{n\dhinterseq w_n}\mathcal{R}=O_P\left(\frac{\dhinterseq^{2}w_n}{n}+\frac{\dhinterseq^3 w_n^2}{n}\right)=O_P\left(\frac{\dhinterseq^3 w_n^2}{n}\right)\;.\end{aligned}$$* In order to prove the above lemma, we need to provide a decomposition of $\mathcal{R}$. Then, will follow from (small blocks case and large blocks cases applied with $\gamma=0$), (small blocks case) and (large blocks case) below. The first summand in the remainder stems from the internal clusters: $$\begin{aligned} \mathcal{R}^{\rm IC}(H) = [ {\rm{SB}}_{1} - {\rm{DB}}_{1}] \mathbbm{1}{\left\{ A_1\cap A_2^c\right\}} +{\rm{SB}}_{m_n - 1} \mathbbm{1}{\left\{ A_{m_n - 1}^c \cap A_{m_n} \right\}}\;. %\label{eq;difference-SB&DB;interior-cluster-part;remainder}\end{aligned}$$ The second summand stems from the boundary clusters: $$\begin{aligned} \mathcal{R}^{\rm BC}(H) &= [{\rm{SB}}_{1} - {\rm{DB}}_{1}] \mathbbm{1}{\left\{ A_1\cap A_2 \right\}} + [{\rm{SB}}_{2} - {\rm{DB}}_{2} ] \mathbbm{1}{\left\{ A_1\cap A_2\cap A_3^c \right\}} \\ &\phantom{=}+ [{\rm{SB}}_{m_n - 2}] \mathbbm{1}{\left\{ A_{m_n - 2}^c\cap A_{m_n - 1} \cap A_{m_n} \right\}} + [{\rm{SB}}_{m_n - 1} - {\rm{DB}}_{m_n - 1}] \mathbbm{1}{\left\{ A_{m_n - 1} \cap A_{m_n} \right\}} \; . %\label{eq:expansion;boundary-terms;remainder;1.4} \end{aligned}$$ Recall the definition [\[eq:def-SBj\]](#eq:def-SBj){reference-type="eqref" reference="eq:def-SBj"} of ${\rm{SB}}_j$. In $\mathcal{R}^{\rm IC}(H)$, drop $\mathbbm{1}{\left\{A_2\right\}}$ and $\mathbbm{1}{\left\{A_{m_n-1}\right\}}$. We use then [\[eq:SBj-1:nojump-jump\]](#eq:SBj-1:nojump-jump){reference-type="eqref" reference="eq:SBj-1:nojump-jump"} and the assumption $H\in \mathcal H(\gamma)$ to get $$\begin{aligned} |\mathcal{R}^{\rm IC}(H)|\leqslant \dhinterseq\left\{ \mathcal L^\gamma(\tepseq^{-1}\boldsymbol{X}_{1,\dhinterseq})\mathbbm{1}{\left\{A_1\right\}}+ \mathcal L^\gamma(\tepseq^{-1}\boldsymbol{X}_{(m_n-1)\dhinterseq+1,m_n\dhinterseq})\mathbbm{1}{\left\{A_{m_n}\right\}}\right\}\;. \end{aligned}$$ The same bound applies to $\mathcal{R}^{\rm BC}(H)$. Hence, and give: **Lemma 51**. *Assume that $H\in \mathcal H(\gamma)$.* - *Assume that [$\mathcal{S}^{(\gamma)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. Then $$\begin{aligned} %\label{eq:R-IC} \mathbb E[|\mathcal{R}^{\rm IC}(H)+\mathcal{R}^{\rm BC}(H)|]=O(\dhinterseq^2 w_n)\;. \end{aligned}$$* - *Assume that $\dhinterseq^{\gamma+1}w_n\to \infty$. Then $$\begin{aligned} %\label{eq:R-IC} \mathbb E[|\mathcal{R}^{\rm IC}(H)+\mathcal{R}^{\rm BC}(H)|]=O(\dhinterseq^{\gamma+3} w_n^2)\;. \end{aligned}$$* The negligible part $\mathcal{R} ^ {\rm NC}$ stems from terms that involve jumps in at least three consecutive blocks: $$\begin{aligned} \mathcal{R} ^ {\rm NC} (H) & = \sum_{j=2}^{m_n - 2} [ \mathcal{R} ^ {\rm NC} _{j}(H;1) + \mathcal{R} ^ {\rm NC} _{j}(H;2) + \mathcal{R} ^ {\rm NC} _{j}(H;3) ] \;, \label{eq:expansion;negligible-cluster-terms;1} \end{aligned}$$ where $$\begin{aligned} \mathcal{R} ^ {\rm NC} _{j}(H;1)& = [{\rm{SB}}_{j-1}(H) +{\rm{SB}}_{j}(H) - {\rm{DB}}_{j}(H)] \mathbbm{1}{\left\{ A_{j-1}^c\cap A_{j}\cap A_{j+1} \cap A_{j+2}\right\}} %\label{eq:expansion;negligible-cluster-terms;2.2} \; , \\ \mathcal{R} ^ {\rm NC} _{j} (H;2) &= [{\rm{SB}}_{j}(H) - {\rm{DB}}_{j}(H) + {\rm{SB}}_{j+1}(H) - {\rm{DB}}_{j+1}(H) ] \cdot \mathbbm{1}{\left\{ A_{j-1}\cap A_{j}\cap A_{j+1} \cap A_{j+2}^c \right\}} %\label{eq:expansion;negligible-cluster-terms;3.2} \; , \\ \mathcal{R} ^ {\rm NC} _{j} (H;3) &= [{\rm{SB}}_{j}(H) - {\rm{DB}}_{j}(H) ] %\label{eq:expansion;negligible-cluster-terms;4.1} \\ \cdot \mathbbm{1}{\left\{ A_{j-1}\cap A_{j} \cap A_{j+1}\cap A_{j+2}\right\}} %\label{eq:expansion;negligible-cluster-terms;4.2} \; . \end{aligned}$$ **Lemma 52**. *Assume that $H\in \mathcal H(\gamma)$ and [$\mathcal{S}^{(2\gamma+3)}(\dhinterseq, \tepseq)$](#SummabilityAC) holds. Then $$\begin{aligned} \mathcal{R} ^ {\rm NC} =O_P(n\dhinterseq^{-(\gamma+3)}w_n)\;. \end{aligned}$$* *Proof.* Each of the terms $\mathcal{R} ^ {\rm NC} _{j} (H;i)$, $i=1,2,3$, can be bounded as follows: $$\begin{aligned} \mathbb E[\mathcal{R} ^ {\rm NC} _{j} (H;i)]\leq \mathrm{cst}\ \dhinterseq^{\gamma+1} \mathbb P(A_1\cap A_3)\;. \end{aligned}$$ Then, the first jump (block 1) and the last (block 3) decomposition, give $$\begin{aligned} \mathbb P(A_1\cap A_3)&=\sum_{j_1=1}^{\dhinterseq}\sum_{j_3=2\dhinterseq+1}^{3\dhinterseq}\mathbb P(t_1(1)=j_1,t_3(N_3)=j_3)\\ &\leq \sum_{j=\dhinterseq+1}^{3\dhinterseq}[(j-\dhinterseq)\wedge (3\dhinterseq-j)]\mathbb P(\norm{\boldsymbol{X}_0}>\tepseq,\norm{\boldsymbol{X}_j}>\tepseq)\\ &\leq \dhinterseq \sum_{j=\dhinterseq+1}^{3\dhinterseq}\mathbb P(\norm{\boldsymbol{X}_0}>\tepseq,\norm{\boldsymbol{X}_j}>\tepseq)= o(\dhinterseq \dhinterseq^{-(2\gamma+3)}w_n)\end{aligned}$$ The last estimate follows from [\[eq:consequence-condition-s\]](#eq:consequence-condition-s){reference-type="eqref" reference="eq:consequence-condition-s"} on account of the anticlustering assumption. Thus $$\begin{aligned} \mathbb E[|\mathcal{R} ^ {\rm NC} (H)|]\leq \mathrm{cst}\ m_n \dhinterseq^{\gamma+1}\dhinterseq^{-(2\gamma+3)}w_n\;. \end{aligned}$$ ◻ **Lemma 53**. *Assume that $\boldsymbol{X}$ is $\ell$-dependent and $H$ is bounded. Then $$\begin{aligned} \mathbb E[|\mathcal{R} ^ {\rm NC} (H)|]=O(\dhinterseq^4 w_n^3)\;.\end{aligned}$$* *Proof.* We have $$\begin{aligned} \mathbb E[|\mathcal{R} ^ {\rm NC} (H)|]\leq \mathrm{cst}\ \dhinterseq \mathbb P(A_1\cap A_2\cap A_3)=O(\dhinterseq \dhinterseq^3w_n^3)\;,\end{aligned}$$ where the bound on $\mathbb P(A_1\cap A_2\cap A_3)$ follows easily from $\ell$-dependence. ◻ # Proofs III - Representations for internal and boundary clusters statistics {#sec:detailed-decomposition} In this section we provide detailed calculations that yield representations for both internal and boundary block statistics, $\mathcal{IC}(H)$, $\mathcal{BC}(H;1)$ and $\mathcal{BC}(H;2)$. Recall that ${\rm{SB}}_{j}(H)=\sum_{i=(j-1)\dhinterseq+1}^{j\dhinterseq}H(\tepseq^{-1}\boldsymbol{X}_{i,i+\dhinterseq-1})$. In order to analyse contribution to ${\rm{SB}}_j$, we need to look for large jumps in $j$th and $(j+1)$th block. A special care has to be taken when analysing what happens at the beginning and the end of each block. ## General case Here, we consider an arbitrary function $H$. #### Explanation for [\[eq:internal-as-functional\]](#eq:internal-as-functional){reference-type="eqref" reference="eq:internal-as-functional"}. Note that $\dhinterseq=\sum_{i=0}^{N_j}\Delta t_{j}{(i)}$. Hence. $$\begin{aligned} &\mathcal{IC}=\mathcal{IC}(H)=\sum_{j=2}^{m_n-1}({\rm{SB}}_{j-1}+{\rm{SB}}_j-{\rm{DB}}_j) \mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}^c\right\}}\\ %&=\sum_{j=2}^{m_n-1}\left[ \sum_{i=0}^{N_j}\Delta t_{j}{(i)}\left\{ H(\mathbb{X}_{j}{(1:i)})+ %H(\mathbb{X}_{j}{(i+1:N_j)}) \right\}-\dhinterseq H(\mathbb{X}_{j}) \right]\ind{A_{j-1}^c\cap A_j\cap %A_{j+1}^c}\notag\\ &=\sum_{j=2}^{m_n-1}\left[\sum_{i=0}^{N_j}\Delta t_{j}{(i)}\left\{ H(\mathbb{X}_{j}{(1:i)}+ H(\mathbb{X}_{j}{(i+1:N_j)})-H(\mathbb{X}_{j}) \right\} \right]\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}^c\right\}}\notag\\ &=\sum_{j=2}^{m_n-1}\left[ \sum_{i=1}^{N_j-1}\Delta t_{j}{(i)}\left\{ H(\mathbb{X}_{j}{(1:i)}+ H(\mathbb{X}_{j}{(i+1:N_j)}) -H(\mathbb{X}_{j}) \right\} \right]\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}^c\right\}}\notag\\ &=\sum_{j=2}^{m_n-1}\mathcal{IC}_{j}(H)\;.\end{aligned}$$ #### Explanation for [\[eq:SBj-1:nojump-jump\]](#eq:SBj-1:nojump-jump){reference-type="eqref" reference="eq:SBj-1:nojump-jump"}. We analyse ${\rm{SB}}_{j-1}(H)$ on $A_{j-1}^c\cap A_j$. The index $i$ comes from the $(j-1)$th block: $(j-2)\dhinterseq+1\leqslant i\leqslant (j-1)\dhinterseq$. A contribution to ${\rm{SB}}_{j-1}(H)$ comes from the jumps in block $j$ only. - At the beginning of the $(j-1)$th block: If $(j-1)\dhinterseq\leqslant i+\dhinterseq-1<t_{j}(1)$, then $H\equiv 0$. - In the middle of the $(j-1)$th block: Let $\ell=1,\ldots,N_j-1$. If $t_{j}(\ell)\leqslant i+\dhinterseq-1<t_{j}(\ell+1)$, then the contribution is $H(\tepseq^{-1}\boldsymbol{X}_{i,i+\dhinterseq-1})=H(\tepseq^{-1}\boldsymbol{X}_{t_{j}(1),t_{j}(\ell)})= H(\mathbb{X}_{j}{(1:\ell)})$. This happens for $\Delta t_{j}(\ell)=t_{j}(\ell+1)-t_{j}(\ell)$ of the indices. - At the end of the $(j-1)$th block: If $t_{j}{(N_j)}\leqslant i+\dhinterseq-1< j\dhinterseq=t_{j}{(N_j+1)}$, then the contribution is $H(\tepseq^{-1}\boldsymbol{X}_{i,i+\dhinterseq-1})=H(\tepseq^{-1}\boldsymbol{X}_{t_{j}(1),t_{j}{(N_j)}})= H(\mathbb{X}_{j})$. This happens for $j\dhinterseq - t_{j}{(N_j)}=t_{j}{(N_j+1)}- t_{j}{(N_j)}=\Delta t_{j}{(N_j)}$ of the indices. In summary, on $A_{j-1}^c\cap A_j$, we have $$\begin{aligned} %\label{eq:explain-1} {\rm{SB}}_{j-1}(H)&=\sum_{\ell=1}^{N_j}\Delta t_{j}(\ell)H(\mathbb{X}_{j}{(1:\ell)})\;.\end{aligned}$$ #### Explanation for [\[eq:SBj:jump-nojump\]](#eq:SBj:jump-nojump){reference-type="eqref" reference="eq:SBj:jump-nojump"}. We analyse ${\rm{SB}}_{j+1}(H)$ on $A_{j+1}\cap A_{j+2}^c$ (then [\[eq:SBj:jump-nojump\]](#eq:SBj:jump-nojump){reference-type="eqref" reference="eq:SBj:jump-nojump"} follows by replacing $j$ with $j-1$.) The index $i$ comes from the $(j+1)$th block: $j\dhinterseq+1\leqslant i\leqslant (j+1)\dhinterseq$. A contribution to ${\rm{SB}}_{j+1}(H)$ comes from the jumps in block $j+1$ only. - At the beginning of the $(j+1)$th block: If $j\dhinterseq+1\leqslant i\leqslant t_{j+1}(1)$, then the contribution is $H(\mathbb{X}_{j+1}{(1:N_{j+1})})$. This happens for $t_{j+1}(1)-j\dhinterseq=t_{j+1}(1)-t_{j+1}(0)=\Delta t_{j+1}(0)$ of the indices. - In the middle of the $(j+1)$th block: Let $\ell=1,\ldots,N_{j+1}-1$. If $t_{j+1}(\ell)< i\leqslant t_{j+1}(\ell+1)$, then the contribution is $H(\mathbb{X}_{j+1}{(\ell+1:N_{j+1})})$. This happens for $\Delta t_{j+1}(\ell)=t_{j+1}(\ell+1)-t_{j+1}(\ell)$ of the indices. - At the end of the $(j+1)$th block: If $t_{j+1}{(N_j+1)}< i$, then $H\equiv 0$. In summary, on $A_{j+1}\cap A_{j+2}^c$, we have $$\begin{aligned} {\rm{SB}}_{j+1}(H)&=\sum_{\ell=0}^{N_{j+1}-1}\Delta t_{j+1}(\ell)H(\mathbb{X}_{j+1}{(\ell+1:N_{j+1})})\;.\end{aligned}$$ #### Explanation for [\[eq:boundary-2-as-functional\]](#eq:boundary-2-as-functional){reference-type="eqref" reference="eq:boundary-2-as-functional"}. #### Contribution from ${\rm{SB}}_j(H)$ on $A_j\cap A_{j+1}$. The index $i$ runs from $(j-1)\dhinterseq+1$ to $j\dhinterseq$. Contribution to ${\rm{SB}}_j$ comes from the jumps in both $j$th and $(j+1)$th block. This contribution can be written in a more concise way, but the more detailed decomposition below will be useful when combining ${\rm{SB}}_j(H)\mathbbm{1}{\left\{A_j\cap A_{j+1}\right\}}$ with other terms. Recall the convention $t_{j}(0)=(j-1)\dhinterseq$ and $t_{j}{(N_j+1)}=j\dhinterseq$. Recall also the notation for adjacent blocks from . Let $q=0,\ldots,N_{j}$ and $s=0,\ldots,N_{j+1}$. Let $$\begin{aligned} \mathcal{J}_{q,s}=\{i=(j-1)\dhinterseq+1,\ldots,j\dhinterseq:t_{j}(q)< i\leqslant t_{j}(q+1),t_{j+1}(s)\leqslant i+\dhinterseq-1< t_{j+1}(s+1)\}\;.\end{aligned}$$ If $i\in \mathcal{J}_{q,s}$ then the contribution is $H(\mathbb{X}_{j,j+1}{(q+1:N_j+s)})$. - If $t_{j}(q+1)\leqslant t_{j+1}(s)-\dhinterseq$, then $\mathcal{J}_{q,s}$ is empty and there is no contribution. - If $t_{j}(q)<t_{j+1}(s)-\dhinterseq< t_{j}(q+1)\leqslant t_{j+1}(s+1)-\dhinterseq$, then we have the contribution for $t_{j}(q+1)-(t_{j+1}(s)-\dhinterseq)$ of the indices. - If $t_{j}(q)<t_{j+1}(s)-\dhinterseq<t_{j+1}(s+1)-\dhinterseq< t_{j}(q+1)$, then we have the contribution for $t_{j+1}(s+1)-t_{j+1}(s)$ of the indices. - If $t_{j+1}(s)-\dhinterseq\leqslant t_{j}(q)< t_{j}(q+1)\leqslant t_{j+1}(s+1)-\dhinterseq$, then we have the contribution for $t_{j}(q+1)-t_{j}(q)$ of the indices. - If $t_{j+1}(s)-\dhinterseq\leqslant t_{j}(q)<t_{j+1}(s+1)-\dhinterseq< t_{j}(q+1)$, then we have the contribution for $t_{j+1}(s+1)-t_{j}(q)-\dhinterseq$ of the indices. #### Joint contribution from ${\rm{SB}}_{j-1}$, ${\rm{SB}}_j$, ${\rm{SB}}_{j+1}$ on $A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c$. We start by observing that the case $q=1,\ldots,N_j$, $s=0,\ldots,N_{j+1}-1$ in (III) simplifies: - The contribution $H( \mathbb{X}_{j,j+1}{(q+1:N_j+s)})$ can occur only if $\mathcal L_{j,j+1}\geqslant \dhinterseq$. The other cases of $q$ and $s$ require a precise analysis. - The contribution (I-a), (I-b) remains the same. Note further that for $\ell=1,\ldots,N_{j-1}$, $\Delta t_{j}(\ell)=\Delta t_{j,j+1}(\ell)$ and $H(\mathbb{X}_{j}{(1:\ell)})=H(\mathbb{X}_{j,j+1}(1:\ell))$. In total, the contribution from ${\rm{SB}}_{j-1}(H)$ in (I-a) and (I-b) can be written as $$\begin{aligned} \label{eq:contribution-to-mixed} \sum_{\ell=1}^{N_j-1}\Delta t_{j,j+1}(\ell)H(\mathbb{X}_{j,j+1}(1:\ell))\;. \end{aligned}$$ - We analyse the contribution $H(\mathbb{X}_{j})=H(\mathbb{X}_{j,j+1}(1:N_j))$. This stems from both ${\rm{SB}}_{j-1}$ and ${\rm{SB}}_j$. We combine (I-c) with (III) for $q=s=0$. - (III-b) and (III-d) are duplicates in this case. If $t_{j}(1)\leqslant t_{j+1}(1)-\dhinterseq$, we have the contribution for $$\begin{aligned} \underbrace{t_{j}(1)-(j-1)\dhinterseq}_{\textrm{from (III)}}+\underbrace{(j\dhinterseq - t_{j}{(N_j)})}_{\textrm{from (I-c)}}=\dhinterseq-(t_{j}{(N_j)}-t_{j}(1)) \end{aligned}$$ of the indices. - (III-c) and (III-e) are duplicates in this case. If $t_{j+1}(1)-\dhinterseq < t_{j}(1)$ we have the contribution for $$\begin{aligned} \underbrace{t_{j+1}(1)-j\dhinterseq}_{\textrm{from (III)}}+\underbrace{(j\dhinterseq - t_{j}{(N_j)})}_{\textrm{from (I-c)}}=t_{j+1}(1)-t_{j}{(N_j)}=\Delta t_{j,j+1}(N_j) \end{aligned}$$ of the indices. - Thus, the contribution $H(\mathbb{X}_{j,j+1}(1:N_j))$ is for $$\begin{aligned} \label{eq:contribution-to-mixed-1} \Delta t_{j,j+1}(N_j) \mathbbm{1}{\left\{t_{j+1}(1)-t_{j}(1)<\dhinterseq\right\}}+ \left(\dhinterseq-(t_{j}{(N_j)}-t_{j}(1))\right)\mathbbm{1}{\left\{t_{j+1}(1)-t_{j}(1)\geqslant \dhinterseq\right\}} \end{aligned}$$ of the indices. Note that the second part vanishes whenever the joint cluster length fulfills $\mathcal L_{j,j+1}< \dhinterseq$. - We take $q=0$ and $s=1,\ldots,N_{j+1}-1$ in (III). Then the contribution is $H(\mathbb{X}_{j,j+1}{(1:N_j+s)})$. It stems from ${\rm{SB}}_j$ only. Set $\ell=N_j+s$. With the re-numeration we have $\Delta t_{j+1}(s)=\Delta t_{j,j+1}(\ell)$. - For (III-b): If $t_{j+1}(s)-\dhinterseq< t_{j}(1)\leqslant t_{j+1}(s+1)-\dhinterseq$, then we have the contribution for $t_{j}(1)+\dhinterseq-t_{j+1}(s)\leqslant t_{j+1}(s+1)-t_{j+1}(s)=\Delta t_{j+1}(s)$ of the indices. - Next, for (III-c), if $t_{j+1}(s+1)-\dhinterseq< t_{j}(1)$, then we have the contribution for $t_{j+1}(s+1)-t_{j+1}(s)=\Delta t_{j+1}(s)$ of the indices. - Thus, $t_{j+1}(s+1)- t_{j}(1)\leqslant \dhinterseq$, the contribution occurs for at most $\Delta t_{j+1}(s)$ of the indices. - The cases (III-d) and (III-e) are empty, since $t_{j+1}(s)-\dhinterseq\leqslant t_{j}(0)=(j-1)\dhinterseq$ cannot happen. - Thus, the contribution for this part is at most $$\begin{aligned} \label{eq:contribution-to-mixed-2} \sum_{\ell=N_j+1}^{N_{j}+N_{j+1}-1}\Delta t_{j,j+1}(\ell)H(\mathbb{X}_{j,j+1}(1:\ell))\;. \end{aligned}$$ Thus, whenever $\mathcal L_{j,j+1}< \dhinterseq$, [\[eq:contribution-to-mixed\]](#eq:contribution-to-mixed){reference-type="eqref" reference="eq:contribution-to-mixed"}, [\[eq:contribution-to-mixed-1\]](#eq:contribution-to-mixed-1){reference-type="eqref" reference="eq:contribution-to-mixed-1"} and [\[eq:contribution-to-mixed-2\]](#eq:contribution-to-mixed-2){reference-type="eqref" reference="eq:contribution-to-mixed-2"} give the joint contribution from ${\rm{SB}}_{j-1}$ and ${\rm{SB}}_j$: $$\begin{aligned} \label{eq:contribution-to-mixed-3} \sum_{i=1}^{N_{j}+N_{j+1}-1} \Delta t_{j,j+1}(i) H( \mathbb{X}_{j,j+1}{(1:i)} )\;.\end{aligned}$$ Similar analysis applies to (II) and (III) combined, and we obtain - the joint contribution from ${\rm{SB}}_j$ and ${\rm{SB}}_{j+1}$: $$\begin{aligned} \label{eq:contribution-to-mixed-4} \sum_{i=1}^{N_{j}+N_{j+1}-1} \Delta t_{j,j+1}(i) H( \mathbb{X}_{j,j+1}{(i+1:N_j+N_{j+1})} )\;.\end{aligned}$$ The case $q=0$ and $s=N_j$ needs a special care. - We take $q=0$ and $s=N_{j+1}$ in (III). Then the contribution is $H(\mathbb{X}_{j,j+1})$. It stems from ${\rm{SB}}_j$ only. - Start with (III-b). If $t_{j+1}{(N_{j+1})}-\dhinterseq< t_{j}(1)$ then we have the contribution for $\dhinterseq-(t_{j+1}{(N_{j+1})}-t_{j}(1))$ of the indices. - The cases (III-c), (III-d) and (III-e) are empty. - Hence, the contribution is $$\begin{aligned} \label{eq:contribution-to-mixed-5} \left(\dhinterseq-\left(t_{j+1}{(N_{j+1})}-t_{j}(1)\right)\right)_+ H( \mathbb{X}_{j,j+1} )\;. \end{aligned}$$ We combine [\[eq:contribution-to-mixed-5\]](#eq:contribution-to-mixed-5){reference-type="eqref" reference="eq:contribution-to-mixed-5"} with ${\rm{DB}}_{j,j+1}=\dhinterseq H( \mathbb{X}_{j,j+1} )$ to get $$\begin{aligned} \label{eq:contribution-to-mixed-6} &\left(\dhinterseq-\left(t_{j+1}{(N_{j+1})}-t_{j}(1)\right)\right)\mathbbm{1}{\left\{t_{j+1}{(N_{j+1})}-t_{j}(1)<\dhinterseq\right\}} H( \mathbb{X}_{j,j+1} )-{\rm{DB}}_{j,j+1}\nonumber\\ &= -\left(t_{j+1}{(N_{j+1})}-t_{j}(1)\right)\mathbbm{1}{\left\{t_{j+1}{(N_{j+1})}-t_{j}(1)<\dhinterseq\right\}}H( \mathbb{X}_{j,j+1} )\\ &\phantom{=}-\dhinterseq \mathbbm{1}{\left\{t_{j+1}{(N_{j+1})}-t_{j}(1)\geqslant \dhinterseq\right\}}H( \mathbb{X}_{j,j+1} )\;. \nonumber\end{aligned}$$ Now, with the re-numeration for the adjacent blocks: $$\begin{aligned} t_{j+1}{(N_{j+1})}-t_{j}(1)=t_{j,j+1}{(N_j+N_{j+1})}-t_{j,j+1}(1)=\sum_{i=1}^{N_j+N_{j+1}-1}\Delta t_{j,j+1}(i)\;. \end{aligned}$$ Combining this with [\[eq:contribution-to-mixed-3\]](#eq:contribution-to-mixed-3){reference-type="eqref" reference="eq:contribution-to-mixed-3"}-[\[eq:contribution-to-mixed-6\]](#eq:contribution-to-mixed-6){reference-type="eqref" reference="eq:contribution-to-mixed-6"} we have $$\begin{aligned} &\big[ {\rm{SB}}_{j-1} + {\rm{SB}}_j + {\rm{SB}}_{j+1} - {\rm{DB}}_{j,j+1} \big]\mathbbm{1}{\left\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\right\}}\\ &= \sum_{i=1}^{N_{j}+N_{j+1}-1} \Delta t_{j,j+1}(i) \left\{ H( \mathbb{X}_{j,j+1}{(1:i)} ) + H( \mathbb{X}_{j,j+1}{(i+1:N_{j}+N_{j+1})} ) - H( \mathbb{X}_{j,j+1} ) \right\}\end{aligned}$$ whenever $\mathcal L_{j,j+1}<\dhinterseq$. ## Extremal index case #### Explanation for . - On $A_{j-1}^c\cap A_j$: for $i\in I_{j-1}$, we have $H(\boldsymbol{X}_{i+1,i+\dhinterseq}/\tepseq)=1$ if and only if $i+\dhinterseq-1\geqslant t_{j}(1)$. Hence, ${\rm{SB}}_{j-1}(H)=(j-1)\dhinterseq-(t_{j}(1)-\dhinterseq)$. - On $A_j\cap A_{j+1}^c$: for $i\in I_{j}$, we have $H(\boldsymbol{X}_{i+1,i+\dhinterseq}/\tepseq)=1$ if and only if $i\leqslant t_{j}{(N_j)}$. Hence, ${\rm{SB}}_{j}(H)=t_{j}{(N_j)}-(j-1)\dhinterseq$. - On $A_j$, ${\rm{DB}}_j(H)=\dhinterseq$. Hence, on $A_{j-1}^c\cap A_j\cap A_{j+1}^c$, we have $$\begin{aligned} {\rm{SB}}_{j-1}(H)+{\rm{SB}}_{j}(H)-{\rm{DB}}_j(H)=t_{j}{(N_j)}-t_{j}(1)\;.\end{aligned}$$ #### Explanation for . - On $A_{j-1}^c\cap A_j$: for $i\in I_{j-1}$, we have $H(\tepseq^{-1}\boldsymbol{X}_{i+1,i+\dhinterseq})=1$ if and only if $i+\dhinterseq-1\geqslant t_{j}(1)$. Hence, ${\rm{SB}}_{j-1}(H)=(j-1)\dhinterseq-(t_{j}(1)-\dhinterseq)=j\dhinterseq-t_{j}(1)$. - On $A_{j+1}\cap A_{j+2}^c$: for $i\in I_{j+1}$, we have $H(\tepseq^{-1}\boldsymbol{X}_{i+1,i+\dhinterseq})=1$ if and only if $i\leqslant t_{j+1}{(N_{j+1})}$. Hence, ${\rm{SB}}_{j+1}(H)=t_{j+1}{(N_{j+1})}-j\dhinterseq$. - On $A_j\cap A_{j+1}$, for $i\in I_j$, $H(\tepseq^{-1}\boldsymbol{X}_{i,i+\dhinterseq-1})\equiv 0$ if and only if $i>t_{j}{(N_j)}$ and $i+\dhinterseq-1<t_{j+1}(1)$. Thus, ${\rm{SB}}_j(H)=\dhinterseq- (t_{j+1}(1)-t_{j}{(N_j)}-\dhinterseq)_+$. Thus, on $\{A_{j-1}^c\cap A_j\cap A_{j+1}\cap A_{j+2}^c\}$ $$\begin{aligned} &{\rm{SB}}_{j-1} + {\rm{SB}}_j + {\rm{SB}}_{j+1} - {\rm{DB}}_{j,j+1}=\left[\left(t_{j+1}{(N_{j+1})}-t_{j}(1)\right)-\left(t_{j+1}(1)-t_{j}{(N_j)}-\dhinterseq\right)_+\right] \;. \end{aligned}$$ # Acknowledgement {#acknowledgement .unnumbered} The authors would like to thank Stanislav Volgushev for indicating the expansion problem. The authors are grateful to Philippe Soulier for an extensive discussion. Both authors were supported by an NSERC grant. A part of this research was conducted during the second author's stay at the Sydney Mathematical Research Institute (October - November 2022). Rafał Kulik thanks for the support and hospitality provided by SMRI. # Bibliography 10 Bojan Basrak, Hrvoje Planinić, and Philippe Soulier. An invariance principle for sums and record times of regularly varying stationary sequences. , 172(3-4):869--914, 2018. Bojan Basrak and Johan Segers. Regularly varying multivariate time series. , 119(4):1055--1080, 2009. Axel Bücher and Johan Segers. Inference for heavy tailed stationary time series based on sliding blocks. , 12(1):1098--1125, 2018. Axel Bücher and Chen Zhou. A horse racing between the block maxima method and the peak over threshold approach. , 36(3):360--378, 2021. Gloria Buriticá, Thomas Mikosch, and Olivier Wintenberger. Large deviations of $\ell^p$-blocks of regularly varying time series and applications to cluster inference. , 161:68--101, 2023. Zaoli Chen and Rafal Kulik. Limit theorems for unbounded cluster functionals of regularly varying time series. In preparation. Youssouph Cissokho and Rafal Kulik. Estimation of cluster functionals for regularly varying time series: sliding blocks estimators. , 15(1):2777--2831, 2021. Youssouph Cissokho and Rafal Kulik. Estimation of cluster functionals for regularly varying time series: runs estimators. , 6(1):3561--3607, 2022. Richard A. Davis and Tailen Hsing. Point process and partial sum convergence for weakly dependent random variables with infinite variance. , 23(2):879--917, 1995. Holger Drees and Sebastian Neblung. . , 27(2):1239 -- 1269, 2021. Holger Drees and Holger Rootzén. Limit theorems for empirical processes of cluster functionals. , 38(4):2145--2186, 2010. Tailen Hsing. Estimating the parameters of rare events. , 37(1):117--139, 1991. Rafał Kulik and Philippe Soulier. . Springer, 2020. Thomas Mikosch and Olivier Wintenberger. Precise large deviations for dependent regularly varying sequences. , 156(3-4):851--887, 2013. Hrvoje Planinić and Philippe Soulier. The tail process revisited. , 21(4):551--579, 2018. Richard L. Smith and Ishay Weissman. Estimating the extremal index. , 56(3):515--528, 1994. [^1]: University of Ottawa [^2]: Corresponding author: University of Ottawa, rkulik\@uottawa.ca

arxiv_math

--- abstract: | For $d$ even, we prove that on a $d$-dimensional Riemannian manifold with constant sectional curvature, the distance set of a Borel set $E$ has positive Lebesgue measure if the Hausdorff dimension of $E$ is greater than $\frac d2+\frac14.$ address: - School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, PR China - Department of Mathematics & International Center for Mathematics, Southern University of Science and Technology, Shenzhen, 518055, PR China - School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, PR China author: - Changbiao Jian - Bochen Liu - Yakun Xi bibliography: - mybibfile.bib title: Falconer distance problem on Riemannian manifolds with constant sectional curvature --- # Introduction {#section1} For a compact subset $E \subseteq \mathbb{R}^d$, where $d \geq 2$, we define its distance set $\Delta(E)$ as: $$\Delta(E)=\{|x-y|:x,y\in E\}.$$ In 1986, Falconer introduced a famously challenging problem in geometric measure theory and harmonic analysis [@Fal]. He was interested in how large the Hausdorff dimension of $E$ needs to be to ensure that the Lebesgue measure of $\Delta(E)$ is positive. Falconer conjectured that the Lebesgue measure of $\Delta(E)$ is positive when $\dim_{\mathcal{H}}(E) > \frac{d}{2}$ and proved this result under a slightly weaker assumption: $\dim_{\mathcal{H}}(E) > \frac{d + 1}{2}$. Over the following decades, this conjecture has drawn significant attention. Many remarkable partial results have been established by Mattila [@87], Bourgain [@94], Wolff [@99], Erdogan [@05; @06], Du and Zhang[@du-zhang], Du, Guth, Ou, Wang, Wilson and Zhang[@du-guth]. Nonetheless, Falconer's conjecture in its full strength is still open in all dimensions. The best result in dimension 2 is due to Guth, Iosevich, Ou, and Wang [@guth2020]. They proved the Lebesgue measure of $\Delta(E)$ is positive if $\dim_{\mathcal{H}}E>\frac{5}{4}$ in the plane. Before long, their method was generalized to all even dimensions by Du, Iosevich, Ou, Wang, and Zhang [@du2021]. They showed that the distance set of $E\subset\mathbb R^d$ will have positive Lebesgue measure if $\dim_{\mathcal{H}}(E)>\frac{d}{2}+\frac14$ for all even $d>2$. In a recent work by Du, Ou, Ren, and Zhang [@kevin], they are able to improve the dimensional exponent in every dimension $d\geq 3$. Exploring this conjecture on Riemannian manifolds is intriguing as well. Let $(M,g)$ be a boundaryless Riemannian manifold of dimension $d\geq2$. For a set $E\subset M$, define its distance set as $$\Delta_g(E)=\{d_g(x,y): x,y\in E\},$$ where $d_g$ is the associated Riemannian distance function. It is then a natural question: how large must the Hausdorff dimension of $E$ be to ensure that the Lebesgue measure of $\Delta_{g}(E)$ is positive? Similar to the work of Falconer [@Fal], the Lebesgue measure of $\Delta_g(E)$ can be shown to be positive if $\dim_{\mathcal{H}}(E)>\frac{d}{2}+\frac{1}{2}$, which is a consequence of the generalized projection theorem of Peres--Schlag [@R00]. Their generalized projection theorem implies a stronger result, that is, the Lebesgue measure of the pinned distance set $$\Delta_{g,x}(E):=\{d_g(x,y): y\in E\}$$ is positive under the same conditions. Several authors have studied this Riemannian version of Falconer's conjecture. Eswarathasan--Iosevich--Taylor [@R11] and Iosevich--Taylor--Uriarte--Tuero [@R21] studied more general distance set problems via Fourier integral operators. Their results imply the same bounds as Peres--Schlag for the Riemannian case. Using local smoothing estimates of Fourier integral operators, Iosevich and Liu [@R19] improved Peres and Schlag's results if the pins are required to lie in the given set itself. Recently, Iosevich, Liu, and Xi [@xi2022] extended the result of Guth, Iosevich, Ou, and Wang [@guth2020] in $\mathbb R^2$ to general Riemannian surfaces. The primary purpose of this paper is to generalize the even-dimensional result of Du--Iosevich--Ou--Wang-Zhang to the Riemannian setting. **Theorem 1**. *Let $d>2$ be an even integer. Let $(M,g)$ be a boundaryless $d$-dimensional Riemannian manifold with constant sectional curvature. If $E\subset M$ and $\dim_{\mathcal{H}}(E)>\frac{d}{2}+\frac{1}{4}$, then the Lebesgue measure of $\Delta_{g}(E)$ is positive.* Similar to recent works on Falconer's distance problem, we establish a slightly stronger version of the main theorem concerning the pinned distance set. **Theorem 2**. *Let $d>2$ be an even integer. Let $(M,g)$ be a boundaryless $d$-dimensional Riemannian manifold with constant sectional curvature. If $E\subset M$ and $\dim_{\mathcal{H}}(E)>\frac{d}{2}+\frac{1}{4}$, then there is a point $x\in E$ so that the Lebesgue measure of its pinned distance set $\Delta_{g,x}(E)$ is positive.* The proof of Theorem [Theorem 1](#t0){reference-type="ref" reference="t0"} and [Theorem 2](#t1){reference-type="ref" reference="t1"} is inspired by key strategies introduced in [@guth2020], [@du2021], and [@xi2022]. In [@guth2020], the authors employ Liu's approach [@L19] and further improve it by removing some "bad\" wave packets at different scales. To do so, they first make use of a radial projection of Orponen [@orponen19] to justify the deletion of such "bad\" wave packets. Then, they prove a refined decoupling inequality, which provides a gain once the "bad\" wave packets are removed. At first, it seems difficult to generalize this result to higher dimensions since Orponen's radial projection theorem [@orponen19] is only effective for sets with dimensions greater than $d-1$. Nonetheless, in the work of Du--Iosevich--Ou--Wang--Zhang [@du2021], they overcome this difficulty by using orthogonal projections to project the original measure onto a suitable lower dimensional space so that Orponen's theorem can be exploited. In [@xi2022], refined microlocal decoupling inequalities are established on Riemannian manifolds, which are inspired by Beltran, Hickman, and Sogge's work on variable coefficient decoupling inequalities [@sogge20]. Moreover, Orponen's radial projection theorem has been extended to the case of Riemannian surfaces. Unlike Euclidean space, where orthogonal projections are readily available, the lack of a natural analog of the orthogonal projection on Riemannian manifolds poses a serious challenge. This is the sole reason we must impose the constant sectional curvature condition. We introduce a family of submanifolds associated with each element $V$ in the Grassmannian ${\bf Gr}(d/2,d)$ on the tangent space of a fixed point $p_0\in M$. Fixing $V$, we consider all totally geodesic submanifolds $\Sigma_{V,p}$ of dimension $d/2$ which are orthogonal to $\exp_{p_0}(V)$ at some point $p\in\exp_{p_0}(V)$. The existence of such totally geodesic submanifolds is the key consequence of our assumption that $M$ has constant sectional curvature. Notice that $\{\Sigma_{V,p}\}_{p\in\exp_{p_0}(V)}$ is a local foliation of $M$ near the point $p_0$. With this foliation, we are able to define a natural analog of the orthogonal projection on $M$ and adopt the approach in [@du2021]. The constant sectional curvature assumption could be a natural condition for the Falconer problem on higher dimensional Riemannian manifolds. In the recent work of Guo, Wang, and Zhang [@guo23] on Hömander-type oscillatory integral operators, they put forward a curvature condition for phase functions in addition to the Carleson--Sjölin condition. They call this condition Bourgain's condition. They show that whenever Bourgain's condition fails, then the $L^{\infty}\to L^q$ boundedness always fails for some $q=q(n)>\frac{2n}{n-1}$. Therefore, Bourgain's condition is a natural additional condition for studying oscillatory integral operator bounds associated with Carleson--Sjölin phase functions, which are unnecessary in dimension 2. The phase function relevant to our problem is the Riemannian distance function $\phi(x,y)=d_g(x,y)$. It is well-known that such a phase function satisfies the Carleson--Sjölin condition. We learned from Guo [@shaoming] that the phase function $d_g(x,y)$ satisfying Bourgain's condition is equivalent to the underlying manifold having constant sectional curvature. This paper is organized as follows. In Section [2](#section2){reference-type="ref" reference="section2"}, we establish the framework for studying Falconer's problem Riemannian manifolds. We perform some standard reductions and outline the two main estimates needed. Sections [3](#section3){reference-type="ref" reference="section3"} and [4](#section4){reference-type="ref" reference="section4"} house the proof of these two main estimates, respectively. In Section [5](#section5){reference-type="ref" reference="section5"}, we prove Lemma [Lemma 4](#l3){reference-type="ref" reference="l3"} and complete the proof of Theorem [Theorem 2](#t1){reference-type="ref" reference="t1"}. **Notation.** We use the notation $A\lesssim B$, if there exists an absolute constant $C$ such that $A\leq CB$. We shall employ the notation $A\sim B$ when both $A\lesssim B$ and $B\lesssim A$. Throughout this article, $\mathrm{RapDec}(R)$ refers to quantities that rapidly decay as $R\to\infty$, that is, there is a constant $C_N$ so that $\mathrm{RapDec}(R)\leq C_NR^{-N}$ for arbitrarily large $N$. $\dim_{\mathcal{H}}$ denotes the Hausdorff dimension. $B^d(x,r)$ denotes the $d$-dimensional geodesic ball centered at $x$ with a radius of $r$. Similarly, $S^{d-1}(x,r)$ denotes the geodesic sphere. We shall omit the dimensions if it is clear from context. **Acknowledgement.** This project is supported by the National Key Research and Development Program of China No. 2022YFA1007200 and NSF China Grant No. 12131011 and 12171424. The authors would like to thank Kevin Ren for helpful conversations. # Standard reductions and the Main estimates {#section2} Let $(M,g)$ be a $d$-dimensional Riemannian manifold, and let $\alpha>0$. Consider a compact set $E\subset M$ with Hausdorff dimension $>\alpha$. By possibly rescaling the metric, we assume that $E$ is contained in the unit geodesic ball centered at a point $p_0$, so that in the geodesic normal coordinate about $p_0$, the metric $g$ is approximately flat, in the sense that for all $i,j$, $|g_{ij}-\delta_{ij}|<\epsilon_0$ for some universal constant $\epsilon_0$ that we can choose to be small. We select two disjoint compact subsets $E_1\subset B_1$ and $E_2\subset B_2$ of $E$ such that $$\dim_{\mathcal{H}}(E_1), \dim_{\mathcal{H}}(E_2)>\alpha$$ and $d_g(E_1,E_2)\sim1$. Here, $B_1$ and $B_2$ denote two geodesic balls with radius $\frac{1}{100}$, and the distance between $E_1$ and $E_2$ is defined as $$d_g(E_1,E_2):=\inf_{x\in E_1,y\in E_2}d_g(x,y)>0.$$ Additionally, each $E_j$ admits a probability measure $\mu_j$ such that $\mathrm{supp}~\mu_j\subset E_j$ and $\mu_j(B(x,r))\lesssim r^{\alpha}$. For any fixed $x\in E_2$, we define the pinned distance function associated with $x$ as $$d^x_g(y) := d_g(x,y).$$ We then define the pushforward measure $d^x_*(\mu) = (d_g)^x_*(\mu)$ of $\mu$ through the relation: $$\int_{\mathbb{R}}h(t)\,d^x_*(\mu)=\int_Mh(d_g(x,y))\,d\mu(y).$$ Next, we establish the framework of our problem for the Riemannian setting and construct a complex-valued measure $\mu_{1, \text{good}}$, which corresponds to the "good\" part of $\mu_1$ concerning $\mu_2$. Then, we shall investigate the pushforward of this measure under $d^x_g(y)$ and reduce the problem to two main estimates. Let us now proceed to set up the problem. We start with a microlocal decomposition, a higher dimensional analog of that in [@xi2022]. We work in the geodesic normal coordinates ${(x', x_d)}\in\mathbb{R}^{d-1}\times\mathbb{R}$ about the point $p_0$. We may assume that $p_0$ lays near the middle point between $B_1$ and $B_2$ so that $E_1\subset B_1$ and $E_2\subset B_2$ are approximately $1$-separated by their $d$-th coordinate. Let $R_0$ be a sufficiently large number, denoted by $R_j=2^jR_0$. Cover the annulus region $R_{j-1}\leq|\xi|\leq R_j$ where $\left\{(\xi',\xi_d): \frac{|\xi_d|}{|\xi|}\geq\frac{1}{10}\right\}$ by rectangular blocks $\tau$. These blocks have dimensions $R_j^{1/2}\times\cdots\times R_j^{1/2}\times R_j$, where the long direction of each block aligns with the radial direction of the annulus. To account for the remaining portion of the annulus, we use two larger blocks to cover them. The corresponding smooth partition of unity subordinate to this cover is the following. $$1=\psi_0+\sum_{j\geq1}\big[\sum_{\tau}\psi_{j,\tau}+\psi_{j,-}+\psi_{j,+}\big],$$ where $\psi_{j,\tau}$ is supported in $\tau$, $\psi_{j,+}$ is a smooth bump function adapted to the set: $$\Big\{(\xi',\xi_d): R_{j-1}\leq|\xi|\leq R_j, 0\leq\frac{\xi_d}{|\xi|}\leq\frac{1}{10}\Big\}.$$ Similarly $\psi_{j,-}$ is a smooth bump function adapted to the set: $$\Big\{(\xi',\xi_d): R_{j-1}\leq|\xi|\leq R_j, -\frac{1}{10}\leq\frac{\xi_d}{|\xi|}\leq0\Big\}.$$ Let $\delta>0$ be a small constant. Denote $T_0$ by the geodesic tube around the $x_d$-axis with cross-section radius $\frac{1}{10}$. For each pair $(j,\tau)$, we consider the collection of geodesics $\gamma$ such that each $\gamma$ intersects the hyperplane $\{x_d=0\}$ and the tangent vector at the intersection point is pointing in the direction of $\tau$ in our coordinate system. For each such $\gamma$, we define the associated tube $T$ to be a geodesic tube with central geodesic $\gamma$ of dimension $R_j^{-1/2+\delta}\times\cdots\times R_j^{-1/2+\delta}\times1$. We then choose a subcollection of such tubes, $\mathbb{T}_{j,\tau}$, so that tubes in $\mathbb{T}_{j,\tau}$ cover $T_0$ with bounded overlap. We then define $\mathbb{T}_j=\cup_{\tau}\mathbb{T}_{j,\tau}$ as well as $\mathbb{T}=\cup_{j}\mathbb{T}_j$. Let $\eta_T$ be a smooth partition of unity subordinate to this covering. That is, for each pair $(j, \tau)$, we have $\sum_{T\in\mathbb{T}_{j,\tau}}\eta_T$ equals to 1 in the unit ball. Similar to [@xi2022], we need a microlocal decomposition for functions with support in $B_1$. Fix $(j,\tau)$, for each $(x,\xi)$, there exists a unique point $z=(z'(x,\xi),0)$ on the hyperplane $\{x_d=0\}$ such that the geodesic connecting $z$ and $x$ has tangent vector $\xi/|\xi|$ at $x$. We then define $\Phi(x,\xi/|\xi|)$ to be the unit tangent vector of this geodesic at the point $z$ and extend it to be a homogeneous function of degree 1 in $\xi$ so that $|\Phi(x,\xi)|=|\xi|$. Let $f$ be a function with $\mathrm{supp}(f)\subset B_1$, and for each $(j,\tau)$, we define $$\mathcal P_{j,\tau}f(x)=\iint_{\mathbb{R}^d\times\mathbb{R}^d}e^{2\pi i(x-y)\cdot\xi}\psi_{j,\tau}(\Phi(x,\xi))f(y)\,dyd\xi.$$ Similarly, we define $$\mathcal P_{j,\tau}f(x)=\iint_{\mathbb{R}^d\times\mathbb{R}^d}e^{2\pi i(x-y)\cdot\xi}\psi_{j,\pm}(\Phi(x,\xi))f(y)\,dyd\xi,$$ and $$\mathcal P_{j,0}f(x)=\iint_{\mathbb{R}^d\times\mathbb{R}^d}e^{2\pi i(x-y)\cdot\xi}\psi_{j,0}(\Phi(x,\xi))f(y)\,dyd\xi.$$ Consequently, $\mathcal P_0+\sum_{j\geq1}[\sum_{\tau}\mathcal P_{j,\tau}+\mathcal P_{j,+}+\mathcal P_{j,-}]$ is the identity operator. Now, for each $T\in\mathbb{T}_{j,\tau}$, we define $$M_Tf:=\eta_T\mathcal P_{j, \tau}f.$$ Similarly, we define $M_{j,\pm}f=\eta_{T_0}\mathcal P_{j,\pm}f$ and $M_0f=\eta_{T_0}\mathcal P_0f$, where $\eta_{T_0}$ is a smooth function satisfying $\eta_{T_0}=1$ on $T_0$ and decays rapidly away from it. With these preparations in place, we now construct the "good\" measure $\mu_{1, good}$ by removing certain "bad\" wave packets from $\mu_1$. We shall show that the error between $d^x_*(\mu_{1,\text{\rm good}})$ and $d^x_*(\mu_1)$ remains sufficiently small in the sense of $L^1$ norm. Denote $4T$ as a concentric tube of $T$ with four times the cross-section radius. Consider a large constant $c(\alpha)=c(\alpha,d)>0$ which shall be determined later. For each tube $T\in\mathbb{T}_{j,\tau}$, we categorize it as a bad tube if $$\mu_2(4T)\geq R_j^{-d/4+c(\alpha)\delta}.$$ Otherwise, we consider $T$ as a good tube. Accordingly, we define $$\mu_{1,\text{\rm good}}:=M_0\mu_1+\sum_{T\in\mathbb{T},T~\text{is good}}M_T\mu_1.$$ In [@guth2020][@du2021], the authors established that good tubes make primary contributions in even-dimensional Euclidean space. We generalize this result to the Riemannian setting. We will prove the following two propositions in the Riemannian framework. Indeed, Theorem [Theorem 2](#t1){reference-type="ref" reference="t1"} is a consequence of these two propositions. **Proposition 1**. *Let $d\geq 2$ be an even integer and $(M,g)$ be a boundaryless Riemannian manifold with constant sectional curvature of dimension $d$. If $\alpha>\frac{d}{2}$ and $R_0$ is large enough, then there exists a subset $E'_2\subset E_2$ such that $\mu_2(E'_2)\geq1-\frac{1}{1000}$ and for any $x\in E'_2$, $$\|d^x_*(\mu_1)-d^x_*(\mu_{1,\text{\rm good}})\|_{L^1}<\frac{1}{1000}.$$* **Proposition 2**. *Let $d\geq 2$ be an even integer. If $\alpha>\frac{d}{2}+\frac{1}{4}$, then for sufficiently small $\delta=\delta(\alpha)$, we have $$\int_{E_2}\|d^x_*(\mu_{1,\text{\rm good}})\|_{L^2}^2\,d\mu_2(x)<+\infty.$$* We now apply the above two propositions to establish the main theorem. We shall give the proof of Proposition [Proposition 1](#p1){reference-type="ref" reference="p1"} and Proposition [Proposition 2](#p2){reference-type="ref" reference="p2"} in Section [3](#section3){reference-type="ref" reference="section3"} and Section [4](#section4){reference-type="ref" reference="section4"} respectively. *Proof of Theorem [Theorem 2](#t1){reference-type="ref" reference="t1"}.* The proof is identical to that in [@guth2020; @xi2022; @du2021]. We include it here for completeness. $$\begin{aligned} \int_{\Delta_{g,x}(E)}|d^x_*(\mu_{1,\text{\rm good}})|&=\int|d^x_*(\mu_{1,\text{\rm good}})|-\int_{\Delta_{g,x}(E)^c}|d^x_*(\mu_{1,\text{\rm good}})|\\ &\geq\int|d^x_*(\mu_{1})|-2\int|d^x_*(\mu_1)-d^x_*(\mu_{1,\text{\rm good}})|\\ &\geq1-\frac{2}{1000}. \end{aligned}$$ Here, we have used the fact that $d^x_*(\mu_1)$ is a probability measure. The final inequality is a direct consequence of Proposition [Proposition 1](#p1){reference-type="ref" reference="p1"}. Conversely, employing Proposition [Proposition 2](#p2){reference-type="ref" reference="p2"} and Cauchy-Schwarz, we obtain $$\Big(\int_{\Delta_{g,x}(E)}|d^x_*(\mu_{1,\text{\rm good}})|\Big)^2\leq |\Delta_{g,x}(E)|\cdot\|d^x_*(\mu_{1,\text{\rm good}})\|^2_{L^2}\lesssim|\Delta_{g,x}(E)|.$$ Hence, we conclude that $|\Delta_{g,x}(E)|$ is positive. ◻ # proof of Proposition [Proposition 1](#p1){reference-type="ref" reference="p1"} {#section3} In this section, we establish Proposition [Proposition 1](#p1){reference-type="ref" reference="p1"} by studying the pushforward measures $d^x_*(\mu_1)$ and $d^x_*(\mu_{1,\text{\rm good}})$. We will apply the geometric properties of bad tubes to control $\|d^x_*(\mu_1)-d^x_*(\mu_{1,\text{\rm good}})\|_{L^1}$. To commence, let us recall the following lemma, which is essentially from [@xi2022]. **Lemma 3**. *For each point $x\in E_2,$ define $$\text{\rm Bad}_j(x):=\mathop{\bigcup}_{T\in\mathbb{T}_j:\, x\in 2T~ \text{and}~T~\text{is}~ \text{bad}}2T,~~~\forall j\geq1.$$ Then one has $$\|d^x_*(\mu_{1,\text{\rm good}})-d^x_*(\mu_1)\|_{L^1}\lesssim\sum_{j\geq1}R^{\delta}_j\mu_1(\text{\rm Bad}_j(x))+\mathrm{RapDec}(R_0).$$* *Proof.* The only differences between the above lemma and [@xi2022 Lemma 7.3] are in the definitions of bad tubes and the dimensions of the underlying manifolds. By the arguments in [@xi2022] (and [@guth2020; @du2021]), it suffices to prove that for $x\in E_2$ and $x\notin 2T$, $$\label{l21} \|d^x_*(M_T\mu_1)\|_{L^1}\lesssim \mathrm{RapDec}(R_j),$$ and for any $x\in E_2$, $$\label{l23} \|d^x_*(M_{j,\pm}\mu_1)\|_{L^1}\lesssim \mathrm{RapDec}(R_j).$$ Now we prove [\[l21\]](#l21){reference-type="eqref" reference="l21"}. Recall that $$d^x_*(M_T\mu_1)(t)=\int_{S^{d-1}(x,t)}M_T\mu_1(y)\,d\sigma(y),$$ where $\sigma(y)$ is the formal restriction of $\mu_2$ on the geodesic sphere $$S^{d-1}(x,t):=\{y: d_g(x,y)=t\}.$$ Noting the separation of $E_1$ and $E_2$, we may assume that $t\approx1$, otherwise $d^x_*(M_T\mu_1)(t)$ is negligible. Denote $$A_{j,\tau}(x,\xi)=\psi_{j,\tau}(\Phi(x,\xi)).$$ Since $|\hat{\mu}_1(\xi)|\leq1$ and $$\begin{aligned} M_T\mu_1&=\eta_T(y)\iint e^{2\pi i(y-z)\cdot\xi}A_{j,\tau}(z,\xi)\,d\mu_1(z)d\xi\\ &=\eta_T(y)\int e^{2\pi iy\xi}\hat{\mu}_1(\xi)A_{i,\tau}(y,\xi)\,d\xi,\end{aligned}$$ It suffices to show that one has, uniformly in $\xi$, $$\label{l22} \bigg|\int_{S^{d-1}(x,t)}\eta_T(y)A_{j,\tau}(y,\xi)e^{2\pi iy\cdot\xi}\,d\sigma(y)\bigg|\lesssim \mathrm{RapDec}(R_j).$$ The tube $T$ intersects $S^{d-1}(x,t)$ in at most two caps, which can be dealt with in similar fashions. It suffices to consider a single cap. Denote by $y_0$ the center of the cap. Since $x\notin 2T,\ y\in T\cap S^{d-1}(x,t)$ and $\xi$ is essentially in $\tau$, we have that the angle between vectors $\frac{y-x}{|y-x|}$ and $\frac{\xi}{|\xi|}$ is greater than $R_j^{-1/2+\delta}.$ By a change of variable $z=y-x$, we see that [\[l22\]](#l22){reference-type="eqref" reference="l22"} equals $$\label{l33} \bigg|e^{2\pi ix\cdot\xi}\int_{S^{d-1}(0,t)}\eta_T(z+x)A_{j,\tau}(z+x,\xi)e^{2\pi iz\cdot\xi}\,d\sigma(z)\bigg|.$$ Without loss of generalities, in geodesic normal coordinates about $x$, we may assume that $z_0=y_0-x$ is given by $$z_0=y_0-x=z(0,\theta_0)=(0,\ldots, 0,t\cos\theta_0,t\sin\theta_0),$$ and $$\frac{\xi}{|\xi|}=(0,\ldots, 0,\cos\omega,\sin\omega),$$ for some $\theta_0,\omega\in[0,2\pi).$ More generally, we employ the following change of coordinates: $$\begin{aligned} B^{d-2}(0,t)\times[0,2\pi)&\to S^{d-1}(0,t):\\(m,\theta)&\to z(m,\theta):=\big(m,\sqrt{t^2-|m|^2}\cos\theta,\sqrt{t^2-|m|^2}\sin\theta\big).\end{aligned}$$ The associated Jacobian is $\approx 1$ in our domain of integration since $t\approx1$, thus [\[l33\]](#l33){reference-type="eqref" reference="l33"} is reduced to $$\label{l44} \bigg|\int_{B^{d-2}(0,t)}\int^{2\pi}_0\eta_T(z(m,\theta)+x)A_{j,\tau}(z(m,\theta)+x, \xi)e^{2\pi i|\xi|\sqrt{t^2-|m|^2}\cos(\omega-\theta)}\,d\theta dm\bigg|.$$ Since the intersection angle between $\frac{y-x}{|y-x|}$ and $\frac{\xi}{|\xi|}$ is greater than $R_j^{-1/2+\delta}$, we have $|\theta_0-\omega|\gtrsim R_j^{-1/2+\delta}$. Furthermore, we have $|\theta-\omega|\gtrsim R_j^{-1/2+\delta}$ if $z\in S^{d-1}(0,t)$ and $z+x\in T$. Since the cap of $T\cap S^{d-1}(x,t)$ has dimensions $R^{-1/2+\delta}\times\cdots\times R^{-1/2+\delta}_j$ and center $z_0=(0,t\cos\theta_0,t\sin\theta_0)$, we have $|m|\lesssim R_j^{-1/2+\delta}$. Note that $t\approx1$, so we get $$\frac{d}{d\theta}(|\xi|\sqrt{t^2-m^2}\cos(\omega-\theta))=|\xi|\sqrt{t^2-m^2}\sin(\theta-\omega)\gtrsim R_j^{1/2+\delta}.$$ Since $$\partial^{(k)}_{\theta}[\eta_T(z(m,\theta)+x)A_{j,\tau}(z(m,\theta)+x,\xi)]\lesssim R_j^{k(1/2-\delta)(d-1)},$$ an integration by parts argument in the $\theta$ variable shows that [\[l44\]](#l44){reference-type="eqref" reference="l44"} decays rapidly in $R_j$. The bound [\[l23\]](#l23){reference-type="eqref" reference="l23"} can be proved similarly if we replace $T$ by $T_0$. ◻ Applying lemma [Lemma 3](#l2){reference-type="ref" reference="l2"}, Proposition [Proposition 1](#p1){reference-type="ref" reference="p1"} is reduced to estimating the measure of $\text{\rm Bad}_j(x)$. We define $$\text{\rm Bad}_j:=\{(x_1,x_2)\in E_1\times E_2: \text{there is a bad tube } T\in\mathbb{T}_j~\text{so that }~2T~\text{contains}~x_1~\text{and}~x_2\},$$ The following lemma holds for a manifold $M$ with constant sectional curvature. **Lemma 4**. *Let $d\geq 2$ be an even integer. If $\alpha>\frac{d}{2}$, then for sufficiently large $c(\alpha)>0$ and every $j\geq1,$ we have $$\mu_1\times\mu_2(\text{\rm Bad}_j)\lesssim R_j^{-{2\delta}}.$$* We shall postpone the proof of Lemma [Lemma 4](#l3){reference-type="ref" reference="l3"} to the last section and show that Proposition [Proposition 1](#p1){reference-type="ref" reference="p1"} follows from Lemma [Lemma 4](#l3){reference-type="ref" reference="l3"}. *Proof of Proposition [Proposition 1](#p1){reference-type="ref" reference="p1"}.* Notice that $$\mu_1\times\mu_2(\text{\rm Bad}_j)=\int\mu_2(\text{\rm Bad}_j(x))\,d\mu_1(x).$$ For every $j\geq1$, one can choose $A_j\subset E_2$ such that $\mu_2(A_j)\leq R_j^{-(1/2)\delta}$ and $$\mu_1(\text{\rm Bad}_j(x))\lesssim R_j^{-(3/2)\delta},\qquad \forall x\in E_2\backslash A_j.$$ We have $$\mu_2(E_2\backslash\cup_{j\geq1}A_j)\geq1-\frac{1}{1000},$$ provided that $R_0$ is large enough. Thus for $x\in E'_2$, we get $$\|d^x_*(\mu_1)-d^x_*(\mu_{1,\text{\rm good}})\|_{L^1}\lesssim\sum_{j\geq1}R^{\delta}_j\mu_1(\text{\rm Bad}_j(x))+{\rm RapDec}(R_0)\lesssim R_0^{-\delta/2}\leq \frac{1}{1000}.$$ ◻ # Proof of Proposition [Proposition 2](#p2){reference-type="ref" reference="p2"} {#section4} To establish Proposition [Proposition 2](#p2){reference-type="ref" reference="p2"}, we need the refined microlocal decoupling inequality proven in [@xi2022]. We shall follow the definitions and notation in [@xi2022 Section 2]. Given $1\le R\le\lambda$ and $\delta>0$. For a phase function $\phi(x,y)$ satisfying the Carleson--Sjölin condition, we denote by $\phi^\lambda({}\cdot{},{}\cdot{}):=\phi({}\cdot/\lambda{},{}\cdot/\lambda{})$ the rescaling of $\phi$. Let $N_{\phi^{\lambda},R}$ and $N^{\tau}_{\phi^{\lambda},R}$ be as defined in [@xi2022 Equation (2.5)]. A function $f$ is said to be microlocalized to a tube $T$ if it has microlocal support essentially in $N^{\tau}_{\phi^{\lambda}, R}$ and is essentially supported in $T$ in physical space as well. We partition $\mathbb{S}^{d-1}$ into $R^{-1/2}$-caps denoted by $\tau$. Now, we define a collection of curved tubes $\mathbb{T}=\mathbb{T}_{\tau}$ associated with the phase function $\phi^{\lambda}$ and the caps $\tau$. A curved tube $T$ with a length of $R$ and a cross-sectional radius of $R^{1/2+\delta}$ is in $\mathbb{T}_{\tau}$ if its central axis is the curve $\gamma^{y_0}_{\tau}$, defined by $$\gamma^{y_0}_{\tau}=\left\{x:-\frac{\nabla_y\phi^{\lambda}(x,(y_0,0))}{|\nabla_y\phi^{\lambda}(x,(y_0,0))|}=\theta\right\},$$ where $y_0$ is chosen from a maximal $R^{1/2+\delta}$ separated subset of $\mathbb{R}^{d-1}$ and $\theta$ is the center of $\tau$. This microlocal decomposition is consistent with our wavepacket decomposition in Section [2](#section2){reference-type="ref" reference="section2"}. Indeed, in the Riemannian setting, $\phi=d_g$, and $\theta$ is the tangent vector of $\gamma^{y_0}_{\tau}$ at $(y_0,0)$. We recall the refined microlocal decoupling theorem established in [@xi2022] below. **Theorem 5** (Theorem 5.1 in [@xi2022]). *Let $2\leq p\leq\frac{2(d+1)}{d-1}$ and $1\leq R\leq\lambda$. Let $f$ be a function with microlocal support in $N_{\phi^{\lambda}, R}$. Let $\mathbb{T}$ denote a collection of tubes associated with $R^{-1/2}$-caps denoted by $\tau$, and $\mathbb{W}\subset\mathbb{T}$. For every $T\in\mathbb{W}$, we assume that $T$ is contained in $B_{R}$. Denote $W$ as the cardinality of $\mathbb{W}$. Additionally, we assume that $f=\sum_{T\in\mathbb{W}}f_T$, where each $f_T$ is microlocalized to $T$, and $\|f_T\|_{L^p}$ is roughly constant among all $T\in\mathbb{W}$. Let $Y$ be a collection of $R^{1/2}$-cubes in $B_R$, each of which intersects at most $L$ tubes $T\in\mathbb{W}$. Then $$\|f\|_{L^p(Y)}\lesssim_{\varepsilon}R^{\varepsilon}\Big(\frac{L}{W}\Big)^{1/2-1/p}\Big(\sum_{T\in\mathbb{W}}\|f_T\|_{L^p}^2\Big)^{1/2}.$$* Now, we are ready to prove Proposition [Proposition 2](#p2){reference-type="ref" reference="p2"}. The proof is very similar to that in [@xi2022]. For completeness and convenience of readers, we include the argument here. *Proof of Proposition [Proposition 2](#p2){reference-type="ref" reference="p2"}.* Recalling the definition of $d^x_*(\mu_{1,\text{\rm good}})$, we have $$d^x_*(\mu_{1,\text{\rm good}})(t)\approx\int_{d_g(x,y)=t}\mu_{1,\text{\rm good}}(y)\,dy=\iint e^{-2\pi i(d_g(x,y)-t)\tau}\,d\tau\mu_{1,\text{\rm good}}(y)\,dy.$$ By Plancherel's theorem, we get $$\|d^x_*(\mu_{1,\text{\rm good}})\|_{L^2}^2\approx\int\Big|\int e^{-2\pi i\lambda d_g(x,y)}\mu_{1,\text{\rm good}}(y)\,dy\Big|^2d\lambda.$$ Recall that $R_j=2^jR_0$. It suffices to prove that for every $j$, $$\label{gg5} \Big|\iint_{\lambda\approx R_j}\Big|\int e^{-2\pi i\lambda d_g(x,y)}\mu_{1,\text{\rm good}}(y)\,dy\Big|^2\,d\lambda\,d\mu_2(x)\Big|\lesssim 2^{-j\varepsilon}$$ holds for some $\varepsilon>0.$ Noting that $\lambda\approx R_j,$ by a standard integration by parts argument, we reduce [\[gg5\]](#gg5){reference-type="eqref" reference="gg5"} to estimating $$\label{b2} \iint_{\lambda\approx R_j}\Big|\int e^{-2\pi i\lambda d_g(x,y)}\mu^j_{1,\text{\rm good}}(y)\,dy\Big|^2\,d\lambda\,d\mu_2(x),$$ where $$\mu^j_{1,\text{\rm good}}:=\sum_{T\in\mathbb{T}_j, T ~\text{is good}}M_T\mu_1.$$ Denote $$F(x):=\int e^{-2\pi i\lambda d_g(x,y)}\mu^j_{1,\text{\rm good}}(y)\,dy,$$ and $$F_T(x):=\int e^{-2\pi i\lambda d_g(x,y)}M_T\mu_1(y)\,dy.$$ It follows that $$F(x)=\sum_{T\in\mathbb{T}_j, T~\text{is good}}F_T(x).$$ Now we choose $R=\lambda^{1-2\delta}\approx R_j^{1-2\delta}.$ One sees that each $T$ is a geodesic tube of dimensions $R^{-1/2}\times\cdots\times R^{-1/2}\times 1$. After rescaling, it is straightforward to check that both $F$ and $F_T$ satisfy the assumptions of Theorem [Theorem 5](#t5){reference-type="ref" reference="t5"}. We refer the readers to [@xi2022] for more detail. Let $p_c:=\frac{2(d+1)}{d-1}$. By dyadic pigeonholing, it suffices to consider tubes $T\in \mathbb{W}$ where $\|F_T\|_{L^{p_c}}\sim\beta>0.$ Let $W$ denote as the cardinality of $\mathbb{W}$. Now we do the integration over $d\mu_2(x)$. To this end, we introduce a non-negative radial bump function $\rho_{j}\in C^{\infty}_0$ such that it equals to $1$ in $B_{10R_j}(0)$ and $0$ outside $B_{20R_j}(0)$. Notice that $F$ is supported in the unit ball with $\widehat{F}(\xi)(1-\rho_{j})(\xi)=\mathrm{RapDec}(|\xi|)$. Therefore $$F=(\widehat{F}\cdot\rho_{j})^{\vee}+\mathrm{RapDec}(R_j)=F*\widehat{\rho_{j}}+\mathrm{RapDec}(R_j).$$ We have $$\begin{aligned} \int|F(x)|^2\,d\mu_2(x)&=\int|F*\widehat{\rho_{j}}(x)|^2\,d\mu_2(x)+\mathrm{RapDec}(R_j)\\ &\lesssim\int |F|^2*\widehat{\rho_{j}}(x)\,d\mu_2(x)+\mathrm{RapDec}(R_j)\\ &=\int_{B_1(0)}|F(x)|^2|\widehat{\rho_{j}}|*\mu_2(x)\,dx+\mathrm{RapDec}(R_j).\end{aligned}$$ Let $\mu_{2,j}:=|\widehat{\rho_{j}}|*\mu_2$. Divide $B_1(0)$ into $R^{-1/2}$-cubes $Q$. Using dyadic pigeonholing again, it suffices to consider $$\mathcal{Q}_{r,M}=\Big\{Q:\mu_{2,j}(Q)\sim r~\text{and}~Q~\text{intersects}~\sim L~\text{tubes}~T\in\mathbb{W}\Big\},$$ and further restrict the integration domain of $|F|^2$ from $B_1(0)$ to $Y_{r,M}=\cup_{Q\in\mathcal{Q}_{r,M}}Q.$ Applying Hölder's inequality, we get $$\begin{aligned} \label{b5} \int_{Y_{r,M}}|F|^2\mu_{2,j}(x)\,dx\lesssim\bigg(\int_{Y_{r,M}}|F(x)|^{p_c}\bigg)^{2/{p_c}} \bigg(\int_{Y_{r,M}}\mu_{2,j}(x)^{{p_c}/({p_c}-2)}\,dx\bigg)^{({p_c}-2)/{p_c}}.\end{aligned}$$ By Theorem [Theorem 5](#t5){reference-type="ref" reference="t5"}, one can bound the first factor by $$\label{b3} \big\|F\big\|_{L^{p_c}(Y_{r,M})}\lesssim R^{\varepsilon}\left(\frac{L}{W}\right)^{\frac{1}{2}-\frac{1}{{p_c}}}\bigg(\sum_{T\in\mathbb{W}_{\beta}}\|F_T\|_{L^{p_c}}^2\bigg)^{\frac{1}{2}}.$$ To estimate the second factor, we note that $|\widehat{\rho_{j}}|$ decays rapidly outside $B_{R_j^{-1}}$, and thus $$\mu_{2,j}(x)=\int|\widehat{\psi}_{R_j}(x-y)|\,d\mu_2(y)\lesssim R^d_j\cdot\mu_2(B_{R_j^{-1}})+\mathrm{RapDec}(R_j)\lesssim R^{d-\alpha}_j.$$ So we have $$\label{b4} \int_{Y_{r,M}}\mu_{2,j}(x)^{{p_c}/({p_c}-2)}\,dx\lesssim R_j^{2(d-\alpha)/({p_c}-2)}\mu_{2,j}(Y_{r,M}).$$ Combining [\[b5\]](#b5){reference-type="eqref" reference="b5"}, [\[b3\]](#b3){reference-type="eqref" reference="b3"} and [\[b4\]](#b4){reference-type="eqref" reference="b4"}, by Lemma 8.1 in [@xi2022] and the definition of the bad tube, we have $$\label{b6} \begin{aligned} \int_{\lambda\approx R_j}\int|F(x)|^2\,d\mu_2(x)\,d\lambda&\lesssim_{\delta} R_j^{O(\delta)+(\frac{5}{2{p_c}}-\frac{1}{4})d-\frac{2\alpha}{{p_c}}}\sum_{T\in \mathbb{T}_j}\int_{\lambda\approx R_j}\|F_T\|^2_{L^{p_c}}\,d\lambda\\ &\lesssim_{\delta} R_j^{O(\delta)+(\frac{5}{2{p_c}}-\frac{1}{4})d-\frac{2\alpha}{{p_c}}}\cdot |T|^{2/{p_c}}\sum_{T\in \mathbb{T}_j}\int_{\lambda\approx R_j}\|F_T\|^2_{L^{\infty}}\,d\lambda. \end{aligned}$$ Here, in the last inequality, we have used the fact that $F_T$ is essentially supported on $2T$. To bound the right-hand side of [\[b6\]](#b6){reference-type="eqref" reference="b6"}, we recall that $$F_T(x)=\int\Big(\int_{2T}e^{-2\pi i(\lambda d_g(x,y)y\cdot\xi)}\,dy\Big)\widehat{M_T \mu_1}(\xi)\,d\xi.$$ This means that $\xi$ essentially lies in the $\lambda/R\approx R_j^{2\delta}$-neighborhood of $\lambda\tau,$ where $\tau \subset\mathbb{S}^{d-1}_y$ is a $R^{-1/2}$-cap. By Cauchy--Schwartz, we get $$\label{ee10} \begin{aligned} \|F_T\|^2_{L^{\infty}}&\lesssim\bigg(\int_{2T}\int|\widehat{M_T\mu_1}(\xi)|\psi^{\tau}_{\phi^{\lambda},R}(\lambda y,\xi /\lambda)\,d\xi dy\bigg)^2\\ &\lesssim\int_{2T}\int|\widehat{M_T\mu_1}(\xi)|^2\psi^{\tau}_{\phi^{\lambda},R}(\lambda y,\xi/\lambda)\,d\xi dy\int_{2T}\int\psi^{\tau}_{\phi^{\lambda},R}(\lambda y,\xi/y)\,d\xi dy. \end{aligned}$$ Note that for fixed $y\in 2T$, $$\int\psi^{\tau}_{\phi^{\lambda},R}(\lambda y,\xi/y)\,d\xi\lesssim R^{O(\delta)}_j\cdot\lambda^{d-1}\cdot R^{-(d-1)/2},$$ So we have $$\label{ee11} \int_{2T}\int\psi^{\tau}_{\phi^{\lambda},R}(\lambda y,\xi/y)\,d\xi dy\lesssim R^{O(\delta)}_j\cdot\lambda^{d-1}\cdot R^{-(d-1)/2}\cdot |T|\lesssim R_j^{O(\delta)}.$$ By the definition of $\psi^{\tau}_{\phi^{\lambda},R}$, we, meanwhile, have the following $$\label{ee12} \int_{\lambda\approx R_j}\psi^{\tau}_{\phi^{\lambda},R}(\lambda y,\xi/\lambda)\,d\lambda\lesssim R_j^{O(\delta)},$$ uniformly holds in $y\in 2T$. Combine [\[ee10\]](#ee10){reference-type="eqref" reference="ee10"},[\[ee11\]](#ee11){reference-type="eqref" reference="ee11"} and [\[ee12\]](#ee12){reference-type="eqref" reference="ee12"} and recall that $\widehat{M_T\mu_1}(\xi)$ is independent of $\lambda$, it follows that $$\int_{\lambda=R_j}\|F_T\|^2_{L^{\infty}}\,d\lambda\lesssim R_j^{O(\delta)}\cdot|T|\cdot\int|\widehat{M_T\mu_1}(\xi)|^2\,d\xi.$$ Recall that $M_T=\eta_T\mathcal P_{j,\tau}$. Since $\mathcal P_{j,\tau}$ behaves like a $0$-th order pseudodifferential operator, $M_T$ is bounded on $L^2$. By invoking Plancherel twice, we have $$\begin{aligned} \sum_{T\in\mathbb{T}_j}\int_{\lambda\approx R_j}\|F_T\|^2_{L^{\infty}}\,d\lambda &\lesssim R_j^{O(\delta)}\cdot R^{-(d-1)/2}\int_{|\xi|\approx R_j}|\hat{\mu}_1(\xi)|^2\,d\xi,\end{aligned}$$ Inserting this bound into [\[b6\]](#b6){reference-type="eqref" reference="b6"} and recalling that $p_c=\frac{2(d+1)}{d-1}$, we have $$\begin{aligned} \int_{\lambda\approx R_j}\int|F(x)|^2\,d\mu_2(x)\,d\lambda&\lesssim R_j^{O(\delta)-\frac{d}{2(d+1)}-\frac{(d-1)\alpha}{d+1}}\int_{|\xi|\approx R_j}|\hat{\mu}_1(\xi)|^2\,d\xi\\ &\lesssim R_j^{O(\delta)+d-\alpha-\frac{d}{2(d+1)}-\frac{(d-1)\alpha}{d+1}}I_{\alpha-\delta}(\mu_1).\end{aligned}$$ Here $$I_{s}(\mu_1)=\iint|x-y|^{-s}\,d\mu_1(x)\,d\mu_2(y)=c_{s,d}\int_{\mathbb{R}^d} |\xi|^{-(d-s)}|\hat{\mu}_1(\xi)|^2\,d\xi,$$ denotes the energy integral of $\mu_1$, which is finite for any $s\in(0,\alpha)$ (see, e.g., [@mattila2015]). We conclude the proof by noting that $d-\alpha-\frac{d}{2(d+1)}-\frac{(d-1)\alpha}{d+1}<0$ if $\alpha$ is greater than $\frac{d}{2}+\frac{1}{4}$ and $\delta$ is sufficiently small . ◻ # Proof of Lemma [Lemma 4](#l3){reference-type="ref" reference="l3"} {#section5} Lemma [Lemma 4](#l3){reference-type="ref" reference="l3"} generalizes the bound on bad tubes in [@du2021] from the Euclidean case to manifolds of constant sectional curvature. In Euclidean space, it follows directly from the classical Marstrand's orthogonal projection theorem and a radial projection estimate due to Orponen [@orponen19]. Here, we would like to pursue a slightly different approach to prove Lemma [Lemma 4](#l3){reference-type="ref" reference="l3"} on manifolds of constant sectional curvature. Consider the origin $p_0=(0,\ldots,0)$ in our coordinate system. Let us introduce a family of submanifolds associated with each element $V$ in the Grassmannian ${\bf Gr}(d/2,d)$ in the tangent space of $p_0$. That is, each $V\in {\bf Gr}(d/2,d)$ is a $d/2$-dimensional linear subspace of $T_{p_0} M$. Fixing $V$, we consider the Fermi coordinates associated to $V$. See [@gray2003tubes] for a detailed discussion on Ferimi coordinates. These coordinates are of the form $$(x^V_1,\ldots,x^V_{d/2},x^V_{d/2+1},\ldots,x^V_d),$$ where $\exp_{p_0}(V)$ is given by $$\{(x^V_1,\ldots,x^V_{d/2},0,\ldots,0):|x^V_i|<1,\,i=1,\ldots, d/2\}.$$ Then for each point $p=(p^V_1,\ldots,p^V_{d/2},0,\ldots,0)\in \exp_{p_0}(V)$, the submanifold $$\Sigma_{V,p}:=\{(p^V_1,\ldots,p^V_{d/2},x^V_{d/2+1},\ldots,x^V_d):|x^V_i|<1,\,i=d/2+1,\ldots, d\},$$ is a submanifold orthogonal to $\exp_{p_0}(V)$ at the point $p$, and $\{\Sigma_{V,p}\}_{p\in\exp_{p_0}(V)}$ is a local foliation of $M$ near the point $p_0$. Notice that for $p_1, p_2\in \exp_{p_0}(V)$, $$\label{equal-distance-foliation} d_g(p_1, p_2)\approx d_g(\Sigma_{V, p_1}, \Sigma_{V, p_2})\approx d_g(p, \Sigma_{V, p_2})$$ uniformly for $p\in \Sigma_{V, p_1}$. That is, $\Sigma_{V, p}$ are locally "parallel" to each other. With this foliation, one can define orthogonal projection by $$\pi_V: (x^V_1,\ldots,x^V_{d/2},x^V_{d/2+1},\ldots,x^V_d)\mapsto(x^V_1,\ldots,x^V_{d/2},0,\ldots,0).$$ Then $\pi_V\mu$ is a well defined measure on $\exp_{p_0}(V)$, and in particular $$\pi_V\mu(p)=\int_{\Sigma_{V,p}}\mu$$ if $\mu$ has a continuous density. In fact, we may always assume our measure has a continuous density. When dealing with $\mu(\Sigma^R)$, where $\Sigma^R$ denotes the $R^{-1}$-neighborhood of a codimension $k$ submanifold $\Sigma\subset M$, one can consider the $R^{-1}$-resolution of $\mu$, namely, $$\mu^R(x):= \frac{\int \phi(R\cdot d_g(x,y))\,d\mu(y)}{\int \phi(R\cdot d_g(x,y))\,dy},$$ where $\phi\in C_0^\infty(\mathbb{R})$, positive. Then one can easily see that $$\label{localization}\mu(\Sigma^R)\lesssim R^{-k}\int_{\Sigma}\mu^{R/2} \lesssim \mu(\Sigma^{R/4}).$$ In other words when working in the scale $R^{-1}$ one can replace $\mu$ by its $R^{-1}$-resolution $\mu^R$. Here are the key geometric features of manifolds with constant sectional curvature. As all $\Sigma_{V,p}$ are totally geodesic, it follows that each geodesic segment $\gamma$ near $p_0$ is contained in some $\Sigma_{V,p}$ for suitable $V$'s and $p$'s. Indeed, suppose that the distance between $\gamma$ and $p_0$ is minimized at the point $p_m\in\gamma$. Then for any $V$ such that $\exp_{p_0}^{-1}(p_m)\in V$ and $\exp_{p_0}(V)\perp \gamma$ at $p_m$, we have $\gamma\subset \Sigma_{V,p_m}$ since $\Sigma_{V,p_m}$ is totally geodesic. It follows that $p_m=\pi_V(\gamma)$ and $$\label{codimension-d/2-submanifold}\{V: \gamma\subset\Sigma_{V, \pi_V(\gamma)}\}=\{V: \exp_{p_0}V\perp\gamma \}$$ is a $\frac{d^2}{4}-\frac{d}{2}$ dimensional submanifold of ${\bf Gr}(d/2,d)$. In addition, there also exists a natural diffeomorphism between the Grassmannian at different points. More precisely, let $p_0'$ be another point in our coordinate system, then for every $V=V_{p_0}$ at $p_0$, since $\Sigma_{V, \pi_V (p_0')}$ is a totally geodesic submanifold containing $p_0'$, one can conclude that $\exp_{p_0'}^{-1}(\Sigma_{V, \pi_V (p_0')})$ is a linear subspace of $T_{p_0'}M$. Hence the map $$\Pi_{p_0,p_0'}:V_{p_0}\mapsto \left(\exp_{p_0'}^{-1}(\Sigma_{V, \pi_V (p_0')})\right)^{\perp}=:V_{p_0'}$$ is the desired diffeomorphism. One can also define orthogonal projections as above with $p_0'$ as the origin, and in particular, [\[equal-distance-foliation\]](#equal-distance-foliation){reference-type="eqref" reference="equal-distance-foliation"} still holds. Recall that in our coordinate chart, $|g_{ij}-\delta_{ij}|<\epsilon_0$. We shall choose $\epsilon_0$ to be small enough such that the implicit constant in [\[equal-distance-foliation\]](#equal-distance-foliation){reference-type="eqref" reference="equal-distance-foliation"} is uniform under different origins. We remark that the foliations given by $V_{p_0}$ at $p_0$ and $V_{p_0'}$ at $p_0'$ are the same if and only if the underlying manifold is flat. Nonetheless, these two foliations always share the submanifold $$\label{equal-submanifold}\Sigma_{V_{p_0}, \pi_{V_{p_0}}(p_0')}=\Sigma_{V_{p'_0}, p_0'}=\exp_{p_0'}(V_{p_0'}^\perp).$$ Now we start the proof of Lemma [Lemma 4](#l3){reference-type="ref" reference="l3"}. For convenience, we denote $R:=R_j^{1/2-\delta}$. We first decompose the manifold ${\bf Gr}(d/2,d)$ into a finitely overlapping union of $R^{-1}$-"balls\", or more precisely, $R^{-1}$-neighborhoods of $R^{-1}$-separated $\frac{d}{2}$-dimensional subspaces. We denote the collection of the centers of these "balls\" by $\mathcal{V}_j$. Notice that $\#(\mathcal{V}_j)\approx R^{\frac{d^2}{4}}$ as the dimension of ${\bf Gr}(d/2,d)$ is $\frac{d^2}{4}$. Next, for each $V\in\mathcal{V}_j$, one can cover the neighborhood of $p_0$ under consideration by $R^{-1}$-neighborhood of $\Sigma_{V, p_l}$, where $p_l$ are $R^{-1}$-separated points in $\exp_{p_0}V$. This is also a finitely overlapping cover. We denote by $\Sigma^R_{V, p}$ the $16 R^{-1}$-neighborhood of $\Sigma_{V, p}$. We say $\Sigma_{V,p}$ is bad, if its $16 R^{-1}$-neighborhood $\Sigma^R_{V, p}$ contains some bad tube $T\in\mathbb{T}_j$. Now, we can estimate $$\mu_1\times\mu_2(\text{\rm Bad}_j)=\int\mu_2(\text{\rm Bad}_j(x))\,d\mu_1(x)=\int\sum_{x\in T\in\mathbb{T}_j,\text{bad}}\mu_2(T)\,d\mu_1(x).$$ By our discussion above, the constant curvature assumption guarantees that, for every tube $T\in\mathbb{T}_j$ with central geodesic $\gamma$, $$\{V: \gamma\subset\Sigma_{V, \pi_V(\gamma)}\}\subset {\bf Gr}(d/2,d)$$ is a $\frac{d^2}{4}-\frac{d}{2}$ submanifold. Therefore, by dimension counting $$\label{dimension-counting} \#\{\Sigma_{V,p_l}: T\subset \Sigma^R_{V,p_l}\}\approx R^{\frac{d^2}{4}-\frac{d}{2}}.$$ We say $\Sigma_{V,p}$, or equivalently $\Sigma^R_{V,p}$, is bad if $\Sigma^R_{V,p}$ contains a bad tube. Then we have $$\label{reduction-to-d/2-dimensional}\begin{aligned} \int\sum_{T:x\in T\in\mathbb{T}_j,\text{bad}}\mu_2(T)\,d\mu_1(x)=&\iint\sum_{T:x\in T\in\mathbb{T}_j,\text{bad}}\chi_T(y)\,d\mu_2(y)\,d\mu_1(x)\\ \lesssim & R^{-\frac{d^2}{4}+\frac{d}{2}}\cdot\iint\sum_{l:x\in \Sigma^R_{V,p_l},\text{bad}}\chi_{\Sigma^R_{V,p_l}}(y)\,d\mu_2(y)\,d\mu_1(x)\\= & R^{-\frac{d^2}{4}+\frac{d}{2}}\cdot\int\sum_{l:x\in \Sigma^R_{V,p_l},\text{bad}}\mu_2(\Sigma^R_{V,p_l})\,d\mu_1(x).\end{aligned}$$ Notice that by our definition, $\Sigma^R_{V,p_l}\subset\Sigma^{R/4}_{V,\pi_V(x)}$ whenever $\Sigma^R_{V,p_l}$ contains $x$. Therefore the last line of [\[reduction-to-d/2-dimensional\]](#reduction-to-d/2-dimensional){reference-type="eqref" reference="reduction-to-d/2-dimensional"} is $$\label{reduction-to-discretie-radial-proj}\lesssim R^{-\frac{d^2}{4}+\frac{d}{2}}\cdot\int\sum_{V\in\mathcal{V}_j: \Sigma_{V,\pi_V(x)} \text{ bad}}\mu_2(\Sigma^{R/4}_{V,\pi_V(x)})\,d\mu_1(x).$$ Now we would like to change from taking sum in $V\in\mathcal{V}_j$ to taking integral in $V\in {\bf Gr}(d/2,d)$ over the Haar measure $\lambda$. By [\[localization\]](#localization){reference-type="eqref" reference="localization"}, for every $V'$ lying in a $R^{-1}$-neighborhood of $V$ in ${\bf Gr}(d/2,d)$, we have $$\mu_2(\Sigma^{R/2}_{V,\pi_V(x)})\lesssim R^{-\frac{d}{2}}\int_{\Sigma^{R/2}_{V',\pi_{V'}(x)}}\mu_2^{R/8}=R^{-\frac{d}{2}}\cdot\pi_{V'}\mu_2^{R/8}(\pi_{V'}(x)).$$ Therefore, by dimension counting $$\mu_2(\Sigma^{R/2}_{V,\pi_V(x)})\lesssim R^{\frac{d^2}{4}-\frac{d}{2}}\cdot\int_{B_{R^{-1}}(V)}\pi_{V}\mu_2^{R/8}(\pi_{V}(x))\,d\lambda(V).$$ Recall elements in $\mathcal{V}_j$ are $R^{-1}$-separated, so $B_{R^{-1}}(V)$, $V\in\mathcal{V}_j$ has finite overlap. This means we have successfully reduce [\[reduction-to-discretie-radial-proj\]](#reduction-to-discretie-radial-proj){reference-type="eqref" reference="reduction-to-discretie-radial-proj"} to estimating $$\label{reduction-to-integral}\int\int_{V: \Sigma_{V,\pi_V(x)\text{ bad}}}\pi_{V}\mu_2^{R}(\pi_{V}(x))\,d\lambda(V)\,d\mu_1(x).$$ We need to understand the measure of bad $\Sigma_{V,\pi_V(x)}$, for each fixed $x$. Recall each bad $T\in \mathbb{T}_j$ satisfies $\mu_2(4T)\geq{R_j}^{-d/4+c(\alpha)\delta}$, and the supports of $\mu_1, \mu_2$ are separated. Therefore for each $x$ the number of bad $T\in\mathbb{T}_j$ containing $x$ is $\lesssim {R_j}^{\frac{d}{4}-c(\alpha)\delta}\cdot R^{\delta}_j$. Also [\[codimension-d/2-submanifold\]](#codimension-d/2-submanifold){reference-type="eqref" reference="codimension-d/2-submanifold"} implies that for each $T\in \mathbb{T}_j$, the set of $V$ such that $\Sigma_{V,\pi_V(x)}$ is bad lies in the $R^{-1}$-neighborhood of a codimension $\frac{d}{2}$ submanifold. Together, one can conclude that $$\lambda\{V\in {\bf Gr}(d/2,d): \Sigma_{V,\pi_V(x)}\text{ is bad}\}\lesssim R^{-\frac{d}{2}}\cdot {R_j}^{\frac{d}{4}-c(\alpha)\delta+\delta}={R_j}^{-(c(\alpha)-\frac{d+2}{2})\delta}.$$ Thus by Hölder's inequality, for every $1<q<\infty$ and $1/q+1/q'=1$, [\[reduction-to-integral\]](#reduction-to-integral){reference-type="eqref" reference="reduction-to-integral"} is bounded from above by $$\label{first-Holder}\begin{aligned} & {R_j}^{\frac{-(c(\alpha)-(d+2)/2)\delta}{q'}}\cdot \int \|\pi_V\mu_2^{R}(\pi_V(x))\|_{L^q(V)}\,d\mu_1(x).\end{aligned}$$ It remains to show that there exists $q>1$ such that the integral $$\label{reduction-to-Lp}\int \|\pi_V\mu_2^{R}(\pi_V(x))\|_{L^q(V)}\,d\mu_1(x)<\infty,$$ and the upper bound is independent of $R$. Then the factor ${R_j}^{\frac{-(c(\alpha)-(d+2)/2)\delta}{q'}}$ is $\lesssim {R_j}^{-2\delta}$ as desired when $c(\alpha)$ is chosen large enough. To prove this $L^q$ estimate, we denote $$f(V, \pi_V(x)):=\frac{\pi_V\mu_2^{R}(\pi_V(x))^{q/q'}}{\|\pi_V\mu_2^{R}(\pi_V(x))\|_{L^q(V)}^{q/q'}}$$ and rewrite [\[reduction-to-Lp\]](#reduction-to-Lp){reference-type="eqref" reference="reduction-to-Lp"} into $$\label{reduction-to-linear}\begin{aligned}&\int \int \pi_V\mu_2^{R}(\pi_V(x))\,f(V, \pi_V(x))\,d\lambda(V)d\mu_1(x) \\=&\int\int_V \pi_V\mu_2^{R}(p)\,f(V, p)\,d\pi_V\mu_1(p)\,d\lambda(V).\end{aligned}$$ The only property we need from the function $f$ is, for every $x$, $$\int |f(V,\pi_V(x))|^{q'}\,d\lambda(V)=1.$$ Recall that for each $V$, under Fermi coordinates the submanifold $\exp_{p_0}(V)$ can be identified with $\mathbb{R}^{\frac{d}{2}}$. We are now ready to run some Fourier analysis. Take $\frac{d}{2}<\alpha'<\alpha$. In $\mathbb{R}^{\frac{d}{2}}$, it is well known that, for every compactly supported continuous function $g$ and compactly supported measure $\nu$, $$\left(\int g\,d\nu\right)^2=\left(\int \hat{g}\cdot\overline{\hat{\nu}}\right)^2\leq \int |\widehat{g}(\xi)|^2|\xi|^{\alpha'-\frac{d}{2}}\,d\xi \cdot \int |\widehat{\nu}(\xi)|^2|\xi|^{\frac{d}{2}-\alpha'}\,d\xi.$$ By applying this inequality with $g=\pi_V\mu_2^R$ and $d\nu=f(V,{}\cdot{})\,d\pi_V\mu_1$, the square of [\[reduction-to-linear\]](#reduction-to-linear){reference-type="eqref" reference="reduction-to-linear"} is bounded from above by $$\iint |\widehat{\pi_V\mu_2^{R}}(\xi)|^2|\xi|^{\alpha'-\frac{d}{2}}\,d\xi\,d\lambda(V)\cdot \iint |\widehat{f(V,\cdot)\,d\pi_V\mu_1}(\xi)|^2|\xi|^{\frac{d}{2}-\alpha'}\,d\xi\,d\lambda(V)=:I\cdot II$$ We consider $I$ and $II$ separately. For $I$, we shall show that the projection map $\pi_V(x)$ satisfies the transversality condition for the generalized projection estimate of Peres--Schlag [@R00]. More precisely, one needs to check that $$|\nabla_V(d_g(\pi_V(x),\pi_V(y))|\gtrsim d_g(x,y)$$ at $V=V'$ if $\pi_{V'}(x)=\pi_{V'}(y)$. To see this, recall there is a natural diffeomorphism $\Pi_{p_0,y}$ between $\frac{d}{2}$-dimensional subspaces $V_{p_0}\subset T_{p_0}M$ and $V_y\subset T_yM$. By definition of $\Pi_{p_0,y}$ we have $\pi_{V'_{p_0}}(x)=\pi_{V'_{p_0}}(y)$ if and only if $\pi_{V'_y}(x)=\pi_{V'_y}(y)$ (see [\[equal-submanifold\]](#equal-submanifold){reference-type="eqref" reference="equal-submanifold"}). With this condition in mind, we work on $V_y\in T_yM$, in which $$\label{equal-distance-at-y}d_g(\pi_{V_y}(x),\pi_{V_y}(y))=|\exp_y^{-1}(\pi_{V_y}(x))|.$$ Then it becomes an linear algebra problem in $T_yM$, and the condition $\pi_{V_y'}(x)=\pi_{V_y'}(y)$ becomes $\exp_y^{-1}(x)\perp V_y'$. One can see that, for every small $\delta>0$, there exists $V_y''$ that is $\lesssim \delta$ close to $V_y'$ but has angle $<\frac{\pi}{2}-\delta$ to $\exp_y^{-1}(x)$, which implies, by [\[equal-distance-foliation\]](#equal-distance-foliation){reference-type="eqref" reference="equal-distance-foliation"}, [\[equal-submanifold\]](#equal-submanifold){reference-type="eqref" reference="equal-submanifold"}, and our definition of $\pi_V$, that $$d_g(x,y)\cdot\delta\lesssim d_g(\exp_y((V_y'')^\perp), x)= d_g(\Sigma_{V'',\pi_{V''}(y)}, x)\approx d_g(\pi_{V''}(y), \pi_{V''}(x)),$$ where $V''\subset T_{p_0}M$ is the image of $V''_y\subset T_yM$ under the diffeomorphism $\Pi^{-1}_{p_0,y}$. Since $V_y', V_y''$ are $\delta$-close, so are $V', V''$. Hence, one can conclude that $$|\nabla_V d_g(\pi_{V}(y), \pi_{V}(x))|\gtrsim d_g(x,y)$$ at $V'$ given $\pi_{V'}(x)=\pi_{V'}(y)$, as desired. Once the transversality condition is satisfied, one can apply the generalized projection estimate of Peres--Schlag to obtain $$I\lesssim I_{\alpha''}(\mu_2^R)=\iint d_g(x,y)^{-\alpha''}\mu_2^R(x)\mu_2^R(y)\,dx\,dy$$ with $\frac{d}{2}<\alpha'<\alpha''<\alpha$. One can easily check that $$\label{ball-condition-mu-R} \int_{B_r} \mu_2^{R}(x)\,dx\lesssim r^\alpha,$$ where the implicit constant is independent of $R$. Therefore, by considering a dyadic decomposition on $d_g(x,y)$, it follows that $I$ is finite uniformly in $R$, as desired. It remains to estimate $II$. By standard dyadic decomposition on $|\xi|$, we obtain $$\begin{aligned} II&\leq \sum_{k\geq 0}2^{-(\alpha'-\frac{d}{2})k}\int\int_{|\xi|\leq 2^k} |\widehat{f\,d\pi_V\mu_1}(\xi)|^2\,d\xi\,d\lambda(V)\\ &\lesssim \sum_{k\geq 0}2^{-(\alpha'-\frac{d}{2})k}\int\int |\widehat{f\,d\pi_V\mu_1}(\xi)|^2\,\psi\Big(\frac{\xi}{2^k}\Big)\,d\xi\,d\lambda(V),\end{aligned}$$ where $\psi$ is a positive function whose Fourier transform has compact support. For each $k$, if we denote $f_V(x):=f(V,\pi_V(x))$, then $$2^{-(\alpha'-\frac{d}{2})k}\int\int |\widehat{f\,d\pi_V\mu_1}(\xi)|^2\,\psi\Big(\frac{\xi}{2^k}\Big)\,d\xi\,d\lambda(V)$$ $$\begin{aligned} &=2^{-(\alpha'-\frac{d}{2})k}\int\int \left|\int e^{-2\pi i \pi_V(x)\cdot\xi}f_V( x)\,d\mu_1(x)\right|^2\,\psi\Big(\frac{\xi}{2^k}\Big)\,d\xi\,d\lambda(V)\\ &=2^{-(\alpha'-\frac{d}{2})k}\iiint\left(\int e^{-2\pi i (\pi_V(x)-\pi_V(y))\cdot\xi}\,\psi\Big(\frac{\xi}{2^k}\Big)\,d\xi\right)f_V(x)\,f_V(y)\,d\lambda(V)\,d\mu_1(x)\,d\mu_1(y)\\ &=2^{(d-\alpha')k}\iint\int\hat{\psi}(2^k(\pi_V(x)-\pi_V(y)))\,f_V(x)\,f_V(y)\,d\lambda(V)\,d\mu_1(x)\,d\mu_1(y).\end{aligned}$$ Since $\hat{\psi}$ has compact support and $\|f_V(x)f_V(y)\|_{L^{q’/2}(V)}\leq 1$ for every $x$ (we may assume $1<q<2$), by Hölder's inequality the above is $$\leq 2^{(d-\alpha')k}\iint\left(\lambda\{V: |\pi_V(x)-\pi_V(y)|\lesssim 2^{-k} \}\right)^{1-\frac{2}{q'}}d\mu_1(x)\,d\mu_1(y).$$ Now, we need to understand the measure of the set $$\{V: |\pi_V(x)-\pi_V(y)|\lesssim 2^{-k} \}.$$ Fix $x\neq y$. By the diffeomorphism $\Pi_{p_0,p_0'}$ between the Grassmannians at different points we may take $y$ as the origin. Then by [\[equal-distance-foliation\]](#equal-distance-foliation){reference-type="eqref" reference="equal-distance-foliation"} we have $$|\pi_V(x)-\pi_V(y)|\approx d_g(\pi_V(x), \pi_V(y))\approx d_g(x, \Sigma_{V,\pi_V(y)})=d_g(x, \exp_y(V_y^\perp)).$$ Then, it becomes a linear algebra problem in $T_yM$. One can easily see that $|\pi_V(x)-\pi_V(y)|\lesssim 2^{-k}$ if and only if $V$ is contained in the $\min\{d_g(x,y)^{-1}2^{-k}, 1\}$-neighborhood of the codimension $\frac{d}{2}$ submanifold in [\[codimension-d/2-submanifold\]](#codimension-d/2-submanifold){reference-type="eqref" reference="codimension-d/2-submanifold"}, with $\gamma$ being the geodesic connecting $x$ and $y$. Therefore $$\lambda\{V: |\pi_V(x)-\pi_V(y)|\lesssim 2^{-k} \}\lesssim \min\{d_g(x,y)^{-1}2^{-k}, 1\}^{\frac{d}{2}}$$ and the estimate of $II$ is reduced to $$\sum_{k\geq 0}2^{(d-\alpha')k}\int\int \min\{d_g(x,y)^{-1}2^{-k}, 1\}^{\frac{d}{2}(1-\frac{2}{q'})}\,d\mu_1(x)\,d\mu_1(y).$$ The next step is standard. Consider cases $d_g(x,y)\leq 2^{-k}$ and $d_g(x,y)\approx 2^{-k+j}$, $j=1,2\dots, k$ separately. By the ball condition on $\mu_1$ one can conclude that $$II \lesssim\sum_{k\geq 0} 2^{(d-\alpha')k}\sum_{0\leq j\leq k}2^{-\frac{d}{2}(1-\frac{2}{q'})j+\alpha (-k+j)}=\sum_{j\geq 0}2^{(\alpha-\frac{d}{2}(1-\frac{2}{q'}))j}\sum_{k\geq j} 2^{(d-\alpha'-\alpha)k}.$$ As $\alpha, \alpha'>\frac{d}{2}$, this is $$\lesssim \sum_{j\geq 0} 2^{d-\frac{d}{2}(1-\frac{2}{q'})-\alpha'},$$ which is finite, as desired, when $q-1>0$ is small enough.

arxiv_math

--- abstract: | Let $R$ be a ring and $\mathcal X = \mathcal{SH}(R)-\{0\}$ be the set of all non-zero strongly hollow ideals (briefly, $sh$-ideals) of $R$. We first study the concept $SH$-topology and investigate some of the basic properties of a topological space with this topology. It is shown that if $\mathcal X$ is with $SH$-topology, then $\mathcal X$ is Noetherian if and only if every subset of $\mathcal X$ is quasi-compact if and only if $R$ has $dcc$ on semi-$sh$-ideals. Finally, the relation between the dual-classical Krull dimension of $R$ and the derived dimension of $\mathcal X$ with a certain topology has been studied. It is proved that, if $\mathcal X$ has derived dimension, then $R$ has the dual-classical krull dimension and in case $R$ is a $D$-ring (i.e., the lattice of ideals of $R$ is distributive), then the converse is true. Moreover these two dimension differ by at most $1$. address: - "Sayed Malek Javdannezhad, Department of Science, Shahid Rajaee Teacher Training University: Tehran, Tehran, IR" - Nasrin Shirali, Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran author: - S.M. Javdannezhad - N. Shirali title: A topological expression for dual-classical Krull dimension of rings --- [^1] # Introduction and preliminaries The classical Krull dimension for a ring have been studied by several authors and for commutative Noetherian rings, it coincides with the Krull dimension as introduced by Gabriel and Rentschler, see[@go-ro]. In $1972$, Krause extended the concepts of both dimensions to arbitrary ordinals and investigated their relationship. He also, showed that a ring $R$ has the classical Krull dimension if and only if it has $acc$ on prime ideals. In $1980$, Karamzedeh [@kar] studied the derived dimension with respect to a certain topology, which he defined on $X=Spec(R)$, the set of all prime ideals of $R$ and showed that $R$ has classical Krull dimension if and only if $X$ has derived dimension and these two dimensions differ by at most $1$. Recently, in [@ja-sh], we introduced the concept of dual-classical Krull dimension of rings for any ordinal number, on the set of strongly hollow ideals. Our definition is in the same vein, as definition of the classical Krull dimension by Krause, on the set of prime ideals, see [@go-ro]. We also, dualized almost all the basic results of classical Krull dimension. At the end of [@ja-sh], we raised a question namely, what is the topological expression for the algebraic concept of dual-classical Krull dimension. In this paper, we answer this question. For this purpose, we first define $SH$-topology and investigate some of the basic properties of a topological space with this topology. Among various findings, it is proved that, $R$ has $dcc$ on $sh$-ideals if and only if $R$ has $dcc$ on finite sum of non-comparable $sh$-ideals. Also, it is shown that if $\mathcal X$ is a topological space with $SH$-topology, then $\mathcal X$ is Noetherian if and only if every subset of $\mathcal X$ is quasi-compact if and only if $R$ has $dcc$ on semi-$sh$-ideals. Next, we define $W$-topology and the connection between derived dimension and dual-classicall dimension is verified. It is proved that, if $R$ is a $D$-ring and $\mathcal X = \mathcal{SH}(R)-\{0\}$ be with $W$-topology, then $\mathcal X$ has derived dimension if and only if $R$ has dual-classical Krull dimension. Moreover, these two dimension differ by at most $1$. In essence, we dualize almost all of the results which Karamzadeh obtained in [@kar]. It is convenient, when we are dealing with classical and dual-classical Krull dimension, to begin our list of ordinals with $1$.\ Throughout this paper, all rings are associative with $1\neq 0$ and by an ideal we mean a two-sided one. An ideal $S$ of a ring $R$ is small, if $S+A\neq R$, for every proper ideal $A$ of $R$. In what follows, we recall some definitions and facts from [@ja-sh], which are needed. For more details and some of the basic facts about $sh$-ideals and the dual-classical Krull dimension of a ring, the reader is referred to [@ja-sh]. **Definition 1**. An ideal $L$ of a ring $R$ is strongly hollow (briefly, $sh$-ideal), if $L\subseteq A+B$ implies that $L \subseteq A$ or $L \subseteq B$, for every ideals $A,~B$ of $R$. The notation $L \subseteq_{sh} R$ means that $L$ is an $sh$-ideal of $R$. Also, $\mathcal{SH}(R)$ denote the set of all $sh$-ideals of $R$. Note that always, $0 \in \mathcal {SH}(R)$. Also, if $R$ is a simple ring, then $\mathcal{SH}(R) = \{0, R\}$. $R$ is an $sh$-ring, if it is an $sh$-ideal or equivalently, $A+B \ne R$, for any proper ideals $A$ and $B$ of $R$. Every simple ring is an $sh$-ring. **Definition 2**. An ideal $I$ is called semi-$sh$-ideal if it is equal to the sum of all $sh$-ideals contained in itself. A semi-$sh$-ring is a non-zero ring $R$, for which $R$ is an $sh$-ideal or equivalently, $R= \sum \mathcal{SH}(R)$. **Definition 3**. Let $R$ be a ring and $\mathcal Y = \mathcal Y(R)= \mathcal{SH}(R)$. Set $\mathcal Y_{-1}= \{0\}$ and for each ordinal number $\alpha \geq 0$, let $\mathcal Y_\alpha$ be the set of all $L \in \mathcal Y$ such that for every $L' \in \mathcal Y$ strictly contained in $L$, there exists an ordinal $\beta < \alpha$ such that $L' \in \mathcal Y_\beta$. Then $\mathcal Y_0 \subsetneq \mathcal Y_1 \subsetneq \mathcal Y_2 \subsetneq \dots$. The smallest ordinal $\alpha$, for which $\mathcal Y_\alpha = \mathcal Y$ is called dual-classical Krull dimension of $R$ and denoted by $d.cl. \mbox{{\it k}-{\rm dim}}\,R$. Note that, $d.cl.\mbox{{\it k}-{\rm dim}}\,R=-1$ if and only if $\mathcal Y= \{0\}$, that is $R$ has no any non-zero $sh$-ideal. Also, $\mathcal Y_0$ consists of $0$ and all minimal $sh$-ideals. **Proposition 4**. *Let $R$ be a ring in which the intersection of maximal ideals is zero. Then $d.cl.\mbox{{\it k}-{\rm dim}}\,R \leq 0$.* **Theorem 5**. *Let $R$ be a ring.* 1. *If $R$ satisfies $dcc$ on $sh$-ideals, then $R$ has dual-classical Krull dimansion.* 2. *If $R$ is a $D$-ring, the converse of $(1)$ holds.* # $SH$-topology Let $\mathcal X = \mathcal X(R)$ be the set of all non-zero $sh$-ideals of $R$, that is $\mathcal X(R) = \mathcal{SH}(R)-\{0\}$. In this section, we introduce and study the concept of $SH$-topology.\ We begin with the following definition. **Definition 6**. Let $R$ be a ring and $I$ be an ideal of $R$. We define $V(I) = \{L \in \mathcal X : L \subseteq I\}$ and $\underline{I} = \Sigma V(I) = \Sigma_{L \in V(I)}L$, that is $\underline{I}$ equals to the sum of all $sh$-ideals, which contained in $I$. In what follows, we give the basic properties of these concepts. **Lemma 7**. *Let $R$ be a ring and $I$, $J$ and $\{I_\gamma\}_{\gamma \in \Gamma}$ be ideals of $R$. Then* 1. *$V(I) = \emptyset$ if and only if $I$ does not contain any non-zero $sh$-ideal of $R$.* 2. *$V(I) = \{I\}$ if and only if $I$ is a minimal $sh$-ideal of $R$.* 3. *$V(I) = \mathcal X$ if and only if $I$ contains all of the $sh$-ideals of $R$.* 4. *$V(I + J)= V(I) \cup V(J)$* 5. *$V(\cap I_\gamma) = \cap V(I_\gamma)$* 6. *$V(I) = V(\underline I)$.* 7. *$I = \underline{I}$ if and only if $I$ is a semi-$sh$-ideal.* 8. *If $V(I) = V(J)$, then $\underline{I} = \underline{J}$.* 9. *$\underline{\cap{I_\gamma}} = \cap \underline{I_\gamma}$.* *Proof.* $(1)$ Since $\underline{I} \subseteq I$, we have $V(\underline{I}) \subseteq V(I)$. Conversely, if $L \in V(I)$, then $L$ is a $sh$-ideal contained in $I$, hence $L \subseteq \underline{I}$ and so $L \in V(\underline{I})$. Consequently, $V(I) \subseteq V(\underline{I})$ and we are done.\ $(2)$ It is clear.\ $(3)$ Clearly, $V(I_1) \cup V(I_2) \subseteq V(I_1+I_2)$. Now, let $L \in V(I_1 + I_2)$, then $L$ is a non-zero $sh$-ideal and $L \subseteq I_1+I_2$. If $L \notin V(I_1)$, then $L \nsubseteq I_1$ and so $L \subseteq I_2$, that is, $L \in V(I_2)$. Hence, $V(I_1+I_2) \subseteq V(I_1) \cup V(I_2)$. Thus, $V(I_1) \cup V(I_2) = V(I_1+I_2)$ ◻ **Theorem 8**. *Let $R$ be a ring. The sets $V(I)$, where $I$ is an ideal of $R$, satisfy the axioms for closed sets in a toplogical space.* *Proof.* Clearly, $V(0) = \emptyset$, so the empty set is closed. Also, $V(R) = \mathcal X$, so the entire space is closed. Now, let $I_1, I_2$ be two ideals of $R$. By Lemma [Lemma 7](#l1){reference-type="ref" reference="l1"}(4), $V(I_1) \cup V(I_2) = V(I_1+I_2)$, so $V(I_1) \cup V(I_2)$ is closed. is a closed set. Finally, for any collection $\{I_\gamma\}$ of ideals, Lemma [Lemma 7](#l1){reference-type="ref" reference="l1"}(5) implies that $\cap V(I_\gamma) = V(\cap I_\gamma)$, hence $\cap V(I_\gamma)$ is closed. ◻ **Definition 9**. The collection of complements of the sets $V(I)$ is a topology on $\mathcal X$ and we call it $SH$-topology. Recall that a topological space $X$ is a $T_0$-space if and only if for every dictinct points $x ,y \in X$, there is an open set containing one and not the other. Also, $X$ is a $T_1$-space if and only if for every dictinct points $x ,y \in X$, there are open sets containing one and not the other. Finally, $X$ is a $T_2$-space (Hausdorff space) if and only if for every dictinct points $x ,y \in X$, there are distinct open sets $U$ and $V$ in $X$ with $x\in U$ and $y\in V$. **Remark 10**. The $SH$-topology on $\mathcal X$ is trivial if and only if it is single point. For this, let $\mathcal X$ is trivial, then, for all ideal $I$ either $V(I)=\emptyset$ or $V(I)=\mathcal X$. Now, if $L_1 \ne L_2$ are $sh$-ideals of $R$, then $V(L_1)=\mathcal X =V(L_2)$. So $L_1\subseteq L_2$ and $L_2\subseteq L_1$, thus $L_1=L_2$, and we are done. Conversely, if $\mathcal X=\{L\}$ then for all ideal $I$ we have $V(I)=\emptyset$ or $V(I)=\{L\}=\mathcal X$. Hence, $\mathcal X$ is a trivial space. **Lemma 11**. *Let $R$ be a ring and $\mathcal X = \mathcal X(R)$ be with $SH$-topology. Then* 1. *$\mathcal X$ is a $T_0$-space.* 2. *$\mathcal X$ is a $T_1$-space if and only if every non-zero $sh$-ideal of $R$ is a minimal $sh$-ideal if and only if every non-trivial $sh$-ideal of $R$ is a maximal $sh$-ideal.* 3. *If $\mathcal X$ is hausdorff, then there exist ideals $I_1$ and $I_2$, such that every $sh$-ideal of $R$ is contained either in $I_1$ or $I_2$.* *Proof.* (1) Assume that $L_1\neq L_2$, so $L_1\nsubseteq L_2$ or $L_2\nsubseteq L_1$. If $L_1\nsubseteq L_2$, then $L_1\in V^c(L_2)$ while $L_2\not\in V^c(L_2)$. Similarlly, if $L_2\nsubseteq L_1$, then $L_2\in V^c(L_1)$ while $L_1\notin V^c(L_1)$ and so $\mathcal X$ is a $T_0$-space.\ (2) Suppose that $\mathcal X$ is a $T_1$-space and $L$ and $L'$ are non-zero $sh$-ideals of $R$. If $L' \subseteq L$, then any open set $G=V^c(I)$ that contains $L'$ will also contains $L$. Because, $L\notin G$ implies that $L\subseteq I$ and so $L'\subseteq I$, the contradiction required. Conversly, let any $sh$-ideal of $R$ be a minimal $sh$-ideal, then for different $sh$-ideals of $L_1, L_2$, we can easy to see that $L_1\in V^c(L_2)$ and $L_2\notin V^c(L_2)$ and vice versa. This means that $\mathcal X$ is a $T_1$-space.\ (3) If $R$ has at most one $sh$-ideal, we are done. Now, let $L_1 \ne L_2$ be two $sh$-ideals of $R$. Since $\mathcal X$ is Hausdorff, then there exist ideals $I_1$ and $I_2$ such that, $L_1 \subseteq V^c(I_1)$ and $L_2 \subseteq V^c(I_2)$ and $V^c(I_1) \cap V^c(I_2) = \emptyset$. This implies that $\mathcal X = V(I_1) \cup V(I_2) = V(I_1+I_2)$, hence every $sh$-ideal of $R$ is contained either in $I_1$ or $I_2$. ◻ **Proposition 12**. *The following statements are equivalent for any ring $R$.* 1. *$R$ has $dcc$ on $sh$-ideals.* 2. *$R$ has $dcc$ on finite sum of non-comparable $sh$-ideals.* *Proof.* $(2)\Rightarrow (1)$ It is evident.\ $(1)\Rightarrow (2)$ Let $I_1 \supseteq I_2 \supseteq \dots \supseteq I_n \supseteq \dots$ be an infinite descending chain of ideals, each of which is of the form $I_n = \varSigma_{L_i \in \Delta_n} L_i$, where $\Delta_n$ is a finite set of non-comparable $sh$-ideals. With out loss of generality, we can assume that $(i)$ $\Delta_n \cap \Delta_{n+1} = \emptyset$ (note, if $L \in \Delta_n \cap \Delta_{n+1}$, then since $\Delta_{n+1}$ is a set of non-comparable $sh$-ideals, then $L$ may not contain any other $sh$-ideal and we can remove it) and also $(ii)$ For every $L \in \Delta_n$, there exists $L' \in \Delta_{n+1}$ such that $L \supsetneq L'$ (note, otherwise $L'$ can be omitted). Hence, we get an infinite chain $L_1 \supsetneq L_2 \supsetneq \dots \supsetneq L_n \supsetneq \dots$ of $sh$-ideals where $L_i \in \Delta_i$ for each $i$, this is a contradiction. ◻ **Corollary 13**. *Let $R$ be a $D$-ring with dual classical Krull dimension. Then, $R$ has $dcc$ on finite sum of non-comparable $sh$-ideals.* Recall that, a topological space $X$ is called Noetherian if it satisfies the ascending chain condition for open subsets. **Proposition 14**. *Let $R$ be a ring and $\mathcal X = \mathcal{X}(R)$ be with $SH$-topology. The following statements are equivalent.* 1. *$\mathcal X$ is Noetherian.* 2. *Every subset of $\mathcal X$ is quasi-compact.* 3. *$R$ has $dcc$ on semi-$sh$-ideals.* *Proof.* $(1)\Rightarrow (2)$ Let $A$ be a subset of $\mathcal X$ and $\{O_\lambda\}$ be an open cover for $A$. Then $A \subseteq \cup_\lambda O_\lambda$. If $\{O_\lambda\}$ has not a finite subcover of $A$, then there exists a sequence $\lambda_1, \lambda_2, \lambda_3, \dots$ such that $O_{\lambda_1} \subsetneq O_{\lambda_1} \cup O_{\lambda_2} \subsetneq \dots \subsetneq \cup_{i=1}^n O_{\lambda_i} \subsetneq \dots$, which is a contradiction.\ $(2)\Rightarrow (3)$ Let $I_1\supseteq I_2 \supseteq \dots \supseteq I_n \supseteq \dots$ be an infinite descending chain of semi-$sh$- ideals, each of which is of the form $I_n = \varSigma \Delta_n$, where $\Delta_n \subseteq \mathcal X$. Then $V(I_1) \supseteq V(I_2) \supseteq \dots \supseteq V(I_n) \supseteq \dots$ and so $\mathcal X - V(I_1) \subseteq \mathcal X - V(I_2) \subseteq \dots \subseteq \mathcal {Y}- V(I_n) \subseteq \dots$. Let $A = \cup (\mathcal X- V(I_n))$. Then by hypothesis, there exists $i_1, \dots i_r \in \mathbb N$ such that $A = \cup_{j=1}^r (\mathcal X- V (I_{i_i}))$ and so $A = \mathcal {Y} - V (I_m)$, where $m = max\{i_1, \dots , i_r\}$. It follows that $V (I_m) = V (I_k)$ for all $k \geq m$. Since $I_m$ and $I_k$ are semi-$sh$-ideals, we have $I_m = V (I_m) = V (I_k) = I_k$ for all $k \geq m$ and hence we are done.\ $(3)\Rightarrow (1)$ Let $V(I_1) \supseteq V(I_2) \supseteq \dots \supseteq V(I_n) \supseteq \dots$ is an infinite descending chain of closed subsets of $\mathcal Y$. Hence we have $\varSigma V(I_1) \supseteq \varSigma V(I_2) \supseteq \dots \supseteq \varSigma V(I_n) \supseteq \dots$. and so $\underline{I_1} \supseteq \underline{I_2} \supseteq \dots \supseteq \underline{I_n} \supseteq \dots$. By $(3)$, there exists $m \in \mathbb N$ such that $\underline{I_m} = \underline{I_k}$, thus $\varSigma V(I_m) = \varSigma V(I_k)$ for all $k \geq m$. Hence $V(I_m) = V(I_k)$ for all $k \geq m$. ◻ **Corollary 15**. *Let $R$ be a ring and $\mathcal X = \mathcal X(R)$ be with $SH$-topology. If $\mathcal X$ is quasi-compact, then $R$ has dual-classical-Krull dimension.* *Proof.* Since $\mathcal X$ is quasi-compact, then $R$ has $dcc$ on semi-$sh$-ideals, by the part $(3)$ of the previous proposition. Thus, it also has $dcc$ on $sh$-ideals and by Theorem [Theorem 5](#t1){reference-type="ref" reference="t1"}, it has dual-classical Krull dimension. ◻ # derived dimension versus dual-classical Krull dimension Recall that if $A$ is a subset of a topological space $X$, then an element $x \in X$ is called a limit point for $A$ if every open set containing $A$ intersects $A$ in at least one point of $A$ distinct of $x$. The set of all limit points of $A$ is called the drived set of $A$ and is denoted by $A'$. Every $x \in A-A'$ is called an isolated point of $A$. Also, the $\alpha$-derivative of $X$ is defined by transfinite induction as follows: $X_0= X$ and $X_{\alpha+1} = X'_\alpha$ and if $\alpha$ is a limit ordinal, $X_\alpha = \bigcap _{\beta < \alpha}X_\beta$. Note that, the sets $X_\alpha$ need not to be closed in general, however, in case $X$ is hausdorff, every $X_\alpha$ is a closed. $X$ is called scattered, if $X_\alpha = \emptyset$ for some ordinal $\alpha$. If $X$ is scattered, then the smallest ordinal $\alpha$ sucah that $X_\alpha = \emptyset$ is called the derived dimension of $X$ and is denoted by $d(X) = \alpha$, see [@kar].\ Let $R$ be a ring and $\mathcal B = \{V(I): I ~\text{is an ideal of}~R\}$. For each $L \in \mathcal X$, clearly $L \in V(L)$, so $\mathcal X = \cup \mathcal B$. Also, by Lemma [Lemma 7](#l1){reference-type="ref" reference="l1"}(2), we have $V(I_1 \cap I_2) = V(I_1) \cap V(I_2)$. Hence, $\mathcal B$ is a base for some toplology on $\mathcal X = \mathcal X(R)= \mathcal{SH}(R)-\{0\}$. The tpolology on $\mathcal X$ with $\mathcal B$ as a base, is called $W$-topology. **Lemma 16**. *Let $R$ be a ring, $\mathcal X = \mathcal X(R)$ be with $W$-topology and $S \subseteq \mathcal X$. Then an element $L \in S$ is an isolated point of $S$ if and only if it is a minimal element of $S$.* *Proof.* If $L \in S$ is a minimal element, then $V(L) \cap S= \{L\}$, hence $L$ is an isolated point of $S$. Conversely, let $L \in S$ be an isolated point. Then, there exists an open subset $G$ of $\mathcal X$ such that $G\cap S = \{L\}$. But there exists $V(L')$ such that $L \in V(L') \subseteq G$. This implies that $V(L') \cap S = \{L\}$. Now, let $L'' \in S$ and $L'' \subseteq L$. Then $L'' \in V(L') \cap S = \{L\}$ and so $L'' = L$. Thus, $L$ is a minimal element of $S$. ◻ **Corollary 17**. *Let $R$ be a ring, $\mathcal X = \mathcal X(R)$ be with $W$-topology and $S \subseteq \mathcal X$. The set of all isolated points of $S$ is open.* *Proof.* Suppose that $L \in S$. By the previous lemma, $L$ is a minimal element of $S$ and then $V(L) = \{L\}\subseteq S$. According to $W$-topology, $V(L)$ is open and so $S$ is a open set. ◻ Let $R$ be a ring and $\mathcal X = \mathcal X(R)$. We set $\mathcal X_0 = \mathcal X$ and for every $\beta$, by transfinite induction, we define $\mathcal X_{\beta + 1} = \mathcal X'_\beta$, the set of limit points of $\mathcal X_\beta$ and $\mathcal X_\beta = \cap_{\gamma < \beta} \mathcal X_\gamma$, for a limit ordinal $\beta$. Note that, $\mathcal X_0 \supseteq \mathcal X_1 \supseteq \dots \supseteq \mathcal X_n \supseteq \dots$. To see this, it sufficies to show that every $\mathcal X_\beta$ is a closed set. For this manner, we procced by transfinite induction on $\beta$. Clearly, $\mathcal X_0 = \mathcal X$ is closed. Now, let $\mathcal X_\gamma$ be closed for every $\gamma < \beta$. If $\beta = \gamma + 1$, then $\mathcal X_\beta = \mathcal X_\gamma - \mathcal S_\gamma = \mathcal X_\gamma \cap \mathcal S_\gamma^c$, where $\mathcal S_\gamma$ is the set of all isolated points of $\mathcal X_\gamma$ which is open, by Corollary [Corollary 17](#ipo){reference-type="ref" reference="ipo"}. Hence, $\mathcal S_\gamma^c$ is closed and by induction hypothesis, so is $\mathcal X_\beta$. If $\beta$ is a limit ordinal, since the intersection of any family of closed sets is closed, $\mathcal X_\beta = \cap_{\gamma < \beta} \mathcal X_\gamma$ is closed.\ We cite the following well-known fact from [@kar Lemma 3]. **Lemma 18**. *The following are equivalent for any toplogical space $X$.* 1. *Every nonempty subset of $X$ contains an isolated point.* 2. *There is an ordinal $\alpha$ such that $X_\alpha = \emptyset$.* It follows by the above lemma that if every non-empty subset of $X$, which has an isolated point, then $X$ has derived dimension. The next result is now immediate. **Corollary 19**. *Let $R$ be a ring and $\mathcal X = \mathcal X(R)$ be with $W$-toplology. Then the following statements are equivalent.* 1. *$R$ has $dcc$ on $sh$-ideals.* 2. *$\mathcal X$ has derived dimension.* We need the following result, too. **Theorem 20**. *Let $R$ be a ring and $\mathcal X = \mathcal X(R)$ be with $W$-toplology. If $\mathcal X$ has derived dimension, then $R$ has dual-classical Krull dimension.* *Proof.* By Theorem [Theorem 5](#t1){reference-type="ref" reference="t1"} and Corollary [Corollary 19](#c3.3){reference-type="ref" reference="c3.3"}, it is evident. ◻ We are now ready to prove the following proposition, which is a crucial step towads proving our main result. **Proposition 21**. *Let $R$ be a ring, $\mathcal X = \mathcal X(R)$ be with $W$-topology and $\alpha \geq 0$ be an ordinal. Then $\mathcal Y_\alpha(R) - \{0\} = \cup_{\beta \leq \alpha} S_\beta$, where $S_\beta$ is the set of all isolated ponits of $\mathcal X_\beta$.* *Proof.* We proceed by induction on $\alpha$. For $\alpha = 0$, since $\mathcal Y_0(R) - \{0\}$ consists of all minimal $sh$-ideals of $R$, by Lemma [Lemma 16](#l3.1){reference-type="ref" reference="l3.1"}, we have $\mathcal Y_0(R) -\{0\} = S_0$. Let us assume that $\mathcal Y_\gamma(R)-\{0\} = \cup_{\beta \leq \gamma} \mathcal S_\beta$ for all $\gamma < \alpha$. We show that $\mathcal Y_\alpha(R)-\{0\} = \cup_{\beta \leq \alpha} \mathcal S_\beta$. For this, let $L \in \cup_{\beta \leq \alpha} \mathcal S_\beta$. Then, $0 \ne L \in \mathcal S_\beta$, for some $\beta \leq \alpha$. If $L \in \mathcal S_\alpha$, then by Lemma [Lemma 16](#l3.1){reference-type="ref" reference="l3.1"}, $L$ is a minimal element of $\mathcal Y_\alpha$. Hence, if $L' \in \mathcal X = \mathcal{SH}(R) - \{0\}$ and $L' \subsetneq L$, then $L' \notin \mathcal X_\alpha = \cap_{\beta < \alpha} \mathcal X_\beta = \mathcal X - \cup_{\beta < \alpha} \mathcal S_\beta$. This implies that $L' \in \mathcal S_\beta$ for some $\beta < \alpha$. Hence, $L' \in \cup_{\gamma \leq \beta}\mathcal S_\gamma$. By induction hypotesis, we have $L' \in \mathcal Y_\beta(R) - \{0\}$. This shows that $L \in \mathcal Y_\alpha(R) - \{0\}$. Now, let $L \notin \mathcal S_\alpha$. Then $L \in \mathcal S_\beta$, for some $\beta < \alpha$. This implies that $L \in \cup_{\gamma \leq \beta} \mathcal S_\gamma = \mathcal Y_\beta(R)- \{0\} \subseteq \mathcal Y_\alpha(R)-\{0\}$. Therfore, $\mathcal Y_\alpha(R)-\{0\} \supseteq \cup_{\beta \leq \alpha} \mathcal S_\beta$.\ Converesely, let $L \in \mathcal Y_\alpha(R) - \{0\}$. If $L \notin \cup_{\beta < \alpha} \mathcal S_\beta$, we show that $L \in \mathcal S_\alpha$. To this end, let $0 \ne L' \in \mathcal Y(R)$ and $L' \subsetneq L$. Then $L' \in \mathcal Y_\beta(R)-\{0\} = \cup_{\gamma \leq \beta} \mathcal S_\gamma$ for some $\beta < \alpha$. This implies that $L' \notin \mathcal X_\alpha = \mathcal X - \cup_{\gamma < \alpha} \mathcal S_\gamma$. But, $L \in \mathcal X_\alpha$ and so $L$ is a minimal $sh$-ideal of $\mathcal X_\alpha$. By Lemma [Lemma 16](#l3.1){reference-type="ref" reference="l3.1"}, $L \in \mathcal S_\alpha$. Therefore, $\mathcal Y_\alpha(R) - \{0\} \subseteq \bigcup_{\beta \leq \alpha} \mathcal S_\beta$ and this complete the proof. ◻ Finally, we conclude the article with the following result, which we were after. **Theorem 22**. *Let $R$ be a $D$-ring and $\mathcal X = \mathcal X(R)$ be with $W$-topology. Then* 1. *$\mathcal X$ has derived dimension if and only if $R$ has dual-classical Krull dimension.* 2. *$d.cl.\mbox{{\it k}-{\rm dim}}\,R \leq d(\mathcal X) \leq d.cl.\mbox{{\it k}-{\rm dim}}\,R + 1$. Moreover, if $d(\mathcal X)$ is a limit ordinal, then $d(\mathcal X) = d.cl.\mbox{{\it k}-{\rm dim}}\,R$, otherwise, $d(\mathcal X) = d.cl.\mbox{{\it k}-{\rm dim}}\,R + 1$* *Proof.* $(1)$ By Theorem [Theorem 5](#t1){reference-type="ref" reference="t1"} and Corollary [Corollary 19](#c3.3){reference-type="ref" reference="c3.3"}, is evident.\ $(2)$ Let $d.cl.\mbox{{\it k}-{\rm dim}}\,R = \alpha$. Then $\mathcal Y =\mathcal Y_\alpha(R)$ and by Proposition [Proposition 21](#p3.4){reference-type="ref" reference="p3.4"} $\mathcal X = \cup_{\beta \leq \alpha} \mathcal S_\beta$. Hence, $\mathcal X_{\alpha + 1} = \mathcal X- \cup _{\beta \leq \alpha}S_\beta = \emptyset$. Thus $d(\mathcal X) \leq \alpha + 1$. Now, let $d(\mathcal X) < \alpha$. Then, there exists $\gamma < \alpha$ such that $\mathcal X_\gamma = \emptyset$. This implies that $\mathcal S_\gamma = \mathcal S_{\gamma + 1} = \dots = \mathcal S_\alpha = \emptyset$. Hence, $\mathcal X = \cup_{\beta \leq \alpha} \mathcal S_\beta = \cup_{\beta \leq \gamma} \mathcal S_\beta$. Consequently, $\mathcal Y(R) = \mathcal Y_\gamma(R)$ and so $d.cl.\mbox{{\it k}-{\rm dim}}\,R \leq \gamma < \alpha$, which is a contradinction. Thus, $\alpha \leq d(\mathcal X)$. Therefor $\alpha \leq d(\mathcal X) \leq \alpha + 1$. For the last part, we note that if $d(\mathcal X)$ is a limit ordinal, then clearly $d(\mathcal X) \ne \alpha + 1$ and so $d(\mathcal X) = \alpha$. Finally, if $d(X) = \gamma + 1$, then $\mathcal X_{\gamma + 1} =\mathcal X- \cup _{\beta \leq \gamma}S_\beta= \emptyset$. Hence, $\mathcal X= \cup _{\beta \leq \gamma}S_\beta$. By Proposition [Proposition 21](#p3.4){reference-type="ref" reference="p3.4"}, $\mathcal Y_\gamma (R) =\mathcal Y$ and so $d.cl.\mbox{{\it k}-{\rm dim}}\,R = \alpha \leq \gamma$. This implies that $\alpha + 1 \leq \gamma + 1 =d(\mathcal X)$. Thus $d(\mathcal X) = \alpha + 1$ and this complete the proof. ◻ 50 Goodearl, K.R., Warfield, R.B. (1989). *An Introduction to Noncommutative Noetherian Rings*. Cambridge university press, Cambridge, UK. Gordon, R., Robson, J.C. (1973). *Krull dimension*. Mem. Amer. Math. Soc. 133. Javdannezhad, S.M., Mousavinasab, S.F., Shirali, N. (2022). *On Dual-classical Krull dimension of rings*. Quaestiones Mathematicae. 45(7):1-13. Karamzadeh, O.A.S. (1983). *On the classical Krull dimension of rings*. Fundamenta Mathematica. 10.4064/fm-117-2-103-108. Stephenson, W. (1999). *Modules whose lattice of submodules is distributive*. Proc. London. Math. Soc. 28(2), 1-15. Wisbauer, R. (1991). *Foundations of Module and Ring Theory*. Gordon and Breach Science Publishers, Reading (1991). Woodward, A. (2007). *Rings And Modules With Krull Dimension* . phd. thesis, Glasgow (2007). [^1]: true cm MSC(2010): Primary: 16P60, 16P20 ; Secondary: 16P40. Keywords: strongly hollow ideals, $SH$-topology, derived dimension, dual-classical Krull dimension.\

arxiv_math

--- abstract: | We systematically study the so-called auto-arc spaces. Auto-arc spaces were originally introduced by Schoutens in [@sch2] and later generalized and studied by the author in [@sto2017], [@auto], and [@sto2019]. In that aforementioned work, only results concerning trivial deformations were explicitly considered because even in that case auto-arc spaces being a subset of generalized jet schemes are difficult to understand. The major advance in this work is obtained by considering auto arc spaces of complete intersections. It is shown that over $k[t]/(t^{n+1})$, these spaces can be viewed as global flat deformations over $\mathbb{A}_k^n$ of the classical jet scheme of order $n$. We also introduce the study of so-called strong/weak deformations of curves in this context, and we show that a motivic volume can be defined in this case. address: Borough of Manhattan Community College, CUNY author: - Andrew R. Stout bibliography: - FlatMap.bib nocite: "[@*]" title: Auto-arcs of complete intersection varieties. --- [^1] # Introduction The study of jet schemes and arc spaces are an important area of algebraic geometry because these spaces encode a large amount of information above the underlying schemes singular points. In this paper, we focus our study on particular types of generalized jet scheme which we term auto-arc spaces. These are defined roughly to be generalized jet schemes of a flat deformation of a scheme over fat point along that same fat point. Some first motivating examples are discussed in detail in Section [2](#sec1){reference-type="ref" reference="sec1"} pertain to the reduced tangent bundle of a scheme over the dual numbers. First, we will describe briefly jet schemes and then then the generalization to auto-arc spaces. The *jet scheme* of $X$ of order $n$ over a field $k$, most commonly denoted in the literature as $\mathcal{L}_n(X)$, is the scheme with its induced reduced structure which represents the functor, $\mathbf{Sch}/{k} \to \mathbf{Sets}$, given by $$Y \mapsto \mbox{Hom}_{k}(Y\times_k \operatorname{Spec}(k[t]/(t^{n+1}), Y\times_k X)$$ where $\mathbf{Sch}/{k}$ denotes schemes over a field $k$. As a consequence of a theorem due to Grothendieck, $\mathcal{L}_n(X)$ exists provided $X$ is separated and locally of finite type over $k$. Classically, the *arc space of* $X$ *over* $k$ in the literature is the projective limit $\mathcal{L}(X) := \varprojlim \mathcal{L}_n(X)$ over the natural truncation maps $\pi^n_{n-1} : \mathcal{L}_n(X) \to \mathcal{L}_{n-1}(X)$ induced by modding out by $(t^n)\cdot k[t]/(t^{n+1})$. This is also a scheme over $k$ since the transition maps $\pi^n_{n-1}$ are affine. The definition of a generalized jet scheme over $k$ is obtained by replacing the linear jet $\operatorname{Spec}(k[t]/(t^n))$ with an arbitrary finitely generated Artinian $k$-algebra. One advantage to this viewpoint is that for an affine $k$-scheme $X=\operatorname{Spec}(R)$, the $k$-points, given by an $a$, on the generalized jet scheme are in one-to-one correspondence with $k$-algebra homomorphisms $\varphi_a : R \to A$ from which an explicit description of the generalized jet scheme as an affine scheme can be obtained. Similarly then to the classical jet scheme case, the generalized jet scheme of $X$ with respect to $Z=\operatorname{Spec}(A)$ exists if $X$ is separated and locally of finite type over $k$. As we discuss below, we will then denote the resulting scheme as $\underline{Hom}_S(Z, X)$ to denote this scheme, and we let $\underline{Hom}_S(Z, X)^{{\text{\rm red}}}$ denote this scheme with its reduced induce structure. We also sometimes prefer to work at the following level of generality. Let $S$ be an arbitrary scheme and let $X$ and $X'$ be $S$-schemes locally of finite presentation. We define the functor from $\mathbf{Sch}/{S} \to \mathbf{Sets}$ defined by $$Y \mapsto \mbox{Hom}_S(X'\times_SY, X\times_SY)$$ When this functor is representable by an algebraic space (or more generally by an algebraic stack) over $S$, we write the resulting space as $\underline{Hom}_S(X', X)$. It is a well-known result due to Artin[^2] that the functor is representable by a separated algebraic space over $S$ provided $X'$ is proper and flat over $S$ and $X$ is $S$-separated. Now, given any connected, finite and flat scheme $Z$ over an arbitrary scheme $S$, we consider an infinitesimal flat deformation $Y \to Z$ of an $S$-scheme $X$ (separated and locally of finite presentation over $S$), and we define the auto-arc space of $Y$ to be the algebraic space over $S$, defined by $\mathcal{A}_Z(Y) := \underline{Hom}_S(Z,Y)^{{\text{\rm red}}}$ where $(-)^{{\text{\rm red}}}$ denotes the reduced structure. When $S$ is the spectrum of a field $k$, then this algebraic space will be a reduced separated scheme locally of finite type over $k$. In Section [3](#sec2){reference-type="ref" reference="sec2"}, we introduce the reader to some general results concerning these spaces. Notably, we borrow some well-known results concerning jet schemes of complete intersections (initially worked out by Mustata in [@mus2001]), and we see that many of these types of results transfer remarkably well. In both Section [3](#sec2){reference-type="ref" reference="sec2"} and Section [4](#sec3){reference-type="ref" reference="sec3"}, it becomes apparent that the flat locus of a particular natural morphism, denoted $\theta : \mathcal{A}_Z(Y) \to \mathcal{A}_Z$, is of paramount importance. More explicitly, one point of this paper is that the underlying "relativized notion\" of those aforementioned results concerning complete intersections as applied to general auto-arc spaces is best viewed as a question concerning the flat locus of the morphism $\theta$. To see this more clearly, in Section [4](#sec3){reference-type="ref" reference="sec3"}, we restrict to the case of linear auto-arc spaces, and then in Section [5](#sec4){reference-type="ref" reference="sec4"}, we restrict further to linear auto-arc spaces of curves. In either case, we see that these spaces can be regarded as flat global deformations of the classical jet scheme. Many properties of classical jet schemes carry over directly into this context, but crucially, some very well-known results do not or become very subtle (i.e., dependent on choice of deformation). For example, it is no longer automatic that the linear auto arc space of a deformed curve will be reducible if the original curve is singular. We therefore, propose the idea to measure the so-called "weakness\" of a deformation as the "normalized dimension\" of the inverse image of the singular locus in $\mathcal{A}_Z(Y)$ as a measurement of degeneracy of this behavior. In the remaining sections, we develop the idea of a motivic volume and we show that motivic milnor fiber and its corresponding zeta function have a natural interpretation in this context. # Tangent bundles of deformations over the dual numbers {#sec1} We let $\mathbf{Sch}$ denote the category of schemes. Given an object $S$ in $\mathbf{Sch}$, we let $\mathbf{Sch}/{S}$ denote the slice category of schemes $X$ over $S$. Moreover, we let $\mathbf{Fat}/{S}$ denote the full subcategory of $\mathbf{Sch}/{S}$ whose objects are formed by all connected schemes $Z$ such that the structure morphism $Z \to S$ is finite and flat. Following [@liu2006], we say that a scheme $S$ is a Dedekind scheme if it is a normal locally Noetherian scheme of dimension strictly less than $2$, and we first start with the following observation. **Theorem 1**. *Let $X$ be reduced scheme in $\mathbf{Sch}/{S}$ with $S$ a Dedekind scheme. The structure morphism $j: X \to S$ is flat if and only if every irreducible component of $X$ dominates $S$.* *Proof.* Note that an irreducible component $Z$ of $X$ dominates $S$ if the image of $Z$ is dense in $S$ -- i.e., the set-theoretic closure $\overline{j(Z)}$ is equal to $S$. This theorem is a restatement of Proposition 3.9 on page 137 of [@liu2006]. ◻ There are a lot of straightforward conclusions one can reach by using Theorem [Theorem 1](#LiuFlat){reference-type="ref" reference="LiuFlat"}. For now, will just need to consider the following drastically simplified case. **Corollary 2**. *Let $k$ be a field, $X$ be an integral scheme, and $j: X \to \mathbb{A}_{k}^{1}$ a surjective morphism of schemes. Then, $j$ is flat.* For now, let $D = S \times \operatorname{Spec}(\mathbb{Z}[t]/(t^2))$ be the dual numbers over the scheme $S$. Then, $D \to S$ is finite and flat, and as such $\underline{Hom}_{S}(D, X)$ exists in $\mathbf{Sch}/{S}$ for all objects $X$ in $\mathbf{Sch}/{D}$ such that the structure morphism $j: X \to D$ is separated and locally of finite presentation[^3]. We let $T(X)$ denote *the reduced tangent bundle over* $S$, which is defined by $T(X) := \underline{Hom}_{S}(D, X)^{{\text{\rm red}}}.$ **Lemma 3**. *Let $S = \operatorname{Spec}(A)$ with $A$ a reduced Noetherian local ring and let $D$ be the dual numbers over $S$. Assume that $X$ is separated and locally of finite type[^4] over $D$. Then, $T(D) \cong \mathbb{A}_{S}^1$ and moreover, the natural morphism from $T(X)$ to $T(D)$ is surjective provided $X$ admits at least one morphism $S \to X$.* *Proof.* The fact that $T(D) \cong \mathbb{A}_{S}^1$ is proven in Lemma 4.3 on page 141 of [@sto2019]. Although it is not difficult, it is slightly less straightforward to show that the natural map $\theta: T(X) \to T(D)$ is surjective. We will provide a proof here as it will be illustrative in regards to other results in this section of the paper. Therefore, for this second fact, we can assume without loss of generality that $X$ is an affine subscheme of $\mathbb{A}_{D}^{N}$ defined by equations $F_i = 0$ (for $i = 1, \ldots, s$) of the form $F_i = G_i + tH_i$ where $G_i, H_i$ are elements of $A[x_1, \ldots, x_N]$. Thus, as a subvariety of $\mathbb{A}_{D}^{2N}$, the equations defining $T(X)$ are given by creating *arc variables* $\wideparen{x}_i = y_i + z_it$ and substituting them into $G_i$ to obtain the *arc equations* $\wideparen{G}_i +t \wideparen{H}_i=0$. This implies that the equations defining the arc space are of the form $$\label{editeq1} 0 = G_i(y_1, \ldots, y_N) = t\cdot H_i(y_1, \ldots, y_N) = t\cdot \partial G_i(y_1, \ldots, y_N, z_1, \ldots, z_N)$$ where $\partial G_i$ the polynomial such that $\wideparen{G}_i = G_i(y_1, \ldots, y_N) + t\cdot\partial G_i$. Any point $\alpha$ in $T(D)$ is given by an $a \in A$ defining an endormorphism $t \mapsto a\cdot t$. Thus, any potential point in the fiber $\theta^{-1}(a)$ is given by performing a substitution $at$ for $t$ in Equation [\[editeq1\]](#editeq1){reference-type="ref" reference="editeq1"}. Therefore, any morphism $\beta: D \to \mathbb{A}_{D}^{2N}$ which induces $\alpha : D \to D$ can potentially give a well-defined morphism $\beta'$ from $D$ to $T(X)$, and it will be sent to $\alpha$ via the morphism $\theta$. A morphism $S \to X$ will give rise to a morphism $h : S \to T(X)$. Otherwise, $X^{{\text{\rm red}}}$ would be $S$-smooth since the Jacobian would have no solutions over $A$, and in that case, the statement would be trivial. Therefore, we let $\beta'$ be any morphism inducing $\alpha$ which factors through $h$. In other words, we have a morphism $\beta'$ which fits into the following commutative diagram: $$\begin{tikzcd} D \arrow[drr, bend left , "{\beta'}"] \arrow[drrr, bend left, "{\beta}"] \arrow[ddrrr, bend right, "{\alpha}"] \arrow[dr, "{t =0 }" ] & & \\ &S \arrow[r,"{h}"] & T(X) \arrow[r, hook, ""] & \mathbb{A}_{D}^{2N} \arrow[d, ""] \\ & &\ &D \end{tikzcd}$$ ◻ In particular, in the case that $S = \operatorname{Spec}(k)$ where $k$ is a field, we have the following theorem. **Theorem 4**. *Let $S =\operatorname{Spec}(k)$ with $k$ a field, let $D$ be the dual numbers over $S$, and assume that $X$ has a $k$-point. Assume further that $X$ is separated and locally of finite type over $D$ and let $T(X)$ be the reduced tangent bundle. If $T(X)$ is irreducible, then the natural surjective morphism $T(X) \to \mathbb{A}_{k}^{1}$ is flat.* *Proof.* This is a direct consequence of Corollary [Corollary 2](#1stcor){reference-type="ref" reference="1stcor"} and Lemma [Lemma 3](#surj){reference-type="ref" reference="surj"}. ◻ **Example 5**. Consider the infinitesimal deformation $$X = \operatorname{Spec}(k[x,y,t]/(y^2 - x^3 - t, t^2))$$ of the cuspidal cubic $C$ given by $y^2 = x^3$ in $\mathbb{A}_k^2$. We directly compute the tangent space $T(X)$ by assigning variables $a,b,c,d,e,f$ over $k$ and forming arc variables $$\wideparen{x} = a + bt, \quad \wideparen{y} = c+ dt ,\quad \wideparen{t} = e + ft$$ Then, the coordinate ring for the tangent space is defined by substituting the arc variables into the defining equations for $X$ and then by setting these equations to zero in $A[t]/(t^2)$, where $A=k[a,b,d,e,f]$. Thus, we have the equations $$\begin{split} 0 &= \wideparen{y}^2 - \wideparen{x}^3 - \wideparen{t} = (c^2-a^3) \cdot 1 + (2cd - 3a^2b - f)\cdot t\\ 0 &= \wideparen{t}^2 = e^2\cdot 1 + 2ef \cdot t \end{split}$$ Since $k[t]/(t^2)$ is a vector space over $k$ with basis $\{1, t\}$, these equations imply that $$\underline{Hom}_k(D, X) \cong \operatorname{Spec}(k[a,b,c,d,e,f]/(c^2 - a^3, 2cd - 3a^2b - f, e^2, 2ef))$$ In the reduced structure $e^2 = 0 \implies e = 0$, which implies $T(X)$ is isomorphic to $\mbox{Spec}(k[a,b,c,d,f]/I)$ where $I = (c^2 - a^3, 2cd - 3a^2b - f)$, which is clearly prime ideal. Therefore, the natural morphism $\theta : T(X) \to \operatorname{Spec}(k[t])$ is flat. **Remark 6**. As we will frequently use Singular in this section, we include code which demonstrates how to check that a particular ideal $I$ of $R$ is prime. The interested reader may want to use this code to test alternative examples for which the ideal $I$ is not so clearly prime. ``` {.objectivec language="C"} LIB "primdecint.lib"; //Loads library with primary decomposition procedures ring R = 0, (a,b,c,d,f), dp; ideal I = (c2 - a3, 2cd - 3a2b - f); ideal J = std(I); primdecZ(I); print("Compare the above with the ideal I in standard basis below:"); print(J); ``` The output of this code is [1]: [1]: _[1]=3bc2-2acd+af _[2]=3a2b-2cd+f _[3]=a3-c2 [2]: _[1]=3bc2-2acd+af _[2]=3a2b-2cd+f _[3]=a3-c2 Compare the above with the ideal I in standard basis below: 3bc2-2acd+af, 3a2b-2cd+f, a3-c2 This explicitly shows that $I$ is a prime ideal and thus $T(X)$ is irreducible. **Remark 7**. The fact that $T(X)$ is irreducible is in stark contrast to the reduced case -- i.e., the central fiber $\theta_f^{-1}(0) \cong T(C)$ is reducible whenever $C$ is a singular curve for simply dimensional reasons! We briefly adapt this line of thinking to the case of first order deformations of singular curves in Section [5](#sec4){reference-type="ref" reference="sec4"}. Note that in general $T(X)$ itself my be either reducible or irreducible for any first order deformation of a singular curve, yet the central fiber will always be reducible for dimensional reasons. We consider some reducible cases below as in reality the morphism $\theta_f$ will still be flat in this case as well. **Example 8**. Here is a "non-example\" to Theorem [Theorem 4](#tanflat){reference-type="ref" reference="tanflat"}. Let $Y$ be a smooth scheme over a field $k$. Then, $Y$ is rigid and hence any deformation $X$ over $D$ is isomorphic to the trivial deformation $Y\times_k D$. Since $Y$ is smooth, we may cover $Y$ by open affines $U$ such that for some $d$ there is an étale morphism $U \to \mathbb{A}_{k}^d$, which implies that there is an isomorphism $T(U) \cong U \times_k \mathbb{A}_k^d$ and therefore that $T(Y) \to \operatorname{Spec}(k)$ is smooth. Since smoothness is stable under base change $T(Y) \times_k \mathbb{A}_{k}^1 \to \mathbb{A}_k^1$ is also smooth. Moreover, one can directly show that $T(X)$ is étale locally isomorphic to $T(Y) \times_k \mathbb{A}_{k}^1$ and therefore $T(X) \to \mathbb{A}_k^1$ is smooth and hence flat. If all we want to show is flatness, then we do not need to assume $Y$ is smooth as long as $X$ is the trivial deformation of $Y$ over $X$. This is because $T(Y) \to \operatorname{Spec}(k)$ is obviously flat, and since flatness is stable under base change (cf. Proposition 9.2 on page 254 of [@har2010]), $T(Y) \times_k \mathbb{A}_{k}^1 \to \mathbb{A}_k^1$ is also flat. Moreover, one can generally show that $T(X)$ is étale locally isomorphic to $T(Y) \times_k \mathbb{A}_{k}^1$ from whence it follows that $T(X) \to \mathbb{A}_{k}^1$ is flat provided $X$ is the trivial deformation of $Y$ over $D$. **Example 9**. Here is a more difficult "non-example\" to Theorem [Theorem 4](#tanflat){reference-type="ref" reference="tanflat"} where, unlike in Remark [Example 8](#fibex){reference-type="ref" reference="fibex"}, we have a non-trivial deformation over the dual numbers. Let $k$ be a field. Consider the infinitesimal deformation $$N_1:= \operatorname{Spec}(k[x,y,t]/(xy-t, t^2))$$ of the node $N_0 = \operatorname{Spec}(k[x,y]/(xy))$ over the dual numbers $D=\operatorname{Spec}(k[t]/(t^2))$. Let $a+bt$, $c+dt$, and $e+ft$ be generic elements of $k[t]/(t^2)$. We substitute these arc variables into the defining equations of $N_1$ to obtain: $$\begin{split} 0 &=(a+bt)(c+dt) - (e+ft) = (ac-e)\cdot1 + (ad+bc - f) \cdot t \\ 0 &= (e+ft)^2 = e^2\cdot 1 + 2ef\cdot t \end{split}$$ This gives the equations defining $\underline{Hom}_{k}(D, N_1)$ as an affine scheme over $k$, and in the reduction, $e^2 = 0 \implies e =0$ so that the coordinate ring for $T(N_1) := \underline{Hom}_{k}(D, N_1)^{{\text{\rm red}}}$ is given by $k[a,b,c,d,f]/(ac, ad+bc - f)$. It is clear that $T(N_1)$ is not irreducible, yet we will still be able to demonstrate that the morphism $\theta: T(N_1) \to \operatorname{Spec}(k[f])$ induced from the flat morphism $N_1 \to D$ is again flat. In fact, we first will show that the smooth locus of this is given by $U = \theta^{-1}(\mathbb{A}_{k}^1\setminus\{0\})$. To show this, let $A =k[f]$ and let $B = A[a,b,c,d]/I$ where $I = (F, G)$ with $F = ac$ and $G = ad+bc - f$, then it is enough to calculate the following determinate: $$\mbox{Jac} := \begin{array}{|cc|} \frac{\partial}{\partial a} F & \frac{\partial}{\partial c}F \\ \frac{\partial}{\partial a} G & \frac{\partial}{\partial c}G\end{array} = \begin{array}{|cc|} c & a \\ d & b \end{array} = cb - ad$$ If $\mbox{Jac} = 0$, then $ad = bc$ and so using the equation $G=0$ gives $2ad = f = 2bc$, but either $a= 0$ or $c = 0$ and in either case then $f = 0$. Therefore, $f \neq 0$ implies $\mbox{Jac} \neq 0$ which implies the map $\theta$ is smooth away from points which map $f$ to $0$. Now, if $f = 0$, then the equation $G= 0$ implies $ad = - bc$, which implies $2ad = \mbox{Jac} = 2bc$. For $\mbox{char}(k) = 2$, this immediately implies $\mbox{Jac} = 0$. Otherwise, we know either $a=0$ or $c=0$ (this is given by the equation $F =0$) and so in either case $\mbox{Jac} =0$. This together with the preceding paragraph show that the smooth locus of $\theta$ is given by $U = \theta^{-1}(\mathbb{A}_{k}^1\setminus\{0\})$, as claimed. Actually, one can quickly see that both of the irreducible components of $T(N_1)$ are isomorphic to $Q \times_k \mathbb{A}_{k}^1$ where $Q = \operatorname{Spec}(k[x,y,t]/(xy-t))$. This is straightforward since on the component given by $a=0$ for example, the coordinate ring reduces to $k[b,c,d,f]/(bc-f)$ where $d$ is now a free variable. The reader may quickly check the other component. Regardless, since the map $\theta$ is smooth away from the fiber above $f = 0$, it has constant relative dimension equal to $2$ on each irreducible component when restricted to the smooth locus $U$. Now, notice that the central fiber given by $f=0$ is actually the reduced tangent space of the node $N_0$: $$\theta^{-1}(0) \cong \underline{Hom}_{k}(D, N_0) \cong T(N_0)$$ as it is the spectrum of the ring $k[a,b,c,d]/(ac, ad+bc)$. Then, both irreducible components of $\theta^{-1}(0)$ are isomorphic to $N_1 \times_k \mathbb{A}_{k}^1$, and so $\mbox{dim}(\theta^{-1}(0)) = 2$. In summary, we have explicitly shown that $\theta : T(N_1) \to \mathbb{A}_{k}^{1}$ has constant relative dimension equal to $2$, and so, by the miracle of flatness (Theorem 23.1 on page 179 of [@mat1987]), $\theta$ must be flat since the base is regular and $T(N_1)$ is a complete intersection and hence it is Cohen-Macaulay. **Remark 10**. Examples [Example 8](#fibex){reference-type="ref" reference="fibex"} and [Example 9](#defnode){reference-type="ref" reference="defnode"} are not really non-examples because we can relax the condition in Theorem [Theorem 4](#tanflat){reference-type="ref" reference="tanflat"} to the case that $T(X)$ is reducible provided the restriction of $\theta : T(X) \to \mathbb{A}_{k}^{1}$ to each of its irreducible components is a dominate morphism as noted Theorem [Theorem 1](#LiuFlat){reference-type="ref" reference="LiuFlat"}. This is clearly the case in Example [Example 9](#defnode){reference-type="ref" reference="defnode"} since, as we noted above, both irreducible components are of the form $N_1 \times_k \mathbb{A}_{k}^{1}$ and are mapping surjectively onto $\operatorname{Spec}(k[f])$. In the case of Example [Example 8](#fibex){reference-type="ref" reference="fibex"}, the statement is immediate. **Example 11**. Going back to Example [Example 5](#cubicA){reference-type="ref" reference="cubicA"}, we will check directly that it is flat in characteristic $2$ but by using either of the technique of [Example 9](#defnode){reference-type="ref" reference="defnode"}. One may check that the Jacobian in this case is given by $\mbox{Jac} = a^4$. This then implies that the open subscheme $\theta^{-1}(\mathbb{A}_k^1 \setminus \{0\})$ is contained in the smooth locus $U$ of $\theta$ (although, they do not agree in this case). Thus, away from $f = 0$, the morphism $\theta$ is of constant relative dimension $2$. Moreover, $\theta^{-1}(0)$ is the union of two subschemes $V$ and $W$ each of dimension $2$. Here, $V$ is the singular locus of $\theta$ given by $a = 0$ (the condition that $a=0$ implies $c = 0$ and $f = 0$), and thus $V$ is isomorphic to $\mathbb{A}_{k}^2$. The subscheme $W$ is cut out by $b = f = 0$, and so $W$ is isomorphic to $C \times_k \mathbb{A}_{k}^1$. Regardless, $\theta$ has fibers of constant dimension equal to $2$ and the base is regular. So, by the miracle of flatness, we just need to make sure that the domain is Cohen-Macaulay. The skeptical reader can run the Singular code provided in Remark [Remark 15](#cmcode){reference-type="ref" reference="cmcode"} to truly check that this ring is Cohen-Macaulay. Interestingly, the morphism $\theta$ in Example [Example 5](#cubicA){reference-type="ref" reference="cubicA"} is flat in every characteristic, and, again, the underlying reason for this behavior has to do with the fact that each irreducible component is dominating $\mathbb{A}_k^1$ via $\theta$. **Remark 12**. The cause of this degenerate behavior in Example [Example 11](#cmex){reference-type="ref" reference="cmex"} is that the embedding of an affine truncated linear $n$-jet scheme into $\mathbb{A}_k^M$ can be alternatively described in terms of the universal derivation $\alpha \mapsto (j!\alpha_j^{(l)})$ for $\mbox{char}(k) = 0$ or $\mbox{char}(k)> n$ (cf. page 5 of [@mus2001]). For more on how derivations relate to jet spaces, we refer the interested reader to [@voj2004]. For simplicity then we will later restrict to the case where the underlying field $k$ is of characteristic zero. Considering Example [Example 9](#defnode){reference-type="ref" reference="defnode"} and the previous remark, it would thus be interesting to know in more generality when the irreducible components dominate $\mathbb{A}_{S}^{1}$ via the natural morphism $\theta: T(X) \to \mathbb{A}_{S}^{1}$, whenever $S$ is a reduced Noetherian local ring in any characteristic. It would also seem to be an interesting yet difficult question to try push these types of questions to the non-reduced structure -- i.e., investigate when the morphism from $\underline{Hom}_{S}(D, X)$ to $\underline{Hom}_{S}(D,D)$ is flat. We will not directly concern ourselves with these general problems in this paper. If however we restrict ourselves to the reduced structure, then it is natural in this context to ask the following general question. **Question 13**. Given an arbitrary $S \in \mathbf{Sch}$, let $X, Y,$ and $Z$ be objects of $\mathbf{Sch}/{S}$ whose structure morphisms are all locally of finite presentation. Assume further that $X$ is finite and flat over $S$ and that $Y$ and $Z$ are both $S$-separated. Given a flat morphism $f: Y \to Z$, when is it the case that the induced natural morphism $$\theta_f: \underline{Hom}_S(X,Y)^{{\text{\rm red}}} \to \underline{Hom}_S(X,Z)^{{\text{\rm red}}}$$ is also flat? It is decidedly not true in general that the morphism $\theta_f$ is flat. We specialize this question to the following simplified case. Let $S = \operatorname{Spec}(k)$ where $k$ is a field, $X = \operatorname{Spec}(k[t]/(t^2))$, and $f: Y \to \mathbb{A}_{k}^{1}$ be a flat family of curves, then it is not always the case that the induced natural morphism $\theta_f$ on the reduced tangent spaces $$\theta_f : T(Y) \to \mathbb{A}_{k}^{2}$$ is flat. **Example 14**. Let $W = \operatorname{Spec}(A)$ with $A = k[x,y,z]/(x^2-y^2z)$ be the flat family over $\mathbb{A}_k^1 = \operatorname{Spec}(k[z])$. Let $\wideparen{x} = a+bt,$ $\wideparen{y} = c+dt,$ and $\wideparen{z}= e+ft$ be arc variables where we compute $\wideparen{x}^2-\wideparen{y}^2\wideparen{z} = 0$ in the reduction over $k[t]/(t^2)$. This gives the ideal $$I = (a^2 - c^2e, 2ab - 2cde - c^2f)$$ in the ring $R = k[a,b,c,d,e,f]$ and an isomoprhism $T(W) \cong \operatorname{Spec}(R/I)$. If $\mbox{char}(k) = 2$, then $I$ reduces to the ideal $(a^2 - c^2e, c^2f)$ and we note that $b$ and $d$ are a free variables now. For $(e,f) = (0,0)$, it is easy to see that the fiber is equal to $\mathbb{A}_{k}^{3} = \operatorname{Spec}(k[b,c,d])$. The dimension of the base $\mathbb{A}_{k}^{2}$ at $(0,0)$ is clearly $2$, so we just need to find the dimension of $T(W)$ at a point in the preimage of $(0,0).$ Actually, at any point in the preimage, we have $f \neq 0 \implies c^2 = 0 \implies c = 0$. Then, $0 = a^2-c^2e = a^2$ implies $a = 0$. Thus, for example, the local ring at the maximal ideal defined by the origin is given by localizing $k[b,d,e,f]$ at $(b,d,e,f)$ which is clearly a local ring of Krull dimension $4$. Therefore, the relative dimension of $\theta_f$ at the origin is $2$, yet the dimension of the fiber is $3$. Thus, $\theta_f$ cannot be flat. **Remark 15**. The ring in $R/I$ in Example [Example 14](#nonflat){reference-type="ref" reference="nonflat"} is even a Cohen-Macaulay ring, which we can check using Singular [@DGPS]. We include this here for illustrative purposes and since the code below is referenced in Example [Example 11](#cmex){reference-type="ref" reference="cmex"}. Following SINGULAR Example 7.7.8 on page 426 of [@gre2002], we will use the following code: ``` {.objectivec language="C"} LIB "homolog.lib"; //Loads library with homological algebra procedures //Procedure which calculates depth of a module proc depth(module M) {ideal m=maxideal(1); int n=size(m); int i; while(i<n) {i++; if(size(KoszulHomology(m,M,i))==0){return(n-i+1);} } return(0); } //Procedure tests whether module is CM: returns 1 if true. proc CohenMacaulayTest(module M) {return(depth(M)==dim(std(Ann(M)))); } //Define module and test for CM ring R = 2, (a, b, c, d, e, f), dp; ideal I = a2 - c2e, c2f; module M = I*freemodule(1); CohenMacaulayTest(M); ``` The output of this code is $1$, which demonstrates that $R/I$ is Cohen-Macaulay. This is not at all surprising since $R$ is automatically a complete intersection by Proposition 1.4 on page 7 of [@mus2001], and hence Cohen-Macaulay. **Remark 16**. As far as checking the ring is Cohen-Macaulay in characteristic $2$, it is enough just to work over the finite field $\mathbb{F}_2$, because, by base change (cf., [@stacks-project [Tag 045P](https://stacks.math.columbia.edu/tag/045P)]), this will also prove that $R/I$ is Cohen-Macaulay over any field $k$ of characteristic equal to $2$. **Example 17**. Let us then consider the case of Example [Example 14](#nonflat){reference-type="ref" reference="nonflat"}, but this time we assume $k = \mathbb{Q}$. Actually, we can use Singular to check for flatness directly in this case by asking it to compute the first torsion module. First note, that $$\mbox{Tor}_1^{\mathbb{Q}[e,f]}(\mathbb{Q},R/I) \cong \mbox{Tor}_1^{R}(\mathbb{Q}[a,b,c,d],R/I).$$ ``` {.objectivec language="C"} LIB "homolog.lib"; ring R = 0, (a,b,c,d,e,f), dp; ideal I = a2-c2e, 2ab - 2cde - c2f; matrix Ph[1][2] = I; matrix Ps[1][2] = e, f; Tor(1,Ps,Ph); ``` The output of this code is // dimension of Tor_1: 3 _[1]=gen(1) _[2]=gen(2) _[3]=gen(3) _[4]=gen(4) _[5]=f*gen(5) _[6]=e*gen(5) _[7]=a*gen(5) # General Auto-Arc Spaces {#sec2} **Definition 18**. Let $i: X \to S$ be an object of $\mathbf{Sch}/{S}$ and $Z$ an object of $\mathbf{Fat}/{S}$ where $S$ is an arbitrary scheme. We say that a scheme $Y$ is an *infinitesimal deformation over* $Z$ of $X$ if there is a commutative diagram $$\begin{tikzcd} X \arrow[d,"{i}"] \arrow[r, hook] & Y\arrow[d, "{j}"]\\ S \arrow[r] &Z \end{tikzcd}$$ such that the structure morphism $j : Y \to Z$ is flat and induces an isomorphism $X \cong Y\times_Z S$. Given any infinitesimal deformation $Y$ over $Z$ such that $Y$ is separated and locally of finite presentation over $Z$, we define $$\mathcal{A}_Z(Y) := \underline{Hom}_S(Z,Y)^{red}$$ and call it *the auto-arc space of* $Y$ *with respect to* $Z$, and in the special case that $Y\cong Z$, we call $\mathcal{A}_Z := \mathcal{A}_Z(Z)$ *the auto-arc space of* $Z$. We therefore have the natural induced morphism $$\theta_j : \mathcal{A}_Z(Y) \to \mathcal{A}_Z$$ Assume now that $S$ is a reduced scheme, then we have an induced isomorphism $$\underline{Hom}_S(Z,X)^{{\text{\rm red}}} \cong \underline{Hom}_S(Z,Y) \times_Z S.$$ **Remark 19**. Note that the morphism $\underline{Hom}_S(Z,Y) \to Z$ is very often non-flat. For instance, if $S = \operatorname{Spec}(k)$ with $\mbox{char}(k) \neq 2$, and if $Y$ and $Z$ are both the dual numbers over $S$, then on the level of coordinate rings, the morphism of schemes above induces a ring monomorphism $k[t]/(t^2) \to k[t,s]/(t^2, 2st)$ defined by $t \mapsto t$, which is clearly not flat. In the category $\mathbf{Sch}/{S}$, any infinitesimal deformation $Y$ over $Z$ gives rise to the following induced commutative diagram $$\begin{tikzcd} \underline{Hom}_S(Z, X)\arrow[d] \arrow[r, hook]&\underline{Hom}_S(Z,Y) \arrow[d] \arrow[r, " "] &\underline{Hom}_S(Z,Z)\arrow[d] \\ X\arrow[d,"i"] \arrow[r, hook] &Y\arrow[d,"j"] \arrow[r, "j"] & Z \\ S \arrow[r] & Z & \ \end{tikzcd}$$ provided $Y$ is separated and locally of finite presentation over $Z$. If we further assume $S$ and $X$ are reduced schemes, then by taking the fiber product with respect to $S \to Z$, we obtain the following commutative diagram $$\label{maindiag} \begin{tikzcd} \underline{Hom}_S(Z,X)^{{\text{\rm red}}} \arrow[d] \arrow[r, hook] &\mathcal{A}_Z(Y) \arrow[d] \arrow[r,"{\theta_j}"] &\mathcal{A}_Z \arrow[d] \\ X \arrow[r, "{\cong}"] &X \arrow[r, ""] & S \end{tikzcd}$$ The case at the germ of a plane curve singularity One special case of the above is given when we assume that $S = \operatorname{Spec}(k)$ where $k$ is an algebraically closed field. Given a closed point $x$ of $X$, we let $Z$ be the $n$*-th order jet* $J_p^nX := \operatorname{Spec}(\mathcal{O}_{X,x}/\mathfrak{m}_x^{n+1})$ which is an object in $\mathbf{Fat}/{k}$. In which case, we have the induced closed immersion $$\underline{Hom}_S(Z,Z) \hookrightarrow \underline{Hom}_S(Z,X) \times_S Z$$ which extends the diagram above to the commutative diagram $$\begin{tikzcd} \underline{Hom}_S(Z,Z) \arrow[d] \arrow[r, hook] &\underline{Hom}_S(Z, X)\arrow[d] \arrow[r, hook]&\underline{Hom}_S(Z,Y) \arrow[d] \arrow[r, " "] &\underline{Hom}_S(Z,Z)\arrow[d] \\ Z \arrow[r, hook] &X\arrow[d,"i"] \arrow[r, hook] &Y\arrow[d,"j"] \arrow[r, "j"] & Z \\ &S \arrow[r] &Z & \ \end{tikzcd}$$ Then, by taking the fiber product with respect to $S \to Z$, we obtain the following commutative diagram $$\begin{tikzcd} \mathcal{A}_Z \arrow[d] \arrow[r,hook] & \underline{Hom}_S(Z,X)^{{\text{\rm red}}} \arrow[d] \arrow[r, hook] &\mathcal{A}_Z(Y) \arrow[d] \arrow[r,"{\theta_j}"] &\mathcal{A}_Z \arrow[d] \\ S \arrow[r, hook] &X \arrow[r, "{\cong}"] &X \arrow[r, ""] & S \end{tikzcd}$$ Thus, $\mathcal{A}_Z$ is a closed subscheme of the fiber above the point given by $S \to X$, or, in other words, we have a closed immersion $$\label{isomor} \mathcal{A}_Z \hookrightarrow \underline{Hom}_S(Z,X)^{{\text{\rm red}}} \times_X S$$ **Proposition 20**. *Let $S = \operatorname{Spec}(k)$ where $k$ is an algebraically closed field. Let $X$ be a reduced, separated $S$-scheme, locally of finite type over $S$. Then, the closed immersion [\[isomor\]](#isomor){reference-type="ref" reference="isomor"} above is an isomorphism.* *Proof.* This is essentially a restatement of Lemma 3.2 of [@auto]. The only difference being that finite type there is being replaced with locally of finite type here. ◻ **Example 21**. Let $X$ be a smooth curve over an algebraically closed field $k$ and let $Y$ be the trivial deformation over the dual numbers $D = J_p^1X$. Then, $\mathcal{A}_D(Y) \cong \underline{Hom}_S(D,X)^{{\text{\rm red}}} \times_k \mathcal{A}_D$. Note that these spaces are all tangent bundles as studied in Section [2](#sec1){reference-type="ref" reference="sec1"}. Thus, we have the following commutative diagram $$\begin{tikzcd} \mathbb{A}_k^1 \arrow[d] \arrow[r,hook] &T(X) \arrow[d] \arrow[r, hook] &T(X) \times_k \mathbb{A}_k^1 \arrow[d] \arrow[r,"{\theta_j}"] &\mathbb{A}_k^1 \arrow[d] \\ \operatorname{Spec}(k) \arrow[r, hook] &X \arrow[r, "{\cong}"] &X \arrow[r, ""] & \operatorname{Spec}(k) \end{tikzcd}$$ where $\theta_j$ is projection onto the second factor. Note that $T(X)$ is étale locally isomorphic to $X \times_k \mathbb{A}_k^1$ in this case since $X$ is smooth. **Example 22**. Let $k$ be an algebraically closed field with $\mbox{char}(k) \neq 2, \ 3$. Let $X$ be the cuspidal cubic $\operatorname{Spec}(k[x,y]/(y^2 -x^3))$ and let $p$ be the singular point given by the origin. Let $Z = J_p^{n-1}X$ and let $\mathcal{L}_m(X) = \underline{Hom}_S(J_p^mX,Y)^{{\text{\rm red}}}$ denote the reduced truncated linear arc space. It is proven in [@sto2017] that $\mathcal{A}_Z \cong \mathcal{L}_{m}(X) \times_k \mathbb{A}_k^7$ where $m = 2(n-3)$ whenever $n \geq 4$. Thus, in this case, for an infinitesimal deformation $Y$ of $X$ over $Z$, we have $$\begin{tikzcd} \mathcal{L}_{m}(X) \times_k \mathbb{A}_k^7 \arrow[d] \arrow[r,hook] & \underline{Hom}_S(Z, X)^{{\text{\rm red}}} \arrow[d] \arrow[r, hook] & \mathcal{A}_Z(Y) \arrow[d] \arrow[r,"{\theta_j}"] &\mathcal{L}_{m}(X) \times_k \mathbb{A}_k^7 \arrow[d] \\ \operatorname{Spec}(k) \arrow[r, hook] &X \arrow[r, "{\cong}"] &X \arrow[r, ""] & \operatorname{Spec}(k) \end{tikzcd}$$ **Remark 23**. It is worth noting that the relationship between $\mathcal{L}_m(X)$ and the auto-arc space $\mathcal{A}_Z$ noticed in Example [Example 22](#planearc){reference-type="ref" reference="planearc"} has been generalized to all plane curve singularities $(X,p)$. This is worked out in significant detail in [@auto]. **Question 24**. Given a plane curve singularity $(X,p)$, it should be somewhat straightforward to extend the above diagram to the so-called *mixed auto-arc spaces* $\underline{Hom}_S(J_p^nX, J_p^mX)^{{\text{\rm red}}}$ for $n, \ m$ sufficiently large. More generally, it should be possible to obtain "closed expressions\" related to the truncated linear arc spaces (say in the Grothendieck ring of varieties) of $\underline{Hom}_S(Z, Z')^{{\text{\rm red}}}$ whenever $Z$ and $Z'$ are jets of different plane curve singularities. For higher embedding dimension, say for germs of surface singularities, computations carried out by the author clearly show a "recursive pattern\" for $\mathcal{A}_Z$ in the Grothendieck ring of varieties. Moreover, they seem to be related to iterated linear jet spaces, but establishing a direct relationship in analogy to the situation for plane curves as mentioned in Remark [Remark 23](#help){reference-type="ref" reference="help"} remains somewhat elusive. Further general statements for auto-arc spaces. We go back to assuming that $X$ is an arbitrary reduced scheme over $S=\operatorname{Spec}(k)$ where $k$ is an algebraically closed field. We let $Z$ be an arbitrary object of $\mathbf{Fat}/{k}$ and we assume that $Y$ is a local deformation of $X$ over $Z$ which is also separated and locally of finite type over $Z$. Then, as noted at the beginning of Section [3](#sec2){reference-type="ref" reference="sec2"}, we have the following commutative diagram $$\begin{tikzcd} \underline{Hom}_S(Z,X)^{{\text{\rm red}}} \arrow[d] \arrow[r, hook] &\mathcal{A}_Z(Y) \arrow[d,"{\pi}"] \arrow[r,"{\theta}"] &\mathcal{A}_Z \arrow[d] \\ X \arrow[r, "{\cong}"] &X \arrow[r, ""] & S \end{tikzcd}$$ Of course, this immediately implies that we have a natural induced morphism $$\mathcal{A}_Z(Y) \xrightarrow{\pi\times\theta} X \times_S\mathcal{A}_Z$$ which is locally a piecewise trivial fibration with affine fiber whenever $X$ is smooth, and in fact we have the following lemma. **Lemma 25**. *Given our basic assumptions of this particular subsection, let $X_{{\text{\rm sm}}}$ denote the smooth locus of $X$. Then, the morphism $\pi\times \theta$ above is a piecewise trivial fibration with affine fibers away from the singular locus $X\setminus X_{{\text{\rm sm}}}$. Moreover, $\underline{Hom}_S(Z,X_{{\text{\rm sm}}})^{{\text{\rm red}}} \times_k \mathcal{A}_Z$ and $\pi^{-1}(X_{{\text{\rm sm}}})$ are étale locally isomorphic.* *Proof.* We can find a covering by affine opens $U_{\alpha} \subset X_{{\text{\rm sm}}}$ with étale morphisms $U_{\alpha} \to \mathbb{A}_k^{r_{\alpha}}$. We therefore have open affines $V_{\alpha} \cong U_{\alpha}\times_k Z$ given by the embedded deformation of $U_{\alpha} \subset X$, which will allow us to cover $\pi^{-1}(X_{{\text{\rm sm}}})$ by $V_{\alpha}$ with étale morphisms $V_{\alpha} \to \mathbb{A}_k^{r_{\alpha}}\times_k Z$. Thus, away from the singular locus $$\mathcal{A}_Z(Y)\times_X V_{\alpha} \cong \mathcal{A}_Z(V_{\alpha}) \cong \underline{Hom}_S(Z, U_{\alpha})^{{\text{\rm red}}} \times_k \mathcal{A}_Z$$ ◻ **Remark 26**. In general, the closed immersion $\iota : \underline{Hom}_S(Z, X)^{{\text{\rm red}}} \hookrightarrow\mathcal{A}_Z(Y)$ admits a section $s$ such that $s\circ \iota$ is the identity morphism, which then by the universal property of fiber products yields an induced natural morphism $$\mathcal{A}_Z(Y) \to \underline{Hom}_S(Z, X)^{{\text{\rm red}}} \times_k \mathcal{A}_Z$$ which, by Lemma [Lemma 25](#etlem){reference-type="ref" reference="etlem"}, is étale locally an isomorphism away from the singular locus $X\setminus X_{{\text{\rm sm}}}$. Actually, the section $s$ is induced by applying the reduction functor to the morphism $$\underline{Hom}_S(Z,Y) \to \underline{Hom}_S(Z,X)$$ which is induced from restricting the image of a morphism $Z \to Y$ to the image of the closed immersion $X \hookrightarrow Y$. **Lemma 27**. *Let $O \in \mathcal{A}_Z$ correspond to the trivial endomorphism of $Z$ and assume our basic assumptions of this particular subsection. Then, $$\underline{Hom}_S(Z,X)^{{\text{\rm red}}} \cong \theta^{-1}(O)$$* *Proof.* Let $Z=\operatorname{Spec}(R)$ where $(R, \mathfrak{m})$ is a finitely generated local Aritinian $k$-algebra, and let $O$ denote the point on $\mathcal{A}_Z$ which corresponds to the ring endomorphism $\varphi_O$ of $R$ given by the map onto the residue field $k$ (i.e., $\varphi_O$ is the composition $R \to R/\mathfrak{m} \hookrightarrow R$). Then, this lemma is clearly true since the problem is local (i.e., we may assume $X$ is an affine scheme) and therefore we may reduce to case where a point on $\underline{Hom}_S(Z,X)^{{\text{\rm red}}}$ gives rise directly to a morphism $h: Z \to X$ which fits into the commutative diagram $$\begin{tikzcd} Z \arrow[r,"h"] \arrow[rd] & X \arrow[d,"{i}"] \arrow[r, hook] & Y\arrow[d, "{j}"]\\ \ & S \arrow[r] &Z \end{tikzcd}$$ Thus, post-composing with the closed immersion $X\hookrightarrow Y$ gives a morphism $\bar{h} : Z \to Y$ and therefore corresponds to a point on $\mathcal{A}_Z(Y)$, which means that the point on $\mathcal{A}_Z$ given by $\theta(\bar{h})$ must be equivalent to the morphism $Z \to S \to Z$ which is precisely the morphism $\operatorname{Spec}(\varphi_O)$ where $\varphi_O$ is the ring endomorphism discussed above. ◻ The situation for locally complete intersections. An object $X$ of $\mathbf{Sch}/{k}$ which is reduced, separated and finite type over $k$ is called a *variety over* $k$. We have the following well-known result concerning deformations of locally complete intersection varieties. We need the following fact. **Proposition 28**. *Let $Z \in \mathbf{Fat}/{k}$ with $k$ a field. Moreover, let $X$ be locally complete intersection variety over $k$. Then, any deformation $Y$ over $Z$ is locally complete intersection.* *Proof.* This is Theorem 9.2 on page 74 of [@hart2009]. ◻ Assume in the rest of this subsection that $X$ is a variety of dimension $d$. Let $\delta_Z := \mbox{dim}{\ \mathcal{A}_Z}$ and $\ell:=\ell(Z)$ be the length of $Z$. In general, we have $$\mbox{dim}{\ \underline{Hom}_S(Z,X)} \geq d\ell$$ so that $$\label{inediteq1} \mbox{dim}{\ \mathcal{A}_Z(Y)} \geq d\ell +\delta_Z$$ by Lemma [Lemma 25](#etlem){reference-type="ref" reference="etlem"} and Remark [Remark 26](#oppmap){reference-type="ref" reference="oppmap"}. Now, following the decomposition of Proposition 1.4 on page 7 of [@mus2001], we consider $$\label{dimauto} \mathcal{A}_Z(Y) = \pi^{-1}(X_{{\text{\rm sing}}}) \cup\overline{\pi^{-1}(X_{{\text{\rm sm}}})}$$ where $X_{{\text{\rm sing}}}$ denotes the singular locus of $X$. Therefore, if we assume moreover that $\mathcal{A}_Z(Y)$ is pure dimensional, then we obviously have $$\label{bediteq1} \mbox{dim}{\ \mathcal{A}_Z(Y)} = d\ell +\delta_Z$$ and if $\mathcal{A}_Z(Y)$ is also irreducible, then $$\mbox{dim}{\ \pi^{-1}(X_{{\text{\rm sing}}})} < d\ell+\delta_Z$$ Notice that Inequality [\[inediteq1\]](#inediteq1){reference-type="ref" reference="inediteq1"} then clearly implies that $$\frac{\delta_{Z}}{\ell} \leq \frac{\mbox{dim}{\ \mathcal{A}_Z(Y)}}{\ell} - d$$ For any closed germ $(Z,P)$ giving rise to a limit of fat points $Z_n$ and a sequence of deformations $Y_n$ over $Z_n$ of $X$, we define $$\begin{split} \delta_{(Z,P)}^{*}(X) &:= \lim_{n\to \infty} \frac{\mbox{dim}{\ \mathcal{A}_{Z_n}(Y_n)}}{\ell(Z_n)} - d \\ e(Z,P) &: = \lim_{n\to \infty} \frac{\mbox{dim}{\ \mathcal{A}_{Z_n}}}{\ell(Z_n)} \end{split}$$ where the former is a slight variant of the *asymptotic defect* defined in Equation 28 on page 26 of [@sch2] and the later is defined without change on page 27 of [@sch2]. We note that we then have the inequality $$e(Z,P) \leq \delta_{(Z,P)}^{*}(X)$$ **Remark 29**. We are suppressing notation here. More explicitly, the asymptotic defect defined above is heavily dependent on the choice of deformations. Regardless, our goal here is just to simply produce the lower bound $e(Z,P)$. Thus, when the limits above exist, we can consider the so called *regulated defect* given by $\delta_{(Z,P)}^{*}(X)/e(Z,P)$. More generally, we define $$R_{(Z,P)}^{\mathcal{Y}}(X) := \limsup_{n}\{\frac{ \mbox{dim}{\ }\mathcal{A}_{Z_n}(Y) -d\ell(Z_n)}{\mbox{dim}{\ }\mathcal{A}_{Z_n}}\}$$ and we call it the *regulated defect of* $X$ *at* $(Z,p)$ *along the formal deformation* $\mathcal{Y}=\varprojlim Y_n$ where $Y_n$ is an infinitesimal deformations of $X$ over $Z_n$. Regardless, we continue by adapting the proof of Proposition 1.4 on page 7 of [@mus2001] to the current situation above, and so we now assume that $\mbox{dim}{\ \mathcal{A}_Z(Y)} = d\ell +\delta_Z$ where $d$ is the dimension of $X$ and $\ell$ is the length of the fat point $Z$. Assume now that $X$ is a locally complete intersection variety over $k$ and let $Y$ be a deformation of $X$ over a fat point $Z$ so that, by Theorem [Proposition 28](#deformlci){reference-type="ref" reference="deformlci"}, $Y$ is also a locally complete intersection variety over $Z$. The problem is local so we can reduce to the case where $X$ and $Y$ are affine, and, of course, we choose these affines so that $Y$ is a complete intersection. Therefore, by assumption $Y \subset \mathbb{A}_{Z}^N$ is defined by $N-d$ equations. We therefore have that $\mathcal{A}_Z(Y)$ is defined by $(N-d)\ell$ equations as subvariety of $\mathcal{A}_{Z}(\mathbb{A}_Z^N) \cong \mathbb{A}_{k}^{N\ell}\times_k\mathcal{A}_Z$. Therefore, any irreducible component of $\mathcal{A}_Z(Y)$ has dimension greater than or equal to $N\ell +\delta_Z - (N-d)\ell = d\ell+\delta_Z$. But, then by assumption on the dimension of $\mathcal{A}_Z(Y)$, it must be of pure dimension and a complete intersection. We therefore have the following result. **Theorem 30**. *Let $X$ be a locally complete intersection variety over a field $k$ and let $Z$ be an object of $\mathbf{Fat}/{k}$. Let $Y$ be a deformation of $X$ over $Z$ and assume that $\mbox{dim}{\ }\mathcal{A}_Z(Y) = d\ell +\delta_Z$. Then, $\mathcal{A}_Z(Y)$ must be of pure dimension and a locally complete intersection over $k$.* **Remark 31**. We note that Example [Example 9](#defnode){reference-type="ref" reference="defnode"} shows that the closed set $\overline{\pi^{-1}(X_{{\text{\rm sm}}})}$ of $\mathcal{A}_Z(Y)$ is not necessarily irreducible, and so it will frequently be the case that the dimension of $\pi^{-1}(X_{{\text{\rm sing}}})$ is strictly less than $d\ell +\delta_Z$, yet $\mathcal{A}_Z(Y)$ will be reducible. This is in stark contrast with the classical truncated linear arc case as was also pointed out in Remark [Remark 7](#redcent){reference-type="ref" reference="redcent"} and Remark [Remark 41](#bigremark){reference-type="ref" reference="bigremark"}. We will need Theorem [Theorem 30](#lci){reference-type="ref" reference="lci"} in the proceeding section when we investigate the linear jet case, but it also has another important implication, which we state below. **Theorem 32**. *Assume that $X$ is a locally complete intersection variety over an algebraically closed field $k$. Let $Z$ be the object of $\mathbf{Fat}/{k}$ given by the $n$th jet scheme $J_p^nX$ at some closed point $p$ of $X$. Then, the auto-arc space $\mathcal{A}_Z$ of $Z$ is a locally complete intersection variety over $k$ whenever $\mbox{dim}{\ }\underline{Hom}_S(Z,X)^{{\text{\rm red}}} = d\ell$.* *Proof.* The key elements of the proof do not change if we apply the adapted argument above to $\underline{Hom}_S(Z,X)^{{\text{\rm red}}}$ as opposed to $\mathcal{A}_Z(Y)$. Here, one only needs to assume that the dimension of this generalized arc space is equal to $d\ell$. Now, we may use Proposition [Proposition 20](#yay){reference-type="ref" reference="yay"}, which states that the space $\mathcal{A}_Z$ is obtained by a flat base change of $\underline{Hom}_S(Z,X)^{{\text{\rm red}}} \to X$, which is a locally complete intersection morphism by [@stacks-project [Tag 09RL](https://stacks.math.columbia.edu/tag/09RL)]. This then implies $\mathcal{A}_Z$ is a locally complete intersection variety over $k$ by [@stacks-project [Tag 069I](https://stacks.math.columbia.edu/tag/069I)]. ◻ **Example 33**. As we noted in Example [Example 22](#planearc){reference-type="ref" reference="planearc"}, $\mathcal{A}_Z \cong \mathcal{L}_{2(n-3)}(X)\times_k\mathbb{A}_k^7$ when $Z$ is the $(n-1)$th jet of the cuspidal cubic $X$ given $y^2=x^3$ at the origin $O$ for $n \geq 4$. In this case, not only is it clearly seen that $\mathcal{A}_Z$ is a locally complete intersection (by Proposition 1.4 on page 7 of [@mus2001]), but it is also reducible directly by Corollary 4.2 on page 19 of [@mus2001]. Note also that $\delta_{(Z,P)}^{*}(X) \geq e(X,O) = \lim_{n\to\infty} \frac{2n-6+7}{2n-1} = 1$. As we noted earlier, this behavior generalizes to all plane curve singularities as discussed in [@auto]. **Question 34**. Let $X$ be a locally complete intersection variety of pure dimension $d$ over an algebraically closed field $k$ and assume that $\mbox{dim}{\ }\underline{Hom}_S(Z,X)^{{\text{\rm red}}} = d\ell$. Let $Z$ be the object $J_p^nW$ of $\mathbf{Fat}/{k}$ where $W$ is a locally complete intersection variety over $k$. Given a deformation $Y$ of $X$ over $Z$ such that $\mbox{dim}{\ }\mathcal{A}_Z(Y) = d\ell +\delta_Z$, what is the flat locus of the induced morphism $\theta: \mathcal{A}_Z(Y) \to \mathcal{A}_Z$? **Remark 35**. For any deformation $Y$ of $X$ over $Z$ such that $\mbox{dim}{\ }\mathcal{A}_Z(Y) = d\ell +\delta_Z$, one can show that the induced morphism $\theta$ is flat by the miracle of flatness (cf. Theorem 23.1 on page 178 of [@mat1987]) provided $\mathcal{A}_Z$ is regular. # The situation over linear jets. {#sec3} Now, we will study this problem over the linear jets $Z_n = \operatorname{Spec}(k[t]/(t^{n+1}))$. We let $X$ be a variety over $k$. We assume that there is a deformation $X_n \to Z_n$. For example, this occurs when $X$ is a complete intersection subvariety of $X=\mathbb{P}_{k}^{N}$ or more generally $X$ is a locally complete intersection in $X$ and the obstruction in $H^1(\mathcal{N}_X)$ vanishes (cf. Theorem 9.2 on page 74 of [@hart2009]). Note then that $\mathcal{A}_{Z_n}(X_n)$ is the same as the reduced truncated linear arc space $\mathcal{L}_n(X_n)$. **Remark 36**. The auto-arc spaces in this context, hereafter always denoted by $\mathcal{L}_n(X_n)$, are similar to the truncations $Gr_n(X_n)$ as studied in [@loe2003] and more recently in [@nic2011] provided the underlying field $k$ has equal characteristic. These later spaces are truncated versions of an infinite arc space, therein denoted by $Gr(\mathcal{X})$, where $\mathcal{X}$ is a smooth formal scheme. **Lemma 37**. *Let $S= \operatorname{Spec}(A)$ where $A$ is a reduced Noetherian local ring with residue field $k$. Let $Z_n$ be the object of $\mathbf{Fat}/{S}$ given by $S \times \operatorname{Spec}(\mathbb{Z}[t]/(t^{n+1}))$. Then, $$\mathcal{A}_{Z_n} \cong \mathbb{A}_{S}^n$$* *Proof.* This is the first part of Lemma 4.3 on page 141 of [@sto2019]. ◻ Of course then $\mathcal{L}_n(Z_n) = \mathcal{A}_{Z_n} \cong \mathbb{A}_{k}^{n}$ where $Z_n=\operatorname{Spec}(k[t]/(t^{n+1}))$ and $k$ is a field. Thus, the morphism introduced in the beginning of Section [3](#sec2){reference-type="ref" reference="sec2"} is of the form $$\theta_n : \mathcal{L}_n(X_n) \to \mathbb{A}_k^{n}$$ **Theorem 38**. *Let $X$ be a locally complete intersection variety over an algebraically closed field $k$ of dimension $d$. Let $X_n$ be a deformation over $k[t]/(t^{n+1})$ such that $\mbox{dim}{\ }\mathcal{L}_n(X_n) = d(n+1) + n$. Then, the natural induced morphism $\theta_n : \mathcal{L}_n(X_n) \to \mathbb{A}_k^{n}$ is flat.* *Proof.* The fibers of $\theta_n$ are clearly seen to be constant. Then, by using the miracle of flatness (Theorem 23.1 on page 178 of [@mat1987]), we know that $\theta_n$ is flat if and only if $\mathcal{L}_n(X_n)$ is a Cohen-Macaulay ring. This is true since under our assumptions $\mathcal{L}_n(X_n)$ is a locally complete intersection by Theorem [Theorem 30](#lci){reference-type="ref" reference="lci"}. ◻ **Corollary 39**. *Given the conditions of Theorem [Theorem 38](#main){reference-type="ref" reference="main"}, and moreover, assume that $X \subset \mathbb{A}_{k}^{N}$. Then, the fibers of $\theta_n$ give rise to a flat morphism $$\mathchoice{\widetilde{\theta}}{\widetilde{\theta}}% {\lower 2pt\hbox{$\textstyle{\tilde{\raise 2pt% \hbox{$\scriptstyle{\theta}$}}}$}}{\tilde{\theta}}_n: \mathbb{A}_k^{n} \to \underline{Hilb}(\mathbb{A}_{k}^M)$$ where $M = (n+1)\cdot N$ and the image at the origin $O$ given by $\mathchoice{\widetilde{\theta}}{\widetilde{\theta}}% {\lower 2pt\hbox{$\textstyle{\tilde{\raise 2pt% \hbox{$\scriptstyle{\theta}$}}}$}}{\tilde{\theta}}_n(O)$ is $\mathcal{L}_n(X)$.* *Proof.* This is immediate. ◻ # Remarks on linear auto-arcs for curves. {#sec4} In this section, we briefly study the linear auto-arcs for deformations of curves. Even in this case, the situation is highly non-trivial. **Proposition 40**. *Let $C$ be a curve over a field $k$, and let $C_1$ be a deformation over $\operatorname{Spec}(k[t]/(t^2))$ such that $T(C_1)$ is irreducible and of pure dimension. Then, the inverse image $\pi^{-1}(C_{{\text{\rm sing}}})$ of the singular locus under the morphism $\pi: T(C_1) \to C$ is always a subvariety of the fiber of the morphism $\theta_2 : T(C_1) \to \mathbb{A}_k^1$ at the origin.* *Proof.* When $C$ is non-singular the statement is trivial. Therefore, assume that $C$ is singular and let $V:= \pi^{-1}(C_{{\text{\rm sing}}})$. The decomposition in Equation [\[dimauto\]](#dimauto){reference-type="ref" reference="dimauto"} implies $\mbox{dim}{\ }V \leq 2$ since $T(C_1)$ is irreducible and of pure dimension. Consider the restriction $\theta_1|_V : V \to \mathbb{A}_k^1$ and assume for the sake of contradiction that it is non-constant. This implies that $\theta_1|_V(V)$ is a dense subset of $\mathbb{A}_k^{1}$, from whence it follows from Theorem [Theorem 1](#LiuFlat){reference-type="ref" reference="LiuFlat"} that the restriction $\theta_1|_V$ is also flat. But, then the central fiber $V_0$ of $\theta_1|_V:V\to \mathbb{A}_k^1$ over the origin is such that $\mbox{dim}{\ }V_0 = \mbox{dim}{\ }V - 1$. Thus, $\mbox{dim}{\ }V_0 \leq 1$. By Lemma [Lemma 27](#basiclem){reference-type="ref" reference="basiclem"}, the central fiber of $\theta_2 : C_1 \to \mathbb{A}_k^1$ is isomorphic to $T(C)$, which by Commutative Diagram [\[maindiag\]](#maindiag){reference-type="ref" reference="maindiag"} implies that $V_0$ is the inverse image over the singular locus of the natural morphism $T(C) \to C$. But, at any singular point $p \in C$, the fiber under $T(C) \to C$, which is just the tangent space $T_p(C)$, has dimension strictly greater than $1$, which is a contradiction. Thus, $\theta_1|_V$ is a constant morphism and thus $V_0 = V$, which proves the claim. ◻ **Remark 41**. Example [Example 9](#defnode){reference-type="ref" reference="defnode"} shows that although the conditions for Proposition [Proposition 40](#curvecor){reference-type="ref" reference="curvecor"} are sufficient, they are not necessary - i.e., the reduced tangent bundle of the versal first order deformation of a node (given by $xy-t = 0$ with $t^2 =0$) is reducible, yet the central fiber of $\theta$ still contains the inverse image of the singular locus. **Example 42**. We lift the deformation of Example [Example 9](#defnode){reference-type="ref" reference="defnode"} to the second order deformation $N_2$ defined by $$N_2: = \operatorname{Spec}(k[x,y,t]/(xy-t^2-t, t^3))$$ we create arc variables $$\wideparen{x} = a_{11} + a_{12}t + a_{13}t^2 , \quad \wideparen{y} = a_{21} + a_{22}t + a_{23}t^2 ,\quad \wideparen{t} = e + ft+gt^2$$ We note that $\wideparen{t}^3 = 0$ implies that $e = 0$ and places no further restrictions on $f$ and $g$, and so without loss of generality we may assume $\wideparen{t} = ft+gt^2$. Now, performing the remaining substitution, we have the arc equation $\wideparen{x}\wideparen{y}-\wideparen{t}^2-\wideparen{t}=0$ which generates the following list of equations defining $\mathcal{L}_3(N_3)$. $$\begin{split} a_{11}a_{21} &= 0 \\ a_{11}a_{22} + a_{12}a_{21} - f &= 0\\ a_{11}a_{23}+a_{21}a_{13}+a_{12}a_{22} - g - f^2 &= 0 \end{split}$$ We note that the fiber over the origin $O$ of the natural morphism $\pi_2 : \mathcal{L}_2(N_2) \to N$ is cut out by $a_{11} = a_{21} = 0$. This then implies that on this fiber, the variables $a_{13}$ and $a_{23}$ are free and $f=0$ . Thus, as a subvariety of $\mathcal{L}_2(N_2)$, the fiber over the singular point in $N$ is given by $$\pi_2^{-1}(O) \cong \operatorname{Spec}(k[x,y,g]/(xy - g))\times_k \mathbb{A}_{k}^2$$ We note that $\mbox{dim}{\ }\pi_2^{-1}(O) = 4$ and an irreducible component. Also, we note that the fiber over the singular locus leaves the central fiber of $\theta_2 : \mathcal{L}_2(N_2) \to \mathbb{A}_k^2$. **Remark 43**. In light of Proposition [Proposition 40](#curvecor){reference-type="ref" reference="curvecor"} and Example [Example 42](#dnode3){reference-type="ref" reference="dnode3"}, we consider the fiber of the singular locus for an irreducible curve such that $\mathcal{L}_2(X_2)$ is irreducible and of pure dimension. Let $V:=\pi_2^{-1}(X_{{\text{\rm sing}}})$. By assumption, $\mbox{dim}{\ }V < \mbox{dim}{\ }\mathcal{L}_2(X_2) = 5$. Assume $\theta_2|_V: V \to \mathbb{A}_k^2$ is not constant, and let $L$ be a line on $\mathbb{A}_k^2$ passing through the origin and contained in the image of $\theta_2|V$. By a change of coordinates if necessary, we can find a surjective map $\mathbb{A}_k^2 \to L$ and consider the composition with $\theta_2|_{V}$ whose image will be dense and hence the composition will be flat. Therefore, the fiber at the origin of this composition, say $V_0$ will have dimension strictly less than $4.$ By Lemma 4.1 on page 18 of [@mus2001], the dimension of any fiber over the singular locus is equal to $3$ or more. Thus, the dimension of $V_0$ must be exactly $3$ in this case. Although it may not be contained in the central fiber of $\theta$, we do have a picture for what the expected dimension should be. We attempt to generalize the behavior noticed in Proposition [Proposition 40](#curvecor){reference-type="ref" reference="curvecor"} and Remark [Remark 43](#rm3){reference-type="ref" reference="rm3"}. For this, we let $\mathcal{A}_Z^{*}$ denote the open subscheme of $\mathcal{A}_Z$ isomorphic to $\underline{Aut}_S(Z)^{{\text{\rm red}}}$ for $Z$ an object in $\mathbf{Fat}/{S}$, and we let $B_Z$ denote the compliment $\mathcal{A}_Z \setminus \mathcal{A}_Z^*$. Considering the behavior above and in that of Corollary [Proposition 40](#curvecor){reference-type="ref" reference="curvecor"}, we make the following definition. **Definition 44**. Let $X$ be a scheme over another scheme $S$ and let $Z$ be an object in $\mathbf{Fat}/{S}$. Let $Y$ be a deformation of $X$ over $Z$. Consider the natural induced morphism $\pi : \mathcal{A}_Z(Y) \to X$ and let $V = \pi^{-1}(X_{sing})$. We say that the deformation $Y \to Z$ is *strong* if $V$ is contained in $\theta^{-1}(B_Z)$ and otherwise we call the deformation *weak*. We say that the deformation is *very strong* if $V \subset \theta^{-1}(O)$ where $O$ is the point given by the trivial endomorphism of $Z$. **Remark 45**. In the case of deformations over $k[t]/(t^{n+1})$ and the corresponding truncated linear arc spaces, the notions of strong and very strong are equivalent, and for this reason, we will always refer to a very strong deformation as merely strong in this case. **Remark 46**. We can see that the first order deformation in Example [Example 9](#defnode){reference-type="ref" reference="defnode"} is a strong deformation, yet its second order cousin found in Example [Example 42](#dnode3){reference-type="ref" reference="dnode3"} is a weak deformation. In general, for a strong deformation, the induced morphism $\pi\times \theta : \mathcal{A}_Z(Y) \to X \times_S\mathcal{A}_Z$ is then a piecewise trivial fibration away from the $\theta^{-1}(\mathcal{A}_S^*)$ over the base $X_{{\text{\rm sm}}} \times_S \mathcal{A}_Z^{*}$. Thus, for a strong deformation for example, we have a commutative diagram $$\begin{tikzcd} \mathcal{A}_Z(Y)\setminus \theta^{-1}(B_Z) \arrow[d,] \arrow[r,hook] & \mathcal{A}_Z(Y) \arrow[d,"{\pi|_{V}\times\theta}" left] \arrow[r] \arrow[rd, "{\pi\times\theta}" description ]&\underline{Hom}_S(Z,X)^{{\text{\rm red}}}\times_S \mathcal{A}_Z \arrow[d] \\ X_{{\text{\rm sm}}}\times_S\mathcal{A}_Z^* \arrow[r, hook] &X_{{\text{\rm sm}}}\times_S \mathcal{A}_Z \arrow[r, hook] &X\times_S\mathcal{A}_Z \end{tikzcd}$$ where the left most vertical arrow is a piecewise trivial fibration with affine fibers onto the base $X_{{\text{\rm sm}}} \times \mathcal{A}_Z^{*}$. In the case of strong $n$th order deformation over linear jets, the above diagram simplifies to $$\begin{tikzcd} \mathcal{L}_n(X_n)\setminus \theta^{-1}(O) \arrow[d,] \arrow[r,hook] & \mathcal{L}_n(X_n) \arrow[d,"{\pi|_{V}\times\theta}" left] \arrow[r] \arrow[rd, "{\pi\times\theta}" description ]&\mathcal{L}_n(X)\times_S \mathbb{A}_k^n \arrow[d] \\ X_{{\text{\rm sm}}}\times_k\mathbb{G}_{m}\times_k\mathbb{A}_k^{n-1}\arrow[r, hook] &X_{{\text{\rm sm}}}\times_S \mathbb{A}_k^n \arrow[r, hook] &X\times_S \mathbb{A}_k^n \end{tikzcd}$$ where $\mathbb{G}_{m}$ is the general multiplicative group over $k$. **Proposition 47**. *Let $C$ be a curve over a field $k$. Consider a weak $n$th order deformation $C_n$ of $C$ over $\operatorname{Spec}(k[t]/(t^{n+1}))$ such that $\mathcal{L}_n(C_n)$ is irreducible and of pure dimension. Then, $n+2\leq \mbox{dim}{\ }\pi_n^{-1}(C_{{\text{\rm sing}}}) \leq 2n$.* *Proof.* The proof will be exactly the same as before, and therefore we will just sketch the proof here. If $C$ is smooth, then there is nothing to prove. Therefore, assume $C$ is singular and let $V = \pi_n^{-1}(C_{{\text{\rm sing}}})$ where $\pi_n : \mathcal{L}_n(C_n) \to C$ is the natural morphism. Let $\mbox{dim}{\ }V =m \leq 2n$ since $\mathcal{L}_n(C_n)$ is of pure dimension and irreducible. Assume for now that $\theta_n|_V$ is not constant, then, by a linear change of coordinates if necessary, we can restrict the target of $\theta_n|_V$ to a copy of the affine line for which the image is dense. Thus, if we let $V_0 \subset \mathcal{L}_n(C)$ be the fiber over the origin, $\mbox{dim}{\ }V_0 \leq m - 1$. For any singular point $p$ on $C$, $\mbox{dim}{\ }T_pC \geq 2$, and therefore, by Lemma 4.1 on page 18 of [@mus2001], $\mbox{dim}{\ }V_0\geq n+1$. Actually, equality is obtained in our case, so we simply solve for $m$ to obtain the lower bound. ◻ **Example 48**. Consider the $2$nd order versal deformation $C_2$ of the node given by $xy - t$, with $t^3 =0$ over a field $k$. Following the computation in Example [Example 42](#dnode3){reference-type="ref" reference="dnode3"}, we obtain in exactly the same way the list of equations $$\begin{split} a_{11}a_{21} &= 0 \\ a_{11}a_{22} + a_{12}a_{21} - f &= 0\\ a_{11}a_{23}+a_{21}a_{13}+a_{12}a_{22} - g &= 0 \end{split}$$ These equations define $\mathcal{L}_2(C_2)$ as a subvariety of $\mathbb{A}_k^8$. Clearly, this is reducible with two irreducible components $W_i$ given by $a_{i1} =0$ for $i = 1, \ 2$. Each irreducible component maps in a natural way surjectively onto $\operatorname{Spec}(k[f])$, and is thus flat over $\operatorname{Spec}(k[f])$. The fiber at $f=0$ of each component $W_{i}$ is isomorphic to $T(N_1)\times_k\mathbb{A}_k^1$ where the space $T(N_1)$ studied in Example [Example 9](#defnode){reference-type="ref" reference="defnode"}. The affine factor is coming from a free variable: $a_{23}$ is free in the case $i=1$ and $a_{13}$ is free in the case that $i=2$. Furthermore, we noted in Example [Example 9](#defnode){reference-type="ref" reference="defnode"} that $T(N_1)$ is of dimension $3$. Thus, $\mathcal{L}_2(C_2)$ is of pure dimension since each irreducible component must be of dimension $5$. One can now also quickly check that $\pi_2^{-1}(O) \cong \operatorname{Spec}(k[x,y,t]/(xy-t)) \times_k \mathbb{A}_k^2$, and thus $\mbox{dim}{\ }\pi_2^{-1}(O) = 4$. **Example 49**. Consider the $2$nd order deformation $X_2$ of the node $N$ given by $xy = t^2(x+y)$, with $t^3 =0$ over a field $k$. The equations defining $\mathcal{L}_2(X_2)$ are $$\begin{split} a_{11}a_{21} &= 0 \\ a_{11}a_{22} + a_{12}a_{21} &= 0\\ a_{11}a_{23}+a_{21}a_{13}+a_{12}a_{22} - f^2(a_{11}+a_{21}) &= 0 \end{split}$$ The fiber over the singular locus $\pi_2^{-1}(O)$ is the fiber product of the inverse image over the singular locus in $\mathcal{L}_2(N)$, which is isomorphic to $N\times_k\mathbb{A}_k^2$, and another copy of the affine plane given by $\operatorname{Spec}(k[f,g])$. This example is as far away from being strong as possible for a locally complete intersection in that the fiber over the singular locus is an irreducible component with $\mbox{dim}{\ }\pi_2^{-1}(O) = \mbox{dim}{\ }\mathcal{L}_2(N_2) =5$. **Example 50**. Consider the cusp $C = \operatorname{Spec}(k[x,y]/(y^2 -x^3))$ and the $3$rd order versal deformation $$C_3 = \operatorname{Spec}(k[x,y,t]/(y^2 - x^3 - t , t^4))$$ In exactly the same manner as the previous calculations, we obtain $$\begin{split} a_{21}^2 - a_{11}^3 &= 0\\ 2a_{21}a_{22} - 3a_{11}^3a_{12} - f & = 0 \\ a_{22}^2 + a_{21}a_{23} - 3a_{11}a_{12}^2 - 2a_{11}^2a_{13} - g &= 0 \\ a_{21}a_{24} + 2a_{22}a_{23} - 2a_{11}^2a_{14} - 5 a_{11}a_{12}a_{13} - a_{12}^3 - h &= 0 \end{split}$$ as equations for the auto-arc space $\mathcal{L}_3(C_3)$ in $\mathbb{A}_{k}^{11}$. The fiber $\pi_3^{-1}(O)$ is given by the ideal $I = (a_{22}^2 - g, 2a_{22}a_{23} - a_{12}^3 - h, a_{11}, a_{21}, f)$ from which one sees that $a_{13}, a_{14},$ and $a_{24}$ are free. Thus, there is an isomorphism $$\pi_3^{-1}(O) \cong \operatorname{Spec}(k[x,y,z,t,s]/ (y^2 - t, 2yz - x^3 - s))\times_k \mathbb{A}_k^3\cong \mathbb{A}_k^6$$ Thus, $\pi_3^{-1}(O)$ obtains the minimum possible dimension for a weak deformation. Considering these examples, we define $$\Phi_n := \frac{\mbox{dim}{\ }\pi^{-1}_{Z_n}(X_{{\text{\rm sing}}})}{\ell(Z_n)} - d$$ for any weak deformation $Y_n$ of a reduced scheme $X$ over a fat point $Z_n$. We expect that the limit of $\Phi_n$ to exist for any sequence of weak deformations $Y_n$ over fat points $Z_n$ given by a closed germ $(Z,P)$, and we expect this limit to fit into the inequality $$e(Z,P) \leq \lim_{n\to \infty} \Phi_n \leq \delta^{*}_{(Z,P)}(X)$$ where $e(Z,P)$ and $\delta^{*}_{(Z,P)}(X)$ are the asymptotic defects defined in Section [3](#sec2){reference-type="ref" reference="sec2"}. # Motivic volumes of auto-arc spaces From now on we restrict our attention to the case where $S = \operatorname{Spec}(k)$ with $k$ a fixed algebraically closed field. We let $\mathbf{Var}$ denote the category of varieties over $k$. We fix a $k$-scheme $Z$ and a closed point $p$ on $Z$. Therefore, we have the $n$th order jets $Z_n:=J_p^nZ := \operatorname{Spec}(\mathcal{O}_{Z,p}/\mathfrak{m}_p^{n+1})$ as a fixed sequenced of infinitesimal neighborhoods of $p$ on $Z$. Let us also fix an arbitrary sequence of infinitesimal deformations $Y_n$ of $X$ over $Z_n$ such that $Y_{n-1} \cong Y_n \times_{Z_n} Z_{n-1}$ for all $n \geq 1$, we may consider the sequence of auto-arc spaces $\mathcal{A}_n := \mathcal{A}_{Z_n}(Y_n)$ together with the natural induced map $\pi^{n}_{m} : \mathcal{A}_n \to \mathcal{A}_{m}$ for $n\geq m \geq 0$. Define $\mathcal{A}:= \varprojlim \mathcal{A}_n$ and let $\alpha_n$ denote the canonical morphism from $\mathcal{A}\to \mathcal{A}_n$ Let ${\mathbf {Gr}(\mathbf{Var})}$ denote the Grothendieck ring of varieties, $\mathcal{G}:= {\mathbf {Gr}(\mathbf{Var})}[\mathbb{L}^{-1}]$ the localized Grothendieck ring by the Leftschetz motive, and let $\widehat{\mathcal{G}}$ be the completion of $\mathcal{G}$ along the dimensional filtration. We define $$\mu(X,\mathcal{Y}) := \lim_{n\to \infty} [\mathcal{A}_n]\mathbb{L}^{-d\ell-\delta_n}$$ provided the limit in $\widehat{\mathcal{G}}$ exists. If we assume $X$ is locally a complete intersection over $k$, we may decompose each term in the limit as $[\pi_n^{-1}(X_{{\text{\rm sing}}})]\mathbb{L}^{-d\ell-\delta_n} + M$ where $M$ is a fixed element (i.e., not dependent on $n$). We noted then that $[\pi_n^{-1}(X_{{\text{\rm sing}}})]\mathbb{L}^{-d\ell-\delta_n} \rightarrow 0$, and so we may consider the motivic measure as $M$. [^1]: Support for this project was provided by a PSC-CUNY Award (PSC-Grant Traditional A, \# 66024-00 54), jointly funded by the Professional Staff Congress and The City University of New York. [^2]: Without adding any additional assumptions, we can let $X$ and $X'$ be merely algebraic spaces over $S$. [^3]: Thus, the fiber of $X$ at zero defined by $X_0 := X \times_{D_S} S$ is separated and locally of finite presentation. [^4]: *Note that a morphism $j : X \to S$ is locally of finite presentation if and only if $S$ is locally noetherian and $j$ is locally of finite type.*

arxiv_math

--- abstract: | In this paper, we investigate the geometry of moduli space $P_d$ of degree $d$ del Pezzo pair, that is, a del Pezzo surface $X$ of degree $d$ with a curve $C \sim -2K_X$. More precisely, we study compactifications for $P_d$ from both Hodge's theoretical and geometric invariant theoretical (GIT) perspective. We compute the Picard numbers of these compact moduli spaces which is an important step to set up the Hassett-Keel-Looijenga models for $P_d$. For $d=8$ case, we propose the Hassett-Keel-Looijenga program ${\mathcal F}_8(s)=\mathop{\mathrm{Proj}}(R({\mathcal F}_8,\Delta(s) )$ as the section rings of certain ${\mathbb Q}$-line bundle $\Delta_8(s)$ on locally symmetric variety ${\mathcal F}_8$, which is birational to $P_8$. Moreover, we give an arithmetic stratification on ${\mathcal F}_8$. After using the arithmetic computation of pullback $\Delta(s)$ on these arithmetic strata, we give the arithmetic predictions for the wall-crossing behavior of ${\mathcal F}_8(s)$ when $s\in [0,1]$ varies. The relation of ${\mathcal F}_8(s)$ with the K-moduli spaces of degree $8$ del Pezzo pairs is also proposed. address: - No.220 Handan Road, Shanghai, Fudan University - Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road Haidian District, Beijing, P.R.China 100871 - No.220 Handan Road, Shanghai, Fudan University author: - Long Pan - Fei Si - Haoyu Wu bibliography: - main.bib title: Birational geometry of moduli space of Del Pezzo pairs --- # Introduction ## Background Recent developments in moduli theory provide many compactifications for moduli space of (polarised or lattice polarised) K3 surfaces from various aspects. For example, there are geometric invariant theoretical(GIT) compactifications by [@Shah80] [@Shah]. From the Hodge theoretical side, the Torelli theorem for K3 surfaces is a bridge to link the moduli space of K3 surfaces as a locally symmetric space. Then it turns out that there are many arithemtic compactifications for the moduli space of lattice polarised K3 surfaces, including Baily-Borel compactifications and Looijenga's semitoric compactifications [@Lo03]. Recently relying on the breakthrough in birational geometry and Kähler-Einstein geometry, the general moduli problems for constructing moduli space for log Fano varieties (see [@Xu21]) and varieties of general type (see [@kollar]) were solved. These works inspire the following results on compactifications of moduli spaces of K3 surfaces: [@ADL22] uses the theory of K-stability to provide a compactification for moduli spaces of quartic K3 surfaces. By constructing the recognizable divisor on a K3 surface, Alexeev-Engel [@AE21] views a polarised K3 surface as a KSBA stable pair and provide a modular compactification for moduli spaces of polarised K3 surfaces. It is a natural question to compare these various compactifications and study what is the relation of these compactifications. Indeed, it has been suggested by Xu in the survey [@Xu21 Question 10.9] to study the relation between the compactifications for moduli spaces of polarised K3 surfaces and compact K-moduli of Fano $3$-fold pairs. The relation of the GIT compactification and arithmetic compactification has been explored for moduli space of quartic K3 surfaces in a series of work by Laza-O'Grady [@LaO18ii] [@LO19] [@LaO18], where they propose the Hasset-Keel-Looijenga(HKL) program for the moduli space of quartic K3 surfaces. They conjecture that HKL model defined in [@LO19] will interpolate the GIT compactification and Baily-Borel compactification for the moduli space of quartic K3 surfaces. This is verified by [@ADL22]. In this paper, we will deal with a typical type of K3 surfaces with symmetry. That is, a K3 surface $X$ with an anti-symplectic involution $\tau: X \rightarrow X$. We call such $(X,\tau)$ a K3 pair. What puts a K3 pair $(X,\tau)$ very interesting is that one can relate it to a del Pezzo surface pair $(X/\tau, C)$ where $C$ is a curve on $X/\tau$ such that $C \sim -2K_{X/\tau}$ and $C$ is isomorphic to $X^\tau$, the fixed locus of $\tau$. Let $P_d$ be the moduli space of the del Pezzo surface pairs, then $P_d$ can also be viewed as the moduli space of K3 pairs due to the double covering construction. We concern the compactification for $P_d$ in this paper. The rich geometric structure for $(X,\tau)$ allow us to study the compactifications of $P_d$ via all the perspective mentioned above. More precisely, we are going to study - GIT compactification $\overline{P}^{GIT}_d$, - Baily-Borel compactification $P_d^\ast=(O(\Lambda_d)\setminus {\mathcal{D}}_{\Lambda_d})^\ast$, - compactifications $\overline{P}^{K}_{d,c}$ via K-stability and compactifications $\overline{P}^{KSBA}_{d,c}$ via KSBA theory. - and the connections of the above various compactifications. ## Main results In this paper, we first study the geometry of Baily-Borel compactifications of moduli space of del pezzo surface pairs of degree $d$. In particular, we obtain the Picard numbers: **Theorem 1**. *Let $P_d^\ast$ be the Baily-Borel compactification of moduli space of del Pezzo pairs of degree $d$, then the Picard number of $P_d^\ast$ is give by the following table* *$d$* *$8$* *$7$* *$6$* *$5$* *$4$* *$3$* *$2$* *$1$* -------------------- ------- ------- ------- ------- ------- ------- ------- ------- -- *$\rho(P_d^\ast)$* *$3$* *$4$* *$3$* *$4$* *$4$* *$4$* *$5$* *$6$* : *Picard numbers of $P_d^\ast$* Our methods is to compare the GIT model $\overline{P}^{GIT}_d$ of $P_d$ and its Baily-Borel compactification $P_d^\ast$. The Picard groups of a GIT quotient space can be computed classically by [@KKV89]. By studying of GIT stability, we construct a big open subset $U$ shared by $\overline{P}^{GIT}_d$ and $P_d^\ast$. And then we analyse the contribution of boundary divisors in $P_d^\ast-U$ to the class group $\mathop{\mathrm{\mathrm{Cl}}}(P_d^\ast)_{\mathbb Q}$ to prove the Theorem [Theorem 1](#mthm1){reference-type="ref" reference="mthm1"}. Here in the Table [1](#tb:Picard ){reference-type="ref" reference="tb:Picard "} $\rho(P_d^\ast):=\dim_{\mathbb Q}\mathop{\mathrm{\mathrm{Cl}}}(P_d^\ast)_{\mathbb Q}$. Next, we focus on the case $d=8$ in more detail. We propose to study the space $${\mathcal F}(s):=\mathop{\mathrm{Proj}}\big (\mathop{\bigoplus} \limits_{m \ge 0}H^0({\mathcal F}_d,ml_s\Delta(s) ) \big )$$ where the ${\mathbb Q}$-Cartier divisor $\Delta(s)$ is given by $$\Delta_d(s)=\lambda+s(H_h+25H_u)$$ and $l_s\in {\mathbb Z}_{>0}$ is smallest positive integer so that $l_s \Delta(s)$ is Cartier divisor. Following [@LaO18], we call it HKL model for ${\mathcal F}_d^\ast$. To compute walls for ${\mathcal F}(s)$, we give an arithmetic stratification for ${\mathcal F}$ from towers of morphism in the following diagram. ### Stratification on hyperelliptic divisor $H_h$: {#stratification-on-hyperelliptic-divisor-h_h .unnumbered} $$\label{diah} \begin{tikzcd} \mathop{\mathrm{\mathrm{Sh}}}(A_7''') \arrow[d,dashed] \arrow[dr] & & & & \\ \mathop{\mathrm{\mathrm{Sh}}}(A_7'') \arrow[r,dashed] & \mathop{\mathrm{\mathrm{Sh}}}(A_6'') \arrow[d,dashed] \arrow[dr] & & & \mathop{\mathrm{\mathrm{Sh}}}(A_1)={\mathcal F}\\ & \mathop{\mathrm{\mathrm{Sh}}}(A_6') \arrow[r,dashed] & \mathop{\mathrm{\mathrm{Sh}}}(A_5') \arrow[d,dashed] \arrow[dr] & & \mathop{\mathrm{\mathrm{Sh}}}(A_2) \arrow[u] \\ & & \mathop{\mathrm{\mathrm{Sh}}}(A_5) \arrow[r,dashed] & \mathop{\mathrm{\mathrm{Sh}}}(A_4) \arrow[r] & \mathop{\mathrm{\mathrm{Sh}}}(A_3) \arrow[u] \\ & & & & \mathop{\mathrm{\mathrm{Sh}}}(D_4) \arrow[u] \\ & \mathop{\mathrm{\mathrm{Sh}}}(E_8) \arrow[r] & \mathop{\mathrm{\mathrm{Sh}}}(E_7) \arrow[r] & \mathop{\mathrm{\mathrm{Sh}}}(E_6) \arrow[r] & \mathop{\mathrm{\mathrm{Sh}}}(D_5) \arrow[u] \\ & & & & \mathop{\mathrm{\mathrm{Sh}}}(D_6) \arrow[u] \\ & \mathop{\mathrm{\mathrm{Sh}}}(D_9') \arrow[r] & \mathop{\mathrm{\mathrm{Sh}}}(D_8') \arrow[r] & \mathop{\mathrm{\mathrm{Sh}}}(D_7') \arrow[r,dashed] \arrow[ur] & \mathop{\mathrm{\mathrm{Sh}}}(D_7) \arrow[u,dashed] \\ \end{tikzcd}$$ ### Stratification on unigonal divisor $H_u$: {#stratification-on-unigonal-divisor-h_u .unnumbered} $$\label{diau} \begin{tikzcd} %& \sh(U_4') \arrow[d,dashed] \arrow[dr] & & & & \\ & \mathop{\mathrm{\mathrm{Sh}}}(U_4'') \arrow[r,dashed] & \mathop{\mathrm{\mathrm{Sh}}}(U_3') \arrow[r] & \mathop{\mathrm{\mathrm{Sh}}}(U_2') \arrow[r,dashed] & \mathop{\mathrm{\mathrm{Sh}}}(U_1) \arrow[r] & {\mathcal F} \end{tikzcd}$$ In this above diagram ([\[diah\]](#diah){reference-type="ref" reference="diah"}) and ([\[diau\]](#diau){reference-type="ref" reference="diau"}), $\mathop{\mathrm{\mathrm{Sh}}}(L)$ is a locally symmetric variety of orthogonal type induced by a lattice $L$. See Definition [Definition 43](#defA){reference-type="ref" reference="defA"}, Definition [Definition 49](#1stmv){reference-type="ref" reference="1stmv"} etc, for example, $\mathop{\mathrm{\mathrm{Sh}}}(A_n)$ is locally symmetric variety given by lattice $(E_7\oplus A_n)^\perp_{I\!I}$. The dotted map means that there is a modification for the wall-crossing of ${\mathcal F}(s)$ at this locus. **Theorem 2**. *The class group of $P_8^\ast$ with ${\mathbb Q}$- coefficient has basis $\lambda, H_h,H_u$ where $\lambda$ is the Hodge line bundle and $H_h,H_u$ are reduced Heegener divisors (see Definition [Definition 32](#Heegner){reference-type="ref" reference="Heegner"}). Moreover, there is a Borcherds relation $$76\lambda = H_n+ 2H_h + 57 H_u.$$ Let $\varphi_h: \mathop{\mathrm{\mathrm{Sh}}}(A_1) \rightarrow P_8^\ast$ and $\varphi_u: \mathop{\mathrm{\mathrm{Sh}}}(U_1) \rightarrow P_8^\ast$ be the morphism with image $H_h$ and $H_u$ as in the ([\[diah\]](#diah){reference-type="ref" reference="diah"}) and ([\[diau\]](#diau){reference-type="ref" reference="diau"}), then we have the pullback formula of Heegner divisors $$\varphi_h^\ast H_h=-\lambda+H_{A_2},\ \ \varphi_u^\ast H_u=-\lambda .$$ Assume the section rings $$R(s)=\mathop{\bigoplus} \limits_{m \ge 0 } H^0(P_8^\ast,m(\lambda+sH_h+25sH_u))$$ are finitely generated for $s\in (0,1) \cap {\mathbb Q}$ and birational contractibility of some Heegner divisors (see $\widetilde{Z}$ in theorem [Theorem 73](#cbf){reference-type="ref" reference="cbf"} ) . When $s\in (0,1) \cap {\mathbb Q}$ varies, the wall-crossing behavior of the projective variety ${\mathcal F}(s)=\text{\rm Proj}(R(s))$ is described in Prediction [Prediction 1](#pre1){reference-type="ref" reference="pre1"} and Prediction [Prediction 2](#pre2){reference-type="ref" reference="pre2"}.* Based on the arithmetic strategy of HKL, the above arithmetic stratification predicts that the walls for ${\mathcal F}(s)$ is given by $$\{\ \frac{1}{n} \, \mid \, n=1,2,3,4, 6,8,10,12,16,25,27,28 ,31\ \}$$ In the sequel paper [@psw2], we will apply the K-moduli theory to studythe above HKL model and verify the above prediction. This can be seen as a new example of HKL program worked out completely. Hopefully, our analysis in arithmetic side will be also helpful to find K-moduli walls in other case, provided that K-moduli space is birational to a ball quotient or locally symmetric variety of orthogonal type. ## Further direction {#further-direction .unnumbered} One direction is to study the topology and intersection theory of moduli spaces of K3 surfaces with an anti-symplectic involution. Combining wall-crossing result in [@psw2], the birational map has explicit resolution which transfers the arithmetic stratification in the paper to the GIT strata. If a wall-crossing type formula of cohomology can be deduced from this explicit resolution of birational period map, the cohomology of Baily-Borel compacficationn should be obtained. Note that the similar strategy to compute Cohomology of moduli space of cubic $3$-folds and $4$-folds have been worked out in [@CGHL] [@Si]. It is also likely to be useful to the computation of for chow ring of moduli spaces ${\mathcal F}$ based on our geometric description. The computation for moduli spaces of quasi-polarised K3 surfaces of genus $2$ based on GIT strata has been worked out in [@COP]. Another direction is to investigate possible arithmetic stratifications for certain toroidal compactification of ${\mathcal F}$ studied in [@AE22], which may give the prediction of wall-crossings KSBA compact moduli sapces of the surface pairs $(X,cC)$ for $c>\frac{1}{2}$. ## Convention {#convention .unnumbered} Throughout the paper, we will use the following convention for lattices. - $G(\Lambda):=\Lambda^\ast/\Lambda$ will be the discriminant group of the lattice; - $\det(\Lambda)=|G(\Lambda)|$ the determinant of a lattice $\Lambda$; - $\mathop{\mathrm{\mathrm{div}}}_\Lambda(v)$ will be the divisibility of a vector $v \in \Lambda$; - $v^\ast:=\frac{v}{\mathop{\mathrm{\mathrm{div}}}_\Lambda(v)} \in G(\Lambda)$ for each primitive vector in $\Lambda$; - $q(v):=\frac{\langle v,v \rangle}{2}$; - $I\!I_{2,26}\cong E_8^3\oplus U^2$ is the Borcherds lattice where $U$ is hyperbolic lattice with gram matrix $$\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} % dis=6 \right).$$ ## Organization of the paper {#organization-of-the-paper .unnumbered} In section [2](#2-K3){reference-type="ref" reference="2-K3"}, we set up the moduli theory $P_d$ of del Pezzo pairs and discuss the relation of $P_d$ to the moduli space of K3 surfaces with anti-symplectic involution via period map. And then we introduce Baily-Borel compatification $P_d^\ast$. In section [3](#sect3){reference-type="ref" reference="sect3"}, we study GIT (partial) compactification for $P_d$. As an application, we compute the class group $\mathop{\mathrm{\mathrm{Cl}}}(P_d^\ast)_{\mathbb Q}$ of $P_d^\ast$. In section [4](#sect4){reference-type="ref" reference="sect4"}, we set up the HKL models ${\mathcal F}(s)$ for $P_8^\ast$ and study the ADE type degenerations of smooth del pezzzo pairs $(X,C)$. This will be the geometric input for the arithmetic stratification in the next section. In section [5](#arithmeticstra){reference-type="ref" reference="arithmeticstra"}, we first study several towers of morphism between locally symmetric varieties. This gives an arithmetic stratifications on hyperelliptic divisor $H_h$ and unigonal divisor $H_u$. Then based on the arithmetic principle, we compute the pullback of $\Delta(s)=\lambda+sH_h+25sH_u$ on the towers and give the predictions for wall-crossings of ${\mathcal F}(s)$. ## Acknowledgement {#acknowledgement .unnumbered} We thank Zhiyuan Li and Yuchen Liu for helpful conversations. Some part of this work was written when the second author visited SCMS and he would like to thank their hospitality. The second author was partially supported by LMNS (the Laboratory of Mathematics for Nonlinear Science, Fudan University). This project was supported by NKRD Program of China (No.2020YFA0713200), NSFC Innovative Research Groups (Grant No.12121001) and General Program (No.12171090). # Moduli space of del Pezzo pairs {#2-K3} We work over ${\mathbb C}$. ## Moduli spaces of del Pezzo pairs **Definition 3**. *Let $X$ be a smooth projective surface with $-K_X$ ample. $(X,D)$ is called a del Pezzo pair of degree $d$ if $(-K_X)^2=d$ and $D \sim -2K_X$ a curve with ADE singularities at worst. Let $\pi: (X,D) \rightarrow B$ be a flat proper smooth morphism over a base scheme $B$. We say $\pi$ is a family of del Pezzo pair of degree $d$ if $-K_{X/B}$ is $\pi$-ample and $(X_t,D_t)$ is a del Pezzo pair of degree $d$ for any $t\in B$.* Consider the moduli stack $\mathcal{P}_d:\ \mathop{\mathrm{\mathcal{S}ch}}({\mathbb C}) \rightarrow \mathop{\mathrm{\mathcal{S}et}}$ defined by $$\begin{split} \mathcal{P}_d(B)=\{ \pi: (X,D) \rightarrow B\ |\ \pi \ \hbox{is a family of del Pezzo pair of degree } \ d\ \} /\sim \end{split}$$ **Theorem 4**. *The moduli stack $\mathcal{P}_d$ is a Deligne-Mumford stack. In particular, it has the coarse moduli space $P_d$.* *Proof.* First note that for any del Pezzo pairs $(X,D)$ of degree $d$ forms a bounded family. Indeed, it is well known that there is a uniform $m\in {\mathbb Z}_{>0}$ only depending on $d$ such that both $-mK_X$ and $-mK_X|_D$ are very ample on $X$ and $D$ respectively. Thus for each pairs $(X,D)\in \mathcal{P}_d({\mathbb C})$ there is a embedding given by the line bundle $-mK_X$ $$(X,D) \hookrightarrow {\mathbb P}^N .$$ Let $\mathop{\mathrm{\mathrm{Hilb}}}_1$ and $\mathop{\mathrm{\mathrm{Hilb}}}_2$ be the Hilbert scheme parametrizing the closed subschemes in ${\mathbb P}^N$ with fixed Hilbert polynomials $$\chi_1(t)=\frac{m^2d}{2} t^2+\frac{md}{2} t+1, \ \ \ \chi_2(t)=2md t-d.$$ Since the Hilbert scheme is representable, there is a universal family of pairs over $\mathop{\mathrm{\mathrm{Hilb}}}_1 \times \mathop{\mathrm{\mathrm{Hilb}}}_2$. Denote $Z \subset \mathop{\mathrm{\mathrm{Hilb}}}_1 \times \mathop{\mathrm{\mathrm{Hilb}}}_2$ the locus of smooth del Pezzo pairs. There is a flat family $({\mathfrak X},{\mathcal{D}}) \rightarrow Z$. And $Z$ is a locally closed subscheme invariant under the natural action $G=\mathop{\mathrm{\mathrm{PGL}}}(N+1)$ induced from the action $G$ on ${\mathbb P}^N$. From the construction of $Z$, there is a natural morphism $$\label{stackmor} \mathcal{P}_d \rightarrow [Z/G]$$ from the moduli stack to the quotient stack $[Z/G]$. As any isomorphism $f: (X,D) \rightarrow (X',D')$ of del Pezzo pairs $(X,D)$ and $(X',D')$ over a base $T$ is deduced from some projective transform $g$ in $\mathop{\mathrm{Aut}}_T({\mathbb P}^N_T)$ under the relative embedding $(X,D) \hookrightarrow {\mathbb P}_T^N$, then morphism ([\[stackmor\]](#stackmor){reference-type="ref" reference="stackmor"}) is injective and thus isomorphism. As the GIT quotient $Z \rightarrow Z/G$ is a categorical quotient, to show the existence of coarse moduli space $P_d$, it remains to show the action $G$ on $Z$ is proper ( see [@Huy Chapter 5, section 2.3]). Note that there is an injective morphism $\mathop{\mathrm{Aut}}(X,D) \rightarrow \mathop{\mathrm{Aut}}(Y,\phi^\ast(-K_X))$ where $\phi:Y \rightarrow X$ is the double cover branched along $C$. It is known the auomorphism group $\mathop{\mathrm{Aut}}(Y,\phi^\ast(-K_X))$ of polarised K3 is finite, then so is $\mathop{\mathrm{Aut}}(X,D)$. This shows smooth group scheme $G$ acts on $Z$ has finite and reduced stabilizer $\mathop{\mathrm{Aut}}(X,D)$. The stabilizer of $G$ for each point in $z\in Z$ can be identified with $\mathop{\mathrm{Aut}}({\mathfrak X}_z,{\mathcal{D}}_z)$, which is finite. This proves $\mathcal{P}_d$ is a Deligne-Mumford stack. ◻ ## Moduli spaces of $\rho$-markable K3 surfaces with automorphisms Let's recall $\rho$-markable K3 with non-symplectic action studied in [@AEH]. Denote $L=U^3\oplus E_8^2$ and fix an isometry $\rho \in O(L)$ with order $n$. **Definition 5**. *A $\rho$-markable K3 is a pair $(X,\sigma)$ where $X$ is a K3 surface and $\sigma: X \rightarrow X$ is a non-symplectic isomorphism so that there is an isometry $\phi: H^2(X,{\mathbb Z}) \rightarrow L$ such that $$\sigma^\ast = \phi^{-1} \circ \rho \circ \phi .$$* Denote ${\mathcal F}_\rho$ the moduli stack of $\rho$-markable K3 with automorphisms. That is, $$\begin{split} {\mathcal F}_\rho(B)=\big\{ \pi: ({\mathfrak X},\sigma_B) \rightarrow{} B \big\}/\sim \end{split}$$ ${\mathcal F}_\rho$ is a Deligne-Mumford stack with coarse moduli space $F_\rho$. Let $\xi_n=\exp(\frac{2\pi\sqrt{-1}}{n})$ be the $n$-th primitive root and $$L^{\xi_n}_{\mathbb C}=\{ z\in L_{\mathbb C}\ |\ \rho(z)=\xi_n z \}$$ the $\xi_n$-eigenspace of $L_{\mathbb C}$. Set $${\mathcal{D}}_\rho:={\mathbb P}L^{\xi_n}_{\mathbb C}\cap {\mathcal{D}}_L \subset {\mathcal{D}}_L$$ the subdomain of the period space ${\mathcal{D}}_L$ of complex K3. Then ${\mathcal{D}}_\rho$ is a Type IV Hermitian domain if $n=2$ and a complex ball if $n \ge 3$. The group $$\Gamma_\rho:=\{ g\in O(L)| g\circ \rho =\rho \circ g \}.$$ acts on ${\mathcal{D}}_\rho$. It is shown in [@AEH Theorem 2.9] that the period map $$\label{eq:fine moduli} p_\rho: F_\rho \rightarrow \Gamma_\rho \setminus {\mathcal{D}}_\rho$$ is an open immersion and the complement consists of the union of hypersurfaces ${\mathcal{D}}_{\rho,v}$ cut out by the $-2$ vectors $v\in L$. Moreover, if $n=2$, i.e. $\rho$ is an involution, there is an isomorphism of quotient space (*cf*. [@AE22 Section 2D]) $$\label{eq:orthogonal quotient} \Gamma_\rho \setminus {\mathcal{D}}_\rho \cong O(\Lambda) \setminus {\mathcal{D}}_\Lambda$$ where $\Lambda \subset L$ is the $(-1)$-eigenspace of $\rho$ with signature $(2,m)$. **Remark 6**. *Note that for a $\rho$-markable K3 $(X,\varphi)$, the Neron-Severi group $\mathop{\mathrm{\mathrm{NS}}}(X)$ of $X$ always contain $L_+$ the invariant sublattice of $\rho$. Generically, $\mathop{\mathrm{\mathrm{NS}}}(X)\cong L_+$. For those $(X,\varphi)$ such that $\mathop{\mathrm{{\text{\rm rank}}}}(\mathop{\mathrm{\mathrm{NS}}}(X)/ L_+) \ge 1$, they form Heegner divisors on the $\Gamma_d \setminus ({\mathcal{D}}_\rho)$ moduli space of $\rho$-markable K3 surfaces.* ### K3 with involution In this paper, we are only interested in the $n=2$ case, where such lattices can be classified. Let's recall the classifications of the possible $\rho$ eigen-sublattice $T$. **Definition 7**. *An even indefinite lattice $N$ is called a *$2$-elementary lattice* if the discriminant group has the form $$G(N):=N^\ast/N\cong ({\mathbb Z}/2{\mathbb Z})^a.$$* **Proposition 8**. *([@Nikulin]) A $2$-elementary lattice $N$ is determined by the invariant $(r,a,\delta)$ uniquely where* 1. *$r$ is the rank of $N$;* 2. *$a\in {\mathbb N}$ is called the index of $N$;* 3. *$\delta: = \begin{cases} 0 & \text{if } q(G(N))\subset {\mathbb Z},\\ 1 & \text{otherwise} . \end{cases}$* Geometrically, for $\delta=1$, the $2$-elementary lattices can be realized as Neron-Severi groups of K3 surface constructed from the double cover of del Pezzo pairs studied in section [Definition 3](#delpezzopair){reference-type="ref" reference="delpezzopair"}. Given a del Pezzo pair $(X,C)$ of degree $d$, one can construct a K3 surface $Y$ from the double covering $$\phi: Y \rightarrow X$$ branched along the curve $C\sim -2K_X$. Such K3 surface has a natural involution $\tau: Y \rightarrow Y$ whose fixed locus is just the smooth curve $C$ of genus $d+1$ by adjunction formula. Recall **Definition 9**. *An involution for K3 surface $\tau: Y \rightarrow Y$ is called anti-symplectic if $\tau^\ast \sigma=-\sigma$ where $\sigma$ is the holomorphic volume form and $H^{2,0}(Y)={\mathbb C}\sigma$.* Clearly, the double covering construction produces a K3 surface. Conversely, let $(Y,\tau)$ be a K3 with anti-symplectic involution $\tau$. Then $\tau$ induces an action on K3 lattice $\tau^\ast: H^2(X,{\mathbb Z}) \rightarrow H^2(X,{\mathbb Z})$. Denote $N_d \vcentcolon=H^2(X,{\mathbb Z})^{\tau^\ast}$ the invariant sublattice and set $\Lambda_d:= N_d^\perp \subset H^2(X,{\mathbb Z})$. **Proposition 10**. *(Chapter 2 [@AN06]) [\[Nikunin\]]{#Nikunin label="Nikunin"} An invariant lattice is uniquely determined by its invariants $(r,a,\delta)$. Moreover, the these invariants satisfy relation $$\label{genus} g=\frac{22-r-a}{2},\ \ k=\frac{r-a}{2}.$$ where the fixed locus $X^\tau=C_g+R_1+\cdots R_k$ is a sum of an irreducible nonsingular curve $C_g$ and $k$ irreducible nonsingular rational curves $R_1,\cdots R_k$.* Note that the invariant lattice and anti-invariant one are mutually determined. In the case of $\delta=1$, these anti-invariant lattices $\Lambda_d$ all come from the double cover of the del Pezzo surfaces and they are classified as follows. **Proposition 11**. *The anti-invariant lattice $\Lambda_d$ is given by the following table.* *$d$* *$a$* *$\delta$* *$\Lambda_d\ \ $* *$A_{\Lambda_d}$* --------- ------- ------------ -------------------------------------------- ---------------------------------- ---------------------- -- *1* *$9$* *$1$* *$U^2 \oplus A_1^9$* *$({\mathbb Z}/2{\mathbb Z})^9$* *$E_7 \oplus A_1^8$* *2* *$8$* *$1$* *$U^2 \oplus D_4 \oplus A_1^6$* *$({\mathbb Z}/2{\mathbb Z})^8$* *$E_7 \oplus A_1^7$* *3* *$7$* *$1$* *$U\oplus A_1(-1)\oplus E_7 \oplus A_1^5$* *$({\mathbb Z}/2{\mathbb Z})^7$* *$E_7 \oplus A_1^6$* *4* *$6$* *$1$* *$U^2\oplus E_7 \oplus A_1^5$* *$({\mathbb Z}/2{\mathbb Z})^6$* *$E_7 \oplus A_1^5$* *5* *$5$* *$1$* *$U^2\oplus E_8 \oplus A_1^5$* *$({\mathbb Z}/2{\mathbb Z})^5$* *$E_7 \oplus A_1^4$* *6* *$4$* *$1$* *$U^2\oplus D_4 \oplus E_8 \oplus A_1^2$* *$({\mathbb Z}/2{\mathbb Z})^4$* *$E_7 \oplus A_1^3$* *7* *$3$* *$1$* *$U^2\oplus D_6 \oplus E_8 \oplus A_1$* *$({\mathbb Z}/2{\mathbb Z})^3$* *$E_7 \oplus A_1^2$* *8* *$2$* *$1$* *$U^2\oplus E_7 \oplus E_8 \oplus A_1$* *$({\mathbb Z}/2{\mathbb Z})^2$* *$E_7 \oplus A_1$* *8 $'$* *$2$* *$0$* *$U^2\oplus D_{16}$* *$({\mathbb Z}/2{\mathbb Z})^2$* *$D_8$* : *List of anti-invariant lattices* *Proof.* As the fixed locus of the involution $\tau: Y \rightarrow Y$ is just the smooth branched curve $C$, there is no rational curve. Thus we have $$g=d+1=11-a=11-r$$ by formula ([\[genus\]](#genus){reference-type="ref" reference="genus"}) in Proposition [\[Nikunin\]](#Nikunin){reference-type="ref" reference="Nikunin"} and adjuction formula $(K_Y+C)|_C=K_C$. Since the $2$-elementary lattice is uniquely determined by the invariant $(r,a,\delta)$, one can check these invariants in the table [2](#table:anti-inv lattice){reference-type="ref" reference="table:anti-inv lattice"} are all the possible values of $(r,a,\delta)$. Then the proof will be finished. ◻ **Remark 12**. *${\mathbb P}^1 \times {\mathbb P}^1$ is the unique del Pezzo surface whose associated anti-invariant lattice has $\delta=0$ and we denote its degree by $8'$.* From the above double covering construction, the moduli space $P_d$ of del Pezzo pairs of degree $d$ is closely related to the moduli space of $\rho$-markable K3 pairs for $\rho$ involution. Here involution $\rho$ has invariant $(r=10-d,a=10-d,\delta =1 )$. To make the notation consistent, we denote ${\mathcal F}_d:={\mathcal F}_\rho$ and $\Gamma_d:=\Gamma_\rho$. Then we have **Proposition 13**. *Let $\rho \in$ be an involution such that $(-1)$-eigenspace $\Lambda$ of $\rho$ on $L$ is a $2$-elementary lattice and has signature $(2, d+10)$, then there is open immersion $$p_d: P_d \rightarrow {\mathcal F}_d$$ defined by sending the isomorphism classes of del Pezzo pair $(X,C)$ of degree $d$ to the $O(\Lambda)$-orbit of period point of its associate K3 surface with anti-involution $(Y,\tau)$.* *Proof.* This follows [@AEH Theorem 4.2] directly. ◻ Now we are going to determine the image of ${\mathcal F}_\rho -P_d$, at least the divisors in ${\mathcal F}_\rho -P_d$. Usually, ${\mathcal F}_\rho -P_d$ will consist of so-called Heegner divisors. To do so, let's first recall the theory of the Heegner divisors on locally symmetric varieties. It will be also used in later sections. ## Heegner divisors Let $\Lambda$ be an even lattice of signature $(2,n)$. The period domain associated to $\Lambda$ is defined as $${\mathcal{D}}_{\Lambda} \vcentcolon=\left\{z\in {\mathbb P}(\Lambda \otimes {\mathbb C}) \,\middle|\, z^2=0, z.\overline{z}>0 \right\}.$$ It is a Hermitian symmetric domain of Type IV and consists of two diffeomorphic components. Denote ${\mathcal{D}}_{\Lambda}^+$ one of them. Let $O^+(\Lambda) \le O(\Lambda)$ be the group of isometries preserving the component and $$\widetilde{O}^+(\Lambda):=\{ \gamma\in O^+(\Lambda) |\ \gamma = \mathop{\mathrm{\mathrm{id}}}\ \hbox{on}\ \Lambda^\ast/\Lambda\ \}.$$ **Theorem 14** (Baily-Borel [@BB66]). *For any finite index subgroup $\widetilde{O}(\Lambda) \le \Gamma \le O^+(\Lambda)$, the orbit space $\Gamma \setminus {\mathcal{D}}_{\Lambda}$ is a quasi-projective variety of dimension $n$.* The quotient space $\Gamma \setminus {\mathcal{D}}_{\Lambda}$ is called locally symmetric space of orthogonal type and it is the ${\mathbb C}$-points of a connected Shimura variety of orthogonal type with certain level structure. As a locally symmetric space, $\Gamma \setminus {\mathcal{D}}_{\Lambda}$ contains many natural sub locally symmetric spaces as effective cycles on $\Gamma \setminus {\mathcal{D}}_{\Lambda}$, these cycles are very interesting in arithmetic geometry. In this paper, we are mainly interested in such codimension $1$ cycles. Let us give a precise definition. **Definition 15**. *Notation as above, denote ${\mathcal{D}}_v \vcentcolon=\{z\in {\mathcal{D}}_{\Lambda}^+ \ |\ \langle z,v \rangle =0\}$ the hyperplane cut out by the vector $v\in \Lambda$.* 1. *Let $\beta \in G(\Lambda)/\{\pm 1 \},m\in {\mathbb Q}_{<0}$. A Heegner divisor on $\Gamma \setminus {\mathcal{D}}_{\Lambda}^+$ is defined by $$H_{\beta,m}:=\Gamma \setminus \bigcup_{\substack{v\in\Lambda^\ast,\, v^2=2m \\ v\in \beta + \Lambda }} {\mathcal{D}}_v .$$ Note that $H_{\beta,m}= H_{-\beta,m}$.* 2. *Let $v\in \Lambda$ be a primitive vector, a modified Heegner divisor $H_v$ on $\Gamma \setminus {\mathcal{D}}_{\Lambda}$ associated to $v$ is defined by $$H_v := \Gamma \setminus \bigcup_{g \in \Gamma } {\mathcal{D}}_{g\cdot v} .$$* 3. *Let $\lambda_\Lambda$ be the line bundle which is descent of line bundle ${\mathcal O}_{ {\mathbb P}(\Lambda \otimes {\mathbb C})}(1)|_{{\mathcal{D}}_\Lambda^+}$ under the natural quotient map ${\mathcal{D}}_\Lambda^+ \rightarrow \Gamma \setminus {\mathcal{D}}_\Lambda ^+$. We call $\lambda_\Lambda$ the Hodge line bundle on $\Gamma \setminus {\mathcal{D}}_{\Lambda}^+$.* In general, a Heegner divisor $H_{\beta,m}$ can be non-reduced and reducible while a modified Heegner divisor is always irreducible. Indeed, $H_{\beta,m}$ can be written as the sum of Heegner divisors $$H_{\beta,m}= \mathop{\sum} \limits_{v} a_vH_v$$ **Definition 16**. *Define $\mathop{\mathrm{\mathrm{Sh}}}(\Lambda):=\widetilde{O}^+(\Lambda) \setminus {\mathcal{D}}_\Lambda^+$. Let $\beta \in G(\Lambda)/\{\pm 1 \},m\in {\mathbb Q}_{<0}$. A primitive Heegner divisor $P_{\beta,m}$ on $\mathop{\mathrm{\mathrm{Sh}}}(\Lambda)$ is defined by the quotient $$P_{\beta,m}:=\Gamma \setminus \bigcup_{\substack{v\in\Lambda^\ast,\ \hbox{primitive} \ \\ q(v)=m,\ v\in \beta + \Lambda }} {\mathcal{D}}_v .$$* **Remark 17**. *By [@BM19], there is a relation (called triangular relation in the literature) $$\label{trirel} H_{\beta,m}=\mathop{\sum} \limits_{\substack{r\in {\mathbb Z}_{>0} \\ r^2| m }} \mathop{\sum} \limits_{\substack{\delta\in G(\Lambda)\\ r\delta=\beta}} P_{\delta, \frac{m}{r^2}}$$ between the Heegner divisors and primitive Heegner divisors. It is convenient to use Heegner divisors $H_{\beta,m}$ in arithmetic while primitive Heegner divisors are convenient for geometric use.* **Lemma 18** (Eichler's criterion). *If $\Lambda$ is an even lattice containing $U^{\oplus 2}$, then two nonzero primitive elements $v,w\in \Lambda$ is in the same $\widetilde{O}^+(\Lambda)$-orbit if and only if $v^2=w^2,\ v^\ast=w^\ast$.* *Proof.* See [@GHS09 Proposition 3.3]. ◻ Using Eichler's criterion, Brunier-Moller [@BM19 Lemma 4.3] proved that the reduced part of primitive Heegner divisor $P_{\beta,m}$ is irreducible if $\Lambda$ contains two copies of the hyperbolic lattice. Thus if further more $G(\Lambda)$ is generated by elements of the form $v^\ast$ for some $v\in \Lambda$, then we have $$P_{\beta,m}=H_v$$ on $\mathop{\mathrm{\mathrm{Sh}}}(\Lambda)$ where $v\in \Lambda$ is a primitive vector with $v^\ast =\beta, \ q(v^\ast)=m$. Note that there is a natural finite morphism $$\mathop{\mathrm{\mathrm{Sh}}}(\Lambda) \rightarrow \Gamma \setminus {\mathcal{D}}_{\Lambda}.$$ for any $\Gamma$ containing $\widetilde{O}^+(\Lambda)$ as a finite index subgroup. In particular, if $\Lambda$ is a $2$-elementary lattice, then by [@Ma Section 4.3], the finite morphism $\mathop{\mathrm{\mathrm{Sh}}}(\Lambda) \rightarrow O(\Lambda) \setminus {\mathcal{D}}_{\Lambda}$ has degree $|O(G(\Lambda))|$. ### Orbits of -2 vectors Now we turn to the situation $\Lambda =\Lambda_d=(N_d)^\perp\subset L$ where $$N_d = {\mathbb Z}l \oplus {\mathbb Z}e_1 \oplus \cdots {\mathbb Z}e_{9-d} \cong A_1(-1)\oplus A_1^{9-d}$$ is the invariant lattice of $\rho: L \rightarrow L$ with $l^2=2,\ e_i^2=-2, \ \langle e_i,e_j\rangle= \langle e_i,l \rangle =0$ for $i \neq j$. By Proposition [Proposition 13](#prop:moduli immersion){reference-type="ref" reference="prop:moduli immersion"} and [\[eq:fine moduli\]](#eq:fine moduli){reference-type="eqref" reference="eq:fine moduli"}, [\[eq:orthogonal quotient\]](#eq:orthogonal quotient){reference-type="eqref" reference="eq:orthogonal quotient"}, We have $$P_d \subset {\mathcal F}_d = \Gamma_d \backslash \big( {\mathcal{D}}_{\Lambda_d} - \bigcup_{\substack{v^2 = -2,\\ v\in \Lambda_d}} {\mathcal{D}}_v \big) \subset \Gamma_d \setminus {\mathcal{D}}_d$$ to compute $\mathop{\mathrm{\mathrm{Pic}}}_{\mathbb Q}(P_d)$ and $\mathop{\mathrm{\mathrm{Pic}}}_{\mathbb Q}({\mathcal F}_d)$, one naturally need to study the $\Gamma_d$-orbits of $(-2)$-vectors in $\Lambda_d$. **Theorem 19**. *The $\Gamma_d$-orbit of $(-2)$ vectors $v$ in $\Lambda_d$ are described as follows* 1. *$d=8$ and $7$, there are two $O(\Lambda_8)$-orbit of $(-2)$ vectors $v$ which corresponds to $$\{ 0\},\ \hbox{or}\ \{ \eta \}\ .$$* 2. *$d=6$, then $O(G(\Lambda_6))\cong S_3 \times {\mathbb Z}/2{\mathbb Z}$ and there are two $O(\Lambda_6)$-orbit of $(-2)$ vectors $v$ where $\epsilon_6(v^\ast)$ corresponds to $$\{ 0\},\ \ \hbox{or}\ \ \{ \xi_1+ \xi_2+\xi_3,\, \eta \}.$$* 3. *$d=5$, $O(G(\Lambda_5)) \cong S_5$. There are three $O(\Lambda_5)$-orbits of $(-2)$-vectors $v$ with $\epsilon_5(v^*)$ belonging to$$\{ 0\},\{\eta,\,\xi_i+\xi_j+\xi_k\}_{1\leq i< j< k\leq 4} \text{~or~} \{\eta+\sum^4_{i=1}\xi_i\}$$* 4. *$d=4$, there are three $O(\Lambda_4)$-orbits of $(-2)$-vectors $v$ with $\epsilon_4(v^*)$ belonging to$$\{ 0\}, ~ \{\eta,\,\xi_i+\xi_j+\xi_k\}_{1\leq i< j< k\leq 5} \text{~or~} \{\eta + \xi_i+\xi_j+\xi_k+\xi_l\}_{1\leq i<j<k<l\leq 5}.$$* 5. *$d=3$, there are three $O(\Lambda_3)$-orbits of $(-2)$-vectors $v$ with $\epsilon_3(v^*)$ belonging to$$\{ 0\}, ~ \{\eta,\,\xi_i+\xi_j+\xi_k\}_{1\leq i< j< k\leq 6} \text{~or~} \{\eta + \xi_i+\xi_j+\xi_k+\xi_l\}_{1\leq i<j<k<l\leq 6}.$$* 6. *$d=2$, there are four $O(\Lambda_2)$-orbits of $(-2)$-vectors $v$ with $\epsilon_2(v^*)$ belonging to$$\{ 0\}, ~ \{\eta,\,\xi_i+\xi_j+\xi_k\}_{1\leq i< j< k\leq 5} ,~ \{\eta + \xi_i+\xi_j+\xi_k+\xi_l\}_{1\leq i<j<k<l\leq 7} , \text{~or~} \{ \sum_{i=0}^7 \xi_i\}$$* 7. *$d=1$, there are five $O(\Lambda_1)$-orbits of $(-2)$-vectors $v$ with $\epsilon_1(v^*)$ belonging to$$\begin{split} \{ 0\}, ~ &\{\eta,\,\xi_i+\xi_j+\xi_k\}_{1\leq i< j< k\leq 5} ,~ \{\eta + \xi_i+\xi_j+\xi_k+\xi_l\}_{1\leq i<j<k<l\leq 8} ,\\ &\big \{ \ \sum_{I} \xi_I\ \big \}_{|I|=7, I\subset \{1,2,3,4,5,6,7,8\} }, \text{~or~} \ \{ \eta +\sum_{i=1}^8 \xi_i\} \end{split}$$* *Here $\epsilon_d\colon (A_{\Lambda_d}, q_{\Lambda_d}) \xrightarrow{\cong} (A_{N_d}, -q_{N_d})$ is the canonical isomorphism.* *Proof.* Recall the discriminant form[^1] on $G(N_d)$ is given by $$q(\eta)=\frac{1}{4} \mod {\mathbb Z},\quad q(\xi_i)=-\frac{1}{4} \mod {\mathbb Z}$$ where $\eta:=\frac{l}{2}$ and $\xi_i=\frac{e_i}{2}$. Let $d_v:=\mathop{\mathrm{\mathrm{div}}}(\Lambda_d)$ for any vector $v\in \Lambda_d$, then $d_v\in \{ 1,2\}$ for $(-2)$-vector. As any $(-2)$-vectors with $d_v=1$, $v^\ast=0\in A_{\Lambda_d}$ and Eichler's criterion in Lemma [Lemma 18](#Eichler){reference-type="ref" reference="Eichler"} shows these vectors are in the same $\widetilde{O}(\Lambda_d)$-orbit and thus $O(\Lambda_d)$-orbit. So it remains to classify the $O(\Lambda_d)$-orbit of $(-2)$-vector with $d_v=2$. Note that by Nikulin's results [@Nikulin Theorem 3.6.3], there is a surjection $O(\Lambda_d) \twoheadrightarrow O(G(\Lambda_d))$. Hence it's enough to classify the orbits of discriminant vector $u\in G(\Lambda)$ with $q(u)=-\frac{1}{4}$ under the action of $O(G(\Lambda_d))$. By the canonical isomorphism $\epsilon_d$, it's equivalent to classify the orbits of $\frac{1}{4}$-norm vectors in $G(N_d)$. For $d=8$ and $7$, as the only $\frac{1}{4}$-norm element in $G(N_7)$ is $\eta$, thus $\phi(\eta)=\eta$ for any $\phi\in O(G(N_7))$. For $d=6$, note that the map $$\begin{split} \eta \mapsto \xi_1+\xi_2+\xi_3,\ \ \xi_1 \mapsto \eta+\xi_2+\xi_3; \\ \xi_2 \mapsto \eta+\xi_1+\xi_3,\ \ \ \xi_3 \mapsto \eta+\xi_1+\xi_2 \end{split}$$ defines an element in $O(G(N_6))$. For $d=5$, Note that the $N_5 \cong A_1(-1) \oplus A_1^{\oplus 4}$ and $\Lambda_5 \cong U^{\oplus 2} \oplus E_8 \oplus A_1^5$. This comes from [@Mkondo11 Lemma 2.3, 2.4, Remark 2.6]. Based on the results of $d=5$ case, the proof for remaining cases are similar but rather lengthy. We give the detailed proof of $d=3$ case as a guiding example. For $d=3$, one only need to show that $\eta$ and "$\eta+4\xi$" type vectors are not in the same orbit. Without losing generality, suppose there is $\phi\in O(G(N_d))$ such that $\phi(\eta) = \eta+\sum_{i=1}^4 \xi_i$. Note that $$\phi(\xi_k) \in \{\xi_i ,\, \sum_{|I|=5}\xi_I,\, \eta+\xi_i+\xi_j,\eta+\sum_{j=1}^6 \xi_j\}$$ Consider $W \vcentcolon=\{\phi(\xi_k)\}_{1\leq k \leq 6}$. First note that $\eta+ \sum^6_{i=1}\xi_i$ are not in $W$ since all elements of $W$ are orthogonal to each other. - $\{\xi_5, \xi_6\}\subset W$: It's easily seen that vectors of type $\sum_{|I|=5} \xi_I$ are not in $W$ by the orthogonality of elements of $W$. Then the remaining vectors of $W$ are of the form $\eta+\xi_i+\xi_j$ which are both orthogonal to $\xi_5,\xi_6$ and $\eta+\sum^4_{k=1}\xi_k$. There is no such possibilities. - $\xi_5 \in W$ and $\xi_6 \notin W$: The only possible vectors that are orthogonal to $\xi_5$ are $\xi_1+\xi_2+\xi_3+\xi_4+\xi_6$ and $\{\eta+\xi_i+\xi_6\}_{1\leq i \leq 4}$ as $W$ is orthogonal to $\phi(\eta)$. But $\xi_1+\xi_2+\xi_3+\xi_4+\xi_6$ is not orthogonal to $\eta + \xi_i +\xi_6, (1\leq i \leq 4)$, a contradiction. - $\xi_5,\xi_6 \notin W$: The only possible vectors are $\{\sum_{i=1}^4\xi_i+\xi_5$, $\sum_{i=1}^4 \xi_i+\xi_6\}$, $\{\eta+\xi_i+\xi_5\}_{1\leq i \leq 4}$ and $\{\eta+\xi_i+\xi_6\}_{1\leq i \leq 4}$ as $W$ is orthogonal to $\phi(\eta)$. Each of the three sets consists of orthogonal vectors. One can see that $\{\sum_{i=1}^4\xi_i+\xi_5$, $\sum_{i=1}^4 \xi_i+\xi_6\}\cap W = \emptyset$ since they are not orthogonal to any vectors in the remaining two sets. Then for the remaining two sets, each vector in one set is only orthogonal to a unique one in the other set, say $$\langle \eta+\xi_i +\xi_5, \eta+\xi_i+\xi_6 \rangle =0.$$ So there are at most 4 possible orthogonal vectors in $W$, a contradiction.  ◻ ## Nodal and multipoint divisor If the pair $(X,C) \in P_d$ has nodal curve $C$, then the double cover $S_0$ of $X$ branched along $C$ has a nodal point, then the minimal resolution $S$ of $S_0$ has Neron-Severi group $$\left(\begin{array}{c|cc} & N_d & E \\ \hline N_d & N_d & 0 \\ E & 0 &-2 \\ \end{array}\right)$$ where we also use $N_d$ to indicate the gram matrix and $E$ is the exceptional divisor of resolution of the nodal point. The period point of such pairs is just the irreducible divisor $H_v$ with $v\in \Lambda_d$ and $\mathrm{div}(v)=1$. We call it nodal divisor and denote it by $H_{\rm n}$. Assume $X$ normal surface with a $A_1$ singularities and $-K_X$ is ample with $d=(-K_X)^2$. It is called nodal del Pezzo surface of degree $d$. Geometrically, it is a blowup of ${\mathbb P}^2$ along $9-d$ points $(x_1,\cdots , x_{8-d},x_{9-d})$ with only two points infinitesimally near, say $x_{8-d}$ and $x_{9-d}$. Its minimal resolution $\widetilde{X}$ is a weak del Pezzo surface, i.e., $-K_{\widetilde{X}}$ is big and nef. Let $S_0$ be the double cover of $\widetilde{X}$ branched along general curve $B \sim -2K_{\widetilde{X}}$ (that is, $B$ does not intersect with the exceptional divisor lying over the $A_1$ point of $Y$). Denote $\mu: Y \rightarrow {\mathbb P}^2$, then $$L=\mu^\ast {\mathcal O}_{{\mathbb P}^2}(1), e_1:=\mu^\ast E_1,\cdots, e_{7-d}:=\mu^\ast E_{7-d}, E_{8-d}, E', E''$$ form a basis of $\mathop{\mathrm{\mathrm{NS}}}(X)$ where $E'$ and $E''$ . Then the Neron-Serveri group of $\mathop{\mathrm{\mathrm{NS}}}(\widetilde{Y})$ is given by $$\left(\begin{array}{c|cccc} & N_{d+2} & E' & E'' & \alpha \\\hline N_{d+2} & N_{d+2}& 0 & 0 & 0 \\ E' & 0 & -2& 0 & 0 \\ E'' & 0 & 0 & -2 & 0 \\ \alpha & 0 & 0 & 0 & -4 \\ \end{array}\right)$$ where $\alpha:=2 E_{8-d}+E'+E''$ is a long root class. Let $v_\alpha \in \Lambda_d$ the vector corresponding the long root class, i.e, ${\mathbb Z}v_\alpha ={\mathbb Q}(\alpha-\alpha_{N_d})\cap (N_d)^\perp_L$ where $\alpha_{N_d}$ is the projection of $\alpha$ on $N$. **Definition 20**. *The irreducible Heegner divisor $(\Gamma_d \setminus \mathop{\cup} \limits_{\gamma \in \Gamma_d} {\mathcal{D}}_{d,\gamma v})^{\rm red}$ is called the multipoint divisor and denote it by $H_m$.* **Remark 21**. *Note that in the case $d=8, a=1$, $H_m$ is empty since a degree $8$ del Pezzo is obtained by blowing up ${\mathbb P}^2$ at one point.* **Proposition 22**. *The divisors in ${\mathcal F}_d-(P_d\cup H_m)$ consists of irreducible Heegner divisors $H_v=\Gamma_d \setminus \bigcup_{g \in \Gamma_d} {\mathcal{D}}_{g\cdot v}$ where $v$ is $(-2)$-vectors in Theorem [Theorem 19](#Torelli){reference-type="ref" reference="Torelli"} with $\mathrm{div}(v)=2$.* *Proof.* First, note that $H_n \subset P_d$ since the period map can be extended to a nodal K3 surface as an injective map. Thus it remains to prove codimension $\le 1$ points in ${\mathcal{D}}_\rho - \mathop{\cup} \limits_{v^2=-2} {\mathcal{D}}_{\rho,v}$ is either from smooth del pezzo or from nodal del pezzo. Let $z\in {\mathcal{D}}_\rho - \mathop{\cup} \limits_{v^2=-2} {\mathcal{D}}_{\rho,v}$ be a point. By surjectivity of period map of K3 surfaces, there is a marked K3 surface $(X_z,\varphi: H^2(X_z,{\mathbb Z}) \rightarrow L)$ such that $z=\varphi(H^{2,0}(X_z))$. Since the period period point avoid divisors $\mathop{\cup} \limits_{v^2=-2} {\mathcal{D}}_{\rho,v}$, then the involution $\varphi(\rho)$ is an isometry on $H^2(X_z,{\mathbb Z})$ preserving ample class and $\rho(z)=-z$. Then the classical Torelli theorem implies $\varphi(\rho)$ can be uniquely lifted as an ani-symplectic involution $\tau_z: X_z \rightarrow X_z$, i.e., $\tau_z^\ast=\varphi(\rho)$. It is known (*cf*. [@AN06 Chapter 2, Section 2.1]) the fixed locus of the involution $C:=X_z^{\tau_z}$ is a smooth curve without rational curve as a connected component. (Otherwise, the rational curve will force the period point $z\in \mathop{\cup} \limits_{v^2=-2} {\mathcal{D}}_{\rho,v}$, a contradiction.) Let $X_z \rightarrow Y:=X/\tau_z$. By [@AN06 Chapter 2,Section 2.1], we have $Y$ is smooth surface and $C\sim -2K_Y$. As a smooth curve $C$ of genus $>1$ on K3 surface $X_z$ is big and nef, so is $-K_Y$, i.e, $Y$ is a weak del pezzo of degree $d=(-K_Y)^2$. If $-K_Y$ is ample, then $Y$ is obtained by blowing up of general $9-d$ points on ${\mathbb P}^2$. If $-K_Y$ is not ample, as we only care about the codimension one point in ${\mathcal{D}}_\rho - \mathop{\cup} \limits_{v^2=-2} {\mathcal{D}}_{\rho,v}$, we may assume $X_z$ has Picard rank $11-d$ (Recall generically $X_z$ has Picard rank $10-d$). Then $Y$ also has Picard rank $11-d$ and there is only one rational curve $E$ on $Y$ such that $-K_Y.E=0$. By the contraction theorem, we obtain an ample model $Y_0$ of $Y$ by contacting $\nu: Y \rightarrow Y_0$ the rational curve $E$. It is not hard to see the morphism $\nu$ is crepant and $E$ is a $(-2)$-curve contacting to a $A_1$ point on $Y_0$. In other words, $Y_0$ is a nodal del pezzo surface. By the classification results [@Brenton1980 Theorem 1], $Y_0$ is exactly the blowup of ${\mathbb P}^2$ along $9-d$ points with only two points infinitesimally near. This shows that the orbit of $z$ lies in $H_m$. Then the proof is finished. ◻ ## Arithmetic compactification for $P_d$. {#subsec:arithmetic cptfy} By Proposition [Theorem 19](#Torelli){reference-type="ref" reference="Torelli"}, $P_d$ is an open subset of locally symmetric space ${\mathcal F}_d$. Thus, each compactification of ${\mathcal F}_d$ from the arithmetic side, for example, Baily-Borel compactification or toroidal compactification will provide a compactification for $P_d$. In this paper we are mainly interested in the Baily-Borel compactification ${\mathcal F}_d^\ast$, which is the minimal arithmetic compactification of ${\mathcal F}_d$. One may refer to [@Lo03 Section 2] for the details on Baily-Borel compactification. Let $$\breve{{\mathcal{D}}}:=\{z\in {\mathbb P}\Lambda \otimes {\mathbb C}\ |\ z^2=0 \}$$ be the compact dual of Hermitian symmetric space of type IV, which is a smooth quadratic hypersurface and $\partial {\mathcal{D}}$ be the boundary component of ${\mathcal{D}}\subset \breve{{\mathcal{D}}}$ in Satake topology. For each primitive isotropic sublattice $I \subset \Lambda$, it will give a rational boundary component $$B_I= \begin{cases} \{ z=x+\sqrt{-1}y \in \partial {\mathcal{D}}| \{x,y\}_{\mathbb R}\cong I\otimes {\mathbb R}\} & \mathop{\mathrm{{\text{\rm rank}}}}(I)=2 \\ \{ \hbox{point} \} & \mathop{\mathrm{{\text{\rm rank}}}}(I)=1 \end{cases}$$ Then there is identification $${\mathcal F}_d^\ast= O(\Lambda)\setminus ({\mathcal{D}}\sqcup \mathop{\cup} \limits_{I\ } B_I)$$ where $I$ runs over $O(\Lambda)$-invariant primitive isotropic sublattice. In particular, the boundaries $${\mathcal F}_d^\ast-{\mathcal F}_d= O(\Lambda)\setminus \mathop{\cup}\limits_{I} B_I \cong \mathop{\cup}\limits_{I} \mathrm{Stab}_{O(\Lambda)}(I) \setminus B_I$$ consist of finitely many - points (also called $0$-cusps) for $\mathop{\mathrm{{\text{\rm rank}}}}(I)=1$ and - modular curves (also called $1$-cusps) for $\mathop{\mathrm{{\text{\rm rank}}}}(I)=2$. **Remark 23**. *Via the modular form construction, Baily-Borel inded showed that $${\mathcal F}^\ast \cong \text{\rm Proj}\big ( \mathop{\bigoplus} \limits_{m \ge 0 } H^0({\mathcal F}, \lambda^{m})\ \big)$$ where the space $H^0({\mathcal F}, \lambda^m)$ of sections of line bundle $\lambda^{m}$ is just space of modular form of weight $m$.* From the perspective of Hodge theoretic degeneration of K3 surfaces, these points in $B_I$ correspond to Type II degeneration if $\mathop{\mathrm{{\text{\rm rank}}}}(I)=2$ and Type III degeneration if $\mathop{\mathrm{{\text{\rm rank}}}}(I)=1$. See [@AE21 Section 3,4] for more details. **Proposition 24**. *Let $d=8$, then the boundaries ${\mathcal F}_d^\ast-{\mathcal F}_d$ consist of four modular curves ( Type II boundaries) intersecting at a unique point (Type III boundary). The root lattices of Type II boundaries are given by $$E_8\oplus E_7\oplus A_1,\ E_7\oplus D_9,\ D_{15}\oplus A_1,\ A_{16}$$* *Proof.* The method is standard. By [@Scattone1987 §2.1], the Type II (rep. Type III) boundaries of ${\mathcal F}^\ast$ correspond to the classes of isotropic planes $J$ (rep. isotropic lines) in $\Lambda$. By lemma [Lemma 18](#Eichler){reference-type="ref" reference="Eichler"}, isotropic lines are in a $\widetilde{O}^+(\Lambda)$-orbit, and thus Type III boundary is a unique point. Let $J\subset \Lambda$ be an isotropic plane, then $(J)^\perp_\Lambda/J$ is a negative definite of rank 16 with the same genus as $E_7\oplus A_1$. Using the embedding of $E_7\oplus A_1$ into a Niemeier lattices [^2] which are classified into $24$ types (for example, see [@Scattone1987 Theorem 3.5.1]), we get the following possibilities $$E_8\oplus E_7\oplus A_1,\ E_7\oplus D_9,\ D_{15}\oplus A_1,\ A_{16}.$$ This classification is a prior up to $O(\Lambda)$. But as $O^+(\Lambda)/\widetilde{O}^+(\Lambda)=\{ \pm \mathop{\mathrm{\mathrm{id}}}\}$, we get the classification for the Type II boundaries. ◻ **Remark 25**. *As the boundaries ${\mathcal F}^\ast-{\mathcal F}$ of Baily-Borel compactification ${\mathcal F}^\ast$ are $1$-dimensional strata, we will identify $\mathop{\mathrm{Pic}}_{{\mathbb Q}}({\mathcal F})=\mathop{\mathrm{Pic}}_{{\mathbb Q}}({\mathcal F}^\ast)$.* To make the notation more convenient, in the next sections, we also denote $P_d^\ast:={\mathcal F}_d^\ast$. # GIT compactification for $P_d$ {#sect3} In this section, we provide various GIT (or partial )compactifications for the moduli space $P_d$ of smooth log del Pezzo pairs and discuss their relations with the arithmetic compactification constructed in §[2.5](#subsec:arithmetic cptfy){reference-type="ref" reference="subsec:arithmetic cptfy"} . In particular, as an application, we compute the Picard number of Baily-Borel compactification $P_d^*$. ## Construction of GIT models {#gitcons} Let $(X,C)$ be a del Pezzzo pair. We are going to construct GIT model $\overline{P}_d ^{GIT}$ for their moduli space $P_d$ separately due to their different geometric structure. In degree $d=8$ case, let $\mathbb{F}_1 \rightarrow {\mathbb P}^2$ be the blowup at $p=[0,0,1]\in {\mathbb P}^2$ and $\Delta \subset |{\mathcal O}_{{\mathbb P}^2}(6)|$ be the discriminant locus of plane sextic curves, then there is a universal nodal curve with two projections $p_1: {\mathcal V}\rightarrow {\mathbb P}^2$ and $p_2: {\mathcal V}\rightarrow \Delta$. So $p_2({\mathcal V}_p)$ is the locus of plane sextic curves singular at $p$ where ${\mathcal V}_p:=p_1^{-1}([0,0,1])$. It is not hard to see $$p_2({\mathcal V}_p) \cong |{\mathcal O}_{{\mathbb P}^2}(6) \otimes \mathfrak{m}_p^2| \cong {\mathbb P}^{24}.$$ Note that $|{\mathcal O}_{{\mathbb P}^2}(6) \otimes \mathfrak{m}_p^2|$ is also the parameter space of the polynomials of the form $$\label{nodalequ} z^4f_2(x,y)+z^3f_3(x,y)+\cdots +f_6(x,y).$$ The non-reductive group $\mathop{\mathrm{Aut}}(\mathbb{F}_1)=\mathop{\mathrm{\mathrm{GL}}}(2,{\mathbb C}) \rtimes {\mathbb C}^2$ acts on $|{\mathcal O}_{{\mathbb P}^2}(6) \otimes \mathfrak{m}_p^2|$ by $$\begin{split} x \mapsto & a_{11}x+a_{12}y \\ y \mapsto & a_{21}x+a_{22}y\\ z\mapsto & m^{-1}z+b_1x+b_2y, (b_1,b_2)\in {\mathbb C}^2 \end{split}$$ where $m = a_{11}a_{22} -a_{12}a_{21}$. Let $U\subset |{\mathcal O}_{{\mathbb P}^2}(6) \otimes \mathfrak{m}_p^2|$ be the locus where $f_2(x,y)$ is a smooth conic, i.e., the sextic curve with a node at $p$, then each $f\in U$ has the normal form $f=t\cdot z^4xy+h$ where $t\in {\mathbb C}$ and $$\label{nf} h= z^3\widetilde{f}_3(x,y)+z^2f_4(x,y)+zf_5(x,y)+f_6(x,y),\ \ \widetilde{f}_3(x,y)=a_1x^2y+a_2xy^2$$ under the action of $\mathop{\mathrm{Aut}}(\mathbb{F}_1)$. Denote the $V$ by the ${\mathbb C}$-vector space spanned by the monomials in the normal form. Thus, $U$ is the orbit $\mathop{\mathrm{Aut}}(\mathbb{F}_1)\cdot \mathbb{P}W$. Moreover, the stabilizer of ${\mathbb P}W$ is a 2-dimensional torus $T = \left\{\mathop{\mathrm{diag}}(a,b,c)\,\middle| \, abc =1 \right\}\cong {\mathbb C}^* \times {\mathbb C}^\ast$. Therefore, we have a reductive GIT space $$\label{gitd8} \overline{P}_8 ^{GIT} \vcentcolon={\mathbb P}V /\!\!/T.$$ For del Pezzo surface $X$ of degree $7$, we identify $\pi: X=Bl_{p_1,p_2} {\mathbb P}^2 \rightarrow {\mathbb P}^2$ the blowup of ${\mathbb P}^2$ at two points $p_1=[1,0,0],p_2=[0,0,1]$ with two exceptional divisor $E_1$ and $E_2$. Suppose $B$ has only $A_1$-singularity at $p_1$ and $A_1$-singularity at $p_2$, that is, $B$ is defined by the equations of the form $$\label{deg7equa} % y^6+y^5 f_1(x,z) + y^4 f_x(x,z) + y^3 f_3(y,z) + y^2f_4(x,z) \sum_{i=0}^4 y^{6-i}f_i(x,z) + xyz g_3(x,z) +x^2 z^2 h_2(x,z) =0$$ where $f_i,g_i,h_i$ are homogeneous polynomials of degree $i$ in $x,z$. There is a natural non-reductive GIT construction for the moduli space for such curves $B$. Indeed, let $V\subset {\mathbb C}[x,y,z]_6$ be the sub-vector space of homogeneous polynomials of degree $6$ in ([\[deg7equa\]](#deg7equa){reference-type="ref" reference="deg7equa"}). Denote $$G:=\big\{ \ \left( \begin{array}{ccc} a_1 & 0 & 0 \\ a_2 & a_3 & a_4\\ 0 & 0 & a_5 \end{array} % dis=6 \right) \ |\ a_1a_3a_5 \neq 0 \ \big\}$$ be the non-reductive subgroup of $\mathop{\mathrm{\mathrm{PGL}}}(3)$, which is just the stabilizer subgroup of $\mathop{\mathrm{\mathrm{PGL}}}(3)$ fixing the two points $p_1$ and $p_2$. Then there is a natural action of $G$ on the projective space ${\mathbb P}V$ induced from the action of $\mathop{\mathrm{\mathrm{PGL}}}(3)$ on ${\mathbb P}{\mathbb C}[x,y,z]_6$. Then there is a non-reductive GIT quotient space ${\mathbb P}/\!\!/G$ in the sense of Doran-Kirwan [@DK]. It has a reductive realization as follows. By the action $x\mapsto x+ay$ and $z \mapsto z+b y$, any equations in ([\[deg7equa\]](#deg7equa){reference-type="ref" reference="deg7equa"}) with $\mathop{\mathrm{{\text{\rm rank}}}}h_2=2$ can be reduced to the following form $$\label{deg7equa'} % x^4(y^2+z^2)+x^3f_3(y,z)+x^2f_4(y,z)+xyg_4(y,z)+y^2h_4(y,z)=0 x^2z^2 h_2(x,z) + xyz g_3(x,z) + \sum_{i=2}^4 y^{6-i}f_i(x,z) + t\cdot y^6$$ with stabilizer subgroup $T=\{ (a_1,a_3,a_5)\,|\, a_1a_3a_5 =1\} \cong {\mathbb C}^*\times {\mathbb C}^*$. In this way, we get a reductive GIT quotient space $$\label{gitd7} \overline{P}_7 ^{GIT}:={\mathbb P}V' /\!\!/T$$ where $V'\subset V$ is $19$-dimensional sub-vector space parametrizing homogeneous polynomials of the form [\[gitd7\]](#gitd7){reference-type="eqref" reference="gitd7"}. If $d=5,6$, then $\mathop{\mathrm{Aut}}(X)$ is reductive and the surface $X$ is rigid, then we set $$\label{gitd56} \overline{P}_d ^{GIT}:=|-2K_X| /\!\!/\mathop{\mathrm{Aut}}(X).$$ If $d=3$, let ${\mathfrak X}\subset {\mathbb P}^3 \times {\mathbb P}H^0({\mathbb P}^3,{\mathcal O}_{{\mathbb P}^3}(3))^\vee$ be the universal cubic surfaces. on $V$ with two natural projections $$\pi_1\colon {\mathfrak X}\rightarrow V, \quad \pi_2 \colon {\mathfrak X}\rightarrow {\mathbb P}^N \vcentcolon={\mathbb P}H^0(V,{\mathcal O}_V(X))^\vee .$$ Then ${\mathcal{E}}:=\pi_{2\ast}(\pi_1^\ast {\mathcal O}_V(-2K_V-2X))$ is a locally free sheaf of rank $3(-K_X)^2+1$ on ${\mathbb P}^N$ as all the higher direct images of $\pi_2$ vanish. Thus we get a projective bundle $p:{\mathbb P}{\mathcal{E}}\rightarrow {\mathbb P}^N$ over ${\mathbb P}^N$ with Picard group generated by ${\mathcal O}_{{\mathbb P}{\mathcal{E}}}(1)$ and $p^\ast {\mathcal O}_{{\mathbb P}^N}(1)$. Let $G \le \mathop{\mathrm{Aut}}(V)$ be a reductive subgroup and it acts on ${\mathbb P}^N$ by the action induced from $V$. By [@Benoist14 Proposition 2.7 ], the ${\mathbb Q}$-line bundle $$L_t:={\mathcal O}_{{\mathbb P}{\mathcal{E}}}(t) \otimes p^*{\mathcal O}_{{\mathbb P}^N}(1),\ \ t\in {\mathbb Q}$$ is ample for $1 \gg t >0$. Hence there is the GIT quotient space $$\label{git} {\mathbb P}{\mathcal{E}}/\!\!/_{L_t} G:=\mathop{\mathrm{Proj}}(R({\mathbb P}{\mathcal{E}},L_t)^G).$$ The the same construction works for $d=4$ where we only need to replace ${\mathbb P}H^0({\mathbb P}^3,{\mathcal O}_{{\mathbb P}^3}(3))^\vee$ by the Grassimanian $Gr(2,H^0({\mathbb P}^4,{\mathcal O}_{{\mathbb P}^4}(2)))$. For the remaining cases $d = 2$, the geometry is a little different. By [@dolgachev_2012 Section 6.3.3], a del Pezzo surface of degree $2$ is a double cover of ${\mathbb P}^2$ branched along a quartic curve. So we may consider the universal quartic curve ${\mathcal{C}}\subset {\mathbb P}^2 \times |{\mathcal O}_{{\mathbb P}^2}(4)|$. Then ${\mathcal{C}}\in |{\mathcal O}(4,2)|$ and the double cover ${\mathfrak X}\xrightarrow{\phi} {\mathbb P}^2 \times |{\mathcal O}_{{\mathbb P}^2}(4)|$ branched along ${\mathcal{C}}$ has two natural projection $$\pi_1: {\mathfrak X}\rightarrow |{\mathcal O}_{{\mathbb P}^2}(4)|={\mathbb P}^{14},\ \ \pi_2: {\mathfrak X}\rightarrow {\mathbb P}^{2}.$$ We consider the locally free sheaf ${\mathcal{E}}:={\pi_1}_\ast \left(\omega_{{\mathfrak X}/{\mathbb P}^{14}}^{\otimes -2}\right)$ of rank $7$ and define the GIT space $$\label{git2} {\mathbb P}{\mathcal{E}}/\!\!/_{L_t} \mathop{\mathrm{\mathrm{PGL}}}(3):=\mathop{\mathrm{Proj}}(R({\mathbb P}{\mathcal{E}},L_t)^{\mathop{\mathrm{\mathrm{PGL}}}(3)}).$$ ## Analysis of GIT stability A fundamental tool to study GIT stability is Hilbert-Mumford's numerical criterion via the Hilbert-Mumford index. Recall for a polarised projective variety $(Z,L)$ under the action of a reductive group $G$, there is a numerical function (HM index ) on the space of one parameter subgroup ($1$-PS) of $G$ $$\mu(,z):\ \mathop{\mathrm{Hom}}(\mathbb{G}_m,G) \rightarrow {\mathbb R}$$ for any given point $z\in Z$. By [@MFK94 Section 2.1], $z\in Z$ is semistable if and only if $\mu(\lambda,z) \ge 0$ for any $1$-PS $\lambda$. $z\in Z$ is stable if and only if $\mu(\lambda,z) > 0$ for any non-trivial $1$-PS $\lambda$. Now we will apply this numerical criterion to study the GIT space constructed in the last subsection §[3.1](#gitcons){reference-type="ref" reference="gitcons"}. **Proposition 26**. *If $d =5\ \hbox{or}\ 6$, Let $U \subset \left|-2 K_X\right|$ be the locus of smooth curve, then $$\operatorname{codim} \Big( {\lvert-2K_X\rvert}/\!\!/\mathop{\mathrm{Aut}}(X) -U /\!\!/\mathop{\mathrm{Aut}}(X) \Big) \ge 2 .$$* *Proof.* (1) For $d=5$, $S$ is a bidegree $(1,2)$-hypersurface in $\mathbb{P}^1\times \mathbb{P}^2$. We claim if $S=\{F=0\}$ is a singular bidegree $(1,2)$ surface and $0< t \ll 1$, then $(S,C)$ is $L_t$-unstable for any curve $C = \{F=G=0\}\in |{\mathcal O}_{S}(2,2)|$. Indeed, suppose $S$ is singular at $[1,0]\times [1,0,0]$, then the equation $F$ is of the form $$F= x_0 q(y_1,y_2) +x_1y_0l(y_1,y_2)+ x_1 \Tilde{q}(y_1,y_2)$$ where $[x_0,x_1]\times [y_0,y_1,y_2]$ is the coordinate of $\mathbb{P}^1\times \mathbb{P}^2$. let $\lambda$ be a 1-PS with weight $(3,-3) \times (4,-2,-2)$. Then $$\begin{split} \mu_t(S,C; \lambda) &= \mu(F;\lambda) + t\cdot \min \big\{\mu(G; \lambda):G \in |{\mathcal O}_S(2,2)|\big\} \\ & \leq -1+ 14t \end{split}$$ So $(S,C)$ is unstable for $0<t<\frac{1}{14}$. This finishes the proof of claim. Thus there is a morphism $\overline{P}^{GIT}_d \rightarrow |-2K_X| /\!\!/\mathop{\mathrm{Aut}}(X)$ . Similar arguments work for $d=6$ and we omit the details. \(2\) For $d=6$, note that $\mathop{\mathrm{Aut}}(X)^0$ is two dimensional torus, then $|-2K_X| /\!\!/\mathop{\mathrm{Aut}}(X)= |-2K_X|/ \mathop{\mathrm{Aut}}(X)$ and observe that the singular locus in $|-2K_X|$ has codimension $\le 2$. ◻ **Proposition 27**. *Let $\overline{P}^{GIT}_d$ be the space defined either in ([\[gitd8\]](#gitd8){reference-type="ref" reference="gitd8"}) or in ([\[gitd7\]](#gitd7){reference-type="ref" reference="gitd7"}). Then $\overline{P}^{GIT}_d$ is a partial compactification of $P_d$ and the Picard number $\rho(\overline{P}^{GIT}_d)=1$. Let $U \subset \overline{P}^{GIT}_d$ be the smooth locus, then $\overline{P}^{GIT}_d-U$ has codimension $\ge 2$.* *Proof.* Note that the GIT space in ([\[gitd8\]](#gitd8){reference-type="ref" reference="gitd8"}) and ([\[gitd7\]](#gitd7){reference-type="ref" reference="gitd7"}) are toric GIT quotient under torus $T$ and thus the all the torus in $T$ are diagnoal. In this case, the unstable locus is cut out by the monials with negative weight with $1$-PS of $T$. Thus to estimate codimension of unstable locus, it is equivalent to estimate the number of monomials with negative weight under $1$-PS of $T$. We leave the simple computation to interested readers. ◻ **Theorem 28**. *Let $\overline{P}^{GIT}_d$ be the space defined either in ([\[git\]](#git){reference-type="ref" reference="git"}) or in ([\[git2\]](#git2){reference-type="ref" reference="git2"}). If $d =2, 3,4$, then $\overline{P}^{GIT}_d$ is a compactification of $P_d$ and the Picard number of $\overline{P}^{GIT}_d$ is $2$.* *Proof.* For $d=3$, ${\mathbb P}{\mathcal{E}}\cong {\mathcal O}_{{\mathbb P}^N}^{\oplus 10}$ and thus ${\mathbb P}^{\mathcal{E}}=|{\mathcal O}_{{\mathbb P}^3}(3)| \times |{\mathcal O}_{{\mathbb P}^3}(2) |\cong {\mathbb P}^{19} \times {\mathbb P}^9$. We claim if $X=\{ c=0\}$ is a smooth del Pezzo surface of degree $4$ and $C=\{c=0,q=0\}\in |-2K_X|$ is a curve with only a $A_1$ singularity, then $(c,q)\in$ is a GIT stable. Note that if the rank of $q$ is less than or equal to $2$, then $C$ is reducible. Thus one can assume $\mathop{\mathrm{{\text{\rm rank}}}}(q)\ge 3$. If $\mathop{\mathrm{{\text{\rm rank}}}}(q)=4$, it is known that the smooth quadrics and cubics in ${\mathbb P}^n$ are GIT stable, then $$\mu_t(c,q\,;\lambda)=\mu(c\,;\lambda)+t\cdot \mu(q\,;\lambda)>0$$ for any 1-PS in $G=\mathop{\mathrm{\mathrm{SL}}}(4)$ and $t>0$. Now we only consider $\mathop{\mathrm{{\text{\rm rank}}}}(q)=3$. We argue by contradiction. If the pair $(c,q)$ is GIT-unstable, there is a 1-PS $\lambda$ in the Weyl chamber of $\mathfrak{g}^\ast$ such that $$\label{constr} \mu(q\,;\lambda)<0, \ \mu(c\,;\lambda)> 0.$$ Note that $c$ has at least $3$ monomials of degree $3$ and $q$ has at least $2$ monomials of degree $2$. By running a computer program [^3], there is no solution for the $\lambda$ satisfying the constraints in ([\[constr\]](#constr){reference-type="ref" reference="constr"}). For $d=4$, recall that a smooth degree 4 del Pezzo surface $X=(Q_1 \cap Q_2)$ is stable in $Gr$ under the natural action of $\mathop{\mathrm{\mathrm{SL}}}(5,\mathbb{C})$ by [@Mabuchi1993 Theorem 6.1]. And a curve $C\in |-2K_X|$ which is the complete intersection $Q_1\cap Q_2\cap Q_3$ with at worst $A_1$ singularity is also stable in $Gr(3,H^0({\mathcal O}_{\mathbb{P}^4}(2))$ by the results of Fedorchuk-Smyth [@Fedorchuk2013 Theorem 3.1]. So for any 1-PS $\lambda$, up to coordinate change one may assume the action is diagonalized, and we have $$\begin{split} \mu(X;\lambda) &=\mu(Q_1 \wedge Q_2 ;\lambda) >0 \\ \mu(X;\lambda) + \mu (C;\lambda) &= \mu (Q_1 \wedge Q_2 ;\lambda)+\mu ([Q_3];\lambda) \\ &= \mu (Q_1 \wedge Q_2 ;\lambda)+\min \left\{\mu (Q_3';\lambda) \,\middle|\, Q_3' \equiv Q_3 \bmod \langle Q_1,Q_2\rangle \right\}\\ &\geq \mu (Q_1 \wedge Q_2 \wedge Q_3;\lambda) > 0 \end{split}$$ by definition. We only need to show that $(X,C)$ is stable for $0<t \ll 1$. One may assume $\mu(C;\lambda) <0$ otherwise the statement holds trivially. In this case, we have $$\mu_t(X,C;\lambda) \vcentcolon=\mu(X;\lambda) + t\cdot \mu(C;\lambda) > \mu(X;\lambda) + \mu(C;\lambda) >0$$ For $d=2$, we show that $(X, D)$ is stable if $X$ and $D$ are both smooth and for generic $D$ with $A_1$-singularity the pair is also stable. By direct computation, one gets $${\mathcal{E}}= H^0({\mathbb P}^2,{\mathcal O}_{{\mathbb P}^2}(2)) \otimes {\mathcal O}_{{\mathbb P}^{14}}(-2) \oplus {\mathcal O}_{{\mathbb P}^{14}}(-3)$$ One can identify the fiber of $p\mathbin{:}{\mathbb P}{\mathcal{E}}\to \lvert {\mathcal O}_{{\mathbb P}^2}(4) \rvert$ over $C\in \lvert {\mathcal O}_{{\mathbb P}^2}(4) \rvert$ with ${\mathbb P}V$, where $$V= H^0(X_C, -2K_{X_C}) \cong H^0({\mathbb P}^2, {\phi_C}_\ast (-2K_{X_C}) )\cong H^0({\mathbb P}^2, {\mathcal O}_{{\mathbb P}^2}(2) \oplus {\mathcal O}_{{\mathbb P}^2}).$$ Given $[(q,a)] \in {\mathbb P}H^0({\mathbb P}^2, {\mathcal O}_{{\mathbb P}^2}(2) \oplus {\mathcal O}_{{\mathbb P}^2})$, we get a $D\in \lvert -2K_{X_C} \rvert$ defined by $a^2 w^2 = q$ which is the double cover of a plane conic $Q \vcentcolon=[q]\in {\mathbb P}H^0({\mathbb P}^2,{\mathcal O}_{{\mathbb P}^2}(2))$ ramified over $Q\cap C$. For $(C,q,a) \in {\mathbb P}{\mathcal{E}}$, the Hilbert-Mumford index is given by $$\begin{split} \mu_t(C,q,a\,;\lambda) &\vcentcolon=\mu(C;\lambda) +t \cdot \mu(q,a\,;\lambda) \\ &{\phantom{:}=~} \mu(C;\lambda) +t \cdot \mu(q\,;\lambda). \end{split}$$ Note that for $(X, D)$ attached to $(C,q,a)$, $X$ is smooth if and only if $C$ is smooth, and $D$ is smooth if and only if $q$ and $q\cap C$ are smooth. So the case that $(X,D)$ are both smooth follows easily from that smooth plane quartics and smooth conics are stable. If $X=X_C$ is smooth and $D$ has an $A_1$-singularity, there are two cases: - (Generic case) The plane conic $q$ is smooth and tangent to $C$ at exactly one point with intersection multiplicity $2$. - The plane conic $q$ has exactly an $A_1$-singularity where $C$ passes through. The statement follows. ◻ ## The Picard group of Baily-Borel compactifications Let us first give the geometric strategy to compute the Picard group of $P_d^\ast$. Similar methods have been used in [@GLT] to compute the Picard group of moduli spaces of quasi-polarised K3 surface with Mukai models. Let $U\subset \overline{P}^{GIT}_d$ be the locus with node at worst in GIT space studied in section 3.2. By Theorem [Theorem 19](#Torelli){reference-type="ref" reference="Torelli"}, the period map induces a birational contraction map $$p^{-1}:\ P_d^\ast \dashrightarrow \overline{P}^{GIT}_d$$ which are isomorphic over a open subset $U \subset \overline{P}^{GIT}_d$. Thus $U\subset P_d^\ast$ is also an open subset of $P_d^\ast$. Let $\# (P^*_d - U)$ be the number of $9+d$-dimensional irreducible components in $P^*_d - U$. Note that irreducible components of $\widetilde{P}^*_d-U$ form divisors on $P^*_d$ which are linearly independent since they are birationally contractible. Then by the localization sequence $$\mathop{\mathrm{\mathrm{CH}}}_{9+d}(P^*_d - U) \rightarrow \mathop{\mathrm{\mathrm{CH}}}_{9+d}(P_d^*) \rightarrow \mathop{\mathrm{\mathrm{CH}}}_{9+d}(U) \rightarrow 0,$$ We have $$\rho (P_d^\ast)= \rho (U)+\# (P^*_d - U)$$ Similarly, let $\# (\overline{P}^{GIT}_d-U)$ be the number of $9+d$-dimensional irreducible components of $(\overline{P}^{GIT}_d-U)$. Applying localization sequence of Chow groups to $\overline{P}^{GIT}_d$, we deduce the following Picard number formula $$\label{picardformula} \rho (P_d^\ast)= \rho(\overline{P}^{GIT}_d)- \# (\overline{P}^{GIT}_d-U)+ \# (P^*_d - U)$$ **Theorem 29**. *The Picard group of $P_d^\ast$ are freely generated by Hodge line bundle $\lambda$ and irreducible Heegner divisors described in Theorem [Theorem 19](#Torelli){reference-type="ref" reference="Torelli"}. Moreover, the Picard numbers are given by* *$d$* *$1$* *$2$* *$3$* *$4$* *$5$* *$6$* *$7$* *$8$* *$8$* *$9$* -------------------------------------------------------------------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -- *$\dim_{\mathbb Q}\mathop{\mathrm{\mathrm{Cl}}}(P_d ^\ast)_{\mathbb Q}$* *$6$* *$5$* *$4$* *$4$* *$4$* *$3$* *$4$* *$3$* *$2$* *$2$* : *Picard numbers of Baily-Borel compactifications $P_d ^\ast$* *Proof.* If $d=9$, then $P_d^\ast$ is Baily-Borel compactification of moduli space quasi-polarised K3 surface of degree $2$. This is classical known (cf. [@GLT]). If $d=8$, We take $U \subset \overline{P}_d ^{GIT}$ the locus with the pair $(X,C)$ with nodal singularities at worst as in Proposition [Proposition 27](#gitd78){reference-type="ref" reference="gitd78"}, then $\rho(U)=\rho(\overline{P}_d ^{GIT})=1$. According to the construction of GIT space, $U \subset P_d$ is a open subset such that the complement $P_d-U$ is a divisor parametrizing the pair $(X,C)$ where the smooth curve $C$ tangent to a exceptional divisor. By part (1) of theorem [Theorem 19](#Torelli){reference-type="ref" reference="Torelli"} and Proposition [Proposition 22](#complement){reference-type="ref" reference="complement"}, $\# (P^*_d - P_d)=2-1$ due to removing the nodal divisor $H_{ \rm n}$ and thus $\# (P^*_d - U)=2$. This proves $\rho(P_d ^\ast)=1-0+2=3$ by formula ([\[picardformula\]](#picardformula){reference-type="ref" reference="picardformula"}). $d=7$ case is the same as $d=8$ in addition to that $H_m$ shows up and it turns out that $\# (P^*_7 - P_7)=2$. This proves $\rho(P^*_7)=4$. If $d=5,6$, we take $U$ in Proposition [Proposition 26](#git56){reference-type="ref" reference="git56"} as the common open subset. By part (2) of theorem [Theorem 19](#Torelli){reference-type="ref" reference="Torelli"} and Proposition [Proposition 22](#complement){reference-type="ref" reference="complement"}, $\#(P_d^\ast-U)=2$ for $d=6$ and $\#(P_d^\ast-U)= 3$ for $d=5$. This implies $\rho(P_6 ^\ast)=1-0+2=3$ by the formula ([\[picardformula\]](#picardformula){reference-type="ref" reference="picardformula"}). Similarly, we have $\rho(P_5 ^\ast)=1-0+3=4$. If $d=2,3,4$, we take $U$ in Theorem [Theorem 28](#git234){reference-type="ref" reference="git234"} as the common open subset. Note that in these case, a nodal surface of del pezzo is allowed by Proposition [Proposition 22](#complement){reference-type="ref" reference="complement"}. Thus, a generic point in $H_m$ come from $U$. Combining $\rho(\overline{P}^{GIT}_d)=2$ and part (4) (5) (6) of theorem [Theorem 19](#Torelli){reference-type="ref" reference="Torelli"}, we have $$\#(P_d^\ast-U)=\begin{cases} 2 & d=4 \\ 2 & d=3 \\ 3 & d=2 \end{cases}$$ ◻ **Remark 30**. *Indeed, the above arguments can be used to find the generators of Picard groups $\mathop{\mathrm{\mathrm{Pic}}}(P_d^*)_{\mathbb Q}$. Let $v_{d,i} \in \Lambda_d$ be a $(-2)$ vector with divisibility $\mathrm{div}(v_{d,i})=2$ and whose $\Gamma_d=O(\Lambda_d)$-orbit is one of those described in Theorem [Theorem 19](#Torelli){reference-type="ref" reference="Torelli"}. Denote $H_{d,i}$ the irreducible Heegner divisor defined by the vector $v_{d,i} \in \Lambda_d$. Then the Picard groups of $P_d^\ast$ have generators as in the following Table [\[tab:my_label\]](#tab:my_label){reference-type="ref" reference="tab:my_label"}.* *$$\begin{array}{|c|c|c|} \hline d & \mathop{\mathrm{\mathrm{Pic}}}(U)_{\mathbb Q}& \text{contractable divisors} \\ \hline 8 & H_{\rm n} & H_h , H_u\\ \hline 7 & H_{\rm n} & H_u ,H_h, H_m \\ \hline 6 & H_{\rm n} & H_m, H_{6,1} \\ \hline 5 & H_{\rm n} & H_m, H_{5,1},H_{5,2} \\ \hline 4 & H_{\rm n}, H_m & H_{4,1},H_{4,2} \\ \hline 3 & H_{\rm n}, H_m & H_{3,1},H_{3,2} \\ \hline 2 & H_{\rm n}, H_m & H_{2,1},H_{2,2},H_{2,3} \\ \hline 1 & H_{\rm n}, H_m & H_{1,1},H_{1,2},H_{1,3}, H_{1,4} \\ \hline \end{array}$$* ## A motivating question Comparing different compactifications for a moduli space is an interesting topic. It dates backs to the work of Shah ([@Shah80]) and Looigenga ([@Loo86]) on the moduli of genus $2$ quasi-polarised K3 surface and Laza's work ([@Laza10]) cubic $4$-folds. And later it is the work of Laza-O'Grady ([@LO19]) and Ascher-DeVleming- Liu ([@ADL21]) on moduli of genus $3$ quasi-polarised K3 surface, also the forthcoming work [@FLLST] on moduli of K3 surface in lower genus. In our situation, a very natural question is **Question 31**. *Can we resolve the birational period map $\overline{P}^{GIT}_d \dashrightarrow \overline{P}^{\ast}_d$ explicitly, even with modular interpretation ?* From the known works mentioned above, it suggests that we can approach the question both by arithmetic methods and K-moduli methods. It is expected that the K-moduli spaces of del Pezzo pairs should be identified with a suitable Hasset-Keel-Looijenga model for $P_d^\ast$ and this Hasset-Keel-Looijenga model has natural connection between GIT (partial) compactification of $P_d$ and Baily-Borel compactification of $P_d$. In other words, wall-crossing of K-moduli spaces should provide the resolution of birational period map $\overline{P}^{GIT}_d \dashrightarrow \overline{P}^{\ast}_d$. In the next sections, we mainly focus on arithmetic methods in the case $d=8$. # HKL model for $P_8^\ast$ {#sect4} Recall the discriminant group of lattice $\Lambda_8= U^2 \oplus E_8 \oplus E_7 \oplus A_1$ associated to a smooth del pezzo pair of degree $8$ is given by $$G(\Lambda_8)={\mathbb Z}\eta \oplus {\mathbb Z}\xi.$$ **Definition 32**. *We call a primitive vector $v\in \Lambda_8$* 1. *nodal if $v^2=-2$ and $\mathop{\mathrm{\mathrm{div}}}_{\Lambda}(v)=1$ .* 2. *unigonal if $v^2=-2$ and $\mathop{\mathrm{\mathrm{div}}}_{\Lambda}(v)=2$, $v^\ast = \xi_i$ .* 3. *hyperelliptic if $$v^2 =-6,\ \ \hbox{and}\ \ \ v^\ast=\xi$$ Then the complement $v^\perp$ of $v$ in $\Lambda_8$ is isomorphic to the complement of the following definite lattice $(E_7\oplus A_2)^\perp$ in $I\!I$.* Note that the points lying in $P_8\cup {\mathcal F}_8^\circ$ correspond to K3 surfaces coming from the double cover of $\mathbb{F}_1$ or its nodal degeneration. Then the locus consisting of the double cover of the admissible degenerations of $\mathbb{F}_1$ will give rise to some arithmetic divisors in ${\mathcal F}_8$ in the following way: Let $X$ be some possible singular degeneration of $\mathbb{F}_1$ and $\widetilde{X}$ be its minimal resolution. Take the double cover of $\widetilde{X}$ branched along a curve $B\in \lvert -2K_{\widetilde{X}}\rvert$ and we get a possible singular K3 surface $S_0$. Then the crepant resolution $S$ of $S_0$ lies in ${\mathcal F}_8$. We first characterize the general period point in $H_h$ and $H_u$ as K3 surfaces $S$ obtained from $X=\mathbb{F}_1$ or $Bl_p{\mathbb P}(1,1,4)$ by computing their Neron-Severi groups. ## Hyperelliptic K3 and its ADE degenerations {#hk3} After identifying $\mathbb{F}_1=Bl_p {\mathbb P}^2\xrightarrow{\pi} {\mathbb P}^2$, we denote $E$ be the the exceptional curve of the blowup of ${\mathbb P}^2$ at $p=[0,0,1]\in {\mathbb P}^2$ and $C\in |-2K_{\mathbb{F}_1}|$ where a general curve tangents to $E$ at only one point. Let $\phi: X \rightarrow \mathbb{F}_1$ be the double cover branched along the curve $C$. Then $X$ is a smooth K3 surface. The tangency condition of $E$ implies $$\phi^{-1}(E)=E_1+E_2$$ where $E_i \cong {\mathbb P}^1$ and $E_1.E_2=1$. Denote $L$ the proper transform of a general line in ${\mathbb P}^2$, that is, $$L\sim \phi^\ast \pi^\ast {\mathcal O}_{{\mathbb P}^2}(1)$$ then $\mathop{\mathrm{\mathrm{NS}}}(X)={\mathbb Z}[L]\oplus {\mathbb Z}[E_1] \oplus{\mathbb Z}[E_2]$ with the the following intersection numbers $$\left(\begin{array}{c|ccc} & L & E_1 & E_2 \\\hline L & 2 & 0 & 0 \\ E_1 & 0 & -2 & 1 \\ E_2 & 0 & 1 &-2 \\ \end{array}\right)$$ Clearly, the K3 surface $X$ with antisymplectic involution from such del Pezzo pair is a general member in the hyperelliptic divisor $H_h$. Now we compute the Neron-Severi groups of the K3 surface of the possible geometric degeneration of pairs $(\mathbb{F}_1, C)$ which will serve as the geometric input for arithmetic stratification in the next sections. First, we make the following observation. If one take a double cover $\phi': X' \longrightarrow {\mathbb P}^2$ branched along a plane sextic curve $B=\pi(C)$ and then a minimal resolution $\mu': \widetilde{X} \longrightarrow X'$, there is a commutative diagram as follows $$\begin{tikzcd} \widetilde{X} \arrow[d,"\mu"] \arrow[r,"="] & \widetilde{X}\arrow[d,"\mu'"] \\ X \arrow[d,"\phi"] & X' \arrow[d,"\phi'"] \\ \mathbb{F}_1=Bl_p{\mathbb P}^2 \arrow[r,"\pi"] & {\mathbb P}^2 \end{tikzcd}$$ Therefore, we can also compute the Neron-Severi group $\mathop{\mathrm{\mathrm{NS}}}(\widetilde{X})$ via a covering of ${\mathbb P}^2$. Note that in this viewpoint a general element in $H_h$ is obtained from a plane sextic curve $B$ with only a $A_2$-singularity. ### Degeneration with $A_n$-singularities. {#degeneration-with-a_n-singularities. .unnumbered} If $B$ degenerate to an irreducible plane sextic curve $B_0=\pi(C_0)$ with only $A_n$ singularity at $p$ which is locally of the form $$x^2+y^n=0,$$ then $X'$ is a singular K3 with $A_n$-surface singularity under the double cover $\phi: X' \rightarrow {\mathbb P}^2$. Thus the Neron-Severi group $\mathop{\mathrm{\mathrm{NS}}}(\widetilde{X})$ is generated by class $H\sim \mu^\ast L$ from a general line in ${\mathbb P}^2$ and exceptional divisors $E_1,\cdots,E_n$. The gram matrix of $\mathop{\mathrm{\mathrm{NS}}}(\widetilde{X})$ is given by $$\label{NSA} \left( \begin{array}{c|c c} & H & \vec{E} \\ \hline H & 2 & 0 \\ \vec{E} &0 &A_n \end{array}\right)$$ where $\vec{E}=(E_1,\cdots,E_n)$ and $\mathop{\mathrm{\mathrm{NS}}}(\widetilde{X})$ has rank $n+1$. Then we consider the following tangency conditions. 1. *$A_n'$ modification:* If $B$ degenerate to the case where there is a line $L=\{ x=0 \}$ tangent to $B$ and $B$ is irreducible, then the preimage $\phi^{-1}(L)$ of $L$ will split into two lines $L_1, L_2 \ \subset X$. Note that the tangency condition shows that the plane curve $B$ is defined by the equation of the following form $$z^4xl(x,y)+z^3xf_2(x,y)+z^2xf_3(x,y)+zxf_4(x,y)+f_6(x,y)=0.$$ with coefficient of $y^6$ is nonzero. Then locally around $p$ the curve germ $(B,p)$ is isomorphic to $$x^2+xy^n=0,\ \ \hbox{for}\ n \le 5$$ This implies locally around $p$, $X$ has singularity $$w^2= x^2+xy^n, \ \hbox{for}\ n \le 5,$$ which is a $A_{2n-1}$-singularity. Then under the minimal resolution $\mu: \widetilde{X} \longrightarrow X$, $$\mu^\ast L_i=l_i+ \frac{E_1+2E_2+\cdots + (n-1)E_{n-1}+ nE_n+(n-1)E_{n+1}+\cdots+ 2E_{2n-2}+E_{2n-1}}{2}$$ where $l_i$ is the proper transform of $L_i$. Then $\mathop{\mathrm{\mathrm{NS}}}(\widetilde{X})$ is generated by $E_1,\cdots, E_{2n-1}, l_1$ and $l_2$ and the gram matrix of $\mathop{\mathrm{\mathrm{NS}}}(\widetilde{X})$ is $$\label{NSA'} \left( \begin{array}{c|c c c} & H & \vec{E} & l_2 \\ \hline H & 2 & 0 & 1 \\ \vec{E} & 0 & A_n & w \\ l_2 &1 & w^t & -2 \end{array}\right)$$ where $w^t=(0,0,\cdots,0,1,0,0)$. Moreover, $\mathop{\mathrm{\mathrm{NS}}}(\widetilde{X})$ has rank $2+n$ and discriminant $(-1)^{n-1}(17-n)$. 2. *$A_n''$ modification:* If $B$ degenerate to the plane sextic curve $B_0$ which is the union of a line $L$ and a plane quintic curve $D$ such that $L$ and $D$ intersects two points with multiplicity $4$ and $1$ respectively, then there is an additional class $\widetilde{L}$ in $\mathop{\mathrm{\mathrm{NS}}}(\widetilde{X})$ which comes from the preimage of $L$ under the covering map. Then the new gram matrix of $\mathop{\mathrm{\mathrm{NS}}}(\widetilde{X})$ is $$\label{NSA''} \left( \begin{array}{c|c c c} & H & \vec{E} & \widetilde{L} \\ \hline H & 2 & 0 & 1 \\ \vec{E} & 0 &A_{n+1} & l \\ \widetilde{L} & 1 &l^t & -2 \end{array}\right)$$ where $l^t=(0,0,\cdots,1,0,0,0)$. Thus, the discriminant group of the transcendental lattice of $\widetilde{X}$ is a finite group of order $17-n$ and $\mathop{\mathrm{\mathrm{NS}}}(\widetilde{X})$ has rank $2+n$. 3. *$A_n'''$ modification:* If in addition $C$ is union of a line $L$ and a plane quintic curve $D$ such that $D$ and $L$ are tangent at $p$, then the $\mathop{\mathrm{\mathrm{NS}}}(\widetilde{X})$ is given by $$\label{NSA'''} \left( \begin{array}{c|c c c} & H & \vec{E} & \widetilde{L} \\ \hline H & 2 & 0 & 1 \\ \vec{E} & 0 &D_{n+2} & l' \\ \widetilde{L} & 1& (l')^t & -2 \end{array}\right)$$ where $(l')^t=(1,0,\cdots,0,1,0,0,0)$ with discriminant group ${\mathbb Z}/12{\mathbb Z}$. (compare to Subsection [5.3](#subsec: arithmetic strata){reference-type="ref" reference="subsec: arithmetic strata"}). ### Degeneration with $D_n$-singularities. {#degeneration-with-d_n-singularities. .unnumbered} Similarly, if $C\in |-2K_{\mathbb{F}_1}|$ has only $D_n$ ($E_n$ ) singularity at $p$, $$\label{NSD} \left( \begin{array}{c|cc} & H & \vec{E} \\ \hline H & 2 & 0 \\ \vec{E} &0 &D_n \end{array}\right)$$ If there is an additional line such that, the gram matrix of $\mathop{\mathrm{\mathrm{NS}}}(\widetilde{X})$ is $$\label{NSD'} \left( \begin{array}{c|c c c} & H & \vec{E} & \widetilde{L} \\ \hline H & 2 & 0 & 1 \\ \vec{E} & 0 &D_n & v \\ \widetilde{L} & 1& v^t & -2 \end{array}\right)$$ where $v^t=(1,0,\cdots,0,1,0)$. For such smooth K3 surface $\widetilde{X}$, $\mathop{\mathrm{\mathrm{NS}}}(\widetilde{X})$ has rank $n+2$ and the discriminant group of its transcendental lattice is a finite group of order $4$. ### Degeneration with $E_n$-singularities. {#degeneration-with-e_n-singularities. .unnumbered} The plane sextic curve $B$ with $A_2$-singularity at $p$ degenerates to the palne sextic with only $E_n$ singularity at $p$, then $$\label{NSE} \left( \begin{array}{c|cc} & H & \vec{E} \\ \hline H & 2 & 0 \\ \vec{E} &0 &E_n \end{array}\right)$$ ## Unigonal K3 and its ADE type degenerations {#uk3} The surface $X:=Bl_p{\mathbb P}(1,1,4)$ has only one singularity $v$, which is a quotient singularity of type $\frac{1}{4}(1,1)$. Let $\mu:\widetilde{X} \rightarrow X$ be the minimal resolution of singularities, i.e., $\widetilde{X} \cong Bl_p \mathbb{F}_4$. Denote $E$ and $F$ the exceptional divisor over $p$ and quotient singularity. Let $H_y\subset \widetilde{X}$ be the proper transform of the line $\{y=0\}$ passing $p$ and the quotient singularity $v$. Now we take a double cover $\psi: S \rightarrow \widetilde{X}$ alone a general smooth curve $B \in |-2K_{\widetilde{X}}|$ and get a smooth K3 surface $S$. Then the classes$$E'=\psi^\ast E,\ 2F':=\psi^\ast F,\ H_y':=\psi^\ast H_y$$ form a basis of $\mathop{\mathrm{\mathrm{NS}}}(Y)$ and by computation we have the following intersection numbers $$\label{nsU1} \left(\begin{array}{c|ccc} & E' & F' & H_y' \\\hline E' & -2 & 0 & 2 \\ F' & 0 & -2 & 1 \\ H_y' & 2 & 1 &-2 \\ \end{array}\right)$$ Note that the linear system $|-2K_{\widetilde{X}}|$ has a fixed component $F$ and generic elements in it has the form $C+F$ where $C\in |\mu^*(-2K_X)|$ a smooth curve not passing through $v$ with $C\cdot F =0$. Hence unlike the hyperelliptic case, for $X = Bl_p{\mathbb P}(1,1,4)$ we can not switch the order of taking the double cover and the minimal resolution as the Picard number of the resulting K3 surfaces are different. Here we provide several degenerations for curve $B$ and compute the Neron-Severi groups of K3 surfaces associated with these degenerate curves. 1. *1st-level degeneration*: $C$ passes through $E\cap H_y$ and $[0,1,0]$ and there are two irreducible singular (nodal) curves $\Sigma_1$ and $\Sigma_2$ in $|-2K_{\widetilde{X}}-F|$ tangent to $C$ at the nodes lying on $E\cap H_y$ and $[0,1,0]$ respectively. In this case, we have $\psi^*\Sigma_i= \Sigma'_i + \Sigma_i^{''}$. Note that $\Sigma_i'$ and $\Sigma^{''}_i$ are both resolution of $\Sigma_i$, we have ${\Sigma'_{i}}^2={\Sigma^{''}_{i}}^2= 14$. And $\Sigma'_1 \cdot \Sigma'_2 =15$ by Hodge index theorem. $$\label{nsU2'} \left(\begin{array}{c|ccccc} & E' & F' & H_y' & \Sigma_1' & \Sigma_2' \\\hline E' & -2 & 0 & 2 &2 &2 \\ F' & 0 & -2 & 1 & 0 & 0 \\ H_y' & 2 &1 &-2 & 1 & 1 \\ \Sigma_1 & 2 & 0 & 1 & 14 & 15 \\ T_1 & 2 & 0 & 1 & 15 & 14\\ \end{array}\right)$$ 2. *2nd-level degeneration*: In addition to case (1), the exceptional divisor $E$ is tangent to the irreducible component $C$ of the branching locus and $E\cap F =\emptyset$. Hence $\psi^*E= E'_1 + E'_2$ are two rational curves and $-2=2E^2 = (\psi^*E)^2 = (E'_1 + E'_2)^2$. Hence $E_1'\cdot E_2'=1$. And $E'_i \cdot D = \frac{1}{2}\psi^*E \cdot D$ for $D = H_y'$ and $\Sigma'_j$ . $$\label{nsU3'} \left(\begin{array}{c|cccccc} & E'_1 & E'_2 & F' & H_y' & \Sigma_1' & \Sigma_2' \\\hline E'_1 & -2 & 1 & 0 & 1 & 1 & 1 \\ E'_2 & 1 & -2 & 0 & 1 & 1 & 1 \\ F' & 0 & 0 & -2 & 1 & 0 & 0 \\ H_y' & 1 & 1 & 1 & -2 & 1 & 1 \\ \Sigma_1' & 1 & 1 & 0 & 1 & 14 & 15 \\ \Sigma_2' & 1 & 1 & 0 & 1 & 15 & 14 \\ \end{array}\right)$$ 3. *3rd-level degeneration*: In addition case (1), the branched curve $B=C\sqcup F$ where $C=C_0\cup E$ has two irreducible components and $C_0$ intersects with $E$ at three distinct points. Then the K3 surface $S$ has three $A_1$ singularities. Let $\widetilde{S}$ be the minimal resolution of $Y$, then there are three exceptional curves $L_1, L_2$ and $L_3$. Clearly $L_i\cdot E'=1 ,\,L_i^2 =-2$ and $L_i$ does not intersect with $F',H_y'$ and $\Sigma'_j$. $$\label{nsU4''} \left(\begin{array}{c|cccccccc} & E' & F' & H_y' & \Sigma_1' & \Sigma_2' & L_1 & L_2 & L_3 \\\hline E' & -2 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ F' & 0 & -2 & 1 & 0 & 0 & 0 & 0 & 0 \\ H_y' & 1 & 1 & -2 & 1 & 1 & 0 & 0 & 0 \\ \Sigma_1' & 1 & 0 & 1 & 14 & 15 & 0 & 0 & 0 \\ \Sigma_2' & 1 & 0 & 1 & 15 & 14 & 0 & 0 & 0 \\ L_1 & 1 & 0 & 0 & 0 & 0 & -2 & 0 & 0 \\ L_2 & 1 & 0 & 0 & 0 & 0 & 0 & -2 & 0 \\ L_3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -2 \end{array}\right)$$ ## HKL models for $P_8^\ast$ Consider the projective scheme ${\mathcal F}_8(s)$ defined by the $\mathop{\mathrm{Proj}}$ of the section rings $$R_8(s):= R({\mathcal F}_8, \lambda+s(H_h+25H_u)),\ \ s\in [0,1] \cap {\mathbb Q}$$ **Remark 33**. *Since the locally symmetric space $\Gamma \setminus {\mathcal{D}}$ is strictly log canonical (see [@Alexeev96 Theorem 3.2]), the finite generation of the section rings $R(s)$ are not known in prior except for $R(0)$ due to Baily-Borel. But we may assume this, then this condition can be applied to give the prediction for wall-crossings.* By adding a coefficient $c\in (0,\frac{1}{2})\cap {\mathbb Q}$ to the boundary curve $C$, the pair $(X,cC)$ can be viewed as a log Fano surface. Recent breakthrough in (log) Fano geometry ensures that the K-semistable log del Pezzo surfaces $(X,cC)$ have a good moduli space $\overline{P}_{d,c}^K$ known as K-moduli space(see [@Xu21] for a survey). It is known a general smooth del Pezzo surface pair $(X,cC)$ is K-semistable for certain $c\in (0,\frac{1}{2})\cap {\mathbb Q}$. Indeed, this can be obtained by interpolation of K-stability and classical results on K-stability of smooth del pezzo surface in [@Tian90] and [@OSS16]. Then we know $\overline{P}_{d,c}^K$ is birational to $P_d$. According to the HKL principle explained in §[4.4](#arith){reference-type="ref" reference="arith"}, we make the following conjecture **Conjecture 34** (HKL for $P_8^\ast$). *Notation as above, let $d=8$.* 1. *The section rings $R_d(s)$ are finitely generated for all $s\in [0,1] \cap {\mathbb Q}$. In particular, ${\mathcal F}(s)$ is a projective variety of dimension $18$.* 2. *${\mathcal F}_d(s)$ will interpolate between $P_d^\ast$ and $\overline{P}^{GIT}_d$ when $s$ varies. That is, $${\mathcal F}_d(0)=P_d^\ast,\ \ {\mathcal F}_d(1)= \overline{P}^{GIT}_d .$$ Moreover, there are finitely many rational numbers $0<w_1 <w_2<\cdots<w_n <1$ (called walls ) such that if $s, s'\in (w_i,w_{i+1}) \cap {\mathbb Q}$ , there is natural isomorphism ${\mathcal F}_d(s)\cong {\mathcal F}_d(s')$. Denote ${\mathcal F}_d(w_i,w_{i+1}):={\mathcal F}_d(s)$ for any $s\in (w_i,w_{i+1}) \cap {\mathbb Q}$, then there is a wall-crossing behavior (called HKL wall-crossing ) $$\xymatrix@R-1em@C-1em{ {\mathcal F}_d(0,w_1)\ar[dr] & & {\mathcal F}_d(w_1,w_2) \ar[dl] \ar[dr] & & \ar[dl]{\mathcal F}_d(w_{n-1},w_n) \ar[dr] & & \ar[dl] {\mathcal F}_d(w_{n},1) \\ & {\mathcal F}_d(w_1) & & \cdots & & {\mathcal F}_d(w_n) }$$ where at each wall $w_i$, the birational maps are flips or divisorial contractions.* 3. *There is isomorphism $\overline{P}_{c}^K \cong {\mathcal F}_d(s)$ under the transformation $$s=s(c)=\frac{1-2c}{56c-4}$$ In particular, the K-moduli wall crossing will coincide with HKL wall-crossing .* **Remark 35**. *The work in [@ADL21] deals with the case of $d=8$ where the generic surface is isomorphic to ${\mathbb P}^1\times {\mathbb P}^1$. In [@psw2], we will prove the above conjecture.* ## Arithmetic principle for HKL {#arith} For simplicity, we explain the principle for boundary only consisting of one prime divisor. Let $H_v=\Gamma \setminus \bigcup_{g \in \Gamma} {\mathcal{D}}_{g\cdot v}$ be the modified Heegner divisor on the locally symmetric variety of orthogonal type ${\mathcal F}$ and $\Delta(s)=\lambda+a(s) H_v$, up to scaling we may assume $s\in [0,1]\cap {\mathbb Q}, a(0)=0$. Suppose the section ring $$R(s):=\mathop{\bigoplus} \limits_{m} H^0({\mathcal F}, m l_s\cdot \Delta(s))$$ is finitely generated ${\mathbb C}$-algebra and thus ${\mathcal F}(s):= \mathop{\mathrm{Proj}}( R(s))$ defines a projective variety where $l_s\in {\mathbb Z}_{>0}$ is sufficiently divisible such that $l_s \Delta(s)$ is Cartier. **Definition 36**. *A self intersections $H_v^{(m)}$ for $H_v$ with depth $m\in {\mathbb Z}_{>0}$ in the sense of Looijenga is defined to be $$\Gamma \setminus \bigcup_{(v_1,\cdots,v_m) } {\mathcal{D}}_{v_1} \cap \cdots \cap {\mathcal{D}}_{v_m}$$ where $(v_1,\cdots,v_m)$ runs over all collection of $m$ linearly independent vectors in the $\Gamma$-orbit of $v$.* Let $m_v$ be the biggest depth of self-intersections for $H_v$ and there is a natural stratification (Looijenga's stratification for $H_v$) $$\label{Looijenga's stratification} H_v^{(m_v)} \subset H_v^{(m_v-1)}\subset \cdots \subset H_v^{(2)} \subset H_v^{(1)}=H_v.$$ A philosophy of HKL is that the birational maps of ${\mathcal F}(s)$ when varying $s$ from $0$ to $1$ should transform the stratification ([\[Looijenga\'s stratification\]](#Looijenga's stratification){reference-type="ref" reference="Looijenga's stratification"}) from the deepest stratum to the less deep one successively. If we expect the birational maps of ${\mathcal F}(s)$ can be obtained by running log MMP, write the restriction of $\Delta(s)$ on the expected flipped locus $H_v^{(n)}$ $$\Delta(s)|_{H_v^{(n)}}=(1-a_n(s)) \lambda+Z_n$$ If $Z_n$ is birationally contractible on $H_v^{(n)}$ (in particular, $Z_n$ is an extrmal in the effective cone of $H_v^{(n)}$), then $\Delta(s)|_{H_v^{(n)}}$ should lose positivity. Hence $1-a_n(s)=0$ and $s_n\in [0,1]$ such that $1-a_n(s_n)=0$ should be a wall. If $Z_n$ fails to be birationally contractible, then one need to modify the stratification ([\[Looijenga\'s stratification\]](#Looijenga's stratification){reference-type="ref" reference="Looijenga's stratification"}) from depth $n$ stratum. Based on this philosophy, [@FLLST Section 10] outlines an algorithm to find all the potential walls. In the §[5](#arithmeticstra){reference-type="ref" reference="arithmeticstra"}, we will apply this algorithm and combine the geometric degeneration of K3 surfaces with involution to give a explicit stratification on ${\mathcal F}_8$ and we conjecture it gives a complete description of HKL for moduli of del Pezzo pairs in case the of degree $8$. **Remark 37**. *Note that in the construction of the unigonal K3 surfaces, the order of the covering map and minimal resolution map can not be reversed.* # Arithmetic prediction for wall-crossings on ${\mathcal F}_8(s)$ {#arithmeticstra} ## Modular forms on locally symmetric varieties Let $\Lambda$ be an even lattice of signature $(2,n)$ containing $U^{\oplus 2}$ as a summand and $$\mathop{\mathrm{\mathrm{Sh}}}(\Lambda):=\widetilde{O}^+(\Lambda) \setminus {\mathcal{D}}_\Lambda.$$ In this subsection, we will apply the tool of modular form to study the Picard group $\mathop{\mathrm{Pic}}(\mathop{\mathrm{\mathrm{Sh}}}(\Lambda))_{\mathbb Q}$ of locally symmetric space $\mathop{\mathrm{\mathrm{Sh}}}(\Lambda)$, in particular for the $2$-elementary lattice. This builds on the work of Borcherds [@Bor], Bruinier [@bruinier] and Bergeron-Li-Millson-Moeglin [@BLMM]. In general, the Heegner divisor $H_{\beta,m}$ can be non-reduced and reducible. In the case that $\Lambda$ is a transcendental lattice of a K3 surface, it can be written as the sum of some irreducible NL divisor. Now we recall the theory of vector-valued modular forms. Let ${\mathbb C}[G(\Lambda)]$ be the complex vector space spanned by the discriminant group $G(\Lambda)$ and denote $\rm Mp_2({\mathbb Z})$ the metaplectic group. Then there is so-called Weil representation $$\rho_\Lambda: \rm Mp_2({\mathbb Z}) \rightarrow \mathop{\mathrm{\mathrm{GL}}}({\mathbb C}[G(\Lambda)]) .$$ Denote the $\mathop{\mathrm{Acusp}}(\rho_\Lambda,k)$ space of almost cuspital ${\mathbb C}[G(\Lambda)]$-valued modular forms of weight $k$, that is, the space of vector valued holomorphic functions $$f: \mathbb{H}\rightarrow {\mathbb C}[G(\Lambda)]$$ satisfying certain modularity axioms. **Theorem 38** ([@Bor], [@bruinier] and [@BLMM]). *Notation as above, then the Picard group $\mathop{\mathrm{\mathrm{Pic}}}(\mathop{\mathrm{\mathrm{Sh}}}(\Lambda)_{\mathbb C}$ is generated by Heegner divisors and there is an isomorphism of ${\mathbb C}$-vector space $$\mathop{\mathrm{\mathrm{Pic}}}(\mathop{\mathrm{\mathrm{Sh}}}(\Lambda)_{\mathbb C}\rightarrow \mathop{\mathrm{Acusp}}(\rho_\Lambda,1+\frac{n}{2})^\vee$$ by sending the Heegner divisor $H_{\xi,m}$ to the linear map $$f= \mathop{\sum} \limits_{m \in {\mathbb Q}, \xi \in G(\Lambda) } c_{l,\zeta}\cdot q^m e_\zeta \ \mapsto \ c_{m,\xi}$$ where $\{e_\xi\}$ the standard basis of ${\mathbb C}[G(\Lambda)]$ given by the elements $\xi$ of the discriminant group $G(\Lambda)$.* In the practice of computation, Raum ([@Rau16]) gives a algorithm based on Jacobi form for definite lattice which can be translated into a computer Program in $\mathbf{Sage}$. Let $V$ be definite even lattice of rank $N$ and $$\Gamma^J:=\mathop{\mathrm{\mathrm{SL}}}_2({\mathbb Z}) \ltimes {\mathbb Z}^2 \otimes V$$ the Jacobi group where semidirect product is defined by natural action of $\mathop{\mathrm{\mathrm{SL}}}_2({\mathbb Z})$ on ${\mathbb Z}^2$. Then $\Gamma^J$ acts on $\mathbb{H}\times V_{\mathbb C}$ by $$(\gamma, (\lambda,\mu))\cdot (\tau,Z)=( \gamma(\tau), \frac{z+\lambda \tau+\mu}{c\tau+d})$$ where $(\gamma, (\lambda,\mu))\in \Gamma$ and $(\tau.z)\in \mathbb{H}\times V_{\mathbb C}$. A Jacobi form of weight $k\in {\mathbb Z}$ and index $V$ is a holomorphic function $$f:\ \mathbb{H}\times V_{\mathbb C}\rightarrow {\mathbb C}$$ satisfying certain conditions (see [@Rau16 Definition 3.1]). Denote $J_{k,V}$ the space of Jacobi forms of weight $k\in {\mathbb Z}$ and index $V$. **Theorem 39**. *There is an isomorphism between the space of almost cuspidal forms and the spaces of Jacobi forms.* **Remark 40**. *If $\widetilde{O}^+(\Lambda)$ is replaced by other arithmetic group $\Gamma$, then the above theorems that works for $\mathop{\mathrm{\mathrm{Sh}}}(\Lambda)=\widetilde{O}^+(\Lambda) \setminus {\mathcal{D}}_\Lambda$ may fail for $\Gamma \setminus {\mathcal{D}}_\Lambda$.* ## Restriction of Heegner divisors Let $\Lambda:=U^{\oplus 2} \oplus \Lambda_0$ be an even lattice of signature $(2,n)$ where $\Lambda_0$ is negative definite. Let $v\in \Lambda$ be a primitive vector with $v^2<0$. Then $${\mathbb Z}v \oplus v^\perp \subset \Lambda$$ is an overlattice and the quotient group $\Lambda/({\mathbb Z}v \oplus v^\perp)$ is a finite group. Denote $N$ be the order of the finite group $\Lambda/({\mathbb Z}v \oplus v^\perp)$ , then $$N^2\cdot |G(\Lambda)|=|v^2|\cdot |\det(v^\perp)|.$$ Thus for any $w \in \Lambda$, there is unique decomposition $$\label{vectorprojection} m\cdot w=a_w\cdot v+ x_w \in {\mathbb Z}v \oplus v^\perp,\ a_w\in {\mathbb Z}$$ where $m\in {\mathbb Z}_{>0}$ is the order of image of $w$ in the quotient group $\Lambda/({\mathbb Z}v \oplus v^\perp)$. **Definition 41**. *We denote $\pi_{v^\perp}(w)$ the projection of vector $w\in \Lambda$ onto $v^\perp$ where $${\mathbb Z}_{>0} \cdot \pi_{v^\perp}(w) = ({\mathbb Q}_{>0}\cdot x_w) \cap v^\perp.$$* By the natural lattice embedding $v^\perp \hookrightarrow \Lambda$, we have natural morphism between two locally symmetric varieties $$\iota_v: \mathop{\mathrm{\mathrm{Sh}}}(v^\perp) \longrightarrow \Gamma_v \setminus D_v \longrightarrow \Gamma \setminus D$$ If the irreducible Heegner divisor $H_w$ on $\Gamma \setminus D$ is different from $H_v$, then we have the pullback formula $$\iota_v^\ast H_w=\mathop{\sum} \limits_{w' } H_{\pi_{v^\perp}(w')}$$ where $w'\in \Lambda$ runs over the $\Gamma$-orbit such that the rank $2$ sublattice $\langle v,w'\rangle$ of $\Lambda$ is negatively definite. **Proposition 42**. *There is commutative diagram $$\begin{tikzcd} \mathop{\mathrm{\mathrm{Sh}}}(v^\perp) \arrow[r] \arrow[d] & \arrow[d] \mathop{\mathrm{\mathrm{Sh}}}(\Lambda) \\ H_v \arrow[r,hook] & \Gamma \setminus {\mathcal{D}} \end{tikzcd}$$* *Proof.* Let $\Gamma_v:=\{ \gamma \in \Gamma | \gamma(v)=v \}$ the subgroup preserving integral vector $v$. Then $\Gamma_v \le \mathop{\mathrm{\mathrm{Stab}}}_\Gamma({\mathcal{D}}_v)$ and thus there is surjetive morphism $$\Gamma_v \setminus {\mathcal{D}}_v \rightarrow H_v=\mathop{\mathrm{\mathrm{Stab}}}_\Gamma({\mathcal{D}}_v) \setminus {\mathcal{D}}_v$$ since $H_v= \Gamma \setminus \bigcup_{\gamma \in \Gamma } {\mathcal{D}}_{\gamma \cdot v}$. Then the proposition follows from the claim: there is an embedding of arithmetic groups $\widetilde{O}(v^\perp) \le \Gamma_v \le O(v^\perp)$. Proof of the claim: Let $\Lambda_v:=v^\perp$ be the orthogonal complement of $v\in \Lambda$, then $\Lambda_{\mathbb C}={\mathbb C}v \oplus (\Lambda_v \otimes {\mathbb C})$. We can write $\gamma(v)=av+b w$ where $w\in \Lambda_v \otimes {\mathbb C}$ for any $\gamma\in \Gamma_v$. For any $x\in v^\perp$, $$0=\langle\gamma^{-1}x,v\rangle=\langle x,\gamma(v)\rangle=b\langle x,w\rangle$$ this implies $b=0$. In particular, $\gamma(v)=a v$ and $a\in {\mathbb Z}$ since $\gamma(v)\in \Lambda$ and $v$ is primitive. Observe that $\langle\gamma(v),\gamma(v)\rangle=|a|^2\langle v,v\rangle$ shows $|a|=1$. Since $\gamma \in \widetilde{O}^+(\Lambda)$, it has already preserved ${\mathcal{D}}^+_{\Lambda}$. On the contrary, any $\gamma \in \widetilde{O}^+(\Lambda)$ such that $\gamma(v) = \pm v$ preserves ${\mathcal{D}}^+_{v}$. Note that $\gamma(v) =-v$ is possible if $v^* = -v^* \in A_\Lambda$ and $v^2 <0$. In this way, we prove $$\Gamma_v = \{ \gamma \in \widetilde{O}^+(\Lambda)\ | \ \gamma (v)= \pm v \} \leq O^+(\Lambda_v).$$ Let $\gamma \in \widetilde{O}^+(\Lambda_v)$. Since $$\Lambda_v \oplus {\mathbb Z}v \subset \Lambda \subset \Lambda^* \subset \Lambda_v^* \oplus {\mathbb Z}v^* \, ,$$ we may first extend $\gamma$ to an automorphism of $\Lambda_v^* \oplus {\mathbb Z}v^*$ by setting $\gamma (v)= v$. By abuse of notation, we still denote it $\gamma$. For any element in $\Lambda \subset \Lambda_v^* \oplus {\mathbb Z}v^*$, we may write it as $xv+w$ where $w\in \Lambda_v^\ast, x\in \frac{1}{\mathrm{div}_\Lambda(v)}{\mathbb Z}$. Since $\gamma |_{\Lambda_v^\ast/\Lambda_v}=id$, there is a $u\in \Lambda_v$ such that $\gamma(w)=w+u$. Thus, $$\gamma (xv+w)=x\gamma(v)+ \gamma(w)= (xv+w)+u \in \Lambda$$ This shows $\gamma(\Lambda) \subset \Lambda$ and preserves orientation. Since $\gamma$ acts trivially on $({\mathbb Z}v^*/{\mathbb Z}v) \oplus (\Lambda_v^\ast/\Lambda_v)$ and $A_\Lambda \le ({\mathbb Z}v^*/{\mathbb Z}v) \oplus (\Lambda_v^\ast/\Lambda_v)$, it also acts trivially on $A_\Lambda$. It is not hard to show such an extension is unique. Thus, we get embedding $$\label{eq:gamma_v} \widetilde{O}^+(\Lambda_v) \hookrightarrow \Gamma_v \hookrightarrow O^+(\Lambda_v)$$ ◻ ## Arithmetic stratification on hyperelliptic divisor and unigonal divisor {#subsec: arithmetic strata} Let ${\mathcal F}= \Gamma \backslash {\mathcal{D}}_\Lambda$ be the locally symmetric space associated to the lattice $\Lambda = \Lambda_8 = U^2 \oplus E_8 \oplus E_7 \oplus A_1$ in section [2](#2-K3){reference-type="ref" reference="2-K3"}. By Nikulin's results (see [@Nikulin Theorem 1.14.4], also [@Dolgachev1983 Proposition 1.4.8]), there is a unique primitive embedding of $$\Lambda \hookrightarrow I\!I_{2,26}\cong U^{\oplus 2} \oplus E_8^{\oplus 3},\ \hbox{and}\ U_1\vcentcolon=(\Lambda)^{\perp}_{I\!I_{2,26}} \cong E_7 \oplus A_1\subset I\!I_{2,26} .$$ Inspired by the prediction in [@LaO18ii], we may add a root vector to the $E_7\oplus A_1$ to get $E_7\oplus A_2$ and continue the procedure. In this way, we have 1. the $A_n$ lattice towers $$\begin{split} E_7\oplus A_1 \subset E_7\oplus A_2 \subset E_7\oplus A_3 \subset \cdots \subset E_7\oplus A_5 \end{split}$$ 2. the $D_n$ lattice towers $$\begin{split} E_7\oplus D_3=E_7\oplus A_3 \subset E_7\oplus D_4 \subset \cdots \subset E_7\oplus D_{10} \end{split}$$ 3. the $E_n$ lattice towers $$\begin{split} E_7\oplus E_6 \subset E_7\oplus E_7 \subset E_7\oplus E_8 \end{split}$$ Suggested the computation in §[4.1](#hk3){reference-type="ref" reference="hk3"} and §[4.2](#uk3){reference-type="ref" reference="uk3"}, we also have modified towers of lattices, where $\diamond$ and $\circ$ are the new roots adding to the Dynkin diagram and $\diamond$ has intersection number $1$ with the fixed polarization class (omitted in the Dynkin diagram). 1. the $A_n'$ lattice towers are the complements of some hyperbolic lattices in K3 lattice, where the hyperbolic lattices are given by the following diagram $$\xymatrix{ & & & & \diamond \ar@{-}[d] \\ \bullet_{{n}} \ar@{-}[r] & \bullet_{{n-1}} \ar@{-}[r]& \cdots \ar@{-}[r] & \bullet_{4} \ar@{-}[r] & \ar@{-}[r] \bullet_3 & \ar@{-}[r] \bullet_2 & \bullet_1 }$$ Similarly, the $A_n''$ lattice towers are determined by the hyperbolic lattices whose diagrams are given by the following $$\xymatrix{ & & & & \diamond \ar@{-}[d] \\ \bullet_{{n}} \ar@{-}[r] & \bullet_{{n-1}} \ar@{-}[r]& \cdots \ar@{-}[r] & \bullet_{4} \ar@{-}[r] & \ar@{-}[r] \bullet_3 & \ar@{-}[r] \bullet_2 & \bullet_1 \ar@{-}[r] & \circ }$$ The $A_n'''$ lattice towers $$\label{fig:A_n'''} \xymatrix{ & \circ \ar@{-}[d] & & & \diamond \ar@{-}[d] \\ \bullet_{{n}} \ar@{-}[r] & \bullet_{{n-1}} \ar@{-}[r]& \cdots \ar@{-}[r] & \bullet_{4} \ar@{-}[r] & \ar@{-}[r] \bullet_3 & \ar@{-}[r] \bullet_2 & \bullet_1 \ar@{-}[r] & \circ }$$ 2. the $D_n'$ lattice towers are given by the complements of some hyperbolic lattices in K3 lattice where the hyperbolic lattices are given by the following diagram $$\xymatrix{ & & \bullet_{{n-1}} \ar@{-}[d] & & & & \\ \diamond \ar@{-}[r] & \bullet_{{n}} \ar@{-}[r] & \bullet_{{n-2}} \ar@{-}[r]& \cdots \ar@{-}[r] & \bullet_{4} \ar@{-}[r] & \ar@{-}[r] \bullet_3 & \ar@{-}[r] \bullet_2 & \bullet_1 }$$ These towers of lattices motivate us to define the following tower of locally symmetric varieties. **Definition 43**. 1. *Let $\Lambda_{A_n}$ be the orthogonal complement of $E_7\oplus A_{n}$ in the Borcherds lattice $I\!I_{2,26}$. The discriminant group $G(\Lambda_{A_n})={\mathbb Z}/2{\mathbb Z}\times {\mathbb Z}/(n+1){\mathbb Z}$, generated by $\xi,\zeta$ with $$\xi^2\equiv -\frac{1}{2} \,(\bmod\,2),\quad \zeta.\xi=0,\quad \zeta^2\equiv \frac{n}{n+1} \,(\bmod\,2).$$ Denote $\mathop{\mathrm{\mathrm{Sh}}}(A_n):= \widetilde{O}(\Lambda_{A_n})\setminus {\mathcal{D}}_{\Lambda_{A_n}}$ the associated $19-n$ dimensional locally symmetric spaces.* *If $n=5,6$, denote $\Lambda_{A_n}'$ the orthogonal complement of the Neron-Serveri lattice ([\[NSA\'\]](#NSA'){reference-type="ref" reference="NSA'"}) in the K3 lattice. We call it $A_n'$ tower. If $n\ge 7$, similarly we define $\Lambda_{A_n}'', \Lambda_{A_n}'''$ and $\mathop{\mathrm{\mathrm{Sh}}}(A_n'')$, $\mathop{\mathrm{\mathrm{Sh}}}(A_n''')$ via Neron-Serveri lattice ([\[NSA\'\'\]](#NSA''){reference-type="ref" reference="NSA''"}) and ([\[NSA\'\'\'\]](#NSA'''){reference-type="ref" reference="NSA'''"}) . We call them $A_n''$-tower and $A_n'''$-tower.* 2. *Let $\Lambda_{D_n}$ be the orthogonal complement of $E_7\oplus D_{n}$ in the Borcherds lattice $I\!I_{2,26}$. The discriminant group $$d(\Lambda_{D_n})= \left\{ \begin{array}{lc} {\mathbb Z}/2{\mathbb Z}\times {\mathbb Z}/4{\mathbb Z}, &\ n \ \hbox{odd}, \\ {\mathbb Z}/2{\mathbb Z}\times ({\mathbb Z}/2{\mathbb Z}\times {\mathbb Z}/2{\mathbb Z}), &\ n \ \hbox{even}. \\ \end{array} \right.$$ is generated by $\xi,\eta$ with $$\xi^2 = -\frac{1}{2} \,(\bmod\, 2), ~ \eta^2=\frac{n}{4} \,(\bmod\,2),~ \xi.\eta=0$$ for odd $n$ and generated by $\xi,\eta_1, \eta_2$ with $$\xi^2 = -\frac{1}{2} \,(\bmod\, 2), ~ \eta_1^2 = \eta_2^2=\frac{n}{4} \,(\bmod\,2),~ (\eta_1 + \eta_2)^2 =1,~\xi.\eta_i=0$$ when $n$ even. Denote $\mathop{\mathrm{\mathrm{Sh}}}(D_n):= \widetilde{O}(\Lambda_{D_n})\setminus {\mathcal{D}}_{\Lambda_{D_n}}$ the associated $19-n$ dimensional locally symmetric variety.* *For $n \ge 6$, we denote $\Lambda_{D_n}'$ the orthogonal complement of the Neron-Serveri lattice ([\[NSD\'\]](#NSD'){reference-type="ref" reference="NSD'"}) in the K3 lattice. We call it $D_n'$-tower and $\mathop{\mathrm{\mathrm{Sh}}}(D_n')$ the associated locally symmetric variety.* 3. *Let $\Lambda_{E_n}$ be the orthogonal complement of $E_7\oplus E_{n}$ in the Borcherds lattice $I\!I_{2,26}$. The discriminant group $$d(\Lambda_{E_n})={\mathbb Z}/2{\mathbb Z}\times {\mathbb Z}/(9-n){\mathbb Z}$$ is generated by $\xi,\eta_E$ with $$\xi^2 = -\frac{1}{2} \,(\bmod\, 2), ~ \xi.\eta_E=0,\ \ \eta_E^2=\begin{cases} \frac{4}{3} \,(\bmod\, 2), & n=6 \\ \frac{3}{2} \,(\bmod\, 2), & n=7 \\ \end{cases}$$ Denote $\mathop{\mathrm{\mathrm{Sh}}}(E_n):= \widetilde{O}(\Lambda_{E_n})\setminus {\mathcal{D}}_{\Lambda_{E_n}}$ the associated $19-n$ locally symmetric variety.* Induced by the primitive embeddings $\Lambda_{A_n} \subset \Lambda_{A_{n-1}},~ \Lambda_{D_n} \subset \Lambda_{D_{n-1}}$ and $\Lambda_{E_n} \subset \Lambda_{E_{n-1}}$, we have natural morphisms between the locally symmetric spaces $$\mathop{\mathrm{\mathrm{Sh}}}(A_n) \rightarrow \mathop{\mathrm{\mathrm{Sh}}}(A_{n-1}),~ \mathop{\mathrm{\mathrm{Sh}}}(D_n) \rightarrow \mathop{\mathrm{\mathrm{Sh}}}(D_{n-1}),~ \mathop{\mathrm{\mathrm{Sh}}}(E_n) \rightarrow \mathop{\mathrm{\mathrm{Sh}}}(E_{n-1})$$ which are just finite onto its images in general. The morphism $\mathop{\mathrm{\mathrm{Sh}}}(A_n) \rightarrow {\mathcal F}$ can be factored into a series of composition of morphism $\mathop{\mathrm{\mathrm{Sh}}}(A_n) \rightarrow \mathop{\mathrm{\mathrm{Sh}}}(A_{n-1})$ from lattice towers. Similar results hold for other lattice towers. This induces diagram ([\[diah\]](#diah){reference-type="ref" reference="diah"}) and ([\[diau\]](#diau){reference-type="ref" reference="diau"}). ### A-type tower and their modification **Definition 44**. *A primitive vector $v\in \Lambda_{A_n}$ is called* 1. *$A_{n+1}$-type if $$v^2=-(n+1)(n+2),\ \ v^\ast=\zeta$$ so that $v^\perp \cong \Lambda_{A_{n+1}}$. In this case, $[\Lambda_{A_n}: {\mathbb Z}v \oplus v^\perp]=n+2$ and $\mathrm{div}_{\Lambda_{A_n}}(v)=n+1$.* 2. *$D_{n+1}$-type if $$v^2= \begin{cases} -(n+1), & n\ \hbox{odd} \\ -4(n+1), & n \ \hbox{even} \end{cases}, \ \ \ v^\ast=2\zeta$$ and $v^\perp \cong \Lambda_{D_{n+1}}$. In this case, $$= \begin{cases} 2, & \ n\ \hbox{odd} \\ 4 , &\ n \ \hbox{even} \end{cases}$$* 3. *$A_{n}'$ type if $n=5,6$ $$v^2= \begin{cases} -4, & n=5\ \hbox{odd} \\ -154, & n=6 \ \hbox{even} \end{cases},\ \ v^\ast= \xi +3\zeta$$ so that $v^\perp \cong \Lambda_{A_n'}$ and $$= \begin{cases} 2, & \ n=5\ \hbox{odd} \\ 11 , &\ n=6 \ \hbox{even} \end{cases}$$ and $\mathrm{div}(v)=2$ for $n=5$.* 4. *unigonal type if $v^2=-2$ and $v^\ast =\xi$ so that sublattice $v^\perp$ has discriminant $(-1)^n\cdot (n+1)$ and $[\Lambda_{A_n}: {\mathbb Z}v \oplus v^\perp]=2$, $\mathrm{div}(v)=2$.* 5. *$E_6$-type for $n=5$ and $v^2=-2,\ v^\ast=3\zeta$ so that $v^\perp= \Lambda_{E_6}$. In particular, $[\Lambda_{A_5}: {\mathbb Z}v \oplus v^\perp]=2$ and $\mathrm{div}(v)=2$.* Denote the divisors $$\begin{split} & H_{\rm {n}}(\Lambda_{A_n}):=(H_{v})^{red} \ \hbox{for}\ v\ \hbox{nodal vector}\ ( \hbox{nodal divisor}) \\ &H_{\rm u}(\Lambda_{A_n}):= (H_{\xi,-\frac{1}{4}})^{red}\ ( \hbox{unigonal divisor}) \\ & H_{A_{n+1}}(\Lambda_{A_n}):= (H_{\zeta,-\frac{n+2}{2(n+1)}})^{red} \\ & H_{D_{n+1}}(\Lambda_{A_n}):= (H_{2\zeta,-\frac{2}{n+1}})^{red} \\ & H_{A_{n}'}(\Lambda_{A_n}):= (H_{\xi+3\zeta,-\frac{17-n}{4(n+1)}})^{red},\ \ n=5,6 \\ & H_{E_6}:=(H_{3\zeta,-\frac{1}{4}})^{red} \text{~for~} \mathop{\mathrm{\mathrm{Sh}}}(A_5) \end{split}$$ on $\mathop{\mathrm{\mathrm{Sh}}}(A_n)$ where $()^{red}$ means taking reduced part. Note that $H_{D_{n+1}}(\Lambda_{A_n})=0$ for $n=1$ and $$H_h=H_{A_{2}}(\Lambda_{A_1}),\ \ H_{D_{3}}(\Lambda_{A_2})=H_{A_{3}}(\Lambda_{A_2}).$$ By the triangular relation ([\[trirel\]](#trirel){reference-type="ref" reference="trirel"}) , we have $$H_{0,-1}(\Lambda_{A_n})= 2(H_{\rm {n}}(\Lambda_{A_n})+H_{\rm u}(\Lambda_{A_n}))$$ and $H_{\rm u}(\Lambda_{A_n})$ is irreducible. **Proposition 45**. *The Picard group $\mathop{\mathrm{Pic}}(\mathop{\mathrm{\mathrm{Sh}}}(A_n))_{\mathbb Q}$ is generated by $H_{\rm {n}}(\Lambda_{A_n})$, $H_{\rm u}(\Lambda_{A_n})$, $H_{A_{n+1}}(\Lambda_{A_n})$ and $H_{D_{n+1}}(\Lambda_{A_n})$. The Picard number of $\mathop{\mathrm{\mathrm{Sh}}}(A_n)$ are giving in the following Table [4](#Picard of Atower){reference-type="ref" reference="Picard of Atower"}* *$n$* *$2$* *$3$* *$4$* *$5$* *$6$* *$7$* *$8$* *$9$* *$10$* --------------------------------------------------- ------- ------- ------- ------- ------- ------- ------- ------- -------- *$\rho(\mathop{\mathrm{\mathrm{Sh}}}(A_n)^\ast)$* *$3$* *$4$* *$4$* *$5$* *$5$* *$6$* *$5$* *$5$* *$4$* : *Picard numbers of $A_n$-towers* *Moreover if $n \le 4$, there is a relation $$\label{hodgerel1} \begin{split} (75+\frac{n(n+1)}{2}) \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(A_n)}=&H_{\rm {n}}(\Lambda_{A_n})+57 H_{\rm u}(\Lambda_{A_n})+(n+1)\cdot H_{A_{n+1}}\\ &+(1-\delta_{2,n})\frac{n(n+1)}{2}H_{D_{n+1}}(\Lambda_{A_n}) \end{split}$$ where $\delta_{i,j} = 1$ for $i=j$ and $0$ for $i\neq j$ the Kronecker symbol.* *If $n=5,6$, in addition to the relation $$\begin{split} 90\lambda_{\mathop{\mathrm{\mathrm{Sh}}}(A_5)}=H_{\rm {n}}(\Lambda_{A_5})+57H_{\rm u}+6H_{A_6}+15H_{D_6}+20H_{E_6}\\ 96\lambda_{\mathop{\mathrm{\mathrm{Sh}}}(A_6)}=H_{\rm {n}}(\Lambda_{A_6})+57H_{\rm u}+7H_{A_7}+21H_{D_7}+35H_{E_7}, \end{split}$$ there is a second relation $$\begin{split} H_{A_6}=2\lambda_{\mathop{\mathrm{\mathrm{Sh}}}(A_5)}-4H_{\rm u}+2H_{D_6}+2H_{E_6}+2H_{A_5'} %\\H_{A_7}=3\lambda_{\sh(A_6)}-2H_{\rm u}+H_{D_6}+2H_{A_6'} \end{split} %2\lambda_{\sh(A_5)}=H_{A_6}-2H_{E_6}+2H_{A_5'}-2H_{D_6}-H_{\xi^{(5)}_1+ 3\xi^{(5)}_2}$$ In particular, $H_{A_6}$ is big on $\mathop{\mathrm{\mathrm{Sh}}}(A_5)$ since the divisor $H_{\rm u},H_{D_6}, H_{E_6},H_{A_5'}$ are birationally contractible.* *Proof.* By putting the lattice $L=\Lambda_{A_n}$ into the Software $\mathbf{Sage}$ package [weilrep](https://github.com/btw-47/weilrep)[^4], then we get the Picard numbers of $\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_n})$ directly by Theorem [Theorem 38](#modularform1){reference-type="ref" reference="modularform1"} and Theorem [Theorem 39](#modularform2){reference-type="ref" reference="modularform2"}. For the relation ([\[hodgerel1\]](#hodgerel1){reference-type="ref" reference="hodgerel1"}), we explain how to deduce them by an example. Suppose $L=\Lambda_{A_1}$, recall its discriminant group is isomorphic to ${\mathbb Z}\eta \oplus {\mathbb Z}\xi$. Denote the basis $$\vec{e}_1=e_0, \ \vec{e}_2=e_\zeta,\ \vec{e}_3=e_\xi,\ \vec{e}_4=e_{\zeta+\xi}$$ for the $4$-dimensional vector space ${\mathbb C}[A_L]$. Here $e_v$ means the basis associated with an element in the discriminant group. The output of $\mathbf{Sage}$ [^5] is the basis of vectored modular form with $q$ expansion $$\label{qexan} \begin{split} f_0=& (1-152q+o(q))\cdot \vec{e}_1+ o(q) \cdot \vec{e}_2+o(q) \cdot \vec{e}_3+(-112q+o(q)) \cdot \vec{e}_4 \\ f_1=&(-56q+o(q))\cdot \vec{e}_1+ o(q) \cdot \vec{e}_2+(q^{\frac{1}{4}}+o(q) )\cdot \vec{e}_3+(56q+o(q)) \cdot \vec{e}_4 \\ f_2=&(-2q+o(q))\cdot \vec{e}_1+(q^{\frac{3}{4}}+o(q)) \cdot \vec{e}_2+ o(q) \cdot \vec{e}_3+(2q+o(q)) \cdot \vec{e}_4 \end{split}$$ By theorem [Theorem 38](#modularform1){reference-type="ref" reference="modularform1"}, the above vectored modular form gives relation $$152 \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(A_1)}=H_{0,-1}+56H_{\xi,-\frac{1}{4}}+2H_{\zeta,-\frac{3}{4}}.$$ Note that $H_{0,-1} = 2H_n +2H_u, H_{\xi,-\frac{1}{4}}=2H_u, H_{\zeta,-\frac{3}{4}}=2H_{A_2}$. This proves relation ([\[hodgerel1\]](#hodgerel1){reference-type="ref" reference="hodgerel1"}) for $n=1$. The same arguments will prove the remaining cases and we omit the repeated computation. ◻ **Remark 46**. *The coefficient of $\lambda$ in relation ([\[hodgerel1\]](#hodgerel1){reference-type="ref" reference="hodgerel1"}) is just the weight of the modular form $\Phi_\Lambda$ constructed via Borcherds product in [@Bor] associated to the lattice $\Lambda=( E_7 \oplus A_n)^\perp_{I\!I}$, i.e., $$12+ \frac{1}{2}R(E_7\oplus A_n))=12+ \frac{126+n(n+1)}{2}=75+\frac{n(n+1)}{2}.$$ Here $12$ comes from the fact that $\Phi_\Lambda$ is the pullback of a weight $12$ modular form $\Phi_{12}$ on ${\mathcal{D}}_{I\!I}$.* **Proposition 47**. *Let $$\iota : \mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_{n+1}}) \rightarrow \mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_n})$$ be the natural morphism induced from lattice embedding, then for $n \le 4$, we have* 1. *$\iota^\ast H_{\rm {n}}(\Lambda_{A_n})=H_{\rm {n}}(\Lambda_{A_{n+1}})+H_{A_{n+2}}(\Lambda_{A_{n+1}})$* 2. *$\iota^\ast H_{\rm u}(\Lambda_{A_n})=H_{\rm u}(\Lambda_{A_{n+1}})$ and $\iota^\ast H_{D_{n+1}}(\Lambda_{D_n})=H_{D_{n+2}}(\Lambda_{D_{n+1}})$* *In particular, we have $$\label{adjuctionAn} \iota^\ast H_{A_{n+1}}= \begin{cases} - \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_{n+1}})}+H_{A_{n+2}}(\Lambda_{A_{n+1}}), & n= 1\\ - \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_{n+1}})}+H_{A_{n+2}}(\Lambda_{A_{n+1}})+2H_{D_{n+2}}(\Lambda_{A_{n+1}}), & n= 2\\ - \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_{n+1}})}+H_{A_{n+2}}(\Lambda_{A_{n+1}})+H_{D_{n+2}}(\Lambda_{A_{n+1}}), & n= 3\\ -\lambda_{\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_5})}+H_{A_6}(\Lambda_{A_5})+H_{D_6}(\Lambda_{A_5}) +4H_{E_6}(\Lambda_{A_5}),& n=4 . \end{cases}$$* *Proof.* We first show the pullback formula for nodal divisor $H_{\rm {n}}(\Lambda_{A_n})$. Fixed a $A_{n+1}$-type vector $v\in \Lambda_{A_n}$ defined in Definition [Definition 44](#Avector){reference-type="ref" reference="Avector"}. Let $w$ be a nodal vector ( i.e., $w^2=-2,\ \mathrm{div}(w)=1$) in $\Lambda_{A_n}$ such that $\langle v,w\rangle$ is negatively definite. We claim the projection $\pi_{v^\perp}(w)\in v^\perp=\Lambda_{A_{n+1}}$ of $w$ is either nodal or $A_{n+2}$-type. By the projection decomposition $m\cdot w=a_w\cdot v+ \pi_{v^\perp}(w)$ in ([\[vectorprojection\]](#vectorprojection){reference-type="ref" reference="vectorprojection"}), $$-2m^2=-(n+1)(n+2)a_w^2+\pi_{v^\perp}(w)^2$$ Since negativity of sublattice $\langle v,w\rangle$ is equivalent to $\pi_{v^\perp}(w)^2<0$, we have either $a_w^2=0$ or $a_w^2=1$ as $m^2 \le N^2=(n+2)^2$. If $a_w^2=0$, then $m\cdot w=\pi_{v^\perp}(w) \in v^\perp$. Actually, $m=1$ since $v^\perp \hookrightarrow \Lambda_{A_n}$ is a primitive embedding. So in this case we obtain a nodal vector $\pi_{v^\perp}(w)$ in $v^\perp$. If $a_w^2=1$, then the only possibility for $m$ is $m=N=n+2$ and thus $$\pi_{v^\perp}(w)^2=-2(n+2)^2+(n+1)(n+2)=-(n+2)(n+3)$$ Moreover, the divisibility of $\pi_{v^\perp}(w)$ is. Thus, $\pi_{v^\perp}(w)\in v^\perp=\Lambda_{A_{n+1}}$ is a $A_{n+2}$-type vector. This finishes the proof of the claim. The claim implies $$\iota^\ast H_{\rm {n}}(\Lambda_{A_n})=a_{\rm n}\cdot H_{\rm {n}}(\Lambda_{A_{n+1}})+a_{A_{n+1}}\cdot H_{A_{n+2}}(\Lambda_{A_{n+1}})$$ To finish the proof of (1), it remains to show coefficient $a_{\rm n}=1$ and $a_{A_{n+1}}=1$. We follow the strategy [@LO19 Section 5.2] and [@FLLST Section 8.4] by computing the intersection multiplicity of germs at a very general point in $\iota( H_{\rm {n}}(\Lambda_{A_n})$. Let $[z] \in {\mathcal{D}}_{\Lambda_{A_n}} \cap v^\perp$ very general point so that there is isomorphism of local germs $$\big (\widetilde{O}(\Lambda_{A_n}) \setminus {\mathcal{D}}_{\Lambda_{A_n}} , \widetilde{O}(\Lambda_{A_n})[z] \big )\ \cong\ \big ( {\mathcal{D}}_{V^\perp},[z] \big ) \times \big ( G \setminus V_{\mathbb C},0 \big )$$ where $G=stab([z])/\{\pm Id\}$ and $V=\text{\rm span}_{\mathbb Q}\{v,w\} \cap \Lambda_{A_n}$ is the rank $2$ sublattice. Let $x_1,x_2$ be the coordinate under the integral basis of $V$. Then the gram matrix of $V$ is the following $$\left( \begin{array}{cc c } -(n+1)(n+2) & a \\ a & -2 \\ \end{array}\right)$$ where $a=0$ if the projection $\pi_{v^\perp}(w)$ of the nodal vector $w\in \Lambda_{A_n}$ onto $v^\perp=\Lambda_{A_{n+1}}$ is still nodal type and $a=$ if $\pi_{v^\perp}(w)$ is a $A_2$-type. Thus by [@FLLST Example 8.13], $a_{\rm n}=1$ and $a_{A_2}=1$. The similar argument works for the pullback of unigonal type divisors $H_{\rm u}(\Lambda_{A_n})$ and $D_{n+1}$-type divisors $H_{D_{n+1}}(\Lambda_{D_n})$ and thus proves (2). We leave the details for these two cases to the interested readers. Finally combining the above pullback results in (1) and (2), the pullback formula ([\[adjuctionAn\]](#adjuctionAn){reference-type="ref" reference="adjuctionAn"}) follows from the difference of Hodge relations in ([\[hodgerel1\]](#hodgerel1){reference-type="ref" reference="hodgerel1"}). ◻ **Remark 48**. *If $\iota$ is a closed embedding and the image $H_{A_{n}}$ is a normal divisor on $\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_n})$, one can also compute the pullback $\iota^\ast H_{A_{n}}$ directly via adjunction formula $$\iota^\ast (K_{\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_n})}+H_{A_{n}})=K_{\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_{n+1}})}$$ combined with canonical bundle formula of $\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_n})$ $$K_{\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_n})}=(19-n) \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_n})}-\frac{1}{2}B_{\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_n})}.$$ In this ideal situation, $\iota^\ast H_{A_{n}}=-\lambda_{\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_n})}+\frac{1}{2}(\iota^\ast B_{\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_n})}-B_{\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_{n+1}})} )$. But in general, it is hard to check the normality and there are also examples that $\iota$ fails to be a closed embedding.* Before we step into the deeper modifications and deal with the more complicated discriminant groups, we unify the notation here. Let $$G = ({\mathbb Z}/ n_1{\mathbb Z}) \times ({\mathbb Z}/n_2{\mathbb Z}) \times \cdots \times ({\mathbb Z}/n_k {\mathbb Z}),$$ we denote the $(\Bar{i_1},\Bar{i_2},\cdots,\Bar{i_k})$ the corresponding element in $G$. The discriminant group of $\Lambda_{A_n}'$ is given by $$G(\Lambda_{A_n}')=\begin{cases} {\mathbb Z}/ 12{\mathbb Z}& n=5, \\ {\mathbb Z}/ 11{\mathbb Z}& n=6 \end{cases}$$ **Definition 49**. *We call* 1. *a primitive vector $v \in \Lambda_{A_5}'$ is $A_6'$-type if $v^2=-132$ and so that $v^\perp = \Lambda_{A_6}'$ and $[\Lambda_{A_5}': {\mathbb Z}v \oplus v^\perp]=11$.* 2. *a primitive vector $v \in \Lambda_{A_6}'$ is $A_7'$-type if $v^2=-110$ and so that $v^\perp = \Lambda_{A_7}'$ and $[\Lambda_{A_6}': {\mathbb Z}v \oplus v^\perp]=10$.* 3. *a primitive vector $v \in \Lambda_{A_6}'$ is $A_6''$-type if $v^2=-88$ and so that $v^\perp = \Lambda_{A_6}''$ and $[\Lambda_{A_6}': {\mathbb Z}v \oplus v^\perp]=8$.* Define the Heegner divisors $$\begin{aligned} H_n &:= (H_{\bar{0},-1})^{red}, & H_{A_6'}(\Lambda_{A_5'})&:=H_{\bar{1},-\frac{11}{24}}, & H_{\rm u'}(\Lambda_{A_5'})&:=H_{\Bar{3},-\frac{1}{8}} \\ H_{A_6''}(\Lambda_{A_6'})&:=H_{\bar{1},\,-\frac{4}{11}},& H_{\rm u'(\Lambda_{A_6'})}&:=H_{\bar{3},-\frac{3}{11}},& H_{A_7'}(\Lambda_{A_6'})&:=H_{\bar{2},\,-\frac{5}{11}}\end{aligned}$$ on $\mathop{\mathrm{\mathrm{Sh}}}(A_5')$ and $\mathop{\mathrm{\mathrm{Sh}}}(A_6')$. **Proposition 50**. *The Picard number of $\mathop{\mathrm{\mathrm{Sh}}}(A_n')$ is given in Table [5](#Picard of A'tower){reference-type="ref" reference="Picard of A'tower"}.* *$n$* *$5$* *$6$* *$7$* ---------------------------------------------------- ------- ------- ------- -- -- -- -- -- -- *$\rho(\mathop{\mathrm{\mathrm{Sh}}}(A_n')^\ast)$* *$4$* *$4$* *$4$* : *Picard numbers of $A_n'$-towers* *Moreover, there are the first relations $$\label{relaA'1} \begin{split} 90 \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(A_5')}=H_{\rm n}+14 H_{A_6'}+13 H_{\bar{4},-\frac{1}{3}}+ 78 H_{\Bar{3},-\frac{1}{8}} \\ 89 \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(A_6')}=H_{\rm n}+91H_{\bar{5},-\frac{1}{11}}+28 H_{\rm u'}+15 H_{A_6''} \end{split}$$ and the second relations $$\label{relaA'2} \begin{split} \lambda= H_{A_7'}-H_{\bar{3},-\frac{3}{11}}-H_{A_6''} % H_{A_7'}=&\lambda+H_{-\frac{3}{11}}+H_{A_6''} \end{split}$$ In particular, $H_{A_7'}$ is big on $\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_6'})$ if $H_{\bar{3},-\frac{3}{11}}$ and $H_{A_6''}$ are birationally contractible on $\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_6'})$.* Denote $$\label{morpull} j: \ \mathop{\mathrm{\mathrm{Sh}}}(A_5') \longrightarrow \mathop{\mathrm{\mathrm{Sh}}}(A_5),\ \ \ \iota': \mathop{\mathrm{\mathrm{Sh}}}(A_6') \longrightarrow \mathop{\mathrm{\mathrm{Sh}}}(A'_{5})$$ **Proposition 51**. *The morphism $j$ and $\iota'$ in ([\[morpull\]](#morpull){reference-type="ref" reference="morpull"}) will induce the pullback $$j^\ast: \mathop{\mathrm{Pic}}(\mathop{\mathrm{\mathrm{Sh}}}(A_5)) \rightarrow \mathop{\mathrm{Pic}}(\mathop{\mathrm{\mathrm{Sh}}}(A_5')),\; \iota'^\ast: \mathop{\mathrm{Pic}}(\mathop{\mathrm{\mathrm{Sh}}}(A_{5}')) \rightarrow \mathop{\mathrm{Pic}}(\mathop{\mathrm{\mathrm{Sh}}}(A_6')).$$ Moreover, the following pullback formulas hold* 1. *$j^\ast H_{\rm n}= H_{\rm n}+ H_{\Bar{3},-\frac{1}{8}} + \frac{1}{2}H_{\bar{6},\frac{1}{2}}$, $\ j^\ast H_{A_6}=2H_{A_6}'$* 2. *$j^\ast H_{D_6}=0,\ \ \ \ j^\ast H_{E_6}=H_{\bar{3},-\frac{1}{8}}$* 3. *$\iota'^\ast H_{\rm n}=H_{\rm n}+H_{A_7'}$* *In particular, the above will imply $$\label{pbA6'} \iota'^\ast H_{A_6'}=-\lambda+H_{A_7'}+Z$$ where $Z$ is linear combination Heegner divisors $\iota'^{\ast}H_{\bar{4},-\frac{1}{3}}, \iota'^{\ast}H_{\bar{3},-\frac{1}{8}}, H_{\bar{3},-\frac{3}{11}}, H_{\bar{5},-\frac{1}{11}}$* *Proof.* We first deal with the pullback $j^\ast$. Fixed a $A_5'$-type vector $v \in \Lambda_{A_5}$, then $v^\perp \cong \Lambda_{A_5}'$ and $[\Lambda_{A_5}: {\mathbb Z}v \oplus v^\perp]= 2$. Let $w$ be $A_6$-type vector, i.e., $w^2=-42$ and $\mathrm{div}(w)=6$. Let $x_w:=\pi_{v^\perp}(w)$ the projection on $v^\perp=\Lambda_{A_5}'$. Then $mw=av+x_w$ for $m=1$ or $m=2$. Note that $m \langle w,x_w \rangle =x_w^2$ implies $x_w^2$ is also a multiple of $\mathrm{div}(w)=6$. Then the only possibility is $m=2$ and $|a|=3$ by considering the divisibility of the vectors $v$ and $w$, and their corresponding elements in the discriminant group. In this case, $x_w^2=-132$ and $x_w$ is $A_6'$-type vector. This shows $j^\ast H_{A_6}$ is supported on $H_{A_6'}$. If $w$ is a nodal vector, then the projection $x_w$ is either nodal vector or primitive vector with $x_w^2=-4$ ($x_w=2w\pm v$) and $\mathop{\mathrm{disc}}((x_w)_{v^\perp}^\perp)=2$ or $4$. This proves (1). Similarly, if $w$ is a $D_6$-type vector, consider the projection $x_w$ of $w$ one can see that the intersection of $H_{D_6} \cap H_{A_5'} = \emptyset$ set-theoretically in $\mathop{\mathrm{\mathrm{Sh}}}(A_5)$. If $w$ is a $E_6$-type vector, then $x_w^2=-4$ with $\mathrm{div}(x_w) =4$. This proves $(2)$. Now we consider the pullback $\iota'^\ast$. Fixed a $A_6'$-type vector $v \in \Lambda_{A_5'}$ so . Let $w \in \Lambda_{A_5'}$ be a nodal vector. Arguments as before will show the projection $x_w$ of $w$ on $v^\perp=\Lambda_{A_6'}$ is either nodal vector ($x_w=w$) or $A_7'$-type vector ($x_w=-11w\pm v$) on $v^\perp=\Lambda_{A_6'}$. The multiplicity of $H_{\rm n}$ and $H_{A_7'}$ on $\iota'^\ast H_{\rm n}$ are argued as in the previous. This shows (3). For the last assertion, by 1st relation in ([\[relaA\'1\]](#relaA'1){reference-type="ref" reference="relaA'1"}) and 2nd relation ([\[relaA\'2\]](#relaA'2){reference-type="ref" reference="relaA'2"}), we have $$(89+15)\lambda= H_{\rm n} +15H_{A_7'}+13H_{\bar{3},-\frac{3}{11}} +91 H_{\bar{5},-\frac{1}{11}}$$ in order to replace $H_{A_6}''$ in ([\[relaA\'1\]](#relaA'1){reference-type="ref" reference="relaA'1"}). Then again using the difference of the above relation with pullback first identity in relation ([\[relaA\'1\]](#relaA'1){reference-type="ref" reference="relaA'1"}), one obtains ([\[pbA6\'\]](#pbA6'){reference-type="ref" reference="pbA6'"}). ◻ **Remark 52**. *Indeed $H_{A_6'}$ is a movable divisor. This can be shown from the K-moduli side by [@psw2] but not from pure arithmetic computation. We only give a heuristic reason from an arithmetic perspective. Indeed, the pullback $\iota'^{\ast}H_{\Bar{4},-\frac{1}{3}}, \iota'^{\ast}H_{\Bar{3},-\frac{1}{8}}$ can also be computed explicitly as Heegner divisors on $\mathop{\mathrm{\mathrm{Sh}}}(A_5')$. But as their discriminant is less than that of divisor $H_{A_6'}$, then they should be birationally contractable. Thus these Heegner divisors will be viewed as irrelevant. For simplicity of computations, we will denote them $Z_{\mathop{\mathrm{\mathrm{Sh}}}}$ on $\mathop{\mathrm{\mathrm{Sh}}}$.* The discriminant group of $\Lambda_{A_n}''$ is given by $$G(\Lambda_{A_n}'')=\begin{cases} {\mathbb Z}/ 8 {\mathbb Z}, & n=6, \\ {\mathbb Z}/ 5 {\mathbb Z}, & n=7,\\ {\mathbb Z}/ 2 {\mathbb Z}, & n=8. \end{cases}$$ **Definition 53**. *We call* 1. *a primitive vector $v \in \Lambda_{A_6}''$ is $A_7''$-type if $v^2=-40$ and so that $v^\perp = \Lambda_{A_7}''$ and $[\Lambda_{A_6}'': {\mathbb Z}v \oplus v^\perp]=5$.* 2. *a primitive vector $v \in \Lambda_{A_7}''$ is $A_8''$-type if $v^2=-10$ and so that $v^\perp = \Lambda_{A_8}''$ and $[\Lambda_{A_7}': {\mathbb Z}v \oplus v^\perp]=2$.* *Let $H_{A''_7} \vcentcolon=H_{\bar{1},-\frac{5}{16}}$ on $\mathop{\mathrm{\mathrm{Sh}}}(A''_6)$ and $H_{A''_8} \vcentcolon=H_{\bar{1},-\frac{1}{5}}$ on $\mathop{\mathrm{\mathrm{Sh}}}(A''_7)$.* **Proposition 54**. *The Picard number of $\mathop{\mathrm{\mathrm{Sh}}}(A_n'')$ are given by $$\rho(\mathop{\mathrm{\mathrm{Sh}}}(A_n'')^\ast)=\begin{cases} 3 & n= 6\\ 2 & n=7 \end{cases}$$ and there are relations $$\label{relaa'''} \begin{split} & 104 \lambda = H_{\rm n}+28 H_{\bar{2},-\frac{1}{4}}+ 16 H_{A_7''} \ \ \ \hbox{on} \ \ \mathop{\mathrm{\mathrm{Sh}}}(A_6'') \\ & 120 \lambda=H_{\rm n}+45 H_{A_8''} \ \ \ \hbox{on} \ \ \ \mathop{\mathrm{\mathrm{Sh}}}(A_7'') \end{split}$$* **Proposition 55**. *Let $j': \mathop{\mathrm{\mathrm{Sh}}}(A_6'') \longrightarrow \mathop{\mathrm{\mathrm{Sh}}}(A_6')$ and $\iota'': \mathop{\mathrm{\mathrm{Sh}}}(A_7'') \longrightarrow \mathop{\mathrm{\mathrm{Sh}}}(A_{6}'')$ be the natural morphism. Then we have* 1. *$j^{'\ast} H_{\rm n}= H_{\rm n} + H_{\bar{2},-\frac{1}{4}}$,  $j^{'\ast} H_{A_7'}= H_{\bar{2},-\frac{1}{4}} + H_{A_7''}$* 2. *$\iota''^\ast H_{\rm n}= H_{\rm n} + H_{A_{8}''}$,  $\iota''^\ast H_{A_{7}''}=-\lambda+Z'$ where $Z'$ is linear combination of Heegner divisors $\iota^{''\ast} H_{\bar{2},-\frac{1}{4}}$ and $H_{A_8''}$ .* *Proof.* Fixed $v\in \Lambda_{A_6'}$ a $A_6''$-type vector in the Definition [Definition 49](#1stmv){reference-type="ref" reference="1stmv"}. If $w$ is a nodal vector, then the projection $\pi_{v^\perp}(w)$ can be of the following forms: - $\pi_{v^\perp}(w) = w$. - $\pi_{v^\perp}(w) = 8w \pm v$, $\pi_{v^\perp}(w)^2 = -40, \mathrm{div}(\pi_{{v}^\perp}(w))=8$. If $w$ is a $A_7'$-type vector, then the projection $\pi_{v^\perp}(w)$ can be of the following forms: - $11\pi_{v^\perp}(w) = 4w \pm 3v$, $\pi_{v^\perp}(w)^2 = -8, \mathrm{div}(\pi_{{v}^\perp}(w))=4$. - $11\pi_{v^\perp}(w) = 8w \pm 5v$, $\pi_{v^\perp}(w)^2 = -40, \mathrm{div}(\pi_{{v}^\perp}(w))=8$. Fixed $v\in \Lambda_{A_6''}$ a $A_7''$-type vector and let $w\in \Lambda_{A_6''}$ be a nodal vector, then it is easy to check the projection of $w$ onto $v^\perp$ is either a nodal vector or $A_8''$-type vector on $v^\perp$. By taking difference of the relation in ([\[relaa\'\'\'\]](#relaa'''){reference-type="ref" reference="relaa'''"}), we get $$\begin{split} -32\lambda=\iota^{''\ast} ( H_{\rm n}+56H_{\bar{2},-\frac{1}{4}}+ 32 H_{A_7''})-(H_{\rm n}+90 H_{A_8''})=32\iota^{''\ast} ( H_{A_7''})+56 \iota^{''\ast} H_{\bar{2},-\frac{1}{4}}-90 H_{A_8''} \end{split}$$ This finishes the proof of (2). ◻ ### D-type tower and their modification **Definition 56**. *A primitive vector $v\in \Lambda_{D_n}$ is called* 1. *unigonal type if $$v^2=-2 , \mathrm{div}(v)=2, v^* =\xi \in G(\Lambda_{D_{n}})$$ so that $(v^\perp)^\perp_{I\!I_{2,26}} \cong E_8\oplus D_n$.* 2. *$D_{n+1}$-type if $$v^2=-4,\ \ v^\ast=\tilde{\eta} \vcentcolon= \begin{cases} 2\eta &\ n\ \hbox{odd} \\ \eta_1+\eta_2 &\ n \ \hbox{even} \end{cases}$$ so that $v^\perp \cong \Lambda_{D_{n+1}}$ and $[\Lambda_{D_n}:{\mathbb Z}v \oplus v^\perp]=2$.* 3. *$D_{n}'$ type if $$v^2=\begin{cases} -12, & n=7\\ -2 , & n=8\\ -4, & n=9 \end{cases},\ \ v^\ast=\eta' \vcentcolon= \begin{cases} \xi+\eta &\ n\ \hbox{odd} \\ \xi+\eta_1 &\ n \ \hbox{even} \end{cases}$$ so that $v^\perp \cong \Lambda_{D_{n}}'$.* 4. *$E_{n+1}$ type if $n \ge 5$ and $$v^2=\begin{cases} -12 , &\ n =5, \\ -2 , &\ n =6, \\ -4 , &\ n =7 \end{cases} , \ \ v^\ast=\eta_u \vcentcolon= \begin{cases} \eta &\ n\ \hbox{odd} \\ \eta_1 \text{~or~}\eta_2 &\ n \ \hbox{even} \end{cases}$$ so that $v^\perp \cong \Lambda_{E_{n+1}}$ and $$[\Lambda_{D_n}:{\mathbb Z}v \oplus v^\perp]= \begin{cases} 8-n, & \ n\ \hbox{odd} \\ 1, & \ n \ \hbox{even} \end{cases} .$$* Denote the divisors $$\begin{aligned} {2} H_{\rm {n}}(\Lambda_{D_n})&:= (H_v)^{red}\textbf{} \text{~for nodal vector~} v ,\quad & H_{D_{n+1}}(\Lambda_{D_n})&:= (H_{\tilde{\eta},-\frac{1}{2}})^{red} \\ H_{\rm u}(\Lambda_{D_n})&:= (H_{\xi,-\frac{1}{4}})^{red}, & H_{E_{n+1}}(\Lambda_{D_n})&:= (H_{v_{E_{n+1}}^*})^{red} \\ H_{D_{n}'}(\Lambda_{D_n})&:= \begin{cases} (H_{\eta',-\frac{3}{8}})^{red} , & n=7\\ (H_{\eta',-\frac{1}{4}})^{red} , & n=8\\ (H_{\eta',-\frac{1}{8}})^{red} , & n=9 \end{cases} & & \end{aligned}$$ on $\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{D_n})$ and $H_{E_{n+1}}(\Lambda_{D_n})=0$ for $n \le 3$. Here note that $$(H_{\eta_u,- \frac{8-n}{8}})^{red}= \begin{cases} H_{\eta}, & n \text{~odd} \\ (H_{\eta_1})^{red} + (H_{\eta_2})^{red}, & n \text{~even} \end{cases} \;= \begin{cases} H_{E_{n+1}}(\Lambda_{D_n}), & n\neq 7 \\ H_{E_8} + H_{D_8} , & n=7 \end{cases}$$ and we make the convention that $H_{E_5} \vcentcolon=(H_{\eta_1})^{red} + (H_{\eta_2})^{red}$. **Proposition 57**. *The Picard numbers of $\mathop{\mathrm{\mathrm{Sh}}}(D_n)$ are given in Table [6](#Picard of Dtower){reference-type="ref" reference="Picard of Dtower"}.* *$n$* *$4$* *$5$* *$6$* *$7$* *$8$* *$9$* *$10$* --------------------------------------------------- ------- ------- ------- ------- ------- ------- -------- -- -- *$\rho(\mathop{\mathrm{\mathrm{Sh}}}(D_n)^\ast)$* *$5$* *$4$* *$5$* *$4$* *$4$* *$3$* *$2$* : *Picard numbers of $D_n$-towers* *Moreover, there is a relation (called Hodge relation) for $n=4 ,5,6,7$ $$\label{relaD} \begin{split} (75+n(n-1)) \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(D_n)} & =H_{\rm n}(\Lambda_{D_n})+57 H_{\rm u}(\Lambda_{D_n})+2nH_{D_{n+1}}(\Lambda_{D_n}) \\ &\phantom{=} +(2^{n-1} + \delta_{n,6} + 2n \cdot\delta_{n,7}) H_{E_{n+1}}(\Lambda_{D_n}) \end{split}$$* **Remark 58**. *$H_{D_{7}}(\Lambda_{D_6})$ is a movable divisor and thus is not contractable on $\lambda_{\mathop{\mathrm{\mathrm{Sh}}}(D_6)}$. This also can not be seen directly from the arithmetic side.* **Proposition 59**. *Let $$\iota : \mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{D_{n+1}}) \rightarrow \mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{D_n})$$ be the natural morphism induced from lattice embedding, then we have* 1. *$$\iota^\ast H_{\rm {n}}(\Lambda_{D_n})= \begin{cases} H_{\rm {n}}(\Lambda_{D_{n+1}})+H_{D_{n+2}}(\Lambda_{D_{n+1}}), & n=3; \\ H_{\rm {n}}(\Lambda_{D_{n+1}})+2H_{D_{n+2}}(\Lambda_{D_{n+1}}) + \delta_{n,5} H_{E_{n+2}}(\Lambda_{D_{n+1}}), & \text{otherwise}. \end{cases}$$* 2. *$\iota^\ast H_{A_{4}}(\Lambda_{D_3})=H_{D_4}(\Lambda_{D_{4}}) + 2 H_{E_{5}}(\Lambda_{D_{4}}), \quad \iota^\ast H_{E_{n+1}}(\Lambda_{D_n})=2 H_{E_{n+2}}(\Lambda_{D_{n+1}})$.* *Moreover, we get the pullback formula $$\label{pullD} \iota^\ast H_{D_{n+1}}(\Lambda_{D_n})=\begin{cases} -\lambda_{\mathop{\mathrm{\mathrm{Sh}}}(D_{n+1})}+ \frac{1}{2} H_{D_{n+2}}, & n=3; \\ -\lambda_{\mathop{\mathrm{\mathrm{Sh}}}(D_n)}+ H_{D_{n+2}}(\Lambda_{D_{n+1}}) + H_{E_{n+2}} (\Lambda_{D_{n+1}}), & n=6; \\ -\lambda_{\mathop{\mathrm{\mathrm{Sh}}}(D_n)}+ H_{D_{n+2}}(\Lambda_{D_{n+1}}) , & \text{otherwise}. \end{cases}$$ for $n \le 6$.* *Proof.* The argument is parallel to the case of $A_n$ towers. We give the proof of (1) here and leave the rest for the interested readers. Fixed a $D_{n+1}$-type vector $v\in \Lambda_{D_n}$ and let $w\in \Lambda_{D_n}$ be a nodal vector, then the projection $x_w$ of $w$ onto $v^\perp=\Lambda_{D_{n+1}}$ is either $w$ itself or a $D_{n+1}$-type vector $x_w=2w\pm v$. In the first case, by [@LO19 Proposition 1.4.6], we have $w$ is a nodal vector and moreover a $E_7$-type vector when $n=5$. The multiplicity computation is identical as in the case of $A_n$-tower in Proposition [Proposition 47](#proppullA){reference-type="ref" reference="proppullA"}. This proves (1). The last pullback formula ([\[pullD\]](#pullD){reference-type="ref" reference="pullD"}) is obtained by relation ([\[relaD\]](#relaD){reference-type="ref" reference="relaD"}) combining (1) and (2). ◻ Recall the discriminant group of $\Lambda_{D_n}'$ is given by $$G(\Lambda_{D_n}')= {\mathbb Z}/2 {\mathbb Z}\times {\mathbb Z}/ (10-n){\mathbb Z},\ 6 \le n\le 9. %\bZ/ 2\bZ \times$$ **Definition 60**. *We call a primitive vector* 1. *$v\in \Lambda_{D_7}'$ is a $\Lambda_{D_8}'$-type vector if $v^2=-6$ and $\mathrm{div}(v) = 3$ so that $v^\perp \cong \Lambda_{D_8}'$ and $[\Lambda_{D_7}':{\mathbb Z}v \oplus v^\perp]=2$.* 2. *$v\in \Lambda_{D_8}'$ is a $\Lambda_{D_9}'$-type vector if $v^2=-2$ and $\mathrm{div}(v)=2$ so that $v^\perp=\Lambda_{D_9}'$ and $[ \Lambda_{D_8}': {\mathbb Z}v \oplus v^\perp ]=2$.* Define the divisor $$\begin{split} H_{D_{n+1}'}:= (H_{v_{D_{n+1}'}})^{red} ,\ \ H_{\rm n}:=(H_v)^{red},\ \hbox{for}\ v \ \hbox{is nodal vector} \ %H_{\rm u'}(\Lambda_{A_6'}):=H_{,-\frac{3}{11}} \end{split}$$ on $\mathop{\mathrm{\mathrm{Sh}}}(D_n)'$ for $n=6,7,8$. **Proposition 61** (Hodge relations on $D_n'$-towers $\mathop{\mathrm{\mathrm{Sh}}}(D_n')$). *The Picard number of $\mathop{\mathrm{\mathrm{Sh}}}(D_n')$ are given by $$\rho(\mathop{\mathrm{\mathrm{Sh}}}(D_n')^\ast)=\begin{cases} 4, &\ n=6\\ 3, &\ n=7\\ 2, &\ n=8\\ 1, & \ n=9 \end{cases}$$ Moreover, there is a relation $$\label{relaD'} \begin{split} & 103 \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(D_6')}=H_{\rm n}+14 H_{D_7'}+ 29 H_{(\bar{1},\bar{2}),-\frac{1}{4}} + 92 H_{(\bar{0},\bar{1}),-\frac{1}{8}} \\ & 117 \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(D_7')}=H_{\rm n}+ 15 H_{D_8'}+ 135 H_{(\bar{1},\bar{1}),-\frac{1}{12}} \\ & 132 \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(D_8')}=H_{\rm n}+33H_{D_9'} \\ & 165 \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(D_9')} = H_{\rm n} \end{split}$$* *Proof.* One only needs to notice that for $n=7$, we have $$H_{(\bar{0},\bar{1}),-\frac{1}{3}} = H_{D_8'} + H_{(\bar{1},\bar{1}),-\frac{1}{12}}.$$ ◻ **Proposition 62**. *Let $$\label{propD'} j: \mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{D_{7}'}) \rightarrow \mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{D_7}),\ \ \ \iota_m' : \mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{D_{m+1}'}) \rightarrow \mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{D_m}')$$ be the natural morphisms induced from lattice embedding, then the pullback* 1. *$j^\ast H_{\rm n}= H_{\rm n}+ H_{D_8'}$ , $j^\ast H_{D_8}=H_{D_8'},j^\ast H_{E_8}=j^\ast H_{\rm u}= H_{(\Bar{0},\Bar{1}),-\frac{1}{12}}$.* 2. *$\iota_7'^\ast H_{\rm n}= H_{\rm n}+3 H_{D_8'}$, $\iota_7'^\ast H_{(\bar{1},\bar{1}),-\frac{1}{12}}=0$.* *In particular, $$\label{pullD'} \iota'^\ast H_{D_{7}'}=-\lambda +H_{D_8'},\ \ \ \iota'^\ast H_{D_{8}'}=-\lambda +2 H_{D_9'},\ \ \ \iota'^\ast H_{D_{9}'}=-\lambda$$* *Proof.* (1). Fix a $D_{7}'$-type vector $v\in \Lambda_{D_7}$ and let $w\in \Lambda_{D_7}$ be a $E_8$ vector. Take $w_1\in \Lambda_{D_7}$ a $E_8$-type vector, and $w_2\in \Lambda_{D_7}$ a unigonal vector then we have $$\begin{split} 2\pi_{v^\perp}(w_1) &= 3w_1 - v, \\ \pi_{v^\perp}(w_2) &= 3w_2 - v, \end{split}$$ and $H_{\pi_{v^\perp}(w_1)}= H_{\pi_{v^\perp}(w_2)}=H_{(\Bar{0},\Bar{1}),-\frac{1}{12}}$. (2). Fix $v\in \Lambda_{D_7'}$ a $D_8'$-vector, and take $w\in\Lambda_{D_7'}$ with $$w^2=-6, \mathrm{div}(w) =6, H_w = H_{(\bar{1},\bar{1}),-\frac{1}{12}}.$$ We have $$mw=av+ l \cdot \pi_{v^\perp}(w)$$ where $m\in\{1,2\}$. Then the only possibility is $m=2$, $a=1$, $l^2 \pi^2 =-18$. Note that $$12 \,\big|\, (2w\cdot \pi = l\cdot \pi^2)\, \big|\, l^2 \pi^2 =-18$$ as $\mathrm{div}(w) =6$. This gives the contradiction and $\iota_7'^\ast H_{(\bar{1},\bar{1}),-\frac{1}{12}}=0$. The remaining cases are similar and we omit here. Combining relation ([\[relaD\'\]](#relaD'){reference-type="ref" reference="relaD'"}), the pullback formula ([\[pullD\'\]](#pullD'){reference-type="ref" reference="pullD'"}) is obtained as before. ◻ ### E-type tower **Definition 63**. *A vector $v\in \Lambda_{E_n}$ is called $E_{n+1}$-type if $$v^2=\begin{cases} -6 &\ n=6\\ -2 &\ n=7 \end{cases},\ \ v^\ast=\eta_E$$ so that $v^\perp \cong \Lambda_{E_{n+1}}$ and $$[\Lambda_{E_n}:{\mathbb Z}v \oplus v^\perp]=\begin{cases} 3, &\ n=6\\ 2, &\ n=7 \end{cases}$$* Denote the integral divisors $$\begin{split} & H_{\rm {n}}(\Lambda_{E_n}):= (H_{v})^{red} \text{~for nodal vectors~} v, \quad H_{E_{n+1}}(\Lambda_{E_n}):= (H_{\eta,-\frac{8-n}{2(9-n)}})^{red} \\ &H_{\rm u}(\Lambda_{E_n}):= H_{\xi,-\frac{1}{4}} . \end{split}$$ on $\mathop{\mathrm{\mathrm{Sh}}}(E_n)$. We have $$(H_{0,-1})^{red}= H_{\rm {n}}(\Lambda_{E_m})+ H_{\rm u}(\Lambda_{E_m}) + \delta_{m,7}H_{E_{m+1}}(\Lambda_{E_m})$$ and $H_{\rm {n}}(\Lambda_{E_n}),H_{\rm u}(\Lambda_{E_n})$ are integral divisors on $\mathop{\mathrm{\mathrm{Sh}}}(E_n)$. **Proposition 64**. *The Picard group of $\mathop{\mathrm{\mathrm{Sh}}}(E_n)$ is generated by the integral divisors $H_{\rm {n}}(\Lambda_{E_n}), H_{\rm u}(\Lambda_{E_n})$ and $H_{E_{n+1}}(\Lambda_{E_n})$. The Picard number of $\mathop{\mathrm{\mathrm{Sh}}}(D_n)$ are given by $$\rho(\mathop{\mathrm{\mathrm{Sh}}}(E_n)^\ast)=\begin{cases} 3, &\ n=6\\ 3, &\ n=7\\ 2, &\ n=8 . \end{cases}$$ Moreover, there are relations $$\label{pullE1} \begin{split} & 111 \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(E_6)}=H_{\rm {n}}(\Lambda_{E_n})+ 57H_{\rm u}(\Lambda_{E_n}) + 27 H_{E_{7}}(\Lambda_{E_6}) \\ & 138 \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(E_7)}=H_{\rm {n}}(\Lambda_{E_n})+57H_{\rm u}(\Lambda_{E_n})+57H_{E_{8}}(\Lambda_{E_7}) \\ & 195 \lambda_{\mathop{\mathrm{\mathrm{Sh}}}(E_8)}=H_{\rm {n}}(\Lambda_{E_n})+57H_{\rm u}(\Lambda_{E_n}) \end{split}$$* **Proposition 65**. *Let $$\iota : \mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{E_{n+1}}) \rightarrow \mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{E_n}),\ \ \nu : \mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{E_{6}}) \rightarrow \mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{D_5})$$ be the natural morphisms induced from lattice embedding, then we have* 1. *$\nu^\ast H_{\rm {n}}(\Lambda_{D_5})=H_{\rm {n}}+H_{E_7}$,    $\nu^\ast H_{D_6}=H_{E_7}\ $ and $\ \nu^\ast H_{\rm u}(\Lambda_{D_5})=H_{\rm u}(\Lambda_{E_6})$* 2. *$\iota^\ast H_{\rm {n}}(\Lambda_{E_6})=H_{\rm {n}}(\Lambda_{E_{7}})+3 H_{E_{8}}(\Lambda_{E_{n+1}})$,    $\iota^\ast H_{\rm u}(\Lambda_{E_n})=H_{\rm u}(\Lambda_{E_{n+1}})$* *In particular, the following pullback formulas hold $$\label{pullE2} \begin{split} &\nu^\ast H_{E_6}(\Lambda_{D_5}) =-\lambda_{\mathop{\mathrm{\mathrm{Sh}}}(E_6)}+ H_{E_7} \\ & \iota^\ast H_{E_{7}}(\Lambda_{E_6})=-\lambda_{\mathop{\mathrm{\mathrm{Sh}}}(E_{7})}+ 2 H_{E_{8}}(\Lambda_{E_{7}}), \\ & \iota^\ast H_{E_{8}}(\Lambda_{E_7})=-\lambda. \end{split}$$* *Proof.* The argument is similar to before. We fix $v\in \Lambda_{D_5}$ be a vector of $E_6$-type in Definition [Definition 56](#vectoronD){reference-type="ref" reference="vectoronD"} (3) so that $v^\perp \cong \Lambda_{E_6}$. Then taking $w\in \Lambda_{D_5}$ nodal vector, i.e., $w^2=-2$ and $\mathrm{div}(w)=1$, then $$m w=a v+ \pi_{v^\perp}(w),\ m |3,\ \ a\in {\mathbb Z}\ $$ since $[ \Lambda_{D_5}:{\mathbb Z}v\oplus v^\perp]=3$. Clearly, $m=1$ implies $\pi_{v^\perp}(w)=w$. If $m=3$, then $-18=-12a^2+(\pi_{v^\perp}(w))^2$. The negativity of $\pi_{v^\perp}(w)$ implies only solution is $a= \pm 1$ and $\pi_{v^\perp}(w)=-6$. One can also show $(\pi_{v^\perp}(w))^\ast =\eta_E$, i.e., $\pi_{v^\perp}(w) \in v^\perp \cong \Lambda_{E_6}$ is $E_7$-type vector. This proves the support of $\nu^\ast H_{\rm {n}}(\Lambda_{D_5})$ is $H_{\rm {n}}$ and $H_{E_7}$. The multiplicity is $1$ by the same argument in proof of Proposition [Proposition 47](#proppullA){reference-type="ref" reference="proppullA"}. The remaining pullback divisors are the same arguments. This proves part (1). The part (2) is the same computation. We leave the details to the interested reader. ◻ ### Unigonal divisor Let $v_h \in {\mathbb L}_{K3}$ (rep. $v_u\in {\mathbb L}_{K3}$) be the classes so that its span with invariant lattice $N_8$ of rank 2 has the gram matrix as follows $$\label{lattice3} \left( \begin{array}{ccc} 2 & 0 & 0 \\ 0 & -2 & 1 \\ 0 & 1 & -2 \end{array} % dis=6 \right) \ \ (\ \text{rep.}\; \left( \begin{array}{ccc} 2 & 0 & 1 \\ 0 & -2 & 2 \\ 1 & 2 & -2 \end{array} % disc=2 \right) ).$$ Denote $\Lambda_h$ (rep. $\Lambda_u$) be the orthogonal complement of the lattice ([\[lattice3\]](#lattice3){reference-type="ref" reference="lattice3"}) in ${\mathbb L}$. By [@Huy Corollary 1.18, Chapter 14], there is a unique primitive embedding $\Lambda_v \hookrightarrow I\!I_{2,26}$ with orthogonal complement $E_7 \oplus A_2$ (rep. $E_8 \oplus A_1$). Let $H_h$ (rep. $H_u$) be the associated NL divisor of $\Lambda_h$ (rep. $\Lambda_u$). **Proposition 66**. *$H_h \cap H_u$ is nonempty and $H_u$ has no self-intersections in the sense of Looijenga, i.e., $H_u^{(2)}$ is empty.* *Proof.* Let $v_h, v_u \in {\mathbb L}_{K3}$ be the hyperelliptic and unigonal vector in ${\mathbb L}_{K3}$, then their span with 2-elementary K3 lattice of rank 2 has the gram matrix of the following form $$\label{hi} \left( \begin{array}{cccc} 2 & 0 & 1 & 0 \\ 0 & -2 & 2 & 1 \\ 1 & 2 & -2 & a \\ 0 & 1 & a & -2 \end{array} \right)$$ where $a\in {\mathbb Z}_{\ge 0}$. Thus, it has discriminant $4a^2+8a+1$. Then by Hodge index theorem, the matrix ([\[hi\]](#hi){reference-type="ref" reference="hi"}) should have a negative discriminant, which is possible only for $a=-1$. The method of the second statement is similar to the first one. Assume we have distinct unigonal vectors $v_u,v_u'$ in ${\mathbb L}_{K3}$, then the self-intersection associated to $v_u,v_u'$ is determined by the lattice spanned by 2-elementary K3 lattice of rank 2 with $v_u,v_u'$. Thus, this lattice has a gram matrix $$\label{uni} \left( \begin{array}{cccc} 2 & 0 & 1 & 1 \\ 0 & -2 & 2 & 2 \\ 1 & 2 & -2 & a \\ 1 & 2 & a & -2 \end{array} \right)$$ with discriminant $4a^2+12a+8\geq 0$ for any $a\in {\mathbb Z}$. This contradicts the Hodge index theorem. Hence $H_u^{(2)}=\emptyset$. ◻ **Remark 67**. *Indeed, the generic period point in $H_u\cap H_h$ comes from the period of the double covering of pairs $(Bl_p {\mathbb P}(1,1,4), C)$ where $C$ is tangent to the exceptional divisor $E$ over $p$.* **Definition 68** (Towers on unigonal divisor). *Let $$U_1 \supset U_2' \supset U_3' \supset U_4''$$ be the complement of Neron-Severi lattice from ([\[nsU1\]](#nsU1){reference-type="ref" reference="nsU1"}), ([\[nsU2\'\]](#nsU2'){reference-type="ref" reference="nsU2'"}), ([\[nsU3\'\]](#nsU3'){reference-type="ref" reference="nsU3'"}) and ([\[nsU4\'\'\]](#nsU4''){reference-type="ref" reference="nsU4''"}) in K3 lattice.* By [@Huy Corollary 1.18, Chapter 14], there are unique primitive embedding $U_1,U_2', U_3', U_4''\hookrightarrow I\!I$ with complement $E_8\oplus A_1 , E_8\oplus A_1 \oplus A_2 , E_8 \oplus A_2^2, E_8 \oplus A_2 \oplus D_4$. This motivates us to consider the following towers $$E_8\oplus A_1 \subset E_8\oplus A_1^2 \subset E_8\oplus A_1 \oplus A_2 \subset E_8 \oplus A_2^2 \subset E_8 \oplus A_2 \oplus A_3 \subset E_8 \oplus A_2 \oplus D_4 \subset I\!I$$ Let $\Lambda_{L} = (E_8\oplus L)^\perp_{I\!I}$ for some root lattice $L$ and we denote the discriminant groups - $G(\Lambda_{A_n\oplus A_m}) \cong {\mathbb Z}\langle\zeta_{n} \rangle \times {\mathbb Z}\langle \zeta_{m} \rangle$ where $\mathrm{ord}_{}(\zeta_i) = i+1$ with $\zeta_i^2 = -\frac{i+2}{i+1}$. - $G(\Lambda_{A_n\oplus D_4}) \cong {\mathbb Z}\langle\zeta_{n} \rangle \times {\mathbb Z}\langle \eta \rangle \times {\mathbb Z}\langle \eta' \rangle$ where $\eta,\eta'$ are similar as in the case of hyperelliptic $D$-type tower. Then as in the study of towers for hyperelliptic divisor, we define **Definition 69**. *We say a primitive vector* 1. *$v \in U_1= (E_8\oplus A_1)_{I\!I}^\perp$ is nodal if $v^2=-2$ and $\mathrm{div}(v)=1$ so that $v^\perp \cong (E_8\oplus A_1^2)_{I\!I}^\perp$ and $[U_1: {\mathbb Z}v \oplus v^\perp]=2$. $v \in (E_8\oplus A_1)_{I\!I}^\perp$ is $A_2$-type if $v^2=-6$ and $\mathrm{div}(v)=2$, $v^\perp \cong (E_8\oplus A_2)_{I\!I}^\perp$.* 2. *$v \in U_2=(E_8\oplus A_1^2 )_{I\!I}^\perp$ is $A_1 \oplus A_2$-type if $v^2= -6$ and $v^\ast =\zeta_1$ or $\zeta_1'$, so that $v^\perp \cong (E_8\oplus A_1 \oplus A_2 )_{I\!I}^\perp$ and $[U_2: {\mathbb Z}v \oplus v^\perp]=3$. Denote $$H_{A_1 \oplus A_2}:=H_v ,\ \ v\ \hbox{is an}\ A_1 \oplus A_2 \hbox{-type vector}$$* 3. *$v \in U_3=(E_8\oplus A_1 \oplus A_2 )_{I\!I}^\perp$ is $A_2 \oplus A_2$-type if $v^2= -6$ and $v^\ast =\zeta_1$, so that $v^\perp \cong (E_8\oplus A_2^2)_{I\!I}^\perp$ and $[U_2: {\mathbb Z}v \oplus v^\perp]=3$. Denote $$H_{A_2 \oplus A_2}:=H_v ,\ \ v\ \hbox{is an}\ A_2 \oplus A_2 \hbox{-type vector}$$* 4. *$v \in U_4=(E_8\oplus A_2 \oplus A_2 )_{I\!I}^\perp$ is $A_2 \oplus A_3$-type if $v^2= -12$ and $v^\ast =\zeta_2$ or $\zeta_2'$, so that $v^\perp \cong (E_8\oplus A_2 \oplus A_3 )_{I\!I}^\perp$ and $[U_4: {\mathbb Z}v \oplus v^\perp]=3$. Denote $$H_{A_1 \oplus A_2}:=H_v ,\ \ v\ \hbox{is an}\ A_2 \oplus A_2 \hbox{-type vector}$$* 5. *$v \in U_5=(E_8\oplus A_2 \oplus A_3 )_{I\!I}^\perp$ is $A_2 \oplus D_4$-type if $v^2= -4$ and $v^\ast =2\zeta_3$ so that $v^\perp \cong (E_8\oplus A_2 \oplus D_4)_{I\!I}^\perp$ and $[U_5: {\mathbb Z}v \oplus v^\perp]=2$. Denote $$H_{A_2 \oplus A_4}:=H_v ,\ \ v\ \hbox{is an}\ A_2 \oplus D_4 \hbox{-type vector}$$* Let $\mathop{\mathrm{\mathrm{Sh}}}(U_n)$ be the locally symmetric variety associated to $U_n$. Thus, we have $$\mathop{\mathrm{\mathrm{Sh}}}(U_3)= \mathop{\mathrm{\mathrm{Sh}}}(U_2'),\ \ \mathop{\mathrm{\mathrm{Sh}}}(U_4)= \mathop{\mathrm{\mathrm{Sh}}}(U_3'),\ \ \mathop{\mathrm{\mathrm{Sh}}}(U_6)= \mathop{\mathrm{\mathrm{Sh}}}(U_4''),$$ **Proposition 70**. *The Picard group of $\mathop{\mathrm{\mathrm{Sh}}}(U_1)$ is generated by $H_{\rm {n}}(U_1)$ and $H_{A_{2}}(U_1)$ and the Picard numbers is $2$. Moreover, there is a relation $$\label{relationU} \begin{split} 133 \lambda_{U_1} &= H_{\rm n}+ 2 H_{A_{2}} \\ %204\lambda_{\sh(A_1 \oplus A_1)}&=H_{\rm n}+ 20H_{,-\frac{1}{2}}\\ 134 \lambda_{ A_1 \oplus A_1 } &= H_n + 4 H_{\zeta_1+\zeta_1',-\frac{1}{2}}^{\mathbf{red}} + 2 H_{A_1\oplus A_2} \\ % 32 \lambda_{ A_1 \oplus A_1 } &=H_{A_1 \oplus A_2}- 8 H_{\zeta_1+\zeta_1',-\frac{1}{2}} \\ % 34 \lambda_{ A_1 \oplus A_2} &=H_{A_2 \oplus A_2}+ 2 H_{A_1\oplus A_3} - 14 H_{\zeta_1+\zeta_2,-\frac{5}{12}}\\ % 36 \lambda_{ A_2 \oplus A_2} &= 2 H_{A_2 \oplus A_3}-12 (H_{\zeta_2+\zeta_2',-\frac{1}{3}} + H_{\zeta_2- \zeta_2',-\frac{1}{3}})\\ % 19\lambda_{A_2 \oplus A_3 } &= H_{A_2 \oplus D_4}+H_{A_3+\oplus A_3}+H_{A_2\oplus A_4}-12H_{\zeta_2+2\zeta_3,-\frac{1}{6}}-5H_{\zeta_2\pm \zeta_3,-\frac{7}{24}} \\ % 21 \lambda_{ A_2 \oplus D_4} &= H_{\zeta_2,-\frac{2}{3}}+3 H_{\eta,-\frac{1}{2}}-16(H_{\eta'+\zeta_2,-\frac{1}{6}}+H_{\eta+\eta'+\zeta_2,-\frac{1}{6}})+2 H_{\eta+\zeta_2,-\frac{1}{6}} \end{split}$$* **Remark 71**. *$H_{A_{2}}$ is big and thus not birationally contractible. Indeed, in [@psw2] we prove K-moduli space is isomorphic to ${\mathcal F}(s)$. The bigness of $H_{A_{2}}$ can be seen from the wall-crossing results on the K-moduli side as long as the isomorphism between ${\mathcal F}(s)$ and K-moduli space.* **Proposition 72**. *Let $\nu: \mathop{\mathrm{\mathrm{Sh}}}(U_1) \longrightarrow {\mathcal F}$ and $\iota: \mathop{\mathrm{\mathrm{Sh}}}(U_{n+1}) \longrightarrow \mathop{\mathrm{\mathrm{Sh}}}(U_n)$ be the morphisms induced by the lattice embedding,. Then* 1. *$\nu^\ast\lambda=\lambda,\ \nu^\ast H_{\rm n}= H_{\rm n}$,   $\nu^\ast H_{h}= H_{A_2}$,  $\nu^\ast (H_{\rm u})=-\lambda$.* 2. *$\iota^\ast H_{A_2}= 2 H_{\zeta^h+\zeta,-\frac{1}{2}}^{\mathbf{red}} +H_{A_1 \oplus A_2}$ on $\ \mathop{\mathrm{\mathrm{Sh}}}(U_2)$;* 3. *$\iota ^\ast H_{A_1 \oplus A_2}= -2\lambda+ H_{A_2 \oplus A_2}+Z_{U_3}\ $ on $\ \mathop{\mathrm{\mathrm{Sh}}}(U_3)$;* 4. *$\iota ^\ast H_{A_2 \oplus A_2}=-\lambda+H_{A_2 \oplus A_3}+Z_{U_4}\ $ on $\ \mathop{\mathrm{\mathrm{Sh}}}(U_4)$;* 5. *$\iota ^\ast H_{A_2 \oplus A_3}=-\lambda+H_{A_2 \oplus D_4}+Z_{U_5}\ $ on $\ \mathop{\mathrm{\mathrm{Sh}}}(U_5)$;* 6. *$\iota ^\ast H_{A_2 \oplus D_4}=-2\lambda+Z_{U_6}\ $ on $\ \mathop{\mathrm{\mathrm{Sh}}}(U_6)$.* *where each $Z_{U_i}$ is a linear combination of possible Heegner divisors with smaller discriminant.* *Proof.* Fixed a unigonal vector $v\in \Lambda_{A_1}=(E_7\oplus A_1)^\perp_{I\!I}$, i.e., $v^2=-2, \mathop{\mathrm{\mathrm{div}}}_{\Lambda_{A_1}}(v)=2$, then $\Lambda_{A_1}={\mathbb Z}v \oplus v^\perp$ since $v^\perp\cong (E_8\oplus A_1)^\perp_{I\!I}$. Thus a nodal vector $w$ in $\Lambda_{A_1}$ such that the sublattice $\langle v,w \rangle$ is negatively definite must be a nodal vector in $v^\perp$. In other words, the projection of nodal vector in $\Lambda_{A_1}$ on $v^\perp$ must be nodal. This shows $\nu^\ast H_{\rm n}= H_{\rm n}$. Similarly, for a hyperelliptic vector $w\in$, i.e., $w^2=-6, \mathop{\mathrm{\mathrm{div}}}_{\Lambda_{A_1}}(w)=2$. Then $$\left( \begin{array}{c|cc} & v & w \\ \hline v & -2 & a \\ w & a & -6 \end{array} \right)$$ So $|a| \le 3$ as the sublattice $\langle v,w \rangle$ is negatively definite. By divisibility of $v$ and $w$, $a=0$ or $|a|=2$. The latter is impossible. Indeed, if $a=2$, then $w':=v+w\in v^\perp$ will be the projection of $w$ onto $v^\perp$ and clearly $\mathop{\mathrm{\mathrm{div}}}_{v^\perp}(w')=2m,\ w'^2=-4$, therefore $(w'^\ast)^2=(\frac{w'}{2m})^2=-\frac{1}{m^2}$, a contradiction since the discriminant group of $v^\perp$ has only two elements $0$ and $\xi$ with $\xi^2=-\frac{1}{2}$. Similar reason shows $a=-2$ is impossible. This proves $a=0$ and thus $w\in v^\perp=U_1$ with $w^2=-6, \mathop{\mathrm{\mathrm{div}}}_{v^\perp}(w)=2$, exactly the $A_2$-type vector in the definition [Definition 69](#uAn){reference-type="ref" reference="uAn"}. Thus we finish the proof of $\nu^\ast H_h=H_{A_2}$. Recall we have relation $$76\lambda=H_{\rm n} +57H_{\rm u}+2H_{h} \ \ \hbox{on}\ {\mathcal F}.$$ Combining 1st identity in ([\[relationU\]](#relationU){reference-type="ref" reference="relationU"}), we get $\nu^\ast (H_{\rm u})=-\lambda$. This proves (1). Fixed a nodal vector $v\in U_1=(E_8\oplus A_1)^\perp_{I\!I}$ and so that $v^\perp =(E_8\oplus A_1 \oplus A_1 )^\perp_{I\!I}$ and in this case, $[U_1:{\mathbb Z}v \oplus v^\perp]=2$. By the similar arguments in (1), we have $$\iota^\ast H_{A_2}=H_{A_1 \oplus A_2}$$ Then combining 2nd identity in ([\[relationU\]](#relationU){reference-type="ref" reference="relationU"}), we get the pullback formula in (2). The remaining cases are similar lattice arguments as before and also apply identities in ([\[relationU\]](#relationU){reference-type="ref" reference="relationU"}). We omit here. ◻ ## Arithmetic predictions for HKL According to the arithmetic principle discussed in Section [4.4](#arith){reference-type="ref" reference="arith"}, we will compute the restriction of ${\mathbb Q}$-line bundle $\Delta(s)= \lambda+s(25H_u+H_h)$ given by Heegner divisors on the arithmetic stratification. ### Arithmetic stratifications Denote $$\begin{split} &\mathop{\mathrm{\mathrm{NL}}}_h(A_n):=\mathop{\mathrm{\text{\rm Im}}}(\mathop{\mathrm{\mathrm{Sh}}}(A_n) \rightarrow {\mathcal F}), \ \ \mathop{\mathrm{\mathrm{NL}}}_h(A_n'):=(\mathop{\mathrm{\mathrm{Sh}}}(A_n') \rightarrow {\mathcal F})\\ &\mathop{\mathrm{\mathrm{NL}}}_h(D_n):=\mathop{\mathrm{\text{\rm Im}}}(\mathop{\mathrm{\mathrm{Sh}}}(D_n) \rightarrow {\mathcal F}), \ \ \mathop{\mathrm{\mathrm{NL}}}_h(D_n)':= (\mathop{\mathrm{\mathrm{Sh}}}(D_n') \rightarrow {\mathcal F}) \\ &\mathop{\mathrm{\mathrm{NL}}}_h(E_n):=\mathop{\mathrm{\text{\rm Im}}}(\mathop{\mathrm{\mathrm{Sh}}}(E_n) \rightarrow {\mathcal F}) \end{split}$$ the NL locus on the moduli space ${\mathcal F}$. Then we get an arithmetic stratification by the NL-locus 1. $A_n$-type stratification $$\mathop{\mathrm{\mathrm{NL}}}_h(A_4) \subset \mathop{\mathrm{\mathrm{NL}}}_h(A_3) \subset \mathop{\mathrm{\mathrm{NL}}}_h(A_2) \subset \mathop{\mathrm{\mathrm{NL}}}_h(A_1)={\mathcal F}^\ast$$ and its modification $$\mathop{\mathrm{\mathrm{NL}}}_h(A_7'') \subset \mathop{\mathrm{\mathrm{NL}}}_h(A_7') \subset \mathop{\mathrm{\mathrm{NL}}}_h(A_6') \subset \mathop{\mathrm{\mathrm{NL}}}_h(A_5') \subset \mathop{\mathrm{\mathrm{NL}}}_h(A_5)$$ 2. $D_n$-type stratification $$\mathop{\mathrm{\mathrm{NL}}}_h(D_7) \subset \cdots \subset \mathop{\mathrm{\mathrm{NL}}}_h(D_4) \subset \mathop{\mathrm{\mathrm{NL}}}_h(D_3)=\mathop{\mathrm{\mathrm{NL}}}_h(A_3) \subset {\mathcal F}^\ast$$ and its modification $$\mathop{\mathrm{\mathrm{NL}}}_h(D_9') \subset \mathop{\mathrm{\mathrm{NL}}}_h(D_8') \subset \mathop{\mathrm{\mathrm{NL}}}_h(D_7') \subset \mathop{\mathrm{\mathrm{NL}}}_h(D_7)$$ 3. $E_n$-type stratification $$\mathop{\mathrm{\mathrm{NL}}}_h(E_8) \subset \mathop{\mathrm{\mathrm{NL}}}_h(E_7) \subset \mathop{\mathrm{\mathrm{NL}}}_h(E_6) \subset {\mathcal F}^\ast$$ On unigonal divisor $H_u$, we similarly define these Noether-Lefschetz loci $$\begin{split} \mathop{\mathrm{\mathrm{NL}}}_u(U_n):= \mathop{\mathrm{\text{\rm Im}}}(\mathop{\mathrm{\mathrm{Sh}}}(U_n) \rightarrow {\mathcal F}). \end{split}$$ ### Prediction of walls Recall the ${\mathbb Q}$-line bundle $\Delta(s)=\lambda+s(25H_u+H_h)$ on $\mathop{\mathrm{\mathrm{Sh}}}(L)$. **Theorem 73**. *Denote $\varphi: \mathop{\mathrm{\mathrm{Sh}}}(L) \rightarrow {\mathcal F}$ the composition of morphism of towers for $L$ being one of the following lattice $$\{ \Lambda_{A_n}, \Lambda_{A_n}',\Lambda_{D_n},\Lambda_{D_n}', \Lambda_{E_n},\ U_n \}.$$* 1. *if  $L=\Lambda_{A_{n+1} },\ n\le 3$, $\varphi^\ast \Delta(s)$ will be $$(1- n s) \lambda+s(H_{A_{n+2}}+n(1-\delta_{1,n}) H_{D_{n+2}} +25\cdot H_{\rm u}(\Lambda_{A_{n+1}}))$$* 2. *if $L=\Lambda_{A_5 }'$, then $\varphi^\ast \Delta(s)$ will be $$(1- 4 s) \lambda+ 2sH_{A_{6}'}+ \widetilde{Z}_A\ $$ where $\widetilde{Z}_A$ is the pullback of Heegner divisor $Z$ in Proposition [Proposition 51](#pullA'){reference-type="ref" reference="pullA'"}* 3. *if $L=\Lambda_{A_6 }''$, then $\varphi^\ast \Delta(s)$ will be $$(1- 6s) \lambda+ 2sH_{A_7''} +\widetilde{Z}_A' \ $$ where $\widetilde{Z}_A'$ is the pullback of Heegner divisor $Z'$ in Proposition [Proposition 55](#pullA''){reference-type="ref" reference="pullA''"}.* 4. *if $L=\Lambda_{A_7 }'''$, $\varphi^\ast \Delta(s)$ will be $$(1- 8s) \lambda+\widetilde{Z}_A''\ $$* 5. *if $L=\Lambda_{D_{n+1} }, 3 \le n \le 5$, $\varphi^\ast \Delta(s)$ will be $$(1- 2(n-1) s) \lambda+s\cdot \left[ 2 H_{D_{n+2}}+ 2^{n-2} H_{E_{n+2}} + 25 H_{\rm u}(\Lambda_{D_{n+1}}) \right]$$* 6. *if $L=\Lambda_{D_{n} }'$, $$\varphi^\ast \Delta(s) = \begin{cases} (1-10s)\lambda + 2sH_{D_8'} +Z_D, & n=7 \\ (1-12s)\lambda + 4sH_{D_9'} , & n=8 \\ (1-16s)\lambda , & n=9 \end{cases}.$$* 7. *if  $L=\Lambda_{E_n },\ n=6,7,8$, $\varphi^\ast \Delta(s)$ will be $$(1- \mu_{E_n} s) \lambda+25s\cdot H_{\rm u}(\Lambda_{E_{n}}).$$* 8. *if $L=U_n$, then $$\begin{split} \varphi^\ast \Delta(s)=&(1-25s) \lambda+s H_{A_2} \ \hbox{on} \ \ \mathop{\mathrm{\mathrm{Sh}}}(U_1) \\ \varphi^\ast \Delta(s)=&(1-25s) \lambda+sH_{A_1 \oplus A_2}+s \widetilde{Z}_{U_2} \ \ \hbox{on} \ \ \mathop{\mathrm{\mathrm{Sh}}}(U_2) \\ \varphi^\ast \Delta(s)=&(1-27s) \lambda+sH_{A_2 \oplus A_2}+s \widetilde{Z}_{U_3} \ \ \ \hbox{on} \ \mathop{\mathrm{\mathrm{Sh}}}(E_8\oplus A_1 \oplus A_2) \\ \varphi^\ast \Delta(s)=&(1-28s) \lambda+sH_{A_2 \oplus A_3}+s \widetilde{Z}_{U_5} \ \hbox{on} \ \mathop{\mathrm{\mathrm{Sh}}}(U_5) \\ \varphi^\ast \Delta(s)=&(1-31s) \lambda+s \widetilde{Z}_{U_6} \ \hbox{on} \ \mathop{\mathrm{\mathrm{Sh}}}(U_6) \\ \end{split}$$ where $\widetilde{Z}_{U_i}$ are linear combinations pullback of $Z_{U_{i-1}}$ and other Heegner divisors in relation ([\[relationU\]](#relationU){reference-type="ref" reference="relationU"}).* *Proof.* We prove the theorem by inductive computations on the towers. If $L=\Lambda_{A_n }$, then using the pullback formula ([\[adjuctionAn\]](#adjuctionAn){reference-type="ref" reference="adjuctionAn"}) repeatedly we will have $$\varphi^\ast(\lambda+s(25H_u+H_h))=(1- n s) \lambda+sH_{A_{n+1}}+s(n-1)H_{D_{n+2}} +25s\cdot H_{\rm u}(\Lambda_{A_{n+1}}) .$$ This proves the item (1). Consider the pullback on $\mathop{\mathrm{\mathrm{Sh}}}(\Lambda_{A_5})$, we have $$(1-4s)\lambda+sH_{A_6}+3sH_{D_6}+2sH_{E_6}+25sH_{\rm u}$$ Since Heegner divisors $H_{A_6}, H_{D_6}, H_{E_6}, H_{\rm u}$ on $\mathop{\mathrm{\mathrm{Sh}}}(A_5)$ are different from $H_{A_5'}$, direct pullback formula in Proposition [Proposition 51](#pullA'){reference-type="ref" reference="pullA'"} shows item (2). Then combining (2) in Proposition [Proposition 51](#pullA'){reference-type="ref" reference="pullA'"}, we get the pullback $$\varphi^\ast \Delta(s)=(1-4s)\lambda+sH_{A_6'}+\widetilde{Z}_A$$ on $\mathop{\mathrm{\mathrm{Sh}}}(A_5')$ by computation by (3) in Proposition [Proposition 51](#pullA'){reference-type="ref" reference="pullA'"}. This finishes the proof of item (3). As $\mathop{\mathrm{\mathrm{Sh}}}(A_3)=\mathop{\mathrm{\mathrm{Sh}}}(D_3)$, we have $$(1- 2 s) \lambda+sH_{A_{4}}+sH_{D_{4}} +25s\cdot H_{\rm u}(\Lambda_{D_{3}})$$ by the above computation in $A_n$-tower. Then we use pullback formula ([\[pullD\]](#pullD){reference-type="ref" reference="pullD"}) for the $D_n$-tower and obtain $$(1- 2(n-1)s) \lambda+sH_{D_{n+2}}+4sH_{E_{n+2}}+25s\cdot H_{\rm u}(\Lambda_{D_{n+1}}).$$ This proves item (5). Now we are going to compute pullback on $\mathop{\mathrm{\mathrm{Sh}}}(D_7')$. By computation in (5), the pullback of $\Delta(s)$ on $\mathop{\mathrm{\mathrm{Sh}}}(D_7)$ is $(1-10s)\lambda+sH_{D_8}+4sH_{E_8}+25sH_{\rm u}$, then Proposition [\[propD\'\]](#propD'){reference-type="ref" reference="propD'"} (1) imply the pullback on $\mathop{\mathrm{\mathrm{Sh}}}(D_7')$ is $$(1-10s)\lambda+2sH_{D_8'}+Z_D$$ where $Z_D$ is linear combination of $\iota^\ast (H_{\rm u})$ and $\iota^\ast H_{D_8}$. Also by formula ([\[pullD\'\]](#pullD'){reference-type="ref" reference="pullD'"}), the pullback on $\mathop{\mathrm{\mathrm{Sh}}}(D_8')$ via $\mathop{\mathrm{\mathrm{Sh}}}(D_8') \rightarrow \mathop{\mathrm{\mathrm{Sh}}}(D_7')$ is given by $$(1-10s)\lambda+2s(-\lambda+H_{D_9'})+Z_D= (1-12s)\lambda+2sH_{D_9'}+Z_D.$$ Continuing computation by last identity in the formula ([\[pullD\'\]](#pullD'){reference-type="ref" reference="pullD'"}), we get the pullback $$(1-12s)+2s(-2\lambda)+\iota'^{\ast} Z_{D_9'}$$ This proves (6). To compute the pullback on $\mathop{\mathrm{\mathrm{Sh}}}(E_6)$, we use the morphism $\nu: \mathop{\mathrm{\mathrm{Sh}}}(E_6) \longrightarrow \mathop{\mathrm{\mathrm{Sh}}}(D_5)$. As we have already determine the pullback of $\lambda+s(25H_u+H_h)$ under $\mathop{\mathrm{\mathrm{Sh}}}(D_5) \longrightarrow {\mathcal F}$ is $$(1-6s)\lambda+sH_{D_{6}}+4sH_{E_{6}}+25s\cdot H_{\rm u}(\Lambda_{D_{5}})$$ Therefore using the pullback formula in formula ([\[pullE2\]](#pullE2){reference-type="ref" reference="pullE2"}) directly we get $$\begin{split} \nu^\ast \Delta(s)=&(1-6s)\lambda+2sH_{E_7}+4s(-\lambda+H_{E_7})+25s\cdot H_{\rm u}(\Lambda_{E_6})\\ =&(1-10s)\lambda+6sH_{E_7}(\Lambda_{E_6})+25s\cdot H_{\rm u}(\Lambda_{E_6}) \end{split}$$ Similarly, the pullback of $\nu^\ast \Delta(s)$ under morphism $\mathop{\mathrm{\mathrm{Sh}}}(E_7) \longrightarrow \mathop{\mathrm{\mathrm{Sh}}}(E_6)$ will be $$\begin{split} (1-10s)\lambda+6s(-\lambda+H_{E_8}(\Lambda_{E_7}))+25s\cdot H_{\rm u}(\Lambda_{E_7})= (1-16s)\lambda+6sH_{E_8}+25s\cdot H_{\rm u}(\Lambda_{E_7}) \end{split}$$ Also under the morphism $\mathop{\mathrm{\mathrm{Sh}}}(E_8) \longrightarrow \mathop{\mathrm{\mathrm{Sh}}}(E_7)$ we get $$\begin{split} (1-16s)\lambda+6s(-2\lambda)+25s\cdot H_{\rm u}(\Lambda_{E_8})=(1-28s)\lambda+25s H_{\rm u}(\Lambda_{E_8}) \end{split}$$ by the 3rd identity in formula ([\[pullE2\]](#pullE2){reference-type="ref" reference="pullE2"}). Then we finish the proof of part (7). By (1) in Proposition [Proposition 72](#unigonalpull){reference-type="ref" reference="unigonalpull"}, we get $$\varphi^\ast (\lambda+s(25H_u+H_h))=(1-25s)\lambda+s H_{A_2}$$ from morphism $\mathop{\mathrm{\mathrm{Sh}}}(E_8\oplus A_1) \rightarrow {\mathcal F}$. The remaining pullback formula is obtained similarly by (2),(3),(4), (5) in Proposition [Proposition 72](#unigonalpull){reference-type="ref" reference="unigonalpull"}. ◻ Based on the above result, we make the following arithmetic prediction for wall crossings of ${\mathcal F}(s)$. **Prediction 1**. *The walls $W_h$ for ${\mathcal F}(s)$ from hyperelliptic divisor $H_h$ can be divided into 5 types* 1. *$A_n$ type walls are given by $W_A=\{ \frac{1}{n} \ |\ n=1,2,3\ \}$;* 2. *$A_n'$ type walls are given by $W_{A'}=\{\ \frac{1}{n} \ |\ n= 4,6,8\ \}$;* 3. *$D_n$ type walls are given by $W_D=\{\ \frac{1}{2n}\ | \ n=2,3,4\ \}$;* 4. *$D_n'$ type walls are given by $W_{D'}=\{\ \frac{1}{2n}\ | \ n=5,6,8\ \}$;* 5. *$E_n$ type walls are given by $W_A=\{\ \frac{1}{\mu_{E_n}} \}$ [^6] where $\mu_{E_n}$ is half of the number of roots in $E_7 \oplus E_n$ that are simply incident to a generator of $A_1$ under the primitive embedding $$E_7 \oplus A_1 \hookrightarrow E_7 \oplus E_n .$$* Assume the divisors $\widetilde{Z}$ in Theorem [Theorem 73](#cbf){reference-type="ref" reference="cbf"} are birationally contractible, by arithmetic principle discussed in §[4.4](#arith){reference-type="ref" reference="arith"}, we get these predictions that walls are exactly the value so that the coefficient of $\lambda$ in $\varphi^\ast \Delta(s)$ vanishes. In this way, we get the Prediction 1. Now let us give an explanation why the modifications appear. - The modification for $A$-type tower appear since $H_{A_5}$ should be a movable divisor on $\mathop{\mathrm{\mathrm{Sh}}}(A_4)$ and thus the corresponding NL locus $\mathop{\mathrm{\mathrm{NL}}}(A_5)$ will be contracted earlier. That is, $\mathop{\mathrm{\mathrm{NL}}}(A_5)$ will be contracted together with $\mathop{\mathrm{\mathrm{NL}}}(A_4)$ and $\mathop{\mathrm{\mathrm{NL}}}(A_5')$ will be contracted after $\mathop{\mathrm{\mathrm{NL}}}(A_4)$. This force the wall-crossing jumps from $A_n$-tower to $A_n'$-tower. There are similar reasons to modification where $A_n''$-tower and $A_n'''$-tower appear. - The modification for the $D$-type tower appear since $H_{D_7}$ on $\mathop{\mathrm{\mathrm{Sh}}}(D_6)$ should be movable. As $\rho(\mathop{\mathrm{\mathrm{Sh}}}(D_9'))=1$, then so is $\rho(\mathop{\mathrm{\mathrm{NL}}}_h(D_9'))=1$, this shows the wall crossing at this stratum will terminate. **Prediction 2**. *The walls for ${\mathcal F}(s)$ from unigonal divisor $H_u$ are given by the NL locus $\mathop{\mathrm{\mathrm{NL}}}_u$. The unigonal walls are $$W_u=\{ \frac{1}{25}, \frac{1}{27}, \frac{1}{28}, \frac{1}{31} \}.$$* As before, we give an explanation for the modification. By remark [Remark 71](#Humodify){reference-type="ref" reference="Humodify"}, $H_{A_2}$ is big on $H_u$ and thus it will be contracted early and thus the modification appears. The walls are predicted by (8) in Theorem [Theorem 73](#cbf){reference-type="ref" reference="cbf"}. **Remark 74**. *We cannot explain the termination of wall-crossing at stratum $\mathop{\mathrm{\mathrm{NL}}}_u(U_6)$ from arithmetic side and we suspect that the morphism $\mathop{\mathrm{\mathrm{Sh}}}(U_4'') \rightarrow {\mathcal F}$ is not isomorphism onto its image $\mathop{\mathrm{\mathrm{NL}}}_u(U_6)$ but just a finite morphism of degree $\ge 2$.* ### Proof of Theorem [Theorem 2](#mthm2){reference-type="ref" reference="mthm2"} {#proof-of-theorem-mthm2 .unnumbered} Under the assumption of birational contractability and finite generation of section ring $R(s)$, Theorem [Theorem 2](#mthm2){reference-type="ref" reference="mthm2"} follows from Theorem [Theorem 73](#cbf){reference-type="ref" reference="cbf"}. [^1]: We warn that $\eta, \xi_i$ are only temporary notations here in the proof and one shall not confuse with the notations in Section [5](#arithmeticstra){reference-type="ref" reference="arithmeticstra"}. [^2]: A unimodular definite lattice of rank $24$ [^3]: The code can be found [here](https://changfeng1992.github.io/SiFei/GIT (3,2) pair in P^3.py). [^4]: This is a Sage package developed by Brandon Williams, for computing with vector-valued modular forms, Jacobi forms and theta lifts. See [@SageDevelopers2022] and [@Williamsa]. [^5]: The data of modular form computation is [here]( https://changfeng1992.github.io/SiFei/Sage notebook for Hodge relations.pdf). [^6]: *$\mu_{E_6}=10, \mu_{E_7}=16, \mu_{E_8}=28$*

arxiv_math

--- abstract: | We are interested in the global bifurcation diagram of radial solutions for the Gelfand problem with the exponential nonlinearity and a radially symmetric weight $0<a(|x|)\in C^2(\overline{B_1})$ in the unit ball. When the weight is constant, it is known that the bifurcation curve has infinitely many turning points if the dimension $N\le 9$, and it has no turning points if $N\ge 10$. In this paper, we show that the perturbation of the weight does not affect the bifurcation structure when $N\le 9$. Moreover, we find specific radial singular solutions with specific weights and study the Morse index of the solutions. As a consequence, we prove that the perturbation affects the bifurcation structure in the critical dimension $N=10$. Moreover, we give an optimal classification of the bifurcation diagrams in the critical dimension. address: Department of Mathematics, Tokyo Institute of Technology author: - Kenta Kumagai bibliography: - bifurcation.bib title: Classification of bifurcation diagrams for semilinear elliptic equations in the critical dimension --- [^1] # Introduction Let $N\ge 3$ and $B_1\subset \mathbb{R}^N$ be the unit ball. We are interested in the global bifurcation diagram for radial solutions for the semilinear elliptic problem $$\label{gelfand} \left\{ \begin{alignedat}{4} -\Delta u&=\lambda a(|x|)e^{u}&\hspace{2mm} &\text{in } B_1,\\ u&>0 & &\text{in } B_1, \\ u&=0 & &\text{on } \partial B_1, \end{alignedat} \right.$$ where $\lambda>0$ is a parameter and $a:[0,1]\to \mathbb{R}$ satisfies the following $$a(|x|)\in C^2(\overline{B_1}), \hspace{2mm} a(r)> 0,\hspace{2mm}\text{and}\hspace{2mm} a(0)=1. \tag{A}$$ ## The case of $a=1$ {#a1} In this case, we remark that each solution of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} is radially symmetric and $\lVert u \rVert_{L^{\infty}(B_1)}=u(0)$ by the symmetric result of Gidas, Ni, and Nirenberg [@Gidas]. Thus we can use ODE techniques and as a result, we verify that the solution set of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} is an unbounded curve described as $(\lambda(\alpha), u(r, \alpha))$ and emanating from $(0,0)$, where $u(r,\alpha)$ is the solution satisfying $\lVert u \rVert_{L^{\infty}(B_1)}=\alpha$. See [@korman; @Mi2014; @Mi2015] for example. We call this set $\{(\lambda(\alpha),\alpha); \alpha>0\}$ the bifurcation curve. A celebrated result of Joseph and Lundgren [@JL] states that the bifurcation structure changes depending on $N$. More precisely, they proved that if $N\le 9$, the curve turns infinitely many times around some $\lambda_*$. We call this property of the curve Type I. On the other hand, if $N\ge 10$, the curve can be parametrized by $\lambda$. In addition, $\alpha (\lambda)$ is increasing and it blows up at some $\lambda_*$. We call this property of the curve Type II. Motivated by their result, a number of attempts have been made in order to determine the bifurcation diagrams for various nonlinearities $f>0$. We refer to [@Mi2015; @Mi2014; @Marius; @KiWei; @Mi2018] for [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} and [@Guowei; @Nor; @Flore; @chend] for related problems. Among them, we note that Miyamoto [@Mi2014] provided an example of $f>0$ for which the bifurcation curve of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} has at least one but finitely many turning points for a sufficiently large $N\ge 10$. We call this property of the curve Type III. Along with the studies of the bifurcation diagrams, many studies on the properties of radial singular solutions have been done for various supercritical nonlinearities $f\ge 0$ (see [@Mi2014; @Merle; @Lin; @Mi2015; @Marius; @Mi2018; @KiWei; @Guowei; @Luo; @chen; @chend; @Mi2020] for details). Here, we say that $(\lambda_{*},U_{*})$ is a radial singular solution of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} if $U_{*}(r)$ is a regular solution of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} in $(0,1]$ satisfying $U_{*}(r)\to\infty$ as $r\to 0$. In addition to the above studies, Miyamoto and Naito [@Mi2023] showed the following properties for a larger class of $f$: there exists the unique radial singular solution $(\lambda_{*}, U_{*})$ such that the bifurcation curve converges to $(\lambda_{*}, U_{*})$. Moreover, they studied the asymptotic behavior of $U_* (r)$ near $r=0$. We point out that the stability of radial singular solutions is deeply connected to the bifurcation structure. More precisely, when $f>0$ is a nondecreasing and convex function in the above class, it follows that the bifurcation diagram is of Type II if and only if the radial singular solution is stable (see [@Br; @Mi2023] and Section [3](#Hardy sec){reference-type="ref" reference="Hardy sec"} for details). Here, we say that a solution $(\lambda,u)$ of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} is stable if the linearized operator $-\Delta-\lambda a(|x|) e^{u}$ is nonnegative in the sense that $$\begin{aligned} \label{stab} Q_{u}(\xi):=\int_{B_1}|\nabla\xi|^2\,dx-\int_{B_1}\lambda a(|x|) e^{u}\xi^2\,dx\ge 0\hspace{6mm}\text{for all $\xi \in C^\infty_0(B_1)$}\notag.\end{aligned}$$ In particular, when $f(u)=e^u$, we remark that the radial singular solution $(2(N-2), -2\log |x|)$ exists and the solution is stable if and only if $N\ge 10$ by the Hardy inequality. Also from the above facts, we can confirm that the bifurcation structure changes at the dimension $10$. Furthermore, Miyamoto [@Mi2014] focused on the the Morse index $m(U_{*})$ of the radial singular solution $(\lambda_{*},U_{*})$ in the space of radial functions and expected that $m(U_{*})$ is equal to the number of turning points of the bifurcation curve. Here, we explain the definition of $m(U_{*})$ precisely: $m(U_{*})$ is the maximal dimension of a subspace $X\subset H^1_{0,\mathrm{rad}}(B_1)$ such that $Q_{U_{*}}(\xi)<0$ for all $\xi\in X\setminus \{0\}$, where $H^1_{0, \mathrm{rad}}(B_1)$ is the space of radially symmetric functions in $H^1_{0}(B_1)$. We remark that $m(U_{*})=0$ if $U_{*}$ is stable. ## Weighted case In this case, in general, we cannot know whether $\lambda$ is parametrized by $\alpha:=\lVert u \rVert_{L^{\infty}(B_1)}$. On the contrary, we prove the following theorem by using the specific change of variables noted in Subsection [2.2](#ababa){reference-type="ref" reference="ababa"}. **Theorem 1**. *Assume that $N\ge 3$ and $a(r)$ satisfies (A). Then, the radial solution set is described as $$\{\left(\lambda(\beta), u(r, \alpha(\beta))\right); \beta\in\mathbb{R}\}\hspace{4mm}\text{with}\hspace{4mm}\alpha(\beta)=\beta-\log \lambda(\beta),$$ where $\alpha(\beta):=\lVert u \rVert_{L^{\infty}(B_1)}$. Moreover, the branch $\mathcal{C}:=\{(\lambda(\beta), \alpha(\beta)); \beta\in \mathbb{R}\}$ is an unbounded analytic curve emanating from $(0,0)$ and there exists $0<\lambda^{*}<\infty$ such that $\lambda(\beta)\le\lambda^{*}$ for all $\beta$. We call this curve the bifurcation curve and we say that $(\lambda(\beta),\alpha(\beta))$ is a turning point if $\lambda(\beta)$ is a local minimum or local maximum.* This change of variables enables us to apply the various methods established for the case where $a=1$. As a result, we prove **Theorem 2**. *Assume that $N\ge 3$ and $a(r)$ satisfies (A). Then, there exists the unique singular radial solution $(\lambda_{*}, U_{*})$ of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} in the sense that if $(\lambda, U)$ is a singular radial solution of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"}, then $\lambda=\lambda_{*}$, $U=U_{*}$. In addition, we have $$\label{convergence schme} \lambda(\beta)\to \lambda_{*} \hspace{4mm} \text{and}\hspace{4mm} u(r,\alpha(\beta))\to U_{*} \hspace{2mm} \text{in $C^2_{\mathrm{loc}}(0,1]$\hspace{4mm} as $\beta\to \infty$}.$$ In particular, it follows that $\alpha(\beta)\to \infty$ if and only if $\beta\to \infty$. Moreover, there exists $\hat{\lambda}>0$ such that the bifurcation curve $\mathcal{C}$ is parametrized by $\lambda$ if $\lambda<\hat{\lambda}$. Furthermore, $U_*\in H^1_{0}(B_1)$ and it satisfies $$\label{asymptotic behavior} U_{*}(r)\to - 2\log r + \log2(N-2)-\log \lambda_{*} \hspace{4mm}\text{as $r\to 0$}.$$* The following theorem indicates that the perturbation of $a$ does not affect the bifurcation structure if $N\le 9$. **Theorem 3**. *Let $3\le N\le 9$ and we assume that $a(r)$ satisfies $(A)$. Then, the bifurcation diagram is of Type I.* ## Main results Motivated from the above fact, we arrive at a fundamental question: does not the perturbation of $a(r)$ make a change to the bifurcation structure in the critical dimension $N=10$? In order to answer this question, we find specific radial singular solutions for specific weighted terms. By studying the stability and the Morse index of the solutions, we show that, for $N=10$, the bifurcation curve exhibits different types depending on the choice of weight $a(r)$. **Theorem 4**. *Let $3\le N\le 10$ and $h>-2(N-2)$. We define $$a_h(r):= \left(1+\frac{h}{2(N-2)}r^2\right)e^{\frac{h}{2N}r^2}\hspace{4mm} and \hspace{4mm} \lambda_{h}=2(N-2)e^{-\frac{h}{2N}}.\notag$$ Then, $$U_h(x):=\frac{h}{2N} -2 \log|x|-\frac{h}{2N}|x|^2\notag$$ is a radial singular solution of ([\[gelfand\]](#gelfand){reference-type="ref" reference="gelfand"}) for $a=a_h$ and $\lambda=\lambda_{h}$. Moreover,* - *$m(U_h)=\infty$ if $N\le 9$,* - *$m(U_h)=0$ and $U_h$ is stable if $N=10$ and $h\le H$,* - *$1\le m(U_h)<\infty$ if $N=10$ and $h>H$,* *where $H$ is the first eigenvalue of the Laplacian in the unit ball in $N=2$.* *As a consequence, when $N=10$ and $a=a_h$, the bifurcation diagram is of Type II if and only if $h\le H$. In particular, the bifurcation structure changes at $a_H$ when $N=10$.* We remark that the exponent $H$ arises from the best constant of the improved Hardy inequality [@Br Theorem 4.1] and this inequality plays a key role in studying the stability/instability of the singular solutions. In addition, thanks to Theorem [Theorem 4](#singularthm){reference-type="ref" reference="singularthm"} and [@Mi2014 Conjecture 1.4], we can expect that the bifurcation diagram is of Type III for $a=a_h$ if $N=10$ and $h>H$. In the following theorem, we show that this expectation is true. Moreover, we give an optimal classification for the bifurcation diagrams in the critical dimension. **Theorem 5**. *Let $N=10$ and we assume that $a(r)$ satisfies $(A)$. Then, the bifurcation diagram is of:* - *Type II if $(a/a_H)'\le 0$ in $(0,1]$,* - *Type III if $(a/a_H)'>0$ in $(0,1]$,* *where $H$ and $a_H$ are those in Theorem [Theorem 4](#singularthm){reference-type="ref" reference="singularthm"}. In particular, when $N=10$ and $a=a_h$, the bifurcation diagram is of Type II if $h\le H$ and of Type III if $h>H$.* We recall that the classical Hardy inequality controls the bifurcation structure when $a=1$. On the other hand, Theorem [Theorem 5](#Mainthm){reference-type="ref" reference="Mainthm"} tells us that the improved Hardy inequality controls it when the weight $a$ is general and $N=10$. Therefore, our results state not only the important difference for the bifurcation structure between the case of $a=1$ and the weighted case, but also the reason why the difference happens. We also remark that it is a challenging problem to show that the bifurcation diagram is of Type III. In addition, to the best of the author's knowledge, the bifurcation of type III was only confirmed in the case where $a=1$ and a cleverly chosen $f$. We not only provide many examples of Type III bifurcation but also give a new method to prove that the bifurcation diagram is of Type III. Finally, we remark that we can prove the change of the bifurcation structure when $N\ge 11$ and a similar classification holds when $N=11$ or $a=a_h$ by using similar methods. ## Application: A change of the regularity property of extremal solutions. {#applisub} In this subsection, we focus on the minimal branch of the bifurcation curve. It is well known [@Ali; @B; @Br; @Dup] that the set of minimal solutions is a continuous curve emanating from $(0,0)$ and there exists $\lambda^* \in(0,\infty)$ such that - For $0<\lambda<\lambda^*$, there exists a minimal classical solution $u_\lambda\in C^2(\overline{B_1})$. In particular, $u_\lambda$ is radially symmetric, stable, and $u_\lambda<u_{\lambda'}$ for $\lambda<\lambda'$. - For $\lambda=\lambda^*$, we define $u^*:=\lim_{\lambda \uparrow \lambda^* } u_\lambda$. Then, $u^*$ is called the extremal solution, which is a radially symmetric stable weak solution in the sense that $u^*\in L^1(B_1)$, $a(|x|)e^{u^*}\mathrm{dist}(\cdot,\partial \Omega)\in L^1(B_1)$, and $$-\int_{B_1}u^*\Delta \xi \,dx = \int_{B_1}\lambda^*a(|x|)e^{u^*}\xi \,dx\hspace{8mm} \text{for all}\hspace{2mm}\xi\in C^2_0(\overline{B_1}).\notag$$ - For $\lambda>\lambda^*$, there exists no weak solution. We remark that when the nonlinearity $f>0$ is nondecreasing and superlinear (i.e., $f(u)/u \to \infty$ as $u\to\infty$), the above properties hold even if the weight $a$ and the domain $\Omega$ are arbitrary.[^2] Thus, the boundedness of extremal solutions is well-studied instead of the bifurcation diagrams. We refer to [@CR; @Ye; @A2016; @AC; @Davdup; @sansan; @Dup; @Ned; @cabre2017; @cabre2019; @cabre2010; @CCS; @CR-O; @Vil; @cabrecapella; @CFRS]. As a consequence of Theorem [Theorem 5](#Mainthm){reference-type="ref" reference="Mainthm"}, we get **Corollary 6**. *Let $3\le N\le10$ and we assume that $a(r)$ satisfies $(A)$. Then, the extremal solution of $(\ref{gelfand})$ is:* - *bounded if $N\le 9$,* - *singular if $(a/a_H)'\le 0$ in $(0,1]$ and $N=10$,* - *bounded if $(a/a_H)'>0$ in $(0,1]$ and $N=10$,* *where $H$ and $a_H$ are those in Theorem [Theorem 4](#singularthm){reference-type="ref" reference="singularthm"}.* We introduce known results in the specific case of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"}. It is known by the method in [@CR] that $u^*$ is bounded when $N\le 9$. Moreover, when $N\ge 10$, Bae [@Bae] proved that $u^*$ is singular if $$\label{bae1} a(r)\le \frac{N-2}{8}\inf_{0\le s \le r} a(s) \hspace{4mm}\text{for $r>0$}.$$ We note that the condition ([\[bae1\]](#bae1){reference-type="ref" reference="bae1"}) means that $a(r)$ is nonincreasing when $N=10$. Thus, Corollary [Corollary 6](#maincor){reference-type="ref" reference="maincor"} is not only a generalization of known results when $N\le 10$ but also the first result which states that the boundedness property of $u^*$ changes depending on the perturbation of $a$. Moreover, we remark that when $a=1$ and $\Omega$ is general, Crandall and Rabinowitz [@CR] proved that $u^*$ is bounded if $N\le 9$. Based on this result, Brezis and Vázquez asked in [@Br] whether $u^*$ is always singular for all $\Omega$ if $N\ge 10$ and $a=1$. This expectation was answered negatively: some examples of $\Omega$ were constructed which $u^*$ is bounded for some $N\ge 10$. We refer to [@Davdup; @AC]. Corollary [Corollary 6](#maincor){reference-type="ref" reference="maincor"} describes a similar property to the above fact. ## Structure of the paper In Section [2](#basic){reference-type="ref" reference="basic"}, we combine the specific change of variables with the method established for the case when $a=1$ and as a result, we prove Theorem [Theorem 1](#diagramth){reference-type="ref" reference="diagramth"} and Theorem [Theorem 2](#singularth){reference-type="ref" reference="singularth"}. In Section [3](#Hardy sec){reference-type="ref" reference="Hardy sec"}, we prove the optimality[^3] of the improved Hardy inequality. By using it, we study the stability and the Morse index of $U_h$ and prove Theorem [Theorem 4](#singularthm){reference-type="ref" reference="singularthm"}. In Section [4](#separate sec){reference-type="ref" reference="separate sec"}, we prove the comparison results. The results work well to prove Theorem [Theorem 5](#Mainthm){reference-type="ref" reference="Mainthm"}. In Section [5](#mainsec){reference-type="ref" reference="mainsec"}, we prove Theorem [Theorem 3](#39th){reference-type="ref" reference="39th"} by using the idea of [@Mi2014] and we prove Theorem [Theorem 5](#Mainthm){reference-type="ref" reference="Mainthm"}. # Basic properties of radial solutions {#basic} In this section, we prove basic properties of radial solutions and as a result, we prove Theorem [Theorem 1](#diagramth){reference-type="ref" reference="diagramth"} and Theorem [Theorem 2](#singularth){reference-type="ref" reference="singularth"}. We first fix the notation. In this paper, we only deal with radial solutions. Thus, from now on, we identify the the operator $\Delta$ with the operator $\frac{d^2}{dr^2}+\frac{N-1}{r}\frac{d}{dr}$, and $r$ with $|x|$. ## Preliminaries In this subsection, we prove some basic properties for radial solutions of $$\label{apriorieq} -\Delta u = \lambda a(r) e^u, \hspace{4mm}0<r<R,$$ where $R>0$ and $\lambda$ are positive constants and $a:[0,R]\to \mathbb{R}$ satisfies the following $$a(|x|)\in C^2(\overline{B_R}), \hspace{2mm} a(r)> 0,\hspace{2mm}\text{and}\hspace{2mm} a(0)=1. \tag{$\hat{A}$}$$ We note that the following results in this subsection are proved as a modification of [@Mi2020]. We begin by introducing an a priori estimate of positive radial solutions. **Lemma 7**. *Let $0<\lambda_0<\lambda_1$ and let $u$ be a positive radial solution of [\[apriorieq\]](#apriorieq){reference-type="eqref" reference="apriorieq"} for some $\lambda_0<\lambda<\lambda_1$. We assume that $a(r)$ satisfies $(\hat{A})$. Then, there exist $C_1>0$ and $C_2>0$ depending only on $\lambda_0, \lambda_1, a(r), R$ such that $$\label{aprioriestimate 2-1} u(r)\le -2\log r + C_1, \hspace{4mm} 0\le -\frac{d}{dr}u(r) \le \frac{C_2}{r} \hspace{4mm} \text{for $0<r\le R$},$$ and $$\label{aprioriestimate 2-2} -r^{N-1}\frac{d}{dr}u(r)= \lambda\int_{0}^{r}s^{N-1}a(s)e^{u(s)}\,ds \hspace{4mm} \text{for $0<r\le R$}.$$* We remark that we can obtain the a priori estimate in the same way as in [@Mi2020 Lemma 2.3] and thus we omit the proof. Then, we quote the result in [@Mi2020]. Let $t_0\in \mathbb{R}$, $p>0$, and $-\infty\le \alpha<\beta\le \infty$. In addition, we take $H\in C^{0}((\alpha,\beta))$, $G\in C^{0}([t_0,\infty)\times (\alpha,\beta))$ and we assume that there exists $\gamma\in (\alpha,\beta)$ such that $$(w-\gamma)H(w)>0\hspace{4mm}\text{for all $w\in (\alpha,\beta)\setminus \{\gamma\}$}.$$ We remark that the assumption implies $H(\gamma)=0$. In [@Mi2020], the authors consider the following ODE $$\label{general} \frac{d^2}{dr^2}w-p\frac{d}{dr}w+H(w)+G(t,w)=0 \hspace{4mm}\text{for $t\ge t_0$}$$ and they prove the following **Lemma 8**. *Let $w\in C^2([t_0,\infty))$ be a solution of [\[general\]](#general){reference-type="eqref" reference="general"} satisfying $\alpha<w(t)<\beta$ for all $t\ge t_0$. Assume that $w$ satisfies $$\alpha<\limsup_{t\to\infty} w(t)<\beta\hspace{4mm}and \hspace{4mm} \lim_{t\to\infty}G(t, w(t))=0.$$ If $\alpha=-\infty$, assume in addition that $$\lim_{t\to\infty} \frac{G(t, w(t))}{e^{w(t)}}=0.$$ Then, $\lim_{t\to\infty}w(t)=\gamma$.* By using Lemma [Lemma 8](#ordlem){reference-type="ref" reference="ordlem"}, we study the asymptotic behavior of positive radial singular solutions of [\[apriorieq\]](#apriorieq){reference-type="eqref" reference="apriorieq"}. **Proposition 9**. *Let $u$ be a positive singular radial solution of [\[apriorieq\]](#apriorieq){reference-type="eqref" reference="apriorieq"}. We assume that $a(r)$ satisfies $(\hat{A})$. Then, $u$ satisfies $$u(r)\to -2\log r -\log \lambda+ \log2(N-2) \hspace{2mm}\text{as $r\to 0$}.$$* *Proof.* We define $$w(t)=u(r)+2\log r + \log \lambda-\log 2(N-2)\hspace{2mm}\text{with $t=-\log r$}.$$ Then, $w$ satisfies $$\frac{d^2}{dt^2}w-(N-2)\frac{d}{dt}w+2(N-2)(e^w-1) + 2(N-2)(a(e^{-t})-1) e^w=0$$ for $t\ge t_0$ with $t_0:=-\log R$. Moreover, we obtain $$\begin{aligned} \label{fact2-1} -\infty<\limsup_{r\to 0}(u(r)+2\log r)\le C_1\end{aligned}$$ by using Lemma [Lemma 7](#apriorilemma){reference-type="ref" reference="apriorilemma"} and the same method as in the proof of [@Mi2020 Lemma 2.4], where $C_1$ is that in [\[aprioriestimate 2-1\]](#aprioriestimate 2-1){reference-type="eqref" reference="aprioriestimate 2-1"}. The existence of this change of variables and the fact [\[fact2-1\]](#fact2-1){reference-type="eqref" reference="fact2-1"} allow for us to apply Lemma [Lemma 8](#ordlem){reference-type="ref" reference="ordlem"} for $p=N-2$, $H(w)=2(N-2)(e^{w}-1)$, $G(t,w)=2(N-2)(a(e^{-t})-1)e^w$, $\alpha=-\infty$, $\beta=\infty$. As a result, it holds $\lim_{t\to \infty} w(t)=0$ and we get the result. ◻ ## Proof of Theorem [Theorem 1](#diagramth){reference-type="ref" reference="diagramth"} and Theorem [Theorem 2](#singularth){reference-type="ref" reference="singularth"} {#ababa} We begin by introducing a specific change of variables and proving Theorem [Theorem 1](#diagramth){reference-type="ref" reference="diagramth"}. We extend $a$ on $[1,\infty)$ a positive function such that $a(r)\in C^2(\mathbb{R})$. For any $\beta\in \mathbb{R}$, we define $v=v(r,\beta)$ as the solution of $$\label{odev} \left\{ \begin{alignedat}{4} &-\Delta v= a(r)e^{v}, \hspace{14mm}0<r<\infty,\\ &v(0,\beta)=\beta, \hspace{4mm} \frac{d}{dr} v(0,\beta)=0. \end{alignedat} \right.$$ *Proof of theorem [Theorem 1](#diagramth){reference-type="ref" reference="diagramth"}.* For any $\beta\in \mathbb{R}$, we define $u(r,\alpha(\beta))=v(r, \beta)-\log \lambda(\beta)$ with $\lambda(\beta):=e^{v(1,\beta)}$ and $\alpha(\beta):=\beta- \log \lambda(\beta)$. Then, we can easily confirm that $(\lambda(\beta), u(r,\alpha(\beta)))$ is a solution of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"}. On the other hand, if $(\lambda, u)$ is a solution of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} which satisfies $\alpha=\lVert u \rVert_{L^{\infty}(B_1)}$ for some $\alpha>0$, then $v(r,\beta):=u+\log \lambda$ is a solution of [\[odev\]](#odev){reference-type="eqref" reference="odev"} for $\beta=\alpha+\log \lambda$. Thus, we have $(\lambda,u)=(\lambda(\beta), u(r,\alpha(\beta)))$. Therefore, every radial solution is parametrized by $\beta$. Moreover, the analyticity of this curve follows from the analyticity of the exponential function. The left part can be proved by using the properties of the minimal branch (see Subsection [1.4](#applisub){reference-type="ref" reference="applisub"}). ◻ Then, based on the idea of [@Lin; @Mi2020], we prove the following **Proposition 10**. *Let $q>\frac{N+2}{N-2}$, $\lambda>0$. We assume that $a(r)$ satisfies $(\hat{A})$ and $u\in C^2(B_R)$ is a positive radial solution of [\[apriorieq\]](#apriorieq){reference-type="eqref" reference="apriorieq"} satisfying $u(R)=0$.* - *Let $u(0)>q+1$ and we assume that $u(r)\ge q+1$ for $0\le r\le r_0$ with some $r_0>0$. Then, $$0<-r\frac{du}{dr}(r)<\frac{2N}{q+1}u(r)+ C\hspace{4mm}\text{for $0<r<r_0$,}$$ where $C$ is a constant depending only on $a$ and $R$.* - *Define $$\tau=\frac{1}{3}\left(1-\frac{2N}{(q+1)(N-2)}\right).$$ Take any $\rho\ge q+1+C$ and define $r_{\rho}$ by $$r_{\rho}:=\left(\frac{2N\rho}{\lambda e^{\rho/\tau}\lVert a\rVert_{L^{\infty}(B_R)} }\right)^{1/2},$$ where $C$ is that in (i). If $u(0)>\rho/\tau$ then $u$ satisfies $$u(r)>\rho \hspace{4mm}\text{for $0\le r \le r_{\rho}$.}$$* In order to prove this proposition, we quote a Pohozaev-type identity. This identity is proved by Ni and Serrin [@Niserrin] when $a=1$. **Lemma 11**. *Let $u$ be a radial solution of [\[apriorieq\]](#apriorieq){reference-type="eqref" reference="apriorieq"} and let $\mu$ be an arbitrary constant. We assume that $a(r)$ satisfies $(\hat{A})$. Then, for any $r\in (0,R)$, we have $$\begin{aligned} &\frac{d}{dr}\left\{r^N\left(\frac{1}{2}\left(\frac{du}{dr}\right)^2 +\lambda a(r) (e^{u(r)}-1) +\frac{\mu}{r}u(r)\frac{du}{dr}\right)\right\}-\lambda \frac{da}{dr}(e^{u(r)}-1)r^N\notag\\ &=r^{N-1}\left\{N\lambda a(r)(e^{u(r)}-1)-\lambda\mu a(r) u(r)e^{u(r)}+\left(\mu+1-\frac{N}{2}\right)\left(\frac{du}{dr}\right)^2\right\}.\notag\end{aligned}$$* *Proof.* By a direct computation, we get the result. ◻ *Proof of Proposition [Proposition 10](#pohopro){reference-type="ref" reference="pohopro"}.* Put $\mu:=N/(q+1)$. Then, we remark that the right-hand side of the inequality in Lemma [Lemma 11](#generalpohozaev){reference-type="ref" reference="generalpohozaev"} is negative for $0\le r \le r_0$ since it holds $N(e^{u(r)}-1)<\mu e^{u(r)}u(r)$ for $0\le r\le r_0$. By integrating the inequality in Lemma [Lemma 11](#generalpohozaev){reference-type="ref" reference="generalpohozaev"} on $[0,r]$ with $0\le r\le r_0$, we obtain $$\frac{1}{2}r^{N} \left(\frac{du}{dr}\right)^2 +\lambda a(r)r^{N} (e^{u(r)}-1) +\mu r^{N-1}\frac{du}{dr}u<\lambda\int_{0}^{r}s^{N} \frac{da}{ds}(e^{u(s)}-1)\,ds.$$ Moreover, it follows by ([\[aprioriestimate 2-2\]](#aprioriestimate 2-2){reference-type="ref" reference="aprioriestimate 2-2"}) that $$\begin{aligned} \lambda\int_{0}^{r} s^{N}\frac{da}{ds}(e^{u(s)}-1)\,ds\le Cr\lambda\int_{0}^{r} s^{N-1}a(s)e^{u(s)}\,ds\le -Cr^{N-1}\frac{du}{dr}, \end{aligned}$$ where $C>0$ is depending only on $a$ and $R$. Thus, we get (i). As a result, we can prove (ii) by using a method similar to the proof of [@Mi2020 Lemma 5.1 (ii)]. ◻ *Proof of Theorem [Theorem 2](#singularth){reference-type="ref" reference="singularth"}..* We divide the proof into the following three steps. **Step 1.** *The bifurcation curve is parametrized by $\lambda$ when $\lambda$ is sufficiently small.* We first claim that there exist $C_1$, $\delta$ depending only on $a$, $N$ such that if $\alpha(\beta)>C_1$, then $\lambda(\beta)>\delta$. Indeed, let $q$ and $\rho$ be constants satisfying the assumption in Proposition [Proposition 10](#pohopro){reference-type="ref" reference="pohopro"} and we define $C_1:= \rho/\tau$. Let $\delta$ be a small constant such that $\tau_{\rho}\ge1$ if $\lambda<\delta$. Then, thanks to Proposition [Proposition 10](#pohopro){reference-type="ref" reference="pohopro"}, if $\lambda(\beta)\le \delta$, then $\alpha(\beta)\le C_1$. Thus, we get the claim. Let $\eta:=\lVert u^*\rVert_{L^\infty(B_1)}+\log \lambda^*$, where $u^*$ is the extremal solution with $\lambda=\lambda^*$. We fix a constant $\hat{\lambda}<\min\{\delta,\lambda^{*}\}$ such that $C_1+\log \hat{\lambda}<\eta$. Let $(\lambda(\beta),u(r,\alpha(\beta))$ be a solution of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} such that $\lambda(\beta)\le \hat{\lambda}$. Since $\hat{\lambda}<\delta$, we get $\alpha(\beta)\le C_1$. Moreover, since $\beta=\alpha(\beta)+\log \lambda(\beta)$, we have $\beta<\eta$. We recall that the bifurcation curve is parametrized by $\beta$. Therefore, the solution $(\lambda(\beta), u(r,\alpha(\beta)))$ belongs to the minimal branch of $C$. Since the minimal branch is parametrized by $\lambda$, we get the result. We recall that $\beta=\alpha(\beta)+\log \lambda(\beta)$ and $\lambda(\beta)\le \lambda^{*}$ for all $\beta$. Thus, as a consequence of Step 1, we verify that $\alpha(\beta)\to\infty$ if and only if $\beta\to\infty$. **Step 2.** *The radial singular solutions $(\lambda_{*},U_{*})$ are at most one.* Let $(\lambda^{1}_{*}, U^1_{*})$ and $(\lambda^{2}_{*}, U^2_{*})$ be radial singular solutions. Thanks to Proposition [Proposition 9](#asymptotic prop){reference-type="ref" reference="asymptotic prop"}, $(\lambda^{1}_{*}, U^1_{*})$ and $(\lambda^{2}_{*}, U^2_{*})$ satisfy [\[asymptotic behavior\]](#asymptotic behavior){reference-type="eqref" reference="asymptotic behavior"}. Let $V^{i}_{*}:=U^{i}_{*}+\log \lambda^{i}_{*}$ with $i=1,2$. Then $V_{*}=V^{i}_*$ satisfies $$\label{singulareq} \left\{ \begin{alignedat}{4} -\Delta V_{*}&= a(r)e^{V_{*}}, &&0<r<\infty,\\ V_{*}(r)&= -2\log r + \log 2(N-2)+o(1) \hspace{6mm} &&\text{as $r\to 0$}. \end{alignedat} \right.$$ Moreover, we have $V^{i}_*(1)=\log \lambda^{i}_{*}$. By [@Bae Theorem 1.6], it holds $V^{1}_*=V^{2}_*$. In particular, since $V^{1}_*(1)=V^{2}_*(1)$, it holds $\lambda^{1}_{*}=\lambda^{2}_{*}$ and $U_*^{1}=U_*^{2}$. **Step 3** *Existence of the radial singular solution and [\[convergence schme\]](#convergence schme){reference-type="eqref" reference="convergence schme"}.* Let $\delta>0$ and let $(\lambda(\beta_{k}), u(r,\alpha(\beta_{k})))\in C$ with $\beta_{k}\to\infty$ as $k\to\infty$. Thanks to Step 1, we get $\alpha(\beta_k)\to \infty$ as $k\to\infty$ and $\hat{\lambda}< \lambda(\beta_k) \le \lambda^*$ for all $k$ sufficiently large. Moreover, by Lemma [Lemma 7](#apriorilemma){reference-type="ref" reference="apriorilemma"} and the elliptic regularity theory (see [@gil]), there exists some $0<\delta_0<1$ such that it holds $\lVert u(r,\alpha(\beta_k))\rVert_{C^{2+\delta_0}(B_1 \setminus B_{\delta})}<C$, where $C$ is a constant independent of $k$. Since $\delta>0$ is arbitrary, by Ascoli-Arzelá theorem and a diagonal argument, there exists a subsequence $k_j$ of $k$ and a solution $U_{*}\in C^2_{\mathrm{loc}}((0,1])$ of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} for $\lambda=\lambda_{*}$ such that $u(r,\alpha(\beta_{k_{j}}))\to U_{*}$ in $C^2_{\mathrm{loc}}((0,1])$ as $j\to\infty$ and $\lambda(\beta_{k_{j}})\to \lambda_{*}$ as $j\to \infty$. Moreover, thanks to Proposition [Proposition 10](#pohopro){reference-type="ref" reference="pohopro"}, we verify that $(\lambda_{*}, U_{*})$ is a radial singular solution. By the uniqueness of the radial singular solution, we have $u(r,\alpha(\beta_{k}))\to U_{*}$ as $\beta_k\to\infty$. We can easily know the left part by using Lemma [Lemma 7](#apriorilemma){reference-type="ref" reference="apriorilemma"} and Proposition [Proposition 9](#asymptotic prop){reference-type="ref" reference="asymptotic prop"}. ◻ ## Some notations From now on, we fix some notations. We define $v(r,\beta)$ and $V_{*}$ as those in [\[odev\]](#odev){reference-type="eqref" reference="odev"} and [\[singulareq\]](#singulareq){reference-type="eqref" reference="singulareq"} respectively. Then, we define $v_h(r,\beta)$ and $V_h$ as those in [\[odev\]](#odev){reference-type="eqref" reference="odev"} and [\[singulareq\]](#singulareq){reference-type="eqref" reference="singulareq"} for $V_{*}=V_{h}$ and $a=a_h$, where $a_h$ is that in Theorem [Theorem 4](#singularthm){reference-type="ref" reference="singularthm"}. In particular, $v_0(r,\beta)$ and $V_0$ mean those for $a=a_0=1$ and we remark that $V_0(r)=-2 \log r$. Moreover, we define $(\lambda(\beta), u(r,\alpha(\beta)))$ as that in Theorem [Theorem 1](#diagramth){reference-type="ref" reference="diagramth"} and we define $a_h, \lambda_h, U_h$ as those in Theorem [Theorem 4](#singularthm){reference-type="ref" reference="singularthm"}. Then, we define $u^*$ as the extremal solution for $\lambda=\lambda^*$. # Stability and Morse index of the specific singular solutions {#Hardy sec} In this section, we study the stability and the Morse index of $U_h$, where $U_h$ is that in Theorem [Theorem 4](#singularthm){reference-type="ref" reference="singularthm"}. We first recall the improved Hardy inequality. **Proposition 12** (see [@Br]). *Let $N\ge 2$ and $\Omega$ be a bounded domain in $\mathbb{R}^{N}$. Then, for every $\xi\in H^{1}_{0}(\Omega)$ we have $$\label{imp} \int_{\Omega}|\nabla \xi|^2\,dx\ge \frac{(N-2)^2}{4}\int_{\Omega} \frac{\xi^2}{|x|^2}\,dx+ H \left(\frac{\omega_N}{|\Omega|}\right)^{\frac{2}{N}}\int_{\Omega} \xi^2\,dx,$$ where $H$ is the first eigenvalue of the Laplacian in the unit ball in $N=2$ and $\omega_N$ is the measure of the $N$-dimensional unit ball.* We consider only the case $\Omega=B_1$. We remark that the optimizer of the inequality ([\[imp\]](#imp){reference-type="ref" reference="imp"}) is $\xi=|x|^{\frac{2-N}{2}}\varphi$, where $\varphi>0$ is a first eigenfunction of the Laplacian in the unit ball in $N=2$ and it does not belong to $H^1_{0}(B_1)$. As mentioned in the introduction, the improved Hardy inequality plays a important role in studying the stability/instability of $U_h$. Thus, it is important to prove the optimality of the best constant $H$ for the test functions in $H^1_{0}(B_1)$. In other words, we find a test function $\hat{\xi}\in H^1_{0}(B_1)$ such that the inequality ([\[imp\]](#imp){reference-type="ref" reference="imp"}) does not hold for $\xi=\hat{\xi}$ if we replace $H$ with $H+\varepsilon$. To be more precise, we prove **Proposition 13**. *Let $N\ge 2$ and let $\varepsilon:[0,1]\to [0,1]$ be a nondecreasing function such that $\varepsilon(\frac{1}{2})>0$. Then, there exists $\xi\in H^{1}_{0}(B_1)$ such that $$\int_{B_1}|\nabla \xi|^2\,dx< \frac{(N-2)^2}{4}\int_{B_1} \frac{\xi^2}{|x|^2}\,dx+ \int_{B_1} \left(H+\varepsilon (|x|)\right)\xi^2\,dx.\notag$$* *Proof.* Let $\varepsilon>0$ and $n\in \mathbb{N}$. We define $$\phi_n(x)=\min\{\frac{n}{1-\log |x|}, 1\}, \hspace{4mm} \xi_{n}(x)= \phi_{n}\varphi|x|^{\frac{2-N}{2}}\notag,$$ where $\varphi>0$ is a first eigenfunction of the Laplacian in the unit ball in $N=2$. Then it holds $\xi_{n}\in H^1_{0}(B_1)$. Moreover, by a direct computation, we have $$\begin{aligned} \int_{B_1}&|\nabla \xi_{n}|^2\,dx-\frac{(N-2)^2}{4}\int_{B_1} \frac{\xi_{n}^2}{|x|^2}\,dx- \int_{B_1} \left(H+\varepsilon(|x|) \right)\xi_{n}^2\,dx\notag\\ &=N\omega_{N}\int_{0}^{1}\left((\phi_{n}\varphi)'\right)^2 r-(N-2)\phi_{n}\varphi(\phi_{n}\varphi)'-\left(H+\varepsilon \right)(\phi_{n}\varphi)^2 r\,dr \notag\\ &=N\omega_{N}\int_{0}^{1}\left((\phi_{n}\varphi)'\right)^2 r -(H+\varepsilon )(\phi_{n}\varphi)^2 r\,dr \notag.\end{aligned}$$ Let $\delta>0$ be a constant which satisfies $$\label{yoikanzi} H\delta^2 \int^{1}_{0}\varphi^2 r\,dr \le\frac{1}{2} \int^{1}_{0}\varepsilon\varphi^2 r\,dr.$$ We note that $0<\phi_{n}\le 1$ for all $0<r<1$ and $\phi_{n}=1$ for $r>e^{1-n}$. In addition, we note that it holds $$\int_{0}^{1}(\varphi')^2 r\,dr= H\int_{0}^{1}\varphi^2 r\,dr.$$ By Young's inequality, ([\[yoikanzi\]](#yoikanzi){reference-type="ref" reference="yoikanzi"}), and the above notation, we get $$\begin{aligned} &\int_{0}^{1}\left((\phi_{n}\varphi)'\right)^2 r -(H+\varepsilon)(\phi_{n}\varphi)^2 r\,dr \notag\\ &\hspace{2mm}\le(1+\delta^2)\int_{0}^{1} \phi_{n}^2(\varphi')^2 r\,dr-\int_{0}^{1}(H+\varepsilon)(\phi_{n}\varphi)^2 r\,dr+\left(1+\frac{1}{\delta^2}\right)\int_{0}^{1} (\phi_{n}')^2\varphi^2 r\,dr\notag\\ &\hspace{2mm}\le H(1+\delta^2)\int_{0}^{1}\varphi^2 r\,dr-\int_{e^{n-1}}^{1}(H+\varepsilon)\varphi^2 r\,dr+\left(1+\frac{1}{\delta^2}\right)\int_{0}^{1} (\phi_{n}')^2\varphi^2 r\,dr\notag\\ &\hspace{2mm}\le \int_{0}^{e^{1-n}} \left(\left(1+\frac{1}{\delta^2}\right)\frac{n^2}{r(1-\log r)^4}+(H+\varepsilon)r\right)\varphi^2\,dr\notag\\ &\hspace{62mm}+H\delta^2\int_{0}^{1}\varphi^2 r\,dr -\int_{0}^{1}\varepsilon\varphi^2 r\,dr\notag\\ &\hspace{2mm}\le \left(\frac{1}{3n}\left(1+\frac{1}{\delta^2}\right)+\frac{(H+1)e^{2-2n}}{2}\right)\lVert \varphi\rVert_{L^{\infty}(B_1)}^2-\frac{1}{2} \int_{e^{1-n}}^{1}\varepsilon\varphi^2 r\,dr\notag\\ &\hspace{2mm}\to -\frac{1}{2} \int_{0}^{1}\varepsilon\varphi^2 r\,dr<0\hspace{4mm}\text{as $n\to \infty$.}\notag\end{aligned}$$ Thus, by letting $n$ sufficiently large, the result holds for $\xi=\xi_{n}$. ◻ In preparation for proving Theorem [Theorem 4](#singularthm){reference-type="ref" reference="singularthm"}, we quote the following proposition that describes the relationship between the bifurcation structure and the stability of radial singular solutions. **Proposition 14** (see [@Br; @B]). *Let $(\lambda_{*}, U_{*})$ be a singular solution of ([\[gelfand\]](#gelfand){reference-type="ref" reference="gelfand"}) such that $U_{*}\in H^1_{0}(B_1)$ and $U_{*}$ is stable. Then, $U_{*}$ is the extremal solution for $\lambda_{*}$. In particular, the bifurcation diagram is of Type II.* We remark that the authors prove Proposition [Proposition 14](#brezis V){reference-type="ref" reference="brezis V"} in [@Br] by applying [@B Corollary 2] when $a=1$. As the same way, we can prove it for the weighted case. Moreover, we remark that the extremal solution of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} is radially symmetric and stable (see Subsection [1.4](#applisub){reference-type="ref" reference="applisub"}). Thus, thanks to Theorem [Theorem 2](#singularth){reference-type="ref" reference="singularth"} and this proposition, we verify that the bifurcation diagram is of Type II if and only if the radial singular solution is stable. *Proof of Theorem [Theorem 4](#singularthm){reference-type="ref" reference="singularthm"}..* By a direct computation, we verify that $U_{h}$ is the radial singular solution of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} for $a=a_{h}$ and $\lambda= \lambda_{h}$. Moreover, we verify that the linearized operator $-\Delta - a_{h} e^{U_h}$ is equal to $-\Delta-2(N-2)|x|^{-2}-h$ and it holds $$\begin{aligned} \label{qnohyouzi} Q_{U_h}(\xi)= \int_{B_1}|\nabla \xi|^2\,dx- 2(N-2)\int_{B_1} \frac{\xi^2}{|x|^2}\,dx- h \int_{B_1} \xi^2\,dx\end{aligned}$$ for all $\xi\in H^1_{0}(B_1)$. We claim that $U_h$ is stable if and only if $m(U_h)=0$. Indeed, we denote by $\xi^*\in H^1_{0}(B_1)$ the symmetric rearrangement of $\xi$. Then it follows by Lemma [Lemma 23](#apenlem1){reference-type="ref" reference="apenlem1"} that $Q_{U_h}(\xi^*)\le Q_{U_h}(\xi)$. Thus, we get the claim. **Case 1.** *The case of $N\le 9$.* In this case, we use a similar method used in [@Mi2014]. Let $\varepsilon>0$ be a small constant such that $$\delta:=2(N-2)-\frac{(N-2)^2+\varepsilon^2}{4}>0$$ and we define $\xi_{j}(r)=r^{\frac{2-N}{2}} \sin(\frac{\varepsilon}{2}\log r) \chi_{[r_{j+1},r_j]}$ with $r_j=e^{-2\pi j/\varepsilon}$. Then, it holds $\xi_{j}\in H^1_{0}(B_1)$ and it satisfies $$-\Delta \xi_j=\left(\frac{(N-2)^2+\varepsilon^2}{4}\right)r^{-2} \xi_j\hspace{4mm}\text{for all $r\in (r_{j+1},r_j)$}.$$ Thus, we get $$\begin{aligned} Q_{U_h}(\xi_j)\le |h|\int_{B_{r_{j}}}\xi_{j}^2\,dx-\delta \int_{B_{r_j}}\frac{\xi_{j}^2}{|x|^2}\,dx<0\end{aligned}$$ for all $j$ sufficiently large. Since $\mathrm{supp}(\xi_j)\cap \mathrm{supp}(\xi_k)=\phi$ if $j\neq k$, we have $m(U_h)=\infty$ for all $h>-2(N-2)$. **Case 2.** *The case of $N=10$.* We first point out that $2(N-2)=\frac{(N-2)^2}{4}$. Thus, thanks to Proposition [Proposition 12](#improved hardy){reference-type="ref" reference="improved hardy"} and Proposition [Proposition 13](#improvedhardy2){reference-type="ref" reference="improvedhardy2"}, we verify that $U_h$ is stable if and only if $h\le H$. As a result, the bifurcation diagram is of Type II if and only if $h\le H$. Moreover, we have $m(U_h)=0$ if $h\le H$ and $m(U_h)\ge 1$ if $h>H$. Thus, it suffices to prove that $m(U_{h})<\infty$. We define $X\subset H^1_{0,\mathrm{rad}}(B_1)$ as a maximal space such that $Q_{U_h}(\xi)<0$ for all $\xi\in X\setminus \{0\}$ and we define $Y\subset H^1_{0,\mathrm{rad}}(B_1^2)$ as a maximal space such that $<\left(-\Delta-h\right)\xi,\xi>_{L^2(B_1^2)}<0$ for all $\xi \in Y\setminus \{0\}$, where $B_1^2$ is the 2-dimensional unit ball. We consider the linear map $\Lambda:X\to Y$ defined for $\xi\in X$ by $\Lambda \xi(|x|)=|x|^{\frac{2-N}{2}}\xi(|x|)$. Since the map is injective, it holds $\dim X\le \dim Y$. We consider the eigenvalue problem $$(-\Delta-h) u = \mu u \hspace{4mm}\text{in $B^2_{1}$}, \hspace{4mm} u\in H^1_{0,\rm{rad}}(B_1^{2}).$$ Then, it is well-known that there are countable eigenvalues of this equation and each of them is discrete. Since $\dim Y$ is equal to the number of negative eigenvalues for the above equation (see [@Dup Proposition 1.5.1] for example), it holds $\dim Y<\infty$ and thus we get the result. ◻ # Comparison results {#separate sec} In this section, we prove comparison results. From now on, we deal with the case where the weight $a$ is more general. The comparison results play an important role in studying the stability of $U_{*}$. We first introduce the following **Proposition 15**. *Assume that $N=10$ and $a(r)$ satisfies $(A)$ and $$\label{atarimae} (a/a_{h})'\le 0,\hspace{4mm}0<r<1$$ for some $h>0$. Then, we have $$v(r,\beta)+\log a(r) <v_{h}(r,\gamma) + \log a_{h}(r),\hspace{4mm} 0<r<r_h\notag$$ for all $\gamma>\beta$, where $r_h=\min\{1,\sqrt{H/h}\}$.* We briefly explain the idea of the proof. We introduce the following comparison result provided in [@Gui; @Guini]. **Lemma 16** (see [@Gui; @Guini]). *Let $R>0$ and let $k(|x|)\in C^0(B_R)$ be a nonnegative radial function. We assume that $w_1(|x|)\in C^2(B_R)$ and $w_2(|x|)\in C^2(B_R)$ are radial functions which satisfy $$-\Delta w_1\le k(x)w_1 \hspace{4mm}\text{in $B_R\cap \{w_1>0\}$}$$ and $$-\Delta w_2 \ge k(x)w_2 \hspace{4mm}\text{in $B_R$}.$$ Moreover, we assume that $w_1(0)>0$ and $w_2>0$ in $B_{R}$. Then, we have $w_1>0$ in $B_R$.* We define $k(r)=e^{v_h(r,\gamma)+\log a_h}$ and we denote by $w_1$ that in [\[kiee\]](#kiee){reference-type="eqref" reference="kiee"}. Then, we wish to provide the function $w_2$ and apply Lemma [Lemma 16](#guilem){reference-type="ref" reference="guilem"}. In [@Gui; @Guini], the authors focused on the equation for which the separation result holds (i.e., no two solutions intersect). As a result, they constructed $w_2$ as the difference between the two solutions. We note that this idea was applied by many authors (see e.g.[@Baeni; @Bae; @Mi2015]). In this case, we point out that the singular solution $V_h$ plays an important role in providing $w_2$. Indeed, we verify that $V_{h}-V_{h}(r_{h})$ is a stable solution of [\[comp11\]](#comp11){reference-type="eqref" reference="comp11"}. The fact means that the bifurcation diagram of [\[comp11\]](#comp11){reference-type="eqref" reference="comp11"} is of Type II. As a consequence, we construct $w_2$ as the difference between two solutions for [\[comp11\]](#comp11){reference-type="eqref" reference="comp11"}. Moreover, we point out that we cannot apply Lemma [\[kiee\]](#kiee){reference-type="ref" reference="kiee"} directly since we do not know the sign of $-\Delta (\log a_h -\log a)$. By combining the idea of the proof of Lemma [Lemma 16](#guilem){reference-type="ref" reference="guilem"} with Green's first identity, we get the result from the assumption ([\[atarimae\]](#atarimae){reference-type="ref" reference="atarimae"}) only. *Proof of Proposition [Proposition 15](#comparison3){reference-type="ref" reference="comparison3"}.* Let $\gamma>\beta$ and we define $$\begin{aligned} \label{kiee} w_1(r):=v_{h}(r,\gamma)+\log a_{h} -v(r,\beta)-\log a\end{aligned}$$ and $$\begin{aligned} w_0(r):=v_h(r,\gamma)-v_h(r_h,\gamma).\notag\end{aligned}$$ Then, we claim that there exist $\mu>e^{v_h(r_h,\gamma)}$ and a stable solution of $$\begin{aligned} \label{comp11} \left\{ \begin{alignedat}{4} -\Delta X&= \mu a_h(|x|)e^{X} &\hspace{2mm} &\text{in } B_{r_h},\\ \ X&> 0 & &\text{in $B_{r_h}$},\\ X&=0 &&\text{on $B_{r_h}$} \end{alignedat} \right.\end{aligned}$$ such that $w_0<X$ in $B_{r_h}$. Indeed, we easily know that $w_0$ is a solution of [\[comp11\]](#comp11){reference-type="eqref" reference="comp11"} for $\mu=e^{v_h(r_h,\gamma)}$. Moreover, we verify that $V_h -V_h(r_h)$ is the radial singular solution of [\[comp11\]](#comp11){reference-type="eqref" reference="comp11"} for $\mu=e^{V_{h}(r_h)}$. Thanks to Proposition [Proposition 12](#improved hardy){reference-type="ref" reference="improved hardy"}, the singular solution is stable. Thus, we verify that the bifurcation diagram of [\[comp11\]](#comp11){reference-type="eqref" reference="comp11"} is of Type II. It means that the bifurcation curve has only the stable branch and thus we get the claim. We define $$w_2(r)=X(r)-w_0(r).$$ Then, it holds $w_2(r)>0$ for $0<r<r_h$. Suppose the contrary, i.e., we assume that there exists $0<R<r_h$ such that $w_1(r)>0$ for $0<r<R$ and $w_1(R)=0$. Then, we have $w_1 '(R)\le 0$. By a direct calculation, it follows $$\begin{aligned} &-\Delta w_1 = e^{v_{h}(r,\gamma)+\log a_{h}}-e^{v(r,\beta)+\log a}-\Delta\left(\log a_h-\log a\right)\notag\\ &\hspace{18mm}\le e^{v_{h}(r,\gamma)+\log a_{h}}w_1-\Delta\left(\log a_h-\log a\right),\notag\\ &-\Delta w_2= e^{X+\log \left(\mu a_h\right)}-e^{v_h(r,\gamma)+\log a_h}\ge(\log\mu-v_h(r_h,\gamma)+w_2)e^{v_{h}(r,\gamma)+\log a_h}\notag\end{aligned}$$ for all $0<r<R$. In particular, since $w_2$ is superharmonic, we have $w_2'\le 0$ for all $0<r<R$ (see Lemma [Lemma 24](#rough lemma){reference-type="ref" reference="rough lemma"}). Moreover, by the assumption [\[atarimae\]](#atarimae){reference-type="eqref" reference="atarimae"}, it holds $(\log a_h-\log a)'\ge 0$ in $B_1$. Thus, it follows by Green's identity that $$\begin{aligned} \omega_{N}R^N w_1 '(R) &w_2(R) = \int_{B_R}w_2\Delta w_1-w_1\Delta w_2\,dx\notag\\ &=\int_{B_R}\log \left(\frac{\mu}{e^{v_h(r_h,\gamma)}}\right) e^{v_{h}(r,\gamma)+\log a_h}w_1+\Delta\left(\log a_h -\log a\right)w_2\,dx\notag\\ &>\int_{\partial B_R} \left(\nabla\left(\log a_h-\log a\right)\cdot\frac{x}{R}\right)w_2\,d\mathcal{H}^{N-1}\notag\\ &\hspace{10mm}-\int_{B_R} \nabla \left(\log a_h-\log a\right)\cdot\nabla w_2\,dx\notag\\ &\ge 0\notag,\end{aligned}$$ which contradicts that $w_1'(R)\le 0$ and $w_2(R)>0$. ◻ Proposition [Proposition 15](#comparison3){reference-type="ref" reference="comparison3"} implies the separation property of $v$ in $B_{r_h}$. To be more precise, the following proposition holds. **Proposition 17**. *Assume that $N=10$ and $a(r)$ satisfies $(A)$ and [\[atarimae\]](#atarimae){reference-type="eqref" reference="atarimae"} for some $h>0$. Then, we have $$v(r,\beta)<v(r,\gamma),\hspace{4mm} 0<r<r_h\notag$$ for all $\gamma>\beta$, where $r_h$ is that in Proposition [Proposition 15](#comparison3){reference-type="ref" reference="comparison3"}.* *Proof.* Let $\gamma>\beta$ and we define $$\begin{aligned} &w_1(r):=v(r,\gamma)-v(r,\beta),\notag\\ &w_0(r,\gamma):=v(r,\gamma)-v(r_h,\gamma).\notag \end{aligned}$$ Then, we claim that there exist $\mu>e^{v(r_h,\gamma)}$ and a stable solution of $$\begin{aligned} \label{comp12} \left\{ \begin{alignedat}{4} -\Delta X&= \mu a(|x|)e^{X} &\hspace{2mm} &\text{in } B_{r_h},\\ \ X&> 0 & &\text{in $B_{r_h}$},\\ X&=0 &&\text{on $B_{r_h}$} \end{alignedat} \right.\end{aligned}$$ such that $w_0(r,\gamma)<X(r)$ in $B_{r_h}$. Indeed, we easily know that $w_0(r,\gamma)$ is a solution of [\[comp12\]](#comp12){reference-type="eqref" reference="comp12"} for $\mu=e^{v(r_h,\gamma)}$. Let $V_{*}$ be the limit function of $v(r,\gamma)$ as $\gamma\to\infty$. By Theorem [Theorem 2](#singularth){reference-type="ref" reference="singularth"}, there exists the unique radial singular solution $(\mu_*,U_*)$ of [\[comp12\]](#comp12){reference-type="eqref" reference="comp12"} such that $w_0(r,\gamma)\to U_{*}$ in $C^2_\mathrm{loc}((0,r_h])$. Thus, we have $\mu_*=e^{V(r_h)}$ and $U_{*}=V_{*}-V(r_h)$. Moreover, thanks to Proposition [Proposition 15](#comparison3){reference-type="ref" reference="comparison3"} and Theorem [Theorem 2](#singularth){reference-type="ref" reference="singularth"}, it follows $$V_{*}+\log a\le V_h + \log a_h, \hspace{4mm} 0<r<r_h.$$ Therefore, thanks to Proposition [Proposition 12](#improved hardy){reference-type="ref" reference="improved hardy"}, we verify that the bifurcation diagram of [\[comp12\]](#comp12){reference-type="eqref" reference="comp12"} is of Type II. It means the bifurcation curve has only the stable branch and thus we get the claim. We define $$w_2(r)=X(r)-w_0(r,\gamma).$$ Then, it holds $w_2(r)>0$ for $0<r<r_h$. Suppose the contrary, i.e., we assume that there exists $0<R<r_h$ such that $w_1(r)>0$ for $0<r<R$ and $w_1(R)=0$. Then, we have $w_1 '(R)\le 0$. By a direct calculation, it follows $$\begin{aligned} &-\Delta w_1 = e^{v(r,\gamma)+\log a}-e^{v(r,\beta)+\log a }\le e^{v(r,\gamma)+\log a}w_1,\notag\\ &-\Delta w_2= e^{X+\log \left(\mu a\right)}-e^{v(r,\gamma)+\log a}\ge(\log \mu-v(r_h,\gamma)+w_2)e^{v(r,\gamma)+\log a}\notag\end{aligned}$$ for all $0<r<R$. Thus, it follows by Green's identity that $$\begin{aligned} \omega_{N}R^N w_1 '(R) w_2(R) &= \int_{B_R}w_2\Delta w_1-w_1\Delta w_2\,dx\notag\\ &=\int_{B_R}\log \left(\frac{\mu}{e^{v(r_h,\gamma)}}\right)e^{v(r,\gamma)+\log a}w_1\,dx\notag\\ &>0\notag,\end{aligned}$$ which contradicts that $w_1'(R)\le 0$ and $w_2(R)>0$. ◻ As a consequence of Proposition [Proposition 15](#comparison3){reference-type="ref" reference="comparison3"} and Proposition [Proposition 17](#comparison2){reference-type="ref" reference="comparison2"}, we get the following **Corollary 18**. *Assume that $N=10$ and $a(r)$ satisfies $(A)$ and the assumption of $(i)$ in Theorem [Theorem 5](#Mainthm){reference-type="ref" reference="Mainthm"}. Then, we have $$v(r,\beta)+\log a(r) <v_{H}(r,\gamma) + \log a_{H}(r)\hspace{4mm}\text{and}\hspace{4mm} v(r,\beta)<v(r,\gamma),$$ for all $\gamma>\beta$ and $0<r<1$.* Proposition [Proposition 15](#comparison3){reference-type="ref" reference="comparison3"} and Corollary [Corollary 18](#cor){reference-type="ref" reference="cor"} imply that $V_{*}+\log a$ stays below $V_{h}+\log a_{h}$ in $B_{r}$ as long as $V_{h}$ is stable in $B_{r}$ if the assumption [\[atarimae\]](#atarimae){reference-type="eqref" reference="atarimae"} holds. In particular, if [\[atarimae\]](#atarimae){reference-type="eqref" reference="atarimae"} holds for some $h\le H$, the function $V_{*}+\log a$ stays below $V_{h}+\log a_{h}$ in $B_{1}$. This fact is useful to study the stability of $U_{*}$. On the other hand, if [\[atarimae\]](#atarimae){reference-type="eqref" reference="atarimae"} holds for some $h>H$, we get the following proposition by focusing on the stable branch. **Proposition 19**. *Assume that $N=10$ and $a(r)$ satisfies $(A)$ and the assumption of $(iii)$ in Theorem [Theorem 5](#Mainthm){reference-type="ref" reference="Mainthm"}. Then, there exists a nondecreasing function $\varepsilon:[0,1]\to [0,1]$ such that $$\label{seisitu epsilon} \varepsilon(|x|)\in C^2(\overline{B_1}),\hspace{4mm} \varepsilon(1/2)>0,$$ and $$v(r,\gamma)+\log a >v_0(r,\beta)+\log \left(1+\frac{H+\varepsilon}{2(N-2)}r^2\right),\hspace{4mm} 0<r<1\notag$$ for any $\lVert u^{*}\rVert_{L^{\infty}(B_1)}+\log \lambda^*>\gamma>\beta>0$, where $\varepsilon$ is depending only on $a$ and $N$.* *Proof.* Thanks to the assumption of (iii) in Theorem [Theorem 5](#Mainthm){reference-type="ref" reference="Mainthm"}, we can take a nondecreasing convex function $\varepsilon:[0,1]\to [0,1]$ such that $$\varepsilon(|x|)\in C^2(\overline{B_1}),\hspace{2mm} \varepsilon>0\hspace{2mm}\text{in $B_1\setminus B_{\frac{1}{4}}$},\hspace{2mm}\text{and} \hspace{2mm}(\log a-\log a_{H+\varepsilon})'\ge 0\hspace{4mm} \text{in $B_1$},\notag$$ where $$a_{H+\varepsilon}(r):=\left(1+\frac{H+\varepsilon(r)}{2(N-2)}r^2\right)e^{\frac{H+\varepsilon(r)}{2N}r^2}.$$ Let $u^* (0)+\log \lambda^{*}>\alpha>\gamma>\beta$ and we define $$\begin{aligned} &w_1(r):=v(r,\gamma)+\log a -v_0(r,\beta)-\log\left(1+\frac {H+\varepsilon}{2(N-2)}r^2\right),\notag\\ &w_2(r):= v(r,\alpha)-v(r,\gamma)-\log \frac{\lambda(\alpha)}{\lambda(\gamma)}.\notag\end{aligned}$$ We claim that $w_2(r)>0$ for $0<r<1$. Indeed, we set $u_1:=v(r,\alpha)-\log \lambda(\alpha)$ and $u_{2}:=v(r,\gamma)-\log \lambda(\gamma)$. Then, $(\lambda(\alpha),u_1)$ and $(\lambda(\gamma), u_2)$ are minimal solutions of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} and it holds $\lambda(\alpha)>\lambda(\gamma)$. Thus, it holds $u_1<u_2$ and we get the claim. Suppose the contrary, i.e., we assume that that there exists $0<R<1$ such that $w_1(r)>0$ for $0<r<R$ and $w_1(R)=0$. Then, we have $w_1 '(R)\le 0$. On the other hand, we verify that $v_0(r,\beta)$ is increasing for $\beta$ by Corollary [Corollary 18](#cor){reference-type="ref" reference="cor"}. Since $v_0(r,\beta)\to -2\log r+\log 2(N-2)$, it holds $e^{v_0 (r,\beta)}<2(N-2)r^{-2}$. Thus, we have for $0<r<R$, $$\begin{aligned} -&\Delta w_1(r) = e^{v(r,\gamma)+\log a}-e^{v_0(r,\beta)+\log \left(1+\frac{H+\varepsilon}{2(N-2)}r^2\right)}+\frac{H+\varepsilon}{2(N-2)}r^2e^{v_0(r,\beta)}\notag\\ &\hspace{40mm}-\Delta\left(\log a-\log(1+\frac{H+\varepsilon}{2(N-2)}r^2)\right)\notag\\ &\le e^{v(r,\gamma)+\log a}w_1+H+\varepsilon-\Delta\left(\log a-\log(1+\frac{H+\varepsilon}{2(N-2)}r^2)\right)\notag\\ &\le e^{v(r,\gamma)+\log a}w_1-\Delta\left(\log a-\log a_{H+\varepsilon}\right)\hspace{4mm}\text{(since $\varepsilon'\ge 0, \varepsilon''\ge 0$) }.\notag\end{aligned}$$ Moreover, we have for $0<r<1$, $$\begin{aligned} -\Delta w_2= e^{v(r,\alpha)+\log a}-e^{v(r,\gamma)+\log a}> \left(\log \frac{\lambda(\alpha)}{\lambda(\gamma)}+w_2\right)e^{v(r,\gamma)+\log a}\notag.\end{aligned}$$ In particular, since $w_2$ is superharmonic, we have $w_2'\le 0$ for all $0<r<1$ (see Lemma [Lemma 24](#rough lemma){reference-type="ref" reference="rough lemma"}). Moreover, we recall that $(\log a-\log a_{H+\varepsilon})'\ge 0$. Thus, it follows by Green's identity that $$\begin{aligned} \omega_{N}R^N w'_1 (R) &w_2(R) = \int_{B_R}w_2\Delta w_1-w_1\Delta w_2\,dx\notag\\ &>\int_{B_R}\Delta\left(\log a -\log a_{H+\varepsilon}\right)w_2\,dx\notag\\ &\ge \int_{\partial B_R} \left(\nabla\left(\log a-\log a_{H+\varepsilon}\right)\cdot\frac{x}{R}\right)w_2\,d\mathcal{H}^{N-1}\notag\\ &\hspace{10mm}-\int_{B_R} \nabla \left(\log a-\log a_{H+\varepsilon}\right)\cdot\nabla w_2\,dx\notag\\ %&\ge \int_{\partial B_R} %\left(\nabla\left(\log a_{H+2\varepsilon}(\lambda_{\gamma}^{-1/2} r)-\log a_{H+\varepsilon}(\lambda_{\gamma}^{-1/2} r)\right)\cdot\frac{x}{R}\right)w_2\,d\mathcal{H}^{N-1}\notag\\ &\ge 0\notag,\end{aligned}$$ which contradicts that $w_1'(R)\le 0$ and $w_2(R)>0$. ◻ # Classification of bifurcation diagrams. {#mainsec} In this section, we classify the bifurcation diagrams. For simplicity, we set $v'(r,\beta):=\frac{d}{d\beta} v(r,\beta)$ and $v''(r,\beta)=\frac{d^2}{d\beta^2}v(r,\beta)$. Moreover, we set $\lambda'(\beta):=\frac{d}{d\beta}\lambda(\beta)$ and $\lambda''(\beta):=\frac{d^2}{d\beta^2}\lambda(\beta)$. ## When $N\le 9$ In this subsection, following the idea of [@Mi2014], we prove Theorem [Theorem 3](#39th){reference-type="ref" reference="39th"}. We define $$\begin{aligned} \hat{v}_{\beta}(r):=v(e^{-\frac{\beta}{2}}r,\beta)-\beta\hspace{4mm}\text{and}\hspace{4mm} \hat{V}_{\beta}(r,\beta)=V_{*}(e^{-\frac{\beta}{2}}r)-\beta.\end{aligned}$$ Thus, $\hat{v}_{\beta}$ satisfies $$\left\{ \begin{alignedat}{4} &-\Delta \hat{v}_{\beta}= \hat{a}_{\beta}(r)e^{\hat{v}_{\beta}}, \hspace{14mm}0<r<\infty,\\ &\hat{v}_{\beta}(0)=0, \hspace{4mm} \frac{d}{dr} \hat{v}_{\beta}(0)=0,\notag \end{alignedat} \right.$$ where $\hat{a}_{\beta}(r)=a(e^{-\frac{\beta}{2}}r)$. We prove the following **Proposition 20**. *Let $N\le 9$. We assume that $a(r)$ satisfies $(A)$. Then $$\hat{v}_{\beta}\to v_{0}(r,0) \hspace{2mm}\text{in $C^1_{\mathrm{loc}}[0,\infty)$}$$ and $$\hat{V}_{\beta}\to -2\log r + \log 2(N-2) \hspace{2mm}\text{in $C^0_{\mathrm{loc}}(0,\infty)$}.$$* *Proof.* By Lemma [Lemma 24](#rough lemma){reference-type="ref" reference="rough lemma"}, it holds $\frac{d}{dr} \hat{v}_{\beta}\le 0$ for all $0<r<\infty$. Moreover, for each bounded interval $I$, there exists $\beta_0$ such that $1\le\hat{a}_{\beta}\le 2$ in $I$ if $\beta>\beta_0$. Since $$\frac{d}{dr}(r^{N-1}\frac{d}{dr}\hat{v}_{\beta})= -\hat{a}_{\beta}(r)e^{\hat{v}_{\beta}}r^{N-1}\ge -2r^{N-1}\hspace{4mm}\text{in $I$},$$ we have $$\frac{d}{dr}\hat{v}_{\beta}(r)\ge -2r^{1-N}\int^{r}_{0}s^{N-1}\,ds=-\frac{2}{N}r \hspace{4mm}\text{in $I$}$$ and thus it holds $-r^2/N\le\hat{v}_{\beta}\le 0$ in $I$. By Ascoli-Arzelá theorem, there exist $\beta_m$ and $\hat{v}_{\infty} \in C^{0}_{\mathrm{loc}}[0,\infty)$ such that $\hat{v}_{\beta_m}\to \hat{v}_{\infty}$ in $C^{0}_{\mathrm{loc}}[0,\infty)$ as $m\to\infty$. Moreover, since $$-\frac{d}{dr} \hat{v}_{\beta_m} (r)= r^{1-N}\int^{r}_{0}s^{N-1} \hat{a}_{\beta_m}(s) e^{\hat{v}_{\beta_m}}\,ds$$ and $\hat{a}_{\beta_m}(r)\to 1$ in $C^{0}_{\mathrm{loc}}[0,\infty)$ as $m\to \infty$, $\frac{d}{dr}v_{\beta_m}$ converges in $C^{0}_{\mathrm{loc}}[0,\infty)$. Since, $\frac{d}{dr}$ is a closed operator with a domain $C^{1}(I)$ in $C^{0}(I)$ for each bounded interval $I$, we get $\hat{v}_{\infty}\in C^{1}_{\mathrm{loc}}[0,\infty)$ and $\hat{v}_{\beta_m}\to \hat{v}_{\infty}$ in $C^{1}_{\mathrm{loc}}[0,\infty)$ as $m\to\infty$. In particular, it holds $$\hat{v}_{\infty}(r)=-\int^{r}_{0}\,ds\int^{s}_{0}\left(\frac{t}{s}\right)^{N-1}e^{\hat{v}_{\infty}}\,dt.$$ Thus, by the uniqueness of the solution of the ODE, we get $\hat{v}_{\infty}(r)=v_0(r,0)$. It implies $\hat{v}_{\beta}\to v_0(r,0)$ in $C^{1}_{\mathrm{loc}}[0,\infty)$ as $\beta\to\infty$. By Proposition [Proposition 9](#asymptotic prop){reference-type="ref" reference="asymptotic prop"}, it holds $V_{*}(r)+2\log r\to \log 2(N-2)$ as $r\to 0$ and thus we have $\hat{V}_{\beta}(r)+2\log r \to \log 2(N-2)$ as $e^{-\frac{\beta}{2}}r\to 0$. Since, for each bounded interval $I$, it holds $e^{-\frac{\beta}{2}}r\to 0$ as $\beta\to\infty$, we get the result. ◻ We denote the zero number of the function $u(r)$ on the interval $I$ by $$\mathcal{Z}_{I}[u(r)]=\sharp\{r\in I: u(r)=0\}.$$ Then it is well known that $$\mathcal{Z}_{[0,\infty)}[v_0(r,0)-2\log r + \log 2(N-2)]=\infty \hspace{4mm}\text{if $N\le 9$}.$$ By using Proposition [Proposition 20](#inter){reference-type="ref" reference="inter"}, we get the following **Proposition 21**. *Under the same assumption of Proposition [Proposition 20](#inter){reference-type="ref" reference="inter"}, we have $$\mathcal{Z}_{[0,1]}[v(r,\beta)-V_{*}(r)]\to\infty\hspace{4mm}\text{as $\beta\to \infty$}.$$* *Proof of Theorem [Theorem 3](#39th){reference-type="ref" reference="39th"}..* We remark that each zero of $v(r,\beta)-V_{*}(r)$ is simple and discrete by the uniqueness of the solution of the ODE. Moreover, since $v(0,\beta)-V_{*}(r)=-\infty$, we have $\mathcal{Z}_{[0,2]}[v(r,\beta)-V_{*}(r)]<\infty$ for all $0\le\beta<\infty$. We set $h(r,\beta)=v(r,\beta)-V_{*}(r)$. In addition, let $r_0\in (0,2]$ and $\beta_0>0$ be constants such that $h(r_0,\beta_0)=0$. Since $\frac{d}{dr} h(r_0,\beta_0)\neq 0$, it follows by the implicit function theorem that there exists a $C^1$-function $r(\beta)$ such that $h(r(\beta),\beta)=0$ for $\beta$ near $\beta_0$ and $r(\beta_0)=r_0$. Let $M\in \mathbb{N}$ and we define $$\beta_{M}:=\sup \{ \beta\ge 0 :\mathcal{Z}_{[0,1]}[h(r,\beta)]<M\}.$$ Thanks to Proposition [Proposition 21](#interprop){reference-type="ref" reference="interprop"} and the above discussion, there exists $M_0\in \mathbb{N}$ such that we can define $\beta_M$ for all $M>M_0$ and it holds $0<\beta_{M}<\infty$ for all $M>M_0$. We fix $\beta_M$ and we denote by $r_{i}(\beta)$ the $i$-th zero of $h(\cdot,\beta)$. Then, we verify that $r_{M}(\beta_M)=1$ and $r_{M}(\beta)>1$ if $\beta<\beta_M$ and $\beta$ is close to $\beta_M$. Moreover, we know that $r_{M}(\beta)\le 1$ for all $\beta>\beta_M$ and $r_{M}(\beta)<1$ for a sufficiently large $\beta$. We define $$\beta_{\hat{M}}:=\sup\{\beta>\beta_{M}: r_{M}(\beta)=1\hspace{2mm} \text{on}\hspace{2mm} [\beta_{M},\beta]\}.$$ Then, we have $r_{M}(\beta_{\hat{M}})=1$ and $\beta_{\hat{M}}<\beta_{M+1}$. Moreover, since $h(1,\beta)$ is analytic and $r_{M+1}(\beta)$ is continuous, there exists $\beta_{\hat{M}}<\hat{\beta}<\beta_{M+1}$ such that $r_{M}(\beta)<1$ and $r_{M+1}(\beta)>1$ for all $\beta_{\hat{M}}<\beta<\hat{\beta}$. By the above facts, it follows that the sign of $h(1,\beta)=\log \lambda(\beta)-\log \lambda_{*}$ changes if $\beta$ crosses $[\beta_{M},\beta_{\hat{M}}]$. Thanks to Proposition [Proposition 21](#interprop){reference-type="ref" reference="interprop"}, we know that $\lambda(\beta)$ oscillates around $\lambda_{*}$ infinity many times as $\beta\to\infty$. Therefore, we verify that the bifurcation is of Type I. ◻ ## When $N=10$ In this subsection, we prove Theorem [Theorem 5](#Mainthm){reference-type="ref" reference="Mainthm"}. We begin by introducing the following proposition, which plays a key role in proving that the bifurcation diagram is of Type III. **Proposition 22**. *Let $N= 10$ and we assume that $a$ satisfies [\[atarimae\]](#atarimae){reference-type="eqref" reference="atarimae"} for some $h>0$. Then there exists $\beta_0>0$ such that if $(\lambda(\beta), u(r,\alpha(\beta)))$ is a radial solution of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} satisfying $\beta>\beta_{0}$, then $\lambda'(\beta)\neq 0$.* *Proof.* Let $i$ be a natural number depending only on $a$ to be defined later. Then, it suffices to prove the following: there exists $\gamma>0$ such that if it follows $\gamma<\beta$ and $\lambda'(\beta)=0$, then $(-1)^{i-1}\lambda''(\beta)>0$. Suppose the contrary, i.e., we assume that there exist a sequence of radial solutions of [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} $(\lambda(\beta_k), u(r,\alpha(\beta_k)))$ such that $\lambda'(\beta_k)=0$, $(-1)^{i-1}\lambda''(\beta_k)\le 0$, and $\beta_k\to \infty$ as $k\to\infty$. We set $v_{k}(r):= u(r,\alpha(\beta_k))+\log \lambda(\beta_k)$. We recall that $v_k$ is the solution of [\[odev\]](#odev){reference-type="eqref" reference="odev"} for $\beta=\beta_k$. By a direct calculation, it follows that $v_{k}'$ and $v_{k}''$ satisfy $$\left\{ \begin{alignedat}{4} &-\Delta v_{k}'= a(r)e^{v_{k}}v_{k}', \hspace{14mm}0<r\le 1,\\ &v_{k}'(0)=1, \hspace{1mm} \frac{d}{dr} v_{k}'(0)=0, \hspace{1mm} v_{k}'(1)=0\notag \end{alignedat} \right.$$ and $$\left\{ \begin{alignedat}{4} -\Delta v_{k}''&= a(|x|)e^{v_{k}}({v_{k}'}^2+v_{k}''), \hspace{4mm}&&\text{in $B_1$},\\ v_{k}''&=\frac{\lambda''(\beta_k)}{\lambda(\beta_k)} &&\text{on $\partial B_1$}.\notag \end{alignedat} \right.$$ We remark that by the assumption [\[atarimae\]](#atarimae){reference-type="eqref" reference="atarimae"} and Proposition [Proposition 17](#comparison2){reference-type="ref" reference="comparison2"}, we have $v'_{k}\ge 0$ for $0<r<r_{h}$, where $r_h$ is that in Proposition [Proposition 15](#comparison3){reference-type="ref" reference="comparison3"}. In addition, we remark that $v_{k}\to V_{*}$ in $C^2_{\mathrm{loc}}(0,1]$ and $\lambda_{*}:=\lim_{k\to \infty} \lambda(\beta_k)>0$, where $V_{*}$ is that in [\[singulareq\]](#singulareq){reference-type="eqref" reference="singulareq"}. We define $$\label{spe} w_k=\frac{|x|^{\frac{N-2}{2}} v_{k}'}{\lVert |x|^{\frac{N-2}{2}} v_{k}' \rVert_{L^2(B^{2}_{1})}},$$ where $B^{2}_{1}$ is the 2-dimensional unit ball. Then, $w_k$ satisfies $$\label{odew} \left\{ \begin{alignedat}{4} -\Delta w_{k}&= \left(a(|x|)e^{v_{k}}-\frac{(N-2)^2}{4|x|^2}\right) w_{k}, \hspace{4mm}&&\text{in $B^{2}_1$},\\ w_{k}&=0 &&\text{on $\partial B^{2}_1$} \end{alignedat} \right.$$ and it satisfies $w_{k}(r)\ge 0$ in $0<r<r_h$. Thanks to Theorem [Theorem 2](#singularth){reference-type="ref" reference="singularth"} and Proposition [Proposition 15](#comparison3){reference-type="ref" reference="comparison3"}, it holds $$a(|x|) e^{V_{*}} \le e^{V_{H}+\log a_{H}}=2(N-2)|x|^{-2}+ H \hspace{4mm}\text{in $B_{r_h}$}.$$ As a result, thanks to Proposition [Proposition 17](#comparison2){reference-type="ref" reference="comparison2"} and the fact that $v_{k}\to V_{*}$ in $C^2_{\mathrm{loc}}(0,1]$, we get $$a(|x|)e^{v_{k}}-2(N-2)|x|^{-2}\le C$$ for some $C$ depending only on $a$. We remark that $2(N-2)=\frac{(N-2)^2}{4}$ when $N=10$. Thus, it follows the energy estimate $$\int_{B^{2}_{1}} |\nabla w_k|^2\,dx= \int_{B^{2}_{1}}\left(a(|x|)e^{v_{k}}-\frac{(N-2)^2}{4|x|^2}\right) w^2_{k}\,dx\le C\int_{B^{2}_1} w^2_{k}\,dx=C.$$ Therefore, there exists $w\in H^{1}_{0}(B^2_{1})$ such that $w_k\rightharpoonup w$ in $H^{1}(B^{2}_1)$ by taking a subsequence if necessary. In particular, we get $w_k\to w$ in $L^q(B^{2}_1)$ for any $1\le q<\infty$. Thus $\lVert w\rVert_{L^2(B^2_{1})}=1$ and $w$ satisfies $$\left\{ \begin{alignedat}{4} -\Delta w&= \left(a(|x|)e^{V_{*}}-\frac{(N-2)^2}{4|x|^2}\right) w, \hspace{4mm}&&\text{in $B^2_{1}$},\\ w&=0 &&\text{on $\partial B^2_{1}$}.\notag \end{alignedat} \right.$$ Moreover, $w$ is radially symmetric. Since $V_{*}$ satisfies [\[singulareq\]](#singulareq){reference-type="eqref" reference="singulareq"}, it holds $$K(|x|):=\left(a(|x|)e^{V_{*}}-\frac{(N-2)^2}{4|x|^2}\right)\in C^{0}(B^{2}_1).$$ Thus, by the elliptic theory [@gil] and Lemma [Lemma 25](#apenlem2){reference-type="ref" reference="apenlem2"}, we verify that there are countable eigenvalues of $-\Delta-K$ and each eigenfunction is in $C^2(\overline{B^{2}_1})$. If $0$ is not an eigenvalue, it holds $w=0$, which contradicts $\lVert w\rVert_{L^2(B^2_{1})}=1$. From now on, we deal with the remaining case where $0$ is an eigenvalue. In this case, we can define $i$ as the number such that $(-1)^{i-1}\frac{d}{dr}w(1)<0$ since it holds $\frac{d}{dr}w(1)\neq 0$. We remark that $i$ is depending only on $a$. Moreover, we define $$\mu_k:= \frac{\lambda''(\beta_k)}{\lambda(\beta_k)}\hspace{4mm}\text{and}\hspace{4mm}z_{k}=r^{\frac{N-2}{2}} v_{k}''- \mu_{k} r^2.$$ Then, it holds $(-1)^{i-1}\mu_k\le 0$. Moreover, $z_k\in C^2_{0}(\overline{B^2_{1}})$ satisfies $$-\Delta z_{k}= \left(a(|x|)e^{v_{k}}-\frac{(N-2)^2}{4|x|^2}\right)(z_{k}+\mu_{k}|x|^2)+a(|x|)|x|^ {\frac{N-2}{2}}e^{v_{k}}{v_{k}'}^2+4\mu_k\hspace{2mm}\text{in $B_1^{2}$}\notag.$$ Thanks to the above equation and [\[odew\]](#odew){reference-type="eqref" reference="odew"}, we get $$\begin{aligned} 0&=\int_{B_{1}^2} \left(-\Delta w_k - \left(a(|x|)e^{v_{k}}-\frac{(N-2)^2}{4|x|^2}\right)w_k\right)z_k\,dx\\ &=\int_{B_{1}^2} \left(-\Delta z_k - \left(a(|x|)e^{v_{k}}-\frac{(N-2)^2}{4|x|^2}\right)z_k\right)w_k\,dx= \mathcal{A}_k+ \mu_k \mathcal{B}_k,\end{aligned}$$ where $$\begin{aligned} &\mathcal{A}_k:= \int_{B^2_{1}}a(|x|)|x|^{\frac{N-2}{2}}e^{v_{k}}{v_{k}'}^2 w_k\,dx,\\ &\mathcal{B}_k:=\int_{B^2_{1}}\left(\left(a(|x|)e^{v_{k}}-\frac{(N-2)^2}{4|x|^2}\right)|x|^2+\Delta |x|^2\right)w_k\,dx.\end{aligned}$$ We claim that there exists $k_0$ such that $A_k>0$, $(-1)^{i-1}B_k<0$ for all $k>K_0$, which contradicts the above equality. Indeed, we recall that $w_k\to w$ in $L^q(B^{2}_1)$ for any $1\le q<\infty$ and $v_k\to V_{*}$ in $C^2_{\mathrm{loc}}(0,1]$. Thanks to the above notations and Fatou's Lemma, it holds $$\begin{aligned} \liminf_{k\to\infty}\frac{\mathcal{A}_k}{\lVert |x|^{\frac{N-2}{2}}v_k'\rVert_{L^2(B^2_{1})}^3}&=\liminf_{k\to\infty}\int_{B^2_{1}}a(|x|)|x|^{\frac{2-N}{2}}e^{v_{k}}w^3_{k}\,dx\\ &\ge \liminf_{k\to\infty} \int_{B^2_{r_0}}a(|x|)|x|^{\frac{2-N}{2}}e^{v_{k}}w^3_{k}\,dx-C\\ &=\infty\end{aligned}$$ and $$\begin{aligned} \lim_{k\to\infty}\mathcal{B}_k&=\int_{B^2_{1}} \left(K(|x|)|x|^2+\Delta |x|^2\right)w\,dx\\ &=\int_{B^2_{1}} K(|x|)w\,dx+\int_{B^2_{1}} \left(K(|x|)(|x|^2-1)+\Delta (|x|^2-1)\right)w\,dx\\ &=\int_{B^2_{1}} -\Delta w\,dx=2\pi \frac{d}{dr}w(1).\end{aligned}$$ Thus, we get the claim. ◻ *Proof of Theorem [Theorem 5](#Mainthm){reference-type="ref" reference="Mainthm"}..* We first assume that $a(r)$ satisfies $(A)$ and the assumption of $(i)$. In this case, thanks to Theorem [Theorem 2](#singularth){reference-type="ref" reference="singularth"} and Corollary [Corollary 18](#cor){reference-type="ref" reference="cor"}, it holds $$V_{*}+ \log a \le V_{H} + \log a_{H},$$ where $a_H$ is that in Theorem [Theorem 4](#singularthm){reference-type="ref" reference="singularthm"}, $V_{*}$ is the solution of [\[singulareq\]](#singulareq){reference-type="eqref" reference="singulareq"}, and $V_H$ is the solution of [\[singulareq\]](#singulareq){reference-type="eqref" reference="singulareq"} for $a=a_H$. We recall $2(N-2)=\frac{(N-2)^2}{4}$ and $a_{H}e^{V_H}=2(N-2)|x|^{-2}+H$. Thus, for any $\xi\in H^1_{0}(B_1)$, we have $$\begin{aligned} Q_{U_{*}}(\xi)&=\int_{B_1} \left(|\nabla \xi|^2-\lambda_{*}ae^{U_{*}}\xi^2\right)\,dx =\int_{B_1}\left(|\nabla \xi|^2-ae^{V_{*}}\xi^2\right)\,dx\notag\\ &\ge \int_{B_1}|\nabla \xi|^2\,dx- 2(N-2)\int_{B_1} \frac{\xi^2}{|x|^2}\,dx- H\int_{B_1} \xi^2\,dx\notag\\ &=\int_{B_1}|\nabla \xi|^2\,dx- \frac{(N-2)^2}{4}\int_{B_1} \frac{\xi^2}{|x|^2}\,dx- H\int_{B_1} \xi^2\,dx.\notag \end{aligned}$$ By Proposition [Proposition 12](#improved hardy){reference-type="ref" reference="improved hardy"}, $U_{*}$ is stable. Thanks to Proposition [Proposition 14](#brezis V){reference-type="ref" reference="brezis V"} and Theorem [Theorem 2](#singularth){reference-type="ref" reference="singularth"}, we verify that the bifurcation diagram is of Type II. Then we assume that $a(r)$ satisfies $(A)$ and the assumption of $(ii)$. We divide the proof into the following three steps. **Step 1.** *$u^*$ is bounded.* We recall the following notation: $u^*$ is the extremal solution for $\lambda=\lambda^*$. Suppose the contrary, i.e., we assume that $\lVert u^{*}\rVert_{ L^{\infty}(B_1)}=\infty$. Since $\lambda^*>0$, it holds $\lVert u^{*}\rVert_{ L^{\infty}(B_1)}+\log \lambda^*=\infty$. Then, thanks to Theorem [Theorem 2](#singularth){reference-type="ref" reference="singularth"} and Proposition [Proposition 19](#comparison1){reference-type="ref" reference="comparison1"}, there exists a nondecreasing function $\varepsilon:[0,1]\to [0,1]$ which satisfies [\[seisitu epsilon\]](#seisitu epsilon){reference-type="eqref" reference="seisitu epsilon"} such that $$V_{*}+\log a\ge -2\log |x|+\log 2(N-2)+\log \left(1+\frac{H+\varepsilon(|x|)}{2(N-2)}|x|^2\right).$$ Thus, for any $\xi\in H^1_{0}(B_1)$, we have $$\begin{aligned} Q_{U_{*}}(\xi)&=\int_{B_1}\left(|\nabla \xi|^2-ae^{V_{*}}\xi^2\right)\,dx\notag\\ &\le\int_{B_1}|\nabla \xi|^2\,dx- \frac{(N-2)^2}{4}\int_{B_1} \frac{\xi^2}{|x|^2}\,dx- \int_{B_1}(H+\varepsilon(|x|)) \xi^2\,dx.\notag \end{aligned}$$ By Proposition [Proposition 13](#improvedhardy2){reference-type="ref" reference="improvedhardy2"}, we know that $U_{*}$ is unstable. Thus, it follows by Proposition [Proposition 14](#brezis V){reference-type="ref" reference="brezis V"} and Theorem [Theorem 2](#singularth){reference-type="ref" reference="singularth"} that $u^* \in L^{\infty}(B_1)$. **Step 2.** *The bifurcation curve turns at $\lambda=\lambda^*$*. We set $\beta_0 := \lVert u^{*} \rVert_{L^\infty(B_1)}+\log \lambda^*$, We recall that equation [\[gelfand\]](#gelfand){reference-type="eqref" reference="gelfand"} has no weak solution for $\lambda>\lambda^{*}=\lambda(\beta_0)$. By the implicit function theorem, it holds $\lambda'(\beta_0)=0$. We recall that $\alpha(\beta)=\beta-\log \lambda(\beta)$. Thus it holds $\alpha'(\beta_0)=1$. Therefore, it suffices to prove $\lambda''(\beta_0)<0$. We denote $v=v(r,\beta_0)$ and $\mu=\lambda''(\beta_0)/\lambda(\beta_0)$. Then, by a direct calculation, it follows that $v'$ and $v''$ satisfy $$\label{eqv'} \left\{ \begin{alignedat}{4} &-\Delta v'= a(r)e^{v}v', \hspace{14mm}0<r\le 1,\notag\\ &v'(0)=1, \hspace{1mm} \frac{d}{dr} v'(0)=0, \hspace{1mm} v'(1)=0 \end{alignedat} \right.$$ and $$\left\{ \begin{alignedat}{4} -\Delta v'' &= ae^{v}({v'}^2+v'') \hspace{4mm}&&\text{in $B_1$},\notag\\ v''&=\mu &&\text{on $\partial B_1$}.\notag \end{alignedat} \right.$$ Thus, by the Green's inequality, it holds $$\begin{aligned} 0&=\int_{B_{1}} (-\Delta v' - a(|x|)e^{v}v')v''\,dx\\ &=\int_{B_1} (-\Delta v'' -a(|x|)e^{v}v'')v'\,dx- \mu\int_{\partial B_1} v'\,d\mathcal{H}^{N-1}\notag\\ &=\int_{B_1} a(|x|)e^v {v'}^3\,dx- \mu Nw_{N}\frac{d}{dr}v'(1)\notag,\end{aligned}$$ where $w_N$ is the measure of the $N$-dimensional unit ball. We claim that $v'>0$ in $B_1$ and $\frac{d}{dr}v'(1)<0$. Indeed, since $u^*$ is stable, $v$ is stable. It means that $-\Delta- ae^v$ is nonnegative. Thus, we verify that $0$ is the first eigenvalue of $-\Delta-ae^v$ and $v'$ is a first eigenfunction of this operator. Moreover, by Proposition [Proposition 17](#comparison2){reference-type="ref" reference="comparison2"}, we have $v'\ge 0$ in $B_{r_h}$. Thus, we get the claim. In particular, we have $$\int_{B_1} a(|x|)e^v {v'}^3\,dx>0.$$ Thus, we get the result. **Step 3.** *Conclusion*. We remark that there exists $h>0$ depending only on $a$ such that [\[atarimae\]](#atarimae){reference-type="eqref" reference="atarimae"} holds. Thus, thanks to Proposition [Proposition 22](#mainprop){reference-type="ref" reference="mainprop"}, we know that every solution is nondegenerate if $\beta$ is sufficiently large. Moreover, thanks to Theorem [Theorem 1](#diagramth){reference-type="ref" reference="diagramth"}, the bifurcation curve is analytic. Therefore, we verify that the bifurcation diagram is of Type III. ◻ # Acknowledgments {#acknowledgments .unnumbered} The author would like to thank Associate Professor Michiaki Onodera and Professor Yasuhito Miyamoto, for their valuable advice. This work was supported by JSPS KAKENHI Grant Number 23KJ0949. # Appendix In this section, we introduce some basic properties. We first define the symmetric rearrangement. Let $f:\mathbb{R}\to \mathbb{R}$ be a measurable function which decays to zero at infinity. Then, we define the radially symmetric function $$f^*(|x|)= \sup \{t>0:|\{y\in \mathbb{R}^N:|f(y)|>t\}|>w_{N}|x|^N\}$$ as the symmetric rearrangement of $f$, where $|G|$ is the Lebesgue measure of a set $G\in \mathbb{R}^N$ and $w_N$ is the volume of the $N$-dimensional unit ball. Then, it is well-known that the following lemma holds. **Lemma 23** (see [@Lieb; @Kaw]). *Let $N\ge 3$ and $f\in H^1_{0}(\mathbb{R}^N)$. Then, the symmetric rearrangement $f^*$ satisfies $$\lVert f^* \rVert_{L^2(\mathbb{R}^N)}=\lVert f\rVert_{L^2(\mathbb{R}^N)},\hspace{4mm} \lVert \nabla f^* \rVert_{L^2(\mathbb{R}^N)}\le \lVert f\rVert_{L^2(\mathbb{R}^N)},$$ and $$\int_{\mathbb{R}^{N}}\frac{|f^*|^2}{|x|^2}\,dx\ge \int_{\mathbb{R}^{N}} \frac{|f|^2}{|x|^2}\,dx.$$* Then, we introduce the following lemma, which is used in Section [4](#separate sec){reference-type="ref" reference="separate sec"}. **Lemma 24**. *Assume that $N\ge 2$ and let $R>0$. Moreover, let $w\in C^2(B_{R})$ be a radially symmetric superharmonic function. Then, it follows $w'(r)\le 0$ in $(0,R)$.* *Proof.* Since $w$ is superharmonic, it holds $(r^{N-1}w'(r))'\le 0$. Suppose the contrary, i.e., we assume that $w'(t)>0$ for some $R>t>0$. Then, we have $t^{N-1}w'(t)\le r^{N-1}w'(r)$ for all $0<r<t$, Since $w\in C^2(B_R)$, we get $t^{N-1}w'(t)\le 0$ by letting $r\to 0$. It is a contradiction. Thus, it holds $w'\le 0$. ◻ Finally, we consider the regularity of radial solution for $$\label{apeneq} -\Delta u = K(|x|) u \hspace{4mm}\text{in $B_1$.}$$ **Lemma 25**. *Let $N\ge 2$ and $K\in C^0(\overline{B_1})$. We assume that $u\in H^1_{0}(B_1)$ is a radial solution of [\[apeneq\]](#apeneq){reference-type="eqref" reference="apeneq"} in the weak sense. Then, $u\in C^2_{0}(\overline{B_1}).$* *Proof.* By the elliptic regularity theory (see [@gil]), we have $u\in C^1_{0}(\overline{B_1})$. Since $u$ is radially-symmetric, it holds $\frac{d}{dr} u(0)=0$. Moreover, for any $0<r_0<r\le 1$, we have $$u(r)=u(r_0)+\int^{r}_{r_0}\left( \left(\frac{r_0}{s}\right)^{N-1} \frac{du}{ds}(r_0)- \int^{s}_{r_0}\left(\frac{t}{s}\right)^{N-1}K(t)u\,dt\right)\,ds.$$ By letting $r_0\to 0$, we get $$u(r)=u(0)-\int^{r}_{0}\,ds \int^{s}_{0}\left(\frac{t}{s}\right)^{N-1}K(t)u\,dt$$ for all $0\le r\le 1$. By using an ODE technique, we get $u\in C^2_{0}(\overline{B_1})$. ◻ [^1]: This work was supported by JSPS KAKENHI Grant Number 23KJ0949 [^2]: To avoid a misunderstanding, we remark that $u$ is not always radially symmetric if $a$ is not radially symmetric or $\Omega$ is not a ball. [^3]: We explain the meaning of optimality in Section [3](#Hardy sec){reference-type="ref" reference="Hardy sec"}.

arxiv_math

--- abstract: | The shear shallow water model is an extension of the classical shallow water model to include the effects of vertical shear. It is a system of six non-linear hyperbolic PDE with non-conservative products. We develop a high-order entropy stable finite difference scheme for this model in one dimension and extend it to two dimensions on rectangular grids. The key idea is to rewrite the system so that non-conservative terms do not contribute to the entropy evolution. Then, we first develop an entropy conservative scheme for the conservative part, which is then extended to the complete system using the fact that the non-conservative terms do not contribute to the entropy production. The entropy dissipative scheme, which leads to an entropy inequality, is then obtained by carefully adding dissipative flux terms. The proposed schemes are then tested on several one and two-dimensional problems to demonstrate their stability and accuracy. author: - Anshu Yadav - Deepak Bhoriya - "Harish Kumar[^1]" - Praveen Chandrashekar bibliography: - main.bib date: "Received: date / Accepted: date" title: Entropy stable schemes for the shear shallow water model Equations --- # Introduction The system of equations describing multi-dimensional shear shallow water (SSW) flows was derived by Teshukov [@Teshukov2007]. This system provides an approximation for shallow water flows by including the effects of vertical shear, which are neglected in the classical shallow water (Saint-Venant) model. It is derived from the incompressible Euler equations by a depth averaging process that gives rise to second-order velocity fluctuations, which are retained in the model but ignored in the classical model. Additional equations that account for the second-order fluctuations are also derived where third-order fluctuations arise but are neglected within the order of the approximations. The resulting system of equations has a very close resemblance to the Ten-moment Gaussian closure model of gas dynamics [@Levermore1998], except for the presence of some additional terms arising from gravitational effects. In particular, the entropy function of the two models is the same since the non-conservative terms in the SSW model, which are purely due to gravitational effects, do not make any contribution to the entropy equation. Being a non-conservative hyperbolic system, the numerical solution of the SSW model is challenging since the notion of weak solution requires the choice of a path which is usually not known. The correct path depends on the physical regularization mechanism and even when the correct path is known, the construction of a numerical scheme that converges to the weak solution is hard since the solution is sensitive to the numerical viscosity [@Abgrall2010]. In practice, a linear path is assumed in state space and some path conservative methods are developed which build some information of the waves present in the Riemann solution. For the SSW model, such methods have been developed following HLL-type ideas in [@Gavrilyuk2018; @bhole2019fluctuation; @Chandrashekar2020]. The first two works split the model into some sub-systems and developed Riemann solver type methods, while the last one treats it in a unified manner by writing it in the form of the Ten-moment system. An exact Riemann solver has been developed in [@Nkonga2022] for the linear path, and comparisons of the path conservative HLL-type numerical methods have been performed. The work in [@Busto2021] proposes a slightly different model of the shear shallow water problem and develops a thermodynamically consistent scheme. In the present work, we take a different approach to the construction of numerical methods, which is based on entropy consistency ideas [@tadmor2003entropy; @ismail2009affordable; @fjordholm2012arbitrarily; @chandrashekar2013kinetic]. The main technique is to first construct an entropy conservative scheme following the ideas of Tadmor and then add dissipative terms [@ismail2009affordable; @chandrashekar2013kinetic] that lead to an entropy inequality. For conservative systems, constructing the entropy conservative scheme is based on finding a central numerical flux that satisfies a certain jump condition [@tadmor2003entropy], see Theorem [\[thm:entconcond\]](#thm:entconcond){reference-type="ref" reference="thm:entconcond"}. The SSW model is non-conservative, but the equation has conservative and non-conservative terms. The conservative terms have the same structure as the Ten-moment equations of gas dynamics. Since the non-conservative terms do not contribute to the entropy, the ideas from conservative systems can be used to construct an entropy conservative scheme. This is the approach taken in the present work in the finite difference context where high-order accuracy is also achieved by following the ideas in LeFloch [@leFloch2002]. For conservation laws, there is a close relationship between the existence of a convex entropy function and the symmetrization of the equations, see [@Godlewski1996], Theorem 3.2. This property does not hold for general non-conservative systems; for the SSW model, we have a convex entropy function and an entropy conservation law for smooth solutions, but the equations cannot be symmetrized. The failure to symmetrize is due to the non-conservative terms related to gravitational effects but since they do not contribute to the entropy equation, we still have an entropy equation satisfied by smooth solutions. A framework to construct entropy stable schemes for non-conservative hyperbolic systems is presented in [@Castro2013], which uses the idea of path-consistent schemes and fluctuation splitting. Our approach is, however, different from this as we exploit the conservation form and the special structure of the non-conservative terms, which do not contribute to the entropy. The scheme is first developed in one dimension and extended to two dimensions on logically rectangular meshes. The stability and accuracy of the proposed schemes are demonstrated on several test cases in one and two dimensions. We have also compared the computed solutions with the exact solutions for several test cases. For the roll wave test cases, we have compared the computed solutions with the roll waves observed in some experimental studies in one and two dimensions. The rest of the paper is organized as follows. Section [2](#sec:ssweq){reference-type="ref" reference="sec:ssweq"} presents the non-conservative SSW model in a form where the conservative terms are similar to the Ten-moment equations . The entropy function and the entropy equation are discussed in Section [3](#sec:entropy){reference-type="ref" reference="sec:entropy"}. The semi-discrete entropy conservative and dissipative schemes are constructed in Section [4](#sec:semid){reference-type="ref" reference="sec:semid"}, which is also extended to higher order accuracy, and the entropy condition is demonstrated. Section [5](#sec:fulld){reference-type="ref" reference="sec:fulld"} discusses the fully discrete scheme obtained by adding a time integration scheme. Section [6](#sec:res){reference-type="ref" reference="sec:res"} presents numerical results obtained from the proposed schemes in one and two dimensions, and Section [7](#sec:sum){reference-type="ref" reference="sec:sum"} provides a summary. In the appendices, we examine the symmetrizability issue of the SSW model and derive the entropy scaled eigenvectors which are used to construct the entropy stable dissipative fluxes. # Equations of shear shallow water model {#sec:ssweq} The shear shallow water model has been recently studied in [@Chandrashekar2020] and expressed in an almost conservative form for the evolution of the water depth $h$, the depth average momentum $h\bm{v}$ and the energy tensor $\mathcal{E}=\{\mathcal{E}_{11},\mathcal{E}_{12},\mathcal{E}_{22}\}$. It is a system of non-linear, non-conservative hyperbolic partial differential equations. In $2-$D, following [@Chandrashekar2020], the governing equations of the shear shallow water model (SSW) can be expressed as, $$\label{eq:ssw} \frac{\partial \bm{U}}{\partial t} + \frac{\partial \bm{F}^x}{\partial x} + \frac{\partial \bm{F}^y}{\partial y} + \bm{B}^x\frac{\partial h}{\partial x} + \bm{B}^y\frac{\partial h}{\partial y} = \bm{S}$$ where $$\bm{U}=\begin{pmatrix} h\\ h v_1\\ h v_2\\ \mathcal{E}_{11}\\ \mathcal{E}_{12}\\ \mathcal{E}_{22} \end{pmatrix}, \quad \bm{F}^x=\begin{pmatrix} h v_1\\ h (v_{1}^2+\mathcal{P}_{11})\\ h (v_1 v_2+\mathcal{P}_{12})\\ \frac{1}{2}h v_1(v_1^2+{3} \mathcal{P}_{11})\\ \frac{1}{2}h (v_1^2 v_2+2 v_1 \mathcal{P}_{12}+v_2\mathcal{P}_{11})\\ \frac{1}{2}h (v_1 v_2^2+2 v_2\mathcal{P}_{12}+v_1 \mathcal{P}_{22}) \end{pmatrix},$$ $$\bm{F}^y=\begin{pmatrix} h v_2\\ h (v_1 v_2+\mathcal{P}_{12})\\ h (v_{2}^2+{\mathcal{P}_{22}})\\ \frac{1}{2}h(v_1^2 v_2+2 v_1 \mathcal{P}_{12}+v_2\mathcal{P}_{11})\\ \frac{1}{2}h(v_1 v_2^2+2v_2\mathcal{P}_{12}+v_1 \mathcal{P}_{22})\\ \frac{1}{2}h (v_2^3+{3}v_2 \mathcal{P}_{22}), \end{pmatrix}, \bm{B}^x=\begin{pmatrix} 0\\ gh\\ 0\\ ghv_1\\ \frac{1}{2}ghv_2\\ 0 \end{pmatrix}, \quad \bm{B}^y=\begin{pmatrix} 0\\ 0\\ gh\\ 0\\ \frac{1}{2} ghv_1\\ ghv_2 \end{pmatrix},$$ $$\bm{S}=\begin{pmatrix} 0\\ -gh\frac{\partial b}{\partial x}-C_f|\bm{v}|v_1\\ -gh\frac{\partial b}{\partial y} -C_f|\bm{v}|v_2\\ -\alpha |\bm{v}|^3 \mathcal{P}_{11}-ghv_1\frac{\partial b}{\partial x}-C_f|\bm{v}|v_1^{2}\\ -\alpha |\bm{v}|^3 \mathcal{P}_{12}-\frac{1}{2}ghv_2\frac{\partial b}{\partial x} -\frac{1}{2}ghv_1\frac{\partial b}{\partial y}-C_f|\bm{v}|v_1 v_{2}\\ -\alpha |\bm{v}|^3\mathcal{P}_{22} -ghv_2\frac{\partial b}{\partial y}-C_f|\bm{v}|v_2^{2} \end{pmatrix}.$$ In the above set of equations, $\bm{v}=(v_1,v_2)$ is the velocity vector, $g>0$ is the acceleration due to gravity, $b\equiv b\left( x, y\right)$ is the bottom topography, $C_f$ is the Chezy coefficient and $\alpha$ is given by the following relation [@Gavrilyuk2018; @richard2013classical], $$\alpha = \max\left(0, C_r \frac{T - \phi h^2}{T^2} \right), \qquad T = \mathop{\mathrm{trace}}(\mathcal{P}) = \mathcal{P}_{11} + \mathcal{P}_{22}, \qquad C_r > 0,$$ where, $\mathcal{P}=\{\mathcal{P}_{11},\mathcal{P}_{12},\mathcal{P}_{22}\}$ is the Reynolds stress tensor, which is symmetric, positive definite, and arises due to depth averaging. The quantities $C_f,~ C_r,~\phi$ are model constants and must be determined from experiments. The above system is closed with the equation of state, $$\mathcal{E}= \frac{h}{2}\left ( \rule{0mm}{4mm} \bm{v}\otimes\bm{v}+ \mathcal{P}\right ).$$ Next, we define the set of primitive variables $\bm{W}$, $$\bm{W}=(h,v_1,v_2,\mathcal{P}_{11},\mathcal{P}_{12},\mathcal{P}_{22})^\top$$ For the solution to be physically acceptable, we need the water depth $h$ and the symmetric stress tensor $\mathcal{P}$ to be positive. Hence, we consider the following set $\Omega$ of physically admissible solutions, $$\begin{aligned} \Omega=\{\bm{U}\in \mathbb{R}^6 |~h>0,~x^\top\mathcal{P}x>0, \forall x\in \mathbb{R}^2 \setminus \{(0,0)\}\}.\end{aligned}$$ Now for the solutions of the homogeneous case (i.e., $\bm{S}=0$) in $\Omega$, the system [\[eq:ssw\]](#eq:ssw){reference-type="eqref" reference="eq:ssw"} is hyperbolic for the states $\bm{U}\in \Omega$ with the following set of eigenvalues, $$\lambda_1 = v_d - \sqrt{gh + 3 \mathcal{P}_{dd}}, \quad \lambda_2 = v_d - \sqrt{\mathcal{P}_{dd}}, \quad \lambda_3 = \lambda_4 = v_d,$$ $$\lambda_5 = v_d+ \sqrt{\mathcal{P}_{dd}}, \quad \lambda_6 = v_d + \sqrt{g h + 3 \mathcal{P}_{dd}}.$$ Here, $d\in \{1,2\}$ and the indices $\{1,2\}$ denote the $x-$direction and $y-$direction respectively. The first and last eigenvalues correspond to genuinely non-linear characteristic fields in the sense of Lax [@Godlewski1996], while the remaining eigenvalues correspond to linearly degenerate characteristic fields [@Gavrilyuk2018]. In the $x$-direction, the matrix of right eigenvectors in terms of primitive variables $\bm{W}$ is given by $$\begin{aligned} {R}_{\bm{W}}^{x}=\begin{pmatrix} h (A^2-C^2) & 0 & -h & 0 & 0 & h (A^2-C^2) \\ -A(A^2-C^2) & 0 & 0 & 0 & 0 & A(A^2-C^2) \\ -2A\mathcal{P}_{12} & -C & 0 & 0 & C & 2A\mathcal{P}_{12} \\ 2C^2(A^2-C^2) & 0 &g h+ \mathcal{P}_{11} & 0 & 0 & 2C^2(A^2-C^2) \\ \mathcal{P}_{12}(A^2+C^2) & C^2 & \mathcal{P}_{12} & 0 & C^2 & \mathcal{P}_{12}(A^2+C^2) \\ 4\mathcal{P}_{12}^2 & 2\mathcal{P}_{12} & 0 & 1 & 2\mathcal{P}_{12} & 4\mathcal{P}_{12}^2 \end{pmatrix}.\end{aligned}$$ where $A = \sqrt{gh + 3 \mathcal{P}_{11}}$ and $C = \sqrt{\mathcal{P}_{11}}$. One can get the matrix of right eigenvectors in conservative variables by pre-multiplying the above matrix by the Jacobian matrix $\dfrac{\partial \mathbf{U}}{\partial \bm{W}}$ for the change of variable. # Entropy analysis {#sec:entropy} Solutions of a nonlinear hyperbolic system can be discontinuous even for very smooth initial data. This leads us to the consideration of weak solutions, which, however, may not be unique. Hence, an additional criterion is considered to select the physically relevant solution among all weak solutions in terms of the entropy condition. For the SSW model [\[eq:ssw\]](#eq:ssw){reference-type="eqref" reference="eq:ssw"}, we follow [@berthon2006numerical; @biswas2021entropy; @sen_entropy_2018; @berthon2015entropy] to define the entropy $\eta$ and the entropy fluxes $(q^x, q^y)$ as follows $$\begin{aligned} \label{entropy-pair} \eta=\eta(\bm{U}) = -h s, \qquad q^x=-h v_1s, \qquad q^y=-hv_2s\end{aligned}$$ where $$s=\log \left( \dfrac{\det\mathcal{P}}{h^2} \right)=\log \left( \dfrac{\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2}{h^2} \right)$$ For the homogeneous case, we will now prove the entropy equation. We proceed in one dimension as the two and three-dimensional cases are similar. The proof is similar to the entropy equality proof for the Ten-Moment equations presented in [@berthon2015entropy; @sen_entropy_2018]. **Proposition 1**. *Smooth solutions of [\[eq:ssw\]](#eq:ssw){reference-type="eqref" reference="eq:ssw"} without the source term satisfy the following entropy equality, $$\partial_t s+v_1 \partial_x s=0. \label{s}$$ As a corollary, for any smooth function H(s), we have, $$\partial_t(h H(s))+ \partial_x (hv_1 H(s))=0. \label{h_s}$$ In particular, smooth solutions will satisfy the entropy equality, $$\partial_t\eta+ \partial_xq^x=0. \label{entropy_pair_equality}$$* *Proof.* First, we will prove the equality [\[s\]](#s){reference-type="eqref" reference="s"}. Assuming $\bm{U}$ is a smooth solution of the system [\[eq:ssw\]](#eq:ssw){reference-type="eqref" reference="eq:ssw"} for the homogeneous case, we subtract the kinetic energy contributions from the energy equations to obtain the following equations in terms of the stress components, $$\begin{aligned} \partial_t \mathcal{P}_{11}+v_1\partial_x \mathcal{P}_{11}+2\mathcal{P}_{11}\partial_x v_1 & =0, & \\ \partial_t \mathcal{P}_{12}+v_1\partial_x \mathcal{P}_{12}+\mathcal{P}_{12}\partial_x v_1+\mathcal{P}_{11}\partial_x v_2 & =0, & \\ \partial_t \mathcal{P}_{22}+v_1\partial_x \mathcal{P}_{22}+2\mathcal{P}_{12}\partial_x v_2 & =0. \end{aligned}$$ Using the definition, $\det\mathcal{P}=\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2$, we apply the chain rule and use the above set of equations to obtain, $$\begin{aligned} \partial_t \det\mathcal{P}+v_1\partial_x\det\mathcal{P}+2\det\mathcal{P}\partial_x v_1=0. \label{detp} \end{aligned}$$ Now using [\[detp\]](#detp){reference-type="eqref" reference="detp"} and the water depth equation, $\partial_t h + v_1 \partial_x h + h \partial_x v_1 = 0$, we get, $$\begin{aligned} \partial_t s +v_1\partial_x s &=\frac{1}{\det \mathcal{P}} \partial_t \textrm{det}(\mathcal{P})- \frac{2}{h}\partial_t h+ v_1\frac{1}{\det \mathcal{P}} \partial_x \textrm{det}(\mathcal{P})- v_1 \frac{2}{h}\partial_x h\\ &=-\frac{1}{\det\mathcal{P}}\left(v_1\partial_x \det\mathcal{P}+2 \det\mathcal{P}\partial_x v_1 \right) + \frac{1}{\det\mathcal{P}}\left(v_1\partial_x \det\mathcal{P}\right) \\ &+\frac{2}{h}\left( h\partial_x v_1 +v_1\partial_x h\right) -v_1\frac{2}{h}\partial_x h\\ &=-2\partial_x v_1 + 2\partial_xv_1\\ &=0 \end{aligned}$$ The relations [\[h_s\]](#h_s){reference-type="eqref" reference="h_s"} and [\[entropy_pair_equality\]](#entropy_pair_equality){reference-type="eqref" reference="entropy_pair_equality"} can now be obtained using a simple application of the chain rule on the Eqn. [\[s\]](#s){reference-type="eqref" reference="s"}. ◻ From the proof of Proposition [Proposition 1](#prop:entropy){reference-type="ref" reference="prop:entropy"}, we observe that the non-conservative terms containing the gravitational effects do not make any contribution to the entropy evolution. In fact, this also follows from the fact that $\eta'(\bm{U}) \bm{B}^x(\bm{U}) = \eta'(\bm{U}) \bm{B}^y(\bm{U}) = 0$. This motivates the following definition of entropy function for non-conservative systems. **Definition 1**. *A convex function $\eta(\bm{U})$ is said to be an entropy function for the system $$\frac{\partial \bm{U}}{\partial t} + \frac{\partial \bm{F}^x}{\partial x} + \frac{\partial \bm{F}^y}{\partial y} + \tilde\bm{B}^x\frac{\partial \bm{U}}{\partial x} + \tilde\bm{B}^y\frac{\partial \bm{U}}{\partial y} = 0$$ if there exist smooth functions $q^x(\bm{U})$ and $q^y(\bm{U})$ such that $${q^x}'(\bm{U}) = \eta'(\bm{U}){\bm{F}^x}'(\bm{U}), \qquad {q^y}'(\bm{U}) = \eta'(\bm{U}){\bm{F}^y}'(\bm{U})$$ and $$\eta'(\bm{U}) \tilde\bm{B}^x(\bm{U}) = \eta'(\bm{U}) \tilde\bm{B}^y(\bm{U}) = 0$$ The functions ($\eta, q^x, q^y$) form an entropy-entropy flux pair.* The SSW model can be put in the above form with the matrix $\tilde\bm{B}^x$ containing the vector $\bm{B}^x$ in its first column and similarly, the matrix $\tilde\bm{B}^y$ containing the vector $\bm{B}^y$ in its first column, and all other columns being zero. We have seen above that the SSW model has the entropy pair $(\eta, q)$ and additionally satisfies the conservation law [\[entropy_pair_equality\]](#entropy_pair_equality){reference-type="eqref" reference="entropy_pair_equality"} for smooth solutions in the absence of source terms, while for discontinuous solutions, we can demand the entropy inequality $$\label{eq:entropyineq} \partial_t\eta+ \partial_xq^x \le 0$$ to hold in the sense of distributions. In the next Section, we will develop semi-discrete numerical schemes that satisfy a discrete entropy inequality [\[eq:entropyineq\]](#eq:entropyineq){reference-type="eqref" reference="eq:entropyineq"}. There is a close connection between the existence of an entropy pair and the symmetrization of a system of conservation laws. Since the SSW model is a non-conservative hyperbolic system, we investigate the symmetrizability of the system in detail in Appendix [8](#symmetrizability){reference-type="ref" reference="symmetrizability"}. Based on the discussion in Appendix [8](#symmetrizability){reference-type="ref" reference="symmetrizability"}, we conclude this section with the following remark. The existence of an entropy pair does not guarantee the symmetrizability of the system in the case of non-conservative hyperbolic systems. In particular, the SSW system [\[eq:ssw\]](#eq:ssw){reference-type="eqref" reference="eq:ssw"} is not symmetrizable. # Semi-discrete numerical schemes {#sec:semid} We can rewrite the SSW model [\[eq:ssw\]](#eq:ssw){reference-type="eqref" reference="eq:ssw"} as follows, $$\label{eq:ssw1} \frac{\partial \bm{U}}{\partial t} + \frac{\partial \bm{F}^x}{\partial x} + \frac{\partial \bm{F}^y}{\partial y} + \bm{B}^{nc} =\bm{S},$$ where $\bm{B}^{nc}= \bm{B}^x\frac{\partial h}{\partial x} + \bm{B}^y\frac{\partial h}{\partial y}$. In this Section, we will first develop semi-discrete schemes for the homogeneous part of the system [\[eq:ssw1\]](#eq:ssw1){reference-type="eqref" reference="eq:ssw1"}. The discretization of the source term is then discussed in Section [4.5](#subsec:semidisscheme){reference-type="ref" reference="subsec:semidisscheme"}. We discretize the domain $D=(x_a,x_b) \times (y_a,y_b)$ uniformly into cells $I_{ij}$ with mesh size of $\Delta x \times \Delta y$, where $\Delta x= \frac{x_b-x_a}{N_x}$ and $\Delta y= \frac{y_b-y_a}{N_y}$. We define the grid points by $x_i=x_a+i \Delta x$, $y_j=y_a+j \Delta y$, , with $0 \le i \le N_x$ and $0 \le j \le N_y$. We also define cell interfaces as $x_{i+1/2}= \frac{x_i+x_{i+1}}{2}$, $y_{j+1/2}= \frac{y_j+y_{j+1}}{2}$. Then a general semi-discrete conservative finite difference scheme has the following form, $$\begin{aligned} \frac{d }{dt}\bm{U}_{i,j}(t) + & \frac{1}{\Delta x}\left({\bm{F}^x}_{i+\frac{1}{2},j}(t)-{\bm{F}^x}_{i-\frac{1}{2},j}(t)\right)\nonumber \\ + & \frac{1}{\Delta y} \left({\bm{F}^y}_{i,j+\frac{1}{2}}(t)-{\bm{F}^y}_{i,j-\frac{1}{2}}(t)\right)+\bm{B}^{nc}_{i,j}(\bm{U})=0, \label{scheme}\end{aligned}$$ where ${\bm{F}^x}_{i+\frac{1}{2},j}$ and ${\bm{F}^y}_{i,j+\frac{1}{2}}$ are the numerical fluxes consistent with the continuous fluxes ${\bm{F}^x}$ and ${\bm{F}^y}$, respectively. The derivatives $\frac{\partial h}{\partial x},~\frac{\partial h}{\partial y}$ in the non-conservative term ${\bm{B}}_{i,j}^{nc}={\bm{B}^x(\bm{U}_{i,j})}\big(\frac{\partial h}{\partial x}\big)_{i,j}+{\bm{B}^y(\bm{U}_{i,j})}\big(\frac{\partial h}{\partial y}\big)_{i,j}$ are approximated by suitable order central difference approximations.\ The semi-discrete scheme [\[scheme\]](#scheme){reference-type="eqref" reference="scheme"} is said to be an entropy stable scheme if the computed solution satisfies the following entropy inequality, $$\frac{d}{dt} \eta(\bm{U}_{i,j}) +\frac{1}{\Delta x} \left( \hat{q^x}_{i+\frac{1}{2},j} - \hat{q^x}_{i-\frac{1}{2},j}\right)+\frac{1}{\Delta y}\left( \hat{q^y}_{i,j+\frac{1}{2}} - \hat{q^y}_{i,j-\frac{1}{2}}\right) \le 0,$$ for some numerical entropy fluxes $\hat{q^y}$ and $\hat{q^y}$ consistent with the fluxes ${q}^x$ and $q^y$, respectively. The procedure for construction of an entropy stable scheme involves first constructing an entropy conservative scheme. We say the semi-discrete scheme [\[scheme\]](#scheme){reference-type="eqref" reference="scheme"} is an entropy conservative scheme if the computed solution satisfies the following entropy equality $$\frac{d}{dt} \eta(\bm{U}_{i,j}) +\frac{1}{\Delta x} \left( \tilde{q^x}_{i+\frac{1}{2},j} - \tilde{q^x}_{i-\frac{1}{2},j}\right)+\frac{1}{\Delta y}\left( \tilde{q^y}_{i,j+\frac{1}{2}} - \tilde{q^y}_{i,j-\frac{1}{2}}\right) = 0,$$ for some numerical entropy fluxes $\tilde{q^x}$ and $\tilde{q^y}$ consistent with the fluxes ${q}^x$ and $q^y$, respectively. Hence, first, we discuss the construction of entropy conservative scheme. ## Entropy conservative schemes For the construction of numerical flux that leads to an entropy conservative scheme, we define the entropy variable $\bm{V}=\frac{\partial \eta}{\partial \bm{U}}$ and entropy potential $\psi^{k}=\bm{V}^\top \bm{F^k}-q^{k},~k\in \{x,y\}$. A simple calculation results in, $$\begin{aligned} \label{entvar} \renewcommand{\arraystretch}{2} \bm{V}:=\frac{\partial \eta}{\partial \bm{U}}=\begin{pmatrix} 4-s- \dfrac{1}{\textrm{det}(\mathcal{P})}\left(\mathcal{P}_{11} v_2^2+\mathcal{P}_{22} v_1^2-2\mathcal{P}_{12}v_1 v_2\right)\\ \dfrac{2 (\mathcal{P}_{22}v_1-\mathcal{P}_{12}v_2)}{\textrm{det}(\mathcal{P})}\\ \dfrac{2 (\mathcal{P}_{11}v_2-\mathcal{P}_{12}v_1)}{\textrm{det}(\mathcal{P})}\\ -\dfrac{2 \mathcal{P}_{22}}{\textrm{det}(\mathcal{P})}\\ \dfrac{4 \mathcal{P}_{12}}{\textrm{det}(\mathcal{P})}\\ -\dfrac{2 \mathcal{P}_{11}}{\textrm{det}(\mathcal{P})} \end{pmatrix}\end{aligned}$$ The entropy potentials are given by, $$\begin{aligned} \psi^x=\bm{V}^\top \bm{F}^x-q^x= 2 h v_1, \qquad \psi^y=\bm{V}^\top \bm{F}^y-q^y= 2 h v_2.\end{aligned}$$ We now recall the following theorem, which provides us a procedure for the construction of entropy conservative fluxes, ${\tilde{\bm{F}^x}}$ and ${\tilde{\bm{F}^y}}$. For a given variable $a$, we introduce the notations $[\![{\cdot}]\!]$ for the jump and $\bar{\cdot}$ for the arithmetic average in the following way, $$\!]_{i+\frac{1}{2},j}=a_{i+1,j}-a_{i,j}, \qquad \bar{a}_{i+\frac{1}{2},j}=\frac{1}{2}(a_{i+1,j}+a_{i,j}),$$ $$\!]_{i,j+\frac{1}{2}}=a_{i,j+1}-a_{i,j}, \qquad \bar{a}_{i,j+\frac{1}{2}}=\frac{1}{2}(a_{i,j+1}+a_{i,j}).$$ [\[thm:entconcond\]]{#thm:entconcond label="thm:entconcond"} Let ${\tilde{\bm{F}^x}}$ and ${\tilde{\bm{F}^y}}$ be the consistent numerical fluxes, which satisfy $$\!]^{\top}_{i+\frac{1}{2},j}\,{\tilde{\bm{F}^x}}_{i+\frac{1}{2},j}=[\![\psi^x]\!]_{i+\frac{1}{2},j}, \ \quad \ [\![\bm{V}]\!]^{\top}_{i,j+\frac{1}{2}}\,{\tilde{\bm{F}^y}}_{i,j+\frac{1}{2}}=[\![\psi^y]\!]_{i,j+\frac{1}{2}}, \label{tadmor_thm}$$ then the scheme [\[scheme\]](#scheme){reference-type="eqref" reference="scheme"} with the numerical fluxes ${\tilde{\bm{F}^x}}$ and ${\tilde{\bm{F}^y}}$ is second-order accurate and entropy conservative, i.e., the computed solutions satisfy the discrete entropy equality $$\frac{d}{dt} \eta(\bm{U}_{i,j}) +\frac{1}{\Delta x} \left( \tilde{q^x}_{i+\frac{1}{2},j} - \tilde{q^x}_{i-\frac{1}{2},j}\right)+\frac{1}{\Delta y}\left( \tilde{q^y}_{i,j+\frac{1}{2}} - \tilde{q^y}_{i,j-\frac{1}{2}}\right) = 0,$$ corresponding to the numerical entropy fluxes, $$\tilde{q^x}_{i+\frac{1}{2},j}=\bar{\bm{V}}_{i+\frac{1}{2},j}^{\top}{\tilde{\bm{F}^x}}_{i+\frac{1}{2},j}-\bar{\psi^x}_{i+\frac{1}{2},j} \qquad \text{and} \qquad \tilde{q^y}_{i,j+\frac{1}{2}}=\bar{\bm{V}}_{i,j+\frac{1}{2}}^{\top}{\tilde{\bm{F}^y}}_{i,j+\frac{1}{2}}-\bar{\psi^y}_{i,j+\frac{1}{2}}.$$ First, we consider the $x$-directional identity [\[tadmor_thm\]](#tadmor_thm){reference-type="eqref" reference="tadmor_thm"} to get the conservative flux in the $x$-direction. Note that we have a single algebraic equation with $6$ unknowns ${\tilde{\bm{F}^x}}=[\tilde{f}_1,\,\tilde{f}_2,\,\tilde{f}_3,\,\tilde{f}_4,\,\tilde{f}_5,\,\tilde{f}_6]^\top$. Therefore, we cannot have a unique solution for the algebraic equation [\[tadmor_thm\]](#tadmor_thm){reference-type="eqref" reference="tadmor_thm"}. In [@ismail2009affordable; @chandrashekar2013kinetic], the authors have presented a procedure to find an inexpensive entropy conservative flux. For the SSW model [\[eq:ssw1\]](#eq:ssw1){reference-type="eqref" reference="eq:ssw1"}, we follow the approach presented in [@chandrashekar2013kinetic] to construct an entropy conservative flux in the next sub-section [\[entropy_con_flux\]](#entropy_con_flux){reference-type="eqref" reference="entropy_con_flux"}. ### Entropy conservative flux {#entropy_con_flux} We first consider the $x-$directional case. Following [@tadmor1987numerical], we need to find an entropy conservative flux ${\tilde{\bm{F}^x}}=[\tilde{f}_1,\,\tilde{f}_2,\,\tilde{f}_3,\,\tilde{f}_4,\,\tilde{f}_5,\,\tilde{f}_6]^\top$ satisfying the identity: $$\begin{aligned} \!]^\top \cdot {\tilde{\bm{F}^x}}=[\![ \psi^x]\!].\label{confluxx}\end{aligned}$$ For simplicity, we ignore the indices $i$ and define $$D = {\det(\mathcal{P})}=\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2, \quad \beta_{11}=\dfrac{\mathcal{P}_{11}}{D}, \quad \beta_{12}=\dfrac{\mathcal{P}_{12}}{D},$$ $$\beta_{22}=\dfrac{\mathcal{P}_{22}}{D}, \quad D_{\beta}=\beta_{11}\beta_{22}-\beta_{12}^2$$ We also define the logarithmic average, $a^{\ln}=\dfrac{[\![a]\!]}{[\![\ln a]\!]}$. As the conservative flux is the same as the flux of Ten-Moment equations (where the water depth is replaced by density), we use the entropy conservative flux derived in [@sen_entropy_2018] for Ten-Moment equations. The expression of the numerical flux is, $${\tilde{\bm{F}^x}}= \begin{pmatrix} h^{ln} \bar{v}_1 \\ \bar{v}_1\tilde{f}_1+\frac{\bar{\beta}_{11} \bar{h}}{\bar{\beta}_{11}\bar{\beta}_{22}-\left(\bar{\beta}_{12}\right)^2} \\ \bar{v}_2\tilde{f}_1+\frac{\bar{\beta}_{12} \bar{h}}{\bar{\beta}_{11}\bar{\beta}_{22}-\left(\bar{\beta}_{12}\right)^2} \\ \frac{1}{2}\left(\dfrac{\bar{\beta}_{11}}{D_\beta^{ln}}-\overline{\left(v_1\right)^2}\right)\tilde{f}_1+\bar{v}_1\tilde{f}_2 \\ \frac{1}{2}\bigg(\left(\dfrac{\bar{\beta}_{12}}{D_\beta^{ln}}-\overline{v_1 v_2}\right)\tilde{f}_1+\bar{v}_1\tilde{f}_3+\bar{v}_2\tilde{f}_2\bigg) \\ \frac{1}{2}\left(\dfrac{\bar{\beta}_{22}}{D_\beta^{ln}}-\overline{\left(v_2\right)^2}\right)\tilde{f}_1+\bar{v}_2\tilde{f}_3 \end{pmatrix}.$$ The $y-$directional entropy conservative flux is given by, $\bm{\tilde{\bm{F}}^y}=[\tilde{g}_1,\,\tilde{g}_2,\,\tilde{g}_3,\,\tilde{g}_4,\,\tilde{g}_5,\,\tilde{g}_6]^\top$ as $${\tilde{\bm{F}^y}}= \begin{pmatrix} h^{ln} \bar{v}_2 \\ \bar{v}_1\tilde{g}_1+\frac{\bar{\beta}_{12} \bar{h}}{\bar{\beta}_{11}\bar{\beta}_{22}-\left(\bar{\beta}_{12}\right)^2} \\ \bar{v}_2\tilde{g}_1+\frac{\bar{\beta}_{22} \bar{h}}{\bar{\beta}_{11}\bar{\beta}_{22}-\left(\bar{\beta}_{12}\right)^2} \\ \frac{1}{2}\left(\dfrac{\bar{\beta}_{11}}{D_\beta^{ln}}-\overline{\left(v_1\right)^2}\right)\tilde{g}_1+\bar{v}_1\tilde{g}_2 \\ \frac{1}{2}\bigg(\left(\dfrac{\bar{\beta}_{12}}{D_\beta^{ln}}-\overline{v_1 v_2}\right)\tilde{g}_1+\bar{v}_1\tilde{g}_3+\bar{v}_2\tilde{g}_2\bigg) \\ \frac{1}{2}\left(\dfrac{\bar{\beta}_{22}}{D_\beta^{ln}}-\overline{\left(v_2\right)^2}\right)\tilde{g}_1+\bar{v}_2\tilde{g}_3 \end{pmatrix}.$$ Note that these are two-point fluxes, i.e., they depend on two states. One can easily observe that the above fluxes ${\tilde{\bm{F}^x}}$ and ${\tilde{\bm{F}^y}}$ are consistent with the exact fluxes $\bm{F}^x$ and $\bm{F}^y$, respectively, when the two states are equal. ## Higher order entropy conservative schemes The entropy conservative fluxes presented above are only second-order accurate. To get higher-order accurate conservative fluxes, we follow the approach of [@leFloch2002]. They have constructed $2p^{th}$, $p\in\mathbb{N}$ , order accurate entropy conservative flux by choosing specific linear combinations of the second-order accurate entropy conservative fluxes. In particular, the $x$-directional entropy conservative flux for the $4^{th}$-order ($p=2$) scheme is given by $$\begin{aligned} {\tilde{\bm{F}^x}}_{i+\frac{1}{2},j}^4=\frac{4}{3}{\tilde{\bm{F}^x}}_{i+\frac{1}{2},j}(\bm{U}_{i,j},\bm{U}_{i+1,j})\nonumber\\-\frac{1}{6} \bigg( {\tilde{\bm{F}^x}}_{i+\frac{1}{2},j} (\bm{U}_{i-1,j},\bm{U}_{i+1,j})+ {\tilde{\bm{F}^x}}_{i+\frac{1}{2},j}(\bm{U}_{i,j},\bm{U}_{i+2,j}) \bigg).\label{4thorder_x} \end{aligned}$$ A similar expression can be derived for the $y$-directional $4^{th}$-order flux $$\begin{aligned} {\tilde{\bm{F}^y}}_{i,j+\frac{1}{2}}^4=\frac{4}{3}{\tilde{\bm{F}^y}}_{i,j+\frac{1}{2}}(\bm{U}_{i,j},\bm{U}_{i,j+1})\nonumber\\-\frac{1}{6} \bigg( {\tilde{\bm{F}^y}}_{i,j+\frac{1}{2}} (\bm{U}_{i,j-1},\bm{U}_{i,j+1})+ {\tilde{\bm{F}^y}}_{i,j+\frac{1}{2}}(\bm{U}_{i,j},\bm{U}_{i,j+2}) \bigg).\label{4thorder_y} \end{aligned}$$ The scheme [\[scheme\]](#scheme){reference-type="eqref" reference="scheme"} with the numerical fluxes ${\tilde{\bm{F}^x}}^4$ and ${\tilde{\bm{F}^y}}^4$ is fourth order accurate and entropy conservative. ## Entropy stable schemes As the entropy needs to decay at shocks, the entropy conservative schemes designed above will produce oscillations at the shock. Hence, we need an appropriate entropy dissipation process, resulting in the entropy inequality. We follow [@tadmor1987numerical] to define the modified fluxes ${\hat{\bm{F}^x}},~{\hat{\bm{F}^y}}$ as follows: $$\begin{aligned} {\hat{\bm{F}^x}}_{i+\frac{1}{2},j} ={\tilde{\bm{F}^x}}_{i+\frac{1}{2},j} - \frac{1}{2} \textbf{D}^{x}_{i+\frac{1}{2},j}[\![ \bm{V}]\!]_{i+\frac{1}{2},j}, % \\ % {\hat{\bm{F}^y}}_{i,j+\frac{1}{2}} = {\tilde{\bm{F}^y}}_{i,j+\frac{1}{2}} - \frac{1}{2} \textbf{D}^{y}_{i,j+\frac{1}{2}}[\![ \bm{V}]\!]_{i,j+\frac{1}{2}}, \label{es_numflux} \end{aligned}$$ where $\textbf{D}^x_{i+\frac{1}{2},j}$ and $\textbf{D}^y_{i,j+\frac{1}{2}}$ are symmetric positive definite matrices. Then we have the following Lemma: The numerical scheme [\[scheme\]](#scheme){reference-type="eqref" reference="scheme"} with the modified numerical fluxes [\[es_numflux\]](#es_numflux){reference-type="eqref" reference="es_numflux"} is entropy stable, i.e., the computed solution satisfies, $$\frac{d}{dt} \eta(\bm{U}_{i,j}) +\frac{1}{\Delta x} \left( \hat{q^x}_{i+\frac{1}{2},j} - \hat{q^x}_{i-\frac{1}{2},j}\right)+\frac{1}{\Delta y}\left( \hat{q^y}_{i,j+\frac{1}{2}} - \hat{q^y}_{i,j-\frac{1}{2}}\right) \le 0,$$ with consistent numerical entropy flux functions, $$\begin{aligned} \hat{q^x}_{i+\frac{1}{2},j}= \tilde{q^x}_{i+\frac{1}{2},j} - \frac{1}{2}\bar{\bm{V}}_{i+\frac{1}{2},j}^{\top} \textbf{D}_{i+\frac{1}{2},j}^{x}[\![ \bm{V}]\!]_{i+\frac{1}{2},j} \end{aligned}$$ and $$\begin{aligned} \qquad \hat{q^y}_{i,j+\frac{1}{2}}= \tilde{q^y}_{i,j+\frac{1}{2}} - \frac{1}{2}\bar{\bm{V}}_{i,j+\frac{1}{2}}^{\top} \textbf{D}_{i,j+\frac{1}{2}}^{y}[\![ \bm{V}]\!]_{i,j+\frac{1}{2}}.\end{aligned}$$ Here, we use *Rusanov's type* diffusion operators for the matrix $\textbf{D}$, given by, $$\label{diffusiontype} \textbf{D}_{i+\frac{1}{2},j}^{x} = \tilde{R}^x_{i+\frac{1}{2},j} \Lambda_{i+\frac{1}{2},j}^x \tilde{R}_{i+\frac{1}{2},j}^{x \top} \qquad \text{and} \qquad \textbf{D}_{i,j+\frac{1}{2}}^y = \tilde{R}_{i,j+\frac{1}{2}}^y \Lambda_{i,j+\frac{1}{2}}^y \tilde{R}_{i,j+\frac{1}{2}}^{y \top},$$ where ${\tilde{R}^d},\, d \in \{x,y\},$ are matrices of the scaled entropy right eigenvectors of the jacobian $\dfrac{\partial \bm{F}^d}{\partial \bm{U}}$, and ${\Lambda^d},\, d \in \{x,y\},$ are $6\times 6$ diagonal matrices of the form $${\Lambda^d}= \left( \max_{1 \leq k \leq 6} |\lambda_k^d|\right) \mathbf{I}_{6 \times 6}, \ \ \, d \in \{x,y\}.$$ Here $\{\lambda_k^d: 1 \leq k \leq 6 \}$ is the set of eigenvalues of the jacobian $\frac{\partial\bm{F}^d}{\partial \bm{U}}$. The procedure to obtain the scaled right eigenvector matrices $\tilde{R}$ is given in [@barth1999numerical]. We follow [@barth1999numerical],[@sen_entropy_2018] to derive expressions for the scaling matrices in Appendix [\[scaledrev\]](#scaledrev){reference-type="eqref" reference="scaledrev"}. Now, with the choice of diffusion operator [\[diffusiontype\]](#diffusiontype){reference-type="eqref" reference="diffusiontype"}, the numerical scheme [\[scheme\]](#scheme){reference-type="eqref" reference="scheme"} with the numerical flux [\[es_numflux\]](#es_numflux){reference-type="eqref" reference="es_numflux"} is entropy stable. ## Higher order entropy stable schemes The entropy stable scheme [\[scheme\]](#scheme){reference-type="eqref" reference="scheme"} discussed above with the numerical flux [\[es_numflux\]](#es_numflux){reference-type="eqref" reference="es_numflux"} contains the jump terms $[\![\bm{V}]\!]_{i+\frac{1}{2},j}$ and $[\![\bm{V}]\!]_{i,j+\frac{1}{2}}$ which are of first order accuracy. Therefore, the resultant scheme cannot be expected to be more than first-order accurate. The natural idea to increase the order of accuracy is to approximate the jump terms using higher-order polynomial reconstructions. However, straightforward reconstruction cannot be shown to preserve entropy stability. Therefore, instead of reconstructing the entropy variable $\bm{V}_{i,j}$ we follow the reconstruction procedure of [@fjordholm2012arbitrarily] to reconstruct the scaled entropy variables $\bm{\mathcal{{V}}}_{i,j}$, defined as $$\bm{\mathcal{{V}}}_{i,j}^{x,\pm}\,=\, {R}^{x^{\top}}_{i\pm\frac{1}{2},j}\bm{V}_{i,j}.$$ If $\bm{\mathcal{\tilde{V}}}_{i,j}^{x,\pm}$ denotes the $k$-th order reconstructed values of $\bm{\mathcal{V}}_{i,j}^{x,\pm}$ in the $x$-direction, then, $$\bm{\tilde{V}}_{i,j}^{x,\pm}\,=\, \left\lbrace \tilde{ R}^{x^{\top}}_{i\pm\frac{1}{2},j}\right\rbrace ^{(-1)}\bm{\mathcal{\tilde{{V}}}}_{i,j}^{x,\pm},$$ are the corresponding $k$-th order reconstructed values for $\bm{V}_{ij}$. Hence, the modified numerical flux is given by, $${\hat{\bm{F}^x}}^{k}_{i+\frac{1}{2},j}\,=\,{\tilde{\bm{F}^x}}^{2p}_{i+\frac{1}{2},j}\,-\,\frac{1}{2}\,\textbf{D}_{i+\frac{1}{2},j}^x[\![ \bm{\tilde{V}}^x]\!]_{i+\frac{1}{2},j}^k \label{eq:entropy_stable_flux}$$ where $[\![ \bm{\tilde{V}}^x]\!]_{i+\frac{1}{2},j}^k$ stands for, $$[\![ \bm{\tilde{V}}^x]\!]_{i+\frac{1}{2},j}^k = \bm{\tilde{V}}^{x,-}_{i+1,j}\,-\,\bm{\tilde{V}}^{x,+}_{i,j}$$ and $p \in \mathbb{N}$ is chosen as - $p=k/2$ if $k$ is even, - $p=(k+1)/2$ if $k$ is odd, where $k$ is the accuracy of the time integration scheme. As in [@fjordholm2012arbitrarily], the sufficient condition for the numerical flux [\[eq:entropy_stable_flux\]](#eq:entropy_stable_flux){reference-type="eqref" reference="eq:entropy_stable_flux"} to be entropy stable is that the reconstruction process for ${\bm{\mathcal{V}}}$ must satisfy the sign preserving property, i.e., the sign of the reconstructed jumps at any face must be same as the sign of the original jumps; for example, for a reconstruction along the $x$-direction, we need the following $$\label{eq:recsign} \textrm{sign} \left( \bm{\mathcal{V}}^{x,-}_{i+1,j} - \bm{\mathcal{V}}^{x,+}_{i,j} \right) = \textrm{sign} \left( \bm{\mathcal{V}}_{i+1,j} - \bm{\mathcal{V}}_{i,j} \right)$$ to hold for each component. Consequently, we use *minmod* reconstruction for the second order scheme, which satisfies this property and denotes it by O2$\_$ES. Following [@fjordholm2013eno], for the higher order schemes, we use the *ENO* based reconstruction. In particular, for the third-order scheme, we use the fourth-order entropy conservative flux [\[4thorder_x\]](#4thorder_x){reference-type="eqref" reference="4thorder_x"} and the third-order $\text{ENO}$ reconstruction to obtain the expression for the $x-$directional flux as $${\hat{\bm{F}^x}}^{3}_{i+\frac{1}{2},j}\,=\,{\tilde{\bm{F}^x}}^4_{i+\frac{1}{2},j}\,-\,\frac{1}{2}\,\textbf{D}_{i+\frac{1}{2},j}^x[\![ \bm{\tilde{V}}^x]\!]_{i+\frac{1}{2},j}^3$$ and denote it by O3$\_$ES. Similarly, for the fourth-order scheme, we use the fourth-order entropy conservative flux [\[4thorder_x\]](#4thorder_x){reference-type="eqref" reference="4thorder_x"} and a fourth-order ENO reconstruction and denote the scheme by O4$\_$ES. Note that the extension to two dimensions is straightforward . ## Semi-discrete entropy stability {#subsec:semidisscheme} We now proceed to show that the scheme [\[scheme\]](#scheme){reference-type="eqref" reference="scheme"} with the numerical flux [\[eq:entropy_stable_flux\]](#eq:entropy_stable_flux){reference-type="eqref" reference="eq:entropy_stable_flux"} is entropy stable. The semi-discrete schemes O2_ES, O3_ES, and O4_ES designed above are entropy stable, i.e., they satisfy, $$\frac{d}{dt} \eta(\bm{U}_{i,j}) +\frac{1}{\Delta x} \left( \hat{q^x}_{i+\frac{1}{2},j} - \hat{q^x}_{i-\frac{1}{2},j}\right)+\frac{1}{\Delta y}\left( \hat{q^y}_{i,j+\frac{1}{2}} - \hat{q^y}_{i,j-\frac{1}{2}}\right) \le 0, \label{eq:semi_disc_ent_inq}$$ where $\hat{q^x}$ and $\hat{q^y}$ are the consistent numerical entropy fluxes. *Proof.* Following [@fjordholm2012arbitrarily; @tadmor1987numerical], we have $$\begin{aligned} & \frac{d}{dt} \eta(\bm{U}_{i,j}) +\frac{1}{\Delta x} \left( \hat{q^x}_{i+\frac{1}{2},j} - \hat{q^x}_{i-\frac{1}{2},j}\right)+\frac{1}{\Delta y}\left( \hat{q^y}_{i,j+\frac{1}{2}} - \hat{q^y}_{i,j-\frac{1}{2}}\right) \\ = & - {\frac{1}{2}}\left( \bm{\mathcal{V}}_{i,j} - \bm{\mathcal{V}}_{i-1,j} \right)^\top \Lambda^x_{i-1/2,j} \left( \bm{\mathcal{V}}^{x,-}_{i,j} - \bm{\mathcal{V}}^{x,+}_{i-1,j} \right) \\ & - {\frac{1}{2}}\left( \bm{\mathcal{V}}_{i+1,j} - \bm{\mathcal{V}}_{i,j} \right)^\top \Lambda^x_{i+1/2,j} \left( \bm{\mathcal{V}}^{x,-}_{i+1,j} - \bm{\mathcal{V}}^{x,+}_{i,j} \right) \\ & - {\frac{1}{2}}\left( \bm{\mathcal{V}}_{i,j} - \bm{\mathcal{V}}_{i,j-1} \right)^\top \Lambda^y_{i,j-1/2} \left( \bm{\mathcal{V}}^{y,-}_{i,j} - \bm{\mathcal{V}}^{y,+}_{i,j-1} \right) \\ & - {\frac{1}{2}}\left( \bm{\mathcal{V}}_{i,j+1} - \bm{\mathcal{V}}_{i,j} \right)^\top \Lambda^y_{i,j+1/2} \left( \bm{\mathcal{V}}^{y,-}_{i,j+1} - \bm{\mathcal{V}}^{y,+}_{i,j} \right) \\ & -\bm{V}_{i,j}^\top\bm{B}^x_{i,j}\left(\frac{\partial h}{\partial x}\right)_{i,j}-\bm{V}_{i,j}^\top\bm{B}^y_{i,j}\left(\frac{\partial h}{\partial y}\right)_{i,j} \\ \color{black}= & - {\frac{1}{2}}\left( \bm{\mathcal{V}}_{i,j} - \bm{\mathcal{V}}_{i-1,j} \right)^\top \Lambda^x_{i-1/2,j} \left( \bm{\mathcal{V}}^{x,-}_{i,j} - \bm{\mathcal{V}}^{x,+}_{i-1,j} \right) \\ & - {\frac{1}{2}}\left( \bm{\mathcal{V}}_{i+1,j} - \bm{\mathcal{V}}_{i,j} \right)^\top \Lambda^x_{i+1/2,j} \left( \bm{\mathcal{V}}^{x,-}_{i+1,j} - \bm{\mathcal{V}}^{x,+}_{i,j} \right) \\ & - {\frac{1}{2}}\left( \bm{\mathcal{V}}_{i,j} - \bm{\mathcal{V}}_{i,j-1} \right)^\top \Lambda^y_{i,j-1/2} \left( \bm{\mathcal{V}}^{y,-}_{i,j} - \bm{\mathcal{V}}^{y,+}_{i,j-1} \right) \\ & - {\frac{1}{2}}\left( \bm{\mathcal{V}}_{i,j+1} - \bm{\mathcal{V}}_{i,j} \right)^\top \Lambda^y_{i,j+1/2} \left( \bm{\mathcal{V}}^{y,-}_{i,j+1} - \bm{\mathcal{V}}^{y,+}_{i,j} \right)\;\; (\text{Using } \bm{V}_{i,j}^\top\bm{B}^x_{i,j}=\bm{V}_{i,j}^\top\bm{B}^y_{i,j}=0) \color{black} \end{aligned}$$ Using the sign property [\[eq:recsign\]](#eq:recsign){reference-type="eqref" reference="eq:recsign"} of the reconstruction process, the jumps in scaled entropy variables and their reconstructed jumps have the same signs. Also, matrices $\Lambda^x$ are $\Lambda^y$ are diagonal with positive entry. Hence, each term on the right side of the above equality is negative. This results in the inequality [\[eq:semi_disc_ent_inq\]](#eq:semi_disc_ent_inq){reference-type="eqref" reference="eq:semi_disc_ent_inq"}. ◻ The general semi-discrete finite difference scheme for the system [\[eq:ssw1\]](#eq:ssw1){reference-type="eqref" reference="eq:ssw1"} has the following form, $$\begin{aligned} \frac{d }{dt}\bm{U}_{i,j}(t) + \frac{1}{\Delta x}\left({\bm{F}^x}_{i+\frac{1}{2},j}(t)-{\bm{F}^x}_{i-\frac{1}{2},j}(t)\right)\nonumber \\ + \frac{1}{\Delta y} \left({\bm{F}^y}_{i,j+\frac{1}{2}}(t)-{\bm{F}^y}_{i,j-\frac{1}{2}}(t)\right)+\bm{B}^{nc}_{i,j}(\bm{U})=\bm{S}_{i,j}, \label{scheme1} \end{aligned}$$ where $\bm{S}_{i,j}=\bm{S}(\bm{U}_{i,j})$. Then, we have the following remark: The semi-discrete scheme [\[scheme1\]](#scheme1){reference-type="eqref" reference="scheme1"} with the numerical flux [\[eq:entropy_stable_flux\]](#eq:entropy_stable_flux){reference-type="eqref" reference="eq:entropy_stable_flux"} satisfies the following inequality, $$\frac{d}{dt} \eta(\bm{U}_{i,j}) +\frac{1}{\Delta x} \left( \hat{q^x}_{i+\frac{1}{2},j} - \hat{q^x}_{i-\frac{1}{2},j}\right)+\frac{1}{\Delta y}\left( \hat{q^y}_{i,j+\frac{1}{2}} - \hat{q^y}_{i,j-\frac{1}{2}}\right) \le 4 \alpha_{i,j}|\bm{v}_{i,j}|^3.$$ Following [@fjordholm2012arbitrarily; @tadmor1987numerical], we have $$\begin{aligned} & \frac{d}{dt} \eta(\bm{U}_{i,j}) +\frac{1}{\Delta x} \left( \hat{q^x}_{i+\frac{1}{2},j} - \hat{q^x}_{i-\frac{1}{2},j}\right)+\frac{1}{\Delta y}\left( \hat{q^y}_{i,j+\frac{1}{2}} - \hat{q^y}_{i,j-\frac{1}{2}}\right) \\ \le & -\bm{V}_{i,j}\top\bm{B}^x_{i,j}\left(\frac{\partial h}{\partial x}\right)_{i,j}-\bm{V}_{i,j}^\top\bm{B}^y_{i,j}\left(\frac{\partial h}{\partial y}\right)_{i,j}+\bm{V}_{i,j}^\top\bm{S}_{i,j}(\bm{U}), & \\ \color{black} \le & \bm{V}_{i,j}^\top\bm{S}_{i,j}(\bm{U})=4 \alpha_{i,j}|\bm{v}_{i,j}|^3. \color{black} \end{aligned}$$ # Fully discrete scheme {#sec:fulld} Let the initial time be $t^0$ and let $\bm{U}^n$ be the discrete solution at time $t^n$. The semi-discrete scheme [\[scheme\]](#scheme){reference-type="eqref" reference="scheme"} can be expressed as $$\frac{d }{dt}\bm{U}_{i,j}(t) = \mathcal{L}_{i,j}(\bm{U}(t)) -{\bm{B}^x(\bm{U}_{i,j}(t))}\left(\frac{\partial h}{\partial x}\right)_{i,j}-{\bm{B}^y(\bm{U}_{i,j}(t))}\left(\frac{\partial h}{\partial y}\right)_{i,j}+\bm{S}(\bm{U}_{i,j}(t)) \label{fullydiscrete}$$ where, $$\mathcal{L}_{i,j}(\bm{U}(t)) = - \frac{1}{\Delta x} \left(\mathbf{F}_{i+\frac{1}{2},j}^x(t)-\mathbf{F}_{i-\frac{1}{2},j}^x(t)\right) - \frac{1}{\Delta y} \left(\mathbf{F}_{i,j+\frac{1}{2}}^y(t)-\mathbf{F}_{i,j-\frac{1}{2}}^y(t)\right).$$ The spatial derivatives are approximated using central differencing of suitable order (see section [\[sec:res\]](#sec:res){reference-type="eqref" reference="sec:res"}). The system of ODE [\[fullydiscrete\]](#fullydiscrete){reference-type="eqref" reference="fullydiscrete"} can be integrated in time in several ways and we use explicit time discretization. ## Explicit schemes We use explicit strong stability preserving Runge Kutta (SSP-RK) methods explained in [@gottlieb2001strong] for the time discretization of the SSW model. The second and third-order accurate SSP-RK schemes have the following structure for one time step. 1. Set $\bm{U}^{0} \ = \ \bm{U}^n$. 2. For $k$ in $\{1,\dots,m+1\}$, compute $$\begin{aligned} \bm{U}_{i,j}^{(k)} \ = \ \sum_{l=0}^{k-1}\gamma_{kl}\bm{U}_{i,j}^{(l)} + \delta_{kl}\Delta t \Big(\mathcal{L}_{i,j}(\bm{U}^{(l)})-{\bm{B}^x}(\bm{U}_{i,j}^{(l)})\left(\frac{\partial h}{\partial x}\right)_{i,j}-{\bm{B}^y(\bm{U}_{i,j}^{(l)})}\left(\frac{\partial h}{\partial y}\right)_{i,j} + \bm{S}(\bm{U}_{i,j}^{(l)}) \Big), \end{aligned}$$ where $\gamma_{kl}$ and $\delta_{kl}$ are given in Table [\[table:ssp\]](#table:ssp){reference-type="eqref" reference="table:ssp"}. 3. Finally, $\bm{U}_{i,j}^{n+1} \ = \ \bm{U}_{i,j}^{(m+1)}$. Order $\gamma_{il}$ $\delta_{il}$ ------- --------------- ----- ----- --------------- ----- ----- 2 1 1 1/2 1/2 0 1/2 3 1 1 3/4 1/4 0 1/4 1/3 0 2/3 0 0 2/3 : Coefficients for Explicit SSP-Runge-Kutta time stepping: [\[table:ssp\]]{#table:ssp label="table:ssp"} The fourth order RK-SSP scheme [@gottlieb2001strong] has the following structure: $$\begin{aligned} \bm{U}^{(1)} &= \textbf{U}^n + 0.39175222700392 \Delta t \big(\mathcal{M}(\bm{U}^n) \big) \\ % \bm{U}^{(2)} &= 0.44437049406734 \bm{U}^n + 0.55562950593266 \bm{U}^{(1)}+0.36841059262959 \Delta t \big( \mathcal{M}(\bm{U}^1) \big) \\ % \bm{U}^{(3)} &= 0.62010185138540 \bm{U}^n + 0.37989814861460 \bm{U}^{(2)} +0.25189177424738 \Delta t \big( \mathcal{M}(\bm{U}^2) \big) \\ % \bm{U}^{(4)} &= 0.17807995410773 \bm{U}^n + 0.82192004589227 \bm{U}^{(3)} + 0.54497475021237 \Delta t \big( \mathcal{M}(\bm{U}^3) \big) \\ % \bm{U}^{n+1} &= 0.00683325884039 \bm{U}^n + 0.51723167208978 \bm{U}^{(2)} + 0.12759831133288 \bm{U}^{(3)}\\ &+ 0.34833675773694 \bm{U}^{(4)}+ 0.08460416338212 \Delta t \big( \mathcal{M}(\bm{U}^3) \big)\\ &+ 0.22600748319395 \Delta t \big( \mathcal{M}(\bm{U}^4) \big). \end{aligned}$$ where $\mathcal{M}(\bm{U}^n)=\mathcal{L}(\bm{U}^n)-{\bm{B}^x}(\bm{U}^n)\left(\frac{\partial h}{\partial x}\right)^n - {\bm{B}^y(\bm{U}^n)}\left(\frac{\partial h}{\partial y}\right)^n + \bm{S}(\bm{U}^n)$. Here, we have ignored the subscripts $\{i,j\}$. # Numerical results {#sec:res} We test the fully discrete schemes on some 1-D and 2-D test cases and present the results for O1_ES, O2_ES, O3_ES, and O4_ES schemes. Here, 1. O1_ES: the Euler time-stepping with first-order entropy stable flux and second-order central difference approximation for the derivatives in the non-conservative terms. 2. O2_ES: the explicit second-order scheme with second-order entropy stable flux and second-order central difference approximation for the derivatives in the non-conservative terms. 3. O3_ES: the third-order explicit scheme with third-order entropy stable flux and fourth-order central difference approximation for the derivatives in the non-conservative terms. 4. O4_ES: the fourth-order explicit SSP RK scheme with fourth-order entropy stable flux and fourth-order central difference approximation for the derivatives in the non-conservative terms. We take the acceleration due to gravity as $g = 9.81 \ m/s^2$. To compute the time step, we use $$\Delta t =\text{CFL}\frac{1}{\max_{ij} \left( \frac{|\lambda^x(\bm{U}^n_{i,j})|}{\Delta x} + \frac{|\lambda^y(\bm{U}^n_{i,j})|}{\Delta y} \right)},$$ from [@Chandrashekar2020]. Here $\lambda^x$ and $\lambda^y$ are the maximum eigenvalues in $x$ and $y$ directions, respectively. We take CFL to be $0.45$. For the Riemann problem test, we consider the Neumann boundary conditions at both boundaries. In effect, we copy the value in the last cell to the ghost cells. The final time is chosen in all the Riemann problem test cases so the waves do not reach the boundary. We set the source term $\bm{S}$ to be zero for all the test cases except for the 1-D roll wave test in Section [6.1.6](#test11){reference-type="ref" reference="test11"} and the 2-D roll wave test in Section [6.2.2](#test13){reference-type="ref" reference="test13"}. ## One-dimensional test problems ### Accuracy test {#test:accuracy} We consider the shear shallow water model without source term $(\bm{S}=0)$ but instead, add an artificial source term $\mathcal{S}$ so that we can manufacture an exact solution. Following [@biswas2021entropy], we add the forcing term $\mathcal{S}(x,t)$ in the right-hand side of the SSW model as follows, $$\frac{\partial \bm{U}}{\partial t}+\frac{\partial\bm{F}^x}{\partial x} +\bm{B}^x\frac{\partial h}{\partial x}= \mathcal{S}(x,t),$$ where, $$\begin{aligned} \mathcal{S}(x,t)=\left(0,2\pi\cos(2\pi(x-t))(1+2 g+g\sin(2 \pi (x-t))), 0, \right.\\ \left. 2\pi\cos(2\pi(x-t))(1+2 g+g\sin(2 \pi (x-t))),0,0 \right)^\top. \end{aligned}$$ The computational domain is $[-0.5,\,0.5]$ with periodic boundary conditions. The exact solution is given by $$h(x,t)=2+\sin(2\pi (x-t)),\qquad v_1(x,t)=1, \qquad v_2(x,t)=0,$$ $$\mathcal{P}_{11}(x,t)=\mathcal{P}_{22}(x,t)=1, \qquad \mathcal{P}_{12}(x,t)=0.$$ The computations are performed up to the final time $T=0.5$. Number of cells O2_ES O3_ES O4_ES ----------------- ------------- ------- ------------- ------- ------------- ------- $L^1$ error Order $L^1$ error Order $L^1$ error Order 50 4.58e-03 -- 2.26e-04 -- 1.92e-05 -- 100 1.39e-03 1.72 2.92e-05 2.94 1.56e-06 3.62 200 4.67e-04 1.57 3.70e-06 2.98 1.14e-07 3.77 400 1.35e-04 1.79 4.63e-07 2.99 7.83e-09 3.86 800 3.67e-05 1.88 5.80e-08 2.99 5.32e-10 3.88 1600 9.71e-06 1.92 7.25e-09 2.99 4.17e-11 3.68 : Accuracy test: $L^1$ errors and order of accuracy for the water depth $h$. We present the $L^1$ errors and order of accuracy for the water depth $h$ in Table [2](#tab:acc2){reference-type="ref" reference="tab:acc2"} using the schemes O2_ES, O3_ES, and O4_ES. We observe that the schemes have reached the designed order of accuracy. ### Dam break problem {#test2} This is a Riemann problem from [@Nkonga2022], which models a dam break problem. The domain is taken to be $[-0.5,0.5]$ with Neumann boundary conditions. The initial discontinuity is placed at $x=0$, and the initial conditions are given by $$(h,\ v_1, v_2, \ \mathcal{P}_{11}, \mathcal{P}_{12}, \mathcal{P}_{22}) = \begin{cases} \big(0.02 ,\ 0,\ 0, \ 4.0\times10^{-2}, \ 0, \ 4.0\times10^{-2} \big) & \text{if } x < 0.0, \\ \big(0.01,\ 0, \ 0, \ 4.0\times10^{-2}, \ 0, \ 4.0\times10^{-2} \big) & \text{if } x > 0.0. \end{cases}$$ The computations are performed up to the final time $T=0.5.$ The numerical solutions for the schemes O1_ES, O2_ES, O3_ES, and O4_ES at 500 and 2000 cells are presented in Fig. [\[fig:test2a\]](#fig:test2a){reference-type="ref" reference="fig:test2a"}. We have plotted the water depth $h$, velocity $v_1$ and stress component $\mathcal{P}_{11}$. The numerical solution has been compared with the exact solution given in [@Nkonga2022]. We can observe the convergence of the schemes. The result in Fig. [\[fig:test2b\]](#fig:test2b){reference-type="ref" reference="fig:test2b"} shows the entropy decay of the proposed numerical scheme. {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} Next, we test another dam break problem [@Nkonga2022], where $\mathcal{P}_{12}$ is set to be $10^{-8}$, and the other initial conditions are kept the same. The numerical solutions are presented in Fig. [\[fig:test3a\]](#fig:test3a){reference-type="ref" reference="fig:test3a"} and Fig. [\[fig:test3b\]](#fig:test3b){reference-type="ref" reference="fig:test3b"} using 500 and 2000 cells. In this test problem, along with the water depth $h$, velocity $v_1$, stress component $\mathcal{P}_{11}$, we have also plotted the stress component $\mathcal{P}_{12}$. The $\mathcal{P}_{12}$ profile is able to capture all the five waves of the SSW model. The numerical solution has been compared with the exact solution from [@Nkonga2022], and we note that all the schemes converge towards the exact solution. The result in Fig. [\[fig:test3c\]](#fig:test3c){reference-type="ref" reference="fig:test3c"} shows the entropy decay for the different numerical schemes using 500 cells; all schemes show monotonic decay of total entropy, with higher-order schemes showing smaller decay. {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} ### Five wave dam break problem {#test6} This is a Riemann problem [@Chandrashekar2020; @Nkonga2022], which gives rise to all five waves in the solution. The computational domain is $[-0.5,0.5]$ with Neumann boundary conditions. The initial discontinuity is placed at $x=0$, and initial conditions are given by $$(h,\ v_1, v_2, \ \mathcal{P}_{11}, \mathcal{P}_{12}, \mathcal{P}_{22}) = \begin{cases} \big(0.01 ,\ 0.1,\ 0.2, \ 4.0\times10^{-2}, \ 10^{-8}, \ 4.0\times10^{-2} \big) & \text{if } x < 0.0, \\ \big(0.02,\ 0.1, \ -0.2, \ 4.0\times10^{-2}, \ 10^{-8}, \ 4.0\times10^{-2} \big) & \text{if } x > 0.0. \end{cases}$$ The numerical solutions are computed up to the final time $T=0.5$. {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} We have plotted all the primitive variables in Fig. [\[fig:test6a\]](#fig:test6a){reference-type="ref" reference="fig:test6a"} and Fig. [\[fig:test6b\]](#fig:test6b){reference-type="ref" reference="fig:test6b"} obtained using 200 and 2000 cells. The numerical solutions have been compared with the exact solution [@Nkonga2022]. We observe that the schemes O1_ES, O2_ES, O3_ES, and O4_ES converge toward the exact solution. The result in Fig. [\[fig:test6c\]](#fig:test6c){reference-type="ref" reference="fig:test6c"} shows the entropy decay for the different numerical schemes at 500 cells, which shows monotonic decay with time. {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} ### 1-D shear test problem {#test1} This is a Riemann problem from [@Gavrilyuk2018; @bhole2019fluctuation; @Chandrashekar2020; @Nkonga2022], which gives rise to two shear waves. The domain is $[-0.5,0.5]$ with Neumann boundary conditions. The initial discontinuity is placed at $x=0$, and the initial conditions are given by $$(h,\ v_1, v_2, \ \mathcal{P}_{11}, \mathcal{P}_{12}, \mathcal{P}_{22}) = \begin{cases} \big(0.01 ,\ 0,\ 0.2, \ 1.0\times10^{-4}, \ 0, \ 1.0\times10^{-4} \big) & \text{if } x < 0.0, \\ \big(0.01,\ 0, \ -0.2, \ 1.0\times10^{-4}, \ 0, \ 1.0\times10^{-4} \big) & \text{if } x > 0.0. \end{cases}$$ The computations are performed up to the final time $T=10.0.$ {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} The numerical solutions for the schemes O1_ES, O2_ES, O3_ES, and O4_ES using 200 and 2000 cells are presented in Fig. [\[fig:test1b\]](#fig:test1b){reference-type="ref" reference="fig:test1b"}. We have plotted the transverse velocity $v_2$, $\mathcal{P}_{12}$ and $\mathcal{P}_{22}$ component of the stress tensor. The numerical solution has been compared with the exact solution from [@Nkonga2022]. The exact solution of this Riemann problem consists of two shear waves. We observe that all the schemes are able to capture shear waves, and as expected, O4_ES, O3_ES, and O2_ES are more accurate than O1_ES. However, there are spurious spikes found at the center in $\mathcal{P}_{22}$, and this behavior is similar to what is observed with other numerical methods [@Gavrilyuk2018; @bhole2019fluctuation; @Chandrashekar2020; @Nkonga2022]. The result in Fig. [\[fig:test1c\]](#fig:test1c){reference-type="ref" reference="fig:test1c"} shows the entropy decay behavior of the numerical scheme, which confirms the entropy stability of the scheme. {width="\\textwidth"} ### Single shock wave problem {#test4} This Riemann problem from [@Nkonga2022] should have a single shock wave according to the exact solution derived there. The computational domain is $[-0.5,0.5]$ with the Neumann boundary conditions. The initial discontinuity is placed at $x=0$, and the initial conditions are given by $$(h,\ v_1, v_2, \ \mathcal{P}_{11}, \mathcal{P}_{12}, \mathcal{P}_{22}) = \begin{cases} \big(0.02 ,\ 0,\ 0, \ 1.0\times10^{-1}, \ 0, \ 1.0\times10^{-1} \big) , \\ \big(0.03,\ -7.010706099, \ 0, \ 16.616666666666658, \ 0, \ 1.0\times10^{-1} \big) \end{cases}$$ The numerical solutions are computed up to the final time $T=0.015811388$ with gravitational constant $g=9.81\times10^{3}$. The numerical solutions for the schemes O1_ES, O2_ES, O3_ES, and O4_ES are presented in Fig. [\[fig:test4a\]](#fig:test4a){reference-type="ref" reference="fig:test4a"} using 500 and 2000 cells. We have plotted the water depth $h$, velocity $v_1$, $\mathcal{P}_{11}$ components of the stress tensor and compare the numerical results with the exact solution provided in [@Nkonga2022]. The exact solution of this Riemann problem consists of a single shock wave but we have observed that the computed numerical solutions exhibit an extra contact wave that is not present in the exact solution, and this is seen even with mesh refinement. Similar results were observed for the HLL-type schemes in [@Nkonga2022], which is a consequence of the sensitivity of solutions of non-conservative systems to numerical dissipation. The result in Fig. [\[fig:test4b\]](#fig:test4b){reference-type="ref" reference="fig:test4b"} shows the entropy decay for the different numerical schemes using 500 cells. {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} ### 1-D roll wave problem {#test11} This problem models the flow of a thin layer of liquid flowing down an inclined bottom and results in the formation of hydraulic jump and roll waves. We use periodic boundary conditions and the initial conditions are taken from [@Gavrilyuk2018; @bhole2019fluctuation; @Chandrashekar2020] and given by $$h(x,0)=h_0[1+a\sin(2\pi x/L_x)],~~~~v_1(x,0)=\sqrt{g h_0 \tan{\theta}/C_f},~~~v_2(x,0)=0,$$ $$\mathcal{P}_{11}(x,0)=\mathcal{P}_{22}(x,0)=\frac{1}{2}\phi h^2(x,0),~~~\mathcal{P}_{12}(x,0)=0.$$ The bottom topography is given by $b=-x\tan{\theta}$ and we consider two sets of parameters as given in [@Ivanova2017]. In case $1$, the parameters are $\theta=0.05011 ,~C_f=0.0036,~h_0=7.98\times 10^{-3}$m, $a=0.05,~\phi=22.7s^{-2}, ~ C_r=0.00035,~L_x=1.3$m. In case $2$, the parameters are $\theta=0.11928 ,~C_f=0.0038,~h_0=5.33\times 10^{-3}$m, $a=0.05,~\phi=153.501s^{-2}, ~ C_r=0.002,~L_x=1.8$m. The computations are performed using 500 cells up to the final time $T=25$. The numerical results are presented in Fig. [\[fig:test11a\]](#fig:test11a){reference-type="ref" reference="fig:test11a"}. We have also plotted the water depth $h$ for both the cases with Brock's experimental data [@brock1969development; @brock1970periodic] in Fig. [\[fig:test11b\]](#fig:test11b){reference-type="ref" reference="fig:test11b"} and observe that the numerical results are comparable with measurements. The classical shallow water model captures the hydraulic jump but is unable to predict the roll wave profile, which is captured by the SSW model. {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} ## Two-dimensional test problems ### 2-D accuracy test This is a two-dimensional extension of the smooth problem [\[test:accuracy\]](#test:accuracy){reference-type="eqref" reference="test:accuracy"}, which was solved in 1-D. The test case is used to check the formal order and accuracy of the proposed scheme in two dimensions. The forcing term $\mathcal{S}(x,y,t)$ is given by, $$\mathcal{S}(x,y,t)=\left(0, 2\alpha, 2\alpha, \alpha, \alpha, \alpha \right)^\top,$$ where $\alpha=\pi\cos(2\pi(x+y-t))(1+2 g+g\sin(2 \pi (x+y-t)))$. The exact solution with domain $[-0.5,\,0.5]\times[-0.5,0.5]$ is as follows, $$\begin{aligned} h(x,y,t) &= 2+\sin(2\pi (x+y-t)), \qquad v_1(x,y,t)=0.5, \qquad v_2(x,y,t)=0.5,&\\ &\mathcal{P}_{11}(x,y,t)=\mathcal{P}_{22}(x,y,t)=1, \qquad \mathcal{P}_{12}(x,y,t)=0.\end{aligned}$$ Periodic boundary conditions are used for the computations, and the error is computed using the exact solution at time $T=0.5$s. We present the $L^1$ errors and order of accuracy for the water depth $h$ in Table [3](#tab:acc3){reference-type="ref" reference="tab:acc3"} using the schemes O2_ES, O3_ES, and O4_ES. We observe that the schemes have reached the designed order of accuracy. Number of cells O2_ES O3_ES O4_ES ----------------- ------------- ------- ------------- ------- ------------- ------- $L^1$ error Order $L^1$ error Order $L^1$ error Order 40 1.10e-02 -- 6.76e-04 -- 4.68e-05 -- 80 2.42e-03 2.19 9.05e-05 2.90 4.29e-06 3.45 160 8.14e-04 1.57 1.16e-05 2.96 3.31e-07 3.70 320 2.40e-04 1.78 1.46e-06 2.992 2.30e-08 3.85 640 6.63e-05 1.86 1.82e-07 2.998 1.54e-09 3.90 1280 1.78e-05 1.90 2.28e-08 2.999 1.01e-10 3.93 : Accuracy test: $L^1$ errors and order of accuracy for the water depth $h$. ### 2-D roll wave problem {#test13} This is a two-dimensional extension of the 1-D roll wave test from Section [6.1.6](#test11){reference-type="ref" reference="test11"}. The initial conditions are given by $$h(x,y,0)=h_0[1+a\sin(2\pi x/L_x)+a\sin(2\pi y/L_y)],$$ $$v_1(x,y,0)=\sqrt{g h_0 \tan{\theta}/C_f}, \qquad v_2(x,y,0)=0,$$ $$p_{11}(x,y,0)=\mathcal{P}_{22}(x,y,0)=\frac{1}{2}\phi h^2(x,0), \qquad \mathcal{P}_{12}(x,y,0)=0.$$ The computational domain is $[0,1.3]\times[0,0.5]$ with the periodic boundary conditions. This problem includes the source term with bottom topography given by $b=-x\tan{\theta}$. Here, $\theta=0.05011 ,~C_f=0.0036,~h_0=7.98\times 10^{-3}$m, $a=0.05,~\phi=22.7s^{-2}, ~ C_r=0.00035,~L_x=1.3$m, $L_y=0.5$m as given in [@Gavrilyuk2018; @bhole2019fluctuation; @Chandrashekar2020]. The computations are performed up to the final time $T=36$s, and the numerical results are presented in Figures [\[fig:test13b\]](#fig:test13b){reference-type="ref" reference="fig:test13b"}, [\[fig:test13c\]](#fig:test13c){reference-type="ref" reference="fig:test13c"}, [\[fig:test13d\]](#fig:test13d){reference-type="ref" reference="fig:test13d"}. The elevation of the water surface shown in Fig. [\[fig:test13b\]](#fig:test13b){reference-type="ref" reference="fig:test13b"} indicates the formation of hydraulic jump and roll waves, but the solutions do not look smooth. This type of solution has been observed in previous studies [@bhole2019fluctuation; @Chandrashekar2020] using different numerical schemes. Fig. [\[fig:test13c\]](#fig:test13c){reference-type="ref" reference="fig:test13c"} shows the projection of the $h$ profile onto the plane $y=0$, and its $y$-average is shown as a red line. While the profile varies in the $y$ direction and looks random/turbulent, the average profile shows the characteristic roll wave and hydraulic jump that is also seen in the 1-D simulations. The higher order schemes exhibit more fluctuations about the average and also give a better resolution of the roll wave than the first order scheme. Fig. [\[fig:test13d\]](#fig:test13d){reference-type="ref" reference="fig:test13d"} shows the contour lines of the $h$ field at time $T=36$ units which show carbuncle-like structures that are seen in some compressible flow problems [@Elling2009]. The first-order scheme shows a somewhat smooth solution similar to [@bhole2019fluctuation; @Chandrashekar2020] , while the higher-order schemes show more small-scale structures which have been observed in previous studies also [@Chandrashekar2020]. The solutions qualitatively look similar to those obtained using the five-wave HLLC solver, while the two-wave and three-wave HLL-type schemes show more smooth solutions [@Chandrashekar2020]. This indicates that the present schemes are able to more accurately model the five waves in the solution, like the sophisticated multi-wave approximate Riemann solvers. The similarity of solutions obtained for this problem from different numerical schemes suggests that they may not be purely numerical artifacts. {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} {width="\\textwidth"} # Summary and conclusions {#sec:sum} We have developed semi-discretely entropy stable schemes for the shear shallow water model which is a non-conservative hyperbolic system modeling shallow flows but including horizontal vorticity effects. The conservative part of the model is identical to the Ten-moment model of gas dynamics, and the non-conservative terms are due to gravity. For conservative systems, the existence of an entropy condition is related to the symmetrizability of the system, but this is not sufficient for non-conservative systems. In fact, the SSW model does not become symmetric when written in terms of entropy variables. However, we can exploit the symmetrizability of the conservative part to construct entropy conservative and entropy stable schemes since the non-conservative terms do not contribute to the entropy equation. We have constructed up to fourth-order finite difference schemes which satisfy the entropy inequality. The inequality is obtained due to the addition of carefully designed dissipative fluxes based on entropy scaled eigenvectors. The fully discrete schemes obtained with RK time stepping have been applied to several test problems like dam break and roll waves and shown to yield stable solutions that compare well with some exact solutions. The fully discrete schemes are observed to satisfy the entropy inequality in the numerical results. The roll wave solutions are able to match the experimental results of Brock. In multi-dimensions, the roll waves also generate turbulent like solutions and carbuncle like features that have been observed from other numerical techniques based on approximate Riemann solvers that include five waves in their model. Thus, the proposed schemes are expected to be similar to such accurate Riemann solver models in their wave resolution capabilities. The work of Praveen Chandrashekar is supported by the Department of Atomic Energy, Government of India, under project no. 12-R&D-TFR-5.01-0520. The work of Harish Kumar is supported in parts by DST-SERB, MATRICS grant with file No. MTR/2019/000380. # Conflict of interest {#conflict-of-interest .unnumbered} The authors declare that they have no conflict of interest. # Data Availability Declaration {#data-availability-declaration .unnumbered} Data will be made available on reasonable request. # A note on non-symmetrizability of shear shallow water model {#symmetrizability} In this section, we will discuss the symmetrizability of the following SSW model in one dimension, i.e., we consider, $$\label{eq:ssw1d} \frac{\partial \bm{U}}{\partial t}+\frac{\partial \bm{F}^x(\bm{U})}{\partial x}+\tilde\bm{B}^x(\bm{U})\frac{\partial \bm{U}}{\partial x}=0,$$ where $\bm{U}, \bm{F}^x$ and $\tilde\bm{B}^x$ are defined in Section [\[sec:entropy\]](#sec:entropy){reference-type="eqref" reference="sec:entropy"}. Additionally, this system has the entropy pair $(\eta,q)$ [\[entropy-pair\]](#entropy-pair){reference-type="eqref" reference="entropy-pair"}, such that in addition to [\[eq:ssw\]](#eq:ssw){reference-type="eqref" reference="eq:ssw"} the following equality holds, $$\frac{\partial \eta}{\partial t}+\frac{\partial q}{\partial x}=0.$$ for smooth solutions. For detailed proof, refer to Lemma [\[prop:entropy\]](#prop:entropy){reference-type="eqref" reference="prop:entropy"}. In the standard symmetrization theory [@godunov1961; @lax1973hyperbolic; @harten1983symmetric; @mock1980systems], one seeks a change of variable $\mathbf{U} \to \mathbf{\bm{V}}$ applied to [\[eq:ssw1d\]](#eq:ssw1d){reference-type="eqref" reference="eq:ssw1d"} so that when transformed $$\begin{aligned} %\frac{\partial \mathbf{U}}{\partial \evar}\frac{\partial \evar}{\partial t}+\frac{\partial \mathbf{F}}{\partial \mathbf{U}}\frac{\partial \mathbf{U}}{\partial \evar}\frac{\partial \evar}{\partial x}+\mathbf{B}\frac{\partial \mathbf{U}}{\partial \evar}\frac{\partial \evar}{\partial x}=0,&\\ \frac{\partial \mathbf{U}}{\partial \bm{V}}\frac{\partial \bm{V}}{\partial t}+\bigg(\frac{\partial \bm{F}^x}{\partial \mathbf{U}}+\tilde\bm{B}^x\bigg)\frac{\partial \mathbf{U}}{\partial \bm{V}}\frac{\partial \bm{V}}{\partial x}=0,\end{aligned}$$ the matrix $\frac{\partial \mathbf{U}}{\partial \bm{V}}$ is symmetric, positive definite and the matrix $\tilde A_1 = \bigg(\frac{\partial \bm{F}^x}{\partial \mathbf{U}}+\tilde\bm{B}^x\bigg)\frac{\partial \mathbf{U}}{\partial \bm{V}}$ is symmetric. For the SSW system [\[eq:ssw1d\]](#eq:ssw1d){reference-type="eqref" reference="eq:ssw1d"}, we calculate the matrix $\tilde A_1$ to check it's symmetry, which yields $$\begin{aligned} \tilde A_1 - \tilde A_1^\top = \begin{pmatrix} 0 & -\alpha & 0 & -\alpha v_1 & -\frac{1}{2} \alpha v_2 & 0 \\ \alpha & 0 & \alpha v_2 & -\beta_1 & \frac{1}{2} \alpha \mathcal{P}_{12} & \beta_2 \\ 0 & -\alpha v_2 & 0 & -\alpha v_1 v_2 & -\frac{1}{2} \alpha v_2^2 & 0 \\ \alpha v_1 & \beta_1 & \alpha v_1 v_2 & 0 & a & v_1 \beta_2 \\ \frac{1}{2}\alpha v_2 & -\frac{1}{2} \alpha \mathcal{P}_{12} & \frac{1}{2} \alpha v_2^2 & -a & 0 & \frac{1}{2}v_2 \beta_2\\ 0 & -\beta_2 & 0 & -\beta_2 & -\frac{1}{2} v_2 \beta_2 & 0 \end{pmatrix}.\end{aligned}$$ where $$\alpha=\frac{gh^2}{2}, \quad \beta_1=\frac{1}{4} g h^2 \left(v_1^2-\mathcal{P}_{11}\right),\quad \beta_2=\frac{1}{4} g h^2 \left(v_2^2+\mathcal{P}_{22}\right)$$ $$a=\frac{1}{8} g h^2 \left(2 \mathcal{P}_{12} v_1+\left(v_1^2-\mathcal{P}_{11}\right) v_2\right)$$ Hence, $\tilde A_1$ is not a symmetric matrix unless $g=0$, in which case the non-conservative terms vanish from the SSW model. Furthermore, we recall the following result presented in [@godlewski1996numerical] which gives the necessary and sufficient condition for a non-linear system of conservation laws to admit a strictly convex entropy. A necessary and sufficient condition for the conservative system, $$\begin{aligned} \label{eq:sys_conservative} \frac{\partial \bm{U}}{\partial t} + \frac{\partial \bm{F}^x}{\partial x}=0, \end{aligned}$$ to posses a strictly convex entropy $\eta$ is that there exists a change of dependent variables $\bm{U}=\bm{U}(\bm{V})$ that symmetrizes [\[eq:sys_conservative\]](#eq:sys_conservative){reference-type="eqref" reference="eq:sys_conservative"}. Analogously, we extend the above result for the case of non-conservative hyperbolic systems of the form [\[eq:ssw1d\]](#eq:ssw1d){reference-type="eqref" reference="eq:ssw1d"}. If $\eta$ is a strictly convex entropy for the non-conservative system of the form [\[eq:ssw1d\]](#eq:ssw1d){reference-type="eqref" reference="eq:ssw1d"} and $\eta{'}(\bm{U})\tilde\bm{B}^x(\bm{U})=0$, then the change of variable $\bm{U}\rightarrow\bm{V}$ with $\bm{V}^\top = \eta{'}(\bm{U})$ symmetrizes the non-conservative system if and only if $\tilde\bm{B}^x(\bm{U})\bm{U}{'}(\bm{V})$ is symmetric. *Proof.* Define the conjugate functions $$\begin{aligned} \eta^{*}(\bm{V})=\bm{V}^\top\bm{U}(\bm{V})-\eta(\bm{U}(\bm{V})), \qquad q^{*}(\bm{V})=\bm{V}^\top\bm{F}(\bm{U}(\bm{V}))-q(\bm{U}(\bm{V})). \end{aligned}$$ Differentiating with respect to $\bm{V}$ gives, $$\begin{aligned} {\eta^{*}}^{'}(\bm{V})=\bm{U}(\bm{V})^\top-\bm{V}^\top\bm{U}'(\bm{V})-\eta'(\bm{U}(\bm{V}))\bm{U}'(\bm{V})=\bm{U}(\bm{V})^\top, \end{aligned}$$ and $$\begin{aligned} {q^{*}}^{'}(\bm{V})&=\bm{F}(\bm{U}(\bm{V}))^\top+\bm{V}^\top\bm{F}'(\bm{U}(\bm{V}))\bm{U}'(\bm{V})-q'(\bm{U}(\bm{V}))\bm{U}'(\bm{V})&\\ &=\bm{F}(\bm{U}(\bm{V}))^\top+[\bm{V}^\top\bm{F}'(\bm{U}(\bm{V}))-q'(\bm{U}(\bm{V}))]\bm{U}'(\bm{V})&\\ &=\bm{F}(\bm{U}(\bm{V}))^\top \end{aligned}$$ since, $$\begin{aligned} \bm{V}^\top\bm{F}'(\bm{U}(\bm{V}))-q'(\bm{U}(\bm{V}))=0. \end{aligned}$$ Hence, the matrices $\bm{U}'(\bm{V})={\eta^{*}}^{''}(\bm{V})$ and $\bm{F}'(\bm{U}(\bm{V}))\bm{U}'(\bm{V})={q^{*}}^{''}(\bm{V})$ are symmetric. Moreover, the matrix $\bm{U}'(\bm{V})=\eta{''}(\bm{U}(\bm{V}))^{-1}$ is positive definite.\ The change of variable yields $$\begin{aligned} \bm{U}'(\bm{V})\bm{V}_t+[\bm{F}'(\bm{U}(\bm{V})) + \tilde\bm{B}^x(\bm{U}(\bm{V}))]\bm{U}'(\bm{V})\bm{V}_x=0 \end{aligned}$$ We need $[\bm{F}'(\bm{U}(\bm{V}))+\tilde\bm{B}^x(\bm{U}(\bm{V}))]\bm{U}'(\bm{V})$ to be symmetric, since, $\bm{F}'(\bm{U}(\bm{V}))\bm{U}'(\bm{V})$ is symmetric we need $\tilde\bm{B}^x(\bm{U}(\bm{V}))\bm{U}'(\bm{V})$ to be symmetric. ◻ The matrix $\tilde\bm{B}^x(\bm{U}(\bm{V}))\bm{U}'(\bm{V})$ for system [\[eq:ssw\]](#eq:ssw){reference-type="eqref" reference="eq:ssw"} is not symmetric, since, $$\begin{aligned} \tilde\bm{B}^x(\mathbf{U}(\textbf{V}))\mathbf{U}'(\textbf{V}) - \left[ \tilde\bm{B}^x(\mathbf{U}(\textbf{V}))\mathbf{U}'(\textbf{V}) \right]^\top = \begin{pmatrix} 0 & -\alpha & 0 & -\alpha v_1 & -\frac{1}{2} \alpha v_2 & 0 \\ \alpha & 0 & \alpha v_2 & -\beta_1 & \frac{1}{2} \alpha \mathcal{P}_{12} & \beta_2 \\ 0 & -\alpha v_2 & 0 & -\alpha v_1 v_2 & -\frac{1}{2} \alpha v_2^2 & 0 \\ \alpha v_1 & \beta_1 & \alpha v_1 v_2 & 0 & a & v_1 \beta_2 \\ \frac{1}{2}\alpha v_2 & -\frac{1}{2} \alpha \mathcal{P}_{12} & \frac{1}{2} \alpha v_2^2 & -a & 0 & \frac{1}{2}v_2 \beta_2\\ 0 & -\beta_2 & 0 & -\beta_2 & -\frac{1}{2} v_2 \beta_2 & 0 \end{pmatrix}, \end{aligned}$$ where $\alpha=\frac{gh^2}{2},~\beta_1=\frac{1}{4} g h^2 \left(v_1^2-\mathcal{P}_{11}\right),~\beta_2=\frac{1}{4} g h^2 \left(v_2^2+\mathcal{P}_{22}\right),$\ $a=\frac{1}{8} g h^2 \left(2 \mathcal{P}_{12} v_1+\left(v_1^2-\mathcal{P}_{11}\right) v_2\right)$. The above matrix is identical to $\tilde A_1 - \tilde A_1^\top$ which we derived explicitly and shown above. From the above discussion, we observe that the existence of entropy pair does not guarantee the symmetrizability of the non-conservative hyperbolic systems. In particular, we have seen that the shear shallow water model has the entropy pair $(\eta,q)$ but it is not symmetrizable. # Entropy scaled right eigenvectors for shear shallow water model {#scaledrev} In this section, we will calculate the entropy scaled right eigenvectors for the case of $x-$direction. Consider the conservative part of the SSW system [\[eq:ssw\]](#eq:ssw){reference-type="eqref" reference="eq:ssw"}, $$\frac{\partial \bm{U}}{\partial t} + \frac{\partial \bm{F}^x}{\partial x} =\frac{\partial \bm{U}}{\partial t} + A_1\frac{\partial \bm{U}}{\partial x} = 0, \label{eq:jacobi}$$ where $A_1$ is jacobian matrix of the flux function $\bm{F}^x$. To derive the eigenvalues and right eigenvectors, it is useful to transform the system [\[eq:jacobi\]](#eq:jacobi){reference-type="eqref" reference="eq:jacobi"} in terms of the primitive variables $\bm{W}$. The eigenvalues of the jacobian matrix $A_1$ [@biswas2021entropy; @sen_entropy_2018] are given by, $$\begin{aligned} v_1-\sqrt{3\mathcal{P}_{11}},\quad v_1-\sqrt{\mathcal{P}_{11}}, \quad v_1,\quad v_1,\quad v_1+\sqrt{\mathcal{P}_{11}}, \quad v_1+\sqrt{3\mathcal{P}_{11}}.\end{aligned}$$ We observe that if $\mathcal{P}_{11}>0$ then all eigenvalues are real. The right eigenvector matrix ${R}^x$ for the matrix $A_1$ is given by the relation $$\begin{aligned} {R}^x=\dfrac{\partial \bm{U}}{\partial \bm{W}}{R}_{\bm{W}}^x,\end{aligned}$$ where $\dfrac{\partial \bm{U}}{\partial \bm{W}}$ is the jacobian matrix for the change of variable, given by $$\begin{aligned} \dfrac{\partial \bm{U}}{\partial \bm{W}}=\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ v_1 & h & 0 & 0 & 0 & 0 \\ v_2 & 0 & h & 0 & 0 & 0 \\ \frac{1}{2}(\mathcal{P}_{11}+v_1^2) & h v_1 & 0 & \frac{h}{2} & 0 & 0 \\ \frac{1}{2}(\mathcal{P}_{12}+v_1 v_2) & \frac{h v_2}{2} & \frac{h v_1}{2} & 0 & \frac{h}{2} & 0 \\ \frac{1}{2}(\mathcal{P}_{22}+v_2^2) & 0 & h v_2 & 0 & 0 & \frac{h}{2} \end{pmatrix},\end{aligned}$$ and the matrix $R^x_{\bm{W}}$ is given by $$\begin{aligned} R_{\bm{W}}^x=\begin{pmatrix} h \mathcal{P}_{11} & 0 & -h & 0 & 0 & h \mathcal{P}_{11} \\ -\sqrt{3\mathcal{P}_{11}}\mathcal{P}_{11} & 0 & 0 & 0 & 0 & \sqrt{3\mathcal{P}_{11}}\mathcal{P}_{11} \\ -\sqrt{3\mathcal{P}_{11}}\mathcal{P}_{12} & -\sqrt{\mathcal{P}_{11}} & 0 & 0 & \sqrt{\mathcal{P}_{11}} & \sqrt{3\mathcal{P}_{11}}\mathcal{P}_{12} \\ 2\mathcal{P}_{11}^2 & 0 & \mathcal{P}_{11} & 0 & 0 & 2\mathcal{P}_{11}^2 \\ 2\mathcal{P}_{11}\mathcal{P}_{12} & \mathcal{P}_{11} & \mathcal{P}_{12} & 0 & \mathcal{P}_{11} & 2\mathcal{P}_{11}\mathcal{P}_{12} \\ 2\mathcal{P}_{12}^2 & 2\mathcal{P}_{12} & 0 & 1 & 2 \mathcal{P}_{12} & 2\mathcal{P}_{12}^2 \end{pmatrix}.\end{aligned}$$ We need to find a scaling matrix $T^x$ such that the scaled right eigenvector matrix $\tilde{R}^x=R^x T^x$ satisfies $$\begin{aligned} \label{eq1} \frac{\partial \bm{U}}{\partial \bm{V}} ={{\tilde{R}}^x} {{}{{\tilde{R}}^x}}^\top.\end{aligned}$$ where $\bm{V}$ is the entropy variable vector as in Eqn. [\[entvar\]](#entvar){reference-type="eqref" reference="entvar"}. We follow Barth scaling process [@barth1999numerical] to scale the right eigenvectors. The scaling matrix $T^x$ is the square root of $Y^x$ where $Y^x$ has the expression $$\begin{aligned} {Y}^x= \left(\tilde{R}^x_{\bm{W}} \right)^{-1} \frac{\partial \bm{W}}{\partial \bm{V}} \left( \frac{\partial \bm{U}}{\partial \bm{W}}\right)^{-\top} \left(\tilde{R}^x_{\bm{W}}\right)^{-\top},\end{aligned}$$ which results in $$\begin{aligned} Y^x=\begin{pmatrix} \frac{1}{12 h \mathcal{P}_{11}^2} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{\mathcal{P}_{11} \mathcal{P}_{22}-\mathcal{P}_{12}^2}{4 h \mathcal{P}_{11}^2} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{3h} & \frac{\mathcal{P}_{12}^2}{3 h \mathcal{P}_{11}} & 0 & 0 \\ 0 & 0 & \frac{\mathcal{P}_{12}^2}{3 h \mathcal{P}_{11}} & \frac{3(\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2)^2+\mathcal{P}_{12}^4}{3 h \mathcal{P}_{11}^2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{\mathcal{P}_{11} \mathcal{P}_{22}-\mathcal{P}_{12}^2}{4 h \mathcal{P}_{11}^2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{12 h \mathcal{P}_{11}^2} \end{pmatrix}.\end{aligned}$$ The matrix $Y^x$ is a block diagonal matrix which contains the blocks of order $1$ and $2$. It is straightforward to write the square root of a block matrix of order $1$. Consider the $2\times 2$ block sub-matrix of the matrix $Y^x$ and denote it by $Y^{x}_b$, $$\begin{aligned} Y^{x}_b=\begin{pmatrix} \frac{1}{3h} & \frac{\mathcal{P}_{12}^2}{3 h \mathcal{P}_{11}} \\ \frac{\mathcal{P}_{12}^2}{3 h \mathcal{P}_{11}} & \frac{3(\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2)^2+\mathcal{P}_{12}^4}{3 h \mathcal{P}_{11}^2} \end{pmatrix}.\end{aligned}$$ We need to find matrix $T^{x}_b=\sqrt{Y^{x}_b}$. To obtain formula for the matrix $T^x_b$ we first consider the characteristic polynomial of $T^{x}_b$, $$\begin{aligned} \label{characteristic1} {T^{x}_b}^{2}-\mathop{\mathrm{trace}}(T^{x}_b)T^{x}_b+\det(T^{x}_b)I=0,\end{aligned}$$ where $\det(T^{x}_b)=\pm\sqrt{det(Y^{x}_b)}=r_1$, say, with $r_1$ being the positive square root, given by $$\begin{aligned} r_1 & =\sqrt{\frac{3(\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2)^2+\mathcal{P}_{12}^4}{9 h^2 \mathcal{P}_{11}^2}-\frac{\mathcal{P}_{12}^4}{9 h^2 \mathcal{P}_{11}^2}} & \\ & =\sqrt{\frac{3(\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2)^2}{9 h^2 \mathcal{P}_{11}^2}} & \\ & =\frac{\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2}{\sqrt{3}h \mathcal{P}_{11}},\end{aligned}$$ and $I_{2\times2}$ is the identity matrix. Observe from Eqn. [\[characteristic1\]](#characteristic1){reference-type="eqref" reference="characteristic1"} that $$\begin{aligned} \label{charateristic2} \mathop{\mathrm{trace}}(T^{x}_b)T^{x}_b={T^{x}_b}^2+r_1I=Y^{x}_b+r_1I,\end{aligned}$$ and, $$\begin{aligned} (\mathop{\mathrm{trace}}(T^{x}_b))^2=\mathop{\mathrm{trace}}(\mathop{\mathrm{trace}}(T^{x}_b)T^{x}_b)=\mathop{\mathrm{trace}}(Y^{x}_b+r_1I)=\mathop{\mathrm{trace}}(Y^{x}_b)+2r_1.\end{aligned}$$ Simultaneously solving Eqns. [\[characteristic1\]](#characteristic1){reference-type="eqref" reference="characteristic1"}, [\[charateristic2\]](#charateristic2){reference-type="eqref" reference="charateristic2"} we obtain $$\begin{aligned} \label{block_T} T^{x}_b=\frac{1}{\sqrt{\mathop{\mathrm{trace}}(Y^{x}_b)+2r_1}}(Y^{x}_b+r_1I).\end{aligned}$$ Observe that $$\begin{aligned} {\mathop{\mathrm{trace}}(Y^{x}_b)+2r_1} & =\frac{1}{3h}+\frac{3(\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2)^2+\mathcal{P}_{12}^4}{3 h \mathcal{P}_{11}^2}+\frac{2(\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2)}{\sqrt{3}h \mathcal{P}_{11}} \\ & =\frac{\mathcal{P}_{11}^2+{3(\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2)^2+\mathcal{P}_{12}^4}+2\sqrt{3}\mathcal{P}_{11}(\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2)}{3 h \mathcal{P}_{11}^2} \\ & =\frac{3(\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2)^2+2\sqrt{3}\mathcal{P}_{11}(\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2)+\mathcal{P}_{11}^2+\mathcal{P}_{12}^4}{3 h \mathcal{P}_{11}^2} & \\ & =\frac{(\sqrt{3}(\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2)+\mathcal{P}_{11})^2+\mathcal{P}_{12}^4}{3 h \mathcal{P}_{11}^2}.\end{aligned}$$ Since $h>0$, we have $\mathop{\mathrm{trace}}(Y^{x}_b)+2r_1>0$. We use notation $\alpha_1 = \sqrt{\mathop{\mathrm{trace}}(Y^{x}_b)+2r_1}$. A long simplification using the block matrix $T^x_b$ as in Eqn. [\[block_T\]](#block_T){reference-type="eqref" reference="block_T"} results in $$\begin{aligned} T^x=\begin{pmatrix} \sqrt{\frac{1}{12 h \mathcal{P}_{11}^2}} & 0 & 0 & 0 & 0 & 0 \\ 0 & \sqrt{\frac{\mathcal{P}_{11} \mathcal{P}_{22}-\mathcal{P}_{12}^2}{4 h \mathcal{P}_{11}^2}} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{\frac{1}{3h}+r_1}{\alpha_1} & \frac{\mathcal{P}_{12}^2}{3 h \mathcal{P}_{11}\alpha_1} & 0 & 0 \\ 0 & 0 & \frac{\mathcal{P}_{12}^2}{3 h \mathcal{P}_{11}\alpha_1} & \frac{\beta_1(\beta_1+\mathcal{P}_{11})+\mathcal{P}_{12}^4}{3 h \mathcal{P}_{11}^2\alpha_1} & 0 & 0 \\ 0 & 0 & 0 & 0 & \sqrt{\frac{\mathcal{P}_{11} \mathcal{P}_{22}-\mathcal{P}_{12}^2}{4 h \mathcal{P}_{11}^2}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{1}{12 h \mathcal{P}_{11}^2}} \end{pmatrix}.\end{aligned}$$ We proceed similarly in the $y-$direction, the eigenvalues for the jacobian matrix $A_2=\frac{\partial \bm{F}_2}{\partial \bm{U}}$ are given by $$\begin{aligned} v_2-\sqrt{3\mathcal{P}_{22}},\quad v_2-\sqrt{\mathcal{P}_{22}},\quad v_2,\quad v_2,\quad v_2+\sqrt{\mathcal{P}_{22}},\quad v_2+\sqrt{3\mathcal{P}_{22}}. \end{aligned}$$ and right eigenvector matrix is given by the relation $$\begin{aligned} {R}^y=\dfrac{\partial \bm{U}}{\partial \bm{W}}{R}_{\bm{W}}^y, \end{aligned}$$ where $$\begin{aligned} R_{\bm{W}}^y=\begin{pmatrix} h \mathcal{P}_{22} & 0 & -h & 0 & 0 & h \mathcal{P}_{22} \\ -\sqrt{3\mathcal{P}_{22}}\mathcal{P}_{12} & -\sqrt{\mathcal{P}_{22}} & 0 & 0 & \sqrt{\mathcal{P}_{22}} & \sqrt{3\mathcal{P}_{22}}\mathcal{P}_{12} \\ -\sqrt{3\mathcal{P}_{22}}\mathcal{P}_{22} & 0 & 0 & 0 & 0 & \sqrt{3\mathcal{P}_{22}}\mathcal{P}_{22} \\ 2\mathcal{P}_{12}^2 & 2 \mathcal{P}_{12} & 0 & 1 & 2\mathcal{P}_{12} & 2\mathcal{P}_{12}^2 \\ 2\mathcal{P}_{22}\mathcal{P}_{12} & \mathcal{P}_{22} & \mathcal{P}_{12} & 0 & \mathcal{P}_{22} & 2\mathcal{P}_{22}\mathcal{P}_{12} \\ 2\mathcal{P}_{22}^2 & 0 & \mathcal{P}_{22} & 0 & 0 & 2\mathcal{P}_{22}^2 \end{pmatrix}. \end{aligned}$$ Accordingly, we obtain the scaling matrix $T^y$ as, $$\begin{aligned} T^y=\begin{pmatrix} \sqrt{\frac{1}{12 h \mathcal{P}_{22}^2}} & 0 & 0 & 0 & 0 & 0 \\ 0 & \sqrt{\frac{\mathcal{P}_{11} \mathcal{P}_{22}-\mathcal{P}_{12}^2}{4 h \mathcal{P}_{22}^2}} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{\frac{1}{3h}+r_2}{\alpha_2} & \frac{\mathcal{P}_{12}^2}{3 h \mathcal{P}_{22}\alpha_2} & 0 & 0 \\ 0 & 0 & \frac{\mathcal{P}_{12}^2}{3 h \mathcal{P}_{22}\alpha_2} & \frac{\beta_1(\beta_1+\mathcal{P}_{22})+\mathcal{P}_{12}^4}{3 h \mathcal{P}_{22}^2\alpha_2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \sqrt{\frac{\mathcal{P}_{11} \mathcal{P}_{22}-\mathcal{P}_{12}^2}{4 h \mathcal{P}_{22}^2}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{1}{12 h \mathcal{P}_{22}^2}} \end{pmatrix}, \end{aligned}$$ where $r_2=\frac{\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2}{\sqrt{3}h \mathcal{P}_{22}}$ and $\alpha_2=\sqrt{\frac{(\sqrt{3}(\mathcal{P}_{11}\mathcal{P}_{22}-\mathcal{P}_{12}^2)+\mathcal{P}_{22})^2+\mathcal{P}_{12}^4}{3 h \mathcal{P}_{22}^2}}$. [^1]: Corresponding Author

arxiv_math

--- abstract: | We prove a number of results on the number of solutions to the asymptotic Plateau problem in $\mathbb H^3$. In the direction of non-uniqueness, we construct an example of a quasicircle that is the asymptotic boundary of uncountably many pairwise distinct stable minimal disks. Moreover, we discuss criteria that ensure uniqueness. Given a Jordan curve $\Lambda$ in the asymptotic boundary of $\mathbb H^3$, we show that uniqueness of the minimal surfaces with asymptotic boundary $\Lambda$ is equivalent to uniqueness in the smaller class of stable minimal disks, and, when $\Lambda$ is invariant by a Kleinian group, to uniqueness in the even smaller class of group invariant stable minimal disks. Finally, we show that if a quasicircle (or more generally, a Jordan curve of finite width) $\Lambda$ is the asymptotic boundary of a minimal surface $\Sigma$ with principal curvatures less than or equal to 1 in absolute value, then uniqueness holds. address: - Department of Mathematics, The City University of New York, Staten Island, NY 10314, USA - The Graduate Center, The City University of New York, 365 Fifth Ave., New York, NY 10016, USA - Department of Mathematics, University of Chicago, Chicago, IL 60637, USA - Univ. Grenoble Alpes, CNRS, IF, 38000 Grenoble, France author: - Zheng Huang - Ben Lowe - Andrea Seppi bibliography: - ref-afd.bib title: Uniqueness and non-uniqueness for the asymptotic Plateau problem in hyperbolic space --- 1ex # Introduction The classical "asymptotic Plateau problem\" asks, given a Jordan curve $\Lambda$ on ${\mathbb{S}}^2_{\infty} = \partial_\infty \mathbb{H}^3$, how to count the number of (properly embedded) minimal surfaces $\Sigma$ in $\mathbb{H}^3$, if any, that are asymptotic to $\Lambda$, in the sense that the closure of $\Sigma$ in ${\mathbb{S}}^2_{\infty}\cup \mathbb{H}^3$ is equal to $\Lambda \cup \Sigma$. The existence of minimal disk solutions to the asymptotic Plateau problem was obtained by Anderson ([@And83]). Using geometric measure theory, Anderson also obtained existence results in higher dimensions. ; the solution, however, lies in the class of locally integral $n$-currents, and may fail to be smoothly embedded hypersurfaces on a singular set of dimension $n-7$. The uniqueness does not hold in general: as shown in [@And83; @HW15], taking advantage of group actions, one can construct a Jordan curve $\Lambda$ in ${\mathbb{S}}^2_{\infty} = \partial_\infty \mathbb{H}^3$ which is the limit set of some quasi-Fuchsian group such that $\Lambda$ spans multiple minimal disks (even an arbitrarily large, but finite, number). Anderson ([@And83]) even constructed a curve $\Lambda$ which spans infinitely many minimal surfaces (the surfaces he constructs have positive genus). On the other hand, when $\Lambda$ is a round circle, the unique minimal surface it spans is a totally geodesic disk. To look for unique solutions, it is therefore natural to consider the class of minimal surfaces that are "close\" to totally geodesic, for which $\Lambda$ is "close\" to a round circle. Related questions with conditions on natural invariants of $\Lambda$ were studied in [@Sep16] (for the quasi-conformal constant of $\Lambda$), and [@HW13; @San18] (for the Hausdorff dimension of $\Lambda$). Some properness questions for the asymptotic Plateau problem solutions for various classes of curves were addressed, for example, in [@GS00; @AM10]. For an overview of this active area of research see the survey paper [@co14]. Motivated by the above results and by many natural questions arising from the study of the asymptotic Plateau problem, in this paper we address two basic questions, which, for the sake of simplicity, we state only in dimension three, but may naturally be extended to hypersurfaces in $\mathbb{H}^{n+1}$: - under what conditions does a Jordan curve $\Lambda$ on ${\mathbb{S}}^2_{\infty}$ span exactly one minimal surface in $\mathbb{H}^3$? - does there exist a Jordan curve $\Lambda$ on ${\mathbb{S}}^2_{\infty}$ that spans infinitely many minimal disks in $\mathbb{H}^3$, and if yes, which cardinality may the set of solutions have? ## Characterizing uniqueness Recall that a (hyper)surface is *minimal* if it is a critical point of the area under compactly supported variations. It is *stable* if moreover the second variation of the area under any compactly supported variation is non-negative. In this introduction, we will always implicitly consider *properly embedded* (hyper)surfaces. Our first theorem shows that it suffices to check uniqueness in the class of *stable minimal disks*. **Theorem 1**. *Let $\Lambda$ be a Jordan curve on ${\mathbb{S}}^2_{\infty} = \partial_{\infty} \mathbb{H}^3$. Then $\Lambda$ spans a unique minimal surface if and only if it spans a unique stable minimal disk.* A statement similar to Theorem [Theorem 1](#thm:char uniqueness){reference-type="ref" reference="thm:char uniqueness"}, but in the context of the *finite* Plateau problem, was proved in [@MY19]. When $\Lambda$ is invariant under the action of a Kleinian group (i.e. a discrete subgroup of isometries of $\mathbb{H}^3$), which is for instance the case for the limit set of a quasi-Fuchsian group, we can prove a stronger statement. **Theorem 2**. *Let $\Lambda$ be a Jordan curve on ${\mathbb{S}}^2_{\infty} = \partial_{\infty} \mathbb{H}^3$, and let $\Gamma$ be any Kleinian group preserving $\Lambda$. Then $\Lambda$ spans a unique minimal surface if and only if it spans a unique $\Gamma$-invariant stable minimal disk.* The main idea in the proof of Theorem [Theorem 1](#thm:char uniqueness){reference-type="ref" reference="thm:char uniqueness"} is an adaptation of an argument in [@And83 Theorem 3.1]: we show that if there is a minimal surface, which is not stable or is not topologically a disk, with asymptotic boundary $\Lambda$, then we can construct two distinct --- actually, disjoint --- stable minimal disks with the same asymptotic boundary $\Lambda$ (Theorem [Theorem 15](#main3){reference-type="ref" reference="main3"}). The proof of Theorem [Theorem 2](#thm:char uniqueness invariant){reference-type="ref" reference="thm:char uniqueness invariant"} then relies on a further improvement of the arguments of [@And83 Theorem 3.1], showing that if there is a non-invariant stable minimal disk, then we can construct two disjoint invariant stable minimal disks (Theorem [Theorem 16](#main3groupaction){reference-type="ref" reference="main3groupaction"}). ## Uniqueness criteria via curvature conditions Next, we turn our attention to sufficient conditions for uniqueness. For an immersed hypersurface in $\mathbb{H}^{n+1}$, or more generally in a hyperbolic ($n+1$)-manifold, we say it has *strongly small curvature* if its principal curvatures $\{\lambda_i\}$ satisfy that $$\label{cc} |\lambda_i| \le 1-\epsilon, \ i=1,\ldots,n, \ \ \ \text{for some small} \ \ \epsilon>0.$$ Similarly we say it has *small curvature* if $|\lambda_i| < 1$, and it has *weakly small curvature* if $|\lambda_i| \le 1$. This definition has some immediate consequences: for instance, a complete immersion of weakly small curvature is in fact a properly embedded topological disk ([@Eps84; @Eps86], see also [@ES22] for a generalization). See Section [4](#sec:small curvatures){reference-type="ref" reference="sec:small curvatures"} and Appendix [7](#appendix){reference-type="ref" reference="appendix"} for more details. Surfaces of small curvature are very special in three-manifold theory: Thurston observed that a closed surface of small curvature in a complete hyperbolic three-manifold is incompressible ([@Thu86; @Lei06]); they are abundant in closed hyperbolic three-manifolds ([@KM12]); many results have been extended to the study of complete noncompact hyperbolic three-manifold of finite volume ([@Rub05; @CF19; @kw21]). It is often favorable to consider canonical representatives within a homotopy class of surfaces, and minimal surfaces are in many ways the most natural choice. a more "representative\" type of surfaces in the homotopy class of surfaces , and often these special surfaces are minimal surfaces.Among hyperbolic three-manifolds, *almost-Fuchsian* manifolds are quasi-Fuchsian manifolds which admit a closed minimal surface of small curvature. This notion was introduced by Uhlenbeck and it played an important role in her study of parametrization of the moduli space of minimal surfaces in hyperbolic three-manifolds ([@Uhl83]). Subsequently many different aspects of this subclass of quasi-Fuchsian manifold have been studied up to recent years (for instance [@KS07; @GHW10; @HL21] and many others). It is known ([@Uhl83]) that any almost-Fuchsian manifold admits a unique closed minimal surface --- in other words, identifying the almost-Fuchsian manifold with a quotient $\mathbb{H}^3/\Gamma$, the limit set $\Lambda$ of the group $\Gamma$ bounds a unique $\Gamma$-invariant minimal disk asymptotic to $\Lambda$. Inspired by this fact, we prove (Corollary [Corollary 4](#cor strongly small){reference-type="ref" reference="cor strongly small"} below) that if a Jordan curve $\Lambda$ (not necessarily group equivariant) spans a minimal disk $\Sigma$ of [strongly]{.ul} small curvature in $\mathbb{H}^3$, then $\Sigma$ is the unique minimal surface asymptotic to $\Lambda$. Our results, however, are more general. The main result we prove in this direction is the following: **Theorem 3**. *Let $\Lambda$ be a topologically embedded $n$-sphere on ${\mathbb{S}}^n_{\infty} = \partial_{\infty} \mathbb{H}^{n+1}$ of finite width, and let $\Sigma$ be a minimal hypersurface in $\mathbb{H}^{n+1}$ of [weakly]{.ul} small curvature asymptotic to $\Lambda$. Then $\Sigma$ is the unique minimal hypersurface in $\mathbb{H}^{n+1}$ asymptotic to $\Lambda$. Moreover, $\Sigma$ is area-minimizing.* Let us explain the terminology of the statement. First, recall that a hypersurface $\Sigma$ is *area-minimizing* if any compact codimension-zero submanifold with boundary has smaller area than any rectifiable hypersurface with the same boundary in the ambient space. This implies that $\Sigma$ is a stable minimal hypersurface. Second, the *width* of a Jordan curve $\Lambda$ in ${\mathbb{S}}^2_{\infty} = \partial_{\infty} \mathbb{H}^3$ has been introduced in [@BDMS21] --- and the definition is immediately extended to higher dimensions --- as the supremum over all points in the convex hull of $\Lambda$ of the sum of the distances from each boundary component of the convex hull. We include two proofs of Theorem [Theorem 3](#main1){reference-type="ref" reference="main1"} as they offer different perspectives. The first proof only works for $n=2$, but it is based on elementary geometric arguments, which we sketch here. The first observation is that the finite width condition implies that every minimal surface $\Sigma'$ has, roughly speaking, finite normal distance from $\Sigma$. Moreover, by Theorem [Theorem 1](#thm:char uniqueness){reference-type="ref" reference="thm:char uniqueness"}, it suffices to show uniqueness among *stable* minimal disks. Now, if the maximum normal distance between $\Sigma$ and another stable minimal surface is realized in the interior, the conclusion follows from a maximum principle argument, taking advantage the properties of the normal flow from $\Sigma$. If the maximum normal distance is not realized in the interior, we use isometries to send the run-away sequence of points back to a fixed point and use the compactness theorems for stable minimal surfaces, so as to reduce essentially to the previous case. The second proof, which works in any dimension, rests on an application of a more general maximum principle proved in [@Whi10] in the context of minimal varifolds (see also [@jt03]). We now derive several corollaries of Theorem [Theorem 3](#main1){reference-type="ref" reference="main1"}. Firstly, observe that if a properly embedded hypersurface $\Sigma$ has [strongly]{.ul} small curvatures, then its asymptotic boundary has finite width (See Lemma [Lemma 34](#lemma finite width){reference-type="ref" reference="lemma finite width"} in Appendix [7](#appendix){reference-type="ref" reference="appendix"}). Hence we obtain: **Corollary 4**. *Let $\Lambda$ be a topologically embedded $n$-sphere on ${\mathbb{S}}^n_{\infty} = \partial_{\infty} \mathbb{H}^{n+1}$, and let $\Sigma$ be a minimal hypersurface in $\mathbb{H}^{n+1}$ of [strongly]{.ul} small curvature asymptotic to $\Lambda$. Then $\Sigma$ is the unique minimal hypersurface in $\mathbb{H}^{n+1}$ asymptotic to $\Lambda$. Moreover, $\Sigma$ is area-minimizing.* Secondly, in dimension $n=2$, quasicircles are an important class of Jordan curves, which are known to have finite width. We thus obtain immediately the following. **Corollary 5**. *Let $\Lambda$ be a quasicircle on ${\mathbb{S}}^2_{\infty} = \partial_{\infty} \mathbb{H}^3$, and let $\Sigma$ be a minimal surface in $\mathbb{H}^3$ of [weakly]{.ul} small curvature asymptotic to $\Lambda$. Then $\Sigma$ is the unique minimal surface in $\mathbb{H}^3$ asymptotic to $\Lambda$. Moreover, $\Sigma$ is area-minimizing.* We remark that the setting of Theorems [Theorem 1](#thm:char uniqueness){reference-type="ref" reference="thm:char uniqueness"} and [Theorem 2](#thm:char uniqueness invariant){reference-type="ref" reference="thm:char uniqueness invariant"} is more general than Theorem [Theorem 3](#main1){reference-type="ref" reference="main1"} and Corollary [Corollary 5](#cor weakly small quasicircles){reference-type="ref" reference="cor weakly small quasicircles"}. Indeed, it follows from [@HL21 Theorem 5.2] that there are examples of quasi-Fuchsian groups $\Gamma$ whose limit set $\Lambda$ bounds a unique $\Gamma$-invariant stable minimal disk $\Sigma$ (hence, by Theorem [Theorem 2](#thm:char uniqueness invariant){reference-type="ref" reference="thm:char uniqueness invariant"}, a unique minimal surface), but $\Sigma$ does *not* have weakly small curvature. ## Strong non-uniqueness Now we turn to the other extreme case: we aim to construct a Jordan curve on $\mathbb{S}^2_\infty$ which spans *a lot* of minimal disks. To give some context, Hass-Thurston conjectured that no closed hyperbolic 3-manifold admits a foliation by minimal surfaces. Anderson even conjectured ([@And83]) that no hyperbolic 3-manifold admits a local 1-parameter family of closed minimal surfaces, and proved this statement for quasi-Fuchsian hyperbolic 3-manifolds. It has been a folklore conjecture that no Jordan curve in ${\mathbb{S}}^2_{\infty}= \partial_\infty \mathbb{H}^3$ asymptotically bounds a 1-parameter family of minimal surfaces. The full extent of these conjectures remain as major open questions in the field. What has been proven up to this point tends to support these conjectures. Huang-Wang [@HW19] and Hass [@has15] made progress on the Hass-Thurston conjecture for certain fibered closed hyperbolic 3-manifolds containing short geodesics; Wolf-Wu ([@ww20]) ruled out so-called geometric local 1-parameter families of closed minimal surfaces; it follows from the work of Alexakis-Mazzeo [@AM10] that a generic $C^{3,\alpha}$ simple closed curve in the boundary at infinity of $\mathbb{H}^3$ bounds only finitely many surfaces of any given finite genus; Coskunuzer proved that generic simple closed curves in $\partial_\infty \mathbb{H}^3$ bound unique area-minimizing surfaces [@c11genericuniqueness]. On the other hand what we prove, while compatible with the folklore conjecture, is in the other direction. Based on the aforementioned results, one might be tempted to strengthen the folklore conjecture to the statement that any Jordan curve in ${\mathbb{S}}^2_{\infty}= \partial_\infty \mathbb{H}^3$ bounds at most countably many minimal surfaces. We show that this stronger statement is false: **Theorem 6**. *There exists a quasicircle in ${\mathbb{S}}^2_{\infty} = \partial_{\infty} \mathbb{H}^3$ spanning uncountably many pairwise distinct stable minimal disks.* Let us emphasize some important features of the construction of this extreme curve $\Lambda$. In ([@And83]), Anderson constructed a Jordan curve which is the limit set of a quasi-Fuchsian group (hence continuous but almost nowhere differentiable) such that it spans infinitely many minimal surfaces, one of which is a minimal disk. In [@HW15], for each integer $N > 1$, also using the limit set of a quasi-Fuchsian group, an extreme curve spanning at least $2^N$ distinct minimal disks was constructed. However, Anderson ([@And83]) has shown that any quasi-Fuchsian manifold only admits finitely many least area closed minimal surfaces diffeomorphic to the fiber, which poses a possible limitation on how much one can improve the aforementioned constructions to find infinitely many minimal disks if one insists on using the limit set of some quasi-Fuchsian group as the curve at infinity. The starting point of our construction is similar to the ideas in [@HW15], but the Jordan curve is constructed in such a way to allow an improvement of the argument, leading to $2^{\mathbb N}$ pairwise distinct minimal disk. Moreover, since this Jordan curve is not invariant under any quasi-Fuchsian group, we must adopt a different approach in order to produce the minimal disks, namely, in a spirit similar to the proofs of Theorems [Theorem 1](#thm:char uniqueness){reference-type="ref" reference="thm:char uniqueness"} and [Theorem 2](#thm:char uniqueness invariant){reference-type="ref" reference="thm:char uniqueness invariant"}, we take the limit of a sequence of solutions of the finite Plateau problems inside $\mathbb{H}^3$. ## Quasiconformal constant We conclude this introduction with an improvement of the curvature estimates obtained in [@Sep16] in terms of quasiconformal constants, by a direct application of Corollary [Corollary 5](#cor weakly small quasicircles){reference-type="ref" reference="cor weakly small quasicircles"}. More concretely, [@Sep16 Theorem A] showed that there exist universal constants $C>0$ and $K_0>1$ such that any stable minimal disk in $\mathbb{H}^3$ with asymptotic boundary a $K$-quasicircle, for $K<K_0$, has principal curvatures bounded in absolute value by $C\log K$. This result has been recently applied in several directions, see [@Bis19; @Low21; @CMN22]. The proof, however, relies on the application of compactness for minimal surfaces, and therefore requires stability. However, when $K$ is sufficiently small, the principal curvatures of the area-minimizing (hence stable) disk whose existence is guaranteed by [@And83] are less than $1-\epsilon$ in absolute value, and therefore, as a consequence of Corollary [Corollary 5](#cor weakly small quasicircles){reference-type="ref" reference="cor weakly small quasicircles"}, $\Sigma$ is the unique minimal surface. Up to taking a smaller constant $K_0$, we can therefore remove the stability assumption: **Corollary 7**. *There exist universal constants $C>0$ and $K_0>1$ such that the principal curvatures $\lambda_i$ of any minimal surface $\Sigma$ in $\mathbb{H}^3$ with asymptotic boundary a $K$-quasicircle with $K\leq K_0$ satisfy $$|\lambda_i|\leq C\log K,\; i=1,2.$$* In particular, we also improve [@Sep16 Theorem B] (up to choosing a smaller constant) by removing the stability assumption. **Corollary 8**. *There exists a universal constant $K'_0>1$ such that any $K$-quasicircle with $K\leq K_0'$ is the asymptotic boundary of a unique minimal surface, which is an area-minimizing disk of strongly small curvature.* ## Organization of the paper In the preliminary section §[2](#prelim){reference-type="ref" reference="prelim"}, we collect and prove some facts in preparation to work towards proofs of our main results introduced above. In §[3](#sec3){reference-type="ref" reference="sec3"} we prove Theorems [Theorem 1](#thm:char uniqueness){reference-type="ref" reference="thm:char uniqueness"} and [Theorem 2](#thm:char uniqueness invariant){reference-type="ref" reference="thm:char uniqueness invariant"}. In §[4](#sec:small curvatures){reference-type="ref" reference="sec:small curvatures"} we work in the case of small curvature conditions and prove Theorem [Theorem 3](#main1){reference-type="ref" reference="main1"} and, using similar methods, we provide an alternative argument for Theorem [Theorem 2](#thm:char uniqueness invariant){reference-type="ref" reference="thm:char uniqueness invariant"} in a special case; in §[5](#sec:higher){reference-type="ref" reference="sec:higher"} we address some generalizations of our results in higher dimensions and codimensions; in §[6](#sec:uncountably){reference-type="ref" reference="sec:uncountably"} we detail a construction of an extreme Jordan curve which spans uncountably many minimal disks in $\mathbb{H}^3$ and hence prove Theorem [Theorem 6](#thm:uncountably){reference-type="ref" reference="thm:uncountably"}. In Appendix [7](#appendix){reference-type="ref" reference="appendix"} we provide the details to extend some well-known arguments for hypersurfaces of small curvature to the setting of weakly small curvature that is of interest here. ## Acknowledgements The first-named author wishes to thank Bill Meeks for his insightful suggestions and Biao Wang for his generous help. Part of this work was done during a visit of the first-named author at the Institut Fourier (Université Grenoble Alpes) in the framework of the "Visiting Scientist Campaign 2023", he wishes to thank the institute for excellent working environment. The second-named author was supported by NSF grant DMS-2202830. The third-named author was partially supported by the ANR JCJC grant GAPR (ANR-22-CE40-0001, Geometry and Analysis in the Pseudo-Riemannian setting). # Preliminaries {#prelim} In this article, we denote by $\mathbb{H}^{n+1}$ the hyperbolic space of dimension $n+1$, and by $\mathbb{S}^n_\infty=\partial_\infty\mathbb{H}^{n+1}$ its visual boundary. ## Hypersurfaces theory Given an immersed hypersurface $\Sigma$ in $\mathbb{H}^{n+1}$, we recall that its first fundamental form is the restriction to $T\Sigma$ of the hyperbolic metric $h$ of $\mathbb{H}^{n+1}$. The second fundamental form is defined as $$A_\Sigma(v,w)=h(\nabla^{h}_{v}W,N_\Sigma)~,$$ where $\nabla^{h}$ is the Levi-Civita connection of $h$, $W$ is a local smooth extension of $w$, and $N_\Sigma:\Sigma\to T\mathbb{H}^{n+1}$ is a continuous choice of a unit normal vector to the immersion. (Since $N_\Sigma$ is uniquely determined only up to a sign, so too is $A_\Sigma$.) The second fundamental form satisfies the identity $$A_\Sigma(v,w)=h(B_\Sigma(v),w)$$ where $B_\Sigma$ is the shape operator, namely the endomorphism of the tangent bundle of $\Sigma$ given by $B_\Sigma(v)=-\nabla^h_vN_\Sigma$. The principal curvatures of $\Sigma$ are the eigenvalues of $B_\Sigma$, denoted by $\lambda_1,\ldots,\lambda_n$. The mean curvature of the immersed hypersurface $\Sigma$ is $$H_\Sigma=\mathrm{tr}(B_\Sigma)=\lambda_1+\ldots+\lambda_n~,$$ and $\Sigma$ is minimal if and only if its mean curvature vanishes identically. Although $H_\Sigma$ depends (but only up to a sign) on the choice of the normal vector $N_\Sigma$, the condition of being minimal does not. Also, the mean curvature vector, which is defined as $H_\Sigma\cdot N_\Sigma$, does not depend on such a choice. Finally, the norm of the second fundamental form is $$\|A_\Sigma\|=\sqrt{\mathrm{tr}(B_\Sigma B_\Sigma^T)}=\sqrt{\lambda_1^2+\ldots+\lambda_n^2}~.$$ ## Convex hull and width of a Jordan curve Let $\Lambda$ be a topologically embedded $n$-sphere in $\mathbb{S}^n_\infty =\partial\mathbb{H}^{n+1}$. We recall that the convex hull $\mathcal{C}(\Lambda) \subset \overline{\mathbb{H}^{n+1}}=\mathbb{H}^{n+1}\cup\mathbb{S}^n_\infty$ of $\Lambda$ is the smallest geodesically convex subset that contains $\Lambda$. When $\Lambda$ is not the boundary of a totally geodesic hyperplane in $\mathbb{H}^{n+1}$, $\mathcal{C}(\Lambda)$ is homeomorphic to a ball, and, by the Jordan-Brouwer separation theorem, its boundary is the union of $\Lambda$ and two properly embedded disks, denoted by $\partial^{+}\mathcal{C}(\Lambda)$ and $\partial^{-}\mathcal{C}(\Lambda)$. When $\Lambda$ is the boundary of a totally geodesic hyperplane $P$, $\mathcal{C}(\Lambda)$ equals $P\cup\Lambda$. In this case, by an abuse of notation, we will still use the symbols $\partial^{+}\mathcal{C}(\Lambda)$ and $\partial^{-}\mathcal{C}(\Lambda)$, meaning that $P=\partial^{+}\mathcal{C}(\Lambda)=\partial^{-}\mathcal{C}(\Lambda)$. Following (and extending to all dimensions) the recent work [@BDMS21], we now define the *width* of $\Lambda$. **Definition 9**. Given a topologically embedded $n$-sphere $\Lambda$ in $\mathbb{S}_\infty^n$, the width of $\Lambda$ is defined as: $$w(\Lambda)=\sup_{x\in\mathcal{C}(\Lambda)}\left(d(x,\partial^{+}\mathcal{C}(\Lambda))+d(x,\partial^{-}\mathcal{C}(\Lambda))\right)\in[0,+\infty]~.$$ The following lemma, that relates minimal hypersurfaces and the convex hull of their asymptotic boundaries, is well-known. **Lemma 10**. *Given a topologically embedded $n$-sphere $\Lambda$ in $\mathbb{S}_\infty^n$, let $\Sigma$ be any properly embedded minimal hypersurface such that $\partial_\infty\Sigma=\Lambda$. Then $\Sigma$ is contained in $\mathcal{C}(\Lambda)$. Moreover, if $\Lambda$ is not the boundary of a totally geodesic hyperplane, then $\Sigma$ is contained in the interior of $\mathcal{C}(\Lambda)$.* *Proof.* Let $P$ be any totally geodesic hyperplane disjoint from $\Lambda$. The signed distance function from $P$, defined in such a way that it goes to $-\infty$ as it approaches $\Lambda$, cannot have a positive maximum by a standard application of the geometric maximum principle (see also Corollary [Corollary 22](#max principle){reference-type="ref" reference="max principle"} for a more general statement). This implies that $\Sigma$ is contained in the half-space bounded by $P$ whose closure contains $\Lambda$. Since $\mathcal{C}(\Lambda)$ is the intersection of all such half-spaces, this concludes the proof of the first assertion. For the second assertion, suppose that $\Sigma$ contains a point $x$ in the boundary of $\mathcal{C}(\Lambda)$. Let $P$ be any support hyperplane for $\mathcal{C}(\Lambda)$ containing $x$. This means that $\Sigma$ is tangent to $P$ and, by the first assertion, it is contained in a half-space bounded by $P$. By the strong maximum principle, $\Sigma=P$, and therefore $\Lambda=\partial_\infty P$. ◻ ## Stable and area-minimizing minimal hypersurfaces A minimal hypersurface $\Sigma$ in a Riemannian manifold $M^{n+1}$ is stable if and only if, for every $u\in C_0^\infty(\Sigma)$, $$\int_{\Sigma} uL_{\Sigma}(u)\mathrm{dVol}_\Sigma\geq 0~,$$ where $L_\Sigma$ is the Jacobi operator: $$\label{eq:jacobi} L_\Sigma(u)=-\Delta_\Sigma u-\left(\|A_\Sigma\|^2+\mathrm{Ric}_M(N,N)\right)u~.$$ Equivalently, integrating by parts, $\Sigma$ is stable if and only if $$\label{eq:stability} \int_\Sigma\left(\|A_\Sigma\|^2+\mathrm{Ric}_M(N,N)\right)u^2\mathrm{dVol}_\Sigma\leq \int_\Sigma\|\nabla u\|^2\mathrm{dVol}_\Sigma$$ for every $u\in C_0^\infty(\Sigma)$. A compact hypersurface $\Sigma_c$ with boundary is a *least area hypersurface* if its area is less than or equal to that of any other compact hypersurface $\Sigma_c'$ such that $\partial\Sigma_c=\partial\Sigma_c'$. For the non-compact case, we say a hypersurface $\Sigma$ is area-minimizing if any compact codimension 0 submanifold with boundary of $\Sigma$ has least area in the sense above. An area-minimizing hypersurface is stable. Indeed, the area-minimizing condition implies that $\Sigma$ is a critical point of the area functional among compactly supported variation (that is, $\Sigma$ is minimal) and the second derivative of the area is non-negative (which is equivalent to the definition of stability given above). When $n=2$, the following theorem provides an important a priori bound on the curvature of a stable minimal surface. We state the theorem here in its version for embedded minimal surfaces in $\mathbb{H}^3$, but it holds more generally for immersed CMC surfaces in a complete Riemannian manifold of bounded sectional curvature. **Theorem 11**. *[@RST10 Main Theorem][\[thm:rosenberg\]]{#thm:rosenberg label="thm:rosenberg"} There exists a constant $c>0$ such that, for any stable embedded two-sided minimal surface $\Sigma$ in $\mathbb{H}^3$, $$\|A_\Sigma(p)\|\leq \frac{c}{\min\{d(p,\partial\Sigma),\pi/2\}}~.$$* Applying Theorem [\[thm:rosenberg\]](#thm:rosenberg){reference-type="ref" reference="thm:rosenberg"} to a properly embedded minimal surface, which is automatically two-sided (see [@Sam69]), we immediately get the following corollary. See also [@Ros06 Corollary 11]. **Corollary 12**. *There exists a constant $c'>0$ such that, for any stable properly embedded minimal surface $\Sigma$ in $\mathbb{H}^3$, $\|A_\Sigma\|\leq c'$.* ## Compactness results We conclude the preliminaries by stating an important compactness theorem for minimal surfaces. Again, we state the theorem here only for embedded minimal surfaces in $\mathbb{H}^3$, but it holds more generally for immersed surfaces in a complete Riemannian manifold (see [@and85 Section 2]). This result is well-known and is often stated under the assumption of bounded Gaussian curvature (in absolute value); however, by the Gauss' equation for minimal surfaces when $n=2$, $K_{\Sigma}=-1-\|A_{\Sigma}\|^2/2$, hence this is equivalent to a bound on the norm of the second fundamental form. **Theorem 13**. *Let $B$ be a bounded domain in $\mathbb{H}^3$ and let $\{\Sigma_n\}_{n\in\mathbb{N}}$ be a sequence of embedded minimal surfaces in $B$ with $\partial\Sigma_n\subset\partial B$. Suppose there exists a constant $C$ such that $\|A_{\Sigma_n}\|\leq C$ and that there exists a compact subset $K\subset \mathrm{int}(B)$ such that $\Sigma_n\cap K\neq \emptyset$, for all $n$. Then, up to extracting a subsequence, $\Sigma_n$ converges smoothly to an embedded minimal surface $\Sigma_\infty$.* In particular, for properly embedded minimal surfaces, we obtain the following compactness result. **Corollary 14**. *Let $\{\Sigma_n\}_{n\in\mathbb{N}}$ be a sequence of properly embedded minimal surfaces in $\mathbb{H}^3$. Suppose there exists a constant $C$ such that $\|A_{\Sigma_n}\|\leq C$ and that there exists a compact subset $K\subset \mathbb{H}^3$ such that $\Sigma_n\cap K\neq \emptyset$, for all $n$. Then, up to extracting a subsequence, $\Sigma_n$ converges smoothly on compact subsets of $\mathbb{H}^3$ to a properly embedded minimal surface $\Sigma_\infty$.* *Proof.* Fix a point $x_o\in\mathbb{H}^3$. Let $m_0\in\mathbb{N}$ be such that $K\subset B(x_o,m_0)$. For any $m\geq m_0$, by applying Theorem [Theorem 13](#thm compactness bounded){reference-type="ref" reference="thm compactness bounded"} on the balls $B(x_o,m)$ of radius $m$ centered at $x_o$, we find a subsequence converging smoothly to a minimal surface in $B(x_o,m)$. By a standard diagonal argument, we can then extract a subsequence converging smoothly on all compact subsets to an embedded minimal surface $\Sigma_\infty$. It only remains to show that $\Sigma_\infty$ is properly embedded. Suppose by contradiction that $x_i\in\Sigma_\infty$ is a diverging sequence in $\Sigma_\infty$, which accumulates to $x_\infty\in\mathbb{H}^3$. Then for $m$ large, $x_\infty$ would be contained in the interior of $B(x_o,m)$, and this would contradict the smooth convergence. ◻ # Uniqueness among stable minimal disks {#sec3} In this section, we prove Theorems [Theorem 1](#thm:char uniqueness){reference-type="ref" reference="thm:char uniqueness"} and [Theorem 2](#thm:char uniqueness invariant){reference-type="ref" reference="thm:char uniqueness invariant"}. ## Outline The key result to prove Theorem [Theorem 1](#thm:char uniqueness){reference-type="ref" reference="thm:char uniqueness"} is the following: **Theorem 15**. *Let $\Lambda$ be a Jordan curve on ${\mathbb{S}}^2_{\infty} = \partial \mathbb{H}^3$. Suppose that there exists a properly embedded minimal surface in $\mathbb{H}^3$, which is not a stable minimal disk, spanning $\Lambda$, then $\Lambda$ spans at least two disjoint stable minimal disks in $\mathbb{H}^3$.* Before proving Theorem [Theorem 15](#main3){reference-type="ref" reference="main3"}, let us explain how this implies Theorem [Theorem 1](#thm:char uniqueness){reference-type="ref" reference="thm:char uniqueness"}. By the results of [@And83], we know that $\Lambda$ spans a stable (actually, area-minimizing) minimal disk. So uniqueness among minimal surfaces clearly implies uniqueness among stable minimal disks. For the converse direction, suppose by contradiction that $\Lambda$ span a unique stable minimal disk, but it spans several minimal surfaces. Hence it spans a minimal surface $U$ which is not a stable minimal disk (that is, either it is not stable, or it is not a disk, or both). By Theorem [Theorem 15](#main3){reference-type="ref" reference="main3"}, $\Lambda$ spans two distinct stable minimal disks, and this gives a contradiction. When $\Lambda$ is invariant by the action of a Kleinian group, in order to achieve Theorem [Theorem 2](#thm:char uniqueness invariant){reference-type="ref" reference="thm:char uniqueness invariant"}, we will first prove the following result in the same spirit of Theorem [Theorem 15](#main3){reference-type="ref" reference="main3"}. **Theorem 16**. *Let $\Lambda$ be a Jordan curve on ${\mathbb{S}}^2_{\infty} = \partial \mathbb{H}^3$ and let $\Gamma$ be a Kleinian group preserving $\Lambda$. Suppose that there exists a properly embedded minimal surface in $\mathbb{H}^3$, which is not $\Gamma$-invariant, spanning $\Lambda$. Then $\Lambda$ spans at least two disjoint $\Gamma$-invariant stable minimal disks in $\mathbb{H}^3$.* Theorem [Theorem 16](#main3groupaction){reference-type="ref" reference="main3groupaction"} implies Theorem [Theorem 2](#thm:char uniqueness invariant){reference-type="ref" reference="thm:char uniqueness invariant"} very similarly to above. Indeed, Anderson proved the existence of a $\Gamma$-invariant stable minimal disk, hence uniqueness among minimal surfaces implies uniqueness among $\Gamma$-invariant stable minimal disks. Conversely, suppose that $\Lambda$ spans a minimal surface which is not a $\Gamma$-invariant stable minimal disk. If it is not a minimal disk, Theorem [Theorem 15](#main3){reference-type="ref" reference="main3"} applies as above. If it is instead not $\Gamma$-invariant, we apply Theorem [Theorem 16](#main3groupaction){reference-type="ref" reference="main3groupaction"} to produce two distinct $\Gamma$-invariant stable minimal disks, and this gives a contradiction. ## Adapting the arguments of Anderson To prove Theorems [Theorem 15](#main3){reference-type="ref" reference="main3"} and [Theorem 16](#main3groupaction){reference-type="ref" reference="main3groupaction"}, we will adapt an argument in [@And83]. We first need the following result. **Lemma 17**. *Let $\Lambda$ be a Jordan curve on ${\mathbb{S}}^2_{\infty} = \partial \mathbb{H}^3$, let $U$ be a properly embedded minimal surface in $\mathbb{H}^3$ spanning $\Lambda$, and let $\Omega$ be a connected component of $\mathbb{H}^3\setminus U$. Then there exists a properly embedded stable minimal disk $\Sigma$ in $\Omega\cup U$ spanning $\Lambda$. Moreover, if $\Lambda$ is invariant under a Kleinian group $\Gamma$, then $\Sigma$ can be taken such that $$\label{eq:intersection} \Sigma\subset\bigcap_{g\in\Gamma}g(\Omega\cup U)~.$$* *Proof.* We can clearly assume that $\Lambda$ is not the boundary of a totally geodesic plane. Then, since $U$ is contained in the interior of $\mathcal{C}(\Lambda)$ by Lemma [Lemma 10](#lemma interior convex hull){reference-type="ref" reference="lemma interior convex hull"}, $\Omega$ contains either $\partial^+\mathcal{C}(\Lambda)$ or $\partial^-\mathcal{C}(\Lambda)$. Suppose it contains $\partial^+\mathcal{C}(\Lambda)$, the other case being analogous. Fix a homeomorphism $\phi:\overline{\mathbb{D}}\to \Lambda\cup\partial^+\mathcal{C}(\Lambda)$ and let $\gamma_n=\phi(\{|z|=1-\frac1n\})$. Then $\{\gamma_n\}_{n\in\mathbb{N}}$ is a sequence of nullhomotopic simple closed curves in $\Omega\cap\mathcal{C}(\Lambda)$ that converge to $\Lambda$ as $n\to+\infty$. Now, fix a point $x_o\in U$. For any $n$, let $R_n$ be a sufficiently large radius so that $\gamma_n\subset B(x_o,R_n)$. By genericity of the transversality of intersection of submanifolds, we can furthermore pick $R_n$ so that $U$ and $\partial B(x_o,R_n)$ intersect transversely. By [@hassscott Theorem 6.3] applied to $M_n:=B(x_o,R_n)\cap\Omega$, which (as a consequence of transversality) is sufficiently convex in the terminology of [@hassscott], we can solve the Plateau problem to find a minimal disk $\Sigma_n$, contained in $M_n$, with $\partial\Sigma_n=\gamma_n$, and of least area (in $M_n$), hence in particular stable. Next we show that the limit of these $\Sigma_n$ provides the desired stable minimal disk $\Sigma$. However, when $\Lambda$ is $\Gamma$-invariant, we first need an additional adjustement which will ensure [\[eq:intersection\]](#eq:intersection){reference-type="eqref" reference="eq:intersection"}. Since $\Gamma$ is a discrete subgroup of $\mathrm{Isom}(\mathbb H^3)$, it is countable, and so too is the coset space $G/\mathrm{Stab}(U)$. Let us enumerate its elements as $$G/\mathrm{Stab}(U)=\{g_0=[\mathrm{id}],g_1,g_2,\ldots\}$$ and let us denote $U_n:=g_n(U)$, which is a properly embedded minimal surface spanning $\Lambda$. Here, we construct $\Sigma_n$ as the solution of the Plateau problem with $\partial\Sigma_n=\gamma_n$, but instead of $M_n$, we apply [@hassscott Theorem 6.3] to the ambient manifold $$M_n':=B(x_o,R_n)\cap\Omega\cap g_1(\Omega)\ldots\cap g_n(\Omega)~,$$ where $R_n$ is chosen so that $\partial B(x_o,R_n)$ intersects transversely all the minimal surfaces $\Omega,g_1(\Omega),\ldots,g_n(\Omega)$. Since the points of tangency of two minimal surfaces are isolated (see [@CM11 Section 5.3]), the boundary of $M_n'$ is piecewise smooth, meaning that it is smooth (and minimal) in the complement of an embedded finite graph. Hence $M_n'$ is is sufficiently convex in the terminology of [@hassscott], which allows us to apply [@hassscott Theorem 6.3]. We now claim that, up to extracting a subsequence, $\Sigma_n$ converges smoothly on compact subsets of $\mathbb{H}^3$ to a properly embedded minimal disk $\Sigma$. For this, first observe that there exists $R_0>0$ such that $\Sigma_n\cap B(x_o,R_0)\neq\emptyset$ for all $n$. Indeed, since $\partial\Sigma_n=\gamma_n$ is contained in $\partial^+\mathcal{C}(\Lambda)$, by the maximum principle (as in the first part of Lemma [Lemma 10](#lemma interior convex hull){reference-type="ref" reference="lemma interior convex hull"}) $\Sigma_n$ is entirely contained in $\mathcal{C}(\Lambda)$, and by the Jordan-Brouwer Theorem, $\Sigma_n$ must disconnect $\mathcal{C}(\Lambda)$ in two components, one of which contains $x_o$, and the other contains $\phi(0)\in \partial^+\mathcal{C}(\Lambda)$. Fixing a path $\eta$ connecting $x_o$ to $\phi(0)$ which is contained in the interior of $\mathcal{C}(\Lambda)$ (except for its endpoints), $\Sigma_n$ must therefore intersect $\eta$. Hence $\Sigma_n$ intersects $B(x_o,R_0)$ for any choice of $R_0$ such that $B(x_o,R_0)$ contains $\eta$. Now, fix any radius $R>0$ and consider the ball $B(x_o,R)$. For $n=n(R)$ sufficiently large, $\gamma_n$ is contained in $\mathbb{H}^3\setminus B(x_o,R+1)$. Hence the distance from any point of $\Sigma_n\cap B(x_o,R)$ and $\partial\Sigma_n=\gamma_n$ is bounded below by $1$. Theorem [\[thm:rosenberg\]](#thm:rosenberg){reference-type="ref" reference="thm:rosenberg"} implies that there is a constant $C$, independent of $n$, such that $\|A_{\Sigma_n}(x)\|\leq C$ for every $x\in \Sigma_n\cap B(x_o,R)$. Applying Theorem [Theorem 13](#thm compactness bounded){reference-type="ref" reference="thm compactness bounded"}, we can extract a subsequence $n_k$ such that $\Sigma_{n_k}\cap B(x_o,R)$ converges smoothly to a minimal disk. By a diagonal argument, we can then find a subsequence of $\{\Sigma_n\}_{n\in\mathbb{N}}$ that converges smoothly on compact subsets to a minimal disk $\Sigma_\infty$. Moreover, the limit $\Sigma_\infty$ is stable as a smooth limit of stable minimal surfaces, and is contained in $\Omega\cup U$ because every $\Sigma_n$ is contained in $\Omega$. It remains to show that $\Sigma_\infty$ is properly embedded and $\partial_\infty\Sigma=\Lambda$. To see that it is properly embedded, the same argument as in the proof of Corollary [Corollary 14](#cor compactness unbounded){reference-type="ref" reference="cor compactness unbounded"} applies. Observe that $\Sigma_n$ is contained in $\mathcal{C}(\Lambda)$ for all $n$, and therefore so too is $\Sigma_\infty$. Since $\mathcal{C}(\Lambda)\cap\partial_\infty\mathbb{H}^3=\Lambda$, we have that $\partial_\infty\Sigma=\Lambda$. This concludes the proof. For the "moreover" part when $\Lambda$ is $\Gamma$-invariant, recall that in that case we have constructed the $\Sigma_n$ to be contained in $M_n'$. In particular, for fixed $m$, $\Sigma_n$ is contained in $g_m(\Omega)$ for all $n\geq m$. Taking limits, $\Sigma_\infty$ is contained in $g(\Omega\cup U)$ for all $g\in\Gamma$. This concludes the proof. ◻ In the case where $\Lambda$ is $\Gamma$-invariant, we improve Lemma [Lemma 17](#lemma Plateau){reference-type="ref" reference="lemma Plateau"} to the following lemma. We include a proof, which follows closely the construction done by Anderson in [@And83 Theorem 3.1], for convenience of the reader. **Lemma 18**. *Let $\Lambda$ be a Jordan curve on ${\mathbb{S}}^2_{\infty} = \partial \mathbb{H}^3$ and let $\Gamma$ be a Kleinian group preserving $\Lambda$. Let $U$ be a properly embedded minimal surface in $\mathbb{H}^3$ spanning $\Lambda$, and let $\Omega$ be a connected component of $\mathbb{H}^3\setminus U$. Then there exists a $\Gamma$-invariant properly embedded stable minimal disk $\Sigma$ in $\Omega\cup U$ spanning $\Lambda$.* *Proof.* We will use the following notation. Choose a connected component $D$ of $\partial_\infty\mathbb{H}^3\setminus\Lambda$. Given a properly embedded (minimal) surface $\Sigma$ with $\partial_\infty\Sigma=\Lambda$, recalling that $\Sigma$ disconnects $\mathbb{H}^3$ by the Jordan-Brower separation theorem, we denote $\Omega^+(\Sigma)$ to be the connected component of $\mathbb{H}^3\setminus\Sigma$ whose closure in $\mathbb{H}^3\cup\partial_\infty\mathbb{H}^3$ contains $D$. Finally, given $\Sigma_1$ and $\Sigma_2$, we write $\Sigma_1\preceq \Sigma_2$ if $\Omega^+(\Sigma_1)\supseteq \Omega^+(\Sigma_2)$. Now, given $U$, we define a sequence $\{\Sigma_n\}_{n\in\mathbb N}$ inductively as follows. Set $\Sigma_0:=U$ and let $\Sigma_{n+1}$, using Lemma [Lemma 17](#lemma Plateau){reference-type="ref" reference="lemma Plateau"}, to be a properly embedded stable minimal surface such that $$\label{eq:intersection induction} \Sigma_{n+1}\subset\bigcap_{g\in\Gamma}g(\Omega^+(\Sigma_n)\cup \Sigma_n)~.$$ As in the proof of Lemma [Lemma 17](#lemma Plateau){reference-type="ref" reference="lemma Plateau"}, there exists a fixed compact set (for example a compact set containing a path connecting the two boundary components of $\mathcal{C}(\Lambda)$) that intersects each $\Sigma_n$. Moreover, by Corollary [Corollary 12](#cor:rosenberg prop emb){reference-type="ref" reference="cor:rosenberg prop emb"}, $\|A_{\Sigma_n}\|\leq C$ for some universal constant $C$. Hence by Corollary [Corollary 14](#cor compactness unbounded){reference-type="ref" reference="cor compactness unbounded"}, $\Sigma_n$ converges up to a subsequence to a properly embedded minimal surface $\Sigma_\infty$, which is moreover stable. Arguing as in the second last paragraph of Lemma [Lemma 17](#lemma Plateau){reference-type="ref" reference="lemma Plateau"}, $\partial_\infty\Sigma_\infty=\Lambda$. The main point of the construction is that $\Sigma_\infty$ is $\Gamma$-invariant. To see this, fix $g\in\Gamma$. By the inductive step, $g(\Sigma_n)\preceq\Sigma_{n+1}$ for all $n$. Taking limits along the chosen subsequence, $g(\Sigma_\infty)\preceq \Sigma_\infty$. Repeating the argument for $g^{-1}$, this shows that $g(\Sigma_\infty)= \Sigma_\infty$ and concludes the proof. ◻ ## Conclusion of the proofs We now prove Theorem [Theorem 15](#main3){reference-type="ref" reference="main3"}. *Proof of Theorem [Theorem 15](#main3){reference-type="ref" reference="main3"}.* Suppose $U$ is a properly embedded minimal surface in $\mathbb{H}^3$ spanned by $\Lambda$, which is not a stable minimal disk. By the Jordan-Brouwer separation theorem, $U$ disconnects $\mathbb{H}^3$. Let $\Omega^{\pm}$ be the two connected components of $\mathbb{H}^3\backslash U$. By Lemma [Lemma 17](#lemma Plateau){reference-type="ref" reference="lemma Plateau"}, there exists a properly embedded stable minimal disk $\Sigma^\pm$ contained in $\Omega^\pm\cup U$ with $\partial_\infty\Sigma^\pm=\Lambda$. We claim that $\Sigma^\pm$ is disjoint from $U$. Indeed, $\Sigma^\pm$ is contained by construction in $\Omega^\pm\cup U$, so by the strong maximum principle, if $\Sigma^\pm$ and $U$ intersect, then they are equal. Since $\Sigma^\pm$ is a stable minimal disk, and $U$ is not, they must be disjoint. This shows that $\Sigma^+\subset\Omega^+$ and $\Sigma^-\subset\Omega^-$, hence they are in particular disjoint. ◻ The proof of Theorem [Theorem 16](#main3groupaction){reference-type="ref" reference="main3groupaction"} is completely analogous, relying on Lemma [Lemma 18](#lemma Plateau group){reference-type="ref" reference="lemma Plateau group"} instead of Lemma [Lemma 17](#lemma Plateau){reference-type="ref" reference="lemma Plateau"}. *Proof of Theorem [Theorem 16](#main3groupaction){reference-type="ref" reference="main3groupaction"}.* If $U$ is a properly embedded minimal surface in $\mathbb{H}^3$ spanned by $\Lambda$, which is not $\Gamma$-invariant, denoting by $\Omega_\pm$ the two connected components of the complement of $U$, then by Lemma [Lemma 18](#lemma Plateau group){reference-type="ref" reference="lemma Plateau group"} there exists a properly embedded $\Gamma$-invariant stable minimal disk $\Sigma^\pm$ contained in $\Omega^\pm\cup U$ with $\partial_\infty\Sigma^\pm=\Lambda$. Since $\Sigma^\pm$ is $\Gamma$-invariant, and $U$ is not, they are distinct, and by the strong maximum principle they are disjoint. Hence $\Sigma^+$ and $\Sigma^-$ are disjoint. ◻ # Small curvatures and uniqueness {#sec:small curvatures} In this section, we show that having some notions ofa small curvature condition can imply uniqueness for solutions to the asymptotic Plateau problem. This will prove Theorem [Theorem 3](#main1){reference-type="ref" reference="main1"} in the case of $n=2$. ## Small curvature conditions We now introduce several conditions of having small curvature. **Definition 19**. Let $\Sigma$ be an immersed hypersurface in $\mathbb H^{n+1}$, and let $\lambda_i$ denote its principal curvatures, for $i=1,\ldots,n$. We say that: - $\Sigma$ has *weakly small curvature* if $|\lambda_i(x)|\leq 1$ for every $x$ in $\Sigma$ and every $i=1,\ldots,n$; - $\Sigma$ has *small curvature* if $|\lambda_i(x)|< 1$ for every $x$ in $\Sigma$ and every $i=1,\ldots,n$; - $\Sigma$ has *strongly small curvature* if there exists $\epsilon>0$ such that $|\lambda_i(x)|\leq 1-\epsilon$ for every $x$ in $\Sigma$ and every $i=1,\ldots,n$. In this article, will need two results on hypersurfaces of weakly small curvatures, namely Theorem [Theorem 20](#thm:weakly small is properly emb){reference-type="ref" reference="thm:weakly small is properly emb"} and Theorem [Theorem 21](#exponential diffeo){reference-type="ref" reference="exponential diffeo"} below. These results are well known under the small curvatures assumptions, see [@Eps84] and [@ES22 Section 4]. In Appendix [7](#appendix){reference-type="ref" reference="appendix"} we will explain how the arguments adapt under the weakly small curvatures assumption. **Theorem 20**. *Let $\Sigma\subset\mathbb H^{n+1}$ be a properly immersed hypersurface of weakly small curvature. Then $\Sigma$ is properly embedded and diffeomorphic to $\mathbb{R}^n$.* By this result, we can hereafter assume that all the hypersurfaces of weakly small curvatures are properly embedded. **Theorem 21**. *Let $\Sigma\subset\mathbb{H}^{n+1}$ be a properly embedded hypersurface of weakly small curvatures. Then the map $F:\Sigma\times\mathbb{R}\to\mathbb{H}^{n+1}$ defined by $$F_\Sigma(x,t)=\exp_x(t N(x))$$ is a diffeomorphism. For every $t\in\mathbb{R}$, $\Sigma_t:=F_\Sigma(\Sigma,t)$ is a properly embedded hypersurface of weakly small curvature.* *If moreover $\Sigma$ is minimal, then the mean curvature of $\Sigma_t$, computed with respect to the unit normal vector field pointing towards the direction where $t$ is increasing, has the same sign as $-t$.* A consequence of Theorem [Theorem 21](#exponential diffeo){reference-type="ref" reference="exponential diffeo"} is that, if $\Sigma$ is a properly embedded hypersurface of weakly small curvature, the signed distance function $f_\Sigma:\mathbb{H}^{n+1}\to\mathbb{R}$ is smooth and has no critical points. Indeed, $f_\Sigma$ is simply the composition $$\label{eq:signed distance} f_\Sigma=\pi\circ F_\Sigma^{-1}$$ where $\pi:\Sigma\times\mathbb{R}\to\mathbb{R}$ is the projection to the second factor. Let us focus again on minimal hypersurfaces. The following statement is an immediate consequence of great importance for this work. **Corollary 22**. *Let $\Sigma\subset\mathbb{H}^{n+1}$ be a properly embedded minimal hypersurface of weakly small curvature. For any embedded minimal hypersurface $\Sigma'\subset\mathbb{H}^{n+1}$, the restriction of $d_{\Sigma}:=d(\cdot,\Sigma)$ to $\Sigma'$ has no positive local maximum.* *Proof.* Assume by contradiction that $(d_{\Sigma})|_{\Sigma'}$ has a positive local maximum. Let $x_{\max}\in\Sigma'$ the maximum point and $t_{\max}=d_\Sigma(x_{\max})>0$ be the corresponding maximum value. Up to switching the sign of $f_\Sigma$, $d_{\Sigma}$ coincides with $f_\Sigma$ in a neighbourhood of $x_{\max}$, where $f_\Sigma$ is the signed distance function as in [\[eq:signed distance\]](#eq:signed distance){reference-type="eqref" reference="eq:signed distance"}. Hence $(f_{\Sigma})|_{\Sigma'}$ also has a positive local maximum at $x_{\max}$. This implies that $\Sigma'$ is, in an embedded neighbourhood of $x_{\max}$, tangent to $\Sigma_{t_{\max}}$ and contained in the region $F_\Sigma(\Sigma\times\{t\leq t_{\max}\})$, which is strictly mean convex by the last part of Theorem [Theorem 21](#exponential diffeo){reference-type="ref" reference="exponential diffeo"}. This contradicts the maximum principle for mean curvature. ◻ Finally, it is immediate to check that minimal hypersurfaces of weakly small curvatures are stable. **Lemma 23**. *Let $\Sigma$ be a minimal hypersurface in $\mathbb{H}^{n+1}$ of weakly small curvatures. Then $\Sigma$ is stable.* *Proof.* If $\Sigma$ has weakly small curvatures, then $$\|A_\Sigma\|^2=\lambda_1^2+\ldots+\lambda_n^2\leq n=-\mathrm{Ric}_{\mathbb{H}^{n+1}}(N,N)~.$$ Hence the left hand side of [\[eq:stability\]](#eq:stability){reference-type="eqref" reference="eq:stability"} is non-positive, and this concludes the proof. ◻ ## Proof of Theorem [Theorem 3](#main1){reference-type="ref" reference="main1"} for $n=2$ {#proof-of-theorem-main1-for-n2} In this subsection, we prove Theorem [Theorem 3](#main1){reference-type="ref" reference="main1"} for surfaces in $\mathbb{H}^3$. By Theorem [Theorem 1](#thm:char uniqueness){reference-type="ref" reference="thm:char uniqueness"}, we know that it will be sufficient to prove the uniqueness in the class of stable minimal disks. The fundamental result is therefore the following theorem. **Theorem 24**. *Let $\Lambda$ be a Jordan curve on ${\mathbb{S}}^2_{\infty} = \partial \mathbb{H}^3$ of finite width, and let $\Sigma$ be a properly embedded minimal surface in $\mathbb{H}^3$ of weakly small curvature asymptotic to $\Lambda$. Then $\Sigma$ is the unique properly embedded stable minimal surface in $\mathbb{H}^3$ asymptotic to $\Lambda$.* Combining Theorem [Theorem 24](#main2){reference-type="ref" reference="main2"} with Theorem [Theorem 1](#thm:char uniqueness){reference-type="ref" reference="thm:char uniqueness"}, we immediately obtain the first part Theorem [Theorem 3](#main1){reference-type="ref" reference="main1"} in the case $n=2$. The "moreover" part of Theorem [Theorem 3](#main1){reference-type="ref" reference="main1"} then follows from the results of [@And83], which show that every Jordan curve bounds at least one area-minimizing minimal disk. We begin by an easy lemma. **Lemma 25**. *Let $\Lambda$ be a Jordan curve on ${\mathbb{S}}^2_{\infty} = \partial \mathbb{H}^3$ of width $w(\Lambda)<+\infty$, and let $\Sigma$ be any properly embedded surface contained in $\mathcal{C}(\Lambda)$ with $\partial_\infty\Sigma=\Lambda$ . Then for every $x\in \mathcal{C}(\Lambda)$, $d(x,\Sigma)\leq w(\Lambda)$.* *Proof.* We may assume that $\Lambda$ is not the boundary of a totally geodesic hyperplane, as in that case the conclusion holds trivially. Let $x\in \mathcal{C}(\Lambda)$. By definition of the width, there exist two geodesic segments joining $x$ to $y_{\pm}\in\mathcal{C}^\pm(\Lambda)$, each of length at most $w(\Lambda)+\epsilon$. Since $\Sigma$ is contained in the interior of $\mathcal{C}(\Lambda)$ by Lemma [Lemma 10](#lemma interior convex hull){reference-type="ref" reference="lemma interior convex hull"}, and disconnects $\mathcal{C}(\Lambda)$ by the Jordan-Brouwer separation theorem, one of these two segments must meet $\Sigma$ at a point $y$, which is on the segment between $x$ and the other endpoint $y_\pm$. Hence $$d(x,\Sigma)\leq d(x,y)\leq d(x,y_\pm)\leq w(\Lambda)+\epsilon~.$$ Since $\epsilon$ was arbitrary, this concludes the proof. ◻ *Proof of Theorem [Theorem 24](#main2){reference-type="ref" reference="main2"}.* Let $\Sigma'$ be a properly embedded stable minimal surface such that $\partial_\infty\Sigma'=\Lambda$. Consider the restriction to the surface $\Sigma'$ of the distance function $d_{\Sigma}:=d(\cdot,\Sigma)$. We want to show that $d_{\Sigma}$ vanishes identically on $\Sigma'$. This will imply $\Sigma'\subseteq \Sigma$ and thus, since $\Sigma'$ is also properly embedded, $\Sigma=\Sigma'$. Suppose by contradiction that $$\label{eq sup contradiction} \sup_{\Sigma'}d_\Sigma>0~.$$ By Lemma [Lemma 25](#lemma finite distance){reference-type="ref" reference="lemma finite distance"}, the supremum of $d_\Sigma$ on $\Sigma'$ is bounded above by $w(\Lambda)$, hence in particular it is finite. Let $\{x_n'\}_{n\in\mathbb N}$ be a maximizing sequence for $(d_\Sigma)|_{\Sigma'}$, namely $$\label{eq sup distance} d(x'_n,\Sigma) \ge \sup_{\Sigma'}d_\Sigma - \frac1n~.$$ Now, fix a point $x_o\in\mathbb{H}^3$. Let $\phi_n$ be isometries of $\mathbb{H}^3$ such that $\phi_n(x'_n) = x_o$. We denote $\Sigma_n = \phi_n(\Sigma)$ and $\Sigma'_n = \phi_n(\Sigma')$. Observe that $\Sigma_n'$ contains $x_o$, and thus it intersects every ball centered at $x_o$. For $\Sigma_n$, we have $$d(x_o,\Sigma_n)=d(\phi_n(x'_n),\phi_n(\Sigma))=d(x_n',\Sigma)\leq w(\Lambda)~.$$ This shows that $\Sigma_n$ intersects every ball centered at $x_o$ of radius $R\geq w(\Lambda)$. Recall that $\Sigma$ has weakly small curvatures, hence $\|A_{\Sigma}\|\leq 2$, and since this condition is preserved by isometries, $\|A_{\Sigma_n}\|\leq 2$. By Corollary [Corollary 14](#cor compactness unbounded){reference-type="ref" reference="cor compactness unbounded"}, $\Sigma_n$ converges up to a subsequence, smoothly on a compact subsets in $\mathbb{H}^3$, to a stable properly embedded minimal surface $\Sigma_\infty$. Moreover, since the weakly small curvatures condition is closed for the topology of smooth convergence on compact domains, $\Sigma_\infty$ has weakly small curvatures. Since $\Sigma'$ is stable by assumption, and stability is preserved by isometries, $\Sigma'_n$ is stable for every $n$. However, we do not have explicit curvature bounds as for $\Sigma_n$. In this case, we use Theorem [Corollary 12](#cor:rosenberg prop emb){reference-type="ref" reference="cor:rosenberg prop emb"} to find that each $\|A_{\Sigma'_n}\|$ is bounded by a universal constant. This enables us to apply Corollary [Corollary 14](#cor compactness unbounded){reference-type="ref" reference="cor compactness unbounded"} again to find that the sequence $\Sigma'_n$ converges, up to a subsequence, smoothly on a compact domains in $\mathbb{H}^3$ to another properly embedded minimal surface $\Sigma'_\infty$. Observe that, since by construction $x_o\in\Sigma_n'$ for all $n$, $x_o\in\Sigma_\infty'$. We now claim that $$\label{eq:global maximum} d(x_o,\Sigma_\infty)=\sup_{\Sigma_\infty'}d_{\Sigma_\infty}$$ where as usual $d_{\Sigma_\infty}=d(\cdot,\Sigma_\infty)$. Clearly we have $d(x_o,\Sigma_\infty)\leq \sup_{\Sigma_\infty'}d_{\Sigma_\infty}$. By contradiction, if the inequality were strict, then one would find $p\in\Sigma_\infty'$ and $\epsilon>0$ such that $d(p,\Sigma_\infty)>d(x_o,\Sigma_\infty)+\epsilon$. By smooth convergence on compact sets, this would imply that, for large $n$, $$d(p_n,\Sigma_n)>d(x_o,\Sigma_n)+\frac{\epsilon}{2}$$ where $p_n\in \Sigma_n'$ and $p_n\to p$. But $d(p_n,\Sigma_n)=d(\phi^{-1}(p_n),\Sigma)$ and $d(x_o,\Sigma_n)=d(x_n',\Sigma)$. Hence we would have $$d(\phi^{-1}(p_n),\Sigma)>d(x_n',\Sigma)+\frac{\epsilon}{2}$$ thus contradicting [\[eq sup distance\]](#eq sup distance){reference-type="eqref" reference="eq sup distance"}. Having established the claim, [\[eq:global maximum\]](#eq:global maximum){reference-type="eqref" reference="eq:global maximum"} implies that $x_o$ is a global maximum for the restriction to $\Sigma'$ of the distance function $d_{\Sigma_\infty}$. This contradicts Corollary [Corollary 22](#max principle){reference-type="ref" reference="max principle"} and thus concludes the proof. ◻ ## Alternative argument for Theorem [Theorem 2](#thm:char uniqueness invariant){reference-type="ref" reference="thm:char uniqueness invariant"} {#alternative-argument-for-theorem-thmchar-uniqueness-invariant} We now explain an alternative proof of Theorem [Theorem 2](#thm:char uniqueness invariant){reference-type="ref" reference="thm:char uniqueness invariant"}, which however only applies when the Kleinian group $\Gamma$ is quasi-Fuchsian and isomorphic to the fundamental group of a closed surface. This is based on the work of Guaraco, Lima and Vargas-Pallete [@GLP21], which we summarize now. Let $M$ be a quasi-Fuchsian manifold. One of their main results is that $M$ admits a foliation by closed surfaces that are either minimal, or have non-vanishing mean curvature (i.e., are either strictly mean-convex, or strictly mean-concave.) The sign of the mean curvature switches at the minimal leaves. Their work implies that if $M$ contains a unique closed stable minimal surface $S$, then $M\setminus S$ has a foliation $\mathcal{F}$ by surfaces that are isotopic to $S$ and have mean-curvature vector everywhere pointing towards $S$. *Proof of Theorem [Theorem 2](#thm:char uniqueness invariant){reference-type="ref" reference="thm:char uniqueness invariant"} for $\Gamma$ quasi-Fuchsian.* Suppose $\Gamma$ admits a unique invariant stable minimal disk $\Sigma$, which thus gives a unique closed stable minimal surface $\Sigma/\Gamma$ in $M:=\mathbb{H}^3/\Gamma$. Let $F:M\to\mathbb{R}$ be a function whose level sets $F^{-1}(t)$ are the leafs of the foliation of $M$ provided by [@GLP21], and such that $F^{-1}(0)=\Sigma/\Gamma$. The foliation $\mathcal{F}$ lifts to a foliation of $\mathbb{H}^3$ by properly embedded disks $\widetilde\Sigma_t:=\widetilde F^{-1}(t)$ (where $\widetilde F$ is the lift of $F$ to $\mathbb{H}^3$, and therefore $\widetilde\Sigma_0=\Sigma$) with asymptotic boundary the limit set $\Lambda$ of $\Gamma$. By Theorem [Theorem 1](#thm:char uniqueness){reference-type="ref" reference="thm:char uniqueness"}, it suffices to show that, if $\Sigma'$ is a properly embedded stable minimal disk in $\mathbb{H}^3$ with $\partial_\infty\Sigma'=\Lambda$, then $\Sigma'=\Sigma$. Consider the function $f:=\widetilde F|_{\Sigma'}:\Sigma'\to\mathbb{R}$. We claim that $f\equiv 0$, which will imply $\Sigma'=\Sigma$ since both are properly embedded. First, observe that $f$ is bounded. Indeed, since $\Gamma$ is quasi-Fuchsian, the convex core $\mathcal{C}(\Lambda)/\Gamma$ of $M$ is compact, and $F$ is thus bounded on the convex core. Lifting to $\mathbb{H}^3$, this shows that $\widetilde F|_{\mathcal{C}(\Lambda)}$ is bounded, and so too is $f$ by Lemma [Lemma 10](#lemma interior convex hull){reference-type="ref" reference="lemma interior convex hull"}. Now, the proof is similar to the proof of Theorem [Theorem 24](#main2){reference-type="ref" reference="main2"}, using the foliation lifted from $\mathcal F$ instead of the equidistant foliation. Suppose by contradiction that $\sup f>0$, the other case being analogous. Let $x_n'$ be a sequence such that $f(x_n')\to \sup f$, let $\phi_n$ be a sequence of isometries sending $x_n'$ to a fixed point $x_o$, and let $\Sigma_n':=\phi_n(\Sigma')$ and $\Sigma_n:=\phi_n(\widetilde\Sigma_{\sup f})$. Using Theorem [Corollary 12](#cor:rosenberg prop emb){reference-type="ref" reference="cor:rosenberg prop emb"} and Corollary [Corollary 14](#cor compactness unbounded){reference-type="ref" reference="cor compactness unbounded"}, the sequence $\Sigma'_n$ converges, up to a subsequence, smoothly on a compact domains in $\mathbb{H}^3$ to another properly embedded minimal surface $\Sigma'_\infty$. We claim that the sequence $\Sigma_n$ also converges up to subsequences to a smooth surface in $\mathbb{H}^3$. For this, first observe that the width $\mathcal{C}(\Lambda)$ is finite because $\Lambda$ is a quasicircle, and, by cocompactness of the action of $\Gamma$ on $\widetilde\Sigma_{\sup f}$, the distance of $\widetilde\Sigma_{\sup f}$ from $\mathcal{C}(\Lambda)$ is bounded by some constant $c$. It follows that $\Sigma_n$ intersects every ball centered at $x_o$ of radius $R\geq w(\Lambda)+c$. Second, by smoothness of $\widetilde\Sigma_{\sup f}$ and cocompactness again, the norm of the second fundamental form of $\widetilde\Sigma_{\sup f}$ and of all its covariant derivative is bounded. Since $\phi_n$ is an isometry, the norm of the second fundamental form of $\Sigma_n$ and of all its covariant derivative is bounded too. This allows to extract a subsequence of $\{\Sigma_n\}_{n\in\mathbb N}$ converging smoothly on compact sets to a surface $\Sigma_\infty$ (see for instance [@Smi07]). As we are assuming $\sup f>0$, $\widetilde\Sigma_{\sup f}$ has negative mean curvature with respect to the unit normal pointing towards $\widetilde F=+\infty$, hence the same holds for the limit $\Sigma_\infty$. It follows from the construction that $x_o\in\Sigma'_\infty$. By cocompactness, the distance between different leaves $\widetilde \Sigma_{\sup f}$ and $\widetilde \Sigma_{t}$ goes to zero as $t\to\sup f$. Using this fact, one can show similarly to the proof of Theorem [Theorem 24](#main2){reference-type="ref" reference="main2"} that $x_o\in\Sigma_\infty$. Moreover, since $\Sigma'$ is contained in the region $\{x\,|\,\widetilde F(x)\leq \sup f\}$ whose boundary is $\widetilde\Sigma_{\sup f}$, after taking limits $\Sigma'_\infty$ is contained in the side of $\Sigma_\infty$ corresponding to decreasing values of $\widetilde F$. This contradicts the geometric maximum principle at the tangency point $x_o$, and therefore shows that $\sup f\leq 0$. Repeating the argument for the infimum of $f$ instead of the supremum shows that $f\equiv 0$ and concludes the proof. ◻ # Higher dimension and codimension {#sec:higher} In this section, we outline some generalization of previous results to higher dimensions and higher codimensions. ## Hypersurfaces in higher dimensions We first provide a proof of Theorem [Theorem 3](#main1){reference-type="ref" reference="main1"} in higher dimensions. The argument is similar to the proof of Theorem [Theorem 24](#main2){reference-type="ref" reference="main2"}. However, in that case it was sufficient to work with a *stable* minimal surface, by Theorem [Theorem 1](#thm:char uniqueness){reference-type="ref" reference="thm:char uniqueness"}. The difference here is that instead of passing to a limit of minimal submanifolds (which we will be unable to do if they are unstable), we pass to a limit of metric balls in the ambient space. We are then able to apply a theorem of White [@Whi10] in the setting of minimal varifolds to get a contradiction. *Proof of Theorem [Theorem 3](#main1){reference-type="ref" reference="main1"}.* Recall that, if $d_{\Sigma}=d(\cdot,\Sigma)$ denotes the distance function from $\Sigma$, by Theorem [Theorem 21](#exponential diffeo){reference-type="ref" reference="exponential diffeo"} the level hypersurfaces of $d_{\Sigma}$ form a strictly mean-convex foliation of $\mathbb{H}^n \backslash \Sigma$, meaning that the mean curvature vector does not vanish and always points towards $\Sigma$. Let $\Sigma'$ be another properly embedded minimal hypersurface with the same asymptotic boundary as $\Sigma$. Since $\Lambda$ has finite width, we know that $d_{\Sigma}$ is bounded on $\Sigma'$ by some constant. (Indeed, the proof of Lemma [Lemma 25](#lemma finite distance){reference-type="ref" reference="lemma finite distance"} works identically in any dimension.) If $(d_\Sigma)|_{\Sigma'}$ attains its supremum at an interior point, then the strong maximum principle implies that $\Sigma'=\Sigma$. If not, we can take a sequence of points $x_k$ on $\Sigma'$ with $d_\Sigma(x_k)$ tending to the supremal distance $D:=\sup_{\Sigma'}d_\Sigma$. Without loss of generality assume that all of the $x_k$ are in the same connected component $\Omega_+$ (say) of $\mathbb{H}^{n+1}\setminus\Sigma$. Then we claim that the intersections of the open metric balls of radius $2D$ centered at the $x_k$ intersected with $d_{\Sigma}^{-1}([0,D])\cup\Omega_-$ (where $\Omega_-$ is the other connected component of $\mathbb{H}^{n+1}\setminus\Sigma$) are a sequence of smooth Riemannian manifolds with boundary, that we call $M_k$, that subsequentially converge to a smooth limit Riemannian manifold with strictly mean convex boundary $M_\infty$. Note that since we are intersecting with open balls, the $M_k$ are incomplete and have boundary contained in $d_{\Sigma}^{-1}(D)\cap\Omega_+$. We just need to check smooth convergence at the boundary. For every $x_k$ there is a point $p_k$ on $\Sigma$ such that the ball of radius $4D$ centered at $p_k$ contains $M_k$. Let $\phi_k$ be an isometry of $\mathbb{H}^{n+1}$ sending $p_k$ to a given point $p$, and fix any radius $R\geq 4D$. Using that $\|A_\Sigma\|\leq n$ by assumption, as in Corollary [Corollary 14](#cor compactness unbounded){reference-type="ref" reference="cor compactness unbounded"} (which holds in any dimensions) we can pass to a subsequential limit $\Sigma_{\infty}$ of the minimal hypersurfaces $\phi_k(\Sigma)$ inside every ball $B(p,R)$. Note that $\Sigma_{\infty}$ has weakly small curvature. By Theorem [Theorem 21](#exponential diffeo){reference-type="ref" reference="exponential diffeo"}, the set of points at signed distance $D$ from $\Sigma_{\infty}$ is a smooth strictly mean convex submanifold, and so is its intersection with the interior of $B(p,4D)$. Since $\phi_k(d_{\Sigma}^{-1}(D)\cap B(p_k,4D))$ converges smoothly to the set of points in $B(p,4D)$ at signed distance $D$ from $\Sigma_{\infty}$, it follows that the boundary of $M_k$ (which is contained in $d_{\Sigma}^{-1}(D)$) subsequentially converges smoothly, after applying $\phi_k$. In other words, we have shown that the $M_k$ subconverge smoothly, in the sense of limits of Riemannian manifolds, to a manifold with strictly mean convex boundary $M_\infty$. Now, since $d_{\Sigma}(x_n)$ tends to $D$, the set of limit points of $\Sigma \cap M_k$ in $M_\infty$ contains a point in the boundary of $M_\infty$. But [@Whi10 Theorem 3] says exactly that this is impossible. For the "moreover" part, Anderson [@and85] proved the existence of an area-minimizing integral current with asymptotic boundary $\Lambda$, which, however, is a smooth hypersurface only in the complement of a singular set of Hausdorff dimension at most $n-7$. Nevertheless, calling this integral current $\Sigma'$, the above argument based on the theorem of White [@Whi10] applies to such a $\Sigma'$ as well, showing that $\Sigma'=\Sigma$ is, in particular, a smooth hypersurace. Hence $\Sigma$ is area-minimizing. ◻ ## Higher codimension submanifolds We briefly discuss some generalizations to higher codimension. By following Uhlenbeck's approach [@Uhl83], Jiang [@j21 Lemma 2.2] defined a notion of almost Fuchsian for quotients $M$ of $\mathbb{H}^{n+1}$ by discrete faithful representations of surface groups in $\text{Isom}(\mathbb{H}^{n+1})$. Such a quotient $M$ is called almost-Fuchsian if it admits a (two-dimensional) minimal surface $\Sigma$ with principal curvatures smaller than 1 in magnitude. Jiang showed that an almost-Fuchsian $M$ with $\Sigma$ removed is foliated by equidistant surfaces to $\Sigma$ that are mean 2-convex (the sum of the smallest two principal curvatures is positive), and thus by the maximum principle that $\Sigma$ is the unique closed minimal surface in $M$. Bronstein recently showed that in four dimensions $M$ could in fact be homeomorphic to a nontrivial disk bundle over a surface [@b23]. The limit set of such an $M$ would then be a self-similar everywhere-non-tame knot in $\partial_\infty \mathbb{H}^4 \cong S^3$ (see [@glt88].) For any almost-Fuchsian $M$, the foliation by mean 2-convex equidistant surfaces to $\Sigma$ lifts to a foliation of the complement of a lift $\widetilde{\Sigma}$ of $\Sigma$ to $\mathbb{H}^{n+1}$. We thus expect that the same argument by contradiction using [@Whi10 Theorem 3] (which applies in any codimension) can be used, to show that the limit set of $M$ bounds a unique minimal disk, namely $\widetilde{\Sigma}$. Jiang's work is for two dimension surfaces in arbitrary codimension, but we expect that similar arguments would work for $k$-dimensional minimal surfaces with small principal curvatures in any dimension. In that case, the same arguments would give a uniqueness result for asymptotic Plateau problems in that setting as well. Finally here we mention that Fine has recently developed a theory for counting minimal surfaces in $\mathbb{H}^4$ with asymptotic boundary a given knot [@f21]. In view of [@b23] and the previous discussion, one should be able to give examples of (wild) knots in $S^3$ that bound a unique minimal surface in $\mathbb{H}^4$. # Uncountably many minimal disks {#sec:uncountably} The aim of this section is to prove Theorem [Theorem 6](#thm:uncountably){reference-type="ref" reference="thm:uncountably"}. ## Constructing the Jordan curve We begin by constructing the Jordan curve $\Lambda$ in $\mathbb{S}_\infty=\partial_\infty\mathbb{H}^3$, that will be the asymptotic boundary of uncountably many stable minimal disks. For this purpose, we will construct an explicit continuous injective map $$f:\mathbb{R}\mathrm P^1=\mathbb{R}\cup\{\infty\}\to \mathbb{C}\mathrm P^1=\mathbb{C}\cup\{\infty\}~.$$ We begin by fixing some notation. Let $\epsilon,\delta>0$ be some constants which will be fixed later. For the moment, we only need to assume that $\epsilon<1/3$, so that the circle of radius $2^n(1+\epsilon)$ is smaller than the circle of radius $2^{n+1}(1-\epsilon)$, and $\delta<\pi/8$. Let $$\centering \begin{aligned} p^n_{+,+}:=&2^n\left(1-{\epsilon}\right)e^{ i\left(\frac{\pi}{2}-\delta\right)} \\ q^n_{+,+}:=&2^n\left(1+{\epsilon}\right)e^{ i\left(\frac{\pi}{2}-\delta\right)} \\ p^n_{-,+}:=&2^n\left(1-{\epsilon}\right)e^{ i\left(\frac{\pi}{2}+\delta\right)} \\ q^n_{-,+}:=&2^n\left(1+{\epsilon}\right)e^{ i\left(\frac{\pi}{2}+\delta\right)} \end{aligned} \qquad \begin{aligned} p^n_{+,-}:=&2^n\left(1-{\epsilon}\right)e^{ i\left(-\frac{\pi}{2}+\delta\right)} \\ q^n_{+,-}:=&2^n\left(1+{\epsilon}\right)e^{ i\left(-\frac{\pi}{2}+\delta\right)} \\ p^n_{-,-}:=&2^n\left(1-{\epsilon}\right)e^{ i\left(-\frac{\pi}{2}-\delta\right)} \\ q^n_{-,-}:=&2^n\left(1+{\epsilon}\right)e^{ i\left(-\frac{\pi}{2}-\delta\right)} \\ \end{aligned}$$ Observe that the points $p^n_{\sigma_1,\sigma_2}$ and $q^n_{\sigma_1,\sigma_2}$ lie in the quadrant determined by the two signs $\sigma_1$ and $\sigma_2$, and that the Euclidean distance between $p^n_{\sigma_1,\sigma_2}$ and $q^n_{\sigma_1,\sigma_2}$ is $2^{n+1}\epsilon$. Moreover, denoting $\alpha(z)=2z$, we see that $p^n_{\sigma_1,\sigma_2}=2^np^0_{\sigma_1,\sigma_2}$ and the same for $q^n_{\sigma_1,\sigma_2}$. Now, we define the map $f$, which will be injective and piecewise smooth when restricted to $\mathbb{R}\setminus\{0\}$, with junctions exactly at the points $p^n_{\sigma_1,\sigma_2}$ and $q^n_{\sigma_1,\sigma_2}$. The explicit formula for $f$ is the following: $$f(t)=\begin{cases} 0 &\textrm{ if } t=0 \\ 2^n\left(1-{\epsilon}\right)e^{ i\left((4\pi-8\delta)t-\frac{9\pi}{2}+9\delta\right)} &\textrm{ if } t\in \left[2^n,2^n\left(1+\frac{1}{4}\right)\right] \\ 2^n\left(1+\epsilon(-8t+7)/3\right)e^{ i\left(\frac{\pi}{2}-\delta\right)} &\textrm{ if } t\in \left[2^n\left(1+\frac{1}{4}\right),2^n\left(1+\frac{1}{2}\right)\right] \\ 2^n\left(1+{\epsilon}\right)e^{ i\left((-4\pi+8\delta)t+\frac{5\pi}{2}-5\delta\right)} &\textrm{ if } t\in \left[2^n\left(1+\frac{1}{2}\right),2^n\left(1+\frac{3}{4}\right)\right] \\ 2^n\left((-4/3+4\epsilon)t+(10/3-6\epsilon)\right)e^{ i\left(-\frac{\pi}{2}+\delta\right)} &\textrm{ if } t\in \left[2^n\left(1+\frac{3}{4}\right),2^{n+1}\right] \\ \infty &\textrm{ if } t=\infty \\ -2^n\left(1-{\epsilon}\right)e^{ i\left((4\pi-8\delta)t+\frac{9\pi}{2}-9\delta\right)} &\textrm{ if } t\in \left[-2^n\left(1+\frac{1}{4}\right),-2^n\right] \\ -2^n\left(1+\epsilon(8t+7)/3\right)e^{ i\left(-\frac{\pi}{2}+\delta\right)} &\textrm{ if } t\in \left[-2^n\left(1+\frac{1}{2}\right),-2^n\left(1+\frac{1}{4}\right)\right] \\ -2^n\left(1+{\epsilon}\right)e^{ i\left((-4\pi+8\delta)t-\frac{5\pi}{2}+5\delta\right)} &\textrm{ if } t\in \left[-2^n\left(1+\frac{3}{4}\right),-2^n\left(1+\frac{1}{2}\right))\right] \\ -2^n\left((4/3-4\epsilon)t+(10/3-6\epsilon)\right)e^{ i\left(\frac{\pi}{2}-\delta\right)} &\textrm{ if } t\in \left[-2^{n+1},-2^n\left(1+\frac{3}{4}\right)\right] \\ \end{cases}$$ {#fig:Jordancurve width=".8\\textwidth"} Let us explain this definition (see also Figure [1](#fig:Jordancurve){reference-type="ref" reference="fig:Jordancurve"}). First, observe that the image of $(0,+\infty)$ lies in the half-space $\{\Re(z)>0\}$, while the image of $(-\infty,0)$ lies in the half-space $\{\Re(z)<0\}$. The intervals $[2^n,2^n(1+1/4)]$ are mapped to the arcs of circles contained in $\{\Re(z)>0\}$ connecting $p^n_{+,-}$ to $p^n_{+,+}$, and similarly the intervals $[2^n(1+1/2),2^n(1+3/4)]$ are mapped to the arcs of circles contained in $\{\Re(z)>0\}$ connecting $q^n_{+,+}$ to $q^n_{+,-}$, parameterized proportionally to arclenght. The intervals $[2^n(1+1/4),2^n(1+1/2)]$ are mapped to straight lines connecting $p^n_{+,+}$ to $q^n_{+,+}$. The intervals $[2^n(1+1/4),2^n(1+1/2)]$ are mapped to straight lines connecting $q^n_{+,-}$ to $p^{n+1}_{+,-}$. For negative values of $t$, the situation is absolutely analogous up to composing with a reflection in the imaginary axis. Namely, $f$ satisfies the symmetry: $$\label{symmetry f} f(-t)=-\overline{f(t)}~.$$ Also, $f$ satisfies the equivariance property $$\label{equivariance f} f\circ \alpha^n=\alpha^n\circ f~,$$ where we recall that $\alpha(z)=2z$. **Lemma 26**. *The image of $f$ is a quasicircle in $\mathbb{C}\mathrm P^1$.* *Proof.* We show that $f$ is continuous and injective, which shows that the image of $f$ is a Jordan curve. One immediately checks from the formula that $f$ is well-defined and continuous (actually, piecewise smooth) when restricted to $\mathbb{R}\setminus\{0\}$. Observe moreover that, if $|t|\in [2^n,2^{n+1}]$, then $|f(t)|\in [2^n(1-\epsilon),2^{n+1}(1-\epsilon)]$. This immediately implies that $f$ is continuous at $t=0$ and $t=\infty$. To check that $f$ is injective, we have already observed that $f$ maps $(0,+\infty)$ to the open right half-space and $(-\infty,0)$ to the open left half-space. Hence, together with [\[symmetry f\]](#symmetry f){reference-type="eqref" reference="symmetry f"}, it suffices to check that $f$ is injective on $(0,+\infty)$. By the condition on $|f(t)|$ in the previous paragraph and [\[equivariance f\]](#equivariance f){reference-type="eqref" reference="equivariance f"} it is actually sufficient to check that $f$ is injective on $[1,2]$, which is an immediate consequence of the construction. Let us denote by $\Lambda$ the image of $f$. We now show that $\Lambda$ is a quasicircle, by using the following characterization of quasicircles given in [@Ahl66 Chapter IV, Theorem 5]. Let $\Lambda$ be a Jordan curve in $\mathbb{C}\mathrm P^1=\mathbb{C}\cup\{\infty\}$ going through $\infty$. Then $\Lambda$ is a quasicircle if and only if there exists a constant $C>0$ such that, for every $z_1,z_2\in\Lambda\cap \mathbb{C}$ and every $z_3$ in the arc of $\Lambda\cap \mathbb{C}$ with endpoints $z_1,z_2$, $$\label{eq ahlfors} |z_3-z_1|\leq C|z_2-z_1|~.$$ To check that [\[eq ahlfors\]](#eq ahlfors){reference-type="eqref" reference="eq ahlfors"} holds for the image of $f$, first observe that the condition [\[eq ahlfors\]](#eq ahlfors){reference-type="eqref" reference="eq ahlfors"} is scale-invariant. Hence, since $\Lambda$ is invariant by the transformations $\alpha^n(z)=2^nz$, we can assume that $\max\{|z_1|,|z_2|\}\in [1/2,1]$. By definition of $f$, the arc of $\Lambda\cap \mathbb{C}$ with endpoints $z_1,z_2$ is then contained in the Euclidean ball $B(0,1)$. It follows that it is enough to check [\[eq ahlfors\]](#eq ahlfors){reference-type="eqref" reference="eq ahlfors"} for $|z_2-z_1|\leq \epsilon_0$, for any fixed $\epsilon_0>0$. Indeed, if $|z_2-z_1|\geq \epsilon_0$, since by the triangle inequality $|z_3-z_1|\leq 2$, we have $|z_3-z_1|\leq (2/\epsilon_0)|z_2-z_1|$. In conclusion, it is sufficient to check that [\[eq ahlfors\]](#eq ahlfors){reference-type="eqref" reference="eq ahlfors"} holds when either $z_1$ or $z_2$ has modulus in $[1/2,1]$ and $|z_2-z_1|\leq \epsilon_0$. This is immediately verified since $\Lambda\cap \{1/2-\epsilon_0\leq |z|\leq 1+\epsilon_0\}$ is a union of finitely many straight line segments or arcs of circles meeting orthogonally at the adjacency points. ◻ *Remark 27*. We remark that $\Lambda$ is not a symmetric quasicircle. Indeed, as a consequence of the invariance under the scaling $\alpha(z)=2z$, $\Lambda$ cannot satisfy the so-called strong reverse triangle property in a neighbourhood of $0$, see [@GS92 Section 6]. Now, in order to fix the suitable values of $\epsilon$ and $\delta$, we need to introduce minimal catenoids. A *minimal catenoid* is a minimal surface in $\mathbb{H}^3$, homeomorphic to an open annulus, whose asymptotic boundary is the union of two disjoint circles $C$ and $C'$ in $\partial_\infty\mathbb{H}^3=\mathbb{C}\mathrm P^1$, and which is obtained as a surface of revolution with respect to the unique geodesic meeting orthogonally both totally geodesic planes $P$ and $P'$ with $\partial_\infty P=C$ and $\partial_\infty P'=C'$. See Figure [2](#fig:catenoid){reference-type="ref" reference="fig:catenoid"}. {#fig:catenoid width=".4\\textwidth"} We will use the following existence result: **Theorem 28** ([@MG87]). *There exists a constant $d>0$ such that, if $C$ and $C'$ are disjoint circles in $\partial_\infty\mathbb{H}^3=\mathbb{C}\mathrm P^1$ and the distance between the totally geodesic planes $P$ and $P'$ that they span is less than $d$, then there exists a minimal catenoid with asymptotic boundary equal to $C\cup C'$.* Now, let us define the following circles in $\mathbb{C}\mathrm P^1=\mathbb{C}\cup\{\infty\}$: $$\begin{aligned} C_{n,0}&=\left\{z\,:\,\left| z-2^n\left(1-{2\epsilon}-\frac{1+\epsilon}{100}\right)\right|=\frac{2^n}{100}\right\} \\ C'_{n,0}&=\left\{z\,:\,\left| z-2^n\left(1+{2\epsilon}+\frac{1+\epsilon}{100}\right)\right|=\frac{2^n}{100}\right\} \\ C_{n,1}&=\{z\,:\,|z-2^n e^{i\left(\frac{\pi}{2}-\delta-(1+\delta)\epsilon\right)}|={\epsilon}\} \\ C'_{n,1}&=\{z\,:\,|z-2^n e^{i\left(\frac{\pi}{2}+\delta+(1+\delta)\epsilon\right)}|=\epsilon\} \\\end{aligned}$$ See Figure [3](#fig:circles){reference-type="ref" reference="fig:circles"}. Observe that $$\label{equivariance circles} C_{n,0}=\alpha^nC_{0,0}\qquad C'_{n,0}=\alpha^nC'_{0,0}\qquad C_{n,1}=\alpha^nC_{0,1}\qquad C'_{n,1}=\alpha^nC'_{0,1}~.$$ and that $C_{n,0}$, $C_{n,0}'$, $C_{n,1}$ and $C_{n,1}'$ are all in the complement of the image of $f$. {#fig:circles width=".6\\textwidth"} **Lemma 29**. *There exists constants $\epsilon,\delta>0$ such that for all $n\in\mathbb Z$ and all $j\in\{0,1\}$, $C_{n,j}\cup C_{n,j}'$ is the asymptotic boundary of a minimal catenoid.* *Proof.* By [\[equivariance circles\]](#equivariance circles){reference-type="eqref" reference="equivariance circles"} and the fact that $\alpha$ is the boundary value of an isometry of $\mathbb{H}^3$ in the upper half-space model, it suffices to prove the statement for $n=0$. We shall first choose $\epsilon$ in such a way that $C_{0,0}\cup C_{0,0}'$ bounds a minimal catenoid. For this, it suffices to observe that $C_{0,0}$ and $C_{0,0}'$ are circles of radii $1/100$, and the Euclidean distance between their centers is $2(2\epsilon+(1+\epsilon)/100)=2/100+402\epsilon/100$. As $\epsilon\to 0$, the circles $C_{0,0}$ and $C_{0,0}'$ tend to a pair of tangent circles, each with radius $1/100$. Since the hyperbolic distance varies continuously, and the hyperbolic distance between two totally geodesic planes tangent at infinity equals zero, the distance between the totally geodesic planes bounded by $C_{0,0}$ and $C_{0,0}'$ tends to zero. Hence by Theorem [Theorem 28](#existence min catenoids){reference-type="ref" reference="existence min catenoids"}, when $\epsilon$ is sufficiently small, there exists a minimal catenoid with asymptotic boundary $C_{0,0}\cup C_{0,0}'$. We shall now keep such $\epsilon$ fixed, and choose $\delta$ in such a way that $C_{0,1}\cup C_{0,1}'$ bounds a minimal catenoid. The reasoning is completely analogous to the previous case. Indeed, the circles $C_{0,1}$ and $C_{0,1}'$ have radius $\epsilon$ and the distance between their centers is bounded above by $2\epsilon+2\delta(1+\epsilon)$. Hence $C_{0,1}$ and $C_{0,1}'$ tend to be tangent when $\delta\to 0$, and we conclude again by Theorem [Theorem 28](#existence min catenoids){reference-type="ref" reference="existence min catenoids"}. ◻ In the rest of the section, we will denote by $\mathrm{Cat}_{n,j}$ the minimal catenoid whose asymptotic boundary is $C_{n,j}\cup C_{n,j}'$, and by $\mathrm{Cat}^{\mathrm{solid}}_{n,j}$ the closure in $\mathbb{H}^3$ of the connected component of $\mathbb{H}^3\setminus \mathrm{Cat}_{n,j}$ that contains the totally geodesic planes spanned by $C_{n,j}$ and $C_{n,j}'$. ## Topological disks We will now need to construct topological disks with asymptotic boundary the Jordan curve $\Lambda=f(\mathbb{R}\mathbb P^1)$, contained in the complement of suitably chosen solid catenoids. The choice of these solid catenoids will be encoded in a function $\iota:\mathbb{Z}\to\{0,1\}$. Each such topological disk will be the starting point to construct pairwise distinct *minimal* disks. **Lemma 30**. *For every $\iota:\mathbb{Z}\to\{0,1\}$, there exists a continuous injective map $F_\iota$ from the upper half-plane $\mathbb H^2$ to $\mathbb{H}^3$, whose image is contained in $$\mathbb{H}^3\setminus\bigcup_{n\in\mathbb{Z}}\mathrm{Cat}^{\mathrm{solid}}_{n,\iota(n)}$$ and which extends continuously to $f:\mathbb{R}\cup\{\infty\}=\partial_\infty\mathbb H^2\to \mathbb{C}\cup\{\infty\}=\partial_\infty\mathbb{H}^3$.* *Proof.* We will divide the construction in several steps. We fix $n\in\mathbb{Z}$, and we will first define $F_\iota$ on the strip $\{z\in\mathbb H^2\,:\,2^n\leq|z|\leq 2^n(1+3/4)\}$ in the upper half-plane. We will distinguish two cases here, according to whether $\iota(n)=0$ or $\iota(n)=1$. Then we will define $F_\iota$ on the strip $\{z\in\mathbb H^2\,:\,2^n(1+3/4)\leq|z|\leq 2^{n+1}\}$ in a way which is independent of $\iota(n)$. Suppose first that $\iota(n)=1$. We divide the strip $\{2^n\leq|z|\leq 2^n(1+3/4)\}$ in three portions, namely $\{2^n\leq|z|\leq 2^n(1+1/4)\}$, $\{2^n(1+1/4)\leq|z|\leq 2^n(1+1/2)\}$ and $\{2^n(1+1/2)\leq|z|\leq 2^n(1+3/4)\}$, and define $F_\iota$ on each of them in such a way that it extends to $f_\iota$ on the corresponding interval of the real axis. Moreover, the image of $F_\iota$ will have to avoid the solid catenoid $\mathrm{Cat}^{\mathrm{solid}}_{n,1}$. The idea is to map $\{2^n\leq|z|\leq 2^n(1+1/4)\}$ to the convex hulls of the arcs of circles $f_\iota([2^n,2^n(1+1/4)])$ and $f_\iota([-2^n(1+1/4),-2^n])$, and similarly $\{2^n(1+1/2)\leq|z|\leq 2^n(1+3/4)\}$ to the convex hull of $f_\iota([2^n(1+1/2),2^n(1+3/4)])$ and $f_\iota([-2^n(1+3/4),-2^n(1+1/2)])$ --- these convex hulls are portions of totally geodesic planes --- and then connect them by a band connecting $f_\iota([2^n(1+1/4),2^n(1+1/2)])$ and $f_\iota([-2^n(1+1/2),-2^n(1+1/4)])$. See Figure [4](#fig:disks1){reference-type="ref" reference="fig:disks1"}. {#fig:disks1 width=".9\\textwidth"} Formally, let us use the upper half-space model $\{(z,h)\,:\,z\in\mathbb{C},h>0\}$ of $\mathbb{H}^3$. As $t$ varies in $[2^n,2^n(1+1/4)]$, let $\gamma(t)$ parameterize the curve in $\mathbb{H}^3$ (actually, a geodesic, although not parameterized by arclength) contained in the vertical plane $\Re(z)=0$, in such a way that $\lim_{x\to\pm\infty}\exp_{\gamma(t)}(xv_{\gamma(t)})=f_\iota(\pm t)$, where $v_p$ is the vector based at $p$ and orthogonal to the vertical plane $\Re(z)=0$, pointing towards $\Re(z)>0$. Then define $$F_\iota(ite^{is})=\exp_{\gamma(t)}(\chi(s)v_{\gamma(t)})$$ for a fixed decreasing diffeomorphism $\chi:[-\pi/2,\pi/2]\to\mathbb{R}$. By construction, $F_\iota$ extends to $f_\iota$ on $\pm[2^n,2^n(1+1/4)]$. Starting from $t\in [2^n(1+1/2),2^n(1+3/4)]$, we repeat exactly the same procedure to define $F_\iota$ on $\{2^n(1+1/2)\leq|z|\leq 2^n(1+3/4)\}$, thus obtaining another portion of totally geodesic plane. We repeat the same again to define $F_\iota$ on $\{2^n(1+1/4)\leq|z|\leq 2^n(1+1/2)\}$, simply by changing the interval of definition of $t$, with the sole irrelevant difference that in this case the image of $\gamma$ is not a geodesic. Putting together the three pieces, $F_\iota$, defined on $\{2^n\leq|z|\leq 2^n(1+3/4)\}$, is continuous and piecewise smooth. Since the minimal catenoids are contained in the convex hull of their two asymptotic circles, it is immediate to see that the image of $F_\iota$ is disjoint from the solid catenoid $\mathrm{Cat}^{\mathrm{solid}}_{n,1}$. Having described this construction in detail when $\iota(n)=1$, we now explain how to modify it for $\iota(n)=0$. Here we divide again the strip $A:=\{2^n\leq|z|\leq 2^{n}(1+3/4)\}$ in three parts, but differently from above. Let $A_\pm=A\cap \{|z\mp 2^n(1+3/8)|=2^n(3/8)\}$, and let $A_0$ be the closure of $A\setminus (A_-\cup A_+)$. We can use again an auxiliary curve $\gamma:[-\pi/2,\pi/2]$ in $\mathbb{H}^3$, having the property of being contained in the totally geodesic plane of symmetry between the two totally geodesic planes with asymptotic boundaries $\{|z|=2^n(1\pm \epsilon)\}$, such that $\gamma(\pm\pi/2)=f_\iota (\pm 2^n(1+3/8))$, and define $F_\iota(ie^s)=\gamma(s)$. Then we "fill\" the strip $A$ in such a way that $F_\iota(A_\pm)$ is sweeped-out by the geodesics with endpoints $f_\iota(\pm2^n(1+3/8+t))$, for $t\in 2^n(-3/8,3/8)$, and that $F_\iota(A_0)$ is sweeped-out by finite geodesic segments, in such a way that $F_\iota(2^nie^s)=\exp_p(\chi(s)v)$ parameterizes, as in the previous case, a geodesic with endpoints $f_\iota(\pm 2^n)$, for $p$ a point in the vertical plane $\Re(z)=0$, and analogously for $F_\iota(2^n(1+3/4)ie^s)$, on the other boundary half-circle of $A$. The convex hull property will then ensure again that the image of $F_\iota$ is in the complement of $\mathrm{Cat}^{\mathrm{solid}}_{n,0}$. See Figure [5](#fig:disks2){reference-type="ref" reference="fig:disks2"}. We leave to the reader the details, which are only slightly more technical than in the case $\iota(n)=1$. {#fig:disks2 width=".9\\textwidth"} Finally, it remains to define $F_\iota$ on the strip $\{2^n(1+3/4)\leq|z|\leq 2^{n+1}\}$. Here we take again a curve $\gamma:[2^n(1+3/4),2^{n+1}]\to\mathbb{H}^3$ with the property that its image is contained in $\Re(z)=0$ and that $\lim_{x\to\pm\infty}\exp_{\gamma(t)}(xv_{\gamma(t)})=f_\iota(\pm t)$. We can then define $F_\iota(ite^{is})=\exp_{\gamma(t)}(\chi(s)v_{\gamma(t)})$. This definition will match the previous definitions of $F_\iota$ on the common boundaries $|z|=2^n(1+3/4)$ or $|z|=2^{n+1}$. See Figure [6](#fig:disks3){reference-type="ref" reference="fig:disks3"}. {#fig:disks3 width=".9\\textwidth"} Hence $F_\iota$ is globally well-defined, continuous and piecewise smooth. As in the proof of Lemma [Lemma 26](#lemma quasicircle){reference-type="ref" reference="lemma quasicircle"}, one can observe that the image of $2^n\leq |z|\leq 2^{n+1}$ under the map $F_\iota$ is contained, in the upper half-space, in $B(0,2^{n+1}(1+\epsilon))\setminus B(0,2^{n}(1+\epsilon))$. Using this property, following the same lines as the proof of Lemma [Lemma 26](#lemma quasicircle){reference-type="ref" reference="lemma quasicircle"} it is easy to verify that $F_\iota$ is injective and $\lim_{z\to 0}F_\iota(z)=0$, $\lim_{z\to \infty}F_\iota(z)=\infty$. By construction $F_\iota$ extends to $f_\iota$ on $\mathbb{R}$, and therefore $F_\iota$ is the desired continuous and injective extension of $f_\iota$. ◻ ## Conclusion of the proof We are now ready to prove Theorem [Theorem 6](#thm:uncountably){reference-type="ref" reference="thm:uncountably"}. *Proof of Theorem [Theorem 6](#thm:uncountably){reference-type="ref" reference="thm:uncountably"}.* Let $\Lambda$ be the image of the continuous injective map $f$, and fix $\iota:\mathbb{Z}\to\{0,1\}$. We will construct a stable minimal disk $\Sigma_\iota$ contained in $$\Omega_\iota:=\mathbb{H}^3\setminus\bigcup_{n\in\mathbb{Z}}\mathrm{Cat}^{\mathrm{solid}}_{n,\iota(n)}$$ with $\partial_\infty\Sigma_\iota=\Lambda$. Let us denote by $U_\iota$ the topological disk constructed in Lemma [Lemma 30](#lemma topo disk){reference-type="ref" reference="lemma topo disk"}, namely the image of the continuous injective map $F_\iota:\mathbb{D}\to\mathbb{H}^3$, where we identify here the upper half-plane model of $\mathbb H^2$ with the unit disc $\mathbb{D}$. We will denote by $\mathcal{C}(U_\iota)$ the convex hull of $\Lambda\cup U_\iota$ in $\mathbb{H}^3\cup\partial_\infty\mathbb{H}^3$. As a preliminary step, observe that $$\label{eq:convex hull topo disk} \partial_\infty \mathcal{C}(U_\iota)=\Lambda~.$$ Indeed, given any point $p$ in $\partial_\infty\mathbb{H}^3\setminus\Lambda$, since the map $F_\iota$ extends to a continuous injective map from $\overline \mathbb{D}$ to $\mathbb{H}^3\cup\partial_\infty\mathbb{H}^3$, $U_\iota\cup\Lambda$ is a closed subset of $\mathbb{H}^3\cup\partial_\infty\mathbb{H}^3$, and one can find a neighbourhood $\mathcal U$ of $p$ in the topology of the closed ball which is disjoint from $U_\iota\cup\Lambda$. Then a totally geodesic plane $P$ contained in $\mathcal U$ disconnects $p$ from the convex hull of $\Lambda\cup U_\iota$, thus showing $p\notin \mathcal{C}(U_\iota)$. Now, let $\gamma_n=F_\iota(\{|z|=1-1/n\})$. Then $\{\gamma_n\}_{n\in\mathbb{N}}$ is a sequence of nullhomotopic simple closed curves in $\mathcal{C}(U_\iota)$ that converge to $\Lambda$ as $n\to+\infty$. The proof of the existence of $\Sigma_\iota$ now roughly follows the strategy of Lemma [Lemma 17](#lemma Plateau){reference-type="ref" reference="lemma Plateau"}. Fix the point $x_o=F_\iota(0)\in U_\iota$. For any $n$, let $R_n$ be a sufficiently large radius so that $\gamma_n\subset B(x_o,R_n)$. Up to a small perturbation of $R_n$, assume moreover, as a consequence of genericity of transverse intersection, that the intersection of $\partial B(x_o,R_n)$ and $\partial\Omega_\iota$ is transverse. By [@hassscott Theorem 6.3] applied to $M_n:=\Omega_\iota\cap B(x_o,R_n)$, which is sufficiently convex in the terminology of [@hassscott], we can solve the Plateau problem to find a least area (hence stable) minimal disk $\Sigma_\iota^n$ contained in $M_n$ with $\partial\Sigma_\iota^n=\gamma_n$. We will show that $\Sigma_\iota^n$ converges, up to extracting a subsequence, smoothly on compact subsets of $\mathbb{H}^3$ to a properly embedded minimal disk $\Sigma_\iota$ contained in $\Omega_\iota$. We first observe that there exists $R_0>0$ such that $\Sigma_\iota^n\cap B(x_o,R_0)\neq\emptyset$ for all $n$. To see this, recall that $\mathcal{C}(U_\iota)$ is homeomorphic to a closed ball, hence $\Lambda$ disconnects $\partial \mathcal{C}(U_\iota)$, which is homeomorphic to a sphere, into two topological disks which we denote $\partial^\pm\mathcal{C}(U_\iota)$. Pick a geodesic segment $\eta$ connecting two points $x_\pm\in \partial^\pm\mathcal{C}(U_\iota)$ and containing $x_o$. Since $\gamma_n$ and $\eta$ are contained in $\mathcal{C}(U_\iota)$ and linked, and the disk $\Sigma_\iota^n$ is contained in $\mathcal{C}(U_\iota)$, it must intersect $\eta$. Hence $\Sigma_\iota^n$ intersects $B(x_o,R_0)$ for any choice of $R_0$ such that $B(x_o,R_0)$ contains $\eta$. Now, fix any radius $R>0$ and consider the ball $B(x_o,R)$. For $n=n(R)$ sufficiently large, $\gamma_n$ is contained in $\mathbb{H}^3\setminus B(x_o,R+1)$. The distance from any point of $\Sigma_\iota^n\cap B(x_o,R)$ and $\partial\Sigma_\iota^n=\gamma_n$ is thus bounded below by $1$. By Theorem [\[thm:rosenberg\]](#thm:rosenberg){reference-type="ref" reference="thm:rosenberg"} there is a constant $C$, independent of $n$, such that the Gaussian curvature of $\Sigma_\iota^n$ is bounded by $C$ in absolute value. Applying Theorem [Theorem 13](#thm compactness bounded){reference-type="ref" reference="thm compactness bounded"}, we can extract a subsequence $n_k$ such that $\Sigma_\iota^{n_k}\cap B(x_o,R)$ converges smoothly to a minimal disc. By a diagonal argument, we can then find a subsequence of $\{\Sigma_\iota^n\}_{n\in\mathbb{N}}$ that converges smoothly on compact subsets to a minimal disk $\Sigma_\iota$. Moreover the limit $\Sigma_\iota$ is stable since it is the smooth limit of the stable minimal surfaces $\Sigma_\iota^n$. An easy argument by contradiction as in the proof of Corollary [Corollary 14](#cor compactness unbounded){reference-type="ref" reference="cor compactness unbounded"} shows that $\Sigma_\iota$ is properly embedded. Moreover, since $\Sigma_\iota^n$ is contained in $\Omega_\iota$, then $\Sigma_\iota$ is contained in the closure of $\Omega_\iota$. This would be sufficient for our purpose; nonetheless the strong maximum principle actually implies that $\Sigma_\iota$ is contained in $\Omega_\iota$ itself. Finally, since $\Sigma_\iota^n$ is contained in $\mathcal{C}(U_\iota)$ for all $n$, so too is $\Sigma_\iota$. Then [\[eq:convex hull topo disk\]](#eq:convex hull topo disk){reference-type="eqref" reference="eq:convex hull topo disk"} implies that $\partial_\infty\Sigma_{\iota}=\Lambda$. It only remains to show that, if $\iota_0\neq \iota_1$, then $\Sigma_{\iota_0}\neq \Sigma_{\iota_1}$. By hypothesis, there exists $m\in\mathbb{Z}$ such that $\iota_0(m)\neq \iota_1(m)$. Up to switching the roles, suppose $\iota_0(m)=0$ and $\iota_1(m)=1$. Then, by construction, $\Sigma_{\iota_0}$ is contained in the complement of $\mathrm{Cat}^{\mathrm{solid}}_{m,0}$, whereas $\Sigma_{\iota_1}$ is contained in the complement of $\mathrm{Cat}^{\mathrm{solid}}_{m,1}$. But the solid catenoids $\mathrm{Cat}^{\mathrm{solid}}_{m,0}$ and $\mathrm{Cat}^{\mathrm{solid}}_{m,1}$ are linked in the sense of [@HW15 Section 3.4]. Then [@HW15 Lemma 3.12] implies that $\Lambda$ is not nullhomotopic in $(\mathbb{H}^3\cup\partial_\infty\mathbb{H}^3)\setminus(\mathrm{Cat}^{\mathrm{solid}}_{m,0}\cup\mathrm{Cat}^{\mathrm{solid}}_{m,1})$. Now, if $\Sigma_{\iota_0}= \Sigma_{\iota_1}=:\Sigma$, then $\Sigma$ would be a disk contained in the complement of $\mathrm{Cat}^{\mathrm{solid}}_{m,0}\cup\mathrm{Cat}^{\mathrm{solid}}_{m,1}$, hence contradicting that $\Lambda$ is not nullhomotopic. This concludes the proof. ◻ # Small curvatures {#appendix} In this appendix, we explain how to obtain Theorems [Theorem 20](#thm:weakly small is properly emb){reference-type="ref" reference="thm:weakly small is properly emb"} and [Theorem 21](#exponential diffeo){reference-type="ref" reference="exponential diffeo"}. These results are well-known under the assumption that $\Sigma$ has small curvatures (see for instance [@Eps84]). However, we need to relax the hypothesis here to assume that $\Sigma$ has only weakly small curvatures. We follow here the approach of [@ES22 Section 4]. The fundamental observation is that, if $\Sigma$ is a hypersurface in $\mathbb{H}^{n+1}$ of weakly small curvatures and $\gamma:I\to\Sigma$ is a geodesic for the first fundamental form of $\Sigma$, then the acceleration of $\gamma$ is $$\label{eq:acceleration} \nabla^h_{\gamma'(t)}\gamma'(t)=A(\gamma'(t),\gamma'(t))N(\gamma(t))$$ and therefore, by the weakly small curvature assumption, $$\label{eq:norm acceleration} \|\nabla^h_{\gamma'(t)}\gamma'(t)\|\leq 1~.$$ This leads us to study curves satisfying the bound [\[eq:norm acceleration\]](#eq:norm acceleration){reference-type="eqref" reference="eq:norm acceleration"} on the acceleration. Let $S$ be a horosphere or a sphere in $\mathbb{H}^{n+1}$. (By sphere, here we always mean the set of points at distance $r>0$ from a given points of $\mathbb{H}^{n+1}$.) The complement of $S$ has two connected components, one of which is geodesically convex, and the other is not. We call the former the (open) convex side of $S$, and the latter the (open) concave side of $S$. The closed convex (resp. concave) side of $S$ is the union of $S$ with the open convex (resp. concave) side. The following lemma is an improvement of [@ES22 Lemma 4.9], following a similar strategy of proof. **Lemma 31**. *Let $\gamma:[a,b]\to\mathbb{H}^{n+1}$ be a smooth curve satisfying [\[eq:norm acceleration\]](#eq:norm acceleration){reference-type="eqref" reference="eq:norm acceleration"}. Then* - *The image of $\gamma$ lies in the open concave side of any sphere tangent to $\gamma$, except for the tangency point.* - *The image of $\gamma$ lies in the closed concave side of any horosphere tangent to $\gamma$.* *Proof.* It suffices to prove the first item, as the second item follows by taking the limit of tangent spheres as the radius goes to infinity. Assume that the tangency point is, in the upper half-space model $\gamma(0)=(0,\ldots,0,1)$, that $\gamma$ is parameterized by arclength, that $\gamma'(0)=(1,0,\ldots,0)$ and that the tangent horosphere is $\{x_n=1\}$. Hence the tangent spheres $S_r$ of radius $r$ are, in a neighbourhood of $\gamma(0)$, graphs of functions $f_r:U_r\to\mathbb{R}$, where $U_r\subset\mathbb{R}^n$ is a sufficiently small ball, and by symmetry $f_r$ only depends on the distance from the origin in $\mathbb{R}^n$. First we show $\gamma$ lies on the open concave side of any tangent sphere $S_r$ for small $t$. We use a subscript $(\cdot)_{n+1}$ to denote the last coordinate in the upper half-space model. Thus $\gamma_{n+1}(0)=\gamma'_{n+1}(0)=0$. By a direct computation of the Christoffel symbol $\Gamma_{11}^n=1$, we see that $$(\nabla^h_{\gamma'}\gamma')_{n+1}(0)=\gamma_{n+1}''(0)+1~.$$ Since $\gamma$ has small acceleration, we obtain $$\gamma_{n+1}''(0)\leq |(\nabla^h_{\gamma'}\gamma')_{n+1}(0)|-1\leq \|(\nabla^h_{\gamma'}\gamma')(0)\|-1\leq 0~.$$ Now, denote $\hat\gamma$ the projection of $\gamma$ to $\mathbb{R}^n$. By an easy computation following similar lines, $(f_r\circ\hat\gamma)(0)=(f_r\circ\hat\gamma)'(0)=0$ and $(f_r\circ\hat\gamma)''(0)=-1+1/\tanh(r)>0$. Hence for $t\in(0,\epsilon)$, $\gamma_{n+1}(t)<f_r(t)$ and this shows that $\gamma$ stays for small times in the open concave side of $S_r$. It remains to show that $\gamma(t)$ stays in in the open concave side of $S_r$ for every $t>0$. Suppose by contradiction that for some $t_0$, $\gamma(t_0)\in S_r$. Let $p$ be the center of $S_r$. Then the function $t\mapsto d(p,\gamma(t))$ is equal to $r$ for $t=0$ and $t=t_0$, and is larger than $r$ for $t\in (0,\epsilon)$. Hence it admits a maximum point $t_{\max}$. This implies that $\gamma$ is tangent to the geodesic sphere centered at $p$ of radius $d(p,\gamma(t_{\max}))$ and thus contradicts the first part of the proof. ◻ Having established Lemma [Lemma 31](#lemma:curve concave side){reference-type="ref" reference="lemma:curve concave side"}, the following lemma follows immediately. **Lemma 32**. *Let $\Sigma\subset\mathbb{H}^{n+1}$ be a properly immersed hypersurface of weakly small curvatures. Then $\Sigma$ is contained in the closed concave side of every tangent horosphere.* *Proof.* Let $p\in\Sigma$ and let $S$ be a tangent horosphere. Given any other $q$, by completeness of $\Sigma$, we can find a geodesic segment $\gamma$ joining $p$ and $q$. As observed in [\[eq:norm acceleration\]](#eq:norm acceleration){reference-type="eqref" reference="eq:norm acceleration"}, $\gamma$ satisfies the hypothesis of Lemma [Lemma 31](#lemma:curve concave side){reference-type="ref" reference="lemma:curve concave side"}, hence $q$ is contained in the closed concave side of $S$. ◻ Now, the proof of Theorem [Theorem 20](#thm:weakly small is properly emb){reference-type="ref" reference="thm:weakly small is properly emb"} follows exactly the proof of [@ES22 Proposition 4.15]. To prove Theorem [Theorem 21](#exponential diffeo){reference-type="ref" reference="exponential diffeo"}, we need the following well-known result, which follows from a direct computation (see for instance [@HW13 formula $(2.2)$] or [@ES22 Lemma 4.5]). **Lemma 33**. *Let $\Sigma$ be an embedded hypersurface in $\mathbb{H}^{n+1}$ of weakly small curvatures, and let $N$ be its unit normal vector. Then for every $t\in\mathbb{R}$, the map $\iota_t:\Sigma\to\mathbb{H}^{n+1}$ defined by $\iota_t(x)=\exp_x(tN(x))$ is an immersion, and its principal curvatures satisfy the identity $$\label{nfcurv} \lambda^t_i(\iota_t(x)) = {\frac{\lambda_i(x)-\tanh(t)}{1-\lambda_i(x)\tanh(t)}}~,$$ where $\lambda_1,\ldots,\lambda_n$ are the principal curvatures of $\Sigma$.* *Proof of Theorem [Theorem 21](#exponential diffeo){reference-type="ref" reference="exponential diffeo"}.* It follows from Lemma [Lemma 33](#lemma principal curvatures){reference-type="ref" reference="lemma principal curvatures"} that the map $F(x,t)=\iota_t(x)$ has injective differential, hence is a local diffeomorphism. We show that it is bijective. To show that $F$ is surjective, let $p$ be any point in $\mathbb{H}^{n+1}$, let $r=d(p,\Sigma)$ and let $x\in\Sigma$ be a point that realizes this distance. Then the geodesic sphere of center $p$ and radius $r$ is tangent to $\Sigma$ at $x$, and $p=F(x,r)$. For injectivity, suppose that $p=F(x_1,t_1)=F(x_2,t_2)$. Let $S_{t_i}$ be the geodesic spheres centered at $p$ of radius $t_i$. As before, $S_{t_i}$ is tangent to $\Sigma$ at $x_i$. By Lemma [Lemma 31](#lemma:curve concave side){reference-type="ref" reference="lemma:curve concave side"}, $\Sigma$ is contained, except for the tangency points, in the open concave side of both $S_{t_1}$ and $S_{t_2}$. This implies that $t_1=t_2$, because otherwise the open concave side of $S_{t_1}$ would intersect the convex side of $S_{t_2}$ (or vice versa), leading to a contradiction. By Lemma [Lemma 31](#lemma:curve concave side){reference-type="ref" reference="lemma:curve concave side"} again, $\Sigma\setminus\{x_1\}$ is contained in the open concave side of $S_{t_1}=S_{t_2}$, and thus $x_2=x_1$. We have thus shown that each $\iota_t$ is a proper embedding, and it has weakly small curvatures by Lemma [Lemma 33](#lemma principal curvatures){reference-type="ref" reference="lemma principal curvatures"}. When $\Sigma$ is minimal, Lemma [Lemma 33](#lemma principal curvatures){reference-type="ref" reference="lemma principal curvatures"} again implies the last statement on the sign of the mean curvature of $\iota_t$, because $\lambda_i^t$ is a decreasing function of $t$. ◻ We have already observed that, for $n=2$, if the Jordan curve $\Lambda$ spans a surface of strongly small curvatures, then $\Lambda$ is a quasicircle, and this in turns implies that $\Lambda$ has finite width. We conclude this appendix by a direct proof, that works in any dimensions. **Lemma 34**. *Let $\Lambda$ be a topologically embedded $n$-sphere $\Lambda$ in $\mathbb{S}_\infty^n$ and $\Sigma$ be a properly embedded hypersurface of strongly small curvatures such that $\partial_\infty\Sigma=\Lambda$. Then $\mathcal{C}(\Lambda)$ has finite width.* *Proof.* Assume that the principal curvatures $\lambda_1,\ldots,\lambda_n$ of $\Sigma$ are smaller than $1-\epsilon$ in absolute value. By Lemma [\[nfcurv\]](#nfcurv){reference-type="ref" reference="nfcurv"}, for $t_0 > \tanh^{-1}(1-\epsilon)$ the hypersurface $\Sigma_{t_0}$ (resp. $\Sigma_{-t_0}$) has negative (resp. positive) principal curvatures, that is, it is locally concave (resp. convex) with respect to the direction of the normal vector $N$. Since $\Sigma_{\pm t_0}$ are properly embedded, they bound a geodesically convex region $U$, which thus contains the $\mathcal{C}(\Lambda)$. By Theorem [Theorem 21](#exponential diffeo){reference-type="ref" reference="exponential diffeo"}, every point in $\mathcal{C}(\Lambda)$ lies on a geodesic segment of the form $t\mapsto F(x,t)$, for $t\in [-t_0,t_0]$. Hence for every $x\in \mathcal{C}(\Lambda)$, $d(x,\partial^{+}\mathcal{C}(\Lambda))+d(x,\partial^{-}\mathcal{C}(\Lambda)\leq 2t_0$. This shows that $w(\Lambda)\leq 2t_0$. ◻

arxiv_math

--- abstract: | Inspired by the construction of the Föllmer process [@follmer1988random], we construct a unit-time flow on the Euclidean space, termed the Föllmer flow, whose flow map at time 1 pushes forward a standard Gaussian measure onto a general target measure. We study the well-posedness of the Föllmer flow and establish the Lipschitz property of the flow map at time 1. We apply the Lipschitz mapping to several rich classes of probability measures on deriving dimension-free functional inequalities and concentration inequalities for the empirical measure. address: - School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China - Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China - Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China - School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China - School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China - School of Data Science, Chinese University of Hong Kong, Shenzhen, China author: - Yin Dai - Yuan Gao - Jian Huang - Yuling Jiao - Lican Kang - Jin Liu bibliography: - Follmer_Flow.bib title: Lipschitz Transport Maps via the Föllmer Flow --- [^1] # Introduction Functional inequalities, such as Poincaré-type and Sobolev-type inequalities, are fundamental tools in studying sampling algorithms [@chewi2020exponential; @li2020riemannian; @lehec2021langevin; @chewi2022analysis; @lu2022explicit; @cattiaux2022functional; @andrieu2022comparison; @andrieu2022poincar], stochastic optimization [@raginsky2017non; @xu2018global; @kinoshita2022improved; @li2022sharp], and score-based generative modeling [@block2020generative; @lee2022convergence; @koehler2022statistical; @wibisono2022convergence] in machine learning, statistics, and applied probability. Establishing functional inequalities with dimension-free constants has attracted widespread attention in various fields of mathematics like probability, geometry, and analysis [@wang2016functional; @bardet2018functional; @courtade2020bounds; @chen2021dimension; @cattiaux2022functional; @lee2018kannan; @chen2021almost; @klartag2022bourgain; @jambulapati2022slightly; @conforti2022weak; @klartag2023logarithmic]. The seminal work [@bakry1985diffusions] introduces the Bakry-Émery criterion to verify dimension-free Poincaré and log-Sobolev inequalities for strongly log-concave measures. Holley-Stroock perturbation principle [@holley1987logarithmic] implies any bounded perturbation of a strongly log-concave measure satisfies dimension-free Poincaré and log-Sobolev inequalities. Up to now, it remains an attractive topic to explore general classes of probability measures that satisfy dimension-free functional inequalities. One of the basic principles for proving functional inequalities regarding probability measures on the Euclidean space is Lipschitz changes of variables between a source and a target probability measures [@caffarelli2000monotonicity; @otto2000generalization; @bobkov2010perturbations; @kim2012generalization; @colombo2017lipschitz; @fathi2020proof; @klartag2021spectral; @mikulincer2021brownian; @mikulincer2022lipschitz; @neeman2022lipschitz; @chewi2022entropic; @shenfeld2022exact]. For instance, provided that $\mu$ and $\nu$ are Borel probability measures defined on $\mathbb R^d$, we would seek a Lipschitz transport map $\varphi: \mathbb R^d \rightarrow \mathbb R^d$ such that $\nu$ can be represented as a push-forward measure under $\varphi$, namely $\nu=\mu \circ \varphi^{-1}$. $\mu$ is transported onto $\nu$ in the sense that for every Borel set $B \subseteq \mathbb R^d$, $\nu(B)=\mu(\varphi^{-1}(B) )$. The Lipschitz nature of transport map $\varphi$ plays a profound impact on transferring desirable analytic results from the source measure $\mu$ to the target measure $\nu$. However, existence of such Lipschitz transport maps is not guaranteed unless proper convexity restrictions are placed on the measures. Our main goal is to extend quantitative Lipschitz regularity of transport maps to measures that are not necessarily strongly log-concave. Let us first recall the celebrated Caffarelli's contraction theorem [@caffarelli2000monotonicity Theorem 2]. Let $\mu (\mathrm{d}x) = \exp(-U(x)) \mathrm{d}x$ and $\nu (\mathrm{d}x) = \exp(-W(x)) \mathrm{d}x$ be two probability measures defined on $\mathbb R^d$ with $U, W \in C^2({\mathbb{R}}^d)$. Suppose that $\nabla^2 U(x) \preceq \beta \mathbf{I}_d$ and $\nabla^2 W(x) \succeq \alpha \mathbf{I}_d \succ 0$. Then the optimal transport map $\varphi_{\mathrm{opt}}:=\nabla \psi$ from $\mu$ to $\nu$ is $\sqrt{\beta/\alpha}$-Lipschitz, where $\psi: \mathbb R^d \rightarrow \mathbb{R}$ is the convex Brenier potential. The optimal transport map $\varphi_{\mathrm{opt}}: \mathbb R^d \rightarrow \mathbb R^d$ pushes forward $\mu$ onto $\nu$ in the sense that $\nu = \mu \circ ( \varphi_{\mathrm{opt}} )^{-1}$. In particular, if $\gamma_d$ is the standard Gaussian measure on $\mathbb R^d$ and probability measure $\mu$ has a log-concave density with respect to $\gamma_d$, then there exists a 1-Lipschitz map $\varphi_{\mathrm{opt}}$ such that $\nu = \gamma_d \circ \left(\varphi_{\mathrm{opt}} \right)^{-1}$. The 1-Lipschitz transport map $\varphi_{\mathrm{opt}}$ enables dimension-free functional inequalities to be transferred from $\gamma_d$ to $\nu$. Recently, [@mikulincer2021brownian] defines a Brownian transport map, based on the Föllmer process defined in Definition [Definition 10](#def:follmer-sde){reference-type="ref" reference="def:follmer-sde"}, that transports the infinite-dimensional Wiener measure onto probability measures on the Euclidean space. Lipschitz properties of the Brownian transport map are investigated extensively while no analogous results for optimal transport maps are known. [@neeman2022lipschitz] and [@mikulincer2022lipschitz] also utilize a Lipschitz transport map along the reverse heat flow, which previously appears in [@otto2000generalization] and is further studied by [@kim2012generalization], to establish functional inequalities, perform eigenvalues comparisons, and study domination of distribution functions. On the Caffarelli's contraction theorem, [@fathi2020proof] and [@chewi2022entropic] provide new proofs using the entropic interpolation between the source and the target measures. In this work, we construct a flow over the unit time interval on the Euclidean space, named the Föllmer flow as in Definition [Definition 3](#def:follmer-flow){reference-type="ref" reference="def:follmer-flow"} and Theorem [Theorem 4](#main-thm1){reference-type="ref" reference="main-thm1"}. Our construction is greatly enlightened by Föllmer's derivation of the Föllmer process. Then we define and analyze a new transport map, along the Föllmer flow, which pushes forward the standard Gaussian measure to a general measure satisfying mild regularity assumptions (see Assumptions [Assumption 1](#condn:well-defined){reference-type="ref" reference="condn:well-defined"}, [Assumption 2](#condn:semi-log-convex){reference-type="ref" reference="condn:semi-log-convex"} and [Assumption 3](#condn:semi-log-concave){reference-type="ref" reference="condn:semi-log-concave"}). The well-posedness of the Föllmer flow and the Lipschitz property of its flow map at time 1 are rigorously investigated under these regularity assumptions. By virtue of the Lipschitz changes of variables principle, we prove dimension-free $\Psi$-Sobolev inequalities, isoperimetric inequalities, $q$-Poincaré inequalities and sharp non-asymptotic concentration bounds for the empirical measure. Furthermore, we shall emphasize that both the Föllmer flow and its flow map possess much computational flexibility in terms of the analytic expression of its velocity field, which we believe may be of independent interest, to develop sampling algorithms and generative models with theoretical guarantees. ## Related work The work is notably relevant to the Brownian transport map built upon the Föllmer process [@mikulincer2021brownian] and the transport map defined via the reverse heat flow [@otto2000generalization; @kim2012generalization; @neeman2022lipschitz; @mikulincer2022lipschitz]. The Brownian transport map, acquired from a strong solution of the Föllmer process, pushes forward the Wiener measure onto probability measures on the Euclidean space. The infinite-dimensional nature of the Brownian transport map is quite different from that of the Föllmer flow which is defined on the finite-dimensional Euclidean space. To produce the randomness within the target measure, the Brownian transport map leverages the randomness of the path while the Föllmer flow makes use of the randomness delivered by the source measure. Meanwhile, [@mikulincer2022lipschitz] studies a transport map along the reverse heat flow from the standard Gaussian measure to a target measure, constructed by [@otto2000generalization] and [@kim2012generalization], as well as its Lipschitz property. The transport map investigated in the work shares a similar Lipschitz property with the transport map associated with the reverse heat flow. Nonetheless, the transport map investigated by [@kim2012generalization] and [@mikulincer2022lipschitz] is deduced via a limiting argument, thus has no explicit expression. Under Assumptions [Assumption 1](#condn:well-defined){reference-type="ref" reference="condn:well-defined"}, [Assumption 2](#condn:semi-log-convex){reference-type="ref" reference="condn:semi-log-convex"} and [Assumption 3](#condn:semi-log-concave){reference-type="ref" reference="condn:semi-log-concave"}, our considered transport map could be expressed as the flow map of the well-posed Föllmer flow at time $t = 1$ in a simple and explicit form. Towards connections between the flows, the Föllmer flow over time interval $[0, 1)$ (without time $1$) is equivalent to the reverse heat flow through a deterministic change of time, as revealed in Lemmas [Lemma 35](#lm:sde-time-change){reference-type="ref" reference="lm:sde-time-change"} and [Lemma 36](#lm:ode-time-change){reference-type="ref" reference="lm:ode-time-change"}. Technically, the equivalence cannot ensure well-posedness of the Föllmer flow at time $1$. We explicitly extend the flow to time $1$ by deriving a uniform lower bound on the Jacobian matrix of the velocity field via the Cramér-Rao inequality. Additionally, it is worth mentioning that [@albergo2023building] and [@albergo2023stochastic] introduce a unit-time normalizing flow, relevant to the Föllmer flow, from the perspective of stochastic interpolation between the Gaussian measure and a target measure. Nonetheless, the well-posedness of this normalizing flow is not studied in the scope of their work. ## Notations For any integer $d\geq 1$, the Borel $\sigma$-algebra of $\mathbb R^d$ is denoted by $\mathcal B(\mathbb R^d)$. For $x,y\in \mathbb R^d$, define $\left<x, y\right> := \sum_{i=1}^d x_iy_i$ and the Euclidean norm $|x|:=\left<x,x \right>^{1/2}$. Denote by $\mathbb S^{d-1} :=\{x\in \mathbb R^d: |x|=1 \}$. The operator norm of a matrix $M \in \mathbb R^{m\times n}$ is denoted by $\|M \|_{\mathrm{op}} := \sup_{x\in \mathbb R^n, |x|=1} |Mx|$ and $M^{\top}$ is the transpose of $M$. Let $f:\mathbb R^d \rightarrow \mathbb R$ be a twice continuously differentiable function. Denote by $\nabla f, \nabla^2 f$ and $\Delta f$ the gradient of $f$, the Hessian of $f$ and the Laplacian of $f$, respectively. Let $\gamma_d$ denote the standard Gaussian measure on $\mathbb R^d$, i.e., $\gamma_d(\mathrm{d}x) := (2\pi)^{-d/2} \exp(-|x|^2 /2) \mathrm{d}x$. Let $N (0, {\mathbf{I}}_d)$ stand for a $d$-dimensional Gaussian random variable with mean $0$ and covariance $\mathbf{I}_d$ being the $d\times d$ identity matrix. Moreover, we use $\phi(x)$ to denote its probability density function with respect to the Lebesgue measure. The set of probability measures defined on a measurable space $(\mathbb R^d, \mathcal B(\mathbb R^d))$ is denoted as $\mathcal P(\mathbb R^d)$. For any $\mathbb R^d$-valued random vector, $\mathbb E[X]$ is used to denote its expectation. We say that $\Pi$ is a transference plan of $\mu$ and $\nu$ if it is a probability measure on $(\mathbb R^d \times \mathbb R^d, \mathcal B(\mathbb R^d) \times \mathcal B(\mathbb R^d))$ such that for any Borel set $A$ of $\mathbb R^d$, $\Pi(A \times \mathbb R^d)=\mu(A)$ and $\Pi(\mathbb R^d \times A)=\nu(A)$. We denote $\mathcal C(\mu,\nu)$ the set of transference plans of $\mu$ and $\nu$. Furthermore, we say that a couple of $\mathbb R^d$-valued random variables $(X,Y)$ is a coupling of $\mu$ and $\nu$ if there exists $\Pi \in \mathcal C(\mu,\nu)$ such that $(X,Y)$ is distributed according to $\Pi$. For two probability measures $\mu,\nu \in \mathcal P(\mathbb R^d)$, the Wasserstein distance of order $p \geq 1$ is defined as $$W_p(\mu,\nu) := \inf_{\Pi \in \mathcal C(\mu,\nu)} \left(\int_{\mathbb R^d \times \mathbb R^d} |x-y|^p \,\Pi(\mathrm{d}x, \mathrm{d}y) \right)^{1/p}.$$ Let $\mu,\nu \in \mathcal P(\mathbb R^d)$. The relative entropy of $\nu$ with respect to $\mu$ is defined by $$H(\nu \mid \mu) = \begin{cases} \int_{\mathbb R^d} \log \left( \frac{\mathrm{d}\nu}{\mathrm{d}\mu} \right) \nu(\mathrm{d}x), & \text{if $\nu \ll \mu $, } \\ +\infty, & \text{otherwise.} \end{cases}$$ # Main results We first present two definitions to characterize convexity properties of probability measures and some useful notations. **Definition 1** ([@cattiaux2014semi; @mikulincer2021brownian]). *A probability measure $\mu (\mathrm{d}x) = \exp(-U(x)) \mathrm{d}x$ is $\kappa$-semi-log-concave for some $\kappa \in {\mathbb{R}}$ if its support $\Omega \subseteq {\mathbb{R}}^d$ is convex and $U \in C^2(\Omega)$ satisfies $$\nabla^2 U(x) \succeq \kappa \mathbf{I}_d, \quad \forall x \in \Omega.$$* **Definition 2** ([@eldan2018regularization]). *A probability measure $\mu (\mathrm{d}x) = \exp(-U(x)) \mathrm{d}x$ is $\beta$-semi-log-convex for some $\beta > 0$ if its support $\Omega \subseteq {\mathbb{R}}^d$ is convex and $U \in C^2(\Omega)$ satisfies $$\nabla^2 U(x) \preceq \beta \mathbf{I}_d, \quad \forall x \in \Omega.$$* Let $\nu(\mathrm{d}x) = p(x) \mathrm{d}x$ be a probability measure on ${\mathbb{R}}^d$ and define an operator $(\EuScript Q_t)_{t\in [0,1]}$, acting on function $f:\mathbb R^d \rightarrow \mathbb R$ by $$\begin{aligned} \EuScript{Q}_{1-t} f (x) := \int_{\mathbb R^d} \varphi^{t x, 1-t^2} (y) f(y) \mathrm{d}y = \int_{\mathbb{R}^d} f \left(t x + \sqrt{1 - t^2} z \right) \mathrm{d}\gamma_d (z)\end{aligned}$$ where $\varphi^{tx, 1-t^2} (y)$ is the density of the $d$-dimensional Gaussian measure with mean $tx$ and covariance $(1-t^2)\mathbf{I}_d$. Our first result is that we construct a flow over the unit time interval, named the Föllmer flow, that pushes forward a standard Gaussian measure $\gamma_d$ to a general target measure $\nu$ at time $t=1$. Before rigorously defining the Föllmer flow, let us specify several regularity assumptions that would ensure well-definedness and well-posedness of the Föllmer flow. **Assumption 1**. *The probability measure $\nu$ has a finite third moment and is absolutely continuous with respect to the standard Gaussian measure $\gamma_d$.* **Assumption 2**. *The probability measure $\nu$ is $\beta$-semi-log-convex for some $\beta >0$.* **Assumption 3**. *Let $D := (1 / \sqrt{2}) \mathrm{diam} (\mathrm{supp} (\nu))$. The probability measure $\nu$ satisfies one or more of the following assumptions:* - *$\nu$ is $\kappa$-semi-log-concave for some $\kappa > 0$ with $D \in (0, \infty]$;* - *$\nu$ is $\kappa$-semi-log-concave for some $\kappa \le 0$ with $D \in (0, \infty)$;* - *$\nu = N(0,\sigma^2 \mathbf{I}_d) * \rho$ where $\rho$ is a probability measure supported on a ball of radius $R$ on $\mathbb R^d$.* Let us move to a formal definition of the Föllmer flow and the exhibition of its well-posedness. A complete exposition would be found in Section [3](#sec:follmer-flow){reference-type="ref" reference="sec:follmer-flow"}. **Definition 3**. *Suppose that probability measure $\nu$ satisfies Assumption [Assumption 1](#condn:well-defined){reference-type="ref" reference="condn:well-defined"}. If $(X_t)_{t \in [0, 1]}$ solves the initial value problem (IVP) $$\label{main-ode} \frac{\mathrm{d}X_t} {\mathrm{d}t} = V(t, X_t), \quad X_0 \sim \gamma_d, \quad t \in [0,1]$$ where the velocity field $V$ is defined by $$\label{eq:vector-field} V(t, x) :=\frac{\nabla \log \EuScript{Q}_{1-t} r(x)}{t}, \qquad \forall t\in (0,1]$$ with $V(0, x) :=\mathbb E_{\nu} [X], r(x) := \frac{\mathrm{d}\nu}{\mathrm{d}\gamma_d} (x)$. We call $(X_t)_{t \in [0, 1]}$ a Föllmer flow and $V(t, x)$ a Föllmer velocity field associated to $\nu$.* **Theorem 4** (Well-posedness). *Suppose that Assumptions [Assumption 1](#condn:well-defined){reference-type="ref" reference="condn:well-defined"}, [Assumption 2](#condn:semi-log-convex){reference-type="ref" reference="condn:semi-log-convex"} and [Assumption 3](#condn:semi-log-concave){reference-type="ref" reference="condn:semi-log-concave"} hold. Then the Föllmer flow $(X_t)_{t \in [0, 1]}$ associated to $\nu$ is a unique solution to the IVP ([\[main-ode\]](#main-ode){reference-type="ref" reference="main-ode"}). Moreover, the push-forward measure $\gamma_d \circ (X_1^{-1}) = \nu$.* The following results show that the Föllmer flow map at time $t = 1$ is Lipschitz when the target measure satisfies either the strong log-concavity assumption or the bounded support assumption. **Theorem 5** (Lipschitz mapping). *Assume that Assumptions [Assumption 1](#condn:well-defined){reference-type="ref" reference="condn:well-defined"}, [Assumption 2](#condn:semi-log-convex){reference-type="ref" reference="condn:semi-log-convex"}, [Assumption 3](#condn:semi-log-concave){reference-type="ref" reference="condn:semi-log-concave"}-(i) or [Assumption 3](#condn:semi-log-concave){reference-type="ref" reference="condn:semi-log-concave"}-(ii) hold.* - *If $\kappa D^2 \ge 1$, then $X_1(x)$ is a Lipschitz mapping with constant $\tfrac{1}{\sqrt{\kappa}}$, i.e., $$\|\nabla X_1(x) \|_{\mathrm{op}} \le \frac{1}{\sqrt{\kappa}}, \quad \forall x \in {\mathbb{R}}^d.$$* - *If $\kappa D^2 <1$, then $X_1(x)$ is a Lipschitz mapping with constant $\exp\left(\frac{1-\kappa D^2}{2} \right)D$, i.e., $$\|\nabla X_1(x) \|_{\mathrm{op}} \le \exp\left(\frac{ 1-\kappa D^2}{2} \right) D, \quad \forall x \in {\mathbb{R}}^d.$$* **Theorem 6** (Gaussian mixtures). *Assume that Assumptions [Assumption 1](#condn:well-defined){reference-type="ref" reference="condn:well-defined"}, [Assumption 2](#condn:semi-log-convex){reference-type="ref" reference="condn:semi-log-convex"} and [Assumption 3](#condn:semi-log-concave){reference-type="ref" reference="condn:semi-log-concave"}-(iii) hold. Then $X_1(x)$ is a Lipschitz mapping with constant $\sigma \exp \left( \frac{R^2}{2 \sigma^2} \right)$, i.e., $$\|\nabla X_1(x) \|_{\mathrm{op}} \leq \sigma \exp\left( \frac{R^2}{2 \sigma^2} \right), \quad \forall x \in {\mathbb{R}}^d.$$* **Remark 7**. *Combining $\mathrm{Lip}(X_1(x)) \leq \|\nabla X_1(x) \|_{\mathrm{op}}$ and Theorem [Theorem 6](#main-thm3){reference-type="ref" reference="main-thm3"}, we get $$\label{eq:ode-gradx-bd-var-gaussian-conv} \mathrm{Lip}(X_1(x)) \leq \sigma \exp \left( \frac{R^2}{2 \sigma^2} \right), \quad \forall x \in {\mathbb{R}}^d.$$* *For Gaussian mixtures, the Lipschitz constants of ([\[eq:ode-gradx-bd-var-gaussian-conv\]](#eq:ode-gradx-bd-var-gaussian-conv){reference-type="ref" reference="eq:ode-gradx-bd-var-gaussian-conv"}) are better than those provided by the Brownian transport map [@mikulincer2021brownian Theorem 1.4] and match those presented in [@mikulincer2022lipschitz]. Meanwhile, the Lipschitz constants of $X_1$ lead to a dimension-free logarithmic Sobolev constant and a dimension-free Poincaré constant $$\label{eq:conv-lsc-pc} C_{\mathrm{LS}} (p) \le 2 \sigma^2 \exp \left( \frac{R^2}{\sigma^2} \right), \quad C_{\mathrm{P}} (p) \le \sigma^2 \exp \left( \frac{R^2}{\sigma^2} \right).$$* *On the one hand, ([\[eq:conv-lsc-pc\]](#eq:conv-lsc-pc){reference-type="ref" reference="eq:conv-lsc-pc"}) implies a Gaussian log-Sobolev constant $2 \sigma^2$ and a Gaussian Poincaré constant $\sigma^2$ as $R$ goes to zero. In fact, Poincaré constant $\sigma^2$ and log-Sobolev constant $2\sigma^2$ are optimal for Gaussian measure $N(0,\sigma^2 \mathbf{I}_d)$ on $\mathbb R^d$. On the other hand, the Poincaré constant obtained by ([\[eq:conv-lsc-pc\]](#eq:conv-lsc-pc){reference-type="ref" reference="eq:conv-lsc-pc"}) is obviously smaller than the result in [@bardet2018functional Theorem 1.2]. In fact, the upper bound of Poincaré constant for distribution $p=N(0,\sigma^2 \mathbf{I}_d) * \rho$ in [@bardet2018functional Theorem 1.2] is $\sigma^2 \exp \left(4R^2 / \sigma^2 \right)$. Similarly, the log-Sobolev constant ([\[eq:conv-lsc-pc\]](#eq:conv-lsc-pc){reference-type="ref" reference="eq:conv-lsc-pc"}) we obtained is slightly better than that in [@chen2021dimension Corollary 1]. Indeed, the upper bound of log-Sobolev constant for distribution $\nu = N(0, \sigma^2 \mathbf{I}_d) * \rho$ in [@chen2021dimension Corollary 1] is $6(4R^2 +\sigma^2) \exp \left(4R^2 / \sigma^2 \right)$. Nonetheless, it is worthwhile to remark that [@chen2021dimension] considers a rich class of probability measures with the convolutional structure, which leads to general results on dimension-free log-Sobolev and Poincaré inequlities.* # The Föllmer flow and its well-posedness {#sec:follmer-flow} Let us present our motivations to derive the Föllmer flow. We are largely inspired by the construction of the Föllmer process [@follmer1988random; @lehec2013representation; @eldan2018regularization; @eldan2020stability], which provides a probabilistic solution to the Schrödinger problem [@schrodinger1931uber; @leonard2014survey], though our construction of the Föllmer flow is partially heuristic using a similar time-reversal argument. ## The Föllmer process In Föllmer's lecture notes at the École d'Été de Probabilités de Saint-Flour in 1986 [@follmer1988random], the Föllmer process is constructed with time reversal of a linear Itô SDE under a finite relative entropy condition, which rigorously determines a Schrödinger bridge from a source Dirac measure $\delta_0$ to a general target measure $\nu$. Let us briefly revisit Föllmer's arguments to derive such a process. **Definition 8** ([@follmer1988random]). *A diffusion process $\overline{P} := \left(\overline{X}_t \right)_{t\in [0, 1]}$ starting with marginal distribution $\nu$ at time $t = 0$ and reaching $0$ at time $t = 1$ is defined by the following Itô SDE $$\label{eq:follmer-sde-tr} \mathrm{d}\overline{X}_t = - \frac{1} {1 - t} \overline{X}_t \mathrm{d}t + \mathrm{d}\overline{W}_t, \ \overline{X}_0 \sim \nu, \ t \in [0, 1)$$ with an extended solution at time $t = 1$, i.e., $\overline{X}_1 \sim \delta_{0}$. The transition probability distribution of ([\[eq:follmer-sde-tr\]](#eq:follmer-sde-tr){reference-type="ref" reference="eq:follmer-sde-tr"}) from $\overline{X}_0$ to $\overline{X}_t$ is given by $\overline{X}_t | \overline{X}_0 \sim N ((1-t) \overline{X}_0, t(1-t)\mathbf{I}_d )$ for every $0 \leq t < 1$.* **Lemma 9** ([@follmer1985entropy]). *Suppose that the diffusion process $Q$ has finite relative entropy with respect to a standard Wiener process $W_t$ over the unit time interval, i.e., $t \in [0, 1]$. Then for almost all $t \in [0, 1]$, the logarithmic derivative of marginal density $\rho_t$ of $Q$ satisfies the duality equation $\nabla \log \rho_t (x) = b (x, t) + \overline{b} (x, 1-t)$ for almost all $x \in {\mathbb{R}}^d$, where $b (x, t)$ and $\overline{b} (x, t)$ are drifts of diffusion process $Q$ and its time-reversed diffusion process $\overline{Q}$, respectively.* **Definition 10** ([@follmer1988random; @lehec2013representation]). *Föllmer process $P = (X_t)_{t \in [0, 1]}$ is defined by the Itô SDE $$\label{eq:follmer-sde} \mathrm{d}X_t = \nabla \log \EuScript{P}_{1-t} r (X_t) \mathrm{d}t + \mathrm{d}W_t, \ X_0 = 0, \ t \in [0, 1]$$ where $W_t$ is a standard Wiener process and $\EuScript{P}_t$ is the heat semigroup defined by $\EuScript{P}_t h(x) := \mathbb{E}\left[ h(x + W_t) \right]$. Moreover, the drift $\nabla \log \EuScript{P}_{1-t} r (X_t)$ is called the Föllmer drift.* **Remark 11**. *According to Lemma [Lemma 9](#lm:follmer-tr){reference-type="ref" reference="lm:follmer-tr"}, the Föllmer process $P$ can be obtained by taking the time reversal of the diffusion process $\overline{P}$ over $t \in [0, 1]$. It implies that the Föllmer drift has an alternative representation, i.e., for any $t \in (0, 1]$, $\nabla \log \EuScript{P}_{1-t} r (X_t) = X_t/t + \nabla \log p_t (X_t)$, where $p_t$ is the marginal density of the Föllmer process $P$.* ## The Föllmer flow via time reversal Since $\delta_0$ is a degenerate distribution in the sense that its mass is concentrated at $0$, we consider constructing a diffusion process that starts with a marginal distribution $\nu$ and would be able to keep the nonzero variance of its marginal distribution at time $t = 1$. Let us present the constructed diffusion process first. For any $\varepsilon \in (0,1)$, we consider a diffusion process $\left(\overline{X}_t \right)_{t\in [0, 1-\varepsilon]}$ defined by the following Itô SDE $$\label{eq:vp-sde-tr} \mathrm{d}\overline{X}_t = - \frac{1} {1 - t} \overline{X}_t \mathrm{d}t + \sqrt{\frac{2} {1-t}} \mathrm{d}\overline{W}_t, \quad \overline{X}_0 \sim \nu$$ for all $t \in [0, 1-\varepsilon]$. By Theorem 2.1 in [@revuz2013continuous Chapter IX], the diffusion process $\overline{X}_t$ defined in ([\[eq:vp-sde-tr\]](#eq:vp-sde-tr){reference-type="ref" reference="eq:vp-sde-tr"}) has a unique strong solution on $[0,1-\varepsilon]$. Moreover, the transition probability distribution of ([\[eq:vp-sde-tr\]](#eq:vp-sde-tr){reference-type="ref" reference="eq:vp-sde-tr"}) from $\overline{X}_0$ to $\overline{X}_t$ is given by $\overline{X}_t | \overline{X}_0 =x_0 \sim N( (1-t) x_0, \ t (2-t) {\mathbf{I}}_d)$ for every $t\in [0,1-\varepsilon]$. It is a straightforward observation that the variance of $\overline{X}_{1 - \varepsilon} | \overline{X}_0$ for SDE ([\[eq:vp-sde-tr\]](#eq:vp-sde-tr){reference-type="ref" reference="eq:vp-sde-tr"}) will approach the identity matrix $\mathbf{I}_d$ when $\varepsilon$ is small enough. That is why we could expect the marginal distribution of $\overline{X}_{1 - \varepsilon}$ would have a nonzero variance. In contrast, for the time-reversed Föllmer process ([\[eq:follmer-sde-tr\]](#eq:follmer-sde-tr){reference-type="ref" reference="eq:follmer-sde-tr"}), the variance of $\overline{X}_{1 - \varepsilon} | \overline{X}_0$ will approach constant $0$ as $\varepsilon \to 0$, which indicates the variance of its marginal distribution vanishes at time $t = 1$. However, SDE ([\[eq:vp-sde-tr\]](#eq:vp-sde-tr){reference-type="ref" reference="eq:vp-sde-tr"}) is not well-defined at time $t = 1$ due to unbounded drift and diffusion coefficients. Then we leverage the fact that the marginal distribution $\overline{\mu}_t$ of the diffusion process $\overline{X}_t$ defined in ([\[eq:vp-sde-tr\]](#eq:vp-sde-tr){reference-type="ref" reference="eq:vp-sde-tr"}) has been determined in the sense that $\overline{X}_t \overset{d}{=} (1-t) X + \sqrt{t(2-t)} Y$ with $X \sim \nu, Y \sim \gamma_d$, and concentrate on an ODEs system sharing the same marginal distribution flow with SDE ([\[eq:vp-sde-tr\]](#eq:vp-sde-tr){reference-type="ref" reference="eq:vp-sde-tr"}) in order to circumvent the singularity of SDE ([\[eq:vp-sde-tr\]](#eq:vp-sde-tr){reference-type="ref" reference="eq:vp-sde-tr"}) at time $t = 1$. Note that the marginal distribution flow $( \overline{\mu}_t )_{t \in [0, 1-\varepsilon]}$ of the diffusion process ([\[eq:vp-sde-tr\]](#eq:vp-sde-tr){reference-type="ref" reference="eq:vp-sde-tr"}) satisfies the Fokker-Planck-Kolmogorov equation in an Eulerian framework [@bogachev2022fokker] $$\label{eq:fke-vp-sde-tr} \partial_t \overline{\mu}_t = \nabla \cdot (\overline{\mu}_t V(1-t, x)) \quad \textrm{on} \ [0, 1-\varepsilon] \times {\mathbb{R}}^d, \ \overline{\mu}_0 = \nu$$ in the sense that $\overline{\mu}_t$ is continuous in $t$ under the weak topology, i.e., $$\begin{aligned} \overline{\mu}_t(f) :=\int_{\mathbb R^d} f(x) \mu_t(\mathrm{d}x) =\nu(f) - \int_0^t \overline{\mu}_s \left( \left<V(1-s, \cdot), \nabla f \right> \right) \mathrm{d}s\end{aligned}$$ for all $f\in C^{\infty}_0(\mathbb R^d)$ and the velocity field is given by $$\label{eq:vp-ode-eps-tr-vf} V(1-t, x) := \frac{1}{1-t} \left[x + S(1-t, x) \right], \quad t \in[0, 1-\varepsilon]$$ and $$S(t, x) := \nabla \log \int_{\mathbb R^d} (2\pi (1-t^2))^{-\frac d 2} \exp\left(-\frac{|x- t y|^2}{2(1-t^2)} \right) p(y) \mathrm{d}y$$ for all $t \in [\varepsilon, 1]$. Due to the classical Cauchy-Lipschitz theory [@ambrosio2014continuity Section 2] with a Lipschitz velocity field or the well-established Ambrosio-DiPerna-Lions theory with lower Sobolev regularity assumptions on the velocity field [@diperna1989ordinary; @ambrosio2004transport], we shall define a flow $( X^{*}_t )_{t \in [0, 1-\varepsilon]}$ in a Lagrangian formulation via the following ODEs system $$\begin{aligned} \label{eq:vp-ode-eps-tr} \mathrm{d}X^{*}_t = - V \left( 1-t, X^{*}_t\right) \mathrm{d}t, \quad X^{*}_0 \sim \nu, \quad t \in [0, 1-\varepsilon].\end{aligned}$$ **Proposition 12**. *Assume the velocity field $V(t, x)$ satisfies $V \in L^1([\varepsilon, 1]; W^{1, \infty}_{\mathrm{loc}}({\mathbb{R}}^d; {\mathbb{R}}^d))$ and $|V|/(1+|x|) \in L^1([\varepsilon, 1]; L^{\infty}({\mathbb{R}}^d))$. Then the push-forward measure associated with the flow map $X^{*}_t$ satisfies $X^{*}_t \overset{d}{=} (1-t) X + \sqrt{t(2-t)} Y$ with $X \sim \nu, Y \sim \gamma_d$. Moreover, the push-forward measure $\nu \circ ({X^{*}_{1 - \varepsilon}})^{-1}$ converges to the Gaussian measure $\gamma_d$ in the sense of Wasserstein-2 distance as $\varepsilon$ tends to zero, i.e., $W_2 (\nu \circ ({X^{*}_{1 - \varepsilon}})^{-1}, \gamma_d) \to 0$.* **Remark 13**. *Suppose that the target measure $\nu$ has a finite third moment. By Lemma [Lemma 26](#lem-4.1){reference-type="ref" reference="lem-4.1"}, we can supplement the definition of velocity field $V(1-t, x)$ at time $t=1$, i.e., $$V(0, x) := \lim_{t \downarrow 0} V(t, x) =\lim_{t \downarrow 0} \frac{x +S(t, x)}{t} =\mathbb E_{\nu}[X].$$ Then we extend the flow $(X^{*}_t)_{t \in [0,1)}$ to time $t =1$ such that $X^{*}_1 \sim \gamma_d$, which solves the IVP $$\begin{aligned} \label{eq:vp-ode-unit-tr} \mathrm{d}X^{*}_t = - V \left( 1-t, X^{*}_t\right) \mathrm{d}t, \quad X^{*}_0 \sim \nu, \quad t \in [0, 1],\end{aligned}$$ where the velocity field $$V(1-t, x) = \frac{1}{1-t} \left[ x + S(1-t, x) \right], \quad \forall t \in [0, 1)$$ and $V(0, x) = \mathbb{E}_{\nu} [X]$.* In order to exploit a time-reversal argument inspired by Föllmer, it remains crucial to establish the well-posedness of a flow $( X^{*}_t )_{t \in [0, 1]}$ that solves the IVP ([\[eq:vp-ode-unit-tr\]](#eq:vp-ode-unit-tr){reference-type="ref" reference="eq:vp-ode-unit-tr"}). We proceed to study regularity properties of the velocity field $V$ on $[0, 1] \times {\mathbb{R}}^d$ by imposing structural assumptions on the target measure $\nu$. By Theorem [Theorem 31](#thm1){reference-type="ref" reference="thm1"}, we know that there exists $0 \leq \theta^{\star}_t <\infty$ such that $$\|- \nabla V(t, x) \|_{\mathrm{op}}= \| \nabla V(t, x) \|_{\mathrm{op}} \leq \theta^{\star}_{t}$$ for any $t\in [0,1]$. Furthermore, the velocity field $-V(1-t, x)$ is smooth and with the bounded derivative for any $t\in [0,1]$ and $x\in \mathbb R^d$. Therefore, the IVP ([\[eq:vp-ode-unit-tr\]](#eq:vp-ode-unit-tr){reference-type="ref" reference="eq:vp-ode-unit-tr"}) has a unique solution and the flow map $x \mapsto X^{*}_t(x)$ is a diffeomorphism from $\mathbb R^d$ onto $\mathbb R^d$ at any time $t \in [0, 1]$. A standard time-reversal argument of ODE would yield a formal definition of the Föllmer flow. **Definition 14**. *Suppose that probability measure $\nu$ satisfies Assumption [Assumption 1](#condn:well-defined){reference-type="ref" reference="condn:well-defined"}. If $(X_t)_{t \in [0, 1]}$ solves the IVP $$\label{eq:vp-ode-unit} \frac{\mathrm{d}X_t} {\mathrm{d}t} = V(t, X_t), \quad X_0 \sim \gamma_d, \quad t \in [0,1]$$ where the velocity field $$V(t, x) = \frac{1}{t} \left[ x + S(t, x) \right], \quad \forall t \in (0, 1], \quad V(0, x) = \mathbb{E}_{\nu} [X],$$ we call $(X_t)_{t \in [0, 1]}$ a Föllmer flow and $V(t, x)$ a Föllmer velocity field associated to $\nu$.* **Remark 15**. *Notice that $$\begin{aligned} \EuScript{Q}_{1-t} r (x) = (2\pi)^{d/2} \exp\left(\frac{ |x |^2}{2} \right) \frac{1}{(2\pi(1-t^2))^{d/2}} \int_{\mathbb R^d} p(y)\exp\left(-\frac{|x-ty |^2 }{2(1-t^2)} \right) \mathrm{d}y\end{aligned}$$ where $\EuScript{Q}_{1-t} r(x)$ is defined in ([\[eq:vp-vf-op-form\]](#eq:vp-vf-op-form){reference-type="ref" reference="eq:vp-vf-op-form"}). We further obtain $\nabla \log \EuScript{Q}_{1-t} r(x) = x + S(t, x), \ \forall t \in [0,1]$. Therefore, we have that ([\[main-ode\]](#main-ode){reference-type="ref" reference="main-ode"}) and ([\[eq:vp-ode-unit\]](#eq:vp-ode-unit){reference-type="ref" reference="eq:vp-ode-unit"}) are equivalent, which satisfy $X_0 \sim \gamma_d$ and $X_1 \sim \nu$.* Finally, let us conclude with the well-posedness properties of the Föllmer flow, which is presented in Theorem [Theorem 4](#main-thm1){reference-type="ref" reference="main-thm1"} and summarized below. **Theorem 16** (Well-posedness). *Suppose that Assumptions [Assumption 1](#condn:well-defined){reference-type="ref" reference="condn:well-defined"}, [Assumption 2](#condn:semi-log-convex){reference-type="ref" reference="condn:semi-log-convex"} and [Assumption 3](#condn:semi-log-concave){reference-type="ref" reference="condn:semi-log-concave"} hold. Then the Föllmer flow $(X_t)_{t \in [0, 1]}$ associated to $\nu$ is a unique solution to the IVP ([\[eq:vp-ode-unit\]](#eq:vp-ode-unit){reference-type="ref" reference="eq:vp-ode-unit"}). Moreover, the push-forward measure $\gamma_d \circ (X_1^{-1}) = \nu$.* # Applications Owing to the Lipschitz transport properties proved in Theorems [Theorem 5](#main-thm2){reference-type="ref" reference="main-thm2"} and [Theorem 6](#main-thm3){reference-type="ref" reference="main-thm3"}, we are motivated to establish a variety of functional inequalities and concentration inequalities for several classes of probability measures on Euclidean space. ## Dimension-free inequalities In this subsection, we provide dimension-free results for the $\Psi$-Sobolev inequalities. For completeness, we incorporate classical results for strongly-log-concave measures ($\kappa > 0$), which have been studied with the optimal transport maps [@caffarelli2000monotonicity]. Compared with [@mikulincer2021brownian], we obtain that the upper bound constants of $\Psi$-Sobolev inequalities, Isoperimetric inequalities and $q$-Poincaré inequalities are the same for $\kappa D^2 \geq 1$. When $\kappa D^2 <1$, our upper bound constants to $\Psi$-Sobolev inequalities and Isoperimetric inequalities are in the same order with the results of Lemmas 5.3-5.5 in [@mikulincer2021brownian]. For the Gaussian mixtures case, we obtain that the constants of these inequalities are slightly better than the result of Lemmas 5.3-5.5 in [@mikulincer2021brownian]. **Definition 17**. *Let $\mathcal I$ be a closed interval (not necessarily bounded) and let $\Psi: \mathcal I \rightarrow \mathbb R$ be a twice differentiable function. We say that $\Psi$ is a divergence if each of the functions $\Psi, \Psi''$ and $-1/\Psi''$ is a convex function. Given a probability measure $\nu(\mathrm{d}x)=p(x) \mathrm{d}x$ on $\mathbb R^d$ and a function $\zeta: \mathbb R^d \rightarrow \mathcal I$ such that $\int_{\mathbb R^d} \zeta(x) p(x) \mathrm{d}x \in \mathcal I$, we define $$\mathrm{Ent}^{\Psi}_{p}(\zeta) :=\int_{\mathbb R^d} \Psi(\zeta(x)) p(x) \mathrm{d}x -\Psi \left(\int_{\mathbb R^d} \zeta(x) p(x) \mathrm{d}x \right).$$* Some examples of the divergences are $\Psi: \mathbb R \rightarrow \mathbb R$ with $\Psi(x)=x^2$ (Poincaré inequality) and $\Psi: \mathbb R_{+} \rightarrow \mathbb R$ with $\Psi(x) =x\log x$ (log-Sobolev inequality). **Theorem 18** ($\Psi$-Sobolev inequalities). *Let Assumptions [Assumption 1](#condn:well-defined){reference-type="ref" reference="condn:well-defined"}, [Assumption 2](#condn:semi-log-convex){reference-type="ref" reference="condn:semi-log-convex"} and [Assumption 3](#condn:semi-log-concave){reference-type="ref" reference="condn:semi-log-concave"} hold.* - *Let $\zeta: \mathbb R^d \rightarrow \mathcal I$ be any continuously differentiable function such that $\int_{\mathbb R^d} \zeta^2(x) p(x) \mathrm{d}x \in \mathcal I$.* - *If $\kappa D^2 \ge 1$, then $$\mathrm{Ent}^{\Psi}_{p}(\zeta) \le \frac{1}{2\kappa} \int_{\mathbb R^d} \Psi''(\zeta(x)) |\nabla \zeta(x)|^2 p(x) \mathrm{d}x.$$* - *If $\kappa D^2 <1$, then $$\begin{aligned} \mathrm{Ent}^{\Psi}_{p}(\zeta) \le \frac{\exp(1-\kappa D^2)}{2} D^2 \int_{\mathbb R^d} \Psi''(\zeta(x)) |\nabla \zeta(x)|^2 p(x) \mathrm{d}x.\end{aligned}$$* - *Fix a probability measure $\rho$ on $\mathbb R^d$ supported on a ball of radius $R$ and let $p:=N(a,\Sigma) * \rho$ and denote $\lambda_{\min} :=\lambda_{\min}(\Sigma)$ and $\lambda_{\max}=\lambda_{\max}(\Sigma)$. Then for any continuously differentiable function $\zeta: \mathbb R^d \rightarrow \mathcal I$ such that $\int_{\mathbb R^d} \zeta^2(x) p(x) \mathrm{d}x \in \mathcal I$, we have $$\begin{aligned} \mathrm{Ent}^{\Psi}_{p}(\zeta) \le \frac{1}{2} \lambda_{\max} \exp\left(\frac{R^2}{\lambda_{\min}} \right) \int_{\mathbb R^d} \Psi''(\zeta(x)) |\nabla \zeta(x)|^2 p(x) \mathrm{d}x.\end{aligned}$$* **Theorem 19** (Isoperimetric inequalities). *Assume that Assumptions [Assumption 1](#condn:well-defined){reference-type="ref" reference="condn:well-defined"}, [Assumption 2](#condn:semi-log-convex){reference-type="ref" reference="condn:semi-log-convex"} and [Assumption 3](#condn:semi-log-concave){reference-type="ref" reference="condn:semi-log-concave"} hold. Let $\Phi$ be the cumulative distribution function of $\gamma_1$ on $\mathbb R$, that is, $$\Phi(x)=\gamma_1(-\infty,x)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^x \exp\left(-\frac{y^2}{2} \right) \mathrm{d}y, \quad -\infty < \forall x < +\infty$$ and $B_2^d:=\{x\in \mathbb R^d: |x| \leq 1 \}$ be the unit ball in $\mathbb R^d$.* - *Let $A_t:=A +tB_2^d$ for any Borel set $A \subseteq \mathbb R^d$ and $t \ge 0$, then $$p\left(A_t \right) \ge \Phi\left(p(A) +\frac{t}{C} \right), \quad C := \begin{cases} 1/\sqrt{\kappa}, \ & \text{if $\kappa D^2 \ge 1$, } \\ \exp\left(\frac{1-\kappa D^2}{2} \right) D, \ & \text{if $\kappa D^2 <1$.} \end{cases}$$* - *Let $p:=N(a,\Sigma) * \rho$ where $\rho$ is a probability measure on $\mathbb R^d$ and is supported on a ball of radius $R$. Set $\lambda_{\min} :=\lambda_{\min}(\Sigma), \lambda_{\max} :=\lambda_{\max}(\Sigma)$ and $$C := (\lambda_{\min} \lambda_{\max})^{1/2} \exp\left(\frac{R^2}{2\lambda_{\min}} \right).$$ Then $$p(A_t) \ge \Phi\left(p(A) +\frac{t}{C} \right), \quad \quad A_t:=A +tB_2^d.$$* Finally, let $\eta: \mathbb R^d \rightarrow \mathbb R$ be any continuously differentiable function such that $\int_{\mathbb R^d} \eta(x) p(x) \, \mathrm{d}x =0.$ **Theorem 20** ($q$-Poincaré inequalities). *Suppose that Assumptions [Assumption 1](#condn:well-defined){reference-type="ref" reference="condn:well-defined"}, [Assumption 2](#condn:semi-log-convex){reference-type="ref" reference="condn:semi-log-convex"} and [Assumption 3](#condn:semi-log-concave){reference-type="ref" reference="condn:semi-log-concave"} hold.* - *Let $q \geq 2$ be an even integer and $\eta \in L^q(\gamma_d)$, then it holds that $$\int_{\mathbb R^d} \eta^q(x) p(x) \, \mathrm{d}x \leq \left( \int_{\mathbb R^d} |\nabla \eta(x)|^q p(x) \, \mathrm{d}x \right) \begin{cases} C^{\star}_1 & \text{if $\kappa D^2 \geq 1$}, \\ C^{\star}_2 & \text{if $\kappa D^2 <1$.} \end{cases}$$ where $$C^{\star}_1 := \left( \frac{q-1}{\kappa} \right)^{q/2}, \quad C^{\star}_2 :=D^q \exp \left(\frac{q(1-\kappa D^2)}{2} \right).$$* - *Fix a probability measure $\rho$ on $\mathbb R^d$ supported on a ball of radius $R$, and let $p:= N(a,\Sigma) * \rho$ and denote $\lambda_{\min} :=\lambda_{\min}(\Sigma)$ and $\lambda_{\max} :=\lambda_{\max}(\Sigma)$. Then for any $\eta \in L^q(\gamma_d)$ with even integer $q \geq 2$, it holds that $$\int_{\mathbb R^d} \eta^q(x) p(x) \, \mathrm{d}x \le (q-1)^{\frac q 2} (\lambda_{\min} \lambda_{\max})^{\frac q 2} \exp\left(\frac{q R^2}{2 \lambda_{\min}} \right) \int_{\mathbb R^d} |\nabla \eta(x)|^q p(x) \, \mathrm{d}x.$$* ## Non-asymptotic bounds for empirical measures Let $\mu$ be a probability distribution on $\mathbb R^d$ and $$\label{eq-empirical-measure} \mu_n := \frac{1}{n} \sum_{i=1}^n \delta_{X_i},$$ be the empirical measure, where $(X_i)_{i = 1}^n$ are i.i.d. samples drawn from $\mu$. Deriving the non-asymptotic convergence rate under the Wasserstein distance of the empirical measure $\mu_n$ and the probability measure $\mu$ on Polish space is one of the most important topics in statistics, probability, and machine learning. In recent years, significant progress has been made on this topic. When $p =1$, the Kantorovich-Rubinstein duality [@kantorovich1958space] implies that $W_1(\mu_n,\mu)$ is equivalent to the supremum of the empirical process indexed by Lipschitz functions. As a consequence, [@dudley1969speed] provides sharp lower and upper bounds of $\mathbb E \left[W_1(\mu_n,\mu) \right]$ for $\mu$ supported on a bounded finite dimensional set. Subsequently, [@talagrand1994transportation] studies the case when $\mu$ is the uniform distribution on a $d$-dimensional unit cube. For general distributions, [@boissard2014mean; @dereich2013constructive; @fournier2015rate] establish sharp upper bounds of $\mathbb E\left[W_p(\mu_n,\mu) \right]$ in finite dimensional Euclidean spaces. Recently, by extending finite dimensional spaces to infinite dimensional functional spaces, [@lei2020convergence] establishes similar results for general distributions. Besides the above mentioned bounds in expectation, [@weed2019sharp] obtains a high probability bound on $W_p(\mu_n,\mu)$ for measures $\mu$ with bounded supports. By applying Sanov's theorem to independent random variables, [@bolley2007quantitative] establishes concentration inequalities for empirical measures on non-compact space. In this subsection, we will give a high probability bound on $W_2(\mu_n,\mu)$ by the Lipschitz transport properties proved in Theorems [Theorem 5](#main-thm2){reference-type="ref" reference="main-thm2"} and [Theorem 6](#main-thm3){reference-type="ref" reference="main-thm3"}. To begin with, we will review the transportation inequality defined in Definition [Definition 21](#eq-def-transport){reference-type="ref" reference="eq-def-transport"}, the non-asymptotic convergence rate of $\mathbb E\left[ W_p(\mu_n,\mu) \right]$ as given in Theorem [Theorem 23](#thm-empirical-bound){reference-type="ref" reference="thm-empirical-bound"}, and its concentration inequality for $p=2$ as stated in Theorem [Theorem 24](#thm-concentration-ineq){reference-type="ref" reference="thm-concentration-ineq"}. Then, we will derive the non-asymptotic convergence rate for $W_p(\mu_n,\mu)$, as stated in Theorem [Theorem 25](#thm-main-concentration-ineq){reference-type="ref" reference="thm-main-concentration-ineq"} by combining the transportation inequality of the Gaussian measure on $\mathbb R^d$ as established in [@talagrand1994transportation] and the transportation inequality of the push-forward measure of the Gaussian measure under Lipschitz mapping, as shown in Lemma [Lemma 22](#lem-lip-function){reference-type="ref" reference="lem-lip-function"}. **Definition 21** (Transportation inequality). *The probability measure $\mu$ satisfies the $L^p$-transportation inequality on $\mathbb R^d$ if there is some constant $C>0$ such that for any probability measure $\nu$, $W_p(\mu,\nu) \leq \sqrt{2C H(\nu \mid \mu)}$. To be short, we write $\mu \in \mathrm{T_p}(C)$ for this relation.* **Lemma 22** ([@djellout2004transportation]). *Assume that $\mu \in \mathrm{T_p} (C)$ on $\mathbb R^d$. If $\Phi: \mathbb R^d \rightarrow \mathbb R^d$ is Lipschitz continuous with constant $\alpha > 0$, then $\nu =\mu \circ \Phi^{-1} \in \mathrm{T_p}(\alpha^2 C)$ on $\mathbb R^d$.* **Theorem 23** ([@fournier2015rate]). *Let $p>0$, assume that for some $r>p$ and $\int_{\mathbb R^d} |x|^r \, \mu(\mathrm{d}x)$ is finite. Then there exists a constant $C >0$ depending only on $p,r,d$ such that for all $n \geq 1$, $$\begin{aligned} \mathbb E \left[ W_p(\mu_n, \mu) \right] \leq C \left( \int_{\mathbb R^d} |x|^r \, \mu(\mathrm{d}x) \right)^{p/r} \begin{cases} n^{-\frac 1 2} + n^{-\frac{r-p}{r} }, & \text{if $p> d/2$ and $r \neq 2p$ } \\ n^{-\frac 1 2 } \log(1+n) + n^{-\frac{r-p}{r} }, & \text{if $p=d/2$ and $r \neq 2p$ } \\ n^{-\frac p d} +n^{-\frac{r-p}{r} }, & \text{if $p < d/2$ and $r \neq \frac{d}{d-p}$ } \end{cases} \end{aligned}$$ where the expectation is taken on the samples $X_1, \cdots, X_n$.* The next result states that a $\mathrm{T_2}(C)$ inequality on $\mu$ implies Gaussian concentration inequality for $W_2(\mu_n,\mu)$. **Theorem 24** ([@gozlan2007large]). *Let a probability measure $\mu$ on $\mathbb R^d$ satisfy the transportation inequality $\mathrm{T_2}(C)$. The following holds: $$\mathbb P\left( W_2(\mu_n,\mu) \geq \mathbb E\left[ W_2(\mu_n,\mu) \right] +t \right) \leq \exp\left(-\frac{nt^2}{C} \right).$$* For any probability measure $\nu$ on $\mathbb R^d$ with a finite fifth moment, let us define $$\begin{aligned} \label{eq-mathsf-const-M} \mathsf M(\nu, d, n) := c_d \left( \int_{\mathbb R^d} |x|^5 \, \nu(\mathrm{d}x) \right)^{2/5} \begin{cases} n^{-1/2} & \text{if $d<4$} \\ n^{-1/2} \log(1+n) & \text{if $d=4$ } \\ n^{-2/d} & \text{if $d>4$ } \end{cases}\end{aligned}$$ where the constant $c_d$ depends only on $d$. On the other hand, for the $L^2$-transportation inequality $\mathrm{T_2}(C)$, recall that Talagrand [@talagrand1996transportation] proved that the standard Gaussian measure $\gamma_1= N(0,1)$ satisfies $\mathrm{T_2}(C)$ on $\mathbb R$ w.r.t. the Euclidean distance with the sharp constant $C = 1$ and found that $\mathrm{T_2}(C)$ is stable for product (or independent) tensorization. Therefore, combining Lemma [Lemma 22](#lem-lip-function){reference-type="ref" reference="lem-lip-function"}, Theorems [Theorem 5](#main-thm2){reference-type="ref" reference="main-thm2"}, [Theorem 6](#main-thm3){reference-type="ref" reference="main-thm3"}, [Theorem 23](#thm-empirical-bound){reference-type="ref" reference="thm-empirical-bound"} and [Theorem 24](#thm-concentration-ineq){reference-type="ref" reference="thm-concentration-ineq"}, we obtain the following results. **Theorem 25** (Concentration for empirical measures). *Suppose that Assumptions [Assumption 1](#condn:well-defined){reference-type="ref" reference="condn:well-defined"}, [Assumption 2](#condn:semi-log-convex){reference-type="ref" reference="condn:semi-log-convex"} and [Assumption 3](#condn:semi-log-concave){reference-type="ref" reference="condn:semi-log-concave"} hold, and let probability measure $\nu$ has a finite fifth moment.* - *If $\kappa D^2 \geq 1$, then $\nu \in \mathrm{T_2} (1/\kappa)$. Moreover, for any $\varepsilon \in (0,1)$, it holds that $$\begin{aligned} W_2(\nu_n,\nu) \leq \left( \frac{\log \varepsilon^{-1} }{n \kappa} \right)^{1/2} + \mathsf{M}(\nu,d,n) \end{aligned}$$ with probability at least $1-\varepsilon$ and constant $\mathsf M(\nu, d, n)$ given in ([\[eq-mathsf-const-M\]](#eq-mathsf-const-M){reference-type="ref" reference="eq-mathsf-const-M"}).* - *If $\kappa D^2 < 1$, then $\nu \in \mathrm{T_2} \left( D^2 \exp(1-\kappa D^2) \right)$. Moreover, for any $\varepsilon \in (0,1)$, it holds that $$\begin{aligned} W_2(\nu_n,\nu) \leq \left\{ \frac{\log \varepsilon^{-1} }{nD^2 \exp(1-\kappa D^2) } \right\}^{1/2} + \mathsf M(\nu, d, n) \end{aligned}$$ with probability at least $1-\varepsilon$ and constant $\mathsf M(\nu, d, n)$ given in ([\[eq-mathsf-const-M\]](#eq-mathsf-const-M){reference-type="ref" reference="eq-mathsf-const-M"}).* - *If $\nu = N(0,\sigma^2 \mathbf{I}_d) * \rho$ where $\rho$ is a probability measure supported on a ball of radius $R$ on $\mathbb R^d$, then $\nu \in \mathrm{T_2} \left( \sigma^2 \exp(R^2/\sigma^2) \right)$. Moreover, for any $\varepsilon \in (0,1)$, it holds that $$\begin{aligned} W_2(\nu_n,\nu) \leq \left\{ \frac{\log \varepsilon^{-1} }{n\sigma^2 \exp(R^2/\sigma^2) } \right\}^{1/2} + \mathsf M(\nu, d, n) \end{aligned}$$ with probability at least $1-\varepsilon$ and constant $\mathsf M(\nu, d, n)$ given in ([\[eq-mathsf-const-M\]](#eq-mathsf-const-M){reference-type="ref" reference="eq-mathsf-const-M"}).* # Conclusion We have constructed the Föllmer flow originating from a standard Gaussian measure and hitting a general target measure. By studying the well-posedness of the Föllmer flow, we have established the Lipschitz property of its flow map at time $t = 1$. Such a Lipschitz transport map enables get functional inequalities with dimension-free constants and derive concentration inequalities for the empirical measure for rich classes of probability measures. It is worthwhile to notice that the Föllmer velocity field has an analytic expression that is compatible with Monte Carlo approximations. Therefore, a possible direction of future research would be to design general-purpose sampling algorithms and score-based generative models using the Föllmer flow. Besides, being limited to scenarios covered in Assumptions [Assumption 2](#condn:semi-log-convex){reference-type="ref" reference="condn:semi-log-convex"} and [Assumption 3](#condn:semi-log-concave){reference-type="ref" reference="condn:semi-log-concave"}, the work could be extended to explore weaker and even minimal regularity assumptions on the target measure. For example, replacing semi-log-concavity with "convexity at infinity\" in [@bolley2012convergence; @cattiaux2014semi] is a potential step. # Proof of Theorem [Theorem 4](#main-thm1){reference-type="ref" reference="main-thm1"} and Proposition [Proposition 12](#prop:vp-ode-eps){reference-type="ref" reference="prop:vp-ode-eps"} {#proof-of-theorem-main-thm1-and-proposition-propvp-ode-eps} ## Well-definedness of the Föllmer flow Recall that the velocity field $V(t, x)$ defined in ([\[eq:vector-field\]](#eq:vector-field){reference-type="ref" reference="eq:vector-field"}) yields $$V(t, x) :=\frac{\nabla \log \EuScript{Q}_{1-t} r(x)}{t}, \quad r(x):=\frac{p(x)}{\phi(x)}$$ where $\nu(\mathrm{d}x)=p(x) \mathrm{d}x$. For any $t \in (0,1]$, then one obtains $$\begin{aligned} \EuScript{Q}_{1-t} r (x) = \int_{\mathbb R^d} \varphi^{tx, 1-t^2} (y) r(y) \mathrm{d}y = \int_{\mathbb{R}^d} \phi (z) r (tx + \sqrt{1 - t^2}z) \mathrm{d}z,\end{aligned}$$ where $\varphi^{tx, 1-t^2} (y)$ is the density of the $d$-dimensional Gaussian measure with mean $tx$ and covariance $(1-t^2)\mathbf{I}_d$. For the convenience of subsequent calculation, we introduce the following symbols: $$S(t, x) :=\nabla \log q_t(x), \quad q_t(x) :=\int_{\mathbb R^d} q(t,x|1,y) p(y) \mathrm{d}y$$ where $q(t,x|1,y) :=(2\pi (1-t^2))^{-\frac d 2} \exp\left(-\frac{|x-ty|^2}{2(1-t^2)} \right)$ for any $t\in [0,1]$. Notice that $$\begin{aligned} \EuScript{Q}_{1-t} r (x) = (2\pi)^{d/2} \exp\left(\frac{ |x |^2}{2} \right) \times \frac{1}{(2\pi(1-t^2))^{d/2}} \int_{\mathbb R^d} p(y)\exp\left(-\frac{|x-ty |^2 }{2(1-t^2)} \right) \mathrm{d}y.\end{aligned}$$ Then we have $\nabla \log \EuScript{Q}_{1-t} r(x) = x + S(t, x)$, for any $t \in [0,1]$. Suppose that the target distribution $p$ satisfies the third moment condition, we can supplement the definition of velocity field $V$ at time $t = 0$, so that $V$ is well-defined on the interval $[0,1]$. Then we have the following result: **Lemma 26**. *Suppose that $\mathbb E_{p}[|X|^3] <\infty$, then $$\lim_{t \downarrow 0} V(t, x) =\lim_{t \downarrow 0} \frac{x +S(t, x)}{t} =\mathbb E_{p}[X].$$* *Proof.* Let $t \to 0$, then it yields $$\begin{aligned} \lim_{t \downarrow 0} V(t, x) =\lim_{t \downarrow 0} \partial_t S(t,x) = \lim_{t \downarrow 0} \left \lbrace \frac{\nabla[\partial_{t} q_t(x)]} {q_t(x)}- \frac{\partial_{t} q_t(x)} {q_t(x)} S(t, x) \right \rbrace.\end{aligned}$$ On the one hand, by simple calculation, it holds that $$\begin{aligned} \partial_{t} q_t(x) & = \partial_{t} \int_{\mathbb R^d} q(t, x | 1, y) p(y) \mathrm{d}y = \partial_{t} \int_{\mathbb R^d} \left[ 2 \pi ( 1 - t^2 ) \right]^{-\frac{d}{2}} \exp \left( - \frac{|x - ty|^2} { 2 (1 - t^2) } \right) p(y) \mathrm{d}y \\ &= \frac { t d} {1 - t^2} q_t(x) - \frac{t} {(1 - t^2)^2} |x|^2 q_t(x) + \frac{1 + t^2} {(1 - t^2)^2} \int_{\mathbb R^d} x^{\top} y q(t,x | 1, y) p(y) \mathrm{d}y \\ & ~~~ - \frac{t} {(1 - t^2)^2} \int_{\mathbb R^d} |y|^2 q(t,x | 1,y) p(y) \mathrm{d}y.\end{aligned}$$ Furthermore, we also obtain $$\begin{aligned} \frac{\partial_{t} q_t(x)} {q_t(x)} = \frac {t d} {1 - t^2} - \frac{t} {(1 - t^2)^2} |x|^2 + \frac{1 + t^2} {(1 - t^2)^2} \int_{\mathbb R^d} x^{\top} y q(1,y | t,x) \mathrm{d}y -\frac{t} {(1 - t^2)^2} \int_{\mathbb R^d} |y|^2 q(1,y | t,x) \mathrm{d}y.\end{aligned}$$ On the other hand, by straightforward calculation, it yields $$\begin{aligned} \nabla [\partial_{t} q_t(x)] &= - \frac { td} {(1 - t^2)^2} x q_t(x) + \frac{t^2 d} {(1 - t^2)^2} \int_{\mathbb R^d} y q(t,x | 1,y) p(y) \mathrm{d}y \\ & \quad - \frac{2 t} {(1 - t^2)^2} x q_t(x) + \frac{t} {(1 - t^2)^3} |x|^2 x q_t(x) - \frac{t^2 |x|^2} {(1 - t^2)^3} \int_{\mathbb R^d} y q(t,x | 1,y) p(y) \mathrm{d}y \\ & \qquad + \frac{1 + t^2} {(1 - t^2)^2} \int_{\mathbb R^d} y q(t,x | 1,y) p(y) \mathrm{d}y \\ &\quad - \frac{1 + t^2} {(1 - t^2)^3} \int_{\mathbb R^d} (x^{\top} y)x q(t,x | 1,y) p(y) \mathrm{d}y + \frac{t (1 + t^2)} {(1 - t^2)^3} \int_{\mathbb R^d} (x^{\top} y)y q(t,x | 1,y) p(y) \mathrm{d}y\\ & \quad + \frac{t} {(1 - t^2)^3} \int_{\mathbb R^d} x |y|^2 q(t,x| 1,y) p(y) \mathrm{d}y - \frac{t^2} {(1 - t^2)^3} \int_{\mathbb R^d} y |y|^2 q(t,x | 1,y) p(y) \mathrm{d}y.\end{aligned}$$ Moreover, we also obtain $$\begin{aligned} \frac{\nabla [\partial_{t} q_t(x)]} {q_t(x)} & = -\frac {t d } {(1 - t^2)^2} x + \frac{t^2 d} {(1 - t^2)^2} \int_{\mathbb R^d} y q(1,y | t,x) \mathrm{d}y - \frac{2 tx} {(1 - t^2)^2} + \frac{t |x|^2 x} {(1 - t^2)^3} \\ & \quad - \frac{t^2} {(1 - t^2)^3} |x|^2 \int_{\mathbb R^d} y q(1,y | t,x) \mathrm{d}y + \frac{1 + t^2} {(1 - t^2)^2} \int_{\mathbb R^d} y q(1,y | t,x) \mathrm{d}y \\ & \quad - \frac{1 + t^2} {(1 - t^2)^3} \int_{\mathbb R^d} (x^{\top} y)x q(1,y | t,x) \mathrm{d}y + \frac{t (1 + t^2)} {(1 - t^2)^3} \int_{\mathbb R^d} (x^{\top} y)y q(1,y | t,x) \mathrm{d}y \\ & \quad + \frac{t} {(1 - t^2)^3} x \int_{\mathbb R^d} |y|^2 q(1,y | t,x) \mathrm{d}y - \frac{t^2} {(1 - t^2)^3} \int_{\mathbb R^d} y |y|^2 q(1,y | t,x) \mathrm{d}y.\end{aligned}$$ Since $\mathbb E_p[|X|^3] <\infty$, it yields $$\lim_{t \downarrow 0} \int_{\mathbb R^d} |y|^3 q(1, y | t, x) \mathrm{d}y = \int_{\mathbb R^d} |y|^3 \lim_{t \downarrow 0} q(1, y | t, x) \mathrm{d}y =\mathbb E_p \left[|X|^3 \right] < + \infty.$$ Furthermore, we have $$\begin{aligned} \lim_{t \downarrow 0} \frac{\partial_{t} q_t(x)} {q_t(x)} S(t, x) = - x x^{\top} \mathbb{E}_{p} [X], \quad \lim_{t \downarrow 0} \frac{\nabla[\partial_{t} q_t(x)]} {q_t(x)} &= \mathbb{E}_{p} [X] - x x^{\top} \mathbb{E}_{p} [X].\end{aligned}$$ Therefore, it yields $\lim_{t \downarrow 0} V(t, x) = \mathbb{E}_{p} [X]$, which completes the proof. ◻ ## Cramér-Rao inequality In order to obtain a lower bound of the Jacobian matrix of velocity field $V$ defined in ([\[eq:vector-field\]](#eq:vector-field){reference-type="ref" reference="eq:vector-field"}), we apply the classical Cramér-Rao bound [@rao1945information; @cramer1946mathematical] in statistical parameter estimation to a special case for location parameter estimation. This particular application is far from being new in information theory and convex geometry (for example, see [@dembo1991information; @lutwak2002cramer; @cianchi2013unified; @saumard2014log]). We include it here for the sake of completeness. A lower bound on the covariance matrix of a prescribed probability measure directly follows the Cramér-Rao bound [@cover2005chapter11 Theorem 11.10.1]. **Lemma 27** (Cramér-Rao bound). *Let $\mu_{\theta}(\mathrm{d}x) = f_{\theta}(x) \mathrm{d}x$ be a probability measure on ${\mathbb{R}}^d$ such that the density $f_{\theta}(x)$ is of class $C^2$ with respect to an unknown parameter $\theta \in \Theta$. Assume that a few mild regularity assumptions hold. Then provided that $\{ X_i \}_{i=1}^n$ are i.i.d. samples from $\mu_{\theta}$ with size $n$, the mean-squared error of any unbiased estimator $g(X_1, X_2, \cdots, X_n)$ for the parameter $\theta$ is lower bounded by the inverse of the Fisher information matrix: $$\begin{aligned} \mathbb{E}_{\mu_{\theta}} \left[ (g(X_1, X_2, \cdots, X_n) - \theta)^{\otimes 2} \right] \succeq \left( - n \mathbb{E}_{\mu_{\theta}} \left[ \frac{\partial^2} {\partial \theta^2} \log f_{\theta}(X_1) \right] \right)^{-1}.\end{aligned}$$* We consider the example of location parameter estimation. Suppose $\theta$ is the location parameter and let $f_{\theta} (x) = f(x - \theta)$ and $g(x) = x$. Specifically, it yields a lower bound on the covariance matrix of the probability measure $\mu_{\theta}$ in the case that $\theta = \mathbb{E}_{\mu_{\theta}} [X]$, i.e., a random sample $X \sim \mu_{\theta}$ is an unbiased estimator of the mean $\theta$. Apart from this implication, an alternative proof of the same lower bound on the covariance matrix is presented in [@chewi2022entropic]. It is worth noting that a compactly supported probability measure $\mu$ would suffice to ensure the Cramér-Rao inequality holds. **Lemma 28**. *Let $\mu(\mathrm{d}x) = \exp(-U(x)) \mathrm{d}x$ be a probability measure on ${\mathbb{R}}^d$ such that $U$ of class $C^2$ on the interior of its domain. Suppose $X$ is a random sample from $\mu$. Then the covariance matrix is lower bounded as $\mathrm{Cov}_{\mu} (X) \succeq \left( \mathbb E_{\mu} \left[ \nabla^2 U(X) \right] \right)^{-1}$.* ## Proof of Propositions [Proposition 12](#prop:vp-ode-eps){reference-type="ref" reference="prop:vp-ode-eps"} {#proof-of-propositions-propvp-ode-eps} *Proof.* By Itô SDE defined in ([\[eq:vp-sde-tr\]](#eq:vp-sde-tr){reference-type="ref" reference="eq:vp-sde-tr"}), we have the distribution $\overline{X}_t$, which is given by $$\label{eq-sp-x} \overline{X}_t|\overline{X}_0=x_0 \sim N((1-t)x_0, t(2-t) \mathbf{I}_d).$$ Due to the Cauchy-Lipschitz theory [@ambrosio2014continuity Section 2], the push-forward map $X^{*}_t$ and process $\overline{X}_t$ have the same distribution for any $t\in [0,1-\varepsilon]$. Then by ([\[eq-sp-x\]](#eq-sp-x){reference-type="ref" reference="eq-sp-x"}), we obtain $$X^{*}_t \overset{d}{=} \overline{X}_{t} \overset{d}{=} (1-t) X + \sqrt{t(2-t)} Y$$ with $X \sim \nu, Y \sim \gamma_d$. Recall that $X^{*}_{1 - \varepsilon}$ and $\varepsilon X + \sqrt{1-\varepsilon^2} Y$ have the same distribution. Therefore, by the definition of $W_2$ and Cauchy-Schwarz's inequality, it yields $$\begin{aligned} W^2_2 (\nu \circ ({X^{*}_{1 - \varepsilon}})^{-1}, \gamma_d) & \leq \int_{\mathbb R^d \times \mathbb R^d} |\varepsilon x +(\sqrt{1-\varepsilon^2} -1) y |^2 p(x) \phi(y) \mathrm{d}x \mathrm{d}y \notag \\ & \leq 2 \varepsilon^2 \int_{\mathbb R^d} |x|^2 p(x) \mathrm{d}x + 2 \left(\sqrt{1-\varepsilon^2} -1 \right)^2 \int_{\mathbb R^d} |y|^2 \phi(y) \mathrm{d}y \notag \\ &= 2 \varepsilon^2 \, \mathbb E_p[|X|^2] + 2d\left(\sqrt{1-\varepsilon^2} -1 \right)^2.\end{aligned}$$ Let $\varepsilon \rightarrow 0$, and it yields $\lim_{\varepsilon \to 0} W_2(\nu \circ ({X^{*}_{1 - \varepsilon}})^{-1}, \gamma_d) = 0$, which completes the proof. ◻ # Proof of Theorem [Theorem 5](#main-thm2){reference-type="ref" reference="main-thm2"} and Theorem [Theorem 6](#main-thm3){reference-type="ref" reference="main-thm3"} {#proof-of-theorem-main-thm2-and-theorem-main-thm3} ## Bound on the Lipschitz constant of the flow map We take a close look at Lipschitz properties of the Föllmer flow ([\[main-ode\]](#main-ode){reference-type="ref" reference="main-ode"}). In order to derive functional inequalities, we deploy the approach of Lipschitz changes of variables from the Gaussian measure $\gamma_d$ to the target measure $\nu$. One key argument is to bound the maximum eigenvalue of the Jacobian matrix of velocity field denoted as $\lambda_{\max} (\nabla V(t, x))$. By integrating both sides of ([\[main-ode\]](#main-ode){reference-type="ref" reference="main-ode"}) w.r.t. time $s \in [0, t]$, we have $$\label{eq:vp-ode-integral} X_t(x) - X_0(x) = \int_0^t V(s, X_s(x)) \mathrm{d}s, \quad X_0(x)=x.$$ Taking the first-order derivative w.r.t $x$ on both sides of ([\[eq:vp-ode-integral\]](#eq:vp-ode-integral){reference-type="ref" reference="eq:vp-ode-integral"}), we get $$\label{eq:vp-map-gradx} \nabla X_t(x) - \nabla X_0(x) = \int_0^t \nabla V(s, X_s(x)) \nabla X_s(x) \mathrm{d}s.$$ Taking the first-order derivative w.r.t $t$ on both sides of ([\[eq:vp-map-gradx\]](#eq:vp-map-gradx){reference-type="ref" reference="eq:vp-map-gradx"}), we get $$\label{eq:vp-map-gradx-gradt} \frac{\partial} {\partial t} \nabla X_t(x) = \nabla V(t, X_t(x)) \nabla X_t(x).$$ Let $a_t = | \nabla X_t(x) r |^2$ with $|r|=1$. Assume $\lambda_{\max} (\nabla V(t, x)) \le \theta_t$. By ([\[eq:vp-map-gradx-gradt\]](#eq:vp-map-gradx-gradt){reference-type="ref" reference="eq:vp-map-gradx-gradt"}) we get $$\begin{aligned} \frac{\partial} {\partial t} a_t = 2 \left<(\nabla X_t)r, \frac{\partial}{\partial t} (\nabla X_t) r \right> = 2 \left<(\nabla X_t)r, \left(\nabla V(t, X_t) \nabla X_t\right) r \right> \leq 2\theta_t a_t.\end{aligned}$$ The above display and Grönwall's inequality imply $$\begin{aligned} \|\nabla X_t (x) \|_{\mathrm{op}}=\sup_{\|r \|_2=1} \sqrt{a_t} \le \sup_{\|r \|_2=1} \sqrt{a_0} \exp \left( \int_0^t \theta_s \mathrm{d}s \right) = \exp \left( \int_0^t \theta_s \mathrm{d}s \right).\end{aligned}$$ Let $t = 1$, then we get $$\label{eq:vp-map-lips} \mathrm{Lip}(X_1(x)) \le \|\nabla X_1 (x) \|_{\mathrm{op}} \le \exp \left( \int_0^1 \theta_s \mathrm{d}s \right).$$ ## Lipschitz properties of transport maps In this subsection, we show that the considered flow map is Lipschitz in various settings. The following is the main result of this subsection and it covers the Lipschitz statements of Theorem [Theorem 5](#main-thm2){reference-type="ref" reference="main-thm2"} and Theorem [Theorem 6](#main-thm3){reference-type="ref" reference="main-thm3"}. **Theorem 29**. - *Suppose that either $p$ is $\kappa$-semi-log-concave for some $\kappa >0$, or $p$ is $\kappa$-semi-log-concave for some $\kappa \in \mathbb R$ and $D <+ \infty$. Then the Föllmer flow ([\[main-ode\]](#main-ode){reference-type="ref" reference="main-ode"}) has a unique solution for all $t\in [0,1]$. Furthermore,* - *If $\kappa D^2 \ge 1$, then $X_1(x)$ is a Lipschitz mapping with constant $\tfrac{1}{\sqrt{\kappa}}$, or equivalently, $$\|\nabla X_1(x) \|^2_{\mathrm{op}} \le \frac{1}{\kappa}, \quad \forall x \in \mathbb{R}^d.$$* - *If $\kappa D^2 <1$, then $X_1(x)$ is a Lipschitz mapping with constant $\exp\left(\frac{1-\kappa D^2}{2} \right)D$, or equivalently, $$\|\nabla X_1(x) \|^2_{\mathrm{op}} \le \exp\left( 1-\kappa D^2 \right)D^2, \quad \forall x \in \mathbb{R}^d.$$* - *Fix a probability measure $\rho$ on $\mathbb R^d$ supported on a ball of radius $R$ and let $p :=N(0,\sigma^2 \mathbf{I}_d) * \rho$. Then the Föllmer flow ([\[main-ode\]](#main-ode){reference-type="ref" reference="main-ode"}) has a unique solution for all $t\in [0,1]$. Furthermore, $X_1(x)$ is a Lipschitz mapping with constant $\sigma \exp \left( \frac{R^2}{2 \sigma^2} \right)$, or equivalently, $$\|\nabla X_1(x) \|^2_{\mathrm{op}} \le \sigma^2 \exp\left( \frac{R^2}{\sigma^2} \right), \quad \forall x \in \mathbb{R}^d.$$* *Proof.* Combine Theorem [Theorem 31](#thm1){reference-type="ref" reference="thm1"}-(4) and Corollaries [Corollary 32](#cor1){reference-type="ref" reference="cor1"}-[Corollary 33](#cor2){reference-type="ref" reference="cor2"} and then complete the proof. ◻ In fact, the existence of a solution to the IVP ([\[main-ode\]](#main-ode){reference-type="ref" reference="main-ode"}) also relies on controlling $\nabla V$. To this end, we represent $\nabla V$ as a covariance matrix. We start by defining a measure $p^{tx, 1-t^2}$ on ${\mathbb{R}}^d$, for fixed $t \in [0, 1)$ and $x \in {\mathbb{R}}^d$, by $$\begin{aligned} \label{eq:measure-cov-vp} p^{tx, 1-t^2} (y) :=& \frac{ \varphi^{tx, 1-t^2} (y) r (y) } {\EuScript{Q}_{1-t} r (x)}, \quad r := \frac{\mathrm{d}\nu} {\mathrm{d}\gamma_d} = \frac{p} {\phi},\end{aligned}$$ where $\varphi^{tx, 1-t^2} (y)$ is the density of the $d$-dimensional Gaussian measure with mean $tx$ and covariance $(1-t^2)\mathbf{I}_d$ and $$\begin{aligned} \label{eq:vp-vf-op-form} \EuScript{Q}_{1-t} r (x) = \int_{\mathbb R^d} \varphi^{tx, 1-t^2} (y) r(y) \mathrm{d}y = \int_{\mathbb{R}^d} \phi (z) r (tx + \sqrt{1 - t^2}z) \mathrm{d}z.\end{aligned}$$ Notice that $$\begin{aligned} \EuScript{Q}_{1-t} r (x)= (2\pi)^{d/2} \exp\left(\frac{ |x |^2}{2} \right) \frac{1}{(2\pi(1-t^2))^{d/2}} \int_{\mathbb R^d} p(y)\exp\left(-\frac{|x-ty |^2 }{2(1-t^2)} \right) \mathrm{d}y.\end{aligned}$$ Hence, we obtain $$V(t, x) =\frac{x + S(t, x)}{t} = \frac{\nabla \log \EuScript{Q}_{1-t} r (x)}{t}, \quad 0< t \le 1.$$ **Lemma 30**. *Suppose velocity field $V$ is defined in ([\[eq:vector-field\]](#eq:vector-field){reference-type="ref" reference="eq:vector-field"}), then $$\label{eq:cov-vp-vf} \nabla V(t, x) = \frac{t} {(1 - t^2)^2} \mathrm{Cov}\left( p^{tx, 1-t^2} \right) - \frac{t} {1 - t^2} {\mathbf{I}}_d, \quad \forall t\in (0,1), \quad \nabla V(0, x) =0.$$* *Proof.* By taking the first-order and the second-order derivatives on both sides in ([\[eq:vp-vf-op-form\]](#eq:vp-vf-op-form){reference-type="ref" reference="eq:vp-vf-op-form"}), we get $$\begin{aligned} \nabla \EuScript{Q}_{1-t} r (x) &= \frac{t} {1 - t^2} \int_{\mathbb R^d} (y - tx) \varphi^{tx, 1-t^2} (y) r(y) \mathrm{d}y, \\ \nabla^2 \EuScript{Q}_{1-t} r (x) &= \frac{t^2} { (1 - t^2)^2 } \int_{\mathbb R^d} (y - tx)^{\otimes 2} \varphi^{tx, 1-t^2} (y) r(y) \mathrm{d}y - \left( \frac{t^2} {1 - t^2} \int_{\mathbb R^d} \varphi^{tx, 1-t^2} (y) r(y) \mathrm{d}y \right) {\mathbf{I}}_d.\end{aligned}$$ Then we obtain $$\begin{aligned} & \nabla^2 \log \EuScript{Q}_{1-t} r (x) \\ =&\frac{ \nabla^2 \EuScript{Q}_{1-t} r (x) } { \EuScript{Q}_{1-t} r (x) } - \left( \frac{ \nabla \EuScript{Q}_{1-t} r (x) } { \EuScript{Q}_{1-t} r (x) } \right)^{\otimes 2} \\ =& \frac{t^2} { (1 - t^2)^2 } \left[ \int_{\mathbb R^d} (y - tx)^{\otimes 2} p^{tx, 1-t^2} (y) \mathrm{d}y - \left( \int_{\mathbb R^d} (y - tx) p^{tx, 1-t^2} (y) \mathrm{d}y \right)^{\otimes 2} \right] - \frac{t^2} {1 - t^2} {\mathbf{I}}_d \\ =& \frac{t^2} { (1 - t^2)^2 } \left[ \int_{\mathbb R^d} y^{\otimes 2} p^{tx, 1-t^2} (y) \mathrm{d}y - \left( \int_{\mathbb R^d} y p^{tx, 1-t^2} (y) \mathrm{d}y \right)^{\otimes 2} \right] - \frac{t^2} {1 - t^2} {\mathbf{I}}_d\\ =& \frac{t^2} { (1 - t^2)^2 } \mathrm{Cov}(p^{tx, 1-t^2} ) - \frac{t^2} {1 - t^2} {\mathbf{I}}_d.\end{aligned}$$ Therefore, we get $$\label{eq:cov-vp-score} \nabla V(t, x) = \frac{t} {( 1 - t^2 )^2} \mathrm{Cov}( p^{tx, 1-t^2} ) - \frac{t} {1 - t^2} {\mathbf{I}}_d.$$ This completes the proof. ◻ Next, we use the representation of ([\[eq:cov-vp-vf\]](#eq:cov-vp-vf){reference-type="ref" reference="eq:cov-vp-vf"}) to estimate the upper bound of $\nabla V(t, x)$. **Theorem 31**. *Let $p$ be a probability measure on $\mathbb R^d$ with $D := (1/\sqrt{2} ) \mathrm{diam} (\mathrm{supp} (p))$.* - *For every $t \in [0, 1)$, $$\label{eq:vp-vf-ubd-bounded} \frac{t}{1-t^2} \mathbf{I}_d \preceq \nabla V(t, x) \preceq \left( \frac{t D^2} {(1 - t^2)^2} - \frac{t} {1 - t^2} \right) {\mathbf{I}}_d.$$* - *Suppose that $p$ is $\beta$-semi-log-convex with $\beta \in (0, +\infty)$. Then for any $t\in [0,1]$, $$\nabla V(t, x) \succeq \frac{t (1 - \beta)} {\beta (1 - t^2) + t^2} {\mathbf{I}}_d.$$ In particular, when $p \sim N\left(0, \frac{1}{\beta} \mathbf{I}_d \right)$, then $$\nabla V(t, x) =\frac{t (1 - \beta)} {\beta (1 - t^2) + t^2} {\mathbf{I}}_d.$$* - *Let $\kappa \in {\mathbb{R}}$ and suppose that $p$ is $\kappa$-semi-log-concave. Then for any $t \in \left[ \sqrt{ \frac{\kappa} {\kappa - 1} \mathds{1}_{\kappa < 0} }, 1 \right]$, $$\label{eq:vp-vf-ubd-log-concave} \nabla V(t, x) \preceq \frac{t (1 - \kappa)} {\kappa (1 - t^2) + t^2} {\mathbf{I}}_d.$$* - *Fix a probability measure $\rho$ on ${\mathbb{R}}^d$ supported on a ball of radius $R$ and let $p := N(0,\sigma^2 \mathbf{I}_d) * \rho$ with $\sigma > 0$. Then for any $t \in [0, 1]$, $$\begin{aligned} \label{eq:vp-vf-ubd-convolution} \frac{(\sigma^2-1) t}{1+ (\sigma^2-1)t^2} \mathbf{I}_d \preceq \nabla V(t, x) \preceq t \left\{ \frac{ (\sigma^2 - 1) [ 1 + (\sigma^2 - 1) t^2 ] + R^2 } { [ 1 + (\sigma^2 - 1) t^2 ]^2 } \right\} {\mathbf{I}}_d. \end{aligned}$$* *Proof.* The proof idea of this theorem follows similar arguments as in [@mikulincer2021brownian Lemma 3.3]. - By [@danzer1963helly Theorem 2.6], there exists a closed ball with radius less than $D := (1/\sqrt{2} ) \mathrm{diam} (\mathrm{supp} (p))$ that contains $\mathrm{supp} (p)$ in ${\mathbb{R}}^d$. Then the desired bounds are a direct result of $0 {\mathbf{I}}_d \preceq \mathrm{Cov}(p^{tx, 1-t^2}) \preceq D^2 {\mathbf{I}}_d$ and ([\[eq:cov-vp-vf\]](#eq:cov-vp-vf){reference-type="ref" reference="eq:cov-vp-vf"}). - For any $t\in (0,1)$, recall that ([\[eq:cov-vp-vf\]](#eq:cov-vp-vf){reference-type="ref" reference="eq:cov-vp-vf"}) reads $$\label{eq-cov-v} \nabla V(t, x) = \frac{t} {(1 - t^2)^2} \mathrm{Cov}\left( p^{tx, 1-t^2} \right) - \frac{t} {1 - t^2} {\mathbf{I}}_d.$$ On the one hand, let $p$ be $\beta$-semi-log-convex for some $\beta >0$. Then for any $t\in [0,1), p^{tx, 1-t^2}$ is $\left(\beta +\frac{t^2}{1-t^2} \right)$-semi-log-convex because $$\begin{aligned} -\nabla^2 \log \left( p^{tx,1-t^2}(y) \right) =-\nabla^2 \log \left(r(y) \phi(y) \right) -\nabla^2 \log \left(\frac{\varphi^{tx,1-t^2}(y)}{\phi(y) } \right) \preceq \left(\beta +\frac{t^2}{1-t^2} \right) \mathbf{I}_d\end{aligned}$$ where we use that $p(y) = r(y) \phi(y)$. On the other hand, by Lemma [Lemma 28](#lem-cramer-rao){reference-type="ref" reference="lem-cramer-rao"}, we obtain $$\mathrm{Cov}\left(p^{tx,1-t^2} \right) \succeq \left(\beta +\frac{t^2}{1-t^2} \right)^{-1} {\mathbf{I}}_d.$$ Furthermore, by ([\[eq-cov-v\]](#eq-cov-v){reference-type="ref" reference="eq-cov-v"}), we obtain $$\begin{aligned} \nabla V(t, x) \succeq \left \lbrace \frac{t}{(1-t^2)^2} \left(\beta +\frac{t^2}{1-t^2} \right)^{-1} -\frac{t}{1-t^2} \right \rbrace \mathbf{I}_d =\frac{t(1-\beta)}{\beta (1-t^2) +t^2} \mathbf{I}_d.\end{aligned}$$ Recall that $(X_t)_{t \in [0, 1]}$ satisfies the IVP ([\[main-ode\]](#main-ode){reference-type="ref" reference="main-ode"}), then we have $$\nabla V(t, x) = \frac{\nabla^2 \log \EuScript{Q}_{1-t} r(x)}{t}, \quad r(x) :=\frac{p(x)}{\phi(x)}.$$ Since $p \sim N\left(0,\frac{1}{\beta} \mathbf{I}_d \right)$, then it yields $$r(x) = \beta^{d/2} \exp\left(-\frac{\beta -1}{2} |x|^2 \right) \propto \exp\left(-\frac{\beta -1}{2} |x|^2 \right),$$ where the symbol $\propto$ signifies equality up to a constant which does not depend on $x$. Then by straightforward calculation for $\EuScript{Q}_{1-t} r(x)$, we obtain $$\begin{aligned} \EuScript{Q}_{1-t} r(x) & \propto \int_{\mathbb R^d} \exp \left \lbrace -\frac{\beta -1}{2} \left|tx +\sqrt{1-t^2} y \right|^2 -\frac{|y|^2}{2} \right \rbrace \mathrm{d}y \\ &= \int_{\mathbb R^d} \exp \left \lbrace -\frac{(\beta -1)t^2}{2}|x|^2 -(\beta -1)t \sqrt{1-t^2} \left<x,y \right> -\frac{\beta(1-t^2) +t^2}{2} |y|^2 \right \rbrace \mathrm{d}y \\ &= \exp\left(-\frac{(\beta-1)t^2 |x|^2 }{2\beta_t} \right) \int_{\mathbb R^d} \exp\left \lbrace -\frac{\beta_t}{2} \left|y +\frac{(\beta -1)t \sqrt{1-t^2} }{\beta_t} x \right|^2 \right \rbrace \mathrm{d}y, \end{aligned}$$ where we denote $\beta_t :=(1-t^2) \beta +t^2$. Considering that the integrand in the last line is proportional to the density of a Gaussian measure, then the value of the integral does not depend on $x$, and $$\begin{aligned} \EuScript{Q}_{1-t} r(x) \propto \exp\left(-\frac{(\beta-1)t^2 |x|^2 }{2\beta_t} \right) =\exp\left(-\frac{|x|^2}{2} \cdot \frac{(\beta -1)t^2}{(1-t^2)\beta +t^2} \right).\end{aligned}$$ So we have $$\nabla V(t, x) =\frac{\nabla^2 \EuScript{Q}_{1-t} r(x)}{t} =\frac{(1-\beta) t}{(1-t^2) \beta +t^2} \mathbf{I}_d.$$ - Let $p$ be $\kappa$-semi-log-concave. Then for any $t\in [0,1)$, $p^{tx, 1-t^2}$ is $\left(\kappa +\frac{t^2}{1-t^2} \right)$-semi-log-concave because $$\begin{aligned} -\nabla^2 \log \left( p^{tx,1-t^2} (y) \right) =-\nabla^2 \log \left(r(y) \phi(y) \right) -\nabla^2 \log \left(\frac{\varphi^{tx,1-t^2}(y)}{\phi(y) } \right) \succeq \left(\kappa +\frac{t^2}{1-t^2} \right) \mathbf{I}_d\end{aligned}$$ where we use $p(y) =r(y) \phi(y)$. If $t\in \left[ \sqrt{ \frac{\kappa} {\kappa - 1} \mathds{1}_{\kappa < 0} }, 1 \right]$, then $\kappa +\frac{t^2}{1-t^2} \ge 0$. By the well-known Brascamp-Lieb inequality [@bakry2014analysis; @brascamp1976extensions], applied to functions of the form $\mathbb R^d \ni x \mapsto f(x)= \left<x, v \right>$ for any $v \in \mathbb S^{d-1}$, we obtain $$\mathrm{Cov}\left(p^{tx,1-t^2} \right) \preceq \left(\kappa +\frac{t^2}{1-t^2} \right)^{-1} {\mathbf{I}}_d$$ and the result follows by ([\[eq:cov-vp-vf\]](#eq:cov-vp-vf){reference-type="ref" reference="eq:cov-vp-vf"}). - On the one hand, we have $$\begin{aligned} p^{tx,1-t^2}(y) =\frac{(N(0,\sigma^2 \mathbf{I}_d) * \rho)(y)}{\varphi^{0,1}(y)} \cdot \frac{\varphi^{tx,1-t^2}(y)}{\EuScript{Q}_{1-t} \left( \frac{N(0,\sigma^2 \mathbf{I}_d) *\rho )}{\varphi^{0,1}} \right) (x) } =A_{x,t} \int_{\mathbb R^d} \varphi^{z,\sigma^2}(y) \varphi^{\frac{x}{t}, \frac{1-t^2}{t^2}} (y) \rho(\mathrm{d}z),\end{aligned}$$ where the constant $A_{x,t}$ depends only on $x$ and $t$. Moreover, we obtain $$\ p^{tx,1-t^2}(y) =\int_{\mathbb R^d} \varphi^{\frac{(1-t^2)z +\sigma^2 tx}{1+(\sigma^2-1)t^2},\frac{\sigma^2(1-t^2)}{1+(\sigma^2-1)t^2}}(y) \tilde \rho(\mathrm{d}z)$$ where $\tilde \rho$ is a probability measure on $\mathbb R^d$ which is a multiple of $\rho$ by a positive function. In particular, $\tilde \rho$ is supported on the same ball as $\rho$. On the other hand, let $Z \sim \gamma_d$ and $Y \sim \tilde{\rho}$ be independent. Then $$\begin{aligned} \sqrt{\frac{\sigma^2(1-t^2)}{1+(\sigma^2-1)t^2}} Z +\frac{(1-t^2)}{1+(\sigma^2-1)t^2}Y +\frac{t \sigma^2}{1+(\sigma^2-1) t^2} x \sim p^{tx,1-t^2}.\end{aligned}$$ Due to $0 {\mathbf{I}}_d \preceq \mathrm{Cov}(Y) \preceq R^2 {\mathbf{I}}_d$, it holds that $$\begin{aligned} \frac{\sigma^2 (1-t^2)}{1 + (\sigma^2-1) t^2} \mathbf{I}_d \preceq \mathrm{Cov}(p^{tx,1-t^2}) \preceq \frac{\sigma^2(1-t^2)[1+(\sigma^2-1)t^2] + (1-t^2)^2 R^2}{[1+(\sigma^2-1)t^2]^2} \mathbf{I}_d.\end{aligned}$$ By applying ([\[eq:cov-vp-vf\]](#eq:cov-vp-vf){reference-type="ref" reference="eq:cov-vp-vf"}) again, it yields $$\begin{aligned} \frac{(\sigma^2 -1)t}{1+(\sigma^2 -1)t^2} \mathbf{I}_d \preceq \nabla V(t, x) \preceq t \left\{ \frac{ (\sigma^2 - 1) [ 1 + (\sigma^2 - 1) t^2 ] + R^2 } { [ 1 + (\sigma^2 - 1) t^2 ]^2 } \right\} {\mathbf{I}}_d.\end{aligned}$$ This completes the proof of Theorem [Theorem 31](#thm1){reference-type="ref" reference="thm1"}. ◻ Next, we present an upper bound on $\lambda_{\max} (\nabla V(t, x))$ and its exponential estimation. **Corollary 32**. *Let $p$ be a probability measure on $\mathbb R^d$ with $D := (1/\sqrt{2}) \mathrm{diam} ( \mathrm{supp} (p))$ and suppose that $p$ is $\kappa$-semi-log-concave with $\kappa \in [0, +\infty)$.* - *If $\kappa D^2 \ge 1$, then $$\label{eq:vp-max-egv-ubd-ge1} \lambda_{\max} (\nabla V(t, x)) \le \theta_t := \frac{t (1 - \kappa)} {t^2 (1 - \kappa) + \kappa}.$$ and $$\label{eq:vp-max-egv-ubd-exp-ge1-time-1} \exp \left( \int_{0}^1 \theta_s \mathrm{d}s \right) = \frac{1} {\sqrt{\kappa}}.$$* - *If $\kappa D^2 < 1$, then $$\label{eq:vp-max-egv-ubd-less1} \lambda_{\max} (\nabla V(t, x)) \le \theta_t := \begin{cases} \frac{t (t^2 + D^2 - 1)} {(1 - t^2)^2}, \ &t \in [0, t_0], \\ \frac{t (1 - \kappa)} {t^2 (1 - \kappa) + \kappa}, \ &t \in [t_0, 1], \end{cases}$$ where $t_0 = \sqrt{ \frac{1 - \kappa D^2} {(1 - \kappa) D^2 + 1}}$ and $$\label{eq:vp-max-egv-ubd-exp-less1} \exp \left( \int_{0}^1 \theta_s \mathrm{d}s \right) = \exp \left( \frac{1-\kappa D^2}{2} \right) D.$$* *Proof.* By Theorem [Theorem 31](#thm1){reference-type="ref" reference="thm1"}, we obtain $$\begin{aligned} \lambda_{\max}(\nabla V(t, x)) \le \frac{tD^2}{(1-t^2)^2} -\frac{t}{1-t^2}, \quad \lambda_{\max}(\nabla V(t, x)) \le \frac{t(1-\kappa)}{\kappa(1-t^2) +t^2}, \quad \forall t\in [0,1].\end{aligned}$$ By simple algebra calculation, it yields $$\begin{aligned} \frac{t(D^2 +t^2-1)}{(1-t^2)^2} \le \frac{t(1-\kappa)}{\kappa(1-t^2)+t^2} \quad \text{if and only if} \quad (1+D^2-\kappa D^2)t^2 \le 1-\kappa D^2.\end{aligned}$$ We consider two cases. - $\kappa D^2 \ge 1$: By considering $\kappa D^2=1$, we see that the bound $(1+D^2-\kappa D^2)t^2 \le 1-\kappa D^2$ cannot hold. So it would be advantageous to use the bound $$\begin{aligned} \lambda_{\max}(\nabla V(t, x)) \le \theta_t := \frac{t(1-\kappa)}{\kappa(1-t^2) +t^2} = \frac{t(1-\kappa)}{t^2(1-\kappa) +\kappa}.\end{aligned}$$ Next, we will compute $\exp\left(\int_0^1 \theta_t \mathrm{d}t\right)$ and we first check that the integral $\int_0^1 \theta_t \mathrm{d}t$ is well-defined. For this reason, we only need to consider whether the sign of the denominator $(1-\kappa)t^2 +\kappa$ is equal to 0. The only case is $(1-\kappa)t^2 +\kappa =0$ that happens when $t^2_0:= \kappa/(\kappa -1)$. If $\kappa \in (0,1]$, $(1-\kappa) t^2 +\kappa \ne 0$. Thus, $\theta_t$ is integrable on $[0,1]$. If $\kappa >1$, $t_0 >1$. Then $\theta_t$ is integrable on $[0,1]$ as well. The only case is $\kappa = 0$ which results in $t_0 =0$. However, in this case, we cannot have $\kappa D^2 \ge 1$ as $\kappa =0$. Then by simple calculation, $$\begin{aligned} \int_0^1 \theta_t \mathrm{d}t =(1-\kappa)\int_0^1 \frac{t \mathrm{d}t}{(1-\kappa)t^2 +\kappa} =-\frac{1}{2}\log \kappa, \quad \exp\left(\int_0^1 \theta_t \mathrm{d}t \right) \le \frac{1}{\sqrt{\kappa}}.\end{aligned}$$ - $\kappa D^2 <1$: The condition $(1+D^2 -\kappa D^2) t^2 \le 1-\kappa D^2$ is equivalent to $$t \le \sqrt{\frac{1-\kappa D^2}{1+(1-\kappa) D^2}}$$ since the denominator is nonnegative as $\kappa D^2 <1$. Hence, we define $$\lambda_{\max} (\nabla V(t, x)) \le \theta_t := \begin{cases} \tfrac{t (t^2 + D^2 - 1)} {(1 - t^2)^2}, \ & 0\le t \le t_0 , \\ \tfrac{t (1 - \kappa)} {t^2 (1 - \kappa) + \kappa}, \ & t_0 \le t \le 1, \end{cases}$$ where $t_0 := \sqrt{ \tfrac{1 - \kappa D^2} {(1 - \kappa) D^2 + 1}}$. In order to compute integral $\int_0^1 \theta_t \mathrm{d}t$, we note that, following the discussion in the case $\kappa D^2 \ge 1$, the denominators $1-t^2$ and $(1-\kappa )t^2 +\kappa$ do not vanish in the intervals $[0,t_0]$ and $[t_0,1]$, respectively. For $t \in [0, t_0]$, using integral by parts, we have $$\begin{aligned} & \int_0^{t_0} \frac{t(t^2 +D^2 -1)}{(1-t^2)^2} \mathrm{d}t =\frac{1}{2} \int_0^{t_0} (t^2+D^2-1) \mathrm{d}\left(\frac{1}{1-t^2} \right) \\ & =\left. \frac{t^2+D^2-1}{2(1-t^2)} \right|^{t=t_0}_{t=0} -\frac{1}{2} \int_0^{t_0} \frac{2t}{1-t^2} \mathrm{d}t = \frac{t^2_0}{2(1-t^2_0)} D^2 +\frac{1}{2} \log(1-t^2_0) \\ & =\frac{1-\kappa D^2}{2} +\frac{1}{2} \log \left(\frac{D^2}{1+(1-\kappa) D^2} \right).\end{aligned}$$ For $t\in [t_0, 1]$, we have $$\begin{aligned} \int_{t_0}^1 \frac{t(1-\kappa)}{\kappa +(1-\kappa)t^2} \mathrm{d}t = -\frac{1}{2} \log \left(t^2_0 +(1-t^2_0) \kappa \right) =-\frac{1}{2} \log \left(\frac{1}{1+(1-\kappa) D^2} \right).\end{aligned}$$ Hence, we obtain $$\int_0^1 \theta_t \mathrm{d}t=\int_0^{t_0} \theta_t \mathrm{d}t +\int_{t_0}^1 \theta_t \mathrm{d}t =\frac{1-\kappa D^2}{2} +\log D.$$ Then $$\exp\left(\int_0^1 \theta_t \mathrm{d}t \right) =\exp\left(\frac{1-\kappa D^2}{2} +\log D \right) =D \exp\left(\frac{1-\kappa D^2}{2} \right).$$ This completes the proof of Corollary [Corollary 32](#cor1){reference-type="ref" reference="cor1"}. ◻ **Corollary 33**. *Let $p$ be a probability measure on $\mathbb R^d$ with $D := (1/\sqrt{2}) \mathrm{diam} ( \mathrm{supp} (p)) < \infty$ and suppose that $p$ is $\kappa$-semi-log-concave with $\kappa \in (-\infty, 0)$. We have $$\label{eq:vp-max-egv-ubd-kappa-nega} \lambda_{\max} (\nabla V(t, x)) \le \theta_t := \begin{cases} \tfrac{t (t^2 + D^2 - 1)} {(1 - t^2)^2}, \ &t \in [0, t_0] \\ \tfrac{t (1 - \kappa)} {t^2 (1 - \kappa) + \kappa}, \ &t \in [t_0, 1] \end{cases}$$ where $t_0 = \sqrt{ \tfrac{1 - \kappa D^2} {(1 - \kappa) D^2 + 1} }$ and $$\label{eq:vp-max-egv-ubd-exp-kappa-nega} \exp \left( \int_{0}^1 \theta_s \mathrm{d}s \right) = \exp \left( \frac{1-\kappa D^2}{2} \right) D.$$* *Proof.* By Theorem [Theorem 31](#thm1){reference-type="ref" reference="thm1"}, we obtain $$\begin{aligned} \lambda_{\max}(\nabla V(t, x)) \le \frac{tD^2}{(1-t^2)^2} -\frac{t}{1-t^2}, \quad \forall t\in [0, 1), \quad \lambda_{\max}(\nabla V(t, x)) \le \frac{t(1-\kappa)}{\kappa(1-t^2) +t^2}, \quad \forall t \in \left[\sqrt{\frac{\kappa}{\kappa -1}}, 1 \right].\end{aligned}$$ Then it yields $$\lambda_{\max}(\nabla V(t, x)) \le \frac{t(t^2 +D^2-1)}{(1-t^2)^2}, \quad \forall t\in \left[0, \sqrt{\frac{\kappa}{\kappa -1}} \right).$$ Next, since $0<\sqrt{\tfrac{\kappa}{\kappa -1}} <\sqrt{\tfrac{1-\kappa D^2}{(1-\kappa)D^2 +1}} \le 1$ and $\kappa(1-t^2) +t^2 \ge 0$ for all $t \ge \sqrt{\frac{\kappa}{\kappa -1}}$, then one obtains $$\frac{t(t^2 +D^2 -1)}{(1-t^2)^2} \le \frac{t(1-\kappa)}{\kappa(1-t^2) +t^2}$$ for all $t\in \left[\sqrt{\frac{\kappa}{\kappa -1}}, \sqrt{\frac{-\kappa D^2}{(1-\kappa)D^2 +1}} \right]$. We define $$\lambda_{\max} (\nabla V(t, x)) \le \theta_t := \begin{cases} \tfrac{t (t^2 + D^2 - 1)} {(1 - t^2)^2}, \ &t \in [0, t_0] \\ \tfrac{t (1 - \kappa)} {t^2 (1 - \kappa) + \kappa}, \ &t \in [t_0, 1] \end{cases}$$ where $t_0 := \sqrt{ \tfrac{1 - \kappa D^2} {(1 - \kappa) D^2 + 1} }$. As in the proof of Corollary [Corollary 32](#cor1){reference-type="ref" reference="cor1"}, it holds that $$\begin{aligned} \int_0^{t_0} \theta_t \mathrm{d}t = \frac{1-\kappa D^2}{2} +\frac{1}{2} \log \left(\frac{D^2}{1+(1-\kappa) D^2} \right), \quad \int_{t_0}^1 \theta_t \mathrm{d}t =-\frac{1}{2} \log \left(\frac{1}{1+(1-\kappa) D^2} \right).\end{aligned}$$ Then we have $$\begin{aligned} \int_0^1 \theta_t \mathrm{d}t =\frac{1-\kappa D^2}{2} +\log D, \quad \exp\left(\int_0^1 \theta_t \mathrm{d}t \right) = D \exp\left(\frac{1-\kappa D^2}{2} \right).\end{aligned}$$ This completes the proof of Corollary [Corollary 33](#cor2){reference-type="ref" reference="cor2"}. ◻ # Proof of Theorems [Theorem 18](#application-thm-1){reference-type="ref" reference="application-thm-1"}, [Theorem 19](#application-thm-2){reference-type="ref" reference="application-thm-2"} and [Theorem 20](#thm-q-poincare){reference-type="ref" reference="thm-q-poincare"} {#proof-of-theorems-application-thm-1-application-thm-2-and-thm-q-poincare} We start with a differential Lipschitz mapping $T: \mathbb R^d \rightarrow \mathbb R^d$ associated with constant $C$. The following result describes the Lipschitz properties of the derivatives of composite mappings. **Lemma 34**. *Let $T: \mathbb R^d \rightarrow \mathbb R^d$ be a differential Lipschitz mapping with constant $C$ and let $\zeta: \mathbb R^d \rightarrow \mathbb R$ be a continuously differentiable function. Then $$\nabla (\zeta \circ T) = [(\nabla \zeta) \circ T ] \nabla T,$$ where $(\nabla T)(x) : \mathbb R^d \rightarrow \mathbb R^{d\times d}$ be a Jacobian matrix for any $x\in \mathbb R^d$. Furthermore, we obtain $$\begin{aligned} |\nabla \zeta(T(x))| \le \|\nabla T(x) \|_{\mathrm{op}} \cdot |(\nabla \zeta) \circ (T(x)) | \le C |(\nabla \zeta) \circ (T(x)) |\end{aligned}$$ for all $x \in \mathbb R^d$.* Since the proof of this result is almost trivial by using the chain rule and the Lipschitz mapping $T$, we omit it here. Through Lemma [Lemma 34](#lem1){reference-type="ref" reference="lem1"}, we can start the proofs of the functional inequalities which follow from Theorems [Theorem 5](#main-thm2){reference-type="ref" reference="main-thm2"} and [Theorem 6](#main-thm3){reference-type="ref" reference="main-thm3"}. We first begin with the $\Psi$-Sobolev inequalities defined in [@chafai2004entropies]. ## Proof of Theorem [Theorem 18](#application-thm-1){reference-type="ref" reference="application-thm-1"} {#proof-of-theorem-application-thm-1} *Proof.* - It can be seen from [@chafai2004entropies Corollary 2.1] that for standard Gaussian measure $\gamma_d$ on $\mathbb R^d$, we have the following $\Psi$-Sobolev inequalities: $$\label{eq:Sobolev ineq} \mathrm{Ent}^{\Psi}_{\gamma_d}(F) \le \frac{1}{2} \int_{\mathbb R^d} \Psi''(F) |\nabla F|^2 \mathrm{d}\gamma_d$$ for any smooth function $F: \mathbb R^d \rightarrow \mathcal I$. Let $(X_t)_{t\in [0,1]}$ be the solution of IVP ([\[main-ode\]](#main-ode){reference-type="ref" reference="main-ode"}) so that $X_1 \sim p$ if $X_0 \sim N(0,\mathbf{I}_d)$. Suppose that $X_1(x): \mathbb R^d \rightarrow \mathbb R^d$ is a Lipschitz mapping with constant $C$ and let $F :=\zeta \circ X_1: \mathbb R^d \rightarrow \mathcal I$ with $\zeta: \mathbb R^d \rightarrow \mathcal I$. Then combining Lemma [Lemma 34](#lem1){reference-type="ref" reference="lem1"}, ([\[eq:Sobolev ineq\]](#eq:Sobolev ineq){reference-type="ref" reference="eq:Sobolev ineq"}) and $p= \gamma_d \circ (X_1)^{-1}$ we have $$\begin{aligned} \mathrm{Ent}^{\Psi}_{p}(\zeta) =\mathrm{Ent}^{\Psi}_{\gamma_d}(F) \le \frac{1}{2} \int_{\mathbb R^d} \Psi''(F) |\nabla F|^2 \mathrm{d}\gamma_d & \le \frac{C^2}{2} \int_{\mathbb R^d} \Psi''(\zeta \circ X_1) |\nabla \zeta \circ X_1|^2 \mathrm{d}\gamma_d \\ & = \frac{C^2}{2} \int_{\mathbb R^d} \Psi''(\zeta(x)) |\nabla \zeta(x)|^2 p(x) \mathrm{d}x.\end{aligned}$$ The proof is complete by Theorems [Theorem 5](#main-thm2){reference-type="ref" reference="main-thm2"} and [Theorem 6](#main-thm3){reference-type="ref" reference="main-thm3"}. - Let random vector $Y \sim \rho$, let $\tilde \rho$ be the law of $\Sigma^{-1/2} Y$, and define $\tilde p:=\gamma_d * \tilde \rho$. Set $\lambda_{\min} :=\lambda_{\min}(\Sigma)$ and $\lambda_{\max} :=\lambda_{\max}(\Sigma)$. Then combining Lemma [Lemma 34](#lem1){reference-type="ref" reference="lem1"}, ([\[eq:Sobolev ineq\]](#eq:Sobolev ineq){reference-type="ref" reference="eq:Sobolev ineq"}) and $\tilde p= \gamma_d \circ (X_1)^{-1}$, we have $$\mathrm{Ent}^{\Psi}_{\tilde p}(\zeta) \le \frac{\exp(\lambda^{-1}_{\min} R^2)}{2} \int_{\mathbb R^d} \Psi''(\zeta(x)) |\nabla \zeta(x)|^2 \tilde p(x) \mathrm{d}x.$$ Let $p=N(a,\Sigma) * \rho$ and let $\tilde X \sim \tilde p$ such that $$\begin{aligned} \Sigma^{1/2} \tilde X +a =\Sigma^{1/2}\left(X+\Sigma^{-1/2} Y \right) +a =\left( \Sigma^{1/2} X +a \right)+Y \sim p=N(a,\Sigma) * \rho,\end{aligned}$$ where $X\sim N(0,\mathbf{I}_d)$. Given $\zeta: \mathbb R^d \rightarrow \mathcal I$ and let $\tilde \zeta(x) :=\zeta(\Sigma^{1/2} x+ a)$ so that $$\begin{aligned} \mathrm{Ent}^{\Psi}_{p}(\zeta) =\mathrm{Ent}^{\Psi}_{\tilde p}(\tilde \zeta) \le \frac{\exp(\lambda^{-1}_{\min} R^2)}{2} \int_{\mathbb R^d} \Psi''(\tilde \zeta(x)) |\nabla \tilde \zeta(x)|^2 \tilde p(x) \mathrm{d}x.\end{aligned}$$ Since $(\nabla \tilde \zeta)(x)=\Sigma^{1/2} \left(\nabla \zeta(\Sigma^{1/2}x +a) \right)$, we get $$|(\nabla \tilde \zeta)(x)|^2 \le \lambda_{\max} \left|\left(\nabla \zeta (\Sigma^{1/2}x +a) \right) \right|^2.$$ Furthermore, it yields that $$\begin{aligned} \mathrm{Ent}^{\Psi}_{p}(\zeta) \le \frac{\lambda_{\max} \exp(\lambda^{-1}_{\min} R^2)}{2} \int_{\mathbb R^d} \Psi''(\zeta(x)) |\nabla \ \zeta(x)|^2 p(x) \mathrm{d}x.\end{aligned}$$ This completes the proof. ◻ ## Proof of Theorem [Theorem 19](#application-thm-2){reference-type="ref" reference="application-thm-2"} {#proof-of-theorem-application-thm-2} *Proof.* - By using [@ledoux1996isoperimetry Theorem 4.3], then the Gaussian measure $\gamma_d$ on $\mathbb R^d$ satisfies the following Gaussian isoperimetric inequality: $$\gamma_d (K_t) \ge \Phi\left(\gamma_d(K) +t \right), \quad t \geq 0$$ for any Borel measurable set $K_t :=K+t B_2^d$ and $K \subseteq \mathbb R^d$. Therefore, suppose $(X_t)_{t\in [0,1]}$ be the solution of IVP ([\[main-ode\]](#main-ode){reference-type="ref" reference="main-ode"}) so that $X_1 \sim p$ if $X_0 \sim N(0,\mathbf{I}_d)$. Moreover, suppose that $X_1(x): \mathbb R^d \rightarrow \mathbb R^d$ is a Lipschitz mapping with constant $C$, then for any fixed $x \in \mathbb R^d$, $$|X_1(x+ y) -X_1(x)| \le C |y|, \quad \forall y \in \mathbb R^d.$$ We first show the following result: $$\label{eq:iso-inequality} X^{-1}_1(E) +\frac{t}{C} B_2^d \subseteq X^{-1}_1(E_t), \quad E_t :=E +t B_2^d$$ for any Borel measurable set $E \subseteq \mathbb R^d$ and $t\geq 0$. To obtain ([\[eq:iso-inequality\]](#eq:iso-inequality){reference-type="ref" reference="eq:iso-inequality"}), we only need to prove that $$X_1\left(X^{-1}_1(E) +\frac{r}{C} B_2^d \right) \subseteq E_t, \quad t\geq 0$$ or, in other words, if $x \in X^{-1}_1(E) +\frac{t}{C} B_2^d$, then $X_1(x) \in E_t$ for any Borel measurable set $K$. Furthermore, if we assume $$x \in X^{-1}_1(E) +\frac{t}{C} B_2^d \quad \text{so that} \quad x=\theta +\frac{t}{C} h$$ for some $\theta \in X^{-1}_1(E)$ and $h \in B_2^d$, we have $X_1\left(x -\frac{t}{C} h \right) \in E$. Then it yields that $$\left|X_1\left( x-\frac{t}{C} h \right) -X_1(x) \right| \leq t, \quad t\geq 0$$ where $x -\frac{t}{C} h \in X^{-1}_1(E)$. Therefore, $X_1(x) \in E_t$ as desired. Finally, combining the Gaussian isoperimetric inequality and ([\[eq:iso-inequality\]](#eq:iso-inequality){reference-type="ref" reference="eq:iso-inequality"}), it yields $$\begin{aligned} p(E_t) =\gamma_d\left(X^{-1}_1(E_t) \right) \ge \gamma_d\left(X^{-1}_1(E) +\frac{t}{C} B_2^d \right) \geq \Phi\left(\gamma_d\left[X^{-1}_1(E) +\frac{t}{C} \right] \right) =\Phi\left(p(E)+ \frac{t}{C} \right).\end{aligned}$$ This proof is completed by Theorems [Theorem 5](#main-thm2){reference-type="ref" reference="main-thm2"} and [Theorem 6](#main-thm3){reference-type="ref" reference="main-thm3"}. - Let random vector $Y \sim \rho$, let $\tilde \rho$ be the law of $\Sigma^{-1/2} Y$, and define measure $\tilde p:= \gamma_d * \tilde \rho$. Set $\lambda_{\min} :=\lambda_{\min}(\Sigma)$ and $\lambda_{\max} :=\lambda_{\max}(\Sigma)$. Similar to the argument of part (1), for any Borel set $E \subset \mathbb R^d$ and $t \ge 0$, we obtain $$\tilde p\left( E_t \right) \ge \Phi\left(\tilde p(E) + \frac{t}{C}\right), C:= \left( \lambda_{\min} \right)^{1/2} \exp\left(\frac{R^2}{2\lambda_{\min}} \right).$$ Let $p=N(a,\Sigma) * \rho$ and let $\tilde X \sim \tilde p$ so that $$\Sigma^{1/2} \tilde X +a =\left( \Sigma^{1/2} X +a \right)+Y \sim p=N(a,\Sigma) * \rho$$ and $X\sim N(0,\mathbf{I}_d)$. Then for any Borel measurable set $E \subset \mathbb R^d$ and $t \ge 0$, it yields that $$\begin{aligned} p(E_t) =\tilde p \left(\Sigma^{-1/2}(E-a) +\Sigma^{-1/2} tB_2^d \right) \ge \tilde p\left(\Sigma^{-1/2} (E-a) +t \lambda^{-1/2}_{\max} B_2^d \right).\end{aligned}$$ Hence, we obtain $$p(E_t) \ge \Phi\left(\tilde p\left[\Sigma^{-1/2} (E-a) \right] +\frac{t \lambda^{-1/2}_{\max}}{C} \right).$$ We obtain the desired result by applying $\tilde p\left[\Sigma^{-1/2}(E-a) \right] = p(E)$. This completes the proof. ◻ ## Proof of Theorem [Theorem 20](#thm-q-poincare){reference-type="ref" reference="thm-q-poincare"} {#proof-of-theorem-thm-q-poincare} *Proof.* - We will use the fact that [@nourdin2009second Proposition 3.1] the $q$-Poincaré inequality holds for the standard Gaussian measure $\gamma_d$ on $\mathbb R^d$: $$\label{eq:Poincare ineq} \mathbb E_{\gamma_d} \left[F^q \right] \le (q-1)^{q/2} \mathbb E_{\gamma_d} \left[|\nabla F|^q \right],$$ for any smooth function $F \in L^q(\gamma_d)$ with $\mathbb E_{\gamma_d}[F] =0$. Let $(X_t)_{t\in [0,1]}$ be the solution of IVP ([\[main-ode\]](#main-ode){reference-type="ref" reference="main-ode"}) so that $X_1 \sim p$ if $X_0 \sim N(0,\mathbf{I}_d)$. Suppose that $X_1(x): \mathbb R^d \rightarrow \mathbb R^d$ is a Lipschitz mapping with constant $C$ and let $F :=\eta \circ X_1$. Then combining Lemma [Lemma 34](#lem1){reference-type="ref" reference="lem1"}, ([\[eq:Poincare ineq\]](#eq:Poincare ineq){reference-type="ref" reference="eq:Poincare ineq"}) and $p= \gamma_d \circ (X_1)^{-1}$ we have $$\begin{aligned} \mathbb E_p[\eta^q] =\mathbb E_{\gamma_d}[F^q] \leq (q-1)^{q/2} \mathbb E_{\gamma_d}[|\nabla F|^q] \leq C^q (q-1)^{q/2} \mathbb E_p[|\nabla \eta|^q].\end{aligned}$$ The proof is complete by Theorems [Theorem 5](#main-thm2){reference-type="ref" reference="main-thm2"} and [Theorem 6](#main-thm3){reference-type="ref" reference="main-thm3"}. - Let $Y \sim \rho$, let $\tilde \rho$ be the law of $\Sigma^{-1/2} Y$, and define $\tilde p:=N(0,\mathbf{I}_d) * \tilde \rho$. Set $\lambda_{\min} :=\lambda_{\min}(\Sigma)$ and $\lambda_{\max} :=\lambda_{\max}(\Sigma)$. The argument of part (1) gives, $$\mathbb E_{\tilde p} [\eta^q] \le \exp\left(\frac{q R^2}{2 \lambda_{\min}} \right) \lambda_{\min}^{q/2} (q-1)^{q/2} \mathbb E_{\tilde p}[ |\nabla \eta|^q].$$ Let $p=N(a,\Sigma) * \rho$ and let $\tilde X \sim \tilde p$ such that $$\Sigma^{1/2} \tilde X +a =\left( \Sigma^{1/2} X +a \right)+Y \sim p=N(a,\Sigma) * \rho$$ and $X \sim N(0,\mathbf{I}_d)$. Let $\tilde \eta(x) :=\eta\left(\Sigma^{1/2} x +a \right)$ so that $$\begin{aligned} \mathbb E_p[\eta^q ] =\mathbb E_{\tilde p} \left[(\tilde \eta )^q \right] \leq \exp\left(\frac{q R^2}{2 \lambda_{\min}} \right) \lambda_{\min}^{q/2} (q-1)^{q/2} \mathbb E_{\tilde p}[ |\nabla \tilde \eta|^q].\end{aligned}$$ Since $(\nabla \tilde \eta)(x)=\Sigma^{1/2} \left( \nabla \eta\left(\Sigma^{1/2} x+ a \right) \right)$ we have $$\left|(\nabla \tilde \eta)(x) \right|^q \le (\lambda_{\max})^{q/2} \left| \nabla \eta \left(\Sigma^{1/2} x + a \right) \right|^q.$$ Further, we obtain $$\begin{aligned} \mathbb E_p[\eta^q] = \mathbb E_{\tilde p} \left[(\tilde \eta)^q \right] \leq (\lambda_{\min} \lambda_{\max})^{q/2} \exp\left(\frac{q R^2}{2 \lambda_{\min}} \right) (q-1)^{q/2} \mathbb E_p[|\nabla \eta |^q].\end{aligned}$$ This completes the proof. ◻ # Time changes **Lemma 35**. *Let $(\overline{X}_t)_{t\in [0,1)}$ be a diffusion process defined by ([\[eq:vp-sde-tr\]](#eq:vp-sde-tr){reference-type="ref" reference="eq:vp-sde-tr"}) with $\varepsilon \to 0$ and let $(\overline{Y}_s)_{s \ge 0}$ be an Ornstein-Uhlenbeck process $(\overline{Y}_s)_{s \ge 0}$ defined by $$\label{eq:ou-sde} \mathrm{d}\overline{Y}_s = - \overline{Y}_s \mathrm{d}s +\sqrt{2} \mathrm{d}\overline{W}_s, \quad \overline{Y}_0 \sim \nu, \quad s \ge 0.$$ Then $(\overline{X}_t)_{t\in [0,1)}$ is equivalent to $(\overline{Y}_s)_{s \ge 0}$ through the change of time formula $t = 1 - e^{-s}$.* *Proof.* Let $s = -\log(1-t)$ for any $t \in [0, 1)$. By applying ([\[eq:vp-sde-tr\]](#eq:vp-sde-tr){reference-type="ref" reference="eq:vp-sde-tr"}), it yields $$\mathrm{d}\overline{X}_{1-e^{-s}} = - \overline{X}_{1-e^{-s}} \mathrm{d}s +\sqrt{2} \mathrm{d}\overline{W}_s, \quad \overline{X}_0 \sim \nu, \quad s \ge 0.$$ On the one hand, since ([\[eq:ou-sde\]](#eq:ou-sde){reference-type="ref" reference="eq:ou-sde"}) has a unique strong solution, it indicates $\overline{Y}_s = \overline{X}_{1-e^{-s}}$ for all $s \ge 0$. On the other hand, the infinitesimal generator of Markov process $(\overline{Y}_s)_{s \ge 0}$ is given by $$\label{eq:gene-ou} L^{\overline{Y}} = \Delta - x \cdot \nabla.$$ By using ([\[eq:vp-sde-tr\]](#eq:vp-sde-tr){reference-type="ref" reference="eq:vp-sde-tr"}), the infinitesimal generator of $(\overline{X}_t)_{t\in [0,1)}$ is given by $$\label{eq:gene-vp} L^{\overline{X}}_t = \frac{1}{1-t} ( \Delta - x \cdot \nabla ).$$ Furthermore, combining the chain rule and straightforward calculation, we obtain that processes $\overline{X}_t$ and $\overline{Y}_s$ have the same infinitesimal generator, which implies $\overline{X}_t =\overline{Y}_s$ for any $t \in [0,1), s = - \log(1 - t)$. ◻ **Lemma 36**. *Let $(X^*_t)_{t\in [0,1)}$ be the time reversal of a Föllmer flow associated to probability measure $\nu$ defined by ([\[eq:vp-ode-eps-tr\]](#eq:vp-ode-eps-tr){reference-type="ref" reference="eq:vp-ode-eps-tr"}) with $\varepsilon \to 0$ and let $(Y^*_s)_{s \ge 0}$ be a heat flow from probability measure $\nu$ to the standard Gaussian measure $\gamma_d$ defined by $$\begin{aligned} \label{eq:ode-heat-flow} \mathrm{d}Y^*_s (x) = - \nabla \log \left\{ \int_{\mathbb{R}^d} r \left( e^{-s} Y^*_s (x) + \sqrt{1 - e^{-2s}} z \right) \mathrm{d}\gamma_d (z) \right\} \mathrm{d}s\end{aligned}$$ where $r(x) := (\mathrm{d}\nu / \mathrm{d}\gamma_d)(x), Y^*_0 \sim \nu$ for all $s \geq 0$. Then $(X^*_t)_{t\in [0,1)}$ is equivalent to $(Y^*_s)_{s \ge 0}$ through the change of time formula $t = 1 - e^{-s}$.* *Proof.* Let $s = -\log(1-t)$ for every $t \in [0, 1)$. By ([\[eq:vp-ode-eps-tr\]](#eq:vp-ode-eps-tr){reference-type="ref" reference="eq:vp-ode-eps-tr"}), it yields $$\begin{aligned} \mathrm{d}X^*_{1-e^{-s}} (x) = - \nabla \log \left\{ \int_{\mathbb{R}^d} r \left( e^{-s} X^*_{1-e^{-s}} (x) + \sqrt{1 - e^{-2s}} z \right) \mathrm{d}\gamma_d (z) \right\} \mathrm{d}s\end{aligned}$$ where $X^*_0 \sim \nu$ for all $s \geq 0$. The expression above indicates that $Y^*_s := X^*_{1-e^{-s}}$ satisfies ([\[eq:ode-heat-flow\]](#eq:ode-heat-flow){reference-type="ref" reference="eq:ode-heat-flow"}). ◻ [^1]: ${}^{\ddagger}$ Authors are listed in alphabetical order.

arxiv_math

--- abstract: | The volume growth function of a Riemannian manifold of bounded geometry is a function having bounded growth of derivative, called a bgd-function. Grimaldi and Pansu showed that any one-ended manifold of finite type admits a metric of bounded geometry such that the volume growth function has the growth type of any given bgd-function. In the case of infinite type, this holds for any superlinear bgd-function. We generalize this result to manifolds with finitely many ends and show that a manifold with infinitely many ends does not admit a metric with some bgd-functions as volume growth, including polynomials. We also prove that on certain manifolds with countably many ends, there exist metrics such that the volume growth functions have a growth type arbitrarily close to that of any predetermined bgd-function. address: - Department of Mathematical Sciences, Indian Institute of Science Education and Research Mohali, Sector 81, SAS Nagar, Punjab- 140306, India. - Department of Mathematical Sciences, Indian Institute of Science Education and Research Mohali, Sector 81, SAS Nagar, Punjab- 140306, India. author: - Anushree Das - Soma Maity title: Volume Growth on manifolds with more than one end --- # Introduction {#intro} Let $(M,g)$ be a complete non-compact smooth Riemannian manifold without boundary. $(M,g)$ is said to have bounded geometry if the injectivity radius $i_g$ is bounded below by $1$ and the sectional curvature $K$ satisfies $|K|\leq 1$. Any non-compact Riemannian covering space of a compact Riemannian manifold satisfies the bounded geometry condition. A volume growth function $v$ of $(M,g)$ is a positive function on $\mathbb{N}$ such that $v(n)$ is the volume of the ball of radius $n$ centered at a point $o$ in $M$. Badura, Funar, Grimaldi and Pansu investigate those functions which are volume growth functions of a Riemannian manifold with bounded geometry, and their relations with the topology of the manifold in [@BM],[@FG], [@GP]. **Definition 1**. A function $v:\mathbb{N} \to \mathbb{R}_+$ is said to have bounded growth of derivative if there exists a positive integer $L$ such that, $\forall n\in \mathbb{N}$, $$\frac{1}{L}\leq v(n+2)-v(n+1)\leq L(v(n+1)-v(n)).$$ We call a function with bounded growth of derivative a *bgd-function* in short. A volume growth function of a Riemannian manifold with bounded geometry is a bgd-function [@GP]. **Definition 2**. Two non-decreasing functions $f,g:\mathbb{N}\rightarrow \mathbb{R}_{+}$ have the same growth type if there exists an integer $A\geq 1$ such that for all $n\in \mathbb{N}$, $$\begin{aligned} f(n)\leq Ag(An+A)+A \ \ \text{and} \ \ g(n)\leq Af(An+A)+A. \end{aligned}$$ We call the constant $A$, in this case, a *growth constant* for $f$ and $g$. Functions of the same growth type define an equivalence relation on the set of non-decreasing functions from $\mathbb{N}$ to $\mathbb{R}_{+}$, and the equivalence classes are denoted by $[.].$ Grimaldi and Pansu prove the following theorem: **Theorem 1** ([@GP]). Let $M$ be a connected manifold. 1. If $M$ has finite topological type, every bgd-function belongs to the growth type of a Riemannian manifold of bounded geometry diffeomorphic to $M$. 2. If $M$ has infinite topological type, a bgd-function $v$ belongs to the growth type of a Riemannian manifold of bounded geometry diffeomorphic to $M$ if and only if $\lim_{n\to\infty} \frac{v(n)}{n}=+\infty$. For the definitions of manifolds of finite and infinite type, we refer to Section [3](#finite){reference-type="ref" reference="finite"}. We observe that the statement of this theorem does not necessarily hold if $M$ has infinitely many ends. In [@CG] Gilles Carron proved that if the volume growth function of a Riemannian manifold satisfies the doubling condition, the manifold must have finitely many ends. Using his techniques we show that if $M$ admits a metric with a polynomial volume growth function then the manifold has finitely many ends. If $M$ has infinitely many ends then it has infinite topological type. Therefore, if $v$ is a polynomial of degree greater than or equal to $2$ then the second part of the above theorem does not hold. We also observe that the proof in [@GP] holds only for manifolds with one end. For a detailed discussion, we refer to Section [3](#finite){reference-type="ref" reference="finite"}. For manifolds with finitely many ends, we have the following result. **Theorem 2**. Let $M$ be a connected manifold with finitely many ends. 1. If each end of $M$ is of finite topological type, every bgd-function $v$ belongs to the growth type of a Riemannian manifold of bounded geometry diffeomorphic to $M$. 2. If some end of $M$ is not of finite topological type, every bgd-function $v$ belongs to the growth type of a Riemannian manifold of bounded geometry diffeomorphic to $M$ if and only if $\lim_{n\to\infty}\frac{v(n)}{n}=\infty$. If $M$ has finitely many ends, say $k$, then there exists a compact manifold $P$ with $k$ boundaries and finitely many one-ended manifolds with boundaries $P_1,P_2,...,P_k$ such that $M$ is the union of $P$ and $P_1,P_2,..,P_k$. From the proof of Theorem [Theorem 1](#t3){reference-type="ref" reference="t3"} we observe that the same result holds for a one-ended manifold with a boundary. Using this observation we define a Riemannian metric of growth type of a given bgd-function $v$ on each one-ended manifold $P_i$ such that the metric is a product metric near the boundary. We stitch the manifold back together by identifying the boundary components of these one-ended manifolds $P_i$ with $P$ and define a metric on the entire $M$. We show that the resulting Riemannian manifold has the volume growth function in our desired growth class. We can partially generalize the above theorem in the case of certain manifolds with countably many ends, which we call *admissible manifolds*. Our motivation to define such manifolds comes from [@BBM]. **Definition 3**. Let $\mathcal{U}$ be a collection of finite compact $n$-dimensional manifolds without boundary. $M$ is called an admissible manifold if there exists a manifold $N$ with countably many boundaries $\partial N_i$, and a sequence of one-ended manifolds $\{M_i\}_{i\in\mathbb{N}}$ with one boundary component $\partial M_i$, such that - $\partial N_i$ and $\partial M_i$ are $S^{n-1}$ for all $i$. - $N=N_1\# N_2 \#N_3\#...$ such that $\partial N_i\in N_i$ and each $N_i$ is an element of $\mathcal{U}$ with one disk removed. - $M_i=M_{i1}\#M_{i2}\#M_{i3}\#...$ for some $M_{ij}\in \mathcal{U}$ for $i\geq 2$, $\partial M_i\in M_{i1}$, and $M_{i1}$ is an element of $\mathcal{U}$ with one disk removed. - $M$ is diffeomorphic to the manifold formed by attaching the boundary of each $M_i$ with $\partial N_i$ by identity diffeomorphism. Define a pre-order $\preceq$ on the set of all bgd-functions as $f\preceq g$ if there exists some constant $A$ such that $f(n)\leq Ag(An+A)+A$ for all $n$. Any equivalence class induced by this pre-order consists of functions of the same growth type. That is, $f$ and $g$ are of the same growth type if and only if $f\preceq g \preceq f$, and we then write $[f]=[g]$. The pre-order $\preceq$ induces an order $\leq$ on the set of all equivalence classes of bgd-functions. We have $$f\preceq g \iff [f]\leq [g].$$ On admissible manifolds, we have the following theorem. **Theorem 3**. Let $M$ be an admissible manifold. For a bgd-function $v$ and an increasing $C^1$ function $f:[1,\infty) \to \mathbb{R}_+$ such that $f'$ is bounded, there exists a Riemannian manifold $(M',g)$ of bounded geometry diffeomorphic to $M$ such that the volume growth function $v_1$ of $(M',g)$ satisfies $[v(x)]\leq[v_1(x)]\leq[f(x)v(x)]$. One can choose $f_k(x)=x^{\frac{1}{k}}$ to obtain a sequence of Riemannian metric $g_k$ with bounded geometry on $M$ such the volume growth functions $v_k$ satisfy $[v(x)]\leq[v_k(x)]\leq[x^{\frac{1}{k}}v(x)]$. As $k\to \infty$, $x^{\frac{1}{k}}\to 1.$ Hence for each $n$ the sequence $\{v_{k}(n)\}$ converges to $v(n)$. In particular, if we choose $v=x^\alpha$ for any $\alpha>0$ then $M$ does not admit any metric such that its volume growth function belongs to $[x^\alpha]$ since it has countably many ends. The above theorem implies that $M$ admits a Riemannian metric of bounded geometry such that its volume growth function $v$ satisfies $[x^\alpha]\leq [v(x)]\leq [x^\beta]$ for any $\alpha$ and $\beta$ with $\beta>\alpha>0$. Note that each end $M_i$ of an admissible manifold $M$ has finite topological type. So, for any bgd-function $v$ each $M_i$ admits a Riemannian metric of bounded geometry such that its volume growth function $v_i$ belongs to $[v]$ and the metric is a product metric near the boundary of $M_i$. We show that the growth constant of $v_i$ and $v$ depends only on $v$ and $\mathcal{U}.$ Therefore, one can choose the same growth constant for all $v_i$ and $v.$ We apply a modified version of Theorem [Theorem 1](#t3){reference-type="ref" reference="t3"} to define a metric with volume growth function of the same growth type as $v$ on $N$ so that the boundaries remain at suitable distances and the metric is a product metric near each boundary. Stitching these one-ended manifolds back together reproduces the original manifold so that the volume growth function of the entire manifold has a growth type between $[v]$ and $[fv]$. # Volume growth and Number of ends {#prelim} In this section we first prove some basic results about functions of the same growth type. Then we establish a relation between the volume growth function of a Riemannian manifold and its number of ends. ** **Lemma** 1**. If two functions $f$ and $g$ are of the same growth type, then $f+g$ is also of the same growth type as $f$ and $g$. *Proof.* Since $f$ and $g$ are of the same growth type, there exists an integer $A\geq 1$ such that for all $n\in \mathbb{N}$, $$\begin{aligned} f(n) \leq Ag(An+A)+A \quad \text{and} \quad g(n) \leq Af(An+A)+A. \end{aligned}$$ Therefore, $$\begin{aligned} (f+g)(n) &\leq f(n)+Af(An+A)+A\\ &\leq f(An+A)+Af(An+A)+A\\ &\leq (A+1)f((A+1)n+A+1)+(A+1). \end{aligned}$$ Here we use the fact that the functions $f$ and $g$ are non-decreasing. Also, since the functions are non-negative, $$\begin{aligned} f(n) \leq (f+g)(n)\leq (A+1)(f+g)((A+1)n+(A+1))+(A+1). \end{aligned}$$ Therefore, $f+g$ is of the same growth type as $f$ and hence also as $g$. ◻ ** **Lemma** 2**. Let $f$ be a non-decreasing function. For some positive integer $K$, define a function $h$ by $$\begin{aligned} h(n)=\begin{cases} 0 &\text{if } n\leq K\\ f(n-K) &\text{otherwise} \end{cases} \end{aligned}$$then $f$ and $h$ are of the same growth type. *Proof.* As $f$ is a non-decreasing function, so is $h$. Therefore, since $K\geq 1$, $$\begin{aligned} f(n) = h(n+K)\leq Kh(Kn+K)+K. \end{aligned}$$ Also, $$\begin{aligned} h(n)\leq f(n)\leq Kf(Kn+K)+K. \end{aligned}$$ So, $f$ and $g$ are of the same growth type. ◻ ** **Lemma** 3**. Let $f$ and $g$ be functions of the same growth type. If $h$ is a non-decreasing function such that $f\leq h \leq g$, then $h$ is also of the same growth type as $f$ and $g$. *Proof.* By definition, there exists an integer $A\geq 1$ such that $$\begin{aligned} f(n) &\leq Ag(An+A)+A \quad \text{and} \quad g(n) \leq Af(An+A)+A. \end{aligned}$$ Now by the given condition, $$\begin{aligned} h(n) \leq g(n) \leq Af(An+A)+A. \end{aligned}$$ And, $$\begin{aligned} f(n) \leq h(n)\leq Ah(An+A)+A. \end{aligned}$$ Hence, $h$ is of the same growth type as $f$ and $g$. ◻ ** **Lemma** 4**. If $p$ is a polynomial of degree $k$ such that $p(x)\geq 0$ for all $x\in \mathbb{N}$ and $p$ is increasing on $\mathbb{R}_+$, then $[p]=[x^k]$. *Proof.* Let $p=\Sigma_{i=0}^ka_ix^i$, and $p(x)\geq 0$, $\forall x\in \mathbb{N}$. $a_k>0$ as $p$ is non-negative on $\mathbb{N}$.\ Let $A=\text{max}\{1,|a_1|,|a_2|,...,|a_k|\}>0.$ Then, $$\begin{aligned} p(x) & = a_kx^k+a_{k-1}x^{k-1}+...+a_1x+a_0 \\ %\leq (a_k+a_{k-1}+...+a_1+a_0)x^k \\ & \leq (k+1)Ax^k \\ & \leq (k+1)A((k+1)Ax+(k+1)A)^k+(k+1)A. \end{aligned}$$ Consider a function $f:\mathbb{N}\to \mathbb{R}$ defined by $f(x)=\frac{p(x)}{x^k}.$ As $\lim_{x\to \infty}f(x)=a_k>0$, there exists $n_0\in \mathbb{N}$ such that $f(x)\geq \frac{a_k}{2}$ for all $x>n_0.$ Let $m$ be the minimum of $f$ on $[1,n_0]$ and $c=\min\{m,\frac{a_k}{2}\}.$ Then $f(x)\geq c$ for all $x\in \mathbb{N}.$ Hence $x^k\leq \frac{1}{c}p(x).$ This proves the reverse direction. ◻ Next, we define the ends of a manifold. **Definition 4**. Consider infinite sequences $U_1\supseteq U_2\supseteq \cdots$ of non-empty connected open subsets of a manifold $M$ such that each $\overline{U}_i$ is compact and $\bigcap_{i\geq 1} \overline{U}_i = \phi$. We say two such sequences $U_1\supseteq U_2 \supseteq \cdots$ and $U_1'\supseteq U_2'\supseteq \cdots$ are equivalent if for every $i$ there exist $j, k$ such that $U_i\supseteq U'_j$ and $U'_i\supseteq U_k$. The equivalence classes of those sequences are the ends of the manifold $M$. In [@CG], Gilles Carron showed that if the volume growth function $v$ of a Riemannian manifold satisfies a doubling condition then the manifold necessarily has finitely many ends. Generalizing his idea we try to see what other constraints on properties of the volume growth functions can be concluded from the number of ends of a manifold. ** **Lemma** 5**. Let $(M,g)$ be a Riemannian manifold with volume growth function $v$. If there exist two non-negative increasing functions $f$ and $F$ such that $f(r)\leq v(r)\leq F(r)$ for all $r>0$, then the number of connected components $C(r)$ of the complement of any ball of radius $r$ satisfies the condition $C(r)\leq \frac{F(3r)}{f(r)}$. *Proof.* Let $o$ be a base point of the volume growth function $v$, and let $\{U_\alpha\}$, $\alpha \in A$ be the connected components of the complement of $M\setminus B(o,r)$. Choose a point $y_\alpha$ from each component at distance $2r$ from $o$. Then $B(o,r)\cup (\bigcup_{\alpha \in A}B(y_\alpha,r))\subseteq B(o,3r)$. Therefore, $$\sum_{\alpha\in A}vol(B({y_\alpha,r}))\leq vol(B(o,3r)).$$ For each $\alpha$, $vol(B({y_\alpha,r}))\geq f(r)$. Thus if $C(r)$ is the number of connected components of $M\setminus B(o,r)$, then $C(r)\times f(r)\leq \sum_{\alpha}vol(B{y_\alpha,r})$. So, $$C(r)\times f(r)\leq vol(B(o,3r)) \leq F(3r).$$ Thus $C(r)\leq \frac{F(3r)}{f(r)}$. ◻ ****Corollary** 1**. If a non-compact manifold $M$ admits a complete Riemannian metric such that the volume growth function lies in the class of a polynomial, then $M$ has finitely many ends. *Proof.* Let the volume growth function $v$ of a Riemannian manifold lie in the class of a polynomial of degree $n$. Then for some constants $\{a_i\}$ and $\{A_i\}$, where $0\leq i\leq n$, $v$ satisfies $$a_nx^n + a_{n-1}x^{n-1}+...+a_1x+a_0\leq v(x)\leq A_nx^n + A_{n-1}x^{n-1}+...+A_1x+A_0,$$ Then for any $r>0$, the number of connected components $C(r)$ of the a ball of radius $r$ satisfies $$\begin{aligned} C(r)&\leq \frac{A_n(3r)^n + A_{n-1}(3r)^{n-1}+...+A_1(3r)+A_0}{a_nr^n + a_{n-1}r^{n-1}+...+a_1r+a_0}. \end{aligned}$$ As $r\to \infty$ the term in the right-hand side converges to $\frac{3A_n}{a_n}$. Thus, $C(r)$ is bounded above by a constant for sufficiently large $r$. This implies that $M$ has finitely many ends. ◻ ****Corollary** 2**. Let $(M,g)$ be a non-compact complete Riemannian manifold with volume growth function $v$. If $M$ has infinitely many ends, and if any two non-negative increasing functions $f$ and $F$ satisfy $f(r)\leq v(r)\leq F(r)$ for all $r>0$, then necessarily, $\lim_{r\to \infty} \frac{F(3r)}{f(r)}=\infty$. Consequently, a manifold with infinitely many ends does not admit a Riemannian metric of volume growth function lying in the class $[x^\alpha]$ for any $\alpha>0$. However, the volume growth function can be bounded by above or below by polynomials. Consider an infinite cylinder $S^1\times \mathbb{R}$ and equip it with a metric of volume growth $x^\alpha$ for any $\alpha>0$. We remove a disc around a point on $S^1\times \{\alpha n\}$ for some $\alpha>0$ and for each $n\in \mathbb{Z}$. Now, take the cylinder $M_1=S^1\times[0,\infty)$ with the same metric, and attach one copy of $M_1$ smoothly along the boundary of each of those discs to create a new smooth manifold $M_2$. Then the volume function $v$ of $M_2$ satisfies $[x^\alpha]\leq[v]\leq [x^{\alpha+1}]$. But it does not admit any metric of volume growth $[x^\alpha].$ # manifolds with finitely many ends {#finite} **Definition 5**. A non-compact manifold is said to be of finite topological type if it admits an exhaustion by compact submanifolds $M_i$ such that $\partial M_i$ are all diffeomorphic. We first summarise the construction of the required Riemannian metric on a one-ended manifold by Grimaldi and Pansu in [@GP]. We shall be modifying their construction and proof in this section. Given a bgd-function $v$, Grimaldi and Pansu show in [@GP] that there exists a function $w$ of the same growth type as $v$ satisfying the following conditions: 1. $w(0)=1.$ 2. For all $n\in N, 2\leq w(n+2)-w(n+1)\leq 2(w(n+1)-w(n)).$ 3. $w(n)=O(\lambda^n)$ for some $\lambda<2.$ Hence, it can be assumed that the required bgd-function $v$ satisfies these conditions without any loss of generality. Given a manifold $M$ with one end and of dimension $m$, consider an exhaustion $\{\mathcal{A}_j\}$ of $M$ such that each $\mathcal{A}_{j+1}\setminus \mathcal{A}_j$ is a connected manifold of dimension $m$ with boundary. Let $Q_j=\mathcal{A}_{j+1}\setminus \mathcal{A}_j$. The boundary components of $Q_j$ are denoted by $\partial^+ Q_j$ and $\partial^- Q_j$, where $\partial^+ Q_j$ is diffeomorphic to $\partial^- Q_{j+1}$ by the identity diffeomorphism. Fix a subset $S$ of $\mathbb{N}$ such that $\liminf_{n\rightarrow \infty}\frac{S\cap \{0,1,...,n\}}{n}=0$. Consider a rooted tree $T$ with growth $v$ such that it has a single trunk, each vertex has at most $2$ branches, and $S$ is the set of those vertices of the trunk of $T$ which only have one branch. We refer to Lemma $10$ in [@GP] for a proof of the existence of such a tree. The pieces $Q_i$ are attached to a rooted tree $T$ with growth $v$ satisfying these rules. Additional pieces attached to the tree are the following. The piece $R_i$ diffeomorphic to $\partial^+Q_j\times [0,1]$ with a disc removed; $K$, an $m$-dimensional cylinder diffeomorphic to $S^{m-1}\times [0,1]$; $J$, diffeomorphic to an $m$-dimensional sphere with $3$ balls removed; $HS$, a half-sphere of dimension $m$. Write $S$ as $S=\bigcup_j [n_j,n_j+t_j-1]$. Attach a single copy of $Q_j$ collectively to all the vertices in the interval $[n_j,n_j+t_j-1]$. Attach the pieces $R_j$ to each vertex $k$ of the trunk where $n_j+t_j\leq k < n_{j+1}$, that is, the vertices of the trunk with $2$ branches. For the vertices which do not lie on the trunk, other pieces are attached as per the number of branches of those vertices. The root has a copy of $HS$ attached. The vertices with $2$ branches are represented with a join $J$, those with a single branch with a cylinder $K$, and the ones with $0$ branches are capped with a half-sphere $HS$. The manifold resulting from attaching the pieces in the prescribed manner is diffeomorphic to $M$, and is denoted by $R_T$. For a piece $P$, let $t_P$ and $T_P$ respectively denote the minimum and maximum of the function distance to $\partial^-P$, restricted to $\partial^+P$. For $k\leq T_P$, let $U_{P,k}$ denote the $k$-tubular neighbourhood of $\partial^-P$, and $v_P(k)=vol(U_{P,k})$, $v_P'(k)=v_P(k)-v_P(k-1)$. Proposition $13$ in [@GP] gives us estimates for the volume and diameter bounds for each piece: ****Proposition** 1**. Let $Q_j$ be a sequence of compact manifolds with boundary. Assume that - $\partial Q_j$ is split into two open and closed subsets $\partial^-Q_j$ and $\partial^+ Q_j$; - $\partial^- Q_{j+1}$ is diffeomorphic to $\partial^+ Q_j.$ Then there exist integers $l,\ h,\ H$, sequences of integers $t_j,\ u_j,\ U_j,\ d_j$ and Riemannian metrics on pieces $Q_j,\ R_j,\ K,\ HS,\ J$ such that 1. For all pieces $P$, the maximal distance of a point of $P$ to $\partial^- P$ is achieved on $\partial^+ P$. In other words, it is equal to $T_P$. 2. $\frac{1}{3}lt_j\leq t_{Q_j} \leq T_{Q_j}\leq lt_j$. 3. For all other pieces $P$, $\frac{1}{3}l\leq t_{P} \leq T_{P}\leq l$. 4. diameter$(\partial^- Q_j)\leq d_j$. 5. All pieces $P$ carry a marked point $y_P\in \partial^- P$. When a piece $P'$ is glued on top of $P$, $d(y_P,y_{P'})\leq l$ (resp. $lt_j$ if $P=Q_j$), unless $P=R_j$ and $P'$ is of type $K,\ HS$, or $J$. In that case, $d(y_P,y_{P'})\leq d_j$. 6. For all pieces $P=K,\ HS,\ J,\ h\leq \textrm{min}v'_{P}\leq \textrm{max}v'_{P}\leq H$. 7. [max]{.roman}$v'_{Q_j}\leq U_j$. 8. [max]{.roman}$v'_{R_j}\leq u_j \leq U_j$. 9. If $\partial^+Q_j$ and $\partial^-Q_j$ are diffeomorphic, then they are isometric, by an isometry that maps $y_j$ to $y_{j+1}$, and $u_{j+1}=u_j$. 10. All pieces have bounded geometry and product metric near the boundary. 11. If two pieces $Q_i$ and $Q_j$ are diffeomorphic, then they are isometric. $t_j,\ u_j,\ d_j$ are respectively called the height, volume, and diameter parameters. The last point in this proposition follows from the proof in [@GP]. To show that the volume growth function of the manifold $R_T$ lies in the same class as the growth function $v$ of the tree, Grimaldi and Pansu define a discrete growth function. First, they define a function $r : R_T \rightarrow \mathbb{N}$ as follows. If $P = Q_j$ and $x\in P$, $r(x)=\lfloor d(x,\partial^-Q_j)\rfloor+n_jl$. If $P$ is any other type of piece, attached at a vertex of $T$ of level $n$, and $x \in P$, $r(x) = \lfloor d(x, \partial^-P )\rfloor + nl$. Then the discrete growth function $z$ is defined, for $n \in \mathbb{N}$, as $$\label{e1} z(n)=\text{vol}\{x\in R_T |r(x)\leq n\}.$$ Proposition $17$ of [@GP] says that the sequence $\{n_j\}$ (where the piece $Q_j$ is attached to vertices starting at $n_j$) can be chosen in such a manner that the discrete growth function $z$ of the resulting $R_T$ is of the same growth type as $v$. This choice of $n_j$ depends on the sequences in Proposition [**Proposition** 1](#p13){reference-type="ref" reference="p13"} and $v$. Finally, it is shown that the volume of $R_T$ is of the same growth type as $z$, with a growth constant $3$. Hence, $R_T$ is the required Riemannian manifold diffeomorphic to $M$ with volume growth in the same class as $v$. In Proposition [**Proposition** 1](#p13){reference-type="ref" reference="p13"}, note that a point $y_{Q_i}$ is fixed in each $\partial Q_i$. In [@GP], to prove that the volume growth of $R_T$ lies in the same growth class as $z$, the authors use the points $y_{Q_i}$ to create a path between the basepoint and any other point $x$. This path is then used to estimate the distance of $x$ from the basepoint. Here, the connectedness (path-connectedness) of each $Q_i$ is implicitly assumed. In the case of a manifold with multiple ends, for any exhaustion $\mathcal{A}_i$, there exists some $i$ such that for all $j>i$, $Q_j$ has multiple components. Then this algorithm chooses a point $y_{Q_i}$ in only one of those components. Hence for any point $x$ lying in a different component of $Q_i$, this process does not give an estimate for the distance of $x$. Therefore it is necessary to choose different points $y_{Q_i}$ in each component of $Q_i$, and to do so, we need to take into account the number of components of $Q_i$ for each $i$, and hence, the number of ends of $M$. We try to modify the above-mentioned construction and extend Theorem [Theorem 1](#t3){reference-type="ref" reference="t3"} in the case of manifolds with boundary. The proof closely follows the proof in [@GP]. ****Proposition** 2**. Let $M$ be a connected manifold of one end and a boundary $\partial M$. 1. If $M$ has finite topological type, every bgd-function belongs to the growth type of a Riemannian manifold diffeomorphic to $M$ with bounded geometry. 2. If $M$ has infinite topological type, a bgd-function $v$ belongs to the growth type of a Riemannian manifold diffeomorphic to $M$ of bounded geometry if and only if $\lim_{x\to\infty} \frac{v(n)}{n}=+\infty$. The Riemannian metric in both cases can be chosen such that in a tubular neighborhood of $\partial M$ it is a product metric of the form $dt^2+g_0$ where $g_0$ is the metric on $\partial M$. *Proof.* Consider a manifold $M$ of dimension $m$ with a single end and a boundary $\partial M$. We can take an exhaustion $\{\mathcal{A}_i\}$ of $M$ such that each $\mathcal{A}_i$ has connected boundary, and set $Q_j=\mathcal{A}_{j}\setminus \mathcal{A}_{j-1}$ such that $\partial M$ lies in the interior of $Q_1$. Since the addition of a boundary component to $Q_1$ does not cause any change in the other types of pieces described in Proposition [**Proposition** 1](#p13){reference-type="ref" reference="p13"} (in particular, no boundary component is added to the other types of pieces), we only need to modify the metric on the $Q_1$ as per our requirements. The pieces can again be attached according to a rooted tree as described earlier, with one possible modification. An additional piece $R_0$ is also added to the root of the tree if $n_1>1$. Here, $R_0$ is diffeomorphic to $\partial M\times[0,1]$ with a disc removed. The metric on $R_0$ is defined in the same way as the metric on the other $R_i$, and in particular, satisfies all the conditions in Proposition [**Proposition** 1](#p13){reference-type="ref" reference="p13"}. We compute the volume with the base point on $\partial R_0\times \{0\}$ which is diffeomorphic to $\partial M$. To put a metric on $Q_1$, first, we choose any metric of bounded geometry on $\partial Q_1=\partial^+Q_1 \sqcup \partial ^-Q_1$ where $\partial ^-Q_1$ is diffeomorphic to $\partial M$. Now, extend the metric to the entire $Q_1$ so that it is a product metric in some tubular neighbourhood of $\partial Q$. It can then be scaled to ensure it is a metric of bounded geometry in the entire $Q_1$. We follow the proof in modifying the metric to ensure that the diameter, distance, and volume parameters of these pieces satisfy the conditions in Proposition [**Proposition** 1](#p13){reference-type="ref" reference="p13"}. In case the root has a piece $R_0$ attached, the metric on it is defined as follows. $\partial^- Q_1$ is diffeomorphic to $\partial^+R_0$ by construction. Use the metric on $\partial^- Q_1$ to put a metric on $\partial^+R_0$, such that they become isometric. Extend it to a product metric, $\partial^-R_0\times [-10,10]$. Then it contains a ball of radius $3$ where we can surge in a handle with the usual metric. This gives the initial metric on $R_0$. Adjust the height, diameter, and volume parameters as per the specifications in Proposition [**Proposition** 1](#p13){reference-type="ref" reference="p13"}, which is possible since the arguments for $R_j$ are independent of the choice of $j$. We get a final metric on $R_0$ of bounded geometry. Hence, we also have the inequalities $\frac{1}{3}l\leq t_{R_0}\leq T_{R_0}\leq l$ where $t_{R_0}$ and $T_{R_0}$ are the minimum and maximum lengths of geodesics joining any $2$ points in $\partial^-R_0$ and $\partial^+R_0$ respectively. Also, diameter$(R_0)=$ diameter$(\partial^-Q_1)\leq d_1$, where $d_1$ is the diameter bound for $Q_1$ as in the proposition. Consider the volume growth function of the ball centered at some point $y_{R_0}$. We claim that a different choice of base point does not change the growth type of this function. To see this, consider another point $y'$ in $\partial^-R_0$ such that $d(y_{R_0},y')=h$. Then $vol(B(y',n))\leq vol(B(y_{R_0},n+h))$ and $vol(B(y_{R_0}),n)\leq vol(B(y',n+h))$. Thus $vol(B(y',x))$ and $vol(B(y_{R_0},x))$ are functions of the same growth type (by taking the constant $A\geq h$). So we can compute the volume growth about the point $y_{R_0}$ without any loss of generality. For any point $x$ in a piece $P$, where $P$ is attached at level $n$ of the tree, $y_{R_0}$ can be connected to $x$ via a minimizing geodesic. Let $\{P_0=R_0, P_1, P_2,...,P_k=P\}$ be the pieces intersected by the geodesic, and let $y_i\in \partial^- P_i$ be the marked points on those pieces (see Proposition [**Proposition** 1](#p13){reference-type="ref" reference="p13"}). Then $$d(y_{R_0},x)\leq \Sigma_{i=0}^{k-1}d(y_i,y_{i+1})+d(\partial^-P,x)+diameter(\partial^-P).$$ Now if $P= Q_a$ for some $a$, then $\Sigma_{i=0}^kd(y_i,y_{i+1})\leq n_al$. In all other cases, $\Sigma_{i=0}^{k-1}d(y_i,y_{i+1})\leq nl$. Similarly, diameter$(\partial^-P)\leq n_al$ or $l$ respectively. Hence, $$d(y_{R_0},x)\leq 2n_al+d(\partial^-P,x)$$ if $x\in Q_a$. Otherwise, $$d(y_{R_0},x)\leq 2nl+d(\partial^-P,x).$$ Therefore the volume estimates given in the proof of Theorem [Theorem 1](#t3){reference-type="ref" reference="t3"} by Grimaldi and Pansu remain unchanged after replacing the base point with $y_{R_0}$. ◻ Next, we study the volume growth of manifolds with finitely many ends. **Theorem 4**. Let $M$ be a connected manifold with finitely many ends. 1. If each end of $M$ is of finite topological type, every bgd-function $v$ belongs to the growth type of a Riemannian manifold of bounded geometry diffeomorphic to $M$. 2. If some end of $M$ is not of finite topological type, every bgd-function $v$ belongs to the growth type of a Riemannian manifold of bounded geometry diffeomorphic to $M$ if and only if $\lim_{n\to\infty}\frac{v(n)}{n}=\infty$. *Proof.* Consider a bgd-function $v$. Let $M$ be a connected manifold with $k$ ends and no boundaries, where $k\in \mathbb{N}$. There exists a compact manifold $P$ with $k$ boundary components such that $M\setminus P$ consists of $k$ connected components, each of which is a one-ended submanifold of $M$ of the same dimension, denoted by $P_i$, for $1\leq i\leq k$. Then $\partial P_i$ consists of a single component for each $i$ and $\partial P_i\subset \partial P$. Using Proposition [**Proposition** 2](#l12){reference-type="ref" reference="l12"} on each $P_i$ we get Riemannian manifolds diffeomorphic to $P_i$ and with volume growth function of the same growth type as $v$. Specifically, Proposition [**Proposition** 2](#l12){reference-type="ref" reference="l12"} provides manifolds $\{P_i', g_i'\}$, such that $P_i$ is diffeomorphic to $P_i'$, $P_i'$ has bounded geometry, and the volume growth function $v_i'$ of $P_i'$ is of the same type as $v$. Note that if each of the $P_i$ is of finite topological type then any bgd-function $v$ works, whereas if there exists some $i$ such that $P_i$ is of infinite topological type, then we need the added constraint on $v$ that $\lim_{n\to \infty} \frac{v(n)}{n}=+\infty$. The proof involves reattaching the $k$ unbounded components to $P$ to get a manifold diffeomorphic to the original manifold, and then computing its volume with the aid of the lemmas proven earlier. Note that $P_i$ and $P_i'$ are diffeomorphic for all $i$, by a diffeomorphism which restricts to the identity diffeomorphism on the boundary and the metric on $P_i'$ is a product metric on a tubular neighbourhood of $\partial P_i'$. $\partial P_i$ is also diffeomorphic to the corresponding component of $\partial P$ by construction, again by the identity diffeomorphism. Hence, $\partial P_i'$ is diffeomorphic to the corresponding component of $\partial P_i$. To define a metric on $P$, for each $i$, define a product metric a $dt_i^2+g_i$ on a tubular neighbourhood of $\partial P_i$ such that $g_i$ is a metric on $\partial P_i$ isometric to $g_i'$ restricted to $\partial P'_i$. Extend it to a metric $g'$ of bounded geometry defined on the entire $P.$ Then attach each $\partial P_i'$ with the corresponding component of $\partial P$ by the isometry to obtain a metric $g''$ of bounded geometry on $M'$ which is diffeomorphic to $M$. The metric $g''$ is smooth at $\partial P_i'$ for each $i$ since it is a product metric on a tubular neighbourhood of $\partial P_i'$ and $\partial P_i$. We now investigate the volume growth of $M'$. With the new metric, consider constants $d$ and $D$ such that $d\leq d_{x\in \partial P}(o,x) \leq D$. Then $\partial P\subset B(o,D)\setminus B(o,d)$. Hence, we get the following volume bound: $$vol(B(o,d))\leq vol(P) \leq vol(B(o,D)).$$ Also, for $n> d$, $$\begin{aligned} B(o,n)=(B(o,n)\cap P)\bigcup_i (B(o,n)\cap P_i).\end{aligned}$$ Denote by $y_i$ the point $y_{R_0}$ fixed earlier on the piece $\partial R_0^-$ on each component $P_i$. Then $d\leq d(o,y_i)\leq D$.\ Let $x\in P_i$ be any point. Then we have the following containment in each $P_i$: $$\begin{aligned} B(o,n)\cap P_i \subseteq B(y_i,n).\end{aligned}$$ Also if $n\geq D$, $$\begin{aligned} B(y_i,n-D)\subseteq B(o,n)\cap P_i.\end{aligned}$$ This gives the following volume bounds. $$\begin{aligned} vol(B(o,n))\cap P = & vol(B(o,n))\leq vol(P) \;\;\;\; &\text{for} \; n\leq d\\ vol(B(o,n))\cap P \leq & vol(B(o,n))\leq vol(P)+\Sigma_{i=1}^m v_i(B(y_i,n)) \;\;\;\; &\text{for } d\leq n\leq D\\ vol(P)+\Sigma_{i=1}^mv_i(B(y_i,n-D))\leq & vol(B(o,n))\leq vol(P)+\Sigma_{i=1}^m v_i(B(y_i,n)) \;\;\;\; &\text{for } n\geq D.\end{aligned}$$ Since $v_i$ has the same growth type as $v$ for each $i$, apply Lemma [ **Lemma** 1](#le1){reference-type="ref" reference="le1"} and Lemma [ **Lemma** 2](#le2){reference-type="ref" reference="le2"} to get that $\Sigma_i^m w_i(n)=\Sigma_i^m v_i(n-D)$ and $\Sigma_i^m v_i(n)$ are also of the same growth type as $v$. Now, apply the Lemma [ **Lemma** 1](#le1){reference-type="ref" reference="le1"} and Lemma [ **Lemma** 3](#le3){reference-type="ref" reference="le3"} to conclude that the function $vol(B(o,n))$ is of the same growth type as $v$. Hence, we get a Riemannian manifold $M'$ which is diffeomorphic to $M$, has bounded geometry, and has volume growth function of the same growth type as $v$. ◻ # Volume growth on admissible manifolds {#countably} **Theorem 5**. Let $\mathcal{U}$ be a finite collection of compact manifolds without boundary. If $M$ is a one-ended manifold such that $M=M_1\#M_2\#...\#M_i\#...$ where each $M_i\in \mathcal{U}$, then given a bgd-function $v$, there exists a Riemannian metric of bounded geometry on $M$ such that its volume growth function lies in $[v]$ and the growth constant depends only on $\mathcal{U}$ and $v$. The same result holds if $M=M_1\#M_2\#...\#M_i...$ for $M_i\in \mathcal{U}$ for $i>1$ and $M_1$ is an element of $\mathcal{U}$ with a disc removed. *Proof.* Let $v$ be any bgd-function, and let $L$ be the constant controlling the growth of its derivative. By Lemma $11$ of [@GP], there exists a bgd-function $w$ of the same growth type as $v$, such that the growth of the derivative of $w$ is controlled by the constant $2$. The growth constant for $v$ and $w$ is $L^{\frac{1}{l}}$ where $l$ is any integer such that $l>log_2(L)$. Let $M$ be a one-ended manifold without boundary formed by taking a connected sum of elements of $\mathcal{U}$. We choose an exhaustion $\{\mathcal{A}_i\}$ of $M$ such that each $\mathcal{A}_j=M_{1}\#M_{2}\#...\#M_{j}$. Then $Q_j=\mathcal{A}_{j+1}\setminus\mathcal{A}_j$ is an element of $\mathcal{U}$ (with boundary discs removed due to the connected sum). For every element $V$ in $\mathcal{U}$, remove two discs from $V$, and put a metric of bounded geometry on the resulting manifold such that it is a product metric in some tubular neighbourhoods of the boundary spheres. Scale the metrics such that the boundary spheres on all the pieces are isometric, while preserving the bounded geometry condition. This is possible since there are only finitely many elements in $\mathcal{U}$. Then the metric on the piece $Q_j$ can be chosen to be isometric to the metric on some piece of $\mathcal{U}$ minus two discs. To attach the pieces to a tree $T$, there needs to be a choice of the sequence $\{n_j\}$ of vertices of the trunk of the tree such that the discrete growth function (defined in ([\[e1\]](#e1){reference-type="ref" reference="e1"})) of the resulting Riemannian manifold lies in the same class as the bgd-function $w$. The sequences $\{U_j\}$, $\{u_j\}$, $\{d_j\}$ and $\{t_j\}$ from Proposition [**Proposition** 1](#p13){reference-type="ref" reference="p13"} are bounded, since there are only finitely many distinct elements in each sequence, one corresponding to each element of $\mathcal{U}$. Again by the finiteness of $\mathcal{U}$, there exists some $\alpha$ such that for all pieces $P$ of the type $Q_j$ or $R_j$, $\alpha\leq$min$v'_P$. Let $t_0=\sup\{t_j\}$. Each piece $Q_j$ can be modified such that $t_j=t_0$ by thickening the boundary as required and using the product metric on the thickened boundary. This change does not affect the bounded geometry of the piece $Q_j$. Choose an $R\geq d_j$ for all $j$, which is possible by the boundedness of $d_j$. Let the sequence $n_j$ be chosen as $n_j=n_0+j(t_0+R)$ for some positive integer $n_0$. Consider the tree $T$ with growth function $w$, where $T$ has a single trunk, each vertex of $T$ has at most $2$ branches, and the vertices $S=\bigcup_j [n_j,n_j+t_0-1]$ of the trunk have a single branch. Recall that such a tree exists by Lemma $10$ of [@GP]. Attach the pieces $Q_j$ in a similar manner to the earlier construction, according to the tree $T$. The resultant manifold is diffeomorphic to $M$, and we again call it $R_T$. Since the $U_j$ are bounded, fix some $H'\geq \{U_j,H\}$. Also, fix some $h'$ such that $h'\leq\{h,\alpha\}$. Then the bounds for $z$ are $$\begin{aligned} \label{e2} { (w(n)-w(n-1)) h'\leq z(n)-z(n-1)\leq H'(w(n)-w(n-1)).} \end{aligned}$$ By the definition of $z$, $z(0)=0$. Also by definition, $w(0)=1$. Therefore, $$z(1)\leq H'(w(1)-1)\leq H'w(1).$$ Iterating ([\[e2\]](#e2){reference-type="ref" reference="e2"}), $$\begin{aligned} \label{e3} z(n)\leq H'w(n). \end{aligned}$$ Similarly, $$(w(1)-1)h'\leq z(1).$$ This implies, $$w(1)\leq \frac{1}{h'}z(1)+1.$$ Again iterating, $$\begin{aligned} \label{e4} w(n)\leq \frac{1}{h'}z(n)+n. \end{aligned}$$ Take $A=\text{max}\{\frac{1}{h'},H'\}$. Then the equations ([\[e3\]](#e3){reference-type="ref" reference="e3"}) and ([\[e4\]](#e4){reference-type="ref" reference="e4"}) can be rewritten as $$\begin{aligned} \label{e6} z(n)\leq Aw(n) \hspace{1cm} \text{and} \hspace{1cm} w(n)\leq Az(n)+An. \end{aligned}$$ Considering the equation $w(n)\leq A(z(n)+n)$, use $\frac{1}{2}\leq w(n)-w(n-1)$ to get $$\frac{h'}{2}\leq z(n)-z(n-1).$$ Hence, on iteration, $$\begin{aligned} z(n)+\frac{kh'}{2}\leq z(n+k).\end{aligned}$$ Putting $k=(A-1)n+A$, $$\begin{aligned} \label{e8} z(n)+\frac{1}{2}((A-1)n+A)h'\leq z(An+A).\end{aligned}$$ Therefore when $n\leq \frac{1}{2}((A-1)n+A)h'$, that is, for all $A\geq 1+\frac{2}{h'}$, ([\[e8\]](#e8){reference-type="ref" reference="e8"}) implies that $$\begin{aligned} z(n)+n\leq z(An+A).\end{aligned}$$ Therefore, the other direction of ([\[e6\]](#e6){reference-type="ref" reference="e6"}) also reduces to $$\begin{aligned} \label{e9} w(n)\leq Az(An+A).\end{aligned}$$ Together, ([\[e6\]](#e6){reference-type="ref" reference="e6"}) and ([\[e9\]](#e9){reference-type="ref" reference="e9"}) imply that $z$ and $w$ are in the same growth class, with a growth constant $A$ where $A=$max$\{1+\frac{2}{h'},H,U_j,3\}$. This $A$ depends only on $h'$, $H$ and $\{U_j\}$ which depend on the collection $\mathcal{U}$. Also, the growth constant for $z$ and the volume of $R_T$ is $3$, and the growth constant for $v$ and $w$ is $L^{\frac{1}{l}}$. Hence, the volume growth function of $R_T$ is in the same class as $v$, with a growth constant given by (max$\{1+\frac{2}{h'},H,U_j,3\})\times3L^{\frac{1}{l}}$. Now consider a manifold $M$ such that $M=M_1\#M_2\#...\#M_i\#...$ where $M_i\in \mathcal{U}$ for $i>1$ and $M_1$ is an element of $\mathcal{U}$ with a disc removed. $M$ has boundary diffeomorphic to a sphere. We choose an exhaustion $\{\Tilde{\mathcal{A}}_j\}$ of $M$ such that each $\Tilde{\mathcal{A}}_j=M_{1}\#M_{2}\#...\#M_{j}$. Then $Q_j=\Tilde{\mathcal{A}}_{j+1}\setminus \Tilde{\mathcal{A}}_j$ is an element of $\mathcal{U}$ with two discs removed. We take that base point at $\partial M$. The rest of the construction follows as above. ◻ Let $\{M_\alpha\}$ be a collection of one-ended manifolds such that for each $\alpha$, $M_{\alpha}=M_{\alpha 1}\#M_{\alpha 2}\#...\#M_{\alpha i}\#...$ for some $M_{\alpha i}\in \mathcal{U}$. Then given a bgd function $v$, each $M_{\alpha}$ admits a Riemannian metric with bounded geometry such that its volume growth function lies in $[v]$ and the growth constants are uniformly bounded, by the above. Next, we prove a technical lemma to prove Theorem [Theorem 3](#t18){reference-type="ref" reference="t18"}. ** **Lemma** 6**. Given a non-decreasing bgd-function $v$, a sequence of bgd-functions $\{v_i\}_{i\geq 1}$ such that $[v_i]=[v]$ for all $i$, and a non-decreasing function $N$ with range $\mathbb{Z}_+$, if the growth constants $A_i$ corresponding to $[v_i]$ can be chosen such that the sequence $\{A_i\}$ has an upper bound, then for $F(r)=\Sigma_{i=1}^{N(r)}(v_i(r))$ and $f(r)=v(r)\times N(r)$, $f$ and $F$ are of the same growth type. *Proof.* Since $[v_i]=[v]$, we have constants $A_i$ such that $$\begin{aligned} & & v(n)& \leq A_iv_i(A_in+A_i)+A_i&\\ &\text{and} & v_i(n)& \leq A_iv(A_in+A_i)+A_i. & \end{aligned}$$ Hence, $$F(n)=\Sigma_{i=1}^{N(n)}v_i(n)\leq \Sigma_{i=1}^{N(n)}(A_iv(A_in+A_i)+A_i).$$ Assume that the sequence $\{A_i\}$ has an upper bound $A$. Then, $\forall i$, $$\begin{aligned} & &v(n)& \leq Av_i(An+A)+A &\\ &\text{and} &v_i(n)& \leq Av(An+A)+A. & \end{aligned}$$ Since $v$ is a bgd-function, it grows at least linearly. If $v(A)<1$, replace $A$ by some larger constant so that $1\leq v(A)$. Then, $$\begin{aligned} F(n) &= \Sigma_{i=1}^{N(n)}v_i(n)\\ &= \Sigma_{i=1}^{N(n)}(Av(An+A)+A)\\ & = N(n)(Av(An+A)+A)\\ & \leq AN(n)(v(An+A)+1)\\ & \leq AN(n)(2v(An+A))\\ & \leq 2AN(2An+2A)v(2An+2A)+2A\\ & = 2Af(2An+2A) + 2A. \end{aligned}$$ Also, $$\begin{aligned} f(n) & = N(n)v(n)\\ & = \Sigma_{i=1}^{N(n)}v_(n)\\ & \leq \Sigma_{i=1}^{N(n)}(v_i(A_in+A_i) +A_i)\\ &\leq \Sigma_{i=1}^{N(An+A)}(v_i(An+A))+A\\ &\leq \Sigma_{i=1}^{N(2An+2A)}(v_i(2An+2A))+2A\\ & \leq 2AF(2An+2A) + 2A.\end{aligned}$$ Hence, $[F]=[f]$ with growth constant $2A$. ◻ We can now proceed with the proof of Theorem [Theorem 3](#t18){reference-type="ref" reference="t18"}. *Proof.* Let $M$ be an $n$-dimensional admissible manifold over some finite collection $\mathcal{U}$ of compact manifolds without boundary. Then there exists a manifold $N$ with countably many boundaries $\partial N_i$ and one-ended manifolds $M_i$ with boundary $\partial M_i$ which satisfies the criterion given in Definition [Definition 3](#d1){reference-type="ref" reference="d1"}. Each $M_i$ is a one-ended manifold with one boundary sphere. Using the second part of Theorem [Theorem 5](#l21){reference-type="ref" reference="l21"} we obtain Riemannian manifold $M_i'$ of bounded geometry diffeomorphic to $M_i$ for all $i$, with volume growth function $v_i$ having the same growth class as $v$, and the same growth constant. By the construction, one can also assume that boundary spheres $M_i'$ are isometric to each other. Now consider $N.$ By definition $N=N_1\# N_2 \#N_3...$ such that $\partial N_i\in N_i$ and each $N_i$ is an element of $\mathcal{U}$ with one disk removed. Take the exhaustion $\{\mathcal{B}_i\}$ of $N$ where $\mathcal{B}_i=N_{1}\#N_2\#..\#N_{i}$. Then $\Tilde{Q}_j=\mathcal{B}_{j}\setminus\mathcal{B}_{j-1}$ is diffeomorphic to an element of $\mathcal{U}$ with $3$ discs removed for all $j\geq1$. Remove three discs from each $V\in\mathcal{U}$, and put a metric of bounded geometry on the resulting manifold with product metric in some tubular neighbourhood of the boundary spheres. Scale the metrics so that all the boundary spheres become isometric to each other and they are also isometric to boundary spheres of all $M_i$. This gives the initial metric on $\Tilde{Q}_j$. We also follow the proof of Proposition [**Proposition** 1](#p13){reference-type="ref" reference="p13"} in modifying the metric to ensure that the diameter, distance, and volume parameters of these pieces satisfy the conditions in the same proposition. In particular, we can thicken the boundary components of $\partial^-Q_j$ and $\partial^+Q_j$ as necessary to ensure that the maximum distance from $\partial^-Q_j$ is achieved on $\partial^+Q_j$. Hence, the addition of the new boundary components does not affect any of the arguments in these parts. Since there are finitely many distinct elements in the sequence $\{\Tilde{Q}_j\}$, the sequences $\{t_j\}$, $\{u_j\}$ and $\{U_j\}$ given in Proposition [**Proposition** 1](#p13){reference-type="ref" reference="p13"} are bounded. Take the function $C:\mathbb{R}_+\to \mathbb{Z}_+$ where $C(r)$ gives the number of connected components of $M$ in the complement of the ball of radius $r$. This is a non-decreasing step function. Given an increasing smooth function $f:[1,\infty)\to \mathbb{R}_+$ such that $f'$ is bounded, we wish to modify the sequence $\{t_j\}$ such that $C(x)$ is of the same growth type as $f(x)$. Increasing the value of $t_j$ for some $j$ has the effect of thickening the boundary spheres of $\Tilde{Q}_j$, and this does not affect the boundedness of the metric on $\Tilde{Q}_j$ or the growth type of the function $v_j$. Recall that the proof of Proposition $17$ in [@GP] provides a formula to choose a sequence $\{n_j\}$ such that the piece $\Tilde{Q}_j$ is simultaneously attached to the vertices $\{n_j,...,n_j+t_j-1\}$ to make the growth function of the resulting Riemannian manifold lie in the same class as $v$. The sequence can be chosen recursively as $n_j=n_{j-1}+t_{j-1}+R$ for some large constant $R$. $M\setminus \mathcal{B}_j$ has $j+1$ connected components by construction, and we can use this to relate the function $C(x)$ to the sequence $n_j$. Since $f$ is an increasing function it is a bijective function from $[1,\infty)$ on its image. Consider the function $g=f^{-1}$. Since $f'(x)< K$ for some positive constant $K$, $g'(x)> \frac{1}{K}$ for all $x\in \mathbb{R}_+$. Therefore, using the Fundamental Theorem of Calculus, $\frac{1}{K}< g(n)-g(n-1)$ for all $n\in \mathbb{N}$. By Proposition [**Proposition** 1](#p13){reference-type="ref" reference="p13"}, each piece $P$ satisfies the inequality $\frac{1}{3}l\leq t_P\leq T_P\leq l$. Hence, the piece $\Tilde{Q}_j$ lies at a distance $\frac{1}{3}n_jl\leq d(o,\Tilde{Q}_j)\leq n_jl$ from the basepoint. Therefore, $C(n_jl)\geq j+1\geq C(\frac{1}{3}n_jl)$. Thus, the map $\eta : \mathbb{N}\to \mathbb{N}$ given by $\eta(j)=n_j$ is such that $C\circ \eta$ is of the same growth type as the identity map. This implies that if $C$ is of the same growth type as $f$, then $\eta$ is of the same growth type as $g$, and vice versa. Thus it suffices to choose a sequence $\{t_j\}$ such that $n_j=\alpha g(j)$ for some constant $\alpha$. The sequence $\{t_j\}$ initially has an upper bound, say $t'$. We can modify any piece $\Tilde{Q}_i$ by thickening the boundary to have a new value $t_i\geq t'$ without affecting the conditions mentioned in Proposition [**Proposition** 1](#p13){reference-type="ref" reference="p13"}. We use the relation $n_j=n_{j-1}+t_{j-1}+R$. Therefore, $$t_{j-1} = n_j - n_{j-1} - R,$$ and substituting for $n_j$ and $n_{j-1}$, $$t_{j-1} = \alpha g(j) - \alpha g(j-1) - R.$$ To ensure that $t_j$ is greater than $t'$, we require $\alpha g(j)-\alpha g(j-1)>R+t'$. Using $g(j)-g(j-1)>K$, it suffices to choose $\alpha>\frac{R+t'}{K}$. Then, $[C]=[f]$. With this choice of sequences $\{n_j\}$ and $\{t_j\}$ and following the same construction as in [@GP], we get a metric on $N$ such that the volume growth function $v'$ of $N$ lies in $[v]$. Next, we attach $M_i$ to $\partial N_i$ using the identity map for all $i$. As a result, we get a smooth Riemannian metric on $M$ as the boundary spheres $\partial M_i$ and $\partial N_i$ are isometric and the metrics on $M_i$ and $N$ are product metrics on tubular neighbourhoods of $\partial M_i$ and $\partial N_i$ respectively. Let us now try to compute the volume growth function of $M$ based at a point $o\in N_1$. We denote the volume of a ball of radius $r$ centered at $o$ as $vol(B(o,r))$ and the distance on $M$ by $d$. Then $$B(o,r)\subset \cup_{i=1}^{C(r)} \{B(o,r)\cap M_i\}\cup \{N\cap B(o,r)\} .$$ Therefore, $$v'(r)\leq vol(B(o,r))\leq \Sigma_{i=1}^{C(r)}(v_i(r))+v'(r).$$ Apply the Theorem [Theorem 5](#l21){reference-type="ref" reference="l21"} to get that the growth constant for $v'$ and for the $v_i$ corresponding to each end is bounded in our case. Combining this with Lemma [ **Lemma** 6](#l48){reference-type="ref" reference="l48"}, we get $$[v(r)]\leq[vol(B(o,r))]\leq[(C(r)+1)v(r)].$$ Substituting $[C+1]=[C]=[f]$ into this, we can conclude that given a positive integer $k$, there exists a metric of bounded geometry on $M$ such that $[v(r)]\leq[vol(B(o,r))]\leq[f(r)v(r)]$. This concludes the proof of the theorem. ◻ ****Corollary** 3**. Given a manifold $M$, if there exists a compact manifold $N$ with boundaries $\partial N_1,$ $\partial N_2,$ $...,$$\partial N_k$ such that $M\setminus N$ is a disjoint union of $k$ admissible manifolds with boundaries diffeomorphic to $\partial N_i$, then given any bgd-function $v$ and any increasing $C^1$ function $f:[1,\infty)\to\mathbb{R}_+$ such that $f'$ is bounded, there exists a metric of bounded geometry on $M$ such that the volume growth function of $M$ has growth type between $[v]$ and $[fv]$. *Proof.* For $1\leq i\leq k$, let $N_i$ be the component of $M\setminus N$ with boundary $\partial N_i$. $N_i$ is an admissible manifold, and therefore, given the functions $v$ and $f$, Theorem [Theorem 3](#t18){reference-type="ref" reference="t18"} can be applied to each $N_i$. This gives Riemannian manifolds $N_i'$ such that $N_i$ is diffeomorphic to $N_i'$, $N_i'$ is of bounded geometry, and the volume growth function $v_i$ of $N_i'$ satisfies $[v]\leq[v_i]\leq [fv]$. The metric can be chosen such that it is a product metric in some tubular neighbourhood of the boundary, which is diffeomorphic to $\partial N_i$. Put a metric of bounded geometry on $N$ such that each boundary component $\partial N_i$ of $N$ is isometric to the boundary component of $N_i'$, and it is a product metric on some neighbourhood of $\partial N_i$. This isometry can be used to attach the $N_i'$ to $N$, resulting in a Riemannian manifold $M'$ diffeomorphic to $M$, with a metric of bounded geometry. Fix any point $o\in N$. Since $N$ has finitely many boundary components, there exist $d$ and $D$ such that $B(o,d)\subset N\subset B(o,D)$. For $n<D$, $$vol(B(o,n))\leq vol(N)+\Sigma_{i=1}^kv_i(n).$$ For $n\geq D$, $$vol(N)+\Sigma_{i=1}^kv_i(n-D)\leq vol(B(o,n))\leq vol(N)+\Sigma_{i=1}^kv_i(n).$$ Thus, $vol(B(o,n))$ is of the same growth type as $\Sigma_{i=1}^kv_i(n)$. Using $v(n)\leq v_i(n)\leq f(n)v(n)$, $kv(n)\leq \Sigma_{i=1}^kv_i(n)\leq kf(n)v(n)$. Since multiplication by a constant does not change growth type, $vol(B(o,n))$ has a growth type such that $[v]\leq[vol(B(o,n))]\leq [fv]$. ◻ HD G. Carron, *Riesz transform on manifolds with quadratic curvature decay*, Rev. Mat. Iberoam **33** (2017), no. 3, 53-79. R. Grimaldi and P. Pansu, *Bounded geometry, growth and topology*, Journal de Mathématiques Pures et Appliquées **95** (2011), 85-98. M. Badura, *Prescribing growth type of complete Riemannian manifolds of bounded geometry*, Annales Polonici Mathematici. **75** (2000) no. 2, 167-175. L. Funar and R. Grimaldi, *The ends of manifolds with bounded geometry, linear growth and finite filling area*, Geom. Dedicata **104** (2004), 139--148. L. Bessières, G. Besson and S. Maillot, *Open 3-manifolds which are connected sums of closed ones*, (2020): hal-02462552.

arxiv_math

--- abstract: | We construct elementary subgroups of all reductive groups over rings of the local isotropic rank $\geq 2$ and prove their basic properties. In particular, our results may be applied to the automorphism groups of any finitely generated projective modules over commutative unital rings of rank $\geq 3$ at every prime ideal. author: - | Egor Voronetsky[^1]\ Chebyshev Laboratory,\ St. Petersburg State University,\ 14th Line V.O., 29B,\ Saint Petersburg 199178 Russia\ bibliography: - references.bib title: Locally isotropic elementary groups --- # Introduction Let $K$ be a unital commutative ring and $\Phi$ be a root system of rank $\geq 2$ (reduced and irreducible). The Chevalley group $G(\Phi, K)$ is the point group of the Chevalley --- Demazure group scheme with the root system $\Phi$ and some choice of the weight lattice, e.g. of adjoint type or simply connected type. This group has so-called root elements $t_\alpha(x)$ for $\alpha \in \Phi$, $x \in K$, and the elementary subgroup $\mathop{\mathrm{E}}(\Phi, K) \leq G(\Phi, K)$ is the subgroup generated by all root elements. It is well-known [@abe-local; @abe; @fuan; @kopeiko; @suslin-kopeiko; @taddei] that the elementary subgroup is normal and perfect (if $\Phi$ is of type $\mathsf B_2$ or $\mathsf G_2$, then the perfectness holds only under the additional assumption that $K$ does not have residue fields isomorphic to $\mathbb F_2$). Moreover, if $K$ is finite-dimensional and the rank of $\Phi$ is at least $3$, then $G(\Phi, K) / \mathop{\mathrm{E}}(\Phi, K)$ is solvable [@k1-nilp], i.e. $\mathop{\mathrm{E}}(\Phi, K)$ is the largest perfect subgroup of $G(\Phi, K)$. Finally, if $G(\Phi, K)$ is simply connected, then Alexey Stepanov showed in [@stepanov] that the width in terms of root elements of a commutator $[x, g]$ for $x \in \mathop{\mathrm{E}}(\Phi, K)$, $g \in G(\Phi, K)$ is bounded uniformly on $K$ (i.e. the bound depends only on the root system $\Phi$). The elementary group is used in algebraic $\mathop{\mathrm{K}}$-theory, the factor-group $\mathop{\mathrm{K}}_1(\Phi, K) = G(\Phi, K) / \mathop{\mathrm{E}}(\Phi, K)$ is an unstable $\mathop{\mathrm{K}}_1$-functor. Also, classifications of normal and subnormal subgroups of $G(\Phi, K)$ essentially use the elementary subgroup. For example, $N \leq \mathop{\mathrm{SL}}(n, K)$ is normalized by the elementary subgroup $\mathop{\mathrm{E}}(\mathsf A_{n - 1}, K)$ for $n \geq 3$ if and only if $\mathop{\mathrm{E}}(\mathsf A_{n - 1}, K, \mathfrak a) \leq N \leq \mathop{\mathrm{SC}}(n, K, \mathfrak a)$ for some explicitly defined groups: the relative elementary subgroup $\mathop{\mathrm{E}}(\mathsf A_{n - 1}, K, \mathfrak a)$ is given by generators and the full congruence subgroup $\mathop{\mathrm{SC}}(n, K, \mathfrak a)$ is given by equations. It is natural to generalize these results to isotropic reductive groups. Over arbitrary unital commutative rings reductive group schemes are considered in [@sga3] (see also [@conrad]). The notion of isotropic reductive groups over fields [@borel-tits] is generalized to local rings (more generally, semi-local rings with connected spectra) in [@sga3 Exp. XXVI]. Over arbitrary rings Victor Petrov and Anastasia Stavrova [@petrov-stavrova] considered reductive groups with globally defined parabolic subgroups (non-trivially intersecting all simple factors of the group scheme over geometric points), they proved the normality of the elementary subgroup in this context. In [@luzgarev-stavrova] Alexander Luzgarev and Stavrova showed that the elementary subgroup is perfect (under a natural additional assumption). Stavrova and Stepanov also proved [@stavrova-stepanov] that such isotropic reductive groups admit a standard classification of subgroups normalized by the elementary group if the structure constants are invertible in $K$. For twisted Chevalley groups normality of the elementary subgroup was already proved in [@bak-vavilov; @suzuki]. See also the survey [@survey] for details. We are interested in the following more general situation. Suppose that $G$ is a reductive group scheme over $K$ such that it is sufficiently isotropic (in the sense of [@sga3] or [@petrov-stavrova]) Zariski locally, i.e. at all localizations at prime ideals. A typical example is the twisted general linear group $G(K) = \mathop{\mathrm{GL}}(P)$, where $P$ is a finite projective $K$-module without direct summands and of the rank at least $3$ at each point. Even the definition of the elementary subgroup for such groups is non-trivial since $P$ does not have any unimodular vectors. We construct the elementary subgroup as an abstract group in theorem [Theorem 1](#e-discr){reference-type="ref" reference="e-discr"} and describe it in an elementary (though not entirely constructive) way. In theorem [Theorem 2](#elem-gen){reference-type="ref" reference="elem-gen"} we show that the elementary subgroup may be generated by a scheme morphism (as in the case of Chevalley groups) and it is normal. The next theorem [Theorem 3](#par-gen){reference-type="ref" reference="par-gen"} states that our elementary groups coincide with the groups defined in [@petrov-stavrova] if the global isotropic rank is positive. Finally, the perfectness of the elementary group is proved in theorem [Theorem 4](#perfect){reference-type="ref" reference="perfect"}. Actually, our ultimate goal is to study Steinberg groups over locally isotropic reductive groups. Even in such generality centrality of $\mathop{\mathrm{K}}_2$-functor may be proved using a suitable localization technique [@linear-k2; @thesis], but we still have to construct the Steinberg group and $\mathop{\mathrm{K}}_2$-functor. This problem it technically more difficult than the case of $\mathop{\mathrm{K}}_1$, so in this paper we work only with the elementary subgroup using the localization-and-patching method from [@yoga]. As in [@linear-k2; @thesis] we use homotopes of the base ring instead of prime ideals and the formalism of ind- and pro-completion of categories in order to apply the method of construction of the elementary subgroup to the level of $\mathop{\mathrm{K}}_2$ in the future. Homotopes also allow us to prove various functorial properties of the elementary subgroups, in particular we are able to use generic elements of various affine schemes. Ind- and pro-completions are necessary to adapt the localization-and-patching method to groups defined by homotopes instead of principal ideals, since such groups are not subgroups of the base group. # Isotropic pinnings All rings and algebras in this paper are commutative and associative, but not necessarily unital. Throughout the paper we fix a unital ring $K$. If $\mathbf C$ is a category, then $X \in \mathbf C$ means that $X$ is an object of $\mathbf C$. The categories of sets, groups, unital $K$-algebras, and non-unital $K$-algebras are denoted by $\mathbf{Set}$, $\mathbf{Grp}$, $\mathbf{CRing}_K$, and $\mathbf{CRng}_K$ respectively. We identify schemes over $K$ and corresponding covariant functors $\mathbf{CRing}_K \to \mathbf{Set}$. A principal open subscheme $\mathop{\mathrm{Spec}}(K_s) \subseteq \mathop{\mathrm{Spec}}(K)$ is also denoted by $\mathcal D(s)$. In this paper root systems are always crystallographic, but neither irreducible nor reduced in general, i.e. we admit components of types $\mathsf{BC}_\ell$ for $\ell \geq 1$. Let us say that roots $\alpha$ and $\beta$ in a root system are *neighbors* if they are linearly independent, non-orthogonal, and there are no roots in the open angle $\mathbb R \alpha + \mathbb R \beta$. Let $\Phi$ and $\Psi$ be two root systems. A map $f \colon \Phi \cup \{0\} \to \Psi \cup \{0\}$ is called a *morphism* of root systems if it is induced by a linear map $\mathbb R \Phi \to \mathbb R \Psi$ of the ambient spaces. A morphism $f$ of root systems is called a *factor-morphism* if it is surjective. The next lemma implies that for any morphism $f \colon \Phi \cup \{0\} \to \Psi \cup \{0\}$ and any irreducible component $\Phi' \subseteq \Phi$ not in the kernel of $f$ there is unique irreducible component $\Psi' \subseteq \Psi$ such that $f(\Phi') \subseteq \Psi' \cup \{0\}$. **Lemma 1**. *Let $\Phi$ be an irreducible root system. Then it cannot be covered by two hyperplanes of the ambient space. If $H \leq \mathbb R \Phi$ is a hyperplane and $\alpha \in \Phi \cap H$, then $\alpha$ has a neighbor in $\Phi \setminus H$.* *Proof.* We show that $\Phi$ cannot be covered by a union of proper subspaces $V_1 \cup V_2$ by induction on $\dim(V_1 \cap V_2)$. Clearly, the subsets $\Phi_1 = \Phi \cap V_1 \setminus V_2$ and $\Phi_2 = \Phi \cap V_2 \setminus V_1$ are orthogonal, so the case $\Phi \cap V_1 \cap V_2 = \varnothing$ (and the induction base $\dim(V_1 \cap V_2) = 0$) follows from the irreducibility. Take any $\alpha \in \Phi \cap V_1 \cap V_2$. If $\alpha$ is not orthogonal to $\Phi_1$, then it lies in the span of $\Phi_1$ and orthogonal to $\Phi_2$. In other words, $\Phi \subseteq V_1 \cup (V_2 \cap \alpha^\perp)$ or $\Phi \subseteq (V_1 \cap \alpha^\perp) \cup V_2$, where $\alpha^\perp$ is the orthogonal complement to $\alpha$. Since $\dim(V_1 \cap V_2 \cap \alpha^\perp) = \dim(V_1 \cap V_2) - 1$, we conclude by the induction assumption. To prove the second assertion, note that $\Phi$ cannot be covered by $H$ and $\alpha^\perp$. ◻ Now let $G$ be a reductive group scheme over $K$. Recall [@sga3 Exp. XXIII, déf. 1.1] (or [@conrad def. 6.1.1]) that a *pinning* of $G$ consists of - a maximal torus $T \leq G$; - an isomorphism $T \cong \mathbb G_{\mathrm m}^k$ such that the roots and coroots are constant elements of $\mathbb Z^k$ and the dual lattice ${^{k}\!{\mathbb Z}}$ respectively, the root subspaces of the Lie algebra $\mathfrak g$ of $G$ are trivial linear bundles; - a basis $\Delta \subseteq \Phi$; - trivializing sections $x_\alpha \in \mathfrak g_\alpha$ for $\alpha \in \Delta$, where $\mathfrak g_\alpha$ is the $\alpha$-th weight submodule of the Lie algebra $\mathfrak g$ of $G$. We say that an *isotropic pinning* of $G$ is a tuple of - a torus $T \leq G$; - an isomorphism $T \cong \mathbb G_{\mathrm m}^k$ such that the roots are constant elements of $\mathbb Z^k$ and form a root system $\Psi$; - isomorphisms $\mathfrak g_\alpha \cong K^{m_\alpha}$ for all $\alpha \in \Psi$ with the following property: there is an fppf extension $K \subseteq K'$ and a pinning of $G_{K'}$ with the maximal torus $T' \cong \mathbb G_{\mathrm m}^{k'}$ and the root system $\Phi$ such that $T_{K'} \leq T'$ and this inclusion is given by a constant homomorphism of the weight lattices $\mathbb Z^{k'} \to \mathbb Z^k$, so the induced map $\Phi \cup \{0\} \to \Psi \cup \{0\}$ is surjective (i.e. a factor-morphism). The *rank* of an isotropic pinning is smallest rank of the components of $\Psi$ if no component of $\Phi$ maps to zero and $0$ otherwise. In other words, the rank is positive if and only if $T$ does not commute with all simple factors of $G / \mathop{\mathrm{C}}(G)$ at every geometric point. An isotropic pinning is usually denoted just by $(T, \Psi)$. By definitions, every pinning may be continued to an isotropic pinning (uniquely up to the choices of bases of $\mathfrak g_\alpha$). An isotropic pinning $(T, \Psi)$ is *contained* in an isotropic pinning $(T', \Psi')$ if $T \leq T'$ and the inclusion is given by a constant homomorphism of the weight lattices $\mathbb Z^{k'} \to \mathbb Z^k$, so the induced map $\Psi' \cup \{0\} \to \Psi \cup \{0\}$ is a factor-morphism and the rank of $(T, \Psi)$ is at most the rank of $(T', \Psi')$. Each isotropic pinning is contained in a pinning after an fppf extension. Recall from [@thesis §2.1] and [@twisted-forms §4] that a *$2$-step nilpotent $K$-module* $(M, M_0)$ consists of - a group $M$ with the group operation $\dotplus$ and a $2$-step nilpotent filtration $M_0 \leq M$ (i.e. $[M, M] \leq M_0$ and $[M, M_0] = \dot 0$); - a left $K$-module structure on $M_0$; - a right action $({-}) \cdot ({=}) \colon M \times K \to M$ of the multiplicative monoid $K^\bullet$ by group endomorphisms such that - $[m \cdot k, m' \cdot k'] = kk' [m, m']$; - $m \cdot (k + k') = m \cdot k \dotplus kk' \tau(m) \dotplus m \cdot k'$ for some (uniquely determined) $\tau(m) \in M_0$; - $m_0 \cdot k = k^2 m_0$ for $m_0 \in M_0$. It easily follows that $m \cdot 0 = \dot 0$, $m \cdot (-k) = k^2 \tau(m) \mathbin{\dot{-}}m \cdot k$, $\tau(m \dotplus m') = \tau(m) \dotplus [m, m'] \dotplus \tau(m')$, $\tau(m \cdot k) = k^2 \tau(m)$, $\tau(\mathbin{\dot{-}}m) = -\tau(m)$, and $\tau(m_0) = 2m_0$ for $m_0 \in M_0$. Every $K$-module $M$ admits two natural structures of a $2$-step nilpotent $K$-module with the minimal nilpotent filtration $0 \leq M$ and the maximal one $M \leq M$. Scalar extensions of $2$-step nilpotent modules are defined in [@thesis §2.1] and [@twisted-forms §4]. Let $(M, M_0)$ and $(N, N_0)$ be $2$-step nilpotent $K$-modules. A map $q \colon M \to N$ is called *$K$-quadratic* if - $q(M_0) \subseteq N_0$ and $q|_{M_0} \colon M_0 \to N_0$ is $K$-linear, - $q(m \dotplus m') = q(m) + b(m, m') + q(m')$ for some (uniquely determined) $K$-bilinear map $b \colon M / M_0 \times M / M_0 \to N_0$, - $q(m \cdot k) = q(m) \cdot k$. It follows that $q(\mathbin{\dot{-}}m) = b(m, m) \mathbin{\dot{-}}q(m)$, $q(\tau(m)) = \tau(q(m)) \mathbin{\dot{-}}b(m, m)$, and $q([m, m']) = [q(m), q(m')] \dotplus b(m, m') \mathbin{\dot{-}}b(m', m)$. Let $(T, \Psi)$ be an isotropic pinning of $G$ and $\alpha \in \Psi$. If $K \subseteq K'$ is an fppf extension with a pinning $(T', \Phi) \supseteq (T_{K'}, \Psi)$, then $\prod_{\substack{ \beta \in \Phi \\ \mathop{\mathrm{Im}}(\beta) \in \{\alpha, 2 \alpha\} } } U_\beta \leq G_{K'}$ descents to a unique group subscheme $U_\alpha \leq G$. We have - If $\alpha \in \Psi \setminus \frac 1 2 \Psi$, then $U_\alpha$ is abelian with the Lie algebra $\mathfrak g_\alpha$ and there is a unique isomorphism $t_\alpha \colon \mathbb G_{\mathrm a}^{m_\alpha} \to U_\alpha$ inducing the coordinatization isomorphism on the Lie algebras, it is $T$-equivariant. - If $\alpha, 2 \alpha \in \Psi$, then $U_\alpha$ is $2$-step nilpotent with the nilpotent filtration $U_{2 \alpha} \leq U_\alpha$ and the Lie algebra $\mathfrak g_\alpha \oplus \mathfrak g_{2 \alpha}$, $U_\alpha / U_{2 \alpha}$ is representable, and there is a unique isomorphism $t_\alpha \colon \mathbb G_{\mathrm a}^{m_\alpha} \to U_\alpha / U_{2 \alpha}$ inducing the coordinatization isomorphism on the Lie algebras, it is $T$-equivariant. - For any $\alpha \in \Psi$ the action of $T$ on $U_\alpha$ factors through $\alpha \colon T \to \mathbb G_{\mathrm m}$ and also through $\mathbb G_{\mathrm m}\to \mathbb A^1$ (in the unique way), where $\mathbb A^1$ is considered as a multiplicative monoid. - If $\alpha \in \Psi \setminus 2 \Psi$, then $U_\alpha = \bigcap_\lambda U_G(\lambda)$, where $\lambda$ runs over all constant coweights of $T$ such that $\langle \alpha, \lambda \rangle > 0$ and $U_G(\lambda)$ are defined in [@conrad thm. 4.1.7]. In particular, $U_\alpha$ is independent on $K'$ and $(T', \Phi)$. - If $\alpha, \frac 1 2 \alpha \in \Psi$, then $U_\alpha \leq U_{\alpha / 2}$ is the locus where the action of $\mathbb A^1$ on $U_{\alpha / 2}$ factors through the squaring map $\mathbb A^1 \to \mathbb A^1$. In particular, $U_\alpha$ is also independent on $K'$ and $(T', \Psi)$. - Consider the fppf sheaf $U_\alpha$ on the category of unital $K$-algebras if $\alpha, 2 \alpha \in \Psi$. By descent, it is a sheaf of $2$-step nilpotent modules. Using [@thesis lemma 13] it is easy to see that the $2$-step nilpotent $K$-module $U_\alpha(K)$ is *universal*, i.e. its scalar extensions are always defined, and such scalar extensions are the values of $U_\alpha$. Since $U_\alpha(K) / U_{2 \alpha}(K)$ is a free $K$-module, $U_\alpha(K)$ splits, i.e. there is an isomorphism $t_\alpha \colon \mathbb G_{\mathrm a}^{m_{2 \alpha}} \mathbin{\dot{\oplus}}\mathbb G_{\mathrm a}^{m_\alpha} \to U_\alpha$ of sheaves of $2$-step nilpotent $K$-modules (such a structure on $\mathbb G_{\mathrm a}^{m_{2 \alpha}} \mathbin{\dot{\oplus}}\mathbb G_{\mathrm a}^{m_\alpha}$ is given by some $K$-bilinear $2$-cocycle $c \colon K^{m_\alpha} \times K^{m_\alpha} \to K^{m_{2 \alpha}}$). Of course, $t_\alpha|_{\mathbb G_{\mathrm a}^{m_{2 \alpha}}} = t_{2 \alpha}$. - The scheme centralizer $L$ of $T$ in $G$ is a reductive closed subgroup of $G$ with the Lie algebra $\mathfrak g_0$. In particular, it is smooth with connected fibers. Moreover, $L$ normalizes all $U_\alpha$. - If $\Pi \subseteq \Psi$ is a subsystem of positive roots, then the product map $\prod_{ \alpha \in \Pi \setminus 2 \Pi } U_{-\alpha} \times L \times \prod_{ \alpha \in \Pi \setminus 2 \Pi } U_\alpha \to G$ is an open embedding. The groups $U^{\pm} = \prod_{ \alpha \in \Pi \setminus 2 \Pi } U_{\pm \alpha}$ are the unipotent radicals of opposite parabolic subgroups $P^{\pm} = L U^{\pm}$ and $L$ is the common Levi subgroup of $P^{\pm}$. We denote the domain of $t_\alpha$ by $P_\alpha$, i.e. $P_\alpha = \mathbb G_{\mathrm a}^{m_\alpha}$ for $\alpha \in \Psi \setminus \frac 1 2 \Psi$ and $P_\alpha = \mathbb G_{\mathrm a}^{m_{2 \alpha}} \mathbin{\dot{\oplus}}\mathbb G_{\mathrm a}^{m_\alpha}$ for $\alpha, 2 \alpha \in \Psi$. The Chevalley commutator formula is $$[t_\alpha(x), t_\beta(y)] = \prod_{\substack{ i \alpha + j \beta \in \Psi \\ i, j > 0 } } t_{i \alpha + j \beta}( f_{\alpha \beta i j}(x, y) )$$ for some uniquely determined maps $f_{\alpha \beta i j}$ if $\alpha$ and $\beta$ are linearly independent (we assume that $f_{\alpha, \beta, 2i, 2j} = 0$ if both $i \alpha + j \beta$ and $2i \alpha + 2j \beta$ lie in $\Psi$). The maps $f_{\alpha \beta i j}$ are $L$-equivariant. Moreover, - If all $\alpha$, $\beta$, $i \alpha + j \beta$ are not ultrashort (i.e. roots of a component of type $\mathsf{BC}_\ell$ with the smallest length), then $f_{\alpha \beta i j}(x, y)$ is a polynomial, homogeneous on $x$ of degree $i$ and on $y$ of degree $j$. - If $\alpha$ and $\beta$ are ultrashort in a common irreducible component of $\Psi$, then $f_{\alpha \beta 1 1}$ factors through a bilinear map $P_\alpha / P_{2 \alpha} \times P_\beta / P_{2 \beta} \to P_{\alpha + \beta}$. - Suppose that $\alpha$ is ultrashort, $\beta$ is short, and $(\alpha, \beta)$ is a base of a root subsystem of type $\mathsf{BC}_2$. Then $f_{\alpha \beta 2 1}(x, y)$ is $K$-quadratic on $x$ (with respect to the maximal nilpotent filtration on $P_{2 \alpha + \beta}$) and $K$-linear on $y$, $f_{\alpha \beta 1 1}(x, y)$ is a homomorphism on $x$ and $K$-quadratic on $y$ (with respect to the minimal nilpotent filtration on $P_\beta$). The canonical morphism $P_\alpha \times \mathbb G_{\mathrm a}\to P_\alpha$ is denoted by $({-}) \cdot ({=})$ (it is the usual $\mathbb G_{\mathrm a}$-module structure in the case $2 \alpha \notin \Psi$), so always $f_{\alpha \beta i j}(x \cdot k, y \cdot k') = f_{\alpha \beta i j}(x, y) \cdot k^i {k'}^j$. **Lemma 2**. *If $K$ is semilocal with connected spectrum, then every parabolic subgroup is contained in a maximal one and every isotropic pinning is contained in a maximal one. If $(T, \Psi)$ and $(T', \Psi')$ are two maximal isotropic pinnings, then there is $g \in U_{T, \Psi}^+(K)\, U_{T, \Psi}^-(K)\, U_{T, \Psi}^+(K)$ such that $T' = {^{g}\!{T}}$ and $\Psi' = {^{g}\!{\Psi}}$. Here $U^{\pm}_{T, \Psi}$ are the unipotent radicals of the canonical parabolic subgroups associated with $(T, \Psi)$ and some subsystem of positive roots. Every maximal parabolic subgroup may be constructed by a maximal isotropic pinning and a family of positive roots.* *Proof.* See [@sga3 Exp. XXVI, cor. 5.2, cor. 5.7, prop. 6.16, thm. 7.13]. ◻ If $K$ is semilocal with connected spectrum, then the *isotropic rank* of $G$ is the common rank of its maximal isotropic pinnings. The *local isotropic rank* of $G$ over any unital ring $K$ is the minimum of the isotropic ranks of $G_{\mathfrak m}$ for all maximal ideals $\mathfrak m \trianglelefteq K$. # Ind-pro-completion Let $\mathbf C$ be a category. Its *ind-completion* $\mathop{\mathrm{Ind}}(\mathbf C)$ is the universal category with a functor from $\mathbf C$ containing all direct limits indexed by small filtered categories (e.g. by directed sets, especially $\mathbb N$). We use the following standard construction of $\mathop{\mathrm{Ind}}(\mathbf C)$: its objects are *direct systems* in $\mathbf C$, i.e. covariant functors $X \colon \mathbf I_X \to \mathbf C$ for small filtered *index categories* $\mathbf I_X$. Morphisms are given by the formula $$\mathop{\mathrm{Ind}}(\mathbf C)(X, Y) = \varprojlim\nolimits_{i \in \mathbf I_X}^\mathbf{Set} \varinjlim\nolimits_{j \in \mathbf I_Y}^\mathbf{Set} \mathbf C(X(i), Y(j)).$$ For the definition of composition see [@kashiwara-schapira def. 6.1.1]. Dually, the *pro-completion* of $\mathbf C$ is $\mathop{\mathrm{Pro}}(\mathbf C) = \mathop{\mathrm{Ind}}(\mathbf C^{\mathrm{op}})^{\mathrm{op}}$, it is the universal category with a functor from $\mathbf C$ containing all projective limits indexed by small filtered categories. Objects of $\mathop{\mathrm{Pro}}(\mathbf C)$ are *inverse systems* in $\mathbf C$, i.e. contravatiant functors $X \colon \mathbf I_X \to \mathbf C$ for small filtered index categories $\mathbf I_X$. Morphisms are given by $$\mathop{\mathrm{Pro}}(\mathbf C)(X, Y) = \varprojlim\nolimits_{j \in \mathbf I_Y}^\mathbf{Set} \varinjlim\nolimits_{i \in \mathbf I_X}^\mathbf{Set} \mathbf C(X(i), Y(j)).$$ If $X \colon \mathbf I_X \to \mathbf C$ is a direct system and $u \colon \mathbf I'_X \to \mathbf I_X$ is a cofinal, then the induced morphism $X \circ u \to X$ is an isomorphism in $\mathop{\mathrm{Ind}}(\mathbf C)$. Dually, if $X$ is an inverse system, then the induced morphism $X \to X \circ u$ is an isomorphism in $\mathop{\mathrm{Pro}}(\mathbf C)$. A *level morphism* in $\mathop{\mathrm{Ind}}(\mathbf C)$ or $\mathop{\mathrm{Pro}}(\mathbf C)$ is an object of the ind-completion or the pro-completion of the category of arrows in $\mathbf C$. Every morphism in $\mathop{\mathrm{Ind}}(\mathbf C)$ or $\mathop{\mathrm{Pro}}(\mathbf C)$ is isomorphic to a level one in the category of arrows [@kashiwara-schapira prop. 6.1.14]. There are canonical fully faithful functors $\mathbf C \to \mathop{\mathrm{Ind}}(\mathbf C)$ and $\mathbf C \to \mathop{\mathrm{Pro}}(\mathbf C)$, they map $X \in \mathbf C$ to the corresponding functor indexed by the terminal category with unique morphism. If $\mathbf C$ is finitely complete or cocomplete, then $\mathop{\mathrm{Ind}}(\mathbf C)$ and $\mathop{\mathrm{Pro}}(\mathbf C)$ are also finitely complete or cocomplete respectively by [@kashiwara-schapira cor. 6.1.17(i) and prop. 6.1.18(iii)]. If $\mathbf C$ is finitely complete, then in $\mathop{\mathrm{Ind}}(\mathbf C)$ direct limits commute with finite limits. A limit and a colimit of a small level diagram may be calculated levelwise. If $F \colon \mathbf C \to \mathbf D$ is a functor, then it continues to unique (up to unique natural isomorphism) functors $\mathop{\mathrm{Ind}}(F) \colon \mathop{\mathrm{Ind}}(\mathbf C) \to \mathop{\mathrm{Ind}}(\mathbf D)$ and $\mathop{\mathrm{Pro}}(F) \colon \mathop{\mathrm{Pro}}(\mathbf C) \to \mathop{\mathrm{Pro}}(\mathbf D)$ commuting with direct limits and projective limits respectively [@kashiwara-schapira prop. 6.1.9, 6.1.10]. If $F$ is fully faithful, then $\mathop{\mathrm{Ind}}(F)$ and $\mathop{\mathrm{Pro}}(F)$ are also fully faithful. Recall that a *regular epimorphism* in a category $\mathbf C$ is a coequalizer of a couple of morphisms. A *kernel pair* of a morphism $f \colon X \to Y$ is the limit $\lim(X \xrightarrow f Y \xleftarrow f X) \rightrightarrows X$. A category $\mathbf C$ is called *regular* if it has all finite limits, all kernel pairs have coequalizers, and the class of regular epimorphisms is closed under base change. If $\mathbf C$ is regular, then any morphism $f \colon X \to Y$ has unique (up to unique isomorphism) *image decomposition* $X \xrightarrow u \mathop{\mathrm{Im}}(f) \xrightarrow v Y$, where $u$ is a regular epimorphism and $v$ is a monomorphism. We usually denote the image $\mathop{\mathrm{Im}}(f)$ just by $f(X)$. The image decomposition is functorial on $f$. The pro-completion and the ind-completion of a regular category are regular by [@barr; @day-street] (see also [@jacqmin-janelidze example 1.11 and §1.8]). Moreover, image decompositions of level morphisms may be computed levelwise. In particular, if the components of a level morphism $f$ are monomorphisms or regular epimorphisms, then $f$ is itself a monomorphism or a regular epimorphism respectively. A regular category $\mathbf C$ is called *coherent* if the families of subobjects $\mathop{\mathrm{Sub}}(X)$ are $\vee$-semilattices for all $X \in \mathbf C$ and for all $f \in \mathbf C(X, Y)$ the base change maps $f^* \colon \mathop{\mathrm{Sub}}(Y) \to \mathop{\mathrm{Sub}}(X)$ are homomorphisms of $\vee$-semilattices. The category $\mathbf{Set}$ is coherent. We denote the upper bound of subobjects $A, B \subseteq X$ by $A \cup B$ and the smallest subobject by $\varnothing \subseteq X$. By [@jacqmin-janelidze the list after example 1.12 and §1.8] if $\mathbf C$ is coherent with finite coproducts, then $\mathop{\mathrm{Ind}}(\mathbf C)$ and $\mathop{\mathrm{Pro}}(\mathbf C)$ also have this property. Alternatively, it is easy to check that ind- and pro-completion preserve the distributivity of binary products over finite coproducts and regular categories with finite coproducts and this property are coherent. It follows that $\mathop{\mathrm{Ind}}(\mathop{\mathrm{Pro}}(\mathbf{Set}))$ is a coherent category with finite colimits (moreover, with binary products distributive over finite coproducts). The categories $\mathbf{Set}$, $\mathop{\mathrm{Ind}}(\mathbf{Set})$, $\mathop{\mathrm{Pro}}(\mathbf{Set})$ are its full subcategories and the embedding functors preserve finite limits, finite colimits, monomorphisms, regular epimorphisms, and the image decomposition. In any category $\mathbf C$ with finite products we often write compositions of morphisms by first-order terms, e.g. the identities $(xy)z = x(yz)$, $x 1 = x = 1 x$, $x x^{-1} = 1 = x^{-1} x$ for $({-}) ({=}) \colon X \times X \to X$, $({-})^{-1} \colon X \to X$, $1 \colon \{*\} \to X$ mean that $X$ is a group object (here $\{*\}$ denotes the terminal object in $\mathbf C$). We write $x \in X$ if a formal variable $x$ has the domain $X \in \mathbf C$. An *action* of a group object $G$ on a group object $H$ is a morphism ${^{({-})}\!{({=})}} \colon G \times H \to H$ such that ${^{gg'}\!{h}} = {^{g}\!{({^{g'}\!{h}})}}$, ${^{1}\!{h}} = h$, and ${^{g}\!{hh'}} = {^{g}\!{h}}\, {^{g}\!{h'}}$. In this case the *semidirect product* $H \rtimes G$ is their usual product $H \times G$ in $\mathbf C$ with the group object structure given by $(h \rtimes g) (h' \rtimes g') = (h\, {^{g}\!{h}}) \rtimes gg'$. A *crossed module* is a morphism $\delta \colon X \to G$ of group objects together with an action of $G$ on $X$ such that $\delta$ is $G$-equivariant and ${^{x}\!{y}} = {^{\delta(x)}\!{y}}$ for $x, y \in X$. By the Yoneda lemma elementary properties of such algebraic objects follow from the usual group theory. **Lemma 3**. *Let $\mathbf C$ be a coherent category with finite colimits (so $\mathop{\mathrm{Ind}}(\mathbf C)$ is also coherent), $G$ be a group object in $\mathop{\mathrm{Ind}}(\mathbf C)$, and $X \subseteq G$ be a subobject of $G$. Then $\langle X \rangle = \varinjlim_{n \in \mathbb N} (X \cup \{1\} \cup X^{-1})^n$ is the smallest group subobject of $G$ containing $X$, where $X^n$ is the image of the multiplication morphism $X^{\times n} \to G$ and $X^{-1}$ is the image of $X$ under the inversion.* *Proof.* Let $H \leq G$ be a group subobject containing $X$. Clearly, $(X \cup e \cup X^{-1})^n \subseteq H$ for any $n \in \mathbb N$, so $\langle X \rangle \leq H$. On the other hand, the group operations of $G$ induce group operations on $\langle X \rangle$ since direct limits commute with finite products in $\mathop{\mathrm{Ind}}(\mathbf C)$. ◻ Non-unital ring objects in a category with finite products are defined in the same way. If $A$ and $B$ are ring objects in $\mathbf C$, then an *action* of $A$ on $B$ is given by a biadditive morphism $({-}) ({=}) \colon A \times B \to B$ such that $a(bb') = (ab)b'$ and $(aa')b = a(a'b)$. A *unital action* of a unital $A$ on $B$ satisfies also the additional axiom $1 b = b$. The term *$A$-algebra* always means a ring object with a unital action of $A$. If $A$ and $B$ are $R$-algebras for some unital ring object $R$, then we also require that $(ra) b = r(ab) = a(rb)$ for $r \in R$, $a \in A$, $b \in B$. If $A$ acts on $B$, then $A \rtimes B$ is their *semi-direct product*, it is their product as abelian group objects together with the multiplication $(a \rtimes b) (a' \rtimes b') = (aa' + ab' + a'b) \rtimes bb'$. A *crossed module* of ring objects is a morphism $\delta \colon X \to A$ together with an action of $A$ on $X$ such that $\delta(ax) = a \delta(x)$ and $xy = \delta(x) y$ for $x, y \in X$ and $a \in A$. # Ind-pro-algebras Recall that $K$ denotes the base unital ring. For any $s, t \in K$ and $A \in \mathbf{CRng}_K$ consider the $(A \rtimes K)$-module $$\tfrac 1 s A^{(t)} = \{\tfrac{a^{(t)}}s \mid a \in A\}$$ with the operations $\frac{a^{(t)}}s + \frac{b^{(t)}}s = \frac{(a + b)^{(t)}}s$ and $\frac{a^{(t)}}s (b \rtimes k) = \frac{(ab + ak)^{(t)}}s$. There are natural homomorphisms $$\tfrac 1 s A^{(tt')} \to \tfrac 1{ss'} A^{(t)},\, \tfrac{a^{(tt')}}s \mapsto \tfrac{(at's')^{(t)}}{ss'}$$ and bilinear multiplication maps $$({-}) ({=}) \colon \tfrac 1s A^{(t)} \times \tfrac 1{s'} A^{(t')} \to \tfrac 1{ss'} A^{(tt')},\, (\tfrac{a^{(t)}}s, \tfrac{b^{(t')}}{s'}) \mapsto \tfrac{(ab)^{(tt')}}{ss'}.$$ We omit the upper index $(1)$ and the denominator $1$ since $\frac 1 1 A^{(1)} \cong a,\, \frac{a^{(1)}}1 \mapsto a$ is an algebra isomorphism. Let the *formal localization* of $A$ at $s$ be $$A_s^{\mathop{\mathrm{Ind}}} = \varinjlim\nolimits^{\mathop{\mathrm{Ind}}(\mathbf{Set})} ( A \to \tfrac 1s A \to \tfrac 1{s^2} A \to \ldots ),$$ so $A_s$ is the ordinary direct limit of the direct system $A_s^{\mathop{\mathrm{Ind}}}$. The object $A_s^{\mathop{\mathrm{Ind}}}$ is a non-unital algebra object over $A \rtimes K$ in $\mathop{\mathrm{Ind}}(\mathbf{Set})$ (it is unital if $A$ is unital), $A_s^{\mathop{\mathrm{Ind}}} \to A_{ss'}^{\mathop{\mathrm{Ind}}}$ are morphisms of algebras. Similarly, the *colocalization* of $A$ at $s$ is $$A^{(s^\infty)} = \varprojlim\nolimits^{\mathop{\mathrm{Pro}}(\mathbf{Set})} ( \ldots \to A^{(s^2)} \to A^{(s)} \to A ),$$ it is a non-unital algebra object over $A \rtimes K$ in $\mathop{\mathrm{Pro}}(\mathbf{Set})$, $A^{(ss')} \to A^{(s)}$ are morphisms of algebras. Actually, $A^{(s^\infty)}$ is a pro-$(A \rtimes K)$-algebra since $A^{(s)}$ are algebras with respect to $a^{(s)} b^{(s)} = (abs)^{(s)}$ called *homotopes* of $A$. Now we construct the categories suitable to our construction of elementary groups. Let $\mathbf D$ be a small category and $\mathbf{Cat}(\mathbf D, \mathbf{Set})$ be the category of functors $\mathbf D \to \mathbf{Set}$ and natural transformations between them, it is also a coherent category with finite limits (limits, colimits, and the image decomposition are calculated componentwise). We usually take $\mathbf D = \{*\}$ in order to study an individual elementary group or $\mathbf D = \mathbf{CRing}^{\mathrm{fp}}_K$ (the category of finitely presented unital $K$-algebras) to study the whole elementary group functor. Let $$\mathbf{IP}_{\mathbf D} = \mathop{\mathrm{Ind}}(\mathop{\mathrm{Pro}}(\mathbf{Cat}(\mathbf D, \mathbf{Set}))).$$ This construction is a contravariant functor on $\mathbf D$, so for any object $D \in \mathbf D$ there is a functor $\mathbf{IP}_{\mathbf D} \to \mathbf{IP}_{\{*\}}$ of *restriction to D*. On the other side, if $\mathbf D$ is non-emtpy then there is a full subcategory of *$\mathbf D$-constant objects* $\mathbf{IP}_{\{*\}} \subseteq \mathbf{IP}_{\mathbf D}$ induced by $\mathbf D \to \{*\}$. Let $\mathbf{IPRng}_{\mathbf D, s}$ be the category of non-unital algebras in $\mathbf{IP}_{\mathbf D}$ over the $\mathbf D$-constant unital ring object $K_s^{\mathop{\mathrm{Ind}}}$, so $\mathbf{IPRng}_{\{*\}, s}$ is the usual category of non-unital algebras over $K_s^{\mathop{\mathrm{Ind}}}$ in $\mathop{\mathrm{Ind}}(\mathop{\mathrm{Pro}}(\mathbf{Set}))$. Note that the set of global elements of $K_s^{\mathop{\mathrm{Ind}}}$ (i.e. $\mathbf{IP}_{\mathbf D}(\{*\}, K_s^{\mathop{\mathrm{Ind}}})$) is canonically isomorphic to $K_s$ if $\mathbf D$ is connected. Actually, we would like to take $\mathbf D = \mathbf{CRing}_K$ to consider elementary groups for all unital $K$-algebras simultaneously. Unfortunately, the "category" $\mathbf{Cat}(\mathbf{CRing}_K, \mathbf{Set})$ is too large (it does not exist in ZFC even as a class). Instead we may take the category of functors $\mathbf{CRing}_K \to \mathbf{Set}$ preserving direct limits and all natural transformations, it is equivalent to $\mathbf{Cat}(\mathbf{CRing}_K^{\mathrm{fp}}, \mathbf{Set})$ by [@kashiwara-schapira cor. 6.3.2] since $\mathbf{CRing}_K$ is an ind-completion of $\mathbf{CRing}_K^{\mathrm{fp}}$ [@kashiwara-schapira cor. 6.3.5]. Hence for any $R \in \mathbf{CRing}_K$ there is a functor $\mathbf{IP}_{\mathbf{CRing}^{\mathrm{fp}}_K} \to \mathbf{IP}_{\{*\}}$ of *restriction to $R$*. We may also work only with small categories by considering sets with bounded rank (sufficiently large for our further constructions) instead of the whole $\mathbf{Set}$ and by taking only countable ind- and pro-completions. For example, let $\mathcal R$ be the inclusion functor $\mathbf{CRing}_K \to \mathbf{Set}$ and $s \in K$. Applying the colocalization construction to every component of $\mathcal R$ (i.e. for every unital $K$-algebra) we obtain an object $\mathcal R^{(s^\infty)} \in \mathbf{IPRng}_{\mathbf{CRing}^{\mathrm{fp}}_K, s}$. The action of $K^{\mathop{\mathrm{Ind}}}_s$ on $\mathcal R^{(s^\infty)}$ is given by the maps $({-}) ({=}) \colon \mathcal R^{(s^{n + m})} \times \frac 1{s^n} K \to \mathcal R^{(s^m)},\, (r^{(s^{n + m})}, \frac 1{s^n} k) \mapsto (rk)^{(s^m)}$. We may consider this object as a functor $\mathbf{CRing}_K \to \mathbf{IPRng}_{\{*\}, s}$, where the pro-structure (and the action of $K^{\mathop{\mathrm{Ind}}}_s$) are "uniform" on the algebra $R$. The value of this functor on $R$ (i.e. the restriction of $\mathcal R^{(s^\infty)}$ to $R$) is precisely $R^{(s^\infty)}$. It is easy to see that if $\mathcal D(s) = \mathcal D(s')$ (i.e. $s$ and $s'$ divide some powers of each other), then $\mathcal R^{(s^\infty)} \cong \mathcal R^{({s'}^\infty)}$ as ring objects in $\mathbf{IP}_{\mathbf{CRing}^{\mathrm{fp}}_K}$. Instead of $\mathcal R$ we may take any functor $\mathbf{CRing}_K \to \mathbf{CRng}_K$ commuting with direct limits such as $\mathcal R^{(t)}$ or $t\mathcal R$. Note that the composition $\mathcal R^{(s^\infty)} \to \mathcal R \to \mathcal R_s$ is $K_s^{\mathop{\mathrm{Ind}}}$-equivariant. The following lemma shows that in the Noetherian case we may consider ideals instead of homotopes and such composition is a monomorphism for every individual algebra $R$. But these properties are not "uniform" on the algebra, i.e. do not hold for the whole $\mathcal R$. **Lemma 4**. *Let $K$ be a Noetherian unital ring, $A \in \mathbf{CRng}_K^{\mathrm{fp}}$ (i.e. $A \rtimes K$ be a finitely generated unital $K$-algebra), and $s \in K$. Then the natural level morphism $$A^{(s^\infty)} \to \varprojlim\nolimits^{\mathop{\mathrm{Pro}}(\mathbf{Set})}_n s^n A,\, a^{(s^n)} \mapsto s^n a$$ is an isomorphism of pro-sets. Moreover, the morphism $$\varprojlim\nolimits^{\mathop{\mathrm{Pro}}(\mathbf{Set})}_n s^n A \to A_s,\, s^n a \mapsto \textstyle \frac{s^n a}1$$ is a monomorphism of pro-sets.* *Proof.* Replacing $A$ by $A \rtimes K$ we may assume that $A = K$. Since the maps $K^{(s^n)} \to s^n K$ are surjective, it suffices to check that $$K^{(s^\infty)} \to K_s,\, x^{(s^n)} \mapsto \textstyle \frac{s^n x}1$$ is a monomorphism of pro-sets. Let $I_n = \{x \in K \mid s^n k = 0\}$, so $0 = I_0 \trianglelefteq I_1 \trianglelefteq\ldots \trianglelefteq K$ is an ascending chain of ideals. Since $K$ is Noetherian it stabilizes at some index $n_*$. It follows that $\mathop{\mathrm{Ker}}(K^{(s^{n + n_*})} \to K_s) = I_{n_*}^{(s^n)}$ maps to zero under $K^{(s^{n + n_*})} \to K^{(s^n)}$ for all $n$, so we may apply [@dydak-portal prop. 2.3]. ◻ Pro-rings $R^{(s^\infty)}$ and the ring object $\mathcal R^{(s^\infty)}$ are not unital. Fortunately, they satisfy a weaker condition sufficient for our purposes. A ring object $R$ in an ind-completion of a coherent category (e.g. in $\mathbf{IP}_{\mathbf D}$) is called *idempotent* if $R = \langle xy \mid x, y \in R \rangle$, the right hand sides denotes the additive subgroup generated by a subobject. An action of such $R$ on a ring object $A$ is *unital* if $A = \langle xa \mid x \in R, a \in A \rangle$. Finally, we say that $R$ is *power idempotent* if $R = \langle xy^k \mid x, y \in R \rangle$ for every $k > 0$. **Lemma 5**. *For any $s \in K$ the ring object $\mathcal R^{(s^\infty)}$ is power idempotent. Moreover, for any functor $\mathcal A \colon \mathbf{CRing}_K \to \mathbf{CRng}_K$ commuting with direct limits and with a structure of an $\mathcal R$-algebra (e.g. $\mathcal A = t \mathcal R$ or $\mathcal A = \mathcal R^{(t)}$ for $t \in K$) the action of $\mathcal R^{(s^\infty)}$ on $\mathcal A^{(s^\infty)}$ is unital.* *Proof.* Let us show that $\mathcal A = \langle x^k a \mid x \in \mathcal R, a \in \mathcal A \rangle$ for every $k \geq 0$. In other words, for all $k \geq 0$ there is $T \geq 0$ such that for all $n \geq 0$ there is $m \geq 0$ with the following property: for every $R \in \mathbf{CRing}_K$ the image $s^m \mathcal A(R)^{(s^n)}$ of $\mathcal A(R)^{(s^{n + m})} \to \mathcal A(R)^{(s^n)}$ is contained in $$\sum_{t = 1}^T (R^{(s^n)})^{[k]}\, \mathcal A(R)^{(s^n)} = s^{kn} \sum_{t = 1}^T (R^{[k]} \mathcal A(R))^{(s^n)} \leq \mathcal A(R)^{(s^n)}$$ where $B^{[k]} = \{b^k \mid b \in B\}$ for any $B \in \mathbf{CRng}_K$. We may take $T = 1$ and $m = kn$. ◻ **Lemma 6**. *Let $\mathcal A \colon \mathbf{CRing}_K \to \mathbf{CRng}_K$ be a functor commuting with direct limits and with a structure of an $\mathcal R$-algebra. Suppose that $\mathcal D(s) = \bigcup_{i = 1}^N \mathcal D(s_i)$. Then $\mathcal A^{(s^\infty)}$ is the sum of the images of $\mathcal A^{(s_i^\infty)} \to \mathcal A^{(s^\infty)}$ for $1 \leq i \leq N$.* *Proof.* First of all, $s$ invertible in all $K_{s_i}$, so without loss of generality $s_i = s a_i$ for some $a_i \in K$. On the other hand, $\sum_i K_s s_i = K_s$, i.e. $s^n \in \sum_i Ks_i$ for sufficiently large $n$. We have $$\sum_i \mathop{\mathrm{Im}}( \mathcal A(R)^{(s_i^m)} \to \mathcal A(R)^{(s^m)} ) = (\sum_i a_i^m \mathcal A(R))^{(s^m)} \supseteq \mathcal A(R)^{ (s^{m + n (Nm - N + 1)}) }. \qedhere$$ ◻ Take $s \in K$. One can define $A$-points of pointed affine $K_s$-schemes of finite presentation for every $A \in \mathbf{IPRng}_{\mathbf D, s}$. Every pointed affine $K_s$-scheme is of the type $X = \mathop{\mathrm{Spec}}(R)$ for some $R \in \mathbf{CRing}_{K_s}$ together with a homomorphism $\varepsilon \colon R \to K_s$ of $K_s$-algebras. If $X$ is of finite presentation, then up to isomorphism $R = (x_1, \ldots, x_n) / (f_1, \ldots, f_m)$ and $\varepsilon(x_i) = 0$ for some $f_i \in (x_1, \ldots, x_n) \trianglelefteq K_s[x_1, \ldots, x_n]$. If $X' = \mathop{\mathrm{Spec}}(R')$ is another such scheme with $R' = (x'_1, \ldots, x'_{n'}) / (f'_1, \ldots, f'_{m'})$ and $h \colon X \to X'$ is a morphism of pointed schemes, then $h$ is given by polynomials $h_1, \ldots, h_{n'} \in (x_1, \ldots, x_n)$ (the images of $x'_i$, so $f'_i(h_1, \ldots, h_{n'}) \in (f_1, \ldots, f_m)$) uniquely determined modulo $(f_1, \ldots, f_m)$. For any $A \in \mathbf{IPRng}_{\mathbf D, s}$ and finitely presented pointed affine $K_s$-scheme $X = \mathop{\mathrm{Spec}}(R)$ with the presentation as above we construct the object $$X(A) = \{ \vec x \in A^n \mid f_1(\vec x) = \ldots = f_m(\vec x) = 0 \} \in \mathbf{IP}_{\mathbf D}$$ of $A$-points of $X$ (the right hand side is the limit of $A^n \xrightarrow f A^m \xleftarrow 0 \{*\}$). Clearly, $X(A)$ is functorial on both $X$ and $A$. Functors $X({-})$ preserve all finite limits and monomorphisms. For example, if $A \in \mathbf{CRng}_K$, then $X(A)$ is just the kernel of $X(A \rtimes K) \to X(K)$, i.e. the limit of $X(A \rtimes K) \to X(K) \leftarrow \{*\}$. # Elementary groups over ind-pro-algebras In this section $G$ be a reductive group scheme over $K$ with an isotropic pinning $(T, \Psi)$ of rank $\geq 2$ over $K_s$ for some $s \in K$. We also fix a category $\mathbf D$ from the definition of $\mathbf{IP}_{\mathbf D}$, e.g. $\mathbf D = \{*\}$ or $\mathbf D = \mathbf{CRing}_K^{\mathrm{fp}}$. For any $A \in \mathbf{IPRng}_{\mathbf D, s}$ we have the group objects $G(A)$ and $P_\alpha(A)$ in $\mathbf{IP}_{\mathbf D}$ for all $\alpha \in \Psi$. The *unrelativized elementary group* is the group subobject $$\mathop{\mathrm{E}}_{G, T, \Psi}(A) = \langle t_\alpha(A) \mid \alpha \in \Psi \rangle \leq G(A)$$ in $\mathbf{IP}_{\mathbf D}$ constructed in lemma [Lemma 3](#subgr-gen){reference-type="ref" reference="subgr-gen"} (in this definition we do not use the rank condition on $(\Psi, T)$). Fix a power idempotent object $R \in \mathbf{IPRng}_{\mathbf D, s}$ and $A \in \mathbf{IPRng}_{\mathbf D, s}$ with a unital action of $R$ (so $A$ is an $R \rtimes K_s^{\mathop{\mathrm{Ind}}}$-algebra with the unitality condition). The *relative elementary group* is $$\mathop{\mathrm{E}}_{G, T, \Psi}(R, A) = \mathop{\mathrm{Ker}}( \mathop{\mathrm{E}}_{G, T, \Psi}(A \rtimes R) \to \mathop{\mathrm{E}}_{G, T, \Psi}(R) ) \leq G(A).$$ We say that a subset $\Sigma \subseteq \Psi$ is *saturated special closed* if it is an intersection of $\Psi$ with a convex cone contained in an open half-space. Let $$\mathop{\mathrm{E}}_{G, T, \Sigma}(A) = \langle \mathop{\mathrm{Im}}(t_\alpha) \mid \alpha \in \Sigma \rangle \leq \mathop{\mathrm{E}}_{G, T, \Psi}(A)$$ and $$z_\Sigma \colon \mathop{\mathrm{E}}_{G, T, -\Sigma}(R) \times \mathop{\mathrm{E}}_{G, T, \Sigma}(A) \to \mathop{\mathrm{E}}_{G, T, \Psi}(R, A),\, (g, h) \mapsto {^{g}\!{h}}$$ for such $\Sigma$. Let also $\mathop{\mathrm{E}}'_{G, T, \Psi}(R, A)$ be the subgroup of $G(A)$ generated by the morphisms $$z_\alpha \colon P_{-\alpha}(R) \times P_\alpha(A) \to G(A),\, (r, a) \mapsto {^{t_{-\alpha}(r)}\!{t_\alpha(a)}}.$$ **Lemma 7**. *$\mathop{\mathrm{E}}_{G, T, \Psi}(R, A) = \langle {^{g}\!{t_\alpha(a)}} \mid g \in \mathop{\mathrm{E}}_{G, T, \Psi}(R), a \in P_\alpha(A), \alpha \in \Psi \rangle$.* *Proof.* Let $H$ be the right hand side of the asserted equality. Clearly, $$\mathop{\mathrm{E}}_{G, T, \Psi}(A) \leq H \leq \mathop{\mathrm{E}}_{G, T, \Psi}(R, A).$$ Also, $H \rtimes \mathop{\mathrm{E}}_{G, T, \Psi}(R) \subseteq G(A \rtimes R)$ is a subgroup (instead of just a subobject) since $H$ is $\mathop{\mathrm{E}}_{G, T, \Psi}(R)$-invariant. It follows that $H \rtimes \mathop{\mathrm{E}}_{G, T, \Psi}(R) = \mathop{\mathrm{E}}_{G, T, \Psi}(A \rtimes R)$, so $H = \mathop{\mathrm{E}}_{G, T, \Psi}(R, A)$. ◻ **Lemma 8**. *Let $(T, \Psi)$ be an isotropic pinning of $G$ of rank $\geq 2$, $\alpha \in \Psi$, $e_k$ be a basis element of $P_\alpha$, and $\alpha \in H \leq \mathbb R \Psi$ be a hyperplane. Then there are $V \geq 0$ and elements $\beta_v, \gamma_v \in \Psi \setminus H$, $p_v \in P_{\beta_v}(K)$, $q_v \in P_{\gamma_v}(x K[x])$ for $1 \leq v \leq V$ such that $\beta_v$ is a neighbor of $\alpha$, $\alpha = i_v \beta_v + \gamma_v$ for some $i_v \geq 1$, and $$e_k \cdot x = \sum^\cdot_{1 \leq v \leq V} f_{\beta_v \gamma_v i_v 1}(p_v, q_v) \in P_\alpha(x K[x]).$$* *Proof.* Consider the $K$-submodule (or $2$-step nilpotent $K$-submodule) $N \leq P_\alpha(x K[x])$ generated by $f_{\beta \gamma i 1}( P_\beta(K), P_\gamma(x K[x]) )$, where $i \beta + \gamma = \alpha$, $i \in \{1, 2\}$, and $\beta \notin H$ is a neighbor to $\alpha$. It it easy to see that $N$ is preserved under scalar extensions. In order to study $N$ we may assume that $(T, \Psi)$ is contained in a pinning $(T', \Phi)$ of $G$. By lemma [Lemma 1](#root-sys-dec){reference-type="ref" reference="root-sys-dec"} any preimage $\alpha' \in \Phi$ of $\alpha$ has a neighbor $\beta'$ with the image in $\Psi \setminus H$, i.e. the subgroup of $P_\alpha(x K[x])$ corresponding to $\alpha'$ lies in $N$, so $N = P_\alpha(x K[x])$ unless $\alpha$ is long in a component of type $\mathsf G_2$. In the exceptional case we have only $3 P_\alpha(x K[x]) \leq N$. Consider the $K$-submodule $N' \leq P_\alpha(x K[x])$ generated by $f_{\beta \gamma i 1}( P_\beta(K), P_\gamma(x K[x]) )$, where $i \beta + \gamma = \alpha$, $i \geq 1$ is arbitrary, and $\beta \notin H$ is a neighbor to $\alpha$. The submodule $N'$ is also preserved under scalar extensions. The same argument as above implies that $N' = P_\alpha(x K[x])$. ◻ **Lemma 9**. *For any $\alpha \in \Psi$ and a hyperplane $\alpha \in H \leq \mathbb R \Psi$ the group object $P_\alpha(A)$ is generated by $f_{\beta \gamma i 1}(P_\beta(R), P_\gamma(A))$ for all $\beta, \gamma \in \Psi$ and $i \geq 1$ such that $i \beta + \gamma = \alpha$ and $\beta \notin H$ is a neighbor of $\alpha$.* *Proof.* By lemma [Lemma 8](#loc-root-gen){reference-type="ref" reference="loc-root-gen"} there are identities $$e_k \cdot ar^3 = \sum^\cdot_{1 \leq v \leq V_k} f_{\beta_{kv}, \gamma_{kv}, i_{kv}, 1}( p_{kv} \cdot r, q_{kv}(a) \cdot r^{3 - i_{kv}} )$$ of morphisms $A \times R \to P_\alpha(A)$, where $\beta_{kv} \notin H$ are neighbors of $\alpha$, $i_v \geq 1$, $\alpha = i_{kv} \beta_{kv} + \gamma_{kv}$, $p_{kv} \in P_{\beta_{kv}}(K_s)$, and $q_{kv} \in P_{\gamma_{kv}}(x K_s[x])$. ◻ **Lemma 10**. *For any $\alpha \in \Psi$ the elementary subgroup $\mathop{\mathrm{E}}_{G, T, \Psi}(R)$ is generated by $t_\beta(R)$ for $\beta \in \Phi \setminus \mathbb R \alpha$.* *Proof.* This easily follows from lemma [Lemma 9](#root-gen){reference-type="ref" reference="root-gen"} applied to $A = R$. ◻ **Lemma 11**. *For any saturated special closed subset $\Sigma \subseteq \Psi$ the morphism $z_\Sigma$ takes values in $\mathop{\mathrm{E}}'_{G, T, \Psi}(R, A)$.* *Proof.* We use induction on $|\Sigma|$, the case $\dim(\mathbb R \Sigma) \leq 1$ is clear. If $\Sigma \neq \varnothing$, then there exists an *extreme root* $\alpha \in \Sigma$, i.e. a root such that $\frac 1 2 \alpha \notin \Sigma$ and $\mathbb R_+ \alpha$ is an extreme ray of the convex cone spanned by $\Sigma$ (then $\Sigma \setminus \mathbb R \alpha$ is also a saturated special closed set). Moreover, if $\dim(\mathbb R \Sigma) \geq 2$, then there is an extreme root $\alpha \in \Sigma$ linearly independent with any given root of $\Sigma$. Since $z_\Sigma$ is a homomorphism on the second variable, it suffice to check that any $z_\Sigma(g, t_\alpha(a))$ lie in $\mathop{\mathrm{E}}'_{G, T, \Psi}(R, A)$. Take an extreme $\beta \in \Sigma$ linearly independent with $\alpha$, so the product map $$\mathop{\mathrm{E}}_{ G, T, \Sigma \setminus \mathbb R \beta }(R) \times P_\beta(R) \to \mathop{\mathrm{E}}_{G, T, \Sigma}(R)$$ is an isomorphism. Now $$z_\Sigma(g t_\beta(r), t_\alpha(a)) = z_{\Sigma \setminus \mathbb R \beta}( g, {^{t_\beta(r)}\!{t_\alpha(a)}} )$$ for $g \in \mathop{\mathrm{E}}_{ G, T, \Sigma \setminus \mathbb R \beta }(R)$ and the right hand side lies in $\mathop{\mathrm{E}}'_{G, T, \Psi}(R, A)$ by the induction hypothesis. ◻ Let $\alpha \in \Psi$. We say that $\Sigma \subseteq \Psi$ is a *thick $\alpha$-series* if is if of type $\Sigma = \Psi \cap ( \mathbb R_{> 0} \beta + \mathbb R \alpha )$ for some $\beta \in \Psi \setminus \mathbb R \alpha$. Clearly, $\Psi \setminus \mathbb R \alpha$ is a disjoint decomposition of thick $\alpha$-series. Every thick $\alpha$-series is a saturated special closed set. **Lemma 12**. *For any $\alpha \in \Psi$ the group object $\mathop{\mathrm{E}}'_{G, T, \Psi}(R, A)$ is generated by the images of $z_\Sigma$ for thick $\alpha$-series $\Sigma$.* *Proof.* Without loss of generality, $\frac 1 2 \alpha \notin \Psi$. Since $t_{\pm \alpha}(P_{\pm \alpha}(R))$ normalize $\mathop{\mathrm{E}}_{G, T, \Sigma}(A)$ for all thick $\alpha$-series $\Sigma$, we only have to check that $t_{\pm \alpha}(a)$ for $a \in P_{\pm \alpha}(A)$ may be expressed in terms of these $z_\Sigma$. This easily follows from lemma [Lemma 9](#root-gen){reference-type="ref" reference="root-gen"}. ◻ **Lemma 13**. *The subgroup $\mathop{\mathrm{E}}_{G, T, \Psi}(R)$ normalizes $\mathop{\mathrm{E}}^\prime_{G, T, \Psi}(R, A) \leq G(A)$.* *Proof.* It suffices to check that for any root $\alpha$ the subgroup $t_\alpha(P_\alpha(R))$ normalizes $\mathop{\mathrm{E}}'_{G, T, \Psi}(R, A)$. This follows from lemma [Lemma 12](#elim-rel){reference-type="ref" reference="elim-rel"}. ◻ **Lemma 14**. *$\mathop{\mathrm{E}}^\prime_{G, T, \Psi}(R, A) = \mathop{\mathrm{E}}_{G, T, \Psi}(R, A) \leq G(A)$.* *Proof.* Clearly, the first subgroup is contained in the second one. Conversely, by lemma [Lemma 13](#expl-conj){reference-type="ref" reference="expl-conj"} all generators of $\mathop{\mathrm{E}}_{G, T, \Psi}(R, A)$ from lemma [Lemma 7](#rel-elem){reference-type="ref" reference="rel-elem"} lie in $\mathop{\mathrm{E}}'_{G, T, \Psi}(R, A)$. ◻ **Lemma 15**. *If $A$ is also power idempotent then $\mathop{\mathrm{E}}_{G, T, \Psi}(A) = \mathop{\mathrm{E}}_{G, T, \Psi}(R, A)$.* *Proof.* By lemma [Lemma 14](#rel-expl){reference-type="ref" reference="rel-expl"} it suffices to check that for any $\alpha \in \Psi$ the subgroup $\mathop{\mathrm{E}}_{G, T, \Psi}(A)$ is normalized by $t_\alpha(P_\alpha(R))$. This easy follows from lemma [Lemma 10](#elim-abs){reference-type="ref" reference="elim-abs"}. ◻ **Lemma 16**. *If $(T', \Psi') \subseteq (T, \Psi)$ is another isotropic pinning, then $\mathop{\mathrm{E}}_{G, T', \Psi'}(R) \leq \mathop{\mathrm{E}}_{G, T, \Psi}(R)$. If the rank of $(T', \Psi')$ is positive, then $\mathop{\mathrm{E}}_{G, T', \Psi'}(R) = \mathop{\mathrm{E}}_{G, T, \Psi}(R)$.* *Proof.* Clearly, $\mathop{\mathrm{E}}_{G, T', \Psi'}(R) \leq \mathop{\mathrm{E}}_{G, T, \Psi}(R)$. If $\alpha \in \Psi$ has non-zero image $\alpha' \in \Psi'$, then $t_\alpha$ factors through $t_{\alpha'}$. Finally, if $\alpha \in \Psi$ has zero image in $\Psi' \cup \{0\}$, then $t_\alpha(P_\alpha(R)) \leq \mathop{\mathrm{E}}_{G, T', \Psi'}(R)$ by lemma [Lemma 9](#root-gen){reference-type="ref" reference="root-gen"}. ◻ **Lemma 17**. *Suppose that $\delta \colon A \to R$ is a crossed module in $\mathbf{IPRng}_{\mathbf D, s}$ and $A = \sum_{i = 1}^n A_i$ for $(R \rtimes K_s^{\mathop{\mathrm{Ind}}})$-submodules $A_i$ unital over $R$. Then $\mathop{\mathrm{E}}_{G, T, \Psi}(R, A) = \prod_{i = 1}^n \mathop{\mathrm{E}}_{G, T, \Psi}(R, A_i)$. If $P \leq G_s$ is a parabolic subgroup, then also ${\mathrm R_{\mathrm u}}(A) = \prod_{i = 1}^n {\mathrm R_{\mathrm u}}(A_i)$, where ${\mathrm R_{\mathrm u}}(P)$ is the unipotent radical of $P$.* *Proof.* Note that $\mathop{\mathrm{E}}_{G, T, \Psi}(R, A_i) \trianglelefteq\mathop{\mathrm{E}}_{G, T, \Psi}(R, A)$ since this subgroup is normalized by $\mathop{\mathrm{E}}_{G, T, \Psi}(R)$, so the Minkowski product $\prod_{i = 1}^n \mathop{\mathrm{E}}_{G, T, \Psi}(R, A_i)$ is a subgroup of $\mathop{\mathrm{E}}_{G, T, \Psi}(A)$. On the other hand, $t_\alpha(P_\alpha(A)) \leq \prod_{i = 1}^n t_\alpha(P_\alpha(A_i))$ for any $\alpha \in \Psi$. For unipotent radicals there is a finite nilpotent filtration ${\mathrm R_{\mathrm u}}(P) = U_1 \geq U_2 \geq \ldots \geq U_m = 0$ such that $U_k \leq {\mathrm R_{\mathrm u}}(P)$ are closed smooth $P$-invariant subschemes, $[U_k, U_l] \leq U_{k + l}$ (i.e. the commutator restricted to $U_k \times U_l$ takes values in $U_{k + l}$), and $U_k / U_{k + 1}$ is isomorphic to a vector bundle over $\mathcal D(s)$ [@sga3 Exp. XXVI, prop. 2.1]. By [@sga3 Exp. XXVI, the proof of cor. 2.5] the morphisms $U_k \to U_k / U_{k + 1}$ have scheme sections, so $(U_k / U_{k + 1})(A) \cong U_k(A) / U_{k + 1}(A)$ for every $k$. It suffices to prove that $V(A) = \sum_i V(A_i)$ for any vector bundle $V$ over $\mathcal D(s)$. In other words, if $e \in \mathop{\mathrm{M}}(N, K_s)$ is an idempotent for some $N \geq 0$, then $e(A^N) = \sum_i e(A_i^N)$. This easily follows from $A = \sum_i A_i$. ◻ # Locally isotropic elementary groups In this section $G$ is a reductive group scheme over $K$ of local isotropic rank at least $2$. Choose a Zariski covering $\mathop{\mathrm{Spec}}(K) = \bigcup_{i = 1}^N \mathcal D(a_i)$ for some $a_i \in K$ (i.e. $a_i$ generate the unit ideal) and isotropic pinnings $(T_i, \Psi_i)$ of $G_{a_i}$ of rank $\geq 2$. Such data exist since the isotropic rank of $G$ is at least $2$ at every point of $\mathop{\mathrm{Spec}}(K)$ and an isotropic pinning of $G_{\mathfrak p}$ for a prime ideal $\mathfrak p \trianglelefteq K$ may be continued to $G_s$ for some $s \in K \setminus \mathfrak p$ by a direct limit argument. For any unital $K$-algebra $R$ the *elementary subgroup* is $$\mathop{\mathrm{E}}_G(R) = \langle \mathop{\mathrm{Im}}( \mathop{\mathrm{E}}_{G, T_i, \Psi_i}(R^{(a_i^\infty)}) \to G(R) ) \mid 1 \leq i \leq N \rangle,$$ it is a group object in $\mathbf{IP}_{\{*\}} = \mathop{\mathrm{Ind}}(\mathop{\mathrm{Pro}}(\mathbf{Set}))$. We are going to show that it is independent on the choices and is actually a normal subgroup of $G(R)$. In order to study functorial properties of $\mathop{\mathrm{E}}_G$ let also $$\mathop{\mathrm{E}}_G(\mathcal R) = \langle \mathop{\mathrm{Im}}( \mathop{\mathrm{E}}_{G, T_i, \Psi_i}( \mathcal R^{(a_i^\infty)} ) \to G(\mathcal R) ) \mid 1 \leq i \leq N \rangle$$ as a group object in $\mathbf{IP}_{\mathbf{CRing}_K^{\mathrm{fp}}}$, so $\mathop{\mathrm{E}}_G(R)$ is the restriction of $\mathop{\mathrm{E}}_G(\mathcal R)$ to not necessarily finitely presented $R$. **Lemma 18**. *Let $s \in K$, $(T, \Psi)$ be an isotropic pinning of $G_s$, and $L$ be the centralizer of $T$ in $G_s$. Then the subgroup $\mathop{\mathrm{GE}}_{G, T, \Psi}(\mathcal R_s^{\mathop{\mathrm{Ind}}}) = L(\mathcal R_s^{\mathop{\mathrm{Ind}}}) \mathop{\mathrm{E}}_{G, T, \Psi}(\mathcal R_s^{\mathop{\mathrm{Ind}}}) \leq G(\mathcal R_s^{\mathop{\mathrm{Ind}}})$ normalizes $\mathop{\mathrm{E}}_{G, T, \Psi}(\mathcal R^{(s^\infty)})$ and $\mathop{\mathrm{E}}_{G, T, \Psi}( \mathcal R^{(s^\infty)}, \mathcal A^{(s^\infty)} )$ for any functor $\mathcal A \colon \mathbf{CRing}_K \to \mathbf{CRng}_K$ commuting with direct limits and with a structure of an $\mathcal R$-algebra.* *Proof.* For $t_\alpha(P_\alpha(\mathcal R^{\mathop{\mathrm{Ind}}}_s))$ the claim follows from lemmas [Lemma 10](#elim-abs){reference-type="ref" reference="elim-abs"}, [Lemma 12](#elim-rel){reference-type="ref" reference="elim-rel"}, [Lemma 14](#rel-expl){reference-type="ref" reference="rel-expl"}. Let us denote by $\widehat L = \Gamma(L, \mathcal O_L) \in \mathbf{CRing}_{K_s}$ the coordinate algebra of $L$. For all $\alpha \in \Psi$ and basis vectors $e_k \in P_\alpha$ there are scheme morphisms $h_{\alpha k} \colon L \times \mathbb A^1 \to P_\alpha$ such that $${^{g}\!{t_\alpha(e_k \cdot r)}} = t_\alpha(h_{\alpha k}).$$ Moreover, $h_{\alpha k}$ are pointed on the second variable, i.e. they may be represented by elements of $P_\alpha(x \widehat L[x])$. It follows that $h_{\alpha k}$ induce morphisms $L(\mathcal R_s^{\mathop{\mathrm{Ind}}}) \times \mathcal A^{(s^\infty)} \to P_\alpha(\mathcal A^{(s^\infty)})$ such that $${^{g}\!{t_\alpha(e_k \cdot a)}} = t_\alpha(h_{\alpha k}(g, a))$$ for $g \in L(\mathcal R_s^{\mathop{\mathrm{Ind}}})$ and $a \in \mathcal A^{(s^\infty)}$. ◻ The next lemma shows that $\mathop{\mathrm{E}}_G(\mathcal R)$ is independent of the Zariski covering and the isotropic pinnings $(T_i, \Psi_i)$. Moreover, for any $R \in \mathbf{CRing}_K$ and $g \in \mathop{\mathrm{Aut}}(G)(R)$ the subgroup $\mathop{\mathrm{E}}_G(R) \leq G(R)$ is $g$-invariant (this does not imply the normality since $\mathop{\mathrm{E}}_G(R)$ is a group object in $\mathop{\mathrm{Ind}}(\mathop{\mathrm{Pro}}(\mathbf{Set}))$ rather than $\mathbf{Set}$). We denote the unipotent radical of a parabolic subgroup $P \leq G$ by ${\mathrm R_{\mathrm u}}(P)$. **Lemma 19**. *Let $s \in K$ and $P$ be a parabolic subgroup of $G_s$. Then the image of ${\mathrm R_{\mathrm u}}(P)(\mathcal R^{(s^\infty)})$ in $G(\mathcal R)$ is contained in $\mathop{\mathrm{E}}_G(\mathcal R)$. In particular, if $(T, \Psi)$ is an isotropic pinning of $G_s$, then the image of $\mathop{\mathrm{E}}_{G, T, \Psi}(\mathcal R^{(s^\infty)})$ in $G(\mathcal R)$ is contained in $\mathop{\mathrm{E}}_G(\mathcal R)$.* *Proof.* For all $1 \leq i \leq N$ choose Zariski coverings $\mathcal D(a_i) \cap \mathcal D(s) = \bigcup_{j = 1}^{M_i} \mathcal D(b_{ij})$, isotropic pinnings $(T_{ij}, \Psi_{ij})$ of $G_{b_{ij}}$ containing $(T_i|_{\mathcal D(b_{ij})}, \Psi_i)$, and elements $g_{ij} \in \mathop{\mathrm{E}}_{G, T_{ij}, \Psi_{ij}}(K_{b_{ij}})$ such that $P|_{\mathcal D(b_{ij})} \subseteq {^{g_{ij}}\!{P_{ij}}}$, where $P_{ij}$ is the parabolic subgroup corresponding to a family of positive roots of $\Psi_{ij}$. Such data exist by lemma [Lemma 2](#loc-iso-pin){reference-type="ref" reference="loc-iso-pin"} and a direct limit argument. By lemmas [Lemma 16](#pin-elim){reference-type="ref" reference="pin-elim"} and [Lemma 18](#levi-action){reference-type="ref" reference="levi-action"} we have $${\mathrm R_{\mathrm u}}(P)(\mathcal R^{(b_{ij}^\infty)}) \leq \mathop{\mathrm{E}}_{ G, {^{g}\!{T_{ij}}}, {^{g}\!{\Psi_{ij}}} }( \mathcal R^{(b_{ij}^\infty)} ) = \mathop{\mathrm{E}}_{G, T_{ij}, \Psi_{ij}}( \mathcal R^{(b_{ij}^\infty)} ) = \mathop{\mathrm{E}}_{G, T_i, \Psi_i}( \mathcal R^{(b_{ij}^\infty)} )$$ (the left inclusion is obvious). From lemmas [Lemma 6](#cozariski){reference-type="ref" reference="cozariski"} and [Lemma 17](#xmod-dec){reference-type="ref" reference="xmod-dec"} we obtain $$\mathop{\mathrm{Im}}(\mathop{\mathrm{E}}_{G, T, \Psi}( \mathcal R^{(s^\infty)} )) \leq \mathop{\mathrm{E}}_G(\mathcal R). \qedhere$$ ◻ **Lemma 20**. *Let $(T, \Psi)$ and $(T', \Psi')$ be isotropic pinnings of $G$ such that the rank of $(T', \Psi')$ is $\geq 2$. Then there is $n \geq 1$ such that $\mathop{\mathrm{E}}_{G, T, \Psi}(x^n K[x]) \leq \mathop{\mathrm{E}}_{G, T', \Psi'}(x K[x])$.* *Proof.* By lemmas [Lemma 15](#rel-to-abs){reference-type="ref" reference="rel-to-abs"} (applied to $R = K[x]$ and $A = \varprojlim^{\mathop{\mathrm{Pro}}(\mathbf{Set})}_n x^n K[x] \cong K[x]^{(x^\infty)}$) and [Lemma 19](#invariance){reference-type="ref" reference="invariance"} we have $$\begin{aligned} \mathop{\mathrm{E}}_{G, T, \Psi}(x^n K[x]) &\leq \mathop{\mathrm{E}}_{G, T, \Psi}(K[x], x^n K[x]) \leq \mathop{\mathrm{Ker}}( \mathop{\mathrm{E}}_G(x^n K[x] \rtimes K[x]) \to \mathop{\mathrm{E}}_G(K[x]) )\\ &= \mathop{\mathrm{E}}_{G, T', \Psi'}(K[x], x^n K[x]) \leq \mathop{\mathrm{E}}_{G, T', \Psi'}(x K[x]) \end{aligned}$$ for sufficiently large $n$. ◻ **Lemma 21**. *Let $s \in K$ and $(T, \Psi)$ be an isotropic pinning of $G_s$. Then there is $m \geq 0$ such that for any $\alpha \in \Psi$ the image of $t_\alpha(P_\alpha(\mathcal R^{(s^m)}))$ is contained in $\mathop{\mathrm{E}}_G(\mathcal R)$ (this subgroup is defined for sufficiently large $m$).* *Proof.* Fix a root $\alpha$ and a basis element $e_k \in P_\alpha$. Choose a Zariski covering $\mathop{\mathrm{Spec}}(K) = \bigcup_{i = 1}^N \mathcal D(a_i)$ and isotropic pinnings $(T_i, \Psi_i)$ of $G_{a_i}$ of rank $\geq 2$. By lemma [Lemma 20](#loc-gluing){reference-type="ref" reference="loc-gluing"} applied to $K_{s a_i}[z]$ there are $n \geq 0$, $M_i \geq 0$, roots $\beta_{ij} \in \Psi_i$ for $1 \leq j \leq M_i$, and elements $p_{ij} \in P_{\beta_{ij}}(x K_{s a_i}[x, z])$ such that the image of $t_\alpha(e_k \cdot x^n z)$ in $G(K_{s a_i}[x, z])$ is $\prod_{j = 1}^{M_i} t_{\beta_j}(p_{ij})$. This is an equality of some elements of $x \widehat G_{s a_i}[x, z]$, where $\widehat G = \Gamma(G, \mathcal O_G) \in \mathbf{CRing}_K$ is the coordinate algebra of $G$. Replacing $x$ by $s^{m_i} y$ for sufficiently large $m_i$ we obtain a similar equality in $\widehat G_{a_i}[y, z]$, i.e. the images of $t_\alpha(e_k \cdot s^{nm_i} y^n z)$ and $\prod_{j = 1}^{M_i} t_{\beta_j}(q_{ij})$ in $G(K_{a_i}[y, z])$ coincide for some $q_{ij} \in P_{\beta_{ij}}(y K_{a_i}[y, z])$. Here $$t_\alpha(e_k \cdot s^{nm_i} y^n z) = t_\alpha(e_k \cdot y^n z^{(s^{nm_i})}) \in G(K[y, z])$$ is well-defined if $m_i$ is large enough. By lemma [Lemma 5](#power-idem){reference-type="ref" reference="power-idem"} we have $t_\alpha( e_k \cdot ( \mathcal R^{(s^{nm_i})} )^{(a_i^\infty)} ) \leq \mathop{\mathrm{E}}_G(\mathcal R)$, so $t_\alpha(e_k \cdot \mathcal R^{(s^m)}) \leq \mathop{\mathrm{E}}_G(\mathcal R)$ for sufficiently large $m$ by lemma [Lemma 6](#cozariski){reference-type="ref" reference="cozariski"}. ◻ **Theorem 1**. *Let $G$ be a reductive group scheme over a unital ring $K$ of local isotropic rank $\geq 2$. Then the elementary group $\mathop{\mathrm{E}}_G(\mathcal R) \leq G(\mathcal R)$ is isomorphic to an object from $\mathop{\mathrm{Ind}}(\mathbf{Cat}(\mathbf{CRing}_K^{\mathrm{fp}}, \mathbf{Set})) \subseteq \mathbf{IP}_{\mathbf{CRing}_K^{\mathrm{fp}}}$. In other words, if $\mathop{\mathrm{Spec}}(K) = \bigcup_{i = 1}^N \mathcal D(a_i)$ is a Zariski covering and $(T_i, \Psi_i)$ are isotropic pinnings of $G_{a_i}$ of rank $\geq 2$, then there is $m \geq 0$ such that the maps $t_\alpha \colon P_\alpha(\mathcal R^{(a_i^m)}) \to G(\mathcal R)$ are well-defined and generate $\mathop{\mathrm{E}}_G(\mathcal R)$ for $\alpha \in \Psi_i$, $1 \leq i \leq N$. Moreover, $P_\alpha(\mathcal R^{(a_i^m)})$ are $\mathcal R$-modules ($2$-step nilpotent if $\alpha$ is ultrashort), $t_\alpha$ are homomorphisms, and $f_{\alpha \beta i j} \colon P_\alpha(\mathcal R^{(a_i^m)}) \times P_\beta(\mathcal R^{(a_i^m)}) \to P_{i \alpha + j \beta}(\mathcal R^{(a_i^m)})$ are homogeneous polynomial maps over $\mathcal R$ (or $\mathcal R$-quadratic on each variable for components of type $\mathsf{BC}_\ell$). If $K$ is Noetherian, then for any fixed algebra $R \in \mathbf{CRing}_K^{\mathrm{fp}}$ we may change the homotopes to the corresponding principal ideals.* *Proof.* By lemma [Lemma 21](#main-lemma){reference-type="ref" reference="main-lemma"} there is $m$ such that $t_\alpha \colon P_\alpha(\mathcal R^{(a_i^m)}) \to G(\mathcal R)$ are well-defined and take values in $\mathop{\mathrm{E}}_G(\mathcal R)$. On the other hand, the generating morphisms $t_\alpha \colon P_\alpha(\mathcal R^{(a_i^\infty)}) \to G(\mathcal R)$ clearly factors through $t_\alpha \colon P_\alpha(\mathcal R^{(a_i^m)}) \to G(\mathcal R)$. The remaining assertions are easy. The $2$-step nilpotent $\mathcal R$-module structure on $P_\alpha(\mathcal R^{(a_i^m)})$ is well-defined for ultrashort $\alpha$ and sufficiently large $m$ since it is given by some $K_{a_i}$-bilinear $2$-cocycle. Actually, every polynomial $f \in K_{a_i}[x_1, \ldots, x_p]$ without terms of degree $\leq 1$ induces a well-defined map $(\mathcal R^{(a_i^m)})^p \to \mathcal R^{(a_i^m)}$ if $m$ is large enough. The last claim follows from lemma [Lemma 4](#noeth-coloc){reference-type="ref" reference="noeth-coloc"}. ◻ By theorem [Theorem 1](#e-discr){reference-type="ref" reference="e-discr"} the elementary group $\mathop{\mathrm{E}}_G(R)$ is a group object in $\mathop{\mathrm{Ind}}(\mathbf{Set})$ for every $R \in \mathbf{CRing}_K$. We may consider it as an ordinary subgroup of $G(R)$ by evaluating the direct limit in $\mathbf{Set}$ (i.e. by taking the union of components of $\mathop{\mathrm{E}}_G(R)$). Let us say that a subfunctor $H \colon \mathbf{CRing}_K \to \mathbf{Grp}$ of $G$ is *scheme generated* if there is an affine $K$-scheme $X$ and a morphism $f \colon X \to G$ such that $H(R) = \langle f(X(R)) \rangle$ as an abstract group for every $R \in \mathbf{CRing}_K$ (here $G$ is any affine $K$-scheme, not necessarily reductive). The next lemma shows that such subfunctor has a canonical ind-structure. **Lemma 22**. *Let $H \colon \mathbf{CRing}_K \to \mathbf{Grp}$ be a subgroup of $G$. If it is scheme generated by $f \colon X \to G$ and a scheme morphism $f' \colon X' \to G$ takes values in $H$ for an affine $K$-scheme $X'$, then there is $n \geq 0$ such that $$f'(X'(R)) \subseteq ( f(X(R)) \cup \{1\} \cup f(X(R))^{-1} )^n$$ for all $R \in \mathbf{CRing}_K$. In particular, if $f'$ also scheme generates $H$, then the subgroups of $G$ in $\mathop{\mathrm{Ind}}(\mathbf{Cat}(\mathbf{CRing}_K^{\mathrm{fp}}, \mathbf{Set}))$ generated by $f$ and $f'$ coincide.* *Proof.* Let $\widehat{X'}$ be the coordinate $K$-algebra of $X'$ and $x'_0 \in X'(\widehat{X'})$ be the universal point corresponding to ${\mathrm{id}}\colon X' \to X'$ by the Yoneda lemma. By definition, there are $n \geq 0$, $a_i \in X(\widehat{X'})$, and $\varepsilon_i \in \{-1, 1\}$ for $1 \leq i \leq n$ such that $f'(x'_0) = \prod_{i = 1}^n f(a_i)^{\varepsilon_i}$. If $x' \in X'(R)$ is any $R$-point, then $x' = u(x'_0)$ for unique homomorphism $u \colon \widehat{X'} \to R$ and $$f'(x') = u_*(f'(x'_0)) = \prod_{i = 1}^n f(u_*(a_i))^{\varepsilon_i}. \qedhere$$ ◻ **Theorem 2**. *Let $G$ be a reductive group scheme over a unital ring $K$ of local isotropic rank $\geq 2$. Then the elementary subgroup $\mathop{\mathrm{E}}_G \leq G$ is scheme generated by a morphism $f \colon \mathbb A^N \to G$ from an affine space and its ind-structure is induced by $f$. In particular, $\mathop{\mathrm{E}}_G(R \times R') = \mathop{\mathrm{E}}_G(R) \times \mathop{\mathrm{E}}_G(R')$, $\mathop{\mathrm{E}}_G(R / I) = \mathop{\mathrm{Im}}(\mathop{\mathrm{E}}_G(R) \to G(R / I))$ for any ideal $I \trianglelefteq R$, and $\mathop{\mathrm{E}}_G$ commutes with direct limits as an abstract group. On the other hand, $\mathop{\mathrm{E}}_G$ is also scheme generated by a $G$-equivariant morphism $u \colon X \to G$ from a finitely presented affine $K$-scheme $X$ with an action of $G$, so $\mathop{\mathrm{E}}_G \leq G$ is normal both as an abstract group subfunctor and as a group object in $\mathbf{IP}_{\mathbf{CRing}_K^{\mathrm{fp}}}$.* *Proof.* The first claim follows from theorem [Theorem 1](#e-discr){reference-type="ref" reference="e-discr"}. In order to prove the second claim it suffice to prove that $$u \colon G \times \mathbb A^N \to G,\, (g, \vec x) \mapsto {^{g}\!{f(\vec x)}}$$ takes values in $\mathop{\mathrm{E}}_G$ as an abstract subgroup functor of $G$. Actually, we only have to check that $u(g_0, \vec x_0) = {^{g_0}\!{f(\vec x_0)}} \in G(R)$, where $R = \widehat G[x_1, \ldots, x_N]$, and $\widehat G$ is the coordinate algebra of $G$, and $(g_0, \vec x_0) \in G(R) \times \mathbb A^N(R)$ is the universal point. This follows from lemma [Lemma 19](#invariance){reference-type="ref" reference="invariance"}. ◻ The next theorem shows that our elementary subgroup coincide with the elementary subgroup constructed in [@petrov-stavrova] if the latter is defined. **Theorem 3**. *Let $G$ be a reductive group scheme over a unital ring $K$ of local isotropic rank $\geq 2$ and $P^+, P^- \leq G$ be opposite parabolic subgroups. Suppose that their unipotent radicals ${\mathrm R_{\mathrm u}}(P^\pm)$ non-trivially intersect all simple factors of $G / \mathop{\mathrm{C}}(G)$ over geometric points. Then $\mathop{\mathrm{E}}_G$ is scheme generated by the subscheme ${\mathrm R_{\mathrm u}}(P^-) \times {\mathrm R_{\mathrm u}}(P^+) \subseteq G$.* *Proof.* The unipotent radicals lie in $\mathop{\mathrm{E}}_G$ by lemma [Lemma 19](#invariance){reference-type="ref" reference="invariance"}. Conversely, by lemma [Lemma 2](#loc-iso-pin){reference-type="ref" reference="loc-iso-pin"} there is a Zariski covering $\mathop{\mathrm{Spec}}(K) = \bigcup_{i = 1}^N \mathcal D(a_i)$ and isotropic pinnings $(T_i, \Psi_i)$ of $G_s$ such that $P_{a_i}^+$ contain the standard parabolic subgroups $P_i \leq G_{a_i}$ constructed by systems of positive roots in $\Psi_i$. By [@petrov-stavrova lemma 12] the subfunctor $\mathop{\mathrm{E}}_{G, T_i, \Psi_i} \leq G_{a_i}$ is generated by ${\mathrm R_{\mathrm u}}(P^{\pm})_{a_i}$. It follows that the image of $\mathop{\mathrm{E}}_{G, T_i, \Psi_i}(\mathcal R^{(a_i^\infty)})$ in $G(\mathcal R)$ is contained in the subgroup generated by ${\mathrm R_{\mathrm u}}(P^\pm)(\mathcal R)$. ◻ Finally, let us prove that $\mathop{\mathrm{E}}_G$ is perfect. It is well-known that $\mathop{\mathrm{E}}(\mathsf B_2, \mathbb F_2)$ and $\mathop{\mathrm{E}}(\mathsf G_2, \mathbb F_2)$ are not perfect (their derived subgroups are simple but of index $2$). Following [@luzragev-stavrova] we impose the following additional condition: if $\mathfrak m \trianglelefteq K$ is a maximal ideal such that $K / \mathfrak m \cong \mathbb F_2$, then the geometric fiber of $G / \mathop{\mathrm{C}}(G)$ at $K / \mathfrak m$ has no simple factors of the types $\mathsf B_2$ and $\mathsf G_2$. Since the residue fields at non-maximal prime ideals of $K$ are infinite, this condition also holds for all unital $K$-algebras. **Theorem 4**. *Let $G$ be a reductive group scheme over a unital ring $K$ of local isotropic rank $\geq 2$. Suppose that the additional condition holds, i.e. the geometric fibers of $G / \mathop{\mathrm{C}}(G)$ over all residue fields of $2$ elements does not have components with root systems of the types $\mathsf B_2$ and $\mathsf C_2$. Then $\mathop{\mathrm{E}}_G$ is perfect, i.e. $\mathop{\mathrm{E}}_G(R) = [\mathop{\mathrm{E}}_G(R), \mathop{\mathrm{E}}_G(R)]$ for any $R \in \mathbf{CRing}_K$.* *Proof.* By [@luzgarev-stavrova lemma 5] and a direct limit argument for every prime ideal $\mathfrak p \trianglelefteq K$ there are $s \in K \setminus \mathfrak p$, an isotropic pinning $(T, \Psi)$ of $G_s$ of rank $\geq 2$, and $n \geq 1$ such that $$\mathop{\mathrm{E}}_{G, T, \Psi}(x^n K_s[x, y]) \leq [ \mathop{\mathrm{E}}_{G, T, \Psi}(x K_s[x, y]), \mathop{\mathrm{E}}_{G, T, \Psi}(x K_s[x, y]) ].$$ Thus by lemma [Lemma 5](#power-idem){reference-type="ref" reference="power-idem"} for every $\alpha \in \Psi$ and basis vector $e_i \in P_\alpha$ we have $$t_\alpha(e_i \cdot \mathcal R^{(s^\infty)}) \leq [ \mathop{\mathrm{E}}_{G, T, \Psi}( \mathcal R^{(s^\infty)} ), \mathop{\mathrm{E}}_{G, T, \Psi}( \mathcal R^{(s^\infty)} ) ].$$ This implies the claim. ◻ [^1]: Research is supported by the Russian Science Foundation grant 19-71-30002.

arxiv_math

--- abstract: | The $\mathrm{K}$-cowaist $\mathop{\mathrm{K-cw_2}}(M)$ and the $\hat{\mathsf{A}}$-cowaist $\mathop{\mathrm{\Hat{A}-cw_2}}(M)$ are two interesting invariants on a manifold $M$, which are closely related to the existence of the positive scalar curvature metric on $M$. In this note, we give a detailed proof of the following inequality due to Gromov: $\mathop{\mathrm{K-cw_2}}(M) \le c \mathop{\mathrm{\Hat{A}-cw_2}}(M)$, where $c$ is a dimensional constant. address: School of Mathematics, Shandong University, Jinan, Shandong 250100, China author: - Xiangsheng Wang title: | On a relation between the $\mathrm{K}$-cowaist\ and the $\hat{\mathsf{A}}$-cowaist --- [^1] # Two invariants associated with the positive scalar curvature The famous Lichnerowicz theorem asserts that for a closed spin manifold $M$ if the $\hat{\mathsf{A}}$-genus of $M$ does not vanish, $M$ cannot carry a Riemannian metric $g$ with the scalar curvature $\kappa_g > 0$. An interesting generalization of this theorem is to find a "quantitative" version of it. Namely, under what conditions, can we obtain an upper bound for $\inf_M \kappa_g$ holding for any Riemannian metric $g$ on $M$? To investigate this problem, Gromov [@Gromov_1996aa] formulates an invariant called $\mathrm{K}$-cowaist.[^2] Let us recall the definition of this invariant. Let $M$ be a closed connected oriented smooth Riemannian manifold of even dimension. Let $E\to M$ be a Hermitian vector bundle with a Hermitian connection $\nabla^E$ and $\mathop{\mathrm{R}}^E$ be the curvature of $\nabla^E$. Note that for any $x\in M$ and $\alpha,\beta \in \mathrm{T}_x M$, $\mathop{\mathrm{R}}^E(\alpha \wedge \beta) \in \operatorname{End}(E_x)$. We define a norm on $\mathop{\mathrm{R}}^E$ in the following way, $$\|\mathop{\mathrm{R}}^{E}\| = \sup_{x\in M} \sup_{\substack{\alpha,\beta\in \mathrm{T}_x{M} \\ \alpha \perp \beta,\ |\alpha\wedge \beta|=1}}|\mathop{\mathrm{R}}^{E}(\alpha\wedge \beta)|.$$ We are interested in the following class of unitary vector bundles over $M$, $$\label{eq:cn} \exists\; i_1,\cdots,i_{l} \in \mathbb{Z}_{> 0} \text{ such that } \int_M \prod_{k=1}^l\mathop{\mathrm{c}}_{i_k}(E) \neq 0,$$ where $\mathop{\mathrm{c}}_{i_k}(E)$ is the $i_k$-th Chern class of $E$. The $\mathrm{K}$-cowaist of $M$ is defined to be $$\mathop{\mathrm{K-cw_2}}(M)=\sup\{\|\mathop{\mathrm{R}}^E\|^{-1}| E \text{ satisfies (\ref{eq:cn})}\}.$$ In Gromov's words, the condition on the Chern numbers of $E$ ensures that $E$ is "homologically non-trivial". In view of Lichnerowicz's theorem, it is also natural to replace the condition ([\[eq:cn\]](#eq:cn){reference-type="ref" reference="eq:cn"}) of $\mathrm{K}$-cowaist with the following condition, $$\label{eq:ahg} \int_M \hat{\mathsf{A}}(M)\mathop{\mathrm{ch}}(E) \neq 0.$$ And the $\hat{\mathsf{A}}$-cowaist[^3] of $M$ is defined to be $$\mathop{\mathrm{\Hat{A}-cw_2}}(M)=\sup\{\|\mathop{\mathrm{R}}^E\|^{-1}| E \text{ satisfies (\ref{eq:ahg})}\}.$$ Using the $\mathrm{K}$-cowaist, Gromov gives the following quantitative version of Lichnerowicz's theorem. Let $(M,g)$ be a spin Riemannian manifold of even dimension. $$\label{eq:g-est} \inf_M \kappa_g\le C (\mathop{\mathrm{K-cw_2}}(M))^{-1},$$ where $C$ is a constant only depending on the dimension of $M$. In [@Gromov_1996aa], although the definition of $\hat{\mathsf{A}}$-cowaist does not appear explicitly, the proof of ([\[eq:g-est\]](#eq:g-est){reference-type="ref" reference="eq:g-est"}) uses it and consists of two steps. The first step is to show ([\[eq:g-est\]](#eq:g-est){reference-type="ref" reference="eq:g-est"}) with $\mathop{\mathrm{K-cw_2}}(M)$ replaced by $\mathop{\mathrm{\Hat{A}-cw_2}}(M)$. The second step is proving the following comparison results between $\mathop{\mathrm{K-cw_2}}(M)$ and $\mathop{\mathrm{\Hat{A}-cw_2}}(M)$. **Theorem 1**. *Let $M$ be a closed Riemannian metric manifold $M$ of even dimension. There is a constant $c$ depending only on the dimension of $M$ such that $$\label{eq:klea} \mathop{\mathrm{K-cw_2}}(M) \le c \mathop{\mathrm{\Hat{A}-cw_2}}(M).$$* In this note, we would like to clarify some arguments used in [@Gromov_1996aa] to prove Theorem [Theorem 1](#thm:com){reference-type="ref" reference="thm:com"}, see Remark [Remark 9](#rk:gromov){reference-type="ref" reference="rk:gromov"}. In other words, we give a detailed proof of this theorem following Gromov's method. Before proving Theorem [Theorem 1](#thm:com){reference-type="ref" reference="thm:com"}, we note that it has the following corollary. **Corollary 2**. *If $\mathop{\mathrm{K-cw_2}}(M) = +\infty$, then $\mathop{\mathrm{\Hat{A}-cw_2}}(M) = +\infty$.* This corollary, as well as its generalization for manifolds with boundary or non-compact manifolds, has found application in the literature, see [@Cecchini_2021sc; @Su_2021ma]. This paper is organized as follows. In Section [2](#sec:pf-com){reference-type="ref" reference="sec:pf-com"}, after some algebraic preliminaries, we prove Theorem [Theorem 1](#thm:com){reference-type="ref" reference="thm:com"}. In Section [3](#sec:so-re){reference-type="ref" reference="sec:so-re"}, we give two remarks about Theorem [Theorem 1](#thm:com){reference-type="ref" reference="thm:com"}, about the its generalization on non-closed manifolds and the reverse direction inequality of ([\[eq:klea\]](#eq:klea){reference-type="ref" reference="eq:klea"}) respectively. # Proof of Theorem [Theorem 1](#thm:com){reference-type="ref" reference="thm:com"} {#sec:pf-com} In the rest of this paper, we assume that the dimension of $M$ is even. Using the definition of $\mathrm{K}$-cowaist, we can see that the following result implies ([\[eq:klea\]](#eq:klea){reference-type="ref" reference="eq:klea"}). **Proposition 3**. *Fix a positive number $m_0$. Let $E$ be a Hermitian vector bundle[^4] over $M$ satisfying ([\[eq:cn\]](#eq:cn){reference-type="ref" reference="eq:cn"}) and $$\label{eq:ba0} \Vert\mathop{\mathrm{R}}^E\Vert^{-1} \ge m_0.$$ Then there exists a Hermitian vector bundle $E'$ over $M$ satisfying ([\[eq:ahg\]](#eq:ahg){reference-type="ref" reference="eq:ahg"}) and $$\label{eq:ca} \Vert\mathop{\mathrm{R}}^{E'}\Vert^{-1} \ge c m_0$$ where $c$ is constant only depending on the dimension of $M$.* ## Two algebraic lemmas **Definition 4**. Let $\mathcal{V}$ be the category of the Hermitian vector bundles over a fixed manifold.We call a functor $J:\mathcal{V}\times \cdots \times \mathcal{V}\rightarrow \mathcal{V}$ *admissible* if $J$ is a finite composition of following functors: 1. $I$, the identify functor of $\mathcal{V}$; 2. $E \rightarrow \mathbb{C}^k$, where $\mathbb{C}^k$ is the trivial bundle; 3. $E \rightarrow E'$, where $E'$ is the dual bundle of $E$; 4. $E \rightarrow \wedge^k E$, where $\wedge^k E$ is the $k$-th wedge product bundle of $E$; 5. $E, F \rightarrow E\oplus F$, the direct sum; 6. $E, F \rightarrow E\otimes F$, the tensor product. Let $E,F$ (resp. $E_1,\cdots,E_k,F$) be Hermitian vectors bundles. We say that *$F$ is constructed from $E$ (resp. $E_1,\cdots,E_k$) in an admissible way*, if there exists an admissible functor $J$ and an isomorphism between $F$ and $J(E)$ (resp. $J(E_1,\cdots,E_k)$), which preserves the metric and the connection. *Remark 5*. Note that the six simple operations on Hermitian vector bundles listed in Definition [Definition 4](#def:ad-f){reference-type="ref" reference="def:ad-f"} exist on every manifold. Therefore, for two manifolds $M$ and $N$, there is a natural bijection between the sets of admissible functors defined by $M$ and $N$. In this sense, the definition of admissible functors does not depend on the ambient manifolds. We would like to remark that in the following, when we say that a constant depends only on an admissible functor $J$, we precisely mean that the constant does not depend on the choice of ambient manifold defining $J$. For example, $\mathbb{C}^n$, $E'$, $E\oplus E$ and $(\otimes^n E) \otimes (\wedge^k E)$ are all constructed from $E$ in an admissible way. **Proposition 6**. *If $J$ is an admissible functor, there is a constant $C_0$ such that $$\label{eq:c0} \Vert \mathop{\mathrm{R}}^{J(E)} \Vert \le C_J \Vert \mathop{\mathrm{R}}^{E} \Vert$$ holds for any Hermitian vector bundle $E$, where $C_0$ is a constant depending only on $J$.* *Proof.* By the definition of the admissible functor, we only need to check ([\[eq:c0\]](#eq:c0){reference-type="ref" reference="eq:c0"}) for the functors listed in Definition [Definition 4](#def:ad-f){reference-type="ref" reference="def:ad-f"}. We check ([\[eq:c0\]](#eq:c0){reference-type="ref" reference="eq:c0"}) for the tensor product as an example. Let $V,W$ be two Hermitian vector spaces and $A,B$ be endomorphisms on $V,W$ respectively. We note that $$(A \otimes \mathop{\mathrm{id}})^*(A \otimes \mathop{\mathrm{id}}) = (A^* \otimes \mathop{\mathrm{id}})(A \otimes \mathop{\mathrm{id}}) = A^*A \otimes \mathop{\mathrm{id}}.$$ By the relation between the singular values and the matrix norm, the above equality implies that $$\Vert A \otimes \mathop{\mathrm{id}}\Vert = \Vert A \Vert.$$ As a result, we have $$\label{eq:m-inq} \Vert A \otimes \mathop{\mathrm{id}}+ \mathop{\mathrm{id}}\otimes B\Vert \le \Vert A \Vert + \Vert B \Vert.$$ By the definition of the tensor product of connections, for Hermitian vector bundles $E,F$, we have $$\mathop{\mathrm{R}}^{E\otimes F} = \mathop{\mathrm{R}}^E \otimes \mathop{\mathrm{id}}+ \mathop{\mathrm{id}}\otimes \mathop{\mathrm{R}}^F.$$ Then, the needed inequality $$\Vert \mathop{\mathrm{R}}^{E\otimes F} \Vert \le \Vert \mathop{\mathrm{R}}^E \Vert + \Vert \mathop{\mathrm{R}}^F \Vert$$ follows from ([\[eq:m-inq\]](#eq:m-inq){reference-type="ref" reference="eq:m-inq"}) immediately. ◻ After Gromov, we use the following algebraic lemma to prove Proposition [Proposition 3](#prop:klea){reference-type="ref" reference="prop:klea"}. **Lemma 7**. *Fix $N\in \mathbb{Z}_{+}$. There is a finite set of admissible functors $\mathcal{C}_N = \{J_i\}$ satisfying the following property. For any positive integer $K\le N$ and any partition of $K$ by positive integers, $K = \sum_{l = 1}^{\nu} a_l$, there exist $\lambda_i\in \mathbb{Q}$ such that $$\label{eq:cal} \prod_{l=1}^{\nu} \mathop{\mathrm{c}}_{a_l}(E) = \sum_i \lambda_i\mathop{\mathrm{ch}}_K(J_i(E))$$ holds for any Hermitian vector bundle $E$, where $\mathop{\mathrm{ch}}_K$ denotes the degree $2K$ component of the Chern character.* This lemma is just a restatement of [@Gromov_1996aa p. 36, Trivial Algebraic Lemma]. For readers' convenience, we also provide a proof for this lemma in Subsection [2.3](#sub:trivial){reference-type="ref" reference="sub:trivial"}. To state another algebraic lemma, let $\psi_k$ denote the $k$-th Adams operation of a complex vector bundle. It is known that $\psi_k(E)$ can be constructed from $E$ in an admissible way,[^5] [@Adams_1962aa; @Atiyah_1969gr]. The Chern character of $E$ and $\psi_k(E)$ have the following relation. $$\label{eq:ch-psi} \mathop{\mathrm{ch}}(\psi_k(E)) = \sum_{i=0}^{\dim M/2} \mathop{\mathrm{ch}}_i(E) k^i.$$ The second algebraic lemma we need is as follows. **Lemma 8**. *Let $E$ be a Hermitian vector bundle over $M$. If for $1\le k \le \dim M /2 + 1$, $$\label{eq:ae} \int_M \hat{\mathsf{A}}(M) \mathop{\mathrm{ch}}(\psi_k(E)) = 0,$$ then $\int_M \mathop{\mathrm{ch}}(E) = 0$.* *Remark 9*. In [@Gromov_1996aa p. 36], Gromov uses a result similar to Lemma [Lemma 8](#lm:ch){reference-type="ref" reference="lm:ch"}. But the condition of the lemma, ([\[eq:ae\]](#eq:ae){reference-type="ref" reference="eq:ae"}), is replaced with the condition that for all $k$, $$\int_M \hat{\mathsf{A}}(M) (\mathop{\mathrm{ch}}(E))^k = 0.$$ In our opinion, it is not very straightforward to see why such a condition also yields the same conclusion of Lemma [Lemma 8](#lm:ch){reference-type="ref" reference="lm:ch"}. Our main motivation to write this note is to clarify this point. *Proof of Lemma [Lemma 8](#lm:ch){reference-type="ref" reference="lm:ch"}.* Let $\dim M = 2n$. Denote the degree $2l$ component of $\hat{\mathsf{A}}(M)$ by $\hat{\mathsf{A}}_l(M)$. Note that if $l$ is an odd number, $\hat{\mathsf{A}}_l(M) = 0$. For $0\le i \le n$, let $$a_i \coloneqq \int_M \hat{\mathsf{A}}_{n-i}(M) \mathop{\mathrm{ch}}_i(E)$$ and $\mathbf{a} \coloneqq [a_0, a_1, \cdots, a_n]^{\mathrm{T}}$. Then by ([\[eq:ch-psi\]](#eq:ch-psi){reference-type="ref" reference="eq:ch-psi"}), the condition ([\[eq:ae\]](#eq:ae){reference-type="ref" reference="eq:ae"}) gives $$\label{eq:ai} \mathbf{L}\mathbf{a} = \mathbf{0},\quad \text{where}\quad \mathbf{L}\coloneqq \begin{bmatrix} 1 & 1 & 1 & \cdots & 1 \\ 1 & 2 & 2^2 & \cdots & 2^n \\ 1 & 3 & 3^2 & \cdots & 3^n \\ \vdots& \vdots & \vdots & \ddots & \vdots \\ 1 & n+1 & (n+1)^2 & \cdots & (n+1)^n \end{bmatrix}.$$ Since $\mathbf{L}$ is a Vandermonde matrix, its determinant is $\det{\mathbf{L}} = \prod_{1\le i< j \le n+1}(j-i) \neq 0$. As a result, ([\[eq:ai\]](#eq:ai){reference-type="ref" reference="eq:ai"}) implies $\mathbf{a} = 0$, which means $\int_M \mathop{\mathrm{ch}}(E) = a_n = 0$. ◻ ## Proof of Proposition [Proposition 3](#prop:klea){reference-type="ref" reference="prop:klea"} {#proof-of-proposition-propklea} We use Lemma [Lemma 7](#lm:alg){reference-type="ref" reference="lm:alg"} and Lemma [Lemma 8](#lm:ch){reference-type="ref" reference="lm:ch"} to show this proposition. Take the constant $N$ in Lemma [Lemma 7](#lm:alg){reference-type="ref" reference="lm:alg"} to be $\dim M/2$. Since $\mathcal{C}_N$ is a finite set, by the estimate ([\[eq:c0\]](#eq:c0){reference-type="ref" reference="eq:c0"}), we have $$\label{eq:est-c} \sup_{J\in \mathcal{C}_N} \Vert \mathop{\mathrm{R}}^{J(E)} \Vert \le A_N \Vert \mathop{\mathrm{R}}^{E} \Vert,$$ where $A_N = \sup_{J\in \mathcal{C}_N} C_{J}$ is a constant depending only on $N$. Since there is a nonvanishing Chern number for $E$, due to Lemma [Lemma 7](#lm:alg){reference-type="ref" reference="lm:alg"}, we can find $J_1 \in \mathcal{C}_N$ and $E_1\coloneqq J_1(E)$ such that $$\int_M \mathop{\mathrm{ch}}(E_1) = \int_M\mathop{\mathrm{ch}}_N(E_1) \neq 0.$$ By ([\[eq:ba0\]](#eq:ba0){reference-type="ref" reference="eq:ba0"}) and ([\[eq:est-c\]](#eq:est-c){reference-type="ref" reference="eq:est-c"}), we also have $$\label{eq:est-e1} \Vert \mathop{\mathrm{R}}^{E_1} \Vert \le A_N /m_0.$$ Then by Lemma [Lemma 8](#lm:ch){reference-type="ref" reference="lm:ch"}, there exists an integer $1 \le k_0 \le \dim M/2 + 1$ satisfying $$\int_M \hat{\mathsf{A}}(M) \mathop{\mathrm{ch}}(\psi_{k_0}(E_1)) \neq 0.$$ As we have noted, there exist admissible functors $G_{k_0}^1,G_{k_0}^2$ such that $\psi_{k_0}(E_1)$ is a virtual bundle $G_{k_0}^1(E_1) - G_{k_0}^2(E_1)$. Therefore, the above inequality implies that at least one of $G_{k_0}^1(E_1)$ and $G_{k_0}^2(E_1)$, say $G_{k_0}^1(E_1)$, satisfies $$\int_M \hat{\mathsf{A}}(M) \mathop{\mathrm{ch}}(G_{k_0}^1(E_1)) \neq 0.$$ Using ([\[eq:c0\]](#eq:c0){reference-type="ref" reference="eq:c0"}) again, we have $$\Vert \mathop{\mathrm{R}}^{G_{k_0}^1(E_1)} \Vert \le C_{k_0} \Vert \mathop{\mathrm{R}}^{E_1} \Vert \le \max_{1 \le k_0 \le N +1}C_{k_0} A_N/m_0,$$ where $C_{k_0} = \max(C_{G_{k_0}^1}, C_{G_{k_0}^2})$. Since the functors $G_{k_0}^1, G_{k_0}^2$ depend only on $k_0$, $C_{k_0}$ is also a constant depending only on $k_0$. Therefore, $G_{k_0}^1(E_1)$ satisfies ([\[eq:ca\]](#eq:ca){reference-type="ref" reference="eq:ca"}) and the proof of Proposition [Proposition 3](#prop:klea){reference-type="ref" reference="prop:klea"} is finished. ## Proof of Lemma [Lemma 7](#lm:alg){reference-type="ref" reference="lm:alg"}. {#sub:trivial} As we will see, the proof of Lemma [Lemma 7](#lm:alg){reference-type="ref" reference="lm:alg"} is similar to the proof of Lemma [Lemma 8](#lm:ch){reference-type="ref" reference="lm:ch"} in some sense. Firstly, by [@Gilkey_1995aa Lemma 2.1.6], we know that $\mathop{\mathrm{c}}_i(E)$ can be represented by a homogeneous polynomial of $\{\mathop{\mathrm{ch}}_j(E)\}$ and such a polynomial is independent of the choice of $E$. Therefore, we only need to prove Lemma [Lemma 7](#lm:alg){reference-type="ref" reference="lm:alg"} with ([\[eq:cal\]](#eq:cal){reference-type="ref" reference="eq:cal"}) replaced by the following equality. $$\label{eq:chal} \prod_{l=1}^{\nu} \mathop{\mathrm{ch}}_{a_l}(E) = \sum_i \lambda_i\mathop{\mathrm{ch}}_K(J_i(E)).$$ We will prove this result by using induction on $\nu$. More specifically, for each $\nu$, we construct a finite set $\mathcal{C}_N^\nu$ inductively such that by choosing $J_i\in \mathcal{C}_N^\nu$, ([\[eq:chal\]](#eq:chal){reference-type="ref" reference="eq:chal"}) holds for $\nu$. As a first step, set $\mathcal{C}_N^1 = \{I\}$. By definition, since $I(E) = E$, ([\[eq:chal\]](#eq:chal){reference-type="ref" reference="eq:chal"}) holds for $\nu = 1$. Supposing that for $\nu\ge 1$, we have constructed a finite set $\mathcal{C}_N^{\nu}$ such that by choosing $J_i\in \mathcal{C}_N^\nu$, ([\[eq:chal\]](#eq:chal){reference-type="ref" reference="eq:chal"}) holds for $\nu$. We will construct $\mathcal{C}_N^{\nu+1}$ using $\mathcal{C}_N^{\nu}$. For $K\le N$, we take an abitrary $\nu+1$ partition of $K$, $K= a_0 + \cdots + a_{\nu}$, $a_i\in \mathbb{Z}_+$. Let $K_1 \coloneqq a_1 + \cdots + a_{\nu} \le N$. Using the induction assumption for $\prod_{l=1}^{\nu} \mathop{\mathrm{ch}}_{a_l}(E)$, we can find $\lambda_j\in \mathbb{Q}$ such that $$\label{eq:ch-nu} \mathop{\mathrm{ch}}_{a_0}(E)\prod_{l=1}^{\nu} \mathop{\mathrm{ch}}_{a_l}(E) = \sum_j \lambda_j\mathop{\mathrm{ch}}_{a_0}(E)\mathop{\mathrm{ch}}_{K_1}(J_j(E)),$$ where $J_j\in \mathcal{C}_N^{\nu}$. To deal with the r.h.s. of ([\[eq:ch-nu\]](#eq:ch-nu){reference-type="ref" reference="eq:ch-nu"}), we use the following result, which is a variation of the $\nu=2$ case of Lemma [Lemma 7](#lm:alg){reference-type="ref" reference="lm:alg"}. **Lemma 10**. *Let $i,j\in \mathbb{Z}_{\ge1}$. There is a finite set of admissible functors $\{G_l\}$ and $\lambda_l\in \mathbb{Q}$ such that $$\mathop{\mathrm{ch}}_i(F_1)\mathop{\mathrm{ch}}_j(F_2) = \sum_l \lambda_l\mathop{\mathrm{ch}}_{i+j}(G_l(F_1,F_2))$$ holds for any two Hermitian vector bundles $F_1,F_2$.* *Proof.* Take $H_l = \psi_l(F_1) \otimes F_2$. Then, there exist admissible functors $G^1_l,G^2_l$ such that $H_l = G_{l}^1(F_1,F_2)-G_{l}^2(F_1,F_2)$. Set $r = i+ j$. By ([\[eq:ch-psi\]](#eq:ch-psi){reference-type="ref" reference="eq:ch-psi"}), we have $$\begin{gathered} \mathop{\mathrm{ch}}_r(G_{l}^1(F_1,F_2))-\mathop{\mathrm{ch}}_r(G_{l}^2(F_1,F_2))= \mathop{\mathrm{ch}}_r(H_l) \\ = \sum_{a = 0}^{r} \mathop{\mathrm{ch}}_a(\psi_l(F_1))\mathop{\mathrm{ch}}_{r-a}(F_2) = \sum_{a = 0}^{r} \mathop{\mathrm{ch}}_a(F_1) \mathop{\mathrm{ch}}_{r-a}(F_1) l^a. \end{gathered}$$ Now, by choosing $l\in \{1,\cdots,r+1\}$, we have an invertible Vandermonde matrix as in the proof of Lemma [Lemma 8](#lm:ch){reference-type="ref" reference="lm:ch"}. Therefore, the set $\{G_{1}^1,G_{1}^2,\cdots,G_{r+1}^1,G_{r+1}^2\}$ satisfies the requirement in the lemma. ◻ By Lemma [Lemma 10](#lm:alg2){reference-type="ref" reference="lm:alg2"}, for any $J\in \mathcal{C}^{\nu}_N$ and fixing $a_0,K_1$ such that $a_0\ge 1$, $K_1 \ge 1$ and $a_0 + K_1 \le N$, we can construct finitely many admissible functors $G_{J,a_0,K_1,l}$ such that $$\label{eq:ch-nv1} \mathop{\mathrm{ch}}_{a_0}(E)\mathop{\mathrm{ch}}_{K_1}(J(E)) = \sum_l \lambda'_{l} \mathop{\mathrm{ch}}_{a_0+ K_1}(G_{J,a_0,K_1,l}(E)).$$ Set $$\mathcal{C}_N^{\nu+1} = \mathcal{C}_N^{\nu} \bigcup \{G_{J,a_0, K_1,l}|{J\in\mathcal{C}^{\nu}_N, a_0\ge 1, K_1 \ge 1, a_0+K_1 \le N}\}.$$ By ([\[eq:ch-nu\]](#eq:ch-nu){reference-type="ref" reference="eq:ch-nu"}) and ([\[eq:ch-nv1\]](#eq:ch-nv1){reference-type="ref" reference="eq:ch-nv1"}), by choosing $J_i\in \mathcal{C}_N^{\nu+1}$, ([\[eq:chal\]](#eq:chal){reference-type="ref" reference="eq:chal"}) holds for $\nu+1$. The proof of Lemma [Lemma 7](#lm:alg){reference-type="ref" reference="lm:alg"} is finished. # Two remarks about Theorem [Theorem 1](#thm:com){reference-type="ref" reference="thm:com"} {#sec:so-re} ## Theorem [Theorem 1](#thm:com){reference-type="ref" reference="thm:com"} for more general manifolds {#theorem-thmcom-for-more-general-manifolds} Till now, we assume that the manifold $M$ is closed. Now, we comment briefly on how to extend Theorem [Theorem 1](#thm:com){reference-type="ref" reference="thm:com"} to compact manifolds with boundary or non-compact manifolds. In [@Cecchini_2021sc; @Gromov_1996aa; @Su_2021ma], the authors define the $\mathrm{K}$-cowaist and $\hat{\mathsf{A}}$-cowaist on compact manifolds with boundary or non-compact manifolds. The idea behind these two kinds of generalization is the same. Let us recall the definition in [@Cecchini_2021sc] as an example. Let $M$ be a compact manifold with boundary. To take the effect of boundary into consideration, we choose a pair of Hermitian bundles $E,F$, called a *compatible pair*, such that there is an isomorphism between $E$ and $F$ near $\partial M$ which preserves the metric and the connection on $E$ and $F$.[^6] To replace the condition ([\[eq:cn\]](#eq:cn){reference-type="ref" reference="eq:cn"}), we use the condition that there exists a polynomial $p$ of Chern forms such that $$\label{eq:cn1} \int_M (p(\mathop{\mathrm{c}}_0(E),\mathop{\mathrm{c}}_1(E),\cdots) - p(\mathop{\mathrm{c}}_0(F),\mathop{\mathrm{c}}_1(F),\cdots)) \neq 0.$$ Accordingly, to replace the condition ([\[eq:ahg\]](#eq:ahg){reference-type="ref" reference="eq:ahg"}), we use the condition that $$\label{eq:ahg1} \int_M \hat{\mathsf{A}}(M)(\mathop{\mathrm{ch}}(E) - \mathop{\mathrm{ch}}(F)) \neq 0.$$ Then the $\mathrm{K}$-cowaist (resp. $\hat{\mathsf{A}}$-cowaist) of $M$ is the supremum of $\|\mathop{\mathrm{R}}^{E \oplus F}\|^{-1}$ with respect to all compatible pairs $E,F$ satisfying ([\[eq:cn1\]](#eq:cn1){reference-type="ref" reference="eq:cn1"}) (resp. ([\[eq:ahg1\]](#eq:ahg1){reference-type="ref" reference="eq:ahg1"})). For the case that $M$ is non-compact, these two definitions remain valid if we modify the definition of the compatible pair a little. Namely, in this non-compact case, we require that the isomorphism between $E$ and $F$ is defined outside a compact set of $M$. Note that for different compatible pairs on $M$, the compact set may vary. The method to show Theorem [Theorem 1](#thm:com){reference-type="ref" reference="thm:com"} in fact also works for these more general cases. The key point is that the equality ([\[eq:ch-psi\]](#eq:ch-psi){reference-type="ref" reference="eq:ch-psi"}), although we treat it as a cohomological equality in Section [2](#sec:pf-com){reference-type="ref" reference="sec:pf-com"}, holds at the differential form level. To check this fact, we can use the explicit construction of $\psi_k(E)$ given in [@Adams_1962aa § 4]. Then one can use the same arguments to show that Lemma [Lemma 7](#lm:alg){reference-type="ref" reference="lm:alg"} and Lemma [Lemma 8](#lm:ch){reference-type="ref" reference="lm:ch"} still hold with the vector bundles replaced by the compatible pairs. ## The reverse direction inequality of ([\[eq:klea\]](#eq:klea){reference-type="ref" reference="eq:klea"}) As another remark for Theorem [Theorem 1](#thm:com){reference-type="ref" reference="thm:com"}, we would to like to point out that an inequality like ([\[eq:klea\]](#eq:klea){reference-type="ref" reference="eq:klea"}) in the reverse direction does not hold in general. Let $N$ be a 4-dimensional simply connected closed manifold with $\int_N\hat{\mathsf{A}}(N) \neq 0$ (e.g. a K$3$ surface) and $\mathbb{S}^2(R)$ be the standard 2-dimensional sphere with the radius $R$. Besides, we denote the Hopf bundle over $\mathbb{S}^2(R)$ by $H$. In [@Gromov_1996aa § 4$\frac{1}{4}$], Gromov shows that $\mathop{\mathrm{K-cw_2}}(N) < +\infty$. In fact, by checking the proof of this result, we know that there exists a constant $a$ depending on $N$ such that for any Hermitian vector bundle $L$ over $N$ with $\Vert \mathop{\mathrm{R}}^L\Vert < a$, $L$ must be a topological trivial bundle (with a possible nontrivial metric and connection). Furthermore, the same proof implies that for any Hermitian vector bundle $E$ over $N \times \mathbb{S}^2(R)$ such that $\Vert \mathop{\mathrm{R}}^E\Vert < a$, $E$ must be topologically isomorphic to a pullback bundle from $\mathbb{S}^2(R)$, which implies that $E$ cannot satisfy ([\[eq:cn\]](#eq:cn){reference-type="ref" reference="eq:cn"}). As a result, we know that $$\label{eq:ns1} \mathop{\mathrm{K-cw_2}}(N \times \mathbb{S}^2(R)) \le a^{-1}.$$ On the other hand, we denote the pullback bundle of $H$ over $N \times \mathbb{S}^2$ by $\bar{H}$. We have $$\int_{N \times \mathbb{S}^2(R)} \hat{\mathsf{A}}(N \times \mathbb{S}^2(R)) \mathop{\mathrm{ch}}(\bar{H}) = \int_{N} \hat{\mathsf{A}}{(N)}\int_{\mathbb{S}^2(R)}\mathop{\mathrm{c}}_1(H) \ne 0,$$ that is, $\bar{H}$ satisfies ([\[eq:ahg\]](#eq:ahg){reference-type="ref" reference="eq:ahg"}). However, by direct calculation, $$\Vert\mathop{\mathrm{R}}^{\bar{H}}\Vert = \frac{1}{2R^2}.$$ As a result, $$\label{eq:ns2} \mathop{\mathrm{\Hat{A}-cw_2}}(N \times \mathbb{S}^2(R)) \ge 2R^2.$$ Combining ([\[eq:ns1\]](#eq:ns1){reference-type="ref" reference="eq:ns1"}) and ([\[eq:ns2\]](#eq:ns2){reference-type="ref" reference="eq:ns2"}), we know that ([\[eq:klea\]](#eq:klea){reference-type="ref" reference="eq:klea"}) in the reverse direction does not hold in general. # Acknowledgments The author would like to thank Prof. Guangxiang Su for helpful discussion about the content of this paper and the anonymous referee for reading the paper carefully and the very inspiring suggestions. 1 J. F. Adams, *Vector fields on spheres*, Ann. of Math. (2) **75** (1962), 603--632. MR 0139178 M. F. Atiyah and D. O. Tall, *Group representations, $\lambda$-rings and the $J$-homomorphism*, Topology **8** (1969), 253--297. MR 244387 S. Cecchini and R. Zeidler, *Scalar and mean curvature comparison via the Dirac operator*, arXiv:2103.06833, (2021), to appear in Geometry & Topology. P. B. Gilkey, *Invariance theory, the heat equation, and the Atiyah-Singer index theorem*, second ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1396308 M. Gromov, *Positive curvature, macroscopic dimension, spectral gaps and higher signatures*, Functional analysis on the eve of the 21st century, Vol. II (New Brunswick, NJ, 1993), Progr. Math., vol. 132, Birkhäuser Boston, Boston, MA, 1996, pp. 1--213. MR 1389019 M. Gromov, *Four lectures on scalar curvature*, Perspectives in scalar curvature. Vol. 1, World Sci. Publ., Hackensack, NJ, 2023, pp. 1--514. MR 4577903 G. Su and X. Wang, *$\mathrm{K}$-cowaist on complete foliated manifolds*, arXiv:2107.08354, (2021). [^1]: The author is partially supported by NSFC Grant No. 12101361. [^2]: In [@Gromov_1996aa], $\mathrm{K}$-cowaist was called $\mathrm{K}$-area. But recently, Gromov [@Gromov_2019fv] suggests that $\mathrm{K}$-cowaist should be a more proper name for this concept. [^3]: In [@Cecchini_2021sc], this invariant is called $\hat{\mathsf{A}}$-area. We choose to rename it as $\hat{\mathsf{A}}$-cowaist to be incoordination with the name of $\mathrm{K}$-cowaist. [^4]: *In the following, every Hermitian vector bundle carries a Hermitian connection implicitly.* [^5]: More precisely, $\psi_k(E)$ is a virtual bundle $F_1 - F_2$, where $F_1, F_2$ can be constructed from $E$ in an admissible way respectively. [^6]: In [@Cecchini_2021sc], $E,F$ are called the admissible pair. We rename it to avoid the possible ambiguities with the admissible functors used in this paper.

arxiv_math

--- abstract: | The future power system is increasingly interconnected via both AC and DC interconnectors. These interconnectors establish links between previously decoupled energy markets. In this paper, we propose an optimal multi-market energy storage arbitrage model that includes emergency service provisions for system operator(s). The model considers battery ramping and capacity constraints and utilizes operating envelopes calculated based on interconnector capacity, efficiency, dynamic energy injection and offshore wind generation in the day-ahead market. The arbitrage model considers two separate electricity prices for buying and selling of electricity in the two regions, connected via an interconnector. Using disjunctive linearization of nonlinear terms, we exactly reformulate the inter-regional energy arbitrage optimization as a mixed integer linear programming problem. We propose two capacity limit selection models for storage owners providing emergency services. The numerical analyses focus on two interconnections linking Belgium and the UK. The results are assessed based on revenue, operational cycles, payback period, shelf life and computation times. author: - - bibliography: - reference.bib title: Multi-market Optimal Energy Storage Arbitrage with Capacity Blocking for Emergency Services --- Energy arbitrage, energy market, interconnector, capacity blocking, emergency services. # Introduction With the large-scale adoption of AC and DC interconnections, the power network worldwide is becoming more and more interconnected, creating a new business case for grid-scale energy storage batteries. With the greater amount of interconnections, batteries can now simultaneously participate in more than one energy market and increase their revenue, compared to participating in just one system. Prior use cases for grid-scale batteries [@link2vse; @link3arena; @bowen2019grid; @schoenung2017green] indicate that they are often used for more than one application. One common responsibility of grid-scale batteries is providing emergency services under extreme operating conditions. Such obligations are often contractual and are defined in a bilateral contract or power purchasing agreement (PPA). They may also be enforced via grid code. The goal of this work is to develop a computationally efficient optimization model for facilitating grid-scale battery participation in more than one energy market. Further, we present models for determining the allocation of battery capacity to dedicate to emergency services. It's important to note that grid-scale batteries, often operate under bilateral agreements, which may require the provision of emergency services. Thus, capacity blocking is crucial for assessing their true potential. There exists limited literature on multi-regional energy market participation for storage. Authors in [@hurta2022impact] observed that the bidding zone separation for the German and Austrian day-ahead power markets led to the decoupling of wholesale electricity prices. They explored a scenario where a battery performs energy arbitrage in both German and Austrian day-ahead power markets operated by the Energy Exchange of Austria. Prior works, [@bunn2010inefficient; @soonee2006novel] discussed the benefits of inter-regional energy arbitrage or flows. The authors in [@soonee2006novel] showcase that the energy trade between the northern and the southern grids of India, connected via HVDC back-to-back stations, led to around 85 million euros of savings for the years 2003 to 2005. Authors in [@bunn2010inefficient] highlight the inefficiencies in the context of energy arbitrage in inter-regional electricity transmission, considering the example of the Anglo-French interconnector (IFA). ## Use cases of grid-scale batteries {#sec:usecases} In this part, we elaborate on several use cases involving grid-scale batteries. These cases reveal that batteries often serve multiple objectives, including provisions for emergency services. *Victorian Big Battery*: A case resembling the system under discussion in this paper is exemplified in Australia. The 170 MW Victoria-New South Wales interconnector, is co-owned by AEMO and Transgrid. On the Victorian side, there is a 300 MW, 450 MWh battery situated in Geelong, owned by Neoen. The battery utilization objectives are [@link2vse]:\ $\bullet$ *Energy Market Participation*: the battery engages in the energy market year-round, with 50 MW allocated for summer (Nov. to March) and the entire 300 MW available otherwise.\ $\bullet$ *Capacity blocking for Emergency Services*: To mitigate the risk of unanticipated power cuts during the summer season, the battery blocks 250 MW to act as a \"virtual transmission line\" between Victoria and New South Wales. This is sufficient to power over 1 million Victorian households for 30 minutes. *The Hornsdale Power Reserve:* at 100 MW/129 MWh, has operated in South Australia since December 2017. The battery utilization objectives are:\ $\bullet$ *Energy Market Participation*: 30 MW and 119 MWh capacity are bid directly into the market.\ $\bullet$ *Emergency services*: It provides backup power during grid outages, ensuring stable frequency until slow generators start. This was first demonstrated in 2017 when the battery rapidly injected power into the grid, preventing a potential cascading blackout after a large coal plant tripped [@link3arena; @bowen2019grid]. *Moss Landing battery* was commissioned in Dec 2020 and is owned by Vistra Energy. The capacity of this grid-scale battery is 400MW and 1600 MWh. In addition to energy market participation, it is bounded by a 10-year resource adequacy contract for 100 MW/ 400 MWh with PG&E [@mosslanding]. In contrast to the above examples, Manatee grid-scale battery (409 MW, 900 MWh) is owned and operated by Florida Power and Light (FPL). This utility-owned and operated battery aims to increase reliability during emergency situations like hurricanes. As observed in the above use cases, the operational objectives of the battery may change based on the ownership. In this work, we assume the battery and the interconnector are owned by different entities. This is also in line with EU regulation 2019/943 [@euregulation]. ## Contributions The key contributions of the paper are:\ $\bullet$ *Optimization formulation*: A novel Mixed-Integer Linear Programming (MILP) based formulation for energy storage, performing energy arbitrage in more than one energy market simultaneously. The optimization formulation considers interconnector capacity limits, storage efficiency, interconnector flows and efficiency and also ensures that under no case the battery is charging and discharging simultaneously.\ $\bullet$ *Energy Storage Profitability Assessment*: We provide a comprehensive framework for assessing the profitability of energy storage, including considerations for reserving a fraction of battery capacity for emergency services. This assessment accounts for the battery's operational and shelf life limitations while projecting annual simulated revenue.\ $\bullet$ *Realistic case studies* are performed for (a) [NEMO Link](https://www.nemolink.co.uk/) (case 1), and (b) the set of interconnections between Belgium and the UK connecting the Princess Elisabeth Energy Island (PEEI) in the North Sea (case 2). In these case studies, we (i) perform the sensitivity of interconnector rent on the performance indices, (ii) provide a solution to how much battery capacity can be blocked for emergency services, and (iii) answer whether the battery owners should consider reserving interconnector capacity for inter-regional energy trades or not. This paper is organized as follows. Section [2](#section2){reference-type="ref" reference="section2"}, describes the energy market models for test cases. Section [3](#section3){reference-type="ref" reference="section3"}, details the inter-regional energy arbitrage model as an MILP formulation. Section [4](#section4){reference-type="ref" reference="section4"} extends the inter-regional arbitrage model to consider the interconnector flows and capacity blocking. Section [5](#section5){reference-type="ref" reference="section5"} details the performance indices used for evaluating the simulations. Section [6](#section6){reference-type="ref" reference="section6"} describes the two case studies for interconnections connecting Belgium and the UK. Section [7](#section7){reference-type="ref" reference="section7"} concludes the paper. # Energy market modelling {#section2} In this section, we outline a data-driven model for the Belgian-UK energy markets to assess the potential advantages of an inter-regional arbitrage strategy. The analysis begins by considering a single interconnector and then introduces a concept known as the Hybrid Offshore Asset (HOA), which combines offshore wind power and an interconnector as a shared resource. We explore two distinct case studies based on the Belgian offshore development plan up to 2030 as presented in the Federal Development Plan of the Belgian transmission system 2024-2034 [@elia_dev_plan]. The offshore grid considered is visualized in Fig. [1](#topo_fig){reference-type="ref" reference="topo_fig"}. The capacities of the transmission lines as well as which lines are considered in which test case are detailed in Table [1](#topo_table){reference-type="ref" reference="topo_table"}. {#topo_fig width="6.5in"} Name Transmission Line from to GVA test case ------------- ------------------- ------ ------ ----- ----------- NEMO link HVDC LINE 1 UK BE 1.0 1 Nautilus-UK HVDC LINE 2 UK PEEI 1.4 2 Nautilus-BE HVDC LINE 3 PEEI BE 1.4 2 \- HVAC LINE 1 PEEI BE 2.1 2 : Test case transmission infrastructure. [\[topo_table\]]{#topo_table label="topo_table"} ## Test Case 1: NEMO Link The first case focuses on the existing infrastructure, i.e., the 1 GW NEMO link interconnector connecting Belgium and the UK. The offshore wind generation is modelled as part of the home market, thus it is lumped in with other Belgian generation assets. Historical data from the years 2019 and 2020, sourced from the ENTSO-E transparency platform, are used to model the energy markets in Zone A (Belgium) and Zone B (UK). These years were selected due to the UK's departure from the common energy market at the beginning of 2020, known as \"Brexit\". The aim is to examine whether this event significantly impacts the feasibility of storage participating in inter-regional arbitrage. The generators in zones A and B, denoted as $g_1$ and $g_3$, are assumed to be infinite sources with per-unit production costs represented by $P_{g_3,i}^{\text{\sc a}} \in \Psi^{\text{\sc p:a}}$ and $P_{g_1,i}^{\text{\sc b}} \in \Psi^{\text{\sc p:b}}$, where $\Psi^{\text{\sc p:a}}$ and $\Psi^{\text{\sc p:b}}$ are the historical day-ahead market clearing price time series for Belgium and the UK [@market_clearing], respectively. Each market zone also includes a demand component with hourly values denoted as $D_{l_2,i}^{\text{\sc a}} \in \Psi^{\text{\sc d:a}}$ and $D_{l_1,i}^{\text{\sc b}} \in \Psi^{\text{\sc d:b}}$, where $\Psi^{\text{\sc d:a}}$ and $\Psi^{\text{\sc d:b}}$ correspond to the demand time series for Belgium and the UK [@trans_demand], respectively. Furthermore, interconnector flow at each hour is defined as $L_{i}^{\text{\sc ab}} \in \Psi^{\text{\sc l:ab}}$, where $\Psi^{\text{\sc l:ab}}$ is the time series data of the net physical flows through the Nemo link from Belgium to the UK [@realized_flows]. ## Test Case 2: NEMO Link and the PEEI HOA In the second case, we introduce the 3.5 GW PEEI HOA, an energy island connected to both the UK and Belgium via the Nautilus link, a 1.4 GW HVDC interconnector. Additionally, the energy island is linked to Belgium through a 2.1 GW HVAC connection. To estimate energy production at the PEEI Offshore Wind Power Plant (OWPP), we employ the CorRES software [@Koivisto2019], which utilizes historical meteorological data from the WRF database [@WRF_data] and introduces stochastic fluctuations and forecast errors over top to capture the inherent variability and uncertainties in renewable energy generation. To ensure temporal compatibility with the existing data, simulations for years 2019 and 2020 are performed in CorRES. With the addition of the PEEI HOA, historical data alone becomes insufficient to model the energy markets. To address this, we introduce additional generators ($g_2$ and $g_4$) to each zone, as illustrated in Fig. [1](#topo_fig){reference-type="ref" reference="topo_fig"}. The hourly peak production of these assets is calculated based on the difference between hourly demand and a fixed \"block size\" parameter, assumed to be 1 GW. This parameter models the step-wise addition or removal of gas turbine-generating blocks from the energy market mix [@block_size]. As gas turbines often represent the marginal generating unit [@gas_is_marginal], a reduction in generation of one block size due to offshore wind generation is assumed to shift the marginal pricing generator from gas to wind. The price of offshore wind results from an assumed bidding strategy of offering at 95% of the Belgian energy market price. This idealized approach results in minimal curtailment for the OWPP while ensuring a healthy profit even when the OWPP becomes the marginal price-setting unit. The hourly pricing of generators $g_2$ and $g_4$ are therefore set at $P_{g_4,i}^{\text{\sc a}}=0.95 P_{g_3,i}^{\text{\sc a}} \in \Psi^{\text{\sc p:a}}$ and $P_{g_2,i}^{\text{\sc b}}=0.95 P_{g_1,i}^{\text{\sc b}} \in \Psi^{\text{\sc p:b}}$. Under these conditions, we calculate energy market prices by solving the optimal dispatch problem. This simulation provides estimates of both the market clearing prices and interconnector power flows after the addition of the PEEI. The market clearing prices for Belgium, the UK and the PEEI are determined from the dual variables ($\lambda_{i}^{\text{\sc be}}$, $\lambda_{i}^{\text{\sc ei}}$ and $\lambda_{i}^{\text{\sc uk}}$) of the nodal power balance equations, often referred to as the shadow prices. ## Flow-based energy market coupling Since the early 2010s, the EU has been working towards single-day-ahead market coupling across the EU via a gradual implementation of flow-based market coupling. Flow-based market coupling takes into account the physical constraints of power flows in interconnected grids, allowing for more efficient and reliable energy trading between different regions. It aims to maximize market efficiency and minimize congestion-related issues in the electricity grid [@entsoE_flow_based]. On January 31, 2020, the UK formally exited the European Union's common energy market, a development commonly referred to as \"Brexit\". Following Brexit, a provisional agreement governing energy markets between the EU and the UK went into effect. The provisional agreement is valid until 2026 [@UK_EU_market_agreement]. This agreement essentially maintained the pre-Brexit status quo, inhibiting any further progress towards the objective of flow-based market coupling. While the energy market landscape beyond 2026 remains uncertain, there is optimism that an agreement can be reached, and the transition towards flow-based market coupling will resume. Consequently, in this study, we operate on the assumption that full implementation of flow-based market coupling will be achieved by the anticipated completion date of the PEEI project in 2030. This assumption carries the implication of improved and more efficient utilization of interconnector capacity in the second test case in the numerical results compared to the first. # Battery for inter-regional arbitrage {#section3} Given the grid configuration, a Battery Energy Storage System (BESS) can be strategically located at one end to capitalize on pricing disparities between the two grids during charge and discharge cycles. Furthermore, this battery can serve additional functions, including ancillary services and enhancing grid reliability. For the purposes of this study, we assume that the battery is located on the Belgian side of the interconnector and engages exclusively in inter-regional energy arbitrage. The battery is assumed to be a price-taker of electricity. ## Battery model The battery model considers the ramping constraint, and the capacity constraint along with charging and discharging efficiencies denoted by $\eta_{\text{ch}}, \eta_{\text{dis}} \in (0,1]$, respectively. The energy optimization considers a change in energy levels of the battery at time $i$ denoted as $x_i$. Change in battery energy level at $i$ is defined as $x_i = h \delta_i$, where $\delta_i \in [\delta_{\min}, \delta_{max}]$ $\forall i$ denotes ramp rate of the battery. $h$ denotes the sampling period. $\delta_i> 0$ when the battery is charging and vice versa. Note, $\delta_i$ is in units of power (MW) and $x_i$ is in units of energy (MWh). The battery charge level is denoted as $$b_i = b_{i-1} + x_i, \quad b_i\in [b_{\min},b_{\max}], \forall i, \label{eq:cap}$$ where $b_{\min}, b_{\max}$ are the minimum and maximum battery capacity within which the battery should be operating. The power consumed by the battery at time $i$ is denoted as $$f(x_i)= \frac{[x_i]^+}{h \eta_{\text{ch}}} - \frac{\eta_{\text{dis}}[x_i]^-}{h}=\frac{\max(0,x_i)}{h\eta_{\text{ch}}} - \frac{\eta_{\text{dis}}\max(0,-x_i)}{h}, \label{finverse}$$ where $x_i$ must lie in the range from $X_{\min}=\delta_{\min}h$ to $X_{\max}={\delta_{\max}h}$. The ramping constraint is given as $$x_i \in [X_{\min}, X_{\max}]. \label{eq:ramp}$$ The battery is interfaced via an inverter. Let the inverter's efficiency be denoted as $\eta_{\text{inv}} \in (0,1]$. The modified battery charging and discharging efficiency is denoted as $$\begin{gathered} \eta_{\text{ch}}^* = \eta_{\text{ch}}\eta_{\text{inv}},~~~~ \eta_{\text{dis}}^* = \eta_{\text{dis}}\eta_{\text{inv}}.\end{gathered}$$ ## Inter-regional energy arbitrage model In this section, we formulate an optimal energy arbitrage model for energy storage participating in inter-regional energy arbitrage in two regions, shown as grid A and B in Fig. [1](#topo_fig){reference-type="ref" reference="topo_fig"}. Since both grids A and B have separate energy markets, the price levels for consumption or injection vary, this is because the size of the interconnector connecting the two grids is small compared to cumulative power needs in grid A and B[^1]. $P_i^{\text{\sc b,a}}$ and $P_i^{\text{\sc s,a}}$ denote the buying (consumption) and selling (injection) prices of electricity in grid A for time instant $i$, respectively. Similarly, $P_i^{\text{\sc b,b}}$ and $P_i^{\text{\sc s,b}}$ denote the buying and selling prices of electricity in grid B for time instant $i$, respectively. As shown in Fig. [1](#topo_fig){reference-type="ref" reference="topo_fig"}, the battery is located in grid A. The efficiency of AC or DC interconnector linking grid A and B is given as $\eta_{\text{line}} \in (0,1]$[^2]. In this work, we assume that the energy storage and the interconnector are owned by different entities, thus, the interconnector owner levies a rent of $\zeta_i > 0, \forall i$ on the energy storage. The unit of $\zeta_i$ is in euros per MWh. The buying price seen by the battery in grid A for charging from grid B is given as $$\widetilde{P}_i^{\text{\sc b,b}} = \frac{P_i^{\text{\sc b,b}} + \zeta_i}{\eta_{\text{line}}}. \label{eqbuypriceredef}$$ The selling price seen by the battery in grid A for discharging from grid B is given as $$\widetilde{P}_i^{\text{\sc s,b}} = {(P_i^{\text{\sc s,b}} - \zeta_i)}{\eta_{\text{line}}}. \label{eqsellpriceredef}$$ Note from [\[eqbuypriceredef\]](#eqbuypriceredef){reference-type="eqref" reference="eqbuypriceredef"} that the adjusted buying price seen in grid A increases due to interconnector line losses and interconnector rent, while the adjusted selling price denoted in [\[eqsellpriceredef\]](#eqsellpriceredef){reference-type="eqref" reference="eqsellpriceredef"} decreases due to interconnector line losses and interconnector rent. ### K1: Using either Grid A or B and not both This case uses either one of the power markets and not both simultaneously. The objective function for the battery in this case is given as $$\min \sum \Big\{ \min (P_i^{\text{\sc b,a}}, {\widetilde{P}_{i}^{\text{\sc b,b}}})\frac{[x_i]^+}{\eta_{\text{ch}}} - \max(P_i^{\text{\sc s,a}}, \widetilde{P}_{i}^{\text{\sc s,b}})[x_i]^-\eta_{\text{dis}} \Big\} \label{eqobjcase1}$$ [\[eqobjcase1\]](#eqobjcase1){reference-type="eqref" reference="eqobjcase1"} can be solved using linear programming formulation proposed in [@hashmi2019optimal; @hashmi2019optimization]. This case assumes that both grids A and B are fully available to charge or discharge, which may not be true. ### K2: Multi-market Charging and Discharging Cost functions for operating the battery using grids A and B are given by $$\begin{aligned} & C_i^{\text{\sc a}} = P_i^{\text{\sc b,a}} \frac{[x_{i}^{\text{ \sc a}}]^+}{\eta_{\text{ch}}^*} - P_i^{\text{\sc s,a}} [x_{i}^{\text{ \sc a}}]^- \eta^*_{\text{dis}}, \\ & C_i^{\text{\sc b}} = \widetilde{P}_{i}^{\text{\sc b,b}} \frac{[x_{i}^{\text{ \sc b}}]^+}{\eta^*_{\text{ch}}} - \widetilde{P}_{i}^{\text{\sc s,b}} [x_{i}^{\text{ \sc b}}]^- \eta^*_{\text{dis}}, \ \text{respectively},\end{aligned}$$ where $x_{i}^{\text{ \sc a}}, x_{i}^{\text{ \sc b}} \in [X_{\min}, X_{\max}]$ denote the change in battery charge level due to grid A and B, respectively. Thus, the battery ramping constraint is redefined as $$(x_{i}^{\text{ \sc a}} + x_{i}^{\text{ \sc b}}) \in [X_{\min}, X_{\max}]. \label{eq:ramp2a}$$ Note that it is crucial that $x_{i}^{\text{ \sc a}}$ and $x_{i}^{\text{ \sc b}}$ have the same sign, i.e., the battery can either charge or discharge at any given moment, but not both simultaneously. This requirement can be modelled via the constraint, $$x_{i}^{\text{ \sc a}} \cdot x_{i}^{\text{ \sc b}} \geq 0 \quad \forall i. \label{eqbilinear}$$ As it is, constraint [\[eqbilinear\]](#eqbilinear){reference-type="eqref" reference="eqbilinear"} is intractable, as it involves a bilinear term and its feasible region is also non-convex. Conventional approaches for handling bilinear terms, such as the McCormick relaxation [@mccormick1976computability], are not suitable as the relaxation can be very weak and provide impractical simultaneous charging and discharging solutions for the battery. Therefore, we instead choose to exactly reformulate the nonlinear constraint [\[eqbilinear\]](#eqbilinear){reference-type="eqref" reference="eqbilinear"} into a set of linear constraints, via the following crucial observation. For every time period, $i$, in the space of variables $(x_{i}^{\text{ \sc a}}, x_{i}^{\text{ \sc b}})$, the feasible region is either the non-negative orthant ($x_{i}^{\text{ \sc a}} \geqslant 0, x_{i}^{\text{ \sc b}} \geqslant 0$) or the non-positive orthant ($x_{i}^{\text{ \sc a}} \leqslant 0, x_{i}^{\text{ \sc b}} \leqslant 0$). Since this feasible region is piecewise convex [@nagarajan2019adaptive; @yang2022optimal], we can equivalently reformulate it to capture the disjunctive union of these two regions by introducing two binary variables $z_i^{\text{ch}}, z_i^{\text{dis}} \in \{0,1\}$, and by satisfying the following inequalities $\forall i$: $$\begin{aligned} & x_{i}^{\text{ \sc a}} \geqslant z_i^{\text{ch}} \cdot X_{\min}, \quad x_{i}^{\text{ \sc a}} \leqslant z_i^{\text{dis}} \cdot X_{\max},\\ & x_{i}^{\text{ \sc b}} \geqslant z_i^{ch} \cdot X_{\min}, \quad x_{i}^{\text{ \sc b}} \leqslant z_i^{\text{dis}} \cdot X_{\max},\\ & z_i^{\text{ch}} + z_i^{\text{dis}} = 1.0. \end{aligned}$$ [\[eq:linearization\]]{#eq:linearization label="eq:linearization"} ## Epigraph-based mathematical formulation The inter-regional energy arbitrage problem discussed so far for K2 is in the form of a constrained minimization problem of a convex piecewise linear cost function with linear constraints and binary variables. This is given as: l C l [\[eq:original_formulation\]]{#eq:original_formulation label="eq:original_formulation"} &,& [\[eqoriOBJ\]]{#eqoriOBJ label="eqoriOBJ"}\ & [\[eq:cap\]](#eq:cap){reference-type="eqref" reference="eq:cap"}, [\[eq:ramp2a\]](#eq:ramp2a){reference-type="eqref" reference="eq:ramp2a"}, [\[eqbilinear\]](#eqbilinear){reference-type="eqref" reference="eqbilinear"} &\ A problem of this form can be transformed into an equivalent MILP formulation by forming the epigraph problem as in: l C l [\[eq:epi_formulation\]]{#eq:epi_formulation label="eq:epi_formulation"} & &\ , & & [\[eqoriOBJ_epi\]]{#eqoriOBJ_epi label="eqoriOBJ_epi"}\ &&\ & &\ P_i\^ t\_i\^, & &    i, [\[eq14\]]{#eq14 label="eq14"}\ P_i\^ x\_i\^ \^\*\_ t\_i\^, & &    i, [\[eq15\]]{#eq15 label="eq15"}\ & &\ \_i\^ t\_i\^, & &    i, [\[eq16\]]{#eq16 label="eq16"}\ \_i\^ x\_i\^ \^\*\_ t\_i\^, & &    i, [\[eq17\]]{#eq17 label="eq17"}\ & &\ x\_i\^ ,& &    i,\ x\_i\^ ,& &    i, [\[eq:rampxB\]]{#eq:rampxB label="eq:rampxB"}\ (x\_i\^ + x\_i\^) ,& &    i, [\[eq:ramp2\]]{#eq:ramp2 label="eq:ramp2"}\ & &\ \_j=1\^i {x\_j\^ + x\_j\^} b\_-b_0, & &   i, [\[eq20\]]{#eq20 label="eq20"} [\[eqcap1\]]{#eqcap1 label="eqcap1"}\ -\_j=1\^i {x\_j\^ + x\_j\^} b_0- b\_, & &   i, [\[eq21\]]{#eq21 label="eq21"} [\[eqcap2\]]{#eqcap2 label="eqcap2"}\  [\[eq:linearization\]](#eq:linearization){reference-type="eqref" reference="eq:linearization"}   x\_i\^x\_i\^,  i & &\ Re-formulating the problem in this manner is beneficial, as $P_{\text{MILP}}$ is composed of two epigraphs for the two regions, for each time instance. This decouples the storage participating in two energy markets simultaneously. Thus, this formulation can be easily modified for more than two energy markets simultaneously, provided [\[eqbilinear\]](#eqbilinear){reference-type="eqref" reference="eqbilinear"} is extended for the desired number of market zones. However, the focus of this work would be to assess two energy markets. In the next section, we extend $P_{\text{MILP}}$ to consider interconnector flows and offshore wind injections while considering the interconnector capacity. # Interconnector flow & Capacity Blocking {#section4} AC or DC interconnections linking two separate energy markets have a limited capacity, denoted as $L_{\max}$. But the inter-regional energy arbitrage formulation, $P_{\text{MILP}}$, assumes that the AC or DC interconnector is not limiting the battery charging and discharging, as shown in [\[eq:rampxB\]](#eq:rampxB){reference-type="eqref" reference="eq:rampxB"}. Thus, $P_{\text{MILP}}$ assumes $\max(|X_{\min}|,X_{\max}) \leq L_{\max}$ and interconnector capacity is assumed to be available for all time instances. This may not be true, as the interconnectors are often used for many other purposes (renewable energy curtailment reduction, security of supply, energy trade, ancillary services [@kaushal2019overview] etc.) and may not be fully available. In this section, we develop *operating envelopes* for battery optimization based on the interconnector flows, and the offshore wind injections (OWI). ## Impact of interconnector flow: test case 1 We denote the flow in the NEMO link as $L^{\text{\sc ab}}_i$, where $L^{\text{\sc ab}}_i>0$ implies power injected from Belgium to the UK and vice versa. The power flow in the interconnector might affect the battery's ability to charge or discharge. Due to the flow, [\[eq:rampxB\]](#eq:rampxB){reference-type="eqref" reference="eq:rampxB"} needs to be updated considering the interconnector flow. The operating envelopes are defined as follows: $$X_{\max}^{\text{adj}} = \begin{cases} \max\big(0, \min(X_{\max}, L_{\max} + L^{\text{\sc ab}}_i)\big), \text{~if $L^{\text{\sc ab}}_i < 0$}, \\ X_{\max}, \text{~if $L^{\text{\sc ab}}_i \geq 0$}. \end{cases} \label{eqlimmax}$$ $$X_{\min}^{\text{adj}} = \begin{cases} X_{\min}, \text{~if $L^{\text{\sc ab}}_i < 0$}, \\ \min\big(0, \max(X_{\min}, -L_{\max} + L^{\text{\sc ab}}_i)\big), \text{~if $L^{\text{\sc ab}}_i \geq 0$}. \end{cases} \label{eqlimmax}$$ [\[eq:rampxB\]](#eq:rampxB){reference-type="eqref" reference="eq:rampxB"} thus gets modified into $$x_{i}^{\text{ \sc b}} \in [X_{\min}^{\text{adj}}, X_{\max}^{\text{adj}}], ~\forall~ i.$$ ## Impact of interconnector: test case 2 The Belgian and the UK transfer capacities to the PEEI have a capacity of 3.5 and 1.4 GW respectively. The different line limits are denoted as $L_{\max}^{\text{\sc be}}$ and $L_{\max}^{\text{\sc uk}}$. The OWPP at the PEEI affects the flows in the lines towards Belgium and the UK. The flow in these lines are denoted as $L^{\text{\sc be}}_i$ and $L^{\text{\sc uk}}_i$. Fig. [2](#fig:casegen){reference-type="ref" reference="fig:casegen"} shows the stylized diagram showcasing the interconnector flow and OWPP. {#fig:casegen width="6.6in"} The operating envelope of the interconnector including the OWPP is the intersection of the operating regions for Belgium and the UK. The operating region at time $i$ in the Belgian portion of the interconnector is given as $$X_{\max}^{\text{adj, BE}} = \begin{cases} \max\big(0, \min(X_{\max}, L_{\max}^{\text{\sc be}} + L^{\text{\sc be}}_i)\big), \text{~if $L^{\text{\sc be}}_i < 0$}, \\ X_{\max}, \text{~if $L^{\text{\sc be}}_i \geq 0$}. \end{cases} \label{eqlimmax}$$ $$X_{\min}^{\text{adj, BE}} = \begin{cases} X_{\min}, \text{~if $L^{\text{\sc be}}_i < 0$}, \\ \min\big(0, \max(X_{\min}, -L_{\max}^{\text{\sc be}} + L^{\text{\sc be}}_i)\big), \text{~if $L^{\text{\sc be}}_i \geq 0$}. \end{cases} \label{eqlimmax}$$ The operating region at time $i$ in the UK portion of the interconnector is given as $$X_{\max}^{\text{adj, UK}} = \begin{cases} \max\big(0, \min(X_{\max}, L_{\max}^{\text{\sc uk}} + L^{\text{\sc uk}}_i)\big), \text{~if $L^{\text{\sc uk}}_i < 0$}, \\ X_{\max}, \text{~if $L^{\text{\sc uk}}_i \geq 0$}. \end{cases} \label{eqlimmax}$$ $$X_{\min}^{\text{adj, UK}} = \begin{cases} X_{\min}, \text{~if $L^{\text{\sc uk}}_i < 0$}, \\ \min\big(0, \max(X_{\min}, -L_{\max}^{\text{\sc uk}} + L^{\text{\sc uk}}_i)\big), \text{~if $L^{\text{\sc uk}}_i \geq 0$}. \end{cases} \label{eqlimmax}$$ [\[eq:rampxB\]](#eq:rampxB){reference-type="eqref" reference="eq:rampxB"} in $P_{\text{MILP}}$ thus gets modified into $$x_{i}^{\text{ \sc b}} \in [X_{\min}^{\text{adj, \sc be}}, X_{\max}^{\text{adj, \sc be}}] \cap [X_{\min}^{\text{adj, \sc uk}}, X_{\max}^{\text{adj, \sc uk}}] , ~\forall~ i.$$ ## Capacity blocking for emergency services Based on the use cases detailed in Section [1.1](#sec:usecases){reference-type="ref" reference="sec:usecases"}, we model the capacity blocking as shown in Fig. [3](#fig:stack){reference-type="ref" reference="fig:stack"}. Capacity blocking typically involves reserving a certain portion of the energy storage system's capacity for specific use cases, such as emergency services. This means that a predetermined amount of the battery's energy would be kept in reserve to ensure that critical services have access to power during emergencies or grid disruptions. {#fig:stack width="2.99in"} For $P_{\text{MILP}}$, capacity blocking for emergency services are modeled by replacing $b_{\max}$ and $b_{\min}$ by $b_{\max}^{'}$ and $b_{\min}^{'}$, respectively, in [\[eqcap1\]](#eqcap1){reference-type="eqref" reference="eqcap1"} and [\[eqcap2\]](#eqcap2){reference-type="eqref" reference="eqcap2"}. The total blocked battery is denoted as $$b_{\text{block}} = (b_{\max} - b_{\max}^{'}) + (b_{\min}^{'} - b_{\min}).$$ # Performance indices {#section5} The performance indices, detailed in the following subsections, are used to evaluate the numerical simulations. In this work, we define the interconnector utilization factor in percentage is defined as $$\texttt{UF}= 100 \times \frac{\sum_{i=1}^T |L^{\text{\sc ab}}_i|}{L_{\text{lim}} T}, i \in \{1,..,T\}.$$ ## Revenue from arbitrage The revenue from performing energy arbitrage in multiple markets is denoted as $R = -\sum_i^N \{C_i^{\text{\sc a}} ~ + ~C_i^{\text{\sc b}}\}.$ ## Cycles of operation of the battery The life of the battery is often defined in terms of cycle life and calendar life. The cycle life is determined by the operational cycles of charging and discharging. The relationship between the cycle of operation and depth of discharge is not a linear one. In this work, we utilize Algorithm 1 in [@hashmi2018long] for calculating the cycles of operation. The input to this algorithm is the charge and discharge trajectory for the whole year calculated by solving $P_{\text{MILP}}$. The cycle counting of a battery is crucial for assessing the financial viability of installing a battery. ## Simple payback period A simple payback period is the number of years required to recover the initial investment. The interest and operational cost is not considered. The revenue for 1 year of simulation is used for calculating a simple payback period denoted as $$\text{SPP} = \frac{\text{Investment cost}}{\text{1 year of revenue}}.$$ ## Tuning optimal value blocking level Finding the ideal battery capacity that can be blocked is a critical question. We present two models for calculating the ideal value of $b_{\text{block}}$. ### M1: Knee point The knee point of the plot of SPP versus $b_{\text{block}}$ (Fig. [9](#fig:stack1){reference-type="ref" reference="fig:stack1"}) is calculated using [@kneepoint]. ### M2: Based on the calendar life For Fig. [9](#fig:stack1){reference-type="ref" reference="fig:stack1"}, the capacity corresponding to the calendar life of the battery is selected as the desired value of $b_{\text{block}}$. ## Computation time For the simulations, the computation time is calculated using 1000 Monte Carlo simulation runs. Simulations are performed on HP Intel(R) Core(TM) i7 CPU, 1.90GHz, 32 GB RAM personal computer on Matlab 2021a. # Numerical case study {#section6} The two case studies first introduced in Section [2](#section2){reference-type="ref" reference="section2"} are now presented. The battery characteristics considered in these case studies are detailed in Tab. [2](#tab:batparameters){reference-type="ref" reference="tab:batparameters"}. Attributes Value ---------------------------------------------- --------------- Cost 100 euros/kWh Rated capacity ($b_{\max}$) 1 MWh Minimum operational capacity ($b_{\min}$) 0.1 MWh Max charging rate ($\delta_{\max}$) 0.5 MW Min discharging rate ($\delta_{\min}$) -0.5 MW Charging efficiency ($\eta_{\text{ch}}$) 0.95 Discharging efficiency ($\eta_{\text{dis}}$) 0.95 Converter efficiency ($\eta_{\text{conv}}$) 0.95 Initial charge level ($b_0$) 0.5 MWh Cycle life (100% DoD) 7200 Calendar life 10 years : Battery parameter used for simulations [\[tab:batparameters\]]{#tab:batparameters label="tab:batparameters"} As mentioned, a data-driven market model is used based on the years 2019 and 2020. It is observed, however, that there are missing dates in the price and interconnector flow time series. To clean the data, any day with 2 or more consecutive hours of missing data (i.e. 2 samples) is ignored from the analysis. Single missing hours are replaced by the average of the previous and subsequent values. Applying this rule, we received 357 days of data for 2019 and 331 days for 2020. The suboptimal model presented in [\[eqobjcase1\]](#eqobjcase1){reference-type="eqref" reference="eqobjcase1"} is not evaluated, as the optimality gap observed between K1 denoted in [\[eqobjcase1\]](#eqobjcase1){reference-type="eqref" reference="eqobjcase1"} generates 24.2% less revenue compared to $P_{\text{MILP}}$ model. Further, we cannot consider interconnector flows in [\[eqobjcase1\]](#eqobjcase1){reference-type="eqref" reference="eqobjcase1"}. Note that the arbitrage mechanism proposed in this work relies on the convexity of the problem. The formulated optimization is no longer convex for negative electricity price levels. However, negative prices are not frequent. For 2019, the electricity prices were negative for 50 and 1 hourly instances for Belgium and the UK respectively. This consists of 0.3% of the total horizon. For 2020, 136 and 91 hourly instances of negative prices were observed for Belgium and the UK respectively, consisting of 1.4%. Since the negative prices are not frequent, the price level used for evaluating the numerical simulation is saturated at the lower limit of zero. The mean electricity prices for 2019 are 37.9 and 47.24 euros/MWh for Belgium and the UK[^3]. The mean electricity prices for 2020 are 31.28 and 40.79 euros per MWh for Belgium and the UK. For traceability of numerical results, we assume that $\zeta_i$ is constant for all time instances. Fig. [\[fig:price\]](#fig:price){reference-type="ref" reference="fig:price"} compares the electricity prices in Belgium and the UK for the years 2019 and 2020. Note that pre- and post-BREXIT, the electricity prices have similar trends, where the prices in the UK are more often higher than in Belgium. This is evident from the fact that most price instances are higher than the red line. The red line denotes the hypothetical line where prices in the two regions are equal. This skewness in the electricity price motivates us to explore the possibility of utilizing energy storage batteries in the two energy markets. The observation made in Fig. [\[fig:price\]](#fig:price){reference-type="ref" reference="fig:price"} is that the interconnector is predominantly used to inject power from Belgium and extract it in the UK. The electricity price distribution for the years 2019 and 2020 is similar. The MILP-based inter-regional energy arbitrage, $P_{\text{MILP}}$, code described in this paper, is publicly available at [github.com/umar-hashmi/Inter-Refional-Energy-Arbitrage](https://github.com/umar-hashmi/Inter-Refional-Energy-Arbitrage) [@gitcode]. ## Case study 1: NEMO Link The NEMO link connects mainland Belgium with the UK grid via the North Sea. The undersea cable length is 140 km long. The capacity of the line is 1 GW, with line losses in the order of 2.5% [@linknemoloss]. The battery is assumed to be placed at the Belgian end of the NEMO link. We have two objectives in performing this case study: 1. A sensitivity analysis on the interconnector rent towards performance indices, and 2. Selecting $b_{\text{block}}$ for emergency services: Evaluating M1, M2. To accomplish this, we consider three scenarios: - **C1 (Lower bound on benefits)**: considers storage participating in only the local Belgian grid. - **C2 (Upper bound on benefits)**: considers the storage participating in both the Belgian and the UK energy markets without any constraint on charging and discharging due to interconnector unavailability. - **C3**: considers the storage participating in both grids, while also considering the interconnector capacity constraints. The battery revenue for C3 should be lower than or at best equal to C2. ### **Objective 1: Sensitivity towards interconnector rent** The interconnector auctioned the capacity and levied a charge for its utilization proportional to the power level [@linkAUCTION]. Fig. [4](#fig:div){reference-type="ref" reference="fig:div"} shows the separated revenue for inter-regional arbitrage. Note that the battery utilizes the Belgian grid for charging and almost entirely discharges into the UK grid. This is expected due to the price difference bias between these regions also highlighted in Fig. [\[fig:price\]](#fig:price){reference-type="ref" reference="fig:price"}. {#fig:div width="6.9in"} Fig. [5](#fig:res1){reference-type="ref" reference="fig:res1"}, [6](#fig:res2){reference-type="ref" reference="fig:res2"}, [7](#fig:res3){reference-type="ref" reference="fig:res3"}, and [8](#fig:res4){reference-type="ref" reference="fig:res4"} shows the performance indices for models C1, C2 and C3 with different levels of interconnector rent depicted on the *x*-axis. {#fig:res1 width="5.8in"} Fig. [5](#fig:res1){reference-type="ref" reference="fig:res1"} (a) and (b) show the profit due to the battery for the scenarios C1, C2 and C3 for the years 2019 and 2020, respectively. Note that model C1 is independent of interconnector rent, as in this case, the battery considers only the Belgian grid. Also, C2 which does not consider interconnector flows generates a higher profit compared to C3. However, the disparity in scenarios C2 and C3 decreases as the interconnector rent increases, thus, making it not profitable to use the interconnector at all. Fig. [5](#fig:res1){reference-type="ref" reference="fig:res1"} (c) shows the impact of interconnector flows on the profitability of the battery. For small values of interconnector rent, the impact of interconnector congestion on profitability exceeds 20% and 24% for the years 2019 and 2020, respectively. {#fig:res2 width="5.8in"} Fig. [6](#fig:res2){reference-type="ref" reference="fig:res2"} shows the total number of 100% DoD cycles the battery performed for the years 2019 and 2020. The cycles of operation for models C2 and C3 converge to C1 for high levels of interconnector rent, as it is no longer profitable to use the interconnector for energy exchange. {#fig:res3 width="5.7in"} Fig. [7](#fig:res3){reference-type="ref" reference="fig:res3"} illustrates the simple payback period computed based on profits from the years 2019 and 2020. Notably, for lower interconnector rent values, the simple payback period for the C3 case can be as short as 8 years, indicating the potential profitability of the battery system (as described in Table [2](#tab:batparameters){reference-type="ref" reference="tab:batparameters"}). It is important to emphasize that these calculations do not account for additional revenue streams that the battery could generate by participating in additional market products within both regions. These findings hold promise for prospective battery owners. However, it is worth noting that as more batteries are integrated into the grid, the marginal value of installing a battery is expected to decline. This reduction in value is attributed to the decrease in electricity price volatility when there is an increased level of energy storage in the system, a phenomenon discussed in [@hashmi2018effect]. Fig. [8](#fig:res4){reference-type="ref" reference="fig:res4"} plots the number of cycles the battery would require to reach its payback investment. The black line shows the cycle life of the battery. Observe that for the year 2019, the battery inter-regional arbitrage is not profitable for model C3 for an interconnector rent of more than 13 euros/MW. For model C1, the battery is not profitable for the year 2019. However, for the year 2020, model C1 is profitable. This is due to the price volatility, which is higher in the year 2020 compared to 2019. The price variance for the year 2020 was 287.6 and 401.9 in Belgium and the UK. However, for the year 2019, it was 137.7 and 139.1 for Belgium and the UK; substantially less than in 2020. {#fig:res4 width="5.7in"} ### **Objective 2: Capacity blocking** In this part of the case study related to the NEMO link, we evaluate two models for selecting capacity blocking level, $b_{\text{block}}$. Tab. [\[tab:capblock\]](#tab:capblock){reference-type="ref" reference="tab:capblock"} lists the key outcomes for models M1 and M2 regarding the capacity blocked, SPP, and cycles of operation. cell21 = r=4, cell22 = r=2, cell42 = r=2, cell61 = r=4, cell62 = r=2, cell82 = r=2, vlines, hline1-2,6,10 = -, hline3,5,7,9 = 3-5, hline4,8 = 2-5, Years & Model & Metric & C2 & C3\ & M1 (SPP = 10) & $b_{\text{block}}$ & 0.27 & 0.04\ & (SPP = 10) & 100% DoD cycles & 6677 & 6666\ & M2 ($b_{\text{block}}$=0.4) & Payback (years) & 11.26 & 13.64\ & & 100% DoD cycles & 6295 & 6780\ 2020 & M1 (SPP = 10) & $b_{\text{block}}$ & 0.41 & 0.12\ & (SPP = 10) & 100% DoD cycles & 5993 & 6399\ & M2 ($b_{\text{block}}$=0.4) & Payback (years) & 9.81 & 12.86\ & & 100% DoD cycles & 6005 & 6676 [\[tab:capblock\]]{#tab:capblock label="tab:capblock"} Fig. [9](#fig:stack1){reference-type="ref" reference="fig:stack1"} plots the SPP and $b_{\text{block}}$ for C2 and C3 models. Note that due to limited battery life, the $b_{\text{block}}$ is lower for M1 compared to M2. For the year 2019 and 2020, the $b_{\text{block}}$ for C3 model are 4 and 12%. {#fig:stack1 width="5.59in"} Fig. [10](#fig:stack2){reference-type="ref" reference="fig:stack2"} plots the 100% DoD cycles the battery performed versus $b_{\text{block}}$. Note that as the battery capacity is reserved for other emergency applications, the number of cycles the battery is performing due to inter-regional arbitrage reduces. {#fig:stack2 width="5.29in"} The key takeaways of this case study are: $\bullet$ *Interconnector Flow:* When evaluating the financial feasibility of energy storage engaged in inter-regional energy arbitrage, it is essential to account for interconnector flows. These flows can significantly affect potential gains.\ $\bullet$ *Interconnector Rent:* A rising level of interconnector rent can diminish the attractiveness of multi-region energy arbitrage as an investment opportunity.\ $\bullet$ *Profitability of Belgian-based Batteries:* Batteries positioned at the Belgian end of the NEMO link have the potential for profitability. However, this is contingent on factors such as price volatility and interconnector flow dynamics.\ $\bullet$ *Battery Lifecycle Consideration:* Evaluating the financial viability of battery installation should incorporate a thorough assessment of both the battery cycle and calendar life.\ $\bullet$ *Capacity Blocking for Emergency Services:* When implementing capacity blocking strategies for emergency services, it is imperative to factor in the battery's cycle and calendar life to ensure sustained profitability. ## Case study 2 The second case study examines interconnectors connecting the PEEI in the North Sea. It is assumed that inter-regional markets operate under a flow-based market design as described in Sec. [2](#section2){reference-type="ref" reference="section2"}. Under such a case, the opportunities for inter-regional arbitrage are significantly reduced as interconnector flow becomes saturated in a single direction. As illustrated in Fig. [11](#fig:feasiblerange){reference-type="ref" reference="fig:feasiblerange"}, it is generally possible for the battery to charge from the UK grid but discharging into the UK grid for the majority of the time is not possible without previously blocked capacity. Given this constraint under the flow-based market design, battery owners must participate in an interconnector capacity auction to reserve interconnector capacity apriori. As such, in this case study, we assume the interconnector rent is awarded at a fixed rate of 5 euros per megawatt-hour (MWh). {#fig:feasiblerange width="7in"} Fig. [12](#fig:case22){reference-type="ref" reference="fig:case22"} presents the marginal increase in storage revenue when compared to participating exclusively in the Belgian energy market. It is evident from the graph that the marginal revenue increase is nearly linear up to 50% of the ramping power limit. However, it diminishes beyond that point. This analysis underscores that storage owners can significantly boost their revenue, surpassing an 88% increase, by reserving interconnector capacity equal to the battery's ramping limit. However, for a comprehensive optimal bidding strategy, further numerical evaluations are necessary to inform battery owners' decisions. {#fig:case22 width="5.5in"} Fig. [13](#fig:computationTime){reference-type="ref" reference="fig:computationTime"} shows the computational time distributions for 1000 Monte Carlo (MC) simulations for different levels of interconnector capacity blocked for the years 2019 and 2020. {#fig:computationTime width="5.5in"} The key takeaways of the case study are:\ $\bullet$ When interconnector flows are calculated based on the flow-based market design, the interconnector saturates in one direction, leaving no room for inter-regional energy arbitrage. Under this condition, the battery owners should bid to reserve interconnector capacity. For interconnector capacity reserved, we observe an increase in battery revenue by more than 88% compared to the battery participating only in the local grid.\ $\bullet$ The median computation time of 1000 MC simulations is less than 21 seconds for one whole year. # Conclusions {#section7} This paper explores the evolving landscape of the future power system, characterized by a growing web of AC and DC interconnectors that bridge previously separate energy markets. The interconnectors serve as a vital link between these distinct energy markets. The authors propose an advanced model for optimal multi-market energy storage arbitrage, augmenting it with emergency services that can be extended to system operators. The proposed modelling includes battery ramping and capacity constraints, and the utilization of operating envelopes derived from interconnector capacity, efficiency, dynamic energy injection, and offshore wind generation forecasts for the day ahead. The key findings of this paper are as follows: Firstly, it develops a Mixed-Integer Linear Programming (MILP) model for energy storage, enabling simultaneous energy arbitrage across multiple markets while adhering to constraints like interconnector capacity, efficiency, and charge-discharge sequencing. The energy arbitrage model is designed to accommodate two distinct electricity price dynamics, with specialized considerations for bilinear terms. Secondly, the paper introduces a profitability assessment framework for energy storage, accounting for a portion reserved for emergency services and the battery's limited lifespan. Lastly, the study conducts realistic case studies, including NEMO Link and interconnections between Belgium and the UK connected to the PEEI, exploring factors such as interconnector rent's impact on performance, the allocation of battery capacity for emergencies, and the potential reservation of interconnector capacity for inter-regional energy trading. The paper introduces two different models for determining the upper limit of capacity that energy storage owners can allocate for emergency services. To validate their approach, the authors conduct numerical case studies, focusing on the interconnections between Belgium and the UK. In summary, we present a comprehensive model for optimizing grid-scale batteries in the context of interconnected energy markets, with a focus on both profitability and emergency services, demonstrated through real-world case studies. # Acknowledgement {#acknowledgement .unnumbered} This project has received funding from the CORDOBA project funded by Flanders Innovation and Entrepreneurship (VLAIO) in the framework of the spearhead cluster for blue growth in Flanders (Blue Cluster) -- Grant number HBC.2020.2722. ## Implementing MILP formulation for inter-regional energy arbitrage {#appendix:lpmatrix} The matrix format for the optimization problem $P_{\text{MILP}}$ is denoted as minimize ${f}^T X$, subject to ${A}X\leq b$, and $X \in [lb, ub]$. The dimension of $A$ is 13Nx6N, $b$ is 13Nx1, $X$ and $f$ are of size 6Nx1, and N denotes the number of samples in the horizon of optimization. $$f\text{=}{\begin{bmatrix} 0\\ :\\ 0\\ 0\\ :\\ 0\\ 1\\ :\\ 1\\ 1\\ :\\ 1\\ 0\\ :\\ 0\\ 0\\ :\\ 0\\ \end{bmatrix}},~~ X \text{=} {\begin{bmatrix} x_{1}^{\text{\sc a}}\\ :\\ x_{N}^{\text{\sc a}}\\ x_{1}^{\text{\sc b}}\\ :\\ x_{N}^{\text{\sc b}}\\ t_{1}^{\text{\sc a}}\\ :\\ t_{N}^{\text{\sc a}}\\ t_{1}^{\text{\sc b}}\\ :\\ t_{N}^{\text{\sc b}}\\ z_{1}^{\text{\sc a}}\\ :\\ z_{N}^{\text{\sc a}}\\ z_{1}^{\text{\sc b}}\\ :\\ z_{N}^{\text{\sc b}}\\ \end{bmatrix}},~~ lb\text{=} {\begin{bmatrix} X_{\min}\\ :\\ X_{\min}\\ \hline X_{\min}\\ :\\ X_{\min}\\ \hline T_{\min}^{\text{\sc a}}\\ :\\ T_{\min}^{\text{\sc a}}\\ \hline T_{\min}^{\text{\sc b}}\\ :\\ T_{\min}^{\text{\sc b}}\\ \hline 0\\ :\\ 0\\ \hline 0\\ :\\ 0\\ \end{bmatrix}} \leq {\begin{bmatrix} x_{1}^{\text{\sc a}}\\ :\\ x_{N}^{\text{\sc a}}\\ \hline x_{1}^{\text{\sc b}}\\ :\\ x_{N}^{\text{\sc b}}\\ \hline t_{1}^{\text{\sc a}}\\ :\\ t_{N}^{\text{\sc a}}\\ \hline t_{1}^{\text{\sc b}}\\ :\\ t_{N}^{\text{\sc b}}\\ \hline z_{1}^{\text{\sc a}}\\ :\\ z_{N}^{\text{\sc a}}\\ \hline z_{1}^{\text{\sc b}}\\ :\\ z_{N}^{\text{\sc b}}\\ \end{bmatrix}} \leq ub\text{=} {\begin{bmatrix} X_{\max}\\ :\\ X_{\max}\\ \hline X_{\max}\\ :\\ X_{\max}\\ \hline T_{\max}^{\text{\sc a}}\\ :\\ T_{\max}^{\text{\sc a}}\\ \hline T_{\max}^{\text{\sc b}}\\ :\\ T_{\max}^{\text{\sc b}}\\ \hline 1\\ :\\ 1\\ \hline 1\\ :\\ 1\\ \end{bmatrix}}, ~ {b} = {\begin{bmatrix} \text{zeros}(N,1) \\ \text{zeros}(N,1) \\ \text{zeros}(N,1) \\ \text{zeros}(N,1) \\ \hline b_{\max} - b_0\\ :\\ b_{\max} - b_0\\ b_0- b_{\min}\\ :\\ b_0- b_{\min}\\ \hline \text{ones}(N,1) X_{\max} \\ -\text{ones}(N,1) X_{\min}\\ \text{zeros}(4N,1) \\ \text{ones}(N,1) \\ \end{bmatrix}}. \label{mateqsame}$$ where $T_{\min}$ and $T_{\max}$ are bounds on $t_i$. Since these bounds are not known to us, we choose $T_{\min}$ to be negative with a large magnitude and $T_{\max}$ to be positive with a large magnitude. $${A} = { \renewcommand\arraystretch{1.3} \mleft[ \begin{array}{c |c |c | c | c|c } \texttt{diag}_N(P^{\text{\sc b,a}}_i/\eta^*_{\text{ch}}) & \texttt{zeros}_N & -1 \times \texttt{diag}_N & \texttt{zeros}_N & \texttt{zeros}_N & \texttt{zeros}_N \\ %eq1 epigraph 1 segment 1 \texttt{diag}_N(P^{\text{\sc s,a}}_i\eta^*_{\text{dis}}) & \texttt{zeros}_N & -1 \times \texttt{diag}_N & \texttt{zeros}_N & \texttt{zeros}_N & \texttt{zeros}_N\\ %eq2 epigraph 1 segment 2 \texttt{zeros}_N &\texttt{diag}_N(\widetilde{P}_i^{\text{\sc b,b}}/\eta^*_{\text{ch}}) & \texttt{zeros}_N & -1 \times \texttt{diag}_N & \texttt{zeros}_N & \texttt{zeros}_N\\ %eq3 epigraph 2 segment 1 \texttt{zeros}_N &\texttt{diag}_N(\widetilde{P}_i^{\text{\sc s,b}}\eta^*_{\text{dis}}) & \texttt{zeros}_N & -1 \times \texttt{diag}_N & \texttt{zeros}_N & \texttt{zeros}_N\\ %eq4 epigraph 2 segment 2 \texttt{trigL}_N & \texttt{trigL}_N & \texttt{zeros}_N& \texttt{zeros}_N & \texttt{zeros}_N & \texttt{zeros}_N \\ %eq5 battery capacity upper limit -1 \times \texttt{trigL}_N & -1 \times \texttt{trigL}_N & \texttt{zeros}_N& \texttt{zeros}_N & \texttt{zeros}_N & \texttt{zeros}_N\\ %eq6 battery capacity lower limit \texttt{diag}_N & \texttt{diag}_N & \texttt{zeros}_N& \texttt{zeros}_N & \texttt{zeros}_N & \texttt{zeros}_N \\ %eq7 battery ramp limit x_A + x_B -1 \times \texttt{diag}_N & -1 \times \texttt{diag}_N & \texttt{zeros}_N& \texttt{zeros}_N & \texttt{zeros}_N & \texttt{zeros}_N\\ %eq8 battery ramp limit x_A + x_B % -1 \times \texttt{diag}_N & \texttt{zeros}_N & \texttt{zeros}_N & \texttt{zeros}_N & X_{\min} \times \texttt{diag}_N & \texttt{zeros}_N \\ \texttt{diag}_N & \texttt{zeros}_N & \texttt{zeros}_N & \texttt{zeros}_N & \texttt{zeros}_N & - X_{\max} \times \texttt{diag}_N \\ \texttt{zeros}_N & -1 \times \texttt{diag}_N & \texttt{zeros}_N & \texttt{zeros}_N & X_{\min} \times \texttt{diag}_N & \texttt{zeros}_N \\ \texttt{zeros}_N & \texttt{diag}_N & \texttt{zeros}_N & \texttt{zeros}_N & \texttt{zeros}_N & - X_{\max} \times \texttt{diag}_N \\ \texttt{zeros}_N & \texttt{zeros}_N & \texttt{zeros}_N & \texttt{zeros}_N & \texttt{diag}_N & \texttt{diag}_N % % X_{\min} \times \texttt{diag}_N & X_{\min} \times \texttt{diag}_N & \texttt{zeros}_N & -1 \times \texttt{diag}_N & \texttt{zeros}_N & \texttt{zeros}_N\\ % X_{\max} \times \texttt{diag}_N & X_{\max} \times \texttt{diag}_N & \texttt{zeros}_N & -1 \times \texttt{diag}_N & \texttt{zeros}_N & \texttt{zeros}_N\\ % -X_{\min} \times \texttt{diag}_N & -X_{\max} \times \texttt{diag}_N & \texttt{zeros}_N & \texttt{diag}_N & \texttt{zeros}_N & \texttt{zeros}_N \\ % -X_{\max} \times \texttt{diag}_N & -X_{\min} \times \texttt{diag}_N & \texttt{zeros}_N & \texttt{diag}_N & \texttt{zeros}_N & \texttt{zeros}_N \end{array} \mright]} \begin{array}{c } \eqref{eq14} \vspace{3pt}\\ \eqref{eq15} \vspace{3pt}\\ \eqref{eq16} \vspace{3pt}\\ \eqref{eq17} \vspace{3pt}\\ \eqref{eq20} \vspace{3pt}\\ \eqref{eq21} \vspace{3pt}\\ \eqref{eq:ramp2} \vspace{3pt}\\ \eqref{eq:ramp2} \vspace{3pt}\\ \eqref{eq:linearization} \vspace{3pt}\\ \eqref{eq:linearization} \vspace{3pt}\\ \eqref{eq:linearization} \vspace{3pt}\\ \eqref{eq:linearization} \vspace{3pt}\\ \eqref{eq:linearization} \end{array},$$ where $\texttt{zeros}_N$ denotes a zero square matrix with $N$ rows and columns, $\texttt{diag}_N$ denotes a diagonal matrix with 1 in the diagonal and zeros otherwise, $\texttt{trigL}_N$ denotes the lower triangular matrix of size $N$, $\texttt{ones}_N$ denotes a square matrix of order $N$ with 1s and $\texttt{diag}_N(J_i)$ denotes a diagonal matrix with $J_i$ in the diagonal and zeros otherwise. Using a mixed integer linear programming solver, we solve the following problem $P_{\text{MILLP}}$. l C [\[eq:epi_formulation\]]{#eq:epi_formulation label="eq:epi_formulation"} & \_X f\^T X\ & A.X b, X , X\_ X {0,1}.\ In this work, we use MATLAB's `intlinprog` [@intlinprog] function for implementing $P_{\text{MILP}}$. [^1]: Interconnectors operated under a flow-based market would lead to a reduction in price fluctuations implying inter-regional energy trades would be marginally less profitable with their growth [@leuven2015cross]. [^2]: This implies $x$ MWh of energy supplied at one end of the interconnector will decrease to $\eta_{\text{line}} x$ MWh at the other end. [^3]: The electricity prices in the UK are in pounds/MWh. This is converted to euros/MWh. The conversion factor used in the work is fixed at 1.16.

arxiv_math

--- abstract: | We study the asymptotic behavior of short cycles of random permutations with cycle weights. More specifically, on a specially constructed metric space whose elements encode all possible cycles, we consider a point process containing all information on cycles of a given random permutation on $\{1,\ldots,n\}$. The main result of the paper is the distributional convergence with respect to the vague topology of the above processes towards a Poisson point process as $n\to\infty$ for a wide range of cycle weights. As an application, we give several limit theorems for various statistics of cycles. author: - Oleksii Galganov - Andrii Ilienko bibliography: - Cycles.bib title: | Short cycles of random permutations with cycle weights:\ point processes approach --- random permutation ,cycle structure ,point process ,Poisson convergence 60C05 ,60G55 # Introduction Random permutations are a classical object of combinatorial probability. Uniform permutations (that is, uniformly distributed random elements of the symmetric group $\mathcal S_n$) have been studied since de Montmort's matching problem. In recent years, there has been an extensive literature on non-uniform permutations $\sigma_n$ with cycle weights, which are defined by the probability distribution $$\label{eq:dist} \mathbb{P}\left\{\sigma_n=\pi\right\} = \frac{1}{h_n n!} \prod_{k=1}^\infty \theta_k^{C_k(\pi)}, \qquad \pi \in \mathcal S_n.$$ Here $\theta_k$ are non-negative parameters, $C_k(\pi)$ stands for the number of $k$-cycles in $\pi$, and $h_n$ is a normalization ensuring that $\sum_{\pi\in\mathcal S_n}\mathbb{P}\left\{\sigma_n=\pi\right\}=1$. These permutations were introduced in [@BU11], motivated by the theory of Bose-Einstein condensate. For other applications and connections, see [@EU14] and references therein. Note that special cases of [\[eq:dist\]](#eq:dist){reference-type="eqref" reference="eq:dist"} are the uniform random permutation (with $\theta_k=1$ for all $k$ and $h_n=1$) and Ewens random permutations (with $\theta_k=\theta$ for all $k$, $h_n=\theta^{(n)}/n!$, and $\theta^{(n)}$ standing for the rising factorial). The latter are based on the Ewens sampling formula which was introduced in population genetics and subsequently found numerous applications, see [@C16]. A rich theory of Ewens permutations was developed in [@ABT03]. An important subject of study of random permutations is their cycle structure and, in particular, the asymptotic statistics of short cycles (that is, cycles of bounded length) as $n\to\infty$. It is well known that, for Ewens permutations $\sigma_n$ on $\mathcal S_n$, $n\ge1$, $$\label{eq:limcounts} \left(C_k(\sigma_n), k \ge 1\right) \xrightarrow d (Z_k, k \ge 1)$$ in $\mathbb Z_+^\infty$, where $Z_k$ are independent and Poisson distributed with means $\theta/k$, see Theorem 5.1 in [@ABT03]. In the case of more general permutations with cycle weights, Corollary 2.2 in [@EU14] shows that [\[eq:limcounts\]](#eq:limcounts){reference-type="eqref" reference="eq:limcounts"} remains true with independent Poisson distributed $Z_k$ with means $\theta_k/k$ provided the stability condition $$\label{eq:stab} \lim_{n\to\infty} \frac{h_{n-1}}{h_n} = 1$$ holds. In the same paper, it is shown that [\[eq:stab\]](#eq:stab){reference-type="eqref" reference="eq:stab"} is satisfied for a wide range of asymptotics of $\theta_k$, from sub-exponential decay to sub-exponential growth. What can be said about the limiting composition of short cycles themselves? Say, for fixed points (that is, $1$-cycles), the invariance of [\[eq:dist\]](#eq:dist){reference-type="eqref" reference="eq:dist"} under relabeling suggests that, conditionally on $C_1(\sigma_n)=c_1$, the set of fixed points of $\sigma_n$ is distributed as a random equiprobable $c_1$-sample from $[n]\vcentcolon=\{1,\ldots,n\}$ without replacement. Similar reasoning can be given for cycles of any fixed length. However, a rigorous description of the limiting behavior is possible only within the framework of convergence of random point measures. Using this approach, in Section [2](#sec:main){reference-type="ref" reference="sec:main"} we state and prove a multivariate point processes version of [\[eq:limcounts\]](#eq:limcounts){reference-type="eqref" reference="eq:limcounts"}. Its main advantage is that it allows us to easily prove further limit theorems for different statistics of short cycles, bypassing involved combinatorial calculations and complicated asymptotic analysis. Various examples of such results are given in Section [3](#sec:appl){reference-type="ref" reference="sec:appl"}. # Preliminaries and main result {#sec:main} We first introduce a metric space appropriate for describing the limiting composition of cycles. For $k\ge1$, let $$\label{eq:Xk} \mathbb X_k = \left\{\boldsymbol x=(x_1,\ldots,x_k) \in [0,1]^k \colon \min\{x_1,\ldots,x_k\} = x_1\right\}$$ and denote by $\rho_k$ the Euclidean metric on $\mathbb X_k$. The last equality in [\[eq:Xk\]](#eq:Xk){reference-type="eqref" reference="eq:Xk"} is due to the fact that any element of a cycle can be regarded as its ''beginning''. Consider now a multi-level space $\mathbb X=\bigcup_{k=1}^\infty\mathbb X_k$ with metric given by $$\rho(\boldsymbol x_1,\boldsymbol x_2)= \begin{cases} \rho_k(\boldsymbol x_1,\boldsymbol x_2), & \boldsymbol x_1, \boldsymbol x_2 \in \mathbb X_k,\\ \sqrt{\max\{k_1, k_2\}}, & \boldsymbol x_1 \in \mathbb X_{k_1}, \, \boldsymbol x_2 \in \mathbb X_{k_2}, \, k_1 \ne k_2. \end{cases}$$ The triangle inequality holds since $\sup_{\boldsymbol x_1,\boldsymbol x_2 \in \mathbb X_k} \rho_k(\boldsymbol x_1, \boldsymbol x_2) = \sqrt k$. The metric $\rho$ makes $\mathbb X$ a Polish space. Moreover, $\mathbb X$ equipped with the Borel $\sigma$-algebra $\mathcal B(\mathbb X)$ turns into a measurable space, and $B\in\mathcal B(\mathbb X)$ if and only if $B\cap\mathbb X_k\in\mathcal B(\mathbb X_k)$ for all $k$. This makes it possible to define a measure on $\left(\mathbb X,\mathcal B(\mathbb X)\right)$ by $$\label{eq:mu} \lambda(B) = \sum_{k=1}^\infty \theta_k \hspace{1pt}\lambda_k(B\cap\mathbb X_k), \qquad B \in \mathcal B(\mathbb X),$$ where $\theta_k$ are defined in [\[eq:dist\]](#eq:dist){reference-type="eqref" reference="eq:dist"} and $\lambda_k$ stands for the $k$-dimensional Lebesgue measure. Let $\delta_{\boldsymbol x} = \mathds 1\{\boldsymbol x\in\cdot\}$ be the Dirac measure at $\boldsymbol x$. We will focus on the limiting behavior of random point measures on $\left(\mathbb X, \mathcal B(\mathbb X)\right)$ given by $$\label{eq:Psin} \Psi_n = \sum_{k=1}^\infty\;\; \sideset{}{^{\ne}}\sum_{i_1,\ldots,i_k \in [n]} \delta_{\left(\frac{i_1}{n}, \ldots, \frac{i_k}{n} \right)} \mathds 1\!\left\{ \sigma_n(i_1)=i_2, \ldots, \sigma_n(i_k)=i_1 \right\},$$ where $\sum^{\ne}$ means that the sum is taken over $k$-tuples with distinct entries. Moreover, since $\Psi_n$ is considered as a measure on $\left(\mathbb X, \mathcal B(\mathbb X)\right)$, the latter sum includes only those tuples in which the minimum element comes first. $\Psi_n$ carries all the information about the cycle structure of $\sigma_n$. Recall that the vague topology on the space of locally finite measures is generated by the integration maps $\nu\mapsto\int_\mathbb Xf\,\mathrm d\nu$ for all continuous functions $f$ with bounded support; see, e.g., @R87 [Section 3.4] or @K17 [Chapter 4] for a general exposition. Denote by $\xrightarrow{vd}$ the distributional convergence of random point measures with underlying vague topology. Let $\Psi$ denote the Poisson random measure on $\left(\mathbb X, \mathcal B(\mathbb X)\right)$ with intensity measure $\lambda$ given by [\[eq:mu\]](#eq:mu){reference-type="eqref" reference="eq:mu"}. The following theorem can be regarded as a point processes extension of [\[eq:limcounts\]](#eq:limcounts){reference-type="eqref" reference="eq:limcounts"}. **Theorem 1**. *Let the stability condition [\[eq:stab\]](#eq:stab){reference-type="eqref" reference="eq:stab"} hold. Then $\Psi_n \xrightarrow{vd} \Psi$ as $n \to \infty$.* [\[rmk:restrict\]]{#rmk:restrict label="rmk:restrict"} Let $\Psi_n^{(k)}$ and $\Psi^{(k)}$ be the restrictions to $\mathbb X_k$ of $\Psi_n$ and $\Psi$, respectively. By the restriction property of Poisson processes (see, e.g., Theorem 5.2 in [@LP18]), $\Psi^{(k)}$ are independent homogeneous Poisson processes with intensities $\theta_k$. Due to the vague continuity of the restriction mapping $\mu\mapsto\mu\hspace{-3.5pt}\restriction_{\mathbb X_k}$, we also have $\Psi_n^{(k)}\xrightarrow{vd}\Psi^{(k)}$. In particular, as $C_k(\sigma_n) = \Psi_n^{(k)}(\mathbb X_k)$, [\[eq:limcounts\]](#eq:limcounts){reference-type="eqref" reference="eq:limcounts"} directly follows from Theorem [Theorem 1](#th:main){reference-type="ref" reference="th:main"}. For the proof, we will first need an asymptotics for probabilities of cycles. **Lemma 2**. *Let $r \ge 1$ and $\boldsymbol i^{(j)} = \bigl(i_1^{(j)},\ldots,i_{k_j}^{(j)}\bigr)$, $j \in [r]$, be disjoint integer tuples with distinct entries such that $\boldsymbol i^{(j)}/n \in \mathbb X_{k_j}$. Then, under [\[eq:stab\]](#eq:stab){reference-type="eqref" reference="eq:stab"}, we have $$\label{eq:Pcycles} \mathbb{P}\left\{\sigma_n \textit{ contains the cycles } \boldsymbol i^{(1)},\ldots,\boldsymbol i^{(r)}\right\} \sim \frac{\prod_{j=1}^r\theta_{k_j}}{n^{k_1+\ldots+k_r}}, \qquad n\to\infty.$$* Let $\mathcal I$ be the set of all entries of $\boldsymbol i^{(j)}$, $j\in[r]$, and $s = \#\mathcal I= k_1 + \ldots + k_r$, where $\#$ stands for the cardinality of a set. By [\[eq:dist\]](#eq:dist){reference-type="eqref" reference="eq:dist"}, the probability in [\[eq:Pcycles\]](#eq:Pcycles){reference-type="eqref" reference="eq:Pcycles"} equals $$\begin{aligned} \label{eq:Pcycles_proof} \sum_{\tilde\pi \in \mathcal S_{[n]\setminus\mathcal I}} \! \mathbb{P}\left\{ \sigma_n = \boldsymbol i^{(1)} \circ \ldots \circ \boldsymbol i^{(r)} \circ \tilde\pi \right\} & = \sum_{\tilde\pi \in \mathcal S_{[n]\setminus\mathcal I}} \! \frac{1}{h_nn!} \prod_{k=1}^\infty \theta_k^{\#\{j\colon k_j=k\} + C_k(\tilde\pi)} \\ = \frac{1}{h_nn!} \prod_{j=1}^r \theta_{k_j} \cdot\! \sum_{\tilde\pi \in \mathcal S_{[n]\setminus\mathcal I}} \prod_{k=1}^\infty \theta_k^{C_k(\tilde\pi)} & = \frac{h_{n-s}\hspace{1pt}(n-s)!}{h_nn!} \prod_{j=1}^r \theta_{k_j}, \end{aligned}$$ where the last equality follows again from [\[eq:dist\]](#eq:dist){reference-type="eqref" reference="eq:dist"}. Hence, the claim follows from [\[eq:stab\]](#eq:stab){reference-type="eqref" reference="eq:stab"}.0◻ Let $\langle$ mean either $($ or $[$, and the same applies to $\rangle$. By Theorem 4.18 in [@K17], it suffices to prove that (i) $\lim_{n\to\infty} \mathbb E\hspace{1pt}\Psi_n(B) = \mathbb E\hspace{1pt}\Psi(B)$ for any box $B = \bigtimes_{i=1}^k\langle a_j,b_j\rangle\in \mathbb X_k$, $k \ge 1$, (ii) $\lim_{n\to\infty} \mathbb{P}\left\{\Psi_n(U)=0\right\} = \mathbb{P}\left\{\Psi(U)=0\right\}$ for any finite union $U$ of boxes from possibly different levels $\mathbb X_k$. Let $B$ be a box with fixed $k\ge1$ and $B_{\ne}$ the set of points in $B$ with distinct coordinates. By [\[eq:Psin\]](#eq:Psin){reference-type="eqref" reference="eq:Psin"}, we have $$\mathbb E\hspace{1pt}\Psi_n(B) = \sideset{}{^{\ne}}\sum_{\left(i_1,\ldots,i_k\right)/n\in B} \mathbb{P}\left\{\sigma_n(i_1) = i_2, \ldots, \sigma_n(i_k) = i_1\right\}.$$ It follows from [\[eq:Pcycles_proof\]](#eq:Pcycles_proof){reference-type="eqref" reference="eq:Pcycles_proof"} and [\[eq:Pcycles\]](#eq:Pcycles){reference-type="eqref" reference="eq:Pcycles"} with $r=1$ that all summands on the right-hand side are equal and asymptotically equivalent to $\theta_k/n^k$. Thus, as $n\to\infty$, $$\mathbb E\hspace{1pt}\Psi_n(B) \sim \frac{\theta_k}{n^k} \cdot \#\left(B_{\ne}\cap(\mathbb Z^k/n)\right) \to \theta_k\hspace{1pt}\lambda_k(B) = \lambda(B) = \mathbb E\hspace{1pt}\Psi(B),$$ which proves (i). We now proceed to (ii). Let us fix a finite union $U$ of boxes and denote $U_m = U \cap \mathbb X_m$. So, $U_m = \varnothing$ for $m$ greater than some $k \ge 1$ (the maximum dimension of boxes in the union), and $U = \bigcup_{m=1}^k U_m$. Let $\boldsymbol i_m$ and $\boldsymbol i_m^{(\cdot)}$ stand for integer $m$-tuples with distinct entries and $\bigcirc_{j_m=1}^{r_m} \boldsymbol i_m^{(j_m)}$ be the composition $\boldsymbol i_m^{(1)} \circ \ldots \circ \boldsymbol i_m^{(r_m)}$ of $r_m$ cycles defined by such tuples. For any $R\ge1$, by Bonferroni's inequality, $$\begin{aligned} \notag & \mathbb{P}\left\{\Psi_n(U)=0\right\} = 1 - \mathbb P\hspace{1pt}\biggl\{\bigcup_{m=1}^k\bigcup_{\boldsymbol i_m/n\in U_m} \{\sigma_n \textit{ contains the cycle } \boldsymbol i_m\}\biggr\} \\ \label{eq:zero_prob_incl_excl} & \le \sum_{r_1, \ldots, r_k = 0}^{2R} (-1)^{r_1 + \ldots + r_k} \sideset{}{^\ast} \sum_{\substack{ \boldsymbol i_1^{(1)}/n, \ldots, \boldsymbol i_1^{(r_1)}/n \in U_1, \\ \vbox{\fontsize{\sf@size}{\sf@size pt}\linespread{0.3}\selectfont \kern 0.2\baselineskip \hbox{.}\hbox{.}\hbox{.}% \kern 0.1\baselineskip }\\ \boldsymbol i_k^{(1)}/n, \ldots, \boldsymbol i_k^{(r_k)}/n \in U_k }} \mathbb{P}\left\{ \sigma_n \textit{ contains } \bigcirc_{j_1=1}^{r_1} \boldsymbol i_1^{(j_1)} \circ \ldots \circ \bigcirc_{j_k=1}^{r_k} \boldsymbol i_k^{(j_k)} \right\}, \end{aligned}$$ where $\sum^\ast$ means that the sum is taken over all unordered sets of disjoint tuples, and a similar lower bound holds with $2R$ replaced by $2R-1$. Due to [\[eq:Pcycles_proof\]](#eq:Pcycles_proof){reference-type="eqref" reference="eq:Pcycles_proof"}, all the summands in $\sum^\ast$ are equal and the right-hand side of [\[eq:zero_prob_incl_excl\]](#eq:zero_prob_incl_excl){reference-type="eqref" reference="eq:zero_prob_incl_excl"} becomes $$\label{eq:sum} \sum_{r_1, \ldots, r_k = 0}^{2R} (-1)^{r_1 + \ldots + r_k}\cdot \frac{ h_{n - \sum_{m=1}^k m r_m} \bigl(n - \sum_{m=1}^k m r_m\bigr)! }{h_n n!} \cdot \frac{ \prod_{m=1}^k\theta_m^{r_m} }{\prod_{m=1}^k r_m!} \cdot S_n(r_1, \ldots, r_k),$$ where $$S_n(r_1, \ldots, r_k) = \hspace{-15pt}\sum_{\substack{ \boldsymbol i_1^{(1)}/n, \ldots, \boldsymbol i_1^{(r_1)}/n \in \mathbb X_1, \\ \vbox{\fontsize{\sf@size}{\sf@size pt}\linespread{0.3}\selectfont \kern 0.2\baselineskip \hbox{.}\hbox{.}\hbox{.}% \kern 0.1\baselineskip }\\ \boldsymbol i_k^{(1)}/n, \ldots, \boldsymbol i_k^{(r_k)}/n \in \mathbb X_k }} \hspace{-15pt}\mathds{1}\{ \textit{all tuples are disjoint}\hspace{1pt}\} \prod_{m=1}^k \mathds{1}\bigl\{ \boldsymbol i_m^{(1)}/n, \ldots, \boldsymbol i_m^{(r_m)}/n \in U_m \bigr\},$$ and division by $\prod_{m=1}^k r_m!$ is due to the fact that the sum in the definition of $S_n$ is taken over ordered sets of tuples. Note that $\frac{S_n(r_1, \ldots, r_k)}{n^{r_1 + \ldots + k r_k}}$ can be viewed as an integral sum for $$\int_{U_1^{r_1} \times \ldots \times U_{k}^{r_k}} \mathds{1}\left\{ \textit{all components of all } \boldsymbol x_{m}^{(j_m)} \textit{ are distinct} \right\} \; \prod_{m=1}^k\prod_{j_m=1}^{r_m}\mathrm d\boldsymbol x_m^{(j_m)} = \prod_{m=1}^k \left( \lambda_m(U_m) \right)^{r_m}.$$ Hence, by [\[eq:stab\]](#eq:stab){reference-type="eqref" reference="eq:stab"}, each summand in [\[eq:sum\]](#eq:sum){reference-type="eqref" reference="eq:sum"} converges as $n\to\infty$ to $$(-1)^{r_1 + \ldots + r_k}\prod_{m=1}^k \frac{\left(\theta_m\lambda_m(U_m) \right)^{r_m}}{r_m!}.$$ It now follows from [\[eq:zero_prob_incl_excl\]](#eq:zero_prob_incl_excl){reference-type="eqref" reference="eq:zero_prob_incl_excl"} and a similar lower bound that $$\begin{aligned} \sum_{r_1, \ldots, r_k = 0}^{2R-1} (-1)^{r_1 + \ldots + r_k}\prod_{m=1}^k \frac{\left(\theta_m\lambda_m(U_m) \right)^{r_m}}{r_m!}&\le\lim_{n\to\infty}\mathbb{P}\left\{\Psi_n(U)=0\right\}\\&\le\sum_{r_1, \ldots, r_k = 0}^{2R} (-1)^{r_1 + \ldots + r_k}\prod_{m=1}^k \frac{\left(\theta_m\lambda_m(U_m) \right)^{r_m}}{r_m!}. \end{aligned}$$ Letting $R\to\infty$ finally yields $$\begin{aligned} \lim_{n\to\infty} \mathbb{P}\left\{\Psi_n(U)=0\right\} & = \sum_{r_1, \ldots, r_k = 0}^{\infty} (-1)^{r_1 + \ldots + r_k}\prod_{m=1}^k \frac{\left(\theta_m\lambda_m(U_m) \right)^{r_m}}{r_m!} \\ & = \exp\hspace{1pt}\Bigl\{ -\sum_{m=1}^k \theta_m \lambda_m(U_m)\Bigr\} = \exp\hspace{1pt}\{-\lambda(U)\}=\mathbb{P}\left\{\Psi(U) = 0\right\}. \end{aligned}$$ This concludes the proof of (ii) and, hence, that of Theorem [Theorem 1](#th:main){reference-type="ref" reference="th:main"}.0◻ # Limit theorems for statistics of short cycles {#sec:appl} Theorem [Theorem 1](#th:main){reference-type="ref" reference="th:main"} allows us to derive limiting distributions for various statistics of short cycles. In what follows, we will always assume that the stability condition [\[eq:stab\]](#eq:stab){reference-type="eqref" reference="eq:stab"} is satisfied. We first give a general result which covers the case of additive statistics. **Proposition 3**. *Let $k\ge1$ and $f_m\colon\mathbb X_m\to[0,\infty)$, $m\in[k]$, be a family of continuous functions. Denote by $\mathcal C_m(\sigma_n)$ the set of all $m$-cycles in $\sigma_n$. Then $$\label{eq:addstat} \sum_{m=1}^k\sum_{c\hspace{1pt}\in\hspace{1pt}\mathcal C_m(\sigma_n)}f_m\left(\frac cn\right)\xrightarrow d S,\qquad n\to\infty,$$ where $c\in\mathcal C_m(\sigma_n)$ is understood as an integer tuple whose minimum element comes first, and the limiting random variable $S$ is defined by its Laplace transform $$\label{eq:Laplace} \mathbb E\exp\hspace{1pt}\{-tS\}= \exp\hspace{1pt}\Bigl\{-\sum_{m=1}^k\theta_m\int_{\mathbb X_m} \left(1-\mathrm e^{-tf_m(\boldsymbol x)}\right)\,\mathrm d\boldsymbol x\Bigr\},\qquad t\ge0.$$* Define $f\colon\mathbb X\to[0,\infty)$ by $f(\boldsymbol x)=\sum_{m=1}^kf_m(\boldsymbol x)\mathds 1\{\boldsymbol x\in\mathbb X_m\}$. The left-hand side of [\[eq:addstat\]](#eq:addstat){reference-type="eqref" reference="eq:addstat"} can be written as $\int_\mathbb Xf(\boldsymbol x)\,\Psi_n(\mathrm d\boldsymbol x)$, and $f$ is continuous with bounded support. By Theorem [Theorem 1](#th:main){reference-type="ref" reference="th:main"} and Lemma 4.12 in [@K17], [\[eq:addstat\]](#eq:addstat){reference-type="eqref" reference="eq:addstat"} holds with $S=\int_\mathbb Xf(\boldsymbol x)\,\Psi(\mathrm d\boldsymbol x)$. The Laplace transform of $S$ is thus of the form $$\mathbb E\exp\hspace{1pt}\{-tS\}=\mathbb E\exp\hspace{1pt}\Bigl\{-\int_\mathbb Xtf(\boldsymbol x)\,\Psi(\mathrm d\boldsymbol x)\Bigr\},$$ which coincides with the right-hand side of [\[eq:Laplace\]](#eq:Laplace){reference-type="eqref" reference="eq:Laplace"} due to the form of the Laplace functional of a Poisson random measure, see, e.g., Proposition 3.6 in [@R87].0◻ As an example, we give a limit theorem for the sum $S_n^{(k)}$ of elements in all $k$-cycles of $\sigma_n$. **Proposition 4**. * * (i) *$\frac{S_n^{(k)}}n\xrightarrow d S^{(k)}$ as $n\to\infty$, where $S^{(k)}$ is defined by its Laplace transform $$\label{eq:LaplaceSk} \mathbb E\exp\left\{-tS^{(k)}\right\}=\exp\hspace{1pt}\biggl\{\frac{\theta_k}k \biggl(\left(\frac{1-\mathrm e^{-t}}t\right)^k-1\biggr)\biggr\},\qquad t>0.$$* (ii) *If $k=1$, that is, for $S_n^{(1)}$ being the sum of fixed points, $$\label{eq:cdfS1} \mathbb{P}\left\{S^{(1)}\le x\right\}=\mathrm e^{-\theta_1}\sum_{j=0}^{\lfloor x\rfloor} \frac{(-1)^j}{j!}(\theta_1(x-j))^{\frac{j}{2}}I_j\!\left(2\sqrt{\theta_1(x-j)}\right),\qquad x\ge0,$$ where $I_j$ is the modified Bessel function of the first kind, see, e.g., §10.25(ii) in [@NIST].* For $\boldsymbol x\in\mathbb X_k$, let $f_k(\boldsymbol x)$ denote the sum of all its components. By Proposition [Proposition 3](#prop:add){reference-type="ref" reference="prop:add"}, (i) holds with $$\mathbb E\exp\left\{-tS^{(k)}\right\}=\exp\hspace{1pt}\Bigl\{-\theta_k\int_{\mathbb X_k} \left(1-\mathrm e^{-tf_k(\boldsymbol x)}\right)\,\mathrm d\boldsymbol x\Bigr\},$$ where the integral on the right-hand side, due to symmetricity of $f_k$, equals $$\frac 1k\int_{[0,1]^k} \left(1-\mathrm e^{-tf_k(\boldsymbol x)}\right)\,\mathrm d\boldsymbol x= \frac 1k\Bigl(1-\Bigl(\int_0^1\mathrm e^{-tx}\,\mathrm dx\Bigr)^k\,\Bigr).$$ This yields [\[eq:LaplaceSk\]](#eq:LaplaceSk){reference-type="eqref" reference="eq:LaplaceSk"}. To prove [\[eq:cdfS1\]](#eq:cdfS1){reference-type="eqref" reference="eq:cdfS1"}, we first note that $$\mathbb E\exp\left\{-tS^{(1)}\right\}= \mathbb E\int_0^\infty t\mathrm e^{-tx}\,\mathds 1\{S^{(1)}\le x\}\,\mathrm dx =t\int_0^\infty \mathrm e^{-tx}\,\mathbb P\hspace{1pt}\{S^{(1)}\le x\}\,\mathrm dx,$$ cf. [@L20]. Hence, $\mathbb{P}\left\{S^{(1)}\le x\right\}$ is the inverse Laplace transform of the right-hand side in [\[eq:LaplaceSk\]](#eq:LaplaceSk){reference-type="eqref" reference="eq:LaplaceSk"} for $k=1$ multiplied by $t^{-1}$, that is, in expanded form, of the function $$\label{eq:Ginf} G(t)=\sum_{j=0}^\infty\frac{(-\theta_1)^j}{j!}\mathrm e^{-\theta_1}\hspace{1pt}t^{-j-1}\mathrm e^{\frac{\theta_1}t}\mathrm e^{-jt},\qquad t>0.$$ By the time shifting property and @E54 [eq. (5.5.35)], the inverse Laplace transform of the $j$-th summand $G_j(t)$ in [\[eq:Ginf\]](#eq:Ginf){reference-type="eqref" reference="eq:Ginf"} equals $$F_j(x)=\frac{(-1)^j}{j!}e^{-\theta_1}\hspace{1pt}(\theta_1(x-j))^{\frac j2}I_j\!\left(2\sqrt{\theta_1(x-j)}\right)\mathds 1\{x\ge j\}.$$ This means that, for all $t>0$ and $R\in\mathbb N$, $$\label{eq:R} \int_0^\infty\!\mathrm e^{-tx}\sum_{j=0}^R F_j(x)\,\mathrm dx=\sum_{j=0}^R G_j(t).$$ Since $I_j(x)$ decreases in $j$ and increases in $x$ (see, e.g., §10.37 in [@NIST]), and $I_0(x)\sim\frac{\mathrm e^x}{\sqrt{2\pi x}}$ as $x\to\infty$ (ibid., §10.30(ii)), we have $$\Bigl|\sum_{j=0}^R F_j(x)\Bigr|\le e^{-\theta_1}I_0\!\left(2\sqrt{\theta_1x}\right) \sum_{j=0}^\infty\frac 1{j!}(\theta_1x)^{\frac j2}\sim e^{-\theta_1}\frac{\mathrm e^{3\sqrt{\theta_1x}}}{2\pi^{\frac 12}(\theta_1x)^{\frac14}},\qquad x\to\infty,$$ which makes it possible to apply dominated convergence to [\[eq:R\]](#eq:R){reference-type="eqref" reference="eq:R"}. Hence, the inverse Laplace transform of $G$ is $\sum_{j=0}^\infty F_j$, which yields [\[eq:cdfS1\]](#eq:cdfS1){reference-type="eqref" reference="eq:cdfS1"}.0◻ We now turn to some examples of non-additive statistics. For $k\ge2$, let us call the range of a cycle the difference between its maximum and minimum elements and denote by $r_n^{(k)}$ (resp., $R_n^{(k)}$) the minimum (maximum) range among all $k$-cycles in $\sigma_n$. If there are no $k$-cycles, we set $r_n^{(k)}=n$ and $R_n^{(k)}=0$. **Proposition 5**. *$\frac{r_n^{(k)}}n\xrightarrow d r^{(k)}$ and $\frac{R_n^{(k)}}n\xrightarrow d R^{(k)}$ as $n\to\infty$, where $r^{(k)}$ (resp., $R^{(k)}$), $k\ge2$, have CDF's of the form $$\begin{gathered} \label{eq:range1} \mathbb{P}\left\{r^{(k)}\le x\right\}=1- \exp\hspace{1pt}\Bigl\{-\frac{\theta_k}k\left(kx^{k-1}-(k-1)x^k\right)\Bigr\},\\ \mathbb{P}\left\{R^{(k)}\le x\right\}= \exp\hspace{1pt}\Bigl\{\frac{\theta_k}k\left(kx^{k-1}-(k-1)x^k-1\right)\Bigr\}, \label{eq:range2} \end{gathered}$$ as $x\in[0,1)$ and $0$ (resp., $1$) to the left (right.* For $\boldsymbol x\in\mathbb X_k$, let $f_k(\boldsymbol x)$ denote the difference between its maximum and minimum components. It follows from the interpretation of vague convergence in Proposition 3.13 of [@R87] that the function which maps a finite point measure $\mu$ on $\mathbb X_k$ into $\min_{\mu\{\boldsymbol x\}\ge1}f_k(\boldsymbol x)$ is vaguely continuous. Hence, by Remark [\[rmk:restrict\]](#rmk:restrict){reference-type="ref" reference="rmk:restrict"} and continuous mapping theorem, $\frac{r_n^{(k)}}n\xrightarrow d r^{(k)}$ holds with $r^{(k)}=\min_{\Psi^{(k)}\{\boldsymbol x\}\ge1}f_k(\boldsymbol x)$, and the CDF of $r^{(k)}$ is of the form $$\begin{aligned} \label{eq:range_prob} \mathbb{P}\left\{r^{(k)}\le x\right\}&=1-\mathbb{P}\left\{\Psi^{(k)}\{\boldsymbol x\in\mathbb X_k\colon f_k(\boldsymbol x)\le x\}=0\right\} \\&=1-\exp\hspace{1pt}\bigl\{-\theta_k\hspace{1pt}\lambda_k \{\boldsymbol x\in\mathbb X_k\colon f_k(\boldsymbol x)\le x\}\bigr\}. \end{aligned}$$ The value of the Lebesgue measure on the right-hand side is $$\int_0^1\biggl(\int_{x_1}^{\min\{x_1+x,1\}}\mathrm dx_2\ldots\mathrm dx_k\biggr)\mathrm dx_1=x^{k-1}(1-x)+\frac{x^k}k,$$ which together with [\[eq:range_prob\]](#eq:range_prob){reference-type="eqref" reference="eq:range_prob"} yields [\[eq:range1\]](#eq:range1){reference-type="eqref" reference="eq:range1"}. The proof of [\[eq:range2\]](#eq:range2){reference-type="eqref" reference="eq:range2"} is similar.0◻ As a final example, we consider some statistics of fixed points. Let $m_n$ be the minimum fixed point ($n+1$ in case there is none), $M_n$ the maximum one ($0$ in that case), $\delta_n$ the minimum spacing between fixed points (the two extreme spacings of lengths $m_n$ and $n+1-M_n$ are also taken into account), and $\Delta_n$ the maximum one. **Proposition 6**. *$\left(\frac{m_n}n, \frac{M_n}n, \frac{\delta_n}n, \frac{\Delta_n}n\right) \xrightarrow d(m,M,\delta,\Delta)$ as $n\to\infty$ with $$\begin{gathered} \mathbb{P}\left\{m\le x\right\}=1-\exp\hspace{1pt}\{-\theta_1x\},\qquad \mathbb{P}\left\{M\le x\right\}=\exp\hspace{1pt}\{\theta_1(x-1)\},\qquad x\in[0,1),\\ \delta\overset d=\frac{X_{\nu+1}}{(\nu+1)\sum_{i=1}^{\nu+1}X_i},\qquad \Delta\overset d=\frac{\sum_{i=1}^{\nu+1}\frac{X_i}i}{\sum_{i=1}^{\nu+1}X_i}, \end{gathered}$$ where $\nu$ is Poisson distributed with parameter $\theta_1$, $X_i$ are exponentially distributed with unit mean, and all these are independent.* As in the proof of Proposition [Proposition 5](#prop:ranges){reference-type="ref" reference="prop:ranges"}, the convergence takes place with $m$ being the leftmost point of $\Psi^{(1)}$, $M$ the rightmost one, $\delta$ the minimum spacing, and $\Delta$ the maximum one. All that remains is to derive the corresponding CDF's and distributional equalities. For $m$ and $M$, this follows from $$\begin{gathered} \mathbb{P}\left\{m\le x\right\} = 1-\mathbb{P}\left\{\Psi^{(1)}[0, x] = 0\right\} = 1-\exp\hspace{1pt}\{-\theta_1x\},\\ \mathbb{P}\left\{M\le x\right\} = \mathbb{P}\left\{\Psi^{(1)}(x, 1] = 0\right\} = \exp\hspace{1pt}\{-\theta_1(1-x)\}. \end{gathered}$$ We now turn to the equalities for $\delta$ and $\Delta$. Since $\Psi^{(1)}$ is a homogeneous Poisson process with intensity $\theta_1$, the conditional distribution of $\delta$ (resp., $\Delta$) given $\Psi^{(1)}(\mathbb X_1) = r$ coincides with that of the minimum (maximum) spacing $d_r$ ($D_r$) for $r$ independent random variables, uniformly distributed on $[0,1]$, see, e.g., Proposition 3.8 in [@LP18]. It follows from the theory of uniform spacings (see, e.g., [@H80], p. 625) that $$\label{eq:Uspac} d_r\overset d=\frac{\min Y_i} {\sum_{i=1}^{r+1}Y_i}, \qquad D_r\overset d=\frac{\max Y_i} {\sum_{i=1}^{r+1}Y_i},$$ where $Y_1,\ldots,Y_{r+1}$ are independent $\mathsf{Exp}(1)$-distributed random variables. Let $Y_{(i)}$, $i\in[r+1]$, be their order statistics, $Y_{(0)}=0$, and $\tau_i=Y_{(i)}-Y_{(i-1)}$. By Sukhatme--Rényi decomposition (see, e.g., Theorem 4.6.1 in [@ABN08]), $\tau_i$ are independent and $\mathsf{Exp}(r-i+2)$-distributed with $$\min Y_i=\tau_1,\qquad \max Y_i=\sum_{i=1}^{r+1}\tau_i,\qquad \sum_{i=1}^{r+1}Y_i=\sum_{i=1}^{r+1}(r-i+2)\tau_i.$$ Denoting $X_{r-i+2}=(r-i+2)\tau_i \sim \mathsf{Exp}(1)$, we can rewrite [\[eq:Uspac\]](#eq:Uspac){reference-type="eqref" reference="eq:Uspac"} as $$d_r \overset d= \frac{X_{r+1}}{(r+1) \sum_{i=1}^{r+1} X_i}, \qquad D_r \overset d= \frac{\sum_{i=1}^{r+1}\frac{X_i}i} {\sum_{i=1}^{r+1} X_i}.$$ Since this holds for any $r\ge0$, the claim follows by deconditioning.

arxiv_math

--- abstract: | In [@ManturovNikonovMay2023; @ManturovWanMay2023] the first named author discovered a very general principle (called *the photography principle*) which allows one: a\) To solve various equations (like pentagon equation) b\) To construct invariants of manfolds. The advantage of that principle is that it deals with a very general notion of *data* and *data transmission* which may be of any kind. In the present paper, we show that the definition of the Dijkgraaf-Witten invariants of manifolds can be thought of as an evidence of the above principle. address: - | Moscow Institute of Physics and Technology, Moscow 141700, Russia\ Nosov Magnitogorsk State Technical University, Zhilyaev Laboratory of mechanics of gradient nanomaterials, 38 Lenin prospect, Magnitogorsk, 455000, Russian Federation\ vomanturov\@yandex.ru - | Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China\ wanzheyan\@bimsa.cn author: - Vassily Olegovich Manturov - Zheyan Wan title: The Dijkgraaf-Witten invariant is a partial case of the photography method --- Keywords: Dijkgraaf-Witten invariants, photography method MSC 2020: 20F36, 13F60, 57K20, 57K31 # Introduction A very general method, *the photography method* for solving the pentagon equation, and other equations, was proposed by the first named author in [@ManturovNikonovMay2023]. Roughly speaking, if one has several *states* (say, triangulations of a pentagon), some *data* for each state (e.g., edge lengths or triangle areas) and a *rule (formula)* for translating data from one state to another (say, Ptolemy equation), then after returning to the initial state, we obtain the same data. Saying more standardly, this means that the photography method gives a solution to some equations in terms of "data" whatever this data means (in geometrical setting one can have lengths, angles, areas, volumes, etc.); even the amount of data may change, so, it is sometimes hard to say what we mean by an "equation". We state the photography method as follows: "We associate some data to each of the triangulations of an $n$-manifold. Then we use some data transmission law: if we know all for one triangulation, then we should know all for an adjacent triangulation, hence for any other triangulation." We may use this method to solve equations or construct invariants. For $n\ge3$, $n$-manifolds can be described as equivalence classes of triangulations modulo Pachner moves [@Pachner1; @Pachner2]. With a triangulation, one can associate some *data* and some *data transmission laws*: some elements in a group $G$ corresponding to 1-simplices and the values of a group $n$-cocycle $\phi\in H^n(G,A)$ ($A$ is an abelian group) corresponding to $n$-simplices, and the way these values are changing under moves. Taking this together, we can get to an equation (a system of equations) needed to get a state-sum associated to triangulations, which will be invariant under, say, (2-3)-Pachner moves and (1-4)-Pachner moves. This equation is exactly the $n$-cocycle condition. There are two versions of the photography method. One of them is for solving equations [@ManturovWanMay2023] and the other one is for constructing invariants [@KauffmanKimManturovNikonov]. In [@ManturovWanMay2023] we solve the pentagon equation in two ways: one by using the realisability of edge-lengths as lambda-lengths in the hyperbolic plane and Ptolemy relation, the other one by using the areas. In [@KauffmanKimManturovNikonov] we construct invariants of manifolds by using the following observation. If we have $(n+1)$ points (say, 5) and all edges between them, and for two $n$-tuples of points some relations hold, then we can restore the last edge so that the relations hold for all possible $n$-tuples. For such relations, one can take *realisability of edge-lengths as edge-lengths of a simplex in some Euclidean space*. The invariants we have constructed before [@KauffmanKimManturovNikonov] and here are not exactly "Turaev-Viro invariants" [@TV] or Dijkgraaf-Witten invariants [@DW]. For example, in the paper [@ManturovNikonovMay2023] instead of taking a finite palette of colours and solving the Biedenharn-Elliot equations, we guess how to get a solution. The main guess is the "geometrical" observation that "if four quadrilaterals ABCD and ABCE are inscribed then any of ABDE, ACDE, BCDE is inscribed". Similarly, in [@ManturovWanMay2023 Sec. 4] we used "areas" of triangles in order to get some matrices satisfying pentagon identities. But these "areas" are not areas in some proper sense: they are just variables which satisfy the only condition: the sum of "areas" of a whole polyhedron equals the sum of its constituent parts. Then we take it more abstractly, and get a solution to the pentagon identity suggested by Korepanov in [@K19]. In the present paper a similar guess answers the question "How to get Dijkgraaf-Witten-like invariants"? We take a triangulation and associate data to its 1-dimensional edges and top-dimensional cells in order to get an invariant under Pachner moves. The data we want to associate to edges are the edge lengths, and the data associate to top-dimensional simplices are "volumes". When we try to figure out what we really need from volumes, we see that this should be nothing but the cocycle condition [\[eq:cocycle\]](#eq:cocycle){reference-type="eqref" reference="eq:cocycle"}. A crucial observation explaining the "naturality" of the Dijkgraaf-Witten invariants is Lemma [Lemma 1](#volume){reference-type="ref" reference="volume"} which allows to axiomatize volumes as cocycles. The paper is organized as follows. In Sec. [2](#sec:definition){reference-type="ref" reference="sec:definition"}, we review the definition of the Dijkgraaf-Witten invariants. In Sec. [3](#sec:main){reference-type="ref" reference="sec:main"}, we show that the Dijkgraaf-Witten invariants are a partial case of the photography method. In Sec. [4](#sec:further){reference-type="ref" reference="sec:further"}, we list some directions of further research. ## Acknowledgements The authors are grateful to L.A.Grunwald, L.H.Kauffman, Seongjeong Kim, and V.G.Turaev for helpful discussion and comments. The study was supported by the grant of Russian Science Foundation (No. 22-19-20073 dated March 25, 2022 "Comprehensive study of the possibility of using self-locking structures to increase the rigidity of materials and structures"). # Dijkgraaf-Witten invariants {#sec:definition} In the present section, we review the Dijkgraaf-Witten invariants for $n$-manifolds. For $n=3$, in [@DW] Dijkgraaf and Witten gave a combinatorial definition for Chern-Simons with finite gauge groups using 3-cocycles of the group cohomology. This was generalised to higher dimensional manifolds in [@Freed]. We follow the description in [@CKS], see also [@Wakui] (we use the Pachner moves). Let $T$ be a triangulation of an oriented closed $n$-manifold $M$, with $a$ vertices and $m$ $n$-simplices. Give an ordering to the set of vertices. Let $G$ be a finite group. Let $\phi: \{\text{oriented edges}\}\to G$ be a map such that 1. for any triangle with vertices $v_0,v_1,v_2$ of $T$, $\phi(\langle v_0,v_2\rangle)=\phi(\langle v_0,v_1\rangle)\phi(\langle v_1,v_2\rangle)$, where $\langle v_i,v_j\rangle$ denotes the oriented edge with endpoints $v_i$ and $v_j$, and 2. $\phi(-e)=\phi(e)^{-1}$. Let $\alpha: G^{\times n}\to A$, $(g_1,g_2,\dots,g_n)\mapsto \alpha[g_1|g_2|\cdots|g_n]\in A$, be an $n$-cocycle valued in a multiplicative abelian group $A$. The $n$-cocycle condition is $$\begin{aligned} \label{eq:cocycle} &&\alpha[g_2|g_3|\cdots|g_{n+1}]\alpha[g_1g_2|g_3|\cdots|g_{n+1}]^{-1}\alpha[g_1|g_2g_3|\cdots|g_{n+1}]\cdots{\nonumber}\\ &&\cdot\alpha[g_1|g_2|\cdots|g_ng_{n+1}]^{(-1)^n}\alpha[g_1|g_2|\cdots|g_n]^{(-1)^{n+1}}=1.\end{aligned}$$ Then the Dijkgraaf-Witten invariant is defined by $$\begin{aligned} \label{eq:DW} Z_M=\frac{1}{|G|^a}\sum_{\phi}\prod_{i=1}^mW(\sigma_i,\phi)^{\epsilon_i}.\end{aligned}$$ Here $a$ denotes the number of the vertices of the given triangulation, $W(\sigma,\phi)=\alpha[g_1|g_2|\cdots|g_n]$ where $\phi(\langle v_0,v_1\rangle)=g_1$, $\phi(\langle v_1,v_2\rangle)=g_2$, $\dots$, $\phi(\langle v_{n-1},v_n\rangle)=g_n$, for the $n$-simplex $\sigma=|v_0v_1v_2\cdots v_n|$ with the ordering $v_0<v_1<v_2<\cdots<v_n$, and $\epsilon=\pm1$ according to whether or not the orientation of $\sigma$ with respect to the vertex ordering matches the orientation of $M$. # Dijkgraaf-Witten invariants are a partial case of the photography method {#sec:main} We associate some data to each of the triangulations of an $n$-manifold. We may associate data to edges and $n$-simplices. Then we use some data transmission law: if we know all for one triangulation, then we should know all for an adjacent triangulation, hence for any other triangulation. The data are the elements of $G$ associated to the edges and the values of an $n$-cocycle on the $n$-simplices and we consider the state sum defined by [\[eq:DW\]](#eq:DW){reference-type="eqref" reference="eq:DW"}. The elements of $G$ associated to the edges of a triangle satisfy the product rule. The data transmission law is the $n$-cocycle condition. Two adjacent triangulations are connected by a Pachner move. {#fig:Pachner width="90%"} {#fig:Pachner2 width="90%"} For $n=3$, the Pachner moves are the Pachner 2-3 and 1-4 moves, see Fig. [1](#fig:Pachner){reference-type="ref" reference="fig:Pachner"} and Fig. [2](#fig:Pachner2){reference-type="ref" reference="fig:Pachner2"}. The Pachner 2-3 move does not create a new vertex. The values of the 3-cocycle on the initial 2 3-simplices and the final 3 3-simplices satisfy the 3-cocycle condition. The 3-cocycle condition yields the invariance of the state sum [\[eq:DW\]](#eq:DW){reference-type="eqref" reference="eq:DW"} under the Pachner 2-3 moves. The Pachner 1-4 move creates a new vertex. Hence there is an additional factor $\frac{1}{|G|}$ in the state sum [\[eq:DW\]](#eq:DW){reference-type="eqref" reference="eq:DW"}. Also there is a new variable on an additional edge which runs through the elements of $G$. For each choice of the new variable, the values of the 3-cocycle on the initial 3-simplex and the final 4 3-simplices satisfy the 3-cocycle condition. Hence each summand in $\frac{1}{|G|}\sum$ equals the initial value of the 3-cocycle. Therefore the state sum [\[eq:DW\]](#eq:DW){reference-type="eqref" reference="eq:DW"} is invariant under the Pachner 1-4 moves. The cocycle condition can be formulated in terms of volumes. One crucial observation is the following obvious lemma. **Lemma 1**. *Given an $(n+1)$-simplex $\Delta^{n+1}$ with vertices belonging to the space ${\mathbb{R}}^{n+1}$. Then the sum of volumes of the faces of codimension 1 of a projection of $\Delta^{n+1}$ onto the subspace ${\mathbb{R}}^n$ with appropriate signs is zero.* Say, for $n=2$ one can consider a quadrilateral whose area can be split into two triangles in two ways (a quadrilateral is a projection of a tetrahedron), hence, the sum of two areas is equal to the sum of the other two areas. We can articulate it as: the "area" function on $n$-simplices (in the latter case, $2$-simplices) gives zero on the "boundary" of an $(n+1)$-simplex ($3$-simplex). In Lemma [Lemma 1](#volume){reference-type="ref" reference="volume"}, we showed that a volume is just a cocycle. Though a cocycle need not be a volume, we may try to axiomatize volumes further and further to get to cocycles finally. Therefore, $n$-cocycles can be treated as volumes of simplices in dimension $n$. **Theorem 2**. *The Dijkgraaf-Witten invariants are a partial case of the photography method.* *Proof.* We associate weights to $n$-simplices. These weights satisfy the cocycle condition which means that the sum of weights of a boundary is zero. We checked the cocycle condition. This means that we checked that the sum of volumes is zero. For example, we consider the case $n=3$. Regarding the values of cocycles as volumes up to a sign corresponding to the orientation, the cocycle condition may be interpreted as the invariance of the sum of volumes, see Fig. [1](#fig:Pachner){reference-type="ref" reference="fig:Pachner"} and Fig. [2](#fig:Pachner2){reference-type="ref" reference="fig:Pachner2"}. In the Pachner 2-3 move, the sum of the volumes of the tetrahedra 0123, 0124, and 0234 equals the sum of the volumes of the tetrahedra 0134 and 1234. This is exactly the cocycle condition (up to a sign) $$\alpha[g|h|k]+\alpha[gh|k|l]+\alpha[g|h|kl]=\alpha[g|hk|l]+\alpha[h|k|l].$$ In the Pachner 1-4 move, the volume of the tetrahedron 0123 equals the sum of the tetrahedra 0124, 0234, 0134, and 1234. This is exactly the cocycle condition (up to a sign) $$\alpha[g|h|k]=\alpha[g|h|kl]+\alpha[gh|k|l]+\alpha[g|hk|l]+\alpha[h|k|l].$$ Therefore, the data satisfy the data transmission law (the $n$-cocycle condition) for two adjacent triangulations connected by a Pachner move. The photography method constructs the Dijkgraaf-Witten invariants. ◻ Recall that in [@ManturovWanMay2023 Sec. 4], we took areas. They are not really areas in some Euclidean space nor hyperbolic space. They are formal variables, maybe from some field. So, we can predict a solution by looking at some geometry in some proper sense. Say, area of the quadrilateral equals sum of areas of constituent parts, and then we can check it by hand. Here we do almost the same. We take volumes, but volumes can not be negative, hence we take cocycles. The only thing we need is that: a cocycle is linear and its coboundary is 0. This is what is guaranteed by volumes, but this may go beyond the volume in the strict sense. # Further research {#sec:further} The authors are excited with the paper by Kevin Walker [@Walker]. In that paper, he presents a "universal" state-sum invariant. In our opinion, one of the greatest advantages of that paper is that it deals with cell decompositions/ handle slides and handle cancellations rather than with Pachner moves. But there are a lot of similarities between that work and our work: we associate some *data* to cells, and this data is *naturally changed* under *moves*. The next natural task is the Crane-Yetter invariant, which we shall do in a subsequent publication. It is important to calculate our invariants. Calculating Dijkgraaf-Witten invariants is hard and some special cases were calculated in [@de; @Hu; @Huang; @Wan; @Wang; @WangWen; @Wen]. 99 J.S. Carter, L.H. Kauffman, and M. Saito, Structures and Diagrammatics of Four Dimensional Topological Lattice Field Theories, Adv. Math. **146**, 39--100 (1999) M. de Wild Propitius. Topological interactions in broken gauge theories. PhD thesis, University of Amsterdam, 1995. R. Dijkgraaf and E. Witten, Topologcal gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), 393--429. D.S. Freed, Higher algebraic structures and quantization, Comm. Math. Phys. 159(2) (1994), 343--398. Y. Hu, Y. Wan, and Y.-S. Wu. Twisted quantum double model of topological phases in two dimensions. Physical Review B, 87(12):125114, 2013. H.-L. Huang, Z. Wan, and Y. Ye. Explicit cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf--Witten invariants. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 150(4), 1937-1964. L. Kauffman, V. O. Manturov, I. M. Nikonov, S. Kim, *Photography principle, data transmission, and invariants of manifolds,* arXiv:2307.03437 I.G. Korepanov, Private communication to V.O. Manturov, 2019. V.O. Manturov, I. Nikonov, The groups $\Gamma_n^4$, braids, and 3-manifolds, arXiv: 2305.06316. V.O.Manturov, Z.Wan, The photography method: solving pentagon, hexagon, and other equations, arXiv:2305.11945 U. Pachner: Konstruktionsmethoden und das kombinatorische Homoomorphieproblem fur Triangulationen kompakter semilinearer Mannigfaltigkeiten, Abh Math Sem Univ Hamburg 57, 69-85 (1986). U. Pachner: P.L. Homeomorphic Manifolds are Equivalent by Elementary Shellings, Europ J Combinatorics 12, 129-145 (1991). V. G. Turaev, O. Ya. Viro, State sum invariants of 3-manifolds and quantum $6j$-symbols, *Topology* **31**:4 (1992), 865--902. M. Wakui, On Dijkgraaf-Witten invariants for 3-manifolds, Osaka J. Math. 29 (1992), 675--696. K. Walker, A universal state sum, arXiv: 2104.02101 Y. Wan, J. C. Wang, and H. He. Twisted gauge theory model of topological phases in three dimensions. Physical Review B, 92(4):045101, 2015. J. Wang, K. Ohmori, P. Putrov, Y. Zheng, Z. Wan, M. Guo, H. Lin, P. Gao, and S.-T. Yau. Tunneling topological vacua via extended operators:(Spin-) TQFT spectra and boundary deconfinement in various dimensions. Progress of Theoretical and Experimental Physics, 2018(5):053A01, 2018. J. C. Wang and X.-G. Wen. Non-Abelian string and particle braiding in topological order: Modular SL(3,Z) representation and (3+1)-dimensional twisted gauge theory. Physical Review B, 91(3):035134, 2015. X.-G. Wen. Exactly soluble local bosonic cocycle models, statistical transmutation, and simplest time-reversal symmetric topological orders in 3+1 dimensions. Physical Review B, 95(20):205142, 2017.

arxiv_math

--- abstract: | We introduce a novel algorithm for solving network utility maximization (NUM) problems that arise in resource allocation schemes over networks with known safety-critical constraints, where the constraints form an arbitrary convex and compact feasible set. Inspired by applications where customers' demand can only be affected through posted prices and real-time two-way communication with customers is not available, we require an algorithm to generate "safe prices". This means that at no iteration should the realized demand in response to the posted prices violate the safety constraints of the network. Thus, in contrast to existing distributed first-order methods, our algorithm, called safe pricing for NUM (SPNUM), is guaranteed to produce feasible primal iterates at all iterations. At the heart of the algorithm lie two key steps that must go hand in hand to guarantee safety and convergence: 1) applying a projected gradient method on a shrunk feasible set to get the desired demand, and 2) estimating the price response function of the users and determining the price so that the induced demand is close to the desired demand. We ensure safety by adjusting the shrinkage to account for the error between the induced demand and the desired demand. In addition, by gradually reducing the amount of shrinkage and the step size of the gradient method, we prove that the primal iterates produced by the SPNUM achieve a sublinear static regret of ${\cal O}(\log{(T)})$ after $T$ time steps. author: - - - "[^1]" bibliography: - references.bib title: A Safe First-Order Method for Pricing-Based Resource Allocation in Safety-Critical Networks --- # Introduction Many applications falling within the scope of resource allocation over networks, e.g., power distribution systems [@samadi2010optimal], congestion control in data networks [@kelly1998rate], wireless cellular networks [@chiang2004balancing], and congestion control in urban traffic networks [@mehr2017joint], deal with a multi-user optimization problem that falls under the general umbrella of *network utility maximization* (NUM) problems. The shared goal in these problems is to *safely* and *efficiently* allocate the shared resources to the users, where safety refers to satisfying the constraints of the system that depend on the resource allocation of all the users, and efficiency refers to the total utility of the users for a given resource allocation. In NUM problems, the user-specific utility functions are assumed to be private to the users and therefore a centralized solution is not possible. Accordingly, distributed optimization methods have become suitable tools thanks to the separable structure of NUM problems [@palomar2006tutorial; @necoara2011parallel]. The idea is to decompose the main problem into sub-problems that can be solved by the individual users and use the solutions of the sub-problems to solve the main problem [@bertsekas1997nonlinear; @bertsekas2015parallel], and this has been advocated for use in different applications, e.g., [@li2011optimal; @kelly1998rate]. Among the two main types of decomposition methods, primal decomposition methods correspond to a direct allocation of the resources by a central coordinator and solve the primal problem, whereas dual decomposition methods based on the Lagrangian dual problem [@shor2012minimization] correspond to resource allocation via pricing and solve the dual problem [@palomar2006tutorial]. Due to the structure of NUM problems, the latter approach has been widely adopted in the literature [@palomar2006tutorial; @nedic2009approximate; @beck20141]. Additionally, it gives users the freedom of determining their own demand based on pricing-type signals. Although there is extensive literature on pricing algorithms based on dual decomposition, the majority of studies focus on linear constraints [@nedic2009approximate; @beck20141; @necoara2013rate; @necoara2015linear; @turan2022safe], or on non-linear constraints with the assumption of separability and full user knowledge of these constraints [@simonetto2016primal; @falsone2017dual; @notarnicola2019constraint]. Furthermore, none of the aforementioned studies propose an iterative pricing algorithm that induces resource demand satisfying the hard constraints of the problem *during* the iterative optimization process. Instead, these studies only provide bounds on the infeasibility amount of the resource demand (e.g., [@beck20141; @necoara2015linear]). Our preliminary work in [@turan2022safe] is an exception, which is limited to problems with linear inequality constraints characterized by binary matrices. Thus, pricing-based solutions can only be realized after convergence to a near-feasible point for resource allocation systems with safety-critical constraints. Therefore, implementation of such solutions requires a negotiation process through a two-way communication network if the system has hard safety-critical constraints, which can be considered impractical in many applications. The research presented in this paper is motivated by the context of network resource allocation applications, which involve a number of key considerations. First, users themselves determine their own resource demand in response to the prices, with the actual demand only becoming observable ex-post. Second, it is essential that the systems in question have safety-critical hard constraints that must not be violated by users' resource demand at any time. Finally, it is important to recognize that the constraints associated with such resource allocation systems can form arbitrary convex and compact feasible sets. One particularly relevant example of this type of application can be seen in the context of price-based demand response, in which users determine their own electricity consumption in response to prices that must be set such that the realized demand does not violate the capacity constraints of the electric grid [@vardakas2014survey]. This is necessary to ensure the safe and reliable operation of the grid, as violating the capacity constraints could have serious physical implications that could compromise the overall integrity of the system. In light of these considerations, it is evident that the resource demand of users must always satisfy the constraints of the system, even as they respond dynamically to pricing information. To this end, in this paper, we develop an iterative pricing algorithm to solve NUM problems with arbitrary convex and compact feasible sets, called safe pricing for NUM (SPNUM). We design our algorithm based solely on the realized demand in response to prices and communicate to the users only the prices for the resources at each iteration. Our contributions can be summarized as follows: - We introduce a novel algorithm, the SPNUM, for solving NUM problems with arbitrary convex and compact feasible sets through pricing. Our algorithm iteratively designs prices and allows users the freedom of determining their own decision variable based on prices according to their own profit maximization problem (without imposing any iterative variable update rule on the users). - We characterize a principled way to choose algorithm parameters to guarantee feasible primal iterates at all iterations. Furthermore, we prove that the static regret incurred by the feasible primal iterates produced by the SPNUM, i.e., the cumulative gap between the optimal objective value and the objective function evaluated at the primal iterates, up to time $T$ is bounded by ${\cal O}(\log{(T)})$. - We numerically evaluate our algorithm to support our theoretical findings and compare its performance to existing first-order distributed methods for NUM problems. To the best of the authors' knowledge, no previous work has studied pricing algorithms for NUM problems on arbitrary convex feasible sets that are unknown to the users, even without consideration of safety. While primal-dual algorithms [@zhu2011distributed; @koshal2011multiuser; @sakurama2017distributed; @turan2020resilient] can handle non-separable arbitrary convex feasible sets, they rely on a primal update rule users need to follow in order to converge as opposed to maximizing their own profit based on observed prices. To this end, our contributions extend beyond safety, since SPNUM solves NUM problems on arbitrary convex feasible sets by iteratively designing prices and allowing the users to determine their own resource demand according to their own profit maximization problem. The primal feasibility and the regret guarantees of the SPNUM result from a combination of two ingredients: 1) given prices and demand at a given instant, we apply a projected gradient method on a shrunk feasible set to get the next desired demand, and 2) we estimate the price response function of the users around the current prices and determine the next prices so that the induced demand is close to the desired demand. To ensure the algorithm behaves as a projected gradient method, the induced demand must be in the strict interior of the feasible set. The algorithm operates on a shrunk feasible set to account for the error between induced and desired demand, and gradually reduces shrinkage and step size to converge to the optimal solution. **Related work:** Besides dual (sub)gradient methods, a few other branches of literature study a similar problem to ours. We highlight how those lines of work do not meet our particular design criteria and what differentiates our work from them. Additional details on distributed optimization algorithms and their classifications can be found in the surveys [@yang2019survey; @zheng2022review]. 1. *Primal-dual methods*: Primal-dual methods tackle multi-user optimization problems with arbitrary convex global constraints by applying a projected gradient descent/ascent on the primal/dual variables of the Lagrangian [@zhu2011distributed; @koshal2011multiuser; @sakurama2017distributed; @turan2020resilient]. The dual variables are updated using the aggregate resource demand information of the users and can be used for pricing of the resources. Therefore the update rule for the dual variables meets our design goals. However, the primal variables, i.e., the resource demand of the users, are updated by applying one step of gradient descent instead of solving for the profit-maximizing optimal demand in response to prices. Accordingly, these algorithms do not resemble the selfish profit-maximizing behavior of the users we adopt in this paper. 2. *Projected gradient methods*: The main goal of the projected gradient methods is to maintain feasibility by projecting the primal variables on the feasible convex set after each update step. Scholars have extensively studied the convergence properties of the projected gradient methods under different assumptions [@bertsekas1997nonlinear; @calamai1987projected; @levitin1966constrained]. On the other hand, the main challenge brought by our setup is that the primal variables are controlled solely by the users and cannot be manipulated (e.g., projected). Even though we can determine a feasible desired resource allocation by means of a projected gradient method, the prices that induce such resource demand are unknown due to the privacy of the utility functions, which brings unique challenges not addressed by the previous literature. 3. *Interior point methods*: Interior point methods are commonly used to solve inequality-constrained problems by using barrier functions to convert them into a sequence of equality-constrained problems, which are then solved using Newton's method [@boyd2004convex]. While producing feasible iterates, the use of Newton's method requires the Hessian, which is often not available in practical applications, such as demand response without two-way communications. To address this limitation, previous works such as [@armand2000feasible] and [@wei2010distributed] have proposed feasible interior point methods that approximate the Hessian using first or second-order information exchange. However, these methods do not match the profit maximization rule we would like to preserve in this paper, which allows users to freely determine their resource consumption in response to posted prices. Closest to our setup and design goals in this paper would be [@athuraliya2000optimization; @necoara2009interior], where separable optimization problems with linear constraints are considered. While [@athuraliya2000optimization] proposes a Newton-like dual update that approximates the Hessian using first-order information, only the asymptotic convergence of the algorithm is proven and the feasibility of primal iterates is not guaranteed. [@necoara2009interior] proposes an interior point method using Lagrangian dual decomposition with theoretical guarantees, but requires the exact Hessian for dual updates. **Paper Organization:** The remainder of the paper is organized as follows. In Section [2](#sec:problem){reference-type="ref" reference="sec:problem"}, we formalize the problem setup. In Section [3](#sec:spnum){reference-type="ref" reference="sec:spnum"}, we describe the SPNUM (Algorithm [\[alg:safenum\]](#alg:safenum){reference-type="ref" reference="alg:safenum"}) and in Section [4](#sec:regret){reference-type="ref" reference="sec:regret"}, we prove its feasibility and regret guarantees. In Section [5](#sec:num){reference-type="ref" reference="sec:num"}, we provide a numerical study demonstrating the efficacy of the SDGM. **Notation and Basic Definitions:** We denote the set of real numbers by ${\mathbb R}$ and the set of non-negative real numbers by ${\mathbb R}_+$. For vectors, $\| \cdot \|$ denotes the standard Euclidean norm and $\|\cdot\|_p$ denotes the $p$-norm. For matrices, $\|\cdot\|$ denotes the matrix norm. Given a positive integer $n>0$, $[n]$ denotes the set of integers $\{1,2,\dots,n\}$. For two vectors $x,y\in{\mathbb R}^d$, $\langle x,y\rangle$ denotes the inner product of $x$ and $y$. Given a vector $x=[x_1^\top,~x_2^\top,~\dots,~x_n^{\top}]^\top\in{\mathbb R}^d$, $x_i\in{\mathbb R}^{d_i}$ denotes the $i$'th block of $x$. For a matrix $A\in\mathbb{R}^{m\times n}$, $A_j$ denotes the $j$'th row of $A$, $A_{:,j}$ denotes the $j$'th column of $A$. Given a matrix ${A}\in {\mathbb R}^{m\times m}$, $\textnormal{diag}(A)\in {\mathbb R}^m$ is the vector of the diagonals of $A$, $\kappa(A)$ is the condition number of $A$, and $\sigma_{\min}(A)$/$\sigma_{\max}(A)$ are the minimum/maximum singular values of $A$. Given a function $f:{\cal X}\subseteq{\mathbb R}^d\rightarrow{\mathbb R}$, $\nabla f$ denotes the gradient of $f$, $\nabla^k f$ denotes the $k$'th order gradient of $f$, and $\textnormal{dom}f$ denotes the domain ${\cal X}$ of $f$. Given two vectors $x,y\in {\mathbb R}^m$, $x\leq y$ implies element-wise inequality. Given a set ${\cal X}\subset\mathbb{R}^{d}$, ${\cal X}^\textnormal{int}$ denotes the interior of ${\cal X}$. Given a convex and compact set ${\cal X}\subset\mathbb{R}^{d}$ and a point $x\in{\mathbb R}^d$, ${\Pi}_{\cal X}(x)$ denotes the Euclidian projection of ${x}$ onto ${\cal X}$. We denote the closed and the open Euclidean ball with radius $r$ centered at origin as $\bar{\mathcal{B}}(r)$ and ${\cal B}(r)$, respectively. $I_d$ denotes the identity matrix of size $d$, $\bm{1}_d$ denotes the vector of all 1's with dimension $d$, and $e_i$ denotes the unit vector with $1$ in $i$'th dimension and $0$ everywhere else. **Definition 1**. *A differentiable function $f(\cdot)$ is said to be **$\boldsymbol{\mu}$-strongly concave** over the domain ${\cal X}$ if there exists $\mu>0$ such that $$\label{eq:strong} \langle \nabla f(x_2)-\nabla f(x_1),x_1-x_2 \rangle\geq \mu\|x_1-x_2\|^2$$ holds for all $x_1,x_2\in \cal X$.* **Definition 2**. *A differentiable function $f(\cdot)$ is said to be **$\boldsymbol{L}$-smooth** over the domain ${\cal X}$ if there exists $L>0$ such that $$\|\nabla f(x_1)-\nabla f(x_2)\|\leq L \|x_1-x_2\|$$ holds for all $x_1,x_2\in \cal X$.* **Definition 3**. *A function $f(\cdot)$ is said to be **$\boldsymbol{M}$-Lipschitz continuous** over the domain ${\cal X}$ if there exists $M>0$ such that $$\|f(x_1)- f(x_2)\|\leq M\|x_1-x_2\|$$ holds for all $x_1,x_2\in \cal X$.* # Problem Setup {#sec:problem} We study the standard NUM problem [@kelly1998rate], where the goal is to allocate resources to $n$ users subject to a set of coupling constraints such that the total utility of the users is maximized. It can be formulated as the following optimization problem: [\[eq:main\]]{#eq:main label="eq:main"} $$\begin{aligned} \label{eq:objective}\underset{x\in \textnormal{dom}f \subseteq \mathbb{R}^d}{\max}&~ f(x)=\sum_{i=1}^n f_i(x_i)\\ % \textnormal{s.t.}&~Ax\leq c,\label{eq:constraint} \textnormal{s.t.}&~x\in {\cal X},\label{eq:constraint}\end{aligned}$$ where $f_i(\cdot)$ is the concave utility function of user $i$ that depends on the $d_i$-dimensional vector of resource consumption, denoted by $x_i\in\textnormal{dom}f_i\subseteq {\mathbb R}^{d_i}$, and ${\cal X}\subset \mathbb{R}^d$ is the convex and compact set of feasible resource allocations. We also have $\sum_{i\in[n]}d_i=d$, $\textnormal{dom}f = \prod_{i\in[n]}\textnormal{dom}f_i$, and define $\bar{d}=\max_{i\in[n]}d_i$. For all users $i\in[n]$, we define the set ${\cal X}_i=\{x_i\in{\mathbb R}^{d_i}:\exists x\in{\cal X} \textnormal{ s.t. } x_i \textnormal{ is the } i\textnormal{'th block of } x\}$ as the set of values that user $i$'s resource demand vector can take in the aggregate feasible set ${\cal X}$. Note that since ${\cal X}$ is convex and compact, ${\cal X}_i$ is convex and compact, ${\forall i\in[n]}$. Furthermore, if $x\in {\cal X}$, then $x_i\in {\cal X}_i$ and if $x\in {\cal X}^{\textnormal{int}}$, then $x_i\in {\cal X}_i^\textnormal{int}$ hold by definition. We make the following assumptions on the feasible set ${\cal X}$, and on the utility functions over ${\cal X}_i$, $\forall i\in[n]$. **Assumption 1**. *The feasible set ${\cal X}$ is a subset of $\textnormal{dom}f$, i.e., ${\cal X}\subseteq \textnormal{dom}f$. The diameter of the feasible set ${\cal X}$ is bounded by $R$, i.e., $\|x-y\|\leq R$, $\forall x,y\in {\cal X}$. There exists a vector $\tilde{x}$ in the interior of ${\cal X}$ such that $\tilde{x}\in{\cal X}^{\textnormal{int}}$.* **Assumption 2**. *For all $i\in[n]$, the utility function $f_i(\cdot)$ is $\mu$-strongly concave, $L$-smooth, $M$-Lipschitz continuous, and has $\beta$-smooth gradient over ${\cal X}_i$.* **Example 1** (Utility function). *For instance, take $f_i(x_i)=f_\alpha(x_i)$ to be an $\alpha$-fair utility function (see [@mo2000fair]) and let ${\cal X}_i=[\stackunder[1.2pt]{$x$}{\rule{.8ex}{.075ex}}_i,\bar{x}_i]$ with $\stackunder[1.2pt]{$x$}{\rule{.8ex}{.075ex}}_i>0$. We have that $\nabla f_i(x_i)\leq 1/{\stackunder[1.2pt]{$x$}{\rule{.8ex}{.075ex}}}_i^{\alpha}$, $-\alpha/{\stackunder[1.2pt]{$x$}{\rule{.8ex}{.075ex}}}_i^{\alpha+1}\leq \nabla^2 f_i^(x_i)\leq-\alpha/{\bar{x}}_i^{\alpha+1}$, and $\alpha(\alpha+1)/\bar{x}_i^{\alpha+2}\leq \nabla ^3f_i(x_i)\leq\alpha(\alpha+1)/\stackunder[1.2pt]{$x$}{\rule{.8ex}{.075ex}}_i^{\alpha+2}$, $\forall x\in{\cal X}_i$. Therefore, $f_i(x_i)$ is $\alpha/{\bar{x}}_i^{\alpha+1}$-strongly concave, $\alpha/{\stackunder[1.2pt]{$x$}{\rule{.8ex}{.075ex}}}_i^{\alpha+1}$-smooth, and $1/{\stackunder[1.2pt]{$x$}{\rule{.8ex}{.075ex}}}_i^{\alpha}$-Lipschitz continuous, and has $\alpha(\alpha+1)/\stackunder[1.2pt]{$x$}{\rule{.8ex}{.075ex}}_i^{\alpha+2}$-smooth gradient over ${\cal X}_i$.* Under Assumption [Assumption 2](#ass:utility){reference-type="ref" reference="ass:utility"}, the objective function [\[eq:objective\]](#eq:objective){reference-type="eqref" reference="eq:objective"} is strongly concave with coefficient $\mu$. Accordingly, the convex optimization problem [\[eq:main\]](#eq:main){reference-type="eqref" reference="eq:main"} has a unique solution denoted by $x^\star$ and an optimal objective value denoted by $f^\star$. Since $f_i(\cdot)$ are private to the users, [\[eq:main\]](#eq:main){reference-type="eqref" reference="eq:main"} cannot be solved centrally. Therefore, distributed optimization methods based on the dual decomposition framework have been proposed in the literature (e.g., [@palomar2006tutorial] for the case when ${\cal X}$ is a polytope) in order to incentivize selfish users with private utility functions to follow the optimal global solution. The common high-level idea is to divide the main problem into subproblems that can be solved by the individual users upon observing a pricing signal, and iteratively design prices $\{p^0,p^1,\dots\}$ to converge to the optimal resource allocation vector $x^\star$. In this framework, upon observing a price $p_i\in {\mathbb R}^{d_i}$, each user $i\in[n]$ determines their own decision variable according to their own profit maximization problem: $$\label{eq:priceresponse} g_i(p_i) = \underset{x_i\in\textnormal{dom} f_i}{\mathop{\mathrm{arg\,max}}}f_i(x_i) - \langle p_i,x_i\rangle.$$ We call $g_i(\cdot)$ the price response function of user $i$ and let $g(p) = [g_1(p_1)^\top,~g_2(p_2)^\top,\dots,~g_n(p_n)^{\top}]$ be the concatenated vector of price responses given a price vector $p\in{\mathbb R}^d$. In the next section, we propose an algorithm to iteratively design $p^t,~\forall t\geq 1$, that produce feasible primal solutions, i.e., $x^t\in{\cal X},~\forall t\geq 1$, where $x_i^t=g_i(p_i^t)$ is determined by user $i$ through [\[eq:priceresponse\]](#eq:priceresponse){reference-type="eqref" reference="eq:priceresponse"}. In addition, the algorithm should produce primal iterates that result in a sublinear static regret per user, which is measured by $$\label{eq:regret} R(T)=\frac{1}{n}\sum_{t=1}^T f^\star-f(x^t).$$ It is worthwhile to highlight that even without the safety criterion, the literature on distributed optimization methods does not provide a distributed solution based on pricing to [\[eq:main\]](#eq:main){reference-type="eqref" reference="eq:main"} with any type of convergence guarantees. Existing works in the literature 1) utilize a pricing algorithm based on the dual decomposition framework but consider linear constraints [@nedic2009approximate; @beck20141; @necoara2013rate; @necoara2015linear; @turan2022safe] or non-linear and separable constraints known by the users [@simonetto2016primal; @falsone2017dual; @notarnicola2019constraint], or 2) solve the Lagrangian dual problem by primal-dual methods [@zhu2011distributed; @koshal2011multiuser; @sakurama2017distributed; @turan2020resilient], which restrict the users to follow a primal update method that cannot be enforced in the setting where users only care about maximizing their own profit dictated by [\[eq:priceresponse\]](#eq:priceresponse){reference-type="eqref" reference="eq:priceresponse"}. Therefore, a pricing algorithm that induces a sequence of primal iterates converging to the optimal solution of [\[eq:main\]](#eq:main){reference-type="eqref" reference="eq:main"} with general convex and compact feasible sets ${\cal X}$ is novel in the distributed optimization literature. Additionally, we note that the definition of regret in [\[eq:regret\]](#eq:regret){reference-type="eqref" reference="eq:regret"} quantifies the difference between the efficiencies of the optimal resource allocation and the proposed algorithm up to time $T$. When the primal iterates $\{x^t\}_{t\in[T]}$ are in the feasible set ${\cal X}$, users' resource demand can actually be realized through the posted prices without waiting for the convergence of the algorithm, and therefore regret is a meaningful measure. On the other hand, although the above sum is computable for many of the existing works mentioned earlier (e.g., [@beck20141; @necoara2013rate] with linear constraints), they do not guarantee feasible primal iterates but only establish bounds on the amount of constraint violation at a given iteration $t$. Therefore, solutions are only realizable after convergence to a near-feasible point for resource allocation systems with safety-critical constraints. As such, they can be viewed as complex negotiations with users over what their potential demand would be in response to different prices in order to converge to the optimal price, which renders regret a less meaningful measure. By incorporating primal feasibility into our design goals, we aim to continually allocate resources to the users through posted prices *during the iterative optimization process* and measure the overall efficiency of this process through regret. # Safe Pricing Algorithm for NUM {#sec:spnum} In this section, we describe the price update algorithm we propose, called Safe Pricing for NUM (SPNUM), that produces feasible primal iterates satisfying a sublinear regret. To do so, we will use some definitions and results from [@spencerl4dc] regarding the geometric properties of convex and compact sets. While the primary focus of [@spencerl4dc] centers on a linear stochastic bandit setup that bears little resemblance to the NUM setup under study, the definitions of the shrunk set outlined in the former are applicable to the present context as well. ## Geometric Properties of the Feasible Set The main ingredient that ensures the safety of SPNUM is that it operates on a shrunk feasible set, which is formally defined as follows: **Definition 4**. *For a compact set $\mathcal{X} \subset \mathbb{R}^d$ and a positive scalar $\Delta\in{\mathbb R}_+$, we define the **shrunk version** of $\mathcal{X}$ as $\mathcal{X}_{\Delta}:= \{ x \in \mathcal{X} : x + v \in \mathcal{X}, \forall v \in \bar{\cal B} (\Delta) \}$.* **Example 2**. *(Shrunk polytope) Let $A\in {\mathbb R}^{m\times d}$ and $\mathcal{X} = \{ x \in \mathbb{R}^d : Ax \leq c\}$ be a polytope. The shrunk version of ${\cal X}$ is defined as ${\cal X}_{\Delta}= \{ x \in \mathbb{R}^d : A_j ^{\top} x \leq c_j - \Delta \| A_j \|,~j\in[m] \}$.* **Remark 1**. *If ${\cal X}$ is convex and compact, then ${\cal X}_{\Delta}$ is also convex and compact.[^2]* Given the above definition of the shrunk version of a set, one can consider the maximum shrinkage that a set can withstand while still being nonempty. We introduce the *maximum shrinkage of a set* in the following definition. **Definition 5**. *For a compact set $\mathcal{X} \subset \mathbb{R}^d$, we define the **maximum shrinkage** of $\mathcal{X}$, as $H_{\mathcal{X}}:= \sup\{ \Delta : \mathcal{X}_{\Delta} \neq \emptyset \}$.* ## Description of the Algorithm $p^0$, $\Delta^t$, $\gamma^t$, $\eta^t$. *(Initialization stage)*: [\[step:initialx\]]{#step:initialx label="step:initialx"} Each user $i\in[n]$ receives $p_i^{0}$ and $p_i^{-t} = p_i^0+\eta^0 e_{1+\mathrm{mod}(t,d_i)}$, $\forall t\in[d_i]$ and solves $$\label{eq:initx} x_i^{t}=g_i(p_i^t),~t=-d_i,-d_i+1,\dots, 0.$$ [\[step:initialjacobian\]]{#step:initialjacobian label="step:initialjacobian"} For all $i\in[n]$, estimate the Jacobian of $g_i$ as: $$\hat{\nabla}g_i^0=\left[\frac{x_i^{-d_i}-x_i^0}{\eta^0},\dots,~\frac{x_i^{-1}-x_i^0}{\eta^0}\right]$$ *(Update stage)* [\[step:updatestage\]]{#step:updatestage label="step:updatestage"} [\[step:xupdate\]]{#step:xupdate label="step:xupdate"} Compute $\hat{x}^{t+1} = \Pi_{{\cal X}_{\Delta^t}}(x^t + \gamma^t p^t)$. [\[step:pupdate\]]{#step:pupdate label="step:pupdate"}Set ${p}_i^{t+1} = p_i^t + [\hat{\nabla}g_i^t]^{-1}(\hat{x}_i^{t+1}-x_i^t)$, for all $i\in [n]$. Each user $i\in[n]$ receives $p_i^{t+1}$ and solves $$x_i^{t+1}=g_i(p_i^{t+1})$$ *(Sampling stage)* [\[step:samplingstage\]]{#step:samplingstage label="step:samplingstage"} [\[step:sampleprices\]]{#step:sampleprices label="step:sampleprices"}Each user $i\in[n]$ receives $p_i^{t+1,s}=p_i^{t+1}+\eta^{t+1}e_{1+\mathrm{mod}(t,d_i)}$ and solves $$x_i^{t+1,s}=g_i(p_i^{t+1,s})$$ [\[step:jacobianupdate\]]{#step:jacobianupdate label="step:jacobianupdate"}For each user $i\in[n]$ $$\begin{aligned} &[\hat{\nabla}g_i^t]_{:,1+\mathrm{mod}(t,d_i)}\leftarrow (x_i^{t+1,s}-x_i^{t+1})/{\eta^{t+1}}\\ &\hat{\nabla}g_i^{t+1} = \hat{\nabla}g_i^t \end{aligned}$$ [\[alg:safenum\]]{#alg:safenum label="alg:safenum"} The proposed method, called safe pricing for NUM (SPNUM) and outlined in Algorithm [\[alg:safenum\]](#alg:safenum){reference-type="ref" reference="alg:safenum"}, consists of two stages at each iteration: 1) update stage (Step [\[step:updatestage\]](#step:updatestage){reference-type="ref" reference="step:updatestage"}) and 2) sampling stage (Step [\[step:samplingstage\]](#step:samplingstage){reference-type="ref" reference="step:samplingstage"}). The update stage proceeds similarly to a projected gradient method on the primal iterates while designing prices that induce realized iterates close to a *desired iterate*. The sampling stage estimates the Jacobians of the price response functions of the users, which are used during the update stage. In the update stage, the algorithm first determines a desired next iterate $\hat{x}^{t+1}$ in Step [\[step:xupdate\]](#step:xupdate){reference-type="ref" reference="step:xupdate"}. However, because the primal variables are not directly controllable, prices that induce $x^{t+1}$ that is close to $\hat{x}^{t+1}$ have to be determined at Step [\[step:pupdate\]](#step:pupdate){reference-type="ref" reference="step:pupdate"}. Accordingly, at the heart of the update stage lie two key steps: 1. At iteration $t$, the central coordinator observes $x^t$ and determines the next desired iterate $\hat{x}^{t+1}$ by means of a projected gradient ascent step in Step [\[step:xupdate\]](#step:xupdate){reference-type="ref" reference="step:xupdate"}. This is because if $x^t\in{\cal X}^\textnormal{int}$, then $x_i\in{\cal X}_i^\textnormal{int}$, which implies that $p_i^t = \nabla f_i(x^t)$ by Assumption [Assumption 2](#ass:utility){reference-type="ref" reference="ass:utility"} and the first order optimality condition for [\[eq:priceresponse\]](#eq:priceresponse){reference-type="eqref" reference="eq:priceresponse"}. Therefore, $p^t=\nabla f(x^t)$. In addition, projection is performed onto a shrunk set ${\cal X}_{\Delta^t}$, where $\Delta^t$ controls the amount of shrinkage at time $t$. This is the key ingredient to ensure the safety of the algorithm because the uncertainty in the price response functions will cause the actual induced iterate $x^{t+1}$ in response to the price vector $p^{t+1}$ to deviate from the desired iterate $\hat{x}^{t+1}$. By adding this safety margin to the constraint, we can ensure safety if $\|x^{t+1}-\hat{x}^{t+1}\|\in\bar{\cal B}(\Delta^t)$. Finally, by utilizing a diminishing safety margin sequence $\{\Delta^t\}_{t\geq 0}$, we can ensure convergence to the optimal solution of [\[eq:main\]](#eq:main){reference-type="eqref" reference="eq:main"}. 2. Once the desired next iterate $\hat{x}^{t+1}$ is determined, the central coordinator has to determine $p_i^{t+1}$ that would ideally induce $\hat{x}_i^{t+1}$, $\forall i\in[n]$. However, the price response function is unknown to the central coordinator, and therefore an exact solution is not possible. Instead, the central coordinator makes a linear approximation of the price response function using the Jacobian estimate of $g_i$, $\forall i\in[n]$. In particular, the central coordinator keeps an estimate of the Jacobian denoted by $\hat{\nabla}g_i^t$ initialized in Steps [\[step:initialx\]](#step:initialx){reference-type="ref" reference="step:initialx"} and [\[step:initialjacobian\]](#step:initialjacobian){reference-type="ref" reference="step:initialjacobian"} of the algorithm, which is constructed by varying the price vector along each dimension and estimating the gradient using the difference equation. This results in the following linear approximation of the price response function around $p_i^t$: $$\label{eq:ghat} \hat{g}_i(p) = x_i^t + \hat{\nabla}g_i^t(p-p_i^{t}).$$ By setting $p={p}_i^{t+1}$, $\hat{g}_i({p}_i^{t+1})=\hat{x}^{t+1}$, and rearranging, we get the price update rule in Step [\[step:pupdate\]](#step:pupdate){reference-type="ref" reference="step:pupdate"}. This requires that the $\hat{\nabla}g_i^t$ is an invertible matrix, which will be proven in Section [4](#sec:regret){reference-type="ref" reference="sec:regret"}. After determining $p^{t+1}$ and $x^{t+1}$, the algorithm proceeds to the sampling stage to update the Jacobian estimates. To achieve this, the central coordinator varies the price vector $p_i^{t+1}$ along the dimension $1+\mathrm{mod}(t,d_i)$ in Step [\[step:sampleprices\]](#step:sampleprices){reference-type="ref" reference="step:sampleprices"} for user $i\in[n]$, resulting in a sampling price of ${p_i^{t+1,s}}$. The response is observed and denoted as $x_i^{t+1,s}$. The difference between $x_i^{t+1,s}$ and $x_i^{t+1}$ divided by the amount of price variation serves as an estimate of the gradient of the price response function along the $1+\mathrm{mod}(t,d_i)$'th principal axis, which becomes the $1+\mathrm{mod}(t,d_i)$'th column of the Jacobian estimate $\hat{\nabla}g_i^{t+1}$ in Step [\[step:jacobianupdate\]](#step:jacobianupdate){reference-type="ref" reference="step:jacobianupdate"}. It is worthwhile to highlight that for a user $i$, the error between $\hat{x}_i^{t+1}$ and $x_i^{t+1}$ has two sources: 1) the difference between the estimated Jacobian and the actual Jacobian, i.e., $\hat{\nabla}g_i^t-\nabla g_i(p_i^t)$, and 2) the high order terms not captured by the linear approximation, i.e., $R_1=g_i(p^t)-\nabla g_i(p^t)(p-p^t)$. It is necessary that there exists an initial price vector $p^0$ such that the demand vectors in response to the initial sampling prices in [\[eq:initx\]](#eq:initx){reference-type="eqref" reference="eq:initx"} are in ${\cal X}^\textnormal{int}$ so that the algorithm can proceed as described above. Since this has to hold before getting any feedback from the users, we make the following assumption: **Assumption 3**. *There exists a known price vector $p^0$ such that $g(p^0)\in{\cal X}^{\textnormal{int}}$ and for all $i\in[n]$, $x_i^{-d_i}\in{\cal X}_i^\textnormal{int}$.* The above assumption guarantees that the initial demand vectors in [\[eq:initx\]](#eq:initx){reference-type="eqref" reference="eq:initx"} are in ${\cal X}^\textnormal{int}_i,~\forall i\in[n]$ and therefore the initial Jacobian estimation is meaningful. **Remark 2**. *One way to satisfy Assumption [Assumption 3](#ass:initialprices){reference-type="ref" reference="ass:initialprices"} is to choose $\eta^0$ such that ${\cal X}_{\frac{\sqrt{n}\eta^0}{\mu}}$ is non-empty and $p^0$ such that $g(p^0)\in {\cal X}_{\frac{\sqrt{n}\eta^0}{\mu}}$, which is proven in Appendix [6.8](#app:initialset){reference-type="ref" reference="app:initialset"}.* In the next section, we characterize a principled way to choose parameters $\Delta^t$, $\gamma^t$, and $\eta^t$ in order to produce feasible primal iterates. Additionally, we prove that the regret incurred by the iterates produced by Algorithm [\[alg:safenum\]](#alg:safenum){reference-type="ref" reference="alg:safenum"} is ${\cal O}(\log(T))$ after $T$ iterations, and the last iterate converges to the optimal solution at the rate ${\cal O}(\log(T)/T)$. # Feasibility and Regret Analysis {#sec:regret} In order to prove the safety and the regret guarantees of our algorithm, we will need to bound the distance between a point in $x\in {\cal X}$ and its projection onto the shrunk set $\Pi_{{\cal X}_{\Delta}}(x)$. The following definition from [@spencerl4dc] formalizes this notion called the *sharpness of a set*, which is defined as the maximum distance from any point in a set to the projection of it onto the shrunk version of that set. **Definition 6**. *For a convex and compact set $\mathcal{X} \subset \mathbb{R}^d$, we define the **sharpness** of $\mathcal{X}$ as $$\mathrm{Sharp}_{\mathcal{X}} (\Delta) \vcentcolon= \sup_{x \in {\cal X}}\|\Pi_{{\cal X}_\Delta}(x)-x\|,$$ for all non-negative $\Delta$ such that $\mathcal{X}_{\Delta}$ is nonempty.* The following proposition establishes a bound on the sharpness of convex and compact sets as a linear function of ${\Delta}$: **Proposition 1**. *[@spencerl4dc Corollary 11] [\[prop:sharp_conv\]]{#prop:sharp_conv label="prop:sharp_conv"} For a convex, compact set $\mathcal{X} \subset \mathbb{R}^d$ with non-empty interior, we have that $\mathrm{Sharp}_{\mathcal{X}} (\Delta) \leq \Gamma_{\mathcal{X}} \Delta$ where $\Gamma_{\mathcal{X}}\geq 1$ is a constant that depends only on the geometry and the dimension of $\mathcal{X}$.* **Example 3** (Sharpness of a polytope [@spencerl4dc]). *Let $\mathcal{X} = \{ x \in \mathbb{R}^d : Ax \leq c\}$ be a polytope with a nonempty interior. Define $\mathcal{I}_A$ to refer to the collection of all sets of $d$ indices such that for each $\{i_1, i_2, ..., i_d\} \in \mathcal{I}_A$ the vectors $A_{i_1}, A_{i_2}, ..., A_{i_d}$ are linearly independent. For each $\ell \in \mathcal{I}_A$ where $\ell = \{i_1, i_2, ..., i_d\}$, we define $A^\ell = [A_{i_1}^\top\ A_{i_2}^\top\ ...\ A_{i_d}^\top]^\top$. We have that $\mathrm{Sharp}_{\mathcal{X}}(\Delta) \leq \sqrt{d} K_{\mathcal{X}} \Delta$, where $K_{\mathcal{X}} := \max_{\ell \in \mathcal{I}_A} \kappa(A^\ell)$.* **Example 4** (Sharpness of a ball in ${\mathbb R}^d$). *Let ${\cal X}=\{x\in\mathbb{R}^d:(x-x_0)^\top (x-x_0)\leq r^2\}$ be a ball in ${\mathbb R}^d$ with radius $r$ centered at $x_0$. We have that $\mathrm{Sharp}_{\mathcal{X}}(\Delta) = \Delta$.* Although we do not specify a closed-form expression of $\Gamma_{\cal X}$ for a general convex and compact set $\cal X$, it relates to the sharpness of polytopes that are contained in ${\cal X}$, which have closed-form bounds as given by Example [Example 3](#ex:sharppolytope){reference-type="ref" reference="ex:sharppolytope"}. We refer the reader to [@spencerl4dc] (Proposition 10) for a detailed discussion. The next lemma characterizes the regularity properties of $g_i(p_i)$ over the set of prices that induce a resource demand in ${\cal X}_i^\textnormal{int}$ for a user $i\in[n]$. This property is crucial for our analysis and for the feasibility of the algorithm, as we need to show that the inverse of the matrix $\hat{\nabla}g_i^t$ for the price update rule in Step [\[step:pupdate\]](#step:pupdate){reference-type="ref" reference="step:pupdate"} is a valid operation. **Lemma 1**. *Let ${\cal P}_i=\{p_i\in{\mathbb R}^{d_i}:g_i(p_i)\in{\cal X}_i^\textnormal{int}\}$ be the set of prices that induce a resource demand in ${\cal X}_i^{\textnormal{int}}$ for a user $i\in[n]$. Over ${\cal P}_i$, $g_i(p_i)$ is bijective, $1/\mu$-Lipschitz continuous, and $\beta/\mu^3$-smooth. Accordingly, $g_i(p_i)$ is invertible and $\nabla g_i(p_i) = [\nabla^2f_i(g_i(p_i))]^{-1}$.* The proof of Lemma [Lemma 1](#lem:gprops){reference-type="ref" reference="lem:gprops"} can be found in Appendix [6.1](#app:gprops){reference-type="ref" reference="app:gprops"}. Lemma [Lemma 1](#lem:gprops){reference-type="ref" reference="lem:gprops"} establishes that the true Jacobian of the price response function for user $i$ is invertible because it corresponds to the inverse of the Hessian of the strongly concave utility function of user $i$. However, this does not imply that the estimated Jacobian $\hat{\nabla}g_i^t$ is invertible since it is constructed by finite difference gradient approximation. The next lemma states that the estimated Jacobian $\hat{\nabla}g_i^t$ is close enough to $\nabla g_i(p^t)$, which allows us to bound the minimum singular value of it and therefore guarantees invertibility with the appropriate choice of algorithm parameters. **Lemma 2**. *Let $\gamma^t=1/(\mu(t+\tau))$, $\Delta^t=\Delta/(t+\tau)^2$, and $\eta^t=\mu\Delta^{t-1}/(4\sqrt{n})$ for some $\Delta>0$ and $$\begin{aligned} \label{eq:taucondition} \nonumber&\tau=\max\Big\{2,2\bar{d}-1,1+{2\mu\Delta\Gamma_{\cal X}}/({M\sqrt{n}}),\sqrt{{\Delta}/{{ H}_{\cal X}}},\\ &\hspace{1cm}{L\beta M\sqrt{\bar{d}}\left(\mu+32L\Gamma_{\cal X}\sqrt{n}(\bar{d}-1)\right)}/({2\mu^4\Gamma_{\cal X}})\Big\}. \end{aligned}$$ Suppose that at iteration $t$, $x^k\in{{\cal X}}_{\frac{\eta^k\sqrt{n}}{\mu}}^\textnormal{int}$, $\forall k\in[\max\{t-\bar{d}+1,0\},t]$. Then, the following holds for all $i\in[n]$: $$\begin{aligned} \|\hat{\nabla}g_i^t-\nabla g_i(p_i^t)\|\leq e_i^t, \end{aligned}$$ where $$\hspace{-.1cm}e_i^t {=} \frac{2\beta\sqrt{d_i}}{\mu^3}\left(\eta^{t}{+}2L(d_i{-}1)(M\sqrt{n}\gamma^t{+}2\Delta^t\Gamma_{\cal X})\right)\leq \frac{1}{2L}.$$ Accordingly, $\sigma_{\min}(\hat{\nabla}g_i^t)\geq \frac{1}{2L}$ and therefore $\hat{\nabla}g_i^t$ is invertible.* The proof of Lemma [Lemma 2](#lem:nablagerror){reference-type="ref" reference="lem:nablagerror"} can be found in Appendix [6.2](#app:nablagerror){reference-type="ref" reference="app:nablagerror"}. Lemma [Lemma 2](#lem:nablagerror){reference-type="ref" reference="lem:nablagerror"} characterizes a principled way to choose the algorithm parameters with respect to a free parameter $\Delta$ in order to bound the difference between $\hat{\nabla}g_i^t$ and $\nabla g_i(p^t)$. In the following subsections, we will first characterize the choice of $\Delta$ that guarantees primal feasibility at all iterations and then prove the regret and convergence guarantees of Algorithm [\[alg:safenum\]](#alg:safenum){reference-type="ref" reference="alg:safenum"} under this choice of parameters. ## Feasibility Analysis The following proposition characterizes the choice of the parameters $\Delta^t$, $\gamma^t$, and $\eta^t$ to ensure feasible primal iterates: **Proposition 2**. *Let $\gamma^t = 1/(\mu(t+\tau))$ and $\Delta^t = \Delta/(t+\tau)^2$, $\eta^t=\mu\Delta^{t-1}/(4\sqrt{n})$, where $\tau$ is given by [\[eq:taucondition\]](#eq:taucondition){reference-type="eqref" reference="eq:taucondition"} and $$\begin{aligned} \Delta&= {\beta LMn^{3/2}(6L+\sqrt{d}(\mu/\sqrt{n}+32L(\bar{d}-1)))}/{\mu^5}.\end{aligned}$$ Then for all $t\geq 0$, $\|\hat{x}^{t+1}-x^{t+1}\|<3\Delta^t/4$ and $\|x^{t+1}-x^{t+1,s}\|\leq\Delta^t/4$. Accordingly, for all $t\geq 0$, the iterates $x^t$ and $x^{t,s}$ produced by Algorithm [\[alg:safenum\]](#alg:safenum){reference-type="ref" reference="alg:safenum"} are feasible and in the strict interior of the feasible set, i.e., $x^t\in{\cal X}_{\frac{\eta^t\sqrt{n}}{\mu}}^\textnormal{int}$ and $x^{t,s}\in{\cal X}^{\textnormal{int}},~\forall t\geq 1$.* The proof of Proposition [Proposition 2](#prop:safety){reference-type="ref" reference="prop:safety"} can be found in Appendix [6.3](#app:safety){reference-type="ref" reference="app:safety"}. Given that under Proposition [Proposition 2](#prop:safety){reference-type="ref" reference="prop:safety"}, $x^t$ for all $t\geq 1$ are feasible and therefore implementable, the static regret [\[eq:regret\]](#eq:regret){reference-type="eqref" reference="eq:regret"} is a valid choice of performance metric. Next, we prove that the regret of Algorithm [\[alg:safenum\]](#alg:safenum){reference-type="ref" reference="alg:safenum"} is ${\cal O}(\log(T))$ and the primal variables converge to the optimal solution at the rate ${\cal O}(\log(T)/T)$. ## Regret and Convergence Analysis As our algorithm alternates between executing one update and one sampling stage, after $T$ iterations it will have executed $T/2$ update stages and $T/2$ sampling stages. In this case, the regret per user is fairly calculated as: $$R(T)=\frac{1}{n} \sum_{t=1}^{T/2} (f(x^\star)-f(x^t)+f(x^\star)-f(x^{t,s})).$$ The following theorem establishes an upper bound on the regret incurred by the primal iterates produced by Algorithm [\[alg:safenum\]](#alg:safenum){reference-type="ref" reference="alg:safenum"}, and the squared distance between last iterate $x^{T/2}$ and the optimum solution $x^\star$: **Theorem 1**. *Let $p^0$, $\gamma^t$, $\Delta^t$, and $\eta^t$ be chosen as in Proposition [Proposition 2](#prop:safety){reference-type="ref" reference="prop:safety"}. Then for all $t\geq 0$, the iterates produced by Algorithm [\[alg:safenum\]](#alg:safenum){reference-type="ref" reference="alg:safenum"} are feasible. Furthermore, the regret $R(T)$ for $T\geq 2$ satisfies $$\label{eq:regretbound} R(T)\leq {\cal O}(\log(T)(1+\Delta\Gamma_{\cal X}/n)),$$ where ${\cal O}(\cdot)$ hides other constants. In addition, the last primal iterate $x^{T/2}$ satisfies $$\|x^{T/2}-x^\star\|^2\leq {\cal O}(\log(T)/T).$$* *Proof outline:* Since the algorithm proceeds similarly to a projected gradient method, the proof is similar to that of a projected gradient ascent for strongly concave functions. We have an additional error term due to $\|x^{t+1}-\hat{x}^{t+1}\|$, which is ${\cal O}(\Delta^t)$. The error term impacts the result as ${\cal O}(\sum_{t=1}^{T/2} \Delta^t/\gamma^t)$, which results in an additive ${\cal O}(\log(T)\Delta\Gamma_{\cal X}/n)$ term. The complete proof of Theorem [Theorem 1](#thm:regret){reference-type="ref" reference="thm:regret"} and the explicit constants of [\[eq:regretbound\]](#eq:regretbound){reference-type="eqref" reference="eq:regretbound"} can be found in Appendix [6.6](#app:regret){reference-type="ref" reference="app:regret"}. According to Theorem [Theorem 1](#thm:regret){reference-type="ref" reference="thm:regret"}, Algorithm [\[alg:safenum\]](#alg:safenum){reference-type="ref" reference="alg:safenum"} produces feasible solutions that achieve a sublinear regret of ${\cal O}(\log(T))$. Furthermore, the primal variables induced by the prices converge to the optimal solution at the rate ${\cal O}(\log(T)/T)$. **Remark 3**. *When $d_i=1,~\forall i\in[n]$, $\Delta = {\cal O(}\beta n^{3/2})$ and $R(T)={\cal O}(\log(T)(1+\sqrt{n}\beta\Gamma_{\cal X}))$.* In the next section, we numerically demonstrate our theoretical results about the primal variables induced by Algorithm [\[alg:safenum\]](#alg:safenum){reference-type="ref" reference="alg:safenum"} and compare its performance to existing pricing algorithms. # Numerical Studies {#sec:num} In this section, we demonstrate the efficacy of SPNUM via three numerical studies: 1) a benchmarking study to compare SPNUM's convergence and feasibility performance to existing pricing methods that solve the NUM problem, 2) a toy NUM problem with a non-linear feasible set to demonstrate the success of SPNUM on non-linear feasible sets, and 3) a parameter study to demonstrate how the regret depends on the second order smoothness parameter $\beta$, sharpness parameter ${\Gamma}_{\cal X}$, and the number of users $n$. ## Benchmarking Study {#fig:binA width="\\textwidth"} {#fig:realA width="\\textwidth"} {#fig:spnum_nonlinear width=".42 \\textwidth"} In this study, our aim is to compare the safety and convergence performance of SPNUM to the existing algorithms on feasible sets characterized by linear inequalities, i.e., ${\cal X}=\{x\in{\mathbb R}^d: Ax\leq c\}$. We compare SPNUM to DG [@necoara2015linear], which can achieve a linear convergence rate, and SDGM [@turan2022safe], which can provide safety when $A$ is a binary matrix. We have implemented all algorithms on two types of $A$ matrices: 1) $A$ is a binary matrix and 2) $A$ is a real matrix. For both cases, we randomly generated a collection of 50 networks with a random number of users $n$ taking (integer) values in range $[5,20]$, and a random number of constraints $m$ taking values in the interval $[5,10]$ (generated independently). Inspired by [@necoara2015linear], for all users $i\in[n]$, we let the utility function be $f_i (x_i) = -0.5 (x_i-3)^2-x_i-\theta_i \log(1+e^{x_i})$, where $\theta_i$ is sampled uniformly from $[0,1]$ for each network configuration (we shifted the quadratic term by $3$ to ensure that the optimal solution is on the boundary of the feasible set). We set $\textnormal{dom}f_i=[0,1]$ for all $i\in[n]$. For each network configuration, we first randomly generated a matrix $\hat{A}$ by sampling $m\times n$ Bernoulli random variables for the binary matrix case, and by sampling $m\times n$ random variables from the continuous uniform distribution in $[-1,1]$ for the real matrix case. We then let $A = [\hat{A}^\top~I_n]^\top$. For the binary case, we let $c=\bm{1}_{m+n}$, and for the real case, we let $c=[0.1\bm{1}_m^\top~\bm{1}_n^\top]^\top$.[^3] We note that ${\cal X}_i \subseteq [0,1],~\forall i\in[n]$. Within ${\cal X}_i$, using bounds on $\theta_i$ and computing the derivatives of $f_i$, we get $M=2$, $L=5/4$, $\mu=1$, $\beta=\sinh(1)/(2(1+\cosh(1))^2)\approx0.0909$. Finally, from Example [Example 3](#ex:sharppolytope){reference-type="ref" reference="ex:sharppolytope"} we have ${\Gamma}_{\cal X} \leq \sqrt{n}\kappa(A)$. For each configuration, we initialized the dual variables and prices to induce $x_i^0=\eta^0/\mu,~\forall i\in[n]$, and ran all three algorithms for a horizon of $T=1000$. We demonstrate the results for the binary matrix case and the real matrix case in Figure [1](#fig:binA){reference-type="ref" reference="fig:binA"} and Figure [2](#fig:realA){reference-type="ref" reference="fig:realA"}, respectively. In Figure [1](#fig:binA){reference-type="ref" reference="fig:binA"} we observe that **1)** while DG converges the fastest, it is not safe, **2)** SDGM and SPNUM converge slower but are safe, and **3)** SDGM converges faster than SPNUM because it is designed specifically for this setting. On the other hand, in Figure [2](#fig:realA){reference-type="ref" reference="fig:realA"} we observe that **1)** SDGM does not provide safety and convergence when $A$ is a real matrix, as its assumptions do not hold anymore (note that the plot for $\|x^t-x^\star\|^2$ flattens for SDGM), **2)** SPNUM successfully provides safety and convergence. {width=".8\\textwidth"} ## SPNUM on Non-linear Feasible Set This study aims to demonstrate numerically the regret and safety guarantees of SPNUM on a problem with a feasible set characterized by non-linear inequalities. We select the feasible set ${\cal X} = \{x\in{\mathbb R}^d: \|x\|\leq 1\}$ as the unit ball in ${\mathbb R}^d$ centered at the origin. At the beginning of each run, we sample the number of users $n$ as an integer from the range $[5,20]$ uniformly at random. For all $i\in[n]$, we let the utility function be $f_i(x_i) = -0.5(x_i-y_i)-x_i-\theta_i\log(1+e^{x_i})$, where $\theta_i$ is sampled uniformly from $[0,1]$ and $y_i$ is sampled uniformly from $[-2,2]$ at random at the beginning of each run. Noting that ${\cal X}_i=[-1,1]$, using bounds on $\theta_i$ and $y_i$ and computing the derivatives of $f_i$, we get $M=4+e/(1+e)$, $L=5/4$, $\mu = 1$, $\beta=\sinh(1)/(2(1+\cosh(1))^2)\approx0.0909$. Finally, from Example [Example 4](#ex:sharpball){reference-type="ref" reference="ex:sharpball"} we have ${\Gamma}_{\cal X} = 1$. We initialize the prices to induce $x_i^0 = \eta^0/\mu,~\forall i\in[n]$, and ran SPNUM 100 times for a horizon of $T=50$. The results are illustrated in Figure [3](#fig:spnum_nonlinear){reference-type="ref" reference="fig:spnum_nonlinear"}. The figure shows that **1)** the regret of SPNUM grows as ${\cal O}(\log(t))$, **2)** SPNUM guarantees feasible iterates at all iterations, and **3)** the primal iterates produced by SPNUM converge to the optimal solution. ## Impact of Sharpness on Regret In this study, our aim is to support our theoretical results about SPNUM with numerical examples. In particular, we study the impact of sharpness parameter ${\Gamma}_{\cal X}$ and the number of users $n$ on regret through $\beta$. We set $d_i=1$, in which case $R(T) = {\cal O}(\log(T)(1+\beta\sqrt{n}\Gamma_{\cal X}))$ as stated in Remark [Remark 3](#rem:di1){reference-type="ref" reference="rem:di1"}. For each user $i$, we set $f_i(x_i) = \theta_i(\cos(\omega(x-1))/\omega^2-10(x-2)^2-x\sin(\omega)/\omega)$, where $\theta_i$ is sampled uniformly from $[1,2]$. This particular choice of $f_i$ allows us to control $\beta$ while keeping the other parameters constant by simply choosing $\omega$. Using the bounds on $\theta_i$ and computing the derivatives of $f_i$, we get $M = 40$, $L=42$, $\mu=19$, and $\beta = 2\omega$. In order to have control over the sharpness parameter ${\Gamma}_{\cal X}$, we study linear constraints of the form ${\cal X}=\{x\in \mathbb{R}^n:x\geq 0, Ax\leq c\}$, where $A_{ij}=(1-K)/(1+K(n-1))$ if $i\neq j$, and $A_{ii} = 1$. This choice of $A$ allows us to parameterize the feasible set as a function of the condition number $K$, since $\kappa(A) = K$ and ${\Gamma}_{\cal X}=\sqrt{n}\kappa(A)$. Finally, since $f_i$ is increasing over ${\cal X}_i$, the optimal solution is given by $x^\star = \bm{1}_n$. For $n=\{2,4,8,16\}$, $\omega = \{0.001,0.1\}$, and $K=\{4/\sqrt{n},8/\sqrt{n},16/\sqrt{n},32/\sqrt{n}\}$, we randomly sampled 10 sets of $\{\theta_i\}_{i\in[n]}$, initialized $p_i^0$ so that $x_i^0 = \eta^0/\mu,~\forall i\in[n]$, and ran SPNUM for a horizon of $T=500$. Note that this corresponds to configurations of $\beta = \{0.002,0.2\}$ and $\Gamma_{\cal X}=\{4,8,16,32\}$. We plot the regret for each configuration in Figure [\[fig:spnum_betagamman\]](#fig:spnum_betagamman){reference-type="ref" reference="fig:spnum_betagamman"}. The results indicate that **1)** when $\beta$ is small, $\Gamma_{\cal X}$ and $n$ have little impact on the regret, and **2)** when $\beta$ is large, regret grows with ${\Gamma}_{\cal X}$ and $n$ as the term proportional to $\sqrt{n}\beta\Gamma_{\cal X}$ dominates. # Conclusion In this work, we introduced a novel algorithm, called the safe pricing for NUM (SPNUM), for solving resource allocation problems over networks with arbitrary convex and compact feasible sets in a distributed fashion. Our algorithm iteratively designs prices for resources and allows the users the determine their own resource demand in response to prices according to their own profit maximization problem. The prices produced by SPNUM ensure that the induced demand satisfies the constraints of the system during the optimization process, which promotes safety. This is done by: 1) shrinking the constraint set and applying a projected gradient method to the primal variables to determine the updated desired demand, and 2) determining the prices that would induce the desired demand by estimating the price response function of the users using the historical data. By carefully controlling the amount of shrinkage to account for the error in the estimated price response, we ensure the safety of the algorithm. In addition, we have proven that the regret incurred by the SPNUM is ${\cal O}(\log(T))$, and the primal variables converge to the optimal solution at the rate of ${\cal O}(\log(T)/T)$. ## Proof of Lemma [Lemma 1](#lem:gprops){reference-type="ref" reference="lem:gprops"} {#app:gprops} By definition, $f_i(x_i)$ is strongly concave over ${\cal X}_i$, therefore the optimization problem ${\max}_{x\in{\textnormal{dom}f_i}} f_i(x_i)-\langle x_i,p_i\rangle$ is strongly concave and has a unique solution for $p_i\in{\cal P}_i$. Since ${\cal X}_i\subseteq \textnormal{dom}f_i$ by Assumption [Assumption 1](#ass:feasibleset){reference-type="ref" reference="ass:feasibleset"}, the optimal solution is in the interior of the feasible set. Therefore the first-order optimality condition implies that the optimal solution $g_i(p_i)$ satisfies $$p_i = \nabla f_i(g_i(p_i)),$$ which implies that $\nabla{f}_i$ is surjective for $p_i\in{\cal P}_i$. We also know that the gradient of a strongly concave function is injective[^4], therefore, $\nabla {f}_i$ is bijective and invertible and ${g}_i(p_i)=\nabla{f}_i^{-1}(p_i)$, which also proves that $g_i(p_i)$ is injective. By the inverse function theorem, we get that: $$\nabla {g}_i(p_i) = [\nabla^2 {f}_i({g}_i(p_i))]^{-1}.$$ Since ${f}_i$ is $L$-smooth and $\mu$-strongly concave, inverse of it's Hessian has eigenvalues in $[-1/\mu,-1/L]$, which results in $$\| \nabla {g}_i(p_i)\| = \|[\nabla^2 {f}_i({g}_i(p_i))]^{-1}\| \leq 1/\mu,$$ proving the Lipschitz property of ${g}_i(p_i)$. To show smoothness, we let $x_i^1= {g}_i(p_i^1)$ and $x_i^2={g}_i(p_i^2)$ and write: $$\begin{aligned} &\hspace*{-.2cm}\|\nabla {g}_i(p_i^1){-}\nabla {g}_i(p_i^2)\|=\|[\nabla^2 {f}_i(x_i^1)]^{-1}{-}[\nabla^2 {f}_i(x_i^2)]^{-1}\|\\ &\hspace*{-.2cm}=\|[\nabla^2 {f}_i(x_i^1)]^{-1}(\nabla^2 {f}_i(x_i^2){-}\nabla^2 {f}_i(x_i^1))[\nabla^2 {f}_i(x_i^2)]^{-1}\|\\ &\hspace*{-.2cm}\leq {\beta}\|x_i^1-x_i^2\|/{\mu^2}\leq {\beta}\|p_i^1-p_i^2\|/{\mu^3},\end{aligned}$$ where the last inequality uses $1/\mu$-Lipschitz continuity of ${g}_i(p_i)$, which proves $\beta/\mu^3$-smoothness of ${g}_i(p_i)$. ## Proof of Lemma [Lemma 2](#lem:nablagerror){reference-type="ref" reference="lem:nablagerror"} {#app:nablagerror} Firstly we note that by the choice of $\tau\geq\sqrt{\Delta/H_{\cal X}}$, we can ensure that $\Delta^t\leq H_{\cal X}$ and that ${\cal X}_{\Delta^t}$ is non-empty. Next, we show that $e_i^t\leq 1/(2L),~\forall t\geq 0$. Note that $e_i^t$ is decreasing with $t$, and therefore is maximized for $t=0$: $$\begin{aligned} \hspace{-.25cm}e_i^t {\leq} e_i^0{=} {2\beta\sqrt{d_i}}\left(\eta^0{+}2L(d_i{-}1)(M\sqrt{n}\gamma^0{+}2\Delta^0\Gamma_{\cal X})\right)/{\mu^3}\end{aligned}$$ For $\tau\geq{2\mu\Delta\Gamma_{\cal X}}/({M\sqrt{n}})$ and $d_i\leq \bar{d}$, we get: $$\begin{aligned} e_i^0&\leq \frac{2\beta\sqrt{\bar{d}}}{\mu^3\tau}\left(\frac{M}{8\Gamma_{\cal X}}+\frac{4L(\bar{d}-1)M\sqrt{n}}{\mu}\right)\\ &={\beta M \sqrt{\bar{d}}\left(\mu+32L\Gamma_{\cal X}\sqrt{n}(\bar{d}-1)\right)}/({4\mu^4\Gamma_{\cal X}\tau}).\label{eq:ei0bound}\end{aligned}$$ Next, using $\tau \geq{L\beta M\sqrt{\bar{d}}\left(\mu+32L\Gamma_{\cal X}\sqrt{n}(\bar{d}-1)\right)}/({2\mu^4\Gamma_{\cal X}})$: $$e_i^t\leq e_i^0\leq 1/(2L).$$ We will prove the lemma by induction that if $\|\hat{\nabla}g_i^k-\nabla g_i(p_i^k)\|\leq e_i^k$ holds for $k\in[\max\{0,t-d_i+1\},t-1]$, then it holds for $k = t$. Using Cauchy-Schwarz inequality: $$\begin{aligned} \|\hat{\nabla}g_i^t{-}\nabla g_i(p_i^t)\|\leq \sqrt{d_i}\max_{j\in[d_i]}\|[\hat{\nabla}g_i^t]_{:,j}{-}[\nabla g_i(p_i^t)]_{:,j}\|.\end{aligned}$$ For a given $j\in[d_i]$, by construction of $\hat{\nabla}g_i^t$ we have $$_{:,j}=({g_i(p_i^{\ell_j}+\eta^{\ell_j} e_j)-g_i(p_i^{\ell_j})})/{\eta^{\ell_j}},$$ for some $\ell_j\in[\max\{0,t-d_i+1\},t]$. Using the Taylor series expansion, we can rewrite the above as: $$_{:,j} = [\nabla g_i(p_i^{\ell_j})]_{:,j} + R_1/\eta^{\ell_j},$$ where $\|R_1\|\leq \beta (\eta^{\ell_j})^2/(2\mu^3)$ follows from [@nesterov2006cubic Lemma 1] using $\beta/\mu^3$-smoothness of $g_i$. Accordingly, $$\begin{aligned} \nonumber\|\hat{\nabla}g_i^t-\nabla g_i(p_i^t)\|&\leq \sqrt{d_i}\max_{j\in[d_i]}\|[\nabla g_i(p_i^{\ell_j})]_{:,j}-[\nabla g_i(p_i^t)]_{:,j}\|\\ &~~~~+\sqrt{d_i}\beta\eta^{\ell_j}/(2\mu^3)\\ &\hspace{-2.5cm}{\leq} \max_{{\ell_j}\in[\max\{0,t-d_i+1\},t]}\frac{\beta\sqrt{d_i}}{\mu^3}\|p_i^{\ell_j}-p_i^t\|+\frac{\sqrt{d_i}\beta\eta^{\ell_j}}{2\mu^3},\label{eq:nablagerrorcheckpt1}\end{aligned}$$ where we used $$\|[\nabla g_i(p_i^{\ell_j})]_{:,j}-[\nabla g_i(p_i^t)]_{:,j}\|\leq\|\nabla g_i(p_i^{\ell_j})-\nabla g_i(p_i^t)\|,$$ for all $j\in[d_i]$, and smoothness of $g_i$. Furthermore, note that for $\tau\geq 2\bar{d}-1$, $\eta^{t-d_i+1}\leq 4\eta^t$ and therefore for $t=0$ we have $$\begin{aligned} \|\hat{\nabla}g_i^0-\nabla g_i(p^0)\|&\leq 2\sqrt{d_i}\beta\eta^0/\mu^3\leq e_i^0.\end{aligned}$$ Accordingly, the statement holds for $t=0$, which covers the base case. For $t>0$, we continue from [\[eq:nablagerrorcheckpt1\]](#eq:nablagerrorcheckpt1){reference-type="eqref" reference="eq:nablagerrorcheckpt1"} and bound $\|p_i^{\ell_j}-p_i^t\|$ as $$\begin{aligned} \|p_i^{\ell_j}-p_i^t\|&\leq\sum_{k={\ell_j}}^{t-1}\|p_i^{k}-p_i^{k+1}\|\\ &=\sum_{k={\ell_j}}^{t-1}\|[\hat{\nabla}g_i^k]^{-1}(\hat{x}_i^{k+1}-x_i^k)\|\\ &\leq \sum_{k={\ell_j}}^{t-1}\|[\hat{\nabla}g_i^k]^{-1}\|\|\hat{x}_i^{k+1}-x_i^k\|.\end{aligned}$$ The following two lemmas, whose proofs can be found in Appendices [6.4](#app:sigmamin){reference-type="ref" reference="app:sigmamin"} and [6.5](#app:xstepamount){reference-type="ref" reference="app:xstepamount"} bound each of the terms in the above summation: **Lemma 3**. *Suppose that $\|\hat{\nabla}g_i^t-\nabla g_i(p_i^t)\|\leq 1/(2L)$ for some $t$. Then $\sigma_{\min}(\hat{\nabla}g_i^t)\geq 1/(2L)$ and $\|[\hat{\nabla}g_i^t]^{-1}\|\leq 2L$.* **Lemma 4**. *For all $t\geq 0$, if $x^t\in{\cal X}^\textnormal{int}$, then for a user $i\in[n]$ the following holds: $$\|\hat{x}_i^{t+1}-x_i^{t}\|\leq M\sqrt{n}\gamma^t+\Delta^t\Gamma_{\cal X}.$$* Using Lemmas [Lemma 3](#lem:sigmamin){reference-type="ref" reference="lem:sigmamin"} and [Lemma 4](#lem:xstepamount){reference-type="ref" reference="lem:xstepamount"}, we get $$\begin{aligned} &\max_{{\ell_j}\in[\max\{0,t-d_i+1\},t]} \|p_i^{{\ell_j}}-p_i^t\|\\ &\leq\max_{{\ell_j}\in[\max\{0,t-d_i+1\},t]}2L\sum_{k={\ell_j}}^{t-1}M\sqrt{n}\gamma^k+\Delta^k\Gamma_{\cal X}\\ &\leq 2L(t-{\ell}_{\min})(M\sqrt{n}\gamma^{\ell_{\min}}+\Delta^{\ell_{\min}}\Gamma_{\cal X}),\end{aligned}$$ where $\ell_{\min}=\max\{0,t-d_i+1\}$. Lastly, note that $t-\ell_{\min}\leq d_i-1$, $\gamma^{\ell_{\min}}/\gamma^t\leq 2$, and $\Delta^{\ell_{\min}}/\Delta^t\leq 4$ for $\tau\geq 2\bar{d}-1$, which gives the final result. ## Proof of Proposition [Proposition 2](#prop:safety){reference-type="ref" reference="prop:safety"} {#app:safety} We will prove by induction that if at iteration $t$, $\forall k\in[\max\{t-\bar{d}+1,0\},t]$, $x^k\in{{\cal X}}_{\frac{\sqrt{n}\eta^k}{\mu}}^\textnormal{int}$, then $x^{t+1}\in{\cal X}^{\textnormal{int}}_{\frac{\sqrt{n}\eta^{t+1}}{\mu}}$ and use Assumption [Assumption 3](#ass:initialprices){reference-type="ref" reference="ass:initialprices"} that $x^{0}\in{\cal X}^{\textnormal{int}}_{\frac{\sqrt{n}\eta^0}{\mu}}$. This will ensure that $x^{t+1,s}\in {\cal X}^\textnormal{int}$ as well by choice of $\Delta^t$ and $\eta^t$. Therefore, we assume that $x^k\in{\cal X}^\textnormal{int}_{\frac{\sqrt{n}\eta^k}{\mu}}$. Note that $\hat{x}^{t+1}\in{\cal X}^{\textnormal{int}}$ by definition. For all $i\in[n]$, we consider a modified utility function $\tilde{f}_i(x_i)$, which is equal to $f_i(x_i)$ if $x_i\in{\cal X}_i$, and an $L$-smooth, $\mu$-strongly concave extension with $\beta$-smooth gradient beyond the set ${\cal X}_i$. Accordingly, $\textnormal{dom}\tilde{f}_i=\mathbb{R}^{d_i}$, and $\tilde{f}_i$ is $L$-smooth and $\mu$-strongly concave over ${\mathbb R}^{d_i}$ with $\beta$-smooth gradient. Using the modified utility function, we define the modified price response function $$\label{eq:modifiedresponse} {\tilde g}_i(p_i)=\underset{x_i\in{\mathbb R}^{d_i}}{\mathop{\mathrm{arg\,max}}}\tilde{f}_i(x_i)-\langle x_i,p_i\rangle.$$ The following Lemma, whose proof can be found in Appendix [6.7](#app:smoothg){reference-type="ref" reference="app:smoothg"}, characterizes the regularity properties of $\tilde{g}_i(p_i)$, $\forall i\in[n]$, under Assumption [Assumption 2](#ass:utility){reference-type="ref" reference="ass:utility"}: **Lemma 5**. *For all $i\in[n]$, let $\tilde{g}_i(p_i)$ be the modified price response function in [\[eq:modifiedresponse\]](#eq:modifiedresponse){reference-type="eqref" reference="eq:modifiedresponse"}. Then, $\tilde{g}_i(p_i)$ is bijective, $1/\mu$-Lipschitz continuous and $\beta/\mu^3$-smooth over ${\mathbb R}^{d_i}$. Furthermore, let ${\cal P}_i=\{p_i\in{\mathbb R}^{d_i}:g_i(p_i)\in{\cal X}_i^\textnormal{int}\}$. The following hold true:* 1. *If $\tilde{g}_i(p_i)\in{\cal X}_i^\textnormal{int}$, then $p_i\in{\cal P}_i$.* 2. *If $p_i\in{\cal P}_i$, then $\tilde{g}_i(p_i) = g_i(p_i)$.* For each user $i\in[n]$, we let $\tilde{x}_i^{t+1}=\tilde{g}_i(p_i^{t+1})$ and we rearrange the price update rule: $$\label{eq:xdiff} \tilde{x}_i^{t+1}-\hat{x}_i^{t+1} = \tilde{x}_i^{t+1}-x_i^t-\hat{\nabla}g_i^t(p^{t+1}-p^t).$$ We can also write the Taylor expansion of the modified price response function $\tilde{g}_i(p)$ around $p_i^{t}$: $$\tilde{g}_i(p_i^{t+1})-\tilde{g}_i(p_i^t)=\nabla\tilde{g}_i(p_i^t)(p_i^{t+1}-p_i^t)+R_1.$$ We replace $\tilde{g}_i(p_i^{t})=g_i(p_i^t)=x_i^t$ and $\nabla\tilde{g}_i(p_i^t)=\nabla g_i(p_i^t)$ (since $p_i^t\in{\cal P}_i$) and plug the above equation into [\[eq:xdiff\]](#eq:xdiff){reference-type="eqref" reference="eq:xdiff"}: $$\tilde{x}_i^{t+1}-\hat{x}_i^{t+1} = (\nabla g_i(p_i^t)-\hat{\nabla}g_i^t)(p_i^{t+1}-p_i^t) + R_1.$$ To bound the norm of the above equation, we use Lemma [Lemma 2](#lem:nablagerror){reference-type="ref" reference="lem:nablagerror"} to bound the norm of the first term and [@nesterov2006cubic Lemma 1] to bound the second term: $$\label{eq:xdiffabs} \|\tilde{x}_i^{t+1}-\hat{x}_i^{t+1}\|\leq e_i^t\|p_i^{t+1}-p_i^t\|+\frac{\beta}{2\mu^3}\|p_i^{t+1}-p_i^t\|^2.$$ Rearranging the price update rule and using Lemmas [Lemma 2](#lem:nablagerror){reference-type="ref" reference="lem:nablagerror"} and [Lemma 4](#lem:xstepamount){reference-type="ref" reference="lem:xstepamount"} we can bound the norm of the price change: $$\begin{aligned} \label{eq:pdiffabs} \begin{split} \|p_i^{t+1}-p_i^{t}\|&\leq\|[\hat{\nabla}g_i^t] ^{-1}\|\|\hat{x}_i^{t+1}-x_i^t\| \end{split}\\ &\leq 2L(M\sqrt{n}\gamma^t+\Delta^t\Gamma_{\cal X}). \end{aligned}$$ Note that both upper bounds for $e_i^t$ and $\|p_i^{t+1}-p_i^t\|$ are decreasing with $t$. We can bound $e_i^t$ using $\tau> \frac{2\mu\Delta\Gamma_{\cal X}}{M\sqrt{n}}$ and $1\leq \Gamma_{\cal X}$ as: $$\begin{aligned} e_i^t&< {\beta M\sqrt{d_i n}(\mu/\sqrt{n}+32L(\bar{d}-1))}/({4\mu^4(t+\tau)})\\ &={\beta M\sqrt{d_i n}\gamma^t(\mu/\sqrt{n}+32L(\bar{d}-1))}/({4\mu^3}), \end{aligned}$$ and further upper bound $\|p_i^{t+1}-p_i^t\|$ as $$\|p_i^{t+1}-p_i^t\|\leq 3LM\sqrt{n}\gamma^t.$$ Plugging the above bounds and $\gamma^t$ into [\[eq:xdiffabs\]](#eq:xdiffabs){reference-type="eqref" reference="eq:xdiffabs"}: $$\begin{aligned} \begin{split} \|\tilde{x}_i^{t+1}-\hat{x}_i^{t+1}\|&< \frac{3\beta LM^2n}{4\mu^5(t+\tau)^2}\Big(6L\\ &+{\sqrt{d_i}\left(\mu/\sqrt{n}+32L(\bar{d}-1)\right)}\Big). \end{split} \end{aligned}$$ Next, using Cauchy-Schwarz inequality, we bound $\|\tilde{x}^{t+1}-x^{t+1}\|$ as $$\begin{aligned} \begin{split} \|\tilde{x}^{t+1}-\hat{x}^{t+1}\|&< \frac{3\beta LM^2n^{3/2}}{4\mu^5(t+\tau)^2}\Big(6L\\ &+{\sqrt{d}\left(\mu/\sqrt{n}+32L(\bar{d}-1)\right)}\Big) \end{split}\\ &={3\Delta^t}/{4}, \end{aligned}$$ where we used the definition of $\Delta^t$ and $\sum_{i\in[n]}\sqrt{d_i}\leq \sqrt{dn}$. This establishes that by definition of a shrunk set and $\Delta^t/4 = \frac{\sqrt{n}\eta^{t+1}}{\mu}$, $\tilde{x}^{t+1}\in{\cal X}^\textnormal{int}_{\frac{\sqrt{n}\eta^{t+1}}{\mu}}$. Furthermore, let $\tilde{x}_i^{t+1,s}=\tilde{g_i}(p_i^{t+1,s})$. Using $1/\mu$-Lipschitz continuity of $\tilde{g_i}(p_i)$: $$\|\tilde{x}_i^{t+1,s}-\tilde{x}_i^{t+1}\|\leq {\Delta^t}/(4\sqrt{n}),$$ and $\|\tilde{x}^{t+1,s}-\tilde{x}^{t+1}\|\leq \Delta^t/4$. Accordingly, we have $$\|\tilde{x}^{t+1,s}-\hat{x}^{t+1}\|<\Delta^t,$$ which establishes that $\tilde{x}^{t+1,s}\in{\cal X}^{\textnormal{int}}$. Lastly, note that if $\tilde{x}^{t+1},\tilde{x}^{t+1,s}\in{\cal X}^\textnormal{int}$, then for all $i\in[n]$, $\tilde{x}_i^{t+1},\tilde{x}_i^{t+1,s}\in {\cal X}_i^\textnormal{int}$, or equivalently, $\tilde{g}_i(p_i^{t+1}),\tilde{g}_i(p_i^{t+1,s})\in {\cal X}_i^\textnormal{int}$. Using Lemma [Lemma 5](#lem:smoothg){reference-type="ref" reference="lem:smoothg"} we have that $p_i^{t+1},p_i^{t+1,s}\in{\cal P}_i$, $\forall i\in[n]$. Hence, $\tilde{g}_i(p^{t+1})=g_i(p^{t+1})$ and $\tilde{x}_i^{t+1}=x_i^{t+1}$ as well as $\tilde{g}_i(p^{t+1,s})=g_i(p^{t+1,s})$ and $\tilde{x}_i^{t+1,s}=x_i^{t+1,s}$ for all $i\in [n]$, which proves the proposition. ## Proof of Lemma [Lemma 3](#lem:sigmamin){reference-type="ref" reference="lem:sigmamin"} {#app:sigmamin} Note that for $p_i^t\in{\cal P}_i$, $\nabla g_i(p^t)=[\nabla^2 f_i(g_i(p^t))]^{-1}$ is symmetric by Schwarz's theorem, since $\nabla^2 f_i(g_i(p_i))$ is $\beta$-Lipschitz continuous for $p_i\in{\cal P}_i$. Accordingly, the minimum singular value of $\nabla g_i(p_i^t)$ is equal to smallest absolute eigenvalue of $[\nabla^2 f_i(g_i(p^t))]^{-1}$, i.e., $\sigma_{\min}(\nabla g_i(p_i^t))=1/L$. This implies that if $\|\hat{\nabla}g_i^t-\nabla g_i(p_i^t)\|\leq 1/(2L)$ holds, then $$\begin{aligned} &\sigma_{\min}(\hat{\nabla}g_i^t)=\underset{\|x\|=1}{\min}\|\hat{\nabla}g_i^t x\|\\ &= \underset{\|x\|=1}{\min}\|\nabla g_i(p_i^t)x+(\hat{\nabla}g_i^t-\nabla g_i(p_i^t))x\|\\ &\geq \underset{\|x\|=1}{\min}\|\nabla g_i(p_i^t)x\|-\underset{\|x\|=1}{\max}\|(\hat{\nabla}g_i^t-\nabla g_i(p_i^t))x\|\\ &=1/L-1/(2L)\geq 1/(2L),\end{aligned}$$ which implies that $\|[\hat{\nabla}g_i^t]^{-1}\|=1/\sigma_{\min}(\hat{\nabla}g_i^t)\leq 2L$. ## Proof of Lemma [Lemma 4](#lem:xstepamount){reference-type="ref" reference="lem:xstepamount"} {#app:xstepamount} To bound $\|\hat{x}_i^{t+1}-x_i^t\|$, we will use the following as an auxiliary result: **Theorem 2**. *[@schneider2014convex Theorem 1.2.1][\[thm:schneider121\]]{#thm:schneider121 label="thm:schneider121"} Let ${\cal X}$ be a convex and compact set in $\mathbb{R}^d$. Then, the metric projection onto ${\cal X}$ is contracting, that is, $$\|\Pi_{\cal X}(x)-\Pi_{\cal X}(y)\|\leq \|x-y\|,~\forall x,y,\in{\mathbb R}^d.$$* Using the above result, we bound $\|\hat{x}_i^{t+1}-x_i^t\|$ as: $$\begin{aligned} &\|\hat{x}_i^{t+1}-x_i^{t}\| \leq \|\hat{x}^{t+1}-x^{t}\|\\ &=\|\Pi_{{\cal X}_{\Delta^t}}(x^t+p^t\gamma^t)-\Pi_{{\cal X}_{\Delta^t}}(x^t)+\Pi_{{\cal X}_{\Delta^t}}(x^t)-x^t\|\\ &\leq \|\Pi_{{\cal X}_{\Delta^t}}(x^t+p^t\gamma^t)-\Pi_{{\cal X}_{\Delta^t}}(x^t)\| {+} \|\Pi_{{\cal X}_{\Delta^t}}(x^t)-x^t\|\\ &\leq \|p^t\gamma^t\| + \Delta^t\Gamma_{\cal X}\leq M\sqrt{n}\gamma^t+ \Delta^t\Gamma_{\cal X}, \end{aligned}$$ where we used $\|p_i^t\|=\|\nabla f_i(x_i^t)\|\leq M$ since $x_i^t\in{\cal X}_i^{\textnormal{int}}$, and Proposition [\[prop:sharp_conv\]](#prop:sharp_conv){reference-type="ref" reference="prop:sharp_conv"}. ## Proof of Theorem [Theorem 1](#thm:regret){reference-type="ref" reference="thm:regret"} {#app:regret} We denote the regret incurred by the update stage as $R_u(T) = \sum_{t=1}^{T/2}f(x^\star)-f(x^t)$ and the regret incurred by the sampling stage as $R_s(T) = \sum_{t=1}^{T/2}f(x^\star)-f(x^{t,s})$. Let $y^{t+1}=x^t+\gamma^tp_t$. By Lemma [Lemma 5](#lem:smoothg){reference-type="ref" reference="lem:smoothg"}, we know that $p^t=\nabla f(x^t)$, $\forall t\geq 0$, since $x^t\in{\cal X}^\textnormal{int}$ by Proposition [Proposition 2](#prop:safety){reference-type="ref" reference="prop:safety"}. For $t\geq 1$, we write using strong concavity: $$\begin{aligned} &f(x^\star)-f(x^t)\leq \langle-\nabla f(x^t), x^t-x^\star\rangle-\frac{\mu}{2}\|x^t-x^\star\|^2\\ &=\frac{1}{\gamma^t}\langle x^t-y^{t+1},x^t-x^\star\rangle-\frac{\mu}{2}\|x^t-x^\star\|^2\\ \begin{split}\label{eq:instaregret} &=\frac{1}{2\gamma^t}\left(\|x^t-y^{t+1}\|^2+\|x^t-x^\star\|^2-\|y^{t+1}-x^\star\|^2\right)\\ &\hspace{1cm}-\frac{\mu}{2}\|x^t-x^\star\|^2. \end{split}\end{aligned}$$ Next, we bound the $\|y^{t+1}-x^\star\|^2$ term using Theorem [\[thm:schneider121\]](#thm:schneider121){reference-type="ref" reference="thm:schneider121"} as follows: $$\begin{aligned} &\|y^{t+1}-x^\star\|^2\geq \|\Pi_{{\cal X}_{\Delta^t}}(y^{t+1})-\Pi_{{\cal X}_{\Delta^t}}(x^\star)\|^2\\ &= \|\hat{x}^{t+1}- \Pi_{{\cal X}_{\Delta^t}}(x^\star)\|^2\\ &=\|\hat{x}^{t+1}-x^{t+1}+x^{t+1}-x^\star+x^\star-\Pi_{{\cal X}_{\Delta^t}}(x^\star)\|^2\\ \nonumber &=\|\hat{x}^{t+1}-x^{t+1}\|^2 + \|x^{t+1}-x^\star\|^2 +\|x^\star-\Pi_{{\cal X}_{\Delta^t}}(x^\star)\|^2\hspace{-1cm}\\ \nonumber &+2\langle\hat{x}^{t+1}{-}x^{t+1},x^{t+1}{-}x^\star\rangle {+} 2\langle x^{t+1}{-}x^\star, x^\star{-}\Pi_{{\cal X}_{\Delta^t}}(x^\star)\rangle\\ &+2\langle x^\star-\Pi_{{\cal X}_{\Delta^t}}(x^\star),\hat{x}^{t+1}-x^{t+1}\rangle\\ \begin{split} &\geq \|x^{t+1}-x^\star\|^2 - 2\|\hat{x}^{t+1}-x^{t+1}\|\|x^{t+1}-x^\star\|\\ &-2\|x^{t+1}-x^\star\|\|x^\star-\Pi_{{\cal X}_{\Delta^t}}(x^\star)\|\\ &-2\|x^\star-\Pi_{{\cal X}_{\Delta^t}}(x^\star)\|\|\hat{x}^{t+1}-x^{t+1}\| \end{split}\\ \begin{split} &\geq \|x^{t+1}-x^\star\|^2-2\Delta^tR(\Gamma_{\cal X}+3/4)-3/2(\Delta^t)^2\Gamma_{\cal X} \end{split}\\ &\vcentcolon=\|x^{t+1}-x^\star\|^2-C_t,\end{aligned}$$ where the last inequality uses $\|{x}^{t+1}-\hat{x}^{t+1}\|<3\Delta^t/4$ given by Proposition [Proposition 2](#prop:safety){reference-type="ref" reference="prop:safety"} and Proposition [\[prop:sharp_conv\]](#prop:sharp_conv){reference-type="ref" reference="prop:sharp_conv"} to bound $\|x^\star{-}\Pi_{{\cal X}_{\Delta^t}}(x^\star)\|$. Plugging this in [\[eq:instaregret\]](#eq:instaregret){reference-type="eqref" reference="eq:instaregret"}: $$\label{eq:thm1checkpt} \begin{split} f(x^\star)-f(x^t)\leq &\frac{M^2n\gamma^t}{2}-\frac{\mu}{2}\|x^t-x^\star\|^2+\frac{C^t}{2\gamma^t}\\ &{+}\frac{1}{2\gamma^t}(\|x^t{-}x^\star\|^2-\|x^{t+1}-x^\star\|^2). \end{split}$$ Summing from $t=1$ to $T/2$ telescopes the $\|x^t-x^\star\|^2$ terms: $$\begin{aligned} \begin{split} & nR_u(T)\leq \frac{M^2n\log(T/2)}{2\mu}+ \frac{\mu\tau}{2}\|x^1-x^\star\|^2\hspace{-1cm}\\ & \hspace{1cm}+\sum_{t=2}^{T/2}\left(\frac{1}{2\gamma^t}-\frac{1}{2\gamma^{t-1}}-\frac{\mu}{2}\right)\|x^t-x^\star\|^2\\ &\hspace{1cm}-\frac{1}{2\gamma^{T/2}}\|x^{T/2+1}-x^\star\|^2+\sum_{t=1}^{T/2}\frac{C^t}{2\gamma^t} \end{split}\\ \label{eq:finalregret} &\leq \frac{M^2n\log(T/2)}{2\mu}+ \frac{\mu\tau}{2}\|x^1-x^\star\|^2+\sum_{t=1}^{T/2}\frac{C^t}{2\gamma^t}.\end{aligned}$$ Finally, note that $C^t={\cal O}(1/t^2)$ because the it consists of terms $\Delta^t$ and $(\Delta^t)^2$. Therefore, we can use the bounds ${\sum}_{t=1}^{T/2}\frac{1}{t+\tau}\leq{\sum}_{t=1}^{T/2}\frac{1}{t+2} \leq \log(T/2)$ and for $k\geq 2$, $\sum_{t=1}^{T/2}\frac{1}{(t+2)^k}\leq 1$ to show that: $$\label{eq:Ctsum} \begin{split} \sum_{t=1}^{T/2}\frac{C^t}{2\gamma^t}\leq& \mu{\Delta R(3/4+\Gamma_{\cal X})\log(T/2)}+{3\mu\Delta^2\Gamma_{\cal X}}/4. \end{split}$$ Plugging [\[eq:Ctsum\]](#eq:Ctsum){reference-type="eqref" reference="eq:Ctsum"} into [\[eq:finalregret\]](#eq:finalregret){reference-type="eqref" reference="eq:finalregret"} and dividing by both sides by $n$, we get the regret incurred by the update stages. For the sampling stages, we note that due to the strong concavity of $f$ $$\begin{aligned} f(x^t){-}f(x^{t,s})&\leq\langle\nabla f(x^{t,s}),x^t{-}x^{t,s}\rangle\leq M\sqrt{n}\frac{\Delta^{t-1}}{4}.\end{aligned}$$ Accordingly $f(x^\star)-f(x^{t,s})\leq f(x^\star)-f(x^t)+M\sqrt{n}\Delta^{t-1}/4$. Summing from $t=1$ to $T/2$, we get $$\begin{aligned} nR_s(T)&{=}nR_u(T){+}\frac{M}{4}\sum_{t=1}^{T}\Delta^{t-1}{\leq} nR_u(T){+}\frac{\Delta M\sqrt{n}}{4},\end{aligned}$$ which gives the final result as $$R(T)\leq 2R_u(T)+{\Delta M}/({4\sqrt{n}}).$$ To get the convergence result, we rearrange [\[eq:thm1checkpt\]](#eq:thm1checkpt){reference-type="eqref" reference="eq:thm1checkpt"}: $$\begin{aligned} \begin{split} \|x^{t+1}-x^\star\|^2&\leq \|x^t-x^\star\|^2(1-\mu\gamma^t) +M^2 n (\gamma^t)^2\\ & + C^t +2\gamma^t (f(x^t)-f(x^\star)) \end{split}\\ \leq& \|x^t-x^\star\|^2(1-\mu\gamma^t) +M^2 n (\gamma^t)^2+ C^t.\end{aligned}$$ We get an equation like the above for all $t\geq 0$. We multiply each by $(1-\mu\gamma^{t+1})$ for $t<T/2-1$ and sum them from $t=0$ to $t= T/2-1$ to get: $$\begin{aligned} \begin{split} &\|x^{T/2}-x^\star\|^2\leq \|x^0-x^\star\|^2\prod_{t=0}^{T/2-1}(1-\mu\gamma^t)\\ &+M^2n\sum_{t=0}^{T/2-1}(\gamma^t)^2\prod_{i=t+1}^{T/2-1}(1-\mu\gamma^i)\\ &+\sum_{t=0}^{T/2-1}C^t\prod_{i=t+1}^{T/2-1}(1-\mu\gamma^i)\hspace{-1cm} \end{split}\\ \begin{split} &\leq \|x^0-x^\star\|^2\frac{\tau-1}{\tau-1+T/2}+\frac{M^2n\log(T/2)}{\mu^2(T/2+\tau-1)}\\ &+\frac{2R(3/4+\Gamma_{\cal X})\Delta\log(T/2)}{(T/2+\tau-1)}+\frac{3\Delta^2\Gamma_{\cal X}}{2(T/2+\tau-1)}. \end{split} % \\ % \begin{split} % &=\|x^1-x^\star\|^2\frac{\tau}{\tau+T}+\frac{M^2n\log(T+1)}{\mu^2(T+\tau)}\\ % &+\frac{36M^2n^{3/2}\beta L^2R(\Gamma_{\cal X})\log(T+1)}{\mu^5(T+\tau)}\\ % &+\frac{648M^4n^3\beta^2L^4\Gamma_{\cal X}}{\mu^ % {10}(T+\tau)}, % \end{split}\end{aligned}$$ which completes the proof. ## Proof of Lemma [Lemma 5](#lem:smoothg){reference-type="ref" reference="lem:smoothg"} {#app:smoothg} The first part of the lemma follows from the same steps as in Lemma [Lemma 1](#lem:gprops){reference-type="ref" reference="lem:gprops"} for $p_i\in{\mathbb R}^{d_i}$ instead of $p_i\in{\cal P}_i$, and using $\tilde{f}_i$ and $\tilde{g}_i$ instead of $f_i$ and $g_i$. Next, we prove the second part of the lemma. For the first statement, given a $p_i\in{\mathbb R}^{d_i}$, suppose that $\tilde{g}_i(p_i)\in{\cal X}_i^\textnormal{int}$. This implies that there exists $x_i\in{\cal X}_i^{\textnormal{int}}$ that satisfies $\nabla \tilde{f}_i(x_i)=p_i$. Since $\tilde{f}_i(x_i)=f_i(x_i)$ for $x_i\in{\cal X}_i^\textnormal{int}$, the same $x_i$ solves the optimization problem in [\[eq:priceresponse\]](#eq:priceresponse){reference-type="eqref" reference="eq:priceresponse"}, which implies $g_i(p_i) = \tilde{g}_i(p_i)$. Therefore, $g_i(p_i)\in{\cal X}_i^\textnormal{int}$, which proves $p_i\in{\cal P}_i$ by definition. To prove the second statement, note that if $p_i\in{\cal P}_i$, then $g_i(p_i)\in{\cal X}_i^\textnormal{int}$. Since ${\cal X}_i\subseteq \textnormal{dom}f_i$ by Assumption [Assumption 1](#ass:feasibleset){reference-type="ref" reference="ass:feasibleset"}, the first order optimality condition of [\[eq:priceresponse\]](#eq:priceresponse){reference-type="eqref" reference="eq:priceresponse"} implies that there exists $x_i=g_i(p_i)\in{\cal X}_i^\textnormal{int}$ such that $\nabla f_i(x_i) = p_i$. The same $x_i$ solves the optimization problem [\[eq:modifiedresponse\]](#eq:modifiedresponse){reference-type="eqref" reference="eq:modifiedresponse"}, since $f_i(x_i)=\tilde{f}_i(x_i)$ for $x_i\in{\cal X}_i^\textnormal{int}$. The optimal solution to [\[eq:modifiedresponse\]](#eq:modifiedresponse){reference-type="eqref" reference="eq:modifiedresponse"} has to be unique due to strong concavity, therefore it must hold true that $\tilde{g}_i(p_i) = g_i(p_i)$. ## Proof of Remark [Remark 2](#rem:initialset){reference-type="ref" reference="rem:initialset"} {#app:initialset} For a user $i\in[n]$, using the modified price response function $\tilde{g}_i(p_i)$ introduced in the proof of Proposition [Proposition 2](#prop:safety){reference-type="ref" reference="prop:safety"}, we have that $$\|\tilde{x}_i^{-t}-x_i^0\|\leq{\eta^0}/{\mu},~\forall t\in[-d_i,-1],$$ which implies that $\tilde{x}_i^{-t}\in{\cal X}_i^\textnormal{int}$ because $x^0\in{\cal X}^\textnormal{int}_{\frac{\eta^0\sqrt{n}}{\mu}}$. As such, $\tilde{x}_i^{-t} = x_i^{-t}$ and $p_i^{-t}= \nabla f_i(x_i^{-t})$. BERKAY TURANis pursuing the Ph.D. degree in Electrical and Computer Engineering at the University of California, Santa Barbara. He received the B.Sc. degree in Electrical and Electronics Engineering as well as the B.Sc. degree in Physics degree from  Boğaziçi University, Istanbul, Turkey, in 2018. The overarching goal of his research is to design network control, optimization, and learning frameworks to promote efficiency and resiliency in societal-scale cyber-physical systems. plus -1fil SPENCER HUTCHINSON received the B.S. degree in electrical engineering from Colorado School of Mines in 2021. He is currently pursuing the Ph.D. degree in electrical and computer engineering from the University of California, Santa Barbara in Santa Barbara, CA, USA. His research interests include the design and analysis of optimization and learning algorithms for the control of human-cyber-physical systems. plus -1fil MAHNOOSH ALIZADEH is an associate professor of Electrical and Computer Engineering at the University of California Santa Barbara. She received the B.Sc. degree ('09) in Electrical Engineering from Sharif University of Technology and the M.Sc. ('13) and Ph.D. ('14) degrees in Electrical and Computer Engineering from the University of California Davis. From 2014 to 2016, she was a postdoctoral scholar at Stanford University. Her research interests are focused on designing network control, optimization, and learning frameworks to promote efficiency and resiliency in societal-scale cyber-physical systems. Dr. Alizadeh is a recipient of the NSF CAREER award. [^1]: B. Turan, S. Hutchinson, and M. Alizadeh are with Dept. of ECE, UCSB, Santa Barbara, CA, USA. This work is supported by NSF grant \#1847096. E-mails: [bturan\@ucsb.edu](bturan@ucsb.edu), [shutchinson\@ucsb.edu](shutchinson@ucsb.edu), [alizadeh\@ucsb.edu](alizadeh@ucsb.edu) [^2]: *We can equivalently define $\mathcal{X}_{\Delta}$ using Minkowski subtraction. The Minkowski subtraction of sets $A,B \subseteq \mathbb{R}^d$ is defined as $A \ominus B := \{a - b : a \in A, b \in B \}$, or equivalently, $A \ominus B = \bigcap_{b\in{B}} (A-b)$. Therefore, $\mathcal{X}_{\Delta} = \mathcal{X} \ominus \mathcal{B}(\Delta)$ is an intersection of convex and closed sets and hence is convex and closed [@schneider2014convex Section 3.1]. By Definition [Definition 4](#def:shrunk_set){reference-type="ref" reference="def:shrunk_set"}, ${\cal X}_{\Delta}$ is a subset of ${\cal X}$, and therefore bounded. A closed and bounded convex set is convex and compact.* [^3]: For SPNUM, we additionally include the constraints $x\geq 0$ in ${\cal X}$ to satisfy Assumption [Assumption 1](#ass:feasibleset){reference-type="ref" reference="ass:feasibleset"}. For the other algorithms, this is not needed. [^4]: To see this, suppose that $x_1\neq x_2$ and therefore $\|x_1-x_2\|>0$. If $\nabla f(x_1)= \nabla f(x_2)$, [\[eq:strong\]](#eq:strong){reference-type="eqref" reference="eq:strong"} results in $0\geq \mu\|x_1-x_2\|^2$, which is a contradiction and $x_1=x_2$ must hold.

arxiv_math

--- abstract: | In this paper, we first construct a *differential graded Lie algebra* that controls deformations of a Lie-Yamaguti algebra. Furthermore, a relative Rota-Baxter operator on a Lie-Yamaguti algebra is characterized as a Maurer-Cartan element in an appropriate *$L_\infty$-algebra* that we build through the graded Lie bracket of Lie-Yamaguti algebra's controlling algebra, and gives rise to a twisted $L_\infty$-algebra that controls its deformation. Next we establish the *cohomology* theory of relative Rota-Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then we clarify the *relationship* between the twisted $L_\infty$-algebra and the cohomology theory. Finally as byproducts, we classify certain deformations on Lie-Yamaguti algebras using the cohomology theory. address: - Jia Zhao, School of Sciences, Nantong University, Nantong 226019, Jiangsu, China - Yu Qiao (corresponding author), School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710119, Shaanxi, China author: - Jia Zhao - Yu Qiao\* title: Maurer-Cartan characterization, $L_\infty$-algebras, and cohomology of relative Rota-Baxter operators on Lie-Yamaguti algebras --- [^1] # Introduction In this paper, we determine a differential graded Lie algebra as controlling algebra for Lie-Yamaguti algebras and an $L_\infty$-algebra for their relative Rota-Baxter operators. Then we establish the cohomology theory to classify deformations of relative Rota-Baxter operators. ## Maurer-Cartan characterizations and deformations In mathematics, informally speaking, a deformation of an object is another object that shares the same structure of the original object after a perturbation. Motivated by the foundational work of Kodaira and Spencer [@Kodaira] for complex analytic structures, the generalization in algebraic geometry of deformation theory was founded [@Hart]. Recently, a number of works referring to deformation quantization turn up in the context of mathematical physics [@Kont1; @Kont2]. As an application in algebra, Gerstenhaber first studied the deformation theory on associative algebras [@Gerstenhaber1; @Gerstenhaber2; @Gerstenhaber3; @Gerstenhaber4; @Gerstenhaber5]. Then Nijenhuis and Richardson extended this idea and established the similar results on Lie algebras [@Nij1; @Nij2]. Deformations of other algebraic structures such as pre-Lie algebras have also been developed [@Burde]. In general, deformation theory was set up for binary quadratic operads by Balavoine [@bala]. As we all know, for an algebraic object, deformation theory and cohomology theory are connected tightly. More precisely, a suitable deformation of an object should obey the following rule: on one hand, there exists a differential graded Lie algebra (or generally an $L_\infty$-algebra), to whose Maurer-Cartan elements the algebraic structures correspond; on the other hand, there should be a suitable cohomology theory that characterizes its deformations. Deligne, Drinfeld, and Kontsevich proposed a celebrated slogan that *every reasonable deformation is controlled by a differential graded Lie algebra, up to quasi-isomorphisms*. More precisely, this slogan means that for an algebraic structure, Maurer-Cartan elements in a suitable graded Lie algebra are used to characterize realizations of its algebraic structure on a vector space. And moreover, for a given realization of the algebraic structure, a Maurer-Cartan element induces a differential making the graded Lie algebra into a differential graded Lie algebra (this graded Lie algebra is also called the controlling algebra). See [@GLST; @TBGS; @THS; @T.S2] for Maurer-Cartan characterizations of some algebraic structures and their relative Rota-Baxter operators. ## Rota-Baxter operators Baxter introduced the notion of Rota-Baxter operators on associative algebras when studying fluctuation theory [@Ba]. Then Kupershmidt introduced the notion of ${\mathcal{O}}$-operators (called relative Rota-Baxter operators in the present paper) on Lie algebras when he found that a relative Rota-Baxter operator is a solution to the classical Yang-Baxter equation in [@Kupershmidt], whereas the classical Yang-Baxter equation plays an important role in many fields such as in integration systems [@CP]. For more details about the classical Yang-Baxter equation and Rota-Baxter operators, one can see [@STS]. Recently, people found that Rota-Baxter operators have many applications. For eaxmple, the first work about Rota-Baxter algebras by Guo had been published [@Gub]. From then on, Rota-Baxter algebra open up a new branch in Lie theory and in mathematical physics. Bai and his collaborators examined Rota-Baxter operators in the context of binary operads in [@Bai-Bellier-Guo-Ni; @PBG]. Moreover, Rota-Baxter operators are closely connected with Hopf algebras. ## Lie-Yamaguti algebras A Lie-Yamaguti algebra is a generalization of a Lie algebra and a Lie triple system, which can be traced back to Nomizu's work on the invariant affine connections on homogeneous spaces in 1950's [@Nomizu]. Inspired by thoughts of Nomizu, in 1960's Yamaguti introduced an algebraic object called a general Lie triple system, defined its representation and established its cohomology theory in [@Yamaguti1; @Yamaguti2]. Kinyon and Weinstein first called this object a Lie-Yamaguti algebra  when studying Courant algebroids in the earlier 21st century [@Weinstein]. Since then, this system is called a Lie-Yamaguti algebra, which has attracted much attention and is widely investigated recently. For instance, Benito, Draper, and Elduque investigated Lie-Yamaguti algebras related to simple Lie algebras of type $G_2$ [@B.D.E]. Afterwards, Benito, Elduque, and Mart$\acute{i}$n-Herce explored irreducible Lie-Yamaguti algebras in [@B.E.M1; @B.E.M2]. More recently, Benito, Bremmer, and Madariaga examined orthogonal Lie-Yamaguti algebras in [@B.B.M]. Deformations and extensions of Lie-Yamaguti algebras were explored in [@L.CHEN; @Zhang1]. Takahashi studied modules over quandles using representations of Lie-Yamaguti algebras in [@Takahashi]. Since a Lie-Yamaguti algebra owns two algebraic structures, by the virtue of this object, the first author has been working on the deformation theory of Lie-Yamaguti algebras these years. For example, Sheng, the first author, and Zhou analyzed product structures and complex structures on Lie-Yamaguti algebras by means of Nijenhuis operators in [@Sheng; @Zhao]. Afterwards, Sheng and the first author introduced the notion of relative Rota-Baxter operators on Lie-Yamaguti algebras and revealed the fact that a pre-Lie-Yamaguti algebra is the underlying algebraic structure of relative Rota-Baxter operators [@SZ1]. Since then, thanks to importance of deformations and relative Rota-Bxter operators on Lie-Yamaguti algebras, we established Lie-Yamaguti bialgebra theory and clarified the relationship between the solution to the classical Lie-Yamaguti Yang-Baxter equation and relative Rota-Baxter operators in [@ZQ1], and studied cohomology and deformations of $\mathsf{LieYRep}$ pairs and then explored several properties of relative Rota-Baxter-Nijenhuis structures in [@ZQ2]. **Motivation.** Since Sheng and the first author defined relative Rota-Baxter operators on Lie-Yamaguti algebras [@SZ1], according to Deligne, Drinfeld, and Kontsevich's slogan, it is natural to consider Maurer-Cartan characterizations for relative Rota-Baxter operators on Lie-Yamaguti algebras and its relationship between cohomology and the Maurer-Cartan elements. However, the first and foremost, we have to consider Maurer-Cartan characterization for Lie-Yamaguti algebras because on the one hand, unlike associative algebras, Lie algebras, Leibniz algebras, or even $3$-Lie algebras, no works of controlling algebras of Lie-Yamaguti algebras appeared before; on the other hand, it is inevitable, for we construct the controlling algebra for relative Rota-Baxter operators. Thus, this step of constructing controlling algebra for Lie-Yamaguti algebras also promotes development of deformation theory for Lie-Yamaguti algebras. Anyway, our project can be transferred to the following questions: - Does there exist a suitable graded Lie algebra whose Maurer-Cartan elements correspond to the Lie-Yamaguti algebra structures on a vector space? - Does there exist a suitable algebra whose Maurer-Catan elemnts are precisely the relative Rota-Baxter operators on Lie-Yamaguti algebras? - Does there exist an appropriate cohomology theory of relative Rota-Baxter operators on Lie-Yamaguti algebras, which can be used to classify certain types of deformations? After solving these problems, we are able to consider deformations spontaneously, perfecting researches about cohomology and deformation theory on Lie-Yamaguti algebras and on their relative Rota-Baxter operators. ## Outline of the paper We tackle these problems as follows. After some preliminaries introduced in Section 2, Section 3 answers the first question. As was stated before, no works of controlling algebras of Lie-Yamaguti algebras appeared, thus constructing a graded Lie algebra whose Maurer-Cartan elements correspond to the Lie-Yamaguti algebra structures is a *difficulty* to overcome. Once the graded Lie algebra is constructed (see Theorem [Theorem 8](#fund){reference-type="ref" reference="fund"}), consequently a Maurer-Cartan element induces a differential making the graded Lie algebra into a differential graded Lie algebra as the controlling algebra for Lie-Yamaguti algebras (see Theorem [Theorem 10](#fund2){reference-type="ref" reference="fund2"}). Moreover, we establish the *relation* between the controlling algebra and the coboundary operator associated to the adjoint representation (see Theorem [Theorem 12](#diffe){reference-type="ref" reference="diffe"}). Section 4 is the solution to the last two questions, which is the core of the present paper. Immediately after the constructing controlling algebra for Lie-Yamaguti algebras, we make use of the graded Lie brackets by the method of derived brackets to construct an $L_\infty$-algebra whose Maurer-Cartan elements are precisely relative Rota-Baxter operators (see Theorem [Theorem 18](#main){reference-type="ref" reference="main"}). Moreover, from a relative Rota-Baxter operator (as a Maurer-Cartan element), we obtain a twisted $L_\infty$-algebra that controls deformations of relative Rota-Baxter operators (see Theorem [Theorem 20](#deformation){reference-type="ref" reference="deformation"}). As an application, we endow a continuous function space and a series ring with Lie-Yamaguti algebra structure respectively and give two integral operators which are treated as relative Rota-Baxter operators on them. And then we build cohomology of relative Rota-Baxter operators as follows. Given a relative Rota-Baxter operator $T:V\longrightarrow\mathfrak g$ with respect to a representation $(V;\rho,\mu)$, we mentioned in [@SZ1] that there is a Lie-Yamaguti algebra structure $([\cdot,\cdot]_T,\left\llbracket \cdot,\cdot,\cdot\right\rrbracket _T)$ on $V$, in which case the Lie-Yamaguti algebra $(V,[\cdot,\cdot]_T,\left\llbracket \cdot,\cdot,\cdot\right\rrbracket _T)$ is called the sub-adjacent Lie-Yamaguti algebra. In order to establish the cohomology theory, we first should build the cohomology of this sub-adjacent Lie-Yamaguti algebra using Yamaguti cohomology. Thus, we have to construct a representation of the sub-adjacent Lie-Yamaguti algebra (see Theorem [Theorem 25](#represent){reference-type="ref" reference="represent"}). However, note that the cochain complex of Yamaguti cohomology starts only from $1$-cochians, *not* from $0$-cochains. We have to point out that $0$-cochains play an import role in our study of deformations. Thus the next step is to choose $0$-cochains appropriately and build a proper coboundary map from the set of $0$-cochains to that of $1$-cochains, which is another *difficulty* (see Proposition [Proposition 26](#0cocy){reference-type="ref" reference="0cocy"}). In this way, we obtain a cochain complex (associated to $V$) starting from $0$-cochains, which gives rise to the cohomology of the relative Rota-Baxter operator $T$ on Lie-Yamaguti algebras $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ (see Definition [Definition 27](#cohomology){reference-type="ref" reference="cohomology"}). We also establish the *relation* between the differential of the twisted $L_\infty$-algebra and the coboundary operator of cohomology of relative Rota-Baxter operators parallel to the case of Lie-Yamaguti algebras (see Theorem [Theorem 28](#diff){reference-type="ref" reference="diff"}). Finally in Section 5, as an application, we make use of the cohomology theory to investigate deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras. We intend to consider two kinds of deformations: linear and higher order deformations. It turns out that our cohomology theory satisfies the rule that is mentioned above and works well (see Theorem [Theorem 34](#thm1){reference-type="ref" reference="thm1"} and Theorem [Theorem 42](#ob){reference-type="ref" reference="ob"}). A Lie-Yamaguti algebra owns two algebraic operations, which makes the operation of its controlling algebra and cochain complex much more complicated than other algebras, such as Lie algebras, pre-Lie algebras, Leibniz algebras or even $3$-Lie algebras, one of which owns only one operation. As a result, the computation is technically difficult in constructing the controlling algebras and defining the cohomology of relative Rota-Baxter operators. Note that a Lie triple system is a spacial case of a Lie-Yamaguti algebra [@Lister], so the conclusions in the present paper can also be adapted to the Lie triple system context. In this paper, all the vector spaces are over $\mathbb{K}$, a field of characteristic $0$. # Preliminaries In this section, we recall some basic notions such as Lie-Yamaguti algebras, representations and their cohomology theory. The notion of Lie-Yamaguti algebras was first defined by Yamaguti in [@Yamaguti1; @Yamaguti2]. **Definition 1**. [@Weinstein][\[LY\]]{#LY label="LY"} A **Lie-Yamaguti algebra** is a vector space $\mathfrak g$, together with a bilinear bracket $[\cdot,\cdot]:\wedge^2 \mathfrak{g} \to \mathfrak{g}$ and a trilinear bracket $\left\llbracket \cdot,\cdot,\cdot\right\rrbracket :\wedge^2\mathfrak g\otimes \mathfrak{g} \to \mathfrak{g}$ such that the following equations are satisfied for all $x,y,z,w,t \in \mathfrak g$, $$\begin{aligned} ~ &&\label{LY1}[[x,y],z]+[[y,z],x]+[[z,x],y]+\left\llbracket x,y,z\right\rrbracket +\left\llbracket y,z,x\right\rrbracket +\left\llbracket z,x,y\right\rrbracket =0,\\ ~ &&\left\llbracket [x,y],z,w\right\rrbracket +\left\llbracket [y,z],x,w\right\rrbracket +\left\llbracket [z,x],y,w\right\rrbracket =0,\\ ~ &&\label{LY3}\left\llbracket x,y,[z,w]\right\rrbracket =[\left\llbracket x,y,z\right\rrbracket ,w]+[z,\left\llbracket x,y,w\right\rrbracket ],\\ ~ &&\left\llbracket x,y,\left\llbracket z,w,t\right\rrbracket \right\rrbracket =\left\llbracket \left\llbracket x,y,z\right\rrbracket ,w,t\right\rrbracket +\left\llbracket z,\left\llbracket x,y,w\right\rrbracket ,t\right\rrbracket +\left\llbracket z,w,\left\llbracket x,y,t\right\rrbracket \right\rrbracket .\label{fundamental}\end{aligned}$$ **Example 2**. *[@Nomizu] Let $M$ be a closed manifold with an affine connection, and denote by $\mathfrak X(M)$ the set of vector fields on $M$. For all $x,y,z\in \mathfrak X(M)$, set $$\begin{aligned} =-T(x,y),\quad \left\llbracket x,y,z\right\rrbracket =-R(x,y)z,\end{aligned}$$ where $T$ and $R$ are torsion tensor and curvature tensor respectively. It turns out that the triple $(\mathfrak X(M),[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ forms an (infinite-dimensional) Lie-Yamaguti algebra.* **Definition 3**. [@Yamaguti2][\[defi:representation\]]{#defi:representation label="defi:representation"} Let $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ be a Lie-Yamaguti algebra and $V$ a vector space. A **representation** of $\mathfrak g$ on $V$ consists of a linear map $\rho:\mathfrak g\to \mathfrak {gl}(V)$ and a bilinear map $\mu:\otimes^2 \mathfrak g\to \mathfrak {gl}(V)$ such that for all $x,y,z,w \in \mathfrak g$, $$\begin{aligned} ~&&\label{RLYb}\mu([x,y],z)-\mu(x,z)\rho(y)+\mu(y,z)\rho(x)=0,\\ ~&&\label{RLYd}\mu(x,[y,z])-\rho(y)\mu(x,z)+\rho(z)\mu(x,y)=0,\\ ~&&\label{RLYe}\rho(\left\llbracket x,y,z\right\rrbracket )=[D_{\rho,\mu}(x,y),\rho(z)],\\ ~&&\label{RYT4}\mu(z,w)\mu(x,y)-\mu(y,w)\mu(x,z)-\mu(x,\left\llbracket y,z,w\right\rrbracket )+D_{\rho,\mu}(y,z)\mu(x,w)=0,\\ ~&&\label{RLY5}\mu(\left\llbracket x,y,z\right\rrbracket ,w)+\mu(z,\left\llbracket x,y,w\right\rrbracket )=[D_{\rho,\mu}(x,y),\mu(z,w)],\end{aligned}$$ where the bilinear map $D_{\rho,\mu}:\otimes^2\mathfrak g\to \mathfrak {gl}(V)$ is given by $$\begin{aligned} D_{\rho,\mu}(x,y):=\mu(y,x)-\mu(x,y)+[\rho(x),\rho(y)]-\rho([x,y]), \quad \forall x,y \in \mathfrak g.\label{rep} \end{aligned}$$ It is obvious that $D_{\rho,\mu}$ is skew-symmetric. We write $D$ in the sequel without ambiguities and we denote a representation of $\mathfrak g$ on $V$ by $(V;\rho,\mu)$. By a direct computation, we have **Proposition 4**. *If $(V;\rho,\mu)$ is a representation of a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$, then we have the following equalities: $$\begin{aligned} \label{RLYc}&&D([x,y],z)+D([y,z],x)+D([z,x],y)=0;\\ \label{RLY5a}&&D(\left\llbracket x,y,z\right\rrbracket ,w)+D(z,\left\llbracket x,y,w\right\rrbracket )=[D(x,y),D(z,w)];\\ ~ &&\mu(\left\llbracket x,y,z\right\rrbracket ,w)=\mu(x,w)\mu(z,y)-\mu(y,w)\mu(z,x)-\mu(z,w)D(x,y).\label{RLY6}\end{aligned}$$* **Example 5**. *Let $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ be a Lie-Yamaguti algebra. We define linear maps $\mathrm{ad}:\mathfrak g\to \mathfrak {gl}(\mathfrak g)$ and $\mathfrak R:\otimes^2\mathfrak g\to \mathfrak {gl}(\mathfrak g)$ to be $x \mapsto \mathrm{ad}_x$ and $(x,y) \mapsto \mathfrak{R}_{x,y}$, where $\mathrm{ad}_xz=[x,z]$ and $\mathfrak{R}_{x,y}z=\left\llbracket z,x,y\right\rrbracket$ for all $z \in \mathfrak g$ respectively. Then $(\mathfrak g;\mathrm{ad},\mathfrak{R})$ forms a representation of $\mathfrak g$ on itself, called the **adjoint representation**. In this case, $\mathfrak L\triangleq D_{\mathrm{ad},\mathfrak R}$ is given by for all $x,y \in \mathfrak g$, $$\begin{aligned} \mathfrak L_{x,y}=\mathfrak{R}_{y,x}-\mathfrak{R}_{x,y}+[\mathrm{ad}_x,\mathrm{ad}_y]-\mathrm{ad}_{[x,y]}.\end{aligned}$$ By [\[LY1\]](#LY1){reference-type="eqref" reference="LY1"}, we have $$\begin{aligned} \mathfrak L_{x,y}z=\left\llbracket x,y,z\right\rrbracket , \quad \forall z \in \mathfrak g.\label{lef}\end{aligned}$$* Representations of a Lie-Yamaguti algebra can be characterized by the semidirect product Lie-Yamaguti algebras. **Proposition 6**. *[@Zhang1] Let $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ be a Lie-Yamaguti algebra and $V$ a vector space. Let $\rho:\mathfrak g\to \mathfrak {gl}(V)$ and $\mu:\otimes^2 \mathfrak g\to \mathfrak {gl}(V)$ be linear maps. Then $(V;\rho,\mu)$ is a representation of $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ if and only if there is a Lie-Yamaguti algebra structure $([\cdot,\cdot]_{\rho,\mu},\left\llbracket \cdot,\cdot,\cdot\right\rrbracket _{\rho,\mu})$ on the direct sum $\mathfrak g\oplus V$ which is defined by for all $x,y,z \in \mathfrak g, ~u,v,w \in V$, $$\begin{aligned} \label{semi1}[x+u,y+v]_{\rho,\mu}&=&[x,y]+\rho(x)v-\rho(y)u,\\ \label{semi2}~\left\llbracket x+u,y+v,z+w\right\rrbracket _{\rho,\mu}&=&\left\llbracket x,y,z\right\rrbracket +D(x,y)w+\mu(y,z)u-\mu(x,z)v,\end{aligned}$$ This Lie-Yamaguti algebra $(\mathfrak g\oplus V,[\cdot,\cdot]_{\rho,\mu},\left\llbracket \cdot,\cdot,\cdot\right\rrbracket _{\rho,\mu})$ is called the **semidirect product Lie-Yamaguti algebra**, and denoted by $\mathfrak g\ltimes_{\rho,\mu} V$.* The cohomology theory of Lie-Yamaguti algebras was established in [@Yamaguti2]. Let $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ be a Lie-Yamaguti algebra and $(V;\rho,\mu)$ a representation. We denote by $C^p_{\rm LieY}(\mathfrak g,V)~(p \geqslant 1)$ the set of $p$-cochains, where $$\begin{aligned} C^{n+1}_{\rm LieY}(\mathfrak g,V)\triangleq \begin{cases} \mathrm{Hom}(\underbrace{\wedge^2\mathfrak g\otimes \cdots \otimes \wedge^2\mathfrak g}_n,V)\times \mathrm{Hom}(\underbrace{\wedge^2\mathfrak g\otimes\cdots\otimes\wedge^2\mathfrak g}_{n}\otimes\mathfrak g,V), & n\geqslant 1,\\ \mathrm{Hom}(\mathfrak g,V), &n=0. \end{cases}\end{aligned}$$ For $p\geqslant 1$, the coboundary operator $\delta:C^p_{\rm LieY}(\mathfrak g,V)\to C^{p+1}_{\rm LieY}(\mathfrak g,V)$ is defined as follows: - If $n\geqslant 1$, for any $F=(f,g)\in C^{n+1}_{\rm LieY}(\mathfrak g,V)$, the coboundary map $$\delta=(\delta_{\rm I},\delta_{\rm II}):C^{n+1}_{\rm LieY}(\mathfrak g,V)\to C^{n+2}_{\rm LieY}(\mathfrak g,V),$$ $$\qquad \qquad\qquad \qquad\qquad \quad F\mapsto(\delta_{\rm I}(F),\delta_{\rm II}(F)),$$ is given by: $$\begin{aligned} ~ &&\nonumber\Big(\delta_{\rm I}(F)\Big)(\mathfrak X_1,\cdots,\mathfrak X_{n+1})\\ ~\label{cohomolo1} &=&(-1)^n\Big(\rho(x_{n+1})g(\mathfrak X_1,\cdots,\mathfrak X_n,y_{n+1})-\rho(y_{n+1})g(\mathfrak X_1,\cdots,\mathfrak X_n,x_{n+1})\\ ~ &&\nonumber-g(\mathfrak X_1,\cdots,\mathfrak X_n,[x_{n+1},y_{n+1}])\Big)\\ ~ &&\nonumber+\sum_{k=1}^{n}(-1)^{k+1}D(\mathfrak X_k)f(\mathfrak X_1,\cdots,\hat{\mathfrak X_k},\cdots,\mathfrak X_{n+1})\\ ~ &&\nonumber+\sum_{1\leqslant k<l\leqslant n+1}(-1)^{k}f(\mathfrak X_1,\cdots,\hat{\mathfrak X_k},\cdots,\mathfrak X_k\circ\mathfrak X_l,\cdots,\mathfrak X_{n+1}),\end{aligned}$$ and $$\begin{aligned} ~ &&\nonumber\Big(\delta_{\rm II}(F)\Big)(\mathfrak X_1,\cdots,\mathfrak X_{n+1},z)\\ ~ \label{cohomolo2}&=&(-1)^n\Big(\mu(y_{n+1},z)g(\mathfrak X_1,\cdots,\mathfrak X_n,x_{n+1})-\mu(x_{n+1},z)g(\mathfrak X_1,\cdots,\mathfrak X_n,y_{n+1})\Big)\\ ~ &&\nonumber+\sum_{k=1}^{n+1}(-1)^{k+1}D(\mathfrak X_k)g(\mathfrak X_1,\cdots,\hat{\mathfrak X_k},\cdots,\mathfrak X_{n+1},z)\\ ~ &&\nonumber+\sum_{1\leqslant k<l\leqslant n+1}(-1)^kg(\mathfrak X_1,\cdots,\hat{\mathfrak X_k},\cdots,\mathfrak X_k\circ\mathfrak X_l,\cdots,\mathfrak X_{n+1},z)\\ ~ &&\nonumber+\sum_{k=1}^{n+1}(-1)^kg(\mathfrak X_1,\cdots,\hat{\mathfrak X_k},\cdots,\mathfrak X_{n+1},\left\llbracket x_k,y_k,z\right\rrbracket ),\end{aligned}$$ where $\mathfrak X_i=x_i\wedge y_i\in\wedge^2\mathfrak g,~(i=1,\cdots,n+1),~z\in \mathfrak g$ and the operation $\circ$ means that $$\mathfrak X_k\circ\mathfrak X_l\triangleq\left\llbracket x_k,y_k,x_l\right\rrbracket \wedge y_l+x_l\wedge\left\llbracket x_k,y_k,y_l\right\rrbracket .$$ - For the case that $n=0$, any element $f \in C^1_{\rm LieY}(\mathfrak g,V)$ given, the coboundary map $$\delta:C^1_{\rm LieY}(\mathfrak g,V)\to C^2_{\rm LieY}(\mathfrak g,V),$$ $$\qquad \qquad \qquad f\mapsto (\delta_{\rm I}(f),\delta_{\rm II}(f)),$$ is given by $$\begin{aligned} \label{1cochain}(\delta_{\rm I}(f))(x,y)&=&\rho(x)f(y)-\rho(y)f(x)-f([x,y]),\\ ~ \label{2cochain}(\delta_{\rm II}(f))(x,y,z)&=&D(x,y)f(z)+\mu(y,z)f(x)-\mu(x,z)f(y)-f(\left\llbracket x,y,z\right\rrbracket ),\quad \forall x,y, z\in \mathfrak g.\end{aligned}$$ Yamaguti showed in [@Yamaguti2] that $\delta$ is a differential, i.e., $\delta\circ\delta=0$. More precisely, for any $f\in C^1_{\rm LieY}(\mathfrak g,V)$, we have $$\delta_{\rm I}\Big(\delta_{\rm I}(f),\delta_{\rm II}(f)\Big)=0\quad\text{and}\quad\delta_{\rm II}\Big(\delta_{\rm I}(f),\delta_{\rm II}(f)\Big)=0.$$ Moreover, for all $F\in C^p_{\rm LieY}(\mathfrak g,V)~(p\geqslant 2)$, we have $$\delta_{\rm I}\Big(\delta_{\rm I}(F),\delta_{\rm II}(F)\Big)=0\quad\text{and}\quad\delta_{\rm II}\Big(\delta_{\rm I}(F),\delta_{\rm II}(F)\Big) =0.$$ Thus the cochain complex $(C^\bullet_{\rm LieY}(\mathfrak g,V)=\bigoplus_{p=1}^\infty C^p_{\rm LieY}(\mathfrak g,V),\delta)$ is well defined, whose cohomology is called the **Yamaguti cohomology** in this paper. **Definition 7**. An $p$-cocycle $(p\geqslant 1)$ is an element $F=(f,g)$ in $C^p_{\rm LieY}(\mathfrak g,V)~(p\geqslant 2)$ (resp. $f\in C^1_{\rm LieY}(\mathfrak g,V)$) such that $\delta(F)=0$ (resp. $\delta(f)=0$). The set of $p$-cocycles is denoted by $Z^p_{\rm LieY}(\mathfrak g,V)$; for an element $F$ in $C^p_{\rm LieY}(\mathfrak g,V)~(p\geqslant 2)$, if there exists $G=(h,s)\in C^{p-1}_{\rm LieY}(\mathfrak g,V)$ (resp. $t\in C^1(\mathfrak g,V)$, if $p=2$) such that $F=\delta(G)$ (resp. $F=\delta(t)$), then $F$ is called an $p$-coboundary. The set of $p$-coboundaries is denoted by $B^p_{\rm LieY}(\mathfrak g,V)$. The resulting $p$-cohomology group is defined by the factor space $$H^p_{\rm LieY}(\mathfrak g,V)=Z^p_{\rm LieY}(\mathfrak g,V)/B^p_{\rm LieY}(\mathfrak g,V)~\quad (p\geqslant 2).$$ # Maurer-Cartan characterizations for Lie-Yamaguti algebras In this section, for a preparation of constructing controlling algebra for relative Rota-Baxter operators, we have to construct the controlling algebra for Lie-Yamaguti algebras first. Let us recall some notions and basic facts in [@Loday]. A degree $1$ element $x\in \mathfrak g_1$ is called a **Maurer-Cartan** element of a differential graded Lie algebra $(\mathfrak g=\oplus_{k\in\mathbb Z}\mathfrak g_k,[\cdot,\cdot],d)$ if it satisfies the Maurer-Cartan equation: $dx+\frac{1}{2}[x,x]=0.$ Note that a graded Lie algebra is a special differential graded Lie algebra with $d=0$. Correspondingly, for a graded Lie algebra $(\mathfrak g=\oplus_{k\in\mathbb Z}\mathfrak g_k,[\cdot,\cdot])$, an element $x\in \mathfrak g_1$ satisfying $[x,x]=0$ is a Maurer-Cartan element of $\mathfrak g$. A permutation $\sigma\in S_n$ is called an **$(i,n-i)$-shuffle** if $\sigma(1)<\cdots <\sigma(i)$ and $\sigma(i+1)<\cdots <\sigma(n)$. If $i=0$ or $i=n$, we assume $\sigma={\rm{Id}}$. The set of $(i,n-i)$-shuffles is denoted by $\mathbb S_{(i,n-i)}$. Let $\mathfrak g$ be a vector space. Denote by $$\begin{aligned} \mathfrak C^p(\mathfrak g,\mathfrak g):= \begin{cases} \mathrm{Hom}(\underbrace{(\wedge^2\mathfrak g)\otimes\cdots(\wedge^2\mathfrak g)}_p,\mathfrak g)\oplus\mathrm{Hom}(\underbrace{(\wedge^2\mathfrak g)\otimes\cdots(\wedge^2\mathfrak g)}_p\otimes\mathfrak g,\mathfrak g),& p\geqslant 1,\\ \mathrm{Hom}(\mathfrak g,\mathfrak g),&p=0. \end{cases}\end{aligned}$$ Then $\mathfrak C^\bullet(\mathfrak g,\mathfrak g)=\oplus_{p\geqslant 0}\mathfrak C^p(\mathfrak g,\mathfrak g)$ is a graded vector space, in which the degree of elements in $\mathfrak C^p(\mathfrak g,\mathfrak g)$ is assumed to be $p$. For $P=(P_{\rm I},P_{\rm II}) \in \mathfrak C^p(\mathfrak g,\mathfrak g),~Q=(Q_{\rm I},Q_{\rm II})\in \mathfrak C^q(\mathfrak g,\mathfrak g)~(p,q\geqslant 1)$, we define $P\circ Q=\Big((P\circ Q)_{\rm I},(P\circ Q)_{\rm II}\Big)\in \mathfrak C^{p+q}(\mathfrak g,\mathfrak g)$ to be $$\begin{aligned} ~ \nonumber&&\Big(P\circ Q\Big)_{\rm I}(\mathfrak X_1,\cdots,\mathfrak X_{p+q})\\ ~ \nonumber&=&\sum_{\sigma\in\mathbb S_{(p,q)}\atop \sigma(p+q)=p+q}(-1)^{pq}sign(\sigma)P_{\rm II}(\mathfrak X_{\sigma(1)},\cdots,\mathfrak X_{\sigma(p)},Q_{\rm I}(\mathfrak X_{\sigma(p+1)}, \cdots,\mathfrak X_{\sigma(p+q)}))\\ \label{gradedbra1}&&+\sum_{k=1}^p(-1)^{(k-1)q}\sum_{\sigma\in \mathbb S_{(k-1,q)}}sign(\sigma)P_{\rm I}(\mathfrak X_{\sigma(1)},\cdots,\mathfrak X_{\sigma(k-1)},x_{q+k}\wedge Q_{\rm II}(\mathfrak X_{\sigma(k)},\cdots,\mathfrak X_{\sigma(k+q-1)},y_{k+q}),\mathfrak X_{k+q+1},\cdots,\mathfrak X_{p+q})\\ ~ \nonumber&&+\sum_{k=1}^p(-1)^{(k-1)q}\sum_{\sigma\in \mathbb S_{(k-1,q)}}sign(\sigma)P_{\rm I}(\mathfrak X_{\sigma(1)},\cdots,\mathfrak X_{\sigma(k-1)}, Q_{\rm II}(\mathfrak X_{\sigma(k)},\cdots,\mathfrak X_{\sigma(k+q-1)},x_{k+q})\wedge y_{k+q},\mathfrak X_{k+q+1},\cdots,\mathfrak X_{p+q}),\end{aligned}$$ and $$\begin{aligned} ~ \nonumber&&\Big(P\circ Q\Big)_{\rm II}(\mathfrak X_1,\cdots,\mathfrak X_{p+q},x)\\ ~ \nonumber&=&\sum_{\sigma\in\mathbb S_{(p,q)}}(-1)^{pq}sign(\sigma)P_{\rm II}(\mathfrak X_{\sigma(1)},\cdots,\mathfrak X_{\sigma(p)},Q_{\rm II}(\mathfrak X_{\sigma(p+1)}, \cdots,\mathfrak X_{\sigma(p+q)},x))\\ \label{gradedbra3}&&+\sum_{k=1}^p(-1)^{(k-1)q}\sum_{\sigma\in \mathbb S_{(k-1,q)}}sign(\sigma)P_{\rm II}(\mathfrak X_{\sigma(1)},\cdots,\mathfrak X_{\sigma(k-1)},x_{q+k}\wedge Q_{\rm II}(\mathfrak X_{\sigma(k)},\cdots,\mathfrak X_{\sigma(k+q-1)},y_{k+q}),\mathfrak X_{k+q+1},\cdots,\mathfrak X_{p+q},x)\\ ~ \nonumber&&+\sum_{k=1}^p(-1)^{(k-1)q}\sum_{\sigma\in \mathbb S_{(k-1,q)}}sign(\sigma)P_{\rm II}(\mathfrak X_{\sigma(1)},\cdots,\mathfrak X_{\sigma(k-1)}, Q_{\rm II}(\mathfrak X_{\sigma(k)},\cdots,\mathfrak X_{\sigma(k+q-1)},x_{k+q})\wedge y_{k+q},\mathfrak X_{k+q+1},\cdots,\mathfrak X_{p+q},x),\end{aligned}$$ In particular, for $f,g\in \mathfrak C^0(\mathfrak g,\mathfrak g)=\mathrm{Hom}(\mathfrak g,\mathfrak g)$ and $P=(P_{\rm I},P_{\rm II})\in \mathfrak C^p(\mathfrak g,\mathfrak g)~(p\geqslant 1)$, we define $$\begin{aligned} \label{gradedbra4}\Big(P\circ f\Big)_{\rm I}(\mathfrak X_1,\cdots,\mathfrak X_p)&=&\sum_{k=1}^pP_{\rm I}\Big(\mathfrak X_1,\cdots,\mathfrak X_{k-1},x_k\wedge f(y_k),\mathfrak X_{k+1},\cdots,\mathfrak X_p\Big)\\ ~\nonumber&& +\sum_{k=1}^pP_{\rm I}\Big(\mathfrak X_1,\cdots,\mathfrak X_{k-1},f(x_k)\wedge y_k,\mathfrak X_{k+1},\cdots,\mathfrak X_p\Big),\\ ~ \nonumber&&\\ \Big(f\circ P\Big)_{\rm I}(\mathfrak X_1,\cdots,\mathfrak X_p)&=&f\Big(P_{\rm I}(\mathfrak X_1,\cdots,\mathfrak X_p)\Big),\label{gradedbra5}\\ \Big(P\circ f\Big)_{\rm II}(\mathfrak X_1,\cdots,\mathfrak X_p,x)&=&\sum_{k=1}^pP_{\rm II}\Big(\mathfrak X_1,\cdots,\mathfrak X_{k-1},x_k\wedge f(y_k),\mathfrak X_{k+1},\cdots,\mathfrak X_{p},x\Big)\label{gradedbra6}\\ ~\nonumber&& +\sum_{k=1}^pP_{\rm II}\Big(\mathfrak X_1,\cdots,\mathfrak X_{k-1},f(x_k)\wedge y_k,\mathfrak X_{k+1},\cdots,\mathfrak X_{p},x\Big)\\ ~ \nonumber&&+P_{\rm II}\Big(\mathfrak X_1,\cdots,\mathfrak X_p,f(x)\Big),\\ \label{gradedbra2}\Big(f\circ P\Big)_{\rm II}(\mathfrak X_1,\cdots,\mathfrak X_p,x)&=&f\Big(P_{\rm II}(\mathfrak X_1,\cdots,\mathfrak X_p,x)\Big). \end{aligned}$$ Moreover, $f\circ g$ means the composition of $f$ and $g$. Let us introduce some notations. Let $\mathfrak g$ and $V$ be vector spaces. Denote a degree $1$ element $(\pi,\omega)\in \mathfrak C^1(\mathfrak g,\mathfrak g)$ in $\mathfrak C^\bullet(\mathfrak g,\mathfrak g)$ by $\Pi$. Correspondingly, for another element $\Pi'=(\pi',\omega')\in \mathfrak C^1(\mathfrak g,\mathfrak g)$, the element $(\pi+\pi',\omega+\omega')$ is written as $\Pi+\Pi'$. Moreover, recall that $$\mathfrak C^1(\mathfrak g\oplus V,\mathfrak g\oplus V)=\mathrm{Hom}(\wedge^2(\mathfrak g\oplus V),\mathfrak g\oplus V)\oplus \mathrm{Hom}(\wedge^2(\mathfrak g\oplus V)\otimes (\mathfrak g\oplus V),\mathfrak g\oplus V).$$ For linear maps $\rho:\mathfrak g\to\mathfrak {gl}(V)$ and $\mu:\otimes^2\mathfrak g\to \mathfrak {gl}(V)$, define the degree $1$ elements $\overline\Pi=(\bar\pi,\bar\omega)$ and $\overline\Theta=(\bar\rho,\bar\mu)\in \mathfrak C^1(\mathfrak g\oplus V,\mathfrak g\oplus V)$ to be for all $x,y,z\in \mathfrak g$ and $u,v,w\in V$, $$\begin{aligned} \begin{cases} \bar{\pi}(x+u,y+v)=\pi(x,y),\\ \bar{\omega}(x+u,y+v,z+w)=\omega(x,y,z), \end{cases}\end{aligned}$$ and $$\begin{aligned} \begin{cases} \bar{\rho}(x+u,y+v)=\rho(x)v-\rho(y)u,\\ \bar{\mu}(x+u,y+v,z+w)=D(x,y)w+\mu(y,z)u-\mu(x,z)v, \end{cases}\end{aligned}$$ respectively. In the following, we denote $\bar \rho$ and $\bar\mu$ by $\rho$ and $\mu$ respectively, then the degree $1$ element $\overline\Pi+\overline\Theta=(\bar\pi+\bar\rho,\bar\omega+\bar\mu)\in \mathfrak C^1(\mathfrak g\oplus V,\mathfrak g\oplus V)$ is written as $\Pi+\Theta=(\pi+\rho,\omega+\mu)$ correspondingly. In the sequel, we also denote a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot\right\rrbracket )$ by $(\mathfrak g,\pi,\omega)$, where $\pi=[\cdot,\cdot]$ and $\omega=\left\llbracket \cdot,\cdot,\cdot\right\rrbracket$. **Theorem 8**. *With the above notations, the graded vector space $\mathfrak C^\bullet(\mathfrak g,\mathfrak g)$ equipped with the graded commutator $$[P,Q]_{\mathsf{LieY}}=P\circ Q-(-1)^{pq}Q\circ P$$ is a graded Lie algebra, where the operation $\circ$ is defined as [\[gradedbra1\]](#gradedbra1){reference-type="eqref" reference="gradedbra1"}-[\[gradedbra2\]](#gradedbra2){reference-type="eqref" reference="gradedbra2"}. Moreover, if $\Pi\in \mathfrak C^1(\mathfrak g,\mathfrak g)$ defines a Lie-Yamaguti algebra structure on $\mathfrak g$, then $\Pi$ is a Maurer-Cartan element of the graded Lie algebra $(C^\bullet(\mathfrak g,\mathfrak g),[\cdot,\cdot]_{\mathsf{LieY}})$.* *Proof.* We first prove the graded bracket is skew-symmetric. Indeed, for all $P \in \mathfrak C^p(\mathfrak g,\mathfrak g),~Q \in \mathfrak C^q(\mathfrak g,\mathfrak g)~(p,q\geqslant 0)$, we write $$[P,Q]_{\mathsf{LieY}}=([P,Q]_{\rm I},[P,Q]_{\rm II})$$ where $$\begin{aligned} ~[P,Q]_{\rm I}&=&\Big(P\circ Q\Big)_{\rm I}-(-1)^{pq}\Big(Q\circ P\Big)_{\rm I}\\ ~[P,Q]_{\rm II}&=&\Big(P\circ Q\Big)_{\rm II}-(-1)^{pq}\Big(Q\circ P\Big)_{\rm II}.\end{aligned}$$ We have $$\begin{aligned} ~[P,Q]_{\mathsf{LieY}}&=&([P,Q]_{\rm I},[P,Q]_{\rm II})\\ ~ &=&(-(-1)^{pq}[Q,P]_{\rm I},-(-1)^{pq}[Q,P]_{\rm II}\\ ~ &=&-(-1)^{pq}\Big([Q,P]_{\rm I},[Q,P]_{\rm II}\Big)\\ ~ &=&-(-1)^{pq}[Q,P]_{\mathsf{LieY}}.\end{aligned}$$ Next we prove that the bracket $[\cdot,\cdot]_{\mathsf{LieY}}$ satisfies the graded Jacobi Identity, which means that both $[\cdot,\cdot]_{\rm I}$ and $[\cdot,\cdot]_{\rm {II}}$ satisfy the graded Jacobi Identity. In fact, that the bracket $[\cdot,\cdot]_{\rm {II}}$ satisfies the graded Jacobi Identity has been proved in [@Rot] using the weight rule. For any element $P\in \mathfrak C^p(\mathfrak g,\mathfrak g)~(p\geqslant 1)$, fix a nonzero element $x\in \mathfrak g$, and define a map $$\Phi:\mathrm{Hom}(\wedge^2\mathfrak g^{\otimes p}\otimes\mathfrak g,\mathfrak g)\to\mathrm{Hom}(\wedge^2\mathfrak g^{\otimes p},\mathfrak g)$$ to be $$\Phi(P)(\mathfrak X_1,\cdots,\mathfrak X_p)=P(\mathfrak X_1,\cdots,\mathfrak X_p,x),\quad \forall P\in \mathrm{Hom}(\wedge^2\mathfrak g^{\otimes p}\otimes\mathfrak g,\mathfrak g),$$ where $\mathfrak X_k\in \wedge^2\mathfrak g~(k=1,2,\cdots,p)$. Then since the graded vector space $\oplus_{p\geqslant 0}\mathrm{Hom}(\wedge^2\mathfrak g^{\otimes p}\otimes\mathfrak g,\mathfrak g)$ endowed with $[\cdot,\cdot]_{\rm II}$ is a graded Lie algebra, the graded vector space $\oplus_{p\geqslant 0}\mathrm{Hom}(\wedge^2\mathfrak g^{\otimes p},\mathfrak g)$ is also a graded Lie algebra via the map $\Phi$, whose graded Lie bracket is exactly the bracket $[\cdot,\cdot]_{\rm I}$ for two variables whose degrees $\geqslant 1$. Furthermore, by a direct computation, if either two variables is of degree $0$, then the bracket $[\cdot,\dot]_{\rm I}$ is also a graded Lie bracket. Thus the bracket $[\cdot,\cdot]_{\mathsf{LieY}}$ satisfies the graded Jacobi Identity. Finally, for all $x,y,z,w,t\in \mathfrak g$, we have $$\begin{aligned} ~ &&[\Pi,\Pi]_{\rm I}(x,y,z,w)\\ ~ &=&2\Big(\big(\Pi\circ\Pi\big)_{\rm I}(x,y,z,w)\Big)\\ ~ &=&2\Big(\omega(x,y,\pi(z,w))-\pi(\omega(x,y,z),w)-\pi(z,\omega(x,y,w))\Big),\end{aligned}$$ and $$\begin{aligned} ~ &&[\Pi,\Pi]_{\rm II}(x,y,z,w,t)\\ ~ &=&2\Big(\big(\Pi\circ\Pi\big)_{\rm II}(x,y,z,w)\Big)\\ ~ &=&2\Big(\omega(x,y,\omega(z,w,t))-\omega(z,w,\omega(x,y,t))-\omega(\omega(x,y,z),w,t)-\omega(z,\omega(x,y,w),t)\Big),\end{aligned}$$ which proves the last statement. This completes the proof. ◻ **Remark 9**. We have to point out that any Maurer-Cartan element $\Pi=(\pi,\omega)\in \mathfrak C^1(\mathfrak g,\mathfrak g)$ in the graded Lie algebra $\mathfrak C^\bullet(\mathfrak g,\mathfrak g)$ does not precisely correspond to a Lie-Yamaguti algebra structure on $\mathfrak g$. By the proof of Theorem [Theorem 8](#fund){reference-type="ref" reference="fund"}, we know that $\Pi$ is a Maurer-Cartan element if and only if $\Pi$ satisfies Eqs. [\[LY3\]](#LY3){reference-type="eqref" reference="LY3"} and [\[fundamental\]](#fundamental){reference-type="eqref" reference="fundamental"}, i.e., for all $x,y,z,w,t\in \mathfrak g$, $\Pi$ satisfies $$\begin{aligned} \omega(x,y,\pi(z,w))&=&\pi(\omega(x,y,z),w)+\pi(z,\omega(x,y,w)),\\ ~\omega(x,y,\omega(z,w,t))&=&\omega(\omega(x,y,z),w,t)+\omega(z,\omega(x,y,w),t)+\omega(z,w,\omega(x,y,t)).\end{aligned}$$ This reveals to us that case of Lie-Yamaguti algebras is more complicated than that of Lie algebras, Leibniz algebras or even $3$-Lie algebras. See [@GLST] for more details about Maurer-Cartan characterizations of other algebraic structures. Let $\Pi$ define a Lie-Yamaguti algebra structure on the vector space $\mathfrak g$. It follows from the graded Jacobi identity that $d_\Pi:=[\Pi,\cdot]_{\rm LieY}$ is a differential on $(\mathfrak C^\bullet(\mathfrak g,\mathfrak g),[\cdot,\cdot]_{\rm LieY})$. More precisely, we have **Theorem 10**. *Let $(\mathfrak g,\pi,\omega)$ be a Lie-Yamaguti algebra. Then with the above notations, the triple $(\mathfrak C^\bullet(\mathfrak g,\mathfrak g),[\cdot,\cdot]_{\rm LieY},d_{\Pi})$ is a differential graded Lie algebra, where $d_{\Pi}$ is defined to be $$d_{\Pi}:=[\Pi,\cdot]_{\rm LieY}.$$ If moreover, for an element $\Pi'\in \mathfrak C^1(\mathfrak g,\mathfrak g)$, $\Pi+\Pi'$ defines a Lie-Yamaguti algebra structure on $\mathfrak g$, then $\Pi'$ is a Maurer-Cartan element of the differential graded Lie algebra $(\mathfrak C^\bullet(\mathfrak g,\mathfrak g),[\cdot,\cdot]_{\rm LieY},d_{\Pi})$.* From Proposition [Proposition 6](#semi){reference-type="ref" reference="semi"} and Theorem [Theorem 8](#fund){reference-type="ref" reference="fund"}, representations of Lie-Yamaguti algebras can be characterized as Maurer-Cartan elements in a graded Lie algebra. More precisely, we have **Proposition 11**. *With the above notations, if $(V;\rho,\mu)$ is a representation of Lie-Yamaguti algebra $(\mathfrak g,\pi,\omega)$, then $\overline\Theta\in \mathfrak C^1(\mathfrak g\oplus V,\mathfrak g\oplus V)$ is a Maurer-Cartan element of the differential graded Lie algebra $(\mathfrak C^\bullet(\mathfrak g\oplus V,\mathfrak g\oplus V),[\cdot,\cdot]_{\rm LieY},d_{\overline\Pi})$.* At the end of this section, we give the relationship between the coboundary operator of a Lie-Yamaguti algebra associated to the adjoint representation and the differential $d_{\Pi}$ induced by the Maurer-Cartan element. **Theorem 12**. *Let $(\mathfrak g,\pi,\omega)$ be a Lie-Yamaguti algebra and $\delta:\mathfrak C^n_{\mathsf{LieY}}(\mathfrak g,\mathfrak g)\to \mathfrak C^{n+1}_{\mathsf{LieY}}(\mathfrak g,\mathfrak g)$ be the coboundary map associated to the adjoint representation. Then we have $$\begin{aligned} \label{cohomo1}\delta(F)&=&(-1)^{n}d_{\Pi}(F)=(-1)^{n}[\Pi,F]_{\rm LieY}, \quad \forall F\in \mathfrak C^n(\mathfrak g,\mathfrak g) ~(n\geqslant 1),\\ \label{cohomo2}\delta(f)&=&d_{\Pi}(f)=[\Pi,f]_{\rm LieY},\quad \forall f\in \mathfrak C^0(\mathfrak g,\mathfrak g)=\mathrm{Hom}(\mathfrak g,\mathfrak g).\end{aligned}$$* *Proof.* For all $F=(f,g)\in \mathfrak C^n(\mathfrak g,\mathfrak g)~(n\geqslant 1)$ and for all $\mathfrak X_1,\cdots,\mathfrak X_{n+1}\in \wedge^2\mathfrak g,~x\in \mathfrak g$, we compute that $$\begin{aligned} ~ &&[\Pi,F]_{\rm I}(\mathfrak X_1,\cdots,\mathfrak X_{n+1})\\ ~ &=&\Big(\big(\Pi\circ F\big)_{\rm I}-(-1)^n\big(F\circ\Pi\big)_{\rm I}\Big)(\mathfrak X_1,\cdots,\mathfrak X_{n+1})\\ ~ &\stackrel{\eqref{gradedbra1}}{=}&\sum_{\sigma\in \mathbb S_{(1,n)}\atop \sigma(n+1)=n+1}(-1)^n sign(\sigma)\left\llbracket \mathfrak X_{\sigma(1)},f(\mathfrak X_{\sigma(2)},\cdots,\mathfrak X_{\sigma(n+1)})\right\rrbracket \\ ~ &&+[x_{n+1},g(\mathfrak X_1,\cdots,\mathfrak X_n,y_{n+1})]-[y_{n+1},g(\mathfrak X_1,\cdots,\mathfrak X_n,x_{n+1})]\\ ~ &&-(-1)^n\Big((-1)^ng(\mathfrak X_{\sigma(1)},\cdots,\mathfrak X_{\sigma(n)},[x_{n+1},y_{n+1}])\\ ~ &&+\sum_{k=1}^n(-1)^{k-1}\sum_{\sigma\in \mathbb S_{(k-1,1)}}sign(\sigma)f(\mathfrak X_{\sigma(1)},\cdots,\mathfrak X_{\sigma(k-1)}, x_{k+1}\wedge\left\llbracket \mathfrak X_{\sigma(k)},y_{k+1}\right\rrbracket ,\mathfrak X_{k+2},\cdots,\mathfrak X_{n+1})\\ ~ &&+\sum_{k=1}^n(-1)^{k-1}\sum_{\sigma\in \mathbb S_{(k-1,1)}}sign(\sigma)f(\mathfrak X_{\sigma(1)},\cdots,\mathfrak X_{\sigma(k-1)}, \left\llbracket \mathfrak X_{\sigma(k)},x_{k+1}\right\rrbracket \wedge y_{k+1},\mathfrak X_{k+2},\cdots,\mathfrak X_{n+1})\Big)\\ ~ &=&(-1)^{n}\Big((-1)^n\big([x_{n+1},g(\mathfrak X_1,\cdots,\mathfrak X_n,y_{n+1})]-[y_{n+1},g(\mathfrak X_1,\cdots,\mathfrak X_n,x_{n+1})]\\ ~ &&-g(\mathfrak X_1,\cdots,\mathfrak X_n,[x_{n+1},y_{n+1}])\big)\\ ~ &&+\sum_{k=1}^{n}(-1)^{k+1}\left\llbracket \mathfrak X_k,f(\mathfrak X_1,\cdots,\hat{\mathfrak X_k},\cdots,\mathfrak X_{n+1})\right\rrbracket \\ ~ &&+\sum_{1\leqslant k<l\leqslant n+1}(-1)^{k}f(\mathfrak X_1,\cdots,\hat{\mathfrak X_k},\cdots,\mathfrak X_k\circ\mathfrak X_l,\cdots,\mathfrak X_{n+1})\Big)\\ ~ &\stackrel{\eqref{cohomolo1}}{=}&(-1)^{n}\delta_{\rm I}(F)(\mathfrak X_1,\cdots,\mathfrak X_{n+1}),\end{aligned}$$ and $$\begin{aligned} ~ &&[\Pi,F]_{\rm II}(\mathfrak X_1,\cdots,\mathfrak X_{n+1},x)\\ ~ &=&\Big(\big(\Pi\circ F\big)_{\rm II}-(-1)^n\big(F\circ\Pi\big)_{\rm II}\Big)(\mathfrak X_1,\cdots,\mathfrak X_{n+1},x)\\ ~ &\stackrel{\eqref{gradedbra3}}{=}&\sum_{\sigma\in \mathbb S_{(1,n)}}(-1)^n sign(\sigma)\left\llbracket \mathfrak X_{\sigma(1)},g(\mathfrak X_{\sigma(2)},\cdots,\mathfrak X_{\sigma(n+1)},x)\right\rrbracket \\ ~ &&+\left\llbracket g(\mathfrak X_1,\cdots,\mathfrak X_n,x_{n+1}),y_{n+1},x\right\rrbracket +\left\llbracket x_{n+1},g(\mathfrak X_{1},\cdots,\mathfrak X_n,y_{n+1}),x\right\rrbracket \\ ~ &&-(-1)^n\Big(\sum_{\sigma\in \mathbb S_{(n,1)}}(-1)^n sign(\sigma)g(\mathfrak X_{\sigma(1)},\cdots,\mathfrak X_{\sigma(n)},\left\llbracket \mathfrak X_{\sigma(n+1)},x\right\rrbracket )\\ ~ &&+\sum_{k=1}^n(-1)^{k-1}\sum_{\sigma\in \mathbb S_{(k-1,1)}}sign(\sigma)g(\mathfrak X_{\sigma(1)},\cdots,\mathfrak X_{\sigma(k-1)},\left\llbracket \mathfrak X_{\sigma(k)},x_{k+1}\right\rrbracket \wedge y_{k+1},\mathfrak X_{k+2},\cdots,\mathfrak X_{n+1},x)\\ ~ &&+\sum_{k=1}^n(-1)^{k-1}\sum_{\sigma\in \mathbb S_{(k-1,1)}}sign(\sigma)g(\mathfrak X_{\sigma(1)},\cdots,\mathfrak X_{\sigma(k-1)},x_{k+1}\wedge\left\llbracket \mathfrak X_{\sigma{(k)}},y_{k+1}\right\rrbracket ,\mathfrak X_{k+2},\cdots,\mathfrak X_{n+1},x)\Big)\\ ~ &=&(-1)^{n}\Big((-1)^n\big(\left\llbracket g(\mathfrak X_1,\cdots,\mathfrak X_n,x_{n+1}),y_{n+1},x\right\rrbracket -\left\llbracket g(\mathfrak X_1,\cdots,\mathfrak X_n,y_{n+1}),x_{n+1},x\right\rrbracket \big)\\ ~ &&+\sum_{k=1}^{n+1}(-1)^{k+1}\left\llbracket x_k,y_k,g(\mathfrak X_1,\cdots,\hat{\mathfrak X_k},\cdots,\mathfrak X_{n+1},x)\right\rrbracket \\ ~ &&+\sum_{1\leqslant k<l\leqslant n+1}(-1)^kg(\mathfrak X_1,\cdots,\hat{\mathfrak X_k},\cdots,\mathfrak X_k\circ\mathfrak X_l,\cdots,\mathfrak X_{n+1},x)\\ ~ &&+\sum_{k=1}^{n+1}(-1)^kg(\mathfrak X_1,\cdots,\hat{\mathfrak X_k},\cdots,\mathfrak X_{n+1},\left\llbracket x_k,y_k,x\right\rrbracket )\Big)\\ ~ &\stackrel{\eqref{cohomolo2}}{=}&(-1)^{n}\delta_{\rm II}(F)(\mathfrak X_1,\cdots,\mathfrak X_{n+1},x).\end{aligned}$$ Thus, we have $$[\Pi,F]_{\rm I}=(-1)^n\delta_{\rm I}(F)\quad \text{and}\quad [\Pi,F]_{\rm II}=(-1)^n\delta_{\rm II}(F),\quad \forall F\in \mathfrak C^n(\mathfrak g,\mathfrak g)~(n\geqslant 1),$$ which proves [\[cohomo1\]](#cohomo1){reference-type="eqref" reference="cohomo1"}. Similarly, for any $f\in \mathrm{Hom}(\mathfrak g,\mathfrak g)$ and for all $x,y,z\in \mathfrak g$, we have $$\begin{aligned} ~ &&[\Pi,f]_{\rm I}(x,y)\\ ~ &=&\Big(\Pi\circ f\Big)_{\rm I}(x,y)-\Big(f\circ \Pi\Big)_{\rm I}(x,y)\\ ~ &\stackrel{\eqref{gradedbra4},\eqref{gradedbra5}}{=}&[x,f(y)]+[f(x),y]-f([x,y])\\ ~ &\stackrel{\eqref{1cochain}}{=}&\delta_{\rm I}(f)(x,y),\end{aligned}$$ and $$\begin{aligned} ~ &&[\Pi,f]_{\rm II}(x,y,z)\\ ~ &\stackrel{\eqref{gradedbra6},\eqref{gradedbra2}}{=}&\left\llbracket f(x),y,z\right\rrbracket +\left\llbracket x,f(y),z\right\rrbracket +\left\llbracket x,y,f(z)\right\rrbracket -f(\left\llbracket x,y,z\right\rrbracket )\\ ~ &\stackrel{\eqref{2cochain}}{=}&\delta_{\rm II}(f)(x,y,z).\end{aligned}$$ Thus we have $$[\Pi,f]_{\rm I}=\delta_{\rm I}(f)\quad\text{and}\quad [\Pi,f]_{\rm II}=\delta_{\rm II}(f), \quad\forall f\in \mathfrak C^0(\mathfrak g,\mathfrak g)=\mathrm{Hom}(\mathfrak g,\mathfrak g),$$ which proves [\[cohomo2\]](#cohomo2){reference-type="eqref" reference="cohomo2"}. The conclusion thus follows. ◻ # Maurer-Cartan characterizations and Cohomology of relative Rota-Baxter operators In this section, we determine an $L_\infty$-algebra as controlling algebra for relative Rota-Baxter operators and then establish their cohomology theory. Finally, we reveal the relation between cohomology and differential in the controlling algebra. ## Maurer-Cartan characterizations for relative Rota-Baxter operators In this subsection, we use the Voronov's derived bracket to construct an $L_{\infty}$-algebra, which characterizes the relative Rota-Baxter operators on a Lie-Yamaguti algebra as Maurer-Cartan elements. At the beginning, we recall some notions and conclusions. The notion of an $L_{\infty}$-algebra was introduced by Stasheff in [@Stasheff]. See also [@Lada2; @Lada1] for more details. **Definition 13**. An **$L_{\infty}$-algebra** is a $\mathbb Z$-graded vector space $\mathfrak g=\oplus_{k\in \mathbb Z}\mathfrak g_k$ equipped with a collection of multilinear maps $l_k:\otimes^k\mathfrak g\to \mathfrak g~(k\geqslant 1)$ of degree $1$ with the property that, for all homogeneous elements $x_1,\cdots,x_n \in \mathfrak g$, we have - for all $\sigma\in S_n$, $$l_n(x_{\sigma(1)},\cdots, x_{\sigma(n)})=\varepsilon(\sigma)l_n(x_1,\cdots,x_n).$$ - for all $n\geqslant 1$, $$\sum_{i=1}^n\sum_{\sigma\in \mathbb S_{(i,n-1)}}\varepsilon(\sigma)l_{n-i+1}(l_i(x_{\sigma(1)},\cdots,x_{\sigma(i)}),x_{\sigma(i+1)},\cdots,x_{\sigma(n)})=0,$$ where $\varepsilon(\sigma)$ means the Koszul sign: $l_n(x_1,\cdots,x_i,x_{i+1},\cdots,x_n)=(-1)^{|x_i||x_{i+1}|}l_n(x_1,\cdots,x_{i+1},x_i,\cdots,x_n)$. In the sequel, we denote an $L_\infty$-algebra by $(\mathfrak g,\{l_k\}_{k=1}^\infty)$. **Definition 14**. Let $(\mathfrak g,\{l_k\}_{k=1}^\infty)$ be an $L_{\infty}$-algebra. - A degree $0$ element $\alpha \in \mathfrak g_0$ is called a **Maurer-Cartan element** of the $L_{\infty}$-algebra $(\mathfrak g,\{l_k\}_{k=1}^\infty)$ if $\alpha$ satisfies the following Maurer-Cartan equation: $$\begin{aligned} \sum_{k=1}^\infty \frac{1}{k!}l_k(\underbrace{\alpha,\cdots,\alpha}_k)=0.\end{aligned}$$ - A Maurer-Cartan element $\alpha\in \mathfrak g_0$ of $(\mathfrak g,\{l_k\}_{k=1}^\infty)$ is called **strict** if $$l_k(\alpha,\cdots,\alpha)=0,\quad k\geqslant 1.$$ Let us recall the method by which we use the Voronov's derived brackets to construct an $L_{\infty}$-algebra. **Definition 15**. ([@Voronov]) A **$V$-data** is a quadruple $(L,\mathfrak h,P,\Delta)$ consisting of the following items: - $(L,[\cdot,\cdot])$ is a graded Lie algebra; - $\mathfrak h$ is an abelian graded Lie subalgebra of $(L,[\cdot,\cdot])$; - $P:L \to L$ is a projection, i.e. $P\circ P=P$, whose image is $\mathfrak h$ and its kernel is a graded Lie subalgebra of $(L,[\cdot,\cdot])$; - $\Delta$ is an element in $\ker(P)^1$ such that $[\Delta,\Delta]=0$. **Theorem 16**. *([@Voronov])[\[construct\]]{#construct label="construct"} Let $(L,\mathfrak h,P,\Delta)$ be a $V$-data. Define multilinear maps $l_k:\otimes^k\mathfrak h\to \mathfrak h$ to be $$\begin{aligned} l_k(x_1,\cdots,x_k)=P[\cdots[[\Delta,x_1],x_2],\cdots,x_k],\label{derived}\end{aligned}$$ for all homogeneous elements $x_1,\cdots,x_k\in \mathfrak h$. Then $(\mathfrak h,\{l_k\}_{k=1}^\infty)$ is an $L_\infty$-algebra, where $\{l_k\}_{k=1}^\infty$ is called the **higher derived brackets** of the $V$-data $(L,\mathfrak h,P,\Delta)$.* **Definition 17**. [@SZ1] Let $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ be a Lie-Yamaguti algebra and $(V;\rho,\mu)$ a representation of $\mathfrak g$. A linear map $T:V\to \mathfrak g$ is called a **relative Rota-Baxter operator** on $\mathfrak g$ with respect to $(V;\rho,\mu)$ if $T$ satisfies $$\begin{aligned} ~\label{Ooperator1}[Tu,Tv]&=&T\Big(\rho(Tu)v-\rho(Tv)u\Big),\\ ~\label{Ooperator2}\left\llbracket Tu,Tv,Tw\right\rrbracket &=&T\Big(D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\Big), \quad \forall u,v,w \in V.\end{aligned}$$ In the sequel, let $\mathfrak g$ and $V$ be vector spaces, and we denote the elements in $\mathfrak g$ by $x,y,z$, and we denote the elements in $V$ by $u,v,w$. Consider the graded vector subspace $C^*(V,\mathfrak g)=\oplus_{p\geqslant 0}C^p(V,\mathfrak g)$ of $\mathfrak C^\bullet(\mathfrak g\oplus V,\mathfrak g\oplus V)$, where $$\begin{aligned} C^p(V,\mathfrak g)= \begin{cases} \mathrm{Hom}(\underbrace{\wedge^2V\otimes\cdots\otimes\wedge^2V}_p,\mathfrak g)\oplus\mathrm{Hom}(\underbrace{\wedge^2V\otimes\cdots\otimes\wedge^2V}_p\otimes V,\mathfrak g),& p\geqslant 1,\\ \mathrm{Hom}(V,\mathfrak g),&p=0. \end{cases}\end{aligned}$$ Recall that $\Pi+\Theta=(\pi+\rho,\omega+\mu)\in \mathfrak C^1(\mathfrak g\oplus V,\mathfrak g\oplus V)$. For all $P\in C^p(V,\mathfrak g),~Q\in C^q(V,\mathfrak g)$, and $R\in C^r(V,\mathfrak g)~(p,q,r\geqslant 0)$, we define a bilinear operation $$l_2:C^p(V,\mathfrak g)\times C^q(V,\mathfrak g) \to C^{p+q+1}(V,\mathfrak g)$$ and a trilinear operation $$l_3:C^p(V,\mathfrak g)\times C^q(V,\mathfrak g) \times C^r(V,\mathfrak g) \to C^{p+q+r+1}(V,\mathfrak g)$$ to be $$\begin{aligned} \label{multi1}l_2(P,Q)&=&[[\Pi+\Theta,P]_{\mathsf{LieY}},Q]_{\mathsf{LieY}},\end{aligned}$$ and $$\begin{aligned} \label{multi2}l_3(P,Q,R)&=&[[[\Pi+\Theta,P]_{\mathsf{LieY}},Q]_{\mathsf{LieY}},R]_{\mathsf{LieY}},\end{aligned}$$ respectively. Note that the standard symbols for $l_2$ and $l_3$ are $\Big((l_2)_{\rm I},(l_2)_{\rm II}\Big)$ and $\Big((l_3)_{\rm I},(l_3)_{\rm II}\Big)$ respectively, if we emphasize their components. Now we give the main theorem in this section which gives the Maurer-Cartan characterization for relative Rota-Baxter operators on Lie-Yamaguti algebras. **Theorem 18**. *With the above notations, $(C^*(V,\mathfrak g),l_2,l_3)$ is an $L_\infty$-algebra. Moreover, its strict Maurer-Cartan elements are precisely the relative Rota-Baxter operators on Lie-Yamaguti algebra $(\mathfrak g,\pi,\omega)$ with respect to the representation $(V;\rho,\mu)$.* *Proof.* Let $(V;\rho,\mu)$ be a representation of the Lie-Yamaguti algebra $(\mathfrak g,\pi,\omega)$. Then we have - a graded Lie algebra $(\mathfrak C^\bullet(\mathfrak g\oplus V,\mathfrak g\oplus V),[\cdot,\cdot]_{\mathsf{LieY}})$; - an abelian subalgebra $C^*(V,\mathfrak g)$; - $\mathrm{pr}_\mathfrak g$ is a projection onto $C^*(V,\mathfrak g)$; - $\Delta=\Pi+\Theta$. These items listed above forms a $V$-data. By Theorem [\[construct\]](#construct){reference-type="ref" reference="construct"}, $(C^*(V,\mathfrak g),\{l_k\}_{k=1}^\infty)$ is an $L_\infty$-algebra, where multilinear maps ${l_k}'s$ are given by [\[derived\]](#derived){reference-type="eqref" reference="derived"}. But we note that $$\begin{aligned} ~\mathrm{pr}_\mathfrak g[\Pi+\Theta,P]_{\mathsf{LieY}}=0,\\ ~\mathrm{pr}_\mathfrak g[[\Pi+\Theta,P]_{\mathsf{LieY}},Q]_{\mathsf{LieY}}\in C^*(V,\mathfrak g),\\ ~\mathrm{pr}_\mathfrak g[[[\Pi+\Theta,P]_{\mathsf{LieY}},Q]_{\mathsf{LieY}},R]_{\mathsf{LieY}}\in C^*(V,\mathfrak g),\end{aligned}$$ for all $P\in C^p(V,\mathfrak g),~Q\in C^q(V,\mathfrak g)$ and $R\in C^r(V,\mathfrak g)$. Thus, the nonzero multilinear maps of $L_\infty$-algebra $(C^*(V,\mathfrak g),\{l_k\}_{k=1}^\infty)$ are only $l_2$ and $l_3$. For a degree $0$ element $T\in \mathrm{Hom}(V,\mathfrak g)$, by [\[multi1\]](#multi1){reference-type="eqref" reference="multi1"} and [\[multi2\]](#multi2){reference-type="eqref" reference="multi2"}, we have for all $u,v,w \in V$, $$\begin{aligned} ~ &&(l_2)_{\rm I}(T,T)(u,v)\\ ~ &=&[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm I}(u,v)\\ ~ &=&\Big([\Pi+\Theta,T]_{\mathsf{LieY}}\circ T\Big)_{\rm I}(u,v)-\Big(T\circ[\Pi+\Theta,T]_{\mathsf{LieY}} \Big)_{\rm I}(u,v)\\ ~ &=&[\Pi+\Theta,T]_{\rm I}(Tu,v)+[\Pi+\Theta,T]_{\rm I}(u,Tv)\\ ~ &=&(\Pi+\Theta)(Tu,Tv)-\Big(T\circ(\Pi+\Theta) \Big)_{\rm I}(Tu,v)\\ ~ &&+(\Pi+\Theta)(Tu,Tv)-\Big(T\circ(\Pi+\Theta) \Big)_{\rm I}(u,Tv)\\ ~ &=&2\Big([Tu,Tv]-T\big(\rho(Tu)v-\rho(Tv)u\big)\Big),\end{aligned}$$ and $$\begin{aligned} ~ &&(l_3)_{\rm II}(T,T,T)(u,v,w)\\ ~ &=&[[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\mathsf{LieY}},T]_{\rm II}(u,v,w)\\ ~ &=&[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm II}(Tu,v,w)+[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm II}(u,Tv,w)\\ ~ &&+[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm II}(u,v,Tw)-\Big(T\circ [[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\mathsf{LieY}}\Big)_{\rm II}(u,v,w)\\ ~ &=&[\Pi+\Theta,T]_{\rm II}(Tu,Tv,w)+[\Pi+\Theta,T]_{\rm II}(Tu,v,Tw)\\ ~ &&-\Big(T\circ [\Pi+\Theta,T]_{\mathsf{LieY}}\Big)_{\rm II}(Tu,v,w)+[\Pi+\Theta,T]_{\rm II}(Tu,Tv,w)\\ ~ &&+[\Pi+\Theta,T]_{\rm II}(u,Tv,Tw)-\Big(T\circ [\Pi+\Theta,T]_{\mathsf{LieY}}\Big)_{\rm II}(u,Tv,w)\\ ~ &&+[\Pi+\Theta,T]_{\rm II}(Tu,v,Tw)+[\Pi+\Theta,T]_{\rm II}(u,Tv,Tw)\\ ~ &&-\Big(T\circ [\Pi+\Theta,T]_{\mathsf{LieY}}\Big)_{\rm II}(u,v,Tw)-\Big(T\circ [\Pi+\Theta,T]_{\mathsf{LieY}}\Big)_{\rm II}(Tu,v,w)\\ ~ &&-\Big(T\circ [\Pi+\Theta,T]_{\mathsf{LieY}}\Big)_{\rm II}(u,Tv,w)-\Big(T\circ [\Pi+\Theta,T]_{\mathsf{LieY}}\Big)_{\rm II}(u,v,Tw)\\ ~ &=&6\Big(\left\llbracket Tu,Tv,Tw\right\rrbracket -T\big(D(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\big)\Big).\end{aligned}$$ To sum up, we compute the nonzero multilinear maps $l_2$ and $l_3$ as follows: $$\begin{aligned} \begin{cases} (l_2)_{\rm I}(T,T)(u,v)=2\Big([Tu,Tv]-T\big(\rho(Tu)v-\rho(Tv)u\big)\Big),\\ (l_2)_{\rm II}(T,T)(u,v,w)=0,\\ (l_3)_{\rm I}(T,T,T)(u,v)=0,\\ (l_3)_{\rm II}(T,T,T)(u,v,w)=6\Big(\left\llbracket Tu,Tv,Tw\right\rrbracket -T\big(D(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\big)\Big). \end{cases}\end{aligned}$$ Thus we have that a linear map $T:V\to \mathfrak g$ is a relative Rota-Bxter operator on the Lie-Yamaguti algebra $(\mathfrak g,\pi,\omega)$ with respect to the representation $(V;\rho,\mu)$ if and only if $T$ is a strict Maurer-Cartan element of the $L_\infty$-algebra $(C^*(V,\mathfrak g),l_2,l_3)$. This completes the proof. ◻ **Remark 19**. From Theorem [Theorem 18](#main){reference-type="ref" reference="main"}, we notice that for a relative Rota-Baxter operator on a Lie-Yamaguti algebra, on the one hand, its realization of Maurer-Cartan characterization is an *$L_\infty$-algebra* owning two nonzero multilinear maps: $l_2$ and $l_3$. In [@TBGS; @THS], authors showed the realizations of Maurer-Cartan characterization for relative Rota-Baxter operators on Lie algebras and $3$-Lie algebras are graded Lie algebras and Lie $3$-algebras respectively, both of which can be seen as a special $L_\infty$-algebra (whose nonzero multilinear maps are $l_2$ and $l_3$ respectively). Whichever is a graded Lie algebra or a Lie $3$-algebra, it (treated as $L_\infty$-algebra) owns one nonzero bracket. On the other hand, a relative Rota-Baxter operator on a Lie-Yamaguti algebra  corresponds to a *strict* Maurer-Cartan element in its $L_\infty$-algebra, not a Maurer-Cartan element. The above two points distinguish from the case of relative Rota-Baxter operators on Lie algebras and on $3$-Lie algebras [@TBGS; @THS; @T.S2], and from proof of Theorem [Theorem 18](#main){reference-type="ref" reference="main"}, it reveals the fact that the case of Lie-Yamaguti algebras is more complicated than other algebras. Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,\pi,\omega)$ with respect to a representation $(V;\rho,\mu)$. Since $T\in C^0(V,\mathfrak g)=\mathrm{Hom}(V,\mathfrak g)$ is a (strict) Maurer-Cartan element of the $L_\infty$-algebra $(C^*(V,\mathfrak g),l_2,l_3)$, by [@Get], we have a twisted $L_\infty$-algebra $(C^*(V,\mathfrak g),l_1^T,l_2^T,l_3^T)$ as follows: $$\begin{aligned} ~l_1^T(P)&=&l_2(T,P)+\frac{1}{2}l_3(T,T,P),\label{diffs}\\ ~l_2^T(P,Q)&=&l_2(P,Q)+l_3(T,P,Q),\label{bracket}\\ ~l_3^T(P,Q,R)&=&l_3(P,Q,R),\label{homotopy}\\ ~l_k^T&=&0,\quad k\geqslant 4,\end{aligned}$$ for all $P\in C^p(V,\mathfrak g),~Q\in C^q(V,\mathfrak g)$ and $R\in C^r(V,\mathfrak g)$. **Theorem 20**. *Let $T:V\to \mathfrak g$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$. Then for a linear map $T':V\to \mathfrak g$, $T+T'$ is still a relative Rota-Baxter operator on $\mathfrak g$ if and only if $T'$ is a Maurer-Cartan element of the twisted $L_\infty$-algebra $(C^*(V,\mathfrak g),l_1^T,l_2^T,l_3^T)$, i.e., $T'$ satisfies the following Maurer-Cartan equation: $$l_1^T(T')+\frac{1}{2}l_2^T(T',T')+\frac{1}{3!}l_3^T(T',T',T')=0.$$* *Proof.* By Theorem [Theorem 18](#main){reference-type="ref" reference="main"}, $T+T'$ is a relative Rota-Baxter operator if and only if it is a strict Maurer-Cartan element, i.e., $$\begin{aligned} \label{Maurer} \begin{cases} l_2(T+T',T+T')=0,\\ l_3(T+T',T+T',T+T')=0. \end{cases}\end{aligned}$$ Since $T$ is a relative Rota-Baxter operator on $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$, then we have $$\begin{aligned} \begin{cases} l_2(T,T)=0,\\ l_3(T,T,T)=0. \end{cases}\end{aligned}$$ Since the degrees of both $T$ and $T'$ are $0$, expanding Eqs.[\[Maurer\]](#Maurer){reference-type="eqref" reference="Maurer"} yields that $$\begin{aligned} \begin{cases} 2l_2(T,T')+l_2(T',T')=0,\\ 3l_3(T,T,T')+3l_3(T,T',T')+l_3(T',T',T')=0.\label{MC} \end{cases}\end{aligned}$$ Then we obtain that $$\begin{aligned} ~ &&l_1^T(T')+\frac{1}{2}l_2^T(T',T')+\frac{1}{3!}l_3^T(T',T',T')\\ ~ &\stackrel{\eqref{diffs},\eqref{bracket},\eqref{homotopy}}{=}&l_2(T,T')+\frac{1}{2}l_3(T,T,T')+\frac{1}{2}\Big(l_2(T',T')+l_3(T,T',T')\Big)+\frac{1}{3!}l_3(T',T',T')\\ ~ &=&\Big(l_2(T,T')+\frac{1}{2}l_2(T',T')\Big)+\frac{1}{2}\Big(l_3(T,T,T')+l_3(T,T',T')+\frac{1}{3}l_3(T',T',T')\Big)\\ ~ &\stackrel{\eqref{MC}}{=}&0,\end{aligned}$$ which implies that $T'$ is a Maurer-Cartan element of the twisted $L_\infty$-algebra $(C^*(V,\mathfrak g),l_1^T,l_2^T,l_3^T)$. This completes the proof. ◻ At the end of this subsection, we give two examples of Rota-Baxter operators on Lie-Yamaguti algebras. **Example 21**. *Let $\mathfrak g=C[0,1]$ endowed with the following operations $$\begin{aligned} (x)&=&f(x)g(x)-g(x)f(x),\\ \left\llbracket f,g,h\right\rrbracket (x)&=&f(x)g(x)h(x)-g(x)f(x)h(x)-h(x)f(x)g(x)+h(x)g(x)f(x),\quad \forall x\in [0,1],\end{aligned}$$ for all $f,g,h\in \mathfrak g$. Then $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ forms a Lie-Yamaguti algebra. For all $f\in C[0,1]$, the *integral operator* $R:\mathfrak g\longrightarrow\mathfrak g$ defined to be $$R(f)(x):=\int_0^xf(t){\mathrm d}t,\quad\forall x\in [0,1]$$ is a Rota-Baxter operator on $\mathfrak g$. For the integral operator $R$, define $l_2$ and $l_3$ to be $$\begin{cases} (l_2)_{\rm I}(R,R)(f,g)=2\Big([R(f),R(g)]-R([R(f),g]-[f,R(g)])\Big),\\ (l_2)_{\rm II}(R,R)(f,g,h)=0,\\ (l_3)_{\rm I}(R,R,R)(f,g)=0,\\ (l_3)_{\rm II}(R,R,R)(f,g,h)=6\Big(\left\llbracket R(f),R(g),R(h)\right\rrbracket -R\big(\left\llbracket R(f),R(g),h\right\rrbracket -\left\llbracket f,R(g),R(h)\right\rrbracket -\left\llbracket R(f),g,R(h)\right\rrbracket \big)\Big), \end{cases}$$ for all $f,g,h\in \mathfrak g$. Obviously, $R$ is a strict Maurer-Cartan element of the $L_\infty$-algebra $(C^*(\mathfrak g,\mathfrak g),l_2,l_3)$ by Theorem [Theorem 18](#main){reference-type="ref" reference="main"}, where $l_2$ and $l_3$ are defined by [\[multi1\]](#multi1){reference-type="eqref" reference="multi1"} and [\[multi2\]](#multi2){reference-type="eqref" reference="multi2"}. Moreover, for another linear operator $R':\mathfrak g\longrightarrow\mathfrak g$, $R+R'$ is still a Rota-Baxter on $\mathfrak g$ if and only if $R'$ satisfies the following Maurer-Cartan equation: $$l_1^R(R')+\frac{1}{2}l_2^R(R',R')+\frac{1}{3!}l_3^R(R',R',R')=0,$$ where $l_1^R,~l_2^R$ and $l_3^R$ are defined by Eqs. [\[diffs\]](#diffs){reference-type="eqref" reference="diffs"}-[\[homotopy\]](#homotopy){reference-type="eqref" reference="homotopy"} respectively.* **Example 22**. *Let $(A,\cdot)$ be an associative algebra and $A[[\nu]]$ an algebra of formal series with coefficients in $A$. Define operations $[\cdot,\cdot]$ and $\left\llbracket \cdot,\cdot,\cdot\right\rrbracket$ to be $$\begin{aligned} &:=&[a_i,a_j]\nu^{i+j},\\ ~\left\llbracket a_i\nu^i,a_j\nu^j,a_k\nu^k\right\rrbracket &:=&[[a_i,a_j],a_k]\nu^{i+j+k},\end{aligned}$$ where $a_i,a_j,a_k\in A$ and $[\cdot,\cdot]$ on the right hand side means the commutator: $[x,y]=x\cdot y-y\cdot x$ for all $x,y\in A$, then $(A[[\nu]],[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ forms a Lie-Yamaguti algebra. Define a formal integral operator $\Omega:A[[\nu]]\longrightarrow A[[\nu]]$ to be $$\Omega(a_i\nu^i):=\int a_i\nu^id\nu=\frac{1}{i+1}a_i\nu^{i+1}, \quad \forall a_i\in A,$$ then $\Omega$ is a Rota-Baxter operator on $A[[\nu]]$. Then we define $l_2$ and $l_3$ by Eqs. [\[multi1\]](#multi1){reference-type="eqref" reference="multi1"} and [\[multi2\]](#multi2){reference-type="eqref" reference="multi2"} respectively to achieve an $L_\infty$-algebra $(C^*(A[[\nu]],A[[\nu]]),l_2,l_3)$, and consequently we obtain that $\Omega$ is a strict Maurer-Cartan element, i.e., $$l_2(\Omega,\Omega)=0, \quad\text{and}\quad l_3(\Omega,\Omega,\Omega)=0.$$* ## Cohomology of relative Rota-Baxter operators In this subsection, we establish the cohomology of relative Rota-Baxter operators on Lie-Yamaguti algebras, and then we show the relation between the differential of the twisting $L_\infty$-algebra and the coboundary operator. We write $D_{\rho,\mu}$ to emphasize that it is relative with $\rho$ and $\mu$ in this subsection. **Proposition 23**. *[@SZ1] Let $T:V\longrightarrow\mathfrak g$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to $(V;\rho,\mu)$. Define $$\begin{aligned} _T&=&\rho(Tu)v-\rho(Tv)u,\\ \left\llbracket u,v,w\right\rrbracket _T&=&D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v,\quad \forall u,v,w\in V.\end{aligned}$$ Then $(V,[\cdot,\cdot]_T,\left\llbracket \cdot,\cdot,\cdot\right\rrbracket _T)$ is a Lie-Yamaguti algebra, which is called the **sub-adjacent Lie-Yamaguti algebra**. Thus $T$ is a Lie-Yamaguti algebra homomorphism[^2] from $(V,[\cdot,\cdot]_T,\left\llbracket \cdot,\cdot,\cdot\right\rrbracket _T)$ to $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$.* Next, we give a representation of the sub-adjacent Lie-Yamaguti algebra $(V,[\cdot,\cdot]_T,\left\llbracket \cdot,\cdot,\cdot\right\rrbracket _T)$ on the vector space $\mathfrak g$. Define two linear maps $\varrho:V\to \mathfrak {gl}(\mathfrak g)$ and $\varpi:\otimes^2V\to \mathfrak {gl}(\mathfrak g)$ to be $$\begin{aligned} \label{repre1}\varrho(u)x&:=&[Tu,x]+T\big(\rho(x)u\big),\\ \label{repre2}\varpi(u,v)x&:=&\left\llbracket x,Tu,Tv\right\rrbracket -T\big(D_{\rho,\mu}(x,Tu)v-\mu(x,Tv)u\big), \quad \forall x\in \mathfrak g,~u,v \in V.\end{aligned}$$ Consequently, we give the precise formula of $D_{\varrho,\varpi}$ first. **Proposition 24**. *Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to $(V;\rho,\mu)$. Then with the notations above, we have $$\begin{aligned} \label{repre3}D_{\varrho,\varpi}(u,v)x=\left\llbracket Tu,Tv,x\right\rrbracket -T\Big(\mu(Tv,x)u-\mu(Tu,x)v\Big), \quad \forall u,v \in V, ~x\in \mathfrak g.\end{aligned}$$* *Proof.* Since $T$ is a relative Rota-Baxter operator, for all $u,v\in V$ and $x\in \mathfrak g$, by a direct computation, we have $$\begin{aligned} ~ &&D_{\varrho,\varpi}(u,v)x\\ &\stackrel{\eqref{rep}}{=}&\varpi(v,u)x-\varpi(u,v)x+[\varrho(u),\varrho(v)]x-\varrho([u,v]_T)x\\ ~ &\stackrel{\eqref{repre1},\eqref{repre2}}{=}&\left\llbracket x,Tv,Tu\right\rrbracket -T\Big(D_{\rho,\mu}(x,Tv)u-\mu(x,Tu)v\Big)-\left\llbracket x,Tu,Tv\right\rrbracket +T\Big(D_{\rho,\mu}(x,Tu)v-\mu(x,Tv)u\Big)\\ ~ &&+[Tu,[Tv,x]]+[Tu,T(\rho(x)v)]+T(\rho([Tv,x])u)+T(\rho(T(\rho(x)v)u))\\ ~ &&-[Tv,[Tu,x]]-[Tv,T(\rho(x)u)]-T(\rho([Tu,x])v)-T(\rho(T(\rho(x)u)v))\\ ~ &&-[T(\rho(Tu)v-\rho(Tv)u),x]-T(\rho(x)\rho(Tu)v)+T(\rho(x)\rho(Tv)u)\\ ~ &\stackrel{\eqref{Ooperator1}}{=}&\left\llbracket x,Tv,Tu\right\rrbracket -\left\llbracket x,Tu,Tv\right\rrbracket +[Tu,[Tv,x]]-[Tv,[Tu,x]]-[[Tu,Tv],x]\\ ~ &&-T\Big(D_{\rho,\mu}(x,Tv)u-\mu(x,Tu)v\Big)+T\Big(D_{\rho,\mu}(x,Tu)v-\mu(x,Tv)u\Big)\\ ~ &&+T(\rho(Tu)\rho(x)v-\rho(T(\rho(x)v)u)-T(\rho(Tv)\rho(x)u-\rho(T(\rho(x)u)v)\\ ~ &&+T(\rho([Tv,x])u)+T(\rho(T(\rho(x)v)u))-T(\rho([Tu,x])v)-T(\rho(T(\rho(x)u)v))\\ ~ &&-T(\rho(x)\rho(Tu)v)+T(\rho(x)\rho(Tv)u)\\ ~ &\stackrel{\eqref{LY1},\eqref{rep}}{=}&\left\llbracket Tu,Tv,x\right\rrbracket -T\Big(\mu(Tv,x)u-\mu(Tu,x)v\Big).\end{aligned}$$ The conclusion thus follows. ◻ **Theorem 25**. *With the above notations, then $(\mathfrak g;\varrho,\varpi)$ is a representation of the sub-adjacent Lie-Yamaguti algebra $(V,[\cdot,\cdot]_T,\left\llbracket \cdot,\cdot,\cdot\right\rrbracket _T)$, where linear maps $\varrho,~\varpi$, and $D_{\varrho,\varpi}$ are given by Eqs. [\[repre1\]](#repre1){reference-type="eqref" reference="repre1"}-[\[repre3\]](#repre3){reference-type="eqref" reference="repre3"} respectively.* We may prove Theorem [Theorem 25](#represent){reference-type="ref" reference="represent"} by checking that linear maps $\varrho$, $\varpi$, and $D_{\varrho,\varpi}$ satisfy conditions in Definition [\[defi:representation\]](#defi:representation){reference-type="ref" reference="defi:representation"}, but here we prove this proposition by another way. In order to do this, we should go back some notions in [@Sheng; @Zhao]. Recall that a Nijenhuis operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ is a linear map $N:\mathfrak g\to \mathfrak g$ satisfying $$\begin{aligned} &=&N\big([Nx,y]+[x,Ny]-N[x,y]\big),\\ ~\left\llbracket Nx,Ny,Nz\right\rrbracket &=&N\Big(\left\llbracket Nx,Ny,z\right\rrbracket +\left\llbracket Nx,y,Nz\right\rrbracket +\left\llbracket x,Ny,Nz\right\rrbracket \\ ~ &&-N\left\llbracket Nx,y,z\right\rrbracket -N\left\llbracket x,Ny,z\right\rrbracket -N\left\llbracket x,y,Nz\right\rrbracket +N^2\left\llbracket x,y,z\right\rrbracket \Big), \quad \forall x,y,z \in \mathfrak g.\end{aligned}$$ Then we get a pair of deformed brackets $([\cdot,\cdot]_N,\left\llbracket \cdot,\cdot,\cdot\right\rrbracket _N)$: $$\begin{aligned} \label{deform1}[x,y]_N&=&[Nx,y]+[x,Ny]-N[x,y],\\ \label{deform2}~\left\llbracket x,y,z\right\rrbracket _N&=&\left\llbracket Nx,Ny,z\right\rrbracket +\left\llbracket Nx,y,Nz\right\rrbracket +\left\llbracket x,Ny,Nz\right\rrbracket \\ \nonumber~ &&-N\left\llbracket Nx,y,z\right\rrbracket -N\left\llbracket x,Ny,z\right\rrbracket -N\left\llbracket x,y,Nz\right\rrbracket +N^2\left\llbracket x,y,z\right\rrbracket , \quad \forall x,y,z \in \mathfrak g.\end{aligned}$$ In [@Sheng; @Zhao], authors showed that $(\mathfrak g,[\cdot,\cdot]_N,\left\llbracket \cdot,\cdot,\cdot\right\rrbracket _N)$ is a Lie-Yamaguti algebra and thus $N$ is a Lie-Yamaguti homomorphism from $(\mathfrak g,[\cdot,\cdot]_N,\left\llbracket \cdot,\cdot,\cdot\right\rrbracket _N)$ to $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$. *Proof of Theorem [Theorem 25](#represent){reference-type="ref" reference="represent"}:* It is direct to see that if $T:V \to \mathfrak g$ is a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$, then $N_T=\begin{pmatrix} 0 & T\\ 0 & 0 \end{pmatrix}$ is a Nijenhuis operator on the semidirect product Lie-Yamaguti algebra $\mathfrak g\ltimes_{\rho,\mu} V$. Then by [\[deform1\]](#deform1){reference-type="eqref" reference="deform1"} and [\[deform2\]](#deform2){reference-type="eqref" reference="deform2"}, we deduce that there is a Lie-Yamaguti algebra structure on $V\oplus\mathfrak g\cong\mathfrak g\oplus V$ given by for all $x,y,z \in \mathfrak g,~u,v,w\in V,$ $$\begin{aligned} ~ &&[x+u,y+v]_{N_T}\\ ~ &=&[N_T(x+u),y+v]_{\rho,\mu}+[x+u,N_T(y+v)]_{\rho,\mu}-N_T[x+u,y+v]_{\rho,\mu}\\ ~ &=&[Tu,y+v]_{\rho,\mu}+[x+u,Tv]_{\rho,\mu}-N_T([x,y]+\rho(x)v-\rho(y)u)\\ ~ &=&[Tu,y]+\rho(Tu)v+[x,Tv]-\rho(Tv)u-T(\rho(x)v-\rho(y)u)\\ ~ &=&[u,v]_T+\varrho(u)y-\varrho(v)x,\\ ~ &&\\ ~&&\left\llbracket x+u,y+v,z+w\right\rrbracket _{N_T}\\ ~ &=&\left\llbracket N_T(x+u),N_T(y+v),z+w\right\rrbracket _{\rho,\mu}+\left\llbracket N_{T}(x+u),y+v,N_{T}(z+w)\right\rrbracket _{\rho,\mu}+\left\llbracket x+u,N_{T}(y+v),N_{T}(z+w)\right\rrbracket _{\rho,\mu}\\ ~ &&-N_T(\left\llbracket N_{T}(x+u),y+v,z+w\right\rrbracket _{\rho,\mu}+\left\llbracket x+u,N_{T}(y+v),z+w\right\rrbracket _{\rho,\mu})+\left\llbracket x+u,y+v,N_{T}(z+w)\right\rrbracket _{\rho,\mu})\\ ~ &=&\left\llbracket Tu,Tv,z+w\right\rrbracket _{\rho,\mu}+\left\llbracket Tu,y+v,Tw\right\rrbracket _{\rho,\mu}+\left\llbracket x+u,Tv,Tw\right\rrbracket _{\rho,\mu}\\ ~ &&-N_T(\left\llbracket Tu,y+v,z+w\right\rrbracket _{\rho,\mu}+\left\llbracket x+u,Tv,z+w\right\rrbracket _{\rho,\mu}+\left\llbracket x+u,y+v,Tw\right\rrbracket _{\rho,\mu})\\ ~ &=&\left\llbracket Tu,Tv,z\right\rrbracket +D_{\rho,\mu}(Tu,Tv)w+\left\llbracket Tu,y,Tw\right\rrbracket -\mu(Tu,Tw)v+\left\llbracket x,Tv,Tw\right\rrbracket +\mu(Tv,Tw)u\\ ~ &&-T\Big(D_{\rho,\mu}(Tu,y)w-\mu(Tu,z)v+D_{\rho,\mu}(x,Tv)w+\mu(Tv,z)u+\mu(y,Tw)u-\mu(x,Tw)v\Big)\\ ~ &=&\left\llbracket u,v,w\right\rrbracket _T+D_{\varrho,\varpi}(u,v)z+\varpi(v,w)x-\varpi(u,w)y,\end{aligned}$$ which implies that $(\mathfrak g;\varrho,\varpi)$ is a representation of Lie-Yamaguti algebra $(V,[\cdot,\cdot]_T,\left\llbracket \cdot,\cdot,\cdot\right\rrbracket _T)$. This finishes the proof.0◻ Next, we construct the $0$-cochians and the corresponding coboundary map. For all $\mathfrak X\in \wedge^2\mathfrak g$, define $\delta(\mathfrak X):V\to \mathfrak g$ to be $$\begin{aligned} \label{delta}\delta(\mathfrak X)v:=TD(\mathfrak X)v-\left\llbracket \mathfrak X,Tv\right\rrbracket , \quad \forall v\in V.\end{aligned}$$ **Proposition 26**. *Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot,],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$. Then $\delta(\mathfrak X)$ is a $1$-cocycle on the Lie-Yamaguti algebra $(V,[\cdot,\cdot]_T,\left\llbracket \cdot,\cdot,\cdot\right\rrbracket _T)$ with coefficients in the representation $(\mathfrak g;\varrho,\varpi)$.* *Proof.* It is sufficient to show that both $\delta_{\rm I}^T(\delta(\mathfrak X))$ and $\delta_{\rm II}^T(\delta(\mathfrak X))$ all vanish. Indeed, for any $u,v,w\in V$, we have $$\begin{aligned} ~ &&\delta_{\rm II}^T\Big(\delta(\mathfrak X)\Big)(u,v,w)\\ ~ &\stackrel{\eqref{2cochain}}{=}&-\delta(\mathfrak X)(\left\llbracket u,v,w\right\rrbracket _T)+D_{\varrho,\varpi}(u,v)\Big(\delta (\mathfrak X)w\Big)+\varpi(v,w)\Big(\delta (\mathfrak X)u\Big)-\varpi(u,w)\Big(\delta (\mathfrak X)v\Big)\\ ~ &\stackrel{\eqref{repre1},\eqref{repre2}}{=}&\left\llbracket Tu,Tv,TD_{\rho,\mu}(\mathfrak X)w-\left\llbracket \mathfrak X,Tw\right\rrbracket \right\rrbracket +\left\llbracket TD_{\rho,\mu}(\mathfrak X)u-\left\llbracket \mathfrak X,Tu\right\rrbracket ,Tv,Tw\right\rrbracket \\ ~ &&+\left\llbracket Tu,TD_{\rho,\mu}(\mathfrak X)v-\left\llbracket \mathfrak X,Tv\right\rrbracket ,Tw\right\rrbracket \\ ~ &&-TD(\mathfrak X)\Big(D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\Big)\\ ~ &&+\left\llbracket \mathfrak X,T\big(D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\big)\right\rrbracket \\ ~ &&-T\Big(D_{\rho,\mu}\big(TD_{\rho,\mu}(\mathfrak X)u-\left\llbracket \mathfrak X,Tu\right\rrbracket ,Tv\big)w-D_{\rho,\mu}\big(TD_{\rho,\mu}(\mathfrak X)v-\left\llbracket \mathfrak X,Tv\right\rrbracket ,Tu\big)w\Big)\\ ~ &&-T\Big(\mu\big(TD_{\rho,\mu}(\mathfrak X)v-\left\llbracket \mathfrak X,Tv\right\rrbracket ,Tw\big)u-\mu\big(TD_{\rho,\mu}(\mathfrak X)u-\left\llbracket \mathfrak X,Tu\right\rrbracket ,Tw\big)v\Big)\\ ~ &&-T\Big(\mu\big(Tv,TD_{\rho,\mu}(\mathfrak X)w-\left\llbracket \mathfrak X,Tw\right\rrbracket \big)u-\mu\big(Tu,TD_{\rho,\mu}(\mathfrak X)w-\left\llbracket \mathfrak X,Tw\right\rrbracket \big)v\Big)\\ ~ &\stackrel{\eqref{Ooperator2}}{=}&\left\llbracket Tu,Tv,TD_{\rho,\mu}(\mathfrak X)w\right\rrbracket -\left\llbracket Tu,Tv,\left\llbracket \mathfrak X,Tw\right\rrbracket \right\rrbracket +\left\llbracket TD_{\rho,\mu}(\mathfrak X)u,Tv,Tw\right\rrbracket \\ ~ &&-\left\llbracket \left\llbracket \mathfrak X,Tu\right\rrbracket ,Tv,Tw\right\rrbracket +\left\llbracket Tu,TD_{\rho,\mu}(\mathfrak X)v,Tw\right\rrbracket -\left\llbracket Tu,\left\llbracket \mathfrak X,Tv\right\rrbracket ,Tw\right\rrbracket \\ ~ &&-TD_{\rho,\mu}(\mathfrak X)\Big(D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\Big)+\left\llbracket \mathfrak X,\left\llbracket Tu,Tv,Tw\right\rrbracket \right\rrbracket \\ ~ &&-T\Big(D_{\rho,\mu}\big(TD_{\rho,\mu}(\mathfrak X)u-\left\llbracket \mathfrak X,Tu\right\rrbracket ,Tv\big)w-D_{\rho,\mu}\big(TD_{\rho,\mu}(\mathfrak X)v-\left\llbracket \mathfrak X,Tv\right\rrbracket ,Tu\big)w\Big)\\ ~ &&-T\Big(\mu\big(TD_{\rho,\mu}(\mathfrak X)v-\left\llbracket \mathfrak X,Tv\right\rrbracket ,Tw\big)u-\mu\big(TD_{\rho,\mu}(\mathfrak X)u-\left\llbracket \mathfrak X,Tu\right\rrbracket ,Tw\big)v\Big)\\ ~ &&-T\Big(\mu\big(Tv,\big(TD_{\rho,\mu}(\mathfrak X)w-\left\llbracket \mathfrak X,Tw\right\rrbracket \big)u-\mu\big(Tu,TD_{\rho,\mu}(\mathfrak X)w-\left\llbracket \mathfrak X,Tw\right\rrbracket \big)v\Big)\\ ~ &\stackrel{\eqref{fundamental}}{=}&\left\llbracket Tu,Tv,TD_{\rho,\mu}(\mathfrak X)w\right\rrbracket +\left\llbracket TD_{\rho,\mu}(\mathfrak X)u,Tv,Tw\right\rrbracket +\left\llbracket Tu,TD_{\rho,\mu}(\mathfrak X)v,Tw\right\rrbracket \\ ~ &&-TD_{\rho,\mu}(\mathfrak X)\Big(D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\Big)\\ ~ &&-T\Big(D_{\rho,\mu}\big(TD_{\rho,\mu}(\mathfrak X)u,Tv\big)w-D_{\rho,\mu}\big(\left\llbracket \mathfrak X,Tu\right\rrbracket ,Tv\big)w-D_{\rho,\mu}\big(TD_{\rho,\mu}(\mathfrak X)v,Tu\big)w\Big) +D_{\rho,\mu}\big(\left\llbracket \mathfrak X,Tv\right\rrbracket ,Tu\big)w\Big)\\ ~ &&-T\Big(\mu\big(TD_{\rho,\mu}(\mathfrak X)v,Tw\big)u-\mu\big(\left\llbracket \mathfrak X,Tv\right\rrbracket ,Tw\big)u-\mu\big(TD_{\rho,\mu}(\mathfrak X)u,Tw\big)v\Big)+\mu\big(\left\llbracket \mathfrak X,Tu\right\rrbracket ,Tw\big)v\Big)\\ ~ &&-T\Big(\mu\big(Tv,TD_{\rho,\mu}(\mathfrak X)w\big)u-\mu\big(Tv,\left\llbracket \mathfrak X,Tw\right\rrbracket \big)u-\mu\big(Tu,TD_{\rho,\mu}(\mathfrak X)w\big)v+\mu\big(Tu,\left\llbracket \mathfrak X,Tw\right\rrbracket \big)v\Big)\\ ~ &\stackrel{\eqref{Ooperator2}}{=}&T\Big(D_{\rho,\mu}(Tu,Tv)D_{\rho,\mu}(\mathfrak X)w+\mu(Tv,TD_{\rho,\mu}(\mathfrak X)w)u-\mu(Tu,TD_{\rho,\mu}(\mathfrak X)w)v\Big)\\ ~ &&+T\Big(D_{\rho,\mu}(TD_{\rho,\mu}(\mathfrak X)u,Tv)w+\mu(Tv,Tw)D_{\rho,\mu}(\mathfrak X)u-\mu(TD_{\rho,\mu}(\mathfrak X)u,Tw)v\Big)\\ ~ &&+T\Big(D_{\rho,\mu}(Tu,TD_{\rho,\mu}(\mathfrak X)v)w+\mu(TD_{\rho,\mu}(\mathfrak X)v,Tw)u-\mu(Tu,Tw)D_{\rho,\mu}(\mathfrak X)v\Big)\\ ~ &&-TD_{\rho,\mu}(\mathfrak X)\Big(D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\Big)\\ ~ &&-T\Big(D_{\rho,\mu}\big(TD_{\rho,\mu}(\mathfrak X)u,Tv\big)w-D_{\rho,\mu}\big(\left\llbracket \mathfrak X,Tu\right\rrbracket ,Tv\big)w-D_{\rho,\mu}\big(TD_{\rho,\mu}(\mathfrak X)v,Tu\big)w +D_{\rho,\mu}\big(\left\llbracket \mathfrak X,Tv\right\rrbracket ,Tu\big)w\Big)\\ ~ &&-T\Big(\mu\big(TD_{\rho,\mu}(\mathfrak X)v,Tw\big)u-\mu\big(\left\llbracket \mathfrak X,Tv\right\rrbracket ,Tw\big)u-\mu\big(TD_{\rho,\mu}(\mathfrak X)u,Tw\big)v)+\mu\big(\left\llbracket \mathfrak X,Tu\right\rrbracket ,Tw\big)v\Big)\\ ~ &&-T\Big(\mu\big(Tv,TD_{\rho,\mu}(\mathfrak X)w\big)u-\mu\big(Tv,\left\llbracket \mathfrak X,Tw\right\rrbracket \big)u-\mu\big(Tu,TD_{\rho,\mu}(\mathfrak X)w\big)v+\mu\big(Tu,\left\llbracket \mathfrak X,Tw\right\rrbracket \big)v\Big)\\ ~ &=&T\Big(D_{\rho,\mu}(Tu,Tv)D_{\rho,\mu}(\mathfrak X)w+\mu(Tv,Tw)D_{\rho,\mu}(\mathfrak X)u-\mu(Tu,Tw)D_{\rho,\mu}(\mathfrak X)v\Big)\\ ~ &&-TD_{\rho,\mu}(\mathfrak X)\Big(D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\Big)\\ ~ &&+T\Big(D_{\rho,\mu}(\left\llbracket \mathfrak X,Tu\right\rrbracket ,Tv)w-D_{\rho,\mu}(\left\llbracket \mathfrak X,Tv\right\rrbracket ,Tu)w+\mu(\left\llbracket \mathfrak X,Tv\right\rrbracket ,Tw)u\\ ~ &&-\mu(\left\llbracket \mathfrak X,Tu\right\rrbracket ,Tw)v+\mu(Tv,\left\llbracket \mathfrak X,Tw\right\rrbracket )u-\mu(Tu,\left\llbracket \mathfrak X,Tw\right\rrbracket )v\Big)\\ &\stackrel{\eqref{RLY5},\eqref{RLY5a}}{=}&0.\end{aligned}$$ Similarly, we deduce that $\delta^T_{\rm I}\Big(\delta(\mathfrak X)\Big)(u,v)=0$ for all $u,v\in V$. This finishes the proof. ◻ Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$. Combining the Yamaguti cohomology with Proposition [Proposition 26](#0cocy){reference-type="ref" reference="0cocy"}, we obtain a well-defined cochain complex $({\mathcal{C}}_T^\bullet(V,\mathfrak g)=\bigoplus_{n=0}^\infty{\mathcal{C}}_T^n(V,\mathfrak g),\mathrm{d})$, where the $n$-cochians ${\mathcal{C}}^n_T(V,\mathfrak g)$ and the coboundary map $\mathrm{d}:{\mathcal{C}}^n_{T}(V,\mathfrak g)\to {\mathcal{C}}^{n+1}_{T}(V,\mathfrak g)$ are defined to be $$\begin{aligned} {\mathcal{C}}^n_T(V,\mathfrak g):= \begin{cases} C^n_{\rm LieY}(V,\mathfrak g),&n\geqslant 1,\\ \wedge^2\mathfrak g,&n=0, \end{cases}\end{aligned}$$ and $$\begin{aligned} \mathrm{d}:= \begin{cases} \delta^T,&n\geqslant 1,\\ \delta,&n=0, \end{cases}\end{aligned}$$ respectively. Here $\delta^T:C_{\rm LieY}^{n}(V,\mathfrak g)\to C_{\rm LieY}^{n+1}(V,\mathfrak g)~(n\geqslant 1)$ is the corresponding Yamaguti coboundary operator on the sub-adjacent Lie-Yamaguti algebra $(V,[\cdot,\cdot]_T,\left\llbracket \cdot,\cdot,\cdot\right\rrbracket _T)$ with coefficients in the representation $(\mathfrak g;\varrho,\varpi)$. **Definition 27**. The cohomology of cochian complex $({\mathcal{C}}_T^\bullet(V,\mathfrak g)=\bigoplus_{n=0}^\infty{\mathcal{C}}_T^n(V,\mathfrak g),\mathrm{d})$ is called the **cohomology of relative Rota-Baxter operator $T$** on Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to the representation $(V;\rho,\mu)$. Denote the set of $n$-cocycles and $n$-coboundaries by $\mathcal{Z}^n_T(V,\mathfrak g)$ and $\mathcal{B}^n_T(V,\mathfrak g)$ respectively. The $n$-th cohomology group of relative Rota-Baxter operator $T$ is taken to be $$\begin{aligned} \mathcal{H}^n_T(V,\mathfrak g):=\mathcal{Z}^n_T(V,\mathfrak g)/\mathcal{B}^n_T(V,\mathfrak g), \quad n\geqslant 1.\end{aligned}$$ Let us give the formula of $\delta^T$ explicitly: - if $n\geqslant 1$, $\delta^T:C_{\rm LieY}^{n+1}(V,\mathfrak g)\to C_{\rm LieY}^{n+2}(V,\mathfrak g)$ is given by for any $F=(f,g)\in C_{\rm LieY}^{n+1}(V,\mathfrak g)$, $$\begin{aligned} ~\nonumber &&\Big(\delta^T_{\rm I}(F)\Big)(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1})\\ ~\label{Ocohomo1} &=&(-1)^{n}\Big([Tu_{n+1},g(\mathcal{V}_1,\cdots,\mathcal{V}_n,v_{n+1})]-[Tv_{n+1},g(\mathcal{V}_1,\cdots,\mathcal{V}_n,u_{n+1})]\\ ~\nonumber &&-g(\mathcal{V}_1,\cdots,\mathcal{V}_n,\rho(Tu_{n+1})v_{n+1}-\rho(Tv_{n+1})u_{n+1})+T\big(\rho(g(\mathcal{V}_1,\cdots,\mathcal{V}_n,v_{n+1}))u_{n +1}\big)\\ ~\nonumber &&-T\big(\rho(g(\mathcal{V}_1,\cdots,\mathcal{V}_n,u_{n+1}))v_{n +1}\big)\Big)\\ ~ &&\nonumber+\sum_{k=1}^{n+1}(-1)^{k+1}\Big(\left\llbracket Tu_{k},Tv_k,f(\mathcal{V}_1,\cdots,\hat{\mathcal{V}_k},\cdots,\mathcal{V}_{n+1})\right\rrbracket +T\big(\mu(Tv_k,f(\mathcal{V}_1,\cdots,\hat{\mathcal{V}_k},\cdots,\mathcal{V}_{n+1}))u_k\big)\\ ~ \nonumber&&-T\big(\mu(Tu_k,f(\mathcal{V}_1,\cdots,\hat{\mathcal{V}_k},\cdots,\mathcal{V}_{n+1}))v_k\big)\Big)\\ ~ \nonumber&&+\sum_{1\leqslant k<l\leqslant n+1}(-1)^kf(\mathcal{V}_1,\cdots,\hat{\mathcal{V}_k},\cdots,\mathcal{V}_k\circ\mathcal{V}_l,\cdots,\mathcal{V}_{n+1}),\end{aligned}$$ and $$\begin{aligned} ~\nonumber &&\Big(\delta^T_{\rm II}(F)\Big)(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1},w)\\ ~\label{Ocohomo2}&=&(-1)^n\Big(\left\llbracket g(\mathcal{V}_1,\cdots,\mathcal{V}_n,u_{n+1}),Tv_{n+1},Tw\right\rrbracket -\left\llbracket g(\mathcal{V}_1,\cdots,\mathcal{V}_n,v_{n+1}),Tu_{n+1},Tw\right\rrbracket \\ ~\nonumber &&+T\big(D(g(\mathcal{V}_1,\cdots,\mathcal{V}_n,u_{n+1}),Tv_{n+1})w-\mu(g(\mathcal{V}_1,\cdots,\mathcal{V}_n,u_{n+1}),Tw)v_{n+1}\\ ~\nonumber &&+D(g(\mathcal{V}_1,\cdots,\mathcal{V}_n,v_{n+1}),Tu_{n+1})w-\mu(g(\mathcal{V}_1,\cdots,\mathcal{V}_n,v_{n+1}),Tw)u_{n+1}\big)\Big)\\ ~ \nonumber&&+\sum_{k=1}^{n+1}(-1)^{k}\Big(\left\llbracket Tu_k,Tv_k,g(\mathcal{V}_1,\cdots,\hat{\mathcal{V}_k},\cdots,\mathcal{V}_{n+1},w)\right\rrbracket +T\big(\mu(Tv_k,g(\mathcal{V}_1,\cdots,\hat{\mathcal{V}_k},\cdots,\mathcal{V}_{n+1},w))u_k\\ ~\nonumber &&-\mu(Tu_k,g(\mathcal{V}_1,\cdots,\hat{\mathcal{V}_k},\cdots,\mathcal{V}_{n+1},w))v_k\big)\Big)\\ ~\nonumber &&+\sum_{1\leqslant k<l\leqslant n+1}(-1)^kg(\mathcal{V}_1,\cdots,\hat{\mathcal{V}_k},\cdots,\mathcal{V}_k\circ\mathcal{V}_l,\cdots,\mathcal{V}_{n+1},w)\\ ~\nonumber &&+\sum_{k=1}^{n+1}(-1)^kg(\mathcal{V}_1,\cdots,\hat{\mathcal{V}_k},\cdots,\mathcal{V}_{n+1},\left\llbracket u_k,v_k,w\right\rrbracket _T),\end{aligned}$$ where $\mathcal{V}_i=u_i\wedge v_i\in \wedge^2V,~ (1\leqslant i\leqslant n+1),~ w\in V$ and $\mathcal{V}_k\circ\mathcal{V}_l=\left\llbracket u_k,v_k,u_l\right\rrbracket _T\wedge v_l+u_l\wedge\left\llbracket u_k,v_k,v_l\right\rrbracket _T$. - if $n=0$, for any $f\in C_{\rm LieY}^1(V,\mathfrak g)=\mathrm{Hom}(V,\mathfrak g)$, $$\delta^T:C_{\rm LieY}^1(V,\mathfrak g)\to C_{\rm LieY}^2(V,\mathfrak g),\quad f \mapsto (\delta^T_{\rm I}(f),\delta^T_{\rm II}(f))$$ is given by $$\begin{aligned} (\delta^T_{\rm I}(f))(u,v)&=&[Tu,f(v)]-[Tv,f(u)]+T\Big(\rho(f(v)u)-\rho(f(u)v)\Big)-f([u,v]_T),\\ (\delta^T_{\rm II}(f))(u,v,w)&=&\left\llbracket Tu,Tv,f(w)\right\rrbracket +\left\llbracket f(u),Tv,Tw\right\rrbracket -\left\llbracket f(v),Tu,Tw\right\rrbracket -f(\left\llbracket u,v,w\right\rrbracket _T)\\ ~ &&-T\Big(D(f(u),Tv)w-D(f(v),Tu)w+\mu(Tv,f(w))u-\mu(Tu,f(w))v\\ ~ &&-\mu(f(u),Tw)v+\mu(f(v),Tw)u\Big), \qquad \forall u,v,w \in V.\end{aligned}$$ At the end of this section, we show that the coboundary map $\mathrm{d}$ coincides (up to sign) with the differential $l_1^T$ defined by Equation [\[diffs\]](#diffs){reference-type="eqref" reference="diffs"} which involves the Maurer-Cartan element $T$ of the twisted $L_\infty$-algebra $(C^*(V,\mathfrak g),l_1^T,l_2^T,l_3^T)$. **Theorem 28**. *Let $T:V\to \mathfrak g$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$. Then we have $$\begin{aligned} \label{Oo1}\mathrm{d}(F)&=&(-1)^{n-1}l_1^T(F),\quad \forall F\in C^n(V,\mathfrak g),~n=1,2,3,\cdots,\\ \label{Oo2}\mathrm{d}(f)&=&l_1^T(f),\quad \forall f\in C^0(V,\mathfrak g)=\mathrm{Hom}(V,\mathfrak g).\end{aligned}$$* *Proof.* First, for all $x,y, z\in \mathfrak g$, $u,v,w \in V$, we compute that $$\begin{aligned} ~ &&[\Pi+\Theta,T]_{\rm I}(x+u,y+v)\\ ~ &=&\Big((\Pi+\Theta)\circ T\Big)_{\rm I}(x+u,y+v)-\Big( T\circ (\Pi+\Theta)\Big)_{\rm I}(x+u,y+v)\\ ~ &=&[Tu,y]+\rho(Tu)v+[x,Tv]-\rho(Tv)u-T\Big(\rho(x)v-\rho(y)u\Big),\\ ~ &&\\ ~ &&[\Pi+\Theta,T]_{\rm II}(x+u,y+v,z+w)\\ ~ &=&\Big((\Pi+\Theta)\circ T\Big)_{\rm II}(x+u,y+v,z+w)-\Big( T\circ (\Pi+\Theta)\Big)_{\rm II}(x+u,y+v,z+w)\\ ~ &=&\left\llbracket Tu,y,z\right\rrbracket +D(Tu,y)w-\mu(Tu,z)v+\left\llbracket x,Tv,z\right\rrbracket +D(x,Tv)w+\mu(Tv,z)u\\ ~ &&+\left\llbracket x,y,Tw\right\rrbracket +\mu(y,Tw)u-\mu(x,Tw)v-T\Big(D(x,y)w+\mu(y,z)u-\mu(x,z)v\Big),\\ ~ &&\\ ~ &&[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm I}(x+u,y+v)\\ ~ &=&\Big([\Pi+\Theta,T]_{\mathsf{LieY}}\circ T\Big)_{\rm I}(x+u,y+v)-\Big( T\circ [\Pi+\Theta,T]_{\mathsf{LieY}}\Big)_{\rm I}(x+u,y+v)\\ ~ &=&2\Big([Tu,Tv]-T\big(\rho(Tu)v-\rho(Tv)u\big)\Big),\\ ~ &&\\ ~ &&[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm II}(x+u,y+v,z+w)\\ ~ &=&[\Pi+\Theta,T]_{\rm II}(Tu,y+v,z+w)+[\Pi+\Theta,T]_{\rm II}(x+u,Tv,z+w)\\ ~ &&+[\Pi+\Theta,T]_{\rm II}(x+u,y+v,Tw)-T\Big([\Pi+\Theta,T]_{\rm II}(x+u,y+v,z+w)\Big)\\ ~ &=&2\Big(\left\llbracket Tu,Tv,z\right\rrbracket +\left\llbracket x,Tv,Tw\right\rrbracket +\left\llbracket Tu,y,Tw\right\rrbracket +D(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\Big)\\ ~ &&-2T\Big(D(Tu,y)w+D(x,Tv)w-\mu(Tu,z)v-\mu(x,Tw)v+\mu(Tv,z)u+\mu(y,Tw)u\Big).\end{aligned}$$ Then, by [\[multi1\]](#multi1){reference-type="eqref" reference="multi1"} and [\[multi2\]](#multi2){reference-type="eqref" reference="multi2"}, for all $F=(f,g)\in C^n(V,\mathfrak g)~(n\geqslant 1)$, let us compute that $$\begin{aligned} ~ &&(l_2)_{\rm I}\Big(T,F\Big)(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1})\\ ~ &=&\mathrm{pr}_\mathfrak g[[\Pi+\Theta,T]_{\mathsf{LieY}},F]_{\rm I}(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1})\\ ~ &=&\Big([\Pi+\Theta,T]_{\mathsf{LieY}}\circ F\Big)_{\rm I}(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1})-(-1)^n\Big(F\circ [\Pi+\Theta,T]_{\mathsf{LieY}}\Big)_{\rm I}(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1})\\ ~ &=&\sum_{\sigma\in\mathbb S_{(1,n)}\atop\sigma(n+1)=n+1}(-1)^n sign(\sigma)[\Pi+\Theta,T]_{\rm II}(\mathcal{V}_{\sigma(1)},f(\mathcal{V}_{\sigma(2)},\cdots,\mathcal{V}_{\sigma(n+1)}))\\ ~ &&+[\Pi+\Theta,T]_{\rm I}(u_{n+1},g(\mathcal{V}_1,\cdots,\mathcal{V}_n,v_{n+1}))+[\Pi+\Theta,T]_{\rm I}(g(\mathcal{V}_1,\cdots,\mathcal{V}_n,u_{n+1}),v_{n+1})\\ ~ &&-(-1)^n\Big((-1)^ng(\mathcal{V}_1,\cdots,\mathcal{V}_n,[\Pi+\Theta,T]_{\rm I}(\mathcal{V}_{n+1}))\\ ~ &&+\sum_{k=1}^n(-1)^{k-1}\sum_{\sigma\in\mathbb S_{(k-1,1)}}sign(\sigma)f(\mathcal{V}_{\sigma(1)},\cdots,\mathcal{V}_{\sigma(k-1)},u_{k+1}\wedge [\Pi+\Theta,T]_{\rm II}(\mathcal{V}_{\sigma(k)},v_{k+1}),\mathcal{V}_{k+2},\cdots,\mathcal{V}_{n+1})\\ ~ &&+\sum_{k=1}^n(-1)^{k-1}\sum_{\sigma\in\mathbb S_{(k-1,1)}}sign(\sigma)f(\mathcal{V}_{\sigma(1)},\cdots,\mathcal{V}_{\sigma(k-1)}, [\Pi+\Theta,T]_{\rm II}(\mathcal{V}_{\sigma(k)},u_{k+1})\wedge v_{k+1},\mathcal{V}_{k+2},\cdots,\mathcal{V}_{n+1})\Big)\\ ~ &=&[Tu_{n+1},g(\mathcal{V}_1,\cdots,\mathcal{V}_n,v_{n+1})]+T(\rho(g(\mathcal{V}_1,\cdots,\mathcal{V}_n,v_{n+1}))u_{n+1})+[g(\mathcal{V}_1,\cdots,\mathcal{V}_n,u_{n+1}),Tv_{n+1}]\\ ~ &&-T(\rho(g(\mathcal{V}_1,\cdots,\mathcal{V}_n,u_{n+1}))v_{n+1})-g(\mathcal{V}_1,\cdots,\mathcal{V}_n,\rho(Tu_{n+1})v_{n+1}-\rho(Tv_{n+1})u_{n+1}),\end{aligned}$$ and $$\begin{aligned} ~ &&(l_3)_{\rm I}\Big(T,T,F\Big)(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1})\\ ~ &=&\mathrm{pr}_\mathfrak g[[[\Pi+\Theta,T]_{\mathsf{LieY}}],T]_{\mathsf{LieY}},F]_{\rm I}(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1})\\ ~ &=&\Big([[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\mathsf{LieY}}\circ F\Big)_{\rm I}(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1})-(-1)^n\Big(F\circ [[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\mathsf{LieY}}\Big)_{\rm I}(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1})\\ ~ &=&\sum_{\sigma\in\mathbb S_{(1,n)}\atop \sigma(n+1)=n+1}(-1)^nsign(\sigma)[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm II}(\mathcal{V}_{\sigma(1)},f(\mathcal{V}_{\sigma(2)},\cdots,\mathcal{V}_{\sigma(n+1)}))\\ ~&&+[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm I}(u_{n+1},g(\mathcal{V}_{1},\cdots,\mathcal{V}_n,v_{n+1}))+[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm I}(g(\mathcal{V}_{1},\cdots,\mathcal{V}_n,u_{n+1}),v_{n+1})\\ ~ &&-(-1)^n\Big(\sum_{k=1}^n(-1)^{k-1}\sum_{\sigma\in\mathbb S_{(k-1,1)}}sign(\sigma)f(\mathcal{V}_{\sigma(1)},\cdots,\mathcal{V}_{\sigma(k-1)},u_{k+1}\wedge[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm II}(\mathcal{V}_{\sigma(k)},v_{k+1}),\mathcal{V}_{k+2},\cdots,\mathcal{V}_{n+1})\\ ~ &&+\sum_{k=1}^n(-1)^{k-1}\sum_{\sigma\in\mathbb S_{(k-1,1)}}sign(\sigma)f(\mathcal{V}_{\sigma(1)},\cdots,\mathcal{V}_{\sigma(k-1)},[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm II}(\mathcal{V}_{\sigma(k)},u_{k+1})\wedge v_{k+1},\mathcal{V}_{k+2},\cdots,\mathcal{V}_{n+1})\Big)\\ ~ &=&2\Big(\sum_{\sigma\in\mathbb S_{(1,n)}\atop \sigma(n+1)=n+1}(-1)^nsign(\sigma)\big(\left\llbracket Tu_{\sigma(1)},Tv_{\sigma(1)},f(\mathcal{V}_{\sigma(2)},\cdots,\mathcal{V}_{\sigma(n+1)})\right\rrbracket +T(\mu(Tv_{\sigma(1)}, f(\mathcal{V}_{\sigma(2)},\cdots,\mathcal{V}_{\sigma(n+1)}))u_{\sigma(1)})\\ ~ &&-T(\mu(Tu_{\sigma(1)}, f(\mathcal{V}_{\sigma(2)},\cdots,\mathcal{V}_{\sigma(n+1)}))v_{\sigma(1)})\big)+\sum_{k=1}^n(-1)^{k-1}\sum_{\sigma\in \mathbb S_{(k-1,1)}}f(\mathcal{V}_{\sigma(1)},\cdots,\mathcal{V}_{\sigma(k-1)},\mathcal{V}_{\sigma(k)}\circ \mathcal{V}_{k+1},\mathcal{V}_{k+2},\cdots,\mathcal{V}_{n+1})\Big).\end{aligned}$$ Thus, we have that $$\begin{aligned} ~ &&(-1)^{n-1}\Big(l_1^T\Big)_{\rm I}(F)(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1})\\ ~ &\stackrel{\eqref{diffs}}{=}&(-1)^{n-1}\Big((l_2)_{\rm I}(T,F)+\frac{1}{2}(l_3)_{\rm I}(T,T,F)\Big)(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1})\\ ~ &\stackrel{\eqref{multi1},\eqref{multi2}}{=}&(-1)^{n}\Big([Tu_{n+1},g(\mathcal{V}_1,\cdots,\mathcal{V}_n,v_{n+1})]-[Tv_{n+1},g(\mathcal{V}_1,\cdots,\mathcal{V}_n,u_{n+1})]\\ ~ &&-g(\mathcal{V}_1,\cdots,\mathcal{V}_n,\rho(Tu_{n+1})v_{n+1}-\rho(Tv_{n+1})u_{n+1})+T\big(\rho(g(\mathcal{V}_1,\cdots,\mathcal{V}_n,v_{n+1}))u_{n +1}\big)\\ ~ &&-T\big(\rho(g(\mathcal{V}_1,\cdots,\mathcal{V}_n,u_{n+1}))v_{n +1}\big)\Big)\\ ~ &&+\sum_{k=1}^{n+1}(-1)^{k+1}\Big(\left\llbracket Tu_{k},Tv_k,f(\mathcal{V}_1,\cdots,\hat{\mathcal{V}_k},\cdots,\mathcal{V}_{n+1})\right\rrbracket +T\big(\mu(Tv_k,f(\mathcal{V}_1,\cdots,\hat{\mathcal{V}_k},\cdots,\mathcal{V}_{n+1}))u_k\big)\\ ~ &&-T\big(\mu(Tu_k,f(\mathcal{V}_1,\cdots,\hat{\mathcal{V}_k},\cdots,\mathcal{V}_{n+1}))v_k\big)\Big)\\ ~ &&+\sum_{1\leqslant k<l\leqslant n+1}(-1)^kf(\mathcal{V}_1,\cdots,\hat{\mathcal{V}_k},\cdots,\mathcal{V}_k\circ\mathcal{V}_l,\cdots,\mathcal{V}_{n+1}),\\ ~ &\stackrel{\eqref{Ocohomo1}}{=}&\delta_{\rm I}^T(F)(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1}).\end{aligned}$$ Hence, we have $$\begin{aligned} \delta_{\rm I}^T(F)=(-1)^{n-1}\Big(l_1^T\Big)_{\rm I}(F), \quad\forall F\in C^n(V,\mathfrak g) ~(n\geqslant 1).\label{Oopcoho1}\end{aligned}$$ Similarly, by a direct computation, we have that for all $F\in C^n(V,\mathfrak g) ~(n\geqslant 1)$ and for all $\mathcal{V}_i\in \wedge^2V~(i=1,\cdots,n+1)$ and $u\in V$ $$\begin{aligned} ~(l_2)_{\rm II}(F)(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1},u)&=&\mathrm{pr}_\mathfrak g[[\Pi+\Theta,T]_{\mathsf{LieY}},F]_{\rm II}(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1},u)=0,\\ (l_3)_{\rm II}(T,T,F)(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1},u)&=&(-1)^{n-1}2\delta_{\rm II}^T(F)(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1},u). \end{aligned}$$ Thus, we have that $$\begin{aligned} (-1)^{n-1}\Big(l_1^T\Big)_{\rm II}(F)(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1},u)=\delta_{\rm II}^T(F)(\mathcal{V}_1,\cdots,\mathcal{V}_{n+1},u).\label{Oopcoho2}\end{aligned}$$ Combining Eqs. [\[Oopcoho1\]](#Oopcoho1){reference-type="eqref" reference="Oopcoho1"} and [\[Oopcoho2\]](#Oopcoho2){reference-type="eqref" reference="Oopcoho2"} yields [\[Oo1\]](#Oo1){reference-type="eqref" reference="Oo1"}. Moreover, for any $f\in \mathrm{Hom}(V,\mathfrak g)$, by a direct computation, we have that for all $u,v,w\in V$, $$\begin{aligned} ~ &&\Big(l_2\Big)_{\rm I}(T,f)(u,v)\\ ~ &=&\mathrm{pr}_\mathfrak g[[\Pi+\Theta,T]_{\mathsf{LieY}},f]_{\rm I}(u,v)\\ ~ &=&[f(u),Tv]+[Tu,f(v)]-T\Big(\rho(f(u))v-\rho(f(v))u\Big)-f([u,v]_T),\\ ~ &&\Big(l_3\Big)_{\rm I}(T,T,f)(u,v)=\mathrm{pr}_\mathfrak g[[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\mathsf{LieY}},f]_{\rm I}=0.\end{aligned}$$ Thus we get $$\begin{aligned} \Big(l_1^T\Big)_{\rm I}(f)(u,v)=\Big(l_2\Big)_{\rm I}(T,f)(u,v)=\delta_{\rm I}^T(f)(u,v). \label{Oopcoho3}\end{aligned}$$ Similarly, $$\begin{aligned} ~ &&\Big(l_2\Big)_{\rm II}(T,f)(u,v,w)=\mathrm{pr}_\mathfrak g[[\Pi+\Theta,T]_{\mathsf{LieY}},f]_{\rm II}(u,v,w)=0,\\ ~ &&\Big(l_3\Big)_{\rm II}(T,T,f)(u,v,w)=\mathrm{pr}_\mathfrak g[[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\mathsf{LieY}},f]_{\rm II}(u,v,w)\\ ~ &=&2\Big(\left\llbracket f(u),Tv,Tw\right\rrbracket -T\big(D(f(u),Tv)w-\mu(f(u),Tw)v+c.p.\big)-f(\left\llbracket u,v,w\right\rrbracket _T)\Big),\end{aligned}$$ where the last equation is equivalent to $\Big(l_3\Big)_{\rm II}(T,T,f)(u,v,w)=2\delta_{\rm II}^T(f)(u,v,w)$. Thus, for all $u,v,w\in V$, we have that $$\begin{aligned} \Big(l_1^T\Big)_{\rm II}(f)(u,v,w)=\delta_{\rm II}^T(f)(u,v,w). \label{Oopcoho4}\end{aligned}$$ Hence, Eqs. [\[Oopcoho3\]](#Oopcoho3){reference-type="eqref" reference="Oopcoho3"} and [\[Oopcoho4\]](#Oopcoho4){reference-type="eqref" reference="Oopcoho4"} give [\[Oo2\]](#Oo2){reference-type="eqref" reference="Oo2"}. This completes the proof. ◻ # Deformations of relative Rota-Baxter operators After establishing cohomology theory of relative Rota-Baxter operators, we study deformations of relative Rota-Baxter operators, i.e., we use this cohomology to characterize deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras. ## Linear deformations of relative Rota-Baxter operators In this subsection, we explore linear deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras, and we show that the infinitesimals of two equivalent linear deformations of a relative Rota-Baxter operator on Lie-Yamaguti algebra are in the same cohomology classes of the first cohomology group. **Definition 29**. Let $T$ and $T'$ be two relative Rota-Baxter operators on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$. A **homomorphism** from $T'$ to $T$ consists of a Lie-Yamaguti homomorphism $\phi_\mathfrak g: \mathfrak g\to \mathfrak g$ and a linear map $\phi_V: V \to V$ such that $$\begin{aligned} \label{homo1}T\circ \phi_V&=&\phi_\mathfrak g\circ T'\\ ~\label{homo2}\phi_V(\rho(x)v)&=&\rho(\phi_\mathfrak g(x))\phi_V(v),\\ ~\label{homo3}\phi_V\mu(x,y)(v)&=&\mu(\phi_\mathfrak g(x),\phi_\mathfrak g(y))(\phi_V(v)), \quad \forall x,y \in \mathfrak g,~v\in V.\end{aligned}$$ In particular, if $\phi_\mathfrak g$ and $\phi_V$ are invertible, then $(\phi_\mathfrak g,\phi_V)$ is called an **isomorphism** from $T'$ to $T$. **Remark 30**. If two relative Rota-Baxter operators $T$ and $T'$ are homomorphic, that is, there exists a pair $(\phi_\mathfrak g,\phi_V)$ such that [\[homo1\]](#homo1){reference-type="eqref" reference="homo1"}-[\[homo3\]](#homo3){reference-type="eqref" reference="homo3"} hold, then by a direct computation we have that $$\begin{aligned} \label{homo4}\phi_VD(x,y)(v)&=&D(\phi_\mathfrak g(x),\phi_\mathfrak g(y))(\phi_V(v)), \quad \forall x,y \in \mathfrak g,~v\in V.\end{aligned}$$ **Definition 31**. Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$ and $\mathfrak T:V \to \mathfrak g$ a linear map. If $T_t=T+t\mathfrak T$ are still relative Rota-Baxter operators on $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ for all $t$, we say that $\mathfrak T$ generates a **linear deformation** of the relative Rota-Baxter operator $T$. **Remark 32**. If $\mathfrak T$ generates a linear deformation of a relative Rota-Baxter operator $T$, then we have that - $\mathfrak T$ is a relative Rota-Baxter operator on the Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to the representation $(V;\rho,\mu)$; - $\mathfrak T\in{\mathcal{C}}^1(V,\mathfrak g)$ is a $1$-cocycle of $\delta^T$. **Definition 33**. Let $T:V\to \mathfrak g$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$. - Two linear deformations $T_t^1=T+t\mathfrak T_1$ and $T_t^2=T+t\mathfrak T_2$ are said to be **equivalent** if there exists an element $\mathfrak X\in \wedge^2\mathfrak g$ such that $({{\rm{Id}}}_\mathfrak g+t\mathfrak L_{\mathfrak X},{{\rm{Id}}}_V+tD(\mathfrak X))$ is a homomorphism from $T_t^2$ to $T_t^1$. - A linear deformation $T_t=T+t\mathfrak T$ of a relative Rota-Baxter operator $T$ is said to be **trivial** if there exists an element $\mathfrak X\in \wedge^2\mathfrak g$ such that $({{\rm{Id}}}_\mathfrak g+t\mathfrak L_{\mathfrak X},{{\rm{Id}}}_V+tD(\mathfrak X))$ is a homomorphism from $T_t$ to $T$. Let $({{\rm{Id}}}_\mathfrak g+t\mathfrak L_{\mathfrak X},{{\rm{Id}}}_V+tD(\mathfrak X))$ be a homomorphism from $T_t^2$ to $T_t^1$. Then ${{\rm{Id}}}_\mathfrak g+t\mathfrak L_{\mathfrak X}$ is a Lie-Yamaguti algebra homomorphism of $\mathfrak g$, which is equivalent to the following conditions: $$\begin{aligned} ~\label{Nije} [\left\llbracket \mathfrak X,x\right\rrbracket ,\left\llbracket \mathfrak X,y\right\rrbracket ]&=&0,\\ ~ \label{Nij1}\left\llbracket \left\llbracket \mathfrak X,x\right\rrbracket ,\left\llbracket \mathfrak X,y\right\rrbracket ,z\right\rrbracket +\left\llbracket \left\llbracket \mathfrak X,x\right\rrbracket ,y,\left\llbracket \mathfrak X,z\right\rrbracket \right\rrbracket +\left\llbracket x,\left\llbracket \mathfrak X,y\right\rrbracket ,\left\llbracket \mathfrak X,z\right\rrbracket \right\rrbracket &=&0,\\ ~ \label{Nij2}\left\llbracket \left\llbracket \mathfrak X,x\right\rrbracket ,\left\llbracket \mathfrak X,y\right\rrbracket ,\left\llbracket \mathfrak X,z\right\rrbracket \right\rrbracket &=&0.\end{aligned}$$ By $T_t^1\big(({{\rm{Id}}_V}+tD(\mathfrak X))v\big)=\big({{\rm{Id}}}_\mathfrak g+t\mathfrak L_{\mathfrak X}\big)T_t^2(v)$, we have $$\begin{aligned} \label{cocycle}(\mathfrak T_2-\mathfrak T_1)(v)&=&T\Big(D(\mathfrak X)v\Big)-\left\llbracket \mathfrak X,Tv\right\rrbracket ,\\ \mathfrak T_1\Big(D(\mathfrak X)v\Big)&=&\left\llbracket \mathfrak X,\mathfrak T_2(v)\right\rrbracket \end{aligned}$$ Finally, by $\Big({{\rm{Id}}}_V+tD(\mathfrak X)\Big)\mu(z,w)v=\mu\Big(({{\rm{Id}}}_\mathfrak g+t\mathfrak L_{\mathfrak X})z,({{\rm{Id}}}_\mathfrak g+t\mathfrak L_{\mathfrak X})w\Big)({{\rm{Id}}}_V+tD(\mathfrak X))v$, we have $$\begin{aligned} \label{Nij3}\mu(z,\left\llbracket \mathfrak X,w\right\rrbracket )D(\mathfrak X)+\mu(\left\llbracket \mathfrak X,z\right\rrbracket ,w)D(\mathfrak X)+\mu(\left\llbracket \mathfrak X,z\right\rrbracket ,\left\llbracket \mathfrak X,w\right\rrbracket )&=&0,\\ ~\label{Nij4}\mu(\left\llbracket \mathfrak X,z\right\rrbracket ,\left\llbracket \mathfrak X,w\right\rrbracket )D(\mathfrak X)&=&0.\end{aligned}$$ Note that [\[cocycle\]](#cocycle){reference-type="eqref" reference="cocycle"} means that there exists $\mathfrak X\in \wedge^2\mathfrak g$, such that $\mathfrak T_2-\mathfrak T_1=\delta(\mathfrak X)$. Thus we have the following **Theorem 34**. *Let $T:V\to \mathfrak g$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$. If two linear deformations $T_t^1=T+t\mathfrak T_1$ and $T_t^2=T+t\mathfrak T_2$ of $T$ are equivalent, then $\mathfrak T_1$ and $\mathfrak T_2$ are in the same class of the cohomology group $\mathcal{H}^1_T(V,\mathfrak g)$.* **Definition 35**. Let $T:V\to \mathfrak g$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$. An element $\mathfrak X\in \wedge^2\mathfrak g$ is called a **Nijenhuis element** with respect to $T$ if $\mathfrak X$ satisfies [\[Nije\]](#Nije){reference-type="eqref" reference="Nije"}-[\[Nij2\]](#Nij2){reference-type="eqref" reference="Nij2"}, [\[Nij3\]](#Nij3){reference-type="eqref" reference="Nij3"}, [\[Nij4\]](#Nij4){reference-type="eqref" reference="Nij4"} and the following equation $$\begin{aligned} \left\llbracket \mathfrak X,T(D(\mathfrak X)v)-\left\llbracket \mathfrak X,Tv\right\rrbracket \right\rrbracket =0, \quad \forall v \in V. \label{Nij5}\end{aligned}$$ It is obvious that a trivial deformation of a relative Rota-Baxter operator on a Lie-Yamaguti algebra gives rise to a Nijenhuis element. Indeed, the converse is also true. **Proposition 36**. *Let $T:V\to \mathfrak g$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$. Then for any Nijenhuis element $\mathfrak X\in \wedge^2 \mathfrak g$, $T_t:=T+t\mathfrak T$ with $\mathfrak T:=\delta(\mathfrak X)$ is a trivial linear deformation of the relative Rota-Baxter operator $T$.* *Proof.* The proof is similar to the case of Lie algebras or Leibniz algebras etc. Thus we omit the details. ◻ **Example 37**. *Let $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ be a 2-dimensional Lie-Yamaguti algebra, whose nontrivial brackets are given by $$[e_1,e_2]=e_1,~\quad \left\llbracket e_1,e_2,e_2\right\rrbracket =e_1,$$ where $\{e_1,e_2\}$ is a basis for $\mathfrak g$. Moreover, $$R= \begin{pmatrix} 0 & a\\ 0 & b \end{pmatrix}$$ is a Rota-Baxter operator on $\mathfrak g$. Then by a direct computation, any element in $\wedge^2\mathfrak g$ is a Nijenhuis element of $R$.* **Example 38**. *Let $\mathfrak g$ be a 4-dimensional Lie-Yamaguti algebra with a basis $\{e_1,e_2,e_3,e_4\}$ defined by $$[e_1,e_2]=2e_4,~\quad \left\llbracket e_1,e_2,e_1\right\rrbracket =e_4.$$ And $$R= \begin{pmatrix} 0 & a_{12}& 0 & 0 \\ 0 & 0& 0 & 0\\ a_{31} &a_{32} & a_{33} & a_{34}\\ a_{41} &a_{42} & a_{43} & a_{44} \end{pmatrix}$$ is a Rota-Baxter operator on $\mathfrak g$. Then any element in $\wedge^2\mathfrak g$ is a Nijenhuis element of $R$. In particular, $$\begin{aligned} \mathfrak X_1=e_1\wedge e_2,\quad \mathfrak X_2=e_1\wedge e_3,\quad \mathfrak X_3=e_1\wedge e_4,\\ \mathfrak X_4=e_2\wedge e_3,\quad \mathfrak X_5=e_2\wedge e_4,\quad \mathfrak X_6=e_3\wedge e_4, \end{aligned}$$ are all Nijenhuis elements of $R$.* ## Higher order deformations of relative Rota-Baxter operators In this subsection, we introduce a cohomology class associated to an order $n$ deformation of a relative Rota-Baxter operator, and show that an order $n$ deformation is extendable if and only if this cohomology class is trivial. Thus we call this cohomology class the obstruction class of an order $n$ deformation being extendable. **Definition 39**. Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$. If $T_t=\sum_{i=0}^n\mathfrak T_it^i$ with $\mathfrak T_0=T$, $\mathfrak T_i\in \mathrm{Hom}_{\mathbb K}(V,\mathfrak g)$, $i=1,2,\cdots,n$, defines a $\mathbb K[t]/(t^{n+1})$-module from $V[t]/(t^{n+1})$ to the Lie-Yamaguti algebra $\mathfrak g[t]/(t^{n+1})$ satisfying $$\begin{aligned} &=&T_t(\rho(T_t)u-\rho(T_tv)u),\\ \left\llbracket T_t,T_tv,T_tw\right\rrbracket &=&T_t(D_{\rho,\mu}(T_tu,T_tv)w+\mu(T_tv,T_tw)u-\mu(T_tu,T_tw)v), \quad \forall u,v,w \in V, \end{aligned}$$ we say that $T_t$ is an **order $n$ deformation** of the relative Rota-Baxter operator $T$. **Remark 40**. The left hand sides of the equations hold in the Lie-Yamaguti algebra $\mathfrak g[t]/(t^{n+1})$ and the right hand sides of the equations above make sense since $T_t$ is a $\mathbb K[t]/(t^{n+1})$-module map. **Definition 41**. Let $T_t=\sum_{i=0}^n\mathfrak T_it^i$ be an order $n$ deformation of a relative Rota-Baxter operator $T$ on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$. If there exists a $1$-cochain $\mathfrak T_{n+1}\in \mathrm{Hom}_{\mathbb K}(V,\mathfrak g)$ such that $\widetilde{T_t}=T_t+\mathfrak T_{n+1}t^{n+1}$ is an order $n+1$ deformation of $T$, then we say that $T_t$ is **extensible**. **Theorem 42**. *Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$, and $T_t=\sum_{i=0}^n\mathfrak T_it^i$ be an order $n$ deformation of $T$. Then $T_t$ is extensible if and only if the cohomology class  $[\mathsf{Ob}^T]\in \mathcal{H}_T^2(V,\mathfrak g)$ is trivial, where $\mathsf{Ob}^T=(\mathsf{Ob}_I^T,\mathsf{Ob}_{II}^T)\in {\mathcal{C}}_T^2(V,\mathfrak g)$ is defined to be $$\begin{aligned} \mathsf{Ob}_I^T(v_1,v_2)&=&\sum_{i+j=n+1,\atop i,j\geqslant 1}\Big([\mathfrak T_iv_1,\mathfrak T_jv_2]-\mathfrak T_i(\rho(\mathfrak T_jv_1)v_2-\rho(\mathfrak T_jv_2)v_1)\Big),\\ \mathsf{Ob}_{II}^T(v_1,v_2,v_3)&=&\sum_{i+j+k=n+1,\atop i,j,k\geqslant 1}\Big(\left\llbracket \mathfrak T_iv_1,\mathfrak T_jv_2,\mathfrak T_kv_3\right\rrbracket -\mathfrak T_i(D(\mathfrak T_jv_1,\mathfrak T_kv_2)v_3\\ ~ \nonumber&&+\mu(\mathfrak T_jv_2,\mathfrak T_kv_3)v_1-\mu(\mathfrak T_jv_1,\mathfrak T_kv_3)v_2\Big), \quad \forall v_1,v_2,v_3 \in V.\end{aligned}$$* *Proof.* Let $\widetilde{T_t}=\sum_{i=0}^{n+1}\mathfrak T_it^i$ be the extension of $T_t$, then for all $u,v,w \in V$, $$\begin{aligned} \label{n order1}[\widetilde{T_t}u,\widetilde{T_t}v]&=&\widetilde{T_t}\Big(\rho(\widetilde{T_t}u)v-\rho(\widetilde{T_t}v)u\Big),\\ \label{n order2}\left\llbracket \widetilde{T_t}u,\widetilde{T_t}v,\widetilde{T_t}w\right\rrbracket &=&\widetilde{T_t}\Big(D(\widetilde{T_t}u,\widetilde{T_t}v)w+\mu(\widetilde{T_t}v,\widetilde{T_t}w)u -\mu(\widetilde{T_t}u,\widetilde{T_t}w)v\Big).\end{aligned}$$ Expanding the Eq. [\[n order1\]](#n order1){reference-type="eqref" reference="n order1"} and comparing the coefficients of $t^n$ yields that $$\begin{aligned} \sum_{i+j=n+1,\atop i,j \geqslant 0}\Big([\mathfrak T_iu,\mathfrak T_jv]-\mathfrak T_i\big(\rho(\mathfrak T_ju)v-\rho(\mathfrak T_jv)u\big)\Big)=0,\end{aligned}$$ which is equivalent to $$\begin{aligned} ~ &&\sum_{i+j=n+1,\atop i,j \geqslant 1}\Big([\mathfrak T_iu,\mathfrak T_jv]-\mathfrak T_i\big(\rho(\mathfrak T_ju)v-\rho(\mathfrak T_jv)u\big)\Big)+[\mathfrak T_{n+1}u,Tv]+[Tu,\mathfrak T_{n+1}v]\\ ~ &&\quad-\big(T(\rho(\mathfrak T_{n+1}u)v-\rho(\mathfrak T_{n+1}v)u)+\mathfrak T_{n+1}(\rho(Tu)v-\rho(Tv)u)\big)=0,\end{aligned}$$ which is also equivalent to $$\begin{aligned} \mathsf{Ob}_I^T+\delta_I^T(\mathfrak T_{n+1})=0.\label{ob:cocy1}\end{aligned}$$ Similarly, expanding the Eq. [\[n order2\]](#n order2){reference-type="eqref" reference="n order2"} and comparing the coefficients of $t^n$ yields that $$\begin{aligned} \mathsf{Ob}_{II}^T+\delta_{II}^T(\mathfrak T_{n+1})=0.\label{ob:cocy2}\end{aligned}$$ From [\[ob:cocy1\]](#ob:cocy1){reference-type="eqref" reference="ob:cocy1"} and [\[ob:cocy2\]](#ob:cocy2){reference-type="eqref" reference="ob:cocy2"}, we get $$\mathsf{Ob}_T=-\delta^T(\mathfrak T_{n+1}).$$ Thus, the cohomology class $[\mathsf{Ob}_T]$ is trivial. Conversely, suppose that the cohomology class $[\mathsf{Ob}_T]$ is trivial, then there exists $\mathfrak T_{n+1}\in {\mathcal{C}}_T^1(V,\mathfrak g)$, such that  $\mathsf{Ob}_T=-\delta^T(\mathfrak T_{n+1}).$ Set $\widetilde{T_t}=T_t+\mathfrak T_{n+1}t^{n+1}$. Then for all $0 \leqslant s\leqslant n+1$,  $\widetilde{T_t}$ satisfies $$\begin{aligned} \sum_{i+j=s}\Big([\mathfrak T_iu,\mathfrak T_jv]-\mathfrak T_i\big(\rho(\mathfrak T_ju)v-\rho(\mathfrak T_jv)u\big)\Big)=0,\\ \sum_{i+j+k=s}\Big(\left\llbracket \mathfrak T_iu,\mathfrak T_jv,\mathfrak T_kw\right\rrbracket -\mathfrak T_i\big(D(\mathfrak T_ju,\mathfrak T_kv)w+\mu(\mathfrak T_jv,\mathfrak T_kw)u-\mu(\mathfrak T_ju,\mathfrak T_kw)v\big)\Big)=0,\end{aligned}$$ which implies that $\widetilde{T_t}$ is an order $n+1$ deformation of $T$. Hence it is a extension of $T_t$. ◻ **Definition 43**. Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$, and $T_t=\sum_{i=0}^n\mathfrak T_it^i$ be an order $n$ deformation of $T$. Then the cohomology class $[\mathsf{Ob}^T]\in \mathcal{H}_T^2(V,\mathfrak g)$ defined in Definition [Theorem 42](#ob){reference-type="ref" reference="ob"} is called the **obstruction class** of $T_t$ being extensible. **Corollary 44**. *Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra $(\mathfrak g,[\cdot,\cdot],\left\llbracket \cdot,\cdot,\cdot\right\rrbracket )$ with respect to a representation $(V;\rho,\mu)$. If $\mathcal{H}_T^2(V,\mathfrak g)=0$, then every $1$-cocycle in $\mathcal{Z}_T^1(V,\mathfrak g)$ is the infinitesimal of some formal deformation of the relative Rota-Baxter operator $T$.* **Acknowledgements:** Qiao was partially supported by NSFC grant 11971282. a C. Bai, O. Bellier, L. Guo, and X. Ni, Splitting of operations, Manin products, and Rota-Baxter operators, *Int. Math. Res. Not. IMRN* (2013), no. 3, 485-524. D. Balavoine, Deformations of algebras over a quadratic operad. 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arxiv_math

--- abstract: | We prove *a priori* bounds for Feigenbaum quadratic polynomials, i.e., infinitely renormalizable polynomials $f_c: z\mapsto z^2+c$ of bounded type. It implies local connectivity of the corresponding Julia sets $J(f_c)$ and MLC (local connectivity of the Mandelbrot set ${\mathcal{M}}$) at the corresponding parameters $c$. It also yields the scaling Universality, dynamical and parameter, for the corresponding combinatorics. The MLC Conjecture was open for the most classical period-doubling Feigenbaum parameter as well as for the complex tripling renormalizations. Universality for the latter was conjectured by Goldberg-Khanin-Sinai in the early 1980s. author: - Dzmitry Dudko and Mikhail Lyubich title: MLC at Feigenbaum points --- =cmr7 '=12 # Introduction ## Brief history The MLC Conjecture on the local connectivity of the Mandelbrot set ${\mathcal{M}}$, put forward by Douady and Hubbard in the mid 19980s [@DH:Orsay], is the central problem of contemporary Holomorphic Dynamics. It would imply a precise topological model for ${\mathcal{M}}$ and the Fatou Conjecture on the density of hyperbolic maps in ${\mathcal{M}}$, and it is intimately related to the Mostow-Thurston Rigidity phenomenon in 3D Hyperbolic Geometry. Around 1990 Yoccoz proved MLC at any parameter $c\in {\mathcal{M}}$ that is not infinitely renormalizable (in the quadratic-like sense) thus linking the problem tightly to the Quadratic-like Renormalization Theory (see [@H; @M]). First advances in this direction appeared in [@puzzle], where MLC was established for infinitely renormalizable parameters of *high type* and the general problem was reduced, under some circumstances, to the problem of *a priori* bounds. Quadratic-like renormalization appears in two flavors, *primitive* and *satellite*. The above results were concerned with the primitive case. Next breakthrough in this direction appeared in the work of Jeremy Kahn [@K] who established *a priori* bounds, and hence MLC, for all infinitely renormalizable parameters of *bounded primitive* type (using the Covering Lemma [@covering; @lemma]). It followed up with the work [@decorations; @molecules] handling the *definitely primitive case*. It took more than 10 years to prove MLC at some infinitely renormalizable parameters of *bounded satellite* type [@DL] (based upon the Pacmen Renormalization Theory developed in [@DLS]). However, the most prominent parameter, the *period-doubling Feigenbaum point* corresponding to the cascade of doubling renormalizations (see Figure [\[Fig:FeigPar\]](#Fig:FeigPar){reference-type="ref" reference="Fig:FeigPar"}) was not covered by this result. In this paper we are filling in this gap by proving MLC at the Feigenbaum point, and in fact, in *all infinitely renormalizable maps of bounded type* (for which we will still preserve the same name). **Remark 1**. *The universality of period-doubling cascade was discovered in the mid 70s by Feigenbaum [@F1; @F2] (the parameter part, see Figure [\[Fig:FeigPar\]](#Fig:FeigPar){reference-type="ref" reference="Fig:FeigPar"}) and independently by Coullet and Tresser [@TC; @CT] (the dynamical part). It is intimately related to the renormalization phenomenon in the Quantum Field Theory and Statistical Mechanics; the discovery opened up a new universality paradigm in Dynamical Systems.* ## Statement of the main result and its consequencses A *Feigenbaum map* is an infinitely renormalizable quadratic-like map $f: U\rightarrow V$ with bounded combinatorics, i.e., all renormalization periods of $f$ are bounded by some $\bar p$ (see §[2.2.5](#sss:infin renorm){reference-type="ref" reference="sss:infin renorm"} for precise definitions). One says that such an $f$ has *a priori* bounds if its quadratic-like renormalizations ${\mathcal R}^n f : U^n\rightarrow V^n$ can be selected so that $$\operatorname{mod}(V^n\setminus U^n)\geq \mu>0.$$ The bounds are called *beau* if $\mu$ depends only on the combinatorial bound $\bar p$ as long as $n$ is big enough (depending on $\operatorname{mod}(V\setminus U)$). **Theorem 1**. *Any Feigenbaum quadratic-like map has a priori beau bounds.* Together with the Rigidity Theorem of [@puzzle], it implies MLC at the corresponding parameters: **Theorem 2**. *The Mandelbrot set is locally connected at any Feigenbaum parameter.* The *Renormalization Conjecture* asserts that for any combinatorial bound $\bar p$ the renormalization transformation ${\mathcal R}$ has a renormalizatrion horseshoe ${\mathcal A}_{\bar p}$ of Feigenbaum maps of type $\bar p$ on which it acts hyperbolically with one-dimensional ustable foliation. Together with [@L:universality] (see also [@S; @McM2; @AL]) we obtain: **Theorem 3**. *For any combinatorial bound $\bar p$, the Renormalization Conjecture is valid in the space of quadratic-like maps.* Let us consider the set ${\mathcal I}_{\bar p}$ of Feigenbaum parameters $c\in {\mathcal{M}}$ of type $\bar p$. Theorem A implie that each $c\in {\mathcal I}_{\bar p}$ is the intersection of the nest $M^n(c)$ of little $M$-copies corresponding to the $n$-fold renormalizations of $f_c$. Let us say that that Mandelbrot set is *self-similar* over ${\mathcal I}_{\bar p}$ if the straightening map $\chi:{\mathcal M}^{n+1} (c) \rightarrow{\mathcal M}^n(\chi(c))$ (see §[2.2.3](#renorm sec){reference-type="ref" reference="renorm sec"}) is $C^{1+{\delta}}$-conformal at any $c\in {\mathcal I}_{\bar p}$ (with uniform constants). The Renormalization Conjecture (Theorem [Theorem 3](#mainthm:C){reference-type="ref" reference="mainthm:C"}) implies: **Theorem 4**. *For any $\bar p$, the Mandelbrot set is self-similar over ${\mathcal I}_{\bar p}$.* For the *satellite triplings*, the last two theorems confirm the Goldberg-Sinai-Khanin Conjecture [@GSK] from the 1980s, see Figure [\[Fig:conj:GSK\]](#Fig:conj:GSK){reference-type="ref" reference="Fig:conj:GSK"}. ## Rough outline of the proof The historical developments of ideas behind the proof are summarized in [@L21]. For a ql map $f\colon U\to V$, its width is $${\mathcal W}(f) = {\mathcal W}(V\setminus {\mathfrak K }_f)=\frac{1}{\operatorname{mod}(V\setminus {\mathfrak K }_f)} ,$$ where ${\mathfrak K }_f$ is the filled Julia set of $f$. Our goal is to bound ${\mathcal W}(f)$ from above. We will first review the main ideas for the bounded primitive case, then we will discuss how to complete the argument in the bounded satellite case. $$\begin{tikzpicture}[] \draw[fill=red,opacity=0, fill opacity=0.2] (0.2,-0.2)--(0.2,0.2) -- (5.8,0.2)--(5.8,-0.2); \node[red,] at (3,0.5) {${\mathcal G}_1$}; \draw[fill=blue,opacity=0, fill opacity=0.2] (6.2,-0.2)--(6.2,0.2) -- (8.8,0.2)--(8.8,-0.2); \node[blue] at (7.5,0.5) {${\mathcal G}_2$}; \begin{scope}[rotate =0] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \end{scope} \begin{scope}[shift={(6,0)}] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \end{scope} \begin{scope}[shift={(9,0)}] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \end{scope} \begin{scope}[shift={(0,-3)}] \draw[fill=blue,opacity=0, fill opacity=0.2] (0.2,-0.2)--(0.2,0.2) -- (2.8,0.2)--(2.8,-0.2); \draw[fill=red,opacity=0, fill opacity=0.2] (6.2,-0.2)--(6.2,0.2) -- (8.8,0.2)--(8.8,-0.2); \draw[fill=red,opacity=0, fill opacity=0.2] (3.2,-0.2)--(3.2,0.2) -- (5.8,0.2)--(5.8,-0.2); \draw[line width=0.3mm] (4.1,0.5) edge[bend left=10,->] (3,2.5); \node[left,red] at (4.8,0.5) {${\mathcal G}'_1$}; \draw[line width=0.3mm] (7.5,0.5) edge[bend left=15,->] (3.5,2.5); \node[right,red] at (7.7,0.5) {${\mathcal G}''_1$}; \draw[line width=0.3mm] (1.5,0.5) edge[,->] (7.5,2.5); \node[blue,left] at (1.3,0.5) {${\mathcal G}'_2$}; \begin{scope}[rotate =0] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \end{scope} \begin{scope}[shift={(3,0)},gray] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \end{scope} \begin{scope}[shift={(6,0)}] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \end{scope} \begin{scope}[shift={(9,0)}] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \end{scope} \end{scope} \end{tikzpicture}$$ ### Pull-off in the primitive case [@K; @covering; @lemma] {#sss:intro:PrimPullOff} Consider the $n$th renormalization $f_n={\mathcal R}^n f$ of $f=f_0$, and let ${\mathfrak K }_0, {\mathfrak K }_1,\dots ,{\mathfrak K }_{p-1}$ be the periodic cycle of little filled Julia sets of $f_n$ in the dynamical plane of $f_0\colon U\to V$. Assuming that the renormalization is primitive, all ${\mathfrak K }_i$ are pairwise disjoint. Consider the thin-thick decomposition of $V'\coloneqq V\setminus \bigcup_i {\mathfrak K }_i\,$: there are finitely many pairwise disjoint rectangles ${\mathcal G}_i$ between components of $\partial V'$ such that, up to $O_p(1)$ of width, the ${\mathcal G}_i$ contain all non-peripheral curves in $V'$ with endpoints in $\partial V'$, see Figure [\[Fg:TTD\]](#Fg:TTD){reference-type="ref" reference="Fg:TTD"} for an illustration of a thin-thick decomposition. A rectangle ${\mathcal G}_i$ is called - *vertical* if it connects the outer boundary $\partial^{\mathrm{out}}V'\coloneqq \partial V$ of $V'$ to one of its inner boundary components; - *horizontal* if it connects two inner (i.e., non-outer) boundary components of $V'$. Below we assume that ${\mathcal W}(f_n)\gg_p 1$. We have: $$\label{eq:outline:1} {\mathcal W}(f)\ge \sum_{\text{ vertical }{\mathcal G}_i} {\mathcal W}({\mathcal G}_i)$$ and, for every ${\mathfrak K }_j$: $$\label{eq:outline:2} {\mathcal W}(f_n) = {\mathcal W}_{\mathrm{loc}}({\mathfrak K }_j) +O_p(1) \coloneqq \sum_{\substack{ {\mathcal G}_i \text{ adjacent}\\ \text{to }{\mathfrak K }_j }}{\mathcal W}({\mathcal G}_i) +O_p(1),$$ i.e. the sum in [\[eq:outline:2\]](#eq:outline:2){reference-type="eqref" reference="eq:outline:2"} is taken over all rectangles ${\mathcal G}_i$ adjacent to ${\mathfrak K }_j$. (A "$\psi$-ql renormalization" is used to achieve [\[eq:outline:2\]](#eq:outline:2){reference-type="eqref" reference="eq:outline:2"}, see §[3.4](#psi-renorm){reference-type="ref" reference="psi-renorm"}, [3.5.2](#sss:WADs){reference-type="ref" reference="sss:WADs"}, and [\[eq:Width is Loc WAD\]](#eq:Width is Loc WAD){reference-type="eqref" reference="eq:Width is Loc WAD"}.) Set also $${\mathcal W}_{\mathrm{loc}}^{\mathrm {ver}}({\mathfrak K }_j)\coloneqq \sum_{\substack{ \text{vertical }{\mathcal G}_i \\ \text{adjacent to }{\mathfrak K }_j }}{\mathcal W}({\mathcal G}_i).$$ A fundamental fact is that the horizontal ${\mathcal G}_i$ are eventually (after several restrictions of the domain) aligned with the Hubbard tree $T_f$ of $f$ as it shown on Figure [\[Fg:Pull off:Air Comb\]](#Fg:Pull off:Air Comb){reference-type="ref" reference="Fg:Pull off:Air Comb"}. Applying the Grötzsch inequality to the horizontal rectangles and their pullbacks (see the caption of Figure [\[Fg:Pull off:Air Comb\]](#Fg:Pull off:Air Comb){reference-type="ref" reference="Fg:Pull off:Air Comb"}), we experience a definite loss of the horizontal weight in favor of the vertical one. The created definite vertical weight can then be pushed forward by means of the Covering Lemma [@covering; @lemma]. Since all the local weights are comparable, [^1] we obtain: $$\label{eq:outline:3} {\mathcal W}_{\mathrm{loc}}^{\mathrm {ver}}({\mathfrak K }_j) \asymp {\mathcal W}_{\mathrm{loc}}({\mathfrak K }_j) {\ \ }{\ \ }{\ \ }\text{ for every }{\mathfrak K }_j,$$ where "$\asymp$" is independent of $p$. We stress that a key combinatorial ingredient used for [\[eq:outline:3\]](#eq:outline:3){reference-type="eqref" reference="eq:outline:3"} is the absence of periodic horizontal rectangles; i.e. a rectangle that has an iterated lift homotopic to itself rel small Julia sets, see §[2.4](#sss:InvArcDiagr){reference-type="ref" reference="sss:InvArcDiagr"}, §[3.5.8](#sss:PeriodRect){reference-type="ref" reference="sss:PeriodRect"}. Combining  [\[eq:outline:1\]](#eq:outline:1){reference-type="eqref" reference="eq:outline:1"}, [\[eq:outline:3\]](#eq:outline:3){reference-type="eqref" reference="eq:outline:3"}, [\[eq:outline:2\]](#eq:outline:2){reference-type="eqref" reference="eq:outline:2"}, and assuming $p\gg 1$, we obtain: $$\label{eq:intro:comp} { \begin{array}{c} {\mathcal W}(f)\ge \sum_{\text{ vertical }{\mathcal G}_i} {\mathcal W}({\mathcal G}_i)\ = \ \sum_{j=1}^p {\mathcal W}_{\mathrm{loc}}^{\mathrm {ver}}({\mathfrak K }_j)\ \asymp \\ \\ \sum_{j=1}^p {\mathcal W}_{\mathrm{loc}}({\mathfrak K }_j)\ \asymp p\ {\mathcal W}(f_n)\ \gg\ 2 {\mathcal W}(f_n); \end{array} }$$ i.e: $$\label{eq:W f vs W f_n} {\mathcal W}(f) \ge 2 {\mathcal W}(f_n)$$ implying a priori bounds in the primitive case. (For instance, by the *Record Argument*: by selecting "record levels", on which the degeneration exceeds all the preceding ones, we immediately arrive at a contradiction.) $$\begin{tikzpicture}[scale=0.8] \draw[fill=red,opacity=0, fill opacity=0.2] (0.2,-0.2)--(0.2,0.2) -- (2.8,0.2)--(2.8,-0.2); \node[red,] at (1.5,0.5) {${\mathcal R}_0$}; \draw[fill=blue,opacity=0, fill opacity=0.2] (3.2,-0.2)--(3.2,0.2) -- (10.8,0.2)--(10.8,-0.2); \node[blue] at (7,0.5) {${\mathcal G}$}; \draw[fill=red,opacity=0, fill opacity=0.2] (11.2,-0.2)--(11.2,0.2) -- (13.8,0.2)--(13.8,-0.2); \node[red,] at (12.5,0.5) {${\mathcal R}_1$}; \begin{scope}[rotate =0] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \node[above] at (-0.2,0.2) {${\mathfrak B }_0$}; \end{scope} \begin{scope}[shift={(3,0)}] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \node[above] at (-0.,0.2) {${\mathfrak B }_1$}; \end{scope} \begin{scope}[shift={(11,0)}] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \node[above] at (-0.,0.2) {${\mathfrak B }_2$}; \end{scope} \begin{scope}[shift={(14,0)}] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \node[above] at (0.2,0.2) {${\mathfrak B }_3$}; \end{scope} \begin{scope}[shift={(0,-3)}] \draw[fill=red,opacity=0, fill opacity=0.2] (0.2,-0.2)--(0.2,0.2) -- (2.4,0.2)--(2.4,-0.2); \node[red,] at (1.5,0.5) {${\mathcal R}'_0$}; \draw (2,0.5) edge[line width=0.2mm,->,bend right] node[right]{ $f^2$}(2, 2.5); \begin{scope}[shift={(2.5,0)}] \draw[fill=red,opacity=0, fill opacity=0.2] (0.7,-0.2)--(0.7,0.2) -- (2.3,0.2)--(2.3,-0.2); \node[red,] at (1.5,0.5) {${\mathcal R}''_0$}; \end{scope} \draw[fill=blue,opacity=0, fill opacity=0.2] (5.2,-0.2)--(5.2,0.2) -- (8.8,0.2)--(8.8,-0.2); \node[blue] at (7,0.5) {${\mathcal G}'$}; \draw (7.7,0.5) edge[line width=0.2mm,->,bend right] node[right]{ $f^2$}(7.7, 2.5); \draw[fill=red,opacity=0, fill opacity=0.2] (11.6,-0.2)--(11.6,0.2) -- (13.8,0.2)--(13.8,-0.2); \node[red,] at (12.5,0.5) {${\mathcal R}'_1$}; \draw (13,0.5) edge[line width=0.2mm,->,bend right] node[right]{ $f^2$}(13, 2.5); \begin{scope}[shift={(-2.5,0)}] \draw[fill=red,opacity=0, fill opacity=0.2] (11.7,-0.2)--(11.7,0.2) -- (13.3,0.2)--(13.3,-0.2); \node[red,] at (12.5,0.5) {${\mathcal R}''_1$}; \end{scope} \begin{scope}[rotate =0] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); %\draw[line width=0.9mm,gray] (0.5,0) -- (0.8,0); %\draw[line width=0.9mm,gray] (0.6,-0.2) -- (0.6,0.2); \end{scope} \begin{scope}[shift={(3,0)}] \draw[line width=0.9mm,gray] (-0.6,-0.2) -- (-0.6,0.2); \draw[line width=0.9mm,gray] (-0.9,0) -- (-0.5,0); \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \end{scope} \begin{scope}[gray,shift={(5,0)}] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \end{scope} \begin{scope}[gray,shift={(9,0)}] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \end{scope} \begin{scope}[shift={(11,0)}] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \draw[line width=0.9mm,gray] (0.5,0) -- (0.8,0); \draw[line width=0.9mm,gray] (0.6,-0.2) -- (0.6,0.2); \end{scope} \begin{scope}[shift={(14,0)}] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); %\draw[line width=0.9mm,gray] (-0.6,-0.2) -- (-0.6,0.2); %\draw[line width=0.9mm,gray] (-0.9,0) -- (-0.5,0); \end{scope} \end{scope} \end{tikzpicture}$$ ### Satellite case {#ss:intro:SatCase} Let us now assume that $f$ is an infinitely renormalizable map with the periodic-doubling combinatorics; i.e., $f$ is hybrid equivalent to a map in the period-doubling Feigenbaum combinatorial class. The general bounded case is similar. Let ${\mathfrak B }\coloneqq {\mathfrak K }^1_0\cup {\mathfrak K }^1_1$ be the bouquet consisting of two level $1$ little Julia sets of $f$. Instead of ${\mathcal W}(f)$, we will measure $${\mathcal W}_\bullet(f)\coloneqq {\mathcal W}(V\setminus {\mathfrak B }).$$ In the dynamical plane of $f=f_0$, consider the periodic cycle of little bouquets ${\mathfrak B }_0,{\mathfrak B }_1,\dots, {\mathfrak B }_{p-1}$ (where $p=2^n$) associated with $f_n={\mathcal R}^n f$. We enumerate these bouquets linearly from left-to-right. Consider the thick-thin decomposition of $V'\coloneqq V\setminus \bigcup_i {\mathfrak B }_i$. The following properties are established similarly as in the primitive case considered above: - Estimates [\[eq:outline:1\]](#eq:outline:1){reference-type="eqref" reference="eq:outline:1"} and [\[eq:outline:2\]](#eq:outline:2){reference-type="eqref" reference="eq:outline:2"}, where little Julia sets ${\mathfrak K }_i$ are replaced with bouquets ${\mathfrak B }_i$; - alignment of the horizontal rectangles with the Hubbard tree of $f$ (after several restrictions of the domain). However, unlike in the primitive case §[1.3.1](#sss:intro:PrimPullOff){reference-type="ref" reference="sss:intro:PrimPullOff"}, $f$ may have periodic horizontal rectangles ${\mathcal R}_k$ aligned with the Hubbard tree between ${\mathfrak B }_{2k}$ and ${\mathfrak B }_{2k+1}$ as shown on Figure [\[Fg:Pull off:Feigen Comb\]](#Fg:Pull off:Feigen Comb){reference-type="ref" reference="Fg:Pull off:Feigen Comb"}. This leads us to the following alternative: either most of the total degeneration is within the ${\mathcal R}_k$ or a substantial part of the total degeneration is vertical (i.e. [\[eq:outline:3\]](#eq:outline:3){reference-type="eqref" reference="eq:outline:3"} holds). More precisely, choose a small $\delta>0$. We have: 1. [\[C1:intro\]]{#C1:intro label="C1:intro"}either there is a periodic rectangle ${\mathcal R}_k$ connecting ${\mathfrak B }_{2k},{\mathfrak B }_{2k+1}$ such that a $(1-\delta)$ part of ${\mathcal R}_k$ overflows its iterated lift ${\mathcal R}'_k$ under $f^{2^{n-1}}$; 2. [\[C2:intro\]]{#C2:intro label="C2:intro"} or else we have: $${\mathcal W}_{\mathrm{loc}}^{\mathrm {ver}}({\mathfrak K }_j) \asymp_\delta {\mathcal W}_{\mathrm{loc}}({\mathfrak K }_j) {\ \ }{\ \ }{\ \ }\text{ for every }{\mathfrak K }_j$$ If [\[C2:intro\]](#C2:intro){reference-type="ref" reference="C2:intro"} holds, then assuming $p\gg_\delta 1$ and ${\mathcal W}_\bullet(f_n)\gg_{p,\delta} 1$ and repeating the argument of [\[eq:intro:comp\]](#eq:intro:comp){reference-type="eqref" reference="eq:intro:comp"}, we obtain 1. [\[C2\':intro\]]{#C2':intro label="C2':intro"} ${\mathcal W}_\bullet(f)>2 {\mathcal W}_\bullet(f_n)$; the Record Argument leads to a contradiction in case [\[C2\':intro\]](#C2':intro){reference-type="ref" reference="C2':intro"}. The dichotomy "Case [\[C1:intro\]](#C1:intro){reference-type="ref" reference="C1:intro"} vs Case [\[C2\':intro\]](#C2':intro){reference-type="ref" reference="C2':intro"}" is stated as Theorem [Theorem 24](#thm:prim pull-off){reference-type="ref" reference="thm:prim pull-off"}. This is a refinement of [@K]. *Consider now Case [\[C1:intro\]](#C1:intro){reference-type="ref" reference="C1:intro"}.* As Figure [\[Fg:Sat Pull off:Feig Comb\]](#Fg:Sat Pull off:Feig Comb){reference-type="ref" reference="Fg:Sat Pull off:Feig Comb"} illustrates, bouquets ${\mathfrak B }_0,{\mathfrak B }_1$ grow under lifting thus the "combinatorial distance" between $\widetilde {\mathfrak B }_0,\widetilde {\mathfrak B }_1$ shrinks. This allows us to detect laminations ${\mathcal S}_i$ (the "differences" between ${\mathcal R}={\mathcal R}_k$ and its lift ${\mathcal R}'$) that are much wider than ${\mathcal W}({\mathcal R})\succeq {\mathcal W}_\bullet(f_n)$; i.e. ${\mathcal W}({\mathcal S}_i)\ge C_\delta {\mathcal W}({\mathcal R})$ with $C_\delta\to \infty$ as $\delta\to 0$. There are two possibilities: either a substantial part of ${\mathcal S}_i$ hits preperiodic bouquets associated with $f_{n+1}$ or a substantial part of ${\mathcal S}_i$ creates a "wave" (see §[5](#waves sec){reference-type="ref" reference="waves sec"}) above such a preperiodic bouquet. In the latter case, the Wave Lemma implies that ${\mathcal W}_\bullet(f_{n-1}) \gg {\mathcal W}_\bullet(f_n)$ and the Record Argument is applicable. In the former case, we obtain ${\mathcal W}_\bullet (f_{n+1}) \succeq {\mathcal W}({\mathcal S}_i) \succeq C_\delta {\mathcal W}(f_n)$; i.e.: $${\mathcal W}_\bullet (f_{n+1}) \ge C'_\delta {\mathcal W}(f_n){\ \ }{\ \ }\text{ with }{\ \ }C'_\delta\to \infty\text{\ \ }{ as }{\ \ }\delta\to 0.$$ For a sufficiently small $\delta$, this eventually contradicts the Teichmüller contraction within hyperbolic classes [@S:Berkeley] (stated as Proposition [Proposition 9](#prop:Teichm contr){reference-type="ref" reference="prop:Teichm contr"}): $${\mathcal W}_\bullet (f_n) = O\left( \Delta^n\right) {\ \ }{\ \ }{\ \ }\text{ for some } \Delta >1.$$ Finally, we will convert in Theorem [Theorem 31](#thm:psi bounds){reference-type="ref" reference="thm:psi bounds"} bounds for ${\mathcal W}_\bullet(f_m)$ into bounds for ${\mathcal W}(f_m)$. **Remark 2**. *In the paper, we will be measuring the degeneration around "bushes" §[2.3](#ss:bushes){reference-type="ref" reference="ss:bushes"}, which are Hubbard continua enhanced with the periodic cycle of level one little Julia sets. An alternative would be to keep the Julia sets for primitive renormalization and use the satellite flowers in the satellite case. We believe that it is also possible to work with Hubbard continua; however, the Wave Lemma §[5](#waves sec){reference-type="ref" reference="waves sec"} and the transition toward classical QL-bounds will be more subtle.* ## Acknowledgments {#acknowledgments .unnumbered} The first author was partially supported by the NSF grant DMS $2055532$ and the ERC grant "HOLOGRAM". The second author has been partly supported by the NSF, the Hagler and Clay Fellowships, the Institute for Theoretical Studies at ETH (Zurich), and MSRI (Berkeley). We would like to thank Jeremy Kahn for many stimulating discussions. Results of this paper were first announced in 2021 at the [Quasiworld Seminar](https://sites.google.com/g.ucla.edu/quasiworld/) and at the CIRM conference ["Advancing Bridges in Complex Dynamics"](https://www.i2m.univ-amu.fr/events/advancing-bridges-in-complex-dynamics/), see the talk by the second author [@L21]. A number of our pictures are made with W. Jung's program *Mandel*. $$\begin{tikzpicture}[xscale=2,yscale=1.5] \draw[fill=red,opacity=0, fill opacity=0.2] (0.2,-0.17)--(0.2,0.17) -- (5.8,0.17)--(5.8,-0.17); \node[red,scale=1.2] at (3,0.5) {${\mathcal R}$}; \begin{scope}[rotate =0] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \node[above,scale=1.2] at(0,0.2){${\mathfrak B }_0$}; \end{scope} \begin{scope}[shift={(6,0)}] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \node[above,scale=1.2] at(0,0.2){${\mathfrak B }_1$}; \end{scope} \begin{scope}[shift={(0,-2)}] \draw[fill=red,opacity=0, fill opacity=0.2] (1.1,-0.17)--(1.1,0.17) -- (4.9,0.17)--(4.9,-0.17); \node[red,scale=1.2] at (3,0.5) {${\mathcal R}'$}; \draw[fill=orange,opacity=0, fill opacity=0.3] (1.1,-0.17)--(1.1,0.17) -- (0.6,0.17)--(0.6,-0.17); \node[orange,scale=1.2] at (0.85,0.5) {${\mathcal S}_1$}; \draw[fill=orange,opacity=0, fill opacity=0.3] (5.4,-0.17)--(5.4,0.17) -- (4.9,0.17)--(4.9,-0.17); \node[orange,scale=1.2] at (5.15,0.5) {${\mathcal S}_2$}; \begin{scope}[rotate =0] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \draw[line width=0.9mm,gray] (0.5,0) -- (1.4,0); \draw[line width=0.9mm,gray] (0.6,-0.2) -- (0.6,0.2); \draw[line width=0.9mm,gray] (1.1,-0.2) -- (1.1,0.2); \node[above,scale=1.2] at(0,0.2){$\widetilde {\mathfrak B }_0$}; \end{scope} \begin{scope}[shift={(6,0)}] \draw[line width=0.9mm,gray] (-1.1,-0.2) -- (-1.1,0.2); \draw[line width=0.9mm,gray] (-0.6,-0.2) -- (-0.6,0.2); \draw[line width=0.9mm,gray] (-1.4,0) -- (-0.5,0); \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \node[above,scale=1.2] at(0,0.2){$\widetilde {\mathfrak B }_1$}; \end{scope} \end{scope} \end{tikzpicture}$$ # QL renormalization, bushes, and invariant arc diagrams {#ql maps} The reader can find most of the needed background material in the book [@L:book]. Let us briefly outline the context of this section with emphasis on the notation. ## Outline Basics aspects of quadratic-like maps are summarized in §[2.2](#ss:QL maps){reference-type="ref" reference="ss:QL maps"}. We will usually denote by - $f\colon X\to Y$ a ql map, - ${\mathfrak K }_f$ its (filled) Julia set, - ${\mathfrak K }^{[n]}_i\subset {\mathfrak K }_f$ its level $n$ little Julia sets, - ${\boldsymbol{\mathfrak K} }^{[n]}\coloneqq \bigcup_i {\mathfrak K }^{[n]}_i$ the union of periodic level $n$ little Julia sets, see [\[eq:bfilled\]](#eq:bfilled){reference-type="eqref" reference="eq:bfilled"}, - ${\mathfrak T}_f\subset {\mathfrak K }$ the Hubbard continuum of $f$, see §[2.2.2](#sss:CombHT){reference-type="ref" reference="sss:CombHT"}, - ${\mathfrak B }_f\coloneqq {\mathfrak T}_f \cup {\boldsymbol{\mathfrak K} }^{[1]}$ the *bush* of $f$, see [\[eq:dfn:bush\]](#eq:dfn:bush){reference-type="eqref" reference="eq:dfn:bush"}, - ${\mathfrak B }_f^{(m)}\coloneqq f^{-m}({\mathfrak B }_f)$ the little bush of height $m$, - ${\mathfrak B }^{[n]}_i \equiv {\mathfrak B }^{[n],(0)}\subset {\mathfrak B }^{[n],(m)}\subset {\mathfrak K }^{[n]}_i$ the associated little bushes, - ${\boldsymbol {\mathfrak B} }^{[n]}\coloneqq \bigcup_i {\mathfrak B }^{[n]}_i\subset {\boldsymbol{\mathfrak K} }^{[n]}$ the union of periodic level $n$ little bushes, - ${\boldsymbol {\mathfrak B} }^{[n],(m)}\coloneqq f^{-1} \left({\boldsymbol {\mathfrak B} }^{[n]}\right)$ the union of level $n$ and height $m$ little periodic and preperiodic bushes, - ${\boldsymbol {\mathfrak B} }^{[n],(m)}_{\operatorname{per}}\coloneqq{\boldsymbol {\mathfrak B} }^{[n],(m)} \cap {\boldsymbol{\mathfrak K} }^{[n]}$ the union of periodic level $n$ and height $m$ little bushes. If ${\mathfrak B }_f ={\boldsymbol{\mathfrak K} }^{[1]}$, i.e., ${\mathfrak T}_f \subset {\boldsymbol{\mathfrak K} }^{[1]}$, then we will also refer to ${\mathfrak B }_f$ as the *satellite flower* of $f$. Preperiodic little Julia sets and bushes are defined accordingly, [2.3.1](#sss:LittleBushes){reference-type="ref" reference="sss:LittleBushes"}. We will usually assume that 1. [\[CrtPoint\]]{#CrtPoint label="CrtPoint"} ${\mathfrak K }^{[n]}_0$ contains the critical point of $f$. The Teichmüller contraction of ql renormalization will be established in §[2.5](#ss:Teichm contra){reference-type="ref" reference="ss:Teichm contra"}. In Lemma [Lemma 4](#lem:Alignment){reference-type="ref" reference="lem:Alignment"}, we will show that invariant horizontal arc diagrams are aligned with the Hubbard dendrite. Figures [\[Fig:exm Inv Arc Diagr\]](#Fig:exm Inv Arc Diagr){reference-type="ref" reference="Fig:exm Inv Arc Diagr"} and [\[Fig:PeriodicArcs\]](#Fig:PeriodicArcs){reference-type="ref" reference="Fig:PeriodicArcs"} demonstrate new subtleties arising in the satellite case. In particular, there are genuine periodic arcs, described in Lemma [Lemma 5](#lem:per arcs){reference-type="ref" reference="lem:per arcs"}; they encode almost periodic rectangles (see §[3.5.8](#sss:PeriodRect){reference-type="ref" reference="sss:PeriodRect"} and Figure [\[Fg:Pull off:Feigen Comb\]](#Fg:Pull off:Feigen Comb){reference-type="ref" reference="Fg:Pull off:Feigen Comb"}) that will have a separate treatment in Section [6](#s:Pulloff for per rectangles){reference-type="ref" reference="s:Pulloff for per rectangles"}. ## Quadratic-like maps {#ss:QL maps} A *quadratic-like map* $f: X\rightarrow Y$, which will also be abbreviated as a *ql map*, is a holomorphic double branched covering between two Jordan disks $X \Subset Y\subset {\Bbb C}$. It has a single critical point that we usually put at the origin $0$. The annulus $A= Y\setminus\overline X$ is called the *fundamental annulus* of $f\colon X\to Y$. We let $\operatorname{mod}f : = \operatorname{mod}A$. The *filled Julia set* ${\mathfrak K }_f$ is the set of non-escaping points: $${\mathfrak K }\equiv {\mathfrak K }_f \equiv {\mathfrak K }(f)\coloneqq \{ z:\, f^n z\in X, \ n=0,1,2,\dots \}.$$ Its boundary is called the *Julia set* ${\mathfrak J }\equiv {\mathfrak J }(f)$. The (filled) Julia set is either connected or Cantor, depending on whether the critical point is non-escaping (i.e., $0\in {\mathfrak K }(f)$) or otherwise. In this paper, filled Julia sets will play bigger role than the Julia sets, and we will often skip the adjective "filled" when it is clear (e.g., from the notation) what we mean. We say that two quadratic-like maps with connected Julia set represent the same *germ* if they have the same filled Julia set and coincide in some neighborhood of it. For a ql germ $f$, we define $$\label{eq:mod(f)} \operatorname{mod}f \coloneqq \sup \operatorname{mod}(Y'\setminus X'),$$ where the supremum is taken over all $f\colon X'\to Y'$ representing the ql germ of $f$. Two quadratic-like maps $f: X\rightarrow Y$ and $\tilde f: \tilde X\rightarrow\tilde Y$ are called *hybrid conjugate* if they are conjugate by a quasiconformal map $h: (Y,X) \rightarrow(\tilde Y, \tilde X)$ such that $\bar\partial h= 0$ a.e. on ${\mathfrak K }$. A simplest example of a quadratic-like map is provided by a quadratic polynomial $f_c: z\mapsto z^2+c$ restricted to a disk $Y=\mathbb{D}_R$ of sufficiently big radius. The Douady and Hubbard *Straightening Theorem* asserts that any quadratic-like map $f$ is hybrid conjugate to some restricted quadratic polynomial $f_c$. Moreover, if $J$ is connected then the parameter $c\in {\mathcal M}$ is unique. As for quadratic polynomials, two fixed points of a quadratic-like maps with connected Julia set have a different dynamical meaning. One of them, called $\beta$, is the landing point of a proper arc $\gamma\subset X \setminus K(f)$ such that $f(\gamma)\supset \gamma$. It is either repelling or parabolic with multiplier one. The other fixed point, called $\alpha$, is either non-repelling or a *cut-point* of the Julia set. ### Combinatorial models A quadratic polynomial $f$ is called *periodically hyperbolic* (resp., *repelling*) if it does not have neutral (resp., non-repelling) cycles. Note that in the peridically repelling case, $\operatorname{int}{\mathfrak K }=\emptyset$, so ${\mathfrak J }={\mathfrak K }$. The filled Julia set of a peridically hyperbolic map admits a locally connected *combinatorial model* ${\mathfrak K }^{\mathrm{com}}= {\mathfrak K }^{\mathrm{com}}(f)$ obtained by taking the quotient of the unit disk by the associated rational lamination (see [@L:book §32.1.3]). The combinatorial model is endowed with the induced dynamics $f_{\mathrm{com}}: {\mathfrak K }^{\mathrm{com}}\rightarrow{\mathfrak K }^{\mathrm{com}}$ which is semi-conjugate to $f$ by a natural projection $\pi: {\mathfrak K }\rightarrow{\mathfrak K }^{\mathrm{com}}$ whose *fibers* are "combinatorial classes" -- sets of points inseparable by preperiodic points. ### Hubbard continua {#sss:CombHT} In the superattracting case (when the crittical point is periodic) the *Hubbard tree* ${\mathfrak T}_f$ is defined as the smallest tree in ${\mathfrak K }$ containing the postcritical set such that ${\mathfrak T}_f$ intersects the components of $\operatorname{int}{\mathfrak K }$ along internal rays. It is invariant under $f$ and is marked with the orbit of $0$. If ${\mathfrak K }$ is peridically repelling and locally connected, we can define the *Hubbard dendrite* ${\mathfrak T}_f$ as the smallest connected closed forward-invariant subset of ${\mathfrak K }$ containing the postcritical set. For a general periodically repelling map, the (combinatorial) Hubbard dendrite ${\mathfrak T}_f^{\mathrm{com}}\subset {\mathfrak K }^{\mathrm{com}}$ is defined for its combinatorial model $f\colon {\mathfrak K }^{\mathrm{com}}\selfmap$. Lifting ${\mathfrak T}_f^{\mathrm{com}}$ via $\pi\colon {\mathfrak K }\to {\mathfrak K }^{\mathrm{com}}$, we obtain the *Hubbard continuum* ${\mathfrak T}_f$. For a periodically repelling $f$ and $x,y \in {\mathfrak J }_f={\mathfrak K }_f$, we define a *geodesic* continuum $$\label{eq:geod contin} [x,y]\coloneqq \pi^{-1}[\pi(x),\pi(y)],$$ where $[\pi(x),\pi(y)]$ is a unique arc in ${\mathfrak J }^{\mathrm{com}}$ connecting $\pi(x),\pi(y)$. ### QL-renormalization {#renorm sec} A quadratic-like map $f: X \rightarrow Y$ is called *DH renormalizable* (after Douady and Hubbard) if there is a quadratic-like restriction $$\label{eq:DH pre renorm} f_1=f_{1,0}= {\mathcal R}f= f^{p_1}: X_{1,0} \rightarrow Y_{1,0}$$ with connected Julia set ${\mathfrak K }_{0}^{[1]}$ such that the little Julia sets $$\label{eq:small J cycle} {\mathfrak K }^{[1]}_i\coloneqq f^i\left({\mathfrak K }_{0}^{[1]}\right), \hspace{1cm} k=0,\dots, p_1-1,$$ are either pairwise disjoint or else touch at their $\beta$-fixed points. In the former case the renormalization is called *primitive*, while in the latter it is called *satellite*. Note that there are many ql maps $f_1\colon X_{1,0}' \rightarrow Y'_{1,0}$ that satisfy the above requirements (if $f$ is renormalizable). However, all of them represent the same *ql germ*. The map $f_1$ in [\[eq:DH pre renorm\]](#eq:DH pre renorm){reference-type="eqref" reference="eq:DH pre renorm"} is called a *pre-renormalization* of $f$. If it is considered up to a linear rescaling, it is called the *renormalization* of $f$. (In what follows, we will often skip the prefix "-pre" as long as it does not lead to a confusion.) For every ${\mathfrak K }^{[1]}_i$, there are $Y_{1,i} \Supset X_{1,i}\Supset {\mathfrak K }^{[1]}_i$ such that $$\label{eq:f_1 i} f_1=f_{1,i}= {\mathcal R}f= f^{p_1}: X_{1,i} \rightarrow Y_{1,i}$$ is a ql map with non-escaping set ${\mathfrak K }^{[1]}_i={\mathfrak K }_{f_{1,i}}$. All maps [\[eq:f_1 i\]](#eq:f_1 i){reference-type="eqref" reference="eq:f_1 i"} represent conformally conjugate germs. We assume that ${\mathfrak K }^{[1]}_0$ contains the critical point of $f$; see [\[CrtPoint\]](#CrtPoint){reference-type="ref" reference="CrtPoint"}. ### Little copies of ${\mathcal M}$ {#sss:LittleCopies} The sets ${\mathfrak K }^{[1]}_i$ in [\[eq:small J cycle\]](#eq:small J cycle){reference-type="eqref" reference="eq:small J cycle"} are referred to as the *little (filled) Julia sets*. Their positions in the big Julia set ${\mathfrak K }_f$ determine the renormalization *combinatorics*. By the Douady-Hubbard Straightening Theorem [@DH:pol-like], the set of parameters $c$ for which the quadratic polynomial $f_c\colon z\mapsto z^2+c$ is renormalizable with a given combinatorics forms a *little Mandelbrot copy* ${\mathcal M}_1\subset {\mathcal{M}}$ (see [@L:book Theorem 43.1]). The renormalization combinatorics can be formally encoded by the Hubbard tree ${\mathfrak T}_{\circ}$ of the superattracting center $c_{\circ}$ of ${\mathcal M}_1$. Each little copy ${\mathcal M}_1$ can be canonically mapped onto the whole Mandelbrot set ${\mathcal M}$ by the *straightening homeomorphism* $\chi_1: {\mathcal M}_1\rightarrow{\mathcal M}$. A little Mandelbrot copy ${\mathcal M}_1$ is called *primitive* or *satellite* depending on the type of the corresponding renormalization. They can be easily distinguished as any satellite copy is attached to some hyperbolic component of $\operatorname{int}{\mathcal M}$ and does not have the cusp at its root point. ### Infinitely renormalizable maps {#sss:infin renorm} The notions of an infinitely DH renormalizable map $f$ with renormalization periods $p_n$, and its renormalizations $f_n= {\mathcal R}^n f$, are defined naturally. By default, we assume that $p_n$ is the *smallest renormalization period after $p_{n-1}$*. We will denote by ${\mathfrak K }^{[n]}_i, i\in \{0,1,\dots, p_n-1\}$ the level $n$ little Julia sets of $f$ enumerated dynamically so that ${\mathfrak K }^{[n]}_0$ contains the critical point of $f$. We will write $$\label{eq:bfilled} {\boldsymbol{\mathfrak K} }^{[n]}\coloneqq \bigcup_i {\mathfrak K }^{[n]}_i$$ The ratios $q_n:= p_n/p_{n-1}$ are called *relative periods*. One says that such a map has *bounded combinatorics of type $\bar p$* if the relative periods are bounded by $\bar p$. In this case, the map $f$ is called *Feigenbaum of type $\bar p$*. We say that a Feigenbaum map is *primitive/satellite* if all its renormalizations are such. A Feigenabaum map has *a priori* bounds if $$\label{eq: ql a priori bounds} \operatorname{mod}{\mathcal R}^n f\geq {\epsilon}>0$$ We say that the family ${\mathcal F}_{\bar p}$ of Feigenbaum maps of type $\bar p$ have *beau bounds* if there exists $\mu>0$ depending only on $\bar p$ such that for any $\nu>0$ there exists $n_0= n_0(\bar p, \nu)$ such that for any $f\in {\mathcal F}_{\bar p}$ with $\operatorname{mod}f\geq \nu$ we have $$\operatorname{mod}{\mathcal R}^n f \geq \mu\quad \text{for all }\ n\geq n_0.$$ ## Bushes {#ss:bushes} Consider a DH renormalizable quadratic-like map $f\colon X\to Y$. The *bush* of $f$ is $$\label{eq:dfn:bush} {\mathfrak B }_f\equiv {\mathfrak B }(f)\coloneqq {\mathfrak T}_f \cup {\boldsymbol{\mathfrak K} }^{[1]},$$ where ${\mathfrak T}_f$ is the Hubbard continuum and ${\boldsymbol{\mathfrak K} }^{[1]}$ is the periodic cycle of level one little Julia sets [\[eq:bfilled\]](#eq:bfilled){reference-type="eqref" reference="eq:bfilled"}. For $m \ge 0$, we define the *bush of height $m$* to be $$\label{eq:dfn:bush:preim} {\mathfrak B }^{(m)}={\mathfrak B }^{(m)}_f\coloneqq f^{-m}({\mathfrak B }_f).$$ ### Little bushes {#sss:LittleBushes} Suppose that $f$ is at least $n+1$ times DH renormalizable and let $f_n={\mathcal R}^n f$ be its $n$th renormalization of $f$. Then $f_n$ has a well defined bush ${\mathfrak B }(f_n)$. Consider the periodic cycle of little level $n$ filled Julia sets ${\mathfrak K }_i^{[n]}$ associated with $f_n$ in the dynamical plane of $f$. Let $f_{n,i}$ be the $n$th prerenormalization around ${\mathfrak K }_i^{[n]}$, compare [\[eq:f_1 i\]](#eq:f_1 i){reference-type="eqref" reference="eq:f_1 i"}. Then ${\mathfrak K }_i^{[n]}$ contains the little bush ${\mathfrak B }_i^{[n]}\equiv{\mathfrak B }(f_{n,i})\simeq {\mathfrak B }(f_n)$ as well as ${\mathfrak B }_i^{[n],(m)}\equiv{\mathfrak B }^{(m)}(f_{n,i})\simeq {\mathfrak B }^{(m)}(f_n)$. Note that ${\mathfrak B }^{[n],(m)}_i$ is the unique periodic lift of ${\mathfrak B }_{i+m}^{[n]}$ under $f^{m}$. We write $${\boldsymbol {\mathfrak B} }^{[n]}\coloneqq \bigcup_i {\mathfrak B }_i^{[n]} ,{\ \ }{\boldsymbol {\mathfrak B} }^{[n],(m)}\coloneqq f^{-1} \left({\boldsymbol {\mathfrak B} }^{[n]}\right),{\ \ }{\boldsymbol {\mathfrak B} }^{[n],(m)}_{\operatorname{per}}\coloneqq{\boldsymbol {\mathfrak B} }^{[n],(m)} \cap {\boldsymbol{\mathfrak K} }^{[n]}.$$ Observe that ${\boldsymbol {\mathfrak B} }^{[n],(m)}_{\operatorname{per}}$ is the union of periodic little bushes of height $m$. A *preperiodic bush* ${\mathfrak B }^{[n],(m)}_a$ is a non-periodic connected component of ${\boldsymbol {\mathfrak B} }^{[n],(m)}$. The *preperiod* of ${\mathfrak B }^{[n],(m)}_a$ is the smallest $s\le m$ such that $f^s \left( {\mathfrak B }^{[n],(m)}_a \right)$ is periodic. Every periodic or preperiodic little bush ${\mathfrak B }_{a}^{[n],(m)}$ is within a unique little periodic or preperiodic filled Julia set ${\mathfrak K }^{[n]}_a$ associated with $f_n$. The ${\mathfrak B }_{t}^{[n],(m)}$ exhaust ${\mathfrak K }^{[n]}_a$: $${\mathfrak K }^{[n]}_a = \overline{\bigcup_{m} {\mathfrak B }^{[n],(m)}_a}\ .$$ By construction, the ${\mathfrak B }^{[n],(m)}_a$ are pairwise disjoint but the ${\mathfrak K }^{[n]}_a$ may touch each other in the satellite case. ### Superattracting model {#sss:psib:super attr model} Consider a ql map $f\colon X\to Y$ and assume it is $n+1$ DH renormalizable. Then $f$ is hybrid equivalent to $z^2+c$, where $c$ is in a level $n+1$ little copy ${\mathcal M}^{[n+1]}_i\subset {\mathcal M}$. A *superattracting* model for $f$ of level $n+1$ is any ql map $f_\circ \colon X_\circ \to Y_\circ$ hybrid equivalent to the center of ${\mathcal M}^{[n+1]}_i$. It is well-known (follows, for instance, from the lamination theory) that the Hubbard continua ${\mathfrak T}_f$ and ${\mathfrak T}_\circ\equiv {\mathfrak T}_{f_\circ}$ are *combinatorially equivalent* up to ${\boldsymbol {\mathfrak B} }^{[n]}$: there is a bijection between - components of ${\mathfrak T}_f\setminus f^{-m}\left({\boldsymbol {\mathfrak B} }^{[n]}_f\right)$ and components of ${\mathfrak T}_\circ\setminus f^{-m}_\circ\left({\boldsymbol {\mathfrak B} }^{[n]}_\circ\right)$ for all $m\ge0$; and - components of $f^{-m}\left({\boldsymbol {\mathfrak B} }^{[n]}_f\right)$ and components of $f^{-m}_\circ\left({\boldsymbol {\mathfrak B} }^{[n]}_\circ\right)$ for all $m\ge0$ that respects the adjacency, natural embedding, and dynamical relations between respective components. In other words, the above components define equivalent Markov partitions for $f, f_\circ$. $$\begin{tikzpicture}[scale=1.3] \begin{scope}[rotate =0] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (ba) at (0.5,0); \node[below] at (0.2,-0.2) {${\mathfrak B }^{[1]}_{a}$}; \draw[dashed,line width=0.8,fill, fill opacity=0.05] (0.6,0) ellipse (2cm and 0.9cm); \node[] at (-0.75,0.2){${\mathfrak B }^{[1], (6p_2)}_{a}$}; \begin{scope}[shift={(1,2)},rotate=90,scale=0.7] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (xup) at (-0.5,0); \node[above] at (0.5,0) {${\mathfrak B }''_1$}; \end{scope} \begin{scope}[shift={(1,-2)},rotate=-90,scale=0.7] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (xdown) at (-0.5,0); \node[left] at (-0.8,0) {$\ell_1$}; \node[below] at (0.5,0) {${\mathfrak B }'_1$}; \end{scope} \draw[] (xup) -- (xdown); \begin{scope}[shift={(-0.05,0)}] \draw [blue, line width=0.8mm] (1.1,0)--(1.4,0) (1.25,-0.1) -- (1.25,0.1) ; \node[blue, below] at(1.29,-0.1) {${\mathfrak B }_i^{[2]}$}; \end{scope} \begin{scope}[shift={(0.55,0)}] \draw [blue, line width=0.8mm] (1.1,0)--(1.4,0) (1.25,-0.1) -- (1.25,0.1) ; \node[blue, below] at(1.25,-0.1) {${\mathfrak B }_{i+1}^{[2]}$}; \end{scope} \begin{scope}[shift={(1,0)}] \begin{scope}[shift={(1,2)},rotate=90,scale=0.7] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (xup) at (-0.5,0); \node[above] at (0.5,0) {${\mathfrak B }''_2$}; \end{scope} \begin{scope}[shift={(1,-2)},rotate=-90,scale=0.7] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (xdown) at (-0.5,0); \node[below] at (0.5,0) {${\mathfrak B }'_2$}; \node[left] at (-0.8,0) {$\ell_2$}; \end{scope} \draw[] (xup) -- (xdown); \end{scope} \end{scope} \begin{scope}[rotate around={45:(4,0)} ,shift={(7,0)},xscale=-1] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (bb) at (0.5,0); \node[below] at (0.15,-0.2) {${\mathfrak B }^{[1]}_{b}$}; \end{scope} \draw[] (ba)--(4,0)--(bb); \begin{scope}[rotate around={-45:(4,0)}] \coordinate (cc) at (6,0); \end{scope} \draw[dashed] (4,0)--(cc); \filldraw (4,0) circle (0.04 cm); \node[above] at (4,0) {$\alpha$}; \end{tikzpicture}$$ ### Satellite combinatorics Suppose that $f$ is at least $3$ times renormalizable and the first renormalization of $f$ is satellite. Then all ${\mathfrak K }_i^{[1]}$ are organized in the *satellite flower* around the $\alpha$ fixed point of $f$: $${\mathfrak B }_f={\boldsymbol{\mathfrak K} }^{[1]}_f\hspace{1cm} \text{ because }\hspace{1cm} {\boldsymbol{\mathfrak K} }^{[1]}_f\supset {\mathfrak T}_f,$$ where ${\mathfrak T}_f$ is the Hubbard continuum of $f$. For $s\ge 0$ we write ${\mathfrak T}^{(s)}_f = f^{-s}({\mathfrak T}_f)$. As before, the $p_k$ are the periods of level $k$ Julia sets. **Lemma 3** (Satellite flower). *For a satellite $f$ as above, let ${\mathfrak B }^{[1]}_{a}, {\mathfrak B }^{[1]}_{b}$ be its two level $1$ bushes. Then the continuum ${\mathfrak T}^{(6p_2)}_f$ contains geodesic continua $\ell_1$ and $\ell_2$ (as in §[2.2.2](#sss:CombHT){reference-type="ref" reference="sss:CombHT"}) connecting preperiodic lifts ${\mathfrak B }'_1,{\mathfrak B }''_1$ and ${\mathfrak B }'_2,{\mathfrak B }''_2$ of ${\mathfrak B }^{[1]}_b$ of preperiods $\le 6 p_2$ such that, see Figure [\[Fg:lem:bush:prim comb\]](#Fg:lem:bush:prim comb){reference-type="ref" reference="Fg:lem:bush:prim comb"}:* - *$\ell_1$ and $\ell_2$ separate ${\mathfrak B }^{[1]}_a$ from $\alpha$ (within ${\mathfrak K }_f$),* - *the geodesic continuum $\ell\subset {\mathfrak T}_f$ connecting $\ell_1, \ell_2$ intersects little preperidic bushes ${\mathfrak B }_i^{[2]},{\mathfrak B }^{[2]}_{i+1}$of level $2$ with preperiods less than $6p_1$;* - *${\mathfrak B }_i^{[2]},{\mathfrak B }^{[2]}_{i+1}$ are disjoint from $\ell_1\cup \ell_2$.* We remark that $\ell,{\mathfrak B }_i^{[2]},{\mathfrak B }^{[2]}_{i+1}$ are within ${\mathfrak B }^{[1],(6 p_2)}_a$. *Proof.* Let ${\mathfrak B }_\gamma^{[2]}\subset {\mathfrak B }^{[1]}_a$ be the level $2$ periodic bush closest to $\alpha$; i.e. $({\mathfrak T}_f\cap {\mathfrak B }^{[1]}_a)\setminus {\mathfrak B }^{[2]}_\gamma$ and $\alpha$ are in different components of ${\mathfrak T}_f\setminus {\mathfrak B }_\gamma^{[2]}$. Let the ${\mathfrak B }_t^{[2]}\subset {\mathfrak B }^{[1],(tp_1)}_a$ be the lifts of ${\mathfrak B }_\gamma^{[2]}$ under $f^{t p_1}$ towards $\alpha$; i.e. each ${\mathfrak B }_t^{[2]}$ separate $\alpha$ from ${\mathfrak B }^{[1],(tp_1-p_1)}_a$. Since $f^{6p_2}$ has critical points in ${\mathfrak B }^{[2]}_1$ and in ${\mathfrak B }_4^{[2]}$, we can select $\ell_1$ and $\ell_2$ passing through these critical points such that $\ell_1,\ell_2$ connect preperiodic lifts ${\mathfrak B }'_1,{\mathfrak B }''_1$ and ${\mathfrak B }'_2,{\mathfrak B }''_2$ of ${\mathfrak B }^{[1]}_b$. Then $\ell_1, \ell_2$ separate ${\mathfrak B }^{[1]}_a$ from $\alpha$, and $\ell_1\cup \ell_2$ separate ${\mathfrak B }^{[2]}_2,{\mathfrak B }^{[2]}_3$ from $\alpha$ and ${\mathfrak B }^{[1]}_a$; i.e. we can take ${\mathfrak B }^{[2]}_i,{\mathfrak B }^{[2]}_{i+1}$ to be ${\mathfrak B }^{[2]}_2,{\mathfrak B }^{[2]}_3$. ◻ ## Invariant arc diagrams {#sss:InvArcDiagr} In this subsection, we will discuss invariant up to homotopy arc diagrams of ql maps. Arc diagrams endowed with weights will naturally appear from the thin-thick decompositions of the dynamical planes of $\psi^\bullet$-ql maps, see §[3.3](#ss:WAD){reference-type="ref" reference="ss:WAD"}, §[3.5](#ss:Width+WAD){reference-type="ref" reference="ss:Width+WAD"}. ### Arc diagrams {#sss:AD} Consider a hyperbolic open Riemann surface $S$ of finite type without cusps. We endow $S$ with its ideal boundary $\partial^i S$. This naturally makes $V\coloneqq \overline S\equiv S\cup \partial^i S$ a compact surface. A *path* (closed or open) $\ell$ in $S$ is an embedded (closed or open) interval $\ell\colon I \to S$. An open path $\gamma\colon (0,1)\to S$ is *proper* if it extends to $\gamma\colon [0,1]\to V\equiv S\cup \partial^i S$ with $\gamma\{0,1\}\subset \partial^i S$. Two proper paths $\gamma_0,\gamma_1$ in $S$ are homotopic if there is a homotopy $\gamma_t$ among proper paths. Similarly, *proper curves* and their homotopy are defined. An *arc* on $S$ is a class of properly homotopic paths, $\alpha=[\gamma]$. A curve $\gamma$ is *trivial* if it can be represented in an arbitrary small neighborhood of $\partial^i S$. Two different arcs are *non-crossing* if they can be represented by non-crossing paths. An *arc diagram* (AD) is a family of non-trivial pairwise non-crossing arcs $A=\{\alpha_i\}$. A *weighted arc diagram* (WAD) ${\mathcal A}=\sum_{ \alpha_i \in A} w_i \alpha_i,\ w_i\in {\Bbb R}_{>0}$ is an arc diagram endowed with positive weights. ### Arc diagrams of ql maps Consider a ql map $f\colon X\to Y$ and assume that it is $n+1$ times DH renormalizable. An arc diagram $A=\{\alpha_i\}$ on $X\setminus {\boldsymbol {\mathfrak B} }^{[n]}$ is *horizontal* if every $\alpha_i$ connects components of ${\boldsymbol {\mathfrak B} }^{[n]}$. A horizontal arc diagram $A=\{\alpha_i=[\gamma_i]\}$ is called *invariant* if every $\alpha_i$ can be represented up to a proper homotopy in $X\setminus {\boldsymbol {\mathfrak B} }^{[n]}$ in an arbitrary small neighborhood of $f^{-1}\left({\boldsymbol {\mathfrak B} }^{[n]}\cup \Gamma \right)$, where $\Gamma=\bigcup_i \gamma_i$. In other words, every $[\gamma]\in A$ can be presented as a concatenation $$\label{eq:form for gamma:base} \gamma = \ell_0\#\gamma_1\# \ell_1 \#\gamma_2\# \dots \#\gamma_s\#\ell_s,$$ where - $[\gamma_j]\in f^*(A)$ are proper arcs in $X\setminus {\boldsymbol {\mathfrak B} }^{[n],(1)}=f^{-1}\left(Y\setminus {\boldsymbol {\mathfrak B} }^{[n]}\right)$; and - every component of $\ell_i\setminus {\boldsymbol {\mathfrak B} }^{[n],(1)}$ is trivial in $X\setminus {\boldsymbol {\mathfrak B} }^{[n],(1)}$ (with respect to a proper homotopy, see §[2.4.1](#sss:AD){reference-type="ref" reference="sss:AD"}). (In other words, the $\ell_i$ are contractible into ${\boldsymbol {\mathfrak B} }^{[n],(1)}$.) Figure [\[Fig:exm Inv Arc Diagr\]](#Fig:exm Inv Arc Diagr){reference-type="ref" reference="Fig:exm Inv Arc Diagr"} gives an example of an invariant arc diagram. $$\begin{tikzpicture}[xscale=2,yscale=1.5] \begin{scope}[rotate =0,shift={(1.3,0)},scale=0.6] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (la1) at (-0.2,0.2); \coordinate (la0) at (-0.2,-0.2); \draw[line width=0.4mm] (4,0.7) edge[->,bend right]node[above]{$f$} (0.7,0.7); \node[above,scale=1.2] at(0,0.2){${\mathfrak B }^{[1]}_0$}; \end{scope} \begin{scope}[rotate =120,shift={(1.3,0)}, scale=0.6] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (lb1) at (-0.2,0.2); \coordinate (lb0) at (-0.2,-0.2); \end{scope} \begin{scope}[rotate =240,shift={(1.3,0)}, scale=0.6] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); %\node[above,scale=1.2] at(0,0.2){$\bush_0$}; \coordinate (lc1) at (-0.2,0.2); \coordinate (lc0) at (-0.2,-0.2); \end{scope} \draw[line width=0.3mm, red] (la1) edge[bend left=10] node[above] {$\alpha$} (lb0); \draw[line width=0.3mm, blue] (lc0) edge[bend right=10] node[left] {$\beta$} (lb1); \begin{scope}[shift={(4.5,0)}] \begin{scope}[rotate =0,shift={(1.3,0)},scale=-0.6] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \draw[line width=0.9mm,gray] (0.5,0) -- (0.9,0); \draw[line width=0.9mm,gray] (0.65,-0.2) -- (0.65,0.2); \coordinate (ra1) at (0.65,-0.2) ; \coordinate (ra0) at (0.65,0.2) ; \node[above,scale=1.2] at(0,-0.2){${\mathfrak B }^{[1],(1)}_0$}; \end{scope} \begin{scope}[rotate =120,shift={(1.3,0)}, scale=0.6] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (rb1) at (-0.2,0.2); \coordinate (rb0) at (-0.2,-0.2); \end{scope} \begin{scope}[rotate =240,shift={(1.3,0)}, scale=0.6] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (rc1) at (-0.2,0.2); \coordinate (rc0) at (-0.2,-0.2); \end{scope} \draw[line width=0.3mm, red] (rc1) edge[bend left=10] node[above] {$\tilde \alpha$} (ra0); \draw[line width=0.3mm, blue] (rb0) edge[bend right=10] node[above] {$\tilde \beta$} (ra1); \end{scope} \end{tikzpicture}$$ **Lemma 4** (Alignment with ${\mathfrak T}_f$, following [@K §4]). *If $A=\{\alpha_i\}$ is an invariant horizontal AD on $X\setminus {\boldsymbol {\mathfrak B} }^{[n]}$, where $f\colon X\to Y$ is a ql map, then $A$ is *aligned with the Hubbard continuum* ${\mathfrak T}_f$: every $\alpha_i$ can be represented in an arbitrary small neighborhood of a geodesic continuum $T_i\subset {\mathfrak T}_f\setminus {\boldsymbol {\mathfrak B} }^{[n]}$ connecting components of ${\boldsymbol {\mathfrak B} }^{[n]}$.* *Proof.* Write $\alpha_i=[\gamma_i]$ and consider $Y'\coloneqq Y\setminus \left({\boldsymbol {\mathfrak B} }^{[n]}\cup \Gamma \right)$, where $\Gamma=\bigcup_i \gamma_i$. Then one of the little bushes ${\mathfrak B }^{[n]}_i$ is accessible from the outermost component of $\partial Y'$. Therefore, we can select a proper path $\ell \subset Y'$ connecting $\partial Y$ and ${\boldsymbol {\mathfrak B} }^n$. For every such a path $\ell$, its *legal pullback* $\widetilde \ell$ is any of its lift connecting ${\boldsymbol {\mathfrak B} }^{[n]}$ and $\partial X$ and concatenated by a path in $Y\setminus X$ so that $\widetilde \ell\subset Y\setminus {\boldsymbol {\mathfrak B} }^{[n]}$ is a proper path connecting $\partial Y$ and ${\boldsymbol {\mathfrak B} }^{n}$. Since $A$ is invariant, $\widetilde \ell$ can be again represented in $Y'$ up to a proper homotopy in $Y\setminus {\boldsymbol {\mathfrak B} }^{[n]}$. It is well-known that iterated pullbacks of $\ell$ can realize all periodic rays landing at ${\boldsymbol {\mathfrak B} }^{[n]}$ up to a proper homotopy in $Y\setminus {\boldsymbol {\mathfrak B} }^{[n]}$. (It is sufficient to verify the property for the superattracting model §[2.3.2](#sss:psib:super attr model){reference-type="ref" reference="sss:psib:super attr model"}.) Therefore, $A$ is aligned with ${\mathfrak T}_f$. ◻ ### Genuine periodic arcs {#sss:periodic arcs} Let $A=\{\alpha_i\}$ be a horizontal invariant AD on $Y\setminus {\boldsymbol {\mathfrak B} }^{[n]}$ of $f\colon X\to Y$. For every $m\ge 1$, we can present $[\gamma]\in A$ as a concatenation $$\label{eq:form for gamma} \gamma = \ell_0\#\gamma_1\# \ell_1 \#\gamma_2\# \dots \#\gamma_s\#\ell_s$$ (compare with [\[eq:form for gamma:base\]](#eq:form for gamma:base){reference-type="eqref" reference="eq:form for gamma:base"}), where - $[\gamma_j]\in\big( f^{m}\big)^*(A)$ are proper arcs in $f^{-m}(Y)\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}=f^{-m}\left(Y\setminus {\boldsymbol {\mathfrak B} }^{[n]}\right)$; and - every component of $\ell_i\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$ is trivial in $X\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$ (with respect to a proper homotopy, see §[2.4.1](#sss:AD){reference-type="ref" reference="sss:AD"}) . We call [\[eq:form for gamma\]](#eq:form for gamma){reference-type="eqref" reference="eq:form for gamma"} the *decomposition of $\gamma$ rel ${\boldsymbol {\mathfrak B} }^{[n],(m)}$.* We define the *expansivity number* $${\operatorname{EN}}_A(f^m, [\gamma])\coloneqq \min \{s \mid \ s \text{ is in \eqref{eq:form for gamma}}\},$$ where the minimum is taking over all paths possible $\gamma$ as above. We call an arc $\alpha\in A$ a *genuine periodic* if ${\operatorname{EN}}_A(f^m,\alpha)=1$ for all $m\ge 1$. $$\begin{tikzpicture}[xscale=2,yscale=1.5] \begin{scope}[rotate =0,shift={(1.3,0)},scale=0.6] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (la1) at (-0.2,0.2); \coordinate (la0) at (-0.2,-0.2); \draw[line width=0.4mm] (4,0.7) edge[->,bend right]node[above]{$f$} (-0.,1.8); \node[above,scale=1.2] at(0,0.2){${\mathfrak B }^{[1]}_0$}; \end{scope} \begin{scope}[rotate =90,shift={(1.3,0)}, scale=0.6] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (lb1) at (-0.2,0.2); \coordinate (lb0) at (-0.2,-0.2); \coordinate (lbb) at (-0.5,0); \end{scope} \begin{scope}[rotate =180,shift={(1.3,0)}, scale=0.6] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); %\node[above,scale=1.2] at(0,0.2){$\bush_0$}; \coordinate (lc1) at (-0.2,0.2); \coordinate (lc0) at (-0.2,-0.2); \end{scope} \begin{scope}[rotate =270,shift={(1.3,0)}, scale=0.6] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); %\node[above,scale=1.2] at(0,0.2){$\bush_0$}; \coordinate (ld1) at (-0.2,0.2); \coordinate (ld0) at (-0.2,-0.2); \coordinate (ldd) at (-0.5,0); \end{scope} \draw[line width=0.3mm, red] (la1) edge[bend left=10] node[above right] {$\alpha_1$} (lb0); \draw[line width=0.3mm, red] (lb1) edge[bend left=10] node[above left] {$\alpha_2$} (lc0); \draw[line width=0.3mm, red] (lc1) edge[bend left=10] node[below left] {$\alpha_3$} (ld0); \draw[line width=0.3mm, red] (ld1) edge[bend left=10] node[below right] {$\alpha_4$} (la0); % \draw[line width=0.3mm, blue] (lc0) edge[bend right=10] node[left] {$\beta$} (lb1); \draw[line width=0.3mm, blue] (lbb) edge[] node[left] {$\beta$} (ldd); \begin{scope}[shift={(4.5,-1)}] \begin{scope}[rotate =0,shift={(1.3,0)},scale=-0.6] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \draw[line width=0.9mm,gray] (0.5,0) -- (0.9,0); \draw[line width=0.9mm,gray] (0.65,-0.2) -- (0.65,0.2); \coordinate (ra1) at (0.65,-0.2) ; \coordinate (ra0) at (0.65,0.2) ; \coordinate (raa) at (0.9,0) ; \node[above,scale=1.2] at(0,-0.2){${\mathfrak B }^{[1],(1)}_0$}; \end{scope} \begin{scope}[rotate =90,shift={(1.3,0)}, scale=0.6] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (rb1) at (-0.2,0.2); \coordinate (rb0) at (-0.2,-0.2); \coordinate (rbb) at (-0.5,0); \end{scope} \begin{scope}[rotate =180,shift={(1.3,0)}, scale=0.6] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (rc1) at (-0.2,0.2); \coordinate (rc0) at (-0.2,-0.2); \coordinate (rcc) at (-0.5,0); \end{scope} \begin{scope}[rotate =270,shift={(1.3,0)}, scale=0.6] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (rd1) at (-0.2,0.2); \coordinate (rd0) at (-0.2,-0.2); \end{scope} \draw[line width=0.3mm, blue] (raa) edge node[blue,above ] {$\tilde \beta$} (rcc); \draw[line width=0.3mm, red] (rb0) edge[bend right=10] node[above right] {$\tilde \alpha_2$} (ra1); \draw[line width=0.3mm, red] (rc0) edge[bend right=10] node[above left] {$\tilde \alpha_3$} (rb1); \draw[line width=0.3mm, red] (rd0) edge[bend right=10] node[below left] {$\tilde \alpha_4$} (rc1); \draw[line width=0.3mm, red] (ra0) edge[bend right=10] node[below right] {$\tilde \alpha_1$} (rd1); \end{scope} \end{tikzpicture}$$ **Lemma 5** (Expansivity of $f\mid {\big[{\mathfrak T}_f\setminus {\boldsymbol {\mathfrak B} }^{[n]}\big]}$). *Consider an invariant AD $A$ and an arc $\gamma\in A$ aligned with a proper geodesic continuum $T_\gamma$ of ${\mathfrak T}_f\setminus {\boldsymbol {\mathfrak B} }^{[n]}$. Then* - *$\gamma$ is genuine periodic if and only if $T_\gamma$ connects two neighboring bushes ${\mathfrak B }^{[n]}_a, {\mathfrak B }^{[n]}_{a+1}$ (with respect to the cyclic order) of a periodic satellite flower ${\mathfrak B }^{[n-1]}_j\supset {\mathfrak B }^{[n]}_a, {\mathfrak B }^{[n]}_{a+1}$ of level $n-1$. In particular, the $n$-th renormalization of $f$ is satellite.* - *if $\gamma$ is not genuine periodic, then $$\label{eq:lem:per arcs} {\operatorname{EN}}_A(f^{p_n}, \gamma) \ge 2.$$* See Figure [\[Fig:PeriodicArcs\]](#Fig:PeriodicArcs){reference-type="ref" reference="Fig:PeriodicArcs"} for illustration. *Proof.* Clearly, if $T_\gamma$ is in a satellite flower ${\mathfrak B }^{[n-1]}_j$ and connects two neighboring bushes ${\mathfrak B }^{[n]}_a,\ {\mathfrak B }^{[n]}_{a+1}$, then $\gamma$ is genuine periodic. It is also clearly that [\[eq:lem:per arcs\]](#eq:lem:per arcs){reference-type="eqref" reference="eq:lem:per arcs"} holds unless $T_\gamma$ is in a satellite flower ${\mathfrak B }^{[n-1]}_j$. (It can be easily proven for a superattracting model §[2.3.2](#sss:psib:super attr model){reference-type="ref" reference="sss:psib:super attr model"}.) Assume $T_\gamma$ is in a satellite flower ${\mathfrak B }^{[n-1]}_j$ but connects two non-neighboring bushes ${\mathfrak B }^{[n]}_a,\ {\mathfrak B }^{[n]}_b,\ |a-b|>1$. Then a certain lift $T'$ of $T_\gamma$ under $f^{m}$ with $m=kp_{n-1}<p_n$ cross-intersects $T_\gamma$. It follows that $\big(f^{m}\big)^* A$ has an arc aligned with $T'$ but has no arc aligned with $T_\gamma\setminus \left( {\mathfrak B }^{[n],(m)}_a\cup {\mathfrak B }^{[n],(m)}_b\right)$. This implies that ${\operatorname{EN}}_A(f^{m}, \gamma) \ge 2$. ◻ ### Arcs inside and outside of satellite flowers {#sss:in out: sat flowers} Consider an invariant AD $A$ on $X\setminus {\boldsymbol {\mathfrak B} }^{[n]}$. Consider an arc $\alpha\in A$ aligned with a geodesic continuum $T_\alpha\subset {\mathfrak T}_f\setminus {\boldsymbol {\mathfrak B} }^{[n]}$. We say that $\alpha$ is in ${\mathfrak B }^{[n-1]}_c$ if $T_\alpha\subset {\mathfrak B }^{[n-1]}_c$. **Lemma 6**. *Consider an invariant arc diagram $A$ on $X\setminus {\boldsymbol {\mathfrak B} }^{[n]}$ and an arc $\alpha\in A$. Assume that $\alpha$ is not in any satellite flower ${\mathfrak B }^{[n-1]}_c$ of level $n-1$. Then there are two arcs $$\alpha_1,\alpha_2\in (f^{2p_n})^* (A)\hspace{0.6cm} \text{ with }{\ \ }f^{2p_n}(\alpha_1)=f^{2p_n}(\alpha_2)=\alpha_{\operatorname{new}}\in A$$ such that $\alpha$ overflows $\alpha_1, \alpha_2$ in the following sense: any decomposition [\[eq:form for gamma\]](#eq:form for gamma){reference-type="ref" reference="eq:form for gamma"} of $\alpha=[\gamma]$ rel ${\boldsymbol {\mathfrak B} }^{[n],(2p_n)}$ contains two paths $\gamma_a, \gamma_b$ representing $\alpha_1=[\gamma_a]$ and $\alpha_2=[\gamma_b]$.* *Proof.* Since $T_\alpha$ is not in any satellite flower ${\mathfrak B }^{[n-1]}_c$, there is a strictly preperiodic component ${\mathfrak B }^{[n],(p_n)}_d$ of ${\boldsymbol {\mathfrak B} }^{[n],(p_n)}$ intersecting $T_\alpha$. There are two components $T_1,T_2\subset T_\alpha\setminus {\boldsymbol {\mathfrak B} }^{[n],(2p_n)}$ adjacent to ${\mathfrak B }^{[n],(2p_n)}_d$. Since there is an $m\le 2p_n$ such that $f^m\big({\mathfrak B }^{[n],(2p_n)}_d\big)$ contains the critical value, $T_1,T_2$ have a common injective image of generation $m\le 2p_n$. There must be two arcs $\alpha_1,\alpha_2\in (f^{2p_n})^* (A)$ aligned with $T_1,T_2$. ◻ Assume that the $n$th renormalization is satellite and consider a satellite flower ${\mathfrak B }^{[n-1]}_c$. Let $A$ be an invariant AD as above. Let $A_c=A({\mathfrak B }^{[n-1]}_c)\subset A$ be the AD consisting of all arcs from $A$ that are in ${\mathfrak B }^{[n-1]}_c$. Similarly, $A^{(p_n)}_c\subset (f^{p_n})^*(A)$ be the AD consisting of all arcs from $(f^{p_n})^*(A)$ that are in ${\mathfrak B }^{[n-1]}_c$. Then $$\label{eq:invarirance:A_c} (f^{p_n})^*\colon A_c^{p_n}\overset{1:1}{\longrightarrow} A_c$$ is a bijection and every arc $\alpha\in A_c$ is homotopic to its unique lift $\tilde \alpha\in A_c^{p_n}, (f^{p_n})^*(\tilde \alpha)=\alpha$ in the following sense: $T_{\alpha}\setminus {\boldsymbol {\mathfrak B} }^{[n],(p_n)}=T_{\tilde \alpha}$. ### Inhomogeneous configuration {#sss:mixed conf} Discussion here will be only used in Section [7](#s:conclusions){reference-type="ref" reference="s:conclusions"}. Consider a ql map $f\colon X\to Y$ with a cycle of bushes ${\boldsymbol {\mathfrak B} }^{[1]}$. Write ${\boldsymbol {\Upsilon} }\coloneqq {\boldsymbol {\mathfrak B} }^{[1]}\cup {\mathfrak K }^{[1]}_0$. Note that $f$ does not permute components of ${\boldsymbol {\Upsilon} }$. An AD $A=\{\alpha_i\}$ on $X\setminus {\boldsymbol {\Upsilon} }$ is *horizontal* if every $\alpha_i$ connects components of ${\boldsymbol {\Upsilon} }$. A horizontal AD $A=\{\alpha_i=[\gamma_i]\}$ is *$f^{p_1}$-invariant* if every $\alpha_i$ can be represented up to a proper homotopy in $X\setminus {\boldsymbol {\Upsilon} }$ in an arbitrary small neighborhood of $f^{-p_1}\left({\boldsymbol {\Upsilon} }\cup \Gamma \right)$, where $\Gamma=\bigcup_i \gamma_i$. By replacing $f$ with its superattracting model (§[2.3.2](#sss:psib:super attr model){reference-type="ref" reference="sss:psib:super attr model"}, compare also wit Lemma [Lemma 4](#lem:Alignment){reference-type="ref" reference="lem:Alignment"}), we obtain: **Lemma 7** (Alignment with ${\mathfrak T}_f$). *If $A=\{\alpha_i\}$ is an $f^{p_1}$-invariant AD on $Y\setminus {\boldsymbol {\Upsilon} }$ as above, then $A$ is *aligned with the Hubbard continuum* ${\mathfrak T}_f$: every $\alpha_i$ can be represented in an arbitrary small neighborhood of a geodesic continnum $T_i$ of ${\mathfrak T}_f\setminus {\boldsymbol {\Upsilon} }$ connecting components of ${\boldsymbol {\Upsilon} }.$0◻* For $m\ge 1$ and $\alpha\in A$, the *expansivity number* ${\operatorname{EN}}_A(f^{mp_1}, \alpha)$ is defined in the same way as in § [2.4.3](#sss:periodic arcs){reference-type="ref" reference="sss:periodic arcs"}; i.e., it is the smallest number of arcs in $f^{-p_1m}\left( A\right)$ overflown by $\alpha$. An arc $\alpha\in A$ is *genuine periodic* if ${\operatorname{EN}}_A(f^{p_1m},\alpha)=1$ for all $m\ge 1$. By the same argument as in Lemma [Lemma 5](#lem:per arcs){reference-type="ref" reference="lem:per arcs"}, we have: **Lemma 8** (Expansivity of $f\mid [{\mathfrak T}_f\setminus {\boldsymbol {\Upsilon} }]$). *Consider an $f^{p_1}$-invariant AD $A$ on $Y\setminus {\boldsymbol {\Upsilon} }$ and an arc $\gamma\in A$ aligned with a proper geodesic continuum $T_\gamma$ of ${\mathfrak T}_f\setminus {\boldsymbol {\Upsilon} }$. If the first renormalization of $f$ is primitive, then $$\label{eq:lem:per arcs:ups} {\operatorname{EN}}_A(f^{p_1}, \gamma) \ge 2.$$0◻* ## Teichmüller contraction {#ss:Teichm contra} The Teichmüller contraction comes from the observation that the restriction of a qc conjugacy to deeper renormalization levels can only decrease the dilatation. Therefore, the renormalization orbits (for bounded-type Feigenbaum families) can escape to infinity with at most linear rate. This fact can be traced back to Sullivan [@S:Berkeley], where structures of the Teichmüller spaces (reminiscent to the Thurston's machinery) were brought into the Renormalization Theory. The proposition below is stated for quadratic maps; the Straightening Theorem easily extends it quadratic-like maps. **Proposition 9**. *For every combinatorial bound $\bar p$, there is a constant $\Delta=\Delta_{\overline p}>1$ such that the following holds. Let $f_c(z)= z^2+c$ be an infinitely renormalizable quadratic polynomial of bounded type $\overline p$, see §[2.2.5](#sss:infin renorm){reference-type="ref" reference="sss:infin renorm"}. Then $g_n={\mathcal R}^n f_c$ has ql prerenormalization $g_n=f_c^{p_n}\colon X_n\to Y_n$, $X_n\Subset Y_n$ such that* 1. *${\mathcal W}(Y_n\setminus X_n)=O( \Delta^n)$; and[\[prop:Teichm contr:Cond 1\]]{#prop:Teichm contr:Cond 1 label="prop:Teichm contr:Cond 1"}* 2. *$Y_n\setminus {\mathfrak K }_{g_n}$ is disjoint from ${\boldsymbol {\mathfrak B} }^{[n]}_{f_c}$.[\[prop:Teichm contr:Cond 2\]]{#prop:Teichm contr:Cond 2 label="prop:Teichm contr:Cond 2"}* *Proof.* Let ${\mathcal M}_i, i \in I$ be the (finite) set of all maximal ${\mathcal M}$-copies in ${\mathcal M}$ of period $\le \bar p$, and let $$R\colon \bigcup _{i\in I}{\mathcal M}_i\to {\mathcal M}$$ be the ql straightening map. Let us remove from every satellite ${\mathcal M}_i$ a small open neighborhood of its cusp such that the remaining ${\mathcal M}^\circ_i$ contains all the preimages of the ${\mathcal M}_j$ under $R\colon {\mathcal M}_i\to {\mathcal M}$. For a primitive copy ${\mathcal M}_i$, set ${\mathcal M}_i^\circ\coloneqq {\mathcal M}_i$. By compactness, there is a $\Delta>1$ such that every $f_c$ with $c\in \bigcup _i {\mathcal M}_i^\circ$ has a ql restriction $g_1=f_c^{p_1}\colon X_{1,c}\to Y_{1,c}$ around its maximal little Julia set containing $0$ such that $g_1$ satisfies [\[prop:Teichm contr:Cond 2\]](#prop:Teichm contr:Cond 2){reference-type="ref" reference="prop:Teichm contr:Cond 2"} and such that $g_1$ is hybrid conjugate via $h_{c}\colon Y_{1,1}\to Y_{R(c)}$ with dilatation $\le \Delta$ to a ql restriction $f_{R(c)}\colon X_{R(c)}\to Y_{R(c)}$. Moreover, we can select the $Y_{1,c}$ and $Y_c$ so that $Y_c\Supset Y_{1,c}$. Wright $c_n \coloneqq R^n(c)$. Then $$\bar h\coloneqq \big(h_{c_{n-1}}\circ \dots \circ h_{c_1}\circ h_{c}\big)^{-1}$$ is a hybrid conjugacy from $f_{c_n}\colon X_{c_n}\to Y_{c_n}$ to a ql restriction $g_n=f^{p_1\dots p_n}_c\colon X_{n,c}\to Y_{n,c}$ satisfying [\[prop:Teichm contr:Cond 2\]](#prop:Teichm contr:Cond 2){reference-type="ref" reference="prop:Teichm contr:Cond 2"}. Since the dilatation of $\overline h$ is $\le \Delta^n$, Property [\[prop:Teichm contr:Cond 1\]](#prop:Teichm contr:Cond 1){reference-type="ref" reference="prop:Teichm contr:Cond 1"} follows. ◻ **Remark 10**. *Property [\[prop:Teichm contr:Cond 2\]](#prop:Teichm contr:Cond 2){reference-type="ref" reference="prop:Teichm contr:Cond 2"} implies that the prerenormalization $g_n\colon X_n\to Y_n$ is unbranched: ${\mathfrak P }(f_c)\cap Y_n\subset {\mathfrak K }(g_n)$, compare with [@McM2]. Moreover, by induction, $Y_n$ is disjoint from $$\Upsilon\coloneqq \bigcup_{m\le n} \left({\boldsymbol {\mathfrak B} }^{[m]}_f\setminus {\mathfrak B }^{[m]}_{f,0} \right).$$ This implies that the quadratic-like prerenormalizations $g_n\colon X_n\to Y_n$ can be univalently lifted to the dynamical plane of any $\psi^\bullet$ renormalization $${\mathcal R}^{n_1\bullet}\circ{\mathcal R}^{n_2\bullet}\dots\dots \circ {\mathcal R}^{n_s\bullet} (f_c)$$of $f_c$, see §[3.4.3](#sss:psi b renorm){reference-type="ref" reference="sss:psi b renorm"}.* # $\psi^\bullet$-ql renormalization and near-degenerate regime In this section, we will introduce $\psi^\bullet$-ql renormalization and discuss tools to detect its degeneration; see [@A; @K; @covering; @lemma; @L:book; @DL2] for details. Given a compact subset $K\Subset S$, we denote by ${\mathcal F}(S,K)$ the family of non-trivial proper curves in $S\setminus K$ emerging from $\partial S$. We write $$\label{eq:width:dfn} {\mathcal W}(S,K) ={\mathcal W}\big({\mathcal F}(S,K)\big).$$ ## Outline Consider a ql map $f\colon X\to Y$ and let ${\boldsymbol {\mathfrak B} }^{[n]}$ be its level $n$ cycle of little bushes. Around every periodic ${\mathfrak K }^{[n]}_i$ there is an associated ql prerenormalization $$\label{eq:f_n,i} f_{n,i}=f^{p_n}\colon X_{n,i}\to Y_{n,i}.$$ In §[3.4](#psi-renorm){reference-type="ref" reference="psi-renorm"}, we will define the $\psi^\bullet$-ql renormalization $$\label{eq:F_n,i} F_{n,i} = (f_{n,i},\ \iota_{n,i})\colon U_{n,i} \rightrightarrows V_{n,i}$$ of $f\colon X\to Y$ by *extending [\[eq:f_n,i\]](#eq:f_n,i){reference-type="eqref" reference="eq:f_n,i"} along all curves* in $Y\setminus {\boldsymbol {\mathfrak B} }^{[n]}$, see §[3.4.3](#sss:psi b renorm){reference-type="ref" reference="sss:psi b renorm"}. This is similar to the $\psi$-ql renormalization [@K], where the extension is performed along all curves in $Y\setminus {\boldsymbol{\mathfrak K} }^{[n]}$ under the assumption that the ${\mathfrak K }^{[n]}_i$ are pairwise disjoint (the primitive case); see §[3.4.1](#sss:psi renorm: motiv){reference-type="ref" reference="sss:psi renorm: motiv"} and Remark [Remark 15](#rem:general constr){reference-type="ref" reference="rem:general constr"}. The correspondence $F_{n,i}$ in [\[eq:F_n,i\]](#eq:F_n,i){reference-type="eqref" reference="eq:F_n,i"} is called a $\psi^\bullet$-ql map. It consists of a degree $2$ branched covering $f_{n,i}$ and an immersion $\iota_{n,i}$, see definitions in §[3.4.2](#sss:qlb:Defn){reference-type="ref" reference="sss:qlb:Defn"}. We define the Julia set and bush of $F_{n,i}$ to be that of the ql map $f_{n,i}$ [\[eq:f_n,i\]](#eq:f_n,i){reference-type="eqref" reference="eq:f_n,i"}: $${\mathfrak K }_{F_{n,i}}\coloneqq {\mathfrak K }_{f_{n,i}}{\ \ }{\ \ }\text{ and }{\ \ }{\ \ }{\mathfrak B }_{F_{n,i}}\coloneqq {\mathfrak B }_{f_{n,i}} .$$ Similarly, ${\mathfrak K }_F$ and ${\mathfrak B }_F$ are defined for any $\psi^\bullet$-ql map, see [\[eq:dfn:non-esc b\]](#eq:dfn:non-esc b){reference-type="eqref" reference="eq:dfn:non-esc b"}. A $\psi^\bullet$-ql renormalization can be naturally defined for a $\psi^\bullet$-ql map; i.e., $\psi^\bullet$-ql renormalization can be iterated. Sup-Chain Rule for renormalization domains is stated in §[3.4.4](#sss:sup chain rule){reference-type="ref" reference="sss:sup chain rule"}. Just like a ql map $f\colon X\to Y$ can be restricted to $f\colon X^{k+1}\to X^k$, where $X^{k+1}=f^{-k}(X)$, a $\psi^\bullet$-ql map $F=(f,\iota )\colon U\rightrightarrows V$ can be restricted to $$F=(f,\iota )\colon U^{k+1}\rightrightarrows U^{k}$$ by considering the fiber product §[3.4.6](#sss:FibProd){reference-type="ref" reference="sss:FibProd"}. We have a natural $2^{k}:1$ branched covering $f^k\colon U^k\to V$ and an immersion $\iota^k\colon U^k\to V$. The *width* of a $\psi^\bullet$-ql map $F\colon U\rightrightarrows V$ is $${\mathcal W}_\bullet(F)\coloneqq {\mathcal W}(V\setminus {\mathfrak B }_F).$$ We denote by ${\mathcal A}^{[n]}\equiv {\mathcal A}^{[n]}_F$ the weighted arc diagram (WAD) of $V\setminus {\boldsymbol {\mathfrak B} }^{[n]}$: it is a formal sum of weighted arcs representing rectangles in the thick-thin decomposition of $V\setminus {\boldsymbol {\mathfrak B} }^{[n]}$; see §[3.3](#ss:WAD){reference-type="ref" reference="ss:WAD"} and Figure [3.3](#ss:WAD){reference-type="ref" reference="ss:WAD"}. The *horizontal part* ${\mathcal A}^{[n]}_{\mathrm {hor}}$ of ${\mathcal A}^{[n]}$ consists of weighted arcs connecting components of ${\boldsymbol {\mathfrak B} }^{[n]}$. The *vertical part* ${\mathcal A}^{[n]}_{\mathrm {ver}}$ of ${\mathcal A}^{[n]}$ consists of weighted arcs connecting $\partial V$ and components of ${\boldsymbol {\mathfrak B} }^{[n]}$. The *local WAD* ${\mathcal A}^{[n]}_i$ consists of arcs in ${\mathcal A}^{[n]}$ adjacent to ${\mathfrak B }^{[n]}_i$, where the weight of self-arcsto ${\mathfrak B }^{[n]}_i$ is doubled. More generally, ${\mathcal A}^{[n],(m),k}$ is the WAD of $U^k\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$, and the WADs ${\mathcal A}^{[n],(m),k}_{\mathrm {hor}}, {\mathcal A}^{[n],(m),k}_i,\dots$ are defined accordingly, see §[3.5.2](#sss:WADs){reference-type="ref" reference="sss:WADs"}. We have $${\mathcal W}(F_{n,i})={\mathcal W}\left({\mathcal A}^{[n]}_i\right)+O_{p_n}(1).$$ By *identifying up to homotopy* $U^k\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$ and $U^{k+1}\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$, we obtain ${\mathcal A}^{[n],(m),k+1}_{\mathrm {hor}}\le{\mathcal A}^{[n],(m),k}_{\mathrm {hor}}$, see §[3.5.6](#sss:monot:AA^k){reference-type="ref" reference="sss:monot:AA^k"}. Since the complexity of $A^{[n],k}_{\mathrm {hor}}\equiv{\operatorname{AD}}\left[{\mathcal A}^{[n],k}_{{\mathrm {hor}}}\right]$ decreases, the $A^{[n],k}_{{\mathrm {hor}}}$ are essentially invariant for $k\ge 3 p_n$. Then either most of ${\mathcal A}^{[n]}_{\mathrm {hor}}$ is in ${\mathcal A}^{[n],k}_{\mathrm {hor}}$ or a substantial part of ${\mathcal A}^{[n]}_{\mathrm {hor}}$ is in ${\mathcal A}^{[n],k}_{\mathrm {ver}}$ -- this is a key dichotomy in Section [4](#s:Pull off for non-periodic rect){reference-type="ref" reference="s:Pull off for non-periodic rect"}; see also the dichotomy "[\[C1:intro\]](#C1:intro){reference-type="ref" reference="C1:intro"} vs [\[C2:intro\]](#C2:intro){reference-type="ref" reference="C2:intro"}" in §[1.3.2](#ss:intro:SatCase){reference-type="ref" reference="ss:intro:SatCase"}. ## Rectangles {#ss:rectangles} A *Euclidean rectangle* is a rectangle $E_x\coloneqq[0,x]\times [0,1] \subset {\Bbb C}$, where: - $(0,0), (x,0), (x,1), (0,1)$ are four vertices of $E_x$, - $\partial^h E_x=[0,x]\times\{0,1\}$ is the horizontal boundary of $E_x$, - $\partial^{h,0} E_x\coloneqq [0,x]\times\{0\}$ is the *base* of $E_x$, - $\partial^{h,1} E_x\coloneqq [0,x]\times\{1\}$ is the *roof* of $E_x$, - $\partial^v E_x=\{0,x\}\times [0,1]$ is the *vertical* boundary of $E_x$, - ${\mathcal F}(E_x)\coloneqq \{\{t\}\times[0,1]\mid t\in [0,x]\}$ is the *vertical foliation* of $E_x$, - ${\mathcal F}^{\operatorname{full}}(E_x)\coloneqq \{\gamma\colon [0,1]\to E_x \mid \gamma(0)\in \partial ^{h,0}E_x,\ \gamma(1)\in \partial ^{h,1}E_x \}$ is the *full family of curves* in $E_x$; - ${\mathcal W}(E_x)={\mathcal W}({\mathcal F}(E_x))={\mathcal W}\big({\mathcal F}^{\operatorname{full}}(E_x)\big)=x$ is the *width* of $E_x$, - $\operatorname{mod}(E_x)=1/{\mathcal W}(E_x)=1/x$ the extremal length of $E_x$. By a *(topological) rectangle* in a Riemann surface we mean a closed Jordan disk ${\mathcal R}$ together with a conformal map $h\colon {\mathcal R}\to E_x$ into the standard rectangle $E_x$. The vertical foliation ${\mathcal F}({\mathcal R})$, the full family ${\mathcal F}^{\operatorname{full}}({\mathcal R})$, the base $\partial^{h,0}{\mathcal R}$, the roof $\partial^{h,1}{\mathcal R}$, the vertices of ${\mathcal R}$, and other objects are defined by pulling back the corresponding objects of $E_x$. Equivalently, a rectangle ${\mathcal R}$ is a closed Jordan disk together with four marked vertices on $\partial {\mathcal R}$ and a chosen base between two vertices. A *genuine subrectangle* of $E_x$ is any rectangle of the form $E'=[x_1,x_2]\times [0,1]$, where $0\le x_1<x_2\le x$; it is identified with the standard Euclidean rectangle $[0,x_2-x_1]\times [0,1]$ via $z\mapsto z- x_1$. A genuine subrectangle of a topological rectangle is defined accordingly. A *subrectangle* of a rectangle ${\mathcal R}$ is a topological rectangle ${\mathcal R}_2\subset {\mathcal R}$ such that $\partial^{h,0}{\mathcal R}_2\subset {\mathcal R}$ and $\partial^{h,1}{\mathcal R}_2\subset {\mathcal R}$. By monotonicity: ${\mathcal W}({\mathcal R}_2)\le {\mathcal W}({\mathcal R})$. Assume that ${\mathcal W}(E_x)>2$. The *left and right $1$-buffers* of $E_x$ are defined $$B^\ell_1\coloneqq [0,1]\times [0,1]{\ \ }\text{ and }{\ \ }B^\rho_1\coloneqq [x-1,x]\times [0,1]$$ respectively. We say that the rectangle $$E^{\operatorname{new}}_x\coloneqq [1,x-1]\times [0,1]=E_x\setminus \big( B^\ell_1 \cup B^\rho_1\big)$$ is obtained from $E_x$ by *removing $1$-buffers*. If ${\mathcal W}(E_x)\le 2$, then we set $E_x^{\operatorname{new}}\coloneqq\emptyset$. Similarly, buffers of any width are defined. ### Monotonicity and Grötzsch inequality {#sss:Monot+Gr} We say a family of curves ${\mathcal S}$ *overflows* a family ${\mathcal G}$ if every curve $\gamma\in{\mathcal S}$ contains a subcurve $\gamma'\in {\mathcal G}$. We also say that - a family of curves ${\mathcal F}$ *overflows* a rectangle ${\mathcal R}$ if ${\mathcal F}$ overflows ${\mathcal F}^{\operatorname{full}}({\mathcal R})$; - a rectangle ${\mathcal R}_1$ overflows another rectangle ${\mathcal R}_2$ if ${\mathcal F}({\mathcal R}_1)$ overflows ${\mathcal F}^{\operatorname{full}}({\mathcal R}_2)$. If ${\mathcal F}$ overflows a family or a rectangle ${\mathcal G}$, then ${\mathcal G}$ is wider than ${\mathcal F}$: $$\label{eq:Width Monot} {\mathcal W}({\mathcal F}) \le {\mathcal W}({\mathcal G}).$$ If ${\mathcal F}$ overflows both ${\mathcal G}_1,{\mathcal G}_2$, and ${\mathcal G}_1,{\mathcal G}_2$ are disjointly supported, then the *Grötzsch* inequality states: $$\label{eq:Grot} {\mathcal W}({\mathcal F}) \le {\mathcal W}({\mathcal G}_1)\oplus {\mathcal W}({\mathcal G}_2),$$ where $x\oplus y =(x^{-1}+y^{-1})^{-1}$ is the harmonic sum. ## Weighted arc diagrams {#ss:WAD} Let us recall the notion of the Weighted Arc Diagram (WAD) describing wide rectangles in the thick-thin decomposition of a Riemann surface with boundary, see Figure [\[Fg:TTD\]](#Fg:TTD){reference-type="ref" reference="Fg:TTD"} for illustration and brief summary. We also recall that WAD were abstractly defined in §[2.4.1](#sss:AD){reference-type="ref" reference="sss:AD"}. $$\begin{tikzpicture} \draw[,line width=0.4mm] (0,0) .. controls (0.7, 1.3) and (1.3,1) .. (2,2) .. controls (2.8, 3) and (4.6,3.5) .. (6,2.5) .. controls (6.8, 2) and (7.2,1.5) .. (8,0.5) .. controls (6, -2) and (5,-2.5) .. (4,-2.5) .. controls (3, -2.5) and (1,-2.5) .. (0,0); \node[above] at (4,-2) {$V$}; \draw[opacity=0, fill=red, fill opacity=0.5] (6,2.5) .. controls (6.8, 2) and (7.2,1.5) .. (8,0.5) .. controls (8,0.5) and (7.8,0.5).. (7.8,0.5) .. controls (7,1.5) and (6.6, 2).. (5.8,2.5); \draw[,line width=0.4mm] (0.2,0) .. controls (0.9, 1.3) and (1.5,1) .. (2.2,2) .. controls (2.9, 2.3) and (3.2,1.6) .. (3.5,1.5) .. controls (3.8, 1.3) and (4.1,1.1) .. (4.5,0.5) .. controls (4, 0.2) and (3.5,0) .. (3,0) .. controls (2.5, 0.2) and (2,0.2) .. (1.5,0) .. controls (1, 0.2) and (0.6,-0.1) .. 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(2,0); \node at (4.5,0.15) {$Y$}; \node[red] at (1.2,-0.28) {${\boldsymbol \Pi}^{\operatorname{can}}_{\alpha_i}$}; \node[red] at (5.8,-0.28) {${\boldsymbol \Pi}^{\operatorname{can}}_{\alpha_j}$}; \draw[opacity=0, fill=red, fill opacity=0.5] (5,0.) .. controls (5.08, 0.1) and (5.08,0.2) .. (5,0.3) .. controls (4,0.3) and (4,0.3) .. (7,0.3) .. controls (7.08, 0.2) and (7.08,0.1) .. (7,0.); \end{scope} \end{tikzpicture}$$ ### Proper rectangles A *proper rectangle* $R$ in an open hyperbolic surface $S$ is a rectangle in its ideal completion $\overline S =S \cup \partial^i S$ such that $\partial^h R\subset \partial^i S$. We naturally view vertical curves of $R$ as proper (and open) paths in $S$. We say that $R$ *connects* boundary components $J_0, J_1\subset \partial^i R$ if $\partial ^{h,0}R\subset J_0,\ \partial^{h,1}R\subset J_1$. If $S=U\setminus K$, where $U$ is a topological disk and $K\Subset U$ is a compact subset with finitely many connected components, then $\partial^i S$ has one *outer* component $\partial^i U$; all remaining *inner* components are parameterized by the components of $K$. If $J_0, J_1$ represent the boundaries of components $K_0, K_1$ of $K$, then we say that the above $R$ connects $K_0,K_1$. ### Universal cover For compact Riemann surface $V$ as in §[2.4.1](#sss:AD){reference-type="ref" reference="sss:AD"}x, write its non-empty boundary as $\partial V = \bigsqcup_k J_k$. We set $\pi: \mathbb{D}\rightarrow\operatorname{int}V$ to be the universal covering, and we let ${\Lambda}\subset {\Bbb T}=\partial \mathbb{D}$ be the limit set for the group ${\Delta}$ of deck transformations of $\pi$. Then $\pi$ extends continuously to ${\Bbb T}\setminus {\Lambda}$, and $\pi: \overline{\Bbb D}\setminus{\Lambda}\rightarrow V$ is the universal covering of $V$. Moreover, $\pi$ restricted to any component of ${\Bbb T}\setminus{\Lambda}$ gives us a universal covering of some component $J_k$ of $\partial V$. Such a component of ${\Bbb T}\setminus{\Lambda}$ will be denoted ${\mathbf{J}}_k$ (usually we will need only one component for each $k$, so we will not use an extra label for it). The stabilizer of ${\mathbf{J}}_k$ in ${\Delta}$ is a cyclic group generated by a hyperbolic Möbius transformation $\tau_k$ with fixed points in $\partial{\mathbf{J}}_k$. ### Covering annuli and local weights {#sss:CovAnn LocWeights} We let $$\label{eq:CovAnn} {\Bbb A}(V, J_k)\coloneqq \mathbb{D}/<\tau_k>$$ be the *covering annulus* of $V$ corresponding to $J_k$. We call its width $${\mathcal W}(J_k)\coloneqq {\mathcal W}\left[ {\Bbb A}(V, J_k)\right]$$ the *local weight* of $J_k$. We let $${\mathcal W}(V) = \sum_k {\mathcal W}(J_k)$$ be the *total weight* of $V$. ### Canonical arc diagram {#sss:WAD} Recall that an arc $\alpha$ is a proper path in $V$ up to proper homotopy. Any arc $\alpha$ connects some components $J_k$ and $J_i$ of $\partial V$ and lifts to an arc ${\mbox{\boldmath$\alpha$} }$ in ${\Bbb D}$ connecting some intervals ${\mathbf{J}}_k$ and ${\mathbf{J}}_i$. We let ${\mathcal W}({{\mbox{\boldmath$\alpha$} }})= {\mathcal W}_{\overline \mathbb{D}}({\mathbf{J}}_k, {\mathbf{J}}_i)$ to be the width of all curves in $\mathbb{D}$ connecting ${\mathbf{J}}_k$ and ${\mathbf{J}}_i$. Then ${\mathcal W}({{\mbox{\boldmath$\alpha$} }})$ is also the width of the rectangle ${\boldsymbol{\Pi}}_{\mbox{\boldmath$\alpha$} }=\overline{\Bbb D}$ with horizontal sides ${\mathbf{J}}_k$ and ${\mathbf{J}}_i$. It is independent of the lift used. If ${\mathcal W}({{\mbox{\boldmath$\alpha$} }})> 2$, then by removing from ${\boldsymbol{\Pi}}_{\mbox{\boldmath$\alpha$} }$ the square buffers we obtain a rectangle ${\boldsymbol{\Pi}}^{{\operatorname{can}}}_{\mbox{\boldmath$\alpha$} }$. In this case we let $${\mathcal W}^{\operatorname{can}}(\alpha) = {\mathcal W}^{\operatorname{can}}({{\mbox{\boldmath$\alpha$} }}) = {\mathcal W}({\boldsymbol{\Pi}}^{\operatorname{can}}_{\mbox{\boldmath$\alpha$} }) = {\mathcal W}({{\mbox{\boldmath$\alpha$} }})-2.$$ By construction, the $\Delta$-orbit of ${\boldsymbol{\Pi}}^{\operatorname{can}}_{\mbox{\boldmath$\alpha$} }$ consists of pairwise disjoint rectangles. Therefore, ${\boldsymbol{\Pi}}^{\operatorname{can}}_{\mbox{\boldmath$\alpha$} }$ projects injectively onto a rectangle ${\boldsymbol{\Pi}}^{\operatorname{can}}_\alpha$ in $V$. Moreover, different ${\boldsymbol{\Pi}}^{\operatorname{can}}_{{\mbox{\boldmath$\alpha$} }_i}$ project to pairwise disjoint ${\boldsymbol{\Pi}}^{\operatorname{can}}_{\alpha_i}$. If ${\mathcal W}({{\mbox{\boldmath$\alpha$} }})\leq 2$ we let ${\mathcal W}^{\operatorname{can}}(\alpha) =0$ and ${\boldsymbol{\Pi}}^{\operatorname{can}}(\alpha)\coloneqq \emptyset$. Arcs with positive weight form the *canonical weighted arc diagram (WAD) on $V$:* $$\label{eq:WAD:dfn} {\mathcal A}_V\equiv {\operatorname{WAD}}(V)\coloneqq\sum_{{\mathcal W}^{\operatorname{can}}(\alpha)>0} {\mathcal W}^{\operatorname{can}}(\alpha)\alpha .$$ Every ${\boldsymbol \Pi}^{\operatorname{can}}(\alpha)$ supports the canonical vertical foliation ${\mathcal F}({\boldsymbol \Pi}^{\operatorname{can}}(\alpha)$. Altogether these foliations form the *canonical lamination* of $V$: $$\label{eq:can laminat} {\mathcal F}({\mathcal A}_V)\coloneqq \bigsqcup_{\alpha \in {\mathcal A}_V} {\mathcal F}({\boldsymbol \Pi}^{\operatorname{can}}(\alpha)),$$ We set $$\label{eq:can laminat:weight}{\mathcal W}({\mathcal A}_V)\coloneqq \sum_{\alpha\in {\mathcal A}_V} {\mathcal W}(\alpha) ={\mathcal W}\big( {\mathcal F}({\mathcal A}_V)\big).$$ The canonical *arc diagram* of $V$ is the set of arcs in ${\mathcal A}_V$ $$A_V \coloneqq \{\alpha\mid {\mathcal W}^{\operatorname{can}}(\alpha)>0\}\equiv {\operatorname{AD}}[{\mathcal A}_V].$$ ### Local WADs and the thick-thin decomposition {#sss:LocalWAD} For a boundary component $J_k$ of $\partial V$, its *local WAD* ${\mathcal A}_{J_k}$ consists of weighted arcs of ${\mathcal A}_V$ adjacent to $J_k$ such that the weights of self-arcs adjacent to $J_k$ are doubled. We have: $$\sum_{J_k} {\mathcal A}_{J_k}= 2{\mathcal A}_V .$$ See [@L:book §7.6.3] for a reference of the following fact. **Theorem 11** (Thin-Thick Decomposition). *For any compact Riemann surface $V$ with boundary $\partial V = \bigsqcup_k J_k$, we have: $${\mathcal W}(J_k) - C \leq {\mathcal W}({\mathcal A}_{J_k}) \leq {\mathcal W}(J_k){\ \ }{\ \ }\text{ for every }J_k$$ and $$\sum_{J_k} {\mathcal W}(J_k) - C \leq 2 {\mathcal W}({\mathcal A}_V) \leq \sum_{J_k} {\mathcal W}(J_k) ={\mathcal W}(V),$$ where $C$ depends only on the topological type of $V$.* ### Sub-diagrams and removing buffers {#sss:sub-diagram} Let ${\mathcal A}\equiv{\mathcal A}_V$ be the canonical WAD of $V$ as in §[3.3.4](#sss:WAD){reference-type="ref" reference="sss:WAD"}, and let $B\subset A$ be a subset of arcs from $A\equiv{\operatorname{AD}}({\mathcal A})$. Then the *WAD induced by $B$* is the formal sum ${\mathcal B}\coloneqq\sum_{\alpha\in B} {\mathcal W}^{\operatorname{can}}(\alpha)\alpha$, compare to [\[eq:WAD:dfn\]](#eq:WAD:dfn){reference-type="eqref" reference="eq:WAD:dfn"}. The weight $W({\mathcal B})$ of ${\mathcal B}$ and its canonical lamination ${\mathcal F}({\mathcal B})$ is defined as for ${\mathcal A}$, see [\[eq:can laminat\]](#eq:can laminat){reference-type="eqref" reference="eq:can laminat"} and [\[eq:can laminat:weight\]](#eq:can laminat:weight){reference-type="eqref" reference="eq:can laminat:weight"}. In §[3.5.2](#sss:WADs){reference-type="ref" reference="sss:WADs"}, we will specify the vertical and horizontal parts of the WAD associated with $\psi^\bullet$-ql maps. For $B,{\mathcal B}$ as above and $c>0$, we define $${\mathcal B}-c \coloneqq \sum_{ \substack{\alpha\in B \\ {\mathcal W}(\alpha)>c} } ({\mathcal W}^{\operatorname{can}}(\alpha)-c)\alpha,$$ $$B-c \coloneqq {\operatorname{AD}}({\mathcal B}-c);$$ i.e., we remove the $c$-weight from each arc. For a rectangle ${\boldsymbol \Pi}^{\operatorname{can}}(\alpha)$ as in §[3.3.4](#sss:WAD){reference-type="ref" reference="sss:WAD"}, we denote by ${\boldsymbol \Pi}^{\operatorname{can}}(\alpha)-c$ the rectangle obtained from ${\boldsymbol \Pi}^{\operatorname{can}}(\alpha)$ by removing the $c/2$-buffers on each side. We call $${\mathcal F}({\mathcal B}-c)\coloneqq \bigsqcup_{\alpha \in {\mathcal B}-c } {\mathcal F}({\boldsymbol \Pi}^{\operatorname{can}}(\alpha)-c)$$ the *canonical lamination of ${\mathcal B}-c$*. We also denote by $${\mathcal F}^{\operatorname{full}}({\mathcal B}-c)\coloneqq \bigsqcup_{\alpha \in {\mathcal B}-c } {\mathcal F}^{\operatorname{full}}({\boldsymbol \Pi}^{\operatorname{can}}(\alpha)-c)$$ the full family of vertical curves within respective rectangles, see §[3.2](#ss:rectangles){reference-type="ref" reference="ss:rectangles"}. ### Transformation rules **Lemma 12**. *Let $i: S' \rightarrow S$ be a holomorphic map between two compact Riemann surfaces with boundary. Then* *$\mathrm{(i)}$ If a boundary component $J'$ of $S'$ is mapped with degree $d$ to a boundary component $J$ of $S$ then ${\mathcal W}(J')\leq d\cdot {\mathcal W}(J)$;* *$\mathrm { (ii) }$ If an arc $a '$ of $S'$ is mapped to an arc $\alpha$ of $S$ then ${\mathcal W}(\alpha') \leq {\mathcal W}(\alpha)$.* **Lemma 13**. *Let ${\mathbf{i}}: {\boldsymbol{\Pi}}'\rightarrow{\boldsymbol{\Pi}}$ be a holomorpic map between two rectangles which extends to homeomorphisms between the respective horizontal sides ${\boldsymbol{I}}'\rightarrow{\boldsymbol{I}}$ and ${\mathbf{J}}'\rightarrow{\mathbf{J}}$. Then $${\mathcal W}( {\boldsymbol{\Pi}}' ) \leq {\mathcal W}( {\boldsymbol{\Pi}}).$$* ### WAD under covering {#sss:WAD:cov} Assume that $f \colon U\to V$ as a finite-degree covering between surfaces with boundaries. Then, see [@K Lemma 3.3] $${\mathcal A}_U=f^*{\mathcal A}_V, {\ \ }\text{ where }{\ \ }f^*{\mathcal A}_V=\sum_{\alpha \in {\mathcal A}_V} W^{\operatorname{can}}(\alpha) f^{-1}(\alpha)$$ and $f^{-1}(\alpha)$ is the set of preimages of $\alpha$. ## $\psi^\bullet$-ql renormalization {#psi-renorm} As in §[2.2.3](#renorm sec){reference-type="ref" reference="renorm sec"}, consider a ql map $f\colon X\to Y$ and its ql prerenormalization $$\label{eq:f n,0} f_{n,0}=f^{p_n}\colon X_n\to Y_n.$$ ### Motivation {#sss:psi renorm: motiv} A major technical issue is that there is no natural choice of $Y_n$ in [\[eq:f n,0\]](#eq:f n,0){reference-type="eqref" reference="eq:f n,0"} so that ${{\mathcal W}(Y_n\setminus {\mathfrak K }_n)}$ is optimal and satisfies, in particular, [\[eq:outline:2\]](#eq:outline:2){reference-type="eqref" reference="eq:outline:2"}. To handle this problem, a $\psi$-ql renormalization was introduced in [@K]: assuming that [\[eq:f n,0\]](#eq:f n,0){reference-type="eqref" reference="eq:f n,0"} is primitive and $Y_n\cap {\boldsymbol{\mathfrak K} }^{[n]}={\mathfrak K }^{[n]}_0$, *extend $f_n\colon X_n \to Y_n$ along all curves in $Y_n\setminus {\boldsymbol{\mathfrak K} }^{[n]}$*; the result is a correspondence $F_n=(f_n,\iota_n)\colon U_n\rightrightarrows V_n$, where $$V_n\coloneqq {\Bbb A}\left(Y\setminus {\boldsymbol{\mathfrak K} }^{[n]}, \ \partial {\boldsymbol{\mathfrak K} }^{[n]}_0\right) \bigcup _{\partial {\boldsymbol{\mathfrak K} }^{[n]}_0} {\mathfrak K }^{[n]}_0$$ is the covering annulus [\[eq:CovAnn\]](#eq:CovAnn){reference-type="eqref" reference="eq:CovAnn"} of $Y\setminus {\boldsymbol{\mathfrak K} }^{[n]}$ rel $\partial {\boldsymbol{\mathfrak K} }^{[n]}_0$ glued with ${\mathfrak K }^{[n]}_0$ and $$U_n\coloneqq {\Bbb A}\left( X\setminus f^{-1}({\boldsymbol{\mathfrak K} }^{[n]}), \ \partial {\boldsymbol{\mathfrak K} }^{[n]}_0\right) \bigcup _{\partial {\boldsymbol{\mathfrak K} }^{[n]}_0} {\mathfrak K }^{[n]}_0$$ is the covering annulus of $X\setminus f^{-1}\left({\boldsymbol{\mathfrak K} }^{[n]}\right)$ rel $\partial {\boldsymbol{\mathfrak K} }^{[n]}_0$ glued with ${\mathfrak K }^{[n]}_0$. The correspondence $F_n$ is called a *$\psi$-ql map* and it is independent of the choice of $Y_n$ in [\[eq:f n,0\]](#eq:f n,0){reference-type="eqref" reference="eq:f n,0"}. A $\psi$-ql renormalization can be naturally iterated. Moreover, $\psi$-ql bounds can be converted into ql bounds, see Lemma [Lemma 17](#lem:from psi-ql to ql){reference-type="ref" reference="lem:from psi-ql to ql"}. In this section, we will introduce $\psi^\bullet$-renormalization by replacing ${\boldsymbol{\mathfrak K} }^{[n]}$ with the periodic cycle ${\boldsymbol {\mathfrak B} }^{[n]}$ of little bushes so that it is also applicable for satellite combinatorics. **Remark 14**. *The viewpoint that various classes of self-correspondences ${(g,h)\colon A \rightrightarrows B}$ form interesting dynamical systems was popularized by Sullivan and later by A. Epstein in the context of deformation spaces. In the 2000s, it became apparent that self-correspondences give a natural framework to study some of the classical dynamical systems [@IS; @K; @N]. See also [@BD; @T] for more recent developments.* ### Definitions {#sss:qlb:Defn} A *pseudo${}^\bullet$-quadratic-like map* ("$\psi^\bullet$-ql map") is a pair of holomorphic maps $$\label{eq:dfn:psib-ql} F=(f,\iota)\colon {\ \ }( U, {\mathfrak B }') \rightrightarrows (V, {\mathfrak B })$$ between two conformal disks $U$, $V$ with the following properties: 1. [\[dfn:psib-ql:1\]]{#dfn:psib-ql:1 label="dfn:psib-ql:1"} $f\colon U\to V$ is a double branched covering (we usually normalize it so that its critical point is located at $0$); 2. [\[dfn:psib-ql:2\]]{#dfn:psib-ql:2 label="dfn:psib-ql:2"} $\iota\colon U\to V$ is an immersion; 3. [\[dfn:psib-ql:3\]]{#dfn:psib-ql:3 label="dfn:psib-ql:3"} ${\mathfrak B }$ and ${\mathfrak B }'$ are hulls (compact connected full sets) such that $$\iota^{-1} ({\mathfrak B })={\mathfrak B }' \subset f^{-1} ({\mathfrak B }) ;$$ 4. [\[dfn:psib-ql:4\]]{#dfn:psib-ql:4 label="dfn:psib-ql:4"} there exist neighborhoods $X'\supset {\mathfrak B }'$ and $X \supset {\mathfrak B }$ with the following property: $\iota: X' \rightarrow X$ is a conformal isomorphism such that $$\label{eq:dfn:ql b domain} f_X\coloneqq f\circ \big(\iota\mid X'\big)^{-1} : X \to f(X')\eqqcolon Y$$ is a quadratic-like map with connected filled Julia set $$\label{eq:dfn:non-esc b} {\mathfrak K }_F\coloneqq {\mathfrak K }\big(f_X : X\to Y\big),$$ and such that ${\mathfrak B }\equiv{\mathfrak B }_F\equiv {\mathfrak B }(f_X)$ is the bush of $f_X$. Since $\iota$ is a conformal isomorphism in a neighborhood of ${\mathfrak K }_F$, we will below identify $${\mathfrak K }_F\simeq \big(\iota\mid X\big)^{-1}({\mathfrak K }_F){\ \ }{\ \ }\text{ and hence }{\ \ }{\ \ }{\mathfrak B }\simeq {\mathfrak B }'\equiv {\mathfrak B }_F,$$ and write $$F\colon ( U, {\mathfrak B }) \rightrightarrows (V, {\mathfrak B }) {\ \ }{\ \ }\text{ or }{\ \ }{\ \ }F\colon U \rightrightarrows V.$$ Let us say that a subset $\Omega\subset {\mathfrak K }_F$ is called *$\iota$-proper* if $$\iota^{-1}(\Omega)=\Omega.$$ Item [\[dfn:psib-ql:3\]](#dfn:psib-ql:3){reference-type="ref" reference="dfn:psib-ql:3"} implies that ${\mathfrak B }_F$ is $\iota$-proper. If ${\mathfrak K }_F$ is $\iota$-proper, then $F$ is $\psi$-ql map [@K]. ### $\psi^\bullet$-ql renormalization {#sss:psi b renorm} Consider a $\psi^\bullet$-ql map $F$ from [\[eq:dfn:psib-ql\]](#eq:dfn:psib-ql){reference-type="eqref" reference="eq:dfn:psib-ql"}, and assume that $f_X$ (see [\[eq:dfn:ql b domain\]](#eq:dfn:ql b domain){reference-type="eqref" reference="eq:dfn:ql b domain"}) is $n+1$ DH renormalizable. Let $f_{n,i}=f^{p_n}\colon X_{n,i}\to Y_{n,i}$ be a ql prerenormalization of $f_X\colon X\to Y$. The associated $\psi^\bullet$-ql renormalization is the extension of $f_{n,i}\colon X_{n,i} \to Y_{n,i}$ along all curves in $V\setminus {\boldsymbol {\mathfrak B} }_n$, compare with §[3.4.1](#sss:psi renorm: motiv){reference-type="ref" reference="sss:psi renorm: motiv"}; the result is a $\psi^\bullet$-ql map $F_{n,i}=(f_{n,i},\iota_{n,i})\colon U_{n,i}\rightrightarrows V_{n,i}$, where $$V_{n,i}\coloneqq {\Bbb A}\left(V\setminus {\boldsymbol {\mathfrak B} }^{[n]}, \ \partial {\boldsymbol {\mathfrak B} }^{[n]}_i\right) \bigcup _{\partial {\boldsymbol {\mathfrak B} }^{[n]}_i} {\mathfrak B }^{[n]}_i$$ is the covering annulus [\[eq:CovAnn\]](#eq:CovAnn){reference-type="eqref" reference="eq:CovAnn"} of $V\setminus {\boldsymbol {\mathfrak B} }^{[n]}$ rel $\partial {\boldsymbol {\mathfrak B} }^{[n]}_i$ glued with ${\mathfrak B }^{[n]}_i$ and $$U_{n,i}\coloneqq {\Bbb A}\left( U\setminus f^{-1}({\boldsymbol{\mathfrak K} }^{[n]}), \ \partial {\boldsymbol{\mathfrak K} }^{[n]}_i\right) \bigcup _{\partial {\boldsymbol{\mathfrak K} }^{[n]}_i} {\mathfrak K }^{[n]}_i$$ is the covering annulus of $U\setminus f^{-1}({\boldsymbol{\mathfrak K} }^{[n]})$ rel $\partial {\boldsymbol{\mathfrak K} }^{[n]}_i$ glued with ${\mathfrak K }^{[n]}_i$. We will write suppress index "$0$" for $F_{n,0}$: $$F_{n,0}\equiv F_{n}=(f_{n},\iota_{n})\colon U_{n}\rightrightarrows V_{n}, \hspace{0.4cm}\text{ where } {\ \ }U_n\equiv U_{n,0},\ V_n\equiv V_{n,0}.$$ We write $$\label{RR bullet f} {\mathcal R}^{n\bullet }( F) \coloneqq F_n \equiv F_{n,0}.$$ **Remark 15**. *The theory works similarly if the bush ${\mathfrak B }$ is replaced with any connected forward invariant set $\Upsilon$ satisfying ${\mathfrak T}\subset \Upsilon\subset {\mathfrak K }$, where ${\mathfrak T}$ is the Hubbard continuum. Namely, let us say that a map $$\label{eq:psi^* ql map} F=(f,\iota)\colon {\ \ }( U, \Upsilon') \rightrightarrows (V, \Upsilon )$$ is *$\psi^\Upsilon$-ql* if it satisfies [\[dfn:psib-ql:1\]](#dfn:psib-ql:1){reference-type="ref" reference="dfn:psib-ql:1"} -- [\[dfn:psib-ql:4\]](#dfn:psib-ql:4){reference-type="ref" reference="dfn:psib-ql:4"} from §[3.4.2](#sss:qlb:Defn){reference-type="ref" reference="sss:qlb:Defn"} so that ${\mathfrak B }'\simeq {\mathfrak B }={\mathfrak B }(f_X)$ are replaced with $$\Upsilon'=\iota^{-1} (\Upsilon)\simeq \Upsilon=\Upsilon(f_X),\hspace{1.5cm}\text{ i.e., }\Upsilon \text{ is $\iota$-proper}.$$ For example, one can take $\Upsilon ={\mathfrak T}$ or $\Upsilon={\mathfrak T}\cup {\boldsymbol {\mathfrak B} }^{[m]}$.* *To define "$\psi^\Upsilon$-renormalization", one should extend a ql prerenormalization $f_{n,i}\colon {X_{n,i} \to Y_{n,i}}$ along all curves in $V\setminus \bigcup_{j} \Upsilon_{f_{n,j}}$. Moreover, $\Upsilon_{f_{n,j}}$ can often be enlarged, see [3.4.5](#sss: psi^b to psi){reference-type="ref" reference="sss: psi^b to psi"}.* ### Sup-Chain Rule for renormalization domains {#sss:sup chain rule} Consider a $\psi^\bullet$-ql map $F$ as in [\[eq:dfn:psib-ql\]](#eq:dfn:psib-ql){reference-type="eqref" reference="eq:dfn:psib-ql"}. Since - ${\mathcal R}^{n_1\bullet} \circ {\mathcal R}^{n_2 \bullet} (F)$, see [\[RR bullet f\]](#RR bullet f){reference-type="eqref" reference="RR bullet f"}, is the extension of $f_{n_1+n_2,0}$ along all curves in $V\setminus \Upsilon$, where $$\Upsilon\coloneqq {\boldsymbol {\mathfrak B} }^{[n_1+n_2]}\cup \left({\boldsymbol {\mathfrak B} }^{[n_2]}\setminus {\mathfrak B }^{[n_2]}_0\right),$$ - ${\mathcal R}^{n_1+n_2 \bullet} (F)$ is the extension of $f_{n_1+n_2,0}$ along all curves in $V\setminus {\boldsymbol {\mathfrak B} }^{[n_1+n_2]},$ - and every connected component of ${\boldsymbol {\mathfrak B} }^{[n_1+n_2]}$ is within a connected component of $\Upsilon$, we obtain the natural embedding of dynamical systems (respecting all the maps) $$\label{eq:sup chain rule} {\mathcal R}^{n_1\bullet} \circ {\mathcal R}^{n_2 \bullet} (F) \ \subset \ {\mathcal R}^{n_1+n_2 \bullet} (F).$$ ### From $\psi^\bullet$ to $\psi$-ql maps {#sss: psi^b to psi} Consider a $\psi^\bullet$-ql map $F$ from [\[eq:dfn:psib-ql\]](#eq:dfn:psib-ql){reference-type="eqref" reference="eq:dfn:psib-ql"}, and assume that $f_X$ (see [\[eq:dfn:ql b domain\]](#eq:dfn:ql b domain){reference-type="eqref" reference="eq:dfn:ql b domain"}) is twice DH renormalizable. Let $f_{1,0}\colon X_{1,0}\to Y_{1,0}$ be a ql prerenormalization of $f_X$. By Item [\[dfn:psib-ql:3\]](#dfn:psib-ql:3){reference-type="ref" reference="dfn:psib-ql:3"} of the definition of $F$, the little Julia set ${\mathfrak K }^{[1]}_{0}\equiv {\mathfrak K }(f_{1,0})$ is $\iota$-proper. Therefore, we can extend $f_{1,0}\colon X_{1,0}\to Y_{1,0}$ along all curves in ${\boldsymbol {\Upsilon} }\coloneqq {\mathfrak K }^{[1]}_{0}\cup {\boldsymbol {\mathfrak B} }^{[1]}$ and obtain $\psi$-ql map $$\label{eq:F^Ups} F_{{\boldsymbol {\Upsilon} },1} = (f_{1,0},\iota_{1,0})\colon U _{{\boldsymbol {\Upsilon} },1}\to V_{{\boldsymbol {\Upsilon} },1}.$$ We recall from §[2.4.5](#sss:mixed conf){reference-type="ref" reference="sss:mixed conf"} that the set ${\boldsymbol {\Upsilon} }$ is inhomogeneous: $f_X$ does not permute components of ${\boldsymbol {\Upsilon} }$ . ### Restrictions {#sss:FibProd} A $\psi^\bullet$-ql map $F$ as in [\[eq:dfn:psib-ql\]](#eq:dfn:psib-ql){reference-type="eqref" reference="eq:dfn:psib-ql"} has the natural *restriction* (also known as the *pullback*, *fiber product*, *graph*) denoted by $$F=(f,\iota)\colon U^2\rightrightarrows U^1=U,{\ \ }{\ \ }\text{ where }{\ \ }U^2=\{(x,y)\in U\times U\mid{\ \ }f(x)=\iota(y)\},$$ where $f,\iota\colon U^2\to U$ are component-wise projections; see [@K §2.2.2]. Note that $F\colon U^2\rightrightarrows U^1$ is also a $\psi^\bullet$-ql map. Repeating the construction, we obtain the sequence $$F\colon U^k\rightrightarrows U^{k-1},\hspace{1cm} n\ge 1, {\ \ }U^0=V ,{\ \ }U^1=U,$$ together with induced iterations denoted by $$\label{eq:F^k} F^k = \big(f^k, \iota^k\big) \colon U^k \to V.$$ Since $\iota\mid U^{k+1}$ is the lift of $\iota\mid U$ under the covering $f^k$, we have: **Lemma 16**. *The set $f^{-k}({\mathfrak B }_F)$ is $\iota$-proper for $F\colon U^{k+1}\rightrightarrows U^k$.0◻* ## Width and WADs of $\psi^\bullet$ maps {#ss:Width+WAD} For a $\psi^\bullet$-ql map $F\colon U\rightrightarrows V$ we write $${\mathcal W}_\bullet(F)\coloneqq {\mathcal W}(V\setminus {\mathfrak B }_F).$$ If $F$ is $\psi$-ql map (i.e., ${\mathfrak K }_F$ is $\iota$-proper), then we can also measure $${\mathcal W}(F)\coloneqq {\mathcal W}(V\setminus {\mathfrak K }_F).$$ The Grötzsch inequality easily implies $$\label{eq:WidthRestr} {\mathcal W}\left(U^k\setminus {\mathfrak B }\right)\le{\mathcal W}\left(U^k\setminus f^{-k}({\mathfrak B })\right)= 2^k{\mathcal W}_\bullet(F).$$ The Sup-Chain Rule [\[eq:sup chain rule\]](#eq:sup chain rule){reference-type="eqref" reference="eq:sup chain rule"} implies that $$\label{eq:width:sup chain rule} {\mathcal W}\left[{\mathcal R}^{n_1\bullet} \circ {\mathcal R}^{n_2 \bullet} (F) \right] \ge {\mathcal W}\left[ {\mathcal R}^{n_1+n_2 \bullet} (F)\right].$$ ### Compactness of $\psi$ and $\psi^\bullet$-ql maps {#sss:compactness} If ${\mathcal W}(F)$ is bounded for $\psi$-ql map $F$, then $F$ has also regular ql bounds: **Lemma 17** ([@K Lemma 2.4]). *There is a positive function $\mu(K)$ with the following property. If $F$ is a $\psi$-ql map with ${\mathcal W}(F)\le K$, then the quadratic-like map $f_X\colon X\to Y$ in [\[eq:dfn:ql b domain\]](#eq:dfn:ql b domain){reference-type="eqref" reference="eq:dfn:ql b domain"} can be selected so that $$\label{eq:lem:from psi-ql to ql}\operatorname{mod}\left(Y\setminus X \right) \ge \mu(K).$$* **Remark 18**. *Let $\gamma$ be the core hyperbolic geodesic of the annulus $V\setminus {\mathfrak K }_F$. Lemma [Lemma 17](#lem:from psi-ql to ql){reference-type="ref" reference="lem:from psi-ql to ql"} (=[@K Lemma 2.4]) constructs the quadratic-like map $f_X\colon X\to Y$ within the subdisk $V_\gamma\Subset V$ bounded by $\gamma$. If $F$ is obtained from a ql map $g\colon X_g\to Y_g$ using finitely many $\psi^\bullet$-renormalizations followed by a renormalization of §[3.4.5](#sss: psi^b to psi){reference-type="ref" reference="sss: psi^b to psi"}, then we naturally have a covering $\rho\colon V\setminus {\mathfrak K }_F\to Y_g\setminus \Upsilon_g$ for a certain $\Upsilon_g\subset {\mathfrak K }_g$ such that $\gamma$ univalently projects to a hyperbolic geodesic of $Y_g\setminus \Upsilon_g$. This implies that the quadratic-like restriction $f_X\colon X\to Y$ from Lemma [Lemma 17](#lem:from psi-ql to ql){reference-type="ref" reference="lem:from psi-ql to ql"} and satisfying [\[eq:lem:from psi-ql to ql\]](#eq:lem:from psi-ql to ql){reference-type="eqref" reference="eq:lem:from psi-ql to ql"} descent univalently to a ql renormalization in the dynamical plane of $g$.* first establishes that $\iota$ is an embedding in a neighborhood of ${\mathfrak K }_F$; this argument is applicable to $\psi^\bullet$ maps: **Lemma 19**. *There is a positive function $\mu_\bullet(K)$ with the following property. If $F$ is a $\psi_\bullet$-ql map with ${\mathcal W}_\bullet(F)\le K$, then ${\mathfrak B }_F$ has a neighborhood $\Omega$ such that $\iota$ is injective on $\Omega$ and $$\operatorname{mod}\left(\Omega\setminus {\mathfrak B }_F \right) \ge \mu_\bullet(K).$$* 0◻ ### WADs of $F$ {#sss:WADs} Assume that ${\boldsymbol {\mathfrak B} }^{[n]}$ is well-defined, i.e., $F$ is $n+1$ renormalizable. Let us consider the canonical WAD of $U^k\setminus {\boldsymbol {\mathfrak B} }^{[n],m}$: $${\mathcal A}^{(m),k}\equiv {\mathcal A}^{[n],(m),k}\coloneqq {\operatorname{WAD}}\left(U^k\setminus {\boldsymbol {\mathfrak B} }^{[n],m} \right).$$ We say (compare with §[1.3.1](#sss:intro:PrimPullOff){reference-type="ref" reference="sss:intro:PrimPullOff"}) that an arc $\alpha\in {\mathcal A}^{[n],(m),k}$ is - *vertical* if it connects the outer boundary $\partial^{\mathrm{out}}\left(U^k\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}\right)\coloneqq \partial U^k$ of $U^k\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$ to one of its inner boundary components; - *horizontal* if it connects two inner (i.e., non-outer) boundary components of $U^k\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$. We set (see §[3.3.6](#sss:sub-diagram){reference-type="ref" reference="sss:sub-diagram"}) - ${\mathcal A}^{(m),k}_{\mathrm {ver}}$ to be *vertical part* of ${\mathcal A}^{(m),k}$: it consists of arcs connecting $\partial U^k$ and one of the components of $\partial {\boldsymbol {\mathfrak B} }^{[n],(m)}$; - ${\mathcal A}^{(m),k}_{\mathrm {hor}}$ to be *horizontal part* of ${\mathcal A}^{(m),k}$: it consists of arcs connecting components of $\partial {\boldsymbol {\mathfrak B} }^{[n],(m)}$; - ${\mathcal A}_i^{(m),k}$ to be the *local WAD* for ${\mathfrak B }^{[n],(m),k}_i$, see §[3.3.5](#sss:LocalWAD){reference-type="ref" reference="sss:LocalWAD"}: ${\mathcal A}_i^{k,(m)}$ consisting of arcs adjacent to ${\mathfrak B }^{[n],(m),k}_i$ such that the weights of self-arcs adjacent to ${\mathfrak B }^{[n],(m),k}_i$ are doubled; - ${\mathcal A}_{i,{\mathrm {ver}}}^{(m),k},{\mathcal A}_{i,{\mathrm {hor}}}^{(m),k}$ to be the local parts of ${\mathcal A}^{(m),k}_{\mathrm {ver}},\ {\mathcal A}^{(m),k}_{\mathrm {hor}}$. We will usually suppress upper zero induces: $${\mathcal A}_i={\mathcal A}^{(0),0}_i, \hspace{0.4cm}{\mathcal A}_i^{(m)}={\mathcal A}^{(m),0}_i, \hspace{0.4cm}{\mathcal A}_i^{k}={\mathcal A}^{(0),k}_i, \dots$$ The following lemma provides a reverse to [\[eq:width:sup chain rule\]](#eq:width:sup chain rule){reference-type="eqref" reference="eq:width:sup chain rule"} estimate. **Lemma 20**. *Assume that in the dynamical plane of $F={\mathcal R}^{n_0 \bullet }\colon U\rightrightarrows V$, there is a non-trivial horizontal lamination of curves ${\mathcal L}\subset V\setminus {\boldsymbol {\mathfrak B} }^{[n]}$ emerging from ${\mathfrak B }^{[n]}_0$ and landing at ${\boldsymbol {\mathfrak B} }^{[n]}$. Then $${\mathcal W}\left( {\mathcal R}^{n_0+n \bullet }G \right)\ge {\mathcal W}({\mathcal L}).$$* *Proof.* By definition §[3.4.2](#sss:qlb:Defn){reference-type="ref" reference="sss:qlb:Defn"}, the ${\mathcal L}$ lifts univalently to a horizontal lamination $\widetilde {\mathcal L}$ in the dynamical plane of $G$ emerging from ${\mathfrak B }^{n_0+n}_{G, 0}$ and landing at ${\boldsymbol {\mathfrak B} }^{[n_0+n]}_{G}$. After that $\widetilde {\mathcal L}$ lifts univalently to a vertical lamination in the dynamical plane of ${\mathcal R}^{n_0+n \bullet }G$. We obtain that ${\mathcal W}\left( {\mathcal R}^{n_0+n \bullet }G \right)\ge {\mathcal W}(\widetilde {\mathcal L})\ge {\mathcal W}({\mathcal L})$. ◻ ### WADs and ${\mathcal W}_\bullet(F_{n,i})$ {#sss:Width:F_ni} It follows from Theorem [Theorem 11](#thin-thick for S){reference-type="ref" reference="thin-thick for S"} that $$\label{eq:Width is Loc WAD} {\mathcal W}_\bullet(F_{n,i}) = {\mathcal W}({\mathcal A}_i)+O_{p_n}(1).$$ As with $\psi$-ql renormalization (compare with [@McM1 Theorem 9.3]), we have $$\label{eq:LocWidth is compar} \frac 12{\mathcal W}(F_{n, 0}) \le {\mathcal W}(F_{n, i}) \le {\mathcal W}(F_{n, 0})\hspace{0.7cm} \text{ for all }\ i.$$ Indeed, let $\gamma_i \subset V\setminus {\boldsymbol {\mathfrak B} }^{[n]}$ be the peripheral hyperbolic geodesic around ${\mathfrak K }^{[n]}_{i}$. Then the hyperoblic length of $|\gamma_i|_{V\setminus {\boldsymbol {\mathfrak B} }^{[n]}}$ proportional to ${\mathcal W}(F_{n,i})$. Let $\gamma^{1}_i$ be the lift of $\gamma_i$ under $f\colon U^{(1)}\setminus {\boldsymbol {\mathfrak B} }^{[n],(1)}\to V\setminus {\boldsymbol {\mathfrak B} }^{[n]}$ so that $\gamma^{m}_i\subset U^{(m)}\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$ is the peripheral hyperbolic geodesic around ${\mathfrak B }^{[n],(1)}_{i-1}$, where the subscript is $\operatorname{mod}p_n$. Counting the degree of $f\colon \gamma^1_i\to \gamma_i$, we obtain - $|\gamma^1_i|_{U^{(1)}\setminus {\boldsymbol {\mathfrak B} }^{[n],(1)}}=|\gamma_i|_{V\setminus {\boldsymbol {\mathfrak B} }^{[n]}}$ if $i>1$; and - $\frac 12|\gamma^1_1|_{U^{(1)}\setminus {\boldsymbol {\mathfrak B} }^{[n],(1)}}=|\gamma_1|_{V\setminus {\boldsymbol {\mathfrak B} }^{[n]}}$. Finally, $|\gamma^1_i|_{U^{(1)}\setminus {\boldsymbol {\mathfrak B} }^{[n],(1)}}\ge |\gamma_{i-1}|_{V\setminus {\boldsymbol {\mathfrak B} }^{[n]}}$; this implies [\[eq:LocWidth is compar\]](#eq:LocWidth is compar){reference-type="eqref" reference="eq:LocWidth is compar"}. ### Covering: from $U^{k}\setminus {\boldsymbol {\mathfrak B} }^{(m)}$ to $U^{k+s}\setminus {\boldsymbol {\mathfrak B} }^{(m+s)}$ {#sss:WAD:covering:psi maps} It follows from §[3.3.8](#sss:WAD:cov){reference-type="ref" reference="sss:WAD:cov"} that the WADs change naturally under the covering: $$\label{eq:WAD:under coverin} {\mathcal A}^{(m+s),k+s}_{\mathrm {hor}}=(f^s)^* {\mathcal A}^{(m),k}_{\mathrm {hor}}, \hspace{0.7cm} {\mathcal A}^{(m+s),k+s}_{i-s} =(f^s)^* {\mathcal A}^{(m),k}_i,$$ and similar for other WAD such as ${\mathcal A}^{(m),k}, {\mathcal A}^{(m),k}_{\mathrm {ver}},\dots$. ### Restriction by $\iota^s$: from $U^{k}\setminus {\boldsymbol {\mathfrak B} }^{(m)}$ to $U^{k+s}\setminus {\boldsymbol {\mathfrak B} }^{(m)}$ {#sss:WAD:imm} Assume that $m \le k$. By Lemma [Lemma 16](#lem:iota proper:preim){reference-type="ref" reference="lem:iota proper:preim"}, the set ${\boldsymbol {\mathfrak B} }^{[n],(m)}\subset {\mathfrak B }_F$ is $\iota$-proper. Let $\gamma$ be a proper curve in $U^{k}\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$ emerging from ${\mathfrak B }^{[n],(m)}_a$. Applying $\iota^{-s}$ along $\gamma$, we construct, see [\[eq:iota to s\]](#eq:iota to s){reference-type="eqref" reference="eq:iota to s"} below, its *restriction* $(\iota^{*})^s[\gamma]$ which is a proper curve in $U^{k+s}\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$. Moreover, - if $\gamma$ is vertical, then so is $(\iota^{*})^s[\gamma]$, - if $\gamma$ is horizontal, then $(\iota^{*})^s[\gamma]$ is either horizontal or vertical. If $(\iota^{*})^s[\gamma]$ is horizontal, then we will also call $(\iota^{*})^s[\gamma]$ the *lift* of $\gamma$ under $\iota^s$. More precisely, write $\gamma\colon (0,1)\to U^k\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$. For every $s>0$, there is a $t_s\in (0,1]$ such that $\iota^{-s}$ extends along $\gamma\mid (0,t_s)$ and the resulting curve $$\label{eq:iota to s} (\iota^{*})^s[\gamma]\coloneqq \iota^{-s}\circ \gamma\colon (0,t_s) \to U^{k+s}$$ is a proper curve in $U^{k+s}\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$ (because ${\boldsymbol {\mathfrak B} }^{[n],(m)}\subset U^k$ is $\iota$-proper). Moreover, - if $t_m=1$, then $(\iota^{*})^n\gamma$ is vertical if and only if $\gamma$ is vertical, - if $t_m<1$, then $(\iota^{*})^n\gamma$ is vertical. The following lemma implies that curves in a horizontal rectangle that restrict under $(\iota^s)^*$ to vertical curves form buffers of the rectangle. **Lemma 21**. *Assume that two disjoint horizontal paths $\gamma_1,\gamma_2\subset U^{k}\setminus {\boldsymbol {\mathfrak B} }^{(m)}$ representing the same arc $\alpha=[\gamma_1]=[\gamma_2]$ lift under $(\iota^s)^*$ to horizontal paths in $U^{k+s}\setminus {\boldsymbol {\mathfrak B} }^{(m)}$. Let $R\subset U^{k}\setminus {\boldsymbol {\mathfrak B} }^{(m)}$ be the proper rectangle between $\gamma_1$ and $\gamma_2$ such that all vertical curves in $R$ also represent $\alpha$. Then all curves in ${\mathcal F}^{\operatorname{full}}(R)$ lift under $(\iota^s)^*$ to homotopic horizontal curves in $U^{k+s}\setminus {\boldsymbol {\mathfrak B} }^{(m)}$.* *Proof.* The curve $\gamma\in {\mathcal F}^{\operatorname{full}}(R)$ becomes vertical under the restriction by $(\iota^s)^*$ if and only if $\gamma$ hits $\iota^s\big(\partial U^{k+s}\big)$. This is impossible because $\gamma_1,\gamma_2$ do not hit $\iota^s\big(\partial U^{k+s}\big)$. ◻ ### Monotonicity of the ${\mathcal A}^{k}_{\mathrm {hor}}$ {#sss:monot:AA^k} Since $\iota\colon U^{k+1}\to U^k$ is an embedding in a neighborhood of ${\mathfrak K }_F$, we *identify up to homotopy* $U^k\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$ and $U^s\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$ for all $k,s$. In particular, horizontal arcs $\alpha$ in $U^k\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$ are naturally viewed as horizontal arcs in $U^s\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$ by realizing the $\alpha$ in a small neighborhood of ${\mathfrak K }_F$ where $\iota^{s-k}$ is an embedding. It follows essentially from Lemma [Lemma 13](#tranform rule for rectangles){reference-type="ref" reference="tranform rule for rectangles"} (by lifting $\iota$ to the universal coverings, see [@K §3.5] for reference) that $$\label{eq:sss:monot:AA^k} {\mathcal A}^{[n],(m),k}_{\mathrm {hor}}\ge {\mathcal A}^{[n],(m),k+1}_{\mathrm {hor}}\hspace{0.5cm} \text{ and } \hspace{0.5cm} {\mathcal A}^{[n],(m),k}_{i,{\mathrm {hor}}} \ge {\mathcal A}_{i+1,{\mathrm {hor}}}^{[n],(m),k+1}.$$ ### Domination: from $U^{k}\setminus {\boldsymbol {\mathfrak B} }^{(m)}$ to $U^{k}\setminus {\boldsymbol {\mathfrak B} }^{(m+1)}$ (following [@K §3.6]) {#sss:domination} Assume that $\gamma\subset U^{k}\setminus {\boldsymbol {\mathfrak B} }^{[n],(m)}$ is a proper horizontal curve. Then it has the decomposition $$\label{eq:sss:domination} \gamma = \ell_0\#\gamma_1\# \ell_1 \#\gamma_2\# \dots \#\gamma_s\#\ell_s$$ (compare with §[2.4.3](#sss:periodic arcs){reference-type="ref" reference="sss:periodic arcs"}) such that - $\gamma_j\subset U^{k}\setminus {\boldsymbol {\mathfrak B} }^{[n],(m+1)}$ are proper horizontal curves; and - every component of $\ell_i\setminus {\boldsymbol {\mathfrak B} }^{[n],(m+1)}$ is trivial in $U^k\setminus {\boldsymbol {\mathfrak B} }^{[n],(m+1)}$ (with respect to a proper homotopy, see §[2.4.1](#sss:AD){reference-type="ref" reference="sss:AD"}). We call $s$ the *expansivity number* of $\gamma$ in rel ${\boldsymbol {\mathfrak B} }^{(m+1)}$. It follows from Decomposition [\[eq:sss:domination\]](#eq:sss:domination){reference-type="eqref" reference="eq:sss:domination"} that there is a constant $C\equiv C_{p_n,m}$ such that for all $M>0$ $${\mathcal A}^{[n],(m),k}_{\mathrm {hor}}-{\operatorname{max}}\{M,1\}C {\ \ }{\ \ }\text{ is \emph{dominated} by }{\ \ }{\ \ }{\mathcal A}^{[n],(m+1),k}_{\mathrm {hor}}-M$$ in the following sense, see §[3.3.6](#sss:sub-diagram){reference-type="ref" reference="sss:sub-diagram"} for notation. For every arc $\alpha$ in ${\mathcal A}^{[n],(m+1),k}_{\mathrm {hor}}-{\operatorname{max}}\{M,1\}C$, there is a vertical sublamination $${\mathcal L}_\alpha\subset {\mathcal F}(e)\equiv {\boldsymbol \Pi}^{\operatorname{can}}(\alpha){\ \ }{\ \ }\text{ with }{\ \ }{\ \ }{\mathcal W}({\mathcal L}_\alpha) \ge {\mathcal W}(\alpha) - {\operatorname{max}}\{M,1\}C$$ such that Decomposition [\[eq:sss:domination\]](#eq:sss:domination){reference-type="ref" reference="eq:sss:domination"} of every $\gamma\in {\mathcal L}_\alpha$ has the additional property that $$\label{eq:dfn:domin:gamma_i} \gamma_i\in {\mathcal F}^{\operatorname{full}}\left({\mathcal A}^{[n],(m+1),k}_{\mathrm {hor}}-C\right).$$ The lamination ${\mathcal L}_a$ is constructed by removing all $\gamma$ from ${\mathcal F}(\alpha)$ that do not satisfy [\[eq:dfn:domin:gamma_i\]](#eq:dfn:domin:gamma_i){reference-type="eqref" reference="eq:dfn:domin:gamma_i"} -- the width of removed curves is bounded by ${\operatorname{max}}\{M,1\}C$. ### Almost periodic rectangles {#sss:PeriodRect} Consider a $\psi^\bullet$-ql map $F\colon U \rightrightarrows V$ as in [\[eq:dfn:psib-ql\]](#eq:dfn:psib-ql){reference-type="eqref" reference="eq:dfn:psib-ql"}. We say that a proper rectangle $R$ in $V\setminus {\boldsymbol {\mathfrak B} }^{[1]}$ is almost periodic if most of the width of $R$ overflows its iterative lift; more precisely: **Definition 22**. *For a $\psi^\bullet$-ql map $F\colon U \rightrightarrows V$, a proper rectangle $R\subset V\setminus {\boldsymbol {\mathfrak B} }^{[1]}$ connecting ${\mathfrak B }^{[1]}_a,{\mathfrak B }^{[1]}_{a+1}$ is called *$\delta$-almost periodic* if $R$ represents a genuine periodic arc (of some AD, see §[2.4.3](#sss:periodic arcs){reference-type="ref" reference="sss:periodic arcs"}) and $R$ has a vertical sublamination ${\mathcal G}\subset {\mathcal F}(R)$ with ${\mathcal W}({\mathcal G})\ge (1-\delta){\mathcal W}(R)$ such that Conditions [\[c1:dfn:s delta inv rect\]](#c1:dfn:s delta inv rect){reference-type="ref" reference="c1:dfn:s delta inv rect"}, [\[c3:dfn:s delta inv rect\]](#c3:dfn:s delta inv rect){reference-type="ref" reference="c3:dfn:s delta inv rect"} stated below hold for all $s\le 10 \overline p$, where $\overline p$ is the combinatorial bound §[2.2.5](#sss:infin renorm){reference-type="ref" reference="sss:infin renorm"}.* 1. *[\[c1:dfn:s delta inv rect\]]{#c1:dfn:s delta inv rect label="c1:dfn:s delta inv rect"} Under the immersion, ${\mathcal G}$ lifts to the lamination $${\mathcal G}^s\coloneqq (\iota^{sp_1})^*({\mathcal G})\subset U^{sp_1}\setminus {\boldsymbol {\mathfrak B} }^{[1]}$$ still connecting ${\mathfrak B }^{[1]}_a,{\mathfrak B }^{[1]}_{a+1}$.* *Let $R^{(s)}\subset U^{(sp_1)}\setminus {\boldsymbol {\mathfrak B} }^{[1],(sp_1)}$ be the periodic lift of $R$ under $f^{s p_1}$; i.e., $R^{(s)}$ connects ${\mathfrak B }^{[1],(sp_1)}_a,{\mathfrak B }^{[1],(sp_1)}_{a+1}$.* 1. *[\[c3:dfn:s delta inv rect\]]{#c3:dfn:s delta inv rect label="c3:dfn:s delta inv rect"} The lamination ${\mathcal G}^s$ overflows $R^{(s)}$ as follows: every curve $\gamma$ in ${\mathcal G}^s$ is the concatenation $\gamma=\ell_a\#\gamma'\#\ell_{a+1}$ such that* - *$\gamma'\in {\mathcal F}^{\operatorname{full}}\big(R^{(s)}\big)$, see §[3.2](#ss:rectangles){reference-type="ref" reference="ss:rectangles"}; and* - *every component of $\ell_a\setminus {\mathfrak B }^{[1],(sp_1)}_a$ and every component of $\ell_{a+1}\setminus {\mathfrak B }^{[1],(sp_1)}_{a+1}$ is trivial in $U^{sp_1}\setminus {\boldsymbol {\mathfrak B} }^{[1],(sp_1)}$ with respect to a proper homotopy, see §[2.4.1](#sss:AD){reference-type="ref" reference="sss:AD"}.* By Lemma [Lemma 5](#lem:per arcs){reference-type="ref" reference="lem:per arcs"}, the first renormalization of $F$ is satellite and ${\mathfrak B }^{[1]}_a, {\mathfrak B }^{[1]}_{a+1}$ are neighboring bushes with respect to the cyclic order. **Remark 23**. *An almost periodic rectangle $R$ between little bushes ${\mathfrak B }^{[n]}_a,{\mathfrak B }^{[n]}_{a+1}$ is defined in the same way as in the case $n=1$. Such a rectangle $R$ can be lifted to the dynamical plane of $F_{n-1,c}$ (via the covering map representing the $\psi^\bullet$-renormalization, see §[3.4.3](#sss:psi b renorm){reference-type="ref" reference="sss:psi b renorm"}) and its lift will be an almost periodic rectangle between level $1$ little bushes as in Definition [Definition 22](#dfn:s delta inv rect){reference-type="ref" reference="dfn:s delta inv rect"} .* # Pull-off for non-periodic rectangles {#s:Pull off for non-periodic rect} In this section we will establish the following theorem that refines a result from [@K]: **Theorem 24**. *For every bound $\bar p$ on renormalization periods as in §[2.2.5](#sss:infin renorm){reference-type="ref" reference="sss:infin renorm"}, every small $\delta>0$, and every $n\gg_{\bar p, \delta} 1$, the following holds.* *Let $F=(f,\iota)\colon U\rightrightarrows V$ be a $\psi^\bullet$-ql map $n+1$ times renormalizable of type $\bar p$, and let $F_{n,i}=(f_{n,i},\iota_{n,i})\colon U_{n,i}\rightrightarrows V_{n,i}$ be its $n$th $\psi^\bullet$-renormalization, see §[3.4.3](#sss:psi b renorm){reference-type="ref" reference="sss:psi b renorm"}. If $${\mathcal W}_\bullet (F_{n,i}) =K\gg_{\bar p ,\delta, n} 1,$$ then* 1. *[\[case:P:thm:prim pull-off\]]{#case:P:thm:prim pull-off label="case:P:thm:prim pull-off"}either ${\mathcal W}_\bullet(F)\ge 2K$;* ```{=html} <!-- --> ``` 1. *[\[case:S:thm:prim pull-off\]]{#case:S:thm:prim pull-off label="case:S:thm:prim pull-off"}or the $n$th renormalization of $F$ is satellite and the $(n-1)$-th $\psi^\bullet$-ql renormalization $F_{n-1}$ has a $\delta$-almost periodic rectangle $R$ with ${\mathcal W}(R) \ge K/20$ (see Definition [Definition 22](#dfn:s delta inv rect){reference-type="ref" reference="dfn:s delta inv rect"}) between two neighboring bushes in its satellite flower.* If the $n$th renormalization of $F$ is primitive, then $F_{n-1}$ has no periodic rectangles; i.e., Case [\[case:P:thm:prim pull-off\]](#case:P:thm:prim pull-off){reference-type="ref" reference="case:P:thm:prim pull-off"} holds. *Proof.* (See also §[1.3.1](#sss:intro:PrimPullOff){reference-type="ref" reference="sss:intro:PrimPullOff"}.) Consider the periodic cycle ${\boldsymbol {\mathfrak B} }^{[n]}$ of level $n$ little bushes in the dynamical plane of $F\colon U\rightrightarrows V$. Recall from [\[eq:F\^k\]](#eq:F^k){reference-type="eqref" reference="eq:F^k"} that $F^k\colon U^k\rightrightarrows V$ represents the $k$th iteration of $F$. The degeneration of $U^k\setminus {\boldsymbol {\mathfrak B} }^{[n]}$ is described by the WAD ${\mathcal A}^k\equiv {\mathcal A}^{[n],k}$; let us consider its horizontal and vertical parts ${\mathcal A}^{k}_{\mathrm {hor}}, {\mathcal A}^k_{\mathrm {ver}}$, see details in §[4.1](#sss:alignment with HT){reference-type="ref" reference="sss:alignment with HT"}. Since the $U^k\setminus {\boldsymbol {\mathfrak B} }^{[n]}$ decrease, so are the arc diagrams representing ${\mathcal A}^k_{\mathrm {hor}}$; i.e., the $A^k_{\mathrm {hor}}\equiv {\operatorname{AD}}\left({\mathcal A}^k_{\mathrm {hor}}\right)$ stabilize with $k\le 3p_n$. And since $f^{-1}(U^k\setminus {\boldsymbol {\mathfrak B} }^{[n]})\subset U^{k+1}\setminus {\boldsymbol {\mathfrak B} }^{[n]}$, the $A^k_{\mathrm {hor}}$ stabilize at an invariant AD; i.e., $A^k_{\mathrm {hor}}$ is aligned with the Hubbard tree of the superattracting model. Consider the local part ${\mathcal A}^k_i$ of ${\mathcal A}^k$ around ${\mathfrak B }^n_i$. Write $q\coloneqq (10 \overline p +3 )p_n$. If for all $i$ we have $$\label{eq:LC:prim pull-off} {\mathcal W}({\mathcal A}^{q}_i)-{\mathcal W}({\mathcal A}^0_i)\le \delta_1 K,{\ \ }{\ \ }{\ \ }\delta_1\ll \delta,$$ then most of the degeneration in ${\mathcal A}^0_i$ is represented by genuine periodic arcs §[2.4.3](#sss:periodic arcs){reference-type="ref" reference="sss:periodic arcs"}; this leads to Case [\[case:S:thm:prim pull-off\]](#case:S:thm:prim pull-off){reference-type="ref" reference="case:S:thm:prim pull-off"}, see §[4.3](#prf:P:thm:prim pull-off){reference-type="ref" reference="prf:P:thm:prim pull-off"}. If [\[eq:LC:prim pull-off\]](#eq:LC:prim pull-off){reference-type="eqref" reference="eq:LC:prim pull-off"} does not hold some $i$, then ${\mathcal W}({\mathcal A}^q_{i, {\mathrm {ver}}})\ge \delta_1 K$ and applying the Covering Lemma we obtain $${\mathcal W}\left(V\setminus {\mathfrak K }_j^{[n]}\right)\asymp_{\delta_1, \overline p}\ K\hspace{0.5cm }\text{ for all }j.$$ Applying the Quasi-Additivity Law and using $p_n\gg_{\delta_1,\overline p} 1$, we obtain $${\mathcal W}(F)\ge \sum_j {\mathcal W}\left(V\setminus {\mathfrak K }_j^{[n]}\right) \succeq_{\delta_1, \overline p} \ p_n K\gg 2K,$$ see §[4.2](#ss:CoverLmm){reference-type="ref" reference="ss:CoverLmm"}; this is Case [\[case:P:thm:prim pull-off\]](#case:P:thm:prim pull-off){reference-type="ref" reference="case:P:thm:prim pull-off"}. Below we will make a more technical exposition. ◻ ## Alignment of WAD with the Hubbard continuum {#sss:alignment with HT} Following §[2.3.1](#sss:LittleBushes){reference-type="ref" reference="sss:LittleBushes"}, let $${\boldsymbol {\mathfrak B} }\equiv{\boldsymbol {\mathfrak B} }^{[n],(0)} {\ \ }{\ \ }\text{ and } {\ \ }{\ \ }{\boldsymbol {\mathfrak B} }^{(m)}\equiv{\boldsymbol {\mathfrak B} }^{[n],(m)}=f_X^{-m} ({\boldsymbol {\mathfrak B} })$$ be the periodic cycle of level-$n$ bushes and its preimage, where $f_X=[f\colon X\to Y]$ is a ql restriction of $f$, see [\[eq:dfn:ql b domain\]](#eq:dfn:ql b domain){reference-type="eqref" reference="eq:dfn:ql b domain"} in §[3.4.2](#sss:qlb:Defn){reference-type="ref" reference="sss:qlb:Defn"}. The similar suppression of induces is applied to WAD from §[3.5.2](#sss:WADs){reference-type="ref" reference="sss:WADs"}. Recall from §[3.5.4](#sss:WAD:covering:psi maps){reference-type="ref" reference="sss:WAD:covering:psi maps"} and §[3.5.6](#sss:monot:AA^k){reference-type="ref" reference="sss:monot:AA^k"} that we have $$\label{eq:covering+monoton:AA} {\mathcal A}^{(m+1),k+1}_{{\mathrm {hor}}}= f^*\left({\mathcal A}^{(m),k}_{{\mathrm {hor}}}\right){\ \ }{\ \ }\text{ and }{\ \ }{\ \ }{\mathcal A}^{(m),k+1}_{\mathrm {hor}}\ge {\mathcal A}^{(m),k}_{\mathrm {hor}}.$$ Let us choose a sufficiently big threshold $C\gg_{p_n} 1$. By [\[eq:covering+monoton:AA\]](#eq:covering+monoton:AA){reference-type="eqref" reference="eq:covering+monoton:AA"}, the ADs forming ${\mathcal A}^k_{\mathrm {hor}}-C^k$ decrease, hence for a certain $$\label{eq:t:main est} {\mathbf t}\le \ \big[\text{maximal number of horizontal arcs on $V\setminus {\boldsymbol {\mathfrak B} }_n$}\big]\ \le 3p_n$$ the arc diagram $$\label{eq:dfn:HH} H={\operatorname{AD}}({\mathcal H}), \hspace{0.5cm} \text{ where }\hspace{0.5cm} {\mathcal H}\coloneqq {\mathcal A}^{\mathbf t}_{\mathrm {hor}}- C^{\mathbf t}$$ coincides with the AD of ${\mathcal A}^{{\mathbf t}+1}_{\mathrm {hor}}- C^{{\mathbf t}+1}$. Since the WAD ${\mathcal A}^{{\mathbf t}+1}_{\mathrm {hor}}- C^{{\mathbf t}+1}$ on $U^{{\mathbf t}+1}\setminus {\boldsymbol {\mathfrak B} }$ is dominated by the WAD $f^* \left({\mathcal H}\right)=f^*\left({\mathcal H}-{\mathcal A}^{\mathbf t}_{\mathrm {hor}}- C^{\mathbf t}\right)$ on $U^{{\mathbf t}+1}\setminus {\boldsymbol {\mathfrak B} }^{(1)}$, see §[3.5.7](#sss:domination){reference-type="ref" reference="sss:domination"}, we obtain that $H$ is an invariant AD, and therefore it is aligned with the Hubbard continuum by Lemma [Lemma 4](#lem:Alignment){reference-type="ref" reference="lem:Alignment"}. ## Case [\[case:P:thm:prim pull-off\]](#case:P:thm:prim pull-off){reference-type="ref" reference="case:P:thm:prim pull-off"} {#ss:CoverLmm} Consider local WAD ${\mathcal A}_{i}^{k}$ as in §[3.5.2](#sss:WADs){reference-type="ref" reference="sss:WADs"}, we have by §[3.5.3](#sss:Width:F_ni){reference-type="ref" reference="sss:Width:F_ni"} $${\mathcal W}({\mathcal A}_{i}^{k}) \ge {\mathcal W}({\mathcal A}_{i}^{0})={\mathcal W}_\bullet(F_{n,i})-O_{p_n}(1)\ge K/2-O_{p_n}(1).$$ Write $q\coloneqq (10 \overline p +3 )p_n$ and fix a sufficiently small $\delta_1>0$. Assume that: 1. [\[case:P:loc:thm:prim pull-off\]]{#case:P:loc:thm:prim pull-off label="case:P:loc:thm:prim pull-off"} there is a $\kappa$ such that $${\mathcal W}({\mathcal A}_{\kappa,{\mathrm {ver}}}^{ q})\ge \delta_1 K.$$ Since all local weights are comparable §[3.5.3](#sss:Width:F_ni){reference-type="ref" reference="sss:Width:F_ni"}, we have the Collar Assumption: $$\label{eq:collar assumpt} {\mathcal W}\left(V\setminus \bigcup_{j\not=i}{\mathfrak B }_j, {\ \ }{\mathfrak B }_i \right)\asymp K\hspace{0.8cm} \text{ for all $i$.}$$ Since ${\mathcal W}({\mathcal A}_{\kappa,{\mathrm {ver}}}^{ q+j})\ge {\mathcal W}({\mathcal A}_{\kappa,{\mathrm {ver}}}^{ q}) \ge \delta_1 K$ and §[\[eq:collar assumpt\]](#eq:collar assumpt){reference-type="ref" reference="eq:collar assumpt"} holds, we obtain from the [@covering; @lemma Covering Lemma] that $${\mathcal W}(V, {\mathfrak B }_{\kappa+i})\succeq_{\delta_1, \overline p} \ {\mathcal W}(U^{q+i}, {\mathfrak B }_{\kappa}) \ge {\mathcal W}({\mathcal A}_{\kappa,{\mathrm {ver}}}^{ q+i}) -O_{p_n}(1) \succeq_{\delta_1, \overline p}\ K$$ for all $i\in \{0,1, \dots, p_n-1\}$. Therefore, $$\sum_{j=0}^{p_{n}}{\mathcal W}(V, {\mathfrak B }_{j})\succeq_{\delta_1, \overline p}\ p_n K.$$ Applying [@covering; @lemma Quasi Additivity Law with separation] together with [\[eq:collar assumpt\]](#eq:collar assumpt){reference-type="eqref" reference="eq:collar assumpt"} and using $p_n\gg _{\delta_1,\overline p}\ 1$, we obtain $${\mathcal W}(V\setminus {\boldsymbol {\mathfrak B} })\succeq_{\delta_1,\overline p }\ \sum_{j=0}^{p_{n}}{\mathcal W}(V, {\mathfrak B }_{j})\succeq_{\delta_1, \overline p}\ p_n K\ge 2K.$$ We conclude that $${\mathcal W}^\bullet(F) = {\mathcal W}( V,\ {\mathfrak B }_F)\ge {\mathcal W}( V,\ {\boldsymbol {\mathfrak B} }) \ge 2K.$$ This establishes Case [\[case:P:thm:prim pull-off\]](#case:P:thm:prim pull-off){reference-type="ref" reference="case:P:thm:prim pull-off"} from [\[case:P:loc:thm:prim pull-off\]](#case:P:loc:thm:prim pull-off){reference-type="ref" reference="case:P:loc:thm:prim pull-off"}. ## Case [\[case:S:thm:prim pull-off\]](#case:S:thm:prim pull-off){reference-type="ref" reference="case:S:thm:prim pull-off"} {#prf:P:thm:prim pull-off} Let us now assume that [\[case:P:loc:thm:prim pull-off\]](#case:P:loc:thm:prim pull-off){reference-type="ref" reference="case:P:loc:thm:prim pull-off"} does not hold: for all $i$ we have ${\mathcal W}({\mathcal A}_{i,{\mathrm {ver}}}^{q})\le \delta_1 K$. This implies that most paths in the canonical lamination ${\mathcal F}({\mathcal A}_{i})$ of ${\mathcal A}_i$ are horizontal and they restrict to horizontal paths under the immersions $\iota^s\colon (U^s,{\boldsymbol {\mathfrak B} })\to (V,{\boldsymbol {\mathfrak B} })$ for $s\le q$. We will use notations of §[2.4.4](#sss:in out: sat flowers){reference-type="ref" reference="sss:in out: sat flowers"}. Let ${\mathcal H}$ be the WAD from [\[eq:dfn:HH\]](#eq:dfn:HH){reference-type="eqref" reference="eq:dfn:HH"} on $(U^{\mathbf t}, {\boldsymbol {\mathfrak B} })$. Consider an arc $e$ in $H$ with maximal ${\mathcal W}(e)$. Since the number of arcs in $H$ is bounded by $3p_n$ and ${\mathcal W}({\mathcal H})\ge p_nK/4 - \delta_1 p_n K$ by [\[eq:LocWidth is compar\]](#eq:LocWidth is compar){reference-type="eqref" reference="eq:LocWidth is compar"}, we obtain that ${\mathcal W}(e)\ge K/14$. We claim that $e$ is within a periodic satellite flower ${\mathfrak B }^{[n-1]}_c$. In particular, the $n$-th renormalization of $f$ is satellite. Assume converse: $e$ is not in any satellite flower ${\mathfrak B }^{[n-1]}_c$. By Lemma [Lemma 6](#lem:sss:in out:sat flower){reference-type="ref" reference="lem:sss:in out:sat flower"}, $e$ overflows arcs $e_1,e_2\in (f^{2p_n})^*(H)$ with $f^{2p_n}(e_1)=f^{2p_n}(e_2)=e_{\operatorname{new}}\in H$. By Lemma [Lemma 21](#lem:restr:buffers){reference-type="ref" reference="lem:restr:buffers"}, $\big(\iota^{2p_n}\big)^*\left[{\mathcal F}(e-\delta_1K)\right]$ is a horizontal family of curves in $U^{{\mathbf t}+2p_n}\setminus {\boldsymbol {\mathfrak B} }$; after removing $O_{p_n}(1)$ curves, this family is dominated by $(f^{2p_n})^*(H)$. Therefore, by the Series Law: $${\operatorname{max}}\left({\mathcal W}(e_1), \ {\mathcal W}(e_2)\right) \ge 2 {\mathcal W}(e) - 2.1\ \delta_1K.$$ Applying $f^{2p_1}$, we obtain the contradiction: ${\mathcal W}(e_{\operatorname{new}})>{\mathcal W}(e)$. Consider now a satellite flower ${\mathfrak B }^{[n-1]}_c$. As in §[2.4.4](#sss:in out: sat flowers){reference-type="ref" reference="sss:in out: sat flowers"}, we denote by ${\mathcal H}_c$ the WAD consisting of arcs of ${\mathcal H}$ that are in ${\mathfrak B }^{[n-1]}_c$. Write $q'=10 \overline p p_n$. Let ${\mathcal H}^{(q')}_c$ be the WAD consisting of arcs of $\big(f^{q'}\big)^*{\mathcal H}$ that are in ${\mathfrak B }^{[n-1]}_c$. By Lemma [Lemma 21](#lem:restr:buffers){reference-type="ref" reference="lem:restr:buffers"}, the restriction $(\iota^{q'})^*\left[{\mathcal H}_c-\delta_1 K \right]$ consists of horizontal curves. Since $H={\operatorname{AD}}({\mathcal H})$ is aligned with the Hubbard dendrite, $(\iota^{q'})^*\left[{\mathcal H}_c-\delta_1 K \right]$ is after removing $O_{p_n}(1)$ curves, dominated by ${\mathcal H}^{(q')}_c$, see §[3.5.7](#sss:domination){reference-type="ref" reference="sss:domination"}. And since $(f^{q'})_*\colon {\mathcal H}^{q'}_c\to {\mathcal H}_c$ is a bijection, as most $\overline p \delta_1K + 2C^2< \delta_2 K$ curves in ${\mathcal H}_c$ can have expansivity number grater than $2$, where $\delta_1\ll \delta_2\ll \delta$. This implies that most curves in ${\mathcal F}({\mathcal H}_c)$ are within rectangles representing genuine periodic arcs §[2.4.3](#sss:periodic arcs){reference-type="ref" reference="sss:periodic arcs"} that connect neighboring level $n$ bushes of ${\mathfrak B }^{[n-1]}_c$, see Lemma [Lemma 5](#lem:per arcs){reference-type="ref" reference="lem:per arcs"}. This also demonstrate that all the ${\mathcal H}_c$ have comparable width -- the difference of their weights are bounded by $\delta_2 K$. Since one of the ${\mathcal H}$ has an edge $e$ with ${\mathcal W}(e)\ge K/14$, the map $F\colon U^{{\mathbf t}+1}\rightrightarrows U^{\mathbf t}$ has a $\delta/2$-almost periodic rectangle $R$ between neighboring level $n$ bushes of ${\mathfrak B }^{[n-1]}_0$ with ${\mathcal W}(R)\ge K/15$ and satisfying Remark [Remark 23](#rem:dfn:s delta inv rect){reference-type="ref" reference="rem:dfn:s delta inv rect"}. ### Pushing forward $R$ into $V\setminus{\boldsymbol {\mathfrak B} }^{n}$ and then into the dynamical plane of $F_{n-1}$ Since ${\mathcal W}({\mathcal A}_{i,{\mathrm {ver}}}^{q})\le \delta_1 K$, curves in ${\mathcal F}\big({\mathcal A}_{i,{\mathrm {hor}}}-2\delta_1 K\big)$ restrict under $(\iota^{\mathbf t})^*$ to horizontal curves in $(U^{\mathbf t},{\boldsymbol {\mathfrak B} })$. Since ${\mathcal A}^{{\mathbf t}}_i \le {\mathcal A}_i$, see [\[eq:sss:monot:AA\^k\]](#eq:sss:monot:AA^k){reference-type="eqref" reference="eq:sss:monot:AA^k"}, we obtain from Lemma [Lemma 21](#lem:restr:buffers){reference-type="ref" reference="lem:restr:buffers"} that $${\mathcal F}\big({\mathcal A}^{\mathbf t}_{i,{\mathrm {hor}}}-2 \delta_1 K -2\big) \subset\ (\iota^{\mathbf t})^*\left[ {\mathcal F}^{\operatorname{full}}\big({\mathcal A}_{i,{\mathrm {hor}}}-2\delta_1 K\big) \right].$$ Therefore, ${\mathcal F}({\mathcal A}^{\mathbf t}_{i,{\mathrm {hor}}}-2 \delta_1 K -2)$ can be univalently pushed forward under $(\iota^{\mathbf t})_*$. By removing $\delta_1K$ buffers from $R$, we push forward $R$ into the dynamical plane of $F\colon U\rightrightarrows V$ and then push forward $R$ into the dynamical plane of $F_{n-1}$, see Remark [Remark 23](#rem:dfn:s delta inv rect){reference-type="ref" reference="rem:dfn:s delta inv rect"}. This establishes Case [\[case:S:thm:prim pull-off\]](#case:S:thm:prim pull-off){reference-type="ref" reference="case:S:thm:prim pull-off"} of the theorem. # Waves {#waves sec} Given a compact connected filled set $X\subset {\Bbb C}$, we denote by $\partial ^c X$ its Carathéodory boundary; i.e., the set of prime ends of $\widehat{\mathbb{C}}\setminus X$. A Riemann map identifies $\partial^c \widetilde X$ with the unit circle ${\Bbb T}$. For a compact connected subset $Y\subset X$, we denote $\partial^c_X Y\subset \partial^c X$ the set of prime ends of $X$ accumulating at $Y$. A *side of $Y$ rel $X$* is a connected component of $\partial^c_X Y$ viewed as a subset of ${\Bbb T}\simeq \partial ^cX$. Let $F=(f,\iota)\colon U\rightrightarrows V$ be a $\psi^\bullet$-ql map. Assume that - $({\mathfrak K }_k)_{k}$ is a forward invariant collection of periodic and preperiodic little filled Julia sets of $f$ of the same level; - $T\supset \bigcup_k {\mathfrak K }_k$ is forward invariant compact connected filled subset of the Julia set of $f$; - $T$ is $\iota$-proper; - every ${\mathfrak K }_k$ has finitely many sides in $T$. A relevant example for us is $T={\mathfrak B }_F^{(m)}$. Let us denote by $M$ the total number of sides of all ${\mathfrak K }_k$. Consider a side ${\mathfrak K }_k^\iota$ of ${\mathfrak K }_k$ in $T$. A *wave ${\mathcal S}$ above* ${\mathfrak K }_k^\iota$ is a lamination of proper paths in $U\setminus T$ such that every curve $\gamma\in {\mathcal S}$ starts and ends at $\partial^cX\setminus {\mathfrak K }_k^\iota$ and goes above ${\mathfrak K }_k^\iota$: the bounded component $O$ of $U\setminus (\gamma\cup T)$ contains ${\mathfrak K }_k^\iota$ on its Carathéodory boundary (i.e., prime ends of ${\mathfrak K }_k^\iota$ are accessible from $O$). **Lemma 25** (Wave Lemma). *Let ${\mathcal S}$ be a wave as above. Then ${\mathcal W}(U\setminus T) \succeq_M {\mathcal W}({\mathcal S}).$* *Proof.* We will use the following fact: **Lemma 26**. *Suppose $f\colon A\to B$ is a degree $m$ covering between closed annuli. Suppose $J\subset \partial A$ is an interval. Let ${\mathcal S}$ be a wave in $A$ above $J$. Then ${\mathcal S}$ contains a genuine subwave ${\mathcal S}^{\operatorname{new}}$ such that $f\mid {\mathcal S}^{\operatorname{new}}$ is injective and $${\mathcal W}({\mathcal S}^{\operatorname{new}})\ge {\mathcal W}({\mathcal S})-O(\ln m).$$* In particular, if $f\mid J$ is not injective, then ${\mathcal W}({\mathcal S})=O(\ln m)$. *Proof.* Consider universal covering maps $X\to A$ and $X\to B$. Their group of deck transformations are isomorphic to $m{\Bbb Z}$ and ${\Bbb Z}$ respectively. Let ${\mathcal S}_{k }$, $k\in m{\Bbb Z}$ be the lifts of ${\mathcal S}$ under $X\to A$; all these lifts are disjoint and permuted by $m{\Bbb Z}$. Let ${\mathcal S}_{k}$, $k\in {\Bbb Z}$ be the orbit of ${\mathcal S}$ under ${\Bbb Z}$. We **claim** that, by removing $O(\ln m)$ outermost curves from ${\mathcal S}$, we obtain ${\mathcal S}^{\operatorname{new}}$ and the new ${\mathcal S}^{\operatorname{new}}_k, k\in {\Bbb Z}$ so that $${\mathcal S}^{\operatorname{new}}_0,{\ \ }{\mathcal S}^{\operatorname{new}}_1,{\ \ }\dots ,{\mathcal S}^{\operatorname{new}}_{m-1}$$ are pairwise disjoint. Then, by the claim, all ${\mathcal S}^{\operatorname{new}}_k,k\in {\Bbb Z}$ are pairwise disjoint, i.e. $$f\colon {\mathcal S}^{\operatorname{new}}\longrightarrow f({\mathcal S}^{\operatorname{new}})={\mathcal S}^{\operatorname{new}}_0/{\Bbb Z}$$ is injective. Let us verify the claim. Let us denote by $A_1$ the component of $\partial A$ containing $J$, by $H\simeq {\Bbb Z}/m$ the group of deck transformations of $f\colon A\to B$. Then $H$ acts on $A_1$. We can decompose ${\mathcal S}={\mathcal S}^0\sqcup {\mathcal S}^1\sqcup \dots {\mathcal S}^T$ into possibly empty pairwise-disjoint laminations such that - curves in ${\mathcal S}^0$ start and end at an interval $I^0\subset A_1$ that is a fundamental interval for the action $H\curvearrowright A_1$; - curves in ${\mathcal S}^t$, $t>0$ start at an interval $I^t_- \subset A_1$ and end at an interval $I^t_+\subset A_1$ such that one of the intervals $I^t_-, I^t_+$ is within a union of $2^t-1$ fundamental intervals for the action $H\curvearrowright A_1$ and the interval $J^t\subset A_1,$ $J^t\cap J\not=\emptyset$ between $I^t_-,I^t_+$ is the union of exactly $2^t-1$ fundamental intervals of the action $H\curvearrowright A_1$; - $T= O(\ln m)$. Then ${\mathcal W}({\mathcal S}^t)=O(1)$ for $t>0$ because ${\mathcal S}^t$ crosses its shift under $H$, see [@DL2 Shift Argument §A.3]. Replacing ${\mathcal S}$ with ${\mathcal S}^0$, we obtain that curves in the ${\mathcal S}^{0}_k$ start and end at pairwise disjoint intervals. Removing an extra $0(1)$ outermost curves, we obtain that the new ${\mathcal S}^{\operatorname{new}}_l$ are pairwise disjoint. ◻ Let us assume first that ${\mathfrak K }_k$ is periodic. Fix an iteration $f^n\mid T,\ n\le M^3$ such that $f^n\mid {\mathfrak K }_k^\iota$ covers $\partial^c_T {\mathfrak K }_k$ at least twice. Consider the associated iteration $$(f^n,\iota^n)\colon U^n\rightrightarrows V,\hspace{1cm}\text{see~\eqref{eq:F^k}}.$$ Under the immersion $\iota^n$, either $\frac 1 3 {\mathcal W}({\mathcal S})$ part of ${\mathcal S}$ lifts to a vertical family of $U^n\setminus T$ or $\frac 2 3 {\mathcal W}({\mathcal S})$ part of ${\mathcal S}$ lifts univalently into the lamination ${\mathcal S}'$. In the former case, the lemma is proven: $$\label{eq:lem:Wave:prf} {\mathcal W}(V\setminus T)\ge \frac{1}{2^n}{\mathcal W}(U^n\setminus T)\ge \frac{1}{2^n 3}{\mathcal W}({\mathcal S}).$$ Assume the latter case. Let $\widetilde T$ be the full preimage of $T$ under $f^n$. Then ${\mathfrak K }_k^\iota$ splits into finitely many sides $X_1,X_2,\dots, X_t$ of ${\mathfrak K }_k$ in $\widetilde T$. Every $X_i$ maps univalently to the side $f^n(X)$ of ${\mathfrak K }_k$ in $T$ under $f^n$. Since $f^n\mid {\mathfrak K }_k^\iota$ covers $\partial^c_T {\mathfrak K }_k$ at least twice, we can choose two sides $X_a,X_b$ that map onto ${\mathfrak K }^\iota_k$. Every curve $\gamma\in {\mathcal S}'$ has first shortest subcurves $\gamma_a, \gamma_b$ (which may coincide) in $U^n\setminus \widetilde T$ above $X_a,X_b$ respectively. By Lemma [Lemma 26](#lem:WL: removing bugfers){reference-type="ref" reference="lem:WL: removing bugfers"}, the width of $\gamma$ with $\gamma_a=\gamma_b$ is $O_M(1)$. Let ${\mathcal F}'$ be the family of all $\gamma$ in ${\mathcal S}$ with $\gamma_a\not =\gamma_b$. Let ${\mathcal S}_a,{\mathcal S}_b$ be family of curves consisting of $\gamma_a$ and $\gamma_b$ with $\gamma$ in ${\mathcal S}'$. Since ${\mathcal S}'$ overflows ${\mathcal S}_a$ and then ${\mathcal S}_b$, either ${\mathcal W}({\mathcal S}_a)$ or ${\mathcal W}({\mathcal S}_b)$ is at least $4/3K-O_M(1)$. After removing $O_M(1)$ buffers and applying Lemma [Lemma 26](#lem:WL: removing bugfers){reference-type="ref" reference="lem:WL: removing bugfers"}, the waves ${\mathcal S}_a,{\mathcal S}_b$ map univalently by onto waves above ${\mathfrak K }_k^\iota$. We obtain a wave with width $\ge \frac 4 3 K-O_M(1)$ and the whole argument can be repeated with a strictly bigger wave leading eventually to the contradiction. Assume now that ${\mathfrak K }_k$ is strictly preperiodic. Let $f^n$ be the smallest iteration so that $f^n({\mathfrak K }_k)$ is periodic. As above, either the $1/3$ part of ${\mathcal S}$ lifts to a vertical family under $\iota^n$ or the $2/3$ part of ${\mathcal S}$ lifts univalently. In the former case, [\[eq:lem:Wave:prf\]](#eq:lem:Wave:prf){reference-type="eqref" reference="eq:lem:Wave:prf"} concludes the proof. In the latter case, we construct the family ${\mathcal S}_a$ as above and then we pushforward ${\mathcal S}_a$ under $f^n$. The result will be a wave above a side of a periodic Julia set. This reduces the preperiodic to the periodic case. ◻ # Pull-off for periodic rectangles {#s:Pulloff for per rectangles} Consider a $\psi$-ql like map $F=(f,\iota)\colon U\rightrightarrows V$. We assume that the first renormalization of $F$ is satellite. **Theorem 27**. *Fix a combinatorial bounds $\overline p$ on the renormalization period §[2.2.5](#sss:infin renorm){reference-type="ref" reference="sss:infin renorm"}. Then for every sufficiently small $\delta>0$ there is a $C_\delta = C_{\delta, \overline p}>1$ with $$C_\delta\to \infty{\ \ }{\ \ }\text{ as }{\ \ }{\ \ }\delta\to 0$$ such that the following holds for every $\psi^\bullet$-ql map $F$, and its $\psi^\bullet$-ql renormalizations $\ F_1={\mathcal R}^\bullet (F),\ F_2={\mathcal R}^{2\bullet }(F)$, see [\[RR bullet f\]](#RR bullet f){reference-type="eqref" reference="RR bullet f"}.* *Suppose that $R$ with ${\mathcal W}(R)\gg_{\delta, \overline p}1$ is a $\delta$-almost periodic rectangle (see Definition [Definition 22](#dfn:s delta inv rect){reference-type="ref" reference="dfn:s delta inv rect"}) in the dynamical plane of $F$ between bushes ${\mathfrak B }_{a}^{[1]}$ and ${\mathfrak B }_{a+1}^{[1]}$. Then $$\label{eq:1:thm:sat pull off} \text{either }{\ \ }{\ \ }{\ \ }{\mathcal W}_\bullet(F_2)\ge C_\delta K {\ \ }{\ \ }\text{ or }{\ \ }{\ \ }{\mathcal W}_\bullet(F)\ge C_\delta K.$$ Moreover, if $F={\mathcal R}^{n\bullet} (G)$, then we also have the alternative $$\label{eq:2:thm:sat pull off} \text{either }{\ \ }{\ \ }{\ \ }{\mathcal W}_\bullet\left[ {\mathcal R}^{n+2 \bullet} (G)\right]\ge C_\delta K {\ \ }{\ \ }\text{ or }{\ \ }{\ \ }{\mathcal W}_\bullet(F)\ge C_\delta K$$ (independent of $n$).* ## Proof of Theorem [Theorem 27](#thm:sat pull off){reference-type="ref" reference="thm:sat pull off"} {#proof-of-theorem-thmsat-pull-off} Let $\Pi\subset {\mathfrak B }_F$ be the geodesic continuum between ${\mathfrak B }^{[n]}_a$ and ${\mathfrak B }^{[1]}_{a+1}$. We will use notations from Definition [Definition 22](#dfn:s delta inv rect){reference-type="ref" reference="dfn:s delta inv rect"} such as ${\mathcal G}, {\mathcal G}^s, R^{(s)}$. ### Spiraling Numbers First we will introduce the spiraling parameters rel $\Pi$ for a curve $\gamma\in R$ shortly summarized in the following remark: $$\begin{tikzpicture}[scale=1.3] \begin{scope}[rotate =0] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (ba) at (0.5,0); \node[below] at (0.2,-0.2) {${\mathfrak B }^{[1]}_{a}$}; \draw[dashed,line width=0.8,fill, fill opacity=0.05] (0.6,0) ellipse (2cm and 0.9cm); \node[] at (-0.75,0.2){${\mathfrak B }^{[1], (q)}_{a}$}; \begin{scope}[shift={(1,2)},rotate=90,scale=0.7] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (xup) at (-0.5,0); \node[above] at (0.5,0) {${\mathfrak B }''_1$}; \end{scope} \begin{scope}[shift={(1,-2)},rotate=-90,scale=0.7] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (xdown) at (-0.5,0); \node[left] at (-0.8,0) {$\ell_1$}; \node[below] at (0.5,0) {${\mathfrak B }'_1$}; \end{scope} \draw[] (xup) -- (xdown); \begin{scope}[shift={(-0.05,0)}] \draw [blue, line width=0.8mm] (1.1,0)--(1.4,0) (1.25,-0.1) -- (1.25,0.1) ; \node[blue, below] at(1.29,-0.1) {${\mathfrak B }_i^{[2]}$}; \end{scope} \begin{scope}[shift={(0.55,0)}] \draw [blue, line width=0.8mm] (1.1,0)--(1.4,0) (1.25,-0.1) -- (1.25,0.1) ; \node[blue, below] at(1.25,-0.1) {${\mathfrak B }_{i+1}^{[2]}$}; \end{scope} \begin{scope}[shift={(1,0)}] \begin{scope}[shift={(1,2)},rotate=90,scale=0.7] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (xup) at (-0.5,0); \node[above] at (0.5,0) {${\mathfrak B }''_2$}; \end{scope} \begin{scope}[shift={(1,-2)},rotate=-90,scale=0.7] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (xdown) at (-0.5,0); \node[below] at (0.5,0) {${\mathfrak B }'_2$}; \node[left] at (-0.8,0) {$\ell_2$}; \end{scope} \draw[] (xup) -- (xdown); \end{scope} \end{scope} \begin{scope}[shift={(8,0)},xscale=-1] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (bb) at (0.5,0); \node[below] at (0.15,-0.2) {${\mathfrak B }^{[1]}_{a+1}$}; \draw[dashed,line width=0.8,fill, fill opacity=0.05] (0.6,0) ellipse (2cm and 0.9cm); \node[] at (-0,0.6){${\mathfrak B }^{[1], (q)}_{a+1}$}; \begin{scope}[shift={(1,2)},rotate=90,scale=0.7] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (xup) at (-0.5,0); \end{scope} \begin{scope}[shift={(1,-2)},rotate=-90,scale=0.7] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (xdown) at (-0.5,0); \end{scope} \draw[] (xup) -- (xdown); \begin{scope}[shift={(1,0)}] \begin{scope}[shift={(1,2)},rotate=90,scale=0.7] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (xup) at (-0.5,0); \end{scope} \begin{scope}[shift={(1,-2)},rotate=-90,scale=0.7] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (xdown) at (-0.5,0); \end{scope} \draw[] (xup) -- (xdown); \end{scope} \end{scope} \draw[] (ba)--(bb); \draw[orange,line width=1mm] (0.5,0) edge[bend left] (1,1); \draw[violet,line width=1mm] (1,1) edge[bend left] (2,0.5); \node[above,violet] at (1.5,1) {${\mathcal G}_a$}; \draw[orange,line width=1mm] (2,0.5) edge[bend right] (2.5,0.3); \draw[red,line width=1.2mm] (2.5,0.3) edge[bend right] (2.5,0.3) .. controls (3, 1) and (4,-1) .. (5.5,-0.3) ; \filldraw (4,0) circle (0.04 cm); \node[above] at (4,0) {$\alpha$}; \node[red,below]at(3.8,-0.25) {$ R^{(q)}$}; \begin{scope}[shift={(8,0)},scale=-1] \draw[orange,line width=1mm] (0.5,0) edge[bend left] (1,1); \draw[orange,line width=1mm] (1,1) edge[bend left] (2,0.5); \draw[orange,line width=1mm] (2,0.5) edge[bend right] (2.5,0.3); \end{scope} \end{tikzpicture}$$ **Remark 28**. *Recall the the pure Mapping Class Group $\operatorname{MCG}$ of a $3$-punctured sphere $S^2\setminus \{a,b,\infty\}$ is trivial. By replacing $a,b$ with small open Jordan disks $D_a,D_b$, we obtain the new group $\operatorname{MCG}(S^2\setminus (D_a\cup D_b\cup \{\infty\}))\simeq {\Bbb Z}^2$ consisting of Dehn-twists around $D_a,D_b$, see [@MCG]. This fact has the following implication.* *Let $O$ be a small disk-neighborhood of ${\mathfrak B }^{[1]}_a\cup \Pi \cup {\mathfrak B }^{[1]}_{a+1}$. Then $O'\coloneqq O\setminus ({\mathfrak B }^{[1]}_a\cup {\mathfrak B }^{[1]}_{a+1})$ is topologically a disk with two holes. Therefore, a path (simple curve) $\beta \subset O'$ from ${\mathfrak B }^{[1]}_a$ to ${\mathfrak B }^{[1]}_{a+1}$ has a simple description: it first spirals $\tilde t_a\in {\Bbb Z}$ times around ${\mathfrak B }^{[1]}_a$ and then spirals $\tilde t_{a+1}\in {\Bbb Z}$ times around ${\mathfrak B }^{[1]}_{a+1}$, where $\Pi$ can be used as a reference "path" of zero spiralings. Below we will ignore the spiraling orientation and introduce the absolute quantity $t_a=|\tilde t_a|$.* Recall §[3.3](#ss:WAD){reference-type="ref" reference="ss:WAD"} that $\gamma$ lands at the ideal boundary of $V\setminus {\boldsymbol {\mathfrak B} }^{[1]}$. Let $\gamma_\Pi\subset V\setminus {\boldsymbol {\mathfrak B} }^{[1]}$ be a path homotopic rel the endpoints to $\gamma$ in $V\setminus {\boldsymbol {\mathfrak B} }^{[1]}$ such that $\gamma_\Pi\cap \Pi$ is the minimal possible number. In other words, $\gamma_\Pi$ and $\Pi$ are in the minimal position -- this is well defined because $\Pi$ has infinitely many cut points. Consider the components $\gamma_0,\gamma_1,\dots, \gamma_{f}, \gamma_{f+1}$ of $\gamma_\Pi\setminus \Pi.$ Since $\gamma$ is properly homotopic into (a small neighborhood of) $\Pi$, there is a $t_a\le f$ such that - $\gamma_t\cup \Pi$ surrounds ${\mathfrak B }^{[1]}_a$ for $t\le t_a$; - $\gamma_t\cup \Pi$ does not surround ${\mathfrak B }^{[1]}_{a}$ for $t>t_a$. (It follows that $\gamma_t\cup \Pi$ surrounds ${\mathfrak B }^{[1]}_{a+1}$ for $t>t_a$, and only $\gamma_{t_a}\cup \Pi$ can surround both ${\mathfrak B }^{[1]}_a$ and ${\mathfrak B }^{[1]}_{a}$.) We say that $t_a=t_a(\gamma)$ is the *spiraling number of $\gamma$ around ${\mathfrak B }^{[1]}_a$.* Similarly, the *spiraling number of $\gamma\in R^{(s)}$ around ${\mathfrak B }^{[1],(sp_1)}_a$* is introduced. Below are some properties of spiraling numbers: 1. [\[prop:A:thm:sat\]]{#prop:A:thm:sat label="prop:A:thm:sat"} The spiraling numbers of $\gamma_1,\gamma_2\in R$ differ by at most $1$. 2. [\[prop:B:thm:sat\]]{#prop:B:thm:sat label="prop:B:thm:sat"} If $\gamma_s\in R^{(s)}$ is the lift of $\gamma\in R$ into $R^{(s)}$, then $t_a(\gamma^s)\le \frac{1}{2^s} t_a(\gamma)$. Indeed, [\[prop:A:thm:sat\]](#prop:A:thm:sat){reference-type="ref" reference="prop:A:thm:sat"} follows from the fact (the disjoint curves) $\gamma_1,\gamma_2$ can be simultaneously put into the minimal position with $\Pi$. And [\[prop:B:thm:sat\]](#prop:B:thm:sat){reference-type="ref" reference="prop:B:thm:sat"} follows from the fact that $f^{sp_1}\colon{\mathfrak B }^{[1],(sp_1)}_a\to {\mathfrak B }^{[1]}_a$ has degree $2^s$. ### The non-spiraling case: $t_a(\gamma)\le 4$ for all $\gamma\in R$; see Figure [\[Fg:no spiral\]](#Fg:no spiral){reference-type="ref" reference="Fg:no spiral"} for illustration. Write $q\coloneqq 10 \overline p$ and note that $qp_1>10 p_2$. Let us consider the objects introduced in Lemma [Lemma 3](#lem:bush:prim comb){reference-type="ref" reference="lem:bush:prim comb"}. It follows from [\[prop:B:thm:sat\]](#prop:B:thm:sat){reference-type="ref" reference="prop:B:thm:sat"} that curves in $R^{(3)}$ do not spiral around ${\mathfrak B }^{[1],(3)}_a$. Therefore, $\ell_2$ separates the base of $R^{(q)}$ from ${\mathfrak B }_a^{[1]}$: the base $\partial ^{h,0} R^{(q)}$ and ${\mathfrak B }_a^{[1]}$ are in different components of ${\mathfrak B }^{[1],(q)}_a\setminus \ell_2$. Let ${\mathcal G}_a$ be the restriction of ${\mathcal G}^{s}$ between $\ell_1$ and $\ell_2$ -- this lamination consists of the first shortest subarcs $\gamma'\subset \gamma$ between $\ell_1$ and $\ell_2$ for all $\gamma\in {\mathcal G}^s$. Since ${\mathcal G}^s$ consequently overflows ${\mathcal G}_a$ and then $R^{(q)}$, we obtain from the Grötzsch inequality that $$\label{eq:Grotzsch:ineq} (1-\delta){\mathcal W}(R)\le {\mathcal W}({\mathcal G}^{q})\le {\mathcal W}( R^{(q)})\oplus {\mathcal W}({\mathcal G}_a)={\mathcal W}( R)\oplus {\mathcal W}({\mathcal G}_a),$$ where ${\mathcal W}( R^{(q)})={\mathcal W}(R)$ because $R^{(q)}$ is a lift of $R$. Therefore, ${\mathcal W}({\mathcal G}_a)\ge C_{0,\delta} {\mathcal W}(R)\ge C_{0,\delta} K,$ where $C_{0,\delta}\to \infty$ as $\delta\to 0$. There are two possibilities. If a substantial part of ${\mathcal G}_a$ travels through both ${\mathfrak B }^{[2]}_i, {\mathfrak B }^{[2]}_{i+1}$, then this substantial part of ${\mathcal G}_a$ restricts to a wide lamination ${\mathcal L}$ between ${\mathfrak B }^{[2]}_i, {\mathfrak B }^{[2]}_{i+1}$. Pushing ${\mathcal L}$ with respect to $f_*$ and $\iota^*$ and using Lemma [Lemma 12](#generaltransform rules){reference-type="ref" reference="generaltransform rules"} (the number of iterations and the degree is bounded in terms of $\overline p$), we obtain a family ${\mathcal F}$ of non-trivial proper curves in $V\setminus {\boldsymbol {\mathfrak B} }^{[2]}$ starting at ${\mathfrak B }^{[2]}_0$ such that ${\mathcal W}({\mathcal F})\succeq_{\overline p} C_{0,\delta} K$. If a substantial part of ${\mathcal F}$ is vertical, then we obtain the second estimate in [\[eq:1:thm:sat pull off\]](#eq:1:thm:sat pull off){reference-type="eqref" reference="eq:1:thm:sat pull off"}, [\[eq:2:thm:sat pull off\]](#eq:2:thm:sat pull off){reference-type="eqref" reference="eq:2:thm:sat pull off"}. If a substantial part of ${\mathcal F}$ is horizontal, then we obtain the first estimate in [\[eq:1:thm:sat pull off\]](#eq:1:thm:sat pull off){reference-type="eqref" reference="eq:1:thm:sat pull off"}; applying Lemma [Lemma 20](#lem:reverse to sup chain rule){reference-type="ref" reference="lem:reverse to sup chain rule"} to ${\mathcal F}$, we obtain the first estimate in [\[eq:2:thm:sat pull off\]](#eq:2:thm:sat pull off){reference-type="eqref" reference="eq:2:thm:sat pull off"}. If a substantial part of ${\mathcal G}_a$ omits either ${\mathfrak B }^{[2]}_i$ or ${\mathfrak B }^{[2]}_{i+1}$, then we obtain a wide wave above one of the sides of ${\mathfrak B }^{[2]}_i, {\mathfrak B }^{[2]}_{i+1}$; Wave Lemma [Lemma 25](#lem:Wave){reference-type="ref" reference="lem:Wave"} implies that ${\mathcal W}_\bullet(F)\ge C_\delta K$. $$\begin{tikzpicture}[scale=1.3] \begin{scope}[rotate =0] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (ba) at (0.5,0); \node[below] at (0.2,-0.2) {${\mathfrak B }^{[1]}_{a}$}; \draw[dashed,line width=0.8,fill, fill opacity=0.05] (0.6,0) ellipse (2cm and 0.9cm); \node[] at (-0.85,0.1){${\mathfrak B }_{a}^{[1],(1)}$}; \end{scope} \begin{scope}[shift={(8,0)},xscale=-1] \draw[line width=0.8mm] (-0.5,0)--(0.5,0); \draw[line width=0.8mm] (-0.2,-0.2)--(-0.2,0.2); \draw[line width=0.8mm] (0.2,-0.2)--(0.2,0.2); \coordinate (bb) at (0.5,0); \node[below] at (0.15,-0.2) {${\mathfrak B }^{[1]}_{a+1}$}; \draw[dashed,line width=0.8,fill, fill opacity=0.05] (0.6,0) ellipse (2cm and 0.9cm); \node[] at (-0.95,-0.1){${\mathfrak B }_{a+1}^{[1],(1)}$}; \end{scope} \draw[] (ba)--(bb); \draw[orange,line width=1mm] (0.2,-0.2)edge[bend right] (0.8,0) ; \draw[violet,line width=1mm] (0.8,0) edge[bend right=60] (-1.5,0); \draw[violet,line width=1mm] (-1.5,0) edge[bend right=40] (1.5,-0.5) ; \draw[violet,line width=1mm](1.5,-0.5) edge[bend left=10] (2,0); \node[above,violet] at (0.7,0.3){${\mathcal G}_a$}; \draw[orange,line width=1mm](2,-0) edge[bend left=10] (2.5,0.3); \draw[red,line width=1.2mm](2.5,0.3) edge[bend right=50] (-1.5,0.5) ; \draw[red,line width=1.2mm] (-1.5,0.5) edge[bend right=30] (-1.5,-0.5) ; \draw[red,line width=1.2mm] (-1.5,-0.5) edge[bend right=40] (5.5,-0.3) ; \node[red,above]at(3.8,-1.2) {$R^{(1)}$}; \begin{scope}[shift={(8,0)},scale=-1] \draw[orange,line width=1mm] (0.5,0) edge[bend left] (1,1); \draw[orange,line width=1mm] (1,1) edge[bend left] (2,0.5); \draw[orange,line width=1mm] (2,0.5) edge[bend right] (2.5,0.3); \end{scope} \filldraw (4,0) circle (0.04 cm); \node[above] at (4,0) {$\alpha$}; \end{tikzpicture}$$ ### The spiraling case: $t_a(\gamma)\ge 4$ for all $\gamma\in R$; see Figure [\[Fg:spiral\]](#Fg:spiral){reference-type="ref" reference="Fg:spiral"} for illustration. It follows from [\[prop:B:thm:sat\]](#prop:B:thm:sat){reference-type="ref" reference="prop:B:thm:sat"} that the spiraling number of any curve in ${\mathcal G}^1$ around ${\mathfrak B }^{[1]}_a$ is strictly less than the spiraling number of any curve in $R^{(1)}$ around ${\mathfrak B }^{[1],(1)}_a$. Therefore, curves in ${\mathcal G}^1$ first spirals around ${\mathfrak B }_a$ before they continue within ${\mathcal F}^{\operatorname{full}}( R^{(1)})$. Applying the Grötzsch inequality as in [\[eq:Grotzsch:ineq\]](#eq:Grotzsch:ineq){reference-type="eqref" reference="eq:Grotzsch:ineq"}, we obtain a wide lamination ${\mathcal G}_a$ with ${\mathcal W}({\mathcal G}_a)\ge C_{2,\delta} {\mathcal W}(R)$ such that ${\mathcal G}_a$ creates a wave above one of the sides of ${\mathfrak B }_a$. Wave Lemma [Lemma 25](#lem:Wave){reference-type="ref" reference="lem:Wave"} implies that ${\mathcal W}_\bullet(f)\ge C_\delta K$.0◻ **Remark 29**. *The argument of this section will be repeated in §[7.0.1](#sss:prf:eq:prf:thm:psi bounds){reference-type="ref" reference="sss:prf:eq:prf:thm:psi bounds"} with ${\mathfrak B }^{[1]}_{a+1}$ being replaced by a little Julia set ${\mathfrak K }^{[1]}_0$. Observe that no specific properties of ${\mathfrak B }^{[1]}_{a+1}$ has been used in this section (only that it is forward invariant set that is disjoint from ${\mathfrak B }^{[1]}$).* # Conclusions {#s:conclusions} In this section, we will now deduce the main theorems. **Theorem 30** (A priori beau $\psi^\bullet$-bounds). *For any combinatorial bound $\bar p$, there is an $n>1$ and $K_{\bar p}>1$ such that the following holds. If $F$ is an inifinitely renormalizable $\psi^\bullet$-ql map of bounded type $\bar p$, then $${\mathcal W}_\bullet \left[ ({\mathcal R}^{n\bullet})^m (F)\right]\le K_{\overline p} {\ \ }{\ \ }{\ \ }\text{ for }{\ \ }m\ \gg_{{\mathcal W}(F)}\ 1,$$ ${\mathcal R}^{n \bullet}$ is a $\psi^\bullet$-ql renormalization [\[RR bullet f\]](#RR bullet f){reference-type="eqref" reference="RR bullet f"}.* *Proof.* Let us choose a sufficiently small $\delta>0$ such that $C_\delta\gg \Delta_{\overline p}$, where $C_\delta$ is from Theorem [Theorem 27](#thm:sat pull off){reference-type="ref" reference="thm:sat pull off"} and $\Delta_{\overline p}$ is from Proposition [Proposition 9](#prop:Teichm contr){reference-type="ref" reference="prop:Teichm contr"}. We next a choose sufficiently big $n\gg _{\overline p, \delta} 1$ such that Theorem [Theorem 24](#thm:prim pull-off){reference-type="ref" reference="thm:prim pull-off"} is applicable as follows: if $${\mathcal W}_\bullet \left[ ({\mathcal R}^{n\bullet})^m (F)\right] = K \gg_{\overline p, \delta, n} 1,$$ then 1. [\[case:1:final\]]{#case:1:final label="case:1:final"} either ${\mathcal W}_\bullet \left[ ({\mathcal R}^{n\bullet})^{(m-1)} (F)\right] \ge 2 K$; 2. [\[case:2:final\]]{#case:2:final label="case:2:final"} or ${\mathcal W}_\bullet \left[ {\mathcal R}^{n+1\bullet} \circ ({\mathcal R}^{n\bullet})^{m-1} (F)\right] \ge C_\delta K$, where [\[case:2:final\]](#case:2:final){reference-type="ref" reference="case:2:final"} follows from Case [\[case:S:thm:prim pull-off\]](#case:S:thm:prim pull-off){reference-type="ref" reference="case:S:thm:prim pull-off"} of Theorem [Theorem 24](#thm:prim pull-off){reference-type="ref" reference="thm:prim pull-off"} combined with Theorem [Theorem 27](#thm:sat pull off){reference-type="ref" reference="thm:sat pull off"}. Write $G\coloneqq ({\mathcal R}^{n\bullet})^{(m-1)} (F).$ Reapplying Theorem [Theorem 24](#thm:prim pull-off){reference-type="ref" reference="thm:prim pull-off"} for $G$ and its renormalization ${\mathcal R}^{n+1 \bullet }G$, we obtain the alternative "[\[case:1:final\]](#case:1:final){reference-type="ref" reference="case:1:final"} vs [\[case:2:b:final\]](#case:2:b:final){reference-type="ref" reference="case:2:b:final"}", where 1. [\[case:2:b:final\]]{#case:2:b:final label="case:2:b:final"} or ${\mathcal W}_\bullet \left[ {\mathcal R}^{n+2\bullet} (G)\right] \ge C^2_\delta K$. Repeating the argument, we eventually obtain the alternative "[\[case:1:final\]](#case:1:final){reference-type="ref" reference="case:1:final"} vs [\[case:2:c:final\]](#case:2:c:final){reference-type="ref" reference="case:2:c:final"}", where 1. [\[case:2:c:final\]]{#case:2:c:final label="case:2:c:final"} or ${\mathcal W}_\bullet \left[ {\mathcal R}^{2n\bullet} (G)\right] \ge C^n_\delta K$. Therefore, if there are no a priori beau $\psi^\bullet$-bounds, then we obtain $${\mathcal W}_\bullet \left[ ({\mathcal R}^{n\bullet})^m (F)\right] \succeq_F C^{nm}_\delta K.$$ This contradicts the Teichmüller contraction: by Proposition [Proposition 9](#prop:Teichm contr){reference-type="ref" reference="prop:Teichm contr"}, see also Remark [Remark 10](#rem:prop:Teichm contr){reference-type="ref" reference="rem:prop:Teichm contr"}, the sequence $({\mathcal R}^{n\bullet})^m (F)$ restricts to ql maps $f_{nm}:X_{nm}\to Y_{nm}$ such that $${\mathcal W}\big(Y_{nm}\setminus X_{nm}\big) =O_F(\Delta_{\overline p}^{nm}) ,$$ where $\Delta_{\overline p}\ll C_\delta.$ ◻ **Theorem 31** (A priori beau ql-bounds). *For any combinatorial bound $\bar p$, there is a $K_{\bar p}>1$ such that the following holds. If $f\colon U\to V$ is an inifinitely renormalizable ql map of bounded type $\bar p$, then for $n\gg_{{\mathcal W}(f)} 1$, the map $f_n$ has a ql restriction $$f_n\colon X_n\to Y_n{\ \ }{\ \ }{\ \ }\text{ with }{\ \ }{\ \ }{\mathcal W}(Y_n\setminus X_n)\le K_{\overline p}.$$* *Proof.* Write $F\coloneqq f$, and consider the sequence $G_m\coloneqq ({\mathcal R}^{n\bullet})^m (F)$ from Theorem [Theorem 30](#thm:psi bullet bounds){reference-type="ref" reference="thm:psi bullet bounds"}. We have $$\label{eq:prf:thm:psi bounds:0} {\mathcal W}_\bullet(G_m)\le K_{\overline p}\hspace{1cm}\text{ for }m\gg_{{\mathcal W}(F)} 1$$ for $K_{\overline p}$ from Theorem [Theorem 30](#thm:psi bullet bounds){reference-type="ref" reference="thm:psi bullet bounds"}. For $m$ satisfying [\[eq:prf:thm:psi bounds:0\]](#eq:prf:thm:psi bounds:0){reference-type="eqref" reference="eq:prf:thm:psi bounds:0"}, consider the dynamical plane of $G_m\colon U\to V$. Let ${\boldsymbol {\mathfrak B} }^{[1]}$ be the cycle of little bushes in the dynamical plane of $G_m$. We ** claim** that there is a space between ${\mathfrak K }^{[1]}_0$ and ${\boldsymbol {\mathfrak B} }^{[1]}\setminus {\mathfrak B }^{[1]}_0$; i.e., that there is a $K_2>0$ depending on $K_{\overline p}, \overline p$ such that $$\label{eq:prf:thm:psi bounds} {\mathcal W}\left(V\setminus \bigcup_{i\not= 0} {\mathfrak B }^{[1]}_i,\ {\mathfrak K }^{[1]}_0\right) \le K_2.$$ Then [\[eq:prf:thm:psi bounds\]](#eq:prf:thm:psi bounds){reference-type="eqref" reference="eq:prf:thm:psi bounds"} together with §[3.4.5](#sss: psi^b to psi){reference-type="ref" reference="sss: psi^b to psi"}, Lemma [Lemma 17](#lem:from psi-ql to ql){reference-type="ref" reference="lem:from psi-ql to ql"}, and Remark [Lemma 17](#lem:from psi-ql to ql){reference-type="ref" reference="lem:from psi-ql to ql"} will imply that $G_m$ has a ql renormalization around ${\mathfrak K }_0^{[1]}$ with definite modulus. ### Proof of [\[eq:prf:thm:psi bounds\]](#eq:prf:thm:psi bounds){reference-type="eqref" reference="eq:prf:thm:psi bounds"} {#sss:prf:eq:prf:thm:psi bounds} Assuming converse, the associated degeneration is arbitrary big: $$K\coloneqq {\mathcal W}\left({\mathcal G}\right)\gg_{K_{\overline p}} 1, \hspace{0.7cm}\text{ where }{\ \ }{\mathcal G}\coloneqq {\mathcal F}\left(V\setminus \bigcup_{i\not= 0} {\mathfrak B }^{[1]}_i,\ {\mathfrak K }^{[1]}_0\right).$$ We will now argue that a substantial part of ${\mathcal G}$ travels through level two little bushes; this will lead to a contradiction to the $\psi^\bullet$-bounds of Theorem [Theorem 31](#thm:psi bounds){reference-type="ref" reference="thm:psi bounds"}. Write ${\boldsymbol {\Upsilon} }\coloneqq {\boldsymbol {\mathfrak B} }^{[1]}\cup {\mathfrak K }^{[1]}_0$, and consider the horizontal and vertical WAD ${\mathcal A}^{[1],k}_{\mathrm {hor}}, {\mathcal A}^{[1],k}_{\mathrm {ver}}$ of $U^k\setminus {\boldsymbol {\Upsilon} }$. By [\[eq:prf:thm:psi bounds:0\]](#eq:prf:thm:psi bounds:0){reference-type="eqref" reference="eq:prf:thm:psi bounds:0"}, $$\label{eq:prf of the claim: last thm} {\mathcal W}({\mathcal A}^{[1],k}_{\mathrm {ver}})= O_{k,\overline p}(K_{\overline p}) \hspace{0.5cm}\text{ and hence }\hspace{0.4cm}{\mathcal W}({\mathcal A}^{[1],k}_{\mathrm {hor}})=K - O_{k,\overline p}(K_{\overline p}).$$ As in the proof of Theorems [Theorem 24](#thm:prim pull-off){reference-type="ref" reference="thm:prim pull-off"}, see §[4.1](#sss:alignment with HT){reference-type="ref" reference="sss:alignment with HT"}, the arc diagrams ${\mathcal A}^{[1],k p_1}_{\mathrm {hor}}$ eventually stabilize: $${\operatorname{AD}}\left( {\mathcal A}^{[1],(t+1) p_1}_{\mathrm {hor}}-C^{t+1} \right)= {\operatorname{AD}}\left( {\mathcal A}^{[1],t p_1}_{\mathrm {hor}}-C^{t} \right) {\ \ }{\ \ }{\ \ }\text{ for }{\ \ }t\le 3p_1,$$ where $C\gg_{p_1} 1$ is fixed. By Lemma [Lemma 7](#lem:Alignment:Ups){reference-type="ref" reference="lem:Alignment:Ups"}, $H={\operatorname{AD}}\left( {\mathcal A}^{[1],(t+1) p_1}_{\mathrm {hor}}-C^{t+1} \right)$ is invariant under $f^{p_1}$. Since most of the horizontal curves restrict to horizontal curves (by [\[eq:prf of the claim: last thm\]](#eq:prf of the claim: last thm){reference-type="eqref" reference="eq:prf of the claim: last thm"}) the same argument as in §[4.3](#prf:P:thm:prim pull-off){reference-type="ref" reference="prf:P:thm:prim pull-off"} provides a $\delta$-almost periodic rectangle $R\subset {V\setminus {\boldsymbol {\Upsilon} }},\ {\mathcal W}(R)\asymp K$ between ${\mathfrak K }^{[1]}_0$ and some ${\mathfrak B }^{[1]}_a$, where $\delta$ is sufficiently small and ${\mathfrak K }^{[1]}_0$ replacing ${\mathfrak B }^{[1]}_{a+1}$ in Definition [Definition 22](#dfn:s delta inv rect){reference-type="ref" reference="dfn:s delta inv rect"}. By Lemma [Lemma 8](#lem:per arcs:ups){reference-type="ref" reference="lem:per arcs:ups"}, the first renormalization of $G_m$ is satellite. We can now repeat the argument of Theorem [Theorem 27](#thm:sat pull off){reference-type="ref" reference="thm:sat pull off"}, see Remark [Remark 29](#rem:about thm:sat case){reference-type="ref" reference="rem:about thm:sat case"}, with ${\mathfrak K }^{[1]}_0$ replacing ${\mathfrak B }^{[1]}_{a+1}$ to obtain $$\text{either }{\ \ }{\ \ }{\ \ }{\mathcal W}_\bullet\left[ {\mathcal R}^{2 \bullet} (G_m)\right]\succeq_{\overline p} K {\ \ }{\ \ }\text{ or }{\ \ }{\ \ }{\mathcal W}_\bullet( G_m)\succeq_{\overline p} K;$$ both estimates leading to a contradiction with $\psi^\bullet$-bounds of Theorem [Theorem 30](#thm:psi bullet bounds){reference-type="ref" reference="thm:psi bullet bounds"}. ◻ \*\*\*\*\* L. Ahlfors. Conformal Invariants: Topics in Geometric Function Theory, McGraw Hill Book Co., New York, 1973. A. Avila and M. Lyubich. The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes. Publ. Math. IHES 114 (2011), 171-223. P. Coullet and C. Tresser, Itérations d'endomorphismes et groupe de renormalisation. J. Phys. Colloque. C 539, C5-25 (1978). L. Bartholdi and D. Dudko. Algorithmic aspects of branched coverings I. Van Kampen's Theorem for bisets. Groups, Geometry, and Dynamics, 12 (1), 121--172, 2018. A. Douady and J.H. Hubbard. Étude dynamique des polynômes complexes. Publication Mathematiques d'Orsay, 84-02 and 85-04. A. Douady & J.H. Hubbard. On the dynamics of polynomial-like maps. Ann. Sc. Éc. Norm. Sup., v. 18 (1985), 287 -- 343. D. Dudko and M. Lyubich. Local connectivity of the Mandelbrot set at some satellite parameter values of bounded type. GAFA (33) 2023, 912--1047. D. Dudko and M. Lyubich. Uniform a priori bounds for neutral renormalization, arXiv:2210.09280 (2022). D. Dudko, M. Lyubich and N. Selinger. Pacman renormalization and self-similarity of the Mandelbrot set near Siegel parameters. JAMS, v. 33 (2020), 653--733. M. J. Feigenbaum. Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), 25--52. M. J. Feigenbaum. The universal metric properties of nonlinear transformations, J. Statist. Phys. 21 (1979), 669--706. Benson Farb and Dan Margalit. A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. A. Goldberg, K. Khanin, and Ya. Sinai. Universal properties of sequences of period-tripling bifurcations. Russian Math. Surveys, v. 38 (1983), 159--160. J.H. Hubbard. Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz. In: "Topological Methods in Modern Mathematics, A Symposium in Honor of John Milnor's 60th Birthday\", Publish or Perish, 1993. J. Kahn. A priori bounds for some infinitely renormalizable quadratics: I. Bounded primitive combinatorics. Preprint Stony Brook, \# 5 (2006). Y. Ishii and J. Smillie. Homotopy shadowing, Amer. J. Math. 132 (2010), no. 4, 987--1029. J. Kahn & M. Lyubich. Quasi-Additivity Law in Conformal Geometry. Annals of Math., v. 169 (2009), 561--593. J. Kahn & M. Lyubich. *A priori bounds* for some infinitely renormalizable quadratics: II. Decorations. Ann. Scient. Éc. Norm. Sup., v. 41 (2008), 57 -- 84. J. Kahn & M. Lyubich. *A priori bounds* for some infinitely renormalizable quadratics: III. Molecules. In: "Complex Dynamics: Families and Friends". Proceeding of the conference dedicated to Hubbard's 60th birthday (ed.: D. Schleicher). Peters, A K, Limited, 2009. M. Lyubich. Dynamics of quadratic polynomials, I-II. Acta Math., v. 178 (1997), 185 -- 297. M. Lyubich. Feigenbaum-Coullet-Tresser Universality and Milnor's Hairiness Conjecture. Ann. Math., v. 149 (1999), 319 - 420. M. Lyubich. Conformal Geometry and Dynamics of Quadratic Polynomials. Book in preparation; www.math.stonybrook.edu/ mlyubich/book.pdf M. Lyubich. Story of the Feigenbaum point. Talk at the CIRM conference ["Advancing Bridges in Complex Dynamics"](https://www.i2m.univ-amu.fr/events/advancing-bridges-in-complex-dynamics/), <https://www.youtube.com/watch?v=YhwPuOei0rI> C. McMullen. Complex dynamics and renormalization. Princeton University Press, 1994. C. McMullen. Renormalization and three manifolds which fiber over the circle. Princeton University Press, 1996. J. Milnor. Local connectivity of Julia sets: expository lectures. In: "The Mandelbrot Set, Themes and Variations", 67-116, ed. Tan Lei. Cambridge University Press, 2000. Volodymyr Nekrashevych, Combinatorial models of expanding dynamical systems, Ergodic Theory Dynam. Systems 34 (2014), no. 3, 938--985. D. Sullivan. Quasiconformal homeomorphisms in dynamics, topology and geometry, Proc. ICM-86, Berkeley II, A.M.S., Providence, RI (1987), 1216-1228. D. Sullivan. Bounds, quadratic differentials and renormalization conjetures. AMS Centennial Publ. 2: Mathematics into Twenty-first century, 1992. D. P. Thurston. A positive characterization of rational maps, Annals of Math 192 (2020), 1--46. P. Coullet and C. Tresser, Itérations d'endomorphismes et groupe de renormalisation. C. R. Acad. Sci. Paris 287A (1978), 577--580. [^1]: This observation goes back to McMullen [@McM1 Theorem 9.3].

arxiv_math

--- abstract: | The $p$-ary linear code $\mathcal{C}_{k}\!\left(n,q\right)$ is defined as the row space of the incidence matrix $A$ of $k$-spaces and points of $\mathrm{PG}\!\left(n,q\right)$. It is known that if $q$ is square, a codeword of weight $q^k\sqrt{q}+\mathcal{O}\!\left(q^{k-1}\right)$ exists that cannot be written as a linear combination of at most $\sqrt{q}$ rows of $A$. Over the past few decades, researchers have put a lot of effort towards proving that any codeword of smaller weight *does* meet this property. We show that if $q\geqslant 32$ is a composite prime power, every codeword of $\mathcal{C}_{k}\!\left(n,q\right)$ up to weight $\mathcal{O}\!\left(q^k\sqrt{q}\right)$ is a linear combination of at most $\sqrt{q}$ rows of $A$. We also generalise this result to the codes $\mathcal{C}_{j,k}\!\left(n,q\right)$, which are defined as the $p$-ary row span of the incidence matrix of $k$-spaces and $j$-spaces, $j < k$. author: - | Sam Adriaensen\ *Vrije Universiteit Brussel* - | Lins Denaux\ *Ghent University* bibliography: - main.bib title: Small weight codewords of projective geometric codes II --- linear codes, incidence matrices, projective spaces, small weight codewords. $05$B$25$, $94$B$05$. # Introduction and overview In this article, we are interested in a particular class of linear codes, which can be defined as follows. Consider a prime power $q\coloneqq p^h$, with $p$ prime. Choose integers $0 \leqslant j < k < n$. Let $\mathrm{PG}\!\left(n,q\right)$ denote the $n$-dimensional projective space over $\mathbb{F}_{q}$, and let $\mathcal{J}$ and $\mathcal{K}$ denote the respective sets of $j$-spaces and $k$-spaces of $\mathrm{PG}\!\left(n,q\right)$. The incidence matrix of $k$- and $j$-spaces is the matrix $A$ whose rows and columns are indexed by $\mathcal{K}$ and $\mathcal{J}$ respectively, which contains a 1 in positions where the corresponding subspaces are incident, and a 0 in all other positions. Symbolically, $$\begin{aligned} A \in \{0,1\}^{\mathcal{K}\times \mathcal{J}}, && A(\kappa,\lambda) \coloneqq \begin{cases} 1 & \text{if } \lambda \subset \kappa, \\ 0 & \text{otherwise.} \end{cases}\end{aligned}$$ The codes we are interested in are the row spaces of these incidence matrices. These codes consist of vectors whose positions are labelled by the $j$-spaces of $\mathrm{PG}\!\left(n,q\right)$. It is therefore more convenient to interpret the codewords as functions from $\mathcal{J}$ to $\{0,1\}$. **Definition 1**. For every $k$-space $\kappa$ of $\mathrm{PG}\!\left(n,q\right)$, define its *characteristic function* with respect to the $j$-spaces as the function $$\kappa^{(j)}: \mathcal{J}\to \{0,1\}: \lambda \mapsto \begin{cases} 1 & \text{if } \lambda \subset \kappa, \\ 0 & \text{otherwise.} \end{cases}$$ Since $\kappa^{(j)}$ only takes the values $0$ and $1$, we can interpret it as a function from $\mathcal{J}$ to any field. We will study these characteristic functions as functions $\mathcal{J}\to \mathbb{F}_{p}$. The vector space consisting of all functions from $\mathcal{J}$ to $\mathbb{F}_{p}$ will be denoted by $\mathbb{F}_{p}^\mathcal{J}$. **Definition 2**. The code $\mathcal{C}_{j,k}\!\left(n,q\right)$ is the vector subspace of $\mathbb{F}_{p}^\mathcal{J}$ generated by the set $\left\{\kappa^{(j)}\,:\,\kappa \in \mathcal{K}\right\}$ of characteristic functions of the $k$-spaces of $\mathrm{PG}\!\left(n,q\right)$ with respect to the $j$-spaces. In case $j=0$, we denote these codes by $\mathcal{C}_{k}\!\left(n,q\right)$. We aim to characterise the small weight codewords of $\mathcal{C}_{j,k}\!\left(n,q\right)$. One way to make codewords of relatively small weight is by taking linear combinations of a small number of characteristic functions. We will say that a codeword $c \in \mathcal{C}_{j,k}\!\left(n,q\right)$ is a "linear combination of (exactly) $m$ $k$-spaces" if it can be written as a linear combination of $m$ characteristic functions of $k$-spaces, each occurring in the linear combination with a non-zero scalar. We remark that the characteristic functions (when seen as $p$-ary functions) are linearly dependent, hence if $c \in \mathcal{C}_{j,k}\!\left(n,q\right)$, there is not a *unique* linear combination of characteristic functions of $k$-spaces equal to $c$. We will begin with an overview of the known results. Most of the notation and terminology are standard. Everything will be defined in . As a first step, the codewords of minimum weight have been characterised. **Result 3** ([@AssmusKey; @DelsarteGoethalsMacWilliams] and [@BagchiInamdar Theorem 1]). *The minimum weight of $\mathcal{C}_{j,k}\!\left(n,q\right)$ is $\genfrac{[}{]}{0pt}{}{k+1}{j+1}_q$. Every minimum weight codeword is a scalar multiple of the characteristic function of a $k$-space.* Stronger characterisations are known. ## The planar case Initially, most attention was paid to the smallest set of parameters, i.e. the codes $\mathcal{C}_{1}\!\left(2,q\right)$. Several results emerged in case $q=p$ is prime, starting with @McGuireWard [@McGuireWard]. They discovered a gap in the weight spectrum by proving that no codeword of $\mathcal{C}_{1}\!\left(2,p\right)$ has weight $w\in\left\{p+2,\dots,\frac{3}{2}(p+1)\right\}$, $p\neq2$ [@McGuireWard Corollary $2.3$]. @Chouinard:PhD [@Chouinard:PhD Proposition $27$] extended this result by showing that no codeword has weight $w\in\left\{p+2,\dots,2p-1\right\}$. A decade later, @FackFancsaliStormeVandeVoordeWinne [@FackFancsaliStormeVandeVoordeWinne] generalised this result by proving that if $p\geqslant 11$, any codeword of $\mathcal{C}_{1}\!\left(2,p\right)$ of weight smaller than $\frac{5}{2}p$ is equal to a linear combination of at most two lines. Add another decade, @Bagchi:FourthWeight [@Bagchi:FourthWeight] extended this result to all codewords of weight smaller than $3p-3$, $p\geqslant 5$. Generally, researchers try to prove that any codeword $c\in\mathcal{C}_{1}\!\left(2,q\right)$ whose weight is upper bounded by some function $W\!\left(q\right)$ is a linear combination of exactly $\left\lceil\frac{\mathrm{wt}\left(c\right)}{q+1}\right\rceil$ lines, which are relatively few. In [-@Key], @Key [@Key] proved that the characteristic function of a Hermitian variety[^1] is a codeword of $\mathcal{C}_{n-1}\!\left(n,q^2\right)$, while @BlokhuisBrouwerWilbrink [@BlokhuisBrouwerWilbrink] showed that any unital $\mathcal{H}$ of $\mathrm{PG}\!\left(2,q^2\right)$ is a non-singular Hermitian curve if and only if its characteristic function $v_\mathcal{H}$ is a codeword of $\mathcal{C}_{1}\!\left(2,q^2\right)$, or, in other words, if and only if $v_\mathcal{H}$ is equal to a $p$-ary linear combination of characteristic functions of lines. One can easily prove that any linear combination of lines equal to $v_\mathcal{H}$ must consist of at least $q^2-q+1$ lines, which is substantially larger than $\left\lceil\frac{\mathrm{wt}\left(v_\mathcal{H}\right)}{q^2+1}\right\rceil=q$ and implies that $W\!\left(q\right)$ cannot be larger than $q\sqrt{q}$ if $q$ is square. @Bagchi:OddCodeword [@Bagchi:OddCodeword Theorem $5.2$] and @DeBoeckVandendriessche [@DeBoeck:PhD Example $10.3.4$], [@DeBoeckVandendriessche Example $1.8$] independently discovered a peculiar codeword $c \in \mathcal{C}_{1}\!\left(2,p\right)$ of weight $3p-3$ that cannot be written as a linear combination of fewer than $p-1$ lines. If $p>3$, then $p-1$ is larger than $\left\lceil\frac{\mathrm{wt}\left(c\right)}{p+1}\right\rceil\leqslant 3$, implying that $W\!\left(p\right)$ is at most $3p-3$ if $p\geqslant 5$ is prime. Using polynomial methods, @SzonyiWeiner contributed considerably to the characterisation of small weight codewords of $\mathcal{C}_{1}\!\left(2,q\right)$ for somewhat larger values of $q$. **Result 4** ([@SzonyiWeiner Theorems $4.3$, $4.8$ and Corollary $4.10$]). *Let $c$ be a codeword of $\mathcal{C}_{1}\!\left(2,q\right)$, $q=p^h$, $p$ prime.* - *If $h=1$, $p\geqslant 19$ and $\mathrm{wt}\!\left(c\right)\leqslant\max\left\{3p+1,4p-22\right\}$, then $c$ is either a linear combination of at most three lines or a certain generalisation of the peculiar codeword described above.* - *If $h\geqslant 2$, $q\geqslant 32$ and $$\mathrm{wt}\!\left(c\right)<\begin{cases} \frac{\left(p-1\right)\left(p-4\right)\left(p^2+1\right)}{2p-1}&\text{if}\ h=2,\\ \left(\left\lfloor\sqrt{q}\right\rfloor+1\right)\left(q+1-\left\lfloor\sqrt{q}\right\rfloor\right)&\text{if}\ h\geqslant 3, \end{cases}$$ then $c$ is a linear combination of exactly $\left\lceil\frac{\mathrm{wt}\left(c\right)}{q+1}\right\rceil$ lines.* Hence, if $q$ is neither small nor prime, the above result characterises all codewords of $\mathcal{C}_{1}\!\left(2,q\right)$ up to weight $W\!\left(q\right)=\mathcal{O}\!\left(q\sqrt{q}\right)$. If $q\geqslant 32$ and if $h\geqslant 4$ is even, then the result is sharp, as illustrated by the characteristic functions of the Hermitian curves. ## The general case Consider a codeword $c\in\mathcal{C}_{1}\!\left(2,q\right)$ and embed $\mathrm{PG}\!\left(2,q\right)$ as a plane $\pi$ in $\mathrm{PG}\!\left(n,q\right)$. By fixing a $\left(k-2\right)$-space $\Pi$ disjoint to $\pi$, one can construct from $c=\sum_{i} \alpha_i \ell_i^{(0)}$ a codeword $c' \coloneqq \sum_i \alpha_i \left\langle\ell_i,\Pi\right\rangle^{(0)} \in \mathcal{C}_{k}\!\left(n,q\right)$ of weight $\mathrm{wt}\!\left(c\right)q^{k-1}$ or $\mathrm{wt}\!\left(c\right)q^{k-1}+\theta_{k-2}$, depending on whether or not $\Pi$ avoids $\mathrm{supp}\!\left(c\right)$, or equivalently whether or not $\sum_i \alpha_i = 0$. Therefore, the observations made in the planar case can be related to the more general case of the codes $\mathcal{C}_{k}\!\left(n,q\right)$. One commonly tries to prove that any codeword $c\in\mathcal{C}_{k}\!\left(n,q\right)$ of weight at most (some function) $W\!\left(k,n,q\right)$ is equal to a linear combination of exactly $\left\lceil\frac{\mathrm{wt}\left(c\right)}{\theta_k}\right\rceil$ $k$-spaces. Moreover, $W\!\left(k,n,q\right)$ must be smaller than $q^k\sqrt{q}+\theta_{k-1}$ if $q$ is square, and smaller than $\left(3q-3\right)q^{k-1}$ if $q\geqslant 5$ is prime. Characterising small weight codewords of $\mathcal{C}_{k}\!\left(n,q\right)$, $n\geqslant 3$, has gained some popularity in recent years. We present a short overview based on the survey article of @LavrauwStormeVandeVoorde:LinearCodes [@LavrauwStormeVandeVoorde:LinearCodes]. While this was already utilized in the planar case, @LavrauwStormeVandeVoorde:PointsHyperplanes [@LavrauwStormeVandeVoorde:PointsHyperplanes; @LavrauwStormeVandeVoorde:PointsKspaces] exploited a strong link between codewords of $\mathcal{C}_{k}\!\left(n,q\right)$ of small weight and blocking sets. One year later, @LavrauwStormeSziklaiVandeVoorde [@LavrauwStormeSziklaiVandeVoorde Theorem $12$] proved that there exist no codewords in $\mathcal{C}_{k}\!\left(n,q\right)\setminus\mathcal{C}_{n-k}\!\left(n,q\right)^\perp$, $p>5$, with weight in the interval $\left]\theta_k,2q^k\right[$. As pointed out in [@LavrauwStormeVandeVoorde:LinearCodes Theorem $3.12$], using a known lower bound on the minimum weight of $\mathcal{C}_{n-k}\!\left(n,q\right)^\perp$ [@BagchiInamdar Theorem $3$], one can show that there exist no codewords of $\mathcal{C}_{k}\!\left(n,q\right)$, $p>5$, having weight in the interval $\left]\theta_k,2\left(\frac{q^n-1}{q^{n-k}-1}\left(1-\frac{1}{p}\right)+\frac{1}{p}\right)\right[$. By analysing what is known about the codewords in $\mathcal{C}_{k}\!\left(n,q\right)\cap\mathcal{C}_{n-k}\!\left(n,q\right)^\perp$ and narrowing their view to the cases $k=n-1$ and $q$ prime, @LavrauwStormeSziklaiVandeVoorde managed to prove that no codewords of $\mathcal{C}_{k}\!\left(n,q\right)$, $p>5$, have weight in the interval $\left]\theta_k,2q^k\right[$ if $k=n-1$ or if $q$ is prime [@LavrauwStormeSziklaiVandeVoorde Corollaries $19$ and $21$]. Roughly a decade later, @PolverinoZullo:Codes characterised all codewords of $\mathcal{C}_{n-1}\!\left(n,q\right)$ up to the second smallest non-zero weight: **Result 5** ([@PolverinoZullo:Codes Theorem $1.4$]). *There are no codewords of $\mathcal{C}_{n-1}\!\left(n,q\right)$ with weight in the interval $\left]\theta_{n-1},2q^{n-1}\right[$. Any codeword of weight $2q^{n-1}$ is a non-zero scalar multiple of the difference of two distinct hyperplanes.* For a shorter and self-contained proof of the above result, see [@Adriaensen Theorem 4.4]. The authors of this paper together with Storme and Weiner [@AdriaensenDenauxStormeWeiner] extended by proving that all codewords of $\mathcal{C}_{n-1}\!\left(n,q\right)$ up to weight roughly $4q^{n-1}$ are linear combinations of hyperplanes through a fixed $\left(n-3\right)$-space if $q$ is large enough, which in turn has been improved slightly in [@Denaux:PhD]. One year later, we [@AdriaensenDenaux] characterised all codewords of $\mathcal{C}_{k}\!\left(n,q\right)$, $q$ large enough, up to weight roughly $3q^{k}$ as being linear combinations of at most two $k$-spaces. In addition, we proved a similar result for the more general family of codes arising from the incidence of $j$- and $k$-spaces. Finally, the second author and Bartoli [@BartoliDenaux] showed that if $q$ is not prime and large enough, then codewords of $\mathcal{C}_{n-1}\!\left(n,q\right)$ up to weight roughly $\frac{1}{2^{n-2}}q^{n-1}\sqrt{q}$ are linear combinations of exactly $\left\lceil\frac{\mathrm{wt}\left(c\right)}{\theta_{n-1}}\right\rceil$ hyperplanes. One of the aims of this paper is to remove the exponential factor $\frac 1 {2^{n-2}}$. ## Outline and main result In this paper, we prove the following theorem. **Theorem 6**. *Suppose that $j,k,n\in\mathbb{N}$, $0\leqslant j < k < n$, and let $q\coloneqq p^h\geqslant 32$ with $p$ prime and $h\in\mathbb{N}\setminus\left\{0,1\right\}$. Consider a codeword $c\in\mathcal{C}_{j,k}\!\left(n,q\right)$ with $\mathrm{wt}\!\left(c\right)\leqslant\Delta_q\genfrac{[}{]}{0pt}{}{k+1}{j+1}_q$, where $$\Delta_q\coloneqq\begin{cases} \frac{1}{2}\sqrt{q}-\frac{7}{2}&\text{if}\ h=2,\\ \left\lfloor\sqrt{q}-\frac{3}{2}\right\rfloor&\text{otherwise}. \end{cases}$$ Then $c$ is a linear combination of exactly $\left\lceil\frac{\mathrm{wt}\left(c\right)}{\genfrac{[}{]}{0pt}{}{k+1}{j+1}_q}\right\rceil$ $k$-spaces.* The paper is structured as follows. In we give the necessary definitions and background. In we prove a crucial intermediary result. This result roughly states that if a point set in $\mathrm{PG}\!\left(n,q\right)$ intersects all spaces of some fixed dimension in either few or many points, then the point set is either small or large. After this, we are ready to start the proof of . This is done by using induction on each of the parameters $n$, $k$ and $j$. In we prove the theorem for the codes $\mathcal{C}_{n-1}\!\left(n,q\right)$, in we prove it for the codes $\mathcal{C}_{k}\!\left(n,q\right)$ and in , we finish the proof for the general case. # Preliminaries {#Sec:Prel} ## Finite projective geometries Throughout this work, we assume that $n\in\mathbb{N}\setminus\left\{0,1\right\}$ and that $q$ is a prime power, i.e. $q\coloneqq p^h$, where $p$ is prime and $h\in\mathbb{N}\setminus\left\{0\right\}$. We will mostly consider the case $h\geqslant 2$. Finally, we assume that $j$ and $k$ are integers satisfying $0 \leqslant j < k < n$. The Galois field of order $q$ will be denoted by $\mathbb{F}_{q}$ and the Desarguesian projective geometry of (projective) dimension $n$ over $\mathbb{F}_{q}$ will be denoted by $\mathrm{PG}\!\left(n,q\right)$. Whenever 'dimension' or '(sub)space' is mentioned, these are implied to be *projective*. When working in $\mathrm{PG}\!\left(n,q\right)$, we denote the set of $j$-spaces incident with a given subspace $\kappa$ by $\mathcal G_j(\kappa)$. The set of all $j$-spaces is denoted by $\mathcal G_j$, or $\mathcal G_j(n,q)$ if we want to emphasise the ambient projective geometry. The number of $k$-spaces through a fixed $j$-space in $\mathrm{PG}\!\left(n,q\right)$ is given by the Gaussian coefficient $$\genfrac{[}{]}{0pt}{}{n-j}{k-j}_q \coloneqq \prod_{i=1}^{k-j} \frac{q^{n-k+i}-1}{q^i-1}.$$ For simplicity's sake, we denote the number of points (or hyperplanes) of $\mathrm{PG}\!\left(n,q\right)$ by $\theta_{n}$, i.e. $$\theta_{n} \coloneqq \genfrac{[}{]}{0pt}{}{n+1}{1}_q = \frac{q^{n+1}-1}{q-1}=q^n+q^{n-1}+\dots+q+1,$$ where we settle on the convention that $\theta_{-1}\coloneqq0$. **Definition 7**. Let $\mathcal{S}$ be a set of points in $\mathrm{PG}\!\left(n,q\right)$. We say that $\mathcal{S}$ is a *blocking set with respect to the $k$-spaces* or that $\mathcal{S}$ *blocks all $k$-spaces* if it intersects every $k$-space. A famous result by Bose and Burton gives a lower bound on the size of a blocking set. **Result 8** ([@boseburton]). *If $\mathcal{S}$ blocks all $k$-spaces, then $|\mathcal{S}| \geqslant\theta_{n-k}$, and equality occurs if and only if $\mathcal{S}$ is an $(n-k)$-space.* ## Codes from projective geometries As mentioned in the introduction, we are interested in the codes $\mathcal{C}_{j,k}\!\left(n,q\right)$ (see ), whose ambient vector space is $\mathbb{F}_{p}^{\mathcal G_j}$. We define the *support* of $v \in \mathbb{F}_{p}^{\mathcal G_j}$ to be the set $$\mathrm{supp}\!\left(v\right) \coloneqq \left\{\lambda \in \mathcal G_j\,:\,v(\lambda) \neq 0\right\}.$$ More generally, we define, for each $i\in\left\{0,1,\dots,j\right\}$, the set $$\mathrm{supp}_{i}\!\left(v\right) \coloneqq \left\{ \iota \in \mathcal G_i\,:\,(\exists \, \lambda \in \mathrm{supp}\!\left(v\right))(\iota \subseteq \lambda)\right\}.$$ In case $j=0$, points having value $0$ with respect to $v$ are called *holes* with respect to $v$. The *weight* $\mathrm{wt}\!\left(v\right)$ of $v$ is equal to the size of its support, i.e. $\mathrm{wt}\!\left(v\right)\coloneqq\left|\mathrm{supp}\!\left(v\right)\right|$. **Proposition 9**. *If $c\in\mathcal{C}_{j,k}\!\left(n,q\right)$ is a linear combination of exactly $m \leqslant\sqrt{q^{j+1}}$ $k$-spaces, then* 1. *$\displaystyle m = \left\lceil \mathrm{wt}\!\left(c\right) / \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q\right\rceil$,* 2. *if $j=0$, every subspace contains either at most $m$ or at least $q-m+2$ points of $\mathrm{supp}\!\left(c\right)$.* *Proof.* (1) Since any two $k$-spaces share at most $\genfrac{[}{]}{0pt}{}{k}{j+1}_q$ $j$-spaces, we know that $$\begin{aligned} m \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q \geqslant\mathrm{wt}\!\left(c\right) & \geqslant m \left( \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q - (m-1) \genfrac{[}{]}{0pt}{}{k}{j+1}_q \right) > m \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q - q^{j+1} \genfrac{[}{]}{0pt}{}{k}{j+1}_q \\ & = \left( m - q^{j+1} \frac{q^{k-j}-1}{q^{k+1}-1} \right) \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q > (m-1) \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q. \end{aligned}$$ \(2\) We prove this statement by induction on $m$. Note that it is trivial for $m=0$. Now assume that the statement holds for all $m' < m$. Suppose that $c = \sum_{i=1}^m \alpha_i \kappa_i^{(0)},$ with all $\kappa_i$ distinct $k$-spaces and $\alpha_i \in \mathbb{F}_{p}^*$. Let $\rho$ be a subspace and let $\sigma$ be an element of $\left\{\kappa_i \cap \rho\,:\,i=1,\dots,m\right\}$ of maximal dimension $s$. If $s \leqslant 0$, then $\rho$ trivially contains at most $m$ points of $\mathrm{supp}\!\left(c\right)$, so we only need to consider the case where $s \geqslant 1$. Define the set $I \coloneqq \left\{i \in \{1,\dots,m\}\,:\,\sigma \subseteq \kappa_i\right\}$. First, suppose that $\sum_{i \in I} \alpha_i = 0.$ Then $\mathrm{supp}\!\left(c\right) \cap \rho = \mathrm{supp}\!\left(c'\right) \cap \rho$, with $c' \coloneqq \sum_{i \notin I} \alpha_i \kappa_i^{(0)}.$ Since $c'$ is a linear combination of less than $m$ $k$-spaces, the statement follows from the induction hypothesis. Next, suppose that $\sum_{i \in I} \alpha_i \neq 0$. Then all points of $\sigma \setminus \bigcup_{i \notin I} \kappa_i$ have the same non-zero coefficient with respect to $c$. It follows that $$\left|\mathrm{supp}\!\left(c\right) \cap \rho\right| \geqslant\left|\sigma \setminus \bigcup_{i \notin I} \kappa_i\right| \geqslant\theta_s - (m-1) \theta_{s-1} = (q-m+1) \theta_{s-1} + 1 \geqslant q-m+2. \qedhere$$ ◻ For any $i$-space $\iota$ of $\mathrm{PG}\!\left(n,q\right)$, we can naturally define the *restriction* of $v\in\mathbb{F}_{p}^{\mathcal G_j(n,q)}$ to $\iota$ as the function ${v}\rvert_{\iota}\in\mathbb{F}_{p}^{\mathcal G_j(\iota)}$ by restricting the domain of $v$ to $\mathcal G_j(\iota)$. Using the fact that scalar multiples of the all-one function are codewords of $\mathcal{C}_{n-1}\!\left(n,q\right)$, the following can easily be proved. **Result 10** ([@PolverinoZullo:Codes Remark $3.1$]). *Suppose that $c\in\mathcal{C}_{n-1}\!\left(n,q\right)$ and let $\iota$ be an $i$-space of $\mathrm{PG}\!\left(n,q\right)$. Then ${c}\rvert_{\iota}\in\mathcal{C}_{i-1}\!\left(i,q\right)$.* The following projection map is originally due to @LavrauwStormeVandeVoorde:PointsKspaces [@LavrauwStormeVandeVoorde:PointsKspaces Lemma $11$] and was generalised to arbitrary $j$-spaces in [@AdriaensenDenaux]. **Definition 11**. Let $R$ be a point and $\Pi \not \ni R$ a hyperplane of $\mathrm{PG}\!\left(n,q\right)$. Given $v \in \mathbb{F}_{p}^{\mathcal G_j(n,q)}$, define $$\mathrm{proj}^{(j)}_{R,\Pi }(v): \mathcal G_j(\Pi) \to \mathbb{F}_{p}: \lambda \mapsto \sum_{\lambda' \in \mathcal G_j(\left\langle R,\lambda\right\rangle)} v(\lambda').$$ Hence, $\mathrm{proj}^{(j)}_{R,\Pi}$ is a map $\mathbb{F}_{p}^{\mathcal G_j(n,q)} \to \mathbb{F}_{p}^{\mathcal G_j(\Pi)}$. We will denote $\mathrm{proj}^{(0)}_{R,\Pi}$ simply by $\mathrm{proj}_{R,\Pi}$. **Result 12** ([@AdriaensenDenaux Lemma $5.2$]). *Assume that $k\leqslant n-2$ and let $\left(R,\Pi\right)$ be a non-incident point-hyperplane pair of $\mathrm{PG}\!\left(n,q\right)$.* 1. *$\mathrm{proj}^{(j)}_{R,\Pi}$ maps $\mathcal{C}_{j,k}\!\left(n,q\right)$ to $\mathcal{C}_{j,k}\!\left(n-1,q\right)$.* 2. *If $R \notin \mathrm{supp}_{0}\!\left(c\right)$, then $\mathrm{wt}\!\left(\mathrm{proj}^{(j)}_{R,\Pi }(c)\right) \leqslant\mathrm{wt}\!\left(c\right)$.* ## The expander mixing lemma We will introduce a helpful tool for counting problems in finite geometry. This lemma is situated in algebraic graph theory, but we can use it in the context of finite geometry without having to use graph theory terminology. As far as we are aware, the earliest occurrence of the expander mixing lemma in the form in which we will use it was in the PhD thesis of Haemers, see e.g. the summary article of his thesis [@haemers Theorem 5.1] (including the paragraph after the proof of Theorem 5.1 for the determination of the relevant eigenvalue). For a statement of the expander mixing lemma more closely resembling the one that we will use, the reader may consult for instance [@dewinterschillewaertverstraete Lemma 8]. Recall that a $2-(v,k,\lambda)$ design is an incidence structure consisting of points and blocks, such that - there are $v$ points, - every block contains $k$ points, and - through any two distinct points, there are exactly $\lambda$ blocks. Every point is contained in the same number of blocks $r$, called the replication number of the design. **Lemma 13** (Expander mixing lemma). *Consider a $2-(v,k,\lambda)$ design. Let $S$ be a set of points and $T$ be a set of blocks. Denote the number of incidences between $S$ and $T$ by $$e(S,T) \coloneqq \left|\left\{(P,B) \in S \times T\,:\, P \in B\right\}\right|.$$ Then $$\left| e(S,T) - \frac k v \left|S\right| \left|T\right| \right| < \sqrt{ (r-\lambda) \left|S\right| \left|T\right|}.$$* Remark that if the design consists of the points and $j$-spaces of $\mathrm{PG}\!\left(n,q\right)$, then $$r - \lambda = \genfrac{[}{]}{0pt}{}{n}{j}_q - \genfrac{[}{]}{0pt}{}{n-1}{j-1}_q = q^j \genfrac{[}{]}{0pt}{}{n-1}{j}_q,$$ hence, in this case, the expander mixing lemma tells us that $$\begin{aligned} \label{Eq:EML} \left| e(S,T) - \frac {\theta_j}{\theta_n} \left|S\right| \left|T\right| \right| < \sqrt{ q^j \genfrac{[}{]}{0pt}{}{n-1}{j}_q \left|S\right| \left|T\right|}.\end{aligned}$$ # Amplifying a gap in the intersection sizes with subspaces {#Sec:Spectrum} In this section, we show that if a point set intersects every $r$-space in either a few or many points, the same should be true for all higher-dimensional subspaces. **Theorem 14**. *Consider a prime power $q \geqslant 16$ and integers $r, n, \delta$ satisfying $1 \leqslant r < n$ and $\delta \leqslant\sqrt q - 1$. Suppose that $S$ is a set of points in $\mathrm{PG}\!\left(n,q\right)$ intersecting every $r$-space in either $$\begin{aligned} \text{at most $\delta$ points} && \text{or} && \text{at least $q-\sqrt q +3$ points.} \end{aligned}$$ Then $$\begin{aligned} \left|S\right| \leqslant\delta \theta_{n-r} && \text{or} && \left|S\right| > \left( q - \sqrt q + \frac 32 \right) \frac{q^n-1}{q^r-1}. \end{aligned}$$* We will prove this theorem throughout this section. The main tools are two useful counting techniques. The first one is sometimes referred to as the standard equations, the second one is the expander mixing lemma. Although they usually yield the same result in the context of finite geometric counting problems, we will use them here to complement each other. We make the following conventions for the remainder of this section: - $q \geqslant 16$, - $1 \leqslant r < n$ are integers, - $\delta$ is an integer satisfying $\delta \leqslant\sqrt q - 1$, - $S$ is a set of points in $\mathrm{PG}\!\left(n,q\right)$ intersecting every $r$-space in either at most $\delta$ points or at least $q-\sqrt q+3$ points, and - $s \coloneqq \left|S\right|$. **Lemma 15**. *Either $s < \displaystyle \left(\sqrt q - \frac 12\right) \frac{q^n-1}{q^r-1}$ or $s > \displaystyle \left( q - \sqrt q + \frac 32 \right) \frac{q^n-1}{q^r-1}$.* *Proof.* We use the standard equations. Let $n_i$ denote the number of $r$-spaces intersecting $S$ in exactly $i$ points. It follows from our assumptions that $S$ intersects every $r$-space in at most $\sqrt q - 1$ points or in at least $q-\sqrt q+3$ points. Hence, $$\label{Eq:StandardEq} \sum_i \left(i-\left(\sqrt q - 1\right)\right) \left(i-\left(q-\sqrt q + 3\right)\right) n_i \geqslant 0.$$ On the other hand, we know that $$\begin{aligned} \sum_i n_i &= \genfrac{[}{]}{0pt}{}{n+1}{r+1}_q = \frac{q^{n+1}-1}{q^{r+1}-1} \frac{q^n-1}{q^r-1} \genfrac{[}{]}{0pt}{}{n-1}{r-1}_q, \\ \sum_i i n_i &= s \genfrac{[}{]}{0pt}{}{n}{r}_q = s \frac{q^n-1}{q^{r}-1} \genfrac{[}{]}{0pt}{}{n-1}{r-1}_q, \\ \sum_i i(i-1) n_i &= s(s-1)\genfrac{[}{]}{0pt}{}{n-1}{r-1}_q. \end{aligned}$$ The first equation should be clear. The second and third equations follow from performing a double count on the following sets, respectively: $$\begin{aligned} \left\{(P,\rho) \in S \times \mathcal G_{r}\,:\,P \in \rho\right\}, && \left\{(P,R,\rho) \in S \times S \times \mathcal G_{r}\,:\,P \neq R, \, P,R \in \rho\right\}. \end{aligned}$$ Plugging this into and dividing by $\genfrac{[}{]}{0pt}{}{n-1}{r-1}_q$ yields $$\begin{aligned} 0 \leqslant s(s-1) - (q+1) \frac{q^n-1}{q^{r}-1} s + \left(\sqrt q - 1\right) (q-\sqrt q+3) \frac{q^{n+1}-1}{q^{r+1}-1} \frac{q^n-1}{q^{r}-1}. \end{aligned}$$ Since $s \geqslant 0$ and $\frac{q^{n+1}-1}{q^{r+1}-1} < \frac{q^n-1}{q^{r}-1}$, we obtain $$\label{Eq:StandardEq2} 0 < s^2 - (q+1) \frac{q^n-1}{q^{r}-1} s + \left(\sqrt q - 1\right) (q-\sqrt q+3) \left( \frac{q^n-1}{q^{r}-1} \right)^2.$$ The right-hand side of is a quadratic polynomial in $s$. We will give estimates of its roots. The discriminant of the polynomial is given by $$\left( \frac{q^n-1}{q^{r}-1} \right)^2 \left( (q+1)^2 - 4 \left(\sqrt q - 1\right) (q-\sqrt q+3) \right) \geqslant\left( \frac{q^n-1}{q^{r}-1} (q-2\sqrt q + 2) \right)^2.$$ Therefore, does not hold when $$\begin{aligned} s &\in \left[ \frac 12 \frac{q^n-1}{q^{r}-1}( (q+1) - (q-2\sqrt q+2) ), \frac 12 \frac{q^n-1}{q^{r}-1}( (q+1) + (q-2\sqrt q+2) ) \right] \\ & = \left[ \left(\sqrt q - \frac 12\right) \frac{q^n-1}{q^{r}-1}, \left( q - \sqrt q + \frac 32 \right) \frac{q^n-1}{q^{r}-1} \right]. \qedhere \end{aligned}$$ ◻ It only remains to exclude the case $$\delta \theta_{n-r} < s < \left(\sqrt q - \frac 12\right) \frac{q^n-1}{q^{r}-1}.$$ We will do this by fixing $r$ and using induction on $n$. Call an $(r+i)$-space $\rho$ *poor* if $\left|\rho \cap S\right| \leqslant\delta \theta_i$, and *rich* if $\left|\rho \cap S\right| \geqslant\left(q-\sqrt q + \frac 3 2\right) \frac{q^{r+i}-1}{q^r-1}$. It follows from our assumptions and the induction hypothesis that, for every $i\in\left\{0,1,\dots,n-r-1\right\}$, every $(r+i)$-space $\rho$ is either poor or rich. Let $T$ denote the set of rich hyperplanes and define $t \coloneqq \left|T\right|$. **Lemma 16**. *Suppose that $s > \delta \theta_{n-r}$. Then $t \geqslant\theta_{r}$.* *Proof.* Let $\rho$ be an $(r-1)$-space. We will prove that $\rho$ lies in a rich hyperplane. If $\rho$ only lies in poor $r$-spaces, then $$s \leqslant\left|\rho \cap S\right| + \theta_{n-r} (\delta - \left|\rho \cap S\right|) \leqslant\delta \theta_{n-r},$$ contradicting our assumptions. Thus, $\rho$ lies in a rich $r$-space. Now we prove, for every $i \in \left\{0,1,\dots,n-r-2\right\}$, that a rich $(r+i)$-space lies in a rich $(r+i+1)$-space. Then it follows by induction that $\rho$ lies in a rich hyperplane. So suppose, to the contrary, that $\sigma$ is a rich $(r+i)$-space lying only in poor $(r+i+1)$-spaces. Then $$\begin{aligned} \delta \theta_{n-r} < s &\leqslant\left|\sigma \cap S\right| + \theta_{n-r-i-1} (\delta \theta_{i+1} - \left|\sigma \cap S\right|) \\ &< \delta \theta_{n-r-i-1} \theta_{i+1} - q \theta_{n-r-i-2} \left(q-\sqrt q + \frac 3 2 \right) q^i. \end{aligned}$$ After multiplying both sides by $(q-1)^2$ and moving some terms around, we obtain $$\begin{aligned} q \left( q^{n-r-i-1}-1 \right) & \left(q-\sqrt q + \frac 3 2 \right) q^i (q-1) \\ & < \delta \left[ \left( q^{n-r-i}-1 \right) \left( q^{i+2}-1 \right) - \left(q^{n-r+1}-1 \right) (q-1) \right] \\ & = \delta \left( q^{n-r+1} - q^{n-r-i} - q^{i+2} + q \right) \\ & = \delta q \left( q^{i+1}-1 \right) \left( q^{n-r-i-1} - 1 \right). \end{aligned}$$ It follows that $$\begin{aligned} \sqrt q - 1 & \geqslant\delta > \frac{\left(q-\sqrt q + \frac 3 2 \right) q^i (q-1)}{q^{i+1}-1} > \left(q-\sqrt q + \frac 3 2 \right) \left( 1 - \frac 1 q \right). \end{aligned}$$ This yields a contradiction for $q \geqslant 2$. Thus, we may conclude that every $(r-1)$-space $\rho$ lies in a rich hyperplane. This means that any duality of the projective space maps $T$ to a blocking set of the $(n-r)$-spaces. Therefore, by , there are at least $\theta_{r}$ rich hyperplanes. ◻ **Lemma 17**. *Suppose that $s \leqslant\left(\sqrt q - \frac 12\right) \frac{q^n-1}{q^{r}-1}$. Then $$t < \frac{\sqrt q}{\left( \sqrt q - 2 \right)^2} q^r.$$* *Proof.* Recall that $e(S,T)$ denotes the number of incidences between the set $S$ of points and the set $T$ of rich hyperplanes. By the definition of rich, we have that $$e(S,T) \geqslant t \left(q-\sqrt q + \frac 3 2\right) \frac{q^{n-1}-1}{q^r-1}.$$ On the other hand, by applying the expander mixing lemma to $S$ and $T$, we have that $$e(S,T) \leqslant\frac{\theta_{n-1}}{\theta_n} st + \sqrt{q^{n-1}st}.$$ Hence, $$\label{Eq:EmlIneq} t \left( \left(q-\sqrt q + \frac 3 2\right) \frac{q^{n-1}-1}{q^r-1} - \frac{\theta_{n-1}}{\theta_n} s \right) \leqslant\sqrt{q^{n-1} s t}.$$ Next, we verify that the left-hand side of is non-negative. This follows from $$\begin{aligned} s \leqslant\left(\sqrt q - \frac 12\right) \frac{q^n-1}{q^{r}-1} < \sqrt q \frac{q^n}{q^r-1} \leqslant\frac{\theta_n}{\theta_{n-1}} \left( q-\sqrt q + \frac 3 2 \right) \frac{q^{n-1}-1}{q^r-1}. \end{aligned}$$ Hence, we may square both sides of and the inequality still holds. From this we obtain $$\begin{aligned} t & \leqslant\frac{ q^{n-1} s}{\left( \left(q-\sqrt q + \frac 3 2\right) \frac{q^{n-1}-1}{q^r-1} - \frac{\theta_{n-1}}{\theta_n} s \right)^2} \\ & \leqslant\frac{ q^{n-1} \left(\sqrt q - \frac 12\right) \frac{q^n-1}{q^r-1} }{ \left( \left(q-\sqrt q + \frac 3 2\right) \frac{q^{n-1}-1}{q^r-1} - \frac{q^n-1}{q^{n+1}-1} \left(\sqrt q - \frac 12\right) \frac{q^n-1}{q^r-1} \right)^2 } \\ & = \frac{ q^{n-1} \left(\sqrt q - \frac 12\right) \left(q^n-1\right) \left(q^r-1\right) }{ \left( \left(q-\sqrt q + \frac 3 2\right) \left(q^{n-1}-1\right) - \frac{q^n-1}{q^{n+1}-1} \left(\sqrt q - \frac 12\right) \left(q^n-1\right) \right)^2 } \\ & < \frac{\sqrt q q^{2n+r-1}}{ \left( \left(q- 2 \sqrt q\right) q^{n-1}\right)^2 } = \frac{\sqrt q}{\left( \sqrt q - 2 \right)^2} q^r. \qedhere \end{aligned}$$ ◻ **Lemma 18**. *It is not possible that $$\delta \theta_{n-r} < s < \left( \sqrt q - \frac 1 2 \right) \frac{q^n-1}{q^{r}-1}.$$* *Proof.* Suppose the contrary. By and , $$q^{r} < \theta_{r} \leqslant t < \frac{\sqrt q}{\left( \sqrt q - 2 \right)^2} q^r.$$ In particular, this implies that $$\sqrt q > \left( \sqrt q - 2 \right)^2$$ which yields a contradiction for $q \geqslant 16$. ◻ and together prove . **Remark 19**. is a rough generalisation of a theorem from the second author's PhD thesis [@Denaux:PhD Theorem $2.2.1$], which considers the case $r=1$. # Codes of points and $\bm{k}$-spaces This section is dedicated to proving in case $j=0$, i.e. the following theorem: **Theorem 20**. *Suppose that $k,n\in\mathbb{N}$, $0<k<n$, and let $q\coloneqq p^h\geqslant 32$ with $p$ prime and $h\in\mathbb{N}\setminus\left\{0,1\right\}$. Consider a codeword $c\in\mathcal{C}_{k}\!\left(n,q\right)$ with $\mathrm{wt}\!\left(c\right)\leqslant\Delta_q\theta_k$, where $$\Delta_q\coloneqq\begin{cases} \frac{1}{2}\sqrt{q}-\frac{7}{2}&\text{if}\ h=2,\\ \left\lfloor\sqrt{q}-\frac{3}{2}\right\rfloor&\text{otherwise}. \end{cases}$$ Then $c$ is a linear combination of exactly $\left\lceil\frac{\mathrm{wt}\left(c\right)}{\theta_k}\right\rceil$ $k$-spaces.* Mimicking , the proof will be given throughout this section. In essence, we use induction on both $n$ and $k$, starting from the plane case described in . As suggests, we will often make use of the following integer value: $$\Delta_q\coloneqq\begin{cases} \frac{1}{2}\sqrt{q}-\frac{7}{2}&\text{if}\ q=p^2,\\ \left\lfloor\sqrt{q}-\frac{3}{2}\right\rfloor&\text{if}\ q=p^h, h\geqslant 3. \end{cases}$$ One can check that $$\label{eq:SzonyiWeiner} \left(\Delta_q+1\right)\left(q+1\right)<\begin{cases} \frac{\left(\sqrt{q}-1\right)\left(\sqrt{q}-4\right)\left(q+1\right)}{2\sqrt{q}-1}&\text{if}\ q=p^2,\\ \left(\left\lfloor\sqrt{q}\right\rfloor+1\right)\left(q+1-\left\lfloor\sqrt{q}\right\rfloor\right)&\text{if}\ q=p^h, h\geqslant 3. \end{cases}$$ Moreover, throughout this section, we - assume that $q\geqslant 32$ is not prime, and - suppose, to the contrary, that there exists some codeword $c\in\mathcal{C}_{k}\!\left(n,q\right)$, $\mathrm{wt}\!\left(c\right)\leqslant\Delta_q\theta_k$, for which is not true. We then choose $n$, $k$ and $\mathrm{wt}\!\left(c\right)$ --- in that order --- to be minimal with respect to this property. **Definition 21**. An $i$-space $\iota$ is called *thick* (with respect to $c$) in case $$\mathrm{wt}\!\left({c}\rvert_{\iota}\right)\geqslant\begin{cases}q-\sqrt{q}+3&\text{if}\ i=1,\\\left(q-\sqrt{q}+1\right)\theta_{i-1}&\text{if}\ i\geqslant 2.\end{cases}$$ **Lemma 22**. *There are no thick $k$-spaces.* *Proof.* Suppose, to the contrary, that $\kappa$ is a thick $k$-space. Then $\kappa$ has at most $$\theta_k - \left(q-\sqrt q + 1\right) \theta_{k-1} = \left(\sqrt q - 1\right) \theta_{k-1} + 1$$ holes. Since there are $p-1 \leqslant\sqrt q - 1$ scalars in $\mathbb{F}_{p}^*$, one of these occurs at least $$\frac{q-\sqrt q + 1}{\sqrt q - 1} \theta_{k-1} > \left(\sqrt q - 1\right) \theta_{k-1} + 1$$ times as the coefficient of a point of $\kappa$ with respect to $c$. Call this scalar $\alpha$. Define $c' \coloneqq c-\alpha \kappa^{(0)}$. Then $\mathrm{wt}\!\left(c'\right) < \mathrm{wt}\!\left(c\right)$. Therefore, due to the minimal weight of $c$, the codeword $c'$ must be equal to a linear combination of at most $\Delta_q$ $k$-spaces, which means that $c=c'+\alpha\kappa^{(0)}$ has to be a linear combination of at most $\Delta_q+1$ $k$-spaces. But then, by (1), $c$ is a codeword for which is true, a contradiction. ◻ As $\mathrm{wt}\!\left(c\right)\leqslant\Delta_q\theta_k<\theta_{k+1}$, there must exist an $\left(n-k-1\right)$-space $\rho$ that avoids $\mathrm{supp}\!\left(c\right)$, cf. . If all $\left(n-k\right)$-spaces would contain at least $\Delta_q+1$ points of $\mathrm{supp}\!\left(c\right)$, then so would all $\left(n-k\right)$-spaces through $\rho$, implying that $\mathrm{wt}\!\left(c\right)\geqslant\left(\Delta_q+1\right)\theta_k$, a contradiction. Therefore, the following is well-defined. **Definition 23**. Define $\delta$ to be the largest number in $\left\{0,1,\dots,\Delta_q\right\}$ for which there exists an $\left(n-k\right)$-space containing precisely $\delta$ points of $\mathrm{supp}\!\left(c\right)$. ## The case $\bm{k=n-1}$ {#Sec:PtHyp} Throughout this subsection, we assume that $k=n-1$. **Definition 24** ($k=n-1$). An $i$-space $\iota$ is called *thin* (with respect to $c$) in case $$\mathrm{wt}\!\left({c}\rvert_{\iota}\right)\leqslant\delta\theta_{i-1}.$$ **Proposition 25**. *Let $\pi$ be a plane that contains an $m$-secant $\ell$ to $\mathrm{supp}\!\left(c\right)$. Then $$\mathrm{wt}\!\left({c}\rvert_{\pi}\right)\geqslant\begin{cases} \left(\Delta_q+1\right)\left(q+1\right)+1&\text{if}\ \Delta_q+2\leqslant m\leqslant q-\Delta_q,\\ m\left(q-m+2\right)&\text{otherwise}. \end{cases}$$* *Proof.* Suppose that $\mathrm{wt}\!\left({c}\rvert_{\pi}\right)\leqslant\left(\Delta_q+1\right)\left(q+1\right)$. Then by and Results [Result 10](#res:Restricted){reference-type="ref" reference="res:Restricted"} and [Result 4](#res:SzonyiWeiner){reference-type="ref" reference="res:SzonyiWeiner"}, there exists a set $\mathcal{L}$ of at most $\Delta_q+1$ lines covering the points of $\mathrm{supp}\!\left({c}\rvert_{\pi}\right)$, each such a line containing at least $q-\left|\mathcal{L}\right|+2\geqslant q-\Delta_q+1$ unique points of $\mathrm{supp}\!\left({c}\rvert_{\pi}\right)$. If $\ell\in\mathcal{L}$, then $m\geqslant q-\Delta_q+1$. If $\ell\notin\mathcal{L}$, then it intersects each line of $\mathcal{L}$ in exactly one point, implying that $m\leqslant\left|\mathcal{L}\right|\leqslant\Delta_q+1$. In conclusion, if $\mathrm{wt}\!\left({c}\rvert_{\pi}\right)\leqslant\left(\Delta_q+1\right)\left(q+1\right)$, then either $m\leqslant\Delta_q+1$ or $m\geqslant q-\Delta_q+1$. Moreover, by the above observation, we know that $$\mathrm{wt}\!\left({c}\rvert_{\pi}\right) \geqslant|\mathcal{L}| \left( q - |\mathcal{L}| + 2 \right) \geqslant m (q-m+2).\qedhere$$ ◻ **Lemma 26**. *Every line is either thin or thick.* *Proof.* Consider an arbitrary $m$-secant $\ell$ with respect to $\mathrm{supp}\!\left(c\right)$ and suppose, to the contrary, that $\delta+1\leqslant m<q-\sqrt{q}+3$, or, equivalently, that $\Delta_q+1\leqslant m\leqslant q-\left\lfloor\sqrt{q}\right\rfloor+2$. Note that not every plane through $\ell$ contains at least $q \Delta_q$ points of $\mathrm{supp}\!\left(c\right) \setminus \ell$. Otherwise, $$\mathrm{wt}\!\left(c\right) \geqslant\left(q \Delta_q\right) \theta_{n-2} + m \geqslant q \Delta_q\theta_{n-2} + \Delta_q+ 1 = \Delta_q\theta_{n-1} + 1,$$ a contradiction. First, suppose that $\Delta_q+ 2 \leqslant m \leqslant q - \Delta_q$. By , every plane through $\ell$ has to contain at least $$\left(\Delta_q+1\right)\left(q+1\right)+1 - m \geqslant\Delta_q(q+2) + 2 > q \Delta_q$$ points of $\mathrm{supp}\!\left(c\right) \setminus \ell$. This leads to a contradiction. Similarly, if $m = \Delta_q+1$, implies that each plane through $\ell$ contains at least $$\left(\Delta_q+ 1\right) \left( q - \Delta_q+ 1 \right) - \left(\Delta_q+ 1\right) = q \Delta_q+ q - \Delta_q^2 - \Delta_q> q \Delta_q$$ points of $\mathrm{supp}\!\left(c\right) \setminus \ell$, again yielding a contradiction. Since $\Delta_q\leqslant\left\lfloor\sqrt q - \frac 32\right\rfloor \leqslant\left\lfloor\sqrt q\right\rfloor - 1$, the only remaining case to exclude is $m = q - \left\lfloor\sqrt q\right\rfloor + 2$ and $\Delta_q= \left\lfloor\sqrt q\right\rfloor - 1$. In particular, this can only happen if $h \geqslant 3$. Not every plane through $\ell$ contains at least $\left(\left\lfloor\sqrt q\right\rfloor + 1\right) \left(q-\left\lfloor\sqrt q\right\rfloor + 1\right)$ points of $\mathrm{supp}\!\left(c\right)$, since $$\begin{aligned} \left(\left\lfloor\sqrt q\right\rfloor + 1\right) \left(q-\left\lfloor\sqrt q\right\rfloor + 1\right) - \left(q - \left\lfloor\sqrt q\right\rfloor + 2\right) &= \left\lfloor\sqrt q\right\rfloor q - \left\lfloor\sqrt q\right\rfloor^2 + \left\lfloor\sqrt q\right\rfloor - 1 \\ \geqslant\left(\left\lfloor\sqrt q\right\rfloor - 1\right) q + \left\lfloor\sqrt q\right\rfloor - 1 &= q\Delta_q+ \left\lfloor\sqrt q\right\rfloor - 1 > q\Delta_q. \end{aligned}$$ Thus there exists a plane $\pi$ through $\ell$ with $\mathrm{wt}\!\left({c}\rvert_{\pi}\right) < \left(\left\lfloor\sqrt q\right\rfloor + 1\right) \left(q-\left\lfloor\sqrt q\right\rfloor + 1\right)$. By , ${c}\rvert_{\pi}$ is a linear combination of $$\left\lceil\frac{\mathrm{wt}\!\left({c}\rvert_{\pi}\right)}{q+1}\right\rceil \leqslant\left\lceil \frac{\left(\left\lfloor\sqrt q\right\rfloor + 1\right) \left(q-\left\lfloor\sqrt q\right\rfloor + 1\right)}{q+1} \right\rceil \leqslant\left\lfloor\sqrt q\right\rfloor + 1$$ lines. Moreover, since ${c}\rvert_{\pi}$ has a $(q-\left\lfloor\sqrt q\right\rfloor + 2)$-secant, it cannot be a linear combination of fewer than $\left\lfloor\sqrt q\right\rfloor$ lines by (2). Since $\binom{\left\lfloor\sqrt q\right\rfloor + 1}2 < q+1$, the intersection points of these lines do not form a blocking set in $\pi$ (see ). In particular, some line of $\pi$ doesn't contain any of these intersection points and hence contains either $\left\lfloor\sqrt q\right\rfloor$ or $\left\lfloor\sqrt q\right\rfloor + 1$ points of $\mathrm{supp}\!\left(c\right)$. But we have already excluded the existence of such a line since $\Delta_q+ 1 = \left\lfloor\sqrt q\right\rfloor < \left\lfloor\sqrt q\right\rfloor + 1 < q - \Delta_q$. This yields a contradiction yet again, concluding the proof. ◻ **Corollary 27**. *Every subspace is either thin or thick. In particular, every hyperplane is thin.* *Proof.* By and ($r = 1$), every subspace is either thick or thin. Moreover, by , there are no thick hyperplanes. ◻ *Proof of in case $k = n-1$.* Because we chose $n$ to be minimal such that $c$ is a supposed counterexample to ($k=n-1$), we know that $n\geqslant 3$ due to and . This choice of minimality also implies --- due to and --- that the codeword ${c}\rvert_{\Pi}$ is a linear combination of exactly $\left\lceil\frac{\mathrm{wt}\left({c}\rvert_{\Pi}\right)}{\theta_{n-2}}\right\rceil \leqslant\delta$ $\left(n-2\right)$-subspaces of $\Pi$, for every hyperplane $\Pi$. We know that $\mathrm{wt}\!\left(c\right)\leqslant\delta\theta_{n-1}$, otherwise $\mathrm{wt}\!\left(c\right) \geqslant\left(q-\sqrt q + 1\right) \theta_{n-1} > \Delta_q\theta_{n-1}$ by , a contradiction. Hence, we may assume that $\delta\geqslant 1$. Consider a $\delta$-secant $\ell$ to $\mathrm{supp}\!\left(c\right)$. By , all planes through $\ell$ contain at least $\delta\left(q-\delta+2\right)$ points of $\mathrm{supp}\!\left(c\right)$, which implies that $$\begin{aligned} \label{eq:LowerBoundWeight} \mathrm{wt}\!\left(c\right)&\geqslant\left(\delta\left(q-\delta+2\right)-\delta\right)\theta_{n-2}+\delta\nonumber\\ &=\delta q^{n-1}-\left(\delta^2-2\delta\right)\theta_{n-2}. \end{aligned}$$ Let $\Pi$ be a hyperplane containing a point of $\mathrm{supp}\!\left(c\right)$. Then ${c}\rvert_{\Pi}$ is a linear combination of at most $\delta$ $(n-2)$-spaces. Let $\Sigma$ be one of these $\left(n-2\right)$-spaces. Then $\mathrm{wt}\!\left({c}\rvert_{\Sigma}\right) \geqslant\theta_{n-2} - (\delta-1) \theta_{n-3} > \delta \theta_{n-3}$. Now take any hyperplane $\Pi'$ through $\Sigma$. Then ${c}\rvert_{\Pi'}$ is a linear combination of at most $\delta$ $(n-2)$-spaces. Every $(n-2)$-space of $\Pi'$ not occurring in the linear combination contains at most $\delta \theta_{n-3}$ points of $\mathrm{supp}\!\left(c\right)$. Therefore, $\Sigma$ is one of the $(n-2)$-spaces occurring in the linear combination, and the points of $\mathrm{supp}\!\left( {c}\rvert_{\Pi'}\right) \setminus \Sigma$ are contained in at most $\delta-1$ other $(n-2)$-spaces. As this holds for every hyperplane $\Pi'$ through $\Sigma$, $$\mathrm{wt}\!\left(c\right)\leqslant\left(q+1\right)\left(\delta-1\right)q^{n-2}+\theta_{n-2}=\left(\delta-1\right)q^{n-1}+\delta q^{n-2}+\theta_{n-3}.$$ Combining this with , we obtain $$\begin{aligned} \left(\delta-1\right)q^{n-1}+\delta q^{n-2}+\theta_{n-3}&\geqslant\delta q^{n-1}-\left(\delta^2-2\delta\right)\theta_{n-2}\\ \iff\quad0&\geqslant q^{n-1}-\left(\delta^2-\delta\right)q^{n-2}-\left(\delta-1\right)^2\theta_{n-3}. \end{aligned}$$ Using that $\delta^2-\delta<\left(\sqrt{q}-1\right)^2-\left(\sqrt{q}-1\right)$ and $\left(\delta-1\right)^2<q-1$, we get $$0>3\left(\sqrt{q}-1\right)q^{n-2}+1,$$ a contradiction. ◻ ## The case $\bm{k\leqslant n-2}$ {#Sec:PtK} Due to the result of the previous subsection and the minimality of $k$, we know that $k\leqslant n-2$. Moreover, due to the minimality of $n$, combined with , $\mathrm{proj}_{R,\Pi}\!\left(c\right)$ is a linear combination of at most $\Delta_q$ $k$-subspaces of $\Pi$ for every non-incident point-hyperplane pair $\left(R,\Pi\right)$, $R\notin\mathrm{supp}\!\left(c\right)$. Keep this observation in mind during the remainder of this subsection. **Definition 28** ($k\leqslant n-2$). An $i$-space $\iota$ is called *thin* (with respect to $c$) in case $$\mathrm{wt}\!\left({c}\rvert_{\iota}\right)\leqslant\Delta_q\theta_{i-1}.$$ **Lemma 29**. *Every line is either thin or thick.* *Proof.* Consider an arbitrary $m$-secant $\ell$ with respect to $\mathrm{supp}\!\left(c\right)$ and suppose, to the contrary, that $\Delta_q+1\leqslant m\leqslant q-\left\lfloor\sqrt{q}\right\rfloor+2$. If every plane through $\ell$ contained at least $\Delta_q$ points of $\mathrm{supp}\!\left(c\right)\setminus\ell$, then $$\mathrm{wt}\!\left(c\right)\geqslant\Delta_q\theta_{n-2}+m>\Delta_q\theta_k,$$ a contradiction. Thus, there must exist a plane $\pi$ through $\ell$ containing at most $\Delta_q-1$ points of $\mathrm{supp}\!\left(c\right)$ not lying in $\ell$, forming a point set $\mathcal{S}$. First, assume that $\Delta_q+1\leqslant m\leqslant\frac{q}{2}$. Fix a point set $\mathcal{P}$ consisting of $\Delta_q+1$ points of $\mathrm{supp}\!\left({c}\rvert_{\ell}\right)$. By connecting each point of $\mathcal{P}$ with each point of $\mathcal{S}$, one obtains a set of lines that cover at most $\left|\mathcal{P}\right|\cdot\left|\mathcal{S}\right|\cdot q\leqslant\left(\Delta_q+1\right)\left(\Delta_q-1\right)q<q^2$ points of $\pi\setminus\ell$. Therefore, there exists a point $R\in\pi\setminus\ell$ that does not lie on a line connecting a point of $\mathcal{P}$ with a point of $\mathcal{S}$. As a consequence, $R\notin\mathrm{supp}\!\left(c\right)$. Choosing a hyperplane $\Pi$ through $\ell$ that does not contain $\pi$. (2) states that $\ell$ contains either at most $\Delta_q$ or at least $q-\Delta_q+2$ points of $\mathrm{supp}\!\left(\mathrm{proj}_{R,\Pi}\!\left(c\right)\right)$. However, by the choice of the point $R$ and the way $\mathrm{proj}_{R,\Pi}$ is defined, we know that - $\mathcal{P}\subseteq\mathrm{supp}\!\left(\mathrm{proj}_{R,\Pi}\!\left(c\right)\right)$, implying, as $\left|\mathcal{P}\right|=\Delta_q+1$, that $\ell$ contains at least $q-\Delta_q+2$ points of $\mathrm{supp}\!\left(\mathrm{proj}_{R,\Pi}\!\left(c\right)\right)$, and - $\ell$ contains at most $m+\left|\mathcal{S}\right|\leqslant\frac{q}{2}+\Delta_q-1$ points of $\mathrm{supp}\!\left(\mathrm{proj}_{R,\Pi}\!\left(c\right)\right)$. This implies $q-\Delta_q+2\leqslant\frac{q}{2}+\Delta_q-1$, a contradiction. The case $\frac{q+1}{2}\leqslant m\leqslant q-\left\lfloor\sqrt{q}\right\rfloor+2$ is similar by choosing a point set $\mathcal{P}$ consisting of $\Delta_q$ holes of $\ell$ with respect to $c$ and proving that $\ell$ contains at most $\Delta_q$ but also at least $\frac{q+1}{2}-\left|\mathcal{S}\right|\geqslant\frac{q+1}{2}-\Delta_q+1$ points of $\mathrm{supp}\!\left(\mathrm{proj}_{R,\Pi}\!\left(c\right)\right)$. ◻ **Corollary 30**. *Every $k$-space is thin.* *Proof.* This follows from , ($r=1$) and . ◻ *Proof of in case $k \leqslant n-2$.* Consider an $\left(n-k\right)$-space $\lambda$ with $\delta<\mathrm{wt}\!\left({c}\rvert_{\lambda}\right)\leqslant q-\Delta_q+2$. Then $\mathrm{wt}\!\left({c}\rvert_{\lambda}\right)\geqslant\Delta_q+1$, so we can select a point set $\mathcal{P}$ in $\mathrm{supp}\!\left({c}\rvert_{\lambda}\right)$ of size $\Delta_q+1$. All lines containing at least two points of $\mathcal{P}$ cover at most $\binom{\Delta_q+1}{2}\left(q+1\right)<\frac{q}{2}\left(q+1\right)<q^2$ points of $\lambda\setminus\mathrm{supp}\!\left(c\right)$. Hence, as $\dim\!\left(\lambda\right)=n-k\geqslant 2$ and as $\mathrm{wt}\!\left({c}\rvert_{\lambda}\right)\leqslant q-\Delta_q+2$, there must exist a point $R\in\lambda\setminus\mathrm{supp}\!\left(c\right)$ through which each line contains at most one point of $\mathcal{P}$. Pick a hyperplane $\Pi\not\ni R$. Then $\mathrm{proj}_{R,\Pi}\!\left(c\right)$ is a linear combination of at most $\Delta_q$ $k$-subspaces of $\Pi$. Due to the choice of $R$, at least $\Delta_q+1$ points of $\mathrm{supp}\!\left(\mathrm{proj}_{R,\Pi}\!\left(c\right)\right)$ lie in $\lambda\cap\Pi$, hence at least two of these points, say $Q_1$ and $Q_2$, originate from the very same $k$-subspace of the linear combination. This means that the line $\left\langle Q_1,Q_2\right\rangle$ must contain at least $q+1-\left(\Delta_q-1\right)=q-\Delta_q+2$ points of $\mathrm{supp}\!\left(\mathrm{proj}_{R,\Pi}\!\left(c\right)\right)$. By (2), the plane $\left\langle Q_1,Q_2,R\right\rangle\subseteq\lambda$ must contain at least $q-\Delta_q+2$ points of $\mathrm{supp}\!\left(c\right)$. We conclude that every $\left(n-k\right)$-space contains either at most $\delta$ or at least $q-\Delta_q+2$ points of $\mathrm{supp}\!\left(c\right)$. By , either $\mathrm{wt}\!\left(c\right)\leqslant\delta\theta_k$ or $\mathrm{wt}\!\left(c\right)>\left(q-\sqrt{q}+1\right)\frac{q^n-1}{q^{n-k}-1}$. The latter implies that $$\begin{aligned} \sqrt{q}\theta_k>\left(q-\sqrt{q}\right)\frac{q^n-1}{q^{n-k}-1}&&\Longleftrightarrow&&\frac{q^{n+1}-q^{n-k}-q^{k+1}+1}{q^{n+1}-q^n-q+1}>{\sqrt{q}-1}\\ &&\Longrightarrow&&\frac{q^{n+1}-2q^2+1}{q^{n+1}-q^n-q+1}>{\sqrt{q}-1}\\ &&\Longleftrightarrow&&\frac{q}{q-1}-\frac{2q+1}{q^n-1}>{\sqrt{q}-1}, \end{aligned}$$ a contradiction. Thus, $\mathrm{wt}\!\left(c\right)\leqslant\delta\theta_k$. Now suppose that $\lambda$ is an $\left(n-k\right)$-space containing precisely $\delta$ points of $\mathrm{supp}\!\left(c\right)$. Define $\mathcal{P}\coloneqq\mathrm{supp}\!\left({c}\rvert_{\lambda}\right)$, hence $\left|\mathcal{P}\right|=\delta$. For any set $\mathcal{S}$ of $x$ points in a projective space, there are at most $x + \binom x 2 (q-1)$ points lying on a line connecting two points of $\mathcal{S}$. If $x \leqslant\sqrt q$, then $x + \binom x 2 (q-1) < \theta_2$. Since $\delta\leqslant\sqrt q - 1$, there exists a point $R_1 \in \lambda$, not lying on any line connecting two points of $\mathcal{P}$. Moreover, there exists a point $R_2$ not lying on any line connecting two points of $\mathcal{P}\cup \{R_1\}$. Take a hyperplane $\Pi$ that misses $R_1$ and $R_2$. Let $\lambda'$ denote $\lambda \cap \Pi$. Note that for $i\in\left\{1,2\right\}$, $c_i \coloneqq \mathrm{proj}_{R_i,\Pi }(c)$ is a codeword of $\mathcal{C}_{k}\!\left(n-1,q\right)$ with $\mathrm{wt}\!\left(c_i\right) \leqslant\mathrm{wt}\!\left(c\right) \leqslant\delta\theta_k$ and hence a linear combination of at most $\delta$ $k$-spaces. In addition, since no line through $R_i$ contains more than one point of $\mathcal{P}$, $\lambda'$ intersects $\mathrm{supp}\!\left(c_i\right)$ in exactly $\delta$ points. It follows from (2) that $c_i$ is a linear combination of exactly $\delta$ $k$-spaces of $\Pi$, each intersecting $\lambda'$ exactly in a point. Let $\mathcal{K}_i$ denote the set of these $k$-spaces. Take a $k$-space $\kappa_1\in\mathcal{K}_1$ and a $k$-space $\kappa_2\in\mathcal{K}_2$. Then $\left\langle R_i,\kappa_i\right\rangle$ intersects $\lambda$ in a line through $R_i$ and a point of $\mathcal{P}$, $i\in\left\{1,2\right\}$. Since the line $\left\langle R_1,R_2\right\rangle$ does not contain a point of $\mathcal{P}$, $\left\langle R_1,\kappa_1\right\rangle$ and $\left\langle R_2,\kappa_2\right\rangle$ intersect $\lambda$ in distinct lines, and in particular are distinct subspaces. As a result, the $\left(k+1\right)$-spaces $\left\langle R_1,\kappa_1\right\rangle$ and $\left\langle R_2,\kappa_2\right\rangle$ either intersect in a subspace of dimension at most $k-1$ (and therefore share at most $\theta_{k-1}$ points of $\mathrm{supp}\!\left(c\right)$), or intersect in a $k$-space. By , the latter $k$-space must be thin, so $\left\langle R_1,\kappa_1\right\rangle$ and $\left\langle R_2,\kappa_2\right\rangle$ share at most ${\Delta_q}\theta_{k-1}$ points of $\mathrm{supp}\!\left(c\right)$. As a consequence, we get $$\begin{aligned} \delta\theta_k\geqslant\mathrm{wt}\!\left(c\right)&\geqslant\mathrm{wt}\!\left(\mathrm{proj}_{R_1,\Pi}\!\left(c\right)\right)+\mathrm{wt}\!\left(\mathrm{proj}_{R_2,\Pi}\!\left(c\right)\right)-\left|\mathcal{K}_1\right|\cdot\left|\mathcal{K}_2\right|\cdot{\Delta_q}\theta_{k-1}\\ &\geqslant 2\cdot\delta\left(\theta_k-\left(\delta-1\right)\theta_{k-1}\right)-\delta^2{\Delta_q}\theta_{k-1}\\ &=\delta\theta_k+\left(q-2\left(\delta-1\right)-\delta{\Delta_q}\right)\delta\theta_{k-1}+\delta. \end{aligned}$$ This implies that $q-2\left(\delta-1\right)-\delta{\Delta_q}<0$, or, in other words, that $$q<2\left(\delta-1\right)+\delta{\Delta_q}\leqslant 2\left(\sqrt{q}-1-1\right)+\left(\sqrt{q}-1\right)^2=q-3,$$ a contradiction. ◻ # Codes of $\bm{j}$- and $\bm{k}$-spaces {#Sec:j&k} In this section, we finish the proof of . The proof works by induction on $j$ and is very similar to the proof of our previous paper [@AdriaensenDenaux §6]. Essentially, we only need to improve the lower bound of Step 1 in the proof of [@AdriaensenDenaux Theorem 6.7]. Let us introduce the proper notation. **Definition 31**. 1. For each integer $i$, $0\leqslant i<j$, and for each $v \in \mathbb{F}_{p}^{\mathcal G_j(n,q)}$, define $\textnormal{\dn l}_i(v) \in \mathbb{F}_{p}^{\mathcal G_i(n,q)}$ as $$\textnormal{\dn l}_i(v): \mathcal G_i(n,q) \to \mathbb{F}_{p}: \iota \mapsto \sum_{\substack{\lambda \in \mathcal G_j \\ \iota \subset \lambda}} v(\lambda).$$ This means that the value of an $i$-space $\iota$ with respect to $\textnormal{\dn l}_i(v)$ is the sum of the values with respect to $v$ of all $j$-spaces $\lambda$ through $\iota$. We will denote $\textnormal{\dn l}_0$ by $\textnormal{\dn l}$. 2. Define $\mathcal{K}_{j,k}(n,q) \coloneqq \ker(\textnormal{\dn l}_{j-1}) \cap \mathcal{C}_{j,k}(n,q)$. We want to prove a good lower bound on the minimum weight of $\mathcal{K}_{j,k}(n,q)$. This will be done by induction on $n$. We recall the most important properties of $\textnormal{\dn l}_i$. **Result 32** ([@AdriaensenDenaux §6]). *Let $i$ be an integer, $0\leqslant i<j$.* 1. *$\textnormal{\dn l}_i$ is a linear map.* 2. *$\textnormal{\dn l}_i$ maps $\mathcal{C}_{j,k}\!\left(n,q\right)$ to $\mathcal{C}_{i,k}\!\left(n,q\right)$, and more specifically maps $\kappa^{(j)}$ to $\kappa^{(i)}$.* 3. *$\textnormal{\dn l}\circ \textnormal{\dn l}_i = \textnormal{\dn l}$.* 4. *If $c \in \mathcal{K}_{j,k}(n,q)$ and $P \in \mathrm{supp}_{0}\!\left(c\right)$, then there are at least $2 \frac{q^{k-1}}{\theta_{j-1}} \genfrac{[}{]}{0pt}{}{k-1}{j-1}_q$ $j$-spaces of $\mathrm{supp}\!\left(c\right)$ incident with $P$.* 5. *If $c \in \mathcal{C}_{j,k}\!\left(n,q\right)$ and $\iota \in \mathrm{supp}_{j-1}\!\left(c\right)$, then there are at least $\theta_{k-j}$ $j$-spaces of $\mathrm{supp}\!\left(c\right)$ incident with $\iota$.* We now establish a lower bound on the minimum weight of $\mathcal{K}_{j,k}(n,q)$ in the base case $n = k+1$. **Lemma 33**. *Suppose that $q \geqslant 8$ and $j > 0$. If $c \in \mathcal{K}_{j,k}(k+1,q) \setminus \{ \bm{0}\}$, then $$\mathrm{wt}\!\left(c\right) > \frac 12 q \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q.$$* *Proof.* Take $c \neq \bm{0}$ in $\mathcal{K}_{j,k}(k+1,q)$. Define $T \coloneqq \mathrm{supp}\!\left(c\right)$ and $S \coloneqq \mathrm{supp}_{0}\!\left(c\right)$. Denote their sizes by $t \coloneqq |T| = \mathrm{wt}\!\left(c\right)$ and $s \coloneqq |S|$. Define $x$ as the average number of $j$-spaces $\lambda \in T$ through a point of $S$. Then the number of incidences between $S$ and $T$ is given by $$e(S,T) = t \theta_j = s x.$$ Now apply the expander mixing lemma (). This tells us $$\begin{aligned} \left| t \theta_j - \frac{\theta_j}{\theta_{k+1}} st \right| < \sqrt{{q^j \genfrac{[}{]}{0pt}{}{k}{j}_q} st} & \implies t \theta_j \left| 1 - \frac s {\theta_{k+1}} \right| < \sqrt{{q^j \genfrac{[}{]}{0pt}{}{k}{j}_q} \frac{t^2 \theta_j} x} \\ & \implies \left | 1 - \frac s {\theta_{k+1}} \right| < \sqrt{\frac {q^j \genfrac{[}{]}{0pt}{}{k}{j}_q} {x \theta_j}}.\end{aligned}$$ Since $s \leqslant\theta_{k+1}$, this implies that $$\begin{aligned} 1 - \frac{t \theta_j}{x \theta_{k+1}} < \sqrt{\frac {q^j \genfrac{[}{]}{0pt}{}{k}{j}_q} {x \theta_j}} && \implies && t > \frac{x \theta_{k+1}}{\theta_j} \left( 1 - \sqrt{\frac {q^j \genfrac{[}{]}{0pt}{}{k}{j}_q} {x \theta_j}} \right).\end{aligned}$$ This lower bound on $t$ is increasing in $x$. By (4), $x \geqslant 2 \frac{q^{k-1}}{\theta_{j-1}} \genfrac{[}{]}{0pt}{}{k-1}{j-1}_q$, so we get that $$\begin{aligned} \label{EqLowerBoundT} t > 2 \frac{q^{k-1}}{\theta_{j-1}} \genfrac{[}{]}{0pt}{}{k-1}{j-1}_q \frac{\theta_{k+1}}{\theta_j} \left( 1 - \sqrt{\frac{q^j \genfrac{[}{]}{0pt}{}{k}{j}_q} {{2 \frac{q^{k-1}}{\theta_{j-1}} \genfrac{[}{]}{0pt}{}{k-1}{j-1}_q \theta_j}}} \right).\end{aligned}$$ The rest of the proof consists of estimations of the above expression. First, consider the expression under the square root. $$\begin{aligned} \frac 12 \frac{q^j}{q^{k-1}} \frac{\genfrac{[}{]}{0pt}{}{k}{j}_q}{\genfrac{[}{]}{0pt}{}{k-1}{j-1}_q} \frac 1 {\frac{\theta_j}{\theta_{j-1}}} &= \frac 12 \frac{q^j}{q^{k-1}} \frac{q^k-1}{q^j-1} \frac{q^j-1}{q^{j+1}-1} = \frac 12 \frac{q^{k+j}-q^j}{q^{k+j}-q^{k-1}} \\ &< \frac 12 \frac{q^{k+j}}{q^{k+j}-q^{k+j-2}} = \frac 1 2 \frac{q^2}{q^2-1}.\end{aligned}$$ Next, we prove that $$\frac{q^{k-1}}{\theta_{j-1}} \genfrac{[}{]}{0pt}{}{k-1}{j-1}_q \frac{\theta_{k+1}}{\theta_j} > (q-1) \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q.$$ Dividing both sides by $\genfrac{[}{]}{0pt}{}{k-1}{j-1}_q$, we need to prove that $$\frac{q^{k-1}(q-1)}{q^j-1} \frac{q^{k+2}-1}{q^{j+1}-1} > (q-1) \frac{q^{k+1}-1}{q^{j+1}-1} \frac{q^k-1}{q^j-1}.$$ This is equivalent to $$q^{k-1}(q^{k+2}-1) > (q^{k+1}-1)(q^k-1),$$ which is easy to check. Combining this with , this yields $$\mathrm{wt}\!\left(c\right) > \underbrace{2 \left(1 - \frac 1 q\right) \left( 1 - \sqrt{\frac 12 \frac{q^2}{q^2-1}} \right)}_{\eqqcolon C_q} q \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q %\underbrace{\left( 2 \frac{q}{q+2} \left( 1 - \sqrt{\frac 12 \frac{q^2}{q^2-q-1}} \right) \right)}_{\coloneqq C_q} q \gauss{k+1}{j+1}.$$ The expression $C_q$ is increasing in $q$. Since we assumed that $q \geqslant 8$, we get that $C_q \geqslant C_8 \geqslant\frac 12$, the last inequality being checked by computer. ◻ For the induction step, we will again make use of $\mathrm{proj}^{(j)}_{R,\Pi}$ (see ). **Lemma 34**. *Consider a point $R$ and a hyperplane $\Pi \not \ni R$.* 1. *For $i<j$, $\textnormal{\dn l}_i \circ \mathrm{proj}^{(j)}_{R,\Pi }= \mathrm{proj}^{(i)}_{R,\Pi }\circ \textnormal{\dn l}_i$.* 2. *$\mathrm{proj}^{(j)}_{R,\Pi}$ maps $\mathcal{K}_{j,k}(n,q)$ to $\mathcal{K}_{j,k}(n-1,q)$.* *Proof.* (1) Take $v \in \mathbb{F}_{p}^{\mathcal G_j(n,q)}$. Choose an $i$-space $\iota$ in $\Pi$. Then $$\begin{aligned} \textnormal{\dn l}_i \left( \mathrm{proj}^{(j)}_{R,\Pi }(v) \right) (\iota) &= \sum_{\substack{\lambda' \in \mathcal G_j(\Pi) \\ \iota \subset \lambda' }} \mathrm{proj}^{(j)}_{R,\Pi }(v) (\lambda') = \sum_{\substack{\lambda' \in \mathcal G_j(\Pi) \\ \iota \subset \lambda' }} \sum_{\lambda \in \mathcal G_j(\left\langle R,\lambda'\right\rangle)} v(\lambda) \\ &= \sum_{\lambda \in \mathcal G_j(n,q)} v(\lambda) \underbrace{\left| \left\{\lambda' \in \mathcal G_j(\Pi)\,:\,\iota \subset \lambda', \, \lambda \subset \left\langle R,\lambda'\right\rangle\right\} \right|}_{\eqqcolon f_1(\lambda)}. \end{aligned}$$ On the other hand, $$\begin{aligned} \mathrm{proj}^{(i)}_{R,\Pi }\left(\textnormal{\dn l}_i(v)\right)(\iota) & = \sum_{\iota' \in \mathcal G_i(\left\langle R,\iota\right\rangle)} \textnormal{\dn l}_i(v) (\iota') = \sum_{\iota' \in \mathcal G_i(\left\langle R,\iota\right\rangle)} \sum_{\substack{\lambda \in \mathcal G_j(n,q) \\ \iota' \subset \lambda}} v(\lambda) \\ & = \sum_{\lambda \in \mathcal G_j(n,q)} v(\lambda) \underbrace{|\left\{\iota' \in \mathcal G_i(\left\langle R,\iota\right\rangle)\,:\, \iota' \subset \lambda \right\}|}_{\eqqcolon f_2(\lambda)}. \end{aligned}$$ Thus, we need to prove that $f_1(\lambda) \equiv f_2(\lambda) \pmod p$, for every $j$-space $\lambda$. Note that $$f_2(\lambda) = |\mathcal G_i(\left\langle R,\iota\right\rangle \cap \lambda)| \equiv \begin{cases} 1 & \text{if } \dim \left(\left\langle R,\iota\right\rangle \cap \lambda\right) \geqslant i, \\ 0 & \text{otherwise} \end{cases} \pmod p.$$ Moreover, $$\begin{aligned} \dim \left(\left\langle R,\iota\right\rangle \cap \lambda\right) = \begin{cases} \dim \left(\iota \cap \lambda\right) + 1 & \text{if } R \in \lambda, \\ \dim \left(\iota \cap \lambda\right) & \text{if } R \notin \lambda. \end{cases} \end{aligned}$$ On the other hand, $$\begin{aligned} f_1(\lambda) &= |\left\{\lambda' \in \mathcal G_j(\Pi)\,:\,\iota \subset \lambda', \left\langle R,\lambda\right\rangle \cap \Pi \subseteq \lambda'\right\}| \\ &\equiv \begin{cases} 1 & \text{if } \dim\left(\left\langle\iota,\left\langle R,\lambda\right\rangle \cap \Pi\right\rangle\right) \leqslant j, \\ 0 & \text{otherwise} \end{cases} \pmod p. \end{aligned}$$ Moreover, $$\begin{aligned} \dim\left(\left\langle\iota,\left\langle R,\lambda\right\rangle \cap \Pi\right\rangle\right) = \dim \left(\left\langle\iota,R,\lambda\right\rangle\right) - 1 = \begin{cases} \dim \left(\left\langle\iota,\lambda\right\rangle\right) - 1 & \text{if } R \in \lambda, \\ \dim \left(\left\langle\iota,\lambda\right\rangle\right) & \text{if } R \notin \lambda. \end{cases} \end{aligned}$$ Thus, we need to prove that $$\begin{cases} \dim \left(\iota \cap \lambda\right) + 1 \geqslant i \iff \dim \left(\left\langle\iota,\lambda\right\rangle\right) - 1 \leqslant j & \text{if } R \in \lambda, \\ \dim \left(\iota \cap \lambda\right) \geqslant i \iff \dim \left(\left\langle\iota,\lambda\right\rangle\right) \leqslant j & \text{if } R \notin \lambda. \end{cases}$$ This follows in both cases from Grassmann's identity: $\dim \left(\iota \cap \lambda\right) + \dim \left(\left\langle\iota,\lambda\right\rangle\right) = i + j.$ \(2\) It follows directly from (1) that $\mathrm{proj}^{(j)}_{R,\Pi}$ maps $\ker(\textnormal{\dn l}_{j-1})$ to $\ker(\textnormal{\dn l}_{j-1})$. By [Result 12](#res:Projection){reference-type="ref" reference="res:Projection"} (1), $\mathrm{proj}^{(j)}_{R,\Pi}$ also maps $\mathcal{C}_{j,k}\!\left(n,q\right)$ to $\mathcal{C}_{j,k}\!\left(n-1,q\right)$. The statement follows. ◻ **Proposition 35**. *Suppose that $q \geqslant 8$ and $j > 0$. If $c \in \mathcal{K}_{j,k}(n,q) \setminus \{ \bm{0}\}$, then $$\mathrm{wt}\!\left(c\right) > \frac 12 q \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q.$$* *Proof.* We prove this proposition by induction on $n$. The base case $n=k+1$ was dealt with in . So assume that $n \geqslant k+2$ and suppose that the proposition holds for $\mathcal{K}_{j,k}(n-1,q)$. Take a non-zero codeword $c \in \mathcal{K}_{j,k}(n,q)$ and a $j$-space $\lambda \in \mathrm{supp}\!\left(c\right)$. We again denote $\mathrm{supp}_{0}\!\left(c\right)$ by $S$. First, suppose that there exists a $(j+1)$-space $\sigma$ through $\lambda$ that contains no other $j$-space of $\mathrm{supp}\!\left(c\right)$ and contains a point $R \notin S$. Choose a hyperplane $\Pi$ intersecting $\sigma$ in $\lambda$. By (2), $\mathrm{proj}^{(j)}_{R,\Pi }(c)$ is a codeword of $\mathcal{K}_{j,k}(n-1,q)$. Moreover, since $\lambda$ is the only $j$-space of $\mathrm{supp}\!\left(c\right)$ in $\sigma$, $\mathrm{proj}^{(j)}_{R,\Pi }(c)(\lambda) = c(\lambda) \neq 0$. In particular, this means that $\mathrm{proj}^{(j)}_{R,\Pi }(c) \neq \bm{0}$. Using (2) and the induction hypothesis, this implies that $$\mathrm{wt}\!\left(c\right) \geqslant\mathrm{wt}\!\left(\mathrm{proj}^{(j)}_{R,\Pi }(c)\right) > \frac 12 q \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q.$$ Now suppose that every $(j+1)$-space through $\lambda$ contains either another $j$-space of $\mathrm{supp}\!\left(c\right)$ or contains no points outside of $S$. Then every $(j+1)$-space through $\lambda$ contains at least $q^j$ points of $S \setminus \lambda$. Therefore, $$|S| \geqslant\theta_j + \theta_{n-j-1} q^j > \theta_{n-1} \geqslant\theta_{k+1}.$$ As in the proof of , we have that $$\mathrm{wt}\!\left(c\right) \geqslant 2 \frac{q^{k-1} \genfrac{[}{]}{0pt}{}{k-1}{j-1}_q}{\theta_{j-1} \theta_j} |S| \geqslant 2 \frac{q^{k-1} \genfrac{[}{]}{0pt}{}{k-1}{j-1}_q}{\theta_{j-1} \theta_j} \theta_{k+1}.$$ It suffices to check that the right-hand side of the above inequality is greater than $\frac 12 q \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q$. This is equivalent to $$2 q^{k-1} \left(q^{k+2}-1\right) (q-1) > \frac 1 2 q \left(q^{k+1}-1\right) \left(q^k-1\right).$$ Since $q^{k+2}-1 > q \left(q^{k+1}-1\right)$, this follows from $$2 q^{k-1}(q-1) > \frac 1 2 \left(q^k-1\right),$$ which can be easily checked to hold for every $q \geqslant 2$. ◻ **Proposition 36**. *Suppose that $q \geqslant 8$ and suppose that $C_q \leqslant\frac 1 4 q$ is a constant such that every codeword $c \in \mathcal{C}_{j-1,k}(n,q)$ with $\mathrm{wt}\!\left(c\right) \leqslant C_q \genfrac{[}{]}{0pt}{}{k+1}{j}_q$ is a linear combination of exactly $\left\lceil\frac{\mathrm{wt}\!\left(c\right)}{\genfrac{[}{]}{0pt}{}{k+1}{j}_q}\right\rceil$ $k$-spaces. Then every codeword $c \in \mathcal{C}_{j,k}(n,q)$ with $\mathrm{wt}\!\left(c\right) \leqslant C_q \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q$ is a linear combination of exactly $\left\lceil\frac{\mathrm{wt}\!\left(c\right)}{\genfrac{[}{]}{0pt}{}{k+1}{j+1}_q}\right\rceil$ $k$-spaces.* *Proof.* Suppose that the condition of the proposition holds for $\mathcal{C}_{j-1,k}(n,q)$. Take a codeword $c \in \mathcal{C}_{j,k}(n,q)$ with $\mathrm{wt}\!\left(c\right) \leqslant C_q \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q$. Consider $c' \coloneqq \textnormal{\dn l}_{j-1}(c)$. Perform a double count on $$\left\{(\iota,\lambda) \in \mathrm{supp}\!\left(c'\right) \times \mathrm{supp}\!\left(c\right)\,:\,\iota \subset \lambda\right\}.$$ If $\iota \in \mathrm{supp}\!\left(c'\right)$, then $\iota$ is incident with at least $\theta_{k-j}$ $j$-spaces of $\mathrm{supp}\!\left(c\right)$ by (5). Hence, $$\mathrm{wt}\!\left(c'\right) \leqslant\mathrm{wt}\!\left(c\right) \frac{\theta_{j} }{\theta_{k-j} } \leqslant C_q \frac{q^{j+1}-1}{q^{k-j+1}-1} \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q = C_q \genfrac{[}{]}{0pt}{}{k+1}{j}_q.$$ By our hypothesis, $c'$ is a linear combination of at most $C_q$ $k$-spaces, i.e. $$c' = \sum_{i=1}^{C_q} \alpha_i \kappa_i^{(j-1)},$$ for some scalars $\alpha_i$ and $k$-spaces $\kappa_i$. Then $$c' = \textnormal{\dn l}_{j-1} \left( \sum_{i=1}^{C_q} \alpha_i \kappa_i^{(j)} \right)$$ by (1,2). This implies that $c'' \coloneqq c - \sum_{i=1}^{C_q} \alpha_i \kappa_i^{(j)}$ is contained in $\mathcal{K}_{j,k}(n,q)$. Since $c''$ is equal to the difference of two functions, each of weight at most $C_q \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q$, we find that $\mathrm{wt}\!\left(c''\right) \leqslant 2 C_q \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q \leqslant\frac 12 q \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q$. By , this means that $c'' = \bm{0}$, hence $c$ is a linear combination of at most $C_q$ characteristic functions of $k$-spaces. By (1), $c$ is a linear combination of exactly $\left\lceil\mathrm{wt}\!\left(c\right) / \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q\right\rceil$ $k$-spaces. ◻ now follows immediately by inductively applying , with as base case. One only needs to check that $\Delta_q\leqslant\frac 1 4 q$, which follows directly from $\Delta_q< \sqrt q$ and $q \geqslant 32$. ---------------------------------------------------------------------------------- -------------------------------------------------------------------- Sam Adriaensen Lins Denaux *Vrije Universiteit Brussel* *Ghent University* Department of Mathematics Department of Mathematics: Analysis, and Data Science Logic and Discrete Mathematics Pleinlaan 2 -- Building G Krijgslaan $281$ -- Building S$8$ $1050$ Elsene $9000$ Ghent BELGIUM BELGIUM `e-mail:` [`sam.adriaensen@vub.be`](mailto:sam.adriaensen@vub.be) `e-mail:` [`lins.denaux@ugent.be`](mailto:lins.denaux@ugent.be) `website:` [`samadriaensen.wordpress.com`](https://samadriaensen.wordpress.com/) `website:` [`cage.ugent.be/ldnaux`](https://cage.ugent.be/~ldnaux) ---------------------------------------------------------------------------------- -------------------------------------------------------------------- [^1]: For any set $\mathcal{S}$ of points in $\mathrm{PG}\!\left(n,q\right)$, we can define its characteristic function $v_\mathcal{S}$ as the function that maps the points of $\mathcal{S}$ to 1, and the other points of $\mathrm{PG}\!\left(n,q\right)$ to 0.

arxiv_math

--- abstract: | In 1953, Carlitz [@Car53] showed that all permutation polynomials over ${\mathbb F}_q$, where $q>2$ is a power of a prime, are generated by the special permutation polynomials $x^{q-2}$ (the inversion) and $ax+b$ (affine functions, where $0\neq a, b\in {\mathbb F}_q$). Recently, Nikova, Nikov and Rijmen [@NNR19] proposed an algorithm (NNR) to find a decomposition of the inverse function in quadratics, and computationally covered all dimensions $n\leq 16$. Petrides [@P23] found a class of integers for which it is easy to decompose the inverse into quadratics, and improved the NNR algorithm, thereby extending the computation up to $n\leq 32$. Here, we extend Petrides' result, as well as we propose a number theoretical approach, which allows us to cover easily all (surely, odd) exponents up to $250$, at least. author: - "Florian Luca$^{1,2}$, Santanu Sarkar$^3$, Pantelimon Stănică$^4$ $^1$ School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits 2050, Johannesburg, South Africa; and $^2$ Centro de Ciencias Matemáticas, UNAM, Morelia, Mexico; `Florian.Luca@wits.ac.za`, $^3$ Department of Mathematics, Indian Institute of Technology Madras, Sardar Patel Road, Chennai TN 600036, INDIA; `santanu@iitm.ac.in`, $^4$ Applied Mathematics Department, Naval Postgraduate School, Monterey 93943, USA; `pstanica@nps.edu`" title: "**Representing the inverse map as a composition of quadratics in a finite field of characteristic $2$**" --- **Keywords:** Permutations, Decompositions, Quadratics, Algorithm, Primes, Sieves **MSC 2020**: 11A41, 11N13, 11N36, 20B99, 94A60, 94D10 # Introduction In [@Car53], Carlitz showed that all permutation polynomials over ${\mathbb F}_q$, where $q>2$ is a power of a prime, are generated by the special permutation polynomials $x^{q-2}$ (the inversion) and $ax+b$ (affine functions, where $0\neq a, b\in {\mathbb F}_q$). The smallest number of inversions in such a decomposition is called the *Carlitz rank*. Here, we ask whether the inverse in ${\mathbb F}_{2^n}$ (the finite field of dimension $n$ over the two-element prime field ${\mathbb F}_2$) can be written as a composition of quadratics, or quadratics and cubics. That is, we ask if there are integers $r\ge 1$ and $a_1\ge 0,\ldots,a_r\ge 0$ such that $$-1\equiv \prod_{i=1}^r (2^{a_i}+1)\pmod {2^n-1}.$$ Nikova, Nikov and Rijmen [@NNR19] proposed an algorithm to find such a decomposition. Via Carlitz [@Car53], they were able to use the algorithm and show that for $n\leq 16$ any permutation can be decomposed in quadratic permutations, when $n$ is not multiple of $4$ and in cubic permutations, when $n$ is multiple of $4$. Petrides [@P23], in addition to a theoretical result, which we will discuss below, improved the complexity of the algorithm and presented a computational table of shortest decompositions for $n\leq 32$, allowing also cubic permutations in addition to quadratics. Here, we find a number theoretical approach which allows us to cover all (surely, odd) exponents up to $250$. # Our results Let $\nu_2$ be the 2-valuation, that is, the largest power of $2$ dividing the argument. We start with a proposition, extending one of Petrides' results [@P23], which states that if $n$ is an odd integer and $\frac{n-1}{2^{\nu_2(n-1)}}\equiv 2^k\pmod {2^n-1}$, for some $k$, then $$\begin{aligned} 2^n-2&=2\left(2^{\left(\frac{n-1}{2^{\nu_2(n-1)}}\right) 2^{\nu_2(n-1)}}-1\right)\cr &=2\left(2^{\frac{n-1}{2^{\nu_2(n-1)}}}-1\right)\prod_{j=1}^{\nu_2(n-1)}\left(2^{\frac{n-1}{2^j}}+1\right)\cr &\equiv 2\left(2^{2^k}-1\right)\prod_{j=1}^{\nu_2(n-1)}\left(2^{\frac{n-1}{2^j}}+1\right)\cr & = 2\prod_{j=0}^{k-1}\left(2^{2^j}+1\right)\prod_{j=1}^{\nu_2(n-1)}\left(2^{\frac{n-1}{2^j}}+1\right).\end{aligned}$$ This implies, via Carlitz [@Car53], that for all odd integers (coined *good integers*, with the counterparts named *bad integers* in [@M97]) satisfying the congruence $\frac{n-1}{2^{\nu_2(n-1)}}\equiv 2^k\pmod {2^n-1}$, one can decompose any permutation polynomial in ${\mathbb F}_{2^n}$ into affine and quadratic power permutations. The smallest odd positive integer that is not *good* is $n=7$. We note however that in that case $$2^7-2=2(2^6-1)=2(2^2-1)(2^4+2^2+1)=2(2+1)(2^4+2^2+1),$$ and so, any permutation in ${\mathbb F}_{2^7}$ can be decomposed into affine, quadratic and cubic permutations. We are ready to generalize this observation. **Theorem 1**. *Let $n$ be an odd integer satisfying $$\frac{n-1}{2^{\nu_2(n-1)}}\equiv 2^k3^s\pmod {2^n-1},$$ for some non-negative integers $r,s$. Then, the inverse power permutation in ${\mathbb F}_{2^n}$ has a decomposition into affine, quadratic and cubic power permutations of length $k+s+\nu_2(n-1)$.* *Proof.* As we have already alluded to above, using the difference of cubes factorization, $a^3-b^3=(a-b)(a^2+ab+b^2)$, we have $$\begin{aligned} 2^n-2&=2\left(2^{\frac{n-1}{2^{\nu_2(n-1)}}}-1\right)\prod_{j=1}^{\nu_2(n-1)}\left(2^{\frac{n-1}{2^j}}+1\right)\cr &\equiv 2 \left(2^{2^k3^s}-1\right) \prod_{j=1}^{\nu_2(n-1)}\left(2^{\frac{n-1}{2^j}}+1\right) \cr &= 2\left(2^{2^k 3^{s-1}}-1\right) \left(2^{2^{k+1}3^{s-1}}+2^{2^k3^{s-1}}+1\right) \prod_{j=1}^{\nu_2(n-1)}\left(2^{\frac{n-1}{2^j}}+1\right) \cr &\cdots\cdots\cdots\cdots\cr &=2 \left(2^{2^k}-1\right)\prod_{j=0}^{s-1} \left(2^{2^{k+1}3^j}+2^{2^k 3^j}+1\right) \prod_{j=1}^{\nu_2(n-1)}\left(2^{\frac{n-1}{2^j}}+1\right) \cr & \equiv 2\prod_{j=0}^{k-1}\left(2^{2^j}+1\right) \prod_{j=0}^{s-1} \left(2^{2^{k+1}3^j}+2^{2^k 3^j}+1\right)\prod_{j=1}^{\nu_2(n-1)}\left(2^{\frac{n-1}{2^j}}+1\right).\end{aligned}$$ The claim is shown. ◻ **Example 1**. *It is natural to investigate the counting function ${\mathcal B}(x)$ of *superbad integers* (that is, integers $n$ such that $\frac{n-1}{2^{\nu_2(n-1)}}\not\equiv 2^k3^s\pmod {2^n-1}$), with ${\mathcal B}(x)=\{n\leq x\,:\, n\text{ is superbad}\}$, or its complement $${\mathcal A}(x)=\{n\leq x\,:\, \frac{n-1}{2^{\nu_2(n-1)}}\equiv 2^k3^s\pmod {2^n-1}\}.$$* *As an example, $|{\mathcal B}(50)|=16$, more precisely, $${\mathcal B}(50)=\{1, 2, 3, 4, 5, 7, 9, 10, 13, 17, 19, 25, 28, 33, 37, 49\}$$ (Petrides [[@P23]]{.upright} noted that $25$ integers up to $50$ are bad, so our extension surely prunes the integers better). In a recent paper [[@LS23]]{.upright}, it was shown that $${\mathcal A}(x)\ll \frac{x}{(\log\log x)^{1+o(1)}}.$$* For most of the remaining work we restrict our attention to $n=p$, a prime. Let $p\ge 3$ be prime, $N:=N_p=2^p-1$. It is known that if $q\mid N_p$, then $q\equiv 1\pmod {p}$. We ask if we can say anything about the number of distinct prime factors $\omega(N_p)$ of $N_p$. We propose the following conjecture. **Conjecture 1**. *There exists $p_0$ such that for $p>p_0$, $\omega(N_p)<1.36\log p$.* Similar heuristics regarding lower bounds for $\Omega(2^n-1)$ and $\omega(2^n-1)$ can be found in [@KoLa] and [@LuSt]. Our Conjecture [Conjecture 1](#conj:1){reference-type="ref" reference="conj:1"}, is based on statistical arguments originating from sieve methods. The Turán-Kubilius inequality asserts that $$\sum_{n\le x} (\omega(n)-\log\log x)^2=O(x\log\log x).$$ So, if $\delta>0$ is fixed, the set of $n\le x$ such that $\omega(n)\ge (1+\delta)\log\log x$ is of counting function $O_{\delta}(x/\log\log x)$. One can do better using sieves. Indeed, Exercise 04 in [@HT] shows that for fixed $\delta>0$ we in fact have that $$\#\{n\le x: \omega(n)\ge (1+\delta)\log\log x\}\ll_{\delta} \frac{x}{(\log x)^{Q(\delta)}},$$ where $Q(\delta):=(1+\delta)\log((1+\delta)/e)+1$. We would like to apply such heuristics to $N_p=2^p-1$. But note that if $q\mid N_p$, then $2^p\equiv 1\pmod q$. In particular, ${\displaystyle{\left(\frac{2}{q}\right)=1}}$, so $q\equiv \pm 1\pmod 8$. But then the same proof as Exercise 04 in [@HT] shows that $$\begin{aligned} & \# & \{n\le x: q\mid n \Rightarrow q\equiv \pm 1\pmod 8 \quad {\text{\rm and}} \quad \omega(n)\ge (1+\delta)\log\log x\}\cr & \le & \frac{x}{(\log x)^{Q_1(\delta)+o(1)}}\end{aligned}$$ as $x\to\infty$, where $Q_1(\delta):=(1+\delta)\log((1+\delta)/(0.5 e))+1$. Taking $\delta=0.36$, we get $Q_1(\delta)=1.00086\ldots$. Thus, the probability that a number having only prime factors congruent to $\pm 1\pmod 8$ to have more than $1.36\log\log n$ distinct prime factors is $$O\left(\frac{1}{(\log n)^{1.00008}}\right).$$ Applying this to $N_p$, we get $$O\left(\frac{1}{(\log(2^p-1))^{1.0008}}\right)\ll \frac{1}{p^{1.0008}},$$ and since the series $$\sum_{p\ge 3} \frac{1}{p^{1.0008}}$$ is convergent, we are led to believe that maybe there are at most finitely many prime numbers $p$ such that $\omega(N_p)\ge 1.36\log p$. It has been noted that perhaps infinitely often $\omega(N_p)\ge 2$. For example, this is the case if $p\equiv 3\pmod 4$ is such that $q=2p+1$ is prime. Indeed, then $2$ is a quadratic residue modulo $q$ so $2^{(q-1)/2}\equiv 1\pmod q$, showing that $q\mid N_p$. Since $N_p$ is never a perfect power, in particular it cannot be a power of $q$, we get the desired conclusion that $\omega(N_p)\ge 2$. **Conjecture 2**. *There exists $p_0$ such that if $p>p_0$, then $N _p$ is squarefree.* We offer some heuristic evidence for Conjecture [Conjecture 2](#conj:2){reference-type="ref" reference="conj:2"}. Knowing that the prime $q$ divides $N_p$, the conditional probability that $N_p$ is divisible by $q^2$ is $1/q$. Thus, the probability that $N_p$ is not squarefree is bounded above by $$\sum_{q: q\mid N_p~{\text{\rm for~some~prime}}~p} \frac{1}{q},$$ and it was shown by Murata and Pomerance in [@MuPo] that the above sum is finite under GRH. So, assuming Conjecture [Conjecture 1](#conj:1){reference-type="ref" reference="conj:1"} and [Conjecture 2](#conj:2){reference-type="ref" reference="conj:2"}, let $N_p:=q_1\cdots q_k$ for some distinct primes $q_1,\ldots,q_k$ with $k\le 1.36\log p$. We take numbers of the form $2^a+1$ with an odd $a\in [5,p-2]$. We want to compute $$\left(\frac{2^a+1}{2^p-1}\right).$$ This was done by Rotkiewicz in [@Ro]. Namely, write the Euclidean algorithm with even quotients and signed remainders: $$\begin{aligned} p & = & (2k_1)a+\varepsilon_1 r_1,\quad \varepsilon_1\in \{\pm 1\},\quad 1\le r_1\le a-1\cr a & = & (2k_2) r_1+\varepsilon_2 r_2,\quad \varepsilon_2\in \{\pm 1\},\quad 1\le r_2\le r_1-1,\cr \ldots & = & \ldots\cr r_{\ell-2} & = & (2k_{\ell}) r_{\ell-1} +\varepsilon_{\ell} r_\ell,\quad \varepsilon_{\ell}\in \{\pm 1\},\quad r_{\ell}=1,\end{aligned}$$ where $\ell:=\ell(a,p)$ is minimal with $r_{\ell}=1$. Note that $r_i$ are all odd for $i=1,\ldots,\ell$. In particular, $r_j\ge 3$ for $j=1,\ldots,\ell-1$. Then $$\left(\frac{2^a+1}{2^p-1}\right)=\left(\frac{2^p-1}{2^a+1}\right)=\left(\frac{(2^a)^{2k_1}\cdot 2^{\varepsilon_1 r_1}-1}{2^a+1}\right)= \left(\frac{2^{\varepsilon_1 r_1}-1}{2^a+1}\right)=\left(\frac{2^{r_1}-1}{2^a+1}\right).$$ The right--most equality is clear if $\varepsilon_1=1$, and if $\varepsilon_1=-1$, then $$\begin{aligned} \left(\frac{2^{-r_1}-1}{2^a+1}\right) & = & \left(\frac{2}{2^a+1}\right)^{r_1} \left(\frac{2^{-r_1}-1}{2^a+1}\right)=\left(\frac{2^{r_1}(2^{-r_1}-1)}{2^a+1}\right)\cr & = & \left(\frac{1-2^{r_1}}{2^a+1}\right)=\left(\frac{-1}{2^a+1}\right)\left(\frac{2^{r_1}-1}{2^a+1}\right) = \left(\frac{2^{r_1}-1}{2^a+1}\right). \end{aligned}$$ The above calculation shows how to transit from the first step to the second step, or more generally from a step with $2^{r_j}-1$ in the bottom to a step with $2^{r_j}-1$ in the top. For the next step, we write $$\left(\frac{2^{r_1}-1}{2^a+1}\right)=\left(\frac{2^a+1}{2^{r_1}-1}\right)=\left(\frac{(2^{r_1})^{2k_2}\cdot 2^{\varepsilon_2 r_2}+1}{2^{r_1}-1}\right)=\left(\frac{2^{\varepsilon_2 r_2}+1}{2^{r_1}-1}\right)=\left(\frac{2^{r_2}+1}{2^{r_1}-1}\right).$$ The last inequality above is clear if $\varepsilon_2=1$. If $\varepsilon_2=-1$, then since $r_1\ge 3$, we have $$\left(\frac{2^{-r_2}+1}{2^{r_1}-1}\right)=\left(\frac{2}{2^{r_1}-1}\right)^{r_2}\left(\frac{2^{-r_2}+1}{2^{r_1}-1}\right)=\left(\frac{2^{r_2}(2^{-r_2}+1)}{2^{r_1}-1}\right)=\left(\frac{2^{r_2}+1}{2^{r_1}-1}\right).$$ At step $\ell$ we end up with $$\left(\frac{2^{r_{\ell}}+(-1)^{\ell}}{2^{r_{\ell-1}}+(-1)^{\ell-1}}\right).$$ If $\ell(a,p)$ is odd, we get $$\left(\frac{2^1-1}{2^{r_{\ell-1}}+1}\right)=1,$$ and if $\ell(a,p)$ is even we get $$\left(\frac{2^1+1}{2^{r_{\ell-1}}-1}\right)=-\left(\frac{2^{r_{\ell-1}}-1}{3}\right)=-\left(\frac{1}{3}\right)=-1.$$ We thus get that $$\left(\frac{2^a+1}{2^p-1}\right)=(-1)^{\ell+1}.$$ We select the subset ${\mathcal A}(p)$ of odd $a$ in the interval $[5,p-2]$ such that $\ell\equiv 0\pmod 2$. We assume that there are a positive proportion of such, namely that there is a constant $c_1>0$ such that for large $p$, there are $>c_1p$ odd numbers $a\in [5,p-2]$ such that $\ell(a,p)\equiv 0\pmod 2$. So, we have $$\prod_{i=1}^k \left(\frac{2^a+1}{q_i}\right)=-1\quad {\text{\rm for}}\quad a\in {\mathcal A}(p).$$ We next conjecture that for such $a$, the values are $$\label{eq:2} \left(\left(\frac{2^a+1}{q_i}\right),1\le i\le k\right)$$ are uniformly distributed among the $2^k$ vectors ${\underbrace{(\pm 1,\pm 1,\cdots,\pm 1)}_{k~{\text{\rm times}}}}$. Indeed, if not, then somehow for some $a$ the value of $2^a+1$ of the Legendre symbol ${\displaystyle{\left(\frac{2^a+1}{p_i}\right)}}$ should be determined in terms of the values of the same symbol for $b\le a-1$. This can happen for example if: - $2^a+1$ is a square. This never happens for $a\ge 4$. - $2^a+1$ is multiplicatively dependent over $\{2^b+1: 0\le b\le a-1\}$. This does not happen for $a\ge 4$ because of the Carmichael's Primitive Divisor Theorem: $2^a+1$ has a prime factor $p_a$ which is primitive in the sense that $p_a$ does not divide $2^b+1$ for any $b\le a-1$. Well, so we fix $i\in \{1,\ldots,k\}$ and search for $a_i$ such that $$\label{eq:3} \left(\frac{2^{a_i}+1}{q_i}\right)=(-1)^{\delta_{ij}},$$ where $\delta_{ij}$ is the Kronecker symbol. That is, $2^{a_i}+1$ is a quadratic residue modulo $p_j$ for all $j\ne i$ but it is not a quadratic residue modulo $q_i$. Do we expect to find it? Well, let us see. Fix $i$ in $\{1,\ldots,k\}$. The probability that $2^{a_i}+1$ verifies the Legendre conditions given by [\[eq:3\]](#eq:3){reference-type="eqref" reference="eq:3"} is $1/2^k$ so it is $(1-1/2^k)$ that they are not satisfied. Note that since ${\displaystyle{\left(\frac{2^{a_i}+1}{N_p}\right)=-1}}$ we know that an odd number of the $p=p_j$'s satisfy that ${\displaystyle{\left(\frac{2^{a_i}+1}{p_j}\right)=1}}$. So, if we assume that this is so for all possible $a_i$'s, and assuming that there events are independent, we get that the probability that this be so is $$\ll \left(1-\frac{1}{2^{k}}\right)^{c_1 p}<\left(1-\frac{1}{p^{1.36\log 2}}\right)^{c_1p}<\left(1-\frac{1}{p^{0.95}}\right)^{c_1 p}\ll \frac{1}{e^{c_1 p^{0.05}}}.$$ In the above, we used that $k<1.36\log p$ and $1.36\cdot \log 2<0.95$. Of course, this is for $i$ fixed and now we sum up over $i$ from $1$ to $k$ introducing another logarithmic factor in the above count. Since the series $$\sum_{p} \frac{\log p}{e^{c_1 p^{0.05}}}$$ converges, so we expect that the above event does not occur when $p>p_0$. So, we have the following conjecture. **Conjecture 3**. *Assume Conjectures [Conjecture 1](#conj:1){reference-type="ref" reference="conj:1"} and [Conjecture 2](#conj:2){reference-type="ref" reference="conj:2"}. Write $2^p-1=q_1\ldots q_k$ for $p>p_0$ with prime factors $q_1<\cdots<q_k$ and $k<1.36\log p$. Then for each $i=1,\ldots,k$, there exists an odd $a_i\in [5,p-2]$ such that equalities [\[eq:3\]](#eq:3){reference-type="eqref" reference="eq:3"} hold.* The rest of the proof is unconditional. We will show that there exist integers $x_i$ such that $$\label{eq:10} (-1)=\prod_{i=1}^k (2^{a_i}+1)^{x_i}\pmod {2^p-1}.$$ Write $q_i-1=:2^{\alpha_i} R_i$ for $i=1,\ldots,k$, where $R_i$ is odd. Let $$R:={\text{\rm lcm}}[R_i:1\le i\le k]$$ and write $x_i=y_ iR$ for $i=1,\ldots,k$. Let $\rho_i$ be a primitive root modulo $q_i$. Write $$\label{eq:100} 2^{a_i}+1=\rho_j^{b_{ij}}\pmod {q_j}.$$ Conditions [\[eq:3\]](#eq:3){reference-type="eqref" reference="eq:3"} show that $b_{ij}\equiv \delta_{ij}\pmod 2$. Equation [\[eq:10\]](#eq:10){reference-type="eqref" reference="eq:10"} holds if and only if it holds one prime $q_j$ at a time. Thus, we want $$\rho_j^{(q_j-1)/2}\equiv \rho_j^{R\sum_{i=1}^k y_i b_{ij}}\pmod {q_j},$$ which holds provided that $$\frac{(q_j-1)}{2}\equiv R\sum_{i=1}^k y_i b_{ij}\pmod {q_j-1}.$$ This in turn is equivalent to $$2^{\alpha_j-1} \equiv (R/R_j) \sum_{i=1}^k y_i b_{ij}\pmod {2^{\alpha_j}}.$$ Since $R/R_j$ is odd, it follows that it is invertible modulo $2^{\alpha_j}$. Writing $(R/R_j)^*$ for the inverse of $(R/R_j)$ modulo $2^{\alpha_j}$, we get that $$2^{\alpha_j-1} (R/R_j)^*\equiv \sum_{i=1}^k y_i b_{ij}\pmod {2^{\alpha_j}}.$$ Since $(R/R_j)^*$ is odd the left--hand side is just $2^{\alpha_j-1}\pmod {2^{\alpha_j}}$. Thus, $$2^{\alpha_j-1} \equiv \sum_{i=1}^k y_i b_{ij}\pmod {2^{\alpha_j}}.$$ This is a linear system of modular equations for $i=1,\ldots,k$. To see that it is nondegenerate note that the coefficient matrix ${\mathcal B}=(b_{ij})_{1\le i,j\le k}$ modulo $2$ is in fact the identity matrix. Hence, its determinant is odd integer, so invertible modulo powers of $2$, which shows that there exist an integer solution $y_1,\ldots,y_k$. To solve it, we can generate for each $i,j$ the number $b_{i,j}\pmod {2^{\alpha_j}}$ appearing in [\[eq:100\]](#eq:100){reference-type="eqref" reference="eq:100"} as an integer in the interval $[0,2^{\alpha_j}-1]$. Having done that, we solve the linear system $$\sum_{i=1}^k y_i b_{ij}=2^{\alpha_j-1}\quad {\text{\rm for}}\quad j=1,2,\ldots,k.$$ This is non-degenerate since the determinant of the coefficient matrix is odd. Thus, $(y_1,\ldots,y_k)$ are some rational numbers. Now we treat them as residue classes modulo $2^{\alpha}$, where $\alpha:=\max\{\alpha_i: 1\le i\le k\}$ (by inverting the odd determinant modulo $2^{\alpha}$). These ones are the $y_i$'s that we are looking for. We implemented this and checked it for all primes $p\le 250$. We present our approach in Algorithm [\[algo1\]](#algo1){reference-type="ref" reference="algo1"}. Factor $2^p-1=q_1\cdots q_k$, where $q_i$ is prime for $1 \leq i \leq k$; Take a primitive root $\rho_i$ modulo $q_i$ for $1 \leq i \leq k$; Find $b_{ij}$ such that $2^{a_i}+1=\rho_j^{b_{ij}}\pmod {q_j}$ for $1 \leq i,j \leq k$; Find largest $\alpha_i$ such that $2^{\alpha_i}$ is a divisior of $q_i-1$ for $1 \leq i \leq k$; Calculate $\alpha=\max\{\alpha_i: 1\le i\le k\}$; Solve the system of linear equations $\sum_{i=1}^k y_i b_{ij}=2^{\alpha_j-1}\quad {\text{\rm for}}\quad j=1,2,\ldots,k.$ in $\mathbb{Z}_{\alpha}$ The factorization of $2^p-1$ is known for all primes $p<1000$. Surely, we can use the same algorithm modulo $2^n-1$ for $n\le 250$ and odd. Note that if $$2^n-1=\prod_{j=1}^k q_j^{\alpha_j},$$ then we only want to find a relation of the form $$\label{eq:M} (-1)\equiv \prod_{i=1}^k (2^{a_i}+1)^{x_i}\pmod {q_1\ldots q_k}.$$ Indeed, if we have found the above relation, then $$q_1\ldots q_k\mid (2^{a_1}+1)^{x_1}\cdots (2^{a_k}+1)^{x_k}+1.$$ Writing $Q:=(2^n-1)/(q_1\ldots q_k)$, we then get easily that $$(2^n-1)\mid (2^{a_1}+1)^{x_1 Q} (2^{a_2}+1)^{x_2 Q}\cdots (2^{a_k}+1)^{x_k Q}+1.$$ Thus, $$(-1)\equiv \prod_{i=1}^k (2^{a_i}+1)^{x_i Q}\pmod {2^n-1}.$$ Thus, we factor $2^n-1$, take $q_1,\ldots,q_k$ to be all its distinct prime factors and attempt to find some numbers $a_1,\ldots,a_k$ in $[5,n-2]$ such that the congruences [\[eq:3\]](#eq:3){reference-type="eqref" reference="eq:3"} are satisfied. If we are successful, then the argument based on the matrix with odd determinant will work to find a solution of [\[eq:M\]](#eq:M){reference-type="eqref" reference="eq:M"}, which in turn can be easily lifted to a solution modulo $2^n-1$. The factorizations of $2^n-2$ with weight 2 factors for odd $33 \leq n \leq 249$ are given in Table [\[tab:1\]](#tab:1){reference-type="ref" reference="tab:1"}. **Remark 1**. *We have checked that Algorithm [[\[algo1\]](#algo1){reference-type="ref" reference="algo1"}]{.upright} works for most primes up to $250$. But there are a few primes like $47$ for which there is no $a_j \in [5, p-2]$ such that $\left(\frac{2^{a_j}+1}{q_i}\right)=(-1)^{\delta_{ij}}$. In these cases, we use the following trick. We first find $a_i$ and calculate $(\frac{2^{a_j}+1}{q_i})=(-1)^{d_{i,j}}$. Ideally, $d_{i,j}$ should be Kronecker symbols, but if they are not, we can just record what they are. Because $d_{i,j}$ are no longer Kronecker symbols, we cannot be certain that the system is solvable because it may have an even determinant and we cannot invert the matrix modulo powers of $2$. However, we checked primes (and odd integers) up to $250$. We observed that in the case of failure, we can use this trick and always get suitable $a_i$'s such that the corresponding matrix has odd determinant, and is therefore invertible.* \|c\|l\| $n=33$ & $(2^5+1)^{599478} \cdot (2^{13}+1)^{299739} \cdot (2^{29}+1)^{1798434}$ $n=35$ & $\left((2+1)(2^{17}+1)\right)^{967995}\cdot (2^{29}+1)^{276570}$ $n=37$ & $(2^5+1)^{77039772} \cdot (2^{13}+1)^{19259943}$ $n=39$ & $\left((2^{11}+1) (2^{21}+1)\right)^{1592955}$ $n=41$ & $(2^{9}+1)^{20111512782} \cdot (2^{13}+1)^{3351918797}$ $n=43$ & $\left((2^{5}+1) (2^{17}+1) (2^{23}+1)\right)^{593211015}$ $n=45$ & $(2+1)^{407925} \cdot (2^{13}+1)^{349650} \cdot \left((2^{25}+1) (2^{33}+1) (2^{41} +1)\right)^{116550}$ $n=47$ & $(2^{11}+1)^{1927501725} \cdot (2^{37}+1)^{435242325} \cdot (2^{41}+1)^{ 1616614350}$ $n=49$ & $(2^9+1)^{34630287489} \cdot (2^{11}+1)^{3393768173922}$ $n=51$ & $(1+2^{ 29 })^{ 150009615 }$ $n=53$ & $(1+2^{ 5 })^{ 6512186850 } \cdot (1+2^{ 15 })^{ 3506562150 } \cdot (1+2^{ 21 })^{ 250468725 }$ & $(1+2)^{ 6588945 } \cdot (1+2^{ 11 })^{ 5856840 } \cdot (1+2^{ 17 })^{ 732105 }\cdot$ & $(1+2^{ 25 })^{ 1464210 } \cdot (1+2^{ 33 })^{ 10249470 } \cdot (1+2^{ 47 })^{ 732105 }$ & $(1+2^{ 5 })^{ 396029391534 } \cdot (1+2^{ 17 })^{ 1188088174602 } \cdot (1+2^{ 21 })^{ 594044087301 }\cdot (1+2^{ 47 })^{ 198014695767 }$ $n=59$ & $(1+2^{ 7 })^{ 3663925098759300 } \cdot (1+2^{ 13 })^{ 305327091563275 }$ $n=61$ & $(1+2^{ 9 })^{ 1152921504606846975 }$ & $(1+2)^{ 42958503 } \cdot (1+2^{ 5 })^{ 3735522 } \cdot (1+2^{ 39 })^{ 56032830 }\cdot (1+2^{ 43 })^{ 44826264 } \cdot (1+2^{ 47 })^{ 29884176 }$ $n=65$ & $(1+2^{ 17 })^{ 72647571779055 } \cdot (1+2^{ 23 })^{ 72647571779055 } \cdot (1+2^{ 29 })^{ 72647571779055 }$ $n=67$ & $(1+2^{ 5 })^{ 15295807610659665 }$ & $(1+2^{ 11 })^{ 36566619637113225 } \cdot (1+2^{ 17 })^{ 2437774642474215 } \cdot (1+2^{ 53 })^{ 19502197139793720 } \cdot (1+2^{ 67 })^{ 21939971782267935 }$ $n=71$ & $(1+2^{ 11 })^{ 3659326099961865 } \cdot (1+2^{ 13 })^{ 14637304399847460 }$ $n=73$ & $(1+2^{ 31 })^{ 1726845200475585 } \cdot (1+2^{ 45 })^{ 107064402429486270 }$ & $(1+2)^{ 36654975 } \cdot (1+2^{ 39 })^{ 17832150 } \cdot (1+2^{ 41 })^{ 9906750 } \cdot$ & $(1+2^{ 43 })^{ 7925400 } \cdot (1+2^{ 53 })^{ 57459150 } \cdot (1+2^{ 55 })^{ 15850800 } \cdot (1+2^{ 63 })^{ 43589700 }$ & $(1+2^{ 25 })^{ 290641821624556479 } \cdot (1+2^{ 31 })^{ 290641821624556479 } \cdot$ & $(1+2^{ 41 })^{ 290641821624556479 } \cdot (1+2^{ 67 })^{ 581283643249112958 }$ $n=79$ & $(1+2^{ 9 })^{ 12102186118644337359 } \cdot (1+2^{ 15 })^{ 12102186118644337359 } \cdot (1+2^{ 41 })^{ 12102186118644337359 }$ & $(1+2)^{ 106331083505919 } \cdot (1+2^{ 25 })^{ 155626336778778 } \cdot (1+2^{ 37 })^{ 105108887143782 } \cdot$ & $(1+2^{ 39 })^{ 155626336778778 } \cdot (1+2^{ 43 })^{ 4073987873790 }$ $n=83$ & $(1+2^{ 11 })^{ 7239076764159456135965 }$ & $(1+2^{ 9 })^{ 4760486403166879215 } \cdot (1+2^{ 13 })^{ 4760486403166879215 } \cdot (1+2^{ 23 })^{ 4760486403166879215 }$ & $(1+2^{ 39 })^{ 3371346107168004 } \cdot (1+2^{ 41 })^{ 280945508930667 } \cdot (1+2^{ 53 })^{ 2809455089306670 } \cdot$ & $(1+2^{ 61 })^{ 4214182633960005 } \cdot (1+2^{ 71 })^{ 1685673053584002 } \cdot (1+2^{ 83 })^{ 280945508930667 }$ $n=89$ & $(1+2^{ 13 })^{ 309485009821345068724781055 }$ & $(1+2^{ 59 })^{ 280368506850705 } \cdot (1+2^{ 67 })^{ 1682211041104230 } \cdot$ & $(1+2^{ 71 })^{ 280368506850705 } \cdot (1+2^{ 73 })^{ 280368506850705 } \cdot (1+2^{ 81 })^{ 3364422082208460 }$ $n=93$ & $(1+2^{ 17 })^{ 2305843010287435773 }$ $n=95$ & $(1+2^{ 43 })^{ 7354378117756963125 } \cdot (1+2^{ 51 })^{ 7354378117756963125 }$ $n=97$ & $(1+2^{ 5 })^{ 612535370185410489825162846 } \cdot (1+2^{ 9 })^{ 102089228364235081637527141 }$ & $(1+2)^{ 160190876329840719 } \cdot (1+2^{ 23 })^{ 160190876329840719 } \cdot (1+2^{ 35 })^{ 58251227756305716 }\cdot$ & $(1+2^{ 57 })^{ 29125613878152858 } \cdot (1+2^{ 59 })^{ 101939648573535003 } \cdot (1+2^{ 75 })^{ 58251227756305716 }$ $n= 101$&$(1+2^{7})^{261479084205457681314981849}\cdot(1+2^{9})^{1045916336821830725259927396}$ $n= 103$ & $(1+2^{5})^{8204858250687037849538541156}\cdot(1+2^{9})^{2051214562671759462384635289}$ & $(1+2^{7})^{736412106675}\cdot(1+2^{29})^{6627708960075}\cdot(1+2^{37})^{1472824213350}\cdot$ & $(1+2^{55})^{6627708960075}\cdot(1+2^{69})^{15464654240175}\cdot(1+2^{79})^{736412106675}\cdot$ & $(1+2^{83})^{4418472640050}\cdot(1+2^{85})^{441847264005}\cdot(1+2^{87})^{13255417920150}$ $n= 107$ & $(1+2^{5})^{27043212804868893898596335048021}$ & $(1+2^{7})^{744308608310570490215126499806}\cdot (1+2^{15})^{372154304155285245107563249903}$ & $(1+2^{17})^{2078233794395472907116}\cdot(1+2^{31})^{742226355141240323970}\cdot (1+2^{39})^{890671626169488388764}$ & $(1+2^{71})^{180254971962872650107}\cdot (1+2^{87})^{519558448598868226779}$ & $(1+2^{15})^{82901226266607482846190}\cdot(1+2^{25})^{13816871044434580474365}\cdot$ & $(1+2^{29})^{37854441217628987601}\cdot(1+2^{75})^{13816871044434580474365}\cdot$ & $(1+2^{97})^{82901226266607482846190}$ &$(1+2^{17})^{23588654041464621525}\cdot(1+2^{23})^{165120578290252350675}\cdot$ &$(1+2^{39})^{23588654041464621525}\cdot(1+2^{45})^{23588654041464621525}\cdot$ &$(1+2^{75})^{188709232331716972200}$ & $(1+2^{5})^{350280341971560}\cdot(1+2^{11})^{481635470210895}\cdot(1+2^{31})^{1225981196900460}\cdot$ & $(1+2^{55})^{1269766239646905}\cdot(1+2^{71})^{1225981196900460}\cdot(1+2^{87})^{744345726689565}\cdot$ & $(1+2^{93})^{1903697510715}\cdot(1+2^{111})^{1094626068661125}\cdot(1+2^{115})^{1182196154154015}$ & $(1+2^{21})^{121807344007626864485535}\cdot(1+2^{25})^{28109387078683122573585}\cdot$ & $(1+2^{51})^{6635419517925198843570}\cdot(1+2^{81})^{5968559856373716359791215}\cdot$ & $(1+2^{97})^{852651408053388051398745}\cdot(1+2^{109})^{6635419517925198843570}$ & $(1+2^{9})^{99244104353509123769903900571}\cdot(1+2^{19})^{893196939181582113929135105139}\cdot$ & $(1+2^{25})^{893196939181582113929135105139}\cdot(1+2^{43})^{1786393878363164227858270210278}$ & $(1+2^{5})^{38263506571610465341512132024}\cdot(1+2^{9})^{19131753285805232670756066012}\cdot$ & $(1+2^{27})^{4782938321451308167689016503}\cdot (1+2^{53})^{28697629928707849006134099018}\cdot$ &$(1+2^{113})^{7726284980805959347805334351}$ & $(1+2^{23})^{2898591397871459238374625}\cdot(1+2^{29})^{644131421749213164083250}\cdot$ & $(1+2^{95})^{1610328554373032910208125}$ $(1+2^{109})^{4186854241369885566541125}\cdot$ & $(1+2^{121})^{1932394265247639492249750}$ $n= 127$ & $(1+2^{5})^{28356863910078205288614550619314017621}$ $n= 129$ & $(1+2^{9})^{8471295533565243108183419405055}\cdot(1+2^{79})^{9529016348217371325290685495}$ & $(1+2^{9})^{1293849303881895298339827404683529321}\cdot (1+2^{15})^{2587698607763790596679654809367058642}$ & $(1+2^{5})^{27256203475454233141905720953493}\cdot (1+2^{17})^{81768610426362699425717162860479}$ & $(1+2^{45})^{81768610426362699425717162860479}$ & $(1+2^{9})^{1390256215369200900}\cdot(1+2^{25})^{9826330930229511961200}\cdot(1+2^{47})^{38551429107264868200}$ & $(1+2^{55})^{2813183451799578021150}\cdot(1+2^{89})^{7686726614776311776100}$ & $(1+2^{101})^{1406591725899789010575}\cdot(1+2^{109})^{4576375896941567062575}$ & $(1+2^{117})^{1042692161526900675}\cdot(1+2^{121})^{2119793164384189072275}$ &$(1+2^{5})^{39741006355730039527321333167397040041}\cdot (1+2^{7})^{79482012711460079054642666334794080082}$ &$(1+2^{9})^{17408530362059304982034022473992637175}\cdot (1+2^{17})^{457116770994992690994328817697837300}$ &$(1+2)^{1216799735702178355508978464575}\cdot(1+2^{51})^{110618157791107123228088951325}\cdot$&$(1+2^{65})^{63210375880632641844622257900}\cdot(1+2^{121})^{56889338292569377660160032110}$ &$(1+2^{41})^{534639083977880631530660485081925505}\cdot(1+2^{49})^{76377011996840090218665783583132215}\cdot$ &$(1+2^{135})^{305508047987360360874663134332528860}\cdot(1+2^{137})^{4144489023084345980857833217689345}\cdot$ &$(1+2^{139})^{534639083977880631530660485081925505}$ &$(1+2^{15})^{17249119260282613026137951811234}\cdot(1+2^{59})^{51747357780847839078413855433702}\cdot$ &$(1+2^{67})^{43122798150706532565344879528085}\cdot(1+2^{71})^{96786724738252439757774062940813}\cdot$ &$(1+2^{73})^{8624559630141306513068975905617}$ $n= 149$&$(1+2^{9})^{29933886172524326364132038117944134026225}$ &$(1+2^{35})^{47657859344287051433215338407025}\cdot(1+2^{53})^{47657859344287051433215338407025}\cdot$ &$(1+2^{55})^{95315718688574102866430676814050}\cdot(1+2^{81})^{95315718688574102866430676814050}\cdot$ &$(1+2^{119})^{47657859344287051433215338407025}$ &$(1+2^{67})^{74105228687928761744692516074690}\cdot(1+2^{85})^{566868663181345103125787385}\cdot$ &$(1+2^{101})^{185263071719821904361731290186725}\cdot(1+2^{115})^{24701742895976253914897505358230}\cdot$ &$(1+2^{125})^{66694705819135885570223264467221}\cdot(1+2^{147})^{22231568606378628523407754822407}$ &$(1+2^{17})^{62671642336461616797239779725}\cdot(1+2^{43})^{35812367049406638169851302700}\cdot$ &$(1+2^{51})^{17906183524703319084925651350}\cdot(1+2^{59})^{2984363920783886514154275225}\cdot$ &$(1+2^{73})^{26859275287054978627388477025}\cdot(1+2^{101})^{17906183524703319084925651350}\cdot$ &$(1+2^{123})^{865014224607504542831738415625}$ &$(1+2^{15})^{1707444675887681902216221662393643900}\cdot (1+2^{17})^{341488935177536380443244332478728780}\cdot$ &$(1+2^{29})^{3841750520747284279986498740385698775}\cdot (1+2^{45})^{30734004165978274239891989923085590200}$ &$(1+2^{57})^{9362516203257056384802075}\cdot(1+2^{65})^{44348760962796582875378250}\cdot$ &$(1+2^{69})^{4434876096279658287537825}\cdot(1+2^{89})^{38435592834423705158661150}\cdot$ &$(1+2^{101})^{79907677410444293469150}\cdot(1+2^{123})^{5913168128372877716717100}\cdot$ &$(1+2^{127})^{2002242486366485713350}\cdot(1+2^{137})^{48783637059076241162916075}$ &$(1+2^{29})^{93343471924356402246389002034385}\cdot(1+2^{43})^{62228981282904268164259334689590}\cdot$ &$(1+2^{47})^{82971975043872357552345779586120}\cdot(1+2^{87})^{217801434490164938574907671413565}\cdot$ &$(1+2^{157})^{248915925131617072657037338758360}$ &$(1+2^{13})^{168486137937535997136381884224759350}\cdot(1+2^{87})^{5616204597917866571212729474158645}\cdot$ &$(1+2^{89})^{1925555862143268538701507248282964}\cdot(1+2^{119})^{78626864370850131996978212638221030}$ &$(1+2^{61})^{39948352158132627380823842541450}\cdot(1+2^{105})^{9321282170230946388858896593005}\cdot$ &$(1+2^{109})^{13316117386044209126941280847150}\cdot(1+2^{113})^{15535470283718243981431494321675}\cdot$ &$(1+2^{119})^{19974176079066313690411921270725}\cdot(1+2^{123})^{79896704316265254761647685082900}\cdot$ &$(1+2^{127})^{3698921496123391424150355790875}\cdot(1+2^{135})^{5326446954417683650776512338860}\cdot$ &$(1+2^{147})^{4438705795348069708980426949050}\cdot(1+2^{157})^{13316117386044209126941280847150}$ &$(1+2^{5})^{350060123390813635242448130256489390771914057740}\cdot$ &$(1+2^{33})^{17503006169540681762122406512824469538595702887}$ &$(1+2^{15})^{7089726406583596958466287242575266870940}\cdot$ & $(1+2^{25})^{7286663251210919096201461888202357617355}\cdot$ & $(1+2^{55})^{29146653004843676384805847552809430469420}$ &$(1+2^{49})^{49149123355767553400835304429946193135}\cdot$ $(1+2^{55})^{9829824671153510680167060885989238627}\cdot$ &$(1+2^{77})^{9829824671153510680167060885989238627}\cdot$ $(1+2^{99})^{9829824671153510680167060885989238627}\cdot$ &$(1+2^{109})^{9829824671153510680167060885989238627}\cdot$ $(1+2^{159})^{9829824671153510680167060885989238627}\cdot$ &$(1+2^{163})^{9829824671153510680167060885989238627}$ &$(1+2^{21})^{752712011377013221558430642567508556008861}\cdot (1+2^{61})^{752712011377013221558430642567508556008861}\cdot$ &$(1+2^{69})^{752712011377013221558430642567508556008861}\cdot (1+2^{77})^{31613904477834555305454086987835359352372162}$ &$(1+2^{19})^{281198684623467689204913686779425}\cdot$ $(1+2^{43})^{540766701198976325394064782268125}\cdot$ &$(1+2^{61})^{12329480787336660218984677035713250}\cdot$ $(1+2^{97})^{129784008287754318094575547744350}\cdot$ &$(1+2^{105})^{6651799969190141587990774087125}\cdot$ $(1+2^{123})^{11139794044698912303117734514723375}\cdot$ &$(1+2^{141})^{4975053651030582193625395996866750}\cdot$ $(1+2^{163})^{1838606784076519506339820259711625}\cdot$ &$(1+2^{165})^{1114982889070054279163020169625}$ &$(1+2^{19})^{102104448867391604403906908898445293975}\cdot (1+2^{57})^{4288386852430447384964090173734702346950}\cdot$ &$(1+2^{93})^{81683559093913283523125527118756235180}\cdot (1+2^{103})^{1286516055729134215489227052120410704085}\cdot$ &$(1+2^{133})^{1837880079613048879270324360172015291550}\cdot (1+2^{163})^{568867643689753224536052778148480923575}$ &$(1+2^{21})^{1713334865061395551905989490523888483510297545}\cdot$ &$(1+2^{29})^{17731656063809998410201669013040877638868476180}$ & $(1+2^{11})^{16825262628616094460214312446168738419261880}\cdot$ & $(1+2^{21})^{21031578285770118075267890557710923024077350}\cdot$ & $(1+2^{31})^{2103157828577011807526789055771092302407735}\cdot$ & $(1+2^{17})^{5301342361191869603932356617428842175355175}\cdot$ & $(1+2^{75})^{171011043909415148513946987658994908882425}\cdot$ & $(1+2^{177})^{1279634363046313352673327459379375697499525}$ & $(1+2^{21})^{24289606148175875174394125504156277451293856980}\cdot$ &$(1+2^{25})^{56675747679077042073586292843031314053018999620}\cdot$ & $(1+2^{61})^{6072401537043968793598531376039069362823464245}\cdot$ & $(1+2^{73})^{28337873839538521036793146421515657026509499810}\cdot$ & $(1+2^{121})^{3333867510533943651387428990766547885471705860}$ & $(1+2^{9})^{52332852605905914814562661586433935229041537035}\cdot$ & $(1+2^{19})^{52332852605905914814562661586433935229041537035}\cdot$ & $(1+2^{41})^{203629776676676711340710745472505584548799755}\cdot$ & $(1+2^{89})^{104665705211811829629125323172867870458083074070}$ & $(1+2^{7})^{954808575327093401067436576289941581}\cdot (1+2^{11})^{76741427691674298191288254054776183774}\cdot$ & $(1+2^{21})^{862835224381042468504831900414614}\cdot (1+2^{33})^{8687708795283882814108104232616171748}\cdot$ & $(1+2^{39})^{6515781596462912110581078174462128811}\cdot (1+2^{73})^{60813961566987179698756729628313202236}\cdot$ & $(1+2^{89})^{53091553748957061641771748088209938460}\cdot (1+2^{125})^{62503238277181268023722194340210791187}\cdot$ & $(1+2^{157})^{121386597889660918208232678583498177479}$ & $(1+2^{55})^{1025444869877616060103489665965652402208725}\cdot$ & $(1+2^{77})^{1860992541629747664632259023419146952156575}\cdot$ & $(1+2^{179})^{1063424309502712951218433727668083972660900}\cdot$ & $(1+2^{183})^{2126848619005425902436867455336167945321800}\cdot$ & $(1+2^{189})^{138245160235352683658396384596850916445917}$ & $(1+2^{5})^{690644713229389686815238348164730529976649425988651}\cdot$ & $(1+2^{9})^{76738301469932187423915372018303392219627713998739}\cdot$ & $(1+2^{13})^{98663530461341383830748335452104361425235632284093}$ & $(1+2^{31})^{397218618589975176651322156679935564968150}\cdot$ & $(1+2^{129})^{42129247426209488432715986314538620526925}\cdot$ & $(1+2^{163})^{66203103098329196108553692779989260828025}$ & $(1+2^{5})^{47788807121282329843547481918370106279929779662675299482}\cdot$ & $(1+2^{13})^{7964801186880388307257913653061684379988296610445883247}$ & $(1+2^{7})^{9012349943070385113930109307619809450853768536749084363}\cdot$ & $(1+2^{27})^{234321098519830012962182841998115045722197981955476193438}$ & $(1+2^{67})^{201}\cdot (1+2^{107})^{915031412652462933978517192307754031960084170}\cdot$ & $(1+2^{119})^{14182986896113175476667016480770187495381304635}\cdot$ & $(1+2^{145})^{3507620415167774580250982570513057122513655985}\cdot$ & $(1+2^{177})^{83184673877496630361683381118886730178189470}$ & $(1+2^{57})^{718674251279934430341052428990271148449812}\cdot$ & $(1+2^{81})^{51205540403695328161799985565556819327049105}\cdot$ & $(1+2^{111})^{6288399698699426265484208753664872548935855}\cdot$ & $(1+2^{127})^{18865199096098278796452626260994617646807565}\cdot$ & $(1+2^{153})^{2695028442299754113778946608713516806686795}\cdot$ &$(1+2^{175})^{64680682615194098730694718609124403360483080}\cdot$ & $(1+2^{193})^{8085085326899262341336839826140550420060385}$ &$(1+2^{43})^{9467961424350347777980448419013907428323430550}\cdot$ & $(1+2^{67})^{29756450190815378730795695031186566203302210300}\cdot$ &$(1+2^{131})^{676282958882167698427174887072421959165959325}\cdot$ &$(1+2^{145})^{7213684894743122116556532128772500897770232800}\cdot$ & $(1+2^{157})^{772894810151048798202485585225625096189667800}\cdot$ & $(1+2^{187})^{16907073972054192460679372176810548979148983125}$ & $(1+2^{5})^{33192619261066535128289058132930982761836775}\cdot$ &$(1+2^{121})^{46469666965493149179604681386103375866571485}\cdot$ & $(1+2^{127})^{102131136187897031163966332716710716190267}\cdot$ & $(1+2^{179})^{19915571556639921076973434879758589657102065}\cdot$ & $(1+2^{187})^{39831143113279842153946869759517179314204130}\cdot$ & $(1+2^{199})^{3983114311327984215394686975951717931420413}$ & $(1+2^{59})^{270250337042154559392793016292662090505143401152585}\cdot$ &$(1+2^{61})^{360333782722872745857057355056882787340191201536780}\cdot$ & $(1+2^{87})^{180166891361436372928528677528441393670095600768390}\cdot$ &$(1+2^{97})^{25738127337348053275504096789777341952870800109770}$ & $(1+2^{9})^{1154929871973565275224108161553051512738585560465529713690}\cdot$& $(1+2^{13})^{64162770665198070845783786752947306263254753359196095205}\cdot$ & $(1+2^{55})^{2309859743947130550448216323106103025477171120931059427380}$ & $(1+2^{11})^{5497736589318700986099894638477660690688115321335}\cdot$ & $(1+2^{41})^{140967604854325666310253708678914376684310649265}\cdot$ & $(1+2^{85})^{1832578863106233662033298212825886896896038440445}\cdot$ & $(1+2^{119})^{2356172823993728994042811987918997438866335137715}\cdot$ & $(1+2^{161})^{1099547317863740197219978927695532138137623064267}\cdot$ & $(1+2^{203})^{1570781882662485996028541325279331625910890091810}$ & $(1+2^{41})^{66443114885376278757699109185773253174500402025}\cdot$ & $(1+2^{97})^{31006786946508930086926250953360851481433520945}\cdot$ & $(1+2^{119})^{186040721679053580521557505720165108888601125670}\cdot$ & $(1+2^{151})^{186040721679053580521557505720165108888601125670}\cdot$ & $(1+2^{203})^{279061082518580370782336258580247663332901688505}$ & $(1+2^{13})^{1126940817087752014462533930787563178937611179075}\cdot$ & $(1+2^{57})^{125215646343083557162503770087507019881956797675}\cdot$ & $(1+2^{89})^{50086258537233422865001508035002807952782719070}\cdot$ & $(1+2^{185})^{751293878058501342975022620525042119291740786050}\cdot$ & $(1+2^{215})^{9537309672323317621714046370301517484866488275}$ & $(1+2^{19})^{2066997275827785054568851212103878502402165134820}\cdot$ &$(1+2^{63})^{15659070271422614049764024334120291684864887385}\cdot$ & $(1+2^{107})^{8267989103311140218275404848415514009608660539280}\cdot$ & $(1+2^{113})^{861248864928243772737021338376616042667568806175}\cdot$ & $(1+2^{115})^{2239247048813433809116255479779201710935678896055}\cdot$ & $(1+2^{139})^{9802832608939360014892925802660670404346311615}\cdot$ & $(1+2^{189})^{574165909952162515158014225584410695111712537450}$ &$(1+2^{9})^{3989486683832532366484265709728099870565523986917322701595}\cdot$ &$(1+2^{19})^{3989486683832532366484265709728099870565523986917322701595}\cdot$ &$(1+2^{29})^{3989486683832532366484265709728099870565523986917322701595}$ &$(1+2^{17})^{1014848095919801109402176209373936431577327592183}\cdot$ & $(1+2^{91})^{1503478660621927569484705495368794713447892729160}\cdot$ &$(1+2^{103})^{1691413493199668515670293682289894052628879320305}\cdot$ &$(1+2^{113})^{563804497733222838556764560763298017542959773435}\cdot$ &$(1+2^{145})^{2631087656088373246598234616895390748533812276030}$ &$(1+2^{29})^{64638048753257449056214060125}\cdot(1+2^{31})^{788456198653595814230254674000}\cdot$ & $(1+2^{49})^{5797472048923498634045990250}\cdot(1+2^{51})^{84063344709390730193666858625}\cdot$ &$(1+2^{85})^{4479864765077248944490083375}\cdot(1+2^{97})^{1809865365091208573573993683500}\cdot$ &$(1+2^{99})^{365703246128755015876741500}\cdot(1+2^{105})^{5302697068866947730212751750}\cdot$ & $(1+2^{115})^{68066717742077353015043250}\cdot (1+2^{131})^{5270429135384998758223627500}\cdot$ & $(1+2^{203})^{152315402012626464112662834750}\cdot(1+2^{207})^{828774981539291054730665424375}\cdot$ &$(1+2^{217})^{438130774024554946771130154075}\cdot(1+2^{219})^{94077160066622227834291750875}$ &$(1+2^{5})^{59383142438657794704063431302361624145812280978139757521155099815}\cdot$ &$(1+2^{7})^{118766284877315589408126862604723248291624561956279515042310199630}$ &$(1+2^{9})^{147385697387219867509609565535361987370219507869771466051}\cdot$ &$(1+2^{21})^{1768628368646638410115314786424343848442634094437257592612}\cdot$ & $(1+2^{121})^{589542789548879470038438262141447949480878031479085864204}$ & $(1+2^{41})^{459189704451443127647362594916155732882807466830218}\cdot$ &$(1+2^{55})^{1567200356489566988557551518485173149770673948226}\cdot$ &$(1+2^{69})^{2755138226708658765884175569496934397296844800981308}$ & $(1+2^{91})^{1147974261128607819118406487290389332207018667075545}\cdot$ &$(1+2^{131})^{535721321860016982255256360735515021696608711301921}\cdot$ &$(1+2^{157})^{918379408902886255294725189832311465765614933660436}$ & $(1+2^{159})^{1147974261128607819118406487290389332207018667075545}\cdot$ &$(1+2^{209})^{2350800534734350482836327277727759724656010922339}$ & $(1+2^{15})^{17038321523200602421013846948443997824061934891243478185488130}\cdot$ &$(1+2^{23})^{2839720253866767070168974491407332970676989148540579697581355}$ & $(1+2^{49})^{2839720253866767070168974491407332970676989148540579697581355}\cdot$ & $(1+2^{113})^{2839720253866767070168974491407332970676989148540579697581355}$ & $(1+2^{73})^{23191222403616672243206263595493280420250247335633250}\cdot$ &$(1+2^{105})^{26891949382917205047973220552220931551141244250893875}$ & $(1+2^{119})^{8141599354461172170487305304800832487960193213573375}\cdot$ &$(1+2^{141})^{363659699007176721748930895080669461243705878250}$ & $(1+2^{167})^{14062762521342024658114436435565074297385788277990375}\cdot$ &$(1+2^{171})^{23684652667523409950508524523056967237702380257668000}$ & $(1+2^{197})^{2556865912971277210566261170102740781342870595998250}$ & $(1+2^{9})^{312620627852416761979153117722703908013332701360090433}\cdot$ &$(1+2^{169})^{3647240658278195556423453040098212260155548182534388385}$ & $(1+2^{173})^{1042068759508055873263843725742346360044442337866968110}\cdot$ & $(1+2^{175})^{173678126584675978877307287623724393340740389644494685}$ & $(1+2^{199})^{429087136268023006638053298835083795312417433239339810}\cdot$ & $(1+2^{225})^{416827503803222349305537490296938544017776935146787244}$ & $(1+2^{233})^{67961006054873209125902851678848675655072326382628355}$ &$(1+2^{59})^{137903930258717580167839711391793163262983138814261779866}\cdot$&$(1+2^{73})^{25172939650400828125875502873105101230544541212127150293}\cdot$&$(1+2^{87})^{206855895388076370251759567087689744894474708221392669799}\cdot$&$(1+2^{197})^{298791848893888090363652708015551853736463467430900523043}\cdot$&$(1+2^{199})^{91935953505811720111893140927862108841988759209507853244}\cdot$&$(1+2^{237})^{6566853821843694293706652923418722060142054229250560946}$ &$(1+2^{7})^{916414411031455987244373856003197176202834172185057465573737668369009}\cdot$&$(1+2^{9})^{5498486466188735923466243136019183057217005033110344793442426010214054}$ &$(1+2^{23})^{785013025560884300793728032156196770104389659275}\cdot$& $(1+2^{31})^{22335098635205076340574688781013129860710668130}\cdot$&$(1+2^{35})^{847419918806310249392392603750204044715198879050}\cdot$&$(1+2^{61})^{132039847814006480719279777793636444176554243945}\cdot$&$(1+2^{151})^{41135370735706828274307725135899531886322869175}\cdot$&$(1+2^{155})^{725890705644164981068677385382926720473096714225}\cdot$&$(1+2^{163})^{160944093106624814807082316216124023996297461525}\cdot$&$(1+2^{177})^{387579652787382207086443128846992547582920417550}\cdot$& $(1+2^{181})^{91968053203785608461189894980642299426455692300}$ &$(1+2^{69})^{404534281273826986829987345146663806009193260698421162909645}\cdot$&$(1+2^{117})^{652474647215849978758044105075264203240634291449066391789750}\cdot$&$(1+2^{125})^{1096157407322627964313514096526443861444265609634431538206780}\cdot$&$(1+2^{141})^{1057008928489676965588031450221928009249827552147487554699395}\cdot$&$(1+2^{151})^{404534281273826986829987345146663806009193260698421162909645}\cdot$&$(1+2^{165})^{1578988646262356948594466734282139371842334985306740668131195}\cdot$&$(1+2^{167})^{404534281273826986829987345146663806009193260698421162909645}$ & $(1+2)^{134057357388441380704540286280333486890775035828909707815645} \cdot$ & $(1+2^{9})^{130787665744820859223941742712520475015390278857472885673800}\cdot$ & $(1+2^{35})^{16348458218102607402992717839065059376923784857184110709225}\cdot$ & $(1+2^{71})^{85011982734133558495562132763138308760003681257357375687970}\cdot$ & $(1+2^{147})^{81742291090513037014963589195325296884618924285920553546125}\cdot$ & $(1+2^{195})^{15040581560654398810753300411939854626769882068609381852487}$ & $(1+2^{97})^{292527702190729434230102491312771097283901482612310325863937852070}\cdot$ & $(1+2^{119})^{204769391533510603961071743918939768098731037828617228104756496449}\cdot$ $n= 249$& $(1+2^{137})^{585055404381458868460204982625542194567802965224620651727875704140}\cdot$ & $(1+2^{173})^{633810021413247107498555397844337377448453212326672372705198679485}\cdot$ & $(1+2^{199})^{536300787349670629421854567406747011687152718122568930750552728795}$ # Further comments One can go further than our choice of bound, namely $250$, by using our method. We have yet to encounter an exponent for which we cannot apply Algorithm [\[algo1\]](#algo1){reference-type="ref" reference="algo1"}. If there is such an exponent $n$ for which we cannot find $a_j$, $d_{i,j}$ as above, then we can involve cubics in the factorization of $-1\pmod{2^n-1}$. More precisely, we do something similar as above using $2^{a_i}+2^{b_i}+1$ and check if we can find such powers $a_i,b_i$ such that the number above is a quadratic nonresidue modulo $q_i$ and quadratic residue modulo $q_j$ for all $j\neq i$. The rest of Algorithm [\[algo1\]](#algo1){reference-type="ref" reference="algo1"} runs unchanged. While, via Carlitz' result, we know that any permutation can be decomposed as a composition of inverses and affine functions, it would also be interesting to check whether one can modify our method in this paper to other exponents, other than the inverse and surely, the Gold $2^k+1$ exponents, to directly find their decomposition in quadratics, or quadratics and cubics, and we leave that to future work and the interested reader. .5cm **Acknowledgements.** The first and the third-named authors worked on this paper during visits to the Max Planck Institute for Software Systems in Saarbrücken, Germany in Spring of 2022 and 2023. They thank Professor J. Ouaknine for the invitation and the Institute for hospitality and support. During the final stages of the preparation of this paper, the first-named author was a fellow at the Stellenbosch Institute for Advanced Study. He thanks this Institution for hospitality and support. 9999 L. Carlitz, "Permutations in a finite field", *Proc. Amer. Math. Soc.* **4** (1953), 538. R. T. Hall and G. Tenenbaum, *Divisors*, Cambridge Tracts in Mathematics, **90**. Cambridge University Press, Cambridge, 1988. A. Kontorovich and J. Lagarias, "On toric orbits in the affine sieve", *Exp. Math*. **30** (2021), 575--587. F. Luca and P. Stănică, "Asymptotics on a class of $\mathcal S$-unit integers", to appear in *Periodica Math. Hungarica*. F. Luca and P. Stănică, "Prime divisors of Lucas sequences and a conjecture of Skałba", *Int. J. Number Theory* **1** (2005), no. 4, 583--591. P. Moree, "On the divisors of $a^k+b^k$", *Acta Arith.* LXXX.3 (1997), 197--212. L. Murata and C. Pomerance, "On the largest prime factor of a Mersenne number", in *Number Theory*, 209--218, CRM Proc. Lecture Notes, 36, Amer. Math. Soc., Providence, RI, 2004. S. Nicoka, V. Nikov, V. Rijmen, "Decomposition of permutations in a finite field", *Cryptogr. Commun.* **11** (2019), 379--384. G. Petrides, "On decompositions of permutation polynomials into quadratic and cubic power permutations", *Cryptogr. Commun.* **15** (2023), 199--207. A. Rotkiewicz, "Applications of Jacobi's symbol to Lehmer's numbers\", *Acta Arith.* **42** (1983), 163--187.

arxiv_math

--- abstract: | It is well-known that the SRB measure of a $C^{1+\alpha}$ Anosov diffeomorphism has exponential decay of correlations with respect to Hölder-continuous observables. We propose a new approach to this phenomenon, based on optimal transport. More precisely, we define a space of measures having absolutely continuous disintegrations with respect to some foliation close to the unstable foliation of the map, endowed with a variant of the Wasserstein metric where mass is only allowed to be transported along the diffeomorphism's stable foliation. We show that this metric is indeed finite on that space, and use that the construction makes the diffeomorphism act as a contraction to deduce two corollaries. First, the SRB measure has exponential decay of correlation with respect to pairs of observable that are only asked to be Hölder-continuous *in the stable, respectively unstable direction*, but can be discontinuous overall. Then, we prove quantitative statistical stability: the map sending a $C^{1+\alpha}$ Anosov diffeomorphism to its SRB measure is locally Hölder-continuous (using the $C^1$ metric for diffeomorphisms and the usual Wasserstein metric for measures). author: - "Houssam Boukhecham [^1]" - Benoı̂t R. Kloeckner bibliography: - SRB.bib title: Mixing speed and stability of SRB measures through optimal transportation --- ***Disclaimer.** This article is an advanced draft: some proofs are somewhat sketchy and the results should therefore be taken with a pinch of salt. Both authors have projects outside academia for the near future, and time did not permit to provide a more complete write-up. We still hope the idea is fruitfull and the proofs detailed enough that the method can be used by others in many cases beyond Anosov diffeomophism.* # Introduction Let $T:M\to M$ be a $C^{1+\alpha}$, topologically mixing Anosov diffeomorphism. The chaotic properties of $T$ make it practically impossible to predict the future of an orbit at any given time, and at the same time enable one to predict accurately how such an orbit distributes over a long enough period of time, for Lebesgue-almost all starting point. This distribution is indeed well-known to be described by the *SRB measure* of $T$, which is also a *physical measure*. The "accuracy" of the distribution of orbits is quantified by the decay of correlations, which is exponential for Hölder observables. An important contemporary line of research consists in extending this to various classes of systems enjoying some flavor of non-uniformly hyperbolicity, see e.g. [@Liverani1995decay; @Baladi1996stochastic; @Dolgopyat1998flows; @Young1998statistical; @Benedicks2000Henon; @deCastro2004attractor; @Varandas2008correlation; @Alves2012statistical; @DeSimoi2016mostly; @Korepanov2019coupling]. The purpose of the present article is to propose a novel method to study the decay of correlations, yielding some new results in the case of Anosov diffeomorphisms and hopefully simple enough to be extended in other settings. The method is by a coupling argument, akin to the one commonly used in finite-states Markov Chains where one creates two realizations of the chain that are coupled one to another: they run independently until they meet at the same state, then they evolve along the same trajectory. Our idea is to do the same for orbits of $T$, with "meeting at the same state" replaced by "lying not too far away on the same stable leaf". The statement of our main result needs a few definitions which we introduce first. ## Wasserstein metric and variants To state our central result, let us first introduce the *Wasserstein metric* $\operatorname{\mathsf{D}}_1$ associated to the distance function $d(\cdot,\cdot)$ defined on $M$ by any Riemannian metric : to any two probability measures $\mu_1,\mu_2$, it associates the distance $$\operatorname{\mathsf{D}}^1(\mu_1,\mu_2) := \inf_{\gamma\in\Gamma(\mu_1,\mu_2)} \int_{M\times M} d(x,y) \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\gamma(x,y)$$ where $\Gamma(\mu_1,\mu_2)$ is the set of *coupling* (aka *transport plans*) between $\mu_1$ and $\mu_2$, i.e. the set of probability measures $\gamma$ on $M\times M$ having $\mu_1,\mu_2$ as marginals. This last condition can be written $p_{i*}\gamma = \mu_i$ for both $i=1,2$, where $p_i:M\times M\to M$ is the projection to the $i$th factor. "Kantorovich's duality" enables one to rewrite this as $$\operatorname{\mathsf{D}}^1(\mu_1,\mu_2) = \sup_{f\in\operatorname{Lip}(d,1)} \Big\lvert\int f\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_1 - \int f\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_2 \Big\rvert$$ where $\operatorname{Lip}(d,k)$ denotes the space of functions $M\to\mathbb{R}$ that are Lipschitz of constant at most $k$ with respect to the metric $d$. This metric induces the weak-$*$ topology on the set of probability measure (this claim uses the compactness of $M$). Now, the same concept can be used for other metrics than $d$. We will combine two variations: first, using $d(\cdot,\cdot)^\beta$ for some $\beta\in(0,1)$, which is also a metric. The corresponding Wasserstein metric is $$\operatorname{\mathsf{D}}^\beta(\mu_1,\mu_2) := \inf_{\gamma\in\Gamma(\mu_1,\mu_2)} \int_{M\times M} d(x,y)^\beta \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\gamma(x,y),$$ it still metrizes the weak-$*$ topology, and the duality will then be against $\beta$-Hölder functions with Hölder constant at most $1$ (it is quite different from $\operatorname{\mathsf{D}}^1(\cdot,\cdot)^\beta$, since small movements of mass are disproportionately penalized compared to large ones). Second, we will use the extended metric $d_s:M\times M\to[0,+\infty]$ which gives the distance along the stable leafs: $d_s(x,y)=\infty$ when $x,y$ are not in the same stable leaf, otherwise $d_s(x,y)$ is the least length of a smooth curve from $x$ to $y$, constrained to stay on the same stable leaf. Then we write $$\operatorname{\mathsf{D}}_s^1(\mu_1,\mu_2) := \inf_{\gamma\in\Gamma(\mu_1,\mu_2)} \int_{M\times M} d_s(x,y)\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\gamma(x,y)$$ which can take the value $+\infty$ but is otherwise well-defined, and has a dual formulation against bounded Borel-measurable function that are Lipschitz *in the stable direction*, i.e. with respect to $d_s$. Last, we combine both variations by introducing for all $\beta\in(0,1]$ $$\operatorname{\mathsf{D}}_s^\beta(\mu_1,\mu_2) := \inf_{\gamma\in\Gamma(\mu_1,\mu_2)} \int_{M\times M} d_s(x,y)^\beta\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\gamma(x,y).$$ Again, we have a duality and we introduce the corresponding space $\operatorname{Hol}_s^\beta(M,T)$ of bounded, Borel-measurable functions $f:M\to\mathbb{R}$ such that for some $C\ge 0$ and all $x,y\in M$, $$\lvert f(x) - f(y)\rvert \le C\cdot d_s(x,y)^\beta.$$ The least such $C$ is then denoted by $\operatorname{Hol}_s^\beta(f)$, and we use on $\operatorname{Hol}_s^\beta(M,T)$ the norm $\lVert\cdot\rVert_{s,\beta} = \lVert\cdot\rVert_\infty + \operatorname{Hol}_s^\beta(\cdot)$, making it a Banach algebra. Then $$\operatorname{\mathsf{D}}_s^\beta(\mu_1,\mu_2) = \sup_{f} \Big\lvert\int f\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_1 - \int f\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_2 \Big\rvert$$ where the supremum is over all $f\in\operatorname{Hol}_s^\beta(M,T)$ such that $\operatorname{Hol}_s^\beta(f)\le 1$. We define similarly $\operatorname{Hol}_W^\beta$, where the stable foliation is replaced by any foliation $W$, and $\operatorname{Hol}_u^\beta:=\operatorname{Hol}_{W^u}^\beta$. ## Regularly Foliated Measures The notion of SRB measure combines two aspects: the absolute continuity of the local disintegrations along the unstable foliation, and the $T$-invariance (the usual definition also includes existence of a positive Lyapunov exponent, which follows here from the Anosov assumption). We will consider non-invariant measures that satisfy the first condition for a foliation close to the unstable foliation of $T$. To define "close", we fix an open cone field $(C_x \subset T_xM)_{x\in M}$ containing a neighborhood of the unstable distribution $E^u$ of $T$ ; we will denote by $U_x$ the corresponding neighborhood of $E^u_x$ in the Grassmanian $G(k_u,T_xM)$ where $k_u := \dim E^u$; i.e. whenever $E$ is a $k_u$-dimensional subspace of $T_xM$, we write indifferently $E\in U_x$ or $E\subset C_x$, and we say that $E$ is *tangent to the cone field*. A $k_u$-dimensional submanifold $N$ of $M$ is said to be *tangent to the cone field* when $T_xN\in U_x$ for all $x\in N$, and a foliation is said to be tangent to the cone field whenever its leaves are. The cone field will be chosen small enough that the action of $T$ on the sections of $(U_x)_x$ is contracting (with $E^u$ as its unique fixed point). We use two parameters $\beta\in(0,1]$ and $K>0$ to bound the regularity of both the foliation and the densities of the disintegration. Foliations to be considered will be tangent to the cone field, and have their leaves written as $\beta$-Hölder graphs with constant at most $K$ in some fixed finite atlas of $M$. Measures to be considered will have absolutely continuous local disintegrations with respect to such a foliation, with $\beta$-Hölder densities, and the log-densities will be asked to have Hölder constant at most $K$. The set of such probability measures is denoted by $\mathscr{R}_K^\beta$ (the precise definition is given as Section [3.3.1](#s:defiR){reference-type="ref" reference="s:defiR"}). ## Results We are now in a position to state our central result. **Theorem 1**. *Let $T:M\to M$ be a $C^{1+\alpha}$ Anosov diffeomorphism. For some $\beta_0\in(0,1)$ and for each $\beta\in(0,\beta_0)$ there exist $K_0,C>0$ and $n_0\in\mathbb{N}$ with the following properties:* 1. *$\mathscr{R}_{K_0}^\beta$ contains a unique $T$-invariant measure, which is the SRB measure $\mu_0$ of $T$,* 2. *$\operatorname{\mathsf{D}}_s^{\beta}(\mu_1,\mu_2) \le C$ for all $\mu_1,\mu_2\in \mathscr{R}_{K_0}^\beta$,* 3. *$T^{n_0}_*\mathscr{R}_{2K}^\beta\subset \mathscr{R}_{K}^\beta$ for all $K\ge K_0$.* In particular, for all $\mu_1,\mu_2\in \mathscr{R}_{K_0}^\beta$ we can pair $\mu_1$ and $\mu_2$ by a couling $\gamma$ that is supported on the set of $(x,y)$ where $x$ and $y$ are in the same stable leaf. Since the stable direction is contracting exponentially, denoting by $\lambda\in(0,1)$ any constant such that $T$ is $\lambda$-contracting along its stable foliation $W^s$, we get $$\operatorname{\mathsf{D}}_s^{\beta}(T^n_*\mu_1,T^n_*\mu_2) \le C\lambda^{\beta n} \qquad \forall n\in\mathbb{N}; \label{eq:decay}$$ and since $\operatorname{\mathsf{D}}^1$ is bounded above by $\operatorname{diam}(M)^{1-\beta} \operatorname{\mathsf{D}}_s^{\beta}$, we also have exponential convergence in the usual Wasserstein metric. Given any $\mu\in \mathscr{R}_K^\beta$, the sequence $(T^n_*\mu)$ will converge to an invariant measure in $\mathscr{R}^\beta_{K_0}$, which must then be the SRB measure of $T$. The following two corollaries are easily deduced from Theorem [Theorem 1](#t:central){reference-type="ref" reference="t:central"}. **Corollary 2** (Exponential decay of correlations for stable/unstable-Hölder pairs of observables). *Let $\mu_0$ be the SRB measure of $T$. For all $\beta\le \beta_0$, there exist $C_\beta\ge 1$ such that for all $n\in\mathbb{N}$, all $f\in \operatorname{Hol}_s^\beta$ and all $g\in \operatorname{Hol}_u^\beta$, $$\Big\lvert \int f\circ T^n \cdot g \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0 -\int f\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0\int g\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0 \Big\rvert \le C_\beta\, \lVert f\rVert_{s,\beta} \, \lVert g\rVert_{u,\beta} \, \lambda^{\beta n}$$* While exponential decay of correlations is long known for Anosov diffeomorphisms, this version applies to observables with only partial continuity requirement: $f$, $g$ can exhibit discontinuities in the unstable, respectively stable, direction; the duality appearing between $\operatorname{Hol}_u^\beta$ and $\operatorname{Hol}_s^\beta$ seems very natural. One could also point out that the rate of decay is directly determined by the contraction factor $\lambda$ of $T$, but beware that $\beta_0$, while constructive, is not made completely explicit in the proof and can in principle be very small. **Corollary 3** (Hölder statistical stability). *There exist $\varepsilon>0$, $C'$ and $\beta'$ such that for any $C^{1,\alpha}$ Anosov map $T_1:M\to M$ which is $\varepsilon$-close from $T$ in the $C^1$ topology, the SRB measures $\mu_0,\mu_1$ of $T$ and $T_1$ are close in the Wasserstein distances: $$\operatorname{\mathsf{D}}_1(\mu_0,\mu_1) \le C' \lVert T-T_1\rVert_{C^0}^{\beta'}$$* Here the main emphasis is on the explicit, Hölder modulus of continuity of the map sending an Anosov diffeomorphism to its SRB measure. Also note that the distance between measure is given in the quite strong Wasserstein metric, and that while we need $T_1$ to be $C^1$-close to $T$, the control we get uses only the uniform distance $$\lVert T-T_1\rVert_{C^0} := \max_{x\in M} d\big(T(x), T_1(x) \big).$$ # Warm-up: expanding maps {#s:expanding} Our goal here is to illustrate one strand of our method in the simplest possible setting by providing a short and elementary proof of the exponential decay of correlation for expanding maps. The setting is as follows: $T:\mathbb{T}^1\to\mathbb{T}^1$ is a uniformly expanding circle map of class $C^{1+\alpha}$ for some $\alpha\in(0,1]$. Identify $\mathbb{T}^1=\mathbb{R}/\mathbb{Z}$ with $[0,1)$ and denote the usual circle distance by $d$. Since $T$ is expanding, $\lambda:=1/\inf\lvert T'\rvert \in (0,1)$. Since $T$ is $C^{1+\alpha}$, there exist $H>0$ such that $\lvert T'(x)\rvert \le \lvert T'(y)\rvert e^{Hd(x,y)^\alpha}$ for all $x,y\in\mathbb{T}^1$. Let $m$ be the Lebesgue measure; searching for an absolutely continuous invariant probability measure (ACIP), one as usual defines the transfer operator $\mathcal{L}$ by $T_*(\rho m) = \mathcal{L}(\rho) m$, so that a non-negative eigenfunction of $\mathcal{L}$ for the eigenvalue $1$ is the density of an ACIP. By the change of variable formula, the transfer operator can be expressed as $$\mathcal{L}\rho(x) := \sum_{z\in T^{-1}(x)} \frac{\rho(z)}{\lvert T'(z) \rvert}$$ and given any $x,y\in \mathbb{T}^1$, we can order $T^{-1}(x)=:\{x_i:1\le i\le k\}$ and $T^{-1}(y)=:\{y_i:1\le i\le k\}$ (where $k=\lvert\deg T\rvert$) in such a way that $d(x_i,y_i)\le \lambda d(x,y)$ for all $i$. Such an ordering will always be assumed whenever we consider a pair of points and their inverse images. It will be convenient to have $\mathcal{L}$ act on $\alpha$-Hölder function, i.e. we view it as a linear, bounded operator on the Banach algebra $\operatorname{Hol}_\alpha(\mathbb{T}^1)$ where the norm is $$\lVert f\rVert_\alpha := \lVert f\rVert_\infty + \operatorname{Hol}_\alpha(f); \qquad \operatorname{Hol}_\alpha(f) := \sup \Big\{\frac{\lvert f(x)-f(y)\rvert}{d(x,y)^\alpha} \colon x\neq y\in \mathbb{T}^1 \Big\}.$$ **Theorem 4** (Folklore[^2]). *There is an $\alpha$-Hölder positive density $\rho_0:\mathbb{T}^1\to(0,+\infty)$ such that $\rho_0 m$ is $T$-invariant, and there exist $C\ge 1$ and $\theta\in(0,1)$ such that for all $\alpha$-Hölder positive density $\rho$ and all $n\in\mathbb{N}$, $$\lVert \mathcal{L}^n \rho -\rho_0 \rVert_\alpha \le C \lVert \rho-\rho_0\rVert_\alpha \theta^n.$$* After the proof, we will briefly recall why this statements contains the decay of correlation, and its relation to a very strong convergence of probability measure, in the total variation norm. This convergence will inspire the metric to be used on measures in the Anosov case. Our proof starts in a usual way, showing that the "stretching" effect stemming from the expansion hypothesis regularizes densities above some determined level. Then we phrase a simple coupling argument in term of densities: regular enough densities stay far from zero, so that any two of them "share" a definite amount of mass. *Proof.* For each $K>0$, consider the following set of Hölder densities: $$\operatorname{\mathcal{H}}_K^\alpha := \Big\{\rho : \mathbb{T}^1\to(0,+\infty) \,\Big|\, \int\rho \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}m= 1, \frac{\rho(x)}{\rho(y)} \le e^{K d(x,y)^\alpha}\ (\forall x,y\in\mathbb{T}^1)\Big\}.$$ For all $\rho\in\operatorname{\mathcal{H}}^\alpha_K$ and all $x,y\in\mathbb{T}^1$ with (suitably ordered) inverse images $\{x_i\}_i$ and $\{y_i\}_i$: $$\begin{aligned} \frac{\rho}{\lvert T'\rvert}(x) &\le \frac{\rho}{\lvert T'\rvert}(y) \cdot e^{(K+H)d(x,y)} \\ \frac{\rho}{\lvert T'\rvert} (x_i) &\le \frac{\rho}{\lvert T'\rvert}(y_i) \cdot e^{(K+H)d(x_i,y_i)^\alpha} \\ \mathcal{L}\rho(x) &\le \mathcal{L}\rho(y) \cdot e^{(K+H)\lambda^\alpha d(x,y)^\alpha}\end{aligned}$$ so that for all $K>0$, $\mathcal{L}(\operatorname{\mathcal{H}}^\alpha_K) \subset \operatorname{\mathcal{H}}^\alpha_{h(K)}$ where $h(K)=(K+H)\lambda^\alpha$. For all $K$ and $n\in\mathbb{N}$: $$\begin{aligned} h^n(K) &= K\lambda^{n\alpha} + \lambda^\alpha H + \lambda^{2\alpha} H + \dots +\lambda^{n\alpha} H \\ &\le K\lambda^{n\alpha} + \frac{\lambda^\alpha}{1-\lambda^\alpha}H.\end{aligned}$$ Set $K_0=2\frac{\lambda^\alpha}{1-\lambda^\alpha} H$ and let $n_0$ be the smallest integer such that $2\lambda^{n_0\alpha} \le \frac{\lambda^\alpha}{1-\lambda^\alpha}$, to get: $$\mathcal{L}^{n_0}(\operatorname{\mathcal{H}}^\alpha_{2K_0}) \subset \operatorname{\mathcal{H}}^\alpha_{K_0} \qquad\text{and}\qquad \mathcal{L}(\operatorname{\mathcal{H}}^\alpha_{K_0}) \subset \operatorname{\mathcal{H}}^\alpha_{K_0}$$ Every $\rho\in \operatorname{\mathcal{H}}^\alpha_K$, being a density, reaches a value at least $1$ at some point $x_0$, and each $x\in\mathbb{T}^1$ is at distance less than $1$ from $x_0$ so that $\rho(x)\ge e^{-Kd(x,x_0)^\alpha} \ge e^{-K}$. For all $\tau\in(0,e^{-K_0})$ and all $\rho\in \operatorname{\mathcal{H}}^\alpha_{K_0}$, we can decompose $$\rho = \tau \boldsymbol{1}+ (1-\tau)\tilde \rho$$ where $\boldsymbol{1}$ is the constant function with value $1$ and $\tilde \rho = (\rho-\tau\boldsymbol{1})/(1-\tau)$ is positive of integral $1$ with respect to $m$. From now on we fix a value of $\tau$ small enough to further ensure that $\tilde \rho\in\operatorname{\mathcal{H}}^\alpha_{2K_0}$ whenever $\rho\in\operatorname{\mathcal{H}}^\alpha_{K_0}$. Then for all $\rho_1,\rho_2\in \operatorname{\mathcal{H}}^\alpha_{K_0}$ and all $n\ge n_0$: $$\begin{aligned} \mathcal{L}^{n_0}(\rho_1-\rho_2) &= \mathcal{L}^{n_0}\big(\tau\boldsymbol{1}+(1-\tau)\tilde\rho_1-\tau\boldsymbol{1}- (1-\tau)\tilde\rho_2\big) \nonumber\\ &= (1-\tau) \mathcal{L}^{n_0} (\tilde \rho_1 -\tilde\rho_2) \nonumber\\ \mathcal{L}^{n}(\rho_1-\rho_2) &= (1-\tau)\mathcal{L}^{n-n_0} \big(\mathcal{L}^{n_0} \tilde \rho_1 -\mathcal{L}^{n_0} \tilde\rho_2\big) \label{e:contraction}\end{aligned}$$ where $\tilde\rho_i\in \operatorname{\mathcal{H}}^\alpha_{2K_0}$ and $\mathcal{L}^{n_0} \tilde \rho_i\in \operatorname{\mathcal{H}}^\alpha_{K_0}$. For each $n\in\mathbb{N}$, define $$\gamma_n = \sup_{\rho_1,\rho_2\in\operatorname{\mathcal{H}}^\alpha_{K_0}} \lVert \mathcal{L}^n(\rho_1-\rho_2) \rVert_\alpha \in [0,+\infty);$$ the above computation shows $\gamma_n \le (1-\tau) \gamma_{n-n_0}$ so that $\gamma_n$ goes to $0$, exponentially fast. The sequence of sets of functions $\mathcal{L}^n(\operatorname{\mathcal{H}}^\alpha_{K_0})$ thus converges to a point $\{\rho_0\}\subset \operatorname{\mathcal{H}}^\alpha_{K_0}$ in the uniform norm (using compactness provided by the Azelà-Ascoli Theorem). Then $\rho_0$ must be a fixed point of $\mathcal{L}$ and $\rho_0m$ is an invariant measure of $T$. Being a fixed point, $\rho_0$ must belong to $\operatorname{\mathcal{H}}^\alpha_{K_0/2}$, and $(\mathcal{L}^n\rho)_n$ converges to $\rho_0$ exponentially fast, uniformly over all $\rho\in \operatorname{\mathcal{H}}^\alpha_{K_0}$. The statement follows. ◻ **Remark 5** (Decay of correlations). Denote by $\mu_0=\rho_0 m$ the ACIP and let $f,g:\mathbb{T}^1\to\mathbb{R}$ be two "observables", with $f\in L^1(m)$ and $g$ $\alpha$-Hölder. For $a$ small enough and $b=1-a\lVert g\rVert_\alpha^{-1}\int g\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0$, the function $\rho=(a\lVert g\rVert_\alpha^{-1} g+b)\rho_0$ is a $\alpha$-Hölder density with a $g$-uniform bound on $\lVert \rho-\rho_0\rVert_\alpha$, so that: $$\begin{aligned} \int f\circ T^n \cdot g \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0 &= \frac{\lVert g\rVert_\alpha}{a}\int f\circ T^n \cdot \big(\frac{\rho}{\rho_0}-b\big) \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0 \\ &= \frac{\lVert g\rVert_\alpha}{a}\Big(\int f \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}T_*^n(\rho m) -b\int f\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}T_*^n\mu_0\Big) \\ &= \frac{\lVert g\rVert_\alpha}{a}\int f \big(\mathcal{L}^n\rho-\rho_0\big)\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}m +\int g\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0 \int f\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0 \\ &=\int g\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0 \int f\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0 + O(\lVert g\rVert_\alpha \lVert f\rVert_{L^1(m)} \theta^n)\end{aligned}$$ i.e. we have exponential decay of correlations. **Remark 6**. As seen in Remark [Remark 5](#r:decay){reference-type="ref" reference="r:decay"}, Theorem [Theorem 4](#t:expanding){reference-type="ref" reference="t:expanding"} implies that the measures $T_*^n(\rho m)$ converge to $\mu_0:=\rho_0m$ in duality with $L^1(m)$ test functions; the convergence thus also holds against Borel bounded test functions, i.e. in the *total variation* distance. This can be rephrased as the existence of positive measures $\nu_n$ of total mass increasing to $1$ such that $\nu_n\le T_*^n(\rho m)$ and $\nu_n\le \mu_0$ for all $n$. Such a strong convergence cannot be expected in the Anosov case, where the measures we will have to work with might be singular with respect to the limit measure. Seeing an expanding map as an Anosov endomorphism with stable dimension $0$ gives a good guess on the natural replacement for the total variation distance: we shall use an optimal transportation distance where mass is only allowed to move along the stable direction. This is the second strand of proof we shall weave with the first one. # Anosov diffeomorphisms ## Coupling measures and deduction of Theorem [Theorem 1](#t:central){reference-type="ref" reference="t:central"} {#coupling-measures-and-deduction-of-theorem-tcentral} For each $\beta\in(0,\alpha_0]$, we shall build a nested family of sets of probability measures $(\mathscr{R}_K^\beta)_K>0$, where $K$ is a regularity bound (the lesser, the more regular), where elements of $\mathscr{R}_K^\beta$ have absolutely continuous local disintegration with respect to some foliation close to the unstable foliation of $T$, and $K$ bounds the Hölder regularity both of the foliation and of the densities on the leafs. For all $L>0$ we define $$\Delta_s(L) := \big\{(x,y)\in M\times M \mid d_s(x,y)\le L \big\}$$ the set of pairs of points at distance at most $L$ along the stable foliation. The slightly technical part of the article (detailed in Section [3.3](#s:technical){reference-type="ref" reference="s:technical"}) leads to the following properties: **Proposition 7**. *There exist $L_0$ and $\tau\in(0,1)$ (depending only on $T$), $K_0>0$ and $n_0\in\mathbb{N}$ (both further depending on $\beta$) such that:* 1. *The SRB measure of $T$ lies in $\mathscr{R}_{K}^\beta$ for some $K>0$,* 2. *$\mathscr{R}_K^\beta\subset\mathscr{R}_{K'}^\beta$ for all $K'>K$,* 3. *[\[enumi:tech2.5\]]{#enumi:tech2.5 label="enumi:tech2.5"} for all $\mu\in \mathscr{R}_K^\beta$ associated with a foliation $W$ and all positive $\rho\in\operatorname{Hol}_W^\beta$ such that $\int \rho\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu=1$, the probability measure $\rho\mu$ lies in $\mathscr{R}_{K'}^\beta$ where $K'=K + \operatorname{Hol}_W^\beta(\log \rho)$,* 4. *$T_*(\mathscr{R}_{K_0}^\beta)\subset \mathscr{R}_{K_0}^\beta$ and $T^{n_0}_*(\mathscr{R}_{2K}^\beta)\subset \mathscr{R}_{K}^\beta$ for all $K\ge K_0$,* 5. *[\[enumi:tech4\]]{#enumi:tech4 label="enumi:tech4"} for all $\mu_1,\mu_2\in \mathscr{R}_{K_0}^\beta$ we can find a probability measure $\eta$ concentrated on $\Delta_s(L_0)$ and two probability measures $\mu'_1,\mu'_2\in \mathscr{R}_{2K_0}$ such that $$\mu_i=\tau p_{i*}\eta + (1-\tau) \mu'_i \qquad (i\in\{1,2\}).$$* *Moreover, for any $\beta\in(0,\alpha_0)$ we can make the map $T\mapsto K_0$ upper semi-continuous in the $C^1$ topology, in particular there exist $\varepsilon>0$ such that to any map $T_1$ that is $\varepsilon$-close to $T$ in the $C^1$ topology we can apply the above with the same $\beta$ and a constant $K_1\le 2K_0$ in the role of $K_0$ (and further constants $L_1, n_1$).* While item [\[enumi:tech2.5\]](#enumi:tech2.5){reference-type="ref" reference="enumi:tech2.5"} might seem difficult to apply, since $W$ might not be known and $\operatorname{Hol}_W^\beta$ is thus quite abstract a space, it can be applied e.g. to a $\beta$-Hölder function $\rho$, and in the case of the SRB measure when $W=W_u$. *Proof of Theorem [Theorem 1](#t:central){reference-type="ref" reference="t:central"}..* We first prove that $\operatorname{\mathsf{D}}_s^\beta$ is uniformly bounded over all $\mu_1,\mu_2\in\mathscr{R}_{K_0}^\beta$, and that the SRB measure is the unique $T$-invariant measure in $\mathscr{R}_K^\beta$. For now $\beta$ is fixed anywhere in $(0,\alpha_0]$, and we will see along the proof how we need to further restrict it. We will apply item [\[enumi:tech4\]](#enumi:tech4){reference-type="ref" reference="enumi:tech4"} of Proposition [Proposition 7](#p:technical){reference-type="ref" reference="p:technical"} repeatedly, defining sequences $(\mu_1^k,\mu_2^k)_{k\ge0}$ and $(\eta^k)_{k\ge 0}$ as follows. First, $\mu_i^0=\mu_i\in \mathscr{R}_{K_0}^\beta$ are the measures we start with. Given $\mu_1^k,\mu_2^k$, we apply to them Proposition [Proposition 7](#p:technical){reference-type="ref" reference="p:technical"} to get a "partial coupling" $\eta^k$ and "residual measures" $(\mu_i^k)'\in \mathscr{R}_{2K_0}$. Then we set $$\mu_i^{k+1} = T^{n_0}_*(\mu_i^k)'\in \mathscr{R}_{K_0}.$$ By induction, for both $i$ and all $k$: $$T^{kn_0}_*\mu_i = \sum_{j=0}^k (1-\tau)^j \tau \big[T^{(k-j)n_0}\circ p_{i}\big]_* \eta_j + (1-\tau)^{k+1}(\mu_i^k)'$$ Applying $T^{-kn_0}_*$ to this equality and letting $k\to\infty$, it follows that $$\gamma := \tau \sum_{j\ge0}(1-\tau)^j (T^{-jn_0},T^{-jn_0})_*\eta_j$$ is a coupling of $(\mu_1,\mu_2)$. Since $\eta_j$ is concentrated on $\Delta_s(L_0)$, letting $\lambda_0 = \inf\{\lVert D_xT u\rVert\colon x\in M, u\in E^s_x, \lVert u \rVert=1\}$ be the maximal contraction of $T$ on the stable direction, $$\begin{aligned} \int d_s(x,y)^\beta \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\gamma(x,y) &= \tau \sum_{j\ge 0} (1-\tau)^j \int d_s(T^{-jn_0} x, T^{-jn_0} y)^\beta \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\eta_j \\ &\le \tau \sum_{j\ge 0} (1-\tau)^j \int \lambda_0^{-j\beta n_0} d_s( x, y)^\beta \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\eta_j \\ &\le \tau L_0^\beta \sum_{j\ge 0} \big(\frac{1-\tau}{\lambda_0^{\beta n_0}}\big)^j\end{aligned}$$ Set $\beta_0=\min\big\{\alpha_0,\frac{\log(1-\tau)}{n_0\log\lambda_0}\big\}$; then whenever $\beta\in(0,\beta_0)$, the above series is convergent and provides the desired uniform bound. Last, by Proposition [Proposition 7](#p:technical){reference-type="ref" reference="p:technical"} the SRB measure $\mu_0$ lies in some $\mathscr{R}_{K}^\beta$. It must be that $\mu_0\in \mathscr{R}_{K_0}^\beta$: if $K\le K_0$ this is direct, otherwise it suffices to apply $T$ sufficiently many times and use invariance of $\mu_0$. Uniqueness of the $T$-invariant measure in this set then follows from [\[e:contraction\]](#e:contraction){reference-type="eqref" reference="e:contraction"}, which only depends on the part of Theorem [Theorem 1](#t:central){reference-type="ref" reference="t:central"} we already proved. ◻ ## Proof of Corollaries [Corollary 2](#c:decay){reference-type="ref" reference="c:decay"} and [Corollary 3](#c:stability){reference-type="ref" reference="c:stability"} {#proof-of-corollaries-cdecay-and-cstability} Let $\mu_0$ be the SRB measure of $T$ and consider any fixed $\beta\in(0,\beta_0)$. By Proposition [Proposition 7](#p:technical){reference-type="ref" reference="p:technical"}, $\mu_0\in\mathscr{R}_{K_0}^\beta$. *Proof of Corollary [Corollary 2](#c:decay){reference-type="ref" reference="c:decay"}.* Let $f\in\operatorname{Hol}_s^\beta$ and $g\in\operatorname{Hol}_u^\beta$. Much like in Remark [Remark 5](#r:decay){reference-type="ref" reference="r:decay"}, we take $a$ small enough (but independent of $g$) and $b=1-a\lVert g\rVert_{u,\beta}^{-1} \int g \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0$ ensuring that $\rho:= a\lVert g\rVert_{u,\beta}^{-1} g+b$ is positive, that $\int \rho\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0=1$, and that $\rho\mu_0\in \mathscr{R}_{2K_0}^\beta$ (item [\[enumi:tech2.5\]](#enumi:tech2.5){reference-type="ref" reference="enumi:tech2.5"} of Proposition [Proposition 7](#p:technical){reference-type="ref" reference="p:technical"}). Then $$\begin{aligned} \int f\circ T^n \cdot g \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0 &= \frac{\lVert g\rVert_\alpha}{a}\int f\circ T^n \cdot (\rho-b) \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0 \\ &= \frac{\lVert g\rVert_\alpha}{a}\Big(\int f \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}T_*^n(\rho \mu_0) -b\int f\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}T_*^n\mu_0\Big) \\ &= \frac{\lVert g\rVert_\alpha}{a}\Big(\int f \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}T_*^n(\rho \mu_0)- \int f\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0\Big) +\int g\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0 \int f\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0 \\ &=\int g\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0 \int f\relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mu_0 + O\Big(\lVert g\rVert_{u,\beta} \lVert f\rVert_{s,\beta} \ \operatorname{\mathsf{D}}_s^\beta\big(T_*^n(\rho \mu_0),\mu_0\big)\Big)\end{aligned}$$ Using Theorem [Theorem 1](#t:central){reference-type="ref" reference="t:central"} and its consequence [\[eq:decay\]](#eq:decay){reference-type="eqref" reference="eq:decay"}, since $T_*^{n_0}(\rho \mu_0)\in\mathscr{R}_{K_0}^\beta$, for all $n\in\mathbb{N}$: $$\begin{aligned} \operatorname{\mathsf{D}}_s^\beta(T_*^{n}(\rho \mu_0),\mu_0) &= \operatorname{\mathsf{D}}_s^\beta(T_*^{n-n_0}(T_*^{n_0}\rho \mu_0)), T_*^{n-n_0}\mu_0)\le C\lambda^{\beta(n-n_0)}.\end{aligned}$$ ◻ *Proof of Corollary [Corollary 3](#c:stability){reference-type="ref" reference="c:stability"}.* We shall denote with an index $1$ instead of $0$ the quantities related to $T_1$ instead of $T$ (e.g. $\mu_1$, $\alpha_1$, $K_1$, etc.) with implied dependency on $T_1$, $\beta$, etc. Fix any $\beta<\beta_0$; taking $\varepsilon$ small enough ensures that the unstable distribution $E^u_1$ of $T_1$ is tangent to the cone field fixed around $E^u$ to define the $(\mathscr{R}^\beta_K)_K$, and that $\alpha_1>\beta$. By Proposition [Proposition 7](#p:technical){reference-type="ref" reference="p:technical"} we can further assume that $K_1\le 2K_0$. Observe that whenever Proposition [Proposition 7](#p:technical){reference-type="ref" reference="p:technical"} holds for some value of $K_0$, it also hold for all larger values. Up to enlarging $K_0$, we can thus assume that $K_1\le K_0$ and that the unstable foliation of $T_1$ is $\beta$-Hölder of constant $K_0$. This enlargement really depends on $\varepsilon$ only, uniformly on all $T_1$. We thus have $\mu_1\in \mathscr{R}_{K_0}^\beta$. Normalize the Riemannian metric to ensure the diameter of $M$ is $\le 1$. Let $A=\lVert T\rVert_{C^1}+\varepsilon$, so that both $T$ and $T_1$ are $A$-Lipschitz. Set $D= \operatorname{\mathsf{D}}_1(\mu_0,\mu_1)$, and fix $n\in\mathbb{N}$ large enough to ensure that $\operatorname{\mathsf{D}}_\beta^s(T^n_*\mu_1,T^n_*\mu_0)\le \frac12 \operatorname{\mathsf{D}}_1(\mu_0,\mu_1)$; we can take $n\simeq \log 1/D$, so that $A^n \simeq D^{-p}$ for some $p>0$ independent of $T_1$. Then $$\begin{aligned} D = \operatorname{\mathsf{D}}_1(\mu_1,\mu_0) &\le \operatorname{\mathsf{D}}_1(\mu_1,T^n_*\mu_1) + \operatorname{\mathsf{D}}_1(T^n_*\mu_1,\mu_0) \\ &= \operatorname{\mathsf{D}}_1(T_{1*}^n \mu_1,T^n_*\mu_1) + \operatorname{\mathsf{D}}_1(T^n_*\mu_1,T^n_*\mu_0) \\ &\le \lVert T_1^n-T^n\rVert_{C^0} + \operatorname{\mathsf{D}}_\beta^s(T^n_*\mu_1,T^n_*\mu_0)\end{aligned}$$ We bound $\lVert T_1^n-T^n\rVert_{C^0}$ by induction: for all $x\in M$, $$\begin{aligned} d(T_1^n x, T^nx) &\le d\big(T_1 (T_1^{n-1}x), T_1(T^{n-1} x)\big) + d\big(T_1(T^{n-1}x), T(T^{n-1} x)\big) \\ &\le A\, d(T_1^{n-1} x, T^{n-1}x) + \lVert T_1-T\rVert_{C_0} \\ & \le (1+A+\dots +A^{n-1}) \lVert T_1-T\rVert_{C_0} \\ &\lesssim A^n\lVert T_1-T\rVert_{C_0}\end{aligned}$$ Plugging this into the previous computation, we get $$\begin{aligned} D &\lesssim A^n \lVert T_1-T\rVert_{C^0} +\frac{D}{2} \\ \frac{D}{2} &\lesssim D^{-p} \lVert T_1-T\rVert_{C^0} \\ D &\lesssim \lVert T_1-T\rVert_{C^0}^{\frac{1}{1+p}}.\end{aligned}$$ ◻ ## Technical part of the proof: $(\mathscr{R}^\beta_K)_{K}$ and its properties {#s:technical} We start be recalling classical facts on Anosov diffeomorphisms and setting up some notations. Recall that we assume $T:M\to M$ to be a $C^{1+\alpha}$, topologically mixing Anosov diffeomorphism. We denote by $E^s,E^u$ the *stable* and *unstable* distributions, which are $T$-invariant, of constant dimensions $k_s$, $k_u$, and such that $E_x^s\oplus E_x^u = T_xM$ for all $x\in M$. By choosing an adapted riemannian metric on $M$, we can assume that there exist $\lambda\in(0,1)$ such that for all $x\in M$, $u\in E_x^s$, $v\in E_x^u$ and $n\in\mathbb{N}$: $$\lVert D_x(T^n)(u) \rVert \le \lambda^n \lVert u \rVert, \quad \lVert D_x(T^{-n})(v) \rVert \le \lambda^n \lVert v \rVert.$$ The stable and unstable distribution can be integrated into the *stable* and *unstable foliations* $(W^s_x)_x$, $(W^u_x)_x$. The Riemannian metric on $M$ induces a canonical Riemannian metric on each Grassmanian $G(k,T_xM)$ (the space of all $k$-dimensional subspaces of $T_xM$), we denote the induced distance function by $d_{G(k,T_xM)}$. We can in particular define for all continuous $k$-dimensional distributions $E,F$: $$d(E,F) := \max_{x\in M} d_{G(k,T_xM)} (E_x,F_x);$$ the Anosov property in particular implies that $T_*$ acts on $G(k_u,T_xM)$ with $E^u$ as an exponentially attracting fixed point: for some $\eta\in (0,1)$ and some $\varepsilon>0$, for all $x\in M$ and all $E_x\in G(k_u,T_xM)$ such that $d_{G(k_u,T_xM)}(E_x,E^u_x)\le \varepsilon$, $$d_{G(k_u,T_{T(x)}M)}(D_xT(E_x),E^u_{T(x)})\le \eta\cdot d_{G(k_u,T_xM)}(E_x,E^u_x).$$ We fix a field of open cones $(C_x)_{x\in M}$ in the tangent bundle $TM$ containing $E_u$, strongly preserved by $T$ (meaning $\overline{T(C_x)}\subset C_{T(x)}$ for all $x\in M$) and in such a way that the above contraction holds on each $U_x := \{E\in G(k_u,T_xM) \mid E\subset C_x\}$. To compare supspaces of $T_xM$ at different nearby base points $x$, we simply fix a finite smooth atlas of $M$; we say that a $k$-dimension distribution $E$ is $\beta$-Hölder if for some constant $K$, in all charts in the fixed atlas, $E$ is $\beta$-Hölder with that constant $K$. The exponent $\beta$ does not depend on the choice of atlas, but the constant does. There exist some $\alpha_0\in(0,\alpha)$ such that: - $E^s,E^u$ are $\alpha_0$-Hölder continuous, and in particular the leafs of $W^s$, $W^u$ are $C^{1+\alpha_0}$ submanifolds; - the holonomies $\pi_s^{N_1\to N_2}$ and $\pi_u^{N_1\to N_2}$ of $W^s$, $W^u$ between $C^{1,\alpha_0}$ transversals $N_1$, $N_2$ to $E^s$ (or $E^u$) are $\alpha_0$-Hölder and absolutely continuous with $\alpha_0$-Hölder jacobian; moreover the $\alpha_0$-Hölder constants can be chosen uniform over all $(N_1,N_2)$ that are bounded in diameter, are written as graphs of bounded $C^{1,\alpha_0}$ norm, and are at bounded distance one from the other along $W^s$ (or $W^u$). While $\alpha_0$ depends on $T$, it does so only through a few parameters ($\alpha$, the $\alpha$-Hölder constants of $E^s$ and $E^u$, $\lambda$, a uniform bound on $DT$, etc.) and it can be chosen such that every $C^{1+\alpha}$ maps $T_1$ that is sufficiently close to $T$ in the $C^1$ topology shares the same $\alpha_0$. From now on we fix any $\beta\in(0,\alpha_0)$. ### Définition of $(\mathscr{R}^\beta_K)_{K}$ {#s:defiR} While elements of $(\mathscr{R}^\beta_K)_{K}$ are probability measures, their crucial defining property is a relation with some foliation, which we thus have to define beforehand. We will need to define a regularity constant to control certain submanifolds; to this end we will again use a specified atlas, but with some additionnal features. We consider smooth charts $\varphi:U_\varphi\subset M \mapsto V_\varphi\subset \mathbb{R}^k$ (where $k$ is the dimension of $M$) whose range $V_\varphi$ are of the form $B^{k_u}(0,1)\times B^{k_s}(0,1)\subset \mathbb{R}^k$, where $B^\bullet(p,r)$ denotes the ball of center $p$ and radius $r$ in a factor $\mathbb{R}^\bullet$ of $\mathbb{R}^k$, with respect to the canonical Euclidean metric. A typical point of $V$ will be denoted by $(p,q)$, with $p\in B^{k_u}(0,1), q\in B^{k_s}(0,1)$. We denote by $x_\varphi := \varphi^{-1}(0,0)$ the "center" of the chart, and we request that the "slices" $\varphi^{-1}(B^{k_u}(0,1)\times \{q\})$ are $C^1$-close to $W^u_{\varphi^{-1}(0,q)}$, in particular their tangent space shall be contained in the cone field $C$ at all point; and similarly the $\varphi^{-1}(\{p\}\times B^{k_s}(0,1))$ shall be $C^1$-close to $W^s_{\varphi^{-1}(p,0)}$, in particular their tangent space shall avoid the closure $\bar C=(\bar C_x)_x)$ of the cone field at all point. We can further assume (taking the domains $U_\varphi$ somewhat tall in the stable direction and thin in the unstable direction) that the image $\varphi_*C$ of the cone field is very narrow; in particular that every $k_u$-dimensional $C^1$ submanifold $N$ whose tangent space is everywhere contained in $C$ and that meets $\mathrm{Core}_\varphi := \varphi^{-1}(B^{k_u}(0,1/2)\times B^{k_s}(0,1/2))$ at some point $x$, can be written locally around $x$ as $\varphi^{-1}(G)$ where $G$ is a graph of a map $B^{k_u}(0,1/2)\to B^{k_s}(0,1)$. In particular, every foliation whose tangent distribution is everywhere contained in $C$ admits a local trivialization that cover the core of $\varphi$. From now on we fix a finite atlas of such charts, where the $\mathrm{Core}_\varphi$ cover $M$. **Definition 8**. Let us say that a $k_u$-dimensional foliation $W$ of $M$ is $K$-*adapated* (to $T,C,\beta$) when the leafs of $W$ are $C^{1+\beta}$ and their tangent space is inside $c$ at all point, and in each chart of the specified atlas the leafs are graphs of $\beta$-Hölder with constant at most $K$. **Definition 9**. Let $\mathscr{R}_K^\beta$ be the set of probability measure $\mu$ on $M$ such that there exist a $K$-adapted foliation $W$ with respect to which $\mu$ has absolutely continuous local disintegrations with positive $\beta$-Hölder densities, and each of those densities $\rho$ satisfy $$\frac{\rho(x)}{\rho(y)} \le e^{K d_W(x,y)^\beta} \label{eq:log-holder}$$ where $d_W$ denotes the distance along the leafs of $W$. We say that $K$ is a *log constant* of *log $\beta$-Hölder constant* for $\rho$, or for $\mu$. Assume $U_1,U_2$ are two intersecting open sets on which $W$ can be trivialized and $N_1,N_2$ are transversals; i.e. $U_i = \cup_{y\in N_i} W_y\cap U_i \simeq N_i\times B^{k_u}(0,1)$ and consider $x\in U_1\cap U_2$. Then there are two different densities $\rho_1$, $\rho_2$ corresponding to the disintegrations of the restrictions of $\mu$ to $U_1$, $U_2$ with respect to $N_1$ and $N_2$; but this densities are proportional to each other along leaves where they are both defined. Indeed, denoting by $p_i:U_i\to N_i$ the holonomic projection to the transversal, $\rho_1(x)/\rho_2(x)$ is the Radon derivative of $\pi^{N_1\to N_2}_*p_{1*}\mu$ with respect to $p_{2*}\mu$ at $y=p_2(x)$. In particular, equation [\[eq:log-holder\]](#eq:log-holder){reference-type="eqref" reference="eq:log-holder"} holds for both $\rho_i$ simultaneously, with the same constant. ### Proof of Proposition [Proposition 7](#p:technical){reference-type="ref" reference="p:technical"} {#proof-of-proposition-ptechnical} We consider $\beta\in(0,\alpha_0)$ and the above specified atlas fixed. For each $r>0$ and $x\in M$, set $$B_\times(x,r) = \{y\in M \mid \exists z\in M, d_s(x,z)<r, d_u(z,y)<r\}.$$ This "product balls" form a basis of the topology of $M$, and we can find $\delta_-,\delta_+>0$ such that for every chart $\varphi$, $B_\times(x_\varphi,\delta_-)\subset \mathrm{Core}_\varphi\subset U_\varphi \subset B_\times(x_\varphi,\delta_+)$. Up to refining the atlas, we can moreover ensure that $\delta_+$ is small enough that both the stable and unstable foliations of $T$ are topologically trivial at scale $\delta_+$ (i.e. the $B_\times(x,\delta_+)$ are domains of trivailizing charts). Since $T$ is assumed to be topologically mixing, its stable leafs are dense and we can find $\tilde L_0$ such that for every pair of charts $\varphi,\psi$, their is a stable path of length at most $\tilde L_0$ from $\varphi^{-1}(0,0)$ to a point $\delta_-/100$-close to $x_\psi$ in the distance $d_u$. We set $L_0=\tilde L_0+2\delta_+$. To see that the SRB measure $\mu_0$ lies in some $\mathscr{R}_{K}^\beta$, it suffices to use the usual expression for the densities of its local disintegrations as an infinite product involving the unstable jacobian of $T$, see section 9.3 in [@Barreira2007book]. By definition, we immediately get $\mathscr{R}_K^\beta\subset\mathscr{R}_{K'}^\beta$ whenever $K'>K$. Moreover if $\mu\in\mathscr{R}_K^\beta$ and $\rho\in \operatorname{Hol}_W^\beta$ is a positive density with respect to $\mu$, then $\rho\mu$ has absolutely continuous disintegrations with respect to the same $K$-adapted foliation $W$ as $\mu$; if $\rho_1$ is a local leaf density of $\mu$, then the local leaf density of $\rho\mu$ is proportional to $\rho\rho_1$, hence $\rho\mu\in \mathscr{R}_{K'}^\beta$ for $K'=K +\lVert\log \rho\rVert_{W,\beta}$. As in Section [2](#s:expanding){reference-type="ref" reference="s:expanding"}, there is $H>0$ and $\lambda'\in (0,1)$ such that whenever $\mu\in \mathscr{R}_K^\beta$, $T_*\mu\in \mathscr{R}_{\lambda'(K+H)}^\beta$; the constant $\lambda'$ is $\beta$th power of the contraction factor of $T$ on every submanifold whose tangent space is contained in $C$ at all point, and $H$ accounts for the unstable jacobian of $T$ and the way $T$ and change of charts distort $\beta$-Hölder graphs. We can thus define $K_0$ and $n_0$ by $$K_0 = 2\frac{\lambda'}{1-\lambda'}H, \qquad 2\lambda'^{n_0} \le \frac{\lambda'}{1-\lambda'}$$ to obtain that $T_*(\mathscr{R}_{K_0}^\beta)\subset \mathscr{R}_{K_0}^\beta$ and $T^{n_0}_*(\mathscr{R}_{2K}^\beta)\subset \mathscr{R}_{K}^\beta$ for all $K\ge K_0$. We now prove the core coupling property, item [\[enumi:tech4\]](#enumi:tech4){reference-type="ref" reference="enumi:tech4"}. Let $\mu_1,\mu_2\in \mathscr{R}_{K_0}^\beta$ and denote by $W^1,W^2$ their $K_0$-adapted foliations. Denote by $N$ the number of charts in the chosen atlas and recall that the $\mathrm{Core}_\varphi:=\varphi^{-1}(B^{k_u}(0,1/2)\times B^{k_s}(0,1/2))$ cover $M$. We can thus find two charts $\varphi_1,\varphi_2$ such that $\mu_i(\mathrm{Core}_{\varphi_i})\ge 1/N$ for both $i$; and since the cores are covered by trivializing charts of the $W^i$, we can locally disintegrate restriction $\tilde\mu_i$ of the $\mu_i$ through the holonomic projections $p_i$ to local stable leaves $D_i\subset W^s_{x_{\varphi_i}}$: $$\mu_i = \int_{D_i} \rho_{i,x}(y) \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mathrm{vol}_{W^i_x}(y) \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\eta_i(x)$$ where $\eta_i$ are positive measures on $D_i$ of mass at least $1/N$ and $\rho_i$ are $\beta$-Hölder with log constant at most $K_0$. Consider a $1/N$-partial coupling $\tilde\gamma$ of $(\eta_1,\eta_2)$, i.e. $\gamma$ is a positive measure of mass $1/N$ on $M\times M$ with marginals $\le \eta_i$; for concreteness, let us take $\tilde\gamma=\frac1N \eta_1\otimes \eta_2$. For each pair of leafs $W^1_x, W^2_y$ we construct a partial coupling of the $\rho_i \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\mathrm{vol}_{W^i_x}$ as follows. Given $h>0$, Let $f=f_{x,h}:W^1_x\to \mathbb{R}_+$ be defined by $f(z) = \max(0,h-\frac{10 h}{\delta_-}d_{W^1_x}(x,z))$. Let $g=g_{y,h}$ be defined as the composition of $f$ with the holonomy $\pi^s_{x,y}$ of $W^s$, along a curve of length at most $L_0$. By choosing $h\le h_0$ with $h_0$ small enough, we can ensure that both $\rho_{1,x}-f_{x,h}$ and $\rho_{1,x}-g_{y,h}$ are positive and $\beta$-Hölder with log constant at most $2K_0$ (indeed, $f$ is Lipschitz with constant only depending on $\delta_-$, hence on $T$, and $g$ is $\alpha_0$-Hölder with constant depending further on $L_0$ and the stable holonomy Hölder constant, thus still only depending on $T$). Now all considered submanifold in the foliations are $C^{1,\beta}$ with bounded Hölder constant in the specified charts, so that there exist $\tilde\tau>0$ such that the function $f_{x,h_0}$ has integral at least $\tilde\tau$ with respect to $\mathrm{vol}_{W^1_x}$. By choosing $h=h_{x}\le h_0$ depending on $x$, we can make $f_{x,h}$ of integral exactly $\tilde\tau$. finally, our partial coupling of $\mu_1,\mu_2$ is given by $$\gamma = \int (\mathrm{Id},\pi^s_{x,y})_* \big(f_{x,h_{x}} \mathrm{vol}_{W^1_x}\big) \relax\ifnum\lastnodetype>0\mskip\medmuskip\fi\mathrm{d}\tilde\gamma(x,y).$$ It has mass $\tau :=\tilde\tau/N$, and its marginals are $\le\mu_i$; moreover $\mu'_i := \mu_i-p_{i*}\gamma$ have local disintegrations with respect to $W^i$ that are $\beta$-Hölder with log constant at most $2K_0$, hence $\mu'_i\in\mathscr{R}_{2K_0}^\beta$. ### Final remarks A similar method can be used to handle hyperbolic attractors instead of Anosov maps, but Theorem [Theorem 1](#t:central){reference-type="ref" reference="t:central"} does not hold anymore in this setting (stable leafs are no longer dense, and most measures to be considered will no longer be at finite $\operatorname{\mathsf{D}}_s^\beta$ distance, for any $\beta$). What can be done is to pair iterates of measures $T^{n_1}_*\mu_1, T^{n_1}_*\mu_2$ where $n_1$ is chosen to ensure that a significant part of the mass of $\mu_1$, $\mu_2$ are at small stable distance one from the other. If we relax uniform hyperbolicity, we can hope to keep weaker results in the same vein. In the case of lack of uniform hyperbolicity in the stable direction, it might be necessary to use $\omega\circ d_s$ instead of $d_s^\beta$, where $\omega$ is a concave function increasing quicker than any Hölder function at $0$. In the case of lack of uniform hyperbolicity in the unstable direction, it might be necessary to use densities and observables with an adapted regularity. In both cases, weaker speed for the decay of correlation is of course expected. It might be possible to adapt the present method to maps of flows with discontinuities, given it relies on a relatively flexible coupling argument: pair together along the stable direction part of any two measures that are regular enough in the unstable direction, then apply the dynamics long enough to recover the initial regularity for the remaining, uncoupled parts. Rince and repeat. [^1]: Univ Paris Est Creteil, Univ Gustave Eiffel, CNRS, LAMA UMR8050, F-94010 Creteil, France [^2]: By this we mean that there are so many variants of such a result that we prefer not attribute it to a particular article or set of authors.

arxiv_math

--- abstract: | Each connected graded, graded-commutative algebra $A$ of finite type over a field $\Bbbk$ of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the *(higher) Koszul modules* of $A$. In this note, we investigate the geometry of the support loci of these modules, called the *resonance schemes* of the algebra. When $A=\Bbbk\langle \Delta \rangle$ is the exterior Stanley--Reisner algebra associated to a finite simplicial complex $\Delta$, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group. address: - "Marian Aprodu: Simion Stoilow Institute of Mathematics P.O. Box 1-764, RO-014700 Bucharest, Romania, and Faculty of Mathematics and Computer Science, University of Bucharest, Romania" - "Gavril Farkas: Institut für Mathematik, Humboldt-Universität zu Berlin Unter den Linden 6, 10099 Berlin, Germany" - "Claudiu Raicu: Department of Mathematics, University of Notre Dame Hurley Notre Dame, IN 46556, USA, and Simion Stoilow Institute of Mathematics, P.O. Box 1-764, RO-014700 Bucharest, Romania" - "Alessio Sammartano: Dipartimento di Matematica, Politecnico di Milano Via Bonardi 9, Milan, 20133, Italy" - "Alexander I. Suciu: Department of Mathematics, Northeastern University Boston, MA, 02115, USA" author: - Marian Aprodu - Gavril Farkas - Claudiu Raicu - Alessio Sammartano - Alexander I. Suciu title: Higher resonance schemes and Koszul modules of simplicial complexes --- # Introduction and statement of results {#sect:intro} Koszul modules are graded modules over a symmetric algebra that are constructed from the classical Koszul complex. They emerged from geometric group theory and topology [@AFPRW; @PS-crelle] and found applications in other fields such as algebraic geometry. One prominent instance is [@AFPRW2], where the effective vanishing in high degrees of some Koszul modules led to a new proof of the celebrated Green's Conjecture on syzygies of generic canonical curves. The argument relies on a connection between the graded pieces of those particular Koszul modules and the Koszul cohomology of the tangent developable surface of a rational normal curve. The non-trivial vehicle that permits the passage in [@AFPRW2] from symmetric powers (Koszul modules) to exterior powers (Koszul cohomology) is an explicit version of the Hermite reciprocity formula. It is the aim to this paper to describe a completely new instance where the passage from Koszul modules to Koszul cohomology of some homogeneous coordinate ring is still possible. The setup is however simpler and more elementary than the one involved with Green's Conjecture. For a ground field $\Bbbk$ of characteristic $0$, a classical construction of Stanley and Reisner associates to every simplicial complex $\Delta$ on $n$ vertices a graded, graded-commutative algebra $\Bbbk\langle \Delta \rangle= E/J_{\Delta}$, where $E=\mbox{\normalsize $\bigwedge$}_{\Bbbk}(e_1,\dots, e_n)$ is the exterior algebra over $\Bbbk$ and $J_{\Delta}$ is the ideal generated by all the monomials $e_{\sigma}=e_{j_1}\wedge \cdots \wedge e_{j_s}$ corresponding to simplices $\sigma=(j_1,\dots, j_s)$ with $1\le j_1<\cdots <j_s \le n$ which do not belong to $\Delta$. Let $S\coloneqq \Bbbk[x_1,\dots, x_n]$ be the polynomial ring in $n$ variables over $\Bbbk$, and consider the cochain complex $(\Bbbk\langle \Delta \rangle^{\bullet}\otimes_{\Bbbk} S, \delta)$ of free, finitely generated, graded $S$-modules obtained by applying the BGG correspondence to the finitely generated, graded $E$-module $\Bbbk\langle \Delta \rangle^{\bullet}$. The Fitting ideals of this complex define the *jump resonance loci* of our simplicial complex, $$\label{eq:res-intro} \mathcal{R}^i(\Delta)\coloneqq V\bigl(\operatorname{Fitt}_{\beta_{i+1}} (\delta^{i-1}\oplus \delta^{i})\bigr),$$ where $\beta_{i+1}$ is the number of faces of dimension $i$ in $\Delta$. It was shown in [@PS-adv09] that the irreducible components of $\mathcal{R}^i(\Delta)$ are coordinate subspaces of $\Bbbk\langle \Delta \rangle^1=\Bbbk^n$, given explicitly in terms of the (simplicial) homology groups of certain subcomplexes of $\Delta$. Now let $\bigl(\Bbbk\langle \Delta \rangle_{\bullet} \otimes_{\Bbbk} S, \partial\bigr)$ be the dual chain complex, and define the *Koszul modules* (in weight $i$) of the simplicial complex $\Delta$ to be the homology $S$-modules of this complex, $$\label{eq:km-intro} W_i(\Delta)\coloneqq H_i\bigl(\Bbbk\langle \Delta \rangle_{\bullet} \otimes_{\Bbbk} S, \partial\bigr).$$ An alternate definition of resonance is given by the support loci of these modules, $$\label{eq:res-tilde-intro} \mathcal{R}_i(\Delta) \coloneqq V\bigl(\operatorname{Ann}(W_i(\Delta))\bigr).$$ These varieties, called the *support resonance loci*, are again finite unions of coordinate subspaces. Though they do not coincide in general with the previously defined sets $\mathcal{R}^i(\Delta)$, it is known that $\mathcal{R}_1(\Delta)=\mathcal{R}^1(\Delta)$ and $\bigcup_{j\le i}\mathcal{R}_j(\Delta)=\bigcup_{j\le i} \mathcal{R}^j(\Delta)$ for all $i\ge 1$. A notable property of the higher Koszul modules associated to simplicial complexes is that they are multigraded as opposed to the general case when they are only graded modules. Using the general theory of multi-graded square-free modules, we prove that the multi-graded pieces of the Koszul modules can be described as multi-graded pieces of some $\operatorname{Tor}$'s over symmetric algebras. It is known (see for example [@PS-crelle]) that the graded pieces of weight-one Koszul modules are graded pieces of $\operatorname{Tor}$'s over exterior algebras; however, their relations with $\operatorname{Tor}$'s over symmetric algebras is quite rare in general. **Theorem 1**. *For any $i\ge 1$ and any square-free multi-index $\mathbf{b}$, there is a natural isomorphism of vector spaces, $$\label{eqn:Mod-Tor-intro} \left[W_i(\Delta)\right]_\mathbf{b} \cong \left[\operatorname{Tor}^S_{|\mathbf{b}|-i} (\Bbbk, \Bbbk[\Delta])\right]_\mathbf{b}^\vee ,$$ where $\Bbbk[\Delta]$ is the polynomial Stanley--Reisner ring of $\Delta$.* We refer to Section [3.1](#subsec:sq-free){reference-type="ref" reference="subsec:sq-free"} for a quick review of multi-graded square-free modules. This multigraded structure of the Koszul modules is captured in the Hilbert series. **Theorem 2**. *For every simplicial complex $\Delta$, the multigraded Hilbert series of the Koszul modules $W_i(\Delta)$ are given by $$\label{eq:hilb-intro} \sum_{\mathbf{a}\in \mathbb{N}^n} \dim_{\Bbbk} [W_i(\Delta)]_\mathbf{a} \, \mathbf{t}^\mathbf{a}= \sum_{\substack{\mathbf{b}\,\in\, \mathbb{N}^n\\ \mathbf{b} \operatorname{square-free}}} \dim_{\Bbbk}(\widetilde{H}_{i-1}(\Delta_\mathbf{b}; \Bbbk)) \frac{\mathbf{t}^\mathbf{b}}{\prod_{j \in \operatorname{Supp}(\mathbf{b})}(1-t_j)}.$$* In Section [5](#sec:simp-comp){reference-type="ref" reference="sec:simp-comp"}, we give a precise description of the irreducible components of the support resonance loci. In each weight $i$, they correspond to maximal subcomplexes with non-vanishing reduced homology in degree $i-1$. **Theorem 3**. *For every simplicial complex $\Delta$ and every $i\ge 1$, the scheme structure on the support resonance $\mathcal{R}_i(\Delta)$ is reduced. Moreover, the decomposition in irreducible components is given by $$\label{eq:res-delta2-intro} \mathcal{R}_i(\Delta)=\bigcup_{\substack{ \mathsf{V'}\subseteq \mathsf{V}\: \mathrm{ maximal\: with }\\[1pt] \widetilde{H}_{i-1}(\Delta_{\mathsf{V'}};\Bbbk)\neq 0 }}\Bbbk^{\mathsf{V'}}.$$* Particularly interesting is the case when $\Delta$ is $1$-dimensional, that is, it may be viewed as a finite simple graph $\Gamma$. It was shown in [@PS-mathann] that all the irreducible components of $\mathcal{R}^1(\Gamma)$ are coordinate subspaces, which correspond to the maximally disconnected full subgraphs of $\Gamma$. This result comes as a direct consequence of our analysis. The statement concerning the reducedness of $\mathcal{R}_i(\Delta)$ can be compared with the detailed study performed in [@AFRS] on the scheme structure of (support) resonance varieties associated to classical Koszul modules. **Acknowledgments 1**. Aprodu was supported by the PNRR grant CF 44/14.11.2022 *Cohomological Hall algebras of smooth surfaces and applications*. Farkas supported by the DFG Grant *Syzygien und Moduli* and by the ERC Advanced Grant SYZYGY. This project has received funding from the European Research Council (ERC) under the EU Horizon 2020 program (grant agreement No. 834172). Raicu was supported by the NSF Grant No. 2302341. Sammartano was supported by the grant PRIN 2020355B8Y *Square-free Gröbner degenerations, special varieties and related topics*. Suciu was supported by Simons Foundation Collaboration Grant for Mathematicians No. 693825. # Graded algebras, Koszul modules, and higher resonance {#sect:kozul-res} We start in a more general context (adapted from the setup in [@PS-mrl; @Su-indam]), that will be used throughout the paper. ## Chain complexes associated to graded algebras {#subsec:cc-ga} Let $A^{\bullet}$ be a graded, graded-commutative algebra over a field $\Bbbk$ of characteristic $0$, with multiplication maps $A^i \otimes_{\Bbbk} A^j \to A^{i+j}$. We will assume that $A$ is connected (that is, $A^0=\Bbbk$) and of finite-type (that is, $\dim_{\Bbbk}A^i<\infty$, for all $i>0$), and we will write $\beta_i(A)=\dim_{\Bbbk}A^i$. To avoid trivialities, we always assume that $\beta_1(A)\neq 0$. For each $a\in A^1$, graded commutativity of multiplication yields $a^2=0$, therefore, we have a cochain complex $$\label{eq:cc} \begin{tikzcd}[column sep=24pt] (A^{\bullet} , \delta_{a})\colon \ A^0 \ar[r, "\delta^0_{a}"] & A^1 \ar[r, "\delta^1_{a}"] & A^2 \ar[r, "\delta^2_{a}"] & \cdots , \end{tikzcd}$$ with differentials $\delta^i_{a} (u)= a \cdot u$, for all $u \in A^i$. The *resonance varieties* of $A$ are the jump loci for the cohomology groups of this complex: for each $i\ge 0$, we put $$\label{eq:res-a} \mathcal{R}^i(A)\coloneqq \bigl\{a \in A^1 \mid H^i(A^{\bullet}, \delta_{a}) \ne 0 \bigr\}.$$ Clearly, these are homogeneous subsets of the affine space $A^1$. Since $A^0$ is $1$-dimensional, generated by $1\in \Bbbk$, and since $\delta_a(1)=a$ for each $a\in A^1$, it follows that $\mathcal{R}^0(A)=\{0\}$. The most studied is the first resonance variety, which can be described as the set $$\label{eq:res1} \mathcal{R}^1(A) =\{ a \in A^1 \mid \text{$\exists\, b \in A^1, \ 0\neq a\wedge b \in K^{\perp} $}\} \cup \{0\},$$ where $K^{\perp}$ denotes the kernel of the multiplication map $A^1\wedge A^1 \to A^2$. Let us now fix a $\Bbbk$-basis $\{ e_1,\dots, e_n \}$ of $A^1$ and let $\{ x_1,\dots, x_n \}$ be the dual basis of the dual $\Bbbk$-vector space $A_1=(A^1)^{\vee}$. This allows us to identify the symmetric algebra $\operatorname{Sym}(A_1)$ with the polynomial ring $S=\Bbbk[x_1,\dots, x_n]$, the coordinate ring of the affine space $A^1\cong \Bbbk^{\beta_1(A)}$. Viewing $A^{\bullet}$ as a graded module over the exterior algebra $E^{\bullet}=\mbox{\normalsize $\bigwedge$}A^1$, the BGG correspondence [@Ei-syz] yields a cochain complex of finitely generated, free $S$-modules, $$\label{eq:cc-s} \begin{tikzcd}[column sep=20pt] \hspace*{-12pt} \bigl(A^{\bullet} \otimes_{\Bbbk} S,\delta_A\bigr)\colon \cdots \ar[r] &A^{i}\otimes_{\Bbbk} S \ar[r, "\delta^{i}_A"] &A^{i+1} \otimes_{\Bbbk} S \ar[r, "\delta^{i+1}_A"] &A^{i+2} \otimes_{\Bbbk} S \ar[r] & \cdots, \end{tikzcd}$$ whose coboundary maps are the $S$-linear maps given by $\delta^{i}_A(u \otimes s)= \sum_{j=1}^{n} e_j u \otimes s x_j$ for $u\in A^i$ and $s\in S$. It is readily seen that this cochain complex is independent of the choice of basis for $A^1$ and that, moreover, the specialization of $(A\otimes_{\Bbbk} S,\delta_A)$ at an element $a\in A^1$ coincides with the complex $(A,\delta_a)$ defined by ([\[eq:cc\]](#eq:cc){reference-type="ref" reference="eq:cc"}). It follows directly from the definition [\[eq:res-a\]](#eq:res-a){reference-type="eqref" reference="eq:res-a"} that a point $a \in A^1$ belongs to $\mathcal{R}^i(A)$ if and only if $\operatorname{rank}\delta^{i-1}_a + \operatorname{rank}\delta^{i}_a < \beta_i(A)$. Therefore, $$\label{eq:res-fitt} \mathcal{R}^i(A)= V \big( \mathrm{Fitt}_{\beta_{i+1}(A)} \big(\delta^{i-1}_A\oplus \delta^{i}_A\big) \big),$$ where $\psi_1\oplus \psi_2$ denotes the block sum of two matrices, $I_r(\psi)$ denotes the ideal of $r\times r$ minors of a matrix $\psi$, and $V(I)$ denotes the zero-set of an ideal $I\subset S$. This shows that the sets $\mathcal{R}^i(A)$ are algebraic subvarieties of the affine space $A^1$ called *jump resonance loci*. ## Koszul modules and their support loci {#subsec:kozul-res} Set $A_i\coloneqq (A^i)^{\vee}$ and $\partial^A_i\coloneqq (\delta_A^{i-1})^{\vee}$ and consider the chain complex of finitely generated $S$-modules $$\label{eq:a-tensor-s} \begin{tikzcd}[column sep=24pt] \bigl(A_{\bullet} \otimes_{\Bbbk} S,\partial\bigr)\colon \cdots \ar[r] &A_{i+1}\otimes_{\Bbbk} S \ar[r, "\partial_{i+1}^A"] &A_{i} \otimes_{\Bbbk} S \ar[r, "\partial_{i}^A"] &A_{i-1} \otimes_{\Bbbk} S \ar[r] & \cdots. \end{tikzcd}$$ We define the *Koszul modules (in weight $i$)* of the algebra $A$ as the homology $S$-modules of this chain complex, that is, $$\label{eq:wi-a} W_i(A) \coloneqq H_i\bigl(A_{\bullet}\otimes_{\Bbbk} S\bigr) .$$ Clearly, these are finitely generated, graded $S$-modules. The degree $d$ component of the Koszul module $W_i(A)$ is computed by the homology of the complex $$\label{eq:les-as} \begin{tikzcd}[column sep=18pt] A_{i+1}\otimes_\Bbbk S_{d-i-1} \ar[r]& A_i\otimes_\Bbbk S_{d-i} \ar[r]& A_{i-1}\otimes_\Bbbk S_{d-i+1}, \end{tikzcd}$$ where we recall that $S=\operatorname{Sym}(A^1)$. It follows straight from the definitions that $W_0(A)=\Bbbk$ is the trivial $S$-module. Setting $E_{\bullet}\coloneqq \mbox{\normalsize $\bigwedge$}A_1$, the first Koszul module also has the following presentation $$\label{eq:pres-w1a} \begin{tikzcd}[column sep=18pt] \big(E_3 \oplus K^{\perp} \big) \otimes_{\Bbbk} S \ar[r, "\partial_3^E +\iota \otimes \operatorname{id}_S"] &[28pt] E_2 \otimes_{\Bbbk} S \ar[r, two heads] & W_1(A), \end{tikzcd}$$ where $\begin{tikzcd}[column sep=18pt] \hspace*{-5pt} K=\bigl\{\varphi \in A_1\wedge A_1=(A^1\wedge A^1)^{\vee} \mid \varphi_{|K^{\perp}}\equiv 0\bigr\} \ar[r, hook, "\iota"] & A_1\wedge A_1 =E_2. \end{tikzcd}$ The *resonance schemes* of the graded algebra $A$ are defined by the annihilator ideals of the Koszul modules of $A$, $$\label{eq:ria-spec} \mathcal{R}_i(A) \coloneqq \operatorname{Spec}\bigl(S/ \operatorname{Ann}W_i(A)\bigr).$$ By slightly abusing notation, we also denote by $\mathcal{R}_i(A)=\operatorname{Supp}W_i(A)$ the underlying sets and call them *support resonance loci*. Note that the algebra structure on $A^\bullet$ is not essential in the discussion above, as the definitions of Koszul modules and support resonance loci only use the $E$-module structure. In particular, the constructions apply for finitely-generated graded $E$-modules, as well. Clearly $\mathcal{R}_0(A)=\mathcal{R}^0(A)=\{0\}$. More generally, suppose $W_j(A)\neq 0$ for all $1\le j\le i$. Then, as shown in [@PS-mrl Theorem 2.5], the support resonance loci are related to the jump resonance loci by the formula [^1] $$\label{eq:union-res} \bigcup_{j\le i} \mathcal{R}_j(A) = \bigcup_{j\le i} \mathcal{R}^j(A).$$ In particular, if $W_1(A)\neq 0$, then $\mathcal{R}_1(A) =\mathcal{R}^1(A)$. ## Quotients of exterior algebras through ideals generated in fixed degree {#subsec:fixed-deg} We now discuss a particularly interesting case of this general construction. Fix integers $d\ge 1$ and $n\ge 3$. Let $V$ be an $n$-dimensional vector space over the field $\Bbbk$ and let $K\subseteq \mbox{\normalsize $\bigwedge$}^{d+1}V$ be a subspace. Set $S\coloneqq \operatorname{Sym}(V)$ and $E^\bullet\coloneqq \mbox{\normalsize $\bigwedge$}V^\vee$, and then consider the linear subspace $$\label{eq:kperp} K^\perp\coloneqq \big( \mbox{\normalsize $\bigwedge$}^{d+1}V/K \big)^\vee = \big\{\varphi\in \mbox{\normalsize $\bigwedge$}^{d+1}V^\vee \mid \varphi_{|K}= 0\big\} \subseteq \mbox{\normalsize $\bigwedge$}^{d+1}V^\vee.$$ Letting $A^\bullet\coloneqq E^\bullet/\langle K^\perp\rangle$ be the quotient of the exterior algebra $E^\bullet$ by the (homogeneous) ideal generated by $K^\perp$, we clearly have $K=A_{d+1}$. Conversely, if $J\subseteq E^\bullet$ is a homogeneous ideal generated in degree $d+1$ and we take $K^\perp\coloneqq J_{d+1}$, then the algebra $A^\bullet=E^\bullet/J$ is obtained as above. Denote by $j$ the inclusion of the dual algebra $A_\bullet$ into $E_\bullet$. Recalling that $\partial_i \colon \mbox{\normalsize $\bigwedge$}^i V\otimes_{\Bbbk} S\rightarrow \mbox{\normalsize $\bigwedge$}^{i-1} V\otimes_{\Bbbk} S$ is the Koszul differential, we have the following characterization. **Proposition 4**. *The Koszul modules $W_i(V,K)=W_i(A)$ verify the following properties:* 1. *[\[wb1\]]{#wb1 label="wb1"} $W_i(A)=0$ for $i\le d-1$.* 2. *[\[wb2\]]{#wb2 label="wb2"} $W_d(A)=\operatorname{coker}\big(\partial_{d+2}+j_{d+1} \big)$.* *Proof.* The first part is quite straightforward, as $J_i=0$ for $i\le d-1$ and hence $A_i=E_i$ for $i\le d-1$. For the second part, first note that the $d$-th Koszul module is in this case the middle homology of the complex $$\label{eq:dK-module} \begin{tikzcd}[column sep=18pt] K\otimes_{\Bbbk} S\ar[r] & \mbox{\normalsize $\bigwedge$}^dV\otimes_{\Bbbk} S \ar[r]& \mbox{\normalsize $\bigwedge$}^{d-1}V\otimes_{\Bbbk} S, \end{tikzcd}$$ and hence $$\label{eq:WdA} \begin{tikzcd}[column sep=18pt] W_d(V,K)=\operatorname{coker}\Bigl\{K\otimes_{\Bbbk} S \ar[r, "\partial_{d+1}"] & \mbox{\normalsize $\bigwedge$}^{d+1}V\otimes_{\Bbbk} S/\operatorname{im}(\partial_{d+2})\Bigr\}. \end{tikzcd}$$ Applying now the Snake Lemma to the diagram $$\label{eq:snake-1} \begin{tikzcd}[column sep=16pt] 0 \ar[r] & \mbox{\normalsize $\bigwedge$}^{d+1}V\otimes_{\Bbbk} S \ar[r] \ar[d, two heads] & \bigl(K\oplus \mbox{\normalsize $\bigwedge$}^{d+2}V\bigr)\otimes_{\Bbbk} S\ar[r]\ar[d] & K\otimes S\ar[r]\ar[d] & 0 \\ 0 \ar[r]& \operatorname{im}(\partial_{d+2}) \ar[r]& \mbox{\normalsize $\bigwedge$}^{d+1}V\otimes_{\Bbbk} S \ar[r] & \mbox{\normalsize $\bigwedge$}^{d+1}V\otimes_{\Bbbk} S/\operatorname{im}(\partial_{d+2})\ar[r]&0 \end{tikzcd}$$ establishes the claim. ◻ **Remark 5**. Note that the Snake Lemma applies also to the diagram $$\label{eq:snake-2} \begin{tikzcd}[column sep=16pt] 0 \ar[r] & K\otimes_{\Bbbk} S\ar[r]\ar[d, equal] & (K \oplus \mbox{\normalsize $\bigwedge$}^{d+2}V) \otimes_{\Bbbk} S\ar[r]\ar[d] & \mbox{\normalsize $\bigwedge$}^{d+2}V\otimes_{\Bbbk} S \ar[r]\ar[d] & 0 \\ 0 \ar[r]& K\otimes_{\Bbbk} S \ar[r]& \mbox{\normalsize $\bigwedge$}^{d+1}V\otimes_{\Bbbk} S \ar[r] &(\mbox{\normalsize $\bigwedge$}^{d+1}V/K)\otimes_{\Bbbk} S\ar[r]&0, \end{tikzcd}$$ leading to the simpler presentation $$\label{eqn:simple-presentation} \begin{tikzcd}[column sep=18pt] W_d(A)=\operatorname{coker}\Bigl\{\mbox{\normalsize $\bigwedge$}^{d+2}V\otimes_{\Bbbk} S\ar[r] & (\mbox{\normalsize $\bigwedge$}^{d+1}V/K)\otimes_{\Bbbk} S\Bigr\}. \end{tikzcd}$$ If $d=1$, in weight $1$ we recover the original Koszul module $W(V,K)$ of a pair $(V,K)$ with $K\subseteq \mbox{\normalsize $\bigwedge$}^2V$ considered in [@AFPRW; @AFRS; @PS-crelle] and elsewhere. However, note the shift by two in degrees, that is, $W(V,K)=W_1(A)(2)$. **Example 6**. Let $X$ be a smooth complex projective variety, and consider a vector bundle $E$ on $X$ of rank $\ge r+1$, for some integer $r\ge 1$. We consider the determinant maps $$\label{eqn:drmap} \begin{tikzcd}[column sep=18pt] d_r\colon \mbox{\normalsize $\bigwedge$}^{r+1}H^0(X,E)\ar[r]& H^0(X,\mbox{\normalsize $\bigwedge$}^{r+1}E) \end{tikzcd}$$ and take $K_r^\perp\coloneqq \ker(d_r)$. Then the above construction applies, producing for each $r$ a series of Koszul modules $W_r(X,E)\coloneqq W_r\bigl(H^0(X, E)^{\vee}, K_r \bigr)$. As in the case $d=1$ (see [@AFPRW2; @PS-crelle]), we have a geometric characterization of vanishing resonance, in which case the corresponding Koszul module is of finite length and hence vanishes in high degrees. For an element $\omega\in \mbox{\normalsize $\bigwedge$}^{d+1}V^\vee$, we denote by $\varphi(\omega)\colon V^\vee\to\mbox{\normalsize $\bigwedge$}^{d+2}V^\vee$ the map $a\mapsto a\wedge\omega$. Consider the projective variety parameterizing the decomposable elements, $$\label{eq:Sigma} \Sigma_d\coloneqq \bigl\{[\omega]\in\mathbb{P} \big(\mbox{\normalsize $\bigwedge$}^{d+1}V^\vee \big) \mid \operatorname{rank}(\varphi(\omega))\le n-1\bigr\}.$$ Standard multilinear algebra proves the following proposition. **Proposition 7**. *If $d\ge 2$, then $\mathcal{R}_d(A)=\{0\}$ if and only if $\mathbb{P}(K^\perp)\cap \Sigma_d=\emptyset$.* Recall that $a\in V^\vee$ divides $\omega$ if and only if $a\wedge\omega=0$. Therefore, $\mathcal{R}_d(A)\neq\{0\}$ if and only if there exist $a\in V^\vee$ and $b\in \mbox{\normalsize $\bigwedge$}^dV^\vee$ such that $0\neq a\wedge b\in K^\perp$. This is equivalent to the existence of a non-zero element $a\in V^\vee$ and of a non-zero element $\omega\in K^\perp$ such that $a\wedge\omega=0$, i.e., $0\neq a\in \ker(\varphi(\omega))$, and hence $[\omega]\in \Sigma_d$. The case $d=1$ is special, since $\varphi(\omega)$ non-injective implies its kernel is at least $2$-dimensional. Indeed, if $\omega =a\wedge b\neq 0$ then $\ker(\varphi(\omega))$ is generated by $a$ and $b$. In this case, $\Sigma_1$ is the Grassmann variety $\operatorname{Gr}_2(V^\vee)\subseteq \mathbb{P}\bigl(\mbox{\normalsize $\bigwedge$}^2 V^{\vee}\bigr)$. **Remark 8**. For the Koszul module $W_r(X,E)$ considered in Example [Example 6](#ex:vb){reference-type="ref" reference="ex:vb"}, we have that the resonance $\mathcal{R}_r(X,E)\coloneqq \operatorname{Supp}W_r(X,E)$ is non trivial if and only if there exists a section $0\neq s\in H^0(X,E)$ such that the determinant map $d_s\colon \mbox{\normalsize $\bigwedge$}^r H^0(X,E)\rightarrow \mbox{\normalsize $\bigwedge$}^{r+1} H^0(X,E)$ given by $\omega\mapsto d_{r+1}(s\wedge \omega)$ is not injective. # Simplicial complexes and their Koszul modules {#sect:res-sc} ## Square-free modules {#subsec:sq-free} We start this section with some algebraic preliminaries regarding square-free modules. We recall from [@Ya00] some basic facts about this type of modules, which will be needed in Sections [4.2](#subsec:hilb){reference-type="ref" reference="subsec:hilb"}, [5](#sec:simp-comp){reference-type="ref" reference="sec:simp-comp"}, and [6.1](#subsec:reg){reference-type="ref" reference="subsec:reg"}. Let $V$ be a $\Bbbk$-vector space of dimension $n$, and identify the symmetric algebra $\operatorname{Sym}(V)$ with the polynomial ring $S=\Bbbk[x_1,\ldots,x_n]$. We consider the standard $\mathbb{N}^n$-multigrading on $S$, defined by $\deg(x_i) = \mathbf{e}_i \in \mathbb{N}^n$, where $\mathbf{e}_i=(0,\ldots,1,\ldots,0)$ is the multi-index with $1$ placed in the $i$-th position. Given a multi-index $\mathbf{a}=(a_1, \ldots, a_n) \in \mathbb{N}$, its support is defined as the set $\operatorname{Supp}(\mathbf{a})\coloneqq \{ i \mid a_i > 0\}$. **Definition 9**. An $\mathbb{N}^n$-graded $S$-module $M$ is said to be *square-free* if for any $\mathbf{a}\in\mathbb{N}^n$ and any $i\in\operatorname{Supp}(\mathbf{a})$, the multiplication map $$\label{eq:mult-map} \begin{tikzcd}[column sep=16pt] x_i \colon M_{\mathbf{a}} \ar[r]& M_{\mathbf{a}+\mathbf{e}_i} \end{tikzcd}$$ is an isomorphism. This definition is a direct generalization of the case of ideals. Indeed, an ideal $I \subseteq S$ is a square-free module if and only if it is a square-free monomial ideal, and this is also equivalent to $S/I$ being a square-free module. Note that a free $\mathbb{N}^n$-graded $S$-module is square-free if and only it is generated in square-free multidegrees. **Proposition 10**. *If $f\colon M\to N$ is a morphism of $\mathbb{N}^n$-graded $S$-modules, and $M$ and $N$ are square-free modules, then $\ker(f)$ and $\operatorname{coker}(f)$ are also square-free. Moreover, if $$\label{eq:MMM} \begin{tikzcd}[column sep=16pt] 0\ar[r]& M' \ar[r]& M \ar[r]& M'' \ar[r]& 0 \end{tikzcd}$$ is an exact sequence of $\mathbb{N}^n$-graded $S$-modules, and $M'$ and $M''$ are square-free, then so is $M$.* Proposition [Proposition 10](#prop:Nn-sqf-ker-coker){reference-type="ref" reference="prop:Nn-sqf-ker-coker"} has a few interesting consequences. **Corollary 11**. *Let $M$ be an $\mathbb{N}^n$-graded square-free $S$-module. Then all the modules in the minimal free $\mathbb{N}^n$-graded resolution of $M$ are square-free.* **Corollary 12**. *If $\mathbf{F}$ is a bounded complex of free square-free $S$-modules, then the homology modules of $\mathbf{F}$ are also square-free.* The following result will be of particular interest for us. **Theorem 13**. *If $M$ is an $\mathbb{N}^n$-graded, square-free $S$-module, then its annihilator is a square-free monomial ideal. In particular, the annihilator of $M$ is a radical ideal.* *Proof.* Since $M$ is an $\mathbb{N}^n$-graded $S$-module, the annihilator $\operatorname{Ann}(M) \subseteq S$ is also $\mathbb{N}^n$-graded, that is, it is a monomial ideal. Let $m=x_{1}^{a_1}\cdots x_{n}^{a_n}\in\operatorname{Ann}(M)$ be a monomial annihilating $M$, and assume $a_k>1$ for some $k$. Then the multiplication map $$\begin{tikzcd}[column sep=16pt] m \colon M_{\mathbf{b}}\ar[r]& M_{{\mathbf{b}}+\deg(m)} \end{tikzcd}$$ is zero for all $\mathbf{b} \in \mathbb{N}^n$. We have $k \in \operatorname{Supp}(\mathbf{b}+\deg(m)-\mathbf{e}_k)$, and so, by hypothesis, the map $$\begin{tikzcd}[column sep=16pt] x_k \colon M_{\mathbf{b}+\deg(m)-\mathbf{e}_k}\ar[r]& M_{\mathbf{b}+\deg(m)} \end{tikzcd}$$ is an isomorphism. Therefore, $$\begin{tikzcd}[column sep=16pt] m/{x_k} \colon M_{\mathbf{b}}\ar[r]& M_{\mathbf{b}+ \deg(m)-\mathbf{e}_k} \end{tikzcd}$$ is the zero map for all $\mathbf{b} \in \mathbb{N}^n$, and thus $x_{1}^{a_1}\cdots x_k^{a_k-1}\cdots x_{n}^{a_n}\in\operatorname{Ann}(M)$. By repeating the argument, we see that $\operatorname{Ann}(M)$ is a square-free monomial ideal. ◻ Finally, we note that Theorem [Theorem 13](#thm:annihilator){reference-type="ref" reference="thm:annihilator"} and [@Ya00 Lemma 2.2] give the following. **Proposition 14**. *Let $M$ be a finitely-generated $\mathbb{N}^n$-graded, square-free $S$-module. Then the annihilator scheme structure on the support of $M$ is reduced. Moreover, the decomposition of the support in irreducible components is given by $$\mathrm{Supp}(M)=\bigcup_{\substack{ \mathbf{b}\ \operatorname{square-free}\\ \operatorname{ maximal\ with }\ M_\mathbf{b}\neq 0 }}\Bbbk^{\mathrm{Supp}(\mathbf{b})},$$ where $\Bbbk^\mathsf{V'}$ denotes the locus $V(x_i|\ i\not\in \mathsf{V}')$.* Proposition [Proposition 14](#prop:sfq-support){reference-type="ref" reference="prop:sfq-support"} will be essential for describing the components of the support resonance loci of a simplicial complex in the next section. We end this section with the following definition: **Definition 15**. For an $\mathbb{N}^n$-graded vector space $M$, the *square-free part* of $M$ is the subspace $\operatorname{sqf}(M)\subseteq M$ concentrated in square-free multidegrees. ## Stanley--Reisner rings {#subsec:SR-ring} Let $S=\Bbbk[x_1,\dots, x_n]$ be the polynomial ring in $n$ variables over a field $\Bbbk$ of characteristic $0$. Given a simplicial complex $\Delta$ on $n$ vertices, we let $\Bbbk[\Delta] \coloneqq S/I_{\Delta}$ be the (polynomial) *Stanley--Reisner ring* of $\Delta$, where $I_{\Delta}$ is the ideal generated by the (square-free) monomials $x_{\sigma}=x_{i_1} \cdots x_{i_s}$ for all simplices $\sigma=(i_1,\dots, i_s)$ with $1\le i_1<\cdots <i_s \le n$ not in $\Delta$. Similarly, we define the *exterior Stanley--Reisner ring* of $\Delta$ as $\Bbbk\langle \Delta \rangle\coloneqq E/J_{\Delta}$, where $E=\mbox{\normalsize $\bigwedge$}(e_1,\dots, e_n)$ is the exterior algebra in $n$ variables over $\Bbbk$ and $J_{\Delta}$ is the ideal generated by the monomials $e_{\sigma}=e_{i_1}\wedge \cdots \wedge e_{i_s}$ for all simplices $\sigma\notin \Delta$. Consider the graded, graded-commutative $\Bbbk$-algebra $A^\bullet\coloneqq \Bbbk\langle \Delta \rangle$. In each degree $d$, the vector space $A^d$ is spanned by multivectors $e_\sigma$, where $\sigma$ is a $(d-1)$-dimensional face of $\Delta$. Indeed, since $\sigma=(i_1,\ldots,i_s)\not\in\Delta$ implies $(i_1,\ldots,i_s,j)\not\in\Delta$ for all $j\not\in\operatorname{Supp}(\Delta)$, it follows that in each degree $d$, the vector space $J_{\Delta,d}$ is spanned by the multivectors $e_\sigma$ with $\sigma\not\in\Delta$ of dimension $d-1$. With the notation of the previous sections, the dual $A_d$ is generated by the vectors $v_\sigma$ with $\sigma\in\Delta$ being of dimension $d-1$. For an element $a=\sum_{i=1}^{n} \lambda_i e_i\in A^1$, let $(A^{\bullet}, \delta_a)$ be the cochain complex from [\[eq:cc\]](#eq:cc){reference-type="eqref" reference="eq:cc"}. As shown in [@AAH Proposition 4.3] (see also [@PS-adv09 Lemma 3.4]), this complex depends only on $\operatorname{Supp}(a)\coloneqq \{i\mid \lambda_i\ne 0\}$; more precisely, $(A^{\bullet}, \delta_a)$ is isomorphic to $(A^{\bullet}, \delta_{\bar{a}})$, where $\bar{a}=\sum_{i\in \operatorname{Supp}(a)} e_i$. The following Hochster-type formula from [@AAH Proposition 4.3], suitably interpreted and corrected in [@PS-adv09 Proposition 3.6], describes the cohomology groups of the cochain complexes $(A^{\bullet}, \delta_a)$. **Proposition 16** ([@AAH; @PS-adv09]). *Let $\Delta$ be a finite simplicial complex on vertex set $\mathsf{V}=[n]$ and $a\in A^1$ as above. Writing $\mathsf{V}'=\operatorname{Supp}(a)$, we have $$\dim_{\Bbbk} H^{i}\bigl(\Bbbk\langle \Delta \rangle,\delta_a\bigr)=\sum_{\sigma\in \Delta_{\mathsf{V}\setminus \mathsf{V}'}} \dim_{\Bbbk} \widetilde{H}_{i-1-\lvert\sigma\rvert} \bigl(\operatorname{lk}_{\Delta_{\mathsf{V}'}}(\sigma); \Bbbk\bigr).$$* Here $\Delta_{\mathsf{V}'}\coloneqq \{\tau\in \Delta\mid \tau \subset \mathsf{V}'\}$ is the simplicial complex obtained by restricting $\Delta$ to $\mathsf{V}'$ and $\operatorname{lk}_{\Delta_{\mathsf{V}'}}(\sigma)\coloneqq \{\tau \in \Delta_{\mathsf{V}'} \mid \tau\cup \sigma \in \Delta\}$ is the link of a simplex $\sigma$ in $\Delta_{V'}$. The range of summation in the above formula includes the empty simplex, with the convention that $\lvert\emptyset\rvert=0$ and $\widetilde{H}_{-1}(\emptyset; \Bbbk)=\Bbbk$. ## Koszul modules of a simplicial complex {#subsec:K-simp} Fix a basis $v_1, \ldots, v_n$ of the $\Bbbk$-vector space $V$. Let $\mathbf{K}_\bullet$ denote the Koszul complex of $x_1, \ldots, x_n$, whose $i$-th free module is $\mathbf{K}_i = \mbox{\normalsize $\bigwedge$}^i V \otimes_{\Bbbk} S$, and set $\deg(v_i) = \mathbf{e}_i \in \mathbb{N}^n$. Then $\mathbf{K}_\bullet$ is a complex of $\mathbb{N}^n$-graded square-free $S$-modules. A simplicial complex $\Delta$ on vertex set $[n]=\{1, \ldots, n\}$ determines a subcomplex $\mathbf{K}^\Delta_\bullet$ of $\mathbf{K}_\bullet$, whose $i$-th module $\mathbf{K}_i^\Delta$ is the free $S$-module generated by the exterior monomials $v_{j_1}\wedge \cdots \wedge v_{j_i}$ such that $\{j_1, \ldots, j_i\}$ is a face of $\Delta$. Applying ([\[eq:wi-a\]](#eq:wi-a){reference-type="ref" reference="eq:wi-a"}), the $i$-th *Koszul module* $W_i(\Delta)$ defined as the $i$-th Koszul module of the exterior Stanley--Reisner ring of $\Delta$ is the $i$-th homology $H_i(\mathbf{K}^\Delta_\bullet)$. **Proposition 17**. *For every simplicial complex $\Delta$ on $n$ vertices and for every $i$, the Koszul module $W_i(\Delta)$ is an $\mathbb{N}^n$-graded square-free $S$-module.* *Proof.* The subcomplex $\mathbf{K}^\Delta_\bullet$ is a complex of $\mathbb{N}^n$-graded square-free $S$-modules. By Corollary [Corollary 12](#cor:Nn-sqf-homology){reference-type="ref" reference="cor:Nn-sqf-homology"}, it follows that each homology vector space $W_i(\Delta)$ is a square-free $S$-module. ◻ # Hilbert series for Koszul modules of simplicial complexes {#sect:reg-hilbert} We fix some notation first. For a multidegree $\mathbf{b}$, we denote the sum of its entries by $\lvert\mathbf{b}\rvert$. For a square-free multidegree $\mathbf{b}\in \mathbb{N}^n$, we denote by $\Delta_\mathbf{b}$ the restriction of the simplicial complex $\Delta$ to the subset of the vertices $\operatorname{Supp}(\mathbf{b}) \subseteq \{1, \ldots, n \}$. We denote by $\Tilde{h}_i(-; \Bbbk)$ and $\Tilde{h}^i(-; \Bbbk)$ the dimensions of the simplicial homology groups $\Tilde{H}_i(-; \Bbbk)$ and of the reduced cohomology groups $\Tilde{H}^i(-; \Bbbk)\cong \Tilde{H}_i(-; \Bbbk)^\vee$ with coefficients in $\Bbbk$, respectively. ## Koszul modules vs. Koszul (co)homology {#subseq:K vs K} We establish a duality result between the Koszul modules associated to a simplicial complex and Koszul (co)homology of the symmetric Stanley--Reisner algebra. **Theorem 18**. *For any $i\ge 1$ and any square-free multi-index $\mathbf{b}$, there are natural isomorphisms of vector spaces $$\label{eqn:Mod-Tor} \left[W_i(\Delta)\right]_\mathbf{b} \cong \left[\operatorname{Tor}^S_{|\mathbf{b}|-i} (\Bbbk, \Bbbk[\Delta])\right]_\mathbf{b}^\vee\cong \widetilde{H}^{i-1}(\Delta_\mathbf{b}; \Bbbk)^\vee\cong \widetilde{H}_{i-1}(\Delta_\mathbf{b}; \Bbbk).$$* *Proof.* For the square-free multidegree $\mathbf{b}$, we denote $j\coloneqq \lvert\mathbf{b}\rvert - i$. We start by proving the first isomorphism. We use the notation from Section [3.2](#subsec:SR-ring){reference-type="ref" reference="subsec:SR-ring"}. Let $A_d\subseteq \mbox{\normalsize $\bigwedge$}^d V$ be the subspace generated by the exterior monomials $v_\sigma$ such that $\sigma$ is a face of $\Delta$. Denote by $S_d$ the graded component of $S=\operatorname{Sym}(V)$ of total degree $d$. Then, the vector space $[W_i(\Delta)]_\mathbf{b}$ is the middle homology of the complex of vector spaces $$\label{eq:seq1} \begin{tikzcd}[column sep=16pt] {[A_{i+1}\otimes S_{j-1}]}_\mathbf{b} \ar[r] & {[A_{i}\otimes S_{j}]}_\mathbf{b} \ar[r] & {[A_{i-1}\otimes S_{j+1}]}_\mathbf{b}. \end{tikzcd}$$ Clearly, this complex is the same as $$\label{eq:seq2} \begin{tikzcd}[column sep=16pt] {[A_{i+1}\otimes \operatorname{sqf}(S_{j-1})]}_\mathbf{b} \ar[r]& {[A_{i}\otimes \operatorname{sqf}(S_{j})]}_\mathbf{b} \ar[r]& {[A_{i-1}\otimes \operatorname{sqf}(S_{j+1})]}_\mathbf{b}. \end{tikzcd}$$ Upon identifying $\operatorname{sqf}(S_{d})= \bigwedge^d V^\vee$, this chain complex may be written as $$\begin{tikzcd}[column sep=18pt] \left[A_{i+1}\otimes \bigwedge^{j-1} V^\vee\right]_\mathbf{b} \ar[r]& \left[A_{i}\otimes \bigwedge^{j} V^\vee\right]_\mathbf{b} \ar[r]& \left[A_{i-1}\otimes \bigwedge^{j+1} V^\vee\right]_\mathbf{b}, \end{tikzcd}$$ which, by dualization gives $$\label{eq:seq3} \begin{tikzcd}[column sep=18pt] \left[A^{i-1}\otimes \bigwedge^{j+1} V\right]_\mathbf{b} \ar[r]& \left[A^{i}\otimes \bigwedge^{j} V\right]_\mathbf{b} \ar[r]& \left[A^{i+1}\otimes \bigwedge^{j-1} V\right]_\mathbf{b}. \end{tikzcd}$$ After having identified $A^d = \operatorname{sqf}(S/I_\Delta)_d$, the sequence [\[eq:seq3\]](#eq:seq3){reference-type="eqref" reference="eq:seq3"} may be written as $$\label{eq:seq4} \begin{tikzcd}[column sep=16pt] \left[\operatorname{sqf}(S/I_\Delta)_{i-1}\otimes \bigwedge^{j+1} V\right]_\mathbf{b} \ar[r]& \left[\operatorname{sqf}(S/I_\Delta)_{i}\otimes \bigwedge^{j} V\right]_\mathbf{b} \ar[r]& \left[\operatorname{sqf}(S/I_\Delta)_{i+1}\otimes \bigwedge^{j-1} V\right]_\mathbf{b}. \end{tikzcd}$$ Since $\mathbf{b}$ is a square-free multidegree, this complex is the same as $$\label{eq:seq5} \begin{tikzcd}[column sep=16pt] \left[(S/I_\Delta)_{i-1}\otimes \bigwedge^{j+1} V\right]_\mathbf{b} \ar[r]& \left[(S/I_\Delta)_{i}\otimes \bigwedge^{j} V\right]_\mathbf{b} \ar[r]& \left[(S/I_\Delta)_{i+1}\otimes \bigwedge^{j-1} V\right]_\mathbf{b}. \end{tikzcd}$$ By the properties of the Koszul complex, the middle cohomology of this complex is isomorphic to $$\left[\operatorname{Tor}^S_{j} (\Bbbk, \Bbbk[\Delta])\right]_\mathbf{b},$$ and this concludes the proof of the first isomorphism. For the second isomorphism, note that $\left[\operatorname{Tor}^S_j(\Bbbk, \Bbbk[\Delta])\right]_\mathbf{b}$ is isomorphic to $\left[\operatorname{Tor}^S_{j-1}(\Bbbk, I_\Delta)\right]_\mathbf{b}$. Indeed, for $j\ge 2$, this is clear, whereas for $j=1$ this follows from the fact the $I_\Delta$ is contained in the ideal generated by the variables, and hence $\Bbbk\cong\Bbbk\otimes_S \Bbbk[\Delta]$. From [@HerzogHibi Proof of Theorem 8.1.1] we obtain an isomorphism $\left[\operatorname{Tor}^S_{j-1}(\Bbbk, I_\Delta)\right]_\mathbf{b}\cong \tilde{H}^{|\mathbf{a}|-j-1}(\Delta_\mathbf{b}; \Bbbk)$. In conclusion, $$_\mathbf{b}\cong \tilde{H}^{i-1}(\Delta_\mathbf{b}; \Bbbk)^\vee,$$ as soon as $|\mathbf{b}|-i\ge 1$. ◻ **Remark 19**. The isomorphism in the statement of Theorem [Theorem 18](#thm:KoszModules-KoszCohom){reference-type="ref" reference="thm:KoszModules-KoszCohom"} does not necessarily hold if we drop the hypothesis that $\mathbf{b}$ is square-free. Indeed, $\left[\operatorname{Tor}^S_{|\mathbf{b}|-i} (\Bbbk, \Bbbk[\Delta])\right]_\mathbf{b}$ is equal to $0$ if $\mathbf{b}$ is not square-free, [@HerzogHibi Theorem 8.1]. On the other hand, since the square-free multi-indices are finitely many, the vanishing of $\left[W_i(\Delta)\right]_\mathbf{b}$ for all $\mathbf{b}$ that is not square-free implies $W_i(\Delta)$ is of finite length. An alternate, less explicit proof of the above theorem can be obtained by applying the Bernstein--Gelfand--Gelfand correspondence to express $[W_i(\Delta)]_\mathbf{b}$ as the (duals) of some $\operatorname{Tor}$ spaces over the exterior algebra, and then apply a theorem of Aramova, Avramov, and Herzog [@AAH], see [@HerzogHibi Corollary 7.5.2]. More precisely, we have the following result. **Proposition 20**. *For any $i\ge 1$ and any square-free multi-index $\mathbf{b}$, there is a natural isomorphism of vector spaces $$\label{eqn:Mod-Tor-E} \left[W_i(\Delta)\right]_\mathbf{b} \cong \left[\operatorname{Tor}^E_{|\mathbf{b}|-i} \bigl(\Bbbk, \Bbbk\langle \Delta \rangle\bigr)\right]_\mathbf{b}^\vee.$$* The proof of the proposition follows from an adaptation to the multi-graded context [@Brown-Erman] of the classical BGG correspondence, as described in [@Ei-syz]. ## Multigraded Hilbert series {#subsec:hilb} Our next goal is to determine the Hilbert series of the Koszul modules $W_i(\Delta)$ associated to a simplicial complex $\Delta$. **Theorem 21**. *For every simplicial complex $\Delta$ and every $i>0$, the Hilbert series of the Koszul module $W_i(\Delta)$ is given by $$\sum_{a\in \mathbb{N}} \dim [W_i(\Delta)]_a \, t^a= \sum_{\substack{\mathbf{b}\,\in\, \mathbb{N}^n\\ \mathbf{b}\: \operatorname{square-free}}} \mathrm{dim}\bigl(\widetilde{H}_{i-1}(\Delta_\mathbf{b}; \Bbbk)\bigr) \left(\frac{t}{1-t}\right)^{\lvert\mathbf{b}\rvert}.$$* *Proof.* In order to prove the theorem, we shall compute the $\mathbb{N}^n$-graded Hilbert series of $W_i(\Delta)$, then specialize the formula to the single $\mathbb{N}$-grading. For a multidegree $\mathbf{a}=(a_1, \ldots, a_n)\in \mathbb{N}^n$, we denote by $\mathbf{t}^\mathbf{a} = t_1^{a_1 }\cdots t_n^{a_n}$. We begin by observing that, by the definition of a square-free module, we have $$\label{eq:hilb-multi} \sum_{\mathbf{a}\in \mathbb{N}^n} \dim [W_i(\Delta)]_\mathbf{a} \, \mathbf{t}^\mathbf{a} = \sum_{\substack{\mathbf{b}\in \mathbb{N}^n\\ \mathbf{b} \text{ square-free}}} \dim [W_i(\Delta)]_\mathbf{b} \, \frac{\mathbf{t}^\mathbf{b}}{\prod_{j \in \operatorname{Supp}(\mathbf{b})}(1-t_j)} \, .$$ Thus, it suffices to determine $\dim [W_i(\Delta)]_\mathbf{b}$ when $\mathbf{b}$ is a square-free multidegree. Fix a square-free multidegree $\mathbf{b}$, and let $j = \lvert\mathbf{b}\rvert - i$. From Theorem [Theorem 18](#thm:KoszModules-KoszCohom){reference-type="ref" reference="thm:KoszModules-KoszCohom"} we know that $$_\mathbf{b}\cong \tilde{H}_{i-1}(\Delta_\mathbf{b}; \Bbbk),$$ and hence we obtain the following formula for the multigraded Hilbert series of $W_i(\Delta)$, $$\label{eq:hilb-multi-bis} \sum_{\mathbf{a}\in \mathbb{N}^n} \dim [W_i(\Delta)]_\mathbf{a} \, \mathbf{t}^\mathbf{a} = \sum_{\substack{\mathbf{b}\in \mathbb{N}^n\\ \mathbf{b} \text{ square-free}}} \tilde{h}_{i-1}\bigl(\Delta_\mathbf{b}, \Bbbk\bigr) \frac{\mathbf{t}^\mathbf{b}}{\prod_{j \in \operatorname{Supp}(\mathbf{b})}(1-t_j)}.$$ Specializing to the single $\mathbb{N}$-grading, this yields the desired formula. ◻ In the particular case when $\Delta$ has dimension at most $1$, that is, when $\Delta$ is equal to a (simplicial) graph $\Gamma$, we recover the Hilbert series of the module $W_\Gamma\coloneqq W_1(\Gamma)(2)$, as computed in [@PS-mathann Theorem 4.1]. **Corollary 22** ([@PS-mathann]). *For a graph $\Gamma$ on vertex set $\mathsf{V}$, we have $$\label{eq:hilb-graph} \operatorname{Hilb}(W_{\Gamma},t )= \frac{1}{t^2}\cdot Q_{\Gamma}\Big( \frac{t}{1-t} \Big),$$ where $Q_{\Gamma}(t)=\sum_{j\ge 2} c_j(\Gamma) t^j$ and $c_j(\Gamma)=\sum_{\mathsf{V}'\subseteq \mathsf{V}\colon \lvert\mathsf{V}'\rvert=j } \tilde{h}_0(\Gamma_{\mathsf{V}'})$.* The significance of the above formula is that it gives the Chen ranks of the right-angled Artin group $G_{\Gamma}$ associated to the graph $\Gamma$. # Resonance varieties of a simplicial complex {#sec:simp-comp} Given an (abstract) simplicial complex $\Delta$ on vertex set $\mathsf{V}$, we define its resonance varieties as those of the corresponding exterior Stanley--Reisner ring. That is, we put $\mathcal{R}^i(\Delta)\coloneqq \mathcal{R}^i(\Bbbk\langle \Delta \rangle)$ for the jump resonance and $\mathcal{R}_i(\Delta)\coloneqq \mathcal{R}_i(\Bbbk\langle \Delta \rangle)$ for the support resonance varieties, respectively. Using Proposition [Proposition 16](#prop:aah-hochster){reference-type="ref" reference="prop:aah-hochster"}, a precise description of the varieties $\mathcal{R}^i(\Delta)$ was given in [@PS-adv09 Theorem 3.8], as follows. **Proposition 23**. *For each $i\ge 1$, the decomposition in irreducible components of the jump resonance variety is given by $$\label{eq:res-delta} \mathcal{R}^i(\Delta)= \bigcup_{\stackrel{\mathsf{V}' \subseteq \mathsf{V}\, \mathrm{maximal\: such\: that}}% {\exists \sigma\in \Delta_{\mathsf{V}\setminus \mathsf{V}'}, \ \widetilde{H}_{i-1-\lvert\sigma\rvert} (\operatorname{lk}_{\Delta_{\mathsf{V}'}}(\sigma);\Bbbk) \ne 0}} \Bbbk^{\mathsf{V}'}.$$* Here $\Bbbk^{\mathsf{V}'}$ denotes the coordinate subspace of $\Bbbk^{\mathsf{V}}=\Bbbk^n$ (where $n=\lvert\mathsf{V}\rvert$) spanned by the vectors $\{\mathbf{e}_i \mid i\in \mathsf{V}'\}$. On the other hand, for the support resonance defined in ([\[eq:ria-spec\]](#eq:ria-spec){reference-type="ref" reference="eq:ria-spec"}), the situation is different in degrees $i>1$. **Theorem 24**. *For each $i\ge 1$, the scheme structure on the support resonance locus $\mathcal{R}_i(\Delta)$ is reduced. Moreover, the decomposition in irreducible components is given by $$\label{eq:res-delta2} \mathcal{R}_i(\Delta)=\bigcup_{\substack{ \mathsf{V'}\subseteq \mathsf{V}\, \mathrm{ maximal\: with }\\ \widetilde{H}^{i-1}(\Delta_{\mathsf{V'}};\Bbbk)\neq 0 }}\Bbbk^{\mathsf{V'}}.$$* *Proof.* The first claim follows from Proposition [Proposition 17](#prop:koszul-sqf){reference-type="ref" reference="prop:koszul-sqf"} and Theorem [Theorem 13](#thm:annihilator){reference-type="ref" reference="thm:annihilator"}. The precise structure of the decomposition in irreducible components is governed by the multi-graded structure detailed in Theorem [Theorem 21](#thm:hilb-wd){reference-type="ref" reference="thm:hilb-wd"} and Proposition [Proposition 14](#prop:sfq-support){reference-type="ref" reference="prop:sfq-support"}. Observe that ([\[eq:res-delta2\]](#eq:res-delta2){reference-type="ref" reference="eq:res-delta2"}) corresponds to the primary decomposition of the ideal $\operatorname{Ann}(W_i(\Delta))$. ◻ Notice the difference at the set level between () and (); in particular, observe that the support resonance loci are easier to describe. Furthermore, whereas Theorem [Theorem 24](#prop:irred-comps-support){reference-type="ref" reference="prop:irred-comps-support"} guarantees that the support resonance schemes $\mathcal{R}_i(\Delta)$ are always reduced, the corresponding jump resonance loci $\mathcal{R}^i(\Delta)$ are not necessarily reduced (with the Fitting scheme structure), even in weight one, as the following example illustrates. **Example 25**. Let $\Gamma$ be a path on $4$ vertices. Then $\operatorname{Fitt}_0(W_1(\Gamma))=(x_2)\cap (x_3) \cap (x_1,x_2^2,x_3^2,x_4)$ is not reduced, although $\operatorname{Ann}(W_1(\Gamma))=(x_2)\cap (x_3)$ is reduced. Therefore, the Fitting scheme structure on $\mathcal{R}^1(\Gamma)$ has an embedded component at $0$. A simplicial complex $\Delta$ of dimension $d$ is said to be a *Cohen--Macaulay complex* over $\Bbbk$ if $\widetilde{H}^{\bullet}(\operatorname{lk}(\sigma);\Bbbk)$ is concentrated in degree $d -\lvert\sigma\rvert$, for all $\sigma\in \Delta$. As shown in [@DSY], the jump resonance varieties of such a simplicial complex *propagate*; that is, $$\label{eq:propagate} \mathcal{R}^1(\Delta) \subseteq \mathcal{R}^2(\Delta) \subseteq \cdots \subseteq \mathcal{R}^{d+1}(\Delta).$$ For arbitrary simplicial complexes, though, the resonance varieties do not always propagate. This phenomenon, first identified in [@PS-adv09], does happen even for graphs. **Example 26** ([@PS-adv09]). Let $\Delta$ be the disjoint union of two edges. Then $\mathcal{R}^1(\Delta)=\Bbbk^4$, whereas $\mathcal{R}^2(\Delta)=\Bbbk^2 \cup \Bbbk^2$, the union of two transversal coordinate planes. Thus, $\mathcal{R}^1(\Delta) \not\subseteq \mathcal{R}^2(\Delta)$. When $\Delta$ is Cohen--Macaulay, propagation and formula [\[eq:union-res\]](#eq:union-res){reference-type="eqref" reference="eq:union-res"} give $\mathcal{R}^i(\Delta)=\bigcup_{j\le i} \mathcal{R}_j(\Delta)$. But it is not known whether the support resonance varieties $\mathcal{R}_i(\Delta)$ propagate when $\Delta$ is Cohen--Macaulay, or, equivalently, whether $\mathcal{R}^i(\Delta)=\mathcal{R}_i(\Delta)$ in this case. In general, though, we can use the previous example to settle the latter question in the negative. **Example 27**. Let $\Delta$ be the disjoint union of two edges. Then $\mathcal{R}_1(\Delta)=\mathcal{R}^1(\Delta)=\Bbbk^4$ but $\mathcal{R}_2(\Delta)=\emptyset$ whereas, as we saw before, $\mathcal{R}^2(\Delta)=\Bbbk^2 \cup \Bbbk^2$. Thus, $\mathcal{R}_2(\Delta)\ne \mathcal{R}^2(\Delta)$. # Regularity and projective dimension for Koszul modules of simplicial complexes {#section:CMreg} ## General bounds {#subsec:reg} We start this section with an upper bound on the Castelnuovo--Mumford regularity and projective dimension of the Koszul modules. **Proposition 28**. *For every simplicial complex $\Delta$ on $n$ vertices and every $i>0$, the Koszul module $W_i(\Delta)$ has regularity at most $n$ and projective dimension at most $n-i-1$.* *Proof.* By definition, the Koszul module $W_i(\Delta)$ is a sub-quotient of the module $Z_i \subseteq \mbox{\normalsize $\bigwedge$}^i V \otimes_{\Bbbk} S$ of $i$-th cycles in the Koszul complex of $x_1, \ldots, x_n$. Since $Z_i$ is generated in degree $i+1$, it follows that the degree of any of the generators of $W_i(\Delta)$ is at least $i+1$. Let $\mathbf{F}_\bullet$ denote the minimal free resolution of $W_i(\Delta)$. By Proposition [Proposition 17](#prop:koszul-sqf){reference-type="ref" reference="prop:koszul-sqf"} and Corollary [Corollary 11](#cor:sqf-res){reference-type="ref" reference="cor:sqf-res"}, $\mathbf{F}_\bullet$ is a complex of $\mathbb{N}^n$-graded $S$-modules generated in square-free multidegrees, hence, the total degree of the generators of each $\mathbf{F}_h$ is at most $n$. The statement on the regularity follows immediately. Since the least degree of the generators of $\mathbf{F}_{h+1}$ is strictly larger than the least degree of the generators of $\mathbf{F}_{h}$, it follows that $\mathbf{F}_{h} = 0$ for $h > n-i-1$. ◻ ## Regularity of Koszul modules for simplicial complexes of special type {#sect:fixed-degree} We fix integers $n\ge 4$ and $1\le d\le n-3$ and assume $\Delta$ is a simplicial complex of dimension $d$ on $n$ vertices whose $(d-1)$-skeleton coincides with that of the full simplex, that is, $$\label{eq:delta-skeleton} \Delta^{(d-1)}=\big(2^{[n]}\big)^{(d-1)}.$$ For instance, if $d=1$, then $\Delta$ is simply a (simplicial) graph on $n$ vertices. If $d=2$, then $\Delta$ is obtained from the complete graph on $n$ vertices by filling in some triangles. For this type of simplicial complexes that generalize graphs, the nature of the Koszul modules can be made more precise, as follows. **Proposition 29**. *For a simplicial complex $\Delta$ as above, the following hold.* 1. *[\[pr1\]]{#pr1 label="pr1"} $W_i(A)=0$ for $i\notin \{d,d+1\}$.* 2. *[\[pr2\]]{#pr2 label="pr2"} $W_d(A)=\operatorname{coker}\big(\partial ^E_{d+2}+j_{d+1}\big)$.* 3. *[\[pr3\]]{#pr3 label="pr3"} $W_{d+1}(A)=\ker\big(\partial ^A_{d+1}\big)$, and hence it is either zero or torsion-free.* Using the explicit presentation of $W_d(\Delta)$ from part [\[pr2\]](#pr2){reference-type="eqref" reference="pr2"}, we can improve the bound on regularity from Proposition [Proposition 28](#prop:reg-pdim){reference-type="ref" reference="prop:reg-pdim"}. **Proposition 30**. *With notation as above, we have $\operatorname{reg}W_d(\Delta) \leq n-2$.* *Proof.* We have a presentation $$\label{eq:pres} \begin{tikzcd}[column sep=16pt] 0 \ar[r]& D \ar[r]& Z_d \ar[r]& W_d(\Delta) \ar[r]& 0 , \end{tikzcd}$$ where $Z_d\subseteq \mbox{\normalsize $\bigwedge$}^dV \otimes_{\Bbbk} S$ is the module of Koszul $d$-cycles, and $D$ is the image of $K \otimes_{\Bbbk} S$ under the Koszul differential. Both $Z_d$ and $D$ are generated in degree $d+1$. The module $Z_d$ has a linear free resolution, consisting of the truncated Koszul complex, so $\operatorname{reg}Z_d =d+1$. Since the module $D$ is square-free, its syzygy modules are also square-free, and hence, they are generated in degrees at most $n$. This implies that $\operatorname{reg}D \leq n-1$, since $D$ does not have generators of degree $n$. Applying the long exact sequence of $\operatorname{Tor}(-,\Bbbk)$ to [\[eq:pres\]](#eq:pres){reference-type="eqref" reference="eq:pres"}, we obtain $$\label{eq:reg-wb} \operatorname{reg}W_d(\Delta) \leq \max(\operatorname{reg}Z_d, \operatorname{reg}D-1)= \max(d+1,n-2) = n-2.$$ and this completes the proof. ◻ If $d\le 1$, that is, if $\Delta=\Gamma$ is a graph on $n$ vertices, taking into account the degree shift, we obtain the bound $$\label{eq:reg-graph} \operatorname{reg}W_\Gamma \le n-4.$$ **Example 31**. If $\Gamma=\mathcal{C}_n$ is the cycle on $n\ge 4$ vertices, then the regularity of $W_\Gamma$ attains the above bound: $$\operatorname{reg}W_\Gamma =n-4 \quad \text{and} \quad \operatorname{pdim}W_\Gamma= n-2.$$ This follows from [\[eq:pres\]](#eq:pres){reference-type="eqref" reference="eq:pres"}, since in this case the module $D$ has only one syzygy, of degree $n$. **Remark 32**. For the Koszul module $W_d(\Delta)$, the simplified presentation [\[eqn:simple-presentation\]](#eqn:simple-presentation){reference-type="eqref" reference="eqn:simple-presentation"} has the following nice interpretation. Let $\widetilde{\Delta}$ be the maximal simplicial complex with the property that $\Delta^{(d)}=\widetilde{\Delta}^{(d)}$. Denote by $F_i$ the set of $i$-dimensional missing faces of $\widetilde{\Delta}$. Then we have an exact sequence, $$\label{eq:spn-f} \begin{tikzcd}[column sep=16pt] \operatorname{Span}(F_{d+2})\otimes_{\Bbbk} S\ar[r]& \operatorname{Span}(F_{d+1})\otimes_{\Bbbk} S\ar[r]& W_d(\Delta)\ar[r]& 0. \end{tikzcd}$$ Using Proposition [Proposition 29](#prop:koszul-d-complex){reference-type="ref" reference="prop:koszul-d-complex"}, together with formula [\[eq:res-delta\]](#eq:res-delta){reference-type="eqref" reference="eq:res-delta"}, we obtain the following immediate corollary. **Corollary 33**. *With $\Delta$ as above, we have:* 1. *$\mathcal{R}_i(\Delta)=\mathcal{R}^i(\Delta)$ for all $i\ne d+1$.* 2. *$\mathcal{R}_{d+1}(\Delta)$ is equal to either $\emptyset$ or $\Bbbk^n$.* 3. *$\mathcal{R}^d(\Delta)= \bigcup_{\substack{\mathsf{V'}\subseteq \mathsf{V}\, \mathrm{maximal}\\[0.5pt] {\widetilde{H}_{d-1}(\Delta_{\mathsf{V'}};\Bbbk)\ne 0}}} \Bbbk^\mathsf{V'}$.* **Example 34**. Let $\Delta$ be the boundary of the tetrahedron, with the face $\sigma=\{1,2,3\}$ missing. Then $\Delta^{(1)}=\big(2^{[4]}\big)^{(1)}$, and so $\Delta$ is a simplicial complex covered by the above corollary, with $d=2$. In this case, we have that $\mathcal{R}_d(\Delta)=\{x_4=0\}$, since $H_1(\Delta_{\sigma};\Bbbk)=\Bbbk$, and $\mathcal{R}^d(\Delta)=\{x_4=0\}$, since $\widetilde{H}_{2-1-1}(\operatorname{lk}_{\Delta_{\sigma}}(\{4\});\Bbbk)=\widetilde{H}_{0}(\emptyset;\Bbbk)=\Bbbk$. As already mentioned before, the loci $\mathcal{R}_{d+1}(\Delta)$ and $\mathcal{R}^{d+1}(\Delta)$ can be different, in general. For example, if we take the graph $\Gamma$ on four vertices with edges $(1,2)$ and $(3,4)$ as in Example [Example 26](#ex:not-prop){reference-type="ref" reference="ex:not-prop"}, then $\mathcal{R}_2(\Gamma)=\emptyset$ while $\mathcal{R}^2(\Gamma)=V(x_1,x_2)\cup V(x_3,x_4)$. 10 Marian Aprodu, Gavril Farkas, Ştefan Papadima, Claudiu Raicu, and Jerzy Weyman, [*Koszul modules and Green's Conjecture*](https://doi.org/10.1007/s00222-019-00894-1), Invent. Math. **218** (2019), no. 3, 657--720. [MR4022070](http://www.ams.org/mathscinet-getitem?mr=4022070) Marian Aprodu, Gavril Farkas, Ştefan Papadima, Claudiu Raicu, and Jerzy Weyman, [*Topological invariants of groups and Koszul modules*](https://doi.org/10.1215/00127094-2022-0010), Duke Math. J. **171** (2022), no. 19, 2013--2046. [MR4484204](http://www.ams.org/mathscinet-getitem?mr=4484204) Marian Aprodu, Gavril Farkas, Claudiu Raicu, and Alexander I. Suciu, *Reduced resonance schemes and Chen invariants*, preprint `arXiv:2303.07855`. Annetta Aramova, Luchezar Avramov, and Jürgen Herzog, [*Resolutions of monomial ideals and cohomology over exterior algebras*](https://doi.org/10.1090/S0002-9947-99-02298-9), Trans. Amer. Math. Soc. **352** (2000), no. 2, 579--594. [MR1603874](http://www.ams.org/mathscinet-getitem?mr=1603874) Michael K. Brown and Daniel Erman, *Linear strands of multigraded free resolutions*, preprint `arXiv:2202.00402`. Graham Denham, Alexander I. Suciu, and Sergey Yuzvinsky, [*Abelian duality and propagation of resonance*](http://dx.doi.org/10.1007/s00029-017-0343-5) Selecta Math. **23** (2017), no. 4, 2331--2367. [MR3703455](http://www.ams.org/mathscinet-getitem?mr=3703455) David Eisenbud, [*The geometry of syzygies*](https://dx.doi.org/10.1007/b137572), a second course in commutative algebra and algebraic geometry, Grad. Texts in Math., vol. 229, Springer-Verlag, New York, 2005. [MR2103875](http://www.ams.org/mathscinet-getitem?mr=2103875) Jürgen Herzog and Takayuki Hibi, [*Monomial ideals*](https://doi.org/10.1007/978-0-85729-106-6_1), Graduate Texts in Mathematics, vol. 260, Springer, London, 2011. Stefan Papadima and Alexander I. Suciu, [*Algebraic invariants for right-angled Artin groups*](http://doi.org/10.1007/s00208-005-0704-9), Math. Annalen, **334** (2006), no. 3, 533--555. [MR2207874](http://www.ams.org/mathscinet-getitem?mr=2207874) Stefan Papadima and Alexander I. Suciu, [*Toric complexes and Artin kernels*](https://dx.doi.org/10.1016/j.aim.2008.09.008), Adv. Math. **220** (2009), no. 2, 441--477. [MR2466422](http://www.ams.org/mathscinet-getitem?mr=2466422) Stefan Papadima and Alexander I. Suciu, [*Jump loci in the equivariant spectral sequence*](http://dx.doi.org/10.4310/MRL.2014.v21.n4.a13), Math. Res. Lett. **21** (2014), no. 4, 863--883. [MR3275650](http://www.ams.org/mathscinet-getitem?mr=3275650) Stefan Papadima and Alexander I. Suciu, [*Vanishing resonance and representations of Lie algebras*](https://dx.doi.org/10.1515/crelle-2013-0073), J. Reine Angew. Math. **706** (2015), 83--101. [MR3393364](http://www.ams.org/mathscinet-getitem?mr=3393364) Alexander I. Suciu, [*Around the tangent cone theorem*](http://dx.doi.org/10.1007/978-3-319-31580-5_1), in: *Configuration Spaces: Geometry, Topology and Representation Theory*, 1--39, Springer INdAM series, vol. 14, Springer, Cham, 2016. [MR3615726](http://www.ams.org/mathscinet-getitem?mr=3615726) Kohji Yanagawa, [*Alexander duality for Stanley--Reisner rings and squarefree $\mathbf{N}^n$-graded modules*](https://doi.org/10.1006/jabr.1999.8130), J. Algebra **225** (2000), no. 2, 630--645. [MR1741555](http://www.ams.org/mathscinet-getitem?mr=1741555) [^1]: We denote by $\mathcal{R}^i(A)$ what in [@PS-mrl] is denoted by $\mathcal{R}^i_1(A)=\mathcal{V}^i_1(A^{\bullet} \otimes_{\Bbbk} S )$ and in [@Su-indam] by $\mathcal{R}^i(A)$, whereas we use the notation $\mathcal{R}_i(A)$ for what in [@PS-mrl] is denoted by $\mathcal{W}^i_1(A)=\operatorname{Supp}H_i(A_{\bullet} \otimes_{\Bbbk} S)$ and in [@Su-indam] by $\widetilde{\mathcal{R}}_i(A)$.

arxiv_math

--- abstract: | We prove that the unit sphere is the only smooth, strictly convex solution to the isotropic $L_p$ dual Minkowski problem $$\begin{aligned} h^{p-1} |D h|^{n+1-q}\mathcal{K}=1, \end{aligned}$$ provided $(p,q)\in (-n-1,-1]\times [n,n+1)$. author: - Yingxiang Hu, Mohammad N. Ivaki title: On the uniqueness of solutions to the isotropic $L_{p}$ dual Minkowski problem --- # Introduction An important question in convex geometry is the uniqueness or the non-uniqueness of the origin-centred spheres as solutions to the isotropic $L_{p}$ dual Minkowski problem $$\begin{aligned} \label{Lpq-Minkowski-problem} h^{p-1} |D h|^{n+1-q} \mathcal{K}=c, \quad c\in (0,\infty). \end{aligned}$$ The $L_p$ dual Minkowski problem was first introduced by Lutwak, Yang and Zhang [@LYZ18], acting as a bridge which connects the $L_p$-Minkowski problem to the dual Minkowski problem. The former, the $L_p$-Minkowski problem, was introduced by Lutwak in his influential paper [@Lut93] three decades ago, and since then has been extensively investigated; e.g., [@LO95; @LYZ04; @Sta02; @Sta03; @CW06; @BLYZ12; @BLYZ13; @JLW15; @Zh15; @HLW16; @BIS19; @BBCY19; @Li19; @CHLL20; @Mil21; @KM22; @IM23b] and [@Sch14]. The latter, the dual Minkowski problem, was proposed recently by Huang et. al in [@HLYZ16], and further studied in [@Zh17; @Zh18; @BHP18; @CL18; @HP18; @HJ19; @LSW20]. There has been significant progress on the $L_p$ dual Minkowski problem after the paper [@LYZ18] such as [@HZ18; @BF19; @CHZ19; @CL21; @LLL22]; however, the complete answer to the uniqueness and non-uniqueness question, as stated above, has been elusive in the most interesting case: without the *origin-symmetry* assumption. Here are the known uniqueness and non-uniqueness results for the isotropic $L_{p}$ dual Minkowski problem: - [@BCD17], uniqueness of solutions for $-(n+1)\leq p< 1$ and $q=n+1$ (see also [@And99; @And03; @Sar22]); - [@HZ18], uniqueness of solutions for $p>q$; - [@CHZ19], uniqueness of origin-symmetric solutions for $$-(n+1)\leq p < q\leq \min\{n+1,n+1+p\};$$ - [@CL21], uniqueness of solutions for $1<p<q\leq n+1$, or $-(n+1)\leq p<q<-1$, or the uniqueness of solutions up to rescaling for $p=q$; - [@LW22], complete classification for $n=1$; - [@CCL21], non-uniqueness of solutions under any of the following assumptions: (i) $q-2(n+1) >p\geq 0$; (ii) $q>0$ and $-q^\ast <p <\min\{ 0,q-2n-2\}$, where $$q^\ast:=\left\{\begin{aligned}&\frac{q}{q-n}, \quad \text{if $q\geq n+1$}\\ &\frac{nq}{q-1}, \quad \text{if $1<q<n+1$}\\ &+\infty, \quad \text{if $0<q\leq 1;$} \end{aligned}\right.$$ (iii) $p+2(n+1) <q\leq 0$. In the recent work [@IM23a], employing the local Brunn-Minkowski inequality, the following uniqueness was proved. **Theorem 1**. *Let $n\geq 2$ and assume $-(n+1)\leq p$ and $q\leq n+1$, with at least one being strict. Suppose $\mathcal{M}^n$ is a smooth, strictly convex, origin-centred hypersurface such that $h^{p-1}|Dh|^{n+1-q}\mathcal{K}=c$ with $c>0$. Then $\mathcal{M}^n$ is an origin-centred sphere.* Here, we also employ the local Brunn-Minkowski inequality as our main tool to establish the following uniqueness result. [\[s1:cor-Lp-dual-Minkowski\]]{#s1:cor-Lp-dual-Minkowski label="s1:cor-Lp-dual-Minkowski"} Let $n\geq 2$. Suppose $\mathcal{M}^n$ is a smooth, strictly convex hypersurface with $h>0$, such that $h^{p-1}|Dh|^{n+1-q}\mathcal{K}=1$. Suppose either 1. $-(n+1)< p\leq -1$ and $n\leq q\leq n+1$, 2. or $-(n+1)\leq p\leq -n$ and $1\leq q< n+1$. Then $\mathcal{M}^n$ is the unit sphere. # Background ## Convex geometry Let $(\mathbb R^{n+1},\delta:=\langle\, ,\rangle,D)$ denote the Euclidean space with its standard inner product and flat connection, and let $(\mathbb S^n,\bar{g},\bar{\nabla})$ denote the unit sphere equipped with its standard round metric and Levi-Civita connection. Suppose $K$ is a smooth, strictly convex body in $\mathbb R^{n+1}$ with the origin in its interior. Write $\mathcal{M}=\mathcal{M}^n=\partial K$ for the boundary of $K$. The Gauss map of $\mathcal{M}$, denoted by $\nu$, takes the point $p\in \mathcal{M}$ to its unique unit outward normal $x=\nu(p)\in \mathbb S^n$. The support function of $K$ is defined by $$\begin{aligned} h(x):=\max\{\langle x,y\rangle:~y\in K\}, \quad x\in \mathbb S^n. \end{aligned}$$ The inverse Gauss map $X=\nu^{-1}:\mathbb S^n\rightarrow\mathcal{M}$ is given by $$\begin{aligned} X(x)=Dh(x)=\bar{\nabla} h(x)+h(x)x, \quad x\in \mathbb S^{n}. \end{aligned}$$ The support function can also be expressed as $$\begin{aligned} h(x)=\langle X(x),x\rangle=\langle \nu^{-1}(x),x\rangle, \quad x\in \mathbb S^n. \end{aligned}$$ The radial function of $K$ is defined by $$\begin{aligned} r(x):=|X(x)|=(|\bar{\nabla} h(x)|^2+h^2(x))^\frac{1}{2}. \end{aligned}$$ Moreover, the Gauss curvature of $\mathcal{M}$ is defined by $$\begin{aligned} \frac{1}{\mathcal{K}(x)}:=\left.\frac{\det(\bar{\nabla}^2 h+\bar{g}h)}{\det(\bar{g})}\right|_x, \quad x\in \mathbb S^n. \end{aligned}$$ Note that the matrix $A[h]:=\bar{\nabla}^2 h+h\bar{g}=D^2 h|_{T\mathbb S^n}$ is positive-definite. The eigenvalues of the matrix $A[h]$ with respect to the metric $\bar{g}$, denoted by $\lambda=(\lambda_1,\ldots,\lambda_n)$, are the principal radii of curvature at the point $X(x)\in \mathcal{M}$. Then $\sigma_n=\mathcal{K}^{-1}=\Pi_i \lambda_i$. The curvature equation [\[Lpq-Minkowski-problem\]](#Lpq-Minkowski-problem){reference-type="eqref" reference="Lpq-Minkowski-problem"} can be reformulated as the following Monge-Ampére equation: $$\begin{alignedat}{2} h^{1-p} |D h|^{q-n-1} \det(\bar{\nabla}^2 h+\bar{g} h)=c. \end{alignedat}$$ The polar body of $K$ is defined by $$\begin{aligned} K^\ast:=\{y\in \mathbb R^{n+1}: \langle x,y\rangle\leq 1\, ~\forall x\in K\}. \end{aligned}$$ It is well-known that $K^\ast$ is also a smooth, strictly convex body $K^\ast$ in $\mathbb R^{n+1}$ with the origin in its interior. Moreover, the following identity holds $$\label{polar-dual} \frac{h^{n+2}(x) (h^\ast(x^{\ast}))^{n+2}}{\mathcal{K}(x)\mathcal{K}^\ast(x^\ast)}=1.$$ Here $h^{\ast}$ and $\mathcal{K}^{\ast}$ denote respectively the support function and Gauss curvature of $K^{\ast}$, and $x^{\ast}:=X(x)/|X(x)|$. Finally, let us introduce the measure $dV:=h\sigma_nd\mu$, where $\mu$ is the spherical Lebesgue measure of the unit sphere $\mathbb S^n$. Then the measure $\sigma_n d\mu$ is the surface-area measure of $K$, and $dV=h\sigma_n d\mu$ is a constant multiple of the cone-volume measure of $K$. We refer to [@Sch14] for additional background. ## Centro-affine geometry In this section, we recall some basics from centro-affine geometry. For the related concepts, we refer the reader to [@LHSZ15; @NS94] and, in particular, to the excellent paper by Milman [@Mil21]. Let $X: \mathbb{S}^n\to \mathcal{M}$ be a smooth embedding of $\mathcal{M}$ (which we consider it to be $Dh$ as in the previous section), and consider the transversal normal field $\xi(x):=X(x)$ (the centro-affine normal). The transversal vector $\xi$ induces the volume form $V$ (as in the previous section), a connection $\nabla$, as well as a metric $g^{\xi}$ on $\mathbb{S}^n$ as follows: $$\begin{aligned} V(e_1,\ldots, e_{n})=\det (dX(e_1),\ldots, dX(e_{n}),\xi),\quad e_i\in T\mathbb{S}^n, \end{aligned}$$ $$\begin{alignedat}{2} \label{nabla structure} D_udX(v)=dX(\nabla_uv)-g^{\xi}(u,v)\xi,\quad u,v\in T\mathbb{S}^n. \end{alignedat}$$ Note that $g^{\xi}$ is symmetric and positive-definite. Moreover, while $\nabla$ is not the Levi-Civita connection of $g$, it is torsion-free and $$\begin{alignedat}{2} \label{integration by parts} \nabla V\equiv 0. \end{alignedat}$$ The conormal field $\xi^\ast: \mathbb{S}^n\to (\mathbb{R}^{n+1})^{\ast}\sim \mathbb{R}^{n+1}$ is the unique smooth vector field to the dual space of $\mathbb{R}^{n+1}$, such that $\langle \xi^\ast,dX\rangle=0$ and $\langle \xi, \xi^\ast\rangle=1$. Moreover, $\xi^\ast$ is an immersion and transversal to its image, and it induces a bilinear form and a torsion-free connection on $\mathbb{S}^n,$ $$\begin{alignedat}{2} D_ud\xi^\ast(v)=d\xi^{\ast}(\nabla_u^{\ast}v)-g^{\xi^\ast}(u,v)\xi^\ast ,\quad u,v\in T\mathbb{S}^n. \end{alignedat}$$ We furnish all geometric quantities associated with $\xi^{\ast}$ with $\ast$. It is known that $g^{\xi}=g^{\xi^{\ast}}$ and that the two connections $\nabla^{\ast}$ and $\nabla$ are conjugate with respect to $g^{\xi}$: $$\begin{alignedat}{2} ug^{\xi}(v_1,v_2)=g^{\xi}(\nabla_uv_1,v_2)+g^{\xi}(v_1,\nabla^{\ast}_uv_2) \quad u,v_1,v_2\in T\mathbb{S}^n. \end{alignedat}$$ Moreover, by [@Mil21 Proposition 4.2] (or taking the inner production of [\[nabla structure\]](#nabla structure){reference-type="eqref" reference="nabla structure"} with $\nu$), we find $$\begin{alignedat}{2} g^{\xi}=g^{\xi^{\ast}}=\frac{A[h]}{h}:=g. \end{alignedat}$$ For a smooth function $f:\mathbb{S}^n\to \mathbb{R}$, the Hessian and Laplacian with respect to $(\nabla,g)$ are defined as $$\operatorname{Hess}f(u,v)=\nabla df(u,v)=v(uf)-df(\nabla_vu)$$ and $\Delta f=\operatorname{div}_g(\nabla f)=\sum_{i} g(\nabla_{e_i}\nabla f,e_i),$ where $\{e_i\}_{i=1}^n$ is a local $g$-orthonormal frame of $T\mathbb{S}^n$. We write $\operatorname{Hess}^\ast$ and $\Delta^{\ast}$ respectively for the Hessian and Laplacian with respect to $(\nabla^{\ast},g)$. Since $\nabla,\nabla^{\ast}$ are conjugate, we have $$\begin{aligned} v(uf)=vg(\nabla f,u)=g(\nabla_v\nabla f,u) +df(\nabla^{\ast}_vu).\end{aligned}$$ Therefore, we obtain $$\begin{alignedat}{2} \Delta f=\operatorname{tr}_g\operatorname{Hess}^{\ast}f,\quad \Delta^{\ast} f=\operatorname{tr}_g\operatorname{Hess}f. \end{alignedat}$$ By [@Mil21 Proposition 4.2], we have $$\begin{alignedat}{2} \operatorname{Hess}^{\ast}f+gf=\frac{1}{h}\left(\bar{\nabla}^2(hf)+\bar{g}hf\right)=\frac{A[hf]}{h}. \end{alignedat}$$ Let us define $$\begin{alignedat}{2} Q(u,v)=\nabla^{\ast}_vu-\nabla_v u \quad \forall u,v \in T\mathbb{S}^n. \end{alignedat}$$ Then by [@LHSZ15 (6.2)], $$\begin{alignedat}{2} \operatorname{tr}_{g}Q=-\nabla\log\left(\frac{h^{n+2}}{\mathcal{K}}\right). \end{alignedat}$$ In particular, we have $$\begin{alignedat}{2} \label{change of laplacian} (\Delta- \Delta^{\ast})f=-\sum_iQ(e_i,e_i)f=d\log \frac{h^{n+2}}{\mathcal{K}}(\nabla f). \end{alignedat}$$ We conclude this section by recalling the local Brunn-Minkowski inequality, reformulated in the language of centro-affine geometry (cf. [@Mil21]): Let $f\in C^1(\mathbb{S}^n)$. Then $$\begin{alignedat}{2} \label{local BM} n\int f^2 dV\leq \int |\nabla f|_g^2 dV+n\frac{(\int fdV)^2}{\int dV}. \end{alignedat}$$ The equality holds if and only if for some $w\in \mathbb{R}^{n+1}$ $$\begin{alignedat}{2} f(x)=\langle \frac{x}{h(x)},w\rangle\quad \forall x\in \mathbb{S}^n. \end{alignedat}$$ Moreover, by [@Mil21 (5.9)] we also have $$\begin{alignedat}{2} \label{local BM 2} n\int |\nabla f|_g^2 dV\leq \int (\Delta f)^2 dV \quad \forall f\in C^2(\mathbb{S}^n). \end{alignedat}$$ # Uniqueness The following identity is at the heart of our approach to employing the local Brunn-Minkowski inequality. [\[main identity\]]{#main identity label="main identity"} There holds $$\begin{alignedat}{2} \label{main inq} \Delta X+nX=h\bar{\nabla}\log \frac{h^{n+2}}{\mathcal{K}}. \end{alignedat}$$ In particular, $$\begin{alignedat}{2} n\int X dV= \int h\bar{\nabla}\log \frac{h^{n+2}}{\mathcal{K}} dV. \end{alignedat}$$ *Proof.* Let $w\in \mathbb{R}^{n+1}$ be a fixed vector. By the centro-affine Gauss equation for $\xi=X$ (cf. [@Mil21 Section 3.8]), we have $$\begin{alignedat}{2} \Delta^{\ast}\langle X,w\rangle+n\langle X,w\rangle=0. \end{alignedat}$$ Now let $\{v_i\}_{i=1}^n$ be a local orthonormal frame of $T\mathbb{S}^n$ that diagonalizes $A[h]$ at $x_0$ and $A[h]|_{x_0}(v_i,v_j)=\delta_{ij}\lambda_i$. Define $e_i=\sqrt{\frac{h}{\lambda_i}}v_i$, $i=1,\ldots, n$. Then we have $g|_{x_0}(e_i,e_j)=\delta_{ij}$. Hence, by [\[change of laplacian\]](#change of laplacian){reference-type="eqref" reference="change of laplacian"}, at $x_0$ we have $$\begin{alignedat}{2} \Delta \langle X,w\rangle+n\langle X,w\rangle&=(\Delta -\Delta^{\ast}) \langle X,w\rangle\\ &= g( \nabla\log \frac{h^{n+2}}{\mathcal{K}},\nabla \langle X,w\rangle)\\ &= g( \nabla\log \frac{h^{n+2}}{\mathcal{K}},\lambda_i \langle e_i,w\rangle e_i)\\ &=\lambda_i \langle e_i,w\rangle d\log \frac{h^{n+2}}{\mathcal{K}}(e_i)\\ &= \langle \bar{\nabla}\log \frac{h^{n+2}}{\mathcal{K}},hw\rangle. \end{alignedat}$$ The second identity follows from integrating [\[main inq\]](#main inq){reference-type="eqref" reference="main inq"} against $dV$. ◻ [\[ineq local BM - a\]]{#ineq local BM - a label="ineq local BM - a"} Let $0<f\in C^2(\mathbb{S}^n)$. Then $$\begin{alignedat}{2} \int f^2\left(\langle \bar{\nabla}\log \frac{h^{n+2}}{\mathcal{K}}, hX\rangle-|X|^2|\nabla\log f|_g^2\right) dV\leq n\frac{|\int fXdV|^2}{\int dV}. \end{alignedat}$$ *Proof.* Let $\{E_k\}_{k=1}^{n+1}$ be an orthonormal basis of $\mathbb{R}^{n+1}$. We define $$\begin{alignedat}{2} f_k=f\langle X,E_k\rangle\quad k=1,\ldots,n+1. \end{alignedat}$$ In view of , we have $$\begin{aligned} \Delta f_k+nf_k=&f\langle \bar{\nabla}\log \frac{h^{n+2}}{\mathcal{K}}, hE_k\rangle+\langle X,E_k\rangle\Delta f+2g(\nabla f,\nabla \langle X,E_k\rangle).\end{aligned}$$ Therefore, $$\begin{aligned} \label{fk local BM} \sum_kf_k(\Delta f_k+nf_k)=&f^2\langle \bar{\nabla}\log \frac{h^{n+2}}{\mathcal{K}}, hX\rangle+f|X|^2\Delta f\\ &+fg(\nabla f,\nabla |X|^2).\end{aligned}$$ Moreover, by integration by parts (cf. [\[integration by parts\]](#integration by parts){reference-type="eqref" reference="integration by parts"}), there holds $$\begin{alignedat}{2} \label{integration by parts-} \int |X|^2f\Delta f+fg(\nabla f, \nabla |X|^2)dV=-\int |X|^2|\nabla f|_g^2 dV. \end{alignedat}$$ By the local Brunn-Minkowski inequality (see [\[local BM\]](#local BM){reference-type="eqref" reference="local BM"}), we have $$\begin{alignedat}{2} \sum_k\int f_k(\Delta f_k+nf_k)dV\leq n \sum_k\frac{\langle\int fXdV,E_k\rangle^2 }{\int dV}. \end{alignedat}$$ Thus the claim follows from [\[fk local BM\]](#fk local BM){reference-type="eqref" reference="fk local BM"} and [\[integration by parts-\]](#integration by parts-){reference-type="eqref" reference="integration by parts-"}. ◻ [\[ineq local BM - b\]]{#ineq local BM - b label="ineq local BM - b"} Suppose $\varphi: (0,\infty)\to (0,\infty)$ is $C^1$-smooth and $f=\varphi(r)$. Then we have $$\begin{alignedat}{2} \int f^2\langle \bar{\nabla}\log \frac{h^{n+2}}{\mathcal{K}}-(r(\log \varphi)')^2\bar{\nabla}\log r, hX\rangle dV\leq n\frac{|\int fXdV|^2}{\int dV}. \end{alignedat}$$ *Proof.* Let $\{v_i\}_{i=1}^n$ and $\{e_i\}_{i=1}^n$ be as in the proof of . We calculate $$\begin{alignedat}{2} e_i(\log f)=(\log \varphi)'e_ir= \frac{(\log \varphi)'}{r}\lambda_i\langle e_i,X\rangle=\frac{(\log \varphi)'}{r}\sqrt{h\lambda_i}\langle v_i,X\rangle, \end{alignedat}$$ and $$\begin{alignedat}{2} r^2|\nabla \log f|_g^2=((\log \varphi)')^2h\lambda_i(v_ih)^2 = (r(\log \varphi)')^2\langle \bar{\nabla}\log r, hX\rangle. \end{alignedat}$$ Now the inequality follows from . ◻ *Proof of .* Let $\alpha=q-n-1$. Due to with $\varphi(r)=r^{q-n-1}$, and our assumption $h^{n+2}\mathcal{K}^{-1}=h^{n+1+p}r^{n+1-q},$ we obtain $$\begin{alignedat}{2} &(n+1+p)\int r^{2\alpha}|\bar{\nabla}h|^2dV\\ \leq~& \alpha(\alpha+1)\int r^{2\alpha} \langle \bar{\nabla}\log r,h\bar{\nabla}h\rangle dV+ n\frac{|\int r^{\alpha}XdV|^2}{\int dV}. \end{alignedat}$$ Assuming $\alpha^2+\alpha\leq 0$ (i.e. $n\leq q\leq n+1$) we obtain $$\begin{alignedat}{2} (n+1+p)\int r^{2\alpha}|\bar{\nabla}h|^2dV\leq n\frac{|\int r^{\alpha}XdV|^2}{\int dV}. \end{alignedat}$$ Moreover, by using $\bar{\Delta} x+nx=0,$ $$\begin{alignedat}{2} \int r^{\alpha}XdV=\int Xh^{p}d\mu=\frac{n+1+p}{n}\int r^{\alpha}\bar{\nabla}h dV. \end{alignedat}$$ Hence, due to $n+1+p>0$, $$\begin{alignedat}{2} \int r^{2\alpha}|\bar{\nabla}h|^2dV\leq \frac{n+1+p}{n}\frac{|\int r^{\alpha}\bar{\nabla}hdV|}{\int dV}. \end{alignedat}$$ We may rewrite this inequality as $$\begin{alignedat}{2} \int \left|r^{\alpha}\bar{\nabla}h-\frac{\int r^{\alpha}\bar{\nabla}h dV}{\int dV}\right|^2dV\leq \frac{p+1}{n}\frac{|\int r^{\alpha}\bar{\nabla}h dV|^2}{\int dV}. \end{alignedat}$$ Thus $h$ is constant, provided $-(n+1)<p\leq -1$ and $n\leq q \leq n+1$. In view of [\[polar-dual\]](#polar-dual){reference-type="eqref" reference="polar-dual"}, the polar body $K^\ast$ satisfies the following isotropic $L_{-q}$ dual Minkowski problem: $$\begin{aligned} (h^{\ast})^{-1-q}|Dh^\ast|^{n+1+p}\mathcal{K}^\ast=1. \end{aligned}$$ Hence, the uniqueness result also holds when $n\leq -p\leq (n+1)$ and $-(n+1)< -q\leq -1$. ◻ # Acknowledgment {#acknowledgment .unnumbered} The first author's work was supported by the National Key Research and Development Program of China 2021YFA1001800 and the National Natural Science Foundation of China 12101027. Both authors were supported by the Austrian Science Fund (FWF): Project P36545. 1 B. Andrews, *Monotone quantities and unique limits for evolving convex hypersurfaces*, Int. Math. Res. Not. IMRN. **20**(1997): 1001--1031. B. Andrews, *Gauss curvature flow: the fate of the rolling stones*, Invent. Math. **138**(1999): 151--161. B. Andrews, *Classification of limiting shapes for isotropic curve flows*, J. Amer. 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Stancu, *On the number of solutions to the discrete two-dimensional $L_0$-Minkowski problem*, Adv. Math. **180**(2003): 290--323. Y. Zhao, *The dual Minkowski problem for negative indices*, Calc. Var. Partial Differential Equations, **58**(2017), 18. Y. Zhao, *Existence of solutions to the even dual Minkowski problem*, J. Differ. Geom. **110**(2018): 543--572. G. Zhu, *The centro-affine Minkowski problem for polytopes*, J. Differ. Geom. **101**(2015): 159--174. [School of Mathematical Sciences, Beihang University, Beijing 100191, China]{.smallcaps} [Institut für Diskrete Mathematik und Geometrie,\ Technische Universität Wien, Wiedner Hauptstraße 8-10,\ 1040 Wien, Austria,]{.smallcaps} [Institut für Diskrete Mathematik und Geometrie,\ Technische Universität Wien, Wiedner Hauptstraße 8-10,\ 1040 Wien, Austria,]{.smallcaps}

arxiv_math

--- abstract: | In this paper, we show that the existence of weak solution to the equation $$\begin{aligned} \begin{split} (-\Delta_g)^su(x)&=f(x)u(x)^{-q(x)}\;\mbox{in}\; \Omega,\\ u&>0\;\mbox{in }\; \Omega,\\ u&=0\;\mbox{in}\;\mathbb{R}^N\setminus\Omega \end{split} \end{aligned}$$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $q\in C^1(\overline{\Omega})$, and $(-\Delta_g)^s$ is the fractional $g$-Laplacian with $g$ is the antiderivative of a Young function and $f$ in suitable Orlicz space. This includes the mixed fractional $(p,q)-$Laplacian as a special case. The solution so obtained are also shown to be locally Hölder continuous. address: - $^1$Indian Institute of Technology Kanpur, India - $^{2,3}$University of Jyväskylä, Finland author: - $^1$Kaushik Bal - $^2$Riddhi Mishra - $^3$Kaushik Mohanta bibliography: - bibliography_Orlicz.bib title: "On the singular problem involving $g$-Laplacian" --- . # Introduction Nonlocal problems have been a subject of immense interest in mathematics recently. Various studies have been published to verify if the results of the Laplace operator can be suitably generalized for problems involving fractional Laplacian and its generalization. Continuing with the spirit of recent developments in the study of nonlocal operators, in this article, we consider the following problem $$\label{eq problem} \begin{split} (-\Delta_g)^su(x)&=f(x)u(x)^{-q(x)}\; \mbox{in } \Omega,\\ u&>0\; \mbox{in }\; \Omega,\\ u&=0\; \mbox{in }\; \mathbb{R}^N\setminus\Omega \end{split}$$ with $\Omega$ being a smooth bounded domain in $\mathbb{R}^N$ and $q$ is a non-negative $C^1$ function in $\overline{\Omega}$, and the fractional $g$-Laplacian operator is defined as $$(-\Delta_g)^su(x):=\int_{\mathbb{R}^N}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{dy}{|x-y|^{N+s}}$$ with $g:[0,\infty)\to\mathbb{R}$ is a right continuous function satisfying the following assumptions: 1. $g(0)=0;\;g(t)>0$ for $t>0$ and $\lim_{t\to+\infty}g(t)=\infty$. 2. $g$ is convex on $(0,\infty)$. 3. $g'$ is nondecreasing on $(0,\infty)$, and hence on $\mathbb{R}\setminus \{0\}$. Given $g:\mathbb{R}\to\mathbb{R}$, we define $G:[0,\infty)\to[0,\infty)$, called the N-function or Young's function by $$G(t):=\int_{0}^t g(\tau)d\tau.$$ We also assume the following additional conditions on $G$ and $g$: 1. $g=G'$ is absolutely continuous, so it is differentiable almost everywhere. 2. $\int_{0}^1\frac{G^{-1}(\tau)}{\tau^{\frac{N+s}{N}}}d\tau<\infty$ and $\int_{1}^\infty\frac{G^{-1}(\tau)}{\tau^{\frac{N+s}{N}}}d\tau=\infty$ 3. There exist $p^+,p^-$ such that $$1<p^--1\leq \frac{tg'(t)}{g(t)}\leq p^+-1\leq \infty \quad t>0.$$ Note that we will always be assuming conditions $(H_a)-(H_g)$ on $g \;\mbox{and}\; G$ throughout the whole paper until otherwise specified. In literature, $G$ is known as a Young function or an $N$-function. *Remark 1*. The following examples of $G$ fits our framework: 1. $G_{p}(t):=\frac{1}{p}t^{p}$, where $p\geq 2$. 2. If one takes $G_{p_1,p_2}(t):=\frac{1}{p_1}t^{p_1}+\frac{1}{p_2}t^{p_2}$, where $p_1,p_2\geq 2$. One gets, $$(-\Delta_{g_{p_1,p_2}})^s=(-\Delta_{p_1})^s+(-\Delta_{p_2})^s.$$ 3. For $a,b,c>0$ and $g(t)=t^a\log(b+ct)$ we get, $$G(t)=\frac{t^{1+a}}{(1+a)^2}[H^2_1(1+a,1,2+a,-\frac{ct}{b})+(1+a)\log(b+ct)-1]$$ with $p^-=1+a,\;p^{+}=2+a$, where $H^2_1$ is a hyper geometric function. Before we start with the preliminaries, let us briefly recall some related literature concerning the singular problems. Singular problems have a long history starting from the seminal work of Crandall-Rabinowitz-Tartar [@CrRaTa], where for a suitably regular $f$, the problem $-\Delta u= f(x)u^{-\delta}$ was considered in a bounded domain and is shown to admit a classical solution irrespective of the sign of $\delta>0$, subject to Dirichlet boundary condition. The classical solution so obtained was shown to be the weak solution provided $0<\delta<3$ in another celebrated work of Lazer-Mckenna [@LaMc]. Singularly perturbed problems were also studied in [@Gia1; @Gia2] and the reference therein. The case of $f\in L^p(\Omega),\;p\geq 1$ was first treated in Boccardo-Orsina [@BoOr], who showed the existence and regularity results for different cases of $m$ and $\delta$. One may find the p-Laplace generalization of Boccardo-Orsina's work in Scuinzi et al. [@CaScTr], where the delicate issue of uniqueness was also addressed. Anisotropic Laplacians with singular nonlinearities have also been dealt with in several papers, see [@Ba1; @Ba2; @BaGa] to name a few. In [@BaBeDe], the fractional problem given by $$\begin{aligned} (-\Delta)^su(x)&=\lambda f(x)u(x)^{-\gamma} +Mu^p\;\mbox{in } \Omega,\nonumber\\ u&>0 \;\mbox{in } \Omega,\label{eq problem-0}\\ u&=0 \;\mbox{in } \mathbb{R}^N\setminus\Omega,\nonumber \end{aligned}$$ was first considered under the condition that $n >2s,M\geq 0, 0<s<1, 1<p<2_s^{*}-1$ and shown to admit a distributional solution for $f\in L^m(\Omega)$ and $\lambda>0$ small. In [@CaMoScSq], the authors studied the problem $$\begin{aligned} (-\Delta_p)^su(x)&=f(x)u(x)^{-\gamma} \;\mbox{in } \Omega,\nonumber\\ u&>0 \;\mbox{in } \Omega,\label{eq problem-2}\\ u&=0 \;\mbox{in } \mathbb{R}^N\setminus\Omega, \nonumber \end{aligned}$$ and proved the existence and uniqueness results. The first instance, to the best of our knowledge of studying variable exponent singularities was in Carmona-Martı́nez-Aparicio [@CaMa] where the rather surprising phenomenon of the restriction of the nonlinearity to $1$ near the boundary of the domain for a weak solution was studied in contrast to the constant exponent where such restriction is imposed on the whole domain. Similar problems involving fractional p-Laplacian with variable exponent may be found in Garain-Tuhina [@GaMu]. However, by considering the Orlicz setup, we can address a large class of interesting problems in one go. In the next section, we give some preliminary results which we are already known. # Preliminaries Let us start by introducing the reader to the functional setup related to the fractional Orlicz-Sobolev spaces. A detailed discussion can be found in [@BMRS; @BoSa; @BaOuTa]. Throughout the section, we shall assume $\Omega$ to be a bounded domain and $s\in(0,1)$. Throughout the rest of the article, $C$ will stand for a generic constant, which may vary in each of its appearances. First, we define the modular functions: $$M_{L^G(\Omega)}(f):= \int_{\Omega} G(|f(x)|) dx \ \textrm{ and } M_{W^{s,G}(\Omega)}(f):= \int_{\Omega}\int_{\Omega}G\left( \frac{| f(x)-f(y) |}{| x-y |^s}\right) \frac{dxdy}{|x-y|^N}.$$ The Banach space $$L^G(\Omega):=\left\{ f: \Omega\to \mathbb{R}\ \textrm{measurable} \ \Big| \ \exists \ \lambda>0 \mbox{ such that } M_{L^G(\Omega)}\left(\frac{f}{\lambda}\right)<\infty\right\}$$ is called the Orlicz space. This space is equipped with the norm $$\|f\|_{L^G(\Omega)}:=\inf\{\lambda>0\ |\ M_{L^G(\Omega)}\left(\frac{f}{\lambda}\right)\leq 1\}.$$ The infimum in the above definition is known to be achieved. The fractional Orlicz-Sobolev spaces are defined as $$W^{s,G}(\Omega):=\left\{ f\in L^G(\Omega) \ \Big| \ \exists \ \lambda>0 \mbox{ such that } M_{W^{s,G}(\Omega)}\left(\frac{f}{\lambda}\right)<\infty\right\}.$$ This space is equipped with the seminorm $$\|f\|_{W^{s,G}(\Omega)}:=\inf\{\lambda>0\ |\ M_{W^{s,G}(\Omega)}\left(\frac{f}{\lambda}\right)\leq 1\}.$$ However, we shall mainly be working with the spaces defined by $$\hat{W}^{s,G}(\Omega):=\left\{ f\in L^{G}_{loc}(\mathbb{R}^N):\; \exists\;U\Subset\Omega\;\mbox{s.t}\; ||u||_{s,G,U}+\int_{\mathbb{R}^N}g(\frac{|u(x)|}{1+|x|^s})\;\frac{dx}{(1+|x|)^{n+s}}<\infty\right\}$$ and, $$W^{s,G}_0(\Omega):=\left\{ f\in W^{s,G}(\mathbb{R}^N) \ \Big| \ f\equiv 0\ \mbox{ on }\mathbb{R}^N\setminus \Omega\right\}.$$ $W^{s,G}_0(\Omega)$ is equipped with the norm $\|\cdot\|_{W^{s,G}(\mathbb{R}^N)}$. Note that for $G(t)=t^p;\;1<p<\infty$, $L^G(\Omega)$ and $W^{s,G}(\Omega)$ are well known Lebesgue space $L^p(\Omega)$ and the fractional Sobolev space $W^{s,p}(\Omega)$ respectively (see [@hhg p. 524]). We now discuss some properties of these spaces which we shall use in the next section. We start by observing that the assumption $(H_g)$ implies $$\label{eq growth} 2<p^-\leq \frac{tg(t)}{G(t)}\leq p^+<\infty, \quad t>0.$$ To see this, note that assumption $(H_g)$ implies $(tg(t))'\leq p^+G(t)'$. The following two lemmas will be used frequently in the rest of the article. **Lemma 2**. *Let $G$ be an $N$-function, and let $g=G'$ satisfy $(H_a)-(H_g)$. Then $$\lambda^{p^-}G(t) \leq G(\lambda t) \leq \lambda^{p^+}G(t)\quad \forall\ \lambda\geq1, \ \forall t>0,$$ where $p^+,p^-$ is the constant as defined in $(H_g)$. The above inequality is equivalent to $$\lambda^{p^-}G(t) \geq G(\lambda t) \geq \lambda^{p^+}G(t)\quad \forall\ 0 \leq \lambda\leq1, \ \forall t>0.$$* *Proof.* For any $\lambda>1$, $$\log(\lambda^{p^-}) =\int_{t}^{\lambda t}\frac{p^-}{\tau}d\tau \leq \int_{t}^{\lambda t}\frac{g(\tau)}{G(\tau)}d\tau \leq \int_{t}^{\lambda t}\frac{p^+}{\tau}d\tau =\log(\lambda^{p^+}).$$ This implies $$\log \left(\lambda^{p^-}\right) \leq \log \left(\frac{G(\lambda t)}{G(t)}\right) \leq \log \left(\lambda^{p^+}\right).$$ The lemma follows. ◻ An immediate consequence of [Lemma 2](#lemma delta2){reference-type="ref" reference="lemma delta2"} is the following **Lemma 3**. *When $\|f\|_{W^{s,G}( \Omega)}\leq1$, $$\|f\|_{W^{s,G}( \Omega)}^{p^+} \leq M_{W^{s,G}( \Omega)}(f) \leq \|f\|_{W^{s,G}( \Omega)}^{p^-},$$ and when $\|f\|_{W^{s,G}( \Omega)}\geq1$, $$\|f\|_{W^{s,G}( \Omega)}^{p^-} \leq M_{W^{s,G}( \Omega)}(f) \leq \|f\|_{W^{s,G}( \Omega)}^{p^+}.$$* **Lemma 4**. *Let $G$ be an N-function satisfying $(H_a)-(H_g)$. For any two real numbers $a$ and $b$, we have $$(g(b)-g(a))(b-a) \geq C(G)\;G(|b-a|).$$ for some constant $C$ depending on the $N-$function $G$.* *Proof.* By the symmetry of the inequality, it is enough to prove this lemma for the cases $0<a\leq b$ and $a<0<b$. In the first case, using Taylor's theorem with an integral form of reminder, we have $$\begin{aligned} G(|b-a|) &=G(0)+g(0)|b-a|+\frac{1}{2}\int_{0}^{b-a}g'(t)(b-a-t)dt\\ &= \frac{b-a}{2}\int_{a}^{b}g'(t-a)\frac{b-t}{b-a}dt \leq \frac{b-a}{2}\int_{a}^{b}g'(t)dt\\ &=\frac{(b-a)(g(b)-g(a))}{2} \end{aligned}$$ the case $0 \geq a \geq b$ follows similarly. Now suppose $a<0<b$. Using convexity of $G$, we get $$\begin{aligned} G(\frac{|b-a|}{2}) &= G(\frac{b+(-a)}{2}) \leq \frac{1}{2}(G(b)+G(-a)) \leq \frac{1}{2}(\frac{bg(b)}{p^-}+\frac{(-a)g(-a)}{p^-})\\ &\leq \frac{1}{2p^-} (bg(b)+ag(a)-ag(b)-bg(a)) = \frac{1}{2p^-} (g(b)-g(a))(b-a). \end{aligned}$$ ◻ **Definition 5**. Let $G$ be an $N$-function. 1. The $N$-function $\overline{G}$ is called the conjugate of $G$, and is defined by $$\overline{G}(t):=\int_{0}^t\overline{g}(\tau)d\tau,$$ where $\overline{g}(t):=\sup \{\tau\ \Big|\ g(\tau)\leq t \}$. 2. The $N$-function $G_*$, defined by $$G_*^{-1}(t):=\int_{0}^t\frac{G^{-1}(\tau)}{\tau^{\frac{N+s}{N}}}d\tau,$$ is called the Sobolev conjugate of $G$. 3. An $N$-function $G$ is said to be essentially stronger than $H$, written as $H\prec\prec G$, if for any $k>0$, $$\lim_{t\to\infty}\frac{H(kt)}{G(t)}=0.$$ **Lemma 6** (Hölder Inequality). *Let $G$ be an $N$-function, $N\geq1$, and $\Omega\subseteq \mathbb{R}^N$. Then, we have for any $u,v:\Omega\to\mathbb{R}$, $$\int_{\Omega}|uv| \leq \|u\|_{L^G(\Omega)} \|v\|_{L^{\overline{G}}(\Omega)}$$* **Lemma 7** ([@BoSa Corollary 6.2]). *Let $G$ be an $N$-function which satisfy the $\Delta_2$-condition. Then there exists a constant $C=C(n,G,\Omega)$ such that for any $u\in W_0^{s,G}(\Omega)$, $$\int_{\Omega}G(u(x))dx \leq C\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{dxdy}{|x-y|^N}.$$* **Lemma 8** ([@BaOu Theorem 1] and [Lemma 7](#poincare){reference-type="ref" reference="poincare"}). *Let $G$ be an $N$-function, and let $\Omega$ be a bounded open subset of $\mathbb{R}^N$ with $C^{0,1}$-regularity. Then we have the following:* 1. *the embedding $W^{s,G}_0(\Omega) \to L^{G_*}(\Omega)$, is continuous.* 2. *Moreover, for any $N$-function $H$, the embedding $W^{s,G}_0(\Omega) \to L^H(\Omega)$ is compact if $H\prec\prec G$.* **Lemma 9** (Weak Harnack Inequality, see [@Bo]). *If $u\in \hat{W}^{s,G}(B_{3^{-1}R})$ satisfies weakly $$\begin{cases} (-\Delta_g)^s u(x) &\geq 0 \quad \mbox{if}\; x \in \Omega\\ \hfill u(x) &\geq 0 \quad \mbox{if } x \in \mathbb{R}^N \end{cases}$$ then there exists $\sigma\in (0,1)$ such that $$\inf_{B_{4^{-1}R}} u \geq \sigma R^sg^{-1} (\mathop{\ooalign{$\int$\cr$-$}}_{B_R\setminus B_{2^{-1}R}}g(R^{-s}|u|)\;dx)$$* # Main Results We begin this section by stating the definition of our weak solution **Definition 10** (Weak solutions). The function $u\in \hat{W}^{s,G}(\Omega)$ is said to be a weak solution of [\[eq problem\]](#eq problem){reference-type="ref" reference="eq problem"} if $u>0$ in $\Omega$, and for any $\varphi\in C_c^{\infty}(\Omega)$ one has, $\frac{f}{u^{q(.)}}\in L^1_{loc}(\Omega)$ and $$\label{eq weak solution} \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{(\varphi(x)-\varphi(y))}{|x-y|^{N+s}}\;dxdy=\int_{\Omega}f(x)u(x)^{-q(x)}\phi(x)\;dx.$$ The boundary condition is understood in the sense that 1. if $q(x)\leq1$ on $\Omega_\delta:=\{ x\in\Omega\ \Big| \ \mbox{dist}(x,\partial\Omega)<\delta \}$, then $u\in W^{s,G}_0(\Omega).$ 2. Elsewhere one has, $\Phi(u)\in W^{s,G}_0(\Omega)$, where $$\Phi(t):=\int_{0}^tG^{-1}\left(G(1)\tau^{q^*-1}\right)d\tau.$$ Furthermore, we say that $u$ is a subsolution (or supersolution) of [\[eq problem\]](#eq problem){reference-type="ref" reference="eq problem"} if, for any $\varphi\in C_c^{\infty}(\Omega)$, $$\label{eq subsolution} \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{(\varphi(x)-\varphi(y))}{|x-y|^{N+s}}\;dxdy\leq \ (\mbox{or } \geq)\ \int_{\Omega}f(x)u(x)^{-q(x)}\varphi(x)\;dx.$$ **Theorem 11**. *Let there exist $\delta>0$ such that $q(x)\leq1$ on $\Omega_\delta:=\{ x\in\Omega\ \Big| \ \mbox{dist}(x,\partial\Omega)<\delta \}$ and $f\in L^{\overline{G_*}}(\Omega)$. Then [\[eq problem\]](#eq problem){reference-type="ref" reference="eq problem"} has a weak solution in $W^{s,G}_0(\Omega)$ with $\mbox{essinf}_K u>0$ for any $K\Subset\Omega$.* **Theorem 12**. *Let $g$ is sub-multiplicative and there exist $q^*>1,\ \delta>0$ such that $\|q\|_{L^\infty(\Omega_\delta)}\leq q^*$ and let $$H(t):=G_*(t^\frac{{p^-}+q^*-1}{p^-q^*})$$ be an $N$-function such that $f\in L^{\overline{H}}(\Omega)$. Then [\[eq problem\]](#eq problem){reference-type="ref" reference="eq problem"} has a weak solution $u\in W^{s,G}_{loc}(\Omega)$ with $\mbox{essinf}_K u>0$ for any $K\Subset\Omega$ such that $\Phi(u)\in W^{s,G}_0(\Omega)$, where $$\Phi(t):=\int_{0}^tG^{-1}\left(G(1)\tau^{q^*-1}\right)d\tau.$$* **Theorem 13**. *Every weak solution of [\[eq problem\]](#eq problem){reference-type="ref" reference="eq problem"} obtained through [\[main 1,main 2\]](#main 1,main 2){reference-type="ref" reference="main 1,main 2"} belongs to $C_{loc}^\alpha(\Omega)$ for some $\alpha \in (0, 1)$.* We now develop some results which are needed to prove [Theorem 11](#main 1){reference-type="ref" reference="main 1"} and [Theorem 12](#main 2){reference-type="ref" reference="main 2"}. **Lemma 14** (Comparison Principle). *Let $u,v\in C(\mathbb{R}^N)$ with $[u]_{W^{s,G}(\mathbb{R}^N)}, [v]_{W^{s,G}(\mathbb{R}^N)}<\infty$, and $D\subseteq \mathbb{R}^N$ be a domain such that $|\mathbb{R}^N\setminus D|> 0$. If $v \geq u$ in $\mathbb{R}^N \setminus D$, and $$\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{v(x)-v(y)}{|x-y|^s}\right)\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}dxdy \geq \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}dxdy$$ for $\varphi=(u-v)^+$, then $v \geq u$ in $\mathbb{R}^N$.* *Proof.* We need to show that $v \geq u$ in $D$. The two integrals can be shown to be finite using Hölder's inequality and the assumptions on $g$. Then we have $$\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\left[ g\left(\frac{v(x)-v(y)}{|x-y|^s}\right)-g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\right]\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}dxdy \geq 0.$$ This, and the identity $$g(t_2)-g(t_1)=(t_2-t_1)\int_{0}^{1}g'((t_2-t_1)\tau+t_1)d\tau,$$ gives $$\label{eq-cp-1} \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\left(v(x)-v(y)-u(x)+u(y)\right)Q(x,y) \frac{\varphi(x)-\varphi(y)}{|x-y|^{N+2s}}dxdy \geq 0,$$ where $$Q(x,y):=\int_{0}^{1}g'\left(\frac{(v(x)-v(y)-u(x)+u(y))\tau+u(x)-u(y)}{|x-y|^s}\right)d\tau.$$ From the assumption on $g$, we know $g'\geq 0$. So $Q(x,y)\geq0$, and $Q(x,y)=0$ if and only if the integrand is identically zero. Again this happens if and only if $v(x)=v(y)$ and $u(x)=u(y)$. Choose $\varphi=(u-v)^+$ and $\psi:=u-v$. , then, becomes $$\label{eq-cp-2} \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}Q(x,y) \frac{(\varphi(x)-\varphi(y))(\psi(y)-\psi(x))}{|x-y|^{N+2s}}dxdy \geq 0.$$ We can see that, after choosing $\varphi:=(u-v)^+$, and using the fact that $\psi^+(y)\psi^-(y)=0$, $$(\varphi(x)-\varphi(y))(\psi(y)-\psi(x))=-(\psi^+(x)-\psi^+(y))^2-\psi^-(y)\psi^+(x)-\psi^-(x)\psi^+(y)\leq 0.$$ This along with [\[eq-cp-2\]](#eq-cp-2){reference-type="ref" reference="eq-cp-2"}, and the fact that $Q(x,y)\geq0$ implies $Q(x,y)=0$ or $-(\psi^+(x)-\psi^+(y))^2-\psi^-(y)\psi^+(x)-\psi^-(x)\psi^+(y)=0$ almost everywhere. In both the cases, we must have $\psi^+(x)=\psi^+(y)$ for a.e. $(x,y)$. Since $(u-v)^+=0$ on $\mathbb{R}^N\setminus D$, by continuity of $u,v$, we conclude that $\psi^+=0$ on $\mathbb{R}^N$. This implies $v\geq u$ on $\mathbb{R}^N$. ◻ **Lemma 15**. *Let $g$ be sub-multiplicative, that is, there is a constant $C>0$ for which $C g(t_1t_2)\leq g(t_1)g(t_2)$ for any $t_1,t_2>0$. Let $F$ and $u$ be such that $$\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}\;dxdy=\int_{\Omega}F\varphi\; dx,$$ for any $\varphi\in W^{s,G}_0(\Omega)$. Then for any convex and Lipschitz function $\Phi$, we have $$\int_{\Omega}F(x)g(\Phi'(u(x)))\Phi(u)\; dx \geq C\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G\left(\frac{|\Phi(u(x))-\Phi(u(y))|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^N}.$$* *Proof.* First, note that, by density argument, we can assume $\Phi$ to be $C^1$. Choose $\varphi=g(\Phi'(u))\psi$. Then we have $$\begin{aligned} 2\iint_{\{u(x)>u(y)\}}&g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{g(\Phi'(u(x)))\psi(x)-g(\Phi'(u(y)))\psi(y)}{|x-y|^{N+s}}dxdy\\ &=\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{g(\Phi'(u(x)))\psi(x)-g(\Phi'(u(y)))\psi(y)}{|x-y|^{N+s}}dxdy\\ &=\int_{\Omega}F(x)g(\Phi'(u(x)))\psi(x) dx. \end{aligned}$$ Set $u(x)=a$, $u(y)=b$, $\psi(x)=A$ and $\psi(y)=B$. Then the integrand in the LHS becomes $$g\left(\frac{a-b}{|x-y|^s}\right)\frac{g(\Phi'(a))A-g(\Phi'(b))B}{|x-y|^{N+s}}.$$ Using the convexity of $\Phi$, we have $$\Phi(a)-\Phi(b)\leq \Phi'(a)(a-b) \mbox{ and } \Phi(a)-\Phi(b)\geq \Phi'(b)(a-b).$$ We then have $$\begin{aligned} &g\left(\frac{a-b}{|x-y|^s}\right)\frac{g(\Phi'(a))A-g(\Phi'(b))B}{|x-y|^{N+s}}\\ &\geq g\left(\frac{a-b}{|x-y|^s}\right)\frac{g\left(\frac{\Phi(a)-\Phi(b)}{a-b}\right)A-g\left(\frac{\Phi(a)-\Phi(b)}{a-b}\right)B}{|x-y|^{N+s}}\\ &= g\left(\frac{a-b}{|x-y|^s}\right)g\left(\frac{\Phi(a)-\Phi(b)}{a-b}\right)\frac{A-B}{|x-y|^{N+s}}\\ &\geq Cg\left(\frac{\Phi(a)-\Phi(b)}{|x-y|^s}\right)\frac{A-B}{|x-y|^{N+s}}. \end{aligned}$$ This, after taking $\psi=\Phi(u)$ (note that $\Phi$ is assumed to be $C^1$), gives $$\int_{\Omega}F(x)g(\Phi'(u(x)))\Phi(u) dx \geq C\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G\left(\frac{|\Phi(u(x))-\Phi(u(y))|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^N}.$$ ◻ **Lemma 16**. *Let $f\in L^\infty(\Omega)$ with $f\geq 0$, and $f$ is not identically zero. Then the problem $$\label{eq sq problem} \begin{cases} (-\Delta_g)^su=f, \quad \mbox{ in } \Omega,\\ u>0, \mbox{ in } \Omega,\\ u=0, \mbox{ in } \mathbb{R}^N\setminus\Omega \end{cases}$$ has a unique solution $u\in W^{s,G}_0(\Omega)\cap L^\infty (\Omega)$.* *Proof.* The existence, uniqueness, and continuity follows from [@BoSa Theorem 6.16], [Lemma 9](#lemma smp){reference-type="ref" reference="lemma smp"}, and the fact that $f\geq0$, so that $(-\Delta_g)^su\geq0$ on $\Omega$, using [Lemma 14](#comparison){reference-type="ref" reference="comparison"}. It remains to show that $u\in L^\infty(\Omega)$. For this, we shall assume, without loss of generality, that $\Omega\subseteq B(0,1)$ and fix $\alpha>1$. Let us consider $$v_\alpha(x)= \begin{cases} \alpha(1-|x|),\quad &\mbox{when } |x|<1,\\ 0, \quad &\mbox{otherwise.} \end{cases}$$ Note that for since $\alpha>1$, for any $0<\lambda<1$ we have, using [\[lemma delta2,eq growth\]](#lemma delta2,eq growth){reference-type="ref" reference="lemma delta2,eq growth"}, the estimate $g(\alpha\lambda t)>\frac{p^-\alpha^{p^--1}\lambda^{p^+-1}G(t)}{ t}$ when $t>0$. Again, for $x\in \Omega\subseteq B(0,1)\subseteq B(x,1+|x|)$ we get $$\begin{aligned} (-\Delta_g)^s v_\alpha(x)&\geq \int_{|y|>1}g\left(\frac{v_\alpha(x)-v_\alpha(y)}{|x-y|^s}\right)\frac{dy}{|x-y|^{N+s}}\\ &=\int_{|y|>1}g\left(\frac{v_\alpha(x)}{|x-y|^s}\right)\frac{dy}{|x-y|^{N+s}}\\ &\geq p^-\alpha^{p^--1}(1-|x|)^{p^+-1} \int_{|y|>1}G\left(\frac{1}{|x-y|^s}\right)\frac{dy}{|x-y|^N}\\ &= p^-\alpha^{p^--1}(1-|x|)^{p^+-1} \int_{|y|>1}G\left(\frac{1}{(1+|y|)^s}\right)\frac{dy}{(1+|y|)^N} \to \infty \end{aligned}$$ uniformly as $\alpha \to \infty$. Thus, as $f$ is bounded, we can choose $\alpha$ large enough to get $(-\Delta)^s_g v_\alpha>(-\Delta)^s_g u$. Applying [Lemma 14](#comparison){reference-type="ref" reference="comparison"} we get $u\leq v_\alpha$ in $\mathbb{R}^N$. Thus, $u$ is bounded. ◻ We consider the following approximated problem of [\[eq problem\]](#eq problem){reference-type="ref" reference="eq problem"}, where we used the notation, $f_n=\min\{f,n\}$ for all $n\in\mathbb{N}$, and assumed $q>0$ is $C^1$, $$\begin{aligned} \label{eq approx problem} (-\Delta_g)^su(x)&=\frac{f_n(x)}{(u(x)+\frac{1}{n})^{q(x)}}\;\mbox{in }\;\Omega,\\ \nonumber u&>0\;\mbox{in }\;\Omega,\\ \nonumber u&=0\;\mbox{in }\;\mathbb{R}^N\setminus\Omega. \end{aligned}$$ **Lemma 17**. *For a fixed $n\in\mathbb{N}$, [\[eq approx problem\]](#eq approx problem){reference-type="ref" reference="eq approx problem"} has a weak solution $u_n\in C^{\alpha (n)}(\Omega)$ where $\alpha(n)\in (0,1)\;\forall n\in\mathbb{N}$.* *Proof.* Note that $\frac{f_n(x)}{(u^+(x)+\frac{1}{n})^{q(x)}}\in L^\infty(\Omega)$. Hence by [Lemma 16](#laplace equation){reference-type="ref" reference="laplace equation"}, there exists a unique solution $w\in W^{s,G}_0(\Omega)\cap L^\infty(\Omega)$ to the problem $$\label{eq approx problem positive} \begin{split} (-\Delta_g)^sw(x)&=\frac{f_n(x)}{(u^+(x)+\frac{1}{n})^{q(x)}}\;\mbox{in }\; \Omega,\\ \nonumber w&>0\;\mbox{in }\; \Omega,\\ \nonumber w&=0\;\mbox{in }\; \mathbb{R}^N\setminus\Omega. \end{split}$$ This allows us to define the operator $S:W^{s,G}_0(\Omega)\to W^{s,G}_0(\Omega)$ by $S(u)=w$ the solution of [\[eq approx problem positive\]](#eq approx problem positive){reference-type="ref" reference="eq approx problem positive"}. Multiplying both sides of [\[eq approx problem positive\]](#eq approx problem positive){reference-type="ref" reference="eq approx problem positive"} by $w$, we get $$\begin{gathered} \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{w(x)-w(y)}{|x-y|^s}\right)\frac{(w(x)-w(y))}{|x-y|^{N+s}}dxdy =\int_{\Omega}\frac{f_n(x)w(x)}{(u(x)^++\frac{1}{n})^{q(x)}}dx \leq n^{1+\|q\|_{L^\infty(\Omega)}}\|w\|_{L^1(\Omega)}. \end{gathered}$$ Applying [\[eq growth\]](#eq growth){reference-type="ref" reference="eq growth"}, we get $$\begin{aligned} \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G\left(\frac{|w(x)-w(y)|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^{N}} &\leq \frac{1}{p^-}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{|w(x)-w(y)|}{|x-y|^s}\right)\frac{|w(x)-w(y)|}{|x-y|^{N+s}}dxdy\\ &= \frac{1}{p^-}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{(w(x)-w(y))}{|x-y|^s}\right)\frac{(w(x)-w(y))}{|x-y|^{N+s}}dxdy\\ &\leq \frac{n^{1+\|q\|_{L^\infty(\Omega)}}}{p^-}\|w\|_{L^1(\Omega)}. \end{aligned}$$ Assume $\|w\|_{W_0^{s,G}(\Omega)}> 1,$ $$\begin{gathered} \frac{1}{\|w\|_{W_0^{s,G}(\Omega)}^{p^-}}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G\left(\frac{|w(x)-w(y)|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^{N}} \geq \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G\left(\frac{|w(x)-w(y)|}{\|w\|_{W_0^{s,G}(\Omega)}|x-y|^s}\right)\frac{dxdy}{|x-y|^{N}} =1. \end{gathered}$$ So, we have $$\|w\|_{W_0^{s,G}(\Omega)}^{p^-}\leq \frac{n^{1+\|q\|_{L^\infty(\Omega)}}}{p^-}\|w\|_{L^1(\Omega)},$$ and consequently, by [Lemma 8](#lemma embedding){reference-type="ref" reference="lemma embedding"}, $$\|w\|_{W_0^{s,G}(\Omega)}^{p^--1}\leq Cn^{1+\|q\|_{L^\infty(\Omega)}}$$ provided $\|w\|_{W_0^{s,G}(\Omega)}>1$. Setting $R:=\max\{1,\left(Cn^{1+\|q\|_{L^\infty(\Omega)}}\right)^\frac{1}{p^--1}\}$, we can see that $S$ maps the ball of radius $R$ in the metric space $W_0^{s,G}(\Omega)$, into itself. The proof will now be complete if we show that $S$ is continuous and compact.\ **Proof of continuity of $S$:** Assume that $u_i\to u$ in $W_0^{s,G}(\Omega)$. Set $w_i=S(u_i)$ and $w=S(u)$. So that we have for any $\varphi\in W^{s,G}_0(\Omega)$, $$\begin{aligned} \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{w_i(x)-w_i(y)}{|x-y|^s}\right)\frac{(\varphi(x)-\varphi(y))}{|x-y|^{N+s}}dxdy &=\int_{\Omega}\frac{f_n(x)\varphi(x)}{(u_i(x)^++\frac{1}{n})^{q(x)}}dx \quad \mbox{and} \label{eq1} \\ \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{w(x)-w(y)}{|x-y|^s}\right)\frac{(\varphi(x)-\varphi(y))}{|x-y|^{N+s}}dxdy &=\int_{\Omega}\frac{f_n(x)\varphi(x)}{(u(x)^++\frac{1}{n})^{q(x)}}dx\label{eq2}. \end{aligned}$$ We have to show that $w_i\to w$ in $W_0^{s,G}(\Omega)$. By [Lemma 8](#lemma embedding){reference-type="ref" reference="lemma embedding"}, passing to a subsequence, $u_i\to u$ in $L^{G_*}(\Omega)$ and $u_i\to u$ a.e. in $\Omega$. Set $\overline{w_i}:=w_i-w$. Subtracting [\[eq2\]](#eq2){reference-type="ref" reference="eq2"} from [\[eq1\]](#eq1){reference-type="ref" reference="eq1"}, with the choice $\varphi=\overline{w_i}$, and then applying [Lemma 4](#lemma lindqvist){reference-type="ref" reference="lemma lindqvist"} for $a=\frac{w(x)-w(y)}{|x-y|^s}$ and, $b=\frac{w_i(x)-w_i(y)}{|x-y|^s}$, we get $$\begin{gathered} C(G)\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G\left(\frac{|\overline{w_i}(x)-\overline{w_i}(y)|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^N}dxdy\\ \leq \int_{\Omega}f_n(x)\left(\frac{1}{(u_i(x)^++\frac{1}{n})^{q(x)}} - \frac{1}{(u(x)^++\frac{1}{n})^{q(x)}}\right)(w_i(x)-w(x))dx. \end{gathered}$$ We apply [Lemma 3](#lemma equivalent norm){reference-type="ref" reference="lemma equivalent norm"} on the left-hand side and Hölder inequality on the right-hand side of this equation to get, $$\begin{aligned} C(G)&\min\left\{\|w_i-w\|_{W^{s,G}}^{p^+}, \|w_i-w\|_{W^{s,G}}^{p^-}\right\}\\ &\leq C\left\| f_n(x)\left(\frac{1}{(u_i(x)^++\frac{1}{n})^{q(x)}} - \frac{1}{(u(x)^++\frac{1}{n})^{q(x)}}\right) \right\|_{L^{G_*'}}\|w_i-w\|_{L^{G_*}}\\ &\leq C\left\| f_n(x)\left(\frac{1}{(u_i(x)^++\frac{1}{n})^{q(x)}} - \frac{1}{(u(x)^++\frac{1}{n})^{q(x)}}\right) \right\|_{L^{G_*'}}\|w_i-w\|_{W^{s,G}}, \end{aligned}$$ where the last inequality follows from [Lemma 7](#poincare){reference-type="ref" reference="poincare"}. This gives $$\min\left\{\|w_i-w\|_{W^{s,G}}^{p^+-1}, \|w_i-w\|_{W^{s,G}}^{p^--1}\right\}\leq C\left\| f_n(x)\left(\frac{1}{(u_i(x)^++\frac{1}{n})^{q(x)}}-\frac{1}{(u(x)^++\frac{1}{n})^{q(x)}}\right)\right\|_{L^{G_*'}}.$$ Now observe that $$\left|f_n(x)\left(\frac{1}{(u_i(x)^++\frac{1}{n})^{q(x)}} - \frac{1}{(u(x)^++\frac{1}{n})^{q(x)}}\right)\right| \leq 2n^{q(x)+1} \leq 2n^{\|q\|_{L^\infty}+1}.$$ Hence, as $u_i\to u$ pointwise a.e., by DCT it follows that $w_i\to w$ in $W^{s,G}_0$. Thus $S$ is continuous.\ **Proof of compactness of $S$:** Assume that $u_i$ is a bounded sequence in $W_0^{s,G}(\Omega)$. As before, denote $w_i:=S(u_i)$. We wish to show that $w_i$ has a convergent subsequence in $W_0^{s,G}(\Omega)$. From [\[eq1,holder,lemma equivalent norm\]](#eq1,holder,lemma equivalent norm){reference-type="ref" reference="eq1,holder,lemma equivalent norm"}, we get $$\begin{aligned} \min\left\{\|w_i\|_{W^{s,G}}^{p^+}, \|w_i\|_{W^{s,G}}^{p^-}\right\} &\leq C(G)\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G\left(\frac{w_i(x)-w_i(y)}{|x-y|^s}\right)\frac{dxdy}{|x-y|^{N}}\\ &\leq C(G)\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{w_i(x)-w_i(y)}{|x-y|^s}\right)\frac{(w_i(x)-w_i(y))}{|x-y|^{N+s}}dxdy\\ &=C(G)\int_{\Omega}\frac{f_n(x)w_i(x)}{(u_i(x)^++\frac{1}{n})^{q(x)}}dx \leq n^{1+\|q\|_{L^\infty(\Omega)}}\|w_i\|_{L^1(\Omega)}\\ &\leq C n^{1+\|q\|_{L^\infty(\Omega)}}\|w_i\|_{W^{s,G}(\Omega)}. \end{aligned}$$ This shows that $w_i$ is a bounded sequence in $W_0^{s, G}(\Omega)$. From the boundedness of the two sequences, $u_i,w_i$, we conclude that there exists $u,w\in W_0^{s,G}(\Omega)$ such that $u_i\rightharpoonup u$ and $w_i\rightharpoonup w$ in $W_0^{s,G}(\Omega)$. We now want to show $S(u)=w$, that is for any $\varphi\in C_c^{\infty}(\Omega)$, $$\label{eq4} \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{w(x)-w(y)}{|x-y|^s}\right)\frac{(\varphi(x)-\varphi(y))}{|x-y|^{N+s}}dxdy\\ =\int_{\Omega}\frac{f_n(x)\varphi(x)}{(u(x)^++\frac{1}{n})^{q(x)}}dx.$$ Note that we already know $$\label{eq3} \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{w_i(x)-w_i(y)}{|x-y|^s}\right)\frac{(\varphi(x)-\varphi(y))}{|x-y|^{N+s}}dxdy\\ =\int_{\Omega}\frac{f_n(x)\varphi(x)}{(u_i(x)^++\frac{1}{n})^{q(x)}}dx.$$ By DCT, it is seen easily that the right-hand side of [\[eq3\]](#eq3){reference-type="ref" reference="eq3"} converges to the right-hand side of [\[eq4\]](#eq4){reference-type="ref" reference="eq4"}. It remains to show the convergence of the left-hand side. Note that, $$\overline{G}(g(t)) = \int_{0}^{g(t)}g^{-1}(\tau)d\tau =\int_{0}^t\tau g'(\tau)d\tau \equiv \int_{0}^tg(\tau)d\tau =G(t).$$ Using this and the fact that $w_i$'s are bounded in $W^{s,G}_0(\Omega)$, we have that $g\left(\frac{|w_i(x)-w_i(y)|}{|x-y|^s}\right)$ is a bounded sequence in $L^{\overline{G}}(\frac{1}{|x-y|^N},\mathbb{R}^N\times\mathbb{R}^N)$ hence it has a weakly convergent subsequence. Thus we conclude that, up to a subsequence, $$g\left(\frac{|w_i(x)-w_i(y)|}{|x-y|^s}\right) \rightharpoonup g\left(\frac{|w(x)-w(y)|}{|x-y|^s}\right)$$ weakly in $L^{\overline{G}}(\frac{1}{|x-y|^N},\mathbb{R}^N\times\mathbb{R}^N)$. Now, since $\frac{|\varphi(x)-\varphi(y)|}{|x-y|^s}\in L^G(\frac{1}{|x-y|^N},\mathbb{R}^N\times\mathbb{R}^N)$, $$\int_{\mathbb{R}^N}\int_{\mathbb{R}^N} g\left(\frac{|w_i(x)-w_i(y)|}{|x-y|^s}\right)\frac{|\varphi(x)-\varphi(y)|}{|x-y|^{N+s}}dxdy\to \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} g\left(\frac{|w(x)-w(y)|}{|x-y|^s}\right)\frac{|\varphi(x)-\varphi(y)|}{|x-y|^{N+s}}dxdy$$ Since the solution so obtained is in $u_n\in W^{s,G}_0(\Omega)\cap L^\infty(\Omega)$ and hence it is $C^{\alpha (n)}(\Omega)$ where $\alpha(n)\in (0,1),\;\forall n\in\mathbb{N}$ by Theorem 1.1 of Bonder et al [@Bo]. ◻ **Lemma 18**. *Assume $g$ to be convex on $(0,1)$. The sequence of functions $\{u_n\}_n$, found in [Lemma 17](#lemma approx problem){reference-type="ref" reference="lemma approx problem"} satisfies $$u_n(x)\leq u_{n+1}(x), \quad \mbox{for almost every } x\in\Omega,$$ and for any compact set $K\subseteq \Omega$, there exists a constant $l=l(K)>0$ such that for any $n$, large enough, $$u_n(x)\geq l \quad \mbox{for almost every }x\in K.$$* *Proof.* Set the notation $w_n(x)=(u_{n}(x)-u_{n+1}(x))^+$. Then we note that, for any $x\in\Omega$, and $f_n(x)\leq f_{n+1}(x)$, $$\begin{aligned} \int_{\Omega} &\frac{f_n(x)}{(u_n(x)+\frac{1}{n})^{q(x)}}w_n(x)dx-\int_{\Omega}\frac{f_{n+1}(x)}{(u_{n+1}(x)+\frac{1}{n+1})^{q(x)}}w_n(x)dx\\ &=\int_{\Omega}\left( \frac{f_n(x)}{(u_n(x)+\frac{1}{n})^{q(x)}}-\frac{f_{n+1}(x)}{(u_{n+1}(x)+\frac{1}{n+1})^{q(x)}} \right)w_n(x)dx\\ &=\int_{\Omega}\left( \frac{f_n(x)}{(u_n(x)+\frac{1}{n})^{q(x)}}-\frac{f_{n+1}(x)}{(u_{n+1}(x)+\frac{1}{n+1})^{q(x)}} \right)(u_n(x)-u_{n+1}(x))^+dx\\ &\leq \int_{\{u_{n}(x)>u_{n+1}(x)\}}f_{n+1}(x)\left( \frac{(u_{n+1}(x)+\frac{1}{n+1})^{q(x)}-(u_n(x)+\frac{1}{n})^{q(x)}}{(u_n(x)+\frac{1}{n})^{q(x)}(u_{n+1}(x)+\frac{1}{n+1})^{q(x)}} \right)(u_{n}-u_{n+1})^+dx\\ &\leq 0. \end{aligned}$$ Then the above calculation and [\[eq approx problem\]](#eq approx problem){reference-type="ref" reference="eq approx problem"} implies $$\begin{aligned} \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}&g\left(\frac{u_{n}(x)-u_{n}(y)}{|x-y|^s}\right)\frac{w_n(x)-w_n(y)}{|x-y|^{N+s}}dxdy\\ &\leq \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u_{n+1}(x)-u_{n+1}(y)}{|x-y|^s}\right)\frac{w_n(x)-w_n(y)}{|x-y|^{N+s}}dxdy. \end{aligned}$$ Now [@BoSaVi Theorem 1.1] implies that both $u_n,u_{n+1}$ are Hölder continuous up to the boundary. So, we can apply [Lemma 14](#comparison){reference-type="ref" reference="comparison"} to get $u_n\leq u_{n+1}$ a.e. on $\mathbb{R}^N$. This concludes the proof of the first part. The second part follows from the continuity of $u_n$, and [Lemma 9](#lemma smp){reference-type="ref" reference="lemma smp"}, which gives $u_n>0$ on $\Omega$. ◻ ***Proof of [Theorem 11](#main 1){reference-type="ref" reference="main 1"}**.* By [Lemma 17](#lemma approx problem){reference-type="ref" reference="lemma approx problem"}, [\[eq approx problem\]](#eq approx problem){reference-type="ref" reference="eq approx problem"} has a weak solution $u_n$. Let $\varphi\in C_c^{\infty}(\Omega)$. We have $$\label{eq 1} \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u_{n}(x)-u_{n}(y)}{|x-y|^s}\right)\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}dxdy =\int_{\Omega}\frac{f_n(x)\varphi(x)}{(u_n(x)+\frac{1}{n})^{q(x)}}dx.$$ First, we claim: $$\begin{gathered} \label{eq 2} \lim_{n\to\infty}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u_{n}(x)-u_{n}(y)}{|x-y|^s}\right)\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}dxdy\\ = \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}dxdy. \end{gathered}$$ **Proof of the claim:** Set $\omega_\delta:=\Omega\setminus\Omega_\delta$. Then by [Lemma 18](#lemma monotone){reference-type="ref" reference="lemma monotone"}, there exists a constant $l>0$ such that $u_n\geq l>0$ on $\omega_\delta$. We get, using [Lemma 8](#lemma embedding){reference-type="ref" reference="lemma embedding"}, and choosing $\varphi=u_n$, $$\begin{aligned} C(G) &\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G\left(\frac{|u_{n}(x)-u_{n}(y)|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^{N}}\\ &\leq \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u_{n}(x)-u_{n}(y)}{|x-y|^s}\right)\frac{u_n(x)-u_n(y)}{|x-y|^{N+s}}dxdy\\ &= \int_{\Omega}\frac{f_n(x)u_n(x)}{(u_n(x)+\frac{1}{n})^{q(x)}}dx\\ &=\int_{\Omega_\delta}\frac{f_n(x)u_n(x)}{(u_n(x)+\frac{1}{n})^{q(x)}}dx +\int_{\omega_\delta}\frac{f_n(x)u_n(x)}{(u_n(x)+\frac{1}{n})^{q(x)}}dx\\ &\leq \int_{\Omega_\delta\cap \{u_n\leq1\}}f_n(x)dx +\int_{\Omega_\delta\cap\{u_n>1\}}f_n(x)u_n(x)dx +\int_{\omega_\delta}\frac{f_n(x)u_n(x)}{l^{q(x)}}dx\\ &\leq \|f\|_{L^1(\Omega)}+(1+\|l^{-q(\cdot)}\|_{L^\infty(\omega_\delta)})\|f\|_{L^{\overline{G_*}}(\Omega)}\|u_n\|_{L^{G_*}(\Omega)}\\ &\leq \|f\|_{L^1(\Omega)}+C_1\|u_n\|_{W^{s,G}_0(\Omega)}. \end{aligned}$$ Assuming $\alpha:=\|u_n\|_{W^{s,G}_0(\Omega)}>1$, we get, using [Lemma 2](#lemma delta2){reference-type="ref" reference="lemma delta2"}, $$\begin{gathered} 1=\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G\left(\frac{|u_{n}(x)-u_{n}(y)|}{\alpha|x-y|^s}\right)\frac{dxdy}{|x-y|^{N}}\\ \leq \frac{1}{\alpha^{p^-}} \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G\left(\frac{|u_{n}(x)-u_{n}(y)|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^{N}}\leq \frac{\|f\|_{L^1(\Omega)}}{\alpha^{p^-}}+C_1\frac{1}{\alpha^{p^--1}} \end{gathered}$$ This shows that $\|u_n\|_{W^{s,G}_0(\Omega)}$ must be bounded. So $u_n\rightharpoonup u$ in $W^{s,G}_0$ weakly. By [Lemma 8](#lemma embedding){reference-type="ref" reference="lemma embedding"}, $u_n\to u$ strongly in $L^{1}(\Omega)$, and hence $u_n\to u$ pointwise a.e. up to a subsequence. Now applying [Lemma 2](#lemma delta2){reference-type="ref" reference="lemma delta2"} $$\overline{G}(g(t))=\int_{0}^{g(t)}\overline{g}(\tau)d\tau =\int_{0}^t\overline{g}(g(\tau))g'(\tau)d\tau =\int_{0}^t\tau g'(\tau)d\tau.$$ This implies $$\label{eq 3} (p^--1)G(t)\leq \overline{G}(g(t)) \leq (P^+-1)G(t).$$ This, along with [Lemma 3](#lemma equivalent norm){reference-type="ref" reference="lemma equivalent norm"}, shows that the sequence of functions $(x,y)\mapsto g\left(\frac{u_{n}(x)-u_{n}(y)}{|x-y|^s}\right)$ is bounded in $L^{\overline{G}}\left(\mathbb{R}^N\times\mathbb{R}^N,\frac{dxdy}{|x-y|^N}\right)$. So it has a weakly convergent subsequence; without loss of generality, we assume it to be itself. It is easy to check that the the function $(x,y)\mapsto\frac{\varphi(x)-\varphi(y)}{|x-y|^s}$ is in $L^{G}\left(\mathbb{R}^N\times\mathbb{R}^N,\frac{dxdy}{|x-y|^N}\right)$. Hence [\[eq 2\]](#eq 2){reference-type="ref" reference="eq 2"} follows and the claim is true. Now, in order to complete the proof, taking into account [\[eq 1\]](#eq 1){reference-type="ref" reference="eq 1"}, we only need to show the convergence of the right-hand side of [\[eq 1\]](#eq 1){reference-type="ref" reference="eq 1"}. Note that $$\left|\frac{f_n(x)\varphi(x)}{(u_n(x)+\frac{1}{n})^{q(x)}}\right|\leq |l^{-q(x)}f(x)\varphi(x)|\in L^1(\Omega),$$ where we get $l$ from applying [Lemma 18](#lemma monotone){reference-type="ref" reference="lemma monotone"} on $\mbox{supp}(\varphi)$. Therefore, we can apply DCT to get $$\lim_{n\to\infty}\int_{\Omega}\frac{f_n(x)\varphi(x)}{(u_n(x)+\frac{1}{n})^{q(x)}}dx =\int_{\Omega}\frac{f(x)\varphi(x)}{u(x)^{q(x)}}dx.$$ Hence the proof is complete. ◻ **Lemma 19**. *For any $a,b\in\mathbb{R}$, we have $$|g(a)-g(b)| \leq C \frac{|a-b|g(|a|+|b|)}{|a|+|b|} \leq C g(|a|+|b|).$$* *Proof.* $$g(b)-g(a)=\int_{0}^1g'(a+(b-a)t)(b-a)dt.$$ Now since $g'$ is increasing one has for $t\in(0,1)$, $|a+(b-a)t|\leq\left||a|+|b|\right|$. So we get $$|g(a)-g(b)|\leq |a-b|g'(|a|+|b|)$$ The results now follow trivially using the hypothesis on $g$. ◻ **Lemma 20**. *Let $\Phi:(0,\infty)\to(0,\infty)$ be a strictly convex, $C^1$-function such that $\Phi'$ is increasing and there exists $\theta_1,\theta_2\geq 0$ such that $\theta_1\frac{\Phi(x)}{x}\leq \Phi'(x)\leq\theta_2\frac{\Phi(x)}{x}$. For $x,y\in \mathbb{R}$ and $\varepsilon>0$, define $S^x_\varepsilon:=\{x\geq\varepsilon\}\cap\{y\geq0\}$, and $S^y_\varepsilon:=\{x\geq0\}\cap\{y\geq\varepsilon\}$. Then for $(x,y)\in S^x_\varepsilon\cup S^y_\varepsilon$, $$|\Phi(x)-\Phi(y)|\geq C\Phi'(\epsilon)|x-y|\;\mbox{with}\;C:=\max(\theta_1,1).$$* **Proof.* By symmetry, without loss of generality, we can assume $x> y$. Now for some $\lambda\in(y,x)$, we have $\Phi(x)-\Phi(y)=\Phi'(\lambda)(x-y)$. If we assume $x\geq y\geq \varepsilon>0$, then we have $$|\Phi(x)-\Phi(y)| \geq \Phi'(\lambda)|x-y| \geq \Phi'(\varepsilon)|x-y|$$ For, $0\leq y<\varepsilon\leq x$, then by strict convexity of $\Phi$, we get $$\frac{\Phi(x)-\Phi(y)}{x-y}\geq \frac{\Phi(x)}{x}\geq \theta_1\Phi'(x)\geq \theta_1\Phi'(\varepsilon)$$ thus concluding the assertion. ◻* **Lemma 21**. *Let $\Phi,\ H,\ f,\ q$ be as in [Theorem 12](#main 2){reference-type="ref" reference="main 2"}, $u_n$ be as in [Lemma 17](#lemma approx problem){reference-type="ref" reference="lemma approx problem"}. Then there is a constant $C>0$, independent of $n$ such that $\|\Phi(u_n)\|_{W^{s,G}_0(\Omega)}, \ \|\Phi(u)\|_{W^{s,G}_0(\Omega)}\leq C$, where $u$ is the pointwise limit of $u_n$.* *Proof.* We have, for $t>0$, $$\Phi(t):=\int_{0}^tG^{-1}\left(G(1)\tau^{q^*-1}\right)d\tau,$$ that is $$\Phi'(t):=G^{-1}\left(G(1)t^{q^*-1}\right),$$ which gives, applying the fact that $\Phi'(t)$ is increasing and hence $\Phi(t)\leq t\Phi'(t)$, $$\label{eq 7} g\left(\Phi'(t)\right)\Phi(t) =\frac{\Phi'(t)g\left(\Phi'(t)\right)}{G\left(\Phi'(t)\right)}\frac{G\left(\Phi'(t)\right)\Phi(t)}{\Phi'(t)} \leq p^+G(1)t^{q^*-1}\frac{\Phi(t)}{\Phi'(t)} \leq p^+G(1)t^{q^*}.$$ Using [\[eq 7,lemma convex\]](#eq 7,lemma convex){reference-type="ref" reference="eq 7,lemma convex"}, and the fact $q^*>1$ we have $$\begin{aligned} \label{eq 4} \nonumber \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}&G\left(\frac{|\Phi(u_n(x))-\Phi(u_n(y))|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^N}\\ \nonumber &\leq C \int_{\Omega}\frac{f_n(x)}{(u_n(x)+\frac{1}{n})^{q(x)}}g(\Phi'(u_n(x)))\Phi(u_n(x)) dx\\ \nonumber &= C \left(\int_{\Omega_\delta,\ u_n<1}+ \int_{\Omega_\delta,\ u_n\geq1}+\int_{\omega_\delta,\ u_n<1}+\int_{\omega_\delta,\ u_n\geq1}\right)\frac{f_n(x)}{(u_n(x)+\frac{1}{n})^{q(x)}}g(\Phi'(u_n(x)))\Phi(u_n(x)) \\ &\leq C\int_{\Omega\cap\{u_n<1\}}f_n(x)+ C\int_{\Omega\cap\{u_n\geq 1\}}f_n(x)u_n(x)^{q^*}. \end{aligned}$$ Set $r:=\frac{p^-}{p^-+q^*-1}$. We have, for large enough $t_0$, and for any $t>t_0$, $$\begin{aligned} \label{eq-r-bound} \nonumber t^\frac{1}{r} &=\frac{1}{r}\int_{0}^{1}\tau^{\frac{1}{r}-1}d\tau + \frac{1}{r}\int_{1}^{t}\tau^{\frac{1}{r}-1}d\tau \leq \frac{2}{r}\int_{1}^{t}\tau^{\frac{1}{r}-1}G^{-1}(G(1))d\tau\\ &\leq \frac{2}{r}\int_{1}^{t}G^{-1}(G(1)\tau^{\frac{p^-(1-r)}{r}})d\tau \leq \frac{2}{r}\int_{0}^{t}G^{-1}(G(1)\tau^{\frac{p^-(1-r)}{r}})d\tau = \frac{2}{r}\Phi(t). \end{aligned}$$ Applying [\[eq-r-bound\]](#eq-r-bound){reference-type="ref" reference="eq-r-bound"} on [\[eq 4\]](#eq 4){reference-type="ref" reference="eq 4"} and then using Hölder's inequality, and finally the fact that $|f_n|\leq |f|$, we get $$\begin{aligned} \label{eq 6} \nonumber \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}&G\left(\frac{|\Phi(u_n(x))-\Phi(u_n(y))|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^N}\\ \nonumber &\leq C\int_{\Omega\cap\{u_n<1\}}f_n(x)+ C\int_{\Omega\cap\{u_n\geq1\}}f_n(x)\Phi(u_n(x))^{rq^*}\\ \nonumber &\leq C\|f_n\|_{L^1(\Omega)}+C\|f_n\|_{L^{\overline{H}}(\Omega)}\|\Phi(u_n)^{rq^*}\|_{L^{H}(\Omega)}\\ &\leq C\|f\|_{L^1(\Omega)}+C\|f\|_{L^{\overline{H}}(\Omega)}\|\Phi(u_n)^{rq^*}\|_{L^{H}(\Omega)}. \end{aligned}$$ Observe that $$\begin{aligned} \left\| \Phi^{rq^*}(u_n) \right\|_{L^H(\Omega)} &= \inf \left\{ \lambda>0\ \Big|\ \int_{\Omega} H\left(\frac{\Phi(u_n)^{rq^*}}{\lambda}\right) \leq 1\right\}\\ &= \inf \left\{ \lambda^{rq^*}>0\ \Big|\ \int_{\Omega} H\left(\frac{\Phi(u_n)^{rq^*}}{\lambda^{rq^*}}\right) \leq 1\right\}\\ &= \left(\inf \left\{ \lambda>0\ \Big|\ \int_{\Omega} H\left(\frac{\Phi(u_n)^{rq^*}}{\lambda^{rq^*}}\right) \right\}\right)^{rq^*}\\ &=\left(\inf \left\{ \lambda>0\ \Big|\ \int_{\Omega} G_*\left(\frac{\Phi(u_n)}{\lambda}\right) \right\}\right)^{rq^*} =\left\| \Phi(u_n) \right\|_{L^{G_*}(\Omega)}^{rq^*}, \end{aligned}$$ to see the last line recall that $G_*(t):=H\left(t^{rq^*}\right)$. Combining this with [\[eq 6\]](#eq 6){reference-type="ref" reference="eq 6"} gives $$\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G\left(\frac{|\Phi(u_n(x))-\Phi(u_n(y))|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^N} \leq C\|f\|_{L^1(\Omega)}+C\|f\|_{L^{\overline{H}}(\Omega)}\|\Phi(u_n)\|_{L^{G_*}(\Omega)}^{rq^*}.$$ From [Lemma 8](#lemma embedding){reference-type="ref" reference="lemma embedding"}, we can write $$\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G\left(\frac{|\Phi(u_n(x))-\Phi(u_n(y))|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^N} \leq C\|f\|_{L^1(\Omega)}+C\|f\|_{L^{\overline{H}}(\Omega)}\|\Phi(u_n)\|_{W^{s,G}_0(\Omega)}^{rq^*}.$$ When $\|\Phi(u_n)\|_{W^{s,G}_0(\Omega)}>t_0$, using [Lemma 3](#lemma equivalent norm){reference-type="ref" reference="lemma equivalent norm"}, we get $$\|\Phi(u_n)\|_{W^{s,G}_0(\Omega)}^{p^-} \leq C\|f\|_{L^1(\Omega)}+C\|f\|_{L^{\overline{H}}(\Omega)}\|\Phi(u_n)\|_{W^{s,G}_0(\Omega)}^{rq^*}.$$ From the hypothesis, we have, ${rq^*}<p^-$. This implies that the norm $\|\Phi(u_n)\|_{W^{s,G}_0(\Omega)}$ cannot increase arbitrarily. So, there exists a constant $C>0$, independent of $n$, such that $\|\Phi(u_n)\|_{W^{s,G}_0(\Omega)}\leq C$. By [Lemma 18](#lemma monotone){reference-type="ref" reference="lemma monotone"}, $u_n$ is a monotone increasing sequence. So, we can define $u$ as the pointwise limit of $u_n$. Direct application of Fatou's lemma and [Lemma 3](#lemma equivalent norm){reference-type="ref" reference="lemma equivalent norm"} implies that $\|\Phi(u)\|_{W^{s,G}_0(\Omega)}\leq C$. ◻ ***Proof of [Theorem 12](#main 2){reference-type="ref" reference="main 2"}**.* By [Lemma 18](#lemma monotone){reference-type="ref" reference="lemma monotone"}, $u_n$ is a monotone increasing sequence. So, we can define $u$ as the pointwise limit of $u_n$. Next, we show that this $u$ is the required solution. We know from [Lemma 17](#lemma approx problem){reference-type="ref" reference="lemma approx problem"} that there are $u_n$ which satisfy $$\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u_n(x)-u_n(y)}{|x-y|^s}\right)\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}dxdy =\int_{\Omega}\frac{f_n(x)\phi(x)}{\left(u_n(x)+\frac{1}{n}\right)^{q(x)}}dx.$$ Note that, on $\mbox{supp}(\phi)$, as $f\in L^1(\Omega)$, $$\left| \frac{f_n(x)\phi(x)}{\left(u_n(x)+\frac{1}{n}\right)^{q(x)}} \right|\leq \|l^{-q(\cdot)}\|_{L^\infty} |f||\phi|\in L^1.$$ Hence, by dominated convergence theorem, we get $$\lim_{n\to \infty}\int_{\Omega}\frac{f_n(x)\phi(x)}{\left(u_n(x)+\frac{1}{n}\right)^{q(x)}}=\int_{\Omega}\frac{f(x)\phi(x)}{u(x)^{q(x)}}.$$ So, we need to show that $$\lim_{n\to \infty}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u_n(x)-u_n(y)}{|x-y|^s}\right)\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}dxdy =\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}dxdy.$$ We have, $\Phi(u)\in W^{s,G}_0(\Omega)$ and by [Lemma 7](#poincare){reference-type="ref" reference="poincare"}, it follows that $\Phi(u)\in L^G(\Omega)$. Comparing integrals, where $u>1$, it follows that $u\in L^G(\Omega)$. We see, using [Lemma 19](#lemma estimate){reference-type="ref" reference="lemma estimate"}, $$\begin{aligned} &\left| \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u_n(x)-u_n(y)}{|x-y|^s}\right)\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}dxdy -\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}dxdy\right| \\ &= \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\left|g\left(\frac{u_n(x)-u_n(y)}{|x-y|^s}\right)-g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\right|\frac{|\varphi(x)-\varphi(y)|}{|x-y|^{N+s}}dxdy\\ &\leq C \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{|u_n(x)-u_n(y)|+|u(x)-u(y)|}{|x-y|^s}\right)\frac{|\varphi(x)-\varphi(y)|}{|x-y|^{N+s}}dxdy\\ &=C \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}I_n\quad \mbox{(assume)} \end{aligned}$$ The proof will be complete if we can show that $\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}I_n\to 0$. To do this, first, set $$\mathcal{S_\phi}:=\mbox{supp}\phi,\quad \mbox{and}\quad \mathcal{Q}_\phi:=(\mathbb{R}^N\times\mathbb{R}^N)\setminus(\mathcal{S_\phi}^c\times\mathcal{S_\phi}^c).$$ Now using Hölder's inequality with respect to the measure $\frac{dxdy}{|x-y|^N}$, we get for any compact set $K\subseteq \mathbb{R}^N\times\mathbb{R}^N$, $$\begin{aligned} \iint_{\mathbb{R}^{2N}\setminus K}I_n &=\iint_{\mathcal{Q}_\phi\setminus K}I_n\\ &\leq C \left\|g\left(\frac{|u_n(x)-u_n(y)|+|u(x)-u(y)|}{|x-y|^s}\right)\right\|_{L^{\overline{G}}(\mathcal{Q}_\phi\setminus K,\frac{dxdy}{|x-y|^N})} \left\|\frac{|\varphi(x)-\varphi(y)|}{|x-y|^s}\right\|_{L^G(\mathcal{Q}_\phi\setminus K,\frac{dxdy}{|x-y|^N})}. \end{aligned}$$ Now, if $\left\|g\left(\frac{|u_n(x)-u_n(y)|+|u(x)-u(y)|}{|x-y|^s}\right)\right\|_{L^{\overline{G}}(\mathcal{Q}_\phi\setminus K,\frac{dxdy}{|x-y|^N})}\leq1$, we get $$\iint_{\mathbb{R}^{2N}\setminus K}I_n\leq C \left\|\frac{|\varphi(x)-\varphi(y)|}{|x-y|^s}\right\|_{L^G(\mathcal{Q}_\phi\setminus K,\frac{dxdy}{|x-y|^N})}.$$ Otherwise, we apply [\[lemma equivalent norm,eq 3\]](#lemma equivalent norm,eq 3){reference-type="ref" reference="lemma equivalent norm,eq 3"} to get $$\begin{aligned} &\iint_{\mathbb{R}^{2N}\setminus K}I_n\\ &\leq C \left(\iint_{\mathcal{Q}_\phi\setminus K}\overline{G}\left(g\left(\frac{|u_n(x)-u_n(y)|+|u(x)-u(y)|}{|x-y|^s}\right)\right)\frac{dxdy}{|x-y|^N}\right)^\frac{1}{p^-} \left\|\frac{|\varphi(x)-\varphi(y)|}{|x-y|^s}\right\|_{L^G(\mathcal{Q}_\phi\setminus K,\frac{dxdy}{|x-y|^N})}\\ &\leq C \left(\iint_{\mathcal{Q}_\phi\setminus K}G\left(\frac{|u_n(x)-u_n(y)|+|u(x)-u(y)|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^N}\right)^\frac{1}{p^-} \left\|\frac{|\varphi(x)-\varphi(y)|}{|x-y|^s}\right\|_{L^G(\mathcal{Q}_\phi\setminus K,\frac{dxdy}{|x-y|^N})}\\ &\leq C \left[\iint_{\mathcal{Q}_\phi\setminus K}G\left(\frac{|u_n(x)-u_n(y)|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^N}+\iint_{\mathcal{Q}_\phi\setminus K}G\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^N}\right]^\frac{1}{p^-}\\ &\hspace{200pt}\hfill\times\left\|\frac{|\varphi(x)-\varphi(y)|}{|x-y|^s}\right\|_{L^G(\mathcal{Q}_\phi\setminus K,\frac{dxdy}{|x-y|^N})} . \end{aligned}$$ By [Lemma 18](#lemma monotone){reference-type="ref" reference="lemma monotone"}, there exists $l=l(\mathcal{S}_\phi)>0$ such that for $n$ large enough, $u_n(x)>l$. We now apply [Lemma 20](#lemma MVT){reference-type="ref" reference="lemma MVT"} on the two integrands of the last line to get $$\begin{aligned} \iint_{\mathbb{R}^{2N}\setminus K}I_n &\leq C \left[\iint_{\mathcal{Q}_\phi\setminus K}G\left(\frac{|\Phi(u_n)(x)-\Phi(u_n)(y)|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^N}\right.\\ &\left. +\iint_{\mathcal{Q}_\phi\setminus K}G\left(\frac{|\Phi(u(x))-\Phi(u(y))|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^N}\right]^\frac{1}{p^+}\left\|\frac{|\varphi(x)-\varphi(y)|}{|x-y|^s}\right\|_{L^G(\mathcal{Q}_\phi\setminus K,\frac{dxdy}{|x-y|^N})} . \end{aligned}$$ By [\[lemma equivalent norm,lemma bound\]](#lemma equivalent norm,lemma bound){reference-type="ref" reference="lemma equivalent norm,lemma bound"}, it is clear that $$\iint_{\mathbb{R}^{2N}\setminus K}I_n\\ \leq C \left\|\frac{|\varphi(x)-\varphi(y)|}{|x-y|^s}\right\|_{L^G(\mathcal{Q}_\phi\setminus K,\frac{dxdy}{|x-y|^N})} .$$ Since $\phi\in C_c^\infty(\Omega)$, for a fixed $\varepsilon>0$, there exists $K=K(\varepsilon)$ such that $$\iint_{\mathbb{R}^{2N}\setminus K}I_n< \frac{\varepsilon}{2}.$$ We, now have to estimate $\iint_{K}I_n$. For this, we use Vitali's convergence theorem. Let $E\subseteq K$. Arguing as above, we can get $$\iint_{E}I_n\\ \leq C \left\|\frac{|\varphi(x)-\varphi(y)|}{|x-y|^s}\right\|_{L^G(E,\frac{dxdy}{|x-y|^N})} .$$ This shows that the integrand in LHS is uniformly integrable, that is $\iint_{E}I_n\to 0$ as $\mathcal{L}^N(E)\to 0$. Applying Vitali's convergence theorem, we get for large enough $n$, $\iint_{E}I_n<\frac{\varepsilon}{2}$. So, from [\[eq 6\]](#eq 6){reference-type="ref" reference="eq 6"}, we get $\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}I_n\to 0$ as $n\to \infty$, hence the proof follows. ◻ ***Proof of [Theorem 13](#main 3){reference-type="ref" reference="main 3"}**.* Let $u$ be a solution of [\[eq problem\]](#eq problem){reference-type="ref" reference="eq problem"} obtained through [\[main 1,main 2\]](#main 1,main 2){reference-type="ref" reference="main 1,main 2"}. Then $u$ is pointwise limit of a sequence of solutions, $u_n$, of [\[eq approx problem\]](#eq approx problem){reference-type="ref" reference="eq approx problem"}. Also, by [Lemma 18](#lemma monotone){reference-type="ref" reference="lemma monotone"}, there exists $l(K)>0$ for any compact set $K\subseteq \Omega$ such that $$u(x)\geq l(K)>0\quad \mbox{for almost all } x\in K.$$ This implies that there exists some $C_K>0$ such that $u^{-q(x)}(x)\leq C_K$ for all $x\in K$. Fix $x_0\in \Omega$ and $r>0$ such that $B:=B(x_0,r)\subset \overline{B(x_0,r)}\subset \Omega$. Again, since $u$ is a weak solution of [\[eq problem\]](#eq problem){reference-type="ref" reference="eq problem"}, this implies that for any $\varphi\in C_c^{\infty}(B(x_0,r))$, where , with $\varphi\geq 0$, $$\begin{gathered} \label{eq 5} \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{(\varphi(x)-\varphi(y))}{|x-y|^{N+s}}dxdy =\int_{B}f(x)u(x)^{-q(x)}\phi(x)dx\\ \leq C_B\int_{B}f(x)\phi(x)dx = \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{v(x)-v(y)}{|x-y|^s}\right)\frac{(\varphi(x)-\varphi(y))}{|x-y|^{N+s}}dxdy, \end{gathered}$$ where $v\in W^{s,G}(B)\cap L^\infty(B)$, is a solution to the problem $$\begin{cases} (-\Delta_g)^sv=C_Bf, \quad \mbox{ in } B,\\ v>0, \mbox{ in } B,\\ v=0, \mbox{ in } \mathbb{R}^N\setminus B \end{cases}$$ obtained through [Lemma 16](#laplace equation){reference-type="ref" reference="laplace equation"}. By using [Lemma 14](#comparison){reference-type="ref" reference="comparison"}, we can conclude that $u\leq v$ in $B$ if $u$ is continuous on $\mathbb{R}^N$. That is $u\in L^\infty_{loc} (\Omega)$ provided $u$ is continuous on $\mathbb{R}^N$. Again, since we have, from [\[eq 5\]](#eq 5){reference-type="ref" reference="eq 5"}, $$\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{(\varphi(x)-\varphi(y))}{|x-y|^{N+s}}dxdy \leq C\int_{B}\phi(x)dx,$$ defining the sets $$\begin{aligned} U_0&:=\left\{ (x,y)\in\mathbb{R}^N\times\mathbb{R}^N\ \Big|\ \frac{|u(x)-u(y)|}{|x-y|^s}\geq 1 \right\},\\ U_j&:=\left\{ (x,y)\in\mathbb{R}^N\times\mathbb{R}^N\ \Big|\ \frac{1}{j+1}\leq\frac{|u(x)-u(y)|}{|x-y|^s}<\frac{1}{j} \right\}\quad \mbox{ for } j\geq 1, \end{aligned}$$ we get from [Lemma 2](#lemma delta2){reference-type="ref" reference="lemma delta2"} that $$\begin{aligned} C\int_{B}\phi(x)dx &\geq \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{(\varphi(x)-\varphi(y))}{|x-y|^{N+s}}dxdy\\ &= \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{u(x)-u(y)}{|x-y|^s}\frac{(\varphi(x)-\varphi(y))}{(u(x)-u(y))|x-y|^N}dxdy\\ &\geq p^-\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right)\frac{(\varphi(x)-\varphi(y))}{(u(x)-u(y))|x-y|^N}dxdy\\ &= p^-\sum_{j=0}^\infty j^{p^+}G(\frac{1}{j+1})\iint_{U_j} \frac{|u(x)-u(y)|^{p^+-2}(u(x)-u(y))(\phi(x)-\phi(y))}{|x-y|^{N+sp^+}}dxdy\\ &\geq p^-\sum_{j=0}^\infty \frac{j^{p^+}}{(j+1)^{p^+}}G(1)\iint_{U_j} \frac{|u(x)-u(y)|^{p^+-2}(u(x)-u(y))(\phi(x)-\phi(y))}{|x-y|^{N+sp^+}}dxdy\\ &\geq C\int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u(x)-u(y)|^{p^+-2}(u(x)-u(y))(\phi(x)-\phi(y))}{|x-y|^{N+sp^+}}dxdy . \end{aligned}$$ We can now apply Corollary 5.5 of [@AnSuSq] to conclude that there is some $\alpha\in(0,1)$ such that $u\in C^\alpha(B)$. This completes the proof. ◻ # Acknowledgement {#acknowledgement .unnumbered} The first author was funded by MATRICS (DST, INDIA) project MTR/2020/000594.\ The second and the third author were funded by Academy of Finland grant: Geometrinen Analyysi (21000046081). # Author Contribution {#author-contribution .unnumbered} All the authors have contributed equally to the article. # Conflict of Interest {#conflict-of-interest .unnumbered} The authors have no competing interests to declare that are relevant to the content of this article.

arxiv_math

--- abstract: | An irredundant base of a group $G$ acting faithfully on a finite set $\Gamma$ is a sequence of points in $\Gamma$ that produces a strictly descending chain of pointwise stabiliser subgroups in $G$, terminating at the trivial subgroup. Suppose that $G$ is $\operatorname{S}_{n}$ or $\operatorname{A}_{n}$ acting primitively on $\Gamma$, and that the point stabiliser is primitive in its natural action on $n$ points. We prove that the maximum size of an irredundant base of $G$ is $O\left(\sqrt{n}\right)$, and in most cases $O\left((\log n)^2\right)$. We also show that these bounds are best possible. author: - Colva M. Roney-Dougal and Peiran Wu title: Irredundant bases for the symmetric group --- **Keywords** irredundant base, symmetric group **MSC2020** 20B15; 20D06, 20E15 # Introduction Let $G$ be a finite group that acts faithfully and transitively on a set $\Gamma$ with point stabiliser $H$. A sequence $(\gamma_1, \ldots, \gamma_l)$ of points of $\Gamma$ is an *irredundant base* for the action of $G$ on $\Gamma$ if $$\label{equation:irredundant-base} G > G_{\gamma_1} > G_{\gamma_1, \gamma_2} > \cdots > G_{\gamma_1, \ldots, \gamma_{l}} = 1.$$ Let $\operatorname{b}(G, H)$ and $\operatorname{I}(G, H)$ denote the minimum and the maximum sizes of an irredundant base in $\Gamma$ for $G$ respectively. Recently, Gill & Liebeck showed in [@gill_liebeck_2023] that if $G$ is an almost simple group of Lie type of rank $r$ over the field $\mathbb{F}_{p^f}$ of characteristic $p$ and $G$ is acting primitively, then $$\operatorname{I}(G, H) \leqslant 177 r^8 + \Omega(f),$$ where $\Omega(f)$ is the number of prime factors of $f$, counted with multiplicity. Suppose now that $G$ is the symmetric group $\operatorname{S}_{n}$ or the alternating group $\operatorname{A}_{n}$. An upper bound for $\operatorname{I}(G, H)$ is the maximum length of a strictly descending chain of subgroups in $G$, known as the *length*, $\ell(G)$, of $G$. Define $\varepsilon(G) \coloneqq \ell(G / \mathop{\mathrm{soc}}G)$. Cameron, Solomon, and Turull proved in [@cameron_solomon_turull_1989] that $$\label{equation:length-symmetric-group} \nonumber \ell(G) = \left\lfloor \frac{3n - 3}{2} \right\rfloor - b_n + \varepsilon(G),$$ where $b_n$ denotes the number of $1$s in the binary representation of $n$. For $n \geqslant 2$, this gives $$\label{equation:length-symmetric-group-inequality} \ell(G) \leqslant \frac{3}{2} n - 3 + \varepsilon(G).$$ This type of upper bound is best possible for such $G$ in general, in that for the natural action of $\operatorname{S}_{n}$ or $\operatorname{A}_{n}$ on $n$ points, the maximum irredundant base size is $n - 2 + \varepsilon(G)$. A recent paper [@gill_lodà_2022] by Gill & Lodà determined the exact values of $\operatorname{I}(G, H)$ when $H$ is maximal and intransitive in its natural action on $n$ points, and in each case $\operatorname{I}(G, H) \geqslant n - 3 + \varepsilon(G)$. In this article, we present improved upper bounds for $\operatorname{I}(G, H)$ in the case where $H$ is primitive. Note that whenever we refer to the "primitivity" of a subgroup of $G$, we do so with respect to the natural action of $G$ on $n$ points. We say that a primitive subgroup $H$ of $G$ is *large* if there are integers $m$ and $k$ such that $H$ is $\left(\operatorname{S}_{m} \wr \operatorname{S}_{k}\right) \cap G$ in product action or there are integers $m$ and $r$ such that $H$ is $\operatorname{S}_{m} \cap \, G$ acting on the $r$-subsets of a set of size $m$. Logarithms are taken to the base $2$. **Theorem 1**. *Suppose $G$ is $\operatorname{S}_{n}$ or $\operatorname{A}_{n}$ ($n \geqslant 7$) and $H \neq \operatorname{A}_{n}$ is a primitive maximal subgroup of $G$.* 1. *Either $\operatorname{I}(G, H) < (\log n)^2 + \log n + 1$, or $H$ is large and $\operatorname{I}(G, H) < 3 \sqrt{n} - 1$.* 2. *There are infinitely many such pairs $(G, H)$ for which $\operatorname{I}(G, H) \geqslant \sqrt{n}$.* 3. *There are infinitely many such pairs $(G, H)$ for which $\operatorname{I}(G, H) > \left(\log n\right)^{2} / (2 (\log 3)^2) + \log n / (2 \log 3)$ and $H$ is not large.* We also state upper bounds for $\operatorname{I}(G, H)$ in terms of the degree $t$ of the action of $G$. It is easy to show that $\operatorname{I}(G, H) \leqslant \operatorname{b}(G, H) \log t$. Burness, Guralnick, and Saxl showed in [@burness_guralnick_saxl_2011] that if $G$ and $H$ are as in , then with a finite number of exceptions, $\operatorname{b}(G, H) = 2$, from which it follows that $$\operatorname{I}(G, H) \leqslant 2 \log t.$$ Similar $O(\log t)$ upper bounds on the maximum irredundant base size were recently shown to hold for all non-large-base subgroups [@gill_lodà_spiga_2022; @kelsey_roney-dougal_2022], raising the question of whether such bounds are best possible in our case. Using , we shall obtain better bounds in terms of $t$. **Corollary 2**. 1. *There exist constants $c_1, c_2 \in \mathbb{R}_{> 0}$ such that, if $G$ is $\operatorname{S}_{n}$ or $\operatorname{A}_{n}$ ($n \geqslant 7$) and $H \neq \operatorname{A}_{n}$ is a primitive maximal subgroup of $G$ of index $t$, then either $\operatorname{I}(G, H) < c_1 (\log \log t)^2$, or $H$ is large and $\operatorname{I}(G, H) < c_2 \left(\log t / \log \log t\right)^{1/2}$.* 2. *There is a constant $c_3 \in \mathbb{R}_{> 0}$ and infinitely many such pairs $(G, H)$ for which $\operatorname{I}(G, H) > c_3 \left(\log t / \log \log t\right)^{1/2}$.* 3. *There is a constant $c_4 \in \mathbb{R}_{> 0}$ and infinitely many such pairs $(G, H)$ for which $\operatorname{I}(G, H) > c_4 (\log \log t)^2$ and $H$ is not large.* **Remark 3**. We may take $c_1 = 3.5$, $c_2 = 6.1$, $c_3 = 1$, $c_4 = 0.097$. If we assume $n > 100$, then $c_1 = 1.2$ and $c_2 = 4.4$ suffice. A sequence $\mathcal{B}$ of points in $\Gamma$ is *independent* if no proper subsequence $\mathcal{B}'$ satisfies $G_{(\mathcal{B}')} = G_{(\mathcal{B})}$. The maximum size of an independent sequence for the action of $G$ on $\Gamma$ is denoted $\operatorname{H}(G, H)$. It can be shown that $\operatorname{b}(G, H) \leqslant \operatorname{H}(G, H) \leqslant \operatorname{I}(G, H)$. Another closely related property of the action is the *relational complexity*, denoted $\operatorname{RC}(G, H)$, a concept which originally arose in model theory. Cherlin, Martin, and Saracino defined $\operatorname{RC}(G, H)$ in [@cherlin_martin_saracino_1996] under the name "arity" and showed that $\operatorname{RC}(G, H) \leqslant \operatorname{H}(G, H) + 1$. **Corollary 4**. *Suppose $G$ is $\operatorname{S}_{n}$ or $\operatorname{A}_{n}$ ($n \geqslant 7$) and $H \neq \operatorname{A}_{n}$ is a primitive maximal subgroup of $G$. Then either $\operatorname{RC}(G, H) < (\log n)^2 + \log n + 2$, or $H$ is large and $\operatorname{RC}(G, H) < 3 \sqrt{n}$.* The maximal subgroups of the symmetric and alternating groups were classified in [@aschbacher_scott_1985; @liebeck_praeger_saxl_1987]. In order to prove statements (i) and (ii) of Theorem 1, we examine two families of maximal subgroups in more detail and determine lower bounds on the maximum irredundant base size, given in the next two results. **Theorem 5**. *Let $p$ be an odd prime number and $d$ a positive integer such that $p^d \geqslant 7$ and let $n = p^d$. Suppose $G$ is $\operatorname{S}_{n}$ or $\operatorname{A}_{n}$ and $H$ is $\operatorname{AGL}_{d}(p) \cap G$. If $d = 1$, then $$\operatorname{I}(G, H) = 1 + \Omega(p - 1) + \varepsilon(G).$$ If $d \geqslant 2$ and $p = 3, 5$, then $$\frac{d (d + 1)}{2} + d - 1 + \varepsilon(G) \leqslant \operatorname{I}(G, H) < \frac{d (d + 1)}{2} (1 + \log p) + \varepsilon(G).$$ If $d \geqslant 2$ and $p \geqslant 7$, then $$\frac{d (d + 1)}{2} + d \, \Omega(p - 1) - 1 + \varepsilon(G) \leqslant \operatorname{I}(G, H) < \frac{d (d + 1)}{2} (1 + \log p) + \varepsilon(G).$$* **Theorem 6**. *Let $m \geqslant 5$ and $k \geqslant 2$ be integers and let $n = m^k$. Suppose $G$ is $\operatorname{S}_{n}$ or $\operatorname{A}_{n}$ and $H$ is $(\operatorname{S}_{m} \wr \operatorname{S}_{k}) \cap G$ in product action. Then $$1 + (m - 1) (k - 1) + \varepsilon(G) \leqslant \operatorname{I}(G, H) \leqslant \frac{3}{2} m k - \frac{1}{2} k - 1.$$* After laying out some preliminary results in , we shall prove in and respectively, before proving and in . # The maximum irredundant base size {#section:preliminaries} In this section, we collect two general lemmas. Let $G$ be a finite group acting faithfully and transitively on a set $\Gamma$ with point stabiliser $H$. If $(\gamma_1, \ldots, \gamma_l)$ is an irredundant base of $G$, then it satisfies [\[equation:irredundant-base\]](#equation:irredundant-base){reference-type="eqref" reference="equation:irredundant-base"}. The tail of the chain in [\[equation:irredundant-base\]](#equation:irredundant-base){reference-type="eqref" reference="equation:irredundant-base"} is a strictly descending chain of subgroups in $G_{\gamma_1}$, which is conjugate to $H$. Therefore, $$\operatorname{I}(G, H) \leqslant \ell(H) + 1 \leqslant \Omega(\left\lvert H \right\rvert) + 1.$$ To obtain a lower bound for $\operatorname{I}(G, H)$, one approach is to look for a large explicit irredundant base. The following lemma says it suffices to find a long chain of subgroups in $G$ such that every subgroup in the chain is a pointwise stabiliser of some subset in $\Gamma$. **Lemma 7**. *Let $l$ be the largest natural number such that there are subsets $\Delta_0, \Delta_1, \ldots, \Delta_l \subseteq \Gamma$ satisfying $$G_{(\Delta_0)} > G_{(\Delta_1)} > \cdots > G_{(\Delta_l)}.$$ Then $\operatorname{I}(G, H) = l$.* *Proof.* Since $l$ is maximal, we may assume that $\Delta_0 = \emptyset$ and $\Delta_l = \Gamma$ and that $\Delta_{i - 1} \subseteq \Delta_i$, replacing $\Delta_i$ with $\Delta_1 \cup \cdots \cup \Delta_i$ if necessary. For each $i \in \{1, \ldots, l\}$, write $\Delta_i \setminus \Delta_{i - 1} = \{\gamma_{i, 1}, \ldots, \gamma_{i, m_i}\}$. Then $(\gamma_{1, 1}, \ldots, \gamma_{1, m_1}, \allowbreak \gamma_{2, 1}, \ldots, \gamma_{2, m_2}, \allowbreak \ldots, \gamma_{l, 1}, \ldots, \gamma_{l, m_l})$ is a base for $G$ and every subgroup $G_{(\Delta_i)}$ appears in the corresponding chain of point stabilisers. Therefore, by removing all redundant points, we obtain an irredundant base of size at least $l$, so $\operatorname{I}(G, H) \geqslant l$. On the other hand, given any irredundant base $(\gamma_1, \ldots, \gamma_m)$ of $G$, we can take $\Delta_i \coloneqq \{\gamma_1, \ldots, \gamma_i\}$. Therefore, $\operatorname{I}(G, H) = l$. ◻ Once we have an upper or lower bound for $\operatorname{I}(G, H)$, we can easily obtain a corresponding bound for the maximum irredundant base size of various subgroups of $G$. **Lemma 8**. *Suppose $M$ is a subgroup of $\operatorname{S}_{n}$ with $M \nleqslant \operatorname{A}_{n}$. Then $$\nonumber \operatorname{I}(\operatorname{S}_{n}, M) - 1 \leqslant \operatorname{I}(\operatorname{A}_{n}, M \cap \operatorname{A}_{n}) \leqslant \operatorname{I}(\operatorname{S}_{n}, M).$$* *Proof.* This follows immediately from [@gill_lodà_spiga_2022 Lemma 2.8] and [@kelsey_roney-dougal_2022 Lemma 2.3]. ◻ # The affine case {#section:affine-case} In this section, we prove . The upper bounds will follow easily from examinations of group orders. Therefore, we focus most of our efforts on the construction of an irredundant base, leading to the lower bounds. Let $p$ be a prime number and $d$ be an integer such that $p^d \geqslant 7$ and let $V$ be a $d$-dimensional vector space over the field $\mathbb{F}_{p}$. Let $G$ be $\operatorname{Sym}(V)$ or $\operatorname{Alt}(V)$. Consider the affine group $\operatorname{AGL}(V)$, the group of all invertible affine transformations of $V$, and let $H \coloneqq \operatorname{AGL}(V) \cap G$. **Theorem 9** ([@liebeck_praeger_saxl_1987]). *The subgroup $H$ is maximal in $G$ (with $p^d \geqslant 7$) if and only if one of the following holds:* 1. *$d \geqslant 2$ and $p \geqslant 3$;* 2. *$G = \operatorname{Sym}(V)$, $d = 1$ and $p \geqslant 7$;* 3. *$G = \operatorname{Alt}(V)$, $d \geqslant 3$ and $p = 2$;* 4. *$G = \operatorname{Alt}(V)$, $d = 1$, and $p = 13, 19$ or $p \geqslant 29$.* In this section, we only consider the case where $p$ is odd. Owing to , we shall assume $G = \operatorname{Sym}(V)$ and $H = \operatorname{AGL}(V)$ for now. In the light of , we introduce a subgroup $T$ of diagonal matrices and look for groups containing $T$ that are intersections of $G$-conjugates of $H$ () and subgroups of $T$ that are such intersections (), before finally proving (). ## Subspace stabilisers and the diagonal subgroup {#section:subspace-stabilisers} Let $T$ be the subgroup of all diagonal matrices in $\operatorname{GL}(V)$ with respect to a basis $\mathbf{b}_1, \ldots, \mathbf{b}_d$. Let $\mu$ be a primitive element of $\mathbb{F}_{p}$. We now find a strictly descending chain of groups from $\operatorname{Sym}(V)$ to $T$ consisting of intersections of $G$-conjugates of $H$. We treat the cases $d = 1$ and $d \geqslant 2$ separately. **Lemma 10**. *Suppose $d = 1$ and $G = \operatorname{Sym}(V)$. Then there exists $x \in G$ such that $H \cap H^{x} = T$.* *Proof.* Since $V$ is $1$-dimensional, $\operatorname{GL}(V) = T$ is generated by the scalar multiplication $m_\mu$ by $\mu$. Let $\mathbf{u}\in V \setminus \{ \mathbf{0}\}$ and let $t_\mathbf{u}$ be the translation by $\mathbf{u}$. Then $H = \left\langle t_\mathbf{u} \right\rangle \rtimes \left\langle m_\mu \right\rangle$ is the normaliser of $\left\langle t_\mathbf{u} \right\rangle$ in $G$ and $\left\langle t_\mathbf{u} \right\rangle$ is a characteristic subgroup of $H$. Hence $H$ is self-normalising in $G$. Define $$x \coloneqq (\mathbf{u}~~\mu^{-1} \mathbf{u}) (\mu \mathbf{u}~~\mu^{-2} \mathbf{u}) \cdots (\mu^{\frac{p - 3}{2}} \mathbf{u}~~\mu^{-\frac{p - 1}{2}} \mathbf{u}) \in G.$$ Then $x \notin H$ and so $x$ does not normalise $H$. But $x$ normalises $\left\langle m_\mu \right\rangle$, as ${m_\mu}^x = {m_\mu}^{-1}$. Therefore, $$T = \left\langle m_\mu \right\rangle \leqslant H \cap H^{x} < H.$$ Since the index $\left\lvert H : T \right\rvert = p$ is prime, $H \cap H^x = T$. ◻ The following two lemmas concern the case $d \geqslant 2$. An affine subspace of $V$ is a subset of the form $\mathbf{v}+ W$, where $\mathbf{v}\in V$ and $W$ is a vector subspace of $V$. The (affine) dimension of $\mathbf{v}+ W$ is the linear dimension of $W$. For an affine transformation $h = g t_\mathbf{u}$ with $g \in \operatorname{GL}(V)$ and $t_\mathbf{u}$ denoting the translation by some $\mathbf{u}\in V$, if $\mathop{\mathrm{fix}}(h)$ is non-empty, then $\mathop{\mathrm{fix}}(h)$ is an affine subspace of $V$, since $\mathop{\mathrm{fix}}(h) = \mathbf{v}+ \ker(g - \mathop{\mathrm{id}}_V)$ for any $\mathbf{v}\in \mathop{\mathrm{fix}}(h)$. **Lemma 11**. *Suppose $d \geqslant 2$, $p \geqslant 3$, and $G = \operatorname{Sym}(V)$. Let $W$ be a proper, non-trivial subspace of $V$ and let $K < \operatorname{GL}(V)$ be the setwise stabiliser of $W$. Then there exists $x \in G$ such that $H \cap H^x = K$.* *Proof.* Let $\lambda \in \mathbb{F}_{p}^\times \setminus \{1\}$ and define $x \in \operatorname{Sym}(V)$ by setting $$\mathbf{v}^x \coloneqq \begin{cases} \lambda \mathbf{v}, & \text{if } \mathbf{v}\in W, \\ \mathbf{v}, & \text{otherwise}. \end{cases}$$ We first show that $K = \operatorname{C}_{H}(x)$ and then that $H \cap H^x = \operatorname{C}_{H}(x)$. Firstly, let $g \in K$. For all $\mathbf{v}\in W$, we calculate that $\mathbf{v}^{g^{x}} = (\lambda^{-1} \mathbf{v})^{g x} = (\lambda^{-1} \mathbf{v}^g)^x = \mathbf{v}^g$. For all $\mathbf{v}\in V \setminus W$, we see that $\mathbf{v}^{g^{x}} = \mathbf{v}^{g x} = \mathbf{v}^g$. Hence $g^x = g$, and so $K \leqslant \operatorname{C}_{H}(x)$. Now, let $h$ be an element of $\operatorname{C}_{H}(x)$ and write $h = g t_\mathbf{u}$ with $g \in \operatorname{GL}(V)$ and $\mathbf{u}\in V$, so that $h^{-1} = t_{-\mathbf{u}} g^{-1}$. Suppose for a contradiction that there exists $\mathbf{v}\in W \setminus \{ \mathbf{0}\}$ with $\lambda \mathbf{v}^g + \mathbf{u}\notin W$. Then $$\mathbf{v}= \mathbf{v}^{x h x^{-1} h^{-1}} = (\lambda \mathbf{v})^{h x^{-1} h^{-1}} = (\lambda \mathbf{v}^g + \mathbf{u})^{x^{-1} h^{-1}} = (\lambda \mathbf{v}^g + \mathbf{u})^{h^{-1}} = \lambda \mathbf{v}.$$ Since $\lambda \neq 1$, this is a contradiction and so for all $\mathbf{v}\in W$, $$\mathbf{v}= (\lambda \mathbf{v}^g + \mathbf{u})^{x^{-1} h^{-1}} = (\mathbf{v}^g + \lambda^{-1} \mathbf{u})^{h^{-1}} = \mathbf{v}+ (\lambda^{-1} - 1) \mathbf{u}^{g^{-1}}.$$ Hence $\mathbf{u}= \mathbf{0}$ and $\mathbf{v}^g \in W$. Therefore, $h = g t_\mathbf{0}$ stabilises $W$, whence $h \in K$. Thus, $\operatorname{C}_{H}(x) = K$. Since $\operatorname{C}_{H}(x) \leqslant H \cap H^x$, it remains to show that $H \cap H^x \leqslant \operatorname{C}_{H}(x)$. Suppose otherwise. Then there is some $h \in H \cap H^x$ such that $h' \coloneqq x h x^{-1} h^{-1} \neq 1$. The set $\mathop{\mathrm{fix}}(h')$ is either empty or an affine subspace of dimension at most $d - 1$. Moreover, for any $\mathbf{v}\in V$, if $\mathbf{v}\notin (W \setminus \{\mathbf{0}\}) \cup W^{h^{-1}}$, then $x$ fixes both $\mathbf{v}$ and $\mathbf{v}^h$, and $\mathbf{v}^{h'} = \mathbf{v}^{h x^{-1} h^{-1}} = \mathbf{v}^{h h^{-1}} = \mathbf{v}$, whence $\mathbf{v}\in \mathop{\mathrm{fix}}(h')$. Therefore, $$V = (W \setminus \{\mathbf{0}\}) \cup W^{h^{-1}} \cup \mathop{\mathrm{fix}}(h').$$ Then $$p^d = \left\lvert V \right\rvert \leqslant \left\lvert W \setminus \{\mathbf{0}\} \right\rvert + \left\lvert W^{h^{-1}} \right\rvert + \left\lvert \mathop{\mathrm{fix}}(h') \right\rvert \leqslant (p^{d - 1} - 1) + p^{d - 1} + p^{d - 1} = 3 p^{d - 1} - 1.$$ This is a contradiction as $p \geqslant 3$, and so $H \cap H^x = \operatorname{C}_{H}(x) = K$. ◻ We now construct a long chain of subgroups of $G$ by intersecting subspace stabilisers. **Lemma 12**. *Suppose $d \geqslant 2$ and $G = \operatorname{Sym}(V)$. Let $l_1 \coloneqq d (d + 1) / 2 - 1$. Then there exist subspace stabilisers $K_1, \ldots, K_{l_1}$ such that $$\label{equation:diagonal-subgroup-as-intersection-of-subspace-stabilisers} G > H > K_1 > K_1 \cap K_2 > \cdots > \bigcap_{i = 1}^{l_1} K_i = T.$$* *Proof.* Let $\mathcal{I} \coloneqq \{(i, j) \mid i, j \in \{1, \ldots, d\}, i \leqslant j\}\setminus \{ (1, d) \}$ be ordered lexicographically. Note that $\left\lvert \mathcal{I} \right\rvert = l_1$. For each $(i, j) \in \mathcal{I}$, let $K_{i, j}$ be the stabiliser in $\operatorname{GL}(V)$ of $\left\langle \mathbf{b}_i, \mathbf{b}_{i + 1} \ldots, \mathbf{b}_j \right\rangle$ and define $\mathcal{I}_{i, j} \coloneqq \{ (k, l) \in \mathcal{I} \mid (k, l) \leqslant (i, j) \}.$ Since $T \leqslant K_{i, j}$ for all $i, j$, we see that $$\nonumber T \leqslant \bigcap_{(i, j) \in \mathcal{I}} K_{i, j} \leqslant \bigcap_{i = 1}^{d} K_{i, i} = T.$$ Hence equality holds, proving the final equality in [\[equation:diagonal-subgroup-as-intersection-of-subspace-stabilisers\]](#equation:diagonal-subgroup-as-intersection-of-subspace-stabilisers){reference-type="eqref" reference="equation:diagonal-subgroup-as-intersection-of-subspace-stabilisers"}. We now show that, for all $(i, j) \in \mathcal{I}$, $$\bigcap_{\mathclap{(k, l) \in \mathcal{I}_{(i, j)} \setminus \{(i, j)\}}} \; K_{k, l} \; > \quad \bigcap_{\mathclap{(k, l) \in \mathcal{I}_{(i, j)}}} \; K_{k, l}.$$ For $1 \leqslant j < d$, let $g_{1, j}$ be the linear map that sends $\mathbf{b}_j$ to $\mathbf{b}_j + \mathbf{b}_{j + 1}$ and fixes $\mathbf{b}_k$ for $k \neq j$. Then $g_{1, j}$ stabilises $\left\langle \mathbf{b}_1 \right\rangle, \ldots, \left\langle \mathbf{b}_{j - 1} \right\rangle$ and any sum of these subspaces, but not $\left\langle \mathbf{b}_1, \ldots, \mathbf{b}_j \right\rangle$. Hence $g_{1, j} \in K_{1, l}$ for all $l < j$ but $g_{1, j} \notin K_{1, j}$. For $2 \leqslant i \leqslant j \leqslant d$, let $g_{i, j}$ be the linear map that sends $\mathbf{b}_j$ to $\mathbf{b}_{i - 1} + \mathbf{b}_j$ and fixes $\mathbf{b}_k$ for $k \neq j$. Then $g_{i, j}$ stabilises $\left\langle \mathbf{b}_1 \right\rangle, \ldots, \left\langle \mathbf{b}_{j - 1} \right\rangle, \left\langle \mathbf{b}_j, \mathbf{b}_{i - 1} \right\rangle, \left\langle \mathbf{b}_{j + 1} \right\rangle, \ldots, \left\langle \mathbf{b}_d \right\rangle$ and any sum of these subspaces, but not $\left\langle \mathbf{b}_i, \ldots, \mathbf{b}_j \right\rangle$. Hence $g_{i, j} \in K_{k, l}$ for all $(k, l) < (i, j)$ but $g_{i, j} \notin K_{i, j}$. Therefore, the $K_{i, j}$'s, ordered lexicographically by the subscripts, are as required. ◻ We have now found the initial segment of an irredundant base of $\operatorname{Sym}(V)$. The next subsection extends this to a base. ## Subgroups of the diagonal subgroup {#section:subgroups-of-diagonal-subgroup} We now show that, with certain constraints on $p$, every subgroup of $T$ is an intersection of $G$-conjugates of $T$, and hence, by , an intersection of $G$-conjugates of $H$. We first prove a useful result about subgroups of the symmetric group generated by a $k$-cycle. **Lemma 13**. *Let $s \in \operatorname{S}_{m}$ be a cycle of length $k < m$ and let $a$ be a divisor of $k$. Suppose that $(k, a) \neq (4, 2)$. Then there exists $x \in \operatorname{S}_{m}$ such that $$\left\langle s \right\rangle \cap \left\langle s \right\rangle^x = \left\langle s^a \right\rangle.$$* *Proof.* Without loss of generality, assume $s = (1~2~\cdots~k)$ and $a > 1$. If $a = k$, then take $x \coloneqq (1~m)$, so that $\left\langle s \right\rangle \cap \left\langle s \right\rangle^x = 1$, as $m \notin \mathop{\mathrm{supp}}(s^i)$ and $m \in \mathop{\mathrm{supp}}((s^i)^x)$ for all $1 \leqslant i < k$. Hence we may assume $a < k$ and $k \neq 4$. We find that $$s^a = (1~~a + 1~~\cdots~~k - a + 1) (2~~a + 2~~\cdots~~k - a + 2) \cdots (a~~2a~~\cdots~~k).$$ Let $$x \coloneqq (1~~2~~\cdots~~a) (a + 1~~a + 2~~\cdots~~2a) \cdots (k - a + 1~~k - a + 2~~\cdots~~k).$$ Then $\left( s^a \right)^x = s^a$. Hence $\left\langle s^a \right\rangle = \left\langle s^a \right\rangle^x \leqslant \left\langle s \right\rangle \cap \left\langle s \right\rangle^x$. To prove that equality holds, suppose $\left\langle s^a \right\rangle < \left\langle s \right\rangle \cap \left\langle s \right\rangle^x$. Then there exists $b \in \{1, \ldots, a - 1\}$ such that $(s^b)^x = s^c$ for some $c$ not divisible by $a$. Computing $$1^{s^c} = 1^{x^{-1} s^b x} = a^{s^b x} = (a + b)^{x} = a + b + 1 = 1^{s^{a + b}}.$$ Therefore, $$\label{equation:cyclic-subgroup-proof-1} 2^{s^c} = 2^{s^{a + b}} = \begin{cases} a + b + 2, & \text{if } b \neq a - 1 \text{ or } k > 2a, \\ 1, & \text{if } b = a - 1 \text{ and } k = 2a. \end{cases}$$ On the other hand, $$\label{equation:cyclic-subgroup-proof-2} 2^{x^{-1} s^b x} = 1^{s^b x} = (b + 1)^{x} = \begin{cases} b + 2, & \text{if } b \neq a - 1, \\ 1, & \text{if } b = a - 1. \end{cases}$$ Comparing [\[equation:cyclic-subgroup-proof-1\]](#equation:cyclic-subgroup-proof-1){reference-type="eqref" reference="equation:cyclic-subgroup-proof-1"} and [\[equation:cyclic-subgroup-proof-2\]](#equation:cyclic-subgroup-proof-2){reference-type="eqref" reference="equation:cyclic-subgroup-proof-2"}, we see that $b = a - 1$ and $k = 2a$. Hence $a^{s^c} = a^{s^{a + b}} = a - 1$, whereas $$a^{x^{-1} s^b x} = (a - 1)^{s^b x} = (2a - 2)^{x} = 2a - 1$$ ($x$ sends $2a - 2$ to $2a - 1$ as $k \neq 4$), a contradiction. The result follows. ◻ Recall from the subgroup $T$ of $\operatorname{GL}(V)$ and the primitive element $\mu$ of $\mathbb{F}_{p}$. For each $i \in \{1, \ldots, d\}$, let $g_i \in \operatorname{GL}(V)$ send $\mathbf{b}_i$ to $\mu \mathbf{b}_i$ and fix $\mathbf{b}_j$ for $j \neq i$. Then $T = \left\langle g_1, \ldots, g_d \right\rangle$. **Lemma 14**. *Suppose $d \geqslant 1$, $p \geqslant 3$, and $G = \operatorname{Sym}(V)$. Let $i \in \{1, \ldots, d\}$ and let $a$ be a divisor of $(p - 1)$ with $(p, a) \neq (5, 2)$. Then there exists $x \in G$ such that $$T \cap T^x = \left\langle g_1, \ldots, g_{i - 1}, {g_i}^a, g_{i + 1}, \ldots, g_d \right\rangle.$$* *Proof.* Up to a change of basis, $i = 1$. The map $g_1 \in \operatorname{GL}(V) < G$ has a cycle $s = (\mathbf{b}_1~\mu \mathbf{b}_1~\mu^2 \mathbf{b}_1 \allowbreak~\cdots~\mu^{p - 2} \mathbf{b}_1)$. Treating $s$ as a permutation on the subspace $\left\langle \mathbf{b}_1 \right\rangle$, we see that, for all $\mathbf{u}\in \left\langle \mathbf{b}_1 \right\rangle$ and $\mathbf{w}\in \left\langle \mathbf{b}_2, \ldots, \mathbf{b}_d \right\rangle$ (if $d = 1$, then consider $\mathbf{w}= \mathbf{0}$), $$(\mathbf{u}+ \mathbf{w})^{g_1} = \mathbf{u}^{g_1} + \mathbf{w}= \mathbf{u}^s + \mathbf{w}.$$ By , since $s$ is a $(p - 1)$-cycle and $(p - 1, a) \neq (4, 2)$, there exists $x \in \operatorname{Sym}(\left\langle \mathbf{b}_1 \right\rangle)$ such that $\left\langle s \right\rangle \cap \left\langle s \right\rangle^x = \left\langle s^a \right\rangle$. Define $\tilde{x} \in G$ by setting $$(\mathbf{u}+ \mathbf{w})^{\tilde{x}} \coloneqq \mathbf{u}^x + \mathbf{w}$$ for all $\mathbf{u}\in \left\langle \mathbf{b}_1 \right\rangle$ and $\mathbf{w}\in \left\langle \mathbf{b}_2, \ldots, \mathbf{b}_d \right\rangle$. Let $g$ be any element of $T$ and write $g = g_1^c g'$ with $c \in \{1, \ldots, p - 1\}$ and $g' \in \left\langle g_2, \ldots, g_d \right\rangle$. Then, with $\mathbf{u}, \mathbf{w}$ as above, $$(\mathbf{u}+ \mathbf{w})^g = \mathbf{u}^{g_1^c} + \mathbf{w}^{g'} = \mathbf{u}^{s^c} + \mathbf{w}^{g'}$$ and similarly $$(\mathbf{u}+ \mathbf{w})^{g^{\tilde{x}}} = \mathbf{u}^{(s^c)^x} + \mathbf{w}^{g'}.$$ Hence $g^{\tilde{x}} \in T$ if and only if $(s^c)^x \in \left\langle s \right\rangle$, which holds if and only if $a \mid c$. Therefore, $T \cap T^{\tilde{x}} = \left\langle g_1^a, g_{2}, \ldots, g_d \right\rangle,$ as required. ◻ **Lemma 15**. *Suppose $d \geqslant 1$, $p \geqslant 3$, and $G = \operatorname{Sym}(V)$. Let $l_2 \coloneqq d$ if $p = 3, 5$, and $l_2 \coloneqq d \, \Omega(p - 1)$ otherwise. Then there are subsets $Y_1, \ldots, Y_{l_2} \subseteq G$ such that $$\nonumber T > \bigcap_{x \in Y_1} T^x > \bigcap_{x \in Y_2} T^x > \cdots > \bigcap_{x \in Y_{l_2}} T^x = 1.$$* *Proof.* First, suppose $p = 3$ or $p = 5$. For all $i \in \{1, \ldots, d\}$, by , there exists $y_i \in G$ such that $$T \cap T^{y_{i}} = \left\langle g_1, \ldots, g_{i - 1}, g_{i + 1}, \ldots, g_d \right\rangle;$$ setting $Y_i \coloneqq \{y_1, \ldots, y_i\}$ gives $$\bigcap_{x \in Y_i} T^x = \left\langle g_{i + 1}, \ldots, g_d \right\rangle.$$ Therefore, $Y_1, \ldots, Y_d$ are as required. Now, suppose $p \geqslant 7$. Let $a_1, \ldots, a_{\Omega(p - 1)}$ be a sequence of factors of $(p - 1)$ such that $a_i \mid a_{i + 1}$ for all $i$. Let $\mathcal{I} \coloneqq \{1, \ldots, d\} \times \{1, \ldots, \Omega(p - 1) \}$ be ordered lexicographically. For each pair $(i, j) \in \mathcal{I}$, by , there exists $y_{i, j} \in G$ such that $$T \cap T^{y_{i, j}} = \left\langle g_1, \ldots, g_{i - 1}, {g_i}^{a_j}, g_{i + 1}, \ldots, g_d \right\rangle;$$ setting $Y_{i, j} \coloneqq \{ y_{i', j'} \mid (i', j') \in \mathcal{I}, (i', j') < (i, j) \}$ gives $$\bigcap_{\mathclap{x \in Y_{i, j}}} \, T^x \; = \left\langle {g_{i}}^{a_j}, g_{i +1 }, \ldots, g_d \right\rangle.$$ Therefore, the $Y_{i, j}$'s, ordered lexicographically by the subscripts, are as required. ◻ This completes our preparations for the proof of . ## Proof of Theorem [Theorem 5](#theorem:affine-case){reference-type="ref" reference="theorem:affine-case"} {#section:proof-of-affine-case-theorem} Recall the assumption that $G$ is $\operatorname{S}_{p^d}$ or $\operatorname{A}_{p^d}$ ($p$ is an odd prime and $p^d \geqslant 7$), which we identify here with $\operatorname{Sym}(V)$ or $\operatorname{Alt}(V)$, and $H = \operatorname{AGL}_{d}(p) \cap G$, which we identify with $\operatorname{AGL}(V) \cap G$. *Proof of .* First, suppose $d \geqslant 2$, $p \geqslant 3$, and $G = \operatorname{Sym}(V)$. Let $K_1, \ldots, K_{l_1}$ be as in . For each $i \in \{1, \ldots, l_1\}$, by , there exists $x_i \in G$ such that $H \cap H^{x_i} = K_i$. Define $X_i \coloneqq \{ 1 \} \cup \{ x_j \mid 1 \leqslant j < i \} \subseteq G$ for all such $i$. Then by , $$\label{equation:affine-case-theorem-1} G > H = \bigcap_{x \in X_1} H^x > \bigcap_{x \in X_2} H^x > \cdots > \bigcap_{x \in X_{l_1 + 1}} H^x = T.$$ Let $Y_1, \ldots, Y_{l_2} \subseteq G$ be as in . For each $i \in \{1, \ldots, l_2\}$, let $Z_i \coloneqq \{ x y \mid x \in X_{l_1 + 1}, y \in Y_i \},$ so that $$\bigcap_{z \in Z_i} H^z = \bigcap_{y \in Y_i} \left( \bigcap_{x \in X_{l_1 + 1}} H^x \right)^y = \bigcap_{y \in Y_i} T^y.$$ Then gives $$\label{equation:affine-case-theorem-2} T > \bigcap_{z \in Z_{1}} H^x > \bigcap_{z \in Z_{2}} H^x > \cdots > \bigcap_{z \in Z_{l_2}} H^x = 1.$$ Concatenating the chains [\[equation:affine-case-theorem-1\]](#equation:affine-case-theorem-1){reference-type="eqref" reference="equation:affine-case-theorem-1"} and [\[equation:affine-case-theorem-2\]](#equation:affine-case-theorem-2){reference-type="eqref" reference="equation:affine-case-theorem-2"}, we obtain a chain of length $l_1 + l_2 + 1$. Now, suppose $d \geqslant 2$, $p \geqslant 3$, and $G$ is $\operatorname{Sym}(V)$ or $\operatorname{Alt}(V)$. By and , since $\operatorname{AGL}(V) \nleqslant \operatorname{Alt}(V)$, the lower bounds in the theorem hold. For the upper bound on $\operatorname{I}(G, H)$, simply compute $$\begin{aligned} \operatorname{I}(G, H) & \leqslant 1 + \Omega(\left\lvert H \right\rvert) \leqslant \Omega(p^d (p^d - 1) (p^d - p) \cdots (p^d - p^{d - 1})) + \varepsilon(G) \\ & < \frac{d (d + 1)}{2} + \log((p^d - 1) (p^{d - 1} - 1) \cdots (p - 1)) + \varepsilon(G) \\ & < \frac{d (d + 1)}{2} (1 + \log p) + \varepsilon(G). \end{aligned}$$ Finally, suppose $d = 1$ and $p \geqslant 7$. Using , we obtain the chain [\[equation:affine-case-theorem-2\]](#equation:affine-case-theorem-2){reference-type="eqref" reference="equation:affine-case-theorem-2"} again. Concatenating the chain $G > H > T$ with [\[equation:affine-case-theorem-2\]](#equation:affine-case-theorem-2){reference-type="eqref" reference="equation:affine-case-theorem-2"} and applying and , we see that $\operatorname{I}(G, H) \geqslant 1 + \Omega(p - 1) + \varepsilon(G)$. In fact, equality holds, as $\operatorname{I}(G, H) \leqslant 1 + \Omega(\left\lvert H \right\rvert) = 1 + \Omega(p - 1) + \varepsilon(G)$. ◻ # The product action case {#section:wreath-case} In this section, we prove . Once again, most work goes into the explicit construction of an irredundant base in order to prove the lower bounds, while the upper bounds will be obtained easily from the length of $\operatorname{S}_{n}$. Throughout this section, let $m \geqslant 5$ and $k \geqslant 2$ be integers, and let $G$ be $\operatorname{S}_{m^k}$ or $\operatorname{A}_{m^k}$. Let $M \coloneqq \operatorname{S}_{m} \wr \operatorname{S}_{k}$ act in product action on $\Delta \coloneqq \{ (a_1, \ldots, a_k) \mid a_1, \ldots, a_k \in \{1, \ldots, m\} \}$ and identify $M$ with a subgroup of $\operatorname{S}_{m^k}$. **Theorem 16** ([@liebeck_praeger_saxl_1987]). *The group $M \cap G$ is a maximal subgroup of $G$ if and only if one of the following holds:* 1. *$m \equiv 1 \pmod{2}$;* 2. *$G = \operatorname{S}_{m^k}$, $m \equiv 2 \pmod 4$ and $k = 2$;* 3. *$G = \operatorname{A}_{m^k}$, $m \equiv 0 \pmod{4}$ and $k = 2$;* 4. *$G = \operatorname{A}_{m^k}$, $m \equiv 0 \pmod{2}$ and $k \geqslant 3$.* The strategy to proving the lower bound in is once again to find suitable two-point stabilisers from which a long chain of subgroups can be built. For each pair of points $\alpha, \beta \in \Delta$, let $d(\alpha, \beta)$ denote the Hamming distance between $\alpha$ and $\beta$, namely the number of coordinates that differ. **Lemma 17**. *Let $x \in M$. Then for all $\alpha, \beta \in \Delta$, $$d(\alpha^x, \beta^x) = d(\alpha, \beta).$$* *Proof.* Write $x$ as $(v_1, \ldots, v_k) w$ with $v_1, \ldots, v_k \in \operatorname{S}_{m}$ and $w \in \operatorname{S}_{k}$. Let $\alpha = (a_1, \ldots, a_k)$ and $\beta = (b_1, \ldots, b_k)$. Write $\alpha^x = (a_1', \ldots, a_k')$ and $\beta^x = (b_1', \ldots, b_k')$. Then for each $i \in \{1, \ldots, k\}$, $$a_i = b_i \Longleftrightarrow {a_i}^{v_i} = {b_i}^{v_i} \Longleftrightarrow a_{i^{w}}' = b_{i^{w}}'.$$ Since $w$ is a permutation of $\{1, \ldots, k\}$, the result holds. ◻ Define $u \in \operatorname{S}_{m}$ to be $(1~2~\cdots~m)$ if $m$ is odd, and $(1~2~\cdots~m - 1)$ if $m$ is even, so that $u$ is an even permutation. Let $U \coloneqq \left\langle u \right\rangle \leqslant \operatorname{S}_{m}$ and note that $\operatorname{C}_{\operatorname{S}_{m}}(u) = U$. The group $U$ will play a central role in the next lemma. **Lemma 18**. *Let $i \in \{2, \ldots, k\}$ and $r \in \{1, \ldots, m\}$. Let $T_r$ be the stabiliser of $r$ in $\operatorname{S}_{m}$ and let $W_i$ be the pointwise stabiliser of $1$ and $i$ in $\operatorname{S}_{k}$. Then there exists $x_{i, r} \in \operatorname{A}_{m^k}$ such that $$M \cap M^{x_{i, r}} = \left( U \times {(\operatorname{S}_{m})}^{i - 2} \times T_r \times {(\operatorname{S}_{m})}^{k - i} \right) \rtimes W_i.$$* *Proof.* Without loss of generality, assume $i = 2$. Define $x = x_{2, r} \in \operatorname{Sym}(\Delta)$ by $$(a_1, a_2, \ldots, a_k)^x = \begin{cases} ({a_1}^u, a_2, \ldots, a_k) & \text{if } a_2 = r, \\ (a_1, a_2, \ldots, a_k) & \text{otherwise}. \end{cases}$$ The permutation $x$ is a product of $m^{k - 2}$ disjoint $\left\lvert u \right\rvert$-cycles and is therefore even. Let $K \coloneqq \left( U \times T_r \times {(\operatorname{S}_{m})}^{k - 2} \right) \rtimes W_2$. We show first that $K \leqslant M \cap M^{x}$. Let $h = (v_1, \ldots, v_m) w^{-1}$ be an element of $K$. Then $v_1 \in U$, $v_2$ fixes $r$, and $w$ fixes $1$ and $2$. Therefore, for all $\alpha = (a_1, a_2, \ldots, a_k) \in \Delta$, if $a_2 = r$, then $$\begin{aligned} \alpha^{h x} & = ({a_1}^{v_1}, a_2, {a_{3^w}}^{v_{3^w}}, \ldots, {a_{k^w}}^{v_{k^w}})^x = ({a_1}^{v_1 u}, a_2, {a_{3^w}}^{v_{3^w}}, \ldots, {a_{k^w}}^{v_{k^w}}) \\ & = ({a_1}^{u v_1}, a_2, {a_{3^w}}^{v_{3^w}}, \ldots, {a_{k^w}}^{v_{k^w}}) = ({a_1}^{u}, a_2, a_{3}, \ldots, a_{k})^h = \alpha^{x h}; \end{aligned}$$ and if $a_2 \neq r$, then $$\begin{aligned} \alpha^{h x} & = ({a_1}^{v_1}, {a_2}^{v_2}, {a_{3^w}}^{v_{3^w}}, \ldots, {a_{k^w}}^{v_{k^w}})^x = ({a_1}^{v_1}, {a_2}^{v_2}, {a_{3^w}}^{v_{3^w}}, \ldots, {a_{k^w}}^{v_{k^w}}) \\ & = (a_1, a_2, a_3, \ldots, a_k)^h = \alpha^{x h}. \end{aligned}$$ Therefore, $x$ and $h$ commute. Since $h$ is arbitrary, $K = K \cap K^x \leqslant M \cap M^x$. Let $B$ be the base group $(\operatorname{S}_{m})^k$ of $M$. Since $K \leqslant M \cap M^x$, we find that $B \cap K \leqslant B \cap M^x$. We now show that $B \cap M^x \leqslant B \cap K$, so let $h_1 = (v_1, \ldots, v_k) \in B \cap M^{x}$. Then ${h_1}^{x^{-1}} \in M$. We show that $v_1 \in U$ and $v_2$ fixes $r$, so that $h_1 \in K$. By letting $g_1 \coloneqq (1, 1, v_3, \ldots, v_k) \in K$ and replacing $h_1$ with $g_1^{-1} h_1$, we may assume $v_3 = \cdots = v_k = 1$. Let $h_2 \coloneqq x h_1 x^{-1} h_1^{-1} = {h_1}^{x^{-1}} h_1^{-1} \in M$, and let $\alpha \coloneqq (a, b, c, \ldots, c)$ and $\beta \coloneqq (a, r, c, \ldots, c)$ be elements of $\Delta$ with $a \neq m$ and $b \notin \{r, r^{v_2^{-1}}\}$. Then $\alpha$ and $\alpha^{h_1}$ are both fixed by $x$, and so $\alpha^{h_2} = \alpha$. On the other hand, $$\beta^{h_2} = \begin{cases} (a^{u v_1 u^{-1} v_1^{-1}}, r, c, \ldots, c), & \text{if } r^{v_2} = r, \\ (a^{u}, r, c, \ldots, c), & \text{otherwise}. \end{cases}$$ Since $d(\alpha^{h_2}, \beta^{h_2}) = d(\alpha, \beta) = 1$ by and $a^u \neq a$, it must be the case that $r^{v_2} = r$ and $a^{u v_1 u^{-1} v_1^{-1}} = a$. Therefore, $v_2 \in T_r$ and, as $a$ is arbitrary in $\{1, \ldots, m - 1\}$, we deduce that $v_1 \in \operatorname{C}_{\operatorname{S}_{m}}(u) = U$ and hence $h_1 \in K$. Thus, $B \cap M^x \leqslant B \cap K$ and so $B \cap M^x = B \cap K$. To show that $M \cap M^x \leqslant K$, let $h_3 \in M \cap M^x$. Now, $B \unlhd M$ and so $B \cap K = B \cap M^x \unlhd M \cap M^x$. Therefore, $$h_3 \in \operatorname{N}_{M}(B \cap K) = \left( \operatorname{N}_{\operatorname{S}_{m}}(U) \times T_r \times {(\operatorname{S}_{m})}^{k - 2} \right) \rtimes W_2.$$ The equality uses the fact that $\operatorname{N}_{\operatorname{S}_{m}}(U) \neq T_r$ (as $m \geqslant 5$). Through left multiplication by an element of $K$, we may assume $h_3 \in \operatorname{N}_{\operatorname{S}_{m}}(U) \times (1_{\operatorname{S}_{m}})^{k - 1}$. Then $h_3 \in B \cap M^x \leqslant K$. Since $h_3$ is arbitrary, $M \cap M^x \leqslant K$. Therefore, $K = M \cap M^x$, as required. ◻ We are now ready to prove the main result for the product action case. Recall the assumption that $G$ is $\operatorname{S}_{m^k}$ or $\operatorname{A}_{m^k}$ and $H = M \cap G$. *Proof of .* Firstly, suppose that $H = M$. Let $\mathcal{I} \coloneqq \{2, \ldots, k\} \times \{1, \ldots, m - 1\}$, ordered lexicographically. For each $(i, r) \in \mathcal{I}$, let $x_{i, r} \in \operatorname{A}_{m^k} \leqslant G$ be as in , and define $$X_{i, r} \coloneqq \{ 1 \} \cup \{ x_{i', r'} \mid (i', r') \in \mathcal{I}, (i', r') \leqslant (i, r) \} \subseteq G.$$ Then for all $(i, r) \in \mathcal{I}$, $$B \cap \bigcap_{x \in X_{i, r}} M^x = U \times (1_{\operatorname{S}_{m}})^{i - 2} \times (\operatorname{S}_{m})_{1, \ldots, r} \times (\operatorname{S}_{m})^{k - i}.$$ Hence, for all $(i, r), (j, s) \in \mathcal{I}$ with $(i, r) < (j, s)$, $\bigcap_{x \in X_{i, r}} M^x > \bigcap_{x \in X_{j, s}} M^x.$ This results in the following chain of stabiliser subgroups, of length $(m - 1)(k - 1) + 2$: $$G > M > \bigcap_{x \in X_{2, 1}} M^x > \cdots > \bigcap_{x \in X_{2, m - 1}} M^x > \bigcap_{x \in X_{3, 1}} M^x > \cdots > \bigcap_{x \in X_{k, m - 1}} M^x > 1.$$ Therefore, by , $\operatorname{I}(G, H) = \operatorname{I}(G, M) \geqslant (m - 1) (k - 1) + 2$. Now, if $H \neq M$, then $G = \operatorname{A}_{m^k}$, and $\operatorname{I}(G, H) \geqslant \operatorname{I}(\operatorname{S}_{m^k}, M) - 1 \geqslant (m - 1) (k - 1) + 1$ by . Finally, for the upper bound on $\operatorname{I}(G, H)$, we use [\[equation:length-symmetric-group-inequality\]](#equation:length-symmetric-group-inequality){reference-type="eqref" reference="equation:length-symmetric-group-inequality"} and [@cameron_solomon_turull_1989 Lemma 2.1] to compute $$\begin{aligned} \operatorname{I}(G, H) & \leqslant 1 + \ell(H) \leqslant 1 + \ell(M) \leqslant 1 + k \, \ell(\operatorname{S}_{m}) + \ell(\operatorname{S}_{k}) \\ & \leqslant 1 + k \left(\frac{3}{2} m - 2\right) + \left(\frac{3}{2} k - 2\right) \leqslant \frac{3}{2} m k - \frac{1}{2} k - 1. \qedhere \end{aligned}$$ ◻ # Proof of Theorem [Theorem 1](#theorem:optimal-upper-bounds){reference-type="ref" reference="theorem:optimal-upper-bounds"} {#section:proof-of-main-result} In this final section, we zoom out for the general case and prove by considering the order of $H$ and assembling results from previous sections. Recall that $G$ is $\operatorname{S}_{n}$ or $\operatorname{A}_{n}$ ($n \geqslant 7$) and $H \neq \operatorname{A}_{n}$ is a primitive maximal subgroup of $G$. Maróti proved in [@maróti_2002] several useful upper bounds on the order of a primitive subgroup of the symmetric group. **Lemma 19**. 1. *$\left\lvert H \right\rvert < 50 n^{\sqrt{n}}$.* 2. *At least one of the following holds:* 1. *$H = S_m \cap \, G$ acting on $r$-subsets of $\{1, \ldots, m\}$ with $n = \binom{m}{r}$ for some integers $m, r$ with $m > 2 r \geqslant 4$;* 2. *$H = \left(\operatorname{S}_{m} \wr \operatorname{S}_{k}\right) \cap G$ with $n = m^k$ for some $m \geqslant 5$ and $k \geqslant 2$;* 3. *$\left\lvert H \right\rvert < n^{1 + \lfloor \log n \rfloor}$;* 4. *$H$ is one of the Mathieu groups $M_{11}, M_{12}, M_{23}, M_{24}$ acting $4$-transitively.* *Proof.* (i) follows immediately from [@maróti_2002 Corollary 1.1]. (ii) follows from [@maróti_2002 Theorem 1.1] and the description of the maximal subgroups of $\operatorname{S}_{n}$ and $\operatorname{A}_{n}$ in [@liebeck_praeger_saxl_1987]. ◻ Equipped with these results as well as , we are ready to prove . *Proof of .* If $H$ is as in case (a) of (ii), then $n = \binom{m}{r} \geqslant \binom{m}{2} = \frac{m (m - 1)}{2}$. Hence $m < 2 \sqrt{n}$ and, by [\[equation:length-symmetric-group-inequality\]](#equation:length-symmetric-group-inequality){reference-type="eqref" reference="equation:length-symmetric-group-inequality"}, $$\operatorname{I}(G, H) \leqslant 1 + \ell(H) \leqslant 1 + \ell(S_m) < 3 \sqrt{n} - 1.$$ If $H$ is as in case (b) of (ii), then $n = m^k$. By , $\operatorname{I}(G, H) \leqslant \frac{3}{2} m k - \frac{1}{2} k - 1$. If $k = 2$, then $$\operatorname{I}(G, H) \leqslant 3 m - 2 < 3 \sqrt{n} - 1.$$ If $k \geqslant 3$, then $$\operatorname{I}(G, H) < \frac{3}{2} m \frac{\log n}{\log m} \leqslant \frac{3}{2} \sqrt[3]{n} \frac{\log n}{\log 5}< 3 \sqrt{n} - 1.$$ If $H$ is as in case (c) of (ii), then $$\operatorname{I}(G, H) \leqslant 1 + \ell(H) \leqslant 1 + \log \left\lvert H \right\rvert < 1 + \log \left(n^{1 + \log n}\right) = (\log n)^2 + \log n + 1.$$ Using the lists of maximal subgroups in [@atlas1985], one can check that $\ell(M_{11}) = 7$, $\ell(M_{12}) = 8$, $\ell(M_{23}) = 11$, and $\ell(M_{24}) = 14$. It is thus easy to verify that $\operatorname{I}(G, H) \leqslant 1 + \ell(H) < (\log n)^2$ in case (d) of (ii). Therefore, part (i) of the theorem holds. We now prove parts (ii) and (iii). By , if $n = 3^d$ for some integer $d \geqslant 2$, then $H = \operatorname{AGL}_{d}(3) \cap G$ is a maximal subgroup of $G$. now gives $$\operatorname{I}(G, H) > \frac{d^2}{2} + \frac{d}{2} = \frac{(\log n)^2}{2 (\log 3)^2} + \frac{\log n}{2 \log 3},$$ as required. By , if $n = m^2$ for some odd integer $m \geqslant 5$, then $H = (\operatorname{S}_{m} \wr \operatorname{S}_{2}) \cap G$ is a maximal subgroup of $G$. now gives $\operatorname{I}(G, H) \geqslant m = \sqrt{n}$, as required. ◻ Finally, we prove an additional lemma. **Lemma 20**. *Let $t$ be the index of $H$ in $G$. There exist constants $c_5, c_6, c_7, c_8 \in \mathbb{R}_{> 0}$ such that* 1. *$c_5 \log t / \log \log t < n < c_6 \log t / \log \log t.$* 2. *$c_7 \log \log t < \log n < c_8 \log \log t.$* *Proof.* It suffices to prove that such constants exist for $n$ sufficiently large, so we may assume $n > 100$. We first note that $\log t < \log \left\lvert G \right\rvert \leqslant n \log n$, from which we obtain $$\nonumber \log \log t < \log n + \log \log n < \log n + (\log n) \frac{\log \log 100}{\log 100} < 1.412 \log n.$$ Hence we may take $c_7 = 1 / 1.412 > 0.708$ for $n > 100$. By (i), $$\begin{aligned} \log t & = \log \left\lvert G : H \right\rvert = \log \left\lvert G \right\rvert - \log \left\lvert H \right\rvert > \log \frac{n!}{2} - \log \left(50 n^{\sqrt{n}}\right) \\ & > \left(n \log n - n \log e - 1\right) - \left(\sqrt{n} \log n + \log 50\right) = n \log n - n \log e - \sqrt{n} \log n - \log 100 \\ & > n \log n - n (\log e )\frac{\log n}{\log 100} - \sqrt{n} (\log n) \frac{\sqrt{n}}{\sqrt{100}} - (\log 100) \frac{n \log n}{100 \log 100} \\ & > 0.672 \, n \log n, \end{aligned}$$ where the second inequality follows from Stirling's approximation and the last inequality follows from the fact that $\log e / \log 100 < 0.218$. We deduce further that $\log \log t > \log n$ and hence take $c_8 = 1$ for $n > 100$. Finally, $\log t / \log \log t < n \log n / \log n = n$ and $\log t / \log \log t > 0.672 \, n \log n / 1.412 \log n = 0.672 \, n / 1.412$. Therefore, for $n > 100$, we may take $c_5 = 1$, $c_6 = 1.412 / 0.672 < 2.11$. ◻ now follows by combining and . **Remark 21**. Verifying all cases with $7 \leqslant n \leqslant 100$ by enumerating primitive maximal subgroups of $\operatorname{S}_{n}$ and $\operatorname{A}_{n}$ in Magma [@magma_MR1484478], we may take $c_5 = 1$, $c_6 = 4.03$, $c_7 = 0.70$, and $c_8 = 1.53$ in the statement of . With these values of the constants and those in the proof of , it is straightforward to obtain the values of the constants $c_2, c_3, c_4$ given in . For the values of $c_1$, we use in addition the fact that, for any $n_0$, if $n \geqslant n_0$, then $(\log n)^2 + (\log n) + 1 = (\log n)^2 \left(1 + 1 / \log n + 1 / (\log n)^2 \right) < c_8^2 \left(1 + 1 / \log n_0 + 1 / (\log n_0)^2 \right) (\log \log t)^2.$ #### Acknowledgement The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for its support and hospitality during the programme *Groups, representations and applications: new perspectives*, when work on this article was undertaken. This work was supported by EPSRC grant N^[o]{.ul}^ EP/R014604/1, and also partially supported by a grant from the Simons Foundation. 10 M. Aschbacher and L. Scott. Maximal subgroups of finite groups. , 92(1):44--80, 1985. W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system. I. The user language. , 24(3-4):235--265, 1997. T. C. Burness, R. M. Guralnick, and J. Saxl. On base sizes for symmetric groups. , 43(2):386--391, 2011. P. J. Cameron, R. Solomon, and A. Turull. Chains of subgroups in symmetric groups. , 127(2):340--352, 1989. G. L. Cherlin, G. A. Martin, and D. H. Saracino. Arities of permutation groups: Wreath products and $k$-sets. , 74(2):249--286, 1996. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson. . Oxford University Press, 1985. N. Gill and M. W. Liebeck. Irredundant bases for finite groups of Lie type. , 322(2):281--300, may 2023. N. Gill and B. Lodà. Statistics for $\operatorname{{S}}_n$ acting on ${k}$-sets. , 607:286--299, 2022. N. Gill, B. Lodà, and P. Spiga. On the height and relational complexity of a finite permutation group. , 246:372--411, 2022. V. Kelsey and C. M. Roney-Dougal. On relational complexity and base size of finite primitive groups. , 318(1):89--108, 2022. M. W. Liebeck, C. E. Praeger, and J. Saxl. A classification of the maximal subgroups of the finite alternating and symmetric groups. , 111(2):365--383, 1987. A. Maróti. On the orders of primitive groups. , 258(2):631--640, 2002.

arxiv_math

--- abstract: | We study the properties of topological spaces $(X,\tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ on $X$ has a basis that is (uniformly) definable. Examples of such spaces include the canonical euclidean topology on definable sets, definable order topologies, definable quotient spaces and definable metric spaces. We use o-minimality to undertake their study in topological terms, focussing here in particular on spaces of dimension one. We present several results, given in terms of piecewise decompositions and existence of definable embeddings and homeomorphisms, for various classes of spaces that are described in terms of classical separation axioms and definable analogues of properties such as separability, compactness and metrizability. For example, we prove that all Hausdorff one-dimensional definable topologies are piecewise the euclidean, discrete, or upper or lower limit topology; we give a characterization of all one-dimensional, regular, Hausdorff definable topologies in terms of spaces that have a lexicographic ordering or a topology generalizing the Alexandrov double of the euclidean topology; and we show that, if the underlying structure expands an ordered field, then any one-dimensional Hausdorff definable topology that is piecewise euclidean is definably homeomorphic to a euclidean space. As applications of these results, we prove definable versions of several open conjectures from set-theoretic topology, due to Gruenhage and Fremlin, on the existence of a 3-element basis for regular, Hausdorff topologies and on the nature of perfectly normal, compact, Hausdorff spaces; we obtain universality results for some classes of Hausdorff and regular topologies; and we characterize when certain metrizable definable topologies admit a definable metric. address: - School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK - Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA author: - Pablo Andújar Guerrero - Margaret E. M. Thomas title: One-dimensional definable topological spaces in o-minimal structures --- # Introduction A topological space $(X,\tau)$ is definable in a first order structure $\mathcal{R}=(R,\ldots)$ if $X\subseteq R^n$, for some natural number $n$, and the topology $\tau$ has a basis that is (uniformly) definable in $\mathcal{R}$. In other words, the topology $\tau$ is explicitly definable in the sense of Flum and Ziegler [@flum_ziegler_80]. This definition generalizes the notion of first order topological structure in [@pillay87], which addresses the case where $X=R$. A basic example of a topological space which is definable in any structure is the discrete topology on a definable set. Other examples of definable topological spaces arise from valued fields and the field of complex numbers with a predicate for the reals [@pillay87]. In this paper, we begin a detailed study of the general theory of topological spaces that are definable in o-minimal structures. A structure is *o-minimal* if it is an expansion of a linear order $(R,<)$ satisfying that every unary definable set is a finite union of points and intervals with endpoints in $R\cup\{-\infty, +\infty\}$. There is a canonical topology in this setting given by the order topology on $R$ and, for every $n>1$, the induced product topology on $R^n$. In the present paper we refer to these definable topologies collectively as the *euclidean topology*. Much of the research in o-minimality as a tame topological setting has centered on the analysis of the euclidean topology, as this has long been recognized as a suitable setting for the "topologie modérée\" of Grothendieck (i.e. a setting that avoids the pathologies of general set-theoretic topology) [@gro84]. Nevertheless, some examples of other o-minimal definable topological spaces have been explored in the literature, including definable manifold spaces [@dries98 Chapter 10, Section 1], which encompass definable groups [@pillay88]; definable metric spaces [@walsberg15]; and definable orders (which generate definable order topological spaces) [@ram13]. We direct the reader to [@dries98] for the requisite background on the theory of o-minimal structures, which will be used extensively throughout this paper. Our perspective is to consider definable topological spaces $(X,\tau)$ in o-minimal structures from a general standpoint. Here we focus predominantly on the case where $\dim X =1$ but, even in this setting, our perspective brings to the o-minimal setting topological spaces that exhibit a wide variety of topological properties (see Appendix [12](#section:examples){reference-type="ref" reference="section:examples"} for a catalogue of examples), including classical spaces such as the Sorgenfrey Line, the Split Interval and the Alexandrov Double Circle (Examples [Example 116](#example:taur_taul){reference-type="ref" reference="example:taur_taul"}, [Example 117](#example:split_interval){reference-type="ref" reference="example:split_interval"} and [Example 126](#example:alex_double_circle){reference-type="ref" reference="example:alex_double_circle"} respectively). These are common counterexamples in topology, displaying properties that the more "well-behaved\" spaces lack. Nevertheless, we argue through the present work that this setting retains some form of tameness. Specifically, the axiom of o-minimality implies that the structure of one-dimensional definable topological spaces is rather restrictive, in particular when compared to spaces of higher dimensions. The main contributions of this paper are a series of decomposition and embedding theorems for one-dimensional o-minimal definable topological spaces satisfying certain classical separation axioms. These results can be understood as classifying such spaces, in the sense that they can be described in terms of a few classical examples. Much of our approach is motivated by partition problems in set-theoretic topology, which seek to understand topological spaces in similar terms under various axioms of set theory (see for example [@todor89]). In order to prove our results and build an extensive theory of o-minimal definable topology, we also introduce to our setting suitable definable analogues of classical topological properties, including separability, metrizability and compactness, and investigate them in depth. We now describe the main results of this paper. We begin with the following decomposition results for $T_1$ and Hausdorff spaces. Note that, in the following statement, the right half-open interval topology (also known as the lower limit topology) is the definable analogue of the topology of the classical Sorgenfrey Line. **Theorem 1** (Corollary [Corollary 44](#cor:general_them_main){reference-type="ref" reference="cor:general_them_main"}, Corollary [Corollary 52](#cor 1.5){reference-type="ref" reference="cor 1.5"}). Let $\mathcal{R}$ be an o-minimal structure and let $(X,\tau)$ be a definable topological space in $\mathcal{R}$. I. If $(X,\tau)$ is infinite and $T_1$, then it has a subspace that is definably homeomorphic to an interval with either the euclidean, right half-open interval, left half-open interval, or discrete topology. [\[itm:introthm:3EBC\]]{#itm:introthm:3EBC label="itm:introthm:3EBC"} II. If $(X, \tau)$ is Hausdorff and $\dim(X) \leq 1$, then there exists a finite definable partition $\mathcal{X}$ of $X$ such that, for every $Y\in\mathcal{X}$, $(Y, \tau)$ is definably homeomorphic to a point or an interval with either the euclidean, discrete, right half-open interval or left half-open interval topology. [\[itm:introthm:T_2decomp\]]{#itm:introthm:T_2decomp label="itm:introthm:T_2decomp"} Theorem [Theorem 1](#introthm:T_1_T_2_spaces){reference-type="ref" reference="introthm:T_1_T_2_spaces"}.[\[itm:introthm:3EBC\]](#itm:introthm:3EBC){reference-type="ref" reference="itm:introthm:3EBC"} can be seen as an o-minimal version of an open conjecture of set-theoretic topology, due to Gruenhage, known as the 3-element basis conjecture (see [@gru90], [@gm07]). This conjecture states that it is consistent with ZFC that every uncountable, first-countable, regular, Hausdorff topological space contains a subspace homeomorphic to either a fixed subset of the reals with the euclidean topology, or a fixed subset of the reals with the Sorgenfrey topology, or contains an uncountable discrete subspace. Some cases of this conjecture are known that are conditional upon various axioms of set theory (see Subsection [5.1](#subsection: 3-el_basis_conj){reference-type="ref" reference="subsection: 3-el_basis_conj"} for further discussion). From Theorem [Theorem 1](#introthm:T_1_T_2_spaces){reference-type="ref" reference="introthm:T_1_T_2_spaces"}.[\[itm:introthm:3EBC\]](#itm:introthm:3EBC){reference-type="ref" reference="itm:introthm:3EBC"} we derive that the conclusion of the conjecture holds unconditionally for all infinite $T_1$ topological spaces definable in any o-minimal expansion of $(\mathbb{R},<)$, and that a definable generalization holds in any o-minimal structure (see Subsection [5.1](#subsection: 3-el_basis_conj){reference-type="ref" reference="subsection: 3-el_basis_conj"}). We also use Theorem [Theorem 1](#introthm:T_1_T_2_spaces){reference-type="ref" reference="introthm:T_1_T_2_spaces"}.[\[itm:introthm:3EBC\]](#itm:introthm:3EBC){reference-type="ref" reference="itm:introthm:3EBC"} to show that no space homeomorphic to the Cantor space $2^{\omega}$ is definable in any o-minimal structure (Corollary [Corollary 50](#cor:no_cantor_set){reference-type="ref" reference="cor:no_cantor_set"}), supporting the thesis that o-minimality provides a tame topological setting in the general definable topological context. Further to the above, we also investigate o-minimal one-dimensional definable topological spaces that are regular as well as Hausdorff, improving the conclusion of Theorem [Theorem 1](#introthm:T_1_T_2_spaces){reference-type="ref" reference="introthm:T_1_T_2_spaces"}.[\[itm:introthm:T_2decomp\]](#itm:introthm:T_2decomp){reference-type="ref" reference="itm:introthm:T_2decomp"} for these spaces by showing that they can be definably decomposed in terms of suitable generalizations of the Split Interval and the Alexandrov Double Circle, the latter with a topology which we call the 'Alexandrov topology' (see Examples [Example 118](#example:n-split){reference-type="ref" reference="example:n-split"} and [Example 119](#example_n_line){reference-type="ref" reference="example_n_line"}). The precise statement in the case of spaces $(X,\tau)$, with $X$ a unary definable set, is as follows. **Theorem 2** (Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"}). Let $\mathcal{R}$ be an o-minimal structure and let $(X,\tau)$, $X\subseteq R$, be a regular and Hausdorff definable topological space in $\mathcal{R}$. Then there exist disjoint definable open sets $Y, Z \subseteq X$ with $X\setminus (Y\cup Z)$ finite, and $n_Y, n_Z>0$, such that $(Y,\tau)$ embeds definably into $R \times \{0,\ldots, n_Y\}$ endowed with the lexicographic order topology, and $(Z,\tau)$ embeds definably into $R \times \{0,\ldots, n_Z\}$ endowed with the Alexandrov topology. These decomposition results, Theorems [Theorem 1](#introthm:T_1_T_2_spaces){reference-type="ref" reference="introthm:T_1_T_2_spaces"} and [Theorem 2](#introthm:ADC_or_DOTS){reference-type="ref" reference="introthm:ADC_or_DOTS"}, allow us to address universality questions in our setting. Our motivation is classical universality literature in Banach space theory (e.g. the Banach--Mazur theorem). We prove that a number of classes of spaces admit an 'almost definably universal' space, in a sense that we make precise (Definition [Definition 66](#dfn:(almost)universal){reference-type="ref" reference="dfn:(almost)universal"}). Such classes include the class of euclidean spaces of dimension at most $n$, the class of one-dimensional regular Hausdorff definable topological spaces, and the class of Hausdorff definable topological spaces $(X,\tau)$, with $X$ a unary definable set, that satisfy the 'frontier dimension inequality' (Definition [Definition 41](#dfn:fdi){reference-type="ref" reference="dfn:fdi"}). We moreover prove some negative results about the existence of definably universal spaces of certain kinds, in particular that there does not exist a one-dimensional, $T_1$ definable topological space that is almost definably universal for the class of one-dimensional regular Hausdorff definable topological spaces. We also use Theorem [Theorem 2](#introthm:ADC_or_DOTS){reference-type="ref" reference="introthm:ADC_or_DOTS"} to study definable Hausdorff compactifications of o-minimal one-dimensional definable topological spaces. Our main result in this respect (Theorem [Theorem 89](#them_compactification){reference-type="ref" reference="them_compactification"}) is that such a Hausdorff space $(X,\tau)$, with $X$ a unary definable set, admits a definable embedding into a one-dimensional Hausdorff definably compact space if and only if $(X,\tau)$ is regular. A main line of research concerning various classes of definable topological spaces has been the study of affineness (the property of being definably homeomorphic to a space with the euclidean topology). In the setting of o-minimal expansions of ordered fields, van den Dries showed that any definable manifold space is affine if and only if it is regular [@dries98], while Walsberg showed that a definable metric space is affine if and only if it does not contain an infinite definable discrete subspace. Using Theorem [Theorem 1](#introthm:T_1_T_2_spaces){reference-type="ref" reference="introthm:T_1_T_2_spaces"}.[\[itm:introthm:T_2decomp\]](#itm:introthm:T_2decomp){reference-type="ref" reference="itm:introthm:T_2decomp"}, as well as our work on existence of definable compactifications, we prove, in the same setting, the following characterization of affineness for Hausdorff one-dimensional definable topological spaces. **Theorem 3** (Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}). Let $\mathcal{R}$ be an o-minimal expansion of an ordered field, and let $(X, \tau)$ be a Hausdorff topological space definable in $\mathcal{R}$ with $\dim(X) \leq 1$. Then $(X,\tau)$ is affine if and only if it does not contain a subspace that is definably homeomorphic to an interval with either the discrete or the right half-open interval topology. As an application of this theorem, in combination with Theorem [Theorem 2](#introthm:ADC_or_DOTS){reference-type="ref" reference="introthm:ADC_or_DOTS"}, we also address in our setting a further set-theoretic question that is closely related to the aforementioned 3-element basis conjecture, namely the possible nature of non-metrizable, perfectly normal, compact, Hausdorff spaces in ZFC. Fremlin asked whether or not it is consistent with ZFC that every perfectly normal, compact, Hausdorff space admits a continuous, at most $2$-to-$1$ map onto a metric space (see [@gm07]), which in turn led Gruenhage to ask if every such space is either metrizable or contains a copy of $A \times \{0,1\}$ equipped with the lexicographic order topology, for some uncountable $A \subseteq [0,1]$ (see [@gru88]). It is indicated in [@gru90] and [@gm07] that positive answers to these questions follow from the 3-element basis conjecture (see also Subsection [9.1](#subsection: Fremlin){reference-type="ref" reference="subsection: Fremlin"}). We consider both of these questions in our setting and show in particular that positive answers to both questions hold, in a definable sense, in any o-minimal expansion of $(\mathbb{R}, +, \cdot, <)$. More specifically, we show that, in such a structure, Fremlin's question has a positive answer for any regular, Hausdorff one-dimensional definable topological space that is either perfectly normal or separable (and, in particular, we do not require that such a space be compact) (Corollary [Corollary 99](#cor_Fremlin_2-1){reference-type="ref" reference="cor_Fremlin_2-1"}). We also show that Gruenhage's question has a positive answer for any one-dimensional, perfectly normal, compact, Hausdorff topological space definable in such a structure, where we can strengthen the 'metrizable' conclusion to 'being affine', and in the other case take $A$ to be a subinterval of $[0,1]$ (Corollary [Corollary 102](#cor:Fremlin-Q2.2){reference-type="ref" reference="cor:Fremlin-Q2.2"}). We also use Theorem [Theorem 3](#introthm:affine){reference-type="ref" reference="introthm:affine"}, this time together with [Theorem 1](#introthm:T_1_T_2_spaces){reference-type="ref" reference="introthm:T_1_T_2_spaces"}.[\[itm:introthm:T_2decomp\]](#itm:introthm:T_2decomp){reference-type="ref" reference="itm:introthm:T_2decomp"}, to investigate a notion of 'definable metrizability' extracted from the work of Walsberg [@walsberg15]. Our main result (Theorem [Theorem 104](#thm:metrizability){reference-type="ref" reference="thm:metrizability"}) is that definable metrizability is equivalent to metrizability for one-dimensional topological spaces definable in certain o-minimal expansions of ordered fields (including o-minimal expansions of the field of reals). Unsurprisingly, studying o-minimal definable topological spaces of higher dimensions is a lot less straightforward than studying those of dimension one. Key results that we present here fail to generalize to higher dimensions. In particular, Theorem [Theorem 3](#introthm:affine){reference-type="ref" reference="introthm:affine"} does not, which we illustrate by providing an example of a two-dimensional Hausdorff definable topological space that is not affine, but which also does not contain a subspace that is definably homeomorphic to an interval with either the discrete or the right half-open interval topology (Example [Example 124](#example:fdi_Hausdorff_not_regular){reference-type="ref" reference="example:fdi_Hausdorff_not_regular"}). Moreover, we provide an example of a two-dimensional Hausdorff definable topological space that fails to be affine, yet all of its one-dimensional subspaces are affine (Example [Example 128](#example_line-wise_euclidean_not_euclidean){reference-type="ref" reference="example_line-wise_euclidean_not_euclidean"}). Moreover, for regular Hausdorff definable topological spaces, it also does not follow in general that admitting a finite definable partition into euclidean subspaces implies being affine (see Example [Example 130](#example_space_cell-wise_euclidean_not_metrizable){reference-type="ref" reference="example_space_cell-wise_euclidean_not_metrizable"}), although these properties are equivalent for one-dimensional Hausdorff spaces. The outline of this paper is as follows. In Section [2](#section:definitions){reference-type="ref" reference="section:definitions"}, we include many of the necessary definitions. In Section [3](#section: metric spaces){reference-type="ref" reference="section: metric spaces"}, we introduce definable metric spaces, studied in [@walsberg15], to our setting, and discuss the properties of definable metrizability and definable separability in further detail. Section [4](#section: prel. results){reference-type="ref" reference="section: prel. results"} contains preliminary results, both for spaces of all dimensions and for spaces of dimension one in particular. In Section [5](#section:T1_T2_spaces){reference-type="ref" reference="section:T1_T2_spaces"}, we focus on $T_1$ and Hausdorff spaces, in particular proving Theorem [Theorem 1](#introthm:T_1_T_2_spaces){reference-type="ref" reference="introthm:T_1_T_2_spaces"} and deducing, from Theorem [Theorem 1](#introthm:T_1_T_2_spaces){reference-type="ref" reference="introthm:T_1_T_2_spaces"}.[\[itm:introthm:3EBC\]](#itm:introthm:3EBC){reference-type="ref" reference="itm:introthm:3EBC"}, both the Gruenhage 3-element basis conjecture in our setting and the fact that the Cantor space is not definable. In Section [6](#section: universal spaces){reference-type="ref" reference="section: universal spaces"}, we consider regular Hausdorff spaces, and in particular prove Theorem [Theorem 2](#introthm:ADC_or_DOTS){reference-type="ref" reference="introthm:ADC_or_DOTS"}. In Section [7](#section:universal_spaces_2){reference-type="ref" reference="section:universal_spaces_2"}, we answer various universality questions, some as applications of Theorems [Theorem 1](#introthm:T_1_T_2_spaces){reference-type="ref" reference="introthm:T_1_T_2_spaces"} and [Theorem 2](#introthm:ADC_or_DOTS){reference-type="ref" reference="introthm:ADC_or_DOTS"}. In Section [8](#section:compactifications){reference-type="ref" reference="section:compactifications"}, we prove, as a consequence of Theorem [Theorem 2](#introthm:ADC_or_DOTS){reference-type="ref" reference="introthm:ADC_or_DOTS"}, that all regular Hausdorff definable topologies in the line can be Hausdorff compactified in a definable sense. In Section [9](#section: affine){reference-type="ref" reference="section: affine"}, we work in an o-minimal expansion of an ordered field, and, using results of previous sections, in particular Theorem [Theorem 1](#introthm:T_1_T_2_spaces){reference-type="ref" reference="introthm:T_1_T_2_spaces"}.[\[itm:introthm:T_2decomp\]](#itm:introthm:T_2decomp){reference-type="ref" reference="itm:introthm:T_2decomp"} and results on definable compactification, we prove Theorem [Theorem 3](#introthm:affine){reference-type="ref" reference="introthm:affine"}. We also then prove various statements that address Fremlin's and Gruenhage's questions on perfectly normal, compact, Hausdorff spaces as applications of Theorems [Theorem 2](#introthm:ADC_or_DOTS){reference-type="ref" reference="introthm:ADC_or_DOTS"} and [Theorem 3](#introthm:affine){reference-type="ref" reference="introthm:affine"}. In Section [10](#section:metrizability){reference-type="ref" reference="section:metrizability"} we prove, as an application of Theorems [Theorem 1](#introthm:T_1_T_2_spaces){reference-type="ref" reference="introthm:T_1_T_2_spaces"}.[\[itm:introthm:T_2decomp\]](#itm:introthm:T_2decomp){reference-type="ref" reference="itm:introthm:T_2decomp"} and [Theorem 3](#introthm:affine){reference-type="ref" reference="introthm:affine"}, a theorem implying that, in an o-minimal expansion of the field of reals, any one-dimensional definable topology that is metrizable (with respect to the structure) also admits a definable metric. Finally, Appendix [12](#section:examples){reference-type="ref" reference="section:examples"} provides a catalogue of relevant examples, both key examples considered throughout the paper, as well as a number of further examples illustrating the necessity of the hypotheses of many of the results presented here. As we were completing this paper, we became aware that Peterzil and Rosel were working independently on similar questions. Section [11](#sec:Peterzil_Rosel){reference-type="ref" reference="sec:Peterzil_Rosel"} is a note addressing their paper [@pet_rosel_18], in which we describe how their main result relates to some of ours, and answer some of their open questions. # Acknowledgements {#acknowledgements .unnumbered} It is a great pleasure to thank Erik Walsberg for many informative discussions and motivating insights that helped to shape the direction of this paper, as well as for feedback on early drafts. Thanks also to Matthias Aschenbrenner for his considered comments on an earlier version of this work. Thanks as well to Ilijas Farah for pointing us towards some additional background on the set-theoretic questions that we consider in our work (especially for alerting us to the reference [@todorfarah95]) and to Niels J. Diepeveen for a useful discussion via StackExchange (see [@diep17]). The support of the following while the work on this paper was carried out is gratefully acknowledged. The first author is supported under the UK Engineering and Physical Sciences Research Council (EPSRC) fellowship EP/V003291/1, and was previously supported under the Canada Natural Sciences and Engineering Research Council (NSERC) Discovery Grant RGPIN-06555-2018 and by the Fields Institute for Research in Mathematical Sciences, Toronto, Canada (during the Thematic Programs on "Trends in Pure and Applied Model Theory" and "Tame Geometry, Transseries and Applications to Analysis and Geometry"). The second author is supported by NSF grant DMS-2154328, and was previously supported by the Ontario Baden--Württemberg Foundation, and under the Canada Natural Sciences and Engineering Research Council (NSERC) Discovery Grant RGPIN 261961. Both authors were supported by German Research Council (DFG) Grant TH 1781/2-1; the Zukunftskolleg, Universität Konstanz; and the Fields Institute for Research in Mathematical Sciences, Toronto, Canada (during the Thematic Program on "Unlikely Intersections, Heights and Efficient Congruencing\"). # Definitions {#section:definitions} We begin by laying out a number of conventions that we will use throughout the paper. Since our goal is to study various topological spaces and their properties from the perspective of definability in o-minimal structures, and we wish to avoid any ambiguity between conventional concepts and those which we will introduce here, we are careful to make our terminology explicit, even in the case of certain notions that can often be taken as read. Throughout this paper, $\mathcal{R}=(R,<,\ldots)$ denotes an o-minimal expansion of a dense linear order without endpoints, possibly with extra assumptions that we make explicit in context. Unless stated otherwise, by "definable\" we will mean "definable in $\mathcal{R}$, possibly with parameters from $R$\". Throughout, $n$ and $m$ denote natural numbers. By the euclidean topology on $R^n$, we refer to the canonical topology in an o-minimal structure, which is given by the order topology when $n=1$ and by its induced product topology when $n>1$. We let $R_{\pm\infty}= R \cup \{+\infty, -\infty\}$, and extend the euclidean topology to $R_{\pm\infty}^n$ in the natural way. Without reference to a particular topology, any topological notion is to be understood with respect to the euclidean topology. Unless stated otherwise, by interval we mean an open interval with endpoints in $R_{\pm\infty}$. We fix infinitely many parameters $0<1<2<\ldots$ in $R$ in such a way that it will be clear from context when these numerals denote elements of $R$ and when they are just natural numbers. At times, we assume that $\mathcal{R}$ expands an ordered group $(R,0,+,<)$ or field $(R,0,1,+,\cdot,<)$, in which case these parameters have their natural interpretations. Throughout, let $\pi: R^{n}\rightarrow R$ denote the projection to the first coordinate. We abuse terminology as follows: we say that a relation $\Phi(x,y)\subseteq R^n\times R_{\pm\infty}^m$ is definable if its restriction to $R^n\times R^m$ is definable and the family of fibers $\{x\in R^n : \Phi(x,y)\}$ for $y\in R_{\pm\infty}^m\setminus R^m$ is definable. Note that in this sense any definable partial function $R\rightarrow R_{\pm\infty}$ satisfies o-minimal monotonicity. We now define the central object of study in this paper. **Definition 1**. A *definable topological space* is a tuple $(X,\tau)$, where $X\subseteq R^n$ is a definable set and $\tau$ is a topology on $X$ such that there exists a definable family of sets $\mathcal{B}_\tau$ that is a basis for $\tau$. We call $\mathcal{B}_\tau$ a *definable basis for $\tau$* and say that the topology $\tau$ is definable. Clearly there is some redundancy in this definition, as the definability of the basis $\mathcal{B}_\tau$ implies the definability of the set $X$. The following are some basic facts about definable topological spaces which are true regardless of the axiom of o-minimality or even the fact that $\mathcal{R}$ expands a linear order. Familiarity with them will be assumed throughout the paper. **Proposition 2**. Let $(X,\tau)$ and $(Y,\mu)$ be definable topological spaces. (a) If $\mathcal{B}_\tau$ is a definable basis for $\tau$, then the family $\mathcal{B}_\tau(x)=\{A\in\mathcal{B}_\tau : x\in A\}$ is a basis of open neighbourhoods of $x$ that is definable uniformly on $x\in X$. (b) Let $Z\subseteq X$ be a definable set. The closure $cl_\tau Z$, interior $int_\tau Z$ and frontier $\partial_\tau Z$ $(:=cl_\tau Z \setminus Z)$ of $Z$ in $(X,\tau)$ are also definable. (c) Let $f:(X,\tau) \rightarrow (Y,\mu)$ be a definable function. The set of points where $f$ is continuous is definable. (d) If $Z\subseteq X$ is a definable set, then the subspace $(Z,\tau|_Z)$ is a definable topological space. (e) The product space $(X\times Y, \tau \times \mu)$ is a definable topological space. When $(X,\tau)$ is a definable topological space and $Y\subseteq X$, we abuse notation by writing $(Y,\tau)$ to mean $(Y,\tau|_Y)$. Given a definable set $X$, we denote the euclidean and discrete topologies on $X$ by $\tau_e$ and $\tau_s$, respectively, in such a way that the notation remains unambiguous. We generally write the letter $e$ in place of $\tau_e$ when used as a prefix or subscript, for example as in $cl_e$ or $e$-neighbourhood, and adopt analogous conventions in writing the letter $s$ in place of $\tau_s$. Moreover, given a definable function $f:X\rightarrow Y$, we say that $f$ is $e$-continuous (respectively, an $e$-homeomorphism) if, as a map $(X,\tau_e)\rightarrow (Y,\tau_e)$, $f$ is continuous (respectively, a homeomorphism). Recall that a topological space $X$ is $T_1$ if every singleton is closed, $T_2$ if it is Hausdorff and $T_3$ if it is Hausdorff and regular, where regular means that any point $x$ $\in X$ and closed set $C \subseteq X$ with $C \not\ni x$ are separated by neighbourhoods, i.e. there exist disjoint open sets $U, V$ $\subseteq X$ with $x\in U$ and $C\subseteq V$. We approach the study of definable topologies in terms of these three separation axioms. We now introduce two definable topologies that are highly relevant to this paper and which are immediate generalizations of classical topologies definable in $(\mathbb{R},<)$. Since there will be no ambiguity, we use some standard terminology to refer to them as understood in our setting. The right half-open interval topology (Appendix [12](#section:examples){reference-type="ref" reference="section:examples"}, Example [Example 116](#example:taur_taul){reference-type="ref" reference="example:taur_taul"}), also called the lower limit topology, on $R$, denoted $\tau_r$, is the topology with definable basis $$[x,y)\,\text{ for } x, y\in R,\, x<y.$$ We reserve the name "Sorgenfrey Line\" to refer to the classical space with this topology, namely $(R,\tau_r)$ when $\mathcal{R}$ expands $(\mathbb{R},<)$. Similarly, the left half-open interval topology (or upper limit topology) on $R$, denoted $\tau_l$, is the topology with definable basis $$(x,y] \, \text{ for } x, y\in R, \, x<y.$$ These spaces are clearly $T_3$. Much as in general topology, they work as counterexamples to a number of otherwise plausible conjectures in our setting. We adopt all notational conventions with respect to $\tau_r$ and $\tau_l$ that were previously set for $\tau_e$ and $\tau_s$ (i.e. we write $r$ and $l$ in place of $\tau_r$ and $\tau_l$, respectively, when used as subscripts or prefixes). **Definition 3**. A definable topological space $(X,\tau)$ is *definably connected* if it is not the union of two disjoint non-empty definable $\tau$-open sets. For a fixed definable topological space $(X,\tau)$, we say that a definable subset $Y\subseteq X$ is definably connected if $(Y,\tau)$ is. A *definably connected component* of $(X,\tau)$ is a maximal definably connected definable subset of $X$. Note that another way of stating that the structure $\mathcal{R}$ is o-minimal is by saying that $(R,\tau_e)$ is definably connected (equivalently, $\mathcal{R}$ is definably complete) and every definable subset of $R$ can be partitioned into finitely many definably connected euclidean spaces. **Remark 4**. Clearly any order reversing bijection $R\rightarrow R$ is a homeomorphism $(R,\tau_r)\rightarrow (R,\tau_l)$. Let $\tau_*$ denote either $\tau_r$ or $\tau_l$. Then, for any distinct pair $\tau,\mu\in \{\tau_e, \tau_*, \tau_s\}$ and intervals $I,J\subseteq X$, there is no definable homeomorphism $(I,\tau)\rightarrow (J,\mu)$. This is obvious if one of the topologies is discrete. If $\{\tau,\mu\}=\{\tau_e,\tau_*\}$, then this follows from the fact that any interval with the euclidean topology is definably connected, while any interval with either the right or left half-open interval topology is totally definably disconnected (i.e. every definably connected subspace is trivial). We now introduce a notion of definable separability that generalizes the one given by Walsberg for definable metric spaces in [@walsberg15]. We provide further justification for our definition and its relationship to that of [@walsberg15] in Subsection [3.2](#subsection: definable separability){reference-type="ref" reference="subsection: definable separability"}. **Definition 5**. We say that a definable topological space $(X,\tau)$ is *definably separable* if there exists no infinite definable family of $\tau$-open pairwise disjoint subsets of $X$. The reader will note the similarity between our definition of definable separability and the countable chain condition (**ccc**, or Suslin's condition) for topological spaces: a topological space has the **ccc** if it does not contain an uncountable family of pairwise disjoint open sets. In general, every separable topological spaces has the **ccc**, but the converse is not true. The main justification for our terminology (we discuss further justifications in Subsection [3.2](#subsection: definable separability){reference-type="ref" reference="subsection: definable separability"}) is that, in o-minimal expansions of $(\mathbb{R},<)$, separability, definable separability, and having the **ccc** are equivalent notions. To make the present paper self-contained we give a proof of this equivalence for one-dimensional spaces. (The proof in the general case is given in forthcoming work by the first author [@ag_separability].) **Proposition 6**. Suppose that $\mathcal{R}$ expands $(\mathbb{R},<)$. Let $(X,\tau)$ be a definable topological space with $\dim X \leq 1$. The following are equivalent. 1. $(X,\tau)$ is definably separable. 2. $(X,\tau)$ is separable. 3. $(X,\tau)$ has the **ccc**. *Proof.* Since $\dim X \leq 1$, applying o-minimal cell decomposition we fix throughout a finite family of triples $\{ (I_i, C_i, f_i) : 1 \leq i \leq m\}$ where, for each $1 \leq i \leq m$, $I_i\subseteq \mathbb{R}$ is a singleton or an interval, $C_i$ is a definable subset of $X$, $f_i \colon I_i \rightarrow C_i$ is a definable bijection, and moreover the family $\{ C_i : 1 \leq i\leq m\}$ is a partition of $X$. Implication $(2)\Rightarrow(3)$ is a routine exercise. We prove $(1)\Rightarrow(2)$ and $(3)\Rightarrow(1)$. **Proof of $(1)\Rightarrow (2)$.** Suppose that $(X,\tau)$ is definably separable. Let $\mathcal{B}$ be a definable basis for $\tau$. Observe that any given $x\in X$ has a finite $\tau$-neighbourhood if and only if there exists a finite set $A\in \mathcal{B}$ with $x\in A$. By o-minimality (uniform finiteness), there exists some $k<\omega$ such that said neighbourhood $A$ is always of size at most $k$. It follows that the set $X_0$ of all $x\in X$ with a finite $\tau$-neighbourhood is definable. Furthermore, observe that, for every $x\in X_0$, there exists a (necessarily unique) $\tau$-neighbourhood of $x$ in $\mathcal{B}$ of minimum size among all $\tau$-neighbourhoods of $x$. We denote this $\tau$-neighbourhood by $A(x)$. Observe that the family $\{A(x) : x\in X_0\}$ is definable. Let $\preccurlyeq_\tau$ denote the classical specialization preorder (reflexive transitive relation) on $X$, which is given by $x \preccurlyeq_\tau y$ whenever $x \in cl_\tau \{y\}$. Let $\sim_\tau^0$ be the equivalence relation on $X$ where $x\sim_\tau^0 y$ whenever $x$ and $y$ are topologically indistinguishable, i.e. they have the same $\tau$-neighbourhoods. Clearly $\sim_\tau^0$ and $\preccurlyeq_\tau$ are both definable. Observe that $x \sim_\tau^0 y$ holds exactly when $x \preccurlyeq_\tau y$ and $y \preccurlyeq_\tau x$ both do. Let us use the notation $[\cdot]_\tau^0$ to denote the equivalence classes by the relation $\sim_\tau^0$. Let $X_{\text{max}} \subseteq X_0$ denote the set of points in $X_0$ that are $\preccurlyeq_\tau$-maximal, in the sense that there does not exist some $y\in X$ with $x \preccurlyeq_\tau y$ and $y \not\preccurlyeq_\tau x$. Since $\preccurlyeq_\tau$ is definable then $X_{\text{max}}$ is definable too. **Claim 1**. For every $x\in X_0$, it holds that $X_{\text{max}} \cap A(x) \neq \emptyset$. *Proof of claim.* The claim follows easily from the observation that, for any $x\in X_0$, it holds that $A(x)=\{ y \in X : x \preccurlyeq_\tau y \}$. In particular the inclusion $A(x)\supseteq \{ y \in X : x \preccurlyeq_\tau y \}$ follows from the definition of $\preccurlyeq_\tau$ and $A(x)\subseteq \{ y \in X : x \preccurlyeq_\tau y \}$ holds by minimality of $A(x)$. ◻ **Claim 2**. $X_{\text{max}}$ is finite. *Proof of claim.* We first show that, for any $x \in X_{\text{max}}$, it holds that $A(x)=[x]_\tau^0$, and hence, in particular, that $[x]_\tau^0$ is $\tau$-open and finite. Let $x\in X_{\text{max}}$. Clearly $[x]_\tau^0 \subseteq A(x)$. On the other hand, suppose, towards a contradiction, that there exists some $y\in A(x)$ with $y \notin [x]_\tau^0$. However, then $x \preccurlyeq_\tau y$ but $y \not\preccurlyeq_\tau x$, contradicting that $x$ is $\preccurlyeq_\tau$-maximal. We derive that the family $\{A(x) : x \in X_{\text{max}}\}=\{ [x]_\tau^0 : x \in X_{\text{max}}\}$ is a definable family of identical or pairwise disjoint $\tau$-open sets. Since $(X,\tau)$ is definably separable, this definable family contains finitely many pairwise disjoint sets, each of which is finite, the union of which contains $X_{\text{max}}$. Hence $X_{\text{max}}$ is finite. ◻ Now recall the notation $\{ (I_i, C_i, f_i) : 1 \leq i \leq m\}$. Let $J\subseteq \{1,\ldots, m\}$ be the set of those $i$ such that $I_i$ is an interval. We define $$D=\bigcup_{i\in J} f_i(\mathbb{Q} \cap I_i) \cup X_{\text{max}}.$$ Since $(X,\tau)$ is definably separable, by Claim [Claim 2](#claim2:sep-prop){reference-type="ref" reference="claim2:sep-prop"} the set $D$ is countable. We show that it is dense in $(X,\tau)$. Let $A$ be a $\tau$-open set. By making $A$ smaller if necessary we may assume that it is definable. If $A$ is infinite, then there is some $i \in \{1,\ldots, m\}$ such that $A\cap C_i$ is infinite, and so $f_i^{-1}(A\cap C_i)$ contains an interval in $I_i$, and thus $\emptyset \neq f_i(\mathbb{Q} \cap I_i) \cap A \subseteq D\cap A$. Now suppose that $A$ is finite and fix $x\in A$. Then $x \in X_0$ and $A(x)\subseteq A$ and so, by Claim [Claim 1](#claim1:sep-prop){reference-type="ref" reference="claim1:sep-prop"}, $\emptyset \neq X_{\text{max}} \cap A(x) \subseteq D \cap A$. **Proof of $(3)\Rightarrow (1)$.** Suppose that $(X,\tau)$ is not definably separable, witnessed by an infinite definable family of pairwise disjoint $\tau$-open sets $\mathcal{A}$. We show that $\mathcal{A}$ is uncountable. Let $i\leq m$ be such that the family $\{ C_i \cap A : A \in \mathcal{A}\}$ is infinite. We show that the infinite definable family of pairwise disjoint sets $\mathcal{Y}=\{f^{-1}_i(C_i \cap A) : A\in \mathcal{A}\}$ is uncountable. For each non-empty set $Y\in \mathcal{Y}$ consider its infimum $\inf Y \in \mathbb{R}\cup\{-\infty\}$. By o-minimality and the fact that the sets in $\mathcal{Y}$ are pairwise disjoint, for any given $t \in \mathbb{R}\cup\{-\infty\}$ at most two sets in $\mathcal{Y}$ can have infinimum equal to $t$. So the set $\mathbb{R}\cap \{ \inf Y : Y\in \mathcal{Y}\setminus \{\emptyset\}\}$ is infinite. Observe that this set is also definable and so, by o-minimality, it contains an interval, which implies that it is uncountable, and it follows that $\mathcal{Y}$ is uncountable too. ◻ We now give a simple characterization of definable separability for $T_1$ one-dimensional definable topological spaces. **Lemma 7**. Let $(X,\tau)$ be a $T_1$ definable topological space with $\dim X\leq 1$. Then $(X,\tau)$ is definably separable if and only if it has finitely many $\tau$-isolated points. *Proof.* The set of all $\tau$-isolated points in $(X,\tau)$ is clearly definable. If it is infinite, then $(X,\tau)$ fails to be definably separable. Conversely, suppose that $(X,\tau)$ is not definably separable, witnessed by an infinite definable family of pairwise disjoint $\tau$-open sets $\mathcal{A}$. Since $\dim X\leq 1$, by the Fiber Lemma for o-minimal dimension [@dries98 Chapter 4, Proposition 1.5 and Corollary 1.6], the family $\mathcal{A}$ contains only finitely many infinite sets, hence infinitely many finite sets. Since $\tau$ is $T_1$, any finite $\tau$-open subset of $X$ must contain only $\tau$-isolated points. It follows that $(X,\tau)$ has infinitely many $\tau$-isolated points. ◻ Following the above lemma, we now consider definable separability in the context of the three fundamental topologies that have been introduced so far. **Proposition 8**. Let $X\subseteq R^n$ be a definable set. (a) [\[itm1:sep-spaces-examples\]]{#itm1:sep-spaces-examples label="itm1:sep-spaces-examples"} The space $(X,\tau_e)$ is definably separable. (b) [\[itm2:sep-spaces-examples\]]{#itm2:sep-spaces-examples label="itm2:sep-spaces-examples"} The space $(X,\tau_s)$ is definably separable if and only if $X$ is finite. (c) [\[itm3:sep-spaces-examples\]]{#itm3:sep-spaces-examples label="itm3:sep-spaces-examples"} Suppose that $n=1$. The spaces $(X,\tau_r)$ and $(X,\tau_r)$ are definably separable. *Proof.* Statement [\[itm2:sep-spaces-examples\]](#itm2:sep-spaces-examples){reference-type="eqref" reference="itm2:sep-spaces-examples"} is obvious and statement [\[itm3:sep-spaces-examples\]](#itm3:sep-spaces-examples){reference-type="eqref" reference="itm3:sep-spaces-examples"} follows immediately from Lemma [Lemma 7](#lem:T1-sep){reference-type="ref" reference="lem:T1-sep"}, o-minimality and the fact that the topologies $\tau_r$ and $\tau_l$ are $T_1$. Statement [\[itm1:sep-spaces-examples\]](#itm1:sep-spaces-examples){reference-type="eqref" reference="itm1:sep-spaces-examples"} also follows immediately in the case $n=1$, by a similar argument to case [\[itm3:sep-spaces-examples\]](#itm3:sep-spaces-examples){reference-type="eqref" reference="itm3:sep-spaces-examples"}. In order to prove [\[itm1:sep-spaces-examples\]](#itm1:sep-spaces-examples){reference-type="eqref" reference="itm1:sep-spaces-examples"} in general, suppose that $(X,\tau_e)$, $X \subseteq R^n$, is not definably separable, witnessed by an infinite definable family of open pairwise disjoint sets $\mathcal{A}$. By o-minimal cell decomposition there exists at least one cell $C\subseteq X$ such that the family $\{ A \cap C : A\in \mathcal{A}\}$ is infinite, and in particular the subspace $(C,\tau_e)$ is not definably separable. Recall that every cell is definably $e$-homeomorphic to an open cell. Hence to prove [\[itm1:sep-spaces-examples\]](#itm1:sep-spaces-examples){reference-type="eqref" reference="itm1:sep-spaces-examples"} it suffices to show that open cells with the euclidean topology are definably separable. Towards a contradiction let $C$ be an open cell such that $(C,\tau_e)$ is not definably separable, witnessed by an infinite definable family of open pairwise disjoint sets $\mathcal{A}$. By o-minimality every definable non-empty open subset of $C$ has dimension equal to $\dim C$, and in particular this holds for every non-empty set in $\mathcal{A}$. Applying the Fiber Lemma for o-minimal dimension [@dries98 Chapter 4, Proposition 1.5 and Corollary 1.6] we conclude that $\dim \bigcup\mathcal{A} > \dim C$, which contradicts that $\bigcup \mathcal{A}\subseteq C$. ◻ We now introduce definable curves, which play a crucial role in the study of o-minimal definable topologies, often taking the role that sequences have in general topology. **Definition 9**. Let $(X,\tau)$ be a definable topological space. A *curve in $X$* is a map $\gamma: (a,b)\rightarrow X$, where $a,b\in R_{\pm\infty}$, $a<b$. We say that *$\gamma$ converges in the $\tau$-topology* (or *converges in $(X,\tau)$*, or *$\tau$-converges*) *as $t$ tends to $a$* to a point $x\in X$ if, for every $\tau$-neighbourhood $U$ of $x$, there exists some $a<t_U<b$ such that $\gamma(t)\in U$ for all $a<t<t_U$. In this case, we say that $x$ is a *$\tau$-limit of $\gamma$ as $t$ tends to $a$* and, if this limit is unique (which will certainly be the case if $\tau$ is Hausdorff), then we write $x=\mathop{\mathrm{\tau{\operatorname{-}}lim}}_{t\rightarrow a}\gamma(t)$. Convergence *as $t$ tends to $b$* is defined analogously. When we say that $\gamma$ *$\tau$-converges* to $x\in X$, without reference to $a$ or $b$, it should be understood that we have already implicitly fixed an endpoint $c\in \{a,b\}$, which we will call the *convergence endpoint* of $\gamma$, and are saying that $\gamma$ $\tau$-converges to $x$ as $t$ tends to $c$. We say that $\gamma$ is *$\tau$-convergent* if it $\tau$-converges to some $x\in X$ (as $t$ tends to its convergence endpoint). **Remark 10**. We adopt some further conventions regarding definable curves. Let $\gamma:(a,b)\rightarrow R^n$ be a definable curve. Frequently, we are only concerned with the behaviour of $\gamma$ near its convergence endpoint $c \in \{a,b\}$. By o-minimality, for any definable set $X\subseteq R^n$ there exists $a'>a$ such that either $\gamma[(a,a')]\subseteq X$ or $\gamma[(a,a')]\subseteq R^n\setminus X$, and the analogous statement holds for $b$. If (say) $c=a$ and there is some $a'>a$ such that $\gamma[(a,a')]\subseteq X$, we may treat $\gamma$ as a curve in $X$ by implicitly identifying it with its restriction $\gamma|_{(a,a')}$. Similarly, we will adopt the convention of saying that $\gamma$ is constant or injective (or some other property) if it has this property when restricting its domain to an appropriate interval as above. By o-minimality, every definable curve $\gamma:(a,b)\rightarrow R^n$ can be assumed to be either constant or injective (strictly monotonic if $n=1$) and $e$-continuous. Whenever we say that $\gamma$ $\tau$-converges and $\mu$-converges, for two definable topologies $\tau$ and $\mu$, and without explicit reference to a convergence endpoint $c\in \{a,b\}$, it should be understood that the same endpoint is being considered in both cases. **Definition 11**. A definable topological space $(X,\tau)$ is *definably compact* if any definable curve $\gamma:(a,b)\rightarrow X$ $\tau$-converges as $t$ tends to $a$ and as $t$ tends to $b$. Definition [Definition 11](#dfn:compact){reference-type="ref" reference="dfn:compact"} is motivated by the definition of definable compactness introduced in [@pet_stein_99]. Adapting the terminology in [@pet_stein_99], we would say that $(X,\tau)$ is definably compact if and only if every definable curve $\gamma$ in $X$ is $\tau$-completable. In our terminology, this corresponds to the property that every definable curve (with any convergence endpoint) $\tau$-converges. It is easy to see that, for a given infinite definable set $X\subseteq R$, the spaces $(X,\tau_r)$, $(X,\tau_l)$ and $(X,\tau_s)$ are not definably compact, and the space $(X,\tau_e)$ is definably compact if and only if $X$ is $e$-closed and bounded. The fact that euclidean spaces are definably compact if and only if they are closed and bounded was proved for sets of all dimensions in [@pet_stein_99 Theorem 2.1]. Other notions of definable compactness besides the one in Definition [Definition 11](#dfn:compact){reference-type="ref" reference="dfn:compact"} have been studied in the o-minimal and other model-theoretic contexts. One of them is the following (recall that a family of sets $\mathcal{B}$ is downward directed if for every pair $B, B'\in\mathcal{B}$ there exists some $B''\in\mathcal{B}$ such that $B''\subseteq B \cap B'$). $$\label{dfn:directed-compact} \parbox{0.9\textwidth}{ Every downward directed definable family of non-empty closed sets has non-empty intersection. }$$ This notion has been studied in the o-minimal context by Johnson [@johnson14], the first author[@ag_FTT], and the authors and Walsberg [@atw1]. It has also been studied in the setting of p-adically closed fields by the first author and Johnson [@ag-j-22]. In the more general model-theoretic context it has been approached by Fornasiero [@fornasiero]. Johnson proved that for o-minimal euclidean spaces it is equivalent to being closed and bounded [@johnson14 Proposition 3.10]. **Remark 12**. Definition [Definition 11](#dfn:compact){reference-type="ref" reference="dfn:compact"} and condition [\[dfn:directed-compact\]](#dfn:directed-compact){reference-type="eqref" reference="dfn:directed-compact"} are equivalent for one-dimensional definable topological spaces. The fact that condition [\[dfn:directed-compact\]](#dfn:directed-compact){reference-type="eqref" reference="dfn:directed-compact"} implies Definition [Definition 11](#dfn:compact){reference-type="ref" reference="dfn:compact"} can be seen from [@atw1 Corollary 44], (3) $\Rightarrow$ (1) (that corollary is presented in the setting of o-minimal expansions of ordered groups, but this part of the statement does not require that assumption). The fact that Definition [Definition 11](#dfn:compact){reference-type="ref" reference="dfn:compact"} implies condition [\[dfn:directed-compact\]](#dfn:directed-compact){reference-type="eqref" reference="dfn:directed-compact"} can be obtained from the results of the present paper (see Remark [Remark 26](#remark_curve_selection){reference-type="ref" reference="remark_curve_selection"}). The details are all presented in [@andujar_thesis Proposition 6.2.4]. For spaces of any dimension, we show together with Walsberg in [@atw1] that the equivalence holds whenever $\mathcal{R}$ expands an ordered group, and more generally in [@ag_FTT] the first author shows that they are equivalent whenever $\mathcal{R}$ has definable choice, and also without any assumption on $\mathcal{R}$ beyond o-minimality whenever the topology of the space is Hausdorff. Since the focus on the present paper is that of definable topological spaces of dimension at most one, we will usually be in a position to assume equivalence of the two definitions. We will largely work with Definition [Definition 11](#dfn:compact){reference-type="ref" reference="dfn:compact"} (but make clear when we are instead working for convenience with condition ([\[dfn:directed-compact\]](#dfn:directed-compact){reference-type="ref" reference="dfn:directed-compact"})). While it is easy to see that compactness implies definable compactness, the converse is not true in general. Nevertheless, both notions are equivalent whenever $\mathcal{R}$ expands the field of reals [@atw1 Corollary 48]. In [@ag_FTT] the first author shows that, if $\mathcal{R}$ expands $(\mathbb{R},<)$, then a definable topological space is compact if and only if it is definably compact in the sense of condition [\[dfn:directed-compact\]](#dfn:directed-compact){reference-type="eqref" reference="dfn:directed-compact"}. By Remark [Remark 12](#rem:compactness){reference-type="ref" reference="rem:compactness"} it follows that, if $\mathcal{R}$ expands $(\mathbb{R},<)$, then definable compactness and compactness are equivalent notions for definable topological spaces of dimension at most one. We will use the following notions frequently throughout this paper. **Definition 13**. Let $(X,\tau)$ be a definable topological space with definable basis $\mathcal{B}$ and let $f:X\rightarrow R^m$ be an injective definable map. We define the *push-forward of $(X,\tau)$ by $f$* to be the definable topological space $(f(X),f(\tau))$, where $f(\tau)$ is the topology on $f(X)$ with definable basis $\{f(A) : A \in \mathcal{B}\}$. Thus $f(\tau)$ is the topology satisfying that $f:(X,\tau)\rightarrow (f(X),f(\tau))$ is a homeomorphism. Given a bijective definable partial map $g:R^n\rightarrow X$, we define the *pull-back of $(X,\tau)$ by $g$* to be its push-forward by $g^{-1}$. Suppose that we have finitely many topological spaces $(X_i,\tau_i)$, where, for each $0\leq i \leq k$, we have $X_i\subseteq R^n$ and $(X_i,\tau_i)$ has basis $\mathcal{B}_i$. Then their disjoint union is the set $\bigcup_{0\leq i \leq k} (\{i\}\times X_i )$ with topology given by the basis $\bigcup_{0\leq i\leq k} (\{i\}\times \mathcal{B}_i )$. Note that, for each $i$, the map $X_i\rightarrow \bigcup_{0\leq i\leq k} (\{i\}\times X_i)$, given by $x\mapsto \langle i,x\rangle$, is an open embedding. The union of finitely many definable topological spaces is clearly definable. If the sets $X_i$ are not all part of the same ambient space $R^n$ and we wish to consider their disjoint union, we first identify them with their product with singletons through the natural push-forward in order to assume that they are. # Definable metric spaces {#section: metric spaces} In this section, we recall the definition of definable metric spaces, introduce the notion of definable metrizability, and discuss further our definition of definable separability (Definition [Definition 5](#dfn:separable){reference-type="ref" reference="dfn:separable"}) in light of the notion with the same name introduced for definable metric spaces by Walsberg [@walsberg15]. Throughout this section, we suppose that $\mathcal{R}$ expands an ordered group $(R,0,+,<)$. In the spirit of the definition of $\mathcal{M}$-norm introduced by the second author in [@thomas12] we include the following definition. **Definition 14**. Let $X$ be a set. An *$\mathcal{R}$-metric on $X$* is a map $d:X\times X\rightarrow R^{\geq 0}$ that satisfies the metric axioms, i.e. identity of indiscernibles, symmetry and subadditivity. We now recall the following definition of a definable metric space from [@walsberg15]. Although Walsberg works under the assumption that $\mathcal{R}$ is an o-minimal expansion of an ordered field, the following definition, as well as any other notion that we borrow from [@walsberg15], still makes sense in the ordered group setting. **Definition 15** ([@walsberg15]). A *definable metric space* is a tuple $(X,d)$, where $X$ is a definable set and $d$ is a definable $\mathcal{R}$-metric on $X$. Any $\mathcal{R}$-metric $d$ generates a topology in the usual way, which we denote by $\tau_d$. Following the conventions set in Section [2](#section:definitions){reference-type="ref" reference="section:definitions"} for the euclidean, discrete, right and left half-open interval topologies, we sometimes abuse notation and write $d$ in place of $\tau_d$. ## Definable metrizability {#subsection: definable metrizability} It is easy to prove that any topology generated by an $\mathcal{R}$-metric is Hausdorff and regular. Any definable $\mathcal{R}$-metric induces a definable topology, and so by this identification every definable metric space is a definable topological space. Much as with the notion of metrizability in general topology, the converse is not true, i.e. there are definable topologies (including Hausdorff regular topologies) that do not arise from definable $\mathcal{R}$-metrics. Hence it is reasonable to investigate which topologies have this property. This motivates the following definitions. **Definition 16**. A topological space $(X,\tau)$ is *$\mathcal{R}$-metrizable* if there exists an $\mathcal{R}$-metric $d$ on $X$ such that $\tau_d=\tau$ and *definably metrizable* if there exists some definable $\mathcal{R}$-metric $d$ on $X$ such that $\tau_d=\tau$. We shall simplify our terminology throughout to refer to metrics, rather than $\mathcal{R}$-metrics, and similarly to metrizability, rather than $\mathcal{R}$-metrizability, without any loss of clarity. Both the euclidean and discrete topologies are definably metrizable (and hence metrizable) on any definable set. A basic example of a definable topological space that is not metrizable would be any non-Hausdorff definable topological space, e.g. the Sierpinski space $X=\{0,1\}$, $\tau=\{\emptyset, \{1\},\{0,1\}\}$. An example of a space that is not definably metrizable but displays all the separation axioms of definable metric spaces would be the space $(R,\tau_r)$, as we now show. **Proposition 17**. The space $(R,\tau_r)$ is not definably metrizable. *Proof.* If $(R,\tau_r)$ were definably metrizable with definable metric $d$, then there would exist, for every $x\in R$, some $\varepsilon_x>0$ such that $B_d(x,\varepsilon_x) \subseteq [x,+\infty)$, where $B_d(x,\varepsilon_x)$ is the $d$-ball of radius $\varepsilon_x$ and center $x$. Let $1$ denote some fixed positive element of $R$ and let $f:X\rightarrow (0,\infty)$ be the definable map given by $f(x)=\sup\{ t\leq 1 : B_d(x,t)\subseteq [x,+\infty)\}$. By o-minimality, there exists an interval $I\subseteq R$ such that, for some $\varepsilon>0$, we have $f(x)>\varepsilon$, for all $x\in I$. Hence, for any distinct $x,y\in I$, it holds that $d(x,y) \geq \varepsilon$, i.e. $(I,\tau_r)$ is a discrete space, which contradicts the definition of $\tau_r$. ◻ The above result still holds if we consider any infinite definable set $X\subseteq R$ in place of $R$ and also if we put $\tau_l$ in place of $\tau_r$. It is worth noting that, in the particular case where $R=\mathbb{R}$, the Sorgenfrey Line is separable but not second countable, and thus it is not even metrizable. On the other hand, if $\mathcal{R}=(\mathbb{Q},+,<)$, then the space $(\mathbb{Q},\tau_r)$ is metrizable (meaning $(\mathbb{Q},+,<)$-metrizable) (see [@diep17]). In Section [10](#section:metrizability){reference-type="ref" reference="section:metrizability"}, we address the question of which metrizable definable topological spaces are definably metrizable, and give a characterization for o-minimal expansions of certain ordered fields including $(\mathbb{R},+,\cdot,<)$ (Theorem [Theorem 104](#thm:metrizability){reference-type="ref" reference="thm:metrizability"}). ## Definable separability {#subsection: definable separability} We now turn to justifying our notion of definable separability (Definition [Definition 5](#dfn:separable){reference-type="ref" reference="dfn:separable"}), in light of a similar definition given by Walsberg for definable metric spaces in [@walsberg15 Section 7.1]. There, it is stated that a definable metric space $(X,d)$ is definably separable if there exists no infinite definable subset $Y\subseteq X$ such that the subspace topology $\tau_d|_Y$ is discrete. First, we note that this definition is similar to ours, in that it simply asks that the property we define be hereditary (i.e. that every definable subspace of a definable metric space is definably separable in our sense). In fact, when restricted to the context of definable metric spaces, Walsberg's definition and the one in this paper are equivalent, as shown by the following. **Lemma 18**. Let $(X,d)$ be a definable metric space. Then $(X,d)$ is definably separable (in the sense of Definition [Definition 5](#dfn:separable){reference-type="ref" reference="dfn:separable"}) if and only if there exists no infinite definable discrete subspace. *Proof.* Let $(X,d)$ be a definable metric space. Let $Y \subseteq X$ be an infinite definable discrete subspace of $X$. Since $\mathcal{R}$ expands an ordered group, it has definable choice, so one may definably select, for each $x\in Y$, some $\varepsilon_x>0$ such that, for every $y\in Y\setminus \{x\}$, $2\varepsilon_x \leq d(x,y)$. From the triangle inequality, it follows that the infinite definable family of open $d$-balls $\{B_d(x,\varepsilon_x) : x\in Y\}$ is pairwise disjoint, hence $(X,d)$ is not definably separable. The other direction follows immediately from definable choice. ◻ From the above lemma it follows that, for the class of definable metric spaces, definable separability is a hereditary property. Moreover note that the proof of the lemma relies solely on definable choice and not on the fact that the structure $\mathcal{R}$ is o-minimal. Further to our discussion of definable separability in Section [2](#section:definitions){reference-type="ref" reference="section:definitions"} (see Proposition [Proposition 6](#prop:sep-equiv){reference-type="ref" reference="prop:sep-equiv"}), our notion of definable separability is further justified by the following observation. If we drop the assumption of definable metrizability, then there are definable topological spaces that are definably separable (according to our definition) but still contain an infinite definable discrete subspace. Some examples are the generalizations to $\mathcal{R}$ of the Moore Plane (defined assuming ordered field structure in $\mathcal{R}$, see Appendix [12](#section:examples){reference-type="ref" reference="section:examples"}, Example [Example 125](#example:Moore_plane){reference-type="ref" reference="example:Moore_plane"}) or the Sorgenfrey Plane (the product of the Sorgenfrey Line with itself); see also Example [Example 123](#example: dfbly sep not hereditary){reference-type="ref" reference="example: dfbly sep not hereditary"} in Appendix [12](#section:examples){reference-type="ref" reference="section:examples"} for an example of dimension one. Hence definable separability is not in general a hereditary property, and Walsberg's definition turns out to be strictly stronger than ours in the general context. This is in accordance with general topology, where every subspace of a separable metric space is separable, but where the Moore Plane and the Sorgenfrey Plane are examples of separable topological spaces with uncountable discrete subspaces. Walsberg showed in [@walsberg15 Theorem 7.1], that, whenever $\mathcal{R}$ expands an ordered field, any definably separable definable metric space is definably homeomorphic to a euclidean space. This result does not generalize to all $T_3$ definable topologies, as witnessed by the right and left half-open interval topologies. In Section [9](#section: affine){reference-type="ref" reference="section: affine"}, we address the question of which one-dimensional definable topological spaces are, up to definable homeomorphism, euclidean. # Preliminary results {#section: prel. results} From now until the end of Section [8](#section:compactifications){reference-type="ref" reference="section:compactifications"}, we return to the general setting in which $\mathcal{R}$ is an o-minimal expansion of a dense linear order without endpoints. ## Spaces of all dimensions {#subsection: prelim_results_alldim} In this section we include some preliminary results concerning definable topologies of all dimensions. The purpose of the following definition is to have a tool that allows us to study definable topologies in the o-minimal setting. This will in particular allow us to provide a description of a basis of neighbourhoods of any given point in a $T_1$ definable topological space (see Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"}). **Definition 19**. Let $(X,\tau)$, with $X\subseteq R^n$, be a definable topological space. Let $x\in X$ and let $\mathcal{B}(x)$ be a basis of $\tau$-neighbourhoods of $x$. We define the *$e$-accumulation set of $x$ in $(X,\tau)$*, namely $E^{(X,\tau)}_x$, to be: $$E^{(X,\tau)}_x:= \bigcap_{A\in\mathcal{B}(x)} \{ y\in R_{\pm\infty}^n : A\cap B\setminus \{y\} \neq \emptyset, \text{ for all } B\in\tau_e \text{ with } y\in B\},$$ where $\tau_e$ refers to the euclidean topology in $R_{\pm\infty}^n$. So $E^{(X,\tau)}_x$ is the intersection of the set of $e$-accumulation points of every $\tau$-neighbourhood of $x$. If $(X,\tau)$ is $T_1$ and $x, y \in X$ with $x\neq y$, then $y\in E^{(X,\tau)}_x$ is equivalent to stating that, for every $\tau$-neighbourhood $A$ of $x$ and every $e$-neighbourhood $B$ of $y$, $A\cap B\neq \emptyset$, i.e. $y \in cl_e A$, for every $\tau$-neighbourhood $A$ of $x$. The definition of $E^{(X,\tau)}_x$ is clearly independent of the choice of basis of neighbourhoods. Note that, if an element $x\in X$ is $\tau$-isolated, then it satisfies $E^{(X,\tau)}_x=\emptyset$. The converse is not true in general; it is however true for $T_1$ spaces. In the case where $X\subseteq R$, this will follow immediately from Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"} and Proposition [Proposition 32](#prop_basic_facts_P_x_2){reference-type="ref" reference="prop_basic_facts_P_x_2"}([\[itm2:basic_facts_P\_x_2\]](#itm2:basic_facts_P_x_2){reference-type="ref" reference="itm2:basic_facts_P_x_2"}). We leave it to the reader to check that the implication holds in higher dimensions, using Remark [Remark 12](#rem:compactness){reference-type="ref" reference="rem:compactness"} and the fact that the space $(R_{\pm\infty}^n, \tau_e)$ for any $n$ is definably compact. For any point $x$ in a euclidean space $(X,\tau_e)$, with $X \subseteq R^n$, it holds that $E^{(X,\tau_e)}_x=\{x\}$. Generally, since there will be no room for confusion, once a definable topological space $(X,\tau)$ is fixed, then, for any $x\in X$, we will write $E_x$ in place of $E^{(X,\tau)}_x$, and will only resort to the latter when we also intend to address the $e$-accumulation set $E^{(Y,\tau)}_x$ for some definable subspace $Y$ containing $x$. The following are facts regarding $e$-accumulation sets that follow immediately from the definition. Recall that the euclidean topology is understood in $R_{\pm\infty}$. **Proposition 20**. Let $(X,\tau)$ be a definable topological space. (a) [\[itm: basic_facts_1\]]{#itm: basic_facts_1 label="itm: basic_facts_1"} $E_x$ is $e$-closed and $E_x \subseteq cl_e X$ for every $x\in X$. (b) [\[itm: basic_facts_2\]]{#itm: basic_facts_2 label="itm: basic_facts_2"} The relation $\{\langle x, y\rangle: y\in E_x \}\subseteq X\times R_{\pm\infty}^m$ is definable. We now prove a bound (Lemma [Lemma 23](#lemma_2){reference-type="ref" reference="lemma_2"}) on the dimension of $e$-accumulation sets (in particular, that they are finite when $\dim(X) \leq 1$). In order to do so, we need two technical lemmas. For the first of these, recall that the dimension of a definable set $X$ is equal to the dimension of the interpretation $X^*$ of $X$ in any elementary extension $\mathcal{R}^*$ of $\mathcal{R}$. **Lemma 21**. Let $\mathcal{R}^*$ be an elementary extension of $\mathcal{R}$. Let $X\subseteq R^n$ be an $\mathcal{R}$-definable set and let $X^*$ denote the interpretation of $X$ in $\mathcal{R}^*$. If $Y\subseteq X^*$ is $\mathcal{R}^*$-definable and $X\subseteq Y$ (in $(R^*)^n$) then $\dim Y = \dim X$ (where $\dim X$ denotes the dimension of $X$ as a definable set in $\mathcal{R}$). *Proof.* Since $Y\subseteq X^*$, we have that $\dim Y \leq \dim X^*=\dim X$. We show, by induction on $n$, that from $X\subseteq Y$ it follows that $\dim X \leq \dim Y$. Since otherwise $X=Y=X^*$, we may assume that $X$ is infinite. Suppose that $n=1$. Since $X$ is infinite, we have that $Y$ is infinite and so, by o-minimality, it must contain an interval. In particular, $\dim Y \geq 1$, and so the result follows. Now suppose that $n>1$. By passing to a cell inside $X$ of maximal dimension if necessary, we may assume that $X$ is a cell. For any given set $S\subseteq R^n$, let $S'$ denote the projection to the first $n-1$ coordinates. For any $x\in Y'$, let $Y_x=\{t : \langle x,t \rangle\in Y\}$. If $\dim X'=\dim X$, then we are done, since $\dim Y' \leq \dim Y$ and by induction hypothesis $\dim X'\leq \dim Y'$. Otherwise, $\dim X= \dim X'+1$ and $X$ is of the form $(f,g)_{X'}$, for continuous functions $f,g:X'\rightarrow R_{\pm\infty}$ with $f<g$. In this case, consider the definable set $Y_{\text{inf}}=\{x\in Y': Y_x \text{ is infinite} \}$. Then $X'\subseteq Y_{\text{inf}}$ and, by induction hypothesis, $\dim X' \leq \dim Y_{\text{inf}}$. By the Fiber Lemma for o-minimal dimension [@dries98 Chapter 4, Proposition 1.5 and Corollary 1.6], $\dim Y_{\text{inf}} + 1 \leq \dim Y$. We conclude that $\dim X= \dim X' +1 \leq \dim Y_{\text{inf}} +1 \leq \dim Y$. ◻ For the next lemma, recall that a family of sets $\mathcal{S}$ has the finite intersection property if $\bigcap \mathcal{F}\neq \emptyset$ for every finite subfamily $\mathcal{F}\subseteq \mathcal{S}$. **Lemma 22**. Let $\{ S_u\subseteq R^n : u\in \Omega\}$ be a definable family with the finite intersection property. Then there exists $\Sigma \subseteq \Omega$ with $\dim \Sigma =\dim \Omega$ such that $\bigcap \{S_u : u\in \Sigma\}\neq \emptyset$. *Proof.* Let $\mathcal{R}^*=(R^*,<,\ldots)$ denote an $|R|^+$-saturated elementary extension of $\mathcal{R}$. Then, by saturation, there exists $x_0\in (R^*)^n$ such that $x_0\in S^*_u$, for every $u\in \Omega$. By Lemma [Lemma 21](#lemma_1){reference-type="ref" reference="lemma_1"}, the $\mathcal{R}^*$-definable set $\Sigma^*_{x_0}=\{u\in \Omega^* : x_0\in S^*_u\}$ has dimension equal to $\dim \Omega$. For each $x\in R^n$, let $\Sigma_x=\{ u\in \Omega : x\in S_u\}$. Since $\dim \Sigma^*_{x_0} = \dim \Omega$ and $\mathcal{R}\preceq \mathcal{R}^*$, there must exist some $x\in R^n$ such that $\dim \Sigma_x=\dim \Omega$. ◻ Let $(X,\tau)$ be a definable topological space. Let $x\in X$ and let $\mathcal{U}$ be a definable basis of neighbourhoods of $x$ in $(X,\tau)$. For the next lemma, we define the *local dimension of $(X,\tau)$ at $x$* to be $$\dim_x(X,\tau)=\min\{\dim A : A\in\mathcal{U}\}.$$ Clearly the definition of local dimension does not depend on the choice of basis of neighbourhoods. This definition generalizes the definition of local dimension of a definable metric space at a point that was introduced by Walsberg in [@walsberg15]. **Lemma 23**. Let $(X,\tau)$ be a $T_1$ definable topological space. For any $x\in X$, $\dim(E_x)<\dim_x(X,\tau)$. In particular, when $\dim X\leq 1$, the set $E_x$ is finite for every $x\in X$. *Proof.* Towards a contradiction, suppose that there exists $x\in X$ such that $\dim E_x \geq\dim_x(X,\tau)$. Let $\{A_u : u\in \Omega\}$ be a definable basis of $\tau$-neighbourhoods of $x$. If $\dim_x(X,\tau)=0$, then, by definition of $E_x$, we have that $E_x=\emptyset$, which contradicts the fact that $\dim E_x \geq\dim_x(X,\tau)$, so we may assume that $\dim_x(X,\tau)>0$. Let $n=\dim_x(X,\tau)$. We have that $\dim E_x \geq n >0$, and in particular that $\dim E_x = \dim (E_x \setminus \{x\})$. For any $y\in E_x \setminus \{x\}$, let $\Omega_y=\{u\in \Omega : y\notin A_u\}$. Since $(X,\tau)$ is $T_1$, the sets $\Omega_y$ are non-empty and in fact the definable family $\{\Omega_y : y\in E_x \setminus \{x\}\}$ has the finite intersection property. By Lemma [Lemma 22](#lemma_FIP){reference-type="ref" reference="lemma_FIP"}, there exists a definable set $B\subseteq E_x\setminus\{x\}$ with $\dim B=\dim E_x$ and there exists $u\in \Omega$ such that $A_u \cap B=\emptyset$. By shrinking $A:=A_u$ if necessary, we may assume that $\dim A = n$. Note however that, by definition of $E_x$, $B \subseteq cl_e A$, and so $B\subseteq \partial_e A$. In particular $\dim E_x = \dim B \leq \dim \partial_e A$. However, by o-minimality, $\dim \partial_e A < \dim A = n$, a contradiction. ◻ We end this subsection with a remark which we will use extensively throughout the paper. In particular, it allows us to make the assumption, whenever $\mathcal{R}$ expands an ordered field, that any definable topological space of dimension at most one is, up to definable homeomorphism, a bounded subset of $R$. **Remark 24**. Let $X$ be a definable set and $n>0$ be such that $\dim(X) \leq n$. If $\mathcal{R}$ expands an ordered group and $X$ is bounded, then there exists a definable injection $f: X \rightarrow R^n$. In particular, if $\tau$ is a definable topology on $X$ then, by passing to the push-forward of $(X,\tau)$ by $f$ if necessary, one may always assume, up to definable homeomorphism, that $X \subseteq R^n$. If, moreover, $\mathcal{R}$ expands an ordered field, then such an injection $f$ exists without the assumption that $X$ is bounded, and with the added condition that $f(X) \subseteq (0,1)^n$. In particular one may always assume, up to definable homeomorphism, that $X \subseteq (0,1)^n$. The existence of such injections in each case can been seen as follows. Suppose that $\mathcal{R}$ expands an ordered group and that $X$ is bounded. Let $\mathcal{X}$ be a finite partition of $X$ into cells. By o-minimality, each cell in $C \in \mathcal{X}$ is in bijection, under an appropriate projection $\pi_C$, with a subset of $R^{\dim C}$. Let $f_C:C\rightarrow R^n$ be the definable map given by $x \mapsto \pi_C(x)$ if $\dim C=n$, and otherwise $x \mapsto \pi_C(x) \times \{ a_C \}$, where $a_C \in R^{n-\dim C}$ is a given fixed parameter. Note that $f_C$ is an injection into $R^n$. Since $X$ is bounded, then so are the sets $f_C(C)$ for $C \in\mathcal{X}$. As $\mathcal{R}$ is an expansion of a group, we can find appropriate translations of these sets such that they do not intersect each other. The union of the image of these translations is then in definable bijection with $X$. Furthermore note that, whenever $\mathcal{R}$ expands an ordered field, the map given coordinate-wise, for every $n$, by $x_i\mapsto \frac{2x_i-1}{(2x_i-1)^2-1}$, for $i=1,\ldots,n$, gives a definable $e$-homeomorphism from $(0,1)^n$ to $R^n$. If $\mathcal{R}$ does not expand an ordered field, then it is not true in general that a definable set $X$ with $\dim X = 1$ is in definable bijection with a subset of $R$. In this case, however, o-minimal cell decomposition and the observations made in Remark [Remark 40](#remark_generalizing_Rx_Lx){reference-type="ref" reference="remark_generalizing_Rx_Lx"} below will suffice to generalize many of the results in this paper concerning spaces $(X,\tau)$ with $X\subseteq R$ to one-dimensional definable topological spaces. ## One-dimensional spaces {#subsection: prelim_results_dim1} We now focus on preliminary results about definable topological spaces $(X,\tau)$ where $\dim X\leq 1$, or at times more specifically when $X\subseteq R$, which we informally refer to as "spaces in the line\". The following lemma shows that definable curve selection is a property of all one-dimensional definable topological spaces (if we drop the requirement that the curves be continuous). In [@atw1], we prove together with Walsberg that it holds for definable topological spaces of all dimensions when $\mathcal{R}$ expands an ordered field, but not in general e.g. when $\mathcal{R}$ expands an ordered group (although in this case one may use definable choice to show that it still holds for definable metric spaces, arguing similarly to the proof of [@atw1 Proposition 41]). **Lemma 25** (Definable curve selection). Let $(X,\tau)$, $\dim X\leq 1$, be a definable topological space. Then $(X,\tau)$ has definable curve selection, that is, for any $x\in X$ and definable set $Y\subseteq X$, $x\in cl_\tau Y$ if and only if there exists a definable curve $\gamma:I \rightarrow Y$ in $Y$ that $\tau$-converges to $x$. *Proof.* It follows readily from the definition of curve convergence that, if there exists a curve in $Y$ $\tau$-converging to $x \in X$, then $x \in cl_{\tau}Y$. We prove the converse. Fix $x\in X$ and a definable set $Y\subseteq X\subseteq R^n$ with $x\in cl_\tau Y$. Let $\mathcal{U}$ denote a definable basis of $\tau$-neighbourhoods of $x$ and set $\mathcal{B}:=\{U\cap Y : U\in\mathcal{U}\}$. It suffices to prove the existence of an interval $I$ with an endpoint $c\in R_{\pm\infty}$ and a definable curve $\gamma: I \rightarrow R^n$ such that, for every $B\in \mathcal{B}$, $\gamma(t)\in B$ for $t\in I$ close enough to $c$. (We may then restrict $\gamma$ to a suitable subinterval close to $c$ to ensure that it maps only into $Y$.) Since otherwise the proof is immediate, we may assume that $x \notin Y$ and hence, for any $y\in Y$, there is $B\in\mathcal{B}$ such that $y\notin B$. First we consider the case where $X\subseteq R$, finding an interval $I$ on which we may take $\gamma$ to be the identity. Consider the definable set $H=\{ t\in R : \exists B\in\mathcal{B}, B\cap (-\infty,t]= \emptyset \}$. If $H$ is empty then, by o-minimality, for every $B\in\mathcal{B}$, there is $t_B\in R$ such that $(-\infty, t_B)\subseteq B$, in which case we may take $I=R$ and $c=-\infty$. Now suppose that $H$ is non-empty. Note that $H$ is an interval in $R$ (possibly right closed) which is unbounded from below. Let $c=\sup H\in R\cup\{+\infty\}$. If $c=\max H$, then there exists $B_c\in \mathcal{B}$ such that $B_c\cap (-\infty,c]= \emptyset$. Let $B\in \mathcal{B}$. If there exists $s_B>c$ such that $(c,s_B)\cap B=\emptyset$, then any set $B'\in \mathcal{B}$ with $B'\subseteq B_c \cap B$ satisfies that $(-\infty, s_B)\cap B' =\emptyset$, contradicting that $c=\sup H < s_B$. Hence, by o-minimality, for any $B\in\mathcal{B}$, there exists $t_B>c$ such that $(c,t_B)\subseteq B$. So let $I=(c,+\infty)$. If $c\notin H$, then, for every $B\in \mathcal{B}$, $B \cap (-\infty, c]\neq \emptyset$. Let $B\in \mathcal{B}$. Suppose that there exists $s_B<c$ such that $(s_B,c)\cap B=\emptyset$. By assumptions on $\mathcal{B}$, we may assume that $c\notin B$. Since $s_B\in H$ there is $B'\in \mathcal{B}$ such that $B' \cap (-\infty,s_B]=\emptyset$. But then any $B''\in\mathcal{B}$ with $B''\subseteq B \cap B'$ satisfies that $(-\infty, c]\cap B'' =\emptyset$, contradicting that $c\notin H$. Hence, by o-minimality, for any $B\in\mathcal{B}$ there exists $t_B<c$ such that $(t_B,c)\subseteq B$. So let $I=(-\infty,c)$. Now, in the case where $X$ is not a subset of $R$, let us pass, by o-minimal cell decomposition, to a cell $Y'\subseteq Y$ such that $x\in cl_\tau Y'$, where $Y'$ is definably homeomorphic to an interval $J$ by a function $f:J\rightarrow Y'$. Then we may apply the above argument taking the family $\{f^{-1}(U\cap Y') : U\in\mathcal{U}\}$ in place of $\mathcal{B}$ to reach an interval $I$ and an endpoint $c$ of $I$ such that, for every $U\in \mathcal{U}$, we have $t\in f^{-1}(U\cap Y')$, for $t\in I$ close enough to $c$. Finally, we take $\gamma=f|_{I\cap J}: I\cap J \rightarrow Y'$. ◻ Note that, in Lemma [Lemma 25](#lemma_curve_selection_R){reference-type="ref" reference="lemma_curve_selection_R"}, we may relax the condition $\dim X \leq 1$ and instead draw the same conclusion, for any $x \in X$, as long as $\dim_x(X, \tau) \leq 1$ (i.e. the local dimension of $(X, \tau)$ at $x$ is at most one). **Remark 26**. Let $\mathcal{B}$ be a downward directed family of subsets of $R^n$ (i.e. for every pair $B, B'\in\mathcal{B}$ there is $B''\in\mathcal{B}$ such that $B''\subseteq B \cap B'$). We say that a curve $\gamma:(a,b)\rightarrow R^n$ with convergence endpoint $c\in\{a,b\}$ is cofinal in $\mathcal{B}$ if, for every $B\in \mathcal{B}$, $\gamma(t)\in B$ for all $t\in (a,b)$ close enough to $c$. Given this terminology, what the proof of Lemma [Lemma 25](#lemma_curve_selection_R){reference-type="ref" reference="lemma_curve_selection_R"} shows is that any definable downward directed family of non-empty sets of dimension at most one admits a definable cofinal curve. In [@atw1], the authors and Walsberg study definable topologies through an analysis of definable directed sets and existence of cofinal maps. We show that definable directed sets of all dimensions admit definable cofinal curves whenever $\mathcal{R}$ expands an ordered field, but that this property does not hold for o-minimal structures in general. These results can be used in characterizing definable compactness as described in Remark [Remark 12](#rem:compactness){reference-type="ref" reference="rem:compactness"}. Specifically, the proof of Lemma [Lemma 25](#lemma_curve_selection_R){reference-type="ref" reference="lemma_curve_selection_R"} can be expanded to show that, for one-dimensional definable topological spaces, definable compactness in the sense of Definition [Definition 11](#dfn:compact){reference-type="ref" reference="dfn:compact"} implies the definition given in condition ([\[dfn:directed-compact\]](#dfn:directed-compact){reference-type="ref" reference="dfn:directed-compact"}) (see [@andujar_thesis Lemma 6.2.1 and Proposition 6.2.4]). Definable curve selection allows us to understand continuity in terms of convergence of definable curves. This was already shown in [@atw1], Proposition 42. We state here explicitly the analogous statement for one-dimensional spaces that we will use throughout this paper. Note, however, that the conclusion of the following statement (as well as that of [@atw1], Proposition 42) holds more generally, in that the proof does not specifically require that $(X,\tau)$ have dimension at most one (or be definable in an expansion of a field), only that it have definable curve selection. **Proposition 27**. Let $(X,\tau)$ and $(Y,\mu)$ be definable topological spaces, where $\dim X \leq 1$. Let $f:(X,\tau) \rightarrow (Y,\mu)$ be a definable map. Then $f$ is continuous at $x\in X$ if and only if, for every definable curve $\gamma:(a,b) \rightarrow X$ and $c\in\{a,b\}$, if $\gamma$ $\tau$-converges to $x$ as $t$ tends to $c$, then $f \circ \gamma$ $\mu$-converges to $f(x)$ as $t$ tends to $c$. *Proof.* The proof is identical to that of [@atw1], Proposition 42, except that, in order to invoke the property of definable curve selection here, the appeal to Proposition 41 therein should be replaced by one to Lemma [Lemma 25](#lemma_curve_selection_R){reference-type="ref" reference="lemma_curve_selection_R"} in this paper. ◻ Definable curve selection also allows us to prove the following lemma, which we will make use of in proving our characterization (Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}) of which one-dimensional definable topological spaces are homeomorphic to euclidean space (see in particular Lemma [Lemma 96](#lemma_walsberg_homeomorphism){reference-type="ref" reference="lemma_walsberg_homeomorphism"}). By Remark [Remark 12](#rem:compactness){reference-type="ref" reference="rem:compactness"}, this is equivalently [@johnson14 Lemma 3.11], but we include this direct proof for the sake of completeness. **Lemma 28**. Let $f:(X,\tau)\rightarrow (Y,\mu)$ be definable continuous bijection between one-dimensional definable topological spaces. If $(X,\tau)$ is definably compact and $(Y,\mu)$ is Hausdorff, then $f$ is a homeomorphism. *Proof.* By Proposition [Proposition 27](#prop_cont_lim){reference-type="ref" reference="prop_cont_lim"} it suffices to prove that, for any definable curve $\gamma$ in $Y$, if $\gamma$ $\mu$-converges to some $f(x)$, then $h^{-1}\circ \gamma$ $\tau$-converges to $x$. Let $\gamma$ be a definable curve in $Y$ $\mu$-converging to some $y\in Y$. By definable compactness of $(X,\tau)$, the curve $f^{-1}\circ \gamma$ $\tau$-converges to some $x\in X$. Then, by Proposition [Proposition 27](#prop_cont_lim){reference-type="ref" reference="prop_cont_lim"} and continuity of $f$, the curve $\gamma=f\circ f^{-1}\circ \gamma$ $\mu$-converges to $f(x)$. Since $\mu$ is Hausdorff, any definable curve can converge to at most one point, so $f(x)=y$. This completes the proof. ◻ Definable curve selection also allows us to prove the following facts regarding $e$-accumulation sets. For completeness and in accordance with the focus of this paper we only prove them for one-dimensional definable topological spaces. Nevertheless, one may show, using [@atw1 Corollary 25], that Proposition [Proposition 29](#basic_facts_P_x_1){reference-type="ref" reference="basic_facts_P_x_1"}([\[itm: basic_facts_3\]](#itm: basic_facts_3){reference-type="ref" reference="itm: basic_facts_3"}) holds for definable topological spaces of all dimensions whenever $\mathcal{R}$ expands an ordered field and, using the fact that the space $(R_{\pm\infty}^m,\tau_e)$ is definably compact for every $m$ and Remark [Remark 12](#rem:compactness){reference-type="ref" reference="rem:compactness"}, that Proposition [Proposition 29](#basic_facts_P_x_1){reference-type="ref" reference="basic_facts_P_x_1"}([\[itm: basic_facts_4\]](#itm: basic_facts_4){reference-type="ref" reference="itm: basic_facts_4"}) holds in general for definable topological spaces of all dimensions. In the next proposition the euclidean closure means closure with respect to the euclidean topology on $R_{\pm\infty}^m$. **Proposition 29**. Let $(X,\tau)$, $X \subseteq R^m$, $\dim X \leq 1$, be a definable topological space. (a) [\[itm: basic_facts_3\]]{#itm: basic_facts_3 label="itm: basic_facts_3"} For any $x\in X$, $y\in R_{\pm\infty}^m$, it holds that $y\in E_x$ if and only if there exists an injective definable curve in $X$ $\tau$-converging to $x$ and $e$-converging to $y$. (b) [\[itm: basic_facts_4\]]{#itm: basic_facts_4 label="itm: basic_facts_4"} Let $Y\subseteq X$ be a definable set and $x\in \partial_\tau Y$. If $\tau$ is $T_1$, then $E_x \cap cl_e Y \neq \emptyset$. *Proof.* The right to left implication in ([\[itm: basic_facts_3\]](#itm: basic_facts_3){reference-type="ref" reference="itm: basic_facts_3"}) is immediate. For the left to right implication, fix $x\in X$ and $y \in E_x$. Consider the definable topology $\mu$ on $X$ where every $z\neq x$ is isolated and where a basis of neighbourhoods of $x$ is given by the family $\{ \{x\} \cup (A \cap B \setminus \{y\}): x\in A\in \tau, \, y\in B\in \tau_e\}$. Clearly, $\mu$ is Hausdorff and finer than $\tau$. Since $y\in E_x$, the sets $(A \cap B \setminus \{y\})$, where $x\in A\in \tau$ and $y\in B\in \tau_e$, are non-empty. In particular, they intersect $X \setminus \{x\}$: if not, then some such set $A \cap B \setminus \{y\}$ must be equal to $\{x\}$, so $y \neq x$, and there is an element $B' \in \tau_e$ with $B' \subseteq B$ that contains $y$ but does not contain $x$; in this case $x \notin A \cap B' \setminus \{y\} \subseteq A \cap B \setminus \{y\} = \{x\}$, so $A \cap B' \setminus \{y\} = \emptyset$, which is a contradication. Hence $x$ is in the $\mu$-closure of $X\setminus \{x\}$. Applying Lemma [Lemma 25](#lemma_curve_selection_R){reference-type="ref" reference="lemma_curve_selection_R"}, there necessarily exists an injective definable curve $\gamma$ in $X \setminus \{x\}$ $\mu$-converging (and thus $\tau$-converging) to $x$. By construction, $\gamma$ must $e$-converge to $y$. To prove ([\[itm: basic_facts_4\]](#itm: basic_facts_4){reference-type="ref" reference="itm: basic_facts_4"}), note that, if $x\in\partial_\tau Y$, then, by Lemma [Lemma 25](#lemma_curve_selection_R){reference-type="ref" reference="lemma_curve_selection_R"}, there is a definable curve $\gamma$ in $Y$ $\tau$-converging to $x$. If the topology is $T_1$, no such curve can be constant, and so, by o-minimality, $\gamma$ can be assumed to be injective. By o-minimality, $\gamma$ $e$-converges in $R_{\pm\infty}^m$ and the result then follows from the right to left implication in ([\[itm: basic_facts_3\]](#itm: basic_facts_3){reference-type="ref" reference="itm: basic_facts_3"}). ◻ We now turn to the notion of $e$-accumulation set for definable topological spaces in the line. While for the rest of the section we deal almost exclusively with spaces in the line, recall (Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"}) that, whenever $\mathcal{R}$ expands an ordered field, any definable topological space of dimension at most one is definably homeomorphic to a space in the line. Furthermore, in Remark [Remark 40](#remark_generalizing_Rx_Lx){reference-type="ref" reference="remark_generalizing_Rx_Lx"} below we describe how the definitions and results for spaces in the line that follow can be generalized to all spaces of dimension at most one, regardless of any assumption on $\mathcal{R}$ beyond o-minimality. **Lemma 30**. Let $(X,\tau)$, $X\subseteq R$, be a definable topological space. (a) [\[itma:lemma_basic_facts_P\_x_2\]]{#itma:lemma_basic_facts_P_x_2 label="itma:lemma_basic_facts_P_x_2"} Given $x, y\in X$, $y\in E_x$ if and only if at least one of the following holds. (i) [\[itm1:lemma_basic_facts_P\_x_2\]]{#itm1:lemma_basic_facts_P_x_2 label="itm1:lemma_basic_facts_P_x_2"} For any $\tau$-neighbourhood $A$ of $x$, there exists $z>y$ such that $(y,z)\subseteq A$. (ii) [\[itm2:lemma_basic_facts_P\_x_2\]]{#itm2:lemma_basic_facts_P_x_2 label="itm2:lemma_basic_facts_P_x_2"} For any $\tau$-neighbourhood $A$ of $x$, there exists $z<y$ such that $(z,y)\subseteq A$. (b) [\[itmb:lemma_basic_facts_P\_x_2\]]{#itmb:lemma_basic_facts_P_x_2 label="itmb:lemma_basic_facts_P_x_2"} It follows from ([\[itma:lemma_basic_facts_P\_x_2\]](#itma:lemma_basic_facts_P_x_2){reference-type="ref" reference="itma:lemma_basic_facts_P_x_2"}) that, if $(X,\tau)$ is Hausdorff then, for any $y\in R$, there exist as most two points $x_0, x_1 \in X$ such that $y$ belongs in both $E_{x_0}$ and $E_{x_1}$ (i.e. for any distinct $x_0,x_1,x_2\in X$, $E_{x_0} \cap E_{x_1} \cap E_{x_2}=\emptyset$). *Proof.* If ([\[itma:lemma_basic_facts_P\_x_2\]](#itma:lemma_basic_facts_P_x_2){reference-type="ref" reference="itma:lemma_basic_facts_P_x_2"})([\[itm1:lemma_basic_facts_P\_x_2\]](#itm1:lemma_basic_facts_P_x_2){reference-type="ref" reference="itm1:lemma_basic_facts_P_x_2"}) fails then, by o-minimality, there exists a $\tau$-neighbourhood $A'$ of $x$ and $z'>y$ such that $(y,z')\cap A'=\emptyset$. Similarly if ([\[itma:lemma_basic_facts_P\_x_2\]](#itma:lemma_basic_facts_P_x_2){reference-type="ref" reference="itma:lemma_basic_facts_P_x_2"})([\[itm2:lemma_basic_facts_P\_x_2\]](#itm2:lemma_basic_facts_P_x_2){reference-type="ref" reference="itm2:lemma_basic_facts_P_x_2"}) fails there is a $\tau$-neighbourhood $A''$ of $x$ and $z''<y$ with $(z'',y)\cap A''=\emptyset$. So $A'\cap A''$ is a $\tau$-neighbourhood of $x$ such that $(z'',z')\cap A'\cap A''\subseteq \{y\}$. This contradicts that $y\in E_x$. The rest of the lemma is immediate. ◻ Lemma [Lemma 30](#lemma:basic_facts_P_x_2){reference-type="ref" reference="lemma:basic_facts_P_x_2"} motivates the following definition. **Definition 31**. Let $(X,\tau)$, $X \subseteq R$, be a definable topological space. For $x\in X$, we define the *right $e$-accumulation set of $x$*, denoted $R_x \subseteq E_x$, to be the set of points $y\in R_{\pm\infty}$ satisfying that, for any $\tau$-neighbourhood $A$ of $x$, there exists $z>y$ such that $(y,z) \subseteq A$. In other words, if $\{A_u: u\in \Omega_x\}$ is a definable basis of $\tau$-neighbourhoods of $x$ in $(X,\tau)$, then $$R_x=\{y\in R_{\pm\infty}: \forall u\in\Omega_x,\, \exists z>y, (y,z)\subseteq A_u\}.$$ So the set $R_x\setminus \{-\infty\}$ is definable, for every $x \in X$. Similarly, the *left $e$-accumulation set of $x$*, denoted $L_x \subseteq E_x$, is defined to be the set of points $y\in R_{\pm\infty}$ satisfying that, for any $\tau$-neighbourhood $A$ of $x$, there exists $z<y$ such that $(z,y) \subseteq A$. In other words, with $\{A_u: u\in \Omega_x\}$ as above, $$L_x=\{y\in R_{\pm\infty}: \forall u\in\Omega_x,\, \exists z<y, (z,y)\subseteq A_u\},$$ and $L_x \setminus \{ + \infty\}$ is likewise a definable set, for every $x \in X$. The following proposition follows from the definition of right and left $e$-accumulation set and Lemma [Lemma 30](#lemma:basic_facts_P_x_2){reference-type="ref" reference="lemma:basic_facts_P_x_2"}. **Proposition 32**. Let $(X,\tau)$ be a definable topological space with $X\subseteq R$ and $x\in X$. Then (a) [\[itm1:basic_facts_P\_x_2\]]{#itm1:basic_facts_P_x_2 label="itm1:basic_facts_P_x_2"} the relations $\{\langle x,y\rangle\in X\times R_{\pm\infty}: y \in R_x\}$ and $\{\langle x,y\rangle\in X\times R_{\pm\infty}: y\in L_x\}$ are definable; (b) [\[itm2:basic_facts_P\_x_2\]]{#itm2:basic_facts_P_x_2 label="itm2:basic_facts_P_x_2"} $E_x= R_x \cup L_x$; (c) [\[itm3:basic_facts_P\_x_2\]]{#itm3:basic_facts_P_x_2 label="itm3:basic_facts_P_x_2"} if $(X,\tau)$ is Hausdorff then, for any $y\in X\setminus \{x\}$, $R_x \cap R_y = \emptyset$ and $L_x \cap L_y = \emptyset$. **Remark 33**. By Lemma [Lemma 23](#lemma_2){reference-type="ref" reference="lemma_2"} and Proposition [Proposition 32](#prop_basic_facts_P_x_2){reference-type="ref" reference="prop_basic_facts_P_x_2"}([\[itm2:basic_facts_P\_x_2\]](#itm2:basic_facts_P_x_2){reference-type="ref" reference="itm2:basic_facts_P_x_2"}), if $(X,\tau)$ is $T_1$, then $R_x$ and $L_x$ are finite for every $x\in X$. **Remark 34**. Let $\gamma$ be a definable curve in $X \subseteq R$. By o-minimality, $\gamma$ $e$-converges to some $y\in R_{\pm\infty}$. If $\gamma$ is injective, then we may assume that it lies in either $(y,+\infty)$ or $(-\infty,y)$ (recall Remark [Remark 10](#remark_assumptions_curves){reference-type="ref" reference="remark_assumptions_curves"}). In the former case, we say that $\gamma$ $e$-converges to $y$ from the right and in the latter that it does so from the left. Let $\tau$ be a definable topology on $X$. Note that, if $\gamma$ $e$-converges to $y$ from the right (respectively left) and $x\in X$, then $\gamma$ $\tau$-converges to $x$ if and only if $y\in R_x$ (respectively $y\in L_x$). The following lemma will be useful in Section [9](#section: affine){reference-type="ref" reference="section: affine"}. It follows easily from Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}. **Lemma 35**. A definable topological space $(X,\tau)$ with $X\subseteq R$ is definably compact if and only if, for every interval $(a,b)\subseteq X$, it holds that $[a,b) \subseteq \bigcup_{x\in X} R_x$ and $(a,b] \subseteq \bigcup_{x\in X} L_x$. It turns out that, if $(X,\tau)$ is $T_1$, then, for any $x\in X$, the sets $R_x$ and $L_x$ characterize a definable basis of neighbourhoods for $x$. We show this in the next lemma. **Lemma 36**. Let $(X,\tau)$, $X\subseteq R$, be a definable $T_1$ topological space. Let $x\in X$. By Remark [Remark 33](#remark:R_L_finite){reference-type="ref" reference="remark:R_L_finite"}, the sets $R_x$ and $L_x$ are finite. Set $R_x:=\{y_1,\ldots, y_n\}$ and $L_x:=\{z_1,\ldots, z_m\}$. Define $\mathcal{U}(x)$ to be the family of sets of the form $$\{x\} \cup \bigcup_{1\leq i \leq n} (y_i,y'_i) \cup \bigcup_{1 \leq j \leq m} (z'_j, z_j),$$ which is a family uniformly definable over $(y'_1, \ldots, y'_n, z'_1,\ldots, z'_m)\in R^{n+m}$, where $y_i<y'_i$ and $z'_j<z_j$. The definable family $\{ U\cap X : U\in \mathcal{U}(x)\}$ is a basis of neighbourhoods of $x$ in $(X,\tau)$. In particular, in the case where $\mathcal{R}$ expands an ordered group and $x$ has a bounded $\tau$-neighbourhood (implying $E_x\cap \{-\infty, +\infty\}=\emptyset$), we may take $\mathcal{U}(x)$ to be of the form $$U(x,\varepsilon):=\{x\} \cup \bigcup_{y\in R_x} (y,y+\varepsilon) \cup \bigcup_{y\in L_x} (y-\varepsilon, y),$$ for $\varepsilon>0$. By passing to a subfamily if necessary, we may always assume that $\mathcal{U}(x)$ is a family of subsets of $X$. *Proof.* Let $\mathcal{U}(x)$ be as in the lemma. By definition of $R_x$ and $L_x$ it clearly holds that, for every $\tau$-neighbourhood $A$ of $x$, there exists $U\in\mathcal{U}(x)$ such that $U \subseteq A \subseteq X$. It therefore remains to prove that all sets in $\mathcal{U}(x)$ are $\tau$-neighbourhoods of $x$. Towards a contradiction, suppose that there exists $U\in\mathcal{U}(x)$ that is not a $\tau$-neighbourhood of $x$. So $x\in \partial_\tau (X\setminus U)$. By Lemma [Lemma 25](#lemma_curve_selection_R){reference-type="ref" reference="lemma_curve_selection_R"}, there exists a definable curve $\gamma: I\rightarrow X\setminus U$ (which is necessarily injective, as $(X,\tau)$ is $T_1$) that $\tau$-converges to $x$ and that, by o-minimality, must $e$-converge to some $a\in R_{\pm\infty}$. By Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}, if $\gamma$ $e$-converges from the right, then $a\in R_x$, and otherwise $a\in L_x$. Either way, by construction of $\mathcal{U}(x)$, it follows that $\gamma(I) \cap U \neq \emptyset$, a contradiction. ◻ From the above lemma, it follows that, if $(X,\tau)$ is a $T_1$ definable topological space with $X\subseteq R$, then a point $x\in X$ is $\tau$-isolated if and only if $E_x=\emptyset$, and the identity map $(X,\tau)\rightarrow (X,\tau_e)$ is continuous at $x\in X$ if and only if $E_x\subseteq \{x\}$. The following lemma will be fundamental in proofs in later sections. **Lemma 37**. Let $(X,\tau)$, $X\subseteq R$, be an infinite definable topological space. Let $f:I\subseteq X \rightarrow R$ be a function on an interval $I=(a,b)$, $a,b\in R_{\pm\infty}$, such that, for every $x\in I$, $f(x)\in E_x$. Suppose that $f$ is $e$-continuous and strictly increasing (respectively decreasing). We extend $f$ to a function $[a,b] \rightarrow R_{\pm\infty}$ by letting $f(a)=\mathop{\mathrm{{\it e}{\operatorname{-}}lim}}_{x\rightarrow a} f(x)$ and $f(b)=\mathop{\mathrm{{\it e}{\operatorname{-}}lim}}_{x\rightarrow b} f(x)$. For all $y\in X$, we have that (a) [\[itma:lemma_f\_Rx_Lx\]]{#itma:lemma_f_Rx_Lx label="itma:lemma_f_Rx_Lx"} for any $x\in [a,b)$, if $x\in R_y$, then $f(x)\in R_y$ (respectively $f(x)\in L_y$); (b) [\[itmb:lemma_f\_Rx_Lx\]]{#itmb:lemma_f_Rx_Lx label="itmb:lemma_f_Rx_Lx"}for any $x\in (a,b]$, if $x\in L_y$, then $f(x)\in L_y$ (respectively $f(x)\in R_y$). Under the additional assumption that $\tau$ is regular, the converse also holds. In other words, (c) [\[itmc:lemma_f\_Rx_Lx\]]{#itmc:lemma_f_Rx_Lx label="itmc:lemma_f_Rx_Lx"} for any $x\in [a,b)$, if $f(x)\in R_y$ (respectively $f(x)\in L_y$), then $x\in R_y$; (d) [\[itmd:lemma_f\_Rx_Lx\]]{#itmd:lemma_f_Rx_Lx label="itmd:lemma_f_Rx_Lx"} for any $x\in (a,b]$, if $f(x)\in L_y$ (respectively $f(x)\in R_y$), then $x\in L_y$. *Proof.* Let $y\in X$ and suppose that $f$ is strictly increasing. We prove ([\[itma:lemma_f\_Rx_Lx\]](#itma:lemma_f_Rx_Lx){reference-type="ref" reference="itma:lemma_f_Rx_Lx"}) and ([\[itmc:lemma_f\_Rx_Lx\]](#itmc:lemma_f_Rx_Lx){reference-type="ref" reference="itmc:lemma_f_Rx_Lx"}) in this case, with all other parts of the lemma proved analogously to one of these cases. We begin with case ([\[itma:lemma_f\_Rx_Lx\]](#itma:lemma_f_Rx_Lx){reference-type="ref" reference="itma:lemma_f_Rx_Lx"}). Suppose that $x\in [a,b) \cap R_y$. If $f(x)\notin R_y$, then, by o-minimality, there is $z>f(x)$ and an open $\tau$-neighbourhood $A$ of $y$ such that $(f(x),z)\cap A =\emptyset$. Since $x\in R_y$, there is $x'>x$ in $I$ such that $(x,x')\subseteq A$. Since $f$ is $e$-continuous and strictly increasing, there is $x'' \in (x,x')$ such that $f(x'')\in (f(x),z)$. So $A$ is a $\tau$-neighbourhood of $x''$ and $f(x'')\notin cl_e A$, which contradicts that $f(x'')\in E_{x''}$. This completes the proof of ([\[itma:lemma_f\_Rx_Lx\]](#itma:lemma_f_Rx_Lx){reference-type="ref" reference="itma:lemma_f_Rx_Lx"}) in the increasing case. To prove ([\[itmc:lemma_f\_Rx_Lx\]](#itmc:lemma_f_Rx_Lx){reference-type="ref" reference="itmc:lemma_f_Rx_Lx"}), suppose that $f(x)\in R_y$, for $x\in [a,b)$, and let $A$ be a $\tau$-neighbourhood of $y$. Then there is some $z>f(x)$ such that $(f(x),z)\subseteq A$. Since $f$ is $e$-continuous and strictly increasing, there is $x'>x$ such that $f[(x,x')]\subseteq (f(x),z)$. For any $x''\in (x,x')$, since $f(x'')\in E_{x''}$ and $f(x'')\in(f(x),z)\subseteq A$, it follows that $x''\in cl_\tau(f(x),z) \subseteq cl_\tau A$. Hence $(x,x')\subseteq cl_\tau A$. So we have shown that, for every $\tau$-neighbourhood $A$ of $y$, there exists $x'>x$ such that $(x,x')\subseteq cl_\tau A$. If $x\notin R_y$, then there must exist some $x''>x$ and some $\tau$-neighbourhood $A'$ of $y$ such that $(x,x'')\cap A'=\emptyset$. But then, by regularity, there is a $\tau$-neighbourhood $A''\subseteq A'$ of $y$ such that $cl_\tau A'' \subseteq A'$, and in particular such that $(x,x'') \cap cl_\tau A'' =\emptyset$, a contradiction by the above. ◻ **Definition 38**. Let $(X,\tau)$ be a definable topological space. We say that $(X,\tau)$ is *definably normal* if, given any pair of disjoint definable $\tau$-closed sets $B,C \subseteq X$, there exist definable disjoint open sets $U,V \subseteq X$ such that $B\subseteq U$ and $C\subseteq V$. We say that $(X,\tau)$ is *definably completely normal* if any definable subspace of $(X,\tau)$ is definably normal. **Proposition 39**. Let $(X,\tau)$, $X\subseteq R$, be a definable topological space. If $(X,\tau)$ is $T_1$ and regular, then it is definably completely normal. *Proof.* We suppose that $(X,\tau)$, $X\subseteq R$, is $T_1$ and regular and prove that it is definably normal. Since being $T_1$ and regular are hereditary properties we conclude that $(X,\tau)$ is definably completely normal. Let $B,C \subseteq X$ be disjoint $\tau$-closed definable sets in $(X,\tau)$. To prove the proposition it suffices to show the existence of a definable $\tau$-neighbourhood $U$ of $B$ such that the $\tau$-closure of $U$ is disjoint from $C$. We proceed by constructing a suitable partition of $B$ into two sets, $B=B'\cup B''$, where $B''$ is finite. By regularity of $(X,\tau)$, there clearly exists a definable $\tau$-neighbourhood $U''$ of $B''$ such that $cl_\tau U'' \cap C=\emptyset$. It is therefore enough to show, with $B'$ and $B''$ defined in this way, the existence of a definable $\tau$-neighbourhood $U'$ of $B'$ such that $cl_\tau U' \cap C=\emptyset$. The proof is then completed by taking $U=U'\cup U''$. First note that, since $(X,\tau)$ is $T_1$ and regular, the space is also Hausdorff. Set $E_B:=\bigcup_{x\in B} E_x$. Let $int_e E_B$ be the euclidean interior of $E_B$ and set $B':=\{x\in B : E_x \subseteq int_e E_B\}$. By o-minimality, $E_B\setminus int_e E_B$ is finite and so, by Hausdorffness and Lemma [Lemma 30](#lemma:basic_facts_P_x_2){reference-type="ref" reference="lemma:basic_facts_P_x_2"}([\[itmb:lemma_basic_facts_P\_x_2\]](#itmb:lemma_basic_facts_P_x_2){reference-type="ref" reference="itmb:lemma_basic_facts_P_x_2"}), $B''=B\setminus B'$ is also finite. Applying Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"}, and using the fact that, by o-minimality, $int_e E_B$ is a finite union of intervals, observe that $B'\cup int_e E_B$ is a $\tau$-neighbourhood of $B'$. Now note that, for any $x\in X$, if $E_x \cap int_e C \neq \emptyset$ then, by definition of the set $E_x$, it holds that $x\in cl_\tau C$, and so $x\in C$ (since $C$ is $\tau$-closed). Since $B\cap C=\emptyset$ it follows that $E_B \cap int_e C =\emptyset$, i.e. $E_B \cap C \subseteq C \setminus int_e C$, and in particular, by o-minimality, $E_B \cap C$ is finite. It follows that $int_e E_B \setminus C$ is cofinite in $int_e E_B$. Now recall from the previous paragraph that $B'\cup int_e E_B$ is a $\tau$-neighbourhood of $B'$. Since $(X,\tau)$ is $T_1$ and $int_e E_B \setminus C$ is cofinite in $int_e B$, we conclude that $U'=B'\cup int_e E_B \setminus C$ is also a $\tau$-neighbourhood of $B'$. We complete the proof by showing that $cl_\tau U' \cap C=\emptyset$. Towards a contradiction, suppose that some $x\in C$ is in the $\tau$-closure of $U'$. Then, since $B$ is $\tau$-closed and disjoint from $C$, $x$ must be in the $\tau$-closure of $int_e E_B \setminus C$. Set $E'_B:=int_e E_B \setminus C$. If there were some $\tau$-neighbourhood $A$ of $x$ such that $A\cap E'_B$ were finite, then, since the space is $T_1$, $A\setminus E'_B$ would also be a $\tau$-neighbourhood of $x$, which contradicts that $x$ is in the $\tau$-closure of $E'_B$. On the other hand, suppose that, for every $\tau$-neighbourhood $A$ of $x$, the intersection $A\cap E'_B$ is infinite. Then, for every such $A$, there exists an interval $I\subseteq A\cap E'_B$. Since, by definition of $E'_B$, every point in $I$ lies in $E_y$, for some $y\in B$, we clearly have that there exists $y \in B$ with $E_y \cap I \neq \emptyset$, hence there exists $y \in B \cap cl_{\tau}I$, by definition of $E_y$, and in particular $y \in B \cap cl_\tau A$. So, in this case, for every $\tau$-neighbourhood $A$ of $x$, it holds that $B \cap cl_\tau A \neq \emptyset$, which contradicts that $(X,\tau)$ is regular. ◻ As we already indicated at the end of Subsection [4.1](#subsection: prelim_results_alldim){reference-type="ref" reference="subsection: prelim_results_alldim"}, the following remark will allow us to generalize many results in this paper about spaces in the line to definable topological spaces of dimension at most one, even when $\mathbb{R}$ does not expand an ordered field. **Remark 40**. Let $(X,\tau)$, $X\subseteq R^n$, $n>1$, be a definable topological space, let $I=(a,b)\subseteq R$, $a,b\in R_{\pm\infty}$, be an interval and let $f:I\rightarrow f(I)\subseteq X$ be an $e$-homeomorphism (which we extend continuously to a function $f\colon [a,b]\rightarrow R_{\pm\infty}^n$). Suppose moreover that $f(a)\neq f(b)$. Consider the definable total order $\prec$ on $cl_e f(I)=f([a,b])$ given by identifying $cl_e f(I)$ with $[a,b]$ through $f$, i.e. for every pair $x,y\in cl_e f(I)$, set $x\prec y$ if and only if $f^{-1}(x) < f^{-1}(y)$. Accordingly, for any $x\prec y$ in $cl_e f(I)$, let $(x,y)_\prec$ denote the corresponding interval with respect to $\prec$. By means of this identification, we may generalise the notion of right and left $e$-accumulation point to points in $cl_e f(I)$. That is, if $x\in X$ and $y\in cl_e f(I)$, then $y\in R_x$ if and only if, for every $\tau$-neighbourhood $U$ of $x$, there is $z\in f(I)$ such that $y \prec z$ and $(y,z)_{\prec}\subseteq U$. Similarly, $y\in L_x$ if and only if, for every $\tau$-neighbourhood $U$ of $x$, there is $z\in f(I)$ such that $z \prec y$ and $(x,y)_{\prec}\subseteq U$. Proposition [Proposition 32](#prop_basic_facts_P_x_2){reference-type="ref" reference="prop_basic_facts_P_x_2"} ([\[itm1:basic_facts_P\_x_2\]](#itm1:basic_facts_P_x_2){reference-type="ref" reference="itm1:basic_facts_P_x_2"}) and ([\[itm3:basic_facts_P\_x_2\]](#itm3:basic_facts_P_x_2){reference-type="ref" reference="itm3:basic_facts_P_x_2"}) generalize to this setting. Now suppose that $\dim X \leq 1$. By o-minimal cell decomposition, there is a finite definable partition $\mathcal{X}$ of $X$ into cells such that, for every $C\in \mathcal{X}$, there is a projection $\pi_C:C\rightarrow I_C\subseteq R$ that is an $e$-homeomorphism onto a cell (i.e. a point or an open interval). Note that, for every one-dimensional cell $C \in \mathcal{X}$, if we set $I_C = (a,b)$ and extend $(\pi_{C})^{-1}$ continuously (as in the first paragraph of this remark) to a function $(\pi_{C})^{-1}:[a,b] \to R_{\pm\infty}^n$, then $(\pi_C)^{-1}(a) \neq (\pi_C)^{-1}(b)$. By passing to a pushforward of $(X,\tau)$ if necessary, we may assume that, for every distinct pair $C,C'\in\mathcal{X}$, we have $cl_e C \cap cl_e C'= \emptyset$. Then, for any $C\in \mathcal{X}$ such that $I_C$ is an interval, let $\prec_C$ be the order on $cl_e C$ given by identifying $cl_e C$ with $cl_e I_C$ through $\pi_C$ (as indicated above). Now let $\{n(C)<\omega : C\in\mathcal{X}\}$ be an enumeration of the cells in $\mathcal{X}$ and let $\prec$ be definable linear order on $cl_e X$ such that, for any $x\in cl_e C$ and $y\in cl_e C'$, where $C, C'\in \mathcal{X}$, we have that $x\prec y$ if and only if $n(C)<n(C')$ or $n(C)=n(C')$ and $x\prec_C y$, that is, $\prec$ is the linear order induced by the lexicographic order given the push-forward $x\mapsto \{n(C)\}\times \pi_C(x)$ for $x\in cl_e C$. Given this convention, the space $(X,\tau)$ behaves very much like a space in the line. The definitions of right and left $e$-accumulation set immediately generalise to points $x\in X$, by saying that $y\in cl_e C$ belongs in $R_x$ or $L_x$ if it does with respect to $\prec_C$. Note that, under this construction, the definitions of sets $R_x$ and $L_x$, for any $x\in X$, are dependent on the choice of cell decomposition $\mathcal{X}$ of $X$. Under this correspondence, the statements and proofs of Lemma [Lemma 30](#lemma:basic_facts_P_x_2){reference-type="ref" reference="lemma:basic_facts_P_x_2"}, Proposition [Proposition 32](#prop_basic_facts_P_x_2){reference-type="ref" reference="prop_basic_facts_P_x_2"}, Lemma [Lemma 35](#lem:RL-compact){reference-type="ref" reference="lem:RL-compact"}, Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"} and Proposition [Proposition 39](#prop_reg_implies_normal){reference-type="ref" reference="prop_reg_implies_normal"} generalise to this setting. Moreover, suppose that, for any $C,C'\in \mathcal{X}$ and partial function $f:C\rightarrow C'$ defined on an interval $(a,b)_{\prec_C}$, we consider that $f$ is increasing or decreasing to mean with respect to $\prec_C$ and $\prec_{C'}$. Then Lemma [Lemma 37](#lemma_f_Rx_Lx){reference-type="ref" reference="lemma_f_Rx_Lx"} and its proof also generalise to $(X,\tau)$. The following definition is not topological in flavour. Nevertheless, we introduce it as a natural property to consider when seeking to prove facts about definable topological spaces via inductive arguments. **Definition 41**. We say a definable topological space $(X,\tau)$ satisfies the *frontier dimension inequality* (**fdi**) if, for every non-empty definable set $Y\subseteq X$, $\dim \partial_\tau Y < \dim Y$. The **fdi** is clearly a hereditary property. The topologies $\tau_e$, $\tau_r$, $\tau_l$ and $\tau_s$ all satisfy the **fdi**; however, we will show that any $T_1$ definable compactification of $(R,\tau)$, where $\tau\in\{\tau_r, \tau_l, \tau_s\}$, does not (see the proof of Corollary [Corollary 97](#cor_them_2){reference-type="ref" reference="cor_them_2"}). Observe that the **fdi** implies in particular that the frontier of any finite set is empty, and so any space with the **fdi** is $T_1$. Walsberg proved [@walsberg15 Lemma 7.15] that every definable metric space satisfies the **fdi**. By an inductive argument on dimension it is easy to show that in any space with this property every definable set is a boolean combination of definable open sets (i.e. property (A) in [@pillay87 Section 2] holds). The next proposition highlights a connection between the **fdi** and regularity. **Proposition 42**. Let $(X,\tau)$, $\dim X \leq 1$, be a Hausdorff definable topological space that satisfies the frontier dimension inequality. Then $(X,\tau)$ is regular. *Proof.* We prove that, for any $x\in X$ and any $\tau$-neighbourhood $A$ of $x$, there exists a $\tau$-neighbourhood $U$ of $x$ such that $cl_\tau U\subseteq A$. Let $x\in X$ and let $A$ be a $\tau$-neighbourhood of $x$. By passing to a subset of $A$ if necessary, we may assume that $A$ is definable. By the frontier dimension inequality, $\partial_\tau A$ is finite. Since $(X,\tau)$ is Hausdorff, there exists, for every $y\in \partial_\tau A$, a $\tau$-neighbourhood $A(y)$ of $x$ such that $y \notin cl_\tau A(y)$. Let $$U= \bigcap_{y\in \partial_\tau A} A(y) \cap A.$$ Then $U$ is a $\tau$-neighbourhood of $x$ and $cl_\tau U\subseteq A$. ◻ Recall that any $T_1$ regular topological space is Hausdorff. Since any definable topological space with the **fdi** is $T_1$, we derive from Proposition [Proposition 42](#prop_T2_frontier_ineq_regular){reference-type="ref" reference="prop_T2_frontier_ineq_regular"} that a one-dimensional definable topological space with the **fdi** is regular if and only if it is Hausdorff. The assumptions of Hausdorffness and the **fdi** in Proposition [Proposition 42](#prop_T2_frontier_ineq_regular){reference-type="ref" reference="prop_T2_frontier_ineq_regular"} are justified respectively by Examples [Example 122](#example: T_1 fdi not regular){reference-type="ref" reference="example: T_1 fdi not regular"} and [Example 123](#example: dfbly sep not hereditary){reference-type="ref" reference="example: dfbly sep not hereditary"} in Appendix [12](#section:examples){reference-type="ref" reference="section:examples"}, which describe non-regular definable topological spaces in the line, the first of which is non-Hausdorff and satisfies the **fdi**, and the second of which is Hausdorff but does not satisfy the **fdi**. In Appendix [12](#section:examples){reference-type="ref" reference="section:examples"}, Example [Example 124](#example:fdi_Hausdorff_not_regular){reference-type="ref" reference="example:fdi_Hausdorff_not_regular"}, we construct a two-dimensional Hausdorff space which satisfies the **fdi** but again is not regular, showing that, in addition, Proposition [Proposition 42](#prop_T2_frontier_ineq_regular){reference-type="ref" reference="prop_T2_frontier_ineq_regular"} does not generalize to spaces of dimension greater than one. # $T_1$ and Hausdorff ($T_2$) spaces. Decomposition in terms of the $\tau_e$, $\tau_c$, $\tau_r$ and $\tau_l$ topologies {#section:T1_T2_spaces} This section focuses on the properties of $T_1$ and Hausdorff definable topological spaces of dimension one. The main results are Corollary [Corollary 44](#cor:general_them_main){reference-type="ref" reference="cor:general_them_main"} and Theorem [Theorem 51](#them 1.5){reference-type="ref" reference="them 1.5"}. The first shows that every infinite $T_1$ definable topological space contains a definable copy of an interval with one of the $\tau_e$, $\tau_s$, $\tau_r$ or $\tau_l$ topologies. We use this to give a positive answer to the Gruenhage 3-element basis conjecture of set-theoretic topology in our setting (see Subsection [5.1](#subsection: 3-el_basis_conj){reference-type="ref" reference="subsection: 3-el_basis_conj"}). Theorem [Theorem 51](#them 1.5){reference-type="ref" reference="them 1.5"} then improves Corollary [Corollary 44](#cor:general_them_main){reference-type="ref" reference="cor:general_them_main"} in the setting of Hausdorff definable topological spaces, by showing that such spaces in the line can be definably partitioned into finitely many subspaces, each of which has one of the $\tau_e$, $\tau_s$, $\tau_r$ or $\tau_l$ topologies, a result which immediately generalizes to all one-dimensional Hausdorff definable topological spaces (Corollary [Corollary 52](#cor 1.5){reference-type="ref" reference="cor 1.5"}). We begin by approaching the first main result of the section only for spaces in the line. **Proposition 43**. Let $(X,\tau)$, $X\subseteq R$, be an infinite $T_1$ definable topological space. Then there exists an interval $J\subseteq X$ such that $(J,\tau)=(J,\tau_\square)$, where $\square$ is one of $e$, $r$, $l$ or $s$. We prove this proposition below. First, however, we present a generalization which follows directly from Proposition [Proposition 43](#them_main){reference-type="ref" reference="them_main"} and the fact that, by o-minimal cell decomposition, every infinite definable set contains a one-dimensional cell. **Corollary 44**. Every infinite $T_1$ definable topological space has a subspace that is definably homeomorphic to an interval with either the euclidean, right half-open interval, left half-open interval, or discrete topology. *Proof of Proposition [Proposition 43](#them_main){reference-type="ref" reference="them_main"}.* By Lemma [Lemma 23](#lemma_2){reference-type="ref" reference="lemma_2"}, for each $x\in X$ the set $E_x$ is finite. Suppose that there exist infinitely many points $x\in X$ satisfying $(x,\infty)\cap E_x=\emptyset$. In that case, let $I\subseteq X$ be a bounded interval containing only such points and fix $C>I$. Otherwise, let $I'\subseteq X$ be an interval such that $(x,\infty)\cap E_x\neq \emptyset$, for every $x\in I'$, and consider the definable map $f$ on $I'$ taking each $x$ to the smallest $y>x$ such that $y\in E_x$. The map $f$ satisfies $x<f(x)$ for all $x\in X$ and so, by o-minimality, after passing if necessary to a subinterval where $f$ is continuous and then applying continuity, there exists an interval $I\subseteq I'$ and $C>I$ such that, for all $x\in I$, $f(x)>C$. In either case, we have that, for all $x\in I$, $(x,C]\cap E_x=\emptyset$. Similarly, we can isolate a bounded subinterval $J\subseteq I$ and some $c<J$ such that, for every $x\in J$, $[c, x)\cap E_x=\emptyset$. Thus we have reached an interval $J$, and $c, C \in R$ with $c<J<C$, such that, for all $x\in J$, we have $[c,C]\cap E_x \subseteq \{x\}$, and so in particular $cl_e J \cap E_x \subseteq \{x\}$. For any $x\in J$, let $\mathcal{U}(x)$ denote a family of $\tau$-neighbourhoods of $x$ as described in Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"}. Note that, by construction of $J$, for any given $x\in J$ and $y<x<z$, there is $U\in\mathcal{U}(x)$ such that, $$\label{eqn_U} U \cap J=\begin{cases} (y, z) & \text{if } x\in R_x \cap L_x, \\ [x,z) & \text{if } x\in R_x \setminus L_x,\\ (y,x] & \text{if } x\in L_x \setminus R_x,\\ \{x\} & \text{if } x\notin R_x \cup L_x. \end{cases}$$ Recall that the families $\{R_x : x\in J\}$ and $\{L_x : x\in J\}$ are definable. Thus we may partition $J$ into four definable sets as follows: $$\begin{aligned} J_1 &=\{ x\in J: x\in R_x \cap L_x\}, \\ J_2 &=\{ x\in J : x \in R_x, x\notin L_x\}, \\ J_3 &=\{ x\in J : x \notin R_x, x\in L_x \}, \\ J_4 &= \{ x\in J : x\notin R_x \cup L_x \}. \\ \end{aligned}$$ By [\[eqn_U\]](#eqn_U){reference-type="eqref" reference="eqn_U"} and the definitions of $R_x$ and $L_x$, the subspace topology on $J_1$ is $\tau_e$. Similarly, the subspace topologies on $J_2$, $J_3$ and $J_4$ are $\tau_r$, $\tau_l$ and $\tau_s$, respectively. At least one of the four definable sets $J_1$, $J_2$, $J_2$ and $J_4$ must be infinite and thus contain an interval, and so the proposition follows. ◻ For a justification for the condition of $T_1$-ness in Proposition [Proposition 43](#them_main){reference-type="ref" reference="them_main"}, see Appendix [12](#section:examples){reference-type="ref" reference="section:examples"}, Example [Example 120](#example: t_0 not t_1){reference-type="ref" reference="example: t_0 not t_1"}, which describes a $T_0$ definable topological space in the line that fails to be $T_1$ and does not contain an interval with one of the $\tau_e$, $\tau_r$, $\tau_l$ or $\tau_s$ topologies. ## 3-element basis conjecture {#subsection: 3-el_basis_conj} We now discuss how Corollary [Corollary 44](#cor:general_them_main){reference-type="ref" reference="cor:general_them_main"} relates to an open conjecture of set-theoretic topology due to Gruenhage known as the 3-element basis conjecture. Given a class of topological spaces $\mathcal{S}$, a basis for $\mathcal{S}$ is a subclass $\mathcal{F} \subseteq \mathcal{S}$ such that every member of $\mathcal{S}$ contains a subspace that is homeomorphic to one of the members of $\mathcal{F}$. When the class $\mathcal{S}$ consists of topological spaces definable in some structure, we will say that $\mathcal{F} \subseteq \mathcal{S}$ is a *definable basis* for $\mathcal{S}$ if every member of $\mathcal{S}$ contains a subspace that is definably homeomorphic to one of the members of $\mathcal{F}$. Consider the following statement. $$\label{star} \parbox{0.9\textwidth}{The class $\mathcal{S}$ of uncountable, first-countable, regular, Hausdorff topological spaces has a basis $\mathcal{F}$ consisting of an uncountable discrete subspace, a fixed uncountable subset of the reals with the euclidean topology, and a fixed uncountable subset of the reals with the Sorgenfrey topology.}$$ The *$3$-element basis conjecture for uncountable, first-countable, regular, Hausdorff spaces* is an open conjecture stating that $\eqref{star}$ is consistent with ZFC. It arose in connection with various questions concerning perfectly normal, compact, Hausdorff spaces, in particular questions due to Fremlin and Gruenhage, which we discuss further in Subsection [9.1](#subsection: Fremlin){reference-type="ref" reference="subsection: Fremlin"}. The $3$-element basis conjecture appears as Question 2 in [@gm07], following a series of works by Gruenhage in which related statements were put forward. In [@gru88] and [@gru89], Gruenhage considered the statement that all uncountable, first-countable, regular, Hausdorff spaces contain either an uncountable metrizable subspace or a copy of an uncountable subspace of the Sorgenfrey Line. He proved under the Proper Forcing Axiom (PFA) that this statement holds for regular cometrizable spaces (spaces that admit a coarser metric topology such that each point has a neighbourhood base consisting of sets closed in the metric topology), by showing that all such spaces either contain an uncountable discrete subspace, contain a copy of an uncountable subset of the reals with the Sorgenfrey topology, or are cosmic (i.e. are the continuous image of a separable metric space). Todorčević then reproved this result under the Open Colouring Axiom (OCA), which follows from PFA [@todor89]. In both cases, the assumption of first-countability was not in fact required. Following these results, Gruenhage put forward the question of the consistency (with ZFC) of the version of [\[star\]](#star){reference-type="eqref" reference="star"} in which the first-countability assumption is not imposed [@gru90]. This stronger statement was shown to fail by Moore in his solution to the $L$-space problem [@moore06], and so the first-countability assumption in [\[star\]](#star){reference-type="eqref" reference="star"} is necessary if the $3$-element basis conjecture is to have a positive answer. (Note that it is known that $L$-spaces, which are hereditarily Lindelöf non-separable spaces, as well as $S$-spaces, which are hereditarily separable non-Lindelöf spaces, provide counterexamples to [\[star\]](#star){reference-type="eqref" reference="star"}; these can be constructed, for example, with the continuum hypothesis (CH), but the existence of $S$-spaces is independent of ZFC [@todor83] and, to the best of our knowledge, to date no *first-countable* $L$-space has been constructed in ZFC. See [@gru90] for further discussion.) Beyond the statements above, we note that Todorčević studied related questions in the context of the class of spaces that can be represented as relatively compact subsets of the class of all Baire class 1 functions on a Polish space, endowed with the topology of pointwise convergence [@todor99]. In addition, the conclusion of the conjecture was shown by Farhat to be consistent under PFA for the class of uncountable subspaces of monotonically normal compacta, and under Souslin's Hypothesis (SH) for any uncountable space having a zero-dimensional monotonically normal compactification [@farhat15]. Recently, Peng and Todorčević gave an analysis of different possible approaches to proving or disproving the conjecture [@peng_todor22]. Here, we indicate how our work provides a number of results related to the 3-element basis conjecture in the context of topological spaces definable in o-minimal structures. In particular, Corollary [Corollary 44](#cor:general_them_main){reference-type="ref" reference="cor:general_them_main"}, shown above, establishes the existence of a $3$-element basis, as is posited by ([\[star\]](#star){reference-type="ref" reference="star"}), for the class of infinite $T_1$ topological spaces definable in any o-minimal expansion of $( \mathbb{R}, < )$. Moreover, in such a basis, the fixed subsets of the reals that form the underlying sets of the basis elements (i.e. those which have either the discrete, euclidean or Sorgenfrey topology) can each be taken to be $\mathbb{R}$ itself. More specifically, Corollary [Corollary 44](#cor:general_them_main){reference-type="ref" reference="cor:general_them_main"} states that, for any o-minimal structure $\mathcal{R}=(R,<,\ldots)$, the collection of definable topological spaces $\{ (I,\tau) : I \subseteq R \text{ is an interval}, \, \tau\in\{\tau_e, \tau_r, \tau_l, \tau_s\} \}$ is a definable basis for the class of infinite $T_1$ topological spaces definable in $\mathcal{R}$. Clearly, if $\mathcal{R}$ defines an order-reversing bijection (e.g. when $\mathcal{R}$ expands an ordered group), then this definable basis can be reduced to the family $\{ (I,\tau) : I \subseteq R \text{ is an interval}, \, \tau\in\{\tau_e, \tau_r, \tau_s\} \}$. In addition, whenever $\mathcal{R}$ expands an ordered field, this can in fact be reduced further to a definable basis consisting only of the three fixed spaces $(R,\tau_e)$, $(R,\tau_r)$ and $(R, \tau_s)$, since, in this case, any interval with one of the $\tau_e$, $\tau_r$, or $\tau_s$ topologies is definably homeomorphic to one of these three spaces. If there is no requirement that the basis be definable (in the sense that the homeomorphisms involved are definable), then, as long as the ordered set $(R,<)$ can be expanded to an ordered field $(R, +, \cdot, <)$, we have that $\{ (R,\tau_e), (R,\tau_r), (R, \tau_s)\}$ serves as a 3-element basis, in the sense of [\[star\]](#star){reference-type="eqref" reference="star"}, for the class of infinite $T_1$ topological spaces definable in any o-minimal expansion of $(R, < )$ (and in particular this holds, as indicated above, in the special case that $(R,<)=(\mathbb{R},<)$). Note that, for none of these results do we require the assumptions of uncountability, first-countability, Hausdorffness or regularity (although it should be noted that, in the special case of o-minimal expansions of $( \mathbb{R}, < )$, every definable topological space is known to be first-countable; see [@atw1 Proposition 38]). ## Non-definability of the Cantor space {#subsection: Cantor_space} We now consider the Cantor Space $2^\omega$ (i.e. the product of countably many copies of the discrete space $\{0,1\}$). Whether or not such a space exists (up to homeomorphism) in the form of a definable topological space in a given structure could be considered a means of assessing the tameness of the structure. We make use of Proposition [Proposition 43](#them_main){reference-type="ref" reference="them_main"} to prove that, in the setting of o-minimal structures, no space homeomorphic to the Cantor Space is definable. Note that there is a notion of a 'Cantor set', as defined in [@fkms-10], which has been studied by various authors in considering notions of tameness expanding o-minimality. A Cantor set in this sense is (in the terminology of the present paper) any non-empty compact subset of $(\mathbb{R}, \tau_e)$ with empty interior and no isolated points. The Cantor space is known to be homeomorphic to such a set (for example, to the classical middle thirds Cantor set with the euclidean topology). On the one hand, in any o-minimal expansion of $(\mathbb{R},<)$, such a Cantor set is clearly not definable. However, on the other hand, there are examples of definable topological spaces in this structure which do have non-empty definable subsets that are compact, have empty interior and do not have isolated points (for example, the subset $[0,1] \times \{0\}$ of the definable Alexandrov $2$-line, which will be defined in Section [6](#section: universal spaces){reference-type="ref" reference="section: universal spaces"}, Definition [Definition 55](#example_n_line_0){reference-type="ref" reference="example_n_line_0"}). We show that, nevertheless, the Cantor space is not homeomorphic to any definable topological space in an o-minimal structure. The proof of this fact will follow from a series of auxiliary results, beginning with two concerning the weights of our key definable topological spaces. Recall the classical definition of the *weight* of a topological space $(X,\tau)$, $w_\tau(X)$, namely the minimum cardinality of a basis for $\tau$. **Lemma 45**. For any set $X\subseteq R$ it holds that $w_r(X)=w_l(X)=w_s(X)=|X|$. Furthermore, in the case where $X$ is infinite and definable and $(R,<)$ can be expanded to an ordered field, it holds that $w_r(X)=w_l(X)=w_s(X)=|R|$. *Proof.* The second sentence of the lemma follows from the first one by applying o-minimality and the fact that any interval in an ordered field is in bijection with the whole field. We prove the first sentence. In the case of the discrete topology $\tau_s$ the statement $w_s(X)=|X|$ is immediate from the definition. We show that $w_{r}(X)=|X|$. (An analogous argument shows that $w_{l}(X)=|X|$.) We may assume that $X$ is infinite, since otherwise the topology is discrete. By the definition of $\tau_r$ (Appendix [12](#section:examples){reference-type="ref" reference="section:examples"}, Example [Example 116](#example:taur_taul){reference-type="ref" reference="example:taur_taul"}) we have that $w_{r}(X)\leq |X|^2 = |X|$. We show the reverse inequality. Let $\mathcal{B}$ be a basis of $(X,\tau_r)$ of minimum cardinality. For every $x\in X$ there exists some $A\in\mathcal{B}$ such that $x\in A \subseteq [x,+\infty)$. A map $X\rightarrow \mathcal{B}$ that takes each $x\in X$ to one such neighbourhood $A$ must be injective, so $|X|\leq |\mathcal{B}|= w_r(X)$. ◻ **Proposition 46**. Let $(X,\tau)$ be an infinite $T_1$ definable topological space. Let $\alpha_R:=\min\{ w_e(I) : I\subseteq R \text{ is an interval}\}$. The following hold. (a) $\alpha_R\leq w_\tau(X) \leq |X|\leq 2^{w_e(X)}$.[\[itm_weight_1\]]{#itm_weight_1 label="itm_weight_1"} (b) If $\mathcal{R}$ expands an ordered field, then $w_e(R)\leq w_\tau(X)$. [\[itm_weight_2\]]{#itm_weight_2 label="itm_weight_2"} *Proof.* The inequality $w_\tau(X) \leq |X|$ is given by [@ag_cardinality Corollary 2]. The inequality $|X|\leq 2^{w_e(X)}$ follows from noticing that, for any basis for the euclidean topology on $X$, due to this topology being $T_1$, the map taking each $x\in X$ to the collection of its basic neighbourhoods is injective. To prove that $\alpha_R\leq w_\tau(X)$, note that, for any topological space $(Z,\mu)$ and subspace $Z'\subseteq Z$, it holds that $w_\mu(Z') \leq w_\mu(Z)$. Now, by Corollary [Corollary 44](#cor:general_them_main){reference-type="ref" reference="cor:general_them_main"}, there exists an interval $I\subseteq R$ and a definable embedding $(I,\mu)\hookrightarrow (X,\tau)$, where $\mu$ is one of $\tau_e$, $\tau_r$, $\tau_l$ or $\tau_s$. In particular we have that $w_\mu(I)\leq w_\tau(X)$. Furthermore by Lemma [Lemma 45](#lem:weight-taur){reference-type="ref" reference="lem:weight-taur"} observe that $\alpha_R \leq w_\mu(I)$, and so we derive that $\alpha_R \leq w_\tau(X)$. Statement ([\[itm_weight_2\]](#itm_weight_2){reference-type="ref" reference="itm_weight_2"}) follows from the inequality $\alpha_R\leq w_\tau(X)$ in ([\[itm_weight_1\]](#itm_weight_1){reference-type="ref" reference="itm_weight_1"}) and from noticing that, if $\mathcal{R}$ expands an ordered field, then any two intervals are definably $e$-homeomorphic, and so $\alpha_R = w_e(R)$. ◻ **Lemma 47**. Let $(X,\tau)$ be a $T_1$ definable topological space, let $I\subseteq R$ be an interval, and suppose that there exists an injective definable curve $\gamma:I \to X$. If $(X,\tau)$ is compact, then the linear order $(I,<)$ is Dedekind complete (i.e. every non-empty subset of $I$ that is bounded above in $I$ admits a supremum). *Proof.* Towards a contradiction suppose that there exists a non-empty set $S\subseteq I$ bounded above in $I$ but with no supremum. Let $S'$ be the set of upper bounds of $S$ in $I$. Consider the following family of closed non-empty subsets of $(X,\tau)$: $$\mathcal{S}= \{ cl_\tau(\gamma[(t,s)]) : t\in S, s\in S'\}.$$ This family clearly has the finite intersection property. By compactness of $(X,\tau)$, there exists $x\in X$ belonging in $\bigcap \mathcal{S}$. We define a curve $\gamma_x \colon I \to X$ as follows. If $x\notin \gamma(I)$, then we fix some $t_x\in I$ and set $$\gamma_{x}(t) = \begin{cases} \gamma(t) &\text{ if } t \neq t_x, \\ x & \text{ if } t=t_x. \end{cases}$$ Otherwise, let $\gamma_x=\gamma$. Let $(I,\tau_x)$ be the pull-back of $(\gamma_x(I),\tau)$ by $\gamma_x$. Now note that, for every $t \in S$, every $s \in S'$ and every $\tau_x$-neighbourhood $U$ of $t_x$, we have $U\cap (t,s) \neq \emptyset$. This is clear if $t_x$ itself also lies in $(t,s)$. If $t_{x} \notin (t,s)$, then $\gamma = \gamma_{x}$ on $(t,s)$, and so $x \in cl_{\tau}(\gamma[(t,s)]) = cl_{\tau}(\gamma_{x}[(t,s)])$, whence $\gamma_{x}(U) \cap \gamma_{x}[(t,s)] \neq \emptyset$, as $\gamma_{x}(U)$ is a $\tau$-neighbourhood of $x$ in $(\gamma_x(I),\tau)$; it follows that $U \cap (t,s) \neq \emptyset$, by injectivity of $\gamma_{x}$. Now, clearly $(I,\tau_x)$ is $T_1$, and so, by Lemma [Lemma 23](#lemma_2){reference-type="ref" reference="lemma_2"}, the set $E^{(I,\tau_x)}_{t_{x}}$ is finite. Since $S$ has no supremum in $I$, there exists $t\in S$ with $t > \{ y \in E^{(I,\tau_x)}_{t_{x}} : \exists z \in S \text{ with } z > y\}$ and $s\in S'$ with $s<S'\cap E^{(I,\tau_x)}_{t_{x}}$. We may also choose them so that $t_x\notin (t,s)$. The fact that every $\tau_x$-neighbourhood of $t_x$ intersects the interval $(t,s)$ clearly contradicts Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"}. ◻ **Remark 48**. Since any two intervals are order-isomorphic in an ordered field, it follows from Lemma [Lemma 47](#lemma_compact_implies_reals){reference-type="ref" reference="lemma_compact_implies_reals"} that, under the assumption that $\mathcal{R}$ expands an ordered field, if there exists an infinite compact $T_1$ topological space definable in $\mathcal{R}$, then $(R,<)$ is Dedekind complete. In particular, since $(\mathbb{R},+,\cdot,<)$ is, up to (unique) field isomorphism, the only Dedekind complete ordered field, it must be that $\mathcal{R}$ is an expansion of the field of reals. **Proposition 49**. There exists no infinite definable topological space $(X,\tau)$ that is compact, totally disconnected, and that satisfies $w_\tau(X)<|Y|$ for every $Y\subseteq X$ that is infinite and definable. *Proof.* Let $(X,\tau)$ be an infinite compact totally disconnected definable topological space satisfying $w_\tau(X)<|Y|$ for every $Y\subseteq X$ that is infinite and definable. We reach a contradiction by showing that $(X,\tau)$ contains an infinite (and in fact definable) connected subspace. First note that, since $(X,\tau)$ is totally disconnected, in particular it is $T_1$. By o-minimality, there exists an interval $I\subseteq R$ and an injective definable curve $\gamma:I \rightarrow X$. Let $(I,\mu)$ be the pull-back of $(\gamma(I),\tau)$ by $\gamma$. Since $(I,\mu)$ is $T_1$, by Proposition [Proposition 43](#them_main){reference-type="ref" reference="them_main"} we may assume, after passing to a subinterval if necessary, that $\mu \in \{\tau_e, \tau_r, \tau_l, \tau_s\}$. Since, by hypothesis, $w_{\mu}(I)=w_\tau(\gamma(I))\leq w_\tau(X) < |f(I)|=|I|$, by Lemma [Lemma 45](#lem:weight-taur){reference-type="ref" reference="lem:weight-taur"} it must be that $\mu=\tau_e$. Now recall that, by compactness and Lemma [Lemma 47](#lemma_compact_implies_reals){reference-type="ref" reference="lemma_compact_implies_reals"}, $(I,<)$ is Dedekind complete, and so $(I,\tau_e)$ is connected. Hence $(\gamma(I),\tau)$ is connected. ◻ **Corollary 50**. The Cantor space $2^\omega$ is not a definable topological space. *Proof.* Let $(X,\tau)$ be (homeomorphic to) the Cantor space and towards a contradiction suppose that it is a definable topological space. Let $\mathfrak{c}=|X|$ denote the cardinality of the continuum. Recall that the Cantor space is compact, totally disconnected (in particular $T_1$) and second countable (i.e. $w_\tau(X)\leq\omega$). Let $Y\subseteq X$ be an infinite definable set. We show that $|Y|=\mathfrak{c}$. The result then follows from Proposition [Proposition 49](#prop_no_cantor_set){reference-type="ref" reference="prop_no_cantor_set"}. Clearly $|Y|\leq \mathfrak{c}$. By o-minimality, there exists an interval $I\subseteq R$ and an injective definable curve $\gamma:I\rightarrow Y \subseteq X$. Since $(X,\tau)$ is compact and $T_1$, by Lemma [Lemma 47](#lemma_compact_implies_reals){reference-type="ref" reference="lemma_compact_implies_reals"} we have that $(I,<)$ is Dedekind complete, so $\mathfrak{c}\leq |I|$, and hence $\mathfrak{c}\leq |\gamma(I)|\leq |Y|$. ◻ It follows from the above corollary that the class of definable topological spaces up to homeomorphism is not closed under countable products. ## Hausdorff ($T_2$) spaces {#subsection: T2_spaces} We now prove a strengthening of Proposition [Proposition 43](#them_main){reference-type="ref" reference="them_main"} for Hausdorff spaces. The assumption of Hausdorffness is necessary here; see Appendix [12](#section:examples){reference-type="ref" reference="section:examples"}, Example [Example 121](#example: t_1 not t_2){reference-type="ref" reference="example: t_1 not t_2"} for a $T_1$, non-Hausdorff space that cannot be decomposed as described in the statement. **Theorem 51**. Let $(X,\tau)$, $X\subseteq R$, be a Hausdorff definable topological space. Then there exists a finite partition $\mathcal{X}$ of $X$ into points and intervals such that, for every $Y\in\mathcal{X}$, $\tau|_Y \in \{\tau_e, \tau_r, \tau_l, \tau_s\}$. *Proof.* We start by proving a simple case. Suppose that, for every $x\in X$, $E^{(X,\tau)}_x\subseteq \{x\}$. We call this condition (). Then let us partition $X$ into four definable sets as follows. $$\begin{aligned} &\{ x\in X: x\in R_x \cap L_x\}, \\ &\{ x\in X: x \in R_x, x\notin L_x\}, \\ &\{ x\in X: x \notin R_x, x\in L_x \}, \\ &\{ x\in X: x\notin R_x \cup L_x \}. \\ \end{aligned}$$ By Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"}, these correspond respectively to spaces with the $\tau_e$, $\tau_r$, $\tau_l$ and $\tau_s$ topologies. By o-minimality, we can partition each of these into a finite number of points and intervals, and the result follows. In order to prove the theorem, it is enough to show that we may partition $(X,\tau)$ into finitely many definable subspaces where () holds. We do so as follows. Note that, for any definable subspace $S\subseteq X$ and any $x\in S$, $E^{(S,\tau)}_x \subseteq E^{(X,\tau)}_x$. We prove the existence of a finite partition of $X$ formed by points and intervals such that, for any interval $I$ in the partition and any $x\in I$, $E^{(X,\tau)}_x \cap cl_e I \subseteq \{x\}$. Since any element in $E^{(I,\tau)}_x$ must belong in $cl_e I$ (Proposition [Proposition 20](#basic_facts_P_x){reference-type="ref" reference="basic_facts_P_x"}([\[itm: basic_facts_1\]](#itm: basic_facts_1){reference-type="ref" reference="itm: basic_facts_1"})), it follows that, for any $x\in I$, $E^{(I,\tau)}_x = E^{(I,\tau)}_x \cap cl_e I \subseteq E^{(X,\tau)}_x \cap cl_e I \subseteq \{x\}$, i.e. () holds in $(I,\tau)$, which completes the proof. From now on, for any $x\in X$, let $E_x=E^{(X,\tau)}_x$. By Lemma [Lemma 23](#lemma_2){reference-type="ref" reference="lemma_2"}, for any $x\in X$, the set $E_x$ is finite. By o-minimality (uniform finiteness), there exists some $n$ such that $|E_x|\leq n$, for every $x\in X$. We may partition $X$ into finitely many definable subspaces $X_0,\ldots, X_n$, where $X_i=\{x\in X : |E_x|=i\}$, for $0\leq i \leq n$. We fix $Y=X_m$, for some $0\leq m \leq n$ and prove the existence of a partition of $Y$ with the desired properties. Since otherwise the result is trivial we assume that $m>0$ and that $Y$ is infinite. For $1\leq i \leq m$, let $f_i:Y\rightarrow R_{\pm\infty}$ be the definable function taking each element in $x\in Y$ to the $i$-th smallest element in $E_x$. Since the family $\{E_x : x\in Y\}$ is definable, these maps are definable. Moreover, by Hausdorffness (see Lemma [Lemma 30](#lemma:basic_facts_P_x_2){reference-type="ref" reference="lemma:basic_facts_P_x_2"}([\[itma:lemma_basic_facts_P\_x_2\]](#itma:lemma_basic_facts_P_x_2){reference-type="ref" reference="itma:lemma_basic_facts_P_x_2"})([\[itm2:lemma_basic_facts_P\_x_2\]](#itm2:lemma_basic_facts_P_x_2){reference-type="ref" reference="itm2:lemma_basic_facts_P_x_2"}), these functions cannot be constant on any interval. By o-minimality, let $\mathcal{Y}$ be a partition of $Y$ into finitely many intervals and points such that, for every interval $I\in \mathcal{Y}$, the functions $f_i$, $1\leq i\leq m$, are $e$-continuous and strictly monotone. Without loss of generality, we fix an interval $I\in \mathcal{Y}$ and show that, for any $x\in I$, $E_x \cap cl_e I\subseteq \{x\}$, completing the proof. Let $x\in I$ and $y\neq x$ be such that $y\in E_x \cap cl_e I$. If $y\in I$, then, by Lemma [Lemma 37](#lemma_f_Rx_Lx){reference-type="ref" reference="lemma_f_Rx_Lx"}, $E_y\subseteq E_x$. Since $|E_y|=|E_x|$, it follows that $E_y =E_x$, contradicting that the functions $f_i$ are injective. Suppose now that $y\in \partial_e I$. As $y \in E_x$, we have that $y=f_i(x)$ for some $1\leq i \leq m$. By $e$-continuity and strict monotonicity of $f_i$ on $I$ there exists a point $x'\in I$ such that $f_i(x')\in I$ and $f_i(x')\neq x'$. A contradiction then follows as before. ◻ By o-minimal cell decomposition, the above theorem can be immediately generalized to all one-dimensional spaces. **Corollary 52**. Let $(X,\tau)$, $\dim{X} \leq 1$, be a Hausdorff definable topological space. Then there exists a finite definable partition $\mathcal{X}$ of $X$ such that, for every $Y\in\mathcal{X}$, $(Y, \tau)$ is definably homeomorphic to a point or an interval with either the euclidean, discrete, right half-open interval or left half-open interval topology. Since the topologies $\tau_r$, $\tau_l$ and $\tau_s$ are all finer that $\tau_e$, it follows by o-minimality that any definable function $(X,\tau)\rightarrow (R,\tau_e)$, where $\dim X\leq 1$ and $\tau$ is a Hausdorff definable topology, is cell-wise continuous. We end this section with a statement noting that, for spaces in the line, having an interval subspace with any one of the euclidean, discrete or half-open interval topologies is a definable topological invariant. This is an easy consequence of the monotonicity theorem of o-minimality, and in particular the observation that the push-forward of an interval with the $\tau_r$ or $\tau_l$ topology by an $e$-continuous strictly monotone definable function is an interval with either the $\tau_r$ or $\tau_l$ topology. This holds in weakly o-minimal structures too, since these have a form of monotonicity (see [@arefiev97]). **Lemma 53**. If $(X,\tau)$ and $(Y,\mu)$, where $X,Y\subseteq R$, are definable topological spaces, and $f:(X,\tau)\rightarrow (Y,\mu)$ is a definable homeomorphism, then (i) [\[itm:tau-copies-1\]]{#itm:tau-copies-1 label="itm:tau-copies-1"} if $(X,\tau)$ contains an interval subspace with the discrete topology, then $(Y,\mu)$ contains an interval subspace with the discrete topology; (ii) if $(X,\tau)$ contains an interval subspace with the right half-open or left half-open interval topology, then $(Y,\mu)$ contains an interval subspace with the right half-open or left half-open interval topology; (iii) [\[itm:tau-copies-3\]]{#itm:tau-copies-3 label="itm:tau-copies-3"} if $(X,\tau)$ contains an interval subspace with the euclidean topology, then $(Y,\mu)$ contains an interval subspace with the euclidean topology. Note that ([\[itm:tau-copies-1\]](#itm:tau-copies-1){reference-type="ref" reference="itm:tau-copies-1"}) and ([\[itm:tau-copies-3\]](#itm:tau-copies-3){reference-type="ref" reference="itm:tau-copies-3"}) hold for spaces of all dimensions if we substitute "interval subspace\" with "definable subspace of dimension $n$\". Hence definable topological spaces in the line can be classified up to definable homeomorphism according to whether or not they contain interval subspaces with the euclidean, discrete or half-open interval topologies. Moreover, by Proposition [Proposition 43](#them_main){reference-type="ref" reference="them_main"}, every infinite $T_1$ space will fall into at least one of these categories. # Hausdorff regular ($T_3$) spaces. Decomposition in terms of the $\tau_{lex}$ and $\tau_{Alex}$ topologies. {#section: universal spaces} In this section, we study Hausdorff regular (i.e. $T_3$) spaces in the line. The main result is Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"}, which states that any such space can be partitioned into a finite set and two definable open subspaces, one of which definably embeds into a space with the lexicographic order topology, and the other into a space which we label the definable Alexandrov $n$-line. In the next section, we use this result and its proof, as well as Theorem [Theorem 51](#them 1.5){reference-type="ref" reference="them 1.5"}, to address universality questions in our setting. In Section [8](#section:compactifications){reference-type="ref" reference="section:compactifications"}, we extend Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"} to show that any $T_3$ definable topological space in the line has a definable compactification. We also combine Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"} and its proof with our affineness result (Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}) in Section [9](#section: affine){reference-type="ref" reference="section: affine"} to address questions of Fremlin and Gruenhage on perfectly normal, compact, Hausdorff spaces (see Subsection [9.1](#subsection: Fremlin){reference-type="ref" reference="subsection: Fremlin"}). We start by introducing the relevant topologies. Given $X\subseteq R^n$, we denote by $<_{lex}$ the lexicographic order on $X$ and by $(X,\tau_{lex})$ the topological space induced by $<_{lex}$ on $X$. Clearly this space is definable whenever $X$ is. **Definition 54** (Definable $n$-split interval, Appendix [12](#section:examples){reference-type="ref" reference="section:examples"}, Example [Example 118](#example:n-split){reference-type="ref" reference="example:n-split"}). We call the space $(R\times\{0,\ldots, n-1\}, \tau_{lex})$ the *definable $n$-split interval*. This space has the property that all the points in $R\times\{i\}$, for $0<i<n-1$, are isolated. In the case that $n>1$, for any $x\in R$ a basis of open neighbourhoods of $\langle x,0\rangle$ is given by sets of the form $$\langle x,0\rangle\cup ((y,x)\times\{0,\ldots, n-1\}) \text{ for } y<x,$$ and a basis of open neighbourhoods of $\langle x, n-1\rangle$ is given by sets of the form $$\langle x,n\rangle\cup ((x,y)\times\{0,\ldots, n-1\}) \text{ for } y>x.$$ If $n=1$, then $(R\times\{0\}, \tau_{lex})=(R\times \{0\}, \tau_e)$. **Definition 55** (Definable Alexandrov $n$-line, Appendix [12](#section:examples){reference-type="ref" reference="section:examples"}, Example [Example 119](#example_n_line){reference-type="ref" reference="example_n_line"}). Let $\tau_{Alex}$ be the topology on $R^2$ where all points in $R^2\setminus R\times\{0\}$ are isolated and, for any $x\in R$, a basis of open neighbourhoods of $\langle x, 0\rangle$ is given by sets of the form $$\{\langle x,0 \rangle\} \cup (((z,y)\setminus \{x\}) \times R), \text{ for } z<x<y.$$ Then, for any $n>0$, we call the space $(R\times\{0,\ldots, n-1\},\tau_{Alex})$ the *definable Alexandrov $n$-line*. Note, in particular, that $(R\times\{0\},\tau_{lex})=(R\times\{0\},\tau_{Alex})=(R\times\{0\},\tau_e)$. We may now state the main theorem of this section. **Theorem 56**. Let $(X,\tau)$, $X\subseteq R$, be a regular and Hausdorff definable topological space. Then there exist disjoint definable open sets $Y, Z \subseteq X$ with $X\setminus (Y\cup Z)$ finite, and $n_Y, n_Z>0$, such that the following holds. 1. There exists a definable embedding $h_Y:(Y,\tau)\hookrightarrow (R \times \{0,\ldots, n_Y\}, \tau_{lex})$. 2. There exists a definable embedding $h_Z:(Z,\tau)\hookrightarrow(R \times \{0,\ldots, n_Z\},\tau_{Alex})$. Lemmas [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"} and [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"} are the bulk of the proof of Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"}. They are also used in Section [8](#section:compactifications){reference-type="ref" reference="section:compactifications"} to prove that all regular Hausdorff definable topological spaces in the line can be definably Hausdorff compactified (Theorem [Theorem 89](#them_compactification){reference-type="ref" reference="them_compactification"}). In Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}, we construct a finite family $\mathcal{X}_{\text{open}}$ of pairwise disjoint definable open subsets of $X$ of a very special form such that $X\setminus \bigcup\mathcal{X}_{\text{open}}$ is finite. In Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}, we construct, for every $A\in\mathcal{X}_{\text{open}}$, a set $A^*$ of the form $I_A\times\{0,\ldots, n_A\}$, for some interval $I_A$ and natural number $n_A$, and a definable embedding $h_A:(A,\tau)\hookrightarrow (A^*,\tau_A)$, where $\tau_A$ is either $\tau_{lex}$ or $\tau_{Alex}$. The construction will be such that $I_{A}\cap I_{A'}=\emptyset$ for distinct $A, A'\in\mathcal{X}_{\text{open}}$. Then $Z$ will be the union of all the sets $A$ in $\mathcal{X}_{\text{open}}$ such that $(A^*,\tau_A)=(A^*,\tau_{Alex})$, and $h_Z$ the union of the respective embeddings $h_A$. The set $Y$ and embedding $h_Y$ are constructed similarly from the remaining sets in $\mathcal{X}_{\text{open}}$. Until the end of the proof of Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"}, we fix a definable topological space $(X,\tau)$ with $X\subseteq R$. We introduce an equivalence relation on $X$ induced by the topology $\tau$ that is defined as follows. Given $x,y\in X$, we write $x\sim_\tau y$ when one of the following holds: (i) $x=y$; (ii) there exists some $z\in X$ such that $\{x,y\}\cap E_z \neq \emptyset$ and, for all $z\in X, \, x\in E_z \Leftrightarrow y\in E_z$. This relation is clearly reflexive and symmetric, and one easily checks that it is transitive. Moreover, by Proposition [Proposition 20](#basic_facts_P_x){reference-type="ref" reference="basic_facts_P_x"}([\[itm: basic_facts_2\]](#itm: basic_facts_2){reference-type="ref" reference="itm: basic_facts_2"}), it is definable. For any $x\in X$, we denote by $[x]$ the equivalence class $\{y\in X: y\sim_\tau x\}$. We prove some preliminary facts regarding this relation. **Lemma 57**. If $(X,\tau)$ is $T_1$, then every equivalence class of $\sim_\tau$ is finite. *Proof.* Let $x\in X$. If $x\in X\setminus \bigcup_{y\in X} E_y$, then $[x]=\{x\}$. If there is some $y\in X$ such that $x\in E_y$, then, from the definition of $\sim_\tau$, it follows that $[x]\subseteq E_y$. If $(X,\tau)$ is $T_1$, then, by Lemma [Lemma 23](#lemma_2){reference-type="ref" reference="lemma_2"}, the set $E_y$ is finite, so $[x]$ is finite. ◻ **Lemma 58**. If $(X,\tau)$ is regular and Hausdorff, then there exists a cofinite set $X'\subseteq X$ with the following properties. (a) [\[itm:properties_cofnite_subset_1.5_a\]]{#itm:properties_cofnite_subset_1.5_a label="itm:properties_cofnite_subset_1.5_a"} For any $x\in X$, either $[x] \subseteq X'$ or $[x]\cap X'= \emptyset$ (i.e. $X'$ is compatible with $\sim_\tau$). (b) [\[itm:properties_cofnite_subset_1.5_b\]]{#itm:properties_cofnite_subset_1.5_b label="itm:properties_cofnite_subset_1.5_b"}For any $x\in X'$ and $y\in X$, if $x\in E_y$, then $y\in [x]$. (c) [\[itm:properties_cofnite_subset_1.5_c\]]{#itm:properties_cofnite_subset_1.5_c label="itm:properties_cofnite_subset_1.5_c"} For any $x\in X'$, $E_x \subseteq [x]$. In particular, if $E_x$ is non-empty, then $E_x=[x]$. *Proof.* Let $H=\{(x,y)\in X^2 : y\in E_x \text{ and } x\not\sim_\tau y\}$. Let $H_1$ and $H_2$ be the projections of $H$ onto the first and second coordinate respectively. We start by showing that these sets are finite. If $H_2$ is finite then, by Hausdorffness (see Proposition [Lemma 30](#lemma:basic_facts_P_x_2){reference-type="ref" reference="lemma:basic_facts_P_x_2"}([\[itm2:basic_facts_P\_x_2\]](#itm2:basic_facts_P_x_2){reference-type="ref" reference="itm2:basic_facts_P_x_2"})), there exist at most two $x \in X$ such that $y \in E_x$, hence $H_1$ is finite. Towards a contradiction, we suppose that $H_2$ is infinite. Let $g:H_2 \rightarrow H_1$ be the function given by $y \mapsto \min\{x : y\in E_x \text{ and } y\not\sim_\tau x\}$. By Hausdorffness, this function is well defined and, by Lemma [Lemma 23](#lemma_2){reference-type="ref" reference="lemma_2"}, it cannot be constant on an interval, so, by o-minimality, there exists an interval $I\subseteq H_2$ such that $g|_I$ is strictly monotonic and $e$-continuous. But then, since $(X,\tau)$ is regular, by Lemma [Lemma 37](#lemma_f_Rx_Lx){reference-type="ref" reference="lemma_f_Rx_Lx"}, for any $y\in I$, it holds that $y \sim_\tau g(y)$, a contradiction. Set $X':=X\setminus (\bigcup_{x\in H_1 \cup H_2} [x])$. By Lemma [Lemma 57](#lemma_finite_classes){reference-type="ref" reference="lemma_finite_classes"} and the finiteness of $H_1\cup H_2$, this set is cofinite in $X$. By definition of $H$, it follows that $X'$ satisfies [\[itm:properties_cofnite_subset_1.5_a\]](#itm:properties_cofnite_subset_1.5_a){reference-type="eqref" reference="itm:properties_cofnite_subset_1.5_a"}-[\[itm:properties_cofnite_subset_1.5_c\]](#itm:properties_cofnite_subset_1.5_c){reference-type="eqref" reference="itm:properties_cofnite_subset_1.5_c"}. ◻ The next lemma strengthens Lemma [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"} and is the core construction in the proofs of Theorems [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"} and [Theorem 89](#them_compactification){reference-type="ref" reference="them_compactification"}. **Lemma 59**. Suppose that $(X,\tau)$ is regular and Hausdorff. Let $X'\subseteq X$ be as given by Lemma [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}. There exists a finite partition $\mathcal{X}$ of $X$ into singletons $\mathcal{X}_{\text{sgl}}$ and infinite definable $\tau$-open sets $\mathcal{X}_{\text{open}}$, the latter being subsets of $X'$, with the following properties. For every $A\in\mathcal{X}_{\text{open}}$, there exists $n>0$, an interval $I\subseteq A$, and definable $e$-continuous strictly monotonic functions $f_0, f_1,\ldots, f_{n-1}:I\rightarrow A$ such that, for every $x\in I$, $[x]=\{ f_i(x) : 0\leq i < n\}$. In particular, $f_0$ is the identity map. Further, the intervals $f_i(I)$, for $0\leq i < n$, are pairwise disjoint and $A=\bigcup_{0\leq i <n} f_i(I)=\bigcup_{x\in I} [x]$. Additionally, for every $x\in I$ and $0<i<n-1$, the point $f_i(x)$ is $\tau$-isolated. Moreover, if we set $[x]^E=\{ y\in [x] : E_y\neq \emptyset\}$ for every $x\in I$, then exactly one of the following conditions holds: $$\begin{split} \forall x\in I, \quad & [x]=\{x\} \text{ and } E_x=\emptyset \text{, so }[x]^E=\emptyset \text{ (and $A=I$ contains only} \\ & \text{$\tau$-isolated points);} \\ \forall x\in I, \quad & [x]^E=\{x\}; \\ \forall x\in I, \quad & [x]^E=\{x, f_{n-1}(x)\} \text{ (this case only applies if $n>1$)}. \end{split}$$ Finally, in each of the latter two cases, exactly one of the following conditions is also satisfied: $$\begin{split} \forall x\in I,\quad &x\in L_x \setminus R_x; \\ \forall x\in I,\quad &x\in R_x \setminus L_x;\\ \forall x\in I,\quad &x\in R_x \cap L_x. \end{split}$$ *Proof.* We construct $\mathcal{X}$ by describing the family $\mathcal{X}_{\text{open}}$ of $\tau$-open subsets of $X'$, while making sure that $\bigcup\mathcal{X}_{\text{open}}$ is cofinite in $X$. Since the set $X'$ is cofinite in $X$ (see Lemma [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}) it suffices to ensure that $\bigcup\mathcal{X}_{\text{open}}$ is cofinite in $X'$. In particular, we consider a finite number of definable sets that partition $X'$ and, for each such set $S$, describe a partition of a cofinite subset of $S$, which becomes the collection of subsets of $S$ in $\mathcal{X}_{\text{open}}$. Let $A_{\text{isol}}=X'\setminus \bigcup_{y\in X} E_y$. Note that $[x]=\{x\}$ for every $x\in A_{\text{isol}}$. By Lemma [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}(c) and definition of $A_{\text{isol}}$, for all $x\in A_{\text{isol}}$, $E_x =\emptyset$, and so, by Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"}, these points are $\tau$-isolated. If $A_{\text{isol}}$ is infinite, let $\mathcal{X}_{\text{isol}}$ be a finite family of disjoint intervals whose union is cofinite in $A_{\text{isol}}$. Otherwise let $\mathcal{X}_{\text{isol}}=\emptyset$. We now consider $X' \setminus A_{\text{isol}}$. By Lemma [Lemma 57](#lemma_finite_classes){reference-type="ref" reference="lemma_finite_classes"} and uniform finiteness, there exists $n'\geq 1$ such that, for every $x\in X$, $|[x]|\leq n'$. For every $1\leq n \leq n'$, set $X_n:=\{ x\in X'\setminus A_{\text{isol}}: |[x]|=n \}$. These sets are definable and partition $X'\setminus A_{\text{isol}}$. We fix $1\leq n\leq n'$. If $X_n$ is finite, let $\mathcal{X}_n=\emptyset$. If $X_n$ is infinite, then the following describes a finite partition $\mathcal{X}_n$ of a cofinite subset of $X_n$ into definable $\tau$-open sets as desired. Recall the notation $[x]^E=\{y\in [x]: E_y \neq \emptyset\}$ for $x\in X$. Since $X'\setminus A_{\text{isol}}\subseteq \bigcup_{y\in X} E_y$ then, by Lemma [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}([\[itm:properties_cofnite_subset_1.5_b\]](#itm:properties_cofnite_subset_1.5_b){reference-type="ref" reference="itm:properties_cofnite_subset_1.5_b"}), for every $x\in X' \setminus A_{\text{isol}}$, and hence for every $x\in X_n$, it holds that $|[x]^E|\geq 1$. Moreover, by Lemma [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}([\[itm:properties_cofnite_subset_1.5_c\]](#itm:properties_cofnite_subset_1.5_c){reference-type="ref" reference="itm:properties_cofnite_subset_1.5_c"}) and Hausdorffness (see Lemma [Lemma 30](#lemma:basic_facts_P_x_2){reference-type="ref" reference="lemma:basic_facts_P_x_2"}([\[itmb:lemma_basic_facts_P\_x_2\]](#itmb:lemma_basic_facts_P_x_2){reference-type="ref" reference="itmb:lemma_basic_facts_P_x_2"})), for every $x\in X' \setminus A_{\text{isol}}$, and hence for every $x\in X_n$, we have that $|[x]^E|\leq 2$. Let $X_n^{(1)}=\{ x\in X_n : |[x]^E|=1 \}$ and $X_n^{(2)}=\{ x\in X_n : |[x]^E|=2 \}$. These sets partition $X_n$. Set $Dom(X_n^{(1)}):=\bigcup\{[x]^E : x\in X_n^{(1)}\}$, $Dom(X_n^{(2)}):=\{\min[x]^E : x \in X_n^{(2)}\}$ and $Dom(X_n):=Dom(X_n^{(1)})\cup Dom(X_n^{(2)})$. Note that, for every $x\in X_n$, it holds that $|Dom(X_n)\cap [x]|=1$. First, let $f_0$ denote the identity map on $Dom(X_n)$. Then, for every $1\leq i < n$, let $f_i:Dom(X_n)\rightarrow X$ be the function defined as follows. For every $x\in Dom(X_n^{(1)})$, $f_i(x)$ is the $i$-th smallest element in $[x]\setminus\{x\}$. For every $x\in Dom(X_n^{(2)})$, if $1\leq i < n-1$, then $f_i(x)$ is the $i$-th smallest element in $[x]\setminus [x]^E$, while $f_{n-1}(x)=\max [x]^E$. By construction, for every $y\in X_n$ there exists a unique $x\in Dom(X_n)$ and unique $0\leq i < n$ such that $f_i(x)=y$. In particular, for every $x \in Dom(X_n)$, $[x]=\{f_i(x) : 0\leq i < n\}$, all functions $f_i$ are injective and the family of images $\{f_i(Dom(X_n)) : 0\leq i < n\}$ is pairwise disjoint and covers $X_n$. Moreover, by construction, for every $x\in Dom(X_n)$ and $0<i<n-1$, $E_{f_i(x)}=\emptyset$, so, by Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"}, $f_i(x)$ is $\tau$-isolated. Since, by definition of $Dom(X_n)$, $E_x \neq \emptyset$ for all $x\in Dom(X_n)$, by Lemma [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}([\[itm:properties_cofnite_subset_1.5_c\]](#itm:properties_cofnite_subset_1.5_c){reference-type="ref" reference="itm:properties_cofnite_subset_1.5_c"}) and definition of $\sim_\tau$ it follows that $x \in E_x$ for all $x\in Dom(X_n)$. So, by Lemma [Proposition 32](#prop_basic_facts_P_x_2){reference-type="ref" reference="prop_basic_facts_P_x_2"}([\[itm2:basic_facts_P\_x_2\]](#itm2:basic_facts_P_x_2){reference-type="ref" reference="itm2:basic_facts_P_x_2"}), we may further partition $Dom(X_n)$ into three definable sets as follows. $$\begin{split} Dom(X_n)'=\{ x\in Dom(X_n) : x\in L_x \setminus R_x\}, \\ Dom(X_n)''=\{ x\in Dom(X_n) : x\in R_x \setminus L_x\},\\ Dom(X_n)'''=\{ x\in Dom(X_n) : x\in R_x \cap L_x\}. \end{split}$$ By o-minimality, there exists a finite partition $\mathcal{D}_n$ of $Dom(X_n)$, compatible with $\{Dom(X_n^{(1)}),Dom(X_n^{(2)}), Dom(X_n)', Dom(X_n)'', Dom(X_n)'''\}$, which contains only singletons and intervals and is such that, for every interval $I\in\mathcal{D}_n$, $f_i|_I$ is $e$-continuous and strictly monotonic, for every $0\leq i < n$. The family of those sets in $\mathcal{X}_{\text{open}}$ which are subsets of $X_n$ is then defined by $$\mathcal{X}_n=\left\{\bigcup_{0\leq i < n} f_i(I) : I\in \mathcal{D}_n, I \text{ is an interval} \right\}.$$ Note that, by construction, $\bigcup_{0\leq i < n} f_i(I)= \bigcup_{x\in I} [x]$, for any interval $I\in \mathcal{D}_n$. Moreover, since the functions $f_i$ are $e$-continuous, the sets $f_i(I)$ are all intervals. It easily follows that any $A \in \mathcal{X}_n$ is $\tau$-open, by observing that, for every $x \in A$, we have $E_x \subseteq [x]$ by Lemma [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}([\[itm:properties_cofnite_subset_1.5_c\]](#itm:properties_cofnite_subset_1.5_c){reference-type="ref" reference="itm:properties_cofnite_subset_1.5_c"}), and using the form of a basis of $\tau$-neighbourhoods for $x$ given by Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"}. Finally, set $\mathcal{X}_{\text{open}}:=\mathcal{X}_{\text{isol}}\cup \mathcal{X}_1 \cup \cdots \cup \mathcal{X}_{n'}$, and let $\mathcal{X}_{\text{sgl}}$ denote the collection of singletons given by the points in $X\setminus \bigcup\mathcal{X}_{\text{open}}$. By construction, this partition satisfies the properties stated in the lemma. To check this, for any $A\in \mathcal{X}_{\text{open}}$, if $A\subseteq X_n$ for some $n$, let $I$ and $f_i$, for $0\leq i <n$, be as described above. If $A\subseteq A_{\text{isol}}$, then simply consider $I=A$ with $f_0$ denoting the identity map on $I$. ◻ Continuing with the construction in Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}, the next lemma describes how each of the sets $A\in \mathcal{X}_{\text{open}}$ definably embeds into a space with either the lexicographic or Alexandrov $n$-line topology. We first require a definition extending the notion of $e$-convergence from the right or from the left (Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}), and a remark on how this convergence relates to convergence with respect to the topologies $\tau_{lex}$ and $\tau_{Alex}$. **Definition 60**. Given a definable set $\tilde{X} \subseteq R \times \{0,1,\ldots\}$ we say that a definable curve in $\tilde{X}$ *$e$-converges to $\langle x,i \rangle\in \tilde{X}$ from the right* (respectively *left*) if it $e$-converges to $\langle x,i \rangle$ and its projection to the first coordinate, namely $\pi \circ \gamma$, $e$-converges to $x$ from the right (respectively left). **Remark 61**. Consider a definable set $\tilde{X}=I\times\{0,\ldots, n\}$, with $I\subseteq R$ an interval, and an injective definable curve $\gamma$ in $\tilde{X}$. For $x\in I$, note that, by o-minimality, if $n>0$, then $\gamma$ converges in $(\tilde{X},\tau_{lex})$ to $\langle x,0 \rangle$ if and only if it $e$-converges to $\langle x,i \rangle$ from the left, for some $0\leq i \leq n$ (recall the basis of open neighbourhoods for $\langle x, 0 \rangle$ described in Definition [Definition 54](#dfn:lex){reference-type="ref" reference="dfn:lex"}). Similarly, maintaining the assumption that $n>0$, we have that $\gamma$ converges in $(\tilde{X},\tau_{lex})$ to $\langle x,n \rangle$ if and only if it $e$-converges to $\langle x,i \rangle$ from the right, for some $0\leq i \leq n$ (likewise consider the basis of open neighbourhoods for $\langle x, n \rangle$ given earlier). Moreover (recalling Definition [Definition 55](#example_n_line_0){reference-type="ref" reference="example_n_line_0"}), by o-minimality, $\gamma$ converges to $\langle x,0 \rangle$ in $(\tilde{X},\tau_{Alex})$ if and only if it $e$-converges to $\langle x,i \rangle$, for some $0\leq i \leq n$ (from the right or from the left). **Lemma 62**. Suppose that $(X,\tau)$ is regular and Hausdorff. Let $\mathcal{X}$ be a finite partition of $X$ as given by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}. For each $A=\bigcup_{0\leq i <n} f_i(I)\in \mathcal{X}_{\text{open}}$, there exists a definable set $A^*\subseteq R^2$ and a definable injection $h_A:A\rightarrow A^*$ such that the following hold. (1) [\[itm1:lemma_proof_two_theorems\]]{#itm1:lemma_proof_two_theorems label="itm1:lemma_proof_two_theorems"} $A^*=I\times \{0,\ldots, m\}$, for some $m\in\{n-1, n, 2\}$. In particular, for every distinct pair $A_0, A_1$ in $\mathcal{X}_{\text{open}}$ we have that $A_0^* \cap A_1^*=\emptyset$. (2) [\[itm1.5:lemma_proof_two_theorems\]]{#itm1.5:lemma_proof_two_theorems label="itm1.5:lemma_proof_two_theorems"} For every $0\leq i <n$ and $x\in f_i(I)$, it holds that $(\pi\circ h_A)(x)= f_i^{-1}(x)$. (3) [\[itm2:lemma_proof_two_theorems\]]{#itm2:lemma_proof_two_theorems label="itm2:lemma_proof_two_theorems"} The map $h_A:(A,\tau)\hookrightarrow (A^*,\tau_A)$ is an embedding, where either 1. $(A^*,\tau_A)=(A^*,\tau_{lex})$ or 2. $(A^*,\tau_A)=(A^*,\tau_{Alex})$. *Proof.* Following Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}, we distinguish different cases of possible sets $A=\bigcup_{0\leq i <n} f_i(I)$ in $\mathcal{X}_{\text{open}}$, based on properties of $I$. In each case, we define $A^*$ and $h_A$ (which for simplicity we denote by $h$) so that ([\[itm1:lemma_proof_two_theorems\]](#itm1:lemma_proof_two_theorems){reference-type="ref" reference="itm1:lemma_proof_two_theorems"}), ([\[itm1.5:lemma_proof_two_theorems\]](#itm1.5:lemma_proof_two_theorems){reference-type="ref" reference="itm1.5:lemma_proof_two_theorems"}) and ([\[itm2:lemma_proof_two_theorems\]](#itm2:lemma_proof_two_theorems){reference-type="ref" reference="itm2:lemma_proof_two_theorems"}) hold. **Case 0:** $I=A$ is a set of isolated points in $(X,\tau)$. Specifically, $[x]=\{x\}$ and $E_x=\emptyset$, for every $x\in I$. Let $A^*=A\times \{0,1,2\}$ and let $h:A\rightarrow A^*$ be given by $x\mapsto \langle x,1 \rangle$. Let $\tau_A$ be the topology induced by the lexicographic order on $A^*$. With this topology all the points in $A\times \{1\}$ are isolated, so $h$ is an embedding. For the remaining cases we will make use of Proposition [Proposition 27](#prop_cont_lim){reference-type="ref" reference="prop_cont_lim"} to prove ([\[itm2:lemma_proof_two_theorems\]](#itm2:lemma_proof_two_theorems){reference-type="ref" reference="itm2:lemma_proof_two_theorems"}), that is, after specifying what $A^*$, $\tau_A$ and $h$ will be in each case in order to satisfy the other requirements, we will fix a point $x\in X$ and prove that an injective definable curve $\gamma$ converges in $(A,\tau)$ to $x$ if and only if $h\circ \gamma$ converges in $(A^*,\tau_A)$ to $h(x)$. To do this we will use Remarks [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"} and [Remark 61](#remark:lex_side_convergence){reference-type="ref" reference="remark:lex_side_convergence"} extensively. **Case 1:** $[x]^E=\{x\}$ and $x\in L_x \setminus R_x$, for every $x\in I$. In this case, for every $x \in I$, we have the following. Since $[x]^E=\{x\}$, by Lemma [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}[\[itm:properties_cofnite_subset_1.5_c\]](#itm:properties_cofnite_subset_1.5_c){reference-type="eqref" reference="itm:properties_cofnite_subset_1.5_c"} it holds that $E_x = [x]$, hence $E_x = \{f_i(x) : 0\leq i <n\}$. Furthermore, any point $y\in [x]\setminus \{x\}$ satisfies that $E_y=\emptyset$ and so, by Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"}, it is $\tau$-isolated. Moreover, we will make use of the following observation, which follows from applying Lemma [Lemma 37](#lemma_f_Rx_Lx){reference-type="ref" reference="lemma_f_Rx_Lx"} and the fact that $x\in L_x \setminus R_x$, for all $x \in I$. For every $0\leq i <n$ and $x \in I$, $$\label{eqn_cases_1} \begin{split} f_i(x)\in L_x \Leftrightarrow f_i \text{ is increasing}; \\ f_i(x) \in R_x \Leftrightarrow f_i \text{ is decreasing}. \end{split}$$ Let $A^*= I \times \{0,\ldots, n\}$ and let $h:A\rightarrow A^*$ be the definable injection given by $h(f_i(x))=\langle x,i \rangle$, for every $x\in I$ and $0\leq i <n$. Note that, for each $0\leq i <n$, $h|_{f_i(I)}$ is given by $x\mapsto \langle f_i^{-1}(x),i \rangle$, and so $h$ is an $e$-embedding. We show (using Proposition [Proposition 27](#prop_cont_lim){reference-type="ref" reference="prop_cont_lim"}) that $h$ is an embedding $(A,\tau)\hookrightarrow (A^*,\tau_{lex})$. Fix $x\in A$. If $x\in f_i(I)$ with $i>0$, then both $x$ and $h(x)$ are isolated, and hence not the limit of any injective definable curve, so we may assume that $x\in I$. Let $\gamma$ be an injective definable curve in $A$ that $\tau$-converges to $x$. By o-minimality, $\gamma$ is $e$-convergent. By the fact that $E_x=\{f_i(x) : 0\leq i <n\}$ and Proposition [Proposition 29](#basic_facts_P_x_1){reference-type="ref" reference="basic_facts_P_x_1"}[\[itm: basic_facts_3\]](#itm: basic_facts_3){reference-type="eqref" reference="itm: basic_facts_3"}, we have that $\gamma$ necessarily $e$-converges to some $f_i(x)$, from either the right or the left (see Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}). Since $h$ is an $e$-homeomorphism, $h\circ\gamma$ $e$-converges to $\langle x,i \rangle$. If $\gamma$ $e$-converges from the left, then, by Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}, it must be that $f_i(x) \in L_x$. In this case it follows from [\[eqn_cases_1\]](#eqn_cases_1){reference-type="eqref" reference="eqn_cases_1"} that $f_i$ is increasing, and consequently $h \circ \gamma$ also $e$-converges to $\langle x,i \rangle$ from the left, and therefore (by Remark [Remark 61](#remark:lex_side_convergence){reference-type="ref" reference="remark:lex_side_convergence"}), $h \circ \gamma$ converges in $(A^*,\tau_{lex})$ to $\langle x,0 \rangle=h(x)$. Similarly, if $\gamma$ $e$-converges from the right, then it must be that $f_i(x)\in R_x$, and so, by [\[eqn_cases_1\]](#eqn_cases_1){reference-type="eqref" reference="eqn_cases_1"}, $f_i$ is decreasing, and thus $h \circ \gamma$ again $e$-converges to $\langle x,i \rangle$ from the left, so again it converges in $(A^*,\tau_{lex})$ to $\langle x,0 \rangle=h(x)$. Conversely, let $\gamma'\subseteq h(A)$ be an injective definable curve converging in $(A^*,\tau_{lex})$ to $\langle x,0 \rangle$, in which case it must $e$-converge from the left to some $\langle x,i \rangle$ (see Remark [Remark 61](#remark:lex_side_convergence){reference-type="ref" reference="remark:lex_side_convergence"}). We may assume that $\gamma'\subseteq I\times\{i\}$, for some $0 \leq i < n$ (see Remark [Remark 10](#remark_assumptions_curves){reference-type="ref" reference="remark_assumptions_curves"}). Note that $h^{-1}\circ \gamma' = f_i \circ \pi \circ \gamma'$. If $f_i$ is increasing, then, by [\[eqn_cases_1\]](#eqn_cases_1){reference-type="eqref" reference="eqn_cases_1"}, $f_i(x)\in L_x$, and moreover $h^{-1}\circ \gamma'$ $e$-converges to $f_i(x)$ from the left, so it $\tau$-converges to $x$ (see Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}). Similarly, if $f_i$ is decreasing, then, by [\[eqn_cases_1\]](#eqn_cases_1){reference-type="eqref" reference="eqn_cases_1"}, $f_i(x)\in R_x$, and moreover $h^{-1}\circ \gamma$ $e$-converges to $f_i(x)$ from the right, so again it $\tau$-converges to $x$. **Case 2:** $[x]^E=\{x\}$ and $x\in R_x \setminus L_x$ for every $x\in I$. This case is very similar to Case 1, so we only indicate here the key details and requisite changes. As in Case 1, it holds that $E_x=[x]=\{f_i(x) : 0\leq i <n\}$ for every $x\in I$, and every point $y\in A\setminus I$ is $\tau$-isolated. On the other hand, in this case we have the following two equivalences arising from Lemma [Lemma 37](#lemma_f_Rx_Lx){reference-type="ref" reference="lemma_f_Rx_Lx"}. For every $0\leq i <n$ and $x\in I$, $$\label{eqn_cases_2} \begin{split} f_i(x)\in R_{x} \Leftrightarrow f_i \text{ is increasing}, \\ f_i(x) \in L_{x} \Leftrightarrow f_i \text{ is decreasing}. \end{split}$$ We let $A^*=I\times \{0,\ldots, n\}$ and let $h:A\rightarrow A^*$ be the definable injection given by $h(f_i(x))=\langle x,n-i \rangle,$ for $x\in I$ and $0\leq i <n$. Again, this is clearly an $e$-embedding. It can be shown that $h:(A,\tau)\hookrightarrow (A^*,\tau_{lex})$ is an embedding by analogy to Case 1. That is, we fix $x\in I$ and a definable curve $\gamma$ in $A$ that $\tau$-converges to $x$. Then we observe that $\gamma$ $e$-converges to some $f_i(x)$ and $h\circ \gamma$ $e$-converges to $h(f_i(x))=\langle x, n-i \rangle$. Finally, we consider separately the cases where $\gamma$ $e$-converges to $f_i(x)$ from the left and from the right and show, using [\[eqn_cases_2\]](#eqn_cases_2){reference-type="eqref" reference="eqn_cases_2"} together with Remarks [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"} and [Remark 61](#remark:lex_side_convergence){reference-type="ref" reference="remark:lex_side_convergence"}, that in both cases $h\circ \gamma$ converges in $(A^*,\tau_{lex})$ to $h(x)$. The converse, i.e. the case of a definable curve $\gamma'$ in $h(A)$, may be similarly argued by analogy to Case 1. **Case 3:** $n>1$, $[x]^E=\{x, f_{n-1}(x)\}$ and $x\in L_x \setminus R_x$ for every $x\in I$. In this case we have that, by Lemma [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}[\[itm:properties_cofnite_subset_1.5_c\]](#itm:properties_cofnite_subset_1.5_c){reference-type="eqref" reference="itm:properties_cofnite_subset_1.5_c"}, for every $x \in I$ it holds that $E_x = E_{f_{n-1}(x)}=[x]=\{f_i(x) : 0\leq i <n\}$. Moreover, every point $y\in A\setminus (I \cup f_{n-1}(I))$ satisfies that $E_y=\emptyset$ and so (by Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"}) is $\tau$-isolated. Furthermore, by Hausdorffness of $\tau$ (see Proposition [Proposition 32](#prop_basic_facts_P_x_2){reference-type="ref" reference="prop_basic_facts_P_x_2"}[\[itm3:basic_facts_P\_x_2\]](#itm3:basic_facts_P_x_2){reference-type="eqref" reference="itm3:basic_facts_P_x_2"}), it holds that, for every $0\leq i <n$ and $x\in I$, exactly one of the following two possibilities is satisfied: $$\label{eqn_cases_3} \begin{split} f_i(x)\in R_x \cap L_{f_{n-1}(x)};\\ f_i(x)\in R_{f_{n-1}(x)} \cap L_x. \end{split}$$ Let $A^*=I\times\{0,\ldots, n-1\}$, and let $h:A\rightarrow A^*$ be defined in a similar manner to Case $1$, namely by $h(f_i(x))=\langle x,i \rangle$, for every $x\in I$ and $0\leq i <n$. In this case, $h:A\rightarrow A^*$ is a bijection. We show that $h$ is a homeomorphism $(A,\tau)\rightarrow (A^*,\tau_{lex})$, by showing that an injective definable curve $\gamma$ in $A$ $\tau$-converges to a point $y\in A$ if and only if $h \circ \gamma$ $\tau_{lex}$-converges to $h(y)$. We fix $y\in A$. The case $y\in f_i(I)$, for $0<i<n-1$, is as usual trivial, since in this case both $y$ and $h(y)$ are isolated in their respective spaces. If $y\in I$, then the required statement in that case follows from the corresponding argument in Case 1. Therefore, suppose that $y\in f_{n-1}(I)$. Let $x \in I$ be such that $y=f_{n-1}(x)$. Let $\gamma$ be an injective definable curve in $A$ $\tau$-converging to $y$. Using that $E_y=\{f_i(x) : 0\leq i <n\}$ and following the arguments in Case 1, we observe that $\gamma$ $e$-converges to some $f_i(x)$. Since $h$ is an $e$-homeomorphism, $h\circ\gamma$ $e$-converges to $\langle x,i \rangle$, a priori from either the right or from the left (see Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}). However, if $h \circ \gamma$ $e$-converges to $\langle x, i \rangle$ from the left -- converging thus in $(A^*,\tau_{lex})$ to $\langle x,0 \rangle$ (by Remark [Remark 61](#remark:lex_side_convergence){reference-type="ref" reference="remark:lex_side_convergence"}) -- then, by continuity of $h^{-1}$ at $h(x)=\langle x,0 \rangle$ (i.e. using that the statement is already established for all points in $I$), it must be that $h^{-1}\circ h\circ \gamma = \gamma$ $\tau$-converges to $x$, a contradiction (by Hausdorffness of $\tau$). So $h\circ \gamma$ must $e$-converge to $\langle x, i \rangle$ from the right, meaning that it converges in $(A^*,\tau_{lex})$ to $\langle x,n-1 \rangle=h(y)$ (see Remark [Remark 61](#remark:lex_side_convergence){reference-type="ref" reference="remark:lex_side_convergence"}). Conversely, let $\gamma'$ be an injective definable curve converging in $(A^*,\tau_{lex})$ to $h(y)=\langle x,n-1 \rangle$. Then it $e$-converges to some $\langle x,i \rangle$, meaning that $h^{-1}\circ \gamma'$ $e$-converges to $f_i(x)$. If $h^{-1}\circ\gamma'$ does not $\tau$-converge to $y$ then, by [\[eqn_cases_3\]](#eqn_cases_3){reference-type="eqref" reference="eqn_cases_3"} and Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}, it $\tau$-converges to $x$, but then, by continuity of $h$ at $x$, it follows that $\gamma'$ converges in $(A^*,\tau_{lex})$ to $h(x)=\langle x,0 \rangle$, a contradiction. So $h^{-1}\circ \gamma'$ $\tau$-converges to $y$. **Case 4:** $n>1$, $[x]^E=\{x, f_{n-1}(x)\}$ and $x\in R_x \setminus L_x$ for every $x\in I$. Again let $A^*=I\times\{0,\ldots, n-1\}$, and now let $h:A\rightarrow A^*$ be given in a similar manner to Case 2, namely by $h(f_i(x))=\langle x,n-1-i \rangle$. Note that $h$ is again a bijection. Moreover we may again in this case show that $h:(A,\tau)\rightarrow (A^*,\tau_{lex})$ is a homeomorphism. The proof of ([\[itm2:lemma_proof_two_theorems\]](#itm2:lemma_proof_two_theorems){reference-type="ref" reference="itm2:lemma_proof_two_theorems"}) here follows from the proofs of the other cases. The argument in Case $2$ shows that, for any $x\in A\setminus f_{n-1}(I)$, both $h$ and $h^{-1}$ are continuous at $x$ and $h(x)$ respectively. Then, for the points in $f_{n-1}(I)$, one may use an argument analogous to the one in Case 3. **Case 5:** $x\in R_x \cap L_x$ for every $x\in I$. Set $A^*= I \times \{0,\ldots,n-1\}$ and let $h:A\rightarrow A^*$ be given by $h(f_i(x))=\langle x,i \rangle$. This map is clearly bijective. We show that it is a homeomorphism $(A,\tau)\rightarrow (A^*,\tau_{Alex})$. If there exists $0 < i < n$ and $x \in I$ such that $E_{f_{i}(x)} \neq \emptyset$, we must have, by Lemma [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}([\[itm:properties_cofnite_subset_1.5_c\]](#itm:properties_cofnite_subset_1.5_c){reference-type="ref" reference="itm:properties_cofnite_subset_1.5_c"}), that $E_{f_{i}(x)} = [f_i(x)] = [x]$, and in particular $x \in R_{f_i(x)} \cup L_{f_i(x)}$, which contradicts Hausdorffness (Proposition [Proposition 32](#prop_basic_facts_P_x_2){reference-type="ref" reference="prop_basic_facts_P_x_2"}[\[itm3:basic_facts_P\_x_2\]](#itm3:basic_facts_P_x_2){reference-type="eqref" reference="itm3:basic_facts_P_x_2"}) and the fact that $x\in R_x \cap L_x$. Thus, for every $x\in I$ and $0<i<n$, we have $E_{f_{i}(x)} = \emptyset$, meaning that the point $f_i(x)$ is $\tau$-isolated. Moreover, by definition of the topology $\tau_{Alex}$, the points $\langle x,i \rangle$ for $x\in I$ and $0<i<n$ are isolated in $(A^*,\tau_{Alex})$. Applying Lemma [Lemma 37](#lemma_f_Rx_Lx){reference-type="ref" reference="lemma_f_Rx_Lx"}, we note that, for every $x\in I$, it follows from $x\in R_x \cap L_x$ that $f_i(x)\in R_x \cap L_x$, for every $0\leq i <n$. Furthermore, by Lemma [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}([\[itm:properties_cofnite_subset_1.5_c\]](#itm:properties_cofnite_subset_1.5_c){reference-type="ref" reference="itm:properties_cofnite_subset_1.5_c"}), we have that $R_x=L_x=\{ f_i(x) : 0\leq i < n\}$. Consequently, for any $x\in I$, any injective definable curve converges in $(A,\tau)$ to $x$ if and only if it $e$-converges to some $f_i(x)$ (see Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}). Similarly, any injective definable curve converges in $(A^*,\tau_{Alex})$ to $\langle x,0 \rangle$ if and only if it $e$-converges to some $\langle x,i \rangle$ (see Remark [Remark 61](#remark:lex_side_convergence){reference-type="ref" reference="remark:lex_side_convergence"}). Thus the result follows from the fact that $h$ is a $e$-homeomorphism, which is clear from the definition. This covers all possible cases for $A$, and thus completes the proof of the lemma. ◻ We may now prove Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"}. *Proof of Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"}.* Let $\mathcal{X}$ be a partition of $X$ as given by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"} and, for each $A\in \mathcal{X}_{\text{open}}$, let $A^*$, $h_A$ and $\tau_A$ be as given by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}. Set $h:=\bigcup\{h_A : A\in\mathcal{X}_{\text{open}}\}$. By Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}([\[itm1:lemma_proof_two_theorems\]](#itm1:lemma_proof_two_theorems){reference-type="ref" reference="itm1:lemma_proof_two_theorems"}), $h$ is an injection $\bigcup\mathcal{X}_{\text{open}}\rightarrow \bigcup_{A\in\mathcal{X}_{\text{open}}} A^*$. Let $$Y=\bigcup\{A\in\mathcal{X}_{\text{open}}: (A^*,\tau_A)=(A^*,\tau_{lex})\neq (A^*,\tau_{Alex})\}$$ and $$Z=\bigcup\{ A\in\mathcal{X}_{\text{open}}: (A^*,\tau_A)=(A^*,\tau_{Alex})\}.$$ By construction, these sets are disjoint, $\tau$-open and definable, and $X\setminus (Y \cup Z)$ is finite. Set $Y^*:=\bigcup\{ A^* : A\in \mathcal{X}_{\text{open}},\, A\subseteq Y\}$ and $Z^*:=\bigcup\{ A^* : A\in \mathcal{X}_{\text{open}},\, A\subseteq Z \}$. We claim that $h|_Y:(Y,\tau)\rightarrow (Y^*,\tau_{lex})$ and $h|_Z:(Z,\tau)\rightarrow (Z^*,\tau_{Alex})$ are embeddings. We show that this claim holds by noting that we may decompose the maps into embeddings between open subspaces of their domains and codomains. Recall that, by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}([\[itm1:lemma_proof_two_theorems\]](#itm1:lemma_proof_two_theorems){reference-type="ref" reference="itm1:lemma_proof_two_theorems"}), for each $A\in \mathcal{X}_{\text{open}}$, the set $A^*$ is of the form $I\times \{0,\ldots, m\}$, for some $m$ and interval $I\subseteq A$. It follows that, for each $A\in \mathcal{X}_{\text{open}}$, if $A\subseteq Z$, then $A^*$ is open in $(Z^*,\tau_{Alex})$. Similarly, if $A \subseteq Y$, then $A^*$ is open in $(Y^*,\tau_{lex})$, and the subspace topology on $A^*$ is precisely the topology induced by the lexicographic order on $A^*$. The claim then follows from Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}([\[itm2:lemma_proof_two_theorems\]](#itm2:lemma_proof_two_theorems){reference-type="ref" reference="itm2:lemma_proof_two_theorems"}). Finally, if $Y\neq \emptyset$, then let $n_Y=\max\{n : (R\times\{n\})\cap Y^* \neq \emptyset\}$. It is easy to see that the map on $Y^*$ given by $\langle x,m \rangle\mapsto \langle x,n_Y \rangle$, if $m=\max\{m' : \langle x, m'\rangle\in Y^*\}$, and the identity otherwise is a definable embedding $(Y^*,\tau_{lex})\hookrightarrow (R\times\{0,\dots,n_Y\},\tau_{lex})$. Hence we conclude that $(Y,\tau)$ embeds definably into $(R\times\{0,\dots,n_Y\},\tau_{lex})$. Furthermore, we may define $n_Z$ analagously to $n_Y$, and then $h|_Z$ is clearly a definable embedding $(Z,\tau) \hookrightarrow (R\times\{0,\dots,n_Z\},\tau_{Alex})$. ◻ **Remark 63**.  [\[remark_general_them_ADC_or DOTS\]]{#remark_general_them_ADC_or DOTS label="remark_general_them_ADC_or DOTS"} If $\mathcal{R}$ expands an ordered field, then, by Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"}, Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"} applies to all $T_3$ one-dimensional spaces. Otherwise, the theorem and its proof may be rewritten to apply to one-dimensional spaces as follows. Let $(X,\tau)$ be a $T_3$ one-dimensional definable topological space. In the context of Remark [Remark 40](#remark_generalizing_Rx_Lx){reference-type="ref" reference="remark_generalizing_Rx_Lx"}, one may prove analogues of Lemmas [Lemma 57](#lemma_finite_classes){reference-type="ref" reference="lemma_finite_classes"} and [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}, and reach a partition $\mathcal{X}=\mathcal{X}_{\text{open}}\cup \mathcal{X}_{\text{sgl}}$ of $X$ analogous to the one described in Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}. Then, analogously to the proof of Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}, it is possible to show that, for any $A\in \mathcal{X}_{\text{open}}$, there exists some $m \geq 0$, an interval $I$, and a definable embedding $h_A:(A,\tau)\hookrightarrow (A^*,\tau_A)$, where $A^*=I \times \{0, \ldots, m\}$, and $\tau_A\in\{\tau_{lex}, \tau_{Alex}\}$. We may then derive that $X$ has a cofinite subset that is definably homeomorphic to the disjoint union of finitely many spaces of the form $(R \times \{0, \ldots, m\},\tau_{lex})$ or $(R \times \{0, \ldots, m\},\tau_{Alex})$, for various $m$. In [@ram13], Ramakrishnan shows that, if $\mathcal{R}$ has definable choice and defines an order-reversing injection (e.g. if $\mathcal{R}$ expands an ordered group), then every definable linear order definably embeds into $(R^n,<_{lex})$, for some $n$. In particular, under these assumptions, for any definable order topological space one may assume that, up to definable homeomorphism, the topology is induced by the lexicographic order. Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"} adds to the understanding of $T_3$ definable spaces in the line by describing how the $\tau_{Alex}$ topology also plays a role describing them. We complete this picture with the next proposition. **Proposition 64**. For any interval $I$ and any $n>0$, the space $(I\times\{0,\ldots, n\},\tau_{Alex})$ does not definably embed into a definable order topological space. *Proof.* It suffices to prove the propostion for $n=1$. Towards a contradiction, assume that there exists an interval $I$ and one such embedding from $(I\times\{0,1\},\tau_{Alex})$ into a definable topological space $(X,\tau)$, where $\tau$ is given by a definable linear order $\preceq$. Let $Y$ denote the image of the aforementioned embedding. We begin by observing that any clopen definable subset of $(I\times\{0,1\},\tau_{Alex})$ is either finite or cofinite. We then complete the proof by contradiction by showing that $(Y,\tau)$ contains an infinite coinfinite definable clopen subset. Let $Z\subseteq I\times\{0,1\}$ denote a $\tau_{Alex}$-clopen definable subset. Note that, by o-minimality, the subspace $(I\times\{0\}, \tau_{Alex})=(I\times\{0\},\tau_e)$ is definably connected, and so it must be that either $Z \cap (I\times \{0\}) = \emptyset$ or $Z \cap (I\times \{0\}) = I\times \{0\}$. We show that in the first case $Z$ is finite and in the second it is cofinite. By passing from $Z$ to $(I\times\{0,1\})\setminus Z$ in the first case it suffices to prove the second case. By o-minimality, in order to prove that $Z$ is cofinite it suffices to show that it contains a point in every set of the form $J\times \{i\}$ for any interval $J\subseteq I$ and $i\in \{0,1\}$. However, this follows directly from the condition $Z \cap (I\times \{0\}) = I\times \{0\}$, and the fact that $Z$ is open in $(I\times\{0,1\}, \tau_{Alex})$. We now show that $(Y,\tau)$ contains an infinite coinfinite definable clopen subset. Let $Y_1=\{ x\in Y : (x,+\infty)_{\preceq}\cap Y \text{ is finite}\}$. Since the intervals $(x,+\infty)_{\preceq}$ are nested note that, by uniform finiteness, the set $Y_1$ is finite. Similarly the set $Y_2=\{x\in Y : (-\infty, x)_{\preceq}\cap Y \text{ is finite}\}$ is also finite. Since $(Y,\tau)$ is homeomorphic to $(I\times\{0,1\},\tau_{Alex})$ and all the points in $I\times\{1\}$ are $\tau_{Alex}$-isolated, it follows that $(Y,\tau)$ has infinitely many isolated points. Let us fix $x\in Y$, an isolated point in $(Y,\tau)$ that does not belong in $Y_1\cup Y_2$. There must exist $y \prec z$ in $X$ such that $$(y,z)_{\preceq} \cap Y=\{x\}.$$ It follows that the set $(x,+\infty)_{\preceq}\cap Y= [z,+\infty)_{\preceq} \cap Y$ is clopen in $(Y,\tau)$. Since $x\notin Y_1\cup Y_2$, then this set is also infinite and coinfinite in $Y$, the desired contradiction. ◻ Finally, the next corollary of Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"} implies that, for any $T_3$ definably separable definable topological space $(X,\tau)$, where $X\subseteq R$, there exists a cofinite subset $Y\subseteq X$ such that $\tau|_Y$ is induced by a definable linear order. **Corollary 65**. Let $(X,\tau)$, $X\subseteq R$, be a regular Hausdorff definable topological space. If $(X,\tau)$ is definably separable, then there exists a cofinite subset $Y\subseteq X$, a definable set $Y^*\subseteq R\times\{0,1\}$ and a definable embedding $(Y,\tau)\rightarrow (Y^*,\tau_{lex})$. *Proof.* Let $\mathcal{X}$ be a finite partition of $X$ as given by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"} and, for each $A \in \mathcal{X}_{\text{open}}$, let $A^*$ and $h_A$ be as given by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}. Our aim is to show that, under the additional assumption that $(X,\tau)$ is definably separable, the construction in these lemmas yields that, for every $A\in\mathcal{X}_{\text{open}}$, we have that $A^* \subseteq R\times \{0,1\}$ and $(A^*,\tau_A)=(A^*,\tau_{lex})$. We will then let $Y=\bigcup \mathcal{X}_{\text{open}}$ and $Y^*=\bigcup\{A^* : A\in\mathcal{X}_{\text{open}}\}$. Following the proof of Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"}, $Y$ is cofinite and $h=\bigcup \{ h_A : A\in\mathcal{X}_{\text{open}}\}$ is an embedding $(Y,\tau)\hookrightarrow (Y^*,\tau_{lex})$. If $(X,\tau)$ is definably separable, then it can have only finitely many isolated points. Following Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}, fix $A=\bigcup_{0\leq i <n} f_i(I) \in \mathcal{X}_{\text{open}}$. For every $x\in I$ and $0<i<n-1$, the point $f_i(x)$ is $\tau$-isolated. It follows that we must have $n\leq 2$. Similarly, for every $x\in I$, it must be that $[x]^E=\{x, f_{n-1}(x)\}$, since otherwise $E_{y}=\emptyset$ (i.e. $y$ is $\tau$-isolated), for every $y\in f_{n-1}(I)$. If $n=1$, this is considered in Cases 1, 2 and 5 in the proof of Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}. In Cases 1 and 2, $A^*=I\times \{0,1\}$ and $(A^*,\tau_A)=(A^*,\tau_{lex})$. In Case 5, we have that $A^*=I\times \{0\}$ and $(A^*,\tau_A)=(A^*,\tau_{Alex})$, and in this case $(A^*,\tau_{Alex})=(A^*,\tau_e)=(A^*,\tau_{lex})$. If $n=2$, this is considered in Cases 3 and 4 in the proof of Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}. In both of these cases, we have that $A^*=I\times \{0,1\}$ and $(A^*,\tau_A)=(A^*,\tau_{lex})$. ◻ # Some universality results {#section:universal_spaces_2} In this section, we consider universality questions in the o-minimal definable setting, building on results from the previous two sections. This will also lead us to introduce several key concepts that will be important for the results in subsequent sections. The main type of question that we examine here is the following. Given a certain class of definable topological spaces $\mathcal{C}$, is there a topological space $(X, \tau)$ that is universal for $\mathcal{C}$ in a definable sense, i.e. such that every space in $\mathcal{C}$ embeds definably into $(X,\tau)$? Moreover, is there such a universal space that lies within the given class $\mathcal{C}$ itself? We also consider a closely related property, which is natural to consider in the context of o-minimality, that we call 'almost definable universality'. Our analysis is very much in the spirit of universality questions considered in the classical study of Banach spaces, where there is an extensive literature on universal spaces for classes of separable Banach spaces, going back to the classical Banach--Mazur Theorem [@banach] (see for example [@szlenk_68], [@bourgain_80], [@bossard_02], [@odell_schlumprecht_06], [@brech_kosz_12]). This question has also classically been studied in the context of topological spaces, where results of this kind include the Menger--Nöbeling Theorem [@hur_wall41 Theorem V.2], which states that $(\mathbb{R}^{2n+1},\tau_e)$ is universal for the class of all $n$-dimensional compact metric spaces. We begin with the main definitions. **Definition 66**. Let $\mathcal{C}$ be a class of definable topological spaces and let $(X,\tau)$ be a definable topological space. We say that $(X,\tau)$ is *definably universal for $\mathcal{C}$* if every space $(Y,\mu)\in\mathcal{C}$ embeds definably into $(X,\tau)$. We say that $(X,\tau)$ is *almost definably universal for $\mathcal{C}$* if, for every $(Y,\mu)\in\mathcal{C}$, there exists a definable subset $Z\subseteq Y$ with $\dim (Y\setminus Z) < \dim Y$ such that $(Z,\mu)$ embeds definably into $(X,\tau)$. Note that, if a space is definably universal for a class, then in particular it is almost definably universal. We begin by observing how o-minimality implies that, if $\mathcal{R}$ expands an ordered field, then $R^n$ is almost definably universal for the class of euclidean spaces of dimension at most $n$. An analogous result can be proved for the class of bounded euclidean spaces of dimension at most $n$ when $\mathcal{R}$ expands an ordered group, using the observations in Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"}. Moreover, a slight weakening of the result can also be obtained in the most general case, which we discuss below in Remark [Remark 68](#remark:universality_euclidean_spaces){reference-type="ref" reference="remark:universality_euclidean_spaces"}. **Proposition 67**. Suppose that $\mathcal{R}$ expands an ordered field and $n>0$. Then $(R^n,\tau_e)$ is almost definably universal for the class of euclidean spaces of dimension less than or equal to $n$. *Proof.* Let $(X,\tau_e)$ be a euclidean space. We prove the case $\dim X = n$. The case $\dim X = m$, with $1\leq m <n$, then follows by the easy fact that, for any such $m$, the space $(R^m,\tau_e)$ embeds definably into $(R^n, \tau_e)$, while in the case $\dim X = 0$, i.e. where $X$ is finite, $(X,\tau_e)$ clearly embeds definably into $(R^n, \tau_e)$. Applying Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"}, let $(Y,\mu)$ denote the push-forward of $(X,\tau_e)$ into $R^n$ by some definable injection $f:X\rightarrow R^n$. We prove that $(Y,\mu)$ contains a definable subspace $Z\subseteq Y$, with $\dim (Y\setminus Z)<\dim Y$, where the subspace topology is euclidean. Observe that, by o-minimal cell decomposition applied to $f$ and $f^{-1}$, the function $f$ is a finite union of definable $e$-homeomorphisms. It follows that $Y$ can be partitioned into finitely many cells $\mathcal{D}$ where the subspace topology is euclidean (this is the property discussed in Section [9](#section: affine){reference-type="ref" reference="section: affine"} of being 'cell-wise euclidean'; see Definition [\[dfn:cell-wise_euclidean\]](#dfn:cell-wise_euclidean){reference-type="ref" reference="dfn:cell-wise_euclidean"}). Let $Z=\bigcup\{int_\mu D : D\in\mathcal{D},\, \dim D=n\}$. Note that, since $(Y,\mu)$ is the push-forward of a euclidean space, it has the frontier dimension inequality. It follows that $\dim (Y\setminus Z) < \dim Y$. Note that, since any cell of dimension $n$ is $e$-open, we have that, for any $D\in\mathcal{D}$ with $\dim D=n$, the set $int_\mu D$ is $e$-open, as well as $\mu$-open, in $Z$. Moreover the subspace topology on $int_\mu D$ is euclidean. We conclude that the subspace topology on $Z$ is euclidean. ◻ **Remark 68**. In the general case where $\mathcal{R}$ does not necessarily expand an ordered field one may still adapt the proof of Proposition [Proposition 67](#prop:universal_euclidean_space){reference-type="ref" reference="prop:universal_euclidean_space"} to show that, if $(X,\tau)$ is a euclidean space of dimension $n$ with cell decomposition $\mathcal{D}$, then the union $Z$ of the interiors in $X$ of cells in $\mathcal{D}$ of dimension $n$ embeds definably into finitely many disjoint copies of $R^n$ (one for each cell). It follows that the space $(R^{n+1}, \tau_e)$ is almost definably universal for the class of euclidean spaces of dimension at most $n$. The question of definable universality (as opposed to almost definable universality) for euclidean spaces is less straightforward. Walsberg has announced to the authors (through private correspondence) that he and C. Miller have obtained a definable version of the classical Menger--Nöbeling Theorem which implies that, whenever $\mathcal{R}$ expands an ordered field, any euclidean space of dimension $n$ embeds definably into ($R^{2n+1}, \tau_e)$. We now illustrate how results from previous sections can be used to derive o-minimal definable universality results. To begin, we show that Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"} may be framed in terms of existence of an almost definably universal space as follows. **Corollary 69**. The disjoint union of $(R\times [0,1],\tau_{lex})$ and $(R\times[0,\infty), \tau_{Alex})$ is Hausdorff, regular and almost definably universal for the class of Hausdorff regular definable topological spaces $(X,\tau)$, where $X\subseteq R$. *Proof.* It is easy to observe that the spaces $(R\times[0,1], \tau_{lex})$ and $(R\times[0,\infty),\tau_{Alex})$ are Hausdorff and regular, from where it follows that their disjoint union is too. Let $(X,\tau)$, where $X\subseteq R$, be a regular Hausdorff definable topological space. By Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"}, there exist definable disjoint open sets $Y,Z\subseteq X$ and $n_Y>0$ such that $X\setminus (Y\cup Z)$ is finite and there are definable embeddings $(Y,\tau)\hookrightarrow (R\times\{0,\ldots, n_Y\},\tau_{lex})$ and $(Z,\tau)\hookrightarrow (R\times [0,\infty),\tau_{Alex})$. Hence it suffices to show that, for any $n>0$, there exists a definable embedding $(R\times\{0,\dots,n\},\tau_{lex}) \hookrightarrow (R\times [0,1],\tau_{lex})$. Fix parameters $0=a_0<a_1<\cdots<a_{n}=1$. Then the map given by $\langle x,i \rangle\mapsto \langle x, a_i \rangle$ does the job. ◻ In the specific case of spaces in the line that embed definably into definable order topological spaces we may refine the above corollary as follows. **Corollary 70**. The disjoint union of $(R\times [0,1], \tau_{lex})$ and $(R,\tau_e)$, with topology given by the lexicographic order, is almost definably universal for the class of definable topological spaces $(X,\tau)$, with $X\subseteq R$, that embed definably into a definable order topological space. *Proof.* Let $(X,\tau)$, with $X\subseteq R$, be a definable topological space that embeds definably into a definable order topological space. Observe first that, since order topological spaces are Hausdorff and regular and these properties are hereditary, then $(X,\tau)$ is Hausdorff and regular. Now note that, in this case, by Proposition [Proposition 64](#prop_n_line_is_not_order_space){reference-type="ref" reference="prop_n_line_is_not_order_space"}, the partition $\mathcal{X}$ of $X$ described in Lemmas [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"} and [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"} is such that, for every $A\in \mathcal{X}_{\text{open}}$, the space $(A^*,\tau_A)$ described in Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"} satisfies $\tau_A=\tau_{Alex}$ if and only if $A^*=I\times \{0\}$, for some interval $I\subseteq X$ (see the cases in the proof of Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}). Following the proof of Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"}, we derive that there exist disjoint definable $\tau$-open sets $Y, Z \subseteq X$, whose union is cofinite in $X$, and some $n_Y$ such that $(Y,\tau)$ embeds definably into $(R\times \{0,\ldots, n_Y\}, \tau_{lex})$ and $(Z,\tau)$ embeds definably into $(R\times \{0\}, \tau_{Alex})=(R \times\{0\},\tau_e)$. The proof then concludes in a similar manner to the proof of Corollary [Corollary 69](#cor_universal_space){reference-type="ref" reference="cor_universal_space"}. ◻ From now on, we let $\mathcal{C}^{T_3}_{\dim 1}$ denote the class of one-dimensional regular Hausdorff definable topological spaces. **Remark 71**. If $\mathcal{R}$ expands an ordered field then, by Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"}, the space described in Corollary [Corollary 69](#cor_universal_space){reference-type="ref" reference="cor_universal_space"} is almost definably universal for $\mathcal{C}^{T_3}_{\dim 1}$. In general, recall that Remark [\[remark_general_them_ADC_or DOTS\]](#remark_general_them_ADC_or DOTS){reference-type="ref" reference="remark_general_them_ADC_or DOTS"} states that any space in $\mathcal{C}^{T_3}_{\dim 1}$ can be partitioned into finitely many points and open subsets each definably homeomorphic to some set $R\times\{0,\ldots, n-1\}$ with either the $\tau_{lex}$ or $\tau_{Alex}$ topology, for various $n$. Consequently, following the arguments in the proof of Corollary [Corollary 69](#cor_universal_space){reference-type="ref" reference="cor_universal_space"}, one may show that a space given by infinitely many copies of $(R\times [0,1],\tau_{lex})$ and $(R\times[0,\infty), \tau_{Alex})$ is almost definably universal for $\mathcal{C}^{T_3}_{\dim 1}$. Such a space exists as a three-dimensional space. That is, consider $(X,\tau)$, where $X=((-\infty, 0) \times R\times[0,1]) \cup ([0,\infty) \times R \times [0,\infty))$. Then let $\tau$ be the topology such that, for every $t\in R$, the fiber of $X$, which is given by either $\{t\}\times R \times [0,1]$ or $\{t\}\times R\times[0,\infty)$, is open and its projection to the last two coordinates is a homeomorphism onto $(R\times [0,1],\tau_{lex})$ or $(R\times[0,\infty), \tau_{Alex})$ respectively. Using Corollary [Corollary 70](#cor_universal_space_embed_ordertopspace){reference-type="ref" reference="cor_universal_space_embed_ordertopspace"}, as well as Remark [\[remark_general_them_ADC_or DOTS\]](#remark_general_them_ADC_or DOTS){reference-type="ref" reference="remark_general_them_ADC_or DOTS"} and Proposition [Proposition 64](#prop_n_line_is_not_order_space){reference-type="ref" reference="prop_n_line_is_not_order_space"}, one may analogously identify an almost definably universal space specifically for the class of one-dimensional definable topological spaces that embed definably into a definable order topological space. Such a space can be found that is two-dimensional if $\mathcal{R}$ expands an ordered field, and three-dimensional in general. Note that Proposition [Proposition 67](#prop:universal_euclidean_space){reference-type="ref" reference="prop:universal_euclidean_space"} states that, when $\mathcal{R}$ expands an ordered field, the class of euclidean spaces of dimension at most $n$ contains an almost definably universal space for itself (namely $(R^n,\tau_e)$). On the other hand, the space described in Corollary [Corollary 69](#cor_universal_space){reference-type="ref" reference="cor_universal_space"}, which, by Remark [Remark 71](#remark:universality_T_3_spaces){reference-type="ref" reference="remark:universality_T_3_spaces"}, is almost definably universal for $\mathcal{C}^{T_3}_{\dim 1}$ whenever $\mathcal{R}$ expands an ordered field, is two-dimensional, and so it does not belong in $\mathcal{C}^{T_3}_{\dim 1}$. In light of these results, and in the context of the classical literature on universal Banach spaces mentioned at the start of the section, it is natural to ask if, in the case that $\mathcal{R}$ expands an ordered field, there exists a space $(X,\tau)\in\mathcal{C}^{T_3}_{\dim 1}$ that is almost definably universal for $\mathcal{C}^{T_3}_{\dim 1}$, as well as to ask more generally which classes of spaces admit an almost definably universal space, and when does the space belong in the class. We first answer the question regarding the class $\mathcal{C}^{T_3}_{\dim 1}$ negatively and then derive, from Theorem [Theorem 51](#them 1.5){reference-type="ref" reference="them 1.5"} and Corollary [Corollary 65](#cor_T3_separable_embedding){reference-type="ref" reference="cor_T3_separable_embedding"}, positive answers for two other classes of one-dimensional spaces. **Proposition 72**. There does not exist a $T_1$ one-dimensional definable topological space $(X,\tau)$ that is almost definably universal for $\mathcal{C}^{T_3}_{\dim 1}$. In order to prove the proposition we first introduce a notion of equivalence for curves and some preliminary results. **Lemma 73**. Let $\gamma:(a,b)\rightarrow R^n$ and $\gamma':(a',b')\rightarrow R^n$ be two definable curves, with convergence endpoints $c\in \{a,b\}$ and $c'\in\{a',b'\}$ respectively. For any $s, t \in R_{\pm\infty}$, let $I(s,t)$ denote the interval with endpoints $s$ and $t$. The following are equivalent. (a) [\[itm0:curve-equiv\]]{#itm0:curve-equiv label="itm0:curve-equiv"} For any $a < t < b$ and $a' < t' < b'$, it holds that $\gamma(I(c,t)) \cap \gamma(I(c',t')) \neq \emptyset$. (b) [\[itm1:curve-equiv\]]{#itm1:curve-equiv label="itm1:curve-equiv"} For any $a < t < b$, there exists some $a' < t' < b'$ such that $\gamma'(I(c',t'))\subseteq \gamma(I(c,t))$. (c) [\[itm2:curve-equiv\]]{#itm2:curve-equiv label="itm2:curve-equiv"} For any $a' < t' < b'$, there exists some $a < t < b$ such that $\gamma(I(c,t))\subseteq \gamma'(I(c',t'))$. (d) [\[itm3:curve-equiv\]]{#itm3:curve-equiv label="itm3:curve-equiv"} For any definable topological space $(X,\tau)$ with $\gamma[(a,b)] \cup \gamma'[(a',b')] \subseteq X\subseteq R^n$, and any $x\in X$, it holds that $\gamma$ $\tau$-converges to $x$ if and only if $\gamma'$ $\tau$-converges to $x$. *Proof.* It is easy to see that $(\ref{itm1:curve-equiv}) \vee (\ref{itm2:curve-equiv}) \Rightarrow (\ref{itm0:curve-equiv})$. Similarly, one may easily check, using the definition of curve convergence (Definition [Definition 9](#dfn:curve){reference-type="ref" reference="dfn:curve"}), that $(\ref{itm1:curve-equiv}) \wedge (\ref{itm2:curve-equiv}) \Rightarrow (\ref{itm3:curve-equiv})$. We show that $(\ref{itm0:curve-equiv})\Rightarrow (\ref{itm1:curve-equiv}) \wedge (\ref{itm2:curve-equiv})$ and $(\ref{itm3:curve-equiv})\Rightarrow(\ref{itm0:curve-equiv})$. **Proof of $(\ref{itm0:curve-equiv})\Rightarrow (\ref{itm1:curve-equiv}) \wedge (\ref{itm2:curve-equiv})$.** By symmetry of the statements ([\[itm1:curve-equiv\]](#itm1:curve-equiv){reference-type="ref" reference="itm1:curve-equiv"}) and ([\[itm2:curve-equiv\]](#itm2:curve-equiv){reference-type="ref" reference="itm2:curve-equiv"}) it suffices to show that $(\ref{itm0:curve-equiv})\Rightarrow(\ref{itm1:curve-equiv})$. Hence suppose that ([\[itm0:curve-equiv\]](#itm0:curve-equiv){reference-type="ref" reference="itm0:curve-equiv"}) holds, and let us fix some $a<t<b$. Consider the definable set $J$ of all $a' < s' < b'$ such that $\gamma'(s')\in \gamma(I(c,t))$. By o-minimality, there exists some $a' < t' < b'$ such that either $I(c',t') \subseteq J$ or $I(c',t') \cap J = \emptyset$. In the second case however we have that $\gamma'(I(c',t')) \cap \gamma(I(c,t)) = \emptyset$, contradicting ([\[itm0:curve-equiv\]](#itm0:curve-equiv){reference-type="ref" reference="itm0:curve-equiv"}). So $I(c',t') \subseteq J$, meaning that $\gamma'(I(c',t'))\subseteq \gamma(I(c,t))$. **Proof of $(\ref{itm3:curve-equiv})\Rightarrow(\ref{itm0:curve-equiv})$.** By contraposition, assume that there exists some $a < t < b$ and $a' < t' < b'$ such that $\gamma(I(c,t)) \cap \gamma'(I(c',t')) = \emptyset$. In particular, we have that $\gamma'(I(c',t')) \subsetneq R^n$. Fix any point $x\in R^n\setminus \gamma'(I(c',t'))$. Consider the definable topology $\tau$ on $R^n$ where every point in $R^n \setminus \{x\}$ is isolated, and a basis of open neighbourhoods for $x$ is given by the definable family of sets $\{ \{x\}\cup\gamma(I(c,s)) : a < s < b\}$. Clearly $\gamma$ $\tau$-converges to $x$ but $\gamma'$ does not, i.e. we reach the negation of ([\[itm3:curve-equiv\]](#itm3:curve-equiv){reference-type="ref" reference="itm3:curve-equiv"}). ◻ **Definition 74**. Let $(X,\tau)$ be a definable topological space. We say that two definable curves $\gamma:(a,b)\rightarrow X$ and $\gamma':(a',b')\rightarrow X$, with fixed convergence endpoints $c\in\{a',b'\}$ and $c'\in\{a,b\}$ respectively, are *equivalent* if any of the equivalent conditions in Lemma [Lemma 73](#lem:curve-equiv){reference-type="ref" reference="lem:curve-equiv"} holds. It is easy to check that two injective definable curves in $R$ are equivalent if and only if they $e$-converge to the same point in $R_{\pm\infty}$ from the same side, i.e. from left or right (see Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}). **Definition 75**. Let $(X,\tau)$ be a definable topological space. For any $x\in X$, let $\textbf{n}(x,X,\tau)$ ($\textbf{n}(x)$ for short, when the underlying topological space $(X,\tau)$ is clear from the context) denote the maximum cardinality of a set of non-equivalent definable curves (together with fixed convergence endpoints) in $X$ $\tau$-converging to $x$. Let $\textbf{n}(X,\tau)$ ($\textbf{n}(X)$ for short) be defined to be $\sup \{ \textbf{n}(x) : x\in X\}$. **Example 76**. For any $n$ and $1 \leq i\leq n$, let $L(i)$ denote the line $$\{0\}\times \overset{i-1}{\cdots} \times \{0\}\times R \times \{0\}\times \overset{n-i}{\cdots} \times \{0\},$$ where $R$ is in the $i$-th coordinate position. Consider the euclidean space $X=\bigcup_{1\leq i \leq n} L(i)\subseteq R^n$. For any $s,t\in R_{\pm\infty}$, let $I(s,t)$ denote the interval with endpoints $s$ and $t$. Observe that, by o-minimality, a definable curve $\gamma:(a,b)\rightarrow X$, with convergence endpoint $c\in \{a,b\}$, $e$-converges to the point $\langle 0,\ldots, 0 \rangle\in R^n$ if and only if there exists some $1\leq i \leq n$ and $a<t<b$ such that $\gamma(I(c,t))\subseteq L(i)$, and moreover the composition $\pi_i \circ \gamma$, where $\pi_i$ denotes the projection to the $i$-th coordinate, $e$-converges (as $t$ tends to $c$) to $0$, from either the left or the right. One may derive from this that $\textbf{n}(\langle 0,\ldots, 0 \rangle, X, \tau_e)=2n$. **Remark 77**. If $\dim X \leq 1$ and $(X,\tau)$ is $T_1$, then one may easily check that $\textbf{n}(x)=1+|R_x|+|L_x|$, for every $x\in X$, using Remarks [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"} and [Remark 40](#remark_generalizing_Rx_Lx){reference-type="ref" reference="remark_generalizing_Rx_Lx"}, as well as the fact that, by $T_1$-ness, for any $x \in X$, there is only one eventually constant definable curve (up to equivalence) that $\tau$-converges to $x$. By Lemma [Lemma 23](#lemma_2){reference-type="ref" reference="lemma_2"}, Proposition [Proposition 32](#prop_basic_facts_P_x_2){reference-type="ref" reference="prop_basic_facts_P_x_2"} ([\[itm1:basic_facts_P\_x_2\]](#itm1:basic_facts_P_x_2){reference-type="ref" reference="itm1:basic_facts_P_x_2"}) and ([\[itm2:basic_facts_P\_x_2\]](#itm2:basic_facts_P_x_2){reference-type="ref" reference="itm2:basic_facts_P_x_2"}), and uniform finiteness, it follows that $\textbf{n}(X) <\omega$. **Lemma 78**. Let $f:(X,\tau)\rightarrow (Y,\mu)$, be a continuous injective definable map between definable topological spaces. For any $x\in X$, it holds that $\textbf{n}(x) \leq \textbf{n}(f(x))$. In particular, $\textbf{n}(X) \leq \textbf{n}(Y)$. *Proof.* By Proposition [Proposition 27](#prop_cont_lim){reference-type="ref" reference="prop_cont_lim"}, if $\gamma$ and $\gamma'$ are two definable curves $\tau$-converging to $x$ (with convergence endpoints $c$ and $c'$, respectively) then $f\circ \gamma$ and $f \circ \gamma'$ (with the same respective convergence endpoints, namely $c$ and $c'$) $\mu$-converge to $f(x)$. It therefore suffices to show that if $\gamma$ and $\gamma'$ are non-equivalent then $f \circ \gamma$ and $f\circ \gamma'$ (taken with the corresponding convergence endpoints) are non-equivalent. This however follows easily from the characterization of curve equivalence given by Lemma [Lemma 73](#lem:curve-equiv){reference-type="ref" reference="lem:curve-equiv"} ([\[itm0:curve-equiv\]](#itm0:curve-equiv){reference-type="ref" reference="itm0:curve-equiv"}) and the injectivity of $f$. ◻ **Remark 79**. Some observations can be derived from Lemma [Lemma 78](#lemma_universality_index){reference-type="ref" reference="lemma_universality_index"} regarding the existence of definably universal spaces. Recall that in Example [Example 76](#ex:cross){reference-type="ref" reference="ex:cross"} we define, for every $n$, a one-dimensional set $X\subseteq R^n$ such that $\textbf{n}(\langle 0,\ldots, 0 \rangle, X, \tau_e)=2n$. By Lemma [Lemma 78](#lemma_universality_index){reference-type="ref" reference="lemma_universality_index"} and Remark [Remark 77](#rem:finite-n(x)){reference-type="ref" reference="rem:finite-n(x)"}, it follows that there does not exist a one-dimensional $T_1$ definable topological space that is definably universal for the class of all one-dimensional euclidean spaces, and in particular neither for $\mathcal{C}^{T_3}_{\dim 1}$. We may now prove Proposition [Proposition 72](#prop_nonexistence_universal_space){reference-type="ref" reference="prop_nonexistence_universal_space"}. *Proof of Proposition [Proposition 72](#prop_nonexistence_universal_space){reference-type="ref" reference="prop_nonexistence_universal_space"}.* Let $Y=R\times\{0,\ldots, n-1\}$ and consider the space $(Y,\tau_{lex})$, which belongs to $\mathcal{C}^{T_3}_{\dim 1}$. If, for every $0\leq i <n$, we identify the subspace $R\times\{i\}$ with $R$ through the projection to the first coordinate (see Remark [Remark 40](#remark_generalizing_Rx_Lx){reference-type="ref" reference="remark_generalizing_Rx_Lx"}) then, for any $x\in R$, it holds that $L_{\langle x,0\rangle}=\{\langle x,0 \rangle,\ldots, \langle x,n-1\rangle\}$ and $R_{\langle x,0\rangle}=\emptyset$. So $\textbf{n}(\langle x,0\rangle)=n$, and in fact it can easily be shown that $\textbf{n}(Y)=n$. Moreover note that, for any cofinite subset $Y'\subseteq Y$, it still holds that $\textbf{n}(Y')=n$, since we may always find an interval $I\subseteq R$ such that $I\times\{0,\ldots, n-1\}\subseteq Y'$. Suppose that $(X,\tau)$ is a $T_1$ one-dimensional definable topological space that is almost definably universal for $\mathcal{C}^{T_3}_{\dim 1}$. Since $(X,\tau)$ is $T_1$, we have $\textbf{n}(X) < \omega$ (see Remark [Remark 77](#rem:finite-n(x)){reference-type="ref" reference="rem:finite-n(x)"}). However, by Lemma [Lemma 78](#lemma_universality_index){reference-type="ref" reference="lemma_universality_index"} and the above observation, we have that $\textbf{n}(X)\geq n$ for every $n$, a contradiction. ◻ The same proof would still have worked if we had considered the space $(R\times \{0,\ldots, n-1\}, \tau_{Alex})$ in place of $(R\times \{0,\ldots, n-1\}, \tau_{lex})$. Ultimately, one may show that there exists no one-dimensional definable topological space that is almost definably universal for either of the following two classes: all one-dimensional spaces with the $\tau_{lex}$ topology and all one-dimensional spaces with the $\tau_{Alex}$ topology. It is our belief that Proposition [Proposition 72](#prop_nonexistence_universal_space){reference-type="ref" reference="prop_nonexistence_universal_space"} can likely be improved by dropping the condition of being $T_1$. In other words, we believe that the following question has a negative answer: **Question 80**. Is there a one-dimensional definable topological space (which is necessarily not $T_1$ by Proposition [Proposition 72](#prop_nonexistence_universal_space){reference-type="ref" reference="prop_nonexistence_universal_space"}) that is almost definably universal for $\mathcal{C}^{T_3}_{\dim 1}$? As a counterpoint to Proposition [Proposition 72](#prop_nonexistence_universal_space){reference-type="ref" reference="prop_nonexistence_universal_space"}, Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"} -- more specifically, Corollary [Corollary 65](#cor_T3_separable_embedding){reference-type="ref" reference="cor_T3_separable_embedding"} -- and Theorem [Theorem 51](#them 1.5){reference-type="ref" reference="them 1.5"} do yield the existence of two classes of one-dimensional spaces, each of which contains a space that is almost definably universal for itself, as shown by the following corollaries (see also Remark [Remark 83](#rem:2dim-univ){reference-type="ref" reference="rem:2dim-univ"}). Let $\mathcal{C}^{T_3,\, \text{sep}}_{\dim 1}$ denote the class of Hausdorff regular definably separable one-dimensional spaces. **Corollary 81**. The disjoint union of $(R,\tau_e)$ and $(R\times\{0,1\}, \tau_{lex})$ is Hausdorff, regular, definably separable and almost definably universal for the class of Hausdorff, regular, definably separable spaces $(X,\tau)$, where $X\subseteq R$. It follows that, whenever $\mathcal{R}$ expands an ordered field, the class $\mathcal{C}^{T_3,\, \text{sep}}_{\dim 1}$ contains an almost definably universal space. *Proof.* The second paragraph of the corollary follows from the first by direct application of Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"}. We prove the first paragraph. Since $(R,\tau_e)$ and $(R\times\{0,1\}, \tau_{lex})$ are regular, Hausdorff and definably separable (see Lemma [Lemma 7](#lem:T1-sep){reference-type="ref" reference="lem:T1-sep"} and Proposition [Proposition 8](#prop:def-sep_eucl_disc_llt){reference-type="ref" reference="prop:def-sep_eucl_disc_llt"}([\[itm1:sep-spaces-examples\]](#itm1:sep-spaces-examples){reference-type="ref" reference="itm1:sep-spaces-examples"})) their disjoint union is too. By Corollary [Corollary 65](#cor_T3_separable_embedding){reference-type="ref" reference="cor_T3_separable_embedding"}, it suffices to show that, for any definable set $X\subseteq R\times\{0,1\}$, there exists a cofinite subspace $Y$ of $(X,\tau_{lex})$ that embeds definably into the disjoint union of $(R,\tau_e)$ and $(R\times\{0,1\}, \tau_{lex})$. We partition $X\subseteq R\times\{0,1\}$ as follows. Let $X_1=\{ \langle x, i \rangle\in X : \langle x, 1-i \rangle\notin X \}$ and $X_2=X\setminus X_1$. By o-minimality, there exists a partition $\mathcal{X}$ of a cofinite subset of $X$ with the following properties. For every $A\in\mathcal{X}$, there exists an interval $I \subseteq R$ such that either $A=I\times \{i\}$, for some $i\in\{0,1\}$, and $A\subseteq X_1$, or $A=I\times\{0,1\}$ (and so $A\subseteq X_2$). Let $\mathcal{X}_1=\{ A\in\mathcal{X}: A\subseteq X_1\}$ and $\mathcal{X}_2=\mathcal{X}\setminus \mathcal{X}_1$. Note that every $A\in\mathcal{X}$ is open in $(X,\tau_{lex})$, and that the subspace topology on $A$ corresponds precisely to the lexicographic order topology on $A$. If $A\subseteq X_1$, then the projection $\langle x,i\rangle\mapsto x$ is an open embedding $(A, \tau_{lex})\hookrightarrow (R,\tau_e)$, and otherwise the identity is an open embedding $(A,\tau_{lex})\hookrightarrow (R\times\{0,1\},\tau_{lex})$. Hence the projection to the first coordinate is an open embedding $(\bigcup \mathcal{X}_1,\tau_{lex})\hookrightarrow (R,\tau_e)$ and the identity is an open embedding $(\bigcup\mathcal{X}_2,\tau_{lex})\rightarrow (R\times\{0,1\},\tau_{lex})$, which completes the proof. ◻ Finally, we consider the class of one-dimensional Hausdorff definable topological spaces satisfying the frontier dimension inequality (**fdi**; see Definition [Definition 41](#dfn:fdi){reference-type="ref" reference="dfn:fdi"}), which we denote $\mathcal{C}^{T_2,\, \text{fdi}}_{\dim 1}$. By Proposition [Proposition 42](#prop_T2_frontier_ineq_regular){reference-type="ref" reference="prop_T2_frontier_ineq_regular"}, we know that these spaces are regular. We show that, whenever $\mathcal{R}$ expands an ordered field, the class contains an almost definably universal space. This is a corollary of Theorem [Theorem 51](#them 1.5){reference-type="ref" reference="them 1.5"}. **Corollary 82**. The disjoint union of the spaces $(R,\tau_e)$, $(R,\tau_r)$, $(R,\tau_l)$ and $(R,\tau_s)$ is Hausdorff, satisfies the frontier dimension inequality, and is almost definably universal for the class of Hausdorff definable topological spaces $(X,\tau)$ with the frontier dimension inequality, where $X\subseteq R$. It follows that, whenever $\mathcal{R}$ expands an ordered field, the class $\mathcal{C}^{T_2,\, \text{fdi}}_{\dim 1}$ contains an almost definably universal space. *Proof.* As in the proof of Corollary [Corollary 81](#cor:universality_CCCsep){reference-type="ref" reference="cor:universality_CCCsep"}, the second paragraph of the corollary follows from the first by direct application of Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"}. We prove the first paragraph. Since the spaces $(R,\tau_e)$, $(R,\tau_r)$, $(R,\tau_l)$ and $(R,\tau_s)$ are Hausdorff and satisfy the **fdi**, their disjoint union has these properties too. We fix $(X,\tau)$, $X\subseteq R$, a Hausdorff definable space which satisfies the **fdi**. By Theorem [Theorem 51](#them 1.5){reference-type="ref" reference="them 1.5"}, there exists a finite partition $\mathcal{X}$ of $X$ into points and intervals such that, for each $I\in \mathcal{X}$, the subspace topology $\tau|_I$ is one of $\tau_e$, $\tau_r$, $\tau_l$ or $\tau_s$. Let $\mathcal{X}'$ be the subfamily of intervals in $\mathcal{X}$ and let $X'=\bigcup\{ int_\tau I : I\in \mathcal{X}'\}$. By the **fdi**, the set $X'$ is cofinite in $X$. Let $X_e=\{ x\in X' : x\in I\in\mathcal{X},\, (I,\tau)=(I,\tau_e)\}$. Then the identity is an open embedding $(X_e,\tau)\hookrightarrow (R, \tau_e)$. By repeating this argument with the topologies $\tau_r$, $\tau_l$ and $\tau_s$, we may conclude that $X'$ can be partitioned into four definable open subspaces on which the identity is an embedding into one of $(R,\tau_e)$, $(R,\tau_r)$, $(R,\tau_l)$ or $(R,\tau_s)$. The corollary follows. ◻ **Remark 83**. Following Remarks [Remark 68](#remark:universality_euclidean_spaces){reference-type="ref" reference="remark:universality_euclidean_spaces"} and [Remark 71](#remark:universality_T_3_spaces){reference-type="ref" reference="remark:universality_T_3_spaces"}, if $\mathcal{R}$ does not expand an ordered field, then one may adapt the proof of Corollaries [Corollary 81](#cor:universality_CCCsep){reference-type="ref" reference="cor:universality_CCCsep"} and [Corollary 82](#cor_universal_space_for_T2_fdi){reference-type="ref" reference="cor_universal_space_for_T2_fdi"} to show the existence of two-dimensional almost definably universal spaces for each of the classes $\mathcal{C}^{T_3,\, \text{sep}}_{\dim 1}$ and $\mathcal{C}^{T_2,\, \text{fdi}}_{\dim 1}$. More precisely, any space containing infinitely many disjoint copies of $(R,\tau_e)$ and $(R\times\{0,1\}, \tau_{lex})$ is almost definably universal for $\mathcal{C}^{T_3,\, \text{sep}}_{\dim 1}$, and any space with infinitely many disjoint copies of the spaces $(R,\tau_e)$, $(R,\tau_r)$, $(R,\tau_l)$ and $(R,\tau_s)$ is almost definably universal for $\mathcal{C}^{T_2,\, \text{fdi}}_{\dim 1}$. Note that, by Lemma [Lemma 7](#lem:T1-sep){reference-type="ref" reference="lem:T1-sep"} and o-minimality, any definable subspace of $(R,\tau_e)$ or $(R\times\{0,1\}, \tau_{lex})$ is definably separable. By Corollary [Corollary 81](#cor:universality_CCCsep){reference-type="ref" reference="cor:universality_CCCsep"} and Remark [Remark 83](#rem:2dim-univ){reference-type="ref" reference="rem:2dim-univ"}, it follows that any definable subspace of a space in $\mathcal{C}^{T_3,\, \text{sep}}_{\dim 1}$ is also definably separable. In other words, definable separability is a hereditary property for $T_3$ one-dimensional spaces. Since being $T_3$ is also a hereditary property, we have that the class $\mathcal{C}^{T_3,\, \text{sep}}_{\dim 1}$ is closed under passing to one-dimensional definable subspaces. # Definable Hausdorff compactifications {#section:compactifications} In this section, we use the decomposition of $T_3$ spaces described in Section [6](#section: universal spaces){reference-type="ref" reference="section: universal spaces"} to address the question of which definable topological spaces can be Hausdorff compactified in a definable sense. Our main result (Theorem [Theorem 89](#them_compactification){reference-type="ref" reference="them_compactification"}) shows that a Hausdorff definable topological space in the line has a definable Hausdorff compactification of dimension at most one if and only if it is $T_3$. This result then generalizes to all one-dimensional definable topological spaces (Remark [Remark 91](#rmk_compactification){reference-type="ref" reference="rmk_compactification"}). Recall that a definable topological space is definably compact (Definition [Definition 11](#dfn:compact){reference-type="ref" reference="dfn:compact"}) if and only if every definable curve in it converges. We present the following definition, and prove that it characterizes the one-dimensional $T_3$ spaces that can be definably one-point Hausdorff compactified. **Definition 84**. A definable topological space $(X,\tau)$, $\dim X\leq 1$, is *definably near-compact* if, up to equivalence, there are only finitely many non-convergent definable curves in $(X,\tau)$. Clearly definable compactness implies definable near-compactness. We say that a definable topological space $(X^*,\tau^*)$, $X^*\subseteq R$, is a *definable near-compactification* of $(X,\tau)$ if $(X^*, \tau^*)$ is definably near-compact and there exists a definable embedding $(X,\tau)\hookrightarrow(X^*,\tau^*)$. We have the following characterization of definably near-compact definable topological spaces in the line, which is a direct consequence of Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}, and can be seen as an analogue of Lemma [Lemma 35](#lem:RL-compact){reference-type="ref" reference="lem:RL-compact"}, a statement about definably compact spaces in the line. **Lemma 85**. A definable topological space $(X,\tau)$, where $X\subseteq R$, is definably near-compact if and only if the set $(\bigcup_{x\in X} R_x) \cap (\bigcup_{x\in X} L_x)$ is cofinite in $cl_e X$. **Remark 86**. Note that, for any $n$ and interval $I$, the spaces $(I\times\{0,\ldots,n\},\tau_{lex})$ and $(I\times\{0,\ldots,n\},\tau_{Alex})$ are definably near-compact, and furthermore they are definably compact if and only if $I$ is a closed interval. It follows that, if $(X, \tau)$, $X \subseteq R$, is a regular Hausdorff definable topological space, then the embedding $h :(Y\cup Z, \tau) \hookrightarrow (Y^*\cup Z^*, \tau_{lex}|_{Y^*}\cup \tau_{Alex}|_{Z^*})$ described in the proof of Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"} is a definable near-compactfication of a cofinite open subspace of $X$. We extract the following observation from the proof of Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}, which can be seen as an improvement of Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"} for $T_3$ definably near-compact spaces in the line. We will use this result in Section [9](#section: affine){reference-type="ref" reference="section: affine"} (Corollary [Corollary 98](#thm:compact_affine){reference-type="ref" reference="thm:compact_affine"}). **Lemma 87**. Let $(X,\tau)$ be a regular Hausdorff definable topological space with $X\subseteq R$. Let $\mathcal{X}$ be a finite partition of $X$ as given by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}. For each $A\in \mathcal{X}_{\text{open}}$, let $A^*$ and $h_A:A\rightarrow A^*$ be as given by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}. If $(X,\tau)$ is definably near-compact, then, for any $A\in\mathcal{X}_{\text{open}}$, the map $h_A$ is a bijection. In particular, by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}([\[itm2:lemma_proof_two_theorems\]](#itm2:lemma_proof_two_theorems){reference-type="ref" reference="itm2:lemma_proof_two_theorems"}), it is a definable homeomorphism $(A,\tau)\rightarrow (A^*,\mu)$, where $\mu$ is one of $\tau_{lex}$ or $\tau_{Alex}$. *Proof.* Recall the proof by cases of Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}. Let $A = \bigcup_{0 \leq i < n}f_{i}(I)$. Observe that, by Lemma [Lemma 85](#lemma_near-compact_characterization){reference-type="ref" reference="lemma_near-compact_characterization"}, if $(X,\tau)$ is definably near-compact then, for all but finitely many $x\in I$, it holds that $x\in R_{y} \cap L_{z}$, where, by Lemma [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}([\[itm:properties_cofnite_subset_1.5_b\]](#itm:properties_cofnite_subset_1.5_b){reference-type="ref" reference="itm:properties_cofnite_subset_1.5_b"}), $y,z\in [x]^E\subseteq \{x, f_{n-1}(x)\}$. It follows that, if $(X,\tau)$ is definably near-compact, Cases 0, 1 and 2 in the proof of Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"} are not possible. In the remaining cases, the function $h_A$ defined is a bijection. ◻ In the context of Remark [\[remark_general_them_ADC_or DOTS\]](#remark_general_them_ADC_or DOTS){reference-type="ref" reference="remark_general_them_ADC_or DOTS"}, Lemma [Lemma 87](#prop_proof_two_thems){reference-type="ref" reference="prop_proof_two_thems"} generalizes in the natural way to all $T_3$ one-dimensional spaces. We now show that if a $T_3$ one-dimensional space is definably near-compact then it can be definably one-point Hausdorff compactified. The converse implication can also be derived, from the observation that, given a Hausdorff definably compact definable topological space $(X^c,\tau^c)$, with $\dim X^c= 1$, and any point $x\in X^c$, by Lemmas [Lemma 23](#lemma_2){reference-type="ref" reference="lemma_2"} and [Lemma 85](#lemma_near-compact_characterization){reference-type="ref" reference="lemma_near-compact_characterization"} and Remark [Remark 40](#remark_generalizing_Rx_Lx){reference-type="ref" reference="remark_generalizing_Rx_Lx"} the subspace $(X^c\setminus\{x\}, \tau^c)$ is definably near-compact. We will use the following result to show that $T_3$ spaces in the line can be definably Hausdorff compactified (Theorem [Theorem 89](#them_compactification){reference-type="ref" reference="them_compactification"}). **Proposition 88**. Let $(X,\tau)$, $\dim X \leq 1$, be a regular Hausdorff definably near-compact definable topological space. Then there exists a Hausdorff definably compact definable topological space $(X^c, \tau^c)$ and a definable embedding $h:(X,\tau)\hookrightarrow (X^c,\tau^c)$, where $X^c\setminus h(X)$ is a singleton. If $\mathcal{R}$ expands the field of reals then we leave it to the reader to check that $(X^c,\tau^c)$ is the classical one-point compactification of $(X,\tau)$. *Proof of Proposition [Proposition 88](#lemma_one_point_compact){reference-type="ref" reference="lemma_one_point_compact"}.* We prove the lemma in the case where $X\subseteq R$. Given the assumptions in Remark [Remark 40](#remark_generalizing_Rx_Lx){reference-type="ref" reference="remark_generalizing_Rx_Lx"}, the proof adapts to a proof of the general case. Let $c=\langle 0,1 \rangle\in R^2$ and let $X^c=(X\times \{0\}) \cup \{c\}$. Let $h:X \to X^c$ be given by $x\mapsto \langle x,0 \rangle$, and let $\tau_h$ be the push-forward topology of $\tau$ by $h$ (see Definition [Definition 13](#dfn:push-forward){reference-type="ref" reference="dfn:push-forward"}). We will define $\tau^c$ as an extension of $\tau_h$ to a topology on $X^c$. If $\tau$ is definably compact then it clearly it suffices to let $\tau^c= \tau_h \cup \{c\}$, and so we assume otherwise. Set $R_c:=\{ x\in R_{\pm\infty}\setminus \bigcup_{x\in X} R_x : \exists \, y>x \; (x,y)\subseteq X\}$ and $L_c:=\{ x\in R_{\pm\infty}\setminus \bigcup_{x\in X} L_x : \exists \, y<x \; (y,x)\subseteq X\}$. Set $E_c:= R_c \cup L_c$. Since $(X,\tau)$ is definably near-compact, $E_c$ is finite, by Lemma [Lemma 85](#lemma_near-compact_characterization){reference-type="ref" reference="lemma_near-compact_characterization"}. Since, by assumption, $(X,\tau)$ is not definably compact, we also have that $E_c\neq \emptyset$. Let $R_c=\{y_1,\ldots, y_n\}$ and $L_c=\{z_1,\ldots, z_m\}$, and let $\mathcal{U}(c)$ be the family of sets $$U=\bigcup_{1\leq i \leq n} (y_i,y'_i) \cup \bigcup_{1 \leq j \leq m} (z'_j, z_j)$$ definable uniformly over those parameters $(y'_1, \ldots, y'_n, z'_1,\ldots, z'_m)\in R^{n+m}$ for which $y_i<y'_i$, $z'_j<z_j$ and moreover $U\subseteq X$. Let $\tau^c$ be the definable topology with basis $\{ (int_\tau U \times \{0\}) \cup \{c\}: U\in\mathcal{U}(c)\} \cup \tau_h$. It is routine to check that this is a well-defined topology and that $h:(X,\tau)\hookrightarrow (X^c, \tau^c)$ is an embedding. Since $(X,\tau)$ is Hausdorff, by definition of $\mathcal{U}(c)$ and Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"} it is immediate that $(X^c,\tau^c)$ is also Hausdorff. It remains to prove that it is definably compact. Let $\gamma'$ be a definable curve in $(X^c,\tau^c)$. We may assume that $\gamma'$ is injective and hence lies in $X\times\{0\}$. Let $\gamma= h^{-1}\circ \gamma'$. Let $x_0\in R_{\pm\infty}$ denote the limit of $\gamma$ in the euclidean topology. Since the remaining case is analogous, we consider only the case where $\gamma$ $e$-converges to $x_0$ from the right. Then clearly there must exist $y>x_0$ such that $(x_0,y)\subseteq X$. If $x_0\notin R_c$, then $x_0\in\bigcup_{x\in X} R_x$ and so, by Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}, $\gamma$ $\tau$-converges to some $x\in X$, and it follows that $\gamma'$ $\tau^c$-converges to $h(x)$. Therefore, it remains to consider the case $x_0\in R_c$. We will show that $\gamma'$ $\tau^c$-converges to $c$. To prove this, it suffices to show that, for every $U\in\mathcal{U}(c)$, there is $x_U>x_0$ such that $(x_0,x_U)\subseteq int_\tau U$. Towards a contradiction, suppose otherwise. Then, by o-minimality, there exists $U_1\in\mathcal{U}(c)$ and $x_1>x_0$ such that $(x_0,x_1)\cap int_\tau U_1=\emptyset$. By definition of $\mathcal{U}(c)$, we may moreover take $x_1$ close enough to $x_0$ to satisfy that $(x_0,x_1)\subseteq U_1$. For every $x_0<x<x_1$, we have that $x\in \partial_\tau (X\setminus U_1)$, and so, from Proposition [Proposition 29](#basic_facts_P_x_1){reference-type="ref" reference="basic_facts_P_x_1"}([\[itm: basic_facts_4\]](#itm: basic_facts_4){reference-type="ref" reference="itm: basic_facts_4"}), it follows that $E_x \setminus U_1 \neq \emptyset$, as $U_1$ is $e$-open. Let $f:(x_0,x_1)\rightarrow R_{\pm\infty}$ be the definable map given by $x\mapsto \min E_x \setminus U_1$. By Hausdorffness (Lemma [Lemma 30](#lemma:basic_facts_P_x_2){reference-type="ref" reference="lemma:basic_facts_P_x_2"}([\[itmb:lemma_basic_facts_P\_x_2\]](#itmb:lemma_basic_facts_P_x_2){reference-type="ref" reference="itmb:lemma_basic_facts_P_x_2"})) and o-minimality, this function is $e$-continuous and strictly monotone on some subinterval $(x_0,x_2)\subseteq (x_0,x_1)$. Let $y_0=\mathop{\mathrm{{\it e}{\operatorname{-}}lim}}_{x\rightarrow x_0} f(x)$. If $f$ is increasing on $(x_0,x_2)$, then, by construction of $\mathcal{U}(c)$ and the fact that $f$ maps into $R_{\pm\infty}\setminus U_1$, it cannot be that $y_0\in R_c$. However, there clearly exists $y'\in R$ such that $(y_0,y')\subseteq X$. So there exists $y\in X$ such that $y_0\in R_y$ and, by Lemma [Lemma 37](#lemma_f_Rx_Lx){reference-type="ref" reference="lemma_f_Rx_Lx"}([\[itmc:lemma_f\_Rx_Lx\]](#itmc:lemma_f_Rx_Lx){reference-type="ref" reference="itmc:lemma_f_Rx_Lx"}) and regularity, it follows that $x_0\in R_y$, a contradiction since $x_0\in R_c$. The case where $f$ is decreasing is analogous. ◻ We now present the main result of this section. **Theorem 89**. Let $(X,\tau)$, $X\subseteq R$, be a Hausdorff definable topological space. Then $(X,\tau)$ is regular if and only if there exists a definably compact Hausdorff definable topological space $(X^c,\tau^c)$, with $\dim X^c\leq 1$, and a definable embedding $(X,\tau)\hookrightarrow (X^c,\tau^c)$. The "if\" direction of the theorem is proven largely by the following lemma. **Lemma 90**. Let $(X,\tau)$, $\dim X\leq 1$, be a definably compact Hausdorff definable topological space. Then $(X,\tau)$ is regular. *Proof.* Let $(X,\tau)$ be as in the lemma and towards a contradiction suppose that it is not regular. Let $x\in X$ and let $C\subseteq X$ be a $\tau$-closed set such that $x\notin C$ and $cl_\tau (A)\cap C\neq \emptyset$, for every $\tau$-neighbourhood $A$ of $x$. By passing to a larger set if necessary (i.e. by passing if necessary to the complement of a definable $\tau$-neighbourhood of $x$ contained in $X\setminus C$), we may assume that $C$ is definable. Let $\mathcal{U}$ denote a definable basis of $\tau$-neighbourhoods of $x$. Note that $\{cl_\tau (U) \cap C : U\in \mathcal{U}\}$ is a definable downward directed family of non-empty sets of dimension at most one, so, by Remark [Remark 26](#remark_curve_selection){reference-type="ref" reference="remark_curve_selection"}, there exists a definable curve $\gamma: (a,b) \rightarrow C$ that is cofinal for this family. By definable compactness, $\gamma$ $\tau$-converges to some point $y\in C\subseteq X\setminus\{x\}$. By definition of $\gamma$, it holds that $y\in cl_\tau (A)$ for every $\tau$-neighbourhood $A$ of $x$. So $x$ and $y$ cannot be separated by $\tau$-neighbourhoods, which contradicts that $(X,\tau)$ is Hausdorff. ◻ In light of the above lemma, the question arises of whether or not definably compact Hausdorff spaces of any dimension are regular. Using [@atw1 Corollary 25], one may show that the answer is positive whenever $\mathcal{R}$ expands an ordered field. On the other hand, it is easy to prove that if a Hausdorff definable topological space is definably compact in the sense of condition [\[dfn:directed-compact\]](#dfn:directed-compact){reference-type="eqref" reference="dfn:directed-compact"} (i.e. every downward directed definable family of non-empty closed sets has non-empty intersection), then it is regular (see [@andujar_thesis Lemma 5.4.7] for a proof). Furthermore, as noted earlier in Remark [Remark 12](#rem:compactness){reference-type="ref" reference="rem:compactness"}, the first author proved in [@ag_FTT] that condition [\[dfn:directed-compact\]](#dfn:directed-compact){reference-type="eqref" reference="dfn:directed-compact"} is equivalent to definable compactness for all Hausdorff definable topological spaces. Consequently, Lemma [Lemma 90](#lemma_Hausdorff_compact_is_regular){reference-type="ref" reference="lemma_Hausdorff_compact_is_regular"} can be generalized to spaces of any dimension. We now prove Theorem [Theorem 89](#them_compactification){reference-type="ref" reference="them_compactification"}. *Proof Theorem [Theorem 89](#them_compactification){reference-type="ref" reference="them_compactification"}.* Let $(X,\tau)$ be a Hausdorff definable topological space with $X \subseteq R$. Since the finite case is trivial, we assume that $\dim X=1$. The "if\" implication of the theorem follows directly from Lemma [Lemma 90](#lemma_Hausdorff_compact_is_regular){reference-type="ref" reference="lemma_Hausdorff_compact_is_regular"} using the observation that regularity is a hereditary property. We prove the "only if\" implication. Assume that $(X,\tau)$ is regular. We will make use of Lemmas [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"} and [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"} and Proposition [Proposition 88](#lemma_one_point_compact){reference-type="ref" reference="lemma_one_point_compact"} to construct a one-dimensional definable Hausdorff compactification for $(X,\tau)$. Recall the embedding $(Y\cup Z, \tau) \hookrightarrow (Y^*\cup Z^*, \tau_{lex}|_{Y^*}\cup \tau_{Alex}|_{Z^*})$ described in the proof of Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"}. As noted in Remark [Remark 86](#remark:ADC_or_DOTS_compactification){reference-type="ref" reference="remark:ADC_or_DOTS_compactification"}, this embedding is a definable near-compactfication of a cofinite $\tau$-open subspace of $X$. The idea of the current proof is to extend this embedding to an embedding of $(X,\tau)$ into a regular Hausdorff definably near-compact space $(X^*,\tau^*)$, with $X^*\subseteq R\times\{0,1,\ldots\}$. Applying Proposition [Proposition 88](#lemma_one_point_compact){reference-type="ref" reference="lemma_one_point_compact"} then completes the proof. Let $\mathcal{X}=\mathcal{X}_{\text{open}}\cup \mathcal{X}_{\text{sgl}}$ be a finite partition of $X$ as given by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}. For each $A\in\mathcal{X}_{\text{open}}$, let $A^*$, $\tau_A$ and $h_A$ be as given by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}. In particular, recall that each set $A^*$ is of the form $I\times\{0,\ldots,n\}$, for some interval $I\subseteq A$ and some $n$, and that $\tau_A$ is either the $\tau_{lex}$ or the $\tau_{Alex}$ topology on $A^*$. Moreover, $h_A:(A,\tau)\hookrightarrow (A^*,\tau_A)$ is a definable embedding. Let $X^*=\bigcup\{A^* : A\in\mathcal{X}_{\text{open}}\} \cup \{\langle x,0\rangle: \{x\}\in \mathcal{X}_{\text{sgl}}\}$ and $h=\bigcup\{ h_A : A\in \mathcal{X}_{\text{open}}\} \cup h'$, where $h'$ is the map with domain $F_{\text{sgl}}:= \bigcup\mathcal{X}_{\text{sgl}}$ given by $x\mapsto \langle x,0 \rangle$. Note that $h$ is injective. We construct a regular Hausdorff topology $\tau^*$ on $X^*$ such that, for every $A\in\mathcal{X}_{\text{open}}$, $A^*$ is $\tau^*$-open and $(A^*,\tau^*)=(A^*,\tau_A)$. Since every space $(A^*,\tau_A)$ is definably near-compact, it follows that $(X^*,\tau^*)$ is definably near-compact. We then prove that $h:(X,\tau)\hookrightarrow (X^*,\tau^*)$ is an embedding. Let $s=|\mathcal{X}_{\text{open}}|$. We define $\tau^*$ as follows. For every $x\in F_{\text{sgl}}$ and $A\in\mathcal{X}_{\text{open}}$, we first construct a downward directed definable family $\mathcal{B}_A(x)$ of $\tau_A$-open subsets of $A^*$. Then we use this to define, for each $x \in F_{\text{sgl}}$, $$\mathcal{B}(x):=\{ \{\langle x,0 \rangle\}\cup V_1\cup\cdots \cup V_s : (V_1,\ldots, V_s)\in \prod_{A\in\mathcal{X}_{\text{open}}} \mathcal{B}_A(x) \}.$$ It is then routine to check that the family $\bigcup\{ \mathcal{B}(x) : x\in F_{\text{sgl}}\} \cup \bigcup\{ \tau_A : A\in \mathcal{X}_{\text{open}}\}$ is a basis for a topology $\tau^*$ on $X^*$, which will clearly satisfy that $\tau^*|_{A^*}=\tau_A$, for every $A\in\mathcal{X}_{\text{open}}$. Since $F_{\text{sgl}}$ is finite and the topologies $\tau_A$ are definable, $\tau^*$ is also definable. It will then remain to check that $(X^*,\tau^*)$ is Hausdorff and regular, and that $h \colon (X,\tau) \hookrightarrow (X^*,\tau^*)$ is an embedding. We fix $x\in F_{\text{sgl}}$ and $A\in\mathcal{X}_{\text{open}}$ and describe $\mathcal{B}_A(x)$. Recall the notation $A=\bigcup_{0\leq i < n} f_i(I)$ from Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}, and let $I=(a,b)$. We define families of sets $V_a(x,y)$ and $V_b(x,y)$, as $y \in I$ varies, as follows, according to whether or not $a \in R_x$ and whether or not $b \in L_x$: $$V_a(x,y)= \begin{cases} ((a, y)\times R) \cap A^* &\text{for all } y \in I, \text{ if } a\in R_x,\\ \emptyset &\text{for all } y \in I, \text{ if } a\notin R_x, \end{cases}$$ $$V_b(x,y)= \begin{cases} ((y, b)\times R) \cap A^* &\text{for all } y \in I, \text{ if } b\in L_x,\\ \emptyset &\text{for all } y \in I, \text{ if } b\notin L_x. \end{cases}$$ Note that these sets are always open in $(A^*,\tau_A)$. Then $\mathcal{B}_A(x)$ is defined to be the family of sets $$V_A(x,y,z)=V_a(x,y) \cup V_b(x,z),$$ definable uniformly in $y$ and $z$ with $a<y<z<b$. Clearly, $\mathcal{B}_A(x)$ is a definable downward directed family of $\tau_A$-open subsets of $A^*$. We now consider the induced topology $\tau^*$ as described above. Since $\bigcap_{a<y<z<b} V_A(x,y,z)=\emptyset$, for each $A \in \mathcal{X}_{\text{open}}$, it is immediate from the definition that $\tau^*$ is $T_1$. We now show that $(X^*,\tau^*)$ is regular. It will follow (since in a $T_1$ topological space singletons are closed) that it is also Hausdorff. Consider the sets $V_A(x,y,z)$, as $x \in F_{\text{sgl}}$ varies, for some fixed $A \in \mathcal{X}_{\text{open}}$ and fixed $a<y<z<b$. By Hausdorffness of $\tau$, for any two distinct $x, x'\in F_{\text{sgl}}$, if $a\in R_x$, then $a\notin R_{x'}$ and, if $b\in L_x$, then $b\notin L_{x'}$, so $V_A(x,y,z)\cap V_A(x',y,z)=\emptyset$, from where it follows that, for all $x \in F_{\text{sgl}}$, $cl_{\tau^*} V_A(x,y,z)\subseteq \{\langle x,0\rangle\}\cup A^*$, and consequently $cl_{\tau^*} V_A(x,y,z)=\{\langle x,0\rangle\}\cup cl_{\tau_A} V_A(x,y,z)$. Moreover, note that, since $\tau_A$ is one of $\tau_{lex}$ or $\tau_{Alex}$, it holds, for all $x \in F_{\text{sgl}}$ and $y,z \in I$, that $cl_{\tau_A} V_a(x,y)\subseteq ((a,y]\times R)\cap A^*$ and $cl_{\tau_A} V_b(x,z)\subseteq ([z,b)\times R) \cap A^*$. So, for any $x \in F_{\text{sgl}}$, $a<y'<y$ and $z<z'<b$, we have that $cl_{\tau_A} V_A(x,y',z') \subseteq V_A(x,y,z)$. It follows that, for all $x \in F_{\text{sgl}}$ and $U\in\mathcal{B}(x)$, there exists $U'\in \mathcal{B}(x)$ such that $cl_{\tau^*} U' \subseteq U$. Since each $A^*$ is $\tau^*$-open with $(A^*,\tau^*)=(A^*,\tau_A)$, it is easy to check that the same property holds among $\tau^*$-neighbourhoods of points in $A^*$. It follows that the topology $\tau^*$ is regular. It remains to show that $h:(X,\tau)\hookrightarrow (X^*,\tau^*)$ is an embedding. We fix $x\in X$ and show continuity of $h$ and $h^{-1}$ at $x$ and $h(x)$ respectively. Since, for every $A\in\mathcal{X}_{\text{open}}$, $h_A:(A,\tau)\hookrightarrow (A^*,\tau_A)$ is an embedding between open subsets of $(X, \tau)$ and $(X^*, \tau^*)$ respectively, where $\tau^*|_{A^*}=\tau_A$, this holds whenever $x\in A$, for some $A\in\mathcal{X}_{\text{open}}$, so we may assume that $x\in F_{\text{sgl}}$. We make use of Proposition [Proposition 27](#prop_cont_lim){reference-type="ref" reference="prop_cont_lim"}. Fix $\gamma$, an injective definable curve in $X$, and set $\gamma':=h\circ \gamma$. We may assume that there is some fixed $A=\bigcup_{0\leq i<n} f_i(I) \in \mathcal{X}_{\text{open}}$ and $0\leq j <n$ such that $\gamma$ is contained in the interval $f_j(I)$. Recall from Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"} that, for every $0\leq i <n$, $f_i : I \rightarrow R$ is a definable $e$-continuous strictly monotonic function. We will prove the case where $f_j$ is increasing. The decreasing case is analogous. Let $I=(a,b)$ and $f_j(I)=(a_j,b_j)$. We require the following simple fact that follows, for each $x \in F_{\text{sgl}}$, from the definition of the definable families $\mathcal{B}_A(x)$ (recall that $\pi:R^2\rightarrow R$ denotes the projection to the first coordinate). $$\label{fact_proof_compactification_theorem_alt} \parbox{0.9\textwidth}{The curve $\gamma'$ $\tau^*$-converges to $h(x)=\langle x,0 \rangle$ if and only if either $a\in R_x$ and $\pi\circ\gamma'$ $e$-converges to $a$, or $b\in L_x$ and $\pi\circ \gamma'$ $e$-converges to $b$.}$$ Suppose that $\gamma$ $\tau$-converges to $x$. Recall that, by Lemmas [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}([\[itm:properties_cofnite_subset_1.5_b\]](#itm:properties_cofnite_subset_1.5_b){reference-type="ref" reference="itm:properties_cofnite_subset_1.5_b"}) and [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}, if $A\cap E_x \neq \emptyset$, then $x\in A$, a contradiction. So, by o-minimality and Proposition [Proposition 29](#basic_facts_P_x_1){reference-type="ref" reference="basic_facts_P_x_1"}([\[itm: basic_facts_3\]](#itm: basic_facts_3){reference-type="ref" reference="itm: basic_facts_3"}), $\gamma$ must $e$-converge to either $a_j$ or $b_j$. Suppose that $\gamma$ $e$-converges to $a_j$, in which case $a_j\in R_x$ (see Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}). Since $f_j$ is increasing, we have that $f_j^{-1}\circ\gamma$ $e$-converges to $a$. By regularity of $\tau$ and Lemmas [Lemma 37](#lemma_f_Rx_Lx){reference-type="ref" reference="lemma_f_Rx_Lx"} and [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}, it follows that $a \in R_x$. Now note that, by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}([\[itm1.5:lemma_proof_two_theorems\]](#itm1.5:lemma_proof_two_theorems){reference-type="ref" reference="itm1.5:lemma_proof_two_theorems"}), $\pi\circ \gamma'= \pi \circ h \circ \gamma =f^{-1}_j \circ \gamma$. Hence $\pi\circ\gamma'$ $e$-converges to $a$. So, by ([\[fact_proof_compactification_theorem_alt\]](#fact_proof_compactification_theorem_alt){reference-type="ref" reference="fact_proof_compactification_theorem_alt"}), we conclude that $\gamma'$ $\tau^*$-converges to $h(x)=\langle x,0\rangle$. Analogously, if $\gamma$ $e$-converges to $b_j$, then $b_j\in L_x$ and, again by regularity of $\tau$ and Lemmas [Lemma 37](#lemma_f_Rx_Lx){reference-type="ref" reference="lemma_f_Rx_Lx"} and [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}, $b\in L_x$. Moreover, again by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}([\[itm1.5:lemma_proof_two_theorems\]](#itm1.5:lemma_proof_two_theorems){reference-type="ref" reference="itm1.5:lemma_proof_two_theorems"}), $\pi \circ \gamma'$ $e$-converges to $b$ and so, by ([\[fact_proof_compactification_theorem_alt\]](#fact_proof_compactification_theorem_alt){reference-type="ref" reference="fact_proof_compactification_theorem_alt"}), $\gamma'$ $\tau^*$-converges to $h(x)=\langle x,0 \rangle$ in this case as well. Hence, by Proposition [Proposition 27](#prop_cont_lim){reference-type="ref" reference="prop_cont_lim"}, $h$ is continuous at $x$. Now suppose that $\gamma'$ $\tau^*$-converges to $h(x)$. By ([\[fact_proof_compactification_theorem_alt\]](#fact_proof_compactification_theorem_alt){reference-type="ref" reference="fact_proof_compactification_theorem_alt"}), there are two cases to consider: either $a \in R_x$ and $\pi \circ \gamma'$ $e$-converges to $a$, or $b \in L_x$ and $\pi \circ \gamma'$ $e$-converges to $b$. In the first case, since $f_j$ is increasing, by Lemmas [Lemma 37](#lemma_f_Rx_Lx){reference-type="ref" reference="lemma_f_Rx_Lx"} and [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"} we have that $a_j\in R_x$. Moreover, by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}([\[itm1.5:lemma_proof_two_theorems\]](#itm1.5:lemma_proof_two_theorems){reference-type="ref" reference="itm1.5:lemma_proof_two_theorems"}) and since $f_j$ is increasing, $\gamma=h^{-1} \circ\gamma'=f_j \circ \pi \circ \gamma'$ $e$-converges to $a_j$. We conclude that $h^{-1}\circ \gamma'$ $\tau$-converges to $x$, by Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}. In the other case it can analogously be shown that $h^{-1}\circ \gamma'$ $\tau$-converges to $x$. Hence $h^{-1}$ is continuous at $h(x)$. This completes the proof of the theorem. ◻ **Remark 91**. Theorem [Theorem 89](#them_compactification){reference-type="ref" reference="them_compactification"} may be generalized to all Hausdorff definable topological spaces of dimension at most one. In particular, using Remark [\[remark_general_them_ADC_or DOTS\]](#remark_general_them_ADC_or DOTS){reference-type="ref" reference="remark_general_them_ADC_or DOTS"}, one may note that every $T_3$ one-dimensional space $(X,\tau)$ has a cofinite subspace that embeds definably into a finite disjoint union of spaces of the form $(I\times\{0,\ldots\}, \tau_{lex})$ or $(I\times\{0,\ldots\}, \tau_{Alex})$, where $I\subseteq R$ is an interval and $m\geq 0$. Then, using a construction similar to the one in the proof of Theorem [Theorem 89](#them_compactification){reference-type="ref" reference="them_compactification"}, one may expand this disjoint union, by adding finitely many points, to a definably near-compact space $(X^*,\tau^*)$, and extend the embedding of a cofinite subspace of $(X,\tau)$ into an embedding $(X,\tau) \hookrightarrow (X^*,\tau^*)$. Finally, by applying Proposition [Proposition 88](#lemma_one_point_compact){reference-type="ref" reference="lemma_one_point_compact"} to $(X^*,\tau^*)$, one reaches a definable compactification of $(X,\tau)$. # Affine topologies {#section: affine} Unless stated otherwise, throughout this section we assume that $\mathcal{R}$ expands an ordered field. We restrict to this setting to be able to assume, using Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"}, that, up to definable homeomorphism, every definable topological space $(X,\tau)$ satisfies that $X$ is a bounded set. We will also use in this section (in the proof of Lemma [Lemma 96](#lemma_walsberg_homeomorphism){reference-type="ref" reference="lemma_walsberg_homeomorphism"}) facts about o-minimal expansions of ordered fields from [@dries98 Chapter 10]. We wish to classify the one-dimensional definable topologies that are, up to definable homeomorphism, euclidean. We will refer to these topologies as *affine*[^1]. Our main result is the following, which will utilize results of the previous sections, in particular the decomposition of Hausdorff one-dimensional definable topological spaces (Corollary [Corollary 52](#cor 1.5){reference-type="ref" reference="cor 1.5"}) and the Hausdorff compactification of certain one-dimensional $T_3$ definable topological spaces (Proposition [Proposition 88](#lemma_one_point_compact){reference-type="ref" reference="lemma_one_point_compact"}). Later in the section we will use all of these results to address in our setting several questions of Fremlin and Gruenhage on the nature of perfectly normal, compact, Hausdorff spaces (see Subsection [9.1](#subsection: Fremlin){reference-type="ref" reference="subsection: Fremlin"}). **Theorem 92**. Suppose that $\mathcal{R}$ expands an ordered field. Let $(X,\tau)$, $\dim X\leq 1$, be a Hausdorff definable topological space. Exactly one of the following holds. (1) $(X,\tau)$ contains a subspace definably homeomorphic to an interval with either the discrete or the right half-open interval topology. [\[itm:them_main_2\_1\]]{#itm:them_main_2_1 label="itm:them_main_2_1"} (2) $(X,\tau)$ is definably homeomorphic to a euclidean space (i.e. $(X,\tau)$ is affine). [\[itm:them_main_2\_2\]]{#itm:them_main_2_2 label="itm:them_main_2_2"} Note that, since the map $x\mapsto -x$ is a homeomorphism $(R,\tau_r)\rightarrow (R,\tau_l)$, Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"} still holds if we replace the right half-open interval topology by the left half-open interval topology in ([\[itm:them_main_2\_1\]](#itm:them_main_2_1){reference-type="ref" reference="itm:them_main_2_1"}). The main theorem (Theorem 7.1) in [@walsberg15] states that if a definable metric space contains no infinite definable discrete subspace (equivalently, by Lemma [Lemma 18](#remark_def_sep){reference-type="ref" reference="remark_def_sep"}, if it is definably separable), then it is affine. Hence, taking into account this result and the fact that the euclidean topology $\tau_e$ is definably separable (Proposition [Proposition 8](#prop:def-sep_eucl_disc_llt){reference-type="ref" reference="prop:def-sep_eucl_disc_llt"}([\[itm1:sep-spaces-examples\]](#itm1:sep-spaces-examples){reference-type="ref" reference="itm1:sep-spaces-examples"})) and definably metrizable, statement ([\[itm:them_main_2\_2\]](#itm:them_main_2_2){reference-type="ref" reference="itm:them_main_2_2"}) in Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"} can be changed to "$(X,\tau)$ is definably separable and definably metrizable\". **Remark 93**. Recall that, for any interval $I\subseteq R$, the space $(I,\mu)$, where $\mu\in \{\tau_r, \tau_s\}$, is totally definably disconnected (i.e. singletons are the only definably connected non-empty subspaces). On the other hand, by o-minimality, every euclidean space has finitely many definably connected components. Hence Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"} implies that a Hausdorff one-dimensional definable topological space $(X,\tau)$ is affine if and only if every definable topological subspace has finitely many definably connected components (or is not totally definably disconnected). A definition, a remark and a lemma precede the proof of Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}. **Definition 94**.  [\[dfn:cell-wise_euclidean\]]{#dfn:cell-wise_euclidean label="dfn:cell-wise_euclidean"} We say that a definable topological space $(X,\tau)$ is *cell-wise euclidean* if there is a finite partition $\mathcal{X}$ of $X$ into cells such that, for each $C\in \mathcal{X}$, $(C,\tau)=(C,\tau_e)$. By o-minimal cell decomposition, we can clearly relax the requirement in Definition [\[dfn:cell-wise_euclidean\]](#dfn:cell-wise_euclidean){reference-type="ref" reference="dfn:cell-wise_euclidean"} that the sets in $\mathcal{X}$ be cells to just that they be any definable subset of $X$. Furthermore, since it also follows by o-minimal cell decomposition that every definable bijection is a finite union of disjoint definable $e$-homeomorphisms, the property of being cell-wise euclidean is maintained by definable homeomorphism, and is equivalent to being cell-wise affine (i.e. the property that $X$ admits a finite partition into cells on which the subspace topology is affine). In particular, any affine space is cell-wise euclidean. Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"} implies that the converse holds for one-dimensional Hausdorff definable topological spaces. This statement cannot be generalized to definable topological spaces of all dimensions, as illustrated by Appendix [12](#section:examples){reference-type="ref" reference="section:examples"}, Example [Example 130](#example_space_cell-wise_euclidean_not_metrizable){reference-type="ref" reference="example_space_cell-wise_euclidean_not_metrizable"}, which describes a two-dimensional definable topological space that is $T_3$ and cell-wise euclidean but not even definably metrizable (and hence not affine). Moreover, in Section [12](#section:examples){reference-type="ref" reference="section:examples"}, Example [Example 122](#example: T_1 fdi not regular){reference-type="ref" reference="example: T_1 fdi not regular"}, we produce a cell-wise euclidean one-dimensional definable topological space that is not Hausdorff, and in particular not affine. **Remark 95**. By Corollary [Corollary 52](#cor 1.5){reference-type="ref" reference="cor 1.5"} (and the fact that the map $x\mapsto -x$ is a definable homeomorphism $(R,\tau_l)\rightarrow (R,\tau_r)$), if a Hausdorff definable topological space $(X,\tau)$, with $\dim X \leq 1$, does not have a subspace that is definably homeomorphic to an interval with either the $\tau_r$ or the $\tau_s$ topology, then $(X,\tau)$ is cell-wise euclidean. By the above remark, in order to prove that the negation of [\[itm:them_main_2\_1\]](#itm:them_main_2_1){reference-type="eqref" reference="itm:them_main_2_1"} implies [\[itm:them_main_2\_2\]](#itm:them_main_2_2){reference-type="eqref" reference="itm:them_main_2_2"} in Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}, it suffices to show that if a Hausdorff one-dimensional space is cell-wise euclidean then it is affine. The following lemma is essentially Lemma 5.7 in [@walsberg15], proved for definable metric spaces, which we extend to one-dimensional spaces using Lemma [Lemma 28](#lemma:compact_homeomorphism){reference-type="ref" reference="lemma:compact_homeomorphism"} and results of van den Dries [@dries98], one of which requires the setting of an o-minimal expansion of an ordered field. Recall that a euclidean space is definably compact if and only if it is closed and bounded. **Lemma 96**. Suppose that $\mathcal{R}$ expands an ordered field. Let $(X,\tau)$ be a definably compact Hausdorff definable topological space. Let $(Y,\tau_e)$ be a definably compact euclidean space that admits a definable continuous surjection $f:(Y,\tau_e) \rightarrow (X,\tau)$. Then there exists a definable set $Z$ and a definable homeomorphism $(Z,\tau_e)\rightarrow (X,\tau)$. *Proof.* Let $E$ be the kernel of $f$, namely $E=\{\langle x,y \rangle\in Y^2 : f(x)=f(y)\}$. By continuity of $f$, $E$ is closed in $Y^2$, and so definably compact. By [@dries98 Chapter 10, Corollary 2.16], there exists a definable set $Z$ and a definable quotient map $g:(Y,\tau_e)\rightarrow (Z,\tau_e)$ of $E$, i.e. $g$ has kernel $E$, is surjective, continuous and, for every $C\subseteq Z$, if $g^{-1}(C)$ is closed in $(Y,\tau_e)$, then $C$ is closed in $(Z,\tau_e)$. Moreover, by [@dries98 Chapter 6, Proposition 1.10], the space $(Z,\tau_e)$ is definably compact. The definable map $h:(Z,\tau_e)\rightarrow (X,\tau)$ given by $h(g(x))= f(x)$ is well-defined, and clearly continuous and bijective. By Lemma [Lemma 28](#lemma:compact_homeomorphism){reference-type="ref" reference="lemma:compact_homeomorphism"}, it is a homeomorphism. ◻ We may now prove Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}. *Proof of Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}.* Let $(X,\tau)$ be a Hausdorff definable topological space. The case where $\dim(X)=0$ is trivial (as ([\[itm:them_main_2\_2\]](#itm:them_main_2_2){reference-type="ref" reference="itm:them_main_2_2"}) trivially holds in this case) and so we assume that $\dim(X)=1$. By Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"}, we may assume that $X\subseteq R$ and is bounded. Note that, by Remark [Remark 4](#remark_no_homeomorphism){reference-type="ref" reference="remark_no_homeomorphism"}, $(X,\tau)$ cannot be both cell-wise euclidean and have a definable copy of an interval with either the $\tau_r$ or the $\tau_s$ topology, so ([\[itm:them_main_2\_1\]](#itm:them_main_2_1){reference-type="ref" reference="itm:them_main_2_1"}) and ([\[itm:them_main_2\_2\]](#itm:them_main_2_2){reference-type="ref" reference="itm:them_main_2_2"}) in the statement of the theorem are mutually exclusive. Applying Remark [Remark 95](#remark_cell-wise_euclidean){reference-type="ref" reference="remark_cell-wise_euclidean"}, we assume that $(X,\tau)$ is cell-wise euclidean and derive that it is affine. Since $(X,\tau)$ is cell-wise euclidean, it is also definably near-compact so, by Proposition [Proposition 88](#lemma_one_point_compact){reference-type="ref" reference="lemma_one_point_compact"}, by passing to $(X^c,\tau^c)$ if necessary, we may assume that $(X,\tau)$ is definably compact. Let $\mathcal{X}$ be a partition of $X$ into points and intervals such that, for each $C \in \mathcal{X}$, the subspace $(C,\tau)$ is euclidean. We define, for each $C\in \mathcal{X}$, a continuous function $f_C: (cl_e C,\tau_e) \rightarrow (cl_\tau C, \tau)$ extending the identity on $C$. Once we have defined these functions, we complete the proof as follows. Let $num:\mathcal{X}\rightarrow \omega$ be an enumeration of the elements in $\mathcal{X}$ and let $Y=\bigcup_{C\in\mathcal{X}} (cl_{e} C \times \{num(C)\})$ be the disjoint union of the euclidean closures of the sets in $\mathcal{X}$. Clearly, $(Y,\tau_e)$ is definably compact. Let $f: (Y,\tau_e) \rightarrow (X,\tau)$ be the function given by $f(x,num(C))=f_C(x)$, where $x\in cl_e C$. This function is clearly definable, surjective and continuous. The result then follows by Lemma [Lemma 96](#lemma_walsberg_homeomorphism){reference-type="ref" reference="lemma_walsberg_homeomorphism"}. It remains to define, for each $C\in \mathcal{X}$, the function $f_C$. If $C\in \mathcal{X}$ is a singleton, let $f_C$ simply be the identity. Now let us fix an interval $C=I=(a_I,b_I)\in \mathcal{X}$. By Hausdorffness (Proposition [Proposition 32](#prop_basic_facts_P_x_2){reference-type="ref" reference="prop_basic_facts_P_x_2"}([\[itm3:basic_facts_P\_x_2\]](#itm3:basic_facts_P_x_2){reference-type="ref" reference="itm3:basic_facts_P_x_2"})) and definable compactness (Lemma [Lemma 35](#lem:RL-compact){reference-type="ref" reference="lem:RL-compact"}), there exists a unique point $x_I\in X$ such that $a_I\in R_{x_I}$ and similarly a unique point $y_I\in X$ such that $b_I\in L_{y_I}$. Note that, since $(I,\tau)=(I,\tau_e)$, the points $x_I$ and $y_I$ do not belong in $I$. Let $f_I$ be defined as $$f_I|_I = id, \, f(a_I)=x_I \text{ and } f(b_I)=y_I.$$ It is routine to check that $f_I$ is continuous as a map $([a_I,b_I], \tau_e) \rightarrow (cl_\tau I,\tau)$ ◻ In Appendix [12](#section:examples){reference-type="ref" reference="section:examples"}, Example [Example 128](#example_line-wise_euclidean_not_euclidean){reference-type="ref" reference="example_line-wise_euclidean_not_euclidean"} we describe a Hausdorff definable topological space of dimension two that has no definable copy of an interval with either the $\tau_s$ or the $\tau_r$ topology but fails to be cell-wise euclidean. In Appendix [12](#section:examples){reference-type="ref" reference="section:examples"}, Example [Example 130](#example_space_cell-wise_euclidean_not_metrizable){reference-type="ref" reference="example_space_cell-wise_euclidean_not_metrizable"}, we describe, as already noted, a $T_3$ definable topological space of dimension two that is cell-wise euclidean but not affine. Hence, although equivalent for one-dimensional definable topological spaces, the following three implications are strict in general. -------- --------------- --------------------- --------------- ----------------------------------------------- Affine $\Rightarrow$ Hausdorff and $\Rightarrow$ Hausdorff and does not contain cell-wise euclidean a definable copy of an interval with either the $\tau_r$ or $\tau_s$ topology -------- --------------- --------------------- --------------- ----------------------------------------------- In fact it is not even the case that being Hausdorff and cell-wise euclidean implies being definably metrizable, since, by o-minimality, a cell-wise euclidean space cannot contain an infinite definable discrete subspace, and so, by [@walsberg15 Theorem 7.1], every cell-wise euclidean definably metrizable space is affine. This complicates the task of generalising Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"} to spaces of all dimensions. The next corollary however offers a possibility. **Corollary 97**. Suppose that $\mathcal{R}$ expands an ordered field. Let $(X,\tau)$, $\dim X\leq 1$, be a definably compact Hausdorff definable topological space. The following are equivalent. (1) $(X,\tau)$ satisfies the frontier dimension inequality. [\[itm:cor_them_2\_1\]]{#itm:cor_them_2_1 label="itm:cor_them_2_1"} (2) $(X,\tau)$ is definably metrizable. [\[itm:cor_them_2\_2\]]{#itm:cor_them_2_2 label="itm:cor_them_2_2"} (3) $(X,\tau)$ is affine. [\[itm:cor_them_2\_3\]]{#itm:cor_them_2_3 label="itm:cor_them_2_3"} *Proof.* We fix $(X,\tau)$, $\dim X\leq 1$, a definably compact Hausdorff definable topological space. $(\ref{itm:cor_them_2_3})\Rightarrow (\ref{itm:cor_them_2_2})$ is trivial. If $(X,\tau)$ is definably metrizable, then, by [@walsberg15 Lemma 7.15], it satisfies the **fdi**, i.e. $(\ref{itm:cor_them_2_2})\Rightarrow (\ref{itm:cor_them_2_1})$. We complete the proof by showing $(\ref{itm:cor_them_2_1})\Rightarrow (\ref{itm:cor_them_2_3})$, that is, if $(X,\tau)$ satisfies the **fdi**, then it is affine. We prove the contrapositive. Suppose that $(X, \tau)$ is not affine. Then, by Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}, there exists an interval with either the $\tau_r$ or the $\tau_s$ topology that definably embeds into $(X, \tau)$. We prove that $(X, \tau)$ does not have the **fdi**. By Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"}, we may assume that $X\subseteq R$. By Lemma [Lemma 53](#remark_classification_spaces_line){reference-type="ref" reference="remark_classification_spaces_line"}, there exists an interval $I\subseteq X$ such that $\tau|_I\in\{\tau_r, \tau_l, \tau_s\}$. Considering the push-forward of $(X,\tau)$ by $x\mapsto -x$ if necessary, we may assume that $\tau|_I\in\{\tau_r, \tau_s\}$. By definable compactness and Hausdorffness, for every $y\in I$, there exists a unique $x \in X$ such that $y\in L_x$ (see Proposition [Proposition 32](#prop_basic_facts_P_x_2){reference-type="ref" reference="prop_basic_facts_P_x_2"}([\[itm3:basic_facts_P\_x_2\]](#itm3:basic_facts_P_x_2){reference-type="ref" reference="itm3:basic_facts_P_x_2"}) and Lemma [Lemma 35](#lem:RL-compact){reference-type="ref" reference="lem:RL-compact"}). Since $\tau|_I\in\{\tau_r, \tau_s\}$, we have that this $x$ must not belong in $I$, and in particular $x\in \partial_\tau I$. By Lemma [Lemma 23](#lemma_2){reference-type="ref" reference="lemma_2"}, it follows that $\partial_\tau I$ is infinite, and so $(X,\tau)$ does not satsify the **fdi**. ◻ Using our work in Sections [6](#section: universal spaces){reference-type="ref" reference="section: universal spaces"} and [8](#section:compactifications){reference-type="ref" reference="section:compactifications"}, we present a second refinement of Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"} for definably compact spaces. We do not know whether or not this result generalizes to spaces of any dimension. **Corollary 98**. Suppose that $\mathcal{R}$ expands an ordered field. Let $(X,\tau)$, $\dim X\leq 1$, be a definably compact Hausdorff definable topological space. Exactly one of the following holds. (1) [\[itm1:compact-affine\]]{#itm1:compact-affine label="itm1:compact-affine"} There exists an interval $I\subseteq R$, some $n>0$, and a definable open embedding $(I\times\{0,\ldots, n\}, \mu)\hookrightarrow (X,\tau)$, where $\mu\in \{\tau_{lex}, \tau_{Alex}\}$. (2) [\[itm2:compact-affine\]]{#itm2:compact-affine label="itm2:compact-affine"} $(X,\tau)$ is affine. Furthermore, if $(X,\tau)$ is definably separable, then we can take $n$ to be $1$ and $\mu$ to be $\tau_{lex}$ in (1). *Proof.* Applying Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"}, we may assume that $X\subseteq R$. On the one hand, if $(X,\tau)$ is affine, then in particular it is cell-wise euclidean (see comments after Definition [\[dfn:cell-wise_euclidean\]](#dfn:cell-wise_euclidean){reference-type="ref" reference="dfn:cell-wise_euclidean"}). On the other hand, note that, for any interval $I$ and $n>0$, the space $(I\times\{0,\ldots, n\}, \mu)$, where $\mu$ is $\tau_{lex}$ (respectively $\tau_{Alex}$), satisfies that the subspace $I\times\{n\}$ is homeomorphic, taking the projection to the first coordinate, to the interval $I$ with the $\tau_r$ (respectively $\tau_s$) topology. Hence, if ([\[itm1:compact-affine\]](#itm1:compact-affine){reference-type="ref" reference="itm1:compact-affine"}) in the corollary holds, then $(X,\tau)$ contains a subspace definably homeomorphic to an interval with the $\tau_r$ or $\tau_s$ topology, and so, by Remark [Remark 4](#remark_no_homeomorphism){reference-type="ref" reference="remark_no_homeomorphism"}, ([\[itm1:compact-affine\]](#itm1:compact-affine){reference-type="ref" reference="itm1:compact-affine"}) and ([\[itm2:compact-affine\]](#itm2:compact-affine){reference-type="ref" reference="itm2:compact-affine"}) are mutually exclusive. By Lemma [Lemma 90](#lemma_Hausdorff_compact_is_regular){reference-type="ref" reference="lemma_Hausdorff_compact_is_regular"}, the space $(X,\tau)$ is regular. Let $\mathcal{X}=\mathcal{X}_{\text{open}}\cup \mathcal{X}_{\text{sgl}}$ be a partition of $X$ as defined by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"} and, for every $A\in \mathcal{X}_{\text{open}}$, let $A^*$, $\tau_A$ and $h_A$ be as given by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}. Since $(X,\tau)$ is definably compact, by Lemma [Lemma 87](#prop_proof_two_thems){reference-type="ref" reference="prop_proof_two_thems"} we have that, for every $A\in \mathcal{X}_{\text{open}}$, $h_A:(A,\tau)\rightarrow (A^*,\tau_A)$ is a homeomorphism. Furthermore, since every set in $\mathcal{X}_{\text{open}}$ is $\tau$-open, we have that, for every $A\in\mathcal{X}_{\text{open}}$, the map $h_A^{-1}:(A^*,\tau_A)\hookrightarrow (X,\tau)$ is an open embedding. Now recall from Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"} that, for every $A\in \mathcal{X}_{\text{open}}$, the set $A^*$ is of the form $I\times\{0,\ldots, n\}$, for some interval $I\subseteq R$ and some $n=n_A\geq 0$, and moreover $\tau_A$ is one of $\tau_{lex}$ or $\tau_{Alex}$. On the one hand, if $n_A>0$ for some $A\in \mathcal{X}_{\text{open}}$, then we clearly have condition ([\[itm1:compact-affine\]](#itm1:compact-affine){reference-type="ref" reference="itm1:compact-affine"}) in the corollary. On the other hand, if $n_A=0$ for every $A\in \mathcal{X}_{\text{open}}$ then, using the fact that, for any interval $I$ it holds that $(I\times\{0\}, \tau_{lex})=(I\times\{0\}, \tau_{Alex})=(I\times\{0\}, \tau_e)$, we derive that $(X,\tau)$ is cell-wise euclidean (see comments after Definition [\[dfn:cell-wise_euclidean\]](#dfn:cell-wise_euclidean){reference-type="ref" reference="dfn:cell-wise_euclidean"}), and so, by the proof of Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}, it is affine. It remains to show that, if $(X,\tau)$ is definably separable, then we may take $n$ to be $1$ and $\mu$ to be $\tau_{lex}$ in ([\[itm1:compact-affine\]](#itm1:compact-affine){reference-type="ref" reference="itm1:compact-affine"}). By the above paragraph, it suffices to show that, if $(X,\tau)$ is definably separable, then $n_A\leq 1$ and $\tau_A=\tau_{lex}$ for every $A\in\mathcal{X}_{\text{open}}$. This however is shown in the proof of Corollary [Corollary 65](#cor_T3_separable_embedding){reference-type="ref" reference="cor_T3_separable_embedding"}. ◻ ## Fremlin's conjecture {#subsection: Fremlin} As indicated in Subsection [5.1](#subsection: 3-el_basis_conj){reference-type="ref" reference="subsection: 3-el_basis_conj"}, the 3-element basis conjecture (which asserts that statement [\[star\]](#star){reference-type="eqref" reference="star"} is consistent with ZFC) arose as a result of interest in various questions about the nature of perfectly normal, compact, Hausdorff spaces. (Recall that a topological space is perfectly normal if and only if it is normal and any open set is a countable union of closed sets.) In particular, this conjecture is closely related to a conjecture of Fremlin, which posits that the following statement is consistent with ZFC (see Question 1 in [@gm07]): $$\label{Fremlin} \parbox{0.9\textwidth}{Every perfectly normal, compact, Hausdorff space admits a continuous, at most $2$-to-$1$ map onto a compact metric space.}$$ The underlying question is whether or not the Split Interval, which in our setting is $([0,1] \times \{0, 1\}, \tau_{lex})$ defined in $(\mathbb{R},<)$ (see Example [Example 117](#example:split_interval){reference-type="ref" reference="example:split_interval"}), is in essence the only example in ZFC of a non-metrizable, perfectly normal, compact, Hausdorff space, and how to make that notion precise. Fremlin's initial conjecture (see [@gm07]) had been that every perfectly normal, compact, Hausdorff space in ZFC is a continuous image of $([0,1] \times \{0, 1\}, \tau_{lex}) \times ([0,1],\tau_e)$ (where likewise, in our setting, this space should be understood as being defined in $(\mathbb{R},<)$), but this does not hold (see [@watsonweiss88]). However, $([0,1] \times \{0, 1\}, \tau_{lex})$ certainly admits a continuous, at most 2-to-1 map onto a compact metric space, as does the counterexample of [@watsonweiss88]. In considering Fremlin's conjecture, Gruenhage put forward the following question that is in the same spirit in [@gru88] (see also Question 2.2 in [@gru90]). He asked if the following statement is consistent with ZFC: $$\label{Fremlin2.2} \parbox{0.9\textwidth}{Every non-metrizable, perfectly normal, compact, Hausdorff space contains a copy of $(A \times \{0,1\}, \tau_{lex})$, for some uncountable $A \subseteq [0,1]$.}$$ It is indicated in [@gru90] and [@gm07] that, under PFA, statement [\[Fremlin\]](#Fremlin){reference-type="eqref" reference="Fremlin"} is equivalent to the existence of a 3-element basis, as posited in [\[star\]](#star){reference-type="eqref" reference="star"}, for the class of subspaces of perfectly normal, compact, Hausdorff spaces, and furthermore that, under PFA, both of these statements imply [\[Fremlin2.2\]](#Fremlin2.2){reference-type="eqref" reference="Fremlin2.2"}. It is also indicated that, even without assuming PFA, statement [\[star\]](#star){reference-type="eqref" reference="star"} implies both [\[Fremlin\]](#Fremlin){reference-type="eqref" reference="Fremlin"} and [\[Fremlin2.2\]](#Fremlin2.2){reference-type="eqref" reference="Fremlin2.2"}. Here, we consider the questions above and show, as a corollary to Theorems [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"} and [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}, that statement [\[Fremlin\]](#Fremlin){reference-type="eqref" reference="Fremlin"} holds in a definable sense for any regular, Hausdorff one-dimensional topological space which is either perfectly normal or separable and which is definable in any o-minimal expansion of $(\mathbb{R}, +, \cdot, <)$ (Corollary [Corollary 101](#cor_Fremlin_2-2){reference-type="ref" reference="cor_Fremlin_2-2"}). Moreover, we address statement [\[Fremlin2.2\]](#Fremlin2.2){reference-type="eqref" reference="Fremlin2.2"} by showing that any one-dimensional, perfectly normal, compact, Hausdorff space definable in any o-minimal expansion of $(\mathbb{R}, +, \cdot, <)$ is either affine or there exists an interval $I\subseteq \mathbb{R}$ and a definable open embedding $(I\times\{0,1\}, \tau_{lex}) \hookrightarrow (X,\tau)$ (Corollary [Corollary 102](#cor:Fremlin-Q2.2){reference-type="ref" reference="cor:Fremlin-Q2.2"}). We also address definable generalizations of some of these statements for o-minimal expansions of ordered fields. We begin by proving a definable result in our setting that is closely related to statement [\[Fremlin\]](#Fremlin){reference-type="eqref" reference="Fremlin"}. We then specifically address statement [\[Fremlin\]](#Fremlin){reference-type="eqref" reference="Fremlin"} in our setting, whenever $\mathcal{R}$ expands the field of reals, through a subsequent lemma and corollary. **Corollary 99**. Suppose that $\mathcal{R}$ expands an ordered field. Let $(X,\tau)$, $\dim \leq 1$, be a regular and Hausdorff definable topological space. If $(X,\tau)$ is definably separable, then there exists a definable continuous map $f:(X,\tau)\rightarrow (R^n,\tau_e)$, for some $n$, where $f$ is at most $2$-to-$1$ (i.e. $|f^{-1}(z)|\leq 2$ for every $z\in R^n$). *Proof.* Applying Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"} we may assume that $X\subseteq R$. We will use the construction in the proof of Theorem [Theorem 89](#them_compactification){reference-type="ref" reference="them_compactification"} of a (definably near-compact) space $(X^*,\tau^*)$ and of a definable embedding $h:(X,\tau) \hookrightarrow (X^*,\tau^*)$. We show that, if $(X,\tau)$ is definably separable, then $(X^*,\tau^*)$ admits a definable at most $2$-to-$1$ continuous map into a Hausdorff cell-wise euclidean space $(Y,\mu)$, with $Y\subseteq R$. Since $(Y,\mu)$ is Hausdorff and cell-wise euclidean it follows from Remark [Remark 4](#remark_no_homeomorphism){reference-type="ref" reference="remark_no_homeomorphism"} and Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"} that it is affine, completing the proof. Let $\mathcal{X}=\mathcal{X}_{\text{open}}\cup \mathcal{X}_{\text{sgl}}$ be a partition of $X$ as defined by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"} and, for every $A\in \mathcal{X}_{\text{open}}$, let $A^*$ and $\tau_A$ be as given by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}. Set $F_{\text{sgl}}:=\bigcup \mathcal{X}_{\text{sgl}}$, and let $(X^*, \tau^*)$ be as given by the proof of Theorem [Theorem 89](#them_compactification){reference-type="ref" reference="them_compactification"}. Recall that $X^*=\bigcup\{A^* : A\in \mathcal{X}_{\text{open}}\} \cup \{\langle x, 0\rangle: x\in F_{\text{sgl}}\}$. Now, in the proof of Corollary [Corollary 65](#cor_T3_separable_embedding){reference-type="ref" reference="cor_T3_separable_embedding"} it is shown that, if $(X,\tau)$ is definably separable, then, for every $A\in \mathcal{X}_{\text{open}}$, it holds that $A^*\subseteq R\times\{0,1\}$ and $(A^*,\tau_A)=(A^*,\tau_{lex})$. In particular, we have that $X^*\subseteq R\times\{0,1\}$. Consider the projection $\pi:X^*\rightarrow R$ to the first coordinate. Since $X^*\subseteq R\times\{0,1\}$, then $\pi$ is at most $2$-to-$1$. Let $Y=\pi(X^*)$. We define a Hausdorff cell-wise euclidean topology $\mu$ on $Y$ such that $\pi:(X^*,\tau^*)\rightarrow (Y,\mu)$ is continuous. For every $A\in \mathcal{X}_{\text{open}}$, recall that the set $A^*$ is of the form $I\times\{0,1\}$ for some interval $I\subseteq A$. We denote this interval by $(a_A,b_A)$. In particular, for every $A\in \mathcal{X}_{\text{open}}$, we have that $\pi(A^*)=(a_A,b_A)$, and $\mathcal{X}_{\text{sgl}}\cup \{ (a_A, b_A) : A\in \mathcal{X}_{\text{open}}\}$ is a partition of $Y$. We define $\mu$ as follows. For every $A\in \mathcal{X}_{\text{open}}$, the interval $(a_A, b_A)$ is $\mu$-open and its subspace topology is euclidean. Furthermore, for every $x\in F_{\text{sgl}}$, we say that a set $V\subseteq Y$ is a $\mu$-neighbourhood of $x$ if and only if $\pi^{-1}(V)$ is a $\tau^*$-neighbourhood of $\langle x, 0\rangle$. We give a precise description of a basis of neighbourhoods in $(Y,\mu)$ for each point in $F_{\text{sgl}}$. Let $x\in F_{\text{sgl}}$ and let $\mathcal{B}(x)$ denote the definable basis of neighbourhoods of the point $\langle x, 0 \rangle$ in $(X^*,\tau^*)$ described in the proof of Theorem [Theorem 89](#them_compactification){reference-type="ref" reference="them_compactification"}. Then a definable basis of neighbourhoods of $x$ in $(Y,\mu)$ is given by the family $\mathcal{U}(x)=$ $\{\pi(V) : V\in \mathcal{B}(x)\}$. We can describe $\mathcal{U}(x)$ more precisely using the definition of $\mathcal{B}(x)$ as follows. Let $\mathcal{X}_{\text{open}}^{R_x} = \{ A \in \mathcal{X}_{\text{open}}: a_A \in R_x\}$ and $\mathcal{X}_{\text{open}}^{L_x} = \{ A \in \mathcal{X}_{\text{open}}: b_A \in L_x\}$. Then $\mathcal{U}(x)$ consists of every set of the form $$\{x\} \cup \bigcup_{A \in \mathcal{X}_{\text{open}}^{R_x}} (a_A,y_A) \cup \bigcup_{A \in \mathcal{X}_{\text{open}}^{L_x}} (z_A, b_A),$$ for parameters $\{y_A : A \in \mathcal{X}_{\text{open}}^{R_x}\}$ and $\{ z_A : A \in \mathcal{X}_{\text{open}}^{L_x}\}$ satisfying that, for every $A \in \mathcal{X}_{\text{open}}^{R_x}$, we have $y_A \in (a_A, b_A)$ and, for every $A \in \mathcal{X}_{\text{open}}^{L_x}$, we have $z_A \in (a_A, b_A)$. It is thus easy to check that the topology $\mu$ is well defined and, by Proposition [Proposition 32](#prop_basic_facts_P_x_2){reference-type="ref" reference="prop_basic_facts_P_x_2"}([\[itm3:basic_facts_P\_x_2\]](#itm3:basic_facts_P_x_2){reference-type="ref" reference="itm3:basic_facts_P_x_2"}), Hausdorff. Furthermore it is cell-wise euclidean by definition. It remains to show that $\pi:(X^*,\tau^*)\rightarrow (Y,\mu)$ is continuous; in other words, that, for every point $\langle x, i\rangle\in X^*$ and every $\mu$-neighbourhood $V$ of $\pi(\langle x,i \rangle)=x$, the set $\pi^{-1}(V)$ is a $\tau^*$-neighbourhood of $\langle x, i \rangle$. We observe that this follows easily from the definitions of $(X^*,\tau^*)$ and $(Y,\mu)$. Specifically, if $x\in F_{\text{sgl}}$ then, by definition of $X^*$, we have that $i=0$, and the observation is explicit in the definition of $\mu$. For the remaining case, suppose that $x\in (a_A,b_A)$ for some $A\in \mathcal{X}_{\text{open}}$. By definition of $\tau_{lex}$ note that every interval $J\subseteq (a_A,b_A)$ satisfies that $\pi^{-1}(J)$ is open in $(A^*,\tau_{lex})$. Hence the observation follows from the facts that $((a_A,b_A),\mu)=((a_A,b_A),\tau_e)$, and $A^*$ is $\tau^*$-open with $(A^*,\tau^*)=(A^*,\tau_{lex})$. ◻ Through the following lemma and corollary we prove a stronger form of statement [\[Fremlin\]](#Fremlin){reference-type="eqref" reference="Fremlin"} in our setting, where we do not require the assumption that the space be compact, and furthermore we may replace perfect normality by separability. Since the proof of Corollary [Corollary 101](#cor_Fremlin_2-2){reference-type="ref" reference="cor_Fremlin_2-2"} is similar to the proof of Corollary [Corollary 99](#cor_Fremlin_2-1){reference-type="ref" reference="cor_Fremlin_2-1"} above, we only sketch its proof. **Lemma 100**. Let $I\subseteq \mathbb{R}$ be an interval and $0 \leq n \leq m$. The subspace $I\times\{0,\ldots,n\}$ of the space $(I\times\{0,\ldots,m\}, \tau_{lex})$ is perfectly normal if and only if either $n=0$ or $n=m=1$. The space $(I\times\{0,\ldots, n\}, \tau_{Alex})$ is perfectly normal if and only if $n=0$. *Proof.* The fact that the euclidean space $(I\times\{0\},\tau_{lex})=(I\times\{0\},\tau_{Alex})=(I\times\{0\},\tau_e)$ and the space $(I\times\{0,1\},\tau_{lex})$ are both perfectly normal is classical and so we omit the proofs. Similarly, it is also classical that the space $(I,\tau_l)$ is perfectly normal, and so, for any $m>0$, the subspace $I\times\{0\}$ of the space $(I\times\{0,\ldots, m\},\tau_{lex})$ is perfectly normal, since this subspace corresponds to the push-forward of $(I,\tau_l)$ by the map $x\mapsto\langle x, 0 \rangle$. Let $(I\times\{0,\ldots,n\},\mu)$ be a space in any of the remaining cases, i.e. such that $n>0$, and either with $\mu=\tau_{Alex}$ or otherwise satisfying that there is $m>n$ such that $\mu$ is the subspace topology inherited from the space $(I\times\{0,\ldots,m\},\tau_{lex})$. Note that, in all of these cases, there exists some $0<i \leq n$ such that the subset $(I\times \{i\}, \mu)$ only contains $\mu$-isolated points, and in particular it is $\mu$-open. We fix such an $i$. Recall that a topological space is perfectly normal if and only if it is normal and any open set is a countable union of closed sets. Hence to show that $(I\times\{0,\ldots,n\},\mu)$ is not perfectly normal it suffices to show that $I\times\{i\}$ is not a countable union of $\mu$-closed sets. Towards a contradiction, suppose that it were, in which case there would exist an uncountable $\mu$-closed subset $C$ of $I\times\{i\}$. Since such a $C$ is uncountable, the projection $\pi(C)$ of $C$ to the first coordinate would satisfy that there exists a point $x\in \pi(C)$ such that, for any $a<x<b$, it holds that $(a,x)\cap \pi(C)\neq \emptyset$ and $(x,b)\cap \pi(C) \neq \emptyset$ (see Lemma [Lemma 107](#lemma_limit_points){reference-type="ref" reference="lemma_limit_points"} for a generalization of this classical fact of the reals). By definition of the $\tau_{lex}$ and $\tau_{Alex}$ topologies, it follows that $\langle x, 0\rangle$ is in the $\mu$-closure of $C$, a contradiction, since $C$ is a $\mu$-closed subset of $I\times \{i\}$ with $i>0$. ◻ **Corollary 101**. Suppose that $\mathcal{R}$ expands the ordered field of reals $(\mathbb{R},+,\cdot,<)$. Let $(X,\tau)$, $\dim X \leq 1$, be a regular and Hausdorff definable topological space. If $(X,\tau)$ is either separable or perfectly normal, then there exists a definable continuous map $f:(X,\tau)\rightarrow (R^n,\tau_e)$, for some $n$, where $f$ is at most $2$-to-$1$. *Proof.* If $(X,\tau)$ is separable then, by Proposition [Proposition 6](#prop:sep-equiv){reference-type="ref" reference="prop:sep-equiv"}, it is also definably separable, and the result follows from Corollary [Corollary 99](#cor_Fremlin_2-1){reference-type="ref" reference="cor_Fremlin_2-1"}. Suppose that $(X,\tau)$ is perfectly normal. Applying Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"} we may assume that $X\subseteq R$. Let $\mathcal{X}=\mathcal{X}_{\text{open}}\cup \mathcal{X}_{\text{sgl}}$ be a partition of $X$ as defined by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"} and, for every $A\in \mathcal{X}_{\text{open}}$, let $h_A$ be as given by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}. Since perfect normality is a hereditary property we have that, for every set $A\in \mathcal{X}_{\text{open}}$, the subspace $(A,\tau)$ is perfectly normal. Applying Lemma [Lemma 100](#lem:T5){reference-type="ref" reference="lem:T5"} to the construction by cases in the proof of Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}, we observe that, for every $A\in\mathcal{X}_{\text{open}}$, it holds that $h_A(A)\subseteq R\times\{0,1\}$. The rest of the proof is now analogous to the proof of Corollary [Corollary 99](#cor_Fremlin_2-1){reference-type="ref" reference="cor_Fremlin_2-1"}, taking $(h(X),\tau^*)$ in place of $(X^*,\tau^*)$ in said proof. ◻ Observe that the proofs of Corollaries [Corollary 99](#cor_Fremlin_2-1){reference-type="ref" reference="cor_Fremlin_2-1"} and [Corollary 101](#cor_Fremlin_2-2){reference-type="ref" reference="cor_Fremlin_2-2"} can be adapted to yield a different version of Corollary [Corollary 99](#cor_Fremlin_2-1){reference-type="ref" reference="cor_Fremlin_2-1"}, which holds for any expansion $\mathcal{R}$ of an ordered field, where the condition that $(X, \tau)$ is definably separable is replaced by the weaker condition that $(X, \tau)$ does not have a subspace definably homeomorphic to any space that is the $\mathcal{R}$-definable analogue of one of the spaces that Lemma [Lemma 100](#lem:T5){reference-type="ref" reference="lem:T5"} shows are not perfectly normal. We now consider statement [\[Fremlin2.2\]](#Fremlin2.2){reference-type="eqref" reference="Fremlin2.2"} in our setting. We present a definable version of this statement, in which the metrizability condition is replaced by the stronger property of being affine. It is an immediate consequence of Lemma [Lemma 100](#lem:T5){reference-type="ref" reference="lem:T5"} applied to the dichotomy described in Corollary [Corollary 98](#thm:compact_affine){reference-type="ref" reference="thm:compact_affine"}. **Corollary 102**. Suppose that $\mathcal{R}$ expands the ordered field of reals $(\mathbb{R},+,\cdot,<)$. Let $(X,\tau)$, $\dim X\leq 1$, be a definably compact (equivalently, by Remark [Remark 12](#rem:compactness){reference-type="ref" reference="rem:compactness"}, compact) and Hausdorff definable topological space. If $(X,\tau)$ is perfectly normal then it is either affine or otherwise there exists an interval $I\subseteq \mathbb{R}$ and a definable open embedding $(I\times\{0,1\}, \tau_{lex}) \hookrightarrow (X,\tau)$. One may in fact replace the assumption of perfect normality in Corollary [Corollary 102](#cor:Fremlin-Q2.2){reference-type="ref" reference="cor:Fremlin-Q2.2"} by the weaker condition that there does not exist an interval $I\subseteq \mathbb{R}$ and a definable embedding $(I\times\{0,1\},\tau_{Alex})\hookrightarrow (X,\tau)$. By Corollary [Corollary 98](#thm:compact_affine){reference-type="ref" reference="thm:compact_affine"} alone, this condition already implies that $(X,\tau)$ is affine or otherwise there exists an interval $I\subseteq \mathbb{R}$ and a definable embedding $(I\times\{0,1\}, \tau_{lex}) \hookrightarrow (X,\tau)$. (To see this observe that, for any interval $I\subseteq \mathbb{R}$ and $n>0$, the subspace $I\times \{0, n\}$ of the space $(I\times \{0,\ldots, n\}, \tau_{lex})$ is definably homeomorphic to $(I\times \{0,1\}, \tau_{lex})$.) Additionally, since Corollary [Corollary 98](#thm:compact_affine){reference-type="ref" reference="thm:compact_affine"} holds in the setting of an arbitrary o-minimal expansion $\mathcal{R}$ of an ordered field, this version of Corollary [Corollary 102](#cor:Fremlin-Q2.2){reference-type="ref" reference="cor:Fremlin-Q2.2"} moreover holds in this more general setting. # Definable metrizability {#section:metrizability} In this section, we use our affineness characterization of the previous section (Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}) to derive the equivalence of metrizability and definable metrizability for one-dimensional definable topological spaces in certain o-minimal expansions of ordered fields. Throughout, we continue to assume that $\mathcal{R}$ expands an ordered field. Recall that in our setting "metric\" refers to an $\mathcal{R}$-metric (Definition [Definition 14](#dfn:R-metric){reference-type="ref" reference="dfn:R-metric"}), including those instances when it appears implicitly in notions such as metrizability and metric space. We will frequently utilize two classical topological notions in stating and proving results throughout this section. The first of these is the weight $w_\tau(X)$ of a topological space $(X,\tau)$, i.e. the minimum cardinality of a basis for $\tau$ (this was discussed earlier in Subsection [5.2](#subsection: Cantor_space){reference-type="ref" reference="subsection: Cantor_space"}). Note that it follows from Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}, Lemma [Lemma 45](#lem:weight-taur){reference-type="ref" reference="lem:weight-taur"} and Proposition [Proposition 46](#prop:weight){reference-type="ref" reference="prop:weight"}([\[itm_weight_2\]](#itm_weight_2){reference-type="ref" reference="itm_weight_2"}) that, whenever $\mathcal{R}$ expands an ordered field, every one-dimensional Hausdorff definable topological space $(X,\tau)$ satisfies that $w_\tau(X)\in\{w_e(R), |R|\}$. The other classical notion that we will use is the density of $(X,\tau)$, i.e. the minimum cardinality of a $\tau$-dense subset of $X$, which we denote by $den_\tau(X)$. Our main result, Theorem [Theorem 104](#thm:metrizability){reference-type="ref" reference="thm:metrizability"}, shows that, whenever $\mathcal{R}$ is an o-minimal expansion of an ordered field satisfying that $den_e(R)<|R|$ (e.g. whenever $\mathcal{R}$ expands the field of reals), every metrizable one-dimensional space is in fact definably metrizable. We begin by deriving the following as a straightforward corollary of Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}. **Corollary 103**. Let $(X,\tau)$, $\dim(X)\leq 1$, be a definable topological space that is metrizable and separable. Suppose that either of the following two conditions holds. (a) $\mathcal{R}$ expands the field of reals. (b) $(X,\tau)$ is compact. Then $(X,\tau)$ is affine. In particular it is definably metrizable. *Proof.* The case where $X$ is finite is trivial, so we assume that $\dim(X)=1$. Recall from Remark [Remark 48](#remark_compact_implies_reals){reference-type="ref" reference="remark_compact_implies_reals"} that, if there exists a compact infinite $T_1$ topological space definable in $\mathcal{R}$, then $\mathcal{R}$ expands the field of reals. Since a metrizable space is $T_1$, the second case here therefore reduces to the first, and we may consequently assume that $\mathcal{R}$ expands the field of reals, i.e. $R=\mathbb{R}$. By Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"}, we assume that $X\subseteq \mathbb{R}$. By Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}, it is enough to show that $(X,\tau)$ does not have a definable copy of an interval with either the discrete or the right half-open interval topology. This follows from the fact that $(X,\tau)$ is a separable metric space, hence second countable, so $w_{\tau}(X)<|\mathbb{R}|$, while the topological weight of an interval with the Sorgenfrey Line or discrete topology is $|\mathbb{R}|$ (see Lemma [Lemma 45](#lem:weight-taur){reference-type="ref" reference="lem:weight-taur"}). ◻ We now state the main theorem of this section, which improves the metrization part of Corollary [Corollary 103](#thm:metrizability_2){reference-type="ref" reference="thm:metrizability_2"}. **Theorem 104**. Suppose that $\mathcal{R}$ expands an ordered field and satisfies that $den_e(R)<|R|$. Let $(X,\tau)$, $\dim X\leq 1$, be a definable topological space. Then $(X,\tau)$ is metrizable if and only if it is definably metrizable. In the case where $den_e(R)=|R|$, we point the reader towards [@diep17] for a proof that the space $(R,\tau_r)$, which is shown in Proposition [Proposition 17](#prop:tau_r not metrizable){reference-type="ref" reference="prop:tau_r not metrizable"} not to be definably metrizable, is metrizable whenever $\mathcal{R}$ expands a countable densely ordered group. Classically, the Sorgenfrey Line $(\mathbb{R},\tau_r)$ is separable but not second countable, and so it is not ($\mathbb{R}$-)metrizable. We show below (Remark [Remark 106](#remark_not_metrizable){reference-type="ref" reference="remark_not_metrizable"}) that the analogous statement holds whenever $den_e(R) < |R|$, i.e. in this case, $(X, \tau_r)$ is not ($\mathcal{R}$)-metrizable when $X \subseteq R$ is any infinite set. In order to prove Theorem [Theorem 104](#thm:metrizability){reference-type="ref" reference="thm:metrizability"}, we require two simple lemmas, whose aim is to generalise basic results in metric topology and the topology of the real line to our setting. In what follows, recall that, since $\mathcal{R}$ expands an ordered field, any two intervals are definably $e$-homeomorphic and in particular, for any interval $I\subseteq R$, we have that $|I|=|R|$ and $den_e(I)=den_e(R)$. **Lemma 105**. Let $(X,d)$ be a metric space. Let $\tau:=\tau_d$, and let $A,C\subseteq X$ satisfy that $A\subseteq cl_\tau C$ (i.e. $C$ is $\tau$-dense in $A$). Let $D$ be an $e$-dense subset of $(0,+\infty)$. Consider the family of $d$-balls $\mathcal{B}=\{B_d(y,\delta) : y\in C, \delta\in D\}$. Then, for every $x\in A$, there exists a subfamily $\mathcal{B}_x$ of $\mathcal{B}$ that is a basis of open $\tau$-neighbourhoods of $x$. In particular $w_\tau(X)\leq den_\tau(X)den_e(R)$. *Proof.* Let $x\in A$ and $\varepsilon>0$. We must show that there exists $y\in C$ and $\delta\in D$ such that $x\in B_d(y, \delta)\subseteq B_d(x,\varepsilon)$. Let $\delta\in D$ be such that $0<\delta< \varepsilon/2$. Since $A\subseteq cl_\tau C$, there exists $y\in C$ such that $d(x,y)<\delta$. Consider the ball $B_d(y,\delta)$. Clearly $x\in B_d(y,\delta)$ and, if $z\in B_d(y,\delta)$, then, by the triangle inequality, $d(x,z) \leq d(x,y) + d(y,z) \leq \delta+\delta < \varepsilon$. Hence $x\in B_d(y,\delta) \subseteq B_d(x,\varepsilon)$, which completes the proof of the first part of the lemma. For the second part, suppose that $C$ is a dense subset of $(X,\tau)$ of cardinality $den_\tau(X)$ and $D$ is an $e$-dense subset of $(0,+\infty)$ of cardinality $den_e(R)$. Then, by the above, the family $\{ B_d(y,\delta) : y\in C, \delta\in D\}$ is a basis for $\tau$ which has cardinality bounded by $den_\tau(X)den_e(R)$. ◻ **Remark 106**. Let $X\subseteq R$ be an infinite definable set. Recall that, by Lemma [Lemma 45](#lem:weight-taur){reference-type="ref" reference="lem:weight-taur"}, because $\mathcal{R}$ expands an ordered field, the space $(X,\tau_*)$, where $\tau_*\in\{\tau_l, \tau_r\}$, has weight $|R|$. Moreover, clearly the density of $(X,\tau_*)$ is equal to $den_e(R)$. From Lemma [Lemma 105](#lemma_tech_2nd_countable){reference-type="ref" reference="lemma_tech_2nd_countable"}, it follows that, if $(X,\tau_*)$ is metrizable, then $|R|=w_{\tau_*}(X)\leq den_{\tau_*}(X) den_e(R)= den_e(R)^2=den_e(R)\leq |R|$, i.e. $den_e(R)=|R|$. It follows that, if in fact we have $den_e(R)<|R|$, then $(X,\tau_*)$ is not metrizable. For the next lemma recall that, for any set $X\subseteq R$, a right (respectively left) limit point of $X$ is a point $x\in R$ satisfying that, for every $y>x$ (respectively $y<x$), $(x,y)\cap X\neq \emptyset$ (respectively $(y,x)\cap X\neq \emptyset$). **Lemma 107**. Suppose that $den_e(R)<|R|$ and let $X\subseteq R$ be a subset of cardinality $|R|$. Then there exist $|R|$-many elements of $X$ that are both right and left limit points of $X$. *Proof.* We show that all but at most $den_e(R)$-many points in $X$ are right limit points of $X$. The same holds for left limit points. The result then follows from the fact that $den_e(R)<|R|=|X|$. Let $Y$ be the set of points in $X$ that are not right limit points of $X$. For every $x\in Y$, there is some $x'>x$ such that the interval $(x,x')=I_x$ is disjoint from $X$. The family $\{I_x : x\in Y\}$ has cardinality $|Y|$ and contains only non-empty pairwise disjoint intervals. It follows that $|Y|\leq den_e(R)$. The proof for the set of left limit points is analogous. ◻ We may now prove Theorem [Theorem 104](#thm:metrizability){reference-type="ref" reference="thm:metrizability"}. *Proof of Theorem [Theorem 104](#thm:metrizability){reference-type="ref" reference="thm:metrizability"}.* Clearly any definably metrizable topological space is metrizable. Fix $(X,\tau)$, with $\dim X \leq 1$, a definable topological space whose topology is induced by a metric $d$. We prove that $(X,\tau)$ is definably metrizable by describing a definable metric $\hat{d}$ that induces $\tau$. Since every finite metric space is discrete, we may assume that $\dim X=1$. Let $D$ be a dense subset of $R$ of cardinality $den_e(R)$. By Remark [Remark 24](#remark_assumption_X){reference-type="ref" reference="remark_assumption_X"}, we assume that $X$ is a bounded subset of $R$. Consider the definable set $S=\{x\in X : E_x \setminus \{x\} \neq \emptyset\}$. We begin by proving the following claim. **Claim 3**. $S$ is finite. *Proof of claim.* Towards a contradiction suppose that $S$ is infinite. Let $f:S\rightarrow R_{\pm\infty}$ be the map given by $x\mapsto \min E_x\setminus \{x\}$, which, by Proposition [Proposition 20](#basic_facts_P_x){reference-type="ref" reference="basic_facts_P_x"}([\[itm: basic_facts_2\]](#itm: basic_facts_2){reference-type="ref" reference="itm: basic_facts_2"}) and Lemma [Lemma 23](#lemma_2){reference-type="ref" reference="lemma_2"}, is definable. By Hausdorffness (Lemma [Lemma 30](#lemma:basic_facts_P_x_2){reference-type="ref" reference="lemma:basic_facts_P_x_2"}([\[itmb:lemma_basic_facts_P\_x_2\]](#itmb:lemma_basic_facts_P_x_2){reference-type="ref" reference="itmb:lemma_basic_facts_P_x_2"})) and o-minimality, there exists an interval $I\subseteq S$ on which $f$ is $e$-continuous and strictly monotonic. Note (see Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"}) that $I$ is in the $\tau$-closure of $D \cap f(I)$. Consider the family of $d$-balls $\mathcal{B}=\{B_d(q,\delta) : q \in D \cap f(I), \delta \in D \cap (0,+\infty) \}$. This family has cardinality bounded by $den_e(R)$ and, by Lemma [Lemma 105](#lemma_tech_2nd_countable){reference-type="ref" reference="lemma_tech_2nd_countable"}, contains, for every $x\in I$, a subfamily that is a basis of open $\tau$-neighbourhoods of $x$. Now, let $h : I \to \mathcal{B}$ be a function with the property that, for every $x \in I$, $h(x) \in \mathcal{B}$ is a $\tau$-neighbourhood of $x$ such that $f(x) \notin h(x)$. Such a function can be defined since, for every $x \in I$, $f(x) \neq x$ and $\tau$ is $T_1$. Since $|I|=|R|$ and $|\mathcal{B}|\leq den_e(R)<|R|$, there must exist, by the pigeonhole principle, some $d$-ball $B\in h(I)$ such that the set $h^{-1}(B)$ has cardinality $|R|$. By Lemma [Lemma 107](#lemma_limit_points){reference-type="ref" reference="lemma_limit_points"}, there exists $x\in h^{-1}(B)$ that is both a right and left limit point of $h^{-1}(B)$. Recall that $f(x)\in E_x$. Suppose that $f(x)\in R_x$. Then, since $B=h(x)$ is a $\tau$-neighbourhood of $x$, there is some $z>f(x)$ such that $(f(x),z)\subseteq B$. If $f$ is increasing then, by $e$-continuity, there is some $y>x$ with $(x,y)\subseteq I$ such that $f[(x,y)]\subseteq (f(x),z)$. Hence, for every $x'\in (x,y)$, it holds that $f(x') \in B$ and so, by definition of $h$, that $h(x')\neq B$. However, this contradicts that $x$ is a right limit point of $h^{-1}(B)$. Similarly, if $f$ is decreasing, there is some $y<x$ with $(y,x)\subseteq I$ such that $(y,x)\cap h^{-1}(B)=\emptyset$, contradicting that $x$ is a left limit point of $h^{-1}(B)$. The argument in the case where $f(x)\in L_x$ is analogous. This completes the proof of the claim. ◻ We now continue the proof of Theorem [Theorem 104](#thm:metrizability){reference-type="ref" reference="thm:metrizability"}. By Theorem [Theorem 51](#them 1.5){reference-type="ref" reference="them 1.5"} and Remark [Remark 106](#remark_not_metrizable){reference-type="ref" reference="remark_not_metrizable"}, there exists a partition $\mathcal{X}$ of $X$ into finitely many points and intervals where each interval subspace in $\mathcal{X}$ has either the euclidean or the discrete topology. Let $E_S=\bigcup_{x\in S} E_x$. By Claim [Claim 3](#claim:S-finite){reference-type="ref" reference="claim:S-finite"} and Lemma [Lemma 23](#lemma_2){reference-type="ref" reference="lemma_2"}, both $S$ and $E_S$ are finite sets. By passing to a finer partition if necessary, we may require that $\mathcal{X}$ has the following two properties. (i) [\[itm1: metrizability\]]{#itm1: metrizability label="itm1: metrizability"} The elements in $S$ and in $E_S$ do not belong in any interval in $\mathcal{X}$. (ii) [\[itm2: metrizability\]]{#itm2: metrizability label="itm2: metrizability"} For any interval $(a,b)\in \mathcal{X}$ with the discrete subspace topology, it holds that, if $a\in \bigcup_{x\in X} R_x$, then $b\notin \bigcup_{x\in X} L_x$ and, if $b\in \bigcup_{x\in X} L_x$, then $a\notin \bigcup_{x\in X} R_x$. Note that ([\[itm2: metrizability\]](#itm2: metrizability){reference-type="ref" reference="itm2: metrizability"}) can be arranged since any discrete interval subspace $I$ that is disjoint from $E_S$ is also disjoint from $\bigcup_{x\in X} E_x$, and so any proper subinterval of $I$ has the desired property. First note that, by ([\[itm1: metrizability\]](#itm1: metrizability){reference-type="ref" reference="itm1: metrizability"}), for any interval $I=(a,b)\in\mathcal{X}$, any $x\in I$ and any $y\in X\setminus I$, it holds that $E_x\subseteq \{x\}$ and $E_y\cap I=\emptyset$. So, by Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"}, $I$ is $\tau$-open and, if $y\in \partial_\tau I$, then it must be that either $a\in R_y$ or $b\in L_y$. In particular, by ([\[itm2: metrizability\]](#itm2: metrizability){reference-type="ref" reference="itm2: metrizability"}) and Hausdorffness (Proposition [Proposition 32](#prop_basic_facts_P_x_2){reference-type="ref" reference="prop_basic_facts_P_x_2"}([\[itm3:basic_facts_P\_x_2\]](#itm3:basic_facts_P_x_2){reference-type="ref" reference="itm3:basic_facts_P_x_2"})), if $I$ is discrete then $|\partial_\tau I|\leq 1$. Now, let $\mathcal{Y}\subseteq \mathcal{X}$ be the family of all discrete interval subspaces in $\mathcal{X}$. Let $|\mathcal{Y}|=n$. We prove the theorem by induction on $n$. If $n=0$, then $X$ is cell-wise euclidean. In particular, by Remark [Remark 4](#remark_no_homeomorphism){reference-type="ref" reference="remark_no_homeomorphism"}, it contains no definable copy of an interval with either the discrete or the right half-open interval topology and so, applying Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}, $(X,\tau)$ is affine, and in particular it is definably metrizable. Suppose that $n>0$ and let $\mathcal{Y}=\{I_1,\ldots, I_n\}$. Let $X'=X\setminus I_n$. By induction hypothesis, the space $(X',\tau)$ is definably metrizable with some definable metric $d'$. We extend $d'$ to a definable metric $\hat{d}$ on $X$ such that $\tau_{\hat{d}}=\tau$. Let $I_n=I=(a,b)$. By the argument above, $\partial_{\tau}I = \emptyset$ or $|\partial_{\tau}I| = 1$. We consider each of these two cases in turn. **Case 0:** $\partial_\tau I =\emptyset$. In this case, $I$ is a $\tau$-clopen subset of $X$. Note that the metric $\min\{1,d'\}$ induces the same topology as $d'$, hence, by passing to the former if necessary, we may assume that $d'\leq 1$. We define the metric $\hat{d}$ on $X$ as follows. - For all $x,y \in X'$, $\hat{d}(x,y)=d'(x,y)$. - For all $x\in I$, $y \in X$, $\hat{d}(x,y)=\hat{d}(y,x)=1$, if $x\neq y$, and $\hat{d}(x,y)=\hat{d}(y,x)=0$ otherwise. Since the $\tau$-topology on $I$ is discrete, it is easy to check that $\hat{d}$ is a metric that induces the topology $\tau$ on $X$. **Case 1:** $|\partial_\tau I|=1$, i.e. $\partial_\tau I=\{x_0\}$ for some $x_0\in X\setminus I$. Recall that, by ([\[itm1: metrizability\]](#itm1: metrizability){reference-type="ref" reference="itm1: metrizability"}), $E_{x_0}\cap (a,b)=\emptyset$, and so (see Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"}) it must be that either $a \in R_{x_0}$ or $b \in L_{x_0}$, but not both (by ([\[itm2: metrizability\]](#itm2: metrizability){reference-type="ref" reference="itm2: metrizability"})). We prove the case where $a\in R_{x_0}$. The remaining case, where $b\in L_{x_0}$, is analogous. Recall that $X$ is bounded, and so $I$ is a bounded interval. Consider the following definable metric $\hat{d}$ in $X$. - For all $x,y \in X'$, $\hat{d}(x,y)=d'(x,y)$. - For all $x, y\in I$, $\hat{d}(x,y)=|x-a|+|y-a|$, if $x\neq y$, and $\hat{d}(x,y)=0$ otherwise. - For all $x\in I$, $y\in X'$, $\hat{d}(x,y)=\hat{d}(y,x)=|x-a|+d'(y,x_0)$. It is routine to check that $\hat{d}$ is a metric. We show that $\tau_{\hat{d}}=\tau$ by proving, using Proposition [Proposition 27](#prop_cont_lim){reference-type="ref" reference="prop_cont_lim"}, that the identity map $(X,\tau)\rightarrow(X,\tau_{\hat{d}})$ is a homeomorphism. Note that, by definition, $\hat{d}$ induces the corresponding subspace topologies of $\tau$ on $X'$ and $I$, and moreover $I$ is $\hat{d}$-open. In particular, since $I$ is $\tau$-open too, we have that $X'$ is both $\tau$-closed and $\hat{d}$-closed in $X$. Since $\tau|_{X'}=\tau_{\hat{d}}|_{X'}$, we derive that any definable curve in $X'$ $\tau$-converges to a point in $X$ (necessarily inside $X'$) if and only if it $\hat{d}$ converges to that same point. Furthermore, note that, by definition of $\hat{d}$, an injective definable curve $\gamma$ in $I$ $\hat{d}$-converges if and only if it $e$-converges to $a$ from the right, $\hat{d}$-converging thus to $x_0$. By the fact that $a\in R_{x_0}$ and Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}, $\gamma$ must then also $\tau$-converge to $x_0$. Conversely, if an injective definable curve $\gamma$ in $I$ $\tau$-converges then, by the facts that the $\tau$-topology on $I$ is discrete and $\partial_\tau I=\{x_0\}$, $\gamma$ must $\tau$-converge to $x_0$. Recall that, by the assumptions on $I$, we have that $E_{x_0}\cap (a,b)=\emptyset$, $a \in R_{x_0}$ and $b \notin L_{x_0}$. Hence, by Remark [Remark 34](#remark_side_convergence){reference-type="ref" reference="remark_side_convergence"}, it must be that $\gamma$ $e$-converges to $a$ from the right, and so, from the definition of $\hat{d}$, it follows that $\gamma$ $\hat{d}$-converges to $x_0$ too. This completes the proof of the theorem. ◻ It remains open whether or not Theorem [Theorem 104](#thm:metrizability){reference-type="ref" reference="thm:metrizability"} can be generalized to spaces of dimension greater than one. **Question 108**. Let $\mathcal{R}$ be an o-minimal expansion of an ordered field satisfying that $den_e(R)<|R|$. Is any $\mathcal{R}$-metrizable topological space definable in $\mathcal{R}$ definably metrizable? # A note on an affiness result by Peterzil and Rosel {#sec:Peterzil_Rosel} The majority of the work in this paper already appeared in the first author's doctoral dissertation [@andujar_thesis]. After that work had been completed, the authors learned that Peterzil and Rosel were working on similar questions. Their work resulted in [@pet_rosel_18]. The main theorem (page $1$) in said paper is the following affiness result, which we present in the terminology of the present paper. **Theorem 109** ([@pet_rosel_18]). Suppose that $\mathcal{R}$ expands an ordered group. Let $(X,\tau)$ be a Hausdorff definable topological space, where $\dim X=1$ and $X$ is a bounded set. The following are equivalent. (1) [\[itm:Pet_Rosel_1\]]{#itm:Pet_Rosel_1 label="itm:Pet_Rosel_1"} $(X,\tau)$ is affine. (2) [\[itm:Pet_Rosel_2\]]{#itm:Pet_Rosel_2 label="itm:Pet_Rosel_2"} There is a finite set $G\subseteq X$ such that the subspace topology $\tau|_{X\setminus G}$ is coarser than the euclidean topology on $X\setminus G$. (3) [\[itm:Pet_Rosel_3\]]{#itm:Pet_Rosel_3 label="itm:Pet_Rosel_3"} Every definable subspace of $(X,\tau)$ has finitely many definably connected components. (4) [\[itm:Pet_Rosel_4\]]{#itm:Pet_Rosel_4 label="itm:Pet_Rosel_4"} $(X,\tau)$ is regular and has finitely many definably connected components. Furthermore, if $\mathcal{R}$ expands an ordered field then the above is true even without the assumption that the set $X$ is bounded. Their work is in some ways parallel to ours. For example, their notion of set of shadows $S(x)$ of a point $x$ is effectively the $e$-accumulation set $E_x$ of $x$ (both notions are equivalent for Hausdorff topologies, which are the only topologies studied in [@pet_rosel_18], while in general it holds that $E_x \subseteq S(x)$). Similarly, *$x$ inhabits the left (respectively right) side of $y$* means $y\in L_x$ (respectively $y\in R_x$). Moreover, for a definable topological space $(X,\tau)$, they refer to the property *almost $\tau \subseteq \tau^{af}|_X$* (where $\tau^{af}$ is their notation for the euclidean topology $\tau_e$) to mean the condition that $(X,\tau)$ has a cofinite subspace on which the $\tau$-topology is coarser than the euclidean topology. This property clearly implies being cell-wise euclidean. The converse implication also holds in the case where $\dim X=1$ as follows. Suppose that $\mathcal{X}$ is a finite definable partition of $X$ into subsets where the subspace $\tau$-topology is euclidean. Consider the set $A=\bigcup\{ C \setminus cl_e(X\setminus C) : C\in \mathcal{X},\, \dim C = \dim X\}$. Since $\tau|_C = \tau_e|_C$ for every $C\in \mathcal{X}$, observe that it must be that that $\tau|_A = \tau_e|_A$. Furthermore, since the euclidean topology satisfies the **fdi**, it holds that $\dim X\setminus A < \dim X$. In particular if $\dim X=1$ then $X\setminus A$ is finite, and so $(X,\tau)$ satisfies almost $\tau \subseteq \tau^{af}|_X$. Statement ([\[itm:Pet_Rosel_2\]](#itm:Pet_Rosel_2){reference-type="ref" reference="itm:Pet_Rosel_2"}) in Theorem [Theorem 109](#them:PR){reference-type="ref" reference="them:PR"} can thus be reformulated as stating either that $(X,\tau)$ is almost $\tau \subseteq \tau^{af}|_X$ or that $(X,\tau)$ is cell-wise euclidean. We remark here that there appears to be an infelicity in the proof of the first assertion in [@pet_rosel_18 Lemma 3.16], of which the authors of [@pet_rosel_18] are aware (per private correspondence). Specifically, the proof relies on the assertion (in our terminology) that, for any Hausdorff one-dimensional definable topological space $(X,\tau)$ and $x\in X$, it holds that $E_x \setminus \{x\} \subseteq \bigcap_{U\in \mathcal{B}(x)} \partial_e U$, where $\mathcal{B}(x)$ denotes a definable basis of $\tau$-neighbourhoods of $x$. However, to see that this statement is false in general suppose that $E_x\setminus \{x\} \neq \emptyset$ and $X\in \mathcal{B}(x)$. The authors of [@pet_rosel_18] have communicated to us (by private correspondence) a correction to their proof, which we understand may be forthcoming. Moreover, observe that the first statement of [@pet_rosel_18 Lemma 3.16] in fact corresponds to our Lemma [Lemma 23](#lemma_2){reference-type="ref" reference="lemma_2"} localised to Hausdorff one-dimensional spaces. Therefore, this issue does not present any reason to suppose that the statement of Theorem [Theorem 109](#them:PR){reference-type="ref" reference="them:PR"} does not hold. Furthermore, it is straightforward to derive the statement of Theorem [Theorem 109](#them:PR){reference-type="ref" reference="them:PR"}, in the case that $\mathcal{R}$ expands an ordered field, from the results of this paper. In this setting, in light of Theorem [Theorem 51](#them 1.5){reference-type="ref" reference="them 1.5"}, the implications $(\ref{itm:Pet_Rosel_1})\Leftrightarrow (\ref{itm:Pet_Rosel_2})\Leftrightarrow (\ref{itm:Pet_Rosel_3})$ in Theorem [Theorem 109](#them:PR){reference-type="ref" reference="them:PR"} are equivalent to our Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"} and Remark [Remark 93](#rem:(3)implies(1)){reference-type="ref" reference="rem:(3)implies(1)"}. The implication $(\ref{itm:Pet_Rosel_1}) \Rightarrow (\ref{itm:Pet_Rosel_4})$ is a classical corollary of o-minimal cell decomposition (see [@dries98 Chapter 3, Proposition 2.18]) and the easy fact that the euclidean topology is regular. Finally, the implication $(\ref{itm:Pet_Rosel_4}) \Rightarrow (\ref{itm:Pet_Rosel_1})$ can be derived using the framework that we introduced to prove Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"} (in particular Lemmas [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"} and [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}). We outline this later in the section (Proposition [Proposition 113](#prop:connectedness-cellwise-euclidean){reference-type="ref" reference="prop:connectedness-cellwise-euclidean"}). Although our approach in this paper leads to an affineness characterization only in the case where $\mathcal{R}$ expands an ordered field, whereas the equivalence in Theorem [Theorem 109](#them:PR){reference-type="ref" reference="them:PR"} can be shown to hold when $\mathcal{R}$ expands an ordered group as long as the underlying set $X$ is bounded, our alternative approach does allow us to give answers to a number of questions left open in [@pet_rosel_18]. Firstly, Peterzil and Rosel ask (remark at the end of Section 2 in [@pet_rosel_18]) if, given a definable topological space $(X,\tau)$ and $x\in X$, the union of all (definable) definably connected sets containing $x$ is itself definable, i.e. if there exists a (definable) definably connected component containing $x$. We answer this question in the positive in the case that $(X,\tau)$ is $T_3$ with $\dim(X) = 1$ in Proposition [Proposition 112](#prop:connected-components){reference-type="ref" reference="prop:connected-components"} below. In order to prove it we require two lemmas. **Lemma 110**. Let $(X,\tau)$ be a Hausdorff regular definable topological space, with $X\subseteq R$. Let $\mathcal{X}_{\text{open}}$ be as defined by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}, and consider a set $A=\bigcup_{0\leq i < n} f_i(I)$ in $\mathcal{X}_{\text{open}}$, with $I=(a,b)$. Let $(A^*,\tau_A)$ and $h_A:(A,\tau)\hookrightarrow (A^*,\tau_A)$ be as given by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}, with $A^*=I\times\{0,\ldots,m\}$. Suppose that $m>0$ and $\tau_A=\tau_{lex}$. For any $a<c<d<b$, if we set $$A(c,d, m)=((c,d]\times \{0\}) \cup ([c,d)\times \{m\}) \cup \bigcup_{0 <i <m} (c,d)\times \{i\},$$ then $h_A^{-1}(A(c,d,m))$ is clopen in $(X,\tau)$. *Proof.* Fix $a<c<d<b$. By definition of the $\tau_{lex}$ topology, the set $A(c,d,m)$ is clopen in $(A^*,\tau_A)=(A^*,\tau_{lex})$ and so, since $h_A$ is continuous, the set $B=h_A^{-1}(A(c,d,m))$ is clopen in $(A,\tau)$. Since $A$ is $\tau$-open in $X$ it immediately follows that $B$ is $\tau$-open in $X$ too. It remains to show that it is $\tau$-closed in $X$. Let $x\in cl_\tau B$. We show that $x\in B$. By Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}([\[itm1.5:lemma_proof_two_theorems\]](#itm1.5:lemma_proof_two_theorems){reference-type="ref" reference="itm1.5:lemma_proof_two_theorems"}) note that $$B\subseteq\bigcup_{0\leq i<n} f_i([c,d]).$$ In particular $cl_e B \subseteq A$. Since $x\in cl_\tau B$ then, by Proposition [Proposition 29](#basic_facts_P_x_1){reference-type="ref" reference="basic_facts_P_x_1"}([\[itm: basic_facts_4\]](#itm: basic_facts_4){reference-type="ref" reference="itm: basic_facts_4"}), we have that $\emptyset\neq E_x \cap cl_e B \subseteq A$. However, by Lemma [Lemma 58](#properties_cofnite_subset_1.5){reference-type="ref" reference="properties_cofnite_subset_1.5"}([\[itm:properties_cofnite_subset_1.5_b\]](#itm:properties_cofnite_subset_1.5_b){reference-type="ref" reference="itm:properties_cofnite_subset_1.5_b"}) and the fact that $A$ is closed under the equivalence relation addressed in said lemma (i.e. $A=\bigcup_{z\in I} [z]$ by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}), this implies that $x\in A$. Since $B$ is closed in $(A,\tau)$, we conclude that $x\in B$. ◻ **Lemma 111**. Let $(X,\tau)$ be a Hausdorff regular definable topological space, with $X\subseteq R$. Let $\mathcal{X}_{\text{open}}$ be as defined by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}. Suppose that there exists $x\in \bigcup \mathcal{X}_{\text{open}}$ with $x\notin R_x \cap L_x$. Then, for every $y\in X\setminus \{x\}$, there exists a definable $\tau$-clopen set $B$ with $x\in B$ and $y \notin B$. In particular $\{x\}$ is a maximal definably connected subspace of $(X,\tau)$. *Proof.* Let us fix $x\in \bigcup \mathcal{X}_{\text{open}}$ with $x\notin R_x \cap L_x$ and $y\in X\setminus \{x\}$. Since otherwise the result is obvious we may assume that neither $x$ nor $y$ is $\tau$-isolated in $X$. Following the terminology in Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}, let $A=\bigcup_{0\leq i < n} f_i(I)$ be the set in $\mathcal{X}_{\text{open}}$ satisfying that $x\in A$, with $I=(a,b)$. Let $(A^*,\tau_A)$ and $h_A:(A,\tau)\hookrightarrow (A^*,\tau_A)$ be as given by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}, with $A^*=I\times\{0,\ldots,m\}$. Now since $x\notin R_x \cap L_x$ and moreover $x$ is not $\tau$-isolated, note that this avoids Case 5 in the proof of Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}, and in all the other cases it holds that $m>0$ and $\tau_A=\tau_{lex}$. Consequently, by Lemma [Lemma 110](#lem:clopen-tau){reference-type="ref" reference="lem:clopen-tau"}, in order to prove the lemma it suffices to show that there exists some $a<c<d<b$ such that the set $B=h_A^{-1}(A(c,d,m))$ contains exactly one point among $\{x,y\}$. If $y\notin A$ then this is obvious, and so we assume that $y\in A$. Let $h_A(x)=\langle x_0, i \rangle$ and $h_A(y)=\langle x_1, j\rangle$, for points $x_0, x_1 \in I$ and $0\leq i, j \leq m$. If $x_0\neq x_1$, then suppose without loss of generality that $x_0 < x_1$. Then it suffices to choose any $a< c < x_0$ and any $x_0 < d < x_1$, and the result clearly follows. Now suppose that $x_0=x_1$. In particular, by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}([\[itm1.5:lemma_proof_two_theorems\]](#itm1.5:lemma_proof_two_theorems){reference-type="ref" reference="itm1.5:lemma_proof_two_theorems"}), note that $\{x,y\} \subseteq \{ f_i(x_0) : 0\leq i <n\}$. Since by assumption both $x$ and $y$ are not $\tau$-isolated then, by Lemma [Lemma 36](#lemma_explaining_neighbourhoods){reference-type="ref" reference="lemma_explaining_neighbourhoods"}, we have that $E_x\neq \emptyset$ and $E_y\neq \emptyset$. In particular, by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}, it holds that $[x_0]^E=\{ x_0, f_{n-1}(x_0)\}=\{x, y\}$. Observe that this corresponds in the proof of Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"} to Cases 3 and 4. In both these cases it holds that $m=n-1$, and moreover $\{h_A(x), h_A(y)\}=\{h_A(x_0), h_A(f_{n-1}(x_0))\} = \{\langle x_0, 0\rangle, \langle x_0, m\rangle\}$. Finally, note that the set $A(x_0, b, m)$ contains the point $\langle x_0, m \rangle$ but does not contain the point $\langle x_0, 0 \rangle$, and so we conclude that the set $B=h_A^{-1}(A(x_0,b,m))$ contains exactly one point among $\{x,y\}$, as desired. ◻ **Proposition 112**. Let $(X,\tau)$ be a Hausdorff regular definable topological space, with $X\subseteq R$. For each $x\in X$ there exists a maximal definably connected definable set $C\subseteq X$ containing $x$. Furthermore, either $C=\{x\}$ or $(C,\tau)$ is an infinite cell-wise euclidean subspace. *Proof.* Let $\mathcal{X}_{\text{open}}$ be as defined by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}. Let $Z=\bigcup\mathcal{X}_{\text{sgl}}\cup \{ x\in \bigcup \mathcal{X}_{\text{open}}: x\in R_x \cap L_x\}$. Note that, by Lemma [Lemma 111](#lem:connected-T3){reference-type="ref" reference="lem:connected-T3"}, any point in $X\setminus Z$ is not contained in any definably connected set in $(X,\tau)$ besides $\{x\}$. We show that $(Z,\tau)$ is cell-wise euclidean. The proposition follows. Since $\mathcal{X}_{\text{sgl}}$ is finite, to prove that $(Z,\tau)$ is cell-wise euclidean it suffices to show that the set $\{ x\in \bigcup \mathcal{X}_{\text{open}}: x\in R_x \cap L_x\}$ with the subspace $\tau$-topology is cell-wise euclidean. Let us fix $x$ in this set. Following the terminology in Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}, let $A=\bigcup_{0\leq i <n} f_i(I) \in \mathcal{X}_{\text{open}}$ be such that $x\in A$. We complete the proof by showing that $x \in I\subseteq \{ x\in \bigcup \mathcal{X}_{\text{open}}: x\in R_x \cap L_x\}$ and $(I,\tau)=(I, \tau_e)$. Let $(A^*, \tau_A)$ and $h_A$ be as given by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}. Since $x\in R_x \cap L_x$, note that, by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"}, $x\in I\cup f_{n-1}(I)$, and moreover this corresponds to Case 5 in the proof of Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}. Observe that, in Case 5, every point in $f_i(I)$ for $0<i<n$ is $\tau$-isolated, and so we derive that $x\in I$. Furthermore, every point $x'\in I$ satisfies that $x'\in R_{x'}\cap L_{x'}$, and so $I\subseteq \{ x\in \bigcup \mathcal{X}_{\text{open}}: x\in R_x \cap L_x\}$. Additionally, again by the fact that we are in Case 5, it holds that $\tau_A=\tau_{Alex}$ and $h_A(f_i(x')) = \langle x', i\rangle$ for every $x'\in I$ and $0\leq i <n$. Since $(I\times\{0\}, \tau_{Alex})=(I\times \{0\}, \tau_e)$, and $h_A:(A,\tau)\hookrightarrow (A^*, \tau_A)$ is an embedding given by $x'\mapsto \langle x', 0\rangle$ for every $x'\in I$, we derive that $(I,\tau)=(I,\tau_e)$. ◻ Using Lemma [Lemma 111](#lem:connected-T3){reference-type="ref" reference="lem:connected-T3"}, we may now also explain how implication $(\ref{itm:Pet_Rosel_4})\Rightarrow(\ref{itm:Pet_Rosel_1})$ in Theorem [Theorem 109](#them:PR){reference-type="ref" reference="them:PR"} (in the case that $\mathcal{R}$ expands an ordered field) can be obtained from our approach to these ideas. Specifically, the implication follows immediately from the proposition below and the proof of Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}, which shows that, if $\mathcal{R}$ expands an ordered field, then every one-dimensional Hausdorff cell-wise euclidean definable topological space is affine. **Proposition 113**. Let $(X,\tau)$ be a Hausdorff regular definable topological space, with $X\subseteq R$. If $(X,\tau)$ has finitely many definably connected components then it is cell-wise euclidean. *Proof.* Let $(X,\tau)$ be a $T_3$ definable topological space with $X\subseteq R$ and suppose that it has finitely many definably connected components. Let $\mathcal{X}_{\text{open}}$ be as defined by Lemma [Lemma 59](#remark_for_two_theorems){reference-type="ref" reference="remark_for_two_theorems"} and let $A=\bigcup_{0\leq i < n} f_i(I)$ be a set in $\mathcal{X}_{\text{open}}$. We show that $(A,\tau)=(I,\tau_e)$. Since $\bigcup \mathcal{X}_{\text{open}}$ is cofinite in $X$ it follows that $(X,\tau)$ is cell-wise euclidean. Let $(A^*,\tau_A)$ and $h_A$ be as defined by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}. Since $(X,\tau)$ has finitely many definably connected components, by Lemma [Lemma 111](#lem:connected-T3){reference-type="ref" reference="lem:connected-T3"} it follows that there can only be finitely many points in $A$ satisfying that $x\notin R_x \cap L_x$. Observe that this rules out Cases 0, 1, 2, 3 and 4 in the proof of Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}. Furthermore observe that, in the only remaining case -- Case 5 -- it holds that every point $x\in f_i(I)$, for $0<i<n-1$, is $\tau$-isolated, and so, since $(X,\tau)$ has finitely many definably connected components (in particular finitely many $\tau$-isolated points), it must be that $n=1$, i.e. $A=f_0(I)=I$. It then follows directly from the fact that we are in Case 5 that $(A^*, \tau_A) = (I \times \{ 0 \}, \tau_{Alex}) = (I \times \{0\}, \tau_e)$. Moreover, $h_A$ is a homeomorphism $(I, \tau) \to (I \times \{0 \}, \tau_e)$ which, by Lemma [Lemma 62](#lemma_proof_two_thems){reference-type="ref" reference="lemma_proof_two_thems"}([\[itm1.5:lemma_proof_two_theorems\]](#itm1.5:lemma_proof_two_theorems){reference-type="ref" reference="itm1.5:lemma_proof_two_theorems"}), is given by $x \mapsto \langle x, 0 \rangle$. So $(A, \tau) = (I, \tau) = (I, \tau_e)$, as desired. ◻ Peterzil and Rosel also raise the question of whether or not an analogue to their result, Theorem [Theorem 109](#them:PR){reference-type="ref" reference="them:PR"} above, could be obtained for Hausdorff definable topological spaces in higher dimensions ([@pet_rosel_18], Section 4.3 (2)). They note that being affine (condition ([\[itm:Pet_Rosel_1\]](#itm:Pet_Rosel_1){reference-type="ref" reference="itm:Pet_Rosel_1"})) cannot be equivalent to condition ([\[itm:Pet_Rosel_2\]](#itm:Pet_Rosel_2){reference-type="ref" reference="itm:Pet_Rosel_2"}), namely having a cofinite subspace whose topology is coarser than the euclidean topology, in arbitary dimensions, but they leave as open questions whether or not being affine is equivalent to condition ([\[itm:Pet_Rosel_3\]](#itm:Pet_Rosel_3){reference-type="ref" reference="itm:Pet_Rosel_3"}), namely that every definable subspace has finitely many definably connected components, or to condition ([\[itm:Pet_Rosel_4\]](#itm:Pet_Rosel_4){reference-type="ref" reference="itm:Pet_Rosel_4"}), namely being regular and having finitely many definably connected components. We can answer both of these questions negatively: the non-equivalence of conditions ([\[itm:Pet_Rosel_1\]](#itm:Pet_Rosel_1){reference-type="ref" reference="itm:Pet_Rosel_1"}), ([\[itm:Pet_Rosel_3\]](#itm:Pet_Rosel_3){reference-type="ref" reference="itm:Pet_Rosel_3"}) and ([\[itm:Pet_Rosel_4\]](#itm:Pet_Rosel_4){reference-type="ref" reference="itm:Pet_Rosel_4"}) in dimension greater than one is given by Examples [Example 128](#example_line-wise_euclidean_not_euclidean){reference-type="ref" reference="example_line-wise_euclidean_not_euclidean"} and [Example 130](#example_space_cell-wise_euclidean_not_metrizable){reference-type="ref" reference="example_space_cell-wise_euclidean_not_metrizable"} in the Appendix. # Examples {#section:examples} In this appendix we compile examples that witness the heterogeneity of definable topological spaces with reference to their (definable) topological properties, and help frame the results in this paper and their limitations when trying to improve or generalize them. The examples are given in the language $(0,1,+,-,\cdot, <)$, where $\mathcal{R}$ is assumed to expand an ordered group $(R,0,+,-,<)$ or an ordered field $(R,0,1,+,-,\cdot, <)$ whenever the corresponding function symbols are involved. Since we are working in the generality of an o-minimal structure, it is important to note that we will not address certain classical topological properties of definable topological spaces, because they are dependent on the specifics of the underlying structure $\mathcal{R}$. These include compactness, connectedness, separability, normality or metrizability. We consider however definable versions of these properties (as per the definitions in this paper). All the examples that are generalizations of classical topological spaces (e.g. definable Split Interval (Example [Example 117](#example:split_interval){reference-type="ref" reference="example:split_interval"}), definable Alexandrov Double Circle (Example [Example 126](#example:alex_double_circle){reference-type="ref" reference="example:alex_double_circle"})) behave, in terms of their definable topological properties, exactly like their classical counterparts. The only exception to this is the Definable Moore Plane (Example [Example 125](#example:Moore_plane){reference-type="ref" reference="example:Moore_plane"}), which is definably normal. We begin by recalling the key examples of definable topological spaces that are used and studied extensively throughout the paper. **Example 114** (Euclidean topology). The euclidean topology $\tau_e$ on $R^n$ has definable basis $$\left\{ \prod_{1\leq i \leq n}(x_i,y_i) : x_i<y_i,\, 1\leq i \leq n\right\}.$$ It is $T_3$, definably separable (Proposition [Proposition 8](#prop:def-sep_eucl_disc_llt){reference-type="ref" reference="prop:def-sep_eucl_disc_llt"}([\[itm1:sep-spaces-examples\]](#itm1:sep-spaces-examples){reference-type="ref" reference="itm1:sep-spaces-examples"})), definably connected and definably metrizable. It is moreover definably compact if and only if it is restricted to a closed and bounded set [@pet_stein_99]. **Example 115** (Discrete topology). The discrete topology $\tau_s$ on $R^n$ has definable basis $$\{\{x\} : x\in R^n\}.$$ Note that this topology is definable on any definable set in any model-theoretic structure. It is $T_3$ and definably metrizable. **Example 116** (Half-open interval topologies). The right half-open interval topology (or lower limit topology) $\tau_r$ has definable basis $$\{ [x,y) : x, y\in R,\, x<y\}.$$ The space $(\mathbb{R},\tau_r)$ is classically called the Sorgenfrey Line. The left half-open interval topology (or upper limit topology) $\tau_l$ has definable basis $$\{ (x,y] : x, y\in R,\, x<y\}.$$ These topologies are $T_3$ and definably separable (Proposition [Proposition 8](#prop:def-sep_eucl_disc_llt){reference-type="ref" reference="prop:def-sep_eucl_disc_llt"}([\[itm3:sep-spaces-examples\]](#itm3:sep-spaces-examples){reference-type="ref" reference="itm3:sep-spaces-examples"})). They are also totally definably disconnected (the only definably connected subspaces are singletons) and not definably metrizable (see Proposition [Proposition 17](#prop:tau_r not metrizable){reference-type="ref" reference="prop:tau_r not metrizable"}). **Example 117** (Definable Alexandrov Double Arrow space or definable Split Interval). Let $X=[0,1]\times \{0,1\}$. The definable Alexandrov Double Arrow space (or definable Split Interval) is the space $(X,\tau_{lex})$, where $\tau_{lex}$ denotes the topology induced by the lexicographic order on $X$. The space is classically called the Alexandrov Double Arrow space (or Split Interval) when $\mathcal{R}$ expands $(\mathbb{R},<)$. It is $T_3$, definably compact, definably separable and totally definably disconnected. It is not definably metrizable since the bottom line $[0,1]\times \{0\}$ is definably homeomorphic to $([0,1], \tau_l)$ and the top line $[0,1]\times \{1\}$ to $([0,1],\tau_r)$. It is also worth noting that $(X,\tau)$ does not satisfy the **fdi**, since $\partial([0,1]\times\{0\})=[0,1)\times\{1\}$. Moreover, one may show that $[0,1]\times \{0\}$ is not a boolean combination of open definable sets, which was a tameness condition for definable topologies considered by Pillay in [@pillay87]. The following two examples, the definable $n$-split interval and definable Alexandrov $n$-line, were already introduced in Definitions [Definition 54](#dfn:lex){reference-type="ref" reference="dfn:lex"} and [Definition 55](#example_n_line_0){reference-type="ref" reference="example_n_line_0"}, and play a crucial role in Theorem [Theorem 56](#them_ADC_or_DOTS){reference-type="ref" reference="them_ADC_or_DOTS"}. They were motivated by the classical Split Interval (see Example [Example 117](#example:split_interval){reference-type="ref" reference="example:split_interval"} above) and Alexandrov Double Circle (see Example [Example 126](#example:alex_double_circle){reference-type="ref" reference="example:alex_double_circle"} below) respectively. **Example 118** (Definable $n$-split interval). Let $n>0$. We call the space $(R\times\{0,\ldots, n-1\}, \tau_{lex})$ the definable $n$-split interval. If $n=1$, then note that $(R\times\{0\},\tau_{lex}$)$=(R\times\{0\}, \tau_e)$. If $n>1$, then, by analogy to the definable Split Interval (Example [Example 117](#example:split_interval){reference-type="ref" reference="example:split_interval"}), the definable $n$-split interval is $T_3$, definably separable, totally definably disconnected, not definably metrizable and it does not have the **fdi**. Furthermore, it is not definably compact but every subspace of the form $I\times \{0,\ldots, n-1\}$, where $I\subseteq R$ is a closed and bounded interval, is definably compact. Moreover, for any $0<n<m<\omega$, the spaces $(R\times\{0,\ldots, n-1\}, \tau_{lex})$ and $(R\times\{0,\ldots, m-1\}, \tau_{lex})$ are not definably homeomorphic. In fact, they are not even in definable bijection, since they have different Euler characteristic (see [@dries98 Chapter 4]). Specifically, observe that every finite cell partition of $R\times\{0,\ldots, n-1\}$ will contain $n$ more cells of dimension one than points. And similarly every cell partition of $R\times\{0,\ldots, m-1\}$ will contain $m$ more cells of dimension one than points. On the other hand, by o-minimal cell decomposition, every definable injection from $R\times\{0,\ldots, n-1\}$ to $R\times\{0,\ldots, m-1\}$ can be decomposed into disjoint definable bijections between singletons and one-dimensional cells, and so any such injection is not surjective. **Example 119** (Definable Alexandrov $n$-line). For any $y<x<z$ in $R$, let $$A(x,y,z)=\{\langle x,0 \rangle\} \cup (((y,z)\setminus\{x\}) \times R).$$ Let $\tau_{Alex}$ be the topology on $R^2$ with definable basis $$\{ A(x,y,z) : y<x<z\} \cup \{ \{\langle x,y \rangle\} : y \neq 0\}.$$ Let $n>0$. The definable Alexandrov $n$-line is the definable topological space $(R\times\{0,\ldots, n-1\},\tau_{Alex})$. In the case that $n=1$, this is a definable analogue of the classical Alexandrov double of the space $(R,\tau_e)$ (see [@engelking68]). This space is $T_3$. If $n=1$, then it is simply the euclidean topology on $R\times\{0\}$. Suppose that $n>1$. The subset $R\times \{i\}$ for any $i>0$ contains only isolated points, and so the space is not definably separable. For any closed and bounded interval $I\subseteq R$, the subspace $I\times\{0,\ldots,n-1\}$ is definably compact but not definably separable. It follows that it is not definably metrizable (see [@walsberg15 Lemma 7.4], which states that any definably compact definable metric space is definably separable). Hence the definable Alexandrov $n$-line, for $n>1$, is not definably metrizable. For any $0<n<m<\omega$, the sets $R\times\{0,\ldots, n-1\}$ and $R\times\{0,\ldots, n-1\}$ are not in definable bijection (see Example [Example 118](#example:n-split){reference-type="ref" reference="example:n-split"}), and so in particular the spaces $(R\times\{0,\ldots, n-1\}, \tau_{Alex})$ and $(R\times\{0,\ldots, m-1\}, \tau_{Alex})$ are not definably homeomorphic. The next five examples are provided in order to illustrate the necessity of certain hypotheses in some of the key results of this paper, in particular Proposition [Proposition 42](#prop_T2_frontier_ineq_regular){reference-type="ref" reference="prop_T2_frontier_ineq_regular"}, Proposition [Proposition 43](#them_main){reference-type="ref" reference="them_main"}, Corollary [Corollary 44](#cor:general_them_main){reference-type="ref" reference="cor:general_them_main"}, Theorem [Theorem 51](#them 1.5){reference-type="ref" reference="them 1.5"}, Corollary [Corollary 52](#cor 1.5){reference-type="ref" reference="cor 1.5"} and Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}. **Example 120**. Consider the following definable basis for a topology on $R$. $$\{ (-\infty,x] : x\in R \}.$$ The resulting space is $T_0$ but not $T_1$. Any subspace with more than one element fails to be $T_1$; in particular, no interval subspace of this space has the euclidean, discrete or half-open interval topologies. Consequently, the $T_1$ assumption in Proposition [Proposition 43](#them_main){reference-type="ref" reference="them_main"} and Corollary [Corollary 44](#cor:general_them_main){reference-type="ref" reference="cor:general_them_main"} (the definable version of the Gruenhage 3-element-basis Conjecture) cannot be weakened to $T_0$. **Example 121**. Consider the definable family of sets $$\{ (-\infty,x) \cup (y,z) : x<y<z\}.$$ It is a basis for a topology $\tau$ on $R$ that is $T_1$ but not Hausdorff. Any finite definable partition of $R$ must include an interval of the form $(-\infty, x)$, whose subspace topology is not Hausdorff. In particular, $(R,\tau)$ cannot be decomposed into finitely many definable subspaces with the euclidean, discrete or half-open interval topologies. It follows that the Hausdorffness assumption in Theorem [Theorem 51](#them 1.5){reference-type="ref" reference="them 1.5"} (and ~~hence~~ in Corollary [Corollary 52](#cor 1.5){reference-type="ref" reference="cor 1.5"}) cannot be weakened to $T_1$-ness. **Example 122**. Let $X=[0,1) \cup \{2\}$ and consider a topology $\tau$ on $X$ such that the subspace topology $\tau|_{[0,1)}$ is euclidean and a basis of open neighbourhoods of $\{2\}$ is given by $$\{ (0,x) \cup \{2\} : 0<x<1\}.$$ This topology is clearly definable and $T_1$ but not Hausdorff, since points $0$ and $2$ fail to have disjoint neighbourhoods. In particular, it is not regular. It is easy to observe that it satisfies the **fdi**. Since it fails to be regular, it illustrates the necessity of the Hausdorffness assumption in Proposition [Proposition 42](#prop_T2_frontier_ineq_regular){reference-type="ref" reference="prop_T2_frontier_ineq_regular"}. Moreover, we note, by considering the partition into subspaces $(0,1)$ and $\{2\}$, that this space is cell-wise euclidean. So it is also not true that every cell-wise euclidean one-dimensional space is Hausdorff, and in particular affine. **Example 123**. Let $X=R \times\{0,1\}$ and consider a topology $\tau$ on $X$ given by the basis $$\{\{\langle x,0\rangle\}\cup ((x,y)\times \{1\}) : x<y\} \cup \{(z,x]\times \{1\} : z<x\}.$$ This space is Hausdorff but not regular, since $R\times \{0\}$ is a closed set and, for any $x\in R$ and any neighbourhood $U=(z,x]\times \{1\}$ of $\langle x,1\rangle$, we have $cl_\tau (U) \cap (R\times\{0\})=[z,x)\times\{0\} \neq \emptyset$. Moreover, note that, since $\partial_\tau (R\times \{1\})=R\times\{0\}$, this space does not satsify the **fdi**. This example therefore illustrates the necessity of the assumption of satisfying the **fdi** in Proposition [Proposition 42](#prop_T2_frontier_ineq_regular){reference-type="ref" reference="prop_T2_frontier_ineq_regular"}. This example also illustrates a key fact about our notion of definable separability (Definition [Definition 5](#dfn:separable){reference-type="ref" reference="dfn:separable"}). Recall that any definable subspace of a definably separable definable metric space is also definably separable (see Lemma [Lemma 18](#remark_def_sep){reference-type="ref" reference="remark_def_sep"}). The space $(X,\tau)$ in the current example is definably separable, but the subspace $R\times \{0\}$ is infinite and discrete, hence $(X,\tau)$ is not definably metrizable, and, moreover, we see that definable separability, much like separability, is not in general a hereditary property. **Example 124**. Let $X=\{ \langle 0,0 \rangle\} \cup [0,1)\times (0,1)$. Consider the topological space $(X,\tau)$, where the subspace $X\setminus \{\langle 0,0 \rangle\}$ is euclidean, and a basis of open neighbourhoods for $\langle 0,0 \rangle$ is given by sets $$A(t)=\{\langle 0,0\rangle\} \cup \left( (0,1)\times (0,t) \right),$$ for $0<t<1$. The topology $\tau$ is clearly definable and Hausdorff. Moreover, for any $0<t<1$, the $\tau$-closure of $A(t)$ is $\{\langle 0,0 \rangle\} \cup [0,1) \times (0, t]$, and so the space is not regular, since the point $\langle 0,0\rangle$ and the closed set $\{0\}\times (0,1)$ are not separated by neighbourhoods. Since $(X,\tau)$ is $T_1$ and the subspace $X\setminus \{\langle 0,0\rangle\}$ is euclidean, it easily follows that $(X,\tau)$ satisfies the **fdi**. Hence $(X,\tau)$ is Hausdorff and satisfies the **fdi** but fails to be regular, thus is a counterexample to the generalization of Proposition [Proposition 42](#prop_T2_frontier_ineq_regular){reference-type="ref" reference="prop_T2_frontier_ineq_regular"} to spaces of dimension greater than one. Moreover, the space can be partitioned into two euclidean subspaces, namely $\{\langle 0,0 \rangle\}$ and $X\setminus \{\langle 0,0\rangle\}$; in particular it contains no definable copy of an interval with either the discrete or the right half-open interval topology. However, it is not metrizable, since it is not regular. Hence it is a counterexample to a generalization of Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"} to spaces of dimension two. For the remaining examples, let $B_2(\langle x,y \rangle,t)$, for $\langle x,y\rangle\in R^2$ and $t>0$, denote the ball in the $2$-norm of center $\langle x, y\rangle$ and radius $t$, namely $$B_2(\langle x,y\rangle,t)=\{\langle x',y'\rangle\in R^2 : (x-x')^2+(y-y')^2<t^2\}.$$ **Example 125** (Definable Moore Plane). Let $X=\{\langle x,y \rangle\in R^2: y\geq 0\}$ be the closed upper half-plane. Let $\mathcal{B}_e$ be a definable basis for the euclidean topology in $\{\langle x,y \rangle\in R^2 : y>0\}$ and, for any $x\in R$ and $\epsilon >0$, let $$A(x,\varepsilon)=B_2(\langle x,\varepsilon \rangle,\varepsilon)\cup\{\langle x,0 \rangle\}.$$ The family $\mathcal{B}=\mathcal{B}_e \cup \{ A(x,t) : x\in R, t>0\}$ is clearly definable and forms a basis for a topology $\tau$. We call the space $(X,\tau)$ the definable Moore Plane. This space is $T_3$ and definably separable but not definably metrizable since the subspace $R\times \{0\}$ is infinite and discrete (see Lemma [Lemma 18](#remark_def_sep){reference-type="ref" reference="remark_def_sep"}). When $\mathcal{R}$ expands the field of reals, the Moore Plane is a classical example of a separable non-normal space (and in particular, it is not metrizable). It is worth noting that, even though our definition of definable normality (Definition [Definition 38](#definition:dfbly_normal){reference-type="ref" reference="definition:dfbly_normal"}) seems the natural adaptation of the classical notion, the classical Moore Plane fails to be normal, but one may show that the definable Moore Plane is definably normal. This suggests that our notion of definable normality might not be adequate. Moreover, Fornasiero also considered this same notion of definable normality in unpublished work [@fornasiero] (seen in private correspondence) where he showed that a definable topological space that is definably compact (in the sense of condition [\[dfn:directed-compact\]](#dfn:directed-compact){reference-type="eqref" reference="dfn:directed-compact"}) and Hausdorff is not necessarily definably normal, in contrast to the classical fact that a compact Hausdorff space is normal. (However, he did also show that if a definably compact, Hausdorff space is given by a definable uniformity (see [@simon_walsberg19] for definitions), then it is indeed definably normal in this sense.) **Example 126** (Definable Alexandrov Double Circle). Let $X=C_1 \cup C_2$, where $C_1$ and $C_2$ denote respectively the unit circle and circle of radius two in $R^2$ centered at the origin. Let $f:C_1\rightarrow C_2$ be the natural $e$-homeomorphism given by $x\mapsto 2x$. Let $$\mathcal{B}_1=\{ \left( B_2(x,t)\cap C_1 \right) \cup f(B_2(x,t)\cap C_1 \setminus\{x\}) : x\in C_1, t>0\}$$ and $\mathcal{B}_2=\{ \{x\} : x\in C_2\}$. The definable Alexandrov Double Circle is the topology on $X$ generated by the basis $\mathcal{B}_1\cup \mathcal{B}_2$. This space is definably compact and Hausdorff, but not definably separable, since $C_2$ is an infinite definable set of isolated points. It follows (see [@walsberg15 Lemma 7.4]) that it is not definably metrizable. It also fails to satisfy the **fdi**, since the outer circle $C_2$ is a dense subset. When $\mathcal{R}$ expands the field of reals this space is simply called the Alexandrov Double Circle and is a classical example of a compact non-separable space (hence one that is not metrizable). The following example shows that there exists a Hausdorff two-dimensional definable topological space that does not contain a definable copy of an interval with either the discrete or the lower limit topology but still fails to be cell-wise euclidean. This shows that Remark [Remark 95](#remark_cell-wise_euclidean){reference-type="ref" reference="remark_cell-wise_euclidean"} cannot be generalized to higher dimensions. In particular, this example is not affine but, by Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}, any one-dimensional subspace is affine (i.e. it is "line-wise\" affine). This is proved below in Proposition [Proposition 127](#prop:line-wise_euclidean_not_euclidean){reference-type="ref" reference="prop:line-wise_euclidean_not_euclidean"}. **Example 128** (The definable hollow plane). We construct a basis for a topology $\hat{\tau}$ on $R^2$ by considering, for each point $x$, a basis of open neighbourhoods given by open euclidean balls without the graph of $f(t)=t^2$, for $0<t$, translated to have its origin at $x$. That is, for a given $x=\langle x_1,x_2 \rangle\in R^2$, let $\Gamma_x:=\{\langle x_1+t, x_2+t^2 \rangle: t>0\}$. Now let $\mathcal{B}$ be given by sets $$A(x,t) =B_2(x,t)\setminus \Gamma_x,$$ for $x\in R^2$ and $t>0$. We call $A(x,t)$ a $\hat{\tau}$-ball of center $x$ and radius $t$. We claim that $\mathcal{B}$ is a basis for a topology on $R^2$. In order to prove this, let $A_0$ and $A_1$ be intersecting sets in $\mathcal{B}$ and let $x\in A_0 \cap A_1$. We show that there exists some $\varepsilon>0$ such that $A=A(x,\varepsilon)$ satisfies that $A(x,\varepsilon)\subseteq A_0 \cap A_1$. For any $y\in R^2$ and $t>0$, let $A^*(y,t)=A(y,t)\setminus \{y\}$. Note that, for any $A\in \mathcal{B}$, the set $A^*$ is $e$-open. Case 1: $x\in A^*_0 \cap A^*_1$. : Since $A^*_0\cap A^*_1$ is $e$-open, there is some $\varepsilon>0$ such that $B_2(x,\varepsilon)\subseteq A^*_0\cap A^*_1\subseteq A_0\cap A_1$. Hence we may take $A=A(x,\varepsilon)\subseteq B_2(x,\varepsilon)$. Case 2: $x\notin A^*_0 \cap A^*_1$. : Without loss of generality, suppose that $A_0=A(x,\varepsilon_0)$, for some $\varepsilon_0>0$. If $A_1=A(x,\varepsilon_1)$, for some $\varepsilon_1>0$, let $\varepsilon=\min\{\varepsilon_0, \varepsilon_1\}$ and $A=A(x,\varepsilon)$. Otherwise, by analogy to Case 1, let $\varepsilon_2>0$ be such that $A(x,\varepsilon_2)\subseteq A^*_1$ and let $A=A(x,\varepsilon)$, where $\varepsilon =\min\{\varepsilon_0,\varepsilon_2\}$. So we may conclude that $\mathcal{B}$ is a topological basis. Let $\hat{\tau}$ be the corresponding topology. We call $(R^2,\hat{\tau})$ the definable hollow plane. Every $e$-open set in $R^2$ is also $\hat{\tau}$-open, i.e. $\tau_e\subsetneq\hat{\tau}$. In particular $(R^2,\hat{\tau})$ is Hausdorff. It fails, however, to be regular, since it is easy to check that, for any $x\in R^2$ and $\varepsilon>0$, $cl_{\hat{\tau}} A(x,\varepsilon) =cl_e B_2(x,\varepsilon)$, and so, for every $\hat{\tau}$-neighbourhood $A$ of $x$, $cl_{\hat{\tau}}A \cap \Gamma_x \neq \emptyset$. This space, however, is definably separable, which follows from ([\[itm:broken_disk_1\]](#itm:broken_disk_1){reference-type="ref" reference="itm:broken_disk_1"}) in the following proposition. One may also show that it is definably connected. **Proposition 127**. The following are properties of the definable hollow plane $(R^2,\hat{\tau})$. (1) [\[itm:broken_disk_1\]]{#itm:broken_disk_1 label="itm:broken_disk_1"} Any one-dimensional subspace of $(R^2,\hat{\tau})$ is affine. (2) [\[itm:broken_disk_2\]]{#itm:broken_disk_2 label="itm:broken_disk_2"} No two-dimensional subspace of $(R^2,\hat{\tau})$ is cell-wise euclidean. In particular, no two-dimensional subspace of $(R^2,\hat{\tau})$ is affine. *Proof.* Statement ([\[itm:broken_disk_2\]](#itm:broken_disk_2){reference-type="ref" reference="itm:broken_disk_2"}) is obvious from the definition. We prove ([\[itm:broken_disk_1\]](#itm:broken_disk_1){reference-type="ref" reference="itm:broken_disk_1"}). By Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}, it suffices to show that $(R^2,\hat{\tau})$ contains no subspace definably homeomorphic to an interval with either the discrete or the right half-open interval topology. Towards a contradiction, let $I\subseteq R$ be an interval and let $f:(I, \mu) \hookrightarrow (R^2, \hat{\tau})$ be a definable embedding, where $\mu\in\{\tau_r,\tau_s\}$. By o-minimality, after restricting $f$ if necessary, we may assume that $f$ is an $e$-embedding too. Since $\mu=\tau_r$ or $\mu=\tau_s$ and $f$ is an embedding, it follows that, for any $t\in I$, $\mathop{\mathrm{\hat{\tau}{\operatorname{-}}lim}}_{s\rightarrow t^-} f(s)\neq f(t)$ (see Proposition [Proposition 27](#prop_cont_lim){reference-type="ref" reference="prop_cont_lim"}). So, by o-minimality, for every $t\in I$, there exists some $\varepsilon_t>0$ and $s'<t$ in $I$ such that, for all $s'<s<t$, $f(s)\notin A(f(t),\varepsilon_t)$. However, since $f$ is an $e$-embedding, there is also some $s'<s''<t$ such that, for all $s''<s<t$, $f(s)\in B_2(f(t),\varepsilon_t)$, and so $f[(s'',t)]\subseteq \Gamma_{f(t)}$. For any $t\in I$, let $s_t=\inf \{ s\in I : \,s<t,\, f[(s,t)]\subseteq \Gamma_{f(t)}\}$. This family is definable uniformly in $t\in I$. Since $s_t < t$ for all $t \in I$, by o-minimality, passing to a subinterval on which $t \mapsto s_t$ is continuous, if necessary, there exists an interval $J\subseteq I$ such that, for every $t\in J$, $s_t < J$. In other words, for every $s<t$ in $J$, it holds that $f(s)\in \Gamma_{f(t)}$. We now claim that, for any two distinct points $y,z\in R^2$, $|\Gamma_y \cap \Gamma_z|=1$. In that case, we have a contradiction, since we have shown that, for any $s, s', t, t'\in J$, if $s<s'<t<t'$, then $\{f(s), f(s')\} \subseteq \Gamma_{f(t)} \cap \Gamma_{f(t')}$. It therefore remains to prove the claim. Let $y= \langle y_1,y_2 \rangle\in R^2$ and $z=\langle z_1,z_2 \rangle\in R^2$, with $y \neq z$. Suppose that there exist $t, s >0$ such that $$\langle y_1+t, y_2+t^2 \rangle= \langle z_1+s, z_2+ s^2 \rangle.$$ If $y_1=z_1$, then we would have $t=s$ and hence $y=z$, so in fact we must have $y_1 \neq z_1$. We then substitute $s=t+y_1-z_1$ into $t^2=z_2-y_2+s^2$ in order to get $$t^2=z_2-y_2+t^2 +2t(y_1-z_1)+(y_1-z_1)^2,$$ and hence $$t=\frac{y_2-z_2-(y_1-z_1)^2}{2(y_1-z_1)}, \qquad s=\frac{y_2-z_2+(y_1-z_1)^2}{2(y_1-z_1)},$$ which gives us the unique point in $\Gamma_y \cap \Gamma_z$, which proves the claim. ◻ In light of Example [Example 128](#example_line-wise_euclidean_not_euclidean){reference-type="ref" reference="example_line-wise_euclidean_not_euclidean"}, a natural question to ask is whether or not we may instead obtain an analogue to Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"} for spaces of all dimensions by substituting the condition of having a definable copy of an interval with either the $\tau_r$ or the $\tau_s$ topology (condition ([\[itm:them_main_2\_1\]](#itm:them_main_2_1){reference-type="ref" reference="itm:them_main_2_1"}) in Theorem [Theorem 92](#them_main_2){reference-type="ref" reference="them_main_2"}) for simply not being cell-wise euclidean. The answer, even if adding the additional assumption that the space be regular, is no, as witnessed by the following, our final example. **Example 130** (Space that is $T_3$ and cell-wise euclidean but not definably metrizable). Let $X=\{\langle x,y \rangle \in R^2 : y\geq 0\}$ be the closed upper half-plane. Let $\mathcal{B}_e$ be a definable basis for the euclidean topology in $\{\langle x, y \rangle\in R^2 : y>0\}$. For any $x\in R$ and $t>0$, let $$A(x,t) = \{\langle x,0 \rangle \}\cup \{ \langle x', y\rangle\in R^2 : |x'-x|<t,\, 0\leq y<t|x'-x| \}.$$ Note that, for every $x\in R$ and $t>0$, there is $t'>0$ such that $A(x,t')\subseteq B_2(\langle x, 0 \rangle,t)$, while the converse is not true (we have that $B_2(\langle x, 0 \rangle,t') \nsubseteq A(x,t)$ for every $x\in R$ and $t,t'>0$). Moreover, for every $x\in R$, the family $\{A(x,t) : t>0\}$ is nested and, for every $t>0$, the set $A(x,t)\setminus\{\langle x,0 \rangle \}$ is $e$-open in $X$. From these three facts it follows, in a manner similar to the case analysis in Example [Example 128](#example_line-wise_euclidean_not_euclidean){reference-type="ref" reference="example_line-wise_euclidean_not_euclidean"} , that the definable family $\mathcal{B}_{\tilde{\tau}}= \mathcal{B}_e \cup \{A(x,t) : x\in R, t>0\}$ is a basis for a topology $\tilde{\tau}$ on $X$. Since we have $\tau_e|_X \subsetneq \tilde{\tau}$, the topology $\tilde{\tau}$ is Hausdorff. Note that, for every $x\in R$ and $t>0$, we have $cl_{\tilde{\tau}} A(x,t)=cl_e A(x,t)$, and so $(X,\tilde{\tau})$ is also regular. Moreover, the disjoint subspaces $\{\langle x,y \rangle : x\in R, y>0\}$ and $\{\langle x,y \rangle: x\in R, y=0\}$ are both euclidean, i.e. the space is cell-wise euclidean. In particular, the space is definably separable. Finally, it is also definably connected. When $\mathcal{R}$ expands the field of reals this space is separable but not second countable and thus not metrizable. From the completeness of the theory of real closed fields it follows that there is no metric on $X$ definable in the language of ordered rings that induces $\tilde{\tau}$. We show that this holds in greater generality. **Proposition 129**. The space $(X,\tilde{\tau})$ is not definably metrizable. *Proof.* Towards a contradiction, suppose that $(X,\tilde{\tau})$ is definably metrizable with definable metric $d$. For every $x\in R$, let $$r_x=\sup\{0<t<1 : B_d(\langle x,0 \rangle,t)\cap (\{x\}\times (0,\infty))=\emptyset\}.$$ Note that, by definition of the neighbourhoods $A(x,t)$, we have necessarily that $r_x>0$, for every $x\in R$. By o-minimality, there exists an interval $I\subseteq R$ and some $r>0$ such that, for every $x\in I$, we have $r\leq r_x$. Now fix $x\in I$ and consider the $d$-ball $B_d(\langle x,0\rangle,r/2)$. By definition of $\tilde{\tau}$, there exists some $y\in I\setminus \{x\}$ and some $s>0$ such that $\{y\}\times [0,s] \subseteq B_d(\langle x,0 \rangle,r/2)$. But then, by the triangle inequality, $d(\langle y,0 \rangle, \langle y,s \rangle)\leq d(\langle y,0 \rangle, \langle x,0 \rangle)+d(\langle x,0 \rangle, \langle y,s\rangle)<r \leq r_y$. This is a contradiction since, for every $0<t<r_y$, $B_d(\langle y,0\rangle,t)\cap (\{y\}\times (0,\infty))=\emptyset$. ◻ 10 Andj́ar Guerrero, P. Cardinality of definable families of sets in o-minimal structures. , 2023. Andújar Guerrero, P. Definable separability and second countability in o-minimal structures. In preparation. Andújar Guerrero, P. . PhD thesis, Purdue University, 2021. Andújar Guerrero, P. Types, transversals, and definable compactness in o-minimal structures. , 2021. Andújar Guerrero, P., and Johnson, W. Around definable types in $p$-adically closed fields. arXiv:2208.05815, 2022. Andújar Guerrero, P., Thomas, M., and Walsberg, E. Directed sets and topological spaces definable in o-minimal structures. , 3 (2021), 989--1010. Arefiev, R. 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arxiv_math

--- abstract: | The $3n+1$ problem is an extremely simple to state, extremely hard to solve, problem. An interconnection dynamical system model, Component Connection Model, is presented to model the $3n+1$ problem. The model consists of a set of equations describing the component dynamics and component interconnections. This paper attempts to prove both Collatz conjecture and Syracuse conjecture. Syracuse conjecture is a $\textbf{\textit{2N+1}}$-version of Collatz conjecture, where $\textbf{\textit{2N+1}}$ is the positive odd integers. $\textbf{\textit{2N+1}}$ is partitioned into 4 disjoint sets yielding that the Syracuse sequences contain the terms either with the values of $6t+1$ or $6t+5$, and the seed of the sequence can be any positive odd integers. Two incoming term matrices are developed to describe the system components, while the component interconnections are described by a connection tree. The connection tree shows that all Syracuse sequences and Collatz sequences can eventually reach the number of 1. This proves that both conjectures are true. author: - Chin-Long Wey title: Proof of Collatz Conjecture --- *Key words:* Collatz Conjecture, Syracuse Conjecture, Component Connection Model, Dynamical System, Component Dynamics, Component Interconnection. # Introduction Let $\textbf{\textit{N}}=\{0,1,2,\cdots \}$ and $\textbf{\textit{N+1}}=\{1,2,\cdots \}$ denote the natural numbers and the positive integers, respectively, while $\textbf{\textit{2N+1}}=\{1,3,5,\cdots \}$ and $\textbf{\textit{2N+2}}=\{2,4,6,\cdots \}$ are the positive odd and even integers, respectively. The *3n+1 problem*, or *Collatz problem*, is one the hardest math problems, yet still unsolved. The Collatz sequence is $COL(n)$=$\{n=n_0, n_1,\cdots, n_c\}$, for all $n_i\in \textbf{\textit{2N+1}}$. $n$ is called the *seed* and $n_i$ is called the *term*, where the Collatz function is defined as $n_{i+1}=Col(n_i)=3n_i+1$, if $n_i$ is odd, and $n_{i+1}=Col(n_i)=n_i/2$, if $n_i$ is even, $i=0, 1,\cdots, c$. The unsolved 3n+1 problem is to prove or disprove that the sequences always eventually reach the number of 1. Syracuse conjecture is a (***2N+1***)-version of Collatz conjecture. The Syracuse sequence $SYR(n)=\{n=n_0, n_1,\cdots , n_s\}$, $n_i \in \textbf{\textit{2N+1}}$, and $n_{i+1}=Syr(n_i)=(3n_i+1)/2^d$, $d \in \textbf{\textit{2N+1}}$. The extensive surveys and historical discussion of Collatz and Syracuse conjectures can refer \[1-5\]. In a Collatz sequence, if $m \notin COL(n)$ and $n=Col(m)$, m is called the *incoming term* of n. $COL(m)=\{m, n=n_0, n_1, n_2,\cdots , n_c\}$ is also a Collatz sequence and COL(n) is a sub-sequence of COL(m). *The 3n+1 problem is an extremely simple to state, extremely hard to solve, problem* \[1\]. This paper presents a system model, namely, *Component Connection Model (CCM)*, to describe the 3n+1 problem. The CCM of a dynamic system consists of a set of equations describing the component dynamics and component interconnections \[6-8\]. The system input vector u yields the system output response vector y, where $a_i$ and $b_i$ are the component input and output vectors, respectively. For the 3n+1 problem, the input u is any positive integer n, and the output is y=1 if the 3n+1 problem is true. If $a_i$ is the *present term*, then $b_i$ is the *next term*, and $b_i=Col(a_i)$ or $b_i=Syr(a_i)$. The term $b_i$ is linked to the next $a_{i+1}$ though the connection block. ![[\[fig:CCM\]]{#fig:CCM label="fig:CCM"}System mode: Component Connection Model.](Fig1.jpg){#fig:CCM width="0.5\\linewidth"} Note that the components in the CCM can be either functions, modules, or subsystems. A simple component model may cause very complicated component interconnections. Conversely, the complexity of component connections can be significantly reduced if modules or subsystems are used as the components. There exists a design trade-off between component and connection complexities. Most of research works on $3n+1$ problem focus on modelling the component by a simple Syracuse or Collatz function causing extremely complicated component connections. In this paper, from the observation of $Syr(1)=1, Syr(5)=1, Syr(21)=1, \cdots$, the numbers $1, 5, 21, \cdots$, are called the *incoming terms* of 1 and they have the same functional value of 1. Let $m=1, V(m)=4m+1$, and $Syr(V^p(m))=1, p\in \textbf{\textit{N}}$. If a component is modelled by a module which consists of $Syr(m), m, V(m), \cdots, V^p(m)$, the complexity of the connection model can be reduced significantly. In the next section, the properties of the incoming term matrices are presented. The construction of the incoming term tree is discussed in Section 3. The Syracuse sequences generated from the tree is also proven to be free of non-trivial cycles and also bounded. In addition, all components will find a path to reach the component connected with the trivial cycle. This asserts the Syracuse conjecture to be true. Section 4 proves the Collatz conjecture. Finally, summary and concluding remarks are given in Section 5. # Incoming Term Matrices for Positive Odd Integers This section presents the component model and connection model of Syracuse sequences. Consider a Syracuse sequence $SYR(n)=\{n=n_0, n_1, n_2, \cdots , n_s\}$, the seed $n=n_0 \in \textbf{\textit{2N+1}}$, and the values of the remaining terms are either $6t+1$ or $6t+5$, where $t\in \textbf{\textit{N}}$. ## Component Model The positive odd integers ***2N+1*** is partitioned into 4 disjoint sets $\{8q+d\}$, d=1,3,5,7, and $q \in N$, i.e., $\{8d+1\} \cup \{8d+3\}\cup \{8d+5\} \cup \{8d+7\}=\textbf{\textit{2N+1}}$. Because of $\{8q+3\}\cup \{8q+7\}=\{4q+3\}$ and $\{12q+5\}\cup\{12q+11\}=\{6q+5\}$, their Syracuse functional values are $Syr(4q+3)=(3(4q+3)+1)/2=6q+5$, $Syr(8q+1)=(3(8q+1)+1)/4=6q+1$, and $Syr(8t+5)=(3n+1)/2^d$, as shown in Figure 2(a). ![[\[fig:Syr\]]{#fig:Syr label="fig:Syr"}Syracuse sequences: (a) Partitioned sets; Incoming term matrices, $I_a(p,q)$, a=1, 5; and (c) Contents of incoming term matrices $I_a(p,q)$.](Fig2.jpg){#fig:Syr width="0.9\\linewidth"} **Lemma 1**. *Let $S_a(t)=Syr(8t+a)$, a=1, 3, 5, 7. $S_5(4t)=S_1(t)$; $S_5(4t+1)=S_3(t)$; $S_5(4t+2)=S_5(t)$; and $S_5(4t+3)=S_7(t)$.* *Proof.* $S_5(t)=Syr(8t+5)=(3(8t+5)+1)/2^r=(24t+16)/2^r$=$(3t+2)/2^{r-3}\in 2N+1$ only if (i) $r=3$ and t is odd, or (ii) $r=4$ and t is even. Cases (i) and (ii) yield $S_5(t)=3t+2$ and $S_5(t)=3t/2+1$, respectively. Thus, $S_5(4t)=3(4t)/2+1=6t+1=S_1(t)$; $S_5(4t+1)=3(4t+1)+2=12t+5=S_3(t)$; and $S_5(4t+3)=12t+11=S_7(t)$. For $S_5(4t+2)=(3(4t+2)+2)/2^{r-3}$=$(3t+2)/2^{r-5} \in 2N+1$ only if (c) $r=5$ and t is odd, or (d) $r=6$ and t is even. In both cases, $S_5(4t+2)=S_5(t)$. ◻ **Remark 2**. *By Lemma 2.1, $\{S_5(t)\}\subseteq\{S_1(t)\}\cup\{S_3(t)\}\cup\{S_7(t)\}$, i.e., the values of $S_5(t)$ are nothing but just the duplicated values of $S_a(t)$, a=1,3,7. This leads to develop two incoming matrices $\{I_a(p,q)\}$, a=1,5. First, $I_1(0,q)=8q+1$ and $I_5(0,q)=4q+3$, then $I_a(p+1,q)=4I_a(p,q)+1$, $p \in \textbf{\textit{N}}$.* **Remark 3**. *Table A (In Appendix) shows the sample describing $\{S_5(t)\}\subseteq\{S_1(t)\}\cup\{S_3(t)\}\cup\{S_7(t)\},$ for interesting readers.* **Lemma 4**. *Let V(m)=4m+1, U(m)=(m-1)/4, Q(m)=4m+2* 1. *$V^p(m)=(m+1/3)4^p-1/3$ and $U^p(m)=(m+1/3)/4^p-1/3$;* 2. *$Q^p(t)=(t+2/3)4^p-2/3$.* *Proof.* (1) $V(m)=4m+1=(m+1/3)*4-1/3$, and $V^2(m)=4((m+1/3)*4-1/3)+1=(m+1/3)*4^2-1/3$. Assuming that $V^{p-1}(m)=(m+1/3)4^{p-1}-1/3$, $V^p(m)=V(V^{p-1}(m))=(m+1/3)*4^p-4/3+1=(m+1/3)*4^p-1/3$; $U(m)=(m-1)/4$, $V(U(m))=m$, $V^p(U^p(m))=m=(U^p(m)+1/3)4^p-1/3$, i.e., $U^p(m)=(m+1/3)/4^p-1/3$; (2) $Q(m)=4m+2=(m+2/3)*4-2/3$, similar to (1), $Q^p(m)=(m+2/3)4^p-2/3$; ◻ **Lemma 5**. *Let $Q(t)=4t+2$, if $m=8t+5$, $Syr(m)=S_5(t)=S_5(Q^p(t))$.* *Proof.* In the proof of Lemma 2.1, $S_5(4t+2)=(3t+2)/2^r-5 \in \textbf{\textit{2N+1}}$ only if (i) $r=5$ and t is odd, or (ii) $r=6$ and t is even. In both cases, $S_5(4t+2)=S_5(t)$, i.e., $S_5(Q(t))=S_5(t)$. Assuming that $S_5(Q^{p-1}(t))=S_5(t)$. Therefore, $S_5(Q^p(t))=S_5(Q^{p-1}(Q(t)))=S_5(Q(t))=S_5(t)$. Thus, $Syr(m)=S_5(t)=S_5(Q^p(t))$. ◻ **Lemma 6**. *Let $Q(t)=4t+2$, if $m=8t+5$, $Syr(m)=S_5(t)=S_5(Q^p(t))$.* *Proof.* In the proof of Lemma 2.1, $S_5(4t+2)=(3t+2)/2^r-5 \in \textbf{\textit{2N+1}}$ only if (i) $r=5$ and t is odd, or (ii) $r=6$ and t is even. In both cases, $S_5(4t+2)=S_5(t)$, i.e., $S_5(Q(t))=S_5(t)$. Assuming that $S_5(Q^{p-1}(t))=S_5(t)$. Therefore, $S_5(Q^p(t))=S_5(Q^{p-1}(Q(t)))=S_5(Q(t))=S_5(t)$. Thus, $Syr(m)=S_5(t)=S_5(Q^p(t))$. ◻ **Theorem 7**. *If m is incoming term of n, $n=Syr(m)$, and $m=8t+5$, then $n=Syr(m)=Syr(V^p(m))$, for $p \in N$, where $V(m)=4m+1$.* *Proof.* $V(m)=4m+1$, $V^p(m)=(m+1/3)4^p-1/3=(8t+5+1/3)4^p-1/3=8(t+2/3)4^p-1/3=8[(t+2/3)4^p-2/3]+5$, by Lemma 2.4(2), $V^p(m)=8Q^p(t)+5$, where $Q(t)=4t+2$. Thus, $Syr(V^p(m))=Syr(8Q^p(t)+5)$. By Lemma 2.5, if $m=8t+5$, $Syr(V^p(m))=S_5(Q^p(t))=S_5(t)=Syr(m)$. ◻ **Remark 8**. *In $\{I_a(p,q)\}$, let $m=I_a(0,q)$, $V(m)=4m+1=4I_a(0,q)+1=I_a(p+1,q)$. By Lemma 2.7, $I_1(p,q)=V^p(8q+1)=[(6q+1)4^{p+1}-1]/3$ and $I_5(p,q)=V^p(4q+3)=[(6q+5)4^{p+1}-2]/6$. This yields that $I_a(p,q)=5 (mod 8)$, for $p \leq 1$. By Theorem 2.6, $Syr(I_1(p,q))= Syr(V^p(8q+1))=Syr(8q+1)=6q+1$, and $Syr(I_5(p,q))= Syr(V^p(4q+3))=6q+5$ for any $p \in \textbf{\textit{N}}$.* **Theorem 9**. *Consider the matrices $\{I_a(p,q)\},$ a=1 and 5,* 1. *$\{I_1(p,q)\} \cup \{I_5(p,q)\}=\textbf{\textit{2N+1}},$ for $p, q \in \textbf{\textit{N}}$, and $\{I_1(p,q)\} \cap \{I_5(p,q)\}=\phi$;* 2. *$\{I_1(p,q)\} \cup \{I_5(p,q)\}=\{8t+5\}, p \in \textbf{\textit{N+1}}.$* *Proof.* (1) Let $R(p,q)=\{I_1(p,q)\} \cup \{I_5(p,q)\}$. For p=0, $R(0,q)=\{8q+1\}\cup \{4q+3\}=\{2q+1\}-\{8q+5\}, R(1,q)=\{32q+5\}\cup \{16q+13\}=\{8q+5\}-\{32q+21\}$, thus $R(0,q)\cup R(1,q)=\{2q+1\}-\{32q+21\}.$ Let $m=8t+5$, by Lemma 2.4(1), $V(m)=4m+1$, $V(8t+5)=32q+21$. i.e., $R(0,q) \cup R(1,q)=\{2q+1\}-\{V(m)\}$. Similarly, $R(2,q)=\{128q+21\} \cup \{64q+53\}=\{32q+21\}-\{128q+85\}$, and $R(0,q)\cup R(1,q) \cup R(2,q)= \{2q+1\}-\{V^2(m)\}.$ For any r, $R(0,q) \cup R(1,q) \cup \cdots \cup R(r,q)=\{2q+1\}-\{V^r(m)\}$ approaches to {2q+1} as r goes to infinity. Therefore, $R(p,q)=\{I_1(p,q)\} \cup \{I_5(p,q)\}=\{2q+1\}=\textbf{\textit{2N+1}}$. $\{Syr(I_a(p,q))\}=\{6q+a\}, a=1,5, \{6q+1\}\cap \{6q+5\}=\phi$ implies that $\{I_1(p,q)\} \cap \{I_5(p,q)\}=\phi$. (2) $R(p,q)-R(0,q)=(\textbf{\textit{2N+1}})-R(0,q)= (\textbf{\textit{2N+1}})-(\{4q+3\} \cup \{8q+1\}=\{8q+5\})$, i.e., $R(p,q)=\{8t+5\}$ ◻ Figure 2(c) shows the two incoming term matrices $\{I_a(p,q)\}, a=1,5$. Each column of $\{I_a(p,q)\}$ represents a component which includes $I_a(0,q)=m, I_a(1,q)= V(m), I_a(r,q)=V^r(m)$, and so on. All $I_a(p,q)$ are the incoming terms of $c_x$, where $Syr(V^p(m))=c_x= 6t+a$, for $p\in \textbf{\textit{N}}$. By Theorem 2.9, $\{I_1(p,q)\}\cup \{I_5(p,q)\}=\textbf{\textit{2N+1}}$, and $\{I_1(p,q)\}\cup \{I_5(p,q)\}=\{6t+1\}\cup \{6t+3\}\cup \{6t+5\}$. This implies that the entries in both $\{I_1(p,q)\}$ and $\{I_5(p,q)\}$ are with the value of $6t+1$, or $6t+3$, or $6t+5$. As shown in Figure 2(c), the entries with the value in grey are $6t+5$, those in boldface are $6t+1$, and the remaining entries are with $6t+3$. For any $n\in \textbf{\textit{2N+1}}$, there exists one and only one entry (p,q) such that $6t+a=I_b(p,q)$, for $a, b\in \{1,5\}$. ![[\[fig:Comp\]]{#fig:Comp label="fig:Comp"}Component description: (a) a Component; (b) Interconnection of component corresponding to $I_1(p,0)$ in Level 0; and (c) Interconnection of component corresponding to $I_5(p,0)$ in Level 1.](Fig31.png){#fig:Comp width="0.9\\linewidth"} Figure 3(a) shows the component model. The component represents a column of $\{I_a(p,q)\}$. All $I_a(p,q)$ are the incoming terms of $c_x$. By Theorem 2.7, $Syr(V^p(m))=c_x=6t+a$, for $p \in \textbf{\textit{N}}$. The node $c_x$ connects to the internal input node $c_y$ of the next component, where $c_x=6t+1=c_y=I_b(p,q), b \in \{1,5\}$. Figure 3(b) presents the component $\{I_1(p,0)\}$, the column q=0 of $\{I_1(p,q)\}$, its node $c_x$ connects a trivial cycle $\{1,1, \cdots \}$. Figure 3(c) describes the component $I_5(p,0)$ which connects to node with the value of 5 in the component $I_1(p,0)$. The component connects will discussed in the following subsection. ## Connection Model This subsection presents the connection model, including the connection rule, construction of the connection tree, and the algorithm of finding a convergent path. The connection rule is summarized in Theorem 2.10. **Theorem 10**. *The internal input nodes of the component corresponding to Ia(p,q), a=1, 5, are connected as follows:* 1. *If $I_a(p,q)=3 \pmod {6}$ it has no external connections;* 2. *If $I_a(p,q)=1 \pmod {6}$, it is connected by $I_1(p,t_1)$, where $t_1=(n-1)/6;$* 3. *If $I_a(p,q)=5 \pmod {6}$, it is connected by $I_5(p,t_5)$, where $t_5=(n-5)/6;$* *Proof.* If $I_a(p,q)=3 \pmod {6}$, i.e., $I_a(p,q)$ is a multiple of 3 serving as the seed and has no external connection. If $n=I_a(p,q)=1 \pmod {6}$, i.e., $n=6t_1+1$, or $t_1=(n-1)/6$, then $I_a(p,q)$ is connected by the component corresponding to $I_1(p,t_1)$. Similarly, $n=I_a(p,q)=5 \pmod {6}$, i.e., $n=6t_5+5$, or $t_5=(n-5)/6$, then $I_a(p,q)$ is connected by the component corresponding to $I_5(p,t_5)$. ◻ The component connection tree is constructed starting from the component corresponding $I_1(p,0)$, at Level $\sharp 0$, which connects to the trivial cycle, as shown in Figure 3(b), where $I_1(p,0)=1, 5, 21, 85$, for $p=0, 1, 2, 3,$ and $Syr(I_1(p,0))=1$ for $p \in \textbf{\textit{N}}$. Because of $I_1(1,0)=5=5 \pmod {6}, t_5=(n-5)/6=0$. By Theorem 2.10(3), $I_1(1,0)$ is connected by $I_5(p,0)$, as shown in Figure 3(b); Because of $I_1(2,0)=21$ and $I_1(5,0)=1365$ are multiples of 3, they are marked by black nodes, and have no connection; $I_1(3,0)=85=1 \pmod {6}, t_1=(n-1)/6=14$, is connected by $I_1(p,14)$. Similarly, Figure 3(c) presents the component $I_5(p,0)$. $I_5(1,0)= 13$ is connected by $I_1(p,2)$; $I_5(2,0)=53$ is connected by $I_5(p,8)$; $I_5(4,0)=853$ is connected by $I_1(p,142)$; and so on. The component $I_1(p,0)$ located at Level $\sharp 0$ is connected by $I_5(p,0), I_1(p,14), I_5(p,56)$, and many more in Level $\sharp 1$. The component $I_5(p,0)$ is connected by $I_1(p,2), I_5(p,8), I_1(p,142)$, $\cdots$ ; the component $I_5(p,14)$ is connected by $I_5(p,18), I_1(p,302), I_5(p,1208)$, $\cdots$ ; and the component $I_1(p,142)$ is connected by $I_5(p,37), I_5(p,3637), I_5(p,2424)$, $\cdots$ , and many more. These components are located at Level $\sharp 2$. After constructing the components in Level $\sharp 2$, the components in Level $\sharp 3$ are constructed in a similar way. The construction process can be extended as many levels as wished. Note that the numbers under the components are the connection point $c_x=6t+a$, or a term of a Syracuse sequence. ![[\[fig:Com2\]]{#fig:Com2 label="fig:Com2"}Component connection tree.](Fig4.jpg){#fig:Com2 width="0.9\\linewidth"} Based on the connection rules, the components at Level $\sharp r$ is connected only by the components at Level $\sharp (r+1)$ and will never accept any connections from other levels. Each component at Level $\sharp r$ can be connected by many components at Level $\sharp (r+1)$, however, each input of a component at $\sharp r$ is connected only one component from Level $\sharp (r+1)$. As mentioned, the numbers under those components in the connection tree are the terms of a Syracuse sequence. Thus, the connection tree can be used to generate the Syracuse sequences. Let $SYR(n)=\{n=n_0, n_1, n_2, \cdots, n_s\}$ be a Syracuse sequence. If the present term is $n_i$, the next term $n_{i+1}$ is generated as follows, ![[\[fig:Algo\]]{#fig:Algo label="fig:Algo"}Algorithm SyrGen.](Fig5.jpg){#fig:Algo width="0.60\\linewidth"} The operation of $m=(n_i+1/3)/4^p-1/3$ , or $m=U^p(n_i)$, solve for m and p, can be easily executed by repeatedly computing $(m-1)/4$ and increment p by 1 at each cycle before $m \notin \textbf{\textit{2N+1}}$. Based on Algorithm SyrGen, in Figure 5, the Syracuse sequence $SYR(35)=\{35,53,5,1\}$ is generated as an example, and the path is $35 \rightarrow I_5(p,8) \rightarrow I_5(p,0) \rightarrow I_1(p,0) \rightarrow$ trivial cycle, as shown in Figure 4, highlighted by boldface lines. #### ***Step 1***. $(35 \rightarrow I_5(p,8))$ Given $n=35, x=3, a=5, p=0$, and $m=n=35. x=3?5, q=(35-3)/4=8, n=35$ connects to $I_a(p,q)$, i.e., $I_5(p,8)$, and $n_1=6q+a=6*8+5=53$; #### ***Step 2***: $(53 \rightarrow I_5(p,0))$ Because of $m \ne 1, q=(m-3)/4=0$, this yields that 53 connects to $I_5(p,0)$ and $n_2=6q+5=5;$ #### ***Step 3***: $(5 \rightarrow I_1(p,0) \rightarrow$trivial cycle) $n_2=5, x=5, a=5, p>0, (5-1)/4=1, p=1$, and $m=1.$ Because of $x=5$ and $m=1, n_2=5$ connects to $I_1(p,0), n_3=1$ is followed by the trivial cycle. ## Completeness of Connection Tree This subsection is to prove the completeness of the connection tree, i.e., all components in the incoming term matrices $\{I_a(p,q)\}$, are included by the connection tree, where $\{I_1(p,q)\}\cup \{I_5(p,q)\}=\textbf{\textit{2N+1}}$. The completeness of the connection tree will Let $m=J_{ab}(x,y), a,b \in \{1,5\}$, denote the component $I_a(p,m)$ connects to $I_b(p,y)$ at p=x, where $J_{ab}(x,y)=(I_a(x,y)-b)/6$. Table 1 shows the matrices $\{J_a(x,y)\}, a=1,5$, where the boldface numbers indicate the entries with the values of $6t+1$, while the numbers in grey are with $6t+5$. The empty entries are those with $6t+3$. Therefore, $J_{a1}(x,y)$ means the entries with the values of $6t+a$ in $\{J_1(x,y)\}, a=1,5$, and $J_{a5}(x,y)$ indicates the entries with the values of $6t+a$ in $\{J_5(x,y)\}$. For example, in level $\sharp 1$, the component $I_1(p,14)$ connects to $I_1(3,0)$, by definition, $m=14, a=1, b=1, p=x=3$, and $y=0$, i.e., $J_{11}(3,0)=(I_1(0,0)-1)/6=(85-1)/6=14$. The entry $J_{11}(3,0)=14$, or $J_1(3,0)=14$ with the number in boldface. The closed-form of four matrices $\{J_{ab}(x,y)\}$ are shown in Table 1. {width="1.0\\linewidth"} **Theorem 11**. *The closed forms of $J_{ab}(x,y$) are* 1. *$J_{11}(p,q)=4^{p+1}k+2[(6y+1)*4^p-1]/9$; and $J_{51}(p,q)=4^{p+1}k+2[(6y+1)*4^p-4]/9;$* 2. *$J_{15}(p,q)=2*4^pk+[(6y+5)*4^p-2]/9;$ and $J_{55}(p,q)=2*4^pk+[(6y+5)*4^p-8]/9.$* *Proof.* $I_1(p,q)=[(6q+1)*4^{p+1}-1]/3, q=y \pmod {3}$, i.e., $q=3k+y$; and $I_5(p,q)=[(6q+5)*4^{p+1}-2]/6$, ${I_1(p,q)}=I_1(p,3k+y)=[(6(3k+y)+1)4^{p+1}-1]/3=(6k+2y)4^{p+1}+(4^{p+1}-1)/3=6k*4^{p+1}+2y*4^{p+1}+(4^{p+1}-1)/3$; $J_{11}(p,q)=(I_1(p,q)-1)/6=[6k*4^{p+1}+2y*4^{p+1}+(4^{p+1}-1)/3-1]/6=k*4^{p+1}+[6y*4^{p+1}+4^{p+1}-1-3]/18=4^{p+1}k+2[(6y+1)4^p-1]/9$; $J_{51}(p,q)=((I_1(p,q)-1)-5)/6=[6k*4^{p+1}+2y*4^{p+1}+(4^{p+1}-1)/3-5]/6=k*4^{p+1}+[6y*4^{p+1}+4^{p+1}-1-15]/18=4^{p+1}k+2[(6y+1)*4^p-4]/9$; $I_5(p,q)=I_5(p,3k+y)=[(6(3k+y)+5)*4^{p+1}-2]/6=3*4^{p+1}k+[(6y+5)*4^{p+1}-2]/6$; $J_15(p,q)=[(I_5(p,q)-1)]/6={3*4^{p+1}k+[(6y+5)*4^{p+1}-2]/6-1}/6=2*4^pk+4[(6y+5)*4^p-2]/36=2*4^pk+[(6y+5)*4^p-2]/9$; $J_55(p,q)=[(I_5(p,q)-5)]/6={3*4^{p+1}k+[(6y+5)*4^{p+1}-2]/6-5}/6=2*4^pk+4[(6y+5)*4^p-8]/36=2*4^pk+[(6y+5)*4^p-8]/9.$ ◻ For proving the completeness of the connection tree, one must assert that both $\{J_{11}(p,q)\} \cup \{J_{15}(p,q)\}=\textbf{\textit{N}}$ and $\{J_{51}(p,q)\} \cup \{J_{55}(p,q)\}=\textbf{\textit{N}}$ are true. The former indicates that all components with the values of $6t+1$ are included by the tree, while the latter one implies that all components with $6t+5$ are also included. Including the components with $6t+3$, all components $\{6t+1\} \cup \{6t+3\} \cup \{6t+5\}=\textbf{\textit{2N+1}}$, i.e., all positive odd integers are included. Table B (in Appendix) lists the $\{J_{ab}(p,q)\}$ matrices with $q=0 \sim 15$ for the interesting readers **Theorem 12**. *${J_{11}(p,q)} \cup {J_{15}(p,q)}=\textbf{\textit{N}}; and$ ${J_{51}(p,q)} \cup {J_{55}(p,q)}=\textbf{\textit{N}}.$* *Proof.* Let $X(p)=\{J_{11}(p,q)\}$ and $Y(p)=\{J_{15}(p,q)\}$. $\{2k\}=\{4k\}\cup \{4k+2\}=X(0) \cup \{4k+2\}$, and $\{4k+2\}=\{8k+2\} \cup \{8k+6\}=Y(1) \cup \{8k+6\}$, i.e., $\{2k\}=X(0) \cup Y(1)\cup \{8k+6\}. \{8k+6\}=\{16k+6\} \cup \{16k+14\}=X(1) \cup \{16k+14\}, \{2k\}=X(0) \cup Y(1) \cup X(1) \cup \{16k+14\}$. The operation is repeatedly processed, $\{2k\}=X(0)\cup (X(1)\cup Y(1)) \cup \cdots \cup (X(r) \cup Y(r)) \cup \cdots =\{J_{11}(p,q)\} \cup \{J_{15}(p,q)\}-Y(0)$, i.e., $\{J_{11}(p,q)\} \cup \{J_{15}(p,q)\}=\{2k\} \cup \{2k+1\}=\textbf{\textit{N}}$. Similarly, let $X(p)=\{J_{51}(p,q)\}$ and $Y(p)=\{J_{55}(p,q). \{2k\}=\{4k\} \cup \{4k+2\}=X(0) \cup \{4k\}=X(0) \cup Y(1) \cup \{8k\}=X(0) \cup Y(1) \cup X(1) \cup \{16k+8\}= \cdots =\{J_{51}(p,q)\} \cup \{J_{55}(p,q)\}-Y(0)$, i.e., ${J_{51}(p,q)} \cup {J_{55}(p,q)}=\textbf{\textit{N}}$. ◻ ## Non-trivial Cycle-free and boundedness of Syracuse Sequences To prove that the Syracuse conjecture is true, the issues of non-trivial cycle-free and boundedness of Syracuse sequences are also very import to assert the convergence of Syracuse sequences. **Lemma 13**. *Let $n_r=I_a(p_r,q)$ and $n_t=I_a(p_t,q)$ be located at the same column q of $\{I_a(p,q)\}$. If $n_t$ is a term of $SYR(n)$, then $n_r$ will never be a term of $SYR(n)$.* *Proof.* Suppose that $n_r$ is a term of $SYR(n)$, without loss of generality, let $t=r+k$ and $p_r=p_t+d, d \in \textbf{\textit{N+1}}$. By Lemma 2.4, $n_t=U^d(n_r)=((n_r+1/3)/4^d-1/3)$, thus, $n_t=((n_r+1/3)/4^d-1/3)=Syr^k(n_r) \approx (3/4)^kn_r$. Let k=d, i.e., $n_r+1/3=3^dn_r$, this implies that $n_r \notin \textbf{\textit{2N+1}}$, i.e., $n_r \notin SYR(n)$. ◻ **Lemma 14**. *Let $n_t=I_1(0,0)=1$ and $n_r=I_1(p_r,0)$, both are located at the same column $\{I_1(p,0)\}$, a Syracuse sequence may include both terms $n_r$ and $n_t$.* *Proof.* Let $p_r=p_t+d, d \in \textbf{\textit{N+1}}$, by Lemma 2.4, $n_t=U^d(n_r)=(n_r+1/3)/4^d-1/3)=1$, i.e., $n_r+1/3=(4/3)4^d$, or $n_r=(4^{d+1}-1)/3$. Thus, $Syr(n_r)=(3n_r+1)/4^{pr+1}=(3((4^{d+1}-1)/3)+1)/4^{pr+1}=1$, where $d=p_r$, i.e., $Syr(n_r)=n_t$. The Syracuse sequence $SYR(n)=\{n=n_0, n_1, \cdots ,n_r, n_t=1,1,1, \cdots \}$ contains a trivial cycle. ◻ **Theorem 15**. *Syracuse sequences are free of non-trivial cycles.* *Proof.* By Lemma 2.13, if both $n_r=I_a(p_r,q)$ and $n_t=I_a(p_t,q)$ are located at the same column q of $\{I_a(p,q)\}$, and if $n_t \in SYR(n)$, then $n_r \notin SYR(n)$. Any Syracuse sequence never has two terms included by the same column of the incoming term matrices $\{I_b(p,q)\}$, except $I_1(p,0)$. By Lemma 2.14, A Syracuse sequence may contain two to incoming term from the same column of $\{I_1(p,0)\}$, which generates the trivial cycle, not non-trivial cycle. ◻ Note that a sequence may end with $\{\cdots ,5,1\}$ or $\{\cdots ,21,1\}$, where (5,1) and (21,1) are located at $I_1(p,0)$, the cycles exist. By Lemma 2.14, these cycles are trivial cycles, not non-trivial cycles. Based on the connection rules, the components at Level $\sharp r$ is connected only by the components at Level $\sharp (r+1)$ and will never accept any connections from other levels. It is virtually impossible for a component to be enabled again to produce another duplicated output for the sequence. By Theorem 2.15, the Syracuse sequences contain no non-trivial cycles. Regarding the boundedness of the Syracuse sequences, based on the connection rules and the connection tree, for any finite integer $n \in \textbf{\textit{2N+1}}$, there exists one and only component, say component A, locate at Level $\sharp r$. Since the tree does not contain the non-trivial cycles, by the connection rules, Component A connects to a component located at Level $\sharp (r-1)$. Further, component generates an output (another term of sequence) and connects to the component in Level $\sharp (r-2)$. The procedure is repeatedly processed until the component located at Level $\sharp 0$ is reached, the Syracuse sequence is generated and converged to the number of 1. In other words, for any finite positive odd integer n, there exists a component in the connection tree and a path starting from the component to the component located at Level $\sharp 0$ and the generated sequences converges to 1. At each path, the terms of the generated sequence may be up and down, but the terms eventually converge to 1. Thus, the generated Syracuse sequences are bounded. ## Proof of Syracuse Conjecture **Theorem 16**. *The Syracuse sequence SYR(n) converges to 1, for all $n \in \textbf{\textit{2N+1}}$.* *Proof.* By Theorem 2.12, the completeness of the connection tree indicates that, for any finite positive odd integer n, there exists a component located at Level $\sharp r$ for n. The connection tree will find a path for n from Level $\sharp r$ down to the component at Level $\sharp 0$. This implies the generated Syracuse sequence converges to the number of 1. This proves the Syracuse conjecture. ◻ # The Proofs of Collatz Conjecture The previous section has proven the Syracuse conjecture, i.e., all Syracuse sequences are non-trivial cycle-free and bounded sequences and they are converged to the number of 1. In this section, the success of Syracuse conjecture using the component connection model is extended to prove the Collatz conjecture. ## Component Connection Model of Collatz Sequence The components in Collatz sequences are modelled as shown in Figure 5. The component has the odd-numbered inputs/output similar to the component model in Figure 3(a) for Syracuse sequences. The odd-number inputs are multiplied by 3 and plus one, i.e., $Col(n)=3n+1$. The result $m=Col(n)$, even number, perform $Col(m)=m/2$. If $Col(m)$ is odd, it goes to node $c_x$, otherwise, the division operation is repeatedly processed, and the even-numbered quotients are sequentially sent to the Even-numbered Output (ENO) node. The component also allows to apply all positive even integers as the seeds of the sequences. The positive even integer is applied to the Even-number Input (EVI) node. The EVI node feeds the number to the divider to derive the outputs to ENO node. ![[\[fig:Algo5\]]{#fig:Algo5 label="fig:Algo5"}Component model for Collatz sequences:(a) Component model; and (b) Component model for $I_5(p,q)$.](Fig6.jpg){#fig:Algo5 width="0.7\\linewidth"} Figure 5(b) shows the component description of $I_5(p,0)$, the odd-number inputs/output are exactly the same as that in Figure 3(c). If the input $I_1(1,2)=13$ of the component for $I_5(p,0)$ is enabled. $3n+1=40$ is simultaneously sent to the divider and the ENO node., i.e., the next term of 13 is 40. The procedure, in turns, generates 20, 10 and 5. Since 5 is odd, it is sent to $c_x$ node and to complete the sequence generation in this component model, where the sequence is $13\rightarrow 40\rightarrow 20\rightarrow 10\rightarrow 5$. On the other hand, if n=40 as an input (seed), even integer, ENI node is enabled and simultaneously sends n to the divider and the ENO. Then, $m=n/2=40/2=20$, the following procedure is exactly the same as described above. Consider the Syracuse sequence $SYR(35)=\{35,53,5,1\}$, the Collatz sequence COL(35) is generated as follows, $n=35=I_5(0,8), m=3n+1=106$, and $n_1=53=I_5(2,0)$; $m_1=3n_1+ 1=160, m_2=80, m_3=40, m_4=20, m_5=10$, and $n_2=5=I_1(1,0), m_6=16, m_7=8, m_8=4, m_9=2$, and $n_3=1$. The Collatz sequence is $$COL(35)=\{35,106,53,160,80,40,20,10,5,16,8,4,2,1\}.$$ The connection tree for Syracuse sequences has shown that, for a given positive odd integer as a seed of the sequence, there exists a component located at Level $\sharp r$ and the connection model will guide a path down link to the component located at Level $\sharp 0$. The path generates a Syracuse sequence which converges to 1. Similarly, the component connection model for Collatz sequences takes the same connection tree. Given a positive even integer as a seed of a Collatz sequence, there exists a component located as Level $\sharp r$, whose even-numbered input (ENI) is enabled to take the seed and produces an odd-numbered output which connects to the component in Level $\sharp (r-1)$. Similarly, the connection model will also guide the path down link to the component at Level $\sharp 0$ and cause the Collatz sequence to converge to 1. The next step is to prove the completeness of the connection tree, i.e., whether or not all positive even integers are included. ## The Proof of Collatz Conjecture To prove the completeness, it is necessary to assert that COL(n) converges to 1 for all $n \in \textbf{\textit{2N+2}}$, and 1 for all $n \in \textbf{\textit{2N+1}}$, this yields that $n \in ( \textbf{\textit{2N+2}}) \cup(\textbf{\textit{2N+1}})= \textbf{\textit{N+1}}$ for proving the Collatz conjecture. **Lemma 17**. *Let $m=Col(n)$, $COL(m)$ converges to 1 if and only if $COL(n)$ converges to 1.* *Proof.* Let $COL(n)=\{n=n_0, n_1, n_2, \cdots ,1\}$. If $m=Col(n)$, then $COL(m)$ is also a Collatz sequence and converges to 1. On the other hand, $COL(n)$ is a sub-sequence of $COL(m)$. If $COL(m)$ converges to 1, so is $COL(n)$. ◻ **Lemma 18**. *The convergence relationship between SYR(m) and COL(m)* 1. *$SYR(m)=\{m,1\}$ converges to 1 and so is $COL(m)$;* 2. *$SYR(m)=\{m,m_1,1\}$ converges to 1 and so is $COL(m)$;* 3. *If $SYR(m)=\{m,m_1,m_2, \cdots ,m_s\}$ converges to 1, so is $COL(m)$.* *Proof.* (1) $SYR(m)=\{m,1\}$, where $m \in \textbf{\textit{2N+1}}$, there exist even integers $e_i$, such that $e_1=3m+1, e_{i+1}=e_i/2, i=1, 2, \cdots , r-1$, and $e_r=1$. Thus, $COL(m)=\{m,e_1,e_2, \cdots ,e_r,1\}$ is a Collatz sequence and converged to 1; (2) $SYR(m)=\{m,m_1,1\}$ , where $m, m_1\in \textbf{\textit{2N+1}}$, there exist even integers $e_i$ and $d_j$ such that $e_1=3*m+1, e_{i+1}=e_i/2, i=1, 2, \cdots , r-1$, and $e_r=1$, and $d_1=3m+1, d_{i+1}=d_i/2, i=1,2, \cdots ,x-1$, and $d_{x+1}=m_1$. Thus, $COL(m)=\{m,d_1,d_2, \cdots , d_r,m_1,e_1,e_2, \cdots, e_r,1\}$ is a Collatz sequence and converged to 1; and (3) if $SYR(m)=\{m,m_1,m_2, \cdots ,m_s\}$ converges to 1, the sequence will yield $m_s=1$, Similar to the proof of (2), $COL(m)$ also converges to 1. ◻ **Lemma 19**. *$\{4t+2\}\cup \{8t+4\}\cup \cdots \cup \{2_r(2t+1)\}=\{2t\}$ as $r \rightarrow \infty$.* *Proof.* $\{2t\}=\{4t\}\cup \{4t+2\}$ and $\{4t\}\cap \{4t+2\}=\phi$, thus $\{4t+2\}=\{2t\}-\{4t\}$; Similarly, $\{4t+2\}\cup \{8t+4\}=\{8t+2\}\cup \{8t+6\} \cup \{8t+4\}=\{2t\}-\{8t\}$, and $\{4t+2\}\cup \{8t+4\}\cup \cdots \cup \{2^r(2t+1)\}=\{2t\}-\{2^{r+1}t\} \approx \{2t\}$ as $r \rightarrow \infty$. ◻ **Theorem 20**. *$COL(n)$ converges to 1 for all $n \in \textbf{\textit{2N+2}}$.* *Proof.* Let $n=2t+1$, i.e., $n \in \textbf{\textit{2N+2}}$, $m_1=2(2t+1)=4t+2$ is even, and $Col(m_1)=n$. By Lemma 3.1, if $COL(n)$ iconverges to 1, so is $COL(m)$. Let $m_2=2^2(2t+1)=8t+4$ is even, and $Col(m_2)=m_1$. By Lemma 3.1, again, $COL(m_2)$ also converges to 1. Suppose that $COL(m_r)$ converges to 1, where $m_r=2^r(2t+1)$, then $Col(m_{r+1})=m_r$, $COL(m_r)$ is a Collatz sequence and converges to 1. In other words, $COL(m)$ converges to 1 for all $m=2^r(2t+1)$ and $t \in \textbf{\textit{N}}$. By Lemma 3.3, $\{4t+2\}\cup \{8t+4\}\cup \cdots \cup \{2_r(2t+1)\}=\{2t\}$ as $r \rightarrow \infty$, i.e., $COL(m)$ converges to 1 for all even m, i.e., $m \in \textbf{\textit{2N+2}}$, because of $m\neq 0$. ◻ **Theorem 21**. *$COL(n)$ converges to 1 for all $n \in \textbf{\textit{2N+1}}$.* *Proof.* By Theorem 2.7, $SYR(n)$ converges to 1 for all $n \in \textbf{\textit{2N+1}}$. By Lemma 3.2 (3), $COL(n)$ is a Collatz sequence and converges to 1. ◻ **Theorem 22**. *(Collatz Conjecture) $COL(n)$ converges to 1 for all $n \in \textbf{\textit{2N+1}}$.* *Proof.* By Theorem 3.5, $COL(n)$ converges to 1 for all $n \in \textbf{\textit{2N+1}}$, and by Theorem 3.6, COL(n) converges to 1 for all $n \in \textbf{\textit{2N+2}}$, this concludes that $n \in (\textbf{\textit{2N+1}})\cup (\textbf{\textit{2N+2}})=(\textbf{\textit{N+1}})$. The Collatz conjecture is proven. ◻ # Summary and Conclusions This paper employs a system model, Component Connection Model (CCM), to describe the Syracuse sequences. The CCM of a dynamical system consists of a set of equations describing the component dynamics and component interconnections \[6\]. Each component is described by a column of the incoming term matrix $I_a(p,q), a=1,5$. Basically, $\textbf{\textit{2N+1}}$ is partitioned into 4 disjoint sets: $\{8t+a\}, a=1, 3, 5, 7$. Let $S_a(t)=Syr(8t+a)$, by Lemma 2.1, $\{S_5(t)\}\cup \{S_1(t)\} \cup \{S_3(t)\} \cup \{S_7(t)\}$. This property yields the following contributions for proving both Collatz and Syracuse conjectures: \(1\) The terms of any Syracuse sequences are either $6t+1$ or $6t+5$, and the term with $6t+3$ serves only as the seed of the sequences; and \(2\) Two incoming term matrices $\{I_a(p,q)\}, a=1, 5$, were constructed from $\{6q+a\}$, results in $\{8t+5\}=\{I_1(p,q)\} \cup \{I_5(p,q)\}, p\in \textbf{\textit{N+1}}$, and $\{I_1(p,q)\} \cup \{I_5(p,q)\}=\textbf{\textit{2N+1}}$, for $p, q \in \textbf{\textit{N}}$; All positive odd integers are all included; Based on the incoming term matrices, a component connection tree is constructed and, by Theorem 2.5, all components find a path well to reach the component in Level $sharp 0$ which connects a trivial cycle. The contributions show all components including all positive odd integers can eventually reach the component $I_1(p,0)$ in Level $sharp 0$ , thus, all Syracuse sequences are converged to the number of 1 and the Syracuse conjecture is proven to be true. The results of Syracuse sequence are also extended to show the convergence of all Collatz sequences to further prove that Collatz conjecture is true. # Appendix ## Table A. $n=8q+a$ and $S_a(n)$, a=1,3,5,7, $q=0\sim 15$ Table A demonstrates that $\{S_5(t)\}\cup \{S_1(t)\}\cup \{S_3(t)]\}\cup \{S_7(t)\}$, where $S_a(t)=Syr(8t+a)$. The values in row $S_5(n)$, in boldface, are the same as those in each column (in boldface), where q=0, $(s_1(n),s_3(n),s_5(n),s_7(n))=(1,5,1,11)$; (7,17,5,23) for q=1, (13,29,1,35) for q=2, (19,41,11,47) for q=3, and so on. {width="1.0\\linewidth"} ## Table B. Interconnection of Components {width="0.95\\linewidth"} Table B shows the interconnection of components. $m=J_{1b}(x,y)=(I_1(x,y)-b)/6$, The component $I_b(p,m)$ connects to $I_1(p,y)$ at $p=x$; For example, $I_1(2,1)=149=5 \pmod {6}$ and $m=J_{15}(2,1)=(149-5)/6=24$, i.e. x=2, y=1, b=5, and m=24. The component $I_5(p,24)$ connects to $I_1(p,1)$ at p=2; $I_5(0,4)=19=1 \pmod {6}, m=(19-1)/6=3$, i.e., x=0, y=4, b=1, and m=3. The component $I_1(p,3)$ connects to $I_5(p,4)$ at p=0; $I_5(1,4)=77=5 \pmod {6}, m=(77-1)/6=12$, i.e., x=1, y=4, b=5, m=12, the component $I_5(p,12)$ connects to $I_5(p,1)$ at p=4. 1 J.C. Lagarias (2010). *The 3x+1 problem: an Overview*. arXiv preprint arXiv: 2111.02635 (2010). J.C. Lagarias (2010). *The ultimate challenge: The 3x+1 problem*. American Mathematical Soc. Providence, RI, 2000. T. Tao, (2019). *Almost all orbits of the Collatz map attain almost bounded values*. arXiv preprint arXiv: 1909.03562 (2019). J.C. Lagarias. (1985). *The 3x+1 problem and its Generalizations.* The American Mathematical Monthly, 92(1), 3-23, 1985. R.A. DeCarlo, and R. Saeks (1981). *Interconnection Dynamical Systems*. Marcel Dekker, New York, 1981. C.L. Wey and R. Saeks, (1989). *On the Implementation of Analog ATPG: The Linear Case.* *Analog Fault Diagnosis*, Edited by R.W. Liu, IEEE PRESS, 1989. C.L. Wey and W.-H. Huang. (1999). *Designability Check for Analog Circuits with Incomplete Implementation Information.* IEEE Trans. on Circuits and Systems, Part I, Fundamental Theory and Applications, vol. 46, No.8, pp. 939-949, 1999 > ::: {.small} > [Chin-Long Wey]{.smallcaps}. Department of Electronics and Electrical Engineering, National Yang Ming Chiao Tung University, Hsinchu, Taiwan\ > E-mail: `wey@nycu.edu.tw`\ > :::

arxiv_math

--- abstract: | In this note, we study the twisted Jacquet modules of sub-quotients of principal series representations of ${\rm GL}_2(D)$ where $D$ is a division algebra over a non-archimedean local field $F$. We begin with a proof of a conjecture due to D. Prasad on twisted Jacquet modules of Speh representations of ${\rm GL}_2(D)$ when $D$ is the quaternionic division algebra. Later, when $D$ is an arbitrary division algebra over $F$, we focus on depth-zero principal series and compute the dimensions of twisted Jacquet modules of generalised Speh representations and investigate their structure explicitly. address: - Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur - 208016, India - Department of Mathematics, Indian Institute of Science, Bangalore - 560012, India author: - Santosh Nadimpalli - Mihir Sheth bibliography: - biblio.bib title: "Twisted Jacquet modules: a conjecture of D. Prasad" --- # Introduction In this note, we study the twisted Jacquet modules of sub-quotients of principal series representations of ${\rm GL}_2(D)$. Here, $D$ is a division algebra over a non-archimedean local field $F$ with index $n$. We begin with a proof of a conjecture due to D. Prasad on twisted Jacquet modules of Speh representations of ${\rm GL}_2(D)$ when $D$ is a quaternionic division algebra over $F$. Later, when $D$ is an arbitrary division algebra over $F$, we focus on depth-zero principal series and compute the dimensions of twisted Jacquet modules of generalised Speh representations. Then we investigate the explicit structure of twisted Jacquet modules as representations of $D^\times$. To fix some notations, let $\tau$ be an irreducible smooth representation of $D^\times$. Let $\nu_\tau$ be an unramified character of $D^\times$ such that the normalised induction $\tau\nu_\tau^{-1/2}\times \tau\nu_\tau^{1/2}$ is reducible and the generalised Steinberg representation ${\rm St}(\tau)$ occurs as the quotient. The irreducible sub-representation of $\tau\nu_\tau^{-1/2}\times \tau\nu_\tau^{1/2}$ is denoted by ${\rm Sp}(\tau)$--the Speh representation associated with $\tau$. Let $B$ be the minimal parabolic subgroup of ${\rm GL}_2(D)$ with unipotent radical $N$. Let $\psi$ be an additive character of $F$, and let $\Psi:N\rightarrow \mathbb{C}^\times$ be the character: $$\Psi\begin{pmatrix}1&x\\0&1\end{pmatrix}=\psi({\rm Tr}_{D/F}(x)),$$ where ${\rm Tr}_{D/F}$ is the reduced trace. The twisted Jacquet module of a smooth representation $(\pi, V)$ of ${\rm GL}_2(D)$ is denoted by $\pi_{N, \psi}$. There is a natural action of $D^\times$ on $\pi_{N, \psi}$. Let $I$ be the standard Iwahori subgroup of $G$ and $I(1)$ be the pro-$p$ radical of $I$. We assume that $\Psi$ is restricts to a non-trivial character on $I(1)\cap N$, to be denoted by $\psi_0$. For any smooth representation $(\sigma, W)$ of $I$, we denote by $W^{I(1), \psi_0}$ the space $\{ w\in W: \sigma(g)w=\psi_0(g)w, \forall g\in I(1)\}$. We first give a proof of the following result conjectured by D. Prasad. **Theorem 1**. *Let $D$ be the quaternionic division algebra and let $\tau$ be a two-dimensional irreducible depth-zero representation of $D^\times$. The $D^\times$-representation ${\rm Sp}(\tau)_{N, \psi}$ is isomorphic to $\omega_\tau\circ {\rm Nr}_{D/F}$, where $\omega_\tau$ is the central character of $\tau$ and ${\rm Nr}_{D/F}$ is the reduced norm map of $D$.* The above theorem is proved using the Jacquet--Langlands correspondence and the comparison of germ expansions of the corresponding representations of ${\rm GL}_2(D)$ and ${\rm GL}_{4}(F)$. This proves that the twisted Jacquet module is one-dimensional and the explicit action of $D^\times$ on this one-dimensional subspace is obtained by using a result of Gan and Takeda on Shalika models of ${\rm GL}_2(D)$. In the depth-zero case of ${\rm GL}_2(D)$ with $D$ is arbitrary, we begin with the following theorem (see Theorem [Theorem 5](#main){reference-type="ref" reference="main"}). **Theorem 2**. *Let $\tau_1$ and $\tau_2$ be two irreducible depth-zero representations of $D^\times$. Then the natural map $$(\tau_1\times \tau_2)^{I(1), \psi_0}\rightarrow (\tau_1\times \tau_2)_{N, \psi}$$ is an isomorphism.* Using the above theorem, we prove that the natural maps $${\rm Sp}(\tau)^{I(1), \psi_0}\rightarrow {\rm Sp}(\tau)_{N, \psi}$$ and $${\rm St}(\tau)^{I(1), \psi_0}\rightarrow {\rm St}(\tau)_{N, \psi}$$ are isomorphisms. The results of Minguez and Secherre on the $k_{max}$ functor describe a crucial part of the space of invariants for the first principal congruence subgroup $K(1)$. We put together the results of Minguez and Secherre with the results of Moy and Prasad on the compatibility of $I(1)$-invariants with Jacquet modules, to show that the dimension of ${\rm Sp}(\tau)_{N, \psi}$ is equal to $d(d-1)/2$, where $d$ is the dimension of $\tau$. Note that $d$ is a factor of $n$. It then follows that the dimension of ${\rm St}(\tau)_{N, \psi}$ is $d(d+1)/2$. In the case where $d$ is odd, we show that, the $D^\times$-representation ${\rm Sp}(\tau)^{I(1), \psi_0}$ is isomorphic to the exterior square representation. The case where $d=2$ is arithmetically more involved and we use some computations on Gauss sums to determine the explicit structure of the twisted Jacquet module of ${\rm Sp}(\tau)$. When $n=2$ and $d=2$, we obtain a different proof of the conjecture of Prasad when $\tau$ has depth-zero (see Theorem [Theorem 12](#explicit_structure){reference-type="ref" reference="explicit_structure"}). In proving our main results, we obtain a better understanding of the space of $K(1)$-invariants of irreducible non-cuspidal representations of ${\rm GL}_2(D)$. Given any irreducible smooth-representation $(\pi, V)$ of ${\rm GL}_2(D)$ with $V^{K(1)}\neq 0$, we observe that $V^{K(1)}$ is a non-trivial representation of $KD^\times$. It is natural to ask the action of $D^\times$ on $V^{K(1)}$. When $V$ is a depth-zero irreducible non-cuspidal representation, we describe this action of $D^\times$ on $V^{K(1)}$ which is the key to all our results.\ \ : The authors would like to thank Dipendra Prasad for his valuable suggestions and for the proof of Theorem 3.1. We also have benefited from the discussions with Guy Henniart and C.S. Rajan and we thank them for their suggestions. The first author thanks DST-INSPIRE for the research grant. # Preliminaries We fix some notation and recall some facts. ## Let $F$ be a non-archimedean local field of residue characteristic $p$, $\mathfrak{o}_F$ be the ring of integers in $F$, $\mathfrak{p}_F\subseteq\mathfrak{o}_F$ be the maximal ideal, and $\mathbb{F}_q$ be the residue field of $F$ of cardinality $q$. Let $D$ be a central division algebra over $F$ index $n$. The maximal order of $D$ is denoted by $\mathfrak{o}_D$ and the maximal ideal of $\mathfrak{o}_D$ is denoted by $\mathfrak{p}_D$. For a central simple algebra $A$ over a field $k$, the reduced norm map (resp. the reduced trace map) is denoted by ${\rm Nr}_{A/k}$ (resp. ${\rm Tr}_{A/k}$). Similarly, for a finite field extension $l/k$, the field norm map (resp. the field trace map) is denoted by ${\rm Nr}_{l/k}$ (resp. ${\rm Tr}_{l/k}$). For $z\in \mathbb{C}^\times$ we denote by $\mu_z$ the unramified character of $D^\times$ which sends a uniformizer $\varpi_{D}$ of $D$ to $z$. Let $\varpi_F=\varpi_{D}^{n}$. Then ${\rm Nr}_{D/F}(\varpi_{D})=(-1)^{n+1}\varpi_{F}$. ## For a divisor $d$ of $n$ with $n=md$, let $F_{d}$ denote the unramified extension of $F$ of degree $d$ viewed as a subfield of $D$, and $D_{m}$ denote the centralizer of $F_{d}$ in $D$. The algebra $D_{m}$ is a central division algebra over $F_{d}$ of index $m$. Let $\theta:F_{d}^{\times}\rightarrow \mathbb{C}^{\times}$ be a tamely ramified character all whose Galois conjugates are distinct. Composing it with the reduced norm $\mathrm{Nr}_{D_{m}/F_{d}}:D_{m}^{\times}\rightarrow F_{d}^{\times}$ and extending it to $D^{\times}_{m}D(1)$ by declaring it to be trivial on $D(1)=1+\varpi_{D}\mathcal{O}_{D}$, we have a character $\tilde{\theta}:D^{\times}_{m}D(1)\rightarrow \mathbb{C}^{\times}$. Inducing $\tilde{\theta}$ to $D^{\times}$, we obtain a smooth tamely ramified irreducible $d$-dimensional representation $\mathrm{Ind}_{D^{\times}_{m}D(1)}^{D^{\times}}\tilde{\theta}$ of $D^{\times}$. In fact, all smooth tamely ramified irreducible representations of $D^{\times}$ are obtained in this fashion [@sz05]. ## Let $G$ be the group ${\rm GL}_2(D)$. Let $B\subseteq G$ be the subgroup of upper triangular matrices (the standard minimal parabolic subgroup), $N\subseteq B$ be the subgroup of upper triangular unipotent matrices (the unipotent radical of $B$), and $T\subseteq B$ be the subgroup of diagonal matrices (the Levi quotient of $B$). The group $D^\times$ is viewed as a subgroup of $T$ sitting diagonally in it. We denote by $K$ the maximal compact subgroup ${\rm GL}_2(\mathfrak{o}_D)$ of $G$. Let $I$ denote the standard Iwahori subgroup of $G$ and $K(1)$ and $I(1)$ be the pro-$p$ radicals of $K$ and $I$ respectively. Let $T_{0}=T\cap K$. A non-trivial additive (smooth) character $\psi_{F}:F \rightarrow \mathbb{C}^\times$ gives rise to a non-trivial additive character $\psi=\psi_{F}\circ\text{Tr}_{D/F}$ on $D$ to be considered as a character of $N$. For a smooth representation $(\pi, V)$ of $G$, the space spanned by the set of vectors $\{\pi(n)v-\psi(n)v:v\in V, n\in N\}$ is denoted by $V(N, \psi)$. The twisted Jacquet module $V_{N,\psi}$ of $V$ is the quotient $V/ V(N, \psi)$ considered as a representation of ${\rm stab}_T(\psi)= D^{\times}$. Recall that the Jacquet-Langlands lemma says that a vector $v\in V(N, \psi)$ if and only if $$\int_{\mathcal{N}}\psi^{-1}(n)\pi(n)vdn=0,$$ for some compact open subgroup $\mathcal{N}$ of $N$ (see [@Bernstein_zelevinsky_0 Lemma 2.33]). ## For an irreducible smooth representation $\tau$ of $D^\times$, there exists an unramified character $\nu_\tau$ such that the normalized principal series representation $\tau\nu_\tau^{-1/2}\times \tau\nu_\tau^{1/2}$ of $G$ is reducible of length 2 and has a unique square-integrable quotient, the Steinberg representation, denoted by ${\rm St}(\tau)$. The subrepresentation $\mathrm{Sp}(\tau)$ of $\tau\nu_\tau^{-1/2}\times \tau\nu_\tau^{1/2}$ is called the Speh representation. We have the following short exact sequences of $G$-representations: $$0\longrightarrow {\rm Sp}(\tau)\longrightarrow \tau\nu_\tau^{-1/2}\times \tau\nu_\tau^{1/2} \longrightarrow {\rm St}(\tau)\rightarrow 0$$ and $$0\longrightarrow {\rm St}(\tau)\longrightarrow \tau\nu_\tau^{1/2}\times \tau\nu_\tau^{-1/2} \longrightarrow {\rm Sp}(\tau)\rightarrow 0.$$ We refer to Tadic for the above results [@tad90]. For a principal series $\tau_{1}\times\tau_{2}$ of $G$, there is a natural isomorphism of $D^{\times}$-representations $$(\tau_{1}\times\tau_{2})_{N,\psi}\cong\tau_{1}\otimes\tau_{2},$$ see [@pr00 Theorem 2.1]. We note that $\mathrm{Sp}(\tau)_{N,\psi}\neq 0$ if and only if $\tau$ has dimension $>1$. # Proof of the conjecture of D. Prasad In a note [@dp07], D. Prasad conjectured that $$\mathrm{Sp}(\tau)_{N,\psi}\cong\omega_{\tau}\circ\mathrm{Nr}_{D/F}$$ as $D^{\times}$-representations when $D$ is the quaternionic division algebra and $\tau$ is a smooth irreducible representation of $D^{\times}$ of dimension $>1$. We first prove this conjecture: **Theorem 3**. *Let $D$ be the quaternionic division algebra over $F$ and let $\tau$ be a smooth irreducible representation of $D^\times$ of dimension $>1$. Then $$\mathrm{Sp}(\tau)_{N,\psi}\cong\omega_{\tau}\circ\mathrm{Nr}_{D/F}$$ as $D^{\times}$-representations.* *Proof.* It is enough to show that $\mathrm{Sp}(\tau)_{N,\psi}$ is one-dimensional because a result [@shalika-periods_gantakeda Theorem 8.6] of Gan and Takeda on the Shalika models of Speh representations then implies that the $D^\times$-representation ${\rm Sp}(\tau)_{N, \psi}$ is isomorphic to $\omega_\tau\circ {\rm Nr}_{D/F}$. Let $\sigma$ be the Jacquet-Langlands lift of the representation $\tau$. Let $\Delta$ be the segment $[\sigma\nu^{-1/2}, \sigma\nu^{1/2}]$ and let $\langle \Delta \rangle$ be the irreducible subrepresentation of $\mathrm{GL}_{4}(F)$ associated with the segment $\Delta$ in [@zelevinsky_2 Section 3]. The Jacquet-Langlands correspondence between $G={\rm GL}_2(D)$ and ${\rm GL}_4(F)$, and its extension to the Grothendieck groups of irreducible smooth representations, takes the representation ${\rm Sp}(\tau)$ to $\langle \Delta \rangle$. Note that the coefficient of the leading term in the germ expansion of ${\rm Sp}(\tau)$, denoted by $c_{\mathcal{O}}({\rm Sp}(\tau))$, is the dimension of ${\rm Sp}(\tau)_{N, \psi}$. Now, comparing the germ expansions of ${\rm Sp}(\tau)$ and $\langle \Delta \rangle$, we get that $$c_{\mathcal{O}}({\rm Sp}(\tau))=c_{\mathcal{O}'}(\langle\Delta\rangle),$$ where $\mathcal{O}'$ is the nilpotent orbit of $\mathfrak{gl}_4(F)$ corresponding to the partition $(2,2)$ (see [@dipendra_germ_expansions Theorem 2]). Using [@zelevinsky_2 Proposition 3.4]), we get that $(2,2)$ is the maximal element in the Whittaker support of $\langle\Delta\rangle$. Then using [@mw_degenerate Theorem I.16], the nilpotent orbit $(2,2)$ is the maximal nilpotent orbit in the germ expansion of $\langle \Delta \rangle$ and thus $c_{\mathcal{O}'}(\langle\Delta\rangle)$ is $1$. ◻ **Remark 4**. *Note that the above argument does not work when $D$ is not the quaternionic division algebra. Since the non-trivial nilpotent orbit of $\mathfrak{gl}_2(D)$ corresponds to the the partition $(n,n)$ of $2n$ and the maximal element in the Whittaker support of $\langle \Delta \rangle$ corresponds to the partition $(2,2,\dots, 2)$ of $2n$.* # Further results in the tame case In order to understand the structure of twisted Jacquet modules of Speh representations for arbitrary division algebra, we restrict ourselves from now on to tamely ramified (depth-$0$) representations. A generalization of Theorem [Theorem 3](#dpthm){reference-type="ref" reference="dpthm"} is obtained for an arbitrary division algebra in the tame case. ## Dimension formulae Fix an additive character $\psi_{F}:F\rightarrow\mathbb{C}^{\times}$ such that $\psi_{F}$ is non-trivial on $\mathfrak{o}_F$ but trivial on $\mathfrak{p}_F$. Then $\psi=\psi_{F}\circ\text{Tr}_{D/F}$ is non-trivial on $\mathfrak{o}_{D}$ and trivial on $\mathfrak{p}_{D}$. The map $$\begin{pmatrix}a&b\\\varpi_{D}c&d\end{pmatrix}\mapsto \psi(b)$$ defines a non-trivial character $\psi_{0}$ on the group $I(1)$. For any smooth representation $V$ of $I(1)$, the space of $\psi_{0}$-semi-invariants is $$V^{I(1), \psi_{0}}=\{v\in V: g.v=\psi_{0}(g)v\ \text{for all}\ g\in I(1)\}.$$ If $V$ is a smooth $G$-representation, then we note that $V^{I(1),\psi_{0}}$ is stable under the action of $D^{\times}$. This is because $\psi$ is trivial on $\mathfrak{p}_{F}$ and factors through ${\rm Tr}_{D/F}$. Let $\tau_1$ and $\tau_2$ be two irreducible smooth depth-zero representations of $D^\times$ of dimensions $d_{1}$ and $d_{2}$ respectively. In this subsection, we prove the following theorem. **Theorem 5**. *The restriction of natural map $\tau_1\times \tau_2\rightarrow (\tau_1\times \tau_2)_{N, \psi}$ to the subspace $(\tau_1\times \tau_2)^{I(1), \psi_{0}}$: $$(\tau_1\times \tau_2)^{I(1), \psi_{0}}\rightarrow (\tau_1\times \tau_2)_{N, \psi}$$ is an isomorphism of $D^{\times}$-representations.* Note that $(\tau_{1}\times\tau_{2})^{I(1),\psi_{0}}= \mathrm{Hom}_{I(1)}(\psi_{0},\tau_{1}\times\tau_{2}) =\mathrm{Hom}_{I(1)}(\psi_{0},(\tau_{1}\times\tau_{2})^{K(1)}) =\mathrm{Hom}_{I(1)}(\psi_{0},\mathrm{Ind}_{I}^{K}(\tau_{1}\otimes\tau_{2}))$. Thus the space $(\tau_{1}\times\tau_{2})^{I(1),\psi_{0}}$ has dimension $d_{1}d_{2}$. Before we begin the proof of the theorem, we prove some lemmas. For an integer $r$, let $N(r)=\begin{pmatrix}1&\mathfrak{p}^{r}_{D}\\0&1\end{pmatrix}$. Note $\psi|_{N(0)}=\psi_{0}|_{N(0)}$. **Lemma 6**. *Let $f$ be a non-zero element of $(\tau_1\times \tau_2)^{I(1), \psi_{0}}$, then we have $$\int_{N(0)}\psi^{-1}(n)f(sn)dn=\mathrm{vol}(N(0))f(s)\neq 0.$$* *Proof.* The function $f$ is nonzero if and only if $f|_{K}$ is so. We have $K=I\sqcup IsI$. Observe that $f(1)=0$ because $\psi|_{N(0)}$ is non-trivial. From this, we get that $f(i)=0$, for all $i\in I=(k_{D}^{\times}\times k_{D}^{\times})I(1)$. The double coset $IsI$ is equal to the set $(I\cap B)sI(1)$. If $f(s)=0$, then the function $f$ is identically zero on the double coset $IsI$, and hence on $K$. Thus $f(s)\neq 0$. ◻ **Lemma 7**. *For any smooth representation $V$ of $N$, and $v\in V$, the image of $v$ in $V_{N, \psi}$ is non-zero if and only if $$\int_{N(-r)}\psi^{-1}(n)\pi(n)vdn$$ is non-zero for all $r>>0$.* *Proof.* Assume that image of a vector $v\in V$ in $V_{N, \psi}$ is zero. Then there exists a compact open subgroup $\mathcal{N}$ such that $$\int_{\mathcal{N}}\psi^{-1}(n)\pi(n)vdn=0.$$ Since $\{N(-r):r>0\}$ is an increasing filtration of $N$, there exists an $r$ such that $\mathcal{N}\subset N(-r)$. Thus, $$\int_{N(-r)}\psi^{-1}(n)\pi(n)vdn= \sum_{g\in N(-r)/\mathcal{N}}\psi(g)\pi(g)\int_{\mathcal{N}}\psi(n)\pi(n)vdn=0,$$ for all $r$ such that $\mathcal{N}\subset N(-r)$. Conversely, if the above integral is zero for any $r>0$, then the image of $v$ in $V_{N, \psi}$ is zero. ◻ *Proof of Theorem [Theorem 5](#main){reference-type="ref" reference="main"}..* For any positive integer $r$ and $u\in \mathfrak{o}_D^\times$, we have the following matrix identity: $$\label{sn_identity} \begin{pmatrix}0&1\\1&0\end{pmatrix} \begin{pmatrix}1&\varpi_D^{-r}u\\0&1\end{pmatrix}= \begin{pmatrix}0&1\\1&\varpi_D^{-r}u\end{pmatrix}= \begin{pmatrix}-\varpi_D^{-r}u&1\\0&\varpi_D^r\end{pmatrix}^{-1} \begin{pmatrix}1&0\\\varpi_D^r&u\end{pmatrix}.$$ Let $f\in (\tau_1\times \tau_2)^{I(1), \psi_{0}}$ be a non-zero function and $f_r:=\int_{N(-r)}\psi^{-1}(n)\pi(n)fdn$. Then $$\begin{aligned} f_{r}(s)=\int_{N(-r)}\psi^{-1}(n)f(sn)dn= &\int_{\mathfrak{p}_D^{-r}}f\left(s \begin{pmatrix}1&y\\0&1\end{pmatrix}\right)dy\\ =&\sum_{\overline{a}\in\mathfrak{p}_D^{-r}/\mathfrak{p}_D^{-r+1}} \int_{\mathfrak{p}_D^{-r+1}}f\left(s \begin{pmatrix}1&(a+y)\\0&1\end{pmatrix}\right)dy\\ =&f_{r-1}(s)+\sum_{\overline{a}\neq\overline{0}} \int_{\mathfrak{p}_D^{-r+1}}f\left(s \begin{pmatrix}1&(a+y)\\0&1\end{pmatrix}\right)dy.\end{aligned}$$ If $r>0$, then using the identity [\[sn_identity\]](#sn_identity){reference-type="eqref" reference="sn_identity"} and that $f(i)=0$ for $i\in I$ (cf. the proof of Lemma [Lemma 6](#s_supp){reference-type="ref" reference="s_supp"}), we get that $$\text{$\int_{\mathfrak{p}_D^{-r+1}} f\left(s \begin{pmatrix}1&(a+y)\\0&1\end{pmatrix}\right)dy=0$ for $\overline{a}\neq\overline{0}$.}$$ Thus, we obtain $f_{r}(s)=f_{r-1}(s)$ for all $r>0$. By Lemma [Lemma 6](#s_supp){reference-type="ref" reference="s_supp"}, we get that $f_r$ is non-zero for all $r\geq 0$. Hence, by Lemma [Lemma 7](#nbds){reference-type="ref" reference="nbds"}, the natural map $$\label{natural_1} (\tau_1\times \tau_2)^{I(1), \psi_{0}}\rightarrow (\tau_1\times \tau_2)_{N, \psi}$$ is injective. However, $(\tau_{1}\times\tau_{2})_{N,\psi} \cong\tau_{1}\otimes\tau_{2}$ [@pr00 Theorem 2.1]. So, $\mathrm{dim}_{\mathbb{C}}(\tau_1\times \tau_2)^{I(1), \psi_{0}} =\mathrm{dim}_{\mathbb{C}}(\tau_1\times \tau_2)_{N, \psi}=d_{1}d_{2}$. Thus, the map in [\[natural_1\]](#natural_1){reference-type="eqref" reference="natural_1"} is an isomorphism. ◻ **Proposition 8**. *Let $\tau$ be a tamely ramified irreducible representation of $D^\times$. The natural maps $${\rm Sp}(\tau)^{I(1), \psi_{0}}\rightarrow {\rm Sp}(\tau)_{N, \psi}$$ and $${\rm St}(\tau)^{I(1), \psi_{0}}\rightarrow {\rm St}(\tau)_{N, \psi}$$ are isomorphisms.* *Proof.* We have the following commutative diagrams: $$\begin{tikzcd} 0 \arrow[r] & {\rm Sp}(\tau)_{N,\psi} \arrow[r] & (\tau\nu_{\tau}^{-1/2}\times\tau{\nu}_{\tau}^{1/2})_{N,\psi} \arrow[r] & {\rm St}(\tau)_{N,\psi} \arrow[r] & 0 \\ 0 \arrow[r] & {\rm Sp}(\tau)^{I(1),\psi_{0}} \arrow[r]\arrow[u,"f"] & (\tau\nu_{\tau}^{-1/2}\times\tau{\nu}_{\tau}^{1/2})^{I(1),\psi_{0}} \arrow[r]\arrow[u,"g"] & {\rm St}(\tau)^{I(1),\psi_{0}} \arrow[r]\arrow[u,"h"] & 0 \end{tikzcd}$$ and $$\begin{tikzcd} 0 \arrow[r] & {\rm St}(\tau)_{N,\psi} \arrow[r] & (\tau\nu_{\tau}^{1/2}\times\tau{\nu}_{\tau}^{-1/2})_{N,\psi} \arrow[r] & {\rm Sp}(\tau)_{N,\psi} \arrow[r] & 0 \\ 0 \arrow[r] & {\rm St}(\tau)^{I(1),\psi_{0}} \arrow[r]\arrow[u,"h"] & (\tau\nu_{\tau}^{1/2}\times\tau{\nu}_{\tau}^{-1/2})^{I(1),\psi_{0}} \arrow[r]\arrow[u,"g'"] & {\rm Sp}(\tau)^{I(1),\psi_{0}} \arrow[r]\arrow[u,"f"] & 0 \end{tikzcd}$$ where $f$, $g$, $g'$ and $h$ are natural maps. Since $g$ and $g'$ are isomorphisms from Theorem [Theorem 5](#main){reference-type="ref" reference="main"}, we get that $f$ is injective and $h$ is surjective from the first diagram and $f$ is surjective and $h$ is injective from the second diagram. ◻ **Corollary 9**. *Let $\tau=\mathrm{Ind}_{D^{\times}_{m}D(1)}^{D^{\times}}\tilde{\theta}$ be a $d$-dimensional tamely ramified irreducible representation of $D^\times$. We then have $$\dim_{\mathbb{C}}{\rm St}(\tau)_{N, \psi}=\dfrac{d(d-1)}{2}+d \hspace{3mm}\text{and}\hspace{3mm}\dim_{\mathbb{C}}{\rm Sp}(\tau)_{N, \psi}=\dfrac{d(d-1)}{2}.$$* *Proof.* From the work of Minguez and Secherre [@ms14], we find that as $K$-representations $$\begin{aligned} &\mathrm{St}(\tau)^{K(1)}\cong\bigoplus_{0\leq i< j\leq d-1}\mathrm{Ind}_{I}^{K}(\tilde{\theta}^{q^{i}}\otimes\tilde{\theta}^{q^{j}})\oplus\bigoplus_{0\leq i\leq d-1}\mathrm{st}(\tilde{\theta}^{q^{i}}) \hspace{2mm}\text{and}\\&\mathrm{Sp}(\tau)^{K(1)}\cong\bigoplus_{0\leq i< j\leq d-1}\mathrm{Ind}_{I}^{K}(\tilde{\theta}^{q^{i}}\otimes\tilde{\theta}^{q^{j}})\oplus\bigoplus_{0\leq i\leq d-1}\tilde{\theta}^{q^{i}}\circ\mathrm{det}(\overline{\hspace{0.5mm}\cdot\hspace{0.5mm}}), \end{aligned}$$ where $\mathrm{det}(\overline{\hspace{0.5mm}\cdot\hspace{0.5mm}})$ is the composition of the determinant character of $\mathrm{GL}_{2}(\mathbb{F}_{q^{n}})$ and the natural surjection $K\twoheadrightarrow\mathrm{GL}_{2}(\mathbb{F}_{q^{n}})$, and $\tilde{\theta}^{q^{i}}\circ\mathrm{det}(\overline{\hspace{0.5mm}\cdot\hspace{0.5mm}})$ and $\mathrm{st}(\tilde{\theta}^{q^{i}})$ are the two simple factors of the reducible induction $\mathrm{Ind}_{I}^{K}(\tilde{\theta}^{q^{i}}\otimes\tilde{\theta}^{q^{i}})$ (see [@ns23 Lemma 4.5]). Hence, $$\dim_{\mathbb{C}}{\rm St}(\tau)^{I(1), \psi_{0}}=\dim_{\mathbb{C}}\mathrm{Hom}_{I(1)}(\psi_{0}, \mathrm{St}(\tau)^{K(1)})=\dfrac{d(d-1)}{2}+d$$ and $$\dim_{\mathbb{C}}{\rm Sp}(\tau)^{I(1), \psi_{0}}= \dim_{\mathbb{C}}\mathrm{Hom}_{I(1)}(\psi_{0},\mathrm{Sp}(\tau)^{K(1)})= \dfrac{d(d-1)}{2}.$$ The corollary now follows from Proposition [Proposition 8](#semiinvariantsforspst){reference-type="ref" reference="semiinvariantsforspst"}. ◻ ## The $D^\times$-action on the twisted Jacquet module Let $\tau=\mathrm{Ind}_{D^{\times}_{m}D(1)}^{D^{\times}}\tilde{\theta}$ be a $d$-dimensional tamely ramified irreducible representation of $D^\times$. Using Proposition [Proposition 8](#semiinvariantsforspst){reference-type="ref" reference="semiinvariantsforspst"}, we now find the explicit structure of the $D^{\times}$-representation $\mathrm{Sp}(\tau)_{N,\psi}$. The analysis depends on the parity of $d$. The situation is more involved when $d$ is even. ### $d=\mathrm{dim}_{\mathbb{C}}(\tau)$ is odd The space of $K(1)$-invariants of the principal series $\tau\nu_\tau^{-1/2}\times \tau\nu_\tau^{1/2}$ as a $KD^\times$-representation is isomorphic to $$\operatorname{Ind}_{ID^\times}^{KD^\times}(\tau\otimes \tau).$$ Note that $$\operatorname{Res}_{ID^\times}(\tau\otimes \tau)=\bigoplus_{y\in \mathbb{Z}/d\mathbb{Z}}W_y$$ where $W_y$ is an irreducible representation of $ID^\times$ such that $\operatorname{Res}_{I}W_y$ is the direct sum of the characters $\tilde{\theta}^{q^x}\otimes \tilde{\theta}^{q^{x+y}}$, where $x\in \mathbb{Z}/d\mathbb{Z}$. **Lemma 10**. *Let $y\in \mathbb{Z}/d\mathbb{Z}$. If $2y\neq 0$, then the representation $\operatorname{Ind}_{ID^\times}^{KD^\times} W_y$ is irreducible; otherwise it has two distinct irreducible subrepresentations $\rho_1$ and $\rho_2$ such that $\operatorname{Res}_K\rho_1\simeq \operatorname{Res}_K\rho_2$. We have $$\operatorname{Ind}_{ID^\times}^{KD^\times}W_y\simeq \operatorname{Ind}_{ID^\times}^{KD^\times}W_{-y}$$ for all $2y\neq 0$.* *Proof.* Applying the Mackey decomposition, we get that $${\rm Hom}_{KD^\times}(\operatorname{Ind}_{ID^\times}^{KD^\times}W_{y}, \operatorname{Ind}_{ID^\times}^{KD^\times}W_{y'}) ={\rm Hom}_{T_0D^\times}(W_{y}, W_{y'})\oplus {\rm Hom}_{T_0D^\times}(W_y, W_{y'}^s)$$ For $y\in \mathbb{Z}/d\mathbb{Z}$, the representations $W_y$ of $T_0D^\times$ are distinct and irreducible. As $W_y^s$ is equal to $W_{-y}$, the lemma follows. ◻ **Proposition 11**. *If $d=\mathrm{dim}_{\mathbb{C}}(\tau)$ is odd, then the $D^{\times}$-representation ${\rm Sp}(\tau)_{N, \psi}$ is isomorphic to $\wedge^2\tau$.* *Proof.* By the proof of Corollary [Corollary 9](#dimformulae){reference-type="ref" reference="dimformulae"} and Lemma [Lemma 10](#oddcaselemma){reference-type="ref" reference="oddcaselemma"}, we have as $KD^{\times}$-representations, $${\rm Sp}(\tau)^{K(1)}\cong V\oplus \bigoplus_{0<y\leq (d-1)/2}\operatorname{Ind}_{ID^\times}^{KD^\times}W_y,$$ where $\operatorname{Res}_{K}=\bigoplus_{0\leq i\leq d-1}\tilde{\theta}^{q^{i}}\circ\mathrm{det}(\overline{\hspace{0.5mm}\cdot\hspace{0.5mm}}).$ Considering the $\psi_{0}$-semi-invariants for the action of $I(1)$, we get that $${\rm Sp}(\tau)^{I(1), \psi_{0}}\cong \bigoplus_{0<y\leq (d-1)/2}{\rm Res}_{D^\times} W_y.$$ Let $\{e_0,e_1,\dots,e_{d-1}\}$ be a basis of $\tau$ such that $x.e_i=\tilde{\theta}^{q^i}(x)e_i$ for all $x\in \mathfrak{o}_D^\times$. The set $\{e_i\otimes e_j-e_j\otimes e_i: i\neq j\}$ is a basis for $\wedge^2(\tau)$. The space $W_y$ is spanned by vectors $e_i\otimes e_{i+y}$. The map $$e_i\otimes e_{i+y}\mapsto e_i\otimes e_{i+y}-e_{i+y}\otimes e_i$$ defines an isomorphism of $\bigoplus_{0<y\leq (d-1)/2}\operatorname{Res}_{D^\times}W_y$ with $\wedge^2\tau$. ◻ ### $d=\mathrm{dim}_{\mathbb{C}}(\tau)$ is $2$ In this subsection, we assume that $p>2$. As $d=2$, the index of $D$ is $n=2m$. We know from Corollary [Corollary 9](#dimformulae){reference-type="ref" reference="dimformulae"} that $\mathrm{Sp}(\tau)_{N,\psi}$ is a character of $D^{\times}$. The following theorem describes this character precisely generalizing Theorem [Theorem 3](#dpthm){reference-type="ref" reference="dpthm"}. **Theorem 12**. *The $D^\times$-representation ${\rm Sp}(\tau)_{N, \psi}$ is the character $(\theta\circ {\rm Nr}_{D/F})\mu_{(-1)^{m+1}}$.* *Proof.* Let $f$ be a non-zero function in $\mathrm{Sp}(\tau)^{K(1)}$ such that $k.f=\tilde{\theta}({\rm det}(\overline{k}))f$ for all $k\in K$; see the proof of Corollary [Corollary 9](#dimformulae){reference-type="ref" reference="dimformulae"}. Let $$t:=\begin{pmatrix}\varpi_D&0\\0&1\end{pmatrix}.$$ Note that $tit^{-1}\in K$ for $i\in I$ and thus, $it^{-1}f=\tilde{\theta}({\rm det}(\overline{tit^{-1}}))t^{-1}f$, which implies that $t^{-1}f\in {\rm Sp}(\tau)^{I, \tilde{\theta}^q\otimes \tilde{\theta}}$. From the decomposition of $\mathrm{Sp}(\tau)^{K(1)}$ given in the proof of Corollary [Corollary 9](#dimformulae){reference-type="ref" reference="dimformulae"}, we find that the $K$-representation $\langle K\cdot t^{-1}f\rangle$ is stable under the action of $D^{\times}$. We are interested in the $D^{\times}$-representation on the space $\langle K\cdot t^{-1}f\rangle^{I(1),\psi_{0}}$. The Frobenius reciprocity induces an isomorphism of $K$-representations $$\Phi: {\rm Ind}_{I}^K(\tilde{\theta}^q\otimes \tilde{\theta})\rightarrow \langle K\cdot t^{-1}f\rangle$$ such that $\Phi(\varphi)=\sum_{k\in\{1,sn_{x}\}}\varphi(k^{-1})kt^{-1}f$. Here, $\{1,sn_{x}=\left( \begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right) \left(\begin{smallmatrix}1&[x]\\0&1\end{smallmatrix}\right):x\in\mathbb{F}_{q^{2m}}\}$ is a set of representatives for $I\backslash K$. Let $\mathbb{1}_{I}\in{\rm Ind}_{I}^K(\tilde{\theta}^q\otimes \tilde{\theta})$ be the function such that $\Phi(\mathbb{1}_{I})=t^{-1}f$. It is the function supported on $I$ mapping $1$ to $1$. Using the operator $$T(\varphi)(k)=\dfrac{\theta(-1)^{m+1}\theta(\varpi_{F})}{q^m} \sum_{y\in\mathbb{F}_{q^{2m}}}\varphi(sn_{y}\varpi_{D}k\varpi_{D}^{-1})$$ we make ${\rm Ind}_{I}^K(\tilde{\theta}^q\otimes \tilde{\theta})$ into a representation of $KD^\times$ such that $\varpi_D$ acts by $T$. We claim that $\Phi$ is then an isomorphism of $KD^{\times}$-representations. Indeed, we note that $T^{2m}=\theta(\varpi_{F})^{2}{\rm Id}$ and $T$ corresponds to an intertwining operator in $\mathrm{Hom}_{K}({\rm Ind}_{I}^K(\tilde{\theta}^q\otimes \tilde{\theta}),{\rm Ind}_{I}^K(\tilde{\theta}\otimes \tilde{\theta}^{q}))\cong\mathbb{C}$. As $\varpi_{D}^{2m}$ acts on $\langle K\cdot t^{-1}f\rangle$ by the scalar-multiplication by $\theta(\varpi_{F})^{2}$, there exists a scalar $\epsilon$ (an $m$-th root of unity) such that $$\Phi(T(\mathbb{1}_I))= \epsilon\varpi_Dt^{-1}f.$$ Expanding the left-hand side of the above, we find that $$\begin{aligned} \Phi(T(\mathbb{1}_I))= &\sum_{k\in\{1,sn_{x}\}}T(\mathbb{1}_{I})(k^{-1})kt^{-1}f\\ =&\dfrac{\theta(-1)^{m+1}\theta(\varpi_{F})}{q^m} \sum_{k\in\{1,sn_{x}\}} \sum_{y\in\mathbb{F}_{q^{2m}}}\mathbb{1}_{I}(sn_{y}\varpi_{D}k^{-1} \varpi_{D}^{-1})kt^{-1}f\\ =&\dfrac{\theta(-1)^{m+1}\theta(\varpi_{F})}{q^m} \sum_{x\in\mathbb{F}_{q^{2m}}} \sum_{y\in\mathbb{F}_{q^{2m}}}\mathbb{1}_{I}(sn_{y}\varpi_{D}n_{x}s \varpi_{D}^{-1})sn_{-x}t^{-1}f\\ =&\dfrac{\theta(-1)^{m+1}\theta(\varpi_{F})}{q^m} \sum_{x\in\mathbb{F}_{q^{2m}}} \sum_{y\in\mathbb{F}_{q^{2m}}}\mathbb{1}_{I}(sn_{y+x^{q}}s)st^{-1}f\\ =&\dfrac{\theta(-1)^{m+1}\theta(\varpi_{F})}{q^m} \sum_{x\in\mathbb{F}_{q^{2m}}} st^{-1}f =\theta(-1)^{m+1}\theta(\varpi_{F})q^{m}st^{-1}f. \end{aligned}$$ Thus, we have $\theta(-1)^{m+1}\theta(\varpi_{F})q^{m}t^{-1}f =\epsilon\varpi_Dt^{-1}f$. Evaluating both sides on $1$, we obtain that $$\theta(-1)^{m+1}\theta(\varpi_{F})q^{m}st^{-1}f(1)= \epsilon\varpi_Dt^{-1}f(1)\ \text{as}\ \mathrm{det}(\overline{s})=1.$$ This gives $$\theta(-1)^{m+1}\theta(\varpi_{F})q^{m} f\left(\begin{pmatrix}1&0\\0&\varpi_D^{-1} \end{pmatrix}\right) = \epsilon f\left(\begin{pmatrix}1&0\\0&\varpi_D\end{pmatrix}\right).$$ Using that $f\in \tau\nu^{-1/2}\times \tau\nu^{1/2}$, we get $$\theta(-1)^{m+1}\theta(\varpi_{F})q^{m} (\operatorname{id}\otimes \tau(\varpi_D^{-1})|\varpi_D|^{1/4})f(1)= \epsilon(\operatorname{id}\otimes \tau(\varpi_D)|\varpi_D|^{-1/4})f(1),$$ and using that $|\varpi_{D}|=q^{-2m}$ and $\tau(\varpi_{D}^{-1}) =\tau(\varpi_D)\theta(-1)^{m+1}\theta(\varpi_{F})^{-1}$, we conclude $$q^{m/2}(\operatorname{id}\otimes \tau(\varpi_D))f(1)= \epsilon q^{m/2}(\operatorname{id}\otimes \tau(\varpi_D))f(1).$$ Hence, $\epsilon=1$ and thus $$\Phi(\varpi_{D}\mathbb{1}_{I}) =\Phi(T(\mathbb{1}_{I}))=\varpi_{D}t^{-1}f=\varpi_{D}\Phi(\mathbb{1}_{I}).$$ It follows that the $KD^\times$-representation $\langle K\cdot t^{-1}f\rangle$ is isomorphic to ${\rm Ind}_{I}^K(\tilde{\theta}^{q}\otimes \tilde{\theta}).$ Therefore, the $D^\times$-representation on the space $\langle K\cdot t^{-1}f\rangle^{I(1), \psi_0}$ is isomorphic to $\tilde{\theta}^{q+1}\mu_c$, where $$c=\dfrac{\theta(-1)^{m+1}\theta(\varpi_{F})}{q^m} G(\tilde{\theta}^{q-1}, \psi_0).$$ and $G(\tilde{\theta}^{q-1},\psi_{0})=\sum_{x\in\mathbb{F}_{q^{2m}}}\tilde{\theta}^{q-1}(x)\psi_{0}(x)$ is the Gauss sum (cf. [@gar14 Proposition 2.0.10]). Note that, in the Gauss sum, $\psi_{0}$ is viewed as a non-trivial additive character on $\mathbb{F}_{q^{2m}}$ factoring as $\psi_{\mathbb{F}_{q}}\circ\mathrm{Tr}_{\mathbb{F}_{q^{2m}}/\mathbb{F}_{q}}$ where $\psi_{\mathbb{F}_{q}}=\psi_{F}|_{\mathfrak{o}_{F}}$. To get the constant $c$, we need to compute this Gauss sum. By Hasse-Davenport lifting relation, $$G(\tilde{\theta}^{q-1},\psi_{0})=(-1)^{m+1}G(\theta^{q-1},\psi_{\mathbb{F}_{q}}\circ\mathrm{Tr}_{\mathbb{F}_{q^{2}}/\mathbb{F}_{q}})^{m}.$$ To compute $G(\theta^{q-1},\psi_{\mathbb{F}_{q}}\circ\mathrm{Tr}_{\mathbb{F}_{q^{2}}/\mathbb{F}_{q}})=\sum_{x\in\mathbb{F}_{q^{2}}}\theta^{q-1}(x)\psi_{\mathbb{F}_{q}}(\mathrm{Tr}_{\mathbb{F}_{q^{2}}/\mathbb{F}_{q}}(x))$, let us fix a set $\{x_i\}$ of coset representatives for $\mathbb{F}_{q^2}^\times/\mathbb{F}_q^\times$. We abbreviate ${\rm Tr}_{\mathbb{F}_{q^2}/ \mathbb{F}_q}$ as ${\rm Tr}$. Then $$\begin{aligned} G(\theta^{q-1},\psi_{\mathbb{F}_{q}}\circ\mathrm{Tr})=&\sum_{x_i}\sum_{y\in \mathbb{F}_q^\times}\theta^{q-1}(x_iy)\psi_{\mathbb{F}_{q}}(\mathrm{Tr}(x_iy))\\ =&\sum_{x_i}\sum_{y\in \mathbb{F}_q^\times}\theta^{q-1}(x_i)\psi_{\mathbb{F}_{q}}({\rm Tr}(x_i)y)\\ =&\sum_{x_i, {\rm Tr}(x_i)=0}(q-1)\theta^{q-1}(x_i)+ \sum_{x_i, {\rm Tr}(x_i)\neq 0}-\theta^{q-1}(x_i)\\ =&\sum_{x_i, {\rm Tr}(x_i)=0}q\theta^{q-1}(x_i)- \sum_{x_i}\theta^{q-1}(x_i)\\ =&\sum_{x_i, {\rm Tr}(x_i)=0}q\theta^{q-1}(x_i). \end{aligned}$$ Note that if $\mathrm{Tr}(x)=\mathrm{Tr}(y)=0$ for some $x,y\in\mathbb{F}_{q^2}^{\times}$ then $x$ and $y$ belong to the same coset in $\mathbb{F}_{q^2}^{\times}/\mathbb{F}_{q}^{\times}$, i.e., $\frac{x}{y}\in\mathbb{F}_{q}^{\times}$. This is clear because if $\mathrm{Tr}(x)=\mathrm{Tr}(y)=0$ then $(\frac{x}{y})^{q-1}=\frac{x^{q}y}{x y^{q}}=\frac{-xy}{-xy}=1$. There always exists an element $x_{0}\in\mathbb{F}_{q^2}^{\times}$ with $\mathrm{Tr}(x_{0})=0$: for $\mathbb{F}_{q^2}^{\times}=\langle\alpha\rangle$, $x_{0}=\alpha^{\frac{q+1}{2}}$. Hence, $\mathbb{F}_{q^2}^{\times}/\mathbb{F}_{q}^{\times}$ has a unique coset of trace $0$ elements. Therefore $$G(\theta^{q-1},\psi_{\mathbb{F}_{q}}\circ\mathrm{Tr})=q\theta(x_{0}^{q}x_{0}^{-1}) =q\theta(-x_{0}x_{0}^{-1})=q\theta(-1).$$ Thus, $$G(\tilde{\theta}^{q-1},\psi_{0})=(-1)^{m+1}q^{m}\theta(-1)^{m} \hspace{2mm} \mathrm{and} \hspace{2mm} c=(-1)^{m+1}\theta(-\varpi_{F}).$$ It follows that the $D^\times$-representation ${\rm Sp}(\tau)_{N, \psi}$ is isomorphic to $\tilde{\theta}^{q+1}\mu_c=(\theta\circ {\rm Nr}_{D/F}) \mu_{(-1)^{m+1}}$. ◻ **Remark 13**. *We remark that in contrast with odd $d$, for $d=\mathrm{dim}_{\mathbb{C}}(\tau)=2$, the above theorem implies that the $D^{\times}$-representation $\mathrm{Sp}(\tau)_{N,\psi}$ is isomorphic to $\wedge^{2}(\tau)$ if and only if $\theta(-1)^{m}=\omega_{\tau}(-1)=(-1)^{m}$.*

arxiv_math

--- author: - - title: On Averaging of a Class of ''Non-Singularly Perturbed\" Control Systems --- # Introduction and the main result {#S1} In this paper, we consider the system $$\begin{aligned} \label{e:fast} \epsilon\frac{dy(t)}{dt} &=&Ay(t)+Bu(t), \ \ \ \ y(0)=y_0, \\ \frac{dz(t)}{dt}&=&g(u(t),y(t),z(t)) , \ \ \ \ z(0)=z_0, \label{e:slow}\end{aligned}$$ where $\epsilon> 0$ is a parameter, the controls $u(t)$ are measurable functions taking values in ${\rm I\kern-0.2em R}^k$, and $A$ and $B$ are $m\times m$ and $m\times k$ matrices that satisfy the rank controllability condition: $$\label{e-rank} {\rm rank} [ B, AB,..., A^{m-1}B]=m.$$ The function $g:{\rm I\kern-0.2em R}^k\times {\rm I\kern-0.2em R}^m \times {\rm I\kern-0.2em R}^n\to {\rm I\kern-0.2em R}^n$ is assumed to be bounded $$\label{e-g-bound} \sup_{(u,y,z)\in{\rm I\kern-0.2em R}^k\times{\rm I\kern-0.2em R}^m\times{\rm I\kern-0.2em R}^n}||g(u,y,z)||:= M_g<\infty$$ and satisfying the Lipschitz condition in $(y,z)$ uniformly with respect to $u\in {\rm I\kern-0.2em R}^k$. If the parameter $\epsilon$ was small, the system ([\[e:fast\]](#e:fast){reference-type="ref" reference="e:fast"})-([\[e:slow\]](#e:slow){reference-type="ref" reference="e:slow"}) would belong to the class of the so-called *singularly perturbed* (SP) systems. SP systems are characterized by the decomposition of the state variables into groups that change their values with rates of different orders of magnitude (so that some of them can be considered as fast/slow with respect to others), which is due to the presence of the small singular perturbations parameter $\epsilon$. Such systems describe processes and interactions in disparate time scales, and they have been extensively studied in the literature (see, e.g., research monographs [@Ben], [@Kab2], [@Kok1], [@Kus3] and surveys [@Dmitriev], [@Nai], [@Reo1], [@Zhang-Naidu]). One of the approaches to SP control systems is the *averaging method* based on the analysis of asymptotic properties of the sets of time averages of the equation describing the dynamics of the slow state variables over the fast control-state trajectories considered on the intervals $[0,S]$ for large values of $S>0$. If the limit of these sets exists as $S$ tends to infinity, then (under certain additional conditions) it defines the right-hand-side of the differential inclusion, the solutions of which approximate the slow components of the solutions of the SP system when the singular perturbation parameter $\epsilon$ tends to zero (see [@Dont-Don], [@Gai0], [@Gai1], [@Gra], [@QW] and also [@Alv], [@Art2], [@AG] for related developments). In this paper, the parameter $\epsilon$ is not assumed to be small, and to distinguish it from the case of singular perturbations, we call system ([\[e:fast\]](#e:fast){reference-type="ref" reference="e:fast"})-([\[e:slow\]](#e:slow){reference-type="ref" reference="e:slow"}) *non-singularly perturbed*. We show that the approximation of the $z$-components of the solution of system ([\[e:fast\]](#e:fast){reference-type="ref" reference="e:fast"})-([\[e:slow\]](#e:slow){reference-type="ref" reference="e:slow"}) by the solution of a certain differential inclusion is still valid, but this result is no longer asymptotic and holds for any $\epsilon>0$. Such an approximation is possible due to the fact that the $y$-components of the state variables of the system ([\[e:fast\]](#e:fast){reference-type="ref" reference="e:fast"})-([\[e:slow\]](#e:slow){reference-type="ref" reference="e:slow"}) can change their values arbitrarily fast (see Lemma [Lemma 1](#Lem-control){reference-type="ref" reference="Lem-control"} in the next section) while the rates of change of the $z$-components are bounded by the constant $M_g$ (see ([\[e-g-bound\]](#e-g-bound){reference-type="ref" reference="e-g-bound"})). To state our main result, let us introduce the differential inclusion $$\label{eq-DI} \frac{dz(t)}{dt}\in V(z(t)) ,\ \ \ \ z(0)=z_0,$$ where $$\label{eq-RHS} V(z):= \bar{\rm co} (g({\rm I\kern-0.2em R}^k, {\rm I\kern-0.2em R}^m, z)), \ \ \ \ g({\rm I\kern-0.2em R}^k, {\rm I\kern-0.2em R}^m, z):= \{v\ | \ v =g(u,y,z), \ u\in{\rm I\kern-0.2em R}^k, \ y\in {\rm I\kern-0.2em R}^m\},$$ with $\bar{co}$ standing for the closure of the convex hull of the corresponding set. Let $T$ be an arbitrary positive number. Denote by $\mathcal{Z}_T(\epsilon)$ the set of the $z$-components of solutions of the system ([\[e:fast\]](#e:fast){reference-type="ref" reference="e:fast"})-([\[e:slow\]](#e:slow){reference-type="ref" reference="e:slow"}) considered on the interval $[0,T]$, that is, $$\mathcal{Z}_T(\epsilon):=\{z(\cdot)\,|\, (u(\cdot),y(\cdot),z(\cdot))\hbox{ satisfies }(\ref{e:fast}),(\ref{e:slow})\}.$$ By $\mathcal{Z}_T$ denote the set of solutions of the differential inclusion ([\[eq-DI\]](#eq-DI){reference-type="ref" reference="eq-DI"}) considered on this interval. Note that, as can be readily understood, $$\label{eq-INCL} \mathcal{Z}_T(\epsilon)\subset \mathcal{Z}_T \ \ \ \ \forall \ \epsilon>0.$$ The main result of the paper is the following theorem. **Theorem 1**. *The equality $$\label{eq-EQ} {\rm cl}(\mathcal{Z}_T(\epsilon))= \mathcal{Z}_T \ \ \ \ \forall \ \epsilon>0$$ is valid, where ${\rm cl}$ in the expression above stands for the closure in the uniform convergence metrics. That is, the set of the $z$-components of solutions of the system ([\[e:fast\]](#e:fast){reference-type="ref" reference="e:fast"})-([\[e:slow\]](#e:slow){reference-type="ref" reference="e:slow"}) is dense in the set of solutions of the differential inclusion ([\[eq-DI\]](#eq-DI){reference-type="ref" reference="eq-DI"}).* *Proof.* The proof of the theorem is given in the next section. ◻ REMARK I. Theorem [Theorem 1](#Th-main){reference-type="ref" reference="Th-main"} resembles results establishing that the Hausdorff distance (induced by the uniform convergence metric) between the set of the slow components of solutions of a SP control system and the set of solutions of the differential inclusion, constructed by averaging of the slow subsystem over the controls and the corresponding solutions of the fast one, tends to zero when the singular perturbations parameter tends to zero (see [@Dont-Don], [@Gai0], [@Gai1], [@Gra], [@QW]). As mentioned above, in contrast to these results, Theorem [Theorem 1](#Th-main){reference-type="ref" reference="Th-main"} is not of asymptotic nature. The equality ([\[eq-EQ\]](#eq-EQ){reference-type="ref" reference="eq-EQ"}) is valid for any $\epsilon>0$, including, e.g., $\epsilon=1$. REMARK II. Note that the statement of Theorem [Theorem 1](#Th-main){reference-type="ref" reference="Th-main"} would look similar to that of Filippov-Wazewski theorem (see [@aubin1]) if $y(t)$ in ([\[e:slow\]](#e:slow){reference-type="ref" reference="e:slow"}) was another control (that is, an arbitrary measurable function taking values in ${\rm I\kern-0.2em R}^m$) instead of being the solution of ([\[e:fast\]](#e:fast){reference-type="ref" reference="e:fast"}). Let us discuss an implication of Theorem [Theorem 1](#Th-main){reference-type="ref" reference="Th-main"} for optimal control. Consider the optimal control problem $$\label{e-SP-problem} \inf_{u(\cdot)}G(z_{\epsilon}(T)):= G_{\epsilon}^*,$$ where $G(\cdot)$ is a continuous function and $inf$ is taken over the controls $u(\cdot)$ and the corresponding solutions $(y_{\epsilon}(\cdot),z_{\epsilon}(\cdot))$ of system ([\[e:fast\]](#e:fast){reference-type="ref" reference="e:fast"})-([\[e:slow\]](#e:slow){reference-type="ref" reference="e:slow"}). Consider also the optimal control problem $$\label{e-averaged-problem} \inf_{u(\cdot), y(\cdot)}G(z(T)):= G^*,$$ where $inf$ is taken over measurable functions $(u(\cdot), y(\cdot))\in {\rm I\kern-0.2em R}^k\times {\rm I\kern-0.2em R}^m$ and the corresponding solutions $z(\cdot)$ of the system $$\label{e-system-free} \frac{dz(t)}{dt}=g(u(t),y(t),z(t)) , \ \ \ \ z(0)=z_0$$ (both $u(\cdot)$ and $y(\cdot)$ are playing the role of controls in this system). **Corollary 1**. *The optimal value in ([\[e-SP-problem\]](#e-SP-problem){reference-type="ref" reference="e-SP-problem"}) is equal to the optimal value in ([\[e-averaged-problem\]](#e-averaged-problem){reference-type="ref" reference="e-averaged-problem"}): $$\label{e-OV-equality} G_{\epsilon}^*= G^* \ \ \ \ \forall \ \epsilon>0.$$* *Proof.* Denote by $\mathcal{Z}_T^0$ the set of solutions of system ([\[e-system-free\]](#e-system-free){reference-type="ref" reference="e-system-free"}) considered on the interval $[0,T]$. As can be readily seen, the following inclusions are valid: $$\mathcal{Z}_T(\epsilon)\subset \mathcal{Z}_T^0\subset \mathcal{Z}_T.$$ Therefore, by ([\[eq-EQ\]](#eq-EQ){reference-type="ref" reference="eq-EQ"}), $${\rm cl}(\mathcal{Z}_T(\epsilon))= {\rm cl}( \mathcal{Z}_T^0)= \mathcal{Z}_T.$$ The latter implies ([\[e-OV-equality\]](#e-OV-equality){reference-type="ref" reference="e-OV-equality"}). ◻ # Proof of the Main Result Consider the system $$\label{e:associate-sys} \frac{dy(\tau)}{d\tau} = Ay(\tau)+Bu(\tau).$$ Note that this system looks similar to the fast subsystem ([\[e:fast\]](#e:fast){reference-type="ref" reference="e:fast"}) but, in contrast to the latter, it evolves in the time scale $\tau=\frac{t}{\epsilon}$ (([\[e:associate-sys\]](#e:associate-sys){reference-type="ref" reference="e:associate-sys"}) will be referred to as the *associated system*). The following lemma is the key element of the subsequent analysis. **Lemma 1**. *For any $y', y''\in {\rm I\kern-0.2em R}^m$ there exists a control $u(\cdot)$ that steers the associated system ([\[e:associate-sys\]](#e:associate-sys){reference-type="ref" reference="e:associate-sys"}) from $y'$ to $y''$ in arbitrarily short period of time.* *Proof.* The proof of the lemma follows a standard argument, and we give it at the end of this section. ◻ Let $y(\tau, u(\cdot),y)$ stand for the solution of the associated system ([\[e:associate-sys\]](#e:associate-sys){reference-type="ref" reference="e:associate-sys"}) obtained with a control $u(\tau)$ and the initial condition $y(0)=y$. For a fixed $z$, denote by $V(S,y,z)$ the set of the time averages: $$\label{e:V(S)} V(S,y ,z) := \bigcup_{u(\cdot)} {\frac{1}{S} \int_0^S g\bigl( u(\tau),y(\tau, u(\cdot),y),z \bigr)d\tau},\quad z={\rm const},$$ where the union is taken over all controls (that is, over all measurable functions $u(\cdot)$ taking values in ${\rm I\kern-0.2em R}^k$). The set of the time averages similar to ([\[e:V(S)\]](#e:V(S)){reference-type="ref" reference="e:V(S)"}) was introduced/used in [@Dont-Don], [@Gai0], [@Gai1], [@Gra], [@QW], where it was shown that, as $S$ tends to infinity, it converges in the Hausdorff metrics to a convex and compact set (provided that the associated system satisfies certain controllability or stability conditions). It was also shown that it is this set that defines the right-hand-side of the differential inclusion, the solutions of which approximate the dynamics of the slow components of the SP control system. The lemma below establishes that in the case under consideration the set ${\rm cl}(V(S,y_0 ,z))$ (the closure of the set $V(S,y_0 ,z)$) is equal to a convex and compact subset of ${\rm I\kern-0.2em R}^m$ for any $S>0$, and it also provides an explicit representation for this set. **Lemma 2**. *For any $\ z\in {\rm I\kern-0.2em R}^n$, any $y \in {\rm I\kern-0.2em R}^m$, and any $S>0$, the following equality is valid: $$\label{e:Z4} {\rm cl}\,(V(S,y ,z))=\bar{\rm co}\, (g({\rm I\kern-0.2em R}^k,{\rm I\kern-0.2em R}^m,z)).$$* *Proof.* Since $z$ is a constant parameter, we will suppress it in the notation, that is, we will write ([\[e:V(S)\]](#e:V(S)){reference-type="ref" reference="e:V(S)"}) in the form $$\label{e:V(S)-no-z} V(S,y) := \bigcup_{u(\cdot)} {\frac{1}{S} \int_0^S g\bigl( u(\tau),y(\tau , u(\cdot), y) \bigr)d\tau}.$$ The inclusion $$\label{Z40} V(S,y)\subset \bar{\rm co}\,(g({\rm I\kern-0.2em R}^k,{\rm I\kern-0.2em R}^m)),$$ follows from the definition of the integral. Indeed, take $v\in V(S,y_0)$, then there exists $u(\cdot)$ such that $v=\frac{1}{S} \int_0^S g\bigl( u(\tau),y(\tau, u(\cdot), y) \bigr)d\tau$. If the integrand was a simple function, that is, it was a finite sum of indicator functions of Borel measurable sets, then it would be clear that $$\frac{1}{S}\int_0^S g\bigl( u(\tau),y(\tau , u(\cdot), y) \bigr)d\tau\in {\rm co}\,(g({\rm I\kern-0.2em R}^k,{\rm I\kern-0.2em R}^m)),$$ that is, $v\in {\rm co}\,g({\rm I\kern-0.2em R}^r,{\rm I\kern-0.2em R}^m)$. For an arbitrary Borel $g$, the inclusion $v\in \bar{\rm co}\,g({\rm I\kern-0.2em R}^r,{\rm I\kern-0.2em R}^m)$ follows from the definition of Lebesgue integral as the limit of integrals of simple functions. Inclusion ([\[Z40\]](#Z40){reference-type="ref" reference="Z40"}) implies that $${\rm cl}\,(V(S,y))\subset \bar{\rm co}\,(g({\rm I\kern-0.2em R}^k,{\rm I\kern-0.2em R}^m)).$$ Let us show that $$\label{Z4} {\rm co}\,(g({\rm I\kern-0.2em R}^k,{\rm I\kern-0.2em R}^m))\subset {\rm cl}\,(V(S,y)).$$ (This will, obviously, imply that $\bar{\rm co}\,(g({\rm I\kern-0.2em R}^k,{\rm I\kern-0.2em R}^m))\subset {\rm cl}\,(V(S,y))$.) Take $v\in {\rm co}\,( g({\rm I\kern-0.2em R}^k,{\rm I\kern-0.2em R}^m))$. Then, due to the Caratheodori Theorem, $$v=\sum_{i=1}^{l} \lambda_i g(u_i,y_i), \ \ \ \ l\leq k+m+1$$ for some $(u_i,y_i) \in {\rm I\kern-0.2em R}^k \times {\rm I\kern-0.2em R}^m$ and some $\lambda_i>0$ with $\ \sum_{i=1}^{l} \lambda_i=1$. Let $$S_0=0,\; \ \ \ S_j:=S\sum_{i=1}^{j} \lambda_i,\ \ j=1,\dots, l,\;$$ which implies that $$\label{Z2} \lambda_j=\frac{S_{j}-S_{j-1}}{S}.$$ Let $\delta > 0$ be arbitrary small and let $\hat\tau > 0$ be defined by the equation $$\label{e-tau} \hat\tau := \frac{\delta}{M_g},$$ where $M_g$ is defined in ([\[e-g-bound\]](#e-g-bound){reference-type="ref" reference="e-g-bound"}). Note that from ([\[e-tau\]](#e-tau){reference-type="ref" reference="e-tau"}) it follows that the solution $y(\tau, u,y)$ of the associated system ([\[e:associate-sys\]](#e:associate-sys){reference-type="ref" reference="e:associate-sys"}) obtained with the constant valued control $u(\tau)\equiv u$ and with the initial condition $y(0)=y$ satisfies the inequality $$\label{e-tau-1} \max_{\tau\in [0,\hat\tau]}\|y(\tau, u, y)-y\|\leq \delta\ \ \ \ \forall \ (u,y)\in {\rm I\kern-0.2em R}^k\times{\rm I\kern-0.2em R}^m.$$ Let us construct a control-state process $(u(\cdot),y(\cdot))$ of the system ([\[e:associate-sys\]](#e:associate-sys){reference-type="ref" reference="e:associate-sys"}) on the interval $[0,S]$ such that, for all $j=1,\dots, l$, the following inequalities hold true (for brevity, the solution of ([\[e:associate-sys\]](#e:associate-sys){reference-type="ref" reference="e:associate-sys"}) is denoted below as $y(\tau)$ instead of $y(\tau, u(\cdot),y)$): $$\left|\frac{1}{S_{j}-S_{j-1}}\int_{S_{j-1}}^{S_{j}} g(u(\tau),y(\tau))\,d\tau-g(u_j, y_j)\right|\le K\delta$$ for some constant $K$ (recall that $S_0:= 0$ and $S_l=S$). Consider the interval $[S_0,S_1]$. Assuming (without loss of generality) that $\delta <S_1$, define the control $u(\cdot)$ on the interval $[0,\delta]\subset [0,S_1]$ in such a way that $y(\delta)=y_1$. (This is possible due to Lemma [Lemma 1](#Lem-control){reference-type="ref" reference="Lem-control"}.) Set $\tau_{11}:=\delta$ and $\tau_{12}:=\delta +\hat \tau .$ Extend the definition of the control $u(\cdot)$ by taking it to be equal to $u_1$ on the interval $(\tau_{11}, \tau_{12}]$ if $\tau_{12}< S_1$. In case $\tau_{12}\geq S_1$, take $u(\cdot)$ to be equal to $u_1$ on the interval $(\tau_{11}, S_1]$. The control $u(\cdot)$ will be, thus, defined on $[S_0,S_1$\]. Note that, by ([\[e-tau-1\]](#e-tau-1){reference-type="ref" reference="e-tau-1"}), $$\max_{\tau\in [\tau_{11}, \tau_{12}]}||y(\tau)-y_1||\leq \delta \ \ {\rm if} \ \ \tau_{12}< S_1$$ and $$\max_{\tau\in [\tau_{11}, S_1]}||y(\tau)-y_1||\leq \delta \ \ {\rm if} \ \ \tau_{12}\geq S_1 .$$ If $\tau_{12}< S_1$, set $\tau_{13}:= \min \{\tau_{12} +\frac{\delta}{2},\tau_{12} + \frac{S_1-\tau_{12}}{2}\}$ and $\tau_{14}:= \tau_{13} +\hat\tau$. Extend the definition of the control $u(\cdot)$ to the interval $(\tau_{12},\tau_{13}]$ in such a way that the corresponding solution $y(\tau)$ of the system ([\[e:associate-sys\]](#e:associate-sys){reference-type="ref" reference="e:associate-sys"}) satisfies the equation $y(\tau_{13})=y_1$. (Again, this is possible due to Lemma [Lemma 1](#Lem-control){reference-type="ref" reference="Lem-control"}.) Also, extend the definition of the control $u(\cdot)$ further by taking it to be equal to $u_1$ on the interval $(\tau_{13}, \tau_{14}]$ if $\tau_{14}< S_1$. In case $\tau_{14}\geq S_1$, take $u(\cdot)$ to be equal to $u_1$ on the interval $(\tau_{13}, S_1]$. The control $u(\cdot)$ will be, thus, defined on $[S_0,S_1$\]. By ([\[e-tau-1\]](#e-tau-1){reference-type="ref" reference="e-tau-1"}), $$\max_{\tau\in [\tau_{11}, \tau_{12}]\cup [\tau_{13}, \tau_{14}]}||y(\tau)-y_1||\leq \delta \ \ {\rm if} \ \ \tau_{14}< S_1$$ and $$\max_{\tau\in [\tau_{11}, \tau_{12}]\cup [\tau_{13}, S_1]}||y(\tau)-y_1||\leq \delta \ \ {\rm if} \ \ \tau_{14}\geq S_1 .$$ In the general case (for an arbitrary small $\delta$), one can proceed in a similar way to: \(i\) Define the sequence of moments of time $\tau_{11}, \tau_{12}, \cdots , \tau_{1\bar s}, \tau_{1\bar s+1}$, where $\bar s$ is odd, $\tau_{\bar s+1} + \bar\tau \geq S_1$, and, for any odd $s$ such that $3\leq s < \bar s$, $$\label{e:Z1-0} \tau_{1 s}:= \min \{\tau_{1s-1} +\frac{\delta}{2^{s-2}},\tau_{1s-1} + \frac{S_1-\tau_{1s-1}}{2}\}, \ \ \ \ \tau_{1 s+1}:= \tau_{1 s} +\hat\tau < S_1;$$ and \(ii\) Construct the control $u(\cdot)$ that along with the corresponding solution $y(\cdot)$ of ([\[e:associate-sys\]](#e:associate-sys){reference-type="ref" reference="e:associate-sys"}) satisfy the relations $$\label{e:Z1} u(\tau)\equiv u_1, \; ||y(\tau)-y_1||\le \delta \; \forall \tau\in D:= (\tau_{11},\tau_{12}] \cup(\tau_{13},\tau_{14}] \cup \ldots \cup (\tau_{1\bar s-2},\tau_{1\bar s-1}]\cup (\tau_{1\bar s}, S_1],$$ with $y(\tau_{1,s})=y_1$ for all $s=1,3,\ldots, \bar s$. By construction (see ([\[e:Z1-0\]](#e:Z1-0){reference-type="ref" reference="e:Z1-0"}) and ([\[e:Z1\]](#e:Z1){reference-type="ref" reference="e:Z1"})), the Lebesgue measure of the set $[0,S_1]\setminus D$ (which we denote as $meas([0,S_1]\setminus D)$ satisfies the inequality $$meas([0,S_1]\setminus D)= \tau_{11} + (\tau_{13}-\tau_{12})+\cdots + (\tau_{1\bar s}-\tau_{1\bar s-1})\leq \delta + \frac{\delta}{2}+\cdots + \frac{\delta}{2^{\bar s-2}} \leq 2\delta.$$ Therefore, $$\left|\frac{1}{S_1}\int_{[0,S_1]\setminus D} (g(u(\tau),y(\tau))-g(u_1,y_1))\,d\tau\right|\leq \frac{2M_g}{S_1} meas ([0,S_1]\setminus D)\leq \frac{4M_g}{S_1}\delta ,$$ where $M_g$ is as in ([\[e-g-bound\]](#e-g-bound){reference-type="ref" reference="e-g-bound"}). Also, due to ([\[e:Z1\]](#e:Z1){reference-type="ref" reference="e:Z1"}), $$\left|\frac{1}{S_1}\int_{D} \big(g(u(\tau),y(\tau))\, -g(u_1,y_1)\big)d\tau\right|=\left|\frac{1}{S_1}\int_{D} g(u_1,y(\tau))\,d\tau-g(u_1,y_1)\right|\leq L\delta,$$ where $L$ is a Lipschitz constant of $g$. Consequently, $$\begin{aligned} &\left|\frac{1}{S_1}\int_0^{S_1} g(u(\tau),y(\tau))\,d\tau-g(u_1,y_1)\right|= \left|\frac{1}{S_1}\int_0^{S_1} (g(u(\tau),y(\tau))-g(u_1,y_1))\,d\tau\right|\le\\ &\left|\frac{1}{S_1}\int_{[0,S_1]\setminus D} (g(u(\tau),y(\tau))-g(u_1,y_1))\,d\tau\right|+ \left|\frac{1}{S_1}\int_{D} (g(u(\tau),y(\tau))-g(u_1,y_1))\,d\tau\right|\le \\ &\left(\frac{4M_g}{ S_1}+L\right)\delta. \end{aligned}$$ Continuing similarly on each subinterval $[S_{j-1},S_j]$, we construct the control-state process $(u(\cdot),y(\cdot))$ on $[0,S]$ such that for all $j=1,\dots,l$ $$\label{Z3} \left|\frac{1}{S_{j}-S_{j-1}}\int_{S_{j-1}}^{S_{j}} g(u(\tau),y(\tau))\, d\tau-g(u_j,y_j)\right|\le \left(\frac{4M_g}{S_{j}-S_{j-1}}+L\right)\delta.$$ For this process, taking into account ([\[Z2\]](#Z2){reference-type="ref" reference="Z2"}) and ([\[Z3\]](#Z3){reference-type="ref" reference="Z3"}), we have $$\begin{aligned} &\left|\frac{1}{S}\int_{0}^{S} g(u(\tau),y(\tau))\,d\tau-v\right|\\ =&\left|\sum_{i=1}^{l}\frac{S_{i}-S_{i-1}}{S}\frac{1}{S_{i}-S_{i-1}} \int_{S_{i-1}}^{S_{i}} g(u(\tau),y(\tau))\,d\tau- \sum_{i=1}^{l}\lambda_ig(u_i,y_i)\right|\\ =&\sum_{i=1}^{l}\lambda_i\left|\frac{1}{S_{i}-S_{i-1}}\int_{S_{i-1}}^{S_{i}}g(u(\tau),y(\tau))\,d\tau-g(u_i,y_i)\right|\\ \le &\,\delta\sum_{i=1}^{l}\lambda_i\left(\frac{4M_g}{S_{i}-S_{i-1}}+L\right) \le\delta\left(\frac{4M_g}{\min_{i}\{S_{i}-S_{i-1}\}}+L\right). \end{aligned}$$ Since $\delta$ is arbitrarily small, this implies that $v\in {\rm cl}\,V(S,y)$, which implies ([\[Z4\]](#Z4){reference-type="ref" reference="Z4"}). The lemma is proved. ◻ **Lemma 3**. *The multivalued function $\ V(z)$ defined in accordance with ([\[eq-RHS\]](#eq-RHS){reference-type="ref" reference="eq-RHS"}) is Lipschitz continuous. That is, $$\label{e:V-Lipschitz-continuity} d_{H}(V(z'),V(z'')) \leq L||z'-z''|| \ \ \ \ \forall z', z'',$$ where $d_{H}$ stands for the Hausdorff distance between sets and $L$ is the Lipschitz constant of $g(u,y,z)$ in $z$.* *Proof.* Let $z', z''$ be arbitrary elements of ${\rm I\kern-0.2em R}^n$. Take some $S>0$ and choose an arbitrary element $\ v'$ from $\ V(z',S,y_0)$. From the definition of $\ V(z',S,y_0)\ $ it follows that there exists a control $\ u(\cdot)$ such that $\ v' = \frac{1}{S}\int_0^S g(z',u(\tau), y(\tau))d\tau.$ Define $\ v''$ by $$v'':= \frac{1}{S}\int_0^S g(z'',u(\tau), y(\tau))d\tau \in V(z'',S,y_0).$$ We have $$||v'-v''||\leq \frac{1}{S}\int_0^S||g(z',u(\tau), y(\tau))-g(z'',u(\tau), y(\tau))||d\tau$$ $$\leq L||z'-z''|| \ \ \ \ \Rightarrow \ \ \ \ \ d(v', V(z'')) \leq L||z'-z''||,$$ where $d(v,V)$ stands for the distance from a vector $v$ to a set $V$ ($d(v,V):= \inf_{w\in V}||v-w||$). Since $\ v'$ is an arbitrary element of $\ V(z',S,y_0)$, it implies that $$\ \sup_{v\in V(z',S,y_0)}d(v, V(z'',S,y_0))\leq L||z'-z''||.$$ Since $V(z)={\rm cl}\,V(S,y_0,z)$, we conclude that $$\sup_{v\in V(z')}d(v, V(z''))\leq L||z'-z''||.$$ Similarly, it is established that $\ $$\ \sup_{v\in V(z'')}d(v, V(z'))\leq L||z'-z''||.$ ◻ *Proof of Theorem [Theorem 1](#Th-main){reference-type="ref" reference="Th-main"}*. Due to ([\[eq-INCL\]](#eq-INCL){reference-type="ref" reference="eq-INCL"}), to prove the theorem it is sufficient to show that $$\mathcal{Z}_T \subset {\rm cl}(\mathcal{Z}_T(\epsilon)) \ \ \ \ \forall \ \epsilon>0.$$ This, in turn, will be shown, if we establish that, for any $\delta>0$, corresponding to any solution $z(t)$ of the differential inclusion ([\[eq-DI\]](#eq-DI){reference-type="ref" reference="eq-DI"}), there exists a control $u_{\epsilon}(t)$, which, being used in the system ([\[e:fast\]](#e:fast){reference-type="ref" reference="e:fast"})-([\[e:slow\]](#e:slow){reference-type="ref" reference="e:slow"}), generates the solution $(y_{\epsilon}(t), z_{\epsilon}(t))$ that satisfies the inequality $$\label{e:Th2.3-3-extra} \max_{t\in [0,T]}||z_{\epsilon}(t)- z(t)|| \leq \delta .$$ To prove the latter statement, take an arbitrary solution $z(t)$ of ([\[eq-DI\]](#eq-DI){reference-type="ref" reference="eq-DI"}), choose an arbitrary $\delta > 0$ and construct the control $u_{\epsilon}(t)$ that insures the validity of ([\[e:Th2.3-3-extra\]](#e:Th2.3-3-extra){reference-type="ref" reference="e:Th2.3-3-extra"}). Take $S\in (0, \frac{T}{\epsilon}]$ and partition the interval $\ [0,T]\ $ by the points $$\label{e:partition} t_l := l \left(\epsilon S\right) \ , \ \ \ \ l = 0,1,..., N_{\epsilon S}:= \left\lfloor \frac{T}{\epsilon S} \right\rfloor ,$$ where $\lfloor\cdot\rfloor$ is the floor function (that is, for any real $x$, $\lfloor x\rfloor$ is the maximal integer that is less or equal than $x$). Let $v_0$ be the projection of the vector $\ \left(\epsilon S\right)^{-1}\int_{0}^{t_{1}}\frac{dz(t)}{dt}dt \ $ onto the set $V(z_0)$. That is, $$v_0:= {\rm argmin}_{v\in V\left(z_0\right)} \left\{\left\| \left(\epsilon S\right)^{-1}\int_{0}^{t_{1}}\frac{dz(t)}{dt}dt -v \right\| \right\}.$$ By Lemma [Lemma 2](#Th-1){reference-type="ref" reference="Th-1"}, ${\rm cl}\,V(S,y_0,z_0)=V(z_0)$. Therefore, there exists a control $u_0(\tau)$ such that, being used in the associated system ([\[e:associate-sys\]](#e:associate-sys){reference-type="ref" reference="e:associate-sys"}) on the interval $\ [0, \frac{t_{1}}{\epsilon}], \ $ ensures that $$\left\|v_0-\frac{1}{S}\int_{0}^{\frac{t_{1}}{\epsilon}} g(u_0(\tau), y_0(\tau), z_0) d\tau\right\|\le \epsilon S,$$ where $y_0(\tau):= y(\tau , u_0(\cdot),y_0)$. Define the control $u_{\epsilon}(t)$ on the interval $[0,t_1)$ by the equation $$u_{\epsilon}(t):= u_0\left(\frac{t}{\epsilon}\right)\ \ \forall \ t\in [0,t_1)$$ and show how this definition can be extended to the interval $[0,T]$. Assume that the control $u_{\epsilon}(t)$ has been defined on the interval $[0,t_l)$ and denote by $(y_{\epsilon}(t), z_{\epsilon}(t))$ the corresponding solution of the system ([\[e:fast\]](#e:fast){reference-type="ref" reference="e:fast"})-([\[e:slow\]](#e:slow){reference-type="ref" reference="e:slow"}) on this interval. Extend the definition of $u_{\epsilon}(t)$ to the interval $[0,t_{l+1})$ by following the steps: (a) Define $v_l\ $ as the projection of the vector $\ \left(\epsilon S\right)^{-1}\int_{t_l}^{t_{l+1}}\frac{dz(t)}{dt}dt\ $ onto the set $V(z_{\epsilon}(t_l))$. That is, $$\label{e-proj-1} v_l:= {\rm argmin}_{v\in V(z_{\epsilon}(t_l))} \left\{\left\| \left(\epsilon S\right)^{-1}\int_{t_l}^{t_{l+1}}\frac{dz(t)}{dt}dt -v \right\| \right\}.$$ (b) Define a control $u_l(\tau)$ such that, being used in the associated system ([\[e:associate-sys\]](#e:associate-sys){reference-type="ref" reference="e:associate-sys"}) on the interval $\ [\frac{t_l}{\epsilon}, \frac{t_{l+1}}{\epsilon}], \ $ ensures that $$\label{e:proof-10-1-extra} \left\|v_l-\frac{1}{S}\int_{\frac{t_l}{\epsilon}}^{\frac{t_{l+1}}{\epsilon}} g(u_l(\tau), y_l(\tau) , z_{\epsilon}(t_l)) d\tau\right\|\le \epsilon S ,$$ where $y_l(\tau)$ is the solution of the associated system ([\[e:associate-sys\]](#e:associate-sys){reference-type="ref" reference="e:associate-sys"}) obtained with the control $u_l(\tau)$ and with the initial condition $y_l(\frac{t_l}{\epsilon})= y_{\epsilon}(t_l)$. The existence of such control follows the fact that, by Lemma [Lemma 2](#Th-1){reference-type="ref" reference="Th-1"}, ${\rm cl}\,V(S,y_{\epsilon}(t_l),z_{\epsilon}(t_l)) = V(z_{\epsilon}(t_l))$. (c) Define the control $u_{\epsilon}(t)$ on the interval $[t_l, t_{l+1})$ as equal to $u_l(\frac{t}{\epsilon})$. That is, $$\label{e:z-aaproximating control} \ u_{\epsilon}(t):= u_l\left(\frac{t}{\epsilon}\right)\ \ \forall \ t\in [t_l, t_{l+1}).$$ Proceeding in a similar way, one can extend the definition of the control $u_{\epsilon}(t)$ to the interval $[0,t_{N_{\epsilon S}})$. On the interval $[t_{N_{\epsilon S}}, T]$, the control $u_{\epsilon}(t)$ can be taken to be equal to an arbitrary element $u$. Denote by $(y_{\epsilon}(t), z_{\epsilon}(t))$ the corresponding solution of the system ([\[e:fast\]](#e:fast){reference-type="ref" reference="e:fast"})-([\[e:slow\]](#e:slow){reference-type="ref" reference="e:slow"}) on the interval $[0,T]$. Denote $$(\bar y_{\epsilon}(\tau), \bar z_{\epsilon}(\tau)):= (y_{\epsilon}(\epsilon\tau), z_{\epsilon}(\epsilon\tau))\ \ \ \ \forall \ \tau\in [0, \frac{T}{\epsilon}].$$ Then $$(\bar y_{\epsilon}(\tau_l), \bar z_{\epsilon}(\tau_l)) = (y_{\epsilon}(t_l), z_{\epsilon}(t_l)) \ \ \ \ \forall \ l=0,1,..., N_{\epsilon},$$ where $\ \tau_l= \frac{t_l}{\epsilon}=lS$ (see ([\[e:partition\]](#e:partition){reference-type="ref" reference="e:partition"})), and $$\label{e:z-epsSeps-estimate} \max_{\tau\in [\tau_l,\tau_{l+1}] }||\bar z_{\epsilon} (\tau)- \bar z_{\epsilon} (\tau_l)|| = \max_{t\in [t_l,t_{l+1}] }|| z_{\epsilon} (t)- z_{\epsilon} (t_l)|| \leq M_g\epsilon S ,$$ where $M_g$ is defined in ([\[e-g-bound\]](#e-g-bound){reference-type="ref" reference="e-g-bound"}). Also notice that $$\label{e:est-z-2-1} \max_{t \in [t_l, t_{l+1}]}||z(t)-z(t_l)|| = \max_{\tau \in [\tau_l, \tau_{l+1}]}||z(\epsilon\tau)-z(t_l)|| \leq M_g \epsilon S,$$ since $$\max\{||v|| \ | \ v\in V(z) ,\ z \in Z \} \leq M_g.$$ Let us verify that thus defined control ensures the validity of ([\[e:Th2.3-3-extra\]](#e:Th2.3-3-extra){reference-type="ref" reference="e:Th2.3-3-extra"}). Subtracting the equation $$z(t_{l+1}) = z(t_l) + \int_{t_l}^{ t_{l+1}}\frac{dz(t)}{dt}dt$$ from the equation $$z_{\epsilon}(t_{l+1}) = z_{\epsilon}(t_l) + \int_{t_l}^{ t_{l+1}}g(u_{\epsilon}(t), y_{\epsilon}(t),z_{\epsilon}(t))dt,$$ one obtains, taking into account ([\[e:z-aaproximating control\]](#e:z-aaproximating control){reference-type="ref" reference="e:z-aaproximating control"}), that $$\label{e:proof-2-17-1} \begin{aligned} &||z_{\epsilon}(t_{l+1})- z( t_{l+1})|| \leq ||z_{\epsilon}(t_{l})- z( t_{l})||\\ &+ \epsilon\int_{\tau_l}^{\tau_{l+1}}||g(u_l(\tau),\bar y_{\epsilon}(\tau), \bar z_{\epsilon}(\tau)) -g(u_l(\tau),\bar y_{\epsilon}(\tau), \bar z_{\epsilon}(\tau_l)) || d\tau\\ &+ \epsilon S \left\|\frac{1}{S}\int_{\tau_l}^{\tau_{l+1}} g(u_l(\tau),\bar y_{\epsilon}(\tau), \bar z_{\epsilon}(\tau_l))d\tau - v_l \right\| + \epsilon S \left\|v_l - \frac{1}{\epsilon S}\int_{t_l}^{ t_{l+1}}\frac{dz(t)}{dt}dt\right\|. \end{aligned}$$ From Lipschitz continuity of $g(u,y,z)$ in $z$ and ([\[e:z-epsSeps-estimate\]](#e:z-epsSeps-estimate){reference-type="ref" reference="e:z-epsSeps-estimate"}) we obtain $$\label{e-estimate-aux-1} \epsilon\int_{\tau_l}^{\tau_{l+1}}||g(u_l(\tau),\bar y_{\epsilon}(\tau), \bar z_{\epsilon}(\tau)) -g(u_l(\tau),\bar y_{\epsilon}(\tau), \bar z_{\epsilon}(\tau_l)) || d\tau \leq \epsilon S L \left(M_g \epsilon S\right).$$ Due to ([\[e:V-Lipschitz-continuity\]](#e:V-Lipschitz-continuity){reference-type="ref" reference="e:V-Lipschitz-continuity"}) and ([\[e:est-z-2-1\]](#e:est-z-2-1){reference-type="ref" reference="e:est-z-2-1"}), $$\begin{aligned} &\frac{dz(t)}{dt} \in V(z(t))\subset V(z(t_l)) + L||z(t)-z(t_l)||\bar B^n\\ &\subset V(z_{\epsilon}(t_l)) + L(||z(t)-z(t_l)||+ ||z(t_l)- z_{\epsilon}(t_l) ||)\bar B^n\\ &\subset V(z_{\epsilon}(t_l)) + (LM_g \epsilon S+ L||z(t_l)- z_{\epsilon}(t_l) ||) \bar B^n \ \ \ \forall \ t \in [t_l,t_{l+1}], \end{aligned}$$ where $\bar B^n$ is the closed unit ball in $R^n$. Consequently, $$\frac{1}{\epsilon S}\int_{t_l}^{t_{l+1}}\frac{dz(t)}{dt}dt \in V(z_{\epsilon}(t_l)) + (LM \epsilon S+ L||z(t_l)- z_{\epsilon}(t_l) ||)\bar B^n,$$ and, by ([\[e-proj-1\]](#e-proj-1){reference-type="ref" reference="e-proj-1"}), $$\begin{aligned} &\left\| \frac{1}{\epsilon S}\int_{t_l}^{ t_{l+1}}\frac{dz(t)}{dt}dt - v_l\right\|= d\left(\frac{1}{\epsilon S}\int_{t_l}^{ t_{l+1}}\frac{dz(t)}{dt}dt, V(z_{\epsilon}(t_l))\right)\\ &\leq LM_g \epsilon S+ L||z(t_l)- z_{\epsilon}(t_l) ||. \end{aligned}$$ Using the latter and ([\[e-estimate-aux-1\]](#e-estimate-aux-1){reference-type="ref" reference="e-estimate-aux-1"}), ([\[e:proof-10-1-extra\]](#e:proof-10-1-extra){reference-type="ref" reference="e:proof-10-1-extra"}), one can obtain from ([\[e:proof-2-17-1\]](#e:proof-2-17-1){reference-type="ref" reference="e:proof-2-17-1"}) $$\begin{aligned} &||z_{\epsilon}(t_{l+1})- z( t_{l+1})|| \leq ||z_{\epsilon}(t_{l})- z( t_{l})|| + \epsilon S L \left(M_g \epsilon S \right)+(\epsilon S)^2\\ &+ LM_g (\epsilon S)^2 + L\epsilon S||z(t_l)- z_{\epsilon}(t_l) || =(1+L\epsilon S)||z(t_l)- z_{\epsilon}(t_l) ||+(\epsilon S)^2(2LM_g+1)\\ &\leq \left(1+\frac{LT}{N_{\epsilon S}}\right)||z(t_l)- z_{\epsilon}(t_l) ||+\left(\frac{T}{N_{\epsilon S}}\right)(\epsilon S) (2LM_g+1), \end{aligned}$$ where the last inequality follows from the definition of $N_{\epsilon S}$ (see ([\[e:partition\]](#e:partition){reference-type="ref" reference="e:partition"})). Using an argument similar to that of Gronwall's lemma (see, e.g., Proposition 5.1 in [@Gai1]), one can now derive that $$\label{e-estimate-aux-2} ||z_{\epsilon}(t_{l})- z( t_{l})||\leq L^{-1}e^{LT}(2LM_g+1) \epsilon S, \ \ \ l=0,1,..., N_{\epsilon S}.$$ Estimate ([\[e-estimate-aux-2\]](#e-estimate-aux-2){reference-type="ref" reference="e-estimate-aux-2"}), along with ([\[e:z-epsSeps-estimate\]](#e:z-epsSeps-estimate){reference-type="ref" reference="e:z-epsSeps-estimate"}) and ([\[e:est-z-2-1\]](#e:est-z-2-1){reference-type="ref" reference="e:est-z-2-1"}), imply that $$||z_{\epsilon}(t)- z( t)||\leq L^{-1}e^{LT}(2LM_g+1) \epsilon S+2M_g\epsilon S \ \ \; \forall t\in [0,T].$$ The right-hand-side in the expression above tends to zero with $S$ tending to zero. Therefore, it can be made less or equal than $\delta$ if $S$ is chosen small enough. This proves ([\[e:Th2.3-3-extra\]](#e:Th2.3-3-extra){reference-type="ref" reference="e:Th2.3-3-extra"}). *Proof of Lemma [Lemma 1](#Lem-control){reference-type="ref" reference="Lem-control"}.* By Cauchy formula, the lemma will be proved if we show that, for any $\tau >0$, there exists a control $u(\cdot)$ such that $$\label{e-L1-1} y''=e^{A\tau}y'+\int_0^{\tau}e^{A(\tau-s)}Bu(s)\,ds\ \ \ \ \Leftrightarrow\ \ \ \ e^{-A\tau}y'' -y' = \int_0^{\tau}e^{-As}Bu(s)\,ds .$$ Take $u(s)=B^Te^{-A^Ts}\xi$ and show that there exists $\xi\in{\rm I\kern-0.2em R}^m$ such that these equalities are satisfied. To this end, substitute $u(s)=B^Te^{-A^Ts}\xi$ into the second equality in ([\[e-L1-1\]](#e-L1-1){reference-type="ref" reference="e-L1-1"}) to obtain $$e^{-A\tau}y'' -y' = \left(\int_0^{\tau}e^{-As}BB^Te^{-A^Ts}\,ds\right)\xi,$$ therefore, $$\left(\int_0^{\tau}e^{-As}BB^Te^{-A^Ts}\,ds\right)^{-1}\Big(e^{-A\tau}y'' -y'\Big)=\xi ,$$ where the latter implication is valid if the matrix $\left(\int_0^{\tau}e^{-As}BB^Te^{-A^Ts}\,ds\right)$ is non-singular. Thus, a sufficient condition for ([\[e-L1-1\]](#e-L1-1){reference-type="ref" reference="e-L1-1"}) to be valid is non-singularity of the matrix $\left(\int_0^{\tau}e^{-As}BB^Te^{-A^Ts}\,ds\right)$. Since this matrix $\left(\int_0^{\tau}e^{-As}BB^Te^{-A^Ts}\,ds\right)$ is non-negative definite, to show that it is non-singular, one needs to show that it is positive definite. Assume the contrary, that is, there exists a nonzero vector $v$ and time $\tau$ such that $$v^T\left(\int_0^{\tau}e^{-As}BB^Te^{-A^Ts}\,ds\right)v=0.$$ This is possible only if $$v^Te^{-As}BB^Te^{-A^Ts}v\equiv 0\quad \hbox{for all }s\in [0,\tau].$$ The latter identity implies that $$v^Te^{-As}B\equiv 0\quad \hbox{for all } s\in [0,\tau].$$ Differentiating this identity with respect to $s$ $j$ times, $j=0,\dots,m-1$ and plugging in $s=0$ we get $$v^TB=0,\,v^TAB=0,\,\dots,v^TA^{m-1}B=0,$$ or, in a matrix form, $$v^T[B, AB,\dots,A^{m-1}B]=0.$$ This implies that the rank of the matrix $[B, AB,\dots,A^{m-1}B]$ is less than $m$, which is a contradiction to our assumption ([\[e-rank\]](#e-rank){reference-type="ref" reference="e-rank"}). abc99xyz Alvarez, O., Bardi, M.: Viscosity Solutions Methods for Singular Perturbations in Deterministic and Stochastic Control. SIAM J. 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arxiv_math

--- abstract: | A classical theorem by Jacobson says that a ring in which every element $x$ satisfies the equation $x^n=x$ for some $n>1$ is commutative. According to Birkhoff's Completeness Theorem, if $n$ is fixed, there must be an equational proof of this theorem. But equational proofs have only appeared for some values of $n$ so far. This paper is about finding such a proof in general. We are able to make a reduction to the case that $n$ is a prime power $p^k$ and the ring has characteristic $p$. We then prove the special cases $k=1$ and $k=2$. The general case is reduced to a series of constructive Wedderburn Theorems, which we can prove in many special cases. Several examples of equational proofs are discussed in detail. author: - Martin Brandenburg title: Equational proofs of Jacobson's Theorem --- # Introduction Jacobson has shown in [@Ja45] that every ring in which every element satisfies an equation of the form $$x^n = x$$ is commutative, where $n > 1$ may depend on $x$. This theorem has been strengthened by various authors. For example, Herstein [@He53] showed that it is enough to require that $x^n - x$ is central. Recently, Anderson and Danchev [@AD20] proved the commutativity of rings satisfying the equation $x^n = x^m$, where $n,m$ are of opposite parity. In the present paper, we are interested in the special case of rings satisfying $x^n = x$ for a fixed number $n > 1$, which we call *$n$-rings*. The case $n=2$ corresponds to the well-studied boolean rings, for which there is a simple equational commutativity proof (see [Example 4](#2case){reference-type="ref" reference="2case"}). More generally, by Birkhoff's Completeness Theorem [@Bi35] (see also [@Ta79] for a modern account), there must be an equational proof of the commutativity of $n$-rings, which thus exists of simple symbolic manipulations and substitutions that derive $xy=yx$ from the ring axioms and $x^n=x$. However, Jacobson's proof is not equational, and indeed quite sophisticated. More simple proofs of Jacobson's Theorem have been published by Forsythe and McCoy [@FM46], Herstein [@He54; @He61], Rogers [@Ro71] and Dolan [@Do76], but arguably the most elementary and slick proofs have been given by Wamsley [@Wa71] and Nagahara and Tominaga [@NT74] (their proofs are the same). Still, these proofs are not equational. Equational proofs have only been found for some values of $n$ so far. Moreover, they can be quite challenging to find. Buckley and MacHale [@BM13] present the classical case $n=2$, several elementary proofs in the case $n=3$, and also some useful lemmas and generalizations. Wavrik [@Wa99] discusses the cases $n=3,4$, several similar commutativity problems, and also presents a general algorithm which is supposed to derive $xy=xy$ from $x^n=x$ for general $n$, but the author does not verify that it always works. Zhang [@Zh90] presents an algorithm for even $n$, which, in numerical experiments, often reduces the problem to the cases $n=2$ and $n=4$. (The present paper shows that this approach is too optimistic, though.) Morita's impressive work [@Mo78] gives equational proofs for even numbers $\leq 50$ (except for $16,22,32,46$, which are much harder) and odd numbers $\leq 25$. Some larger numbers are covered as well. MacHale [@Ma86] covers the case that $n$ is the sum of two distinct proper powers of $2$, even under the weaker assumption that $x^n-x$ is central. Buckley and MacHale have continued this study in [@BM12]. Forsythe and McCoy [@FM46] give an equational proof that $p$-rings of characteristic $p$ are commutative, where $p$ is a prime number. The general case, however, seems to be an open problem, and no systematic approach has been taken so far. If we follow Birkhoff's proof in our case, we get the following evidence for the existence of an equational proof: Consider the free ring in two variables $\mathds{Z}\langle X,Y\rangle$, consisting of non-commuting polynomials in $X$ and $Y$. Then the quotient ring $\mathds{Z}\langle X,Y\rangle / \langle f^n - f : f \in \mathds{Z}\langle X,Y \rangle\rangle$ is an $n$-ring. By Jacobson's Theorem, this ring is commutative. In particular, the image of the commutator $XY-YX$ vanishes. Hence, there must be an equation $$\label{birkhoffliko} XY - YX = \sum_{i=1}^{s} g_i \cdot (f_i^n -f_i) \cdot h_i$$ for $f_i,g_i,h_i \in \mathds{Z}\langle X,Y \rangle$ for $i=1,\dotsc,s$. Once it has been found, this non-commutative polynomial equation can be verified from the ring axioms alone. Evaluating this equation in any $n$-ring yields an equational proof that it is commutative, since the right hand side vanishes. However, that approach does not tell us at all how $f_i,g_i,h_i$ look like explicitly, and the existing proofs of Jacobson's Theorem do not *construct* $f_i,g_i,h_i$. In fact, the proofs use non-constructive methods such as the axiom of choice (with the exception of [@Wa71; @NT74; @Do76]) and the law of the excluded middle (LEM). The LEM states that for every statement $P$, either $P$ or $\neg P$ holds. Equivalently, $\neg \neg P$ implies $P$. For example, notice that [@Wa71] merely shows $\neg \neg P$ when $P$ is the statement that $n$-rings are commutative. This also means that Jacobson's Theorem has not been proven so far in intuitionistic logic, which is a type of constructive logic that does not include LEM [@Da01]. For an equational proof, only constructive methods are allowed, and we need to *find* an equation like [\[birkhoffliko\]](#birkhoffliko){reference-type="eqref" reference="birkhoffliko"}. On the other hand, such a single equation is quite complicated in practice, and an equational proof should better be divided into multiple steps that are easier to motivate and to digest. For example, the equation $$\begin{aligned} XY - YX & = \bigl((X+Y)^2 - (X+Y)\bigr) - \bigl(X^2-X\bigr) - \bigl(Y^2-Y\bigr) \\ & \quad + \bigl((YX)^2 - (YX)\bigr) - \bigl((-YX)^2 - (-YX)\bigr),\end{aligned}$$ in $\mathds{Z}\langle X,Y \rangle$ proves that every $2$-ring is commutative. But it is better to present the proof in a more structured way (as in [Example 4](#2case){reference-type="ref" reference="2case"}). The case of $3$-rings illustrates this even better (see [Remark 13](#3caselong){reference-type="ref" reference="3caselong"}). Our goal is to develop a method that works for every value of $n$. We also prove several reduction results, which generalize observations from [@Mo78; @Ma86; @Zh90]. They are motivated by the well-known classification of commutative $n$-rings, which we recall in [3](#sec:struct){reference-type="ref" reference="sec:struct"}. Our first result is a reduction to the case of prime characteristic. For reasons to be explained in [Remark 28](#charmeaning){reference-type="ref" reference="charmeaning"}, we write $p=0$ instead, which also includes the zero ring. **Theorem 1** ([Theorem 32](#proofdecomp){reference-type="ref" reference="proofdecomp"}). *Let $n > 1$ and assume that for every prime number $p$ with $p-1 \mid n-1$ there is an equational proof that every $n$-ring with $p=0$ is commutative. Then there is an equational proof that every $n$-ring is commutative.* The reader might object that this theorem is true since Jacobson's Theorem in conjunction with Birkhoff's Theorem shows that there is an equational proof anyway. But we have already observed that this approach does not *construct* a proof. The actual meaning of our theorem is that its proof is a method that, given equational proofs with the additional assumption $p=0$, outputs an equational proof of the general case. The goal of the theorem is not to verify that a certain statement is true in the classical sense. Its goal is to construct equational proofs. And of course, our theorem does not assume Jacobson's Theorem to be true. The next result concerns a useful reduction to prime powers. **Theorem 2** ([Theorem 35](#reduc){reference-type="ref" reference="reduc"}). *Let $n>1$ and let $p$ be a prime number with $p-1 \mid n-1$. Let $k$ be the $\mathop{\mathrm{lcm}}$ of all $d \geq 1$ such that $p^{d}-1 \mid n-1$. Then there is an equational proof that every $n$-ring with $p=0$ is a $p^k$-ring.* Hence, in order to give an equational proof that $n$-rings are commutative, one may restrict to $p^k$-rings with $p=0$. Again, the actual content of the theorem is the method that results from its proof, which we then apply in several examples. For example, every $9$-ring with $5=0$ is a $5$-ring, and every $10$-ring is a $4$-ring. It is not easy to prove this directly without any guidance, but our proof is the outcome of a general method, so that no individual ideas are necessary to find it. Morita [@Mo78] proved several instances of [Theorem 35](#reduc){reference-type="ref" reference="reduc"}: Direct and clever calculations show that $n$-rings coincide with $2$-rings for $n = 6,12,14,18,20,24,26,30,38,42,44,48$, that $n$-rings coincide with $4$-rings for $n=10,28,34,40$, that $n$-rings coincide $8$-rings for $n=36,50$, and that $46$-rings coincide with $16$-rings. (Some of these calculations are omitted, though.) In contrast, our approach does not depend on a specific value of $n$. Incidentally, the classification of the numbers $n$ such that $n$-rings with $2=0$ coincide with $2$-rings can be found in [@HLY94]. The next result concerns the special case $k=1$. **Theorem 3** ([Theorem 56](#pcase){reference-type="ref" reference="pcase"}). *Let $p$ be a prime number. If $A$ is $p$-ring with $p=0$, then every element of $A$ can be explicitly written as a $\mathds{Z}$-linear combination of idempotent elements. In particular, $A$ is commutative by an equational proof.* The last conclusion comes from the basic fact that in a reduced ring, idempotent elements are central. We also provide a version of that theorem that works over any finite field. Sometimes the case $k=1$ is enough: Let us call a natural number $n > 1$ *simple* if every prime power $q$ with $q-1 \mid n-1$ is actually a prime number. There are many simple numbers, the first ones are $2, 3, 5, 6, 11, 12, 14, 18, 20$, and for these, Jacobson's Theorem is easy to prove ([Theorem 53](#jacsimple){reference-type="ref" reference="jacsimple"}). We can turn it into an equational proof as well: **Theorem 4** ([Corollary 58](#speceq){reference-type="ref" reference="speceq"}). *For every simple number $n>1$ there is an equational proof that every $n$-ring is commutative.* Even though the recipe is always the same, these equational proofs are not "uniform" in $n$, though. They have to be written down for every single value of $n$, and for large values of $n$ this can be quite cumbersome. We do not know if there is a uniform proof. For non-simple numbers, our method is often able to reduce the problem to smaller numbers. For example, the commutativity of $2023$-rings can be reduced to the commutativity of $4$-rings (see [Example 63](#2023case){reference-type="ref" reference="2023case"}). We also give a proof of the case $k=2$, inspired by Morita's proofs for $n=9,25$. **Theorem 5** ([Theorem 67](#p2case){reference-type="ref" reference="p2case"}). *Let $p$ be a prime number. There is an equational proof that every $p^2$-ring with $p=0$ is commutative.* In conjunction with the previous reduction results, this covers many more examples. For example, we find an equational proof that $73$-rings are commutative ([Example 68](#73case){reference-type="ref" reference="73case"}). As for the general case, we take inspiration from the elementary proof of Jacobson's Theorem in [@Wa71; @NT74]. It reduces the problem to finite rings, where Wedderburn's Theorem comes into play, which states that finite division rings are commutative. We are able to make the reduction constructive and even equational. We just have to replace Wedderburn's Theorem by the following constructive variant: If $f \in \mathds{F}_p[T]$, then $W_{p,k,f}$ states that in a $p^k$-ring with $p=0$, we have $$ba=f(a)b \implies ba=ab.$$ In classical logic, Wedderburn's Theorem implies $W_{p,k,f}$ ([Lemma 71](#Wc){reference-type="ref" reference="Wc"}). We then prove: **Theorem 6** ([\[generalcase-eq,weddermonomial\]](#generalcase-eq,weddermonomial){reference-type="ref" reference="generalcase-eq,weddermonomial"}). *Let $p$ be a prime number and $k \geq 1$. Assume that for all $1 \leq m < k$ there is an equational proof of the constructive Wedderburn Theorem $\smash{W_{p,k,T^{p^m}}}$. Then there is an equational proof that every $p^k$-ring with $p=0$ is commutative. The same conclusion holds when there are equational proofs of all $W_{p,k,f}$ where $f$ is of degree $<k$.* Thus, the problem reduces to finding proofs of certain constructive Wedderburn Theorems. We do not succeed in general, but the following special case already covers many cases. **Theorem 7** ([Theorem 90](#gcdmain){reference-type="ref" reference="gcdmain"}). *Let $p$ be a prime number and $k \geq 1$. If $\gcd(k,p^k-1)=1$, then there is an equational proof that every $p^k$-ring with $p=0$ is commutative.* Table [1](#resulttable){reference-type="ref" reference="resulttable"} summarizes our results for ${2 \leq n \leq 101}$. Writing $\mathop{\mathrm{{\to}}}m_1,m_2,\dotsc$ means that commutativity can be reduced to the classes of $m_1$-rings, $m_2$-rings, etc. (by [\[proofdecomp,reduc,pcase\]](#proofdecomp,reduc,pcase){reference-type="ref" reference="proofdecomp,reduc,pcase"}). Writing $=\!m$ means that the classes of $n$-rings and $m$-rings coincide. An empty cell means that the equational proof is not complete yet. We plan to fill these gaps in future work. $n$ proof $n$ proof $n$ proof $n$ proof $n$ proof ------ --------------------------------------------------------------------- ------ --------------------------------------------------------------------- ------ ------------------------------------------------------------------ --------------- ------------------------------------------------------------------ ------- ------------------------------------------------------------------ $2$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $22$ $\mathop{\mathrm{{\to}}}64$ $42$ $=2$ $62$ $=2$ $82$ $=4$ $3$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $23$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $43$ $\mathop{\mathrm{{\to}}}64$ $63$ $\mathop{\mathrm{{\to}}}32$ $83$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $4$ [\[p2case\]](#p2case){reference-type="eqref" reference="p2case"} $24$ $=2$ $44$ $=2$ $\mathbf{64}$ $84$ $=2$ $5$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $25$ [\[p2case\]](#p2case){reference-type="eqref" reference="p2case"} $45$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $65$ $\mathop{\mathrm{{\to}}}9$ $85$ $\mathop{\mathrm{{\to}}}64$ $6$ $=2$ $26$ $=2$ $46$ $=16$ $66$ $=2$ $86$ $=2$ $7$ $\mathop{\mathrm{{\to}}}4$ $27$ [\[gcdmain\]](#gcdmain){reference-type="eqref" reference="gcdmain"} $47$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $67$ $\mathop{\mathrm{{\to}}}4$ $87$ $=3$ $8$ [\[8case\]](#8case){reference-type="eqref" reference="8case"} $28$ $=4$ $48$ $=2$ $68$ $=2$ $88$ $=4$ $9$ [\[p2case\]](#p2case){reference-type="eqref" reference="p2case"} $29$ $\mathop{\mathrm{{\to}}}8$ $49$ [\[p2case\]](#p2case){reference-type="eqref" reference="p2case"} $69$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $89$ $\mathop{\mathrm{{\to}}}9$ $10$ $=4$ $30$ $=2$ $50$ $=8$ $70$ $=4$ $90$ $=2$ $11$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $31$ $\mathop{\mathrm{{\to}}}16$ $51$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $71$ $\mathop{\mathrm{{\to}}}8$ $91$ $\mathop{\mathrm{{\to}}}16$ $12$ $=2$ $32$ [\[gcdmain\]](#gcdmain){reference-type="eqref" reference="gcdmain"} $52$ $=4$ $72$ $=2$ $92$ $=8$ $13$ $\mathop{\mathrm{{\to}}}4$ $33$ $\mathop{\mathrm{{\to}}}9$ $53$ $\mathop{\mathrm{{\to}}}27$ $73$ $\mathop{\mathrm{{\to}}}4,9,25$ $93$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $14$ $=2$ $34$ $=4$ $54$ $=2$ $74$ $=2$ $94$ $\mathop{\mathrm{{\to}}}1024$ $15$ $\mathop{\mathrm{{\to}}}8$ $35$ $=3$ $55$ $\mathop{\mathrm{{\to}}}4$ $75$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $95$ $=3$ $16$ [\[gcdmain\]](#gcdmain){reference-type="eqref" reference="gcdmain"} $36$ $=8$ $56$ $=2$ $76$ $=16$ $96$ $=2$ $17$ $\mathop{\mathrm{{\to}}}9$ $37$ $\mathop{\mathrm{{\to}}}4$ $57$ $\mathop{\mathrm{{\to}}}8,9$ $77$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $97$ $\mathop{\mathrm{{\to}}}4,9,25,49$ $18$ $=2$ $38$ $=2$ $58$ $=4$ $78$ $=8$ $98$ $=2$ $19$ $\mathop{\mathrm{{\to}}}4$ $39$ $=3$ $59$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $79$ $\mathop{\mathrm{{\to}}}4,27$ $99$ $\mathop{\mathrm{{\to}}}8$ $20$ $=2$ $40$ $=4$ $60$ $=2$ $80$ $=2$ $100$ $=4$ $21$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} $41$ $\mathop{\mathrm{{\to}}}9$ $61$ $\mathop{\mathrm{{\to}}}16$ $\mathbf{81}$ $101$ [\[speceq\]](#speceq){reference-type="eqref" reference="speceq"} : Equational proofs for $2 \leq n \leq 101$ and their reductions # Preliminaries Following Jacobson [@Ja12 Section 2.17], rings are assumed to be unital, and we explicitly speak of rings without a required unit when necessary, which we call *rngs*. See also [@Po14]. Of course, rings are not assumed to be commutative unless otherwise stated. The set of prime numbers is denoted by $\mathds{P}$. We need to briefly formalize the notion of an equational proof in order to clarify what is allowed in our proofs and what not. For more details, we refer to [@Bi35] and [@Ta79]. **Definition 1**. If $\mathcal{T}$ is an algebraic theory and $\sigma,\tau$ are terms in $\mathcal{T}$, by an *equational proof of $\sigma = \tau$* we mean a deduction $\mathcal{T}\vdash \sigma = \tau$ based on the following rules: 1. [\[rule1\]]{#rule1 label="rule1"} If $\sigma = \tau$ is an axiom of $\mathcal{T}$, then $\mathcal{T}\vdash \sigma = \tau$. 2. [\[rule2\]]{#rule2 label="rule2"} We have $\mathcal{T}\vdash \sigma = \sigma$. 3. [\[rule3\]]{#rule3 label="rule3"} If $\mathcal{T}\vdash \sigma = \tau$, then $\mathcal{T}\vdash \tau = \sigma$. 4. [\[rule4\]]{#rule4 label="rule4"} If $\mathcal{T}\vdash \sigma = \tau$ and $\mathcal{T}\vdash \tau = \rho$, then $\mathcal{T}\vdash \sigma = \rho$. 5. [\[rule5\]]{#rule5 label="rule5"} If $\rho$ is a term of arity $n$ in $\mathcal{T}$ and $\sigma_1,\dotsc,\sigma_n$, $\tau_1,\dotsc,\tau_n$ are terms in $\mathcal{T}$ with $\mathcal{T}\vdash \sigma_i = \tau_i$ for all $i=1,\dotsc,n$, then $\mathcal{T}\vdash \rho(\sigma_1,\dotsc,\sigma_n) = \rho(\tau_1,\dotsc,\tau_n)$. 6. [\[rule6\]]{#rule6 label="rule6"} If $\sigma,\tau$ are terms of arity $n$ in $\mathcal{T}$ with $\mathcal{T}\vdash \sigma = \tau$, then $\mathcal{T}\vdash \sigma(\rho_1,\dotsc,\rho_n) = \tau(\rho_1,\dotsc,\rho_n)$ for all terms $\rho_1,\dotsc,\rho_n$ in $\mathcal{T}$. Rule [\[rule1\]](#rule1){reference-type="eqref" reference="rule1"} is self-explanatory. Rules [\[rule2\]](#rule2){reference-type="eqref" reference="rule2"},[\[rule3\]](#rule3){reference-type="eqref" reference="rule3"},[\[rule4\]](#rule4){reference-type="eqref" reference="rule4"} state that equational equality is an equivalence relation. Rule [\[rule5\]](#rule5){reference-type="eqref" reference="rule5"} means that we may replace equationally equal terms in expressions. Rule [\[rule6\]](#rule6){reference-type="eqref" reference="rule6"} allows substitutions of variables with more complex expressions. In practice, we will just write $\sigma = \tau$ instead of $\mathcal{T}\vdash \sigma = \tau$. Notice that there are no quantors and no negation. **Example 2**. There is an equational proof that every group $G$ with $g^2=1$ for all $g \in G$ is commutative. More precisely, in the theory of groups with the additional axiom $g^2=1$, we derive $gh=hg$ as follows: We have $(gh)^2=1$ by rule [\[rule6\]](#rule6){reference-type="eqref" reference="rule6"}, and $1=1 \cdot 1 = g^2 \cdot h^2$ by rule [\[rule5\]](#rule5){reference-type="eqref" reference="rule5"}. (Of course, we have also applied the other, more trivial rules.) Hence, $(gh)^2=g^2 h^2$, which means $ghgh=gghh$. Multiplying with $g^{-1}$ on the left and with $h^{-1}$ on the right, that is, plugging this equation into the term $g^{-1} x h^{-1}$ and using rule [\[rule5\]](#rule5){reference-type="eqref" reference="rule5"} again, shows $hg=gh$. **Remark 3**. Notice that an equational proof does not talk about elements of a set. It is a purely *syntactical* deduction. Even when we write "For all elements $x,y$, \..." (for the sake of readability of the proof), these are actually just variables. Alternatively, we may interpret an equational proof as a classical proof where every statement has the form "For all elements \..., the equation \... holds", and only a few basic deduction rules are allowed, namely the ones in [Definition 1](#eqlogic){reference-type="ref" reference="eqlogic"}. In the case of the algebraic theory of rings, we are not allowed to use ideals, subrings, quotient rings, direct products of rings, let alone more advanced techniques from commutative or non-commutative algebra. The LEM is not allowed either, so a commutativity proof by contradiction ("If the ring is not commutative, then \...") is not possible. The absence of LEM also entails that the axiom of choice is not available (Diaconescu's Theorem). But in our equational proofs we allow every concept that can easily be reduced to a "pure" equational proof. For rings, this includes the arithmetic of integers, modulo arithmetic, and calculations with polynomials. The statement "$x$ is central" abbreviates $xy=yx$ for a (free) variable $y$. Therefore, equational proofs can be seen as the most direct and elementary proofs possible, even though they can be tough to find. **Example 4**. Every boolean ring is commutative by an equational proof. We start with $$-x=(-x)^2=x^2=x.$$ Then we calculate $$x+y=(x+y)^2 = x^2+xy+yx+y^2=x+xy+yx+y$$ and hence $xy=-yx=yx$. In contrast, the following proof of the same fact is *not* equational: Every reduced ring is a subdirect product of rings without zero-divisors [@Kl80]. The only boolean rings without zero divisors are $0$ and $\mathds{F}_2$, which are commutative. So the subdirect product must be commutative as well. (Incidentally, this approach reduces Jacobson's theorem to the case of division rings.) The second proof shows that every *model* of the theory of boolean rings is commutative. That this implies the existence of an equational proof is the content of Birkhoff's Completeness Theorem [@Bi35]. But as mentioned in the introduction, this approach does not allow us to *find* an equational proof. From a constructive point of view, it is pretty much useless. **Definition 5**. Let $A$ be a rng. 1. We call $A$ *potent* when for every $x \in A$ there is some natural number $n>1$ with $x^n=x$. 2. Let $n > 1$. We call $A$ an *$n$-rng* (or *$n$-ring* in the unital case) when $x^n=x$ holds for every $x \in A$. **Remark 6**. 1. The class of $n$-rings has been studied for example in [@MM37], [@Sch10 Appendix B] (under the name *$n$-boolean rings*) and [@Pi67 Section I.12]. If $p \in \mathds{P}$, some authors require $pA=0$ in the definition of a $p$-ring, but we will not do that. 2. There is an algebraic theory $\mathcal{T}_n$ whose models are $n$-rings: just add the axiom $x^n=x$ to the ring axioms. In particular, [Definition 1](#eqlogic){reference-type="ref" reference="eqlogic"} applies and we can speak of equational proofs in the theory of $n$-rings. 3. The definition of an $n$-ring only uses the multiplicative structure, so that *$n$-monoids* can be defined in the obvious way. But they are not automatically commutative. The smallest counterexample is the unitalization of the semigroup $\{a,b\}$ with $x \cdot y \coloneqq x$. 4. Recently, Oman [@Om23] has shown that a rng is potent if and only if every non-zero subrng contains a non-zero idempotent. 5. Jacobson's Theorem states that every potent rng, and hence every $n$-rng, is commutative [@Ja45]. We refer to [@Wa71; @NT74; @Do76] for elementary (albeit non-equational) proofs. We will see in a moment that it is actually sufficient to consider the unital case. **Remark 7**. Let us gather some basic observations on potent rings and $n$-rings. 1. For every prime power $q$ the finite field $\mathds{F}_q$ is a $q$-ring. This follows from Lagrange's Theorem applied to $\smash{\mathds{F}_q^{\times}}$. 2. Every potent rng is reduced. This is because $x^2 = 0 \implies x=0$ holds. 3. In particular, there is an equational proof that every $n$-rng is reduced (meaning that for every term $\tau$ an equational proof of $\tau^2=0$ leads to an equational proof of $\tau=0$). 4. The class of $n$-rings is closed under product rings, quotient rings and subrings. 5. The class of potent rings is closed under finite products, but not under infinite products. For instance, $\smash{\prod_{p \in \mathds{P}} \mathds{F}_p}$ is not a potent ring. One can even show that $(\mathds{F}_p)_{p \in \mathds{P}}$ has no product in the category of potent rings, which also shows that potent rings are not modelled by an algebraic theory. 6. Every potent ring that is an integral domain is a field. It follows that in a commutative potent ring every prime ideal is maximal, i.e. it has Krull dimension $\leq 0$. (Recall that the zero ring has Krull dimension $-\infty$.) 7. When $n$ is even, there is an equational proof that every $n$-rng satisfies $-x=x$. **Lemma 8**. *In a reduced rng, idempotents are central. In an $n$-rng, an equational proof of this claim is available.* Notice that we cannot claim an equational proof in the case of reduced rings, since they do not form an algebraic theory. *Proof.* Let $e^2 = e$ and $x$ be any element. We compute $$(exe - ex)^2 = exeexe - exeex - exexe + exex =exexe - exex - exexe + exex = 0.$$ Since the rng is reduced, we get $exe = ex$. A similar calculation shows $exe = xe$. Hence, $e$ commutes with $x$. In an $n$-rng, we can make this argument equational: From $(exe-ex)^2=0$ we conclude $exe-ex = (exe-ex)^n = 0$ (since $n > 1$), and proceed exactly as before. ◻ **Lemma 9**. *Let $A$ be a rng, $n>1$ and $x \in A$ be an element with $x^n=x$. If $k,k' \geq 1$ satisfy $k \equiv k' \bmod n-1$, then $x^k = x^{k'}$ by an equational proof. In particular, $x^{n-1}$ is idempotent by an equational proof.* *Proof.* Based on $x^{n-1} x = x$, a simple induction shows $x^{(n-1)i + j} = x^j$ for all $i \geq 0$, $j \geq 1$. ◻ **Corollary 10**. *If $n,m > 1$ are such that $m-1 \mid n-1$, then every $m$-ring is an $n$-ring by an equational proof. $\square$* The last three results will be used a lot in the following. Let us give two sample applications: **Example 11**. Here is an equational proof that every $3$-rng is commutative, taken from [@BM13]. Let $x$ be any element. By [\[idemcentral,idempower\]](#idemcentral,idempower){reference-type="ref" reference="idemcentral,idempower"} every square $y^2$ is central. In particular, $(x^2+x)^2$ is central. It simplifies to $x^4+2x^3+x^2=x^2+2x+x^2=2x^2+2x$. Since $x^2$ is central, it follows that $2x$ is central. Applying $(x^2+x)^2=2(x^2+x)$ twice, we get $x^2+x = (x^2+x)^3 = 4(x^2+x)$, hence $3x^2+3x=0$. Since $x^2$ is central, it follows that $3x$ is central. Thus, $x = 3x-2x$ is central as well. **Example 12**. The following equational proof that every $4$-rng is commutative is inspired from [@Wa99]. Let $x$ be any element. Then $-x=x$, since $4$ is even. Also, every cube $x^3$ is central by [\[idemcentral,idempower\]](#idemcentral,idempower){reference-type="ref" reference="idemcentral,idempower"}. We claim that $f(x) \coloneqq x + x^2$ is idempotent, and hence central. In fact, it squares to $x^2 + 2x^3 + x^4 = x^2 + 0 + x = x + x^2$. For all elements $x,y$ then also $f(x+y)-f(x)-f(y)$ is central, which simplifies to $xy+yx$. In particular, $x(xy+yx)=(xy+yx)x$, which simplifies to $x^2 y = y x^2$. So $y$ commutes with $x^2$. Since $x + x^2$ is central, $y$ also commutes with $x$. (We will generalize this method in [Theorem 67](#p2case){reference-type="ref" reference="p2case"}.) **Remark 13**. As explained in the introduction, every equational proof of the commutativity of $n$-rings leads to a representation of the commutator $XY-YX$ as a sum of expressions of the form $g \cdot (f^n - f) \cdot h$ in the free ring $\mathds{Z}\langle X,Y \rangle$. By unwinding what happens in [Example 11](#3case){reference-type="ref" reference="3case"}, we get the following equation, where $p(f) \coloneqq f^3-f$. $$\begin{aligned} & XY-YX = \\ & + \bigl(p(X+X^2) - p(X) (4+4 X+3 X^2+X^3)\bigr) Y - Y \bigl(p(X+X^2) - p(X) (4+4 X+3 X^2+X^3)\bigr) \\ & + 3 p\bigl(X^2 Y X^2-Y X^2\bigr) - 3 p\bigl(X^2 Y X^2-X^2 Y\bigr) \\ & + 3 \bigl(X^2 Y X^2-X^2 Y\bigr) X^2 Y X p(X) Y (X^2-1) - 3 \bigl(X^2 Y X^2-Y X^2\bigr) (X^2-1) Y X p(X) Y X^2 \\ & + p\bigl((X+X^2)^2 Y (X+X^2)^2 - Y (X+X^2)^2 \bigr) - p\bigl((X+X^2)^2 Y (X+X^2)^2 - (X+X^2)^2 Y \bigr) \\ & + \bigl((X+X^2)^2 Y (X+X^2)^2 - (X+X^2)^2 Y\bigr) (X+X^2)^2 Y (X+X^2) p(X+X^2) Y \bigl((X+X^2)^2-1\bigr) \\ & - \bigl((X+X^2)^2 Y (X+X^2)^2 - Y (X+X^2)^2\bigr) \bigl((X+X^2)^2-1\bigr) Y (X+X^2) p(X+X^2) Y (X+X^2)^2 \\ & + 2 p\bigl(X^2 Y X^2-X^2 Y\bigr) - 2 p\bigl(X^2 Y X^2-Y X^2\bigr) \\ & + 2 \bigl(X^2 Y X^2-Y X^2\bigr) (X^2-1) Y X p(X) Y X^2 - 2 \bigl(X^2 Y X^2-X^2 Y\bigr) X^2 Y X p(X) Y (X^2-1) \\ & + (2 + X) p(X) Y - Y (2 + X) p(X)\end{aligned}$$ It shows that every $3$-ring is commutative. Even though this is a very bad way of writing down the proof, it is interesting that such a single equation always exists. And it also shows for $d \in \mathds{Z}$ that a ring satisfying $d x= d x^3$ for all $x$ satisfies $dxy=dyx$ for all $x,y$. **Proposition 14**. *Commutativity can be reduced to the unital case. More precisely:* 1. *If every potent ring is commutative, then every potent rng is commutative.* 2. *If every $n$-ring is commutative by an equational proof, then every $n$-rng is commutative by an equational proof.* *Proof.* 1. Let $A$ be a potent rng. Let $x,y \in A$ and choose $n,m>1$ with $x^n=x$ and $y^m=y$. By [\[idemcentral,idempower\]](#idemcentral,idempower){reference-type="ref" reference="idemcentral,idempower"} the elements $x^{n-1}$ and $y^{m-1}$ are central. Consider the element $$e \coloneqq x^{n-1} + y^{m-1} - x^{n-1} y^{m-1} = y^{n-1} + x^{m-1} - y^{n-1} x^{m-1}.$$ We calculate $$e x = x^{n-1} x + y^{m-1} x - x^{n-1} y^{m-1} x = x^n + y^{m-1} x - x^n y^{m-1} = x + y^{m-1} x - x y^{m-1} = x.$$ A similar calculation shows $e y = y$. By symmetry we also get $x e = x$ and $y e = y$. Hence, if $B$ denotes the subrng of $A$ generated by $x,y$, which consists of $\mathds{Z}$-linear combinations of finite non-empty products of $x$ and $y$, then $e$ is multiplicative identity of $B$, and clearly $B$ is potent. By assumption, $B$ is commutative, so that $xy=yx$. 2\. In the case of $n$-rngs, we can take $m=n$ and define $e$ as above. There is an equational proof of $xy=yx$ in the theory of $n$-rings. We may assume that $x,y$ are the only variables in the proof. Since $e$ is neutral for $x$ and $y$, every appearance of the multiplicative identity can be replaced by $e$. This yields an equational proof of $xy=yx$ in the theory of $n$-rngs. ◻ Let us demonstrate this reduction in another special case: **Lemma 15**. *In an $n$-rng, there is an equational proof of $$\sum_{k=1}^{n-1} \binom{n}{k} x^k = 0.$$* *Proof.* If we had a multiplicative identity, then this follows simply by expanding $1+x=(1+x)^n$ with the binomial theorem and then cancelling $1$ and $x^n=x$. If not, we just do the same with $x^{n-1} + x = (x^{n-1} + x)^n$ and use that $x^{n-1}$ is neutral for all powers of $x$. ◻ Because of [Proposition 14](#unitalred){reference-type="ref" reference="unitalred"} we will only work with rings (i.e. unital rings) from now on. **Remark 16**. The following are equivalent for an element $x$ of an $n$-ring. 1. $x$ is a unit. 2. $x$ is no zero-divisor. 3. $x^{n-1} = 1$. **Lemma 17**. *Let $A$ be an $n$-ring. Then every element $x \in A$ can be written as $x = e \cdot u$, where $e \in A$ is idempotent and $u \in A$ is a unit. Namely, we have $e = x^{n-1}$ and $u = x + (1-x^{n-1})$.* *Proof.* The element $e \coloneqq x^{n-1}$ is idempotent by Lemma [Lemma 9](#idempower){reference-type="ref" reference="idempower"}. Let $u \coloneqq x + (1-e)$. Then we have $eu = ex =x^n = x$. We need to prove that $u$ is a unit. We use [Remark 16](#unitchar){reference-type="ref" reference="unitchar"}. Notice that the product of $x$ and $1-e$ is zero, and $1-e$ is idempotent. Hence, the binomial theorem shows $u^{n-1} = x^{n-1} + (1-e)^{n-1} = x^{n-1} + (1-e) = 1$. Alternatively, one may verify directly that $x^{n-2} + (1-e)$ is inverse to $u$. ◻ **Corollary 18**. *Let $A$ be an $n$-ring. If every unit of $A$ commutes with every unit of $A$ (by an equational proof), then $A$ is commutative (by an equational proof).* *Proof.* This follows from [\[unitdecomp,idemcentral\]](#unitdecomp,idemcentral){reference-type="ref" reference="unitdecomp,idemcentral"}. ◻ # Structure of $n$-rings {#sec:struct} In this section we will investigate the well-known structure of (commutative) $n$-rings. We also find an equational proof of the reduction to prime characteristic. **Remark 19**. The $n$-fields are those $\mathds{F}_q$, where $q$ is a prime power satisfying $q-1 \mid n-1$. This is because $\smash{\mathds{F}_q^{\times}}$ is cyclic of order $q-1$. See Table [2](#fieldtable){reference-type="ref" reference="fieldtable"} for a list of examples. It has been generated programmatically with a bit of SageMath[^1] code, see [9.1](#app:powers){reference-type="ref" reference="app:powers"}. It follows in particular that every $n$-ring of characteristic $p$ satisfies $p-1 \mid n-1$ (since it contains $\mathds{F}_p$ which is then also an $n$-ring). $n$ $n$-fields $n$ $n$-fields $n$ $n$-fields ------ -------------------------------------------------------------------------------------------------------- ------ ------------------------------------------------------------------------------------------------------------------------------------------- ------ -------------------------------------------------------------------------------------------------------------------------------------------- $2$ $\mathds{F}_{2}$ $18$ $\mathds{F}_{2}$ $34$ $\mathds{F}_{2}, \mathds{F}_{4}$ $3$ $\mathds{F}_{2}, \mathds{F}_{3}$ $19$ $\mathds{F}_{2}, \mathds{F}_{3}, \mathds{F}_{4}, \mathds{F}_{7}, \mathds{F}_{19}$ $35$ $\mathds{F}_{2}, \mathds{F}_{3}$ $4$ $\mathds{F}_{2}, \mathds{F}_{4}$ $20$ $\mathds{F}_{2}$ $36$ $\mathds{F}_{2}, \mathds{F}_{8}$ $5$ $\mathds{F}_{2}, \mathds{F}_{3}, \mathds{F}_{5}$ $21$ $\mathds{F}_{2}, \mathds{F}_{3}, \mathds{F}_{5}, \mathds{F}_{11}$ $37$ $\mathds{F}_{2},\!\mathds{F}_{3},\!\mathds{F}_{4},\!\mathds{F}_{5},\!\mathds{F}_{7},\!\mathds{F}_{13},\!\mathds{F}_{19},\!\mathds{F}_{37}$ $6$ $\mathds{F}_{2}$ $22$ $\mathds{F}_{2}, \mathds{F}_{4}, \mathds{F}_{8}$ $38$ $\mathds{F}_{2}$ $7$ $\mathds{F}_{2}, \mathds{F}_{3}, \mathds{F}_{4}, \mathds{F}_{7}$ $23$ $\mathds{F}_{2}, \mathds{F}_{3}, \mathds{F}_{23}$ $39$ $\mathds{F}_{2}, \mathds{F}_{3}$ $8$ $\mathds{F}_{2}, \mathds{F}_{8}$ $24$ $\mathds{F}_{2}$ $40$ $\mathds{F}_{2}, \mathds{F}_{4}$ $9$ $\mathds{F}_{2}, \mathds{F}_{3}, \mathds{F}_{5}, \mathds{F}_{9}$ $25$ $\mathds{F}_{2},\!\mathds{F}_{3},\!\mathds{F}_{4},\!\mathds{F}_{5},\!\mathds{F}_{7},\!\mathds{F}_{9},\!\mathds{F}_{13},\!\mathds{F}_{25}$ $41$ $\mathds{F}_2,\!\mathds{F}_3,\!\mathds{F}_5,\!\mathds{F}_9,\!\mathds{F}_{11},\!\mathds{F}_{41}$ $10$ $\mathds{F}_{2}, \mathds{F}_{4}$ $26$ $\mathds{F}_{2}$ $42$ $\mathds{F}_2$ $11$ $\mathds{F}_{2}, \mathds{F}_{3}, \mathds{F}_{11}$ $27$ $\mathds{F}_{2}, \mathds{F}_{3}, \mathds{F}_{27}$ $43$ $\mathds{F}_2,\!\mathds{F}_3,\!\mathds{F}_4,\!\mathds{F}_7,\!\mathds{F}_8,\!\mathds{F}_{43}$ $12$ $\mathds{F}_{2}$ $28$ $\mathds{F}_{2}, \mathds{F}_{4}$ $44$ $\mathds{F}_2$ $13$ $\mathds{F}_{2},\!\mathds{F}_{3},\!\mathds{F}_{4},\!\mathds{F}_{5},\!\mathds{F}_{7},\!\mathds{F}_{13}$ $29$ $\mathds{F}_{2}, \mathds{F}_{3}, \mathds{F}_{5}, \mathds{F}_{8}, \mathds{F}_{29}$ $45$ $\mathds{F}_2,\mathds{F}_3,\mathds{F}_5,\mathds{F}_{23}$ $14$ $\mathds{F}_{2}$ $30$ $\mathds{F}_{2}$ $46$ $\mathds{F}_2,\mathds{F}_4,\mathds{F}_{16}$ $15$ $\mathds{F}_{2}, \mathds{F}_{3}, \mathds{F}_{8}$ $31$ $\mathds{F}_{2}, \mathds{F}_{3}, \mathds{F}_{4}, \mathds{F}_{7}, \mathds{F}_{11}, \mathds{F}_{16}, \mathds{F}_{31}$ $47$ $\mathds{F}_2,\mathds{F}_3,\mathds{F}_{47}$ $16$ $\mathds{F}_{2}, \mathds{F}_{4}, \mathds{F}_{16}$ $32$ $\mathds{F}_{2}, \mathds{F}_{32}$ $48$ $\mathds{F}_2$ $17$ $\mathds{F}_{2}, \mathds{F}_{3}, \mathds{F}_{5}, \mathds{F}_{9}, \mathds{F}_{17}$ $33$ $\mathds{F}_{2}, \mathds{F}_{3}, \mathds{F}_{5}, \mathds{F}_{9}, \mathds{F}_{17}$ $49$ $\mathds{F}_2, \mathds{F}_3, \mathds{F}_4, \mathds{F}_5, \mathds{F}_7, \mathds{F}_9,$ $\mathds{F}_{13}, \mathds{F}_{17}, \mathds{F}_{25}, \mathds{F}_{49}$ : List of $n$-fields for $2 \leq n \leq 49$ It turns out that $n$-fields generate all commutative $n$-rings in a suitable sense: **Proposition 20**. *A commutative ring is an $n$-ring if and only if it is a subring of a direct product of fields of the form $\mathds{F}_q$, where $q - 1 \mid n-1$.* *Proof.* The direction $\Leftarrow$ is clear. To prove $\Rightarrow$, let $A$ be a commutative $n$-ring. Then $A$ is reduced, which means that the intersection of all prime ideals is zero. Hence, the canonical map $\smash{A \to \prod_{\mathfrak{p}} Q(A/\mathfrak{p})}$ is injective, where $\mathfrak{p}$ runs through all prime ideals of $A$. Since $Q(A/\mathfrak{p})$ (which is just $A/\mathfrak{p}$, since $\dim(A) \leq 0$) is clearly an $n$-field, the claim follows. ◻ Practically, this means that the class of commutative $n$-rings is already determined by the class of $n$-fields. For example, we have the following improvement of [Corollary 10](#inclusion){reference-type="ref" reference="inclusion"}: **Corollary 21**. *Let $n,n' > 1$. Every commutative $n$-ring is a commutative $n'$-ring if and only if $q-1 \mid n-1 \implies q-1 \mid n'-1$ holds for all prime powers $q$.* *Proof.* The direction $\Leftarrow$ follows from [Proposition 20](#nchar){reference-type="ref" reference="nchar"}. The direction $\Rightarrow$ follows by considering the finite field $\mathds{F}_q$. ◻ **Example 22**. Every $10$-ring is a $4$-ring, and every $14$-ring is a $2$-ring; see also Table [2](#fieldtable){reference-type="ref" reference="fieldtable"}. **Corollary 23**. *Every commutative $n$-ring of prime characteristic $p$ is actually a $p^k$-ring for some $k \geq 1$. Namely, $k$ is the $\mathop{\mathrm{lcm}}$ of all $d \geq 1$ with $p^d-1 \mid n-1$.* *Proof.* Let $A$ be a commutative $n$-ring of characteristic $p$. We have seen in [Proposition 20](#nchar){reference-type="ref" reference="nchar"} that $A$ embeds into a product of fields of the form $\mathds{F}_{p^d}$ with $p^d - 1 \mid n - 1$. But then $d \mid k$ and therefore $\mathds{F}_{p^d} \subseteq \mathds{F}_{p^k}$ is also a $p^k$-ring. ◻ **Remark 24**. If $n,p,k$ are as in [Corollary 23](#pow){reference-type="ref" reference="pow"}, it is not true that, conversely, every $p^k$-ring of characteristic $p$ is an $n$-ring, since we do not necessarily have $p^k - 1 \mid n-1$. The smallest counterexample is $n=22$ and $p=2$. Then $k=\mathop{\mathrm{lcm}}(1,2,3)=6$. Thus, $\mathds{F}_{2^6}$ is an example of a $2^6$-ring that is not a $22$-ring. We can turn any ring into an $n$-ring as follows. **Definition 25**. If $R$ is a ring, we define $$R_{[n]} \coloneqq R/ \langle x^n - x : x \in R \rangle.$$ This is clearly an $n$-ring with a surjective homomorphism $R \to R_{[n]}$. The construction is universal: If $A$ is an $n$-ring and $R \to A$ is a homomorphism of rings, then it lifts uniquely to a homomorphism of $n$-rings $R_{[n]} \to A$. Therefore, we may call $R_{[n]}$ the universal $n$-ring quotient of $R$. **Example 26**. Let us compute the universal $n$-ring quotient of $\mathds{Z}$. We have $$\mathds{Z}_{[n]} \coloneqq \mathds{Z}/\langle z^n - z : z \in \mathds{Z}\rangle = \mathds{Z}/{\gcd(z^n - z : z \in \mathds{Z})}\mathds{Z}.$$ Let $c$ be the $\gcd$ in question. Of course, $c > 0$. Since $\mathds{Z}_{[n]}$ is an $n$-ring and hence reduced, $c$ is square-free. Let $p \mid c$ be a prime factor. This means $z^n \equiv z \bmod p$ for all $z \in \mathds{Z}$. Equivalently, $\mathds{F}_p$ is an $n$-ring, which means $p-1 \mid n-1$ by [Remark 19](#nfields){reference-type="ref" reference="nfields"}. Therefore, $c$ is the product of all primes $p$ with $p-1 \mid n-1$. It follows that if $A$ is an $n$-ring, then $c=0$ holds in $A$, since the unique homomorphism $\mathds{Z}\to A$ lifts to a homomorphism $\mathds{Z}/c\mathds{Z}= \mathds{Z}_{[n]} \to A$. In other words, the characteristic of $A$ divides $c$. **Corollary 27**. *Every $n$-ring is a finite product of $n$-rings of prime characteristic.* *Proof.* Let $A$ be an $n$-ring. By [Example 26](#charcompute){reference-type="ref" reference="charcompute"} its characteristic is $p_1 \cdots p_s$ with distinct prime numbers $p_1,\dotsc,p_s$. Then the Chinese Remainder Theorem (which holds for all rings and sequences of pairwise coprime two-sided ideals) tells us that the canonical homomorphism $A = A/p_1 \cdots p_s A \to A/p_1 A \times \cdots \times A/p_s A$ is an isomorphism. ◻ As a consequence, the commutativity problem can be reduced to prime characteristic. We need to convince ourselves, though, that there is also an equational proof of this reduction. **Remark 28**. Even though it is tempting to say that every boolean ring has characteristic $2$, this is not true: The trivial ring is a boolean ring of characteristic $1$. Every non-trivial boolean ring has characteristic $2$. But since the LEM is not available for equational proofs, we cannot (and do not need to!) decide if a given ring is trivial or not. There is no need to exclude the trivial ring. Therefore, in the following, we will not speak of rings of characteristic $p$, and instead only add $p=0$ as an axiom (a shorthand for $p \cdot 1_A = 0$). Equivalently, we work with $\mathds{F}_p$-algebras. Moreover, notice that in the proof of [Corollary 27](#decomp){reference-type="ref" reference="decomp"} the rings $A/p_i A$ do not necessary have characteristic $p_i$, since they could be trivial. They just satisfy $p_i=0$. Of course, [Corollary 27](#decomp){reference-type="ref" reference="decomp"} remains true (in classical logic). **Lemma 29**. *Let $A$ be an $n$-ring. Let $p_1,\dotsc,p_s$ be the prime numbers $p$ with $p-1 \mid n-1$. Then there is an equational proof of $p_1 \cdots p_s =0$ in $A$.* *Proof.* We saw in [Example 26](#charcompute){reference-type="ref" reference="charcompute"} that $p_1 \cdots p_s$ is the $\gcd$ of all $z^n -z$ with $z \in \mathds{Z}$. The argument used finite fields, which is not an equational concept, but if $n$ is fixed we can ignore this and just compute the $\gcd$ of the numbers $z^n - z$ for $z=1,2,3,\dotsc$ with the Euclidean algorithm until we reach $p_1 \cdots p_s$. It suffices to consider prime numbers for $z$. Then we use the extended Euclidean algorithm to write $p_1 \cdots p_s$ as a $\mathds{Z}$-linear combination of the numbers $z^n - z$. These numbers vanish in $A$, so that $p_1 \cdots p_s$ vanishes as well. ◻ SageMath can do the computation for us for every value of $n$, see [9.2](#app:char){reference-type="ref" reference="app:char"} for the code. **Example 30**. Consider $n=7$. The prime numbers $p$ with $p-1 \mid 7-1$ are $p=2,3,7$. We find $\gcd(2^7-2,3^7-3)=42 = 2 \cdot 3 \cdot 7$ and $42 = -17 (2^7-2) + (3^7-3)$. Hence, $2 \cdot 3 \cdot 7 = 0$ holds in any $7$-ring. For $n=3$, the primes are $2,3$, and we already have $2^3-2 = 6 = 2 \cdot 3$. **Lemma 31**. *Let $\mathcal{T}$ be an algebraic theory extending ring theory by some axioms, and let $p \in \mathds{Z}$. If $\sigma,\tau$ are terms in $\mathcal{T}$ of the same arity such that $\mathcal{T},\, p = 0 \vdash \sigma=\tau$, then there is a term $u$ of the same arity with $\mathcal{T}\vdash \sigma = \tau + p \cdot u$.* *Proof.* This is a simple induction on the structure of an equational proof as in [Definition 1](#eqlogic){reference-type="ref" reference="eqlogic"}. We only show the most interesting one, that is rule [\[rule5\]](#rule5){reference-type="eqref" reference="rule5"}. Assume that $\mathcal{T}$ and $p=0$ prove the equation $\rho(\sigma_1,\dotsc,\sigma_n) = \rho(\tau_1,\dotsc,\tau_n)$ by means of $\sigma_i=\tau_i$ for all $i$. By induction hypothesis, there are terms $u_1,\dotsc,u_n$ such that $\mathcal{T}$ proves $\sigma_i = \tau_i + p \cdot u_i$. Then $\mathcal{T}$ proves the equation $\rho(\sigma_1,\dotsc,\sigma_n) = \rho(\tau_1 + p \cdot u_1,\dotsc,\tau_n + p \cdot u_n)$. Expanding the right hand side yields a term $v$ with $\rho(\tau_1 + p \cdot u_1,\dotsc,\tau_n + p \cdot u_n) = \rho(\tau_1 ,\dotsc,\tau_n ) + p \cdot v$, as desired. Formally, this requires an induction on the structure of $\rho$. ◻ **Theorem 32**. *Assume that for every $p \in \mathds{P}$ with $p-1 \mid n-1$ there is an equational proof that every $n$-ring with $p=0$ is commutative. Then there is an equational proof that every $n$-ring is commutative.* Because of this result, we may restrict our attention to rings with $p=0$ from now on. *Proof.* Let $A$ be an $n$-ring. Let $p_1,\dotsc,p_s$ be the set of $p \in \mathds{P}$ satisfying $p-1 \mid n-1$. Then $p_1 \cdots p_s = 0$ holds in $A$ by [Lemma 29](#chareq){reference-type="ref" reference="chareq"}. If $x,y \in A$, by assumption $xy-yx=0$ can be derived with the additional axiom $p_i=0$. Hence, [Lemma 31](#modproof){reference-type="ref" reference="modproof"} yields $u_i \in A$ with $xy - yx = p_i \cdot u_i$. In classical logic, this step of the proof would be easier: just use that $[x]$ and $[y]$ commute in the quotient ring $A/p_i A$. Therefore, it suffices to prove the following in equational logic: If $p_1,\dotsc,p_s$ are pairwise coprime integers and $x \in A$, $u_i \in A$ satisfy $x = p_i \cdot u_i$ for $i=1,\dotsc,s$, then we can construct an element $v$ with $x = p_1 \cdots p_s \cdot v$. The case $s=1$ is easy. If $s=2$, the extended Euclidean algorithm yields $q_1,q_2 \in \mathds{Z}$ with $q_1 p_1 + q_2 p_2 = 1$. Then $$x = q_1 p_1 x + q_2 p_2 x = q_1 p_1 p_2 u_2 + q_2 p_2 p_1 u_1 = p_1 p_2 (q_1 u_2 + q_2 u_1).$$ For the general case, we use induction and know $x = p_1 \cdots p_{s-1} \cdot v$ for some element $v$. Since $p_1 \cdots p_{s-1}$ and $p_s$ are coprime, the case $s=2$ yields an element $w$ with $x = p_1 \cdots p_{s-1} p_s \cdot w$, and we are done. ◻ **Example 33**. Consider $n = 5$. The primes $p$ with $p-1 \mid 5-1$ are $2,3,5$, and $2^5 - 2 = 2 \cdot 3 \cdot 5$. Hence, $2 \cdot 3 \cdot 5 = 0$ holds in any $5$-ring. Assume that there is a commutativity proof modulo each of these primes. Then we get elements $u_1,u_2,u_3$ with $xy - yx = 2 u_1$, $xy-yx = 3 u_2$, $xy-yx = 5 u_3$. Hence, $xy-yx = 30 (u_3 - u_1 + u_2) = 0$. **Remark 34**. By Stone duality [@St36], the category of boolean rings is anti-equivalent to the category of totally disconnected compact Hausdorff spaces (Stone spaces). There is a similar classification for $n$-rings, which is thus much more precise than [Proposition 20](#nchar){reference-type="ref" reference="nchar"}. Namely, if $p^{d_1},\dotsc,p^{d_s}$ are the powers of $p$ with $p^{d_i} - 1 \mid n - 1$ and $k \coloneqq \mathop{\mathrm{lcm}}(d_1,\dotsc,d_s)$, then the category of $n$-rings with $p=0$ is anti-equivalent to the category of Stone spaces $X$ with a continuous $C_k$-action such that $\smash{X = \bigcup_{i=1}^{s} X^{C_{k/d_i}}}$. The proof of this anti-equivalence will appear elsewhere. # Reduction to prime powers The goal of this section is to find an equational proof of [Corollary 23](#pow){reference-type="ref" reference="pow"}. **Theorem 35**. *Let $n>1$ and let $p \in \mathds{P}$ with $p-1 \mid n-1$. Let $k$ be the $\mathop{\mathrm{lcm}}$ of all $d \geq 1$ such that $p^{d}-1 \mid n-1$. Then there is an equational proof that every $n$-ring with $p=0$ is a $p^k$-ring. Moreover, if $p^k - 1 \mid n-1$, the converse is also true.* This is actually a meta-theorem. The proof is a method that enables us to write down, for every fixed pair $(n,p)$, an equational proof that every $n$-ring with $p=0$ is a $p^k$-ring. We will demonstrate this with several examples. Recall from [Remark 28](#charmeaning){reference-type="ref" reference="charmeaning"} why we write $p=0$ instead of "characteristic $p$". The theorem is a generalization and better explanation of Morita's reduction results [@Mo78] mentioned in the introduction. It also includes MacHale's observation in [@Ma86] that $2^m+1$-rings with $2=0$ are boolean as a special case. *Proof of [Theorem 35](#reduc){reference-type="ref" reference="reduc"}.* Since we merely want to show $\smash{x^{p^k}=x}$, it is clear that only polynomials in one variable over $\mathds{F}_p$ appear in the proofs, so that commutativity is not an issue here. The first step is to use the Euclidean algorithm to compute the polynomial $$\label{def:g} g \coloneqq \gcd(f^n - f : f \in \mathds{F}_p[T], \, \deg(f) < n) \in \mathds{F}_p[T].$$ This computation can be quite cumbersome, but see [Remark 37](#perf){reference-type="ref" reference="perf"} below for performance issues. Then one uses the extended Euclidean algorithm to write $g$ as a linear combination of polynomials of the form $f^n - f$. We claim that $$\label{div} g \mid T^{p^k} - T.$$ The following proof is abstract and certainly not equational, but in practice, for every fixed pair $(n,p)$, [\[div\]](#div){reference-type="ref" reference="div"} can just be verified via polynomial division. This why an equational proof still does exist. The universal $n$-ring with $p=0$ on one generator is the quotient $\mathds{F}_p[T]_{[n]} = \mathds{F}_p[T]/I$, where $I \coloneqq \langle f^n - f : f \in \mathds{F}_p[T] \rangle$ (see [Definition 25](#univquotient){reference-type="ref" reference="univquotient"}). We claim that $I$ is generated by those $f^n - f$ with $\deg(f) < n$. In fact, if $f \in \mathds{F}_p[T]$ is arbitrary, then polynomial division yields a polynomial $f'$ of degree $<n$ with $f \equiv f' \bmod T^n-T$, which also implies $f^n-f \equiv f'^n-f' \bmod T^n-T$. Since $\mathds{F}_p[T]/I$ is a $p^k$-ring by [Corollary 23](#pow){reference-type="ref" reference="pow"}, we have $\smash{T^{p^k}-T \in I = \langle g \rangle}$, which proves $\smash{g \mid T^{p^k}-T}$. Since we already wrote $g$ as a linear combination of polynomials of the form $f^n-f$, we can now do the same for $\smash{T^{p^k}-T}$. So, we have found an equation of the form $$T^{p^k}- T = \sum_{i=1}^{s} u_i \cdot (f_i^n - f_i)$$ in $\mathds{F}_p[T]$. Now, if $A$ is any ring with $p=0$, then for $x \in A$ we get $$x^{p^k}- x = \sum_{i=1}^{s} u_i(x) \cdot (f_i(x)^n - f_i(x)).$$ For a pure equational proof that is independent from the proof above, just expand the right hand side of this equation and then simplify. The result will be $\smash{x^{p^k}-x}$. If $A$ is an $n$-ring with $p=0$, the right hand side vanishes, so that $\smash{x^{p^k}=x}$. Finally, if $p^k - 1 \mid n-1$, then every $p^k$-ring is an $n$-ring by [Corollary 10](#inclusion){reference-type="ref" reference="inclusion"}. ◻ **Corollary 36**. *In order to give an equational proof that $n$-rings are commutative, one may restrict to $p^k$-rings with $p=0$.* *Proof.* This follows from [Theorem 32](#proofdecomp){reference-type="ref" reference="proofdecomp"} and [Theorem 35](#reduc){reference-type="ref" reference="reduc"}. ◻ **Remark 37**. The definition of the $\gcd$ in [\[def:g\]](#def:g){reference-type="ref" reference="def:g"} requires all $p^n$ polynomials $f \in \mathds{F}_p[T]$ of degree $<n$, which is not practical for computations. But a subset of these polynomials is usually sufficient. Let us explain this in more detail. First we may assume that $f$ is monic (since $p-1 \mid n-1$, constants $u \in \mathds{F}_p$ satisfy $u^n=u$ anyway) and $\deg(f) > 0$. Choose an enumeration $f_1,f_2,f_3,\dotsc$ of all monic polynomials of degree $<n$ with $\deg(f_i) \leq \deg(f_{i+1})$. It is recommended to start with $f_1 = T$ and $f_2 = T + 1$. Then the "partial $\gcd$" $$g_i \coloneqq \gcd(f_1^n-f_1,\dotsc,f_i^n-f_i),$$ can be computed recursively via $g_{i+1} = \gcd(g_i,f_{i+1}^n-f_{i+1})$. If some $g_i$ already satisfies $\smash{g_i \mid T^{p^k}-T}$, we are done, even when maybe $g_i \neq g$. If not, compute $g_{i+1}$ and continue. It turns out (see the examples below and [Remark 51](#lineargcd){reference-type="ref" reference="lineargcd"}) that in many cases a small $i$ will do the job, and that only a few, mostly linear polynomials are required for the $f_1,\dotsc,f_i$. It would be very much desirable to give a "uniform" equational proof of [Theorem 35](#reduc){reference-type="ref" reference="reduc"}, which thus can be written down without any specific choice of $(n,p)$ and also without any tedious calculations. But it is not clear if it exists. In any case, using a computer algebra system does help to find the equations. For the examples below, we have again used SageMath, the code can be found in [9.3](#app:reduc){reference-type="ref" reference="app:reduc"}. **Remark 38**. Because of [Remark 24](#converse){reference-type="ref" reference="converse"}, the polynomial $g$ is not always equal to $\smash{T^{p^k}- T}$, even though in the examples below it will often be the case. In general, one can show that $g$ is the product of all monic irreducible polynomials in $\mathds{F}_p[T]$ whose degree $d$ satisfies $p^d - 1 \mid n-1$. For $k=1$ this means $g = T^p - T$. We start with some examples where linear polynomials are sufficient for computing $g$. **Example 39**. Consider $n=3$ and $p=2$. Then $k=1$. The $\gcd$ of $T^3-T$ and $(T+1)^3 - (T+1)$ in $\mathds{F}_2[T]$ is $T^2-T$ with the linear combination $$T^2-T = (T^3-T) + ((T+1)^3-(T+1)).$$ If $A$ is any ring with $2=0$ and $x \in A$, the equation $$x^2-x = (x^3-x) + ((x+1)^3-(x+1))$$ can simply be verified by hand. This shows that any $3$-ring with $2=0$ is a $2$-ring. Here is a more compact version: We have $x+1=(x+1)^3=x^3+3x^2+3x+1=x+x^2+x+1$, hence $x^2=x$. Alternatively, [Lemma 15](#binomrel){reference-type="ref" reference="binomrel"} gives directly $3x + 3x^2 = 0$, hence $x^2=x$. **Example 40**. Consider $n=5$ and $p=2$. Then $k=1$ and the extended Euclidean algorithm yields $$T^2-T = (T+1) \cdot (T^5-T) + T \cdot ((T+1)^5-(T+1)) \in \mathds{F}_2[T].$$ This produces the following equational proof that every $5$-ring with $2=0$ is a $2$-ring: We have $x+1=(x+1)^5 = x^5+x^4+x+1$, which gives $x^4=x^5=x$, hence $x=x^5=x^2$. Alternatively, we may argue with [Lemma 15](#binomrel){reference-type="ref" reference="binomrel"} again and get $0=5x+10x^2+10x^3+5x^4 = x+x^4$, so that $x^4=x$ and then $x=x^2$. **Example 41**. Consider $n = 5$ and $p = 3$. Then $k=1$ and $$T^3 - T = T \cdot (T^5-T) - (T+1) \cdot ((T+1)^5-(T+1)) \in \mathds{F}_3[T].$$ This leads to the following equational proof that every $5$-ring with $3=0$ is a $3$-ring: We compute $x+1=(x+1)^5 = x^5 + 2x^4 + x^3 + x^2 + 2x + 1 = 2x^4+x^3+x^2+1$, hence $x^4=x^3+x^2-x$ ($\dagger$). Multiplying with $x$ and using $x^5=x$ gives $x=x^4+x^3-x^2$. Plugging this back into ($\dagger$) gives $x^4=x^2$ and hence $x=x^5=x^3$. **Example 42**. Consider $n=6$, hence $p=2$ and $k=1$. Then $$T^2-T = T^2 \cdot (T^6-T) + (T^2+1) \cdot ((T+1)^6 - (T+1)) \in \mathds{F}_2[T]$$ leads to the following equational proof that every $6$-ring is a $2$-ring: Since $6$ is even, we have $-x=x$. Next, we compute $x+1 = (x+1)^6 = x^6 + x^4 + x^2 + 1 = x + x^4 + x^2 + 1$, hence $x^4=x^2$. Thus, $x=x^6 = x^2 x^4 = x^2 x^2 = x^4 = x^2$. Conversely, every $2$-ring is a $6$-ring by [Corollary 10](#inclusion){reference-type="ref" reference="inclusion"}. **Example 43**. Consider $n = 9$ and $p = 5$. Then $k = 1$, and $$T^5-T = 2T \cdot (T^9 - T) - (2T + 2) \cdot ((T+1)^9 - (T+1)) \in \mathds{F}_5[T]$$ can be used to give an equational proof that every $9$-ring with $5=0$ is a $5$-ring. **Example 44**. Consider $n=10$, hence $p=2$ and $k=2$. Then $$T^4-T = (T^2+1) \cdot (T^{10}-T) + T^2 \cdot ((T+1)^{10}-(T+1)) \in \mathds{F}_2[T]$$ shows that every $10$-ring is a $4$-ring. Conversely, every $4$-ring is a $10$-ring by [Corollary 10](#inclusion){reference-type="ref" reference="inclusion"}. **Example 45**. Consider $n = 17$ and $p=3$. Then $k=2$ and $$T^9 - T = T \cdot (T^{17}-T) - (T+1) \cdot ((T+1)^{17}-(T+1)) \in \mathds{F}_3[T]$$ shows that every $17$-ring with $3=0$ is a $9$-ring. It produces the following equational proof: [Lemma 15](#binomrel){reference-type="ref" reference="binomrel"} together with $\smash{\binom{17}{i} \equiv (-1)^i \bmod 3}$ for $0 \leq i \leq 8$ yields $$-x^{16}+x^{15}-x^{14}+x^{13}-x^{12}+x^{11}-x^{10}+x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x=0.$$ Multiplying this with $x$ and using $x^{17}=x$ gives $$+x^{16}-x^{15}+x^{14}-x^{13}+x^{12}-x^{11}+x^{10}+x^9-x^8+x^7-x^6+x^5-x^4+x^3-x^2-x=0.$$ Adding these two equations gives $2x^9-2x=0$, so that $x^9=x$. **Example 46**. Consider $n = 35$. Then $p=3$ and $k=1$. In $\mathds{F}_3[T]$ we find that $T^3-T$ is the $\gcd$ of $T^{35}-T$ and $(T+1)^{35}-(T+1)$, so that every $35$-ring is a $3$-ring. The converse also holds by [Corollary 10](#inclusion){reference-type="ref" reference="inclusion"}. **Example 47**. Consider $n=7$ and $p=2$. Then $k=2$. Here, the $\gcd$ with linear polynomials does *not* suffice. But we are lucky with a quadratic one: $$\begin{aligned} T^4 - T & = (T^8+T^6+T^5+T^3+1) \cdot (T^7-T) \\ & \quad + (T^8+T^6+T^5+T^3+1) \cdot ((T+1)^7-(T+1)) \\ & \quad + ((T^2+T+1)^7-(T^2+T+1))\end{aligned}$$ There is a simpler expression with a cubic polynomial: $$T^4 - T = (T^{15}+T^{12} + T^3 + 1) \cdot (T^7-T) + T \cdot ((T^3+1)^7 - (T^3+1))$$ We get the following equational proof that every $7$-ring with $2=0$ is a $4$-ring: We compute $$\begin{aligned} x^3+1 & = (x^3+1)^7 \\ & = x^{21} + x^{18} + x^{15} + x^{12} + x^9 + x^6 + x^3 + 1 \\ & = x^3 + x^6 + x^3 + x^6 + x^3 + x^6 + x^3 + 1 \quad \text{(by \eqref{idempower})} \\ & = x^6 + 1,\end{aligned}$$ so that $x^3=x^6$. Multiplying this with $x$ and using $x^7=x$, we get $x^4=x$. **Example 48**. Consider $n = 13$ and $p = 2$. Then $k=2$. Here, we also need linear and quadratic polynomials: In $\mathds{F}_2[T]$, one finds that $T^4-T$ is the $\gcd$ of $T^{13}-T$, $(T+1)^{13} - (T+1)$, $(T^2+T+1)^{13} - (T^2+T+1)$. This shows that every $13$-ring with $2=0$ is a $4$-ring. **Example 49**. Every $31$-ring with $2=0$ is a $16$-ring (and vice versa). In this example even polynomials of degree $4$ are required: $T^{16}-T$ is the $\gcd$ of the six polynomials $f^{31}-f \in \mathds{F}_2[T]$, where $f$ belongs to $\{T,\, T+1,\, T^2+T+1,\, T^4+T+1,\, T^4+T^3+1,\, T^4+T^3+T^2+T+1\}$. In all the examples so far, the polynomial $g$ was just $\smash{T^{p^k}-T}$. But we already mentioned in [Remark 38](#gform){reference-type="ref" reference="gform"} that this does not have to be the case. Here is the smallest counterexample: **Example 50**. Let $n = 22$, hence $p=2$ and $k=6$. The $\gcd$ of $T^{22} - T$, $(T^2+T)^{22} - (T^2+T)$ in $\mathds{F}_2[T]$ is equal to $g' \coloneqq T^{10}+T^9+T^8+T^3+T^2+T$, and this divides $T^{64}-T$. This yields an equational proof that every $22$-ring with $2=0$ is a $64$-ring. (But this is clear anyway because of $21 \mid 63$, and it is unclear if this helps to prove commutativity of $22$-rings.) Incidentally, we have $g'=g$, because [Remark 38](#gform){reference-type="ref" reference="gform"} implies that $g = T (T+1)(T^2+T+1)(T^3+T+1)(T^3+T^2+1)=g'$. The complexity of $22$-rings has been already noted by Morita [@Mo78]. There is no $m < 22$ such that every $22$-ring is an $m$-ring, as can be seen by considering $\mathds{F}_4 \times \mathds{F}_8$. **Remark 51**. When $k=1$, i.e. $p$ is the only power $p^d$ with $p^d - 1 \mid n-1$, let us call $n$ *simple at* $p$. The previous examples and numerical experiments suggest that then linear polynomials are often enough to compute $g$, i.e. that $T^p-T = \gcd((T+u)^n - (T+u) : u \in \mathds{F}_p)$. Let us call $n$ *nice* for $p$ in this case, otherwise *unpleasant*. See [9.4](#app:linear){reference-type="ref" reference="app:linear"} for the SageMath code to find unpleasant numbers. For $p=2$, there are $5484$ numbers $n \leq 10\,000$ that are simple at $2$, and only $142$ of these (that is $2.6\%$) are unpleasant, the smallest example being $n=74$. For $p=3$, all of the simple numbers $n \leq 100\,000$ are nice, and Peter Müller found that the smallest unpleasant number is $n = 1 + (3^{16}-1)/32 = 1\,345\,211$. For $p=5$, the smallest unpleasant number is $n = 1+(5^8 - 1)/3 = 130\,209$. For other prime numbers, the behavior seems to be similar: most simple numbers are nice. # The simple case In this section, we will show how to produce, for every fixed $n > 1$ with a special property (see below), an equational proof of the commutativity of $n$-rings. As before, we will not be able to write down a single equational proof that works for all $n$. The proofs are not "uniform" and have to be written down for each $n$ individually. But the recipe is always the same. **Definition 52**. Let us call $n \in \mathds{N}$ *simple* if $n > 1$ and every prime power $q$ with $q-1 \mid n-1$ is actually a prime number. The simple numbers $\leq 100$ are (see [9.5](#app:simple){reference-type="ref" reference="app:simple"} for the SageMath code to generate them) $$\begin{aligned} &2,3,5,6,11,12,14,18,20,21,23,24,26,30,35,38,39,42,44,45,47,48,51,54,\\ &56,59,60,62,66,68,69,72,74,75,77,80,83,84,86,87,90,93,95,96,98.\end{aligned}$$ They form the OEIS sequence A366343[^2]. The natural density of the set of simple numbers (if it exists) is approximately $$\lim_{N \to \infty} \frac{\# \{n \leq N : n \text{ is simple}\}}{N} \approx 0.462118,$$ meaning that almost every second number is simple. By [Remark 19](#nfields){reference-type="ref" reference="nfields"}, a number $n$ is simple precisely when every $n$-field is a prime field. We start by giving a quick proof of Jacobson's Theorem for simple numbers. **Theorem 53**. *Let $A$ be a ring such that for every $a \in A$ there is some simple number $n > 1$ such that $a^n=a$. Then every element of $A$ is a $\mathds{Z}$-linear combination of idempotent elements. In particular, $A$ is commutative.* *Proof.* Let $a \in A$. Choose some simple number $n>1$ with $a^n=a$. Let $c$ be the characteristic of $A$, which satisfies $c>0$ (since $2^m=2$ for some $m>0$). The ring $\mathds{Z}[a] = (\mathds{Z}/c\mathds{Z})[a]$ is finite, because it is generated by $1,\dotsc,a^{n-1}$ as a $\mathds{Z}/c\mathds{Z}$-module. By applying the Chinese Remainder Theorem to the finite reduced commutative ring $\mathds{Z}[a]$, we can write $\mathds{Z}[a]$ as a finite direct product of finite fields. If $\mathds{F}_q$ is one of these fields, choose some generator $\zeta$ of its multiplicative group and some simple number $m>1$ such that $\zeta^m=\zeta$. Then $\zeta^{m-1}=1$, so that $q-1 \mid m-1$. Since $m$ is simple, $q$ is a prime number. Therefore, $\mathds{Z}[a]$ is actually a finite direct product of prime fields. If $e_1,\dotsc,e_s \in \mathds{Z}[a]$ are the corresponding idempotent elements, it follows that $a$ is a $\mathds{Z}$-linear combination of $e_1,\dotsc,e_s$. Hence, $a$ is central by [Lemma 8](#idemcentral){reference-type="ref" reference="idemcentral"}. ◻ In particular, we have a short proof that for simple $n$ every $n$-ring is commutative. In fact, $n$ is simple if and only if every $n$-ring is additively generated by idempotents. The goal of this section is to make this proof even more constructive. Because of [Theorem 32](#proofdecomp){reference-type="ref" reference="proofdecomp"}, it suffices to look at $n$-rings with $p=0$. We look at an example first. **Example 54**. The free $5$-ring with $5=0$ on one generator is $\mathds{F}_5[T]/(T^5-T)$. First we will write $[T]$ as a linear combination of idempotent (and hence central) elements. The Chinese Remainder Theorem implies that $$\mathds{F}_5[T]/(T^5-T) \to (\mathds{F}_5)^5, \quad [f] \mapsto \bigl(f(0),f(1),f(2),f(3),f(4)\bigr)$$ is an isomorphism. The image of $[T]$ is $(0,1,2,3,4)$, which is the integer linear combination $1 \cdot (0,1,0,0,0) + 2 \cdot (0,0,1,0,0) + 3 \cdot (0,0,0,1,0) + 4 \cdot (0,0,0,0,1)$ of the canonical idempotents. Their preimages are the classes of $$e_i(T) := 1-(T-i)^4 \in \mathds{F}_5[T]$$ for $0 \leq i \leq 4$. This is because for $u \in \mathds{F}_5$ we have $e_i(u)=1$ for $u=i$ and $e_i(u)=0$ for $u \neq i$. Hence, $$[T] = 1 \cdot [e_1] + 2 \cdot [e_2] + 3 \cdot [e_3] + 4 \cdot [e_4].$$ This leads to the following equational proof that $5$-rings with $5=0$ are commutative: Let $x$ be any element. A calculation (using only $5=0$) shows that $$1 \cdot (1-(x-1)^4) + 2 \cdot (1-(x-2)^4) + 3 \cdot (1-(x-3)^4) + 4\cdot (1-(x-4)^4) = x,$$ or equivalently $$4 \cdot (x-1)^4 + 3 \cdot (x-2)^4 + 2 \cdot (x-3)^4 + 1 \cdot (x-4)^4 = x.$$ Since each $y^4$ is central by [\[idemcentral,idempower\]](#idemcentral,idempower){reference-type="ref" reference="idemcentral,idempower"}, it follows that $x$ is central. We will now generalize this example. **Lemma 55**. *Let $q$ be a prime power. Let $A$ be an $\mathds{F}_q$-algebra and $x \in A$. Then there is an equational proof of $$\label{idemliko} x = \sum_{u \in \mathds{F}_q^\times} u \cdot \bigl(1-(x-u)^{q-1}\bigr).$$* The equational proof takes place in the theory of $\mathds{F}_q$-algebras. This means that the arithmetic of $\mathds{F}_q$ is taken for granted. *Proof.* The abstract proof of [\[idemliko\]](#idemliko){reference-type="ref" reference="idemliko"} works as follows: It is enough to verify the equation $$X = \sum_{u \in \mathds{F}_q^\times} u \cdot \bigl(1-(X-u)^{q-1}\bigr)$$ in the universal example $\mathds{F}_q[X]$. Both sides are polynomials of degree $<q$ in $X$, so that it suffices to prove that the corresponding polynomial functions $\mathds{F}_q \to \mathds{F}_q$ agree. This follows from the observation that for $\alpha \in \mathds{F}_q$ we have $(\alpha-u)^{q-1}=\delta_{u,\alpha}$. The equational proof looks as follows: The case $q=2$ is easy to check. Now let $q>2$. We first prove that for all $j \in \mathds{Z}$ we have $$\label{sum} \sum_{u \in \mathds{F}_q^\times} u^j = \begin{cases} -1, & q-1 \mid j, \\ \phantom{-}0, & \text{else} \end{cases}$$ in $\mathds{F}_q$. The proof is well-known. Choose a generator $\zeta$ of $\smash{\mathds{F}_q^\times}$, and let $s$ be the sum in [\[sum\]](#sum){reference-type="ref" reference="sum"}. Then $\zeta^j s = s$. If $\zeta^j \neq 1$, we get $s = 0$. Otherwise, we have $q-1 \mid j$ and then each summand of $s$ is $1$, so that $s = q-1 = -1$. This proves [\[sum\]](#sum){reference-type="ref" reference="sum"}. For $j=1$ we get in particular $\smash{\sum_{u \in \mathds{F}_q^\times} u = 0}$ (here, we use $q > 2$). Now, the binomial theorem and [\[sum\]](#sum){reference-type="ref" reference="sum"} give us $$\begin{aligned} \sum_{u \in \mathds{F}_q^\times} u \cdot \bigl(1-(x-u)^{q-1}\bigr) & = \sum_{u \in \mathds{F}_q^\times} u \,-\, \sum_{u \in \mathds{F}_q^\times} \, \sum_{i=0}^{q-1} \binom{q-1}{i} \cdot u \cdot (-u)^{q-1-i} \cdot x^i \\ & = - \sum_{i=0}^{q-1} \binom{q-1}{i} (-1)^{q-1-i} \cdot \left(\sum_{u \in \mathds{F}_q^\times} u^{q-i}\right) \cdot x^i \\ & = -\binom{q-1}{1} (-1)^{q-1-1} (-1) \cdot x = x. \qedhere\end{aligned}$$ ◻ **Theorem 56**. *Let $q$ be a prime power. Let $A$ be an $\mathds{F}_q$-algebra and $x \in A$. If $x^q = x$, then $x$ can be explicitly written as an $\mathds{F}_q$-linear combination of idempotent elements as in [\[idemliko\]](#idemliko){reference-type="ref" reference="idemliko"}. In particular, if $A$ is a $q$-ring, $A$ is commutative by an equational proof.* *Proof.* We apply [Lemma 55](#idemgen){reference-type="ref" reference="idemgen"}. If $u \in \mathds{F}_q$, then $(x-u)^q = x^q - u^q = x - u$ since $x,u$ commute, and then $(x-u)^{q-1}$ is idempotent by [Lemma 9](#idempower){reference-type="ref" reference="idempower"}. Then $x$ is central by [Lemma 8](#idemcentral){reference-type="ref" reference="idemcentral"}. ◻ **Remark 57**. A different equational proof of the commutativity of $p$-rings with $p=0$ has been found by Forsythe and McCoy [@FM46 Section 3]. It uses the Vandermonde determinant and is similar to our proof for $p^2$-rings below ([Theorem 67](#p2case){reference-type="ref" reference="p2case"}). **Corollary 58**. *For every simple number $n>1$ there is an equational proof that every $n$-ring is commutative.* *Proof.* By [Theorem 32](#proofdecomp){reference-type="ref" reference="proofdecomp"} it is enough to find an equational proof that $n$-rings with $p=0$ are commutative, where $p \in \mathds{P}$ satisfies $p-1 \mid n-1$. By [Theorem 35](#reduc){reference-type="ref" reference="reduc"} and the definition of a simple number every such ring is actually a $p$-ring by an equational proof. Now [Theorem 56](#pcase){reference-type="ref" reference="pcase"} finishes the proof. ◻ Let us check how the equational proof looks like in some examples. **Example 59**. Consider $n=3$. First we will use some non-equational concepts, but we will get rid of them afterwards. Let $A$ be a $3$-ring. Then $0 = 2^3 - 2 = 6 = 2 \cdot 3$ holds in $A$. So, $$A \cong A/2A \times A/3A$$ by the Chinese Remainder Theorem. The factor $A/2A$ is a $3$-ring with $2=0$, hence a $2$-ring by [Example 39](#reduc-3-2){reference-type="ref" reference="reduc-3-2"} and therefore commutative. The factor $A/3A$ is a $3$-ring with $3=0$, hence commutative by [Theorem 56](#pcase){reference-type="ref" reference="pcase"}. So $A$ is a product of two commutative rings, and we are done. Just by unwinding what happens in the proofs, the equational proof looks as follows: It is sufficient to show $xy \equiv yx \bmod 2$ and $xy \equiv yx \bmod 3$, since then $xy \equiv yx \bmod 6$ by the proof in [Theorem 32](#proofdecomp){reference-type="ref" reference="proofdecomp"} and hence $xy=yx$. In fact, if $xy-yx=2u$ and $xy-yx=3v$, then $xy-yx=6(u-v)=0$. We have $$x+1 = (x+1)^3 = x^3 + 3x^2 + 3x + 1 = 3x^2 + 4x + 1 \equiv x^2+1 \bmod 2$$ and hence $x^2 \equiv x \bmod 2$ for all $x$. Then (we now repeat [Example 4](#2case){reference-type="ref" reference="2case"}) $$x + y \equiv (x+y)^2 = x^2 + xy + yx + y^2 \equiv x + xy - yx + y \bmod 2$$ and hence $xy \equiv yx \bmod 2$. Now for the modulo $3$ part: We have $$1 \cdot (1-(x-1)^2) + 2 \cdot (1-(x-2)^2) = -3x^2+10x-6 \equiv x \bmod 3.$$ So it suffices to show that $xy^2 \equiv y^2 x \bmod 3$ for all $x,y$. But $y^2$ is central by [Lemma 8](#idemcentral){reference-type="ref" reference="idemcentral"} and [Lemma 9](#idempower){reference-type="ref" reference="idempower"} (we don't need to repeat the proofs here). Of course, there are simpler proofs of the commutativity of $3$-rings (such as the one in [Example 11](#3case){reference-type="ref" reference="3case"}), but the advantage is that our proof has been produced with a *general method*. It was not necessary to come up with clever ideas, and the same method works for larger numbers as well, as long as they are simple. Let us also mention that the reduction to $p$-rings will often be rather simple because of [Remark 51](#lineargcd){reference-type="ref" reference="lineargcd"}. **Example 60**. Let us show with our general method that every $5$-ring is commutative. We already saw in [Example 33](#5decomp){reference-type="ref" reference="5decomp"} that it is enough to prove this modulo the primes $2,3,5$. For the primes $2$ and $3$, [\[reduc-5-2,reduc-5-3\]](#reduc-5-2,reduc-5-3){reference-type="ref" reference="reduc-5-2,reduc-5-3"} reduce the problem to $2$- and $3$-rings, which we have covered before. For the prime $5$, we can use [Theorem 56](#pcase){reference-type="ref" reference="pcase"}. Explicitly, the equation $$1 \cdot (1-(x-1)^4) + 2 \cdot (1-(x-2)^4) + 3 \cdot (1-(x-3)^4) + 4 \cdot (1-(x-4)^4) \equiv x \bmod 5$$ shows that $x$ is central modulo $5$ because each $y^4$ is central by [\[idemcentral,idempower\]](#idemcentral,idempower){reference-type="ref" reference="idemcentral,idempower"}. **Example 61**. Let us show with our general method that every $21$-ring is commutative. The primes $p$ with $p-1 \mid 21-1$ are $p=2,3,5,11$, we must have $2 \cdot 3 \cdot 5 \cdot 11 = 0$ in any $21$-ring: This is because $2 \cdot 3 \cdot 5 \cdot 11 = 330 = \gcd(2^{21}-2,3^{21}-3,5^{21}-5)$. Thus, it suffices to show commutativity modulo each of these primes. For $p \in \{2,3,11\}$ we find $$\gcd\bigl(T^{21}-T,(T+1)^{21}-(T+1)\bigr) = T^p-T$$ in $\mathds{F}_p[T]$, showing that every $21$-ring with $p=0$ is a $p$-ring and hence commutative by [Theorem 56](#pcase){reference-type="ref" reference="pcase"}. In $\mathds{F}_5[T]$ we find $$\gcd\bigl((T+u)^{21}-(T+u) : u \in \mathds{F}_4\bigr) = T^5-T,$$ showing that every $21$-ring with $5=0$ is a $5$-ring and hence commutative by [Theorem 56](#pcase){reference-type="ref" reference="pcase"}. The reduction method can be applied to non-simple numbers as well. Let us look at some examples. More results of this type are summarized in Table [1](#resulttable){reference-type="ref" reference="resulttable"}. **Example 62**. Let us show with our general method that every $7$-ring is commutative, even though $7$ is not simple. The primes $p$ with $p-1 \mid 7-1$ are $p=2,3,7$, and since $2 \cdot 3 \cdot 7 = 42 = \gcd(2^7-2,3^7-3)$, it suffices to prove commutativity modulo each of these primes. We already saw in [Example 47](#7char2){reference-type="ref" reference="7char2"} that a $7$-ring with $2=0$ is a $4$-ring, hence commutative by [Example 12](#4case){reference-type="ref" reference="4case"}. In $\mathds{F}_3[T]$ we find $$\gcd\bigl(T^{7}-T,(T+1)^{7}-(T+1)\bigr) = T^3-T,$$ showing that every $7$-ring with $3=0$ is a $3$-ring and hence commutative. Finally, every $7$-ring with $7=0$ is commutative by [Theorem 56](#pcase){reference-type="ref" reference="pcase"}. **Example 63**. Let us show with our general method that every $2023$-ring is commutative. The primes $p$ with $p-1 \mid 2023-1$ are $p=2,3,7$, and since $2 \cdot 3 \cdot 7 = 42 = \gcd(2^{2023}-2,3^{2023}-3)$, it suffices to prove commutativity modulo each of these primes. In $\mathds{F}_2[T]$ we find $$\gcd\bigl(T^{2023}-T,(T+1)^{2023}-(T+1),(T^2+T+1)^{2023}-(T^2+T+1)\bigr) = T^4 - T,$$ so that every $2023$-ring with $2=0$ is a $4$-ring and hence commutative. In $\mathds{F}_3[T]$ we find $$\gcd\bigl(T^{2023}-T,(T+1)^{2023}-(T+1),(T+2)^{2023}-(T+2)\bigr) = T^3 - T,$$ so that every $2023$-ring with $3=0$ is a $3$-ring and hence commutative. Finally, in $\mathds{F}_7[T]$ we find $$\gcd\bigl(T^{2023}-T,(T+1)^{2023}-(T+1)\bigr) = T^7 - T \in \mathds{F}_7[T],$$ so that every $2023$-ring with $7=0$ is a $7$-ring and hence commutative by [Theorem 56](#pcase){reference-type="ref" reference="pcase"}. # Commutativity of $p^2$-rings Morita [@Mo78] has shown by direct and rather long calculations that $p^2$-rings are commutative for $p \in \{2,3,5\}$. He also mentioned that the same method (but with much more effort) works for $p=7$. In this section, we will generalize Morita's approach to show the commutativity of $p^2$-rings with $p=0$. The restriction to $p=0$ (and the fact that $p$ will be arbitrary) allows for a considerable simplification of the calculations. **Lemma 64**. *In a $p^k$-ring with $p=0$ there is an equational proof that for every element $x$ the element $\smash{e \coloneqq x + x^p + \cdots + x^{p^{k-1}}}$ is central.* *Proof.* Observe that $e^p = e$. Hence, [Theorem 56](#pcase){reference-type="ref" reference="pcase"} tells us that $e$ is an $\mathds{F}_p$-linear combination of idempotent and hence central elements. ◻ **Lemma 65**. *Let $V$ be an $\mathds{F}_p$-vector space, $U \subseteq V$ be a subspace, and $x_0,\dotsc,x_{p-1} \in V$ such that $\smash{\sum_{i=0}^{p-1} \lambda^i x_i \in U}$ for all $\lambda \in \mathds{F}_p$. Then $x_0,\dotsc,x_{p-1} \in U$.* *Proof.* Consider the Vandermonde matrix $(\lambda^i)_{\lambda \in \mathds{F}_p,\, 0 \leq i < p}$ over $\mathds{F}_p$. It has an explicit inverse matrix $(u_{i,\lambda})_{0 \leq i < p,\, \lambda \in \mathds{F}_p}$, so that $$\sum_{\lambda \in \mathds{F}_p} u_{i,\lambda} \lambda^j = \delta_{i,j}.$$ Hence, for all $0 \leq j < p$ we have $$U \ni \sum_{\lambda \in \mathds{F}_p} u_{j,\lambda} \sum_{i=0}^{p-1} \lambda^i x_i = \sum_{i=0}^{p-1} \left(\sum_{\lambda \in \mathds{F}_p} u_{j,\lambda} \lambda^i \right) x_i = \sum_{i=0}^{p-1} \delta_{j,i} x_i = x_j. \qedhere$$ ◻ **Definition 66**. Let $A$ be a ring, $x,y \in A$ and $i,j \in \mathds{N}$. Following [@FM46], $[x^i y^j]$ denotes the sum of all monomials in $x,y$ in which $x$ appears $i$ times and $y$ appears $j$ times. There are $\smash{\binom{i+j}{i}}$ of these monomials. For example, $[x y^2]$ equals $x y^2 + yxy + x y^2$. Notice that $$\label{generalbinomi} (x+y)^n = \sum_{i=0}^{n} [x^i y^{n-i}]$$ for all $n \in \mathds{N}$. Of course, if $A$ is commutative, then $\smash{[x^i y^j] = \binom{i+j}{i} x^i y^j}$, and [\[generalbinomi\]](#generalbinomi){reference-type="ref" reference="generalbinomi"} is just the binomial theorem. **Theorem 67**. *There is an equational proof that every $p^2$-ring with $p=0$ is commutative.* *Proof.* In the proof, all variables are quantified over all elements of the ring. By [Lemma 64](#invol){reference-type="ref" reference="invol"}, the element $f(x) \coloneqq x + x^p$ is central. Hence, $f(x+y)-f(x)-f(y)$ is central as well. By [\[generalbinomi\]](#generalbinomi){reference-type="ref" reference="generalbinomi"} this simplifies to $[x y^{p-1}] + \cdots + [x^{p-1} y]$. Substituting $x$ with $\lambda x$ for $\lambda \in \mathds{F}_p$ shows that $$\lambda [x y^{p-1}] + \cdots + \lambda^{p-1} [x^{p-1} y]$$ is central. By applying [Lemma 65](#vandermonde){reference-type="ref" reference="vandermonde"} to the subspace $Z(A) \subseteq A$, all $[x y^{p-1}],\dotsc,[x^{p-1} y]$ are central. In an equational proof, we cannot write $Z(A) \subseteq A$ or speak of subspaces, but it is clear that the proof of the Lemma still applies: we just have to replace the formula $a \in Z(A)$ by $ab=ba$ for a variable $b$. In particular, $[x y^{p-1}]$ commutes with $y$, meaning $$y(x y^{p-1} + y x y^{p-2} + \cdots + y^{p-2} x y + y^{p-1} x) = (x y^{p-1} + y x y^{p-2} + \cdots + y^{p-2} x y + y^{p-1} x) y.$$ The $i$th summand on the left side is equal to the $(i+1)$th summand on the right side. Hence, $$y^p x = x y^p,$$ showing that $y^p$ is central. Since $y + y^p$ is central, $y$ must be central. ◻ **Example 68**. There is an equational proof that every $73$-ring is commutative: The set of $p \in \mathds{P}$ with $p-1 \mid 73-1$ is $2,3,5,7,13,19,37,73$, so by [Theorem 32](#proofdecomp){reference-type="ref" reference="proofdecomp"} it suffices to prove commutativity modulo these primes. We use [Theorem 35](#reduc){reference-type="ref" reference="reduc"}. The exponent $k$ is $1$ for the primes $7,13,19,37,73$ and $2$ for the primes $2,3,5$. So the claim follows from [Theorem 56](#pcase){reference-type="ref" reference="pcase"} for the first set of primes and from [Theorem 67](#p2case){reference-type="ref" reference="p2case"} for the second set of primes. # The general case Next, we aim for the general case. Our methods for $p$-rings and $p^2$-rings do not generalize to $p^3$-rings, so that different ideas are required. The (non-equational) proofs of Jacobson's Theorem in [@Wa71; @NT74] derive it from Wedderburn's Theorem, which states that every finite division ring is commutative. More precisely, they are structured as follows: 1. Reduce the general case to the case of finite rings. 2. Reduce the finite case to the case of division rings. 3. For division rings, cite (or prove again) Wedderburn's Theorem. We will follow a similar approach here, but the proofs need to be made constructive (thus, avoiding LEM) and then transformed into equational logic. The common proofs of the Wedderburn Theorem such as the one by Witt [@Wi31] are not constructive, though. They start with a finite non-commutative division ring of minimal cardinality and derive a contradiction. To solve this, we need to formulate the Wedderburn Theorem in a different way. But first, let us observe the following interesting relationship: **Proposition 69**. *In classical logic, the following are equivalent:* 1. *Wedderburn's Theorem is true.* 2. *Finite reduced rings are commutative.* 3. *For every $n > 1$, finite $n$-rings are commutative.* *Proof.* (1) $\Rightarrow$ (2): It suffices to prove that every finite reduced ring $A$ is a product of division rings. If $A$ is trivial or a division ring, we are done. Otherwise (here we are using LEM) we can choose a non-unit $x \neq 0$. Since $A$ is finite, there are $n,m > 0$ with $x^n = x^{n+m}$. Then $e \coloneqq x^{nm}$ is idempotent $\neq 0,1$ and central by [Lemma 8](#idemcentral){reference-type="ref" reference="idemcentral"}. Thus, $A \cong eA \times (1-e)A$ and induction on the cardinality of $A$ proves the claim. (2) $\Rightarrow$ (3): This is clear since $n$-rings are reduced. (3) $\Rightarrow$ (1): If $D$ is a finite division ring, say with $n$ elements, then $D$ is an $n$-ring because of Lagrange's Theorem applied to $D^{\times}$. ◻ The idea is now to formulate a constructive version of the Wedderburn Theorem for very special kinds of finite $n$-rings. We already know that it is enough to consider $p^k$-rings with $p=0$ ([Corollary 36](#fullred){reference-type="ref" reference="fullred"}). **Definition 70**. Let $p \in \mathds{P}$, $k \geq 1$, and $f \in \mathds{F}_p[T]$ be a polynomial. The *constructive Wedderburn Theorem* $W_{p,k,f}$ is the following statement: Let $A$ be a $p^k$-ring with $p=0$ and $a,b \in A$ two elements with $$\label{commrel} ba = f(a) b.$$ Then $$ba = ab.$$ This theorem will be an intermediate step in proving commutativity of $A$. It is more easy than just showing $ba=ab$ directly because of the additional assumption in [\[commrel\]](#commrel){reference-type="ref" reference="commrel"}, which roughly says that $a$ and $b$ "almost" commute. The statement $W_{p,k,T}$ is trivial. **Lemma 71**. *The Wedderburn Theorem implies the constructive Wedderburn Theorem $W_{p,k,f}$ for all $p,k,f$. In particular, $W_{p,k,f}$ holds in classical logic.* *Proof.* With the notation from [Definition 70](#wedderdef){reference-type="ref" reference="wedderdef"}, consider the subalgebra $\mathds{F}_p\langle a,b\rangle \subseteq A$ generated by $a,b$. Define $\smash{E \coloneqq \{a^i b^j : 0 \leq i,j < p^k\}}$ and let $M$ be the $\mathds{F}_p$-submodule of $\mathds{F}_p\langle a,b\rangle$ generated by $E$. Then $\smash{a^{p^k}=a}$, $\smash{b^{p^k} = b}$ and [\[commrel\]](#commrel){reference-type="ref" reference="commrel"} imply that $a \cdot E$ and $b \cdot E$ are contained in $M$. This implies $E \cdot E \subseteq M$, so that $M \cdot M \subseteq M$ and therefore $M = \mathds{F}_p \langle a,b \rangle$. Hence, $\mathds{F}_p\langle a,b \rangle$ is a finite $p^k$-ring. So the claim follows from [Proposition 69](#finite){reference-type="ref" reference="finite"}. ◻ Trivially, when all $p^k$-rings with $p=0$ are commutative, $W_{p,k,f}$ holds. We will prove next that the converse also holds. The proof is an adaptation of the proofs of Jacobson's Theorem in [@Wa71; @NT74]. The difference is that we will not use a proof by contradiction, and indeed the proof is (almost) constructive. We will use it to find an equational proof afterwards. **Theorem 72**. *Let $p \in \mathds{P}$ and $k \geq 1$. Assume that the constructive Wedderburn Theorem $W_{p,k,f}$ holds for all $f \in \mathds{F}_p[T]$ of degree $<k$. Then every $p^k$-ring with $p=0$ is commutative.* *Proof.* Let $A$ be a $p^k$-ring with $p=0$, and let $a \in A$. Then $\mathds{F}_p[a]$ is a finite reduced commutative ring, hence a finite product of finite fields $\mathds{F}_{p^{m_i}}$ by the proof of [Proposition 69](#finite){reference-type="ref" reference="finite"}. Hence, there are idempotent elements $e_i \in \mathds{F}_p[a]$ such that $\sum_i e_i = 1$, $e_i e_j = \delta_{i,j}$ and such that the $\mathds{F}_p$-algebra $e_i \cdot \mathds{F}_p[a]$ with multiplicative identity $e_i$ is isomorphic to $\mathds{F}_{p^{m_i}}$. They are $p^k$-rings, so that $m_i \mid k$. Notice that $e_i \cdot \mathds{F}_p[a]$ is equal to the $\mathds{F}_p$-subalgebra of $e_i \cdot A$ generated by $e_i \cdot a$. Since $a = \sum_i e_i \cdot a$, it suffices to prove that every $e_i \cdot a$ is central in $e_i \cdot A$. Therefore, we may assume that $\mathds{F}_p[a]$ is a finite field, say $\mathds{F}_{p^m}$ with $m \mid k$. Consider the $\mathds{F}_p$-linear map $$d : A \to A, \quad x \mapsto ax - xa.$$ We need to show $d=0$, because then $a$ is central. One proves by induction on $n \in \mathds{N}$ that $$d^n(x) = \sum_{i=0}^{n} (-1)^i \binom{n}{i} \cdot a^{n-i} x a^i.$$ Apply this to $n=p^m$ and use $\binom{p^m}{i} \equiv 0 \bmod p$ for $0 < i < p^m$ and $\smash{a^{p^m}=a}$. Then we get $$\label{derivpower} d^{p^m} = d.$$ Next, consider for every $u \in \mathds{F}_p[a]$ the map $$\lambda_u : A \to A,\quad x \mapsto ux.$$ It is $\mathds{F}_p$-linear and commutes with $d$ (since $u$ commutes with $a$). In $\mathds{F}_{p^m}[T]$ we have the well-known equation $$T^{p^m}-T = \prod_{u \in \mathds{F}_{p^m}} (T-u).$$ Hence, the same equation holds in $\mathds{F}_p[a][T]$: $$\label{powereq} T^{p^m}-T = \prod_{u \in \mathds{F}_p[a]} (T-u).$$ Consider the $\mathds{F}_p$-algebra homomorphism $$\alpha : \mathds{F}_p[a][T] \to \mathop{\mathrm{End}}_{\mathds{F}_p}(A), \quad u \mapsto \lambda_u, \quad T \mapsto d.$$ Here, $\mathop{\mathrm{End}}_{\mathds{F}_p}$ refers to the $\mathds{F}_p$-algebra of $\mathds{F}_p$-module endomorphisms with $\circ$ as the multiplication. Here, $\alpha$ is well-defined since $d$ commutes with each $\lambda_u$. Applying $\alpha$ to [\[powereq\]](#powereq){reference-type="ref" reference="powereq"} yields $$d^{p^m}-d = \prod_{u \in \mathds{F}_p[a]} (d-\lambda_u)$$ in $\mathop{\mathrm{End}}_{\mathds{F}_p}(A)$. The left side vanishes because of [\[derivpower\]](#derivpower){reference-type="ref" reference="derivpower"}. Therefore, we arrive at $$\label{diffprod} 0 = \prod_{0 \neq u \in \mathds{F}_p[a]} (d-\lambda_u) ~ \circ ~ d.$$ We claim that $d - \lambda_u : A \to A$ is injective for all $0 \neq u \in \mathds{F}_p[a]$. In fact, if $(d - \lambda_u)(b) = 0$, then $ab-ba = ub$, so that $ba = (a-u)b$. Since $a-u \in \mathds{F}_p[a]$, there is a polynomial $f \in \mathds{F}_p[T]$ of degree $<m \leq k$ with $a-u = f(a)$. Thus, $ba = f(a) b$. Since the constructive Wedderburn Theorem $W_{p,k,f}$ holds by assumption, it follows $ab=ba$, which means $ub = 0$. Since $u \neq 0$ and $\mathds{F}_p[a]$ is a field, $u$ is a unit, so that $b = 0$, and we are done. Since all $d - \lambda_u$ ($u \neq 0$) are injective, [\[diffprod\]](#diffprod){reference-type="ref" reference="diffprod"} implies $d = 0$, and $a$ is central. ◻ The next step is to transform the proof into an equational proof. **Theorem 73**. *Let $p \in \mathds{P}$ and $k \geq 1$. Assume that for all $f \in \mathds{F}_p[T]$ of degree $<k$ there is an equational proof of the constructive Wedderburn Theorem $W_{p,k,f}$. Then there is an equational proof that every $p^k$-ring with $p=0$ is commutative.* *Proof.* The first step is to describe a product decomposition $\smash{\mathds{F}_p[T]/\langle T^{p^k}-T \rangle \cong \prod_i \mathds{F}_{p^{m_i}}}$ in more explicit terms. Let $S$ be the finite set of all monic irreducible polynomials over $\mathds{F}_p$ whose degree divides $k$. This can be computed with standard methods. For example, when $p=2$ and $k=2$ we have $S = \{T,\, T+1,\, T^2+T+1\}$. Let $g \in S$. Then $g$ and $\prod_{g' \in S \setminus \{g\}} g'$ are coprime, so that the extended Euclidean algorithm yields polynomials $u,v$ with $$\textstyle u \cdot g + v \cdot \prod_{g' \in S \setminus \{g\}} g' = 1.$$ Define $e_g \coloneqq v \cdot \prod_{g' \in S \setminus \{g\}} g'$. Then $e_g \equiv 1 \bmod g$ and $e_g \equiv 0 \bmod g'$ for all $g' \in S \setminus \{g\}$. We have $$\label{longprod} \textstyle T^{p^k}-T = \prod_{g \in S} g.$$ The abstract reason is that both sides equal the product of all $T-u$ with $u \in \mathds{F}_{p^k}$, but in practice, for fixed $p$ and $k$ we can just verify this by expanding the product. It follows that the images $\smash{[e_g] \in \mathds{F}_p[T]/\langle T^{p^k}-T \rangle}$ are idempotent, pairwise orthogonal[^3], and their sum is $1$. Let $A$ be a $p^k$-ring with $p=0$, and let $a \in A$. We want to show that $a$ is central. There is a homomorphism of $\mathds{F}_p$-algebras $$\mathds{F}_p[T] / \langle T^{p^k}-T \rangle \to A$$ mapping $[T]$ to $a$, and thus $[e_g]$ to $e_g(a)$. It follows that all the elements $e_g(a) \in A$ are idempotent, pairwise orthogonal, and their sum is $1$, meaning $$\label{idemrels} \textstyle e_g(a)^2 = e_g(a), \quad e_g(a) \cdot e_{g'}(a) = 0 ~ (g \neq g'), \quad \sum_{g \in S} e_g(a) = 1.$$ For an equational proof of these equations in $A$, we cannot use homomorphisms and quotients of polynomial rings, but we can just verify them by hand, only using the relation $\smash{a^{p^k}=a}$. In order to show that $\smash{a = \sum_{g \in S} a \cdot e_g(a)}$ is central, it is then enough to show that each $a \cdot e_g(a)$ commutes with each $b \cdot e_g(a)$, where $b \in A$ and $g \in S$. Both elements stay the same when multiplied with $e_g(a)$, which is central by [Lemma 8](#idemcentral){reference-type="ref" reference="idemcentral"}. Hence, we may assume that $e_g(a)$ is the multiplicative identity of our ring. In fact, we just have to replace every occurrence of $1 \in A$ below by $e_g(a)$, and the calculations remain valid. Of course, this corresponds to the product decomposition in our proof of [Theorem 72](#generalcase){reference-type="ref" reference="generalcase"}, but what we do here is an equational version of this argument. Strictly speaking, we are not allowed to speak of "irreducible polynomials" in an equational proof. The construction above, however, just explains the *method* how to write down an equational proof for a fixed pair $(p,k)$, in which we are just using the polynomial expressions $e_g(a)$ and $g(a)$ in $A$. This will become more clear in the examples below. So, we have a polynomial $g \in S$ with $e_g(a)=1$, and this implies $e_{g'}(a)=0$ for $g' \neq g$. Next, we claim that $$\label{zero} g(a) = 0$$ holds in $A$. To see this, notice that for all $g' \neq g$ we have $1 = e_g(a) \equiv 0 \bmod g'(a)$, i.e. $g'(a)$ is a unit. On the other hand, by [\[longprod\]](#longprod){reference-type="ref" reference="longprod"} we have $\smash{0 = a^{p^k}-a = \prod_{g' \in S} g'(a)}$. Since all factors $g'(a)$ with $g' \neq g$ are units, we must have $g(a) = 0$. Conversely, $g(a)=0$ implies $e_g(a)=1$ since $e_g \equiv 1 \bmod g$. The whole preceding discussion was just about finding some $g \in S$ with $g(a)=0$. This is the only thing we will need from it in the following. In classical logic, this means that $\mathds{F}_p[a]$ is either zero or a field. Let $m \coloneqq \deg(g)$. If $m=1$, then $a = u \cdot 1$ for some $u \in \mathds{F}_p$ and we are done. So assume $m > 1$ (and hence $k > 1$). Notice that we are *not* using the LEM here. What we are writing down is one proof for every $g \in S$. When $S = \{T,\, T+1,\, \dotsc\}$, we start with $g = T$, then $g = T + 1$, etc. But for linear polynomials the proof is easy. So we switch directly to the next polynomials. Incidentally, this means that the proof is easy for $k=1$, namely exactly the one in [Theorem 56](#pcase){reference-type="ref" reference="pcase"}. Since $g$ is irreducible of degree $> 1$, the constant term of $g$ is a unit in $\mathds{F}_p$. So clearly, $a \in A$ is a unit. Since $g$ is irreducible of degree $m$, we have $\smash{g \mid T^{p^m}-T}$ (this can be verified manually with a polynomial division), so that $\smash{a^{p^m}=a}$ by [\[zero\]](#zero){reference-type="ref" reference="zero"}. Let $b \in A$. We want to show that $a$ and $b$ commute. Choose an enumeration $f_0,f_1,\dotsc,f_{p^m-1}$ of the set $\mathds{F}_p[T]_{<m}$ of polynomials over $\mathds{F}_p$ of degree $<m$, starting with $f_0=0$. We define a sequence of elements $b_0,b_1,b_2,\dotsc,b_{p^m}$ in $A$ recursively by $$\label{bndef} b_0 \coloneqq b, \quad b_{n+1} \coloneqq a b_n - b_n a - f_n(a) b_n.$$ For example, $b_1 = a b - b a$, and the goal is to prove $b_1 = 0$. We claim that $$\label{end} b_{p^m}=0.$$ Unfortunately, our proof is not equational, but it is constructive, and [\[end\]](#end){reference-type="ref" reference="end"} can be verified in every example of $p,k,g$ manually as well, just by using $g(a)=0$ (see the examples below). Define the map $d : A \to A$ by $d(x) \coloneqq ax - xa$. As before, we prove $\smash{d^{p^m}=d}$, only using $\smash{a^{p^m}=a}$. The definition of $b_n$ becomes $$b_n = \bigl((d - f_{n-1}(a)) \circ \dotsc \circ (d - f_0(a))\bigr)(b).$$ Hence, $$b_{p^m} = \prod_{f \in \mathds{F}_p[T]_{<m}} (d - f(a))(b),$$ so that it suffices to prove the equation $$T^{p^m} - T = \prod_{f \in \mathds{F}_p[X]_{<m}} (T - f(a))$$ in $A[T]$. Since $g(a)=0$, there is a homomorphism of $\mathds{F}_p$-algebras $\mathds{F}_p[X] / \langle g(X) \rangle \to A$, $[X] \mapsto a$, and it suffices to prove $$T^{p^m} - T = \prod_{f \in \mathds{F}_p[X]_{<m}} (T - [f])$$ in $(\mathds{F}_p[X]/\langle g(X) \rangle)[T]$. This follows from the fact that $\mathds{F}_p[X]/\langle g(X) \rangle$ is a field with $p^m$ elements, namely those $[f]$ with $f \in \mathds{F}_p[X]_{<m}$. This finishes the proof of [\[end\]](#end){reference-type="ref" reference="end"}. Next, we claim that for $n \geq 1$ there is an equational proof of $$\label{down} b_{n+1} = 0 \implies b_n = 0.$$ If $b_{n+1} = 0$, this means $b_n a = (a - f_n(a)) b_n = f'(a) b_n$ with $f' \coloneqq T - f_n$. Notice that $f'$ has degree $<k$ since $k > 1$ and $f_n$ has degree $<m \leq k$. By assumption, there is an equational proof of the constructive Wedderburn Theorem $W_{p,k,f'}$, so that $b_n a = a b_n$, meaning $f_n(a) b_n = 0$. We claim that $f_n(a)$ is a unit in $A$, which leads to $b_n=0$. In fact, $f_n$ and $g$ are coprime (since $f_n \neq 0$ has degree $< m = \deg(g)$ and $g$ is irreducible), so that $u f_n + v g = 1$ for some computable polynomials $u,v$. Then $1 = u(a) f_n(a) + v(a) g(a) = u(a) f_n(a)$, proving our claim. Combining [\[end\]](#end){reference-type="ref" reference="end"} and [\[down\]](#down){reference-type="ref" reference="down"} gives $b_1 = 0$, so that $a,b$ commute. ◻ **Remark 74**. If two polynomials $g,g' \in S$ only differ by a linear substitution, then the proof for $g$ in [Theorem 73](#generalcase-eq){reference-type="ref" reference="generalcase-eq"} (that $g(a)=0$ implies that $a$ is central) leads to a proof for $g'$, so that $g'$ may be omitted. In fact, if $g' = w \, g(uT+v)$ for $v \in \mathds{F}_p$ and $u,w \in \mathds{F}_p^{\times}$, then the assumption $g'(a)=0$ means $g(ua + v)=0$. If there is a proof for $g$, then $ua + v$ is central and hence $a$ too. Similarly, if $g,g'$ are reciprocal polynomials (up to a unit), then the proof for $g$ leads to a proof for $g'$. In fact, if $g'(a)=0$, then $a$ is a unit with $a^m g(a^{-1})=0$, so that $g(a^{-1})=0$. If there is a proof for $g$, then $a^{-1}$ is central, so that $a$ is central as well. Again, one can write some SageMath code that computes a sufficient set of polynomials for us. We will now show how the equational proof provided by [Theorem 73](#generalcase-eq){reference-type="ref" reference="generalcase-eq"} looks like in the example $p=2$ and $k=2$. It will be considerably more complicated than ours in [Theorem 67](#p2case){reference-type="ref" reference="p2case"}, but it explains the method (that works for *all* $k \geq 1$) quite well. **Example 75**. Consider $p=2$ and $k=2$. So we will show (again) that every $4$-ring $A$ is commutative. We have $S = \{T,\, T+1,\, T^2+T+1\}$. The extended Euclidean algorithm yields the equations $$\begin{aligned} T^2 \cdot (T) + 1 \cdot (T+1)(T^2+T+1) & = 1 \\ (T^2+1) \cdot (T+1) + 1 \cdot T(T^2+T+1) & = 1 \\ 1 \cdot (T^2+T+1) + 1 \cdot T(T+1) & = 1 \end{aligned}$$ Therefore, we define $$\begin{aligned} e_T & \coloneqq (T+1)(T^2+T+1) = T^3+1,\\ e_{T+1} & \coloneqq T(T^2+T+1) = T^3+T^2+T,\\ e_{T^2+T+1} & \coloneqq T(T+1) = T^2+T.\end{aligned}$$ Let $a \in A$. Then the elements $e_T(a)$, $e_{T+1}(a)$, $e_{T^2+T+1}(a)$ are idempotent (hence central), pairwise orthogonal, and their sum is $1$. We can verify this by a direct calculation, just using $a^4=a$ (and of course $2=0$ in $A$). We only show two examples: $a^3+1$ is idempotent since $(a^3+1)^2 = a^6+1 = a^3+1$, and $(a^3+1)(a^2+a) = (a^4+a)(a+1)=0(a+1)=0$. It suffices to prove that every $a \cdot e_g(a)$ commutes with all $b \cdot e_g(a)$, where $g \in S$ and $b \in A$. For $g = T$ we find $$a \cdot e_T(a) = a \cdot (a^3+1) = a^4+a = 0,$$ so this case is trivial. For $g = T+1$ we find $$a \cdot e_{T+1}(a) = a^4+a^3+a^2 = e_{T+1}(a),$$ so this case is also trivial. Linear polynomials will always be trivial. The only interesting case is $g = T^2+T+1$. Here, $$a \cdot e_{g}(a) = a^3+a^2.$$ Now, we do not necessarily have $g(a)=a^2+a+1=0$ as in [\[zero\]](#zero){reference-type="ref" reference="zero"}, but here is why we may assume it: A direct calculation shows $$(a^3+a^2)^2 + (a^3+a^2) + (a^2+a) = 0.$$ So $c \coloneqq a \cdot e_g(a)$ satisfies $c^2 + c + 1 = 0$ in the $4$-ring $e_g(a) A$ with multiplicative identity $e_g(a)$, and it suffices to prove that $c$ is central in $e_g(e) A$. Formally, this ring does not even exist in equational logic, but any equational proof in $e_g(e) A$ can be turned into an equational proof in $A$ after we replace the multiplicative identity with $e_g(a)$. Alternatively, we may argue as in the proof of [Theorem 72](#generalcase){reference-type="ref" reference="generalcase"} and assume $1 = e_{T^2+T+1}(a)= a^2+a$. So we will assume $$a^2 + a + 1 = 0$$ from now on. We enumerate the polynomials $f_0,f_1,f_2,f_3$ over $\mathds{F}_2$ of degree $<2$ by $0,1,T,T+1$. If $b \in A$, we define a sequence recursively by $b_0 \coloneqq b$ and $b_{n+1} \coloneqq a b_n + b_n a + f_n(a) b_n$. We have $b_0 = b$, $b_1 = ab+ba$ and then $$\begin{aligned} b_2 & = a b_1 + b_1 a + b_1 \\ & = a(ab+ba) + (ab+ba)a + (ab+ba) \\ & = a^2 b + b a^2 + ab + ba \quad (\text{using } a^2=a+1) \\ & = (a+1)b + b (a+1) + ab + ba = 0.\end{aligned}$$ So we are lucky, the sequence terminates quite early. (We only expected $b_4=0$ from the general proof in [Theorem 73](#generalcase-eq){reference-type="ref" reference="generalcase-eq"}.) Assume that we can show $$\label{w2} ax + xa = x \implies x = 0.$$ Then we can derive $b_1 = 0$ from $b_2 = 0$ and we are done. This is exactly the constructive Wedderburn Theorem $W_{2,2,T+1}$. Let us prove it here: Assume $ax+xa = x$. We calculate $$(a+x)^2 = a^2 + ax+xa + x^2 = a^2 + (x + x^2).$$ Since $x + x^2$ is idempotent, it is central. Hence, we can calculate $$a+x = (a+x)^4 = (a^2 + (x + x^2))^2 = a^4 + (x + x^2)^2 = a + x + x^2.$$ From this we conclude $x^2=0$, i.e. $x = 0$, and we are done. One could remove all references to the more general method from this proof, and also get rid of all remarks outside of equational logic. We will not do that here, also because it will be less clear what is actually happening (and why). **Example 76**. For $p = 2$ and $k = 3$, one can proceed in a similar way, and we only briefly indicate how it works (we give a better proof in a moment). It suffices to consider $g = T^3 + T + 1$, since the other irreducible polynomial of degree $3$ is $g(T+1)$. So we may assume $a^3+a+1=0$. We enumerate the polynomials $f_i$ of degree $<3$ by $0,\, 1,\, T,\, T+1,\, T^2,\, T^2+1,\, T^2+T,\, T^2+T+1$. Define the sequence $b_n$ recursively by $b_0 \coloneqq b$ and $b_{n+1} \coloneqq a b_n + b_n a + f_n(a) b_n$. Then a routine calculation gives $b_7 = 0$ (even though we only expected $b_8=0$). Hence, if $W_{2,3,f}$ has been verified for all $f$ of degree $<3$, we get an equational proof that $8$-rings are commutative. In many examples, one already has $b_n = 0$ for some $n < p^m$. This means that the commutativity proof does not require all $W_{p,k,f}$. Choosing a different enumeration of the $f_i$ will also help to reduce the number $n$ with $b_n=0$. For example, in [Example 76](#8brief){reference-type="ref" reference="8brief"} we will get $b_3 = 0$ with a different enumeration. Actually, certain monomials are always sufficient: **Theorem 77**. *Let $p \in \mathds{P}$ and $k \geq 1$. Assume that for all $1 \leq i < k$ there is an equational proof of the constructive Wedderburn Theorem $\smash{W_{p,k,T^{p^i}}}$. Then there is an equational proof that every $p^k$-ring with $p=0$ is commutative.* *Proof.* Let $A$ be a $p^k$-ring with $p=0$, and let $a,b \in A$. We will tweak the proof of [Theorem 73](#generalcase-eq){reference-type="ref" reference="generalcase-eq"}. Let us first simplify the definition of $b_n$ a bit. The old definition was $b_{n+1} = (a - f_n(a)) b_n - b_n a$. So we might as well define $$\label{bndefnew} b_0 \coloneqq b, \quad b_{n+1} \coloneqq b_n a - f_n(a) b_n,$$ and start the enumeration with $f_0 \coloneqq T$. Again, $W_{p,k,f_n}$ proves $b_{n+1}=0 \implies b_n=0$ for $n \geq 1$ (using that $f_n(a) - a$ is a unit then), and we have $b_1 = b a - a b$. Hence, our goal is to find an equational proof of $b_n=0$ for some $n \geq 1$. As before, we may assume $g(a)=0$ for some monic irreducible polynomial $g \in \mathds{F}_p[T]$ whose degree $m$ divides $k$. Now, even though the computation of $b_n$ takes place inside a ring that is non-commutative a priori, we shall see now that it can be encoded by a computation in a commutative ring. By induction, it follows from [\[bndefnew\]](#bndefnew){reference-type="ref" reference="bndefnew"} that we can express $b_n$ as $$\textstyle b_n = \sum_{i,j} \lambda_{i,j} a^i b a^j$$ with $\lambda_{i,j} \in \mathds{F}_p$. We encode this with the commutative polynomial $$\textstyle B_n \coloneqq \sum_{i,j} \lambda_{i,j} X^i Y^j \in \mathds{F}_p[X,Y].$$ More formally, we define $B_n \in \mathds{F}_p[X,Y]$ recursively by $$B_0 \coloneqq 1, \quad B_{n+1} \coloneqq B_n Y - f_n(X) B_n .$$ Then $b_n$ results from $B_n$ by replacing any monomial $\lambda X^i Y^j$ with $\lambda a^i b a^j$. This describes a map $\mathds{F}_p[X,Y] \mapsto A$ that is clearly additive (but not multiplicative). In particular, $B_n=0$ will imply $b_n=0$. Since $\mathds{F}_p[X,Y]$ is commutative, there is a very simple formula for $B_n$: With $B_0 = 1$ and $\smash{B_{n+1} = (Y - f_n(X)) B_n}$ we find $$\label{Bclosed} B_n = \prod_{i < n} (Y - f_i(X)).$$ Consider the finite field $K \coloneqq \mathds{F}_p[X]/\langle g(X) \rangle$. (This concept is not equational, but wait for it.) The irreducible polynomial $g(Y) \in \mathds{F}_p[Y] \subseteq K[Y]$ has a root in $K$, namely $[X]$. Basic finite field theory tells us that the roots are formed by the orbits under the Frobenius. Thus, we have $\smash{g(Y) = \prod_{i < m} (Y - [X]^{p^i})}$ in $K[Y]$. This means that in $\mathds{F}_p[X,Y]$ there is an equation of the form $$\label{gtrans} g(Y) = \prod_{i < m} (Y - X^{p^i}) + g(X) \cdot h$$ for some $h \in \mathds{F}_p[X,Y]$. For every specific choice of $p,g$, we can simply find this equation by calculating $\smash{\prod_{i < m} (Y - X^{p^i})}$, so that no field theory is required. Define $\smash{f_i \coloneqq T^{p^i}}$ for $i < m$; the rest of the polynomials do not matter anymore. Then [\[Bclosed,gtrans\]](#Bclosed,gtrans){reference-type="ref" reference="Bclosed,gtrans"} imply $$g(Y) = B_m(Y) + g(X) \cdot h.$$ But since $g(a)=0$, we then get $b_m=0$. For a purely equational proof, when $p,g$ are fixed, one can just compute $b_m=0$ directly, just by using $g(a)=0$ (see the examples below). ◻ **Example 78**. From [Theorem 77](#weddermonomial){reference-type="ref" reference="weddermonomial"} we get the following equational proof that $8$-rings are commutative, for which only Morita [@Mo78] gave an equational proof so far. We may assume $g(a)=0$ for $g \coloneqq T^3+T+1$. For any element $b$, we define a sequence recursively by $b_0 \coloneqq b$ and $\smash{b_{n+1} \coloneqq b_n a + a^{2^n} b_n}$. We calculate: $$\begin{aligned} b_1 & = b a + a b, \\ b_2 & = (ba + ab)a + a^2 (ba+ab) \\ & = ba^2 + aba + a^2 ba + a^3 b \quad (\text{use } a^3=a+1) \\ & = b + ab + ba^2 + aba + a^2 ba,\\ b_3 & = (b + ab + ba^2 + aba + a^2 ba)a + (a+a^2) (b + ab + ba^2 + aba + a^2 ba) \\ & = ba + aba + ba^3 + aba^2 + a^2 ba^2 + ab + a^2 b + aba^2 + a^2 ba + a^3 ba \\ & \quad + a^2 b + a^3 b + a^2 ba^2 + a^3 ba + a^4 ba \\ & = ba + ab + aba + ba^3 + a^2 ba + a^3 b + a^4 ba \quad (\text{use } a^3=1+a, ~ a^4=a+a^2) \\ & = ba + ab + aba + b + ba + a^2 ba + b + ab + aba + a^2 ba \\ & = 0.\end{aligned}$$ If $W_{2,3,T^4}$ holds, $b_3 = 0$ implies $a^2 b_2 = 0$, hence $b_2 = 0$. If $W_{2,3,T^2}$ holds, then $b_2=0$ implies $(a^2+a) b_1 = 0$, hence $b_1=0$ and $a,b$ commute. We will prove these constructive Wedderburn Theorems in the next section as part of more general results. We can also replace the proof of $b_3=0$ with a "commutative computation", as explained in the general proof in [Theorem 77](#weddermonomial){reference-type="ref" reference="weddermonomial"}. Define polynomials $B_n \in \mathds{F}_2[X,Y]$ recursively by $B_0 \coloneqq 1$ and $\smash{B_{n+1} \coloneqq B_n Y + X^{2^n} B_n}$, so that $b_n$ results from $B_n$ by replacing any monomial $X^i Y^j$ with $a^i b a^j$. We compute $$\begin{aligned} B_3 & = (Y + X)(Y + X^2)(Y + X^4) \\ & = Y^3 + (X + X^2 + X^4)Y^2 + (X^3+X^4 + X^6)Y + X^7 \\ & = Y^3 + X g(X) Y^2 + X^3 g(X) Y + (X^4+X^2+X+1) g(X) + 1 \\ & \equiv Y^3 + Y + 1 \bmod g(X) \\ & \equiv 0 \bmod g(X),g(Y)\end{aligned}$$ and hence $b_3 = 0$. **Example 79**. With [\[generalcase-eq,weddermonomial\]](#generalcase-eq,weddermonomial){reference-type="ref" reference="generalcase-eq,weddermonomial"} we get the following equational proof that every $27$-ring with $3=0$ is commutative. (There seems to be no equational proof in the literature yet.) The list of irreducible polynomials over $\mathds{F}_3$ whose degree divides $3$ is $$\begin{aligned} &T,\, T+1,\, T+2,\, T^3+2T+1,\, T^3+2T+2,\, T^3+T^2+2,\, T^3+T^2+T+2,\\ &T^3+T^2+2T+1,\, T^3+2T^2+1,\, T^3+2T^2+T+1, \,T^3+2T^2+2T+2.\end{aligned}$$ We may remove the linear polynomials and those polynomials which are linear substitutions of others. This leaves us with $T^3+2T+1$ and $T^3+T^2+2$. So we need to handle two cases. Assume first that $a^3+2a+1 = 0$. Let us choose an enumeration starting with $T,T+1,T+2$. We calculate: $$\begin{aligned} b_1 & = ba - ab,\\ b_2 & = (ba-ab)a - (a+1)(ba-ab) \\ & = ba^2 + aba + a^2 b - ba + ab,\\ b_3 & = (ba^2 + aba + a^2 b - ba + ab)a - (a+2)(ba^2 + aba + a^2 b - ba + ab) \\ & = ba^3 + aba^2 + a^2 ba - ba^2 + aba - aba^2 - a^2 ba - a^3 b + aba - a^2 b \\ & \quad + ba^2 + aba + a^2 b - ba + ab \\ & = ba^3 - a^3 b - ba + ab \quad (\text{use } a^3=a-1) \\ & = b(a-1) - (a-1) b - ba + ab = 0.\end{aligned}$$ Thus, if $W_{3,3,T+1}$ and $W_{3,3,T+2}$ hold, we get $b_1 = 0$, and we are done. Now assume $a^3+a^2+2=0$. Let us choose an enumeration starting with $T,T^3,T^9$. We calculate (we omit the details): $$\begin{aligned} b_1 & = ba - ab,\\ b_2 & = b_2 a - a^3 b_2 = b_2 a - (1-a^2) b_2 \\ & = -b + ab - ba + a^2 b - aba + b a^2 + a^2 ba, \\ b_3 & = b_3 a - a^9 b_3 = b_3 a - (1-a+a^2)b_3 = 0.\end{aligned}$$ Thus, if $W_{3,3,T^3}$ and $W_{3,3,T^9}$ hold, we get $b_1 = 0$, and we are done. We will prove the required Wedderburn Theorems in the next section. **Example 80**. Let us prove that $16$-rings are commutative with [Theorem 77](#weddermonomial){reference-type="ref" reference="weddermonomial"}. (Again, there seems to be no equational proof in the literature yet.) The irreducible polynomials over $\mathds{F}_2$ whose degree divides $4$ are $$T^2+T+1,\, T^4+T+1,\, T^4+T^3+1,\, T^4+T^3+T^2+T+1.$$ But $T^4+T+1$ is reciprocal to $T^4+T^3+1$, and $(T+1)^4+(T+1)^3+1 = T^4+T^3+T^2+T+1$. Hence, it is enough to consider the two polynomials $T^2+T+1$ and $T^4+T+1$. If $a^2+a+1=0$, we proceed as in [Example 75](#4casenew){reference-type="ref" reference="4casenew"}. Namely, we find that $x \coloneqq ab+ba$ satisfies $xa = (1+a)x$. Assuming $W_{2,4,T+1}$, we get $x=0$ and we are done. Now let $a^4+a+1=0$. If we define $\smash{b_{n+1} \coloneqq b_n a - a^{2^i} b_n}$ for $i=0,\dotsc,4$, then a routine computation, which we omit, shows $b_4 = 0$. Assuming $W_{2,4,2^i}$, we get $b_1 = 0$ and we are done. We will prove the required constructive Wedderburn Theorems in the next section. # Constructive Wedderburn Theorems In this section, we will study more closely the constructive Wedderburn Theorems $W_{p,k,f}$ from [Definition 70](#wedderdef){reference-type="ref" reference="wedderdef"}. Recall that they state that $ba = f(a) b$ holds in a $p^k$-ring with $p=0$ only when $ba=ab$. Because of [Theorem 73](#generalcase-eq){reference-type="ref" reference="generalcase-eq"}, it would be very desirable to find equational proofs in general. Here, we will just cover some partial results and special cases. We also indicate how the general case could theoretically be handled by a computer program. Even though we already saw in [Theorem 77](#weddermonomial){reference-type="ref" reference="weddermonomial"} that it is enough to consider $\smash{f = T^{p^m}}$ for $0 \leq m < k$, we will consider arbitrary polynomials $f \in \mathds{F}_p[T]$ for the moment. **Lemma 81**. *If $f \in \mathds{F}_p[T]$ is constant, then there is an equational proof of $W_{p,k,f}$.* *Proof.* In a $p^k$-ring $A$ with $p=0$, assume $ba=ub$ holds for some $u \in \mathds{F}_p$. Then, since $u$ and $b$ commute, we have $b(a-u)=0$. Since $A$ is reduced, this implies $(a-u)b=0$, i.e. $ab=ub$. Thus, $ab=ba$. ◻ For a polynomial $f$ in one variable, we denote by $f^{\circ j}$ the $j$-fold composition of $f$ with itself. **Lemma 82**. *If $f \in \mathds{Z}[T]$ is a polynomial and the equation $ba = f(a)b$ holds in a ring, then for all $i,j \in \mathds{N}$ we have $$b^{\, j} a^i = f^{\circ j}(a)^i \, b^{\, j}.$$ More generally, for every $g \in \mathds{Z}[T]$ we have $$b^{\, j} g(a) = g(f^{\circ j}(a)) \, b^{\,j}.$$* *Proof.* The second equation follows from the first. The proof of the first is routine, so a sketch should be enough. One first proves the case $j=1$ by induction on $i$. Then one proves the case $i=1$ by induction on $j$. Then one proves the general case by induction on $i$. ◻ **Lemma 83**. *For $u \in \mathds{F}_p$ there is an equational proof of $W_{p,k,T+u}$.* *Proof.* Assume that $ba = (a+u)b$ holds in a $p^k$-ring with $p=0$. By [Lemma 82](#powerrel){reference-type="ref" reference="powerrel"}, we get $$ba = b^{p^k} a = (a+up^k) b^{p^k} = ab.\qedhere$$ ◻ **Lemma 84**. *To give an equational proof of $W_{p,k,f}$, one may assume that $a$ and $b$ are units.* *Proof.* Let $ba = f(a)b$. Assume that there is an equational proof of $ba=ab$ in case $b$ is a unit. By [Lemma 17](#unitdecomp){reference-type="ref" reference="unitdecomp"} we can write $b = ue$, where $u$ is a unit and $\smash{e = b^{p^{k}-1}}$ is idempotent, hence central. It suffices to prove that $b$ and $ea$ commute, since $b$ commutes with $(1-e)a$ anyway (the products vanish). Consider the $p^k$-ring $eA$ with multiplicative identity $e$ and the homomorphism $\alpha : A \to eA$, $x \mapsto ex$. Then we get $\alpha(b) \alpha(a) = f(\alpha(a)) \alpha(b)$ in $eA$. If $W_{p,k,f}$ has been proven in $eA$, where $b$ is a unit, then $\alpha(b)=b$ commutes with $\alpha(a)=ea$ and we are done. We need to make this argument equational, though, and this works as in the proof of [Theorem 73](#generalcase-eq){reference-type="ref" reference="generalcase-eq"}. Observe that $e$ is neutral for both $ea$ and $b$. Any proof of $W_{p,k,f}$ can be applied to the elements $ea$ and $b$ and the multiplicative "pseudo-identity" $e$. Thus, $b$ and $ea$ commute, and we are done. The reduction to the case that $a$ is a unit works exactly the same. ◻ When $b$ is a unit, then $W_{p,k,f}$ states that $ba = f(a)b$ implies $f(a) = a$. Thus, we need to show that $a$ is a fixed point of $f$. We will see in a moment that at least $a$ is a fixed point of a certain power $f^{\circ d}$. **Definition 85**. Let $f,g \in \mathds{F}_p[T]$ and $k \geq 1$. We write $f \equiv_k g$ or just $f \equiv g$ (when $k$ is clear from the context) when there is an equational proof of $f(a)=g(a)$ for all elements $a$ of a $p^k$-ring with $p=0$. The set of equivalence classes becomes a monoid with respect to the composition of polynomials: $$(f,g) \mapsto f \circ g.$$ Notice that it is well-defined, and the multiplicative identity is $T$. The underlying set can be identified with the one of $\smash{\mathds{F}_p[T] / \langle T^{p^k} - T \rangle}$, since $f \equiv g$ holds if and only if $\smash{T^{p^k}-T \mid f-g}$. In particular, this monoid is finite, it has exactly $\smash{p^{p^k}}$ elements. **Remark 86**. If $f$ is an element of a finite monoid, whose multiplication we write as $\circ$, then there must be a pair of numbers $i \geq 0$ and $\pi > 0$ such that $f^{\circ i} = f^{\circ (i+\pi)}$. Then the powers of $f$ are eventually periodic: $$1,f,\dotsc,f^{\circ i},f^{\circ (i+1)},\dotsc,f^{\circ (i+\pi-1)},f^{\circ i},f^{\circ (i+1)},\dotsc.$$ We say that $f$ is *of period* $\pi$ and *index* $i$. If $\pi$ is minimal, we call $\pi$ *the* period, similarly for the index. In particular, every $f \in \mathds{F}_p[T]$ has some period and index with respect to the finite monoid $\smash{(\mathds{F}_p[T]/\langle T^{p^k}-T \rangle,\circ,T)}$ from [Definition 85](#monoiddef){reference-type="ref" reference="monoiddef"}. Both can be efficiently computed for every specific choice of $p,k,f$ because of [Lemma 9](#idempower){reference-type="ref" reference="idempower"}. See also the following example. **Example 87**. 1. Constant polynomials are of period $1$ and index $1$. 2. The polynomial $-T$ is of period $2$ and index $0$ (also of period $1$ when $p=2$). 3. For $p=2$, $k=3$, $f = T^2+T$, we compute $f^{\circ 2} = T^4+T$, $f^{\circ 3} = T^4+T^2$ and $f^{\circ 4} = T^8 + T^2 \equiv T^2+T$, so that $f$ is of period $3$ and index $1$. 4. For $p=2$, $k=4$, $f = T^2+1$, we compute $f^{\circ 2} = (T^2+1)^2+1 = T^4$, $f^{\circ 3} = T^8+1$ and $f^{\circ 4}=T^{16} \equiv T$, so that $f$ is of period $4$ and index $0$. **Proposition 88**. *Let $f \in \mathds{F}_p[T]$ be a polynomial of period $\pi > 0$. Assume that $a,b$ are two elements of a $p^k$-ring with $p=0$ with $ba = f(a)b$, and $b$ is a unit. Then we have $f^{\circ d}(a)=a$, where $d \coloneqq \gcd(\pi,p^k - 1)$. In particular, if $d=1$, there is an equational proof of $W_{p,k,f}$.* Notice that this result generalizes both [Lemma 81](#wedderconst){reference-type="ref" reference="wedderconst"} and [Lemma 83](#wedderlin){reference-type="ref" reference="wedderlin"}, and that the assumption that $b$ is a unit is harmless because of [Lemma 84](#wedderunit){reference-type="ref" reference="wedderunit"}. Without this assumption, we can only conclude that $b$ commutes with $a^d$. *Proof.* Choose $i \geq 0$ with $f^{\circ i} \equiv f^{\circ (i+\pi)}$. [Lemma 82](#powerrel){reference-type="ref" reference="powerrel"} allows us to compute $$b^i b^\pi a = b^{i+\pi} a = f^{\circ (i+\pi)}(a) b^{i+\pi} = f^{\circ i}(a) b^i b^\pi = b^i a b^\pi.$$ Since $b^i$ is a unit, this shows that $a$ and $b^\pi$ commute. But then $$f^{\circ \pi}(a) b^\pi = b^\pi a = a b^\pi.$$ Since $b^\pi$ is a unit, we get $$\label{potj} f^{\circ \pi}(a) = a.$$ Next, we compute $$f(a) b = ba = b^{p^k} a = f^{\circ p^k}(a) b^{p^k} = f^{\circ p^k}(a) b.$$ Since $b$ is a unit, we get $\smash{f(a) = f^{\circ p^k}(a)}$. Apply $f^{\circ (\pi-1)}$ to both sides and use $a = f^{\circ \pi}(a)$. Then we get $$\label{potp} f^{\circ (p^k - 1)}(a) = a.$$ If $d \coloneqq \gcd(\pi,p^k-1)$, then [\[potj\]](#potj){reference-type="eqref" reference="potj"}, [\[potp\]](#potp){reference-type="eqref" reference="potp"} and the Bézout identity imply $f^{\circ d}(a) = a$. ◻ **Lemma 89**. *Let $m \in \mathds{N}$ and let $\pi$ be the order of $[m]$ in the additive group $\mathds{Z}/k\mathds{Z}$. Then the monomial $\smash{T^{p^m}}$ is of period $\pi$ and index $0$.* *Proof.* Let $\smash{f \coloneqq T^{p^m}}$. Write $\pi m = s k$ with $s \in \mathds{N}$. Then $\smash{f^{\circ \pi} = T^{p^{m \pi}} = T^{p^{s k}} \equiv T}$. ◻ **Theorem 90**. *Let $p \in \mathds{P}$ and $k \geq 1$. If $\gcd(k,p^k-1)=1$, then there is an equational proof that every $p^k$-ring with $p=0$ is commutative.* *Proof.* This follows from [Theorem 77](#weddermonomial){reference-type="ref" reference="weddermonomial"}, [Proposition 88](#periodlemma){reference-type="ref" reference="periodlemma"} and [Lemma 89](#monomialperiod){reference-type="ref" reference="monomialperiod"}. ◻ **Remark 91**. For the lack of a better name, let us call $k \geq 1$ *good* for $p \in \mathds{P}$ if we have $$\gcd(k,p^k-1)=1,$$ otherwise *bad* for $p$. With [Theorem 90](#gcdmain){reference-type="ref" reference="gcdmain"} we have solved the equational commutativity problem for good exponents, so let us analyze how many numbers are good. For $p=2$, the $22$ bad numbers $k \leq 100$ are $$6, 12, 18, 20, 21, 24, 30, 36, 40, 42, 48, 54, 60, 63, 66, 72, 78, 80, 84, 90, 96, 100.$$ For the other $78$ numbers $k \leq 100$, we thus have an equational proof that $2^k$-rings are commutative. As before, this then leads to many more cases: By [Theorem 35](#reduc){reference-type="ref" reference="reduc"}, every $94$-ring is a $2^{\mathop{\mathrm{lcm}}(1,2,5)}=2^{10}$-ring, hence commutative. For $p > 2$, every good number must be odd. For $p = 3$ and $p = 5$, most odd numbers are also good. For example, the only odd numbers $k \leq 100$ that are bad for $3$ are $39,55$, and the only odd numbers $k \leq 100$ that are bad for $5$ are $55, 93$. But for $p > 5$ there are much more bad odd numbers. If $k \mid k'$ and $k$ is bad, then $k'$ is bad as well, since $\gcd(k,p^k-1) \mid \gcd(k',p^{k'}-1)$. The following characterization of bad numbers has been found by Max Alekseyev. **Lemma 92**. *Let $p \in \mathds{P}$. The set of bad numbers $k$ for $p$ is equal to the union $$\bigcup_{q \in \mathds{P},\, q \neq p} q \cdot \mathop{\mathrm{ord}}_{q}(p) \cdot \mathds{N}^+.$$* Here, $\mathop{\mathrm{ord}}_q$ refers to the order in the group $(\mathds{Z}/q\mathds{Z})^{\times}$. *Proof.* Let $k \geq 1$. If $k$ belongs to the union, there is a prime $q \neq p$ with $q \cdot \mathop{\mathrm{ord}}_{q}(p) \mid k$. Then $q \mid k$ and $q \mid p^k - 1$, so that $\gcd(k,p^k-1) > 1$. Conversely, if $\gcd(k,p^k-1) > 1$, there is a prime $q$ with $q \mid k$ and $q \mid p^k - 1$. The second condition implies $q \neq p$ and then becomes equivalent to $\mathop{\mathrm{ord}}_{q}(p) \mid k$. Since the order divides $q-1$, it is coprime to $q$. Hence, $q \cdot \mathop{\mathrm{ord}}_{q}(p) \mid k$. ◻ With [Theorem 90](#gcdmain){reference-type="ref" reference="gcdmain"}, $2^6$ and $3^4$ are the only prime powers $\leq 100$ for which we don't have an equational commutativity proof yet, see also Table [1](#resulttable){reference-type="ref" reference="resulttable"}. For the moment, the general case seems to be out of reach. But it turns out that $W_{p,k,f}$ can be verified by a computer program, at least theoretically, since it is a *finite* problem: **Proposition 93**. *There is a finite $\mathds{F}_p$-algebra $U_{p,k,f}$ with a concrete finite presentation such that $U_{p,k,f}$ is commutative if and only if $W_{p,k,f}$ holds.* *Proof.* Because of [Lemma 84](#wedderunit){reference-type="ref" reference="wedderunit"}, it suffices to work with units. (This is not necessary for the proof, but it will decrease the dimension of the algebra below.) We define the $\mathds{F}_p$-algebra by the finite presentation $\smash{V_{p,k,f} \coloneqq \mathds{F}_p\langle X,Y : X^{p^k-1} = 1,~ Y^{p^k-1} = 1,~ YX = f(X) Y \rangle}$. We saw in the proof of [Lemma 71](#Wc){reference-type="ref" reference="Wc"} that $V_{p,k,f}$ is finite, namely $\{X^i Y^j : 0 \leq i,j < p^k - 1\}$ generates $V_{p,k,f}$ as an $\mathds{F}_p$-module. Then its universal $p^k$-ring quotient $\smash{U_{p,k,f} \coloneqq (V_{p,k,f})_{[p^k]}}$ from [Definition 25](#univquotient){reference-type="ref" reference="univquotient"} does the job: If two units $a,b$ of a $p^k$-ring with $p=0$ satisfy $ba = f(a)b$, then there is a unique homomorphism of rings $U_{p,k,f} \to A$ that maps $X \mapsto a$ and $Y \mapsto b$. Thus, if $U_{p,k,f}$ commutative, we have $ab=ba$. ◻ **Remark 94**. With the notation above, $\{X^i Y^j : 0 \leq i,j < p^k-1\}$ is not necessarily an $\mathds{F}_p$-basis of $V_{p,k,f}$. For example, $X = 1$ holds in $V_{2,3,T^2}$. More generally, when $p,k$ are arbitrary and $f = T^m$ is any monomial (which is sufficient by [Theorem 77](#weddermonomial){reference-type="ref" reference="weddermonomial"}), then we have $X^d = 1$ for $\smash{d \coloneqq \gcd(m^{p^k-1}-1,p^k-1)}$, and we see that $V_{p,k,f} = \mathds{F}_p[G]$ is a group algebra for $$G \coloneqq \langle X,Y : X^d = 1, \, Y^{p^k-1}=1,\, YXY^{-1} = X^m \rangle \cong C_d \rtimes_m C_{p^k-1}.$$ This description should help to calculate $\smash{U_{p,k,T^m} = \mathds{F}_p[G]_{[p^k]}}$, but in practice a brute force attack is still too inefficient, even for small numbers. # SageMath code ## Computation of $n$-primes and $n$-powers {#app:powers} The code below has been used to generate the list of $n$-fields in [Remark 19](#nfields){reference-type="ref" reference="nfields"}. def n_primes(n): """Returns the list of prime numbers p with p-1 | n-1""" return [x+1 for x in divisors(n-1) if (x+1).is_prime()] def n_powers(n): """Returns the list of prime powers q with q-1 | n-1""" return [x+1 for x in divisors(n-1) if (x+1).is_prime_power()] def all_powers(limit): """Returns the lists of n-powers for 2 <= n <= limit""" return [[n,n_powers(n)] for n in range(2,limit+1)] ## Reduction to prime characteristic {#app:char} The code below implements the algorithm in [Lemma 29](#chareq){reference-type="ref" reference="chareq"}. def get_numbers_for_characteristic(n): """Returns a list of numbers z such that p_1 * ... * p_s is the gcd of the z^n - z, where p_i are the prime numbers with p_i - 1 | n - 1""" c = prod(n_primes(n)) z, g, numbers = 0, 0, [] while (g != c): z = z.next_prime() numbers.append(z) g = gcd(g, z^n - z) return numbers def characteristic_proof(n): """Returns an equational proof that p_1 * ... * p_s = 0 holds in any n-ring, where p_i are the prime numbers with p_i - 1 | n - 1""" numbers = get_numbers_for_characteristic(n) myprimes = n_primes(n) relations_string = ", ".join([f"{z}^{n}-{z}" for z in numbers]) factor_string = " * ".join(map(str,myprimes)) return f"gcd({relations_string}) = {str(prod(myprimes))} = {factor_string}" To get the linear combination, one can use the extended_gcd function below. ## Reduction to prime powers {#app:reduc} The code below implements the reduction algorithm from [Remark 37](#perf){reference-type="ref" reference="perf"}. For example, the command reduction_proof(7,2) returns a proof that every $7$-ring with $2=0$ is a $4$-ring. ``` {.python language="Python"} def extended_gcd(args): """Extension of Sage's xgcd function from two to several arguments. Given args = [a_1,....a_n], returns [gcd(a_1,....,a_n),[u_1,....,u_n]] where u_1 * a_1 + ... + u_n * a_n = gcd(a_1,...,a_n).""" if (len(args) == 0): return [0,[]] if (len(args) == 1): return [args[0],[1]] if (len(args) == 2): g, u, v = xgcd(*args) return [g,[u,v]] first, *rest = args rest_gcd, v = extended_gcd(rest) full_gcd, u = extended_gcd([first, rest_gcd]) w = [u[0]] + [u[1] * x for x in v] return [full_gcd, w] def get_exponent(p,n): """Returns the least common multiple of all d > 0 with p^d-1 | n-1""" limit = floor(log(n,p)) powers = [d for d in range(1,limit+1) if (p^d-1).divides(n-1)] return lcm(powers) def reduction_proof(n,p,only_polynomials=False): """Returns a linear combination of the form g = sum_i u_i * (f_i^n - f_i) in GF(p)[T], where g divides T^(p^k) - T, thus proving the reduction theorem. When the flag 'only_polynomials' is set, returns only the list of the f_i.""" if not (p.is_prime() and (p-1).divides(n-1)): raise Exception("Invalid arguments") k = get_exponent(p,n) R.<T> = PolynomialRing(GF(p)) polynomials = [] g = 0 target = T^(p^k) - T found = False # compute gcd of polynomials of the form f^n - f until it divides T^(p^k) - T for f in R.polynomials(max_degree = n-1): if f.degree() < 1 or (g != 0 and g.divides(f^n-f)): continue polynomials.append(f) g = gcd(g, f^n-f) if (g.divides(target)): found = True break if not found: raise Exception("No linear combination found") if (only_polynomials): return polynomials # print the linear combination proof = str(g) coeffs = extended_gcd([f^n-f for f in polynomials])[1] for i in range(len(polynomials)): prefix = " +" if i > 0 else " =" coeff = coeffs[i] poly = polynomials[i] proof += f"{prefix} ({coeff}) * (({poly})^{n} - ({poly}))" return proof ``` ## Finding unpleasant numbers {#app:linear} The code below has been used in [Remark 51](#lineargcd){reference-type="ref" reference="lineargcd"} to demonstrate how many simple numbers are nice. We use the get_exponent function from above. ``` {.python language="Python"} def is_simple_at(n, p): """Checks if n is simple at p""" return n>1 and (p-1).divides(n-1) and get_exponent(p,n) == 1 def unpleasant_numbers(p,start,end): """Returns the number of p-simple numbers within a range and the list of unpleasant numbers among them""" R.<T> = PolynomialRing(GF(p)) unpleasants = [] simple_count = 0 for n in range(start,end+1): if not (is_simple_at(n,p)): continue simple_count += 1 g = gcd([(T+u)^n - (T+u) for u in GF(p)]) if (g != T^p - T): unpleasants.append(n) return [simple_count, unpleasants] ``` ## Simple numbers {#app:simple} The code below generates the first simple numbers as in [Definition 52](#def:simple){reference-type="ref" reference="def:simple"}. def is_simple(n): """Checks if n is a simple number""" return n>1 and all(q.is_prime() for q in n_powers(n)) def first_simple_numbers(limit): """Returns the list of simple numbers below a given limit""" return [n for n in range(2,limit+1) if is_simple(n)] HLY94 D. D. Anderson, P. V. Danchev, *A note on a theorem of Jacobson related to periodic rings*, Proc. Am. Math. Soc. 148.12 (2020), 5087--5089. G. Birkhoff, *On the structure of abstract algebras*, Proc. Cambridge Philos. Soc. 31 (1935), 433--454. S. M. Buckley, D. MacHale, *Centrifiers and ring commutativity*. J. Math. Sci. Adv. Appl. 15 (2012), 139--161. S. M. Buckley, D. MacHale, *Variations on a theme: rings satisfying $x^3=x$ are commutative*, Amer. Math. Monthly 120.5 (2013), 430--440. D. van Dalen, *Intuitionistic Logic*, in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell (2001). S. W. Dolan, *A proof of Jacobson's Theorem*, Canad. Math. Bull. 19.1 (1976), 59--61. A. Forsythe, N. H. McCoy, *On the commutativity of certain rings*, Bull. Amer. Math. Soc. 52.6 (1946), 523--526. I. N. Herstein, *A generalization of a theorem of Jacobson III*, Amer. J. Math. 75 (1953), 105--111. I. N. Herstein, *An elementary proof of a theorem of Jacobson*, Duke Math. 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Univ. Hambg. 41 (1974), 72--74. R. S. Pierce, *Modules over commutative regular rings*, Mem. Amer. Math. Soc. 70 (1967). K. Rogers, *An elementary proof of a theorem of Jacobson*, Abh. Math. Semin. Univ. Hambg. 35.3 (1971), 223--229. H. Schoutens, *The use of ultraproducts in commutative algebra*, Lecture Notes in Math. 1999 (2010). M. H. Stone, *The theory of representation for Boolean algebras*, Trans. Amer. Math. Soc. 40.1 (1936), 37--111. W. Taylor, *Equational logic*, Houston J. Math. (1979). G. Oman, *A characterization of potent rings*, Glasg. Math. J. 65 (2023), 324--327. B. Poonen, *Why all rings should have a 1*, [arXiv:1404.0135 \[math.RA\]](https://arxiv.org/abs/1404.0135) (2014). J. W. Wamsley, *On a condition for commutativity of rings*, J. Lond. Math. Soc. 2.2 (1971), 331--332. J. J. Wavrik, *Commutativity theorems: examples in search of algorithms*, ISSAC (1999), 31--36. W. Witt, *Über die Kommutativität endlicher Schiefkörper*, Abh. Math. Semin. Univ. Hambg. 8 (1931). H. Zhang, *Automated proof of ring commutativity problems by algebraic methods*, J. Symbolic Comput. 9.4 (1990), 423--427. [^1]: <https://www.sagemath.org/> [^2]: <https://oeis.org/A366343> [^3]: Two idempotents $e,f$ are called orthogonal if $ef = 0$.

arxiv_math

--- abstract: | Ball quotients, hyperelliptic varieties, and projective spaces are characterized by their Chern classes, as the varieties where the Miyaoka-Yau inequality becomes an equality. Ball quotients, Abelian varieties, and projective spaces are also characterized topologically: if a complex, projective manifold $X$ is homeomorphic to a variety of this type, then $X$ is itself of this type. In this paper, similar results are established for projective varieties with klt singularities that are homeomorphic to singular ball quotients, quotients of Abelian varieties, or projective spaces. address: - Daniel Greb, Essener Seminar für Algebraische Geometrie und Arithmetik, Fakultät für Mathematik, Universität Duisburg--Essen, 45117 Essen, Germany - Stefan Kebekus, Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Straße 1, 79104 Freiburg im Breisgau, Germany - Thomas Peternell, Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany author: - Daniel Greb - Stefan Kebekus - Thomas Peternell title: Miyaoka-Yau inequalities and the topological characterization of certain klt varieties --- # Introduction {#sec:1} ## The Miyaoka-Yau inequality for projective manifolds Let $X$ be an $n$-dimensional complex-projective manifold and let $D$ be any divisor on $X$. Recall that $X$ is said to "satisfy the Miyaoka-Yau inequality for $D$" if the following Chern class inequality holds, $$\big( 2(n+1)·c_2(X) - n·c_1(X)² \big)·[D]^{n-2} ≥ 0.$$ It is a classic fact that $n$-dimensional projective manifolds $X$ whose canonical bundles are ample or trivial satisfy Miyaoka-Yau inequalities. In case of equality, the universal covers are of particularly simple form. **Theorem 1** (Ball quotients and hyperelliptic varieties). *Let $X$ be an $n$-dimensional complex projective manifold.* - *If $K_X$ is ample, then $X$ satisfies the Miyaoka-Yau inequality for $K_X$. In case of equality, the universal cover of $X$ is the unit the ball $𝔹^n$.* - *If $K_X$ is trivial and $D$ is any ample divisor, then $X$ satisfies the Miyaoka-Yau inequality for $D$. In case of equality, the universal cover of $X$ is the affine space $ℂ^n$. 0◻* We refer the reader to [@GKT16] for a full discussion and references to the original literature. In the Fano case, where $-K_X$ is ample, the situation is more complicated, due to the fact that the tangent bundle $𝒯_X$ and the canonical extension $ℰ_X$ need not be semistable[^1]. If $ℰ_X$ is semistable, then analogous results hold, see [@GKP22 Theorem 1.3], as well as further references given there. **Theorem 2** (Projective space). *Let $X$ be an $n$-dimensional projective manifold. If $-K_X$ is ample and if the canonical extension is semistable with respect to $-K_X$, then $X$ satisfies the Miyaoka-Yau inequality for $-K_X$. In case of equality, $X$ is isomorphic to the projective space $ℙ^n$. 0◻* In each of the three settings, the equality cases are characterized topologically: if $M$ is any projective manifold homeomorphic to a ball quotient, a finite étale quotient of an Abelian variety or the projective space, then $M$ itself is biholomorphic to a ball quotient, to a finite étale quotient of an Abelian variety, or to the projective space. For ball quotients, this is a theorem of Siu [@MR584075]. The torus case is due to Catanese [@MR1945361], whereas the Fano case is due to Hirzebruch-Kodaira [@MR92195] and Yau [@MR480350]. ## Spaces with MMP singularities In general, it is rarely the case that the canonical bundle of a projective variety has a definite \"sign\". Minimal model theory offers a solution to this problem, at the expense of introducing singularities. It is therefore natural to extend our study from projective manifolds to projective varieties with Kawamata log terminal (= klt) singularities. For klt varieties whose canonical sheaves are ample, trivial or negative, analogues of Theorems [Theorem 1](#thm:1-1){reference-type="ref" reference="thm:1-1"} and [Theorem 2](#thm:1-2){reference-type="ref" reference="thm:1-2"} have been found in the last few years. We refer the reader to [@GKPT19 Thm. 1.5] for a characterization of singular ball quotients among projective varieties with klt singularities - see Definition [Definition 8](#def:2-1){reference-type="ref" reference="def:2-1"} for the notion of singular ball quotients. Characterizations of torus quotients and quotients of the projective space can be found in [@LT18], [@MR4263792 Thm. 1.2] and [@GKP22 Thm. 1.3]. In each case, we find it striking that the Chern class equalities imply that the underlying space has no worse than quotient singularities. ## Main results of this paper This paper asks whether the topological characterizations of ball quotients, Abelian varieties and the projective spaces have analogues in the klt settings. Section [2](#sec:2){reference-type="ref" reference="sec:2"} establishes a topological characterization of singular ball quotients. The main result of this section, Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"}, can be seen as a direct analogue of Siu's rigidity theorems. **Theorem 3** (Rigidity in the klt setting, see Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"}). *Let $X$ be a singular quotient of an irreducible bounded symmetric domain and let $M$ be a normal projective variety that is homeomorphic to $X$. If $\dim X ≥ 2$, then, $M$ is biholomorphic or conjugate-biholomorphic to $X$.* Using somewhat different methods, Section [3](#sec:3){reference-type="ref" reference="sec:3"} generalizes Catanese's result to the klt setting. **Theorem 4** (Varieties homeomorphic to torus quotients, see Theorem [Theorem 18](#thm:3-3){reference-type="ref" reference="thm:3-3"}). *Let $M$ be a compact complex space with klt singularities. Assume that $M$ is bimeromorphic to a Kähler manifold. If $M$ is homeomorphic to a singular torus quotient, then $M$ is a singular torus quotient.* In both cases, we find that certain Chern classes equalities are invariant under homeomorphisms. Varieties homeomorphic to projective spaces are harder to investigate. Section [4](#sec:4){reference-type="ref" reference="sec:4"} gives a full topological characterization of $ℙ³$, but cannot fully solve the characterization problem in higher dimensions. **Theorem 5** (Topological $ℙ³$, see Theorem [Theorem 40](#thm:4-20){reference-type="ref" reference="thm:4-20"}). *Let $X$ be a projective klt variety that is homeomorphic to $ℙ³$. Then, $X ≅ ℙ³$.* However, we present some partial results that severely restrict the geometry of potential exotic varieties homeomorphic to $ℙ^n$. These allow us to show the following. **Theorem 6** ($ℚ$-Fanos in dimension 4 and 5, see Theorem [Theorem 41](#thm:4-21){reference-type="ref" reference="thm:4-21"}). *Let $X$ be a projective klt variety that is homeomorphic to $ℙ^n$ with $n = 4$ or $n = 5$. Then, $X ≅ ℙ^n$, unless $K_X$ is ample.* ## Dedication {#dedication .unnumbered} We dedicate this paper to the memory of Jean-Pierre Demailly. His passing is a tremendous loss to the mathematical community and to all who knew him. ## Acknowledgements {#acknowledgements .unnumbered} We thank Markus Banagl, Sebastian Goette, Wolfgang Lück, Jörg Schürmann and Michael Weiss for providing detailed guidance regarding Pontrjagin classes of topological manifolds. Igor Belegradek kindly answered our questions on MathOverflow. We also thank the referee, who suggested, among other improvements, to generalize the results of Section [3](#sec:3){reference-type="ref" reference="sec:3"} to the Kähler case. # Mostow Rigidity for singular quotients of symmetric domains {#sec:2} Consider a compact Kähler manifold $X$ whose universal cover is a bounded symmetric domain. Siu has shown in [@MR584075 Thm. 4] and [@Siu81 Main Theorem] that any compact Kähler manifold $M$ which is homotopy equivalent to $X$ is biholomorphic or conjugate-biholomorphic[^2] to $X$. We show an analogous result for homeomorphisms between *singular* varieties $M$ and $X$. The following notion will be used. **Definition 7** (Quasi-étale cover). *A finite, surjective morphism between normal, irreducible complex spaces is called *quasi-étale cover* if it is unbranched in codimension one.* **Definition 8** (Singular quotient of bounded symmetric domtain). *Let $Ω$ be an irreducible bounded symmetric domain. A normal projective variety $X$ is called a *singular quotient of $Ω$* if there exists a quasi-étale cover $\widehat{X} → X$, where $\widehat{X}$ is a smooth variety whose universal cover is $Ω$.* *Remark 9* (Singular quotients are quotients). Let $X$ be a singular quotient of an irreducible bounded symmetric domain $Ω$. Passing to a suitable Galois closure, one finds a quasi-étale *Galois* cover $\widehat{X} → X$, where $\widehat{X}$ is a smooth variety whose universal cover is $Ω$. In particular, it follows that $X$ is a quotient variety and that it has quotient singularities. Moreover, it can be shown as in [@GKPT19b Sect. 9] that $X$ is actually a quotient of $Ω$ by the fundamental group of $X_{\mathop{\mathrm{reg}}}$, which acts properly discontinously on $Ω$. In addition, the action is free in codimension one. **Theorem 10** (Mostow rigidity in the klt setting). *Let $X$ be a singular quotient of an irreducible bounded symmetric domain and let $M$ be a normal projective variety that is homeomorphic to $X$. If $\dim X ≥ 2$, then, $M$ is biholomorphic or conjugate-biholomorphic to $X$.* *Remark 11* (Varieties conjugate-biholomorphic to ball quotients). We are particularly interested in the case where the bounded symmetric domain of Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"} is the unit ball. For this, observe that the set of (singular) ball quotients is invariant under conjugation. It follows that if the variety $M$ of Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"} is biholomorphic or conjugate-biholomorphic to a (singular) ball quotient $X$, then $M$ is itself a (singular) ball quotient. Before proving Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"} in Sections [2.1](#sec:2-1){reference-type="ref" reference="sec:2-1"}--[2.3](#sec:2-3){reference-type="ref" reference="sec:2-3"} below, we note a first application: the Miyaoka-Yau Equality is a topological property. The symbols $\widehat{c}_•(X)$ in Corollary [Corollary 12](#cor:2-5){reference-type="ref" reference="cor:2-5"} are the $ℚ$-Chern classes of the klt space $X$, as defined and discussed for instance [@GKPT19b Sect. 3.7]. **Corollary 12** (Topological invariance of the Miyaoka-Yau equality). *Let $X$ be a projective klt variety with $K_X$ ample. Assume that the Miyaoka-Yau equality holds: $$\bigl( 2(n+1)· \widehat{c}_2(𝒯_X) - n · \widehat{c}_1(𝒯_X)² \bigr) · [K_X]^{n-2} =0.$$ Let $M$ be a normal projective variety homeomorphic to $X$. Then $M$ is klt, $K_M$ is ample and $$\bigl( 2(n+1)· \widehat{c}_2(𝒯_M) - n · \widehat{c}_1(𝒯_M)² \bigr) · [K_M]^{n-2} =0.$$* *Proof.* Since the Miyaoka-Yau Equality holds on $X$, there is a quasi-étale cover $\widetilde{X} → X$ such that the universal cover of $\widetilde{X}$ is the ball, [@GKPT19b]. By Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"}, there is a quasi-étale cover $\widetilde{M} → M$ such that $\widetilde{M} ≅ \widetilde{X}$ biholomorphically or conjugate-biholomorphically. Hence, the universal cover of $\widetilde{M}$ is the ball. It follows that $M$ is klt, $K_M$ is ample, and that the Miyaoka-Yau Equality holds on $M$. ◻ ## Preparation for the proof of Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"} {#sec:2-1} The following lemma of independent interest might be well-known. We include a full proof for lack of a good reference. **Lemma 13**. *Let $X$ be a normal complex space. Then, the set $X_{\mathop{\mathrm{sing,top}}} ⊂ X$ of topological singularities is a complex-analytic set.* *Proof.* Recall from [@GoreskyMacPherson Thm. on p. 43] that $X$ admits a Whitney stratification where all strata are locally closed complex-analytic submanifolds of $X$. Recall from [@MR1131081 Chapt. IV.8] that the closures of the strata are complex-analytic subsets of $X$. Since Whitney stratifications are locally topologically trivial along the strata[^3], it follows that $X_{\mathop{\mathrm{sing,top}}}$ is locally the union of finitely many strata. The additional observation that the set of topologically smooth points, $X ∖ X_{\mathop{\mathrm{sing,top}}}$, is open in the Euclidean topology implies that $X_{\mathop{\mathrm{sing,top}}}$ is locally the union of the closures of finitely many strata, hence analytic. ◻ ## Proof of Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"} if $X$ is smooth {#sec:2-2} We maintain the notation of Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"} in this section and assume additionally that $X$ is smooth. To begin, fix a homeomorphism $f: M → X$ and choose a resolution of singularities, say $π: \widetilde{M} → M$. The composed map $g = f◦π$ is continuous and induces an isomorphism $$\label{eq:2-7-1} g_*: H_{2n} \bigl(\widetilde{M},\, ℤ\bigr) → H_{2n}\bigl(X,\, ℤ\bigr).$$ Hence, by Siu's general rigidity result [@MR584075 Thm. 6] in combination with the curvature computations for the classical, respectively exceptional Hermitian symmetric domains done in [@MR584075; @Siu81], the continuous map $g$ is homotopic to a holomorphic or conjugate-holomorphic map $\widetilde{g}: \widetilde{M} → X$. Replacing the complex structure on $X$ by the conjugate complex structure, if necessary, we may assume without loss of generality that $\widetilde{g}$ is holomorphic and hence in particular algebraic. The isomorphism [\[eq:2-7-1\]](#eq:2-7-1){reference-type="eqref" reference="eq:2-7-1"} maps the fundamental class of $\widetilde{M}$ to the fundamental class of $X$, and $\widetilde{g}$ is hence birational. We claim that the bimeromorphic morphism $\widetilde{g}$ factors via $π$. To begin, observe that since $g$ contracts the fibres of $π$ and since $\widetilde{g}$ is homotopic to $g$, the map $\widetilde{g}$ contracts the fibres of $π$ as well. In fact, given any curve $\widetilde{C} ⊂ \widetilde{M}$ with $π(\widetilde{C})$ a point, consider its fundamental class $[\widetilde{C}] ∈ H_2\bigl(\widetilde{M},\, ℝ\bigr)$. By assumption, we find that $$\widetilde{g}_* \bigl([\widetilde{C}]\bigr) = g_* \bigl([\widetilde{C}]\bigr) = 0 ∈ H_2\bigl(X,\, ℝ\bigr).$$ Given that $X$ is projective, this is only possible if $\widetilde{g}(\widetilde{C})$ is a point. Since $M$ is normal and since $\widetilde{g}$ contracts the (connected) fibres of the resolution map $π$, we obtain the desired factorisation of $\widetilde{g}$, as follows $$\begin{tikzcd}[column sep=1.5cm] \widetilde{M} \ar[r, "π"'] \ar[rr, bend left=15, "\widetilde{g}"] & M \ar[r, "∃! \widetilde{f}"'] & X. \end{tikzcd}$$ We claim that the birational map $\widetilde{f}$ is biholomorphic.[^4] By Zariski's Main Theorem, [@Ha77 V Thm. 5.2], it suffices to verify that it does not contract any curve $C ⊂ M$. Aiming for a contradiction, assume that there exists a curve $\widetilde{C} ⊂ \widetilde{M}$ whose image $C := π(\widetilde{C})$ is a curve in $M$, while $\widetilde{g}(\widetilde{C}) = \widetilde{f}(C) = (*)$ is a point in $X$. Let $d >0$ be the degree of the restricted map $π|_{\widetilde{C}}: \widetilde{C} → C$. Then, on the one hand, $$f_*\bigl(d·[C]\bigr) = f_*\bigl(π_*[\widetilde{C}]\bigr) = g_* [\widetilde{C}] = \widetilde{g}_* [\widetilde{C}] = 0 ∈ H_2\bigl(X,\, ℝ\bigr).$$ On the other hand, projectivity of $M$ implies that $d·[C]$ is a non-trivial element of $H_2\bigl(M,\, ℝ\bigr)$, which therefore must be mapped to a non-trivial element of $H_2\bigl(X,\, ℝ\bigr)$, since $f$ is assumed to be a homeomorphism. This finishes the proof of Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"} in the case where $X$ is smooth. 0◻ ## Proof of Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"} in general {#sec:2-3} Maintain the setting of Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"}. ### Step 1: Setup {#step-1-setup .unnumbered} By assumption, there exists a bounded symmetric domain $Ω$ and a quasi-étale cover $τ_X :\widehat{X} → X$ such that the universal cover of $\widehat{X}$ is $Ω$. Choose a homeomorphism $f: M → X$ and let $\widehat{M} := \widehat{X} ⨯_X M$ be the topological fibre product. The situation is summarized in the following commutative diagram, $$\label{eq:2-8-1} \begin{tikzcd}[column sep=2cm] \widehat{M} \ar[r, two heads, "τ_M"] \ar[d, "\simeq"'] & M \ar[d, "f", "\simeq"'] \\ \widehat{X} \ar[r, two heads, "τ_X\text{, quasi-étale}"'] & X, \end{tikzcd}$$ in which the vertical maps are homeomorphisms and the horizontal maps are surjective with finite fibres. ### Step 2: A complex structure on $\widehat{M}$ {#step-2-a-complex-structure-on-widehatm .unnumbered} The spaces $M$, $\widehat{X}$ and $X$ all carry complex structures. We aim to equip $\widehat{M}$ with a structure so that all horizontal arrows in [\[eq:2-8-1\]](#eq:2-8-1){reference-type="eqref" reference="eq:2-8-1"} become holomorphic. *Claim 14*. There exists a normal complex structure on $\widehat{M}$ that makes $τ_M$ a finite, holomorphic, and quasi-étale cover. *Proof of Claim [Claim 14](#claim:2-9){reference-type="ref" reference="claim:2-9"}.* Let $X_0$ be the smooth locus of $X$, set $M_0 := f^{-1}(X_0)$ and $\widehat{M}_0 := τ_M^{-1}(M_0)$. The map $τ_M|_{\widehat{M}_0}$ being a local homeomorphism, there is a uniquely determined complex structure on $\widehat{M}_0$ such that $τ_M|_{ \widehat{M}_0}: \widehat{M}_0 → M_0$ is a finite holomorphic cover. Since $X$ has quotient singularities, the topological and holomorphic singularities agree, $X_{\mathop{\mathrm{sing,top}}} = X_{\mathop{\mathrm{sing}}}$. Hence, $f$ being a homeomorphism, we note that $$M_{\mathop{\mathrm{sing,top}}} = f^{-1}\left(X_{\mathop{\mathrm{sing}}}\right)% \quad\text{and}\quad M ∖ M_0 = M_{\mathop{\mathrm{sing,top}}}.$$ We have seen in Lemma [Lemma 13](#lem:2-6){reference-type="ref" reference="lem:2-6"} that $M_{\mathop{\mathrm{sing,top}}}$ is an analytic set. Therefore, by [@DethloffGrauert Thm. 3.4] and [@MR83045 Satz 1], the complex structure on $\widehat{M}_0$ uniquely extends to a normal complex structure on the topological manifold $\widehat{M}$, making $τ_M$ holomorphic and finite. The branch locus of $τ_M$ has the same topological dimension as the branch locus of $τ_X$, so that $τ_M$ is quasi-étale, as claimed.  (Claim [Claim 14](#claim:2-9){reference-type="ref" reference="claim:2-9"}) ◻ Note that as a finite cover of the projective variety $M$, the normal complex space $\widehat{M}$ is again projective. ### Step 3: $\widehat{M}$ as a quotient of $Ω$ {#step-3-widehatm-as-a-quotient-of-ω .unnumbered} The homeomorphic varieties $\widehat{X}$ and $\widehat{M}$ reproduce the assumptions of Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"}. The partial results of Section [2.2](#sec:2-2){reference-type="ref" reference="sec:2-2"} therefore apply to show that the complex spaces $\widehat{M}$ and $\widehat{X}$ are biholomorphic or conjugate-biholomorphic. Replacing the complex structures on $M$ and $\widehat{M}$ by their conjugates, if necessary, we assume without loss of generality for the remainder of this proof that $\widehat{M}$ and $\widehat{X}$ are biholomorphic. This has two consequences. 1. The projective variety $\widehat{M}$ is smooth. The universal cover of $\widehat{M}$ is biholomorphic to $Ω$. 2. [\[il:2-10-2\]]{#il:2-10-2 label="il:2-10-2"} Its quotient $M$ is a singular quotient of $Ω$ and has only quotient singularities. Recalling that quotient singularities are not topologically smooth, Item [\[il:2-10-2\]](#il:2-10-2){reference-type="ref" reference="il:2-10-2"} implies that the homeomorphism $f: M → X$ restricts to a homeomorphism between the smooth loci, $X_{\mathop{\mathrm{reg}}}$ and $M_{\mathop{\mathrm{reg}}}$. The situation is summarized in the following commutative diagram, $$\begin{tikzcd}[column sep=2.2cm] Ω \ar[r, two heads, "u_X\text{, univ.~cover}"] \ar[d, "\simeq"'] & \widehat{M} \ar[r, two heads, "τ_M\text{, quasi-étale}"] \ar[d, "\simeq"'] & M \ar[d, "f", "\simeq"'] & M_{\mathop{\mathrm{reg}}} \ar[l, hook, "\text{inclusion}"'] \ar[d, "f|_{X_{\mathop{\mathrm{reg}}}}", "\simeq"'] \\ Ω \ar[r, two heads, "u_M\text{, univ.~cover}"'] & \widehat{X} \ar[r, two heads, "τ_X\text{, quasi-étale}"'] & X & X_{\mathop{\mathrm{reg}}}, \ar[l, hook, "\text{inclusion}"] \end{tikzcd}$$ where all horizontal maps are holomorphic, and all vertical maps are homeomorphic. The description of $M$ as a singular quotient of $Ω$ can be made precise. The argument in [@GKPT19b Sect. 9.1] shows that the fundamental group $π_1(M_{\mathop{\mathrm{reg}}})$ acts properly discontinously on $Ω$ with quotient $M$. In particular, we have an injective homomorphism from $π_1(M_{\mathop{\mathrm{reg}}})$ into the holomorphic automorphism group $\mathop{\mathrm{Aut}}(Ω)$ of $Ω$, with image a discrete cocompact subgroup $Γ_M ⊆ \mathop{\mathrm{Aut}}(Ω)$. The same reasoning also applies to $X$ and presents $X$ as a quotient $X = Ω/π_1(X_{\mathop{\mathrm{reg}}})$, where $π_1(X_{\mathop{\mathrm{reg}}})$ again acts via an injective homomorphism $π_1(X_{\mathop{\mathrm{reg}}}) ↪ \mathop{\mathrm{Aut}}(Ω)$, with image a cocompact, discrete subgroup $Γ_X$ of $\mathop{\mathrm{Aut}}(Ω)$. As we have seen above, $f$ induces a homeomorphism from $M_{\mathop{\mathrm{reg}}}$ to $X_{\mathop{\mathrm{red}}}$, from which we obtain an abstract group isomorphism $θ: Γ_M → Γ_X$. ### Step 4: End of proof {#step-4-end-of-proof .unnumbered} In the remainder of the proof we will show that not only $\widehat{M}$ and $\widehat{X}$ are (conjugate-)-biholomorphic, but that this actually holds for $M$ and $X$. This will be a consequence of Mostow's rigidity theorem for lattices in connected semisimple real Lie groups. As the groups appearing in our situation are not necessarily connected, we have to do some work to reduce to the connected case[^5]. Given that $Ω$ is an irreducible Hermitian symmetric domain of dimension greater than one, the identity component $\mathop{\mathrm{Aut}}°(Ω) ⊆ \mathop{\mathrm{Aut}}(Ω)$ coincides with the identity component $I°(Ω)$ of the isometry group $I(Ω)$ of the Riemannian symmetric space $Ω$, [@MR1834454 VIII.Lem. 4.3][^6], which is a non-compact simple Lie group without non-trivial proper compact normal subgroups and with trivial centre, [@MR1441541 Prop. 2.1.1 and bottom of p. 379]. We also note that a Bergman-metric argument shows that $\mathop{\mathrm{Aut}}(Ω)$ is contained in $I(Ω)$. Furthermore, both Lie groups have only finitely many connected components. *Claim 15*. There exists an isometry $F ∈ I(Ω)$ such that $$\label{eq:2-11-1} F◦γ = θ(γ)◦F, \quad \text{for every } γ ∈ Γ_M.$$ *Proof of Claim [Claim 15](#claim:2-11){reference-type="ref" reference="claim:2-11"}.* If the rank of $Ω$ is equal to one, then $Ω ≅ 𝔹_n$, the unit ball in $𝔹^n$, see [@MR1834454 Sect. X, §6.3/4]. Consequently, the group $\mathop{\mathrm{Aut}}(Ω)$ is connected, and we may apply [@MR0385004 Thm. A' on p. 4] to obtain an automorphism of real Lie groups, $Θ : \mathop{\mathrm{Aut}}(Ω) → \mathop{\mathrm{Aut}}(Ω)$ such that $Θ|_{Γ_M} = θ$. The desired isometry is then produced by an application of [@MR1441541 Prop. 3.9.11]. We consider the case $\mathop{\mathrm{rank}}(Ω) ≥ 2$ for the remainder of the present proof, where the automorphism group may be non-connected. To deal with this slight difficulty, we proceed as in [@MR1441541 p. 379f]: as $\mathop{\mathrm{Aut}}(Ω)$ has finitely many connected components, we may assume that the subgroups $Γ_{\widehat{M}} ⊆ Γ_M$ and $Γ_{\widehat{X}} ⊆ Γ_X$ corresponding to the deck transformation groups of $u_M$ and $u_X$, respectively, are contained in the identity component $I°(Ω) = \mathop{\mathrm{Aut}}°(Ω)$. Again, apply [@MR0385004 Thm. A' on p. 4] to obtain an automorphism of real Lie groups $Θ : I°(Ω) → I°(Ω)$ such that $Θ|_{Γ_{\widehat{M}}} = θ|_{Γ_{\widehat{M}}}$ and then [@MR1441541 Prop. 3.9.11] to obtain an isometry $F ∈ I(Ω)$ such that $$F ◦ g = Θ(g) ◦ F, \quad \text{for every } g ∈ I°(Ω).$$ This in particular yields [\[eq:2-11-1\]](#eq:2-11-1){reference-type="eqref" reference="eq:2-11-1"} for all $γ$ contained in the finite index subgroup $Γ_{\widehat{M}}$ of $Γ_M$. This is not yet enough. However, noticing that for any finite index subgroup $Γ'_M < Γ_M$, every $Γ'_M$-periodic vector in the sense of [@MR1441541 Def. 4.5.13] by definition is also $Γ_M$-periodic, we see with the argument given in [@MR1441541 p. 379], which uses essentially the same notation as we have introduced here, that the set of $Γ_M$-periodic vectors is dense in the unit sphere bundle $SΩ$ of $Ω$. The subsequent argument in [@MR1441541 bottom of p. 379 and upper part of p. 380] then applies verbatim to yield the desired relation [\[eq:2-11-1\]](#eq:2-11-1){reference-type="eqref" reference="eq:2-11-1"} for all $γ ∈ Γ_M$; this is [@MR1441541 equation (5) on p. 380].  (Claim [Claim 15](#claim:2-11){reference-type="ref" reference="claim:2-11"}) ◻ Now, since the Hermitian symmetric domain $Ω$ is assumed to be irreducible, the $Γ$-equivariant isometry $F ∈ I(Ω)$ is either holomorphic or conjugate-holomorphic, as follows for example from [@363432] together with [@MR1834454 VIII.Prop. 4.2]. By the universal property of the quotient map $π$ with respect to $Γ$-invariant holomorphic maps, $F$ hence descends to a holomorphic or conjugate-holomorphic isomorphism from $M$ to $X$. This completes the proof of Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"}. 0◻ # Topological characterization of torus quotients {#sec:3} In line with the results of Section [2](#sec:2){reference-type="ref" reference="sec:2"}, we show that a Kähler space with klt singularities is a singular torus quotient if and only if it is homeomorphic to a singular torus quotient. In the smooth case, this was shown by Catanese [@MR1945361], but see also [@MR2295551 Thm. 2.2]. The following notion is a direct analogue of Definition [Definition 8](#def:2-1){reference-type="ref" reference="def:2-1"} above. **Definition 16** (Singular torus quotient). *A normal complex space $X$ is called a *singular torus quotient* if there exists a quasi-étale cover $\widehat{X} → X$, where $\widehat{X}$ is a compact complex torus.* *Remark 17* (Singular torus quotients are quotients). Let $X$ be a singular torus quotient. Passing to a suitable Galois closure, one finds a quasi-étale *Galois* cover $\widehat{X} → X$, where $\widehat{X}$ is a compact torus. **Theorem 18** (Varieties homeomorphic to torus quotients). *Let $M$ be a compact complex space with klt singularities. Assume that $M$ is bimeromorphic to a Kähler manifold. If $M$ is homeomorphic to a singular torus quotient, then $M$ is a singular torus quotient.* Theorem [Theorem 18](#thm:3-3){reference-type="ref" reference="thm:3-3"} will be shown in Sections [3.1](#sec:3-1){reference-type="ref" reference="sec:3-1"}--[3.2](#sec:3-2){reference-type="ref" reference="sec:3-2"} below. In analogy to Corollary [Corollary 12](#cor:2-5){reference-type="ref" reference="cor:2-5"} above, we note that vanishing of $ℚ$-Chern classes is a topological property among compact Kähler spaces with klt singularities. **Corollary 19** (Topological invariance of vanishing Chern classes). *Let $X$ be a compact Kähler space with klt singularities. Assume that the canonical class vanishes numerically, $K_X \equiv 0$, and that the second $ℚ$-Chern class of $𝒯_X$ satisfies $$\widehat{c}_2 (𝒯_X)· α_1 ⋯ α_{\dim X-2}= 0,$$ for every $(\dim X-2)$-tuple of Kähler classes on $X$. If $M$ is any compact Kähler space with klt singularities that is homeomorphic to $X$, then $K_M \equiv 0$, and the second $ℚ$-Chern class of $𝒯_M$ satisfies $$\widehat{c}_2 (𝒯_M)· β_1 ⋯ β_{\dim M-2} = 0,$$ for every $(\dim X-2)$-tuple of Kähler classes on $M$.* *Proof.* The characterization of singular torus quotients in terms of Chern classes by Claudon, Graf and Guenancia, [@CGG23 Cor. 1.7], guarantees that $X$ is a torus quotient[^7]. By Theorem [Theorem 18](#thm:3-3){reference-type="ref" reference="thm:3-3"}, then so is $M$. ◻ ## Proof of Theorem [Theorem 18](#thm:3-3){reference-type="ref" reference="thm:3-3"} if $M$ is homeomorphic to a torus {#sec:3-1} As before, we prove Theorem [Theorem 18](#thm:3-3){reference-type="ref" reference="thm:3-3"} first in case where the (potentially singular) space $M$ is homeomorphic to a torus. Recalling that klt singularities are rational, see [@KM98 Thm. 5.22] for the algebraic case and [@FujinoMMP Thm. 3.12] (together with the vanishing theorems proven in [@FujinoVanishing]) for the analytic case, we show the following, slightly stronger statement. **Proposition 20**. *Let $M$ be a compact complex space with rational singularities. Assume that $M$ is bimeromorphic to a Kähler manifold. If $M$ is homotopy equivalent to a compact torus, then $M$ is a compact torus.* *Proof.* We follow the arguments of Catanese, [@MR1945361 Thm. 4.8], and choose a resolution of singularities, $π : \widetilde{M} → M$, which owing to the assumptions on $M$ we may assume to be a compact Kähler manifold. Using the assumption that $M$ has rational singularities together with the push-forward of the exponential sequence, we observe that the pull-back map $H¹\bigl( M,\, ℤ \bigr) → H¹\bigl( \widetilde M,\, ℤ \bigr)$ is an isomorphism. In particular, first Betti numbers of $M$ and $\widetilde{M}$ agree. As a next step, consider the Albanese map of $\widetilde{M}$, observing that $\widetilde{M}$ is bimeromorphic to a Kähler manifold since $M$ is. Again using that $M$ has rational singularities, recall from [@Reid83a Prop. 2.3] that the Albanese factors via $M$, $$\begin{tikzcd}[row sep=1cm] \widetilde{M} \ar[r, "\mathop{\mathrm{alb}}"] \ar[d, two heads, "π\text{, resolution}"'] & \mathop{\mathrm{Alb}}. \\ M \ar[ur, bend right=10, "α"'] \end{tikzcd}$$ Since the pull-back morphisms $$\begin{matrix} \mathop{\mathrm{alb}}^* &=& π^* ◦ α^* &:& H¹\bigl( \mathop{\mathrm{Alb}},\, ℤ \bigr) &→& H¹\bigl( \widetilde{M},\, ℤ \bigr) \\ && π^* &:& H¹\bigl( M,\, ℤ \bigr) &→& H¹\bigl( \widetilde{M},\, ℤ \bigr) \end{matrix}$$ are both isomorphic, we find that $α^*: H¹\bigl( \mathop{\mathrm{Alb}},\, ℤ \bigr) → H¹\bigl( M,\, ℤ \bigr)$ must likewise be an isomorphism. There is more that we can say. Since the topological cohomology ring of a torus is an exterior algebra, $$H^*\bigl( \mathop{\mathrm{Alb}},\, ℤ\bigr) = Λ^* H¹\bigl( \mathop{\mathrm{Alb}},\, ℤ\bigr) \quad\text{and}\quad H^*\bigl(M,\, ℤ \bigr) = Λ^* H¹\bigl( M,\, ℤ\bigr),$$ we find that all pull-back morphisms are isomorphisms, $$α^*: H^q\bigl( \mathop{\mathrm{Alb}},\, ℤ \bigr) \xrightarrow{≅} H^q\bigl( M,\, ℤ \bigr), \quad \text{for every } 0 ≤ q ≤ 2·\dim M.$$ Applying this to $q = 2·\dim M$, we see $α$ is surjective of degree one, hence birational. Again, more is true: if $α$ failed to be isomorphic, Zariski's Main Theorem would guarantee that $α$ contracts a positive-dimensional subvariety, so $b_2(M) > b_2(\mathop{\mathrm{Alb}})$. But we have seen above that equality holds and hence reached a contradiction. ◻ ## Proof of Theorem [Theorem 18](#thm:3-3){reference-type="ref" reference="thm:3-3"} in general {#sec:3-2} By assumption, there exists a homeomorphism $f : M → X$, where $X$ is a singular torus quotient. Choose a quasi-étale cover $τ_X:~\widehat{X} → X$, where $\widehat{X}$ is a complex torus, and proceed as in the proof of Theorem [Theorem 10](#thm:2-3){reference-type="ref" reference="thm:2-3"}, in order to construct a diagram of continuous mappings between normal complex spaces, $$\begin{tikzcd}[column sep=2.2cm] \widehat{M} \ar[r, two heads, "τ_M\text{, quasi-étale}"] \ar[d, "≅"'] & M \ar[d, "f", "≅"'] \\ \widehat{X} \ar[r, two heads, "τ_X\text{, quasi-étale}"'] & X, \end{tikzcd}$$ where - the vertical maps are homeomorphisms, and - the horizontal maps are holomorphic, surjective, and finite. Since $M$ is bimeromorphic to a Kähler manifold, so is $\widehat{M}$. Recalling from [@KM98 Prop. 5.20] that also $\widehat{M}$ has no worse than klt singularities, Proposition [Proposition 20](#prop:3-5){reference-type="ref" reference="prop:3-5"} will then guarantee that $\widehat{M}$ is a complex torus, as claimed. 0◻ # Rigidity results for projective spaces {#sec:4} Recall the classical theorem of Hirzebruch-Kodaira, which asserts that the projective space carries a unique structure as a Kähler manifold. **Theorem 21** (). *Let $X$ be a compact Kähler manifold. If $X$ is homeomorphic to $ℙ^n$, then $X$ is biholomorphic to $ℙ^n$. 0◻* *Remark 22*. Strictly speaking, Hirzebruch-Kodaira proved a somewhat weaker result: $X$ is biholomorphic to $ℙ^n$ if either $n$ is odd, or if $n$ is even and $c_1(X) ≠ -(n+1)·g$, where $g$ is a generator of $H²\bigl(X,\, ℤ\bigr)$ and the fundamental class of a Kähler metric on $X$. The second case was later ruled out by Yau's solution to the Calabi conjecture, which implies that then the universal cover of $X$ is the ball, contradicting $π_1(X) = 0$. Since the topological invariance of the Pontrjagin classes, [@Nov65], was not known at that time, Hirzebruch-Kodaira also had to assume that $X$ is diffeomorphic to $ℙ^n$ rather than merely homeomorphic. We ask whether an analogue of Hirzebruch-Kodaira's theorem remains true in the context of minimal model theory. *Question 23*. Let $X$ be a projective variety with klt singularities. Assume that $X$ is homeomorphic to $ℙ^n$. Is $X$ then biholomorphic to $ℙ^n$? ## Varieties homeomorphic to projective space We do not have a full answer to Question [Question 23](#q:4-3){reference-type="ref" reference="q:4-3"}. The following proposition will, however, restrict the geometry of potential varieties substantially. It will later be used to answer Question [Question 23](#q:4-3){reference-type="ref" reference="q:4-3"} in a number of special settings. **Proposition 24** (Varieties homeomorphic to $ℙ^n$). *Let $X$ be a projective klt variety. If $X$ is homeomorphic to $ℙ^n$, then the following holds.* 1. *[\[il:4-4-1\]]{#il:4-4-1 label="il:4-4-1"} We have $H^q \bigl(X,\, 𝒪_X \bigr) = 0$ for every $1 ≤ q$.* 2. *[\[il:4-4-2\]]{#il:4-4-2 label="il:4-4-2"} The Chern class map $c_1 : \mathop{\mathrm{Pic}}(X) → H²(X,ℤ) ≅ ℤ$ is an isomorphism.* 3. *[\[il:4-4-3\]]{#il:4-4-3 label="il:4-4-3"} The variety $X$ is smooth in codimension two.* 4. *[\[il:4-4-4\]]{#il:4-4-4 label="il:4-4-4"} The maps $r_q : H^q\bigl( X,\, ℤ \bigr) → H^q\bigl(X_{\mathop{\mathrm{reg}}},\, ℤ\bigr)$ are isomorphic, for every $0 ≤ q ≤ 4$. The same statement holds for $ℤ_2$ coefficients.* 5. *[\[il:4-4-5\]]{#il:4-4-5 label="il:4-4-5"} Every Weil divisor on $X$ is Cartier, i.e., $X$ is factorial. In particular, $X$ is Gorenstein.* 6. *[\[il:4-4-6\]]{#il:4-4-6 label="il:4-4-6"} The canonical divisor $K_X$ is ample or anti-ample.* *Proof.* We prove the items of Proposition [Proposition 24](#prop:4-4){reference-type="ref" reference="prop:4-4"} separately. ### Item [\[il:4-4-1\]](#il:4-4-1){reference-type="ref" reference="il:4-4-1"} {#item-il4-4-1 .unnumbered} This is a consequence of the rationality of the singularities of $X$ and the isomorphisms $H^q\bigl(X,\,ℂ \bigr) \simeq H^q\bigl(ℙ^n,\, ℂ\bigr)$. In fact, since $X$ has rational singularities, the morphisms $$\varphi_q: H^q\bigl(X,\, ℂ\bigr) → H^q\bigl(X,\, 𝒪_X\bigr)$$ induced by the canonical inclusion $ℂ → 𝒪_X$, are surjective, [@MR1341589 Thm. 12.3]. If $q$ is odd, this already implies that $H^q(X,𝒪_X) = 0$. If $q$ is even, it suffices to note that $\varphi_q$ has a non-trivial kernel. For this, choose an ample line bundle $ℒ ∈ \mathop{\mathrm{Pic}}(X)$ and observe that $$\varphi_q\bigl(c_1(ℒ)^{q/2}\bigr) = 0 ∈ H^q\bigl(X,\, 𝒪_X\bigr).$$ To prove the observation, pass to a desingularisation and use the Hodge decomposition there. ### Item [\[il:4-4-2\]](#il:4-4-2){reference-type="ref" reference="il:4-4-2"} {#item-il4-4-2 .unnumbered} The description of $c_1$ follows from [\[il:4-4-1\]](#il:4-4-1){reference-type="ref" reference="il:4-4-1"} and the exponential sequence. ### Item [\[il:4-4-3\]](#il:4-4-3){reference-type="ref" reference="il:4-4-3"} {#item-il4-4-3 .unnumbered} Recall that klt varieties have quotient singularities in codimension two, [@GKKP11 Prop. 9.3]. Smoothness follows because quotient singularities have non-trivial local fundamental groups and are hence not topologically smooth. ### Item [\[il:4-4-4\]](#il:4-4-4){reference-type="ref" reference="il:4-4-4"} {#item-il4-4-4 .unnumbered} We describe the relevant cohomology groups in terms of Borel-Moore homology, [@MR131271] and also refer to the reader to [@Fulton98 Sect. 19.1] for a summary of the relevant facts (over $ℤ$). The assumption that $X$ is homeomorphic to an oriented, connected, real manifold implies that singular cohomology and Borel-Moore homology agree, [@MR131271 Thm. 7.6] and [@Fulton98 p. 371]. The same holds for the non-compact manifold $X_{\mathop{\mathrm{reg}}}$, i.e., for $R = ℤ, ℤ_2$ we have $$H^q\bigl( X,\, R \bigr) = H^{BM}_{2·n-q}\bigl( X,\, R \bigr) % \quad\text{and}\quad % H^q\bigl( X_{\mathop{\mathrm{reg}}},\, R \bigr) = H^{BM}_{2·n-q}\bigl( X_{\mathop{\mathrm{reg}}},\, R \bigr), % \quad\text{for every }q.$$ The isomorphisms identify the restriction maps $r_q$ with the pull-back maps for Borel-Moore homology. These feature in the localization sequence for Borel-Moore homology, [@MR131271 Thm.3.8], $$⋯ → H^{BM}_{2·n-q}\bigl( X_{\mathop{\mathrm{sing}}},\, R \bigr) → H^{BM}_{2·n-q}\bigl( X,\, R \bigr) \xrightarrow{r_q} H^{BM}_{2·n-q}\bigl( X_{\mathop{\mathrm{reg}}},\, R \bigr) → H^{BM}_{2·n-q-1}\bigl( X_{\mathop{\mathrm{sing}}},\, R \bigr) → ⋯$$ Recalling from [@Fulton98 Lem. 19.1.1] that $H^{BM}_i\bigl( X_{\mathop{\mathrm{sing}}},\, ℤ \bigr) = 0$ for every $i > 2·\dim_{ℂ} X_{\mathop{\mathrm{sing}}}$ and noticing that the inductive argument employed in the proof also works for $ℤ_2$-coefficients, the claim of Item [\[il:4-4-4\]](#il:4-4-4){reference-type="ref" reference="il:4-4-4"} thus follows from smoothness in codimension two, Item [\[il:4-4-3\]](#il:4-4-3){reference-type="ref" reference="il:4-4-3"}. ### Item [\[il:4-4-5\]](#il:4-4-5){reference-type="ref" reference="il:4-4-5"} {#item-il4-4-5 .unnumbered} Remaining in the analytic category, writing down the exponential sequences for $X$ and $X_{\mathop{\mathrm{reg}}}$, $$\begin{tikzcd}[column sep=0.35cm] H¹\bigl(X,\, ℤ \bigr) \ar[d, "r_1"] \ar[r] & H¹\bigl(X,\, 𝒪_X \bigr) \ar[d] \ar[r] & \mathop{\mathrm{Pic}}(X) \ar[d, hook] \ar[r, "c_1"] & H²\bigl(X,\, ℤ \bigr) \ar[d, "r_2"] \ar[r] & H²\bigl(X,\, 𝒪_X \bigr) \ar[d] \\ H¹\bigl(X_{\mathop{\mathrm{reg}}},\, ℤ \bigr) \ar[r] & H¹\bigl(X_{\mathop{\mathrm{reg}}},\, 𝒪_{X_{\mathop{\mathrm{reg}}}} \bigr) \ar[r] & \mathop{\mathrm{Pic}}\bigl(X_{\mathop{\mathrm{reg}}}\bigr) \ar[r, "c_1"'] & H²\bigl(X_{\mathop{\mathrm{reg}}},\, ℤ \bigr) \ar[r] & H²\bigl(X_{\mathop{\mathrm{reg}}},\, 𝒪_{X_{\mathop{\mathrm{reg}}}}\bigr), \end{tikzcd}$$ and filling in what we already know, we find a commutative diagram with exact rows, as follows, $$\begin{tikzcd}[column sep=1cm, row sep=1cm] 0 \ar[r] & 0 \ar[d] \ar[r] & \mathop{\mathrm{Pic}}(X) \ar[d, hook] \ar[r, hook, two heads, "c_1\text{, iso.}"] & H²\bigl(X,\, ℤ \bigr) \ar[d, hook, two heads, "r_2\text{, iso.}"] \ar[r] & 0 \\ 0 \ar[r] & H¹\bigl( X_{\mathop{\mathrm{reg}}},\, 𝒪_{X_{\mathop{\mathrm{reg}}}} \bigr) \ar[r] & \mathop{\mathrm{Pic}}(X_{\mathop{\mathrm{reg}}}) \ar[r, two heads, "c_1"'] & H²\bigl( X_{\mathop{\mathrm{reg}}},\, ℤ \bigr) \ar[r] & 0. \end{tikzcd}$$ The snake lemma now asserts that $$\label{eq:4-4-7} H¹ \bigl(X_{\mathop{\mathrm{reg}}},\, 𝒪_{X_{\mathop{\mathrm{reg}}}} \bigr) ≅ \left. \raise 2pt\hbox{$\mathop{\mathrm{Pic}}(X_{\mathop{\mathrm{reg}}})$} \right/\hskip -2pt\raise -2pt\hbox{$\mathop{\mathrm{Pic}}(X)$}.$$ We claim that $H¹ \bigl(X_{\mathop{\mathrm{reg}}},\, 𝒪_{X_{\mathop{\mathrm{reg}}}} \bigr)$ vanishes. For this, recall that the singularities of $X$ are rational, so every local ring $𝒪_{X,x}$ of the (holomorphic) structure sheaf has depth equal to $n$. Since the singular set of $X$ has codimension at least $3$ in $X$ by Item [\[il:4-4-3\]](#il:4-4-3){reference-type="ref" reference="il:4-4-3"}, we may apply [@MR0148941 Sec. 5, Korollar after Satz III] or alternatively [@BS76 Chap. II, Cor. 3.9 and Thm. 3.6] to see that the restriction homomorphism $$H¹ \bigl(X,\, 𝒪_X \bigr) → H¹ \bigl(X_{\mathop{\mathrm{reg}}},\, 𝒪_{X_{\mathop{\mathrm{reg}}}} \bigr)$$ is bijective. However, the cohomology group on the left side was shown to vanish in Item [\[il:4-4-1\]](#il:4-4-1){reference-type="ref" reference="il:4-4-1"} above. In summary, we find that every invertible sheaf on $X_{\mathop{\mathrm{reg}}}$ extends to an invertible sheaf on $X$. If $D ∈ \mathop{\mathrm{Div}}(X)$ is any Weil divisor, the invertible sheaf $𝒪_{X_{\mathop{\mathrm{reg}}}}(D)$ will therefore extend to an invertible sheaf on $X$, which necessarily equals the (reflexive) Weil divisorial sheaf $𝒪_X(D)$. It follows that $D$ is Cartier. This applies in particular to the canonical divisor, so $X$ is $ℚ$-Gorenstein of index one. Since $X$ is Cohen-Macaulay, we conclude that $X$ is Gorenstein. ### Item [\[il:4-4-6\]](#il:4-4-6){reference-type="ref" reference="il:4-4-6"} {#item-il4-4-6 .unnumbered} Given that $\mathop{\mathrm{Pic}}(X) = ℤ$, every line bundle is ample, anti-ample, or trivial; we need to exclude the case that $K_X$ is trivial. But if $K_X$ were trivial, use that $X$ is Gorenstein and apply Serre duality to find $$h^n\bigl(X,\, 𝒪_X\bigr) = h⁰\bigl(X,\, ω_X \bigr) = h⁰\bigl(X,\, 𝒪_X \bigr) = 1.$$ This contradicts Item [\[il:4-4-1\]](#il:4-4-1){reference-type="ref" reference="il:4-4-1"} above. ◻ *Notation 25* (Line bundles on varieties homeomorphic to $ℙ^n$). If $X$ is a projective klt variety that is homeomorphic to $ℙ^n$, Item [\[il:4-4-2\]](#il:4-4-2){reference-type="ref" reference="il:4-4-2"} shows the existence of a unique ample line bundle that generates $\mathop{\mathrm{Pic}}(X) ≅ ℤ$. We refer to this line bundle as $𝒪_X(1)$. Items [\[il:4-4-5\]](#il:4-4-5){reference-type="ref" reference="il:4-4-5"} equips us with a unique number $r ∈ ℕ$ and such that $ω_X ≅ 𝒪_X(r)$. Item [\[il:4-4-6\]](#il:4-4-6){reference-type="ref" reference="il:4-4-6"} guarantees that $r ≠ 0$. *Remark 26* (Pull-back of line bundles). The cohomology rings of $X$ and $ℙ^n$ are isomorphic. If $φ : X → ℙ^n$ is any homeomorphism, then $φ^* c_1 \bigl( 𝒪_{ℙ^n}(1) \bigr) = c_1 \bigl( 𝒪_X(± 1) \bigr)$. The cup products $c_1 \bigl( 𝒪_X(1) \bigr)^q$ generate the groups $H^{2q}\bigl( X,\, ℤ\bigr) ≅ ℤ$. ## Characteristic classes {#sec:4-2} We have seen in Proposition [Proposition 24](#prop:4-4){reference-type="ref" reference="prop:4-4"} that $X$ is smooth away from a closed set of codimension $≥ 3$. This allows defining a number of characteristic classes. *Notation 27* (Chern classes on varieties homeomorphic to $ℙ^n$). If $X$ is a projective klt variety that is homeomorphic to $ℙ^n$, Item [\[il:4-4-4\]](#il:4-4-4){reference-type="ref" reference="il:4-4-4"} allows defining first and second Chern classes, as well as a first Pontrjagin class and a second Stiefel-Whitney class $$\begin{aligned} c_1(X) & = r_2^{-1} c_1(X_{\mathop{\mathrm{reg}}}) ∈ H²\bigl( X,\, ℤ\bigr) \\ c_2(X) & = r_4^{-1} c_2(X_{\mathop{\mathrm{reg}}}) ∈ H⁴\bigl( X,\, ℤ\bigr) \\ p_1(X) & = r_4^{-1} p_1(X_{\mathop{\mathrm{reg}}}) ∈ H⁴\bigl( X,\, ℤ\bigr)\\ w_2(X) & = r_2^{-1} w_2(X_{\mathop{\mathrm{reg}}}) ∈ H²\bigl( X,\, ℤ_2\bigr). \end{aligned}$$ *Remark 28* (Pontrjagin and Chern classes). If $X$ be a projective klt variety that is homeomorphic to $ℙ^n$, the restriction maps $r_• : H^•\bigl( X,\, ℤ \bigr) → H^•\bigl(X_{\mathop{\mathrm{reg}}},\, ℤ \bigr)$ commute with the cup products on $X$ and $X_{\mathop{\mathrm{reg}}}$, which implies in particular that $$p_1(X) = r_4^{-1} p_1(X_{\mathop{\mathrm{reg}}}) % = r_4^{-1} \Bigl( c_1(X_{\mathop{\mathrm{reg}}})²-2·c_2(X_{\mathop{\mathrm{reg}}}) \Bigr) % = c_1(X)²-2·c_2(X) % ∈ H⁴\bigl( X,\, ℤ\bigr).$$ *Remark 29* (Stiefel-Whitney class and first Chern class). By definition and the well-known relation in the smooth case, we have $$w_2(X) = c_1(X) \ {\rm mod} 2.$$ Novikov's result on the topological invariance of Pontrjagin classes extends to the generalized Pontrjagin class defined in Notation [Notation 27](#not:4-7){reference-type="ref" reference="not:4-7"}. **Proposition 30** (Topological invariance of Pontrjagin classes). *Let $X$ be a projective klt variety. If $φ : X → ℙ^n$ is any homeomorphism, then $φ^* p_1(ℙ^n) = p_1(X)$ in $H⁴\bigl( X,\, ℤ\bigr)$.* *Proof.* Consider the open set $ℙ^n_{\mathop{\mathrm{reg}}} := φ(X_{\mathop{\mathrm{reg}}})$ and the restricted homeomorphism $φ_{\mathop{\mathrm{reg}}} : X_{\mathop{\mathrm{reg}}} → ℙ^n_{\mathop{\mathrm{reg}}}$. Recalling from Item [\[il:4-4-4\]](#il:4-4-4){reference-type="ref" reference="il:4-4-4"} of Propositions [Proposition 24](#prop:4-4){reference-type="ref" reference="prop:4-4"} that the restriction maps $$r_4 : H⁴\bigl( X,\, ℤ \bigr) → H⁴\bigl(X_{\mathop{\mathrm{reg}}},\, ℤ \bigr)% \quad\text{and}\quad r_4 : H⁴\bigl( ℙ^n,\, ℤ \bigr) → H⁴\bigl(ℙ^n_{\mathop{\mathrm{reg}}},\, ℤ \bigr)%$$ are isomorphic, it suffices to show that the restricted classes in rational cohomology agree. More precisely, $$\begin{aligned} && φ^* p_1(ℙ^n) & = p_1(X) && \text{in } H⁴\bigl( X,\, ℤ \bigr) \\ ⇔ && r_4 φ^* p_1(ℙ^n) & = r_4 p_1(X) && \text{in } H⁴\bigl( X_{\mathop{\mathrm{reg}}},\, ℤ \bigr) \text{, since $r_4$'s are iso.} \\ ⇔ && φ^*_{\mathop{\mathrm{reg}}} p_1(ℙ^n_{\mathop{\mathrm{reg}}}) & = p_1(X_{\mathop{\mathrm{reg}}}) && \text{in } H⁴\bigl( X_{\mathop{\mathrm{reg}}},\, ℤ \bigr) \text{, definition, functoriality} \\ ⇔ && φ^*_{\mathop{\mathrm{reg}}} p_1(ℙ^n_{\mathop{\mathrm{reg}}}) & = p_1(X_{\mathop{\mathrm{reg}}}) && \text{in } H⁴\bigl( X_{\mathop{\mathrm{reg}}},\, ℚ \bigr) \text{, since } H⁴\bigl( X_{\mathop{\mathrm{reg}}},\, ℤ \bigr) = ℤ \end{aligned}$$ The last equation is Novikov's famous topological invariance of Pontrjagin classes, [@Nov65][^8]. ◻ **Corollary 31** (Relation between Chern classes on varieties homeomorphic to $ℙ^n$). *If $X$ is a projective klt variety that is homeomorphic to $ℙ^n$, then $$2·c_2(X) = \bigl[r²-(n+1)\bigr]·c_1\bigl(𝒪_X(1)\bigr)² \quad \text{in } H⁴\bigl( X,\, ℤ \bigr).$$* *Proof.* Choose a homeomorphism $φ : X → ℙ^n$, in order to compare the Pontrjagin class of $ℙ^n$ with that of $X$. $$\begin{aligned} && p_1(ℙ^n) & = (n+1)·c_1\bigl(𝒪_{ℙ^n}(1)\bigr)² && \text{in } H⁴\bigl( ℙ^n,\, ℤ \bigr) \\ ⇔ && φ^* p_1(ℙ^n) & = (n+1)·φ^*c_1\bigl(𝒪_{ℙ^n}(1)\bigr)² && \text{in } H⁴\bigl( X,\, ℤ \bigr) \\ ⇔ && p_1(X) & = (n+1)·c_1\bigl(𝒪_X(± 1)\bigr)² && \text{Prop.~\ref{prop:4-10} and Rem.~\ref{rem:4-6}} \\ ⇔ && c_1\bigl( 𝒪_X(r)\bigr)² - 2·c_2(X) & = (n+1)·c_1\bigl(𝒪_X(1)\bigr)² && \text{Rem.~\ref{rem:4-8}} \end{aligned}$$ The claim thus follows. ◻ Corollary [Corollary 31](#cor:4-11){reference-type="ref" reference="cor:4-11"} allows reformulating the $ℚ$-Miyaoka-Yau inequality and $ℚ$-Bogomolov-Gieseker inequality as inequalities between the index $r$ and the dimension $n$. The first remark will be relevant for varieties of general type, whereas the second one will be used for Fano varieties. *Remark 32* (Reformulation of the $ℚ$-Miyaoka-Yau inequality). Let $X$ be a projective klt variety that is homeomorphic to $ℙ^n$. Since $X$ is smooth in codimension two, the Miyaoka-Yau inequality for $ℚ$-Chern classes, $$\big( 2(n+1)·\widehat{c}_2(X) - n·\widehat{c}_1(X)² \big)·[H]^{n-2} ≥ 0, \quad\text{for one ample }H,$$ is equivalent to the assertion that there exists a non-negative constant $c ∈ ℝ^{≥ 0}$ such that $$\begin{aligned} && \big( 2(n+1)·c_2(X) - n·c_1(X)² \big) & ≥ c·c_1\bigl( 𝒪_X(1) \bigr)² && \text{in } H⁴\bigl( X,\, ℤ\bigr) \\ ⇔ && \big( (n+1)(r²-(n+1)) - n·r² \big)·c_1\bigl( 𝒪_X(1) \bigr)² & ≥ c·c_1\bigl( 𝒪_X(1) \bigr)² && \text{Cor.~\ref{cor:4-11}} \\ ⇔ && \big( r²-(n+1)² \big)·c_1\bigl( 𝒪_X(1) \bigr)² & ≥ c·c_1\bigl( 𝒪_X(1) \bigr)² \\ ⇔ && |r| & ≥ n+1. \end{aligned}$$ The Miyaoka-Yau inequality is an equality if and only if $|r| = n+1$. *Remark 33* (Reformulation of the $ℚ$-Bogomolov-Gieseker inequality). Let $X$ be a projective klt variety that is homeomorphic to $ℙ^n$. Since $X$ is smooth in codimension two, the Bogomolov-Gieseker inequality for $ℚ$-Chern classes, $$\big( 2n·\widehat{c}_2(X) - (n-1)·\widehat{c}_1(X)² \big)·[H]^{n-2} ≥ 0, \quad\text{for one (equiv.~every) ample }H,$$ is equivalent to the assertion that $|r| > n$. We will also need the topological invariance of the second Stiefel-Whitney class $w_2$. **Proposition 34** (Topological invariance of the second Stiefel-Whitney class). *Let $X$ be a projective klt variety. If $φ : X → ℙ^n$ is any homeomorphism, then $φ^* w_2(ℙ^n) = w_2(X)$ in $H²\bigl( X,\, ℤ/2ℤ\bigr)$.* *Proof.* We can argue as in the proof of Proposition [Proposition 30](#prop:4-10){reference-type="ref" reference="prop:4-10"}, replacing Novikov's Theorem by the corresponding invariance result for Stiefel-Whitney classes due to Thom, [@Th52 Thm. III.8]. ◻ **Corollary 35** (Parity of the first Chern class of varieties homeomorphic to $ℙ^n$). *If $X$ is a projective klt variety that is homeomorphic to $ℙ^n$, then $r - (n+1)$ is even.* *Proof.* This follows from the topological invariance established just above together with Remark [Remark 29](#rem:4-9){reference-type="ref" reference="rem:4-9"} and the relation $\varphi^*(c_1(𝒪_{ℙ^n}(1))) = c_1(𝒪_X(± 1))$. ◻ ## Partial answers to Question [Question 23](#q:4-3){reference-type="ref" reference="q:4-3"} {#partial-answers-to-question-q4-3} We conclude the present Section [4](#sec:4){reference-type="ref" reference="sec:4"} with three partial answers to Question [Question 23](#q:4-3){reference-type="ref" reference="q:4-3"}: for threefolds, we answer Question [Question 23](#q:4-3){reference-type="ref" reference="q:4-3"} in the affirmative. In dimension four and five, we give an affirmative answer for Fano manifolds. In higher dimensions, we can at least describe and restrict the geometry of potential exotic klt varieties homeomorphic to $ℙ^n$. **Proposition 36** (Topological $ℙ^n$ with ample canonical bundle). *Let $X$ be a projective klt variety that is homeomorphic to $ℙ^n$. If $K_X$ is ample, then $r > n+1$.* *Proof.* Recall from [@GKPT19b Thm. 1.1] that $X$ satisfies the $ℚ$-Miyaoka-Yau inequality. We have seen in Remark [\[rem:4-12\]](#rem:4-12){reference-type="ref+page" reference="rem:4-12"} that this implies $r = |r| ≥ n+1$, with $r = n+1$ if and only if equality holds in $ℚ$-Miyaoka-Yau inequality. In the latter case, recall from [@GKPT19b Thm. 1.2] that $X$ has no worse than quotient singularities. Since quotient singularities are not topologically smooth, it turns out that $X$ cannot have any singularities at all. By Yau's theorem (or again by [@GKPT19b Thm. 1.2]), $X$ must then be a smooth ball quotient, contradicting $π_1(X) = π_1(ℙ_n) = \{1\}$. ◻ **Proposition 37** (Topological $ℙ^n$ with ample anti-canonical bundle). *Let $X$ be a projective klt variety that is homeomorphic to $ℙ^n$. If $-K_X$ is ample, then either $X ≅ ℙ^n$ or $𝒯_X$ is unstable.* *Remark 38*. Recall from [@KST07 Cor. 32] that Fano varieties with unstable tangent bundles admit natural sequences of rationally connected foliations. These might be used to study their geometry further. If in the situation of Proposition [Proposition 37](#prop:4-17){reference-type="ref" reference="prop:4-17"} we additionally assume that the index is one, i.e., that $r = -1$, then $Ω^{[1]}_X$ is always semistable: if $𝒮 ⊊ Ω^{[1]}_X$ was destabilizing, then $\det 𝒮 ⊆ Ω_X^{[\mathop{\mathrm{rank}}𝒮]}$ is either trivial (hence violating the non-existence of reflexive forms, [@Zh06 Thm. 1] and [@GKKP11 Thm. 5.1]) or ample (hence violating the Bogomolov-Sommese vanishing theorem for klt varieties, [@GKKP11 Thm. 7.2]). *Proof of Proposition [Proposition 37](#prop:4-17){reference-type="ref" reference="prop:4-17"}.* If $𝒯_X$ is semistable, then the $ℚ$-Bogomolov-Gieseker inequality holds, and we have seen in Remark [Remark 33](#rem:4-13){reference-type="ref" reference="rem:4-13"} that $-r = |r| > n$. Fujita's singular version of the Kobayashi-Ochiai theorem, [@MR946238 Thm. 1], will then apply to show that $X ≅ ℙ^n$. ◻ While the Bogomolov-Gieseker inequality does not necessarily hold on a Fano variety with unstable tangent sheaf, we still get some restriction on the index from the following result. **Proposition 39**. *Let $X$ be a projective klt variety that is homeomorphic to $ℙ^n$. If $-K_X$ is ample, then $r² ≥ n+1$. In particular, if $n ≥ 4$, then $r ≥ 3$.* *Proof.* Since $X$ is factorial by [\[il:4-4-5\]](#il:4-4-5){reference-type="ref" reference="il:4-4-5"} and non-singular in codimension two by [\[il:4-4-3\]](#il:4-4-3){reference-type="ref" reference="il:4-4-3"}, we may apply [@Ou Cor. 1.5] to obtain the bound $c_2(X)·c_1(𝒪_X(1))^{n-2} ≥ 0$. Then, we conclude by Corollary [Corollary 31](#cor:4-11){reference-type="ref" reference="cor:4-11"}. ◻ In dimension three we can now fully answer Question [Question 23](#q:4-3){reference-type="ref" reference="q:4-3"}. **Theorem 40** (Topological $ℙ³$). *Let $X$ be a projective klt variety that is homeomorphic to $ℙ³$. Then, $X ≅ ℙ³$.* *Proof.* Since $X$ is a threefold with isolated, rational Gorenstein singularities, Riemann-Roch takes a particularly simple form: $$\begin{aligned} 1 \overset{\ref{il:4-4-1}}{=} χ(𝒪_X) = \frac{1}{24}·[-K_X]·c_2(X). \end{aligned}$$ With Corollary [Corollary 31](#cor:4-11){reference-type="ref" reference="cor:4-11"}, this reads $$-48 = r·(r²-4).$$ This equation has only one real solution: $r = -4$; in particular, $-K_X$ is ample. As before, Fujita's theorem [@MR946238 Thm. 1] applies to show that $X ≅ ℙ^n$. ◻ Finally, in dimensions four and five we show the following. **Theorem 41** ($\mathbb{Q}$-Fano $4$- and $5$-folds homeomorphic to projective spaces). *Let $X$ be a projective klt variety homeomorphic to $ℙ^n$, with $n=4$ or $5$. Assume that $K_X$ is not ample. Then, $X ≅ ℙ^n$.* *Proof.* Recall that $X$ is a Gorenstein Fano variety of index $i = -r$, with canonical singularities, smooth in codimension two. By [@MR946238 Thm. 1 and 2], we may assume that $i ≤ \dim X - 1$. Further, from Proposition [Proposition 39](#prop:4-19){reference-type="ref" reference="prop:4-19"}, we see that $i ≥ 3$. These cases have to be excluded. If $i = \dim X-1$, then by [@Fuj90], $X$ is a hypersurface of weighted degree $6$ embedded in the smooth part of the weighted projective space $ℙ(3,2,1^{n})$. 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[DOI:10.1515/CRELLE.2006.006](https://doi.org/10.1515/CRELLE.2006.006). Preprint [arXiv:math/0408301](https://arXiv.org/abs/math/0408301). Stanisław Łojasiewicz. . Birkhäuser Verlag, Basel, 1991. Translated from the Polish by Maciej Klimek. [DOI:10.1007/978-3-0348-7617-9](https://doi.org/10.1007/978-3-0348-7617-9). [^1]: Recall that the canonical extension $ℰ_X$ is defined as the middle term of the exact sequence $0 → 𝒪_X → ℰ_X → 𝒯_X → 0$ whose extension class equals $c_1(X) ∈ H¹\bigl(X,\, Ω¹_X\bigr)$. [^2]: See also [@MR3404712 Sect. 7] and [@MR1379330 Chapt. 5 and 6] as general references for the main ideas behind Siu's results and for related topics. [^3]: See [@GoreskyMacPherson Part I, Sect. 1.4] for a detailed discussion. [^4]: Cf. [@MR3404712 Rem. 86(2)] [^5]: Alternatively, one could trace the finite group actions through the proof of the results used in Section [2.2](#sec:2-2){reference-type="ref" reference="sec:2-2"}. [^6]: As $Ω$ is irreducible, the compatible Riemannian metric on $Ω$ is unique up to a positive real multiple that does not change the isometry group. [^7]: See [@LT18 Thm. 1.2] for the projective case and see [@GKP13 Thm. 1.17] for the case where $X$ is projective and smooth in codimension two. [^8]: See [@MR1610975 Thm. 0] for the precise result used here and see [@MR2721630 Appendix] for a history of the result. Igor Belegradek explains on [MathOverflow](https://mathoverflow.net/q/442025) why compactness assumptions are not required.

arxiv_math

--- abstract: | This paper is concerned with finding bounds on betti numbers and describing combinatorially and topologically (minimal) free resolutions of powers of ideals generated by a fixed number $q$ of square-free monomials. Among such ideals, we focus on a specific ideal $\mathcal{E}_q$, which we call *extremal*, and which has the property that for each $r\ge 1$ the betti numbers of ${\mathcal{E}_q}^r$ are an upper bound for the betti numbers of $I^r$ for any ideal $I$ generated by $q$ square-free monomials (in any number of variables). We study the Scarf complex of the ideals ${\mathcal{E}_q}^r$ and use this simplicial complex to extract information on minimal free resolutions. In particular, we show that ${\mathcal{E}_q}^r$ has a minimal free resolution supported on its Scarf complex when $q\le 4$ or when $r\le 2$, and we describe explicitly this complex. For any $q$ and $r$, we also show that $\beta_1({\mathcal{E}_q}^r)$ is the smallest possible, or in other words equal to the number of edges in the Scarf complex. These results lead to effective bounds on the betti numbers of $I^r$, with $I$ as above. For example, we obtain that $\mathrm{pd}(I^r)\le 5$ for all ideals $I$ generated by $4$ square-free monomials and any $r\ge 1$. address: - Department of Mathematics, American University of Beirut, Bliss Hall 315, P.O. Box 11-0236, Beirut 1107-2020, Lebanon - | Department of Mathematics & Statistics\ Dalhousie University\ 6316 Coburg Rd.\ PO BOX 15000\ Halifax, NS\ Canada B3H 4R2 - | Liana M. Şega\ Division of Computing, Analytics and Mathematics\ School of Science and Engineering\ University of Missouri-Kansas City\ MO 64110\ U.S.A. - Department of Mathematics, University of Mississippi, Hume Hall 335, P.O. Box 1848, University, MS 38677 USA author: - Sabine El Khoury - Sara Faridi - Liana M. Şega - Sandra Spiroff title: | The Scarf complex and betti numbers of\ powers of extremal ideals --- # **Introduction** Understanding and bounding free resolutions of ideals is a central problem in commutative algebra, and one that has led to the creation of a broad set of tools, many in combinatorics, to better understand them. Powers of a fixed ideal and their resolutions, though indispensable in many areas of commutative algebra and algebraic geometry, are much less understood. Extremal ideals, introduced in [@Lr], were designed so that their powers would have the largest minimal free resolutions among powers of all square-free monomial ideals. As a result, many of the homological invariants of their powers provide effective bounds for the same invariants of powers of any square-free monomial ideal. More precisely, an extremal ideal $\mathcal{E}_q$ has $q$ square-free monomial generators, and if $I$ is *any* ideal in a polynomial ring, minimally generated by $q$ square-free monomials, then we proved, with our coauthors in [@Lr Theorem 7.9], that $$\label{betti} \beta_i(I^r) \le \beta_i({\mathcal{E}_q}^r) \quad \mbox{for all} \quad r\geq 1.$$ Thus, the problem of finding an effective upper bound for the betti numbers of $I^r$ can be reduced to finding the betti numbers of the ideals ${\mathcal{E}_q}^r$. However, despite the combinatorial construction of extremal ideals, understanding the structure of the minimal free resolution of their powers has proved to be a challenging problem. This paper takes the first steps in this direction. To provide some context to our work, we give a summary of some of the known bounds. The largest minimal free resolution for a monomial ideal $I$ is the *Taylor resolution* [@T], which is built on the chain complex of a simplex. Based on the generators of the ideal, one can build a subcomplex of the Taylor complex called the *Scarf complex* [@BPS], whose chain complex is included in the minimal free resolution of $I$. In other words, $$\label{e:guide} \mbox{Scarf (chain) complex of }I \subseteq \mbox{ Minimal free resolution of } I \subseteq \mbox{ Taylor (chain) complex of }I.$$ Note that these bounds are sharp: there are monomial ideals whose minimal free resolutions are Scarf or Taylor. The Taylor resolution, whose size only depends on the number of generators $q$, therefore provides effective binomial upper bounds for betti numbers of monomial ideals. If $I$ is generated by $q$ monomials, this bound is $$\beta_i(I)\le \binom{q}{i+1}\qquad\text{for all \, $i$ \, with \,\, $0\le i\le q-1$}.$$ But once powers are taken, the Taylor resolution is never minimal when $q>1$, and as the powers grow, the binomial bound becomes unreasonably large. This is where extremal ideals come in: for integers $r> 1$ [\[e:guide\]](#e:guide){reference-type="eqref" reference="e:guide"} can be refined (and abbreviated) as $$\label{e:guide-2} \mathrm{Scarf}(I^r) \subseteq \mbox{ \begin{tabular}{c}Minimal free \\ resolution of $I^r$ \end{tabular}}\subseteq \mbox{ \begin{tabular}{c}Minimal free \\ resolution of ${\mathcal{E}_q}^r$ \end{tabular}} \subseteq \mathrm{Taylor}({\mathcal{E}_q}^r)$$ where the last inclusion is strict when $r,q >1$, and the middle inclusion should be understood in a sense made more precise in [Theorem 15](#t:upperbound){reference-type="ref" reference="t:upperbound"}, where we also show that if a simplicial chain complex gives a resolution of ${\mathcal{E}_q}^r$, then it also gives a resolution of $I^r$. The statement in [\[e:guide-2\]](#e:guide-2){reference-type="eqref" reference="e:guide-2"} leads to the following questions. *Is there a combinatorial description of the minimal free resolution of ${\mathcal{E}_q}^r$? Can it be described as the chain complex of a simplicial complex?* Most importantly, computational evidence points to the following questions. *Do the ideals ${\mathcal{E}_q}^r$ have Scarf resolutions (i.e. the smallest possible)?\ Does the Scarf complex of ${\mathcal{E}_q}^r$ have a nice combinatorial description?* This first of these two questions is the premise of the current paper, and we provide evidence towards an affirmative answer. Our focus is finding faces of the Scarf complex of ${\mathcal{E}_q}^r$, which we denote by $\mathbb{S}^r_q$. If ${\mathcal{E}_q}^r$ is indeed Scarf, the complex $\mathbb{S}^r_q$ can be considered as the natural $r$-th power of the Taylor complex on $q$ generators. The ideal $\mathcal{E}_q$ itself (the case $r=1$) is essentially a *nearly Scarf ideal* (see [@PV]) corresponding to a $q$-simplex, and hence has a Scarf resolution. In fact, for $\mathcal{E}_q$ the Scarf and Taylor resolutions coincide. In this paper we prove that the ideal ${\mathcal{E}_q}^2$ is also Scarf for any $q$, as are ${\mathcal{E}_q}^r$ for any $q \leq 4$ and $r \geq 1$ ([Corollary 31](#c:r=2-Scarf){reference-type="ref" reference="c:r=2-Scarf"}, [Theorem 44](#t:Morse-small-q){reference-type="ref" reference="t:Morse-small-q"}). We give precise combinatorial formulas for the betti numbers of ${\mathcal{E}_q}^r$ in these cases ([Theorem 52](#upper bounds){reference-type="ref" reference="upper bounds"}, [Theorem 53](#t:L2r){reference-type="ref" reference="t:L2r"}). Most of the input for the work in the case $r=2$ comes from [@L2; @Lr], while the case $q\le 4$ brings new techniques. We also prove that the first betti numbers of ${\mathcal{E}_q}^r$ are always Scarf for any $r,q \geq 1$: in [\[t:first-betti\]](#t:first-betti){reference-type="ref" reference="t:first-betti"} we prove that $\beta_1({\mathcal{E}_q}^r)$ is equal to the number of $1$-dimensional faces of $\mathbb{S}^r_q$. When $r=3$, we provide in [Theorem 50](#t:betti-1){reference-type="ref" reference="t:betti-1"} a combinatorial count of such faces, and thus a formula for $\beta_1({\mathcal{E}_q}^3)$ for any $q$. Most of this paper is written with a view towards understanding the Scarf complex $\mathbb{S}^r_q$. We give a characterization of the faces of $\mathbb{S}^r_q$ in terms of polyhedral geometry, and we identify a large number of facets of $\mathbb{S}^r_q$ via a subcomplex $\mathbb{U}^r_q$ which we introduce in [Definition 25](#d:Ur){reference-type="ref" reference="d:Ur"}. The cases mentioned above where ${\mathcal{E}_q}^r$ is a Scarf ideal coincide with situations when $\mathbb{S}^r_q=\mathbb{U}^r_q$. While in general we know that $\mathbb{S}^r_q$ has more faces than those in $\mathbb{U}^r_q$, the simplicial complex $\mathbb{U}^r_q$ allows us to characterize a large number of multigraded betti numbers of ${\mathcal{E}_q}^r$ and provide general lower bounds for the total betti numbers of ${\mathcal{E}_q}^r$ ([Corollary 32](#c:scarf-betti){reference-type="ref" reference="c:scarf-betti"} and [Theorem 30](#t:f-vector){reference-type="ref" reference="t:f-vector"}). As a step towards showing that $\mathbb{S}^r_q$ supports a resolution of ${\mathcal{E}_q}^r$ - in the sense that its simplicial chain complex can be homogenized into a free resolution of ${\mathcal{E}_q}^r$ - in [Theorem 33](#t:german){reference-type="ref" reference="t:german"} we use discrete Morse theory to show that for each facet $U$ of $\mathbb{S}^r_q$ which comes from $\mathbb{U}^r_q$, one can construct a CW complex supporting a resolution of ${\mathcal{E}_q}^r$ by eliminating all faces containing $U$. In view of [\[betti\]](#betti){reference-type="eqref" reference="betti"}, the computations of the betti numbers described above also provide effective bounds for the betti numbers of $I^r$, where $I$ is any ideal generated by $q$ square-free monomials. While these bounds are too lengthy to be displayed here, we mention two bounds on projective dimension that come from [\[upper bounds\]](#upper bounds){reference-type="ref" reference="upper bounds"}: If $I$ is any monomial ideal generated by $q$ square-free monomials, then for any $r\ge 1$, $$\mathrm{pd}(I^r)\le 3\quad \mbox{when} \quad q=3 \quad \mbox{and} \quad \mathrm{pd}(I^r) \le 5 \quad \mbox{when} \quad q=4.$$ Extremal ideals were introduced during our collaborative work [@Lr] with Susan Cooper, Susan Morey, and Sarah Mayes-Tang. The BIRS2023  [@juniper] working group have since started working on developing a proper definition of $\mathbb{S}^r_q$ using the tools of discrete geometry and discrete topology. # **Basic Definitions** {#s:basic-definitions} In this section, basic definitions and background are provided. Throughout the section, let $\mathsf k$ be a field, let $S$ be the polynomial ring $\mathsf k[x_1, \dots, x_n]$, and let $I = (m_1, \dots, m_q)$ be an ideal minimally generated by $q$ square-free monomials in $S$. ## Simplicial complexes and resolutions Given a vertex set $V$, a **simplicial complex** $\Delta$ on a vertex set $V$ is a set of subsets of $V$ satisfying the following property: if $\sigma \in \Delta$ and $\tau \subseteq \sigma$, then $\tau \in \Delta$. An element $\sigma$ of $\Delta$ is called a **face**, and a maximal face of $\Delta$ (under inclusion) is called a **facet**. Since a simplicial complex $\Delta$ can be uniquely determined by its facets $\sigma_1,\ldots,\sigma_q$, one writes $\Delta=\langle \sigma_1,\ldots,\sigma_q\rangle$. The **dimension** of a face $\sigma$ is defined as $\dim(\sigma)=|\sigma|-1$, where the vertical bars denote cardinality. If a simplicial complex has only one facet, then it is called a **simplex**. In particular, the **Taylor complex** $\mathrm{Taylor}(I)$ of $I$ is a simplex on $q$ vertices, each of which is labeled by a monomial generator of $I$, and each face is labeled with the lcm of the monomial labels of its generators. More generally, let $\Delta$ be a simplicial complex on $q$ vertices $v_1,\ldots,v_q$, and label each vertex $v_i$ with the monomial $m_i$, corresponding to the minimal generators of the ideal $I$. Label each face of $\Delta$ with the least common multiple of the labels of its vertices. More precisely, if $\sigma\in \Delta$, then the monomial label of $\sigma$ is $$\mathbf{m}_\sigma=\mathrm{lcm}(m_i\colon i\in \sigma).$$ An exact sequence $F_\bullet$ of free $S$-modules is a **graded free resolution of $I$** if it has the form: $$\label{graded free} 0 \to F_d \stackrel{\partial_d}{\longrightarrow} \cdots \to F_i \stackrel{\partial_i}{\longrightarrow} F_{i-1} \to\cdots \to F_1 \stackrel{\partial_1}{\longrightarrow} F_0$$ where $I \cong F_0/\mathrm{im}(\partial_1)$, and each map $\partial_i$ is graded, in the sense that it preserves the degrees of homogeneous elements. The free resolution in [\[graded free\]](#graded free){reference-type="eqref" reference="graded free"} is called **minimal** if $\partial_i(F_i) \subseteq (x_1,\ldots,x_n) F_{i-1}$ for every $i>0$. To further refine the grading on $F_i$, each free module can be written as direct sum of one dimensional free $S$-modules of the form $S(m)$, indexed by the monomials $m$. Denote by $\mathrm{LCM}(I)$ the poset consisting of the lcm's of the generating set of $I$ ordered by divisibility; i.e., the lcm-lattice of $I$. Thus, when [\[graded free\]](#graded free){reference-type="eqref" reference="graded free"} is minimal, one has $$F_i \cong \bigoplus_{m \in \mathrm{LCM}(I)} S(m)^{\beta_{i,m}}$$ where the $\beta_{i,m}$ are the **multigraded betti numbers** of $I$, which are invariants of $I$. The **projective dimension**, denoted by $\mathrm{pd}_R(I)$, is the length of a minimal free resolution of $I$, which is $d$ in the case of $F_\bullet$ in [\[graded free\]](#graded free){reference-type="eqref" reference="graded free"}, with $F_d\ne 0$. Given a simplicial complex $\Delta$ with $q$ vertices labelled with the monomial generators of $I$, the simplicial chain complex of $\Delta$ can be "homogenized" using the monomial labels on the faces to give a graded complex of free $S$-modules. For details on the homogenization construction, see [@PV]. Let $F^\Delta_\bullet$ denote the homogenized complex obtained from $\Delta$. If $F^\Delta_\bullet$ is a free resolution of $I$, $\Delta$ is said to **support a free resolution** of $I$ and the resulting free resolution is called a **simplicial resolution** of $I$. The **Scarf complex** of $I$, denoted by $\mathrm{Scarf}(I)$, is a subcomplex of $\mathrm{Taylor}(I)$ obtained by removing all faces of $\mathrm{Taylor}(I)$ that share a monomial label with another face. The relevance of Scarf complexes comes from the fact that a minimal free resolution of $I$ contains an isomorphic copy of $F^{\mathrm{Scarf}(I)}_\bullet$ and, in particular, for any face $\sigma\in \mathrm{Scarf}(I)$ the minimal free resolution of $I$ has a unique generator with multidegree equal to that of the monomial label of $\sigma$. An ideal $I$ is said to be a **Scarf ideal** if $\mathrm{Scarf}(I)$ supports a free resolution of $I$ (which is necessarily minimal). The next lemma will be useful in our arguments involving Scarf complexes. **Lemma 1**. *Let $I$ be a monomial ideal with Taylor complex $\mathbb{T}$ and Scarf complex $\mathbb{S}$, and let $\sigma \in \mathbb{T}$. Then $\sigma\in \mathbb{S}$ if and only if both of the following statements hold:* 1. *$\mathbf{m}_\sigma\ne \mathbf{m}_{\sigma\smallsetminus\{v\}}$ for all vertices $v\in \sigma$;* 2. *$\mathbf{m}_{\sigma\cup\{v\}}\ne \mathbf{m}_\sigma$ for all vertices $v\in \mathbb{T}\smallsetminus\sigma$.* *In particular, if $\sigma$ is an edge (that is, $\dim(\sigma)=1$), then $\sigma\in \mathbb{S}$ if and only if (2) holds.* *Proof.* If $\sigma\in \mathbb{S}$, then (1) and (2) clearly hold. Assume now that $\sigma\notin \mathbb{S}$. We show that (1) or (2) fails to hold. Since $\sigma\notin \mathbb{S}$, there exists a face $\tau \in \mathbb{T}$, such that $\tau \neq \sigma$ and $\mathbf{m}_\sigma=\mathbf{m}_\tau$. If $\tau \subsetneq \sigma$, then there exists $v \in \sigma \smallsetminus\tau$. So $$\tau \subseteq \sigma \smallsetminus \{v\} \subset \sigma \quad \mbox{and} \quad \mathbf{m}_\tau=\mathbf{m}_\sigma \quad \Longrightarrow \quad \mathbf{m}_\tau=\mathbf{m}_{\sigma\smallsetminus\{v\}}=\mathbf{m}_\sigma.$$ If $\tau \not \subseteq \sigma$, then there exists $v \in \tau$ with $v \notin \sigma$. The fact that $\mathbf{m}_\sigma=\mathbf{m}_\tau$ now guarantees that $$\mathbf{m}_{\sigma\cup \{v\}}=\mathbf{m}_\sigma.$$ Thus (1) or (2) do not hold. Lastly, when $\dim(\sigma)=1$, $\sigma=\{m_1,m_2\}$ where $m_1$ and $m_2$ are minimal monomial generators of $I$, and so $\mathbf{m}_\sigma \neq m_1$ and $\mathbf{m}_\sigma \neq m_2$. Thus (1) holds trivially, and one only needs to check (2) to establish $\sigma\in \mathbb{S}$. ◻ The next criterion can be used to prove that the Scarf complex supports a minimal free resolution. **Lemma 2**. *[@BPS Lemma 3.1] [\[l:BPS\]]{#l:BPS label="l:BPS"} If $I$ is a monomial ideal for which the nonzero betti numbers are concentrated in the multidegrees of the monomial labels of $\mathrm{Scarf}(I)$, then $\mathrm{Scarf}(I)$ supports a minimal free resolution of $I$.* A main technique in this paper relies on an algebraic version of discrete Morse theory, as developed by E. Batzies and V. Welker[@BW], described below. ## CW complexes and Morse matchings {#s:Morse} In general terms, a **CW complex** is a topological space that can be built inductively by a process of attaching $n$-disks along their boundary. The interiors of the $n$-disks are the $n$-**cells** of the CW complex, which can be viewed as a disjoint union of cells. We refer the reader to [@OW] for a precise definition. As described there, one can define a notion of free resolution supported on a CW complex, generalizing the concepts discussed above. We present below the basic notions and results that will allow us to use discrete Morse theory to construct free resolutions. Let $V$ denote a finite set. We let $2^V$ denote the set of subsets of $V$. Let $Y$ be a subset of $2^V$. Then $G_Y$ denotes the directed graph whose vertex set is $Y$, and whose set of directed edges is $$E(G_Y)=\{\sigma \to \sigma\smallsetminus\{v\} \colon v\in \sigma, \sigma\in Y, \sigma\smallsetminus\{v\}\in Y\}.$$ Suppose $\mathcal{A}$ is a matching on $G_Y$, that is, a set of pairwise disjoint edges of $G_Y$, and let $G_Y^\mathcal{A}$ be a directed graph with the same vertex set as $G_Y$, and with edges in $\mathcal{A}$ reversed, that is $$E(G_Y^\mathcal{A})= \big( E(G_Y)\smallsetminus\mathcal{A}\big ) \cup \{\sigma \smallsetminus\{v\} \to \sigma \colon\sigma \to \sigma \smallsetminus\{v\} \in \mathcal{A}\}.$$ The matching $\mathcal{A}$ is **acyclic** if the graph $G_Y^\mathcal{A}$ contains no directed cycle. A vertex of $G_Y$ that is not in $\mathcal{A}$ is called an **$\mathcal{A}$-critical vertex** of $G_Y$, or an $\mathcal{A}$-**critical cell** of $Y$. **Lemma 3**. *[@Morse Lemma 3.2(1)] [\[inclusions\]]{#inclusions label="inclusions"} Let $Y,Y'$ be subsets of $2^V$ such that $Y\subseteq Y'$. If $\mathcal{A}\subseteq E_Y$, then $\mathcal{A}$ is an acyclic matching of $G_{Y'}$ if and only if $\mathcal{A}$ is an acyclic matching of $G_Y$.* **Lemma 4** (**Cluster Lemma** [@Jo Lemma 4.2]). *Let $V\subseteq 2^V$, $Q$ a poset and $\{Y_q\}_{q\in Q}$ a partition of $Y$ such that $$\text{If $\sigma\in Y_q$ and $\sigma'\in Y_{q'}$ satisfy $\sigma' \subseteq \sigma$, then $q'\le q$.}$$ Let $\mathcal{A}_q$ be an acyclic matching on $G_{Y_q}$ for each $q$. Then $\mathcal{A}=\bigcup_{q\in Q}\mathcal{A}_q$ is an acyclic matching on $G_{Y}$.* **Lemma 5**. *[@Morse Lemma 3.3] [\[matching-lemma\]]{#matching-lemma label="matching-lemma"} Let $Y\subseteq 2^V$ and let $v\in V$. Then $$\begin{aligned} A_Y^{v} = \big\{\sigma\to \sigma'\in E_Y\colon v\in \sigma {\text{ and }} \sigma'=\sigma\smallsetminus\{v\}\big\} \end{aligned}$$ is an acyclic matching on $G_Y$.* Assume now that $Y$ is a subset of $\mathrm{Taylor}(I)$. If $\mathcal{A}$ is a matching on $G_Y$, we say that $\mathcal{A}$ is **homogeneous** if $\mathbf{m}_\sigma=\mathbf{m}_{\sigma'}$ for all $\sigma\to \sigma'\in \mathcal{A}$. Batzies and Welker used homogeneous acyclic matchings to build cellular resolutions of monomial ideals. **Theorem 6** ([@BW]). *Let $I$ be a monomial ideal and $\mathcal{A}$ be a homogeneous acyclic matching on $G_{\mathrm{Taylor}(I)}$. Then there is a CW complex $\mathcal{X}_\mathcal{A}$ supporting a free resolution of $I$, where for each $i\geq 0$, the $i$-cells of $\mathcal{X}_\mathcal{A}$ are in one-to-one correspondence with the $i$-dimensional $\mathcal{A}$-critical cells of $\mathrm{Taylor}(I)$.* The one-to-one correspondence in [Theorem 6](#t:BW){reference-type="ref" reference="t:BW"} preserves monomial labels. More precisely, if $\sigma$ is an $\mathcal{A}$-critical face of $\mathrm{Taylor}(I)$, and $\sigma_\mathcal{A}$ is the unique cell of $\mathcal{X}_\mathcal{A}$ corresponding to $\sigma$, then $$\mathbf{m}_\sigma=\mathbf{m}_{\sigma_\mathcal{A}}.$$ Moreover by ([@BW Proposition 7.3]), for $\mathcal{A}$-critical $i$ and $(i-1)$-faces $\sigma$ and $\sigma'$ of $\mathrm{Taylor}(I)$, the cell $\sigma'_\mathcal{A}$ is a subcell of $\sigma_\mathcal{A}$ (or in other words $\sigma'_\mathcal{A}\leq \sigma_\mathcal{A}$ ) if and only if there is a directed path as in [\[e:gradient\]](#e:gradient){reference-type="eqref" reference="e:gradient"} in $G^\mathcal{A}_{\mathrm{Taylor}(I)}$ starting from $\sigma=\sigma_0$ and ending at $\sigma'=\sigma_t\smallsetminus\{v_t\}$. $$\label{e:gradient} \begin{array}{ccccccccccccc} \sigma_0 &&&& \sigma_1 &\ldots&\sigma_{t-1}&&&& \sigma_{t} && \\ &\searrow&&\nearrow &&&&\searrow &&\nearrow &&\searrow & \\ &&\sigma_0\smallsetminus\{v_0\}&&&&&& \sigma_{t-1} \smallsetminus\{v_{t-1}\}&&&& \sigma_{t} \smallsetminus\{v_t\} \\ \end{array}$$ From the construction of this path it follows that all faces $\sigma_i$ and $\sigma_i\smallsetminus\{v_i\}$ in [\[e:gradient\]](#e:gradient){reference-type="eqref" reference="e:gradient"}, except for the start face $\sigma$ and the end face $\sigma'$, must be in the matching $\mathcal{A}$ ([@Morse Discussion 6.3]). **Remark 7**. With assumptions as in [Theorem 6](#t:BW){reference-type="ref" reference="t:BW"}, if the set of $\mathcal{A}$-critical cells of $\mathrm{Taylor}(I)$ is equal to $\mathrm{Scarf}(I)$, it follows from [\[l:BPS\]](#l:BPS){reference-type="ref" reference="l:BPS"} that $\mathrm{Scarf}(I)$ supports a minimal free resolution of $I$. # **The extremal ideals $\mathcal{E}_q$** {#s:extremal-ideals} In this section, we recall the definition introduced in [@Lr] of a class of ideals $\mathcal{E}_q$, minimally generated by $q$ square-free monomials, for which the powers ${\mathcal{E}_q}^r$ have maximal betti numbers among the ideals $I^r$ where $I$ is minimally generated by $q$ square-free monomials. We then recall and expand some of the relevant results in [@Lr]. **Definition 8** (**Extremal ideals** Definition 7.1 [@Lr]). Let $q$ be a positive integer. For every set $A$ with $\emptyset \neq A \subseteq [q]$, introduce a variable $x_A$, and then set $S_{\mathcal{E}}$ to be the polynomial ring over a field $\mathsf k$ shown below: $$S_{\mathcal{E}}=\mathsf k\big [ x_A \colon\emptyset \neq A \subseteq [q] \big ].$$ For each $i \in [q]$, define a square-free monomial $\epsilon_i$ in $S_{\mathcal{E}}$ as $$\epsilon_i= \prod_{\substack{A \subseteq [q]\\ i \in A}} x_A.$$ The square-free monomial ideal $\mathcal{E}_q = (\epsilon_1,\ldots, \epsilon_q)$ is called a **$\pmb{q}$-extremal ideal**. As demonstrated in the example below, when it is unlikely to lead to confusion, we will simplify the notation by writing $x_1$ for $x_{\{1\}}$, $x_{12}$ for $x_{\{1,2\}}$, etc., and refer to a $q$-extremal ideal simply as an extremal ideal. **Example 9**. When $q=4$, the ideal $\mathcal{E}_4$ is generated by the monomials $$\begin{array}{ll} \epsilon_1&=x_1 x_{12} x_{13} x_{14} x_{123} x_{124} x_{134} x_{1234}, \\ \epsilon_2&=x_{2} x_{12} x_{23} x_{24} x_{123} x_{124} x_{234} x_{1234}, \\ \epsilon_3&=x_{3} x_{13} x_{23} x_{34} x_{123} x_{134} x_{234} x_{1234}, \\ \epsilon_4&=x_{4} x_{14} x_{24} x_{34} x_{124} x_{134} x_{234} x_{1234} \\ \end{array}$$ in the ring $\mathsf k[x_1, x_2, x_3, x_4, x_{12}, x_{13}, x_{14}, x_{23}, x_{24}, x_{34}, x_{123}, x_{124}, x_{134}, x_{234}, x_{1234}]$. In general, the ring $S_{\mathcal{E}}$ has $2^{q} - 1$ variables, corresponding to the power set of $[q]$ (excepting $\emptyset$), and each $\epsilon_i$ is the product of $2^{q-1}$ variables; in particular, those corresponding to the subsets of $[q]$ that contain $i$. Using the terminology of [@PV], one can easily verify that $\mathcal{E}_q$ is precisely the *nearly Scarf ideal* of a $q$-simplex, by corresponding the variable $x_{[q]\smallsetminus B}$ to a face $B$ of the simplex, and therefore $\mathcal{E}_q$ has the Taylor resolution as its minimal free resolution. The Taylor resolution is an upper bound for the minimal free resolution of *any* monomial ideal. It turns out, as stated in [Theorem 15](#t:upperbound){reference-type="ref" reference="t:upperbound"} below, that all powers of $\mathcal{E}_q$ also have a similar property. The rest of this section is dedicated to explaining that simplicial resolutions of powers of extremal ideals bound minimal free resolutions of powers of all square-free monomial ideals, and to discussing first steps (as initiated in [@Lr]) towards finding such resolutions. We recall below some of the results of [@Lr], starting with necessary notation. **Notation 10**. For a field $\mathsf k$, let $S = \mathsf k[x_1, \dots, x_n]$ and $I$ be an ideal of $S$ minimally generated by square-free monomials $m_1,\ldots,m_q$. Let $$r, q \in \mathbb{N}\quad \mbox{and} \quad \mathbf{a}= (a_1, \dots, a_q), \mathbf{b}= (b_1, \dots, b_q) \in \mathbb{N}^q.$$ Define - $[q] =\{1, 2, \dots, q\}$ - $|\mathbf{a}|=a_1 + \cdots + a_q$ - ${\pmb{\epsilon}}^{\mathbf{a}}={\epsilon_1}^{a_1}\cdots {\epsilon_q}^{a_q}$ and $\mathbf{m}^{\mathbf{a}}={m_1}^{a_1}\cdots {m_q}^{a_q}$ - $\mathrm{Supp}(\mathbf{a}) = \{j \, | \,a_j \neq 0 \}$ - $\mathcal{N}^r_q= \{\mathbf{c}\in \mathbb{N}^q \colon|{\mathbf{c}}| =r\}$ - $\epsilon\text{-}\mathrm{gcd}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}) = \epsilon_1^{\min(a_1, b_1)} \cdots \epsilon_q^{\min(a_q, b_q)}$ - $\mathbb{T}^r_q=\mathrm{Taylor}({\mathcal{E}_q}^r)$ - $\mathbb{S}^r_q=\mathrm{Scarf}({\mathcal{E}_q}^r)$. The following statement, which follows directly from [@Lr Equation (7.3.1)], allows us to use $\mathbf{a}\in \mathcal{N}^r_q$ and ${\pmb{\epsilon}}^{\mathbf{a}}$ interchangeably to identify a vertex of $\mathbb{T}^r_q$. **Proposition 11** (**The vertices of the Taylor complex $\mathbb{T}^r_q$**). *If $\mathbf{a}$, $\mathbf{b}\in \mathcal{N}^r_q$, then ${\pmb{\epsilon}}^{\mathbf{a}}={\pmb{\epsilon}}^{\mathbf{b}}$ if and only if $\mathbf{a}=\mathbf{b}$. In particular, the vertices of $\mathbb{T}^r_q$ are labeled with distinct monomials ${\pmb{\epsilon}}^{\mathbf{a}}$ with $\mathbf{a}\in \mathcal{N}^r_q$.* The comparison of a free resolution of $I^r$, when $I$ is generated by square-free monomials $m_1,\ldots,m_q$ in $S$, to a free resolution of ${\mathcal{E}_q}^r$ in ${S_\mathcal{E}}$ is done using the ring homomorphism $\psi_I$ presented below. **Definition 12** (**The ring homomorphism $\psi_I$, [@Lr Definition 7.5]**). Let $I$ be an ideal of the polynomial ring $S=\mathsf k[x_1,\ldots,x_n]$ minimally generated by square-free monomials $m_1,\ldots,m_q$. For each $k \in [n]$ set $$A_k= \{j \in [q] \colon x_k \mid m_j\}$$ and define $\psi_I$ to be the ring homomorphism $$\psi_I \colon {S_\mathcal{E}}\to S \quad \mbox{where} \quad \psi_I(x_A)=\begin{cases} \displaystyle \prod_{\substack{k\in[n] \\ A=A_k}} x_k & \mbox{if } A=A_k \mbox{ for some } k \in [n],\\ 1 & \mbox{otherwise}. \end{cases}$$ **Lemma 13** ([@Lr Lemma 7.7]). *Let $I$ be an ideal of the polynomial ring $S=\mathsf k[x_1,\ldots,x_n]$, minimally generated by square-free monomials $m_1,\ldots,m_q$. Then* 1. *$\psi_I({\pmb{\epsilon}}^{\mathbf{a}})=\mathbf{m}^\mathbf{a}$ for each $\mathbf{a}\in \mathcal{N}^r_q$;* 2. *$\psi_I({\mathcal{E}_q}^r)S =I^r$ for every $r\geq 1$;* 3. *$\psi_I$ preserves least common multiples, that is: $$\psi_I(\mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}_1}, \dots, {\pmb{\epsilon}}^{\mathbf{a}_t}))=\mathrm{lcm}(\mathbf{m}^{\mathbf{a}_1}, \dots, \mathbf{m}^{\mathbf{a}_t}) \quad \mbox{for all} \quad \mathbf{a}_1,\ldots,\mathbf{a}_t \in \mathcal{N}^r_q, \quad t\ge 1.$$* An immediate consequence of [Lemma 13](#l:psi){reference-type="ref" reference="l:psi"} is the following statement, that shows the Scarf complex of ${\mathcal{E}_q}^r$ contains the Scarf complex of any $I^r$ where $I$ is minimally generated by $q$ square-free monomials, using the $\psi$ function. For this statement to make sense we consider both Scarf complexes as labeled by vectors $\mathbf{a}\in \mathcal{N}^r_q$, rather than the corresponding monomials $\mathbf{m}^\mathbf{a}$ or ${\pmb{\epsilon}}^{\mathbf{a}}$. **Proposition 14** (**$\mathrm{Scarf}(I^r) \subseteq \mathrm{Scarf}({\mathcal{E}_q}^r)$**). *Let $r, q \geq 1$ and let $I$ be an ideal minimally generated by $q$ square-free monomials $m_1,\ldots,m_q$. Then $\mathrm{Scarf}(I^r) \subseteq \mathbb{S}^r_q$.* *Proof.* Let $\sigma=\{\mathbf{a}_1,\ldots, \mathbf{a}_t\} \in \mathrm{Scarf}(I^r)$. If for some $\mathbf{b}_1,\ldots, \mathbf{b}_s \in \mathcal{N}^r_q$, $$\mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{b}_1},\ldots, {\pmb{\epsilon}}^{\mathbf{b}_s})=\mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}_1},\ldots, {\pmb{\epsilon}}^{\mathbf{a}_t})$$ then by [Lemma 13](#l:psi){reference-type="ref" reference="l:psi"} $$\mathrm{lcm}(\mathbf{m}^{\mathbf{b}_1},\ldots, \mathbf{m}^{\mathbf{b}_s})=\mathrm{lcm}(\mathbf{m}^{\mathbf{a}_1},\ldots, \mathbf{m}^{\mathbf{a}_t})$$ which implies that $s=t$ and $$\{\mathbf{b}_1,\ldots,\mathbf{b}_s\}=\{\mathbf{a}_1, \ldots,\mathbf{a}_t\}.$$ Hence $\{\mathbf{a}_1,\ldots, \mathbf{a}_t\} \in \mathbb{S}^r_q=\mathrm{Scarf}({\mathcal{E}_q}^r)$. ◻ Now consider a simplicial complex $\Delta$ supporting a free resolution of ${\mathcal{E}_q}^r$. Since $\Delta$ is a subcomplex of $\mathbb{T}^r_q$, the vertices of $\Delta$ are identified by the elements $\mathbf{a}\in \mathcal{N}^r_q$, where each vertex $\mathbf{a}$ has monomial label ${\pmb{\epsilon}}^{\mathbf{a}}$ ([Proposition 11](#p:vertices){reference-type="ref" reference="p:vertices"}). We let $\psi_I(\Delta)$ be the simplicial complex $\Delta$, with each vertex $\mathbf{a}$ relabeled by the monomial $\mathbf{m}^\mathbf{a}$. While $\psi_I(\Delta)$ may have multiple vertices sharing the same label, we will see below that its homogenized chain complex supports a free resolution of $I^r$. In particular, (the simplicial chain complex of) $\Delta$ is an upper bound for the minimal free resolution of $I^r$ for any $I$ generated by $q$ square-free monomials. To make this point in a precise manner, we need to introduce a (non-standard) grading on ${S_\mathcal{E}}$, which we call the **$\psi_I$-grading**. Under the standard grading the ring $S$ is $\mathbb{Z}^n$-graded, and the ring ${S_\mathcal{E}}$ is $\mathbb{Z}^{2^q-1}$-graded. The the $\psi_I$-grading adjusts the standard grading on ${S_\mathcal{E}}$ to make $\psi_I$ into a homogeneous map. In other words, assuming the standard grading on $S$ (where the degree of each variable is $1$), the multidegree of the variable $x_A$ is the same as the multidegree of $\psi_I(x_A)$, and thus the multidegree of ${\pmb{\epsilon}}^{\mathbf{a}}$ becomes the same as the the multidegree of $\mathbf{m}^{\mathbf{a}}$ for $\mathbf{a}\in \mathcal{N}^r_q$. To distinguish between the standard and nonstandard gradings, we let $\widetilde{{S_\mathcal{E}}}$ denote the ring ${S_\mathcal{E}}$ with the $\psi_I$-grading. Using the one-to-one correspondence between $\mathbb{Z}^n$ and the set of monomials $\mathcal{M}(S)$ in the polynomial ring $S$, we can use the elements of $\mathcal{M}(S)$ to index the $\mathbb{Z}^n$-grading on $S$, and similarly for $\mathcal{M}({S_\mathcal{E}})$. Therefore, as graded objects, the decompositions of ${S_\mathcal{E}}$ and $\widetilde{{S_\mathcal{E}}}$ are $${S_\mathcal{E}}=\bigoplus_{\mathbf{m}\in \mathcal{M}({S_\mathcal{E}})} \mathsf k\mathbf{m}\quad \mbox{and} \quad \widetilde{{S_\mathcal{E}}} =\bigoplus_{\mathbf{m}'\in \mathcal{M}(S)} \Big( \bigoplus_{\tiny \begin{array}{c}\mathbf{m}\in \mathcal{M}({S_\mathcal{E}})\\ \psi_I(\mathbf{m})=\mathbf{m}' \end{array}} \mathsf k\mathbf{m}\Big ).$$ If $W=\oplus_{\mathbf{m}\in \mathcal{M}({S_\mathcal{E}})}W_\mathbf{m}$ is a $\mathbb{Z}^{2^q-1}$-graded ${S_\mathcal{E}}$-module, we denote by $\widetilde{W}$ the $\mathbb{Z}^n$-graded $\widetilde{{S_\mathcal{E}}}$-module $$\widetilde{W}= \bigoplus_{\mathbf{m}'\in \mathcal{M}(S)}\Big( \bigoplus_{\tiny \begin{array}{c} \mathbf{m}\in \mathcal{M}({S_\mathcal{E}})\\ \psi_I(\mathbf{m})=\mathbf{m}' \end{array}} W_\mathbf{m}\Big ).$$ A homomorphism $\varphi\colon V\to W$ of graded ${S_\mathcal{E}}$-modules then naturally induces a homomorphism $\widetilde \varphi\colon \widetilde V\to \widetilde W$ of graded $\widetilde{{S_\mathcal{E}}}$-modules. To summarize, we have described an exact functor from the category of $\mathbb{Z}^{2^q-1}$-graded ${S_\mathcal{E}}$-modules to the category of $\mathbb{Z}^n$-graded $\widetilde{{S_\mathcal{E}}}$-modules. In particular, if $F_\bullet$ denotes a graded free resolution of $W$ over ${S_\mathcal{E}}$, then $\widetilde F_\bullet$ denotes the corresponding graded free resolution of $\widetilde W$ over $\widetilde{{S_\mathcal{E}}}$. The $\psi_I$-grading allows us to extend the proof of [@Lr Theorem 7.9] to [Theorem 15](#t:upperbound){reference-type="ref" reference="t:upperbound"} and give bounds on multigraded betti numbers of powers of square-free monomial ideals. **Theorem 15** (**Extremal ideals bound betti numbers**). *Let $q$ and $r$ be positive integers, and let $I$ be an ideal of the polynomial ring $S=\mathsf k[x_1,\ldots,x_n]$ minimally generated by $q$ square-free monomials.* 1. *If $F_\bullet$ is a multigraded free resolution of ${\mathcal{E}_q}^r$ over ${S_\mathcal{E}}$, then $\widetilde F_\bullet\otimes_{\widetilde{{S_\mathcal{E}}}}S$ is a multigraded free resolution of $I^r$ over $S$, where, in the tensor product, $S$ is regarded as a graded $\widetilde{{S_\mathcal{E}}}$-module via the homogeneous homomorphism $\psi_I$.* 2. *If a simplicial complex $\Delta$ supports a free resolution of ${\mathcal{E}_q}^r$, then $\psi_I(\Delta)$ supports a free resolution of $I^r$.* 3. *If $\mathbf{m}\in \mathrm{LCM}(I^r)$, then $$\displaystyle \beta_{i, \mathbf{m}}(I^r) \le \sum_{\tiny \begin{array}{ll}{\pmb{\epsilon}}\in \mathrm{LCM}({\mathcal{E}_q}^r)\\ \psi_I({\pmb{\epsilon}})=\mathbf{m} \end{array}} \beta_{i,{\pmb{\epsilon}}}({\mathcal{E}_q}^r).$$* 4. *([@Lr Theorem 7.9]) If $i\ge 0$ then $$\beta_i(I^r)\le \beta_i({\mathcal{E}_q}^r).$$* *Proof.* As in [@Lr Lemma 7.8], with the added improvement of working in the appropriate $\mathbb{Z}^n$-graded setting given by the $\psi_I$-grading, we have $$\label{e:Tors} \widetilde{S_{\mathcal{E}}}/\widetilde{{\mathcal{E}_q}^r}\otimes_{\widetilde S_{\mathcal{E}}}S\cong S/I^r\quad\text{and}\quad \mathrm{Tor}_i^{\widetilde S_{\mathcal{E}}}(\widetilde{S_{\mathcal{E}}}/{\widetilde{\mathcal{E}_q}^r}, S)=0 \quad \mbox{for all} \quad i>0.$$ Then (1) follows from here, by computing the $\mathrm{Tor}$ modules using $\widetilde F_\bullet$. \(2\) Let $F^\Delta_\bullet$, respectively $F^{\psi_I(\Delta)}_\bullet$, denote the homogenization of the chain complex of $\Delta$, respectively $\psi_I(\Delta)$. Then [Lemma 13](#l:psi){reference-type="ref" reference="l:psi"} shows that $\widetilde{F^\Delta}_\bullet\otimes_{\widetilde{{S_\mathcal{E}}}}S\cong F^{\psi_I(\Delta)}_\bullet$. Then (1) implies that $F^{\psi_I(\Delta)}_\bullet$ is a free resolution of $I^r$. Parts (3) and (4) follow directly from (1), by using the free resolution $\widetilde F_\bullet\otimes_{\widetilde{{S_\mathcal{E}}}}S$ of $I^r$ to give upper bounds on the betti numbers of $I^r$. Note that the summation on the right-hand side of the inequality in (3) is equal to $\beta_{i,\mathbf{m}}(\widetilde{{\mathcal{E}_q}^r})$. ◻ [Theorem 15](#t:upperbound){reference-type="ref" reference="t:upperbound"} and [Proposition 14](#p:psi-Scarf){reference-type="ref" reference="p:psi-Scarf"} provide motivation to focus our attention on finding simplicial complexes that support resolutions of ${\mathcal{E}_q}^r$. All evidence points to ${\mathcal{E}_q}^r$ being a Scarf ideal for all $q$ and $r$, which means that we expect $\mathbb{S}^r_q$ to support a minimal free resolution of ${\mathcal{E}_q}^r$. For this reason, significant effort will be spent in later sections on describing the faces of $\mathbb{S}^r_q$. In [@Lr], the authors described a simplicial complex $\mathbb{L}^r_q$ which supports a free resolution of $I^r$ for any ideal $I$ generated by $q$ square-free monomials in a polynomial ring. In particular, $\mathbb{L}^r_q$ supports a free resolution of ${\mathcal{E}_q}^r$, and therefore we have the following inclusions of labelled simplicial complexes (see, for example, [@Mermin]). $$\label{e:first-inclusions} \mathbb{S}^r_q\subseteq \mathbb{L}^r_q\subseteq \mathbb{T}^r_q.$$ The complex $\mathbb{L}^r_q$ is significantly smaller than $\mathbb{T}^r_q$, and supports a minimal resolution when $r=2$. **Definition 16** (**$\mathbb{L}^2_q$ [@L2; @Lr]**). $\mathbb{L}^2_q$ is the subcomplex of $\mathbb{T}^{2}_{q}$ with facets $$\{\epsilon_i\epsilon_j\colon i\ne j, i,j\in [q]\}\quad \mbox{and} \quad \{\epsilon_i\epsilon_j\colon j\in [q]\} \quad \mbox{for each} \quad i \in [q] \quad \mbox{when} \quad q>2,$$ and facets $$\{\epsilon_i\epsilon_j\colon j\in [q]\} \quad \mbox{for each} \quad i \in [q] \quad \mbox{when} \quad q\le 2.$$ **Example 17** ($\mathbb{L}^2_3$ vs $\mathbb{T}^2_3$). The complex $\mathbb{L}^2_3$ corresponding to $q=3$, $r = 2$, and shown in [\[f:L32\]](#f:L32){reference-type="ref" reference="f:L32"} supports a minimal free resolution of $\mathcal{E}_3^2 = (\epsilon_1, \epsilon_2, \epsilon_3)^2$, where $\epsilon_i =x_i x_{ij} x_{ik} x_{ijk}$, for $i,j,k$ with $\{i, j, k\}=\{1,2,3\}$. In contrast, the 5-simplex $\mathbb{T}^2_3$ is a five-dimensional polytope with six vertices, fifteen edges, and twenty triangles. **Proposition 18** ([@Lr]). *$\mathbb{T}^{1}_{q}=\mathrm{Taylor}(\mathcal{E}_q)$ supports a minimal free resolution of $\mathcal{E}_q$ and $\mathbb{L}^2_q$ supports a minimal free resolution of ${\mathcal{E}_q}^2$.* However, as $r$ and $q$ grow, the free resolution supported on $\mathbb{L}^r_q$, while still much smaller than the Taylor resolution, veers farther away from a minimal resolution. # **Faces of the Scarf complex $\mathbb{S}^r_q$** {#s:faces} The focus of this paper is understanding the Scarf complex $\mathbb{S}^r_q$ of ${\mathcal{E}_q}^r$. The first step, as taken in this section, is to understand the monomial labels of the Taylor complex, and describe the faces of the Scarf complex. We will be using notation from [Notation 10](#n:rqs){reference-type="ref" reference="n:rqs"}. **Lemma 19** (**Divisibility conditions I**). *Let $d, q, r \geq 1$ and $\mathbf{a}_k=(a_{k1}, \dots, a_{kq}),\mathbf{b}=(b_1, \dots, b_q)\in \mathcal{N}^r_q$, for $0\le k\le d$. Then $$\label{lcm} \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}_1}, {\pmb{\epsilon}}^{\mathbf{a}_2}, \dots, {\pmb{\epsilon}}^{\mathbf{a}_d})=\prod_{\emptyset\ne A\subseteq [q]} {(x_A)}^{\max \left(\sum_{i\in A}a_{1i}, \sum_{i\in A}a_{2i}, \dots, \sum_{i\in A}a_{di}\right)}$$ and $$\label{equiv} {\pmb{\epsilon}}^{\mathbf{b}}\mid \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}_1}, {\pmb{\epsilon}}^{\mathbf{a}_2}, \dots, {\pmb{\epsilon}}^{\mathbf{a}_d})\iff \sum_{i\in A} b_i\le \max \left(\sum_{i\in A}a_{1i}, \dots, \sum_{i\in A}a_{di}\right)\quad \text{ for all\, $\emptyset \ne A\subseteq [q]$.}$$ In particular, if ${\pmb{\epsilon}}^{\mathbf{b}}\mid \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}_1}, {\pmb{\epsilon}}^{\mathbf{a}_2}, \dots, {\pmb{\epsilon}}^{\mathbf{a}_d})$, then $$\label{ineq} \min(a_{1j}, a_{2j}, \dots, a_{dj})\le b_j\le \max(a_{1j}, a_{2j}, \dots, a_{dj})\qquad\text{for all $j\in [q]$.}$$* *Proof.* Formula [\[lcm\]](#lcm){reference-type="eqref" reference="lcm"} follows directly from the definition of the monomials $\epsilon_i$, and [\[equiv\]](#equiv){reference-type="eqref" reference="equiv"} is an immediate consequence of [\[lcm\]](#lcm){reference-type="eqref" reference="lcm"}. We now prove [\[ineq\]](#ineq){reference-type="eqref" reference="ineq"}, assuming that ${\pmb{\epsilon}}^{\mathbf{b}}\mid \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}_1}, {\pmb{\epsilon}}^{\mathbf{a}_2}, \dots, {\pmb{\epsilon}}^{\mathbf{a}_d})$. Let $j\in [q]$. Note that the inequality on the right in [\[ineq\]](#ineq){reference-type="eqref" reference="ineq"} follows by using [\[equiv\]](#equiv){reference-type="eqref" reference="equiv"} with $A=\{j\}$. The inequality on the left follows by using [\[equiv\]](#equiv){reference-type="eqref" reference="equiv"} with $A=[q]\smallsetminus\{j\}$, and noting that $$b_j=r-\sum_{i\in [q]\smallsetminus\{j\}}b_i\qquad\text{and}\qquad a_{kj}=r-\sum_{i\in [q]\smallsetminus\{j\}}a_{ki}\,. \qedhere$$ ◻ As a result, we can characterize all the faces of $\mathbb{S}^r_q$. **Proposition 20** (**The faces of $\mathbb{S}^r_q$**). *Let $d, q, r \geq 1$ and $\sigma=\{{\pmb{\epsilon}}^{\mathbf{a}_1}, {\pmb{\epsilon}}^{\mathbf{a}_2}, \dots, {\pmb{\epsilon}}^{\mathbf{a}_d}\}$ be an $d$-dimensional face of $\mathbb{T}^r_q$. Let $P_{\sigma}$ denote the polyhedron consisting of all $(b_1, \dots, b_q)\in \mathbb{R}^q$ such that $$\begin{cases} \sum_{i=1}^q b_i=r\\ \sum_{i\in A} b_i\le \max(\sum_{i\in A}a_{1i}, \sum_{i\in A}a_{2i}, \dots, \sum_{i\in A}a_{di}) \qquad \text{for all $ A\subseteq [q]$.} \end{cases}$$ Then $\sigma\in \mathbb{S}^r_q$ if and only if $P_{\sigma'}\cap \mathbb{N}^q=\{\mathbf{a}_j\colon{\pmb{\epsilon}}^{\mathbf{a}_j}\in \sigma'\}$ for all $\sigma'\subseteq \sigma$.* *Proof.* Use [\[equiv\]](#equiv){reference-type="ref" reference="equiv"} to write $$P_{\sigma'}\cap \mathbb{N}^q=\{\mathbf{b}\in \mathcal{N}^r_q\colon{\pmb{\epsilon}}^{\mathbf{b}}\mid \mathbf{m}_{\sigma'}\}\,.$$ Thus the condition that $P_{\sigma'}\cap \mathbb{N}^q=\{\mathbf{a}_j\colon{\pmb{\epsilon}}^{\mathbf{a}_j}\in \sigma'\}$ for all $\sigma'\subseteq \sigma$ is equivalent to: $$\label{e:rewrite} \text{If $\sigma'\subseteq \sigma$ and ${\pmb{\epsilon}}^{\mathbf{b}}\mid \mathbf{m}_{\sigma'}$ for some $\mathbf{b}\in \mathcal{N}^r_q$, then ${\pmb{\epsilon}}^\mathbf{b}\in \sigma'$. }$$ Since ${\pmb{\epsilon}}^\mathbf{b}\mid \mathbf{m}_{\sigma'}$ if and only if $\mathbf{m}_{\sigma'}=\mathbf{m}_{\sigma'\cup \{{\pmb{\epsilon}}^\mathbf{b}\}}$, we then use [Lemma 1](#l:expansion){reference-type="ref" reference="l:expansion"} to see that [\[e:rewrite\]](#e:rewrite){reference-type="eqref" reference="e:rewrite"} is equivalent to $\sigma\in \mathbb{S}^r_q$. ◻ Next we show that $\mathbb{S}^r_q$ inherits the faces of $\mathbb{S}^{r'}_q$ for $r'<r$. For $r'\in [r]$ and $\mathbf{a}\in \mathcal{N}_q^{r'}$ and $\sigma\in \mathbb{T}^{r-r'}_{q}$ we define $$\label{e:e-sigma} {\pmb{\epsilon}}^{\mathbf{a}}\sigma=\{{\pmb{\epsilon}}^{\mathbf{a}}\cdot v\colon v\in \sigma\}\in \mathbb{T}^r_q.$$ If $Y$ is a subcomplex of $\mathbb{T}^r_q$, then ${\pmb{\epsilon}}^{\mathbf{a}}Y$ denotes the subcomplex of $\mathbb{T}^r_q$ with faces ${\pmb{\epsilon}}^{\mathbf{a}}\sigma$, where $\sigma\in Y$. **Proposition 21** (**Scarf faces**). *Let $q\ge 1$, $1 \le r'< r$, and $\mathbf{a}\in \mathcal{N}_q^{r'}$. Then $$\sigma\in \mathbb{S}^{r-r'}_{q} \iff {\pmb{\epsilon}}^{\mathbf{a}}\sigma\in \mathbb{S}^r_q.$$ In particular ${\pmb{\epsilon}}^{\mathbf{a}}\mathbb{S}^{r-r'}_{q} \subseteq \mathbb{S}^r_q.$* *Proof.* We will prove the statement in the case when ${\pmb{\epsilon}}^{\mathbf{a}}=\epsilon_i$ for some $i$. The general statement will then follow by induction. Without loss of generality, assume $i=1$. Let $\sigma = \{{\pmb{\epsilon}}^{\mathbf{a}_1}, \dots, {\pmb{\epsilon}}^{\mathbf{a}_t}\}$, where each $\mathbf{a}_i \in \mathcal{N}_q^{r-1}$. Then $$\epsilon_1\sigma = \{\epsilon_1 {\pmb{\epsilon}}^{\mathbf{a}_1}, \dots, \epsilon_1{\pmb{\epsilon}}^{\mathbf{a}_t}\} \quad \mbox{and} \quad \mathbf{m}_{\epsilon_1 \sigma} = \epsilon_1 \mathbf{m}_{\sigma}.$$ Assume that $\sigma\in \mathbb{S}^{r-1}_{q}$. Let $\mathbf{b}_1,\ldots,\mathbf{b}_h \in \mathcal{N}^r_q$ and $$\rho = \{{\pmb{\epsilon}}^{\mathbf{b}_1}, \dots, {\pmb{\epsilon}}^{\mathbf{b}_h}\} \in \mathbb{T}^r_q\quad\text{such that}\quad \mathbf{m}_{\rho} = \mathbf{m}_{\epsilon_1\sigma}=\epsilon_1\mathbf{m}_{\sigma}.$$ To argue $\epsilon_1\sigma\in \mathbb{S}^r_q$, one must show $\rho=\epsilon_1\sigma$. We claim that $$\epsilon_1 \mid {\pmb{\epsilon}}^{\mathbf{b}_j} \quad \mbox{for each} \, j \in [h].$$ Suppose this is not the case, and for some $j \in [h]$, $\epsilon_1 \nmid {\pmb{\epsilon}}^{\mathbf{b}_j}$. In other words, for some positive integers $b_1,\ldots,b_k$, $${\pmb{\epsilon}}^{\mathbf{b}_j}={\epsilon_{d_1}}^{b_1} \cdots {\epsilon_{d_k}}^{b_k} \quad \mbox{where} \quad 1 < d_1 < \ldots <d_k \leq q.$$ Since $x_{\{d_1,\ldots,d_k\}}\mid \epsilon_{d_z}$ for each $z\in [k]$, and since $b_1 + \cdots + b_k=r$, we must then have $${x_{\{d_1,\ldots,d_k\}}}^{r} \mid {\pmb{\epsilon}}^{\mathbf{b}_j} \mid \mathbf{m}_{\rho}=\epsilon_1\mathbf{m}_\sigma.$$ On the other hand, since $1 \notin \{d_1,\ldots,d_k\}$, we have $x_{\{d_1,\ldots,d_k\}}\nmid \epsilon_1$ and hence $${x_{\{d_1,\ldots,d_k\}}}^r \nmid \epsilon_1\mathbf{m}_\sigma,$$ a contradiction. Therefore, for each $z\in [h]$ one can write ${\pmb{\epsilon}}^{\mathbf{b}_z}=\epsilon_1{\pmb{\epsilon}}^{\mathbf{b}'_z}$ where $\mathbf{b}'_z \in \mathcal{N}_q^{r-1}$, and $$\rho=\epsilon_1 \rho' \quad \mbox{where} \quad \rho' = \{{\pmb{\epsilon}}^{\mathbf{b}'_1}, \cdots, {\pmb{\epsilon}}^{\mathbf{b}'_h}\} \in \mathbb{T}^{r-1}_{q}.$$ We have $$\mathbf{m}_{\epsilon_1 \sigma}=\mathbf{m}_{\rho}=\mathbf{m}_{\epsilon_1 \rho'} \Longrightarrow \epsilon_1\mathbf{m}_{\sigma}=\epsilon_1\mathbf{m}_{\rho'} \Longrightarrow \mathbf{m}_{\sigma}=\mathbf{m}_{\rho'}.$$ As $\sigma \in \mathbb{S}^{r-1}_{q}$, we conclude that $\sigma=\rho'$ and hence $\epsilon_1\sigma=\epsilon_1\rho'=\rho$, which concludes our argument. Assume now $\epsilon_1\sigma\in\mathbb{S}^r_q$. Assume $\mathbf{m}_{\sigma}=\mathbf{m}_{\rho}$ for some $\rho\in \mathbb{T}^{r-1}_{q}$. It follows that $$\mathbf{m}_{\epsilon_1\sigma}=\epsilon_1\mathbf{m}_{\sigma}=\epsilon_1\mathbf{m}_{\rho}=\mathbf{m}_{\epsilon_1\rho}$$ and hence $\epsilon_1\sigma=\epsilon_1\rho$, implying $\sigma=\rho$. Therefore, $\sigma$ is in $\mathbb{S}^{r-1}_{q}$. ◻ # **Facets of the Scarf complex $\mathbb{S}^r_q$** {#f:facets} This section aims to identify a large number of facets of $\mathbb{S}^r_q$ by defining a new simplicial complex $\mathbb{U}^r_q$ contained in $\mathbb{S}^r_q$ ([Theorem 30](#t:f-vector){reference-type="ref" reference="t:f-vector"}), and whose facets are also facets of $\mathbb{S}^r_q$. Using the notation in [\[e:e-sigma\]](#e:e-sigma){reference-type="eqref" reference="e:e-sigma"}, we first identify specific faces. **Definition 22** (**$\mathcal{U}^r_q$ and $\mathcal{U}_{\mathbf{a}}$**). For $r,q\geq 1$, let $\mathcal{U}^r_q\in \mathbb{T}^r_q$ denote the set of all square free $r$-tuples of the $\epsilon_i$. In other words $$\begin{aligned} \mathcal{U}^r_q& =\{ \epsilon_{i_1}\epsilon_{i_2}\cdots \epsilon_{i_r} : 1 \leq i_1 < i_2 <\ldots<i_r\leq q \}\\ &= \{ {\pmb{\epsilon}}^{\mathbf{a}}: \mathbf{a}\in \mathcal{N}^r_q,\ a_i \in \{0, 1\} {\text{ for all }}i\in [q] \}.\end{aligned}$$ For $\mathbf{a}=(a_1, \dots, a_q)\in \mathbb{N}^q$ with $|\mathbf{a}| < r$, set $$\begin{aligned} \label{e:Ua} \mathcal{U}_{\mathbf{a}}& ={\pmb{\epsilon}}^{\mathbf{a}}\mathcal{U}_q^{r-|\mathbf{a}|} \\ \nonumber & =\{{\pmb{\epsilon}}^{\mathbf{c}}\colon\mathbf{c}=(c_1, \dots, c_q)\in \mathcal{N}^r_q, a_i\le c_i\le a_{i}+1\quad\text{for all $i\in [q]$}\}.\end{aligned}$$ **Example 23**. Suppose $q = 3$ and $r = 6$, and suppose $$\mathbf{a}= (1, 2, 0), \quad \mathbf{b}= (2, 2, 0) \quad \mbox{and} \quad \mathbf{c}= (1, 0, 1).$$Then $|\mathbf{a}| = 3$, $|\mathbf{b}| = 4$, and $|\mathbf{c}| = 2$. Note that $\mathcal{U}_3^4=\emptyset$ because $4>3$, and hence $$\mathcal{U}_{\mathbf{c}}={\pmb{\epsilon}}^{\mathbf{c}}\mathcal{U}_q^{r-|\mathbf{c}|} =\epsilon_1\epsilon_3 \mathcal{U}_3^4=\emptyset\,.$$ Next, note that $$\mathcal{U}_{\mathbf{a}}={\pmb{\epsilon}}^{\mathbf{a}}\mathcal{U}_q^{r-|\mathbf{a}|}= \epsilon_1\epsilon_2^2 \mathcal{U}_3^3 = \epsilon_1\epsilon_2^2\{\epsilon_1\epsilon_2\epsilon_3\} = \{\epsilon_1^2\epsilon_2^3\epsilon_3\},$$ and $$\mathcal{U}_{\mathbf{b}}={\pmb{\epsilon}}^{\mathbf{b}}\mathcal{U}_q^{r-|\mathbf{b}|} = \epsilon_1^2\epsilon_2^2 \mathcal{U}_3^2 = \epsilon_1^2\epsilon_2^2\{\epsilon_1\epsilon_2, \epsilon_1\epsilon_3, \epsilon_2\epsilon_3\} = \{\epsilon_1^3\epsilon_2^3, \epsilon_1^3\epsilon_2^2\epsilon_3, \epsilon_1^2\epsilon_2^3\epsilon_3\}.$$ In particular, $\mathcal{U}_{\mathbf{a}}\subseteq \mathcal{U}_{\mathbf{b}}$. The next result explains, in particular, which of the sets $\mathcal{U}_{\mathbf{a}}$ are maximal with respect to inclusion. **Lemma 24** (**The faces $\mathcal{U}_{\mathbf{a}}$**). *Let $r$ and $q$ be positive integers integers. Then* 1. *$|\mathcal{U}^r_q| = \binom{q}{r}$.* 2. *If $r>q$ then $\mathcal{U}^r_q= \emptyset$.* 3. *If $q\ge 2$, then the sets $\mathcal{U}_{\mathbf{a}}$ with $\mathbf{a}\in \mathbb{N}^q$ and $r-q<|\mathbf{a}| < r$ are the maximal elements with respect to inclusion among all sets $\mathcal{U}_{\mathbf{a}}$ with $\mathbf{a}\in \mathbb{N}^q$ and $|\mathbf{a}|<r$.* *Proof.* Statements (1) and (2) are straightforward. Assume now $q\ge 2$. For (3) we show that if $$\mathbf{a}=(a_1, \dots, a_q),\ \mathbf{b}=(b_1, \dots, b_q) \in \mathbb{N}^q \quad \mbox{with} \quad |\mathbf{a}|, \ |\mathbf{b}|<r,$$ then $\mathcal{U}_{\mathbf{a}}\subseteq \mathcal{U}_{\mathbf{b}}$ if and only one of the following conditions holds: - $|\mathbf{a}|<r-q$; - $|\mathbf{a}|=r-q$ and $a_i\le b_i\le a_i+1$ for all $i\in [q]$; - $|\mathbf{a}|>r-q$ and $\mathbf{a}=\mathbf{b}$. Indeed, if $|\mathbf{a}|<r-q$, then $\mathcal{U}_{\mathbf{a}}=\emptyset$. If $|\mathbf{a}|=r-q$, then $\mathcal{U}_{\mathbf{a}}=\{{\pmb{\epsilon}}^{\mathbf{a}}\epsilon_1\epsilon_2\dots \epsilon_q\}$, and $$\mathcal{U}_{\mathbf{a}}\subseteq \mathcal{U}_{\mathbf{b}}\iff {\pmb{\epsilon}}^{\mathbf{b}}\mid {\pmb{\epsilon}}^{\mathbf{a}}\epsilon_1\epsilon_2\dots \epsilon_q \text{ and } r-q\le |\mathbf{b}|\iff a_i\le b_i\le a_i+1 \quad \mbox{for all} \quad i\in [q].$$ In particular, $\mathcal{U}_{\mathbf{a}}\subseteq \mathcal{U}_{\mathbf{a}'}$, where $\mathbf{a}'=(a_1+1, a_2, \dots, a_q)$, with $|\mathbf{a}'|=r-q+1<r$ (since we assumed $q>1$) and thus $\mathcal{U}_{\mathbf{a}}$ is not maximal. Assume now $|\mathbf{a}|>r-q$ and $\mathcal{U}_{\mathbf{a}}\subseteq \mathcal{U}_{\mathbf{b}}$. We need to show $\mathbf{a}=\mathbf{b}$. Let $i\in [q]$. There exists ${\pmb{\epsilon}}^{\mathbf{c}}\in \mathcal{U}_{\mathbf{a}}$ with such that $c_i=a_{i}+1$. Since ${\pmb{\epsilon}}^{\mathbf{c}}\in \mathcal{U}_{\mathbf{b}}$, [\[e:Ua\]](#e:Ua){reference-type="eqref" reference="e:Ua"} implies $c_i\le b_i+1$ and hence $a_i\le b_i$. There also exists ${\pmb{\epsilon}}^{\mathbf{d}}\in \mathcal{U}_{\mathbf{a}}$ with $\mathbf{d}=(d_1, \dots, d_q)$ such that $d_i=a_i$. Since ${\pmb{\epsilon}}^{\mathbf{d}}\in \mathcal{U}_{\mathbf{b}}$, we must have $b_i\le d_i$ and hence $b_i \le a_i$. We conclude $a_i=b_i$ for all $i\in [q]$, so $\mathbf{a}=\mathbf{b}$, as desired. ◻ We want to define now a simplicial complex $\mathbb{U}^r_q$ whose faces are all the subsets of the sets $\mathcal{U}_{\mathbf{a}}$ defined above. We describe this complex by identifying its facets. [Lemma 24](#l:facets){reference-type="ref" reference="l:facets"} shows that our definition below makes sense. **Definition 25** (**The simplicial complex $\mathbb{U}^r_q$**). For integers $r\geq 1$ and $q\ge 2$ let $\mathbb{U}^r_q$ denote the simplicial complex described by its facets, as follows: $$\mathbb{U}^r_q=\left\langle \mathcal{U}_{\mathbf{a}}\colon\mathbf{a}\in \mathbb{N}^q, \ r-q<|\mathbf{a}| < r \right\rangle \,.$$ If $q=1$ , we set $\mathbb{U}^r_q=\langle \mathcal{U}_{(r-1)}\rangle$, where $\mathcal{U}_{(r-1)}=\{\epsilon_1^r\}={\epsilon_1}^{r-1}\mathcal{U}_1^1$. **Example 26**. We write an explicit description of the facets of $\mathbb{U}^r_q$ in a few cases. If $r=2$ and $q\ge 3$, then $$\label{e:U2} \mathbb{U}^2_q=\left \langle \mathcal{U}^{2}_q, \epsilon_i\,\mathcal{U}^{1}_q\colon i\in [q]\right\rangle=\left\langle \{\epsilon_j\epsilon_k\colon 1\le j<k\le q\}, \{\epsilon_i\epsilon_l\colon l\in [q]\} \colon i\in [q]\right\rangle.$$ When $q\le 2$, then $\mathbb{U}^2_q=\left \langle \epsilon_i\,\mathcal{U}^{1}_q\colon i\in [q]\right\rangle$. If $r=3$ and $q\ge 4$, then $$\begin{aligned} \mathbb{U}^3_q&=\left \langle \mathcal{U}^{3}_q, \epsilon_i\,\mathcal{U}^{2}_q, \epsilon_j\epsilon_k\,\mathcal{U}^1_q\colon i,j,k\in [q]\right\rangle\\ &=\left\langle \{\epsilon_a\epsilon_b\epsilon_c\colon 1\le a<b<c\le q\}, \{\epsilon_i\epsilon_u\epsilon_v\colon u,v\in [q], u<v\}, \{\epsilon_j\epsilon_k\epsilon_w\colon w\in [3]\} \colon i,j,k\in [q]\right\rangle. \end{aligned}$$ Also, $\mathbb{U}^3_3=\left \langle \epsilon_i\,\mathcal{U}^{2}_q, \epsilon_j\epsilon_k\,\mathcal{U}^1_q\colon i,j,k\in [q]\right\rangle$ and $\mathbb{U}^3_q=\left \langle \epsilon_j\epsilon_k\,\mathcal{U}^1_q\colon j,k\in [q]\right\rangle$ when $q\le 2$. If $r=4$ and $q=3$, then $$\begin{aligned} \mathbb{U}^4_3&=\left \langle \epsilon_i\epsilon_j\,\mathcal{U}^{2}_3, \epsilon_a\epsilon_b\epsilon_c\,\mathcal{U}^{1}_3\colon i,j ,a,b,c\in [3]\right\rangle\\ &=\left\langle \{\epsilon_i\epsilon_j\epsilon_u\epsilon_v\colon 1\le u<v\le 3\}, \{\epsilon_a\epsilon_b\epsilon_c\epsilon_k\colon k\in [3] \} \colon i,j,a,b,c\in [q]\right\rangle. \end{aligned}$$ A geometric realization of the simplicial complex $\mathbb{U}^4_3$ in [\[f:U-pic\]](#f:U-pic){reference-type="ref" reference="f:U-pic"} shows that $\mathbb{U}^4_3$ has $16$ triangle facets. **Remark 27** (**$\mathbb{U}_q^2=\mathbb{L}_q^2$**). Comparing [Definition 16](#L2){reference-type="ref" reference="L2"} and [\[e:U2\]](#e:U2){reference-type="eqref" reference="e:U2"}, we see $\mathbb{U}_q^2=\mathbb{L}_q^2$. Then [Proposition 18](#L2-supports){reference-type="ref" reference="L2-supports"} implies that $\mathbb{U}_q^2$ supports a minimal free resolution of ${\mathcal{E}_q}^2$. We now start building the steps towards [Theorem 30](#t:f-vector){reference-type="ref" reference="t:f-vector"}, which shows that the facets of $\mathbb{U}^r_q$ are also facets of $\mathbb{S}^r_q$, and as a result we establish effective lower bounds on the betti numbers of ${\mathcal{E}_q}^r$. **Lemma 28** (**Monomial labels on $\mathbb{U}^r_q$**). *Assume $r\leq q$ so that $\mathcal{U}^r_q\neq \emptyset$. Then the following hold.* 1. *The monomial label of the square-free face $\mathcal{U}^r_q$ is $$\label{e:murq} \mathbf{m}_{\mathcal{U}^r_q}=\prod_{\emptyset\ne A \subseteq [q]} {x_A}^{\min(|A|,r)}=\frac{\epsilon_1\epsilon_2 \ldots \epsilon_q}{\displaystyle \prod_{\tiny\begin{array}{c}A \subseteq [q]\\ r <|A| \end{array}} {x_A}^{|A|-r}}.$$* 2. *If $v=\epsilon_{i_1}\epsilon_{i_2}\cdots \epsilon_{i_r} \in \mathcal{U}^r_q$ for $1 \leq i_1 < i_2 <\ldots<i_r\leq q$, then $$\mathbf{m}_{\mathcal{U}^r_q\setminus\{v\}}=\displaystyle \frac{\mathbf{m}_{\mathcal{U}^r_q}}{x_{\{i_1,\ldots, i_r\}}}\,.$$* 3. *If $\mathbf{a}\in \mathbb N^q$ with $r-q<|\mathbf{a}|<r$, then $$\mathbf{m}_{\mathcal{U}_{\mathbf{a}}} ={\pmb{\epsilon}}^{\mathbf{a}}\cdot \prod_{\tiny \emptyset\ne A \subseteq [q]} {x_A}^{\min(|A|,r-|\mathbf{a}|)} =\frac{{\pmb{\epsilon}}^{\mathbf{a}}\epsilon_1\epsilon_2 \ldots \epsilon_q}{\displaystyle \prod_{\tiny\begin{array}{c}A \subseteq [q]\\ r-|\mathbf{a}| <|A| \end{array}} {x_A}^{|A|-(r-|\mathbf{a}|)}}.$$* *Proof.* Let $\mathcal{J}$ denote the set of all subsets of $[q]$ with $r$ elements. Then $\mathcal{J}$ is in one-to-one correspondence to $\mathcal{U}^r_q$. If $J\in \mathcal{J}$ with $J=\{j_1, \dots, j_r\}$, then $$\epsilon_{j_1}\dots \epsilon_{j_r}= \prod_{\emptyset\ne A\subseteq [q]}x_A^{|A\cap J|}\,.$$ We have then $$\begin{aligned} \mathbf{m}_{\mathcal{U}^r_q} &= \mathrm{lcm}\Big ( \prod_{\emptyset\ne A\subseteq [q]}x_A^{|A\cap J|} \colon J\in \mathcal{J}\Big ) =\prod_{\tiny \emptyset\ne A \subseteq [q]} {x_A}^{\min(|A|,r)} =\prod_{\tiny \begin{array}{c} \emptyset\ne A \subseteq [q]\\ |A| \leq r \end{array}} {x_A}^{|A|} \cdot \prod_{\tiny \begin{array}{c} A \subseteq [q]\\ |A| > r \end{array}} {x_A}^{r}\\ &=\frac{\displaystyle \prod_{\tiny \emptyset\ne A \subseteq [q]} {x_A}^{|A|}} {\displaystyle \prod_{\tiny \begin{array}{c} A \subseteq [q]\\ |A| > r \end{array}} {x_A}^{|A|-r}} =\frac{\epsilon_1\epsilon_2 \ldots \epsilon_q}{\displaystyle \prod_{\tiny \begin{array}{c} A \subseteq [q]\\ |A| > r \end{array}} {x_A}^{|A|-r}}. \end{aligned}$$ Next, given $v$ as in the hypothesis, set $B=\{i_1, \dots, i_r\}$. Since $q>r$, for every nonempty $A \subseteq [q]$ we have $$\max\{|A\cap J|\colon J\in \mathcal{J}, J\ne B\}=\begin{cases} \min(|A|,r) &\text{if $A\ne B$}\\ r-1 &\text{if $A=B$} \end{cases}$$ and then $$\begin{aligned} \mathbf{m}_{\mathcal{U}^r_q\smallsetminus\{v\}} &=\mathrm{lcm}\Big ( \prod_{\emptyset\ne A\subseteq [q]}x_A^{|A\cap J|} \colon J\in\mathcal{J}, J\ne B \Big ) = \prod_{\tiny \emptyset\ne A \subseteq [q]} {x_A}^{\max\{|A\cap J|\colon J\in \mathcal{J}, J\ne B\}} \\ &= {x_B}^{r-1} \prod_{\tiny \begin{array}{c}\emptyset\ne A \subseteq [q]\\A\ne B\end{array}} {x_A}^{\min(|A|,r)}. \end{aligned}$$ The last statement follows now from [\[e:murq\]](#e:murq){reference-type="eqref" reference="e:murq"}. ◻ **Proposition 29**. *Let $q, r\geq 1$. If ${\pmb{\epsilon}}^{\mathbf{b}}\notin \mathcal{U}_{\mathbf{a}}$ for some $\mathbf{a} \in \mathbb{N}^q$ with $r-q < |\mathbf{a}| < r$ and $\mathbf{b}\in \mathcal{N}^r_q$, then then there exists $\mathbf{c}\in \mathcal{N}^r_q$ such that $$\label{e:german} {\pmb{\epsilon}}^{\mathbf{c}}\in \mathcal{U}_{\mathbf{a}}\quad \mbox{and} \quad \mathbf{m}_{\mathcal{U}_{\mathbf{a}}\cup \{{\pmb{\epsilon}}^{\mathbf{b}}\}}=\mathbf{m}_{(\mathcal{U}_{\mathbf{a}}\cup \{{\pmb{\epsilon}}^{\mathbf{b}}\})\setminus\{{\pmb{\epsilon}}^{\mathbf{c}}\}}.$$* *Proof.* Suppose $\mathbf{a}=(a_1,\dots, a_q)$ and $\mathbf{b}=(b_1,\dots, b_q)$. Set $r'=r-|\mathbf{a}|$. The hypothesis implies $r'<q$. Assume first $\epsilon\text{-}\mathrm{gcd}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}})=1$, and hence $\mathrm{Supp}(\mathbf{a})\cap \mathrm{Supp}(\mathbf{b})=\emptyset$. After a possible reordering, without loss of generality assume for some $t$ and $h$ with $1 \leq t < h \le q$ $${\pmb{\epsilon}}^{\mathbf{b}}={\epsilon_1}^{b_1}\cdots {\epsilon_t}^{b_t} \quad \mbox{and} \quad {\pmb{\epsilon}}^{\mathbf{a}}={\epsilon_{h}}^{a_{h}}\cdots {\epsilon_{q}}^{a_q},$$ where $a_i, b_j >0$ when $j\leq t$ and $h\le i\le q$, and $$b_1 + \cdots + b_t = r \quad \mbox{and} \quad a_{h} + \cdots + a_q = r-r'.$$ Set ${\pmb{\epsilon}}^{\mathbf{c}}={\pmb{\epsilon}}^{\mathbf{a}}w$ where $$w=\epsilon_1\epsilon_2\ldots \epsilon_{r'}\quad \mbox{and} \quad A=\{1, \ldots, r'\}.$$ We need to show that $${\pmb{\epsilon}}^{\mathbf{a}}w\mid \mathrm{lcm}\left(\mathbf{m}_{\mathcal{U}_{\mathbf{a}}\setminus\{ {\pmb{\epsilon}}^{\mathbf{a}}w\}}, {\pmb{\epsilon}}^{\mathbf{b}}\right).$$ In other words, for every nonempty subset $B$ of $[q]$, and $s\ge 0$, we need to show the implication $$\label{implication} {x_B}^s\mid {\pmb{\epsilon}}^{\mathbf{a}}w \implies {x_B}^s\mid \mathrm{lcm}\left(\mathbf{m}_{\mathcal{U}_{\mathbf{a}}\setminus\{ {\pmb{\epsilon}}^{\mathbf{a}}w\}}, {\pmb{\epsilon}}^{\mathbf{b}}\right)$$ where by [Lemma 28](#l:necessary){reference-type="ref" reference="l:necessary"} $$\label{e:fraction} \mathbf{m}_{\mathcal{U}_{\mathbf{a}}\setminus\{ {\pmb{\epsilon}}^{\mathbf{a}}w\}} = \frac{{\pmb{\epsilon}}^{\mathbf{a}}\mathbf{m}_{\mathcal{U}_q^{r'}}}{x_A}.$$ Assume ${x_B}^s\mid {\pmb{\epsilon}}^{\mathbf{a}}w$. If $B\ne A$, then ${x_B}^s$ divides the fraction in [\[e:fraction\]](#e:fraction){reference-type="eqref" reference="e:fraction"}, and hence [\[implication\]](#implication){reference-type="eqref" reference="implication"} holds. Assume now $B=A$. The largest $s$ such that ${x_A}^s\mid {\pmb{\epsilon}}^{\mathbf{a}} w$ is $$s=r'+a, \quad \mbox{where} \quad a=\begin{cases} a_{h}+\dots +a_{r'} &\text{if $r'\ge h$}\\ 0 &\text{if $r'<h$.} \end{cases}$$ If $r'\le t$, then $r'<h$, hence $s=r'$. We have ${x_A}^{r'} \mid {\pmb{\epsilon}}^{\mathbf{b}}$, hence [\[implication\]](#implication){reference-type="eqref" reference="implication"} holds. Assume now $r' >t$. Since $b_1+\dots +b_t=r$, we have ${x_A}^r\mid {\pmb{\epsilon}}^{\mathbf{b}}$. Since $a\le r-r'$, it follows that ${x_A}^{r'+a}\mid {\pmb{\epsilon}}^{\mathbf{b}}$, and thus [\[implication\]](#implication){reference-type="eqref" reference="implication"} is established, finishing the proof under the assumption that $\epsilon\text{-}\mathrm{gcd}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}})=1$. Next, assume $\epsilon\text{-}\mathrm{gcd}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}})={\pmb{\epsilon}}^{\mathbf{d}}\ne 1$, where $\mathbf{d}\in \mathcal{N}_q^d$ for some $d$ with $0 < d \le r-r'$. Write ${\pmb{\epsilon}}^{\mathbf{a}}= {\pmb{\epsilon}}^{\mathbf{d}}\cdot {\pmb{\epsilon}}^{\mathbf{a}'}$ and ${\pmb{\epsilon}}^{\mathbf{b}}= {\pmb{\epsilon}}^{\mathbf{d}}\cdot {\pmb{\epsilon}}^{\mathbf{b}'}$, with $\mathbf{a}' \in \mathcal{N}_q^{r-r'-d}$, $\mathbf{b}' \in \mathcal{N}_q^{r-d}$ and $\epsilon\text{-}\mathrm{gcd}({\pmb{\epsilon}}^{\mathbf{a}'}, {\pmb{\epsilon}}^{\mathbf{b}'})=1$. Using the argument above, choose $w\in \mathcal{U}_q^{r'}$ such that $$\mathbf{m}_{\mathcal{U}_{\mathbf{a}'} \cup \{{\pmb{\epsilon}}^{\mathbf{b}'}\}}=\mathbf{m}_{(\mathcal{U}_{\mathbf{a}'} \cup \{{\pmb{\epsilon}}^{\mathbf{b}'}\})\setminus\{{\pmb{\epsilon}}^{\mathbf{a}'} w\}}.$$ The equality [\[e:german\]](#e:german){reference-type="eqref" reference="e:german"} follows from here. ◻ We are now ready to state the main theorem in this section. Recall that the $\mathbf{f}$-vector of a simplicial complex $\Delta$ is the tuple $\mathbf{f}(\Delta)=(f_i)_{i\ge 0}$ with $f_i$ denoting the number of $i$-dimensional faces of $\Delta$. **Theorem 30** (**Facets$(\mathbb{U}^r_q) \subseteq$ Facets$(\mathbb{S}^r_q)$**). *The set $\mathcal{U}_{\mathbf{a}}$ is a facet of of the Scarf complex $\mathbb{S}^r_q$ for all $\mathbf{a}\in \mathbb{N}^q$ such that $r-q<|\mathbf{a}|<r$. In particular, $\mathbb{U}^r_q\subseteq \mathbb{S}^r_q$.* *Furthermore, for all $i\ge 0$, $$f_i\le \beta_i^S({\mathcal{E}_q}^r)$$ where $\mathbf{f}(\mathbb{U}^r_q)=(f_i)_{i\ge 0}$.* *Proof.* We first show $\mathcal{U}^r_q$ is a face of $\mathbb{S}^r_q$. If $u, v$ are vertices of $\mathbb{T}^r_q$ with $u \in \mathcal{U}^r_q$ and $v \notin \mathcal{U}^r_q$, then we observe that by [Lemma 28](#l:necessary){reference-type="ref" reference="l:necessary"} $$\mathbf{m}_{\mathcal{U}^r_q\setminus\{u\}}\ne \mathbf{m}_{\mathcal{U}^r_q}.$$ Moreover we claim $$\mathbf{m}_{\mathcal{U}^r_q\cup \{v\}}\ne \mathbf{m}_{\mathcal{U}^r_q}.$$ To see this assume, without loss of generality, that $$v ={\epsilon_{1}}^{a_1}\cdots {\epsilon_{t}}^{a_t} \in \mathbb{T}^r_q\setminus\mathcal{U}^r_q, \quad a_1>1.$$ Since $a_1>1$, it must be that $t<r$. It suffices to show that $v\nmid \mathbf{m}_{\mathcal{U}^r_q}$. Indeed, ${x_{[t]}}^r\mid v$, but the largest power of $x_{[t]}$ that divides $\mathbf{m}_{\mathcal{U}^r_q}$ is ${x_{[t]}}^t$. The conclusion follows thus from the inequality $t<r$. Using [Lemma 1](#l:expansion){reference-type="ref" reference="l:expansion"}, it follows that $\mathcal{U}^r_q\in \mathbb{S}^r_q$. The fact that $\mathcal{U}_{\mathbf{a}}\in \mathbb{S}^r_q$ now follows from [Proposition 21](#p:Scarf-face){reference-type="ref" reference="p:Scarf-face"}. To see that it is a facet, apply [Proposition 29](#p:german){reference-type="ref" reference="p:german"} with $r'=r-|\mathbf{a}|$, noting that $r'\le q-1$ because $r-|\mathbf{a}|<q$. ◻ An immediate consequence is that ${\mathcal{E}_q}^2$ is a Scarf ideal. **Corollary 31** (**${\mathcal{E}_q}^2$ is Scarf**). *If $q\ge 1$, then $\mathbb{U}^2_q=\mathbb{S}^2_q$. In particular the ideal ${\mathcal{E}_q}^2$ is Scarf.* *Proof.* By [Remark 27](#U=L){reference-type="ref" reference="U=L"}, we know that $\mathbb{U}^2_q$ supports a minimal free resolution of $\mathcal{E}_q^2$. Since $\mathbb{U}^2_q\subseteq \mathbb{S}^{2}_{q}$, it follows that $\mathbb{U}^{2}_{q}=\mathbb{S}^{2}_{q}$. ◻ Another consequence of [Theorem 30](#t:f-vector){reference-type="ref" reference="t:f-vector"} is [Corollary 32](#c:scarf-betti){reference-type="ref" reference="c:scarf-betti"} below, which identifies some of the multi-graded betti numbers of ${\mathcal{E}_q}^r$. **Corollary 32** (**Multigraded betti numbers from $\mathbb{U}^r_q$**). *Let $r\ge 1$, $q\ge 1$ and $\mathbf{a}\in \mathbb{N}^q$ with $r-q<|\mathbf{a}|<r$. Then $$\beta_{i,\mathbf{m}_{\mathcal{U}_{\mathbf{a}}}}({\mathcal{E}_q}^r)=1 \quad \mbox{where} \quad i=\binom{q}{r-|\mathbf{a}|}.$$* As mentioned in the introduction, we believe that $\mathbb{S}^r_q$ always supports a minimal free resolution of ${\mathcal{E}_q}^r$. The next result provides support in this direction, by showing that for every facet of $\mathbb{S}^r_q$ of the form $\mathcal{U}_{\mathbf{a}}$, there is a CW complex that supports a free resolution of ${\mathcal{E}_q}^r$ with a cell corresponding to $\mathcal{U}_{\mathbf{a}}$ and no cells corresponding to faces of $\mathbb{T}^r_q$ which strictly contain $\mathcal{U}_{\mathbf{a}}$. **Theorem 33** (**A cellular resolution of ${\mathcal{E}_q}^r$ based on $\mathcal{U}_{\mathbf{a}}$**). *If $q,r \geq 1$, and $\mathbf{a}\in \mathbb{N}^q$ with $r-q<|\mathbf{a}|<r$, then there is a matching $\mathcal{A}$ on the faces of $\mathbb{T}^r_q$ whose critical cells form a CW complex $\mathcal{X}$ such that* - *$\mathcal{X}$ supports a resolution of ${\mathcal{E}_q}^r$;* - *$\mathcal{X}$ contains (an isomorphic copy of) the simplex $\mathcal{U}_{\mathbf{a}}$;* - *the cells of $\mathcal{X}$ are in (monomial label-preserving) one-to-one correspondence with the set of faces of $\mathbb{T}^r_q$ that do not strictly include $\mathcal{U}_{\mathbf{a}}$.* *Proof.* Let $\mathbf{a}\in \mathbb{N}^q$ with $r-q<|\mathbf{a}|<r$. Set $$\mathbb{T}_\mathbf{a}=\{\sigma \in \mathbb{T}^r_q\colon\mathcal{U}_{\mathbf{a}}\subsetneq \sigma\}.$$ *Claim 1*. For $\sigma, \tau \in \mathbb{T}_\mathbf{a}$ and $w\in \mathcal{U}_{\mathbf{a}}$ and $w'\in \tau$, if $\sigma\smallsetminus\{w\}=\tau\smallsetminus\{w'\}$, then $\sigma=\tau$. To prove the Claim, consider the fact that both $\sigma$ and $\tau$ contain $\mathcal{U}_{\mathbf{a}}$, so $\sigma \cap \mathcal{U}_{\mathbf{a}}=\tau \cap \mathcal{U}_{\mathbf{a}}=\mathcal{U}_{\mathbf{a}}$ and $$\mathcal{U}_{\mathbf{a}}\smallsetminus\{w'\}=(\tau\smallsetminus\{w'\})\cap \mathcal{U}_{\mathbf{a}}=(\sigma\smallsetminus\{w\})\cap \mathcal{U}_{\mathbf{a}}=\mathcal{U}_{\mathbf{a}}\smallsetminus\{w\}\,.$$ Since $w\in \mathcal{U}_{\mathbf{a}}$, we must have $w=w'$ and hence $\sigma=\tau$. This proves the claim. For each $\sigma \in \mathbb{T}_\mathbf{a}$, use [Proposition 29](#p:german){reference-type="ref" reference="p:german"} to pick $w_\sigma \in \mathcal{U}_{\mathbf{a}}$ such that $$\label{e:homogeneous} \mathbf{m}_{\sigma}=\mathbf{m}_{\sigma \setminus\{w_\sigma\}}.$$ We will prove the desired statement by constructing a homogeneous acyclic matching on $G_{\mathbb{T}^r_q}$. We set $$\mathcal{A}=\{\sigma\to \sigma\smallsetminus\{w_\sigma\}\in G_{\mathbb{T}^r_q}\colon \sigma\in \mathbb{T}_\mathbf{a}\}.$$ The Claim shows that $\mathcal{A}$ is a matching, and [\[e:homogeneous\]](#e:homogeneous){reference-type="eqref" reference="e:homogeneous"} ensures that the matching $\mathcal{A}$ is homogeneous. We now show that $\mathcal{A}$ is acyclic. A cycle $\mathcal{C}$ will have edges alternating between those in $\mathcal{A}$ and those outside $\mathcal{A}$. In this picture, the up arrows are edges in $\mathcal{A}$ (reversed) and the down arrows are of the type $\sigma\to \sigma\smallsetminus\{v\}\notin \mathcal{A}$. We have thus $\sigma_t\smallsetminus\{v\}=\sigma_1\smallsetminus\{w_{\sigma_1}\}$ for some $v\in \sigma_t$. Then the Claim implies $\sigma_1=\sigma_t$, a contradiction to $\mathcal{C}$ being a cycle. As a result, $\mathcal{A}$ is a homogeneous acyclic matching, and therefore by [Theorem 6](#t:BW){reference-type="ref" reference="t:BW"} there is a CW complex $\mathcal{X}_{\mathcal{A}}$ which supports a resolution of ${\mathcal{E}_q}^r$, and its cells are in (label-preserving) one-to-one correspondence with the $\mathcal{A}$-critical cells of $\mathbb{T}^r_q$. Now note that since $\mathcal{U}_{\mathbf{a}}$ and all of its faces have Scarf monomial labels, they cannot be vertices of any edges in $\mathcal{A}$. Therefore for every $\sigma \subseteq \mathcal{U}_{\mathbf{a}}$, $\sigma$ is $\mathcal{A}$-critical and every subface of $\sigma$ is $\mathcal{A}$-critical by the same reasoning. Also since no Scarf face can appear in a path as in [\[e:gradient\]](#e:gradient){reference-type="eqref" reference="e:gradient"} the only subcells of $\sigma_{\mathcal{A}}$ are $\tau_{\mathcal{A}}$ where $\tau \subseteq \sigma$. This proves that $(\mathcal{U}_{\mathbf{a}})_{\mathcal{A}}$ is (isomorphic to) a simplex in $\mathcal{X}_{\mathcal{A}}$. ◻ **Remark 34**. It is possible to extend the matching in [Theorem 33](#t:german){reference-type="ref" reference="t:german"} and remove faces of the Taylor complex containing multiple $\mathcal{U}_{\mathbf{a}}$'s at the same time, or even almost all of them. This is somewhat beyond the scope of this paper, so we do not include it. However, it provides more evidence that the ideals ${\mathcal{E}_q}^r$ may be Scarf ideals. # **${\mathcal{E}_q}^r$ is a Scarf ideal when $q\le 4$** {#s:small-q} [\[s:small-q\]]{#s:small-q label="s:small-q"} This section is dedicated to proving in [Theorem 44](#t:Morse-small-q){reference-type="ref" reference="t:Morse-small-q"} the statement in the title. We also obtain a concrete description of the Scarf complex of ${\mathcal{E}_q}^r$ when $q\le 4$, namely $\mathbb{S}^r_q=\mathbb{U}^r_q$. We start with an application of [Lemma 19](#l:when-divide){reference-type="ref" reference="l:when-divide"} in establishing divisibility relations that are ingredients in our main result. **Lemma 35** (**Divisibility conditions II**). *Let $r,q\ge 1$ and $\mathbf{a}_1, \mathbf{a}_2\in \mathcal{N}^r_q$. The following hold:* 1. *If $u\in \mathrm{Supp}(\mathbf{a}_1)$, $k\in\mathrm{Supp}(\mathbf{a}_2)$ and $\mathrm{Supp}(\mathbf{a}_1)=\{u\}$ or $\mathrm{Supp}(\mathbf{a}_2)=\{k\}$ , then $$\frac{{\pmb{\epsilon}}^{\mathbf{a}_1}\epsilon_k}{\epsilon_u}\mid \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}_1}, {\pmb{\epsilon}}^{\mathbf{a}_2})\,.$$* 2. *Assume $\mathrm{Supp}(\mathbf{a}_1)=\{u,v\}$ and $\mathrm{Supp}(\mathbf{a}_2)=\{k,l\}$ with $u\ne v$ and $k\ne l$. Then $$\frac{{\pmb{\epsilon}}^{\mathbf{a}_1}\epsilon_k\epsilon_l}{\epsilon_u\epsilon_v}\mid \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}_1}, {\pmb{\epsilon}}^{\mathbf{a}_2})\,.$$* *Proof.* Write $\mathbf{a}_1=(a_{11}, a_{12},\dots, a_{1q})$ and $\mathbf{a}_2=(a_{21}, a_{22}, \dots, a_{2q})$. Recall from [Lemma 19](#l:when-divide){reference-type="ref" reference="l:when-divide"} that, if $\mathbf{b}=(b_1, \dots, b_q)\in \mathcal{N}^r_q$, then $$\label{e:show} {\pmb{\epsilon}}^{\mathbf{b}}\mid \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}_1}, {\pmb{\epsilon}}^{\mathbf{a}_2})\iff \sum_{i\in A}b_i\le \max(\sum_{i\in A}a_{1i}, \sum_{i\in A}a_{2i})$$ for all $A$ with $\emptyset\ne A\subseteq [q]$. Let $A$ be as above. We proceed to prove the inequality in [\[e:show\]](#e:show){reference-type="eqref" reference="e:show"} under the given hypotheses. \(1\) We may assume $k\ne u$, since the statement is trivial otherwise. Let $\mathbf{b}=(b_1, \dots, b_q)\in \mathcal{N}^r_q$ so that ${\pmb{\epsilon}}^{\mathbf{b}}=\displaystyle \frac{{\pmb{\epsilon}}^{\mathbf{a}_1}\epsilon_k}{\epsilon_u}$. We have then $$\sum_{i\in A}b_i=\sum_{i\in A}a_{1i}-|A\cap\{u\}|+|A\cap\{k\}|\,.$$ This expression is less than or equal to $\sum_{i\in A}a_{1i}$ when $u\in A$ or $k\notin A$, and hence it suffices to assume $u\notin A$ and $k\in A$. We further observe $$\begin{aligned} \mathrm{Supp}(\mathbf{a}_1)=\{u\}&\,\implies\,\sum_{i\in A}b_i=1\le \sum_{i\in A}a_{2i}\\ \mathrm{Supp}(\mathbf{a}_2)=\{k\}&\,\implies\, \sum_{i\in A}b_i\le r=\sum_{i\in A}a_{2i}\,,\end{aligned}$$ establishing thus the inequality in [\[e:show\]](#e:show){reference-type="eqref" reference="e:show"}. \(2\) We may assume that $u,v,k,l$ are all distinct, since otherwise the statement reduces, after factoring out a common factor, to (1). Let $\mathbf{b}\in \mathcal{N}^r_q$ such that ${\pmb{\epsilon}}^{\mathbf{b}}={\displaystyle\frac{{\pmb{\epsilon}}^{\mathbf{a}_1}\epsilon_k\epsilon_l}{\epsilon_u\epsilon_v}}$. We have then $$\sum_{i\in A}b_i=\sum_{i\in A} a_{1i}-|A\cap \{u,v\}|+|A\cap \{k,l\}|\,.$$ This expression is less than or equal to $\sum_{i\in A}a_{1i}$ when $|A\cap \{k,l\}|\le |A\cap\{u,v\}|$ and hence it suffices to assume $|A\cap \{k,l\}|> |A\cap\{u,v\}|$. We further have: $$\begin{aligned} |A\cap \{k,l\}|=2 &\,\implies\, \sum_{i\in A}b_i\le r= \sum_{i\in A}a_{2i}\\ |A\cap \{k,l\}|=1 \quad\text{and}\quad |A\cap \{u,v\}|=0 &\,\implies\, \sum_{i\in A}b_i=1\le \sum_{i\in A}a_{2i}\,,\end{aligned}$$ establishing thus the inequality in [\[e:show\]](#e:show){reference-type="eqref" reference="e:show"}. ◻ Using the same notation as in [Notation 10](#n:rqs){reference-type="ref" reference="n:rqs"}, we introduce a lexicographic order on $\mathcal{N}^r_q$, with $$\label{e:order} \mathbf{a}> \mathbf{b}\iff \quad\text{there exists $i\in[q]$ with $a_j=b_j$ for all $j<i$ and $a_i > b_i$}.$$ We induce this order on the generators of ${\mathcal{E}_q}^r$ by $${\pmb{\epsilon}}^{\mathbf{a}}> {\pmb{\epsilon}}^{\mathbf{b}}\iff \mathbf{a}>\mathbf{b}.$$ **Lemma 36**. *Assume $\mathbf{a},\mathbf{b}\in \mathcal{N}^r_q$ are such that $$\label{e:conditions} |\mathrm{Supp}(\mathbf{a})\cup\mathrm{Supp}(\mathbf{b})|\le 4,\qquad \mathbf{a}>\mathbf{b}\quad \mbox{and} \quad |a_s-b_s|>1 \quad \mbox{for some} \quad s\in [q].$$ Then there exists $\mathbf{c}\in \mathcal{N}^r_q$ such that the following hold:* 1. *$\mathbf{c}\ne \mathbf{b}$ and $\mathbf{a}>\mathbf{c}$;* 2. *$\{{\pmb{\epsilon}}^{\mathbf{c}}, {\pmb{\epsilon}}^{\mathbf{a}}\}\in \mathbb{S}^r_q$;* 3. *${\pmb{\epsilon}}^{\mathbf{c}}\mid\mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}})$.* *In particular, $\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}\notin \mathbb{S}^r_q$.* *Proof.* First note that the hypothesis implies $r\geq 2$, otherwise $|a_s-b_s|\leq 1$ for all $s \in [q]$. Assume first $\epsilon\text{-}\mathrm{gcd}({\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}})=1$, so that $\mathrm{Supp}(\mathbf{a})\cap \mathrm{Supp}(\mathbf{b})=\emptyset$. *Case 1 1*. ${\pmb{\epsilon}}^{\mathbf{a}}=\epsilon_i^r$ for some $i\geq 1$. The assumption and the hypothesis [\[e:conditions\]](#e:conditions){reference-type="eqref" reference="e:conditions"} imply $$b_k=0 \quad \mbox{for} \quad k\le i, \qquad b_j>0 \quad \mbox{for some} \quad j>i.$$ Let $\mathbf{c}\in \mathcal{N}^r_q$ so that $${\pmb{\epsilon}}^{\mathbf{c}}=\epsilon_i^{r-1}\epsilon_j.$$ Observe ${\pmb{\epsilon}}^{\mathbf{a}}>{\pmb{\epsilon}}^{\mathbf{c}}$, since $j>i$. Also, ${\pmb{\epsilon}}^{\mathbf{c}}\neq {\pmb{\epsilon}}^{\mathbf{b}}$ since $b_i=0$ and $r\ge 2$. We have $$\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{c}}\}=\epsilon_i^{r-1} \{\epsilon_i,\epsilon_j\}\in\mathbb{U}^r_q$$ and hence (2) holds, by using [Theorem 30](#t:f-vector){reference-type="ref" reference="t:f-vector"}. Moreover, ${\pmb{\epsilon}}^{\mathbf{c}}\mid \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}})$ by [Lemma 35](#l:some-divisibility){reference-type="ref" reference="l:some-divisibility"} , so (3) follows. *Case 2 1*. ${\pmb{\epsilon}}^{\mathbf{b}}=\epsilon_i^r$ for some $i\ge 1$. The assumption and the hypothesis [\[e:conditions\]](#e:conditions){reference-type="eqref" reference="e:conditions"} imply $$i \geq 2, \qquad a_j>0 \quad \mbox{for some} \quad j<i, \qquad a_i=0.$$ Let $\mathbf{c}\in \mathcal{N}^r_q$ so that $${\pmb{\epsilon}}^{\mathbf{c}}=\frac{\epsilon_i{\pmb{\epsilon}}^{\mathbf{a}}}{\epsilon_j}\,.$$ Observe ${\pmb{\epsilon}}^{\mathbf{a}}>{\pmb{\epsilon}}^{\mathbf{c}}$, since $j<i$. Also, ${\pmb{\epsilon}}^{\mathbf{c}}\neq {\pmb{\epsilon}}^{\mathbf{b}}$ since $c_i =1\leq r-1 <b_i$. We have $$\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{c}}\}=\frac{{\pmb{\epsilon}}^{\mathbf{a}}}{\epsilon_j} \{\epsilon_i,\epsilon_j\}\in \mathbb{U}^r_q$$ and hence (2) holds, by using [Theorem 30](#t:f-vector){reference-type="ref" reference="t:f-vector"}. Moreover, ${\pmb{\epsilon}}^{\mathbf{c}}\mid \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}})$ by [Lemma 35](#l:some-divisibility){reference-type="ref" reference="l:some-divisibility"} , so (3) follows. *Case 3 1*. $|\mathrm{Supp}(\mathbf{a})|>1$ and $|\mathrm{Supp}(\mathbf{b})|>1$. Since $|\mathrm{Supp}(\mathbf{a})\cup\mathrm{Supp}(\mathbf{b})|\le 4$, we must have $|\mathrm{Supp}(\mathbf{a})|=|\mathrm{Supp}(\mathbf{b})|=2.$ By [\[e:conditions\]](#e:conditions){reference-type="eqref" reference="e:conditions"} we must therefore have $${\pmb{\epsilon}}^{\mathbf{a}}=\epsilon_i^{a_i}\epsilon_j^{a_j} \quad \mbox{and} \quad {\pmb{\epsilon}}^{\mathbf{b}}= \epsilon_k^{b_k}\epsilon_l^{b_l} \quad \mbox{where} \quad a_i,a_j,b_k,b_l>0,$$ $i,j,k,l$ are distinct, $i>k,j,l$ and $\max(a_i,a_j,b_k,b_l)\ge 2$. Take $\mathbf{c}\in \mathcal{N}^r_q$ so that $${\pmb{\epsilon}}^{\mathbf{c}}=\frac{{\pmb{\epsilon}}^{\mathbf{a}}\cdot \epsilon_k\epsilon_l}{\epsilon_i\epsilon_j}=\epsilon_i^{a_i-1}\epsilon_j^{a_j-1}\epsilon_k\epsilon_l\,.$$ Observe $\mathbf{a}>\mathbf{c}$, since $a_i-1<a_i$. Since $\max(a_i,a_j,b_k,b_l)\ge 2$, we also have $\mathbf{c}\ne \mathbf{b}$, hence (1) holds. It follows that $$\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{c}}\}= \epsilon_i^{a_i-1}\epsilon_j^{a_j-1}\{\epsilon_i\epsilon_j,\epsilon_k\epsilon_l\}\in \mathbb{U}^r_q$$ and hence (2) holds, by using [Theorem 30](#t:f-vector){reference-type="ref" reference="t:f-vector"}. Finally, (3) follows from [Lemma 35](#l:some-divisibility){reference-type="ref" reference="l:some-divisibility"}. This finishes the proof in the case $\epsilon\text{-}\mathrm{gcd}({\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}})=1$. In general, let $\epsilon\text{-}\mathrm{gcd}({\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}})={\pmb{\epsilon}}^{\mathbf{d}}$ so that $${\pmb{\epsilon}}^{\mathbf{a}}={\pmb{\epsilon}}^{\mathbf{d}}{\pmb{\epsilon}}^{{\mathbf{a}}'}\quad \mbox{and} \quad {\pmb{\epsilon}}^{\mathbf{b}}={\pmb{\epsilon}}^{\mathbf{d}}{\pmb{\epsilon}}^{{\mathbf{b}}'}\qquad \text{for}\quad \mathbf{a}',\mathbf{b}'\in \mathcal{N}_q^{r-|\mathbf{d}|}\,.$$ It follows that $\epsilon\text{-}\mathrm{gcd}({\pmb{\epsilon}}^{{\mathbf{a}}'},{\pmb{\epsilon}}^{{\mathbf{b}}'})=1$. The hypothesis [\[e:conditions\]](#e:conditions){reference-type="eqref" reference="e:conditions"} implies $|\mathrm{Supp}(\mathbf{a}')\cup\mathrm{Supp}(\mathbf{b}')|\le 4$, $\mathbf{a}'>\mathbf{b}'$ and $|a'_s-b'_s|>1$ for some $s\in [q]$. Applying the first part of the proof for $\mathbf{a}'$ and $\mathbf{b}'$ implies there exists $\mathbf{c}'\in \mathcal{N}_q^{r-|\mathbf{d}|}$ such that $$\mathbf{c}'\ne \mathbf{b}', \qquad \mathbf{a}'>\mathbf{c}', \qquad \{{\pmb{\epsilon}}^{{\mathbf{c}}'},{\pmb{\epsilon}}^{{\mathbf{a}}'}\}\in \mathbb{S}_q^{r-|\mathbf{d}|},\quad \text {and}\quad {\pmb{\epsilon}}^{{\mathbf{c}}'}\mid \mathrm{lcm}({\pmb{\epsilon}}^{{\mathbf{a}}'}, {\pmb{\epsilon}}^{{\mathbf{b}}'})\,.$$ Taking $\mathbf{c}\in \mathcal{N}^r_q$ so that ${\pmb{\epsilon}}^{\mathbf{c}}={\pmb{\epsilon}}^{\mathbf{d}}{\pmb{\epsilon}}^{{\mathbf{c}}'}$ and using [Proposition 21](#p:Scarf-face){reference-type="ref" reference="p:Scarf-face"} for (2), we see that this choice for $\mathbf{c}$ satisfies the desired conclusions. ◻ **Proposition 37**. *Assume $1\le q\le 4$, $r\ge 1$. The following are equivalent for $\sigma\in \mathbb{T}^r_q$:* 1. *$\sigma\in \mathbb{S}^r_q$;* 2. *$\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}\in \mathbb{S}^r_q$ for all ${\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}}\in \sigma$;* 3. *$|a_i-b_i|\le 1$ for all $i\in [q]$ and all ${\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}}\in \sigma$;* 4. *$\sigma\subseteq \mathcal{U}_{\mathbf{c}}$, where ${\pmb{\epsilon}}^{\mathbf{c}}=\epsilon\text{-}\mathrm{gcd}(\sigma)$.* *Proof.* The implication (1)$\implies$(2) is clear. The implication (2)$\implies$(3) follows directly from [Lemma 36](#l:find-c){reference-type="ref" reference="l:find-c"}. The implication (4)$\implies$(1) follows from [Theorem 30](#t:f-vector){reference-type="ref" reference="t:f-vector"}. We show now (3)$\implies$(4). Assume ${\pmb{\epsilon}}^{\mathbf{a}}\in \sigma$. To show ${\pmb{\epsilon}}^{\mathbf{a}}\in \mathcal{U}_{\mathbf{c}}$, we need to show ${\pmb{\epsilon}}^{\mathbf{a}-\mathbf{c}}\in \mathcal{U}_q^{r-|\mathbf{c}|}$, or, equivalently, $a_i-c_i\in \{0,1\}$ for all $i\in [q]$. This follows from (3), since $c_i=\min\{a_i\colon{\pmb{\epsilon}}^{\mathbf{a}}\in \sigma\}$. ◻ Recall that when $q>1$, $\mathbb{U}^r_q$ was defined as the cell complex with facets $\mathcal{U}_{\mathbf{c}}$, for ${\pmb{\epsilon}}^{\mathbf{c}}\in \mathbb{N}^q$ and $r-q<|\mathbf{c}|<r$. We now have a full characterization of the Scarf complex when $q\le 4$: **Corollary 38**. *If $r \geq 1$ and $1 \leq q \leq 4$, then $\mathbb{S}^r_q=\mathbb{U}^r_q$.0◻* **Remark 39**. When $q>4$, it is no longer true that $\mathbb{U}_q^r=\mathbb{S}^r_q$. We check below that, when $q\ge 5$, then $$\{\epsilon_1^2\epsilon_2, \epsilon_3\epsilon_4\epsilon_5\}\in \mathbb{S}^{3}_{q} \smallsetminus\mathbb{U}^{3}_{q}.$$ Indeed, set $\mathbf{m}=\mathrm{lcm}( \epsilon_1^2\epsilon_2, \epsilon_3\epsilon_4\epsilon_5)$. We have $$\begin{aligned} \epsilon_1^2\epsilon_2&=\prod_{\tiny \{1,2\}\subseteq A\subseteq [q]} {x_A}^3 \prod_{\tiny \begin{array}{c} A \subseteq [q]\\1\in A, 2\notin A\end{array}} {x_A}^2 \prod_{\tiny \begin{array}{c}A \subseteq [q]\\2\in A, 1\notin A \end{array}}x_A\\ \epsilon_3\epsilon_4e_5&=\prod_{\tiny \{3,4,5\}\subseteq A\subseteq [q]} {x_A}^3 \prod_{\tiny \begin{array}{c} A \subseteq [q]\\|A\cap \{3,4,5\}|=2\end{array}} {x_A}^2 \prod_{\tiny \begin{array}{c} A \subseteq [q]\\|A\cap \{3,4,5\}|=1\end{array}}x_A\,. \end{aligned}$$ Observe $$\begin{aligned} \label{a1}x_A^3\mid \mathbf{m}& \iff \{1,2\} \subseteq A\quad \mbox{or} \quad \{3,4,5\}\subseteq A\\ \label{a2} x_A^2\mid \mathbf{m}&\iff 1\in A\quad \mbox{or} \quad |A\cap \{3,4,5\}|\ge 2\\ \label{a3}x_i\mid \mathbf{m}&\iff i\in [5]\,.\end{aligned}$$ Assume that $\epsilon_i\epsilon_j\epsilon_k\mid \mathbf{m}$ with $\epsilon_i\epsilon_j\epsilon_k\notin \{\epsilon_1^2\epsilon_2, \epsilon_3\epsilon_4\epsilon_5\}$ and $i\le j\le k$. Using [\[a3\]](#a3){reference-type="eqref" reference="a3"}, we see $i,j,k\in [5]$. We cannot have $i=j=k$, because ${x_i}^3$ does not divide $\mathbf{m}$, cf. [\[a1\]](#a1){reference-type="eqref" reference="a1"}. If $i,j,k$ are all distinct, then ${x_{ijk}}^3\mid \mathbf{m}$ and it follows from [\[a1\]](#a1){reference-type="eqref" reference="a1"} that $i=1$, $j=2$ and $k\in \{3,4,5\}$. But then ${x_{2k}}^2\mid \mathbf{m}$, contradicting [\[a2\]](#a2){reference-type="eqref" reference="a2"}. If exactly two of $i,j,k$ are equal, we must have $i=j=1$ and $k\in \{3,4,5\}$, because $x_s^2\mid \mathbf{m}$ implies $s=1$ by [\[a2\]](#a2){reference-type="eqref" reference="a2"}. But then ${x_{1k}}^3\mid \mathbf{m}$, again a contradiction by [\[a1\]](#a1){reference-type="eqref" reference="a1"}. In view of [Lemma 1](#l:expansion){reference-type="ref" reference="l:expansion"}, this establishes the desired conclusion. We now proceed to build up preliminaries towards proving that $\mathbb{S}^r_q$ supports a (minimal) free resolution of ${\mathcal{E}_q}^r$, when $q\le 4$. **Lemma 40**. *Assume $r\geq 1$ and $1\le q\le 4$, $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}\in \mathcal{N}^r_q$ and $$\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{d}}\}, \{{\pmb{\epsilon}}^{\mathbf{b}}, {\pmb{\epsilon}}^{\mathbf{d}}\}\in \mathbb{S}^r_q\quad \mbox{and} \quad {\pmb{\epsilon}}^{\mathbf{c}}\mid \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}).$$ Then $\{{\pmb{\epsilon}}^{\mathbf{c}}, {\pmb{\epsilon}}^{\mathbf{d}}\}\in \mathbb{S}^r_q$.* *Proof.* Assume $\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{d}}\}, \{{\pmb{\epsilon}}^{\mathbf{b}}, {\pmb{\epsilon}}^{\mathbf{d}}\}\in \mathbb{S}^r_q$. Then by [Proposition 37](#p:U=Scarf){reference-type="ref" reference="p:U=Scarf"} $$|a_i-d_i|\le 1\quad \mbox{and} \quad |b_i-d_i|\le 1\quad \mbox{for all} \quad i\in[q].$$ Equivalently, $$d_i-1\le a_i, b_i\le d_i+1\quad \mbox{for all} \quad i\in [q].$$ Since ${\pmb{\epsilon}}^{\mathbf{c}}\mid \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}})$, [Lemma 19](#l:when-divide){reference-type="ref" reference="l:when-divide"} implies that $$\min(a_i,b_i)\le c_i\le \max(a_i,b_i) \quad \mbox{for all} \quad i\in [q],$$ and hence $$d_i-1\le c_i\le d_i+1\quad \mbox{for all} \quad i\in [q].$$ Therefore, $|c_i-d_i|\le 1$ for all $i\in [q]$ and thus $\{{\pmb{\epsilon}}^{\mathbf{c}}, {\pmb{\epsilon}}^{\mathbf{d}}\}\in \mathbb{S}^r_q$. ◻ We are now ready to introduce a homogeneous acyclic matching on the face poset of $\mathbb{T}^r_q$, when $q\leq 4$, which will result in a minimal free resolution of ${\mathcal{E}_q}^r$ supported on $\mathbb{S}^r_q$. For this purpose, we will need the following notation. **Notation 41**. Let $$\label{e:S} \mathcal{S}=\{(\mathbf{a}, \mathbf{b})\colon\mathbf{a}, \mathbf{b}\in \mathcal{N}^r_q, \mathbf{a}>\mathbf{b}, \{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}\notin \mathbb{S}^r_q\}.$$ On this set, we introduce the following order relation, recalling [\[e:order\]](#e:order){reference-type="eqref" reference="e:order"} $$\label{e:pair-order} (\mathbf{a}, \mathbf{b})\ge (\mathbf{c}, \mathbf{d})\iff \mathbf{a}> \mathbf{c}\text \quad \mbox{or} \quad \mathbf{a}=\mathbf{c}\mbox{ and }\mathbf{b}\ge \mathbf{d}.$$ - For $\sigma\notin \mathbb{S}^r_q$ (note that $|\sigma| \geq 2$) let $$\label{e:nu} \nu(\sigma)=\max \{(\mathbf{a}, \mathbf{b})\in \mathcal{S}\colon{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\in \sigma\}.$$ - For $(\mathbf{a}, \mathbf{b})\in \mathcal{S}$ let $$\label{e:Sab}\mathcal{S}_{\mathbf{a}, \mathbf{b}}=\{\sigma\in \mathbb{T}^r_q\smallsetminus\mathbb{S}^r_q\colon \nu(\sigma)=(\mathbf{a}, \mathbf{b})\}.$$ - For $\sigma\in \mathbb{T}^r_q\smallsetminus\mathbb{S}^r_q$ and $\nu(\sigma)=(\mathbf{a}, \mathbf{b})$, let $\iota(\sigma)$ be the largest (under the order in [\[e:order\]](#e:order){reference-type="eqref" reference="e:order"}) $\mathbf{c}\in \mathcal{N}^r_q$ satisfying the conclusion of [Lemma 36](#l:find-c){reference-type="ref" reference="l:find-c"}, that is, $$\label{e:iota}\iota(\sigma)=\max \{\mathbf{c}\in \mathcal{N}^r_q\colon\mathbf{a}> \mathbf{c}\ne \mathbf{b}, \quad \{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{c}}\}\in \mathbb{S}^r_q\quad \mbox{and} \quad {\pmb{\epsilon}}^{\mathbf{c}}\mid \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}})\}.$$ **Remark 42**. Note that by [Proposition 37](#p:U=Scarf){reference-type="ref" reference="p:U=Scarf"} the sets $\mathcal{S}_{\mathbf{a},\mathbf{b}}$, with $\mathbf{a},\mathbf{b}\in \mathcal{N}^r_q$, $\mathbf{a}>\mathbf{b}$ and $\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}\notin \mathbb{S}^r_q$ form a partition of $\mathbb{T}^r_q\smallsetminus\mathbb{S}^r_q$, $$\label{e:partition} \mathbb{T}^r_q\smallsetminus\mathbb{S}^r_q= \bigcup_{\tiny \begin{array}{c}\mathbf{a},\mathbf{b}\in \mathcal{N}^r_q, \mathbf{a}>\mathbf{b}\\ \{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}\notin \mathbb{S}^r_q\end{array}} \mathcal{S}_{\mathbf{a},\mathbf{b}}.$$ **Proposition 43**. *Assume $r\ge 1$, $1 \leq q \leq 4$, and $\sigma \in \mathbb{T}^r_q\smallsetminus\mathbb{S}^r_q$.* *If $\mathbf{c}=\iota(\sigma)$, then $\sigma\cup\{{\pmb{\epsilon}}^{\mathbf{c}}\}\notin \mathbb{S}^r_q$ and $\sigma\smallsetminus\{{\pmb{\epsilon}}^{\mathbf{c}}\}\notin \mathbb{S}^r_q$ and* 1. *$\nu(\sigma\cup\{{\pmb{\epsilon}}^{\mathbf{c}}\})= \nu(\sigma)=\nu(\sigma\smallsetminus\{{\pmb{\epsilon}}^{\mathbf{c}}\})$;* 2. *$\iota(\sigma\cup\{{\pmb{\epsilon}}^{\mathbf{c}}\})= \iota(\sigma)=\iota(\sigma\smallsetminus\{{\pmb{\epsilon}}^{\mathbf{c}}\})$;* 3. *$\mathbf{m}_{\sigma\cup\{{\pmb{\epsilon}}^{\mathbf{c}}\}}=\mathbf{m}_\sigma=\mathbf{m}_{\sigma\smallsetminus\{{\pmb{\epsilon}}^{\mathbf{c}}\}}$.* *Proof.* Suppose $$\nu(\sigma)=(\mathbf{a}, \mathbf{b})\in \mathcal{S}, \quad \iota(\sigma)=\mathbf{c}\quad \mbox{and} \quad \sigma'=\sigma\cup\{{\pmb{\epsilon}}^{\mathbf{c}}\}.$$ Since $\sigma\notin \mathbb{S}^r_q$, we also have $\sigma'\notin \mathbb{S}^r_q$. Also, $\mathbf{c}\ne \mathbf{a}$, $\mathbf{c}\ne \mathbf{b}$ by [\[e:iota\]](#e:iota){reference-type="eqref" reference="e:iota"}, and, since $\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}\notin \mathbb{S}^r_q$, it follows that $\sigma\smallsetminus\{{\pmb{\epsilon}}^{\mathbf{c}}\}\notin \mathbb{S}^r_q$. \(1\) Suppose $\nu(\sigma')=(\mathbf{a}', \mathbf{b}')$, so that by [\[e:S\]](#e:S){reference-type="eqref" reference="e:S"} and [\[e:nu\]](#e:nu){reference-type="eqref" reference="e:nu"} we have $\{{\pmb{\epsilon}}^{{\mathbf{a}}'}, {\pmb{\epsilon}}^{{\mathbf{b}}'}\} \notin \mathbb{S}^r_q$. If $\mathbf{a}'<\mathbf{a}$, then $\nu(\sigma')<(\mathbf{a}, \mathbf{b})$, a contradiction. Assume $\mathbf{a}'> \mathbf{a}$. Since $\mathbf{a}>\mathbf{c}$, we have $\mathbf{a}'\ne \mathbf{c}$. If $\mathbf{b}' \ne \mathbf{c}$, then ${\pmb{\epsilon}}^{{\mathbf{a}}'}, {\pmb{\epsilon}}^{{\mathbf{b}}'}\in \sigma$ and thus $\nu(\sigma)= (\mathbf{a}',\mathbf{b}') \neq (\mathbf{a}, \mathbf{b})$, a contradiction. We must have $\mathbf{b}' = \mathbf{c}$, and therefore $$\nu(\sigma')=(\mathbf{a}', \mathbf{c}) \quad \mbox{with} \quad \mathbf{a}'>\mathbf{a}, \quad {\pmb{\epsilon}}^{{\mathbf{a}}'}\in \sigma.$$ Using the order [\[e:pair-order\]](#e:pair-order){reference-type="eqref" reference="e:pair-order"}, we see $(\mathbf{a}',\mathbf{a})>(\mathbf{a},\mathbf{b})$ and $(\mathbf{a}',\mathbf{b})>(\mathbf{a}, \mathbf{b})$. Further, since ${\pmb{\epsilon}}^{{\mathbf{a}}'}\in \sigma$ and $\nu(\sigma)=(\mathbf{a},\mathbf{b})$, the definitions in [\[e:Sab\]](#e:Sab){reference-type="eqref" reference="e:Sab"} and [\[e:S\]](#e:S){reference-type="eqref" reference="e:S"} give $$\{{\pmb{\epsilon}}^{{\mathbf{a}}'}, {\pmb{\epsilon}}^{\mathbf{a}}\}, \{{\pmb{\epsilon}}^{{\mathbf{a}}'}, {\pmb{\epsilon}}^{\mathbf{b}}\}\in\mathbb{S}^r_q.$$ Then [Lemma 40](#l:d){reference-type="ref" reference="l:d"} implies $\{{\pmb{\epsilon}}^{{\mathbf{a}}'}, {\pmb{\epsilon}}^{\mathbf{c}}\}\in \mathbb{S}^r_q$, a contradiction. We conclude $\mathbf{a}'=\mathbf{a}$. If ${\pmb{\epsilon}}^{{\mathbf{b}}'}\notin\sigma$, then $\mathbf{b}'=\mathbf{c}$ and $\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{c}}\}\notin\mathbb{S}^r_q$, a contradiction. So ${\pmb{\epsilon}}^{{\mathbf{b}}'}\in \sigma$, implying $\mathbf{b}'=\mathbf{b}$, and this concludes the proof of the first equality in (1). The second equality in (1) follows directly from [\[e:nu\]](#e:nu){reference-type="eqref" reference="e:nu"}, since $\mathbf{c}\ne \mathbf{a}$ and $\mathbf{c}\ne \mathbf{b}$. \(2\) Since $\iota(\sigma')$ only depends on $\nu(\sigma')=\nu(\sigma)$, this equality follows immediately. \(3\) This statement follows directly from [\[e:iota\]](#e:iota){reference-type="eqref" reference="e:iota"}. ◻ **Theorem 44** (**A Scarf resolution for ${\mathcal{E}_q}^r$ when $q \leq 4$**). *Assume $1 \le q\le 4$ and $r\geq 1$. Using the notation as in [\[e:cab\]](#e:cab){reference-type="eqref" reference="e:cab"} and [2.2](#s:Morse){reference-type="ref" reference="s:Morse"}, let $G=G_{\mathbb{T}^r_q}$ and let $\mathcal{A}$ denote the following subset of $E(G)$ $$\mathcal{A}=\big\{\sigma \to \sigma \smallsetminus\{{\pmb{\epsilon}}^{\iota(\sigma)}\} \colon\sigma\in \mathbb{T}^r_q\smallsetminus\mathbb{S}^r_q\quad \mbox{and} \quad {\pmb{\epsilon}}^{\iota(\sigma)}\in\sigma\big\}$$ Then $\mathcal{A}$ is a homogeneous acyclic matching of $G$ and its set of critical cells is $\mathbb{S}^r_q$.* *Consequently, $\mathbb{S}^r_q$ supports a minimal free resolution on ${\mathcal{E}_q}^r$.* *Proof.* We first show that $\mathcal{A}$ is a matching. By [Proposition 43](#p:inS){reference-type="ref" reference="p:inS"}, for every $\sigma \in \mathbb{T}^r_q\smallsetminus\mathbb{S}^r_q$ with ${\pmb{\epsilon}}^{\iota(\sigma)} \in \sigma$ we have $$\iota(\sigma)=\iota(\sigma')\qquad\text{where}\quad\sigma'=\sigma \smallsetminus\{{\pmb{\epsilon}}^{\iota(\sigma)}\}).$$ In particular, ${\pmb{\epsilon}}^{\iota(\sigma')} \notin \sigma'$, and hence each face of $\mathbb{T}^r_q\smallsetminus\mathbb{S}^r_q$ gets matched exactly once, thus $\mathcal{A}$ is a matching. The fact that $\mathcal{A}$ is homogeneous follows also from [Proposition 43](#p:inS){reference-type="ref" reference="p:inS"}. It remains to show that $\mathcal{A}$ is acyclic. For this purpose we will use [\[matching-lemma\]](#matching-lemma){reference-type="ref" reference="matching-lemma"} applied to the partition of $\mathbb{T}^r_q\smallsetminus \mathbb{S}^r_q$ in [\[e:partition\]](#e:partition){reference-type="eqref" reference="e:partition"} into sets $\mathcal{S}_{\mathbf{a},\mathbf{b}}$, with $(\mathbf{a}, \mathbf{b})\in \mathcal{S}$. For $(\mathbf{a},\mathbf{b})\in \mathcal{S}$ as above, define $$\mathcal{A}_{\mathbf{a},\mathbf{b}}=\big\{\sigma \to \sigma \smallsetminus\{{\pmb{\epsilon}}^{\iota(\sigma)}\} \colon\sigma\in \mathcal{S}_{\mathbf{a},\mathbf{b}} \quad \mbox{and} \quad {\pmb{\epsilon}}^{\iota(\sigma)}\in \sigma \big\}.$$ We proceed with two claims. *Claim 1 1*. $\mathcal{A}_{\mathbf{a},\mathbf{b}}$ is a homogeneous acyclic matching on the induced subgraph $G_{\mathcal{S}_{\mathbf{a},\mathbf{b}}}$ of $G$ on the vertices in $\mathcal{S}_{\mathbf{a},\mathbf{b}}$ and its set of critical cells is empty. If $\sigma \in \mathcal{S}_{\mathbf{a},\mathbf{b}}$, then $\mathbf{c}=\iota(\sigma)$ only depends on $(\mathbf{a}, \mathbf{b})$. In other words if $\tau \in \mathcal{S}_{\mathbf{a},\mathbf{b}}$, $\iota(\tau)=\mathbf{c}$ as well. With the notation in [\[matching-lemma\]](#matching-lemma){reference-type="ref" reference="matching-lemma"}, we see that $$\mathcal{A}_{\mathbf{a}, \mathbf{b}}=\mathcal{A}^{v}_{\mathcal{S}_{\mathbf{a},\mathbf{b}}} \quad\text{with}\quad v={\pmb{\epsilon}}^{\mathbf{c}}.$$ Then [\[matching-lemma\]](#matching-lemma){reference-type="ref" reference="matching-lemma"} shows that $\mathcal{A}_{\mathbf{a}, \mathbf{b}}$ is an acyclic matching. It is also homogeneous, since $\mathcal{A}$ is. Moreover, observe that every $\sigma \in \mathcal{S}_{\mathbf{a},\mathbf{b}}$ is matched as $$\begin{cases} \sigma \to \sigma \smallsetminus\{{\pmb{\epsilon}}^{\mathbf{c}}\} \in \mathcal{A}_{\mathbf{a}, \mathbf{b}} & \quad \mbox{if} \quad {\pmb{\epsilon}}^{\mathbf{c}}\in \sigma, \\ \sigma \cup \{{\pmb{\epsilon}}^{\mathbf{c}}\} \to \sigma \in \mathcal{A}_{\mathbf{a}, \mathbf{b}} & \quad \mbox{if} \quad {\pmb{\epsilon}}^{\mathbf{c}}\notin \sigma. \end{cases}$$ Indeed, to see that $\sigma \cup \{{\pmb{\epsilon}}^{\mathbf{c}}\} \to \sigma \in \mathcal{A}_{\mathbf{a}, \mathbf{b}}$, observe that $\iota(\sigma\cup \{{\pmb{\epsilon}}^{\mathbf{c}}\})=\mathbf{c}$ by [Proposition 43](#p:inS){reference-type="ref" reference="p:inS"}, and hence $(\sigma\cup \{{\pmb{\epsilon}}^{\mathbf{c}}\})\smallsetminus\{{\pmb{\epsilon}}^{\iota(\sigma\cup \{{\pmb{\epsilon}}^{\mathbf{c}}\})}\}=\sigma$. Therefore, the set of $\mathcal{A}_{\mathbf{a}, \mathbf{b}}$-critical cells of this matching is empty. *Claim 2 1*. If $\sigma'\in \mathcal{S}_{\mathbf{a}, \mathbf{b}}$ and $\sigma\in \mathcal{S}_{\mathbf{c}, \mathbf{d}}$ with $\sigma' \subseteq \sigma$, then $(\mathbf{a}, \mathbf{b})\le (\mathbf{c}, \mathbf{d})$. Indeed, the hypotheses in the Claim 2 imply $$(\mathbf{a},\mathbf{b})=\nu(\sigma') \leq \nu(\sigma)=(\mathbf{c}, \mathbf{d}).$$ Now we apply [Lemma 4](#clusterlemma){reference-type="ref" reference="clusterlemma"}, observing that $$\mathcal{A}=\bigcup_{(\mathbf{a}, \mathbf{b})\in \mathcal{S}}\mathcal{A}_{\mathbf{a}, \mathbf{b}}\qquad\text{and} \qquad\mathbb{T}^r_q\smallsetminus\mathbb{S}^r_q=\bigcup_{(\mathbf{a}, \mathbf{b})\in \mathcal{S}}\mathcal{S}_{\mathbf{a},\mathbf{b}}\,.$$ Claim 1 and Claim 2 show that $\mathcal{A}$ is a homogeneous acyclic matching of $G_{X'}$, where $X'=\mathbb{T}^r_q\smallsetminus\mathbb{S}^r_q$, and its set of critical cells is empty. Finally, from [\[inclusions\]](#inclusions){reference-type="ref" reference="inclusions"} it follows that $\mathcal{A}$ is a homogeneous acyclic matching of $G$, with the set of $\mathcal{A}$-critical cells equal to $\mathbb{S}^r_q$. The conclusion that $\mathbb{S}^r_q$ supports a minimal free resolution of ${\mathcal{E}_q}^r$ follows then from [Remark 7](#r:BW){reference-type="ref" reference="r:BW"}. ◻ A direct consequence of [Theorem 44](#t:Morse-small-q){reference-type="ref" reference="t:Morse-small-q"} and [Corollary 38](#c:U=S){reference-type="ref" reference="c:U=S"} is that the $\bf f$-vector of $\mathbb{U}^r_q$ gives the betti numbers of ${\mathcal{E}_q}^r$ when $q\le 4$. **Corollary 45**. *If $1\le q\le 4$ and $r\ge 1$, then $\beta_i({\mathcal{E}_q}^r)=f_i$ for all $i\ge 0$, where ${\bf f}(\mathbb{U}^r_q)=(f_i)_{i\ge 0}$.* In [\[s:betti\]](#s:betti){reference-type="ref" reference="s:betti"}, we will record explicit formulas for these betti numbers. The example below is a preview. **Example 46**. When $q=3$ and $r=4$, the picture in [\[f:U-pic\]](#f:U-pic){reference-type="ref" reference="f:U-pic"} is a geometric realization of the simplicial complex $\mathbb{S}^r_q=\mathbb{U}^r_q$. By [Theorem 44](#t:Morse-small-q){reference-type="ref" reference="t:Morse-small-q"}, we know that this complex supports a minimal free resolution of ${\mathcal{E}_3}^4$. In particular, we see that $\mathrm{pd}({\mathcal{E}_3}^4)=2$ and, counting the number of vertices, edges and triangles, we get: $$\beta_0({\mathcal{E}_3}^4)=15 \qquad \beta_1({\mathcal{E}_3}^4)=30\qquad \beta_2({\mathcal{E}_3}^4)=16\,.$$ If desired, the multi-graded betti numbers can also be explicitly described, along with the differentials in the minimal free resolution supported on this complex. # **The first betti numbers of ${\mathcal{E}_q}^r$ are Scarf for all $r$ and $q$** {#s: first} We will prove in [\[t:first-betti\]](#t:first-betti){reference-type="ref" reference="t:first-betti"} that there is a multigraded free resolution of ${\mathcal{E}_q}^r$ with only Scarf multidegrees appearing in the first homological degree. Since the Scarf multidegrees appear in *every* free resolution, this will imply that $$\beta_{1,\mathbf{m}}({\mathcal{E}_q}^r) \neq 0 \iff \mathbf{m}\mbox{ is the monomial label of an edge of } \mathbb{S}^r_q.$$ To find such a complex, we describe in [\[t:first-betti\]](#t:first-betti){reference-type="ref" reference="t:first-betti"} a matching that removes all the non-Scarf edges in $\mathbb{T}^r_q$. We first need some preliminary steps, using [Notation 10](#n:rqs){reference-type="ref" reference="n:rqs"}. **Proposition 47**. *Let $r,q\ge 1$ and $\mathbf{a}, \mathbf{b}, \mathbf{c}\in \mathcal{N}^r_q$. If $\mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}})=\mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{c}})$, then ${\pmb{\epsilon}}^{\mathbf{b}}={\pmb{\epsilon}}^{\mathbf{c}}$.* *Proof.* Assume $\mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}})=\mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{c}})$, and let $i\in [q]$ be such that $b_i \neq c_i$. In view of [Lemma 19](#l:when-divide){reference-type="ref" reference="l:when-divide"}, ${\pmb{\epsilon}}^{\mathbf{c}}\mid \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}})$ gives $$\label{e:1} \min(a_i,b_i)\le c_i\le \max(a_i,b_i),$$ and ${\pmb{\epsilon}}^{\mathbf{b}}\mid\mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{c}})$ gives $$\label{e:2} \min(a_i,c_i)\le b_i\le \max(a_i,c_i).$$ Without loss of generality, assume $b_i \gneq c_i$. Then by [\[e:2\]](#e:2){reference-type="eqref" reference="e:2"} $\max(a_i,c_i)=a_i$ and hence $b_i\le a_i$. Therefore $\min(a_i,b_i)=b_i$ and [\[e:1\]](#e:1){reference-type="eqref" reference="e:1"} implies $b_i\le c_i$, a contradiction. Hence $b_i=c_i$ for all $i \in [q]$, and therefore ${\pmb{\epsilon}}^{\mathbf{b}}={\pmb{\epsilon}}^{\mathbf{c}}$. ◻ When $\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\} \in \mathbb{T}^r_q\smallsetminus\mathbb{S}^r_q$, [Lemma 1](#l:expansion){reference-type="ref" reference="l:expansion"} guarantees the existence of an element $\mathbf{c}_{\mathbf{a},\mathbf{b}} \in \mathcal{N}^r_q$ such that $$\label{e:cab} \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}},{\pmb{\epsilon}}^{\mathbf{c}_{\mathbf{a},\mathbf{b}}})=\mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}}) \quad \mbox{and} \quad \mathbf{c}_{\mathbf{a},\mathbf{b}} \notin \{\mathbf{a}, \mathbf{b}\}$$ We next show that there is a CW complex supporting a free resolution of ${\mathcal{E}_q}^r$, whose edges correspond to the edges of the $\mathbb{S}^r_q$. This is done by defining a matching on $G=G_{\mathbb{T}^r_q}$ (see [2.2](#s:Morse){reference-type="ref" reference="s:Morse"} for notation). While we need to make a choice of $\mathbf{c}_{\mathbf{a},\mathbf{b}}$ in order to define a matching, the argument below holds for any set of choices that one makes. For the theorem below, we choose a fixed element $\mathbf{c}_{\mathbf{a},\mathbf{b}}$ satisfying [\[e:cab\]](#e:cab){reference-type="eqref" reference="e:cab"} for each $\mathbf{a},\mathbf{b}\in \mathcal{N}^r_q$. **Theorem 48**. ***(The 1st betti numbers of ${\mathcal{E}_q}^r$ are all Scarf.)** [\[t:first-betti\]]{#t:first-betti label="t:first-betti"} Assume $r,q \geq 1$, and using notation as in [\[e:cab\]](#e:cab){reference-type="eqref" reference="e:cab"} and [2.2](#s:Morse){reference-type="ref" reference="s:Morse"}, let $G=G_{\mathbb{T}^r_q}$ and let $\mathcal{A}$ denote the following subset of $E(G)$ $$\mathcal{A}=\left\{\{{\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}},{\pmb{\epsilon}}^{\mathbf{c}_{\mathbf{a},\mathbf{b}}}\}\to \{{\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}}\} \in E(G)\colon\{{\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}}\}\notin \mathbb{S}^r_q\right\}.$$ Then* 1. *The set $\mathcal{A}$ is a homogeneous acyclic matching of $G$;* 2. *The set of $\mathcal{A}$-critical edges of $\mathbb{T}^r_q$ is precisely the set of edges of $\mathbb{S}^r_q$;* 3. *There exists a CW complex $\mathcal{X}_{\mathcal{A}}$ that supports a resolution of ${\mathcal{E}_q}^r$ whose edges are in one-to-one correspondence with the edges of $\mathbb{S}^r_q$;* 4. *The first betti number $\beta_1({\mathcal{E}_q}^r)$ is equal to the number of edges of $\mathbb{S}^r_q$;* 5. *$\mathbf{m}$ is a multidegree with $\beta_{1,\mathbf{m}}({\mathcal{E}_q}^r)\ne 0$ if and only if $\mathbf{m}=\mathrm{lcm}(\mathbf{m}',\mathbf{m}'')$, where $\mathbf{m}'$ and $\mathbf{m}''$ are monomial labels of an edge of $\mathbb{S}^r_q$.* *Proof.* (1) We first show that $\mathcal{A}$ is a matching. Given the fixed choice of $\mathbf{c}_{\mathbf{a},\mathbf{b}}$, the vertices in $\mathcal{A}$ of the form $\{{\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}}\}$ each belong to only one edge of $\mathcal{A}$. Assume now that a $2$-dimensional face $\sigma =\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}, {\pmb{\epsilon}}^{\mathbf{d}}\}$ of $\mathbb{T}^r_q$ is a vertex of two edges of $\mathcal{A}$, say $$\sigma\to \{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\} \quad \mbox{and} \quad \sigma\to \{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{d}}\}$$ which implies, by [\[e:cab\]](#e:cab){reference-type="eqref" reference="e:cab"}, that $$\mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}})=\mathrm{lcm}(\sigma)=\mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{d}}).$$ Then by [Proposition 47](#p:edges){reference-type="ref" reference="p:edges"} ${\pmb{\epsilon}}^{\mathbf{b}}={\pmb{\epsilon}}^{\mathbf{d}}$, a contradiction. By [\[e:cab\]](#e:cab){reference-type="eqref" reference="e:cab"} the matching $\mathcal{A}$ is homogeneous. Finally, assume that there is a cycle $\mathcal{C}$ in $G$, by [@FFGY Lemma 3.3], $\mathcal{C}$ (shown below) will have at least 6 edges alternating between those in $\mathcal{A}$ and those outside $\mathcal{A}$, and all having the same monomial label. Therefore, in the picture above we will have $\mathbf{b}\in \{\mathbf{a}_{1}, \mathbf{a}_{2}\}$, and $\{{\pmb{\epsilon}}^{\mathbf{a}_1} , {\pmb{\epsilon}}^{\mathbf{a}_2}\}$ and $\{{\pmb{\epsilon}}^{\mathbf{b}} , {\pmb{\epsilon}}^{\mathbf{a}_3}\}$ are distinct edges of $\mathbb{T}^r_q$ with the same monomial label $$\mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{a}_1} , {\pmb{\epsilon}}^{\mathbf{a}_2})= \mathrm{lcm}({\pmb{\epsilon}}^{\mathbf{b}} , {\pmb{\epsilon}}^{\mathbf{a}_3}),$$ which contradicts [Proposition 47](#p:edges){reference-type="ref" reference="p:edges"}. \(2\) is clear from the definition of $\mathcal{A}$. \(3\) follows directly from [Theorem 6](#t:BW){reference-type="ref" reference="t:BW"}. \(4\) The fact that $\beta_1({\mathcal{E}_q}^r)$ is at most the number of edges of $\mathbb{S}^r_q$ follows from (3). The reverse inequality is true in general, as every free resolution contains the Scarf faces. \(5\) is a reformulation of (4). ◻ # **Bounds on betti numbers of powers of monomial ideals**[\[s:betti\]]{#s:betti label="s:betti"} In this section we discuss bounds on betti numbers that can be deduced from our results throughout the paper. In view of [Theorem 15](#t:upperbound){reference-type="ref" reference="t:upperbound"}, knowledge of the betti numbers of ${\mathcal{E}_q}^r$ provides an upper bound for the betti numbers of $I^r$ for any ideal $I$ generated by $q$ square-free monomials. [\[t:first-betti\]](#t:first-betti){reference-type="ref" reference="t:first-betti"} shows that the first betti number of ${\mathcal{E}_q}^r$ equals the number of Scarf edges of ${\mathcal{E}_q}^r$. Thus, to explicitly find this betti number, one would like to have a combinatorial count of the Scarf edges. It turns out that a full characterization of these edges for large values of $r$ and large $q$ is quite difficult, but small values of these indices are manageable. We provide below a count when $r=3$. **Lemma 49**. *Let $q\ge 2$ and ${\pmb{\epsilon}}^{\mathbf{a}}\ne {\pmb{\epsilon}}^{\mathbf{b}}$ in ${\mathcal{E}_q}^3$. Then $\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}\in \mathbb{S}^3_q$ if and only if one of the following holds:* 1. *$\{{\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}}\}=\epsilon_u\epsilon_v\{\epsilon_i, \epsilon_j\}$ with $u,v,i,j\in [q]$, $i\ne j$.* 2. *$\{{\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}}\}=\epsilon_u\{\epsilon_i\epsilon_j, \epsilon_k\epsilon_l\}$ with $u,i,j,k,l\in [q]$, $i,j,k,l$ distinct.* 3. *$\{{\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}}\}=\{\epsilon_i\epsilon_j\epsilon_k, \epsilon_u\epsilon_v\epsilon_w\}$ with $i,j,k,u,v,w\in [q]$ distinct.* 4. *$\{{\pmb{\epsilon}}^{\mathbf{a}},{\pmb{\epsilon}}^{\mathbf{b}}\}=\{\epsilon_i\epsilon_j\epsilon_k, \epsilon_u^2e_v\}$ with $i,j,k,u,v\in [q]$ distinct.* *Proof.* Set ${\pmb{\epsilon}}^{\mathbf{d}}=\epsilon\text{-}\mathrm{gcd}({\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}})$ and write ${\pmb{\epsilon}}^{\mathbf{a}}={\pmb{\epsilon}}^{\mathbf{d}}\cdot{\pmb{\epsilon}}^{{\mathbf{a}}'}$ and ${\pmb{\epsilon}}^{\mathbf{b}}={\pmb{\epsilon}}^{\mathbf{d}}\cdot{\pmb{\epsilon}}^{{\mathbf{b}}'}$. By [Proposition 21](#p:Scarf-face){reference-type="ref" reference="p:Scarf-face"}, we have $$\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}\in \mathbb{S}^3_q \iff \{{\pmb{\epsilon}}^{{\mathbf{a}}'}, {\pmb{\epsilon}}^{{\mathbf{b}}'}\}\in \mathbb{S}^{3-|\mathbf{d}|}_{q}\,.$$ If $|\mathbf{d}|>0$, then $\mathbb{S}^{3-|\mathbf{d}|}_{q}=\mathbb{U}^{3-|\mathbf{d}|}_{q}$, in view of [Corollary 31](#c:r=2-Scarf){reference-type="ref" reference="c:r=2-Scarf"}. Thus, if $|\mathbf{d}|=1$, then $\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}\in \mathbb{S}^3_q$ if and only (2) holds and if $|\mathbf{d}|=2$, then $\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}\in \mathbb{S}^3_q$ if and only (1) holds. Assume now $\mathbf{d}=0$. It remains to show that $\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}\in \mathbb{S}^3_q$ if and only if (3) or (4) hold. If (3) holds, then $\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}\in \mathbb{U}^3_q$, and $\mathbb{U}^3_q\subseteq \mathbb{S}^3_q$ by [Theorem 30](#t:f-vector){reference-type="ref" reference="t:f-vector"}. If (4) holds, then $\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}\in \mathbb{S}^3_q$ by [Remark 39](#r:q>4){reference-type="ref" reference="r:q>4"}. Assume now neither (3), nor (4) holds. Since $\mathbf{d}=0$, we must have either $\epsilon_i^3\in\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}$ for some $i\in [q]$ or else $\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}=\{\epsilon_i^2\epsilon_j, e_u^2\epsilon_v\}$ with $i,j,u,v\in[q]$ distinct. In both cases, we observe that $|\mathrm{Supp}(\mathbf{a})\cup \mathrm{Supp}(\mathbf{b})|\le 4$, and we conclude $\{{\pmb{\epsilon}}^{\mathbf{a}}, {\pmb{\epsilon}}^{\mathbf{b}}\}\notin \mathbb{S}^3_q$ by [Lemma 36](#l:find-c){reference-type="ref" reference="l:find-c"}. ◻ For the next results, we recall the convention that $\binom{u}{v}=0$ when $u<v$. **Theorem 50**. *Let $q\ge 1$ and let $I$ be a monomial ideal minimally generated by $q$ square-free monomials. Then $$\label{e:betti1} \beta_1(I^3)\le \binom{q+1}{2}\binom{q}{2}+3q\binom{q}{4}+20\binom{q}{5}+10\binom{q}{6}\,.$$ Moreover, equality holds when $S={S_\mathcal{E}}$ and $I={\mathcal{E}_q}$.* *Proof.* We first show $\beta_1({\mathcal{E}_q}^3)$ is equal to the right-hand side of [\[e:betti1\]](#e:betti1){reference-type="eqref" reference="e:betti1"}. In view of [\[t:first-betti\]](#t:first-betti){reference-type="ref" reference="t:first-betti"}, $\beta_1({\mathcal{E}_q}^3)$ is equal to the number of edges in $\mathbb{S}_q^3$. We can use thus [Lemma 49](#l:types){reference-type="ref" reference="l:types"} to count the number of edges in each of the cases (1)-(4). The number of edges that satisfy (1) in [Lemma 49](#l:types){reference-type="ref" reference="l:types"} is $\binom{q+1}{2}\binom{q}{2}$. The number of edges that satisfy (2) in [Lemma 49](#l:types){reference-type="ref" reference="l:types"} is $3q\binom{q}{4}$. The number of edges that satisfy (3) in [Lemma 49](#l:types){reference-type="ref" reference="l:types"} is $10\binom{q}{6}$. The number of edges that satisfy (4) in [Lemma 49](#l:types){reference-type="ref" reference="l:types"} is $20\binom{q}{5}$. Putting these numbers together, we obtain that $\beta_1({\mathcal{E}_q}^3)$ is equal to the right-hand side of [\[e:betti1\]](#e:betti1){reference-type="eqref" reference="e:betti1"}. Finally, to justify the inequality in [\[e:betti1\]](#e:betti1){reference-type="eqref" reference="e:betti1"}, apply [Theorem 15](#t:upperbound){reference-type="ref" reference="t:upperbound"}. ◻ **Example 51**. Using [Theorem 50](#t:betti-1){reference-type="ref" reference="t:betti-1"}, we see that $$\beta_1({\mathcal{E}_4}^3)=72\qquad \beta_1({\mathcal{E}_5}^3)=245 \qquad \beta_1({\mathcal{E}_6}^3)=715 \qquad \beta_1({\mathcal{E}_7}^3)=1813 \qquad \beta_1({\mathcal{E}_8}^3)=4088\,.$$ (This computation can also be verified using Macaulay2. ) These numbers are thus effective bounds for $\beta_1(I^3)$ for any ideal $I$ generated by $q$ square-free monomials when $q=4,5,6,7$, respectively $8$. **Theorem 52** (**Effective bounds on betti numbers**). *Let $q, r\ge 1$ and let $I$ be an ideal of $S$ minimally generated by $q$ square-free monomials, and set $$\gamma_i=\binom{r-i+q-1}{q-1}.$$* 1. *If $q=2$, then $\mathrm{pd}_S(I^r)\le 1$ and $$\beta_0(I^r)\le r, \qquad \beta_1(I^r)\le r-1\,.$$* 2. *If $q=3$, then $\mathrm{pd}_S(I^r)\le 2$ and $$\begin{array}{lll} \beta_0(I^r)\le \gamma_0, & \beta_1^S(I^r)\le 3\gamma_1, & \beta_2^S(I^r)\le \gamma_1+\gamma_2. \end{array}$$* 3. *If $q=4$, then $\mathrm{pd}_S(I^r) \le 5$ and $$\begin{array}{lll} \beta_0(I^r) \le \gamma_0, & \beta_1^S(I^r) \le 6 \gamma_1+3\gamma_2, & \beta _2^S(I^r) \le 4\gamma_1 + 16 \gamma_2,\\ &&\\ \beta_3^S(I^r)\le \gamma_1+15\gamma_2+\gamma_3, & \beta_4^S(I^r)\le 6\gamma_2, & \beta_5(I^r)\le \gamma_2. \end{array}$$* *Furthermore, all inequalities above become equalities when $S=S_{\mathcal{E}}$ and $I=\mathcal{E}_q$.* *Proof.* In view of [Corollary 45](#c:f-betti){reference-type="ref" reference="c:f-betti"}, the $\mathbf{f}$-vector of the complex $\mathbb{U}^r_q$ gives the betti numbers $\beta_i({\mathcal{E}_q}^r)$ when $q\le 4$. In turn, [Theorem 15](#t:upperbound){reference-type="ref" reference="t:upperbound"} gives the desired inequalities. We proceed now to compute $\mathbf{f}(\mathbb{U}^r_q)$. In what follows, the convention is that $\mathcal{N}_q^i=\emptyset$ when $i<0$. Observe that $\gamma_i=|\mathcal{N}_q^{r-i}|$. Clearly, when $i=0$, $\gamma_0$ is exactly the number of vertices of $\mathbb{U}^r_q$, which is $|\mathcal{N}^r_q|$. Assume $q=2$. The edges of $\mathbb{U}^r_q$ are of the form $${\pmb{\epsilon}}^{\mathbf{a}}\{\epsilon_1, \epsilon_2\}, \qquad \mathbf{a}\in \mathcal{N}^{r-1}_2\,.$$ Thus the number of edges is $|\mathcal{N}^{r-1}_2|=\gamma_1$. There are no higher dimensional faces in $\mathbb{U}_2^r$, and this finishes the explanation for (1). Assume $q=3$. The facets are $${\pmb{\epsilon}}^{\mathbf{a}}\{\epsilon_j\colon j\in [3]\}, \quad \mathbf{a}\in \mathcal{N}_q^{r-1}\quad \mbox{and} \quad {\pmb{\epsilon}}^{\mathbf{a}}\{\epsilon_j\epsilon_k\colon j\ne k\}, \quad \mathbf{a}\in \mathcal{N}_q^{r-2}\,.$$ There are $\gamma_1$ facets of the first type and $\gamma_2$ facets of the second type, hence $\gamma_1+\gamma_2$ choices in total for the $2$-dimensional faces. To count the edges, note that they take the form below: $${\pmb{\epsilon}}^{\mathbf{a}}\{\epsilon_i,\epsilon_j \}, \quad \mathbf{a}\in \mathcal{N}_q^{r-1},\quad i\ne j\,.$$ Since $|\mathcal{N}_q^{r-1}|=\gamma_1$ and there are $3$ choices for $i,j$ with $i\ne j$, there are $3\gamma_1$ choices in total. This finishes the explanation for (2). Assume $q=4$. The edges are of two types: $$\begin{aligned} & {\pmb{\epsilon}}^{\mathbf{a}}\{\epsilon_i, \epsilon_j \}, \quad \mathbf{a}\in \mathcal{N}_q^{r-1} , \quad i\ne j\\ & {\pmb{\epsilon}}^{\mathbf{a}}\{\epsilon_i\epsilon_j, \epsilon_ke_l \}, \quad \mathbf{a}\in \mathcal{N}_q^{r-2} \quad \{i,j,k,l\}=[4].\end{aligned}$$ There are $6\gamma_1$ edges of the first type and $3\gamma_2$ edges of the second type. The 2-dimensional faces are of the form: $$\begin{aligned} & {\pmb{\epsilon}}^{\mathbf{a}}\{\epsilon_i, \epsilon_j , \epsilon_k \}, \quad \mathbf{a}\in \mathcal{N}_q^{r-1}, \quad |\{i,j,k\}|=3\\ & {\pmb{\epsilon}}^{\mathbf{a}}\{\epsilon_i\epsilon_j, \epsilon_ke_l , \epsilon_ue_v\}, \quad \mathbf{a}\in \mathcal{N}_q^{r-2},\quad i\ne j, \quad k\ne l, \quad u\ne v, \quad \{i,j\}\cap \{k,l\}\cap \{u,v\}=\emptyset.\end{aligned}$$ There are $4\gamma_1$ faces of the first type and $(\binom{6}{3}-4)\gamma_2=16\gamma_2$ faces of the second type. The $3$-dimensional faces are of the form: $$\begin{aligned} & {\pmb{\epsilon}}^{\mathbf{a}}\{\epsilon_1, \epsilon_2 , \epsilon_3, \epsilon_4 \},\quad \mathbf{a}\in \mathcal{N}_q^{r-1} \\ & {\pmb{\epsilon}}^{\mathbf{a}}\{\epsilon_i\epsilon_j, \epsilon_ke_l , \epsilon_ue_v, \epsilon_pe_q\},\, \mathbf{a}\in \mathcal{N}_q^{r-2} , \, i\ne j, \, k\ne l, \, u\ne v, \, p\ne q, \, \{i,j\}\!\cap \!\{k,l\} \!\cap \!\{u,v\} \!\cap \!\{p,q\}=\emptyset \\ & {\pmb{\epsilon}}^{\mathbf{a}}\{\epsilon_1\epsilon_2\epsilon_3, \epsilon_1\epsilon_2\epsilon_4 , \epsilon_2\epsilon_3e_4\}, \quad \mathbf{a}\in \mathcal{N}_q^{r-3} .\end{aligned}$$ There are $\gamma_1$ faces of the first type, $\gamma_3$ of the third type and $\binom{6}{4}\gamma_2=15\gamma_2$ faces of the second type. The $4$-dimensional faces are of the form: $$\begin{aligned} {\pmb{\epsilon}}^{\mathbf{a}}\{\epsilon_i\epsilon_j, \epsilon_k\epsilon_l , \epsilon_u\epsilon_v, \epsilon_p\epsilon_q, \epsilon_b\epsilon_c\}, \, \mathbf{a}\in \mathcal{N}_q^{r-2} ,&\, i\ne j, \, k\ne l, \, u\ne v, \, p\ne q, \, b\ne c\,,\\ & \, \{i,j\}\cap \{k,l\}\cap \{u,v\}\cap \{p,q\}\cap \{b,c\}=\emptyset\,.\end{aligned}$$ There are $6\gamma_2$ faces of this form. Finally, the $5$-dimensional faces are of the type $${\pmb{\epsilon}}^{\mathbf{a}}\{\epsilon_1\epsilon_2, \epsilon_1\epsilon_3 , \epsilon_1\epsilon_4, \epsilon_2e_3, \epsilon_2\epsilon_4 , \epsilon_3\epsilon_4 \}, \quad \mathbf{a}\in \mathcal{N}_q^{r-2}$$ hence there are $\gamma_2$ of them. ◻ For the sake of completeness, we also include below the bounds obtained in [@L2] in the case when $r=2$. The fact that the bounds are achieved when $I=\mathcal{E}_q$ is established in [@Lr]. **Theorem 53** ([@L2 Theorem 4.1],[@Lr Proposition 7.9]). *Let $q\ge 1$ and let $I$ be a monomial ideal minimally generated by $q$ square-free monomials. Then $$\beta_i(I^2)\le{{\frac{1}{2}(q^2-q)}\choose{i+1}}+q {{q-1}\choose{i}}\,.$$ Furthermore, equality holds when $S=S_{\mathcal{E}}$ and $I=\mathcal{E}_q$.* While we succeeded in providing effective bounds on the betti numbers of powers of square-free monomials in certain cases, as recorded above, it remains to be established whether such bounds can be given for all $r$ and $q$. In particular, this paper raises the question: **Question 54**. Is the minimal free resolution of ${\mathcal{E}_q}^r$ supported on a simplicial (or cellular) complex for all $q\ge 1$ and all $r\ge 1$? In particular, are the ideals ${\mathcal{E}_q}^r$ Scarf? If this is true, then one would also want to explicitly describe the faces of the simplicial (or cellular) complex in [Question 54](#q1){reference-type="ref" reference="q1"}, with the hope that one can obtain explicit formulas for bounds on betti numbers, as given in this section. While we were able to provide a clear description of $\mathbb{S}^r_q$ when $q\le 4$ and also when $r\le 2$, such a description seems to become a lot more complex for larger $q$ or $r$. We believe one can show that the ideals $\mathbb{S}^r_q$ are Scarf at least when $q=5$ or when $r=3$ as well, but the difficulty of understanding the Scarf complex with our current techniques is increasing sharply for large values of $q$ and $r$. ### Acknowledgements {#acknowledgements .unnumbered} The research for this paper took place in May 2022, during a two-week stay of the authors at the Mathematisches Forschungsinstitut in Oberwolfach (MFO), as Oberwolfach Research Fellows. We are grateful to MFO for providing us with a wonderful environment and facilities to do concentrated work. We are grateful to Susan Cooper and Susan Morey with whom we have an on-going collaboration on products of simplicial complexes. Their insights will have inevitably influenced us while writing the current paper. Sara Faridi's research is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant program (\#2023-05929). Liana Şega was supported in part by a grant from the Simons Foundation (\#354594). For Sandra Spiroff, this material is based upon work supported by and while serving at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The computations for this paper were made using the computer algebra softeware Macaulay2 [@M2]. 2020 D. Bayer, I. Peeva, B. Sturmfels, *Monomial resolutions*, Math. Res. Lett. **5** (1998), 31--46. E. Batzies and V. Welker, *Discrete Morse theory for cellular resolutions*, J. Reine Angew. Math., **543**, 147-168 (2002). T. Chau, A. Duval, S. Faridi, T. Holleben, S. Morey, L. M. Şega, *Powers of a simplex*, work in progress (2023). S. M. Cooper, S. El Khoury, S. Faridi, S. Mayes-Tang, S. Morey, L. M. Şega and S. Spiroff, *Simplicial resolutions of powers of square-free monomial ideals*, to appear in Algebraic Combinatorics, arXiv:2204.03136 (2023). S. M. Cooper, S. El Khoury, S. Faridi, S. Mayes-Tang, S. Morey, L. M. Şega and S. Spiroff, *Morse resolutions of powers of square-free monomial ideals of projective dimension one*, J. Algebraic Combinatorics, **55**, no. 4 (2022) 1105-1122. S. M. Cooper, S. El Khoury, S. Faridi, S. Mayes-Tang, S. Morey, L. M. Şega and S. Spiroff, *Simplicial resolutions for the second power of square-free monomial ideals*, Women in Commutative Algebra, 193-205,Assoc. Women Math. Ser., 29, Springer, Cham, 2021. S. Faridi, M. Farrokhi D. G., R. Ghorbani, A. Yazdan Pour, *Cellular resolutions of monomial ideals and their Artinian reductions*, submitted, arXiv:2209.10338 (2022). D. R. Grayson, M. E. Stillman, *Macaulay2, a software system for research in algebraic geometry*, Available at ` http://www.math.uiuc.edu/Macaulay2/`. J. Jonsson, *Simplicial Complexes of Graphs*, Lecture Notes in Mathematics, **1928**, Springer, Berlin (2008), no. 12 (2017) 5453--5464. J. Mermin, *Three simplicial resolutions*, Progress in commutative algebra 1, 127--141, de Gruyter, Berlin, 2012. P. Orlik, V. Welker, *Algebraic combinatorics. Lectures from the Summer School held in Nordfjordeid, June 2003*, Universitext. Springer, Berlin, (2007). I. Peeva, M. Velasco, *Frames and degenerations of monomial ideals*, Trans. Amer. Math. Soc. **363** (2011), no. 4, 2029--2046. D. Taylor, *Ideals generated by monomials in an $R$-sequence,* Thesis, University of Chicago (1966).

arxiv_math

--- abstract: | The lexicographical ordering of hypergraphs via spectral moments is called the $S$-order of hypergraphs. In this paper, the $S$-order of hypergraphs is investigated. We characterize the first and last hypergraphs in an $S$-order of all uniform hypertrees and all linear unicyclic uniform hypergraphs with given girth, respectively. And we give the last hypergraph in an $S$-order of all linear unicyclic uniform hypergraphs. address: College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, PR China author: - Hong Zhou - Changjiang Bu bibliography: - pbib.bib title: Lexicographical ordering of hypergraphs via spectral moments --- 1.15 hypergraph, spectral moment, adjacency tensor\ *AMS classification (2020):* 05C65, 15A18 # Introduction Let $G$ be a simple undirected graph with $n$ vertices and $A$ be the adjacency matrix of $G$. The *$d$th order spectral moment* of $G$ is the sum of $d$ powers of all the eigenvalues of $A$, denoted by $\mathrm{S}_{d}(G)$ [@1980Spectra]. For two graphs $G_{1}, G_{2}$ with $n$ vertices, if $\mathrm{S}_{i}(G_{1})=\mathrm{S}_{i}(G_{2})$ for $i=0,1,2,\ldots,n-1$, then adjacency matrices of $G_{1}$ and $G_{2}$ have the same spectrum. Therefore, $\mathrm{S}_{i}(G_{1})=\mathrm{S}_{i}(G_{2})$ for $i=0,1,2,\ldots$. We write $G_{1}\prec_{s} G_{2}$ ($G_{1}$ comes before $G_{2}$ in an $S$-order) if there exists a $k\in\{1,2,\ldots,n-1\}$ such that $\mathrm{S}_{i}(G_{1})=\mathrm{S}_{i}(G_{2})$ for $i=0,1,2,\ldots,k-1$ and $\mathrm{S}_{k}(G_{1})<\mathrm{S}_{k}(G_{2})$. We write $G_{1}=_{s} G_{2}$, if $\mathrm{S}_{i}(G_{1})=\mathrm{S}_{i}(G_{2})$ for $i=0,1,2,\ldots,n-1$. In 1987, Cvetković and Rowlinson [@cvetkovic1987spectra] characterized the first and last graphs in an $S$-order of all trees and all unicyclic graphs with given girth, respectively. Other works on the $S$-order of graphs can be referred to [@WU20101707; @PAN20111265; @CHENG20121123; @CHENG2012858; @SHUCHAO2013ON; @10.1216/RMJ-2016-46-1-261]. The $S$-order of graphs had been used in producing graph catalogues [@Drago1984A]. In this paper, the $S$-order of hypergraphs is defined. We characterize the first and last hypergraphs in an $S$-order of all uniform hypertrees and all linear unicyclic uniform hypergraphs with given girth, respectively. And we give the last hypergraph in an $S$-order of all linear unicyclic uniform hypergraphs. Next, we introduce some notations and concepts for tensors and hypergraphs. For a positive integer $n$, let $[n]=\{1,2,\ldots,n\}$. An $m$-order $n$-dimension complex *tensor* $\mathcal{A}=\left( {a_{i_{1}\cdots i_{m}} } \right)$ is a multidimensional array with $n^m$ entries on complex number field $\mathbb{C}$, where $i_{j}\in [n], j=1,\ldots,m$. Let $\mathbb{C}^{n}$ be the set of $n$-dimension complex vectors and $\mathbb{C}^{[m,n]}$ be the set of $m$-order $n$-dimension complex tensors. For $x=\left({x_1 ,\ldots ,x_n}\right)^\mathrm{T}\in\mathbb{C}^n$, $\mathcal{A}x^{m-1}$ is a vector in $\mathbb{C}^n$ whose $i$th component is $$\begin{aligned} (\mathcal{A}x^{m-1})_i=\sum\limits_{i_{2},\ldots,i_{m}=1}^{n}a_{ii_{2}\cdots i_{m}}x_{i_{2}}\cdots x_{i_{m}}.\end{aligned}$$ A number $\lambda\in\mathbb{C}$ is called an *eigenvalue* of $\mathcal{A}$ if there exists a nonzero vector $x\in\mathbb{C}^n$ such that $$\mathcal{A}x^{m-1}=\lambda x^{[m-1]},$$ where $x^{\left[ {m - 1} \right]} = \left( {x_1^{m - 1} ,\ldots,x_n^{m - 1} } \right)^\mathrm{T}$. The number of eigenvalues of $\mathcal{A}$ is $n(m-1)^{n-1}$ [@qi2005eigenvalues; @lim2005singular]. A hypergraph $\mathcal{H}=(V(\mathcal{H}), E(\mathcal{H}))$ is called *$m$-uniform* if $|e|=m\geq2$ for all $e\in E(\mathcal{H})$. For an $m$-uniform hypergraph $\mathcal{H}$ with $n$ vertices, its *adjacency tensor* is the order $m$ dimension $n$ tensor $\mathcal{A}_\mathcal{H}=(a_{i_{1}i_{2}\cdots i_{m}})$, where $$a_{i_{1}i_{2}\cdots i_{m}}=\begin{cases} \frac{1}{(m-1)!},& \text{if } \{i_{1},i_{2},\ldots,i_{m}\}\in E(\mathcal{H}),\notag \\ 0,& \text{otherwise}. \end{cases}$$ Clearly, $\mathcal{A}_\mathcal{H}$ is the adjacency matrix of $\mathcal{H}$ when $\mathcal{H}$ is $2$-uniform [@cooper2012spectra]. The *degree* of a vertex $v$ of $\mathcal{H}$ is the number of edges containing the vertex, denoted by $d_{\mathcal{H}}(v)$ or $d_{v}$. A vertex of $\mathcal{H}$ is called a *core vertex* if it has degree one. An edge $e$ of $\mathcal{H}$ is called a *pendent edge* if it contains $|e|-1$ core vertices. Sometimes a core vertex in a pendent edge is also called a *pendent vertex*. The *girth* of $\mathcal{H}$ is the minimum length of the hypercycles of $\mathcal{H}$, denoted by $g(\mathcal{H})$. $\mathcal{H}$ is called *linear* if any two different edges intersect into at most one vertex. The *$m$-power hypergraph* $G^{(m)}$ is the $m$-uniform hypergraph which obtained by adding $m-2$ vertices with degree one to each edge of the graph $G$. In 2005, the concept of eigenvalues of tensors was proposed by Qi [@qi2005eigenvalues] and Lim [@lim2005singular], independently. The eigenvalues of tensors and related problems are important research topics of spectral hypergraph theories [@cooper2; @doi:10.1137/21M1404740; @2017Tensor; @clark2021harary], especially the trace of tensors [@clark2021harary; @2011Analogue; @hu2013determinants; @shao2015some; @doi:10.1080/03081087.2021.1953431]. Morozov and Shakirov gave an expression of the $d$th order trace $\mathrm{Tr}_{d}(\mathcal{A})$ of a tensor $\mathcal{A}$ [@2011Analogue]. Hu et al. proved that $\mathrm{Tr}_{d}(\mathcal{A})$ is equal to the sum of $d$ powers of all eigenvalues of $\mathcal{A}$ [@hu2013determinants]. For a uniform hypergraph $\mathcal{H}$, the sum of $d$ powers of all eigenvalues of $\mathcal{A}_\mathcal{H}$ is called the *$d$th order spectral moment* of $\mathcal{H}$, denoted by $\mathrm{S}_{d}(\mathcal{H})$. Then $\mathrm{Tr}_{d}(\mathcal{\mathcal{A}_\mathcal{H}})=\mathrm{S}_{d}(\mathcal{H})$. Shao et al. established some formulas for the $d$th order trace of tensors in terms of some graph parameters [@shao2015some]. Clark and Cooper expressed the spectral moments of hypergraphs by the number of Veblen multi-hypergraphs and used this result to give the "Harary-Sachs" coefficient theorem for hypergraphs [@clark2021harary]. Chen et al. gave a formula for the spectral moment of a hypertree in terms of the number of some sub-hypertrees [@doi:10.1080/03081087.2021.1953431]. This paper is organized as follows. In Section 2, the $S$-order of hypergraphs is defined. We introduce $4$ operations of moving edges on hypergraphs and give changes of the Zagreb index after operations of moving edges. In Section 3, we give the first and last hypergraphs in an $S$-order of all uniform hypertrees. In Section 4, the expressions of $2m$th and $3m$th order spectral moments of linear unicyclic $m$-uniform hypergraphs are obtained in terms of the number of sub-hypergraphs. We characterize the first and last hypergraphs in an $S$-order of all linear unicyclic uniform hypergraphs with given girth. And we give the last hypergraph in an $S$-order of all linear unicyclic uniform hypergraphs. # Preliminaries For two $m$-uniform hypergaphs $\mathcal{H}_{1},\mathcal{H}_{2}$ with $n$ vertices, if $\mathrm{S}_{i}(\mathcal{H}_{1})=\mathrm{S}_{i}(\mathcal{H}_{2})$ for $i=0,1,2,\ldots,n(m-1)^{n-1}-1$, then adjacency tensors of $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ have the same spectrum. Therefore, $\mathrm{S}_{i}(\mathcal{H}_{1})=\mathrm{S}_{i}(\mathcal{H}_{2})$ for $i=0,1,2,\ldots$. We write $\mathcal{H}_{1}\prec_{s} \mathcal{H}_{2}$ ($\mathcal{H}_{1}$ comes before $\mathcal{H}_{2}$ in an $S$-order) if there exists a $k\in\{1,2,\ldots,n(m-1)^{n-1}-1\}$ such that $\mathrm{S}_{i}(\mathcal{H}_{1})=\mathrm{S}_{i}(\mathcal{H}_{2})$ for $i=0,1,2,\ldots,k-1$ and $\mathrm{S}_{k}(\mathcal{H}_{1})<\mathrm{S}_{k}(\mathcal{H}_{2})$. We write $\mathcal{H}_{1}=_{s} \mathcal{H}_{2}$ if $\mathrm{S}_{i}(\mathcal{H}_{1})=\mathrm{S}_{i}(\mathcal{H}_{2})$ for $i=0,1,2,\ldots,n(m-1)^{n-1}-1$. In this paper, $\mathrm{S}_i(\mathcal{H})$ is also written $\mathrm{S}_i, i=0,1,2,\ldots$. Let $\textbf{H}_{1}$ and $\textbf{H}_{2}$ be two sets of hypergraphs. We write $\textbf{H}_{1}\prec_{s}\textbf{H}_{2}$ ($\textbf{H}_{1}$ comes before $\textbf{H}_{2}$ in an $S$-order) if $\mathcal{H}_{1}\prec_{s}\mathcal{H}_{2}$ for each $\mathcal{H}_{1}\in \textbf{H}_{1}$ and each $\mathcal{H}_{2} \in \textbf{H}_{2}$. For an $m$-uniform hypergraph $\mathcal{H}$ with $n$ vertices, let $\mathrm{S}_{0}(\mathcal{H})=n(m-1)^{n-1}.$ In [@cooper2012spectra], the $d$th order traces of the adjacency tensor of an $m$-uniform hypergraph were given for $d=1,2,\ldots,m$. **Lemma 1**. *[@cooper2012spectra][\[L2\]]{#L2 label="L2"} Let $\mathcal{H}$ be an $m$-uniform hypergraph with $n$ vertices and $q$ edges. Then* *(1) $\mathrm{Tr}_d(\mathcal{A}_\mathcal{H})=0$ for $d=1,2,\ldots, m-1$;* *(2) $\mathrm{Tr}_{m}(\mathcal{A}_\mathcal{H})=qm^{m-1}(m-1)^{n-m}$.* Next, we introduce $4$ operations of moving edges on hypergraphs and give changes of the Zagreb index after operations of moving edges. The sum of the squares of the degrees of all vertices of a hypergraph $\mathcal{H}$ is called the *Zagreb index* of $\mathcal{H}$, denoted by $M(\mathcal{H})$ [@Kau2020Energies]. Let $E'\subseteq E(\mathcal{H})$, we denote by $\mathcal{H}-E'$ the sub-hypergraph of $\mathcal{H}$ obtained by deleting the edges of $E'$. $\textbf{Transformation 1}$: Let $e=\{u,v,v_{1},v_{2},\ldots,v_{m-2}\}$ be an edge of an $m$-uniform hypergraph $\mathcal{H}$, $e_{1},e_{2},\ldots,e_{t}$ be the pendent edges incident with $u$, where $t\geq 1$, $d_{\mathcal{H}}(u)=t+1$ and $d_{\mathcal{H}}(v)\geq 2$. Write $e_{i}^{'}=(e_{i}\setminus \{u\})\bigcup\{v\}$. Let $\mathcal{H}^{'}=\mathcal{H}-\{e_{1},\ldots,e_{t}\}+\{e'_{1},\ldots,e'_{t}\}$. **Lemma 2**. *Let $\mathcal{H}'$ be obtained from $\mathcal{H}$ by transformation 1. Then $M(\mathcal{H}')>M(\mathcal{H})$.* *Proof.* By the definition of the Zagreb index, we have $$\begin{aligned} M(\mathcal{H}')-M(\mathcal{H})&=d^{2}_{\mathcal{H}'}(v)-d^{2}_{\mathcal{H}}(v)+d^{2}_{\mathcal{H}'}(u)-d^{2}_{\mathcal{H}}(u)\\ &=(d_{\mathcal{H}}(v)+t)^{2}-d_{\mathcal{H}}^{2}(v)+1-(t+1)^{2}\\ &=2t(d_{\mathcal{H}}(v)-1)>0.\end{aligned}$$ ◻ $\textbf{Transformation 2}$: Let $u$ and $v$ be two vertices in a uniform hypergraph $\mathcal{H}$, $e_{1},e_{2},\ldots,e_{r}$ be the pendent edges incident with $u$ and $e_{r+1},e_{r+2},\ldots,e_{r+t}$ be the pendent edges incident with $v$, where $r\geq 1$ and $t\geq 1$. Write $e_{i}^{'}=(e_{i}\setminus \{u\})\bigcup\{v\}, i\in[r]$, $e_{i}^{'}=(e_{i}\setminus \{v\})\bigcup\{u\}, i=r+1,\ldots,r+t$. If $d_{\mathcal{H}}(v)\geq d_{\mathcal{H}}(u)$, let $\mathcal{H}'=\mathcal{H}-\{e_{1},\ldots,e_{r}\}+\{e'_{1},\ldots,e'_{r}\}$. If $d_{\mathcal{H}}(v)<d_{\mathcal{H}}(u)$, let $\mathcal{H}'=\mathcal{H}-\{e_{r+1},\ldots,e_{r+t}\}+\{e'_{r+1},\ldots,e'_{r+t}\}$. **Lemma 3**. *Let $\mathcal{H}'$ be obtained from $\mathcal{H}$ by transformation 2. Then $M(\mathcal{H}')>M(\mathcal{H})$.* *Proof.* By the definition of the Zagreb index, if $d_{\mathcal{H}}(v)\geq d_{\mathcal{H}}(u)$, we have $$\begin{aligned} M(\mathcal{H}')-M(\mathcal{H})&=d^{2}_{\mathcal{H}'}(v)-d^{2}_{\mathcal{H}}(v)+d^{2}_{\mathcal{H}'}(u)-d^{2}_{\mathcal{H}}(u)\\ &=(d_{\mathcal{H}}(v)+r)^{2}-d_{\mathcal{H}}^{2}(v)+(d_{\mathcal{H}}(u)-r)^{2}-d_{\mathcal{H}}^{2}(u)\\ &=2r(r+d_{\mathcal{H}}(v)-d_{\mathcal{H}}(u))>0.\end{aligned}$$ If $d_{\mathcal{H}}(v)<d_{\mathcal{H}}(u)$, we have $$\begin{aligned} M(\mathcal{H}')-M(\mathcal{H})&=d^{2}_{\mathcal{H}'}(v)-d^{2}_{\mathcal{H}}(v)+d^{2}_{\mathcal{H}'}(u)-d^{2}_{\mathcal{H}}(u)\\ &=(d_{\mathcal{H}}(v)-t)^{2}-d_{\mathcal{H}}^{2}(v)+(d_{\mathcal{H}}(u)+t)^{2}-d_{\mathcal{H}}^{2}(u)\\ &=2t(t+d_{\mathcal{H}}(u)-d_{\mathcal{H}}(v))>0.\end{aligned}$$ ◻ The $m$-uniform hypertree with a maximum degree of less than or equal to $2$ is called the *binary $m$-uniform hypertree*. For two vertices $u,v$ of an $m$-uniform hypergraph $\mathcal{H}$, the *distance* between $u$ and $v$ is the length of a shortest path from $u$ to $v$, denoted by $d_{\mathcal{H}}(u, v)$ [@LIN2016564]. Let $d_{\mathcal{H}}(u, u)=0$. Let $\mathcal{H}_{0},\mathcal{H}_{1},\ldots,\mathcal{H}_{p}$ be pairwise disjoint connected hypergraphs with $v_{1},\ldots,v_{p}\in V(\mathcal{H}_{0})$ and $u_{i}\in V(\mathcal{H}_{i})$ for each $i\in[p]$, where $p\geq 1$. Denote by $\mathcal{H}_{0}(v_{1},\ldots,v_{p})\bigodot(\mathcal{H}_{1}(u_{1}),\ldots,\mathcal{H}_{p}(u_{p}))$ the hypergraph obtained from $\mathcal{H}_{0}$ by attaching $\mathcal{H}_{1},\ldots,\mathcal{H}_{p}$ to $\mathcal{H}_{0}$ with $u_{i}$ identified with $v_{i}$ for each $i\in[p]$ [@FAN202389]. Let $P_{q}$ be a path of length $q$. $\textbf{Transformation 3}$: Let $\mathcal{H}\neq P_{0}^{(m)}$ be an $m$-uniform connected hypergraph with $u\in{V(\mathcal{H})}$. Let $\mathcal{T}$ be a binary $m$-uniform hypertree with $v_{k}, v_{n}, u_{1}, u_{2} \in V(\mathcal{T})$ and $e_{k}, e_{k+1}\in E(\mathcal{T})$ such that $d_{\mathcal{T}}(v_{k})=2$, $v_{k}, u_{1}\in e_{k}, v_{k}, u_{2}\in e_{k+1}$, $u_{1}, u_{2}\neq v_{k}$, $v_{n}$ be a pendent vertex and $d_{\mathcal{T}}(u_{1},v_{n})>d_{\mathcal{T}}(u_{2},v_{n})$. Let $\mathcal{H}_{1}=\mathcal{H}(u)\bigodot\mathcal{T}(v_{k})$. $\mathcal{H}_{2}$ is obtained from $\mathcal{H}_{1}$ by deleting $e_{k}$ and adding $(e_{k}\setminus \{v_{k}\})\bigcup\{v_{n}\}$. **Lemma 4**. *Let $\mathcal{H}_{2}$ be obtained from $\mathcal{H}_{1}$ by transformation 3. Then $M(\mathcal{H}_{1})>M(\mathcal{H}_{2})$.* *Proof.* By the definition of the Zagreb index, we have $$\begin{aligned} M(\mathcal{H}_{1})-M(\mathcal{H}_{2})&=d^{2}_{\mathcal{H}_{1}}(v_{k})+d^{2}_{\mathcal{H}_{1}}(v_{n})-d^{2}_{\mathcal{H}_{2}}(v_{k})-d^{2}_{\mathcal{H}_{2}}(v_{n})\\ &=(d_{\mathcal{H}}(u)+2)^{2}+1-(d_{\mathcal{H}}(u)+1)^{2}-4\\ &=2d_{\mathcal{H}}(u)>0.\end{aligned}$$ ◻ $\textbf{Transformation 4}$: Let $\mathcal{H}$ be an $m$-uniform connected hypergraph with $u, v\in{V(\mathcal{H})}$ such that $u\neq v$, $d_{\mathcal{H}}(u)>1$ and $d_{\mathcal{H}}(u)\geq d_{\mathcal{H}}(v)$. Let $\mathcal{T}_{1},\mathcal{T}_{2}$ be two binary $m$-uniform hypertrees, where $|E(\mathcal{T}_{1})|>0$. $\mathcal{H}_{1}$ denotes the hypergraph that results from identifying $u$ with the pendent vertex $u_{0}\in e_{0}$ of $\mathcal{T}_{1}$ and identifying $v$ with the pendent vertex $v_{0}$ of $\mathcal{T}_{2}$. Suppose that $v_{t}\in V(\mathcal{T}_{2})$ is a pendent vertex of $\mathcal{H}_{1}$, let $\mathcal{H}_{2}$ be obtained from $\mathcal{H}_{1}$ by deleting $e_{0}$ and adding $(e_{0}\setminus\{u\})\bigcup\{v_{t}\}$. **Lemma 5**. *Let $\mathcal{H}_{2}$ be obtained from $\mathcal{H}_{1}$ by transformation 4.* *(1). If $|E(\mathcal{T}_{2})|>0$, then $M(\mathcal{H}_{1})>M(\mathcal{H}_{2})$;* *(2). If $|E(\mathcal{T}_{2})|=0, d_{\mathcal{H}}(u)> d_{\mathcal{H}}(v)$, then $M(\mathcal{H}_{1})>M(\mathcal{H}_{2})$.* *Proof.* By the definition of the Zagreb index, if $|E(\mathcal{T}_{2})|>0$, we have $$\begin{aligned} M(\mathcal{H}_{1})-M(\mathcal{H}_{2})&=d^{2}_{\mathcal{H}_{1}}(u)+d^{2}_{\mathcal{H}_{1}}(v_{t})-d^{2}_{\mathcal{H}_{2}}(u)-d^{2}_{\mathcal{H}_{2}}(v_{t})\\ &=(d_{\mathcal{H}}(u)+1)^{2}+1-d_{\mathcal{H}}^{2}(u)-4\\ &=2d_{\mathcal{H}}(u)-2>0.\end{aligned}$$ If $|E(\mathcal{T}_{2})|=0$, $d_{\mathcal{H}}(u)> d_{\mathcal{H}}(v)$, we have $$\begin{aligned} M(\mathcal{H}_{1})-M(\mathcal{H}_{2})&=d^{2}_{\mathcal{H}_{1}}(u)+d^{2}_{\mathcal{H}_{1}}(v_{t})-d^{2}_{\mathcal{H}_{2}}(u)-d^{2}_{\mathcal{H}_{2}}(v_{t})\\ &=(d_{\mathcal{H}}(u)+1)^{2}+d_{\mathcal{H}}^{2}(v)-d_{\mathcal{H}}^{2}(u)-(d_{\mathcal{H}}(v)+1)^{2}\\ &=2d_{\mathcal{H}}(u)-2d_{\mathcal{H}}(v)>0.\end{aligned}$$ ◻ # The $S$-order in hypertrees In this section, we give the first and last hypergraphs in an $S$-order of all uniform hypertrees. In [@doi:10.1080/03081087.2021.1953431], the first $3k$th order spectral moments of uniform hypertrees were given. Let $N_{\mathcal{H}}(\widehat{\mathcal{H}})$ be the number of sub-hypergraphs of $\mathcal{H}$ isomorphic to $\widehat{\mathcal{H}}$ and $S_{q}$ be a star with $q$ edges. **Lemma 6**. *[@doi:10.1080/03081087.2021.1953431][\[w1\]]{#w1 label="w1"} Let $\mathcal{T}=(V(\mathcal{T}),E(\mathcal{T}))$ be an $m$-uniform hypertree. Then $$\begin{aligned} {\mathrm{S}_{m}}(\mathcal{T})&= {m^{m - 1}}{(m - 1)^{(|E(\mathcal{T})| - 1)(m - 1)}}{N_{\mathcal{T}}(P^{(m)}_1)} ,\\ \mathrm{S}_{2m}(\mathcal{T})&=m^{m-1}(m-1)^{(|E(\mathcal{T})|-1)(m-1)}N_{\mathcal{T}}(P_{1}^{(m)})+2m^{2m-3}(m-1)^{(|E(\mathcal{T})|-2)(m-1)}N_{\mathcal{T}}(P_{2}^{(m)}),\\ \mathrm{S}_{3m}(\mathcal{T})&=m^{m-1}(m-1)^{(|E(\mathcal{T})|-1)(m-1)}N_{\mathcal{T}}(P_{1}^{(m)})+6m^{2m-3}(m-1)^{(|E(\mathcal{T})|-2)(m-1)}N_{\mathcal{T}}(P_{2}^{(m)})\\&+3m^{3m-5}(m-1)^{(|E(\mathcal{T})|-3)(m-1)}N_{\mathcal{T}}(P_{3}^{(m)})+6m^{3m-5}(m-1)^{(|E(\mathcal{T})|-3)(m-1)}N_{\mathcal{T}}(S_{3}^{(m)}),\\ \mathrm{S}_{d}(\mathcal{T})&=0, \text{~for~} d=1,\ldots, m-1, m+1,\ldots, 2m-1, 2m+1,\ldots, 3m-1.\\\end{aligned}$$* Let $\textbf{T}_{q}$ be the set of all $m$-uniform hypertrees with $q$ edges. The following theorem gives the last hypergraph in an $S$-order of all $m$-uniform hypertrees. **Theorem 7**. *In an $S$-order of $\textbf{T}_{q}$, the last hypergraph is the hyperstar $S_{q}^{(m)}$.* *Proof.* Since in all $m$-uniform hypertrees with $q$ edges the spectral moments $\mathrm{S}_{0},\mathrm{S}_{1},\\\ldots,\mathrm{S}_{2m-1}$ are the same, the first significant spectral moment is the $2m$th. By Lemma [\[w1\]](#w1){reference-type="ref" reference="w1"}, $\mathrm{S}_{2m}$ is determined by the number of $P_{2}^{(m)}$. The number of vertices of $m$-uniform hypertrees with $q$ edges is $qm-q+1$. For any hypertree $\mathcal{T}$ in $\textbf{T}_{q}$, we have $$N_{\mathcal{T}}(P_{2}^{(m)})=\sum\limits_{i=1}^{qm-q+1}{d_{i}\choose2}=\frac{1}{2}\sum\limits_{i=1}^{qm-q+1}d_{i}^{2}-\frac{qm}{2}=\frac{1}{2}M(\mathcal{T})-\frac{qm}{2},$$ where $d_{1}+d_{2}+\cdots+d_{qm-q+1}=mq$. Repeating transformation 1, any $m$-uniform hypertree with $q$ edges can changed into $S_{q}^{(m)}$. And by Lemma [Lemma 2](#sp1){reference-type="ref" reference="sp1"}, each application of transformation 1 strictly increases the Zagreb index. Therefore, in an $S$-order of $\textbf{T}_{q}$, the last hypergraph is the hyperstar $S_{q}^{(m)}$. ◻ Let $\textbf{T}$ be the set of all binary $m$-uniform hypertrees with $q$ edges. We characterize the first few hypergraphs in the $S$-order of all $m$-uniform hypertrees. **Theorem 8**. *$\textbf{T}\prec_{s}\textbf{T}_{q}\setminus \textbf{T}$.* *Proof.* As in the proof of Theorem [Theorem 7](#sp6){reference-type="ref" reference="sp6"} we pay attention to the Zagreb index. Repeating transformation 3, any $m$-uniform hypertree with $q$ edges can changed into a binary $m$-uniform hypertree with $q$ edges. And from Lemma [Lemma 4](#sp4){reference-type="ref" reference="sp4"}, each application of transformation 3 strictly decreases the Zagreb index. Hence, $\textbf{T}\prec_{s}\textbf{T}_{q}\setminus \textbf{T}$. ◻ Let $P_{3}(\mathcal{H})$ be the set of all sub-hyperpaths length $3$ of an $m$-uniform hypergraph $\mathcal{H}$. **Lemma 9**. *Let $e=\{u,v,w_{1},\ldots,w_{m-2}\}$ be an edge and $\mathcal{H}_{1},\ldots,\mathcal{H}_{p}$ be pairwise disjoint connected $m$-uniform hypergraphs with $\mathcal{H}_{i}\neq P_{0}^{(m)}$ and $\widetilde{w}_{i}\in V(\mathcal{H}_{i})$ for each $i\in[p]$, where $m\geq 3$, $1\leq p\leq m-2$. Let $\mathcal{H}=e(w_{1},\ldots,w_{p})\bigodot(\mathcal{H}_{1}(\widetilde{w}_{1}),\ldots,\mathcal{H}_{p}(\widetilde{w}_{p}))$. Let $\mathcal{H}_{r,s}^{e}=\mathcal{H}(u,v)\bigodot(P_{r}^{(m)}(\widetilde{u}),P_{s}^{(m)}(\widetilde{v}))$, where $\widetilde{u},\widetilde{v}$ are respectively the pendent vertices of $P_{r}^{(m)}$ and $P_{s}^{(m)}$. If $r\geq s\geq 1$, then $$N_{\mathcal{H}_{r,s}^{e}}(P_{3}^{(m)})> N_{\mathcal{H}_{r+s,0}^{e}}(P_{3}^{(m)}).$$* *Proof.* Since $p\geq1$, let $e_{1}\in E(\mathcal{H}_{1})$ be an edge incident with $\widetilde{w}_{1}$. Let $e_{2}\in E(P_{r}^{(m)})$ be an edge incident with $\widetilde{u}$ and $e_{3}\in E(P_{s}^{(m)})$ be an edge incident with $\widetilde{v}$. We have $P_{3}(\mathcal{H}_{r,0}^{e})\subseteq P_{3}(\mathcal{H}_{r,s}^{e})$ and $P_{3}(\mathcal{H}_{r,0}^{e})\subseteq P_{3}(\mathcal{H}_{r+s,0}^{e})$. For a hyperpath $\mathcal{P}_{1}$ with $E(\mathcal{P}_{1})=\{e,e',e''\}$, $\mathcal{P}_{1}$ is also written $ee'e''$ in this paper. If $s=1$, there are hyperpaths $e_{2}ee_{3},e_{3}ee_{1}$ in $P_{3}(\mathcal{H}_{r,1}^{e})$ and not in $P_{3}(\mathcal{H}_{r,0}^{e})$. Since $p\geq1$, $N_{\mathcal{H}_{r,1}^{e}}(P_{3}^{(m)})- N_{\mathcal{H}_{r,0}^{e}}(P_{3}^{(m)})\geq2.$ There is only one hyperpath $\mathcal{P}$ in $P_{3}(\mathcal{H}_{r+1,0}^{e})$ and not in $P_{3}(\mathcal{H}_{r,0}^{e})$. And the edges of $\mathcal{P}$ are not in $E(\mathcal{H}_{i}), i=1,2,\ldots,p$. We have $N_{\mathcal{H}_{r+1,0}^{e}}(P_{3}^{(m)})- N_{\mathcal{H}_{r,0}^{e}}(P_{3}^{(m)})=1.$ So, $N_{\mathcal{H}_{r,1}^{e}}(P_{3}^{(m)})> N_{\mathcal{H}_{r+1,0}^{e}}(P_{3}^{(m)})$. If $s=2$, let $e_{4}\neq e_{3}\in E(P_{s}^{(m)})$. There are hyperpaths $e_{2}ee_{3},e_{3}ee_{1}, ee_{3}e_{4}$ in $P_{3}(\mathcal{H}_{r,2}^{e})$ and not in $P_{3}(\mathcal{H}_{r,0}^{e})$. Since $p\geq1$, $N_{\mathcal{H}_{r,2}^{e}}(P_{3}^{(m)})- N_{\mathcal{H}_{r,0}^{e}}(P_{3}^{(m)})\geq3.$ There are only two hyperpaths $\mathcal{P}'$, $\mathcal{P}''$ in $P_{3}(\mathcal{H}_{r+2,0}^{e})$ and not in $P_{3}(\mathcal{H}_{r,0}^{e})$. And the edges of $\mathcal{P}'$ and $\mathcal{P}''$ are not in $E(\mathcal{H}_{i}), i=1,2,\ldots,p$. We have $N_{\mathcal{H}_{r+2,0}^{e}}(P_{3}^{(m)})- N_{\mathcal{H}_{r,0}^{e}}(P_{3}^{(m)})=2.$ So, $N_{\mathcal{H}_{r,2}^{e}}(P_{3}^{(m)})> N_{\mathcal{H}_{r+2,0}^{e}}(P_{3}^{(m)})$. If $s>2$, similar to $s=2$, there are hyperpaths $e_{2}ee_{3},e_{3}ee_{1}, ee_{3}e_{4}$ in $P_{3}(\mathcal{H}_{r,s}^{e})$ and not in $P_{3}(\mathcal{H}_{r,0}^{e})$. For an $m$-uniform hyperpath with $q~ (q>2)$ edges, the number of the sub-hyperpaths with $3$ edges is $q-2$. Since $p\geq1$, $$N_{\mathcal{H}_{r,s}^{e}}(P_{3}^{(m)})- N_{\mathcal{H}_{r,0}^{e}}(P_{3}^{(m)})\geq 3+s-2=s+1.$$ Since $r\geq s>2$, there are only $s$ hyperpaths in $P_{3}(\mathcal{H}_{r+s,0}^{e})$ and not in $P_{3}(\mathcal{H}_{r,0}^{e})$. We have $N_{\mathcal{H}_{r+s,0}^{e}}(P_{3}^{(m)})- N_{\mathcal{H}_{r,0}^{e}}(P_{3}^{(m)})=s.$ So, if $s>2$, $N_{\mathcal{H}_{r,s}^{e}}(P_{3}^{(m)})> N_{\mathcal{H}_{r+s,0}^{e}}(P_{3}^{(m)})$. Therefore, if $r\geq s\geq 1$, we have $N_{\mathcal{H}_{r,s}^{e}}(P_{3}^{(m)})> N_{\mathcal{H}_{r+s,0}^{e}}(P_{3}^{(m)}).$ ◻ The following theorem gives the first hypergraph in an $S$-order of all $m$-uniform hypertrees. **Theorem 10**. *In an $S$-order of $\textbf{T}_{q}$, the first hypergraph is the hyperpath $P_{q}^{(m)}$.* *Proof.* In an $S$-order of $\textbf{T}_{q}$, by Theorem [Theorem 8](#zsy1){reference-type="ref" reference="zsy1"}, the first hypergraph is in $\textbf{T}$. When $m=2$, $\textbf{T}=\{P_{q}\}.$ Therefore, in an $S$-order of $\textbf{T}_{q}$, the first graph is the path $P_{q}$. When $m>2$, since the spectral moments $\mathrm{S}_{0},\mathrm{S}_{1},\ldots,\mathrm{S}_{3m-1}$ are the same in $\textbf{T}$, the first significant spectral moment is the $3m$th. By Lemma [\[w1\]](#w1){reference-type="ref" reference="w1"}, $\mathrm{S}_{3m}$ is determined by the number of $S_{3}^{(m)}$ and $P_{3}^{(m)}$. For any hypertree $\mathcal{T}$ in $\textbf{T}$, $N_{\mathcal{T}}(S_{3}^{(m)})=0$. Let $e(\mathcal{T})$ denote the set of all edges of $\mathcal{T}$ that contain at least 3 vertices whose degree is equal to 2. Fix a vertex $v$ of degree $2$ as a root. Let $\mathcal{T}_{1}, \mathcal{T}_{2}$ be the hypertrees attached at $v$. We can repeatedly apply the transformation from Lemma [Lemma 9](#z1){reference-type="ref" reference="z1"} at any two vertices $u_{1}, u_{2}\in e\in e(\mathcal{T})$ with largest distance from the root in every hypertree $\mathcal{T}_{i}$ and $d_{u_{1}}=d_{u_{2}}=2$, as long as $\mathcal{T}_{i}$ does not become a hyperpath. From Lemma [Lemma 9](#z1){reference-type="ref" reference="z1"}, each application of this transformation strictly decreases the number of sub-hyperpaths with $3$ edges. In the end of this process, we arrive at the hyperpath $P_{q}^{(m)}$. Therefore, in an $S$-order of $\textbf{T}_{q}$, the first hypergraph is the hyperpath $P_{q}^{(m)}$. ◻ # The $S$-order in unicyclic hypergraphs In this section, the expressions of $2m$th and $3m$th order spectral moments of linear unicyclic $m$-uniform hypergraphs are obtained in terms of the number of sub-hypergraphs. We characterize the first and last hypergraphs in an $S$-order of all linear unicyclic $m$-uniform hypergraphs with given girth. And we give the last hypergraph in an $S$-order of all linear unicyclic $m$-uniform hypergraphs. Let $\mathcal{H}(\omega)$ be a weighted uniform hypergraph, where $\omega: E(\mathcal{H})\rightarrow \mathbb{Z}^{+}$. Let $\omega(\mathcal{H})=\sum_{e\in E(\mathcal{H})}\omega(e)$ and $d_{v}(\mathcal{H}(\omega))=\sum_{e\in E_{v}(\mathcal{H})}\omega(e)$, where $E_{v}(\mathcal{H}):=\{e\in E(\mathcal{H})|v\in e\}$. Let $C_{n}$ be a cycle with $n$ edges. In [@ocestrada], the formula for the spectral moments of linear unicyclic $m$-uniform hypergraphs was given. **Theorem 11**. *[@ocestrada][\[sp8\]]{#sp8 label="sp8"} Let $\mathcal{U}$ be a linear unicyclic $m$-uniform hypergraph with girth $n$. If $m\mid d~(d\neq0)$ , then $$\label{888} \mathrm{S}_{d}(\mathcal{U})=d(m-1)^{|V(\mathcal{U})|}(\sum\limits_{\mathcal{\widehat{T}}\in \mathcal{B}_{tree}(\mathcal{U})}tr_{d}(\mathcal{\widehat{T}})+\sum\limits_{\mathcal{G}\in \mathcal{B}_{cycle}(\mathcal{U})}tr_{d}(\mathcal{G}))$$ and $$tr_{d}(\mathcal{\widehat{T}})=\sum\limits_{\omega:\omega(\mathcal{\widehat{T}})=d/m}(m-1)^{-|V(\mathcal{\widehat{T}})|}m^{(m-2)|E(\mathcal{\widehat{T}})|}\prod\limits_{v\in V(\mathcal{\widehat{T}})}(d_{v}(\mathcal{\widehat{T}}(\omega))-1)!\prod\limits_{e\in E(\mathcal{\widehat{T}})}\frac{\omega(e)^{m-1}}{(\omega(e)!)^{m}},$$ $$tr_{d}(\mathcal{G})=\sum\limits_{\omega:\omega(\mathcal{G})=d/m}2(m-1)^{-|V(\mathcal{G})|}m^{(m-2)|E(\mathcal{G})|-1}\prod\limits_{v\in V(\mathcal{G})}(d_{v}(\mathcal{G}(\omega))-1)!\prod\limits_{e\in E(\mathcal{G})}\frac{\omega(e)^{m-1}}{(\omega(e)!)^{m}}\Omega_{C_{n}^{(m)}(\omega^{0})},$$ where $$\Omega_{C_{n}^{(m)}(\omega^{0})}=\sum\limits_{x=0}^{2\omega_{min}^{0}}\prod\limits_{i=1}^{n}\frac{(\omega_{i}^{0}!)^{2}}{(\omega_{i-1}^{0}+\omega_{min}^{0}-x)!(\omega_{i}^{0}-\omega_{min}^{0}+x)!}\sum\limits_{l=0}^{n-1}\prod\limits_{i=1}^{l}(\omega_{i}^{0}+\omega_{min}^{0}-x) \prod\limits_{i=l+2}^{n}(\omega_{i}^{0}-\omega_{min}^{0}+x),$$ $\omega_{min}^{0}=\mathrm{min}_{i\in n}\omega_{i}^{0}$, $\omega_{i}^{0}=\omega^{0}(e_{i}), i\in[n]$, $\mathcal{B}_{tree}(\mathcal{U})$ denotes the set of connected sub-hypergraphs of $\mathcal{U}$ which are hypertrees, $\mathcal{B}_{cycle}(\mathcal{U})$ denotes the set of connected sub-hypergraphs of $\mathcal{U}$ which contain the hypercycle.* *If $m\nmid d$, then $\mathrm{S}_{d}(\mathcal{U})=0$.* We give expressions of $2m$th and $3m$th order spectral moments of a linear unicyclic $m$-uniform hypergraph in terms of the number of some sub-hypergraphs. **Corollary 12**. *Let $\mathcal{U}$ be a linear unicyclic $m$-uniform hypergraph. Then we have $$\mathrm{S}_{2m}(\mathcal{U})=m^{(m-1)}(m-1)^{|V(\mathcal{U})|-m}N_{\mathcal{U}}(P_{1}^{(m)})+2m^{2m-3}(m-1)^{|V(\mathcal{U})|-2m+1}N_{\mathcal{U}}(P_{2}^{(m)}).$$* *Proof.* Since $2m/m<g(\mathcal{U})$, the second summand in ([\[888\]](#888){reference-type="ref" reference="888"}) does not appear. By Theorem [\[sp8\]](#sp8){reference-type="ref" reference="sp8"}, we have $$\begin{aligned} \mathrm{S}_{2m}(\mathcal{U})&=2m(m-1)^{|V(\mathcal{U})|}\sum\limits_{\mathcal{\widehat{T}}\in \mathcal{B}_{tree}(\mathcal{U})}\sum\limits_{\omega:\omega(\mathcal{\widehat{T}})=2}(m-1)^{-|V(\mathcal{\widehat{T}})|}m^{(m-2)|E(\mathcal{\widehat{T}})|}\\ &\prod\limits_{v\in V(\mathcal{\widehat{T}})}(d_{v}(\mathcal{\widehat{T}}(\omega))-1)!\prod\limits_{e\in E(\mathcal{\widehat{T}})}\frac{\omega(e)^{m-1}}{(\omega(e)!)^{m}}.\end{aligned}$$ Since $\omega(\mathcal{\widehat{T}})=\sum_{e\in E(\mathcal{\widehat{T}})}\omega(e)=2$, $\widehat{T}$ is an edge $e$ with $\omega(e)=2$ or $\widehat{T}$ is a hyperpath of length $2$ with $\omega(e_{i})=1, i\in[2]$, where $E(\mathcal{\widehat{T}})=\{e_{1},e_{2}\}$. So $$\begin{aligned} \mathrm{S}_{2m}(\mathcal{U})&=2m(m-1)^{|V(\mathcal{U})|}((m-1)^{-m}m^{(m-2)}\frac{2^{m-1}}{2^{m}}N_{\mathcal{U}}(P_{1}^{(m)})+(m-1)^{1-2m}m^{2(m-2)}N_{\mathcal{U}}(P_{2}^{(m)}))\\ &=m^{(m-1)}(m-1)^{|V(\mathcal{U})|-m}N_{\mathcal{U}}(P_{1}^{(m)})+2m^{2m-3}(m-1)^{|V(\mathcal{U})|-2m+1}N_{\mathcal{U}}(P_{2}^{(m)}).\end{aligned}$$ ◻ **Corollary 13**. *Let $\mathcal{U}$ be a linear unicyclic $m$-uniform hypergraph with girth $g$ $(g>3)$. Then we have $$\begin{aligned} \mathrm{S}_{3m}(\mathcal{U}) &=(m-1)^{|V(\mathcal{U})|-m}m^{m-1}N_{\mathcal{U}}(P_{1}^{(m)})+6m^{2m-3}(m-1)^{|V(\mathcal{U})|+1-2m}N_{\mathcal{U}}(P_{2}^{(m)})\\ &+3m^{3m-5}(m-1)^{|V(\mathcal{U})|+2-3m}N_{\mathcal{U}}(P_{3}^{(m)})+6m^{3m-5}(m-1)^{|V(\mathcal{U})|+2-3m}N_{\mathcal{U}}(S_{3}^{(m)}).\end{aligned}$$ Let $\mathcal{U}$ be a linear unicyclic $m$-uniform hypergraph with girth $3$. Then we have $$\begin{aligned} \mathrm{S}_{3m}(\mathcal{U}) &=(m-1)^{|V(\mathcal{U})|-m}m^{m-1}N_{\mathcal{U}}(P_{1}^{(m)})+6m^{2m-3}(m-1)^{|V(\mathcal{U})|+1-2m}N_{\mathcal{U}}(P_{2}^{(m)})\\ &+3m^{3m-5}(m-1)^{|V(\mathcal{U})|+2-3m}N_{\mathcal{U}}(P_{3}^{(m)})+6m^{3m-5}(m-1)^{|V(\mathcal{U})|+2-3m}N_{\mathcal{U}}(S_{3}^{(m)})\\ &+24m^{3m-6}(m-1)^{|V(\mathcal{U})|-3m+3}.\end{aligned}$$* *Proof.* When $g>3$, since $3m/m<g$, the second summand in ([\[888\]](#888){reference-type="ref" reference="888"}) does not appear. By Theorem [\[sp8\]](#sp8){reference-type="ref" reference="sp8"}, we have $$\begin{aligned} \mathrm{S}_{3m}(\mathcal{U})&=3m(m-1)^{|V(\mathcal{U})|}\sum\limits_{\mathcal{\widehat{T}}\in \mathcal{B}_{tree}(\mathcal{U})}\sum\limits_{\omega:\omega(\mathcal{\widehat{T}})=3}(m-1)^{-|V(\mathcal{\widehat{T}})|}m^{(m-2)|E(\mathcal{\widehat{T}})|}\\ &\prod\limits_{v\in V(\mathcal{\widehat{T}})}(d_{v}(\mathcal{\widehat{T}}(\omega))-1)!\prod\limits_{e\in E(\mathcal{\widehat{T}})}\frac{\omega(e)^{m-1}}{(\omega(e)!)^{m}}.\end{aligned}$$ Since $\omega(\mathcal{\widehat{T}})=\sum_{e\in E(\mathcal{\widehat{T}})}\omega(e)=3$, we have\ (1). $\widehat{T}$ is an edge $e$ with $\omega(e)=3$;\ (2). $\widehat{T}$ is a hyperpath of length $2$ with $\omega(e_{1})=1$, $\omega(e_{2})=2$ or $\omega(e_{1})=2$, $\omega(e_{2})=1$, where $E(\mathcal{\widehat{T}})=\{e_{1},e_{2}\}$;\ (3). $\widehat{T}$ is a hyperpath of length $3$ with $\omega(e_{i})=1, i\in[3]$, where $E(\mathcal{\widehat{T}})=\{e_{1},e_{2},e_{3}\}$;\ (4). $\widehat{T}$ is a hyperstar with $3$ edges and $\omega(e_{i})=1, i\in[3]$, where $E(\mathcal{\widehat{T}})=\{e_{1},e_{2},e_{3}\}$. Therefore, $$\begin{aligned} \mathrm{S}_{3m}(\mathcal{U})&=3m(m-1)^{|V(\mathcal{U})|}((m-1)^{-m}m^{(m-2)}(2!)^{m}\frac{3^{m-1}}{(3!)^{m}}N_{\mathcal{U}}(P_{1}^{(m)})\\ &+(m-1)^{1-2m}m^{2(m-2)}2!\frac{2^{m-1}}{(2!)^{m}}2N_{\mathcal{U}}(P_{2}^{(m)})\\ &+(m-1)^{2-3m}m^{3(m-2)}N_{\mathcal{U}}(P_{3}^{(m)})+(m-1)^{2-3m}m^{3(m-2)}2!N_{\mathcal{U}}(S_{3}^{(m)}))\\ &=(m-1)^{|V(\mathcal{U})|-m}m^{m-1}N_{\mathcal{U}}(P_{1}^{(m)})+6m^{2m-3}(m-1)^{|V(\mathcal{U})|+1-2m}N_{\mathcal{U}}(P_{2}^{(m)})\\ &+3m^{3m-5}(m-1)^{|V(\mathcal{U})|+2-3m}N_{\mathcal{U}}(P_{3}^{(m)})+6m^{3m-5}(m-1)^{|V(\mathcal{U})|+2-3m}N_{\mathcal{U}}(S_{3}^{(m)}).\end{aligned}$$ When $g=3$, since $\omega(\mathcal{\widehat{T}})=\sum_{e\in E(\mathcal{\widehat{T}})}\omega(e)=3$ , we have\ (1). $\widehat{T}$ is an edge $e$ with $\omega(e)=3$;\ (2). $\widehat{T}$ is a hyperpath of length $2$ with $\omega(e_{1})=1$, $\omega(e_{2})=2$ or $\omega(e_{1})=2$, $\omega(e_{2})=1$, where $E(\mathcal{\widehat{T}})=\{e_{1},e_{2}\}$;\ (3). $\widehat{T}$ is a hyperpath of length $3$ with $\omega(e_{i})=1, i\in[3]$, where $E(\mathcal{\widehat{T}})=\{e_{1},e_{2},e_{3}\}$;\ (4). $\widehat{T}$ is a hyperstar with $3$ edges and $\omega(e_{i})=1, i\in[3]$, where $E(\mathcal{\widehat{T}})=\{e_{1},e_{2},e_{3}\}$. Since $\omega(\mathcal{G})=\sum_{e\in E(\mathcal{G})}\omega(e)=3$, $\mathcal{G}$ is a hypercycle with girth $3$, $\omega_{i}^{0}=\omega^{0}(e_{i})=1, i\in[3]$ and $\Omega_{C_{3}^{(m)}(\omega^{0})}=4$, where $E(\mathcal{G})=\{e_{1},e_{2},e_{3}\}$. By Theorem [\[sp8\]](#sp8){reference-type="ref" reference="sp8"}, we have $$\begin{aligned} \mathrm{S}_{3m}(\mathcal{U})&=3m(m-1)^{|V(\mathcal{U})|}((m-1)^{-m}m^{(m-2)}(2!)^{m}\frac{3^{m-1}}{(3!)^{m}}N_{\mathcal{U}}(P_{1}^{(m)})\\ &+(m-1)^{1-2m}m^{2(m-2)}2!\frac{2^{m-1}}{(2!)^{m}}2N_{\mathcal{U}}(P_{2}^{(m)})+(m-1)^{2-3m}m^{3(m-2)}N_{\mathcal{U}}(P_{3}^{(m)})\\ &+(m-1)^{2-3m}m^{3(m-2)}2!N_{\mathcal{U}}(S_{3}^{(m)}) +2(m-1)^{-3m+3}m^{3(m-2)-1}4) \\ &=(m-1)^{|V(\mathcal{U})|-m}m^{m-1}N_{\mathcal{U}}(P_{1}^{(m)})+6m^{2m-3}(m-1)^{|V(\mathcal{U})|+1-2m}N_{\mathcal{U}}(P_{2}^{(m)})\\ &+3m^{3m-5}(m-1)^{|V(\mathcal{U})|+2-3m}N_{\mathcal{U}}(P_{3}^{(m)})+6m^{3m-5}(m-1)^{|V(\mathcal{U})|+2-3m}N_{\mathcal{U}}(S_{3}^{(m)})\\ &+24m^{3m-6}(m-1)^{|V(\mathcal{U})|-3m+3}.\end{aligned}$$ ◻ The set of all linear unicyclic $m$-uniform hypergraphs with $e+f$ edges which contain a hypercycle $C_{e}^{(m)}$ will be denoted by $\textbf{U}^{m}_{ef}$. Let $F^{(m)}_{ef}$ be the linear unicyclic $m$-uniform hypergraph obtained from the hypercycle $C^{(m)}_{e}$ by attached $f$ pendant edges to one of non core vertices on $C^{(m)}_{e}$. The following theorem gives the last hypergraph in an $S$-order of all linear unicyclic $m$-uniform hypergraphs with given girth. **Theorem 14**. *In an $S$-order of $\textbf{U}^{m}_{ef}$ the last hypergraph is $F^{(m)}_{ef}$.* *Proof.* Since in $\textbf{U}^{m}_{ef}$ the spectral moments $\mathrm{S}_{0},\mathrm{S}_{1},\ldots,\mathrm{S}_{2m-1}$ are the same, the first significant spectral moment is the $2m$th. By Corollary [Corollary 12](#w3){reference-type="ref" reference="w3"}, $\mathrm{S}_{2m}$ is determined by the number of $P_{2}^{(m)}$. The number of vertices of linear unicyclic $m$-uniform hypergraphs with $e+f$ edges is $(e+f)(m-1)$. For any $\mathcal{U}\in \textbf{U}^{m}_{ef}$, we have $$N_{\mathcal{U}}(P_{2}^{(m)})=\sum\limits_{i=1}^{em+fm-e-f}{d_{i}\choose2}=\frac{1}{2}\sum\limits_{i=1}^{em+fm-e-f}d_{i}^{2}-\frac{em+fm}{2}=\frac{1}{2}M(\mathcal{U})-\frac{em+fm}{2},$$ where $d_{1}+d_{2}+\cdots+d_{em+fm-e-f}=em+fm$. Repeating transformation 1, any linear unicyclic $m$-uniform hypergraph in $\textbf{U}^{m}_{ef}$ can be changed into a linear unicyclic $m$-uniform hypergraph such that all the edges not on $C^{(m)}_{e}$ are pendant edges and incident with non core vertices of $C^{(m)}_{e}$. After repeating transformation 1, if we repeat transformation 2, any linear unicyclic $m$-uniform hypergraph in $\textbf{U}^{m}_{ef}$ can be changed into a linear unicyclic $m$-uniform hypergraph obtained from the hypercycle $C^{(m)}_{e}$ by attached $f$ pendant edges to one of non core vertices on $C^{(m)}_{e}$. From Lemma [Lemma 2](#sp1){reference-type="ref" reference="sp1"} and Lemma [Lemma 3](#sp2){reference-type="ref" reference="sp2"}, each application of transformation 1 or 2 strictly increases the Zagreb index. Hence, in an $S$-order of $\textbf{U}^{m}_{ef}$ the last hypergraph is $F^{(m)}_{ef}$. ◻ The set of all linear unicyclic $m$-uniform hypergraphs with $q$ edges will be denoted by $\textbf{U}_{q}$. The following theorem gives the last hypergraph in an $S$-order of all linear unicyclic $m$-uniform hypergraphs. **Theorem 15**. *In an $S$-order of $\textbf{U}_{q}$ the last hypergraph is $F^{(m)}_{3(q-3)}$.* *Proof.* By Theorem [Theorem 14](#sp9){reference-type="ref" reference="sp9"}, we get that in an $S$-order of $\textbf{U}^{m}_{l(q-l)}$ the last hypergraph is $F^{(m)}_{l(q-l)}$. By the definition of the Zagreb index, we have $M(F_{l(q-l)}^{(m)})=(m-2)l+(q-l)(m-1)+4(l-1)+(q-l+2)^{2}=l^{2}-l-2ql+qm+3q+q^{2}, 3\leq l\leq q$. Since the derivative of $M(F_{l(q-l)}^{(m)})$ over $l$ is equal to $2l-1-2q<0$, $M(F_{l(q-l)}^{(m)})\leq M(F^{(m)}_{3(q-3)})$ for $3\leq l\leq q$ with the equality if and only if $l=3$. Hence, in an $S$-order of $\textbf{U}_{q}$ the last hypergraph is $F^{(m)}_{3(q-3)}$. ◻ For $m\geq3,$ let $\textbf{U}$ be the set of all linear unicyclic $m$-uniform hypergraphs with $e+f$ edges and girth $e$ such that the degree of all the vertices is less than or equal to $2$. We characterize the first few hypergraphs in the $S$-order of all linear unicyclic $m$-uniform hypergraphs with given girth. **Theorem 16**. *For $m\geq3,$ $$\textbf{U}\prec_{s}\textbf{U}^{m}_{ef}\setminus \textbf{U}.$$* *Proof.* As in the proof of Theorem [Theorem 14](#sp9){reference-type="ref" reference="sp9"} we pay attention to the Zagreb index. Repeating transformation 3, any $m$-uniform hypertree attached to an $m$-uniform hypergraph $\mathcal{H}$ can be changed into a binary $m$-uniform hypertree. After repeating transformation 3, if we repeat transformation 4, then any linear unicyclic $m$-uniform hypergraph in $\textbf{U}^{m}_{ef}$ can be changed into a linear unicyclic $m$-uniform hypergraph in $\textbf{U}$. And from Lemma [Lemma 4](#sp4){reference-type="ref" reference="sp4"} and Lemma [Lemma 5](#sp5){reference-type="ref" reference="sp5"}, the Zagreb indices decrease. Hence, we have $\textbf{U}\prec_{s}\textbf{U}^{m}_{ef}\setminus \textbf{U}.$ ◻ We give a transformation which will decrease the number of sub-hyperpaths with $3$ edges of hypergraphs as follows: $\textbf{Transformation 5}$: Let $\mathcal{P}_{i}\neq P_{0}^{(m)}$ be an $m$-uniform hyperpath, $u_{i}$ be a pendent vertex of $\mathcal{P}_{i}$ for each $i\in[p]$ and $v_{1},v_{2},\ldots,v_{e(m-2)}$ be core vertices of a linear $m$-uniform hypercycle $C^{(m)}_{e}$, where $m\geq 3$ and $2\leq p\leq e(m-2)$. Let $\mathcal{H}_{1}=C^{(m)}_{e}(v_{1},\ldots,v_{p})\bigodot(\mathcal{P}_{1}(u_{1}),\ldots,\mathcal{P}_{p}(u_{p}))$. Suppose that $u_{1}\in e_{1}$ in $\mathcal{P}_{1}$, $w_{1}\in V(\mathcal{P}_{2})$ is a pendent vertex of $\mathcal{H}_{1}$, let $\mathcal{H}_{2}$ be obtained from $\mathcal{H}_{1}$ by deleting $e_{1}$ and adding $(e_{1}\setminus\{u_{1}\})\bigcup\{w_{1}\}$. **Lemma 17**. *Let $\mathcal{H}_{2}$ be obtained from $\mathcal{H}_{1}$ by transformation 5. Then $N_{\mathcal{H}_{2}}(P_{3}^{(m)})<N_{\mathcal{H}_{1}}(P_{3}^{(m)})$.* *Proof.* Let $\mathcal{H}_{3}=C^{(m)}_{e}(v_{2},\ldots,v_{p})\bigodot(\mathcal{P}_{2}(u_{2}),\ldots,\mathcal{P}_{p}(u_{p}))$ and $\mathcal{P}_{1}'=\mathcal{P}_{1}-e_{1}+(e_{1}\setminus\{u_{1}\})\bigcup\{w_{1}\}$. So $P_{3}(\mathcal{H}_{1})=P_{3}(\mathcal{H}_{3})+P_{3}(\mathcal{P}_{1})+P_{\mathcal{H}_{1}}$ and $P_{3}(\mathcal{H}_{2})=P_{3}(\mathcal{H}_{3})+P_{3}(\mathcal{P}_{1}' )+P_{\mathcal{H}_{2}}$, where $P_{\mathcal{H}_{1}}~(P_{\mathcal{H}_{2}})$ is the set of all the sub-hyperpaths with $3$ edges of $\mathcal{H}_{1}(\mathcal{H}_{2})$, each of them contains both at least one edge in $E(\mathcal{H}_{3})$ and at least one edge in $E(\mathcal{P}_{1})~(E(\mathcal{P}_{1}')).$ We have $|E(\mathcal{P}_{1})|=|E(\mathcal{P}_{1}')|$ and $N_{\mathcal{P}_{1}'}(P_{3}^{(m)})=N_{\mathcal{P}_{1}}(P_{3}^{(m)})$. If $|E(\mathcal{P}_{1})|=1$, since $p\geq2$, in $P_{\mathcal{H}_{1}}$ there are 2 hyperpaths at least which contain $e_{1}$ and two edges in $E(\mathcal{H}_{3})$. In $P_{\mathcal{H}_{2}}$ there is a hyperpath which contain $(e_{1}\setminus\{u_{1}\})\bigcup\{w_{1}\}$ and two edges in $E(\mathcal{H}_{3})$. Therefore, we have $|P_{\mathcal{H}_{1}}|-|P_{\mathcal{H}_{2}}|\geq1$. Hence, $N_{\mathcal{H}_{1}}(P_{3}^{(m)})-N_{\mathcal{H}_{2}}(P_{3}^{(m)})\geq1$. So, $N_{\mathcal{H}_{2}}(P_{3}^{(m)})<N_{\mathcal{H}_{1}}(P_{3}^{(m)})$. If $|E(\mathcal{P}_{1})|\geq2$, since $p\geq2$, in $P_{\mathcal{H}_{1}}$ there are 2 hyperpaths at least which contain $e_{1}$ and two edges in $E(\mathcal{H}_{3})$ and there is a hyperpath which contain two edges in $E(\mathcal{P}_{1})$ and an edge in $E(\mathcal{H}_{3})$. In $P_{\mathcal{H}_{2}}$ there is a hyperpath which contain $(e_{1}\setminus\{u_{1}\})\bigcup\{w_{1}\}$ and two edges in $E(\mathcal{H}_{3})$ and there is a hyperpath which contain two edges in $E(\mathcal{P}_{1}')$ and an edge in $E(\mathcal{H}_{3})$. Therefore, we have $|P_{\mathcal{H}_{1}}|-|P_{\mathcal{H}_{2}}|\geq1$. Hence, $N_{\mathcal{H}_{1}}(P_{3}^{(m)})-N_{\mathcal{H}_{2}}(P_{3}^{(m)})\geq1$. So, $N_{\mathcal{H}_{2}}(P_{3}^{(m)})<N_{\mathcal{H}_{1}}(P_{3}^{(m)})$. ◻ Let $E_{ef}^{m}$ be the linear unicyclic $m$-uniform hypergraph obtained by the coalescence of $C^{(m)}_{e}$ at one of its core vertices with $P^{(m)}_{f}$ at one of its pendent vertices. The following theorem gives the first hypergraph in an $S$-order of all linear unicyclic $m$-uniform hypergraphs with given girth. **Theorem 18**. *For $m\geq 3$, in an $S$-order of $\textbf{U}^{m}_{ef}$ the first hypergraph is $E_{ef}^{m}$.* *Proof.* In an $S$-order of $\textbf{U}^{m}_{ef}$, by Theorem [Theorem 16](#sp10){reference-type="ref" reference="sp10"}, the first hypergraph is in $\textbf{U}$. Since the spectral moments $\mathrm{S}_{0},\mathrm{S}_{1},\ldots,\mathrm{S}_{3m-1}$ are the same in $\textbf{U}$, the first significant spectral moment is the $3m$th. By Corollary [Corollary 13](#sp11){reference-type="ref" reference="sp11"}, $\mathrm{S}_{3m}$ is determined by the number of $S_{3}^{(m)}$ and $P_{3}^{(m)}$. For any $\mathcal{H}\in \textbf{U}$, $N_{\mathcal{H}}(S_{3}^{(m)})=0$. Let $\mathcal{T}_{1},\ldots,\mathcal{T}_{p}$ be pairwise disjoint binary $m$-uniform hypertrees, $u_{i}$ be a pendent vertex of $\mathcal{T}_{i}$ for each $i\in[p]$ and $v_{1},\ldots,v_{p}$ be core vertices of $C^{(m)}_{e}$, where $1\leq p\leq e(m-2)$ and $\sum_{i=1}^{p}|E(\mathcal{T}_{i})|=f$. For any $\mathcal{H}=C^{(m)}_{e}(v_{1},\ldots,v_{p})\bigodot(\mathcal{T}_{1}(u_{1}),\ldots,\mathcal{T}_{p}(u_{p}))\in \textbf{U}$, let $e(\mathcal{H})$ denote the set of all edges of $\mathcal{H}-E(C^{(m)}_{e})$ that contain at least 3 vertices whose degree is equal to 2. Let the vertex $u_{i}$ as a root in $\mathcal{T}_{i}$. We can repeatedly apply the transformation from Lemma [Lemma 9](#z1){reference-type="ref" reference="z1"} at any two vertices $u, v\in e\in e(\mathcal{H})$ with largest distance from the root in every hypertree $\mathcal{T}_{i}$ and $d_{u}=d_{v}=2$, as long as $\mathcal{T}_{i}$ does not become a hyperpath. By Lemma [Lemma 9](#z1){reference-type="ref" reference="z1"}, each application of this transformation strictly decreases the number of sub-hyperpaths with $3$ edges. When all hypertrees $\mathcal{T}_{1},\ldots,\mathcal{T}_{p}$ turn into hyperpaths, we can repeatedly apply the transformation 5, as long as there exist at least two hyperpaths of length at least one, By Lemma [Lemma 17](#zsy2){reference-type="ref" reference="zsy2"}, each application of transformation 5 strictly decreases the number of sub-hyperpaths with $3$ edges. In the end of this process, we arrive at the $E_{ef}^{m}$. ◻ **Acknowledgments** This work is supported by the National Natural Science Foundation of China (No. 11801115, No. 12071097, No. 12042103 and No. 12242105), the Natural Science Foundation of the Heilongjiang Province (No. QC2018002) and the Fundamental Research Funds for the Central Universities. # References {#references .unnumbered}

arxiv_math

--- abstract: | This paper devotes to the study of the classical Abel equation $\frac{dx}{dt}=g(t)x^{3}+f(t)x^{2}$, where $g(t)$ and $f(t)$ are trigonometric polynomials of degree $m\geq1$. We are interested in the problem that whether there is a uniform upper bound for the number of limit cycles of the equation with respect to $m$, which is known as the famous Smale-Pugh problem. In this work we generalize an idea from the recent paper (Yu, Chen and Liu, arXiv:$2304.13528$, $2023$) and give a new criterion to estimate the maximum multiplicity of limit cycles of the above Abel equations. By virtue of this criterion and the previous results given by Álvarez et al. and Bravo et al., we completely solve the simplest case of the Smale-Pugh problem, i.e., the case when $g(t)$ and $f(t)$ are linear trigonometric, and obtain that the maximum number of limit cycles, is three. author: - | Xiangqin Yu$^{a}$, Jianfeng Huang$^{b}$, Changjian Liu$^{a,*}$\ $^{a}$ School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, 519082, P.R. China \ $^{b}$ Department of Mathematics, Jinan University, Guangzhou, 510632, P.R. China title: Maximum number of limit cycles for Abel equation having coefficients with linear trigonometric functions --- **Key words**. Trigonometric Abel equations, Limit cycles, Maximum number. # Introduction and main result[\[introduction\]]{#introduction label="introduction"} The theories on the Abel differential equations provide tools to study the problems in many areas of mathematics, especially in the qualitative theory of differential equations. It initially arose in the studies of Abel [@abel1829] on the theory of elliptic functions, and appeared in the reduction of order for the high-order differential equations, therefore were frequently found in the varied real models, such as the Liénard equations [@gine2011], the evolution equations from the cosmological models [@yurov2014application], and the reaction-diffusion equation that describes the evolution of glioblastomas [@harko2015bio]. In the past decades, the Abel equations have gained increasing attentions because they not only are applied to the characterizations of real periodic phenomena (see e.g. the tracking control problem [@fossas2008iterative], the perturbed pendulums [@gasull2016number; @gasull2020chebyshev] and the seasonal prey harvesting [@xu2005harvesting]), but also play an important role in the study of the Hilbert's $16$th problem (see e.g. [@cherkas1976number; @li2003hilbert]). The second part of the Hilbert's $16$th problem essentially asks that whether there exists a finite number $\mathscr H(m)$ such that each planar polynomial differential system of degree $\leq m$ has no more than $\mathscr H(m)$ limit cycles. An Abel equation of generalized type is a one-dimensional non-autonomous differential equation that is written as $$\label{eq1} \displaystyle\frac{dx}{dt}=\displaystyle\sum_{i=0}^{n} A_{i}(t)x^{i},$$ where $x\in \mathbb R,t\in \mathbb R$ and $A_{i}:\mathbb R\rightarrow \mathbb R,i=1,...,n,$ are analytic $2\pi$-periodic functions. For the cases that $n=3, 2$ and $1$, the equation is called the (classical) Abel equation, the Riccati equation and the linear equation, respectively. Let $x(t,x_{0})$ be the solution of equation [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"} satisfying $x(0,x_{0})=x_{0}$. We say that $x(t,x_{0})$ is a periodic solution of the equation if $x(2\pi,x_{0})=x_{0}$. And an orbit $x=x(t,x_{0})$ is called a periodic orbit (resp. limit cycle) of the equation, if $x(t,x_{0})$ is a periodic solution (resp. isolated periodic solution). The studies of the Hilbert's $16$th problem via the equation [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"} mainly goes back to the 70's of the last century. It is shown that there is a series of planar polynomial differential systems that can be reduced to the Abel equations, using e.g. the polar coordinates and Cherkas¡¯ transformation [@devlin1998cubic]. For more details we refer to the works on the quadratic systems (see e.g. [@lins1980number; @gasull1990limit]), the rigid systems, and some other general planar polynomial differential systems (see e.g. [@gasull1990limit; @huang2017estimate]). For this reason, estimating the number of limit cycles of the Abel equation [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"}, becomes a significant problem and is extensively studied by the researchers. The first progress in this research area is motivated by Lins-Neto [@lins1980number] and Lloyd [@lloyd1973number; @lloyd1975class; @lloyd1979note]. They proved that there exists at most $n$ limit cycle(s) of the equation [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"} for $n=1, 2$ (i.e., for the linear type and the Riccati type). However, the situation becomes extremely complicated for $n=3$ (i.e., for the classical Abel type). In [@lins1980number] Lins-Neto shown that the maximum number of limit cycles of the classical Abel equations, is unbounded without additional conditions (see also [@panov1999diversity]). More concretely, by using the bifurcation methods he presented that the equation of the form $$\begin{aligned} \label{eq2} \frac{dx}{dt}=A_3(t)x^3+A_2(t)x^2 \end{aligned}$$ with $A_3(t)$ and $A_2(t)$ being the trigonometric polynomials of degree $m$, can possesses at least $m$ limit cycles. Such result was generalized later to the case $n>3$ in the work [@gasull2006limit]. Due to these facts and the above backgrounds, a more specific related version of the Hilbert's $16$th problem arises (see for instance [@lins1980number; @ilyashenko2000hilbert]): *Whether the maximum number of limit cycles for the equation [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"} with the coefficients $A_i$'s being trigonometric polynomials of degree $m$, is bounded in terms of $m$*? For the initial case, i.e., the equation of the form [\[eq2\]](#eq2){reference-type="eqref" reference="eq2"}, it is known as the Smale-Pugh problem [@smale2000mathematical]. In fact, this is a very difficult problem that is still open for even the linear trigonometric classical Abel equations. So far, there are two lines of nice works, exploring the problem from different perspectives. The first line of works is based on the definite sign(s) hypothesis for some coefficients or their linear combinations of the equation, which essentially come from the ¡°transversality¡± of some curve(s) and the orbits of the equation. To illustrate, Lloyd [@lloyd1979note] and Pliss [@pliss1966nonlocal] (resp. Gasull and Llibre [@gasull1990limit]) obtained that the equation [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"} with $n=3$ has at most three limit cycles if $A_{3}(t)\neq0$ (resp. $A_2(t)\neq0$ and $A_0(t)\equiv0$). Later Gasull and Guillamon [@gasull2006limit] extended such result, showing that the number of limit cycles for the equation $dx/dt=A_{n_1}(t)x^{n_1}+A_{n_2}(t)x^{n_2}+A_{1}(t)x$ with $n_1>n_2>1$ is bounded if either $A_{n_1}(t)$ or $A_{n_2}(t)$ has definite sign. In [@alvarez2007new] Álvarez et al. proved for the first time that the equation [\[eq2\]](#eq2){reference-type="eqref" reference="eq2"} can only possess at most one non-zero limit cycle if there exists a linear combination of $A_3(t)$ and $A_2(t)$ with fixed sign. This yields one of the key results that will solve the linear trigonometric case of the smale-pugh problem, which is presented in Theorem [Theorem 1](#AT1){reference-type="ref" reference="AT1"} below. Huang and Liang [@huang2020geometric] developed the idea in [@alvarez2007new] and provided a criterion for the linear trigonometric generalized Abel equations having at most $n$ limit cycles. For more works of this line, see [@alvarez2015limit; @ilyashenko2000hilbert; @huang2012periodic; @huang2017estimate]. The second line of works mainly focuses on the hypothesis for the symmetries of the coefficients of the equation [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"}, see for instance [@alvarez2013existence; @bravo2009limit; @bravo2008nonexistence] and the references therein. It is notable that under such symmetry hypothesis the authors in [@bravo2009limit] provide the estimate for the number of limit cycles of the linear trigonometric classical Abel equations, which will be the second key result applied in our paper. We state it in Theorem [Theorem 2](#AT2){reference-type="ref" reference="AT2"}. For the other related works, the readers are referred to papers [@alvarez2017centers; @huang2020number]. Let us come back to the initial case of the problem, i.e., the Smale-Pugh problem. In this paper, we focus on the simplest unsolved case $$\label{eq3} \frac{dx}{dt}=(a_0+a_1\sin t+a_2\cos t)x^3+(b_0+b_1\sin t+b_2\cos t)x^2,$$ where $a_0, a_1, a_2, b_0, b_1, b_2\in\mathbb R$. The equation [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"} was first systematically studied in [@alvarez2007new] and [@bravo2009limit]. Following the different ideas (as introduced above) the authors of these two works gave several important criteria to estimate the maximum number of limit cycles of the equation, which are summarized as below: **Theorem 1** ([@alvarez2007new]). *Suppose that the equation [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"} has not a center at $x=0$. If there exists $\lambda\in\mathbb R$ such that $\lambda(a_0+a_1\sin t+a_2\cos t)+(b_0+b_1\sin t+b_2\cos t)$ does not vanish identically and does not change sign, i.e., one of the conditions $a_{0}^{2}\geq a_{1}^{2}+a_{2}^{2}$, $b_{0}^{2}\geq b_{1}^{2}+b_{2}^{2},$ or $(a_{1}b_{0}-a_{0}b_{1})^{2}+(a_{0}b_{2}-a_{2}b_{0})^{2}\geq (a_{1}b_{2}-a_{2}b_{1})^{2}$ is satisfied, then it has at most one non-zero periodic orbit. Furthermore, when this periodic orbit exists, it is hyperbolic.* **Theorem 2** ([@bravo2009limit]). *If $a_{0}b_{0}=0$, then equation [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"} has at most three limit cycles, including $x=0$. Moreover, the upper bound is sharp and in case of having three limit cycles, one is in the region $x>0$ and the other is in the region $x<0$.* We remark that the result in [@bravo2009limit] is improved recently by Bravo, Fernández and Ojeda [@bravo2023stability]. The authors show that the maximum number of limit cycles of the equation [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"} is still three when $a_{0}b_{0}$ is sufficiently small. Another new work is given by Yu, Chen and Liu [@yu2023number], in which the equation [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"} with $a_2=b_1=0$ reduced from the Josephson equations is studied. The authors also prove that the number of limit cycles of such equation, does not exceed three. On the other hand, by means of the Hopf (resp. Poincaré) bifurcation it was obtained in [@alvarez2007new] (resp. [@huang2020number]) that the equation [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"} can has at least three limit cycles. Throughout these works all the evidences yield the following specific version of the simplest case of the Smale-Pugh problem, which is proposed by Gasull and stated as the $6$th of the $33$ open problems in [@gasull2021some]: *Whether the maximum number of limit cycles of the equation [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"}, is three*? The purpose of this paper is to study the cases of the equation [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"} which are not considered in the previous works [@alvarez2007new; @bravo2009limit], and then completely solve the above Smale-Pugh-Gasull problem in [@gasull2021some]. By generalizing the recent work in [@yu2023number], we give a criterion for the maximum multiplicity of the non-zero limit cycles of the Abel equation [\[eq2\]](#eq2){reference-type="eqref" reference="eq2"} with general coefficients (see section [\[mA1\]](#mA1){reference-type="ref" reference="mA1"} for details). Then, applying this criterion, and taking Theorem [Theorem 1](#AT1){reference-type="ref" reference="AT1"} and Theorem [Theorem 2](#AT2){reference-type="ref" reference="AT2"} into account, we finally provide a positive answer as below: **Theorem 3**. *The equation [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"}, with arbitrary parameters $a_0,a_1,a_2,b_0,b_1,b_2\in\mathbb R$, has at most three limit cycles (including $x=0$). Moreover, this upper bound is sharp.* The layout of the rest of this paper is as follows: In Section [\[pre\]](#pre){reference-type="ref" reference="pre"}, we present some preliminary results. In Section [\[mA1\]](#mA1){reference-type="ref" reference="mA1"}, we give a new criterion to estimate the maximum multiplicity of limit cycles of the Abel equation [\[eq2\]](#eq2){reference-type="eqref" reference="eq2"}. Finally, the proof of Theorem [Theorem 3](#T2){reference-type="ref" reference="T2"} is given in Section [\[mA2\]](#mA2){reference-type="ref" reference="mA2"}. # Preliminaries [\[pre\]]{#pre label="pre"} We recall some basic tools in this section. All of them will be necessary in our argument. The first one characterizes the derivatives of the Poincaré map for the one-dimensional differential equations, which is initially provided in [@lloyd1979note]. **Proposition 1** ([@lloyd1979note]). *Suppose that $x(t,x_{0})$ is the solution of the one-dimensional differential equation ${\frac{dx}{dt}}=S(x,t)$ that satisfies $x(t_0,x_{0})=x_{0}$. Denote by $L(x_0)=x(t_0+2\pi, x_0)$. Then* 1. *$L^{'}(x_0)=\exp\bigg[\displaystyle{\int_{t_0}^{t_0+2\pi} \frac{\partial S}{\partial x}(x(t, x_0),t)\, dt}\bigg];$* 2. *$L^{''}(x_0)=L^{'}(x_0)\bigg[\displaystyle{\int_{t_0}^{t_0+2\pi} \frac{\partial^{2} S}{\partial x^{2}}(x(t, x_0),t)}\cdot \exp{\displaystyle{\bigg\{\int_{t_0}^{t} \frac{\partial S}{\partial x}(x(\tau, x_0),\tau)} \, d\tau \bigg\}}\, dt\bigg].$* The followings present the concept and several properties of rotated one-dimensional differential equations, which are directly adapted from the classical theory of rotated vector fields (see for instance [@duff1953number]). For more details of these adaptions we also refer readers to the works [@bravo2015stability; @bravo2023stability; @han2018theory; @yu2023number]. **Definition 1**. *Consider a family of equations $$\label{eq4} \frac{dx}{dt}=S(x,t;\lambda),$$ where $t\in E\subset\mathbb R$, $x\in J\subset\mathbb R$ and the parameter $\lambda\in \mathbb R$. Then we say that [\[eq4\]](#eq4){reference-type="eqref" reference="eq4"} defines a family of rotated equations in $E\times J$, if $\frac{\partial{S}}{\partial{\lambda}}(x,t;\lambda)>0$ for $\lambda\in\mathbb R$.* **Proposition 2**. *Suppose that the equation [\[eq4\]](#eq4){reference-type="eqref" reference="eq4"} defines a family of rotated equations in the region $[0,2\pi]\times J\subset[0,2\pi]\times\mathbb R$.* 1. *If $x=x(t)$ is a stable or an unstable limit cycle of the equation [\[eq4\]](#eq4){reference-type="eqref" reference="eq4"}$|_{\lambda=\lambda_0}$ in the region, then there exists $\delta>0$ such that for $\lambda\in(\lambda_0-\delta,\lambda_0+\delta)$ the equation [\[eq4\]](#eq4){reference-type="eqref" reference="eq4"} has a limit cycle $x=\hat x(t;\lambda)$ with $\hat x(t;\lambda_0)=x(t)$. Moreover:* - *When $x=x(t)$ is stable (resp. unstable), $x=\hat x(t;\lambda)$ is lower-stable and increases (resp. upper-unstable and decreases) as ${\lambda}$ increases.* - *When $x=x(t)$ is stable (resp. unstable), $x=\hat x(t;\lambda)$ is upper-stable and decreases (resp. lower-unstable and increases) as ${\lambda}$ decreases.* 2. *If $x=x(t)$ is a semi-stable limit cycle of the equation [\[eq4\]](#eq4){reference-type="eqref" reference="eq4"}$|_{\lambda=\lambda_0}$ in the region, then the following statements hold.* - *When $x=x(t)$ is upper-unstable and lower-stable (resp. upper-stable and lower-unstable), there exists $\delta>0$ such that for $\lambda\in(\lambda_0-\delta,\lambda_0)$ (resp. $\lambda\in(\lambda_0,\lambda_0+\delta)$) the equation [\[eq4\]](#eq4){reference-type="eqref" reference="eq4"} has two limit cycles locating on the distinct sides of $x=x(t)$.* - *When $x=x(t)$ is upper-unstable and lower-stable (resp. upper-stable and lower-unstable), the equation [\[eq4\]](#eq4){reference-type="eqref" reference="eq4"} has no limit cycle near $x=x(t)$ for $\lambda>\lambda_0$ (resp. $\lambda<\lambda_0$).* # The maximum multiplicity of limit cycles of equation [\[eq2\]](#eq2){reference-type="eqref" reference="eq2"} [\[mA1\]]{#mA1 label="mA1"} This section is devoted to giving a criterion for the maximum multiplicity of the non-zero limit cycles of the Abel equation [\[eq2\]](#eq2){reference-type="eqref" reference="eq2"} with general coefficients. This will be the last key tool to prove our main result. For the sake of brevity and clarity, in this section we rewrite the equation as $$\label{eq5} \frac{dx}{dt}=g(t)x^{3}+f(t)x^{2},$$ with $g(t)$ and $f(t)$ being the smooth $2\pi$-periodic functions (which can be non-trigonometric). Since the case when $g(t)$ or $f(t)$ has definite sign is already clear from [@alvarez2007new Theorem A] (see also section [\[introduction\]](#introduction){reference-type="ref" reference="introduction"} and the works [@gasull1990limit; @lloyd1979note; @pliss1966nonlocal]), we mainly analyze the case that both $g(t)$ and $f(t)$ have indefinite signs here. More precisely, for equation [\[eq5\]](#eq5){reference-type="eqref" reference="eq5"} we propose the following two hypotheses: 1. $f(t)$ and $g(t)$ have no common zeros, and $g(t)$ has exactly two zeros in any $2\pi$-period. 2. The function $u(t):=-\frac{f(t)}{g(t)}$ has the same strict monotonicity in all connected component of $\{t|g(t)\neq 0\}$. Now let us consider a non-zero limit cycle $x=x(t)$ of the equation [\[eq5\]](#eq5){reference-type="eqref" reference="eq5"}. For abbreviation we use the notations $$\begin{aligned} \label{eq6} x_{*}=\min \limits_{t\in \mathbb R} x(t),\ x^{*}=\max \limits_{t\in \mathbb R} x(t),\ t_{*}\in \{t_{0}|x(t_{0})=x_*\},\ t^{*}\in \{t_{0}|x(t_{0})=x^*\}\cap[t_{*},t_{*}+2\pi),\end{aligned}$$ and also denote by $t_{1}$ and $t_{2}$ the points satisfying $$\begin{aligned} \label{eq7} t_1,t_2\in\{t|g(t)=0\}\cap[t_{*},t_{*}+2\pi),\indent t_1<t_2.\end{aligned}$$ We give the following result. **Lemma 1**. *Suppose that $x=x(t)$ is a non-zero limit cycle of the equation [\[eq5\]](#eq5){reference-type="eqref" reference="eq5"}. Then under hypotheses **(C.1)** and **(C.2)**, the following statements hold.* 1. *$x(t)$ has exactly one maximum point, one minimum point and no other stationary point in any $2\pi$-period.* 2. *The point $t^*$ is contained in $(t_1,t_2)$. Moreover, when $u(t)=-\frac{f(t)}{g(t)}$ is strictly monotonically increasing, $$\mbox{$x(t)-u(t)$ $\begin{cases} <0, t\in (t_{*},t_{1})\cup (t^{*},t_{2}),\\ >0, t\in (t_{1},t^{*})\cup (t_{2},t_{*}+2\pi); \end{cases}$}$$ and when $u(t)=-\frac{f(t)}{g(t)}$ is strictly monotonically decreasing, $$\mbox{$x(t)-u(t)$ $\begin{cases} >0, t\in (t_{*},t_{1})\cup (t^{*},t_{2}),\\ <0, t\in (t_{1},t^{*})\cup (t_{2},t_{*}+2\pi). \end{cases}$}$$* (i) It is sufficient to show that the conclusion holds in the period $[t_{*},t_{*}+2\pi)$. As defined in [\[eq6\]](#eq6){reference-type="eqref" reference="eq6"}, it is clear that $g(t_*)x_*+f(t_*)=0.$ Then, by means of hypothesis **(C.1)**, we have that $t_{*} \in \{t |g(t)\neq 0\}$, and there are $3$ connected components of $g(t)x+f(t)=0$ in the region $[t_{*},t_{*}+2\pi]\times \mathbb R$, that is, $x=u(t)$ with $t \in [t_{*},t_{1}), (t_{1},t_{2})$ and $(t_{2}, t_{*}+2\pi]$. On the other hand, one can get from hypothesis **(C.2)** that $$\left | \begin{matrix} 1 & g(t)x^{3}+f(t)x^{2} \\ 1 &u^{'}(t) \\ \end{matrix} \right |_{x=u(t)} =u^{'}(t)\geq 0\ ( {\rm or}\leq0),$$ which implies that all the orbits of the equation [\[eq5\]](#eq5){reference-type="eqref" reference="eq5"} cross the curve $x=u(t)$ from a same side to the other. Thus, the limit cycle $x=x(t)$ can only intersects each connected component of $g(t)x+f(t)=0$ once, and these intersections in the region $[t_{*},t_{*}+2\pi]\times \mathbb R$ must be $(t_{*},x_*)$, $(t^*,x^*)$ and $(t_{*}+2\pi,x_*)$. As a result, $x(t)$ has a unique minimum point $t_*$, a unique maximum point $t^*$ and no other stationary point in $[t_{*},t_{*}+2\pi)$ (more precisely, $t^*\in(t_1,t_2)$). The statement is verified. Also, we additionally obtain that $x(t)$ is increasing (resp. decreasing) in $(t_{*},t^{*})$ (resp. $(t^{*},t_{*}+2\pi)$). (ii) We only prove the case that $u(t)$ is strictly monotonically increasing, and the other case follows from a similar argument. According to statement (i), $x=x(t)$ and $x=u(t)$ do not intersect when $t\in(t_{*},t_{*}+2\pi)\backslash \{t_{1},t^{*},t_{2}\}$. Since $x(t_{*})=u(t_{*})$ and $x^{'}(t_{*})=0\leq u^{'}(t_{*})$, we get that $x(t)-u(t)<0$ for $t\in (t_{*},t_{1})$. Similarly, $x(t^{*})=u(t^{*})$ with $x^{'}(t^{*})=0\leq u^{'}(t^{*})$ yields that $x(t)-u(t)>0$ for $t\in (t_{1},t^{*})$ and $x(t)-u(t)<0$ for $t\in (t^{*},t_{2})$. Moreover, note that $x(t_{*}+2\pi)=u(t_{*}+2\pi)$ and $x^{'}(t_{*}+2\pi)=0\leq u^{'}(t_{*}+2\pi)$, we also have $x(t)-u(t)>0$ for $t\in (t_{2},t_{*}+2\pi)$. This completes the proof.$\Box$ In the next we provide our main theorem of this section. We remark that this result is essentially obtained by an approach established in the work [@yu2023number] which studies the equation [\[eq5\]](#eq5){reference-type="eqref" reference="eq5"} with some specific $f(t)$ and $g(t)$. **Theorem 4**. *Under hypotheses **(C.1)** and **(C.2)**, the multiplicity of each non-zero limit cycle of the equation [\[eq5\]](#eq5){reference-type="eqref" reference="eq5"} in the region $x\neq0$, is at most two. Moreover, when the non-hyperbolic limit cycle exists, it must be lower-stable and upper-unstable (resp. lower-unstable and upper-stable) if $u(t)$ is strictly monotonically increasing (resp. decreasing).* We only consider below the case that the function $u(t)$ in Hypothesis **(C.2)** is strictly monotonically increasing, and the opposite case follows exactly in the same way. Assume that $x=x(t)$ is a non-zero limit cycle of the equation [\[eq5\]](#eq5){reference-type="eqref" reference="eq5"}. Let $x_*$, $x^*$, $t_*$ and $t^*$ be the notations defined as in [\[eq6\]](#eq6){reference-type="eqref" reference="eq6"}. According to Proposition [Proposition 1](#pro11){reference-type="ref" reference="pro11"}, the multiplicity and stability of $x=x(t)$ can be determined by $$\label{eq8} L'(x_{*})=\exp\big(\int_{t_{*}}^{t_{*}+2\pi} (2f(t)x(t)+3g(t)x^2(t))\, dt\big)$$ and $$\label{eq9} L''(x_{*})=L'(x_{*})\int_{t_{*}}^{t_{*}+2\pi}(2f(t)+6g(t)x(t))\exp\big(\int_{t_{*}}^{t} (2f(s)x(s)+3g(s)x^2(s))\, ds\big)\, dt$$ if $L'(x_{*})-1$ and $L''(x_{*})$ do not simultaneously vanish, where $L$ represents the Poincaré map of the equation [\[eq5\]](#eq5){reference-type="eqref" reference="eq5"} from the initial time $t_*$ to the ending time $t_*+2\pi$. In the following we focus on the case when $x=x(t)$ is non-hyperbolic. First, we shall simply the expression of $L''(x_{*})$. It is clear that $L'(x_*)=1$ in this case. Then, $$\label{eq10} \int_{t_{*}}^{t_{*}+2\pi} (2f(t)x(t)+3g(t)x^2(t))\, dt=0,$$ and $L''(x_{*})$ is decomposed into $L''(x_{*})=4W_{1}+2W_{2}$, where $$\begin{aligned} &W_{1}=\int_{t_{*}}^{t_{*}+2\pi}g(t)x(t)\exp\big(\int_{t_{*}}^{t} (2f(\tau)x(\tau)+3g(\tau)x^2(\tau))\, d\tau\big)\, dt, \\ &W_{2}=\int_{t_{*}}^{t_{*}+2\pi}(f(t)+g(t)x(t))\exp\big(\int_{t_{*}}^{t} (2f(\tau)x(\tau)+3g(\tau)x^2(\tau))\, d\tau\big)\, dt.\end{aligned}$$ Furthermore, note that $$\label{eq11} \int_{t_{*}}^{t} \left(f(\tau)x(\tau)+g(\tau)x^2(\tau)\right) d\tau =\int_{t_{*}}^{t} \frac{dx(\tau)}{x(\tau)} =\ln \frac{x(t)}{x(t_{*})}.$$ Setting $h(t):=\int_{t_{*}}^{t} g(\tau)x^2(\tau)\, d\tau$, we get by [\[eq10\]](#eq10){reference-type="eqref" reference="eq10"} and [\[eq11\]](#eq11){reference-type="eqref" reference="eq11"} that $$\begin{aligned} \label{eq12} h(t_{*}+2\pi) =\int_{t_{*}}^{t_{*}+2\pi} (2f(t)x(t)+3g(t)x^2(t))\, dt -2\int_{t_{*}}^{t_{*}+2\pi} \big(f(t)x(t)+g(t)x^{2}(t)\big)\,dt =0.\end{aligned}$$ Therefore, the equalities [\[eq11\]](#eq11){reference-type="eqref" reference="eq11"} and [\[eq12\]](#eq12){reference-type="eqref" reference="eq12"} yield that $$\begin{aligned} W_{1}&=\int_{t_{*}}^{t_{*}+2\pi} g(t)x(t)\exp\big(2 \ln \frac{x(t)}{x(t_{*})}+h(t)\big)\, dt =\frac{1}{x_{*}^{2}}\int_{t_{*}}^{t_{*}+2\pi} g(t)x^{3}(t)\exp h(t)\, dt\\ & =\frac{1}{x_{*}^{2}}\int_{t_{*}}^{t_{*}+2\pi} x(t) d \exp h(t) =-\frac{1}{x_{*}^{2}}\int_{t_{*}}^{t_{*}+2\pi} \exp h(t) d x(t), \\ W_{2}& =\int_{t_{*}}^{t_{*}+2\pi} (f(t)+g(t)x(t))\exp\big(2 \ln \frac{x(t)}{x(t_{*})}+h(t)\big)\, dt\\ & =\frac{1}{x_{*}^{2}}\int_{t_{*}}^{t_{*}+2\pi} (f(t)x^{2}(t)+g(t)x^{3}(t))\exp h(t)\, dt\\ & =\frac{1}{x_{*}^{2}}\int_{t_{*}}^{t_{*}+2\pi} \exp h(t) dx(t). \label{eq11} \end{aligned}$$ Together with statement (i) of Lemma [Lemma 1](#lem3.1){reference-type="ref" reference="lem3.1"}, the expression of $L''(x_{*})$ can be reduced to $$\label{eq13} \begin{split} L''(x_{*})& =-\frac{2}{x_{*}^{2}}\int_{t_{*}}^{t_{*}+2\pi} \exp h(t) d x(t)\\ & =-\frac{2}{x_{*}^{2}}\big(\int_{t_{*}}^{t^{*}} \exp h(t)d x(t)+\int_{t^{*}}^{t_{*}+2\pi} \exp h(t)d x(t)\big) \\ & =-\frac{2}{x_{*}^{2}}\big(\int_{x_{*}}^{x^{*}} \exp h(\tau_{1}(x))d x+\int_{x^{*}}^{x_{*}} \exp h(\tau_{2}(x))d x\big) \\ & =-\frac{2}{x_{*}^{2}}\big(\int_{x_{*}}^{x_{*}} (\exp h(\tau_{1}(x))-\exp h(\tau_{2}(x)))d x\big), \end{split}$$ where $\tau_{1}$ and $\tau_{2}$ represent the inverse functions of $x|_{[t_{*},t^{*}]}$ and $x|_{[t^{*},t_{*}+2\pi]}$ respectively, satisfying $$\label{eq14} \tau_{1}'|_{(x_*,x^*)}>0,\ \tau_{2}'|_{(x_*,x^*)}<0,\ \tau_{1}(x_{*})=t_{*},\ \tau_{2}(x_{*})=t_{*}+2\pi\text{ and } \tau_{1}(x^{*})=\tau_{2}(x^{*})=t^{*}.$$ For convenience of readers, the graph of the limit cycle $x=x(t)$ when it is positive is provided and depicted in Fig. [\[Fig. 1\]](#Fig. 1){reference-type="ref" reference="Fig. 1"}. Next we ascertain the sign of $L''(x_{*})$ by [\[eq13\]](#eq13){reference-type="eqref" reference="eq13"}. This can be done once the function $$\label{eq15} W(s):=h(\tau_{1}(s))-h(\tau_{2}(s))=\int_{\tau_{2}(s)}^{\tau_{1}(s)} g(\tau)x^2(\tau)\, d\tau$$ has definite sign in the interval $(x_{*},x^{*})$. Since, $W(x_*)=W(x^*)=0$ due to [\[eq12\]](#eq12){reference-type="eqref" reference="eq12"} and [\[eq14\]](#eq14){reference-type="eqref" reference="eq14"}, the problem becomes analyzing the first-order derivative of $W(s)$. We emphasize that $W'(s)$ is continuous in $(x_{*},x^{*})$ with expression $$\begin{split} W'(s)&=g(\tau_{1}(s))x^{2}(\tau_{1}(s))\tau_{1}^{'}(s)-g(\tau_{2}(s))x^{2}(\tau_{2}(s))\tau_{2}^{'}(s)\\ &=\frac{g(\tau_{1}(s))}{f(\tau_{1}(s))+g(\tau_{1}(s))x(\tau_{1}(s))}-\frac{g(\tau_{2}(s))}{f(\tau_{2}(s))+g(\tau_{2}(s))x(\tau_{2}(s))} . \end{split}$$ In addition, let $t_1$ and $t_2$ be defined as in [\[eq7\]](#eq7){reference-type="eqref" reference="eq7"}. Observe that $t_{1}\in(t_*,t^{*})$ and $t_{2}\in(t^*,t_*+2\pi)$ from statement (ii) of Lemma [Lemma 1](#lem3.1){reference-type="ref" reference="lem3.1"}. Then, $g(\tau_1(s))\neq0$ and $g(\tau_2(s))\neq0$ when $s\not\in\{x(t_1), x(t_2)\}$. Accordingly, for $s\in (x_{*},x^{*})\backslash \{x(t_{1}),x(t_{2})\}$, $$\label{eq16} \begin{split} W'(s) =\hspace{1.5mm} \frac{1}{x(\tau_{1}(s))-u(\tau_{1}(s))}-\frac{1}{x(\tau_{2}(s))-u(\tau_{2}(s))}. \end{split}$$ Now denote by $x_{1}=\min \{x(t_{1}),x(t_{2})\}$ and $x_{2}=\max \{x(t_{1}),x(t_{2})\}$. According to the monotonicity of $\tau_1$ and $\tau_2$, we have $$\begin{aligned} \left\{ \begin{aligned} \tau_{1}(x_{1})\leq t_{1},\\ \tau_{1}(x_{2})\geq t_{1}, \\ \end{aligned} \right. \text{ and } \left\{ \begin{aligned} \tau_{2}(x_{1})\geq t_{2},\\ \tau_{2}(x_{2})\leq t_{2},\\ \end{aligned} \right.\end{aligned}$$ respectively. These yield that $$\begin{aligned} \left\{ \begin{aligned} &\tau_{1}(s)\in (t_{*},t_{1}),\ \tau_{2}(s)\in (t_{2},t_{*}+2\pi), &\text{when } s\in (x_{*},x_{1}),\\ &\tau_{1}(s)\in (t_{1},t^{*}),\ \tau_{2}(s)\in (t^{*},t_{2}), &\text{when } s\in (x_{2}, x^{*}). \end{aligned} \right.\end{aligned}$$ Hence, taking statement (ii) of Lemma [Lemma 1](#lem3.1){reference-type="ref" reference="lem3.1"} into account, we get that $$\begin{aligned} \label{eq17A} x(\tau_1(s))-u(\tau_{1}(s)) \left\{ \begin{aligned} <0, s\in (x_{*},x_{1}),\\ >0, s\in (x_{2},x^{*}), \end{aligned} \right.\ \text{ and }\ x(\tau_2(s))-u(\tau_{2}(s)) \left\{ \begin{aligned} >0, s\in (x_{*},x_{1}),\\ <0, s\in (x_{2},x^{*}). \end{aligned} \right.\end{aligned}$$ Let us come back to the analysis of the sign of $W^{'}(s)$. The argument is divided into two cases: Case 1. $s\in (x_{*},x^{*})\backslash[x_{1},x_{2}]$. It follows directly from [\[eq16\]](#eq16){reference-type="eqref" reference="eq16"} and [\[eq17A\]](#eq17A){reference-type="eqref" reference="eq17A"} that $W^{'}(s)< 0$ for $s\in (x_{*},x_{1})$ and $W^{'}(s)> 0$ for $s\in (x_{2},x^{*})$. Case 2. $s\in (x_{1},x_{2})$. In this case, we write [\[eq16\]](#eq16){reference-type="eqref" reference="eq16"} as $$W'(s) =\frac{V(s)}{\big(x(\tau_{1}(s))-u(\tau_{1}(s))\big)\big(x(\tau_{2}(s))-u(\tau_{2}(s))\big)}$$ with $V(s)=u(\tau_{1}(s))-u(\tau_{2}(s))$. We stress that this expression is well-defined in the interval and therefore $\big(x(\tau_{1}(s))-u(\tau_{1}(s))\big)\big(x(\tau_{2}(s))-u(\tau_{2}(s))\big)\neq0$. Also, on account of the monotonicity of $\tau_1$, $\tau_2$ and $u$, one can easily see that both the composite functions $u\circ\tau_1$ and $-u\circ\tau_2$ are strictly monotonically increasing in $(x_{1},x_{2})$, and so does $V(s)$. As a result, $W'(s)$ has at most one zero, i.e., it can only change sign at most once, in $(x_{1},x_{2})$. In summary, according to the continuity of $W'$ and the arguments for the above cases, we confirm that there exists a unique ${x_{0}}\in [x_{1},x_{2}]$, such that $W'({x_{0}})=0$, $W'(x)<0$ for $x\in (x_{*},{x_{0}})$ and $W'(x)>0$ for $x\in ({x_{0}},x^{*})$. This together with $W(x_{*})=W(x^{*})=0$ implies that $W(s)<0$ for $s\in (x_{*},x^{*})$. As a consequence, due to [\[eq13\]](#eq13){reference-type="eqref" reference="eq13"}, we finally obtain that $L''(x_{*})>0$. That is to say, the non-zero limit cycle of the equation [\[eq5\]](#eq5){reference-type="eqref" reference="eq5"} has multiplicity at most $2$, and it is lower-stable and upper-unstable when it is non-hyperbolic. The proof of Theorem [Theorem 4](#T1){reference-type="ref" reference="T1"} is concluded. $\Box$ # Proof of Theorem [Theorem 3](#T2){reference-type="ref" reference="T2"}[\[mA2\]]{#mA2 label="mA2"} {#proof-of-theorem-t2ma2} The goal of this section is to prove Theorem [Theorem 3](#T2){reference-type="ref" reference="T2"} and give a positive answer to the Smale-Pugh-Gasull problem. Let us start from the simplifications for the equation [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"}. By using the transformation $t\mapsto t+\theta$ with $\theta=\arctan\frac{a_2}{a_1}$ (resp. $\theta=\frac{\pi}{2}$) when $a_1\neq 0$ (resp. $a_1=0$), the equation can be reduced to the following type: $$\label{eq18} \frac{dx}{dt}=(p_0+p_1\sin t)x^3+(q_0+q_1\sin t+q_2\cos t)x^2,$$ where $p_0=a_0$, $p_1=\sqrt{a_{1}^{2}+a_{2}^{2}}\geq0$, $q_0=b_0$, $q_1=\frac{a_1b_1+a_2b_2}{\sqrt{a_{1}^{2}+a_{2}^{2}}}$ and $q_2=\frac{a_1b_2-a_2b_1}{\sqrt{a_{1}^{2}+a_{2}^{2}}}$. Moreover, observe that the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} is invariant under the transformations $(t,p_0,q_0,q_2)\mapsto (-t,-p_0,-q_0,-q_2)$ and $(x,t,p_0,q_1)\mapsto (-x,-t,-p_0,-q_1)$, respectively. Thus, without loss of generality, our consideration can be restricted to the case that $q_0, p_0\geq 0$. On the other hand, thanks to the Theorem [Theorem 1](#AT1){reference-type="ref" reference="AT1"}, it is sufficient to deal with the case when the condition of the third inequality in the theorem is invalid for the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"}, that is, $(p_0q_1-p_1q_0)^{2}+p_{0}^{2}q_{2}^{2}<p_{1}^{2}q_{2}^{2}$ (which immediately yields $p_1\neq0$ and $q_2\neq0$). In summary, we only need to focus on the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} with the following hypothesis: $(p_0q_1-p_1q_0)^{2}+p_{0}^{2}q_{2}^{2}<p_{1}^{2}q_{2}^{2},\ p_0\geq0,\ q_0\geq0,\ p_1>0,\ q_1\in\mathbb R$ and $q_2\neq0$. In what follows, we divide our argument into several steps. They are presented as the auxiliary results in the next subsection. ## Some auxiliary results[\[Some\]]{#Some label="Some"} First of all, directly applying Theorem [Theorem 4](#T1){reference-type="ref" reference="T1"} to the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} yields the following lemma. **Lemma 2**. *Under Hypothesis **(H)**, each non-zero limit cycle of the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} has multiplicity at most two. In particular, if such limit cycle is exactly non-hyperbolic, then it must be lower-stable and upper-unstable (resp. lower-unstable and upper-stable) when $q_{2}>0$ (${\rm resp.}\ q_{2}<0$).* We write the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} as [\[eq5\]](#eq5){reference-type="eqref" reference="eq5"} with $g(t)=p_0+p_1\sin t$ and $f(t)=q_0+q_1\sin t+q_2\cos t$. Since the inequality $(p_0q_1-p_1q_0)^{2}+p_{0}^{2}q_{2}^{2}<p_{1}^{2}q_{2}^{2}$ in Hypothesis **(H)** implies that $|p_0|<|p_1|$, the function $g(t)$ has exactly two zeros in any $2\pi$-period. Furthermore, it follows from a direct calculation that $$\begin{aligned} \label{eq19} \begin{split} \text{sgn}\left(g'(t)f(t)-f'(t)g(t)\right) &=\text{sgn}\left(p_{1}q_{2}+p_{0}q_{2}\sin t+(p_{1}q_{0}-p_{0}q_{1})\cos t\right)\\ &=\text{sgn}\left(p_{1}q_{2}+\sqrt{p_{0}^{2}q_{2}^{2}+(p_{1}q_{0}-p_{0}q_{1})^{2}}\sin(t+\phi)\right)\\ &=\text{sgn}\left(q_2\right)\\ &\neq0, \end{split}\end{aligned}$$ where $\phi=\arctan{\frac{p_{1}q_{0}-p_{0}q_{1}}{p_{0}q_{2}}}$ (resp. $\phi=\frac{\pi}{2}$) when $p_0>0$ (resp. $p_0=0$). Hence, there are no common zeros of $g(t)$ and $f(t)$. The equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} satisfies the condition **(C.1)** of Theorem [Theorem 4](#T1){reference-type="ref" reference="T1"}. Observe that [\[eq19\]](#eq19){reference-type="eqref" reference="eq19"} also implies that $$\label{eq17} \text{sgn}\left(\bigg(-\frac{f(t)}{g(t)}\bigg)^{'}\right) =\text{sgn}\left(q_2\right).$$ Thus $-\frac{f(t)}{g(t)}$ is strictly monotonically increasing (resp. decreasing) when $q_{2}>0$ (resp. $q_2<0$). The equation also satisfies the condition **(C.2)** of Theorem [Theorem 4](#T1){reference-type="ref" reference="T1"}. Consequently, our conclusion is obtained on account of Theorem [Theorem 4](#T1){reference-type="ref" reference="T1"}. $\Box$ Next we illustrate the stability of the orbit $x=0$ of the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"}. Let $x(t,x_{0})$ be the solution of the equation satisfying the initial condition $x(0,x_{0})=x_{0}$. It is known that the displacement function $H(x_0):=x(2 \pi ,x_0)-x_0$ is well-defined in a neighbourhood of zero. Then it can be expanded into power series with respect to $x_0$, that is: $$%\label{eq16} H(x_0) =\sum_{i=2}^{\infty} L_ix_0^i,$$ where the coefficients $L_i$'s are called the *Lyapunov constants* of the orbit $x=0$ (see for instance [@alvarez2007new]). Following the approach in [@lloyd1973number; @lloyd1983small] or directly the result in [@alvarez2007new], one can easily get that $L_2=2\pi q_{0}$; $L_3=2\pi p_{0}$ when $L_2=0$; and $L_4=p_{1}q_{2}\pi$ when $L_2=L_3=0$. Hence, under Hypothesis **(H)**, the orbit $x=0$ has multiplicity at most $4$, and its stability with respect to the parameters is summarized in the Table [1](#Tab-zero){reference-type="ref" reference="Tab-zero"}. Classification of parameters Multiplicity Stability ------------------------------ -------------- ----------------------------------- $q_0>0$ 2 upper-unstable and lower-stable $q_0=0, p_0>0$ 3 upper-unstable and lower-unstable $p_0=q_0=0, q_2>0$ 4 upper-unstable and lower-stable $p_0=q_0=0, q_2<0$ 4 upper-stable and lower-unstable : Multiplicity and stability of the orbit $x=0$ for equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} Now by using Lemma [Lemma 2](#lem4.1){reference-type="ref" reference="lem4.1"} and the stability of the orbit $x=0$, we provide a preliminary estimate for the number of limit cycles of the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"}. **Lemma 3**. *Assume that Hypothesis **(H)** holds with $q_0\neq0$ for equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"}.* - *When $q_2>0$, equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} has at most two limit cycles (taking into account multiplicities) in $x>0$ and $x<0$, respectively.* - *When $q_2<0$, equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} has at most one limit cycles (taking into account multiplicities) in $x>0$ and $x<0$, respectively.* First we rewrite the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} into the form $\frac{dx}{dt}=S(x, t; q_{0})$ with $S(x, t; q_{0}):=(p_0+p_1\sin t)x^3+(q_0+q_1\sin t+q_2\cos t)x^2$. It is easy to check that $\frac{\partial S}{\partial q_{0}}=x^2>0$ for $x\neq0$. Then from Definition [Definition 1](#def1){reference-type="ref" reference="def1"}, the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} forms a rotated equation with respect to the parameter $q_{0}$. In the following we verify the statement (i) and statement (ii) one by one. We only give the estimates for the number of positive limit cycles of the equation because the conclusions for the negative limit cycles follows exactly from a same argument. \(i\) Assume for a contradiction that there exists $q>0$ such that the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"}$|_{q_0=q}$ has $m\geq3$ positive limit cycles (taking into account multiplicities). Without loss of generality, our analysis can be restricted to the fact that all these limit cycles are hyperbolic. Indeed, according to Lemma [Lemma 2](#lem4.1){reference-type="ref" reference="lem4.1"} and statement (ii.1) of Proposition [Proposition 2](#pro12){reference-type="ref" reference="pro12"}, all the non-hyperbolic limit cycles of the equation split into two hyperbolic limit cycles respectively in a small decrease of $q_0$. Then the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"}$|_{q_0=q-\varepsilon}$ can possess only hyperbolic positive limit cycles as $0<\varepsilon\ll q$ and their number is at least $m$, which becomes the case of our concerns. Now we are able to denote by $x=x_1(t;q_0),\ x=x_2(t;q_0),\ \cdots,\ x=x_m(t;q_0)$ the $m$ positive limit cycles of the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"}, with $q_0$ being in some neighborhood of $q$ and $0<x_1(t;q)<\cdots<x_m(t;q)$. Since $q>0$, we know by Table [1](#Tab-zero){reference-type="ref" reference="Tab-zero"} that the orbit $x=0$ is upper-unstable for the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"}$|_{q_0=q}$. This yields that both $x=x_{1}(t;q)$ and $x=x_{3}(t;q)$ are stable, and $x=x_{2}(t;q)$ is unstable. Therefore, when $q_{0}$ decreases from $q$, we have by statement (i.2) of Proposition [Proposition 2](#pro12){reference-type="ref" reference="pro12"} that $x=x_{2}(t;q_0)$ is lower-unstable and increases, and $x=x_{3}(t;q_0)$ is upper-stable and decreases, respectively. Furthermore, since Lemma [Lemma 2](#lem4.1){reference-type="ref" reference="lem4.1"} tells us that the non-hyperbolic limit cycles of [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} must be lower-stable and upper-unstable when $q_2>0$ (including the critical case with $q_0=0$), the limit cycles $x=x_{2}(t;q_0)$ and $x=x_{3}(t;q_0)$ keep their hyperbolicity and stabilities, and do exist for $q_0\in[0,q]$. This means that, the critical case for the equation, [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"}$|_{q_0=0}$, has at least two positive limit cycles, which contradicts to Theorem [Theorem 2](#AT2){reference-type="ref" reference="AT2"}. As a result, $m\leq2$ and therefore the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} has at most two positive limit cycles (counting with multiplicities). \(ii\) Similar to the argument in statement (i), it is sufficient to prove that the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} can not possess three consecutive limit cycles $x=0$, $x=x_1(t;q_0)$ and $x=x_2(t;q_0)$ such that the later two are positive and hyperbolic. In fact, if this case occurs, then it is known by the Table [1](#Tab-zero){reference-type="ref" reference="Tab-zero"} that $x=x_1(t;q_0)$ and $x=x_2(t;q_0)$ are stable and unstable, respectively. Thus, on account of statement (i.1) of Proposition [Proposition 2](#pro12){reference-type="ref" reference="pro12"} and Lemma [Lemma 2](#lem4.1){reference-type="ref" reference="lem4.1"}, the limit cycles $x=x_1(t;q_0)$ and $x=x_2(t;q_0)$ get close to each other with their hyperbolicity and stabilities being retained as $q_0$ increases and satisfies $(p_0q_1-p_1q_0)^{2}+p_{0}^{2}q_{2}^{2}<p_{1}^{2}q_{2}^{2}$. This yields that [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"}$|_{(p_0q_1-p_1q_0)^{2}+p_{0}^{2}q_{2}^{2}=p_{1}^{2}q_{2}^{2}}$ has at least two limit cycles (counting with multiplicities), which contradicts to Theorem [Theorem 1](#AT1){reference-type="ref" reference="AT1"}. Accordingly, the conclusion for the number of positive limit cycles of equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} is valid. The proof is finished. $\Box$ So far, by virtue of Theorem [Theorem 2](#AT2){reference-type="ref" reference="AT2"} and Lemma [Lemma 3](#lem4.2){reference-type="ref" reference="lem4.2"}, we can actually assert that the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} has at most $4$ non-zero limit cycles under Hypothesis **(H)**. In order to optimize this upper bound and finally prove Theorem [Theorem 3](#T2){reference-type="ref" reference="T2"}, we provide the last auxiliary result which shows the non-existence of the limit cycles of the equation in some critical cases. **Lemma 4**. *Assume that $p_0=q_{0}=0$, $p_1>0$ and $q_2\neq0$. Then the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} has no limit cycle in the region $x\neq0$.* **Proof:  **We only prove the case with $q_2>0$ and the rest case with $q_2<0$ follows from a same argument. By assumption the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} is reduced to $$\label{eq20} \frac{dx}{dt}=S(x,t):=p_1\sin t\cdot x^3+(q_1\sin t+q_2\cos t)\cdot x^2.$$ Suppose that $x=x(t)$ is a non-zero limit cycle of the equation [\[eq20\]](#eq20){reference-type="eqref" reference="eq20"}. Then we have $\int_{0}^{2 \pi} \big(p_{1}\sin t\cdot x^{2}(t)+(q_{1}\sin t+q_{2}\cos t)\cdot x(t)\big)dt=0$. Therefore, $$\begin{split} \int^{2\pi}_{0}\frac{\partial S}{\partial x}(x(t),t)\ dt & =\int_{0}^{2 \pi} \left(3p_{1}\sin t \cdot x^{2}(t)+2(q_{1}\sin t+q_{2}\cos t)\cdot x(t)\right)\ dt\\ & =p_{1} \int_{0}^{2 \pi} \sin t \cdot x^{2}(t)\,dt\\ & =p_{1} \left(\int_{0}^{\pi} \sin t \cdot x^{2}(t)\,dt+\int_{\pi}^{2\pi} \sin t \cdot x^{2}(t)\,dt\right)\\ & =p_{1} \left(\int_{0}^{\pi} \sin t\cdot \left(x^{2}(t)-x^{2}(2\pi-t)\right)dt\right).\\ \end{split}$$ It can be verified that $v(t):=x(2\pi-t)$ is a solution of the equation $$\frac{dx}{dt}=-S(x,-t)=p_1\sin t\cdot x^3+(q_1\sin t-q_2\cos t)\cdot x^2.$$ Observe that $q_2>0$ implies that $S(x,t)>-S(x,-t)$ (resp. $S(x,t)<-S(x,-t)$) for $t\in[0,\frac{\pi}{2})$ (resp. $t\in(\frac{\pi}{2},\pi]$). Then $x(t)>\nu(t)$ for $t\in(0,\frac{\pi}{2})$ (resp. $t\in(\frac{\pi}{2},\pi)$) taking into account $\nu(0)=x(0)$ (resp. $\nu(\pi)=x(\pi)$), as depicted in Fig. [\[Fig. 2\]](#Fig. 2){reference-type="ref" reference="Fig. 2"}. Consequently, we get that $\int^{2\pi}_{0}\frac{\partial S}{\partial x}(x(t),t)\ dt>0$ (resp. $<0$) when $x(t)>0$ (resp. $<0$). This is sufficient to show that the limit cycle $x=x(t)$ is unstable (resp. stable) when it is positive (resp. negative). However, it follows from Table [1](#Tab-zero){reference-type="ref" reference="Tab-zero"} that the orbit $x=0$ is upper-unstable and lower-stable when $p_0=q_{0}=0$ and $q_{2}>0$. This contradicts to the stability of $x=x(t)$. As a result, there is no limit cycle in the region $x\neq0$ of the equation [\[eq20\]](#eq20){reference-type="eqref" reference="eq20"} and the Theorem is proved (see Fig. [\[Fig. 3\]](#Fig. 3){reference-type="ref" reference="Fig. 3"}(a) and Fig. [\[Fig. 4\]](#Fig. 4){reference-type="ref" reference="Fig. 4"}(a)). $\Box$ ## Proof of Theorem [Theorem 3](#T2){reference-type="ref" reference="T2"}[\[mA2\]]{#mA2 label="mA2"} {#proof-of-theorem-t2ma2-1} We are ready to prove Theorem [Theorem 3](#T2){reference-type="ref" reference="T2"}, and provide a positive answer to the Smale-Pugh-Gasull problem now. As has been explained at the beginning of this section, it is sufficient to prove the conclusion for the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} under the Hypothesis **(H)**. Furthermore, for the subcase $q_2<0$, it is clearly known from Theorem [Theorem 2](#AT2){reference-type="ref" reference="AT2"} (resp. statement (ii) of Lemma [Lemma 3](#lem4.2){reference-type="ref" reference="lem4.2"}) that the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} has at most three limit cycles (including $x=0$) when $q_0=0$ (resp. $q_0\neq0$). Hence, in the following it only remains to consider the subcase $q_2>0$. Let $x(t,x_0;p_0,q_0)$ be the solution of the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} with initial value $x_0>0$ and the parameters $p_0$ and $q_0$. According to Lemma [Lemma 4](#lem4.3){reference-type="ref" reference="lem4.3"} and the stability of $x=0$ from the Table [1](#Tab-zero){reference-type="ref" reference="Tab-zero"}, we have that $x(2\pi,x_0;0,0)>x_0$ when $x(2\pi,x_0;0,0)$ is well-defined. In addition, if we denote by $S(x,t;p_0,q_0)$ the right hand side of the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"}, then $$\begin{aligned} S(x,t;p_0,q_0)-S(x,t;0,0)=p_0x^3+q_0x^2\geq0,\indent (t,x)\in[0,2\pi]\times\mathbb R^+.\end{aligned}$$ Thus, $x(2\pi,x_0;p_0,q_0)\geq x(2\pi,x_0;0,0)>x_0$ always holds when $x(2\pi,x_0;p_0,q_0)$ is well-defined, which implies that no positive limit cycle of the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"} exists. Consequently, taking Theorem [Theorem 2](#AT2){reference-type="ref" reference="AT2"} and statement (i) of Lemma [Lemma 3](#lem4.2){reference-type="ref" reference="lem4.2"} into account, the total number of limit cycles of the equation [\[eq18\]](#eq18){reference-type="eqref" reference="eq18"}, including $x=0$, is at most three. Finally we show that the upper bound of the limit cycles can be achieved. It is sufficient to notice again from the Table [1](#Tab-zero){reference-type="ref" reference="Tab-zero"} that, the changes of the parameters $p_0$ and $q_0$ yield Hopf bifurcations from $x=0$. The distributions of the limit cycles from these bifurcations are summarized in the following Table [2](#Tab A1){reference-type="ref" reference="Tab A1"}. For more details see also Fig. [\[Fig. 3\]](#Fig. 3){reference-type="ref" reference="Fig. 3"}(b,c) and Fig. [\[Fig. 4\]](#Fig. 4){reference-type="ref" reference="Fig. 4"}(b,c). when $q_2>0$ when $q_2<0$ ------- -------------- -------------- $x>0$ 0 1 $x<0$ 2 1 : Distributions of the limit cycles bifurcating from $x=0$ Our proof is finished. $\Box$ # Acknowledgements {#acknowledgements .unnumbered} The first and third authors are supported by NNSF of China (No. 12171491). The second author is supported by the NNSF of China (No. 12271212), and NSF of Guangdong Province (No.2021A1515010029). 99 N. H. 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arxiv_math

--- abstract: | In this article, we determine the loops and neighborhoods of the vertices $E_{1728} \times E_{1728}$ and $E_0\times E_0$ (endowed with canonical principal divisors) in the $(\ell,\ell)$-isogeny graph of principally polarized superspecial abelian surfaces. We also present a simple new proof of the main theorem in [@LOX20]. address: - $^1$Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, Anhui, China - $^2$School of Mathematical Sciences, Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui, China - $^{3}$College of Sciences, National University of Defense Technology, Changsha 410073, Hunan, China author: - Zheng Xu$^{1}$ - Yi Ouuyang$^{1,2}$$^*$ - Zijian Zhou$^{3}$ title: Neighborhood of two vertices in the isogeny graph of principally polarized superspecial abelian surfaces --- [^1] # Introduction The supersingular isogeny-based cryptography is a relatively new suggestion for post quantum cryptosystems and is based on the assumption that it is computationally hard even for a quantum computer to find a path in the $\ell$-isogeny graph between two given vertices or equivalently to compute the endomorphism rings of supersingular elliptic curves. Known instantiations for the isogeny-based cryptography include the key exchange protocols SIDH [@jf], CSIDH [@clm] and OSIDH [@ck], and the signature schemes GPS-sign [@gps] and SQI-sign [@fkl]. As isogenies between supersingular curves found success in post quantum cryptography, it is natural to generalize the isogeny-based cryptosystem to abelian varieties of higher dimensions. Flynn-Ti [@ft] constructed a SIDH-like key exchange cryptosystem based on supersingular hyperelliptic curves over finite fields. In [@cds], Castryck-Decru-Smith constructed hash functions by Richelot isogenies between superspecial abelian varieties of dimension two as a generalization of hash functions in [@cgl]. As the security of isogeny-based cryptosystems depends on finding paths between vertices in the supersingular isogeny graphs, it is important to study the structure of these graphs, especially the distribution of isogenies from a fixed vertex. The graph of supersingular elliptic curves over $\ensuremath{\mathbb F}_p$ has been constructed in [@dg]. Adj-Ahmadi-Menezes [@aam] described the subgraphs with trace $0$ or $\pm p$. The authors (with S. Li) clarified the local structure of the isogeny graph at $\ensuremath{\mathbb F}_p$-vertices (i.e. whose $j$-invariants are inside $\ensuremath{\mathbb F}_p$) in [@OX; @LOX201; @LOX20]. As there is no effective way to describe explicitly the endomorphism rings of supersingular elliptic curves defined over $\ensuremath{\mathbb F}_{p^2}$ but not over $\ensuremath{\mathbb F}_p$, we are unable to compute the loops and neighborhoods of vertices whose $j$-invariants are in $\ensuremath{\mathbb F}_{p^2}\backslash \ensuremath{\mathbb F}_p$. Now let us move on to the principally polarized abelian varieties case, in particular the abelian surfaces case (the dimension $2$ case). Ionica and Thomé [@it] described the structure of the isogeny graph of ordinary abelian surfaces. Katsura and Takashima obtained the number of Richelot isogenies from superspecial abelian surfaces to products of supersingular curves [@kt1; @kt2]. Florit and Smith classified the Richelot isogenies between superspecial abelian surfaces[@fs]. In [@jz1; @jz2], Jordan and Zaytman summarized the relationships between matrices of maximal orders of quaternion algebras and isogenies of principally polarized superspecial abelian varieties ($\mathrm{PPSSAV}$ in short), defined the big and little isogeny graphs of $\mathrm{PPSSAV}$ and studied their structures. Moreover, the method to find a path and the concept of multiradical isogenies have been extended to the case of abelian surfaces in [@cs; @cd2]. Note that the recent broken of SIDH [@cd1; @MM; @R] is based on the fact that the secret key can be recovered by constructing an isogeny from an abelian surface with information extracted from the given torsion points. This indicates that understanding the isogeny (graph) of abelian surfaces could also be very useful in studying the supersingular isogeny cryptosystems. In this article, we determine the loops and neighborhoods of the vertices $E_{1728} \times E_{1728}$ and $E_0\times E_0$ in the $(\ell,\ell)$-isogeny graph of principally polarized superspecial abelian surfaces. We obtain our results by classifying the action of automorphism groups in the kernels of isogenies. # Preliminaries Throughout this paper we assume that $p$ is a prime number and $k$ is a finite field of characteristic $p$. Let $\Bar{\mathbb{F}}_p$ be an algebraic closure of $k$. We assume the abelian varieties are defined over $\Bar{\mathbb{F}}_p$. ## Principally polarized abelian varieties We recall a few facts about principally polarized abelian varieties. Let $A$ be an abelian variety defined over $k$. Then a divisor $D$ determines an isogeny $\lambda_D: A \to \hat{A}$, the dual abelian variety of $A$. If $D$ is an ample divisor, then $\lambda_D$ is a polarization on $A$ . If moreover $\deg(\lambda_D)=1$, then $\lambda_D$ is a principally polarization of $A$ and $\mathcal{A}= (A,D)$ is called a principally polarized abelian variety ($\mathrm{PPAV}$ in short). For $m\in \ensuremath{\mathbb Z}_+$, let $A[m]$ be the $m$-torsion subgroup of $A$. If $m$ is prime to $p$, a subgroup $S$ of $A[m]$ is called maximal $m$-isotropic if it is maximal among subgroups $T$ of $A[m]$ such that the restriction of the Weil pairing $e_m: A[m]\times A[m]\rightarrow \mu_m$ on $T\times T$ is trivial. The following result in [@mu] implies that the kernel of an isogeny of principally polarized abelian varieties is a maximal isotropic subgroup: **Theorem 1**. *Let $(A, D)$ be a principally polarized abelian variety over $\mkern 1.5mu\overline{\mkern-1.5mu\ensuremath{\mathbb F}\mkern-1.5mu}\mkern 1.5mu_p$ and $S$ be a subgroup of $A[m]$. Denote by $\phi:A \to A'=A/S$ the isogeny with kernel $S$. Then there exists a principally polarized divisor $D'$ of $A'$ such that $\phi^{*}D'\sim mD$ if and only if $S$ is a maximal $m$-isotropic subgroup. Particularly, if $S$ is a maximal $m$-isotropic subgroup of $A[m]$, then $(A', D')$ is also a principally polarized abelian variety.* By the above result, if $\varphi$ is an isogeny between a principally polarized abelian variety $(A, D)$ and an abelian variety $A'$, and $\ker(\varphi)$ is a maximal $m$-isotropic subgroup, then $A'$ also has a structure of principal polarization. Suppose $p>3$ and $\ell$ is a prime different from $p$. The following result in [@ft] presents the types of maximal $\ell^n$-isotropic subgroups. **Proposition 2**. *Let $\mathcal{A}= (A, D)$ be a principally polarized abelian surface. Then there are two types of maximal $\ell^n$-isotropic subgroups in $A[\ell^n]$:* 1. *$\mathbb{Z}/\ell^n\mathbb{Z}\times \mathbb{Z}/\ell^n\mathbb{Z}$,* 2. *$\mathbb{Z}/\ell^n\mathbb{Z}\times \mathbb{Z}/\ell^{n-k}\mathbb{Z}\times \mathbb{Z}/\ell^{k}\mathbb{Z}$ with $0\leq k\leq \lfloor\frac{n}{2}\rfloor$.* *The number of maximal $\ell^n$-isotropic subgroup is equal to $\ell^{2n-3}(\ell^2+1)(\ell+1)(\ell^n+ \frac{\ell^{n-1}-1}{\ell-1})$.* *In particular, there are $(\ell+1)(\ell^2+1)$ maximal $\ell$-isotropic subgroups, all of which are of the form $\ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z}\times \ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z}$.* ## Superspecial abelian varieties A supersingular abelian variety $A$ over $\Bar{\ensuremath{\mathbb F}}_p$ is an abelian variety isogenous to a product of supersingular elliptic curves. A superspecial abelian variety $A$ is a supersingular abelian variety that is isomorphic to a product of supersingular elliptic curves. By definition, superspecial = supersingular in the elliptic curve case. There are many (if $p$ is large enough) non-isomorphic supersingular elliptic curves over $\Bar{\ensuremath{\mathbb F}}_p$. However, there is only one (up to isomorphism) superspecial abelian variety for each dimension $g>1$ by the following famous result: **Theorem 3** (Deligne, Ogus[@Ogus], Oort, Shioda[@Shioda]). *Any superspecial abelian variety $A/\Bar{\ensuremath{\mathbb F}}_p$ of dimension $g>1$ is isomorphic to $E^g$, where $E$ is an arbitrary supersingular elliptic curve over $\Bar{\ensuremath{\mathbb F}}_p$.* ## Principally polarized superspecial abelian varieties Assigning a principal polarizations to the superspecial abelian variety (when the dimension $g>1$ is fixed), we obtain a principally polarized superspecial abelian variety ($\mathrm{PPSSAV}$ in short) and in particular a principally polarized superspecial abelian surface ($\mathrm{PPSSAS}$ in short) when $g=2$. Different principal polarization gives different $\mathrm{PPSSAV}$. For example, the product $E_1 \times E_2 \times \cdots \times E_g$ of supersingular elliptic curves, with the principal polarization $\{0\}\times E_2\times \cdots \times E_g+ \cdots + E_1\times E_2\times \cdots \times \{0\}$, is a $\mathrm{PPSSAV}$. For abelian surfaces, the following is well-known: **Theorem 4**. *There are two types of $\mathrm{PPSSAS}$ over $\Bar{\ensuremath{\mathbb F}}_p$:* 1. *Jacobian type $\ensuremath{\mathcal J}_p$, consisting of Jacobians of superspecial hyperelliptic curve of genus $2$ with the canonical principal polarization, whose number is $$\#\ensuremath{\mathcal J}_p= \begin{cases} 0, & \text{ if }p=2,3,\\ 1, & \text{ if }p=5,\\ \dfrac{p^3+24p^2+141p-346}{2880}, & \text{ if } p>5. \end{cases}$$* 2. *Product type $\ensuremath{\mathcal E}_p$: consisting of products of two supersingular elliptic curves with the above principal polarization, whose number is $$\# \ensuremath{\mathcal E}_p=\begin{cases} 1, & \text{ if }p=2,3,5,\\ \frac{1}{2}{S_{p^2}}({S_{p^2}}+1), & \text{ if } p>5, \end{cases}$$ where $S_{p^2}$ is the number of isomorphism classes of supersingular elliptic curves over $\Bar{\ensuremath{\mathbb F}}_p$.* ## Relationship between isogenies and matrices Let $E$ be a fixed supersingular elliptic curve. It is well-known that $\mathcal{O}:=\mathop{\mathrm{\mathrm{End}}}(E)$ is a maximal order in the quaternion algebra $B_{p,\infty}$ over $\ensuremath{\mathbb Q}$ ramified only at $p$ and $\infty$. Suppose $g>1$. Then $E^g$ is a superspecial abelian variety of dimension $g$. We equip $E^g$ with the principal polarization $\{0\}\times E^{g-1}+\cdots + E^{g-1}\times \{0\}$. Then $\mathop{\mathrm{\mathrm{End}}}(E^g)=M_g(\mathcal{O})$ and $$\mathop{\mathrm{\mathrm{Aut}}}(E^g)=\mathop{\mathrm{\mathrm{GL}}}_g(\mathcal{O})=\{M\in M_g(\mathcal{O})\mid M \ \text{is invertible}\}.$$ The reduced norm $\mathop{\mathrm{\mathrm{Nrd}}}: \mathcal{O}\rightarrow\ensuremath{\mathbb Z}$ generalizes to the reduced norm $\mathop{\mathrm{\mathrm{Nrd}}}: M_g(\mathcal{O}) \rightarrow\ensuremath{\mathbb Z}$. We also have $$\mathop{\mathrm{\mathrm{GL}}}_g(\mathcal{O})= \{M\in M_g(\mathcal{O})\mid \mathop{\mathrm{\mathrm{Nrd}}}(M)= 1\}.$$ Let $A$ be a superspecial abelian variety of dimension $g$. By Theorem [Theorem 3](#thm:doos){reference-type="ref" reference="thm:doos"}, $E^g$ and $A$ are isomorphic. Let $\iota_A: A\rightarrow E^g$ be an isomorphism which induces $\iota_A: \mathop{\mathrm{\mathrm{End}}}(A)\cong M_g(\mathcal{O})$. Note that another isomorphism $\iota_A'$ is uniquely determined by $\iota_A'\iota_A^{-1}\in \mathop{\mathrm{\mathrm{GL}}}_g(\mathcal{O})$. For $M\in M_g(\mathcal{O})$, let $M^+$ denote the conjugate transpose of $M$. Suppose $M$ is associated to the endomorphism $\alpha\in \mathop{\mathrm{\mathrm{End}}}(A)$. Then $M^{+}$ is the matrix associated to the Rosati involution $\alpha^{\dagger}$ of $\alpha$. Suppose $X$ is a principal polarized divisor of $A$. The map $$\mathop{\mathrm{\mathrm{Pic}}}(A)\rightarrow \mathop{\mathrm{\mathrm{End}}}(A),\quad L\mapsto \lambda_{X}^{-1} \circ\lambda_L$$ factors through the Néron-Severi group $\mathop{\mathrm{\mathrm{NS}}}(A)= \mathop{\mathrm{\mathrm{Pic}}}(A)/{\mathop{\mathrm{\mathrm{Pic}}}^{0}(A)}$. Let $$j: \mathop{\mathrm{\mathrm{NS}}}(A) \rightarrow {\mathop{\mathrm{\mathrm{End}}}} (A) \cong M_g(\mathcal{O}); \qquad \overline{L} \rightarrow \iota_A(\lambda_{X}^{-1} \circ \lambda_L).$$ This extends to $j: \mathop{\mathrm{\mathrm{NS}}}(A)\otimes \ensuremath{\mathbb Q}\rightarrow\mathop{\mathrm{\mathrm{End}}}(A) \otimes \ensuremath{\mathbb Q}\cong M_g(\mathcal{O})\otimes \ensuremath{\mathbb Q}$. **Proposition 5**. *[@m Proposition 14.2] [\[rosati1\]]{#rosati1 label="rosati1"} $j$ is invariant under the Rosati involution, which implies that $$j(\bar{L})=j(\bar{L})^{+}.$$* The following result allows us to determine whether the divisor of an abelian variety corresponds to a (principal) polarization. **Proposition 6**. *[@iko Proposition 2.8] Let $L$ be a divisor of an abelian variety $A$. Then $$\frac{L^g}{g!}=\chi(L)=\mathop{\mathrm{\mathrm{Nrd}}}(j(\bar{L})), \qquad \chi(L)^2=\deg(\lambda_L),$$ and* 1. *$L$ is associated to a polarization (i.e. $L$ is an ample divisor) if and only if $j(\bar{L})$ is positive definite;* 2. *$L$ is associated to a principal polarization if and only if $j(\bar{L})$ is positive definite with reduced norm $1$.* For different choices of $\iota_A$, we have different $j(\Bar{L})\in M_g(\mathcal{O})$, but they are related by the following result: **Proposition 7**. *[@jz1 Proposition 31] Let $$\mathcal{H}= \{H \in M_n(\mathcal{O}) \mid H \ \text{is positive-definite Hermitian of reduced norm} \ 1\}.$$ Two matrices $H$ and $H'$ in $\mathcal{H}$ correspond to the same polarized divisor if and only if they are in the same orbit under the action of $\mathop{\mathrm{\mathrm{GL}}}_g(\mathcal{O})$ on the set $\mathcal{H}$: $$\mathop{\mathrm{\mathrm{GL}}}_g(\mathcal{O}) \times \mathcal{H}\rightarrow \mathcal{H};\ \ (M, H) \mapsto M^{+}HM.$$ Moreover, there is a one-to-one correspondence between $\mathcal{H}\big/\mathop{\mathrm{\mathrm{GL}}}_g(\mathcal{O})$ and the set of isomorphism classes of $\mathrm{PPSSAS}$ of dimension $g$.* Applying the above results to the case $g=2$, we have **Proposition 8**. *[@iko Corollary 2.9][\[corre_L\_M\]]{#corre_L_M label="corre_L_M"} For $g=2$ and $d\in \ensuremath{\mathbb Z}_+$, there is a one-to-one correspondence $$\begin{aligned} \{ \bar{L}\in \mathop{\mathrm{\mathrm{NS}}}(A) \mid L>0, L^2=2d\} &\rightarrow \left\{ \begin{pmatrix} a & b \\ \bar{b} & c \end{pmatrix} \in M_2({\mathcal{O}}) \mid a,c \in \ensuremath{\mathbb Z}_+, ac-b\bar{b}=d \right \}\\ \bar{L} \qquad &\mapsto \qquad j(\bar{L}). \end{aligned}$$* ## The $(\ell, \ell)$-isogeny graph of $\mathrm{PPSSAS}$ From now on, suppose $p>3$ and $\ell$ is a prime different from $p$. Suppose $g=2$, i.e. we are dealing with abelian surfaces. Let $\mathcal{A}_1= (A, D_1)$ and $\mathcal{A}_2= (A, D_2)$ be two principally polarized abelian surfaces over $\overline{\ensuremath{\mathbb F}}_p$. An $(\ell, \ell)$-isogeny is an isogeny $\phi:\mathcal{A}_1\to \mathcal{A}_2$ such that $\ker(\phi)=\ensuremath{\mathbb Z}/{\ell \ensuremath{\mathbb Z}} \times \ensuremath{\mathbb Z}/{\ell \ensuremath{\mathbb Z}}$. For an $(\ell, \ell)$ isogeny $\phi$, there exists a dual isogeny $\hat{\phi}:\mathcal{A}_2\rightarrow \mathcal{A}_1$ such that $\hat{\phi} \circ \phi=[\ell]$. We can describe $(\ell,\ell)$ isogenies using matrices in $M_2(\mathcal{O})$, and the proof of the following proposition is in [@jz1] Proposition $31$. **Proposition 9**. *Let $A$ be superspecial abelian surface, $P_1$ and $P_2$ be principal polarizations of $A$. Let $H_1=j(\Bar{P_1})$ and $H_2=j(\Bar{P_2})$, $\alpha : A \rightarrow A$ be an isogeny given by $M\in M_g(\mathcal{O})$ of degree $\ell^{2m}$. Then $\alpha^*(P_2)= \ell^m P_1$ if and only if $M^{+}H_2M=\ell^m H_1$.* *In this case, $\alpha$ is an isogeny from $(A, P_1)$ to $(A, P_2)$.* **Definition 10**. The $(\ell, \ell)$-isogeny graph of principally polarized superspecial abelian surfaces, denoted as $\ensuremath{\mathcal G}_p=\ensuremath{\mathcal G}_{p,\ell}$, is the graph whose vertices set $V$ is the set of $\overline{\ensuremath{\mathbb F}}_p$-isomorphism classes of $\mathrm{PPSSAS}$ and whose edge set $E$ is the set of equivalence classes of $(\ell,\ell)$-isogenies. Note that two $(\ell,\ell)$-isogenies are equivalent if they have the same kernel. By Theorem [Theorem 1](#theorem:3. 13){reference-type="ref" reference="theorem:3. 13"} and Proposition [Proposition 2](#prop:lniso){reference-type="ref" reference="prop:lniso"}, the number of non-equivalent $(\ell,\ell)$-isogenies from a principally polarized superspecial abelian surface $A$ to another is equal to the number of maximal $\ell$-isotropic subgroups of $A$, which is $(\ell+1)(\ell^2+1)$. Hence **Lemma 11**. *The out-degree of every vertex in $\ensuremath{\mathcal G}_{p}$ is $(\ell+1)(\ell^2+1)$.* **Definition 12**. Let $\varphi: E_1\times E_2 \rightarrow A_1$ be an edge in $\mathcal{G}_p$. If $\{(P,Q), (P',Q')\}$ is an $\ensuremath{\mathbb F}_{\ell}$-basis of $\ker(\varphi)$, we call $\begin{pmatrix}P & P'\\ Q & Q' \end{pmatrix}$ a generator matrix of $\ker(\varphi)$. The isogeny $\varphi$ is called diagonal if $\ker(\varphi)$ has a diagonal generator matrix, and is called non-diagonal if otherwise. **Lemma 13**. *Let $\varphi: E_1\times E_2 \rightarrow A_1$ be an edge in $\mathcal{G}_p$. For $i=1, 2$, let $K_i =\mathop{\mathrm{\mathrm{im}}}(\ker(\varphi)\hookrightarrow (E_1\times E_2)[\ell]\rightarrow E_i[\ell])$. Then the followings are equivalent:* 1. *$\varphi$ is diagonal.* 2. *$\dim K_1=1$.* 3. *$\dim K_2=1$.* 4. *There exists some $0\neq P\in E_1[\ell]$ such that $(P,0)\in \ker(\varphi)$.* 5. *There exists some $0\neq Q\in E_2[\ell]$ such that $(0,Q)\in \ker(\varphi)$.* *In this case $\ker(\varphi)=K_1\times K_2$.* *Proof.* If $\varphi$ is diagonal, clearly $\dim K_1= \dim K_2=1$ and $\ker(\varphi)=K_1\times K_2$ by definition. Suppose $\{(P, Q), (P', Q')\}$ is a basis of $\ker(\varphi)$ over $\ensuremath{\mathbb F}_\ell$. If $\dim K_1=1$, then $P$ and $P'$ are linearly dependent and not both zero. We may assume $P'=aP$ and replace $(P', Q')$ by $(0, Q'-aQ)$, then $\ker(\varphi)$ has a basis of the form $\{(P, Q), (0, Q')\}$ and in particular $Q'\neq 0$. Since $\ker(\varphi)$ is a maximal isotropic subgroup, the trivial Weil pairing $e_{\ell}((P, Q), (0, Q'))= e_{\ell}(P, 0)e_{\ell}(Q, Q')=1$ means $e(Q, Q')=1$. Then $Q$ and $Q'$ must be linearly dependent too. Thus $K_2=\langle Q'\rangle$, $\ker(\varphi)=K_1\times K_2$ has a basis of the form $\{(P,0), (0,Q')\}$ and $\varphi$ is diagonal. By symmetry, if $\dim K_2=1$, we also have $\dim K_1=1$, $\ker(\varphi)=K_1\times K_2$ and $\varphi$ is diagonal. Thus (i), (ii), (iii) are equivalent. Clearly (i) $\Rightarrow$ (iv). On the other hand, we extend $(P,0)$ to a basis $\{(P,0), (P', Q')\}$ of $\ker(\varphi)$. Then $K_2$ is $1$-dimensional. Hence we have (iv) $\Rightarrow$ (i). Similarly we have (i) $\Leftrightarrow$ (v). ◻ **Corollary 14**. *Among isogenies from $E_1\times E_2$ in $\ensuremath{\mathcal G}_p$, $(\ell+1)^2$ are diagonal and $\ell^3-\ell$ are non-diagonal.* *Proof.* There are $\ell+1$ choices for the $1$-dimensional subspaces $K_1$ of $E_1[\ell]$ and $K_2$ of $E_1[\ell]$ respectively, so there are $(\ell+1)^2$ diagonal isogenies and $(\ell+1)(\ell^2+1)- (\ell+1)^2=\ell^3-\ell$ non-diagonal ones. ◻ As in [@cds], we can define the extension of $(\ell, \ell)$- isogenies: **Definition 15**. Let $\mathcal{A}_i=(A, D_i)$ ($i=0, 1, 2$), $\varphi_1: \mathcal{A}_0 \rightarrow\mathcal{A}_1$ and $\varphi_2: \mathcal{A}_1 \rightarrow\mathcal{A}_2$ be edges in $\mathcal{G}_p$. 1. The isogeny $\varphi_2$ is called a dual extension of $\varphi_1$ if $\ker(\varphi_2\circ\varphi_1)\cong (\ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z})^4$, in this case, $\ker(\varphi_2) = \varphi_1(A[\ell])$. 2. The isogeny $\varphi_2$ is called a bad extension of $\varphi_1$ if $\ker(\varphi_2\circ\varphi_1)\cong \ensuremath{\mathbb Z}/\ell^2\ensuremath{\mathbb Z}\times (\ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z})^2$, in this case, $\ker(\varphi_2) \cap \varphi_1(A[\ell])\cong \ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z}$. 3. The isogeny $\varphi_2$ is called a good extension of $\varphi_1$ if $\ker(\varphi_2\circ\varphi_1)\cong (\ensuremath{\mathbb Z}/\ell^2\ensuremath{\mathbb Z})^2$, in this case, $\ker(\varphi_2) \cap \varphi_1(A[\ell])= 0$. # Loops and neighbors of $[E_{1728}\times E_{1728}]$ ## Basic facts In this section, let $E=E_{1728}$ be the supersingular elliptic curve defined over $\ensuremath{\mathbb F}_p$ with $j$-invariant $1728$ (which implies that $p\equiv 3\bmod{4}$). We know its endomorphism ring is $$\mathcal{O}=\mathcal{O}_{1728} ={\ensuremath{\mathbb Z}}+{\ensuremath{\mathbb Z}} i+{\ensuremath{\mathbb Z}}\frac{1+j}{2}+{\ensuremath{\mathbb Z}}\frac{i+k}{2},$$ where $i^2=-1,\ j^2=-p,\ ij=-ji=k$. Note that the reduced norm on $\mathcal{O}$ is given by $$\label{eq:nrd1} \mathop{\mathrm{\mathrm{Nrd}}}\left(x+yi+z \frac{1+j}{2}+ w\frac{i+k}{2}\right )= \left(x+\frac{z}{2}\right)^2+ \left(y+\frac{w}{2}\right)^2+\frac{p(z^2+w^2)}{4}.$$ Let $[E\times E]=[E_{1728}\times E_{1728}]$ be the superspecial abelian surface $E\times E$ with the principal polarization $\{0\}\times E+ E\times \{0\}$ in the isogeny graph $\ensuremath{\mathcal G}_p$. For $n\in \ensuremath{\mathbb Q}^\times$, let $\sigma(n)=\sigma_1(n)$ be the sum of positive divisors of $n$ if $n\in \ensuremath{\mathbb Z}_+$ and $\sigma(n)=0$ if otherwise. Recall **Lemma 16** (Jacobi). *For $n\in \ensuremath{\mathbb Z}_+$, the number of integer solutions of $x^2+y^2+z^2+w^2=n$ is $$8\sigma(n)-32\sigma(\frac{n}{4})=8\sum_{d\mid n,\ 4\nmid d} d.$$ Particularly, there are $24$ integer solutions of $x^2+y^2+z^2+w^2=2$.* By simple computation, we have **Lemma 17**. *The group $G=\mathop{\mathrm{\mathrm{Aut}}}(E\times E)=\{ g\in M_2(\mathcal{O})\mid g^+g=I\}$ is nothing but the following group of order $32$: $$\begin{aligned} \left\{ \begin{pmatrix} \pm 1, \pm i & 0 \\ 0 & \pm 1, \pm i \end{pmatrix}, \quad \begin{pmatrix} 0 & \pm 1, \pm i \\ \pm 1, \pm i & 0 \end{pmatrix} \right\} . \label{matrix_M_unitsE1728} \end{aligned}$$* We shall need the following notations in this section: 1. If $\ell\equiv 1\bmod{4}$, let 1. $\lambda:=x_{\ell}+ y_{\ell} i\in \ensuremath{\mathbb Z}[i]$ such that $x^2_{\ell}+ y^2_{\ell}=\ell$; 2. $L_1:= \ker(\lambda: E[\ell]\rightarrow E[\ell])$ and $L_2:= \ker(\bar{\lambda}: E[\ell]\rightarrow E[\ell])$; 3. $R\in L_1\backslash\{0\}$ and $R'\in L_2\backslash\{0\}$; 4. $t:=-x_{\ell}/y_{\ell}\in \ensuremath{\mathbb F}_\ell$. Hence $t^2+1=0$. 2. Let $S=R+R'\in E[\ell]$ if $\ell\equiv 1\bmod{4}$ and $S$ be any fixed nonzero $P\in E[\ell]$ if $\ell\equiv 3\bmod{4}$. Let $S^*=i(S)$. 3. Let $G=\mathop{\mathrm{\mathrm{Aut}}}(E\times E)$. 4. For an isogeny $\varphi$ starting from $E\times E$, let $G_\varphi=\{g\in G: \varphi g=\varphi\}$ be the stabilizer of $\varphi$ by the $G$-action, and $O_\varphi=\{\varphi g: g\in G\}$ be the $G$-orbit of $\varphi$. Note that $L_1$ and $L_2$ are the only $1$-dimensional invariant $\ensuremath{\mathbb F}_\ell$-subspaces of the operator $i$ on $E[\ell]$, with eigenvalues $t$ and $-t$ respectively. ## Kernels of $(\ell,\ell)$-isogenies from $E_{1728}\times E_{1728}$ **Lemma 18**. *The set $\{S, S^*\}$ is an $\ensuremath{\mathbb F}_{\ell}$-basis of $E[\ell]$.* *Proof.* If $\ell\equiv 3\mod{4}$, we claim that $0\neq P$ and $i(P)$ are linearly independent. Indeed, if $i(P)= cP$ for some $c\in \ensuremath{\mathbb F}_\ell$, then $1+ c^2=0$, which is impossible. If $\ell\equiv 1\bmod{4}$, then $S=R+R'$ and $S^*=tR-tR'$. One can show easily that they are linearly independent. ◻ **Proposition 19**. *[\[prop:diag1\]]{#prop:diag1 label="prop:diag1"} There is a one-to-one correspondence of the set of non-diagonal $(\ell, \ell)$-isogenies from $E\times E$ and the set of generator matrices $$\left\{ \begin{pmatrix} S & S^*\\ aS+ bS^* & cS+dS^* \end{pmatrix}: a, b, c, d\in \ensuremath{\mathbb F}_\ell, ad-bc=-1\right\}.$$* *Proof.* Since both sets are of order $\ell(\ell^2-1)=\ell^3-\ell$, it suffices to show a non-diagonal isogeny has a generator matrix of the above form. Let $\varphi$ be a non-diagonal $(\ell,\ell)$-isogeny. Suppose $\{(P, Q), (P', Q')\}$ is a basis of $\ker(\varphi)$ over $\ensuremath{\mathbb F}_\ell$. Then $\{P, P'\}$, $\{Q, Q'\}$ and $\{S, S^*\}$ are all bases of $E[\ell]$ by Lemma [Lemma 13](#lemma:diagonal){reference-type="ref" reference="lemma:diagonal"} and Lemma [Lemma 18](#lemma:basis){reference-type="ref" reference="lemma:basis"}. Suppose $S=a_1 P+ b_1 P'$ and $i(S)=c_1 P + d_1 P'$. Write $a_1 Q+ b_1 Q'= aS+ b S^*$, $c_1 Q+d_1 Q'= cS+ d S^*$. Then $\{(S,aS+ b S^*), (S^*, cS+ d S^*)\}$ is a new basis of $\ker(\varphi)$. To ensure $\ker(\varphi)$ is a maximal isotropic subgroup, we have $e_{\ell}(S, S^*)e_{\ell}(aS+ b S^*, cS+ d S^*)=1$. This implies $ad- bc+ 1 \equiv 0 \pmod{\ell}$. ◻ **Definition 20**. For a non-diagonal $(\ell, \ell)$-isogeny $\varphi$ from $E\times E$, we call $\{a,b,c,d\}$ given above the quadruple associated to $\varphi$. We now describe the action of $G=\mathop{\mathrm{\mathrm{Aut}}}(E\times E)$ on the isogenies explicitly. For the diagonal isogenies, by easy computation, we have **Lemma 21**. *Suppose $\ker(\varphi)=K_1\times K_2$. Then $g\ker(\varphi)=\ker (\varphi g^{-1})$ for each $g\in G$ is given in the following table:* *$g$* *$g\ker(\varphi)$* *$g$* *$g\ker(\varphi)$* ------------------------------------------------------------------ ---------------------- ------------------------------------------------------------------ ------------------------- *$\begin{psmallmatrix}\pm 1 & 0\\ 0 & \pm 1 \end{psmallmatrix}$* *$K_1\times K_2$* *$\begin{psmallmatrix}\pm 1 & 0\\ 0 & \pm i \end{psmallmatrix}$* *$K_1\times i(K_2)$* *$\begin{psmallmatrix}\pm i & 0\\ 0 & \pm 1 \end{psmallmatrix}$* *$i(K_1)\times K_2$* *$\begin{psmallmatrix}\pm i & 0\\ 0 & \pm i \end{psmallmatrix}$* *$i(K_1)\times i(K_2)$* *$\begin{psmallmatrix}0 & \pm 1\\ \pm 1 & 0 \end{psmallmatrix}$* *$K_2\times K_1$* *$\begin{psmallmatrix}0 & \pm 1\\ \pm i & 0 \end{psmallmatrix}$* *$K_2\times i(K_1)$* *$\begin{psmallmatrix}0 & \pm i\\ \pm 1 & 0 \end{psmallmatrix}$* *$i(K_2)\times K_1$* *$\begin{psmallmatrix}0 & \pm i\\ \pm i & 0 \end{psmallmatrix}$* *$i(K_1)\times i(K_2)$* Consequently, we have **Proposition 22**. *Diagonal isogenies can be divided into the following classes, which are unions of $G$-orbits:* 1. *$(K_1, K_2)=(L_1, L_1)$ or $(L_2, L_2)$, where $G_\varphi= G$ and $O_\varphi=\{\varphi\}$.* 2. *$(K_1, K_2)=(L_1, L_2)$ or $(L_2, L_1)$, which form $1$ orbit if $\ell\equiv 1\bmod{4}$.* 3. *Exactly one of $K_1,K_2$ is in $\{L_1, L_2\}$, where $|G_\varphi|=8$ and $|O_\varphi|=4$. This class contains $4(\ell-1)$ isogenies if $\ell\equiv 1\bmod{4}$.* 4. *$K_1\notin \{L_1, L_2\}$ and $K_2\in \{K_1, i(K_1)\}$, where $|G_\varphi|=8$ and $|O_\varphi|=4$. This class contains $2(\ell-1)$ isogenies if $\ell\equiv 1\bmod{4}$, and $2(\ell+1)$ isogenies if $\ell\equiv 3\bmod{4}$.* 5. *$K_1\notin \{L_1, L_2\}$ and $K_2\notin \{L_1, L_2, K_1, i(K_1)\}$, where $G_\varphi= \{\begin{psmallmatrix} \pm 1 & 0\\ 0 &\pm 1 \end{psmallmatrix}\}$ and $|O_\varphi|=8$. This class contains $(\ell-1)(\ell-3)$ isogenies if $\ell\equiv 1\bmod{4}$, and $\ell^2-1$ isogenies if $\ell\equiv 3\bmod{4}$.* For the non-diagonal isogenies, we have **Lemma 23**. *Suppose $\varphi$ is non-diagonal associated to the quadruple $\{a,b,c,d\}$. Then for each $g\in G$, the associated quadruple of $\pm \varphi g^{-1}$ is given in the following table:* *$g$* *quadruple* *$g$* *quadruple* *$g$* *quadruple* ----------------------------------------------------------- --------------------- ----------------------------------------------------------- ------------------- ----------------------------------------------------------- --------------------- *$\begin{psmallmatrix}1 & 0\\ 0 & -1 \end{psmallmatrix}$* *$\{-a,-b,-c,-d\}$* *$\begin{psmallmatrix}i & 0\\ 0 & i \end{psmallmatrix}$* *$\{d,-c,-b,a\}$* *$\begin{psmallmatrix}i & 0\\ 0 & -i \end{psmallmatrix}$* *$\{-d,c,b,-a\}$* *$\begin{psmallmatrix}0 & 1\\ -1 & 0\end{psmallmatrix}$* *$\{d,-b,-c,a\}$* *$\begin{psmallmatrix}0 & 1\\ 1 & 0\end{psmallmatrix}$* *$\{-d,b,c,-a\}$* *$\begin{psmallmatrix}0 & i\\ i & 0\end{psmallmatrix}$* *$\{-a,-c,-b,-d\}$* *$\begin{psmallmatrix}0 & i\\ -i & 0 \end{psmallmatrix}$* *$\{a,c,b,d\}$* *$\begin{psmallmatrix}1 & 0\\ 0 & i \end{psmallmatrix}$* *$\{-b,a,-d,c\}$* *$\begin{psmallmatrix}1 & 0\\ 0 & -i \end{psmallmatrix}$* *$\{b,-a,d,-c\}$* *$\begin{psmallmatrix}i & 0\\ 0 & 1 \end{psmallmatrix}$* *$\{-c,-d,a,b\}$* *$\begin{psmallmatrix}i & 0\\ 0 & -1 \end{psmallmatrix}$* *$\{c,d,-a,-b\}$* *$\begin{psmallmatrix}0 & i\\ 1 & 0 \end{psmallmatrix}$* *$\{-c,a,-d,b\}$* *$\begin{psmallmatrix}0 & i\\ -1 & 0 \end{psmallmatrix}$* *$\{c,-a,d,-b\}$* *$\begin{psmallmatrix}0 & 1\\ i & 0 \end{psmallmatrix}$* *$\{-b,-d,a,c\}$* *$\begin{psmallmatrix}0 & 1\\ -i & 0 \end{psmallmatrix}$* *$\{b,d,-a,-c\}$* *Proof.* We first show the case for $g=\begin{psmallmatrix}i & 0\\ 0 & i \end{psmallmatrix}$. In this case, $$\begin{pmatrix}i & 0\\ 0 & i \end{pmatrix} \begin{pmatrix} S & S^*\\ aS+bS^* & cS+dS^* \end{pmatrix}= \begin{pmatrix} S^* & -S\\ -bS+aS^* & -dS+cS^* \end{pmatrix},$$ hence $\{(S, dS-cS^*), (S^*.-bS+aS^*) \}$ is a basis for $\begin{psmallmatrix}i & 0\\ 0 & i \end{psmallmatrix}\varphi$, thus the associated quadruple is $\{d,-c,-b,a\}$. We then show the case for $g=\begin{psmallmatrix}0 & i\\ i & 0 \end{psmallmatrix}$. In this case, $$\begin{pmatrix}0 & i\\ i & 0 \end{pmatrix} \begin{pmatrix} S & S^*\\ aS+bS^* & cS+dS^* \end{pmatrix}= \begin{pmatrix} aS^*-bS & cS^*-dS\\ S^* & -S \end{pmatrix}.$$ Note that $$a\begin{pmatrix} cS^*-dS\\ -S \end{pmatrix} - c\begin{pmatrix} aS^*-bS\\ S^* \end{pmatrix} = \begin{pmatrix} S\\ -aS-CS^* \end{pmatrix},$$ $$b\begin{pmatrix} cS^*-dS\\ -S \end{pmatrix} - d\begin{pmatrix} aS^*-bS\\ S^* \end{pmatrix} = \begin{pmatrix} S^*\\ -bS-dS^* \end{pmatrix}.$$ hence $\{(S, -aS-cS^*), (S^*.-bS-dS^*) \}$ is a basis for $\begin{psmallmatrix}0 & i \\ i & 0 \end{psmallmatrix}\varphi$, thus the associated quadruple is $\{-a,-c,-b,-d\}$. All other cases are similar. ◻ For every $g$, by looking for all $\varphi$ fixed by $g$ in the above table, we get **Proposition 24**. *Non-diagonal isogenies can be divided into the following classes, which are unions of $G$-orbits:* 1. *$\{a,b,c,d\}=\{a,b,-b,a\}$:* 1. *$b=\pm a$, where $|G_\varphi|=8$ and $|O_\varphi|=4$,* 2. *$a=0$ or $b=0$, where $|G_\varphi|=8$ and $|O_\varphi|=4$,* 3. *other cases of this form, where $G_\varphi=\{\pm I_2, \pm i I_2\}$ and $|O_\varphi|=8$.* *This class contains $\ell-1$ isogenies if $\ell\equiv 1\bmod{4}$ and $\ell+1$ isogenies if $\ell\equiv 3\bmod{4}$.* 2. *$\{a,b,c,d\}=\{a,b,b,-a\}$, where $|G_\varphi|=8$ and $|O_\varphi|=4$. This class contains $\ell-1$ isogenies if $\ell\equiv 1\bmod{4}$ and $\ell+1$ isogenies if $\ell\equiv 3\bmod{4}$.* 3. *$\{a,b,c,d\}=\{a,b,c,-a\}$, $c\neq b$ and $c\neq -b$ if $a=0$, or $\{a,b,c,d\}=\{a,b,b,d\}$, $d\neq -a$ and $d\neq a$ if $b=0$, where $|G_\varphi|=4$ and $|O_\varphi|=8$. This class contains $2(\ell^2-1)$ isogenies.* 4. *all other cases, where $G_\varphi=\{\pm I_2\}$ and $|O_\varphi|=16$. This class contains $(\ell-1)(\ell^2-\ell-4)$ isogenies if $\ell\equiv 1\bmod{4}$ and $\ell(\ell+1)(\ell-3)$ isogenies if $\ell\equiv 3\bmod{4}$.* *Proof.* We only need to count the number of isogenies in each class. First for a fixed $c\in \ensuremath{\mathbb F}_{\ell}^\times$, the equation $a^2+b^2=c$ has $\ell-1$ pairs of solutions in $\ensuremath{\mathbb F}_\ell$ if $\ell\equiv 1\bmod 4$ and $\ell+1$ pairs if $\ell\equiv 3\bmod 4$. Take $c=\pm 1$, we get the orders of Classes $\mathbf{N1}$ and $\mathbf{N2}$. For Class $\mathbf{N3}$, we need to find the solutions of $a^2+bc=1 (b\neq c, b\neq -c\ \text{if}\ a=0)$ and $b^2-ad=1 (a\neq -d, a\neq d\ \text{if}\ b=0)$. Consider the first one: 1. if $a=\pm 1$, then we can take either $b=0$ or $c=0$, there are $4\ell-2$ solutions of this type; 2. if $a\neq \pm 1$, then $1-a^2\neq 0$, there are $\ell-1$ pairs of $b, c$ such that $bc=1-a^2$, so there are $(\ell-2)(\ell-1)$ solutions of this type; 3. we need to exclude the $b=c$ case, which counts for $\ell-1$ solutions if $\ell\equiv 1\bmod{4}$ and $\ell+1$ solutions if $\ell\equiv 3\bmod{4}$; 4. we also need to exclude the $\{0,b,-b,0\}$ case, then $b^2=-1$, which has two solutions if $\ell\equiv 1\bmod{4}$ and none if $\ell\equiv 3\bmod{4}$; 5. in conclusion, it has $\ell^2-1$ solutions. Similarly the number of solutions for the second one is also $\ell^2-1$. Moreover, these two sets are disjoint, so Class $\mathbf{N3}$ has $2(\ell^2-1)$ elements. Class $\mathbf{N4}$ follows from results for Classes $\mathbf{N1}$-$\mathbf{N3}$ and Corollary [Corollary 14](#cor:isogenies){reference-type="ref" reference="cor:isogenies"} that the number of non-diagonal isogenies is $\ell^3-\ell$. ◻ ## Loops at $[E_{1728}\times E_{1728}]$ **Theorem 25**. *Suppose $p>4\ell$.* *$(1)$ If $\ell \equiv 1 \bmod 4$, then the set of loops of $E \times E$ is the union of Classes $\mathbf{D1}, \mathbf{D2}$ and $\mathbf{N1}$.* *$(2)$ If $\ell \equiv 3 \bmod 4$, then the set of loops of $E \times E$ is Class $\mathbf{N1}$.* *$(3)$ If $\ell=2$, then $E \times E$ has $3$ loops.* *Proof.* Note that the principal polarized divisor $E\times \{0\}+ \{0\} \times E$ corresponds to the identity matrix $I\in M_2(\mathcal{O})$. To determine the loops at $E \times E$ in the isogeny graph $\ensuremath{\mathcal G}_p$, it is equivalent to determine matrices $M\in M_2(\mathcal{O})$ such that $M^{+}M=\ell I$. Moreover, if $M$ corresponds to $\varphi$, then $\ker\varphi=\{(P,Q)\in E\times E: M(P,Q)^T=0\}$. Thus $gM$, where $g\in G$ given in [\[matrix_M\_unitsE1728\]](#matrix_M_unitsE1728){reference-type="eqref" reference="matrix_M_unitsE1728"}, determines the same loop as $M$ does. \(I\) First assume $\ell$ is odd. Write $$M= \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad (a, b, c, d \in \mathcal{O}).$$ The equation $M^{+}M=\ell I$ implies $$\begin{aligned} \mathop{\mathrm{\mathrm{Nrd}}}(a)+\mathop{\mathrm{\mathrm{Nrd}}}(b)=\ell,\quad \mathop{\mathrm{\mathrm{Nrd}}}(c)+\mathop{\mathrm{\mathrm{Nrd}}}(d)=\ell. \label{1728 ab_ell} \end{aligned}$$ Under the condition $p>4\ell$, by [\[eq:nrd1\]](#eq:nrd1){reference-type="eqref" reference="eq:nrd1"}, similar to the proof of [@OX], we have $$a, b, c, d \in {\ensuremath{\mathbb Z}}[i].$$ Replacing $b, c$ by $bi$, $ci$, we may assume $$\label{eq:M1728} M= \begin{pmatrix} a & bi \\ ci & d \end{pmatrix} \quad (a, b, c, d \in {\ensuremath{\mathbb Z}}[i]).$$ Now we have $M^{+}M=\ell I= M M^+$. Comparing the coefficients, we get $$\begin{aligned} a\bar{a}+ b\bar{b} = c\bar{c}+d\bar{d}= a\bar{a}+c\bar{c} =\ell,\quad a\bar{c}=b\bar{d}. \label{ac=bd} \end{aligned}$$ This implies $$a\bar{a}=d\bar{d},\quad b\bar{b} = c\bar{c},\ a\bar{c}=b\bar{d}. \label{nrda=nrdd}$$ \(i\) If one of $a$,$b$ is $0$, up to an element $g\in G$, we may assume $b=0$, then $c=0$ and $a\bar{a}=d\bar{d}=\ell$, which can only happen if $\ell\equiv 1\bmod{4}$. In this case, $M$ is one of the following matrices $$\begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix},\ \begin{pmatrix} \lambda & 0 \\ 0 & \bar{\lambda} \end{pmatrix},\ \begin{pmatrix} \bar{\lambda} & 0 \\ 0 & \lambda \end{pmatrix},\ \begin{pmatrix} \bar{\lambda} & 0 \\ 0 & \bar{\lambda} \end{pmatrix},$$ which correspond to the $4$ isogenies in Classes $\mathbf{D1}$ and $\mathbf{D2}$. \(ii\) If $ab\neq 0$, then $c,d$ are also not $0$. Note that ${\ensuremath{\mathbb Z}}[i]$ is a principal ideal domain. Let $A=\gcd(a,d)$ and $B=a/A$. Let $C=\gcd(b,c)$ and $D=b/C$. Then [\[nrda=nrdd\]](#nrda=nrdd){reference-type="eqref" reference="nrda=nrdd"} implies that $$a=AB,\ d=A\bar{B},\ b=CD,\ c=C\bar{D}.$$ The equation $a\bar{c}=b\bar{d}$ implies that $A\bar{C}=\bar{A}C$ and hence $A\bar{C} \in {\ensuremath{\mathbb Z}}$. If $\pi$ is a Gauss prime above a rational prime $q$ such that the $\pi$-adic valuation of $A$ is larger than the $\bar{\pi}$-adic valuation of $A$, then $\pi\mid C$, hence $\pi$ is a common divisor of $a,b,c,d$ and then $q\mid a\bar{a}+b\bar{b}=\ell$, which means $q=\ell$ and one of $a, b$ must be $0$, which is a contradiction to the condition $ab\neq 0$. Hence $A=\varepsilon A'$ for some $A'\in \ensuremath{\mathbb Z}$ and $\varepsilon=\pm 1,\pm i$, so $C=\varepsilon C'$ for some $C'\in \ensuremath{\mathbb Z}$. In this case, up to $g\in G$ we may assume $a=\bar{d}$, $b=\bar{c}$, then $$\label{eq:loopmatrix} M=\begin{pmatrix} a & bi \\ \bar{b}i & \bar{a} \end{pmatrix}.$$ Write $a=a_1+a_2i$, $b=b_1+b_2i$. The equation $a\bar{a}+ b\bar{b}=\ell$ implies $${a_1}^2+{a_2}^2+{b_1}^2+{b_2}^2=\ell, (a_1, a_2)\neq (0,0),\ (b_1, b_2)\neq (0,0).$$ By Lemma [Lemma 16](#dio1){reference-type="ref" reference="dio1"}, the equation above has $8(\ell+1)$ solutions if $\ell\equiv 3\bmod{4}$ and $8(\ell-1)$ solutions if $\ell\equiv 1\bmod{4}$. Now if $M$ is of the form [\[eq:loopmatrix\]](#eq:loopmatrix){reference-type="eqref" reference="eq:loopmatrix"}, then $gM$ is also of this form exactly when $$g\in \left\{\pm I_2,\ \pm \begin{pmatrix} i & 0\\ 0 & -i \end{pmatrix},\ \pm \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix},\ \pm \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix} \right\}.$$ Thus the number of non-diagonal loops is $\ell+1$ if $\ell\equiv 3\bmod{4}$ and $\ell-1$ if $\ell\equiv 1\bmod{4}$. Finally let us compute the kernels in this case. Suppose $M$ corresponds to the loop $\varphi$, then $(P,Q)\in \ker\varphi$ means $M (P,Q)^T=0$. If $0\neq P\in E[\ell]$, then $aP\neq 0$, otherwise $a\bar{a} P=0$ and hence $P=0$ as $\ell\nmid a\bar{a}$. So $(P,0)\notin \ker\varphi$. Thus $\ker\varphi$ is non-diagonal. By Proposition [\[prop:diag1\]](#prop:diag1){reference-type="ref" reference="prop:diag1"}, suppose $\ker\varphi$ is associated with $\{u,v,w,t\}$ with $ut-vw=-1$. Then $$\begin{pmatrix} a& bi\\ \bar{b}i & \bar{a} \end{pmatrix} \begin{pmatrix} S & S^*\\ uS+ v S^* & wS+ t S^* \end{pmatrix}= \begin{pmatrix} 0 & 0 \\ 0& 0 \end{pmatrix}.$$ Hence $(a+ ub i- vb)(S)= (ai+ wbi- tb)(S)= 0$. This implies $b(-u- vi- wi+ t)(S)=0$. Again by $\ell\nmid b\bar{b}$, we have $(-u- vi- wi+ t)(S)=0$. Since $\{S, i(S)=S^*\}$ is a basis of $E[\ell]$, we have $t=u$ and $w=-v$ in $\ensuremath{\mathbb F}_{\ell}$. In fact, $$(u,v)=\left(\frac{a_1 b_2 -a_2b_1}{b_1^2+b_2^2}, \frac{a_1 b_1+a_2 b_2}{b_1^2+b_2^2} \right).$$ Thus the loops here are exactly those contained in Class $\mathbf{N1}$. \(II\) Assume $\ell=2$. By calculation there are $3$ loops which correspond to the three matrices $$\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}, \quad \begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}, \quad \begin{pmatrix} 1+i & 0 \\ 0 & 1+i \end{pmatrix}. \qedhere$$ ◻ ## Neighbors of $[E_{1728}\times E_{1728}]$ If $\ell\equiv 1\bmod{4}$, we denote the loops $\lambda\times \lambda$ and $\bar{\lambda}\times \bar{\lambda}$ on $E\times E$ by $[\lambda]$ and $[\bar{\lambda}]$ respectively. Hence $$\ker[\lambda]=L_1\times L_1,\quad \ker[\bar{\lambda}]=L_2\times L_2.$$ **Lemma 26**. *Suppose $\ell\equiv 1\bmod{4}$ and $p>4\ell$. For a loop $\alpha\neq [\lambda]$, $[\bar{\lambda}]$ of $E\times E$ of degree $\ell^2$, let $$L_{i}^\alpha=(L_i\times L_i)\cap \ker\alpha\ \ (i=1, 2).$$ Then $\alpha\mapsto L_i^{\alpha}$ gives a one-to-one correspondence $$\{\text{degree $\ell^2$ loops}\ \neq[\lambda]\ \text{or}\ [\bar{\lambda}]\}\leftrightarrow \{\text{$1$-dimensional subspaces of } L_i\times L_i\}.$$ Consequently, $\ker\alpha=L_1^\alpha\times L_2^\alpha$, and if $L_1^\alpha=\{(P, kP): P\in L_1\}$ for some $k\neq 0$, then $L_2^\alpha=\{(-kQ, Q): Q\in L_2\}$.* *Proof.* We prove the one-to-one correspondence for $\alpha\leftrightarrow L_1^\alpha$. The case for $\alpha\leftrightarrow L_2^\alpha$ is parallel. If $\ker\alpha=L_1\times L_2$ or $L_2\times L_1$, clearly $L_1^\alpha=L_1\times \{0\}$ or $\{0\}\times L_1$ which is $1$-dimensional. Conversely these two lines correspond to these two diagonal kernels. If $\alpha$ is non-diagonal, suppose it is associated to the quadruple $(u,v,-v,u)$ such that $u^2+v^2=-1$, then $(R, (u+vt) R)\in \ker\alpha\cap (L_1\times L_1)$ which must be one dimensional. On the other hand, we may assume the $1$-dimensional subspace is generated by $(R, kR)$ for some $k\neq 0$. By Theorem [Theorem 25](#thmE1728){reference-type="ref" reference="thmE1728"}, we need to find a unique quadruple $(u,v,-v,u)$ such that $u^2+v^2=-1$ and $(R, kR)\in \langle(S, uS+vS^*), (S^*, -vS+uS^*) \rangle$. Note that $iR=tR$, $iR'=-tR'$, $S=R+R'$ and $S^*=tR-tR'$, where $t^2=-1$. Then $$\begin{pmatrix} R \\ uR+vtR \end{pmatrix}=\frac{1}{2} \begin{pmatrix} S \\ uS+v S^* \end{pmatrix}-\frac{t}{2} \begin{pmatrix} S^* \\ -vS+ uS^* \end{pmatrix}.$$ By Lemma [Lemma 13](#lemma:diagonal){reference-type="ref" reference="lemma:diagonal"}, a non-diagonal kernel can not contain two elements of the form $(R, *)$, hence $u+vt=k$. Then $u^2+v^2=(u+vt)(u-vt)=-1$ and hence $u-vt=-k^{-1}$. There is a unique $(u,v)=(\frac{k-k^{-1}}{2}, \frac{k+k^{-1}}{2t})$ satisfying these conditions. We also see $L_2^\alpha$ in this case is the $1$-dimensional subspace generated by $(R', -k^{-1}R')$ or equivalently by $(-kR', R')$. ◻ **Proposition 27**. *Suppose $p>4\ell^2$. Every loop $\varphi$ of $E\times E$ of degree $\ell^4$ corresponds to a matrix of the following form:* 1. *$\ell I_2$. In this case $\ker\varphi\cong (\ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z})^4$.* 2. *$\begin{pmatrix} a & bi \\ \bar{b}i & \bar{a} \end{pmatrix}$, where $a, b\in \ensuremath{\mathbb Z}[i]$, $a\bar{a}+ b\bar{b}= \ell^2$ and $ab\neq 0$. In this case $\ker\varphi\cong (\ensuremath{\mathbb Z}/\ell^2\ensuremath{\mathbb Z})^2$.* 3. *$\lambda \begin{pmatrix} a & bi\\ \bar{b}i & \bar{a} \end{pmatrix}$ or $\bar{\lambda} \begin{pmatrix} a & bi\\ \bar{b}i & \bar{a} \end{pmatrix}$, where $a, b\in \ensuremath{\mathbb Z}[i]$ and $a\bar{a}+ b\bar{b}= \ell$. In this case $\ker\varphi\cong (\ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z})^2\times \ensuremath{\mathbb Z}/\ell^2\ensuremath{\mathbb Z}$ which occurs only if $\ell\equiv 1\bmod{4}$.* *As a consequence, every loop of degree $\ell^4$ can be factorized as the product of two loops of degree $\ell^2$. Moreover, loops in Case $(2)$ is uniquely factorized as the composition of two edges of degree $\ell^2$.* *Proof.* Suppose $M$ corresponds to $\varphi$. Then $M^+M=\ell^4 I$. If $p>4\ell^2$, by following the same argument in the proof of Theorem [Theorem 25](#thmE1728){reference-type="ref" reference="thmE1728"}, we can deduce that $M\in M_2(\ensuremath{\mathbb Z}[i])$ has the form of I, II or III. Clearly a loop $\varphi$ in Case I is the composition of $\alpha$ and $\hat{\alpha}$ where $\alpha$ is a loop of degree $\ell^2$. It is also clear $\ker(\alpha)\cong (\ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z})^4$. A loop in Case III is the extension of $[\lambda]$ or $[\bar{\lambda}]$ of a non-diagonal loop $\alpha$ of degree $\ell^2$. By Lemma [Lemma 26](#goodiso){reference-type="ref" reference="goodiso"}, $L_1^\alpha\cong L_2^\alpha\cong \ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z}$, thus this extension is bad and $\ker\varphi=\ensuremath{\mathbb Z}/\ell^2\ensuremath{\mathbb Z}\times (\ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z})^2$. Now let $\varphi= \alpha\circ \beta$ be a loop of degree $\ell^4$ where $\alpha, \beta$ are two loops of degree $\ell^2$ such that $\alpha, \beta \notin \{[\lambda],\ [\bar{\lambda}]\}$ and $\beta\ne \hat{\alpha}$. We first claim that $\alpha$ is a good extension of $\beta$. If not, then $\alpha$ is a bad extension of $\beta$. Hence $\beta((E\times E)[\ell]) \cap \ker(\alpha)\cong \ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z}$ is an $\ensuremath{\mathbb F}_{\ell}$-line generated by some $(P, Q) \in (E\times E)[\ell]$. Since $i\circ \beta= \beta\circ i$, we have $$(i(P), i(Q)) \in i\circ \beta((E\times E)[\ell])= \beta\circ i((E\times E)[\ell])= \beta((E\times E)[\ell]).$$ Similarly, by $i\circ \alpha= \alpha\circ i$, we have $(i(P), i(Q)) \in \ker(\alpha)$. Hence $(i(P), i(Q))= c (P, Q)$ for some $c\in \ensuremath{\mathbb F}_\ell$. This is impossible if $\ell\equiv 3\bmod{4}$. If $\ell\equiv 1\bmod{4}$, then both of $P$ and $Q$ are in either $L_1$ or $L_2$. Since $\beta((E\times E)[\ell]) \subseteq \ker(\hat{\beta})$, we have $(P, Q)\in \ker(\hat{\beta})\cap \ker(\alpha)$. However, by Lemma [Lemma 26](#goodiso){reference-type="ref" reference="goodiso"}, there is only one loop of degree $\ell^2$ not of the form $[\lambda]$ whose kernel contains $(P,Q)$, thus $\hat{\beta}=\alpha$, which is a contradiction. We then show the factorization of $\varphi$ as $\alpha\circ \beta$, where $\alpha$ and $\beta$ are two edges of degree $\ell^2$, is unique. Indeed, $\ker(\varphi)\cong (\ensuremath{\mathbb Z}/\ell^2\ensuremath{\mathbb Z})^2$ has a unique subgroup which is isomorphic to $(\ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z})^2$. This subgroup must be $\ker(\beta)$. Thus $\beta$ is unique and so is $\alpha$. In conclusion, loops which is the composition of two loops $\neq [\lambda],\ [\bar{\lambda}]$ and without backtracking are all in Case II, with kernels $\cong (\ensuremath{\mathbb Z}/\ell^2\ensuremath{\mathbb Z})^2$, and the number of such loops is $(\ell+1)\ell$. However, following the proof of Theorem [Theorem 25](#thmE1728){reference-type="ref" reference="thmE1728"}, the number of loops of degree $\ell^4$ in Case II is $\sigma(\ell^2)- 1= \ell^2+\ell$. Thus loops of degree $\ell^4$ in Case II are all products of loops of degree $\ell^2$. ◻ **Lemma 28**. *Suppose $p> 4\ell^2$ and $\ell \equiv 1 \pmod 4$.* *If $\alpha$ and $\beta$ are two edges from $E\times E$ to a vertex $V\neq E\times E$ such that the loop $\varphi=\hat{\alpha}\circ \beta$ has a factor of $[\lambda]$ or $[\bar{\lambda}]$. Then* 1. *$\alpha$ and $\beta$ are not in the same $G$-orbit.* 2. *If $\varphi=\tau\circ [\lambda]$, then $\hat{\varphi}=\hat{\beta}\circ \alpha= \hat{\tau}\circ [\bar{\lambda}]$.* 3. *If $\varphi=\tau\circ [\lambda]$, then $$\label{eq:kerg1} \ker(\beta)\cap (L_1\times L_1)= L_1^\tau,\quad \ker(\alpha)\cap (L_2\times L_2)= L_2^{\hat{\tau}}.$$ Hence $\tau$, $\alpha$ (resp. $\tau$, $\beta$) and $\varphi$ are uniquely determined by $\beta$ (resp. $\alpha$).* *On the other hand, if an edge $\beta: E\times E\rightarrow V$ of degree $\ell^2$ satisfies $$\label{eq:kerg} \dim \ker(\beta)\cap (L_i\times L_i)=1,$$ for $i=1$ (resp. $i=2$), then there exists $\alpha: E_{1728}\times E_{1728}\rightarrow V$ such that $\varphi=\hat{\alpha}\circ \beta=\tau\circ [\lambda]$ (resp. $\varphi=\hat{\alpha}\circ \beta=\tau\circ [\bar{\lambda}]$) for some loop $\tau$.* *Proof.* (1) is easy, since if $\alpha=\beta g$ for $g\in G$, then $\hat{\beta}\circ \alpha=\ell g$ belongs to Case (1) in Proposition [Proposition 27](#llc){reference-type="ref" reference="llc"}. (2) follows from the fact that $[\lambda]$ commutes with $\tau$. For (3), we only need to study $\ker(\beta)$. Note that 1. $\varphi$ belongs to Case (3) in Proposition [Proposition 27](#llc){reference-type="ref" reference="llc"}, thus $\ker(\varphi)\cong (\ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z})^2\times \ensuremath{\mathbb Z}/\ell^2\ensuremath{\mathbb Z}$; 2. $L_2^\tau= (L_2\times L_2)\cap \ker(\tau)\subseteq \ker(\varphi)$; 3. $L_1\times L_1=\ker [\lambda]\subseteq \ker(\varphi)$. Thus the $\ell$-part of $\ker(\varphi)$, which contains $\ker(\beta)$, is generated by $L_1\times\{0\}=\langle(R, 0)\rangle$, $\{0\}\times L_1=\langle(0, R)\rangle$ and $L_2^\tau$. 1. If $L_2^\tau=L_2\times \{0\}=\langle(R', 0)\rangle$, then $L_1^\tau=\{0\}\times L_1$. By computation, $\ker(\beta)= \langle(U, 0), (0, R)\rangle$ where $U\in E_{1728}[\ell]$, $U\notin L_1\cup L_2$. Thus $$\ker(\beta)\cap (L_1\times L_1)= L_1^\tau,\quad \ker(\beta)\cap (L_2\times L_2)=0.$$ 2. If $L_2^\tau=\{0\}\times L_2=\langle(0,R')\rangle$, then $L_1^\tau=L_1\times \{0\}$. The proof is similar to (i). 3. If $L_2^\tau=\langle(R', aR')\rangle$ for some $a\neq 0$, by Lemma [Lemma 26](#goodiso){reference-type="ref" reference="goodiso"}, then $L_1^\tau=\langle(R, -a^{-1}R)\rangle$. In this case $\langle(R, 0), (0, R), (R', aR')\rangle\supset \ker(\beta)$. We assume $\ker(\beta)= \langle(R, cR), (R', bR+ aR')\rangle$. By computing the Weil pairing, we get $c=-a^{-1}$ and $\ker(\beta)\cap (L_1\times L_1)= L_1^\tau$ and $\ker(\beta)\cap (L_2\times L_2)=0$. This finishes the proof of (3). Now suppose $\beta$ satisfies [\[eq:kerg\]](#eq:kerg){reference-type="eqref" reference="eq:kerg"} for $i=1$, we let $\tau$ be the loop such that $L_1^\tau=L$. Checking the argument above we see that $\ker(\beta)\subset \ker( \tau\circ [\lambda])$, so $\varphi= \tau\circ [\lambda]$ factors through $\beta$. ◻ **Lemma 29**. *Suppose $\ell\equiv 1\bmod{4}$ and $p>4\ell^2$.* 1. *The diagonal non-loop isogenies satisfying [\[eq:kerg\]](#eq:kerg){reference-type="eqref" reference="eq:kerg"} are exactly those in Class $\mathbf{D3}$, whose number is $4(\ell-1)$.* 2. *The quadruples $\{a,b,c,d\}$ associated to non-diagonal non-loop isogenies satisfying [\[eq:kerg\]](#eq:kerg){reference-type="eqref" reference="eq:kerg"} for $i=1$ are parameterized by $$a= k-k^{-1} -d,\ b= dt-kt,\ c=dt+ k^{-1} t,$$ where $k\in \ensuremath{\mathbb F}_{\ell}^\times, d\in \ensuremath{\mathbb F}_{\ell}, k- k^{-1}\neq 2d$. If $k=\pm 1$ and $d\neq 0$ or $d\neq k=\pm t$, then the isogenies belong to Class $\mathbf{N3}$ , other cases belong to Class $\mathbf{N4}$. The number of isogenies satisfying [\[eq:kerg\]](#eq:kerg){reference-type="eqref" reference="eq:kerg"} for $i=1,2$ is $2(\ell-1)^2$, of which $8(\ell-1)$ are in Class $\mathbf{N3}$ and $2(\ell-1)(\ell-5)$ in Class $\mathbf{N4}$.* *Proof.* (1) is easy. For (2), if $\ker(\beta)$ contains an element $(R,uR)$ for some $u\neq 0$, then $(R,uR)=\frac{1}{2}[(S, aS+bS^*)-t(S^*, cS+dS^*) ]$, which means $$ad=bc-1,\ a-d= bt+ct.$$ Hence $(a+d)^2= (a-d)^2+4ad=-(b+c)^2+4bc-4=-(b-c)^2-4$, and $$(a+d)^2+(b-c)^2= (a+d+bt-ct)(a+d-bt+ct)=-4.$$ Plug in $a=d+bt+ct$, we get $$d+bt=k,\ d+ct=-k^{-1}.$$ This gives the parametrization. However, we need to exclude the loop case, which means $a=d$ or equivalently $k- k^{-1}=2d$. ◻ **Theorem 30**. *Suppose $p>4\ell^2$. Let $V$ be a vertex adjacent to $E\times E$.* 1. *If there exists one isogeny $\beta: E\times E\rightarrow V$ satisfying [\[eq:kerg\]](#eq:kerg){reference-type="eqref" reference="eq:kerg"}, then all isogenies from $E\times E$ to $V$ satisfy [\[eq:kerg\]](#eq:kerg){reference-type="eqref" reference="eq:kerg"} and they form two $G$-orbits.* 2. *For all other cases, $(\ell,\ell)$-isogenies from $E\times E$ to $V$ form a $G$-orbit.* *Proof.* Suppose $\alpha, \beta$ are two edges from $E\times E$ to $V$, then $\varphi=\hat{\alpha}\circ \beta$ is a loop of degree $\ell^4$. Moreover, by the unique factorization of loops of Case II in Proposition [Proposition 27](#llc){reference-type="ref" reference="llc"}, $\varphi$ doesn't belong to this case. By Lemma [Lemma 28](#llc2){reference-type="ref" reference="llc2"}, $\varphi$ belongs to Case III only if $\beta$ satisfies [\[eq:kerg\]](#eq:kerg){reference-type="eqref" reference="eq:kerg"}. For all other cases, $\varphi$ must be in Case I and $\alpha$ and $\beta$ are in the same $G$-orbit. Now suppose $\varphi$ belongs to Case III, then by Lemma [Lemma 28](#llc2){reference-type="ref" reference="llc2"}, $\alpha$ and $\beta$ are in two different $G$-orbits. Moreover $\hat{ g_1}\beta g_2$ for all $g_1, g_2\in G$ are in Case III, which in turn means that $\alpha g_1$ and $\beta g_2$ all satisfy [\[eq:kerg\]](#eq:kerg){reference-type="eqref" reference="eq:kerg"}. We only need to show there is no other $G$-orbit. If not, suppose $\alpha'$ is an edge from $E\times E$ to $V$ not in $\alpha G$ and $\beta G$. Then both $\varphi=\hat{\alpha}\circ\beta$ and $\varphi'= \hat{\alpha'}\circ\beta$ factor through $[\lambda]$ or $[\bar{\lambda}]$, however, there is only one such $\varphi$ by Lemma [Lemma 28](#llc2){reference-type="ref" reference="llc2"}. ◻ **Theorem 31**. *Suppose $p>4\ell^2$.* *$(1)$ If $\ell \equiv 1 \bmod 4$, we have the following table:* *Type* *\#Vertices* *Edges/Vertex* *Type* *\#Vertices* *Edges/Vertex* ----------------- -------------------------------- ---------------- ------------------- ----------------------------------------- ---------------- *$\mathbf{D3}$* *$\frac{\ell-1}{2}$* *8* *$\mathbf{N3}$-1* *$\frac{(\ell-1)(\ell-3)}{4}$* *8* *$\mathbf{D4}$* *$\frac{\ell-1}{2}$* *4* *$\mathbf{N3}$-2* *$\frac{\ell-1}{2}$* *16* *$\mathbf{D5}$* *$\frac{(\ell-1)(\ell-3)}{8}$* *8* *$\mathbf{N4}$-1* *$\frac{(\ell-1)(\ell^2-3\ell+6)}{16}$* *16* *$\mathbf{N2}$* *$\frac{\ell-1}{4}$* *4* *$\mathbf{N4}$-2* *$\frac{(\ell-1)(\ell-5)}{16}$* *32* *$(2)$ If $\ell \equiv 3 \bmod 4$, we have the following table:* *Type* *\#Vertices* *Edges/Vertex* *Type* *\#Vertices* *Edges/Vertex* ----------------- ------------------------ ---------------- ----------------- ------------------------------------- ---------------- *$\mathbf{D4}$* *$\frac{\ell+1}{2}$* *4* *$\mathbf{N2}$* *$\frac{\ell+1}{4}$* *4* *$\mathbf{D5}$* *$\frac{\ell^2-1}{8}$* *8* *$\mathbf{N3}$* *$\frac{\ell^2-1}{4}$* *8* *$\mathbf{N4}$* *$\frac{\ell(\ell+1)(\ell-3)}{16}$* *16* *$(3)$ If $\ell=2$, there are $3$ vertices adjacent to $[E_{1728}\times E_{1728}]$ , each connecting with $4$ edges, $2$ vertices with diagonal and $1$ with non-diagonal kernels.* *Proof.* For $\ell$ odd, this is a consequence of Proposition [Proposition 37](#diagker){reference-type="ref" reference="diagker"}, Proposition [Proposition 24](#nondiagker){reference-type="ref" reference="nondiagker"}, Lemma [Lemma 29](#lemma:kerg){reference-type="ref" reference="lemma:kerg"} and Theorem [Theorem 30](#thm:trans){reference-type="ref" reference="thm:trans"}. Now we consider the case $\ell= 2$. Suppose $P\in E[2]$ such that $P\ne i(P)$. Then $E[2]=\{O,P, Q=i(P), S=P+Q\}$. Moreover $i(S)= S$. Let $K_1=\{O,P\}$, $K_2=\{O,Q\}$ and $L=\{O,S\}$. Then the only diagonal loop kernel is $L\times L$. The other $8$ diagonal isogenies form two $G$-orbits: $$\{K_1\times K_1, K_1\times K_2, K_2\times K_1, K_2\times K_2\},\ \{L\times K_1, L\times K_2, K_1\times L, K_2\times L\}.$$ The kernels of the $2$ non-diagonal loops are $\langle(P, P), (Q, Q)\rangle$ and $\langle(P, Q), (Q, P)\rangle$. The other $4$ non-diagonal isogenies belong to one orbit. ◻ # Loops and neighbors of $[E_{0}\times E_{0}]$ In this section, let $E_{0}$ be the supersingular elliptic curve defined over $\ensuremath{\mathbb F}_p$ with $j$-invariant $0$ (which implies that $p\equiv 2\bmod{3}$). We know its endomorphism ring is $$\mathcal{O}_{0} ={\ensuremath{\mathbb Z}}+{\ensuremath{\mathbb Z}}\frac{1+i}{2}+{\ensuremath{\mathbb Z}}\frac{i+k}{3}+{\ensuremath{\mathbb Z}}\frac{j+k}{2},\ (i^2=-3,\ j^2=-p,\ ij=-ji=k).$$ Note that the reduced norm is given by $$\label{eq:nrd2} \begin{split} &\mathop{\mathrm{\mathrm{Nrd}}}\left(x+y\frac{1+i}{2}+z \frac{i+k}{3} + w\frac{j+k}{2}\right )\\ &= \left(x+\frac{y}{2}\right)^2+ 3\left(\frac{y}{2}+\frac{z}{3}\right)^2+\frac{p(z^2+3zw+3w^2)}{3}. \end{split}$$ Let $[E_{0}\times E_{0}]$ be the superspecial abelian surface $E_{0}\times E_{0}$ with the principal polarization $\{0\}\times E_{0}+ E_{0}\times \{0\}$ in the isogeny graph $\ensuremath{\mathcal G}_p$. Now results in the following are parallel to those in the previous section, whose proofs are almost identical and will be omitted. **Lemma 32**. *The number of integer solutions of Diophantine equation $$x^2+xy+y^2+z^2+zw+w^2=n$$ is $12\sigma(n)-36\sigma(\frac{n}{3})$.* **Lemma 33**. *The group $G=\mathop{\mathrm{\mathrm{Aut}}}(E_0\times E_0)=\{ g\in M_2(\mathcal{O}_0)\mid g^+g=I\}$ is the following group of order $72$: $$\begin{aligned} \left\{ \begin{pmatrix} \pm 1, \pm \frac{1+i}{2}, \pm \frac{1-i}{2} & 0 \\ 0 & \pm 1, \pm \frac{1+i}{2}, \pm\frac{1-i}{2} \end{pmatrix}, \quad \begin{pmatrix} 0 & \pm 1, \pm \frac{1+i}{2}, \pm \frac{1-i}{2} \\ \pm 1, \pm \frac{1+i}{2}, \pm \frac{1-i}{2} & 0 \end{pmatrix}\right\}.\label{matrix_M_unitsE0} \end{aligned}$$* We shall need the following notation in this section: 1. If $\ell\equiv 1\bmod{3}$, let 1. $\lambda:=x_{\ell}+ y_{\ell} \frac{1+ i}{2}\in \ensuremath{\mathbb Z}[\frac{1+ i}{2}]$ such that $x^2_{\ell}+ x_{\ell}y_{\ell}+ y^2_{\ell}=\ell$; 2. $L_1:= \ker(\lambda: E_{0}[\ell]\rightarrow E_{0}[\ell])$ and $L_2:= \ker(\bar{\lambda}: E_{0}[\ell]\rightarrow E_{0}[\ell])$; 3. $R\in L_1\backslash\{0\}$ and $R'\in L_2\backslash\{0\}$; 4. $t:=-x_{\ell}/y_{\ell}\in \ensuremath{\mathbb F}_\ell$. Note that $t^2- t+1=0$. 2. Let $S=R+R'\in E_{0}[\ell]$ if $\ell\equiv 1\bmod{3}$ and $S=P$ be a fixed nonzero $P\in E_{0}[\ell]$ if $\ell\equiv 2\bmod{3}$. Let $S^*=\frac{1+ i}{2}(S)$. 3. Let $G=\mathop{\mathrm{\mathrm{Aut}}}(E_{0}\times E_{0})$. 4. For an isogeny $\varphi$ starting from $E_{0}\times E_{0}$, let $G_\varphi=\{g\in G: \varphi g=\varphi\}$ be the stabilizer of $\varphi$ by the $G$-action, and $O_\varphi=\{\varphi g: g\in G\}$ be the $G$-orbit of $\varphi$. Note that $L_1$ and $L_2$ are the only $1$-dimensional invariant $\ensuremath{\mathbb F}_\ell$-subspaces of the operator $\frac{1+ i}{2}$ on $E_{0}[\ell]$, with eigenvalues $t$ and $-t$ respectively; similarly, $L_1$ and $L_2$ are the only $1$-dimensional invariant $\ensuremath{\mathbb F}_\ell$-subspaces of the operator $\frac{1- i}{2}$ on $E_{0}[\ell]$, with eigenvalues $1- t$ and $1+ t$ respectively. ## Kernels of $(\ell,\ell)$-isogenies from $E_{0}\times E_{0}$ **Lemma 34**. *The set $\{S, S^*\}$ is an $\ensuremath{\mathbb F}_{\ell}$-basis of $E_{0}[\ell]$.* **Proposition 35**. *[\[prop:diag11\]]{#prop:diag11 label="prop:diag11"} There is a one-to-one correspondence of the set of non-diagonal $(\ell, \ell)$-isogenies from $E_{0}\times E_{0}$ and the set of generator matrices $$\left\{ \begin{pmatrix} S & S^*\\ aS+ bS^* & cS+dS^* \end{pmatrix}: a, b, c, d\in \ensuremath{\mathbb F}_\ell, ad-bc=-1\right\}.$$* **Definition 36**. For a non-diagonal $(\ell, \ell)$-isogeny $\varphi$ from $E_{0}\times E_{0}$, we call $\{a,b,c,d\}$ given above the quadruple associated to $\varphi$. We now describe the action of $G=\mathop{\mathrm{\mathrm{Aut}}}(E_{0}\times E_{0})$ on the isogenies explicitly. For the diagonal isogenies, we have **Proposition 37**. *Suppose $\ker(\varphi)=K_1\times K_2$. Diagonal isogenies can be divided into the following classes, which are unions of $G$-orbits:* 1. *$(K_1, K_2)=(L_1, L_1)$ or $(L_2, L_2)$, where $G_\varphi= G$ and $O_\varphi=\{\varphi\}$.* 2. *$(K_1, K_2)=(L_1, L_2)$ or $(L_2, L_1)$, which form $1$ orbit if $\ell\equiv 1\bmod{3}$.* 3. *Exactly one of $K_1,K_2$ is in $\{L_1, L_2\}$, where $|G_\varphi|=12$ and $|O_\varphi|=6$. This class contains $4(\ell-1)$ isogenies if $\ell\equiv 1\bmod{3}$.* 4. *$K_1\notin \{L_1, L_2\}$ and $K_2\in \{K_1, \frac{1+ i}{2}(K_1), \frac{1- i}{2}(K_1)\}$, where $|G_\varphi|=8$ and $|O_\varphi|=9$. This class contains $3(\ell-1)$ isogenies if $\ell\equiv 1\bmod{3}$, and $3(\ell+1)$ isogenies if $\ell\equiv 2\bmod{3}$.* 5. *$K_1\notin \{L_1, L_2\}$ and $K_2\notin \{L_1, L_2, K_1, \frac{1+i}{2}(K_1), \frac{1- i}{2}(K_1)\}$, where $G_\varphi= \{\begin{psmallmatrix} \pm 1 & 0\\ 0 &\pm 1 \end{psmallmatrix}\}$ and $|O_\varphi|=18$. This class contains $(\ell-1)(\ell-4)$ isogenies if $\ell\equiv 1\bmod{3}$, and $\ell^2-\ell-2$ isogenies if $\ell\equiv 2\bmod{3}$.* For the non-diagonal isogenies, we have **Proposition 38**. *Non-diagonal isogenies can be divided into the following classes, which are unions of $G$-orbits:* 1. *$\{a,b,c,d\}=\{a,b,-b,a+ b\}$: This class contains $\ell- 1$ isogenies if $\ell\equiv 1\bmod{3}$ and $\ell+1$ isogenies if $\ell\equiv 2\bmod{3}$.* 2. *$\{a,b,c,d\}=\{a,b, a+b, -a\}$, where $|G_\varphi|=12$ and $|O_\varphi|=6$. This class contains $\ell-1$ isogenies if $\ell\equiv 1\bmod{3}$ and $\ell+1$ isogenies if $\ell\equiv 2\bmod{3}$.* 3. *if $3d^2\ne -1$, $\{a,b,c,d\}=\{a,b,c,-a\}$, $c\neq a+ b$, or $\{a,b,c,d\}=\{a,b,a+ b,d\}$, $d\neq -a$, or $\{a,b,c,d\}=\{a,b,c,b- c\}$, $d\neq -a$ where $|G_\varphi|=4$ and $|O_\varphi|=18$. This class contains $3(\ell^2-1)$ isogenies.* 4. *all other cases, where $G_\varphi=\{\pm I_2\}$ and $|O_\varphi|=36$. This class contains $\ell^3- 3\ell^2- 3\ell+ 5$ isogenies if $\ell\equiv 1\bmod{3}$ and $\ell^3- 3\ell^2- 3\ell+ 1$ isogenies if $\ell\equiv 2\bmod{3}$.* ## Loops at $[E_{0}\times E_{0}]$ **Theorem 39**. *Suppose $p>3\ell$.* *$(1)$ If $\ell \equiv 1 \bmod 3$, then the set of loops of $E_{0} \times E_{0}$ is the union of Classes $\mathbf{D1}, \mathbf{D2}$ and $\mathbf{N1}$.* *$(2)$ If $\ell \equiv 2 \bmod 3$, then the set of loops of $E_{0} \times E_{0}$ is Class $\mathbf{N1}$.* *$(3)$ If $\ell=3$, then $E_{0} \times E_{0}$ has $1$ loop.* ## Neighbors of $[E_{0}\times E_{0}]$ If $\ell\equiv 1\bmod{3}$, we denote the loops $\lambda\times \lambda$ and $\bar{\lambda}\times \bar{\lambda}$ on $E_{0}\times E_{0}$ by $[\lambda]$ and $[\bar{\lambda}]$ respectively. Hence $$\ker[\lambda]=L_1\times L_1,\quad \ker[\bar{\lambda}]=L_2\times L_2.$$ **Lemma 40**. *Suppose $\ell\equiv 1\bmod{3}$ and $p>3\ell$. For a loop $\alpha\neq [\lambda]$, $[\bar{\lambda}]$ of $E_{0}\times E_{0}$ of degree $\ell^2$, let $$L_{i}^\alpha=(L_i\times L_i)\cap \ker\alpha\ \ (i=1, 2).$$ Then $\alpha\mapsto L_i^{\alpha}$ gives a one-to-one correspondence $$\{\text{degree $\ell^2$ loops}\ \neq[\lambda]\ \text{or}\ [\bar{\lambda}]\}\leftrightarrow \{\text{$1$-dimensional subspaces of } L_i\times L_i\}.$$ Consequently, $\ker\alpha=L_1^\alpha\times L_2^\alpha$, and if $L_1^\alpha=\{(P, kP): P\in L_1\}$ for some $k\neq 0$, then $L_2^\alpha=\{(-kQ, Q): Q\in L_2\}$.* **Proposition 41**. *Suppose $p>3\ell^2$. Every loop $\varphi$ of $E_{0}\times E_{0}$ of degree $\ell^4$ corresponds to a matrix of the following form:* 1. *$\ell I_2$. In this case $\ker\varphi\cong (\ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z})^4$.* 2. *$\begin{pmatrix} a & b\frac{1+ i}{2} \\ \bar{b}\frac{{-1+ i}}{2} & \bar{a} \end{pmatrix}$, where $a, b\in \ensuremath{\mathbb Z}[\frac{1+ i}{2}]$, $a\bar{a}+ b\bar{b}= \ell^2$ and $ab\neq 0$. In this case $\ker\varphi\cong (\ensuremath{\mathbb Z}/\ell^2\ensuremath{\mathbb Z})^2$.* 3. *$\lambda \begin{pmatrix} a & b\frac{1+ i}{2}\\ \bar{b}\frac{-1+ i}{2} & \bar{a} \end{pmatrix}$ or $\bar{\lambda} \begin{pmatrix} a & b\frac{1+ i}{2}\\ \bar{b}\frac{-1+ i}{2} & \bar{a} \end{pmatrix}$, where $a, b\in \ensuremath{\mathbb Z}[\frac{1+ i}{2}]$ and $a\bar{a}+ b\bar{b}= \ell$. In this case $\ker\varphi\cong (\ensuremath{\mathbb Z}/\ell\ensuremath{\mathbb Z})^2\times \ensuremath{\mathbb Z}/\ell^2\ensuremath{\mathbb Z}$ which occurs only if $\ell\equiv 1\bmod{3}$.* *As a consequence, every loop of degree $\ell^4$ can be factorized as the product of two loops of degree $\ell^2$. Moreover, loops in Case $(2)$ is uniquely factorized as the composition of two edges of degree $\ell^2$.* **Lemma 42**. *Suppose $p> 3\ell^2$ and $\ell \equiv 1 \pmod 3$.* *If $\alpha$ and $\beta$ are two edges from $E_{0}\times E_{0}$ to a vertex $V\neq E_{0}\times E_{0}$ such that the loop $\varphi=\hat{\alpha}\circ \beta$ has a factor of $[\lambda]$ or $[\bar{\lambda}]$. Then* 1. *$\alpha$ and $\beta$ are not in the same $G$-orbit.* 2. *If $\varphi=\tau\circ [\lambda]$, then $\hat{\varphi}=\hat{\beta}\circ \alpha= \hat{\tau}\circ [\bar{\lambda}]$.* 3. *If $\varphi=\tau\circ [\lambda]$, then $$\label{eq:kerg111} \ker(\beta)\cap (L_1\times L_1)= L_1^\tau,\quad \ker(\alpha)\cap (L_2\times L_2)= L_2^{\hat{\tau}}.$$ Hence $\tau$, $\alpha$ (resp. $\tau$, $\beta$) and $\varphi$ are uniquely determined by $\beta$ (resp. $\alpha$).* *On the other hand, if an edge $\beta: E_{0}\times E_{0}\rightarrow V$ of degree $\ell^2$ satisfies $$\label{eq:kerg11} \dim \ker(\beta)\cap (L_i\times L_i)=1,$$ for $i=1$ (resp. $i=2$), then there exists $\alpha: E_{0}\times E_{0}\rightarrow V$ such that $\varphi=\hat{\alpha}\circ \beta=\tau\circ [\lambda]$ (resp. $\varphi=\hat{\alpha}\circ \beta=\tau\circ [\bar{\lambda}]$) for some loop $\tau$.* **Lemma 43**. *Suppose $\ell\equiv 1\bmod{3}$ and $p>3\ell^2$.* 1. *The diagonal non-loop isogenies satisfying [\[eq:kerg11\]](#eq:kerg11){reference-type="eqref" reference="eq:kerg11"} are exactly those in Class $\mathbf{D3}$, whose number is $4(\ell-1)$.* 2. *The quadruples $\{a,b,c,d\}$ associated to non-diagonal non-loop isogenies satisfying [\[eq:kerg11\]](#eq:kerg11){reference-type="eqref" reference="eq:kerg11"} for $i=1$ belong to Class $\mathbf{N3}$ and $\mathbf{N4}$. The number of isogenies satisfying [\[eq:kerg11\]](#eq:kerg11){reference-type="eqref" reference="eq:kerg11"} for $i=1,2$ is $2(\ell-1)^2$, of which $12(\ell-1)$ are in Class $\mathbf{N3}$ and $2(\ell-1)(\ell-7)$ in Class $\mathbf{N4}$.* **Theorem 44**. *Suppose $p>3\ell^2$. Let $V$ be a vertex adjacent to $E_{0}\times E_{0}$.* 1. *If there exists one isogeny $\beta: E_{0}\times E_{0}\rightarrow V$ satisfying [\[eq:kerg11\]](#eq:kerg11){reference-type="eqref" reference="eq:kerg11"}, then all isogenies from $E_{0}\times E_{0}$ to $V$ satisfy [\[eq:kerg11\]](#eq:kerg11){reference-type="eqref" reference="eq:kerg11"} and they form two $G$-orbits.* 2. *For all other cases, $(\ell,\ell)$-isogenies from $E_{0}\times E_{0}$ to $V$ form a $G$-orbit.* **Theorem 45**. *Suppose $p>3\ell^2$.* *$(1)$ If $\ell \equiv 1 \bmod 3$, we have the following table:* *Type* *\#Vertices* *Edges/Vertex* *Type* *\#Vertices* *Edges/Vertex* ----------------- --------------------------------- ---------------- ------------------- ----------------------------------------- ---------------- *$\mathbf{D3}$* *$\frac{\ell-1}{3}$* *12* *$\mathbf{N3}$-1* *$\frac{(\ell-1)(\ell-3)}{6}$* *18* *$\mathbf{D4}$* *$\frac{\ell-1}{3}$* *9* *$\mathbf{N3}$-2* *$\frac{\ell-1}{3}$* *36* *$\mathbf{D5}$* *$\frac{(\ell-1)(\ell-4)}{18}$* *18* *$\mathbf{N4}$-1* *$\frac{(\ell-1)(\ell^2-4\ell+9)}{36}$* *36* *$\mathbf{N2}$* *$\frac{\ell-1}{6}$* *6* *$\mathbf{N4}$-2* *$\frac{(\ell-1)(\ell-7)}{36}$* *72* *$(2)$ If $\ell \equiv 2 \bmod 3$, we have the following table:* *Type* *\#Vertices* *Edges/Vertex* *Type* *\#Vertices* *Edges/Vertex* ----------------- -------------------------------- ---------------- ----------------- ------------------------------------------ ---------------- *$\mathbf{D4}$* *$\frac{\ell+1}{3}$* *9* *$\mathbf{N2}$* *$\frac{\ell+1}{6}$* *6* *$\mathbf{D5}$* *$\frac{\ell^2- \ell- 2}{18}$* *18* *$\mathbf{N3}$* *$\frac{\ell^2-1}{6}$* *18* *$\mathbf{N4}$* *$\frac{\ell^3- 3\ell^2- 3\ell+ 1}{36}$* *36* *$(3)$ if $\ell=2$, then there is one vertex adjacent to $[E_{0}\times E_{0}]$ with diagonal kernel, and each connecting $[E_{0}\times E_{0}]$ with $9$ edges. There is one vertex adjacent to $[E_{0}\times E_{0}]$ with nondiagonal kernel, and each connecting $[E_{0}\times E_{0}]$ with $3$ edges, other three isogenies are loops.* *$(4)$ if $\ell=3$, then there are two vertices adjacent to $[E_{0}\times E_{0}]$ with diagonal kernel, and connecting $[E_{0}\times E_{0}]$ with $6$ and $9$ edges. There are three vertices adjacent to $[E_{0}\times E_{0}]$ with nondiagonal kernel, and each connecting $[E_{0}\times E_{0}]$ with $8$ edges, the last isogeny is a loop.* # A simple proof of Main Theorem in [@LOX20] In this section, we will give an alternative proof of the following theorem in [@LOX20] without using Deuring's correspondence in [@d]. Our new proof is similar to the proof of Proposition  [Proposition 27](#llc){reference-type="ref" reference="llc"}. **Theorem 46**. *Suppose $\ell>3$. Consider the $\ell$-isogeny graph $\ensuremath{\mathcal G}_{\ell}(\ensuremath{\mathbb F}_{p^2},-2p)$ of supersingular elliptic curves over $\ensuremath{\mathbb F}_{p^2}$ of trace $-2p$.* *$(1)$ If $p\equiv 3\bmod{4}$ and $p>4\ell^2$, there are $\frac{1}{2}\bigl(\ell-(-1)^{\frac{\ell-1}{2}}\bigr)$ vertices adjacent to $[E_{1728}]$ in the graph, each connecting $[E_{1728}]$ with $2$ edges. Moreover, $1+(\frac{\ell}{p})$ of the vertices are of $j$-invariants in $\ensuremath{\mathbb F}_p-\{1728\}$.* *$(2)$ If $p\equiv 2\bmod{3}$ and $p>3\ell^2$, there are $\frac{1}{3}(\ell-(\frac{\ell}{3}))$ vertices adjacent to $[E_{0}]$ in the graph, each connecting $[E_{0}]$ with $3$ edges. Moreover, $1+(\frac{-p}{\ell})$ of the vertices are of $j$-invariants in $\ensuremath{\mathbb F}_p^*$.* *Proof.* We will show the case $E_{1728}$, the case $E_{0}$ follows by a parallel argument. Suppose $p \equiv 3 \pmod 4$. Then $\mathop{\mathrm{\mathrm{Aut}}}(E_{1728})=\{\pm 1, \pm i\}$ where $i^2=-1$. Suppose $\ell$ is another prime. If $p>4\ell^2$, then elements in $\mathcal{O}$ of reduced norm $\ell^2$ are actually inside $\ensuremath{\mathbb Z}[i]$. Thus loops from $E_{1728}$ of degree $\ell^2$ lies in $\ensuremath{\mathbb Z}[i]$. If $\ell\equiv 3\pmod 4$, the only elements in $\ensuremath{\mathbb Z}[i]$ of reduced norm $\ell^2$ are $\sigma \ell$ where $\sigma\in \mathop{\mathrm{\mathrm{Aut}}}(E_{1728})$, so $\ell$ is only one loop of degree $\ell^2$. If there are two different isogenies $\varphi, \psi$ from $E_{1728}$ of degree $\ell$ to the same adjacent vertex, then $\widehat{\psi}\circ \varphi$ is a loop from $E_{1728}$, which is $\ell$. Hence $\varphi= \psi\circ\sigma$ with $\sigma\in \mathop{\mathrm{\mathrm{Aut}}}(E_{1728})$. Since $\varphi, \psi$ are not loops, we have $\ker(\psi)= i(\ker(\varphi)) \ne \ker(\varphi)$. It means an adjacent vertex connects $E_{1728}$ with two edges. If $\ell\equiv 1\pmod 4$, let $x,y\in \ensuremath{\mathbb Z}$ such that $x^2+y^2=\ell$. Then elements of reduced norm $\ell$ in $\ensuremath{\mathbb Z}[i]$ are $\sigma(x\pm yi)$ where $\sigma\in \mathop{\mathrm{\mathrm{Aut}}}(E_{1728})$, so the loops with degree $\ell$ from $E_{1728}$ are $x\pm yi$. Elements of reduced norm $\ell^2$ are $\sigma (x\pm yi)^2$ and $\sigma \ell$, corresponding to the three loops $(x\pm yi)^2$ and $\ell$ of degree $\ell^2$ on $E_{1728}$. Furthermore, the first two loops are compositions of a loop of degree $\ell$ with itself. Hence if there are two different isogenies $\varphi, \psi$ from $E_{1728}$ with degree $\ell$ to the same adjacent vertex, then $\widehat{\psi}\circ \varphi$ is a loop of $E_{1728}$. Since $(x\pm yi)^2 \ne \ell$, we have $\ker((x\pm yi)^2)$ is cyclic (i.e $\ker((x \pm yi)^2)\cong \ensuremath{\mathbb Z}/\ell^2\ensuremath{\mathbb Z}$). If $\widehat{\psi} \circ \varphi= (x+ yi)$ or $(x- yi)^2$, then $\ker(\varphi)= \ker(x+ yi)$ or $\ker(x-yi)$ which means $\varphi$ is a loop. Hence we have $\widehat{\psi} \circ \varphi$ is not equal to $(x+ yi)^2$ or $(x- yi)^2$. This implies the loop is $\ell$, and $\varphi= \psi\circ\sigma$ where $\sigma\in \mathop{\mathrm{\mathrm{Aut}}}(E_{1728})$. Since $\varphi, \psi$ are not loops, we have $\ker(\psi)= i(\ker(\varphi)) \ne \ker(\varphi)$. It means an adjacent vertex connects $E_{1728}$ with two edges. ◻ 100 G. Adj, O. Ahmadi, A. Menezes, *On isogeny graphs of supersingular elliptic curves over finite fields*, Finite Fields Appl. **55** (2019), 268-283. W. Castryck, T. Decru, *An efficient key recovery attack on SIDH*. Annual International Conference on the Theory and Applications of Cryptographic Techniques. Cham: Springer Nature Switzerland, 2023: 423-447. W. Castryck, T. Decru, *Multiradical isogenies*. Proceedings of AGC2T18, Contemporary Mathematics, 2022, 779: 57-89. W. Castryck, T. Decru, B. Smith, *Hash functions from superspecial genus-2 curves using Richelot isogenies*. Journal of Mathematical Cryptology, 2020, 14(1): 268-292. D. Charles, K. Lauter, E. Goren, *Cryptographic Hash Functions from Expander Graphs*. Journal of Cryptology, 2009, 22(1). L. Colò, D. Kohel, *Orienting supersingular isogeny graphs*. Journal of Mathematical Cryptology, 2020, 14(1): 414-437. W. Castryck, T. Lange, C. Martindale, et al, *CSIDH: an efficient post-quantum commutative group action* Advances in Cryptology--ASIACRYPT 2018: 24th International Conference on the Theory and Application of Cryptology and Information Security, Brisbane, QLD, Australia, December 2--6, 2018, Proceedings, Part III 24. Springer International Publishing, 2018: 395-427. C Costello , B Smith, *The supersingular isogeny problem in genus 2 and beyond*. Post-Quantum Cryptography: 11th International Conference, PQCrypto 2020, Paris, France, April 15--17, 2020, Proceedings. Cham: Springer International Publishing, 2020: 151-168. M. Deuring, *Die Typen der Multiplikatorenringe elliptischer Funktionenkörper*, Abh. Math. Sem. Hamburg **14** (1941), 197-272. C. Delfs, S. Galbraith, *Computing isogenies between supersingular elliptic curves over $\ensuremath{\mathbb F}_p$*, Des. Codes Cryptogr. **78** (2016), 425-440. L. De Feo, D. Kohel, A. Leroux, et al, *SQISign: compact post-quantum signatures from quaternions and isogenies*. Advances in Cryptology--ASIACRYPT 2020: 26th International Conference on the Theory and Application of Cryptology and Information Security, Daejeon, South Korea, December 7--11, 2020, Proceedings, Part I 26. Springer International Publishing, 2020: 64-93. E. V. Flynn, Y. B. Ti, *Genus two isogeny cryptography*. Post-Quantum Cryptography: 10th International Conference, PQCrypto 2019, Chongqing, China, May 8--10, 2019 Revised Selected Papers 10. Springer International Publishing, 2019: 286-306. E. Florit, B. Smith, *Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph* Arithmetic, Geometry, Cryptography, and Coding Theory 2021. American Mathematical Society, 2021, 779. S. D. Galbraith, C. Petit, J. Silva, *Identification protocols and signature schemes based on supersingular isogeny problems*. Advances in Cryptology--ASIACRYPT 2017: 23rd International Conference on the Theory and Applications of Cryptology and Information Security, Hong Kong, China, December 3-7, 2017, Proceedings, Part I 23. Springer International Publishing, 2017: 3-33. T. Ibukiyama, T. Katsura, F. Oort, *Supersingular curves of genus two and class numbers*, Compositio Mathematica, 1986, 57(2): 127-152. S. Ionica, E. Thomé, *Isogeny graphs with maximal real multiplication*. Journal of Number Theory, 2020, 207: 385-422. D. Jao, De Feo, *Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies*. In: Yang, B.-Y. (ed.) PQCrypto 2011, Springer LNCS **7071**, pp. 19-34, 2011. B. W. Jordan, Y. Zaytman, *Isogeny graphs of superspecial abelian varieties and Brandt matrices*. arXiv preprint arXiv:2005.09031, 2020. B. W. Jordan, Y. Zaytman, *Isogeny complexes of superspecial abelian varieties*. arXiv preprint arXiv:2205.07383, 2022. T. Katsura, K. Takashima, *Counting richelot isogenies between superspecial abelian surfaces*. Open Book Series, 2020, 4(1): 283-300. T. Katsura, K. Takashima, *Decomposed Richelot isogenies of Jacobian varieties of hyperelliptic curves and generalized Howe curves*. arXiv preprint arXiv:2108.06936, 2021. S. Li, Y. Ouyang, Z. Xu, *Neighborhood of the supersingular elliptic curve isogeny graph at $j=0$ and $1728$*, Finite Fields Appl. **61** (2020), 101600. S. Li, Y. Ouyang, Z. Xu, *Endomorphism rings of supersingular elliptic curves over $\mathbb{F}_p$*, Finite Fields Appl. **62** (2020), 101619. L. Maino, C. Martindale, L. Panny, G. Pope, B. Wesolowski, *A direct key recovery attack on SIDH*. Annual International Conference on the Theory and Applications of Cryptographic Techniques. Cham: Springer Nature Switzerland, 2023: 448-471. J.S. Milne, *Abelian varieties*, Springer (1986). D. Mumford, *Abelian Varieties*. Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Tata Institute of Fundamental Research, Bombay (2008). A. Ogus, *Supersingular K3 crystals*. Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. II, pp. 3--86, Astérisque, 64, Soc. Math. France, Paris, 1979. Y. Ouyang, Z. Xu, *Loops of isogeny graphs of supersingular elliptic curves at $j=0$*, Finite Fields Appl. **58** (2019), 174-176. D. Robert, *Breaking SIDH in polynomial time*. Annual International Conference on the Theory and Applications of Cryptographic Techniques. Cham: Springer Nature Switzerland, 2023: 472-503. T. Shioda, *Supersingular K3 surfaces*. Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), pp. 564--591, Lecture Notes in Math., 732, Springer, Berlin, 1979. [^1]: Partially supported by Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302904), NSFC(Grant No. 12371013), Anhui Initiative in Quantum Information Technologies (Grant No. AHY150200) and The National Natural Science Foundation of China (No.62202475).

arxiv_math

arxiv_math

--- author: - "Iulia Cătălina Pleşca[^1] and Marius Tărnăuceanu[^2]" date: September 9, 2023 title: Finite groups with integer harmonic mean of element orders --- **Abstract.**  In this paper, we introduce a new function computing the harmonic mean of element orders of a finite group. We present a series of properties for this function, and then we study groups for which the value of the function is an integer. # Introduction Throughout the article, let $G$ be a finite group and denote by $o(a)$ the order of an element $a\in G$. In the last three decades, an increasing number of functions and concepts from number theory have been adapted to group theory. For example, Leinster introduced what would later be called Leinster groups [@Leinster], a group analogue of perfect numbers. Subsequently, a plethora of arithmetic functions that involve the orders of elements/subgroups of groups have been introduced. We refer the reader to [@article4] for a recent survey on this topic. Special attention has also been given to various means of element orders: arithmetic in [@Silviu; @article5] and geometric in [@article5]. In our paper, we will focus on the harmonic mean. Let us recall the functions that represent the starting point for our work. Two of the most common functions in number theory associated to a positive integer $n\in\mathbb{N}^*$ are the number of its divisors $\tau(n)$ and the sum of its divisors $\sigma(n)$. These two functions have been adapted to group theory as follows. Given a finite group, $G$, $\tau(G)$ represents the number of its normal subgroups and $\sigma(G)$ the sum of their orders. These latter two appear in the definition of a new function $$H(G)=|G|\frac{\tau(G)}{\sigma(G)}\,,$$ introduced in [@article1] that was meant to generalize the arithmetic function $H(n)=n\tau(n)/\sigma(n)$. The last function that inspired us, the sum of inverses of the orders of elements $$m(G)=\sum_{a\in G}\frac{1}{o(a)}\,,$$ was introduced in [@article2] and, similarly to other sum functions, can give characterizations for commutativity, ciclicity and nilpotency. Inspired by these, we introduce the function "the harmonic mean of element orders of a finite group", i.e. $$h_m(G)=\frac{|G|}{m(G)}\,.$$ Throughout this article we put a dent in the following problem: *Question 1*. Which are the finite groups $G$ with $h_m(G)\in\mathbb{N}$? We start by giving some immediate properties of the newly introduced function including a lower bound that is reached only by finite $p$-groups. We continue by characterizing the finite $p$-groups with integer $h_m$. In the end, we observe that $h_m^{-1}(2)=\{C_4, D_8\}$ and study $h_m^{-1}(3)$. We use the following notations for some of the most common groups. For $m,n\in\mathbb{N}^*$, $C_n$ denotes the cyclic group of order $n$, $D_n$ the dihedral group of order $n$, $S_n$ the symmetric group of order $n$, $SD_{n}$ the semidihedral group of order $n$, $SL(m,n)$ the special linear group of degree $m$ over a field of $n$ elements. In addition, $Q_8$ is the quaternion group. The rest of the notations are standard. Basic notions and results on groups can be found in [@book2]. # Main results A series of inequalities for the function $m$ are given in [@article2] (Lemmas 1.3, 2.2, 2.3, 2.4, 2.6). We summarize these in the first proposition: **Proposition 1** ([@article2]). *The following properties hold for the function $m$:* a) *If $|G|=n$, then $m(C_n)\leq m(G)$, with equality if and only if $G\cong C_n$.* b) *If $H\leq G$, then $m(H)\leq m(G)$, with equality if and only if $H=G$.* c) *If $N\unlhd G$, then $m(G/N)\leq m(G)$, with equality if and only if $N=1$.* d) *If $P$ is a normal cyclic Sylow $p$-subgroup of $G$: $G \in \mathop{\mathrm{Syl}}_p(G)$, then $m(Px) \geq m(P)/o(Px)$, where $Px \in G/P$. Equality holds if and only if x centralizes $P$. Also, $m(G) \geq m(P)m(G/P)$ with equality if and only if $P$ is central in $G$.* e) *If $G_1, G_2$ are finite groups, then $m(G_1 \times G_2) \geq m(G_1)m(G_2)$, with equality if and only if the orders of the groups are coprime: $\gcd(|G_1|, |G_2|) = 1$.* We deduce a corresponding proposition for $h_m$: **Proposition 2**. *The following properties hold:* a) *If $|G|=n$, then $h_m(G)\leq h_m(C_n)$. Equality holds if and only if $G\cong C_n$.* b) *For a subgroup $H\leq G$, it follows that $h_m(G)\leq [G:H]h_m(H)$. Equality holds if and only if $H=G$.* c) *For a normal subgroup $H\lhd G$, it follows that $h_m(G)\leq |H|h_m(\frac{G}{H})$. Equality holds if and only if $H$ is the trivial subgroup.* d) *If $P$ is a normal cyclic Sylow $p$-subgroup of $G$: $G \in \mathop{\mathrm{Syl}}_p(G)$, then $h_m(G)\leq h_m(P)h_m(\frac{G}{P})$. Equality holds if and only if $P$ is central in $G$.* e) *$h_m$ is multiplicative: for all finite groups $G_1, G_2$ of coprime orders, we have $h_m(G_1\times G_2)=h_m(G_1)h_m(G_2)$. This shows that the study of the function $h_m$ for finite nilpotent groups can be reduced to $p$-groups.* Throughout this study, a lower bound for the function $h_m$ is needed and can be obtained from the following lemma. **Lemma 3**. *Let $G$ be a finite group, $C(G)=\{H\leq G\mid H \text{ cyclic}\}$ and $p$ the smallest prime divisor of $|G|$. Then the following inequality holds: $$\label{eq:1} h_m(G)\geq \frac{p|G|}{(p-1)|C(G)|+1}.$$ Equality holds if and only if $G$ is a $p$-group.* *Proof.* Let $d_1=1, d_2=p,\dots, d_r$ be the orders of the elements in $G$. For all $i\in\overline{1,r}$, we introduce the following notations: $$n_i=|\{a\in G\mid o(a)=d_i\}| \mbox{ and } n_i'=|\{H\in C(G)\mid |H|=d_i\}|.$$ Then the following relations hold: $$\begin{aligned} m(G)&=\sum_{i=1}^r \frac{n_i}{d_i}=\sum_{i=1}^r n_i'\frac{\varphi(d_i)}{d_i}=1+\sum_{i=2}^r n_i'\frac{\varphi(d_i)}{d_i}\\ &=1+\sum_{i=2}^r n_i'\prod_{q|d_i, q \text{ prime}}\left(1-\frac{1}{q}\right)\\&\leq 1+\frac{p-1}{p}(n_2'+\dots+n_r')=\frac{(p-1)|C(G)|+1}{p}\,, \end{aligned}$$ which give [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"}. The equality case happens if and only if $p$ is the only prime divisor of $|G|$, which holds if and only if $G$ is a $p$-group. ◻ *Remark 1*. Since $|C(G)|\leq |G|$, [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"} gives the anticipated lower boundary for $h_m$ in terms of the smallest prime divisor $p$ of $|G|$: $$h_m(G)\geq \frac{p|G|}{(p-1)|G|+1}\,.$$ For finite $p$-groups $G$ we can establish when $h_m(G)\in\mathbb{N}$ through the following result. **Theorem 4**. *Let $G$ be a finite $p$-group. Then $h_m(G)\in\mathbb{N}$ if and only if $G$ is cyclic of order $p^{\sum_{i=1}^s p^i}$, with $s\in\mathbb{N}^*$, or $G\simeq D_8$.* *Proof.* Let $|G|=p^n$, $\exp(G)=p^m$ and $$n_i=|\{a\in G\mid o(a)=p^i\}| \mbox{ and } n_i'=|\{H\in C(G)\mid |H|=p^i\}|, \,\forall i\in\overline{0,m}.$$ Lemma [Lemma 3](#lemma:2.1){reference-type="ref" reference="lemma:2.1"} implies that $$h_m(G)=\frac{p^{n+1}}{(p-1)|C(G)|+1}\,,$$ which leads to the following equivalences: $$\label{hm(G)} \begin{aligned} h_m(G)\in\mathbb{N}&\text{ if and only if } (p-1)|C(G)|+1=p^t \text{ with }t\leq n+1\\& \text{ if and only if } |C(G)|=p^{t-1}+p^{t-2}+\dots+p+1, \text{ with }t\leq n.\end{aligned}$$ For $p=2$, this formula becomes $$\label{p2} |C(G)|=2^t-1.$$ Going back to the proof, we distinguish the following two cases: Case 1: $p$ : odd. There are two possibilities: $G$ can be cyclic or not.   a) Suppose $G$ is not cyclic Using Theorem 1.10 from [@book1], it follows that $\begin{cases}n_1'\equiv p+1 \pmod{p^2}\\ n_2',\dots,n_m'\equiv 0\pmod p\end{cases}$ and so, recalling that the trivial subgroup is an element of $C(G)$, we obtain $|C(G)|\equiv 2 \pmod p$. Thus there are no solutions. b) [\[case 1:cyclic\]]{#case 1:cyclic label="case 1:cyclic"} Suppose $G$ is cyclic From [\[hm(G)\]](#hm(G)){reference-type="eqref" reference="hm(G)"}, it follows that $n+1= p^{t-1}+p^{t-2}+\dots+p+1, \text{ with }t\leq n\xRightarrow{s=t-1} n=\sum_{i=1}^s p^i.$ Case 2: $p=2$ : [\[case:2\]]{#case:2 label="case:2"}   a) Suppose $G$ is not cyclic and $G$ is not of maximal class Using Theorem 1.17 from [@book1], it follows that $\begin{cases}n_1'\equiv 3 \pmod{4}\\ n_2',\dots,n_m'\equiv 0\pmod 2\end{cases}$ and so $|C(G)|\equiv 0 \pmod 2$. Thus there are no solutions. b) Suppose $G$ is cyclic Therefore we have $|C(G)|=n+1$. Using [\[p2\]](#p2){reference-type="eqref" reference="p2"}, it follows that $n=2^t-2\xlongequal{s=t-1}\sum_{i=1}^s 2^i$ with $1\leq s\leq n+1$. c) Suppose $G$ is of maximal class It follows that $G\in\{D_{2^n}, Q_{2^n}, SD_{2^n}\}$. Using [@article3], the following analysis is obtained: i) $G\cong D_{2^n}\Rightarrow |C(G)|=2^{n-1}+n\Rightarrow 2^{n-1}+n+1|2^{n+1}$. It follows that the left hand side has to be a power of $2$: $2^{n-1}+n+1=2^t$. This happens when $2^{n-1}=n+1$. Since $2^{n-1}> n+1, \forall n>3$ (it can be shown inductively), it can be proven by direct computation that the only solution is $n=3$ and therefore $G\cong D_8$. ii) $G\cong Q_{2^n}\Rightarrow |C(G)|=2^{n-2}+n\Rightarrow 2^{n-2}+n+1|2^{n+1}$. Reasoning similarly as above, it follows that there are no solutions. iii) $G \cong SD_{2^n}\Rightarrow |C(G)|=3\cdot 2^{n-3}+n\Rightarrow 3\cdot 2^{n-3}+n+1|2^{n+1}$. The left hand side must be a power of $2$: $3\cdot 2^{n-3}+n+1=2^s$, therefore $n+1$ is divisible by $2^{n-3}$. Using $2^{n-3}> n+1, \forall n>5$ (it can be again shown inductively), it can be proved that there are no solutions. The proof of Theorem 2.3 is now complete. ◻ We note that Theorem 2.3 gives the following characterization for $D_8$. **Corollary 5**. *$D_8$ is the only non-cyclic $p$-group with integer harmonic mean of element orders.* An alternative characterization is the following: **Proposition 6**. *$D_8$ is the only dihedral group with integer harmonic mean of element orders.* *Proof.* $$\label{ass} \text{Let }D_{2n}\text{ be a dihedral group of order }2n\text{ such that }h_m(D_{2n})\in\mathbb{N}^*.$$ Let $n=p_1^{n_1}\cdot p_2^{n_2}\cdots p_k^{n_k}$ be the decomposition of $n$ as a product of prime factors, where $p_1<p_2<\dots<p_k$. Let us write $D_{2n}=\{1,r,\dots,r^{n-1}, s, sr, \dots, sr^{n-1}\}$, where $r$ is the rotation of order $\frac{2\pi}{n}$ and $s$ is the reflection around a vertex and the center of a regular polygon with $n$-sides. Obviously, $\langle r\rangle \cong C_n$ and $o(sr^i)=2, \forall i\in\overline{0,n-1}$. If we compute $h_m(D_{2n})$, we get $$h_m(D_{2n})=\frac{|G|}{\sum_{i=0}^{n-1} \frac{1}{o(r^i)}+\sum_{i=0}^{n-1}\frac{1}{o(sr^i)}}=\frac{2n}{m(C_n)+\frac{n}{2}}.$$ Let us denote $$\alpha:=\frac{2n}{m(C_n)+\frac{n}{2}}\in\mathbb{N}^*$$ because of assumption [\[ass\]](#ass){reference-type="eqref" reference="ass"}. Since $m(C_n)>0$, it follows that $\alpha<4$ and therefore three cases can occur: a) $\alpha=1$ Then $m(C_n)=\frac{3n}{2}$ , contradicting $m(C_n)=\sum_{a\in C_n}\frac{1}{o(a)}\leq\sum_{a\in C_n}1=n$. b) $\alpha=2$ It follows that $$\label{n2} m(C_n)=\frac{n}{2}.$$ Clearly $C_n\cong \prod_{i=1}^k C_{p_i^{n_i}}$. According to Proposition [Proposition 1](#propm){reference-type="ref" reference="propm"} e), it follows that $$\label{prod} m(C_n)=\prod_{i=1}^k m(C_{p_i^{n_i}}).$$ Each factor in the right-hand side can be computed. Fix $i\in\overline{1,k}$. For each $j\in\overline{1,n_i}$, there are $\varphi(p_i^j)$ elements of order $p_i^j$. Therefore $\displaystyle m(C_{p_i^{n_i}})=1+\sum_{j=1}^{n_i}\frac{\varphi(p_i^j)}{p_i^j}=1+\sum_{j=1}^{n_i}\frac{p_i^j(1-\frac{1}{p_i})}{p_i^j}=1+\sum_{j=1}^{n_i}\left(1-\frac{1}{p_i}\right)=\frac{p_i-1+1}{p_i}+\frac{n_i}{p_i}\left(p_i-1\right)=\frac{(n_i+1)(p_i-1)+1}{p_i}.$ Using [\[n2\]](#n2){reference-type="eqref" reference="n2"} and [\[prod\]](#prod){reference-type="eqref" reference="prod"}, it follows that $$\prod_{i=1}^k \left((p_i-1)(n_i+1)+1\right)=\frac{1}{2} p_1^{n_1+1}\cdot \dots\cdot p_n^{n_k+1}.$$ Since the left-hand side is an integer, it follows that $p_1=2$, so $$(n_1+2)\prod_{i=2}^k((p_i-1)(n_i+1)+1)=2^{n_1}p_2^{n_2+1}\dots p_k^{n_k+1}$$ Bernoulli's inequality gives: $$p_i^{n_i+1}=\left(1+(p_i-1)\right)^{n_i+1}>(p_i-1)(n_i+1)+1, \forall i=\overline{2,k},$$ therefore $n_1+2>2^{n_1}$, i.e. $n_1=1$. If $k>1$ it follows that $$3\prod_{i=2}^k \left((p_i-1)(n_i+1)+1\right)=2p_2^{n_2+1}\cdot\dots\cdot p_k^{n_k+1},$$ therefore $p_2=3$ and $$3(2n_2+3)\prod_{i=3}^k \left((p_i-1)(n_i+1)+1\right)=2\cdot 3^{n_2+1}p_3^{n_3+1}\cdot \dots p_k^{n+1}.$$ Then $2n_2+3\geq 2\cdot 3^{n_2}$, which does not yield solutions. Thus the assumption that $k>1$ does not hold, and consequently the only solution is $k=1$ and $n_1=2$, which gives $n=4$. c) $\alpha=3$ The analysis is analogous to the previous case. Alternatively, the result follows from Theorem 2.6.  ◻ *Remark 2*. Let us note that we can build nilpotent groups $G$ with $h_m(G)\in\mathbb{N}$ as direct products of $p$-groups of the type of the ones in Theorem [Theorem 4](#th:2.2){reference-type="ref" reference="th:2.2"}. In addition, we can construct non-nilpotent groups $G$ with this property, for example $G=\mathop{\mathrm{SL}}(2,3)\times C_{7^7}$. The idea is to start with a non-nilpotent group $G_1$ and to do a direct product with a cyclic group $G_2$ such that the denominator of $h_m(G_1)$ reduces. In the example above: $$h_m(G)=h_m(\mathop{\mathrm{SL}}(2,3))\cdot h_m(C_{7^7})=\frac{24}{7}\cdot 7^6=24\cdot 7^5.$$ In what follows, we will study the integer values of the function $h_m$. Obviously, we have $h_m(G)=1$ if and only if $G$ is the trivial group. In addition, Proposition [Proposition 6](#D8){reference-type="ref" reference="D8"} and its proof give that $h_m(C_4)=h_m(D_8)=2$. We can show that these are the only groups $G$ with $h_m(G)=2$. **Theorem 7**. *Let $G$ be a finite group. Then $h_m(G)=2$ if and only if $G\cong C_4$ or $G\cong D_8$.* *Proof.* Since $h_m(G)=2$, it follows that $2||G|$, therefore [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"} becomes $$2\geq \frac{2|G|}{|C(G)|+1}\,,$$ i.e. $$\label{eq:2} |C(G)|\geq |G|-1.$$ Using the same notations as in Lemma [Lemma 3](#lemma:2.1){reference-type="ref" reference="lemma:2.1"}, it follows that $d_2=2$ and $$|G|=\sum_{i=1}^r n_i=\sum_{i=1}^r n_i'\varphi(d_i)$$ $$|C(G)|=\sum_{i=1}^r n_i',$$ therefore [\[eq:2\]](#eq:2){reference-type="eqref" reference="eq:2"} becomes $$\label{9} \sum_{i=1}^r n_i'(\varphi(d_i)-1)\leq 1.$$ Since $\varphi(d_i)>1$, for $d_i>2$, we identify two possible cases: a) $r=2$ It follows that $G$ is an elementary abelian $2$-group. Then $h_m(G)\not\in\mathbb{N}$ by Theorem [Theorem 4](#th:2.2){reference-type="ref" reference="th:2.2"}, which is a contradiction. b) $r=3$ Clearly, we have $n_3'=1$ and $\varphi(d_3)=2$, and so $d_3=3$ or $d_3=4$. If $d_3=3$, then $G$ has $1$ element of order $1$, $2$ elements of order $3$ and $|G|-3$ elements of order $2$. Thus $h_m(G)=\frac{6|G|}{3|G|+1}\notin\mathbb{N}$, which is a contradiction. If $d_3=4$, then $h_m(G)=2$. This means that $G$ is a $2$-group of exponent $4$ with a single cyclic subgroup of order $4$. Using Theorem [Theorem 4](#th:2.2){reference-type="ref" reference="th:2.2"}, we obtain $G\cong C_4$ or $G\cong D_8$, as desired.  ◻ **Theorem 8**. *The finite non-trivial groups $G$ with $h_m(G)\leq 2$ are $C_2^n, n\in\mathbb{N}, C_3, S_3, C_4$ and $D_8$.* *Proof.* Using the same ideas and notations as in the proof of Theorem [Theorem 7](#th:2.5){reference-type="ref" reference="th:2.5"}, we can classify the finite groups $G$ with $h_m(G)\leq 2$. We identify the following cases: a) $|G|$ is odd It follows that $$\begin{aligned} h_m(G) &=\frac{|G|}{m(G)}\geq \frac{|G|}{1+\frac{|G|-1}{p_1}}\geq \frac{|G|}{1+\frac{|G|-1}{3}}=\frac{3|G|}{|G|+2}\Rightarrow\\ \frac{3|G|}{|G|+2}\leq & 2\Rightarrow |G|\leq 4\Rightarrow |G|=3\Rightarrow G\cong C_3. & \end{aligned}$$ b) $|G|$ is even It follows that [\[eq:2\]](#eq:2){reference-type="eqref" reference="eq:2"} holds. There are two possibilities: i) $r=2$ Then $G\cong C_2^n$, $n\in\mathbb{N}$. ii) $r=3$ Then there are two possibilities: $\mathbf{d_3=3}$ : Inequality [\[9\]](#9){reference-type="eqref" reference="9"} gives that $n_3'=1$ which means that $G$ has a unique cyclic subgroup of order $3$, let us denote this by $H$. Since $G$ does not contain cyclic subgroups of order $>3$, it follows that $H$ is the only $3$-Sylow subgroup of $G$. It is also normal. Since $|G|$ even, it follows that there is also a $2$-Sylow subgroup of order $2$. Thus $G=HK$. Since $G$ does not have cyclic subgroups of order $6$, it follows that $C_K(H)=1$, therefore $|K|=|{\rm Aut}(H)|=2$. We conclude that $G\cong S_3$. $\mathbf{d_3=4}$ : $G\cong C_4 \text{ or }G\cong D_8, \text{ for }d_3=4.$ This gives the conclusion. Moreover, we have $$\min\{ h_m(G)| G\text{ finite non-trivial}\}=\frac{4}{3}\,.$$ The minimum is obtained for $C_2$. ◻ Next we will focus on finite groups $G$ with $h_m(G)=3$. Note that the smallest example of such a group is $SmallGroup(12,1)$. We are not able to determine all these groups, but we can prove that they have even order and are not nilpotent. **Proposition 9**. *There are no finite groups $G$ of odd order with $h_m(G)=3$.* *Proof.* Let $G$ be a finite group of odd order such that $h_m(G)=3$. Then $3||G|$. Let $H\leq G$ with $|H|=3$. We identify the following cases: Case 1: : $\exp(G)=3$ Then $$h_m(G)=\frac{|G|}{1+\frac{|G|-1}{3}}=\frac{3|G|}{|G|+2}\neq 3,$$which is a contradiction. Case 2: : $\exp(G)\neq 3$ We will prove that $$\label{eq:3} G\text{ has at least }6\text{ elements of order }\geq 5$$ We identify the cases: a) $3^2|\exp(G)$ Then $G$ has at least a cyclic subgroup of order $3^2$ and so at least $6=\varphi(3^2)$ elements of order $9$. b) $3^2\nmid \exp(G)$. Let $p$ the smallest prime $\neq 3$ such that $p|\exp(G)$. If $p\geq 7$, then $G$ has at least a cyclic subgroup of order $p$ and consequently at least $\varphi(p)=p-1$ elements of order $p$. If $p=5$, there are two sub-cases: i) $G$ has only one subgroup $K$ with $|K|=5$. It follows that $K\lhd G$, therefore $KH\leq G$ and $|KH|=15$. Then $KH$ is cyclic and it possesses $\varphi(15)=8$ elements of order $15$. ii) $G$ has at least $3$ subgroups of order $5$. Then $G$ has at least $3\varphi(5)=12$ elements of order $5$. This concludes the proof of [\[eq:3\]](#eq:3){reference-type="eqref" reference="eq:3"}. We get $$h_m(G)\geq \frac{|G|}{1+\frac{6}{5}+\frac{|G|-7}{3}}=\frac{15|G|}{5|G|-2}>3,$$ a contradiction which completes the proof.  ◻ **Proposition 10**. *There are no finite nilpotent groups $G$ with $h_m(G)=3$.* *Proof.* Assume that $G$ is a finite nilpotent group such that $h_m(G)=3$. If $G$ is a $p$-group, then $p=3$ and the conclusion follows from Proposition 2.9. If $G$ is not a $p$-group, then it can be written as a direct product of at least two $p$-groups, say $G=G_1\times\cdots\times G_k$ with $k\geq 2$. Since $$h_m(G)=h_m(G_1)\cdots h_m(G_k),\nonumber$$we get $h_m(G_i)<2$, $\forall\, i=1,...,k$. Now Theorem 2.8 implies that $G_i=C_2^n$ for some $n\in\mathbb{N}$ or $G_i=C_3$, and therefore $h_m(G_i)=\frac{2^{n+1}}{2^n+1}$ for some $n\in\mathbb{N}$ or $h_m(G_i)=\frac{9}{5}$ . We remark that any product of these numbers is not $3$, contradicting our assumption. ◻ Finally, we note that the results so far leave the following open question: *Question 2*. Which are the integer values contained in $\mathop{\mathrm{Im}}(h_m)$? **Acknowledgements.**  The authors are grateful to the reviewer for their remarks which improved the previous version of the paper. **Funding.**  The authors did not receive support from any organization for the submitted work. **Conflicts of interests.**  The authors declare that they have no conflict of interest. 99 M.B. Azad, B. Khosravi and H. Rashidib, *On the sum of the inverses of the element orders in finite groups*, Comm. Algebra **51** (2023), 694-698. S.J. Baishya and A. Kumar, *Harmonic numbers and finite groups*, Rend. Sem. Mat. Univ. Padova **132** (2014), 33-43. Y. Berkovich, *Groups of Prime Power Order*, Volume 1, de Gruyter Expositions in Mathematics 46, 2008. V. Grazian, C. Monetta and M. Noce, *On the structure of finite groups determined by the arithmetic and geometric means of element orders*, arXiv:2212.13770. M. Herzog, P. Longobardi and M. Maj, *New criteria for solvability, nilpotency and other properties of finite groups in terms of the order elements or subgroups*, Int. J. Group Theory **12** (2023), 35-44. I.M. Isaacs, *Finite Group Theory*, Amer. Math. Soc., Providence, R.I., 2008. M.S. Lazorec and M. Tărnăuceanu, *On the average order of a finite group*, J. Pure Appl. Algebra **227** (2023), article ID 107276. T. Leinster, *Perfect numbers and groups*, Eureka **55** (2001), 17--27. M. Tărnăuceanu and L. Tóth, *Ciclicity degrees of finite groups*, Acta Math. Hung. **145** (2015), 489-504. The GAP Group, GAP -- groups, algorithms, and programming, version 4.11.0, https://www.gap-system.org, 2020. [^1]: Faculty of Mathematics of \"Al. I. Cuza\" University of Iaşi, Romania, e-mail: dankemath\@yahoo.com ORCID: 0000-0001-7140-844X [^2]: Faculty of Mathematics of \"Al. I. Cuza\" University of Iaşi, Romania, e-mail: tarnauc\@uaic.ro

arxiv_math

--- abstract: | In recent years, a growing number of method and application works have adapted and applied the causal-graphical-model framework to time series data. Many of these works employ time-resolved causal graphs that extend infinitely into the past and future and whose edges are repetitive in time, thereby reflecting the assumption of stationary causal relationships. However, most results and algorithms from the causal-graphical-model framework are not designed for infinite graphs. In this work, we develop a method for projecting infinite time series graphs with repetitive edges to marginal graphical models on a finite time window. These finite marginal graphs provide the answers to $m$-separation queries with respect to the infinite graph, a task that was previously unresolved. Moreover, we argue that these marginal graphs are useful for causal discovery and causal effect estimation in time series, effectively enabling to apply results developed for finite graphs to the infinite graphs. The projection procedure relies on finding common ancestors in the to-be-projected graph and is, by itself, not new. However, the projection procedure has not yet been algorithmically implemented for time series graphs since in these infinite graphs there can be infinite sets of paths that might give rise to common ancestors. We solve the search over these possibly infinite sets of paths by an intriguing combination of path-finding techniques for finite directed graphs and solution theory for linear Diophantine equations. By providing an algorithm that carries out the projection, our paper makes an important step towards a theoretically-grounded and method-agnostic generalization of a range of causal inference methods and results to time series. author: - "Andreas Gerhardus[^1]" - "Jonas Wahl[^2]" - "Sofia Faltenbacher[^3]" - "Urmi Ninad[^4]" - "Jakob Runge[^5]" bibliography: - library.bib title: Projecting infinite time series graphs to finite marginal graphs using number theory --- # Introduction Many research questions, from the social and life sciences to the natural sciences and engineering, are inherently causal. *Causal inference* provides the theoretical foundations and a variety of methods to combine statistical or machine learning models with domain knowledge in order to quantitatively answer causal questions based on experimental and/or observational data, see for example @pearl2009causality, @imbens2015causal, @Spirtes2000, @peters2017elements and @hernan2020causal. Since domain knowledge often exists in the form causal graphs that assert qualitative cause-and-effect relationships, the *causal-graphical-model framework* [@pearl2009causality] has become increasingly popular during the last few decades. By now, there is a vast body of literature on causal-graphical modeling. Broadly speaking, the framework subsumes the subfields *causal discovery* and *causal effect identification*. In causal discovery, see for example @Spirtes2000 and @peters2017elements, the goal is to learn qualitative cause-and-effect relationships (that is, the causal graph) by leveraging appropriate enabling assumptions on the data-generating process. In causal effect identification, the goal is to predict the effect of *interventions* [@pearl2009causality], which are idealized abstractions of experimental manipulations of the system under study, by leveraging knowledge of (or assumptions on) the causal graph. There are diverse methods and approaches for this purpose, such as the famous *backdoor criterion* [@pearl1993bayesian], the *generalized adjustment criterion* [@shpitser2010validity; @perkovic2018complete], the *instrumental variables* approach [@sargan1958estimation; @angrist2009mostly], the *$do$-calculus* [@pearl1995causal; @huang2006pearls; @shpitser2006identification; @shpitser2008complete] and *causal transportability* [@bareinboim2016causal], to name a few. In recent years, *causal representation learning*, see for example @schoelkopf2021toward, has emerged as another branch and gained increasing popularity in the machine learning community. In causal representation learning, the goal is to learn causally meaningful variables that can serve as the nodes of a causal graph at an appropriate level of abstraction. As one of its major achievements, causal-graphical modeling does not necessarily require temporal information for telling apart cause and effect. In fact, most of the causal-graphical-model framework was originally developed without reference to time [@pearl2009causality]. However, as many research fields specifically concern causal questions about dynamic phenomena, such as in Earth Sciences [@runge2019inferring], ecology [@runge2023modern] or neuroscience [@danks2023causal], in recent years there is a growing interest in adapting the framework to time series. More specifically, in this paper we draw our motivation from adaptations of causal-graphical modeling to the discrete-time domain; see for example @runge2023causal and @camps2023discovering for recent reviews of this setting. In the discrete-time setting, there are, broadly, two different approaches to causal-graphical modeling. The first approach uses time-collapsed graphs, also known as *summary graphs*, which represent each component time series by a single vertex and summarize causal influences across all time lags by a single edge. *Granger causality* [@granger1969investigating] uses this approach, and various works refined and extended Granger's work, see for example @dahlhaus2003causality [see the notion of *causality graphs*], @eichler2007causal, @eichler2010granger, @eichler2010graphical, and further works discussed in @assaad2022survey. This first approach is ideally suited to problem settings in which one is not interested in the specific time lags of the causal relationships. The second approach uses time-resolved graphs, which represent each time step of each component time series by a separate vertex and thus explicitly resolve the time lags of causal influences. Examples of works that employ this approach are @dahlhaus2003causality [see the notion of *time series chain graphs*], @chu2008search, @hyvarinen2010estimation, @runge2019detecting, @runge2020discovering, and @thams2022identifying. This second approach is ideally suited to problem settings in which the specific time lags of the causal relationships are of importance. As opposed to time-collapsed graphs, time-resolved graphs extend to the infinite past and future and thus have an infinite number of vertices. However, adopting the assumption of time-invariant qualitative causal relationships (often referred to as *causal stationarity*), the edges of the infinite time-resolved graphs are repetitive in time. As a result, despite being infinite, causally stationary time-resolved graphs admit a finite description. Throughout this paper, unless explicitly stated otherwise, we only consider causally stationary time-resolved graphs. Time-resolved graphs are useful for a number of reasons: 1) knowledge of time lags is crucial for a deeper process understanding, for example, in the case of time delays of atmospheric teleconnections [@runge2019inferring], and is relevant for tasks such as climate model evaluation [@nowack2020causal]; 2) knowledge of specific time lags allows for a more parsimonious and low-dimensional representation as opposed to modeling all past influences up to some maximal time lag; 3) causally-informed forecasting models benefit from precise time lag information [@runge2015optimal]; and 4) the lag-structure can render particular causal effect queries identifiable. This paper draws its motivation from the time-resolved approach to causal-graphical modeling. Since this setting is conceptually close to the non-temporal modeling framework, one can in principle hope to straightforwardly generalize the wealth of causal effect identification methods developed for non-time-series data. This task is much less straightforward in the time-collapsed approach, where it is harder to in the first place define and then interpret causal effects between time-collapsed nodes (see for example @reiter2023formalising for a recent work in this direction). However, for causally stationary time-resolved graphs one still faces the technical complication of having to deal with infinite graphs, whereas most causal effect identification methods are designed for finite graphs. Therefore, these methods still need specific modifications for making them applicable to infinite graphs. In this paper, we provide a new approach that resolves this complication of having to deal with infinite graphs: A method for projecting infinite causally stationary time-resolved graphs to finite marginal graphs on arbitrary finite time windows. Since this projection preserves *$m$-separations* [@richardson2002ancestral; @richardson2003markov] (cf. @pearl1988 for the related notion of *$d$-separations*) as well as causal ancestral relationships, one can equivalently check the graphical criteria of many causal effect identification methods on appropriate finite marginal graphs instead of on the infinite time-resolved graph itself. In particular, one can answer $m$-separation queries with respect to an infinite time-resolved graph by asking the same query for any of its finite marginal graphs for given finite time-window lengths that contain all vertices involved in the query. Figure [1](#fig:introduction-figure){reference-type="ref" reference="fig:introduction-figure"} illustrates one example of an infinite time-resolved graphs together with one of its finite marginal graphs. Intuitively speaking, the projection method implicitly takes care of the infiniteness of the time-resolved graphs and thereby relieves downstream applications (such as methods for causal effect identification and answering $m$-separation queries) from having to deal with infinite graphs. By providing this reformulation, our paper makes an important step towards a theoretically-grounded and method-agnostic generalization of causal effect identification methods to time series. ![Part **(a)** shows an infinite time-resolved graph for a time series with three components $X$, $Y$ and $Z$. By the assumption of causal stationarity, the edges in this graph are repetitive in time, that is, the graph is invariant under shifts along its temporal dimension. Moreover, as the dashed light gray edges indicate, the graph extends infinitely both into the past and into the future. The contribution of this paper is the development of an algorithmic method that projects such infinite graphs to finite marginal graphs on arbitrary finite time windows. To illustrate, part **(b)** shows the finite marginal graph that results from projecting the infinite graph in part (a) to the time window $[t-2, t]$. Note that this finite marginal graph does not extend to the infinite past or future, as the vertical dashed lines indicate. Since the projection preserves relevant graphical properties, one can use the finite marginal graph in part (b) to reason about the infinite graph in part (a).](figures/00_finite_infinite.jpeg){#fig:introduction-figure} Besides causal effect identification and $m$-separation queries, our projection method is also useful for causal discovery. The reason is that (equivalence classes of) finite marginal graphs of infinite time-resolved graphs are the natural targets of time-resolved time series causal discovery, see @gerhardus2021characterization for more details on this matter. Therefore, in order to obtain a conceptual understanding of the very targets of time-resolved causal discovery, one needs a method for constructing these finite marginal graphs---which our work provides. As the projection procedure, we here employ the widely-used ADMG latent projection [@pearl1995theory] (see also for example @richardson2023nested), thereby giving rise to what we below call *marginal time series ADMGs (marginal ts-ADMGs)*, see Def. [Definition 3](#def:tsADMG-marginal){reference-type="ref" reference="def:tsADMG-marginal"} below. In Section [6](#sec:generalize-to-DMAGs){reference-type="ref" reference="sec:generalize-to-DMAGs"}, we then extend our results to the DMAG latent projection [@richardson2002ancestral; @zhang2008causal], which too is widely-used and gives rise to what we call *marginal time series DMAGs (marginal ts-DMAGs)*, see Definition [Definition 29](#def:tsDMAG){reference-type="ref" reference="def:tsDMAG"} (which is adapted and generalized from @gerhardus2021characterization). While both of these projection procedures themselves are not new, their practical application to infinite time-resolved graphs is non-trivial and is, to the authors' knowledge, not yet solved in generality. The issue is that, when applied to infinite time-resolved graphs, the projection procedures require a search over a potentially infinite number of paths. In this paper, we show how to circumvent this issue by making use of the repetitive structure of the infinite time-resolved graphs in combination with the evaluation of a finite number of number-theoretic solvability problems. Thus, as an important point to note, our solution crucially relies on the assumption of causal stationarity. #### Related works {#related-works .unnumbered} The work of @gerhardus2021characterization already defines marginal ts-DMAGs and utilizes these finite graphs for the purpose of time-resolved time series causal discovery. This work also presents ts-DMAGs for several examples of infinite time-resolved graphs, but does not give a general method for their construction. The work of @thams2022identifying already considers finite marginal graphs obtained by the ADMG latent projection of infinite time series graphs and utilizes these finite graphs for the purpose of causal effect identification in time series. However, this work presents the finite marginal graphs of only a single infinite time-resolved graph (which, in addition, is of a certain special type that comes with significant simplifications) and it too does not give a general method for their construction. As opposed to these two works, we here consider a general class of infinite time-resolved graphs and derive an algorithmic method for constructing their finite marginals---both for the ADMG latent projection, see the main paper, and the DMAG latent projection, see Section [6](#sec:generalize-to-DMAGs){reference-type="ref" reference="sec:generalize-to-DMAGs"}. #### Structure and main results of this work {#structure-and-main-results-of-this-work .unnumbered} In Section [2](#sec:preliminaries){reference-type="ref" reference="sec:preliminaries"}, we summarize the necessary preliminaries. In Section [3](#sec:problem-formulation){reference-type="ref" reference="sec:problem-formulation"}, we first elaborate in more detail on the above-mentioned reasons for why finite marginal graphs of infinite time-resolved graphs are useful in the context of causal inference. We then formally define the finite marginal graphs obtained by the ADMG latent projection and reduce their construction to the search for common ancestors of pairs of vertices in certain infinite DAGs. This common-ancestor search is non-trivial because, in general, there is an infinite number of paths that might give rise to common ancestors. In Section [4](#sec:refined-common-ancestor){reference-type="ref" reference="sec:refined-common-ancestor"}, we first solve the common-ancestor search in a significantly simplified yet important special case and also show why the same strategy does not work in the general case. We then, for the general case, map the common-ancestor search to the number-theoretic problem of deciding whether at least one of a finite collection of linear Diophantine equations has a non-negative integer solution (**Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"}**). This result establishes an intriguing connection between graph theory and number theory, which might be of interest in its own right. Intuitively speaking, the mapping reformulates the problem of searching over the potentially infinite number of to-be-considered paths to a geometric intersection problem for still infinite but *finite-dimensional* affine cones over non-negative integers. Deciding whether such cones intersect is then equivalent to deciding whether a certain linear Diophantine equation admits a non-negative integer solution. Next, building on well-established results from number theory, we provide a criterion that answers the resulting number-theoretic solvability problem in finite time (**Theorem [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"}**). Thereby, we obtain an algorithmic and provably correct finite-time solution to the task of constructing finite marginal graphs of infinite time-resolved graphs. As a corollary, we also present an upper bound on a finite time window to which one can restrict the infinite time-resolved graphs before projecting them to the finite marginal graphs (**Theorem [Theorem 3](#thm.upper-bound-main-theorem){reference-type="ref" reference="thm.upper-bound-main-theorem"}**). This result provides a second solution to the problem of constructing finite marginal graphs, which appears conceptually simpler as it conceals the underlying number-theoretic problem but might computationally more expensive. In Section [5](#sec:summary){reference-type="ref" reference="sec:summary"}, we present the conclusions. In Sections [6](#sec:generalize-to-DMAGs){reference-type="ref" reference="sec:generalize-to-DMAGs"} and [7](#sec:counterexamples){reference-type="ref" reference="sec:counterexamples"}, we respectively extend our results to the finite marginal graphs obtained by the DMAG latent projection and present two examples that we omit from the main paper for brevity. In Sections [8](#sec:proofs){reference-type="ref" reference="sec:proofs"} and [9](#sec:pseudocode){reference-type="ref" reference="sec:pseudocode"}, we provide the proofs of all theoretical claims and pseudocode for our number-theoretic solution to the projection task. # Preliminaries {#sec:preliminaries} In this section, we summarize graphical terminology and notation that we use and build upon throughout this work. Our notation takes inspiration from @mooij2020constraint and @gerhardus2021characterization, among others. ## Basic graphical concepts and notation A *directed mixed graph (DMG)* is a triple $\mathcal{G}= (\mathbf{V}, \mathbf{E}_{\rightarrow}, \mathbf{E}_{\leftrightarrow})$ where $\mathbf{V}$ is the set of *vertices* (also referred to as *nodes*), $\mathbf{E}_{\rightarrow}\subseteq \mathbf{V}\times \mathbf{V}$ is the set of *directed edges*, and $\mathbf{E}_{\leftrightarrow}\subseteq \left(\mathbf{V}\times \mathbf{V}\right) \!/ \,\mathbb{Z}_2$ is the set of *bidirected edges*. Here, the $\mathbb{Z}_2$ action identifies the bidirected edges $(i, j) \in \mathbf{E}_{\leftrightarrow}$ and $(j, i) \in \mathbf{E}_{\leftrightarrow}$ with each other. We denote a directed edge $(i, j) \in \mathbf{E}_{\rightarrow}$ as $i {\,\rightarrow\,}j$ or $j {\,\leftarrow\,}i$ and a bidirected edge $(i, j) \in \mathbf{E}_{\leftrightarrow}$ as $i {\,\leftrightarrow\,}j$ or $j {\,\leftrightarrow\,}i$. We use $i {\ast\!\!\!\rightarrow}j$ resp. $i {\leftarrow\!\!\!\ast}j$ as a wildcard for $i {\,\rightarrow\,}j$ or $i {\,\leftrightarrow\,}j$ resp. $i {\,\leftarrow\,}j$ or $i {\,\leftrightarrow\,}j$, and use $i {\ast\!{-}\!\ast}j$ as a wildcard for $i {\ast\!\!\!\rightarrow}j$ or $i {\leftarrow\!\!\!\ast}j$. We say that two vertices $i, j \in \mathbf{V}$ are *adjacent* in a DMG $\mathcal{G}$ if there is an edge of any type between them, that is, if $i {\ast\!{-}\!\ast}j$ in $\mathcal{G}$. This definition allows *self edges*, that is, edges of the form $i {\,\rightarrow\,}i$ and $i {\,\leftrightarrow\,}i$. Moreover, the definition allows a pair of vertices $i, j \in \mathbf{V}$ to be connected by more than one edge. The *induced subgraph* $\mathcal{G}_{sub}(\mathbf{V}^\prime, \mathcal{G})$ of a DMG $\mathcal{G}= (\mathbf{V}, \mathbf{E}_{\rightarrow}, \mathbf{E}_{\leftrightarrow})$ on a subset $\mathbf{V}^\prime \subseteq \mathbf{V}$ of the vertices is the DMG $\mathcal{G}_{sub}(\mathbf{V}^\prime, \mathcal{G}) = (\mathbf{V}^\prime, \mathbf{E}_{\rightarrow}^\prime, \mathbf{E}_{\leftrightarrow}^\prime)$ where $\mathbf{E}_{\rightarrow}^\prime = \mathbf{E}_{\rightarrow}\cap \left( \mathbf{V}^\prime \times \mathbf{V}^\prime \right)$ and $\mathbf{E}_{\leftrightarrow}^\prime = \mathbf{E}_{\leftrightarrow}\cap \left( \mathbf{V}^\prime \times \mathbf{V}^\prime \right) \!/ \,\mathbb{Z}_2$. Intuitively, $\mathcal{G}_{sub}(\mathbf{V}^\prime, \mathcal{G})$ contains all and only those vertices in the subset $\mathbf{V}^\prime$ as well as the edges between them. A *walk* is a finite ordered sequence $\pi = (\pi(1), e_1, \pi(2), e_2, \pi(3), \ldots, e_{n-1}, \pi(n))$ where $\pi(1), \ldots, \pi(n)$ are vertices and where for all $k = 1, \ldots, n-1$ the edge $e_k$ connects $\pi(k)$ and $\pi(k+1)$. We say that $\pi$ is *between $\pi(1)$ and $\pi(n)$*. The integer $n$ is the *length* of the walk $\pi$, which we also denote as $len(\pi)$. We call the vertices $\pi(1)$ and $\pi(n)$ the *endpoint vertices* on $\pi$ and call the vertices $\pi(k)$ with $1 < k < n$ the *middle vertices* on $\pi$. The definition allows *trivial* walks, that is, walks which consist of a single vertex and no edges. If all vertices $\pi(1), \ldots, \pi(n)$ are distinct, then $\pi$ is a *path*. We can represent and specify a walk graphically, for example $v_1 {\,\leftarrow\,}v_2 {\,\leftrightarrow\,}v_3 {\,\rightarrow\,}v_4$. For $k$ and $l$ with $1 \leq k < l \leq len(\pi)$, we let $\pi(k, l)$ denote the walk $\pi^\prime = (\pi(k), e_k, \pi(k+1), \ldots, e_{l-1}, \pi(l))$. We say that a walk (path) $\pi^\prime$ is a *subwalk (subpath)* of a walk (path) $\pi$ if there are $k$ and $l$ with $1 \leq k < l \leq len(\pi)$ such that $\pi^\prime = \pi(k, l)$. A subwalk (subpath) $\pi^\prime$ of a walk (path) $\pi$ is *proper* if $\pi^\prime \neq \pi$. A walk is *into* its first vertex $\pi(1)$ if its first edge has an arrowhead at $\pi(1)$, that is, if $\pi(1, 2)$ is of the form $\pi(1) {\leftarrow\!\!\!\ast}\pi(2)$. If a walk is not into its first vertex, then it is *out of* its first vertex. Similarly, a walk is into (resp. out of) its last vertex $\pi(len(\pi))$ if its last edge has (resp. does not have) an arrowhead at $\pi(len(\pi))$. A middle vertex $\pi(k)$ on a walk $\pi$ is a *collider on $\pi$* if the edges on $\pi$ meet head-to-head at $\pi(k)$, that is, if the subwalk $\pi(k-1, k+1)$ is of the form $\pi(k-1) {\ast\!\!\!\rightarrow}\pi(k) {\leftarrow\!\!\!\ast}\pi(k+1)$. A middle vertex on a walk is a *non-collider on $\pi$* if it is not a collider on $\pi$. A walk is *directed* if it is non-trivial and takes the form $v_1 {\,\leftarrow\,}v_2 {\,\leftarrow\,}\ldots {\,\leftarrow\,}v_n$ or $v_1 {\,\rightarrow\,}v_2 {\,\rightarrow\,}\ldots {\,\rightarrow\,}v_n$. A non-trivial walk $\pi$ is a *confounding walk* if, first, no middle vertex on $\pi$ is a collider and, second, $\pi$ is into both its endpoint vertices. A walk is a *cycle* if it is non-trivial and $\pi(1) = \pi(len(\pi))$. A cycle is *irreducible* if it does not have a proper subwalk that is also a cycle. Thus, a cycle is irreducible if and only if, first, no vertex other than $\pi(1)$ appears twice and, second, $\pi(1)$ appears not more than twice. A cycle that is not irreducible is *reducible*. A walk is *cycle-free* if it does not have a subwalk which is a cycle. A directed *path* is always cycle-free. Two cycles $c_1$ and $c_2$ are *equivalent to each other* if and only if $c_2$ can be obtained by i) revolving the vertices on $c_1$ or by ii) reversing the order of the vertices on $c_1$ or by iii) by a combination of these two operations. If in a DMG $\mathcal{G}$ there is an edge $i {\,\rightarrow\,}j$, then $i$ is a *parent* of $j$ and $j$ is a *child* of $i$. We denote the sets of parents and children of $i$ as, respectively, $pa(i, \mathcal{G})$ and $ch(i, \mathcal{G})$. If there is a directed walk from $i$ to $j$ or $i = j$, then $i$ is an *ancestor* of $j$ and $j$ is a *descendant* of $i$. We denote the sets of ancestors and descendants of $i$ as, respectively, $an(i, \mathcal{G})$ and $de(i, \mathcal{G})$. If $i {\,\leftrightarrow\,}j$, then $i$ and $j$ are *spouses* of each other. We denote the set of spouses of $i$ as $sb(i, \mathcal{G})$. A path $\pi$ between $i$ and $j$ is an *inducing path* if, first, all its middle vertices are ancestors of $i$ or $j$ and, second, all its middle vertices are colliders on $\pi$. An *acyclic directed mixed graph (ADMG)*, which we here typically denote as $\mathcal{A}$, is a DMG without directed cycles. A *directed graph* is a DMG $(\mathbf{V}, \mathbf{E}_{\rightarrow}, \mathbf{E}_{\leftrightarrow})$ without bidirected edges, that is, a DMG with $\mathbf{E}_{\leftrightarrow}= \emptyset$. For simplicity, we identify a directed graph with the pair $(\mathbf{V}, \mathbf{E}_{\rightarrow})$. A *directed acyclic graph (DAG)*, which we here typically denoted as $\mathcal{D}$, is an ADMG that is also a directed graph. An ADMG $\mathcal{A}$ is *ancestral* if it satisfies two conditions: First, $\mathcal{A}$ does not have self edges. Second, $i \notin sb(j, \mathcal{A})$ if $i \in an(j, \mathcal{A})$. It follows that an ancestral ADMG has most one edge between any pair of vertices. The $m$-separation criterion [@richardson2002ancestral] extends the $d$-separation criterion [@pearl1988] from DAGs to ADMGs: A path $\pi$ between the vertices $i$ and $j$ in an ADMG with vertex set $\mathbf{V}$ is *$m$-connecting* given a set $\mathbf{Z}\subseteq \mathbf{V}\setminus \{i, j\}$ if, first, no non-collider on $\pi$ is in $\mathbf{Z}$ and, second, every collider on $\pi$ is an ancestor of some element in $\mathbf{Z}$. If a path is not $m$-connecting given $\mathbf{Z}$, then the path is *$m$-blocked*. The vertices $i$ and $j$ are *$m$-connected* given $\mathbf{Z}$ if there is at least one path between $i$ and $j$ that is $m$-connecting given $\mathbf{Z}$. If the vertices $i$ and $j$ are not $m$-connected given $\mathbf{Z}$, then they are *$m$-separated* given $\mathbf{Z}$. Let $\mathcal{A}= (\mathbf{V}, \mathbf{E}_{\rightarrow}, \mathbf{E}_{\leftrightarrow})$ be an ADMG without self edges. Then, its *canonical DAG* $\mathcal{D}_{c}(\mathcal{A})$ is the directed graph $(\mathbf{V}^\prime, \mathbf{E}_{\rightarrow}^\prime)$ where $\mathbf{V}^\prime = \mathbf{V}\cup \mathbf{L}$ with $\mathbf{L}= \{l_{ij} ~|~ (i, j) \in \mathbf{E}_{\leftrightarrow}\}$ and $\mathbf{E}_{\rightarrow}^\prime = \mathbf{E}_{\rightarrow}\cup \{(l_{ij}, i) ~|~ l_{ij} \in \mathbf{L}\} \cup \{(l_{ij}, j) ~|~ l_{ij} \in \mathbf{L}\}$ (cf. Section 6.1 of @richardson2002ancestral).[^6] Intuitively, we obtain $\mathcal{D}_{c}(\mathcal{A})$ from $\mathcal{A}$ by replacing each bidirected edge $i {\,\leftrightarrow\,}j$ in $\mathcal{A}$ with $i {\,\leftarrow\,}l_{ij} {\,\rightarrow\,}j$. It follows that acyclicity of $\mathcal{A}$ carries over to $\mathcal{D}_{c}(\mathcal{A})$, which means that $\mathcal{D}_{c}(\mathcal{A})$ is indeed a DAG. ## The ADMG latent projection {#sec:latent-projections} In many applications, some of the vertices $\mathbf{V}$ of a graph $\mathcal{G}$ serving as a graphical model might correspond to unobserved variables. We can formalize this situation by a partition $\mathbf{V}= \mathbf{O}\,\dot{\cup}\, \mathbf{L}$ of the vertices $\mathbf{V}$ into the *observed vertices $\mathbf{O}$* and the *unobserved / latent vertices $\mathbf{L}$*. If one is predominantly interested in reasoning about the observed vertices, then it is often convenient to project $\mathcal{G}$ to a *marginal graph* on the observed vertices only---provided the projection preserves certain graphical properties of interest. In this paper, we consider the following widely-used projection procedure. *Definition 1* (ADMG latent projection [@pearl1995theory], see also for example @richardson2023nested). Let $\mathcal{A}$ be an ADMG with vertex set $\mathbf{V}= \mathbf{O}\,\dot{\cup}\, \mathbf{L}$ that has no self edges. Then, its *marginal ADMG $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A})$ on $\mathbf{O}$* is the graph with vertex set $\mathbf{O}$ such that 1. there is a directed edge $i {\,\rightarrow\,}j$ in $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A})$ if and only if in $\mathcal{A}$ there is at least one directed path from $i$ to $j$ such that all middle vertices on this path are in $\mathbf{L}$, and 2. there is a bidirected edge $i {\,\leftrightarrow\,}j$ in $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A})$ if and only if in $\mathcal{A}$ there is at least one confounding path between $i$ and $j$ such that all middle vertices on this path are in $\mathbf{L}$. It follows that $i \in an(j, \sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A}))$ if and only if $i, j \in \mathbf{O}$ and $i \in an(j, \mathcal{A})$. The acyclicity of $\mathcal{A}$ thus carries over to $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A})$, so that $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A})$ is an ADMG indeed. Moreover, the definitions imply that $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A})$ does not have self edges. There can be more than one edge between a pair of vertices in $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A})$, namely $i {\,\leftrightarrow\,}j$ plus $i {\,\rightarrow\,}j$ or $i {\,\leftarrow\,}j$. Thus, in particular, the marginal ADMG $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A})$ is not necessarily ancestral. Two observed vertices $i$ and $j$ are $m$-separated given $\mathbf{Z}$ in $\mathcal{A}$ if and only if $i$ and $j$ are $m$-separated given $\mathbf{Z}$ in $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A})$, see Proposition 1 in @richardson2023nested.[^7] In Section [6](#sec:generalize-to-DMAGs){reference-type="ref" reference="sec:generalize-to-DMAGs"}, we further consider the DMAG latent projection [@richardson2002ancestral; @zhang2008causal], which is a different widely-used projection procedure. We also show how our results adapt to that case. ## Causal graphical time series models {#sec:background-time-series-models} This paper draws its motivation from causally stationary structural vector autoregressive processes with acyclic contemporaneous interactions (see for example @malinsky2018causal and @gerhardus2021characterization for formal definitions). We employ these potentially multivariate and potentially non-linear $\mathbb{Z}$-indexed stochastic processes with causal meaning by considering them as structural causal models [@bollen1989structural; @pearl2009causality; @peters2017elements]. Moreover, we allow for dependence between the noise variables (also known as innovation terms). The property of causal stationarity then requires that both the qualitative cause-and-effect relationships as well as the qualitative dependence structure between the noise variables are invariant in time. In particular, we are interested in the causal graphs (see for example @pearl2009causality) of such processes. These causal graphs represent the processes' qualitative cause-and-effect relationships by directed edges and non-zero dependencies between the noise variables of the processes by bidirected edges, and they have the following three special properties: 1. Since every vertex $(i, s) \in \mathbf{V}$ corresponds to a particular time step $s$ of a particular component time series $X^i$, the vertex set factorizes as $\mathbf{V}= \mathbf{I}\times \mathbf{T}$ where $\mathbf{I}$ is the *variable index set* and $\mathbf{T}\subseteq \mathbb{Z}$ the *time index set*. Thus, using the terminology of @gerhardus2021characterization, the causal graphs have *time series structure*. The *lag* of an edge $(i, s) {\ast\!{-}\!\ast}(j, u)$ is the non-negative integer $|u - s|$. An edge is *lagged* if its lag is greater or equal than one, else it is *contemporaneous*. We sometimes write $X^i_s$ instead of $(i, s)$ for clarity. 2. Since causation cannot go back in time, the directed edges do not point back in time. That is, $(i, s) {\,\rightarrow\,}(j, u)$ only if $s \leq u$. Thus, using the terminology of @gerhardus2021characterization, the causal graphs are *time ordered*.[^8] 3. Due to causal stationarity, the edges are repetitive in time. Thus, using the terminology of @gerhardus2021characterization, the causal graphs have *repeating edges*. Consequently, the causal graphs of interest are of the following type. *Definition 2* (Time series ADMG, generalizing Definition 3.4 of @gerhardus2021characterization). A *time series ADMG (ts-ADMG)* is an ADMG with time series structure (that is, $\mathbf{V}= \mathbf{I}\times \mathbf{T}$), that has time index set $\mathbf{T}= \mathbb{Z}$, that is time ordered, and that has repeating edges. {#fig:tsADMG} Figure [2](#fig:tsADMG){reference-type="ref" reference="fig:tsADMG"} shows two ts-ADMGs for illustration. *Time series DAGs (ts-DAGs)* [@gerhardus2021characterization] are ts-ADMGs without bidirected edges, see part (b) of Figure [2](#fig:tsADMG){reference-type="ref" reference="fig:tsADMG"} for an example. The absence of bidirected edges corresponds to the assumption of independent noise variables. In the literature, ts-DAGs are also known as *time series chain graphs* [@dahlhaus2003causality], *time series graphs* [@runge2012escaping] and *full time graphs* [@peters2017elements]. Throughout this paper, unless explicitly stated otherwise, we only consider ts-ADMGs that do not have self edges. To stress the repetitive edge structure of ts-ADMGs, we often specify the time index of a vertex in relation to an arbitrary reference time step $t$, that is, we often write $(i, t-\tau)$ instead of, say, $(i, s)$. Let $\mathbf{E}_{\rightarrow}^t = \{(i, t-\tau) {\,\rightarrow\,}(j, t) \, \in \, \mathbf{E}_{\rightarrow}\} \subsetneq \mathbf{E}_{\rightarrow}$ be the subset of directed edges pointing into a vertex at time $t$, and let $\mathbf{E}_{\leftrightarrow}^t = \{(i, t-\tau) {\,\leftrightarrow\,}(j, t) \, \in \, \mathbf{E}_{\leftrightarrow}~|~ \tau \geq 0 \} \subsetneq \mathbf{E}_{\leftrightarrow}$ be the subset of bidirected edges between a vertex at time $t$ and a vertex at or before $t$. Then, the triple $(\mathbf{I}, \mathbf{E}_{\rightarrow}^t, \mathbf{E}_{\leftrightarrow}^t)$ uniquely specifies a ts-ADMG. Throughout this paper, unless explicitly stated otherwise, we impose two mild conditions on all ts-ADMGs: First, we require the variable index set $\mathbf{I}$ to be finite. On the level of the modeled time series processes, this requirement restricts to processes with finitely many component time series. Second, we require both $\mathbf{E}_{\rightarrow}^t$ and $\mathbf{E}_{\leftrightarrow}^t$ to be finite sets (equivalently, we require all vertices to have finite in-degree). Given a finite variable index set, this second requirement is equivalent to the *maximal lag $p_{\mathcal{A}}$* defined as $p_{\mathcal{A}}= \sup \, \{ |\tau_i - \tau_j| ~|~ \text{$(i,t -\tau_i) {\ast\!{-}\!\ast}(j, t-\tau_j)$ in $\mathcal{G}$}\} = \sup \, \{ |\tau_i| ~|~ \text{$(i,t -\tau_i) {\ast\!\!\!\rightarrow}(j, t)$ in $\mathcal{G}$}\}$ being finite. On the level of the modeled time series processes, the second requirement thus restricts to processes of finite order. The *weight* $w(\pi)$ of a walk $\pi = ((i_1, s_1) {\ast\!{-}\!\ast}(i_2, s_2) {\ast\!{-}\!\ast}\ldots {\ast\!{-}\!\ast}(i_n, s_n))$ is the non-negative integer $|s_1 - s_n|$. If $\pi$ is of the form $\pi = ((i_1, s_1) {\,\leftarrow\,}(i_2, s_2) {\,\leftarrow\,}\ldots {\,\leftarrow\,}(i_n, s_n))$, then its weight $w(\pi) = s_1 - s_n$ equals the sum $w(\pi) = \sum_{k = 1}^{n-1} s_{k} - s_{k+1}$ of the lags $s_k - s_{k+1}$ of its edges; similarly for walks that are directed from $(i_1, s_1)$ to $(i_n, s_n)$. If $\pi_1$ is a directed walk from $(i, s_i)$ to $(j, s_j)$ and $\pi_2$ is a directed walk from $(j, s_j)$ to $(k, s_k)$, then $w(\pi_1) + w(\pi_2) = w(\pi)$ where $\pi$ is the directed walk from $(i, s_i)$ to $(k, s_k)$ obtained by appending $\pi_2$ to $\pi_1$ at their common vertex $(j, s_j)$. # Finite marginal time series graphs {#sec:problem-formulation} In this section, we motivate, define and start to approach the projection of ts-ADMGs to finite marginal graphs. To begin, Section [3.1](#sec:projection-motivation){reference-type="ref" reference="sec:projection-motivation"} explains why the finite marginal graphs are useful for answering $m$-separation queries in the ts-ADMGs as well as for causal discovery and causal effect estimation in time series. Section [3.2](#sec:projection-definition){reference-type="ref" reference="sec:projection-definition"} then follows up with a formal definition of the finite marginal graphs, and Section [3.3](#sec:reduction-past-confounding-paths){reference-type="ref" reference="sec:reduction-past-confounding-paths"} reduces the involved projection to the search for common ancestors in ts-DAGs. ## Motivation {#sec:projection-motivation} When interpreted as a causal graph, a ts-ADMG entails various claims about the associated multivariate structural time series process. An important type of claims are **independencies corresponding to m-separations** [@richardson2002ancestral; @richardson2003markov]. For a *finite* ADMG $\mathcal{A}$, the *causal Markov condition* [@spirtes2000causation] says that an $m$-separation $\mathbf{X} \perp\!\!\!\perp_{\mathcal{A}} \mathbf{Y} ~\vert~ \mathbf{Z}$ in the graph $\mathcal{A}$ implies the corresponding independence $\mathbf{X} \perp\!\!\!\perp \mathbf{Y} ~\vert~ \mathbf{Z}$ in all associated probability distributions [@verma1990causal; @geiger1990identifying; @richardson2003markov]. As opposed to that, for a ts-ADMG $\mathcal{A}$, which is an *infinite* graph, the same implication does not immediately follow: As an additional complication of the time series setting, it is non-trivial to say whether a given structural vector autoregressive process (cf. first paragraph of Section [2.3](#sec:background-time-series-models){reference-type="ref" reference="sec:background-time-series-models"}) specifies a well-defined probability distribution---in the terminology of @bongers2018causal, whether the process admits a *solution*---and what the properties of such solutions are. Thus, many works *assume* the existence of a solution with the desired properties, see for example @entner2010causal and @malinsky2018causal in the context of causal discovery. According to @dahlhaus2003causality [Theorem 3.3. combined with Definition 2.1 and Example 2.2], the causal Markov condition provably holds for the special case of stationary linear vector autoregressive processes with Gaussian innovation terms, see also @thams2022identifying [Theorem 1]. What is important for our work here, independent of how one argues for the causal Markov condition, one still deals with the task of asserting $m$-separations $\mathbf{X} \perp\!\!\!\perp_{\mathcal{A}} \mathbf{Y} ~\vert~ \mathbf{Z}$ in an infinite ts-ADMG $\mathcal{A}$. To make this assertion, one must assert that in $\mathcal{A}$ there is no path between $\mathbf{X}$ and $\mathbf{Y}$ that is active given $\mathbf{Z}$. However, as the ts-ADMG is infinite, there might be an infinite number of paths that could potentially be active. It is thus a non-trivial task to decide whether a given $m$-separation holds in a given ts-ADMG, and the authors are not aware of an existing general solution to it (also not in the special case of ts-DAGs). This task was the authors' original motivation for the presented study. In this paper, we solve this task as follows: We develop an algorithm that performs the ADMG projection of (infinite) ts-ADMGs to finite marginal ADMGs on a finite time window $\mathbf{T}_{\mathbf{O}}= \{t-\tau ~|~ 0 \leq \tau \leq p\} \subsetneq \mathbf{T}= \mathbb{Z}$ where $0 \leq p< \infty$. This projection algorithm implicitly solves the $m$-separation task because, if all vertices in $\mathbf{X} \cup \mathbf{Y} \cup \mathbf{Z}$ are within the time window $\mathbf{T}_{\mathbf{O}}$, then the $m$-separation $\mathbf{X} \perp\!\!\!\perp_{\mathcal{A}} \mathbf{Y} ~\vert~ \mathbf{Z}$ holds in the (infinite) ts-ADMG if and only if it holds in the finite marginal ADMG. Of course, this approach merely shifts the difficulty to an equally non-trivial task: constructing the finite marginal ADMGs. The ability to decide about $m$-separations is also necessary for causal reasoning. For example, most methods for **causal effect identification** in ADMGs---such as the (generalized) backdoor criterion [@pearl1993bayesian; @pearl2009causality; @maathuis2015generalized], the ID-algorithm [@tian2002general; @shpitser2006identification; @shpitser2006identification_2; @huang2006pearls] or graphical criteria to choose optimal adjustment sets [@runge2021necessary]---require the evaluation of $m$-separation statements in specific subgraphs of the causal graph. In order to apply these methods to ts-ADMGs with the goal of identifying time-resolved causal effects in structural time series processes,[^9] one first needs a solution for evaluating $m$-separations in certain (still infinite) subgraphs of ts-ADMGs. More generally speaking, most of the graphical-model based causal inference framework applies to finite graphs, whereas specific modifications might be necessary for application to the (infinite) ts-ADMGs. Our approach of projecting ts-ADMGs to finite marginal ADMGs solves the graphical part of this problem, since one can equivalently check the relevant graphical criteria in the finite marginal ADMGs instead of the ts-ADMG. Therefore, our results constitute one step towards making large parts of the causal-graphical-model literature directly applicable to time series. The second step towards this goal, which is independent of our contribution and in general still open (cf. the second paragraphs in the current subsection), is to more generally understand and prove in which cases the causal Markov condition holds for structural time series process. @thams2022identifying is an example of a work that already uses finite marginals of an infinite time series graph for the purpose of causal effect identification. Specifically, that work uses finite ADMG latent projections of a ts-DAG (not a ts-ADMG) in the context of instrumental variable regression [@bowden1990instrumental] for time series. However, that work i) gives the finite marginals of only *one* specific ts-DAG (which, in addition, is of the restricted special type discussed in Section [4.1](#sec:special-case-all-lag-$1$-auto){reference-type="ref" reference="sec:special-case-all-lag-$1$-auto"}) and ii) presents these finite marginal in an ad-hoc way. Contrary to that, in our paper, we i) consider general ts-ADMGs and ii) present a provably correct algorithm that constructs the finite marginal graphs. Moreover, when using **causal discovery** approaches to learn (Markov equivalence classes of) ts-ADMGs from data, in practice, one is always restricted to a finite number of time steps. The natural targets of time-resolved time series causal discovery thus are (equivalence classes of) finite marginals of ts-ADMGs or ts-DAGs on finite time windows. In these finite marginal graphs, the time steps before the considered time window inevitably act as confounders.[^10] Indeed, the causal discovery algorithm tsFCI [@entner2010causal] aims to infer equivalence classes of finite marginal graphs that arise as DMAG projections of ts-DAGs, see @gerhardus2021characterization for a detailed explanation. Similarly, the causal discovery algorithms SVAR-FCI [@malinsky2018causal] and LPCMCI [@gerhardus2020high] aim to infer equivalence classes of subgraphs of these marginal DMAGs. A method for constructing finite marginal DMAGs from a given ts-ADMG or ts-DAG is, thus, needed to formally understand the target graphs of such time series causal discovery algorithms, which is the basis for an analysis of these algorithms. Further, with the ability to construct finite marginal DMAGs of ts-ADMGs one can even improve on the identification power of these state-of-the-art causal discovery algorithms, see Algorithm 1 in @gerhardus2021characterization. While here we only consider finite marginal ADMGs, in Section [6](#sec:generalize-to-DMAGs){reference-type="ref" reference="sec:generalize-to-DMAGs"} we extend our results to finite marginal DMAGs. ## Definition of marginal ADMGs {#sec:projection-definition} The considerations in Section [3.1](#sec:projection-motivation){reference-type="ref" reference="sec:projection-motivation"} motivate us to consider the ADMG projections of (infinite) ts-ADMGs to finite time windows $\mathbf{T}_{\mathbf{O}}= \{t-\tau ~|~ 0 \leq \tau \leq p\}$. In these projections, all vertices outside the observed time window $\mathbf{T}_{\mathbf{O}}$ are treated as unobserved. Hence, adopting the terminology of @gerhardus2021characterization, we call a vertex $(i, t-\tau)$ *temporally observed* if $t-\tau \in \mathbf{T}_{\mathbf{O}}$ and else we call it *temporally unobserved*. In many applications where a ts-ADMG serves as a causal graphical model for a multivariate structural time series process, some of the component time series of the process might be unobserved altogether. We formalize this situation by introducing a partition $\mathbf{I}= \mathbf{I}_{\mathbf{O}}\,\dot{\cup}\, \mathbf{I}_{\mathbf{L}}$ of the ts-ADMG's variable index set $\mathbf{I}$ into the indices $\mathbf{I}_{\mathbf{O}}$ of observed component time series and the indices $\mathbf{I}_{\mathbf{L}}$ of unobserved component time series. Following @gerhardus2021characterization, we say that the component time series $X^i$ with $i \in \mathbf{I}_{\mathbf{O}}$ and all corresponding vertices $(i, t-\tau)$ are *observable*, whereas the component time series $X^i$ with $i \in \mathbf{I}_{\mathbf{L}}$ and all corresponding vertices $(i, t-\tau)$ are *unobservable*. For intuition, we often write $O^i$ (respl $L^i$) instead of $X^i$ if $X^i$ is observable (resp. unobservable). We require the set $\mathbf{I}_{\mathbf{O}}$ to be non-empty, because else there would be no observed vertices, whereas $\mathbf{I}_{\mathbf{L}}$ can but does not need to be empty. Putting together these notions, the sets of observed and unobserved vertices respectively are $\mathbf{O}= \mathbf{I}_{\mathbf{O}}\times \mathbf{T}_{\mathbf{O}}$ and $\mathbf{L}= \mathbf{V}\setminus \mathbf{O}= \left(\mathbf{I}\times \mathbf{T}\right) \setminus \left(\mathbf{I}_{\mathbf{O}}\times \mathbf{T}_{\mathbf{O}}\right)$. That is, a vertex is observed if and only if it is observable and temporally observed. We thus arrive at the following definition. *Definition 3* (Marginal time series ADMG, adapting Definition 3.6 of @gerhardus2021characterization). Let $\mathcal{A}$ be a ts-ADMG with variable index set $\mathbf{I}$, let $\mathbf{I}_{\mathbf{O}}\subseteq \mathbf{I}$ be non-empty, let $\mathbf{T}_{\mathbf{O}}$ be $\mathbf{T}_{\mathbf{O}}= \{t-\tau ~|~ 0 \leq \tau \leq p\}$ and let $\mathbf{O}= \mathbf{I}_{\mathbf{O}}\times \mathbf{T}_{\mathbf{O}}$. Then, its *marginal time series ADMG (marginal ts-ADMG) $\mathcal{A}_{\mathbf{O}}(\mathcal{A})$ on $\mathbf{O}$* is the ADMG $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A})$. We call the non-negative integer $p$ the *observed time window length*. *Remark 4*. To avoid confusion between ts-ADMGs (which are infinite graphs, see Definition [Definition 2](#def:tsADMG-infinite){reference-type="ref" reference="def:tsADMG-infinite"}) and marginal ts-ADMGs (which are finite graphs, see Definition [Definition 3](#def:tsADMG-marginal){reference-type="ref" reference="def:tsADMG-marginal"}), we often attach the attribute "infinite" to the former ("infinite ts-ADMG") and the attribute "finite" to the latter ("finite marginal ts-ADMG"). To avoid confusion, we stress that the *number of temporally observed time steps* is $p+1$; for example, the observed time window $[t, t]$ has the observed time window length $p=0$ and $1 = p+1$ temporally observed time steps. {#fig:example-marginal-tsADMG} Figure [3](#fig:example-marginal-tsADMG){reference-type="ref" reference="fig:example-marginal-tsADMG"} shows an example of a ts-ADMG and a corresponding finite marginal ts-ADMG for illustration. We are now ready to formally state the goal and contribution of this paper. *Problem 1*. Develop an algorithm that, for a given triple of - an arbitrary infinite ts-ADMG $\mathcal{A}$, - a given non-empty set $\mathbf{I}_{\mathbf{O}}$ of the observable component time series' variable indices and - a given length $p$ of the observed time window, provably determines the finite marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{A})$ in finite time. Here, $\mathcal{A}$ is only subject to the following requirements: absence of self edges, finiteness of its variable index set $|\mathbf{I}| < \infty$ and finiteness of its maximal lag $p_{\mathcal{A}}< \infty$. *Remark 5*. For a given triple $(\mathcal{A}, \mathbf{I}_{\mathbf{O}}, p)$ it is sometimes possible to manually find $\mathcal{A}_{\mathbf{O}}(\mathcal{A})$ by "looking at" the paths in $\mathcal{A}$. However, Problem [Problem 1](#problem:eventual-goal){reference-type="ref" reference="problem:eventual-goal"} asks for a general algorithmic procedure that works for every choice of $(\mathcal{A}, \mathbf{I}_{\mathbf{O}}, p)$. We stress that we do not impose any restrictions other than those stated. In particular, we do not restrict $p$ in relation to the maximal lag $p_{\mathcal{A}}$ of the ts-ADMG. Moreover, we do not impose connectivity assumptions on the ts-ADMG $\mathcal{A}$. In particular, every component time series (including the unobserved component time series) is allowed but not required to be auto-dependent at any time lag. To the author's knowledge, Problem [Problem 1](#problem:eventual-goal){reference-type="ref" reference="problem:eventual-goal"} remained unsolved prior to our work. Problem [Problem 1](#problem:eventual-goal){reference-type="ref" reference="problem:eventual-goal"} is non-trivial because the set $\mathbf{L}= \left(\mathbf{I}\times \mathbf{T}\right) \setminus \left(\mathbf{I}_{\mathbf{O}}\times \mathbf{T}_{\mathbf{O}}\right)$ of latent vertices includes the set $\mathbf{L}^{temp}= \mathbf{I}\times \left\{(-\infty, t-p-1] \cup [t+1, \infty)\right\}$ of temporally unobserved vertices and, hence, is infinite. Therefore, for any given pair of observed vertices $(i, t-\tau_i)$ and $(j, t-\tau_j)$ there might be infinitely many paths in the infinite ts-ADMG $\mathcal{A}$ that could potentially induce an edge $(i, t-\tau_i) {\ast\!{-}\!\ast}(j, t-\tau_j)$ in the finite marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{A})$. As we will see below, the way around this complication is the repeating edges property of ts-ADMGs. This property allows to effectively restrict to a finite search space when combined with an evaluation of number-theoretic solvability problems. ## Reduction to common-ancestor search {#sec:reduction-past-confounding-paths} In this section, we identify the missing ingredient for solving Problem [Problem 1](#problem:eventual-goal){reference-type="ref" reference="problem:eventual-goal"} to be an algorithm for the exhaustive search of common ancestors in infinite ts-ADMGs. To this end, we reduce the marginal ts-ADMG projection in three steps. First, in Section [3.3.1](#subsec:tsADMG-to-tsDAG){reference-type="ref" reference="subsec:tsADMG-to-tsDAG"}, we reduce the marginal ts-ADMG projection task for infinite ts-ADMGs to the projection task for the simpler infinite ts-DAGs. Second, in Section [3.3.2](#subsec:ADMGtosimpleADMG){reference-type="ref" reference="subsec:ADMGtosimpleADMG"}, we reduce the marginal ts-ADMG projection task for infinite ts-DAGs with arbitrary subsets of unobservable component time series to the projection task for infinite ts-DAGs without unobservable components. We refer to the map from an infinite ts-DAGs without unobservable component time series to its marginal ts-ADMG projection as the *simple marginal ts-ADMG* projection. Third, in Section [3.3.3](#sec:reduction-to-common-ancestor-search){reference-type="ref" reference="sec:reduction-to-common-ancestor-search"}, we reduce the simple marginal ts-ADMG projection task to the exhaustive search for common ancestors in infinite ts-DAGs. Figure [4](#fig:reduction-steps){reference-type="ref" reference="fig:reduction-steps"} illustrates these reduction steps. We conclude with brief remarks on extensions in Section [3.3.4](#subsec:projection-generalization){reference-type="ref" reference="subsec:projection-generalization"}. {reference-type="ref" reference="subsec:tsADMG-to-tsDAG"}, we determine the infinite canonical ts-DAG $\mathcal{D}^{ts}_{c}(\mathcal{A})$ of the infinite ts-ADMG $\mathcal{A}$. Second, see part **(d)** and Section [3.3.3](#sec:reduction-to-common-ancestor-search){reference-type="ref" reference="sec:reduction-to-common-ancestor-search"}, we determine the simple finite marginal ts-ADMG $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D}^{ts}_{c}(\mathcal{A}))$ of the canonical ts-DAG $\mathcal{D}^{ts}_{c}(\mathcal{A})$. This steps involves a common-ancestor search in the canonical ts-DAG $\mathcal{D}^{ts}_{c}(\mathcal{A})$, see part **(c)**, which is the topic of Section [4](#sec:refined-common-ancestor){reference-type="ref" reference="sec:refined-common-ancestor"}. Third, see Section [3.3.2](#subsec:ADMGtosimpleADMG){reference-type="ref" reference="subsec:ADMGtosimpleADMG"}, we determine the finite marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{A})$ from the simple marginal ts-ADMG $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D}^{ts}_{c}(\mathcal{A}))$ according to the equality of graphs $\mathcal{A}_{\mathbf{O}}(\mathcal{A}) = \sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D}^{ts}_{c}(\mathcal{A})))$. ](figures/04_reduction_steps.jpeg){#fig:reduction-steps} ### Reduction to the marginal ts-ADMG projection of infinite ts-DAGs {#subsec:tsADMG-to-tsDAG} To replace infinite ts-ADMGs with infinite ts-DAGs, we employ the following definition. *Definition 6* (Canonical time series DAG, adapted from Definition 4.13 in @gerhardus2021characterization). Let $\mathcal{A}= (\mathbf{I}\times \mathbb Z, \mathbf{E}_{\rightarrow}, \mathbf{E}_{\leftrightarrow})$ be a ts-ADMG. The *canonical time series DAG (canonical ts-DAG) $\mathcal{D}^{ts}_{c}(\mathcal{A})$ of $\mathcal{A}$* is the ts-DAG $\mathcal{D}^{ts}_{c}(\mathcal{A}) = (\mathbf{I}^{ca}\times \mathbb Z, \mathbf{E}_{\rightarrow}^{ca})$ where - $\mathbf{I}^{ca} = \mathbf{I}\cup \mathbf{J}$ with $\mathbf{J} = \{(i, j, \tau)~\vert~ \text{$X^{i}_{t-\tau} {\,\leftrightarrow\,}X^j_t$ in $\mathcal{A}$ with $\tau > 0$ or ($i < j$ and $\tau = 0$)}\}$ and - $\mathbf{E}_{\rightarrow}^{ca} = \mathbf{E}_{\rightarrow}\cup \{X^{(i, j, \tau)}_{s-\tau} {\,\rightarrow\,}X^j_s ~\vert~ s \in \mathbb Z, (i, j, \tau)\in \mathbf J\} \cup \{X^{(i, j, \tau)}_{s} {\,\rightarrow\,}X^i_s ~\vert~ s \in \mathbb Z, (i, j, \tau)\in \mathbf J\}$. {#fig:canonical_tsDAG} Figure [5](#fig:canonical_tsDAG){reference-type="ref" reference="fig:canonical_tsDAG"} shows a canonical ts-DAG and the corresponding ts-ADMG for illustration. Intuitively, we obtain the canonical ts-DAG $\mathcal{D}^{ts}_{c}(\mathcal{A})$ by replacing all bidirected edges $X^j_{s-\tau} {\,\leftrightarrow\,}X^i_s$ of the ts-ADMG $\mathcal{A}$ with paths $X^j_{s-\tau} {\,\leftarrow\,}X^{(i,j,\tau)}_{s-\tau} {\,\rightarrow\,}X^i_s$ where the $X^{(i,j,\tau)}$ are auxiliary unobservable component time series. It follows that $\mathcal{D}^{ts}_{c}(\mathcal{A}) = \mathcal{A}$ if and only if $\mathcal{A}$ is a ts-DAG. Moreover, acyclicity, time order and the property of repeating edges carry over from $\mathcal{A}$ to $\mathcal{D}^{ts}_{c}(\mathcal{A})$, so the canonical ts-DAG $\mathcal{D}^{ts}_{c}(\mathcal{A})$ is indeed a ts-DAG. The definition also implies that, first, $\mathcal{A}$ and $\mathcal{D}^{ts}_{c}(\mathcal{A})$ have the same directed paths and, second, there is a one-to-one correspondence between confounding paths through unobserved vertices in $\mathcal{A}$ and the same type of paths in $\mathcal{D}^{ts}_{c}(\mathcal{A})$. Lastly, $\mathcal{A}$ and $\mathcal{D}^{ts}_{c}(\mathcal{A})$ have the same $m$-separations among their shared vertices. By combining these observations, we arrive at the following result. **Proposition 7**. *Let $\mathcal{A}_{\mathbf{O}}(\mathcal{A})$ be the finite marginal ts-ADMG of the infinite ts-ADMG $\mathcal{A}$ on the set $\mathbf{O}$ of observed vertices. Then, $\mathcal{A}_{\mathbf{O}}(\mathcal{A}) = \mathcal{A}_{\mathbf{O}}(\mathcal{D}^{ts}_{c}(\mathcal{A}))$ where $\mathcal{A}_{\mathbf{O}}(\mathcal{D}^{ts}_{c}(\mathcal{A}))$ is the finite marginal ts-ADMG of the infinite canonical ts-DAG $\mathcal{D}^{ts}_{c}(\mathcal{A})$ of $\mathcal{A}$.* ### Reduction to the simple marginal ts-ADMG projection {#subsec:ADMGtosimpleADMG} The projection of an infinite ts-DAG $\mathcal{D}$ to its finite marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{D})$ on the set $\mathbf{O}= \mathbf{I}_{\mathbf{O}}\times \mathbf{T}_{\mathbf{O}}$ of observed vertices marginalizes out all unobserved vertices $\mathbf{L}= \mathbf{V}\setminus \mathbf{O}$. This set of unobserved vertices consists of, first, all vertices that are either strictly before time $t-p$ or strictly after time $t$ (temporally unobserved) and, second, all unobservable vertices within the observed time window $[t-p, t]$ (temporally observed but unobservable). Accordingly, $\mathbf{L}= \mathbf{L}^{temp}\,\dot{\cup}\, \mathbf{L}^{unob}$ where $\mathbf{L}^{temp}= \mathbf{I}\times \left\{(-\infty, t-p-1] \cup [t+1, \infty)\right\}$ and $\mathbf{L}^{unob}= \mathbf{I}_{\mathbf{L}}\times [t-p, t]$. This partition of $\mathbf{L}$ is useful for our purpose because the ADMG latent projection commutes with partitioning the set of unobserved vertices. Thus, to determine $\mathcal{A}_{\mathbf{O}}(\mathcal{D})$, we can first marginalize $\mathcal{D}$ over $\mathbf{L}^{temp}$ and then marginalize the resulting graph over $\mathbf{L}^{unob}$. **Proposition 8**. *Let $\mathcal{D}= (\mathbf{I}\times \mathbb{Z}, \mathbf{E}_{\rightarrow})$ be an infinite ts-DAG, let $\mathbf{O}= \mathbf{I}_{\mathbf{O}}\times \mathbf{T}_{\mathbf{O}}$ with $\mathbf{I}_{\mathbf{O}}\subseteq \mathbf{I}$ non-empty and $\mathbf{T}_{\mathbf{O}}= \{t-\tau ~|~ 0 \leq \tau \leq p\}$ where $p< \infty$, and let $\mathbf{O}^\prime = \mathbf{I}\times \mathbf{T}_{\mathbf{O}}$. Then, $\mathcal{A}_{\mathbf{O}}(\mathcal{D}) = \sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D}))$ where $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D}))$ is the ADMG latent projection to $\mathbf{O}$ of the finite marginal ts-ADMG $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ of $\mathcal{D}$ on $\mathbf{O}^\prime$.* Crucially, the marginal ts-ADMG $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ is a finite graph. Hence, finding the ADMG latent projection $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D}))$ of $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ is a solved problem. To solve Problem [Problem 1](#problem:eventual-goal){reference-type="ref" reference="problem:eventual-goal"}, it is thus sufficient to find an algorithm for projecting infinite ts-DAGs $\mathcal{D}$ to finite marginal ts-ADMGs $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ with $\mathbf{O}^\prime = \mathbf{I}\times \mathbf{T}_{\mathbf{O}}$, that is, for the special case of no unobservable component time series. We refer to this simplified projection as the *simple marginal ts-ADMG projection*. ### Reduction to common-ancestor search in ts-DAGs {#sec:reduction-to-common-ancestor-search} In the following, we first consider the directed edges and then the bidirected edges of the simple marginal ts-ADMG $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$. To recall, the projection of an infinite ts-DAG $\mathcal{D}$ to its simple finite marginal ts-ADMG $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ marginalizes over the set $\mathbf{L}^{temp}= \mathbf{I}\times \left\{(-\infty, t-p-1] \cup [t+1, \infty)\right\}$ of temporally unobserved vertices only. By point 1 in Definition [Definition 1](#def:ADMG-latent-projection){reference-type="ref" reference="def:ADMG-latent-projection"}, there is a directed edge $(i, t-\tau_i) {\,\rightarrow\,}(j, t-\tau_j)$ in $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ if and only if in $\mathcal{D}$ there is at least one directed path $\pi = ((i, t-\tau_i) {\,\rightarrow\,}\ldots {\,\rightarrow\,}(j, t-\tau_j))$ such that all middle vertices on $\pi$, if any, are in $\mathbf{L}^{temp}$. Since time order of $\mathcal{D}$ restricts such $\pi$ to the time window $[t-\tau_i, t-\tau_j] \subseteq \mathbf{T}_{\mathbf{O}}$ and since $\mathbf{L}^{temp}$ contains only vertices outside of $\mathbf{T}_{\mathbf{O}}$, we see that $\pi = ((i, t-\tau_i) {\,\rightarrow\,}(j, t-\tau_j))$ and thus get the following result. **Proposition 9**. *Let $(i, t-\tau_i)$ and $(j, t-\tau_j)$ be vertices in $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ with $\mathbf{O}^\prime = \mathbf{I}\times \mathbf{T}_{\mathbf{O}}$. Then, $(i, t-\tau_i) {\,\rightarrow\,}(j, t-\tau_j)$ in $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ if and only if $(i, t-\tau_i) {\,\rightarrow\,}(j, t-\tau_j)$ in $\mathcal{D}$.* Thus, we can directly read off the directed edges of $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ from $\mathcal{D}$. The remaining task is to find the bidirected edges of $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$. By point 2 in Definition [Definition 1](#def:ADMG-latent-projection){reference-type="ref" reference="def:ADMG-latent-projection"}, there is a bidirected edge $(i, t-\tau_i) {\,\leftrightarrow\,}(j, t-\tau_j)$ in $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ if and only if in $\mathcal{D}$ there is at least one confounding path $\pi = ((i, t-\tau_i) {\leftarrow\!\!\!\ast}\ldots {\ast\!\!\!\rightarrow}(j, t-\tau_j))$ such that all middle vertices on $\pi$, if any, are in $\mathbf{L}^{temp}$. Recall that $\mathbf{L}^{temp}$ consists of, first, all vertices strictly before time $t-p$ and, second, all vertices strictly after time $t$. Due to time order of $\mathcal{D}$ and the definitional requirement that confounding paths do not have colliders, such $\pi$ cannot contain vertices strictly after time $\max(t-\tau_i, t-\tau_j) \leq t$. We thus find that all middle vertices of $\pi$, if any, are strictly before $t-p$ and arrive at the following result. **Proposition 10**. *Let $(i, t-\tau_i)$ and $(j, t-\tau_j)$ be vertices in $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ with $\mathbf{O}^\prime = \mathbf{I}\times \mathbf{T}_{\mathbf{O}}$. Then, $(i, t-\tau_i) {\,\leftrightarrow\,}(j, t-\tau_j)$ in $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ if and only if there are (not necessarily distinct) vertices $(k, t-\tau_k)$ with $\tau_k > p$ and $(l, t-\tau_l)$ with $\tau_l > p$ that have a common ancestor in $\mathcal{D}$.* *Remark 11*. Recall that every vertex is its own ancestor. Consequently, $(k, t-\tau_k)$ and $(l, t-\tau_l)$ have a common ancestor if at least one of the following conditions is true: - The vertices $(k, t-\tau_k)$ and $(l, t-\tau_l)$ are equal. - There is a directed path from $(k, t-\tau_k)$ to $(l, t-\tau_l)$. - There is a directed path from $(l, t-\tau_l)$ to $(k, t-\tau_k)$. - There is a confounding path between $(k, t-\tau_k)$ and $(l, t-\tau_l)$. Due to the repeating edges property of ts-DAGs, the vertices $(k, t-\tau_k)$ and $(l, t-\tau_l)$ have a common ancestor if and only if the time-shifted vertices $(k, t-\tau_k+\Delta t)$ and $(l, t-\tau_l+ \Delta t)$ with $\Delta t \in \mathbb Z$ have a common ancestor. Choosing $\Delta t = \min(\tau_k, \tau_l)$, we can place at least one of the time-shifted vertices at time $t$. We thus reduced the simple ts-ADMG projection and, by extension, Problem [Problem 1](#problem:eventual-goal){reference-type="ref" reference="problem:eventual-goal"} to the following problem. *Problem 2*. Let $(i, t-\tau_i)$ and $(j, t)$, where $\tau_i$ is a non-negative integer, be two vertices in an arbitrary infinite ts-DAG $\mathcal{D}$ (subject only to $p_{\mathcal{A}}< \infty$ and $\vert \mathbf{I}\vert < \infty$). Decide in finite time whether $(i, t-\tau_i)$ and $(j, t)$ have a common ancestor in $\mathcal{D}$. Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"} is still non-trivial because ts-DAGs extend to the infinite past and, hence, there might be infinitely many paths that can potentially give rise to a common ancestor of a given pair of vertices. In Section [4](#sec:refined-common-ancestor){reference-type="ref" reference="sec:refined-common-ancestor"}, we present a method that, despite this complication, answers common-ancestor queries in ts-DAGs in finite time. To aid computational efficiency, we note the following: Proposition [Proposition 10](#prop:bidirected-edges-tsADMG-simple){reference-type="ref" reference="prop:bidirected-edges-tsADMG-simple"} together with the repeating edges property of ts-DAGs implies that $(i,t-\tau_i-\Delta t) {\,\leftrightarrow\,}(j, t-\tau_j-\Delta t)$ with $0 < \Delta t \leq p- \max(\tau_i, \tau_j)$ is in $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ if $(i,t-\tau_i) {\,\leftrightarrow\,}(j, t-\tau_j)$ is in $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$. Hence, it is advantageous to first apply Proposition [Proposition 10](#prop:bidirected-edges-tsADMG-simple){reference-type="ref" reference="prop:bidirected-edges-tsADMG-simple"} to pairs of vertices $(i,t-\tau_i)$ and $(j, t-\tau_j)$ with small $\max(\tau_i, \tau_j)$: If we find that these vertices are connected by a bidirected edge, then we can automatically infer the existence of $p- \max(\tau_i, \tau_j)$ further bidirected edges. ### Extensions {#subsec:projection-generalization} We can extend the above findings in two ways: First, in Section [6](#sec:generalize-to-DMAGs){reference-type="ref" reference="sec:generalize-to-DMAGs"} we show how to determine the finite marginal DMAG projection of an infinite ts-ADMG $\mathcal{A}$ from the finite marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{A})$. Second, given the ability to determine arbitrary finite marginal ts-ADMGs, we can also determine the ADMG latent projection of ts-ADMGs to *arbitrary finite* sets $\mathbf{O}^{\prime\prime}$ of observed vertices. Indeed, let $\mathbf{O}$ be any set of the form $\mathbf{O}= \mathbf{I}_{\mathbf{O}}\times \mathbf{T}_{\mathbf{O}}$ with $\mathbf{O}^{\prime\prime} \subseteq \mathbf{O}$. Then, because the ADMG latent projection commutes with partitioning the set of latent vertices, we can determine $\sigma_{\text{ADMG}}(\mathbf{O}^{\prime\prime}, \mathcal{A})$ as $\sigma_{\text{ADMG}}(\mathbf{O}^{\prime\prime}, \mathcal{D}) = \sigma_{\text{ADMG}}(\mathbf{O}^{\prime\prime}, \mathcal{A}_{\mathbf{O}}(\mathcal{A}))$. Since $\mathcal{A}_{\mathbf{O}}(\mathcal{A})$ is a finite graph, finding its projection $\sigma_{\text{ADMG}}(\mathbf{O}^{\prime\prime}, \mathcal{A}_{\mathbf{O}}(\mathcal{D}))$ is a solved problem. # Common-ancestor search in ts-DAGs {#sec:refined-common-ancestor} In this section, we present the three main results of this paper. These results solve Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"} (that is, the common-ancestor search in infinite ts-DAGs) and, hence, by extension also Problem [Problem 1](#problem:eventual-goal){reference-type="ref" reference="problem:eventual-goal"} (that is, the construction of finite marginal ts-ADMGs of infinite ts-ADMGs). To begin, in Section [4.1](#sec:special-case-all-lag-$1$-auto){reference-type="ref" reference="sec:special-case-all-lag-$1$-auto"}, we restrict ourselves to ts-DAGs that have lag-$1$ auto-dependencies in every component and present a simple solution to Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"} in this special case. In Section [4.2](#subsec:approaches-do-not-work){reference-type="ref" reference="subsec:approaches-do-not-work"}, we explain why the same approach does not work for general ts-DAGs and discuss two simple heuristics that do not work either. In Section [4.3](#subsec:general-solution){reference-type="ref" reference="subsec:general-solution"}, we present our solution to Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"}. To this end, we first reformulate Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"} in terms of *multi-weighted directed graphs* (see Definition [Definition 15](#Definitionm-w-graph){reference-type="ref" reference="Definitionm-w-graph"} and Problem [Problem 3](#prob1){reference-type="ref" reference="prob1"}). Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"} then reduces Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"} to the problem of deciding whether any of a finite number of *linear Diophantine equations* has a non-negative integer solution. Moreover, utilizing standard results from number theory, we show how to answer these solvability questions in finite time by Theorem [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"}. The combination of Theorems [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"} and [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"} thus solves Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"}. As a corollary, in Theorem [Theorem 3](#thm.upper-bound-main-theorem){reference-type="ref" reference="thm.upper-bound-main-theorem"} we present an easily computable bound $p^\prime(\mathcal{D})$ such that the common-ancestor searches of Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"} can be restricted to the segment of the ts-DAG $\mathcal{D}$ inside the finite time window $[t-p^\prime(\mathcal{D}),t]$. Thus, Theorem [Theorem 3](#thm.upper-bound-main-theorem){reference-type="ref" reference="thm.upper-bound-main-theorem"} constitutes an alternative solution to Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"}. ## Special case of all lag-1 auto-dependencies {#sec:special-case-all-lag-$1$-auto} As an important special case, we first restrict to ts-DAGs $\mathcal{D}$ that have lag-$1$ auto-dependencies everywhere, that is, to ts-DAGs $\mathcal{D}$ which for all $i \in \mathbf{I}$ have the edge $(i, t-1) {\,\rightarrow\,}(i,t)$. To solve Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"} in this special case, we make use of the following well-known concept. *Definition 12* (Summary graph, cf. for example @peters2017elements). Let $\mathcal{D}= (\mathbf{I}\times \mathbb{Z}, \mathbf{E}_{\rightarrow})$ be a ts-DAG. The *summary graph $\mathcal{S}(\mathcal{D})$ of $\mathcal{D}$* is the directed graph with vertex set $\mathbf{I}$ such that $i {\,\rightarrow\,}j$ in $\mathcal{S}(\mathcal{D})$ if and only if there is at least one $\tau$ such that $(i, t-\tau) {\,\rightarrow\,}(j, t)$ in $\mathcal{D}$.[^11] Figure [6](#fig:heuristic_summary_graph){reference-type="ref" reference="fig:heuristic_summary_graph"} illustrates the concept of summary graphs, which above we also referred to as time-collapsed graphs. Intuitively, we obtain the summary graph $\mathcal{S}(\mathcal{D})$ by collapsing the ts-DAG $\mathcal{D}$ along its temporal dimension. Thus, although the ts-DAG $\mathcal{D}$ is acyclic by definition, its summary graph $\mathcal{S}(\mathcal{D})$ can be cyclic. Due to our standing assumption that $|\mathbf{I}|<\infty$, the summary graph $\mathcal{S}(\mathcal{D})$ is a finite graph and all path searches in it will terminate in finite time. This fact makes summary graphs a promising tool for answering Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"}. Indeed, whenever there is a confounding path (resp. a directed path) between $(i, t-\tau_i)$ and $(j, t)$ in the ts-DAG $\mathcal{D}$, then the projection of that path to $\mathcal{S}(\mathcal{D})$---obtained by forgetting the time indices of the vertices on the path---is a confounding walk (resp. a directed walk) from $i$ to $j$ in $\mathcal{S}(\mathcal{D})$. In the special case of all lag-$1$ auto-dependencies, also the converse is true and we obtain the following result. **Proposition 13**. *Let $\mathcal{D}$ be a ts-DAG that for all $i \in \mathbf{I}$ has the edge $(i,t-1) {\,\rightarrow\,}(i,t)$, and let $(i, t-\tau_i)$ with $\tau_i \geq 0$ and $(j, t)$ be two vertices in $\mathcal{D}$. Then, $(i, t-\tau_i)$ and $(j, t)$ have a common ancestor in $\mathcal{D}$ if and only if the vertices $i$ and $j$ have a common ancestor in $\mathcal{S}(\mathcal{D})$.* We can understand the if-part of Proposition [Proposition 13](#prop:common-ancestors-all-lag-$1$-are-there){reference-type="ref" reference="prop:common-ancestors-all-lag-$1$-are-there"} as follows: Suppose there is a confounding path between $i$ and $j$ in $\mathcal{S}(\mathcal{D})$. By splitting this confounding path at its (unique) root vertex $k$, we obtain a directed path $\pi_{ki}$ from $k$ to $i$ and a directed path $\pi_{kj}$ from $k$ to $j$ in $\mathcal{S}(\mathcal{D})$. By definition of the summary graph and the repeating edges property of ts-DAGs, in the ts-DAG $\mathcal{D}$ there thus are a directed path $\rho_{ki}$ from $(k, t-\tau_{ki})$ to $(i,t-\tau_i)$ for some $\tau_{ki} \geq \tau_i$ and a directed path $\rho_{kj}$ from $(k, t-\tau_{kj})$ to $(j, t)$ for some $\tau_{kj} \geq 0$. If $\tau_{kj} = \tau_{ki}$, then we can directly join $\rho_{ki}$ and $\rho_{kj}$ at their common vertex $(k, t-\tau_{ki}) = (k, t-\tau_{kj})$ to obtain a confounding walk $\rho$. If $\tau_{kj} \neq \tau_{ki}$, then an additional step is needed: Due to the assumption that $\mathcal{D}$ has all lag-$1$ auto-dependencies, there is the path $\rho_{kk} = ((k, t-\max(\tau_{ki},\tau_{ki})) {\,\rightarrow\,}(k, t-\max(\tau_{ki},\tau_{ki})+1) {\,\rightarrow\,}\ldots {\,\rightarrow\,}(k, t-\min(\tau_{ki},\tau_{ki}))$ in the ts-DAG $\mathcal{D}$. By appropriately concatenating the paths $\rho_{ki}$, $\rho_{kj}$ and $\rho_{kk}$, we obtain a confounding walk $\rho$. If $\rho$ is a path, we thus showed that $(i, t-\tau_i)$ and $(j, t)$ have a common ancestor in $\mathcal{D}$. If $\rho$ is not a path, then we can remove a sufficiently large subwalk from $\rho$ to obtain a path $\tilde{\rho}$ which is either a confounding path or directed from $(i, t-\tau_i)$ to $(j, t)$ or directed from $(j, t)$ to $(i, t-\tau_i)$ or trivial. Thus, $(i, t-\tau_i)$ to $(j, t)$ have a common ancestor in $\mathcal{D}$ (see Remark [Remark 11](#remark:bidirected-edges-tsADMG-simple){reference-type="ref" reference="remark:bidirected-edges-tsADMG-simple"} for clarification of what a common ancestor is). The cases in which $i=j$ or in which $i$ and $j$ are connected by a directed path (cf. Remark [Remark 11](#remark:bidirected-edges-tsADMG-simple){reference-type="ref" reference="remark:bidirected-edges-tsADMG-simple"}) follow similarly. While ts-DAGs with lag-$1$ auto-dependencies in all component time series are an important special case, restricting to that case is *not* standard in the literature. For example, none of the causal discovery works @entner2010causal, @malinsky2018causal or @gerhardus2020high makes this assumption. Conversely, @mastakouri2020necessary even specifically assumes a subset of the components to not have lag-$1$ auto-dependencies. Thus, we are not satisfied with Proposition [Proposition 13](#prop:common-ancestors-all-lag-$1$-are-there){reference-type="ref" reference="prop:common-ancestors-all-lag-$1$-are-there"} and move to the general case. ## Approaches that do not work in general {#subsec:approaches-do-not-work} We now turn to general ts-DAGs, that is, we no longer require that $(i,t-1) {\,\rightarrow\,}(i, t)$ for all $i \in \mathbf{I}$. In Sections [4.2.1](#subsec:summary-graph-insufficient){reference-type="ref" reference="subsec:summary-graph-insufficient"} and [4.2.2](#subsect:simple-heuristics-insufficient){reference-type="ref" reference="subsect:simple-heuristics-insufficient"}, we consider two approaches that might appear promising at first but are in general not sufficient to decide about common ancestorship in ts-DAGs. ### Common-ancestor search in the summary graph {#subsec:summary-graph-insufficient} The following example shows that, for general ts-DAGs, common ancestorship of $i$ and $j$ in the summary graph $\mathcal{S}(\mathcal{D})$ does not necessarily imply common ancestorship of $(i,t-\tau_i)$ and $(j, t)$ in the ts-DAG $\mathcal{D}$. *Example 14*. In the summary graph in part (b) of Figure [6](#fig:heuristic_summary_graph){reference-type="ref" reference="fig:heuristic_summary_graph"}, the vertex $Y$ is a common ancestor of $X$ and $Z$. However, in the corresponding ts-DAG in part (a) of Figure [6](#fig:heuristic_summary_graph){reference-type="ref" reference="fig:heuristic_summary_graph"}, the vertices $X_t$ and $Z_{t-1}$ do not have a common ancestor. {reference-type="ref" reference="example:with-summary-graph"}.](figures/05_summary_graph_heuristic.jpeg){#fig:heuristic_summary_graph} To understand why Proposition [Proposition 13](#prop:common-ancestors-all-lag-$1$-are-there){reference-type="ref" reference="prop:common-ancestors-all-lag-$1$-are-there"} does not generalize to general ts-DAGs, we recall its justification from the last paragraph of Section [4.1](#sec:special-case-all-lag-$1$-auto){reference-type="ref" reference="sec:special-case-all-lag-$1$-auto"}: The argument there involved the path $\rho_{kk} = ((k, t-\max(\tau_{ki},\tau_{ki})) {\,\rightarrow\,}(k, t-\max(\tau_{ki},\tau_{ki})+1) {\,\rightarrow\,}\ldots {\,\rightarrow\,}(k, t-\min(\tau_{ki},\tau_{ki}))$. This path only exists if there is the edge $(k,t-1) {\,\rightarrow\,}(k,t)$, which need not be the case for general ts-DAGs. Put differently, the existence of a confounding path between $i$ and $j$ in $\mathcal{S}(\mathcal{D})$ does imply the existence of a path $\rho_{ki}$ from $(k, t-\tau_{ki})$ to $(i,t-\tau_i)$ for some $\tau_{ki} \geq \tau_i$ and a directed path $\rho_{kj}$ from $(k, t-\tau_{kj})$ to $(j, t)$ for some $\tau_{kj} \geq 0$. But, in general, there is no pair of such paths with $\tau_{ki} = \tau_{kj}$, that is, a pair of such paths which start at the same time step. The opposite direction, however, still holds: Common ancestorship of $(i,t-\tau_i)$ and $(j, t)$ in $\mathcal{D}$ does imply common ancestorship of $i$ and $j$ in $\mathcal{S}(\mathcal{D})$. Thus, common ancestorship in $\mathcal{S}(\mathcal{D})$ is a necessary but not sufficient condition for common ancestorship in $\mathcal{D}$. ### Simple heuristics for a finite time window {#subsect:simple-heuristics-insufficient} A different approach would proceed as follows: Given a pair of vertices $(i,t-\tau_i)$ and $(j, t)$ with $0 \leq \tau_i \leq p$, search for common ancestors of that pair within a sufficiently large but finite time window $[t-p^\prime(\mathcal{D}),t]$ of the ts-DAG $\mathcal{D}$. While for every fixed $\mathcal{D}$ there indeed is some non-negative integer $p^\prime(\mathcal{D})$ such that $(i,t-\tau_i)$ and $(j, t)$ have a common ancestor in $\mathcal{D}$ if and only if they have a common ancestor within the time window $[t-p^\prime(\mathcal{D}),t]$, this fact itself is of little practical use: To solve Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"}, one would need to know such an integer $p^\prime(\mathcal{D})$ *a priori*. Therefore it might be tempting to come up with heuristics for choosing $p^\prime(\mathcal{D})$ for a given $\mathcal{D}$, such as the following: - Let $p^\prime(\mathcal{D}) = p+ p^\prime_{sum}(\mathcal{D})$, where $p$ is the observed time window length and $p^\prime_{sum}(\mathcal{D})$ is the sum of all lags of edges in $\mathbf{E}_{\rightarrow}^t \cup \mathbf{E}_{\leftrightarrow}^t$. - Let $p^\prime(\mathcal{D}) = p+ p^\prime_{prod}(\mathcal{D})$, where $p$ is the observed time window length and $p^\prime_{prod}(\mathcal{D})$ is the product of all non-zero lags of edges in $\mathbf{E}_{\rightarrow}^t \cup \mathbf{E}_{\leftrightarrow}^t$. However, Examples [Example 33](#ref:counterexample-1){reference-type="ref" reference="ref:counterexample-1"} and [Example 34](#ref:counterexample-2){reference-type="ref" reference="ref:counterexample-2"} in Section [7](#sec:counterexamples){reference-type="ref" reference="sec:counterexamples"} show that neither of these simple heuristics work in general. What is more, even if one were to come up with a heuristic for which one would not find a counterexample, then one would still have to prove this heuristic. In Section [4.3.4](#sec:formula-cutoff){reference-type="ref" reference="sec:formula-cutoff"}, we will eventually prove a formula for a sufficiently large $p^\prime(\mathcal{D})$ by considering bounds for minimal non-negative integer solutions of linear Diophantine equations. However, this formula is far from obvious and we expect an explicit common-ancestor search in the corresponding time window to be computationally more expensive than a direct application of the intermediate number-theoretic result from which the formula is derived. ## General solution to the common-ancestor search in ts-DAGs {#subsec:general-solution} In this section, we present the main results of Section [4](#sec:refined-common-ancestor){reference-type="ref" reference="sec:refined-common-ancestor"}. These results provide a general solution to Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"}. ### Time series DAGs as multi-weighted directed graphs {#subsubsec:multi-weighted} In order to express our results on the common-ancestor search in ts-DAGs efficiently, we reformulate the information contained in ts-DAGs and, correspondingly, Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"} in the language of *multi-weighted directed graphs*. This reformulation gives us access to standard techniques of discrete graph combinatorics. We begin with the definition of multi-weighted directed graphs. *Definition 15* (Multi-weighted directed graph). A *multi-weighted directed graph (MWDG)* is a tuple $(\mathcal{G},\mathbf{w})$ of a directed graph $\mathcal{G}= (\mathbf{I},\mathbf{E}_{\rightarrow})$ and a collection $\mathbf{w}= (\mathbf{w}(e))_{e \in \mathbf{E}_{\rightarrow}}$ of multi-weights $\mathbf{w}(e)$. The multi-weights $\emptyset \neq \mathbf{w}(e) \subsetneq \mathbb N_0$ are *finite* non-empty sets of non-negative integers. We call $w(e) \in \mathbf{w}(e)$ a *weight* of $e$. A multi-weighted directed graph $(\mathcal{G},\mathbf{w})$ is *weakly acyclic* if it satisfies both of the following conditions: 1. If $e = i {\,\rightarrow\,}i$, then $0 \notin \mathbf{w}(e)$. 2. The directed graph $\mathcal{G}^0 = (\mathbf{I}, \mathbf{E}_{\rightarrow}^0)$ with $\mathbf{E}_{\rightarrow}^0 = \{e \in \mathbf{E}_{\rightarrow}~\vert~ 0 \in \mathbf{w}(e) \}$ is acyclic. For $\pi$ a trivial or directed walk in a MWDG $\mathcal{G}$, we define its multi-weight $\mathbf{w}(\pi)$ as $$\begin{aligned} \mathbf{w}(\pi) = \begin{cases} \left\{ \sum_{e \in \mathbf{E}_{\pi}} w(e) ~\middle\vert~ w(e) \in \mathbf{w}(e)\right\} \equiv \sum_{e \in \mathbf{E}_{\pi}} \mathbf{w}(e) \quad &\text{if $\pi$ is directed} \\ \{0\} \quad &\text{if $\pi$ is trivial} \end{cases}\end{aligned}$$ where $\mathbf{E}_{\pi}$ denotes the sequence of edges on $\pi$. We call $w(\pi) \in \mathbf{w}(\pi)$ a *weight* of $\pi$. To abbreviate notation below, given two sets $A,B \subseteq \mathbb N_0$ and a non-negative integer $n$ we define $$\begin{aligned} A + B = \left\{ a +b ~\vert~ \in A, \ b \in B \right\} \quad \text{and} \quad n \cdot A = \left\{n \cdot a ~\vert~ a \in A \right\} \, .\end{aligned}$$ In order to understand the connection between ts-DAGs and weakly acyclic MWDGs, we employ the following definition. *Definition 16*. Let $\mathcal{D}= (\mathbf{I}\times \mathbb{Z}, \mathbf{E}_{\rightarrow})$ be a ts-DAG. Then, the *multi-weighted summary graph $\mathcal{S}_{\mathbf{w}}(\mathcal{D})$ of $\mathcal{D}$* is the multi-weighted directed graph $\mathcal{S}_{\mathbf{w}}(\mathcal{D}) = (\mathcal{S}(\mathcal{D}), \mathbf{w})$ where - $\mathcal{S}(\mathcal{D})$ is the summary graph of $\mathcal{D}$ in the sense of Definition [Definition 12](#def:summary-graph){reference-type="ref" reference="def:summary-graph"} and - $\mathbf{w}= (\mathbf{w}(i {\,\rightarrow\,}j))_{i {\,\rightarrow\,}j \in \mathbf{E}_{\rightarrow}}$ with $\mathbf{w}(i {\,\rightarrow\,}j) = \{ \tau ~\vert~ (i, t-\tau) {\,\rightarrow\,}(j, t) \in \mathcal{D}\}$. {reference-type="ref" reference="example:running-example-1"}, [Example 21](#example:running-example-2){reference-type="ref" reference="example:running-example-2"}, [Example 22](#example:running-example-3){reference-type="ref" reference="example:running-example-3"} and [Example 26](#example:running-example-4){reference-type="ref" reference="example:running-example-4"}.](figures/06_weighted_summary_graph.jpeg){#fig:multi_weighted_summary_graph} Parts (a) and (b) of Figure [7](#fig:multi_weighted_summary_graph){reference-type="ref" reference="fig:multi_weighted_summary_graph"} illustrate the concept of multi-weighted summary graphs. Due to acyclicity and the absence of self edges, the multi-weighted summary graph $\mathcal{S}_{\mathbf{w}}(\mathcal{D})$ of a ts-DAG $\mathcal{D}$ is a weakly acyclic MWDG. Conversely, a weakly acyclic MWDG $(\mathcal{G}, \mathbf{w})$ where $\mathcal{G}= (\mathbf{I}, \mathbf{E}_{\rightarrow}^{\mathcal{G}})$ induces the ts-DAG $\mathcal{D}= (\mathbf{I}\times \mathbb{Z}, \mathbf{E}_{\rightarrow})$ where $\mathbf{E}_{\rightarrow}= \{(i, t-\tau) {\,\rightarrow\,}(j, t) ~\vert~ (i, j) \in \mathbf{E}_{\rightarrow}^{\mathcal{G}}, \tau \in \mathbf{w}(i{\,\rightarrow\,}j), t\in \mathbb{Z}\}$. These mappings between ts-DAGs and weakly acyclic MWDGs are one-to-one and inverses of each other. Consequently, we can re-express every statement about ts-DAGs on the level of weakly acyclic MWDGs. The following result gives the corresponding reformulation of common ancestorship in ts-DAGs. **Proposition 17**. *Let $\mathcal{D}$ be a ts-DAG, and let $(i,t-\tau)$ and $(j,t)$ with $\tau \geq 0$ be two vertices in $\mathcal{D}$. Then, $(i,t-\tau)$ and $(j,t)$ have a common ancestor in $\mathcal{D}$ if and only if in the multi-weighted summary graph $\mathcal{S}_{\mathbf{w}}(\mathcal{D})$ of $\mathcal{D}$ there are trivial or directed walks $\pi$ and $\pi^\prime$ that satisfy all of the following conditions:* 1. *Their first vertex is equal, that is, $\pi(1) = \pi'(1)$.* 2. *They respectively end at the vertices $i$ and $j$, that is, $\pi(len(\pi)) = i$ and $\pi'(len(\pi')) = j$.* 3. *They have weights $w(\pi) \in \mathbf{w}(\pi)$ and $w(\pi') \in \mathbf{w}(\pi')$ with $w(\pi)+\tau = w(\pi')$.* Using Proposition [Proposition 17](#prop:common-ancestorship-reexpression){reference-type="ref" reference="prop:common-ancestorship-reexpression"}, we can reformulate Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"} on the level of MWDGs as follows. *Problem 3*. Let $(\mathcal{G},\mathbf{w})$ be a weakly acyclic MWDG, let $\tau \in \mathbb{N}_0$, and let $i$, $j$ and $k$ be (not necessarily distinct) vertices in $\mathcal{G}$. Decide in finite time whether $$\begin{aligned} \underbrace{\left(\bigcup_{\pi \in \mathcal{W}(k,i)} \mathbf{w}(\pi) + \{ \tau \} \right) }_{\equiv \mathbf{W}_{\tau}(k,i)}\cap \underbrace{\left(\bigcup_{\pi' \in \mathcal{W}(k,j)} \mathbf{w}(\pi')\right)}_{\equiv \mathbf{W}_{0}(k,j)} \neq \emptyset.\end{aligned}$$ Here, $\mathcal{W}(k,i\neq k)$ is the set of directed walks in $\mathcal{G}$ from $k$ to $i$ and $\mathcal{W}(k,i=k)$ is the set of directed walks in $\mathcal{G}$ from $k$ to $i=k$ plus the trivial walk $(i)$; similarly for $\mathcal{W}(k,j)$. The following example illustrates Proposition [Proposition 17](#prop:common-ancestorship-reexpression){reference-type="ref" reference="prop:common-ancestorship-reexpression"} and Problem [Problem 3](#prob1){reference-type="ref" reference="prob1"}. *Example 18*. Consider the ts-DAG in part (a) of Figure [7](#fig:multi_weighted_summary_graph){reference-type="ref" reference="fig:multi_weighted_summary_graph"}. In this graph, $X_{t-4}$ is an ancestor of $X_t$ through the red-colored directed path $\rho = (X_{t-4} {\,\rightarrow\,}X_{t-2} {\,\rightarrow\,}X_t)$ and an ancestor of $Z_t$ through the green-colored directed path $\rho^\prime = (X_{t-4} {\,\rightarrow\,}Y_{t-3} {\,\rightarrow\,}X_{t-1} {\,\rightarrow\,}Y_t {\,\rightarrow\,}Z_t)$. Consequently, $X_{t-4}$ is a common ancestor of $X_t$ and $Z_t$ in the ts-DAG. Now turn to the corresponding multi-weighted summary graph in part (b) of the same figure. In this graph, as parts (c) and (d) of the same figure illustrate, there are the corresponding directed walks $\pi = (X {\,\rightarrow\,}X {\,\rightarrow\,}X)$ and $\pi^\prime = (X {\,\rightarrow\,}Y {\,\rightarrow\,}X {\,\rightarrow\,}Y {\,\rightarrow\,}Z)$. Note that we can respectively map $\rho$ and $\rho^\prime$ to $\pi$ and $\pi^\prime$ by removing the time indices from the vertices on $\rho$ and $\rho^\prime$. For the multi-weights of $\pi$ and $\pi^\prime$ we find $\mathbf{w}(\pi) = 2\cdot \mathbf{w}(X {\,\rightarrow\,}X) = 2 \cdot \{2\} = \{4\}$ and $\mathbf{w}(\pi^\prime) = 2 \cdot \mathbf{w}(X{\,\rightarrow\,}Y) + 2 \cdot \mathbf{w}(Y{\,\rightarrow\,}X) + 2 \cdot \mathbf{w}(Y {\,\rightarrow\,}Z) = 2 \cdot \{2\} + 2 \cdot \{1\} + 1 \cdot \{0, 5\} = \{4, 9\}$. Thus, setting $i = X$, $j = Z$ and $\tau = 0$, we see that $\pi$ and $\pi^\prime$ satisfy all three properties listed in Proposition [Proposition 17](#prop:common-ancestorship-reexpression){reference-type="ref" reference="prop:common-ancestorship-reexpression"} (with $w(\pi) = w(\pi) +\tau = w(\pi^\prime) = 4$). Moreover, setting $k = X$, we see that the set intersection in Problem [Problem 3](#prob1){reference-type="ref" reference="prob1"} is indeed non-empty. ### Solution of Problem [Problem 3](#prob1){reference-type="ref" reference="prob1"} {#subsubsec.solution-to-problem} To express our solution of Problem [Problem 3](#prob1){reference-type="ref" reference="prob1"}, given a tuple $(a_0,a_1,\dots,a_{\mu})$ where $0 \leq \mu < \infty$ and $a_0 \in \mathbb{N}_0$ and $a_\alpha \in \mathbb{N}$ for all $1 \leq \alpha \leq \mu$, we define the affine convex cone over non-negative integers of that tuple as $$\begin{aligned} \label{eq:cones} \mathrm{con}(a_0;a_1,\dots,a_{\mu}) = \left\{ a_0 + \sum_{\alpha=1}^{\mu} n_{\alpha} \cdot a_{\alpha} \ \bigg| \ n_{\alpha} \in \mathbb{N}_0 \text{ for all } 1 \leq \alpha \leq \mu \right\}.\end{aligned}$$ Our solution of Problem [Problem 3](#prob1){reference-type="ref" reference="prob1"} rests on the following lemma that specifies a finite decomposition of the sets $\mathbf{W}_{\tau}(k,i)$ into such cones. **Lemma 19**. *Let $(\mathcal{G},\mathbf{w})$ be a weakly acyclic MWDG, let $\tau \in \mathbb{N}_0$, and let $i$ and $k$ be (not necessarily distinct) vertices in $\mathcal{G}$. Then, there exist *finite* sets* - *$\mathcal{W}_0(k,i)$,* - *$\mathcal{M}_{\pi}$ for every $\pi \in \mathcal{W}_0(k,i)$,* - *and $D_{\tau}(\pi,\mathcal S)$ for every $\pi \in \mathcal{W}_0(k,i)$ and every $\mathcal{S} \in \mathcal{M}_{\pi}$* *such that $$\begin{aligned} \mathbf{W}_{\tau}(k,i) = \bigcup_{\pi\in \mathcal{W}_0(k,i)}\, \bigcup_{\mathcal{S} \in \mathcal{M}_{\pi}} \,\bigcup_{(a_0,\dots,a_{\mu}) \in D_{\tau}(\pi,\mathcal S)}\mathrm{con}(a_0;a_1,\dots,a_{\mu}) \, . \label{eq:main-decomposition}\end{aligned}$$* Here, $\mathcal{W}_0(k, i \neq k )$ is the finite set of cycle-free directed walks from $k$ to $i$ in the finite graph $\mathcal{G}$ and $\mathcal{W}_0(k,i=k) = \{(k)\}$ is the single-element set consisting of the trivial walk $(k)$. As their construction is more involved, we postpone the definitions of the sets $\mathcal{M}_{\pi}$ and $D_{\tau}(\pi,\mathcal S)$ to the proof of Lemma [Lemma 19](#thm.main1){reference-type="ref" reference="thm.main1"} in Section [8.2](#subsec.proof-of-decomposition-theorem){reference-type="ref" reference="subsec.proof-of-decomposition-theorem"}. Instead, we here give a intuitive justification by means of Remark [Remark 20](#remark:theorem-1){reference-type="ref" reference="remark:theorem-1"} (readers can skip this remark without loosing the conceptual flow of this section). *Remark 20*. By definition of the set $\mathbf{W}_{\tau}(k,i)$, we see that $w \in \mathbf{W}_{\tau}(k,i)$ if and only if there is a directed walk $\pi_{cyc}$ from $k$ to $i$ (or, if $k=i$, a trivial walk $\pi_{cyc}$ consisting of $k=i$ only) such that $w = \tau + w(\pi_{cyc})$ with $w(\pi_{cyc}) \in \mathbf{w}(\pi_{cyc})$. We now face the complication that, since the graph $\mathcal{G}$ can be cyclic (recall that the summary graph $\mathcal{S}(\mathcal{D})$ of a ts-DAG $\mathcal{D}$ can be cyclic), there potentially is an infinite number of directed walks. This potential infiniteness is the manifestation of the potentially infinite number of paths that might give rise to common ancestorship in ts-DAGs as faced when approaching the problem in the formulation of Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"}. The idea to work around this complication is as follows: Given a in general cyclic directed walk $\pi_{cyc}$ from $k$ to $i$, we can map $\pi_{cyc}$ to a cycle-free path $\pi \in \mathcal{W}_0(k,i)$ from $k$ to $i$ by "collapsing" the cycles on $\pi_{cyc}$.[^12] Conversely, we can obtain $\pi_{cyc}$ from $\pi$ by "inserting" certain cycles a certain number of times in a certain order. Consequently, there are non-negative integers $n_{1}, \ldots, n_{\mu}$ where $0 \leq \mu < \infty$ such that $\mathbf{w}(\pi_{cyc}) \subseteq \mathbf{w}(\pi) + n_1 \cdot \mathbf{w}(c_1) + \dots + n_\mu \cdot \mathbf{w}(c_\mu)$ where $c_1, \ldots, c_\mu$ are cycles in $\mathcal{G}$. Since we can obtain every cycle by appropriately combining irreducible cycles, we can without loss of generality assume that the $c_1, \ldots, c_\mu$ are irreducible. We now wish to set $a_0 = \tau + w(\pi)$ where $w(\pi) \in \mathbf{w}(\pi)$ and $a_\alpha = w(c_{\alpha})$ where $w(c_{\alpha}) \in \mathbf{w}(c_{\alpha})$ for all $1 \leq \alpha \leq \mu$ and take the union over $\mathrm{con}(a_0;a_1,\dots,a_{\mu})$ for all such tuples $(a_0;a_1,\dots,a_{\mu})$. There are at most finitely many of such tuples because every walk has a finite set of multi-weights, and $a_\alpha \neq 0$ due to weak acyclicity. This choice of tuples is correct if all of the irreducible cycles $c_1, \ldots, c_\mu$ intersect $\pi$. In general, however, some of the $c_1, \ldots, c_\mu$ might not intersect $\pi$. For example, for $\mu = 2$ it could be that $c_2$ intersects $c_1$ but not $\pi$, thus translating to the constraint that $n_2 > 0$ only if $n_1 > 0$. In order to avoid such constraints, we instead extend $\pi$ to the walk $\pi^{ext}$ by "inserting" the cycle $c_1$ once into $\pi$. Then, the above argument applies with $\pi^{ext}$ replacing $\pi$, and we let both $\pi$ and $\pi^{ext}$ be elements of $\mathcal{M}_{\pi}$.[^13] We might need to add other such extensions of $\pi$ to $\mathcal{M}_{\pi}$ and might even need to build extensions of $\pi^{ext}$ (for example, if $\mu = 3$ and $c_3$ intersects $c_2$ but neither of $c_1$ and $\pi$, and $c_2$ intersects $c_1$ but not $\pi$), which too are added to $\mathcal{M}_{\pi}$. However, since there are only finitely many irreducible cycles, this process terminates and the set $\mathcal{M}_{\pi}$ remains finite. To illustrate Lemma [Lemma 19](#thm.main1){reference-type="ref" reference="thm.main1"}, the following examples continue Example [Example 18](#example:running-example-1){reference-type="ref" reference="example:running-example-1"}. *Example 21*. For the multi-weighted summary graph in part (b) of Figure [7](#fig:multi_weighted_summary_graph){reference-type="ref" reference="fig:multi_weighted_summary_graph"}, the set $\mathcal{W}_0(k = X, j = Z)$ contains the element $\tilde{\pi}^\prime = (X {\,\rightarrow\,}Y {\,\rightarrow\,}Z)$. This walk intersects the irreducible cycles $c_1 = (X {\,\rightarrow\,}X)$ and $c_2 = (X {\,\rightarrow\,}Y {\,\rightarrow\,}X)$. We can "insert" these cycles an arbitrary number of times into $\tilde{\pi}^\prime$, say $n^\prime_1$ and $n^\prime_2$ times, to obtain a directed walk $\tilde{\pi}_{n^\prime_1, n^\prime_2}^\prime \in \mathcal{W}(k = X, j = Z)$ from $X$ to $Z$ with $\mathbf{w}(\tilde{\pi}^\prime_{n^\prime_1, n^\prime_2}) = \mathbf{w}(\tilde{\pi}^\prime) + n^\prime_1 \cdot \mathbf{w}(c_1) + n^\prime_2 \cdot \mathbf{w}(c_2)$. We thus see that $\mathbf{w}(\tilde{\pi}_{n^\prime_1, n^\prime_2}^\prime) \in \mathbf{W}_{\tau = 0}(k = X, j = Z)$ for all $(n^\prime_1, n^\prime_2) \in \mathbb{N}_0 \times \mathbb{N}_0$ and, hence, find the inclusion $\mathrm{con}(a_0; a_1, a_2) \subseteq \mathbf{W}_{\tau = 0}(k = X, j = Z)$ for all $(a_0; a_1, a_2) \in (\{\tau = 0\} + \mathbf{w}(\tilde{\pi}^\prime)) \times \mathbf{w}(c_1) \times \mathbf{w}(c_2) = \{1, 6\} \times \{2\} \times \{3\} = \{(1; 2, 3), (6; 2, 3)\}$. Also note that $\tilde{\pi}^\prime_{0, 1} = \pi^\prime$ and $w(\rho^\prime) = 4 \in \mathbf{w}(\pi^\prime) = \mathbf{w}(\tilde{\pi}^\prime_{0, 1})$ with $\rho^\prime = (X_{t-4} {\,\rightarrow\,}Y_{t-3} {\,\rightarrow\,}X_{t-1} {\,\rightarrow\,}Y_t {\,\rightarrow\,}Z_t)$ and $\pi^\prime = (X {\,\rightarrow\,}Y {\,\rightarrow\,}X {\,\rightarrow\,}Y {\,\rightarrow\,}Z)$ as in Example [Example 18](#example:running-example-1){reference-type="ref" reference="example:running-example-1"}. Conversely, $\pi^\prime$ maps to $\tilde{\pi}^\prime$ by "collapsing" the irreducible cycle $c_2 = (X {\,\rightarrow\,}Y {\,\rightarrow\,}X)$ on $\pi^\prime$ to the single vertex $(X)$. The set $\mathcal{W}_0(k = X, i = X)$ contains the trivial walk $\tilde{\pi} = (X)$. Using arguments along the same lines as before, we see that $\mathrm{con}(a_0; a_1, a_2) \subseteq \mathbf{W}_{\tau = 0}(k = X, i = X)$ for all $(a_0; a_1, a_2) \in (\{\tau = 0\} + \mathbf{w}(\tilde{\pi})) \times \mathbf{w}(c_1) \times \mathbf{w}(c_2) = \{0\} \times \{2\} \times \{3\} = \{(0; 2, 3)\}$. The importance of Lemma [Lemma 19](#thm.main1){reference-type="ref" reference="thm.main1"} for our purpose lies in two facts: First, the union on the right-hand-side of eq. [\[eq:main-decomposition\]](#eq:main-decomposition){reference-type="eqref" reference="eq:main-decomposition"} is over a *finite* number of affine cones $\mathrm{con}(a_0;a_1,\dots,a_{\mu})$. Second, while the cones $\mathrm{con}(a_0;a_1,\dots,a_{\mu})$ themselves can be infinite (the cone is infinite if and only if $\mu > 0$), they take the particular form as specified by the right-hand-side of eq. [\[eq:cones\]](#eq:cones){reference-type="eqref" reference="eq:cones"}. Intuitively speaking, we have thus found regularity in the potentially infinite set $\mathbf{W}_{\tau}(k,i)$. Utilizing this regularity, we can solve Problem [Problem 3](#prob1){reference-type="ref" reference="prob1"} as follows. **Theorem 1** (Solution of Problem [Problem 3](#prob1){reference-type="ref" reference="prob1"}). *Let $(\mathcal{G},\mathbf{w})$ be a weakly acylic MWDG, let $\tau \in \mathbb{N}_0$ be a non-negative integer, and let $i, j, k$ be (not necessarily distinct) vertices in $\mathcal{G}$. Then, $$\begin{aligned} \mathbf{W}_{\tau}(k,i) \cap \mathbf{W}_{0}(k,j) \neq \emptyset\end{aligned}$$ if and only if there exist* - *$(a_0,\dots,a_{\mu}) \in D_{\tau}(\pi,\mathcal S)$ where $\mathcal{S} \in \mathcal{M}_{\pi}$ and $\pi \in \mathcal{W}_0(k,i)$, as well as* - *$(a_0',\dots,a_{\nu}') \in D_{0}(\pi',\mathcal S')$ where $\mathcal{S}' \in \mathcal{M}_{\pi'}$ and $\pi' \in \mathcal{W}_0(k,j)$* *such that the linear Diophantine equation $$\begin{aligned} \label{eq.new_Diophantine} a_0 + \sum_{\alpha=1}^{\mu} n_{\alpha} \cdot a_{\alpha} = a'_0 + \sum_{\beta=1}^{\nu} n'_{\beta} \cdot a'_{\beta},\end{aligned}$$ has a non-negative integer solution $(n_1,\dots,n_{\mu};n'_1,\dots,n'_{\nu})\in \mathbb{N}_0^{\mu}\times \mathbb{N}_0^{\nu}$. Moreover, we can decide about the existence versus non-existence of such a solution in finite time.* Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"} reduces Problem [Problem 3](#prob1){reference-type="ref" reference="prob1"}, and by extension Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"} (that is, the common-ancestor search in infinite ts-DAGs) and Problem [Problem 1](#problem:eventual-goal){reference-type="ref" reference="problem:eventual-goal"} (that is, the construction of finite marginal ts-ADMGs of infinite ts-ADMGs), to a number theoretic problem: deciding whether one of the finitely many linear Diophantine equations in Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"} admits a solution that consists entirely of *non-negative* integers. Theorem [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"}, in Section [4.3.3](#sec:solvability-check){reference-type="ref" reference="sec:solvability-check"} below, shows how to make these decisions in finite time. Combining Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"} and Theorem [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"} thus solves Problem [Problem 3](#prob1){reference-type="ref" reference="prob1"} and, by extension, also Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"} and Problem [Problem 1](#problem:eventual-goal){reference-type="ref" reference="problem:eventual-goal"}. In Section [9](#sec:pseudocode){reference-type="ref" reference="sec:pseudocode"}, we also provide pseudocode for this solution. *Proof of Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"}.* Using the decomposition of $\mathbf{W}_{\tau}(k,i)$ and $\mathbf{W}_{0}(k,j)$ as stated by Lemma [Lemma 19](#thm.main1){reference-type="ref" reference="thm.main1"}, we see that $$\begin{aligned} \mathbf{W}_{\tau}(k,i) \cap \mathbf{W}_{0}(k,j) \neq \emptyset\end{aligned}$$ if and only if there are $(a_0,\dots,a_{\mu}) \in D_{\tau}(\pi,\mathcal S)$ where $\mathcal{S} \in \mathcal{M}_{\pi}$ and $\pi \in \mathcal{W}_0(k,i)$ as well as $(a_0',\dots,a_{\nu}') \in D_{0}(\pi',\mathcal S')$ where $\mathcal{S}' \in \mathcal{M}_{\pi'}$ and $\pi^\prime \in \mathcal{W}_0(k,j)$ such that $$\begin{aligned} \mathrm{con}(a_0;a_1,\dots,a_{\mu}) \cap \mathrm{con}(a'_0;a'_1,\dots,a'_{\nu}) \neq \emptyset \, .\end{aligned}$$ By definition of the affine cones $\mathrm{con}(a_0;a_1,\dots,a_{\mu})$ and $\mathrm{con}(a'_0;a'_1,\dots,a'_{\nu})$, their intersection is non-empty if and only if eq. [\[eq.new_Diophantine\]](#eq.new_Diophantine){reference-type="eqref" reference="eq.new_Diophantine"} has a solution in $\mathbb{N}_0^{\mu}\times \mathbb{N}_0^{\nu}$. Theorem [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"} in Section [4.3.3](#sec:solvability-check){reference-type="ref" reference="sec:solvability-check"} below shows the second part of Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"}, namely that we can answer these solvability queries in finite time. ◻ To illustrate Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"}, the following examples continues Examples [Example 18](#example:running-example-1){reference-type="ref" reference="example:running-example-1"} and [Example 21](#example:running-example-2){reference-type="ref" reference="example:running-example-2"}. *Example 22*. In Example [Example 21](#example:running-example-2){reference-type="ref" reference="example:running-example-2"} above, we found that $\mathrm{con}(a_0; a_1, a_2) \subseteq \mathbf{W}_{\tau = 0}(k = X, i = X)$ with $(a_0; a_1, a_2) = (0; 2, 3)$ and $\mathrm{con}(a^\prime_0; a^\prime_1, a^\prime_2) \subseteq \mathbf{W}_{\tau = 0}(k = X, j = Z)$ with $(a^\prime_0; a^\prime_1, a^\prime_2) = (1; 2, 3)$. This combination of tuples gives rise to the linear Diophantine equation $$0 + n_1 \cdot 2 + n_2 \cdot 3 = 1 + n_1^\prime \cdot 2 + n_2^\prime \cdot 3 \, ,$$ which has the non-negative integer solution $(n_1, n_2; n_1^\prime, n_2^\prime) = (2, 0; 0, 1)$ that corresponds to the common ancestor $X_{t-4}$ of $X_t$ and $Z_t$ in the ts-DAG in part (a) of Figure [7](#fig:multi_weighted_summary_graph){reference-type="ref" reference="fig:multi_weighted_summary_graph"}, cf. Example [Example 18](#example:running-example-1){reference-type="ref" reference="example:running-example-1"}. Further solutions are $(n_1, n_2; n_1^\prime, n_2^\prime) = (2 + n, 0; n, 1)$ and $(n_1, n_2; n_1^\prime, n_2^\prime) = (n, 1; 1+n, 0)$ for all $n \in \mathbb{N}_0$, from which we conclude that $X_{t-\tau^\prime}$ is a common ancestor of $X_t$ and $Z_t$ for all $\tau^\prime \in \{3, 4, 5, \ldots, \} \subseteq \mathbf{W}_{\tau=0}(k=X,i=X) \cap \mathbf{W}_{0}(k=X,j=Z)$. We remark that we discuss these explicit solutions solely for illustration, whereas for Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"} we only need to know whether at least one non-negative integer solution exists. ### Existence of non-negative integers solutions of linear Diophantine equations {#sec:solvability-check} Our solution of Problem [Problem 3](#prob1){reference-type="ref" reference="prob1"} through Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"} depends on a method for deciding whether a linear Diophantine equation admits a solution that consists of non-negative integers only. Importantly, we only need to decide whether a solution *exists or not*. If a solution exists, then we *do not necessarily need to explicitly find a solution*, although in some subcases we will do so. Not having to find explicit solutions reduces the required computational effort significantly. To derive conditions for deciding whether or not eq. [\[eq.new_Diophantine\]](#eq.new_Diophantine){reference-type="eqref" reference="eq.new_Diophantine"} admits a non-negative integer solution, we make use of the following two standard results from the number theory literature. **Lemma 23** (See for example @andreescu_introduction_2010). *Let $a_0$ be an integer and let $a_1, \, \ldots, \, a_{\rho}$ with $\rho \geq 1$ be non-zero integers. Then, the linear Diophantine equation $a_0 = \sum_{\alpha \, = \, 1}^\rho n_{\alpha} \cdot a_\alpha$ has an integer solution $(n_1, \, \ldots, \, n_\rho) \in \mathbb Z^{\rho}$ if and only if $a_0 \!\!\mod \!\gcd(a_1, \, \ldots, \, a_\rho) = 0$ where $\gcd(\cdot)$ denotes the greatest common divisor. $\qedsymbol$* **Lemma 24** (See for example @ramirez_alfonsin_diophantine_2005). *Let $\rho \geq 1$ and let $a_1, \, \ldots, \, a_{\rho}$ be positive integers with $\gcd(a_1, \, \ldots, \, a_\rho) = 1$. Then, there is a unique largest integer $f(a_1, \, \ldots, \, a_{\rho})$, known as the *Frobenius number* of $a_1, \, \ldots, \, a_{\rho}$, such that there are no non-negative integers $n_1, \, \ldots, \, n_\rho$ with $\sum_{\alpha \, = \, 1}^\rho n_{\alpha} \cdot a_\alpha= f(a_1, \, \ldots, \, a_{\rho})$. In particular, for all integers $a_0 > f(a_1, \, \ldots, \, a_{\rho})$ *there are* non-negative integers $n_1, \, \ldots, \, n_\rho$ with $a_0 = \sum_{\alpha \, = \, 1}^\rho n_{\alpha} \cdot a_\alpha$. $\qedsymbol$* By itself, Lemma [Lemma 23](#lemma:linear-diophantine-general){reference-type="ref" reference="lemma:linear-diophantine-general"} is not sufficient to decide whether eq. [\[eq.new_Diophantine\]](#eq.new_Diophantine){reference-type="eqref" reference="eq.new_Diophantine"} has a non-negative integer solution since Lemma [Lemma 23](#lemma:linear-diophantine-general){reference-type="ref" reference="lemma:linear-diophantine-general"} deals with all integer solutions rather than only the non-negative ones. However, in combination with Lemma [Lemma 24](#lemma:frobenius-number){reference-type="ref" reference="lemma:frobenius-number"} we arrive at the following result. **Theorem 2**. *Consider the linear Diophantine equation [\[eq.new_Diophantine\]](#eq.new_Diophantine){reference-type="eqref" reference="eq.new_Diophantine"} and write $c = a_0-a_0^\prime$, $g_a = \gcd(a_1, \, \ldots, \, a_\mu)$, $g_{a^\prime} = \gcd(a^\prime_1, \, \ldots, \, a^\prime_\nu)$ and $g_{aa^\prime} = \gcd(g_a,g_{a^\prime})$. Then, the following mutually exclusive and collectively exhaustive cases answer the question whether this equation has at least one non-negative integer solution for the unknowns $n_1, \, \ldots, \, n_\mu, \, n^\prime_1, \, \ldots, \, n^\prime_\nu$:* 1. *[$\mu = 0$ and $\nu = 0$.]{.ul} There is a non-negative integer solution if and only if $c = 0$.* 2. *[$\mu = 0$ and $\nu \neq 0$ and $c > 0$.]{.ul} If $c \!\! \mod \! g_{a^\prime} \neq 0$, then there is no non-negative integer solution. If $c \!\! \mod \! g_{a^\prime} = 0$, then there is a non-negative integer solution if and only if there is a solution within the finite search space $\{0, \, \ldots, \lfloor{\tfrac{c}{a^\prime_1}}\rfloor\} \times \dots \times \{0, \, \ldots, \lfloor{\tfrac{c}{a^\prime_\nu}}\rfloor\}$.* 3. *[($\mu = 0$ and $\nu \neq 0$ and $c \leq 0$) or ($\mu \neq 0$ and $\nu = 0$ and $c \geq 0$).]{.ul} There is a non-negative integers solution if and only if $c = 0$.* 4. *[$\mu \neq 0$ and $\nu = 0$ and $c < 0$.]{.ul} If $-c \!\! \mod \! g_a \neq 0$, then there is no non-negative integer solution. If $-c \!\! \mod \! g_a = 0$, then there is a non-negative integer solution if and only if there is a solution within the finite search space $\{0, \, \ldots, \lfloor{-\tfrac{c}{a_1}}\rfloor\} \times \dots \times \{0, \, \ldots, \lfloor{-\tfrac{c}{a_\mu}}\rfloor\}$.* 5. *[$\mu \neq 0$ and $\nu \neq 0$.]{.ul} There is a non-negative integer solution if and only $c \! \! \mod g_{aa^\prime} = 0$.* *Remark 25*. In the if-and-only-if subcases of cases 2 and 4 of Theorem [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"}, it is not always necessary to run an explicit search for a solution within the described finite search spaces. Focusing on case 2 because case 4 is similar, if $c \!\! \mod \! g_{a^\prime} = 0$ and moreover $c \geq g_{a^\prime} \cdot f^\prime(\tfrac{a^\prime_1}{g_{a^\prime}} , \, \ldots, \, \tfrac{a^\prime_\nu}{g_{a^\prime}} )$ where $f^\prime(d_1, \, \ldots, \, d_\rho) = \left( \min_\alpha d_{\alpha} -1 \right) \cdot \left( \max_\alpha d_{\alpha} -1 \right)$, then there always exists a non-negative integer solution and an explicit search can thus be avoided. This claim holds because of the upper bound $f^\prime(d_1, \, \ldots, \, d_\rho) \geq f(d_1, \, \ldots, \, d_\rho) + 1$ on the Frobenius number [@brauer_problem_1942] that is attributed to Schur and Brauer. Theorem [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"} immediately translates into a finite-time algorithm for deciding whether or not the linear Diophantine equation [\[eq.new_Diophantine\]](#eq.new_Diophantine){reference-type="eqref" reference="eq.new_Diophantine"} has a non-negative integer solution. Of the five cases in Theorem [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"}, cases 2 and 4 are the computationally most expensive ones as they potentially require to run an explicit solution search within the specified finite search spaces; for example, in case 2, whether the reduced equation $c = \sum_{\beta \,=\, 1}^{\nu} n^\prime_\beta \cdot a^\prime_\beta$ has a solution within the finite search space $\{0, \, \ldots, \lfloor{\tfrac{c}{a^\prime_1}}\rfloor\} \times \dots \times \{0, \, \ldots, \lfloor{\tfrac{c}{a^\prime_\nu}}\rfloor\}$. One can, in principle, perform this explicit search in a brute-force way. However, using that the search for a non-integer solution to the reduced equation is a special case of the *subset sum problem* [@bringmann_near-linear_2017], there are also more refined search algorithms. The subset sum problem is a well-studied NP-complete combinatorial problem that dynamic programming algorithms can solve in pseudo-polynomial time [@pisinger_linear_1999]. For example, the [R]{.sans-serif}-package [nilde]{.sans-serif} [@arnqvist2019nilde] provides an implementation of such a more refined algorithm. Moreover, in order to altogether avoid the explicit searches if possible, one can use Remark [Remark 25](#rem.search-avoiding-condition){reference-type="ref" reference="rem.search-avoiding-condition"} as follows: Focusing on case 2, if the necessary condition $c \!\! \mod \! g_{a^\prime} = 0$ is met, then one can subsequently check the sufficient condition $c \geq g_{a^\prime} \cdot f^\prime(\tfrac{a^\prime_1}{g_a^\prime}, \, \ldots, \, \tfrac{a^\prime_\nu}{g_{a^\prime}})$ before moving to the explicit search. If this sufficient condition is met, then there is a non-negative integer solution and one does not need to run the explicit search at all.[^14] To illustrate Theorem [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"} and continue with our running example, the following example continues Examples [Example 18](#example:running-example-1){reference-type="ref" reference="example:running-example-1"}, [Example 21](#example:running-example-2){reference-type="ref" reference="example:running-example-2"} and [Example 22](#example:running-example-3){reference-type="ref" reference="example:running-example-3"}. *Example 26*. In Example [Example 22](#example:running-example-3){reference-type="ref" reference="example:running-example-3"}, we considered the linear Diophantine equation $$0 + n_1 \cdot 2 + n_2 \cdot 3 = 1 + n_1^\prime \cdot 2 + n_2^\prime \cdot 3 \, .$$ Comparing with the general form of eq. [\[eq.new_Diophantine\]](#eq.new_Diophantine){reference-type="eqref" reference="eq.new_Diophantine"}, we see that $\mu = 2$ and $\nu = 2$, so case 5 of Theorem [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"} applies. Since $c = a_0 - a_0^\prime = 0 - 1 = -1$ and $g_{aa^\prime} = \gcd(\gcd(a_1, a_2), \gcd(a_1^\prime, a_2^\prime)) = \gcd(\gcd(2, 3), \gcd(2, 3)) = 1$, the condition $c \! \! \mod g_{aa^\prime} = 0$ is fulfilled, from which we re-discover that the equation has a non-negative integer solution. Note that, to draw this conclusion, there is no need to explicitly find a solution. ### A formula for the cutoff point {#sec:formula-cutoff} As a corollary to Proposition [Proposition 17](#prop:common-ancestorship-reexpression){reference-type="ref" reference="prop:common-ancestorship-reexpression"}, Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"}, Theorem [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"} and Remark [Remark 25](#rem.search-avoiding-condition){reference-type="ref" reference="rem.search-avoiding-condition"}, we can now derive a formula for a finite cutoff point $p^\prime(\mathcal{D}) < \infty$ with the property that one can restrict the common-ancestor search in ts-DAGs $\mathcal{D}$ to the finite interval $[t-p^\prime(\mathcal{D}),t]$ (cf. the discussion in Section [4.2.1](#subsec:summary-graph-insufficient){reference-type="ref" reference="subsec:summary-graph-insufficient"}). **Theorem 3**. *Let $\mathcal{D}= (\mathbf{I}\times \mathbb{Z}, \mathbf{E}_{\rightarrow})$ be a ts-DAG and let $\mathcal{S}_{\mathbf{w}}(\mathcal{D}) = (\mathcal{S}(\mathcal{D}), \mathbf{w})$ be its multi-weighted summary graph. Denote the (finite) set of equivalence classes of irreducible cycles in $\mathcal{S}(\mathcal{D})$ as $\mathcal{C}$ and define the quantities $$\begin{aligned} &K = \max_{\mathbf{c} \in \mathcal{C}} \max \mathbf{w}(\mathbf{c}) \,\, &&\text{(maximal weight of any irreducible cycle in $\mathcal{S}(\mathcal{D})$)\, ,} \\ &L = \max_{k,i \in \mathbf{I}} \max_{\pi\in \mathcal{W}_0(k,i)} \max \mathbf{w}(\pi) \,\, &&\text{(maximal weight of any directed or trivial path in $\mathcal{S}(\mathcal{D})$)\, ,} \\ &M = \sum_{\mathbf{c} \in \mathcal{C}} \max \mathbf{w}(\mathbf{c}) \,\, &&\text{(sum over the maximal weights of all irreducible cycles)\, .} \end{aligned}$$ Note that the multi-weight of an equivalence class of cycles is well-defined since any representative of the class has the same multi-weight.* *Let $(i,t-\tau)$ and $(j,t)$ with $0 \leq \tau \leq p$ be two vertices in $\mathcal{D}$. Then, $(i,t-\tau)$ and $(j,t)$ have a common ancestor in the infinite ts-DAG $\mathcal{D}$ if and only if $(i,t-\tau)$ and $(j,t)$ have a common ancestor in the finite segment of $\mathcal{D}$ on the time window $[t-p^\prime(\mathcal{D}), t]$ where $$\begin{aligned} \label{eq:cutoff} p^\prime(\mathcal{D}) = \left(K^2+1\right) \cdot \left(p+ L+M \right) + K\cdot\left[\left(K-1\right)^2 +1\right] \, .\end{aligned}$$* Theorem [Theorem 3](#thm.upper-bound-main-theorem){reference-type="ref" reference="thm.upper-bound-main-theorem"} provides a direct solution to Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"} and, by extension, to Problem [Problem 1](#problem:eventual-goal){reference-type="ref" reference="problem:eventual-goal"}. Specifically, given a ts-ADMG $\mathcal{A}$, we can determine its marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{A})$ by the equality $\mathcal{A}_{\mathbf{O}}(\mathcal{A}) = \sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A}_{[t-p^\prime(\mathcal{D})-p, t]})$ where $\mathcal{A}_{[t-p^\prime(\mathcal{D})-p, t]}$ is the finite segment of $\mathcal{A}$ on the finite time window $[t-p^\prime(\mathcal{D})-p, t]$ with $p^\prime(\mathcal{D})$ as in eq. [\[eq:cutoff\]](#eq:cutoff){reference-type="eqref" reference="eq:cutoff"}. Since $\mathcal{A}_{[t-p^\prime(\mathcal{D}), t]}$ is a finite graph, its projection to $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A}_{[t-p^\prime(\mathcal{D}), t]})$ is a solved problem. While conceptually simple, we expect this approach of using Theorem [Theorem 3](#thm.upper-bound-main-theorem){reference-type="ref" reference="thm.upper-bound-main-theorem"} to be computationally more expensive than the approach of using Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"} and Theorem [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"}. The rational behind this expectation is that the bound $p^\prime(\mathcal{D})$ is rather rough, such that $\mathcal{A}_{[t-p^\prime(\mathcal{D})-p, t]}$ can potentially be large. *Example 27*. Consider once more the ts-DAG $\mathcal{D}$ and its multi-weighted summary graph in parts (a) and (b) of Figure [7](#fig:multi_weighted_summary_graph){reference-type="ref" reference="fig:multi_weighted_summary_graph"}. There are exactly two equivalence classes of irreducible cycles, namely $\mathbf{c}_1 = [X {\,\rightarrow\,}X]$ and $\mathbf{c}_2 = [X {\,\rightarrow\,}Y {\,\rightarrow\,}X]$. The multi-weights of these equivalence classes are $\mathbf{w}(\mathbf{c}_1) = \{2\}$ and $\mathbf{w}(\mathbf{c}_2) = \{3\}$, such that $K = 3$ and $M = 5$. Moreover, $L = 6$ corresponding to the directed path $\pi = (X {\,\rightarrow\,}Y {\,\rightarrow\,}Z)$ with $6 \in \mathbf{w}(\pi) = \{1, 6\}$. Inserting these numbers into eq. [\[eq:cutoff\]](#eq:cutoff){reference-type="eqref" reference="eq:cutoff"}, we get $p^\prime(\mathcal{D}) = 10\cdot p + 125$. # Conclusions {#sec:summary} #### Summary {#summary .unnumbered} In this paper, we considered the projection of infinite time series graphs with latent confounders (infinite ts-ADMGs, see Definition [Definition 2](#def:tsADMG-infinite){reference-type="ref" reference="def:tsADMG-infinite"}) to marginal graphs on finite time windows by means of the ADMG latent projection (finite marginal ts-ADMGs, see Definition [Definition 3](#def:tsADMG-marginal){reference-type="ref" reference="def:tsADMG-marginal"}). While the projection procedure itself is not new, its practical execution on infinite graphs is non-trivial and had previously not been approached in generality. To close this conceptual gap, we first reduced the considered projection task (see Problem [Problem 1](#problem:eventual-goal){reference-type="ref" reference="problem:eventual-goal"}) to the search for common ancestors in infinite ts-DAGs (see Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"}). We then further reduced this common-ancestor search to the task of deciding whether any of a finite number of linear Diophantine equations has a non-negative integer solution (see Problem [Problem 3](#prob1){reference-type="ref" reference="prob1"} and Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"}). Thus, we established an intriguing connection between the theory of infinite graphs with repetitive edges and number theory. Building on standard results from number theory, we then derived criteria with which one can answer the corresponding solvability queries in finite time (see Theorem [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"}). Thus, by the combination of Theorems [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"} and [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"}, we provided a solution to both the common-ancestor search in infinite ts-DAGs (Problem [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"}) and the task of constructing finite marginal ts-ADMGs (Problem [Problem 1](#problem:eventual-goal){reference-type="ref" reference="problem:eventual-goal"}). In Section [9](#sec:pseudocode){reference-type="ref" reference="sec:pseudocode"}, we also provide pseudocode that implements this solution. As a corollary to this solution, we derived a finite upper bound on a time window to which one can restrict the common ancestor searches relevant for the projection task (see Theorem [Theorem 3](#thm.upper-bound-main-theorem){reference-type="ref" reference="thm.upper-bound-main-theorem"}). This result constitutes an alternative and conceptually simple solution to Problems [Problem 1](#problem:eventual-goal){reference-type="ref" reference="problem:eventual-goal"} and [Problem 2](#problem:common-ancestor){reference-type="ref" reference="problem:common-ancestor"}, but we expect it to be computationally disadvantageous as compared to the number theoretic solution by means of Theorems [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"} and [Theorem 2](#thm:our-linear-diophantine){reference-type="ref" reference="thm:our-linear-diophantine"}. In Section [6](#sec:generalize-to-DMAGs){reference-type="ref" reference="sec:generalize-to-DMAGs"}, we further show how to execute the DMAG latent projection of infinite ts-ADMGs by utilizing the finite marginal ts-ADMGs. #### Significance {#significance .unnumbered} The finite marginal graphs are useful tools for answering $m$-separation queries in infinite time series graphs as well as for causal effect identification and causal discovery in time series, see Section [3.1](#sec:projection-motivation){reference-type="ref" reference="sec:projection-motivation"}. In particular, provided the causal Markov condition holds with respect to the infinite time series graph, the entirety of causal effect identification results for finite graphs directly applies to finite marginal graphs, whereas specific modifications might be necessary for applying these methods to infinite time series graphs. Therefore, we envision our results to be widely applicable in future research on causal inference in time series. #### Limitations {#limitations .unnumbered} As we stated in the previous paragraph, the applicability of causal effect estimation methods to the finite marginal graphs is contingent on the causal Markov condition holding with respect to the infinite time series graphs. Moreover, the projection to the finite marginal graphs inherently comes with the choice of an observed time window length, and any derived statement about (non-)identifiability with respect to the finite marginal graphs will be contingent on that choice. Lastly, our projection methods make the assumption of causal stationarity (note, however, that we really need this assumption only for the "half-infinite" graph that extends from the infinite past to some arbitrary finite time step). # Acknowledgments {#acknowledgments .unnumbered} The authors thank Christoph Käding for helpful discussions at early stages of this project. J.W., U.N., and J.R. received funding from the European Research Council (ERC) Starting Grant CausalEarth under the European Union's Horizon 2020 research and innovation program (Grant Agreement No. 948112). S.F. was enrolled at Technische Universität Berlin while working on this project. # Generalization to the DMAG latent projection {#sec:generalize-to-DMAGs} In the main paper, we considered the projection of infinite ts-ADMGs (see Definition 2.2) to finite marginal ts-ADMGs (see Definition 3.1) by means of the ADMG latent projection (see Definition 2.1, due to @pearl1995theory, see also for example @richardson2023nested). Here, we extend our results to the DMAG latent projection [@richardson2002ancestral; @zhang2008causal], which is another widely-used projection procedure for representing causal knowledge in the presence of unobserved confounders. *Definition 28* (DMAG latent projection [@richardson2002ancestral; @zhang2008causal]). Let $\mathcal{A}$ be an ancestral ADMG with vertex set $\mathbf{V}= \mathbf{O}\,\dot{\cup}\, \mathbf{L}$. Then, its *marginal DMAG $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{A})$ on $\mathbf{O}$* is the bidirected graph with vertex set $\mathbf{O}$ such that 1. there is an edge $i {\ast\!{-}\!\ast}j$ in $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{A})$ if and only if $i \neq j$ and there is no set $\mathbf{Z}\subseteq \mathbf{O}\setminus \{i, j\}$ that $m$-separates $i$ and $j$ in $\mathcal{A}$; 2. an edge $i {\ast\!{-}\!\ast}j$ in $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{A})$ is of the form $i {\,\rightarrow\,}j$ if and only if $i \in an(j, \mathcal{A})$ and, thus, $i {\,\leftrightarrow\,}j$ if and only if $i \notin an(j, \mathcal{A})$ and $j \notin an(i, \mathcal{A})$. It follows that $i \in an(j, \sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{A}))$ if and only if $i, j \in \mathbf{O}$ and $i \in an(j, \mathcal{A})$, so $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{A})$ is an ADMG. The definition also readily implies that $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{A})$ is ancestral, and @richardson2002ancestral shows that in $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{A})$ there is no inducing path $\pi$ between non-adjacent vertices. Since a *directed maximal ancestral graph (DMAG)* by definition is an ancestral ADMG that does not have inducing paths between non-adjacent vertices [@richardson2002ancestral; @mooij2020constraint], we thus see that $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{A})$ is a DMAG indeed. Moreover, two observed vertices $i$ and $j$ are $m$-separated given $\mathbf{Z}$ in $\mathcal{A}$ if and only if $i$ and $j$ are $m$-separated given $\mathbf{Z}$ in $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{A})$, see Theorem 4.18 in @richardson2002ancestral. For an explanation of the difference between the ADMG and DMAG latent projections see, for example, Section 3.3 of @triantafillou2015constraint. In complete analogy to Definition 3.1, we now define *marginal time series DMAGs* [@gerhardus2021characterization] as DMAG latent projections of infinite ts-ADMGs. *Definition 29* (Marginal time series DMAG, generalizing Definition 3.6 of [@gerhardus2021characterization]). Let $\mathcal{A}$ be a ts-ADMG with variable index set $\mathbf{I}$, let $\mathbf{I}_{\mathbf{O}}\subseteq \mathbf{I}$ be non-empty, let $\mathbf{T}_{\mathbf{O}}$ be $\mathbf{T}_{\mathbf{O}}= \{t-\tau ~\vert~ 0 \leq \tau \leq p\}$ and let $\mathbf{O}= \mathbf{I}_{\mathbf{O}}\times \mathbf{T}_{\mathbf{O}}$. Then, its *marginal time series DMAG (marginal ts-DMAG) $\mathcal{M}_{\mathbf{O}}(\mathcal{A})$ on $\mathbf{O}$* is the DMAG $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{D}^{ts}_{c}(\mathcal{A}))$, where $\mathcal{D}^{ts}_{c}(\mathcal{A})$ is the (infinite) canonical ts-DAG of $\mathcal{A}$. *Remark 30*. [@gerhardus2021characterization] calls marginal ts-DMAGs simply "ts-DMAGs" (that is, does not use the attribute "marginal"). However, since in the main paper we use the term "marginal ts-ADMG" to distinguish these finite graphs from the infinite ts-ADMGs, we here use the terminolgy "marginal ts-DMAGs" for consistency. Moreover, Definition 3.6 of [@gerhardus2021characterization] only applies to the special case of infinite ts-DAGs instead of infinite ts-ADMGs. Noting that $\mathcal{D}^{ts}_{c}(\mathcal{A}) = \mathcal{A}$ if $\mathcal{A}$ is a ts-DAG, we see that Definition [Definition 29](#def:tsDMAG){reference-type="ref" reference="def:tsDMAG"} is indeed a proper generalization of Definition 3.6 of [@gerhardus2021characterization]. Lastly, we need to define $\mathcal{M}_{\mathbf{O}}(\mathcal{A})$ as $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{D}^{ts}_{c}(\mathcal{A}))$ rather than as $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{A})$ because Definition [Definition 28](#def:DMAG-latent-projection){reference-type="ref" reference="def:DMAG-latent-projection"} requires the input graph to be *ancestral*, and hence $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{A})$ is in general formally undefined. In analogy to marginal ts-ADMGs, the construction of marginal ts-DMAGs is non-trivial because there might be infinitely many paths in the infinite ts-ADMG $\mathcal{A}$ that could potentially induce an edge $(i, t-\tau_i) {\ast\!{-}\!\ast}(j, t-\tau_j)$ in the finite marginal ts-DMAG $\mathcal{M}_{\mathbf{O}}(\mathcal{A})$. To the authors' knowledge, the construction of finite marginal ts-DMAGs has not yet been solved in the literature. Here, as a corollary to the results of the main paper, we solve this non-trival task by means of the following result. **Proposition 31**. *Let $\mathcal{A}$ be an infinite ts-ADMG with variable index set $\mathbf{I}$, and let $\mathbf{O}= \mathbf{I}_{\mathbf{O}}\times \mathbf{T}_{\mathbf{O}}$ with $\mathbf{I}_{\mathbf{O}}\subseteq \mathbf{I}$ non-empty and $\mathbf{T}_{\mathbf{O}}= \{t-\tau ~\vert~ 0 \leq \tau \leq p\}$ where $p< \infty$. Then, $\mathcal{M}_{\mathbf{O}}(\mathcal{A}) = \sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{D}_{c}(\mathcal{A}_{\mathbf{O}}(\mathcal{A})))$ where $\mathcal{D}_{c}(\mathcal{A}_{\mathbf{O}}(\mathcal{A}))$ is the (finite) canonical DAG of the finite marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{A})$ of $\mathcal{A}$.* *Remark 32*. Proposition [Proposition 31](#prop:get-ts-DMAG){reference-type="ref" reference="prop:get-ts-DMAG"} is similar to a known relation between the ADMG and DMAG latent projections as specified, for example, by Theorem 13 and Algorithm 1 of [@triantafillou2015constraint]. Note that $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{A}_{\mathbf{O}}(\mathcal{A}))$ is in general ill-defined because $\mathcal{A}_{\mathbf{O}}(\mathcal{A})$ need not be ancestral, hence we use $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{D}_{c}(\mathcal{A}_{\mathbf{O}}(\mathcal{A})))$ instead. Importantly, the results of the main paper enable us to algorithmically construct the finite marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{A})$, and the construction of its finite canonical DAG $\mathcal{D}_{c}(\mathcal{A}_{\mathbf{O}}(\mathcal{A}))$ as well as the projection of this finite graph to $\sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{D}_{c}(\mathcal{A}_{\mathbf{O}}(\mathcal{A})))$ by means of the DMAG latent projection are solved problems. Thus, we solved the task of algorithmically constructing finite marginal ts-DMAGs. Lastly, similar to the discussion in Section 3.3.4, we can further project $\mathcal{M}_{\mathbf{O}}(\mathcal{A})$ to $\sigma_{\text{DMAG}}(\mathbf{O}^{\prime\prime}, \mathcal{M}_{\mathbf{O}}(\mathcal{A})) = \sigma_{\text{DMAG}}(\mathbf{O}^{\prime\prime}, \mathcal{A})$ with $\mathbf{O}^{\prime\prime} \subseteq \mathbf{O}$ arbitrary. Hence, we also solved the construction arbitrary finite DMAG latent projections $\sigma_{\text{DMAG}}(\mathbf{O}^{\prime\prime}, \mathcal{A})$ of infinite ts-ADMGs. # Counterexamples {#sec:counterexamples} Here, we provide the counterexamples to the simple heuristics considered in Section 4.2.2. *Example 33*. Consider the infinite ts-DAG $\mathcal{D}$ in part (a) of Figure [8](#fig:heuristic_sums){reference-type="ref" reference="fig:heuristic_sums"}, which for $\mathbf{I}_{\mathbf{O}}= \mathbf{I}= \{X, Y\}$ and $p= 1$ projects to the finite marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{D})$ in part (b) of the same figure. We show that this case is a counterexample to the first of the two heuristics considered in Section 4.2.2. To this end, we first note that this heuristic prescribes restricting to the time window $[t-10, t]$, where $10$ is the observed time window length $p= 1$ plus the sum of lags $p^\prime_{sum}(\mathcal{D}) = 9 = 5 + 3 + 1$ of all edges in $\mathbf{E}_{\rightarrow}^t \cup \mathbf{E}_{\leftrightarrow}^t = \{X_{t-5} {\,\rightarrow\,}X_t, \, Y_{t-3} {\,\rightarrow\,}Y_t, \, Y_{t-1} {\,\rightarrow\,}X_t\}$. Now consider the blue-colored vertices $X_{t-1}$ and $Y_t$. These vertices have the common ancestor $Y_{t-12}$ by means of the path $X_{t-1} {\,\leftarrow\,}X_{t-6} {\,\leftarrow\,}X_{t-11} {\,\leftarrow\,}Y_{t-12} {\,\rightarrow\,}Y_{t-9}{\,\rightarrow\,}Y_{t-6} {\,\rightarrow\,}Y_{t-3} {\,\rightarrow\,}Y_t$, but no common ancestor within the time window $[t-10, t]$. Thus, while there is the bidirected edge $X_{t-1} {\,\leftrightarrow\,}Y_t$ in the marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{D})$, this edge is not in the ADMG latent projection $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{D}_{[t-10, t]})$ of the segment $\mathcal{D}_{[t-10, t]}$ of $\mathcal{D}$ on $[t-10, t]$ as shown in part (c) of Figure [8](#fig:heuristic_sums){reference-type="ref" reference="fig:heuristic_sums"}. Also the edges $X_{t-1} {\,\leftrightarrow\,}X_t$ and $Y_{t-1} {\,\leftrightarrow\,}X_t$ are in $\mathcal{A}_{\mathbf{O}}(\mathcal{D})$ but not in $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{D}_{[t-10, t]})$. ![**(a)** A ts-DAG $\mathcal{D}$. **(b)** The marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{D})$ for $\mathbf{I}_{\mathbf{O}}= \mathbf{I}$ and $p= 1$. **(c)** The ADMG latent projection $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{D}_{[t-10, t]})$ of the finite segment $\mathcal{D}_{[t-10, t]}$ of $\mathcal{D}$ on $[t-10, t]$. See also Example [Example 33](#ref:counterexample-1){reference-type="ref" reference="ref:counterexample-1"}.](figures/supplement_heuristic_sum.jpeg){#fig:heuristic_sums} *Example 34*. Consider the infinite ts-DAG $\mathcal{D}$ in part (a) of Figure [9](#fig:heuristic_product_lags){reference-type="ref" reference="fig:heuristic_product_lags"}, which for $\mathbf{I}_{\mathbf{O}}= \mathbf{I}= \{X^1, X^2, X^3, X^4, X^5\}$ and $p= 1$ projects to the finite marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{D})$ in part (c) of the same figure. We show that this case is a counterexample to the second of the two heuristics considered in Section 4.2.2. To this end, we first note that this heuristic prescribes restricting to the time window $[t-2, t]$, where $2$ is the observed time window length $p= 1$ plus the product of all non-zero lags $p^\prime_{prod}(\mathcal{D}) = 1 = 1^9$ of edges in $\mathbf{E}_{\rightarrow}^t \cup \mathbf{E}_{\leftrightarrow}^t = \{X^1_{t-1} {\,\rightarrow\,}X^1_t, \, \ldots, X^5_{t-1} {\,\rightarrow\,}X^5_t, \, X^2_{t-1}{\,\rightarrow\,}X^1_t, \, X^3_{t-1} {\,\rightarrow\,}X^2_t, \, X^3_{t-1} {\,\rightarrow\,}X^4_t, \, X^4_{t-1} {\,\rightarrow\,}X^5_t\}$. Now consider the blue-colored vertices $X^1_{t-1}$ and $X^5_{t-1}$. These vertices have the common ancestor $X^3_{t-3}$ by means of the path $X^1_{t-1} {\,\leftarrow\,}X^2_{t-2} {\,\leftarrow\,}X^3_{t-3} {\,\rightarrow\,}X^4_{t-2} {\,\rightarrow\,}X^5_{t-1}$, but no common ancestor within the time window $[t-2, t]$. Consequently, while there is the bidirected edge $X^1_{t-1} {\,\leftrightarrow\,}X^5_{t-1}$ in the marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{D})$, this edge is not in the ADMG latent projection $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{D}_{[t-2, t]})$ of the segment $\mathcal{D}_{[t-2, t]}$ of $\mathcal{D}$ on $[t-2, t]$ as shown in part (b) of Figure [9](#fig:heuristic_product_lags){reference-type="ref" reference="fig:heuristic_product_lags"}. Moreover, there are several other bidirected edges that are in $\mathcal{A}_{\mathbf{O}}(\mathcal{D})$ but not in $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{D}_{[t-2, t]})$. ![**(a)** A ts-DAG $\mathcal{D}$. **(b)** The ADMG latent projection $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{D}_{[t-2, t]})$ of the finite segment $\mathcal{D}_{[t-2, t]}$ of $\mathcal{D}$ on $[t-2, t]$ for $\mathbf{I}_{\mathbf{O}}= \mathbf{I}$ and $p= 1$. **(c)** The marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{D})$. See also Example [Example 34](#ref:counterexample-2){reference-type="ref" reference="ref:counterexample-2"}.](figures/supplement_heuristic_product.jpeg){#fig:heuristic_product_lags} # Proofs {#sec:proofs} Here, we provide proofs for all theoretical claims of the main paper and of Section [6](#sec:generalize-to-DMAGs){reference-type="ref" reference="sec:generalize-to-DMAGs"}. We also state and prove several auxiliary results needed for this task. We split this section in three parts: First, the proofs of all propositions. Second, the proof of Lemma 4.8. Third, the proof of all theorems. ## Proofs of all propositions *Proof of Proposition 3.5.* See the explanations in the paragraph between Definition 3.4 and Proposition 3.5. ◻ **Lemma 35**. *\[See also Remark [Remark 36](#remark:not-novel){reference-type="ref" reference="remark:not-novel"}\] Let $\mathcal{D}$ be a DAG with vertex set $\mathbf{V}= \mathbf{O}\,\dot{\cup}\, \mathbf{L}_1 \,\dot{\cup}\, \mathbf{L}_2$. Then, the ADMG latent projection commutes with partitioning the latent vertices, that is, the equality of graphs $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{D}) = \sigma_{\text{ADMG}}(\mathbf{O}, \sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D}))$ holds.* *Remark 36*. The statement in Lemma [Lemma 35](#lemma:AMDG-projection-partitioning-of-latents){reference-type="ref" reference="lemma:AMDG-projection-partitioning-of-latents"} seems to be well known in the field, so we do not claim novelty in this regard. However, since we did not find a formal proof in the literature, we here include such a formal proof. We stress that $\mathcal{D}$ is allowed to be an infinite graph and that $\mathbf{L}_1$, $\mathbf{L}_2$ are allowed to be infinite sets. *Proof of Lemma [Lemma 35](#lemma:AMDG-projection-partitioning-of-latents){reference-type="ref" reference="lemma:AMDG-projection-partitioning-of-latents"}.* Let $i {\,\rightarrow\,}j$ be in $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{D})$. Then, $i, j \in \mathbf{O}$ and there is a directed path $\pi$ from $i$ to $j$ in $\mathcal{D}$ such that no middle vertex on $\pi$, if any, is in $\mathbf{O}$. Among all vertices on $\pi$, let $v_1, v_2, \ldots v_n$, where $n \geq 2$ and $v_1 = i$ and $v_n = j$, be the ordered sequence of vertices on $\pi$ that are in $\mathbf{O}\cup \mathbf{L}_1$. Then, for all $1 \leq k \leq n-1$ the subpath $\pi(k, k+1)$ is a directed path from $v_k$ to $v_{k+1}$ in $\mathcal{D}$ such that all middle vertices on $\pi(k, k+1)$, if any, are in $\mathbf{L}_2$. Consequently, for all $1 \leq k \leq n-1$ there is the edge $v_k {\,\rightarrow\,}v_{k+1}$ in $\sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D})$. By appending these edges, we obtain the concatenation $\pi^\prime = (v_1 {\,\rightarrow\,}\ldots {\,\rightarrow\,}v_n)$, which is a path in $\sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D})$. That the concatenation $\pi^\prime$ is a path rather than a walk follows because all vertices on $\pi^\prime$ are also on $\pi$, which is a path. The path $\pi^\prime$ is a directed path from $v_1 = i$ to $v_n = j$ in $\sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D})$ and no middle vertex on $\pi^\prime$ is in $\mathbf{O}$. Therefore, there is the edge $i {\,\rightarrow\,}j$ in $\sigma_{\text{ADMG}}(\mathbf{O}, \sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D}))$. Let $i {\,\rightarrow\,}j$ be in $\sigma_{\text{ADMG}}(\mathbf{O}, \sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D}))$. Then, $i, j \in \mathbf{O}$ and there is a directed path $\pi$ from $i$ to $j$ in $\sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D})$ such that all middle vertices on $\pi$, if any, are in $\mathbf{L}_1$. Let $v_1, v_2, \ldots v_n$, where $n \geq 2$ and $v_1 = i$ and $v_n = j$, be the ordered sequence of all vertices on $\pi$. Then, for all $1 \leq k \leq n-1$ there is the edge $v_k {\,\rightarrow\,}v_{k+1}$ in $\sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D})$. Hence, for all $1 \leq k \leq n-1$ there is a directed path $\pi^\prime_k$ from $v_k$ to $v_{k+1}$ in $\mathcal{D}$ such that all middle vertices on $\pi^\prime_k$, if any, are in $\mathbf{L}_2$. Let $\pi^\prime$ be the walk in $\mathcal{D}$ obtained by appending the sequence of paths $\pi^\prime_1, \ldots, \pi^\prime_{n-1}$ at the respective common vertices $v_2, \ldots, v_{n-1}$. This walk $\pi^\prime$ is directed from $v_1 = i$ to $v_j = j$. Thus, due to acyclicity of $\mathcal{D}$, the walk $\pi^\prime$ is a path. Moreover, all middle vertices on $\pi^\prime$, if any, are in $\mathbf{L}_1 \,\dot{\cup}\, \mathbf{L}_2$. Therefore, there is the edge $i {\,\rightarrow\,}j$ in $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{D})$. Let $i {\,\leftrightarrow\,}j$ be in $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{D})$. Then, $i, j \in \mathbf{O}$ and there is a confounding path $\pi$ between $i$ to $j$ in $\mathcal{D}$ such that no middle vertex on $\pi$, if any, is in $\mathbf{O}$. Among all vertices on $\pi$, let $v_1, v_2, \ldots v_n$, where $n \geq 2$ and $v_1 = i$ and $v_n = j$, be the ordered sequence of vertices on $\pi$ that are in $\mathbf{O}\cup \mathbf{L}_1$. We now distinguish two mutually exclusive and collectively exhaustive cases: - *Case 1: The unique root vertex on $\pi$ is in $\mathbf{O}\cup \mathbf{L}_1$.* Then, there is $l$ with $2 \leq l \leq n-1$ such that $v_l$ is the unique root vertex on $\pi$. Moreover, for all $1 \leq k \leq l-1$ the subpath $\pi(k, k+1)$ is a directed path from $v_{k+1}$ to $v_k$ such that all middle vertices on $\pi(k, k+1)$, if any, are in $\mathbf{L}_2$, and for all $l \leq k \leq n-1$ the subpath $\pi(k, k+1)$ is a directed path from $v_{k}$ to $v_{k+1}$ such that all middle vertices on $\pi(k, k+1)$, if any, are in $\mathbf{L}_2$. Consequently, for all $1 \leq k \leq l-1$ there is the edge $v_k {\,\leftarrow\,}v_{k+1}$ in $\sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D})$ and for all $l \leq k \leq n-1$ there is the edge $v_k {\,\rightarrow\,}v_{k+1}$ in $\sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D})$. - *Case 2: The unique root vertex on $\pi$ is not in $\mathbf{O}\cup \mathbf{L}_1$*. Then, there is a $l$ with $1 \leq l \leq n-1$ such that the subpath $\pi(l, l+1)$ is a confounding path between $v_l$ and $v_{l+1}$ such that all middle vertices on $\pi(l, l+1)$, if any, are in $\mathbf{L}_2$. Consequently, there is the edge $v_l {\,\leftrightarrow\,}v_{l+1}$ in $\sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D})$. Moreover, for all $1 \leq k \leq l-1$ the subpath $\pi(k, k+1)$ is a directed path from $v_{k+1}$ to $v_k$ such that all middle vertices on $\pi(k, k+1)$, if any, are in $\mathbf{L}_2$, and for all $l+1 \leq k \leq n-1$ the subpath $\pi(k, k+1)$ is a directed path from $v_{k}$ to $v_{k+1}$ such that all middle vertices on $\pi(k, k+1)$, if any, are in $\mathbf{L}_2$. Consequently, for all $1 \leq k \leq l-1$ there is the edge $v_k {\,\leftarrow\,}v_{k+1}$ in $\sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D})$ and for all $l+1 \leq k \leq n-1$ there is the edge $v_k {\,\rightarrow\,}v_{k+1}$ in $\sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D})$. Let $\pi^\prime$ be the path in $\sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D})$ obtained by appending these edges at the vertices $v_2, \ldots v_{n-1}$, which is of the form $\pi^\prime = (v_1 {\,\leftarrow\,}\ldots {\,\leftarrow\,}v_l {\,\rightarrow\,}\ldots {\,\rightarrow\,}v_n)$ or of the form $\pi^\prime = (v_1 {\,\leftarrow\,}\ldots {\,\leftarrow\,}v_l {\,\leftrightarrow\,}v_{l+1} {\,\rightarrow\,}\ldots {\,\rightarrow\,}v_n)$ and where $v_{l} = v_1$ and $v_{l+1} = v_n$ are allowed. That the concatenation $\pi^\prime$ is a path rather than a walk follows because all vertices on $\pi^\prime$ are also on $\pi$, which is a path. The path $\pi^\prime$ is a confounding path between $v_1 = i$ and $v_n = j$ in $\sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D})$ such that no middle vertex on $\pi^\prime$, if any, is in $\mathbf{O}$. Therefore, there is the edge $i {\,\leftrightarrow\,}j$ in $\sigma_{\text{ADMG}}(\mathbf{O}, \sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D}))$. Let $i {\,\leftrightarrow\,}j$ be in $\sigma_{\text{ADMG}}(\mathbf{O}, \sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D}))$. Then, $i, j \in \mathbf{O}$ and there is a confounding path $\pi$ between $i$ to $j$ in $\sigma_{\text{ADMG}}(\mathbf{O}\cup \mathbf{L}_1, \mathcal{D})$ such that all middle vertices on $\pi$, if any, are in $\mathbf{L}_1$. Let $v_1, v_2, \ldots v_n$, where $n \geq 2$ and $v_1 = i$ and $v_n = j$, be the ordered sequence of all vertices on $\pi$. We now distinguish two mutually exclusive and collectively exhaustive cases: - *Case 1: There is no bidirected edge on $\pi$.* Then, there is $l$ with $2 \leq l \leq n-1$ such that for all $1 \leq k \leq l-1$ the subpath $\pi(l, l+1)$ is $v_k {\,\leftarrow\,}v_{k+1}$ and for all $l \leq k \leq n-1$ the subpath $\pi(l, l+1)$ is $v_k {\,\rightarrow\,}v_{k+1}$. Hence, for all $1 \leq k \leq l-1$ there is a directed path $\pi^\prime_k$ from $v_{k+1}$ to $v_k$ in $\mathcal{D}$ such that all middle vertices on $\pi^\prime_k$, if any, are in $\mathbf{L}_2$, and for all $l \leq k \leq n-1$ there is a directed path $\pi^\prime_k$ from $v_{k}$ to $v_{k+1}$ in $\mathcal{D}$ such that all middle vertices on $\pi^\prime_k$, if any, are in $\mathbf{L}_2$. - *Case 2: There is a bidirected edge on $\pi$.* Then, there is $l$ with $1 \leq l \leq n-1$ such that the subpath $\pi(l, l+1)$ is $v_l {\,\leftrightarrow\,}v_{l+1}$. Consequently, there is a confounding path $\pi^\prime_l$ between $v_l$ and $v_{l+1}$ in $\mathcal{D}$ such that all middle vertices on $\pi^\prime_l$, if any, are in $\mathbf{L}_2$. Moreover, for all $1 \leq k \leq l-1$ the subpath $\pi(k, k+1)$ is $v_k {\,\leftarrow\,}v_{k+1}$ and for all $l+1 \leq k \leq n-1$ the subpath $\pi(k, k+1)$ is $v_k {\,\rightarrow\,}v_{k+1}$. Hence, for all $1 \leq k \leq l-1$ there is a directed path $\pi^\prime_k$ from $v_{k+1}$ to $v_k$ in $\mathcal{D}$ such that all middle vertices on $\pi^\prime_k$, if any, are in $\mathbf{L}_2$, and for all $l+1 \leq k \leq n-1$ there is a directed path $\pi^\prime_k$ from $v_{k}$ to $v_{k+1}$ in $\mathcal{D}$ such that all middle vertices on $\pi^\prime_k$, if any, are in $\mathbf{L}_2$. Let $\pi^\prime$ be the walk in $\mathcal{D}$ obtained by appending the sequence of paths $\pi^\prime_1, \ldots, \pi^\prime_{n-1}$ at the respective common vertices $v_2, \ldots, v_{n-1}$. Not all vertices on the concatenation $\pi^\prime$, which is a confounding walk, are necessarily also on $\pi$. Thus, there can be vertices that appear more than once on $\pi^\prime$. However, all vertices on $\pi^\prime$ that are not also on $\pi$ are necessarily in $\mathbf{L}_2$. Therefore, neither $v_1 = i$ nor $v_n = j$ appears more than once on $\pi^\prime$. We now distinguish two mutually exclusive and collectively exhaustive cases: - *Case 1: No vertex appears more than once in $\pi^\prime$.* Then, $\pi^\prime$ is a path. Let $\tilde{\pi}$ be $\pi^\prime$. - *Case 2: At least one vertex appears more than once in $\pi^\prime$.* Let $w_1, \ldots, w_m$, where $m \geq n$ and $w_1 = i$ and $w_m = j$, be the ordered sequence of all vertices on $\pi^\prime$. Let $a_1$ be the minimum over all $b$ with $1 \leq b \leq m$ such that $w_b$ appears more than once on $\pi^\prime$. Then, $a_1 \neq 1$ and $a_1 \neq m$ because neither $w_1 = i$ nor $w_m = j$ appears more than once on $\pi^\prime$, and $a_1 \neq m-1$ because else $w_a$ could, given the definition of $a_1$, only appear once on $\pi^\prime$. Thus, in summary, $2 \leq a_1 \leq m-2$. Moreover, $\pi^\prime(1, a_1)$ is either a directed path from $w_{a_1}$ to $w_1$ (namely if $w_{a_1}$ is on $\pi^\prime$ between $w_1$ and (including) the unique root vertex on $\pi$) or a confounding path between $w_{1}$ and $w_{a_1}$ (namely if $w_{a_1}$ is on $\pi^\prime$ between $w_m$ and (excluding) the unique root vertex on $\pi$). Let $a_2$ be the maximum over all $b$ with $a_1 < b \leq m-1$ such that $w_{b} = w_{a_1}$. Then $a_1 < a_2 \leq m-1$. Moreover, $\pi^\prime(a_2, m)$ is either a directed path from $w_{a_2}$ to $w_m$ (namely if $w_{a_2}$ is on $\pi^\prime$ between $w_m$ and (including) the unique root vertex on $\pi$) or a confounding path between $w_{m}$ and $w_{a_2}$ (namely if $w_{a_2}$ is on $\pi^\prime$ between $w_1$ and (excluding) the unique root vertex on $\pi$). Note that $\pi^\prime(1, a_1)$ and $\pi^\prime(a_2, m)$ cannot both be a confounding walk at the same time. Let $\tilde{\pi}$ be the walk obtained by appending $\pi^\prime(1, a_1)$ and $\pi^\prime(a_2, m)$ at their common vertex $w_{a_1} = w_{a_2}$. By definition of $a_1$ and $a_2$, this walk $\tilde{\pi}$ is a path. The path $\tilde{\pi}$ is a confounding path between $i$ and $j$ in $\mathcal{D}$ such that no middle vertex on $\tilde{\pi}$, if any, is in $\mathbf{O}$. Therefore, there is the edge $i {\,\leftrightarrow\,}j$ in $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{D})$. ◻ *Proof of Proposition 3.6.* Use Lemma [Lemma 35](#lemma:AMDG-projection-partitioning-of-latents){reference-type="ref" reference="lemma:AMDG-projection-partitioning-of-latents"} with $\mathbf{L}_1 = \mathbf{L}^{unob}$ and $\mathbf{L}_2 = \mathbf{L}^{temp}$. ◻ *Proof of Proposition 3.7.* See the explanations in the paragraph directly above Proposition 3.7. ◻ *Proof of Proposition 3.8.* See the explanations in the paragraph directly above Proposition 3.8. ◻ *Proof of Proposition 4.2.* **Only if**. The premise is that the vertices $(i, t-\tau_i)$ and $(j,t)$ with $\tau_i \geq 0$ have a common ancestor in the ts-DAG $\mathcal{D}$. Thus, there is a path $\rho$ between in $(i,t-\tau_i)$ and $(j, t)$ in $\mathcal{D}$ that is i) a confounding walk or ii) directed from $(i,t-\tau_i)$ to $(j,t)$ or iii) directed from $(j, t)$ to $(i,t-\tau_i)$ or iv) trivial. Let $\pi$ be the projection of $\rho$ to the summary graph $\mathcal{S}(\mathcal{D})$, which is obtained by "removing" the time indices of the vertices on $\rho$. Then, $\pi$ is a walk between $i$ and $j$ in $\mathcal{S}(\mathcal{D})$ that is i) a confounding path or ii) directed from $i$ to $j$ or iii) directed from $j$ to $i$ or iv) trivial. If $\pi$ is a path, we have thus shown that $i$ and $j$ have a common ancestor in $\mathcal{S}(\mathcal{D})$. If $\pi$ is not a path, then let $k$ be the vertex closest to $i$ on $\pi$ (including $i$ itself) that appears more than once on $\pi$. Let $\tilde{\pi}$ be the subwalk of $\pi$ obtained by collapsing the subwalk of $\pi$ between the first and last appearance of $k$ to the single vertex $k$. Then, $\tilde{\pi}$ is a path between $i$ and $j$ in $\mathcal{S}(\mathcal{D})$ that is i) a confounding path or ii) directed from $i$ to $j$ or iii) directed from $j$ to $i$ or iv) trivial. Hence, $i$ and $j$ have a common ancestor in $\mathcal{S}(\mathcal{D})$. **If.** The premise is that the vertices $i$ and $j$ have a common ancestor in the summary graph $\mathcal{S}(\mathcal{D})$. Thus, there is a path $\pi$ between in $i$ and $j$ in $\mathcal{D}$ that is i) a confounding path or ii) directed from $i$ to $j$ or iii) directed from $j$ to $i$ or iv) trivial. We now distinguish these four cases: - *Case i).* Then, the arguments in the paragraph directly below Proposition 4.2 show that $(i,t-\tau_i)$ and $(j, t)$ have a common ancestor in $\mathcal{D}$. - *Case ii).* Then, in the ts-DAG $\mathcal{D}$ there is a directed path $\rho_{ij}$ from $(i, t-\tau_{ij})$ to $(j, t)$ for some $\tau_{ij} \geq 0$. Moreover, since $\mathcal{D}$ has all lag-$1$ autocorrelations, there is the (potentially trivial) path $\rho_{ii} = ((i, t-\max(\tau_i, \tau_{ij})) {\,\rightarrow\,}(i, t-\max(\tau_i, \tau_{ij})+1) {\,\rightarrow\,}\ldots {\,\rightarrow\,}(i, t-\min(\tau_i, \tau_{ij})))$. By concatenating the paths $\rho_{ij}$ and $\rho_{ii}$ at their common endpoint vertex, which is $(i, t-\max(\tau_i, \tau_{ij}))$ if $\tau_{ij} > \tau_i$ and $(i, t-\min(\tau_i, \tau_{ij}))$ if $\tau_{ij} \leq \tau_i$, we obtain a walk $\rho$ between $(i, t-\tau_i)$ and $(j, t)$ that is a confounding walk (if $\tau_{ij} > \tau_i$) or a directed walk from $(i, t-\tau_i)$ to $(j, t)$ (if $\tau_{ij} \leq \tau_i$). If $\rho$ is a path, we have thus shown that $(i, t-\tau_i)$ and $(j, t)$ have a common ancestor in $\mathcal{D}$. If $\rho$ is not a path, then let $(i, t-\tilde{\tau}_i)$ be the vertex closest to $(i, t-\tau_i)$ on $\rho$ (including $(i, t-\tau_i)$ itself) that appears more than once on $\rho$. Let $\tilde{\rho}$ be the walk obtained by collapsing the subwalk of $\rho$ between the first and last appearance of $(i,t-\tilde{\tau}_i)$ to the single vertex $(i,t-\tilde{\tau}_i)$. Then, $\tilde{\rho}$ is a path between $(i,t-\tau_i)$ and $(j, t)$ in $\mathcal{D}$ that is a confounding path or directed from $(i,t-\tau_i)$ to $(j, t)$ or trivial. Hence, we have shown that $(i,t-\tau_i)$ and $(j,t)$ have a common ancestor in $\mathcal{D}$. - *Case iii).* Then, in the ts-DAG $\mathcal{D}$ there is a directed path $\rho_{ji}$ from $(j, t-\tau_{ji})$ to $(i, t-\tau_i)$ for some $\tau_{ji} \geq \tau_i \geq 0$. Moreover, since $\mathcal{D}$ has all lag-$1$ autocorrelations, there is the path $\rho_{jj} = ((j, t-\tau_{ji}) {\,\rightarrow\,}(j, t-\tau_{ji} +1) {\,\rightarrow\,}\ldots {\,\rightarrow\,}(j, t))$. By concatenating the paths $\rho_{ji}$ and $\rho_{jj}$ at their common endpoint vertex $(j, t-\tau_{ji})$, we obtain a walk $\rho$ between $(i, t-\tau_i)$ and $(j, t)$ that is a confounding walk (if $\tau_{ji} > 0$) or a directed walk from $(j, t)$ to $(i, t-\tau_i)$ (if $\tau_{ij} = 0$, which also implies $\tau_i = 0$). If $\rho$ is a path, we have thus shown that $(i, t-\tau_i)$ and $(j, t)$ have a common ancestor in $\mathcal{D}$. If $\rho$ is not a path, then let $(j, t-\tilde{\tau}_j)$ be the vertex closest to $(j, t)$ on $\rho$ (including $(j,t)$ itself) that appears more than once on $\rho$. Let $\tilde{\rho}$ be walk obtained by collapsing the subwalk of $\rho$ between the first and last appearance of $(j, t)$ to the single vertex $(j,t)$. Then, $\tilde{\rho}$ is a path between $(i,t-\tau_i)$ and $(j, t)$ in $\mathcal{D}$ that is a confounding path or directed from $(j,t)$ to $(i, t-\tau_i)$ or trivial. Hence, we have shown that $(i,t-\tau_i)$ and $(j,t)$ have a common ancestor in $\mathcal{D}$. - *Case iv).* Then, $i = j$ and, using the fact that the ts-DAG $\mathcal{D}$ has all lag-$1$ autocorrelations, in $\mathcal{D}$ there is the directed path $\rho = ((i,t-\tau_i) {\,\rightarrow\,}(i,t-\tau_i+1) {\,\rightarrow\,}\ldots {\,\rightarrow\,}(i, t))$ from $(i, t-\tau_i)$ to $(i, t) = (j, t)$ if $\tau_i > 0$. Hence, $(i,t-\tau_i)$ and $(j,t)$ have a common ancestor in $\mathcal{D}$. We have thus proven the claim. ◻ *Proof of Proposition 4.6.* **Only if**. The premise is that $(i,t-\tau)$ and $(j, t)$ with $\tau \geq 0$ have a common ancestor in $\mathcal{D}$. Thus, in $\mathcal{D}$ there is a path $\rho$ between $(i,t-\tau)$ and $(j, t)$ that is i) a confounding path or ii) directed from $(i, t-\tau)$ to $(j, t)$ or iii) directed from $(j, t)$ to $(i, t-\tau)$ or iv) trivial. By splitting $\rho$ at its (unique) root vertex $(k,t-\tau_k)$,[^15] we obtain a path $\rho_{ki}$ from $(k,t-\tau_k)$ to $(i,t-\tau)$ and a path $\rho_{kj}$ from $(k, t-\tau_k)$ to $(j, t)$. Then, $\rho_{ki}$ is the trivial path consisting of $(k,t-\tau_k) = (i, t-\tau)$ or directed from $(k, t-\tau_k)$ to $(i, t-\tau)$. Similarly, $\rho_{kj}$ is the trivial path consisting of $(k,t-\tau_k) = (j, t)$ or directed from $(k, t-\tau_k)$ to $(j, t)$. Let $\pi$ and $\pi^\prime$ respectively be the projections of $\rho_{ki}$ and $\rho_{kj}$ to the summary graph $\mathcal{S}(\mathcal{D})$, which are obtained by "removing" the time indices of the vertices. Then, $\pi$ is the trivial path consisting of $k=i$ or a directed walk from $k$ to $i$, and $\pi^\prime$ is the trivial path consisting of $k=j$ or a directed walk from $k$ to $j$. Consequently, $\pi$ and $\pi^\prime$ satisfy the first two of the three conditions in Proposition 4.6. To show that $\pi$ and $\pi^\prime$ also satisfy the third condition, let $e^{1}, \ldots ,e^{n_{ki}}$ be the (possibly empty) ordered sequence of edges on $\rho_{ki}$, and let $f^{1}, \ldots ,f^{n_{kj}}$ be the (possibly empty) ordered sequence of edges on $\rho_{kj}$. Since $\rho_{ki}$ is directed if it is non-trivial, we then get $w(\rho_{ki}) = \sum_{a=1}^{n_{ki}} w(e^{a})$ in terms of the lags $w(e^{a})$ of the edges $e^a$ if $n_{ki} \geq 1$ and $w(\rho_{ki}) = 0$ if $n_{ki} = 0$ (note that $\rho_{ki}$ is trivial if and only if $n_{ki} = 0$). Similarly, $w(\rho_{kj}) = \sum_{b=1}^{n_{kj}} w(f^{b})$ in terms of the lags $w(f^{b})$ of the edges $f^b$ if $n_{kj} \geq 1$ and $w(\rho_{kj}) = 0$ if $n_{kj} = 0$. Moreover, $w(\rho_{ki}) + \tau = w(\rho_{kj})$ because $\rho_{ki}$ and $\rho_{kj}$ have $(k,t-\tau_k)$ as their common root vertex. Let $e_{\mathcal{S}}^{1}, \ldots ,e_{\mathcal{S}}^{n_{ki}}$ be the (possibly empty) ordered sequence of edges on $\pi$, and let $f_{\mathcal{S}}^{1}, \ldots ,f_{\mathcal{S}}^{n_{kj}}$ be the (possibly empty) ordered sequence of edges on $\pi^\prime$. Then, $e_{\mathcal{S}}^{a}$ is for all $1 \leq a \leq n_{ki}$ the projection of $e^a$ to $\mathcal{S}(\mathcal{D})$ and $f_{\mathcal{S}}^{b}$ is for all $1 \leq b \leq n_{kj}$ the projection of $f^b$ to $\mathcal{S}(\mathcal{D})$. Thus, $w(e^a) \in \mathbf{w}(e_{\mathcal{S}}^{a})$ for all $1 \leq a \leq n_{ki}$ and $w(f^b) \in \mathbf{w}(f_{\mathcal{S}}^{b})$ for all $1 \leq b \leq n_{kj}$, where $\mathbf{w}(\cdot)$ denotes a multi-weight. Then, by definition of the multi-weight of directed walks, $w(\pi) = \sum_{a=1}^{n_{ki}} w(e^{a}) = w(\rho_{ki}) \in \mathbf{w}(\pi)$ if $n_{ki} \geq 1$. If $n_{ki} = 0$, then $0 \in \mathbf{w}(\pi) = \{0\}$ by definition of the multi-weight of a trivial walk. Similarly, $w(\pi^\prime) = \sum_{b=1}^{n_{kj}} w(f^{b}) = w(\rho_{kj}) \in \mathbf{w}(\pi^\prime)$ if $n_{kj} \geq 1$, and $0 \in \mathbf{w}(\pi^\prime) = \{0\}$ if $n_{kj} = 0$. Hence, $\pi$ and $\pi^\prime$ also satisfy the third condition in Proposition 4.6. **If.** The premise is that there are paths $\pi$ and $\pi^\prime$ in the summary graph $\mathcal{S}(\mathcal{D})$ that satisfy all of the three conditions in Proposition 4.6. Let $w(\pi) \in \mathbf{w}(\pi)$ and $w(\pi^\prime) \in \mathbf{w}(\pi^\prime)$ with $w(\pi) + \tau = w(\pi^\prime)$, which exist due to the third condition in Proposition 4.6. Further, let $e_{\mathcal{S}}^{1}, \ldots ,e_{\mathcal{S}}^{n_{ki}}$ be the (possibly empty) ordered sequence of edges on $\pi$, and let $f_{\mathcal{S}}^{1}, \ldots ,f_{\mathcal{S}}^{n_{kj}}$ be the (possibly empty) ordered sequence of edges on $\pi^\prime$. Then, by definition of the multi-weight of directed walks, for all $1 \leq a \leq n_{ki}$ there is $w(e_{\mathcal{S}}^a) \in \mathbf{w}(e_{\mathcal{S}}^a)$ such that $w(\pi) = \sum_{a=1}^{n_{ki}} w(e_{\mathcal{S}}^a)$ if $n_{ki} \geq 1$. If $n_{ki} = 0$, then $w(\pi) = 0$ by definition of the multi-weight of a trivial walk. Similarly, for all $1 \leq b \leq n_{kj}$ there is $w(f_{\mathcal{S}}^b) \in \mathbf{w}(f_{\mathcal{S}}^b)$ such that $w(\pi^\prime) = \sum_{b=1}^{n_{kj}} w(f_{\mathcal{S}}^b)$ if $n_{kj} \geq 1$, and $w(\pi^\prime) = 0$ if $n_{kj} = 0$. Suppose, for the moment, that $n_{ki} \geq 1$. Since $w(e_{\mathcal{S}}^a) \in \mathbf{w}(e_{\mathcal{S}}^a)$ for all $1 \leq a \leq n_{ki}$, according to the definition of the multi-weighted summary graph for all $1 \leq a \leq n_{ki}$ there is the edge $(k_a, t-w(e_{\mathcal{S}}^a)) {\,\rightarrow\,}(k_{a+1}, t)$ in $\mathcal{D}$. Here, $k_1, \ldots, k_{n_{ki}}$ is the ordered sequence of vertices on $\pi$. We denote $k_1 =k$ and infer from the second condition in Proposition 4.6 that $k_{n_{ki}} = i$. Using the repeating edges property of ts-DAGs to appropriately shift these edges backwards in time and recalling that the weight of a directed walk is the sum the lags of its edges, we see that in $\mathcal{D}$ there is a directed path $\rho_{ki}$ from $(k_1, t-w(\pi)-\tau) = (k, t-w(\pi)-\tau)$ to $(k_{n_{ki}}, t-\tau) = (i,t-\tau)$. If $n_{ki} = 0$, then we let $\rho_{ki}$ be the trivial path consisting of $(i,t-\tau)$ only. Similarly, if $n_{kj} \geq 1$, then in $\mathcal{D}$ there is a directed path $\rho_{kj}$ from $(l_1, t-w(\pi^\prime))$ to $(j, t)$. Here, $l_1$ is the fist vertex on $\pi^\prime$, for which $l_1 = k$ according to first condition of Proposition 4.6. If $n_{kj} = 0$, then we let $\rho_{kj}$ be the trivial path consisting of $(j, t)$ only. Since $w(\pi) + \tau = w(\pi^\prime)$, the paths $\rho_{ki}$ and $\rho_{kj}$ have $(k,t-w(\pi)-\tau) = (k, t-w(\pi^\prime)) = (l_1, t-w(\pi^\prime))$ as a common endpoint vertex. Let $\rho$ be the walk obtained by concatenating $\rho_{ki}$ and $\rho_{kj}$ at this common endpoint vertex. This walk $\rho$ is i) a confounding walk between $(i,t-\tau)$ and $(j, t)$ or ii) directed from $(i,t-\tau)$ to $(j, t)$ or iii) directed from $(j, t)$ to $(i,t-\tau)$ or iv) the trivial path consisting of $(i,t-\tau) = (j, t)$ only. If $\rho$ is a walk, we have thus shown that $(i,t-\tau)$ and $(j, t)$ have a common ancestor in $\mathcal{D}$. If $\rho$ is not a path, then we can remove a sufficiently large subwalk from $\rho$ to obtain a path $\tilde{\rho}$ which is i) a confounding path between $(i, t-\tau)$ and $(j, t)$ or ii) directed from $(i, t-\tau)$ to $(j, t)$ or iii) directed from $(j, t)$ to $(i, t-\tau_i)$ or iv) trivial. Hence, $(i,t-\tau)$ and $(j, t)$ have a common ancestor in $\mathcal{D}$. ◻ *Proof of Proposition [Proposition 31](#prop:get-ts-DMAG){reference-type="ref" reference="prop:get-ts-DMAG"}.* According to Proposition 1 in @richardson2023nested, the $m$-separations in the marginal ADMG $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A}) = \mathcal{A}_{\mathbf{O}}(\mathcal{A})$ are in one-to-one correspondence with the $d$-separations between vertices in $\mathbf{O}$ in the ADMG $\mathcal{A}$.[^16] Moreover, as straightforwardly follows from the definition of the ADMG latent projection (see Definition 2.1, due to @pearl1995theory, see also for example @richardson2023nested), for all $i, j \in \mathbf{O}$ we have that $i$ is an ancestor of $j$ in $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A})$ if and only if $i$ is an ancestor of $j$ in $\mathcal{A}$. The definition of canonical DAGs (see the last paragraph of Section 2.2, cf. also Section 6.1 of @richardson2002ancestral) straightforwardly implies that $m$-separations in the ADMG $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A})$ are in one-to-one correspondence with $d$-separations between vertices in $\mathbf{O}$ in its canonical DAG $\mathcal{D}_{c}(\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A})) = \mathcal{D}_{c}(\mathcal{A}_{\mathbf{O}}(\mathcal{A}))$. Moreover, as another implication of the definition of canonical DAGs, for all $i, j \in \mathbf{O}$ we have that $i$ is an ancestor of $j$ in $\mathcal{D}_{c}(\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{D}))$ if and only if $i$ is an ancestor of $j$ in $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{D})$. Putting together the previous observations, we see that $m$-separations $\mathcal{D}_{c}(\mathcal{A}_{\mathbf{O}}(\mathcal{A}))$ between vertices in $\mathbf{O}$ are in one-to-one correspondence with $m$-separations in $\mathcal{A}$ between vertices in $\mathbf{O}$. Moreover, for all $i, j \in \mathbf{O}$ we see that $i$ is an ancestor of $j$ in $\mathcal{D}_{c}(\mathcal{A}_{\mathbf{O}}(\mathcal{A}))$ if and only if $i$ is an ancestor of $j$ in $\mathcal{A}$. The equality $\mathcal{M}_{\mathbf{O}}(\mathcal{A}) = \sigma_{\text{DMAG}}(\mathbf{O}, \mathcal{D}_{c}(\mathcal{A}_{\mathbf{O}}(\mathcal{A})))$ now follows because the DMAG latent projection (see Definition [Definition 28](#def:DMAG-latent-projection){reference-type="ref" reference="def:DMAG-latent-projection"}, due to @richardson2002ancestral and @zhang2008causal) only requires knowledge of $m$-separations and ancestral relationships between observed vertices. ◻ ## Proof of Lemma 4.8 {#subsec.proof-of-decomposition-theorem} In order to prove Lemma 4.8, we first prove the following slightly adjusted version of it. **Lemma 37**. *There exist *finite* sets* - *$\mathcal{W}_0(k,i)$,* - *$\mathcal{N}_{\pi^\prime}$ for every $\pi^\prime \in \mathcal{W}_0(k,i)$,* - *and $E_{\tau}(\pi^\prime,\mathcal S)$ for every $\pi^\prime \in \mathcal{W}_0(k,i)$ and every $\mathcal{S} \in \mathcal{N}_{\pi^\prime}$* *such that $$\begin{aligned} \mathbf{W}_{\tau}(k,i) = \bigcup_{\pi' \in \mathcal{W}_0(k,i)}\, \bigcup_{\mathcal{S} \in \mathcal{N}_{\pi'}}\, \bigcup_{(a_0,\dots,a_{\mu}) \in E_{\tau}(\pi',\mathcal S)}\mathrm{con}_+(a_0;\dots,a_{\mu})\, . \end{aligned}$$* As opposed to Lemma 4.8, Lemma [Lemma 37](#thm.support-thm-1){reference-type="ref" reference="thm.support-thm-1"} uses cones $\mathrm{con}_+(a_0;\dots,a_{\mu})$ over positive integers, instead of cones over non-negative integers, defined as $$\begin{aligned} \mathrm{con}_+(a_0;a_1,\dots,a_{\mu}) = \left\{ a_0 + \sum_{\alpha=1}^{\mu} n_{\alpha} \cdot a_{\alpha} \ \bigg| \ n_{\alpha} \in \mathbb{N} \text{ for all } 1 \leq \alpha \leq \mu \right\} \, ,\end{aligned}$$ that is, with positive integers $n_{\alpha}$. From this adjusted version of the statement, we will then derive Lemma 4.8 by sorting the involved cones over positive integers more economically into cones over non-negative integers. Before we define the sets in Lemmas 4.8 and [Lemma 37](#thm.support-thm-1){reference-type="ref" reference="thm.support-thm-1"} in detail, we need to introduce additional notation on walks in directed graphs and discuss a few results on cycles. ### Operations on walks in directed graphs {#subsubsec.operations-walks} Given a finite directed graph $\mathcal{G}$, let $\mathcal{W}(\mathcal{G})$ be the union of the set of walks $\mathcal{W}(i, j)$ for all pairs of (not necessarily distinct) vertices $i$ and $j$ in $\mathcal{G}$, with $\mathcal{W}(i, j)$ as defined in Problem 3. We define the following operations on elements in $\mathcal{W}(\mathcal{G})$. - **Concatenation**: Given walks $\pi, \pi' \in \mathcal{W}(\mathcal{G})$ with $\pi(len(\pi)) = \pi'(1)$, we define their *concatenation* as $\pi \circ \pi' = (\pi(1),\dots,\pi(len(\pi)),\pi'(2),\dots, \pi'(len(\pi')))$. - **Removal**: Given a walk $\pi \in \mathcal{W}(\mathcal{G})$ with subcycle $c = \pi(a,b)$ (that is, $a < b$ and $\pi(a) = \pi(b)$), we denote by $\pi \backslash c = (\pi(1),\dots, \pi(a),\pi(b+1),\dots, \pi(len(\pi)))$ the walk obtained by collapsing $c$ on $\pi$ to the single vertex $\pi(a) = \pi(b)$. We call the index $a$ the *removal point* of $c$ from $\pi$. - **Insertion**: Given a walk $\pi \in \mathcal{W}(\mathcal{G})$, an integer $i \in \{ 1,\dots, len(\pi) \}$ and a directed cycle $c = (c(1),\dots, c(m)) \in \mathcal{W}(\mathcal{G})$ with $c(1) = c(m) = \pi(i)$, we let $\pi \cup_i c$ denote the walk $(\pi(1),\dots,\pi(i-1),c(1),\dots, c(m), \pi(i+1),\dots, \pi(len(\pi)))$ obtained by inserting $c$ into $\pi$ at point $i$. ### Cycles in directed graphs {#subsubsec.cycles} Recall that the finite group $\mathbb{Z}_n$ acts on the set of cycles of length $n$ by revolving the vertices, that is, through the group action $$\alpha_n(\Bar{k})\left((c(1),\dots,c(n))\right) = (c(\overline{1+k}),\dots,c(\overline{n+k}))\,$$ where $c$ is a cycle, $\Bar{k} \in \mathbb{Z}_n$ and $\Bar{\ell} = \ell\!\! \mod n \in \mathbb{Z}_n$. These group actions $\alpha_n$ induce an equivalence relation on the set of irreducible cycles such that two irreducible cycles $c_1$ and $c_2$ are equivalent (denoted as $c_1 \sim c_2$) if and only if, first, $c_1$ and $c_2$ have the same length $n$ and, second, there is $\Bar{k} \in \mathbb{Z}_n$ such that $c_2 = \alpha_n(\Bar{k})(c_1)$. We denote the set of all equivalence classes of *irreducible* cycles in $\mathcal{W}(\mathcal{G})$ by $\mathcal{C}$ and the corresponding equivalence classes themselves in bold font, for example $\mathbf{c} \in \mathcal{C}$. We denote the set of nodes on an equivalence class $\mathbf{c}$ by $\mathrm{nodes}(\mathbf{c})$, which is well-defined because all elements in the equivalence class $\mathbf{c}$ share the same nodes.[^17] It is convenient to define another graphical object, the *graph of cycles*, which encodes how "far" different equivalence classes in $\mathcal{C}$ are from each other. *Definition 38*. Let $\mathcal{G}$ be a directed graph and let $\mathcal{C}$ be the set of its irreducible cycle classes as defined in the previous paragraph. Then, we define the *graph of cycles* $\mathcal{G}_{\mathcal{C}} = (\mathcal{C},\mathbf{U})$ as the *undirected* graph with node set $\mathcal{C}$ and undirected edges such that $$\begin{aligned} (\mathbf{c},\mathbf{c}') \in \mathbf{U} \qquad \text{if and only if } \qquad \mathbf{c}\neq \mathbf{c}' \quad \text{and} \quad \mathrm{nodes}(\mathbf{c}) \cap \mathrm{nodes}(\mathbf{c}') \neq \emptyset \, .\end{aligned}$$ *Example 39*. Let $\mathcal{G}$ be the directed graph with vertex set $\mathbf{V}= \{1, 2, 3, 4\}$ given by $1 \to 2 \to3 \to4 \to3 \to2$. Then, $\mathcal{C} = \{\mathbf{c}_1, \mathbf{c}_2\}$ with $\mathbf{c}_1 = [(3,4,3)]$ (and $(4,3,4) \sim (3,4,3)$) and $\mathbf{c}_2 = [(2,3,2)]$ (and $(3,2,3) \sim (2,3,2)$). The graph of cycles $\mathcal{G}_{\mathcal{C}}$ is $\mathbf{c}_1 - \mathbf{c}_2$. The geodesic distance on the graph of cycles $\mathcal{G}_{\mathcal{C}}$ equips the set $\mathcal{C}$ with a metric $d$, that is, $$\begin{aligned} d(\mathbf{c},\mathbf{c}') = \begin{cases} \infty \quad \text{if} \quad \mathbf{c}, \mathbf{c}' \text{ are not connected by a path in $\mathcal{G}$}\, ; \\ \text{length of the shortest (possibly trivial) path on } \mathcal{G}_{\mathcal{C}} \text{ between } \mathbf{c}\text{ and } \mathbf{c}' \, . \end{cases}\end{aligned}$$ Given a subset $\mathcal{C}^\prime \subseteq \mathcal{C}$, we define $d(\mathbf{c},\mathcal{D})= \min_{\mathbf{c}^\prime \in \mathcal{C}^\prime} d(\mathbf{c},\mathbf{c}')$. Then, we extend $d$ to a notion of distance between a cycle class $\mathbf{c}$ and a walk $\pi \in \mathcal{W}(\mathcal{G})$ on the original graph $\mathcal{G}$ by defining $$\begin{aligned} d(\pi,\mathbf{c}) = 1 + d(\mathbf{c}, \mathrm{touch}(\pi))\, , \end{aligned}$$ where we define the subset $\mathrm{touch}(\pi) \subseteq \mathcal{C}$ by $$\begin{aligned} \mathbf{c}' \in \mathrm{touch}(\pi) \qquad \text{if and only if} \qquad \mathrm{nodes}(\mathbf{c}') \cap \mathrm{nodes}(\pi) \neq \emptyset\, .\end{aligned}$$ In other words, $\mathrm{touch}(\pi)$ is the set of cycle classes that "touch" the walk $\pi$ in the sense of sharing a node with $\pi$. For $\pi = (1 {\,\rightarrow\,}2 {\,\rightarrow\,}3)$ in $\mathcal{G}$ we have $\mathrm{touch}(\pi) = \{\mathbf{c}_1, \mathbf{c}_2\}$ and $d(\pi, \mathbf{c_2}) = 1$. For $\pi^\prime = (1{\,\rightarrow\,}2)$ we have $\mathrm{touch}(\pi^\prime) = \{\mathbf{c}_1\}$ and $d(\pi^\prime, \mathbf{c_2}) = 2$. Next, for given $\pi \in \mathcal{W}(\mathcal{G})$, we define the set $\mathcal{P}_{\pi}$ to consist of - all (possibly trivial) undirected paths $\xi$ on the graph of cycles $\mathcal{G}_{\mathcal{C}}$ that start at a point $\xi(1)\in \mathrm{touch}(\pi)$ and - the empty path, and then consider the set projection $$\begin{aligned} \Lambda_{\pi}: \,\, &\mathcal{P}_{\pi} \to 2^{\mathcal{C}}\\ &(\xi(1),\dots, \xi(len(\xi))) \mapsto \{ \xi(1),\dots, \xi(len(\xi)) \} \subseteq \mathcal{C}\end{aligned}$$ that maps an element of $\mathcal{P}_{\pi}$ to its (possibly empty) set of nodes (which is consequently a subset of $\mathcal{C}$). The range of this map generates the *set monoid* $\mathcal{N}_{\pi}$ defined as the unique smallest subset $\mathcal{N}_{\pi} \subseteq 2^{\mathcal{C}}$ such that - $\Lambda_{\pi}(\mathcal{P}_{\pi}) \subseteq \mathcal{N}_{\pi}$ and - if $\mathcal A, \mathcal B \in \mathcal{N}_{\pi}$, then $\mathcal A \cup \mathcal B \in \mathcal{N}_{\pi}$. Note that $\emptyset \in \mathcal{N}_{\pi}$ because the empty path is in $\mathcal{P}_{\pi}$. To prove Lemma [Lemma 37](#thm.support-thm-1){reference-type="ref" reference="thm.support-thm-1"}, we would like to define a procedure that iteratively removes irreducible cycles from walks until the remaining walk is free of cycles (or trivial). Since the choice of cycle that is to be removed is not unique, we choose to always remove the first irreducible cycle by convention. To see that this choice is well-defined, consider a directed walk $\pi$ from $i$ to $j$ on a directed graph $\mathcal{G}$ such that $\pi$ contains a cycle. The number $b= \min \{b' ~\vert~ \pi(1,b') \text{ contains an irreducible cycle}\}$ is then well-defined. Moreover the subwalk $\pi(1,b)$ contains a unique irreducible cycle that ends at $\pi(b)$. Indeed, by definition of $b$ any subcycle of $\pi(1,b)$ must end at $\pi(b)$ and if $c = \pi(a,b)$ and $c' = \pi(a',b)$ with $a<a'$ were different irreducible subcycles then $c$ could not be irreducible as it must contain $c'$. Therefore, the following procedure is well-defined: *Definition 40*. Let $\pi$ be a directed walk from $i$ to $j$ on a directed graph $\mathcal{G}$. Then, the *cycle resolution* of $\pi$ is the finite (possibly empty) sequence of irreducible cycles $(c_1,\dots, c_{\nu})$ defined as follows: - $c_1$ is the first irreducible subcycle of $d_0=\pi$, that is, $c_1 = \pi(a_1,b_1)$ for some $a_1, b_1 \in \{1,\dots, len(d_0) \}$ with $a_1 < b_1$ is an irreducible subcycle of $d_0(1, b_1)$ and there is no irreducible subcycle $c'$ of $d_0(1,b_1)$ besides $c_1$. - $c_i$ for $i=2,\dots, \nu$ is the first irreducible subcycle of $d_{i} = d_{i-1} \backslash c_{i-1}$, that is, $c_i = d_i(a_i,b_i)$ for some $a_i, b_i \in \{1,\dots, len(d_i) \}$ with $a_i < b_i$ is an irreducible subcycle of $d_i(1, b_i)$ and there is no irreducible subcycle $c'_i$ on $d_i(1,b_i)$ besides $c_i$. - $d_{\nu} \backslash c_{\nu}$ is a cycle-free directed walk from $i$ to $j$ if $i \neq j$ and the trivial walk $(i)$ without edges if $i=j$. We call the resulting cycle-free walk $\mathrm{bs}(\pi) = d_{\nu} \backslash c_{\nu}$ the *base walk* of the resolution and say that $\pi$ *resolves to* $\pi'$ if $\pi' = \mathrm{bs}(\pi)$. The walk $\pi = (1 {\,\rightarrow\,}2 {\,\rightarrow\,}3 {\,\rightarrow\,}4 {\,\rightarrow\,}3 {\,\rightarrow\,}2)$ in $\mathcal{G}$ has the cycle resolution $(c_1,c_2)$ with $c_1 = (3,4,3)$ and $c_2 = (2,3,2)$ and the base walk $\mathrm{bs}(\pi) = (1,2)$. Let us record how insertion of cycles interacts with cycle resolutions. **Lemma 41**. *Let $\pi$ be a directed walk in $\mathcal{W}(\mathcal{G})$ with cycle resolution $(c_1,\dots, c_{\nu})$ and in-between walks $(d_1,\dots, d_{\nu})$ as in Definition [Definition 40](#def.nesting-resolution-walk){reference-type="ref" reference="def.nesting-resolution-walk"}. Let $\mathbf{c}$ be an irreducible cycle class with $\mathrm{nodes}(\pi) \cap \mathrm{nodes}(\mathbf{c}) \neq \emptyset$, let $j = \min \{ k ~\vert~ \pi(k) \in \mathrm{nodes}(\mathbf{c}) \}$ be the first node of $\pi$ that lies on $\mathbf{c}$ and let $c \in \mathbf{c}$ be the unique member of $\mathbf{c}$ starting and ending at $\pi(j)$.* *Consider the walk $\tilde{\pi} = \pi \cup_j c$ and its cycle resolution $(\tilde{c}_1,\dots, \tilde{c}_{\mu})$. Then, both of the following statements hold:* - *$\mu = \nu+1$ and for any $\tilde{c}_l$ we have that $\tilde{c}_l = c_k$ for some $k=1,\dots, \nu$ or $\tilde{c}_l = c$.* - *$\mathrm{bs}(\tilde{\pi}) = \mathrm{bs}(\pi)$.* *Proof of Lemma [Lemma 41](#lem.cycle_insertion){reference-type="ref" reference="lem.cycle_insertion"}.* Write $c = (c(1),\dots, c(s))$ and $\pi = (\pi(1),\dots, \pi(m))$ and $\tilde{\pi} = (\tilde{\pi}(1) = \pi(1),\dots, \tilde{\pi}(j) = c(1),\dots, \tilde{\pi}(j+s) = c(s),\dots, \tilde{\pi}(m+s) = \pi(m))$. Moreover, denote the in-between walks of $\tilde{\pi}$ by $\tilde{d}_0,\dots, \tilde{d}_{\mu}$. We prove the claim by induction over $\nu$. **Induction base case:** Suppose that $\nu = 0$. In this case, $\pi$ does not have any cycles and hence $\pi = \mathrm{bs}(\pi)$. Consider the first irreducible cycle $\tilde{c}_1 = \tilde{\pi}(a,b)$ of $\tilde{\pi}$. Then $b \leq j + s$ as otherwise $\tilde{c}_1$ would not be the first irreducible subcycle of $\tilde{\pi}$. We distinguish two cases. - Case 1: If $b <j+s$, then we must have $b>j$ as otherwise $\tilde{c}_1$ would be a subcycle of $\pi$, which was assumed to be cycle-free. In addition, we must have $a < j$ as otherwise $\tilde{c}_1$ would be a proper subcycle of $c$ which was assumed to be irreducible. But if $a<j$ and $j\leq b < j+s$, then $\tilde{\pi}(a) = \tilde{\pi}(b) \in \mathrm{nodes}(\mathbf{c})$ contradicts the definition of $j$ as the first touch point. In other words, $b <j+s$ is not possible. - Case 2: If $b = j+s$, then we must have $a \leq j$ as otherwise $\tilde{c}_1$ would be a proper subcycle of $c$ which was assumed to be irreducible. Similarly, we cannot have $a < j$ as in this case $c$ would be a proper subcycle of $\tilde{c}_1$ again contradicting irreducibility. Thus, $a = j$ such that $\tilde{c}_1 = c$ and $\tilde{d}_1 = \pi = \mathrm{bs}(\pi)$ is cycle-free. The cycle resolution of $\tilde{\pi}$ is thus $(c)$ and has length $\nu +1 = 1$. **Induction Hypothesis:** Suppose we have proven the claim for all $\nu$ with $0 \leq \nu \leq \nu^\prime$. **Induction Step:** Suppose that $\nu = \nu^\prime + 1$. Consider the first irreducible cycle $\tilde{c}_1 = \tilde{\pi}(a,b)$ of $\tilde{\pi}$. Then, as for the induction base case, $b \leq j + s$ as otherwise $\tilde{c}_1$ would not be the first irreducible subcycle of $\tilde{\pi}$. We distinguish three cases. - Case 1: If $j< b <j+s$, then we must have $a<j$ by irreducibility of $c$. But then we again find that $\tilde{\pi}(a) = \tilde{\pi}(b)$ violates the defining property of $j$. Thus, the case $j< b <j+s$ is impossible. - Case 2: If $b = j +s$, then we can argue exactly as for $\nu = 0$ that $a=j$ and thus $\tilde{c}_1 = c_1$ and $\tilde{d}_1 = \pi$, such that the cycle resolution of $\tilde{\pi}$ is $(c,c_1,\dots,c_{\nu})$ and $\mathrm{bs}(\tilde{\pi}) = \mathrm{bs}(\pi)$. - Case 3: If $b \leq j$, then $\tilde{c}_1$ must have been a subcycle of $\pi$ already, more precisely the first subcycle of $\pi$, that is, $\tilde{c}_1 = c_1$. Thus, the path $\pi_{\circ} = d_1$ has a cycle resolution of length $\nu - 1 = \nu^\prime$, its first touch point with $\mathbf{c}$ is $j_{\circ} = j- len(c_1)$ and $\mathrm{bs}(\pi) = \mathrm{bs}(\pi_{\circ})$ and $\tilde{\pi}\backslash c_1 = \pi_{\circ} \cup_{j_{\circ}} c$. Applying the induction hypothesis with $\pi_{\circ}$ replacing $\pi$ then completes this case. We have thus proven the claim. ◻ Cycle resolutions offer a convenient way to decompose the multi-weight of a directed walk. The proof of the following lemma is immediate. **Lemma 42**. *Let $\pi$ be a directed walk from $i$ to $j$ with cycle resolution $(c_1,\dots, c_{\nu})$ and base walk $\pi'$. Then, $$\begin{aligned} \mathbf{w}(\pi) = \mathbf{w}(\pi') + \sum_{i=1}^{\nu} \mathbf{w}(c_i)\, .\end{aligned}$$ In addition, for any $\mathbf{c}\in \mathcal{C}$ and any $c,c' \in \mathbf{c}$, we have $\mathbf{w}(c) = \mathbf{w}(c')$, so that $\mathbf{w}(\mathbf{c})= \mathbf{w}(c)$ is well-defined, that is, $\mathbf{w}(\mathbf{c})$ does not depend on the choice of $c \in \mathbf{c}$. $\qedsymbol$* Consider now the following map on the set of directed walks on a directed graph $\mathcal{G}$: $$\begin{aligned} &\Gamma: \mathcal{W}(\mathcal{G}) \to 2^{\mathcal{C}} \, \text{, where} \\ &\Gamma(\pi) = \{ \mathbf{c}\in \mathcal{C}~\vert~ \ \text{there is } c \in \mathbf{c}\text{ that appears on the cycle resolution of } \pi \} \subseteq \mathcal{C}\, .\end{aligned}$$ Then, with $\mathcal{W}(i, j)$ as in Problem 3 and $\mathcal{W}_0(i,j)$ as in Lemma 4.8, we readily obtain the set decomposition $$\begin{aligned} \mathcal{W}(i,j) &= \bigcup_{\pi' \in \mathcal{W}_0(i,j)} \{ \pi \in \mathcal{W}(i,j) ~\vert~ \mathrm{bs}(\pi) = \pi'\} \\ &= \bigcup_{\pi' \in \mathcal{W}_0(i,j)}\, \bigcup_{\mathcal S \subseteq \mathcal{C}} \{ \pi \in \mathcal{W}(i,j) ~\vert~ \mathrm{bs}(\pi) = \pi' \text{ and } \Gamma(\pi) = \mathcal S \}\\ &= \bigcup_{(\pi',\mathcal S) \in \mathcal{W}_0(i,j) \times 2^{\mathcal{C}}} \{ \pi \in \mathcal{W}(i,j) ~\vert~ \mathrm{bs}(\pi) = \pi' \text{ and } \Gamma(\pi) = \mathcal S \} \, .\end{aligned}$$ Note that the index set of this union is finite, so with this decomposition of $\mathcal{W}(i,j)$ we are starting to get close to the claim of Lemma [Lemma 37](#thm.support-thm-1){reference-type="ref" reference="thm.support-thm-1"}. However, some values $(\pi',\mathcal S ) \in \mathcal{W}_0(i,j) \times 2^{\mathcal{C}}$ might never appear when resolving walks. In other words, the restriction of $\Gamma$ to the set $\{ \pi \in \mathcal{W}(i,j) ~\vert~ \mathrm{bs}(\pi) = \pi'\}$ for a fixed $\pi' \in \mathcal{W}_0(i,j)$ might not be surjective. We therefore call a pair $(\pi',\mathcal S ) \in \mathcal{W}_0(i,j) \times 2^{\mathcal{C}}$ *admissible* if $$\mathcal S \in \mathrm{range}\left(\Gamma|_{\{ \pi \in \mathcal{W}(i,j) \vert \ \mathrm{bs}(\pi) = \pi'\}}\right) \equiv \mathrm{adm(\pi')} \, ,$$ and our next step is to compute the set $\mathrm{adm(\pi')} \subseteq 2^{\mathcal{C}}$ for any given element $\pi' \in \mathcal{W}_0(i,j)$. **Lemma 43**. *For given $\pi' \in \mathcal{W}_0(i,j)$, the equality of sets $\mathrm{adm(\pi')} = \mathcal{N}_{\pi'}$ holds.* *Proof of Lemma [Lemma 43](#lem.monoids){reference-type="ref" reference="lem.monoids"}.* **Inclusion $\mathrm{adm(\pi')} \subseteq \mathcal{N}_{\pi'}$:** Let $\mathcal{S} \in \mathrm{adm(\pi')}$. If $\mathcal{S} = \emptyset$, then $\mathcal{S} \in \mathcal{N}_{\pi'}$ because, as we noted above, $\mathcal{N}_{\pi'}$ contains the empty set. Thus, suppose that $\mathcal{S} \neq \emptyset$. We need to show that $\mathcal S$ is the union of nodes on one or multiple paths on the graph of cycles $\mathcal{G}_{\mathcal{C}}$ that start at the touch set $\mathrm{touch}(\pi')$. By definition of $\mathrm{adm(\pi')}$, there must be $\pi \in \mathcal{W}(i, j)$ that resolves to $\pi'$ and such that $\Gamma(\pi) = \mathcal{S}$. Consider the cycle resolution $(c_1,\dots,c_\mu)$ of such $\pi$ with in-between walks $d_{i+1}= d_{i}\backslash c_i$ and cycle equivalence classes $\mathbf{c}_i \ni c_i$. Note that $\mu \geq 1$ because $\mathcal{S}$ is non-empty. When deleting $c_i$ from $d_i$, the endpoints of $c_i$ are glued together at the incision point $j_i$. We now show that for any $c_i$ there exists a path $\chi_i$ on the graph of cycles such that 1. $\chi_i$ starts at $\mathbf{c}_i$ and ends at $\mathrm{touch}(\pi')$ and 2. all nodes of $\chi_i$ are elements of $\{ \mathbf{c}_1,\mathbf{c}_2,\dots, \mathbf{c}_{\mu}\} = \Gamma(\pi) = \mathcal{S}$. If this claim is proven, then $\cup_{i=1}^{\mu} \mathrm{nodes}(\overline{\chi_i}) = \Lambda_{\pi^\prime}(\overline{\chi_i}) = \mathcal{S} \in \mathcal{N}_{\pi^\prime}$, where $\overline{\chi_i}$ is $\chi_i$ read in the reverse direction. Indeed, the inclusion from left to right holds by property (b) whereas the inclusion from right to left holds because for all $i$ we have $\mathbf{c}_i \in \mathrm{nodes}(\overline{\chi_i})$ according to property (b). To prove the claim we used, consider the cycle $c_i$ and its incision point $j_i = c_i(1)$. Then, there are two options: 1. Either $j_i$ is a node of $\pi'$, in which case $\mathbf{c}_i \in \mathrm{touch}(\pi')$ and we can define $\chi'_i$ as the trivial path $\chi'_i = (\mathbf{c}_i)$ consisting of one node. 2. Or, $j_i \notin \mathrm{nodes}(\pi')$, in which case $j_i$ must have been removed at some later steps of the cycle resolution. Hence, there must be a cycle $c_{k_1}$ with $k_1 > i$ such that $j_i \in \mathrm{nodes}(c_{k_1})$. In particular, $c_i$ and $c_{k_1}$ share a node, so that either $\mathbf{c}_i = \mathbf{c}_{k_1}$ or $\mathbf{c}_i - \mathbf{c}_{k_1}$ is an edge in the graph of cycles. We can now repeat this argument for the incision point $j_{k_1}$ of $c_{k_1}$, leading us 1. either to a path $\chi_i' = (\mathbf{c}_i - \mathbf{c}_{k_1})$, in case $c_{k_1}(1) = j_{k_1} \in \mathrm{nodes}(\pi')$, 2. or to another cycle $c_{k_2}$ with $k_2 > k_1$ such that $j_{k_1} \in \mathrm{nodes}(c_{k_2})$, in case $j_{k_1} \notin \mathrm{nodes}(\pi')$. In the latter case, we continue recursively. This recursion terminates because the cycle resolution is a finite sequence and we obtain a walk $\chi_i' = (\mathbf{c}_i - \mathbf{c}_{k_1} - \cdots - \mathbf{c}_{k_l})$ with $\mathbf{c}_{k_l} \in \mathrm{touch}(\pi')$. These walks $\chi_i'$ satisfy both property (a) and (b), and by removing cycles from these walks we obtain paths $\xi_i$ that too satisfy both (a) and (b). **Converse inclusion $\mathrm{adm(\pi')} \supseteq \mathcal{N}_{\pi'}$:** Since $\pi'$ resolves to itself, we get that $\emptyset \in \mathcal{N}_{\pi'}$. Therefore, we can start with (possibly trivial) paths $\xi_1,\dots,\xi_r$ on the graph of cycles $\mathcal{G}_{\mathcal{C}}$ with $\xi_k(1) \in \mathrm{touch}(\pi')$ and such that $\cup_{i = 1}^r \Lambda_{\pi^\prime}(\xi_i) = \mathcal{T} \in \mathcal{N}_{\pi^\prime}$. Let us write the nodes of $\xi_1,\dots,\xi_r$, which are cycle classes, sequentially as $$L = (\xi_1(1),\dots, \xi_1(len(\xi_1)),\dots,\xi_r(1),\dots, \xi_r(len(\xi_r)))\, .$$ We insert the cycle class $\xi_1(1)$ into $\psi_0 = \pi'$ at position $i_1 = \min \left\{ i ~\vert~ \pi'(i) \in \mathrm{nodes}(\xi_1(1))\right\}$ to get $\psi_1 = \psi_0 \cup_{i_1} \xi_1(1)$.[^18] Similarly, we insert the $k$-th element $L_k$ of $L$ into $\psi_{k-1}$ at position $i_1 = \min \left\{ i ~\vert~ \psi_{k-1}(i) \in \mathrm{nodes}(L_k)\right\}$. This insertion is well-defined as $L_k$ always touches $\pi'$ or shares a node with $L_{k-1}$. By Lemma [Lemma 41](#lem.cycle_insertion){reference-type="ref" reference="lem.cycle_insertion"}, we have $\mathrm{bs}(\psi_k) = \mathrm{bs}(\psi_{k-1})$ and that the set $\mathcal{S}_k$ of cycles in the cycle resolution of $\psi_k$ is $\mathcal{S}_{k-1} \cup \{ L_k \}$ (starting with $\mathcal{S}_0 = \emptyset$). Therefore, after running through all of $L$, we end up with a path $\psi_N$ with $\mathrm{bs}(\psi_N) = \mathrm{bs}(\psi_{N-1}) = \dots = \mathrm{bs}(\psi_0) = \pi'$ and $\Gamma(\psi_N) = \mathcal S_N = \bigcup_{i=1}^r \mathrm{nodes}(\xi_i) = \mathcal{T}$ as desired. ◻ We are now ready to prove Lemma [Lemma 37](#thm.support-thm-1){reference-type="ref" reference="thm.support-thm-1"}. *Proof of Lemma [Lemma 37](#thm.support-thm-1){reference-type="ref" reference="thm.support-thm-1"}.* Lemma [Lemma 43](#lem.monoids){reference-type="ref" reference="lem.monoids"} and the discussions preceding Lemma [Lemma 43](#lem.monoids){reference-type="ref" reference="lem.monoids"} show that $\mathcal{W}(i,j)$ admits the decomposition $$\begin{aligned} \mathcal{W}(i,j) = \bigcup_{(\pi',\mathcal S) \in \mathcal{W}_0(i,j) \times \mathcal{N}_{\pi'}} \{ \pi \in \mathcal{W}(i,j) ~\vert~ \mathrm{bs}(\pi) = \pi' \text{ and } \Gamma(\pi) = \mathcal S \} \, .\end{aligned}$$ Thus, the set $$\begin{aligned} \mathbf{W}_{\tau}(i,j) = \bigcup_{\pi \in \mathcal{W}(i,j) } \mathbf{w}(\pi) + \{ \tau \}\end{aligned}$$ decomposes as $$\begin{aligned} \mathbf{W}_{\tau}(i,j) = \bigcup_{(\pi',\mathcal S) \in \mathcal{W}_0(i,j) \times \mathcal{N}_{\pi'}} B_{\tau}((\pi',\mathcal S))\end{aligned}$$ where $$\begin{aligned} B_{\tau}((\pi',\mathcal S)) = \bigcup_{ \{\pi \in \mathcal{W}(i,j) ~\vert~ \mathrm{bs}(\pi) = \pi' \text{ and } \Gamma(\pi) = \mathcal S \}} \mathbf{w}(\pi) + \{ \tau \} \, .\end{aligned}$$ Recall from Lemma [Lemma 42](#lem.weight-decomp){reference-type="ref" reference="lem.weight-decomp"} that the multi-weight of a walk in $\mathcal{W}(i,j)$ is the sum of the multi-weight of its base walk and the multi-weights of the cycles in its cycle resolution. For $\pi \in \mathcal{W}(i,j)$ such that $\mathrm{bs}(\pi) = \pi' \text{ and } \Gamma(\pi) = \mathcal S$, both the base walk and the equivalence classes that appear in the cycle resolution are fully specified, and the only free parameters are the non-negative integers $n_{\mathbf{c}}$ for $\mathbf{c}\in \mathcal{S}$ that specify how often the class $\mathbf{c}$ appears in the cycle resolution. Note also that once $\mathbf{c}\in \mathcal{S}$ appears, inserting $\mathbf{c}$ arbitrarily many times into the path $\pi$ immediately after its first appearance will lead to a new path $\tilde{\pi}$ such that $\mathrm{bs}(\pi) = \mathrm{bs}(\tilde{\pi})$ and $\Gamma(\pi) = \Gamma(\tilde{\pi})$. Thus, $$\begin{aligned} B_{\tau}((\pi',\mathcal S)) &= \{ \tau \}+ \mathbf{w}(\pi') + \sum_{\mathbf{c}\in \mathcal{S}} \{ n_{\mathbf{c}} \cdot w(\mathbf{c}) ~\vert~ n_{\mathbf{c}} \geq 1, \ w(\mathbf{c}) \in \mathbf{w}(\mathbf{c}) \}\\ &= \bigcup_{(a_0,\dots,a_{\mu}) \in E_{\tau}((\pi',\mathcal S))}\mathrm{con}_+(a_0;\dots,a_{\mu}) \end{aligned}$$ where the indexing tuple $(a_0;\dots,a_{\mu})$ in the second line runs over the finite set $$\begin{aligned} E_{\tau}((\pi',\mathcal S)) = (\{ \tau \} + \mathbf{w}(\pi') )\times \prod_{\mathbf{c}\in \mathcal{S}} \mathbf{w}(\mathbf{c}) \, .\end{aligned}$$ We have thus completed the proof. ◻ ### Deriving Lemma 4.8 from Lemma [Lemma 37](#thm.support-thm-1){reference-type="ref" reference="thm.support-thm-1"} {#subsec.deriving-Lemma} While Lemma [Lemma 37](#thm.support-thm-1){reference-type="ref" reference="thm.support-thm-1"} is sufficient to provide a finitary decomposition of $\mathbf{W}_{\tau}(i,j)$, this decomposition in terms of positive cones is not necessarily economical as the set $\mathcal{N}_{\pi'}$ is rather large. For instance, if the graph of cycles does not have any edges and, say, $k$ nodes all of which touch $\pi'$, then $\mathcal{N}_{\pi'}$ contains $2^k$ elements, each with one positive cone associated to it. In this example, we can efficiently summarize all of these positive cones into a single non-negative cone, which significantly reduces the number of Diophantine equations to be considered at a later stage. To implement this and similar efficiency improvement, we will introduce new sets $\mathcal{M}_{\pi'}$ that replace the sets $\mathcal{N}_{\pi'}$. Before doing so, we will need the concept of access points (defined below). *Definition 44*. Let $\mathcal{G}$ be an undirected graph with vertex set $\mathcal V$, let $\mathcal{S} \subseteq \mathcal V$, and let $v \in \mathcal V$ and $w \in \mathcal V\backslash\mathcal{S}$ be two nodes. We say that - $v$ is an $\mathcal{S}$-access point for $w$ if there exists a path $\xi(1) - \ldots - \xi(r-1) -\xi(r)$, where $r \geq 2$, such that $\xi(1) \in \mathcal{S}$ and $\xi(r-1) = v$ and $\xi(r) = w$. - $v$ is an $\mathcal{S}$-access point if it is an $\mathcal{S}$-access point for some $w \in \mathcal V\backslash\mathcal{S}$. We denote the set of all $\mathcal{S}$-access points by $\mathcal{A}_{(\mathcal{G},\mathcal{S})}$. Note that elements of $\mathcal{S}$ can, but do not have to be, $\mathcal{S}$-access points. Recall that, given a graph of cycles $\mathcal{G}_{\mathcal{C}}$ and a touch set $\mathrm{touch}(\pi')$, we defined $\mathcal{P}_{\pi'}$ as - the set of all paths on $\mathcal{G}_{\mathcal{C}}$ that start at the touch set $\mathrm{touch}(\pi')$ - plus the empty path. We now define $\mathcal{Q}_{\pi'} \subseteq \mathcal{P}_{\pi'}$ such that $\xi \in \mathcal{P}_{\pi^\prime}$ is in $\mathcal{Q}_{\pi'}$ if and only if $\xi$ is the empty path or - all nodes of $\xi$, including its starting node $\xi(1)$, are $\mathrm{touch}(\pi')$-access points - and the starting node $\xi(1)$ is the *only* node on $\xi$ that belongs to the touch set $\mathrm{touch}(\pi')$. In particular, $\mathcal{Q}_{\pi'}$ contains the empty path as well as all trivial paths through nodes in the intersection of $\mathrm{touch}(\pi')$ and the $\mathrm{touch}(\pi')$-access points. Moreover, if $\xi \in \mathcal{Q}_{\pi'}$, then $\xi(1)$ is an element of the touch set $\mathrm{touch}(\pi')$. Since $\mathcal{Q}_{\pi'} \subseteq \mathcal{P}_{\pi'}$, we can restrict the node set projection $\Lambda_{\pi^\prime}$ defined above to $\mathcal{Q}_{\pi'}$. Thus, we can consider the set monoid $\mathcal{M}_{\pi'}$ generated by $\mathcal{Q}_{\pi'}$ that is, the set monoid defined as the unique smallest subset $\mathcal{M}_{\pi'} \subseteq 2^{\mathcal{C}}$ such that - $\Lambda_{\pi'}(\mathcal{Q}_{\pi'}) \subseteq \mathcal{M}_{\pi'}$ and - if $\mathcal A, \mathcal B \in \mathcal{M}_{\pi'}$, then $\mathcal A \cup \mathcal B \in \mathcal{M}_{\pi'}$. *Remark 45*. The second condition in the definition of the set $\mathcal{Q}_{\pi'}$, which ensures that $\xi$ does not return to the touch set, is solely there to keep the generating set of the monoid $\mathcal{M}_{\pi'}$ as small as possible. If we would drop this condition and instead work with a set of paths $\mathcal{Q}_{\pi'}'$ without this requirement, then the same set monoid $\mathcal{M}_{\pi'}$ would be generated. This claim follows because, given a path that returns to the touch set, we can split this path into subpaths starting at every return point and ending at the predecessor of the next return point. For instance, if $\mathbf{c}_1 - \mathbf{c}_2 - \mathbf{c}_3 - \mathbf{c}_4 - \mathbf{c}_5$ is a path of access points with $\mathbf{c}_1,\mathbf{c}_3,\mathbf{c}_4 \in \mathrm{touch}(\pi')$, then we can split this path into its subpaths $\mathbf{c}_1 - \mathbf{c}_2$ and $(\mathbf{c}_3)$ and $\mathbf{c}_4-\mathbf{c}_5$. In addition, we define the *closure* $\mathrm{cl}(\mathcal S)$ of a set $\mathcal{S} \in \mathcal{M}_{\pi'}$ as $$\begin{aligned} \mathrm{cl}(\mathcal S) = \Big[\mathcal S &\cup \mathrm{touch}(\pi')\\ &\cup \{ \mathbf{c}\in \mathcal{C}\backslash\mathrm{touch}(\pi') ~\vert~ \mathcal S \text{ contains a } \mathrm{touch}(\pi')\text{-access point for } \mathbf{c}\}\Big] \, .\end{aligned}$$ In particular, the closure of the empty set is $\mathrm{cl}(\emptyset) = \mathrm{touch}(\pi').$ The following observation is straightforward. **Lemma 46**. *Given arbitrary $\mathcal{S}_i \in \mathcal{M}_{\pi'}$ sets for $i=1,\ldots,m$, the equality of sets $\cup_i \mathrm{cl}(\mathcal{S}_i) = \mathrm{cl} \left( \cup_i \mathcal{S}_i \right)$ holds. $\qedsymbol$* To prove Lemma 4.8, we use the following auxiliary result. **Lemma 47**. *Let $\psi \in \mathcal{P}_{\pi'}$ be a non-empty path. Then, one of the two following statements is true:* 1. *$\mathrm{nodes}(\psi) \subseteq \mathrm{touch}(\pi') = \mathrm{cl}(\emptyset)$.* 2. *There are non-empty subpaths $\chi^1,\dots, \chi^m \subseteq \psi$ of $\psi$ such that $\chi^1,\dots, \chi^m \in \mathcal{Q}_{\pi'}$ and $\mathrm{nodes}(\psi) \subseteq \mathrm{cl}\left( \cup_{k=1}^m \mathrm{nodes}(\chi^i) \right)$.* *Proof of Lemma [Lemma 47](#lem.advanced-path-decomposition){reference-type="ref" reference="lem.advanced-path-decomposition"}.* Suppose that $\mathrm{nodes}(\psi) \centernot\subseteq \mathrm{touch}(\pi')$, that is, suppose for $\psi$ the first of the two statements in Lemma [Lemma 47](#lem.advanced-path-decomposition){reference-type="ref" reference="lem.advanced-path-decomposition"} is not true, and write $\psi=(\psi(1),\dots, \psi(r))$. Then, we first note that the set $\mathrm{nodes}(\psi)$ contains at least one $\mathrm{touch}(\pi')$-access point. This claim follows from the following argument: By assumption, there must be $1 \leq k \leq r$ such that $\psi(k) \notin \mathrm{touch}(\pi')$. Moreover since $\psi$ starts in $\mathrm{touch}(\pi')$ by definition of $\mathcal{P}_{\pi^\prime}$, we must have that $k \geq 2$ and that $\psi(k-1)$ is a $\mathrm{touch}(\pi')$-access point for $\psi(k)$.\ **Claim 1**: Let $\psi(k_1)$ be the first $\mathrm{touch}(\pi')$-access point on $\psi$. Then, all of the nodes $\psi(1),\dots, \psi(k_1-1), \psi(k_1)$ are elements of $\mathrm{touch}(\pi')$.\ By definition, $\psi(1) \in \mathrm{touch}(\pi')$. Thus, the claim follows if $k_1=1$, otherwise let $2 \leq l \leq k_1$. If we had that $\psi(l) \notin \mathrm{touch}(\pi')$, then $\psi(l-1)$ were a $\mathrm{touch}(\pi')$-access point with $l-1 < k_1$, thus contradicting the fact that $\psi(k_1)$ is the first such access point on $\psi$. Consequently, we have proven the above Claim 1.\ Next, returning to the overall proof of Lemma [Lemma 47](#lem.advanced-path-decomposition){reference-type="ref" reference="lem.advanced-path-decomposition"}, consider the set $$\begin{aligned} A = \{ l ~\vert~ \ k_1 < l \leq r, \ \psi(l) \in \mathrm{touch}(\pi') \text{ or } \psi(l) \text{ is not a $\mathrm{touch}(\pi')$-access point} \} \, . \end{aligned}$$ There are two options: - **Case 1**: $A = \emptyset$. In this case, the path $\chi^1 = (\psi(k_1),\dots, \psi(r))$ is an element of $\mathcal{Q}_{\pi'}$ and $\mathrm{nodes}(\psi) \subseteq \mathrm{cl}(\mathrm{nodes}(\chi^1) )$. Indeed, $\chi^1$ starts in the touch set by definition of $k_1$ and, since $A$ is empty, all nodes on $\chi^1$ are $\mathrm{touch}(\pi')$-access and $\chi^1$ never returns to $\mathrm{touch}(\pi')$ after the first node, so that $\chi^1 \in \mathcal{Q}_{\pi'}$. Moreover, since all nodes of $\psi$ up to $\psi(k_1)$ belong to the touch set by the above Claim 1 and since all nodes on $\psi$ after $\psi(k_1)$ belong to $\mathrm{nodes}(\chi^1)$, the inclusion $\mathrm{nodes}(\psi) \subseteq \mathrm{cl}(\mathrm{nodes}(\chi^1) )$ follows by definition of the closure. Therefore, in this case, the second statement in Lemma [Lemma 47](#lem.advanced-path-decomposition){reference-type="ref" reference="lem.advanced-path-decomposition"} holds for $\psi$. - **Case 2**: $A \neq \emptyset$. In this case, set $l_1 = \min A$, implying that $l_1 > k_1$ and that $\psi(l_1)$ is a touch point or not a $\mathrm{touch}(\pi')$-access point. Consider the path $\chi^1 = (\psi(k_1),\dots, \psi(l_1-1))$, which is non-empty since $l_1 > k_1$. Since $l_1$ is the minimal element of $A$, all nodes of $\chi_1$ must be $\mathrm{touch}(\pi')$-access points and none of these nodes other than $\psi(k_1)$ belongs to the touch set. Therefore, $\chi^1 \in \mathcal{Q}_{\pi'}$. Moreover, observe that $\psi(l_1) \in \mathrm{cl}(\mathrm{nodes}(\chi^1))$: either $\psi(l_1) \in \mathrm{touch}(\pi')$ and the claim follows, or not, in which case $\psi(l_1-1)$ is a $\mathrm{touch}(\pi')$-access point for $\psi(l_1)$. As a consequence, using that all nodes on $\psi$ up to $\psi(k_1)$ belong to the touch set, we have that $\{ \psi(1),\dots, \psi(l_1) \} \subseteq \mathrm{cl}(\mathrm{nodes}(\chi^1))$. If $l_1 = r$, then we have thus proven that the second statement in Lemma [Lemma 47](#lem.advanced-path-decomposition){reference-type="ref" reference="lem.advanced-path-decomposition"} holds for $\psi$. Therefore, we suppose that $l_1<r$ and consider the remaining path $(\psi(l_1),\dots, \psi(r))$. We consider two subcases: - **Case 2a**: If $\psi(l_1) \in \mathrm{touch}(\pi')$, then we set $\psi^1 = (\psi(l_1),\dots, \psi(r))$. This non-empty path $\psi^1$ is in $\mathcal{P}_{\pi'}$ because $\psi(l_1) \in \mathrm{touch}(\pi')$. Moreover, $\psi^1$ is strictly shorter than $\psi$. - **Case 2b**: If $\psi(l_1) \notin \mathrm{touch}(\pi')$, then $\psi(l_1)$ is not a $\mathrm{touch}(\pi')$-access point according to the above definition of the set $A$ and $l_1 \in A$. We then argue that $\psi(l_1+1)$ must in the touch set. Indeed, if $\psi(l_1+1)$ was not in the touch point, then the path $(\psi(k_1),\dots, \psi(l_1),\psi(l_1+1))$ would make $\psi(l_1)$ a $\mathrm{touch}(\pi')$-access point for $\psi(l_1+1)$, a contradiction. Thus, the non-empty path $\psi^1 = (\psi(l_1+1),\dots, \psi(r))$ is an element $\mathcal{P}_{\pi'}$ and strictly shorter than $\psi$. To summarize, in both subcases we get that $\mathrm{nodes}(\psi) \subseteq \mathrm{cl}(\mathrm{nodes}(\chi^1)) \cup \mathrm{nodes}(\psi^1)$ with $\chi^1 \in \mathcal{Q}_{\pi'}$ and $\psi^1 \in \mathcal{P}_{\pi'}$ with $len(\psi^1) < len(\psi)$. We can now reapply the whole argument with $\psi^1$ replacing $\psi$ and continue recursively to obtain paths $\chi^1,\dots, \chi^m \in \mathcal{Q}_{\pi'}$ such that $\mathrm{nodes}(\psi) \subseteq \mathrm{cl}\left(\cup_{k=1}^m \mathrm{nodes}(\chi^i) \right)$, thus showing that the second statement of Lemma [Lemma 47](#lem.advanced-path-decomposition){reference-type="ref" reference="lem.advanced-path-decomposition"} holds for $\psi$. We have thus completed the proof. ◻ Returning to our eventual goal of proving Lemma 4.8, for any non-negative integer $\tau \geq 0$ and any $\mathcal{S} \in \mathcal{M}_{\pi'}$ with any $\pi' \in \mathcal{W}_0(k,i)$, we define the set $$\label{def.ctau} \begin{split} C_{\tau}((\pi',\mathcal S)) &= \{ \tau \} + \mathbf{w}(\pi') + \mathbf{w}(\mathcal{S}) + \sum_{\mathbf{c}\in \mathrm{cl}(\mathcal{S})} \{ n_{\mathbf{c}} \cdot w(\mathbf{c}) ~\vert~ n_{\mathbf{c}} \geq 0, \ w(\mathbf{c}) \in \mathbf{w}(\mathbf{c}) \} \\ &= \bigcup_{(a_0,\dots,a_{\mu}) \in D_{\tau}((\pi',\mathcal S))}\mathrm{con}(a_0;\dots,a_{\mu}) \end{split}$$ where $$\begin{aligned} \label{def.tuple-sets} D_{\tau}((\pi',\mathcal S))= \left( \{ \tau \} + \mathbf{w}(\pi') + \mathbf{w}(\mathcal{S}) \right) \times \prod_{\mathbf{c}\in \mathrm{cl}(\mathcal{S})} \mathbf{w}(\mathbf{c}) \end{aligned}$$ and $\mathbf{w}(\mathcal{S}) = \sum_{\mathbf{c}\in \mathcal{S}} \mathbf{w}(\mathbf{c})$. We have now defined all sets involved in the statement of Lemma 4.8 and can finish its proof. *Proof of Lemma 4.8.* To deduce Lemma 4.8 from Lemma [Lemma 37](#thm.support-thm-1){reference-type="ref" reference="thm.support-thm-1"}, we only need to show that for every $\pi' \in \mathcal{W}_0(k,i)$ and $\tau \geq 0$ the equality of sets $$\begin{aligned} \bigcup_{\mathcal{T} \in \mathcal{N}_{\pi'}} \,\,\bigcup_{(a_0,\dots,a_{\mu}) \in E_{\tau}(\pi',\mathcal T)}\mathrm{con}_+(a_0;\dots,a_{\mu}) = \bigcup_{\mathcal{S} \in \mathcal{M}_{\pi'}}\,\, \bigcup_{(a_0,\dots,a_{\mu}) \in D_{\tau}(\pi',\mathcal S)}\mathrm{con}(a_0;\dots,a_{\mu})\end{aligned}$$ or, equivalently, that the equality of sets $$\begin{aligned} \bigcup_{\mathcal{T} \in \mathcal{N}_{\pi'}} B_{\tau}((\pi',\mathcal T)) = \bigcup_{\mathcal{S} \in \mathcal{M}_{\pi'}} C_{\tau}((\pi',\mathcal S))\end{aligned}$$ holds.We show the set inclusions $\subseteq$ and $\supseteq$ separately. **Inclusion $\subseteq$**: Let $\mathcal{T} \in \mathcal{N}_{\pi'}$. First, we observe that $B_{\tau}((\pi',\emptyset)) = \{\tau \} + \mathbf{w}(\pi')$ and that $C_{\tau}((\pi',\emptyset)) = \{\tau \} + \mathbf{w}(\pi') +\sum_{\mathbf{c}\in \mathrm{touch}(\pi')} \{ n_{\mathbf{c}} \cdot w(\mathbf{c}) ~\vert~ n_{\mathbf{c}} \geq 0, \ w(\mathbf{c}) \in \mathbf{w}(\mathbf{c}) \}$, so that $B_{\tau}((\pi',\emptyset)) \subseteq C_{\tau}((\pi',\emptyset))$. Therefore, we can now assume that $\mathcal{T} \neq \emptyset$. Our goal is to construct a set $\mathcal{S} \in \mathcal{M}_{\pi'}$ such that $B_{\tau}((\pi',\mathcal T)) \subseteq C_{\tau}((\pi',\mathcal S))$. By definition of the monoid $\mathcal{N}_{\pi'}$, we can write $\mathcal{T}$ as $\mathcal{T} = \cup_{j=1}^r \Lambda_{\pi'}(\psi_j) = \cup_{j=1}^r \mathrm{nodes}(\psi_j)$ for some non-empty paths $\psi_1,\dots,\psi_r \in \mathcal{P}_{\pi'}$ on the graph of cycles $\mathcal{G}_{\mathcal{C}}$ that start at the touch set. Consider the path $\psi_j$. By Lemma [Lemma 47](#lem.advanced-path-decomposition){reference-type="ref" reference="lem.advanced-path-decomposition"}, we have that $\mathrm{nodes}(\psi_j) \subseteq \mathrm{touch}(\pi')$ or there exists a finite sequence of non-empty paths $\chi_1^j,\dots,\chi_{m_j}^j \in \mathcal{Q}_{\pi'}$, all of which are subpaths of $\psi_j$, such that $\mathrm{nodes}(\psi_j) \subseteq \mathrm{cl}( \cup_{k=1}^{m_j} \mathrm{nodes}(\chi_k^j))$. We set $$\begin{aligned} \mathcal{S}= \bigcup_{j} \, \bigcup_{k=1}^{m_j} \mathrm{nodes}(\chi_k^j) \in \mathcal{M}_{\pi'} \, ,\end{aligned}$$ where $j$ runs over all indices for which $\mathrm{nodes}(\psi_j) \centernot\subseteq \mathrm{touch}(\pi')$ (if no such indices exist, then we set $\mathcal{S} = \emptyset \in \mathcal{M}_{\pi'}$). Using Lemma [Lemma 46](#lem.closures){reference-type="ref" reference="lem.closures"}, it then follows that $$\begin{aligned} \mathrm{cl}(\mathcal{S}) &= \mathrm{touch}(\pi') \cup \mathrm{cl}(\mathcal{S})\\ &= \mathrm{touch}(\pi') \cup \bigcup_j\, \mathrm{cl} \left( \bigcup_{k=1}^{m_j} \mathrm{nodes}(\chi_k^j)\right) \supseteq \mathrm{touch}(\pi') \cup \bigcup_j \,\mathrm{nodes}(\psi_j) \supseteq \mathcal{T} \, .\end{aligned}$$ Moreover, we find that $$\begin{aligned} \mathcal{S} = \bigcup_{j} \, \bigcup_{k=1}^{m_j} \mathrm{nodes}(\chi_k^j) \subseteq \bigcup_{j} \, \mathrm{nodes}(\psi^j) \subseteq \mathcal{T} \, .\end{aligned}$$ Now, recalling the notation $n_\mathbf{c}\cdot \mathbf{w}(\mathbf{c}) = \{ n_{\mathbf{c}} \cdot w(\mathbf{c}) ~\vert~ w(\mathbf{c}) \in \mathbf{w}(\mathbf{c}) \}$, any element $y \in B_{\tau}((\pi',\mathcal T))$ is an element of $$\begin{aligned} \{\tau \} + \mathbf{w}(\pi') + \sum_{\mathbf{c}\in \mathcal{T}} n_\mathbf{c}\cdot \mathbf{w}(\mathbf{c})\end{aligned}$$ for some combination $(n_{\mathbf{c}})_{\mathbf{c}\in \mathcal{T}}$ of positive integers $n_\mathbf{c}\geq 1.$ Hence, we can conclude that $$\begin{aligned} \{\tau \} + \mathbf{w}(\pi') + \sum_{\mathbf{c}\in \mathcal{T}} n_\mathbf{c}\cdot \mathbf{w}(\mathbf{c}) &= \{\tau \} + \mathbf{w}(\pi') + \mathbf{w}(\mathcal{S}) \\ &+ \sum_{\mathbf{c}\in \mathcal{S}} (n_\mathbf{c}-1) \cdot \mathbf{w}(\mathbf{c}) + \sum_{\mathbf{c}\in \mathcal{T}\backslash\mathcal{S}} n_\mathbf{c}\cdot \mathbf{w}(\mathbf{c}) \\ &\subseteq C_{\tau}((\pi',\mathcal S))\end{aligned}$$ and thus $B_{\tau}((\pi',\mathcal T)) \subseteq C_{\tau}((\pi',\mathcal S))$. Here, the '$=$' is just a rewriting of the left-hand-side that uses $\mathcal{S} \subseteq \mathcal{T}$ (as shown above), and the '$\subseteq$' follows because $\mathcal{S} \subseteq \mathrm{cl}(\mathcal{S})$ (by definition of the closure) and $\mathcal{T} \subseteq \mathrm{cl}(\mathcal{S})$ (as shown above). **Converse Inclusion $\supseteq$**: Let $\mathcal{S} \in \mathcal{M}_{\pi'}$ and write $C_{\tau}((\pi',\mathcal S))$ as the union of sets of the form $\{\tau \} + \mathbf{w}(\pi') + \mathbf{w}(\mathcal{S}) + \sum_{\mathbf{c}\in \mathrm{cl}(\mathcal{S})} n_\mathbf{c}\cdot \mathbf{w}(\mathbf{c})$ where, this time, $(n_{\mathbf{c}})_{\mathbf{c}\in \mathcal{S}}$ is a finite sequence of non-negative integers $n_\mathbf{c}\geq 0$ (instead of positive integers). We can rewrite such a set as $$\begin{aligned} \{\tau \} + \mathbf{w}(\pi') + \sum_{\mathbf{c}\in \mathcal{S}} (n_\mathbf{c}+1) \cdot \mathbf{w}(\mathbf{c}) + \sum_{\mathbf{c}\in \mathrm{cl}(\mathcal{S}) \backslash\mathcal{S}} n_\mathbf{c}\cdot \mathbf{w}(\mathbf{c}) \, .\end{aligned}$$ Consider the set $\mathcal{T} = \mathcal{S} \cup \{\mathbf{c}\in \mathrm{cl}(\mathcal{S})\backslash\mathcal{S} ~\vert~ n_{\mathbf{c}} \geq 1 \}$ and set $m_{\mathbf{c}} = n_\mathbf{c}+1$ if $\mathbf{c}\in \mathcal{S}$ and $m_{\mathbf{c}} = n_\mathbf{c}$ if $\mathbf{c}\in \mathrm{cl}(\mathcal{S})\backslash\mathcal{S}$ and $n_{\mathbf{c}} \geq 1$. In other words, we simply drop all those summands in the second sum on the right-hand-side for which $n_\mathbf{c}= 0$. Thus, the above set is equal to $\{\tau \} + \mathbf{w}(\pi') + \sum_{\mathbf{c}\in \mathcal{T}} m_\mathbf{c}\cdot \mathbf{w}(\mathbf{c})$ with positive coefficients $m_\mathbf{c}\geq 1$. Therefore, we have finished the proof if we can argue that $\mathcal{T}$ is an element of the monoid $\mathcal{N}_{\pi'}$, as we do now. If $\mathcal{S}$ is empty, then $\mathcal{T}$ is a subset of $\mathrm{touch}(\pi') = \mathrm{cl}(\emptyset)$, such that we can write $\mathcal{T}$ as a finite union of trivial paths in $\mathcal{P}_{\pi'}$, and hence $\mathcal{T} \in \mathcal{N}_{\pi^\prime}$. If $\mathcal{S} \neq \emptyset$, then, by definition of $\mathcal{M}_{\pi^\prime}$, we can write $\mathcal{S} = \cup_{j=1}^r \mathrm{nodes}(\xi_j)$ for some non-empty paths $\xi_j \in \mathcal{Q}_{\pi'} \subseteq \mathcal{P}_{\pi'}$ with $j=1,\dots,r$. For $\mathbf{c}' \in \mathrm{cl}(\mathcal{S})\backslash\mathcal{S}$ there are two cases: - If $\mathbf{c}' \in \mathrm{touch}(\pi')$, then we let $\psi^{\mathbf c'}$ be the trivial path $(\mathbf c')$, which is an element of $\mathcal{P}_{\pi'}$. - If $\mathbf{c}' \notin \mathrm{touch}(\pi')$, then, using the definition of the closure, we see that $\mathcal{S}$ contains a $\mathrm{touch}(\pi')$-access point $\tilde{\mathbf{c}}$ for $\mathbf{c}'$. Because $\mathcal{S} = \cup_{j=1}^r \mathrm{nodes}(\xi_j)$, this access point $\tilde{\mathbf c}$ lies on $\xi_j$ for some $1 \leq j \leq r$, say $\tilde{\mathbf{c}} = \xi_j(k)$. Consequently, the path $\psi^{\mathbf{c}'} = (\xi_j(1),\dots, \xi_j(k),\mathbf{c}')$ exists and is an element of $\mathcal{P}_{\pi'}$. Finally, we obtain that $\mathcal{T} = [\cup_{j=1}^r \mathrm{nodes}(\xi_j)] \cup [\cup_{\mathbf{c}' \in \mathcal{T}\backslash\mathcal{S}} \mathrm{nodes}(\psi^{\mathbf{c}'})] \in \mathcal{N}_{\pi'}$. ◻ ## Proofs of all theorems *Proof of Theorem 1.* We have already proven Theorem 1 in the main paper under the assumption that Theorem 2 holds. Below, we prove Theorem 2 without making use of Theorem 1 in that proof. ◻ *Proof of Theorem 2.* We prove the statement by considering the five cases separately: 1. [$\mu = 0$ and $\nu = 0$.]{.ul} In this case, eq. (3) reduces to the trivial equation $c = 0$. 2. [$\mu = 0$ and $\nu \neq 0$ and $c > 0$.]{.ul} If $c \!\! \mod g_{a^\prime} \neq 0$ with $\gcd(a^\prime_1, \, \ldots, \, a^\prime_\nu)$, then according to Lemma 4.12 there is no integer solution. In particular, there is then no non-negative integer solution, thus proving the "only-if" statement in the first sentence. For the "if-and-only-if" statement in the second sentence, observe that eq. (3) reduces to $c = \sum_{\beta \,=\, 1}^{\nu} n^\prime_\beta \cdot a^\prime_\beta$ where $c > 0$ and $a^\prime_\beta > 0$ for all $1 \leq \beta \leq \nu$. The statement follows because every term in the sum on the right-hand-side of this equation is positive. 3. [($\mu = 0$ and $\nu \neq 0$ and $c \leq 0$) or ($\mu \neq 0$ and $\nu = 0$ and $c \geq 0$).]{.ul} We first consider the subcase $\mu = 0$ and $\nu \neq 0$ and $c \leq 0$. Then, eq. (3) reduces to $c = \sum_{\beta \,=\, 1}^{\nu} n^\prime_\beta \cdot a^\prime_\beta$ where $c \leq 0$ and $a^\prime_\beta > 0$. Recalling that every term in the sum on the right-hand-side of this equation is positive, there is a non-negative solution only if $c = 0$. Conversely, if $c = 0$, then $n^\prime_1 = \dots = n^\prime_\nu = 0$ is a non-negative solution. The subcase $\mu \neq 0$ and $\nu = 0$ and $c \geq 0$ is equivalent to the first subcase up to replacing $c$ with $-c$ and replacing $\sum_{\beta \,=\, 1}^{\nu} n^\prime_\beta \cdot a^\prime_\beta$ with $\sum_{\alpha \,=\, 1}^{\mu} n_\beta \cdot a_\alpha$. 4. [$\mu \neq 0$ and $\nu = 0$ and $c < 0$.]{.ul} This case is equivalent to case 2 up to replacing $c$ with $-c$ and replacing $\sum_{\beta \,=\, 1}^{\nu} n^\prime_\beta \cdot a^\prime_\beta$ with $\sum_{\alpha \,=\, 1}^{\mu} n_\beta \cdot a_\alpha$. 5. [$\mu \neq 0$ and $\nu \neq 0$.]{.ul} If $c \!\! \mod \! g_{aa^\prime} \neq 0$ with $g_{aa^\prime} = \gcd(a_1, \, \ldots, \, a_\mu, a^\prime_1, \, \ldots, \, a^\prime_\nu,)$, then according to Lemma 4.12 there is no integer solution. In particular, there is then no non-negative integer solution, thus proving the "only-if" part. For the "if" part, suppose that $c \!\! \mod \! g_{aa^\prime} = 0$. We can then divide eq. (3) by $g_{aa^\prime}$ to get the modified linear Diophantine equation $$\label{eq:linear-diophantine-gcd-1-in-proof} \tilde{c} + \sum_{\alpha \,=\, 1}^{\mu} n_\alpha \cdot \tilde{a}_\alpha = \sum_{\beta \,=\, 1}^{\nu} m_\beta \cdot \tilde{a}_\beta^\prime \, ,$$ where $\tilde{c} = \tfrac{c}{g_{aa^\prime}}$ is an integer and $\tilde{a}_\alpha = \tfrac{a_\alpha}{g_{aa^\prime}}$ for all $1 \leq \alpha \leq \mu$ as well as $\tilde{a}_\beta^\prime = \tfrac{a^\prime_\beta}{g_{aa^\prime}}$ for all $1 \leq \beta \leq \nu$ are positive integers. Note that $\gcd(\tilde{a}_1, \, \ldots, \, \tilde{a}_\mu, \, \tilde{a}^\prime_1, \, \ldots, \, \tilde{a}^\prime_\nu) = 1$ by definition of the $\tilde{a}_\alpha$ and $\tilde{a}^\prime_\beta$, and that $\gcd(\tilde{a}_1, \, \ldots, \, \tilde{a}_\mu, \, \tilde{a}^\prime_1, \, \ldots, \, \tilde{a}^\prime_\nu)$ by associativity of the greatest common divisor equals $\gcd(\gcd(\tilde{a}_1, \, \ldots, \, \tilde{a}_\mu), \, \gcd(\tilde{a}^\prime_1, \, \ldots, \, \tilde{a}^\prime_\nu))$. Thus, letting $\tilde{g}_a = \gcd(\tilde{a}_1, \, \ldots, \, \tilde{a}_\mu)$ and $\tilde{g}_{a^\prime} = \gcd(\tilde{a}^\prime_1, \, \ldots, \, \tilde{a}^\prime_\nu)$, we get $\gcd(\tilde{g}_a, \, \tilde{g}_{a^\prime}) = 1$. In addition, since all $\tilde{a}_1, \, \ldots, \, \tilde{a}_\mu$ and all $\tilde{a}^\prime_1, \, \ldots, \, \tilde{a}^\prime_\nu$ are positive, also both $\tilde{g}_a$ and $\tilde{g}_{a^\prime}$ are positive. Now consider the linear Diophantine equation $$\label{eq:simple-linear-diophantine-in-proof} \tilde{c} + \tilde{g}_a \cdot k_a = \tilde{g}_{a^\prime} \cdot k_b$$ with unknowns $k_a$ and $k_b$. Since $\gcd(\tilde{g}_a, \, \tilde{g}_{a^\prime}) = 1$, according to Lemma 4.12 there is at least one integer solution $(k_a, k_b) = (k_a^0, k_b^0)$ to this equation. Then, for any integer $q$, also $(k_a, k_b) = (k_a^0 + q \cdot \tilde{g}_{a^\prime}, k_b^0 + q \cdot \tilde{g}_a)$ is an integer solution to eq. [\[eq:simple-linear-diophantine-in-proof\]](#eq:simple-linear-diophantine-in-proof){reference-type="eqref" reference="eq:simple-linear-diophantine-in-proof"}. Since both $\tilde{g}_a$ and $\tilde{g}_{a^\prime}$ are positive, for any pair of integers $k_a^{\text{min}}$ and $k_b^{\text{min}}$, we can choose $q$ sufficiently large such that $k_a^0 + q \cdot \tilde{g}_{a^\prime} \geq k_a^{\text{min}}$ and at the same time $k_b^0 + q \cdot \tilde{g}_a \geq k_b^{\text{min}}$. Thus, there is an integer $\tilde{q}$ such that $k_a^0 + \tilde{q} \cdot \tilde{g}_{a^\prime} \geq \tilde{g}_a \cdot f(\tfrac{\tilde{a}_1}{\tilde{g}_a}, \, \ldots, \, \tfrac{\tilde{a}_\mu}{\tilde{g}_a})$ and at the same time $k_b^0 + \tilde{q} \cdot \tilde{g}_a \geq \tilde{g}_{a^\prime} \cdot f(\tfrac{\tilde{a}^\prime_1}{\tilde{g}_{a^\prime}}, \, \ldots, \,\tfrac{\tilde{a}^\prime_\nu}{\tilde{g}_{a^\prime}})$. Since $\gcd(\tfrac{\tilde{a}_1}{\tilde{g}_a}, \, \ldots, \, \tfrac{\tilde{a}_\mu}{\tilde{g}_a}) = 1$ and $\gcd(\tfrac{\tilde{a}^\prime_1}{\tilde{g}_{a^\prime}}, \, \ldots, \,\tfrac{\tilde{a}^\prime_\nu}{\tilde{g}_{a^\prime}}) = 1$, the Frobenius numbers $f(\tfrac{\tilde{a}_1}{\tilde{g}_a}, \, \ldots, \, \tfrac{\tilde{a}_\mu}{\tilde{g}_a})$ and $f(\tfrac{\tilde{a}^\prime_1}{\tilde{g}_{a^\prime}}, \, \ldots, \,\tfrac{\tilde{a}^\prime_\nu}{\tilde{g}_{a^\prime}})$ exist according to Lemma 4.13. Again according to Lemma 4.13, there are non-negative integer $n_1, \, \ldots, \, n_\mu$ such that $k_a^0 + \tilde{q} \cdot \tilde{g}_{a^\prime} = \sum_{\alpha \, = \,1}^\mu n_\alpha \cdot \tfrac{\tilde{a}_\alpha}{\tilde{g}_{a}}$ as well as non-negative integers $n^\prime_1, \, \ldots, \, n^\prime_\nu$ such that $k_b^0 + \tilde{q} \cdot \tilde{g}_a = \sum_{\beta \, = \,1}^\nu n^\prime_\beta \cdot \tfrac{\tilde{a}^\prime_\beta}{\tilde{g}_{a^\prime}}$. With these choices we arrive at $$\begin{aligned} \tilde{c} + \tilde{g}_a \cdot \left(\sum_{\alpha \, = \,1}^\mu n_\alpha \cdot \frac{\tilde{a}_\alpha}{\tilde{g}_a}\right) &= \tilde{g}_{a^\prime} \cdot \left(\sum_{\beta \, = \,1}^\nu n^\prime_\beta \cdot \frac{\tilde{a}^\prime_\alpha}{\tilde{g}_{a^\prime}}\right) \\ % \Leftrightarrow \qquad c +\sum_{\alpha \, = \,1}^\mu n_\alpha \cdot a_\alpha &= \sum_{\beta \, = \,1}^\nu n^\prime_\beta \cdot a^\prime_\beta\, .\end{aligned}$$ The equivalence of these two equations follows by multiplication with $g_{aa^\prime}$ because $g_{aa^\prime} \cdot \tilde{a}_\alpha = a_\alpha$ and $g_{aa^\prime} \cdot \tilde{a}^\prime_\beta = a^\prime_\beta$ and $c = g_{aa^\prime} \cdot \tilde{c}$. Thus, $n_1, \, \ldots , \, n_\mu, \, m_1, \, \ldots , \, m_\nu$ is a non-negative integer solution to original linear Diophantine equation (3). We have thus proven all of the five cases and thereby completed the proof. ◻ *Definition 48* (Notation for the below statements and proofs). Given a tuple $(a_0; a_1,\ldots,a_{\mu})$ with $\mu \geq 1$ where $a_0 \in \mathbb{N}_0$ and $a_\alpha \in \mathbb{N}$ for all $1 \leq \alpha \leq \mu$, we write $\mathbf{a} = (a_0;a_1,\dots,a_{\mu})$ and $g(\mathbf{a}) = \gcd(a_1,\ldots,a_{\mu})$. Note that the indices inside the $\gcd$ start with $1$, that is, the $\gcd$ does not include $a_0$ as argument. **Lemma 49**. *For a tuple $\mathbf{a} = (a_0,a_1,\ldots,a_{\mu})$ as in Definition [Definition 48](#def:notation-proof-theorem-3){reference-type="ref" reference="def:notation-proof-theorem-3"}, let the positive integer $D \in \mathbb N$ be such that $$\begin{aligned} (D-a_0) \!\!\!\!\!\mod g(\mathbf{a}) = 0 \quad \text{and} \quad D > a_0 + g(\mathbf{a})\cdot f\left(\tfrac{a_1}{g(\mathbf{a})},\ldots, \tfrac{a_{\mu}}{g(\mathbf{a})}\right)\end{aligned}$$ where $f(\cdot)$ denotes the Frobenius number (cf. Lemma 4.13). Then, $D \in \mathrm{con}(a_0;a_1,\ldots,a_{\mu})$, with $\mathrm{con}(\cdot)$ as defined by eq. (1).* *Proof of Lemma [Lemma 49](#lem.first-estimates){reference-type="ref" reference="lem.first-estimates"}.* This claim is an immediate consequence of the defining property of the Frobenius number: By assumption, $\tfrac{D-a_0}{g(\mathbf{a})}$ is a positive integer larger than $f(\tfrac{a_1}{g(\mathbf{a})},\ldots, \tfrac{a_{\mu}}{g(\mathbf{a})})$. Therefore, by the definition of the Frobenius property, there are non-negative integers $n_1,\ldots,n_{\mu}$ such that $\tfrac{D-a_0}{g(\mathbf{a})} = \sum_{\alpha=1}^{\mu} n_{\alpha} \cdot \tfrac{a_{\alpha}}{g(\mathbf{a})}$, which rearranges to $D = a_{0}+ \sum_{\alpha=1}^{\mu} n_{\alpha} \cdot a_{\alpha} \in \mathrm{con}(a_0;a_1,\ldots,a_{\mu})$. ◻ **Corollary 50**. *For tuples $\mathbf{a} = (a_0,a_1,\ldots,a_{\mu})$ and $\mathbf{a}' = (a'_0,a'_1,\ldots,a'_{\nu})$ as in Definition [Definition 48](#def:notation-proof-theorem-3){reference-type="ref" reference="def:notation-proof-theorem-3"}, suppose there is a non-negative integer $D \in \mathbb N$ such that $$\begin{aligned} (D-a_0) \!\!\!\!\!\mod g(\mathbf{a}) = 0 \quad \text{and} \quad (D-a^\prime_0) \!\!\!\!\!\mod g(\mathbf{a}^\prime) = 0\end{aligned}$$ as well as $$\begin{aligned} D > \max \left[ a_0 + g(\mathbf{a})\cdot f\left(\tfrac{a_1}{g(\mathbf{a})},\ldots, \tfrac{a_{\mu}}{g(\mathbf{a})}\right), \, a_0' + g(\mathbf{a}')\cdot f\left(\tfrac{a'_1}{g(\mathbf{a}')},\ldots, \tfrac{a'_{\nu}}{g(\mathbf{a})}\right) \right] \, .\end{aligned}$$ Then, $D \in \mathcal{B}_1(\mathbf{a},\mathbf{a}') \equiv \mathrm{con}(a_1;a_1,\ldots,a_{\mu}) \cap \mathrm{con}(a'_0;a'_1,\ldots,a'_{\nu})$.* *Proof of Corollary [Corollary 50](#cor:b1aaprime){reference-type="ref" reference="cor:b1aaprime"}.* This claim immediately follows from Lemma [Lemma 49](#lem.first-estimates){reference-type="ref" reference="lem.first-estimates"}. ◻ **Lemma 51**. *For tuples $\mathbf{a} = (a_0,a_1,\ldots,a_{\mu})$ and $\mathbf{a}' = (a'_0,a'_1,\ldots,a'_{\nu})$ as in Definition [Definition 48](#def:notation-proof-theorem-3){reference-type="ref" reference="def:notation-proof-theorem-3"}, suppose $(a_0'-a_0)\!\!\! \mod g(\mathbf{a},\mathbf{a'})= 0$ where $g(\mathbf{a},\mathbf{a'}) = \gcd(a_1,\ldots,a_{\mu},a'_1,\ldots,a'_{\nu})$. Then, there exists $D \in \mathcal{B}_1(\mathbf{a},\mathbf{a}')$ such that $$\begin{aligned} \label{ineq.D-bound} D \leq \max(a_0,a_0') + \frac{g(\mathbf{a}')\cdot g(\mathbf{a})}{g(\mathbf{a},\mathbf{a'})} &\cdot \Bigg[\frac{|a_0'-a_0|}{g(\mathbf{a},\mathbf{a'})} + \nonumber \\ &\max\left[ f\left(\tfrac{a_1}{g(\mathbf{a})},\ldots, \tfrac{a_{\mu}}{g(\mathbf{a})}\right),f\left(\tfrac{a'_1}{g(\mathbf{a}')},\ldots, \tfrac{a'_{\nu}}{g(\mathbf{a})}\right)\right] +1 \Bigg] \, . \end{aligned}$$* *Proof of Lemma [Lemma 51](#lem.D-upper-estimate){reference-type="ref" reference="lem.D-upper-estimate"}.* Due to symmetry, we can without loss of generality assume that $a_0 \leq a_0'$. Then, $\max(a_0,a_0') =a_0'$. By Lemma [Lemma 49](#lem.first-estimates){reference-type="ref" reference="lem.first-estimates"}, we have $$\begin{aligned} \underbrace{(D_0(\mathbf{a}),\infty)}_{\text{open interval}} \cap \left\{a_0 + g(\mathbf{a}\} \cdot \mathbb Z \right\} \subseteq \mathrm{con}(a_0;a_1,\ldots,a_{\mu})\, ,\end{aligned}$$ where $$\begin{aligned} D_0(\mathbf{a}) = a_0 + g(\mathbf{a})\cdot \left[ f\left(\tfrac{a_1}{g(\mathbf{a})},\ldots, \tfrac{a_{\mu}}{g(\mathbf{a})}\right) +1 \right] \, ,\end{aligned}$$ and an analogous result holds for $\mathbf{a}'$ instead of $\mathbf{a}$. Therefore, $\mathcal{B}_1(\mathbf{a},\mathbf{a}')$ contains the set $$\begin{aligned} (D_0(\mathbf{a}) ,\infty) \cap (D_0(\mathbf{a}') ,\infty) \cap \mathcal{B}_2(\mathbf{a},\mathbf{a}) \subseteq \mathcal{B}_1(\mathbf{a},\mathbf{a}') \, ,\end{aligned}$$ where $$\begin{aligned} \mathcal{B}_2(\mathbf{a},\mathbf{a}) = \left\{a_0 + g(\mathbf{a}) \cdot \mathbb Z \right\} \cap \left\{a'_0 + g(\mathbf{a}') \cdot \mathbb Z \right\}\, . \end{aligned}$$ Using the assumption that $\gcd(g(\mathbf{a}),g(\mathbf{a}')) = g(\mathbf{a},\mathbf{a'})$ divides $(a_0'-a_0)$, we see that the set $\mathcal{B}_2(\mathbf{a},\mathbf{a})$ is non-empty because the equation $$\begin{aligned} \label{eq.help-short-Diophantine} g(\mathbf{a}) \cdot n - g(\mathbf{a}') \cdot m = a_0'-a_0.\end{aligned}$$ admits a solution $(n,m) \in \mathbb{Z} \times \mathbb{Z}$. Namely, if $u,v \in \mathbb Z$ are any integers for which $$\begin{aligned} g(\mathbf{a}) \cdot u - g(\mathbf{a}') \cdot v = g(\mathbf{a},\mathbf{a'})\end{aligned}$$ (noting that $g(\mathbf{a},\mathbf{a'}) = \gcd(g(\mathbf a), g(\mathbf a'))$, such a pair of integers exists according to Bézout's identity), then the solutions of eq. [\[eq.help-short-Diophantine\]](#eq.help-short-Diophantine){reference-type="eqref" reference="eq.help-short-Diophantine"} are pairs $(n,m)$ of integers of the form $$\begin{aligned} n = \frac{a_0'-a_0}{g(\mathbf{a},\mathbf{a'})}\cdot u + t \cdot \frac{g(\mathbf{a'})}{g(\mathbf{a},\mathbf{a'})} \, , \quad m = \frac{a_0'-a_0}{g(\mathbf{a},\mathbf{a'})}\cdot v + t \cdot \frac{g(\mathbf{a})}{g(\mathbf{a},\mathbf{a'})}\end{aligned}$$ for arbitrary $t \in \mathbb{Z}$. Note that for $v$ we can choose a value $v_0$ such that $0 \leq v_0 \leq \tfrac{g(\mathbf{a})}{g(\mathbf{a},\mathbf{a'})}$, which we will do going forward. Now choose $$\begin{aligned} t_0 &= \max\left[ f\left(\tfrac{a_1}{g(\mathbf{a})},\ldots, \tfrac{a_{\mu}}{g(\mathbf{a})}\right),\,f\left(\tfrac{a'_1}{g(\mathbf{a}')},\ldots, \tfrac{a'_{\nu}}{g(\mathbf{a})}\right)\right] +1 \, ,\\ m_0 &= \frac{a_0'-a_0}{g(\mathbf{a},\mathbf{a'})}\cdot v_0 + t_0 \cdot \frac{g(\mathbf{a})}{g(\mathbf{a},\mathbf{a'})}\, , \\ D &= a_0' + g(\mathbf{a}')\cdot m_0 \, .\end{aligned}$$ Then, the choice of $m_0$ implies that $D \in \mathcal{B}_2(\mathbf{a},\mathbf{a})$. Moreover, since $m_0 \geq t_0 \cdot \tfrac{g(\mathbf{a})}{g(\mathbf{a},\mathbf{a'})}$, we have $$\begin{aligned} D \geq a_0' + \frac{g(\mathbf{a}')\cdot g(\mathbf{a})}{g(\mathbf{a},\mathbf{a'})} \cdot t_0 \geq D_0(\mathbf{a'})\end{aligned}$$ and $$\begin{aligned} D \geq a_0' + \frac{g(\mathbf{a}')\cdot g(\mathbf{a})}{g(\mathbf{a},\mathbf{a'})} \cdot t_0 \geq a_0 + \frac{g(\mathbf{a}')\cdot g(\mathbf{a})}{g(\mathbf{a},\mathbf{a'})} \cdot t_0 \geq D_0(\mathbf{a}).\end{aligned}$$ We have thus shown that $$\begin{aligned} D \in (D_0(\mathbf{a}) ,\infty) \cap (D_0(\mathbf{a}') ,\infty) \cap \mathcal{B}_2(\mathbf{a},\mathbf{a}) \subseteq \mathcal{B}_1(\mathbf{a},\mathbf{a}')\, .\end{aligned}$$ Finally, we need to show that $D$ satisfies inequality [\[ineq.D-bound\]](#ineq.D-bound){reference-type="eqref" reference="ineq.D-bound"}. Indeed, since $v_0 \leq \tfrac{g(\mathbf{a})}{g(\mathbf{a},\mathbf{a'})}$, we have $$\begin{aligned} g(\mathbf{a}')\cdot m_0 &\leq \frac{g(\mathbf{a})g(\mathbf{a}')}{g(\mathbf{a},\mathbf{a'})} \cdot \left( \frac{a_0'-a_0}{g(\mathbf{a},\mathbf{a'})} + t_0 \right) \\ &= \frac{g(\mathbf{a})g(\mathbf{a}')}{g(\mathbf{a},\mathbf{a'})} \cdot \left\{ \frac{a_0'-a_0}{g(\mathbf{a},\mathbf{a'})} + \max\left[ f\left(\tfrac{a_1}{g(\mathbf{a})},\ldots, \tfrac{a_{\mu}}{g(\mathbf{a})}\right),f\left(\tfrac{a'_1}{g(\mathbf{a}')},\ldots, \tfrac{a'_{\nu}}{g(\mathbf{a})}\right)\right] +1 \right\}.\end{aligned}$$ Since $\max(a_0,a_0') =a_0'$ and $|a_0'-a_0| = a_0'-a_0$, inequality [\[ineq.D-bound\]](#ineq.D-bound){reference-type="eqref" reference="ineq.D-bound"} follows. ◻ **Lemma 52**. *Let $\tau,\tau' \geq 0$, let $i,j,k$ be (not necessarily distinct) vertices of a multi-weighted graph $(\mathcal{G},\mathbf{w})$, let $\pi \in \mathcal{W}_0(k,i)$ and $\pi' \in \mathcal{W}_0(k,j)$ be cycle-free walks, and let $\mathcal{S}\in \mathcal{M}_{\pi}$ and $\mathcal{S}' \in \mathcal{M}_{\pi'}$. Consider tuples $\mathbf{a} = (a_0,a_1,\ldots,a_{\mu}) \in D_{\tau}(\pi,\mathcal{S})$ and $\mathbf{a}' = (a'_0,a'_1,\ldots,a'_{\nu}) \in D_{\tau'}(\pi',\mathcal{S}')$ for which $\mu,\nu \geq 1$ and $\mathcal{B}_1(\mathbf{a},\mathbf{a}') \neq \emptyset$. Then, there exists an element $D \in \mathcal{B}_1(\mathbf{a},\mathbf{a}')$ such that $$\begin{aligned} D \leq \left(K^2+1\right) \cdot \left[\max\left(\tau,\tau'\right) + L +M \right] + K\left[\left(K-1\right)^2 +1\right] \, .\end{aligned}$$* *Proof of Lemma [Lemma 52](#lem.main-p-estimate){reference-type="ref" reference="lem.main-p-estimate"}.* Using Lemma 4.12, we see that the assumption $\mathcal{B}_1(\mathbf{a},\mathbf{a}') \neq \emptyset$ implies that $g(\mathbf{a},\mathbf{a}')$ divides $(a_0'-a_0)$. Then, by Lemma [Lemma 51](#lem.D-upper-estimate){reference-type="ref" reference="lem.D-upper-estimate"}, there exists $D \in \mathcal{B}_1(\mathbf{a},\mathbf{a}')$ that satisfies inequality [\[ineq.D-bound\]](#ineq.D-bound){reference-type="eqref" reference="ineq.D-bound"}. We use the bounds (using the quantities defined in Theorem 3) $$\begin{aligned} g(\mathbf{a}) \leq \max_{i=1,\ldots, \mu} a_i \leq K \qquad &\text{and} \qquad g(\mathbf{a}') \leq \max_{i=1,\ldots, \nu} a'_i \leq K \quad \text{and}\\ a_0 \leq \tau + L + M \qquad &\text{and} \qquad a_0' \leq \tau' + L +M \, ,\end{aligned}$$ which directly follow from the definition of the sets $D_{\tau}(\pi,\mathcal{S})$ and $D_{\tau'}(\pi',\mathcal{S}')$. The latter two inequalities in particular imply that $$\begin{aligned} \max(a_0,a_0') \leq \max(\tau,\tau') + L +M\end{aligned}$$ and thus $$\begin{aligned} &\max(a_0,a_0') + \frac{g(\mathbf{a}')\cdot g(\mathbf{a})}{g(\mathbf{a},\mathbf{a'})} \cdot \frac{|a_0'-a_0|}{g(\mathbf{a},\mathbf{a'})} \\ &\leq (K^2 +1) \cdot \max(a_0,a_0') \\ &\leq (K^2+1) \cdot \left(\max(\tau,\tau') + L+M\right)\, .\end{aligned}$$ Using the Brauer-Schur bound on the Frobenius number, see Remark 4.14, we get $$\begin{aligned} &\frac{g(\mathbf{a}')\cdot g(\mathbf{a})}{g(\mathbf{a},\mathbf{a'})} \cdot \left[f\left(\tfrac{a_1}{g(\mathbf{a})},\ldots, \tfrac{a_{\mu}}{g(\mathbf{a})}\right) +1 \right] \\ &\leq g(\mathbf{a}') \cdot g(\mathbf{a}) \cdot \left[ \left( \max_{i=1,\ldots, \mu} \tfrac{a_i}{g(\mathbf{a})} -1\right) \left( \min_{i=1,\ldots, \mu} \tfrac{a_i}{g(\mathbf{a})} -1\right) +1\right] \\ &\leq g(\mathbf{a}') \cdot \left[ \left( \max_{i=1,\ldots, \mu} a_i -1\right) \left( \min_{i=1,\ldots, \mu} \tfrac{a_i}{g(\mathbf{a})} -1 \right) +1\right] \\ &\leq g(\mathbf{a}') \cdot \left[ \left( \max_{i=1,\ldots, \mu} a_i -1 \right)^2 +1\right] \\ &\leq K \cdot \left[\left(K-1\right)^2 +1\right] \, . \end{aligned}$$ Combining inequality [\[ineq.D-bound\]](#ineq.D-bound){reference-type="eqref" reference="ineq.D-bound"} and the previous bounds, the claim follows. ◻ *Proof of Theorem 3.* The "if" direction is trivial. For the "only-if" direction, assume that $(i, t-\tau)$ and $(j,t)$ have a common ancestor $(k,t-\tau_k)$ in $\mathcal{D}$. Let $\mathcal{S}_{\mathbf{w}}(\mathcal{D})$ be the multi-weighted summary graph of $\mathcal{D}$. According to Theorem 1, there are tuples $\mathbf{a} = (a_0,\ldots,a_{\mu}) \in D_{\tau}(\pi_0, \mathcal{S})$ and $\mathbf{a}' = (a'_0,\ldots,a'_{\nu}) \in D_{0}(\pi_0', \mathcal{S}')$ with $\pi_0$, $\mathcal{S}$, $\pi_0'$ and $\mathcal{S}'$ as in that theorem, such that $\mathcal{B}_1(\mathbf{a},\mathbf{a}') \neq \emptyset$. Here, $\mu =0$ and/or $\nu = 0$ is allowed, and we need to distinguish the following three mutually exclusive and collectively exhaustive cases: 1. [$\nu =0$:]{.ul} In this case, $\tau_k = a_0' \in \mathbf{w}(\pi_0')+\mathbf{w}(\mathcal{S}')$. Thus, $\tau_k \leq L+M \leq p'(\mathcal{D})$. 2. [$\nu \neq 0$ and $\mu = 0$:]{.ul} In this case, $\tau_k = \tau + a_0 \in \{\tau\} + \mathbf{w}(\pi_0)+\mathbf{w}(\mathcal{S})$. Thus, $\tau_k \leq \tau + L+M \leq p'(\mathcal{D})$. 3. [$\nu \neq 0$ and $\mu \neq 0$:]{.ul} In this last case, Lemma [Lemma 52](#lem.main-p-estimate){reference-type="ref" reference="lem.main-p-estimate"} applies (setting $\tau' = 0$), and we get the existence of a common ancestor $(k,t-\tau_k^\prime)$ of $(i, t-\tau)$ and $(j,t)$ with $\tau_k^\prime \leq p'(\mathcal{D})$.  ◻ # Pseudocode {#sec:pseudocode} Here, we provide pseudocode for our solution of Problem 1 (construction of finite marginal ts-ADMGs) and Problem 2 (common-ancestor search in ts-DAGs) by means of the number-theoretic approach of Theorem 1 and Theorem 2. The pseudocode stays close to the theoretical results that it is based on and, for the purpose of simplicity, ignores practically important aspects such as memoization. Time series ADMG $\mathcal{A}$ with variable index set $\mathbf{I}$, non-empty subset $\mathbf{I}_{\mathbf{O}}\subseteq \mathbf{I}$ corresponding to the observable component time series, length of the observed time window $p\in \mathbb{N}_0$ Marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{A})$ where $\mathbf{O}= \mathbf{I}_{\mathbf{O}}\times \mathbf{T}_{\mathbf{O}}$ with $\mathbf{T}_{\mathbf{O}}= \{t-\tau ~\vert~ 0 \leq \tau \leq p\}$ $\mathbf{O}$ $\leftarrow$ $\mathbf{I}_{\mathbf{O}}\times \mathbf{T}_{\mathbf{O}}$ with $\mathbf{T}_{\mathbf{O}}= \{t-\tau ~\vert~ 0 \leq \tau \leq p\}$ $\mathcal{D}$ $\leftarrow$ $\mathcal{D}^{ts}_{c}(\mathcal{A})$ $\mathcal{A}^\prime$ $\leftarrow$ [simple-ts-admg($\mathcal{D}$, $p$)]{.smallcaps} $\mathcal{A}_{out}$ $\leftarrow$ $\sigma_{\text{ADMG}}(\mathbf{O}, \mathcal{A}^\prime)$ $\mathcal{A}_{\mathbf{O}}(\mathcal{A}) = \mathcal{A}_{out}$ Algorithm [\[algo:ts-ADMG\]](#algo:ts-ADMG){reference-type="ref" reference="algo:ts-ADMG"} provides pseudocode for the top-level function [ts-admg]{.smallcaps} that returns the finite marginal ts-ADMG $\mathcal{A}_{\mathbf{O}}(\mathcal{A})$ of an infinite ts-ADMG $\mathcal{A}$, where $\mathbf{O}= \mathbf{I}_{\mathbf{O}}\times \mathbf{T}_{\mathbf{O}}$ with $\mathbf{T}_{\mathbf{O}}= \{t-\tau ~\vert~ 0 \leq \tau \leq p\}$. This function implements the reduction steps explained in Sections 3.3.1 and 3.3.2, see also Figure 4, and calls the function [simple-ts-admg]{.smallcaps} that performs the simple marginal ts-ADMG projection. Time series DAG $\mathcal{D}$ with variable index set $\mathbf{I}$, observed time window length $p\in \mathbb{N}_0$ Simple marginal ts-ADMG $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ with $\mathbf{O}^\prime = \mathbf{I}_{\mathbf{O}}\times \mathbf{T}_{\mathbf{O}}$ with $\mathbf{T}_{\mathbf{O}}= \{t-\tau ~\vert~ 0 \leq \tau \leq p\}$ $\mathcal{G}$ $\leftarrow$ copy of the segment of $\mathcal{D}$ on $[t-p, \ldots, t]$ $v_1, v_2$ $\leftarrow$ $(i, t-\tau_j-\Delta\tau), (j, t-\tau_j)$ $w_1, w_2$ $\leftarrow$ $(k, t-\tau_k+\tau_l), (l, t)$ $w_1, w_2$ $\leftarrow$ $(k, t), (l, t-\tau_l+\tau_k)$ $\tilde{v}_1, \tilde{v}_2$ $\leftarrow$ $(i, t-\tilde{\tau}_j-\Delta\tau), (j, t-\tilde{\tau}_j)$ $\mathcal{G}$ $\leftarrow$ Add $\tilde{v}_1 {\,\leftrightarrow\,}\tilde{v}_2$ to $\mathcal{G}$ $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D}) = \mathcal{G}$ Algorithm [\[algo:simple-ts-ADMG\]](#algo:simple-ts-ADMG){reference-type="ref" reference="algo:simple-ts-ADMG"} provides pseudocode for the function [simple-ts-ADMG]{.smallcaps} that returns the simple finite marginal ts-ADMG $\mathcal{A}_{\mathbf{O}^\prime}(\mathcal{D})$ of an infinite ts-DAG $\mathcal{D}$, where $\mathbf{O}^\prime = \mathbf{I}\times \mathbf{T}_{\mathbf{O}}$. This function implements the results in Section 3.3.3 and calls a function [have-common-ancestor]{.smallcaps} that decides about common-ancestorship in ts-DAGs. Time series DAG $\mathcal{D}$ with variable index set $\mathbf{I}$, vertices $(i,t-\tau)$ and $(j, t)$ with $\tau \geq 0$ in $\mathcal{D}$ (not necessarily distinct) As Boolean value: Whether or not $(i,t-\tau)$ and $(j, t)$ have a common ancestor $\mathcal{G}_{w}$ $\leftarrow$ $\mathcal{S}_{\mathbf{w}}(\mathcal{D})$ $\mathcal{C}$ $\leftarrow$ [get-cycle-classes]{.smallcaps}($\mathcal{G}_{w}$) $\mathbf{w}$-$\mathrm{dict}$ $\leftarrow$ [get-cycle-weights]{.smallcaps}($\mathcal{C}, \mathcal{G}_{w}$) $\mathcal{W}_0(k,i) \ \leftarrow$ [get-cycle-free-paths]{.smallcaps}($k,i, \mathcal{G}_{w}$) $\mathcal{W}_0(k,j) \ \leftarrow$ [get-cycle-free-paths]{.smallcaps}($k,j, \mathcal{G}_{w}$) $\mathbf{w}(\pi)$ $\leftarrow$ [get-weightset]{.smallcaps}($\pi, \mathcal{G}_{w}$) $\mathbf{w}(\pi')$ $\leftarrow$ [get-weightset]{.smallcaps}($\pi', \mathcal{G}_{w}$) $\mathcal{M}_{\pi} \ \leftarrow$ [get-monoid]{.smallcaps}($\pi, \mathcal{C}$) $\mathcal{M}_{\pi'} \ \leftarrow$ [get-monoid]{.smallcaps}($\pi', \mathcal{C}$) $D_{\tau}(\pi,\mathcal S)$ $\leftarrow$ [tuple-set]{.smallcaps}($\tau$, $\pi$, $\mathcal{S}, \mathbf{w}$-$\mathrm{dict}$, $\mathbf{w}(\pi)$) $D_{0}(\pi',\mathcal S')$ $\leftarrow$ [tuple-set]{.smallcaps}($0$, $\pi'$, $\mathcal{S}', \mathbf{w}$-$\mathrm{dict}$, $\mathbf{w}(\pi)$) [True]{.smallcaps} [False]{.smallcaps} Cycle-free path $\pi$, set of irreducible cycle classes $\mathcal{C}$. Set monoid $\mathcal{M}_{\pi}$, a subset of the power set of $\mathcal{C}$. $\mathrm{touch}(\pi)$ $\leftarrow$ [compute-touch-set]{.smallcaps}$(\pi,\mathcal{C})$ $\mathcal{G}_{\mathcal{C}}$ $\leftarrow$ [compute-graph-of-cycles]{.smallcaps}$(\pi,\mathcal{C})$ $\mathcal{A}_{\mathcal{C},\pi}$ $\leftarrow$ [compute-access-points]{.smallcaps}$(\mathcal{G}_{\mathcal{C}},\mathrm{touch}(\pi))$ $\mathcal{Q}_{\pi}$ $\leftarrow$ [compute-generating-set]{.smallcaps}$(\mathcal{G}_{\mathcal{C}},\mathrm{touch}(\pi),\mathcal{A}_{\mathcal{C},\pi})$ $\mathcal{Q}_{\pi}'$ $\leftarrow$ [set-project]{.smallcaps}$(\mathcal{Q}_{\pi})$ $\mathcal{M}_{\pi}$ $\leftarrow$ [compute-monoid-from-generating-set]{.smallcaps}$(\mathcal{Q}_{\pi}')$ $\mathcal{M}_{\pi}$ Algorithm [\[algo:have-common-ancestor\]](#algo:have-common-ancestor){reference-type="ref" reference="algo:have-common-ancestor"} provides pseudocode for the function [have-common-ancestor]{.smallcaps} that returns whether or not the vertices $(i,t-\tau)$ and $(j, t)$ have a common ancestor in an infinite ts-DAG $\mathcal{D}$. Here, common-ancestorship is understood in the sense of Remark 3.9. This function jointly implements Proposition 4.6 and Theorem 1 and relies on the subroutines [get-cycle-classes]{.smallcaps}, [get-cycle-free-paths]{.smallcaps}, [get-cycle-weights]{.smallcaps}, [get-weightset]{.smallcaps}, [get-monoid]{.smallcaps}, [tuple-set]{.smallcaps} and [has-nni-solution]{.smallcaps}. The first function [get-cycle-classes]{.smallcaps} computes a list of all equivalence classes of irreducible cycles in the multi-weighted summary graph, and the second function [get-cycle-free-paths]{.smallcaps} takes two indices $k$ and $i$ and the multi-weighted summary graph as an input and returns all cycle-free directed paths from $k$ to $i$.[^19] Since implementations of irreducible cycle enumeration and search routines for cycle-free paths are freely available, we will not provide further details on these two subroutines. Next, the function [get-cycle-weights]{.smallcaps} computes a dictionary that assigns the weight-set $\mathbf{w}(\mathbf{c})$ to every $\mathbf{c}\in \mathcal{C}$ from the edge weight-sets of the multi-weighted graph $\mathcal{G}_{w} = \mathcal{S}_{\mathbf{w}}(\mathcal{D})$. Since this procedure amounts to a simple execution of the definition of $\mathbf{w}(\mathbf{c})$ as the sums over the weight-sets of the edges on $\mathbf{c}$, we omit the pseudocode for this subroutine. The function [get-weightset]{.smallcaps} is very similar to [get-cycle-weights]{.smallcaps} as it computes the weight-set of a cycle-free path $\pi$ as the sum of the weight-sets of the path's edges. Again, we we do not provide pseudocode for this simple operation. The function [get-monoid]{.smallcaps} is the most complicated subroutine, and we provide pseudocode for it in Algorithm [\[algo:get-monoid\]](#algo:get-monoid){reference-type="ref" reference="algo:get-monoid"} below. The function [tuple-set]{.smallcaps} takes the delay $\tau$, a cycle-free path $\pi$, a set of irreducible cycle classes $\mathcal{S}$ as well as the weight-set dictionary for cycle classes $\mathbf{w}$-$\mathrm{dict}$ and the weight-set $\mathbf{w}(\pi)$ as input and computes the direct product set $D_{\tau}(\pi,\mathcal S)$ from its defining equation [\[def.tuple-sets\]](#def.tuple-sets){reference-type="eqref" reference="def.tuple-sets"}. Again, this procedure is a simple computation of a product set from simpler sets and is straightforward to execute, which is why we do not provide pseudocode. Finally, the function [has-nni-solution]{.smallcaps} returns (as Boolean value) whether a linear Diophantine equation as in eq. (3) as non-negative integer solution. We omit pseudocode for this function, since, at least in its most basic implementation, such pseudocode amounts to a one-to-one translation of Theorem 2. See also the paragraph below Theorem 2 for a discussion on how to improve on that basic implementation. Thus, we are left with describing pseudocode for the subroutine [get-monoid]{.smallcaps}, which we do in Algorithm [\[algo:get-monoid\]](#algo:get-monoid){reference-type="ref" reference="algo:get-monoid"}. This algorithm reduces to the subroutines [compute-touch-set]{.smallcaps}, [compute-graph-of-cycles]{.smallcaps}, [compute-access-points]{.smallcaps}, [compute-generating-set]{.smallcaps} as well as [compute-monoid-from-generating-set]{.smallcaps}. The function [compute-touch-set]{.smallcaps} for every cycle class $\mathbf{c}$ checks where $\mathbf{c}$ shares a node with $\pi$ and, if this is the case, then adds $\mathbf{c}$ to the touch set. Similarly, [compute-graph-of-cycles]{.smallcaps} checks for all pairs $\mathbf{c}\neq \mathbf{c}' \in \mathcal{C}$ whether the two cycles share any nodes and, if this is the case, then adds an edge $(\mathbf{c},\mathbf{c}')$ to the graph of cycles. The function [set-project]{.smallcaps} takes in a set of paths $\{ \xi \}$ and computes the set of node sets $\{ \mathrm{nodes}(\xi) \}$, which is a straighforward operation for which we do not provide pseudocode. Undirected Graph $\mathcal{G}$ with vertex set $\mathcal{V}$, subset $\mathcal{S} \subseteq \mathcal{V}$. Set of $\mathcal{S}$-access points on $\mathcal{G}$. $\mathcal{A}(\mathcal{G},\mathcal{S})$ $\leftarrow$ $\emptyset$ $\mathcal{A}(\mathcal{G},\mathcal{S})$ $\leftarrow$ Add $v$ to $\mathcal{A}(\mathcal{G},\mathcal{S})$ **break loop starting in line 5** $\mathcal{A}(\mathcal{G},\mathcal{S})$. Algorithm [\[algo:compute-access-points\]](#algo:compute-access-points){reference-type="ref" reference="algo:compute-access-points"} implements [compute-access-points]{.smallcaps}. This algorithms assumes (1) the availability of a function [neighbors]{.smallcaps}, which computes the set of direct neighbors of a node $v$ on an undirected graph $\mathcal{G}$, (2) a function [subset-delete]{.smallcaps} that inputs a subset of nodes $\mathcal{V}'$ on an undirected graph $\mathcal{G}$ with vertex set $\mathcal{V} \supseteq \mathcal{V}'$ and returns the subgraph $\mathcal{G}\backslash\mathcal{V}'$ from which all nodes in $\mathcal{V}'$ and their adjacent edges have been deleted, and (3) the availability of a function [is-path]{.smallcaps}, which decides whether there is a (possibly trivial) path between two given nodes $v$ and $v'$ in an undirected graph $\mathcal{G}$. All of these functions are available in standard graph theory packages and easily implementable, which is why we do not provide pseudocode. An undirected graph $\mathcal{G}$ with vertex set $\mathcal{V}$, a subset of nodes $\mathcal S \subseteq \mathcal{V}$, the set $\mathcal{A}(\mathcal{G},\mathcal{S})$ of $\mathcal{S}$-access points on $\mathcal{G}$. The set of paths $\mathcal{Q}$ starting in and not returning to $\mathcal{S}$ with nodes in $\mathcal{A}(\mathcal{G},\mathcal{S})$ . $\mathcal{H}$ $\leftarrow$ [subset-delete]{.smallcaps}($\mathcal{V}\backslash\mathcal{A}(\mathcal{G},\mathcal{S}),\mathcal{G}$) $\mathcal{Q}$ $\leftarrow$ $\{\emptyset \}$ add trivial path $(v)$ to $\mathcal{Q}$ $\mathcal{H}_v$ $\leftarrow$ [subset-delete]{.smallcaps}($\left(\mathcal{S} \cap \mathcal{A}(\mathcal{G},\mathcal{S})\right)\backslash\{ v \},\mathcal{H}$) $\mathcal{Q}$ $\leftarrow$ $\mathcal{Q} \cup$ [get-paths]{.smallcaps}($v,v',\mathcal{H}_v$ ) $\mathcal{Q}$ Next, in Algorithm [\[algo:compute-generating-set\]](#algo:compute-generating-set){reference-type="ref" reference="algo:compute-generating-set"}, we provide pseudocode for the function [compute-generating-set]{.smallcaps}. This algorithm employs a subroutine [get-paths]{.smallcaps}, which takes in two nodes $v$ and $v'$ and a finite undirected graph $\mathcal{G}$ (to which the nodes belong) and returns the set of all paths from $v$ to $v'$ on $\mathcal{G}$. Again, such a routine is available in standard packages, which is why we do not provide pseudocode. $\mathcal{Q}$, a finite set of sets. The set monoid generated by $\mathcal{Q}$. $\mathcal{M} = \{\emptyset \}$ $\mathcal{M}^1$ $\leftarrow$ $\mathcal{Q}$ $\mathcal{M}^{i+1}$ $\leftarrow$ $\mathcal{M}^{i}$ add $\mathcal{S}\cup \mathcal{T}$ to $\mathcal{M}^{i+1}$ $\{\emptyset \} \cup \mathcal{M}^{i+1}$ **break** $\{\emptyset \} \cup \mathcal{M}^{|\mathcal{Q}|}$ We finish by describing pseudocode for the last remaining subroutine [compute-monoid-from-generating-set]{.smallcaps} of Algorithm [\[algo:get-monoid\]](#algo:get-monoid){reference-type="ref" reference="algo:get-monoid"}. This subroutine computes a set monoid from a given generating set, for which we provide pseudocode in Algorithm [\[algo:compute-monoid-from-generating-set\]](#algo:compute-monoid-from-generating-set){reference-type="ref" reference="algo:compute-monoid-from-generating-set"}. [^1]: `andreas.gerhardus@dlr.de` [^2]: `wahl@tu-berlin.de` [^3]: `sofia.faltenbacher@dlr.de` [^4]: `urmi.ninad@tu-berlin.de` [^5]: `jakob.runge@dlr.de` [^6]: The definition of canonical DAGs in @richardson2002ancestral requires the ADMG to be ancestral. However, we can use the very same definition also for non-ancestral ADMGs without self edges. [^7]: Proposition 1 in @richardson2023nested only applies to ADMG latent projections of DAGs rather than of ADMGs without self edges, but the proof in @richardson2023nested also works for the latter more general case. [^8]: Contemporaneous edges are *not* in conflict with the intuitive notion of time order: A cause in reality still precedes its effect, but the true time difference can be smaller than the observed time resolution. [^9]: Recall that the alternative approach of causal effect identification based on time-collapsed graphs is conceptually less straightforward and typically has less identification power. [^10]: To avoid this confounding by past time steps, one needs stronger assumptions. For example, as the causal discovery algorithms PCMCI [@runge2019detecting] and PCMCI$^+$ [@runge2020discovering] use, the assumption that no component time series are unobserved altogether (such that, effectively, one deals with a ts-DAG) in combination with a known or assumed upper bound on the maximal lag $p_{\mathcal{A}}$ of that ts-DAG. [^11]: Sometimes, see for example the definition in @peters2017elements, summary graphs specifically exclude self edges $i {\,\rightarrow\,}i$. Here, we allow for self edges. Note that self-edges $i {\,\rightarrow\,}i$ in the summary graph correspond to lagged auto-dependencies $(i,t-\tau) {\,\rightarrow\,}(i, t)$ in the time-resolved graph (here, the ts-DAG). [^12]: In general, $\pi$ depends on the order in which we "collapse" the cycles on $\pi_{cyc}$. This non-uniqueness is, however, not relevant to the argument. [^13]: Formally, $\mathcal{M}_{\pi}$ is not a set of walks. Thus, to be formally correct, we should rather say "we let both $\pi$ and $\pi^{ext}$ correspond to an element of $\mathcal{M}_{\pi}$", but we use the above informal formulation for simplicity. [^14]: To further reduce the number of explicit searches, one can use the following approach: Let $\mathcal{L}_1, \ldots, \mathcal{L}_{A}$ be the finite collection of linear Diophantine equations specified by Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"}. The most straightforward approach is to consider these equations in the sequential order in which they are given. However, if case 2 or 4 of Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"} applies to $\mathcal{L}_1$, then this sequential approach entails to potenially run the respective explicit searches before even looking at $\mathcal{L}_2$; and similar for the other equations. Instead, we might in a first phase only consider those equations to which case 1, 3 or 5 of Theorem [Theorem 1](#cor.prob1){reference-type="ref" reference="cor.prob1"} applies and then, only if we do not find a solution in the first phase, move to considering the other equations in a second phase. Various further computational improvements seem possible, but here we content ourselves with the explained conceptual solution. [^15]: We define the root vertex of a trivial path to be the single vertex on that trivial path. [^16]: Strictly speaking, Proposition 1 in @richardson2023nested only applies to ADMG latent projections of DAGs rather than of ADMGs without self edges. However, the proof of Proposition 1 in @richardson2023nested equally works for this more general case. [^17]: In Section 2.1, we defined a slightly different equivalence relation on the set of cycles according to which two cycles are also equivalent if one of them is the other read in the reverse direction (for example, $i{\,\rightarrow\,}j {\,\rightarrow\,}i$ and $i {\,\leftarrow\,}j {\,\leftarrow\,}i$). However, here we only consider cycles in the set of walks $\mathcal{W}(\mathcal{G})$, which by definition only contains walks of the form ${\,\rightarrow\,}\ldots {\,\rightarrow\,}$ (and not ${\,\leftarrow\,}\ldots {\,\leftarrow\,}$). Thus, given the restriction to cycles in $\mathcal{W}(\mathcal{G})$, the two definitions are in fact equivalent. [^18]: To be more exact, we insert the unique representative of $\xi_1(1)$ that starts at $\pi'(i_1)$, but we will hide this detail to ease notation. [^19]: Note that both of these function disregard the multi-weights because the summary graph itself is sufficient for their purposes.

arxiv_math

--- author: - - bibliography: - ref.bib title: Cyclic Analytic $2$-isometry of Finite Rank and Cauchy Dual Subnormality Problem --- # Introduction and Preliminaries {#sec1} The notion of Cauchy dual of an operator was introduced by Shimorin in [@Sh]. The notion of Cauchy dual turns out to be interesting, especially while simultaneously studying two classes of operators which are in some sense antithetical to each other. An excellent illustration of such classes is provided by contractive subnormal operators and completely hyperexpansive operators. The interplay between these two classes of operators can be thought of as an operator theoretic manifestation of two special classes of functions viz. completely monotone functions and completely alternating functions. Also, both the classes of operators have been described in terms of operator inequalities. This fruitful synthesis of harmonic analysis with operator theory has been extensively explored in the works of Athavale and co-workers [@athavale_vms1999; @athavale_ranjekar2002_1; @athavale_ranjekar2002_2; @athavale2003; @sholapurkar2000] as well in the works of Jablonski and Stochel [@jablonski2001; @jablonski2002; @jablonski2003; @jablonski2004; @jablonski2006]. It turns out that the notion of Cauchy dual fits naturally in these circumstances and helps in understanding the subtle interconnections. The said theme begins with a simple observation :- If $T:\{\alpha_n\}$ is a weighted shift operator, then the Cauchy dual $T'$ is the weighted shift with weight sequence $\Big\{\displaystyle\frac{1}{\alpha_n}\Big\}$. Combining this fact together with a result of [@bcr1984 Chapter 4, Proposition 6.10] enabled Athavale [@At] to prove that the Cauchy dual of a completely hyperexpansive weighted shift is a contractive subnormal weighted shift. As a generalization of this result, it is natural to ask the following question:\ **Question 1**. *Is the Cauchy dual of a completely hyperexpansive operator subnormal?\ * This question appears in [@Ch Question 2.11]. Though the question still remains open in this full generality, some special cases of this problem have been dealt with in recent years. Chavan et. al. have extensively contributed in solving the problem in some special cases. The phrase 'Cauchy dual subnormality problem' has evolved through these considerations. In order to set up the context, we make a brief mention of these special cases here, and elaborate on these results in the next section. - An affirmative answer to CDSP for a family of multiplication operators on Dirichlet-type spaces has been given in [@cgr2022]. The work relies on the results on operator theory in de Branges-Rovnyak spaces. - First counterexample to CDSP has been found in [@sc2019]. A class of $2$-isometric weighted shifts on directed trees such that their Cauchy dual is not subnormal, has been constructed in this paper. - A class of cyclic 2-isometric composition operators has been exhibited in [@sc2020] whose Cauchy dual are not subnormal. In this paper, capitalizing on the works of [@costara2016] and [@cgr2022], we produce a counter example to the CDSP by way of constructing an example of cyclic, analytic $2$-isometry on a Dirichlet-type space, whose Cauchy dual is not subnormal. The said Dirichlet-type space is obtained by choosing a finitely supported measure on the unit circle $\mathbb{T}$. Our construction relies on the fact that such a Dirichlet-type space coincides with a suitable de Branges-Rovnyak space. Moreover, the construction involves the computation of the roots of a polynomial of degree four. In this process, we employ some numerical techniques by using the SageMath tool. ## Preliminaries Let $\mathbb{R}$ denote the set of real numbers, $\mathbb{C}$ denote the set of complex numbers, $\mathbb{D}$ denote the open unit disc and $\mathbb{T}$ denote the unit circle. The notion of a reproducing kernel Hilbert space (RKHS) is well known and has been extensively studied, explored and crucially used in the theory of operators on Hilbert spaces. We quickly recall the definition of RKHS for a ready reference.\ Let $S$ be a non-empty set and $\cal F$$(S,\mathbb{C})$ denote set of all functions from $S$ to $\mathbb{C}$. We say that $\cal H\subseteq \cal F$$(S,\mathbb{C})$ is a ***Reproducing Kernel Hilbert Space***, briefly *RKHS*, if 1. $\cal H$ is a vector subspace of $\cal F$$(S,\mathbb{C})$ 2. $\cal H$ is equipped with an inner product $\langle , \rangle$ with respect to which $\cal H$ is a Hilbert space 3. For each $x\in S$, the linear evaluation functional $E_x:\cal H$$\to \mathbb{C}$ defined by $E_x(f)=f(x)$, is bounded. If $\cal H$ is an RKHS on $S$ then for each $x\in S$, by Riesz representation theorem, there exists unique vector $k_x\in \cal H$ such that, for each $f\in \cal H$, $f(x)=E_x(f)=\langle f,k_x\rangle$.\ The function $K:S\times S \to \mathbb{C}$ defined by $K(x,y)=k_y(x)$ is called as the reproducing kernel for $\cal H$. A classical reference for reproducing kernel Hilbert spaces is [@paulsen2016RKHS].\ We now recall the definitions of three special types of RKHS which are relevant in the present context. 1. **Hardy Space:** Let Hol$(\mathbb{D})$ denote the set of all holomorphic functions on $\mathbb{D}$. The Hardy space $H^2$ is defined as $$H^2=\Bigg\{f(z)=\displaystyle\sum_{n=0}^{\infty}a_nz^n\in \text{Hol}(\mathbb{D}) : \displaystyle\sum_{n=0}^{\infty}|a_n|^2<\infty\Bigg\}$$ It is a well known fact that $H^2$ is a reproducing kernel Hilbert space with the inner product defined as $\langle f,g \rangle :=\displaystyle\sum_{n=0}^{\infty}a_n\overline{b_n}$, if $f(z)=\displaystyle\sum_{n=0}^{\infty}a_nz^n$ , $g(z)=\displaystyle\sum_{n=0}^{\infty}b_nz^n$ and a kernel(called as *Szego Kernel*) function $K_\lambda(z)=\displaystyle\frac{1}{1-\overline{\lambda}z}$. The interested reader may consult [@douglas2012] .\ 2. **Dirichlet-type spaces :** For a finite positive Borel measure $\mu$ on unit circle $\mathbb{T}$, the Dirichlet-type space $D(\mu)$ is defined by $$D(\mu):=\Bigg\{f\in \text{Hol}(\mathbb{D}) : \displaystyle\int_{\mathbb{D}}|f'(z)|^2P_\mu(z)dA(z)<\infty \Bigg\}$$ Here, $P_\mu(z)$ is the Poisson integral for measure $\mu$ given by $\displaystyle\int_{\mathbb{T}}\displaystyle\frac{1-|z|^2}{|z-\zeta|^2}d\mu(\zeta)$, $dA$ denotes the normalised Lebesgue area measure on the open unit disc $\mathbb{D}$ and Hol$(\mathbb{D})$ denotes the set of all holomorphic functions on $\mathbb{D}$. The space $D(\mu)$ is a reproducing kernel Hilbert space and multiplication by the coordinate function $z$ turns out to be a bounded linear operator (refer [@aleman1993]) on $\mathbb D(\mu)$. A celebrated result of [@richter1991] gives a characterization of cyclic, analytic $2$-isometry. Indeed, a cyclic, analytic $2$-isometry can be represented as multiplication by $z$ on a Dirichlet-type space $D(\mu)$ for some finite positive Borel measure $\mu$ on unit circle $\mathbb{T}$. (refer [@dirichlet_sp2014], Definition 7.1.1) The study of Dirichlet type spaces by way of identifying such a space with a de Branges-Rovnyak space was initiated by D. Sarason. This association has been further strengthened in recent years in the works of [@cgr2022; @sarason97; @cgr2010]. The identification of a Dirichlet space as a de Branges-Rovnyak space allows one to compute the reproducing kernel for the Dirichlet space.\ Here, we include a brief description of de Branges-Rovnyak space for a ready reference.\ 3. **de Branges-Rovnyak spaces:**\ For complex, separable Hilbert spaces $\cal U$, $\cal V$, let $\cal B(U,V)$ denote the Banach space of all bounded linear transformations from $\cal U$ to $\cal V$. The **Schur class** $S(\cal U,V)$ is given by $$S({\cal U,V})=\Big\{B:\mathbb{D} \to {\cal B(U,V)} : B \text{ is holomorphic, } \displaystyle\sup_{z\in \mathbb{D}}\|B(z)\|_{B({\cal U,V})}\leq 1\Big\}$$ Observe that when ${\cal U}=\mathbb{C}=\cal V$ then the Schur class is nothing but the closed unit ball of $H^\infty(\mathbb{D})$, the set of all bounded holomorphic functions on $\mathbb{D}$. For any $B\in S(\cal U,V)$, the de-Branges-Rovnyak space, $H(B)$ is the reproducing kernel Hilbert space associated with the $\cal B(V)$-valued semidefinite kernel given by $$K_B(z,w)=\displaystyle\frac{I_{\cal V}-B(z)B(w)^*}{1-z\overline{w}},~~z,w\in \mathbb{D}$$ For equivalent formulations of de Branges-Rovnyak spaces, the reader is referred to an excellent article by J. Ball [@Ball2015]. The kernel $K_B$ is normalised if $K_B(z,0)=I_{\cal V}$ for every $z\in \mathbb{D}$. This is equivalent to the condition $B(0)=0$. Further, when ${\cal U}=\mathbb{C}=\cal V$, we denote $H(B)$ by $H(b)$, the classical de Branges-Rovnyak space (Refer [@fricain_mashreghi_2016vol2] for the basic theory of the classical de Branges-Rovnyak spaces). Let $\cal H$ be a complex, infinite dimensional and separable Hilbert space and $\cal B(\cal H)$ denotes the $C^*$-algebra of bounded linear operators on $\cal H$. Here, we record the relevant definitions for a ready reference. Let $T\in {\cal B(H)}$. We say that $T$ is *cyclic* if there exists a vector $f\in \cal H$ such that ${\cal H}=\vee \{T^nf : n\geq 0\}$(vector $f$ is known as *cyclic vector*). An operator $T$ is said to be *analytic* if $\cap_{n\geq 0}T^n{\cal H}=\{0\}$. The *Cauchy dual* $T'$ of a left invertible operator $T\in \cal B(\cal H)$ is defined as $T'=T(T^*T)^{-1}$. Following Agler\....ref, an operator $T\in \cal B(\cal H)$ is said to be a *$2$-isometry* if $$I-2T^*T+T^{*2}T^2=0.$$ An operator $T\in \cal B(H)$ is said to be **subnormal** if there exist a Hilbert space $\cal K$ and an operator $S\in \cal B(K)$ such that $\cal H \subseteq K$, $S$ is normal and $S|_{\cal H}=T$. Readers are encouraged to refer [@conway1991] for the detailed study of subnormal operators. Agler proved in [@agler1985] that $T\in \cal B(H)$ is a subnormal contraction (means $T$ is subnormal and $\|T\|\leq 1$) if and only if $$B_n(T)\equiv \sum_{k=0}^{n}(-1)^k \binom{n}{k}T^{*k}T^k\geq 0 \text{ for all integers }n\geq 0.$$ Following [@At], an operator $T\in B(\cal H)$ is said to be *completely hyperexpansive* if $$B_n(T)\leq 0~~ \text{for all integers }n\geq 1.$$ Now we are ready to state and describe the problem that has been tackled in this article. ## The Problem We begin our discussion with the fact [@cgr2022 Theorem 6.1] which states that for a cyclic, analytic $2$-isometry $T$ in $\cal B(H)$, the rank of $T^*T-I$ is finite if and only if there exist a finitely supported measure $\mu$ on the unit circle $\mathbb{T}$ such that $T$ is unitarily equivalent to a multiplication by the coordinate function $z$ on $D(\mu)$. In fact, if the rank of $T^*T-I$ is a positive integer $k,$ then the corresponding measure $\mu$ is supported at exactly $k$ points of unit circle $\mathbb{T}.$ In view of this result, a construction of a cyclic, analytic 2-isomtery reduces to choosing finitely many points on the unit circle and looking at the multiplication operator $M_z$ on the Dirichlet space $D(\mu)$, where $\mu$ is supported at the chosen points. Interestingly, in a recent work, it is asserted that such a space $D(\mu)$ coincides with the de Branges-Rovnyak space $H(B)$ for some $B$, with the equality of norms. We now state the problem under consideration: **Problem 1**. *Characterize finitely supported, positive, Borel measures $\mu$ on unit circle $\mathbb{T}$ such that the Cauchy dual of $M_z$ on $D(\mu)$ is subnormal.* A solution to the problem in the case when $\mu$ is supported at a single point is given by the following result (see [@badea2019 Corollary 3.6] and [@cgr2022 Corollary 5.4]). Let $\lambda\in \mathbb{T}$ and $\gamma>0$. The Cauchy dual of $M_z$ on the Dirichlet-type space $D(\gamma \delta_\lambda)$ is a subnormal contraction. Further, it is proved in [@cgr2022 Theorem 2.4] that for a positive, Borel measure $\mu$ supported at any two antipodal points on unit circle $\mathbb{T}$, the Cauchy dual $M_z'$ of $M_z$ is subnormal. In view of these two results, it is natural to investigate the case of a measure supported at two non-antipodal points and address the problem in that case. Here, we consider the measures supported at two points of unit circle $\mathbb{T}$ such that the angle between them is $90^\circ$. Indeed, the main result of this paper is as given below.\ **Theorem 1**. *Let $\zeta_1$ and $\zeta_2$ be any two points on the unit circle $\mathbb{T}$ such that the angle between them is $90^\circ$(considered as vectors in $\mathbb{R}^2$). If $\mu=\delta_{\zeta_1}+\delta_{\zeta_2}$ is a measure on unit circle $\mathbb{T}$ then the Cauchy dual of $M_z$ on the Dirichlet-type space $D(\mu)$ is not subnormal.* The rest of the paper is devoted to the proof of this result. # Proof of the main result As a special case, we choose points $\zeta_1=1$ and $\zeta_2=i$ and begin with the computation of reproducing kernel of the space $D(\mu)$ where $\mu$ is supported at $\{1,i\}$. ## Reproducing kernel of $D(\mu)$ {#sec2} As mentioned earlier, the space $D(\mu)$ is equipped with reproducing kernel. In this section, we take up the task of computing the reproducing kernel for the space $D(\mu)$ where $\mu$ is the regular, positive, Borel measure on the unit circle $\mathbb{T}$ supported on the points $\{1,i\}$. The technique used in obtaining the reproducing kernel heavily relies on the work of Costara [@costara2016]. The computation has been included in the following theorem:\ **Theorem 2**. *Let $\mu$ be a positive Borel measure on unit circle $\mathbb{T}$ supported on $\{1,i\}$ defined as $\mu=\delta_1+\delta_i$. The reproducing kernel for $D(\mu)$ is given by: $$\begin{aligned} K(z,\lambda)& =\displaystyle\frac{b}{\overline{q(\lambda)}q(z)}\Bigg[\displaystyle\frac{(\bar \lambda -1)(\bar\lambda +i)(z-1)(z-i)}{1-\bar\lambda z}\\ &+(a-1)\left((\bar\lambda +i)(z-i)+(\bar\lambda -1)(z-1)\right)\\ &+\displaystyle\frac{\bar s(\bar \lambda -1)(z-i)}{-i(q-pi)^2}+\frac{s(\bar \lambda +i)(z-1)}{i(q+pi)^2}\Bigg] ~~~~~~~(\lambda, z\in \mathbb{D}) \end{aligned}$$ where $a,b,p,q,s$ are constants.\ * We first proceed with the following lemma. **Lemma 3**. *For $z\in \mathbb{T}\setminus\{1,i\}$, if $V(z)=\displaystyle\frac{1}{|z-1|}+\frac{1}{|z-i|}$ then there exists a polynomial $q(z)$ of degree $2$ and a constant $d>0$ such that $1+V(z)=\displaystyle\frac{|\sqrt{d}~q(z)|^2}{|z-1|^2|z-i|^2}$* *Proof.* We know that $1+V(z)=\displaystyle\frac{|z-1|^2|z-i|^2+|z-1|^2+|z-i|^2}{|z-1|^2|z-i|^2}, z\in \mathbb{T} \setminus \{1,i\}$. We have, $$\begin{aligned} &|z-1|^2|z-i|^2+|z-1|^2+|z-i|^2\\ &=(z-1)(\bar z-1)(z-i)(\bar z+i)+(z-1)(\bar z-1)+(z-i)(\bar z+i)\\ &=8+3iz-3i\bar z-3z-3\bar z-iz^2+i\bar z^2\\ &=8+3iz-\displaystyle\frac{3i}{z}-3z-\displaystyle\frac 3z-iz^2+\displaystyle\frac{i}{z^2}\\ % =\displaystyle\frac{1}{z^2}\left[8z^2+3iz^3-3iz-3z^3-3z-iz^4+i\right]\\ % =-\displaystyle\frac{i}{z^2}\left[8iz^2-3z^3+3z-3iz^3-3iz+z^4-1\right]\\ &=-\displaystyle\frac{i}{z^2}\left[z^4-(3+3i)z^3+8iz^2+(3-3i)z-1\right]\\ &=-\displaystyle\frac{i}{z^2}\left[(z-\alpha_1)(z-\alpha_2)(z-\beta_1)(z-\beta_2)\right] \end{aligned}$$ Where, $\alpha_1=2.32798295504488 + 0.207814156136787i$,\ $\alpha_2=0.207814156136787 + 2.32798295504488i$,\ $\beta_1=0.0380424487395548 + 0.426160440078774i$ and\ $\beta_2=0.426160440078774 + 0.0380424487395548i$\ (This can be obtained or verified using sagemath) (See Appendix [4](#secA1){reference-type="ref" reference="secA1"}.[\[a1\]](#a1){reference-type="ref" reference="a1"})\ Let $q(z)=(z-\alpha_1)(z-\alpha_2)$ then we write it as $q(z)=z^2-a(1+i)z+bi$ where $a=2.53579711118167;b=5.46269136247034$.\ Further, we write $(z-\beta_1)(z-\beta_2)=z^2-c(1+i)z+di$\ where $c=0.464202888818329;d=0.183059948594236$.\ We observe that $$\displaystyle\frac{c}{di}=-ai, ~~\displaystyle\frac{1}{di}=-bi$$ Thus, after simplifying the above expression, we get, $$|z-1|^2|z-i|^2+|z-1|^2+|z-i|^2=d q(z)\overline{q(z)}$$ Hence we get, $1+V(z)=\displaystyle\frac{|\sqrt{d}q(z)|^2}{|z-1|^2|z-i|^2}$ ◻ Following Costara [@costara2016], we define $$\label{eq1} O_\mu(z)=\displaystyle\frac{(z-1)(z-i)}{\sqrt{d}q(z)}$$ Observe that $O_\mu(0)>0$. It is also observed in [@costara2016] that $O_\mu H^2$ is a closed subset of $D(\mu)$. By applying [@costara2016 Theorem 3.1], we obtain the reproducing kernel for $O_\mu H^2$ as follows: $$\label{eq2} \tilde{K_\mu}(z,\lambda)=\displaystyle\frac{(\bar \lambda -1)(\bar \lambda +i)(z-1)(z-i)}{d~(1-\bar \lambda z)q(z)\overline{q(\lambda)}}~~~~~~ (\lambda, z\in \mathbb{D})$$ Now, we obtain the reproducing kernel of $(O_\mu H^2)^\perp$. **Proposition 4**. *The reproducing kernel for $(O_\mu H^2)^\perp$ is given by $$\begin{aligned} \hat{K_\mu}(z,\lambda)=\displaystyle\frac{b}{q(z)\overline{q(\lambda)}}&\Bigg[(a-1)\left((\overline{\lambda}+i)(z-i)+(\overline{\lambda}-1)(z-1)\right)\\ &+\frac{\overline{s}(\overline{\lambda}-1)(z-i)}{-i(q-pi)^2}+\frac{s(\overline{\lambda}+i)(z-1)}{i(q+pi)^2}\Bigg]~~~~(\lambda, z\in \mathbb{D}) \end{aligned}$$ where $a,p,q,s$ are constants.* *Proof.* We have, $O_\mu(z)=\displaystyle\frac{(z-1)(z-i)}{\sqrt{d}q(z)}$. Using Sagemath, we get, $O_\mu'(1)=-p-qi$ and $O_\mu'(i)=q+pi$, where,\ $p=0.954692530486206;q=0.297593972106043$.\ By equation 4.1 of [@costara2016], define\ $f_1(z)=\displaystyle\frac{O_\mu(z)}{O_\mu'(1)(z-1)}=\displaystyle\frac{z-i}{\sqrt{d}~q(z)(-p-qi)}$ and\ $f_2(z)=\displaystyle\frac{O_\mu(z)}{O_\mu'(i)(z-i)}=\displaystyle\frac{z-1}{\sqrt{d}~q(z)(q+pi)}$\ Thus,\ $f_1'(1)=1.10401859575639 - 2.66453525910038\times 10^{-15}i\approx 1.10401859575639=m$ (say) and\ $f_2'(i)=2.77555756156289\times 10^{-15} - 1.10401859575639~i\approx - 1.10401859575639~i=-mi$\ Thus, $\|f_1\|_\mu^2=f_1'(1)= m$ and $\|f_2\|_\mu^2=i\times f_2'(i)=i\times (-im)= m$ (See Appendix [4](#secA1){reference-type="ref" reference="secA1"}.[\[a3\]](#a3){reference-type="ref" reference="a3"})\ Using Lemma 4.3 of [@costara2016], we get,\ $\langle f_1,f_2\rangle_\mu =\displaystyle\frac{1}{O_\mu'(1)\overline{O_\mu'(i)}(1+i)}=\displaystyle\frac{1}{(-p-qi)\overline{(q+pi)}(1+i)}=-u-vi$\ where $u=0.695548570053500$ and $v=0.127327085478790$.\ Let $B=\begin{bmatrix} \langle f_1,f_1\rangle & \langle f_1,f_2\rangle\\ \langle f_2,f_1\rangle & \langle f_2,f_2\rangle \end{bmatrix}^{-1}$\ Thus, $B=\begin{bmatrix} a-1 & s\\ \bar s & a-1 \end{bmatrix}$, where $s=0.967575626606929 + 0.177124344467703~i$ and\ $`a$' as obtained in Lemma [Lemma 3](#l1){reference-type="ref" reference="l1"}.\ For $\lambda \in \mathbb{D}$, define $\begin{bmatrix} g_1(\lambda)\\ g_2(\lambda) \end{bmatrix}=\overline{B} \times \begin{bmatrix} \overline{f_1(\lambda)}\\ \overline{f_2(\lambda)} \end{bmatrix}=\begin{bmatrix} a-1 & \bar s\\ s & a-1 \end{bmatrix} \begin{bmatrix} \displaystyle\frac{\overline{\lambda}+i}{\sqrt{d}~\overline{q(\lambda)}(-p+qi)}\\ \displaystyle\frac{\overline{\lambda}-1}{\sqrt{d}~\overline{q(\lambda)}(q-pi)} \end{bmatrix}$\ (See Appendix [4](#secA1){reference-type="ref" reference="secA1"}.[\[a4\]](#a4){reference-type="ref" reference="a4"}).\ This gives, $g_1(\lambda)=\displaystyle\frac{1}{\sqrt{d}~\overline{q(\lambda)}}\left[\displaystyle\frac{(a-1)(\bar \lambda +i)}{-p+qi}+\displaystyle\frac{\bar s (\bar \lambda -1)}{q-pi}\right]$ and\ $g_2(\lambda)=\displaystyle\frac{1}{\sqrt{d}~\overline{q(\lambda)}}\left[\frac{s(\bar \lambda +i)}{-p+qi}+\frac{(a-1)(\bar \lambda -1)}{q-pi}\right]$\ Finally applying Lemma 4.4 from [@costara2016], we get, $$\begin{aligned} \hat{K_\mu}(z,\lambda) &=g_1(\lambda)f_1(z)+g_2(\lambda)f_2(z)\\ &=\displaystyle\frac{b}{q(z)\overline{q(\lambda)}}\Bigg[(a-1)\left((\overline{\lambda}+i)(z-i)+(\overline{\lambda}-1)(z-1)\right)\\ &+\frac{\overline{s}(\overline{\lambda}-1)(z-i)}{-i(q-pi)^2}+\frac{s(\overline{\lambda}+i)(z-1)}{i(q+pi)^2}\Bigg]~~~~(\lambda, z\in \mathbb{D}) \end{aligned}$$ ◻ *Proof.* By [@costara2016 Theorem 5.1], the reproducing kernel of $D(\mu)$ is given as follows :\ For $\lambda, z \in \mathbb{D}$, $K(z,\lambda)=\tilde{K_\mu}(z,\lambda)+\hat{K_\mu}(z,\lambda)$.\ Thus, by ([\[eq2\]](#eq2){reference-type="ref" reference="eq2"}) and proposition [Proposition 4](#p1){reference-type="ref" reference="p1"} we get, $$\begin{aligned} K(z,\lambda)& =\displaystyle\frac{b}{\overline{q(\lambda)}q(z)}\Bigg[\displaystyle\frac{(\bar \lambda -1)(\bar\lambda +i)(z-1)(z-i)}{1-\bar\lambda z}\\ &+(a-1)\left((\bar\lambda +i)(z-i)+(\bar\lambda -1)(z-1)\right)\\ &+\displaystyle\frac{\bar s(\bar \lambda -1)(z-i)}{-i(q-pi)^2}+\frac{s(\bar \lambda +i)(z-1)}{i(q+pi)^2}\Bigg] ~~~~~~~(\lambda, z\in \mathbb{D})\end{aligned}$$ ◻ The above computations for reproducing kernel of $D(\mu)$ further helps to discuss CDSP for $M_z$ on $D(\mu)$. ## Cauchy Dual Subnormality Problem {#sec3} As promised earlier, we now present a counter example to CDSP. We show that the Cauchy dual of $M_z$ on $D(\mu)$ is not subnormal, where $\mu$ is a finite, positive, Borel measure supported at $\{1,i\}$.\ We first find $B=(b_1,b_2)$ such that $D(\mu)$ coincides with $H(B)$ with equality of norms.\ **Theorem 5**. *Let $\mu$ be a positive Borel measure on unit circle $\mathbb{T}$ supported on $\{1,i\}$. Then the Dirichlet type space $D(\mu)$ coinsides with de Branges-Rovnyak space $H(B)$ with $B=(b_1,b_2)$ and $b_j=\displaystyle\frac{p_j}{q}$ where $$p_1(z)=4.13925355z+(-1.31972862 -1.31972862~i)z^2, ~p_2(z)=1.11084575z^2 ~~and$$ $$q(z)=(z-\alpha_1)(z-\alpha_2)$$ with $\alpha_1=x+iy$, $\alpha_2=y+ix$ where $x=2.32798295504488$ and $y=0.207814156136787$.* *Proof.* Let $\xi_1=1$, $\xi_2=i$, $B=(b_{ij})_{1\leq i,j\leq 2}$ be the matrix obtained in proposition [Proposition 4](#p1){reference-type="ref" reference="p1"}. Also, let $O_\mu(z)$ as defined by equation ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"}).\ In tune with the proof of [@cgr2022 Theorem 6.4], we consider the following expression:\ $q(z)\overline{q(w)}-p(z)\overline{p(w)}\left(1+(1-z\overline{w})\displaystyle\sum_{i,j=1}^{2}\frac{\overline{b_{ij}}}{O_\mu'(\xi_j)O_\mu'(\xi_i)}\frac{1}{(z-\xi_j)(\overline{w}-\overline{\xi_i})}\right)$.\ Since, this is a polynomial in $z$ and $\overline{w}$, there exists a matrix $\hat A=(a_{ij})_{0\leq i,j\leq 2}$ such that the above expression is equal to $\displaystyle\sum_{i,j=0}^{2}a_{ij}z^i\overline{w}^j$, $z,w\in \mathbb{D}$.\ But, this expression at $w=0$ is $0$ for each $z\in \mathbb{D}$, we obtain a matrix $A$ from $\hat A$ by deleting first row and first column of matrix $\hat A$.\ After performing all the computations(See Appendix [4](#secA1){reference-type="ref" reference="secA1"}.[\[a6\]](#a6){reference-type="ref" reference="a6"}), we get,\ $$a_{11}=17.1334199164530,~~a_{12}=-5.46269136247035 - 5.46269136247035i$$ $$a_{21}=-5.46269136247035 + 5.46269136247035 i,~~a_{22}=4.71734553342817$$ Matrix $A$ is positive semi-definite and hence by applying Cholesky's decomposition (using Sagemath) we get matrix $P$ such that $A=P^*P$, where\ $$P=\begin{bmatrix} 4.13925355 & ~~~-1.31972862 - 1.31972862i\\ 0 & 1.11084575 \end{bmatrix}$$\ Once again using [@cgr2022 Theorem 6.4], we get,\ $p_1(z)=\langle P (z,z^2)^t,(1,0)\rangle=4.13925355z+(-1.31972862 -1.31972862~i)z^2$\ $p_2(z)=\langle P (z,z^2)^t,(0,1)\rangle=1.11084575 z^2$\ \ By virtue of [@cgr2022 Theorem 6.4], we conclude that the Dirichlet type space $D(\mu)$ coinsides with de Branges-Rovnyak space $H(B)$ with $B=(b_1,b_2)$ and\ $b_j=\displaystyle\frac{p_j}{q}, j=1,2$. ◻ These computations lead to a striking counter example of a finite rank analytic, cyclic $2$-isometry whose Cauchy dual is not subnormal. we consider $M_z$ on the above $D(\mu)$ and claim that its Cauchy dual $M_z'$ is not subnormal.\ *Proof.* By using Sagemath, we compute the following : 1. For $1\leq r\neq t\leq 2$, $\displaystyle\sum_{j=1}^{2}p_j(\alpha_r)\overline{p_j(\alpha_t)}=-230.719263940288\neq 0$ (see Appendix [4](#secA1){reference-type="ref" reference="secA1"}.[\[a8\]](#a8){reference-type="ref" reference="a8"}) 2. $\alpha_1\overline{\alpha_2}=0.967575626606951 - 5.37631791548865i\notin [1, \infty)$ and\ $\alpha_2\overline{\alpha_1}=0.967575626606951 + 5.37631791548865i \notin [1,\infty)$ (See Appendix [4](#secA1){reference-type="ref" reference="secA1"}.[\[a7\]](#a7){reference-type="ref" reference="a7"}). Hence, by [@cgr2022 Corollary 4.3], we conclude that the Cauchy dual $M_z'$ of the multiplication by $z$ operator $M_z$ on $D(\mu)$ where $\mu$ is a positive, Borel measure on the unit circle $\mathbb{T}$ given by $\mu=\delta_1+\delta_i$ is **not subnormal**.\ In view of [@cgr2022 Proposition 7.1], we conclude the proof of the theorem. ◻ **Remark 1**. *This provides a subclass of cyclic, analytic, 2-isometries of finite rank whose Cauchy dual is not subnormal.* # Conclusion {#sec13} The work carried out in this paper asserts that if a finitely supported measure $\mu$ on the unit circle $\mathbb{T}$ is of the form $\mu=\delta_{\zeta_1}+\delta_{\zeta_2}$ where $\zeta_1, \zeta_2\in \mathbb{T}$ are points such that the angle between them is $90^\circ$, then the Cauchy dual of the multiplication operator $M_z$ on the corresponding Dirichlet space $D(\mu)$ is not subnormal. It provides a class of operators which are cyclic, analytic, $2$-isometric of finite rank, such that their Cauchy dual is not subnormal. In this context, the following points deserve further exploration : 1. If the measure $\mu$ is supported at points $\zeta_1$, $\zeta_2$ on the unit circle $\mathbb{T}$ having some other symmetry condition such as the angle between them is $120^\circ$, then is the Cauchy dual of $M_z$ on $D(\mu)$ subnormal? 2. In view of a positive result in case of the measure supported at antipodal points and negative result in case of the measure supported at the points perpendicular to each other, it is interesting to find a symmetry condition on the support of the measure so that we have a complete solution to CDSP in this case. 3. The authors have a feeling that the Cauchy dual is subnormal only in the case when the support of the measure is at antipodal points. The authors would like to thank Prof. Sameer Chavan, IIT Kanpur, India for several useful discussions throughout the preparation of this paper. # Declarations {#declarations .unnumbered} - Funding : Not applicable - Conflict of interest/Competing interests (check journal-specific guidelines for which heading to use) : Not applicable - Ethics approval : Not applicable - Consent to participate : Not applicable - Consent for publication : Yes - Availability of data and materials : Not applicable - Code availability : Yes - Authors' contributions : Equal The present work is carried out at the research center at the Department of Mathematics, S. P. College, Pune, India(autonomous). # Program codes in Sagemath {#secA1} The following codes can be executed using CoCalc. 1. [\[a1\]]{#a1 label="a1"} z = PolynomialRing(ComplexField(), 'z').gen() pol=z^4-(3+3*I)*z^3+8*I*z^2+3*(1-I)*z-1 pol.roots() 2. [\[a2\]]{#a2 label="a2"} a=2.53579711118167;b=5.46269136247034;c=0.464202888818329;d=0.183059948594236 p=0.954692530486206;q=0.297593972106043 alpha1=2.32798295504488 + 0.207814156136787*i; alpha2=0.207814156136787 + 2.32798295504488*i 3. [\[a3\]]{#a3 label="a3"} O(z)=((z-1)*(z-I))/(sqrt(d)*(z^2-a*(1+I)*z+b*I)) Od(z)=derivative(O(z),z) Od(1) Od(i) f1(z)=(z-i)/(sqrt(d)*(z^2-a*(1+I)*z+b*I)*(-p-q*I)) f1d(z)=derivative(f1(z),z) f2(z)=(z-1)/(sqrt(d)*(z^2-a*(1+I)*z+b*I)*(q+p*I)) f2d(z)=derivative(f2(z),z) 4. [\[a4\]]{#a4 label="a4"} 1/((i-1)*(-p-q*I)^2) w=-0.695548570053500 - 0.127327085478790*I; wbar=-0.695548570053500 + 0.127327085478790*I A=matrix(2,2,[1.10401859575639,w,wbar,1.10401859575639]) B=A.inverse() print(B) s=0.967575626606929 + 0.177124344467703*I 5. [\[a5\]]{#a5 label="a5"} qq(z)=(z-alpha1)*(z-alpha2) h1(z)=b*(1+I)/(qq(z)*conjugate(qq(1)))*((a-1)*(z-I)+(s*(z-1)/(I*(q+p*I)^2))) h1d(z)=derivative(h1(z),z) u1(z)=(h1(z)-h1(1))/(z-1) u2(z)=(h1(z)-h1(I))/(z-I) conjugate(h1d(0))+conjugate(u1(0))+conjugate(u2(0)) 6. [\[a6\]]{#a6 label="a6"} m=s/((q+p*I)*(-p+q*I)) n=conjugate(s)/((-p-q*I)*(q-p*I)) a11=2*a^2+(-2-m-n-I*(m-n))/d a12=-a*(1+I)+(2+2*I-a*I-m-n*I-a)/d a21=-a*(1-I)+(2-2*I+a*I+m*I-n-a)/d a22=1+(2*a-3+m+n)/d D=matrix(2,2,[a11,a12,a21,a22]) print(D) import numpy as np from scipy.linalg import cho_factor, cho_solve T=np.linalg.cholesky(D) P=matrix(2,2, [4.13925355+0.j,-1.31972862-1.31972862j,0,1.11084575+0.j]) Q=matrix(2,2,[4.13925355,0,-1.31972862+1.31972862*I,1.11084575]) print(P) print(Q*P) p1(z)=4.13925355*z+(-1.31972862000000 -1.31972862000000*I)*z^2 p2(z)=1.11084575*z^2 7. [\[a7\]]{#a7 label="a7"} a1=alpha1-alpha2;a2=alpha2-alpha1; k11=1/(a1*conjugate(a1))*(p1(alpha1)*conjugate(p1(alpha1))+ p2(alpha1)*conjugate(p2(alpha1)))*(1-1/(alpha1*conjugate(alpha1)))^2 k12=1/(a1*conjugate(a2))*(p1(alpha1)*conjugate(p1(alpha2))+ p2(alpha1)*conjugate(p2(alpha2)))*(1-1/(alpha1*conjugate(alpha2)))^2 k13=1/(a2*conjugate(a1))*(p1(alpha2)*conjugate(p1(alpha1))+ p2(alpha2)*conjugate(p2(alpha1)))*(1-1/(alpha2*conjugate(alpha1)))^2 k14=1/(a2*conjugate(a2))*(p1(alpha2)*conjugate(p1(alpha2))+ p2(alpha2)*conjugate(p2(alpha2)))*(1-1/(alpha2*conjugate(alpha2)))^2 m = matrix(CC, 1,1, lambda i, j: k11*(1/(alpha1^(i+2)*(conjugate(alpha1))^(j+2)))+ k12*(1/(alpha1^(i+2)*(conjugate(alpha2))^(j+2)))+ k13*(1/(alpha2^(i+2)*(conjugate(alpha1))^(j+2)))+ k14*(1/(alpha2^(i+2)*(conjugate(alpha2))^(j+2)))); #print(m); det(m) 8. [\[a8\]]{#a8 label="a8"} alpha1*conjugate(alpha2) alpha2*conjugate(alpha1) p1(alpha1)*conjugate(p1(alpha2))+p1(alpha2)*conjugate(p1(alpha1))+ p2(alpha1)*conjugate(p2(alpha2))+p2(alpha2)*conjugate(p2(alpha1))

arxiv_math

--- abstract: | This work introduces the School Bus Routing Problem with Open Offer Policy (SBRP-OOP) motivated by the Williamsville Central School District (WCSD) of New York. School districts are often mandated to provide transportation, but students assigned to the buses commonly do not use the service. Consequently, buses frequently run with idle capacity over long routes. The SBRP-OOP seeks to improve capacity usage and minimize the bus fleet by openly offering a monetary incentive to students willing to opt-out of using a bus. Then, students who certainly will not take a bus due to acceptance of the incentive are not included in the routing generation. Mathematical formulations are proposed to determine a pricing strategy that balances the trade-off between incentive payments and savings obtained from using fewer buses. The effectiveness of the Open Offer Policy approach is evaluated for instances from a real operational context in New York. author: - "Hernan Caceres[^1]" - "Macarena Duran[^2]" - "Hernán Lespay[^3]" - "Juan Pablo Contreras[^4]" - "Rajan Batta[^5]" title: "School Bus Routing Problem with Open Offer Policy: incentive pricing strategy for students that opt-out using school bus" --- **Keywords:** School bus routing; Pricing strategy; Overbooking, Simulation, Integer programming; Column generation; Heuristics # Introduction School Bus Routing Problem (SBRP) is part of the family of Vehicle Routing Problem (VRP), in which a set of school bus routes has to be determined in order to pick up a given set of students from a set of potential bus stops and transporting them to the school. The SBRP is cost-intensive for school administrators [@Ellegood2020]; therefore, improving transportation efficiency releases school resources that can be used to enhance educational programs, build new infrastructure, and increase teachers' wages, among others. Most SBRP publications focus on minimizing the length or cost of routes, constrained to all students to be assigned to a bus [@Park2010]. However, recent works have shifted the characteristics being examined to emphasize the real issues of the SBRP [@Ellegood2020]. For example, although overbooking strategies are typically used in practical school bus routing problems, they have yet to be approached in the literature. Recently, in [@Caceres2017] has been introduced an overbooking strategy for the SBRP to improve the seat utilization of buses due to some students not using the service. A commonly used policy in school districts is to assign every student to a bus route, regardless of whether they decide to ride the bus. Also, there is no incentive to actively opt-out, i.e., to notify the school district that a student will not use a bus. However, if students not using the school bus system were to opt-out at the beginning of the school year, motivated by a monetary incentive, the district could design a smaller set of routes only for those students who actually use the bus. In this paper, we introduce a new family of SBRP called School Bus Routing Problem with Open Offer Policy (SBRP-OOP). The SBRP-OOP seeks to determine a pricing strategy to balance incentive payments to the students willing to opt-out of using a bus and the savings obtained from using fewer buses to pick up students from their assigned stops. Students that accept the incentive are not considered in the routing generation process. Then, we consider an SBRP with a single school whose objective is to simultaneously: i) find the set of stops to visit from a set of potential stops, ii) assign each student to a stop, considering a maximum total distance that a student has to walk and a maximum number of students per stop, and iii) generate routes to visit the selected stops; such that the total distance traveled by buses is minimized. ## Problem motivation The SBRP-OOP is motivated by the Williamsville Central School District (WCSD) operational context, belonging to the New York State Education Department. School bus transportation in New York State is free and must be provided to all students by law. WCSD policy states that all students must be assigned to a stop, and all stops must be visited despite the uncertainty of students not showing up. Consequently, the district finds itself reliant on a large fleet of buses, many of which are underutilized. This situation incurs significant expenses in terms of bus maintenance and driver salaries, leading to excessive costs. Moreover, the WCSD is grappling with a severe shortage of bus drivers, a problem that the COVID-19 pandemic has exacerbated, according to local media [@MikeDesmond2022]. Prospective candidates, typically retirees, harbor skepticism about working in close proximity to children and often prefer work-from-home or package delivery positions [@lando2023]. As a result, the district is confronted with great challenges in both fleet reduction to achieve cost savings and ensuring an adequate number of personnel to meet the demands of its bus transportation system. Students can choose on any day whether they want to access bus transportation. We know that up to 80% of the students use the transportation provided by the district [@Caceres2017]. By studying the data gathered daily at the district, we found that the likelihood of a student using the bus is highly correlated to i) the kind of school attended (elementary school, middle school, or high school), and ii) the bus schedule (morning or afternoon). We define ridership as the ratio between the number of students riding a bus and the total number assigned to it. Figure [1](#tax:ridership){reference-type="ref" reference="tax:ridership"} shows the ridership for each school, separating the AM and PM cases. We can see how no more than 80% of the students use the transportation the school district provides. The fact that these students are still assigned to a bus accounts for longer routes and low capacity usage. {#tax:ridership width="0.8\\columnwidth"} The WCSD uses an overbooking policy to utilize bus capacities better. By reviewing routes yearly, the district determines whether to update overbooking limits, provided the information on actual ridership is available. With this information, the number of assigned students to a bus is determined so that it is improbable have an event of overcrowding. However, if such an event occurs, it would do in the school's proximity, near the end of the route and close to the bell time; therefore, the students would simply be squeezed into any seat. Finally, saving opportunities for the WCSD come mainly from reducing the maximum number of buses being used simultaneously at any given time during the day. The need for an additional bus implies either hiring a driver and purchasing a new bus or paying the contractor for another bus. Both these options are expensive. Like in Boston, MA [@Bertsimas2020], in Williamsville, NY, most transportation costs are driven by the total number of buses operating every day. Given this operational context, we propose a different approach to improve capacity usage. We explore the effect of encouraging students to opt-out of school buses to reduce the fleet size used for the school district. We call this problem the School Bus Routing Problem with Open Offer Policy (SBRP-OOP). An *Open Offer Policy* will try to reduce the number of students using the school buses through a monetary incentive. This policy aims to offer an incentive openly, extended as a check or a tax return, to any student willing to opt-out of using a bus. This incentive is determined before students decide whether to accept the deal. In this policy, the school district does not control which students would accept the offer but only controls the incentive's value. Thus, the incentive value needs to be determined in such a manner that it should reduce the possibility of having to pay more for incentives than what would be saved by using fewer buses. Figure [2](#tax:Motivation){reference-type="ref" reference="tax:Motivation"} illustrates the fleet reduction. Figure [2](#tax:Motivation){reference-type="ref" reference="tax:Motivation"}(a) describes an instance of the problem with one school (represented by the black square), a set of potential stops (represented by the small white square), and the set of students with the stops that they are able to reach (represented by the black dots and the dotted lines, respectively). Figure [2](#tax:Motivation){reference-type="ref" reference="tax:Motivation"}(b) describes a possible solution for the traditional SBRP with bus stop selection (SBRP-BSS) when the bus capacity is equal to 6. This potential solution uses three routes (represented by the blue lines), visiting seven stops to ride the students. Finally, Figure [2](#tax:Motivation){reference-type="ref" reference="tax:Motivation"}(c) describes a possible solution for the SBRP-OOP, considering the same bus capacity. We can observe that three students accepted the incentive for opt-out of using the buses (represented by the red dots). Therefore, these three students are not considered in the route planning process. In comparison to the traditional SBRP-BSS approach, it is evident that only two routes are utilized to serve five stops and transport the students. Consequently, the school district can accomplish this with just two buses, reducing the idle capacity. It is important to note that this scenario proves advantageous for the school district only if the cost savings resulting from using fewer buses outweigh the total amount spent on incentives provided to the students who have accepted the opt-out offer. {#tax:Motivation width="1.0\\columnwidth"} ## Our contribution This paper makes three distinct contributions. Firstly, we introduce the SBRP with an open offer policy, which aims to reduce the fleet size required for transporting students to their schools by providing economic incentives for students to opt-out of the system. Additionally, we define key performance indicators, specifically focusing on the probability of failure, to evaluate the effectiveness of pricing policies. Secondly, we propose a simulation-based solution method to estimate the indicators associated with a single school. To achieve this, we present two optimization problems that allocate the ridership level and the opt-out probability among the students while maintaining the school's original average parameter. Subsequently, we simulate the opt-out process based on the obtained probabilities and solve the routing problem to determine the minimum fleet required for students who decline the economic incentive. Lastly, we conduct tests using both synthetic and real instances. With synthetic data, we discover that the open offer policy proves advantageous when students are concentrated in proximity to their schools. When examining real data from the WCSD, we observe that the open offer policy has potential implementation in middle schools, even with substantial economic incentives of up to \$1500 USD from the students that decide to opt-out the system. ## Roadmap The remainder of this paper is organized as follows. Section [2](#literature){reference-type="ref" reference="literature"} gives a literature review of the SBRP with bus stop selection. Section [3](#formulation){reference-type="ref" reference="formulation"} presents a mathematical formulation of the problem considering overbooking. Section [4](#solution){reference-type="ref" reference="solution"} describes our solution framework for the SBRP-OOP. Section [5](#experiments){reference-type="ref" reference="experiments"} reports the results of computational experiments. Finally, Section [6](#discussion){reference-type="ref" reference="discussion"} presents the conclusions. # Literature review {#literature} The School Bus Routing Problem (SBRP) was introduced in [@newton1969] and has been studied extensively. It consists of four interrelated sub-problems: the tactical bus stop selection sub-problem, the operational bus route generation, bus route scheduling, and school bell time adjustment sub-problems. The bus stop selection sub-problem seeks to determine a set of bus stops to visit and allocate each student to one of these bus stops, limiting the walking distance of the students by a given value. The bus route generation sub-problem is the central issue of the SBRP. It covers the decision to design the bus routes to visit the bus stops at the minimum cost without exceeding the bus capacity. The bus route scheduling sub-problem specifies each route's starting and ending time and forms a chain of routes that can be executed sequentially by the same bus. Finally, the school bell time adjustment sub-problem considers school's starting and ending times as decision variables and seeks to find the optimal times to maximize the number of routes served sequentially by the same bus and reduce the number of buses used. In the literature, it is difficult to determine a dominant approach to solve the SBRP since all works propose different considerations, differing mainly in the sub-problems that are solved and the real-world characteristics considered. It is possible to observe that the latest studies emphasize real-world issues such as heterogeneous fleets, mixed loading, serving multiple schools, and ridership uncertainty ([@Ellegood2020]). The present review mainly focuses on works that solve the bus stop selection and bus route generation sub-problems, and explores how these works emphasize different real-world aspects. For a general review of the different classifications, objectives, and contemporary trends on SBRP, we recommend the reader to [@Park2010; @Ellegood2020]. Because of the complexity of solving bus stop selection and bus route generation sub-problems simultaneously, these problems have been solved sequentially. The heuristic solution approaches can be classified into: the location-allocation-routing (LAR) strategy and the allocation-routing-location (ARL) strategy ([@Park2010]). The LAR strategy determines a set of bus stops for a school and allocates students to these stops. Then, routes are generated for the selected stops. The main drawback of this strategy is that it tends to create excessive routes because vehicle capacity constraints are ignored in the location-allocation phase ([@Bowerman1995]). The main papers that use a heuristic approach based on the LAR strategy are [@bodin1979; @dulac1980; @li2021; @farzadnia2021; @calvete2021]. In [@bodin1979] the bus route scheduling sub-problem with mixed loads is additionally considered. The objective is to minimize the transportation cost, which consists of a fixed cost for each vehicle, the total routing cost, and the transportation time for each student. An ad-hoc heuristic approach is proposed to solve the problem with constraints on the maximum travel time for each student, a time window for arrival at the school, and the classical constraint on the maximum bus capacity. In [@dulac1980] multiple schools are considered, but mixed loads are not allowed; therefore, the problem is solved for each school independently. The objective is to minimize the product between the number of routes and the sum of route lengths. An ad-hoc heuristic approach is proposed to solve the problem with constraints on the number of visited stops, the transportation time for each vehicle, and the distance walked by the student from the current potential stop to the selected bus stop. In [@Parvasi2017] the student's choice to use alternative transportation systems is considered. A bi-level mathematical model is proposed considering the students are reluctant to choose any bus stops that are visited by a route and decide to use an alternative transportation system. At the upper-level, the decision concerning bus stops location is made, and routes buses to these stops are generated. Subsequently, the decision regarding the allocation of students to transportation systems or outsourcing them will be made at the lower level. Two hybrid metaheuristic approaches based on genetic algorithms, simulated annealing, and tabu search, are proposed to solve the model. In [@li2021] a mixed ride approach, where general and special education students are served on the same bus simultaneously while allowing heterogeneous fleets and mixed loads, is proposed. Here, mixed rides differ from mixed loads, which refers to buses serving students at different schools using the same bus. The goal is to reduce the route redundancy generated in a separate ride system when a general student and a special education student are at the same address but served by two different buses. Schools are divided into three types: (i) schools that only have general students; (ii) schools that only have special students; and (iii) mixed schools that have both general and special students. Then different strategies for bus stop selection are implemented for each type of school. ESRI Location-Allocation tools and Google OR-Tools are used for bus stop selection and route generation. In [@farzadnia2021] the service-oriented single-route SBRP is considered. The approach focuses on providing good quality service for students who are sensitive and need more care and safety. Because walking distance is generally considered a quality service measure, the proposed model minimizes the total walking distance by students to the stop points, while treating the maximum total routing distance as a constraint. The problem considers a single school and a single route. In the bus route generation, it is assumed that a single vehicle with an unlimited capacity can pick up all students of one school who are eligible to ask for the bus. An exact and heuristic algorithm was developed. In [@calvete2021] the SBRP with student choice is introduced. The work is an extension of [@calvete2020] to study student's reaction to the selection of bus stops when they are allowed to choose the bus stop that best suits them. A bilevel optimization model is formulated, considering the selection of the bus stops among the set of potential bus stops and the construction of the bus routes, considering both bus capacity and student preferences. A metaheuristic algorithm is developed to solve the problem in two steps. First, a subset of the set of potential bus stops is selected and allows the students to choose their preferred bus stop among the selected ones. Second, after knowing the number of students who take the bus at each bus stop, the routes are computed by solving the corresponding Capacited VRP and applying local search procedures. The ARL strategy attempts to overcome the LAR strategy's drawback, which tends to create excessive routes. First, the students are allocated into clusters while satisfying vehicle capacity constraints. Then, subsequently, the bus stops are selected, and a route is generated for each cluster. Finally, the students in a cluster (route) are assigned to a bus stop that satisfies all the problem requirements. The main drawback of this strategy is that the route length cannot be explicitly controlled because each route is generated after the allocation phase and depends on student's dispersion within the individual clusters ([@Bowerman1995]). The main papers that use a heuristic approach based on the ARL strategy are [@chapleau1985; @Bowerman1995]. In [@chapleau1985], as in [@dulac1980], multiple schools are considered, but routes for each school are determined independently. The objective is to minimize the number of routes. An ad-hoc heuristic approach is proposed to solve the problem with constraints on the distance walked by the students, the number of stops on each route, and the length of each route. In [@Bowerman1995] a multi-objective optimization approach is introduced, which considers efficiency, effectiveness, and equity criteria. The objective is to minimize the number of routes and total bus route length (efficiency), student walking distance (effectiveness), and optimize both load and length balancing (equity). An ad-hoc heuristic approach is proposed to solve the problem with constraints on the bus capacity and maximum walking distance. Bus stop selection and bus route generation are highly interrelated problems. They are treated separately in the previous works due to the complexity and size of the problems, but this generates sub-optimal solutions ([@salhi1989]). For this reason, recent studies have attempted to solve the problem through an integrated strategy ([@Riera-Ledesma2012; @Riera-Ledesma2013; @Schittekat2013; @Kinable2014]). In [@Riera-Ledesma2012] the multiple vehicle traveling purchaser problem (MV-TPP) is introduced. The MV-TPP is a generalization of the vehicle routing problem and models a family of routing problems combining stop selection and bus route generation. The goal of the problem is to assign each user to a potential stop to find the least cost routes and choose a vehicle to serve each route so that a route serves each stop with assigned users. The objective is to minimize the total length of all routes plus the total user-stop assignment cost subject to capacity bus constraints. A Mixed Integer Programming (MIP) formulation is proposed, and a branch-and-cut algorithm is developed to solve the MIP. Later, an extension of this work is proposed in [@Riera-Ledesma2013]. Here, additional resource constraints on each bus route are considered: bounds on the distances traveled by the students, the number of visited bus stops, and the minimum number of students that a vehicle has to pick up. A branch-and-price algorithm with a set partitioning formulation is developed to solve the problem. In [@Schittekat2013] a matheuristic to solve large instances is proposed. The matheuristic consists of a construction phase, based on a GRASP metaheuristic, and an improvement phase based on a VND metaheuristic. The student allocation sub-problem is solved by modeling the problem as a transportation problem. Then, the problem is decomposed into a master problem and a sub-problem. The master problem is a school bus routing problem with bus stop selection, where the objective is to minimize the total traveled distance. Once the stops have been selected and the routes have been fixed, the sub-problem of allocating students to stops is solved so that the bus capacity is not exceeded. Additionally, a MIP formulation is proposed to compare the effectiveness of the matheuristic and a column generation approach is developed to compute lower bounds on the optimal solution. In [@Kinable2014] a branch-and-price framework based on a set covering formulation is proposed. The location of students is not provided but only is known for each student the set of stops to which this student can be assigned. Thus, no optimization of the student walking distances is required, but a maximum walking distance constraint is considered (propose an exact branch-and-price algorithm for the same problem). In [@calvete2020] the same problem configuration as in [@Schittekat2013] is considered. But two main differences in the solution approach there exist: i) in the construction phase, it is proposed to integrate the allocation of the students and the route construction, while in [@Schittekat2013] the integrated sub-problems are the bus stop selection and the route generation, and ii) the procedure includes the solution of four purpose-built Mixed Integer Linear Programming (MILP) problems for selecting the bus stops, together with a more general random selection, which contributes to diversifying the considered solutions. The matheuristic developed is based on an approach that first partially allocates the students to a set of active stops they can reach and computes a set of routes that minimizes the routing cost. Then, a refining process is performed to complete the allocation and adapt the routes until a feasible solution is obtained. All the articles cited have in common that they formulate the SBRP with deterministic demand; this means that the number of students requiring school bus service is known in advance and is the same every day. Under this assumption, bus capacity management is carried out considering different criteria, such as heterogeneous fleet, mixed loads, and mixed trips, in order to minimize the school bus's travel distance and the cost of the fleet. In contrast to these deterministic models, we propose a stochastic approach in which the number of students traveling on a bus is unknown each day, taking into account the behavior of students not riding the bus on a given day. Therefore, with unknown ridership levels, it is necessary to extend the traditional criteria used in the literature for bus capacity management to avoid high idle capacity levels, unnecessarily increasing the transportation system's costs. To reach this goal, we consider a strategy where students are compensated for giving up the option to ride a bus to reduce the overall cost of the system. # Mathematical formulations {#formulation} Consider a school that offers a predefined incentive to any student willing to opt-out of using the school bus. We will focus on a pricing strategy to find a value for a monetary incentive $\tau$ such that the school district observes an overall reduction in transportation costs. That is to say, the saving on costs from reducing the number of routes must be more significant than the cost of paying the incentive $\tau$ to those students that decided to opt-out. For each student $i \in A$, where $A = \{1,...,N \}$ is the set of all students, the variables $\mathscr{Y}_i$ are equal to 1 if student $i$ opt-out from riding the bus, and 0 otherwise. The variables $\mathscr{Y}_i$ are independently and identically distributed random variables that follow a Bernoulli distribution with parameter $r_i$, which represents the opt-out probability for student $i$. $\mathscr{R}\left(\forall i\in A : \mathscr{Y}_i=0\right)$ is the cost associated with routing the remaining students. Notice that $\mathscr{R}$ is also a random variable since it is a function of the random variables $\mathscr{Y}_i, \forall i \in A$. Then, we aim to determine the value of the incentive $\tau$ to minimize the total expected transportation cost: [\[tax:main\]]{#tax:main label="tax:main"} $$% \mathscr{P}_\theproblemno: \quad \min_{\tau} \quad E\left[\sum_{i\in A} \tau \mathscr{Y}_i + \mathscr{R}\left(\forall i\in A : \mathscr{Y}_i=0\right)\right]$$ An undesirable scenario for a district would be offering a certain incentive and observing students who opt-out not producing any routing savings. Alternatively, having reduced the fleet to a certain number of buses, another unfavorable event is realizing that those savings do not compensate for the total incentive given to the students that opted out. Let $\mathscr{K}$ represent the event "the savings due to fleet reduction do not compensate the cost of incentives for students". Then, if the likelihood $r_i$ that each student accept an incentive $\tau$ is known, we would want to estimate the risk of a failed open offer policy given a particular value for $\tau$ as, $$P\left(\mathscr{K} \;\middle|\; \tau \, , \, r_i \ \forall i\in A \right) = P\left( \sum_{i\in A} \tau \mathscr{Y}_i > \mathscr{R}\left( A \right) -\mathscr{R}\left(\forall i\in A : \mathscr{Y}_i=0\right) \;\middle|\; \tau \, , \, r_i \ \forall i\in A \right) \label{probfail}$$ We refer to this as the probability of failre. Moreover, it would be of value to find a range of $\tau$ for which the risk of a failed open offer policy is under a threshold. Let $R_0$ be the number of buses needed to route all students, and $R$ the number of buses to route students that do not opt-out. Let $\kappa$ be the total cost of operating one bus. And, let $X$ be the number of students that opted out. Then, $$\begin{aligned} P\left(\mathscr{K} \;\middle|\; \tau \right) & = \sum_r P\left( \kappa \left(R_0-R\right) < \tau X \;\middle|\; \tau, r \right) P(r) \\ & = P\left( \kappa \Delta < \tau X \;\middle|\; r \right) \\ & = P\left( \frac{X}{\Delta} > \frac{\kappa}{\tau} \;\middle|\; r \right) \\ & \approx \frac{1}{N}\sum_{i=1}^N \left[ \frac{x_i}{\delta_i} > \frac{\kappa}{\tau} \;\middle|\; r_i \right]\end{aligned}$$ where $\frac{\kappa}{\tau}$ represents the number of students the school district can afford to pay with the saving cost of one bus. ## Bus stop location problem Once the set of students requiring transportation is known, the next step is to determine the location of the bus stops. Students are expected to walk from their homes to their assigned bus stop. School district keep policies regarding this process such as a maximum walking distance allowed and a maximum number of students per stop. We aim to find the minimum number of bus stops that meet all the district policies. However, once a minimum is determined, there are still many combination of bus stop locations that yield the same total number; then, to break the tie between these multiple solutions, we consider a second objective that minimizes the total distance students would walk. Let $P$ and $C$ represent the set of potential stops and the set of students, respectively. Let $d_{ij}$ be the walking distance from $i\in C$ to $j\in P$, $\delta$ the maximum distance a student can walk, $p$ the maximum number of students a stop can be assigned. Let $\phi\left(i\right)=\{j \in P : d_{ij}\leq\delta\}$ be the set of stops within reach of student $i\in C$ and $\theta\left(j\right)=\{i \in C : d_{ij}\leq\delta\}$ the set of students within reach of stop $j\in P$. Let $z_j$ be a binary decision variable that is equal to one when a potential stop $j\in P$ is chosen and $y_{ij}$ a binary decision variable that is equal to one when student $i\in C$ is assigned to stop $j\in \phi\left(i\right)$. Then, the location model reads as follows [\[locationmodel\]]{#locationmodel label="locationmodel"} $$\begin{aligned} % \mathscr{P}_\theproblemno: \quad \min \quad & \sum_{j\in P} z_j \label{locationmodel:obj1} \\ \min \quad & \sum_{i\in C}\sum_{j\in \phi\left(i\right)} d_{ij}y_{ij} \label{locationmodel:obj2} \\ \text{s.t.} \quad & \sum_{j\in \phi\left(i\right)} y_{ij} = 1 && i \in C \label{locationmodel:co1} \\ & y_{ij} \leq z_j && i \in C ,\ j\in \phi\left(i\right) \label{locationmodel:co2} \\ & \sum_{i\in \theta\left(j\right)} y_{ij} \leq p && j\in P \label{locationmodel:co3} \\ & y_{ij} \in \{0,1\} && i \in C ,\ j\in \phi\left(i\right) \label{locationmodel:co4} \\ & z_j \in \{0,1\} && j\in P\end{aligned}$$ where [\[locationmodel:obj1\]](#locationmodel:obj1){reference-type="eqref" reference="locationmodel:obj1"} minimizes the number of bus stops for the school and [\[locationmodel:obj2\]](#locationmodel:obj2){reference-type="eqref" reference="locationmodel:obj2"} minimizes the total distance students would walk. Constraints [\[locationmodel:co1\]](#locationmodel:co1){reference-type="eqref" reference="locationmodel:co1"} ensure every student is assigned one and only one bus stop, constraints [\[locationmodel:co2\]](#locationmodel:co2){reference-type="eqref" reference="locationmodel:co2"} allow students to be assigned a bus stop that has been chosen as such, and constraints [\[locationmodel:co3\]](#locationmodel:co3){reference-type="eqref" reference="locationmodel:co3"} limit the number of students per stop. To solve [$\mathscr{P}_{\ref{locationmodel}}$](#locationmodel) we need to know he sets $P$ and $C$. For simplicity, in this work we will assume that the set of potential bus stop locations $P$ is the same as the set of student requiring transportation. Let $A^\prime$ represent the set of students requiring transportation, i.e., those students that decided not to opt-out. Then, $P=C=A^\prime$. ## SBRP with stochastic heterogeneous demand and duration constraints We propose the following conceptual formulation for the bus route generation sub-problem: [\[routingmodel\]]{#routingmodel label="routingmodel"} $$\begin{aligned} % \mathscr{P}_\theproblemno: \quad \min \quad & \text{number of buses} + \varepsilon \times \text{total travel time} \label{general.obj}\\ \text{s.t.} \quad & \text{Routing Constraints} \label{general.route}\\ & P\left(\text{overcrowding bus}\right) \leq \alpha && \forall \ \text{bus} \label{general.cap}\\ & P\left(\text{being late to school}\right) \leq \beta && \forall \ \text{bus} \label{general.time}\\ & E\left(\text{maximum ride time}\right) \leq t^* && \forall \ \text{bus} \label{general.ride}\end{aligned}$$ The objective function ([\[general.obj\]](#general.obj){reference-type="ref" reference="general.obj"}) minimizes the total number of buses used and the total travel time of the routing plan. The parameter $\varepsilon$ is set as the inverse of an upper of such time. This would make $\varepsilon \left(\text{total travel distance}\right)\leq 1$, implying the number of buses is the main objective. Thus, the weighted travel times encourage the generation of smoother routes. Constraints ([\[general.route\]](#general.route){reference-type="ref" reference="general.route"}) ensures a feasible routing generation. Constraints ([\[general.cap\]](#general.cap){reference-type="ref" reference="general.cap"}) provides an upper bound for the likelihood of overcrowding the bus, constraints ([\[general.time\]](#general.time){reference-type="ref" reference="general.time"}) provide an upper bound for the likelihood of a bus being late to school, and constraint ([\[general.ride\]](#general.ride){reference-type="ref" reference="general.ride"}) provides an upper bound for the expected maximum ride time of a student on any bus. This formulation is proposed in [@Caceres2017] considering a homogeneous student ridership. Therefore, we modify the set of constraints [\[general.cap\]](#general.cap){reference-type="eqref" reference="general.cap"} to account for individual ridership. ### Routing constraints Let us denote by $D$, $P$, and $S$ the set of depots, stops, and schools such that they are disjoint and $D\cup P \cup S=L$ is the set of all locations. Let $\mu_{T_{ij}}$ be the expected value of the travel time between locations $i$ and $j$ where $\left(i,j\right)\in L^2$, $\mu_{T_{i}}$ the expected value of the waiting time or delay at location $i$ where $i\in P$, $w_i$ the number of students assigned to stop $i\in P$, $a_{ij}$ equal to 1 if students at stop $i\in P$ go to school $j\in S$. And $\kappa_i$ equal to 1 if depot $i\in D$ is a depot where buses are still idle and $0$ if that depot represents a school with buses ready to continue picking up students. Let $B$ denote the set of buses and $b_{ik}$ be equal to 1 if depot $i\in D$ contains bus $k\in B$ and 0 otherwise. Let $x_{ijk}$ be a binary decision variable equal to 1 when the edge $\left(i,j\right)\in L^2$ is covered by bus $k\in B$ and 0 otherwise. Then, the single bell-time routing problem reads as follows: $$\begin{aligned} \text{Min} \quad & \sum_{k\in B} \sum_{i\in D}\sum_{j\in P} x_{ijk} + \varepsilon \sum_{k\in B} \sum_{i\in L}\sum_{j\in L} \left(\mu_{T_{ij}}+\mu_{T_{i}}\right) x_{ijk} \label{mo4:obj}\\ \text{s.t. }\quad& \sum_{k\in B} \sum_{i\in D \cup P} x_{ijk} =1 ,\enspace j \in P \label{mo4:002}\\ & \sum_{k\in B} \sum_{j\in P \cup S} x_{ijk} =1 ,\enspace i \in P \label{mo4:003}\\ & \sum_{k\in B} \sum_{i\in L} \left( x_{iik} + \sum_{j\in D} x_{ijk} +\sum_{j\in S} x_{jik} \right) =0 \label{mo4:022}\\ & \sum_{i \in D\cup P} x_{ijk} = \sum_{i \in P \cup S } x_{jik} ,\enspace k \in B, j \in P \label{mo4:004}\\ & \sum_{i \in D\cup P} x_{ijk} \leq \sum_{g\in S} \sum_{i \in P} a_{jg} x_{igk} ,\enspace k\in B, j \in P\label{mo4:nocombine}\\ & \sum_{i \in D \cup P} x_{ijk} \leq \sum_{i \in D}\sum_{j\prime \in P} x_{ij\prime k} ,\enspace k \in B, j\in P \label{mo4:008}\\ & \sum_{j \in L} x_{ijk} \leq b_{ik} ,\enspace k\in B, i \in D \label{mo4:businitialposition}\\ & 1 \leq u_{ik} \leq m+2 ,\enspace k \in B , i \in L \label{mo4:st_elim2}\\ & u_{ik} - u_{jk} +\left(m+2\right) x_{ijk} \leq m+1 ,\enspace k \in B , i \in L ,j \in L \label{mo4:st_elim4}\\ & P\left(\text{overcrowding the bus}\right)\leq\alpha ,\enspace k \in B \label{mo4:g2} \\ & P\left(\text{being late to school}\right)\leq\beta ,\enspace k \in B \label{mo4:g3} \\ & E\left(\text{maximum ride time}\right) \leq \Delta t,\enspace k \in B \label{mo4:g1} \\ & x_{ijk} \text{ binary} \label{mo4:variables}\end{aligned}$$ where ([\[mo4:obj\]](#mo4:obj){reference-type="ref" reference="mo4:obj"}) minimizes the number of buses needed $\sum_{k\in B}\sum_{i\in D}\sum_{j\in P} x_{ijk}$ while maintaining the total length of the routes $\sum_{k\in B}\sum_{i\in L}\sum_{i\in L} \left(\mu_{T_{ij}}+\mu_{T_{i}}\right) x_{ijk}$ to a minimum, $\varepsilon$ is set as the inverse of an upper bound for such length or an initial solution generated with a constructive heuristic. The constraints ensure conditions as follow: ([\[mo4:002\]](#mo4:002){reference-type="ref" reference="mo4:002"}) one and only one bus arrives at every stop, ([\[mo4:003\]](#mo4:003){reference-type="ref" reference="mo4:003"}) one and only one bus departures from every stop, ([\[mo4:022\]](#mo4:022){reference-type="ref" reference="mo4:022"}) no bus stays at the same location nor arrives at a depot nor departures from a school, ([\[mo4:004\]](#mo4:004){reference-type="ref" reference="mo4:004"}) same bus that arrives at a location departures from that location, ([\[mo4:nocombine\]](#mo4:nocombine){reference-type="ref" reference="mo4:nocombine"}) a bus only picks up students attending the same school, ([\[mo4:008\]](#mo4:008){reference-type="ref" reference="mo4:008"}) a location can be visited by a bus only if that bus leaves the depot, ([\[mo4:businitialposition\]](#mo4:businitialposition){reference-type="ref" reference="mo4:businitialposition"}) all buses start their route on their corresponding depot, ([\[mo4:st_elim2\]](#mo4:st_elim2){reference-type="ref" reference="mo4:st_elim2"}) and ([\[mo4:st_elim4\]](#mo4:st_elim4){reference-type="ref" reference="mo4:st_elim4"}) are the sub-tour elimination constraints where $m$ is the maximum number of stops a bus can have, ([\[mo4:g2\]](#mo4:g2){reference-type="ref" reference="mo4:g2"}) to ([\[mo4:g1\]](#mo4:g1){reference-type="ref" reference="mo4:g1"}) are the stochastic constraints which will be developed in detail in the following section and ([\[mo4:variables\]](#mo4:variables){reference-type="ref" reference="mo4:variables"}) is the integrality condition. ### Constraint on the likelihood of overcrowding the bus On each route, a bus will serve one and only one school. In practice, students do not always ride the bus, and their decisions on whether to ride are highly influenced by their grades, the school they attend, and whether the route is done in the morning or afternoon. Also, the capacity of a bus depends on the student's grades (e.g., a bus can hold up to 71 elementary students, whereas the capacity is set to 47 for middle and HI students). Under such circumstances, though it is assumed to be using a homogeneous fleet, bus capacity is dynamic and depends on the grade in which students attend and their choice of whether to ride the bus; the less willing the students are to ride the bus, the more students can be assigned to a bus, i.e., overbooking its capacity [@Caceres2017]. **Definition 1**. Let $\mathscr{X}_i$ be equal to one if a students decides to ride the bus and equal to zero otherwise. Then, $\mathscr{X}_i$ is a random variable following a Bernoulli distribution $\mathscr{X}_i \sim \text{Bernoulli}(\rho_i)$ where $\rho_i$ is the individual ridership of student $i\in A$ (the probability that a student will use their assigned bus). **Definition 2**. Let $\mathscr{W}_k=\sum_{i\in L}\sum_{j\in L} \mathscr{X}_i x_{ij}$ be the actual number of students riding bus $k\in B$. Then, $\mathscr{W}_k$ is a random variable with mean and variance given by $$\mu_{\mathscr{W}_k}=\sum_{i\in L}\sum_{j\in L} \rho_i x_{ij}$$ $$\sigma_{\mathscr{W}_k}^2=\sum_{i\in L}\sum_{j\in L} \rho_i \left(1-\rho_i\right) x_{ij}$$ respectively. Thus, the capacity constraints represented by [\[general.cap\]](#general.cap){reference-type="eqref" reference="general.cap"} is given by $$\label{capacityconstraint} P\left(\mathscr{W}_k > c_k\right) \leq \alpha \qquad k \in B$$ where $c_k$ is the capacity of bus $k\in B$. Notice that $P\left(\mathscr{W}_k > c_k\right)>0$ only when more students than the capacity of the bus are assigned. Then, $\mathscr{W}_k$ will be the summation of more than $c_k$ random variables for when it's relevant, that for this case, will be either 47 or 71 depending on the school. **Conjecture 1**. *The probability distribution of $\mathscr{W}_k$ can be approximated to a normal distribution with a mean of $\mu_{\mathscr{W}_k}$ and a variance of $\sigma_{\mathscr{W}_k}^2$ by means of the Central Limit Theorem.* Thus, we use the aforementioned conjecture in the following proposition to reformulate [\[capacityconstraint\]](#capacityconstraint){reference-type="eqref" reference="capacityconstraint"} into a set of linear inequalities. **Proposition 1**. *For all $k\in B$, the constraints $$\begin{aligned} & \mu_{\mathscr{W}_k}+\Phi^{-1}\left(1-\alpha\right) \tilde{\sigma}_{\mathscr{W}_k} \leq c_k + \frac{1}{2} \label{prop-co01} \\ & \sum_{v=1}^{v^+} v^2 \zeta_v^k \geq \sigma_{\mathscr{W}_k}^2 \label{prop-co02} \\ & \sum_{v=1}^{v^+} v \zeta_v^k = \tilde{\sigma}_{\mathscr{W}_k} \label{prop-co03} \\ & \sum_{v=1}^{v^+} v = 1 \label{prop-co04} \end{aligned}$$ are valid inequalities for [\[capacityconstraint\]](#capacityconstraint){reference-type="eqref" reference="capacityconstraint"}, where $\zeta_v^k$ is a binary variable and $v^+$ is the maximum possible integer value for $\sigma_{\mathscr{W}_k}$.* *Proof.* From [\[capacityconstraint\]](#capacityconstraint){reference-type="eqref" reference="capacityconstraint"} we can reformulate the probability as $$P\left(\mathscr{W}_k > c_k + \frac{1}{2}\right) \leq \alpha$$ since $\mathscr{W}_k$ is now a continuous random variable (see Conjecture [Conjecture 1](#normalaproximation){reference-type="ref" reference="normalaproximation"}). Then, $$P\left(\mathscr{W}_k < c_k + \frac{1}{2}\right) \geq 1 -\alpha$$ which by standardizing becomes $$\Phi\left( \frac{c_k + \frac{1}{2} - \mu_{\mathscr{W}_k}}{\sqrt{\sigma_{\mathscr{W}_k}^2}} \right) \geq 1 -\alpha$$ and by taking the inverse $$\mu_{\mathscr{W}_k}+\Phi^{-1}\left(1-\alpha\right) \sqrt{\sigma_{\mathscr{W}_k}^2} \leq c_k + \frac{1}{2}$$ where $c_k$ and $\Phi^{-1}\left(1-\alpha\right)$ are constant numbers, and $\mu_{\mathscr{W}_k}$ and $\sigma_{\mathscr{W}_k}^2$ are obtained as stated in Definition [Definition 2](#rv-definition-students-riding-bus){reference-type="ref" reference="rv-definition-students-riding-bus"}. Notice that the previous inequality is nonlinear. Then, the square root of $\sigma_{\mathscr{W}_k}^2$ must be computed while maintaining linearity. Since $\zeta_v^k$ is a binary variable, the assignment constraints [\[prop-co03\]](#prop-co03){reference-type="eqref" reference="prop-co03"} and [\[prop-co04\]](#prop-co04){reference-type="eqref" reference="prop-co04"} ensure that the variable $\tilde{\sigma}_{\mathscr{W}_k}$ will only take integer values between 1 and $v^+$, the maximum round-up integer value the standard deviation can take. ◻ ### Constraint on the likelihood of being late to school Since this SBRP considers transportation of students to their schools, the chance of arriving late to school must be assessed. At the same time, the buses are used to serve more than one school in different time spans; a bus picks up students from one school, drop them off, and then starts a new route serving the second school and so on. The following definitions are made to account for these conditions. **Proposition 2**. *Let $\tau_f$ and $\tau_v$ represent the fixed and variable time when picking students up at each stop such that if $r$ students are to be picked up, it will take $\tau_f+\tau_v r$ to do so. Then, given a stop location $i\in A$ where there are $w_i$ students assigned to go to school $j\in S$ with a probability of showing up $p_j$, the expected value and variance of the time required by a bus to pick them up are: $$\begin{aligned} \mu_{T_i}&= \tau_f- \tau_f\left(1-p_j\right)^{w_i}+\tau_v w_i p_j \label{taumu}\\ \sigma_{T_i}^2&=\tau_f^2\left(1-p_j\right)^{w_i}\left(1-\left(1-p_j\right)^{w_i}\right)+2\tau_f \tau_v w_i p_j \left(1-p_j\right)^{w_i}+\tau_v^2 w_i p_j \left(1-p_j\right)\label{tauvar}\end{aligned}$$* We know that $R_i$, the actual number of students showing up at stop $i\in A$, is a r.v. such that $R_i\sim Bin\left(w_i,p_j\right)$. Then, let $T_i\left(R_i\right)= \left\{ \begin{array}{l l} \tau_f+\tau_v R_i & \text{if $R_i>0$}\\ 0 & \text{if $R_i=0$} \end{array} \right.$ be the time that takes making a stop at node $i\in A$. Then, the probability mass function (*pmf*) for $T_i$ is given by $p_{T_i}\left(t_i\right)= \left\{ \begin{array}{l l} p_{R_i}\left(0\right) & \text{if $t_i=0$}\\ p_{R_i}\left(r_i\right) & \text{if $t_i=\tau_f+\tau_v r_i$}\\ 0 & \text{otherwise} \end{array} \right.$ and the expected value and variance of $T_i$ are then derived as follows: $$\begin{aligned} \nonumber \mu_{T_i}&=E\left[T_i\left(R_i\right)\right]=\sum_{r=0}^{w_i} T_i\left(r\right) p_{R_i}\left(r\right) =0\cdot p_{R_i}\left(0\right) + \sum_{r=1}^{w_i} \left(\tau_f+\tau_v r\right) p_{R_i}\left(r\right)\\ \nonumber&=\tau_f\sum_{r=1}^{w_i} p_{R_i}\left(r\right)+\tau_v\sum_{r=1}^{w_i} rp_{R_i}\left(r\right) =\tau_f\left[\sum_{r=0}^{w_i} p_{R_i}\left(r\right)-p_{R_i}\left(0\right) \right]+\tau_v\sum_{r=0}^{w_i} r \thinspace p_{R_i}\left(r\right)\\ \nonumber&=\tau_f\left[1-\left(1-p_j\right)^{w_i} \right]+\tau_v E\left[R_i\right]=\tau_f - \tau_f\left(1-p_j\right)^{w_i}+\tau_v w_i p_j\\ \nonumber \sigma_{T_i}^2&=V\left[T_i\left(R_i\right)\right]=E\left[T_i\left(R_i\right)^2\right]-\left[E\left[T_i\left(R_i\right)\right]\right]^2 =\sum_{r=0}^{w_i} \left[T_i\left(r\right)\right]^2 p_{R_i}\left(r\right)-\mu_{T_i}^2\\ \nonumber &= 0^2 \cdot p_{R_i}\left(0\right) + \sum_{r=1}^{w_i} \left(\tau_f+\tau_v r\right)^2 p_{R_i}\left(r\right) - \mu_{T_i}^2 \\ \nonumber &=\tau_f^2\sum_{r=1}^{w_i}p_{R_i}\left(r\right) +2\tau_f \tau_v \sum_{r=1}^{w_i} r \thinspace p_{R_i}\left(r\right) + \tau_v^2\sum_{r=1}^{w_i} r^2 \thinspace p_{R_i}\left(r\right) - \mu_{T_i}^2 \\ \nonumber &=\tau_f^2\left[\sum_{r=0}^{w_i} p_{R_i}\left(r\right)-p_{R_i}\left(0\right) \right]+2\tau_f \tau_v \sum_{r=0}^{w_i} r \thinspace p_{R_i}\left(r\right) + \tau_v^2\sum_{r=0}^{w_i} r^2 \thinspace p_{R_i}\left(r\right) - \mu_{T_i}^2 \\ \nonumber &=\tau_f^2\left[1-\left(1-p_j\right)^{w_i} \right]+2\tau_f \tau_v E\left[R_i\right]+ \tau_v^2E\left[R_i^2\right] - \mu_{T_i}^2 \\ \nonumber &=\tau_f^2\left[1-\left(1-p_j\right)^{w_i} \right]+2\tau_f \tau_v w_ip_j+ \tau_v^2\left[V\left[R_i\right]+E\left[R_i\right]^2\right] - \mu_{T_i}^2 \\ \nonumber &=\tau_f^2\left[1-\left(1-p_j\right)^{w_i} \right]+2\tau_f \tau_v w_ip_j+ \tau_v^2\left[w_i p_j \left(1-p_j\right)+w_i^2 p_j^2\right] - \left(\tau_f - \tau_f\left(1-p_j\right)^{w_i}+\tau_v w_i p_j\right)^2 \\ \nonumber&=\tau_f^2\left(1-p_j\right)^{w_i}\left(1-\left(1-p_j\right)^{w_i}\right)+2\tau_f \tau_v w_i p_j \left(1-p_j\right)^{w_i}+\tau_v^2 w_i p_j \left(1-p_j\right)\end{aligned}$$ [@Braca1997] provide an estimation for the fixed and variable time for picking up students, where they show that $\tau_f=19$ and $\tau_v=2.6$ (both in seconds). **Definition 3**. Let $T_{ij}$ be the random travel time from location $i \in L$ to location $j\in L$ with expected value and variance given by $\mu_{T_{ij}}$ and $\sigma_{T_{ij}}^2$ respectively. **Definition 4**. Let $\mathcal{T}_k=\sum_{i\in L} \sum_{j\in L} \left( T_{ij} + T_{i} \right) x_{ijk}$ be the total travel time for bus $k\in B$ with expected value and variance given by $$\begin{aligned} \nonumber \mu_{\mathcal{T}_k} &= \sum_{i\in L} \sum_{j\in L} \left( \mu_{T_{ij}} + \mu_{T_i} \right) x_{ijk} \\ \nonumber \sigma_{\mathcal{T}_k}^2&=\sum_{i\in L}\sum_{j \in L} \left(\sigma_{T_{ij}}^2 +\sigma_{T_i}^2\right) x_{ijk}\end{aligned}$$ Then, the travel time constraint that represents ([\[mo4:g3\]](#mo4:g3){reference-type="ref" reference="mo4:g3"}) is given by: $$P\left(t_{\text{avl}}^k +\mathcal{T}_k>t_{\text{bell}}\right)\leq \beta \quad \forall k \in B \label{traveltime_constraint}$$ where $t_{\text{avl}}^k$ represents the time instant at which bus $k\in B$ becomes available, $t_{\text{bell}}$ the latest time instant at which the bus has to be at school and $\beta$ the given upper bound for the probability of bus $k\in B$ not making it on time to school. We now need to reformulate ([\[traveltime_constraint\]](#traveltime_constraint){reference-type="ref" reference="traveltime_constraint"}) such that it can be included in the single bell-time mixed integer linear program. Given the previous definitions, $\mathcal{T}_k$ represents the summation of the driving time $T_{ij}$ and the waiting time at stops $T_i$ of a particular bus. This is $$\mathcal{T}_k=T_{0,1}+T_{1}+T_{1,2}+...+T_{m-1,m}+T_{m}+T_{m,m+1}$$ where $m$ is the number of stops to be made by a bus. Then, $\mathcal{T}_k$ is a summation of $2m+1$ random variables. **Conjecture 2**. *The probability density function of $\text{ }\mathcal{T}_k$ can be approximated to a normal distribution with mean $\mu_{\mathcal{T}_k}$ and variance $\sigma_{\mathcal{T}_k}^2$ by means of the Central Limit Theorem.* Thus, we use the above conjecture in the following proposition in order to reformulate ([\[traveltime_constraint\]](#traveltime_constraint){reference-type="ref" reference="traveltime_constraint"}) into a set of linear inequalities. **Proposition 3**. *For all $k \in B$ the constraints $$t_{\text{avl}}^k + \mu_{\mathcal{T}_k} + \Phi^{-1}\left(1-\beta\right)\tilde{\sigma}_{\mathcal{T}_k} \leq t_{\text{bell}} \label{traveltime_constraint2}$$ $$\sum_{h=1}^{h^{+}} h^2 \gamma_h^k \geq \sigma_{\mathcal{T}_k}^2\label{choosesigma1}$$ $$\sum_{h=1}^{h^{+}} h \gamma_h^k = \tilde{\sigma}_{\mathcal{T}_k}\label{choosesigma2}$$ $$\sum_{h=1}^{h^{+}} \gamma_h^k = 1\label{choosesigma3}$$ are valid inequalities for ([\[traveltime_constraint\]](#traveltime_constraint){reference-type="ref" reference="traveltime_constraint"}), where $\gamma_h^k$ is a binary variable and $h^{+}$ is the maximum possible integer value for $\sigma_{\mathcal{T}_k}$.* From ([\[traveltime_constraint\]](#traveltime_constraint){reference-type="ref" reference="traveltime_constraint"}) it is obtained that $$P\left(t_{\text{avl}}^k + \mathcal{T}_k<t_{\text{bell}}\right)\geq 1-\beta$$ which by standardizing becomes $$\Phi\left(\frac{t_{\text{bell}}-\left(t_{\text{avl}}^k + \mu_{\mathcal{T}_k}\right)}{\sqrt{\sigma_{\mathcal{T}^k}^2}} \right)\geq 1-\beta$$ and by taking the inverse $$t_{\text{avl}}^k + \mu_{\mathcal{T}_k} + \Phi^{-1}\left(1-\beta\right)\sqrt{\sigma_{\mathcal{T}^k}^2} \leq t_{\text{bell}}$$ where $t_{\text{bell}}$, $t_{\text{avl}}^k$ and $\Phi^{-1}\left(1-\beta\right)$ are constant numbers, and $\mu_{\mathcal{T}_k}$ and $\sigma_{\mathcal{T}^k}^2$ are obtained as stated in Definition [Definition 4](#def4){reference-type="ref" reference="def4"}. Notice that, as it is, the previous inequality is not linear. Then, the square root of $\sigma_{\mathcal{T}_k}^2=\sum_{i\in L}\sum_{j \in L} \left(\sigma_{T_{ij}}^2 +\sigma_{T_i}^2\right) x_{ijk}$ must be calculated while maintaining linearity. Since $\gamma_h^k$ is a binary variable, the assignment constraints ([\[choosesigma2\]](#choosesigma2){reference-type="ref" reference="choosesigma2"}) and ([\[choosesigma3\]](#choosesigma3){reference-type="ref" reference="choosesigma3"}) ensure that the variable $\tilde{\sigma}_{\mathcal{T}_k}$ will only take an integer value between 1 and $h^{+}=\lceil \left(t_{\text{bell}}-min \{t_{\text{avl}}^k\}\right)/2\rceil$ the maximum round up integer value the standard deviation can take. Then, the inequality in ([\[choosesigma1\]](#choosesigma1){reference-type="ref" reference="choosesigma1"}) constraints $\tilde{\sigma}_{\mathcal{T}_k}$ to be at least the round-up integer of $\sqrt{\sigma_{\mathcal{T}_k}^2}$. Since now $\sqrt{\sigma_{\mathcal{T}_k}^2}\leq \tilde{\sigma}_{\mathcal{T}_k}$, the following inequality holds: $$t_{\text{avl}}^k + \mu_{\mathcal{T}_k} + \Phi^{-1}\left(1-\beta\right)\sqrt{\sigma_{\mathcal{T}_k}^2} \leq t_{\text{avl}}^k + \mu_{\mathcal{T}_k} + \Phi^{-1}\left(1-\beta\right)\tilde{\sigma}_{\mathcal{T}_k}$$ Therefore, if ([\[traveltime_constraint2\]](#traveltime_constraint2){reference-type="ref" reference="traveltime_constraint2"}) is satisfied then ([\[traveltime_constraint\]](#traveltime_constraint){reference-type="ref" reference="traveltime_constraint"}) will also be satisfied. ### Constraint on the expected maximum ride time As part of the school district's policy, it is expected that the average time a student spends on the bus should not be greater than a certain threshold $\Delta t_{\text{max}}$. For this case, if we assure this condition to the first student who gets picked up, then the condition will also apply to the rest of the students on that bus. Thus, the constraint that represents ([\[mo4:g1\]](#mo4:g1){reference-type="ref" reference="mo4:g1"}) reads as follows: $$\mu_{\mathcal{T}_k} - \sum_{i\in D}\sum_{j\in A} \mu_{T_{ij}}x_{ijk}\leq \Delta t_{\text{max}} \enspace \forall k \in B \label{maxridetime}$$ where $\sum_{i\in D}\sum_{i\in A} \mu_{T_{ij}}x_{ijk}$ represents the expected time from the depot to the first stop. ### Individual ridership estimation In [@Caceres2017], the estimation of student ridership is the average ridership per school, i.e., students from the same school have the same estimate. When considering that students can opt-out, the former assumption becomes troublesome. If students opt-out based on their likelihood of using the bus, the students that continue to use the bus system will be those that use it more often; hence, the remaining group will have a greater overall ridership than the original group. If we assume that all students share the same ridership, after students opt-out the remaining group will continue to have the original ridership. This contradicts our previous analysis. Then, to avoid underestimating ridership, we need to assign each student a ridership estimate that varies depending on how far each one lives from their school. Let $\rho_i$ be the individual ridership estimate for student $i\in A$ and $\bar{\rho}$ the observed average ridership for one school [@Caceres2017]. We aim to find a function $f(d_i)$ that estimates $\rho_i$ such that the average of the resulting estimates equals the original ridership of the school, i.e., $\frac{\sum f(d_i)}{n}=\bar{\rho}$. Let us assume that $f(d_i)$ is a linear function of $d_i$. Then, we have that $\rho_i=f(d_i)=\hat{\rho}_0+\hat{\rho}_1 \times d_i$ where $\hat{\rho}_0$ and $\hat{\rho}_1$ are unknown constants that can be found for each school. Now, notice that there are infinite combinations for $\hat{\rho}_0$ and $\hat{\rho}_1$ that meet the requirement of keeping the overall ridership the same. Also notice that, when $\hat{\rho}_1>0$, $\rho_i$ is increasing with respect to $d_i$. Notice that $d_i$ can be replace by any function $g(d_i)$ as long as $\frac{\partial g(d_i)}{\partial d_i} >0$. Then, we can formulate an optimization problem to find the best combination between $\hat{\rho}_0$ and $\hat{\rho}_1$ as follows [\[ridershipmodel\]]{#ridershipmodel label="ridershipmodel"} $$\begin{aligned} % \mathscr{P}_\theproblemno: \quad \max_{\{\hat{\rho}_0 , \hat{\rho}_1\}} \quad & \hat{\rho}_1 \label{ridershipmodel:objective} \\ \text{s.t.} \quad & \rho_i=\hat{\rho}_0+\hat{\rho}_1 \times g(d_i) && i \in A \label{ridershipmodel:co1} \\ & \frac{\sum_{i} \rho_i}{n} = \bar{\rho} \label{ridershipmodel:co2} \\ & 0 \leq \rho_i \leq 1 && i \in A \label{ridershipmodel:co3} \\ & \hat{\rho}_1 \geq 0 \label{ridershipmodel:co4}\end{aligned}$$ where [\[ridershipmodel:objective\]](#ridershipmodel:objective){reference-type="eqref" reference="ridershipmodel:objective"} aims to maximize the contribution of $d_i$ to the explanation of $\rho_i$. The constraints in [\[ridershipmodel:co1\]](#ridershipmodel:co1){reference-type="eqref" reference="ridershipmodel:co1"} are the linear relation for the individual student ridership, constraint [\[ridershipmodel:co2\]](#ridershipmodel:co2){reference-type="eqref" reference="ridershipmodel:co2"} controls that the average among all ridership continues to be the original overall ridership for the school, constraints [\[ridershipmodel:co3\]](#ridershipmodel:co3){reference-type="eqref" reference="ridershipmodel:co3"} ensure individual riderships are valid probability values, and constraint [\[ridershipmodel:co4\]](#ridershipmodel:co4){reference-type="eqref" reference="ridershipmodel:co4"} ensures that the contribution of $d_i$ to the explanation of $\rho_i$ is non-negative. **Proposition 4**. *The problem [$\mathscr{P}_{\ref{ridershipmodel}}$](#ridershipmodel) admits the solution $\hat{\rho}_1 = \min\left\{\frac{\bar{\rho}}{\bar{d}-\min_i g(d_i) - \bar{d}}, \frac{1-\bar{\rho}}{\max_i g(d_i)-\bar{d}} \right\}$, and $\hat{\rho}_0 = \bar{\rho} + \bar{d}\min\left\{\frac{\bar{\rho}}{\bar{d}-\min_i g(d_i) - \bar{d}}, \frac{1-\bar{\rho}}{\max_i g(d_i)-\bar{d}} \right\}$, where $\bar{d} = \frac{1}{n}\sum_i g(d_i)$.* *Proof.* We note first that constraints [\[ridershipmodel:co1\]](#ridershipmodel:co1){reference-type="eqref" reference="ridershipmodel:co1"} and [\[ridershipmodel:co2\]](#ridershipmodel:co2){reference-type="eqref" reference="ridershipmodel:co2"} implies $\hat{\rho}_0 = \bar{\rho} - \hat{\rho}_1\bar d$. Now, as $\hat{\rho}_1$ is non-negative, the constraints [\[ridershipmodel:co3\]](#ridershipmodel:co3){reference-type="eqref" reference="ridershipmodel:co3"} can be replaced by the following two constraints $$0 \le \hat{\rho}_0 + \hat{\rho}_1 \min_i g(d_i), \qquad \hat{\rho}_0 + \hat{\rho}_1 \max_i g(d_i) \le 1.$$ Replacing the value of $\hat{\rho}_0$ and reordering we find that $$\hat{\rho}_1(\bar{d}-\min_i g(d_i)) \le \bar{\rho}, \qquad \hat{\rho}_1(\max_i g(d_i) - \bar{d}) \le 1-\bar{\rho}.$$ Since we are looking for the maximum value for $\hat{\rho}_1$, the result follows. ◻ Then, the solution for [$\mathscr{P}_{\ref{ridershipmodel}}$](#ridershipmodel) yields the coefficients $\hat{\rho}_0$ and $\hat{\rho}_1$ that allows us to find the estimate for individual ridership $\rho_i$. Once obtained, these values never change because they constitute a characteristic of each student and therefore are not affected by any optimization regarding routing. Now, the overall ridership of any subset of students will be different from the school's original average, and therefore any routing optimization considering a subset of students will need to consider such variation. # Solution approach {#solution} ## Simulation based solution method In this section, we present our simulation-based approach for estimating the probability of failure associated with the open offer policy. Initially, it is important to note that the number of students who will choose to decline the bus system before a school offers an incentive $\tau$ remains uncertain. We can consider an overall estimation, for instance, assuming that 25% of students would opt-out if given the choice. This implies an overall opt-out probability of 0.25. By utilizing this information, we can determine individual opt-out probabilities $r_i$ for each student, such that the cumulative opt-out probability equals 0.25. Furthermore, we can hypothesize that $r_i$ is dependent on the distance between the school and the respective student, we further explain this idea in the next subction. Once we have calculated the overall and individual opt-out probabilities, we can determine the maximum incentive that can be distributed per student, ensuring that the school district does not incur losses. To achieve this, we randomly assign decisions to all students based on their respective opt-out probabilities. Subsequently, we establish bus stop locations and routes for the remaining group by excluding the students who have opted out. To address the routing problem described in [$\mathscr{P}_{\ref{routingmodel}}$](#routingmodel), we employ a decomposition technique utilizing column generation, as outlined in the method presented by[@Caceres2017]. At this stage, we have information on the resulting number of routes and the count of students who have opted out. The potential savings can be determined by calculating the difference between the base routing cost and the cost associated with this particular scenario. Finally, dividing the savings by the number of students who have opted out allows us to determine the maximum amount that the school district could offer without incurring financial losses. By iterating through this process, we can generate a wide range of outcomes for each average opt-out probability level. Furthermore, a similar approach can be employed to estimate the probability of failure, as defined in equation [\[probfail\]](#probfail){reference-type="eqref" reference="probfail"}, for any given overall opt-out probability and incentive value $\tau$. ## Individual opt-out probability estimation {#sec:individualOOP} Let us assume that the closer a student lives to their school, the more likely they are to opt-out of the bus system if offered an incentive, regardless of its value. Then, similarly to the previous section, if we assume the individual opt-out probability $r_i$ as a linear function of $d_i$ we have that $r_i=f(d_i)=\hat{r}_0-\hat{r}_1 d_i$ where $\hat{r}_0$ and $\hat{r}_1$ are unknown constants that can be found for each school. Unlike ridership, there is no observable overall or average opt-out probability at the school level. However, with the simulation based approach, for each level of overall opt-out probability $\bar{r}$ provided in every replicate, we can determine opt-out probability at the individual level by solving the following linear program [\[oopmodel\]]{#oopmodel label="oopmodel"} $$\begin{aligned} % \mathscr{P}_\theproblemno: \quad \max_{\{\hat{r}_0 , \hat{r}_1\}} \quad & \hat{r}_1 \label{oopmodel:obj}\\ \text{s.t.} \quad & r_i= \hat{r}_0 - \hat{r}_1 d_i && i \in A \label{oopmodel:co1}\\ & \frac{\sum_{i} r_i}{n} = \bar{r}\label{oopmodel:co2}\\ & 0 \leq r_i \leq 1 && i \in A \label{oopmodel:co3}\\ & \hat{r}_1 \geq 0 \label{oopmodel:co4}\end{aligned}$$ where [\[oopmodel:obj\]](#oopmodel:obj){reference-type="eqref" reference="oopmodel:obj"} maximizes the contribution of the distance to the explanation of $r_i$ as modeled in constraints [\[oopmodel:co1\]](#oopmodel:co1){reference-type="eqref" reference="oopmodel:co1"}. Constraint [\[oopmodel:co2\]](#oopmodel:co2){reference-type="eqref" reference="oopmodel:co2"} ensures the average opt-out probability matches the value given for the replicate, constraints [\[oopmodel:co3\]](#oopmodel:co3){reference-type="eqref" reference="oopmodel:co3"} warrant that the values for $r_i$ are valid probabilities, and constraint [\[oopmodel:co4\]](#oopmodel:co4){reference-type="eqref" reference="oopmodel:co4"} keeps the coefficient positive so that the contribution $-\hat{r}_1 d_i$ stays negative. **Proposition 5**. *The problem [$\mathscr{P}_{\ref{oopmodel}}$](#oopmodel) admits an analytic solution given by $\hat{r}_1 = \min\left\{\frac{\bar{r}}{\max_i d_i - \bar{d}}, \frac{1-\bar{r}}{\bar{d}-\min_i d_i} \right\}$, and $\hat{r}_0 = \bar{r} + \bar{d}\min\left\{\frac{\bar{r}}{\max_i d_i - \bar{d}}, \frac{1-\bar{r}}{\bar{d}-\min_i d_i} \right\}$, where $\bar{d} = \frac{1}{n}\sum_i d_i$ is the average distance.* *Proof.* First, we note that constraints [\[oopmodel:co1\]](#oopmodel:co1){reference-type="eqref" reference="oopmodel:co1"} and [\[oopmodel:co2\]](#oopmodel:co2){reference-type="eqref" reference="oopmodel:co2"} implies $\hat{r}_0 = \bar{r} + \hat{r}_1\bar d$. Second, as $\hat{r}_1$ is non-negative, the constraints [\[oopmodel:co3\]](#oopmodel:co3){reference-type="eqref" reference="oopmodel:co3"} can be replaced by the following two constraints $$0 \le \hat{r}_0 - \hat{r}_1 \max_i d_i, \qquad \hat{r}_0 - \hat{r}_1 \min_i d_i \le 1.$$ Replacing the value of $\hat{r}_0$ and reordering we find that $$\hat{r}_1(\max_i d_i - \bar{d}) \le \bar{r}, \qquad \hat{r}_1(\bar{d}-\min_i d_i) \le 1-\bar{r}.$$ Cause we are looking for the maximum value for $\hat{r}_1$, the result follows. ◻ Then, the solution for [$\mathscr{P}_{\ref{oopmodel}}$](#oopmodel) yields the coefficients $\hat{r}_0$ and $\hat{r}_1$ that allows us to determine individual opt-out probabilities $r_i$. # Computational experiments {#experiments} ## Synthetic data {#sec:synth} Our numerical example begins with exploring the behavior in one school, with simulations on synthetic data. In order to do so, we proceed as follows: 1. Create sets of students for simulation 2. Determinate individual ridership for each student using problem [$\mathscr{P}_{\ref{ridershipmodel}}$](#ridershipmodel) 3. Generate random ridership $\bar r$ 4. Determinate the individual opt-out probability solving problem [$\mathscr{P}_{\ref{oopmodel}}$](#oopmodel) for $\bar r$ 5. Sample opting-out students and remove them from the system 6. Choose stop location based on the remaining students 7. Create routes solving the problem [$\mathscr{P}_{\ref{routingmodel}}$](#routingmodel) To generate synthetic sets of students for the simulation, we considered three crucial factors to introduce variation. Firstly, we focused on different types of geographic distribution, emphasizing students living near the school, students living far from the school, and uniform distribution of students around the school. To achieve this, we employed a Beta distribution with parameters $\alpha$ and $\beta$, randomly assigning coordinates within a square area with a side length of 3 miles. We positioned this area over part of Amherst, NY for illustration purposes only, as shown in Figure [3](#fig:geography){reference-type="ref" reference="fig:geography"}. For each student, the $x$ and $y$ coordinates were generated independently with respective parameters: $\left(\alpha=1,\ \beta=1\right)$, $\left(\alpha=0.45,\ \beta=0.45\right)$, and $\left(\alpha=6.5,\ \beta=6.5\right)$. The resulting spatial distribution of students across the area of interest can be observed in Figure [4](#fig:geography-sets){reference-type="ref" reference="fig:geography-sets"}. Secondly, we recognized the importance of considering the number of students in the school as another variable. To thoroughly explore the impact of the student population on the simulation outcomes, we conducted tests with two different sample sizes: one comprising 400 students and another comprising 800 students. Lastly, we consider the average ridership per school. To gain comprehensive insights into student transportation demands, we specifically examined scenarios with an average ridership of $\bar{r} = 0.3$ and $\bar{r} = 0.8$. {#fig:geography width="0.55\\columnwidth"} {#fig:geography-sets width="0.85\\columnwidth"} Overall, by deliberately manipulating these three factors --- geographic distribution, student population, and average ridership- we were able to create diverse and representative sets of synthetic students for the simulation. This approach provided us with valuable insights into the dynamics of student transportation and its associated challenges under various scenarios. Table [1](#tab:experimentsetting){reference-type="ref" reference="tab:experimentsetting"} shows the 12 experimental settings. Set Students ($n$) Ridership ($\bar{\rho}$) Geography ----- ---------------- -------------------------- -------------------------------- 1 400 0.3 $\alpha=1.00 \quad \beta=1.00$ 2 400 0.8 $\alpha=1.00 \quad \beta=1.00$ 3 800 0.3 $\alpha=1.00 \quad \beta=1.00$ 4 800 0.8 $\alpha=1.00 \quad \beta=1.00$ 5 400 0.3 $\alpha=0.45 \quad \beta=0.45$ 6 400 0.8 $\alpha=0.45 \quad \beta=0.45$ 7 800 0.3 $\alpha=0.45 \quad \beta=0.45$ 8 800 0.8 $\alpha=0.45 \quad \beta=0.45$ 9 400 0.3 $\alpha=6.50 \quad \beta=6.50$ 10 400 0.8 $\alpha=6.50 \quad \beta=6.50$ 11 800 0.3 $\alpha=6.50 \quad \beta=6.50$ 12 800 0.8 $\alpha=6.50 \quad \beta=6.50$ : Experiment setting The risk of implementing the open offer policy is represented by $P\left(\mathscr{K} \middle| \tau \right)$, the conditional probability that the savings, if any, due to fleet reduction do not compensate the cost of incentives for students given a certain incentive $\tau$. We do not know the exact relation between the incentive $\tau$ and the overall opt-out rate $\bar{r}$; though, one can conjecture that they are positively correlated. Then, we want to study the risk conditional to the incentive and the overall opt-out rate $P\left(\mathscr{K} \middle| \tau , \bar{r}\right)$. To do so, we simulate 500 replicates for each one of the 12 sets described in Table [1](#tab:experimentsetting){reference-type="ref" reference="tab:experimentsetting"}. The result for each set is shown in Figure [5](#fig:pic_failing_prob_set_all){reference-type="ref" reference="fig:pic_failing_prob_set_all"}, where the red area represents the combinations for $\tau$ and $\bar{r}$ that yielded in losses for the school district, i.e., the open policy failed. The color green indicates where the combinations for $\tau$ and $\bar{r}$ yielded gains or overall savings for the school district. {#fig:pic_failing_prob_set_all width="0.9\\columnwidth"} We have identified notable variations in risk levels among the different sets. The instance with the highest risk among these sets is set number 3. A larger number of students, a lower average ridership, and a uniform spatial distribution characterize this particular set. The combination of these factors poses a higher risk regarding student transportation dynamics. With more students and lower ridership, the transportation system may need help in efficiently accommodating the transportation needs of the students. Additionally, it appears that the uniform spatial distribution adds complexity as students are spread out across the area, potentially leading to longer travel distances and increased transportation time. On the other hand, the set with the lowest risk is set number 11. This set exhibits a similar student population size and average ridership as other sets, but the key difference lies in the spatial distribution of students. In set number 11, students are more concentrated around the school. This concentrated spatial distribution potentially reduces the overall travel distances and may promote shorter travel times for students. As a result, the transportation system in this scenario is more efficient and less prone to potential disruptions or delays. To aggregate the risk into one metric per data set, let us define $$P_{\bar \tau}\left(\mathscr{K}\right) = \sum_{\tau}^{\bar\tau} \sum_{\bar{r}} P\left(\mathscr{K} \middle| \tau , \bar{r}\right) p(\bar{r}) p(\tau) \label{agg-risk}$$ to represent the overall mean of the risk for each data set. For computing this metric, we assume the possible values for $\tau$ are uniformly distributed between 0 and $\bar \tau$ US dollars, and $\bar{r}$ ranges uniformly between 0 and 1. {#fig:interaction width="0.7\\columnwidth"} Based on the analysis of the instances described in Table [1](#tab:experimentsetting){reference-type="ref" reference="tab:experimentsetting"} and the corresponding mean values of aggregated risk displayed in Figure [6](#fig:interaction){reference-type="ref" reference="fig:interaction"}, it can be tentatively concluded that there is a notable reduction in risk when students are concentrated around the school, particularly in the context of implementing the open offer policy. This finding holds particular promise for schools with larger student populations and higher ridership. When students are more concentrated around the school, the number of students requiring long trips decreases, resulting in routes with fewer students. This concentration is especially advantageous in schools with a larger student population, as a significant number of students living far from the school are more likely to opt-out of transportation services. Consequently, the need for underutilized routes is reduced. ## Case study: the Williamsville Central School District {#casestudy} The Williamsville Central School District (WCSD) is an esteemed public school district situated in Williamsville, New York, in the United States. The WCSD serves the suburban communities of Williamsville, Amherst, and Clarence in Erie County, comprising a total of 13 schools in the district. It includes six Elementary Schools (ELs), four Middle Schools (MIs), and three High Schools (HIs). The ELs are strategically located throughout the district, providing education from kindergarten through fifth grade. The MIs cater to students in grades six through eight, while the HIs are dedicated to students in grades nine through twelve. Encompassing an area of approximately 40 square miles, WCSD accommodates over 10,000 students, nurturing their academic growth and development. The district strongly emphasizes student transportation, recognizing the importance of safe and efficient travel. To facilitate this, WCSD operates a fleet of nearly 100 buses, maintaining a ratio of approximately 2:3 between their own buses and those provided by a contractor. This robust transportation system ensures that students can access the various schools within the district effectively. The dispersion of students and their corresponding schools was visually represented in Figure [19](#fig:dispersion){reference-type="ref" reference="fig:dispersion"} following a normalization process. This process involved calculating the distances between the students and a common reference point, enabling a fair comparison across all schools. The positions of the students were then adjusted based on this normalization. To quantify the dispersion level, the standard deviation of the student's positions relative to the center of mass was computed. This measure provided a comprehensive assessment of the student's spread out within each school, indicating the degree of dispersion. {#fig:dispersion} {#fig:dispersion} {#fig:dispersion} {#fig:dispersion}\ {#fig:dispersion} {#fig:dispersion} {#fig:dispersion} {#fig:dispersion} {#fig:dispersion} {#fig:dispersion} {#fig:dispersion} {#fig:dispersion} {#fig:dispersion} Upon analyzing the data, it was observed that the level of dispersion varied among the schools, as shown in Table [2](#tab:case-study){reference-type="ref" reference="tab:case-study"}. School 9 exhibited the lowest level of dispersion, with a value of 0.33. This indicates that the students in school 9 were relatively closer to each other and formed a more compact cluster. On the other hand, school 8 displayed the highest level of dispersion among all the schools, with a value of 0.54. This suggests that the students in school 8 were more widely distributed, with larger distances between individual students and the center of mass. Table [2](#tab:case-study){reference-type="ref" reference="tab:case-study"} also shows the values for the ridership. These values were obtained through a comprehensive data collection effort to estimate the likelihood of students not appearing at their assigned bus stops. The data collection process involved meticulous recording of student headcounts on each bus during morning and afternoon periods. The analysis revealed that the probability of a student not showing up at their designated bus stop was inconsistent across all schools. Instead, it exhibited significant variations depending on the specific educational institution, ranging from 22% to 72%. This finding underscores the importance of considering individual school dynamics and factors that may impact student attendance rates. {#fig:results} Figure [20](#fig:results){reference-type="ref" reference="fig:results"} depicts the relationship between the probability of failure and the incentive per student. As expected, the likelihood of failure increases with higher incentives. However, there are significant variations in behavior among different schools. Middle schools exhibit a slower growth rate compared to High schools and Elementary schools, suggesting their ability to offer larger incentives while maintaining a low probability of failure. On the other hand, Elementary schools such as schools 5 and 7 experience probabilities of failure exceeding 0.2 even with minimal incentives. A common characteristic across all curves is the rapid increase in the probability of failure once the incentive surpasses a certain threshold, resembling the probability function of a logistic regression. Based on its unique characteristics, this implies that each school has a point at which implementing the Open Offer Policy becomes unfeasible due to the associated risk. Finally, Table [2](#tab:case-study){reference-type="ref" reference="tab:case-study"} presents the aggregate risk defined in equation [\[agg-risk\]](#agg-risk){reference-type="eqref" reference="agg-risk"} for various maximum incentive levels, denoted as $\bar \tau$. Cases where the aggregate risk is below 10% are highlighted. As previously mentioned, Middle schools (MI) perform well, achieving a low risk even with relatively high incentives. Among Middle schools, the one with the lowest level of dispersion demonstrates the best performance, consistent with the findings from Section [5.1](#sec:synth){reference-type="ref" reference="sec:synth"}. Aggregate risk $P_{\bar \tau}(\mathcal{K})$ -------- ------- ------------ ----------- ---------- --------------------------------------------- ------------------ ------------------- ------------------- ------------------- School group dispersion ridership students $\bar\tau = 300$ $\bar\tau = 500$ $\bar\tau = 1000$ $\bar\tau = 2000$ $\bar\tau = 3000$ 1 MI 0.43 53% 1219 **0.046** **0.047** **0.062** 0.439 0.620 2 MI 0.48 52% 647 **0.039** **0.040** **0.041** 0.278 0.505 3 MI 0.45 62% 676 **0.065** **0.067** **0.069** 0.295 0.526 4 MI 0.39 69% 984 **0.020** **0.021** **0.021** **0.089** 0.366 5 EL 0.50 69% 956 0.215 0.256 0.442 0.696 0.794 6 EL 0.48 72% 467 0.217 0.233 0.385 0.660 0.771 7 EL 0.37 72% 517 0.226 0.262 0.481 0.730 0.818 8 EL 0.54 67% 655 0.059 0.068 0.254 0.578 0.710 9 EL 0.33 62% 660 0.158 0.207 0.458 0.701 0.793 10 EL 0.47 64% 842 **0.088** **0.099** 0.263 0.594 0.722 11 HI 0.43 28% 863 **0.067** **0.077** 0.346 0.649 0.760 12 HI 0.48 36% 544 **0.065** 0.103 0.421 0.694 0.792 13 HI 0.40 22% 712 0.117 0.157 0.499 0.744 0.828 : Data and aggregated risk for the schools in the WCSD in 2014-2015 # Discussion and Future Research {#discussion} In this paper, we introduce a new variant of the School Bus Routing Problem called School Bus Routing Problem with Open Offer Policy (SBRP-OOP), which focuses on finding a pricing strategy that balances incentive payments for students who choose not to use the bus with the savings achieved by operating fewer buses. By excluding students who accept the incentive from the routing process, SBRP-OOP aims to address multiple objectives simultaneously for a single school. This involves selecting the optimal set of stops from a pool of potential stops, assigning students to stops while considering factors like maximum walking distance and student capacity, and generating routes that minimize bus travel distance. To evaluate the effectiveness of SBRP-OOP, we conducted simulations using both synthetic and real data. Our initial analysis suggests that when students concentrates around the school the risk associated with implementing the Open Offer Policy in student transportation is reduced, especially for schools with larger student populations and higher ridership. When considering real instances, we found that the Open Offer Policy can be implemented with low risk and promising benefits in certain schools, particularly Middle schools. However, for some Elementary schools, implementing the policy seems impractical. Additionally, among the schools that benefit from the Open Offer Policy, those with a more compact distribution of students around the school experience greater gains. While these observations indicate potential benefits and risk reduction associated with concentrating students around the school, it is important to approach these conclusions cautiously. Other factors such as local infrastructure, geographical constraints, and specific school transportation policies may influence the outcomes. Therefore, further research and analysis are necessary to validate and explore these findings in diverse contexts. # Acknowledgment {#acknowledgment .unnumbered} This research was supported in part by the Science and Technology National Council (CONICYT) of Chile, grant FONDECYT 11181056. 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A metaheuristic for the school bus routing problem with bus stop selection. *European Journal of Operational Research*, 229 (2): 518--528, 2013. [^1]: hcaceres\@ucn.cl (corresponding author) [^2]: mdp006\@alumnos.ucn.cl [^3]: hernan.lespay\@ucn.cl [^4]: juan.contreras01\@ucn.cl [^5]: batta\@buffalo.edu

arxiv_math

--- author: - | Cunxiang Duan$^{a}$[^1], Ligong Wang$^{b}$, Yulong Wei$^{a}$ \ $^{a}$School of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, P.R. China\ $^{b}$School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710129, P.R. China\ \ E-mail: cxduanmath\@163.com; lgwangmath\@163.com;weiyulong\@tyut.edu.cn\ title: "The characteristic polynomials of uniform hypercycles with length four [^2]" --- **Abstract** Let $C_{m}$ be a cycle with length $m.$ The $k$-uniform hypercycle with length $m$ obtained by adding $k-2$ new vertices in every edge of $C_{m},$ denoted by $C_{m,k}.$ In this paper, we obtain some trace formulas of uniform hypercycles with length four. Moreover, we give the characteristic polynomials of uniform hypercycles with length four. :  Characteristic polynomial, trace formula, eigenvalue, hypercycle :  05C65, 12D05, 15A18. # Introduction {#sec:ch6-introduction} Let $\mathcal{T}=(t_{i_{1}i_{2}\ldots i_{k}})$ be a $k$-order and $n$-dimensional tensor over copmplex $\mathbb{C},$ that is, a multidimensional array, $1\leq i_{1},i_{2},\ldots,i_{k}\leq n.$ For a vertex set $V,$ if any edge $e$ in the edge set $E$ is a subset of $V,$ then $H=(V,E)$ is called a hypergraph. Further, $H$ is $k$-uniform if every edge $e\in E$ such that $|e|=k.$ Note that the $2$-uniform hypergraph is a graph. Recently, the study of spectral hypergraph via eigenvalues of tensors has attracted the attention and research of many researchers. At the same time, it has also achieved rich results about the spectral theory of hypergraphs [@Bret; @DW; @GCH; @KLQY]. The characteristic polynomials of hypergraphs are an important research topic in spectra of hypergraphs. At present, one of the most commonly used method to study the characteristic polynomials of hypergraphs is the Poisson Formula. In 2015, Cooper and Dutle [@CoD], by using the Poisson Formula, gave the spectra of \"all ones\" tensors and the characteristic polynomials of adjacency tensors of 3-uniform hyperstars. In 2018, Bao et al. [@BFWZ], by using the Poisson Formula, obtained the characteristic polynomials of adjacency tensors of $k$-uniform hyperstars. In 2019, Chen and Bu [@CB], by using the Poisson Formula, obtained a reduced formula of the characteristic polynomials of adjacency tensors of $k$-uniform hypergraphs with pendant edges. Further, they presented the characteristic polynomials of adjacency tensors of $k$-uniform hyperpaths. In 2021, Zheng [@Z] gave the characteristic polynomials of adjacency tensors of complete 3-uniform hypergraphs. Moreover, some scholars have also studied the characteristic polynomial coefficients of uniform hypergraphs. In 2012, Cooper and Dutle [@CoDu] researched the characteristic polynomial first $k+1$ coefficients of the adjacency tensor of the $k$-uniform hypergraph and the characteristic polynomial of the adjacency tensor of a single hyperedge by using the characteristic equation, respectively. In 2014, Zhou et al. [@ZSWB] gave the first $k$ coefficient expression of the characteristic polynomial of the signless (Laplacian) tensor of a $k$-uniform hypergraph. Note that it is not easy to get an explicit characteristic polynomial expression of uniform hypergraphs even if some special uniform hypergraphs. In fact, the spectra of hypergraphs can be studied by tensor traces. In 2015, Shao, Qi and Hu [@SQH] gave some new trace formulas and obtained the characterization of $k$-symmetric spectra of adjacency tensors of $k$-uniform hypergraphs. In 2021, Clark and Cooper [@CC] presented the Harary-Sachs theorem of $k$-uniform hypergraphs, which generalized the result of graphs. More results, see [@CBZ; @HHLQ]. Motivated by above papers, we mainly consider the characteristic polynomials of adjacency tensors of uniform hypercycles with length four. Let $H=(V,E)$ be a $k$-uniform hypergraph, where $V=\{v_1, v_{2}, \ldots, v_{m(k-1)+1}\}$ and $E=\{e_1, e_2, \ldots,e_m\}.$ If $e_i=\{v_{(i-1)(k-1)+1},v_{(i-1)(k-1)+2},\ldots, v_{i(k-1)+1}\}\in E$ for $i=1,2,\ldots,m,$ and $v_{1}=v_{m(k-1)+1},$ then $H$ is a $k$-uniform hypercycle with length $m,$ denoted by $C_{m,k}.$ In Section 2, we introduce some basic definitions and properties of tensors and hypergraphs. We also give some useful lemmas which will be used in Section 3. In Section 3, we give some traces of adjacency tensors of $k$-uniform hypercycles with length four, and present the characteristic polynomials of adjacency tensors of $k$-uniform hypercycles with length four by using these traces. # Preliminaries {#sec:ch-sufficient} In this section, we mainly introduce some basic definitions and properties of tensors and hypergraphs. Further, we present some useful Lemmas. For a $k$-order and $n$-dimensional tensor $\mathcal{T}=(t_{i_{1}i_{2}\ldots i_{k}})$ and a vector $x =(x_{1}, x_{2}, \ldots, x_{n})^{\top}$ over $\mathbb{C}^{n}$, $\mathcal{T}x$ is a vector and $$(\mathcal{T}x)_{i}=\sum\limits^{n}_{i_{2},\ldots,i_{k}=1}t_{ii_{2}\ldots i_{k}}x_{i_{2}}\cdots x_{i_{k}},~1\leq i \leq n.$$ **Definition 1**. *[@Qi] Let $\mathcal{T}$ be a $k$-order and $n$-dimensional nonzero tensor, $x\in \mathbb{C}^{n}$ be a nonzero vector and $x^{[k-1]}=(x_{1}^{k-1}, x_{2}^{k-1}, \ldots, x_{n}^{k-1})^{\top}$. If there exists a number $\lambda\in \mathbb{C}$ and $$\mathcal{T} x=\lambda x^{[k-1]},$$ then $\lambda$ and $x$ respectively is an eigenvalue of $\mathcal{T}$ and an eigenvector of $\mathcal{T}$ corresponding to $\lambda.$* For a $k$-order and $n$-dimensional tensor $\mathcal{T}=(t_{i_{1}i_{2}\cdots i_{k}}),$ the characteristic polynomial $\phi_{\mathcal{T}}(\lambda)$ of $\mathcal{T}$ is the resultant $\mathop{\rm Res }\nolimits(\lambda x^{[k-1]}-\mathcal{T} x^{k-1}),$ and $\phi_{\mathcal{T}}(\lambda)$ is a monic polynomial in $\lambda$ of degree $n(k-1)^{n-1}.$ The $j$-th order trace $\mathop{\rm Tr }\nolimits_{j}(\mathcal{T})$ [@MS] of $\mathcal{T}$ is $$\mathop{\rm Tr }\nolimits_{j}(\mathcal{T})=(k-1)^{n-1}\sum_{j_{1}+j_{2}+\cdots+j_{n}=j}[\prod_{i=1}^{n}\frac{1}{(j_{i}(k-1))!}\big(\sum_{l \in [n]^{k-1}}t_{il}\frac{\partial}{\partial x_{il}}\big)^{j_{i}}] \mathop{\rm tr }\nolimits(X^{j(k-1)}),$$ where the auxiliary $n\times n$ matrix $X = (x_{ij})$, $\frac{\partial}{\partial x_{il}}$ equals $\frac{\partial}{\partial x_{il_{2}}}\frac{\partial}{\partial x_{il_{3}}}\cdots \frac{\partial}{\partial x_{il_{k}}}$ for $l = l_{2}\cdots l_{k},$ and $j_{1}, \ldots, j_{n}$ run over all nonnegative integers with $j_{1} +\cdots+j_{n} =j.$ Moreover, Morozov and Shakirov [@MS] gave a formula for calculating the characteristic polynomial of $\mathcal{T}$ by using \"Schur polynomials\" in the generalized traces of $\mathcal{T},$ that is, $$\phi_{\mathcal{T}}(\lambda)=\sum_{j=0}^{s}P_{j}(-\frac{\mathop{\rm Tr }\nolimits_{1}(\mathcal{A})}{1}, -\frac{\mathop{\rm Tr }\nolimits_{2}(\mathcal{A})}{2}, \ldots, -\frac{\mathop{\rm Tr }\nolimits_{j}(\mathcal{A})}{j})\lambda^{s-j},$$ where $s=n(k-1)^{n-1}$ and the Schur function $P_{j}(p_{1}, \ldots, p_{j})=\sum\limits_{i=1}^{j}\sum\limits_{h_{1}+h_{2}+\cdots+h_{i}=j}\frac{p_{h_{1}} \cdots p_{h_{i}}}{i!}~(P_{0}=1).$ Note that the $j$-th order trace of $\mathcal{T}$ is the sum of $j$ power of all eigenvalues of $\mathcal{T}$ [@HHLQ]. **Definition 2**. *[@CoDu] Let $H = (V, E)$ be a $k$-uniform hypergraph with $n$ vertices. The adjacency tensor of $H$ is an $n$-dimensional tensor $\mathcal{A}_{H}=(a_{i_{1}i_{2}\ldots i_{k}})$ of order $k$ and $$a_{i_{1}i_{2}\ldots i_{k}}=\left\{ \begin{array}{ll} \frac{1}{(k-1)!},& \mbox {if} ~\{i_{1},i_{2},\ldots, i_{k}\} \in E, \\ 0,& \mbox {otherwise}. \end{array} \right.$$* The degree $d_{v}$ of a $k$-uniform hypergraph $H$ equals $|e_{v}|$, where $e_{v}$ is a set of edges that incident with the vertex $v.$ If $d_{v}$ is the multiple of $k,$ for all vertex $v\in V,$ then $H$ is called $k$-valent. Let $[n]=\{1,2,\ldots,n\}.$ For a positive integer $j,$ we define $$\mathcal{F}_{j}=\{(i_{1}\alpha_{1},\ldots,i_{j}\alpha_{j}) \mid 1 \leq i_{1} \leq i_{2} \leq\cdots \leq i_{j}\leq n,\alpha_{1}, \ldots, \alpha_{j}\in [n]^{k-1}\}.$$ Let denote $\pi_{F}(\mathcal{T})=t_{i_{1}\alpha_{1}}\cdots t_{i_{j}\alpha_{j}}$ for $F=(i_{1}\alpha_{1},\ldots,i_{j}\alpha_{j})\in \mathcal{F}_{j}$ and a tensor $\mathcal{T}=(t_{i_{1}i_{2}\cdots i_{k}}).$ For a digraph $D=(V,A)$, the in-degree (out-degree) of $v_{i}$ in $D$ is the number of arcs incident to (from) $v_{i},$ denoted by $d^{-}_{v_{i}}~(d^{+}_{v_{i}}).$ **Definition 3**. *[@SQH] For $F=(i_{1}\alpha_{1},\ldots,i_{j}\alpha_{j})\in \mathcal{F}_{j},$ let $E(F)=\bigcup\limits_{h=1}^{j}E_{h}(F),$ where $E_{h}(F)=\{(i_{h},v_{1}), \ldots, (i_{h},v_{k-1}) \}$ denotes the arc multi-set if $\alpha_{h}=v_{1}\ldots v_{k-1}.$ Let $D=(V,A)$ denote the (multi-)digraph corresponding to $E(F),$ where $V$ and $A$ is the vertex set and arc set, respectively. Then\ $(1)$ $b(F)=\prod\limits_{a\in A} m(a)!$ and $c(F)=\prod\limits_{v\in V}d^{+}_{v}!,$ where $m(a)$ is the multiplicity of the arc $a.$\ $(2)$ Denote $W(F)$ the set of all Eulerian closed walks with $E(F).$* Note that multiple arcs of $W(F)$ are not distinguished and $W(F)=\emptyset$ if $F$ is not $k$-valent. **Lemma 4**. *[@SQH] Let $\mathcal{T}=(t_{i_{1}i_{2}\cdots i_{k}})$ be an $n$-dimensional tensor of order $k$. Then $$\mathop{\rm Tr }\nolimits_{j}(\mathcal{T})=(k-1)^{n-1}\sum_{F\in \mathcal{F}'_{j}}\frac{b(F)}{c(F)}\pi_{F}(\mathcal{T})\lvert W(F)\rvert,$$ where $\mathcal{F}'_{j}=\{F\in \mathcal{F}_{j} \mid F$ is $k$-valent$\}.$* Let $G=(V,E)$ be a graph. If each edge of $G$ adds $k-2$ new vertices, then the $k$-unifrom hypergraph is $k$-power of $G,$ denoted by $G^{k}.$ If the edge sign function $\pi: E\rightarrow \{+1,-1\},$ then $(G,\pi)$ is a signed graph, denoted by $G_{\pi}.$ If a graph is an (induced) subgraph of $G_{\pi},$ then the garaph is a signed (induced) subgraphs of $G.$ **Lemma 5**. *[@CDB] $\lambda\in \mathbb{C}$ is an eigenvalue of $G^{k}$ if and only if\ $(1)$ for $k = 3,$ $\beta$ is an eigenvalue of some signed induced subgraph of $G$ and $\beta^{2}= \lambda^{k}$;\ $(2)$ for $k\geq 4,$ $\beta$ is an eigenvalue of some signed subgraph of $G$ and $\beta^{2}= \lambda^{k}.$* # The characteristic polynomial of $k$-uniform hypercycles with length four {#sec:ch-inco} In this section, we present the characteristic polynomials of $k$-uniform hypercycles with length four. Note that the number of spanning trees $\tau(D)$ and the number of Eulerian cycles $\varepsilon(D)$ of a digraph $D$ can be calculate by using the Matrix-Tree Theorem [@CvRS] and BEST Theorem [@AB], respectively. **Theorem 6**. *Let $C_{4,k}$ be a $k(\geq 3)$-uniform hypercycle with length four, and $\mathcal{A}_{C_{4,k}}$ be its adjacency tensor. Then $$\mathop{\rm Tr }\nolimits_{k}(\mathcal{A}_{C_{4,k}})=4k^{k-1}(k-1)^{3k-4}.$$* **Proof.** By Lemma [Lemma 4](#le:4-4){reference-type="ref" reference="le:4-4"}, we consider $F=(i_{1}\alpha_{1}, \ldots, i_{k}\alpha_{k})\in \mathcal{F}'_{k}.$ If $\pi_{F}(\mathcal{A}_{C_{4,k}})\neq 0,$ then we know that all elements of $F$ correspond to edges of $C_{4,k}.$ Since $F\in \mathcal{F}'_{k}$ is $k$-valent, each vertex of the edge in $F$ occurs the times that is the multiple of $k.$ Thus, all elements $i_{h}\alpha_{h}$ of $F$ only correspond to some edge of $C_{4,k},$ $1 \leq h \leq k.$ If $|W(F)|\neq 0,$ then the out-degree is equal to in-degree of every vertex in the directed graph corresponding to $F$, that is, every vertex as the first entry occurs one time in $F$. For an edge of $C_{4,k},$ the total number of such $F$ is $[(k-1)!]^{k}.$ For each such $F,$ $E(F)$ induces a complete digraph $D_{1}$ on $k$ vertices. Hence, $$b(F)=1, c(F)= [(k-1)!]^{k} , ~\pi_{F}(\mathcal{A}_{C_{4,k}})=\big[\frac{1}{(k-1)!}\big]^{k}.$$ By the Matrix-Tree Theorem [@CvRS], we have $\tau(D_{1})=k^{k-2}$. By BEST Theorem [@AB], we have $\varepsilon(D_{1})=[(k-2)!]^{k}k^{k-2}$. Because each Eulerian cycle has $k(k-1)$ arcs, we have $$\lvert W(F)\rvert=k(k-1)[(k-2)!]^{k}k^{k-2}.$$ Since $C_{4,k}$ has four edges, we have $$\begin{aligned} \mathop{\rm Tr }\nolimits_{k}(\mathcal{A}_{C_{4,k}})&=4(k-1)^{n-1}\cfrac{[(k-1)!]^{k}}{[(k-1)!]^{k}}\big[\cfrac{1}{(k-1)!}\big]^{k}k(k-1)[(k-2)!]^{k}k^{k-2} \\&=4k^{k-1}(k-1)^{n-k}=4k^{k-1}(k-1)^{3k-4}.\end{aligned}$$ **Theorem 7**. *Let $C_{4,k}$ be a $k(\geq 3)$-uniform hypercycle with length four, and $\mathcal{A}_{C_{4,k}}$ be its adjacency tensor. Then $$\mathop{\rm Tr }\nolimits_{2k}(\mathcal{A}_{C_{4,k}})=4k^{k-1}(k-1)^{3k-4}+8k^{2k-3}(k-1)^{2k-3}.$$* **Proof.** By Lemma [Lemma 4](#le:4-4){reference-type="ref" reference="le:4-4"}, we consider $F=(i_{1}\alpha_{1}, \ldots, i_{2k}\alpha_{2k})\in \mathcal{F}'_{2k}.$ If $\pi_{F}(\mathcal{A}_{C_{4,k}})\neq 0,$ then we know that all elements of $F$ correspond to edges of $C_{4,k}.$ since $F\in \mathcal{F}'_{2k}$ is $k$-valent, each vertex of the edge in $F$ occurs the times that is the multiple of $k.$ Thus, $F$ has the following two cases. **Case 1.** All elements $i_{h}\alpha_{h}$ of $F$ correspond to some edge of $C_{4,k},$ $1 \leq h \leq 2k.$ Similar to Theorem [Theorem 6](#th:3-1){reference-type="ref" reference="th:3-1"}, we know that $E(F)$ induces a complete multi-digraph $D'_{1}$ on $k$ vertices and the multiplicity of each arc of $E(F)$ is 2. Hence, the total number of such $F$ is $[(k-1)!]^{2k},$ and $$b(F)=(2!)^{k(k-1)}, c(F)= [(2(k-1))!]^{k} , ~\pi_{F}(\mathcal{A}_{C_{4,k}})=\big[\frac{1}{(k-1)!}\big]^{2k}.$$ By the Matrix-Tree Theorem [@CvRS], we have $\tau(D'_{1})=2^{k-1}k^{k-2}$. By BEST Theorem [@AB], we have $\varepsilon(D'_{1})=[(2(k-1)-1)!]^{k}2^{k-1}k^{k-2}$. Because each Eulerian cycle has $2k(k-1)$ arcs and multi-arcs of $W(F)$ are not labelled, we have $$\lvert W(F)\rvert=\cfrac{2k(k-1)[(2(k-1)-1)!]^{k}2^{k-1}k^{k-2}}{(2!)^{k(k-1)}}.$$ Consider all edges, for all such $F,$ we know that the total contribution to $\mathop{\rm Tr }\nolimits_{2k}(\mathcal{A}_{C_{4,k}})$ is $$\begin{aligned} &4(k-1)^{n-1}\cfrac{[(k-1)!]^{2k}(2!)^{k(k-1)}}{[(2(k-1))!]^{k}}\big[\cfrac{1}{(k-1)!}\big]^{2k}\cfrac{2k(k-1)[(2(k-1)-1)!]^{k}2^{k-1}k^{k-2}}{(2!)^{k(k-1)}} \\&=4k^{k-1}(k-1)^{n-k}=4k^{k-1}(k-1)^{3k-4}.\end{aligned}$$ **Case 2.** All elements $i_{h}\alpha_{h}$ of $F$ correspond to two edges of $C_{4,k},$ $1 \leq h \leq 2k.$ Since $F\in \mathcal{F}'_{2k}$ is $k$-valent, we know that $k$ elements of $F$ correspond to the same edge and $k$ elements of $F$ correspond to the same other edge. If $|W(F)|\neq 0,$ then every vertex of each edge as the first entry occurs the same number of times in $F$. We only consider two incident edges since the multi-digraph corresponding to $E(F)$ is connected. For two incident edges of $C_{4,k},$ the number of orderings for the first entry is 2, and the number of orderings for $\alpha_{h}$ is $[(k-1)!]^{2k}$. The total number of such $F$ is $2[(k-1)!]^{2k}.$ Thus, $E(F)$ induces a digraph $D_{2}$ on $2k-1$ vertices. Since $D_{2}$ must be connected, we know $F=(i_{1}\alpha_{1}, \ldots,i_{k}\alpha_{k},i_{k}\alpha_{k+1},\ldots,i_{2k-1}\alpha_{2k-1}).$ Hence, $$b(F)=1,~c(F)= [(k-1)!]^{2k-2}(2(k-1))!,~ \pi_{F}(\mathcal{A}_{C_{4,k}})=\big[\cfrac{1}{(k-1)!}\big]^{2k},$$ By the Matrix-Tree Theorem [@CvRS], we have $\tau(D_{2})=k^{2k-4}$. By BEST Theorem [@AB], we have $\varepsilon(D_{2})=[(k-2)!]^{2k-2}(2(k-1)-1)!k^{2k-4}$. Because each Eulerian cycle has $2k(k-1)$ arcs, we have $$\lvert W(F)\rvert=2k(k-1)[(k-2)!]^{2k-2}(2(k-1)-1)!k^{2k-4}.$$ Consider all edges, for all such $F,$ we know that the total contribution to $\mathop{\rm Tr }\nolimits_{2k}(\mathcal{A}_{C_{4,k}})$ is $$\begin{aligned} &4(k-1)^{n-1}\cfrac{2[(k-1)!]^{2k}}{[(k-1)!]^{2k-2}(2(k-1))!}\big[\cfrac{1}{(k-1)!}\big]^{2k} 2k(k-1)[(k-2)!]^{2k-2}(2(k-1)-1)!k^{2k-4} \\&=8k^{2k-3}(k-1)^{n-2k+1}=8k^{2k-3}(k-1)^{2k-3}.\end{aligned}$$ Thus, we have $$\begin{aligned} \mathop{\rm Tr }\nolimits_{2k}(\mathcal{A}_{C_{4,k}})=4k^{k-1}(k-1)^{3k-4}+8k^{2k-3}(k-1)^{2k-3}.\end{aligned}$$ **Theorem 8**. *Let $C_{4,k}$ be a $k(\geq 3)$-uniform hypercycle with length four, and $\mathcal{A}_{C_{4,k}}$ be its adjacency tensor. Then $$\mathop{\rm Tr }\nolimits_{3k}(\mathcal{A}_{C_{4,k}})=4k^{k-1}(k-1)^{3k-4}+24k^{2k-3}(k-1)^{2k-3}+12k^{3k-5}(k-1)^{k-2}.$$* **Proof.** By Lemma [Lemma 4](#le:4-4){reference-type="ref" reference="le:4-4"}, we consider $F=(i_{1}\alpha_{1}, \ldots, i_{3k}\alpha_{3k})\in \mathcal{F}'_{3k}.$ If $\pi_{F}(\mathcal{A}_{C_{4,k}})\neq 0,$ then we know that all elements of $F$ correspond to edges of $C_{4,k}.$ Since $F\in \mathcal{F}'_{3k}$ is $k$-valent, each vertex of the edge in $F$ occurs the times that is the multiple of $k.$ Thus, $F$ has the following three cases. **Case 1.** All elements $i_{h}\alpha_{h}$ of $F$ correspond to some edge of $C_{4,k},$ $1 \leq h \leq 3k.$ Similar to Case 1 of Theorem [Theorem 7](#th:3-2){reference-type="ref" reference="th:3-2"}, we know that $E(F)$ corresponds a complete multi-digraph with $k$ vertices and the multiplicity of each arc of $E(F)$ is 3. Hence, we know that the total number of such $F$ is $[(k-1)!]^{3k},$ and $$b(F)=(3!)^{k(k-1)}, c(F)= [(3(k-1))!]^{k} , ~\pi_{F}(\mathcal{A}_{C_{4,k}})=\big[\frac{1}{(k-1)!}\big]^{3k},$$ $$\lvert W(F)\rvert=\cfrac{3k(k-1)[(3(k-1)-1)!]^{k}3^{k-1}k^{k-2}}{(3!)^{k(k-1)}}.$$ Since $C_{4,k}$ has four edges, for all such $F,$ we know that the total contribution to $\mathop{\rm Tr }\nolimits_{3k}(\mathcal{A}_{C_{4,k}})$ is $$\begin{aligned} &4(k-1)^{n-1}\cfrac{[(k-1)!]^{3k}(3!)^{k(k-1)}}{[(3(k-1))!]^{k}}\big[\cfrac{1}{(k-1)!}\big]^{3k}\cfrac{3k(k-1)[(3(k-1)-1)!]^{k}3^{k-1}k^{k-2}}{(3!)^{k(k-1)}} \\&=4k^{k-1}(k-1)^{n-k}=4k^{k-1}(k-1)^{3k-4}.\end{aligned}$$ **Case 2.** All elements $i_{h}\alpha_{h}$ of $F$ correspond to two edges of $C_{4,k},$ $1 \leq h \leq 3k.$ Assume that $ak$ elements of $F$ correspond to the same edge and $bk$ elements of $F$ correspond to the same other edge, $a+b=3.$ If $|W(F)|\neq 0,$ then the in-degree is equal to out-degree of every vertex of the multi-digraph corresponding to $F.$ We only consider two incident edges since the multi-digraph corresponding to $E(F)$ is connected. For two incident edges of $C_{4,k},$ the number of orderings for the first entry is $\binom{t}{a}$, and the number of orderings for $\alpha_{h}$ is $[(k-1)!]^{3k}$. The total number of such $F$ is $\sum\limits_{a=1}^{2}\binom{3}{a}[(k-1)!]^{3k}.$ We know that $E(F)$ induces a multi-digraph $D'_{2}$ on $2k-1$ vertices. Since $D'_{2}$ must be connected, $F$ is an appropriate ordering of $(i_{1}\alpha^{1}_{1}, \ldots,i_{1}\alpha^{a}_{1},\ldots,i_{k}\alpha^{1}_{k},\ldots,i_{k}\alpha^{a}_{k},i_{k}\alpha^{1'}_{k}\ldots,i_{k}\alpha^{b'}_{k},\cdots,\\ i_{2k-1}\alpha^{1'}_{2k},\cdots,i_{2k-1}\alpha^{b'}_{2k+1},\ldots, i_{3k-2}\alpha^{b'}_{3k}),$ where $\alpha^{j}_{h}$ (resp. $\alpha^{l'}_{h}$) has the same elements regardless of ordering, for $j=1,2,\cdots,a,$ $l=1,2, \cdots, b.$ Hence, $$b(F)=(a!)^{k(k-1)}(b!)^{k(k-1)},~c(F)= [(a(k-1))!]^{k-1}[(b(k-1))!]^{k-1}(3(k-1))!,$$ and $\pi_{F}(\mathcal{A}_{C_{4,k}})=\big[\cfrac{1}{(k-1)!}\big]^{3k}.$ By the Matrix-Tree Theorem [@CvRS], we have $\tau(D'_{2})=a^{k-1}b^{k-1}k^{2k-4}$. By BEST Theorem [@AB], we have $\varepsilon(D'_{2})=[(a(k-1)-1)!]^{k}[(b(k-1)-1)!]^{k}((a+b)(k-1)-1)!a^{k-1}b^{k-1}k^{2k-4}.$ Because each Eulerian cycle has $3k(k-1)$ arcs and multi arcs of $W(F)$ are not labelled, we have $$\lvert W(F)\rvert=\cfrac{3k(k-1)[(a(k-1)-1)!]^{k-1}[(b(k-1)-1)!]^{k-1}(3(k-1)-1)!a^{k-1}b^{k-1}k^{2k-4}}{(a!)^{k(k-1)}(b!)^{k(k-1)}}.$$ Consider all edges, for all such $F,$ we know that the total contribution to $\mathop{\rm Tr }\nolimits_{3k}(\mathcal{A}_{C_{4,k}})$ is $$\begin{aligned} &4(k-1)^{n-1}\cfrac{\sum_{a=1}^{2}\binom{3}{a}[(k-1)!]^{3k}(a!)^{k(k-1)}(b!)^{k(k-1)}}{[(a(k-1))!]^{k-1}[(b(k-1))!]^{k-1}(3(k-1))!}\big[\cfrac{1}{(k-1)!}\big]^{3k} \\&~~~~\cfrac{3k(k-1)[(a(k-1)-1)!]^{k-1}[(b(k-1)-1)!]^{k-1}(3(k-1)-1)!a^{k-1}b^{k-1}k^{2k-4}}{(a!)^{k(k-1)}(b!)^{k(k-1)}} \\&=24k^{2k-3}(k-1)^{n-2k+1}=24k^{2k-3}(k-1)^{2k-3}.\end{aligned}$$ **Case 3.** All elements $i_{h}\alpha_{h}$ of $F$ correspond to three edges of $C_{4,k},$ $1 \leq h \leq 3k.$ If $|W(F)|\neq 0$, then the in-degree is equal to out-degree of every vertex of the multi-digraph corresponding to $F.$ Thus, $F$ is an appropriate ordering of $(i_{1}\alpha_{1}, i_{2}\alpha_{2},\ldots,i_{k}\alpha_{k},i_{k}\alpha_{k+1},\ldots,\\i_{2k-1}\alpha_{2k},i_{2k-1}\alpha_{2k+1},\ldots, i_{3k-2}\alpha_{3k}).$ We know that $E(F)$ induces a digraph $D_{3}$ on $3k-2$ vertices. For each such $F,$ the number of orderings for the first entry is 4, and the number of orderings for the $\alpha_{h}$ is $[(k-1)!]^{3k}$. Hence, the total number of such $F$ is $4[(k-1)!]^{3k},$ and $$b(F)=1, ~c(F)= [(k-1)!]^{3k-4}[(2(k-1))!]^{2},~\pi_{F}(\mathcal{A}_{C_{4,k}})=\big[\cfrac{1}{(k-1)!}\big]^{3k}.$$ By the Matrix-Tree Theorem [@CvRS], we have $\tau(D_{3})=k^{3k-6}.$ By BEST Theorem [@AB], we have $\varepsilon(D_{3})=[(k-2)!]^{3k-4}[(2(k-1)-1)!]^{2}k^{3k-6}.$ Because each Eulerian cycle has $3k(k-1)$ arcs, we have $$\begin{aligned} \lvert W(F)\rvert&=3k^{3k-5}(k-1)[(k-2)!]^{3k-4}[(2(k-1)-1)!]^{2}.\end{aligned}$$ Consider all edges, for all such $F,$ we know that the total contribution to $\mathop{\rm Tr }\nolimits_{3k}(\mathcal{A}_{C_{4,k}})$ is $$\begin{aligned} &4(k-1)^{n-1}\cfrac{4[(k-1)!]^{3k}}{[(k-1)!]^{3k-4}[(2(k-1))!]^{2}}\big[\cfrac{1}{(k-1)!}\big]^{3k}3k^{3k-5}(k-1)[(k-2)!]^{3k-4}[(2(k-1)-1)!]^{2} \\&=12k^{3k-5}(k-1)^{n-3k+2}=12k^{3k-5}(k-1)^{k-2}.\end{aligned}$$ Thus, we have $$\begin{aligned} \mathop{\rm Tr }\nolimits_{3k}(\mathcal{A}_{C_{4,k}})=4k^{k-1}(k-1)^{3k-4}+24k^{2k-3}(k-1)^{2k-3}+12k^{3k-5}(k-1)^{k-2}\end{aligned}$$ **Theorem 9**. *Let $C_{4,k}$ be a $k(\geq 3)$-uniform hypercycle with length four, and $\mathcal{A}_{C_{4,k}}$ be its adjacency tensor. Then $$\mathop{\rm Tr }\nolimits_{4k}(\mathcal{A}_{C_{4,k}})=4k^{k-1}(k-1)^{3k-4}+56k^{2k-3}(k-1)^{2k-3}+64k^{3k-5}(k-1)^{k-2}+40k^{4k-8}.$$* **Proof.** By Lemma [Lemma 4](#le:4-4){reference-type="ref" reference="le:4-4"}, we consider $F=(i_{1}\alpha_{1}, \ldots, i_{4k}\alpha_{4k})\in \mathcal{F}'_{4k}.$ If $\pi_{F}(\mathcal{A}_{C_{4,k}})\neq 0,$ then all elements of $F$ correspond to edges of $C_{4,k}.$ since $F\in \mathcal{F}'_{4k}$ is $k$-valent, each vertex of the edge in $F$ occurs the times that is the multiple of $k.$ Thus, $F$ has the following four cases. **Case 1.** All elements $i_{h}\alpha_{h}$ of $F$ correspond to some edge of $C_{4,k},$ $1 \leq h \leq 4k.$ Similar to Case 1 of Theorem [Theorem 7](#th:3-2){reference-type="ref" reference="th:3-2"}, we know that $E(F)$ corresponds a complete multi-digraph with $k$ vertices and the multiplicity of each arc of $E(F)$ is 4. Hence, the total number of such $F$ is $[(k-1)!]^{4k},$ and $$b(F)=(4!)^{k(k-1)}, c(F)= [(4(k-1))!]^{k} , ~\pi_{F}(\mathcal{A}_{C_{4,k}})=\big[\frac{1}{(k-1)!}\big]^{4k},$$ $$\lvert W(F)\rvert=\cfrac{4k(k-1)[(4(k-1)-1)!]^{k}4^{k-1}k^{k-2}}{(4!)^{k(k-1)}}.$$ Consider all edges, for all such $F,$ we know that the total contribution to $\mathop{\rm Tr }\nolimits_{4k}(\mathcal{A}_{C_{4,k}})$ is $$\begin{aligned} &4(k-1)^{n-1}\cfrac{[(k-1)!]^{4k}(4!)^{k(k-1)}}{[(4(k-1))!]^{k}}\big[\cfrac{1}{(k-1)!}\big]^{4k}\cfrac{4k(k-1)[(4(k-1)-1)!]^{k}4^{k-1}k^{k-2}}{(4!)^{k(k-1)}} \\&=4k^{k-1}(k-1)^{n-k}=4k^{k-1}(k-1)^{3k-4}.\end{aligned}$$ **Case 2.** All elements $i_{h}\alpha_{h}$ of $F$ correspond to two edges of $C_{4,k},$ $1 \leq h \leq 4k.$ Similar to Case 2 of Theorem [Theorem 8](#th:3-3){reference-type="ref" reference="th:3-3"}, assume that $ak$ elements of $F$ correspond to the same edge and $bk$ elements of $F$ correspond to the same other edge, $a+b=4.$ We only consider two incident edges since the multi-digraph corresponding to $E(F)$ is connected. For two incident edges of $C_{4,k},$ the number of orderings for the first entry is $\binom{t}{a}$, and the number of orderings for the $\alpha_{h}$ is $[(k-1)!]^{4k}$. Thus, we know the total number of such $F$ is $\sum_{a=1}^{3}\binom{4}{a}[(k-1)!]^{4k},$ and $$b(F)=(a!)^{k(k-1)}(b!)^{k(k-1)}, \pi_{F}(\mathcal{A}_{C_{4,k}})=\big[\cfrac{1}{(k-1)!}\big]^{3k},$$ $$c(F)= [(a(k-1))!]^{k-1}[(b(k-1))!]^{k-1}(3(k-1))!,$$ $$\lvert W(F)\rvert=\cfrac{4k(k-1)[(a(k-1)-1)!]^{k-1}[(b(k-1)-1)!]^{k-1}(4(k-1)-1)!a^{k-1}b^{k-1}k^{2k-4}}{(a!)^{k(k-1)}(b!)^{k(k-1)}}.$$ Consider all edges, for all such $F,$ we know that the total contribution to $\mathop{\rm Tr }\nolimits_{4k}(\mathcal{A}_{C_{4,k}})$ is $$\begin{aligned} &4(k-1)^{n-1}\cfrac{\sum_{a=1}^{3}\binom{4}{a}[(k-1)!]^{4k}(a!)^{k(k-1)}(b!)^{k(k-1)}}{[(a(k-1))!]^{k-1}[(b(k-1))!]^{k-1}(4(k-1))!}\big[\cfrac{1}{(k-1)!}\big]^{4k} \\&~~~~\cfrac{4k(k-1)[(a(k-1)-1)!]^{k-1}[(b(k-1)-1)!]^{k-1}(4(k-1)-1)!a^{k-1}b^{k-1}k^{2k-4}}{(a!)^{k(k-1)}(b!)^{k(k-1)}} \\&=56k^{2k-3}(k-1)^{n-2k+1}=56k^{2k-3}(k-1)^{2k-3}.\end{aligned}$$ **Case 3.** All elements $i_{h}\alpha_{h}$ of $F$ correspond to three edges of $C_{4,k},$ $1 \leq h \leq 4k.$ Since $F\in \mathcal{F}'_{4k}$ is $k$-valent, there are $2k$ elements of $F$ corresponding to the same edge. If $|W(F)|\neq 0$, then the in-degree is equal to out-degree of every vertex of the multi-digraph corresponding to $F.$ Thus, $F$ is an appropriate ordering of $(i_{1}\alpha_{1}, i_{1}\alpha'_{1},\ldots,i_{k}\alpha_{k},i_{k}\alpha'_{k},i_{k}\alpha_{k+1},\\\ldots,i_{2k-1}\alpha_{2k-1},i_{2k-1}\alpha_{2k},\ldots, i_{3k-2}\alpha_{3k})$ or $(i_{1}\alpha_{1},\ldots,i_{k}\alpha_{k},i_{k}\alpha_{k+1},i_{k}\alpha'_{k+1},\ldots,i_{2k-1}\alpha_{2k},i_{2k-1}\alpha'_{2k},\\i_{2k-1}\alpha_{2k+1},\ldots, i_{3k-2}\alpha_{3k}),$ where $\alpha_{h}$ and $\alpha'_{h}$ have the same elements regardless of ordering, for $h=k+1,\ldots,2k.$ **Subcase 3.1.** $F$ is an appropriate ordering of $(i_{1}\alpha_{1}, i_{1}\alpha'_{1},\ldots,i_{k}\alpha_{k},i_{k}\alpha'_{k},i_{k}\alpha_{k+1}\ldots,i_{2k-1}\alpha_{2k-1},\\i_{2k-1}\alpha_{2k},\ldots, i_{3k-2}\alpha_{3k}).$ For each such $F,$ the number of orderings for the first entry is 6, and the number of orderings for the $\alpha_{h}$ is $[(k-1)!]^{4k}$. Hence, the total number of such $F$ is $6[(k-1)!]^{4k},$ and $$b(F)=(2!)^{k}, ~c(F)= [(k-1)!]^{2k-3}[(2(k-1))!]^{k}(3(k-1))!,~\pi_{F}(\mathcal{A}_{C_{4,k}})=\big[\cfrac{1}{(k-1)!}\big]^{4k}.$$ Let $D_{3}$ be the multi-digraph induced by $E(F).$ By the Matrix-Tree Theorem [@CvRS], we have $\tau(D_{3})=2^{k-1}k^{3k-6}$. By BEST Theorem [@AB], we have $\varepsilon(D_{3})=2^{k-1}k^{3k-6}[(k-2)!]^{2k-3}[(2(k-1)-1)!]^{k}(3(k-1)-1)!.$ Because each Eulerian cycle has $4k(k-1)$ arcs and multi-arcs of $W(F)$ are not labelled, we have $$\begin{aligned} \lvert W(F)\rvert=2k^{3k-5}(k-1)[(k-2)!]^{2k-3}[(2(k-1)-1)!]^{k}(3(k-1)-1)!.\end{aligned}$$ Consider all edges, for all such $F,$ we know that the total contribution to $\mathop{\rm Tr }\nolimits_{4k}(\mathcal{A}_{C_{4,k}})$ is $$\begin{aligned} &4(k-1)^{n-1}\cfrac{2\times6[(k-1)!]^{4k}(2!)^{k}}{[(k-1)!]^{2k-3}[(2(k-1))!]^{k}(3(k-1))!}\big[\cfrac{1}{(k-1)!}\big]^{4k}2k^{3k-5}(k-1) \\&~~~~[(k-2)!]^{2k-3}[(2(k-1)-1)!]^{k}(3(k-1)-1)! \\&=32k^{3k-5}(k-1)^{k-2}.\end{aligned}$$ **Subcase 3.2.** $F$ is an appropriate ordering of $(i_{1}\alpha_{1},\ldots,i_{k}\alpha_{k},i_{k}\alpha_{k+1},i_{k}\alpha'_{k+1},\ldots,i_{2k-1}\alpha_{2k},\\i_{2k-1}\alpha'_{2k},i_{2k-1}\alpha_{2k+1},\ldots, i_{3k-2}\alpha_{3k}).$ For each such $F,$ the number of orderings for the first entry is 9, and the number of orderings for the $\alpha_{h}$ is $[(k-1)!]^{4k}$. Hence, the total number of such $F$ is $9[(k-1)!]^{4k},$ and $$b(F)=(2!)^{k},~c(F)= [(k-1)!]^{2k-2}[(2(k-1))!]^{k-2}[(3(k-1))!]^{2},~\pi_{F}(\mathcal{A}_{C_{4,k}})=\big[\cfrac{1}{(k-1)!}\big]^{4k}.$$ By calculating, we know $\lvert W(F)\rvert$ is the same as the above Subcase 3.1, i.e., $$\lvert W(F)\rvert=2k^{3k-5}(k-1)[(k-2)!]^{2k-2}[(2(k-1)-1)!]^{k-2}[(3(k-1)-1)!]^{2}.$$ Consider all edges, for all such $F,$ we know that the total contribution to $\mathop{\rm Tr }\nolimits_{4k}(\mathcal{A}_{C_{4,k}})$ is $$\begin{aligned} &4(k-1)^{n-1}\cfrac{9[(k-1)!]^{4k}(2!)^{k}}{[(k-1)!]^{2k-3}[(2(k-1))!]^{k}(3(k-1))!}\big[\cfrac{1}{(k-1)!}\big]^{4k}2k^{3k-5}(k-1) \\&~~~~[(k-2)!]^{2k-2}[(2(k-1)-1)!]^{k-2}[(3(k-1)-1)!]^{2} \\&=32k^{3k-5}(k-1)^{k-2}.\end{aligned}$$ **Case 4.** All elements $i_{h}\alpha_{h}$ of $F$ correspond to all edges of $C_{4,k},$ $1 \leq h \leq 4k.$ If $|W(F)|\neq 0$, then the in-degree is equal to out-degree of every vertex of the (multi-)digraph corresponding to $F.$ Thus, $F$ is an appropriate ordering of $(i_{1}\alpha_{1}, i_{1}\alpha_{2},\ldots,i_{k}\alpha_{k+1},i_{k}\alpha_{k+2},\\\ldots,i_{2k-1}\alpha_{2k+1},i_{2k-1}\alpha_{2k+2},\ldots, i_{4k-4}\alpha_{4k})$ or $(i_{1}\alpha_{1},i_{1}\alpha'_{1},\ldots,i_{k}\alpha_{k},i_{k}\alpha'_{k},\ldots,\i_{2k-1}\alpha_{2k-1},\i_{2k-1}\alpha'_{2k-1},\\\ldots,i_{3k-2}\alpha_{3k-2},i_{3k-2}\alpha'_{3k-2},\ldots, i_{4k-4}\alpha_{4k-4}),$ where $\alpha_{h}$ and $\alpha'_{h}$ have the same elements regardless of ordering, for $h=1,k,2k-1,3k-2.$ **Subcase 4.1.** $F$ is an appropriate ordering of $(i_{1}\alpha_{1}, i_{1}\alpha_{2},\ldots,i_{k}\alpha_{k+1},i_{k}\alpha_{k+2},\ldots,i_{2k-1}\alpha_{2k+1},\\i_{2k-1}\alpha_{2k+2},\ldots, i_{4k-4}\alpha_{4k}).$ For each such $F,$ the number of orderings for the first entry is 16, and the number of orderings for the $\alpha_{h}$ is $[(k-1)!]^{4k}$. Hence, the total number of such $F$ is $16[(k-1)!]^{4k},$ and $$b(F)=1, ~c(F)= [(k-1)!]^{4k-8}[(2(k-1))!]^{4},~\pi_{F}(\mathcal{A}_{C_{4,k}})=\big[\cfrac{1}{(k-1)!}\big]^{4k}.$$ Let $D_{4}$ be the multi-digraph induced by $E(F).$ By the Matrix-Tree Theorem [@CvRS], we have $\tau(D_{4})=8k^{4k-9}.$ By BEST Theorem [@AB], we have $\varepsilon(D_{4})=8k^{4k-9}[(k-2)!]^{4k-8}[(2(k-1)-1)!]^{4}.$ Because each Eulerian cycle has $4k(k-1)$ arcs, we have $$\begin{aligned} \lvert W(F)\rvert&=32k^{4k-8}(k-1)[(k-2)!]^{4k-8}[(2(k-1)-1)!]^{4}.\end{aligned}$$ For all such $F,$ we know that the total contribution to $\mathop{\rm Tr }\nolimits_{3k}(\mathcal{A}_{C_{4,k}})$ is $$\begin{aligned} &(k-1)^{n-1}\cfrac{16[(k-1)!]^{4k}}{[(k-1)!]^{4k-8}[(2(k-1))!]^{4}}\big[\cfrac{1}{(k-1)!}\big]^{4k}32k^{4k-8}(k-1) \\&[(k-2)!]^{4k-8}[(2(k-1)-1)!]^{4} \\&=32k^{4k-8}.\end{aligned}$$ **Subcase 4.2.** $F$ is an appropriate ordering of $(i_{1}\alpha_{1},i_{1}\alpha'_{1},\ldots,i_{k}\alpha_{k},i_{k}\alpha'_{k},\ldots,i_{2k-1}\alpha_{2k+1},\\i_{2k-1}\alpha'_{2k+1},\ldots,i_{3k-2}\alpha_{3k+2},i_{3k-2}\alpha'_{3k+2},\ldots, i_{4k-4}\alpha_{4k-4}).$ For each such $F,$ the number of orderings for the first entry is 2, and the number of orderings for the $\alpha_{h}$ is $[(k-1)!]^{4k}$. Hence, the total number of such $F$ is $2[(k-1)!]^{4k},$ and $$b(F)=(2!)^{4},~c(F)= [(k-1)!]^{4k-8}[(2(k-1))!]^{4},~\pi_{F}(\mathcal{A}_{C_{4,k}})=\big[\cfrac{1}{(k-1)!}\big]^{4k}.$$ Let $D'_{4}$ be the multi-digraph induced by $E(F).$ By the Matrix-Tree Theorem [@CvRS], we have $\tau(D'_{4})=16k^{4k-9}.$ By BEST Theorem [@AB], we have $\varepsilon(D'_{4})=16k^{4k-9}[(k-2)!]^{4k-8}[(2(k-1)-1)!]^{4}.$ Because each Eulerian cycle has $4k(k-1)$ arcs and multi arcs of $W(F)$ are not labelled, we have $$\begin{aligned} \lvert W(F)\rvert&=4k^{4k-8}(k-1)[(k-2)!]^{4k-8}[(2(k-1)-1)!]^{4}.\end{aligned}$$ For all such $F,$ we know that the total contribution to $\mathop{\rm Tr }\nolimits_{4k}(\mathcal{A}_{C_{4,k}})$ is $$\begin{aligned} &(k-1)^{n-1}\cfrac{2[(k-1)!]^{4k}(2!)^{4}}{[(k-1)!]^{4k-8}[(2(k-1))!]^{4}}\big[\cfrac{1}{(k-1)!}\big]^{4k}4k^{4k-8}(k-1) \\&[(k-2)!]^{4k-8}[(2(k-1)-1)!]^{4} =8k^{4k-8}.\end{aligned}$$ Therefore, we have $$\begin{aligned} \mathop{\rm Tr }\nolimits_{4k}(\mathcal{A}_{C_{4,k}})=4k^{k-1}(k-1)^{3k-4}+56k^{2k-3}(k-1)^{2k-3}+64k^{3k-5}(k-1)^{k-2}+40k^{4k-8}.\end{aligned}$$ $\square$ **Theorem 10**. *Let $C_{4,k}$ be a $k(\geq 4)$-uniform hypercycle with length four. Then $$\phi_{C_{4,k}}(\lambda)=\lambda^{m_{0}}(\lambda^{k}-1)^{m_{1}}(\lambda^{k}-2)^{m_{2}}(\lambda^{k}-4)^{m_{4}}(\lambda^{k}-\frac{3+\sqrt{5}}{2})^{m'}(\lambda^{k}-\frac{3-\sqrt{5}}{2})^{m'},$$ where $$m_{0}=4(k-1)^{4k-4}-4k^{k-1}(k-1)^{3k-4}+4k^{2k-3}(k-1)^{2k-3}-4k^{3k-5}(k-1)^{k-2}+5k^{4k-8},$$ $$m_{1}=4k^{k-2}(k-1)^{3k-4}-8k^{2k-4}(k-1)^{2k-3}+4k^{3k-6}(k-1)^{k-2},$$ $$m_{2}=4k^{2k-4}(k-1)^{2k-3}-8k^{3k-6}(k-1)^{k-2}+10k^{4k-9},$$ $$m_{4}=k^{4k-9}, ~m'=4k^{3k-5}(k-1)^{k-2}-8k^{4k-9}.$$* **Proof.** By Lemma [Lemma 5](#le:4-5){reference-type="ref" reference="le:4-5"}, we know the $k$-power of all different eigenvalues of $C_{4,k}$ are $0,~1,~2,~4,\\\frac{3+\sqrt{5}}{2},\frac{3-\sqrt{5}}{2}.$ By Lemma [Lemma 4](#le:4-4){reference-type="ref" reference="le:4-4"}, we know the trace is a real number. So we know that the multiple of $\frac{3+\sqrt{5}}{2}$ and $\frac{3-\sqrt{5}}{2}$ are equal. Let $m_{0}, m_{1}, m_{2}, m_{4},m'$ be the multiplicities of 0,1,2,4, $\frac{3+\sqrt{5}}{2},$ respectively. By Theorems [Theorem 6](#th:3-1){reference-type="ref" reference="th:3-1"}-[Theorem 9](#th:3-4){reference-type="ref" reference="th:3-4"} and $$\mathop{\rm Tr }\nolimits_{k}(\mathcal{A}_{C_{4,k}})=k(m_{1}+2m_{2}+4m_{4}+3m'), \mathop{\rm Tr }\nolimits_{2k}(\mathcal{A}_{C_{4,k}})=k(m_{1}+4m_{2}+16m_{4}+7m'),$$ $$\mathop{\rm Tr }\nolimits_{3k}(\mathcal{A}_{C_{4,k}})=k(m_{1}+8m_{2}+64m_{4}+18m'), ~\mathop{\rm Tr }\nolimits_{4k}(\mathcal{A}_{C_{4,k}})=k(m_{1}+16m_{2}+256m_{4}+47m'),$$ $$m_{0}+k(m_{1}+m_{2}+m_{4}+2m')=4(k-1)^{4k-4},$$ we have $$m_{0}=4(k-1)^{4k-4}-4k^{k-1}(k-1)^{3k-4}+4k^{2k-3}(k-1)^{2k-3}-4k^{3k-4}(k-1)^{k-2}+5k^{4k-8},$$ $$m_{1}=4k^{k-2}(k-1)^{3k-4}-8k^{2k-4}(k-1)^{2k-3}+4k^{3k-5}(k-1)^{k-2},$$ $$m_{2}=4k^{2k-4}(k-1)^{2k-3}-8k^{3k-5}(k-1)^{k-2}+10k^{4k-9},$$ $$m_{4}=k^{4k-9}, ~m'=4k^{3k-6}(k-1)^{k-2}-8k^{4k-9}.$$ $\square$ # Declaration of competing interest {#declaration-of-competing-interest .unnumbered} The authors declare that they have no conflict of interest. # Data availability {#data-availability .unnumbered} The study has no associated data. 99 T. van Aardenne-Ehrenfest, N. de Bruijn, Circuits and trees in oriented linear graphs, Simon Stevin : Wis-en Natuurkundig Tijdschrift 28 (1951) 203-217. 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Rowlinson, S. Simić, An introduction to the theory of graph spectra, Cambridge University Press, Cambridge, 2010. C. Duan, L. Wang, The $\alpha$-spectral radius of $f$-connected general hypergraphs, Appl. Math. Comput. 382 (2020) 125336. G. Gao, A. Chang, Y. Hou, Spectral radius on linear $r$-graphs without expanded $K_{r+1},$ SIAM J. Discrete Math. 36(2) (2022) 1000-1011. S. Hu, Z. Huang, C. Ling, L. Qi, On determinants and eigenvalue theory of tensors, J. Symbolic Comput. 50 (2013) 508-531. L. Kang, L. Liu, L. Qi, X. Yuan, Spectral radii of two kinds of uniform hypergraphs, Appl. Math. Comput. 338 (2018) 661-668. A. Morozov, Sh. Shakirov, Analogue of the identity Log Det=Trace Log for resultants, J. Geom. Phys. 61(3) (2011) 708-726. L. Qi, $H^{+}$-eigenvalues of Laplacian and signless Laplacian tensors, Commun. Math. Sci. 12 (2014) 1045-1064. J. Shao, L. Qi, S. Hu, Some new trace formulas of tensors with applications in spectral hypergraph theorey, Linear Multilinear Algebra 63(5) (2015) 971-992. Y. Zheng, The characteristic polynomial of the complete 3-uniform hypergraph, Linear Algebra Appl. 627 (2021) 275-286. J. Zhou, L. Sun, W. Wang, C. Bu, Some spectral properties of uniform hypergraphs, Electron. J. Combin. 21(4) (2014) \#P4.24. [^1]: Corresponding author. [^2]: Supported by the National Natural Science Foundation of China (Nos. 12271439 and 12301452).

arxiv_math

--- abstract: | We provide a concise, self-contained proof that the Silver Stepsize Schedule proposed in Part I [@alt23hedging1] directly applies to smooth (non-strongly) convex optimization. Specifically, we show that with these stepsizes, gradient descent computes an $\varepsilon$-minimizer in $O(\varepsilon^{-\log_{\rho} 2}) = O(\varepsilon^{-0.7864})$ iterations, where $\rho = 1+\sqrt{2}$ is the silver ratio. This is intermediate between the textbook unaccelerated rate $O(\varepsilon^{-1})$ and the accelerated rate $O(\varepsilon^{-1/2})$ due to Nesterov in 1983. The Silver Stepsize Schedule is a simple explicit fractal: the $i$-th stepsize is $1 + \rho^{\nu(i)-1}$ where $\nu(i)$ is the $2$-adic valuation of $i$. The design and analysis are conceptually identical to the strongly convex setting in [@alt23hedging1], but simplify remarkably in this specific setting. author: - | Jason M. Altschuler\ UPenn\ `alts@upenn.edu` - | Pablo A. Parrilo\ LIDS - MIT\ `parrilo@mit.edu` bibliography: - hedging.bib title: | Acceleration by Stepsize Hedging II:\ Silver Stepsize Schedule for Smooth Convex Optimization --- # Introduction {#sec:intro} We revisit the classical problem of smooth convex optimization: solve $\min_{x \in \mathbb{R}^d} f(x)$ where $f$ is convex and $M$-smooth (i.e., its gradient is $M$-Lipschitz). A celebrated result is that with a prudent choice of stepsizes $\{\alpha_t\}$, the gradient descent algorithm (GD) $$\begin{aligned} x_{t+1} = x_t - \frac{\alpha_t}{M} \nabla f(x_t) \end{aligned}$$ solves such a convex optimization problem to arbitrary accuracy from any initialization $x_0$. How quickly does GD converge? The mainstream approach (see e.g., the textbooks [@BertsekasNonlinear; @Nocedal; @nesterov-survey; @vishnoi2021algorithms; @bubeck-book; @BNO; @lan2020first; @bertsekas2015convex] among many others) is to use a constant stepsize schedule $\alpha_t \equiv \bar{\alpha} \in (0,2)$ since this ensures $$\begin{aligned} f(x_n) - f^* \leqslant\frac{cM\|x_0 - x^*\|^2}{n} \label{eq:rate-std}\end{aligned}$$ where $x^*$ denotes any minimizer of $f$, $f^* := f(x^*)$ denotes the corresponding minimal value, and $c$ is a small constant, e.g., $c = \tfrac{1}{4}$ for $\bar{\alpha} = 1$ [@DT14]. The main question posed in Part I [@alt23hedging1] was: can we accelerate the convergence of GD without changing the algorithm---just by judiciously choosing the stepsizes? Here we continue to investigate this question, now in the setting of smooth convex optimization. Note that this is markedly different from classical approaches to acceleration---starting from Nesterov's seminal result of 1983 [@nesterov-agd], those approaches modify the basic GD algorithm by adding momentum, internal dynamics, or other additional building blocks beyond just changing the stepsizes. For this reason, we do not discuss that line of work in detail, and instead refer to [@d2021acceleration] for a recent survey of this mainstream approach to acceleration, and to [@alt23hedging1] for a full discussion of the relations between these approaches. ## Contribution {#ssec:cont} ![Silver Stepsize schedule $\{\alpha_0,\alpha_1,\alpha_2,\ldots\}$. See [\[eq:steps\]](#eq:steps){reference-type="eqref" reference="eq:steps"} for the definition. Only the first $n=63$ values are shown (i.e., $k=8$). The fractal-like stepsizes are non-monotonic and have increasingly large "spikes" $\alpha_{2^k-1} = 1+\rho^{k-1}$.](figures/silverstepsize-3.pdf){#fig:silverstepsize width="0.5\\linewidth"} This paper provides a concise, self-contained proof that the Silver Stepsize Schedule proposed in Part I [@alt23hedging1] directly applies to smooth (non-strongly) convex optimization. This leads to faster convergence rates of GD for smooth convex optimization, as already pointed out in [@alt23hedging1 §1.1.4]. In this setting, the Silver Stepsize Schedule is particularly simple. For any integer $n = 2^{k}-1$, we recursively construct the schedule $h_{2n+1}$ of length $2n+1$ from the schedule $h_n$ of length $n$ via $$\begin{aligned} h_{2n+1} := [h_n, \; 1 + \rho^{k-1}, \; h_n] \label{eq:steps-recursive}\end{aligned}$$ where $\rho := 1 + \sqrt{2}$ denotes the silver ratio, and $h_1 := [\sqrt{2}]$. This results in the simple pattern $[\sqrt{2}, \; 2, \; \sqrt{2}, \; 1+\sqrt{2}, \dots]$ as depicted in Figure [1](#fig:silverstepsize){reference-type="ref" reference="fig:silverstepsize"}. This schedule is exactly the Silver Stepsize Schedule from [@alt23hedging1] in the limit that the strong convexity parameter vanishes (see Remark [Remark 3](#rem:steps-limit){reference-type="ref" reference="rem:steps-limit"} for details), and bears similarities to those in [@gupta22; @grimmer23; @GrimmerShuWang]; see §[1.2](#ssec:rel){reference-type="ref" reference="ssec:rel"}. We show that these stepsizes yield an improved convergence rate [\[eq:rate-std\]](#eq:rate-std){reference-type="eqref" reference="eq:rate-std"} where $\tfrac{c}{n}$ is replaced by $$\begin{aligned} r_k := \frac{1}{1 + \sqrt{4\rho^{2k} - 3}} \leqslant \frac{1}{2 \rho^{\log_2 n}} = \frac{1}{2n^{\log_2 \rho}} \approx \frac{1}{2 n^{1.2716}} \,. \label{eq:r-asymptotics}\end{aligned}$$ This bound gives the correct asymptotic scaling of $r_k$ since the inequality is asymptotically tight. **Theorem 1** (Main result). *For any horizon $n = 2^k - 1$, any dimension $d$, any $M$-smooth convex function $f : \mathbb{R}^d \to \mathbb{R}$, and any initialization $x_0 \in \mathbb{R}^d$, $$\begin{aligned} f(x_n) - f^* \leqslant r_k\, M\|x_0 - x^*\|^2 \,, \end{aligned}$$ where $x^*$ denotes any minimizer of $f$, and $x_n$ denotes the output of $n$ steps of GD using the Silver Stepsize Schedule. In particular, in order to achieve error $f(x_n) - f^* \leqslant\varepsilon$, it suffices to run GD for $$\begin{aligned} n \leqslant \left( \frac{M \|x_0 - x^*\|^2}{2 \varepsilon} \right)^{\log_{\rho} 2} \approx \left( \frac{M\|x_0 - x^*\|2}{2 \varepsilon} \right)^{0.7864} \;\; \text{iterations}. \end{aligned}$$* We make several remarks. 1) This rate $n^{-\log_2 \rho} \approx n^{-1.2716}$ is intermediate between the textbook unaccelerated rate $n^{-1}$ and the accelerated rate $n^{-2}$ due to Nesterov in 1983 [@nesterov-agd]; see Figure [2](#fig:threerates){reference-type="ref" reference="fig:threerates"} for a visualization. 2) The Silver Stepsize Schedule is independent of the horizon, see §[2](#sec:construction){reference-type="ref" reference="sec:construction"}. 3) We conjecture that, up to a constant factor, the rate $r_k$ is optimal among all possible stepsize schedules. This will be addressed in the forthcoming Part III. ![Upper bound on the optimality gap $f(x_n)-f^*$, as a function of the number of iterations $n$. The three plots correspond to the rates $O(n^{-1})$ for the standard constant stepsize $1/M$ [@nesterov-survey Corollary 2.1.2], $O(n^{-\log_2 \rho}) \approx O({n^{-1.2716}})$ for the Silver Stepsize Schedule (Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}), and $O(n^{-2})$ for Nesterov acceleration [@nesterov-survey Theorem 2.2.2]. ](figures/ThreeRates.pdf){#fig:threerates width="0.45\\linewidth"} ## Related work {#ssec:rel} Several time-varying stepsize schedules have been considered, e.g., Armijo-Goldstein rules, Polyak-type schedules, Barzilai-Borwein-type schedules, etc. However, until recently, no convergence analyses improved over the textbook unaccelerated rate except in the special case of minimizing convex quadratics. Here we discuss the recent line of work on accelerated GD via time-varying stepsizes. For brevity, we refer to Paper I [@alt23hedging1] for a more complete discussion of adjacent bodies of literature. Beginning with Altschuler's 2018 MS thesis [@altschuler2018greed], a line of work designed time-varying stepsize schedules to achieve faster convergence rates for (non-quadratic) convex optimization. The thesis [@altschuler2018greed] showed that time-varying stepsizes can lead to improvements over the textbook unaccelerated GD rate---by an optimal asymptotic factor in the separable setting by using random stepsizes, and by a constant factor in the strongly convex setting by giving optimal stepsizes for $n=2,3$. A key difficulty in the general non-separable setting is that the search for optimal stepsizes is non-convex and computationally difficult for larger horizons $n$. In 2022, @gupta22 combined Branch & Bound techniques with the PESTO SDP of [@pesto] to develop algorithms that perform this search numerically, and as an example computed good approximate schedules in the convex setting for larger values of $n$ up to $50$. They observed a fit of roughly $O(n^{-1.178})$ and suggested from this that the asymptotic rate may be faster than the textbook rate $O(n^{-1})$. In July 2023, @grimmer23 showed how to prove asymptotic rates for the (non-strongly) convex setting by periodically cycling through finite schedules. While in the strongly convex setting composing progress from different cycles just amounts to multiplying contraction rates, in the non-strongly convex setting this can be more subtle depending on the approach.[^1] To deal with this, he introduced the notion of straightforward stepsize patterns and obtained a constant-factor improvement over the textbook unaccelerated rate by cycling through approximate schedules of length $n=127$. From these numerics, he conjectured that further optimizing stepsizes might lead to an asymptotic rate of $O((n \log n)^{-1})$, a milder improvement than that conjectured by [@gupta22]. In September 2023, two concurrent papers appeared [@GrimmerShuWang; @alt23hedging1]. @alt23hedging1 was Part I: there we proposed the Silver Stepsize Schedule of arbitrary size to prove asymptotic acceleration for the strongly convex setting. This improved the textbook unaccelerated rate $\Theta(\kappa)$ to $\Theta(\kappa^{\log_2 \rho}) \approx \Theta(\kappa^{0.7864})$ where $\kappa$ is the condition number. We conjectured and provided partial evidence that these rates are optimal among all possible stepsize schedules. This result was achieved by introducing the technique of recursive gluing to establish multi-step descent, which we make use of here. The other paper was @GrimmerShuWang. By utilizing a certain non-periodic sequence of increasingly large stepsizes and building upon the "straightforwardness" machinery in [@grimmer23], they proved the rate $O(n^{-1.0245})$ for the convex setting in v1, later improved to $O(n^{-1.0564})$ in v2. The results in Part I [@alt23hedging1] were stated for the strongly convex setting. As pointed out throughout that paper, this is not a restriction because on the one hand those results immediately imply analogously accelerated rates $\Theta(n^{-\log_2 \rho}) \approx \Theta(n^{-1.2716})$ for the convex setting via a standard black-box reduction; and on the other hand, this reduction can be bypassed by re-doing the analysis for the convex setting since all the core conceptual ideas extend directly [@alt23hedging1 §1.1.4]. The present paper provides these details. ## Notation {#ssec:notation} #### Indexing. {#indexing. .unnumbered} Throughout, the horizon is $n = 2^k-1$. This correspondence between $n \in \{1,3,7,15,\dots\}$ and $k \in \{1,2,3,4,\dots\}$ allows re-indexing in a way that simplifies notation for the recursion. #### Rescaling. {#rescaling. .unnumbered} For simplicity, we normalize $M = 1$, i.e., $\| \nabla f(x) - \nabla f(y) \| \leqslant\|x-y\|$ for all $x,y$. This is without loss of generality since if $h$ is $M$-smooth and convex, then $f := h/M$ is $1$-smooth and convex, thus the result we establish $f_n - f^* \leqslant r_k\|x_0 - x^*\|^2$ implies $h_n - h^* \leqslant r_k M \|x_0 - x^*\|^2$. Note that running GD on $f$ simply amounts to rescaling the Silver Stepsize Schedules for $h$ by $1/M$. # Silver Stepsize Schedule {#sec:construction} The Silver Stepsize Schedule is defined recursively in [\[eq:steps-recursive\]](#eq:steps-recursive){reference-type="eqref" reference="eq:steps-recursive"}. Here we mention an equivalent direct expression and make several remarks. Let $\mathop{\mathrm{\nu}}(t)$ denote the $2$-adic valuation of $t$, i.e., the smallest non-negative integer $i$ such that $2^i$ is in the binary expansion of $t$. For example, $\mathop{\mathrm{\nu}}(1) = 0$, $\mathop{\mathrm{\nu}}(2) = 1$, $\mathop{\mathrm{\nu}}(3) = 0$, $\mathop{\mathrm{\nu}}(4) = 2$, etc. **Definition 2** (Silver Stepsize Schedule for smooth convex optimization). *For $t \in \{0,1, 2,\dots\}$, the $t$-th stepsize of the Silver Stepsize Schedule is $$\begin{aligned} \alpha_t := 1 + \rho^{\mathop{\mathrm{\nu}}(t+1)-1}\,. \label{eq:steps} \end{aligned}$$* This schedule is non-monotonic, fractal-like, and has increasingly large spikes that grow exponentially (by a factor of $\rho$) yet become exponentially less frequent (by a factor of $2$). See Figure [1](#fig:silverstepsize){reference-type="ref" reference="fig:silverstepsize"}. It can also be easily implemented[^2] in any computer language. **Remark 3** (Limit of Silver Stepsize Schedules in the strongly convex case). *The schedule [\[eq:steps\]](#eq:steps){reference-type="eqref" reference="eq:steps"} is simply the Silver Stepsize Schedule for smooth *strongly*-convex optimization in Part I [@alt23hedging1], in the limit as the strong convexity parameter tends to $0$. The stepsizes simplify in the limit: they increase by a factor of $\rho$, shorter schedules are prefixes of longer schedules, neither $a_1$ nor any of the $b_n$ sequence in [@alt23hedging1] is needed, and they are non-periodic (hence why there is no rate saturation here).* We also record a simple closed-form expression for the sum of the first $n=2^k -1$ Silver Stepsizes. **Lemma 4** (Sum of Silver Stepsizes). *$\sum_{t=0}^{n-1} \alpha_t = \rho^k - 1$ for any $k \in \mathbb{N}$.* *Proof.* The base case $k=1$ is trivial. The inductive step follows from the recursion [\[eq:steps-recursive\]](#eq:steps-recursive){reference-type="eqref" reference="eq:steps-recursive"}. ◻ # Recursive gluing {#sec:gluing} Here we prove that the Silver Stepsize Schedule has convergence rate $r_k$ for smooth convex optimization. The analysis closely mirrors the strongly convex setting in Part I [@alt23hedging1]: we prove the advantage of time-varying stepsizes via multi-step descent rather than iterating the greedy 1-step bound, show multi-step descent by exploiting long-range consistency conditions along the GD trajectory, certify multi-step descent via recursive gluing, and recursively glue by combining the same three components in the same way. In the interest of brevity, we refer to [@alt23hedging1] for a detailed discussion of all these concepts. Briefly, the idea behind multi-step descent is that it is essential to capture how different iterations affect other iterations' progress. We do this by exploiting long-range consistency conditions between the iterates along GD's trajectory, as encoded by the co-coercivities $$\begin{aligned} Q_{ij} := 2(f_i - f_j) + 2\langle g_j, x_j - x_i \rangle - \|g_j - g_i\|^2\,. \label{eq:cocoercivity}\end{aligned}$$ The significance of these co-coercivities is that the constraints $\{Q_{ij} \geqslant 0\}_{i \neq j \in \{0,1,\dots,n,*\}}$ are necessary and sufficient for the existence of a $1$-smooth convex function $f$ satisfying $f_i = f(x_i)$ and $g_i = \nabla f(x_i)$ for each $i \in \{0,1,\dots,n,*\}$ [@pesto]. In other words, the co-coercivity conditions $\{Q_{ij} \geqslant 0 \}_{i \neq j \in \{0,1,\dots,n,*\}}$ generate all possible long-range consistency constraints on the objective function $f$. Or, said another way, the co-coercivity conditions generate all possible valid inequalities with which one can prove convergence rates for GD. For a further discussion, see [@alt23hedging1 §2.2]. Concretely, to prove Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}, we exhibit explicit non-negative multipliers $\lambda_{ij}$ satisfying $$\begin{aligned} \sum_{i \neq j \in \{0, \dots, n,*\}} \lambda_{ij} Q_{ij} = \|x_0 - x^*\|^2 - \|x_n - c_k g_n - x^*\|^2 + \frac{f^* - f_n}{r_k}\,, \label{eq:cert}\end{aligned}$$ where we use the shorthand $c_k := \frac{1}{2r_k}$. Since $\sum_{ij}\lambda_{ij} Q_{ij} \geqslant 0$ for any $1$-smooth convex function, and since $\|x_n - c_k g_n - x^*\|^2 \geqslant 0$ trivially as it is a square, this immediately implies the desired rate $$\begin{aligned} f_n - f^* \leqslant r_k \|x_0 - x^*\|^2\,. \label{eq:cert-cor}\end{aligned}$$ **Remark 5** (The importance of a composable formulation). *We emphasize that while there are several alternative formulations to [\[eq:cert\]](#eq:cert){reference-type="eqref" reference="eq:cert"} that imply a rate like [\[eq:cert-cor\]](#eq:cert-cor){reference-type="eqref" reference="eq:cert-cor"} after dropping terms, our formulation [\[eq:cert\]](#eq:cert){reference-type="eqref" reference="eq:cert"} is "canonical" because it composes well under recurrence (Theorem [Theorem 8](#thm:cert){reference-type="ref" reference="thm:cert"}). This enables a quite short and simple proof. Another reason this formulation is particularly nice is because the base case $n=0$ amounts to a re-writing of the definition of $Q_{*0}$, which is an improvement over the standard bound $f_0 - f^* \leqslant\mathop{\mathrm{\frac{1}{2}}}\|x_0 - x^*\|^2$. In fact, this improvement is precisely what makes recursive gluing work so seamlessly.* **Example 6** ($n=0$). *For $n=0$, the identity [\[eq:cert\]](#eq:cert){reference-type="eqref" reference="eq:cert"} is $$\begin{aligned} Q_{*0} = \|x_0 - x^*\|^2 - \|x_0 - g_0 - x^*\|^2 + 2(f^* - f_0)\,. \end{aligned}$$ Here, $r_0 = \mathop{\mathrm{\frac{1}{2}}}$, $c_0 = 1$, and the only non-zero multiplier is $\lambda_{*0} = 1$.* **Example 7** ($n=1$). *For $n=1$, the identity [\[eq:cert\]](#eq:cert){reference-type="eqref" reference="eq:cert"} is $$\begin{aligned} \sum_{i \neq j \in \{0, 1, *\}} \lambda_{ij} Q_{ij} = \|x_0 - x^*\|^2 - \|x_1 - c_1 g_1 - x^*\|^2 + \frac{f^* - f_1}{r_1}\,. \end{aligned}$$ Here, $x_1 = x_0 - \sqrt{2} g_0$, $r_1 = \frac{1}{1+\sqrt{4\rho^2-3}} \approx 0.1816$, $c_1 = \frac{1}{2r_1} \approx 2.7535$, and $$\begin{aligned} \begin{bmatrix} \lambda_{00} & \lambda_{01} & \lambda_{0*} \\ \lambda_{10} & \lambda_{11} & \lambda_{1*} \\ \lambda_{*0} & \lambda_{*1} & \lambda_{**} \end{bmatrix} = \begin{bmatrix} 0 & 1+\sqrt{2} & 0 \\ 1 & 0 & \sqrt{2} \\ \sqrt{2} & \frac{1}{2r_1} & 0 \end{bmatrix} \,. \end{aligned}$$* For larger horizons $n$, we construct $\lambda_{ij}$ via the recursive gluing technique of [@alt23hedging1]. See Figure [3](#fig:gluing){reference-type="ref" reference="fig:gluing"}. Below, we say that the multipliers $\{\sigma_{ij}\}_{i,j \in \{0,\dots,n,*\}}$ satisfy the $*$-sparsity property if $\sigma_{i*} = 0$ for all $i < n$. This is satisfied by construction and simplifies part of the proof (isolated in Lemma [Lemma 9](#lem:opt){reference-type="ref" reference="lem:opt"}). {reference-type="ref" reference="thm:cert"}, illustrated here for combining two copies of the $n=3$ certificate (shaded) to create the $2n+1 = 7$ certificate. The structure of this recursive gluing is identical to the strongly convex setting from Part I [@alt23hedging1], modulo re-indexing for horizons of the form $n = 2^k -1$ rather than $2^k$.](figures/gluing_convex.png){#fig:gluing width="0.4\\linewidth"} **Theorem 8** (Recursive gluing). *Let $n = 2^k-1$. Suppose $\{\sigma_{ij}\}_{i,j \in \{0,\dots,n,*\}}$ satisfies $*$-sparsity and certifies the $n$-step rate, i.e., $$\begin{aligned} \sum_{i \neq j \in \{0, \dots, n,*\}} \sigma_{ij} Q_{ij} = \|x_0 - x^*\|^2 - \|x_n - c_k g_n - x^*\|^2 + \frac{f^* - f_n}{r_k}\ . \label{eq:cert:n} \end{aligned}$$ Then there exists $\{\lambda_{ij}\}_{i, j \in \{0,\dots,2n+1,*\}}$ that satisfies $*$-sparsity and certifies the $2n+1$-step rate, i.e., $$\begin{aligned} \sum_{i \neq j \in \{0, \dots, 2n+1,*\}} \lambda_{ij} Q_{ij} = \|x_0 - x^*\|^2 - \|x_{2n+1} - c_{k+1} g_{2n+1} - x^*\|^2 + \frac{f^* - f_{2n+1}}{r_{k+1}}\ . \label{eq:cert:2n+1} \end{aligned}$$ Moreover, this certificate is explicitly given by $$\begin{aligned} \lambda_{ij} = \underbrace{\Theta_{ij}}_{\text{gluing}} + \underbrace{\Xi_{ij}}_{\text{rank-one correction}} + \underbrace{\Delta_{ij}}_{\text{sparse correction}} \label{eq:lambda} \end{aligned}$$ The "gluing component" $\Theta$ is defined as $$\begin{aligned} \Theta_{ij} := \underbrace{\sigma_{i,j} \cdot \mathds{1}_{i,j \in \{0,\dots,n,*\}}}_{\text{recurrence for first $n$ steps}} \;\;+\;\; \underbrace{ (1 + 2\rho)\; \sigma_{i-n-1,j-n-1} \cdot \mathds{1}_{i,j \in \{n+1,\dots,2n+1,*\}}}_{\text{recurrence for last $n$ steps}}\,. \label{eq:Theta} \end{aligned}$$ The "rank-one correction" $\Xi$ is zero except the entries $\{\Xi_{ij}\}_{i \in \{n,2n+1,*\},\; j \in \{n+1, \dots, 2n\}}$ which are $$\begin{aligned} \begin{bmatrix} \Xi_{n,n+1} & \Xi_{n,n+2} & \cdots & \Xi_{n,2n} \\ \Xi_{2n+1,n+1} & \Xi_{2n+1,n+2} & \cdots & \Xi_{2n+1,2n} \\ \Xi_{*,n+1} & \Xi_{*,n+2} & \cdots & \Xi_{*,2n} \end{bmatrix} := \rho \begin{bmatrix} 1 \\ 1 \\ -2 \end{bmatrix} \begin{bmatrix} \alpha_{n+1} & \alpha_{n+2} & \cdots & \alpha_{2n} \end{bmatrix} \label{eq:Xi} \end{aligned}$$ The "sparse correction" $\Delta$ is zero except the entries $\{\Delta_{ij}\}_{i \neq j \in \{n,2n+1,*\}}$ which are $$\begin{aligned} \begin{bmatrix} \Delta_{n,n} & \Delta_{n,2n+1} & \Delta_{n,*} \\ \Delta_{2n+1,n} & \Delta_{2n+1,2n+1} & \Delta_{2n+1,*} \\ \Delta_{*,n} & \Delta_{*,2n+1} & \Delta_{*,*} \end{bmatrix} := \begin{bmatrix} 0 & \rho & 1 - \rho^k \\ \rho^k & 0 & 2\rho - \sqrt{2} \rho^{k+1} \\ 1 + \rho^{k-1} - \frac{1}{2r_k} & \frac{1}{2r_{k+1}} - \frac{1+2\rho}{2r_k} & 0 \end{bmatrix} \label{eq:Del} \end{aligned}$$* We remark that using this recursion, one can also write out the $n$-step certificate directly. The value of each multiplier then depends on the binary expansion of its indices. This recursive gluing immediately implies the main result of the paper, Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}. *Proof of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}.* It suffices to prove [\[eq:cert\]](#eq:cert){reference-type="eqref" reference="eq:cert"}; we prove this by induction. The base case $n=1$ is Example [Example 7](#ex:n=1){reference-type="ref" reference="ex:n=1"}. The inductive step is Theorem [Theorem 8](#thm:cert){reference-type="ref" reference="thm:cert"}. ◻ The rest of the section is dedicated to proving Theorem [Theorem 8](#thm:cert){reference-type="ref" reference="thm:cert"}. We first isolate two helper lemmas in §[3.1](#ssec:gluing:helper){reference-type="ref" reference="ssec:gluing:helper"}, and then we combine them to prove the result in §[3.2](#ssec:gluing:pf){reference-type="ref" reference="ssec:gluing:pf"}. For notational simplicity, henceforth we assume $x^* = 0$; this is without loss of generality after translating. ## Helper lemmas {#ssec:gluing:helper} Here we provide three helper lemmas for the proof of Theorem [Theorem 8](#thm:cert){reference-type="ref" reference="thm:cert"}. The first lemma explicitly computes all multipliers involving $x^*$ for the $n$-step certificate. **Lemma 9** (Multipliers involving $x^*$). *$\sigma_{n*} = \rho^k - 1$, $\sigma_{*n} = \frac{1}{2r_k}$, and $\sigma_{*t} = \alpha_t$ for $t \in \{0,\dots,n-1\}$.* *Proof.* Expand both sides of the identity [\[eq:cert:n\]](#eq:cert:n){reference-type="eqref" reference="eq:cert:n"} using the $*$-sparsity assumption, the definition of the co-coercivities, and the definition of GD, i.e., $x_t = x_0 - \sum_{s=0}^{t-1} \alpha_s g_s$. Matching coefficients for the $\langle x_0, g_t \rangle$ terms yields the claimed formulas for $\sigma_{*t}$. Matching coefficients for the $f^*$ term gives the identity $\sigma_{n*} = \sum_{j=0}^n \sigma_{*j}- \tfrac{1}{2r_k}$. This equals $\rho^k - 1$ by Lemma [Lemma 4](#lem:sum-steps){reference-type="ref" reference="lem:sum-steps"}. ◻ The next two lemmas are more substantial. These help us verify [\[eq:cert:2n+1\]](#eq:cert:2n+1){reference-type="eqref" reference="eq:cert:2n+1"}---which consists of a linear form in all $2n+3$ function values and a quadratic form in all $2n+3$ gradients and iterates. Naïvely verifying such an identity requires checking $\Theta(n)$ coefficients for the linear form and $\Theta(n^2)$ coefficients for the quadratic form. The following two lemmas show that due to the recursive construction of the Silver Stepsize Schedule and the gluing, these forms only affect the indices $n,2n+1,*$. This reduces verifying the rate certificate [\[eq:cert:2n+1\]](#eq:cert:2n+1){reference-type="eqref" reference="eq:cert:2n+1"} to checking only $\Theta(1)$ coefficients, as detailed below in §[3.2](#ssec:gluing:pf){reference-type="ref" reference="ssec:gluing:pf"}. Below, for shorthand, let $F_{ij} := 2(f_i - f_j)$ and $P_{ij} := 2\langle g_j, x_j - x_i \rangle - \|g_i - g_j\|^2$ denote the linear and quadratic components of $Q_{ij}$, respectively. For bookkeeping purposes, we use $3$-dimensional vectors and $4 \times 4$ matrices to denote the coefficients of these forms. **Lemma 10** (Succinct linear forms). *Let $u := [f_n, f_{2n+1}, f^*]^T$, and let $e, s, \ell \in \mathbb{R}^3$ be the vectors defined in Appendix [4.1](#app:succinct-linear){reference-type="ref" reference="app:succinct-linear"}.* - *[Gluing error.]{.ul} $\frac{f^* - f_{2n+1}}{r_{k+1}} - \sum_{ij} \Theta_{ij} F_{ij} = \langle e, u \rangle$* - *[Sparse correction.]{.ul} $\sum_{ij} \Delta_{ij} F_{ij} = \langle s, u \rangle$* - *[Rank-one correction.]{.ul} $\sum_{ij} \Xi_{ij} F_{ij} = \langle \ell, u \rangle$* *Proof.* Expand the definition of co-coercivities, simplify the rank-one correction using Lemma [Lemma 4](#lem:sum-steps){reference-type="ref" reference="lem:sum-steps"}, and simplify the sparse correction using the Pell recurrence $\rho^{k+1} = 2\rho^k + \rho^{k-1}$. ◻ **Lemma 11** (Succinct quadratic forms). *Let $v := [x_n,g_n,x_{2n+1},g_{2n+1}]^T$, and let $E$, $S$, $L$ be the $4 \times 4$ matrices defined in Appendix [4.2](#app:succinct){reference-type="ref" reference="app:succinct"}.* - *[Gluing error.]{.ul} $\|x_0\|^2 - \|x_{2n+1} - c_{k+1} g_{2n+1}\|^2 - \sum_{ij} \Theta_{ij} P_{ij} = \langle E, vv^T \rangle$* - *[Sparse correction.]{.ul} $\sum_{ij} \Delta_{ij} P_{ij} = \langle S, vv^T \rangle$* - *[Rank-one correction.]{.ul} $\sum_{ij} \Xi_{ij} P_{ij} = \langle L, vv^T \rangle$* *Proof.* Deferred to Appendix [4.2](#app:succinct){reference-type="ref" reference="app:succinct"} for brevity. ◻ ## Proof of recursive gluing (Theorem [Theorem 8](#thm:cert){reference-type="ref" reference="thm:cert"}) {#ssec:gluing:pf} #### Non-negativity. {#non-negativity. .unnumbered} We verify $\lambda_{ij} \geqslant 0$ for all $i \neq j \in \{0, \dots, 2n+1,*\}$. For nearly all entries, this is obvious since $\lambda_{ij}$ is constructed by adding and multiplying non-negative numbers. For the remaining entries where either $\Xi$ or $\Delta$ is negative, compute the corresponding entry of $\lambda$ by summing the corrections and using Lemma [Lemma 9](#lem:opt){reference-type="ref" reference="lem:opt"}. This gives $\lambda_{n*} = 0$, $\lambda_{2n+1,*} = \rho^{k+1} - 1$, $\lambda_{*,n} = \rho^{k-1} + 1$, $\lambda_{*,2n+1} = \tfrac{1}{2r_{k+1}}$, and $\lambda_{*,t} = \alpha_t$ for all $t \in \{n+1,\dots,2n\}$. All these entries are clearly non-negative. #### Rate certificate. {#rate-certificate. .unnumbered} The identity [\[eq:cert:2n+1\]](#eq:cert:2n+1){reference-type="eqref" reference="eq:cert:2n+1"} has two components: a linear form in the function values and a quadratic form in the iterates and gradients. For the linear form, it suffices to verify $e-s-\ell = 0$ by Lemma [Lemma 10](#lem:succinct-linear){reference-type="ref" reference="lem:succinct-linear"}, where $e,s,\ell \in \mathbb{R}^3$ are the vectors defined in Appendix [4.1](#app:succinct-linear){reference-type="ref" reference="app:succinct-linear"}. This is obvious by inspection. For the quadratic form, it suffices to verify $E - S - L = 0$, where $E,S,L \in \mathbb{R}^{4 \times 4}$ are the matrices in Lemma [Lemma 11](#lem:succinct){reference-type="ref" reference="lem:succinct"} defined in Appendix [4.2](#app:succinct){reference-type="ref" reference="app:succinct"}. By plugging in the explicit values for $r_k$ and $\Delta$, this is straightforward to check by hand. For brevity, we provide a simple Mathematica script that verifies these identities at the URL [@MathematicaURL]. # Deferred proof details {#app:deferred} ## Succinct linear forms (Lemma [Lemma 10](#lem:succinct-linear){reference-type="ref" reference="lem:succinct-linear"}) {#app:succinct-linear} The vectors $e, s, \ell$ in Lemma [Lemma 10](#lem:succinct-linear){reference-type="ref" reference="lem:succinct-linear"} are defined as follows: $$\begin{aligned} \small e := \begin{bmatrix} \frac{1}{r_k} \\ -\frac{1}{r_{k+1}} + \frac{1+2\rho}{r_k} \\ \frac{1}{r_{k+1}} - \frac{4\rho}{r_k} \end{bmatrix} \,, \qquad s := 2\rho(\rho^{k} - 1) \begin{bmatrix} 1 \\ 1 \\ -2 \end{bmatrix} \,, \qquad \ell := \begin{bmatrix} \frac{1}{r_k} - 2\rho(\rho^k - 1)\\ -\frac{1}{r_{k+1}} + \frac{1+2\rho}{r_k} - 2\rho(\rho^k - 1) \\ \frac{1}{r_{k+1}} - \frac{4\rho}{r_k} + 4\rho(\rho^k - 1) \end{bmatrix}\,.\end{aligned}$$ ## Succinct quadratic forms (Lemma [Lemma 11](#lem:succinct){reference-type="ref" reference="lem:succinct"}) {#app:succinct} The matrices in Lemma [Lemma 11](#lem:succinct){reference-type="ref" reference="lem:succinct"} are defined as follows. For brevity, we write $\sim$ for the entries below the diagonal since these coefficient matrices are symmetric. $$\begin{aligned} \small E := \begin{bmatrix} -2\rho & (1 + 2\rho) (1 + \rho^{k-1}) -\frac{1}{2r_k} & 0 & 0 \\ \sim & \frac{1}{4r_k^2} - (1 + 2\rho) (1+\rho^{k-1})^2 & 0 & 0 \\ \sim & \sim & 2\rho & \frac{1}{2r_{k+1}} - \frac{1+2\rho}{2r_k} \\ \sim & \sim & \sim & \frac{1+2\rho}{4r_k^2} - \frac{1}{4r_{k+1}^2} \end{bmatrix}\end{aligned}$$ $$\begin{aligned} \small S := \begin{bmatrix} 0 & \Delta_{2n+1,n} + \Delta_{*,n} & 0 & -\Delta_{n,2n+1} \\ \sim & -\left( \Delta_{n,*} + \Delta_{*,n} + \Delta_{2n+1,n} + \Delta_{n,2n+1} \right) & -\Delta_{2n+1,n} & \Delta_{n,2n+1} + \Delta_{2n+1,n} \\ \sim & \sim & 0 & \Delta_{n,2n+1} + \Delta_{*,2n+1} \\ \sim & \sim & \sim & - \left( \Delta_{n,2n+1} + \Delta_{2n+1,n} + \Delta_{2n+1,*} + \Delta_{*,2n+1}\right) \end{bmatrix}\end{aligned}$$ $$\begin{aligned} \small L := \rho \begin{bmatrix} -2 & 2 + \rho^{k-1} & 0 & 1 \\ \sim & -1 - \rho^k - 2 \rho^{k-1} & \rho^{k-1} & - 1 - \rho^{k-1} \\ \sim & \sim & 2 & -1 \\ \sim & \sim & \sim & 1 - \rho^k \end{bmatrix}\end{aligned}$$ *Proof of Lemma [Lemma 11](#lem:succinct){reference-type="ref" reference="lem:succinct"}.* [Gluing component.]{.ul} Recall that $h_{2n+1} = [h_n, \alpha_n, h_n]$ by the recursive construction [\[eq:steps-recursive\]](#eq:steps-recursive){reference-type="eqref" reference="eq:steps-recursive"} of the Silver Stepsize Schedule. Thus, since $\sigma$ certifies the $n$-step rate, we have $$\begin{aligned} \sum_{i \neq j \in \{0, \dots, n,*\}} \sigma_{ij} P_{ij} &= \|x_0\|^2 - \|x_n - c_k g_n\|^2 \\ \sum_{i \neq j \in \{n+1, \dots, 2n,*\}} \sigma_{ij} P_{ij} &= \|x_{n+1}\|^2 - \|x_{2n+1} - c_k g_{2n+1}\|^2 \end{aligned}$$ By definition, $\sum_{ij} \Theta_{ij} P_{ij}$ is the former plus $(1 + 2\rho)$ times the latter. Thus the gluing error $\|x_0\|^2 - \|x_{2n+1} - c_{k+1} g_{2n+1}\|^2 - \sum_{ij} \Theta_{ij} P_{ij}$ is equal to $$\begin{aligned} (1+2\rho) \|x_{2n+1} - c_k g_{2n+1}\|^2 - \|x_{2n+1} - c_{k+1} g_{2n+1}\|^2 + \|x_n - c_k g_n\|^2 - (1 + 2\rho) \|x_{n+1}\|^2 \end{aligned}$$ The key point is that after expanding $x_{n+1} = x_n - \alpha_n g_n$, this is a quadratic form in only the $4$ variables $x_n, g_n, x_{2n+1},g_{2n+1}$. Collecting terms, substituting $c_k = \tfrac{1}{2r_k}$, and using the definition of the Silver Stepsize $\alpha_n = 1 + \rho^{k-1}$ yields $\langle E, vv^T \rangle$, as desired. [Sparse component.]{.ul} Expand the definition of the co-coercivity and collect terms. Note also that this equals the matrix $S$ in Part I [@alt23hedging1], in the limit that the strong convexity tends to $0$. [Low-rank component.]{.ul} Expanding the co-coercivities, using the identity $x_{2n+1} - x_{n+1}= - \sum_{t=n+1}^{2n} \alpha_t g_t$ by definition of GD, and using the identity $\sum_{t=n+1}^{2n} \alpha_t = \sum_{t=0}^{n-1} \alpha_t = \rho^k - 1$ from Lemma [Lemma 4](#lem:sum-steps){reference-type="ref" reference="lem:sum-steps"}, $$\begin{aligned} \sum_{t=n+1}^{2n} \Big( \Xi_{n,t} P_{n,t} - \Xi_{*,t} P_{*,t} \Big) = \rho \sum_{t=n+1}^{2n} \alpha_t \Big( 2 \langle g_t, g_n - x_n \rangle - \|g_n\|^2 \Big) = 2\rho \langle x_{2n+1} - x_{n+1}, x_n - g_n \rangle - \rho(\rho^k - 1 ) \|g_n\|^2 \,. \end{aligned}$$ An identical calculation yields $$\begin{aligned} \sum_{t=n+1}^{2n} \Big( \Xi_{2n+1,t} P_{2n+1,t} - \Xi_{*,t} P_{*,t} \Big) &= 2\rho \langle x_{2n+1} - x_{n+1}, x_{2n+1} - g_{2n+1} \rangle - \rho(\rho^k - 1 ) \|g_{2n+1}\|^2 \,. \end{aligned}$$ Now sum these two displays, use the definition of GD to expand $x_{n+1} = x_n - \alpha_n g_n$, and use the definition of the Silver Stepsize $\alpha_n = 1 + \rho^{k-1}$. This gives $\langle L, vv^T \rangle$, as desired. ◻ [^1]: Cf., our recursive gluing approach, which is a simple way of composing progress that unifies the convex (in §[3](#sec:gluing){reference-type="ref" reference="sec:gluing"}) and strongly convex settings (in Paper I). This is why our analysis is so compact and completely bypasses the machinery of straightforward patterns. See Remark [Remark 5](#rem:formulation){reference-type="ref" reference="rem:formulation"}. [^2]: For instance, in Python the command `[1+rho**((k & -k).bit_length()-2) for k in range(1,64)]` generates the first $63$ steps of the Silver Schedule shown in Figure [1](#fig:silverstepsize){reference-type="ref" reference="fig:silverstepsize"}.

arxiv_math

--- address: - Department of Mathematics, University of California, Riverside CA, 92521, USA - University of Western Ontario, London, Ontario, Canada - 16 Place Saint Clément, Luz St Sauveur, France author: - John C. Baez - J. Daniel Christensen - Sam Derbyshire date: October 1, 2023 title: The Beauty of Roots --- {width="6in"} 0.5em Figure 1. Roots of all polynomials of degree 23 whose coefficients are $\pm 1$. The brightness shows the number of roots per pixel. 1em One of the charms of mathematics is that simple rules can generate complex and fascinating patterns, which raise questions whose answers require profound thought. For example, if we plot the roots of all polynomials of degree $23$ whose coefficients are all $1$ or $-1$, we get an astounding picture, shown in Figure 1. More generally, define a **Littlewood polynomial** to be a polynomial $p(z) = \sum_{i=0}^d a_i z^i$ with each coefficient $a_i$ equal to $1$ or $-1$. Let ${\mathbf X}_n$ be the set of complex numbers that are roots of some Littlewood polynomial with $n$ nonzero terms (and thus degree $n-1$). The 4-fold symmetry of Figure 1 comes from the fact that if $z \in {\mathbf X}_n$ so are $-z$ and $\overline{z}$. The set ${\mathbf X}_n$ is also invariant under the map $z \mapsto 1/z$, since if $z$ is the root of some Littlewood polynomial then $1/z$ is a root of the polynomial with coefficients listed in the reverse order. It turns out to be easier to study the set $${\mathbf X}= \bigcup_{n = 1}^\infty {\mathbf X}_n = \{ z \in {\mathbb C}| \; z \; \textrm{is the root of some Littlewood polynomial} \} .$$ If $n$ divides $m$ then ${\mathbf X}_n \subseteq {\mathbf X}_{m}$, so ${\mathbf X}_n$ for a highly divisible number $n$ can serve as an approximation to ${\mathbf X}$, and this is why we drew ${\mathbf X}_{24}$. Some general properties of ${\mathbf X}$ are understood. It is easy to show that ${\mathbf X}$ is contained in the annulus $1/2 < |z| < 2$. On the other hand, Thierry Bousch showed [@Bou1] that the closure of ${\mathbf X}$ contains the annulus $2^{-1/4} \le |z| \le 2^{1/4}$. This means that the holes near roots of unity visible in the sets ${\mathbf X}_d$ must eventually fill in as we take the union over all degrees $d$. More surprisingly, Bousch showed in 1993 that the closure $\overline{{\mathbf X}}$ is connected and locally path-connected [@Bou3]. It is worth comparing the work of Odlyzko and Poonen [@OP], who previously showed similar result for roots of polynomials whose coefficients are all $0$ or $1$. {width="3in"} 0.5em Figure 2. The region of ${\mathbf X}_{24}$ near the point $z = \frac{1}{2}e^{i/5}$. The big challenge is to understand the diverse, complicated and beautiful patterns that appear in different regions of the set ${\mathbf X}$. There are websites that let you explore and zoom into this set online [@C; @Egan; @V]. Different regions raise different questions. For example, what is creating the fractal patterns in Figure 2 and elsewhere? An anonymous contributor suggested a fascinating line of attack which was further developed by Greg Egan [@Egan]. Define two functions from the complex plane to itself, depending on a complex parameter $q$: $$f_{+q}(z) = 1 + q z , \qquad f_{-q}(z) = 1 - q z .$$ When $|q| < 1$ these are both contraction mappings, so by a theorem of Hutchinson [@H] there is a unique nonempty compact set $D_q \subseteq {\mathbb C}$ with $$D_q = f_{+q}(D_q) \cup f_{-q}(D_q) .$$ We call this set a **dragon**, or the to be specific. And it seems that *for $|q| < 1$, the portion of the set ${\mathbf X}$ in a small neighborhood of the point $q$ tends to look like a rotated version of $D_q$*. Figure 3 shows some examples. To precisely describe what is going on, much less prove it, would take real work. We invite the reader to try. A heuristic explanation is known, which can serve as a starting point [@Baez; @Egan]. Bousch [@Bou3] has also proved this related result: **Theorem 1**. *For $q \in {\mathbb C}$ with $|q| < 1$, we have $q \in \overline{{\mathbf X}}$ if and only if $0 \in D_q$. When this holds, the set $D_q$ is connected.* {height="2in"} {height="2in"} 0.1em {height="2in"} {height="2in"} em Figure 3. Top: the set ${\mathbf X}$ near $q = 0.594 + 0.254i$ at left, and the set $D_q$ at right.\ Bottom: the set ${\mathbf X}$ near $q = 0.375453 + 0.544825i$ at left, and the set $D_q$ at right. 10 J. C. Baez, The beauty of roots. Available at <http://math.ucr.edu/home/baez/roots>. T. Bousch, Paires de similitudes $Z \to SZ + 1, Z \to SZ - 1$, January 1988. Available at <https://www.imo.universite-paris-saclay.fr/~thierry.bousch/preprints/>. T. Bousch, Connexité locale et par chemins hölderiens pour les systèmes itérés de fonctions, March 1993. Available at <https://www.imo.universite-paris-saclay.fr/~thierry.bousch/preprints/>. J. D. Christensen, Plots of roots of polynomials with integer coefficients. Available at <http://jdc.math.uwo.ca/roots/>. G. Egan, Littlewood applet. Available at <http://www.gregegan.net/SCIENCE/Littlewood/Littlewood.html>. J. E. Hutchinson, Fractals and self similarity, *Indiana Univ. Math. J. ***30** (1981), 713--747. Also available at <https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf>. A. M. Odlyzko and B. Poonen, Zeros of polynomials with 0,1 coefficients, *L'Enseignement Math. ***39** (1993), 317--348. Also available at <http://dx.doi.org/10.5169/seals-60430>. R. Vanderbei, Roots of functions $F(z) = \sum_{j = 0}^n \alpha_j f_j(z)$ where $\alpha_j \in \{-1,1\}$. Available at <https://vanderbei.princeton.edu/WebGL/roots_PlusMinusOne.html>.

arxiv_math

--- abstract: | Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual and let $(\Omega, \mathbb{P})$ be a probability space. A bounded positive random linear operator on $L^1(M,\tau)$ is a map $\gamma\colon \Omega \times L^1(M,\tau)\to L^1(M,\tau)$ so that $\tau(\gamma_\omega(x)a)$ is measurable for all $x\in L^1(M,\tau)$ and $a\in M$, and $x\mapsto \gamma_\omega(x)$ is bounded, positive, and linear almost surely. Given an ergodic $T\in \text{Aut}(\Omega, \mathbb{P})$, we study quantum processes of the form $\gamma_{T^n\omega}\circ \gamma_{T^{n-1}\omega}\circ \cdots \circ \gamma_{T^m\omega}$ for $m,n\in \mathbb{Z}$. Using the Hennion metric introduced in [@jeff], we show that under reasonable assumptions such processes collapse to replacement channels exponentially fast almost surely. Of particular interest is the case when $\gamma_\omega$ is the predual of a normal positive linear map on $M$. As an example application, we study the clustering properties of normal states that are generated by such random linear operators. These results offer an infinite dimensional generalization of the theorems in [@jeff]. author: - "Brent Nelson[^1]  and Eric B. Roon[^2]" bibliography: - bibliography.bib title: | **Ergodic Quantum Processes\ on Finite von Neumann Algebras** --- # Introduction {#introduction .unnumbered} In quantum information theory, changes in a quantum system are modelled by quantum channels. In the Schrödinger picture, quantum channels take the form of trace-preserving completely positive maps on states. Dually, in the Heisenberg picture of quantum mechanics, observable quantities are represented by operators in a C\*-algebra (or von Neumann algebra), and quantum channels are then unital completely positive (normal) maps on that algebra. Physically, a quantum channel represents the dynamics of observables when the quantum system is *weakly coupled* to an environment (or reservoir) into which information about the evolving system can escape [@BreuerPetruccione]. Asymptotic properties of these dynamics play an important role throughout the mathematical physics literature. This dynamical interpretation of quantum channels (ucp maps) is not the only way of incorporating disorder into a quantum mechanical model. Recent years have seen an advance in quantum mechanical models that incorporate *large disorder*, by randomizing the quantum channel that evolves the system (see the non-exhaustive list [@BougronJoyePillet; @GGJN18; @jeff; @pathirana2023law]). It is natural, then, to analyze the asymptotic evolution of the system up to probability. That is, given a family of *random* quantum channels $\{\phi_n\colon n\in \mathbb{N}\}$, one seeks to analyze the asymptotic behavior of $$\phi_n\circ \cdots \circ\phi_1,$$ or in the dual Schrödinger picture one would instead consider $\phi_1^*\circ\cdots \circ \phi_n^*$. In the work [@jeff], the authors use ergodicity to obtain almost sure asymptotic estimates for non-independent homogeneously distributed quantum channels. More precisely, given an ergodic transformation $T$ on a probability space $(\Omega,\mathbb{P})$ and a random variable $\gamma_\omega$ valued in quantum channels on states, Movassagh and Schenker prove a number of results for the family $\{\gamma_{T^n\omega}\colon n\in \mathbb{N}\}$, including almost sure convergence of $\gamma_{\omega}\circ \gamma_{T\omega}\circ \cdots \circ \gamma_{T^n\omega}$ to a replacement channel (see [@jeff Theorem 2]). As an application of these results, they also show that certain matrix product states exhibit an almost sure clustering estimate (see [@jeff Theorem 3]). This work aims to provide an effective generalization of the work of [@jeff] to the case when the observables form a finite von Neumann algebra; that is, a von Neumann algebra admitting a faithful normal tracial state. A basic but infinite dimensional example is the hyperfinite $\mathrm{II}_1$ factor $\mathcal{R}$, which is physically relevant because it arises as the weak operator topology closure of the local algebra associated to a spin chain where the on-site observable algebras consist of the $2\times 2$ matrices (or more generally, the $n\times n$ matrices for any integer $n\geq 2$). Other sources of examples include representations of discrete groups, actions of groups on probability spaces, and measurable equivalence relations. The principal tool we use to carry out our analysis is the metric first introduced by Hennion in his 1997 paper [@Hennion]. Hennion was originally concerned with infinite products of random positive definite matrices and their convergence properties, and he used his metric as a means to study their rates of convergence. Movassagh and Schenker [@jeff] provide a finite dimensional noncommutative version of Hennion's results on the $n\times n$ complex matrices $\mathbb{M}_{n}$. Both papers rely on what we call the $m$-*quantity* of two positive matrices (or, in the case of [@Hennion], vectors) $X$ and $Y$ given by $m(X,Y) := \max \{ \lambda \in \mathbb{R} \colon \lambda Y \le X\}$. In the case that $M$ is finite with faithful normal trace $\tau$, we recall that the normal state space $S\subset L^1(M,\tau)$ can be canonically identified with the set of unit-trace, positive, closed operators affiliated to $M$ [@ap Chapter 7]. Thus, from the $m$-quantity one can form a bounded metric $d$ on the normal state space of $M$ via $$d(x,y) = \frac{1 - m(x,y)m(y,x)}{1+m(x,y)m(y,x)} \qquad\qquad x,y\in S\subset L^1(M,\tau).$$ We call this *Hennion's metric*, and we study its geometric properties in Section [2](#sec:dtop){reference-type="ref" reference="sec:dtop"}. In addition to extending known results to the infinite dimensional case, we exhibit new results about the disconnected components of $S$ (see Theorem [Theorem 24](#thm:geometry){reference-type="ref" reference="thm:geometry"}). To each positive linear map $\gamma$ on $L^1(M,\tau)$, one can induce a projective action on $S$ by $\gamma \cdot x := \frac{1}{\tau(\gamma(x))} \gamma(x)$. Provided that $\tau\circ \gamma$ is non-zero on $S$, one can associate to $\gamma$ the Lipschitz constant $$c(\gamma) := \sup_{\substack{x,y\in S\\x\neq y}} \frac{d(\gamma \cdot x, \gamma \cdot y)}{d(x,y)}.$$ When $c(\gamma)<1$, we say $\gamma$ is a *strict Hennion contraction*. Many properties of these maps are established in Section [3](#sec:cmap){reference-type="ref" reference="sec:cmap"}, including a complete classification (see Theorem [Theorem 38](#thm:strict_Hennion_projective_actions){reference-type="ref" reference="thm:strict_Hennion_projective_actions"}). The duality $L^1(M,\tau)^*\cong M$ implies strict Hennion contractions can also arise from normal positive linear maps on $M$, and indeed we determine precisely when this occurs in Subsection [3.2.1](#subsubsec:SHCfromnormal){reference-type="ref" reference="subsubsec:SHCfromnormal"}. In Section [4](#sec:EQP){reference-type="ref" reference="sec:EQP"}, we consider *ergodic quantum processes*: compositions of random quantum channels on $L^1(M,\tau)$ evolving under an ergodic transformation. Our first main result roughly says that such processes collapse to a replacement channel almost surely, provided there is a chance that the process eventually contracts in Hennion's metric: **Theorem 1** (Theorem [Theorem 60](#thm:right_convergence){reference-type="ref" reference="thm:right_convergence"}). *Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual, let $(\Omega, \mathbb{P})$ be a probability space equipped with ergodic $T\in \mathop{\mathrm{Aut}}(\Omega, \mathbb{P})$, and let $\gamma_\omega\colon L^1(M,\tau)\to L^1(M,\tau)$ be a bounded positive faithful random linear operator. Suppose that $$\mathbb{P}[\exists m \colon c(\gamma_{\omega}\circ \gamma_{T\omega }\circ \cdots \circ \gamma_{T^m \omega})<1]>0.$$ Then there is a state-valued random variable $X_\omega\in S$ so that for all $x\in S$ one has $$\lim_{m\to - \infty} \|\gamma_{\omega}\circ \gamma_{T\omega }\circ \cdots \circ \gamma_{T^m \omega} \cdot x - X_\omega\|_1 = 0$$ almost surely.* Separability of the predual $L^1(M,\tau)$ in the above theorem is used extensively to avoid measurability issues, and the probabilistic assumption is analogous to [@jeff Assumption 1] (see the discussion preceding Lemma [Lemma 52](#lem:a_not_so_mild_hypothesis){reference-type="ref" reference="lem:a_not_so_mild_hypothesis"}). The rate of convergence is controlled by $c(\gamma_{\omega}\circ \gamma_{T\omega }\circ \cdots \circ \gamma_{T^m \omega})$, and so it depends on $\omega\in \Omega$ but is independent of $x\in S$. To show that these Lipschitz constants tend to zero almost surely, we use Kingman's ergodic theorem (see Theorem [Theorem 10](#thm:kingman){reference-type="ref" reference="thm:kingman"}), and in fact the rate of convergence is exponentially fast almost surely (see Lemma [Lemma 55](#lem:randomconstproperties){reference-type="ref" reference="lem:randomconstproperties"}). Our second main result concerns ergodic quantum processes on $M$ rather than $L^1(M,\tau)$ and is essentially dual to Theorem [Theorem 1](#thmx:A){reference-type="ref" reference="thmx:A"}. Under similar assumptions, such processes also collapse to a replacement channel almost surely: **Theorem 2** (Theorem [Theorem 65](#thm:left_convergence_on_M){reference-type="ref" reference="thm:left_convergence_on_M"}). *Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual, let $(\Omega, \mathbb{P})$ be a probability space equipped with ergodic $T\in \mathop{\mathrm{Aut}}(\Omega, \mathbb{P})$, and let $\phi_\omega\colon M\to M$ be a normal unital positive random linear operator. Suppose that $$\mathbb{P}[\exists n \colon c((\phi_{T^n\omega} \circ \cdots \circ \phi_\omega)_*)<1]>0.$$ Then there is a state-valued random variable $Y_\omega \in S$ so that for all $a\in M$ one has $$\lim_{n\to \infty} \| \phi_{T^n\omega}\circ\cdots\cdot \circ\phi_{\omega}(a) - \tau(a Y_\omega)\|_\infty = 0.$$ almost surely.* In fact, we are able to prove the above theorem when $\phi_\omega(1)$ is only assumed to be almost surely invertible. In this case, one must instead consider a normalized process of the form $$a\mapsto \left(\phi_{T^n\omega}\circ\cdots\cdot \circ\phi_{\omega}(1)^{-\frac12} \right)\phi_{T^n\omega}\circ\cdots\cdot \circ\phi_{\omega}(a) \left(\phi_{T^n\omega}\circ\cdots\cdot \circ\phi_{\omega}(1)^{-\frac12} \right).$$ Theorems [Theorem 1](#thmx:A){reference-type="ref" reference="thmx:A"} and [Theorem 2](#thmx:B){reference-type="ref" reference="thmx:B"} recover the asymptotic results in [@jeff Theorem 1] for the finite dimensional algebra $(\mathbb{M}_n,\frac{1}{n}\text{Tr})$. Compared to [@jeff Theorem 1], the above theorems have left-right asymmetries that are a consequence of $L^1(M,\tau)\cong M$ *failing* in the infinite dimensional case. The deterministic versions of Theorems [Theorem 1](#thmx:A){reference-type="ref" reference="thmx:A"} and [Theorem 2](#thmx:B){reference-type="ref" reference="thmx:B"} can be compared with Yeadon's mean ergodic theorems for semifinite von Neumann algebras (see [@Yea77; @Yea80]). Indeed, suppose $\gamma\colon L^1(M,\tau)\to L^1(M,\tau)$ is a unital $\tau$-preserving positive linear map, and denote its dual map by $\phi:=\gamma^*$. Then [@Yea80 Theorem 4.2] implies that for every $x\in L^1(M,\tau)$ and $a\in M$ there exists $\hat{x}\in L^1(M,\tau)$ and $\hat{a}\in M$ satisfying $$\lim_{n\to\infty} \left\| \frac1n \sum_{k=0}^{n-1} \gamma^k(x) - \hat{x}\right\|_1=0 \qquad\qquad \text{ and }\qquad\qquad \lim_{n\to\infty} \left\| \frac1n \sum_{k=0}^{n-1} \phi^k(a) - \hat{a} \right\|_\infty =0.$$ If $\inf_{n\geq 0} c(\gamma^n)<1$, then Theorems [Theorem 1](#thmx:A){reference-type="ref" reference="thmx:A"} and [Theorem 2](#thmx:B){reference-type="ref" reference="thmx:B"} imply there exists $X\in S$ so that $\hat{x}=X$ for all $x\in S$ and $\hat{a} = \tau(aX)$ for all $a\in M$. Moreover, in this case the above convergences can be upgraded to $$\lim_{n\to\infty} \| \gamma^n(x) - X\|_1 =0 \qquad\qquad \text{ and }\qquad\qquad \lim_{n\to\infty} \| \phi^n(a) - \tau(aX)\|_\infty =0.$$ (Note that $\gamma(x)=\gamma\cdot x$ here since $\gamma$ is $\tau$-preserving.) Part of the work done in [@jeff] is to understand the clustering properties of certain matrix product states which are generated by a family of homogeneously-distributed random matrices. Physically, these correspond to *random* states on a spin chain. Therefore it is an interesting question to understand what happens in the case when the on-site algebras are infinite dimensional. Given a von Neumann algebra $M$, let $\{M_n\colon n\in \mathbb{Z}\}$ be isomorphic copies of $M$ and for a finite subset $\Lambda\subset \mathbb{Z}$ we denote $$M_{\Lambda}:=\overline{\bigotimes_{n\in \Lambda}} M_n.$$ Inclusions $\Lambda\subset \Pi$ of finite subsets of $\mathbb{Z}$ induce embeddings $M_\Lambda \subset M_\Pi$ so that one can consider the inductive limit *C\*-algebra* $$\mathscr{A}_{\mathbb{Z}}:= \varinjlim M_\Lambda,$$ which is called the *quasi-local algebra* associated to the spin chain with on-site algebras $M_n=M$ for all $n\in \mathbb{Z}$ (see [@BratteliRobinson1 Definition 2.6.3 and Example 4.2.12]). This algebra admits a canonical translation action $\mathbb{Z}\overset{\alpha}{\curvearrowright}\mathscr{A}_{\mathbb{Z}}$, and a state on $\mathscr{A}_\mathbb{Z}$ is said to be *locally normal* if its restriction to each $M_{\Lambda}$ is normal (see [@BratteliRobinson1 Definition 2.6.6] or [@HudsonMoody]). Taking inspiration from the classification of translation invariant states in [@FannesNachtergaeleWerner], we construct a class of random variables taking values in locally normal states which obey a kind of *translation covariance* property relative to the ergodic transformation. As an application of our previous main results, we establish the following clustering estimate for our class of translation covariant states: **Theorem 3** (Theorem [Theorem 71](#thm:Rclustering){reference-type="ref" reference="thm:Rclustering"}). *Let $(M,\tau_M)$ and $(W,\tau_W)$ be tracial von Neumann algebras with separable preduals, let $(\Omega, \mathbb{P})$ be a probability space equipped with ergodic $T\in \mathop{\mathrm{Aut}}(\Omega, \mathbb{P})$, and let $\mathcal{E}_\omega \colon M\bar\otimes W\to W$ be a normal unital positive random linear operator. Define $\phi_\omega(x):=\mathcal{E}_\omega(1\otimes x)$ and suppose that $$\mathbb{P}[\exists n \colon c((\phi_{T^n\omega} \circ \cdots \circ \phi_\omega)_*)<1]>0.$$ Then $\mathcal{E}_\omega$ determines (see Theorem [Theorem 70](#thm:thermo){reference-type="ref" reference="thm:thermo"}.(4)) a random locally normal state $\Psi_\omega$ on the quasi-local algebra $\mathscr{A}_{\mathbb{Z}}$ associated to the spin chain whose on-site algebras are isomorphic to $M$ that satisfies $$\Psi_\omega \circ \alpha_k = \Psi_{T^k \omega} \qquad\qquad \forall k\in \mathbb{Z}.$$ Moreover, there is a constant $\kappa \in (0,1)$ and a random variable $E_\omega\in [0,+\infty)$ such that $$|\Psi_\omega(ab) - \Psi_\omega(a) \Psi_\omega(b)| \le E_\omega \kappa^{\mathop{\mathrm{dist}}(\Lambda, \Pi)-1} \|a\|_\infty \|b\|_\infty \qquad \qquad \forall a\in M_{\Lambda},\ b\in M_{\Pi}$$ almost surely for finite subsets $\Lambda \subset (-\infty,-1)$ and $\Pi \subset [1,+\infty)$.* In words, we construct a family of *random* locally normal states that exhibit almost sure exponential clustering. Our approach is modelled after [@FannesNachtergaeleWerner Proposition 2.3 and 2.5] and gives $\Psi_\omega(a)$ as an almost sure operator norm limit of $$\mathcal{E}_{T^{-N}\omega}\circ(1\otimes \mathcal{E}_{T^{-N+1}\omega})\circ \cdots \circ (1\otimes \cdots \otimes 1\otimes \mathcal{E}_{T^N\omega})(a\otimes 1_W)$$ for local observables $a\in M_{[m,n]}\subset M_{[-N,N]}$ (see Theorem [Theorem 70](#thm:thermo){reference-type="ref" reference="thm:thermo"}.(4)). This result is an important step towards a better understanding of many-body systems subject to both information loss to a reservoir and large on-site disorder. # Acknowledgements {#acknowledgements .unnumbered} The authors would like to thank Jeffrey Schenker for bringing the question of an infinite dimensional generalization of [@jeff] to their attention, and for many productive conversations on the topic. The authors would also like to thank Stefaan Vaes for his helpful suggestions. EBR would like to thank BN and Jeffrey Schenker for their hospitality during his visits to Michigan State University in the summer of 2022 and spring of 2023, and for their continued mentoring in this area of study. EBR would also like to thank his advisor, Robert Sims, for his support and patience while EBR worked on this project, a helpful discussion of [@CarLa Section IV.1], and for lending EBR his hard copy of [@CarLa] . BN was supported by NSF grants DMS-1856683 and DMS-2247047. # Preliminaries {#sec:prelims} ## Finite von Neumann algebras {#sec:radonnikodym} Throughout, $M$ will be a finite von Neumann algebra equipped with a faithful normal tracial state $\tau$, and we will refer to the pair $(M,\tau)$ simply as a *tracial von Neumann algebra*. We will identify $M$ with its standard representation on $L^2(M,\tau)$; that is, the Gelfand--Naimark--Segal construction associated to $\tau$. For a closed subspace $\mathcal{H}\leq L^2(M,\tau)$ we write $[\mathcal{H}]$ for the projection onto $\mathcal{H}$. We denote by $J$ the conjugate linear map on $L^2(M,\tau)$ determined by $Jx=x^*$, and we recall that $JMJ=M'$ the commutant of $M$ in $B(L^2(M,\tau))$. One says that a closed, densely-defined operator $x$ on $L^2(M, \tau)$ is *affiliated* to $M$ if and only if for the polar decomposition $x = v|x|$ one has $v,1_{[0,t]}(|x|)\in M$ for all $t\geq 0$. In this case one writes $x\ \widetilde{\in}\:M$. In particular, given $x\ \widetilde{\in}\:M$, one has $x\in M$ if and only if $x$ is bounded. Recall that every operator affiliated to $M$ is *maximally extended* in the sense that if $x\ \widetilde{\in}\:M$ and $x\subset y$, then $x=y$. The set of affiliated operators $\widetilde{M}$ forms a $*$-algebra under the operations of closing linear combinations and products, with the usual adjoint as the involution (see [@ap Theorem 7.2.8]). We adopt the notation $\mathop{\mathrm{dom}}(x)$ for the domain of $x$ in $L^2(M,\tau)$. Equip $M$ with the norm $\|\cdot \|_1:= \tau( |\cdot |)$. The completion of $M$ with respect to $\|\cdot \|_1$ is written $L^1(M,\tau)$. It is nontrivially isometrically isomorphic to the predual $M_*$ (see [@ap Theorem 7.4.5]). This isomorphism is implemented via $x\mapsto \tau_x(\cdot) := \tau(x\,\cdot\,)$. One can also identify $L^1(M,\tau)$ with the set of $x\ \widetilde{\in}\:M$ such that $\|x\|_1:=\sup_{t>0} \tau(|x|1_{[0,t]}(|x|)) <\infty$, so we will frequently view elements of $L^1(M,\tau)$ as unbounded operators on $L^2(M,\tau)$. More generally, for $1\leq p <\infty$ one defines $L^p(M,\tau)$ as the set of $x\ \widetilde{\in}\:M$ such that $\|x\|_p:= \sup_{t>0} \tau(|x|^p 1_{[0,t]})^{1/p}<\infty$. For $p=2$ one obtains a Hilbert space that is naturally isomorphic to the standard representation of $M$, and so there is no conflict of notation. Moreover, as unbounded operators $L^2(M,\tau)$ corresponds to those $x\ \widetilde{\in}\:M$ with $1\in \mathop{\mathrm{dom}}(x)$ (see [@ap Theorem 7.3.2]). It follows that $M\subset L^2(M,\tau)$ is a core for such $x$. Note that for $x\in L^1(M,\tau)$, $|x|^{1/p}\in L^p(M,\tau)$ for all $1\leq p <\infty$, and so in particular $M\subset L^2(M,\tau)$ is a core for $|x|^{1/2}$. Recall that we say $M$ has a **separable predual** if $L^1(M,\tau)$ is separable as a Banach space. Examples include finite dimensional von Neumann algebras, the hyperfinite $\mathrm{II}_1$ factor, and group von Neumann algebras for countable discrete groups. There are several equivalent formulations of this which will be useful. **Theorem 1** ([@olesen Theorem 1.3.11]). *For a tracial von Neumann algebra $(M,\tau)$, the following are equivalent:* 1. *The predual $(L^1(M,\tau), \|\cdot\|_1)$ is a separable Banach space.* 2. *$(M,\tau)$ is separable in the $\sigma$-WOT.* 3. *$L^2(M,\tau)$ is a separable Hilbert space.* The following lemma, which is well-known to experts, will be useful in analyzing Hennion's metric $d$. **Lemma 2**. *For a tracial von Neumann algebra $(M,\tau)$, the following are equivalent:* 1. *$M$ is infinite dimensional.* 2. *There exists a family $\{p_n\colon n\in \mathbb N\}\subset M$ of non-zero pairwise orthogonal projections satisfying $\sum p_n =1$.* 3. *There exists a sequence $(p_n)_{n\in \mathbb N}\subset M$ of non-zero projections satisfying $\tau(p_n)\to 0$.* *Proof.* $(i)\Rightarrow (ii)$: Let $\mathcal{Z}(M)$ be the center of $M$, which is isomorphic to $L^\infty(X,\mu)$ for some probability space. If $(X,\mu)$ has any diffuse subsets or infinitely many inequivalent atoms, then we are done. Otherwise, $L^\infty(X,\mu)\cong \mathbb C^n$ for some $n\in \mathbb N$ and $M$ is a finite direct sum of factors. One of these factors is necessarily infinite dimensional (lest $M$ be finite dimensional) and hence contains such a family because it lacks minimal projections.\ $(ii)\Rightarrow(iii)$: Since $\sum \tau(p_n) =1 <\infty$, we have $\tau(p_n)\to 0$.\ $(iii)\Rightarrow (i)$: We proceed by contrapositive. If $M$ is finite dimensional, then it is necessarily a multimatrix algebra and $\tau$ is a convex combination of traces. It follows that the traces of projections in $M$ form a finite discrete subset of $[0,1]$, and in particular $0$ is isolated. ◻ ### Positivity Recall that $a\in M$ is *positive* if $a=b^*b$ for some $b\in M$; equivalently, if $a$ is positive semidefinite as an operator on $L^2(M,\tau)$. In this case, we write $a\geq 0$ and we denote the *positive cone* of $M$ by $M_+:=\{a\in M\colon a\geq 0\}$. The positive cone induces an ordering on the self-adjoint elements of $M$: for $a,b\in M_{s.a.}$ we write $a\leq b$ if $b-a\geq 0$. More generally, for unbounded self-adjoint operators $x,y$ on $L^2(M,\tau)$ we write $x\leq y$ when $y-x$ is positive semidefinite on $\mathop{\mathrm{dom}}(x)\cap \mathop{\mathrm{dom}}(y)$. Note that for affiliated operators $x,y\ \widetilde{\in}\:M$, $x\leq y$ is equivalent to saying the closure of $y-x$ is positive. In particular, $x\leq y$ and $y\leq x$ imply $x$ and $y$ agree on $\mathop{\mathrm{dom}}(x)\cap \mathop{\mathrm{dom}}(y)$ and therefore $x=y$. For each $1\leq p <\infty$, we denote $L^p(M,\tau)_+:=\{x\in L^p(M,\tau)\colon x\geq 0\}$, which we recall is the $\|\cdot\|_p$-closure of $M_+$. **Remark 3**. If $0\leq x\leq y$ and $y$ is bounded, then $x$ is necessarily bounded since $$\langle x\xi,\xi\rangle \leq \langle y\xi,\xi\rangle\leq \|y\|\|\xi\|^2$$ holds for $\xi$ in the dense subspace $\mathop{\mathrm{dom}}(y-x)$.$\hfill\blacksquare$ Also recall that we say a densely defined linear operator $x$ on $L^2(M,\tau)$ is *boundedly invertible* if there exists $b\in B(L^2(M,\tau))$ satisfying $xb=1$ and $bx\subset 1$, in which case we write $x^{-1}:=b$. Note that if $x\ \widetilde{\in}\:M$ then $x^{-1}\in M$. **Remark 4**. A positive densely defined operator $x$ on $L^2(M,\tau)$ is boundedly invertible if and only if $x\geq \delta$ for some scalar $\delta>0$, and in this case one can choose $\delta=\|x^{-1}\|^{-1}$. In particular, if $x,y\ \widetilde{\in}\:M$ satisfy $0\leq x \leq y$, then $x$ being boundedly invertible implies $y$ is boundedly invertible.$\hfill\blacksquare$ A linear map $\phi\colon M\to N$ between von Neumann algebras is said to be *positive* if $\phi(M_+)\subset N_+$, in which case we write $\phi\geq 0$. More generally, we say $\phi$ is *$n$-positive* for $n\in \mathbb{N}$ if the map $$\begin{aligned} \phi\otimes I_n \colon \quad M_n(M) &\to M_n(N)\\ (a_{ij})_{1\leq i,j\leq n} &\mapsto (\phi(a_{ij}))_{1\leq i,j\leq n}, \end{aligned}$$ is positive. We say $\phi$ is *completely positive* if it is $n$-positive for all $n\in \mathbb{N}$. Recall the following classical result. **Theorem 5** (Russo--Dye Theorem). *Let $A$ be a unital $C^*$-algebra and $\phi:A\to A$ be a linear mapping. Then $$\|\phi\|:= \sup_{x\colon \|x\|=1} \|\phi(x)\| = \sup_{u\in \mathcal{U}(A)} \|\phi(u)\|,$$ where $\mathcal{U}(A)$ denotes the unitary group of $A$. In particular, if $\phi$ is positive then $\|\phi\|=\|\phi(1)\|$.* Under the identification of $M_*$ with $L^1(M,\tau)$, the positive linear maps in $M_*$ correspond to $L^1(M,\tau)_+$. In fact, one has $$\begin{aligned} \label{eqn:positive_iff} \begin{split} x\in L^1(M,\tau)_+ \qquad &\Longleftrightarrow \qquad \tau(xa)\geq 0 \qquad \forall a\in M_+,\\ a\in M_+ \qquad &\Longleftrightarrow \qquad \tau(xa) \geq0 \qquad \forall x\in L^1(M,\tau)_+. \end{split} \end{aligned}$$ One says a linear map $\gamma\colon L^1(M,\tau)\to L^1(M,\tau)$ is *positive* if $\gamma(L^1(M,\tau)_+) \subset L^1(M,\tau)_+$, and writes $\gamma\geq 0$. More generally, one can define *$n$-postivitiy* and *complete positivity* for such maps by considering $\gamma\otimes I_n$ defined on $M_n(L^1(M,\tau))\cong L^1( M_n(M), \tau\otimes(\frac1n \text{Tr}))$. There is a well-known correspondence between normal linear maps $\phi\colon M\to M$ and bounded linear maps $\gamma\colon L^1(M,\tau)\to L^1(M,\tau)$. Indeed, using that $L^1(M,\tau)\cong (M,\text{weak}*)^*$, one has that $\phi_*\colon L^1(M,\tau)\to L^1(M,\tau)$ is a bounded linear map satisfying $$\tau(\phi_*(x) a) = \tau(x\phi(a)),$$ for all $x\in L^1(M,\tau)$ and $a\in M$. From Equation ([\[eqn:positive_iff\]](#eqn:positive_iff){reference-type="ref" reference="eqn:positive_iff"}), it follows that $\phi_*$ is positive if and only if $\phi$ is positive. Similarly for $n$-positivity and complete positivity. Conversely, given a bounded linear map $\gamma$ on $L^1(M,\tau)$, using $M\cong L^1(M,\tau)^*$ one has that $\gamma^*\colon M\to M$ is a normal linear map satisfying $$\tau( x \gamma^*(a)) = \tau( \gamma(x) a),$$ for all $x\in L^1(M,\tau)$ and $a\in M$. The positivity (resp. $n$-positivity or complete positivity) of $\gamma^*$ again follows from that of $\gamma$ via Equation ([\[eqn:positive_iff\]](#eqn:positive_iff){reference-type="ref" reference="eqn:positive_iff"}). For future reference, we record these observations in Lemma [Lemma 6](#lem:normalbddextension){reference-type="ref" reference="lem:normalbddextension"} below. Toward refining the above correspondence, we recall a bit more terminology. We say a positive linear map $\phi\colon M\to M$ is *$\tau$-bounded* if there exists a constant $c>0$ so that $\tau(\phi(a)) \leq c\tau(a)$ for all $a\in M_+$. We say a positive linear map $\gamma\colon L^1(M,\tau)\to L^1(M,\tau)$ is *$M$-preserving* if $\gamma(M)\subset M$. Since each element of $M$ can be decomposed as a linear combination of four positive elements, this is equivalent to $\gamma(M_+)\subset M_+$. Using $a\leq \|a\| 1$ for all $a\in M_+$, this is further equivalent to $\gamma(1)\in M_+$ by Remark [Remark 3](#rmk:boundedimpliesbounded){reference-type="ref" reference="rmk:boundedimpliesbounded"}. Observe that if $\phi\colon M\to M$ is normal, positive, and $\tau$-bounded with constant $c>0$, then $$\tau((c-\phi_*(1)) a) = c \tau(a) - \tau(\phi(a)) \geq 0$$ for all $a\in M_+$ so that $\phi_*(1)\leq c$ by Equation ([\[eqn:positive_iff\]](#eqn:positive_iff){reference-type="ref" reference="eqn:positive_iff"}). Thus $\phi_*$ is $M$-preserving. Conversely, if $\gamma\colon L^1(M,\tau)\to L^1(M,\tau)$ is $M$-preserving, bounded, and positive, then for $a\in M_+$ one has $$\tau(\gamma^*(a)) = \tau(\gamma(1) a) \leq \|\gamma(1)\| \tau(a) = \|\gamma\| \tau(a).$$ Thus $\gamma^*$ is $\tau$-bounded with constant $\|\gamma\|$. Thus the correspondence from before restricts to a correspondence between $\tau$-bounded normal positive linear maps on $M$ and $M$-preserving bounded positive linear maps on $L^1(M,\tau)$. We also record this in Lemma [Lemma 6](#lem:normalbddextension){reference-type="ref" reference="lem:normalbddextension"} below. Finally, we note that for $\tau$-bounded normal positive linear maps $\phi$ on $M$ (resp. $M$-preserving bounded positive linear maps $\gamma$ on $L^1(M,\tau)$) there is another correspondence given by extending (resp. restricting) the maps. Indeed, for $a\in M$ let $\phi(a)=v|\phi(a)|$ be the polar decomposition. Then one has $$\| \phi(a)\|_1 = |\tau( v^* \phi(a))| = |\tau( \phi_*(v^*) a)| \leq \| \phi_*(v^*)\| \|a\|_1,$$ where we have used that $\phi_*$ is $M$-preserving. It follows that $\phi$ admits a unique bounded linear extension $\phi|^{L^1(M,\tau)}$ to $L^1(M,\tau)$. Since $L^1(M,\tau)_+ = \overline{M_+}^{\|\cdot\|_1}$ one further has that the extension is positive, and it is $M$-preserving since $\phi(M)\subset M$. Conversely, if $\gamma$ is an $M$-preserving bounded positive linear map on $L^1(M,\tau)$, then $\gamma|_M$ defines a positive linear map on $M$. For $a\in M_+$, one has $$\tau(\gamma(a)) = \| \gamma(a)\|_1 \leq \|\gamma\| \|a\|_1 = \|\gamma\| \tau(a)$$ so that $\gamma|_M$ is $\tau$-bounded and normal (see [@ap Proposition 2.5.11]). This is also recorded in the following lemma as well as an interaction between the above two correspondences (whose proof is left to the reader). **Lemma 6**. *Let $(M,\tau)$ be a tracial von Neumann algebra. There is a one-to-one correspondence between normal positive (resp. completely positive) linear maps $\phi$ on $M$ and bounded positive (resp. completely positive) linear maps $\phi_*$ on $L^1(M,\tau)$ determined by $$\tau(\phi_*(x) a)=\tau(x\phi(a)) \qquad\qquad x\in L^1(M,\tau),\ a\in M.$$ This correspondence restricts to a one-to-one correspondence between $\tau$-bounded normal positive (resp. completely positive) linear maps on $M$ and $M$-preserving bounded positive (resp. completely positive) linear maps on $L^1(M,\tau)$. The former maps $\phi$ also admit unique extensions $\phi|^{L^1(M,\tau)}$ to $L^1(M,\tau)$ that are $M$-preserving bounded and positive (resp. completely positive), and the latter maps $\phi_*$ also have restrictions $\phi_*|_M$ to $M$ that are $\tau$-bounded normal and positive (resp. completely positive). In this case, one has $(\phi_*|_M)_*= \phi|^{L^1(M,\tau)}$.* As with the operators themselves, for linear maps $\phi,\psi\colon M\to M$ we write $\phi\leq \psi$ if $\psi- \phi\geq 0$. Similarly, for linear maps $\gamma,\rho\colon L^1(M,\tau)\to L^1(M,\tau)$ we write $\gamma\leq \rho$ if $\rho - \gamma\geq 0$. **Lemma 7**. *Let $\alpha,\beta \colon M\to M$ be positive normal linear maps. Then $\alpha \leq \beta$ if and only if $\alpha_*\leq \beta_*$.* *Proof.* For $x\in L^1(M,\tau)_+$ and $a\in M_+$ we have $$\tau(x (\beta - \alpha)(a)) = \tau((\beta_* - \alpha_*)(x) a).$$ So $\alpha(a)\leq \beta(a)$ for all $a\in M_+$ if and only if the above quantity is non-negative for all $x\in L^1(M,\tau)_+$ and $a\in M_+$, which is in turn equivalent $\alpha_*(x) \leq \beta_*(x)$ for all $x\in L^1(M,\tau)_+$. ◻ ## Probability theory Throughout $(\Omega,\mathcal{F}, \mathbb{P})$ will denote a probability space. An *automorphism* of $(\Omega,\mathcal{F}, \mathbb{P})$ is a bijection $T\colon \Omega\to \Omega$ such that $T$ and $T^{-1}$ are measurable and measure preserving. Denote the automorphisms of $(\Omega,\mathcal{F}, \mathbb{P})$ by $\text{Aut}(\Omega, \mathbb{P})$. One says $T\in \text{Aut}(\Omega,\mathbb{P})$ is *ergodic* if whenever $A\in\mathcal{F}$ satisfies $T^{-1}(A)\subset A$ then $\mathbb{P}[A]\in \{0,1\}$. Note that in this case $T^{-1}$ is also necessarily ergodic. Indeed, if $T(A)\subset A$, then for $$B:= \bigcup_{n=0}^\infty T^{-n}(A),$$ we have $T^{-1}(B)\subset B$ and $\displaystyle \mathbb{P}[B]=\lim_{n\to \infty} \mathbb{P}[T^{-n}(A)]$ by continuity from below. Thus if $T$ is measure preserving and ergodic, it follows that $\mathbb{P}[A]=\mathbb{P}[B]\in \{0,1\}$. **Remark 8**. Any $T\in \text{Aut}(\Omega, \mathbb{P})$ also induces an automorphism of the von Neumann algebra $L^\infty(\Omega,\mathbb{P})$ via precomposition: $f\mapsto f\circ T$. In fact, one can induce such an automorphism using bijections of the form $T\colon \Omega\setminus N_1\to \Omega\setminus N_2$ where $N_1,N_2$ are null sets and $T$ and $T^{-1}$ are measurable and measure preserving. Also, if $T_1, T_2$ are two such bijections which agree almost surely then they induce the same automorphism. Thus after identifying maps modulo null sets, these bijections form a group $\text{Aut}_0(\Omega,\mathbb{P})$ that embeds as a subgroup of $\text{Aut}(L^\infty(\Omega,\mathbb{P}))$. This embedding is a surjection when $(\Omega, \mathcal{F},\mathbb{P})$ is a *standard probability space* (see [@ap Section 3.3]); that is, if there exists a bijection $S\colon \Omega\setminus N_1\to \Omega_0\setminus N_2$ where $(\Omega_0,\mathcal{F}_0,\mathbb{P}_0)$ is the disjiont uniont of the Lebesgue measure on $[0,1]$ and countably many atoms, $N_1\subset \Omega$ and $N_2\subset \Omega_0$ are null sets, and $S$ and $S^{-1}$ are measurable and measure preserving.$\hfill\blacksquare$ ### Kingman's ergodic theorem We will be interested in multiplicative stochastic processes and their ergodic properties. So we present the following: **Lemma 9**. *Let $(\Omega, \mathbb{P})$ be a probability space equipped with an ergodic automorphism $T$ and suppose that $X_n:\Omega\to [0,1]$ is a submultiplicative stochastic process in the sense that $$X_{n+m}\le X_nX_m\circ T^n.$$ Let $\mathcal{F}_n := \sigma(X_0, \dots, X_n)$ for $n\ge 0$, denote the natural filtration taken with respect to $(X_n)_{n\ge0}$. Suppose that $\mathbb{P}[\exists n_*: X_{n_*}<1]>0$. Then $\nu(X):= \inf\{n: X_n <1\}$ is a finite almost-surely stopping time with respect to $(\mathcal{F}_n)_{n\ge 0}$.* *Proof.* Since $X_n$ is a decreasing sequence, observe that $$\{\omega: \nu(\omega)\le n\}=\{\omega:X_n <1\} \in \mathcal{F}_n.$$ Ergodicity of $T$ informs us that $\mathbb{P}[ \bigcup_{k=0}^\infty \{X_{n_*}\circ T^k <1\}]=1$ and there is a finite almost surely random variable $K(\omega)$ so that $X_{n_*+K}\le X_K X_{n_*}\circ T^{K} <1$. Whence $\nu \le n_* +K$ is finite almost surely. ◻ Furthermore, recall the Kingman ergodic theorem: **Theorem 10** (Kingman's Ergodic Theorem). *Let $(\Omega, \mathbb{P})$ be a probability space equipped with ergodic $T\in \mathop{\mathrm{Aut}}(\Omega,\mathbb{P})$. Assume that a stochastic process $\{X_n:\Omega \to \mathbb{R} \}_{n=1}^\infty$ satisfies:* 1. *$X_{n+m}(\omega) \le X_n(\omega) + X_{m}(T^n\omega)$ almost surely;* 2. *the positive part $\mathbb{E}[(X_n)^+] <\infty$ for all $n$.* *Then, $\displaystyle Z(\omega) := \lim_{n\to \infty} n^{-1} X_n(\omega)\in [-\infty, \infty)$ exists almost surely, and $\displaystyle Z = \inf_{n\in \mathbb N} n^{-1} \mathbb{E}[X_n]$; that is, $Z$ is constant almost surely.* This is a corollary to [@king Theorem 2]. See [@AvilaBochi Theorem 1] for a concise proof. One can find alternative formulations in [@CarLa; @steele; @lalley] among others. The observation underpinning our Lemma [Lemma 55](#lem:randomconstproperties){reference-type="ref" reference="lem:randomconstproperties"}, is that one may apply Theorem [Theorem 10](#thm:kingman){reference-type="ref" reference="thm:kingman"} to the sub-additive process $(\log X_n)_{n\ge0}$, when $(X_n)_{n\ge0}$ is sub-multiplicative. ### Random linear operators {#sec:RLO} We recall some general notions about the theory of random variables in a Banach space and random linear operators. The material in this section is largely based upon the treatment given in [@Bharucha-Reid Chapters 1 and 2], although an equally well-written treatment in the case $\mathfrak{X}$ is a Hilbert space may be found in [@Skorohod]. Let $\mathfrak{X}$ be a Banach space with norm $\|\cdot\|$. By $\mathfrak{B}$ we mean the $\sigma$-algebra generated by all the closed subsets of $\mathfrak{X}$. Let $(\Omega, \mathcal{F}, \mu)$ be a measure space. Given a finite collection of sets $A_1, A_2, \dots, A_n\in \mathcal{F}$ and $b_1, b_2, \dots, b_n \in \mathfrak{X}$, a function of the form $$\phi(\omega) = \sum_{j=1}^n b_j\, \chi_{A_j}(\omega)$$ will be called *simple function*. A *strongly measurable function* is one $f:\Omega \to \mathfrak{X}$ for which there is a sequence of simple functions $\phi_n$ so that $\displaystyle \lim_{n\to \infty} \|f(\omega)-\phi_n(\omega)\|= 0$ almost surely (see [@DiestelUhl Chapter II]). Writing $\mathfrak{X}^*$ for the Banach dual space of $\mathfrak{X}$, we say that $g:\Omega \to \mathfrak{X}$ is a *weakly measurable function* if for every $x^*\in \mathfrak{X}^*$, one has that $x^*\circ g (\omega)$ is a $\mathbb C$-valued Borel-measurable function. The following is originally due to Pettis [@pettis] but can be found in [@TakesakiI Proposition IV.7.2] or [@DiestelUhl Theorem 2]: **Theorem 11** ([@pettis Corollary 1.11]). *Let $\mathfrak{X}$ be a separable Banach space with Borel $\sigma$-algebra $\mathfrak{B}$, then a function $f:\Omega \to \mathfrak{X}$ is strongly measurable if and only if it is weakly measurable.* In the case that $\mu=\mathbb{P}$ is a probability measure, we shall refer to a strongly measurable function in $\mathfrak{X}$ as a *random variable in $\mathfrak{X}$*. A mapping $L:\Omega \times \mathfrak{X}\to \mathfrak{X}$ is an (everywhere-defined) *random linear operator* if: $\omega\mapsto L_{\omega}(x)$ is a random variable in $\mathfrak{X}$ for all $x\in \mathfrak{X}$; and $\mathbb{P}[L_\omega (\alpha x + \beta y) = \alpha L_\omega x + \beta L_\omega y]=1$ for all $x,y\in \mathfrak{X}$ and $\alpha, \beta \in \mathbb C$. Recall from [@Bharucha-Reid Definition 2.24] that a random linear operator $L$ is bounded if there is a random variable $M(\omega)$ so that $\mathbb{P}[ M(\omega)<\infty , \|L_\omega x\|\le M(\omega)\|x\| \text{ all }x\in \mathfrak{X}]=1$. In Sections [4](#sec:EQP){reference-type="ref" reference="sec:EQP"} and [5](#sec:FCS){reference-type="ref" reference="sec:FCS"} we will be concerned with dynamics arising from iterations of bounded random linear operators. It is not obvious that the composition of bounded random linear operators is, however, measurable. We follow the elegant proof given in Bharucha-Reid's [@Bharucha-Reid Theorem 2.14] below: **Theorem 12** ([@Bharucha-Reid Theorem 2.14]). *Let $\mathfrak{X}$ be a separable Banach space and $x_\omega$ a random variable in $\mathfrak{X}$. If $L$ is a bounded random linear operator then the function $$y_\omega := L_\omega x_\omega$$ is a random variable in $\mathfrak{X}$. In particular, the composition of bounded random linear operators is a bounded random linear operator.* *Proof.* Let $(\phi_n)_{n\in \mathbb{N}}$ be a sequence of simple functions approximating $x_\omega$. By reducing to a subsequence if necessary, we assume $\|\phi_n(\omega) - x_\omega\|\to 0$ almost surely. Let $E:=\{\phi_n(\omega)\colon n\in \mathbb{N},\ \omega\in \Omega\}$, which we note is a countable subset of $\mathfrak{X}$. For each $n\in \mathbb{N}$, define $y_n(\omega) := L_\omega \phi_n(\omega)$. Then, for any Borel subset $S\in \mathfrak{B}$, we obtain the decomposition $$[y_n\in S]= \bigcup_{b\in E} [L(b) \in S]\cap [\phi_n(\omega)=b],$$ thus demonstrating that $y_n$ is a sequence of random variables in $\mathfrak{X}$. Using that $L_\omega$ is almost surely bounded, we get that the following limit exists almost surely: $$y_\omega = \lim_{n\to \infty} y_n(\omega) = \lim_{n\to \infty} L_\omega \phi_n(\omega) = L_\omega x_\omega.$$ That composition of random linear operators is a random linear operator is now immediate. ◻ # Metric Geometry of the Normal State Space. {#sec:dtop} Let $(M,\tau)$ be a tracial von Neumann algebra. We write $$S:=\{x\in L^1(M)_+\colon \tau(x)=1\}.$$ Note that $S$ corresponds to the normal states on $M$. Following [@jeff] we introduce a non-standard metric $d$ on the set of normal states $S\subset L^1(M,\tau)_+$ and investigate its properties. We shall show that $d$ admits the same formula (Lemma [Lemma 18](#lem:dformula){reference-type="ref" reference="lem:dformula"}) as was shown in [@jeff Lemma 3.6], and use this to show that $(S,d)$ is complete and is finer than $S$ with the trace-metric (Theorem [Theorem 19](#thm:tracemetricinequality_and_completeness){reference-type="ref" reference="thm:tracemetricinequality_and_completeness"}). Unlike [@jeff], whenever $M$ is infinite dimensional $S$ admits two additional disconnected components corresponding to affiliated operators that are of separate interest and we investigate their properties in Theorem [Theorem 24](#thm:geometry){reference-type="ref" reference="thm:geometry"}. ## Hennion's Metric **Lemma 13**. *For $x,y\in L^1(M,\tau)_+\setminus\{0\}$ one has $\{\lambda\in \mathbb{R} \colon \lambda y \leq x\}=(-\infty,\lambda_0]$ for some $0\leq \lambda_0\leq \frac{\tau(x)}{\tau(y)}$.* *Proof.* Certainly $0\in \Lambda:= \{ \lambda \in \mathbb{R}: \lambda y \le x\}$, and if $\lambda\in \Lambda$ then $(-\infty, \lambda]\subset \Lambda$ since $t y\leq \lambda y\leq x$ for all $t\leq \lambda$. Note that if $\lambda y\le x$ then by applying $\tau$ we obtain $\lambda \le \tau(x)/ \tau(y)$. In particular, $\Lambda$ is bounded and we can find an increasing sequence $(\lambda_n)_{\in \mathbb N}\subset \Lambda$ converging to $\lambda_0:=\sup{\Lambda}\leq \tau(x)/\tau(y)$. Then for all $a\in M_+$ we have $$\tau(a(x-\lambda_0 y)) = \lim_{n\to\infty} \tau(a(x-\lambda_n y)) \geq 0.$$ Hence $x-\lambda_0 y\geq 0$ and $\lambda_0\in \Lambda$ so that $\Lambda=(-\infty,\lambda_0]$. ◻ **Definition 14**. Let $x,y\in L^1(M,\tau)_+$. We define their **m-quantity** to be the number $$m(x,y) := \max\{ \lambda\in \mathbb R : \lambda y \le x\}.\tag*{$\blacksquare$}$$ **Theorem 15** (Properties of $m(x,y)$). *Let $x,y,z\in L^1(M,\tau)_+\setminus\{0\}$.* 1. *[\[part:mrange\]]{#part:mrange label="part:mrange"} $0\le m(x,y) \le \frac{\tau(x)}{\tau(y)}$.* 2. *[\[part:mscaling\]]{#part:mscaling label="part:mscaling"} $m(a x, b y) = \frac{a}{b} m(x,y)$ for scalars $a,b>0$.* 3. *[\[part:mtriangle_ineq\]]{#part:mtriangle_ineq label="part:mtriangle_ineq"} $m(x,z)m(z,y) \le m(x,y)$.* 4. *$m(x,y)m(y,x) = 1$ if and only if $\frac{1}{\tau(x)} x=\frac{1}{\tau(y)}y$.* 5. *If $x\leq y$ then one has $$\begin{aligned} m(x,z) &\leq m(y,z) \\ m(z,x)&\geq m(z,y). \end{aligned}$$* 6. *[\[part:mstrictlypositive\]]{#part:mstrictlypositive label="part:mstrictlypositive"} $m(x,y)>0$ if and only if there exists non-zero $a\in M$ satisfying $y=x^{\frac12}a^*ax^{\frac12}$ with $a=a[x^{\frac12} M]$, in which case $m(x,y) = \|a\|^{-2}$.* 7. *[\[part:mzeroapprox\]]{#part:mzeroapprox label="part:mzeroapprox"} $m(x,y)=0$ if and only if there exists a sequence of vectors $(\xi_n)_{n\in \mathbb N}\subset \mathop{\mathrm{dom}}(x)\cap \mathop{\mathrm{dom}}(y)$ so that $\|x^{1/2}\xi_n\|_2 \to 0$ and $\|y^{1/2}\xi_n\|_2\equiv 1$.* 8. *[\[part:m_inf_formula\]]{#part:m_inf_formula label="part:m_inf_formula"} $$\begin{aligned} m(x,y) %&= \inf \left\{ \left(\frac{\|x^{1/2}\xi\|_2}{\|y^{1/2}\xi\|_2}\right)^2\colon \xi\in \dom(x)\cap \dom(y) \text{ such that }y^{1/2}\xi\neq 0\right\}\\ =\inf \left\{ \frac{\tau(xa)}{\tau(ya)}\colon a\in M_+,\ \tau(ya)> 0 \right\}. \end{aligned}$$* *Proof.* 1. This follows from Lemma [Lemma 13](#lem:mismax){reference-type="ref" reference="lem:mismax"}. 2. The inequality $m(ax,by) by \leq ax$ immediately implies $m(ax,by)\frac{b}{a}\leq m(x,y)$. One the other hand, $m(x,y)y\leq x$ is equivalent to $m(x,y)\frac{a}{b} by \leq ax$, which gives $m(x,y)\frac{a}{b}\leq m(ax,by)$. 3. We have $$m(x,z)m(z,y) y \leq m(x,z) z\leq x,$$ whence $m(x,z)m(z,y)\leq m(x,y)$. 4. If $\frac{1}{\tau(x)}x=\frac{1}{\tau(y)}y$, then $m(x,y)$ and $m(y,x)$ achieve their maximum values of $\frac{\tau(x)}{\tau(y)}$ and $\frac{\tau(y)}{\tau(x)}$, respectively, and hence their product gives one. On the other hand, if $m(x,y)m(y,x)=1$ then by part [\[part:mrange\]](#part:mrange){reference-type="ref" reference="part:mrange"} one necessarily has $m(x,y) =\frac{\tau(x)}{\tau(y)}$ and $m(y,x)=\frac{\tau(y)}{\tau(x)}$. Therefore $\frac{\tau(x)}{\tau(y)}y\leq x$ and $\frac{\tau(y)}{\tau(x)} x\le y$ so that $\frac{1}{\tau(x)} x= \frac{1}{\tau(y)} y$. 5. Suppose $x\leq y$. Then $m(x,z) z\leq x \leq y$ so that $m(x,z)\leq m(y,z)$. Similarly, $z\geq m(z,y) y \geq m(z,y)x$ so that $m(z,x) \geq m(z,y)$. 6. First suppose $y=x^{1/2}a^*ax^{1/2}$ for some non-zero $a\in M$. Then $y\leq \|a\|^2 x$, and so $m(x,y)\geq \|a\|^{-2}>0$. Conversely, if $m(x,y)>0$ then for any $b\in M$ we have $$\|b y^{1/2}\|_2^2 = \tau(b y b^*) \leq \frac{1}{m(x,y)} \tau(b x b^*) = \frac{1}{m(x,y)} \| b x^{1/2}\|_2^2.$$ Hence $bx^{1/2}\mapsto by^{1/2}$ extends to a bounded operator $T\colon \overline{M x^{1/2}} \to \overline{ M y^{1/2}}$ with $\|T\|\leq m(x,y)^{-1/2}$. Observe that $T$ is non-zero by virtue of $x^{1/2}$ and $y^{1/2}$ being non-zero. Let $p,q\in M$ be the support projections of $\tau_x$ and $\tau_y$, respectively, which we note satisfy $JpJ = [ M x^{1/2}]$ and $JqJ = [M y^{1/2}]$. Trivially extend $T$ to $L^2(M)$ so that $JqJ T JpJ=T$, and observe that for $b,c,d\in M$ we have $$\langle T b(cx^{1/2}), d y^{1/2}\rangle_2 = \langle bc y^{1/2}, dy^{1/2}\rangle_2 = \langle c y^{1/2}, b^* y^{1/2}\rangle_2 = \langle T cx^{1/2}, b^* d y^{1/2}\rangle_2 = \langle bT cx^{1/2}, dy^{1/2}\rangle_2.$$ It follows that $$0=JqJ (Tb - bT) JpJ = JqJ T JpJ b - b JqJ T JpJ = Tb - bT.$$ Hence $T\in M'$ and so is of the form $T=JaJ$ for some $a\in M$. Note that $a$ is non-zero since $T$ is non-zero. Additionally, we have for all $b\in M$ $$\tau(yb)=\langle b y^{1/2}, y^{1/2}\rangle_2 =\langle b Tx^{1/2}, T x^{1/2}\rangle_2= \langle b x^{1/2}a^*, x^{1/2}a^*\rangle_2 = \tau(x^{1/2}a^*a x^{1/2} b).$$ Since $y$ is determined by $\tau_y\in M_*$, it follows that $y=x^{\frac12}a^*ax^{\frac12}$. Note that $ap=J(TJpJ)J = JTJ=a$ and $p=J[Mx^{1/2}]J= [x^{1/2} M]$. We saw above that $m(x,y)\geq \|a\|^{-2}$ and $\|T\| \leq m(x,y)^{-1/2}$. Since $\|T\|=\|a\|$, this gives $m(x,y)=\|a\|^{-2}$. 7. Assume $m(x,y)=0$ so that the closed operator $x-\lambda y$ is not positive for any $\lambda>0$. Since $x-\lambda y$ is self-adjoint, this implies there exists some $\xi\in \mathop{\mathrm{dom}}(x-\lambda y)$ such that $$\langle(x-\lambda y)\xi, \xi\rangle_2 <0.$$ Because $\mathop{\mathrm{dom}}(x)\cap\mathop{\mathrm{dom}}(y)$ is a core for all $x-\lambda y$, we can in fact find $\xi\in \mathop{\mathrm{dom}}(x)\cap \mathop{\mathrm{dom}}(y)$ satisfying the above, and by scaling we can further assume $\|y^{1/2}\xi\|_2=1$. For each $n\in \mathbb N$, let $\xi_n$ be the vector obtained in this way for $\lambda = \frac1n$. Then $$\|x^{1/2}\xi_n\|_2^2 = \langle x\xi_n, \xi_n\rangle_2 < \frac{1}{n} \langle y\xi_n, \xi_n\rangle_2 = \frac1n \|y^{1/2} \xi_n \|_2^2 = \frac1n \to 0,$$ as desired. Conversely, if $m(x,y)>0$ then $y=x^{\frac12} a^* a x^{\frac12}$ for some $a\in M$ by part [\[part:mstrictlypositive\]](#part:mstrictlypositive){reference-type="ref" reference="part:mstrictlypositive"}. So for any $\xi \in \mathop{\mathrm{dom}}(x)\cap \mathop{\mathrm{dom}}(y)$ we have $$\| y^{1/2}\xi\|_2^2 = \langle y\xi, \xi\rangle_2 = \langle x^{\frac12}a^* a x^{\frac12} \xi, \xi\rangle_2 = \|ax^{1/2}\xi\|^2 \leq \|a\|^2 \|x^{1/2}\xi\|^2.$$ Consequently, for $(\xi_n)_{n\in \mathbb N}\subset \mathop{\mathrm{dom}}(x)\cap \mathop{\mathrm{dom}}(y)$ the condition $\|x^{1/2}\xi_n\|_2\to 0$ precludes $\|y^{1/2}\xi_n\|_2\equiv 1$. 8. If $m(x,y)=0$ then the equality follows from part [\[part:mzeroapprox\]](#part:mzeroapprox){reference-type="ref" reference="part:mzeroapprox"}. Indeed, for $\epsilon>0$ let $\xi\in \mathop{\mathrm{dom}}(x)\cap \mathop{\mathrm{dom}}(y)$ be such that $\|x^{1/2}\xi\|_2<\frac{\epsilon}{2}$ and $\|y^{1/2}\xi\|_2=1$. Since $M\subset L^2(M,\tau)$ is a core for $x^{1/2}$ and $y^{1/2}$, we can find $b\in M$ satisfying $\|x^{1/2}b\|_2 < \epsilon$ and $\|y^{1/2} b\|_2 > 1- \epsilon$. Letting $a=bb^*\in M_+$, we have $\tau(ya)=\|y^{1/2}b\|_2^2 > 1-\epsilon>0$ and $$\frac{\tau(xa)}{\tau(ya)} = \frac{\|x^{1/2}b\|_2^2}{\|y^{1/2}b\|_2^2} < \frac{\epsilon^2}{(1-\epsilon)^2}.$$ Thus the infinimum is also zero. Now suppose, $m(x,y)>0$, then $y=x^{\frac12}c^*c x^{\frac12}$ for some non-zero $c\in M$ with $c=c[x^{\frac12}M]$ by (4). Moreover, $$m(x,y) = \frac{1}{\|c\|^2} = \frac{1}{\sup\{\|c\xi\|_2/\|\xi\|_2\colon \xi\in L^2(M)\setminus\{0\} \}^2} = \inf\left\{ \frac{\|\xi\|_2^2}{\|c\xi\|_2^2} \colon \xi\in L^2(M),\ \|c\xi\|_2> 0\right\}.$$ The condition $c=c[x^{\frac12}M]$ implies that in the infimum one can restrict to $\xi = x^{1/2}\eta$ for $\eta\in\mathop{\mathrm{dom}}(x)$ with $\|cx^{1/2}\eta\|_2> 0$. Since $M\subset L^2(M)$ is a core for $x^{1/2}$, we can further restrict to $\xi= x^{1/2} b$ for $b\in M$ with $\|cx^{1/2}b\|_2>0$. Using $\|x^{1/2}b\|_2^2 = \tau(xbb^*)$ and $\|cx^{1/2}b\|_2^2 = \|y^{1/2}b\|_2^2 = \tau(ybb^*)$, we obtain the claimed equality.  ◻ **Definition 16**. For $x,y\in S$, let $$d(x,y) = \frac{1 - m(x,y)m(y,x)}{1+m(x,y)m(y,x)}.$$ **Theorem 17**. *The function $d$ forms a metric on $S$ such that $\mathop{\mathrm{diam}}_d(S) =1$ when $M\neq \mathbb C$.* *Proof.* The proof that $d$ is a metric is the same as in [@jeff]. Note that $d$ is valued in $[0,1]$ since $m$ is, and so $\mathop{\mathrm{diam}}_d(S)\le 1$. Also for $M\neq \mathbb C$ there exists a non-trivial projection $p\in M$, and one has $\lambda 1 \not\leq \frac{1}{\tau(p)}p$ for all $\lambda >0$. Hence $m(\frac{1}{\tau(p)} p, 1)=0$ and so $d(\frac{1}{\tau(p)} p, 1)=1$. ◻ The following lemma provides some alternate formulas for the Hennion metric $d$ that are more geometric in nature. **Lemma 18**. *Let $x,y\in S$ be distinct. Then $$\begin{aligned} \label{eqn:dformula} d(x,y) = \frac{t_+-t_-}{t_-+t_+-2t_-t_+} , \end{aligned}$$ where $$\begin{aligned} t_+&:= \sup\{t\in\mathbb{R}: tx+(1-t)y \in S\} \in \left[1,\frac{2}{\|x-y\|_1}\right],\\ t_- &:= \inf \{t\in \mathbb{R}: tx+(1-t)y \in S\} \in \left[\frac{-2}{\|x-y\|_1},0\right]. \end{aligned}$$ Equivalently, if $A_{\pm}=t_{\pm}x + (1-t_{\pm})y$ are the extreme points of the convex set $\{tx+(1-t)y\colon t\in \mathbb{R}\}\cap S$ then $$\begin{aligned} \label{eqn:eqn:conversedformula} d(x,y) = \left|\frac{r(1-s) - (1-r)s}{r(1-s) + (1-r)s}\right| = \frac{|r-s|}{r+s - 2rs}, \end{aligned}$$ where $r,s\in [0,1]$ are determined by $x=rA_- + (1-r)A_+$ and $y=sA_- + (1-s) A_+$.* *Proof.* Since $S$ is convex we immediately have $t_+\geq 1$ and $t_-\leq 0$. To see their other bounds, let $x-y=v|x-y|$ be the polar decomposition. Since $v^*\in (M)_1$, if $tx+(1-ty)\in S$ then we have $$\begin{aligned} |t|\|x-y\|_1 &= | \tau(tv^*(x-y))| = |\tau(v^*[tx+(1-t)y]) - \tau(v^*y)| \leq 2. \end{aligned}$$ Hence $t_+\leq 2/\|x-y\|_1$ and $t_- \geq -2/\|x-y\|_1$. Consider $A_+:= t_+ x + (1-t_+)y$ and $A_-:= t_-x + (1-t_-) y$, which both belong to $S$ since it is $\|\cdot\|_1$-closed. Additionally, $A_{\pm}$ are the extreme points of the $\|\cdot\|_1$-compact convex set $\{tx + (1-t)y\colon t_-\leq t \leq t_+\}$, and so $x=rA_- + (1-r) A_+$ and $y=sA_- + (1-s)A_+$ for some $r,s\in [0,1]$. In fact, one can explicitly solve a linear system as in [@jeff] to show that $$\begin{aligned} r&= \frac{t_+-1}{t_+-t_-} & 1-r&= \frac{1-t_-}{t_+-t_-} \\ s&= \frac{t_+}{t_+ - t_-} & 1-s&=\frac{-t_-}{t_+-t_-}. \end{aligned}$$ We claim that $m(A_+,A_-)=0$ and $m(A_-, A_+)=0$. Indeed, if there existed $\lambda >0$ so that $\lambda A_- \le A_+$ then we would further have $\lambda A_- \leq A_+ +\lambda x$. Hence $$0\leq A_+ + \lambda(x- A_-)= (t_+ + \lambda[1- t_-])x + (1- (t_+ + \lambda[1- t_-]))y,$$ and $t_+ +\lambda[1-t_-] > t_+$ contradicts the supremacy of $t_+$. Thus we must have $m(A_+,A_-)=0$, and a similar argument using $t_-$ shows $m(A_-,A_+)=0$. Now, $m(A_+,A_-)=0$ implies by Theorem [Theorem 15](#thm:mproperties){reference-type="ref" reference="thm:mproperties"}.[\[part:mzeroapprox\]](#part:mzeroapprox){reference-type="ref" reference="part:mzeroapprox"} that there is $(\xi_n)_{n\in \mathbb N}\subset \mathop{\mathrm{dom}}(x) \cap \mathop{\mathrm{dom}}(y)$ so that $\|A^{1/2}_-\xi_n\|_2 \equiv 1$ while $\|A^{1/2}_+\xi_n\|_2 \to 0$ as $n\to \infty$. Since $$\begin{aligned} \label{eqn:A_plus_minus_inequality} m(x,y)(sA_- + (1-s) A_+)= m(x,y)y \leq x = r A_- + (1-r) A_+, \end{aligned}$$ for all $\epsilon >0$ there is $N\in \mathbb N$ so that for all $n\ge N$ one has $$m(x,y) s = m(x,y) \langle s A_- \xi_n, \xi_n\rangle \leq \langle rA_-\xi_n,\xi_n\rangle + \epsilon = r + \epsilon.$$ Letting $\epsilon\to 0$ we see that $m(x,y)\leq \frac{r}{s}$. A similar argument using $m(A_-,A_+)=0$ yields $$m(x,y) \le \min\left\{ \frac{r}{s}, \frac{1-r}{1-s} \right\}.$$ In fact, the above is an equality: using $y=sA_-+(1-s)A_+$ one has $$\min\left\{\frac{r}{s}, \frac{1-r}{1-s}\right\} y \leq r A_- + (1-r) A_+ = x.$$ Reversing the roles of $x$ and $y$ and using the resulting version of ([\[eqn:A_plus_minus_inequality\]](#eqn:A_plus_minus_inequality){reference-type="ref" reference="eqn:A_plus_minus_inequality"}) gives $$m(y,x) = \min\left\{ \frac{s}{r}, \frac{1-s}{1-r} \right\}.$$ Thus one has $$0\le m(x,y)m(y,x) =\min\left\{ \frac{r}{s}, \frac{1-r}{1-s} \right\}\min\left\{ \frac{s}{r}, \frac{1-s}{1-r} \right\} = \min \left\{ \frac{r(1-s)}{s (1-r)}, \frac{(1-r) s}{(1-s) r} \right\}.$$ One then explicitly calculates $$\begin{aligned} d(x,y) = \frac{1-m(x,y)m(y,x)}{1+m(x,y)m(y,x)} = \left| \frac{r(1-s) - (1-r)s}{r(1-s) + (1-r)s} \right| = \frac{t_+-t_-}{t_-+t_+-2t_+t_-}, \end{aligned}$$ as claimed. ◻ **Theorem 19**. *One has for all $x,y\in S$ $$\label{eqn:dinequality} \frac{1}{2}\|x-y\|_1 \le d(x,y).$$ Furthermore, $(S,d)$ is a complete metric space.* *Proof.* Let $A_{\pm}\in S$ and $r,s\in[0,1]$ be as in Lemma [Lemma 18](#lem:dformula){reference-type="ref" reference="lem:dformula"}. Note that $$r-s = (1-s) - (1-r) = r(1-s) - (1-r)s,$$ and $r(1-s) + (1-r)s\leq 1$. Using these observations we have $$\|x-y\|_1 = \|(r - s)A_- + ((1-r) - (1-s))A_+\|_1 = |r-s| \|A_- - A_+\|_1 \leq 2 |r-s| \leq 2 \left|\frac{r(1-s) - (1-r)s}{r(1-s) + (1-r) s}\right|,$$ which equals $2d(x,y)$ by Equation ([\[eqn:eqn:conversedformula\]](#eqn:eqn:conversedformula){reference-type="ref" reference="eqn:eqn:conversedformula"}). To see that $(S,d)$ is complete, let $(x_n)_{n\in \mathbb N}\subset S$ be a Cauchy sequence with respect to $d$. The first part of the proof implies $(x_n)_{n\in \mathbb N}$ is also Cauchy with respect to $\|\cdot\|_1$ and hence converges to some $x\in S$ with respect to $\|\cdot\|_1$. In particular, one has $\tau(x_na)\to \tau(xa)$ for all $a\in M_+$. Given $\epsilon>0$, set $\eta:=(1-\epsilon)/(1+\epsilon)$. Let $N\in \mathbb N$ be such that for $m,n\geq N$ one has $d(x_m,x_n)<(1-\eta^{1/4})/(1+\eta^{1/4})$, which implies $$\min\{m(x_m,x_n),m(x_n,x_m)\} \geq m(x_m,x_n)m(x_n,x_m)>\eta^{1/4}.$$ Fix $n\geq N$. For $a\in M_+$ with $\tau(xa)>0$, let $m\geq N$ be large enough so that $|\tau(x_m a) - \tau(xa)| < (1-\eta^{1/4})\tau(xa)$ and $\tau(x_m a)\neq 0$. Then using Theorem [Theorem 15](#thm:mproperties){reference-type="ref" reference="thm:mproperties"}.[\[part:m_inf_formula\]](#part:m_inf_formula){reference-type="ref" reference="part:m_inf_formula"} one has $$\frac{\tau(x_na)}{\tau(xa)} = \frac{\tau(x_na)}{\tau(x_ma)}\frac{\tau(x_ma)}{\tau(xa)} \geq m(x_n,x_m) \frac{\tau(xa) - (1-\eta^{1/4})\tau(xa)}{\tau(xa)} = m(x_n,x_m) \eta^{1/4}> \eta^{1/2}.$$ Taking an infimum over $a\in M_+$ with $\tau(xa)>0$ yields $m(x_n,x)\geq \eta^{1/2}$. Next, for $a\in M_+$ with $\tau(x_na)>0$ let $m\geq N$ be large enough so that $|\tau(x_m a) - \tau(xa)| < (\eta^{-1/4} -1)\tau(xa)$ (and note $\tau(x_ma)\neq 0$ lest $m(x_m,x_n)=0$ and $d(x_m,x_n)=1$). Then $$\frac{\tau(xa)}{\tau(x_n a)} = \frac{\tau(x_ma)}{\tau(x_n a)}\frac{\tau(xa)}{\tau(x_m a)} \geq m(x_m,x_n) \frac{\tau(xa)}{\tau(xa) + (\eta^{-1/4} -1)\tau(xa)} = m(x_m,x_n) \eta^{1/4} > \eta^{1/2}.$$ Taking an infimum gives $m(x,x_n) \geq \eta^{1/2}$. Altogether this gives $d(x_n,x) \leq (1-\eta)/(1+\eta)=\epsilon$, so that $x_n\to x$ with respect to the metric $d$. ◻ **Remark 20**. Superficially, there is striking visual resemblance between Inequality ([\[eqn:dinequality\]](#eqn:dinequality){reference-type="ref" reference="eqn:dinequality"}) and the famous Pinsker's inequality [@HiaiOhyaTsukuda Theorem 3.1] for the relative entropy: $\frac{1}{2}\|\phi - \psi\|_1^2 \le S(\phi|\psi)$. Moreover the norm of the Radon-Nikodym derivatives of states (when defined) satisfies $\|\frac{d\phi}{d\psi}\|^2=\inf\{ \lambda: \psi \le \lambda \phi\}$ [@araki Theorem 12], [@hiai Appendix 7] which is inversely proportional to $m( \frac{d\phi}{d\tau}, \frac{d\psi}{d\tau})$. In light of Inequality ([\[eqn:dinequality\]](#eqn:dinequality){reference-type="ref" reference="eqn:dinequality"}), it is natural to ask if the metric $d$ is equivalent to the metric induced by $\|\cdot\|_1$. This is always false when $M$ is infinite dimensional (see Remark [Remark 25](#rmk:d_not_homeo_to_1-norm){reference-type="ref" reference="rmk:d_not_homeo_to_1-norm"}), but nevertheless the Hennion metric is jointly lower semicontinuous with respect to $\|\cdot\|_1$: **Theorem 21**. *Suppose $(x_n)_{n\in \mathbb N}, (y_n)_{n\in \mathbb N}\subset S$ satisfy $$\lim_{n\to\infty} \|x_n - x\|_1 =0 \qquad \text{ and } \qquad \lim_{n\to\infty} \|y_n - y\|_1=0,$$ for some $x,y\in S$. Then $$\limsup_{n\to\infty} m(x_n,y_n) \leq m(x,y),$$ and $$\liminf_{n\to\infty} d(x_n,y_n) \geq d(x,y).$$* *Proof.* The definition of $d$ implies its joint lower semicontinuity will follow from the joint upper semicontinuity of $m$. To see the latter, let $\eta> m(x,y)$ and use Theorem [Theorem 15](#thm:mproperties){reference-type="ref" reference="thm:mproperties"}.[\[part:m_inf_formula\]](#part:m_inf_formula){reference-type="ref" reference="part:m_inf_formula"} to find $a\in M_+$ such that $$m(x,y)\leq \frac{\tau(xa)}{\tau(ya)} < \eta.$$ Then we can find $N\in \mathbb N$ so that $$m(x_n,y_n) \leq \frac{\tau(x_na)}{\tau(y_na)} <\eta$$ holds for all $n\geq N$. Thus $$\limsup_{n\to\infty} m(x_n,y_n) \leq \eta,$$ and letting $\eta\to m(x,y)$ completes the proof. ◻ **Remark 22**. This is another striking similarity between the Hennion metric and the relative entropy. We recall that the relative entropy is jointly lower semicontinuous in its arguments [@Kosaki Theorem 4.1]. ## On components of $S$ {#subsec:Components} We demonstrate below that $S$ is the disjoint union of at least four connected components. Recall that we say a densely defined linear operator $x$ on $L^2(M,\tau)$ is *boundedly invertible* if there exists $b\in B(L^2(M,\tau))$ satisfying $xb=1$ and $bx\subset 1$, in which case we write $x^{-1}:=b$. Note that if $x\in L^1(M,\tau)$ (and hence affiliated with $M$), then necessarily $x^{-1}\in M$. **Notation 23**. We let $S^\times$ denote the set of operators $x\in S$ that are boundedly invertible, and we denote $S^\circ:=S\setminus S^\times$. We also set the following notation: $$\begin{aligned} S^\times_b &:=S^\times \cap M & S^\times_u&:=S^\times \setminus M\\ S^\circ_b&:= S^\circ\cap M & S^\circ_u&:= S^\circ\setminus M. \end{aligned}$$ Observe that $$S=S^\times_b\sqcup S^\times_u\sqcup S^\circ_b \sqcup S^\circ_u,$$ and $S^\times= S^\times_b \sqcup S^\times_u$ and $S^\circ = S^\circ_b \sqcup S^\circ_u$. We shall also write $S_b := S\cap M (= S^\times _b \sqcup S^\circ _b)$ and $S_u :=S\setminus M (= S^\times_u \sqcup S^\circ_u)$. Also note the sets $S^\times$, $S_b$, and $S_b^\times$ are convex.$\hfill\blacksquare$ **Theorem 24**. *Let $(M,\tau)$ be a tracial von Neumann algebra. We have the following:* 1. *Suppose $x,y\in S$ satisfy $d(x,y)<1$. Then, $x$ is bounded if and only if $y$ is bounded; and $x$ is boundedly invertible if and only if $y$ is boundedly invertible. In particular, $S_b^\times, S_u^\times, S_b^\circ, S_u^\circ$ are disjoint $d$-clopen sets that are distance one apart (provided they are non-empty).* 2. *For all $x,y\in S^\times_b$, one has $d(x,y)<1$.* 3. *If $M\neq \mathbb C$, then $\mathop{\mathrm{diam}}(S_b^\times)=1$ and $\mathop{\mathrm{diam}}(S_b^\circ)=1$, and the latter is achieved.* 4. *If $M$ is infinite dimensional, then $\mathop{\mathrm{diam}}(S_u^\times)=1$ and $\mathop{\mathrm{diam}}(S_u^\circ)=1$, and both are achieved.* 5. *If $M$ is infinite dimensional, $S_b^\times$ is not $d$-totally bounded.* *Proof.* 1. Notice that $d(x,y)<1$ implies that $m(x,y),m(y,x)>0$. We have $m(x,y) y \leq x \leq m(y,x)^{-1} y$, hence $x$ is bounded if and only if $y$ is bounded by Remark [Remark 3](#rmk:boundedimpliesbounded){reference-type="ref" reference="rmk:boundedimpliesbounded"}, and $x$ is boundedly invertible if and only if $y$ is boundedly invertible by Remark [Remark 4](#rmk:boundedlyinvert_implies_boundedlyinvert){reference-type="ref" reference="rmk:boundedlyinvert_implies_boundedlyinvert"}. Now, if $x,y\in S$ belong to distinct subsets $S_b^\times$, $S_b^\circ$, $S_u^\times$, or $S_u^\circ$, then the above implies $d(x,y)=1$. Consequently, one has $$S_b^\times = \bigcup_{x\in S_b^\times} B_d(x,\frac12),$$ and similarly for $S_b^\circ$, $S_u^\times$, and $S_u^\circ$. Thus each of these sets are open, and since they partition $S$ they are also closed. 2. Note that $m(x,y)\geq \frac{1}{\|y\|\|x^{-1}\|} >0$ (by Remark [Remark 4](#rmk:boundedlyinvert_implies_boundedlyinvert){reference-type="ref" reference="rmk:boundedlyinvert_implies_boundedlyinvert"}) and similarly $m(y,x)>0$. Hence $d(x,y)<1$. 3. Let $p\in M$ be a non-trivial projection, and for $\epsilon\geq 0$ let $$x_\epsilon:= \frac{1}{1+\epsilon}(p + \epsilon(1-p)) \qquad \qquad y_\epsilon:= \frac{1}{1+\epsilon}(\epsilon p + (1-p)).$$ For $\epsilon>0$, $x_\epsilon,y_\epsilon \in S_b^\times$ with $m(x_\epsilon, y_\epsilon)= m(y_\epsilon, x_\epsilon)=\epsilon$, and therefore $d(x_\epsilon, y_\epsilon)\to 1$ as $\epsilon \to 0$. For $\epsilon=0$, $x_0,y_0\in S_b^\circ$ with $m(x_0, y_0)=m(y_0,x_0)=0$, and therefore $d(x_0, y_0)=1$. 4. Since $M$ is infinite dimensional, Lemma [Lemma 2](#lem:infinite_dimensional){reference-type="ref" reference="lem:infinite_dimensional"} yields a family $\{p_n\colon n\in \mathbb N\}\subset M$ of non-zero pairwise orthogonal projections mutually orthogonal projections satisfying $\sum p_n =1$. Note that $\tau(p_n) \to 0$ as $n\to \infty$. Pick a subsequence $(p_{n_k})_{k\in \mathbb N}$ so that $\tau(p_{n_k})<(k2^k)^{-1}$ and $n_k$ is even. Let $A = \{n \in \mathbb N: n \neq n_k \text{ for all } k\}$. Set $x_k = kp_{n_k}$ and observe that $$\tau \left(\sum_{k=1}^\infty x_k\right) = \sum_{k=1}^\infty k\tau(p_{n_k}) < \sum_{k=1}^\infty \frac{1}{2^k} = 1,$$ while by pairwise orthogonality, $$\left \| \sum_{k=1}^d x_k\right\| = d.$$ Now, set $\alpha := \sum_{k=1}^\infty \tau(x_k)<1$ and $\delta := (1-\alpha) (\sum_{n\in A} \tau(p_n))^{-1}$. Note that $$\alpha > \sum_{k=1}^\infty \tau(p_{n_k}) = 1 - \sum_{n\in A} \tau(p_n)$$ implies $\delta<1$. Thus if we let $$x:= \sum_{k=1}^\infty x_k + \delta \sum_{n\in A} p_n,$$ then $x\in S_u^\times$ with $x\geq \delta$. Repeat the above construction but this time choosing the subsequence $(p_{m_k})_{k\in \mathbb N}$ so that $m_k$ is odd to obtain $y\in S_u^\times$ with $y\geq \delta$. Then for any $\lambda>0$, $\lambda y \leq x$ fails since multiplying by $p_{m_k}$ with $k> \frac{\delta}{\lambda}$ gives the contradiction $\lambda k p_{m_k} \leq \delta p_{m_k}$. Hence $d(x,y)=1$ and $\mathop{\mathrm{diam}}(S_u^\times)=1$ and is achieved. The same construction with $\delta=0$ shows $\mathop{\mathrm{diam}}(S_u^\circ)=1$ and is achieved. 5. Let $\{p_n\colon n\in \mathbb N\}\subset M$ be as in the previous part. Put $s_n = \sum_{k> n}p_k$ and observe that $\tau(s_n) \to 0$ as $n\to \infty$. Pick a subsequence $s_{n_k}$ so that $\tau(s_{n_k}) \le (2k)^{-1}$, and set $x_k = \frac12 \sum_{j=1}^{n_k} p_j + k s_{n_k}$. Note that $\alpha_k:= \frac{1}{\tau(x_k)}\in (1,2)$. Now, $(\alpha_k x_k)_{k\in \mathbb N}$ is a sequence of states that are bounded below by $\frac12$ and above by $2k$, and so $(\alpha_k x_k)_{k\in \mathbb N} \subset S_b^\times$. For $k>k'$ one has $m(x_k, x_{k'})\le \frac{1}{2k'}$ and $m(x_{k'}, x_k) \le \frac{k'}{k}$. Indeed, we see $m(x_k,x_{k'}) x_{k'} \le x_k$ demands $m(x_k,x_{k'}) k'\sum_{j=n_{k'}+1}^{n_k} p_j\le \frac12\sum_{j=n_{k'}+1}^{n_k} p_j$, while $m(x_{k'},x_k)x_k \le x_{k'}$ implies $m(x_{k'},x_k) k s_{n_k} \leq k' s_{n_k}$. Therefore by Theorem [Theorem 15](#thm:mproperties){reference-type="ref" reference="thm:mproperties"}.[\[part:mscaling\]](#part:mscaling){reference-type="ref" reference="part:mscaling"}, $$\begin{aligned} d(\alpha_k x_k, \alpha_{k'} x_{k'})&\ge \frac{1-\frac{k'}{k}\frac{1}{2k'}}{1 + \frac{k'}{k}\frac{1}{2k'}}\\ &= \frac{2k-1}{2k+1}\ge 1/3, \end{aligned}$$ uniformly in $k$. Thus $S_b^\times$ does not admit a finite cover by balls of radius $\frac16$ or less.  ◻ **Remark 25**. Note that unlike in the finite dimensional case (see [@jeff Lemma 3.9]), for infinite dimensional $M$ the $d$-topology and $\|\cdot\|_1$-topologies are not homeomorphic on $S^\times$ or even on $S^\times_b$. Indeed, let $(p_n)_{n\in \mathbb N}\subset M$ be as in Lemma [Lemma 2](#lem:infinite_dimensional){reference-type="ref" reference="lem:infinite_dimensional"}.(iii). Then for $0<\alpha<1$ and $$x_n:= \frac{1}{\alpha + (1-\alpha)\tau(p_n)}(p_n + \alpha(1-p_n)),$$ $x_n\to 1$ in $L^1(M,\tau)$ but it is not a Cauchy sequence with respect to $d$.$\hfill\blacksquare$ **Lemma 26**. *Let $(M,\tau)$ be a tracial von Neumann algebra with Hennion metric $d$. Let $(x_n)_{n\in \mathbb N}\subset S_b$ and $x\in S$.* 1. *If $d(x_n,x)\to 0$ then $x\in S_b$ and $\|x_n - x\|\to 0$.* 2. *Let $(x_n)_{n\in \mathbb N}\subset S_b^\times$ with $\|x_n -x\| \to 0$ and suppose $x\in S_b^\times$. Then, $d(x_n,x)\to 0$.* *Consequently, $(S_b^\times,d)$ and $(S_b^\times,\|\cdot\|)$ are homeomorphic.* *Proof.* 1. Suppose $(x_n)_{n\in \mathbb N}\subset S_b$ converges to some $x$ with respect to $d$. Note that necessarily $x\in S_b$ by Theorem [Theorem 24](#thm:geometry){reference-type="ref" reference="thm:geometry"}. Let $N_0\in \mathbb N$ be sufficiently large so that $d(x_n,x)<\frac{1}{2}$ for all $n\ge N_0$. Then, we see from the definition of $d$ that, $$\frac{1}{3} < m(x_n, x)m(x, x_n) \le \min\{m(x_n, x),m(x, x_n)\}$$ using the fact that $m(x,y)\le 1$ for states. In particular, this means that for any $n\ge N_0$, we have $x_n \le 3 x$ whence $\| x_n\| \le 3 \|x\|$. Let $R = \max_{1\le j \le N_0-1}\{ \|x_j\|, 3 \| x\|\}$ so that $(x_n)_{n\in\mathbb N}\subset (M)_R$. Now, for $1>\epsilon>0$ let $N\in \mathbb N$ be sufficiently large so that for $n\geq N$, we have $d(x_n, x)<\epsilon$. In particular, this implies that $$\frac{1-\epsilon}{1+\epsilon}\le m(x_n, x)m(x, x_n)\le \min\{ m(x_n,x), m(x,x_n) \}.$$ This yields the following inequalities $$\frac{1-\epsilon}{1+\epsilon} x \le x_n \qquad \text{ and }\qquad \frac{1-\epsilon}{1+\epsilon} x_n \le x,$$ for all $n \ge N$. Rearranging, we get $$x_n \le x + 2 \epsilon R \qquad \text{ and }\qquad x\leq x_n + 2\epsilon R,$$ for any $n\geq N$. Therefore $$-2\epsilon R = x-2\epsilon R - x \le x_n - x \le 2\epsilon R + x - x = 2\epsilon R,$$ whence for any $\xi \in L^2(M,\tau)$ with $\|\xi\|_2 =1$ we have $$|\langle (x_n-x) \xi, \xi \rangle | \le 2\epsilon R,$$ for any $n,k\ge n_0$. Taking the supremum over $\xi$ this yields $\|x_n - x\| < 2\epsilon R$. Hence $x_n\to x$ in operator norm. 2. Suppose $(x_n)_{n\in \mathbb N}\subset S_b^\times$ converges to $x\in S_b^\times$ with respect to the operator norm. Let $\epsilon>0$. Since $x\in S_b^\times$, there exists $\delta>0$ so that $x\geq \delta$. So for sufficiently large $n\in \mathbb N$ we have $$x_n \leq x + \|x_n-x\| < x + \epsilon\delta \leq (1+\epsilon)x,$$ and hence $m(x_n,x) \geq (1+\epsilon)^{-1}$. Next, for sufficiently large $n$ one has $$x_n \geq x - \|x-x_n\| \geq \frac{\delta}{2}.$$ Increasing $n$ if necessary, we then obtain $$x\leq x_n + \|x-x_n\| < x_n + \epsilon\frac{\delta}{2} \leq (1+\epsilon)x_n,$$ so that $m(x,x_n) \geq (1+\epsilon)^{-1}$. Thus $$\lim_{n\to\infty} d(x_n,x) \leq \frac{1-(1+\epsilon)^{-2}}{1+(1+\epsilon)^{-2}},$$ and letting $\epsilon\to 0$ completes the proof.  ◻ Part (2) of Lemma [Lemma 26](#thm:cauchytocauchy){reference-type="ref" reference="thm:cauchytocauchy"} fails when the limit is not boundedly invertible---even in the finite dimensional case---as the following example demonstrates. Moreover, this example also shows that the Hennion metric is *not* equivalent to metric induced by the operator norm on $S_b^\times$, despite inducing the same topology by Lemma [Lemma 26](#thm:cauchytocauchy){reference-type="ref" reference="thm:cauchytocauchy"}. **Example 27**. Let $M = \mathbb{M}_{2\times 2}$ be the algebra of $2\times 2$ matrices and consider the family of states (with respect to the normalized tracial state) $$X_\eta := \frac{2}{1+\eta} \begin{bmatrix} 1 & 0 \\ 0 & \eta \\ \end{bmatrix}.$$ As $\eta\to 0$, the state $X_\eta$ converges in operator norm to the re-scaled projection $P = \begin{bmatrix} 2 & 0\\ 0 & 0\end{bmatrix}$. However, for any $\eta' < \eta$, one can calculate that $$m(X_\eta, X_\eta')m(X_{\eta'}, X_\eta)= \frac{\eta'}{\eta}.$$ Therefore by choosing $\eta \ge 3\eta '$, one obtains $d(X_\eta, X_{\eta'}) \ge \frac12$, thus $(X_\eta)_{\eta>0}$ has no subsequences which are Cauchy with repsect to the Hennion metric. We also mention that because of [@jeff Lemma 3.3(4)], $d(X_\eta, P) \equiv 1$ for all $\eta>0$.$\hfill\blacksquare$ # Contraction Mappings on $S$ {#sec:cmap} In this section we will consider linear maps $\gamma$ on $L^1(M,\tau)$ and their contraction properties with respect to the Hennion metric. We completely characterize faithful maps that contract on $S$ in terms of an operator inequality (see Theorem [Theorem 38](#thm:strict_Hennion_projective_actions){reference-type="ref" reference="thm:strict_Hennion_projective_actions"}). Of particular interest for application are those $\gamma$ which arise as the preduals of normal positive maps on $M$ which are investigated in Section [3.2.1](#subsubsec:SHCfromnormal){reference-type="ref" reference="subsubsec:SHCfromnormal"}. ## The class of mappings and projective actions **Lemma 28**. *For a positive linear map $\gamma$ on $L^1(M,\tau)$, $\ker\gamma\cap S$ is open in the Hennion metric. If $\gamma$ is bounded, then this set is also closed.* *Proof.* For any $x\in \ker \gamma \cap S$, the open ball $B_d(x, 1)$ is contained in $\ker \gamma \cap S$. Indeed, since $y\in B_d(x,1)$ implies that $m(x,y)m(y,x)>0$, we then have $y \le \frac{1}{m(x,y)}x$ and applying $\gamma$ gives $y\in \ker \gamma \cap S$. If $\gamma$ is bounded, then $\ker\gamma\cap S$ is closed with respect to $\|\cdot\|_1$, and so Theorem [Theorem 19](#thm:tracemetricinequality_and_completeness){reference-type="ref" reference="thm:tracemetricinequality_and_completeness"} implies it is also closed in the Hennion metric. ◻ **Definition 29**. Let $\gamma$ be a positive map on $L^1(M,\tau)$. The **projective action** of $\gamma$ on $x\in S$ is the map $\gamma\, \cdot\, \colon S\setminus \ker \gamma \to S$ given by $$\gamma \cdot x = \frac{1}{\tau(\gamma(x))} \gamma(x).$$ **Remark 30**. Although the projective action is non-linear, it does preserve lines in the following sense. Given $x,y\in S\setminus \ker\gamma$, for any $t\in \mathbb{R}$ satisfying $tx+(1-t)y\in S\setminus \ker\gamma$ there exists $s\in \mathbb{R}$ so that $$\gamma\cdot (tx + (1-t)y) = s\gamma\cdot x + (1-s)\gamma\cdot y;$$ namely, $s:=\tau(\gamma(tx))\tau(\gamma(tx+(1-t)y))^{-1}$. In particular, if $t\in [0,1]$ then $s\in [0,1]$.$\hfill\blacksquare$ The following lemma implies projective actions of positive maps are always Lipschitz continuous with respect to the Hennion metric. We explore the associated Lipschitz constants in greater detail in Section [3.2](#sec:contraction_mappings){reference-type="ref" reference="sec:contraction_mappings"}. **Lemma 31**. *For a positive linear map $\gamma$ on $L^1(M,\tau)$, $$\label{eqn:dantitone}d(\gamma \cdot x, \gamma \cdot y) \le d(x,y).$$ for all $x,y\in S\setminus \ker\gamma$.* *Proof.* First observe that $m(\gamma\cdot x,\gamma\cdot y)m(\gamma\cdot y, \gamma\cdot x) = m(\gamma(x),\gamma(y))m(\gamma(y),\gamma(x))$ by Theorem [Theorem 15](#thm:mproperties){reference-type="ref" reference="thm:mproperties"}.[\[part:mscaling\]](#part:mscaling){reference-type="ref" reference="part:mscaling"}. Thus $$\begin{aligned} d(\gamma \cdot x, \gamma \cdot y) &= \frac{1 - m(\gamma\cdot x, \gamma \cdot y)m(\gamma \cdot y, \gamma \cdot x)}{1+m(\gamma\cdot x, \gamma \cdot y)m(\gamma \cdot y, \gamma \cdot x)} = \frac{1 - m(\gamma( x), \gamma(y))m(\gamma(y), \gamma(x))}{1+m(\gamma(x), \gamma(y))m(\gamma(y), \gamma(x))}. \end{aligned}$$ Next, applying $\gamma$ to $m(x,y) y \leq x$ yields $m(x,y) \gamma(y)\leq \gamma(x)$ so that $m(x,y)\leq m(\gamma(x),\gamma(y))$. Since $\frac{1-t}{1+t}$ is decreasing, we can continue the above computation with $$d(\gamma \cdot x, \gamma \cdot y)\le \frac{1 - m(x,y)m(y,x)}{1+m(x,y)m(x,y)} = d(x,y). \tag*{\qedhere}$$ ◻ For faithful maps, the above inequality can be refined as follows. **Lemma 32**. *Let $\gamma$ be a faithful positive linear map on $L^1(M,\tau)$. For any $x,y\in S$, we have $$\label{eqn:jeffsinequality} d(\gamma \cdot x, \gamma \cdot y) \le d(\gamma \cdot A_-, \gamma \cdot A_+) d(x,y)$$ where $A_{\pm}$ are defined as in Lemma [Lemma 18](#lem:dformula){reference-type="ref" reference="lem:dformula"}.* *Proof.* The proof is essentially the same as in [@jeff Lemma 3.10(1)]. Let $x,y\in S$ be distinct and note that we can assume $\gamma\cdot x, \gamma\cdot y$ are also distinct since otherwise the inequality is trivially true. Applying Lemma [Lemma 18](#lem:dformula){reference-type="ref" reference="lem:dformula"} to the pairs $x,y$ and $\gamma\cdot x, \gamma\cdot y$ gives $A_{\pm},B_{\pm}\in S$, $t_{\pm}, w_{\pm}\in \mathbb{R}$, and $r,s, u,v\in [0,1]$ satisfying $$\begin{aligned} \label{eqn:convex_relations_for_xygammaxgammay} \begin{split} A_{\pm} = t_{\pm} x + (1-t_{\pm})y \qquad\qquad x &= r A_- + (1-r) A_+ \qquad\qquad y = s A_- + (1-s) A_+\\ B_{\pm} = w_{\pm} \gamma\cdot x + (1-w_{\pm})\gamma\cdot y \qquad\qquad \gamma\cdot x &= u B_- + (1-u) B_+ \qquad\qquad \gamma\cdot y = v B_- + (1-v) B_+. \end{split} \end{aligned}$$ Additionally, we have $$d(x,y) = \left|\frac{r(1-s)-(1-r)s}{r(1-s)+(1-r)s}\right| \qquad\qquad d(\gamma\cdot x,\gamma\cdot y)= \left|\frac{u(1-v)-(1-u)v}{u(1-v)+(1-u)v}\right|.$$ By Remark [Remark 30](#rmk:projective_actions_preserve_lines){reference-type="ref" reference="rmk:projective_actions_preserve_lines"}, we have $$\begin{aligned} \gamma\cdot A_{\pm} = \frac{\tau(\gamma(t_\pm x))}{\tau(\gamma(t_\pm x+ (1-t_\pm) y))} \gamma\cdot x + \frac{\tau(\gamma((1-t_\pm) y))}{\tau(\gamma(t_\pm x+ (1-t_\pm) y))} \gamma\cdot y. \end{aligned}$$ In particular, $\gamma\cdot A_{\pm}$ lie in the convex set $$\{w\gamma\cdot x + (1-w)\gamma\cdot y\in S\colon w_-\leq w\leq w_+\}\cap S,$$ and therefore $\{p\gamma\cdot A_- + (1-p)\gamma\cdot A_+\colon p\in \mathbb{R}\}\cap S$ has the same extreme points; namely, $B_{\pm}$. Thus there exist $p,q\in [0,1]$ satisfying $$\begin{aligned} \gamma\cdot A_- &= p B_- + (1-p)B_+\\ \gamma\cdot A_+ &= q B_- + (1-q)B_+, \end{aligned}$$ and $$d(\gamma\cdot A_-, \gamma\cdot A_+) = \left|\frac{p(1-q)-(1-p)q}{p(1-q)+(1-p)q}\right|,$$ by Lemma [Lemma 18](#lem:dformula){reference-type="ref" reference="lem:dformula"}. Using the above formulae we have $$\begin{aligned} u B_- + (1-u) B_+ &= \gamma\cdot x \\ &= \frac{1}{\tau(\gamma(x))}(r \gamma(A_-) + (1-r) \gamma(A_+))\\ &= \frac{\tau(\gamma(A_-)) rp + \tau(\gamma(A_+)) (1-r)q}{\tau(\gamma(x))} B_- + \frac{\tau(\gamma(A_-)) r(1-p) + \tau(\gamma(A_+)) (1-r)(1-q)}{\tau(\gamma(x))}B_+. \end{aligned}$$ This determines $u,1-u$ since $B_{\pm}$ are distinct states (the line connecting them contains the distinct states $\gamma\cdot x$ and $\gamma\cdot y$) and are therefore linearly independent. A similar computation yields $$v = \frac{\tau(\gamma(A_-)) sp + \tau(\gamma(A_+)) (1-s)q}{\tau(\gamma(y))} \qquad\qquad 1-v = \frac{\tau(\gamma(A_-)) s(1-p) + \tau(\gamma(A_+)) (1-s)(1-q)}{\tau(\gamma(y))}.$$ Writing $\alpha:=\tau(\gamma(A_-))$ and $\beta:=\tau(\gamma(A_+))$, we have: $$\begin{aligned} d(\gamma\cdot x , \gamma\cdot y) &= \left|\frac{u(1-v)-(1-u)v}{u(1-v)+(1-u)v}\right| \\ &= \frac{\alpha \beta |p(1-q)-(1-p)q||r(1-s)- (1-r)s|}{2\alpha^2 p(1-p)rs+\alpha\beta (p(1-q)+(1-p)q)(r(1-s)+(1-r)s) + 2\beta^2q(1-q)(1-r)(1-s)}\\ &\leq \frac{|p(1-q)-(1-p)q||r(1-s)- (1-r)s|}{(p(1-q)+(1-p)q)(r(1-s)+(1-r)s)} = d(\gamma\cdot A_-, \gamma\cdot A_+)d(x,y), \end{aligned}$$ where the second equality follows from an abundance of arithmetic and the inequality follows from $2\alpha^2 p (1-p)rs + 2\beta^2 q(1-q)(1-r)(1-s)\geq 0$. ◻ ## Contraction mappings {#sec:contraction_mappings} Given a metric space $(X,\rho)$, recall that a mapping $T:X\to X$ is said to be a *contraction mapping* if there is a constant $0\le q<1$ so that $\rho(T(p_1), T(p_2))\le q \rho(p_1, p_2)$ for all $p_1,p_2\in X$. Since we have shown (Theorem [Theorem 19](#thm:tracemetricinequality_and_completeness){reference-type="ref" reference="thm:tracemetricinequality_and_completeness"}) that the normal state space $S\subset L^1(M,\tau)$ is complete in Hennion's Metric, we shall aim to give a characterization of a large class of contractions with respect to Hennion's metric. **Definition 33**. The **Hennion contraction constant** of a faithful positive linear map $\gamma$ on $L^1(M,\tau)$ is the quantity $$\label{eqn:contraction} c(\gamma):=\sup_{x,y\in S} \frac{d(\gamma\cdot x,\gamma\cdot y)}{d(x,y)}.$$ We say $\gamma$ is a **strict Hennion contraction** if $c(\gamma)< 1$, and we denote by $SHC(M)$ the family of all such maps. Lemma [Lemma 31](#lem:dantitone){reference-type="ref" reference="lem:dantitone"} implies $c(\gamma)\leq 1$ for all faithful positive linear maps $\gamma$ on $L^1(M,\tau)$. Using Lemma [Lemma 32](#lem:jeffinequality){reference-type="ref" reference="lem:jeffinequality"} we actually have $$\frac{d(\gamma\cdot x, \gamma\cdot y)}{d(x,y)} \leq d(\gamma\cdot A_-, \gamma\cdot A_+),$$ and consequently $c(\gamma)\leq \mathop{\mathrm{diam}}(\gamma\cdot S)$. The reverse inequality is a consequence of $d(x,y)\leq 1$ for all $x,y\in S$, and so we have proven the following: **Proposition 34**. *For a faithful positive linear map $\gamma$ on $L^1(M,\tau)$ one has $$c(\gamma)=\mathop{\mathrm{diam}}(\gamma\cdot S).$$* If we further assume $\gamma$ is bounded as linear operator on $L^1(M,\tau)$, then the joint lower semicontinuity of $d$ with respect to $\|\cdot\|_1$ (see Theorem [Theorem 21](#thm:lower_semicontinuity_of_d){reference-type="ref" reference="thm:lower_semicontinuity_of_d"}) implies the contraction constant can be witnessed on any $\|\cdot\|_1$-dense subset of $L^1(M,\tau)$. **Proposition 35**. *Let $\gamma$ be a bounded faithful positive linear map on $L^1(M,\tau)$. For any $\|\cdot\|_1$-dense subset $S_0\subset S$, one has $$c(\gamma) = \mathop{\mathrm{diam}}(\gamma\cdot S_0).$$* *Proof.* Let $\eta < c(\gamma)$ and let $x,y\in S$ be such that $$\eta < d(\gamma\cdot x, \gamma\cdot y) \leq c(\gamma).$$ Letting $(x_n)_{n\in \mathbb N}, (y_n)_{n\in \mathbb N}\subset S_0$ be sequences converging to $x$ and $y$, respectively, with respect to $\|\cdot\|_1$. Then $\gamma\cdot x_n \to \gamma\cdot x$ and $\gamma\cdot y_n\to \gamma\cdot y$ with respect to $\|\cdot\|_1$, and so Theorem [Theorem 21](#thm:lower_semicontinuity_of_d){reference-type="ref" reference="thm:lower_semicontinuity_of_d"} implies $$\mathop{\mathrm{diam}}(\gamma\cdot S_0) \geq \liminf_{n\to\infty} d(\gamma\cdot x_n , \gamma\cdot y_n) \geq d(\gamma\cdot x, \gamma\cdot y ) > \eta.$$ Letting $\eta\to c(\gamma)$ completes the proof. ◻ **Example 36**. Let $(M,\tau) = (\mathbb{M}_n, \frac1n \text{Tr})$ be the $n\times n$ matrices equipped with its normalized trace. If $\gamma\colon \mathbb{M}_n\to \mathbb{M}_n$ is *strictly positive* in the sense that $\gamma\cdot S\subset S^\times$, then it is a strict Hennion contraction. Indeed, $S$ is compact in this case and therefore $$c(\gamma)= \mathop{\mathrm{diam}}(\gamma\cdot S)=d(\gamma\cdot x, \gamma\cdot y),$$ for some $x,y\in S$. Using Theorem [Theorem 24](#thm:geometry){reference-type="ref" reference="thm:geometry"}.(2), we see that $c(\gamma)<1$. Conversely, if $\gamma$ is a strict Hennion contraction and $\gamma\cdot 1\in S^\times$, then $\gamma$ is strictly positive by Theorem [Theorem 24](#thm:geometry){reference-type="ref" reference="thm:geometry"}. $\hfill\blacksquare$ **Example 37** (A Non-Example). Let $(M,\tau) = (\mathbb{M}_n, \frac1n \text{Tr})$ be the $n\times n$ matrices equipped with its normalized trace. The transpose map $x\mapsto x^T$ is a unital and tracial map that is well known to be positive but not completely positive (see [@paulsen]). Moreover, for $x,y\in (\mathbb{M}_n)_+$ one has $x\leq y$ iff $x^T\leq y^T$. Thus $S\ni x\mapsto x^T$ is an isometry with respect to $d$ and is therefore not a strict Hennion contraction. $\hfill\blacksquare$ We have seen in Theorem [Theorem 19](#thm:tracemetricinequality_and_completeness){reference-type="ref" reference="thm:tracemetricinequality_and_completeness"} that $(S,d)$ is a complete metric space, and so the Banach Fixed Point Theorem implies a strict Hennion contraction $\gamma$ has a unique fixed point: $\gamma\cdot x_0 = x_0$ for some $x_0\in S$. In fact, this along with the partial order on the normal state space can be used to characterize when one has a strict Hennion contraction: **Theorem 38**. *For a faithful positive linear map $\gamma$ on $L^1(M,\tau)$, the following are equivalent:* 1. *$\gamma$ is a strict Hennion contraction.* 2. *[\[part:fixed_point_dilation_constant\]]{#part:fixed_point_dilation_constant label="part:fixed_point_dilation_constant"} For some $x_0\in S$ there exists $\eta\in (0,1]$ so that $$\eta x_0 \leq \gamma\cdot x \leq \eta^{-1} x_0 \qquad \forall x\in S.$$* *In this case, $x_0$ can be taken to be the unique fixed point of the projective action. Moreover, if $(\gamma\cdot S)\cap S_b^\times \neq \varnothing$ then the above are further equivalent to* 1. *For each $y_0\in S_b^\times$ there exists $\kappa\in (0,1]$ so that $$\kappa y_0 \leq \gamma\cdot x \leq \kappa^{-1} y_0 \qquad \forall x\in S.$$* *In this case, one has $\gamma\cdot S \subset S_b^\times$.* *Proof.* $(i)\Rightarrow (ii)$: By the discussion preceding the statement of the theorem, there exists a unique $x_0\in S$ so that $\gamma\cdot x_0 = x_0$. Then Proposition [Proposition 34](#prop:cont_const_is_diam){reference-type="ref" reference="prop:cont_const_is_diam"} gives $$d(\gamma \cdot x, x_0) = d(\gamma\cdot x, \gamma\cdot x_0) \leq c(\gamma)<1,$$ for all $x\in S$. Arguing as in Theorem [Theorem 19](#thm:tracemetricinequality_and_completeness){reference-type="ref" reference="thm:tracemetricinequality_and_completeness"}, we see that $$\min\{ m(\gamma\cdot x, x_0), m(x_0, \gamma\cdot x)\}\ge \frac{1-c(\gamma)}{1+c(\gamma)} =:\eta,$$ for all $x\in S$. Thus $$\eta x_0 \leq m(\gamma\cdot x,x_0) x_0 \leq \gamma\cdot x \leq m(x_0,\gamma\cdot x)^{-1}x_0 \leq \eta^{-1} x_0,$$ as claimed.\ $(ii)\Rightarrow (i)$: It follows from $\eta x_0 \leq \gamma\cdot x\leq \eta^{-1}x_0$ that $m(\gamma\cdot x, x_0)m(x_0,\gamma\cdot x) \geq \eta^2$ for all $x\in S_0$. Using Theorem [Theorem 15](#thm:mproperties){reference-type="ref" reference="thm:mproperties"}.[\[part:mtriangle_ineq\]](#part:mtriangle_ineq){reference-type="ref" reference="part:mtriangle_ineq"}, we then have $$m(\gamma\cdot x,\gamma\cdot y)m(\gamma\cdot y,\gamma\cdot x) \geq \eta^4$$ for all $x,y\in S$, and therefore $$d(\gamma\cdot x,\gamma\cdot y) \leq \frac{1-\eta^4}{1+\eta^4}.$$ Thus $$c(\gamma)=\mathop{\mathrm{diam}}(\gamma \cdot S)\leq \frac{1-\eta^4}{1+\eta^4} < 1,$$ where the first equality follows from Proposition [Proposition 34](#prop:cont_const_is_diam){reference-type="ref" reference="prop:cont_const_is_diam"}.\ Now suppose $(\gamma\cdot S)\cap S_b^\times \neq \varnothing$. If $\gamma$ is a strict Hennion contraction, then $\gamma\cdot S\subset S_b^\times$ holds by Theorem [Theorem 24](#thm:geometry){reference-type="ref" reference="thm:geometry"}. Consequently, for the fixed point $x_0=\gamma\cdot x_0$ if we let $\delta := \min\{\|x_0\|^{-1}, \|x_0^{-1}\|^{-1}\}\in (0,1]$ then $\delta 1 \leq x_0 \leq \delta^{-1} 1$. Let $y_0\in S_b^\times$, let $\eta\in (0,1)$ be as in $\ref{part:fixed_point_dilation_constant}$, and set $\kappa:=\delta \eta\min\{\|y_0\|^{-1},\|y_0^{-1}\|^{-1}\}$. Then one has $$\kappa y_0 \leq \delta \eta \leq \eta x_0 \leq \gamma\cdot x \leq \eta^{-1} x_0 \leq (\delta\eta)^{-1} 1 \leq \kappa^{-1} y_0,$$ for all $x\in S$. The converse (namely, $(iii)\Rightarrow (ii)$) is immediate. ◻ **Example 39**. Fix a non-zero $m\in L^1(M,\tau)_+$ and $\eta\in (0,1]$. Let $\{(a_i,m_i)\in M_+\times L^1(M,\tau)_+\colon i\in I\}$ be a family satisfying: 1. $a:=\sum_{i\in I} a_i$ converges in the strong operator topology; 2. $\tau(xa)>0$ for all $x\in L^1(M,\tau)_+\setminus \{0\}$; 3. $\eta m \leq m_i\leq \eta^{-1} m$ for all $i\in I$. For $x\in L^1(M,\tau)$, we claim that $$\gamma(x):= \sum_{i\in I} \tau(xa_i) m_i$$ converges. Indeed, let $x=v|x|$ be the polar decomposition and let $\epsilon>0$. The strong summability of the $a_i$ implies there exists a finite $F_0\subset I$ so that whenever a finite subset $F\subset I$ satisfies $F\cap F_0 = \varnothing$ then $$\left\| \sum_{i\in F} a_i |x|^{\frac12} \right\|_2,\ \left\| \sum_{i\in F} a_i v|x|^{\frac12} \right\|_2 <\epsilon.$$ Let $F,G\subset I$ be finite subsets both containing $F_0$ so that $F\Delta G$ is disjoint from $F_0$. Let $w$ be the polar part of $$\sum_{i\in F} \tau(xa_i)m_i - \sum_{i\in G} \tau(xa_i) m_i.$$ Then using $\|m_i\|_1 =\tau(m_i)\leq \eta^{-1} \tau(m)$ we have $$\begin{aligned} \left\| \sum_{i\in F} \tau(xa_i)m_i - \sum_{i\in G} \tau(xa_i) m_i\right\|_1 &\leq \sum_{i\in F\Delta G} |\tau(a_ix)| |\tau(w^*m_i)|\\ &\leq \sum_{i\in F\Delta G} |\langle a_i^{\frac12} v|x|^{\frac12}, a_i^{\frac12} |x|^{\frac12}\rangle_2| \eta^{-1} \tau(m)\\ &\leq \left( \sum_{i\in F\Delta G} \|a_i^{\frac12} v|x|^{\frac12}\|_2^2 \right)^{\frac12} \left( \sum_{i\in F\Delta G} \|a_i^{\frac12} |x|^{\frac12}\|_2^2 \right)^{\frac12} \eta^{-1} \tau(m)\\ &= \left\langle \sum_{i\in F\Delta G} a_i v|x|^{\frac12},v|x|^{\frac12}\right\rangle_2^{\frac12} \left\langle \sum_{i\in F\Delta G} a_i |x|^{\frac12}, |x|^{\frac12} \right\rangle_2^{\frac12} \eta^{-1} \tau(m)\\ &< \epsilon \|x\|_1^{\frac12} \eta^{-1} \tau(m). \end{aligned}$$ Thus the net of partial sums is Cauchy and converges in $L^1(M,\tau)$. We therefore have a positive linear map $\gamma$ on $L^1(M,\tau)$, which is faithful by (2). One can also show $\gamma$ is bounded using similar estimates as above: $$\begin{aligned} |\tau(\gamma(x) b)| &\leq \sum_{i\in I} |\tau(xa_i)\tau(m_ib)| \leq \sum_{i\in I} |\tau(xa_i)| \eta^{-1}\tau(m)\|b\| \\ &\leq \langle a v|x|^{\frac12}, v|x|^{\frac12}\rangle_2^{\frac12} \langle a|x|^{\frac12}, |x|^\frac12\rangle_2^{\frac12} \eta^{-1} \tau(m) \|b\| \leq \|a\| \|x\|_1 \eta^{-1} \tau(m)\|b\|, \end{aligned}$$ for all $b\in M$. Hence $\|\gamma\| \leq \|a\| \eta^{-1} \tau(m)$. Now, by applying $\tau$ to the inequalities in (3), one can show that $$\tau(m_i) \eta^2 \frac{m}{\tau(m)} \leq m_i \leq \tau(m_i) \eta^{-2} \frac{m}{\tau(m)}.$$ Since $$\tau(\gamma(x)) = \sum_{i\in I} \tau(xa_i) \tau(m_i),$$ for $x\in L^1(M,\tau)_+$, it follows that for $x_0:= \frac{m}{\tau(m)}\in S$ one has $$\gamma(x) \leq \sum_{i\in I} \tau(xa_i) \tau(m_i) \eta^{-2} x_0 = \tau(\gamma(x)) \eta^{-2} x_0,$$ and similarly one has $\gamma(x)\geq \tau(\gamma(x))\eta^2 x_0$. This shows that $\gamma$ is a strict Hennion contraction by Theorem [Theorem 38](#thm:strict_Hennion_projective_actions){reference-type="ref" reference="thm:strict_Hennion_projective_actions"}. In particular, there exists a fixed point; that is, $\gamma(x) = \tau(\gamma(x)) x$ for some $x\in L^1(M,\tau)$. Note that $m$ controls the component of $S$ in which the projective action of $\gamma$ is valued. $\hfill\blacksquare$ The following lemma will be needed in later sections once we start considering quantum processes, which in the context of this article are compositions of positive maps. **Lemma 40**. *For faithful positive linear maps $\alpha,\beta$ on $L^1(M,\tau)$, one has $$c(\alpha\circ \beta) \leq c(\alpha)c(\beta).$$ Consequently, the family of strict Hennion contractions is invariant under (pre or post) composition with faithful positive linear maps on $L^1(M,\tau)$.* *Proof.* For $x\in L^1(M,\tau)_+\setminus\{0\}$ one has $$(\alpha \circ \beta)\cdot x = \frac{1}{\tau(\alpha(\beta(x)))} \alpha(\beta(x)) = \frac{1}{\tau\left(\alpha(\beta\cdot x\right))} \alpha(\beta\cdot x)=\alpha \cdot (\beta \cdot x),$$ so that $$\begin{aligned} d((\alpha\circ \beta)\cdot x, (\alpha \circ \beta) \cdot y)) = d(\alpha\cdot (\beta\cdot x), \alpha\cdot (\beta\cdot y)) \leq c(\alpha) d(\beta \cdot x, \beta \cdot y)\leq c(\alpha) c(\beta)d(x,y). \end{aligned}$$ Hence $c(\alpha\circ\beta)\leq c(\alpha)c(\beta)$. ◻ ### Strict Hennion contractions from normal maps {#subsubsec:SHCfromnormal} A positive normal map $\phi\colon M\to M$ can induce bounded positive maps on $L^1(M,\tau)$ in two ways: by the predual map $\phi_*$, or by extending $\phi$ itself $L^1(M,\tau)$ provided it is $\tau$-bounded (see Lemma [Lemma 6](#lem:normalbddextension){reference-type="ref" reference="lem:normalbddextension"}). We will investigate when these induced maps are strict Hennion contractions, but we must first characterize when they are faithful: **Lemma 41**. *Let $\phi\colon M\to M$ be a normal completely positive map.* 1. *$\phi_*$ is faithful if and only if $\phi(M)M$ is weak\* dense in $M$.* 2. *$\phi$ admits a bounded faithful extension to $L^1(M,\tau)$ if and only if $\phi_*$ is $M$-preserving and $\phi_*(M)M$ is weak\* dense in $M$.* *Proof.* 1. The weak\* closure of $\phi(M)M$ is a weak\* closed right ideal in $M$ and hence of the form $pM$ for $p:=[\phi(M)M]$. Thus $\phi(M)M$ is weak\* dense if and only if $p=1$. Now, suppose $\phi_*$ is not faithful and let $x\in L^1(M,\tau)_+$ be a non-zero element in its kernel. For all $a,b\in M$ we then have $$\begin{aligned} \|x^{\frac12} \phi(a)b\|_2^2 &= \tau(b^*\phi(a)^* x\phi(a)b) = \tau(x^{1/2} \phi (a^*)^*b^*b \phi(a^*) x^{1/2}) \\ &\leq \|b\|^2\|\phi(1)\| \tau(x^{1/2} \phi(aa^*) x^{1/2})= \|b\|^2\|\phi(1)\| \tau(\phi_*(x)aa^*)=0. \end{aligned}$$ Thus $\overline{\phi(M)M}^{\|\cdot\|_2} \subset \ker{x^{\frac12}}$. Since $x$ is non-zero, so is $x^{\frac12}$, and therefore $\overline{\phi(M)M}^{\|\cdot\|_2}$ must be a proper subspace of $L^2(M,\tau)$. The projection onto this subspace is $p$ by definition, so $p\neq 1$. Conversely, if $p\neq 1$ then for all $a\in M$ we have $$\tau(\phi_*(1-p) a)= \tau( (1-p)\phi(a)) = \langle(1-p) \phi(a), 1\rangle_2 = 0.$$ Since this holds for all $a\in M$, it follows that $\phi_*(1-p)=0$. 2. The equivalence of $\phi$ admitting a bounded extension to $L^1(M,\tau)$ and $\phi_*$ being $M$-preserving follows from Lemma [Lemma 6](#lem:normalbddextension){reference-type="ref" reference="lem:normalbddextension"}. The rest is then a consequence of part (1), with the roles of $\phi$ and $\phi_*$ reversed.  ◻ Note that if $\phi$ is unital or even satisfies that $1$ belongs to the weak\* closure of $\phi(M)$, then $\phi_*$ is faithful by the previous lemma. Another way to guarantee faithfulness of $\phi_*$ (which avoids assuming that $\phi$ is completely positive) is to assume that $\phi(1)$ is invertible. Indeed, this implies $\phi(1)\geq \delta 1$ for some $\delta>0$, which is equivalent via duality to $\tau(\phi_*(x)) \geq \delta \tau(x)$ for all $x\in L^1(M,\tau)_+$. **Remark 42**. In the proof of Lemma [Lemma 41](#lem:when_is_predual_faithful){reference-type="ref" reference="lem:when_is_predual_faithful"}.(1), complete positivity was only used for the Schwarz inequality: $$\phi(x)^*\phi(x) \leq \|\phi(1)\| \phi(x^*x) \qquad \forall x\in M.$$ Thus it would be sufficient to assume $\phi$ was merely $2$-positive (see [@paulsen Proposition 3.3]).$\hfill\blacksquare$ As a consequence of the following lemma, it will turn out one of $\phi_*$ or the extension of $\phi$ is a strict Hennion contraction if and only if the other is, provided both maps exist and are faithful (see Corollaries [Corollary 44](#cor:when_do_preduals_give_SHC){reference-type="ref" reference="cor:when_do_preduals_give_SHC"} and [Corollary 45](#cor:when_do_extensions_gives_SHC){reference-type="ref" reference="cor:when_do_extensions_gives_SHC"}). **Lemma 43**. *Let $\phi\colon M\to M$ be a positive normal map with a bounded faithful extension to $L^1(M,\tau)$ and a faithful predual $\phi_*$. Then $c(\phi)=c(\phi_*)$.* *Proof.* By Proposition [Proposition 34](#prop:cont_const_is_diam){reference-type="ref" reference="prop:cont_const_is_diam"} it suffices to show $\mathop{\mathrm{diam}}(\phi\cdot S) = \mathop{\mathrm{diam}}(\phi_*\cdot S)$, and by the formula for $d$ in terms of $m$ along with Theorem [Theorem 15](#thm:mproperties){reference-type="ref" reference="thm:mproperties"}.[\[part:m_inf_formula\]](#part:m_inf_formula){reference-type="ref" reference="part:m_inf_formula"} it further suffices to show $$\begin{aligned} \inf&\left\{\frac{\tau(a\phi(x))}{\tau(a\phi(y))}\frac{\tau(b\phi(y))}{\tau(b\phi(x))}\colon x,y\in L^1(M,\tau)_+,\ a,b\in M_+,\ \tau(b\phi(x)), \tau(a\phi(y))>0 \right\}\\ &=\inf\left\{\frac{\tau(a\phi_*(x))}{\tau(a\phi_*(y))}\frac{\tau(b\phi_*(y))}{\tau(b\phi_*(x))}\colon x,y\in L^1(M,\tau)_+,\ a,b\in M_+,\ \tau(b\phi_*(x)), \tau(a\phi_*(y))>0 \right\}. \end{aligned}$$ Since $\phi$ and $\phi_*$ are both bounded on $L^1(M,\tau)$, neither of the above infima is changed if we restrict to $x,y\in M_+$. But then $$\frac{\tau(a\phi(x))}{\tau(a\phi(y))}\frac{\tau(b\phi(y))}{\tau(b\phi(x))} = \frac{\tau(\phi_*(a)x)}{\tau(\phi_*(a)y)}\frac{\tau(\phi_*(b)y)}{\tau(\phi_*(b)x)} = \frac{\tau(x\phi_*(a))}{\tau(x\phi_*(b))}\frac{\tau(y\phi_*(b))}{\tau(y\phi_*(a))}$$ implies the two infima are equal. ◻ The next results are corollaries to Theorem [Theorem 38](#thm:strict_Hennion_projective_actions){reference-type="ref" reference="thm:strict_Hennion_projective_actions"}. **Corollary 44**. *Let $\phi\colon M\to M$ be a positive normal map with faithful predual $\phi_*$. Then $\phi_*$ is a strict Hennion contraction if and only if for some $x_0\in S$ there exists $\eta\in (0,1]$ so that $$\eta \tau(x_0a)\phi(1) \leq \phi(a) \leq \eta^{-1} \tau(x_0 a)\phi(1) \qquad \forall a\in M_+.$$ Moreover, $\phi_*\cdot S\subset S_b^\times$ if and only if one can choose $x_0=1$, and in this case $\phi$ extends to a bounded faithful map on $L^1(M,\tau)$ which is also a strict Hennion contraction.* *Proof.* Noting that $$(a\mapsto \tau(x_0a)\phi(1))_* = x\mapsto \tau(\phi_*(x)) x_0,$$ we see that from Lemma [Lemma 7](#lem:adjoint&predual){reference-type="ref" reference="lem:adjoint&predual"} that the inequalities on $\phi$ are equivalent to $$\eta \tau(\phi_*(x)) x_0 \leq \phi_*(x) \leq \eta^{-1} \tau(\phi_*(x)) x_0 \qquad \forall x\in L^1(M,\tau)_+.$$ These are in turn equivalent to $\phi_*$ being a strict Hennion contraction by Theorem [Theorem 38](#thm:strict_Hennion_projective_actions){reference-type="ref" reference="thm:strict_Hennion_projective_actions"}. The last part of this theorem also implies the second if and only if statement. Now, suppose $\eta \tau(a) \phi(1) \leq \phi(a) \leq \eta^{-1} \tau(a) \phi(1)$ holds for all $a\in M_+$. Taking the trace yields $\eta \tau(\phi(1)) \tau(a)\leq \tau\circ \phi(a) \leq \eta^{-1} \tau(\phi(1)) \tau(a)$. This shows $\phi$ admits a bounded faithful extension to $L^1(M,\tau)$ and it is necessarily a strict Hennion contraction by Lemma [Lemma 43](#lem:contraction_constant_of_duals){reference-type="ref" reference="lem:contraction_constant_of_duals"}. ◻ **Corollary 45**. *Let $\phi\colon M\to M$ be a positive normal map with a bounded faithful extension to $L^1(M,\tau)$. Then $\phi$ is a strict Hennion contraction if and only if for some $b_0\in S_b$ there exists $\eta\in (0,1]$ so that $$\eta \tau(\phi(a)) b_0 \leq \phi(a) \leq \eta^{-1} \tau(\phi(a)) b_0 \qquad \forall a\in M_+.$$ Moreover, $\phi\cdot S\subset S_b^\times$ if and only if one can choose $b_0=1$, and in this case $\phi_*$ is a bounded faithful map on $L^1(M,\tau)$ which is also a strict Hennion contraction.* *Proof.* The first if and only if statement follows immediately from Theorem [Theorem 38](#thm:strict_Hennion_projective_actions){reference-type="ref" reference="thm:strict_Hennion_projective_actions"}, and the last part of this theorem also implies the second if and only if statement. Now, suppose $\eta\tau(\phi(a))\leq \phi(a)\leq \eta^{-1}\tau(\phi(a))$ holds for all $a\in M_+$. Invoking Lemma [Lemma 7](#lem:adjoint&predual){reference-type="ref" reference="lem:adjoint&predual"} we obtain $$\eta \tau(x)\phi_*(1) \leq \phi_*(x) \leq \eta^{-1} \tau(x) \phi_*(1) \qquad \forall x\in L^1(M,\tau)_+.$$ This implies $\phi_*$ is faithful, and hence is a strict Hennion contraction by Lemma [Lemma 43](#lem:contraction_constant_of_duals){reference-type="ref" reference="lem:contraction_constant_of_duals"}. ◻ **Corollary 46**. *Let $\phi\colon M\to M$ be a positive normal map. Then the following are equivalent.* 1. *There exists $\eta\in (0,1]$ so that $\eta \tau\leq \phi \leq \eta^{-1} \tau$.* 2. *There exists $\delta\in (0,1]$ so that $\delta\tau\leq \tau\circ \phi\leq \delta^{-1} \tau$, and $\phi$ has a bounded faithful extension to $L^1(M,\tau)$ which is a strict Hennion contraction valued in $S_b^\times$.* 3. *There exists $\delta_*\in (0,1]$ so that $\delta_*\tau \leq \tau\circ \phi_*\leq \delta_*^{-1} \tau$, and $\phi_*$ is a faithful map on $L^1(M,\tau)$ which is a strict Hennion contraction valued in $S_b^\times$.* *Proof.* $(i)\Rightarrow (ii),(iii):$ By Lemma [Lemma 7](#lem:adjoint&predual){reference-type="ref" reference="lem:adjoint&predual"}, $\eta\tau \leq \phi\leq \eta^{-1}\tau$ is equivalent to $\eta\tau \leq \phi_*\leq \eta^{-1} \tau$, since $\tau_*=\tau$. Applying $\tau$ to these inequalities gives $\eta\tau\leq \tau\circ \phi\leq \eta^{-1} \tau$, so that $\phi$ extends to a bounded faithful map on $L^1(M,\tau)$, and $\eta\tau\leq \tau\circ \phi_*\le \eta^{-1} \tau$, so that $\phi_*$ is faithful. Since $$\eta^2 \tau(\phi(a)) \leq \eta \tau(a) \leq \phi(a) \leq \eta^{-1} \tau(a)\leq \eta^{-2} \tau(\phi(a)),$$ for all $a\in M_+$, Corollary [Corollary 45](#cor:when_do_extensions_gives_SHC){reference-type="ref" reference="cor:when_do_extensions_gives_SHC"} implies $\phi$ and $\phi_*$ are strict Hennion contractions. They are both necessarily valued in $S_b^\times$ since $\eta \leq \phi(1), \phi_*(1) \leq \eta^{-1}$.\ $(ii)\Rightarrow(i):$ Theorem [Theorem 38](#thm:strict_Hennion_projective_actions){reference-type="ref" reference="thm:strict_Hennion_projective_actions"} yields a $\kappa\in (0,1]$ satisfying $\kappa \tau\circ \phi \leq \phi \leq \kappa^{-1}\tau\circ \phi$, and therefore $$\kappa \delta \tau \leq \kappa \tau\circ \phi \leq \phi \leq \kappa^{-1} \tau\circ \phi \leq \kappa^{-1} \delta^{-1} \tau.$$ So we take $\eta:=\kappa\delta$.\ $(iii)\Rightarrow (i):$ Since $\phi_*\cdot 1\in M$, we have that $\phi_*|_M\colon M\to M$ is a positive normal map (see [@ap Proposition 2.5.11]). So the same argument as in $(ii)\Rightarrow(i)$ gives $\eta \tau\leq \phi_* \leq \eta^{-1} \tau$ for some $\eta\in (0,1]$, and appealing to Lemma [Lemma 7](#lem:adjoint&predual){reference-type="ref" reference="lem:adjoint&predual"} completes the proof. ◻ **Theorem 47**. *Let $\phi\colon M\to M$ be a normal completely positive map such that $\tau\circ\phi\leq \tau$ ($\phi$ is subtracial), $\phi(1)\leq 1$ ($\phi$ is subunital), and $1-\phi(1)$ belongs to the weak\* closure of $\phi(M_+)$. If $\phi_*$ is a strict Hennion contraction then the unital tracial map $$\tilde{\phi}_*(x):= \phi_*(x) + \frac{(\tau-\tau\circ \phi_*)(x)}{(\tau- \tau\circ \phi_*)(1)}(1-\phi_*(1))$$ is a strict Hennion contraction valued in $S_b^\times$. In this case, the extension of $$\tilde{\phi}(x):= \phi(x) + \frac{(\tau-\tau\circ \phi)(x)}{(\tau- \tau\circ \phi)(1)}(1-\phi(1))$$ to $L^1(M,\tau)$ is also a strict Hennion contraction valued in $S_b^\times$.* *Proof.* We first observe that the assumption on $\phi$ implies $1=(1-\phi(1)) + \phi(1)$ belongs to the weak\* closure of $\phi(M)M$ so that $\phi_*$ is faithful by Lemma [Lemma 41](#lem:when_is_predual_faithful){reference-type="ref" reference="lem:when_is_predual_faithful"}. We also claim that for all $x,y\in L^1(M,\tau)_+\setminus\{0\}$ one has $$\begin{aligned} m(\phi_*(x), \phi_*(y))(\tau- \tau\circ\phi_*)(y) \leq (\tau - \tau\circ\phi_*)(x). \end{aligned}$$ Indeed, if $(\tau- \tau\circ\phi_*)(y)=0$ then this holds trivially, so suppose $(\tau- \tau\circ \phi_*)(y)>0$. If we let $(a_i)_{i\in I}\subset M_+$ be a net such that $\phi(a_i)\to 1-\phi(1)$ weak\*, then $$\frac{ (\tau - \tau\circ \phi_*)(x)}{(\tau - \tau\circ \phi_*)(y)} = \frac{\tau(x(1-\phi(1))}{\tau(y(1-\phi(1))} = \lim_{i\to\infty} \frac{\tau(x \phi(a_i))}{\tau(y\phi(a_i))} = \lim_{i\to\infty} \frac{\tau(\phi_*(x) a_i)}{\tau(\phi_*(y) a_i)} \geq m(\phi_*(x), \phi_*(y)),$$ where we have used Theorem [Theorem 15](#thm:mproperties){reference-type="ref" reference="thm:mproperties"}.[\[part:m_inf_formula\]](#part:m_inf_formula){reference-type="ref" reference="part:m_inf_formula"} in the last inequality. Using this we have $$m(\phi_*(x),\phi_*(y)) \tilde{\phi}_*(y) = m(\phi_*(x),\phi_*(y)) \phi_*(y) + \frac{m(\phi_*(x),\phi_*(y))(\tau-\tau\circ \phi_*)(y)}{(\tau- \tau\circ \phi_*)(1)}(1-\phi_*(1)) \leq \tilde{\phi}_*(x).$$ Thus $m(\phi_*(x),\phi_*(y)) \leq m(\tilde{\phi}_*(x), \tilde{\phi}_*(y))$, and consequently $$m(\tilde{\phi}_*(x), \tilde{\phi}_*(y)) m(\tilde{\phi}_*(y), \tilde{\phi}_*(x)) \geq m(\phi_*(x), \phi_*(y)) m(\phi_*(y), \phi_*(x)) = m(\phi_*\cdot x,\phi_*\cdot y) m(\phi_*\cdot y, \phi_*\cdot x),$$ where the equality is due to Theorem [Theorem 15](#thm:mproperties){reference-type="ref" reference="thm:mproperties"}.[\[part:mscaling\]](#part:mscaling){reference-type="ref" reference="part:mscaling"}. From this we obtain $d(\tilde{\phi}_*(x), \tilde{\phi}_*(y))\leq d(\phi_*\cdot x, \phi_*\cdot y)$ for all $x,y\in S$, and hence $$\mathop{\mathrm{diam}}(\tilde{\phi}_*(S))\leq \mathop{\mathrm{diam}}(\phi_*\cdot S).$$ Thus if $\phi_*$ is a strict Hennion contraction, then Proposition [Proposition 34](#prop:cont_const_is_diam){reference-type="ref" reference="prop:cont_const_is_diam"} implies $\tilde{\phi}_*$ is as well, and in fact we have $\tilde{\phi}_*(S) \subset S_b^\times$ because it is unital. Since $\tilde{\phi}_*$ is the predual map of $\tilde{\phi}$, Corollary [Corollary 46](#cor:tau_interval_gives_SHC){reference-type="ref" reference="cor:tau_interval_gives_SHC"} gives that the extension of $\tilde{\phi}$ is also a strict Hennion contraction valued in $S_b^\times$. ◻ Recall [@farenik] that a positive normal map $\phi\colon M\to M$ is said to be *reducible* if there is a nontrivial projection $p\in M$ and a constant $\lambda>0$ so that $\phi(p)\le \lambda p$. This is equivalent [@farenik Proposition 1] to the statement that $\phi(pMp) \subset pMp$. If there is no such nontrivial projection, $\phi$ is said to be *irreducible*. **Corollary 48**. *If $\gamma$ is a bounded strict Hennion contraction valued in $S_b^\times$, then $\gamma|_M$ is irreducible.* *Proof.* Note that $\gamma(M)\subset M$ by virtue of $\gamma\cdot S\subset S_b^\times$. Thus if $\gamma$ is bounded as a map on $L^1(M,\tau)$ then $\gamma|_M\colon M\to M$ is normal. Theorem [Theorem 38](#thm:strict_Hennion_projective_actions){reference-type="ref" reference="thm:strict_Hennion_projective_actions"} then gives a $\kappa\in (0,1]$ so that $\gamma(p)\geq \kappa \tau(\gamma(p))1$ for all projections $p\in M$. Thus if $\gamma(p)\leq \lambda p$ for some $\lambda>0$, then necessarily $p=1$. ◻ # Ergodic Quantum Processes {#sec:EQP} **Definition 49**. Let $(M,\tau)$ be a tracial von Neumann algebra, let $(\Omega,\mathbb{P})$ be a probability space equipped with $T\in \mathop{\mathrm{Aut}}(\Omega,\mathbb{P})$, and let $\gamma_\omega \colon L^1(M,\tau)\to L^1(M,\tau)$ be a bounded random linear operator. We call the family of bounded random linear operators $$\Gamma_{n,m}^T(\omega) = \gamma_{T^n\omega}\circ \gamma_{T^{n-1}\omega}\circ \cdots \circ \gamma_{T^m \omega} \qquad\qquad n,m\in \mathbb{Z},\ n\geq m,$$ an (**interval**) **quantum process** on $L^1(M,\tau)$. If $T$ is ergodic, then we call the above an **ergodic** quantum process on $L^1(M,\tau)$. When no confusion can arise, we will suppress the superscript $T$. In order to avoid measurability issues, we will from now on assume our von Neumann algebras all have separable predual. In Section [4.3](#subsec:convergence_properties_of_EQP){reference-type="ref" reference="subsec:convergence_properties_of_EQP"} we establish a number of convergence properties for ergodic quantum processes under the assumptions that $\gamma_\omega$ is bounded, positive, and faithful almost surely and that with positive probability $\Gamma_{n,m}$ is eventually a strict Hennion contraction. These results are infinite dimensional generalizations of [@jeff Theorems 1 and 2]. Of course, here one lacks the reflexivity $M_n(\mathbb C)_*\cong M_n(\mathbb C)$ present in the finite dimensional case, so in Section [4.2](#subsec:right_EQP_from_normal_maps){reference-type="ref" reference="subsec:right_EQP_from_normal_maps"} we consider quantum processes on $M$. ## Contraction constant asymptotics This first lemma addresses some technical aspects of measurability. Recall that in Section [3](#sec:cmap){reference-type="ref" reference="sec:cmap"} we rarely required boundedness of the positive maps on $L^1(M,\tau)$. However, it will be essential in this section in order to apply Proposition [Proposition 35](#prop:contr_const_for_bounded_maps){reference-type="ref" reference="prop:contr_const_for_bounded_maps"} and leverage our assumption that $L^1(M,\tau)$ is separable. **Lemma 50**. *Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual, let $(\Omega, \mathbb{P})$ be a probability space, and let $\gamma_\omega:L^1(M,\tau) \to L^1(M,\tau)$ be a random linear operator that is positive and faithful almost surely. For any $x,y\in S$ one has $m(\gamma_\omega\cdot x, \gamma_\omega \cdot y)\in L^\infty(\Omega, \mathbb{P})$. Furthermore, if $\gamma_\omega$ is bounded almost surely then $c(\gamma_\omega)\in L^\infty(\Omega,\mathbb{P})$.* *Proof.* Since $L^1(M,\tau)$ is separable, there is a countable $\sigma$-WOT dense subset $\{a_n\}_{n\in \mathbb{N}}\subset M_+$ by Theorem [Theorem 1](#thm:sep){reference-type="ref" reference="thm:sep"}. By setting $a_n \mapsto \frac{\tau(\gamma(y))}{\tau(\gamma(x))} \frac{\tau(a_n\gamma(x))}{\tau(a_n \gamma(y))}$ equal to $+\infty$ whenever $\tau(\gamma_\omega(x))=0$ or $\tau(a_n\gamma(y))=0$, we see that $$m(\gamma \cdot x, \gamma \cdot y) = \frac{\tau(\gamma(y))}{\tau(\gamma(x))} \inf\left\{\frac{\tau(a\gamma(x))}{\tau(a \gamma(y))}: a\in M_+ \setminus \{0\}\right\} = \frac{\tau(\gamma(y))}{\tau(\gamma(x))} \inf_{n\ge 1} \frac{\tau(a_n\gamma(x))}{\tau(a_n \gamma(y))}.$$ We have therefore expressed $m(\gamma \cdot x, \gamma \cdot y)$ as the infimum of a sequence of random variables, hence it is also a random variable. Now suppose $\gamma_\omega$ is bounded almost surely. Fix a countable $\|\cdot\|_1$-dense subset $S_0\subset S$. Then, $$c(\gamma_\omega) = \mathop{\mathrm{diam}}(\gamma_\omega \cdot S_0)= \sup_{x,y\in S_0} d(\gamma_\omega\cdot x, \gamma_\omega\cdot y)$$ almost surely by Proposition [Proposition 35](#prop:contr_const_for_bounded_maps){reference-type="ref" reference="prop:contr_const_for_bounded_maps"}. The first part of the proof implies $d(\gamma_\omega\cdot x, \gamma_\omega\cdot y)$ is measurable for each $x,y\in S_0$, and hence the right-most expression above is measurable as the supremum of a countable set of measurable functions. Since the contraction constant is always bounded, we get $c(\gamma_\omega)\in L^\infty(\Omega,\mathbb{P})$. ◻ **Remark 51**. For $\gamma_\omega$ as in Lemma [Lemma 50](#lem:mismeasurable){reference-type="ref" reference="lem:mismeasurable"}, when $\gamma_\omega$ is bounded almost surely the separability of $L^1(M,\tau)$ also implies that $\omega\mapsto \|\gamma_\omega\|$ is measurable. Indeed, let $A_0$ and $X_0$ be countable subsets of the unit balls of $M$ and $L^1(M,\tau)$, respectively, that are dense in the $\sigma$-weak operator and $\|\cdot\|_1$ topologies, respectively. Then, $$\|\gamma_\omega\| = \sup_{\substack{a\in A_0\\ x\in X_0}} |\tau(a \gamma_\omega(x))|,$$ almost surely, and thus $\|\gamma_\omega\|$ is measurable since each $\tau(a \gamma_\omega(x))$ is measurable by assumption. It follows that $$\mathbb{P}[\|\gamma_\omega\|<\infty \text{ and } \|\gamma_\omega(x)\|_1\leq \|\gamma_\omega\| \|x\|_1\ \forall x\in L^1(M,\tau)]=1.$$ In this case, one says that $\gamma_\omega$ is a *bounded* random linear operator (see [@Bharucha-Reid Definition 2.24]). $\hfill\blacksquare$ In light of the above remark, we henceforth adopt the convention of saying a random linear operator $\gamma\colon \Omega\times L^1(M,\tau)\to L^1(M,\tau)$ has a property associated to linear maps on $L^1(M,\tau)$ if for almost every $\omega\in \Omega$ the map $L^1(M,\tau)\ni x\mapsto \gamma_\omega(x)$ has the corresponding property (e.g. bounded, positive, completely positive, faithful etc.). Provided the list of properties is countable, the event that $\gamma_\omega$ has all of the properties still occurs with probability one. Recall from Definition [Definition 33](#def:Hennion_contraction){reference-type="ref" reference="def:Hennion_contraction"} that $SHC(M)$ denotes the set of all strict Hennion contractions on $M$. The following lemma analyzes a standard hypothesis in our convergence results. It tells us that as long as the event $[\Gamma_{n,m}\in SHC(M)]$ occurs with positive probability for *some* $n\geq m$, then for *any* $m\in\mathbb{Z}$ the sequence $\Gamma_{m,m}, \Gamma_{m+1,m},\ldots$ will almost surely land in $SHC(M)$ (and by Lemma [Lemma 40](#lem:compositions_of_contractions){reference-type="ref" reference="lem:compositions_of_contractions"} remain there forever after), and likewise for the sequence $\Gamma_{n,n}, \Gamma_{n,n-1},\ldots$ for any $n\in \mathbb{Z}$. This is comparable to [@jeff Assumption 1] by Example [Example 36](#exmp:strictly_positive){reference-type="ref" reference="exmp:strictly_positive"} (see also [@jeff Lemma 2.1]). **Lemma 52**. *Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual, let $(\Omega, \mathbb{P})$ be a probability space equipped with ergodic $T \in \mathop{\mathrm{Aut}}(\Omega,\mathbb{P})$, let $\gamma_\omega:L^1(M, \tau) \to L^1(M,\tau)$ be a bounded positive faithful random linear operator, and let $\Gamma_{n,m}$ be the associated interval quantum process. Suppose $$\mathbb{P}[\exists n,m\in \mathbb{Z} \text{ such that } n\geq m \text{ and } \Gamma_{n,m}\in SHC(M)]>0.$$ Then $$\mathbb{P}[\forall m\in \mathbb{Z}\ \exists n\geq m \text{ such that } \Gamma_{n,m}\in SHC(M)]=\mathbb{P}[\forall n\in \mathbb{Z}\ \exists m\leq n \text{ such that } \Gamma_{n,m}\in SHC(M)] =1.$$* *Proof.* Since $$0< \mathbb{P}[\exists n,m\in \mathbb{Z} \text{ such that } n\geq m \text{ and } \Gamma_{n,m}\in SHC(M)] \leq \sum_{m\in \mathbb{Z}} \mathbb{P}[\exists n\geq m \text{ such that } \Gamma_{n,m}\in SHC(M)],$$ it follows that $$\mathbb{P}[\exists n\geq m_0 \text{ such that } \Gamma_{n,m_0}\in SHC(M)]>0$$ for some $m_0\in \mathbb{Z}$. Observe that $\Gamma_{n,m_0}(\omega) = \Gamma_{n+m-m_0, m}(T^{m_0-m}\omega)$ for each $m\in \mathbb{Z}$ and consequently $$\begin{aligned} \label{eqn:T-invariant_tails} [\exists n\geq m \text{ such that } \Gamma_{n,m}\in SHC(M)] = T^{m_0-m}[\exists n\geq m_0 \text{ such that } \Gamma_{n,m_0}\in SHC(M)]. \end{aligned}$$ Also note that for any $m\geq m'$ one has $$[\exists n\geq m \text{ such that } \Gamma_{n,m}\in SHC(M)] \subset [\exists n\geq m' \text{ such that } \Gamma_{n,m'}\in SHC(M)]$$ by Lemma [Lemma 40](#lem:compositions_of_contractions){reference-type="ref" reference="lem:compositions_of_contractions"}. In particular, we have $$\begin{aligned} T^{-1}[\exists n\geq m_0 \text{ such that } \Gamma_{n,m_0}\in SHC(M)] &= [\exists n\geq m_0+1 \text{ such that } \Gamma_{n,m_0+1}\in SHC(M)]\\ &\subset [\exists n\geq m_0 \text{ such that } \Gamma_{n,m_0}\in SHC(M)]. \end{aligned}$$ Thus this event occurs with probability one by the ergodicity of $T$. Since $T$ is measure preserving, this along with Equation ([\[eqn:T-invariant_tails\]](#eqn:T-invariant_tails){reference-type="ref" reference="eqn:T-invariant_tails"}) yields $$\mathbb{P}[\exists n\geq m \text{ such that } \Gamma_{n,m}\in SHC(M)] = \mathbb{P}[\exists n\geq m_0 \text{ such that } \Gamma_{n,m_0}\in SHC(M)] =1$$ for all $m\in \mathbb{Z}$. Finally, it follows from continuity from above that $$\mathbb{P}[\forall m\in \mathbb{Z}\ \exists n\geq m \text{ such that } \Gamma_{n,m}\in SHC(M)] = 1.\qedhere$$ ◻ **Remark 53**. If one removes the ergodicity assumption in Lemma [Lemma 52](#lem:a_not_so_mild_hypothesis){reference-type="ref" reference="lem:a_not_so_mild_hypothesis"}, the same proof shows $$\mathbb{P}[\exists n\geq m \text{ such that } \Gamma_{n,m}\in SHC(M)] = \mathbb{P}[\exists n\geq m_0 \text{ such that } \Gamma_{n,m_0}\in SHC(M)]>0$$ for all $m\in \mathbb{Z}$, and consequently $$\mathbb{P}[\forall m\in \mathbb{Z}\ \exists n\geq m \text{ such that } \Gamma_{n,m}\in SHC(M)] = \mathbb{P}[\exists n\geq m_0 \text{ such that } \Gamma_{n,m_0}\in SHC(M)]>0.$$ Similarly, $\mathbb{P}[\forall n\in \mathbb{Z}\ \exists m\leq n \text{ such that } \Gamma_{n,m}\in SHC(M)]>0$.$\hfill\blacksquare$ These processes induce a natural filtration on the probability space. We will see that one can define a stopping time with respect to the following filtration: **Notation 54**. Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual, let $(\Omega, \mathbb{P})$ be a probability space equipped with ergodic $T \in \mathop{\mathrm{Aut}}(\Omega,\mathbb{P})$, let $\gamma_\omega:L^1(M, \tau) \to L^1(M,\tau)$ be a bounded positive faithful random linear operator, and let $\Gamma_{n,m}$ be the associated interval quantum process. 1. Fix an integer $m\in \mathbb{Z}$. For $n\ge m$ let $$\mathcal{F}^+(n,m) := \sigma( c(\Gamma^T_{n,m}), c(\Gamma_{n-1,m}^T), \dots, c(\Gamma_{m,m}^T)),$$ so that the family $\mathcal{F}^+(\bullet,m):=(\mathcal{F}^+(n,m))_{n\in [m,\infty)}$ is the natural filtration with respect to the stochastic process $c(\Gamma_{\bullet,m}):=(c(\Gamma_{n,m}))_{n\in [m,\infty)}$. 2. Similarly, we will denote by $\mathcal{F}^-(n,\bullet):=(\mathcal{F}^-(n,m))_{m\in (-\infty, n]}$ the natural filtration associated with $c(\Gamma_{n,\bullet}):=(c(\Gamma_{n,m}))_{m\in (-\infty,n]}$ given by $$\mathcal{F}^-(n,m) := \sigma( c(\Gamma^T_{n,n}), c(\Gamma_{n,n-1}^T), \dots, c(\Gamma_{n,m}^T)).$$ The following lemma is similar to [@jeff Lemma 3.11] and uses many of the ideas present in its proof as well as the proof of [@jeff Lemma 2.1]. **Lemma 55**. *Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual, let $(\Omega, \mathbb{P})$ be a probability space equipped with ergodic $T \in \mathop{\mathrm{Aut}}(\Omega,\mathbb{P})$, let $\gamma_\omega:L^1(M, \tau) \to L^1(M,\tau)$ be a bounded positive faithful random linear operator, and let $\Gamma_{n,m}$ be the associated interval ergodic quantum process. Suppose that $$\mathbb{P}[\exists n,m\in \mathbb{Z}\colon n\geq m \text{ and }\Gamma_{n,m}\in SHC(M)] >0.$$* 1. *There exists a constant $C\in [0,1)$ so that for all $m\in \mathbb{Z}$ $$\lim_{n\to +\infty} c(\Gamma_{n,m})^{\frac{1}{n-m+1}} = C$$ almost surely, and for all $n\in \mathbb{Z}$ $$\lim_{m\to-\infty} c(\Gamma_{n,m})^{\frac{1}{n-m+1}} = C$$ almost surely.* 2. *For $\kappa\in (C,1)$ and each $k\in \mathbb{Z}$, there exist finite almost surely random variables $D_{\bullet, k}$ and $D_{k,\bullet}$ satisfying $$D_{\bullet,k}(T^{\ell} \omega) = D_{\bullet, k+\ell}(\omega) \qquad \text{ and } \qquad D_{k,\bullet}(T^\ell \omega) = D_{k+\ell,\bullet}(\omega)$$ for all $\ell\in \mathbb{Z}$, and $$\begin{aligned} c(\Gamma_{n,k}) &\leq D_{\bullet, k}\kappa^{n-k+1}\\ c(\Gamma_{k,m)} &\leq D_{k,\bullet} \kappa^{k-m+1}, \end{aligned}$$ almost surely for all $n\geq k\geq m$. In particular, $c(\Gamma_{n,k}), c(\Gamma_{k,m})\to 0$ exponentially fast almost surely as $n\to +\infty$ and $m\to -\infty$.* 3. *For each $k\in \mathbb{Z}$, with respect to $\mathcal{F}^+(\bullet,k)$ (resp. $\mathcal{F}^{-}(k,\bullet)$) the random variable $\nu(\Gamma_{\bullet,k}):= \inf\{ n\geq k \colon \Gamma_{n,k}\in SHC(M)\}$ (resp. $\nu(\Gamma_{k,\bullet}):= \inf\{ m\leq k \colon \Gamma_{k,m}\in SHC(M)\}$) is a stopping time that is finite almost surely.* *Proof.* 1. Fix $m \in \mathbb{Z}$ and let $X_k:=\log[c(\Gamma_{m+k-1,m})]$ for $k\in \mathbb{N}$. By Lemma [Lemma 40](#lem:compositions_of_contractions){reference-type="ref" reference="lem:compositions_of_contractions"}, $$X_{k+\ell}(\omega) \leq X_k(\omega)+ \log [ c(\Gamma_{m+k+\ell-1,m+k}(\omega)] =X_k(\omega) + X_{\ell}(T^k\omega),$$ holds almost surely. Additionally, $X_k^{+}\equiv 0$ since $c(\Gamma_{m+k-1,m})\le 1$. Thus, by writing $n=m+k-1$, an application of Theorem [Theorem 10](#thm:kingman){reference-type="ref" reference="thm:kingman"} tells us that the sequence $(c(\Gamma_{n,m})^{\frac{1}{n-m+1}})_{n\colon n\geq m}$ converges almost surely to the constant $$C_m:= \exp\left(\inf_{n : n\geq m} \frac{1}{n-m+1} \mathbb{E}[\log(c(\Gamma_{n,m}))] \right).$$ Now, Lemma [Lemma 52](#lem:a_not_so_mild_hypothesis){reference-type="ref" reference="lem:a_not_so_mild_hypothesis"} tells us that $$\bigcup_{n\geq m} [c(\Gamma_{n,m})<1] = [\exists n\geq m \text{ such that } \Gamma_{n,m}\in SHC(M)],$$ occurs with probability one. Therefore there exists $n_1\geq m$ so that $\mathbb{P}[c(\Gamma_{n_1,m})<1]>0$. Consequently, $$\log(C_m) = \inf_{n\geq m} \frac{1}{n-m+1} \mathbb{E}[\log(c(\Gamma_{n,m}))] \leq \frac{1}{n_1-m+1}\mathbb{E}[\log(c(\Gamma_{n_1,m}))]<0,$$ and so $C_m<1$. Also, since $T$ is measure preserving for any $m,m'\in \mathbb{Z}$ we have $$\begin{aligned} C_m &= \inf_{k\geq 0} \frac{1}{k+1} \mathbb{E}[\log(c(\Gamma_{m+k,m}))]\\ &= \inf_{k\geq 0} \frac{1}{k+1} \mathbb{E}[\log(c(\Gamma_{m+k,m}\circ T^{m'-m}))] \\ &= \inf_{k\geq 0} \frac{1}{k +1}\mathbb{E}[\log(c(\Gamma_{m'+k,m'}))] = C_{m'}. \end{aligned}$$ So we set $C:= C_0$. Now, using $Y_k:= \log[c(\Gamma_{n,n-k-1})]$, $k\in \mathbb N$ and the fact that $T^{-1}$ is also ergodic, the same argument as above yields another constant $C'\in [0,1)$ such that for all $n\in \mathbb{Z}$ $$C'= \exp\left(\inf_{m: m\leq n}\frac{1}{n-m+1} \mathbb{E}[\log(c(\Gamma_{n,m}))]\right)$$ and $$\lim_{m\to-\infty} c(\Gamma_{n,m})^{\frac{1}{n-m+1}} = C'$$ almost surely. Given $\epsilon>0$, let $n\geq 1$ be large enough so that $$\exp\left(\frac{1}{n} \mathbb{E}[\log(c(\Gamma_{n,1}))]\right) \leq C+\epsilon.$$ Then $$C' = \exp\left(\inf_{m\colon m\leq n} \frac{1}{n-m+1} \mathbb{E}[\log(c(\Gamma_{n,m}))]\right) \leq \exp\left(\frac{1}{n} \mathbb{E}[\log(c(\Gamma_{n,1}))]\right) \leq C+ \epsilon.$$ Letting $\epsilon\to 0$ yields $C'\leq C$, and reversing the roles of $n$ and $m$ gives $C'=C$. 2. Fix $\kappa\in (C,1)$ and $k\in \mathbb{Z}$ and define $$D_{\bullet,k}:= 1 \vee \sup_{n: n\geq k} \frac{c(\Gamma_{n,k})}{\kappa^{n-k+1}} \qquad \text{ and } \qquad D_{k,\bullet}:= 1 \vee \sup_{m: m\leq k} \frac{c(\Gamma_{k,m})}{\kappa^{k-m+1}},$$ which are random variables by Lemma [Lemma 50](#lem:mismeasurable){reference-type="ref" reference="lem:mismeasurable"}. By the previous part, for almost every $\omega\in \Omega$ $$\lim_{n\to+\infty} c( \Gamma_{n,k})^{\frac{1}{n-k+1}} = \lim_{m\to -\infty} c(\Gamma_{k,m})^{\frac{1}{k-m+1}}=C< \kappa.$$ Consequently there is an $\ell_0\geq 0$ (depending on $\kappa$, $k$, and $\omega$) so that $$\frac{c(\Gamma_{n,k})}{\kappa^{n-k+1}}, \frac{c(\Gamma_{k,m})}{\kappa^{k-m+1}} \leq 1$$ for all $n> k+\ell_0$ and $m < k- \ell_0$, which implies $$\begin{aligned} D_{\bullet,k}(\omega) &= 1 \vee \max_{n: k \leq n \leq k+\ell_0} \frac{c(\Gamma_{n,k}(\omega))} {\kappa^{n-k+1}} \leq \frac{1}{\kappa^{\ell_0+1}} <\infty,\\ D_{k,\bullet}(\omega) &= 1 \vee \max_{m: k-\ell_0\leq m \leq k} \frac{c(\Gamma_{k,m}(\omega))}{\kappa^{k-m+1}} \leq \frac{1}{\kappa^{\ell_0+1}}<\infty. \end{aligned}$$ Hence $D_{\bullet,k}$ and $D_{k,\bullet}$ are finite almost surely, and the remaining properties follow from their definition. 3. This is an immediate consequence of Lemma [Lemma 9](#lem:submultiplicativeStoppingtime){reference-type="ref" reference="lem:submultiplicativeStoppingtime"}.  ◻ ## Ergodic quantum processes from normal maps {#subsec:right_EQP_from_normal_maps} As noted above, in order to emulate the proofs of [@jeff] it is necessary to consider the duals of quantum processes, which according to Lemma [Lemma 6](#lem:normalbddextension){reference-type="ref" reference="lem:normalbddextension"} should correspond to *random* normal positive linear maps on $M$. In fact, from the perspective of von Neumann algebras, such maps may be even more natural than those on $L^1(M,\tau)$. The goal of this section is to formalize this notion and relate it to random linear operators on $L^1(M,\tau)$. **Definition 56**. Let $(M,\tau)$ be a tracial von Neumann algebra and $(\Omega, \mathbb{P})$ be a probability space. A **weak\* random variable** is a function $f\colon \Omega\to M$ such that $\tau(f(\omega)x)$ is measurable for all $x\in L^1(M,\tau)$. We say that a mapping $\phi\colon \Omega\times M\to M$ is a **weak\* random linear operator** if $\omega\mapsto \phi_\omega(a)$ is a weak\* random variable for all $a\in M$ and $\mathbb{P}[\phi(\alpha a + b)= \alpha \phi(a) + \phi(b)]=1$ for all $a,b\in M$ and $\alpha\in \mathbb C$. As we did with random linear operators on $L^1(M,\tau)$, we adopt the convention of saying a weak\* random linear operator $\phi\colon \Omega\times M\to M$ has a property associated to linear maps on $M$ if for almost every $\omega\in \Omega$ the map $M\ni a\mapsto \phi_\omega(a)$ has the corresponding property (e.g. normal, positive, completely positive, $\tau$-bounded). We first establish the random version of the correspondence in Lemma [Lemma 6](#lem:normalbddextension){reference-type="ref" reference="lem:normalbddextension"} between normal linear maps on $M$ and bounded linear maps on $L^1(M,\tau)$. **Lemma 57**. *Let $(M,\tau)$ be a tracial von Neumann algebra with separable predual, and let $(\Omega,\mathbb{P})$ be a probability space. Up to almost sure equality, there is a one-to-one correspondence between normal positive (resp. completely positive) weak\* random linear operators $\phi_\omega$ on $M$ and bounded positive (resp. completely positive) random linear maps $(\phi_\omega)_*$ on $L^1(M,\tau)$ determined by $$\begin{aligned} \label{eqn:random_tau_relation} \tau((\phi_\omega)_*(x) a)=\tau(x\phi_\omega(a)) \qquad\qquad \omega\in \Omega,\ x\in L^1(M,\tau),\ a\in M. \end{aligned}$$ This correspondence restricts to a one-to-one correspondence between $\tau$-bounded normal positive (resp. completely positive) weak\* random linear operators on $M$ and $M$-preserving bounded positive (resp. completely positive) random linear operators on $L^1(M,\tau)$. The former maps $\phi_\omega$ also admit unique extensions $\phi_\omega|^{L^1(M,\tau)}$ to $L^1(M,\tau)$ that are $M$-preserving bounded and positive (resp. completely positive), and the latter maps $(\phi_\omega)_*$ also have restrictions $(\phi_\omega)_*|_M$ to $M$ that are $\tau$-bounded normal and positive (resp. completely positive). In this case, one has $((\phi_\omega)_*|_M)_*= \phi_\omega|^{L^1(M,\tau)}$.* *Proof.* By Lemma [Lemma 6](#lem:normalbddextension){reference-type="ref" reference="lem:normalbddextension"}, it suffices to check measurability. For the first correspondence, this is a consequence of Equation ([\[eqn:random_tau_relation\]](#eqn:random_tau_relation){reference-type="ref" reference="eqn:random_tau_relation"}). (Note that we are invoking the separability of $L^1(M,\tau)$ here to reduce checking that $(\phi_\omega)_*(x)$ is a random variable to checking that it is weakly measurable.) For the measurability of extensions and restrictions, note that if $(b_n)_{n\in \mathbb N}\subset M$ converges to $x\in L^1(M,\tau)$ in $\|\cdot\|_1$-norm, then for any $a\in M$ one has $$\tau(\phi_\omega(x) a) = \lim_{n\to\infty} \tau(\phi_\omega(b_n) a) = \lim_{n\to\infty} \tau(b_n (\phi_\omega)_*(a)) = \tau(x (\phi_\omega)_*(a))$$ almost surely. Each $\tau(\phi_\omega(b_n) a)$ is measurable, from which it follows that the extension $\phi_\omega|^{L^1(M,\tau)}$ is a random linear operator and the restriction $(\phi_\omega)_*|_M$ is a weak\* random linear operator. ◻ **Remark 58**. Note that if $c_\omega$ denotes the norm of $\phi_\omega$ on $L^1(M,\tau)$ whenever the extension exists, then $\omega\to c_\omega$ is a random variable by Remark [Remark 51](#rem:almost_surely_bounded_is_bounded_by_measurable){reference-type="ref" reference="rem:almost_surely_bounded_is_bounded_by_measurable"}. Thus when $L^1(M,\tau)$ is separable, $\phi_\omega$ being $\tau$-bounded almost surely is equivalent to there existing a (finite almost surely) random variable $c\colon \Omega\to [0,\infty]$ so that $\mathbb{P}[\tau\circ \phi_\omega(x^*x) \leq c_\omega \tau(x^*x)\ \forall x\in M]=1$. $\hfill\blacksquare$ Of course, the first correspondence in Lemma [Lemma 57](#lem:measurabilityTest){reference-type="ref" reference="lem:measurabilityTest"} is still true without any positivity assumptions, and consequently compositions of normal weak\* random linear operators on $M$ give normal weak\* random linear operators. Indeed, the predual maps are bounded random linear operators on $L^1(M,\tau)$ whose composition (in the reverse order) is also a bounded random linear operator by Lemma [Theorem 12](#thm:RLOcomp){reference-type="ref" reference="thm:RLOcomp"}. The dual of this then gives a weak\* random linear operator that is almost surely equal to the composition of the original weak\* random linear operators. **Definition 59**. Let $(M,\tau)$ be a tracial von Neumann algebra, let $(\Omega,\mathbb{P})$ be a probability space equipped with $T\in \mathop{\mathrm{Aut}}(\Omega,\mathbb{P})$, and let $\phi_\omega \colon M\to M$ be a normal weak\* random linear operator. We call the family of normal weak\* random linear operators $$\Phi_{n,m}^T(\omega):= \phi_{T^n\omega}\circ \cdots \circ \phi_{T^m\omega} \qquad\qquad n,m\in \mathbb{Z},\ n\geq m$$ an (**interval**) **quantum process** on $M$. If $T$ is ergodic, then we call the above an **ergodic** quantum process on $M$. When no confusion can arise, we will supress the superscript $T$. The most common example of a quantum process on $M$ that we shall consider is the following. Suppose $\Gamma_{n,m}^T$ is a quantum process on $L^1(M,\tau)$ associated to some bounded positive random linear operator $\gamma_\omega$ and $T\in \mathop{\mathrm{Aut}}(\Omega,\mathbb{P})$. Denote $\phi_\omega:=\gamma_\omega^*$, which is a normal positive weak\* random linear operator on $M$ by Lemma [Lemma 57](#lem:measurabilityTest){reference-type="ref" reference="lem:measurabilityTest"}. Then quantum process $\Phi_{n,m}^{T^{-1}}$ on $M$ associated to $\phi_\omega$ and $T^{-1}$ satisfies $$\begin{aligned} \label{eqn:dual_quantum_process} \begin{split} (\Phi_{n,m}^{T^{-1}}(\omega))_* &= \Gamma_{-m,-n}^{T}(\omega),\\ (\Gamma_{n,m}^T(\omega))^* &= \Phi_{-m,-n}^{T^{-1}}(\omega), \end{split} \end{aligned}$$ for all $n\geq m$ and $\omega\in \Omega$. ## Convergence properties {#subsec:convergence_properties_of_EQP} We now prove the first main result of this section. This is the analogue of [@jeff Lemma 3.14], which gives roughly half the proof of [@jeff Theorem 1] (see Remark [Remark 63](#rmk:recovering_jeff_results){reference-type="ref" reference="rmk:recovering_jeff_results"} below). The key observation is that $\Gamma_{n,m-1}\cdot S \subset \Gamma_{n,m}\cdot S$ and that the diameter of these sets tends to zero almost surely as $m\to -\infty$ by Lemma [Lemma 55](#lem:randomconstproperties){reference-type="ref" reference="lem:randomconstproperties"}. This is more or less the same proof as in the finite dimensional case, where one can also treat the limit $n\to +\infty$ by considering the dual process $\Gamma_{n,m}^*(\omega)$. This limit is analyzed in [@jeff Lemma 3.12], which forms the other half of the proof of [@jeff Theorem 1]. However, in the infinite dimensional setting $\Gamma_{n,m}^*$ is a process on $M$ rather than $L^1(M,\tau)$ and therefore requires a separate argument that we present as Theorem [Theorem 62](#thm:right_convergence_on_M){reference-type="ref" reference="thm:right_convergence_on_M"} (see also the proof of Corollary [Corollary 64](#cor:analogue_of_jeff2){reference-type="ref" reference="cor:analogue_of_jeff2"}). **Theorem 60** (Theorem [Theorem 1](#thmx:A){reference-type="ref" reference="thmx:A"}). *Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual, let $(\Omega, \mathbb{P})$ be a probability space equipped with ergodic $T\in \mathop{\mathrm{Aut}}(\Omega, \mathbb{P})$, let $\gamma_\omega\colon L^1(M,\tau)\to L^1(M,\tau)$ be a bounded positive faithful random linear operator, and let $\Gamma_{n,m}$ be the associated interval ergodic quantum process. Suppose that $$\mathbb{P}[\exists n,m\in \mathbb{Z}\colon n\geq m \text{ and }\Gamma_{n,m}\in SHC(M)] >0.$$ Then there exists a family of random variables $X_n\colon \Omega\to S$, $n\in \mathbb{Z}$, satisfying $$\begin{aligned} \label{eqn:right_process_limit_properties} \gamma_{T^{n+1}\omega}\cdot X_n(\omega) = X_{n+1}(\omega) \qquad \text{ and } \qquad X_n(T^{\pm1}\omega) = X_{n\pm 1}(\omega) \end{aligned}$$ almost surely, and for all $x\in S$ $$\lim_{m\to -\infty} \|\Gamma_{n,m}\cdot x - X_n\|_1 =0$$ almost surely for all $n\in \mathbb{Z}$.* *Proof.* For each $\omega\in \Omega$ define the family of (random) sets $S_{n,m}(\omega):= \Gamma_{n,m}(\omega)\cdot S$, $m\leq n$, and observe that $S_{n,m-1}\subset S_{n,m}$ by construction. By Proposition [Proposition 34](#prop:cont_const_is_diam){reference-type="ref" reference="prop:cont_const_is_diam"} and Lemma [Lemma 55](#lem:randomconstproperties){reference-type="ref" reference="lem:randomconstproperties"}, we know that $\mathop{\mathrm{diam}}(S_{n,m})= c(\Gamma_{n,m}) \to 0$ almost surely as $m\to - \infty$. Moreover, since $S$ is complete by Therorem [Theorem 19](#thm:tracemetricinequality_and_completeness){reference-type="ref" reference="thm:tracemetricinequality_and_completeness"}, we may invoke the Cantor intersection theorem to conclude $$\bigcap_{m\colon m\leq n} S_{n,m}(\omega)$$ consists of a single element $X_n(\omega)$ for almost every $\omega$. Note that the relations in Equation ([\[eqn:right_process_limit_properties\]](#eqn:right_process_limit_properties){reference-type="ref" reference="eqn:right_process_limit_properties"}) follow from the relations $$\gamma_{T^{n+ 1}\omega} \cdot S_{n,m}(\omega) = S_{n+ 1,m}(\omega) \qquad { and } \qquad S_{n,m}(T^{\pm 1}\omega) = S_{n\pm 1,m\pm 1}(\omega)$$ for each $\omega\in \Omega$. Now, fix $n\in \mathbb{Z}$ and $x\in S$. By Theorem [Theorem 19](#thm:tracemetricinequality_and_completeness){reference-type="ref" reference="thm:tracemetricinequality_and_completeness"} we have $$\|\Gamma_{n,m}\cdot x - X_n\|_1 \leq 2 d(\Gamma_{n,m}\cdot x, X_n) \leq 2 \mathop{\mathrm{diam}}(S_{n,m}) = 2 c(\Gamma_{n,m}),$$ which tends to zero almost surely as $m\to-\infty$ by Lemma [Lemma 55](#lem:randomconstproperties){reference-type="ref" reference="lem:randomconstproperties"}. Consequently, for all $a\in M$ we have $$\tau(X_n a) = \lim_{m\to -\infty} \tau((\Gamma_{n,m}\cdot x) a),$$ almost surely so that $X_n$ is weakly measurable, and hence a random variable by Theorem [Theorem 11](#thm:Pettis){reference-type="ref" reference="thm:Pettis"}. ◻ **Example 61**. Let $(M,\tau)$ be a tracial von Neumann algebra with separable predual and let $(\Omega,\mathbb{P})$ be a locally compact Hausdorff space with a Radon probability measure. Denote $N:=M\bar\otimes L^\infty(\Omega,\mathbb{P})$ and $\varphi:=\tau\otimes \int_\Omega\cdot\ d\mathbb{P}$. By [@TakesakiI Theorem IV.7.17] $L^1(N,\varphi)$ can be identified with functions $f\colon \Omega\to L^1(M,\tau)$ such that $\omega\mapsto f_\omega(a)$ is measurable for all $a\in M$ and $$\int_{\Omega} \|f_\omega\|_1\ d\mathbb{P}(\omega) <\infty$$ (see also [@TakesakiI Propostions IV.7.2 and IV.7.4]). In particular, if $f\in L^1(N,\varphi)_+$ then $f_\omega\in L^1(M,\tau)_+$ almost surely. Fix $f\in L^1(N,\varphi)_+$ which is non-zero almost surely and $\eta\in (0,1]$. Let $\{(a_i, f^{(i)})\in M_+\times L^1(N,\varphi)_+\colon i\in I\}$ be a countable family satisfying: 1. $a:= \sum_{i\in I} a_i$ converges in the strong operator topology; 2. $\tau(xa)>0$ for all $x\in L^1(M,\tau)_+\setminus\{0\}$; 3. $0<f^{(i)} \leq \eta^{-1} f$ almost surely for all $i\in I$; 4. $\mathbb{P}[\forall i\in I\ \eta f \leq f^{(i)}]>0$. Then by Example [Example 39](#exmp:strongly_summable){reference-type="ref" reference="exmp:strongly_summable"}, $$\gamma(x):= \sum_{i\in I} \varphi(x(a_i\otimes 1)) f^{(i)}$$ defines a bounded positive faithful linear map on $L^1(N,\varphi)$ satisfying $\gamma\cdot x\leq \kappa^{-1} f^{(0)}$, where $f^{(0)} = \frac{f}{\varphi(f)}$ and $\kappa = \eta^2$. Consequently, $$\gamma_\omega(x):=\gamma(x\otimes 1) = \sum_{i\in I} \tau(x a_i) f^{(i)}_\omega$$ defines a bounded random linear operator on $L^1(M,\tau)$. Moreover, $\gamma$ is positive and faithful almost surely. Indeed, recalling that $I$ is countable we see that $f^{(i)} \in L^1(M,\tau)_+$ almost surely implies $\gamma(x)$ is positive almost surely for $x\in L^1(M,\tau)_+$. Next (3) implies $f^{(i)}$ is non-zero almost surely so that $$\mathbb{P}[ \exists i\in I \text{ such that } f_\omega^{(i)}=0]=0.$$ Thus for $x\in L^1(M,\tau)_+\setminus\{0\}$ $$\mathbb{P}[\gamma(x)=0]=\mathbb{P}[\forall i\in I\ \tau(xa_i)=0]=\mathbb{P}[\tau(xa) =0]=0$$ by (2). We also note that $\kappa f^{(0)} \leq \gamma\cdot x\leq \kappa^{-1} f^{(0)}$ holds with positive probability by (3) and (4), which tells us that $\gamma_\omega$ is a strict Hennion contraction with positive probability by Theorem [Theorem 38](#thm:strict_Hennion_projective_actions){reference-type="ref" reference="thm:strict_Hennion_projective_actions"}. Now, let $T\in \text{Aut}(\Omega,\mathbb{P})$ be an ergodic automorphism. Then the associated interval ergodic quantum process is given by $$[\Gamma_{n,m}(\omega)](x) = \sum_{i_m,\ldots, i_n\in I} \tau(xa_{i_m})\tau(f_{T^m\omega}^{(i_m)}a_{i_{m+1}}) \cdots \tau(f_{T^{n-1}\omega}^{(i_{n-1})} a_{i_n}) f_{T^n\omega}^{(i_n)},$$ and satisfies $$\mathbb{P}[\exists n,m\in \mathbb{Z}\colon n\geq m \text{ and } \Gamma_{n,m}\in SHC(M)] >0$$ since we noted above that $\Gamma_{0,0}=\gamma$ is a strict Hennion contraction with positive probability. Theorem [Theorem 60](#thm:right_convergence){reference-type="ref" reference="thm:right_convergence"} therefore yields a family of random variables $F_n\colon \Omega \to S$ (which we can identify with elements of $L^1(N,\varphi)_+$) so that $$\lim_{m\to-\infty} \frac{1}{\tau([\Gamma_{n,m}(\omega)](x)} \sum_{i_m,\ldots, i_n\in I} \tau(xa_{i_m})\tau(f_{T^m\omega}^{(i_m)}a_{i_{m+1}}) \cdots \tau(f_{T^{n-1}\omega}^{(i_{n-1})} a_{i_n}) f_{T^n\omega}^{(i_n)} = F_n(\omega)$$ in $\|\cdot\|_1$-norm almost surely. Additionally, one has $$F_{n+1}(\omega) = \gamma_{T^{n+1}\omega}\cdot F_n(\omega) = \frac{1}{\tau(\gamma_{T^{n+1}\omega} (F_n(\omega)))} \sum_{i\in I} \tau(F_n(\omega) a_i) f^{(i)}_\omega$$ almost surely by Equation ([\[eqn:right_process_limit_properties\]](#eqn:right_process_limit_properties){reference-type="ref" reference="eqn:right_process_limit_properties"}).$\hfill\blacksquare$ The following uses the second part of Lemma [Lemma 57](#lem:measurabilityTest){reference-type="ref" reference="lem:measurabilityTest"} to extend a $\tau$-bounded weak\* random linear operator on $M$ to a bounded random linear operator on $L^1(M,\tau)$. **Theorem 62**. *Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual, let $(\Omega, \mathbb{P})$ be a probability space equipped with ergodic $T\in \mathop{\mathrm{Aut}}(\Omega, \mathbb{P})$, let $\phi_\omega\colon M\to M$ be a $\tau$-bounded normal positive weak\* random linear operator. Suppose that the extension of $\phi_\omega$ to $L^1(M,\tau)$ is faithful almost surely and that the associated ergodic quantum process $\Phi_{n,m}$ on $L^1(M,\tau)$ satisfies $$\mathbb{P}[\exists n,m\in \mathbb{Z}\colon n\geq m \text{ and }\Phi_{n,m}\in SHC(M)] >0.$$ Then there exists a family of random variables $A_n\colon \Omega\to S_b$, $n\in \mathbb{Z}$, satisfying $$\gamma_{T^{n+ 1}\omega}\cdot A_n(\omega) = A_{n+1}(\omega) \qquad \text{ and }\qquad A_n(T^{\pm 1}\omega)= A_{n\pm 1}(\omega)$$ almost surely, and for all $x\in S$ $$\lim_{m\to -\infty} \left\|\Phi_{n,m}\cdot x - A_n \right\|_1 =0$$ almost surely for all $n\in \mathbb{Z}$. Moreover, the above convergence holds in $\|\cdot\|_\infty$-norm for $x\in S_b$.* *Proof.* Applying Theorem [Theorem 60](#thm:right_convergence){reference-type="ref" reference="thm:right_convergence"} gives us this family of random variables $A_n$, $n\in \mathbb{Z}$, though we must argue they are almost surely valued in $S_b$. Using Lemma [Lemma 55](#lem:randomconstproperties){reference-type="ref" reference="lem:randomconstproperties"} for almost every $\omega$, $c(\Phi_{n,m}) <1$ for sufficiently small $m$ (depending on $\omega$). When this occurs, $\Phi_{n,m}(M)\subset M$ implies $\Phi_{n,m}\cdot S \subset S_b$ by Theorem [Theorem 24](#thm:geometry){reference-type="ref" reference="thm:geometry"}. Recalling from the proof of Theorem [Theorem 60](#thm:right_convergence){reference-type="ref" reference="thm:right_convergence"} that $A_n\in \bigcap_{m\leq n} \Phi_{n,m}\cdot S$, we thus have $A_n\in S_b$ almost surely. The final statement then follows from Lemma [Lemma 26](#thm:cauchytocauchy){reference-type="ref" reference="thm:cauchytocauchy"}.(1): $\Phi_{n,m}\cdot S_b\subset S_b$ since $\phi_\omega$ is $M$-preserving and the proof of Theorem [Theorem 60](#thm:right_convergence){reference-type="ref" reference="thm:right_convergence"} in fact shows $d(\Phi_{n,m}\cdot x, A_n)\to 0$ as $m\to -\infty$. ◻ **Remark 63**. Theorems [Theorem 60](#thm:right_convergence){reference-type="ref" reference="thm:right_convergence"} and [Theorem 62](#thm:right_convergence_on_M){reference-type="ref" reference="thm:right_convergence_on_M"} can be used to recover [@jeff Lemmas 3.12 and 3.14], respectively, which together yields [@jeff Theorem 1]. Indeed, the hypotheses of Theorems [Theorem 60](#thm:right_convergence){reference-type="ref" reference="thm:right_convergence"} and [Theorem 62](#thm:right_convergence_on_M){reference-type="ref" reference="thm:right_convergence_on_M"} follow for $\Gamma_{n,m}$ and $\Gamma_{n,m}^*$, respectively, from [@jeff Assumption 1 and Lemma 2.1] (or are automatic in the finite dimensional case). Consequently, any fixed points of the projective actions of $\Gamma_{n,m}$ and $\Gamma_{n,m}^*$ converge to $X_n$ as $m \to -\infty$ and $A_m$ as $n\to+\infty$, respectively. In fact, our results are slightly more general than those of [@jeff] because our hypotheses allow $X_n$ and $A_m$ to be valued outside of $S_b^\times$. $\hfill\blacksquare$ As a corollary to Theorems [Theorem 60](#thm:right_convergence){reference-type="ref" reference="thm:right_convergence"} and [Theorem 62](#thm:right_convergence_on_M){reference-type="ref" reference="thm:right_convergence_on_M"}, we also obtain an analogue of [@jeff Theorem 2]. The discrepancy between that result and the one below are due to the inequivalence of the $\|\cdot\|_1$-norm and $\|\cdot\|_\infty$-norm in the general case, but it is clear how to reconcile the difference in the finite dimensional case. **Corollary 64**. *Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual, let $(\Omega, \mathbb{P})$ be a probability space equipped with ergodic $T\in \mathop{\mathrm{Aut}}(\Omega, \mathbb{P})$, let $\gamma_\omega\colon L^1(M,\tau)\to L^1(M,\tau)$ be an $M$-preserving bounded positive faithful random linear operator, and let $\Gamma_{n,m}$ be the associated interval ergodic quantum process. Suppose that the extension of $\gamma_\omega^*$ to $L^1(M,\tau)$ is faithful almost surely and that $$\mathbb{P}[\exists n,m\in \mathbb{Z}\colon n\geq m \text{ and }\Gamma_{n,m}\in SHC(M)] >0.$$ Fix $k\in \mathbb{Z}$ and let $C$ be as in Lemma [Lemma 55](#lem:randomconstproperties){reference-type="ref" reference="lem:randomconstproperties"}. Then for any $\kappa\in (C,1)$ and $n\geq k> m$, there exists random variables $X_n, B_m\colon \Omega\to S_b$ and $E_{ k}\colon\Omega\to [0,\infty)$ such that for all $a\in M$ $$\left\|\frac{1}{\tau(\Gamma_{n,m}(1))} \Gamma_{n,m}(a) - \tau(B_m a) X_n \right\|_1 \leq E_{k} \kappa^{n-m+1}\|a\|_\infty$$ almost surely.* *Proof.* Applying Theorem [Theorem 60](#thm:right_convergence){reference-type="ref" reference="thm:right_convergence"} gives the random variables $X_n$, which we note are valued in $S_b$ almost surely since $\gamma_\omega$ is $M$-preserving (see the proof of Theorem [Theorem 62](#thm:right_convergence_on_M){reference-type="ref" reference="thm:right_convergence_on_M"}). Also recall from the proof of Theorem [Theorem 60](#thm:right_convergence){reference-type="ref" reference="thm:right_convergence"} that we actually have $$\begin{aligned} \label{eqn:first_factor_bound} \|\Gamma_{n,m}\cdot x - X_n \|_1 \leq 2 c(\Gamma_{n,m}) \end{aligned}$$ for all $x\in S$ almost surely. Next, denote $\phi_\omega:=\gamma_\omega^*$, which by Lemma [Lemma 57](#lem:measurabilityTest){reference-type="ref" reference="lem:measurabilityTest"} is a $\tau$-bounded normal positive weak\* random linear operator. It also almost surely has a faithful extension to $L^1(M,\tau)$ be assumption. Denote by $\Phi_{n,m}^{T^{-1}}$ the ergodic quantum processes associated to $\phi_\omega$ and $T^{-1}$ so that one has $(\Gamma_{n,m}^T)^* = \Phi_{-m,-n}^{T^{-1}}$. Using Lemma [Lemma 43](#lem:contraction_constant_of_duals){reference-type="ref" reference="lem:contraction_constant_of_duals"}, it follows that $$\mathbb{P}[\exists n,m\in \mathbb{Z}\colon n\geq m \text{ and }\Phi_{n,m}^{T^{-1}}\in SHC(M)] = \mathbb{P}[\exists n,m\in \mathbb{Z}\colon n\geq m \text{ and }\Gamma_{n,m}^{T}\in SHC(M)]>0.$$ Thus we can apply Theorem [Theorem 62](#thm:right_convergence_on_M){reference-type="ref" reference="thm:right_convergence_on_M"} to obtain random variables $A_n\colon \Omega\to S_b$ satisfying $$\| \Phi_{n,m}\cdot x - A_n \|_1 \leq 2 c(\Phi_{n,m}) = 2 c(\Gamma_{-m,-n}).$$ for all $x\in S$ almost surely. If we denote $B_m:=A_{-m}$, then by the above we have $$\begin{aligned} \label{eqn:second_factor_bound} | \tau([\Gamma_{n,m}^*\cdot 1 - B_m]a) | \leq 2 c(\Gamma_{m,n}) \|a\|_\infty \end{aligned}$$ for $a\in M$ almost surely. Now, observe that $$\frac{1}{\tau(\Gamma_{n,m}(1))} \Gamma_{n,m}(a) = \frac{\tau(\Gamma_{n,m}(a))}{\tau(\Gamma_{n,m}(1))} \Gamma_{n,m}\cdot a = \tau( [\Gamma_{n,m}^*\cdot 1]a) \Gamma_{n,m}\cdot a.$$ Thus combining Estimates ([\[eqn:first_factor_bound\]](#eqn:first_factor_bound){reference-type="ref" reference="eqn:first_factor_bound"}) and ([\[eqn:second_factor_bound\]](#eqn:second_factor_bound){reference-type="ref" reference="eqn:second_factor_bound"}), for $a\in S_b$ we have $$\begin{aligned} \left\| \frac{1}{\tau(\Gamma_{n,m}(1))} \Gamma_{n,m}(a) - \tau(B_m a) X_n \right\|_1 &\leq |\tau([\Gamma_{n,m}^*\cdot 1]a)| \| \Gamma_{n,m}\cdot a - X_n\|_1 + | \tau([\Gamma_{n,m}^*\cdot 1 - B_m]a) | \|X_n\|_1\\ &\leq 4 c(\Gamma_{n,m}) \|a\|_\infty. \end{aligned}$$ Applying the above estimate to arbitrary $a\in M$ by decomposing it into a linear combination of four positive elements and scaling gives an upper bound of $16 c(\Gamma_{n,m}) \|a\|_\infty$. For $n\geq k> m$, using Lemmas [Lemma 40](#lem:compositions_of_contractions){reference-type="ref" reference="lem:compositions_of_contractions"} and [Lemma 55](#lem:randomconstproperties){reference-type="ref" reference="lem:randomconstproperties"}, we can further bound this by $$16 c(\Gamma_{n,k}) c(\Gamma_{k-1,m}) \|a\| \leq 16 D_{\bullet, k} \kappa^{n-k+1} D_{k-1,\bullet} \kappa^{k -m} \|a\|_\infty = 16 D_{\bullet, k} D_{k-1,\bullet} \kappa^{n-m+1}\|a\|_\infty.$$ So Taking $E_k(\omega):= 16 D_{\bullet, k}(\omega) D_{k-1,\bullet}(\omega)$ completes the proof. ◻ We next prove the second main result of the section, which is essentially a dual version of Theorem [Theorem 60](#thm:right_convergence){reference-type="ref" reference="thm:right_convergence"} for ergodic quantum processes on $M$. Interestingly, this result and the next do *not* have analogues in [@jeff]. **Theorem 65** (Theorem [Theorem 2](#thmx:B){reference-type="ref" reference="thmx:B"}). *Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual, let $(\Omega, \mathbb{P})$ be a probability space equipped with ergodic $T\in \mathop{\mathrm{Aut}}(\Omega, \mathbb{P})$, let $\phi_\omega\colon M\to M$ be a normal positive weak\* random linear operator, and let $\Phi_{n,m}$ be the associated ergodic quantum process. Suppose that $\phi_\omega(1)$ is invertible almost surely and $$\mathbb{P}[\exists n,m\in \mathbb{Z}\colon n\geq m \text{ and }\eta \tau_{x_0} \leq \Phi_{m,n} \leq \eta^{-1} \tau_{x_0} \text{ for some }\eta\in (0,1],\ x_0\in S] >0.$$ Then there exists a family of random variables $Y_m\colon \Omega\to S$, $m\in \mathbb{Z}$, satisfying $$(\phi_{T^{m- 1}\omega})_*\cdot Y_m(\omega) = Y_{m-1}(\omega) \qquad \text{ and } \qquad Y_m(T^{\pm 1}\omega)= Y_{m\pm 1}(\omega)$$ almost surely, and for all $a\in M$ $$\lim_{n\to\infty } \left\|\Phi_{n,m}(1)^{-\frac12}\Phi_{n,m}(a)\Phi_{n,m}(1)^{-\frac12} - \tau(a Y_m) \right\|_\infty =0$$ almost surely for all $m\in \mathbb{Z}$.* *Proof.* Denote $\gamma_\omega:= (\phi_\omega)_*$, which is a bounded positive random linear operator on $L^1(M,\tau)$ by Lemma [Lemma 57](#lem:measurabilityTest){reference-type="ref" reference="lem:measurabilityTest"}. Moreover, $\gamma_\omega$ is faithful almost surely by the discussion following Lemma [Lemma 41](#lem:when_is_predual_faithful){reference-type="ref" reference="lem:when_is_predual_faithful"}. Letting $\Gamma_{n,m}^{T^{-1}}$ denote the associated ergodic quantum process on $L^1(M,\tau)$, we have $$(\Phi_{n,m}^T(\omega))_* = \Gamma_{-m,-n}^{T^{-1}}(\omega),$$ by Equation ([\[eqn:dual_quantum_process\]](#eqn:dual_quantum_process){reference-type="ref" reference="eqn:dual_quantum_process"}). Lemma [Corollary 44](#cor:when_do_preduals_give_SHC){reference-type="ref" reference="cor:when_do_preduals_give_SHC"} implies $\Gamma_{-m,-n}\in SHC(M)$ if and only if there exists $x_0\in S$ and $\eta\in (0,1]$ so that $\eta \tau(x_0 a)\Phi_{m,n}(1) \leq \Phi_{m,n}(a) \leq \eta^{-1} \tau(x_0 a) \Phi_{m,n}(1)$ for all $a\in M_+$. Since $\Phi_{n,m}(1)$ is invertible almost surely by assumption, the latter is almost surely equivalent to $\eta \tau_{x_0} \leq \Phi_{n,m} \leq \eta^{-1} \tau_{x_0}$ after adjusting $\eta$ as necessary. Thus, altogether our hypotheses imply $$\mathbb{P}[\exists n,m\in \mathbb{Z}\colon n\geq m \text{ and } \Gamma_{n,m}^{T^{-1}}\in SHC(M)] >0.$$ Let $X_n\colon \Omega\to S$, $n\in \mathbb{Z}$, be the family of random variables obtained by applying Theorem [Theorem 60](#thm:right_convergence){reference-type="ref" reference="thm:right_convergence"} to $\Gamma_{n,m}^{T^{-1}}$. Set $Y_m:= X_{-m}$ for each $m\in \mathbb{Z}$ so that the claimed relations follow from Equation ([\[eqn:right_process_limit_properties\]](#eqn:right_process_limit_properties){reference-type="ref" reference="eqn:right_process_limit_properties"}) with $T$ replaced by $T^{-1}$. Now, fix $a\in M$ and $x\in S$ and denote $y:= \Phi_{n,m}(1)^{-\frac12} x \Phi_{n,m}(1)^{-\frac12}\in L^1(M,\tau)_+$. We have $$\begin{aligned} \tau\left( \left[\Phi_{n,m}(1)^{-\frac12}\Phi_{n,m}(a)\Phi_{n,m}(1)^{-\frac12} - \tau(a Y_m) \right] x \right) &= \tau( a [\Gamma_{-m,-n}(y) - Y_m])\\ &= \tau( a [ \Gamma_{-m,-n}\cdot y - X_{-m}]). \end{aligned}$$ where in the last equality we have used $\tau(\Gamma_{-m,-n}(y)) = \tau(\Phi_{n,m}(1) y) = \tau(x)=1$. Denote $y_0:= \frac{y}{\tau(y)}$, which satisfies $y_0\in S$ and $\Gamma_{-m,-n}\cdot y_0 = \Gamma_{-m,-n}\cdot y$. We can therefore use the above computation to obtain the following estimate: $$\left| \tau\left( \left[\Phi_{n,m}(1)^{-\frac12}\Phi_{n,m}(a)\Phi_{n,m}(1)^{-\frac12} - \tau(a Y_m) \right] x \right)\right| \leq \|a\| \| \Gamma_{-m,-n}\cdot y_0 - X_{-m} \|_1.$$ Recall from the proof of Theorem [Theorem 60](#thm:right_convergence){reference-type="ref" reference="thm:right_convergence"} that the second factor in the last expression is bounded almost surely by $2 c(\Gamma_{-m,-n})$, and the $\omega$ for which this fails does not depend on $y_0$. Consequently, decomposing an arbtitrary $x\in L^1(M,\tau)$ into a linear combination of four positive elements and scaling gives $$\left| \tau\left( \left[\Phi_{n,m}(1)^{-\frac12}\Phi_{n,m}(a)\Phi_{n,m}(1)^{-\frac12} - \tau(a Y_m) \right] x \right)\right| \leq \|a\| 4 \|x\|_1 2 c(\Gamma_{-m,-n}).$$ almost surely. Therefore $$\| \Phi_{n,m}(1)^{-\frac12}\Phi_{n,m}(a)\Phi_{n,m}(1)^{-\frac12} - \tau(a Y_m)\| \leq 8 \|a\| c(\Gamma_{-m,-n})$$ almost surely by the duality $M\cong L^1(M,\tau)^*$, and as $n\to \infty$ the above tends to zero almost surely by Lemma [Lemma 55](#lem:randomconstproperties){reference-type="ref" reference="lem:randomconstproperties"}. ◻ Finally, we conclude with a dual version of Theorem [Theorem 62](#thm:right_convergence_on_M){reference-type="ref" reference="thm:right_convergence_on_M"}. **Theorem 66**. *Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual, let $(\Omega, \mathbb{P})$ be a probability space equipped with ergodic $T\in \mathop{\mathrm{Aut}}(\Omega, \mathbb{P})$, let $\gamma_\omega\colon L^1(M,\tau) \to L^1(M,\tau)$ be an $M$-preserving bounded positive faithful random linear operator, and let $\Gamma_{n,m}$ be the associated ergodic quantum process. Suppose that $\gamma_\omega(1)$ is boundedly invertible almost surely and $$\mathbb{P}[\exists n,m\in \mathbb{Z}\colon n\geq m \text{ and }\Gamma_{n,m}\in SHC(M)] >0.$$ Then there exists a family of random variables $B_m\colon \Omega\to S_b$, $m\in \mathbb{Z}$, satisfying: $$\gamma_{T^{m-1}\omega}\cdot B_m(\omega)= B_{m-1}(\omega) \qquad \text{ and } \qquad B_m(T^{\pm 1}\omega)= B_{m\pm 1}(\omega)$$ almost surely, and for all $a\in M$ $$\lim_{n\to +\infty} \left\| \Gamma_{n,m}(1)^{-\frac12} \Gamma_{n,m}(a) \Gamma_{n,m}(1)^{-\frac12} - \tau(a B_m) \right\|_\infty =0$$ almost surely for all $m\in \mathbb{Z}$. Furthermore, whenever $m\in \mathbb{Z}$ satisfies $\sup_n \| \Gamma_{n,m}^*(\Gamma_{n,m}(1)^{-1})\|_\infty <\infty$ almost surely, then for all $x\in S$ $$\lim_{n\to +\infty} \left\| \Gamma_{n,m}(1)^{-\frac12} \Gamma_{n,m}(x) \Gamma_{n,m}(1)^{-\frac12} - \tau(x B_m) \right\|_1 =0$$ almost surely.* *Proof.* Denote $\phi_\omega:= (\gamma_\omega)^*$, which is a normal positive weak\* random linear operator on $M$ by Lemma [Lemma 57](#lem:measurabilityTest){reference-type="ref" reference="lem:measurabilityTest"}. Moreover, the assumptions on $\gamma_\omega(1)$ imply $\phi_\omega$ is $\tau$-bounded with a faithful extension to $L^1(M,\tau)$ almost surely. Letting $\Phi_{n,m}^{T^{-1}}$ denote the associated ergodic quantum process on $M$, we have that $$(\Gamma_{n,m}^{T}(\omega))^* =\Phi_{-m,-n}^{T^{-1}}(\omega).$$ by Equation ([\[eqn:dual_quantum_process\]](#eqn:dual_quantum_process){reference-type="ref" reference="eqn:dual_quantum_process"}). Lemma [Lemma 43](#lem:contraction_constant_of_duals){reference-type="ref" reference="lem:contraction_constant_of_duals"} implies $c(\Gamma_{n,m}^{T})=c(\Phi_{-m,-n}^{T^{-1}})$ and hence $$\mathbb{P}[\exists n,m\in \mathbb{Z}\colon n\geq m \text{ and }\Phi_{n,m}^{T^{-1}}\in SHC(M)]=\mathbb{P}[\exists n,m\in \mathbb{Z}\colon n\geq m \text{ and }\Gamma_{n,m}^{T}\in SHC(M)] >0.$$ Let $A_n\colon \Omega\to S$, $n\in \mathbb{Z}$, be the family of random variables obtained by applying Theorem [Theorem 62](#thm:right_convergence_on_M){reference-type="ref" reference="thm:right_convergence_on_M"} to $\Phi_{n,m}^{T^{-1}}$. Set $B_m:=A_{-m}$ for each $m\in \mathbb{Z}$. Arguing exactly as in the proof of Theorem [Theorem 65](#thm:left_convergence_on_M){reference-type="ref" reference="thm:left_convergence_on_M"}, for $a\in M$ we have $$\lim_{n\to +\infty} \left\| \Gamma_{n,m}(1)^{-\frac12} \Gamma_{n,m}(a) \Gamma_{n,m}(1)^{-\frac12} - \tau(a B_m) \right\|_\infty =0$$ almost surely for all $m\in \mathbb{Z}$. Finally, suppose $R:=\sup_n \| \Gamma_{n,m}^*(\Gamma_{n,m}(1)^{-1})\|_\infty <\infty$ almost surely. Then for $x\in L^1(M,\tau)$ and $a\in M$ we have $$\begin{aligned} |\tau\left( \Gamma_{n,m}(1)^{-\frac12} \Gamma_{n,m}(x) \Gamma_{n,m}(1)^{-\frac12} a\right)| &= |\tau (x \Gamma_{n,m}^* \left(\Gamma_{n,m}(1)^{-\frac12}a \Gamma_{n,m}(1)^{-\frac12} \right) )|\\ &\leq \|x\|_1 \left\|\Gamma_{n,m}^* (\Gamma_{n,m}(1)^{-\frac12}a \Gamma_{n,m}(1)^{-\frac12}) ) \right\|_\infty. \end{aligned}$$ Since $a\mapsto \Gamma_{n,m}^* (\Gamma_{n,m}(1)^{-\frac12}a \Gamma_{n,m}(1)^{-\frac12}) )$ is a positive linear map on $M$, its norm is given by $\| \Gamma_{n,m}^*(\Gamma_{n,m}(1)^{-1})\|_\infty \leq R$. Thus, the last expression above is further bounded by $R\|x\|_1\|a\|_\infty$, implying that $$\left\|\Gamma_{n,m}(1)^{-\frac12} \Gamma_{n,m}(x) \Gamma_{n,m}(1)^{-\frac12}\right\|_1 \leq R \|x\|_1.$$ Now, fix $x\in S$, and given $\epsilon>0$ let $a\in M_+$ be such that $\|x-a\|_1<\epsilon$. Then using that the $\|\cdot\|_1$-norm is dominated by the $\|\cdot\|_\infty$-norm we have $$\begin{aligned} \limsup_{n\to\infty} &\| \Gamma_{n,m}(1)^{-\frac12} \Gamma_{n,m}(x) \Gamma_{n,m}(1)^{-\frac12} - \tau(x B_m) \|_1 \\ &\leq \limsup_{n\to\infty} \| \Gamma_{n,m}(1)^{-\frac12} \Gamma_{n,m}(x-a) \Gamma_{n,m}(1)^{-\frac12}\|_1 + |\tau((a-x) B_m)| \leq \epsilon (R + \|B_m\|_\infty) \end{aligned}$$ almost surely. Hence the limit is zero almost surely. ◻ Note that the hypotheses in the previous theorem strong are enough that both Theorems [Theorem 60](#thm:right_convergence){reference-type="ref" reference="thm:right_convergence"} and Corollary [Corollary 64](#cor:analogue_of_jeff2){reference-type="ref" reference="cor:analogue_of_jeff2"} can be applied. In fact, the identity $$\begin{aligned} \frac{1}{\tau(\Gamma_{n,m}(1))}& \Gamma_{n,m}(a) - \tau(a B_m)X_n\\ &= (\Gamma_{n,m}\cdot 1)^{\frac12} \left[\Gamma_{n,m}(1)^{-\frac12} \Gamma_{n,m}(a) \Gamma_{n,m}(1)^{-\frac12} - \tau(a B_m) \right](\Gamma_{n,m}\cdot 1)^{\frac12} + [\Gamma_{n,m}\cdot 1 - X_n] \tau(aB_m) \end{aligned}$$ offers an alternative proof of Corollary [Corollary 64](#cor:analogue_of_jeff2){reference-type="ref" reference="cor:analogue_of_jeff2"} in this case. **Remark 67**. In the finite dimensional case of $(\mathbb{M}_N, \frac{1}{N} \text{Tr})$ the condition $\sup_n \| \Gamma_{n,m}^*(\Gamma_{n,m}(1)^{-1})\|_\infty <\infty$ is always satisfied since $$\begin{aligned} \| \Gamma_{n,m}^*(\Gamma_{n,m}(1)^{-1})\|_\infty \leq \text{Tr}(\Gamma_{n,m}^*(\Gamma_{n,m}(1)^{-1})) = N (\frac1N \text{Tr})\left(\Gamma_{n,m}(1) \Gamma_{n,m}(1)^{-1}\right) = N. \end{aligned}$$ That is, the condition is automatic because the $\|\cdot\|_1$-norm and $\|\cdot\|_\infty$-norm are equivalent. Another assumption that guarantees the condition (even in the infinite dimensional case) is that $\gamma_\omega$ is unital. Indeed, then $\Gamma_{n,m}^*$ is tracial and $$\|\Gamma_{n,m}^*(\Gamma_{n,m}(1)^{-1})\|_\infty \leq \|\Gamma_{n,m}^*\cdot 1 - B_m\|_\infty + \|B_m\|_\infty\to 0,$$ as $n\to\infty$ almost surely by the final statement in Theorem [Theorem 62](#thm:right_convergence_on_M){reference-type="ref" reference="thm:right_convergence_on_M"}. $\hfill\blacksquare$ # Application to Locally Normal States {#sec:FCS} In [@FannesNachtergaeleWerner], Fannes, Nachtergaele, and Werner characterize translation invariant states on spin chains and establish clustering properties of finitely correlated states for local observables. In particular, this characterization offers a way to construct translate invariant states. Taking inspiration from this, we find a wide class of random variables $\Psi_\omega$ taking values in the locally normal states of a spin chain that satisfy a *translation covariance* condition. Moreover, these states also exhibit clustering properties for local observables, and through averaging can yield deterministic translation invariant states. ## Spin chains and their quasi-local algebras and locally normal states {#sec:appendix:SpinChainsLNS} The theory of spin chains arises as a class of quantum mechanical models from quantum statistical mechanics [@BratteliRobinson1; @BratteliRobinson2]. We shall recall some basic facts about spin chains formed from tracial von Neumann algebras and translation invariant states. Let $(M,\tau)$ be a fixed tracial von Neumann algebra. Consider an isomorphic copy of $M$ for each **site** $n\in \mathbb{Z}$, written $(M_n, \tau_n)$. These algebras represent the observable algebras of some physical quantity localized to $n$. For each finite subset $\Lambda\subset \mathbb{Z}$ we denote the von Neumann algebraic tensor product $$M_{\Lambda} := \overline{\bigotimes_{n\in \Lambda}} M_n,$$ which is equipped with the tensor product trace $\tau_{\Lambda}$. Set inclusions in $\mathbb{Z}$ naturally induce inclusions in the corresponding von Neumann algebras so that we may consider the inductive limit *C\*-algebra* $$\mathscr{A}_{\mathbb{Z}}:= \varinjlim M_{\Lambda},$$ which we call the **quasi-local algebra** associated to the spin chain with on-site algebras $M_n$. Note that this algebra can be faithfully represented in the infinite von Neumann algebra tensor product $(M_\mathbb{Z}, \tau_{\mathbb{Z}}):=\overline{\bigotimes}_{n\in \mathbb{Z}} (M_n, \tau_n)$, and consequently $\mathscr{A}_\mathbb{Z}$ admits a faithful tracial state $\tau_\mathbb{Z}|_{\mathscr{A}_\mathbb{Z}}$ (see [@TakesakiIII Chapter XIV]). Identifying $M_{\Lambda}\subset \mathscr{A}_{\mathbb{Z}}$ for each finite subset $\Lambda\subset\mathbb{Z}$, we call the unital $*$-subalgebra $$\mathscr{A}_{\mathbb{Z}}^{\text{loc}}:=\bigcup_{\Lambda \subset \mathbb{Z}} M_\Lambda$$ the **local algebra** and its elements are called **local observables**. The **support** of a local observable $a$ is the smallest $\Lambda\subset \mathbb{Z}$ such that $a\in M_{\Lambda}$. Given a state $\psi$ on $\mathscr{A}_\mathbb{Z}$, after [@HudsonMoody] we say it is **locally normal** if $\psi|_{M_\Lambda}$ is normal for all finite subsets $\Lambda\subset \mathbb{Z}$. For $n,k\in \mathbb{Z}$, the map $M_n\ni a\mapsto a\in M_{n+k}$ extends to a group action $\mathbb{Z}\overset{\alpha}{\curvearrowright} \mathscr{A}_{\mathbb{Z}}$. We say a state $\psi$ on $\mathscr{A}_\mathbb{Z}$ is **translation invariant** if $\psi\circ\alpha_k = \psi$ for all $k\in \mathbb{Z}$. **Theorem 68** (Proposition 2.3 and 2.5 [@FannesNachtergaeleWerner]). *Let $\psi$ be a locally normal state on $\mathscr{A}_{\mathbb{Z}}$. Then, the following are equivalent:* 1. *$\psi$ is translation invariant* 2. *there exists a finite von Neumann algebra $W$, a normal unital completely positive map $\mathcal{E}:M\bar \otimes W \to W$, and a normal state $\varrho$ on $W$ so that for all $a_m\otimes\cdots \otimes a_n\in M_{[m,n]}$, $$\psi( a_m \otimes \cdots \otimes a_{n} ) = \varrho\circ\mathcal{E}\circ (1\otimes \mathcal{E})\circ\cdots \circ (\underbrace{1\otimes \cdots \otimes 1}_{(n-m) \text{ times }} \otimes \mathcal{E})(a_m\otimes\cdots \otimes a_{n}\otimes 1_W).$$* *Proof.* $(i)\Rightarrow (ii)$: Put $W = M_{\mathbb{N}}$, viewed as a subalgebra of $M_{\mathbb{Z}}$. Observe that the automorphism $\alpha_1$ extends to a normal automorphism of $M_\mathbb{Z}$ because it preserves $\tau_{\mathbb{Z}}$. In particular, $\alpha_1(W)\subset M_{[2,+\infty)}$ so that we can define $\mathcal{E}:= id_M\otimes \alpha_1$. Then, taking $\varrho = \psi|_{W}$, the claimed identity holds since the translation invariance of $\psi$ means it is index agnostic.\ $(ii)\Rightarrow (i)$: Mimicking [@FannesNachtergaeleWerner Proposition 2.3], we see that the family of maps $$\label{eqn:quoteiteratesendquote} \mathcal{E}^{(n+1)}:M \bar \otimes\underbrace{M\bar \otimes \cdots M}_{n \text{ times }} \bar \otimes W \to W$$ via $\mathcal{E}^{(n+1)} = \mathcal{E} \circ (\mathop{\mathrm{id}}_{M}\otimes \mathcal{E}^{(n)})$ and $\mathcal{E}^{(1)}:= \mathcal{E}$ is completely positive and normal for each $n$ and moreover $$\psi(a_1 \otimes \cdots \otimes a_n ) := \varrho( \mathcal{E}^{(n)}(a_1 \otimes \cdots \otimes a_n \otimes 1_W)),$$ is positive and normal. Translation invariance of $\psi$ follows from the fact that $\mathcal{E}$ is a function of the observable and is independent of its index (translation does not change the operator itself, just its index). ◻ **Remark 69**. Assuming $M\neq \mathbb{C}$, the $W$ constructed in the above proof is necessarily infinite dimensional. However, Fannes, Nachtergaele, and Werner considered conditions on $\psi$ that were necessary and sufficient to guarantee that $W$ could be chosen to be a finite dimensional algebra (see [@FannesNachtergaeleWerner Propositions 2.1 and 2.3]). $\hfill\blacksquare$ ## Locally normal states with random generating map Throughout this section, let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual, and let $(\Omega, \mathbb{P})$ be a probability space equipped with ergodic $T\in \mathop{\mathrm{Aut}}(\Omega,\mathbb{P})$. We will write $(W,\tau_W)$ for an auxiliary finite von Neumann algebra possessing a separable predual. When it is clear from context, we shall drop the subscript on $\tau_W$. Consider the von Neumann tensor product $M\bar \otimes W$ and let $\mathcal{E}_\omega:M\bar \otimes W \to M \bar \otimes W$ be a normal unital positive weak\* random linear operator such that $$\mathcal{E}_\omega(M\bar\otimes W) \subset \mathbb{C}\bar\otimes W \cong W$$ almost surely. Associated to such a weak\* random linear operator, there is a family of normal weak\* random linear operators $E_{\omega,a}:W\to W$ indexed by $a\in M$ and given by $$W\ni x\mapsto E_{\omega, a}(x):= \mathcal{E}_\omega(a\otimes x).$$ We denote $\phi_\omega:=E_{\omega,1}$, which is a normal unital positive weak\* random linear operator on $W$, and we let $\Phi_{n,m}^{T^{-1}}$ be the associated quantum process on $W$. Similarly, we denote $\gamma_\omega:= (\phi_\omega)_*$, which is a bounded tracial (hence faithful) positive random linear operator on $L^1(W,\tau_W)$ by Lemma [Lemma 57](#lem:measurabilityTest){reference-type="ref" reference="lem:measurabilityTest"}, and we let $\Gamma_{n,m}^T$ be the associated quantum process on $L^1(W,\tau_W)$. Recall that these two processes are dual to each other by Equation ([\[eqn:dual_quantum_process\]](#eqn:dual_quantum_process){reference-type="ref" reference="eqn:dual_quantum_process"}). Given integers $m<n$ and $\omega\in \Omega$, we define a map $\mathcal{E}_\omega^{[m,n]}\colon M_{[m,n]}\to W$ by $$\mathcal{E}^{[m,n]}_\omega(a):= \mathcal{E}_{T^m\omega}\circ (1\otimes \mathcal{E}_{T^{m+1}\omega})\circ\cdots \circ (1\otimes \cdots \otimes 1\otimes \mathcal{E}_{T^n\omega})(a\otimes 1_W),$$ which is almost surely normal unital and positive. Using our previous notation, for $a=a_m\otimes \cdots \otimes a_n\in M_{[m,n]}$ we have $$\mathcal{E}^{[m,n]}_\omega(a) = E_{T^m\omega, a_m}\circ \cdots \circ E_{T^n\omega, a_n}(1_W).$$ Consequently, for $m<k<\ell<n$ and $a\in M_{[k,\ell]}\subset M_{[m,n]}$ one has $$\begin{aligned} \label{eqn:hidden_quantum_process} \mathcal{E}^{[m,n]}_\omega(a) = \Phi_{-m,-k+1}^{T^{-1}}\circ \mathcal{E}_{\omega}^{[k,\ell]}(a) \end{aligned}$$ almost surely. **Theorem 70** (Thermodynamic Limit). *With the assumptions and notation as above, suppose that $$\mathbb{P}[\exists n,m\in \mathbb{Z}\colon n\geq m \text{ and }\eta \tau_{x_0} \leq \Phi_{m,n}^{T^{-1}} \leq \eta^{-1} \tau_{x_0} \text{ for some }\eta\in (0,1],\ x_0\in S] >0.$$ Then there exists a map $\Psi\colon \Omega\times \mathscr{A}_{\mathbb{Z}}\to \mathbb C$ satisfying:* 1. *$\Psi_\omega$ is a locally normal state on $\mathscr{A}_{\mathbb{Z}}$ for almost every $\omega\in \Omega$;* 2. *$\Psi_\omega(a) \in L^\infty(\Omega,\mathbb{P})$ for all $a\in \mathscr{A}_{\mathbb{Z}}$;* 3. *$\Psi_\omega\circ \alpha_k= \Psi_{T^k\omega}$ almost surely for all $k\in \mathbb{Z}$;* 4. *and for any local observable $a\in M_{\Lambda}$ one has $$\lim_{N\to\infty} \|\mathcal{E}^{[-N,N]}_\omega(a) - \Psi_\omega(a)\|_\infty =0$$ almost surely.* *Proof.* Let $Y_m\colon \Omega\to S(W)$, $m\in \mathbb{Z}$, be the family of random variables obtained by applying Theorem [Theorem 65](#thm:left_convergence_on_M){reference-type="ref" reference="thm:left_convergence_on_M"} to the ergodic quantum process $\Phi_{m,n}^{T^{-1}}$ on $W$. Then the family $Z_m:= Y_{-m+1}$, $m\in \mathbb{Z}$, satisfies $$(\phi_{T^{m}\omega})_*(Z_m(\omega))= Z_{m+1}(\omega) \qquad \text{ and } \qquad Z_m(T^{\pm 1}\omega)= Z_{m\pm 1}(\omega)$$ almost surely. Using Equation ([\[eqn:hidden_quantum_process\]](#eqn:hidden_quantum_process){reference-type="ref" reference="eqn:hidden_quantum_process"}), we also have for $a\in M_{[m,n]}$ that $$\lim_{N\to\infty} \left\| \mathcal{E}^{[-N,N]}(a) - \tau_W(\mathcal{E}^{[m,n]}(a) Z_m) \right\|_\infty = \lim_{N\to\infty} \left\| \Phi_{N,-m+1}^{T^{-1}}(\mathcal{E}^{[m,n]}(a)) - \tau_W(\mathcal{E}^{[m,n]}(a) Y_{-m+1}) \right\|_\infty =0$$ almost surely. For $m<k<\ell<n$ and $a\in M_{[k,\ell]}\subset M_{[m,n]}$, using Equation ([\[eqn:hidden_quantum_process\]](#eqn:hidden_quantum_process){reference-type="ref" reference="eqn:hidden_quantum_process"}) again and the above properties of $Z_m$ we have $$\begin{aligned} \tau_W(\mathcal{E}_\omega^{[m,n]}(a) Z_m(\omega)) &= \tau_W( [\Phi_{-m,-k+1}^{T^{-1}}(\omega)](\mathcal{E}_{\omega}^{[k,\ell]}(a)) Z_m(\omega))\\ &= \tau_W( \mathcal{E}_\omega^{[k,\ell]}(a) (\phi_{T^{k-1}\omega})_*\circ\cdots \circ (\phi_{T^m\omega})_*(Z_m(\omega))) = \tau_W( \mathcal{E}_\omega^{[k,\ell]}(a) Z_k(\omega)) \end{aligned}$$ almost surely. Noting that the $\omega$ where the above fails is independent of $a$, it follows that $$\Psi_\omega(a):= \tau_W(\mathcal{E}_\omega^{[-n,n]}(a) Z_{-n}(\omega)) \qquad a\in M_{[-n,n]}$$ almost surely gives a well-defined state on the $*$-subalgebra of local observables. As the local observables are norm dense in the quasi-local algebra, $\Psi_\omega$ almost surely admits a unique extension to a locally normal state on $\mathscr{A}_{\mathbb{Z}}$. Thus (1) holds and (4) follows from our limit computation above. To see (2), recall that the separability of $L^1(W,\tau_W)$ implies $Z_m\colon \Omega\to S(W)$ is strongly measurable and can therefore be approximated by simple functions. Consequently, $\Psi_\omega(a)\in L^\infty(\Omega,\mathbb{P})$ with $\|\Psi_\omega(a)\|_{L^\infty(\Omega,\mathbb{P})} \leq \|a\|$ for local observables $a$ by definition of $\Psi_\omega$. This then holds for all elements of the quasi-local algebra through approximation by sequences of local observables. Finally, towards showing (3) we first observe that for $a\in M_{[m,n]}$ and $k\in \mathbb{Z}$ one has $$\mathcal{E}_\omega^{[m+k,n+k]}(\alpha_k(a)) = \mathcal{E}_{T^k\omega}^{[m,n]}(a).$$ Combined with the properties of $Z_m$ above, we therefore have $$\Psi_\omega(\alpha_k(a)) = \tau_W( \mathcal{E}_\omega^{[m+k,n+k]}(a) Z_{m+k}(\omega)) = \tau_W(\mathcal{E}_{T^k\omega}^{[m,n]}(a) Z_m(T^k\omega)) = \Psi_{T^k \omega}(a).$$ By density, this extends to $a\in \mathscr{A}_\mathbb{Z}$. ◻ **Theorem 71** (Theorem [Theorem 3](#thmx:C){reference-type="ref" reference="thmx:C"}). *Let $\Psi\colon \Omega\times \mathscr{A}_{\mathbb{Z}}\to \mathbb C$ be as in Theorem [Theorem 70](#thm:thermo){reference-type="ref" reference="thm:thermo"}. There exists $\kappa\in [0,1)$ and a family of random variables $E_k\colon \Omega\to [0,\infty)$, $k\in \mathbb{Z}$, satisfying $$E_k(T^\ell\omega) = E_{k+\ell}(\omega)$$ almost surely for all $\ell\in \mathbb{Z}$, and $$|\Psi_\omega(ab) - \Psi_\omega(a) \Psi_\omega(b)| \leq E_k(\omega) \kappa^{\text{dist}(\Lambda,\Pi)-1} \|a\|_\infty \|b\|_\infty \qquad \qquad a\in M_{\Lambda},\ b\in M_{\Pi}$$ almost surely for finite subsets of integers $\Lambda \subset (-\infty,k-1)$ and $\Pi\subset [k+1,+\infty)$.* *Proof.* Let $C\in [0,1)$ be the constant obtained from applying Lemma [Lemma 55](#lem:randomconstproperties){reference-type="ref" reference="lem:randomconstproperties"}.(1) to the dual quantum process $\Gamma^{T}_{n,m} = (\Phi_{-m,-n}^{T^{-1}})_*$. Set $\kappa$ to be any number in $(C,1)$. For $k\in \mathbb{Z}$, we set $$E_k(\omega):= 8 D_{\bullet,k}(\omega) D_{k-1,\bullet}(\omega),$$ where $D_{\bullet,k}, D_{k-1,\bullet}$ are the random variables from Lemma [Lemma 55](#lem:randomconstproperties){reference-type="ref" reference="lem:randomconstproperties"}.(2). Now, for finite subsets of integers $\Lambda\subset (-\infty, k-1)$ and $\Pi\subset [k+1,+\infty)$, let $a\in M_\Lambda$ and $b\in M_\Pi$. Denote $c:=b - \Psi_\omega(b)\in M_\Pi$ so that $$\Psi_\omega(ab) - \Psi_\omega(a)\Psi_\omega(b)= \Psi_\omega(ac)$$ almost surely. Let $N\in \mathbb{N}$ be large enough so that $\Lambda\cup \Pi\subset [-N,N]$ and denote $m:=\max{\Lambda}$ and $n:=\min{\Pi}$. Note that $m<(m+1)\leq k-1<k \leq (n-1)< n$ and $\mathop{\mathrm{dist}}(\Lambda,\Pi)=n-m$. We have $$\begin{aligned} |\Psi_\omega(ab) - \Psi_\omega(a)\Psi_\omega(b)| &= |\Psi_\omega(ac)|\\ &= \left| \tau_W(\mathcal{E}_\omega^{[-N,N]}(ac) Z_{-N}(\omega) ) \right|\\ &= \left| \tau_W(\mathcal{E}_\omega^{[-N,m]}\left(a\otimes \Phi_{-(m+1),-(n-1)}^{T^{-1}}\circ \mathcal{E}_{\omega}^{[n,N]}(c)\right) Z_{-N}(\omega) ) \right|\\ &\leq \|a\|_\infty \|\Phi_{-(m+1),-(n-1)}^{T^{-1}}\circ \mathcal{E}_{\omega}^{[n,N]}(c)\|_\infty \| (\mathcal{E}_\omega^{[-N,m]})_*(Z_{-N}(\omega)) \|_1\\ &\leq \|a\|_\infty \|\Phi_{-(m+1),-(n-1)}^{T^{-1}}\circ \mathcal{E}_{\omega}^{[n,N]}(b) - \tau_W(\mathcal{E}_{\omega}^{[n,N]}(b) Z_n(\omega))\|_\infty, \end{aligned}$$ where in the last inequality we have used that $\Phi_{-(m+1),-(n-1)}^{T^{-1}}\circ \mathcal{E}_{\omega}^{[n,N]}$ is unital and $(\mathcal{E}_\omega^{[-N,m]})_*$ is tracial, almost surely. By the proof of Theorem [Theorem 65](#thm:left_convergence_on_M){reference-type="ref" reference="thm:left_convergence_on_M"}, we can estimate the second term in the last expression by $$\|\Phi_{-(m+1),-(n-1)}^{T^{-1}}\circ \mathcal{E}_{\omega}^{[n,N]}(b) - \tau_W(\mathcal{E}_{\omega}^{[n,N]}(b) Z_n(\omega))\|_\infty \leq 8\|b\|_\infty c(\Gamma_{ n-1,m+1}^T)$$ where we have used that $\mathcal{E}_{\omega}^{[n,N]}$ is unital and positive almost surely. Returning to our original estimate, the above and the properties of $D_{\bullet,k}, D_{k-1,\bullet}$ from Lemma [Lemma 55](#lem:randomconstproperties){reference-type="ref" reference="lem:randomconstproperties"}.(2) yield $$\begin{aligned} |\Psi_\omega(ab) - \Psi_\omega(a)\Psi_\omega(b)| &\leq 8 c(\Gamma_{ n-1,m+1}^T) \|a\|_\infty \|b\|_\infty\\ &\leq 8 c(\Gamma_{ n-1,k}^T)c(\Gamma_{ k-1,m+1}^T) \|a\|_\infty \|b\|_\infty\\ &\leq 8 D_{\bullet,k}(\omega) \kappa^{(n-1)-k+1} D_{k-1,\bullet}(\omega) \kappa^{(k-1)-(m+1)+1} \|a\|_\infty \|b\|_\infty\\ &= 8 D_{\bullet,k}(\omega) D_{k-1,\bullet}(\omega) \kappa^{n-m-1} \|a\|_\infty \|b\|_\infty = E_k(\omega) \kappa^{\text{dist}(\Lambda,\Pi) -1} \|a\|_\infty \|b\|_\infty, \end{aligned}$$ almost surely. ◻ **Corollary 72**. *Let $\Psi\colon \Omega\times \mathscr{A}_{\mathbb{Z}}\to \mathbb C$ be as in Theorem [Theorem 70](#thm:thermo){reference-type="ref" reference="thm:thermo"}. Then $$\bar\Psi(a):= \mathbb{E}[\Psi_\omega(a)]$$ defines a locally normal translation invariant state on $\mathscr{A}_\mathbb{Z}$ such that $$\bar\Psi(a) = \lim_{N\to\infty} \frac{1}{2N+1} \sum_{n=-N}^N \Psi_{T^n\omega}(a)$$ almost surely for all $a\in \mathscr{A}_{\mathbb{Z}}$.* *Proof.* That $\bar\Psi$ is well-defined and a state follows from parts (1) and (2) of Theorem [Theorem 70](#thm:thermo){reference-type="ref" reference="thm:thermo"}, while part (3) and the fact that $T$ is measure preserving gives that $\bar\Psi$ is translation invariant. The limit formula follows from Birkhoff's strong ergodic theorem. It remains to show that $\bar\Psi$ is locally normal, and it suffices to show its restriction to $M_{[m,n]}$ is normal. Recall that for $a\in M_{[m,n]}$ that one has $$\Psi_\omega(a) = \tau_W( \mathcal{E}_\omega^{[m,n]}(a)Z_m(\omega)),$$ which is almost surely normal since $\mathcal{E}_\omega$ is almost surely normal and $Z_m(\omega)\in L^1(M,\tau_W)$. When this is the case, set $X_m(\omega)\in S(M)$ so that $$\tau(a X_m(\omega)) = \Psi_\omega(a),$$ and otherwise let $X_m(\omega)=1$. Part (2) of Theorem [Theorem 70](#thm:thermo){reference-type="ref" reference="thm:thermo"} implies $X_m\colon \Omega\to S(M)$ is weakly measurable, and hence a random variable by Theorem [Theorem 11](#thm:Pettis){reference-type="ref" reference="thm:Pettis"}. Consequently, there exists a a sequence of simple functions $\phi_k\colon \Omega\to L^1(M,\tau)$ satisfying $\|X_m(\omega) - \phi_k(\omega)\|_1 \to 0$ as $k\to\infty$ almost surely. Note that we may assume $\phi_k(\omega)\in S(M)$ almost surely, and therefore the dominated convergence theorem implies $$\lim_{k\to\infty} \int_\Omega \|X_m(\omega) - \phi_k(\omega)\|_1\ d\mathbb{P}(\omega) =0.$$ It follows that $(\int_\Omega \phi_k\ d\mathbb{P})_{k\in \mathbb{N}}$ is a Cauchy sequence in $L^1(M,\tau)$, and if we denote the limit by $\mathbb{E}[X_m]$ then $$\bar\Psi(a) = \int_\Omega \tau(a X_m)\ d\mathbb{P}= \lim_{k\to\infty} \int_\Omega \tau(a \phi_k)\ d\mathbb{P}= \lim_{k\to\infty} \tau(a \int_\Omega \phi_k \ d\mathbb{P})=\tau( a \mathbb{E}[X_m]),$$ for all $a\in M_{[m,n]}$. ◻ [^1]: Department of Mathematics, Michigan State University [brent\@math.msu.edu](brent@math.msu.edu) [^2]: Department of Mathematics, The University of Arizona [ebroon\@math.arizona.edu](ebroon@math.arizona.edu)

arxiv_math

--- abstract: | We study compact locally homogeneous plane waves. Such a manifold is a quotient of a homogeneous plane wave $X$ by a discrete subgroup of its isometry group. This quotient is called standard if the discrete subgroup is contained in a connected subgroup of the isometry group that acts properly cocompactly on $X$. We show that compact quotients of homogeneous plane waves are "essentially\" standard; more precisely, we show that they are standard or 'semi-standard'. We find conditions which ensure that a quotient is not only semi-standard but even standard. As a consequence of these results, we obtain that the flow of the parallel isotropic vector field of a compact locally homogeneous plane wave is equicontinuous. address: - Institut für Mathematik und Informatik Walther-Rathenau-Str. 47, 17489 Greifswald - Institut für Mathematik und Informatik Walther-Rathenau-Str. 47, 17489 Greifswald - École Normale Supérieure de Lyon, allée d'Italie 69364 LYON Cedex 07, FRANCE - UMPA, CNRS, École Normale Supérieure de Lyon, allée d'Italie 69364 LYON Cedex 07, FRANCE author: - Malek Hanounah - Ines Kath - Lilia Mehidi - Abdelghani Zeghib bibliography: - Bibliography.bib date: - - title: Topology and Dynamics of compact plane waves --- # Introduction The general theme of the present article is the study of the fundamental group and the isometry group of compact locally homogeneous Lorentzian manifolds. More precisely, the Lorentzian metrics that we consider here are plane waves (see Definition [Definition 6](#defpw){reference-type="ref" reference="defpw"}). Compact locally homogeneous plane waves are quotients of a homogeneous plane wave by a discrete subgroup of its isometry group. So they fit into the following more general situation. Let $X = G/H$ be a homogeneous space (not necessarily endowed with a metric), and let $\Gamma \subset G$ be a discrete subgroup acting properly, cocompactly and freely on $X$. Then $\Gamma\backslash X$ is a manifold. It is called a compact quotient of $X$. This leads to the problem of describing the discrete subgroups $\Gamma \subset G$ acting properly and cocompactly on $X$. Of particular interest is such a description if $X$ is a semi-Riemannian homogeneous space. The Riemannian case has a long history, especially in the case of constant sectional curvature. The pseudo-Riemannian situation is comparatively less studied and involves a lot of additional difficulties. In the following we give a little insight into the problems that we want to deal with and thereby review some classical results. We also present our results in brief. We start with flat compact Riemannian and Lorentzian manifolds before turning to our actual object of study, compact locally homogeneous plane waves. **Convention 1**. *We say that a group has a virtual property P if it contains a finite index subgroup which has property P.* ## Flat case The flat Riemannian case, that is when $X$ is the Euclidean space, corresponds to the crystallographic problem handled by Bieberbach's theorem, which states that the fundamental group $\Gamma \subset {\sf Isom}(\mathbb{R}^n) = {\sf{O}}(n) \ltimes \mathbb{R}^n$ is in fact contained in $\mathbb{R}^n$ (up to finite index). The Lorentzian flat case is much more complicated. We can summarize the state of current research as follows. **Theorem 2**. *Let $M^{n+1}$ be a connected compact flat Lorentzian manifold. Then:* 1. *Completeness: $M$ is the quotient of the Minkowski space ${\sf{Mink}} ^{1, n}$ by a discrete subgroup $\Gamma$ of ${\sf Isom}({\sf{Mink}} ^{1, n})$ acting properly and freely.* 2. *Fundamental group: $\Gamma$ is virtually polycyclic. More exactly, $\Gamma$ is either virtually nilpotent or virtually an abelian extension of $\mathbb{Z}$, i.e. virtually $\mathbb{Z}\ltimes \mathbb{Z}^n$.* 3. *Standardness: Up to a finite index, $\Gamma$ is a cocompact lattice in a solvable connected Lie subgroup $L$ of ${\sf Isom}({\sf{Mink}} ^{1, n})$ acting simply transitively on ${\sf{Mink}} ^{1, n}$. (In other words, $M = \Gamma \setminus L$, where $L$ is a Lie group endowed with a complete and flat left invariant Lorentzian metric).* Item (1) was proved by Carrière [@Carriere]. The first part of item (2) is proved by Goldman and Kamishima [@goldman1984fundamental], and a classification up to abstract commensurability is achieved by Grunewald and Margulis [@grunewald1988transitive]. Item (3) is done by Fried, Goldman, and Kamishima [@fried1983three §1] [@goldman1984fundamental]. For an excellent survey on this topic see [@carriere1_Bieberbach]. We are going to prove the following result (see Section [2](#Section: Flat case){reference-type="ref" reference="Section: Flat case"}). **Theorem 3** (Parallel fields). *Up to taking a finite cover, $M$ has a parallel vector field $V$.\ $\bullet$ If $V$ is timelike, then $M$ is a flat Lorentzian torus, that is $\Gamma$ consists of translations, and $V$ is a linear flow.\ $\bullet$ If $V$ is isotropic, then its flow is equicontinuous.\ $\bullet$ If $V$ is spacelike, then its dynamics is encoded in the linear part of $L$, that is, its image under the linear part projection ${\sf Isom}({\sf{Mink}} ^{1, n}) = {\sf{SO}}(1, n) \ltimes \mathbb{R}^{1+n} \to {\sf{SO}}(1, n)$. In particular, there are examples where the flow of $V$ is equicontinuous, Anosov or more generally partially hyperbolic.\ $\bullet$ In all cases, up to a finite cover, $M$ admits a parallel null direction.* To our best knowledge, existence of parallel vector fields was never stated in the literature. It can be deduced by carefully reading the more general results in [@grunewald1988transitive]. Our proof is only targeted on the existence of such a vector and is therefore simpler. Example [Example 17](#Example: SOL-Anosov){reference-type="ref" reference="Example: SOL-Anosov"} is an interesting example of a non-equicontinuous spacelike parallel flow. **Remark 4** (Kundt spacetimes). *It is worth pointing out that by Theorem [Theorem 2](#Introduction-Theorem: flat case){reference-type="ref" reference="Introduction-Theorem: flat case"}, any flat compact Lorentzian manifold is "weakly" plane wave, i.e. a plane wave in the sense of Definition [Definition 6](#defpw){reference-type="ref" reference="defpw"}, but with a parallel null line field instead of a parallel null vector field. This class belongs to the so-called "weakly\" Brinkmann manifolds, i.e. Lorentzian manifolds with a parallel null line field. The latter class fits in a larger class of manifolds called locally Kundt spacetimes, defined as those having a codimension one lightlike geodesic foliation. These spacetimes are of a great importance in general relativity, see [@boucetta2022kundt] for more details on the subject.* **Remark 5** (Constant curvature). *Klingler [@klingler1996completude] extended Carrière's result by showing that any compact Lorentzian manifold of constant curvature is complete. Namely, in curvature $-1$ it is the quotient of anti-de-Sitter space ${\sf{AdS}} ^{1,n}$ by a discrete subgroup o $\mathsf{Isom}({\sf{AdS}} ^{1,n})$. In the positive curvature, due to a classical result known as the Calabi-Markus phenomena [@markus-calabi], there are no compact Lorentzian manifolds with positive constant curvature. On the other hand, compact quotients exist in negative curvature only if $n$ is even. In dimension $3$, Goldman shows in [@goldman1985nonstandard] that there are non-standard (for the definition of standard see Section [5](#Section: syndetic hull, standard semi-Standrad){reference-type="ref" reference="Section: syndetic hull, standard semi-Standrad"}) compact quotients. However, it is conjectured [@zeghib_AntideSitter] that for $n>3$, all compact quotients are standard. For recent developments on proper actions in the constant curvature case, see [@Kassel-Gueritaud-2015; @Kassel-Danciger-2016].* Our aim in the present article is to generalize Theorem [Theorem 2](#Introduction-Theorem: flat case){reference-type="ref" reference="Introduction-Theorem: flat case"} and Theorem [Theorem 3](#Introduction-Theorem: parallel fields){reference-type="ref" reference="Introduction-Theorem: parallel fields"} to the case where Minkowski space is replaced by a plane wave spacetime. Plane waves can be thought of as a generalization as well as a deformation of Minkowski spacetime. They are of great mathematical and physical interests, which can be seen from the amount of recent research on the topic. However, results on compact plane waves are rare since most of research is from the physical point of view, and compact Lorentzian manifolds have a bad causal behavior. More precisely, as stated in the beginning of the section, we consider here compact locally homogeneous plane waves. Since compact plane waves are complete by [@Leis-Sch Theorem B] (see also [@mehidi2022completeness Theorem 1.2] for completeness in the more general context of compact Brinkmann spacetimes), the universal cover of a compact locally homogeneous plane wave is homogeneous. ## Homogeneous plane waves A Lorentzian manifold with an isotropic parallel vector field $V$ is called a Brinkmann manifold. The orthogonal distribution $V^\perp$ is integrable and defines a foliation denoted by $\mathcal{F}$, having lightlike geodesic leaves. Plane waves are particular Brinkmann spaces, defined as follows: **Definition 6**. *A plane wave is a Brinkmann manifold such that the leaves of $\mathcal {F}$ are flat, and $\nabla_X R = 0$, for any $X$ tangent to $V^{\perp}$, where $R$ is the Riemannian tensor.* The $(2n+1)$-dimensional Heisenberg group ${\sf{Heis}} _{2n+1}=\mathbb{R}^n \ltimes \mathbb{R}^{n+1}$ is the subgroup of $\sf{Aff}(\mathbb{R}^{n+1})$ defined by $${\sf{Heis}} _{2n+1}=\left\{\begin{pmatrix} 1 &\alpha \\ 0 &I_n \end{pmatrix} | \ \alpha \in \mathbb{R}^{n} \right\}\ltimes \mathbb{R}^{n+1}.$$ Denote by $A^+=\mathbb{R}^n$ the abelian subgroup of unipotent matrices, and by $A^-$ the subgroup $\{0\} \times \mathbb{R}^n$ of the translation part. **Definition 7** (Affine unimodular lightlike group). *Let $\mathbb{R}^{n+1}$ with coordinates $(x_0,x_1,..,x_n)$ be endowed with the lightlike quadratic form $q_0 := x_1^2+..+ x_n^2$.* *The group of affine isometries of $q_0$ preserving an isotropic vector is $\mathsf{L}_{{\mathfrak{u}}}:= ({\sf{O}}(n) \ltimes \mathbb{R}^n) \ltimes \mathbb{R}^{n+1}$. $$\mathsf{L}_{{\mathfrak{u}}}:=\left\{\begin{pmatrix} 1 & \alpha^{\top}\\ 0 & A \end{pmatrix} | \ \alpha\in \mathbb{R}^{n}, A\in {\sf{O}}(n) \right\}\ltimes \mathbb{R}^{n+1},$$ and $\mathbb{R}^n \ltimes \mathbb{R}^{n+1} \subset \mathsf{L}_{{\mathfrak{u}}}$ is the Heisenberg group. It will be called the affine unimodular lightlike group.* *A manifold modeled on $(\mathsf{L}_{{\mathfrak{u}}},\mathbb{R}^{n+1})$ will be said to have an affine unimodular lightlike geometry in the sense of geometric structures $(\text{see \cite[Chapter 3]{thurston2022geometry} and \cite{goldman2022geometric}})$.\ * Plane waves admit an isometric infinitesimal action of some subalgebra of the Heisenberg algebra (see [@Blau Section 3.2]), whose action preserves individually the leaves of $\mathcal{F}$ and is locally transitive on each $\mathcal{F}$-leaf. Thus, general plane waves have already local cohomogeneity $1$. Moreover, the $\mathcal {F}$-leaves have an affine unimodular lightlike geometry. Indeed, $\mathsf{L}_{{\mathfrak{u}}}$ is the group of diffeomorphisms of $\mathbb{R}^{n+1}$ preserving $q_0$, a flat affine connection and a lightlike direction. Let $X$ be a non-flat simply connected homogeneous plane wave of dimension $n+2$. The connected component of the isometry group of $X$ is computed in [@Content1] and has the following form $$G_{\rho}= (\mathbb{R}\times K) \ltimes_{\rho} H,$$ where $K$ is a closed subgroup of ${\sf{SO}}(n)$, $H$ is some subgroup of ${\sf{Heis}} _{2n+1}$ and $\rho$ is a suitable homomorphism from $\mathbb{R}\times K$ to ${\sf{Aut}}(H)$. The space $X$ identifies with the quotient $X_{\rho} =G_{\rho}/I$, with $I = K \ltimes A^+$. More details are given in Section [3](#Section: Preliminary facts){reference-type="ref" reference="Section: Preliminary facts"}. There are many works on general homogeneous plane waves, for instance [@Blau], [@Leis], but none of these is interested in compact quotients of such spaces. A systematic study of compact quotients of Cahen-Wallach spaces is carried out by Kath and Olbrich in [@KO]. These spaces, first introduced in [@cahen1970lorentzian], are exactly the indecomposable symmetric plane waves. In this case, the $\rho$-action in the semi-direct product $G_{\rho}$ is semi-simple. For general locally homogeneous plane waves, this is not necessarily the case. Compact quotients of general homogeneous plane waves are considered in the recent paper [@allout2022homogeneous], in dimension $3$. ## Fundamental groups of compact quotients Let $X:=G/I$ be a homogeneous pseudo-Riemannian manifold. Let $M=\Gamma \backslash X$ be a compact quotient of $X$. In the flat Riemannian setting, Bieberbach [@bieberbach1911bewegungsgruppen] (see also [@milnor1977fundamental Theorem 1.3]) shows that for a discrete subgroup $\Gamma$ to occur as the fundamental group of a compact Riemannian flat manifold, it is necessary and sufficient that $\Gamma$ is torsion-free, finitely generated and virtually isomorphic to $\mathbb{Z}^{n}$. In the affine case, i.e. $M=\Gamma\backslash X$, where $G=\sf{Aff}(\mathbb{R}^n)$ and $X=\sf{Aff}(\mathbb{R}^n)/{\sf{GL}}_{n}(\mathbb{R})$, a conjecture of Auslander states that the fundamental group of a compact complete affine flat manifold is virtually solvable. It is shown to be true in dimension $3$ by Goldman and Fried [@fried1983three] after a full classification of all possible fundamental groups. And also in the flat Lorentzian case by Goldman and Kamishima [@goldman1984fundamental]. Later on, Grunewald and Margulis [@grunewald1988transitive] sharpened the result in the flat Lorentzian case: they show that $\Gamma$ is either virtually nilpotent or virtually an abelian extension of $\mathbb{Z}$ [@grunewald1988transitive Theorem 1.10]. For compact quotients of Cahen-Wallach spaces, a classification of fundamental groups up to finite index is achieved in [@KO Proposition 8.3]. For a compact locally homogeneous plane wave we show the following. **Theorem 8**. *The fundamental group $\Gamma$ of a compact quotient $M=\Gamma\backslash X_{\rho}$ is virtually nilpotent, or virtually a nilpotent extension of $\mathbb{Z}$ by a discrete subgroup of Heisenberg, i.e. virtually $\mathbb{Z}\ltimes \Gamma_0$ where $\Gamma_0\subset {\sf{Heis}} _{2n+1}$.* ## **Standardness and semi-standardness of compact quotients** When $G$ preserves a Riemannian metric on $X=G/I$ (which is equivalent to the isotropy $I$ being compact), the discrete subgroups of $G$ acting properly and cocompactly on $X$ are exactly the uniform lattices of $G$. For general homogeneous spaces, the isotropy is not compact. Then the first source of examples of compact quotients are the standard quotients, see Definition [Definition 34](#Definition: standard){reference-type="ref" reference="Definition: standard"}. In this case, $\Gamma$ is a uniform lattice in some connected Lie subgroup $N$ of $G$ acting properly and cocompactly on $X$. In particular, $N$ (which is necessarily closed in $G$) is a syndetic hull of $\Gamma$, see Definition [Definition 26](#Definition: syndetic hull){reference-type="ref" reference="Definition: syndetic hull"}. For some $X=G/I$, all compact quotients are standard and can even be obtained by using the same $N$ (up to conjugacy) for all discrete subgroups $\Gamma\subset G$ that act properly and cocompactly on $G/I$ (up to taking a finite index group). This happens for instance in the flat Riemannian case by Bieberbach's theorem. In general, not all quotients are standard, and for the standard ones, $N$ depends on $\Gamma$. Finding such an $N$ turns out to be an easier problem, and the existence of compact quotients reduces to an existence theorem for lattices. As stated in Theorem [Theorem 2](#Introduction-Theorem: flat case){reference-type="ref" reference="Introduction-Theorem: flat case"}, compact quotients in the flat Minkowski space are virtually standard [@goldman1984fundamental], i.e., they become standard if we replace the discrete group $\Gamma$ by a suitable finite index group. More generally, a theorem of Fried and Goldman [@fried1983three Section 1.4] states that any virtually solvable subgroup of the affine group, acting properly discontinuously on the affine space, has a syndetic hull (up to finite index). In simply connected nilpotent groups, the existence of a syndetic hull is due to Malcev [@mal1949class], and is called the Malcev closure. Unlike the previous cases, homogeneous plane waves are not affine manifolds. However, they are foliated by codimension one affine leaves, which are the orbits of the affine unimodular lightlike group. The group $G_{\rho}$ may sometimes be solvable, but generically, it is a Lie group which is not even solvable. However, even in the solvable case, there is no construction analogous to the Malcev closure, and in this case we have non-standard examples [@maeta2022] (see also Section [6](#Section: Non Standard){reference-type="ref" reference="Section: Non Standard"}). We prove the following theorem. **Theorem 9**. *Any compact quotient of $X_{\rho}$ is standard or semi-standard. In the standard case, there is a syndetic hull $N$ which is nilpotent or an extension of $\mathbb{R}$ by a subgroup of ${\sf{Heis}}$. Moreover, $N$ acts transitively.* For the definition of semi-standard, see Definition [Definition 35](#Definition: semi-standard){reference-type="ref" reference="Definition: semi-standard"}. Note that in dimension three, all compact quotients are standard [@allout2022homogeneous Theorem 12.4]. ## More general homogeneous structures In the study of the fundamental group and standardness question, we did not make any use of the geometry of $X_{\rho}$. In particular, $G_{\rho}$ does not have to preserve any Lorentzian metric on $X_{\rho}$. So Theorem [Theorem 8](#Introduction-Theorem: fundamental group){reference-type="ref" reference="Introduction-Theorem: fundamental group"} extends to more general homogeneous structures (see Theorem [Theorem 47](#Theorem: General homogeneous){reference-type="ref" reference="Theorem: General homogeneous"}). The property of being standard or semi-standard in Theorem [Theorem 9](#Introduction-Theorem: standarness and semi-standardness){reference-type="ref" reference="Introduction-Theorem: standarness and semi-standardness"} also remains true, the only thing we lose is the transitivity of the action of the syndetic hull. Based on this remark, even if the initial motivation was to study the compact quotients of homogeneous plane waves, one can consider more general groups $$G = (\mathbb{R}\times K) \ltimes {\sf{Heis}} , \ \ X = G/I,$$ with no restriction on the action, and the isotropy given by $I = C \ltimes A^+$, where $C$ is a subgroup of $K$ preserving $A^+$. A natural question in future work would be to see which of these homogeneous spaces are Lorentzian, or, more generally, which geometries on $X$ are preserved by $G$. ## Equicontinuity of the parallel flow The isometry groups considered here are noncompact Lie groups. Let $(M,g)$ be a compact Lorentzian manifold. A $1$-parameter group of ${\sf Isom}(M,g)$ is equicontinuous if its closure in ${\sf Isom}(M,g)$ is compact. This is equivalent to the fact that it is isometric for some Riemannian metric on $M$. In a previous work [@mehidi2022completeness], the third and fourth authors asked the question of the equicontinuity for the parallel flow of a compact Brinkmann manifold: **Question 10**. *Let $(M,g,V)$ be a compact Brinkmann spacetime. Is the flow $\phi^t$ of $V$ equicontinuous?* The condition that ${\sf Isom}(M,g)$ contains a one-parameter group which is not relatively compact in ${\sf Isom}(M,g)$ amounts to the non-compactness of the connected component ${\sf Isom}^{\circ}(M,g)$. The results of [@mehidi2022completeness] show that the flow $\phi^t$ of an arbitrary compact Brinkmann space is equicontinuous if it is equicontinuous for any compact locally homogeneous one. We think it would be interesting to ask this question first in the case of plane waves. This would be an important step towards the general Brinkmann case. As a corollary of Theorem [Theorem 9](#Introduction-Theorem: standarness and semi-standardness){reference-type="ref" reference="Introduction-Theorem: standarness and semi-standardness"}, we obtain that the following: **Theorem 11**. *Let $(M,V)$ be a compact locally homogeneous plane wave. The action of the parallel flow $V$ is equicontinuous.* [In the locally symmetric indecomposable case (Cahen-Wallach spaces), the flow is proved to be periodic [@KO Proposition 8.2]]{style="color: black"}. In the homogeneous general plane wave case, there are non-periodic examples (Appendix [10](#Appendix B){reference-type="ref" reference="Appendix B"}). ## Organization of the article {#organization-of-the-article .unnumbered} The article is organized as follows: in Section [2](#Section: Flat case){reference-type="ref" reference="Section: Flat case"}, we prove existence of parallel vector fields for any flat compact Lorentzian manifold. In Section [3](#Section: Preliminary facts){reference-type="ref" reference="Section: Preliminary facts"} we give a description of the isometry group of a non-flat simply connected homogeneous plane wave. Section [4](#Section: Fundamental group){reference-type="ref" reference="Section: Fundamental group"} deals with the fundamental group of compact quotients of such manifolds, where we prove Theorem [Theorem 8](#Introduction-Theorem: fundamental group){reference-type="ref" reference="Introduction-Theorem: fundamental group"}. In Section [5](#Section: syndetic hull, standard semi-Standrad){reference-type="ref" reference="Section: syndetic hull, standard semi-Standrad"}, we prove Theorem [Theorem 9](#Introduction-Theorem: standarness and semi-standardness){reference-type="ref" reference="Introduction-Theorem: standarness and semi-standardness"} about standardness and semi-standardness of compact quotients, and we consider more general homogeneous structures in Section [8](#Section: More homogeneous structures){reference-type="ref" reference="Section: More homogeneous structures"}. This allows to prove equicontinuity of the parallel flow in Section [7](#Section: Equicontinuity){reference-type="ref" reference="Section: Equicontinuity"} (see also Appendix [10](#Appendix B){reference-type="ref" reference="Appendix B"} for an example of a non-periodic action). Appendix [9](#Appendix A){reference-type="ref" reference="Appendix A"} is related with Section [5](#Section: syndetic hull, standard semi-Standrad){reference-type="ref" reference="Section: syndetic hull, standard semi-Standrad"}: we show that a cocompact proper action on a contractible space of any connected Lie group admitting a torsion free uniform lattice is transitive. In Section [6](#Section: Non Standard){reference-type="ref" reference="Section: Non Standard"}, we study the non-standard phenomena, starting with a concrete example of a non-standard compact quotient. # Flat case {#Section: Flat case} This section is devoted to the proof of the first part of Theorem [Theorem 3](#Introduction-Theorem: parallel fields){reference-type="ref" reference="Introduction-Theorem: parallel fields"}, namely, the existence of a parallel vector field and parallel null direction on any compact flat Lorentzian manifold. For the proof of the equicontinuity statement, see Section [7](#Section: Equicontinuity){reference-type="ref" reference="Section: Equicontinuity"}. Let $M$ be a compact flat Lorentzian manifold, and $\Gamma$ its fundamental group. By item (3) of Theorem [Theorem 2](#Introduction-Theorem: flat case){reference-type="ref" reference="Introduction-Theorem: flat case"} we have $M=\Gamma \backslash L$ up to a finite covering, where $L$ is a solvable connected subgroup of the Poincaré group ${\sf{O}}(1, n) \ltimes \mathbb{R}^{n+1}$ that acts simply transitively on $\mathbb{R}^{n+1}$ and $\Gamma$ is a lattice in $L$. Let $\pi: {\sf{O}}(1, n) \ltimes \mathbb{R}^{1+n} \to {\sf{O}}(1, n)$ be the linear part projection, and let $L^\prime$ be the projection of $L$. Observe that $M$ has a parallel vector field and a parallel null direction if and only if $L^\prime$ preserves some vector $v\not=0$ and a null direction $l$ in $\mathbb{R}^{1+n}$. In [@grunewald1988transitive], Grunewald and Margulis give a precise description of all simply transitive groups of affine Lorentz motions on $\mathbb{R}^{1+n}$ using theorems by Auslander. Propositions 5.1 and 5.3 in their paper ensure the existence of $v$ and $l$ for all such groups. In our situation, $L$ is unimodular. We will use this additional information to give a simpler and more direct proof of the existence of $v$ and $l$. Before we start, let us introduce some notation. Let $\mathsf{Pol}$ be the subgroup of ${\sf{O}}(1, n) \ltimes \mathbb{R}^{1+n}$ preserving a fixed isotropic direction. It has the form $\mathsf{Pol} = L(\mathsf{Pol}) \ltimes \mathbb{R}^{1+n}$, where the linear part $L(\mathsf{Pol})$ is the group $\sf{Sim}_{n-1}$ of similarities of $\mathbb{R}^{n-1}$. This is a semi-direct product $L(\mathsf{Pol}) =(\mathbb{R}\times {\sf{O}}(n-1)) \ltimes \mathbb{R}^{n-1}$, whose radical $\mathbb{R}\ltimes \mathbb{R}^{n-1}$ is the group of affine homotheties of $\mathbb{R}^{n-1}$. The linear part of $\mathsf{Pol}$ is equal to $$L(\mathsf{Pol})=\left\{\begin{pmatrix} e^{\alpha} & \beta^{\top} & -\frac{\vert \beta \vert^2}{2}\\ 0 & A & - A\beta\\ 0 & 0 & e^{-\alpha} \end{pmatrix} |\;\; \alpha \in \mathbb{R}, A \in {\sf{O}}(n-1), \beta \in \mathbb{R}^{n-1}\right\}.$$ **Definition 12**. *$\mathsf{(1)}$ The group $\mathsf{Pol}:=((\mathbb{R}\times {\sf{O}}(n-1)) \ltimes \mathbb{R}^{n-1}) \ltimes \mathbb{R}^{1+n}$ above will be referred to as the *polarized Poincaré* group.* *$\mathsf{(2)}$ The subgroup $\mathsf{SPol} :=({\sf{O}}(n-1) \ltimes \mathbb{R}^{n-1})\ltimes \mathbb{R}^{n+1}$ of $\mathsf{Pol}$ preserving an isotropic vector will be referred to as the special polarized Poincaré group.* We introduce the following notations, which we will keep throughout this section: - $H:=\{{\rm diag}(e^t, I_{n-1}, e^{-t})\mid t\in\mathbb{R}\}$ - $K:= {\sf{O}}(n-1)$ - $U:= \mathbb{R}^{n-1} \subset L(\mathsf{Pol})$ the unipotent radical of $L(\mathsf{Pol})$. - $T:=\mathbb{R}^{n+1}$ the translation part of $\mathsf{Pol}$. Then $\mathsf{Pol}= ((H \times K) \ltimes U) \ltimes T$. In the proof of Theorem [Theorem 14](#Theorem: solvable implies invariant vector){reference-type="ref" reference="Theorem: solvable implies invariant vector"} we will need the following fact on a $1$-parameter subgroup $g^t$ of $\mathsf{Pol}$ which has non-trivial projection to $H$. Let $g^t= h^t k^t u^t$ be its Jordan decomposition, where $h^t$ the hyperbolic part, $k^t$ the elliptic part, $u^t$ the unipotent part, are $1$-parameter groups in $\mathsf{Pol}$. Up to conjugacy in $\mathsf{Pol}$, we can assume that $h^t$ is in $H$, and since $k^t$ and $h^t$ commute, $k^t$ is then a $1$-parameter group in $K$. Define $\mathfrak{p}:= \mathrm{Vect}(W, Z)$ the timelike $2$-plane in $T=\mathbb{R}^{n+1}$ where $h^t$ acts by a hyperbolic matrix. Since $U \ltimes \mathbb{R}^{n+1}$ is normal, the unipotent part $u^t$ is always there, and since $u^t$ and $h^t$ commute, it is contained in $\mathfrak{p}^\perp \subset \mathbb{R}^{n+1}$. **Theorem 13**. *Up to a finite cover, any compact flat Lorentzian manifold $M$ admits a parallel vector field, and a parallel null direction.* The first part of Theorem [Theorem 13](#Theorem section flat: parallel fields){reference-type="ref" reference="Theorem section flat: parallel fields"} follows from Theorem [Theorem 14](#Theorem: solvable implies invariant vector){reference-type="ref" reference="Theorem: solvable implies invariant vector"} below (which is stronger since we don't assume that $L$ has a lattice). **Theorem 14**. *Let $L$ be a unimodular solvable Lie subgroup of ${\sf{O}}(1, n) \ltimes \mathbb{R}^{n+1}$ acting simply transitively on $\mathbb{R}^{n+1}$. Then $L$ preserves some vector field on $\mathbb{R}^{n+1}$.* The following lemma will be used in the proof of Theorem [Theorem 14](#Theorem: solvable implies invariant vector){reference-type="ref" reference="Theorem: solvable implies invariant vector"}. **Lemma 15**. *[@grunewald1988transitive Propositions 4.2, 4.3] Let $S$ be a subgroup of ${\sf{O}}(n) \ltimes \mathbb{R}^n$. Then $S$ acts simply transitively on $\mathbb{R}^n$ if and only if it is generated by pure translations $E:=S \cap \mathbb{R}^n$ and a graph $\phi: E^\perp \to {\sf{O}}(E) \times E^\perp, \phi(v)=(k_v,v)$. In particular, $S$ fixes some vector in $\mathbb{R}^n$.* *Proof of Theorem [Theorem 14](#Theorem: solvable implies invariant vector){reference-type="ref" reference="Theorem: solvable implies invariant vector"}.* [Step 1: existence of an invariant line:]{.smallcaps} We know that $L$ is solvable, hence $L^\prime \subset {\sf{O}}(1, n)$ is also solvable, and thus $L^\prime$ preserves some direction $l$, or a 2-plane $p$. If $L^\prime$ preserves a 2-plane $p$, and $p$ is Lorentzian or degenerate, then it has two or one isotropic direction, which will be $L^\prime$-invariant. If $p$ is spacelike, then we consider the $L^\prime$-action on $p^\perp$ which is Lorentzian. We repeat the process, and surely arrive to either some $2$-plane which is non-spacelike, or to a line. In all cases, $L^\prime$ preserves a direction $l$.\ \ [Step 2: existence of an invariant vector:]{.smallcaps} If $l$ is non-null, then $L^\prime$ preserves a unit vector on it. It then remains to consider the case where $l$ is isotropic. If $L$ is contained in $\mathsf{SPol}$, then it obviously preserves an isotropic vector. Henceforth, we assume that $L$ is not contained in $\mathsf{SPol}$. Then (up to conjugacy in $\mathsf{Pol}$) $L$ contains a $1$-parameter group $g^t=h^t k^t u^t$, with $h^t$ in $H$. The adjoint action of $h^t$ on the Lie algebra of $\mathsf{Pol}$ is as follows: - $\mathsf{Ad}_{h^t}(W)=e^{-t} W, \;\;\mathsf{Ad}_{h^t}(Z) = e^t Z$, where $\mathfrak{p}:= \mathrm{Vect}(W, Z)$. - $\mathsf{Ad}_{h^t}(X)=e^t X$, for any $X \in \mathfrak{u}$. - $\mathsf{Ad}_{h^t}(Y)=Y$, for any $Y \in \mathfrak{p}^{\perp}$. - $\mathsf{Ad}_{h^t}(B)=B$ for any $B \in \mathfrak{k}$. We have that $\mathsf{Ad}_{g^t}$ preserves $\mathfrak{l}:=\mathrm{Lie}(L)$, hence $\mathsf{Ad}_{h^t}$ preserves $\mathfrak{l}$ too. Moreover, since $L$ is unimodular, the adjoint action of $h^t$ restricted to $\mathfrak{l}$ is unimodular. This gives restrictions on the possible eigenvectors of $\mathsf{Ad}_{h^t}$ inside $\mathfrak{l}$, and hence allows to have some precise information on $L$, which we unroll as follows: 1\. $\mathfrak{p}=\mathrm{Vect}(W,Z)$ is contained in $\mathfrak{l}$. Indeed, by assumption $e^{-t}$ is an eigenvalue of $h^t$ whose eigenspace has dimension $1$, hence $W \in \mathfrak{l}$ (this can also be deduced from the fact that $L$ acts transitively). Then $e^t$ is also an eigenvalue of $h^t$ of multiplicity $1$, hence $Z+X \in \mathfrak{l}$ for some $X \in \mathfrak{u}$. But $X$ is necessarily zero, for the action of the $1$-parameter group in $U \ltimes \mathbb{R}^{n+1}$ determined by $Z+X$ on $\mathbb{R}^{n+1}$ has a fixed point, and contradicts the simple transitivity assumption on the $L$-action. 2\. $L$ is contained in $(H \times K) \ltimes T$, i.e. has no element with non-trivial projection to $U$. The only other eigenvalue of $h^t$ is $1$, and the eigenspace has dimension $n-1$, and is contained in $(H \times K) \ltimes P^\perp$ ($\mathfrak{p}$ is invariant by the adjoint action of $H\times K$, hence also $\mathfrak{p}^\perp$). This, together with the first point, implies that $\mathfrak{l}$ is contained in $\mathfrak{h} \oplus \mathfrak{k} \oplus \mathbb{R}^{n+1}$, hence claim 2. 3\. $L$ preserves a spacelike vector in $\mathfrak{p}^\perp$. The eigenspace in $L$ associated to the eigenvalue $1$ of $h^t$ is in $(H \times {\sf{O}}(n-1)) \ltimes P^\perp$, denote it by $Q$. Since $H$ is in the isotropy, $Q$ does not intersect $H$, and its projection to ${\sf{O}}(n-1)) \ltimes P^\perp$ acts simply transitively on $P^\perp$, which implies, using Lemma [Lemma 15](#Lemma: solvable in O(n)xR^n){reference-type="ref" reference="Lemma: solvable in O(n)xR^n"}, that $Q$ fixes some vector in $P^\perp$. Now, 2) and 3) imply that this vector is also fixed by $L$. ◻ *Proof of Theorem [Theorem 13](#Theorem section flat: parallel fields){reference-type="ref" reference="Theorem section flat: parallel fields"}.* The existence of a parallel vector field follows from Theorem [Theorem 14](#Theorem: solvable implies invariant vector){reference-type="ref" reference="Theorem: solvable implies invariant vector"} and the observation just before. To prove the existence of a parallel null direction, observe first that when there is a timelike vector which is $L$-invariant, the claim is a direct consequence of Bieberbach's theorem. Indeed, up to finite index, $\Gamma$ is contained in the translation part $\mathbb{R}^{n+1}$ of the isometry group, hence any constant vector field of ${\sf{Mink}} ^{n,1}$ induces a parallel vector field on the quotient. To conclude, we proceed as in Step 1 of the previous proof. If the linear part $L'$ preserves a lightlike or timelike line or plane, then it preserves a null direction, up to taking a finite cover. Otherwise, it preserves a maximal spacelike subspace, but repeating the process again on its orthogonal gives an invariant line or plane, which is lightlike or timelike. ◻ **Remark 16**. *We proved that any compact flat Lorentzian manifold $M= \Gamma \backslash {\sf{Mink}}$ admits a parallel vector field. When the latter is lightlike, $M$ is a flat plane wave and $\Gamma$ preserves an isotropic vector. So compact flat plane waves are the compact quotients of ${\sf{Mink}}$ with fundamental group contained in $\mathsf{SPol}$.* In the following we give examples of tori with a non-equicontinuous spacelike parallel flow (see Theorem [Theorem 3](#Introduction-Theorem: parallel fields){reference-type="ref" reference="Introduction-Theorem: parallel fields"}), which is in fact Anosov in dimension $3$ and partially hyperbolic in higher dimension. These examples admit a lightlike parallel line field, but no lightlike (neither timelike) parallel vector field. **Example 17**. *Let $q = \Sigma a_{ij} x^i x^j$ be a Lorentzian quadratic form on $\mathbb{R}^{n}$. Consider the flat Lorentzian torus $(\mathbb{T}^n= \mathbb{R}^n/ \mathbb{Z}^n, q)$. Its isometry group is generated by translations together with linear transformations ${\sf{O}}_\mathbb{Z}(q)= {\sf{O}}(q) \cap {\sf{GL}}(n, \mathbb{Z})$, where ${\sf{O}}(q)$ is the orthogonal group of $q$. Let $h \in {\sf{O}}_\mathbb{Z}(q)$ be a partially hyperbolic matrix, i.e. there exists a $2$-dimensional $h$-invariant space of signature $(1,1)$ where the restriction of $h$ has eigenvalues different from $\pm1$. Let $h^t$ be a one parameter group $\subset {\sf{O}}(q)$ such that $h^1 = h$. Consider the suspension $M$ of $h$, that is $\mathbb{T}^n \times [0, 1]$, where $(x, 1)$ is identified with $(h(x), 0)$. Endow it with the product Lorentzian (flat) metric $q + dt^2$. Then $\frac{\partial}{\partial t}$ is a spacelike parallel vector field. When $n=2$, the $\frac{\partial}{\partial t}$-flow is Anosov. For $n\geq 3$, the $\frac{\partial}{\partial t}$-flow is partially hyperbolic. In both cases, the $\frac{\partial}{\partial t}$-flow is non-equicontinuous.* # Preliminary facts on the isometry group {#Section: Preliminary facts} Plane waves (and Brinkmann spacetimes in general) admit locally what is called Brinkmann coordinates, in which the metric has a particular form. When such coordinates exist globally on $J \times \mathbb{R}^{n+1}$, for an open interval $J$, we refer to it as a "plane wave in standard form\". It is known that the Lie algebra of Killing fields of an indecomposable plane wave in standard form contains the Heisenberg algebra ${\mathfrak{heis}} _{2n+1}$, which acts locally transitively on $\{u\}\times \mathbb{R}^{n+1}$ for all $u\in J$. In the homogeneous case, Blau and O'Loughlin [@Blau] determined the Lie algebra of Killing fields of a plane wave in standard form by analysis of the Killing equation, and classified plane waves in standard form that are homogeneous. Note however, that they only consider infinitesimal isometries. In [@Content1], we show that for (general) simply connected indecomposable and homogeneous plane waves, the infinitesimal action of the Heisenberg algebra integrates to an isometric action of the Heisenberg group. Moreover, we compute the identity component of their isometry group. It turns out [@Content1] that these spaces coincide with those found in [@Blau], i.e. these spaces admit global Brinkmann coordinates. Let $(X,V)$ be a non-flat simply connected homogeneous plane wave of dimension $n+2$. Since it is non-flat, $V$ is the unique parallel isotropic vector field of $X$. Then $X$ can be written as a product $X := X_0 \times \mathbb{R}^k$, where $X_0$ is an indecomposable plane wave, and $\mathbb{R}^k$ a (maximal) flat Riemannian factor. Its isometry group is then a product ${\sf Isom}(X)= {\sf Isom}(X_0) \times {\sf Isom}(\mathbb{R}^k)$. The identity component of the isometry group of an indecomposable simply connected homogeneous plane wave of dimension $d+2$ has the form $$(\mathbb{R}\times K_0) \ltimes {\sf{Heis}} .$$ Here $\mathbb{R}$ acts on ${\sf{Heis}} ={\sf{Heis}} _{2d+1}$ via a morphism $\rho: \mathbb{R}\to {\sf{Aut}}({\sf{Heis}} )$, with $\rho(t)= e^{tL}$, $L \in {\sf{Der}}({\mathfrak{heis}} )$, and $K_0$ is a closed subgroup of ${\sf{O}}(d)$ acting on ${\sf{Heis}}$, trivially on the center, and by standard action on $A^+$ and $A^-$. Thus, the identity component of the isometry group of $X$ can be written as $$G_{\rho} = (\mathbb{R}\times K_0 \times {\sf{O}}(k)) \ltimes ({\sf{Heis}} _{2(n-k)+1} \times \mathbb{R}^k),$$ where ${\sf{Heis}} _{2(n-k)+1} \times \mathbb{R}^k$ is the subgroup of ${\sf{Heis}} _{2n+1}$ containing the normal subgroup of translations, and whose linear part consists of the unipotent matrices in $A^+$ acting trivially on the $\mathbb{R}^k$ factor. So $G_{\rho}$ has the form $(\mathbb{R}\times K) \ltimes H$, where $K$ is a closed subgroup of ${\sf{O}}(n)$ and $H$ some subgroup of ${\sf{Heis}} _{2n+1}$ containing the center. Then $X$ identifies with a quotient $X_{\rho}:= G_{\rho} /I$, with the isotropy given by $I:= K \ltimes A^+_0$, where $A^+_0$ is the subgroup of ${\sf{Heis}} _{2n+1}$ acting trivially on the $\mathbb{R}^k$ factor inside $G_{\rho}$. Conversely, the $\rho$-actions for which $G_{\rho}$ preserves a Lorentzian metric on $G_{\rho}/I$ are characterized in [@Content1], and in this case, the Lorentzian space is necessarily a plane wave. Accordingly, throughout this article, we will refer to a homogeneous plane wave as a homogeneous space of the form $X_{\rho}$. **Fact 18**. *The parallel vector field $V$ is a generator of the centre of ${\mathfrak{heis}}$.* *Proof.* Let $z$ be a generator of the centre of ${\mathfrak{heis}}$. The Lorentzian scalar product on $T_{o}(G_{\rho}/I)$ is $\mathsf{Ad}_h$-invariant for any $h \in A^+$, a unipotent matrix of the isotropy, and $z$ is the eigenvector of eigenvalue $1$ of $\mathsf{Ad}_h$. This implies that $z$ is necessarily isotropic. For the fact that $z$ is parallel, observe that ${\sf{Heis}}$ contains an abelian subgroup (whose Lie algebra contains $z$) acting transitively on the leaves of $V^\perp$. Then $z$ is parallel by [@Leis Theorem 3]. ◻ Let us now make the following observation. Let $G={{\sf Isom}}(X_{\rho})$ denote the full isometry group of a non-flat simply connected homogeneous plane wave $X_{\rho}$. The identity component of $G$ is ${{\sf Isom}}^{\sf{o}}(X_{\rho})=G_{\rho}$. We have that $G_{\rho}$ has finite index in $G$ (see [@Content1]). So for a compact $(G,X_{\rho})$-manifold, there is a finite cover which is a $(G_{\rho},X_{\rho})$-manifold. Hence, when studying compact quotients, one restricts to $(G_{\rho}, X_{\rho})$-manifolds. # Discrete subgroups {#Section: Fundamental group} As above, the Heisenberg group ${\sf{Heis}} _{2n+1}$ will be denoted simply by ${\sf{Heis}}$. In order to simplify the notation, in Sections [4](#Section: Fundamental group){reference-type="ref" reference="Section: Fundamental group"} and [5](#Section: syndetic hull, standard semi-Standrad){reference-type="ref" reference="Section: syndetic hull, standard semi-Standrad"}, we consider $G_{\rho}$ to be the isometry group of an indecomposable plane wave. But all the proofs apply straightforwardly to the decomposable case if we replace ${\sf{Heis}}$ by its subgroup $H$ defined in Section [3](#Section: Preliminary facts){reference-type="ref" reference="Section: Preliminary facts"}, and $A^+$ by $A^+_0$. Although we introduced ${\sf{Heis}}$ as a linear group, we will also use the common realisation of ${\sf{Heis}}$ as an extension of $\mathbb{C}^n$ by $\mathbb{R}$ and we will write ${\sf{Heis}} \cong \mathbb{R}\times \mathbb{C}^n$. Under this identification, we have $A^+\cong \mathbb{R}^n=\{0\}\times \mathbb{R}^n\subset \mathbb{R}\times \mathbb{C}^n$ and $A^-\cong \mathbb{R}^n=\{0\}\times (i\mathbb{R})^n\subset \mathbb{R}\times \mathbb{C}^n$. Let $G_{\rho}= (\mathbb{R}\times K) \ltimes_{\rho} {\sf{Heis}}$, $I = K \ltimes A^+$, and $X_{\rho}:= G_{\rho}/I$. Unless otherwise stated, all over the paper, $\mathbb{R}$ acts on ${\sf{Heis}}$ via a morphism $\rho: \mathbb{R}\to {\sf{Aut}}({\sf{Heis}} )$, with $\rho(t)= e^{t{\bf L}}$, ${\bf L} \in {\sf{Der}}({\mathfrak{heis}} )$, and $K$ acts on ${\sf{Heis}}$, trivially on the center, and by standard action on $A^+$ and $A^-$. We can suppose (up to adding $\mathsf{ad}(h)$ for a suitable $h\in {\mathfrak{heis}}$) that ${\bf L}$ preserves $\mathfrak{a}^+ \oplus \mathfrak{a}^- \subset {\mathfrak{heis}}$, where ${\mathfrak a}^\pm={\rm Lie}(A^\pm)$.\ In this section, we generalise results from [@KO Section 3] to the case where $\rho$ is not semi-simple. We will use a different, more conceptual approach, providing the general case directly.\ **Notations.** Define the projections $p: G \to \mathbb{R}$, $p_K: G \to K$, and $r: G \to \mathbb{R}\times K$. Observe that $G_{\rho}$ is a connected amenable Lie group, i.e. a Lie group having a normal solvable subgroup with compact quotient. Hence any discrete subgroup $\Gamma$ of $G_{\rho}$ is virtually polycyclic (this follows from general properties of connected amenable Lie groups (see [@milnor1977fundamental Lemma 2.2]). Moreover, it is well known that polycylic groups are finitely generated, therefore, so is any discrete subgroup of $G_{\rho}$. The latter property will be used in the proof of the next theorem. **Theorem 19**. *Let $\Gamma$ be a discrete subgroup of $G_{\rho}$. Then* - *If the projection of $\Gamma$ to the $\mathbb{R}$-factor is dense, then $\Gamma$ is virtually nilpotent.* - *If the projection of $\Gamma$ to the $\mathbb{R}$-factor is discrete, then either $\Gamma\cong \mathbb{Z}\ltimes \Gamma_0$ or $\Gamma = \Gamma_0$, where $\Gamma_0\subset K\ltimes{\sf{Heis}}$ is virtually nilpotent.* First we state the following two Lemmas. **Lemma 20** (Zassenhaus Lemma, Theorem 4.1.6 [@Thurston], Proposition 8.16 [@raghunathan1972discrete]). *Let $G$ be any Lie group. Then there exists a neighborhood $U$ of the identity such that any discrete subgroup $\Gamma$ generated by $\Gamma\cap U$ is nilpotent. We call such a neighborhood a *Zassenhaus neighborhood** **Lemma 21**. *Let $\Lambda$ be a subgroup of a Lie group $G$. Define $\Lambda_0:=\Lambda\cap {\overline{\Lambda}}^{\mathsf{o}}$, where ${\overline{\Lambda}}^{\mathsf{o}}$ denotes the identity component of the (topological) closure of $\Lambda$. Then the subgroup $\Lambda_{0}$ can be generated by $\Lambda_{0}\cap V$ for any $V$ neighborhood of identity in ${\overline{\Lambda}}^{\mathsf{o}}$.* *Proof.* Let $V$ be a neighborhood of the identity in $\overline {\Lambda}^{\mathsf{o}}$. Define $\Lambda_1:= \langle \Lambda_0 \cap V \rangle$. Then $\Lambda_1$ is dense in $\overline {\Lambda}^{\mathsf{o}}$ since $V\subset\overline{\Lambda_1}\subset\overline{\Lambda_0}=\overline {\Lambda}^{\mathsf{o}}$ and $\overline {\Lambda}^{\mathsf{o}}$ is connected. Now, let $\lambda_0\in \Lambda_0$, by density of $\Lambda_1$ there is $\lambda_1$ such that $\lambda_0\lambda_1^{-1}\in V$. ◻ *Proof of Theorem $\ref{Theorem Gamma}$.* (1) The restriction of the projection $\mathbb{R}\times K\to \mathbb{R}$ to $\overline {r(\Gamma)}$ is a Lie group homomorphism $\overline {r(\Gamma)}\to \overline {p(\Gamma)}=\mathbb{R}$. It is surjective since $K$ is compact. Thus we obtain a fibration $$\overline{r(\Gamma)}\cap K \hookrightarrow \overline{r(\Gamma)}\to \mathbb{R}.$$ Its fibre has only finitely many components since it is compact. Now we see from the long exact homotopy sequence of the fibration that also $\pi_0(\overline{r(\Gamma)})$ is finite. This implies that $\Lambda_0:={\overline{r(\Gamma)}}^{\mathsf{o}}\cap r(\Gamma)$ has finite index in $r(\Gamma)$. Consequently, also $\Gamma^0:=r^{-1}(\Lambda_0)\cap \Gamma$ has finite index in $\Gamma$. Hence it is sufficient to show that $\Gamma^0$ is nilpotent. Since $\Gamma^0$ has finite index in $\Gamma$, it is also finitely generated. We choose $U_1\subset\mathbb{R}\times K$ and $U_2\subset {\sf{Heis}}$ such that $U_1\times U_2$ is a Zassenhaus neighborhood in $G_\rho$. The above Lemma applied to $\Lambda=r(\Gamma)$ yields that $\Lambda_0$ is generated by $\Lambda_0\cap U_1$. Let $rh$ be one of finitely many generators of $\Gamma^0$, where $r\in\mathbb{R}\times K$, and $h\in{\sf{Heis}}$. Then $r=\lambda_1\cdot\ldots\cdot \lambda_k$ for $\lambda_j\in\Lambda_0$, $j=1,\dots,k$. Choose elements $\gamma_j\in\Gamma^0$ such that $r(\gamma_j)=\lambda_j$, $j=1,\dots,k$. Then $$rh=\lambda_1\cdot\ldots\cdot\lambda_k h=\gamma_1\cdot\ldots\cdot\gamma_k h'$$ for some $h'\in \Gamma^0\cap{\sf{Heis}}$ since ${\sf{Heis}}$ is normal in $G_\rho$. Thus we may replace $rh$ by $\gamma_1,\dots,\gamma_k$ and $h'$. Doing so for every generator of $\Gamma^0$ we obtain a set of generators $\{r_ih_i\}_{i=1}^m$, where $r_i\in U_1$ and $h_i\in{\sf{Heis}}$. Let $A$ be a $K$-invariant complement of the center in ${\sf{Heis}}$. We consider the automorphism $\Psi$ of $G_\rho$ which is the identity on $\mathbb{R}\times K$, multiplication by $a\in\mathbb{R}_{>0}$ on $A$ and multiplication by $a^2$ on the center of ${\sf{Heis}}$. Choose $a$ so small that $\Psi(h_i)\in U_2$ for all $i=1,\dots,m$. Then $\Psi(\Gamma^0)=\langle r_i\Psi(h_i),\ i=1,\dots,m\rangle$ is generated by elements of $U_1\times U_2$, thus it is nilpotent. Consequently, also $\Gamma^0$ is nilpotent. \(2\) If the projection of $\Gamma$ to the $\mathbb{R}$-factor is discrete, it is either trivial or isomorphic to $\mathbb{Z}$, and in both cases, the exact sequence $1 \to \Gamma \cap (K\ltimes{\sf{Heis}} ) \to \Gamma \to p(\Gamma) \to 1$ splits. Hence, $\Gamma$ is isomorphic to $p(\Gamma)\ltimes \Gamma_0$ for $\Gamma_0:= \Gamma \cap (K\ltimes{\sf{Heis}} )$. The group $\Gamma_0$ is virtually nilpotent. To see this, one can proceed as in the proof of (1) replacing $r$ by the projection $p_K$ to $K$: The closure $\overline{p_K(\Gamma_0)}$ has finitely many connected components. Thus $\Lambda_0:={\overline{p_K(\Gamma_0)}}^{\mathsf{o}}\cap p_K(\Gamma_0)$ has finite index in $p_K(\Gamma_0)$ and $\Gamma^0:=p_K^{-1}(\Lambda_0)\cap \Gamma_0$ has finite index in $\Gamma_0$. Now we use a Zassenhaus neighborhood $U_1\times U_2$, $U_1\subset K$, $U_2\subset {\sf{Heis}}$ to see that $\Gamma^0$ is nilpotent. ◻ **Remark 22**. *Part $(2)$ of Theorem $\ref{Theorem Gamma}$ can also be deduced from Auslander's theorem [@Auslander1 Theorem 3]. Note that the formulation in Auslander's paper is slightly incorrect.* Let $\Gamma$ be a torsion free discrete subgroup of $\mathsf{Isom}(X_{\rho})$ which acts properly and cocompactly on $X_{\rho}$. The quotient space $M:=\Gamma\backslash X_{\rho}$ is a compact manifold foliated by a codimension $1$ foliation, given by the ${\sf{Heis}}$-action. **Terminology.** Let $\Gamma$ be a discrete subgroup of $\mathsf{Isom}(X_{\rho})$. We say that $\Gamma$ is *straight* if $p(\Gamma)$ is discrete. Otherwise, it is *non-straight*. **Remark 23**. *Let $\Gamma$ be a torsion free discrete subgroup of $\mathsf{Isom}(X_{\rho})$ which acts properly and cocompactly on $X_{\rho}$. Since the ${\sf{Heis}}$-foliation is defined by a closed $1$-form, the leaves are either all closed or all dense, and they are closed exactly when $p(\Gamma)$ is discrete. In the straight case, the ${\sf{Heis}}$-leaves are complete compact affine manifolds, modeled on the so-called affine unimodular lightlike geometry (see Definition [Definition 7](#Definition-Introduction: Affine unimodular lightlike group){reference-type="ref" reference="Definition-Introduction: Affine unimodular lightlike group"}). Namely, they have a $(\mathsf{L}_{{\mathfrak{u}}}, \mathbb{R}^{n+1})$-structure.* **Corollary 24**. *Let $\Gamma$ be a discrete subgroup of $G_{\rho}$ which acts properly and cocompactly on $X_{\rho}$. Then:* - *If $\Gamma$ is non-straight, then $\Gamma$ is virtually nilpotent.* - *If $\Gamma$ is straight, then we have $\Gamma \cong \mathbb{Z} \ltimes \Gamma_0$ and $\Gamma_0\subset K\ltimes {\sf{Heis}}$ is virtually nilpotent. Moreover, $\Gamma_0$ has a finite projection to $K$. In particular, $\Gamma$ is virtually a nilpotent extension of the integers $\mathbb{Z}$ by a discrete subgroup of ${\sf{Heis}}$.* **Fact 25** (Hirsch, Goldman, Fried [@fried1981affine] Theorem A). *Let $M$ be a compact affine manifold with nilpotent holonomy group. Then completeness of $M$ in the sense of $(\sf{Aff}(\mathbb{R}^{n}), \mathbb{R}^{n})$-structure is equivalent to the linear holomomy being unipotent.* *Proof of Corollary $\ref{Corollary: structure of Gamma}$.* Point $(1)$ and the first part of point $(2)$ are direct consequences of Theorem [Theorem 19](#Theorem Gamma){reference-type="ref" reference="Theorem Gamma"}. To complete the proof of (2), we will show that $\Gamma_0 \cap {\sf{Heis}}$ has finite index in $\Gamma_0$. Since $K\ltimes {\sf{Heis}}$ is a linear group, by Selberg's lemma, $\Gamma_0$ contains a torsion free subgroup $\Gamma_0'$ of finite index. The leaf $\Gamma_0' \backslash K\ltimes {\sf{Heis}} /K\ltimes A^+$ is a compact manifold having a complete affine structure with nilpotent holonomy (Remark [Remark 23](#Remark: Heis-leaves have unimodular lightlike structure){reference-type="ref" reference="Remark: Heis-leaves have unimodular lightlike structure"}). It follows from Fact [Fact 25](#fact2.7){reference-type="ref" reference="fact2.7"} that the linear part of $\Gamma_0'$ is unipotent, meaning that $\Gamma_0'$ has a trivial $K$-part. Thus, $p_K(\Gamma_0)$ is finite. Let $\Gamma_1:= \Gamma_0\cap {\sf{Heis}}$, since ${\sf{Heis}}$ is preserved by the $\mathbb{Z}$-action, we can define $\mathbb{Z}\ltimes \Gamma_1$ which is a nilpotent extension of $\mathbb{Z}$ and clearly of a finite index in $\Gamma$. ◻ # Standardness and semi-standardness {#Section: syndetic hull, standard semi-Standrad} ## Existence of a syndetic hull {#Subsection:Definition of syndetic hull?} **Definition 26** (Syndetic hull). *Let $G$ be a Lie group and let $\Gamma$ be a discrete subgroup. A *syndetic hull* of $\Gamma$ in $G$ is a closed connected Lie subgroup $N \subset G$ containing $\Gamma$ such that $\Gamma\backslash N$ is compact.* **Definition 27** ($\mathbb{Z}$-Syndetic hull). *A *$\mathbb{Z}$-syndetic hull* of $\Gamma$ in $G$ is a closed Lie subgroup $N\subset G$ having an infinite cyclic component group such that $\Gamma\subset N$ and $\Gamma\backslash N$ is compact.* So far we proved, using the existence of Zassenhaus neighborhoods, that up to finite index, any discrete subgroup of $G_{\rho}$ is either virtually nilpotent, or is an extension of $\mathbb{Z}$ by a virtually nilpotent group. The same proofs, using now what we call here 'strong Zassenhaus neighborhood' instead of a Zassenhaus neighborhood, lead to a stronger result, namely the existence of a nilpotent syndetic hull (in the non-straight case), or of a $\mathbb{Z}$-syndetic hull with a nilpotent identity component (in the straight case) for a finite index subgroup. Here we are especially interested in the situation where the discrete group acts properly and cocompactly on $X_\rho$. We will see that for such a group in the straight case the identity component of the $\mathbb{Z}$-syndetic hull is contained in ${\sf{Heis}}$. **Fact 28**. *[@Thurston Theorem 4.1.7][\[Thurston: strongly Zassenhaus neighborhood\]]{#Thurston: strongly Zassenhaus neighborhood label="Thurston: strongly Zassenhaus neighborhood"} Let $G$ be a Lie group. There exists a neighborhood $U$ of $1$ in $G$ such that any discrete subgroup $\Gamma$ of $G$ generated by $U \cap \Gamma$ is a cocompact subgroup of a connected, closed, nilpotent subgroup $N$ of $G$.* We call such an identity neighborhood $U$ a *strong Zassenhaus neighborhood*. **Proposition 29**. *Let $\Gamma$ be a discrete subgroup of $G_{\rho}:=(\mathbb{R}\times K) \ltimes_{\rho} {\sf{Heis}}$. If $\Gamma$ is non-straight, there exists a closed connected and nilpotent Lie group $N$ containing a finite index subgroup of $\Gamma$ as a cocompact lattice. Furthermore, $\Gamma$ acts properly and cocompactly on $X_{\rho}=G_{\rho}/I$ if and only if $N$ acts properly and cocompactly on $X_{\rho}$.* **Proposition 30**. *Let $\Gamma = \langle\hat{\gamma}\rangle \ltimes \Gamma_0$ be a straight discrete subgroup of $G_{\rho}:=(\mathbb{R}\times K) \ltimes_{\rho} {\sf{Heis}}$, acting properly and cocompactly on $X_\rho=G_{\rho}/I$. Then there exists a closed, connected and nilpotent Lie subgroup $N_0$ of ${\sf{Heis}}$ containing a finite index subgroup of $\Gamma_0$ as a cocompact lattice, and which is $\hat{\gamma}$-invariant.* *Proof.* The same proof as in Theorem [Theorem 19](#Theorem Gamma){reference-type="ref" reference="Theorem Gamma"} yields that $\Gamma$ (resp. $\Gamma_0$) is generated by a strongly Zassenhaus neighborhood in the non-straight (resp. straight) case. The claim then follows from Fact [\[Thurston: strongly Zassenhaus neighborhood\]](#Thurston: strongly Zassenhaus neighborhood){reference-type="ref" reference="Thurston: strongly Zassenhaus neighborhood"}. In the straight case, up to finite index, we have $\Gamma = \langle \hat{\gamma} \rangle \ltimes \Gamma_1$, with $\Gamma_1$ a subgroup of ${\sf{Heis}}$ (proof of Corollary [Corollary 24](#Corollary: structure of Gamma){reference-type="ref" reference="Corollary: structure of Gamma"}). The Malcev closure of $\Gamma_1$ in ${\sf{Heis}}$ is a syndetic hull for $\Gamma_1$ which is $\hat{\gamma}$-invariant. ◻ ## Transitivity of cocompact actions of Lie groups {#Subsection:Transitivity of cocompact actions of Lie groups} Let $\Gamma$ be a discrete subgroup of $G_{\rho}$ acting properly on $X_{\rho}\cong \mathbb{R}^{n+2}$. The aim of this section is to prove that the syndetic hull $N$ (resp. $N_0$) of $\Gamma$ (resp. $\Gamma_0$) obtained in the previous section acts transitively on $X_{\rho}$ (resp. on an $\mathcal{F}$-leaf). One could actually ask a more general question: let $G$ be a Lie group acting properly cocompactly on a contractible manifold $X$, does $G$ act transitively? In Proposition [Proposition 51](#Proposition-Appendix A: transitive action under existence of a lattice){reference-type="ref" reference="Proposition-Appendix A: transitive action under existence of a lattice"}, we prove that the action is transitive if we further assume that $G$ has a torsion free uniform lattice. In this subsection we do not assume that $G$ has a lattice, and prove transitivity when $G$ is a connected nilpotent Lie subgroup of $G_{\rho}$. **Proposition 31**. *Let $N$ be a connected nilpotent Lie subgroup of $G:= K \ltimes {\sf{Heis}}$, acting cocompactly on the homogeneous space $Y := K \ltimes {\sf{Heis}} / K \ltimes A^+$. Then* - *$N$ acts transitively,* - *$N$ is contained in ${\sf{Heis}}$, and contains the center of Heisenberg.* *Proof.* The group $N$ acts also cocompactly on $Y/Z$. Let $p':K\ltimes {\sf{Heis}} \to A^-$ be defined by $z(a^++a^-)k\mapsto a^-$, where $k\in K$, $a^\pm\in A^\pm$ and $z$ is in the centre of ${\sf{Heis}}$. Then $p'$ induces a bijection from $Y/Z$ to $A^-$. Under this identification, $n\in N$ acts on $A^-$ by its projection to ${K\ltimes A^-}$. Hence the projection $\hat N$ of $N$ to $K\ltimes A^- \subset {\sf{O}}(n)\ltimes \mathbb{R}^n$ acts cocompactly on $A^-\cong \mathbb{R}^n$. The group $\hat K:=p_K (\hat N)\subset {\sf{O}}(n)$ acts trivially on $A_0:=\hat N\cap A^-$ since $\hat N$ is nilpotent. Let $A_1$ be the orthogonal complement of $A_0$ in $A^-$ and define $\phi:\hat K \to A_1$ by $(k,\phi(k))\in\hat N$. Then $K':={\rm graph}(\phi)$ is a subgroup of $\hat N$. Moreover, $\hat N=K'\times A_0$. Since $K'$ is compact, $A_0$ acts cocompactly on $A^-$. This implies $A_0=A^-$ and $\hat K=\{1\}$. Thus $N$ is contained in ${\sf{Heis}}$ and its projection to $A^-$ is surjective. If $N \cap A^+ \neq \{0\}$, then $N$ contains the center, hence its action on $Y$ is transitive. Otherwise, i.e. if $N \cap A^+ = \{0\}$, then $N$ acts freely properly on $Y$, defining a fibration $N \simeq \mathbb{R}^k \to Y \simeq \mathbb{R}^{n+1} \to N \backslash Y.$ It follows from the long exact sequence of homotopy groups that all the homotopy groups of the compact manifold $N \backslash Y$ are trivial. This implies that it is a point, hence $N$ acts transitively on $Y$ and has dimension $n+1$. We claim that $N$ contains the center. Indeed, if $N$ is not abelian, then it contains $[N,N] = Z$, the center of ${\sf{Heis}}$. Otherwise, $N$ is an abelian subgroup of dimension $n+1$ of ${\sf{Heis}}$. It necessarily contains the center, since an abelian subgroup of ${\sf{Heis}}$ has maximal dimension $n+1$. ◻ **Lemma 32**. *Let $N$ be a connected Lie subgroup of $G:=(\mathbb{R}\times K) \ltimes {\sf{Heis}}$ acting cocompactly on the homogeneous space $X := (\mathbb{R}\times K) \ltimes {\sf{Heis}} / K \ltimes A^+$. Then $N_0 := N \cap (K \ltimes {\sf{Heis}} )$ acts cocompactly on each $(K \ltimes {\sf{Heis}} )$-leaf.* *Proof.* Let $W\subset X$ be a compact set such that $N\cdot W=X$. It is sufficient to consider the ($K\ltimes {\sf{Heis}}$)-leaf $\mathcal {F}_0:=p^{-1}(0)$. Since $N$ is connected and acts cocompactly, it contains a one-parameter group $\gamma(s)$ such that $(p\circ\gamma)(s)=s$. Since $p(W)$ is compact, we may assume $p(W)\subset [-a,a]$. Now take $q\in\mathcal {F}_0$. We can choose an element $n\in N$ such that $nq\in W$. Note that $p(n)\in [-a,a]$. Then $n_0:=\gamma(p(n))^{-1} n\in N_0$ and $n_0q\in (\gamma(p(n))^{-1}\cdot W)\cap \mathcal {F}_0\subset W_0:=\gamma([-a,a])\cdot W \cap \mathcal {F}_0,$ thus $N_0\cdot W_0= \mathcal {F}_0$. Moreover, $\mathcal {F}_0$ is compact since $[-a,a]$ and $W$ are compact and and multiplication is continuous. ◻ **Proposition 33**. *Let $N$ be a connected nilpotent Lie subgroup of $G:=(\mathbb{R}\times K) \ltimes {\sf{Heis}}$ acting cocompactly on the homogeneous space $X := (\mathbb{R}\times K) \ltimes {\sf{Heis}} / K \ltimes A^+$. Then $N$ acts transitively. Moreover, $N_0 := N \cap (K \ltimes {\sf{Heis}} )$ is contained in ${\sf{Heis}}$, and contains the center of Heisenberg.* *Proof.* This is a straightforward consequence of Proposition [Proposition 31](#Proposition: transitive action along the Heis-leaves, without lattice){reference-type="ref" reference="Proposition: transitive action along the Heis-leaves, without lattice"} and Lemma [Lemma 32](#Lemma: N co-compact implies N_0 co-compact){reference-type="ref" reference="Lemma: N co-compact implies N_0 co-compact"}. ◻ ## Standardness of compact quotients **Definition 34** (Standard quotient). *Let $X = G/I$ be a simply connected homogeneous space, and $M=\Gamma \backslash X$ a compact quotient of $X$ by some discrete subgroup $\Gamma$ of $G$, acting properly and freely on $X$. We say that $\Gamma$ (or the quotient manifold $M$) is standard if (up to finite index) $\Gamma$ is contained in some connected Lie subgroup $N$ of $G$ acting properly on $X$ (in this case, $N$ necessarily acts cocompactly on $X$ and $\Gamma$ is a uniform lattice in $N$.)* **Definition 35** (Semi-standard quotient). *Let $X = G/I$ be a simply connected homogeneous space, and $M=\Gamma \backslash X$ a compact quotient of $X$ by some discrete subgroup $\Gamma$ of $G$, acting properly and freely on $X$. We say that $\Gamma$ (or the quotient manifold $M$) is semi-standard if there exists a normal Lie subgroup $G_0$ of $G$ containing $I$ such that* - *$\Gamma\cap G_0$ is standard for $G_0/I$.* - *The projection of $\Gamma$ to $G/G_0$ is a lattice.* Let $\Gamma$ be a discrete subgroup of $G_{\rho}$ acting properly cocompactly on $X_{\rho}$. The following two theorems are corollaries of Sections [5.1](#Subsection:Definition of syndetic hull?){reference-type="ref" reference="Subsection:Definition of syndetic hull?"} and [5.2](#Subsection:Transitivity of cocompact actions of Lie groups){reference-type="ref" reference="Subsection:Transitivity of cocompact actions of Lie groups"}. **Theorem 36** (Non-straight case). *Any non-straight compact quotient of $X_{\rho}$ is standard. More precisely, let $\Gamma$ be a non-straight discrete subgroup of $G_{\rho}$ acting properly and cocompactly on $X_{\rho}$. Then, up to finite index, $\Gamma$ is a cocompact lattice in a connected nilpotent closed Lie subgroup $N$ of $G_{\rho}$. Moreover :* - *$N$ acts simply transitively on $X_{\rho}$,* - *$N = \mathbb{R}\ltimes N_0$, where $\mathbb{R}$ is a $1$-parameter group of $N$ with non-trivial projection to the $\mathbb{R}$-factor in $G_{\rho}$, and $N_0 := N \cap (K \ltimes {\sf{Heis}} )$. Moreover, $N_0 \subset {\sf{Heis}}$, and contains the center of Heisenberg.* **Theorem 37** (Straight case). *Any straight compact quotient of $X_{\rho}$ is semi-standard. More precisely, let $\Gamma$ be a straight discrete subgroup of $G_{\rho}$ acting properly and cocompactly on $X_{\rho}$ (by Corollary [Corollary 24](#Corollary: structure of Gamma){reference-type="ref" reference="Corollary: structure of Gamma"}, up to finite index, $\Gamma=\mathbb{Z}\ltimes \Gamma_0$, with $\Gamma_0:= \Gamma \cap {\sf{Heis}}$). Then there exists a connected closed Lie subgroup $N_0$ of ${\sf{Heis}}$ in which $\Gamma_0$ is a cocompact lattice. Moreover, $N_0$ acts transitively on the ${\sf{Heis}}$-leaves, and contains the center of Heisenberg.* We want to study under which conditions a straight compact quotient is even standard. We proceed similarly to [@KO Prop. 8.18]. Let us first recall the following notation. The Heisenberg group is written as ${\sf{Heis}} = Z \cdot A$, where $Z=\mathbb{R}$, and $A:=\mathbb{C}^n=A^+\oplus A^-$. Any connected subgroup $N_0$ of ${\sf{Heis}}$ containing the centre can be written as $N_0=Z\cdot A'$ for a subspace $A'\subset A$. Let $(\mathbb{R}\oplus {\frak k})\ltimes {\mathfrak{heis}}$ be the Lie algebra of $G_\rho$. As described in Section [3](#Section: Preliminary facts){reference-type="ref" reference="Section: Preliminary facts"}, we have $\rho(t)=e^{t{\bf L}}$ for some derivation ${\bf L}$ of ${\mathfrak{heis}}$ that preserves $A$. **Proposition 38**. *Let $\Gamma=\langle \hat \gamma \rangle \ltimes \Gamma_0$, $\Gamma_0\subset {\sf{Heis}}$, be a straight discrete subgroup of $G_{\rho}$ acting properly and cocompactly on $X_{\rho}$. Let $N_0=Z\cdot A'\subset {\sf{Heis}}$ be the Malcev closure of $\Gamma_0$ in ${\sf{Heis}}$. Then $\Gamma\backslash X_\rho$ is standard if and only if there are elements $(1,\phi)\in \mathbb{R}\times {\frak k}$, $X\in {\frak n}_0:={\rm Lie}(N_0)$, $\hat t\in\mathbb{R}\setminus\{0\}$, and $\hat n\in N_0$ such that $\hat \gamma = \hat n \exp\big(\hat t\,(1,\phi,X)\big)$ and $A'$ is invariant under ${\bf L}+\phi$.* *Proof.* The "if" part is clear. We show the "only if" part. Let $S$ be a syndetic hull of $\Gamma$. The subgroup $r(S)\subset \mathbb{R}\times K$ is connected. Since $K$ is compact, there is a one parameter subgroup $c(t)$, $t\in\mathbb{R}$, of $r(S)$ containing $r(\hat \gamma)$. Let $\phi\in \frak k$ be such that $c(t)=(t,e^{t\phi})$ and $\hat t\in\mathbb{R}$ be such that $c(\hat t)=r(\hat \gamma)$. Let $\tilde N_0$ be the unique connected subgroup of ${\sf{Heis}}$ for which $(S\cap {\sf{Heis}} )\backslash \tilde N_0$ is compact, see [@raghunathan1972discrete Prop. 2.5.]. Because $S\cap {\sf{Heis}}$ acts properly on ${\sf{Heis}} / A^+$, also $\tilde N_0$ acts properly on ${\sf{Heis}} / A^+$. Since $\Gamma_0\subset S\cap{\sf{Heis}} \subset \tilde N_0$ and $\tilde N_0$ is nilpotent, $\Gamma_0$ admits a unique Malcev hull in $\tilde N_0$, which is also a Malcev hull in ${\sf{Heis}}$. On the other hand, $N_0$ is a Malcev hull of $\Gamma_0$ in ${\sf{Heis}}$, which implies $N_0\subset \tilde N_0$. Moreover, $\tilde N_0$ cannot be larger than $N_0$. Indeed, $A'+A^+ =A$ since $N_0$ acts transitively on the Heisenberg leaves. On the other hand, we have $\tilde N_0=Z\cdot \tilde A$, where $A'\subset \tilde A\subset A$, and $\tilde N_0$ acts properly on ${\sf{Heis}} / A^+$. In particular, $\tilde A\cap A^+=0$, thus $A'=\tilde A$. We obtain $N_0=\tilde N_0$. Since the subgroup $S\cap {\sf{Heis}}$ is normal in $S$, also $N_0=\tilde N_0$ is normalised by $S$. Consequently, $A'$ is invariant under ${\bf L}+\phi$. Since $c(t)$ is contained in $r(S)$, the vector $(1,\phi)$ is in the projection of the Lie algebra of $r(S)\ltimes N_0$ to $\mathbb{R}\times{\frak k}$. Choose $X\in {\frak n}_0$ such that $(1,\phi,X)$ is in the Lie algebra of $r(S)\ltimes N_0$. Then the projection of $\exp\big(\hat t\,(1,\phi,X)\big)$ equals $r(\hat \gamma)$. Since $\hat \gamma$ belongs to $S\subset r(S)\ltimes N_0$, it differs from $\exp\big(\hat t\,(1,\phi,X)\big)$ by an element of $N_0$. ◻ As a consequence of Proposition [Proposition 38](#Proposition : semi/standard){reference-type="ref" reference="Proposition : semi/standard"}, we obtain that in the standard case, there is a syndetic hull which is an extension of $\mathbb{R}$ by a subgroup of ${\sf{Heis}}$. **Corollary 39**. *Let $\Gamma=\langle \hat \gamma \rangle \ltimes \Gamma_0$, $\Gamma_0\subset {\sf{Heis}}$, be a straight discrete subgroup of $G_{\rho}$ acting properly and cocompactly on $X_{\rho}$. Let $N_0$ be the Malcev closure of $\Gamma_0$ in ${\sf{Heis}}$. If $\Gamma \backslash X_{\rho}$ is standard, then there is a syndetic hull of $\Gamma$ in $G_{\rho}$ of the form $\mathbb{R}\ltimes N_0$, where $\mathbb{R}$ is a $1$-parameter group of $\mathbb{R}\times K \subset G_{\rho}$ with non-trivial projection to the $\mathbb{R}$-factor.* *Proof.* The $1$-parameter group is given by $c(t)$ in the proof of the previous proposition. ◻ # Non-standard phenomena {#Section: Non Standard} In this section, we try to understand the non-standard compact quotients $\Gamma \backslash X_{\rho}$ of plane waves $X_{\rho}=G_{\rho}/I$. In this case, the fundamental group $\Gamma$ is necessarily straight: in Example [Example 42](#Example: non standard C-W){reference-type="ref" reference="Example: non standard C-W"}, we give an explicit example of such a situation. The main motivation here is to see whether $\Gamma$ is standard up to embedding $G_{\rho}$ in some bigger group.   Let $\Gamma$ be a straight discrete subgroup of $G_{\rho}$ acting properly and cocompactly on $X_{\rho}$, then up to finite index, $\Gamma = \langle \hat{\gamma} \rangle \ltimes \Gamma_0$, with $\Gamma_0:=\Gamma \cap {\sf{Heis}}$. Let $N_0$ be the Malcev closure of $\Gamma_0$ in ${\sf{Heis}}$. There are two cases:\ \ **Case 1:** $\hat{\gamma}$ is not contained in a $1$-parameter group of $G_{\rho}$.\ Non-existence of a $1$-parameter group containing $\hat{\gamma}$ does not imply non-standardness. This is already suggested by Proposition [Proposition 38](#Proposition : semi/standard){reference-type="ref" reference="Proposition : semi/standard"}. Indeed, it is proved there that in the standard case, $n_0^{-1} \hat{\gamma}$ is contained in a $1$-parameter group of $G_{\rho}$ for some $n_0 \in N_0$, but not a priori $\hat{\gamma}$. Here, we give an example where this occurs. **Example 40**. *Let $\widetilde{\mathsf{Euc}}_2 :=\mathbb{R}\ltimes \mathbb{R}^{2}$, where $\mathbb{R}$ acts by rotations on $\mathbb{R}^2$. Let $G_{\rho}=(\mathbb{R}\times {\sf{SO}}(2))\ltimes {\sf{Heis}} _5$ be the special polarized Poincaré group in dimension $4$ (recall that this is the subgroup of ${\sf Isom}({\sf{Mink}} ^{1,3})$ preserving an isotropic vector). Let $c(t)=(t, e^{it})$ be a $1$-parameter group in $\mathbb{R}\times {\sf{SO}}(2)$. Since it preserves $A^- \simeq \mathbb{R}^2$, one can consider $S_1:= c(t) \ltimes A^-$, which is isomorphic to $\widetilde{\mathsf{Euc}}_2$. Define $S:= S_1 \times \mathbb{R}$, where $\mathbb{R}$ is the center of ${\sf{Heis}} _5$. Clearly $S$ acts properly cocompactly on $X_{\rho}={\sf{Mink}} ^{1,3}$. Moreover, $S$ contains straight lattices which are not contained in the image of the exponential map. Namely, take any lattice $\Gamma_1$ in the abelian subgroup $2\pi\mathbb{Z}\times \mathbb{R}^2 \subset \widetilde{\mathsf{Euc}}_2$ which does not intersect the $2 \pi \mathbb{Z}$-factor, and consider $\Gamma= \Gamma_1 \times \mathbb{Z}$.\ * **Case 2:** $\hat{\gamma}$ is contained in a $1$-parameter group of $G_{\rho}$.\ \ When $\Gamma$ is non-standard, we ask the following question:\ **Question:** Can we find a compact group $C$ acting on ${\sf{Heis}}$, and construct an embedding $G\hookrightarrow \widehat G := (\mathbb{R}\times C \times K) \ltimes {\sf{Heis}}$, and a $G$-equivariant embedding $X\hookrightarrow \widehat X=\widehat G/ \widehat I$ into a homogeneous space, such that the quotient $\Gamma\backslash \widehat X$ is standard? Here, $\widehat{I}:= (C' \times K) \ltimes A^+$, where $C'$ is a closed subgroup of $C$ preserving $A^+$. In particular, we obtain the semi-standard quotient $\Gamma\backslash X$ as a submanifold of the standard quotient $\Gamma\backslash \widehat X$. Here, $\widehat X$ is not assumed to be simply connected, but one can adapt Definition [Definition 34](#Definition: standard){reference-type="ref" reference="Definition: standard"} and define "standard quotients\" of non-simply connected homogeneous spaces. The answer to this question is provided in the following theorem. **Theorem 41**. *Let $G_{\rho}= (\mathbb{R}\times K) \ltimes_{\rho} {\sf{Heis}}$, $I=K \ltimes A^+$, and $X_{\rho}= G_{\rho}/ I$. Let $\Gamma=\langle \hat{\gamma} \rangle \ltimes \Gamma_0$ be a non-standard straight discrete subgroup acting properly and cocompactly on $X_{\rho}$. If $\hat{\gamma}$ is contained in a $1$-parameter group, there is a $\hat G$-homogeneous space $\hat X$, which is a torus bundle over $X_{\rho}$, such that $\Gamma$ lifts to a standard discrete subgroup of $\hat G$.* In order to explain the construction of $\hat X$ let us look first at the following non-standard quotient of a Cahen-Wallach space: **Example 42** (**A non-standard Example**). *Let $K := {\sf{SO}}(2)$ act on ${\sf{Heis}} _5 = \mathbb{R}\times \mathbb{C}^2$ trivially on the center and by the standard representation on $A^+$ and $A^-$. Furthermore, let $\mathbb{R}$ act on ${\sf{Heis}} _5$ by $t \cdot (v,z_1,z_2)=(v,e^{it} z_1, e^{it} z_2)$ and consider $G_{\rho}= (\mathbb{R}\times K)\ltimes_{\rho} {\sf{Heis}} _5$. Take $\alpha\in\mathbb{R}^*$ and define a three-dimensional subgroup $N:=\mathbb{R}\cdot A'$, where $A':={\sf{Span}} _{\mathbb{R}}\{c_1:=(1,\alpha i),\, c_2:=(-\alpha i,1)\}\subset {\sf{Heis}} _5$. Then $N$ acts properly on $X_{\rho}=G_{\rho} / K \ltimes A^+$ and cocompactly on the ${\sf{Heis}}$-leaves. Take a lattice $\Lambda$ in it and define $\Gamma := \langle \hat{\gamma} \rangle \times \Lambda$, with $\hat{\gamma}=(2 \pi, 1, 0)$. Then $\Gamma$ is a discrete subgroup of $G_{\rho}$ acting properly and cocompactly on $X_{\rho}$. Indeed, by [@KO Proposition 4.8] it suffices to show that $e^{it}A'\cap A^+=\{0\}$ for all $t\in\mathbb{R}$. Since $A^+=\mathbb{R}^2$, this is satisfied if and only if the linear equation $\Im (r_2 e^{it}c_1+ r_2 e^{it}c_2)=0$ for $r_1,r_2\in\mathbb{R}$ admits only the trivial solution. Thus the assertion is equivalent to ${\sf{det}}( \Im (e^{it}c_1,e^{it}c_2))=1+(\alpha^2-1)\cos^2t\not=0$ for all $t$, which is obviously satisfied. We want to show that it is non-standard for $\alpha \neq \pm1$. By Proposition [Proposition 38](#Proposition : semi/standard){reference-type="ref" reference="Proposition : semi/standard"}, it suffices to show that $A'$ is not invariant under ${\bf L}+\phi$ for any $\phi\in{\frak k}$. Since ${\bf L}|_A$ equals multiplication by $i$ and $K={\sf{SO}}(2)$ preserves $A'$, the condition $({\bf L}+\phi)A'\subset A'$ would give ${\sf{Span}} _{\mathbb{R}}\{c_1,\, c_2\}={\sf{Span}} _{\mathbb{R}}\{i c_1,\, ic_2\}$, which implies $\alpha=\pm1$.* #### **How to make Example [Example 42](#Example: non standard C-W){reference-type="ref" reference="Example: non standard C-W"} standard** {#how-to-make-example-example-non-standard-c-w-standard} If one defines $\widehat{G} := (\mathbb{R}\times \mathbb{S}^1 \times {\sf{SO}}(2)) \ltimes {\sf{Heis}} _5$, where $\mathbb{R}$ acts like before, and $\mathbb{S}^1$ acts like $\mathbb{R}/ \ker \rho$, then the diagonal $1$-parameter group $D:=(t, e^{-i t})$ in $\mathbb{R}\times \mathbb{S}^1$ acts trivially on ${\sf{Heis}} _5$, hence preserves $N$. The new homogeneous space $\widehat{X}=\widehat{G} / {\sf{SO}}(2) \ltimes A^+$ is a plane wave which is an $\mathbb{S}^1$-bundle over $X_{\rho}$. We have a natural embedding by a diagonal morphism map $d: G \to \widehat{G}$, $d(t,k,h)=(t,e^{-it},k,h)$, which induces an embedding $X \to \widehat{X}$, such that $\widehat{\Gamma} := d(\Gamma)$ is standard in $\widehat{G}$. Its syndetic hull $D\times N$ in $\widehat{G}$ does not act transitively on $\widehat{X}$. **Some comments :** 1. The compact $\mathbb{S}^1$-factor added here does not preserve $A^+$, this is why it cannot be added to the isotropy. So the dimension here is increased. 2. In this example, $\mathbb{R}$ acts through an elliptic matrix, but in general there may be a hyperbolic and a unipotent part. The problem is formulated in the sequel for general homogeneous plane waves.\ #### **Modifying the group to have standard quotients** Let $G_{\rho}= (\mathbb{R}\times K) \ltimes_{\rho} {\sf{Heis}}$, $I=K \ltimes A^+$, and $X_{\rho}= G_{\rho}/ I$. Let $\Gamma=\langle \hat{\gamma} \rangle \ltimes \Gamma_0$ be a straight non-standard discrete subgroup acting properly and cocompactly on $X_{\rho}$. Assume that $\hat{\gamma}$ is contained in some $1$-parameter group $g(t)$ of $G_\rho$. We identify the image of $\rho$ with a subgroup of ${\sf Sp}_{2n}(\mathbb{R})$ and consider the Jordan decomposition $l(t):=\rho(r(g(t)))= d(t) \circ u(t)$, where $d(t)$ is the semi-simple part and $u(t)$ the unipotent part. Write also $d(t) = e(t) \circ h(t)$, where $e(t)$ is the elliptic part (with complex eigenvalues), and $h(t)$ the hyperbolic part (with real eigenvalues). **Lemma 43**. *Let $l(t)$ be a $1$-parameter group in ${\sf Sp}_{2n}(\mathbb{R})\subset {\sf{Aut}}({\sf{Heis}} )$, and $l(t) = e(t) h(t) u(t)$ be as above. Assume that $l(1)$ preserves a subgroup $N$ of ${\sf{Heis}}$. Then $h(t)$ and $u(t)$ preserve $N$ for any $t$, and therefore, $l(t)$ preserves $N$ if and only if $e(t)$ preserves $N$.* *Proof.* It follows from elementary linear algebra that $l(1)$ preserves $N$ if and only if $e(1)$, $h(1)$ and $u(1)$ do. Moreover, if $h(1)$ preserves $N$, then $h(t)$ preserves $N$ for any $t$. Similarly, if $u(1)$ preserves $N$, then $u(t)$ preserves $N$ for any $t$. Thus $l(t)$ preserves $N$ if and only if $e(t)$ preserves $N$. ◻ Let $N$ be the Malcev closure of $\Gamma_0$ in ${\sf{Heis}}$. Then $\rho(\hat{\gamma}) =l(1)$ preserves $\Gamma_0$, hence also $N$. The elliptic part $e(t)$ of $l(t)$ is a $1$-parameter subgroup of ${\sf{Aut}}({\sf{Heis}} )$, whose closure is a torus $\mathbb{T}^d$ in ${\sf{Aut}}({\sf{Heis}} )$. It follows from Lemma [Lemma 43](#Lemma: elliptic part of l(t)){reference-type="ref" reference="Lemma: elliptic part of l(t)"} above that the $1$-parameter group $D:=(t, e(t)^{-1})$ in $\mathbb{R}\times \mathbb{T}^d$ preserves $N$. As in the previous example, one can add the $\mathbb{T}^d$-factor to the group and define $\widehat{G}= (\mathbb{R}\times \mathbb{T}^d \times K) \ltimes {\sf{Heis}}$. Define a new homogeneous space $\widehat{X}=\widehat{G}/\widehat{I}$, where the isotropy is given by $\widehat{I} = (C \times K) \ltimes A^+$, with $C$ any compact subgroup of $\mathbb{T}^d$ preserving $A^+$. We have a natural injective morphism $d: G_{\rho} \to \widehat{G}$, $d(t,k,h)=(t, e(t)^{-1}, k,h)$ inducing an embedding $X_{\rho} \to \widehat{X}$. Then $\widehat{\Gamma} := d(\Gamma)$ is standard in $\widehat{G}$, and $D \ltimes N$ (note that the $D$-action on $N$ here is generically non-trivial) is a syndetic hull of $\hat \Gamma$ in $\widehat{G}$. **Remark 44**. *In general, the new homogeneous space $\widehat{X}$ does not admit a left $\widehat G$-invariant Lorentzian (even pseudo-Riemannian) metric.* # Equicontinuity of the parallel flow {#Section: Equicontinuity} Let $(M,V)$ be a compact locally homogeneous plane wave. Since $M$ is complete, it is a quotient $\Gamma \backslash G_{\rho}/I$ by some subgroup $\Gamma$ of $G_{\rho}$ acting properly discontinuously on $G_{\rho} /I$. In this case, the action of the flow of $V$ is given by the $Z$-action, where $Z$ is the center of Heisenberg (Fact [Fact 18](#Fact: parallel flow is Z){reference-type="ref" reference="Fact: parallel flow is Z"}). Recall that the flow of $V$ is said to be equicontinuous if it is relatively compact, considered as a one-parameter subgroup of the isometry group endowed with its Lie group topology. Equivalently, the flow is equicontinuous if it preserves a Riemannian metric. The key point for this equivalence is the fact that the isometry group of a Lorentzian metric $g$ on $M$ is closed in $\mathsf{Homeo}(M)$ with respect to the compact-open topology (which in fact coincides on ${\sf Isom}(M,g)$ with the Lie group topology), see [@nomizu]. ## Equicontinuity **Theorem 45**. *Let $(M,V)$ be a compact locally homogeneous plane wave. The action of the parallel $V$-flow is equicontinuous.* *Proof.* Let $X$ be the universal cover of $M$, we have $M=\Gamma\backslash X$, where $\Gamma\subset \mathsf{Isom}(X)$ is a subgroup acting freely, properly discontinuously and cocompactly on $X$. In the non-flat case, $\mathsf{Isom}^{\circ}(X)$ is isomorphic to $G_{\rho}$ for some $\rho$. In the flat case, since $\Gamma$ preserves the lift of $V$ to $X$, we can reduce to the subgroup of the Poincaré group preserving an isotropic vector, namely, $\mathsf{SPol}$ which is also isomorphic to $G_{\rho}$ for some $\rho$ (see Definition [2](#Section: Flat case){reference-type="ref" reference="Section: Flat case"}). And we have $\mathsf{Isom}{(M)}= \Gamma\backslash N_{G_{\rho}}(\Gamma)$. Moreover, by Fact [Fact 18](#Fact: parallel flow is Z){reference-type="ref" reference="Fact: parallel flow is Z"}, the flow of $V$ on $M$ is given by the left $Z$-action on the double quotient $\Gamma\backslash G_{\rho}/I$ (the action is well defined, since $\Gamma$ centralizes $Z$). The equicontinuity of the flow of $V$ is equivalent to $\pi(Z)$ being relatively compact in $\Gamma\backslash N_{G_{\rho}}(\Gamma)$, where $\pi: N_{G_{\rho}}(\Gamma) \to \Gamma \backslash N_{G_{\rho}}(\Gamma)$ is the natural projection. So to show that the action is equicontinuous, it is enough to show that $\Gamma\backslash\overline{\Gamma Z}$ (here the closure is in $N_{G_{\rho}}(\Gamma)$) is compact. There are two cases:\ **1)** Straight case: We know from Theorem [Theorem 37](#Theorem: standarness in straight case){reference-type="ref" reference="Theorem: standarness in straight case"} that $\Gamma$ contains a finite index subgroup $\Gamma'$, which is a cocompact lattice of a closed subgroup $\mathbb{Z}\ltimes N_0$ of $G_{\rho}$, with $N_0$ a closed subgroup containing the center $Z$. On the one hand, $\overline{Z\Gamma'}\subset \mathbb{Z}\ltimes N_0$. On the other hand, the quotient $\Gamma'\backslash(\mathbb{Z}\ltimes N_0)$ is compact. This implies that the $V$-flow is equicontinuous on $\Gamma' \backslash X$, i.e. it preserves a Riemannian metric $h$ on $X$ which is $\Gamma'$-invariant. Now, averaging $h$ over the representatives of the (finite) quotient space $\Gamma'\backslash\Gamma$ defines a new Riemannian metric $h^* :=\sum_{i=1}^{r} \gamma_i^*(h)$, where $\{\gamma_1,\dots,\gamma_r\}$ is a set of such representatives, which is $\Gamma$-invariant and preserved by the flow of $V$.\ **2)** Non-straight case: Similarly, $\Gamma$ contains a finite index subgroup $\Gamma'$ which is a cocompact lattice in some closed Lie subgroup $N$ that contains the center $Z$ (Theorem [Theorem 36](#Theorem: standarness in non-straight case){reference-type="ref" reference="Theorem: standarness in non-straight case"}). This implies in the same way as in the straight case that the $V$-flow preserves a Riemannian metric on $M$. ◻ Now we can prove all statements in Theorem [Theorem 3](#Introduction-Theorem: parallel fields){reference-type="ref" reference="Introduction-Theorem: parallel fields"}. *Proof of Theorem [Theorem 3](#Introduction-Theorem: parallel fields){reference-type="ref" reference="Introduction-Theorem: parallel fields"}.* The existence of a parallel vector field $V$ and the last statement (existence of a parallel line field) are proved in Theorem [Theorem 13](#Theorem section flat: parallel fields){reference-type="ref" reference="Theorem section flat: parallel fields"}. When $V$ is timelike, the isometry group preserving $V$ is ${\sf{SO}}(n) \ltimes \mathbb{R}^{1+n}$, so the claim follows from Bieberbach Theorem. The case where $V$ is isotropic follows from Theorem [Theorem 45](#Theorem: equicontinuity){reference-type="ref" reference="Theorem: equicontinuity"}. For $V$ spacelike, examples of non-equicontinuous Anosov as well as partially hyperbolic flows are given in Example [Example 17](#Example: SOL-Anosov){reference-type="ref" reference="Example: SOL-Anosov"}. ◻ ## Equicontinuity from standardness In the standard case, we can give a more general proof of equicontinuity of the lightlike parallel flow. In this case, $X_{\rho}$ identifies with a Lie group $S$ endowed with a left-invariant Lorentzian metric, and $M$ is finitely covered by $M':=\Gamma \backslash S$, where $\Gamma$ is a uniform torsion free discrete subgroup of $S$. The action of the $V$-flow on $X_{\rho}$ induces an action on $S$ which commutes with all left translations. This defines a left invariant parallel null vector field on $S$. So equicontinuity of the $V$-flow on $M$ is equivalent to the equicontinuity of the induced vector field on $\Gamma \backslash S$. And the latter is a consequence of Theorem [Theorem 46](#Theorem: equicontinuity on Lie group){reference-type="ref" reference="Theorem: equicontinuity on Lie group"} below. **Theorem 46**. *Let $G$ be a Lie group with a left invariant Lorentzian metric. Let $V$ be a left-invariant parallel null vector field on $G$. Then $\mathsf{ad}_V$ is skew-symmetric for a Riemannian scalar product on ${\mathfrak{g}}$.* *Proof.* Let $\langle \cdot\,,\cdot\rangle$ denote the metric on $G$ and also the induced scalar product on ${\mathfrak{g}}$. The vector field $V$ is left invariant, so its flow corresponds to the right action of a one parameter group, say $v^t$ of $G$. Since the Lorentzian metric on $G$ is left invariant, left multiplication by $v^t$ is isometric. On the other hand, $V$ is parallel, hence Killing. Thus, right multiplication by $v^t$ is also isometric. In particular, conjugacy by $v^t$ is isometric, hence $\mathsf{ad}_V: \mathfrak{g} \to \mathfrak{g}$ is skew symmetric. Now we use that $\ker(\mathsf{ad}_V)$ contains the isotropic element $V$. We obtain that ${\mathfrak{g}}$ decomposes into the orthogonal sum ${\mathfrak{g}}={\mathfrak{q}}_1\oplus{\mathfrak{q}}_2$ of $\mathsf{ad}_V$-invariant subspaces, where ${\mathfrak{q}}_1$ is Euclidean and ${\mathfrak{q}}_2$ is either contained in $\ker(\mathsf{ad}_V)$ or ${\mathfrak{q}}_2={\sf{Span}} \{e_1=V, e_2, e_3\}$ such that $\mathsf{ad}_Ve_1=0$, $\mathsf{ad}_Ve_2=e_1$, $\mathsf{ad}_Ve_3=e_2$ and $\langle e_i,e_j\rangle=1$ if $i+j=4$, $\langle e_i,e_j\rangle=0$ otherwise. It suffices to show that the second case cannot occur. Assume that we were in the second case. Then $\langle V,[e_2,e_3]\rangle =\langle \mathsf{ad}_Ve_2, [e_2,e_3]\rangle =-\langle e_2,\mathsf{ad}_V[e_2,e_3]\rangle=-\langle e_2, [\mathsf{ad}_Ve_2,e_3]+[e_2,\mathsf{ad}_V e_3]\rangle =-\langle e_2,e_2\rangle=-1.$ On the other hand, by the Koszul formula, $[{\mathfrak{g}},{\mathfrak{g}}]^\perp$ consists of all elements $Y\in{\mathfrak{g}}$ for which $\nabla Y:{\mathfrak{g}}\to{\mathfrak{g}}$ is symmetric. Hence $V$ belongs to $[{\mathfrak{g}},{\mathfrak{g}}]^\perp$, which gives a contradiction. ◻ # On standardness of more general locally homogeneous structures {#Section: More homogeneous structures} In the study of standardness of compact quotients $\Gamma \backslash X_{\rho}$, we did not use the Lorentzian nature of the homogeneous spaces $X_\rho=G_{\rho}/I$ provided by the restrictions on the $\rho$-action (the $\rho$-actions for which $G_{\rho}$ preserves a Lorentzian metric on $G_{\rho}/I$ are characterized in [@Content1]). In this section, we give an analogous statement to that in Sections [4](#Section: Fundamental group){reference-type="ref" reference="Section: Fundamental group"} and [5](#Section: syndetic hull, standard semi-Standrad){reference-type="ref" reference="Section: syndetic hull, standard semi-Standrad"}, without assuming these spaces to be Lorentzian. Let $G_{\rho} := (\mathbb{R}\times K) \ltimes_{\rho} {\sf{Heis}}$, where $\rho$ is any action, and let $I=C\ltimes A^+$, where $C$ is a closed subgroup of $K$ that preserves $A^+$. We consider $X_{\rho}:=G_{\rho}/I$.\ In the following theorem, when $C \neq K$, Definition [Definition 34](#Definition: standard){reference-type="ref" reference="Definition: standard"} of standardness and Definition [Definition 35](#Definition: semi-standard){reference-type="ref" reference="Definition: semi-standard"} of semi-standardness are adapted to the non-simply connected case. **Theorem 47**. *Let $\rho:\mathbb{R}\times K\to {\sf{Aut}}({\sf{Heis}} )$ be such that $\rho h= h\rho$ for some Heisenberg homothety $h\not=1$. Let $\Gamma$ be a discrete subgroup of $G_{\rho}$ acting properly cocompactly on $X_{\rho}$. Then* - *$\Gamma$ is virtually nilpotent, and $X_{\rho}$ is standard,* - *or $\Gamma\cong \mathbb{Z}\ltimes \Gamma_0$, where $\Gamma_0$ is virtually nilpotent, and $X_{\rho}$ is semi-standard.* *Moreover, when $C=K$, the syndetic hull of $\Gamma$ (resp. $\Gamma_0$) acts transitively on $X_{\rho}$ (resp. on the $K \ltimes {\sf{Heis}}$-leaves).* *Proof.* We proceed analogously to the proofs of Theorem [Theorem 36](#Theorem: standarness in non-straight case){reference-type="ref" reference="Theorem: standarness in non-straight case"} and Theorem [Theorem 37](#Theorem: standarness in straight case){reference-type="ref" reference="Theorem: standarness in straight case"}. Either $\Gamma$ is non-straight or straight. In the non-straight case, since the representation $\rho$ commutes with ${\sf{Heis}}$ homotheties, the same proof as in Theorem $\ref{Theorem Gamma}$ allows to get that $\Gamma$ is generated by elements in a strong Zassenhaus neighborhood. Hence the existence of a (nilpotent) syndetic hull by Fact [\[Thurston: strongly Zassenhaus neighborhood\]](#Thurston: strongly Zassenhaus neighborhood){reference-type="ref" reference="Thurston: strongly Zassenhaus neighborhood"}, and the standardness of $\Gamma \backslash X_{\rho}$. In the straight case, we get that $\Gamma\cong \mathbb{Z}\ltimes \Gamma_0$, where $\Gamma_0 \subset K\ltimes {\sf{Heis}}$. Here, we discuss two cases: either $\Gamma_{0}$ has a finite projection to $K$, or the projection is not discrete. In the first case, $\Gamma_0 \cap {\sf{Heis}}$ has finite index in $\Gamma_0$, and its Malcev closure in ${\sf{Heis}}$ acts properly and cocompactly on the ${\sf{Heis}}$-leaves. In the second case, we proceed as in the non-straight case, and show that $\Gamma_0$ is generated by elements in a strong Zassenhaus neighborhood of $K \ltimes {\sf{Heis}}$, hence the existence of a (nilpotent) syndetic hull of $\Gamma_0$ in $K \ltimes {\sf{Heis}}$ and the semi-standardness of $\Gamma \backslash X_{\rho}$. Now, when $C=K$, the transitive action of the syndetic hull of $\Gamma$ (resp. $\Gamma_0$) on $X_{\rho}$ (resp. on a $\mathcal{F}$-leaf) follows from Proposition [Proposition 51](#Proposition-Appendix A: transitive action under existence of a lattice){reference-type="ref" reference="Proposition-Appendix A: transitive action under existence of a lattice"} and Proposition [Proposition 53](#Proposition-Appendix A: G_rho linear){reference-type="ref" reference="Proposition-Appendix A: G_rho linear"}. ◻ **Example 48** (Non-transitive action). *In the modification of Example [Example 42](#Example: non standard C-W){reference-type="ref" reference="Example: non standard C-W"}, $C={\sf{SO}}(2)$ and $K=\mathbb{S}^1\times {\sf{SO}}(2)$. The syndetic hull $H$ of $\Gamma$ does not act transitively on $\widehat{X}$. Indeed, $H$ acts freely on $\widehat{X}$ and we have $H\backslash \widehat{X}\cong \mathbb{S}^{1}$.* **Remark 49** (Carnot groups). *One can replace ${\sf{Heis}}$ with the so-called Carnot groups, since they also have homotheties. An analogous version of Theorem [Theorem 47](#Theorem: General homogeneous){reference-type="ref" reference="Theorem: General homogeneous"} holds for discrete subgroups of $(\mathbb{R}\times K)\ltimes_{\rho} N$, where $N$ is a Carnot group.* **Remark 50**. *As stated in the beginning of the section, the homogeneous spaces in Theorem [Theorem 47](#Theorem: General homogeneous){reference-type="ref" reference="Theorem: General homogeneous"} are not necessarily Lorentzian. For instance, when the $\rho$-action is trivial, i.e. $G_{\rho}=(\mathbb{R}\times K)\times {\sf{Heis}}$, $X_{\rho}$ does not admit a left $G_{\rho}$-invariant Lorentzian metric (in fact it is not even pseudo-Riemannian). Indeed, $G_{\rho}$ preserves a Lorentzian metric on $G_{\rho}/I$ if and only if the $\mathsf{ad}(I)$-action on $\mathfrak{g}/\mathfrak{i}$ is an infinitesimal isometry of some Lorentzian scalar product. Here, $\mathsf{ad}(I)$ is nilpotent of degree $2$, and a nilpotent non-zero endomorphism of a vector space is an infinitesimal isometry of some Lorentzian scalar product if and only if its nilpotency order equals $3$. This is a well-known fact in pseudo-Riemannian geometry, for a proof see [@Content1].* # Cocompact proper actions of Lie groups {#Appendix A} In this appendix we consider cocompact proper actions of Lie groups admitting a torsion free uniform lattice. **Proposition 51**. *Let $G$ be a connected Lie group which admits a torsion free uniform lattice $\Gamma$. Assume that $G$ acts properly cocompactly on a contractible manifold $X$. Then $G$ acts transitively.* The proof uses techniques from cohomology theory of discrete groups. *Proof.* Since $\Gamma$ is torsion free, $\Gamma \backslash X$ is a closed $K(\Gamma,1)$ manifold. We have by [@brown2012cohomology VIII, (8.1)] that the cohomological dimension of $\Gamma$ is equal to $\dim (\Gamma \backslash X)$. Now, let $K$ be a maximal compact subgroup of $G$. We know that $G$ is diffeomorphic to $K\times\mathbb{R}^{k}$ see [@iwasawa1949some]. We claim that $\Gamma\backslash G/K$ is a $K(\Gamma, 1)$ manifold. Indeed, $\Gamma$ acts freely (torsion free) and cocompactly. Moreover, $\Gamma$ acts properly, since the fiber bundle map $\pi: G\longrightarrow G/K$ is a proper map. Hence, [@brown2012cohomology] we get that $\dim(\Gamma\backslash G/K)=\dim(\Gamma\backslash X)$, i.e. $\dim(G)=\dim(K)+\dim(X)$. Because the $G$-action is proper (in particular the stabilizer of any point is compact), we conclude that the $G$-orbits are open. The claim follows from the connectedness of $X$. ◻ When $G$ is linear, Selberg's Lemma applies. Namely, any finitely generated subgroup of $G$ is virtually torsion free. For linear groups we get the following corollary. **Corollary 52**. *Let $G$ be a connected linear Lie group which admits a uniform lattice $\Gamma$. Assume that $G$ acts properly cocompactly on a contractible manifold $X$. Then $G$ acts transitively.* **Proposition 53**. *$G_\rho$ is linear.* *Proof.* We have a natural morphism $f: G_{\rho} \to {\sf{Aut}}({\sf{Heis}} )\ltimes {\sf{Heis}}$ given by $f(r,k,h)= (\rho(r,k),h)$ (which is not necessarily injective). Moreover, the group ${\sf{Aut}}({\sf{Heis}} )\ltimes {\sf{Heis}}$ is linear. Indeed, it has a trivial center (due to existence of homotheties), hence the adjoint representation $\mathsf{Ad}:{\sf{Aut}}({\sf{Heis}} )\ltimes {\sf{Heis}} \to {\sf{GL}}({\mathfrak{g}})$, where ${\mathfrak{g}}$ is the Lie algebra of ${\sf{Aut}}({\sf{Heis}} )\ltimes {\sf{Heis}}$, is an embedding (faithful). Define now $\Phi: G_{\rho} \longrightarrow (\mathbb{R}\times K) \times {\sf{GL}}({\mathfrak{g}})$, $\Phi(r,k,h) = (r,k, \mathsf{Ad}(f (r,k,h)))$. Then $\Phi$ is clearly a faithful morphism into $(\mathbb{R}\times K) \times {\sf{GL}}({\mathfrak{g}})$, which is a linear group. The claim follows. ◻ # A non-periodic example {#Appendix B} Let $L=\mathbb{R}\ltimes \mathbb{R}^{n+1}$, with coordinates $(v,y) \in \mathbb{R}^{n+1}=\mathbb{R}\times \mathbb{R}^n$, and the $\mathbb{R}$-action defined by $t \cdot (v,y) = (v, R_t(y))$, where $R_t$ is some periodic elliptic action on $\mathbb{R}^n$. Define a Lorentzian left-invariant metric $g$ on $L$, such that the induced metric on $\mathbb{R}^{n+1}$ is degenerate, and $V:= \partial_v$ is isotropic. Then $(L,g)$ is a homogeneous plane wave, by [@Leis Theorem 3]. Let $\Gamma := \langle \hat{\gamma} \rangle \times \Gamma_0$, with $\Gamma_0$ a lattice in $\mathbb{R}^{n+1}$ and $\hat{\gamma}$ generates a lattice in the $\mathbb{R}$-factor acting trivially on $\Gamma_0$. Suppose further that $\Gamma_0$ does not intersect the subgroup generated by the $v$-translations. Then $\Gamma$ is a (uniform) lattice in $L$, and the flow of $V$ is not periodic in $\Gamma \backslash L$.

arxiv_math

--- abstract: | A matchstick graph is a plane graph with edges drawn as unit distance line segments. This class of graphs was introduced by Harborth who conjectured that a matchstick graph on $n$ vertices can have at most $\lfloor 3n - \sqrt{12n - 3}\rfloor$ edges. Recently his conjecture was settled by Lavollée and Swanepoel. In this paper we consider $1$-planar unit distance graphs. We say that a graph is a $1$-planar unit distance graph if it can be drawn in the plane such that all edges are drawn as unit distance line segments while each of them are involved in at most one crossing. We show that such graphs on $n$ vertices can have at most $3n-\sqrt[4]{n}/10$ edges. author: - "Panna Gehér[^1]" - "Géza Tóth[^2]" title: 1-planar unit distance graphs --- # Introduction A graph is called a matchstick graph if it can be drawn in the plane with no crossings such that all edges are drawn as unit distance line segments. This graph class was introduced by Harborth in 1981 [@Harborth1; @Harborth2]. He conjectured that the maximum number of edges of matchstick graph with $n$ vertices is $\lfloor 3n - \sqrt{12n - 3}\rfloor$. He managed to prove it in a special case where the unit distance is also the smallest distance among the points [@Harborth74]. Recently his conjecture was settled by Lavollée and Swanepoel [@LS22]. Other interesting classes of graphs are the classes of $k$-planar graphs. For any $k\ge 0$, a graph $G$ is called $k$-planar if $G$ can be drawn in the plane such that each edge is involved in at most $k$ crossings. Let $e_k(n)$ denote the maximum number of edges of a $k$-planar graph of $n$ vertices. Since $0$-planar graphs are the well known planar graphs, $e_0(n)=3n-6$ for $n\ge 3$. We have $e_1(n)=4n-8$ for $n\ge 4$ [@PT97], $e_2(n)\le 5n-10$, which is tight for infinitely many $n$ [@PT97], $e_3(n)\le 5.5n-11$, which is tight up to an additive constant [@PRTT04] and $e_4(n)\le 6n-12$, which is also tight up to an additive constant [@A19]. For general $k$ we have $e_k(n)\le c\sqrt{k}n$ for some constant $c$, which is tight apart from the value of $c$ [@PT97; @A19].\ In this paper we investigate $k$-planar unit distance graphs. Let $u_k(n)$ be the maximum number of edges of a $k$-planar unit distance graph. Since $0$-planar unit distance graphs are exactly the matchstick graphs, by the result of Lavollée, Swanepoel we have $u_0(n)=\lfloor 3n - \sqrt{12n - 3}\rfloor$. We do not have any better lower bound for $u_1(n)$ than the value of $e_0(n)$. That is, allowing to use one crossing on each edge does not seem to help, still a proper piece of the triangular grid is the best known construction. We also prove an almost matching upper bound. **Theorem 1**. *For the maximum number of edges of a $1$-planar unit distance graph $u_1(n)$ we have $$\lfloor 3n - \sqrt{12n - 3}\rfloor\le u_1(n)\le 3n-\sqrt[4]{n}/10.$$* For general $k$, the best known lower bound is due to Günter Rote (personal communication, 2023). **Theorem 2**. *(Rote) For the maximum number of edges of a $k$-planar unit distance graph $u_k(n)$ we have $$u_k(n) \geq 2^{\Omega\left(\log k/\log\log k \right)}n.$$* We include the proof in this note. We have the following upper bound. **Theorem 3**. *For any $n, k\ge 0$ we have $$u_k(n)\le c\sqrt[4]{k}n$$ for some $c>0$.* In Section [2](#sec2){reference-type="ref" reference="sec2"} we prove Theorem [Theorem 1](#1-planar){reference-type="ref" reference="1-planar"} and in Section [3](#sec3){reference-type="ref" reference="sec3"} we prove Theorems [Theorem 2](#rote){reference-type="ref" reference="rote"} and [Theorem 3](#k-planar){reference-type="ref" reference="k-planar"}. # 1-planar unit distance graphs {#sec2} The lower bound follows directly from Harborth's lower bound construction for matchstick graphs [@Harborth74]. We prove the upper bound. Let $G$ be a $1$-planar unit distance graph with $n$ vertices and consider a $1$-plane unit distance drawing of $G$. Let $E$ be the set of edges, $|E|=e$. We will prove that $e\le 3n-c\sqrt[4]{n}$. Let $G_0$ be a plane subgraph of $G$ with maximum number of edges, and among those one with the minimum number of triangular faces. We have to introduce some notation. Let $E_0\subset E$ denote the set of edges of $G_0$ and $E_1=E\setminus E_0$ denote the set of remaining edges, $|E_0|=e_0$, $|E_1|=e_1$. Let $f$ be the number of faces of $G_0$, including the unbounded face and let $\Phi_1, \Phi_2 \dots \Phi_f$ be the faces of $G_0$. For any face $\Phi_i$, $|\Phi_i|$ is the number of bounding edges of it with multiplicity. That is, if an edge bounds $\Phi_i$ from both sides, then it is counted twice. Due to the maximality of $G_0$, every edge $\alpha\in E_1$ crosses an edge in $E_0$, $\alpha$ connects two vertices that belong to neighbouring faces of $G_0$. Therefore, we can partition every edge $\alpha\in E_1$ into two *halfedges* at the unique crossing point on $\alpha$. Each halfedge is contained in a face $\Phi$, one of its endpoint is a vertex of $\Phi$ the other endpoint is an interior point of a bounding edge. {width="5.5cm"} **Claim 1**. *A triangular face of $G_0$ does not contain any halfedge.* *Proof.* Let $\Phi=uvw$ be a triangular face of $G_0$ that contains a halfedge $\alpha_1$, which is part of the edge $\alpha=ux$. Then $\alpha$ crosses the edge $vw$. Replace the edge $vw$ by $\alpha$ in $G_0$. Since $vw$ is the only edge of $G$ that crosses $\alpha$, we obtain another plane subgraph of $G$. It has the same number of edges. {width="11cm"} We claim that it has fewer triangular faces. The triangular face $\Phi$ disappeared. Suppose that we have created a new triangular face. Then $\alpha$ should be a side of it. But then either $uv$ or $uw$ is also a side, without loss of generality suppose that $uv$ is another side of the new triangular face. But then $uvx$ is also a unit equilateral triangle. If the two equilateral triangles $uvw$ and $uvx$ are on the same side of $uv$, then $x=w$, if they are on opposite sides then $vw$ and $ux$ can not cross. ◻ Assign $1/2$ weight to each halfedge. For any face $\Phi_i$, let $s(\Phi_i)$ be the sum of the weights of its halfedges. Clearly we have $$\sum_{i=1}^fs(\Phi_i)=|E_1|.$$ For any face $\Phi$ of $G_0$, let $t(\Phi)$ denote the number of edges in a triangulation of $\Phi$. A straightforward consequence of Euler's formula is the following statement. If the boundary of $\Phi_i$ has $m$ connected components, then $t(\Phi_i)=|\Phi_i|+3m-6$. **Claim 2**. *For any face $\Phi$ of $G_0$ we have (a) $$s(\Phi)\le t(\Phi),$$ and if $|\Phi|\ge 5$, then (b) $$s(\Phi)\le t(\Phi)-|\Phi|/10.$$* *Proof.* Suppose first that the boundary of $\Phi$ is not connected, that is, $m\ge 2$. Each of the $|\Phi_i|$ edges on the boundary of $\Phi_i$ is crossed by at most one halfedge, therefore, $s(\Phi)\le |\Phi|/2$. On the other hand, $t(\Phi)\ge |\Phi|$. Therefore, $$t(\Phi)\ge |\Phi|\ge |\Phi|/2+|\Phi|/10 \ge s(\Phi)+|\Phi|/10$$ and we are done in this case. Suppose now that the boundary of $\Phi_i$ is connected, that is, $m=1$. If $|\Phi|=3$, then $\Phi$ is a triangle. Then $t(\Phi)=0$ and by Claim [Claim 1](#claim:haromszog){reference-type="ref" reference="claim:haromszog"} $s(\Phi)=0$, so we are done. If $|\Phi|=4$, then $\Phi$ is a quadrilateral (actually, a rhombus). Then $t(\Phi)=1$. Figure [1](#4gons){reference-type="ref" reference="4gons"} shows all possible cases when $\Phi$ has two halfedges. On the other hand, it is shown in [@PT97] that no more halfedges can be added. Therefore, $s(\Phi)\le 1=t(\Phi)$. This finishes part (a). {#4gons width="11cm"} Suppose that $|\Phi_i|\ge 5$. We can assume that $\Phi_i$ has at least two halfedges, otherwise we are done. A halfedge $\alpha$ in $\Phi$ divides $\Phi$ into two parts. Let $a(\alpha)$ and $b(\alpha)$ be the number of vertices of $\Phi$ in the two parts. If a vertex appears on the boundary more than once, then it is counted with multiplicity. Since the halfedges in $\Phi$ do not cross each other, all other halfedges are entirely in one of these two parts. If one part does not contain any halfedges, then $\alpha$ is called a *minimal halfedge*. Let $\alpha$ be a halfedge for which $M=\min \{ a(\alpha), b(\alpha) \}$ is minimal. Then there are $M$ vertices of $\Phi_i$ on one side of $\alpha$. Clearly, this part can not contain any halfedge, so $\alpha$ is minimal. Now for any other halfedge $\beta\neq\alpha$, let $c(\beta)$ be the number of vertices of $\Phi_i$ on the side of $\beta$ not containing $\alpha$. Take a halfedge $\beta$ for which $c(\beta)$ is minimal. Then $\beta$ is also a minimal halfedge. So, we can conclude that there at least two minimal halfedges in $\Phi$, say, $\alpha$ and $\beta$. Then $\alpha$ and $\beta$ together partition $\Phi$ into three parts, two parts contain no other halfedges but both parts contain an edge of $\Phi$. So, at most $|\Phi|-2$ edges of $\Phi$ are crossed by a halfedge, therefore, there are at most $|\Phi|-2$ halfedges in $\Phi$, consequently $s(\Phi)\le (|\Phi|-2)/2$. On the other hand, $t(\Phi)=|\Phi|-3$. Since $|\Phi|\ge 5$, we have $$t(\Phi)=|\Phi|-3\ge (|\Phi|-2)/2+|\Phi|/10 \ge s(\Phi)+|\Phi|/10.$$ This concludes the proof of the Claim. ◻ Return to the proof of Theorem [Theorem 1](#1-planar){reference-type="ref" reference="1-planar"}. For $i\ge 3$, let $f_i$ denote the number of faces $\Phi$ of $G_0$ with $|\Phi|=i$. By definition, $\sum_{i=3}^{\infty}f_i=f$ and $\sum_{i=3}^{\infty}if_i=2e_0$. Let $F_{\ge 5}=\sum_{i=5}^{\infty}if_i$. We want to show that $e=e_0+e_1\le 3n-\sqrt[4]{n}$. By the maximality of $G_0$, every edge in $E_1$ crosses an edge in $E_0$, and by $1$-planarity, every edge in $E_0$ is crossed by at most one edge in $E_1$. Consequently, $|E_0|=e_0\ge |E_1|=e_1$. If $e_0\le n$, then $e=e_0+e_1\le 2e_0\le 2n<3n-c\sqrt[4]{n}$, so we are done. Therefore, for the rest of the proof we can assume that $e_0\ge n$. It follows that $$3f_3+4f_4+F_{\ge 5}=2e_0\ge 2n.$$ **Claim 3**. *Suppose that $F_{\ge 5}\ge m$. Then $e=e_0+e_1\le 3n-m/10$.* *Proof.* By the previous observations, $$e=e_0+e_1=e_0+\sum_{\scriptsize \begin{array}{cc} \alpha\mbox{ is a} \\ \mbox{halfedge} \end{array}} 1/2 = e_0+\sum_{i=1}^f s(\Phi_i) = e_0 + \sum_{|\Phi|=3}s(\Phi) + \sum_{|\Phi|=4}s(\Phi)+ \sum_{|\Phi|\ge 5}s(\Phi)$$ $$\le e_0 + \sum_{|\Phi|=3}t(\Phi) + \sum_{|\Phi|=4}t(\Phi) + \sum_{|\Phi|\ge 5}(t(\Phi)-|\Phi|/10)$$ $$\le e_0 + \sum_{|\Phi|}t(\Phi) - \sum_{|\Phi|\ge 5}|\Phi|/10 \le 3n-6-F_{\ge 5}/10 \leq 3n-m/10.$$ ◻ **Claim 4**. *Suppose that $f_{3}\ge m$. Then $e=e_0+e_1\le 3n-\sqrt{m}/5$.* *Proof.* We can assume that $\Psi$, the unbounded face of $G_0$ has at least 5 edges. If not, the statement holds trivially. Since we have $m$ equilateral triangles in $G_0$, the union of all bounded faces, $R$, has area at least $\sqrt{3}m/4$. The Isoperimetric inequality states that if a polygon has perimeter $l$ and area $A$, then $l^2\ge 4\pi A$ [@C93]. It implies that $R$ has perimeter at least $\sqrt[4]{3}\sqrt{\pi m}>2\sqrt{m}$. That is $|\Psi|\ge 2\sqrt{m}$. Therefore, $$e=e_0+e_1=e_0+\sum_{\scriptsize \begin{array}{cc} \alpha\mbox{ is a} \\ \mbox{halfedge} \end{array}} 1/2 =e_0+\sum_{i=1}^f s(\Phi_i) =e_0 + \sum_{\Phi\neq \Psi}s(\Phi) + s(\Psi)$$ $$\le e_0 + \sum_{\Phi\neq \Psi}t(\Phi) + t(\Psi) - |\Psi|/10 = 3n-6 - |\Psi|/10 \le 3n-6-\sqrt{m}/5.$$ ◻ If $F_{\ge 5}\ge n/2$, then by Claim [Claim 3](#claim:f>=5){reference-type="ref" reference="claim:f>=5"}, $e\le 3n-n/20\le 3n-c\sqrt[4]{n}$ and we are done. If $f_{3}\ge n/9$, then by Claim [Claim 4](#claim:f3){reference-type="ref" reference="claim:f3"}, $e\le 3n-\sqrt{n}/15\le 3n-c\sqrt[4]{n}$ and we are done again. So, we can assume that $F_{\ge 5}\le n/2$, $f_{3}\le n/9$. Since $3f_3+4f_4+F_{\ge 5} = 2e_0\ge 2n$, it follows that $f_4\ge n/4$. Suppose without loss of generality that none of the edges of $G$ are vertical. Otherwise apply a rotation. Define an auxiliary graph $H$ as follows. The vertices represent the quadrilateral faces of $G_0$. Since all edges are of unit length, all these faces are rhombuses. Two vertices are connected by an edge if the corresponding rhombuses have a common edge. The edges of $H$ correspond to the edges of $G_0$ with a rhombus face on both sides. For every edge of $H$ define its weight as the slope of the corresponding edge of $G_0$. A path in $H$, such that all of its edges have the same weight $w$, is called a *$w$-chain*, or briefly a *chain*. A chain corresponds to a sequence of rhombuses such that the consecutive pairs share a side and all these sides are parallel. A chain, with at least two vertices (rhombuses) is called *maximal* if it cannot be extended. With one-vertex chains we have to be careful. Suppose that $v$ is a vertex of $H$, $R$ is the corresponding rhombus, and let $w_1$, $w_2$ be the slopes of its sides. The one-vertex chain $v$ is *maximal* if it cannot be extended to a larger $w_1$-chain or a larger $w_2$-chain. Each vertex of $H$ is in exactly two maximal chains. **Claim 5**. *Suppose that $A$ and $B$ are chains in $H$ whose intersection is not a chain. Then they have at most one common vertex.* *Proof.* Suppose on the contrary that $A$ and $B$ are chains, whose intersection is not a chain and they have at least two common vertices. Let $A=v_1, v_2, \ldots v_a$. We can assume without loss of generality that $v_1, v_a\in B$ but no other vertex of $A$ is in $B$. Otherwise we can delete some vertices of $A$ to obtain this situation. Delete all vertices of $B$ which are not between $v_1$ and $v_a$. Now $B=u_1, u_2, \ldots u_b$ where $v_1=u_1$, $v_a=u_b$ and these are the only common points of $A$ and $B$. Let $R$ be the rhombus that represents $v_1=u_1$ in $G_0$. Its sides have slopes $w_1$ and $w_2$ such that $A$ is a $w_1$-chain, $B$ is a $w_2$-chain. Apply an affine transformation so that $R$ is a unit square, $w_1$ is the horizontal, $w_2$ is the vertical direction. Suppose that $Q$ is the rhombus that represents $v_a=u_b$. Then its sides also have slopes $w_1$ and $w_2$, so $Q$ is also an axis parallel unit square. Represent each vertex $v_1, v_2, \ldots v_a, u_1, u_2, \ldots u_b$ by the center of the corresponding rhombus. For simplicity we call these points also $v_1, v_2, \ldots v_a, u_1, u_2, \ldots u_b$, respectively. Assume without loss of generality that the point $v_a=u_b$ has larger $x$ and $y$ coordinates, than $v_1=u_1$. Connect the consecutive points in both chains by straight line segments. Since $A$ is a $w_1$-chain and $w_1$ is the horizontal direction, the polygonal chain $P_A=v_1, v_2, \ldots v_a$ is $y$-monotone, and similarly, the polygonal chain $P_B=u_1, v_2, \ldots v_b$ is $x$-monotone. Let $l_1$ be the horizontal halfline from $v_1=u_1$, pointing to the left and let $l_2$ be the horizontal halfline from $v_a=u_b$, pointing to the right. The bi-infinite curve $l_1\cup P_B\cup l_2$ is simple, because $P_B$ is $x$-monotone. It divides the plane into two regions, $R_{\text{down}}$, which is below it and its complement, $R_{\text{up}}$, see Figure [2](#chains){reference-type="ref" reference="chains"}. {#chains width="11cm"} Observe, that the initial part of $P_A$, near $v_1=u_1$ is in $R_{\text{up}}$ while the final part, near $v_a=u_b$ is in $R_{\text{down}}$. On the other hand, $P_A$ does not intersect the boundary of $R_{\text{down}}$ and $R_{\text{up}}$: it does not intersect $l_1$ and $l_2$ since it is $y$-monotone, and does not intersect $P_B$ by assumption. This is clearly a contradiction which proves the Claim. ◻ **Claim 6**. *There are at least $\sqrt{n}/\sqrt{2}$ disjoint maximal chains.* *Proof.* For any vertex of $H$ (that is, for any rhombus face in $G_0$) there are exactly two maximal chains containing it. Therefore, the total length of all the maximal chains is $f_4\ge n/2$. If there are less than $\sqrt{n}/\sqrt{2}$ disjoint maximal chains, then one of them, say $C$ has length at least $\sqrt{n}/\sqrt{2}$. Through each of its vertices, there is another maximal chain and by Claim [Claim 5](#claim:2chains){reference-type="ref" reference="claim:2chains"} all of these chains are different. ◻ By Claim [Claim 6](#claim:manychains){reference-type="ref" reference="claim:manychains"}, we have at least $\sqrt{n}/\sqrt{2}$ disjoint maximal chains. Each of them has two ending rhombuses, which bounds a face of size different than $4$. All of these bounding edges are different, therefore, $3f_3+F_{\ge 5}\ge \sqrt{2}\sqrt{n}$, which implies that either $3f_3 \ge \sqrt{n}/\sqrt{2}$, or $F_{\ge 5}\ge \sqrt{n}/\sqrt{2}$. In the first case, by Claim [Claim 4](#claim:f3){reference-type="ref" reference="claim:f3"} we have $e\le 3n-\sqrt[4]{n}/10$. In the second case, by Claim [Claim 3](#claim:f>=5){reference-type="ref" reference="claim:f>=5"} we have $e\le 3n-\sqrt{n}/10$. This concludes the proof of Theorem [Theorem 1](#1-planar){reference-type="ref" reference="1-planar"}. $\Box$ # *k*-planar unit distance graphs {#sec3} Suppose that $n, k >100$. Erdős [@E46] proved the following statement. For any $m$, there is an $r<m$ such that $r$ can be written as $a^2+b^2$ in $2^{\Omega(\log m/\log\log m)}$ different ways where $a$ and $b$ are integers. For any fixed $m$ let $r$ be the product of the first $l$ primes congruent to $1$ mod $4$, such that $l$ is maximal with the property that $r<m$. Erdős proved that this $r$ satisfies the requirements. He used it to construct a set of $n$ points that determine $n2^{\Omega(\log n/\log\log n)}$ unit distances. Clearly, $r$ is square-free, therefore, whenever $r=a^2+b^2$, $(a,b)=1$. Apply the above result for $m=\sqrt{k}/5$. We obtain $r<\sqrt{k}/5$ that can be written as the sum of two integer squares, $r=a^2+b^2$ in $2^{\Omega(\log m/\log\log m)}= 2^{\Omega(\log k/\log\log k)}$ different ways. Take a $\sqrt{n}\times\sqrt{n}$ unit square grid and connect two points by a straight line segment if they are at distance $\sqrt{r}$. Then each vertex has degree $2^{\Omega(\log k/\log\log k)}$, so our graph has $n2^{\Omega(\log k/\log\log k)}$ edges. Observe that no edge contains a vertex in its interior. Let $uv$ be an edge. Consider all vertices adjacent to an edge that crosses $uv$. All these vertices are at distance at most $\sqrt{r}$ from $uv$. This region has area $(2+\pi)r$, so the number of vertices in this region is less than $6r$. Each of these vertices have degree at most $4r$, so $uv$ is crossed by at most $24r^2<k$ edges. Scale the picture by a factor of $1/\sqrt{r}$ and we obtain a $k$-planar unit distance graph of $n$ vertices and $2^{\Omega(\log k/\log\log k)}$ edges. $\Box$ For the proof we need some preparation. The *crossing number* ${\mbox {\sc cr}}(G)$ is the minimum number of edge crossing over all drawings of $G$ in the plane. According to the Crossing Lemma [@ACNS82; @L84], for every graph $G$ with $n$ vertices and $e\ge 4n$ edges, ${{\mbox {\sc cr}}}(G)\ge \frac{1}{64}\frac{e^3}{n^2}$. It is asymptotically tight in general for simple graphs [@PT97]. However, there are better bounds for graphs satisfying some monotone property [@PST00], or for monotone drawing styles [@KPTU21]. A drawing style $\cal D$ is a subset of all drawings of a graph $G$, so some drawings belong to $\cal D$, others don't. A drawing style is monotone if removing edges retains the drawing style, that is, for every graph $G$ in drawing style $\cal D$ and any edge removal, the resulting graph with its inherited drawing is again in drawing style $\cal D$. A vertex split is the following operation. (a) Replace a vertex of $G$ by two vertices, $v_1$ and $v_2$, both very close to $v$. Connect each edge of $G$ incident to $v$ either to $v_1$ or $v_2$ by locally modifying them such that no additional crossing is created. Or as a limiting case, (b) place both $v_1$ and $v_2$ to the same point where $v$ was, connect each edge incident to $v$ either to $v_1$ or $v_2$ without modifying them, such that the edges incident to $v$ in $G$ that are connected to $v_1$ (resp. $v_2$) after the split form an interval in the clockwise order from $v$. A drawing style $\cal D$ is split-compatible if performing vertex splits retains the drawing style. For any graph $G$, the bisection width $b(G)$ of a graph $G$ is the smallest number of edges whose removal splits $G$ into two graphs, $G_1$ and $G_2$, such that $|V(G_1)|$, $|V(G_2)| \geq |V(G)|/5$. For a drawing style $\cal D$ the $\cal D$-bisection width $b_{\cal D}(G)$ of a graph $G$ in drawing style $\cal D$ is the smallest number of edges whose removal splits $G$ into two graphs, $G_1$ and $G_2$, both in drawing style $\cal D$ such that $|V(G_1)|$, $|V(G_2)| \geq |V(G)|/5$. Let $\Delta(G)$ denote the maximum degree in $G$. **Theorem 4** (Kaufmann-Pach-Tóth-Ueckerdt [@KPTU21]). *Suppose that $\cal D$ is a monotone and split-compatible drawing style, and there are constants $k_1$, $k_2$, $k_3 >0$ and $b > 1$ such that each of the following holds for every $n$-vertex $e$-edge graph $G$ in drawing style $\cal D$:* - *If ${{\mbox {\sc cr}}}_{\cal D}(G) = 0$, then $e \leq k_1 \cdot n$.* - *The $\cal D$-bisection width satisfies $b_{\cal D}(G) \leq k_2 \sqrt{{{\mbox {\sc cr}}}_{\cal D}(G) + \Delta(G) \cdot e + n}$.* - *$e \leq k_3 \cdot n^b$.* *Then there exists a constant $\alpha > 0$ such that for any $n$-vertex $e$-edge graph $G$ in drawing style $\cal D$ we have* *$${{\mbox {\sc cr}}}_{\cal D}(G) \geq\alpha\frac{e^{1/(b-1)+2}}{n^{1/(b-1)+1}} \text{ \qquad provided } e > (k_1 + 1)n.$$* In [@KPTU21] only vertex split of type (a) was allowed, but the proof goes through also for type (b). **Theorem 5** (Spencer-Szemerédi-Trotter [@SST84]). *Let $G$ be a unit distance graph on $n$ verities. The number of edges in $G$ is at most $c n^{4/3}$ where $c>0$ is a constant.* The best known constant is due to Ágoston and Pálvölgyi [@AP22] who proved that the statement holds with $c=1.94$.\ Consider now the following drawing style $D$ for a graph $G$. - Vertices are represented by not necessarily distinct points. - Edges are represented by unit segments between the corresponding points. - Two edges cannot overlap. - If a point $p$ represents more than one vertex, say, $v_1, \ldots, v_m$, then the sets of edges incident to $v_1, \ldots, v_m$, respectively, form an interval in the clockwise order from point $p$. Clearly, ${\cal D}$ satisfies the following properties. - the drawing style $D$ is monotone and split compatible. - If ${{\mbox {\sc cr}}}(G) = 0$, then $e \leq 3n-6$. In fact, by [@LS22], $e\le \lfloor 3n - \sqrt{12n - 3}\rfloor$. - By the result of Pach, Shahrokhi and Szegedy [@PSS94] for any graph $G$, $b(G)\le 10\sqrt{{{\mbox {\sc cr}}}(G) + \Delta(G) \cdot e + n}$. But if $G$ is drawn in drawing style $\cal D$, then all of its subgraphs are also drawn in drawing style $D$. Therefore, $b_{\cal D}(G)\le 10\sqrt{{{\mbox {\sc cr}}}(G) + \Delta(G) \cdot e + n}$. - By [@AP22], any $n$-vertex graph in drawing style $\cal D$ has less than $1.94n^{4/3}$ edges. Summarizing, we can apply Theorem [Theorem 4](#Kaufmann-Pach-Toth-Ueckerdt){reference-type="ref" reference="Kaufmann-Pach-Toth-Ueckerdt"} with $k_1=3$, $k_2=10$, $k_3=1.94$, $b=4/3$ and obtain the following. For any graph $G$ in drawing style $\cal D$ with $n$ vertices and $e>4n$ edges we have $${\mbox {\sc cr}}_{\cal D}(G) \geq \alpha \frac{e^{1/(b-1)+2}}{n^{1/(b-1)+1}}=\alpha\frac{e^5}{n^4}$$ for some $\alpha >0$. Consider now a $k$-plane drawing of a unit distance graph $G$ with $n$ vertices and $e$ edges. If $e\le 4n$, we are done, suppose that $e\ge 4n$. Since each edge contains at most $k$ crossings, the total number of crossing $c(G)$ satisfies $c(G)\le ek/2$. On the other hand, we have $c(G)\ge \alpha\frac{e^5}{n^4}$. Therefore, $ek/2\ge \alpha\frac{e^5}{n^4}$ so $e\le \beta\sqrt[4]{k}n$ for some $\beta>0$. $\Box$ # Open questions In this paper we proved that a 1-planar unit distance graph of $n$ vertices can have at most $u_1(n)\le 3n-\sqrt[4]{n}/10$ edges. However, the best known lower bound construction for $u_1(n)$ is the same as for $u_0(n)$. **Problem 1**. *Is it true that $u_0(n)= u_1(n)$?* Even more surprisingly, we do not have a better construction even for $k=2$. For $k=3$, two triangular grids placed on top of each other (shifted by a unit vector) gives us $u_3(n) \geq 3.5n - c\sqrt{n}$. As we have seen, for a larger $k$ our lower and upper bounds for $u_k(n)$ are very far. **Problem 2**. *Determine $u_k(n)$, the maximum number of edges of a k-planar unit distance graph.* It is easy to construct a $k$-regular 1-planar unit distance graph (or even a matchstick graph) for $k \leq 3$. Also, there are known examples of 4-regular matchstick graphs [@Harborth2; @WD17]. However, due to Theorem [Theorem 1](#1-planar){reference-type="ref" reference="1-planar"} there are no $6$-regular 1-planar unit distance graphs. In the case of matchstick graphs, it is known that even 5-regularity can not be achieved [@B14; @KP11]. **Problem 3**. *Are there 5-regular 1-planar unit distance graphs?* **Acknowledgements.** Panna Gehér was supported by the Lendület Programme of the Hungarian Academy of Sciences -- grant number LP2021-1/2021. Géza Tóth was supported by ERC Advanced Grant 'GeoScape' No. 882971 and by the National Research, Development and Innovation Office, NKFIH, K-131529. 30 Ackerman, E. (2019). *On topological graphs with at most four crossings per edge.* Computational Geometry, 85, 101574. Ágoston, P., Pálvölgyi, D. (2022). *An improved constant factor for the unit distance problem.* Studia Scientiarum Mathematicarum Hungarica, 59(1), 40-57. Ajtai, M., Chvátal, V., Newborn, M. M., Szemerédi, E. (1982). *Crossing-free subgraphs.* In North-Holland Mathematics Studies (Vol. 60, pp. 9-12). North-Holland. Blokhuis, A. (2014). *Regular finite planar maps with equal edges.* arXiv preprint arXiv:1401.1799. Do Carmo, M. P. (2013) *Differentialgeometrie von Kurven und Flächen.* Flächen (Vol. 55). Springer-Verlag. Erdős, P. (1946).*On sets of distances of $n$ points.* The American Mathematical Monthly 53(5), 248-250. Harborth, H. (1974). *Solution to problem 664A.* Elemente der Mathematik 29, 14-15. Harborth, H. (1981). *Point sets with equal numbers of unit-distant neighbors.* Discrete Geometry, 12-18. Harborth, H. (1994). *Match sticks in the plane.* In: The Lighter Side of Mathematics. Proceedings of the Eug$\grave{e}$ne Strens Memorial Conference on Recreational Mathematics and its History held at the University of Calgary, Calgary, Alberta, August 1986. edited by R. K. Guy and R. E. Woodrow, 281--288. Mathematical Association of America, Washington, D.C. Kaufmann, M., Pach, J., Tóth, G., Ueckerdt, T. (2018). *The number of crossings in multigraphs with no empty lens.* In: Proc. Graph Drawing and Network Visualization: 26th International Symposium, GD 2018, Barcelona, Spain, September 26-28, 2018, pp. 242-254 Springer, Cham. Also in: Journal of Graph Algorithms and Applications 25 (2021) 383-396. Kurz, S., Pinchasi, R. (2011). *Regular matchstick graphs.* The American Mathematical Monthly, 118(3), 264-267. Lavollée, J., Swanepoel, K. (2022). *A tight bound for the number of edges of matchstick graphs.* arXiv preprint arXiv:2209.09800. Leighton, F. T. (1984). * New lower bound techniques for VLSI. Mathematical systems theory, 17, 47-70.* Pach, J., Agarwal, P. K. (2011). *Combinatorial geometry.* John Wiley $\&$ Sons. Pach, J., Radoicić, R., Tardos, G., Tóth, G. (2004). *Improving the crossing lemma by finding more crossings in sparse graphs.* In Proceedings of the twentieth annual symposium on Computational geometry (pp. 68-75). Pach J., Shahrokhi F., Szegedy M. (1994). *Applications of the crossing number.* In: Proceedings of the tenth annual symposium on Computational Geometry 1994 Jun 10 (pp. 198-202). Pach, J., Spencer J., Tóth G. (2000). *New bounds for crossing numbers.* Proceedings of the 15th Annual ACM Symposium on Computational Geometry 1999, 124-133. Also in: Discrete and Computational Geometry 24, 623-644. Pach, J. Tóth, G. (1997). *Graphs drawn with few crossings per edge* Combinatorica, 17(3), 427-439. Spencer J., Szemerédi E., Trotter W. T. (1984) *Unit Distances in the Euclidean Plane.* Graph Theory and Combinatorics, 293--303. Winkler, M., Dinkelacker, P., Vogel, S. (2017). *On the existence of 4-regular matchstick graphs.* arXiv preprint arXiv:1705.00293. [^1]: Eötvös Loránd University, Budapest, Hungary. Email: [`geherpanni@student.elte.hu`](mailto:geherpanni@student.elte.hu). [^2]: Alfréd Rényi Institute of Mathematics, Budapest, Hungary. Email: [`geza@renyi.hu`](mailto:geza@renyi.hu).

arxiv_math

--- abstract: | We prove the existence of a liquid-gas phase transition for continuous Gibbs point process in $\mathbbm{R}^d$ with Quermass interaction. The Hamiltonian we consider is a linear combination of the volume $\mathcal{V}$, the surface measure $\mathcal{S}$ and the Euler-Poincaré characteristic $\chi$ of a halo of particles (i.e. an union of balls centred at the positions of particles). We show the non-uniqueness of infinite volume Gibbs measures for special values of activity and temperature, provided that the temperature is low enough. Moreover we show the non-differentiability of the pressure at these critical points. Our main tool is an adaptation of the Pirogov-Sinaï-Zahradnik theory for continuous systems with interaction exhibiting a saturation property. author: - D. Dereudre, C. Renaud Chan bibliography: - Liquid_Gas_Transition_for_Quermass_model.bib title: Liquid-Gas phase transition for Gibbs point process with Quermass interaction --- ***Keywords---*** Gibbs measure, DLR equations, Widom-Rowlinson model, Pirogov-Sinaï-Zahradnik theory, cluster expansion # Introduction Spatial point processes with interaction are models describing the locations of objects, particles in a domain and for which the interaction can be of different nature; attractive, repulsive or a mix of both at different scales of distance between the points. The most popular point process is surely the Poisson point process [@last_penrose] which describes random objects without interaction between each other. There are different models for point processes with interaction as for instance Cox point processes [@last_penrose], determinantal point processes [@macchi], zeros of random polynomials or analytic functions [@Hough], Gibbs point processes [@MiniCours; @ruelle_livre], etc. Among the field of applications of such models we have plant ecology, telecommunication, astronomy, data science and statistical physics. A large class of point process coming from statistical physics is the family of the Gibbs point processes. The finite volume Gibbs point process on a bounded set $\Delta \subset \mathbbm{R}^d$ is defined via an unnormalized density, with respect to the Poisson point process in $\Delta$, of the form $z^{N_\Delta} e^{- \beta H}$. The parameters $z$ and $\beta$ are positive numbers (respectively called the activity and the inverse temperature), $N_\Delta$ is the number of points inside $\Delta$ and $H$ an energy function (also called the Hamiltonian). By taking the thermodynamic limit (i.e $\Delta\to \mathbbm{R}^d$) we obtain the infinite volume Gibbs point processes which are characterized by the equilibrium equations also known as the Dobrushin-Landford-Ruelle (DLR) equations. The type of interaction we study in this paper is the Quermass interaction. The energy function is defined as a linear combination of the $d+1$ Minkowski functionals of the halo of the configuration, which is the union of closed balls centred at the position of the particles with random radii. This type of interaction is a natural extension of the Widom-Rowlinson interaction for penetrable spheres [@WidomRowlinson]. Hadwiger's characterization Theorem [@Hadwiger] ensures that any functional $F$ on finite union of convex compact spaces, continuous for the Haussdorff topology, invariant under isometric transformations and additive (i.e. $F(A \cup B) = F(A) + F(B)- F(A \cap B)$) can be written as a linear combination of Minkowski functionals. This representation justifies the study of the Quermass interaction for modelling a large class of morphological interactions. Furthermore, this family of energy function is used to describe the physics of micro-emulsions and complex fluids [@likos_emulsion; @Mecke_complex_fluid; @mecke_integralgeometrie]. The stability condition in the sense of Ruelle has been established in [@Kendall] and the existence of the infinite volume Gibbs measures with random radii has been tackled in [@dereudre_existence_quermass]. In general a phase transition phenomenon occurs when for some special value of $\beta$ and $z$, the boundary conditions at infinity influence some macroscopic statistics in the bulk of the system. A phase transition phenomenon can also be defined by uniqueness/non-uniqueness of solutions to the DLR equilibrium equations. More specifically we call a liquid-gas phase transition of first order when several solutions have different intensities. The Gibbs measure with the lowest density is associated to the distribution of a pure gas phase and in contrary the highest density to the distribution of a pure liquid phase. If we consider models of particles with different spins, there exists an abundance of results on phase transition mainly based on the existence of a dominant spin. We can cite for instance the phase transition results on the continuous Potts model [@georgii_hagstrom] and the non symmetrical multiple colour Widom-Rowlinson model [@bricmont_kuroda]. Without spin, we need to rely only on the self arrangement, the geometry and the density of particles for proving the phase transition. It is more difficult and there exist only few known results in this setting. The first result of phase transition without spin has been proved for the Widom-Rowlinson model. It is a Gibbs point process where the Hamiltonian is given by $H(\omega) = \mathcal{V}(\cup_{x \in \omega} B(x,R))$ where $\mathcal{V}$ if the volume functional. The first proofs are due to Widom-Rowlinson [@WidomRowlinson] and Ruelle [@RuelleWR] and use extensively the symmetry of the associated two-colour Widom-Rowlinson process. They prove the existence of a critical activity $z_c$ such that for $z>z_c$ and $\beta = z$ the liquid-gas phase transition occurs. Later another proof using Fortuin-Kastelyn representation has been given [@2ChayesKotecky]. Recently, the full phase diagram have been almost entirely mapped by proving an OSSS inequality [@HoudebertDereudre]. Outside of a region around the critical point $(z_c, z_c)$, the phase transition occurs if and only if $\beta = z$ for $z > z_c$. Numerical studies have shown that the unmapped region is actually small [@Houdebert]. Among continuous models without any form of special symmetry, there is the beautiful result of liquid gas phase transition for an attractive pair and repulsive four body potential of the Kac type [@LebowitzMazelPresutti]. They manage to prove, using the Pirogov-Sinaï-Zahradnik theory (PSZ), the existence of a phase transition for finite but long range interaction compared to typical distance between particles. Indeed the finite range interaction is obtained as a perturbation of the mean field interaction. In a more recent work using a similar strategy, it has been proven that the liquid gas phase transitions persists if a hard-core interaction is considered [@Pulvirenti]. Until now, these results were the only proofs of phase transition for continuum systems without spins. In the present paper, we are interested in the phase transition phenomenon for the Quermass interaction with the particular form $H(\omega) = \mathcal{V}(\cup_{x \in \omega } B(x,R)) + \theta_1 \mathcal{S}(\cup_{x \in \omega } B(x,R)) - \theta_2 \chi(\cup_{x \in \omega } B(x,R))$ where $\mathcal{S}$ is the measure of the surface and $\chi$ is the Euler-Poincaré characteristic. Contrary to the Widom Rowlinson model we cannot benefit from any symmetry of a two spin process due to the contribution of the surface measure and Euler-Poincaré characteristic. Moreover, in our work we do not use mean-field approximation and the range of interaction is at the same order than the distance between particles. We achieve to prove, for some parameter $\theta_1$ and $\theta_2$, the existence of two infinite Gibbs point processes with distinct intensities when $\beta$ is sufficiently large (i.e. at low temperature) and the activity $z$ equals to a critical activity $z_\beta^c$. The main tool of our proof is an adaptation of the Pirogov-Sinaï-Zahradnik (PSZ) theory [@Zahradnik; @PirogovSinai; @PirogovSinai2] for continuous systems with interaction satisfying a saturation property. For a modern presentation of the PSZ theory we advise the lecture of Chapter 7 of [@Velenik]. Let us describe succinctly what the saturation property is. First we consider a discretisation of the space $\mathbbm{R}^d$ with cubic tiles. Then we introduce two types of \"ground states\", which are not usual ground states as minimisers of the energy, but rather as extreme idealizations of the point process. One ground state is the empty state and the other corresponds to a dense and homogeneous distribution of particles (i.e. at least one particle in any tile). These states have the property to provide explicit and tractable energy for an extra point emerging in a ground state. That is what we call saturation property. Using the PSZ theory we are able to show that for $\beta$ large enough and at the critical activity $z_\beta^c$ the pressure of the model is not differentiable with respect to $z$. Furthermore we construct two different Gibbs measures with the \"ground states\" as boundary conditions and we make the relation between the intensity of these Gibbs measures and the left and right derivatives of the pressure. We believe that our method is robust enough to deal with other interactions with similar saturation property. Our paper is organized as follows. In Section [2](#Section2){reference-type="ref" reference="Section2"}, we introduce the notations, the Quermass interaction and the associated Gibbs point processes. In Section [3](#Section3){reference-type="ref" reference="Section3"}, we present the main results of the paper. In Section [4](#Section4){reference-type="ref" reference="Section4"}, we develop the tools and we prove the main results. Annex A contains results on Cluster Expansion and Annex B contains the technical proof of Proposition [Proposition 4](#prop_tau_stability_truncated_weights){reference-type="ref" reference="prop_tau_stability_truncated_weights"} at the heart of the PSZ theory. # Quermass interaction model {#Section2} ## State spaces and notations We denote by $\mathcal{B}_b(\mathbbm{R}^d)$ the set of bounded Borel sets of $\mathbbm{R}^d$ with positive Lebesgue measure. For any sets $A$ and $B$ in $\mathcal{B}_b(\mathbbm{R}^d)$, $A \oplus B$ stands for the Minkowski sum of these sets. Let $R_0, R_1$ be two real numbers such that $0<R_0 \leq R_1$. We denote by $E$ the state space of a single marked point defined as $\mathbbm{R}^d \times [R_0, R_1]$. For any $(x,R)\in\mathbf{E}$, the first coordinate $x$ is for the location of the point and the second coordinate $R$ is the mark representing the radius of a ball. For any set $\Delta \in \mathcal{B}_b(\mathbbm{R}^d)$, $E_\Delta$ is the local state space $\Delta \times [R_0, R_1]$. A configuration of marked points $\omega$ is a locally finite set in $E$; i.e. $N_\Delta(\omega) := \#(\omega \cap E_\Delta)$ is finite for any $\Delta \in \mathcal{B}_b(\mathbbm{R}^d)$. We denote by $\Omega$ the set of all marked point configurations and by $\Omega_f$ its restriction to finite configurations. For any $\omega\in\Omega$, its projection in $\Delta\subset \mathbbm{R}^d$ is defined by $\omega_\Delta:=\omega \cap E_\Delta$. As usual we equipped the state space $\Omega$ with the $\sigma$-algebra $\mathcal{F}$ generated by the counting functions on $E$. The halo of a configuration $\omega\in \Omega$ is defined as $$L(\omega) = \bigcup_{(x,R) \in \omega} B(x,R)$$ where $B(x,R)$ is the closed ball centred at $x$ with radius $R$. ## Interaction Let us introduce the Quermass interaction as in [@Kendall]. Usually it is defined as a linear combination of $d+1$ Minkowski functionals of the halo $L(\omega)$. Here we consider only the volume $\mathcal{V}$, the surface $\mathcal{S}$ and the Euler-Poincaré characteristic $\chi$ (in dimension $d=2$). This restriction is due to statistical physics considerations since we need the stability of the energy. **Definition 1**. *Let $\theta_1 \in \mathbbm{R}$ and $\theta_2 \geq 0$. The energy of a finite configuration $\omega \in \Omega_f$ is given by $$\begin{aligned} H(\omega) &= \mathcal{V}(L(\omega)) + \theta_1 \mathcal{S}(L(\omega)) - \theta_2 \chi(L(\omega)) & (d=2)\\ &= \mathcal{V}(L(\omega)) + \theta_1 \mathcal{S}(L(\omega)) & (d\ge 3), \end{aligned}$$ where $\mathcal{V}(L(\omega))$ is the volume of $L(\omega)$ defined as the Lebesgue measure of $L(\omega)$, $\mathcal{S}(L(\omega))$ is the surface of $L(\omega)$ defined as the $d-1$-dimensional Hausdorff measure of the boundary $\partial L(\omega)$ and $\chi(L(\omega))$ is the Euler-Poincaré characteristic of $L(\omega)$ defined as the difference between the number of connected components and the number of holes in $L(\omega)$ (in dimension $d=2$).* The energy is parametrized with two parameters $\theta_1$ and $\theta_2$. We discuss below why we impose $\theta_2$ to be non negative. Note that we do not introduce a third parameter in the front of the volume $\mathcal{V}(L(\omega))$ since it is indirectly given by the inverse temperature $\beta\ge 0$ in the Definition [Definition 3](#defGibbs){reference-type="ref" reference="defGibbs"} of Gibbs measures. With this choice of parameters the energy is stable which means that there exists a constant $C \geq 0$ such that for any finite configuration $\omega\in\Omega_f$, $$H(\omega) \geq - C N(\omega).$$ The volume and the surface are clearly stable since the radii are uniformly bounded. The Euler-Poincaré characteristic is more delicate to study. In dimension 2, it is shown by Kendall et al. [@Kendall] that for the union of N closed balls, the number of holes is bounded above by $2N-5$, and the number of connected components is clearly bounded by $N$. Therefore the Euler-Poincaré characteristic is stable for any parameter $\theta_2\in\mathbbm{R}$. In higher dimension $d\geq 3$, for some configurations, the maximum number of holes is of order $N^2$ and thus the Euler-Poincaré characteristic is not stable if $\theta_2<0$. More generally, the stability of this statistic is not obvious even if $\theta_2$ is strictly positive. Therefore the existence of the infinite volume Gibbs point process is not well established. It is for this reason that we impose $\theta_2 =0$ in the case $d\ge3$. Let us turn to the definition of the local energy which provides the cost of energy we need to introduce points in a domain given the configuration outside this domain. **Definition 2**. *Let $\Delta \in \mathcal{B}_b(\mathbbm{R}^d)$ and $\omega \in \Omega_f$ be a finite configuration. We define the local energy of $\omega$ in $\Delta$ as $$H_\Delta (\omega) := H(\omega) - H(\omega_{\Delta^c}).$$ From the additivity of Minkowski functionals, we observe a finite range property. Indeed we have that $H_\Delta ( \omega) = H_\Delta \big(\omega_{\Delta \oplus B(0,2R_1)}\big)$. Therefore, we can extend the definition of the local energy to any configuration $\omega$ in $\Omega$ by $$H_\Delta (\omega) := H_\Delta \left(\omega_{\Delta \oplus B(0,2R_1)}\right).$$* ## Gibbs measures Let $Q$ be a reference measure on $[R_0, R_1]$ for the distribution of the radii and let $z$ be a non-negative real number called the activity parameter. We denote by $\lambda$ the Lebesgue measure on $\mathbbm{R}^d$ and by $\Pi^z$ the distribution of a Poisson point process on $E$ with intensity measure $z \lambda \otimes Q$ [@last_penrose]. Similarly, for any $\Delta \in \mathcal{B}_b(\mathbbm{R}^d)$, $\Pi_\Delta^z$ stands for the Poisson point process on $E_\Delta$ with intensity $z \lambda_\Delta \otimes Q$, where $\lambda_\Delta$ is the Lebesgue measure on $\Delta$. **Definition 3**. *A probability measure $P$ on $\Omega$ is a Gibbs measure for the Quermass interaction with parameters $\theta_1,\theta_2$, the activity $z >0$ and the inverse temperature $\beta \geq 0$ if $P$ is stationary in space and if for any $\Delta \in \mathcal{B}_b(\mathbbm{R}^d)$ and any bounded positive measurable function $f : \Omega \rightarrow \mathbbm{R}$, $$\label{eq_dlr} \int f(\omega) P(d\omega) = \int \int \frac{1}{Z_\Delta(\omega_{\Delta^c})} f(\omega_\Delta' \cup \omega_{\Delta^c}) e^{-\beta H_\Delta(\omega_\Delta' \cup \omega_{\Delta^c})} \Pi_\Delta^z(d\omega'_\Delta) P(d\omega)$$ where $Z_\Delta(\omega_{\Delta^c})$ is the partition function given the outer configuration $\omega_{\Delta^c}$ $$Z_\Delta(\omega_{\Delta^c}) = \int e^{-\beta H_\Delta(\omega_\Delta' \cup \omega_{\Delta^c})} \Pi_\Delta^z(d\omega').$$* These equations are called DLR equations (for Dobrushin, Lanford and Ruelle). It is equivalent to the following conditional probability definition: For all $\Delta \in \mathcal{B}(\mathbbm{R}^d)$ the distribution of $\omega_\Delta$ under $P$ given the outer configuration $\omega_{\Delta^c}$ is absolutely continuous with respect to $\Pi_\Delta^z$ with the following density $$P(d\omega_\Delta' | \omega_{\Delta^c} ) = \frac{1}{Z_\Delta(\omega_{\Delta^c})} e^{-\beta H_\Delta(\omega_\Delta' \cup \omega_{\Delta^c})} \Pi_\Delta^z(d\omega'_\Delta).$$ For the collection of parameters $\theta_1, \theta_2, \beta$ and $z$, we denote by $\mathcal{G}(\theta_1, \theta_2, \beta, z)$ the set of all Gibbs measures for these parameters. The existence, the uniqueness or non-uniqueness (phase transition) of such Gibbs point processes are old and difficult questions in statistical physics. In this present setting, since the interaction is finite range and stable, the existence is a direct application of Theorem 1 in [@MiniCours]. In other words, for any parameters $\theta_1\in\mathbbm{R}, \theta_2\ge 0, \beta\ge 0$ and $z>0$, the set $\mathcal{G}(\theta_1, \theta_2, \beta, z)$ is not empty. # Results {#Section3} We say that a liquid-gas phase transition occurs at $\theta_1,\theta_2, \beta$ and $z$ when the corresponding set of Gibbs measures $\mathcal{G}(\theta_1, \theta_2, \beta, z)$ contains at least two Gibbs measures $P,Q$ with different intensities, i.e. $$\rho (P) := \mathbf{E}_P(N_{[0,1]^d}) > \rho(Q).$$ In particular the set $\mathcal{G}(\theta_1, \theta_2, \beta, z)$ is not reduced to a singleton. This phenomenon is also called first order of phase transition since the non-uniqueness of Gibbs measures is coupled with a discontinuity of the intensity. Other kind of phase transition are possible. Our main result states that such liquid-gas phase transition occurs for a large range of parameters. We denote by $\theta_1^*$ the following constant $R_0 \frac{\mathcal{V}(B(0,1))}{\mathcal{S}(B(0,1))}$ = $\frac{R_0}{d}$. In dimension $d=2$, for any $\theta_1 > -\theta_1^*$ we define the constant $\theta_2^*(\theta_1) >0$ as in the equation [\[eq_theta_2\_lim\]](#eq_theta_2_lim){reference-type="eqref" reference="eq_theta_2_lim"} below. At a first glance, the explicit value for $\theta_2^*(\theta_1)$ is not so important and its existence with a non-null value is already interesting in the next theorem. **Theorem 1**. *Let $\theta_1,\theta_2$ be two parameters such that $\theta_1 > -\theta_1^*$ and $0 \leq \theta_2 < \theta_2^*(\theta_1)$ (recall that $\theta_2=0$ if $d\ge 3$). Then there exists $\beta_c(\theta_1, \theta_2) > 0$ such that for all $\beta > \beta_c(\theta_1, \theta_2)$, there exists $z_\beta^c > 0$ for which a liquid-gas phase transition occurs: i.e. there exist $P,Q \in \mathcal{G}(\theta_1, \theta_2, \beta, z_\beta^c)$ with $\rho(P) > \rho(Q)$. The critical activity $z_\beta^c$ is not explicit but we know that $|z_\beta^c- \beta|$ tends to zero exponentially fast when $\beta$ tends to infinity.* It is important to note that the liquid-gas phase transition is proved only for a special value $z_\beta^c > 0$, the other parameters $\theta_1,\theta_2, \beta$ being fixed. We are not able to prove that $z_\beta^c$ is the only one value for which the phase transition occurs; it is probably not true in general. But, as a corollary of the Pirogov-Sinaï-Zahradnik technology we used, in a neighbourhood of $z_\beta^c$ it is the case. This local uniqueness is typical for a first order liquid-gas phase transition. Note that in the setting of Widom-Rowlinson model (i.e $R_0=R_1$ and $\theta_1=\theta_2=0$), the critical activity $z_\beta^c$ is unique and equal to $\beta$ [@RuelleWR; @WidomRowlinson]. The proof of Theorem [Theorem 1](#thm_quermass_transition){reference-type="ref" reference="thm_quermass_transition"} relies on the study of the regularity of the so-called pressure $\psi$ defined by the following thermodynamic limit $$\psi (z, \beta) : = \lim\limits_{n \rightarrow +\infty} \frac{1}{\beta |\Delta_n|} \ln(Z_{\Delta_n}),$$ where $(\Delta_n)$ is the sequence of the following boxes $[-n,n]^d$ and $Z_{\Delta_n}$ is the partition function with free boundary condition (i.e. $Z_{\Delta_n}(\emptyset)$). This limit always exists as a consequence of sub-additivity of the sequence $(\ln Z_{\Delta_n})_{n\in \mathbbm{N}}$. As a corollary of the Pirogov-Sinaï-Zahradnik technology we used, we obtain the non regularity of the pressure at the critical point. **Proposition 1**. *Under the same assumptions as Theorem [Theorem 1](#thm_quermass_transition){reference-type="ref" reference="thm_quermass_transition"}, for all $\theta_1,\theta_2, \beta$ such that $\theta_1 > -\theta_1^*$, $0 \leq \theta_2 < \theta_2^*(\theta_1)$ (recall that $\theta_2=0$ if $d\ge 3$) and $\beta > \beta_c(\theta_1, \theta_2)$ $$\begin{aligned} \frac{\partial \psi}{\partial z^+}(z_\beta^c, \beta) > \frac{\partial \psi}{\partial z^-}(z_\beta^c, \beta). \end{aligned}$$ Furthermore, there exist two Gibbs measures $P^+, P^- \in \mathcal{G}(\theta_1, \theta_2, \beta, z_\beta^c)$ such that $$\rho(P^-) = z + z\beta\frac{\partial \psi}{\partial z^-}(z_\beta^c, \beta) \quad \text{and } \quad \rho(P^+) = z + z\beta \frac{\partial \psi}{\partial z^+}(z_\beta^c, \beta).$$* # Proof {#Section4} We start by giving the sketch of proof and the plan of the development. Our aim is to prove Proposition [Proposition 1](#prop_first_order_transition){reference-type="ref" reference="prop_first_order_transition"}, as it is clear that this result implies Theorem [Theorem 1](#thm_quermass_transition){reference-type="ref" reference="thm_quermass_transition"}. In section [4.1](#subsection4.1){reference-type="ref" reference="subsection4.1"}, we do a coarse graining of the Hamiltonian. We do a spatial decomposition of the energy function along the lines of tiles that pave $\mathbbm{R}^d$ such that we have $$H = \sum_{i \in \mathbbm{Z}^d} \boldmath{H}_i.$$ We can observe that for small tiles the presence of at least one particle inside a tile $i \in \mathbbm{Z}^d$ would imply that the spatial energy $\boldmath{H}_i = \delta^d$. We call this property the saturation of the energy as no matter how much points we put in it this value remains constant. On the other hand, the spatial energy of an empty tile, when the other tiles around are empty, is equal to $0$. Therefore, when the status of the tiles are homogeneous it becomes easy to compute the energy of a configuration. Hence the study of the behaviour in the mixing areas is necessary. In section [4.2](#subsection4.2){reference-type="ref" reference="subsection4.2"}, we will properly define the contours, $\Gamma = \{ \gamma_1, \cdots, \gamma_n\}$, in order to match with the intuition of the interface areas between homogeneous regions. In section [4.3](#subsection4.3){reference-type="ref" reference="subsection4.3"} we prove that the partition function with boundary condition associated to either state ( $\#= 1$ wired, $\#= 0$ free) can be written as a polymer development, i.e. $$Z_\Lambda^\#= \sum_\Gamma \prod_{\gamma \in \Gamma} w_\gamma^\#$$ where $w_\gamma^\#$ are weights associated to each contours. In section [4.4](#subsection4.4){reference-type="ref" reference="subsection4.4"}, for a large set of parameters $(\theta_1, \theta_2)$ we prove that the spatial energy in any contour $\gamma$ and any configuration $\omega$ achieving the contour verifies $$\boldmath{H}_\gamma (\omega) \geq |\gamma_1| \delta^d + \rho_0 |\gamma|$$ where $\gamma_1 \subset \gamma$ corresponds to the tiles of the contours that contain a particle and $\rho_0$ is a strictly positive constant. This type of result, is in the same spirit of Peierls condition for contours on the lattice and comes from our spatial decomposition of the energy. If the size of the tiles are well tuned some empty tiles would have the same spatial energy as a non-empty tile. We call this property the energy from vacuum. Using this inequality we will prove, in section [4.5](#subsection4.5){reference-type="ref" reference="subsection4.5"}, that for $\beta$ large enough and for $z$ in the vicinity of $\beta$ the weights are $\tau$-stable, i.e. $$w_\gamma^\#\leq e^{-\tau |\gamma|}$$ where $\tau >0$ depends on the value of $\beta$ and can be larger as we want provided that $\beta$ is large. We prove this using truncated weights and pressure as usual with the PSZ theory. This exponential decay of the weights is needed to use results on cluster expansion and write the partition function as follow $$Z_\Lambda^\#= e^{\psi |\Lambda| + \Delta_\Lambda}$$ where $\psi$ is the pressure and $\Delta_\Lambda$ is a perturbation term that is of the order of the surface. Finally in section [4.6](#subsection4.6){reference-type="ref" reference="subsection4.6"} we deduce the non-differentiability of the pressure for a critical activity and we deduce the liquid-gas phase transition as it is stated in Proposition [Proposition 1](#prop_first_order_transition){reference-type="ref" reference="prop_first_order_transition"}. ## Coarse Graining decomposition of the energy {#subsection4.1} Let $\delta > 0$ and for all integers $i \in \mathbbm{Z}^d$, we call a tile $$T_i := \tau_{i\delta} \left( [ -\frac{\delta}{2}, \frac{\delta}{2}[^d\right),$$ where $\tau_{i\delta}$ is the translation by vector $i\delta$. For any $\Lambda \subset \mathbbm{Z}^d$ we denote by $\widehat{\Lambda} := \bigcup_{i \in \Lambda} T_i$. We call a facet $F$ any non-empty intersection between two closed tiles in $(\overline{T}_i)_{i\in \mathbbm{Z}}$. Clearly the dimension of a facet can be any integer between $0$ and $d$. We denote by $\mathcal{F}$ the set of all facets. The energy of a tile $H_i$ is given by $$\forall \omega \in \Omega_f, \quad H_i (\omega) = \mathcal{V}(L(\omega) \cap T_i) + \theta_1 \mathcal{S}_i(L(\omega)) + \theta_2 \chi_i (L(\omega)),$$ where $$\begin{aligned} \mathcal{S}_i (A) = \sum_{k = d-1}^d \sum_{\substack{F \in \mathcal{F} \\ \dim(F)=k}, F \cap T_i \neq \emptyset} (-1)^{d-k} \mathcal{S}(A \cap F) \\ \chi_i(A) = \sum_{k = 0}^d \sum_{\substack{F \in \mathcal{F} \\ \dim(F)=k} , F \cap T_i \neq \emptyset} (-1)^{d-k} \chi(A \cap F). \end{aligned}$$ Furthermore, we can observe that $S_i(\overline{T}_i) =0$ and $\chi_i(\overline{T}_i) = 0$. This is due to the fact that for a $d-1$-dimensional facet $F$, we have $\mathcal{S}(F) = 2 \lambda^{(d-1)}(F)$ where $\lambda^{(d-1)}$ is the $(d-1)$ Lebesgue measure. Therefore, for any configuration $\omega \in \mathcal{C}_f$ such that $T_i\subset L(\omega)$ we have $H_i(\omega) = \mathcal{V}(T_i) = \delta^d$. **Lemma 2**. *For every finite configuration $\omega \in \Omega_f$ $$H(\omega) = \sum_{i \in \mathbbm{Z}^d} H_i(\omega).$$* The proof is a direct consequence of the additivity of the Minkowski functionals. For every $\Lambda \subset \mathbbm{Z}^d$, we use the following notation $$H_\Lambda := \sum_{i \in \Lambda} H_i.$$ ## Contours {#subsection4.2} Let us consider the lattice $\mathbbm{Z}^d$ underlying the tiles $(T_i)_{i \in \mathbbm{Z}^d}$, where two sites $i,j \in \mathbbm{Z}^d$ are connected if $\| i-j \|_\infty =1$. We call the spin configuration the application $$\begin{aligned} \sigma :& \Omega_f \times \mathbbm{Z}^d \rightarrow \{ 0, 1\} \\ & (\omega, i) \mapsto \begin{cases} 0 & \text{if } \omega_{T_i} =\emptyset \\ 1 & \text{otherwise} \end{cases}. \end{aligned}$$ In the following we use the notation $\#$ for either $0$ or $1$. **Definition 4**. *Let $L>0$ and $\omega \in \Omega$, a site $i \in \mathbbm{Z}^d$ is said to be $\#$-correct if for all sites $j$ such that $\| i-j \| \leq L$, we have $\sigma(\omega, j) = \#$. A site $i$ is non-correct when it fails to be $\#$-correct for $\#\in \{0,1\}$. The set of all non-correct sites is denoted by $\overline{\Gamma}$. We can partition $\overline{\Gamma}$ into its maximum connected components that we denote by $\overline{\gamma}$ and we call it contour without types.* Since we are considering only finite configurations, the number of connected components is finite and for any $\overline{\gamma}$, the complementary set has a finite amount of maximum connected components that we denote by $A$ and in particular we have only one unbounded connected component and we call it the exterior of $\overline{\gamma}$ that we denote by $ext(\overline{\gamma})$. **Definition 5**. *Let $\Lambda \subset \mathbbm{Z}^d$ and $L>0$, we define the exterior boundary $\partial_{ext} \Lambda$ and the interior boundary $\partial_{int} \Lambda$ of $\Lambda$ as $$\begin{aligned} \partial_{ext} \Lambda = \{j \in \Lambda^c, d_2(j, \Lambda) \leq L \} \\ \partial_{int} \Lambda = \{i \in \Lambda, d_2(i, \Lambda^c) \leq L+1 \}, \end{aligned}$$ where $d_2$ is the Euclidean distance in $\mathbbm{R}^d$.* **Lemma 3**. *Let $\omega \in \Omega_f$ be a finite configuration and $\overline{\gamma}$ any associated contour without type. Let $A$ be a maximum connected component of $\overline{\gamma}^c$, then there is an unique $\#\in \{0, 1\}$ such that for all $i \in \partial_{ext} A \cup \partial_{int}A, \sigma(\omega, i) = \#$. The value of the spin in the boundary is called the label of $A$ and is denoted by $\mathop{\mathrm{Label}}(A)$.* The proof of this lemma is classical and it corresponds to Lemma 7.23 in [@Velenik]. It relies on the fact that each set $\partial_{int}A$ and $\partial_{ext}A$ are connected and that the sites directly in contact with the contours are correct. Therefore there can be only one spin $\#\in \{0,1\}$ otherwise we would have two correct sites of opposite spin directly connected. **Definition 6**. *Let $\omega \in \Omega_f$, we call a contour $\gamma$ the pair $(\overline{\gamma}, (\#_i)_{i \in \overline{\gamma}})$ where for all sites $i \in \overline{\gamma}$, $\sigma(\omega, i) = \#_i$. We denote by $\Gamma(\omega)$ the set of all contours that appear with the configuration $\omega$.* Furthermore for a contour $\gamma = (\overline{\gamma}, (\#_j)_{j \in \overline{\gamma}})$ we call the type of $\gamma$ the label of $ext(\overline{\gamma})$, $\mathop{\mathrm{Type}}(\gamma) := \mathop{\mathrm{Label}}(ext(\overline{\gamma}))$. And we call the interiors of a contour $\gamma$ the sets $$\mathop{\mathrm{Int}}_\#\gamma = \bigcup_{\substack{A \neq ext(\overline{\gamma}) \\ \mathop{\mathrm{Label}}(A) = \#}} A \quad \text{ and } \quad \mathop{\mathrm{Int}}\gamma = \mathop{\mathrm{Int}}_0 \gamma \cup \mathop{\mathrm{Int}}_1 \gamma.$$ Let $\omega \in \Omega_f$ be a finite configuration, a contour $\gamma \in \Gamma(\omega)$ is said to be external when for any other contour $\gamma' \in \Gamma(\omega)$, $\overline{\gamma} \subset ext(\overline{\gamma}')$. We denote by $\Gamma_{ext}$ the subset of $\Gamma$ comprised only of external contours. Until now we have only considered collection of contours that can be achieved by a finite configuration of points. But classically in the Pirogov-Sinaï-Zahradnik theory we need to introduce abstract collection of contours which are not achievable by any configuration. This is due to the cluster expansion development of the partition function using geometrically compatible collection of contours. **Definition 7**. *An abstract set of contours is a set of contours $\{ \gamma_i = (\overline{\gamma}_i, (\#_j)_{j \in \overline{\gamma_i}}) , i \in I \subset \mathbbm{N}^*\}$ for which each contour $\gamma_i$ is achievable for some configuration $\omega_i$. We do not assume the global achievability. We denote by $\Gamma$ such set of contours. Moreover this set $\Gamma$ is called geometrically compatible if for all $\{i,j\} \subset I$, $d_\infty(\gamma_i, \gamma_j)>1$ and they all have the same type. Let $\Lambda \in \mathbbm{Z}^d$ we denote by $\mathcal{C}^\#(\Lambda)$ the collection of geometrically compatible sets of contours of the type $\#$ such that $d_\infty(\gamma_i, \Lambda^c)>1$.* We allow the set $\Gamma = \{ (\emptyset, \emptyset)\}$ to belong to the collection $\mathcal{C}^\#(\Lambda)$ for any $\Lambda$ which corresponds to the event where not a single contour appears in $\Lambda$. There are several interesting sub-collections of $\mathcal{C}^\#(\Lambda)$, one of them being the collection of sets such that all contours are external. **Definition 8**. *Let $\Lambda \in \mathbbm{Z}^d$ finite, we denote by $\mathcal{C}_{ext}^\#(\Lambda) \subset \mathcal{C}^\#(\Lambda)$ the sub-collection of sets $\Gamma$ where any contours $\gamma \in \Gamma$ is external.* In a way, in the collection $\mathcal{C}_{ext}^\#(\Lambda)$ we are considering sets of contours where we have only one layer. In general, if we take a geometrically compatible abstract sets of contours $\Gamma$, a particular contour in this set can be encapsulated in the interior of another creating layers upon layers of contours. One method of exploration of the contours is by proceeding from the external layer and peel each layer to discover the other contours hidden under. Another sub-collection of $\mathcal{C}^\#(\Lambda)$ is the collection of sets such that for all contours the size of the interior is bounded. **Definition 9**. *A contour $\gamma$ is of the class $k \in \mathbbm{N}$ when $|\mathop{\mathrm{Int}}\gamma | =k$. Let $n \in \mathbbm{N}$ and $\Lambda \subset \mathbbm{Z}^d$ finite, we denote by $\mathcal{C}_n^\#(\Lambda) \subset \mathcal{C}^\#(\Lambda)$ the collection of contours $\Gamma$ such that $\forall \gamma \in \Gamma$, $\gamma$ is of the class $k\leq n$.* ## Partition function and boundary condition {#subsection4.3} We consider two Quermass point process with a free boundary and a wired boundary conditions. The probability measures $P_{\Lambda}^\#$ associated to each one of them is given by $$\label{def_mesure_gibbs_condition_bord} P_\Lambda^\#(d\omega) = \frac{1}{Z_\Lambda^\#} e^{-\beta H_\Lambda(\omega)} \mathbbm{1}_{\{\forall i \in \partial_{int} \Lambda, \sigma(\omega, i) = \#\}} \Pi_{\widehat{\Lambda}}^z(d\omega)$$ where $$Z_\Lambda^\#= \int e^{-\beta H_\Lambda(\omega)} \mathbbm{1}_{\{\forall i \in \partial_{int} \Lambda, \sigma(\omega, i) = \#\}} \Pi_{\widehat{\Lambda}}^z(d\omega).$$ We prove that the two infinite Gibbs measure obtained by taking the thermodynamic limit for each boundary condition yield different intensity. First we have a standard lemma which ensures that the pressure does not depend on the boundary conditions. **Lemma 4**. *For $\delta \le \nicefrac{R_0}{2\sqrt{d}}$, $L\ge \nicefrac{2R_1}{\delta}$ and any $\#\in \{0,1\}$ we have $\psi = \psi^\#$ where $\psi^\#:= \underset{n \to \infty}{\lim} \frac{\ln Z_{\Lambda_n}^\#}{\beta |\Lambda_n| \delta^d}$ and $\Lambda_n = [\![-n,n]\!]^d$.* *Proof.* For any configuration $\omega \in \Omega_f$ such that for all sites $i \in \partial_{int} \Lambda_n, \sigma(\omega,i) = 1$ and $\omega_{\Lambda_n^c} = \emptyset$ we know that no holes are created by the halo outside of $\Lambda_n$ and therefore $$H_{\partial_{ext} \Lambda_n} (\omega) \leq \begin{cases} |\partial_{ext} \Lambda_n | \delta^d & \text{ when } \theta_1 \leq 0 \\ |\partial_{ext} \Lambda_n | \delta^d + \theta_1 \mathcal{S}_{\partial_{ext}\Lambda_n}(L(\omega))& \text{ when } \theta_1 >0 \end{cases}.$$ The boundary of the halo $L(\omega)$ outside $\Lambda_n$, appearing in the computation of $\mathcal{S}_{\partial_{ext}\Lambda_n}(L(\omega))$, is the union of spherical caps built via some marked points $(x_1,r_1), \dots, (x_m, r_m) \in \omega$. We denote by $\alpha_i\in[0,1]$ the ratio of the surface of the $i$th spherical cap with respect to the total surface of its sphere. We have $$\begin{aligned} \mathcal{S}_{\partial_{ext} \Lambda_n}(L(\omega)) &= \sum_{i = 1}^m \alpha_i \mathcal{S}(B(x_i, r_i)) \\ & = \sum_{i = 1}^m \alpha_i \frac{\mathcal{S}(B(x_i, r_i))}{\mathcal{V}(B(x_i, r_i))} \mathcal{V}(B(x_i, r_i)) \\ & \leq \frac{\mathcal{S}(B(0, 1))}{R_0\mathcal{V}(B(0, 1))} \sum_{i = 1}^m \alpha_i \mathcal{V}(B(x_i, r_i)) \\ & \leq \frac{\mathcal{S}(B(0, 1))}{R_0\mathcal{V}(B(0, 1))} |\partial_{int} \Lambda_n \cup \partial_{ext} \Lambda_n |\delta^d. \end{aligned}$$ As a consequence there exists $c>0$ such that we have $$\begin{aligned} Z_{\widehat{\Lambda}_n} & \geq \int e^{-\beta H_{\partial_{ext} \Lambda_n} (\omega) } e^{-\beta H_{\Lambda_n} (\omega) } \mathbbm{1}_{\{\forall i \in \partial_{int}\Lambda_n, \sigma(\omega, i)= 1\}} \Pi_{\widehat{\Lambda}_n}^z(d\omega) \\ & \geq e^{-c |\partial_{ext} \Lambda_n \cup \partial_{int} \Lambda_n| \delta^d } Z_{\Lambda_n}^{(1)}. \end{aligned}$$ Therefore we have $\psi \geq \psi^{(1)}$. Let us consider the following event $$E_n = \{ \omega \in \Omega_f, \forall i \in \Lambda_n, L < d_2(i, \Lambda_n^c) \leq 2L, \sigma(\omega,i) = 0\}.$$ Since $\delta \le \nicefrac{R_0}{2 \sqrt{d}}$ and $L \ge \frac{2R_1}{\delta}$ we have $$\begin{aligned} Z_{\Lambda_n}^{(1)} \geq \int e^{-\beta (|\partial_{int} \Lambda_n| + H_{\partial_{int} \Lambda_{n-L}}(\omega_{\widehat{\partial_{int} \Lambda_n}}))} \mathbbm{1}_{\{\forall i \in \partial_{int}\Lambda_n, \sigma(\omega, i)= 1\}} e^{-\beta H_{\Lambda_{n-L}}(\omega_{\widehat{\Lambda}_{n-L}})} \mathbbm{1}_{E_n} \Pi_{\widehat{\Lambda}_n}^z(d\omega). \end{aligned}$$ Similarly as in the previous case we can find $c>0$ such that $$H_{\partial_{int} \Lambda_{n-L}}(\omega_{\widehat{\partial_{int} \Lambda_n}}) \leq c |\partial_{int} \Lambda_n \cup \partial_{ext} \Lambda_n| \delta^d$$ and therefore we get $$Z_{\Lambda_n}^{(1)} \geq g_1^{|\partial_{int} \Lambda_n|} e^{-\beta c |\partial_{int} \Lambda_n \cup \partial_{ext} \Lambda_n| \delta^d } Z_{\Lambda_{n-L}}^{(0)}.$$ This implies that $\psi^{(1)} \geq \psi^{(0)}$. Finally we simply have that $$Z_{\Lambda_n}^{(0)} \geq g_0^{|\partial_{int} \Lambda_n|} Z_{\Lambda_{n-L}}.$$ Therefore $\psi^{(0)} \geq \psi$ and this finishes the proof of the lemma. ◻ We have also a crucial proposition which provides a representation of the partition function as a polymer development. **Proposition 2** (Polymer development). *Let $R_0 \geq 2\delta \sqrt{d}>0$ and $\delta L>2R_1$, for all $\Lambda \subset \mathbbm{Z}^d$ finite we have $$\Phi_\Lambda^\#:= g_\#^{-|\Lambda|} Z_\Lambda^\#= \sum_{\Gamma \in \mathcal{C}^\#(\Lambda)} \prod_{\gamma \in \Gamma} w_{\gamma}^\#$$ where* - *$g_0 = e^{-z\delta^d}$ and $g_1 = e^{-\beta \delta^d}(1-e^{-z\delta^d})$\ * - *$w_{\gamma}^\#= g_\#^{-|\overline{\gamma}|} I_{\gamma} \frac{Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#^*}}{Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#}}$ and it's called the weight of the contour $\gamma$,\ * - *$I_{\gamma} = \int e^{-\beta H_{\overline{\gamma}} (\omega)} \mathbbm{1}_{\left(\forall i \in \overline{\gamma}, \sigma(\omega,i)=\sigma_i \right)} \Pi_{\widehat{\gamma}}^z(d\omega)$.* *Proof.* We follow a similar development done in Chapter 7 in [@Velenik] with an adaptation to the setting of our model where the main difference is that the states of sites are random and have to be integrated under the Poisson measure. We can decompose the partition function $Z_\Lambda^\#$ according to the external contours $\mathcal{C}_{ext}^\#(\Lambda)$ and we have $$\begin{aligned} Z_\Lambda^\#= \sum_{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda)} \int e^{-\beta H_{\Lambda}(\omega)} \mathbbm{1}_{\{\forall i \in \partial_{int}\Lambda, \sigma(\omega, i)= \#\}} \mathbbm{1}_{\{\Gamma_{ext}(\omega) = \Gamma\}} \Pi_{\widehat{\Lambda}}^z(d\omega). \end{aligned}$$ For any $\Gamma \in \mathcal{C}_{ext}^\#(\Lambda)$ we can do a partition of $\Lambda$ in the following way $$\Lambda = \Lambda_{ext} \bigcup_{\substack{\gamma \in \Gamma}} \left( \gamma \cup \mathop{\mathrm{Int}}_{0}\gamma \cup \mathop{\mathrm{Int}}_{1}\gamma \right)$$ where $\Lambda_{ext} = \bigcap_{{\gamma} \in \Gamma} ext(\overline{\gamma}) \cap \Lambda$. For any finite configuration $\omega \in \Omega_f$ such that $\Gamma_{ext}(\omega) = \Gamma$, we have $$\begin{aligned} &H_{\Lambda_{ext}} (\omega) = H_{\Lambda_{ext}} ( \omega_{{\widehat{\Lambda}}_{ext}}) = \mathcal{V}({{\widehat{\Lambda}}_{ext}}) \mathbbm{1}_{(\#=1)} \label{energy_local_gamma_1}\\ &H_{\overline{\gamma}} (\omega) = H_{\overline{\gamma}} (\omega_{\widehat{\gamma}}) \label{energy_local_gamma_2}\\ &H_{\mathop{\mathrm{Int}}_\#\gamma}(\omega) = H_{\mathop{\mathrm{Int}}_\#\gamma}(\omega_{\widehat{\mathop{\mathrm{Int}}_\#\gamma}}). \label{energy_local_gamma_3} \end{aligned}$$ By construction, we know that for all $i \in \Lambda_{ext}, \sigma(\omega, i) = \#$ and $i$ is $\#$-correct. First case $\#= 1$, since $R_0 \geq \delta \sqrt{d}$ if $\omega_{T_i} \neq \emptyset$ implies that $H_i(\omega) = H_i(\omega_{T_i}) = \mathcal{V}(T_i)$. Second case $\#= 0$, since the sites $i$ in $\Lambda_{ext}$ are $0$-correct and that $\delta L > 2 R_1$ then $H_i(\omega) = H_i(\omega_{T_i}) = 0$. Furthermore, we know that a tile adjacent to an external contour $\gamma$ is correct. For $A= \mathop{\mathrm{Int}}_0 \gamma$ or $A = \Lambda_{ext}$ when $\#= 0$ , because of the previous fact we know that $d_2(\omega_{\widehat{A}}, \widehat{\gamma}) > 2R_1$ and $d_2(\omega_{\widehat{\gamma}}, \widehat{A}) > 2R_1$. Therefore we have $L(\omega_{\widehat{\gamma}}) \cap \widehat{A} = \emptyset$ and $L(\omega_{\widehat{A}}) \cap \widehat{\gamma}= \emptyset$. And if we consider $B = \mathop{\mathrm{Int}}_1 \gamma$ or $B = \Lambda_{ext}$ when $\#=1$, the energy of the tiles in $\partial_{int}B$ and $\partial_{ext}B$ are determined since we have a point in those tiles and that $R_0 \geq \delta \sqrt{d}$. Furthermore, since $\delta L > 2R_1$ what is happening in the tiles $\overline{\gamma} \backslash \partial_{ext}B$ doesn't have any consequence on $B$ and vice versa what is happening in $B \backslash \partial_{int}B$ doesn't affect the tiles in $\overline{\gamma}$. Using these observations on the way the energy behaves according the contours and using the independence of Poisson point process in disjoint areas we have $$\begin{aligned} Z_\Lambda^\#=& \sum_{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda)} \left( \int e^{-\beta H_{\Lambda_{ext}}(\omega)} \mathbbm{1}_{\{ \forall i \in \Lambda_{ext}, \sigma(\omega,i)=\#\}} \Pi_{\widehat{\Lambda_{ext}}}^z(d\omega) \right) \nonumber \\ &\times \prod_{\gamma \in \Gamma} \left( \int e^{-\beta H_{\overline{\gamma}} (\omega)} \mathbbm{1}_{\{ \forall i \in \overline{\gamma}, \sigma(\omega, i) = \sigma_i \}} \Pi_{\widehat{\overline{\gamma}}}^z(d\omega) \right) \nonumber \\ & \times \prod_{\#^*} \left( \int e^{-\beta H_{\mathop{\mathrm{Int}}_{\#^*} \gamma}(\omega)} \mathbbm{1}_{\{ \forall i \in \partial_{int} \mathop{\mathrm{Int}}_{\#^*} \gamma, \sigma(\omega, i) = \#^* \}} \Pi_{\widehat{\mathop{\mathrm{Int}}_{\#^*} \gamma}}^z(d\omega) \right). \label{eq_decomposition_product_integrals} \end{aligned}$$ In this development we recognize the partition function $Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#^*}$ and the term $I_\gamma$. At the end, we have by independence of Poisson point process in disjoint areas $$\begin{aligned} \int e^{-\beta H_{\Lambda_{ext}}(\omega)} \mathbbm{1}_{\{ \forall i \in \Lambda_{ext}, \sigma(\omega,i)=\#\}} \Pi_{\widehat{\Lambda_{ext}}}^z(d\omega) = \left( \int e^{-\beta H_0(\omega)} \mathbbm{1}_{\{ \sigma(\omega,0)=\#\}} \Pi_{T_0}^z(d\omega)\right)^{|\Lambda_{ext}|}. \end{aligned}$$ According to the value $\#\in \{ 0,1\}$ we have $$\begin{aligned} &\int e^{-\beta H_0(\omega)} \mathbbm{1}_{\{ \sigma(\omega,0)=1 \}} \Pi_{T_0}^z(d\omega) = e^{-\beta \delta^d}(1-e^{-z\delta^d}) = g_1\\ &\int e^{-\beta H_0(\omega)} \mathbbm{1}_{\{ \sigma(\omega,0)=0 \}} \Pi_{T_0}^z(d\omega) = e^{-z \delta^d} = g_0. \end{aligned}$$ In summary, we have the following $$\begin{aligned} Z_\Lambda^\#&= \sum_{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda)} g_{\#}^{|\Lambda_{ext}|} \prod_{\gamma \in \Gamma} I_{\gamma} Z_{\mathop{\mathrm{Int}}_\#\gamma}^{\#} Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#^*} \\ &= \sum_{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda)} g_{\#}^{|\Lambda_{ext}|} \prod_{\gamma \in \Gamma} I_{\gamma} \frac{Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#^*}}{Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#}} Z_{\mathop{\mathrm{Int}}_\#\gamma}^{\#} Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#} \\ &= \sum_{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda)} g_{\#}^{|\Lambda_{ext}|} \prod_{\gamma \in \Gamma} g_{\#}^{|\overline{\gamma}|} w_\gamma^\#Z_{\mathop{\mathrm{Int}}_\#\gamma}^{\#} Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#}. \end{aligned}$$ From the properties our energy, we know that $0 <Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#} < +\infty$ and thus the quotient introduced above is allowed. In regards of the quantity $\Phi_\Lambda^\#$ we have $$\begin{aligned} \Phi_\Lambda^\#= \sum_{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda)} \prod_{\gamma \in \Gamma} w_\gamma^\#\Phi_{\mathop{\mathrm{Int}}_\#\gamma}^{\#} \Phi_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#}. \end{aligned}$$ We can continue to iterate the same computation for $\Phi_{\mathop{\mathrm{Int}}_\#\gamma}$ and for $\Phi_{\mathop{\mathrm{Int}}_{\#^*} \gamma}$ until we empty the interior of the contours and thus we have $$\begin{aligned} \Phi_\Lambda^\#= \sum_{\Gamma \in \mathcal{C}^\#(\Lambda)} \prod_{\gamma \in \Gamma} w_\gamma^\#. \end{aligned}$$ ◻ ## Energy and Peierls condition {#subsection4.4} The weights of any contours $\gamma$ is a difficult object to evaluate and our goal is to have a good exponential bound with respect the volume of the contour. Recall the expression of the weight $$\begin{aligned} w_{\gamma}^\#= g_\#^{-|\overline{\gamma}|} I_{\gamma} \frac{Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#^*}}{Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#}}. \end{aligned}$$ **Definition 10**. *Let $\tau > 0$, the weight of a contour $\gamma$ is said to be $\tau$-stable if $$w_{\gamma}^\#\leq e^{-\tau |\overline{\gamma}|}.$$* The ratio of the partition functions is the most difficult part to handle at this stage and will be further developed in the next section. For the moment, we want to control the quantity $I_{\gamma}$ where $\ln{I_{\gamma}}$ can be interpreted as the mean energy of the contour. The aim is to display some sort of Peierls condition on the mean energy of the contour and afterwards to prove the $\tau$-stability of the weights. **Proposition 3**. *For any $\theta_1 > - \theta_1^*:= - R_0 \frac{\mathcal{V}(B(0,1))}{\mathcal{S}(B(0,1))}$ (and, in dimension $d=2$, any $\theta_2$ such that for $0 \leq \theta_2 < \theta_2^*(\theta_1)$, where $\theta_2^*>0$ is defined in [\[eq_theta_2\_lim\]](#eq_theta_2_lim){reference-type="eqref" reference="eq_theta_2_lim"}), there exist $\delta \in \left] 0 , \nicefrac{R_0}{2 \sqrt{d}} \right[$, $K>0$ and $\rho_0>0$ such that for all contour $\gamma$ and $\beta>0$ $$\begin{aligned} I_{\gamma} &\leq g_0^{|\overline{\gamma}_0|} g_1^{|\overline{\gamma}_1|} e^{-\beta \rho_0 |\overline{\gamma}|} \end{aligned}$$ and $$\begin{aligned} \left| \frac{\partial I_\gamma}{\partial z} \right| \leq \left(1 + \frac{K}{1-e^{-z\delta^d}}\right) |\overline{\gamma}|\delta^d g_0^{|\overline{\gamma}_0|} g_1^{|\overline{\gamma}_1|} e^{- \beta \rho_0 |\overline{\gamma}|}. \end{aligned}$$* Before going to the proof of Proposition [Proposition 3](#prop_peierls){reference-type="ref" reference="prop_peierls"}, we need some intermediate geometrical results given in the four next lemmas. We start with the following observation. For any configuration $\omega \in \Omega$, if there exist $(i,j) \in (\mathbbm{Z}^d)^2$ where $d_\infty(i,j)=1$ such that $\sigma(\omega, i)=1$ and $\sigma(\omega,j)=0$, we call this pair a domino. We assume that $R_0 \geq 2\delta \sqrt{d} >0$. Therefore for any configuration $\omega$ and any domino $(i,j)$ we have $H_i(\omega)= H_j(\omega) = \delta^d$ because of the presence of a point inside the tile $T_i$ ensuring that the tiles $T_i$ and $T_j$ are covered. In particular the energy of this empty tile is positive and we call this property the energy from vacuum. We need to show that the proportion of such pair of tiles in the contour is at least proportional to the volume of the contour. **Lemma 5**. *There exists $r_0 >0$ such that for any contour $\gamma$, the set of dominoes $$D(\gamma) := \{ (i,j) \in \overline{\gamma}^2, d_\infty(i,j)=1, \#_i = 1, \#_j=0 \}$$ satisfies $$| D(\gamma) | \geq r_0 |\overline{\gamma}|.$$* *Proof.* We start by choosing randomly in a contour $\gamma$ a site $k$ such that $\#_k= 1$. Since it is in a contour, it is non-correct, meaning that there is a site $j \in \gamma$, where $\#_j = 0$ and $d_2(k,j) \leq L$. We choose such $j$ such as it is the closest to $k$. Forcibly we have a site $i$ adjacent to $j$ such that $\#_i= 1$ ( at least in the direction of $k$). And we assign $S_1 =\{ k \}$ and $D_1 = \{ (i,j) \}$. We repeat the process to build $S_{n+1}$ and $D_{n+1}$ by choosing the points inside $\gamma \backslash \bigcup_{k \in S_n} B(k, 4L)$. There is $p \in \mathbbm{N}$, the number of step until the process stops because there is a finite number of sites with the spin equal to 1 in a contour. We define $S(\gamma) = S_p$ and $D(\gamma) = D_p$. We know that at this point that $$\overline{\gamma}_1 := \{ i \in \overline{\gamma}, \#_i=1\} \subset \bigcup_{k \in S(\gamma)} B(k,4L) \cap \mathbbm{Z}^d$$ and that by non-correctness of sites with spin 0 in the contour we have $$\overline{\gamma}_0 := \{ i \in \overline{\gamma}, \#_i=0\} \subset \bigcup_{k \in S(\gamma)} B(k, 5L) \cap \mathbbm{Z}^d.$$ In summary we have $$\overline{\gamma} \subset \bigcup_{k \in S} B(k, 5L) \cap \mathbbm{Z}^d.$$ Therefore the cardinals verify the following inequalities $$|\overline{\gamma}| \leq |S| |B(0,5L) \cap \mathbbm{Z}^d|.$$ By construction we have $|S(\gamma)| = |D(\gamma)|$. Hence the inequality we are interested in $$|D(\gamma)| \geq r_0 |\overline{\gamma}| \quad \text{where } \: r_0 = \frac{1}{|B(0,5L) \cap \mathbbm{Z}^d|}.$$ ◻ For any contour $\gamma$ and any configuration $\omega$ that achieves this contour, using the dominoes we are able to find a non-negligible amount of empty tiles that is covered by the halo of the configuration. Another way to find such tiles in the contour is by counting the tiles covered by the halo which are close to the boundary of the halo. Indeed those tiles are guaranteed to be empty otherwise the boundary would be further away. **Lemma 6**. *Let $R_0 \geq 2\delta \sqrt{d} >0$, and let us define $\theta_1^\delta$ such as $$\theta_1^\delta := \inf_{\substack{\omega \in \Omega_f \\ \gamma : \mathcal{S}_{\overline{\gamma}}(L(\omega)) >0 }} \left\{ \frac{ V_{\omega, \gamma, \delta}}{\mathcal{S}_{\overline{\gamma}}(L(\omega))} \right\}$$ and $$V_{\omega, \gamma, \delta} = \max \left\{\mathcal{V}(T_I), I \subset \overline{\gamma}, \forall i \in I, T_i \subset \partial L(\omega) \oplus B(0,R_0) \cap L(\omega) \right\}.$$ We have $\theta_1^\delta >0$ and $\theta_1^\delta \underset{\delta \rightarrow 0}{\rightarrow} \theta_1^*$ where $\theta_1^* = R_0 \frac{\mathcal{V}(B(0,1))}{\mathcal{S}(B(0,1))}$.* *Proof.* For any finite configuration $\omega \in \Omega_f$ and $\gamma$ a contour that is created by this configuration such that $\mathcal{S}_{\overline{\gamma}}(L(\omega)) >0$, we have the following inequalities $$V_{\omega, \gamma, \delta}^{+} \geq V_{\omega, \gamma, \delta} \geq V_{\omega, \gamma, \delta}^{-}$$ where $$\begin{aligned} &V_{\omega, \gamma, \delta}^{+} = \mathcal{V}(\partial L(\omega) \oplus B(0,R_0) \cap L(\omega) \cap \widehat{\gamma}) \\ &V_{\omega, \gamma, \delta}^{-} = \mathcal{V}((\partial L(\omega) \oplus B(0,R_0- \delta)) \backslash (\partial L(\omega) \oplus B(0,\delta)) \cap L(\omega) \cap \widehat{\gamma}). \end{aligned}$$ The boundary of the halo $L(\omega)$ inside $\overline \gamma$, appearing in the computation of $\mathcal{S}_{\overline{\gamma}}(L(\omega))$ , is the union of spherical caps built via some marked points $(x_1,r_1), \dots, (x_m, r_m) \in \omega$. We denote by $\alpha_i\in[0,1]$ the ratio of the surface of the $i$th spherical cap with respect to the total surface of its sphere. Therefore, by a simple geometrical argument $$\begin{aligned} \frac{V_{\omega, \gamma, \delta}^{-}}{\mathcal{S}_{\overline{\gamma}}(L(\omega))} &\geq \frac{\mathcal{V}(B(0,1))}{\mathcal{S}(B(0,1))} \left( \frac{\sum_{i=1}^{m} \alpha_i (r_i^d - (r_i-R_0)^d)}{\sum_{i=1}^{m} \alpha_i r_i^{d-1}} - \epsilon_{\omega,\gamma}(\delta) \right) \\ & \geq \frac{\mathcal{V}(B(0,1))}{\mathcal{S}(B(0,1))} ( R_0 - \epsilon_{\omega,\gamma}(\delta)) \end{aligned}$$ where $$\begin{aligned} \epsilon_{\omega,\gamma}(\delta) = \frac{\sum_{i=1}^{m} \alpha_i (r_i^d - (r_i-\delta)^d + (r_i - R_o + \delta)^d - (r_i-R_0)^d)}{\sum_{i=1}^{m} \alpha_i r_i^{d-1}}. \end{aligned}$$ Therefore for all $\omega \in \Omega_f$ and the associated contour $\gamma$ we have $$\begin{aligned} \liminf_{\delta \to 0} \frac{V_{\omega, \gamma, \delta}}{\mathcal{S}_{\overline{\gamma}}(L(\omega))} \geq \frac{R_0 \mathcal{V}(B(0,1))}{\mathcal{S}(B(0,1))} \implies \liminf_{\delta \to 0} \theta_1^\delta \geq \theta_1^*. \end{aligned}$$ Note that for $\omega = \{(0,R_0)\}$ we have $\frac{V_{\omega, \gamma, \delta}^{+}}{\mathcal{S}_{\overline{\gamma}}(L(\omega))} = \frac{R_0 \mathcal{V}(B(0,1))}{\mathcal{S}(B(0,1))}$. So $$\begin{aligned} \frac{R_0 \mathcal{V}(B(0,1))}{\mathcal{S}(B(0,1))} \geq \inf_{\substack{\omega \in \Omega_f \\ \gamma : \mathcal{S}_{\overline{\gamma}}(L(\omega)) >0 }} \left\{ \frac{ V_{\omega, \gamma, \delta}^{+}}{\mathcal{S}_{\overline{\gamma}}(L(\omega))} \right\} \end{aligned}$$ which implies that $\theta_1^* \geq \limsup_{\delta \to 0} \theta_1^\delta$. ◻ **Lemma 7**. *Let $\delta L \geq 2 R_1$ and $d=2$. For any contour $\gamma$ and any configuration $\omega$ that achieves this contour we have $$\chi_{\overline{\gamma}}(\omega) \leq \frac{|\overline{\gamma}| \delta^d}{R_0^{d}\mathcal{V}(B(0,1))}.$$* *Proof.* By definition of contours and the conditions on $\delta$ and $L$ we have $$\begin{aligned} \chi_{\overline{\gamma}}(L(\omega)) = \chi_{\overline{\gamma}}(L(\omega_{\widehat{\gamma}})) \leq N_{cc}(L(\omega_{\widehat{\gamma}})). \end{aligned}$$ Now the aim is to find for each connected components $C$ of $L(\omega_{\widehat{\gamma}})$ a single point $(x,R) \in \omega_{\widehat{\gamma}}$ such that the ball $B(x,R) \subset C\cap \widehat{\gamma}$. Note that a ball $B(x,R)\subset C$ is not necessary included inside the contour. If there exists such a ball not including in the contour, then there is $A \subset \overline{\gamma}^c$ a connected component such that $B(x,R) \cap \widehat{A} \neq \emptyset$. It gives the information that the site $i \in \mathbbm{Z}^d$ such that $x \in T_i$ is included inside $\partial_{ext} A$ and by Lemma [Lemma 3](#lem_partial_type){reference-type="ref" reference="lem_partial_type"} for all $j \in \partial_{ext}A$ we have $\sigma(j, \omega_{\omega_{\widehat{\gamma}}}) = 1$. Therefore all balls that are in the tiles corresponding to $\partial_{ext}A$ belong to the same connected component of the halo. We choose a site $j \in \partial_{ext}A$ such that $d_2(j, A) = \lceil \frac{2R_1}{\delta} \rceil$ and so there exists $(y,R') \in \omega_{T_j}$ such that $B(y,R') \subset \widehat{\gamma}$. Therefore we can replace the original representative of the connected component with one that is more suitable. With this procedure we have now built, for each connected components $C$ of $L(\omega_{\widehat{\gamma}})$, a single point $(x,R) \in \omega_{\widehat{\gamma}}$ such that the ball $B(x,R) \subset C\cap \widehat{\gamma}$. We define $I(\omega_{\widehat{\gamma}})$ as the set of all these points which represent the connected components of $L(\omega_{\widehat{\gamma}})$. By construction for any $(x,R) \neq (y,R') \in I(\omega_{\widehat{\gamma}}), B(x,R) \cap B(y,R') = \emptyset$ and therefore we have $$N_{cc}(\omega_{\widehat{\gamma}}) = |I(\omega_{\widehat{\gamma}})| \leq \frac{|\overline{\gamma}| \delta^d}{\mathcal{V}(B(0,R_0))}.$$ ◻ **Lemma 8**. *For any $\#\in \{0,1\}$ and any contour $\gamma$ we have $r_1 |\overline{\gamma} | \leq |\overline{\gamma}_\#| \leq (1-r_1) |\overline{\gamma} |$ where $$r_1 = \frac{1}{|B(0, 2L)\cap \mathbbm{Z}^d|} \quad \text{and } \quad \overline{\gamma}_\#= \{ i \in \overline{\gamma}, \#_i = \#\}.$$* *Proof.* Let us set a contour $\gamma$. Let us define $\phi : \overline{\gamma}_1 \rightarrow \overline{\gamma}_0$ such that for all $i \in \overline{\gamma}_1$, $\phi(i)=j$ where $j$ is the closest site in $\overline{\gamma}_0$ from $i$. A lexicographical procedure is used if equality. $$\begin{aligned} \alpha_\gamma^1 := \frac{|\overline{\gamma}_1|}{|\overline{\gamma}|} \iff \frac{\alpha_\gamma^1}{1-\alpha_\gamma^1} = \frac{|\overline{\gamma}_1|}{|\overline{\gamma}_0|} = \frac{\sum_{j \in \overline{\gamma}_0 } |\phi^{-1}(j)| }{|\overline{\gamma}_0|} \leq \max |\phi^{-1}(j)|. \end{aligned}$$ It is easy to see that $\max |\phi^{-1}(j)| \leq |B(0, 2L)\cap \mathbbm{Z}^d|-1$ and using the fact that $\alpha_\gamma^1 + \alpha_\gamma^0 =1$ we have that $$\begin{aligned} \alpha_\gamma^1 \leq 1 - \frac{1}{|B(0, 2L)\cap \mathbbm{Z}^d|} \\ \alpha_\gamma^0 \geq \frac{1}{|B(0, 2L)\cap \mathbbm{Z}^d|}. \end{aligned}$$ By symmetry of the roles we can exchange the value of the spin. ◻ We have now all the lemmas for proving the Proposition [Proposition 3](#prop_peierls){reference-type="ref" reference="prop_peierls"}. *Proof.* We detail the proof of Proposition [Proposition 3](#prop_peierls){reference-type="ref" reference="prop_peierls"} in dimension $d=2$. In higher dimension, the proof works in the same manner but we assume that $\theta_2 = 0$. Let us set $L = \left \lceil \frac{2R_1}{\delta} \right \rceil$, therefore the constant $r_0$ in Lemma [Lemma 5](#lem-domino-domino){reference-type="ref" reference="lem-domino-domino"} has the following expression $$\begin{aligned} r_0(\delta) = \frac{1}{\left|B\left(0,5\left \lceil \frac{2R_1}{\delta} \right \rceil\right) \cap \mathbbm{Z}^d \right|}. \end{aligned}$$ Let us consider $\theta_1 >- \theta_1^*$, we define $\theta_2^*(\theta_1)$ and $\theta_2^\delta(\theta_1)$ as $$\begin{aligned} \label{eq_theta_2_lim} &\theta_2^*(\theta_1) = \begin{cases} r_0\left(\frac{R_0}{2 \sqrt{d}}\right) \mathcal{V}(B(0,R_0)) & \text{when } \theta_1 \geq 0 \\ \sup_{\delta \in \left]0, \frac{R_0}{2 \sqrt{d}}\right[ : \theta_1 > -\theta_1^\delta} \left\{r_0(\delta) \mathcal{V}(B(0,R_0))(1 + \frac{\theta_1}{\theta_1^\delta})\right\} & \text{when } \theta_1 < 0 \end{cases} \\ &\theta_2^\delta(\theta_1) = \begin{cases} r_0\left(\frac{R_0}{2 \sqrt{d}}\right) \mathcal{V}(B(0,R_0)) & \text{when } \theta_1 \geq 0 \\ \left\{r_0(\delta) \mathcal{V}(B(0,R_0))(1 + \frac{\theta_1}{\theta_1^\delta})\right\} & \text{when } \theta_1 < 0 \nonumber \end{cases}. \end{aligned}$$ We know by Lemma [Lemma 6](#lem-surface-domino){reference-type="ref" reference="lem-surface-domino"} that for sufficiently small $\delta$ we have $\theta_1 \geq -\theta_1^\delta > -\theta_1^*$ and $\theta_2 \leq \theta_2^\delta(\theta_1) < \theta_2^*(\theta_1)$. In the case where $\theta_1 <0$ we need to consider a threshold t such that $$\begin{aligned} \frac{\theta_2}{\theta_1^\delta + \theta_1} \frac{\delta^d}{\mathcal{V}(B(0,R_0))} < t < \frac{1}{\theta_1}\left(\frac{\theta_2 \delta^d}{\mathcal{V}(B(0,R_0))} - r_0(\delta)\delta^d\right). \end{aligned}$$ We define the quantity $\rho_0$ as such $$\begin{aligned} \rho_0 = \begin{cases} r_0(\delta) \delta^d - \frac{\theta_2 \delta^d}{\mathcal{V}(B(0,R_0))} & \text{when } \theta_1 \geq 0 \\ \min \left\{ (\theta_1^\delta + \theta_1)t - \frac{\theta_2 \delta^d}{\mathcal{V}(B(0,R_0))} , \; r_0(\delta) \delta^d + \theta_1 t - \frac{\theta_2 \delta^d}{\mathcal{V}(B(0,R_0))} \right\} & \text{when } \theta_1 < 0 \end{cases}. \end{aligned}$$ With the conditions on $\delta$ and $t$, it guarantees that $\rho_0>0$. Under our assumptions, for any configuration $\omega \in \Omega_f$ that achieves the contour $\gamma$ we claim that $$\begin{aligned} \label{eq_peierls_energy} H_{\overline{\gamma}} (\omega) \geq |\overline{\gamma}_1| \delta^d + \rho_0 |\overline{\gamma}|. \end{aligned}$$ First let us prove the claim in the case where $\theta_1 <0$. With the condition $R_0 > 2\delta \sqrt{d}$ we know that we have a non negligible amount of empty tiles that are completely covered by the halo. Therefore using Lemmas [Lemma 5](#lem-domino-domino){reference-type="ref" reference="lem-domino-domino"} and [Lemma 6](#lem-surface-domino){reference-type="ref" reference="lem-surface-domino"} we have the following two lower bounds on the energy of the contour $$\begin{aligned} H_{\overline{\gamma}} (\omega) & \geq \begin{cases} |\overline{\gamma}_1| \delta^d + r_0 |\overline{\gamma}| \delta^d + \theta_1 \mathcal{S}_{\overline{\gamma}}(L(\omega)) - \theta_2 \chi_{\overline{\gamma}}(L(\omega)) \\ |\overline{\gamma}_1| \delta^d + (\theta_1^\delta + \theta_1 )\mathcal{S}_{\overline{\gamma}}(L(\omega)) - \theta_2 \chi_{\overline{\gamma}}(L(\omega)) \end{cases} \\ & \geq \begin{cases} |\overline{\gamma}_1| \delta^d + r_0 |\overline{\gamma}| \delta^d + \theta_1 \mathcal{S}_{\overline{\gamma}}(L(\omega)) - \theta_2 \frac{|\overline{\gamma}| \delta^d}{\mathcal{V}(B(0,R_0))} \\ |\overline{\gamma}_1| \delta^d + (\theta_1^\delta + \theta_1 )\mathcal{S}_{\overline{\gamma}}(L(\omega)) - \theta_2 \frac{|\overline{\gamma}| \delta^d}{\mathcal{V}(B(0,R_0))} \end{cases} \text{ by Lemma \ref{lem-chi}}. \end{aligned}$$ Depending on the value of the surface inside the contour, one lower bound will be more preferable than the other. Since $\theta_1^\delta + \theta_1>0$ and given the threshold $t$ that verifies our assumption we have $$\begin{aligned} H_{\overline{\gamma}} (\omega) & \geq \begin{cases} |\overline{\gamma}_1| \delta^d + \left(r_0 \delta^d + \theta_1 t - \frac{\theta_2 \delta^d}{\mathcal{V}(B(0,R_0))} \right) |\overline{\gamma}| & \text{ if } \mathcal{S}_{\overline{\gamma}}(L(\omega)) \leq t |\overline{\gamma}|\\ |\overline{\gamma}_1| \delta^d + (\theta_1^\delta + \theta_1 ) t |\overline{\gamma}| - \theta_2 \frac{|\overline{\gamma}| \delta^d}{\mathcal{V}(B(0,R_0))} & \text{ if } \mathcal{S}_{\overline{\gamma}}(L(\omega)) > t |\overline{\gamma}| \end{cases}. \end{aligned}$$ In either cases, we have the desired lower boundary on the energy of a contour. Let us turn to the second case where $\theta_1 \geq 0$. It is even easier since we can simply drop the contribution of the surface in the energy and therefore we have $$\begin{aligned} H_{\overline{\gamma}} (\omega) & \geq |\overline{\gamma}_1| \delta^d + \left( r_0 \delta^d - \frac{\theta_2 \delta^d}{\mathcal{V}(B(0,R_0))}\right) |\overline{\gamma}| = |\overline{\gamma}_1| \delta^d + \rho_0 |\overline{\gamma}|. \end{aligned}$$ This finishes the proof of the claim. We can now use inequality [\[eq_peierls_energy\]](#eq_peierls_energy){reference-type="eqref" reference="eq_peierls_energy"} to get the following upper bound $$\begin{aligned} I_\gamma &\leq e^{-\beta(\delta^d |\overline{\gamma}_1| + \rho_0|\overline{\gamma}|) } \int \mathbbm{1}_{\left(\forall i \in \overline{\gamma}, \sigma(\omega,i)=\sigma_i \right)} \Pi_{\widehat{\gamma}}^z(d\omega) \\ & \leq e^{-\beta(\delta^d |\overline{\gamma}_1| + \rho_0|\overline{\gamma}|) } (1-e^{z\delta^d})^{|\overline{\gamma}_1|} e^{z\delta^d |\overline{\gamma}_0|} \\ & \leq g_0^{|\overline{\gamma}_0|} g_1^{|\overline{\gamma}_1|} e^{-\beta \rho_0 |\overline{\gamma}|}. \end{aligned}$$ By a direct computation we have $$\begin{aligned} \frac{\partial I_\gamma}{\partial z} = -|\overline{\gamma}|\delta^d I_\gamma + \frac{1}{z} \int N_{\widehat{\gamma}}(\omega) e^{-\beta H_{\overline{\gamma}}(\omega)} \mathbbm{1}_{\{ \forall i \in \overline{\gamma}, \sigma(\omega, i) = \sigma_i \}} \Pi_{\widehat{\gamma}}^z (d\omega). \end{aligned}$$ Using the bound on $I_\gamma$ we have the desired control on the first term. For the second term we need to control the mean number of points in the contour. By using the lower bound on the energy in a contour [\[eq_peierls_energy\]](#eq_peierls_energy){reference-type="eqref" reference="eq_peierls_energy"} and Lemma [Lemma 8](#lem-ratio-0-1){reference-type="ref" reference="lem-ratio-0-1"} we have $$\begin{aligned} \int N_{\widehat{\gamma}}(\omega) e^{-\beta H_{\overline{\gamma}}(\omega)}\mathbbm{1}_{\left(\forall i \in \overline{\gamma}, \sigma(\omega,i)=\sigma_i \right)} \Pi_{\widehat{\gamma}}^z(d\omega) & \leq e^{-\beta ( \rho_0|\overline{\gamma}| + |\overline{\gamma}_1|\delta^d ) } \int N_{\widehat{\gamma}}(\omega)\mathbbm{1}_{\left(\forall i \in \overline{\gamma}, \sigma(\omega,i)=\sigma_i \right)} \Pi_{\widehat{\gamma}}^z(d\omega) \\ & \leq e^{-\beta ( \rho_0 |\overline{\gamma}| + |\overline{\gamma}_1|\delta^d) } \sum_{i \in \overline{\gamma}_1} \int N_{T_i}(\omega) \mathbbm{1}_{\left(\forall i \in \overline{\gamma}, \sigma(\omega,i)=\sigma_i \right)} \Pi_{\widehat{\gamma}}^z(d\omega) \\ & \leq e^{-\beta ( \rho_0|\overline{\gamma}| + |\overline{\gamma}_1|\delta^d) } g_0^{|\overline{\gamma}_0|} |\overline{\gamma}_1| z \delta^d (1-e^{-z\delta^d})^{|\overline{\gamma}_1|-1} \\ & \leq \frac{(1-r_1)z \delta^d}{1- e^{-z\delta^d}}|\overline{\gamma}| g_0^{|\overline{\gamma}_0|} g_1^{|\overline{\gamma}_1|} e^{- \beta \rho_0 |\overline{\gamma}|}. \end{aligned}$$ By combining those inequalities we get $$\begin{aligned} \left| \frac{\partial I_\gamma}{\partial z} \right| \leq \left(1 + \frac{1-r_1}{1-e^{-z\delta^d}}\right) |\overline{\gamma}|\delta^d g_0^{|\overline{\gamma}_0|} g_1^{|\overline{\gamma}_1|} e^{- \beta \rho_0 |\overline{\gamma}|}. \end{aligned}$$ ◻ ## Truncated weights and pressure {#subsection4.5} In order to have $\tau$-stable contours, it is needed that the ratio of partition functions of the interior has to be small. This ratio is more likely to be large when the volume of the interior is large. Therefore depending on the parameters $z,\beta$ the weights of some contours might be unstable because the volume of the interior is too large. In order to deal with those instability we will truncate the weights. We follow mainly the ideas and the presentation given in Chapter 7 of [@Velenik]. We introduce an arbitrary cut-off function $\kappa: \mathbbm{R}\rightarrow [0,1]$ that satisfy the following properties : $\kappa(s) = 1$ if $s \leq \frac{\rho_0}{8}$, $\kappa(s) = 0$ if $s \geq \frac{\rho_0}{4}$ and $\kappa$ is $\mathcal{C}^1$. Therefore such cut-off function $\kappa$ satisfies $\| \kappa' \| = \sup_{\mathbbm{R}} |\kappa'(s)| < + \infty$. We construct step by step the truncated weights and pressure according to the volume of the interior. Contours with no interior does not have a ratio of partition functions introduced in the expression of its weight, thus those contours are stable. We can define the truncated quantities associated to $n=0$ starting by the truncated pressure $$\begin{aligned} \widehat{\psi}_0^\#:= \frac{\ln(g_\#)}{\beta \delta^d} . \end{aligned}$$ We define the truncated weights for contour $\gamma$ of class $0$ as $$\begin{aligned} \widehat{w}_{\gamma}^\#= w_{\gamma}^\#= g_\#^{-|\overline{\gamma}|} I_{\gamma}. \end{aligned}$$ Now we suppose that the truncated weights are well defined for contours $\gamma$ of class $k \leq n$. We define $\widehat{\Phi}_n^\#$ with the following polymer development $$\begin{aligned} \widehat{\Phi}_n^\#(\Lambda) := \sum_{\Gamma \in \mathcal{C}_n^\#(\Lambda) } \prod_{\gamma \in \Gamma} \widehat{w}_{\gamma}^\#. \end{aligned}$$ We can define the truncated partition function as $$\begin{aligned} \widehat{Z}_n^\#(\Lambda) := g_\#^{|\Lambda|} \widehat{\Phi}_n^\#(\Lambda). \end{aligned}$$ With $\Lambda_k = [-\nicefrac{k}{2}, \nicefrac{k}{2}]^d$, the truncated pressure at rank n is given by $$\begin{aligned} \widehat{\psi}_n^\#& := \lim_{k \rightarrow +\infty} \frac{1}{\beta \delta^d |\Lambda_k|} \ln(\widehat{Z}_n^\#(\Lambda_k)) \\ & = \widehat{\psi}_0^\#+ \lim_{k \rightarrow +\infty} \frac{1}{\beta \delta^d |\Lambda_k|} \ln(\widehat{\Phi}_n^\#(\Lambda_k)). \end{aligned}$$ This limit exists as it is shown in Section 7.4.3 of [@Velenik]. We can also note that $\widehat{\psi}_n^\#\in [\widehat{\psi}_0^\#, \psi]$ and that the sequence of truncated pressure $(\widehat{\psi}_n^\#)_{n \in \mathbbm{N}}$ is increasing. **Definition 11**. *Given the truncated weight of contours $\gamma$ of class $k$ smaller than $n$ (and so the truncated pressure $\widehat{\psi}_n^\#$), the truncated weight of a contour $\gamma$ of class $n+1$ is defined by $$\begin{aligned} \widehat{w}_{\gamma}^\#= g_\#^{-|\overline{\gamma}|} I_{\gamma} \, \kappa \left( (\widehat{\psi}_n^{\#^*} - \widehat{\psi}_n^\#) \delta^d |\mathop{\mathrm{Int}}_{\#^*} \gamma|^{\frac{1}{d}} \right) \frac{Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#^*}}{Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#}}. \end{aligned}$$* Intuitively, the reason we do this cut-off is to remove contours that are unstable. Since the cut-off function verify $0 \leq \kappa \leq 1$ we have $\widehat{w}_{\gamma}^\#\leq w_{\gamma}^\#$. Other quantities that are of interest for what will come later are, $\widehat{\psi}_n := \max( \widehat{\psi}_n^0, \widehat{\psi}_n^1)$ and $a_n^\#:= \widehat{\psi}_n - \widehat{\psi}_n^\#$. By definition, we have $a_n^\#\geq 0$ and for all contour $\gamma$ of class $n+1$ we have the following implication $$\begin{aligned} a_n^\#\delta^d (n+1)^{\frac{1}{d}} \leq \frac{\rho_0}{8} \implies \widehat{w}_{\gamma}^\#= w_{\gamma}^\#. \end{aligned}$$ Before we go further we will proceed into a re-parametrisation of the model for fixed $\theta_1$ and $\theta_2$ that verify the assumption of Proposition [Proposition 3](#prop_peierls){reference-type="ref" reference="prop_peierls"}. We change the parameters from $(\beta, z)$ to $(\beta, s)$ where $s = z/\beta$. We set $a = \min \{ \frac{2}{1-r_1}, e^{-c \beta}\}$ where $r_1$ is given in Lemma [Lemma 8](#lem-ratio-0-1){reference-type="ref" reference="lem-ratio-0-1"} and $c>0$. For all $\beta > 0$ we set $$s_\beta = \frac{\ln(1+e^{\beta \delta^d})}{\beta \delta^d}$$ such that we have $g_0 =g_1$. Furthermore, we define the following open interval $$U_\beta = \left( \frac{\ln(1+e^{\beta \delta^d-a})}{\beta \delta^d}, \frac{\ln(1+e^{\beta \delta^d + a})}{\beta \delta^d} \right)$$ such that for all $s \in U_\beta$ we have $e^{-a} \leq \frac{g_0}{g_1} \leq e^{a}$. So, according to Proposition [Proposition 3](#prop_peierls){reference-type="ref" reference="prop_peierls"}, for any $\beta>0$ and $s \in U_\beta$ we have $$w_\gamma^\#\leq e^{-(\beta \rho_0 -2)|\overline{\gamma}|} \frac{Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#^*}}{Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#}}.$$ We can already see that for contours of class $0$ the weights are $\tau$-stable as long as $\beta \rho_0 > 2$. In the following proposition we will show that truncated weights are $\tau$-stable and that we have a good bound on the derivative of the truncated weights that are useful in the study of the regularity of the pressure. **Proposition 4**. *Let $\tau := \frac{1}{2}\beta \rho_0 - 8$. There exists $D\geq 1$ and $0<\beta_0< \infty$ such that all $\beta > \beta_0$ there exist $C_1>0$ and $C_2>0$ where the following statements hold for any $\#$ and $n\ge 0$.* 1. *(Bounds on the truncated weights) For all $k \leq n$, the truncated weights of each contours $\gamma$ of class $k$ is $\tau$-stable uniformly on $U_\beta$ : $$\widehat{w}_{\gamma}^\#\leq e^{-\tau |\overline{\gamma}|} \label{truncated_weights_tau_stability}$$ and $$\label{implication_bound_a_n_to_stability} a_n^\#\delta^d |\mathop{\mathrm{Int}}\gamma|^{\frac{1}{d}} \leq \frac{\rho_0 }{16} \implies \widehat{w}_{\gamma}^\#= w_{\gamma}^\#.$$ Moreover, $s \mapsto \widehat{w}_{\gamma}^\#$ is $\mathcal{C}^1$ and uniformly on $U_\beta$ it verifies $$\left|\frac{\partial \widehat{w}_{\gamma}^\#}{\partial s} \right| \leq D |\overline{\gamma}|^{\frac{d}{d-1}} e^{-\tau |\overline{\gamma}|}. \label{bound_truncated_weight_derivative}$$* 2. *(Bounds on the partition functions) Let us assume that $\Lambda \subset \mathbbm{Z}^d$ and $|\Lambda| \leq n+1$. Then uniformly on $U_\beta$ we have $$\begin{aligned} Z_\Lambda^\#& \leq e^{\beta \delta^d \widehat{\psi}_{n} |\Lambda| + 2 |\partial_{ext} \Lambda|}, \label{bound_partition_function}\\ \left| \frac{\partial Z_\Lambda^\#}{\partial s} \right| & \leq \left( C_1 \beta \delta^d |\Lambda| + C_2 |\partial_{ext} \Lambda| \right) e^{\beta \delta^d \widehat{\psi}_n |\Lambda| + 2 |\partial_{ext} \Lambda|}. \label{bound_derivative_partition_function} \end{aligned}$$* This proposition is the key tool for applying the PSZ theory and is very similar to Proposition 7.34 in [@Velenik] with some adaptations to the present setting. Due to these small adaptations and also to be self-contained, the proof of the proposition is given in the Annex B. ## Proof of Proposition [Proposition 1](#prop_first_order_transition){reference-type="ref" reference="prop_first_order_transition"} {#subsection4.6} According to Proposition [Proposition 4](#prop_tau_stability_truncated_weights){reference-type="ref" reference="prop_tau_stability_truncated_weights"} we know that the weights are $\tau$-stable with $\tau$ as large as we want provided that $\beta$ is large enough. So for $\beta$ large enough, for any $\Lambda \subset \mathbbm{Z}^d$ we can define $$\widehat{\Phi}^\#(\Lambda) := \sum_{\Gamma \in \mathcal{C}^\#(\Lambda)} \prod_{\gamma \in \Gamma} \widehat{w}^\#_\gamma$$ and based on standard cluster expansion results, recalled in Annex A, it is also possible to have a polymer development of $\log(\widehat{\Phi}^\#(\Lambda))$. In particular by Theorem [Theorem 11](#theorem_cluster_expansion){reference-type="ref" reference="theorem_cluster_expansion"} we have $$\begin{aligned} \widehat{\Phi}^\#(\Lambda) = e^{\beta \delta^d f^\#|\Lambda| + \Delta^\#(\Lambda)} \end{aligned}$$ where $f^\#$ and $\Delta^\#_\Lambda$ are $\mathcal{C}^1$ in $U_\beta$. Furthermore, according to Theorem [Theorem 11](#theorem_cluster_expansion){reference-type="ref" reference="theorem_cluster_expansion"} we also have $$\begin{aligned} &|f^\#| \leq \eta(\tau, l_0),& &|\Delta^\#_\Lambda| \leq \eta(\tau, l_0) |\partial_{ext} \Lambda | \\ &\left| \frac{\partial f^\#}{\partial s} \right| \leq D \eta(\tau, l_0),& &\left| \frac{\partial \Delta^\#_\Lambda}{\partial s} \right| \leq D \eta(\tau, l_0) |\partial_{ext} \Lambda|, \end{aligned}$$ where $D>0$ comes from Proposition [Proposition 4](#prop_tau_stability_truncated_weights){reference-type="ref" reference="prop_tau_stability_truncated_weights"} and the constant $\eta(\tau, l_0)$, used intensively in Annex A, is equal to $2e^{- \nicefrac{\tau l_0}{3}}$, where $l_0$ is the smallest size of a non-empty contour (it is a positive integer). Therefore the truncated partition function with boundary $\#$ is given by $$\begin{aligned} \label{eq_Z_cluster} \widehat{Z}_\Lambda^\#:= g_\#^{|\Lambda|} \widehat{\Phi}^\#(\Lambda) = e^{\beta \delta^d ( \widehat{\psi}_0^\#+ f^\#)|\Lambda| + \Delta^\#(\Lambda) } \end{aligned}$$ and the truncated pressure associated to the boundary $\#$ is $$\begin{aligned} \widehat{\psi}^\#:= \widehat{\psi}_0^\#+ f^\#. \end{aligned}$$ We can also obtain $\widehat{\psi}^\#$ as the limit of the sequence of truncated pressure $(\widehat{\psi}_n^\#)_{n \in \mathbbm{N}}$ when $n$ is going to infinity. So we define the truncated pressure as $\widehat{\psi} : = \max\{ \widehat{\psi}^{(0)}, \widehat{\psi}^{(1)}\}$ and we have that $a^\#: = \widehat{\psi} - \widehat{\psi}^\#= \lim_{n \to \infty} a_n^\#$. Similarly to the statement [\[implication_bound_a\_n_to_stability\]](#implication_bound_a_n_to_stability){reference-type="eqref" reference="implication_bound_a_n_to_stability"} in Proposition [Proposition 4](#prop_tau_stability_truncated_weights){reference-type="ref" reference="prop_tau_stability_truncated_weights"}, for any contour $\gamma$ we have the following implication $$a^\#\delta^d |\mathop{\mathrm{Int}}\gamma |^{\frac{1}{d}} \leq \frac{\rho_0}{16} \implies \widehat{w}_\gamma^\#= w_\gamma^\#.$$ Therefore when $a^\#=0$ all the truncated weights are equal to the actual weight of the model and thus for all $\Lambda \subset \mathbbm{Z}^d$ we have $\widehat{Z}_\Lambda^\#= Z_\Lambda^\#$ and with Lemma [Lemma 4](#lem_pression_free_boundary){reference-type="ref" reference="lem_pression_free_boundary"} we have $$\psi = \widehat{\psi} = \max \{ \widehat{\psi}^{(1)}, \widehat{\psi}^{(0)}\}.$$ Therefore we can extract properties of the pressure from the study of the truncated pressures. For $s \in U_\beta$ let us define the gap $G(s)$ between the two truncated pressures $$G(s):= \widehat{\psi}^{(1)} - \widehat{\psi}^{(0)} = (s-1) + \frac{\ln(1-e^{-s\beta \delta^d})}{\beta \delta^d} + f^{(1)} - f^{(0)}.$$ Previously, we have set $a =\min \{\frac{2}{1-r_1}, e^{-c\beta} \}$ with the condition that $c>0$. We are going to be more precise and take $c < \frac{1}{6} \rho_0 l_0$. Hence for sufficiently large $\beta$ we have $$\begin{aligned} G(s_\beta^{-}) = -\frac{a}{\delta^d} + f^{(1)} - f^{(0)} \leq -\frac{a}{\delta^d} + 2 \eta(\tau, l_0) < 0 \\ G(s_\beta^{+}) = \frac{a}{\delta^d} + f^{(1)} - f^{(0)} \geq \frac{a}{\delta^d} - 2 \eta(\tau, l_0) > 0 \end{aligned}$$ where $s_\beta^{-} = \frac{\ln(1+e^{\beta \delta^d - a})}{\beta \delta^d}$ and $s_\beta^{+} = \frac{\ln(1+e^{\beta \delta^d + a})}{\beta \delta^d}$. Furthermore, for $s \in U_\beta$ and for sufficiently large $\beta$ we have $$\begin{aligned} \frac{\partial G}{\partial s}(s) = \frac{1}{e^{s\beta \delta^d}-1} + \frac{\partial f^{(1)}}{\partial s} - \frac{\partial f^{(0)}}{\partial s} > 1 - 2D\eta(\tau, l_0) > 0. \end{aligned}$$ This ensures the existence of an unique $s_\beta^c \in U_\beta$ such that $$\begin{aligned} \widehat{\psi} = \begin{cases} \widehat{\psi}^{(0)} & \text{ when } s \in [s_\beta^{-}, s_\beta^c] \\ \widehat{\psi}^{(1)} & \text{ when } s \in [s_\beta^c, s_\beta^{+}] \end{cases} \end{aligned}$$ and also for all $s \in U_\beta$ (and thus also for $s_\beta^c$) we have $$\label{ineq_derivee_pression} \frac{\partial \widehat{\psi}^{(1)}}{\partial s}(s) > \frac{\partial \widehat{\psi}^{(0)}}{\partial s}(s).$$ Furthermore, we can observe that $$\begin{aligned} z_\beta^- - \beta := \beta s_\beta^- - \beta = -\frac{a}{ \delta^d } + \frac{1}{\delta^d}\ln(1 + e^{-\beta \delta^d + a }) = -\frac{a}{\delta^d } + o(a(\beta)) \\ z_\beta^+ - \beta := \beta s_\beta^+ - \beta = \frac{a}{ \delta^d } + \frac{1}{\delta^d}\ln(1 + e^{-\beta \delta^d - a }) = \frac{a}{\delta^d } + o(a(\beta)) \end{aligned}$$ Therefore we have that $|z_\beta^c - \beta | = O(e^{-c\beta})$ when $\beta$ tends to infinity where $z_\beta^c := \beta s_\beta^c$ and $0<c<\nicefrac{\rho_0 l_0}{6}$. This last part proves the statement in Theorem [Theorem 1](#thm_quermass_transition){reference-type="ref" reference="thm_quermass_transition"} about the exponential decay of the difference $|\beta - z_\beta^c|$. For all $n\geq1$, we define the empirical field $\overline{P}_{\Lambda_n}^\#$ as the probability measure on $\Omega$ such that for any positive test function $f$ $$\int f(\omega) \overline{P}^\#_{\Lambda_n}(d\omega) = \frac{1}{|\widehat{\Lambda}_n| } \int_{\widehat{\Lambda}_n} \int f(\tau_u(\omega)) \widehat{P}^\#_{\Lambda_n}(d\omega)du \quad \text{where } \widehat{P}_{\Lambda_n}^\#= \bigotimes_{i \in \mathbbm{Z}^d} P_{\tau_{2ni}(\Lambda_n)}^\#.$$ The empirical field can be seen as a stationarization of $P^\#_{\Lambda_n}$ and therefore any accumulation point of the sequence $(\overline{P}_{\Lambda_n}^\#)_{n \in \mathbbm{N}}$ is necessarily stationary. Furthermore, we know that the Quermass interaction is stable and has a finite range interaction. Therefore there exists a sub-sequence of the empirical field $(\overline{P}_{\Lambda_n}^\#)_{n \in \mathbbm{N}}$ that converges to $P^\#$ for the local convergence topology defined as the smallest topology on the space of probability measures on $\Omega$ such that for all tame functions $f$ (i.e. $f : \Omega \rightarrow \mathbbm{R}$ such that $\exists \Delta \subset \mathbbm{R}^d$ bounded, $\exists A \geq 0, \forall \omega \in \Omega, f(\omega)= f(\omega_\Delta)$ and $|f| \leq A(1+N_\Delta)$) the map $P \rightarrow \int fdP$ is continuous. Moreover this limit is a Gibbs measure as it satisfies the DLR equations. The proof of these statements is similar to the proof of Theorem 1 in [@MiniCours]. For all $\beta \geq \beta_0$ and for $z = z_\beta^c$ we have by a direct computation $$\label{eq_derivee_fct_partition_1} \frac{\partial}{\partial z} \log \widehat{Z}_{\Lambda_n}^\#= \frac{\partial}{\partial z} \log Z_{\Lambda_n}^\#= - \delta^d |{\Lambda_n}| + \frac{1}{z}E_{P_{\Lambda_n}^\#}(N_{\widehat{\Lambda}_n}).$$ On the other hand we know using [\[eq_Z\_cluster\]](#eq_Z_cluster){reference-type="eqref" reference="eq_Z_cluster"} that $$\label{eq_derivee_fct_partition_2} \frac{\partial}{\partial z} \log \widehat{Z}_{\Lambda_n}^\#= \beta |\Lambda_n| \delta^d \frac{\partial \widehat{\Psi}^\#}{\partial z} + \frac{\partial \Delta^\#_{\Lambda_n}}{\partial z}.$$ By combining [\[eq_derivee_fct_partition_1\]](#eq_derivee_fct_partition_1){reference-type="eqref" reference="eq_derivee_fct_partition_1"} and [\[eq_derivee_fct_partition_2\]](#eq_derivee_fct_partition_2){reference-type="eqref" reference="eq_derivee_fct_partition_2"} we get the following relationship $$\frac{E_{P_{\Lambda_n}^\#}(N_{\widehat{\Lambda}_n})}{|\Lambda_n| \delta^d} = z + z\beta\frac{\partial \psi^\#}{\partial z}(z_\beta^c) + \frac{z}{|\Lambda_n| \delta^d} \frac{\partial \Delta^\#_{\Lambda_n}}{\partial z}.$$ Furthermore, using Theorem [Theorem 11](#theorem_cluster_expansion){reference-type="ref" reference="theorem_cluster_expansion"} we know that $$\left| \frac{\partial \Delta^\#_{\Lambda_n}}{\partial z} \right| \leq D \eta(\tau, l_0) |\partial_{ext} \Lambda_n | \implies \frac{1}{|\Lambda_n|} \frac{\partial \Delta^\#_{\Lambda_n}}{\partial z} \underset{n \to \infty}{\rightarrow} 0.$$ The empirical field is stationary and using the local convergence we have $$\frac{E_{\overline{P}_{\Lambda_n}^\#}(N_{\widehat{\Lambda}_n})}{|\Lambda_n| \delta^d} = E_{\overline{P}_{\Lambda_n}^\#}(N_{[0,1]^d}) \underset{n \to \infty}{\rightarrow} \rho(P^\#).$$ On the other hand, we have by construction of the empirical field that $$E_{\overline{P}_{\Lambda_n}^\#}(N_{\widehat{\Lambda}_n}) = E_{P_{\Lambda_n}^\#}(N_{\widehat{\Lambda}_n}).$$ In summary, when we take limit when $n$ goes to infinity we have that $$\rho(P^\#) = z + z \beta \frac{\partial \psi^\#}{\partial z}(z_\beta^c)$$ and therefore using [\[ineq_derivee_pression\]](#ineq_derivee_pression){reference-type="eqref" reference="ineq_derivee_pression"} we have that $\rho(P^{(1)}) > \rho(P^{(0)})$. # Acknowledgement {#acknowledgement .unnumbered} This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01), the ANR projects PPPP (ANR-16-CE40-0016) and RANDOM (ANR-19-CE24-0014) and by the CNRS GdR 3477 GeoSto. # Annex A : Cluster Expansion {#Annex1} We consider a collection of contours $\mathcal{C}$ and for all $\Lambda \subset \mathbbm{Z}^d$, $\mathcal{C}(\Lambda)$ a sub-collection of contours $\gamma$ such that $\overline{\gamma} \subset \Lambda$. For each contour $\gamma$ in such collection there are weights $w_\gamma$ invariant by translation. We set $l_0 = \min \{ |\overline{\gamma}|, \gamma \in \mathcal{C} \}$ and $\eta (\tau, l_0) = 2 \exp(-\nicefrac{\tau l_0}{3})$. A set of contours $\Gamma = \{ \gamma_1, \dots, \gamma_n\} \in \mathcal{C}$, is said to be geomtrically compatible if for all $i,j \in \{1,\dots, n\}, i \neq j$ we have $d_\infty(\gamma_i, \gamma_j)\geq 1$. We define the polymer development associated to those weights as, for all $\Lambda \subset \mathbbm{Z}^d$ $$\begin{aligned} \Phi(\Lambda) = \sum_{\substack{\Gamma \in \mathcal{C}(\Lambda) \\ \text{geometrically compatible}}} \prod_{\gamma \in \Gamma} w_\gamma. \end{aligned}$$ A collection $C=\{\gamma_1, \cdots, \gamma_n \}$ is said to be decomposable if the support $\overline{C} = \bigcup_{\gamma \in C} \overline{\gamma}$ is not simply connected. A cluster, denoted by $X$, is a non-decomposable finite multiset of contours such that a same contour can appear multiple times and we define $\overline{X}: = \bigcup_{\gamma \in X} \overline{\gamma}$. The cluster expansion for $\ln \Phi(\Lambda)$, if it converges, is given by $$\ln \Phi(\Lambda) = \sum_{\substack{X : \overline{X} \subset \Lambda}} \Psi(X)$$ where $\Psi(X) := \alpha(X) \prod_{\gamma \in X} w_\gamma$. We have this combinatorial term $\alpha$ whose expression is given by $$\alpha(X)= \left\{\prod_{\gamma \in \mathcal{C}(\Lambda)} \frac{1}{n_X(\gamma)!} \right\} \left\{ \sum_{\substack{G \subset G_n \\ \text{connected} }} \prod_{\{i,j\} \in G} \zeta(\gamma_i, \gamma_j) \right\}$$ where $n_X(\gamma)$ is the number of times a contour $\gamma$ appears in the cluster $X$, $G_n = (V_n, E_n)$ is an undirected complete graph on $V_n = \{1, \cdots, n\}$ and $\zeta$ is defined as$$\zeta(\gamma, \gamma') = \begin{cases} 0 & \text{if } \gamma, \gamma' \text{ are geometrically compatible} \\ -1 & \text{otherwise} \end{cases}.$$ Under the assumption that the weights are $\tau$-stable, a sufficient condition for the convergence of the cluster expansion is $$\sum_{\substack{\gamma \in \mathcal{C} \\ 0 \in \overline{\gamma}}} e^{-\tau |\overline{\gamma}|} e^{3^d |\overline{\gamma}|} \leq 1.$$ In practice we will take a larger value for $\tau$ such that the stronger assumption of Lemma [Lemma 9](#lem_condition_convergence_derivative_serie){reference-type="ref" reference="lem_condition_convergence_derivative_serie"} is verified. The following lemmas correspond to Lemma 7.30 and Lemma 7.31 in [@Velenik]. **Lemma 9**. *There exists $\tau_0 >0$ such that when $\tau > \tau_0$ $$\sum_{\gamma \in \mathcal{C} : 0 \in \overline{\gamma}} |\overline{\gamma}|^{\nicefrac{d}{d-1}} e^{-(\nicefrac{\tau}{2}-1) |\overline{\gamma}|} e^{3^d |\overline{\gamma}|} \leq \eta(\tau, l_0) \leq 1.$$* **Lemma 10**. *Let us assume that the weights are $\tau$-stable for $\tau > \tau_0$. Then for $L \geq l_0$ $$\sum_{\substack{X : 0 \in \overline{X} \\ |\overline{X}| \geq L} } |\Psi(X)| \leq e^{-\frac{\tau L}{2}}.$$* Finally we have the following theorem which plays a crucial role in the proofs of Proposition [Proposition 1](#prop_first_order_transition){reference-type="ref" reference="prop_first_order_transition"} and [Proposition 4](#prop_tau_stability_truncated_weights){reference-type="ref" reference="prop_tau_stability_truncated_weights"}. **Theorem 11**. *Assume that, for all $\gamma \in \mathcal{C}$, the weight $w_{\gamma}$ is $\mathcal{C}^1$ in a parameter $s\in (a,b)$ and that uniformly on $(a,b)$, $$w_{\gamma} \leq e^{-\tau |\overline{\gamma}|}, \quad \quad \left| \frac{ d w_{\gamma}}{d s} \right| \leq D|\overline{\gamma}|^{\nicefrac{d}{d-1}} e^{-\tau |\overline{\gamma}|},$$ where $D \geq 1$ is a constant. Then there exists $\tau_1 = \tau_1(D, d) < \infty$ such that the following holds. If $\tau > \tau_1$, the pressure $g$ is given by the following absolutely convergent series, $$g = \sum_{X : 0 \in \overline{X}} \frac{1}{|\overline{X}|} \Psi(X)$$ where the sum is over clusters $X$ made of contours $\gamma \in \mathcal{C}$ and $\overline{X} = \bigcup_{\gamma \in X} \overline{\gamma}$. Moreover, $$|g| \leq \eta(\tau, l_0) \leq 1$$ and for all $\Lambda \subset \mathbbm{Z}^d$ finite, $g$ provides the volume contribution to $\log \Phi(\Lambda)$, in the sense that $$\Phi(\Lambda) = \exp(g|\Lambda| + \Delta_\Lambda)$$ where $\Delta_\Lambda$ is a boundary term : $$|\Delta_\Lambda| \leq \eta(\tau, l_0)|\partial_{ext} \Lambda|.$$ Finally, $g$ and $\Delta_\Lambda$ are also $\mathcal{C}^1$ in $s \in (a,b)$; its derivative equals $$\frac{dg}{ds} = \sum_{X : 0 \in \overline{X}} \frac{1}{|\overline{X}|} \frac{d\Psi(X)}{ds}$$ and $$\left| \frac{dg}{ds}\right| \leq D \eta(\tau, l_0), \quad \quad \left| \frac{d\Delta_\Lambda}{ds}\right| \leq D \eta(\tau, l_0) |\partial_{ext} \Lambda|.$$* This theorem is similar to Theorem 7.29 in [@Velenik], the only difference being is that the following statement is not included $$\left|\frac{d \Delta_\Lambda}{d s} \right| \leq D \eta(\tau, l_0) |\partial_{ext} \Lambda|.$$ Following the computations in [@Velenik] we find $$\frac{d \Delta_\Lambda}{d s} = \sum_{i \in \Lambda} \sum_{X : i \in \overline{X} \not\subset \Lambda} \frac{1}{|\overline{X}|} \frac{d\Psi}{d s}(X).$$ Whenever $i \in \overline{X} \not\subset \Lambda$ we know that $\overline{X} \cap \partial_{ext} \Lambda \neq \emptyset$ and since the weights are invariant by translation we have $$\begin{aligned} \left|\frac{d \Delta_\Lambda}{d s}\right| \leq |\partial_{ext} \Lambda| \max_{j \in \partial_{ext} \Lambda} \sum_{X : j \in \overline{X}} \left|\frac{d\Psi}{d s}(X) \right| = |\partial_{ext} \Lambda| \sum_{X : 0 \in \overline{X}} \left|\frac{d\Psi}{d s}(X) \right|. \end{aligned}$$ We can show that for any cluster $X$ we have $$\left|\frac{d \Psi}{d s} (X)\right| \leq |\overline{\Psi}(X)|$$ where $$\overline{\Psi}(X) = \alpha(X) \prod_{\gamma \in X} \overline{w}_\gamma \quad \text{and} \quad \overline{w}_\gamma = D|\overline{\gamma}|^{\nicefrac{d}{d-1}} e^{-(\tau - 1 )|\overline{\gamma}|}.$$ With classical results on cluster expansion we can show that $$\sum_{X : 0 \in \overline{X}} | \overline{\Psi}(X)| \leq \sum_{\gamma \in \mathcal{C} : 0 \in \overline{\gamma}} \overline{w}_\gamma e^{3^d |\overline{\gamma}|}.$$ Therefore with Lemma [Lemma 9](#lem_condition_convergence_derivative_serie){reference-type="ref" reference="lem_condition_convergence_derivative_serie"} we have the desired control on the derivative of the boundary term. # Annex B : Proof of Proposition [Proposition 4](#prop_tau_stability_truncated_weights){reference-type="ref" reference="prop_tau_stability_truncated_weights"} {#Annex2} Before starting the proof by induction, we need to fix the some constants and in particular the quantity $\beta$ which has to be sufficiently large. Recall that $\eta(\tau, l_0) := 2e^{-\frac{\tau l_0}{3}}$, where $l_0$ is the minimum size of non empty contour. We set $D := (2+2K)\delta^d \beta +4\beta + 4 \delta^d \| \kappa' \| + C_1\beta \delta^d + C_2$ where $C_1(\beta) := e + \sup_{s \in U_\beta} |\nicefrac{\widehat{\psi}_0^\#}{s}|$ and $C_2(\beta) := \sup_{s \in U_\beta} \nicefrac{1}{s}$ and we choose $\beta \geq 1$ sufficiently large such that $$\begin{aligned} &\tau > \tau_0(d) \text{ where $\tau_0$ is defined as in Lemma \ref{lem_condition_convergence_derivative_serie}.} \label{condition_beta_1}\\ &D \eta(\tau, l_0) \leq 1 \label{condition_beta_2}\\ &\forall k \in \mathbbm{N}, \; 2 \beta^{-1} k^{\nicefrac{1}{d}} \exp{(-\frac{\tau k^{\nicefrac{d-1}{d}}}{2})} \leq \frac{\rho_0 }{16} \label{condition_beta_3}\\ &\forall x>0, \; \beta^{-1} \delta^{-d} \exp{(-\max\{(\nicefrac{\rho_0 }{16 \delta^d x})^{d-1}, l_0 \} \frac{\tau}{2})} \leq \frac{x}{2}. \label{condition_beta_5} \end{aligned}$$ Let us prove the proposition for $n=0$. Let $\overline{\gamma}$ be a contour of class 0. For $s \in U_\beta$ and using the upper bound on $I_{\gamma}$ in Proposition [Proposition 3](#prop_peierls){reference-type="ref" reference="prop_peierls"} we have $$\begin{aligned} \widehat{w}_{\gamma}^\#\leq e^{-\tau |\overline{\gamma}|}. \end{aligned}$$ For the derivative we have $$\frac{\partial \widehat{w}_\gamma^\#}{\partial z} = \left( -\frac{|\overline{\gamma}|}{g_\#} \frac{\partial g_\#}{\partial z} I_{\gamma} + \frac{\partial I_{\gamma}}{\partial z}\right) g_\#^{-|\overline{\gamma}|}.$$ Therefore using the upper bounds in Proposition [Proposition 3](#prop_peierls){reference-type="ref" reference="prop_peierls"} $$\label{ineq_derivative_weights_class0} \left| \frac{\partial \widehat{w}_\gamma^\#}{\partial s} \right| \leq (2 + 2K )\beta \delta^d |\overline{\gamma}| e^{-(\beta \rho_0 - 2) |\overline{\gamma}|} < D |\overline{\gamma}|^{\nicefrac{d}{d-1}} e^{-\tau |\overline{\gamma}|}.$$ For the bound on the partition function with $|\Lambda| = 1$ we have $$Z_\Lambda^\#= g_\#\leq e^{\beta \delta^d \widehat{\psi}_0}.$$ Finally concerning the derivative of the partition function with respect to $s$ for sufficiently large $\beta$ we obtain directly by computing that $$\left|\frac{\partial Z_\Lambda^\#}{\partial s}\right| \leq \beta \delta^d e^{\beta \delta^d \widehat{\psi}_0} \leq C_1 \beta \delta^d e^{\beta \delta^d \widehat{\psi}_0}.$$ Now we start the induction. Let us assume that the statements have been proved up to $n$. We have to prove that they hold also for $n+1$. Since all the contours appearing in $\widehat{\Phi}_n^{\#}$ are at most of class $n$, by the induction hypothesis, all these weights are $\tau$-stable and satisfied the bound [\[bound_truncated_weight_derivative\]](#bound_truncated_weight_derivative){reference-type="eqref" reference="bound_truncated_weight_derivative"} on the derivatives. Therefore using Theorem [Theorem 11](#theorem_cluster_expansion){reference-type="ref" reference="theorem_cluster_expansion"} we get $$\begin{aligned} \label{bound_terme_perturbatif_pression} \left| \frac{\partial f_n^\#}{\partial s} \right| \leq \frac{D \eta(\tau, l_0)}{\beta \delta^d} \leq 1 \end{aligned}$$ where $f_n^\#$ appears in $\widehat{\psi}_n^\#= \widehat{\psi}_0^\#+ f_n^\#$ and can also be defined as $$\begin{aligned} f_n^\#:= \lim_{k \rightarrow \infty} \frac{1}{\beta \delta^d |\Lambda_k|} \ln\Phi_n^{\#}(\Lambda_k). \end{aligned}$$ Before proceeding properly into the induction we have the following lemma. **Lemma 12**. *For $n \ge 1$, if all contours of class at most $n$ the truncated weights are $\tau$-stable then for any $k\le n$* *$$|\widehat{\psi}^\#_{n} - \widehat{\psi}^\#_{k}| \leq \frac{1}{\beta \delta^d} e^{-\frac{\tau}{2}k^{\nicefrac{d-1}{d}}} \quad \text{and } \quad |\widehat{\psi}_{n} - \widehat{\psi}_{k}| \leq \frac{1}{\beta \delta^d} e^{-\frac{\tau}{2}k^{\nicefrac{d-1}{d}}}.$$* *Proof.* We have $|\widehat{\psi}^\#_{n} - \widehat{\psi}^\#_{k}| = |f_{n}^\#- f_k^\#|$. Since all contour of class at most $n$ are $\tau$-stable we know that the cluster expansion for $f_k^\#$ converges for $k \leq n$. We can notice that the clusters $X$ that contributes to $f_{n}^\#- f_k^\#$ must have at least one contour $\gamma$ of class greater than $k$ and thus by the isoperimetric inequality $|\overline{\gamma}| \geq k^{\nicefrac{d-1}{d}}$ which in turn implies that $|\overline{X}| \geq k^{\nicefrac{d-1}{d}}$ and thus by Lemma [Lemma 10](#lemma_cluster_expansion){reference-type="ref" reference="lemma_cluster_expansion"} $$\label{eq_inegalite_difference_pression_troncquee_ordre_k_n+1} |\widehat{\psi}^\#_{n} - \widehat{\psi}^\#_{k}| \leq \frac{1}{\beta \delta^d} \sum_{\substack{X : 0 \in \overline{X} \\ |\overline{X}| \geq k^{\nicefrac{d-1}{d}}} } |\widehat{\Psi}^\#(X)| \leq \frac{1}{\beta \delta^d} e^{-\frac{\tau}{2}k^{\nicefrac{d-1}{d}}}.$$ Now we want to have the same kind of estimate for $|\widehat{\psi}_{n} - \widehat{\psi}_{k}|$. If $\widehat{\psi}_{n} = \widehat{\psi}_{n}^\#$ and $\widehat{\psi}_{k} = \widehat{\psi}_{k}^\#$, we get the same estimate since the difference is the same. In the case where $\widehat{\psi}_{n} = \widehat{\psi}_{n}^\#$ and $\widehat{\psi}_{k} = \widehat{\psi}_{k}^{\#^*}$, where $\#\neq \#^*$, we have on one side $$\widehat{\psi}_{n} - \widehat{\psi}_k = \widehat{\psi}_{n}^\#-\widehat{\psi}_{k}^\#+\widehat{\psi}_{k}^\#- \widehat{\psi}_{k}^{\#^*} \leq \widehat{\psi}_{n}^\#- \widehat{\psi}_{k}^\#$$ since by definition $\widehat{\psi}_{k}^\#- \widehat{\psi}_{k}^{\#^*} = \widehat{\psi}_{k}^\#- \widehat{\psi}_{k} \leq 0$. On the other side, we have $$\widehat{\psi}_{n} - \widehat{\psi}_k = \widehat{\psi}_{n}^\#-\widehat{\psi}_{n}^{\#^*} +\widehat{\psi}_{n}^{\#^*} - \widehat{\psi}_{k}^{\#^*} \geq 0$$ since $(\widehat{\psi}_{i}^{\#^*})_{i \in \mathbbm{N}}$ is increasing and $\widehat{\psi}_{n}^\#-\widehat{\psi}_{n}^{\#^*} \geq 0$ by definition. In any case, we obtain $$|\widehat{\psi}_{n} - \widehat{\psi}_k | \leq \frac{1}{\beta \delta^d} e^{-\frac{\tau}{2}k^{\nicefrac{d-1}{d}}}.$$ ◻ We move on the proof that [\[bound_partition_function\]](#bound_partition_function){reference-type="eqref" reference="bound_partition_function"} holds if $|\Lambda|= n+1$. Note that any contour that appears inside of $\Lambda$ is at most of class $k\leq n$. We say that a contour $\gamma$ is stable if $$\begin{aligned} a_n^\#\delta^d |\mathop{\mathrm{Int}}\gamma|^{\nicefrac{1}{d}} \leq \frac{\rho_0 }{16}. \end{aligned}$$ This property is hereditary, in the sense that for all contours $\gamma'$ that can appear inside $\mathop{\mathrm{Int}}\gamma$ are stable as well. Since, we know that all contours of class at most $n$ are $\tau$-stable we can apply Lemma [Lemma 12](#lemma_estimate_truncated_pressure_difference){reference-type="ref" reference="lemma_estimate_truncated_pressure_difference"} and by [\[condition_beta_3\]](#condition_beta_3){reference-type="eqref" reference="condition_beta_3"} we have for any contour $\gamma$ of class $k \leq n$ $$\begin{aligned} a_k^\#\delta^d |\mathop{\mathrm{Int}}\gamma |^{\nicefrac{1}{d}} & = a_{n}^\#\delta^d |\mathop{\mathrm{Int}}\gamma|^{\nicefrac{1}{d}} + (a_k^\#- a_{n}^\#) \delta^d |\mathop{\mathrm{Int}}\gamma|^{\nicefrac{1}{d}} \\ & \leq a_{n}^\#\delta^d |\mathop{\mathrm{Int}}\gamma|^{\nicefrac{1}{d}} + 2\beta^{-1} k^{\nicefrac{1}{d}}e^{-\nicefrac{\tau k^{\nicefrac{d-1}{d}}}{2}} \\ & \leq a_{n}^\#\delta^d |\mathop{\mathrm{Int}}\gamma|^{\nicefrac{1}{d}} + \frac{\rho_0 }{16}. \end{aligned}$$ Therefore, when the contours are stable it implies that $a_k^\#\delta^d |\mathop{\mathrm{Int}}\gamma |^{\nicefrac{1}{d}} \leq \nicefrac{\rho_0 }{8}$ and thus $\widehat{w}_{\gamma}^\#= w_{\gamma}^\#$. In contrast, we would call contours that doesn't satisfy this condition unstable. The stability of a contour depends on the parameter $s$ as it affects the value of $a_n^\#$. Thus we have two cases to consider. The first case is $a_n^\#=0$. Consequently all contours are stable, therefore according to Theorem [Theorem 11](#theorem_cluster_expansion){reference-type="ref" reference="theorem_cluster_expansion"} we have $$\begin{aligned} Z_\Lambda^\#& = \widehat{Z}_\Lambda^\#= e^{\beta \delta^d \widehat{\psi}_n^\#|\Lambda| \delta^d + \Delta} \\ & \leq e^{\beta \delta^d \widehat{\psi}_n^\#|\Lambda| + |\partial_{ext} \Lambda| } \end{aligned}$$ and [\[bound_partition_function\]](#bound_partition_function){reference-type="eqref" reference="bound_partition_function"} is proved. Now let us consider $a_n^\#>0$, in this case some contours must be unstable. Therefore we can partition the configurations that generate among the external contours those that are unstable $$\begin{aligned} Z_\Lambda^\#& = \sum_{\substack{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda) \\ \text{unstable}}} \int e^{-\beta H_\Lambda(\omega) } \mathbbm{1}_{\{\forall i \in \partial_{int} \Lambda, \sigma(\omega,i)=\#\}} \mathbbm{1}_{\{\Gamma \subset \Gamma_{ext}(\omega)\}} \Pi_{\widehat{\Lambda}}^{s\beta}(d\omega).\\ \end{aligned}$$ Similar to what we did in [\[eq_decomposition_product_integrals\]](#eq_decomposition_product_integrals){reference-type="eqref" reference="eq_decomposition_product_integrals"}, we can write each integral as a product of integrals with respect to Poisson point process distribution on different domains using the properties [\[energy_local_gamma_1\]](#energy_local_gamma_1){reference-type="eqref" reference="energy_local_gamma_1"}, [\[energy_local_gamma_2\]](#energy_local_gamma_2){reference-type="eqref" reference="energy_local_gamma_2"} and [\[energy_local_gamma_3\]](#energy_local_gamma_3){reference-type="eqref" reference="energy_local_gamma_3"}. The only difference is that we do consider for the moment only unstable contours and so inside $\Lambda_{ext}$ we have to account for stable contours. Furthermore, these stable contours that cannot encircle any external unstable contour due to the hereditary property of stable contours. $$\begin{aligned} Z_\Lambda^\#= \sum_{\substack{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda) \\ \text{unstable}}} Z_{\Lambda_{ext}, stable}^\#\prod_{\gamma \in \Gamma} I_{\gamma} Z_{\mathop{\mathrm{Int}}_0 \gamma}^{(0)} Z_{\mathop{\mathrm{Int}}_1 \gamma}^{(1)}, \end{aligned}$$ where $Z_{\Lambda_{ext}, stable}^\#$ denotes the partition function restricted to configurations for which all contours are stable and by construction they are of class at most $n$. Since all those contours are of class at most $n$ they are also $\tau$-stable therefore they can be studied using a convergent cluster expansion according to Theorem [Theorem 11](#theorem_cluster_expansion){reference-type="ref" reference="theorem_cluster_expansion"} and thus $$\begin{aligned} Z_{\Lambda_{ext}, stable}^\#& = g_\#^{|\Lambda_{ext}|} \widehat{\Phi}_{ n, stable}^\#(\Lambda_{ext}) \\ & \leq g_\#^{|\Lambda_{ext}|} e^{\beta \delta^d f_{n, stable}^\#|\Lambda_{ext}| + |\partial_{ext} \Lambda_{ext}|} \\ & \leq e^{\beta \delta^d (\widehat{\psi}_0^\#+f_{n,stable}^\#) |\Lambda_{ext}| + |\partial_{ext} \Lambda_{ext}|} \end{aligned}$$ where $f_{n,stable}^\#=\lim_{k \to \infty} \frac{1}{\beta |\Lambda_k| \delta^d} \ln\widehat{\Phi}_{n, stable}^\#(\Lambda_{ext})$. According to the induction hypothesis we have that $$\begin{aligned} Z_{\mathop{\mathrm{Int}}_0 \gamma}^{(0)} Z_{\mathop{\mathrm{Int}}_1 \gamma}^{(1)} & \leq e^{\beta \delta^d \widehat{\psi}_n|\mathop{\mathrm{Int}}\gamma|} e^{2(|\partial_{ext} \mathop{\mathrm{Int}}_1 \gamma| + |\partial_{ext} \mathop{\mathrm{Int}}_0 \gamma|)} \\ & \leq e^{\beta \delta^d \widehat{\psi}_n|\mathop{\mathrm{Int}}\gamma|} e^{2|\overline{\gamma}|}. \end{aligned}$$ For any $\Gamma \in \mathcal{C}_{ext}^\#(\Lambda)$ we have $|\partial_{ext} \Lambda_{ext}| \leq |\partial_{ext} \Lambda| + \sum_{\gamma \in \Gamma} |\overline{\gamma}|$, thus we get that $$\begin{aligned} Z_\Lambda^\#& \leq \sum_{\substack{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda) \\ \text{unstable}}} e^{\beta \delta^d (\widehat{\psi}_0^\#+f_{n,stable}^\#) |\Lambda_{ext}|} e^{ |\partial_{ext} \Lambda| + \sum_{\gamma \in \Gamma} |\overline{\gamma}|} \prod_{\gamma \in \Gamma} I_{\gamma} e^{\beta \delta^d \widehat{\psi}_n |\mathop{\mathrm{Int}}\gamma|} e^{2|\overline{\gamma}|} \\ & \leq e^{\beta \delta^d \widehat{\psi}_n |\Lambda| + |\partial_{ext} \Lambda|} \sum_{\substack{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda) \\ \text{unstable}}} e^{-\beta \delta^d (\widehat{\psi}_n - \widehat{\psi}_{n, stable}^\#) |\Lambda_{ext}|} \prod_{\gamma \in \Gamma} I_{\gamma} e^{(3-\beta \delta^d \widehat{\psi}_n)|\overline{\gamma}| } \end{aligned}$$ where we define $\widehat{\psi}_{n, stable}^\#:= \widehat{\psi}_0^\#+ f_{n, stable}^\#$. Furthermore when we use the following inequalities $\widehat{\psi}_n \geq \widehat{\psi}_n^\#\geq \widehat{\psi}_0^\#= \frac{\ln(g_\#)}{\beta \delta^d}$ we get that $I_{\gamma} e^{-\beta \delta^d \widehat{\psi}_n |\overline{\gamma}|} \leq e^{-\tau |\overline{\gamma}|}$. Therefore we have $$\begin{aligned} Z_\Lambda^\#& \leq e^{\beta \delta^d \widehat{\psi}_n |\Lambda| + |\partial_{ext} \Lambda|} \sum_{\substack{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda) \\ \text{unstable}}} e^{-\beta \delta^d (\widehat{\psi}_n - \widehat{\psi}_{n, stable}^\#) |\Lambda_{ext}|} \prod_{\gamma \in \Gamma} e^{-(\tau - 3) |\overline{\gamma}|}. \end{aligned}$$ It remains to prove that the sum is bounded by $e^{|\partial_{ext} \Lambda|}$. First we note that $\widehat{\psi}_n - \widehat{\psi}_{n, stable}^\#= a_n^\#+ f_n^\#- f_{n, stable}^\#$. By construction, the clusters that appear in the cluster expansion of $f_n^\#- f_{n, stable}^\#$ contain at least one unstable contour $\gamma$. Therefore $$\begin{aligned} |\overline{\gamma}| \geq |\mathop{\mathrm{Int}}\gamma|^{\nicefrac{d-1}{d}} \geq \left( \frac{\rho_0 }{16 a_n^\#\delta^d}\right)^{(d-1)} \end{aligned}$$ and by Lemma [Lemma 10](#lemma_cluster_expansion){reference-type="ref" reference="lemma_cluster_expansion"} and [\[condition_beta_5\]](#condition_beta_5){reference-type="eqref" reference="condition_beta_5"} $$\begin{aligned} &|f_n^\#- f_{n, stable}^\#| \leq \beta^{-1} \delta^{-d} \exp{(-\max\{(\nicefrac{\rho_0 }{16 a_n^\#\delta^d})^{d-1}, l_0 \} \frac{\tau}{2})} \leq \frac{a_n^\#}{2}.\label{toto} \end{aligned}$$ At the end we obtain $$\begin{aligned} \label{eq_borne_inf_psi_n_psi_n_stable} \widehat{\psi}_n - \widehat{\psi}_{n, \text{stable}}^\#\geq \frac{a_n^\#}{2}. \end{aligned}$$ Now let us define new weights $w_\gamma^*$ as follow $$\begin{aligned} w_{\gamma}^* = \left\{ \begin{aligned} &e^{-(\tau-5)|\overline{\gamma}|} &\quad \text{if $\gamma$ is unstable} \\ &0 & \quad \text{otherwise}. \end{aligned} \right. \end{aligned}$$ We denote by $\Phi^*$ the associated polymer development and have $$g^* = \lim_{k \rightarrow \infty} \frac{1}{\beta \delta^d |\Lambda_k|} \ln \Phi^*(\Lambda_k).$$ For sufficiently large $\beta$ we can assure by Theorem [Theorem 11](#theorem_cluster_expansion){reference-type="ref" reference="theorem_cluster_expansion"} that it is a convergent cluster expansion. Since all contours that contribute to $g^*$ are all unstable, we obtain an inequality similar to [\[toto\]](#toto){reference-type="eqref" reference="toto"} $$\begin{aligned} \label{eq_borne_inf_g_star} |g^*| \leq \beta^{-1} \delta^{-d} \exp{(-\max\{(\nicefrac{\rho_0 }{16 a_n^\#\delta^d})^{d-1}, l_0 \} \frac{\tau}{2})} \leq \frac{a_n^\#}{2}. \end{aligned}$$ Therefore with [\[eq_borne_inf_psi_n\_psi_n\_stable\]](#eq_borne_inf_psi_n_psi_n_stable){reference-type="eqref" reference="eq_borne_inf_psi_n_psi_n_stable"} and [\[eq_borne_inf_g\_star\]](#eq_borne_inf_g_star){reference-type="eqref" reference="eq_borne_inf_g_star"} we have $\widehat{\psi}_n - \widehat{\psi}_n^\#\geq g^*$ and thus $$\begin{aligned} \sum_{\substack{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda) \\ \text{unstable}}} e^{-\beta \delta^d (\widehat{\psi}_n - \widehat{\psi}_{n, stable}^\#) |\Lambda_{ext}|} \prod_{\gamma \in \Gamma} e^{-(\tau - 3) |\overline{\gamma}|} & \leq \sum_{\substack{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda) \\ \text{unstable}}} e^{-\beta \delta^d g^* |\Lambda_{ext}|} \prod_{\gamma \in \Gamma} e^{-(\tau - 3) |\overline{\gamma}|}\\ & \leq e^{-\beta \delta^d g^* |\Lambda|} \sum_{\substack{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda) \\ \text{unstable}}} \prod_{\gamma \in \Gamma} e^{-(\tau - 3) |\overline{\gamma}|} e^{\beta \delta^d g^* (|\overline{\gamma}|+ |\mathop{\mathrm{Int}}\gamma|)}. \end{aligned}$$ By [\[eq_borne_inf_g\_star\]](#eq_borne_inf_g_star){reference-type="eqref" reference="eq_borne_inf_g_star"} we know that $\beta \delta^d g^* \le 1$, and again with Theorem [Theorem 11](#theorem_cluster_expansion){reference-type="ref" reference="theorem_cluster_expansion"} we know that $\Phi_{\mathop{\mathrm{Int}}\gamma}^* \geq e^{\beta \delta^d g^* |\mathop{\mathrm{Int}}\gamma| - |\overline{\gamma}|}$ and so $$\begin{aligned} \sum_{\substack{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda) \\ \text{unstable}}} e^{-\beta \delta^d (\widehat{\psi}_n - \widehat{\psi}_{n, stable}^\#) |\Lambda_{ext}|} \prod_{\gamma \in \Gamma} e^{-(\tau - 3) |\overline{\gamma}|} & \leq e^{-\beta \delta^d g^* |\Lambda|} \sum_{\substack{\Gamma \in \mathcal{C}_{ext}^\#(\Lambda) \\ \text{unstable}}} \prod_{\gamma \in \Gamma} e^{-(\tau - 5) |\overline{\gamma}|} \Phi_{\mathop{\mathrm{Int}}\gamma}^* \\ & = e^{-\beta \delta^d g^* |\Lambda|} \Phi_{\Lambda}^* \\ & \leq e^{|\partial_{ext} \Lambda|}. \end{aligned}$$ In summary, we have $$Z_\Lambda^\#\leq e^{\beta \delta^d \widehat{\psi}_n |\Lambda| + 2|\partial_{ext} \Lambda|}$$ which is exactly [\[bound_partition_function\]](#bound_partition_function){reference-type="eqref" reference="bound_partition_function"} in the case where $|\Lambda|=n+1$. If $|\Lambda|\le n$ it is sufficient to notice that $(\widehat{\psi}_n^\#)_{n \in \mathbbm{N}}$ is increasing. Let us now prove that [\[bound_derivative_partition_function\]](#bound_derivative_partition_function){reference-type="eqref" reference="bound_derivative_partition_function"} holds. We start with the case $|\Lambda| = n+1$ and by a similar argument it is true for any smaller $\Lambda$. By a direct computation we have $$\begin{aligned} \frac{\partial Z_\Lambda^\#}{\partial z} &= -|\Lambda| \delta^d Z_\Lambda^\#+ \frac{1}{z} \int N_{\widehat{\Lambda}}(\omega) e^{-\beta H_\Lambda(\omega)} \mathbbm{1}_{\{\forall i \in \partial_{int} \Lambda, \sigma(\omega, i) = \#\}} \Pi_{\widehat{\Lambda}}^z(d\omega) \\ &= -|\Lambda| \delta^d Z_\Lambda^\#+ \frac{1}{z} E_{P_\Lambda^\#}(N_{\widehat{\Lambda}}) Z_\Lambda^\#. \end{aligned}$$ We can observe that for any configuration $\omega \in \Omega_f$ and for any contour $\gamma \in \Gamma(\omega)$ created by this configuration we have $H_{\overline{\gamma}}(\omega) \geq |\overline{\gamma}_1|\delta^d + \rho_0 |\overline{\gamma}| >0$, thus $H_\Lambda(\omega) \geq 0$. By Donsker-Varadhan inequality for the Kullback-Liebler divergence, denoted by $I(\cdot | \cdot)$, we have $$\begin{aligned} E_{P_\Lambda^\#}(N_{\widehat{\Lambda}}) &\leq I(P_\Lambda^\#| \Pi_{\widehat{\Lambda}}^z) + \ln E_{\Pi_{\widehat{\Lambda}}^z}(e^{N_{\widehat{\Lambda}}}) \nonumber \\ & \leq \int -\beta H_\Lambda dP_\Lambda^\#- \ln Z_\Lambda^\#+ (e-1)z \delta^d |\Lambda| \nonumber \\ & \leq - \ln Z_\Lambda^\#+ (e-1)z \delta^d |\Lambda|. \label{choupi} \end{aligned}$$ Furthermore we know that the contours which appear in $|\Lambda|$ are at most of the class $n$. Therefore we know that their truncated weights are $\tau$-stable and by Theorem [Theorem 11](#theorem_cluster_expansion){reference-type="ref" reference="theorem_cluster_expansion"} $$\label{eq_inegalite_inf_fonction_partition_par_cluster} Z_\Lambda^\#\geq \widehat{Z}_\Lambda^\#\geq e^{\beta \delta^d \widehat{\psi}_n^\#|\Lambda| - |\partial_{ext} \Lambda|}.$$ From inequalities [\[eq_inegalite_inf_fonction_partition_par_cluster\]](#eq_inegalite_inf_fonction_partition_par_cluster){reference-type="eqref" reference="eq_inegalite_inf_fonction_partition_par_cluster"} and [\[choupi\]](#choupi){reference-type="eqref" reference="choupi"} and by using the fact that $(\widehat{\psi}_n^\#)_{n \in \mathbbm{N}}$ is increasing we obtain $$E_{P_\Lambda^\#}(N_{\widehat{\Lambda}}) \leq \left((e-1)z - \beta \widehat{\psi}_0^\#\right) \delta^d |\Lambda| + |\partial_{ext} \Lambda|.$$ At the end we have $$\left| \frac{\partial Z_\Lambda^\#}{\partial s} \right| \leq \left[ \left( e + \left|\frac{\widehat{\psi}_0^\#}{s} \right| \right) \beta \delta^d |\Lambda| + \frac{1}{s} |\partial_{ext} \Lambda| \right] Z_\Lambda^\#.$$ Now using [\[bound_partition_function\]](#bound_partition_function){reference-type="eqref" reference="bound_partition_function"} we obtain for $s \in U_\beta$ $$\left| \frac{\partial Z_\Lambda^\#}{\partial s} \right| \leq \left( C_1 \beta \delta^d |\Lambda| + C_2 |\partial_{ext} \Lambda| \right) e^{\beta \delta^d \widehat{\psi}_n |\Lambda| + 2|\partial_{ext} \Lambda|}$$ and [\[bound_derivative_partition_function\]](#bound_derivative_partition_function){reference-type="eqref" reference="bound_derivative_partition_function"} is proved. Now let us prove [\[truncated_weights_tau_stability\]](#truncated_weights_tau_stability){reference-type="eqref" reference="truncated_weights_tau_stability"} which is the $\tau$-stability of truncated weights for contours of class $n+1$. We consider a contour $\gamma$ of class $n+1$. First of all, we can observe that $\widehat{w}_{\gamma}^\#= 0$ whenever $(\widehat{\psi}_n^{\#^*} - \widehat{\psi}_n^\#) \delta^d |\mathop{\mathrm{Int}}_{\#^*} \gamma|^{\nicefrac{1}{d}} > \nicefrac{\rho_0 }{4}$. So we can assume that $$\begin{aligned} (\widehat{\psi}_n^{\#^*} - \widehat{\psi}_n^\#) \delta^d |\mathop{\mathrm{Int}}_{\#^*} \gamma|^{\nicefrac{1}{d}} \leq \frac{\rho_0}{4}. \label{eq_hypothesis_cutoff_value} \end{aligned}$$ Since $|\mathop{\mathrm{Int}}\gamma| = n+1$ we can apply the induction hypothesis on the partition functions that appears in the truncated weights particularly we can use [\[bound_partition_function\]](#bound_partition_function){reference-type="eqref" reference="bound_partition_function"} and have $$\begin{aligned} \label{ineq_induction_hypothesis_partition_function} Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#^*} \leq e^{\beta \delta^d \widehat{\psi}_n |\mathop{\mathrm{Int}}_{\#^*} \gamma| + 2|\partial_{ext} \mathop{\mathrm{Int}}_{\#^*} \gamma|}. \end{aligned}$$ By combining the previous inequalities [\[ineq_induction_hypothesis_partition_function\]](#ineq_induction_hypothesis_partition_function){reference-type="eqref" reference="ineq_induction_hypothesis_partition_function"} and [\[eq_inegalite_inf_fonction_partition_par_cluster\]](#eq_inegalite_inf_fonction_partition_par_cluster){reference-type="eqref" reference="eq_inegalite_inf_fonction_partition_par_cluster"} we have $$\begin{aligned} \frac{Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#^*}}{Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#}} & \leq e^{\beta \delta^d (\widehat{\psi}_n - \widehat{\psi}_n^\#) |\mathop{\mathrm{Int}}_{\#^*} \gamma| + 3|\partial_{ext} \mathop{\mathrm{Int}}_{\#^*} \gamma|} \\ & \leq e^{\beta \delta^d (\widehat{\psi}_n - \widehat{\psi}_n^\#) |\mathop{\mathrm{Int}}_{\#^*} \gamma|^{\nicefrac{1}{d}} |\mathop{\mathrm{Int}}_{\#^*} \gamma|^{\nicefrac{(d-1)}{d}} + 3|\partial_{ext} \mathop{\mathrm{Int}}_{\#^*} \gamma|}. \end{aligned}$$ Furthermore, applying hypothesis [\[eq_hypothesis_cutoff_value\]](#eq_hypothesis_cutoff_value){reference-type="eqref" reference="eq_hypothesis_cutoff_value"} and the isoperimetric inequality we have $$\begin{aligned} \frac{Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#^*}}{Z_{\mathop{\mathrm{Int}}_{\#^*} \gamma}^{\#}} & \leq e^{ \nicefrac{\beta \rho_0 }{4} |\mathop{\mathrm{Int}}_{\#^*} \gamma|^{\nicefrac{(d-1)}{d}} + 3|\partial_{ext} \mathop{\mathrm{Int}}_{\#^*} \gamma|}\nonumber \\ & \leq e^{ (\frac{1}{4} \beta \rho_0 + 3) |\partial_{ext} \mathop{\mathrm{Int}}_{\#^*} \gamma|}. \label{ineq_upper_bound_ratio_partition_function} \end{aligned}$$ Therefore with Proposition [Proposition 3](#prop_peierls){reference-type="ref" reference="prop_peierls"}, the upper bound on the ratio of partition functions in inequality [\[ineq_upper_bound_ratio_partition_function\]](#ineq_upper_bound_ratio_partition_function){reference-type="eqref" reference="ineq_upper_bound_ratio_partition_function"} and the fact that $|\partial_{ext} \mathop{\mathrm{Int}}_{\#^*} \gamma| \leq |\overline{\gamma}|$ we have $$\begin{aligned} \widehat{w}_{\gamma}^\#&\leq e^{-(\beta \rho_0 - 2)|\overline{\gamma}|} e^{(\frac{1}{4} \beta \rho_0 + 3) |\overline{\gamma}|} \\ &\leq e^{-(\frac{3}{4} \beta \rho_0 - 5) |\overline{\gamma}|} \\ &\leq e^{-\tau |\overline{\gamma}|} \end{aligned}$$ and the weight $\widehat{w}_{\gamma}^\#$ is $\tau$-stable. Let us now show that [\[bound_truncated_weight_derivative\]](#bound_truncated_weight_derivative){reference-type="eqref" reference="bound_truncated_weight_derivative"} holds for a contour $\gamma$ of class $n+1$. Similar to the proof of [\[truncated_weights_tau_stability\]](#truncated_weights_tau_stability){reference-type="eqref" reference="truncated_weights_tau_stability"} we consider only the case when $(\widehat{\psi}_n^{\#^*} - \widehat{\psi}_n^\#) \delta^d |\mathop{\mathrm{Int}}_{\#^*} \gamma|^{\nicefrac{1}{d}} \leq \nicefrac{\rho_0 }{4}$. By a direct computation $$\begin{aligned} \frac{\partial \widehat{w}_{\gamma}^\#}{\partial z} = & \left( -\frac{|\overline{\gamma}|}{g_\#} \frac{\partial g_\#}{\partial z} I_{\gamma} + \frac{\partial I_{\gamma}}{\partial z}\right) \left(g_\#^{-|\overline{\gamma}|} \kappa \frac{Z^{\#^*}_{\mathop{\mathrm{Int}}_{\#^*} \gamma}}{Z^\#_{\mathop{\mathrm{Int}}_{\#^*} \gamma}}\right) + \frac{\partial}{\partial z} \left( \kappa \frac{Z^{\#^*}_{\mathop{\mathrm{Int}}_{\#^*} \gamma}}{Z^{\#}_{\mathop{\mathrm{Int}}_{\#^*} \gamma}}\right) g_\#^{-|\overline{\gamma}|} I_{\gamma}. \end{aligned}$$ The first term of the derivative can be bounded in a similar fashion as in [\[ineq_derivative_weights_class0\]](#ineq_derivative_weights_class0){reference-type="eqref" reference="ineq_derivative_weights_class0"} using the facts that $s \in U_\beta$ and by Proposition [Proposition 3](#prop_peierls){reference-type="ref" reference="prop_peierls"} $$\label{ineq_bound_first_term_of_derivative} \left| -\frac{|\overline{\gamma}|}{g_\#} \frac{\partial g_\#}{\partial z} I_{\gamma} + \frac{\partial I_{\gamma}}{\partial z}\right| \left(g_\#^{-|\overline{\gamma}|} \kappa \frac{Z^{\#^*}_{\mathop{\mathrm{Int}}_{\#^*} \gamma}}{Z^\#_{\mathop{\mathrm{Int}}_{\#^*} \gamma}}\right) \leq (2+2K) \delta^d |\overline{\gamma}| e^{-\tau |\overline{\gamma}|}.$$ For the other term, the derivative of the cut-off function satisfies $$\begin{aligned} \left| \frac{\partial \kappa}{\partial z} \right| \leq \left( \left|\frac{\partial \widehat{\psi}_n^\#}{\partial z}\right| + \left|\frac{\partial \widehat{\psi}_n^{\#^*}}{\partial z}\right| \right) \delta^d |\mathop{\mathrm{Int}}_{\#^*} \gamma|^{\frac{1}{d}} \| \kappa' \|. \end{aligned}$$ For $s \in U_\beta$ and by [\[bound_terme_perturbatif_pression\]](#bound_terme_perturbatif_pression){reference-type="eqref" reference="bound_terme_perturbatif_pression"} we have $$\frac{\partial \psi_n^\#}{\partial z} = \frac{1}{\beta \delta^d g_\#} \frac{\partial g_\#}{\partial z } + \frac{1}{\beta} \frac{\partial f_n^\#}{\partial s} \implies \left| \frac{\partial \psi_n^\#}{\partial z} \right| \leq \frac{2}{\beta}.$$ Therefore with the isoperimetric inequality we have $$\label{ineq_bound_derivative_cutoff} \left| \frac{\partial \kappa}{\partial z} \right| \leq \frac{4}{\beta} \delta^d |\overline{\gamma}| \| \kappa' \|.$$ For the derivative of the ratio of partition functions $$\left| \frac{\partial}{\partial z} \frac{Z_{\mathop{\mathrm{Int}}_{\#^*}\gamma}^{\#^*}}{Z_{\mathop{\mathrm{Int}}_{\#^*}\gamma}^{\#}} \right| \leq \left|\frac{\nicefrac{\partial Z_{\mathop{\mathrm{Int}}_{\#^*}\gamma}^{\#^*}}{\partial z}}{Z_{\mathop{\mathrm{Int}}_{\#^*}\gamma}^{\#}} \right| + \left|\frac{\nicefrac{\partial Z_{\mathop{\mathrm{Int}}_{\#^*}\gamma}^{\#}}{\partial z}}{Z_{\mathop{\mathrm{Int}}_{\#^*}\gamma}^{\#}} \right| \frac{Z_{\mathop{\mathrm{Int}}_{\#^*}\gamma}^{\#^*}}{Z_{\mathop{\mathrm{Int}}_{\#^*}\gamma}^{\#}}.$$ Similarly to how we bounded the ratio of partition function and using [\[bound_derivative_partition_function\]](#bound_derivative_partition_function){reference-type="eqref" reference="bound_derivative_partition_function"} we have for $\#^* \in \{0,1\}$ $$\left|\frac{\nicefrac{\partial Z_{\mathop{\mathrm{Int}}_{\#^*}\gamma}^{\#^*}}{\partial z}}{Z_{\mathop{\mathrm{Int}}_{\#^*}\gamma}^{\#}} \right| \leq \left(C_1 \delta^d |\mathop{\mathrm{Int}}_{\#^*} \gamma| + \frac{C_2}{\beta} |\partial_{ext}\mathop{\mathrm{Int}}_{\#^*} \gamma| \right) e^{(\nicefrac{1}{4}\beta \rho_0 +3)|\overline{\gamma}|}.$$ Therefore using the isoperimetric inequality and the fact that $|\partial_{ext}\mathop{\mathrm{Int}}_{\#^*} \gamma| \leq |\overline{\gamma}|$ we have $$\begin{aligned} \label{ineq_bound_derivative_partition_function_ratio} \left| \frac{\partial}{\partial z} \frac{Z_{\mathop{\mathrm{Int}}_{\#^*}\gamma}^{\#^*}}{Z_{\mathop{\mathrm{Int}}_{\#^*}\gamma}^{\#}} \right| \leq \left( C_1 \delta^d + \frac{C_2}{\beta} \right) |\overline{\gamma}|^{\nicefrac{d}{(d-1)}} e^{(\nicefrac{1}{2}\beta \rho_0 +6)|\overline{\gamma}|}. \end{aligned}$$ When combining inequalities [\[ineq_bound_first_term_of_derivative\]](#ineq_bound_first_term_of_derivative){reference-type="eqref" reference="ineq_bound_first_term_of_derivative"}, [\[ineq_bound_derivative_cutoff\]](#ineq_bound_derivative_cutoff){reference-type="eqref" reference="ineq_bound_derivative_cutoff"}, [\[ineq_bound_derivative_partition_function_ratio\]](#ineq_bound_derivative_partition_function_ratio){reference-type="eqref" reference="ineq_bound_derivative_partition_function_ratio"} and Proposition [Proposition 3](#prop_peierls){reference-type="ref" reference="prop_peierls"} we have $$\begin{aligned} \left| \frac{\partial \widehat{w}_{\gamma}^\#}{\partial s} \right| \leq \left((2+2K)\delta^d \beta + 4 \delta^d \| \kappa' \| + C_1\beta \delta^d + C_2 \right) |\overline{\gamma}|^{\nicefrac{d}{(d-1)}} e^{-\tau |\overline{\gamma}|}. \end{aligned}$$ To finish the proof let us show that [\[implication_bound_a\_n_to_stability\]](#implication_bound_a_n_to_stability){reference-type="eqref" reference="implication_bound_a_n_to_stability"} holds at the order $n+1$. Since we have proven this far that the truncated weights of class at most $n+1$ are $\tau$-stable we can apply use Lemma [Lemma 12](#lemma_estimate_truncated_pressure_difference){reference-type="ref" reference="lemma_estimate_truncated_pressure_difference"}. Let $\gamma$ of class $n+1$ if we have $a_{n+1}^\#\delta^d |\mathop{\mathrm{Int}}\gamma|^{\nicefrac{1}{d}} \leq \frac{\rho_0 }{16}$ then by definition of truncated weights we would have $\widehat{w}_{\gamma}^\#= w_{\gamma}^\#$. Now let's consider a contour $\gamma$ of class $k\leq n$, according to Lemma [Lemma 12](#lemma_estimate_truncated_pressure_difference){reference-type="ref" reference="lemma_estimate_truncated_pressure_difference"} and [\[condition_beta_3\]](#condition_beta_3){reference-type="eqref" reference="condition_beta_3"} we have $$\begin{aligned} a_k^\#\delta^d |\mathop{\mathrm{Int}}\gamma|^{\nicefrac{1}{d}} & = a_{n+1}^\#\delta^d |\mathop{\mathrm{Int}}\gamma|^{\nicefrac{1}{d}} + (a_k^\#- a_{n+1}^\#) \delta^d |\mathop{\mathrm{Int}}\gamma|^{\nicefrac{1}{d}} \\ & \leq a_{n+1}^\#\delta^d |\mathop{\mathrm{Int}}\gamma|^{\nicefrac{1}{d}} + 2\beta^{-1} k^{\nicefrac{1}{d}}e^{-\nicefrac{\tau k^{\nicefrac{d-1}{d}}}{2}} \\ & \leq a_{n+1}^\#\delta^d |\mathop{\mathrm{Int}}\gamma|^{\nicefrac{1}{d}} + \frac{\rho_0 }{16}. \end{aligned}$$ Therefore if $a_{n+1}^\#\delta^d |\mathop{\mathrm{Int}}\gamma|^{\nicefrac{1}{d}} \leq \nicefrac{\rho_0 }{16}$ it implies that $a_k^\#\delta^d |\mathop{\mathrm{Int}}\gamma|^{\nicefrac{1}{d}} \leq \nicefrac{\rho_0}{8}$ which in turn would imply $\widehat{w}_{\gamma}^\#= w_{\gamma}^\#$ by definition of the truncated weights.

arxiv_math

--- abstract: | In a previous paper [@BoTo20b] the authors employed the fiber join construction of Yamazaki [@Yam99] together with the admissible construction of Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman [@ACGT08] to construct new extremal Sasaki metrics on odd dimensional sphere bundles over smooth projective algebraic varieties. In the present paper we continue this study by applying a recent existence theorem [@BHLT22] that shows that under certain conditions one can always obtain a constant scalar curvature Sasaki metric in the Sasaki cone. Moreover, we explicitly describe this construction for certain sphere bundles of dimension 5 and 7. address: - Charles P. Boyer, Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA. - Christina W. Tønnesen-Friedman, Department of Mathematics, Union College, Schenectady, New York 12308, USA author: - Charles P. Boyer - Christina W. Tønnesen-Friedman title: "Sasakian Geometry on Sphere Bundles II: Constant Scalar Curvature" --- [^1]   # Introduction {#intro} A central problem in Riemannian geometry is to determine conditions for a metric to have constant scalar curvature. This is particularly true in Kähler geometry as well as in its odd dimensional sister Sasaki geometry. Specifically, we combine our construction of extremal Sasaki metrics on odd dimensional sphere bundles [@BoTo20b] using Yamazaki's fiber join [@Yam99] with the admissible conditions [@ACGT08] as applied in [@BHLT22] to obtain constant scalar curvature (CSC) Sasaki metrics. This involves the introduction of a refinement of the admissibility conditions that we call *strongly admissible* whose precise definition is given below in [Definition 9](#stradmdef){reference-type="ref" reference="stradmdef"}. Previously [@BoTo20b], we gave a stronger condition called *super admissible*; however, we show here that the less stringent condition, strongly admissible, is enough. Explicitly, in Section [3.4](#mainthm){reference-type="ref" reference="mainthm"} we prove our main theorem which is Theorem [Theorem 14](#2ndroundexistence){reference-type="ref" reference="2ndroundexistence"} and is restated here for the convenience of the reader. **Theorem 1**. *Let $M_{\mathfrak w}$ be a strongly admissible Yamazaki fiber join whose regular quotient is a ruled manifold of the form ${\mathbb P}(E_0\oplus E_\infty)\longrightarrow N$ where $E_0 ,E_\infty$ are projectively flat hermitian holomorphic vector bundles on $N$ of complex dimension $(d_0+1),(d_\infty+1)$ respectively, and $N$ is a local Kähler product of non-negative CSC metrics. Then ${\mathfrak t}^+_{sph}$ has a $2$-dimensional subcone of extremal Sasaki metrics (up to isotopy) which contains at least one ray of CSC Sasaki metrics.* Then in Section [4](#Exsect){reference-type="ref" reference="Exsect"} we present detailed descriptions of strongly admissible sphere bundles of dimension 5 and 7 which are obtained by Yamazaki's fiber join construction. We obtain existence results (Propositions [Proposition 16](#Mkprop){reference-type="ref" reference="Mkprop"}, [Proposition 17](#2highergenus){reference-type="ref" reference="2highergenus"} and [Proposition 18](#polyprop){reference-type="ref" reference="polyprop"}) of CSC and extremal Sasaki metrics even when the conditions of Theorem [Theorem 1](#2ndroundexistenceintro){reference-type="ref" reference="2ndroundexistenceintro"} are not all met. **Acknowledgements 1**. The authors would like to thank Claude LeBrun for pointing out [@Shu]. We also would like to thank Eveline Legendre and Hongnian Huang for fruitful conversations. # Brief Review of K-Contact and Sasaki Geometry Recall that an oriented and co-oriented contact manifold $M^{2n+1}$ has a contact metric structure ${\oldmathcal S}=(\xi,\eta,\Phi,g)$ where $\eta$ is a contact form with contact bundle ${\mathcal D}=\ker\eta$, $\xi$ is its Reeb vector field, $J=\Phi |_{\mathcal D}$ is an almost complex structure on ${\mathcal D}$, i.e. $({\mathcal D},J)$ is an almost CR contact structure, and $g=d\eta\circ (\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}\times\Phi) +\eta\otimes\eta$ is a compatible Riemannian metric. ${\oldmathcal S}$ is *K-contact* if $\xi$ is a Killing vector field for $g$, and it is *Sasakian* if in addition the almost CR structure is integrable. A manifold with a K-contact (Sasakian) structure is called a *K-contact (Sasaki) manifold*. Unless otherwise stated we shall assume that our contact manifolds $M^{2n+1}$ are oriented, co-oriented, compact, connected, and without boundary. We refer to [@BG05] for the fundamentals of Sasaki geometry. ## The Sasaki Cone Let $(M,{\oldmathcal S})$ be a K-contact manifold. Within the underlying contact almost CR structure $({\mathcal D},J)$ there is a conical family of K-contact structures known as the Sasaki cone and denoted by ${\mathfrak t}^+({\mathcal D},J)$ or just ${\mathfrak t}^+$ when the underlying almost CR structure is understood. We are also interested in a variation within this family. To describe the Sasaki cone we fix a K-contact structure ${\oldmathcal S}_o=(\xi_o,\eta_o,\Phi_o,g_o)$ on $M$ whose underlying CR structure is $({\mathcal D},J)$ and let ${\mathfrak t}$ denote the Lie algebra of a maximal torus ${\mathbb T}$ in the automorphism group of ${\oldmathcal S}_o$. Since for K-contact (Sasakian) structures, the Reeb vector field $\xi$ is a Killing vector field, we have $\dim{\mathfrak t}^+({\mathcal D},J)\geq 1$. Moreover, it follows from contact geometry that $\dim{\mathfrak t}^+({\mathcal D},J)\leq n+1$. The *(unreduced) Sasaki cone* [@BGS06] is defined by $$\label{sascone} {\mathfrak t}^+({\mathcal D},J)=\{\xi\in{\mathfrak t}~|~\eta_o(\xi)>0~\text{everywhere on $M$}\},$$ which is a cone of dimension $k\geq 1$ in ${\mathfrak t}$ under the transverse scaling operation defined by $$\label{transscale} {\oldmathcal S}=(\xi,\eta,\Phi,g)\mapsto {\oldmathcal S}_a=(a^{-1}\xi,a\eta,\Phi,g_a),\quad g_a=ag+(a^2-a)\eta\otimes\eta, \quad a\in{\mathbb R}^+.$$ We remark also that ${\oldmathcal S}_a$ is a K-contact or Sasakian structure for all $a\in{\mathbb R}^+$. The reduced Sasaki cone $\kappa({\mathcal D},J)$ is ${\mathfrak t}^+({\mathcal D},J)/{\mathcal W}$ where ${\mathcal W}$ is the Weyl group of the maximal compact subgroup of ${\mathfrak C}{\mathfrak R}({\mathcal D},J)$ which, as described in [@BGS06], is the moduli space of K-contact (Sasakian) structures with underlying CR structure $({\mathcal D},J)$. However, it is more convenient to work with the unreduced Sasaki cone ${\mathfrak t}^+({\mathcal D},J)$. Note that each choice of Reeb vector field $\xi\in{\mathfrak t}^+({\mathcal D},J)$ gives rise to an infinite dimensional contractible space ${\mathcal S}(M,\xi)$ of Sasakian structures [@BG05], and we often have need to obtain a particular element of ${\mathcal S}(M,\xi)$ by deforming the contact structure ${\mathcal D}\mapsto {\mathcal D}_\varphi$ by a contact isotopy $\eta\mapsto \eta +d^c\varphi$ where $\varphi\in C^\infty(M)^{\mathbb T}$ is a smooth function invariant under the torus ${\mathbb T}$. We note that the Sasaki cone ${\mathfrak t}^+({\mathcal D},J)$ is invariant under such contact isotopies in the sense that ${\mathfrak t}^+({\mathcal D}_\varphi,J_\varphi)\approx {\mathfrak t}^+({\mathcal D},J)$. We shall often make such a choice ${\oldmathcal S}=(\xi,\eta,\Phi,g)\in{\mathcal S}(M,\xi)$ and identify it with the element $\xi\in{\mathfrak t}^+({\mathcal D},J)$. **Remark 2**. When $\dim{\mathfrak t}^+({\mathcal D},J)=n+1$ we have what in [@BG00b] was called a toric contact manifold of Reeb type. This is actually a toric K-contact manifold, and in [@BG00b] a Delzant [@Del88] type theorem was proved, that is, any toric K-contact manifold is Sasaki. Moreover, as in the symplectic case there is a strong connection between the geometry and topology of $(M,{\oldmathcal S})$ and the combinatorics of ${\mathfrak t}^+({\mathcal D},J)$ [@BM93; @Ler02a; @Ler04; @Leg10; @Leg16][^2]. Much can also be said in the complexity 1 case ($\dim{\mathfrak t}^+({\mathcal D},J)=n$) [@AlHa06]. It is important to realize that there are two types of Reeb orbits, those that are closed (i.e periodic orbits) and those that are not. On a closed K-contact manifold a Reeb vector field in the Sasaki cone ${\mathfrak t}^+$ is $C^\infty$-close to a Reeb vector field all of whose orbits are periodic. What can one say about Reeb vector fields in the complement of ${\mathfrak t}^+$? The famous Weinstein conjecture says that every Reeb vector field on a compact contact manifold has a periodic orbit, and this is known to hold on a compact simply connected K-contact manifold [@Ban90]. See also [@Gin96; @AGH18]. We end this section with the following observation that applies to our examples. **Proposition 3**. *Let ${\mathbb C}{\mathbb P}^1\rightarrow S_{\bf n}\rightarrow N$ be a projective bundle where $N$ is a smooth projective algebraic variety of complex dimension $d_N\geq 2$, and let $M^{2d_N+3}$ be the total space of a Sasaki $S^1$ bundle over $S_{\bf n}$. Then $M^{2d_N+3}$ is a nontrivial lens space bundle (with fiber $F$) over $N$. Furthermore, $F=S^3$ if and only if the natural induced map $\pi_2(M)\longrightarrow\pi_2(N)$ is an epimorphism, and the natural induced map $\pi_1(M)\longrightarrow\pi_1(N)$ is a monomorphism. In particular, if $N$ is simply connected there is a choice of Kähler class on $S_{\bf n}$ such $F=S^3$.* *Proof.* By composition we have a smooth bundle $F\rightarrow M^{2d_N+3}\rightarrow N$, and by construction the $S^1$ action on $M^{2d_N+3}$ only acts on the fibers $F$. Moreover, since the total space of this bundle is Sasaki, the bundle is nontrivial. So its restriction to $F$ is also nontrivial. It follows that $F$ is a lens space and we have the commutative diagram $$\label{commdiag2} \begin{matrix} S^1 &\longrightarrow&F& \longrightarrow&{\mathbb C}{\mathbb P}^1 \\ \phantom{\hbox{$\scriptstyle{id}$}} \left\downarrow\vbox{\vskip 15pt\hbox{$\scriptstyle{id}$}}\right.&&\phantom{\hbox{$\scriptstyle{}$}} \left\downarrow\vbox{\vskip 15pt\hbox{$\scriptstyle{}$}}\right.&&\phantom{\hbox{$\scriptstyle{}$}} \left\downarrow\vbox{\vskip 15pt\hbox{$\scriptstyle{}$}}\right.\\ S^1 &\longrightarrow&M^{2d_N+3}& \longrightarrow&S_{\bf n}\\ &&\phantom{\hbox{$\scriptstyle{}$}} \left\downarrow\vbox{\vskip 15pt\hbox{$\scriptstyle{}$}}\right.&&\phantom{\hbox{$\scriptstyle{}$}} \left\downarrow\vbox{\vskip 15pt\hbox{$\scriptstyle{}$}}\right.\\ &&N &\raise 4pt\hbox{$ id \atop \longrightarrow $} & N. \end{matrix}$$ Now since $N$ is Kähler its third Betti number $b_3(N)$ is even. Furthermore, since $M^{2d_N+3}$ is Sasaki of dimension at least 7, its third Betti number $b_3(M)$ must also be even which implies that the Euler class of the lens space bundle cannot vanish implying that the bundle is nontrivial. Since $F$ is a lens space, $\pi_2(F)=0$, so the long exact homotopy sequence becomes $$\label{homexseq} \mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}\longrightarrow\pi_2(M)\longrightarrow\pi_2(N)\longrightarrow\pi_1(F)\longrightarrow\pi_1(M)\longrightarrow\pi_1(N)\longrightarrow\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}.$$ So when the induced map $\pi_2(M)\longrightarrow\pi_2(N)$ is an epimorphism, and the induced map $\pi_1(M)\longrightarrow\pi_1(N)$ is a monomorphism we have $\pi_1(F)=\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}$ which gives $F=S^3$ in this case. The converse is also clear from the homotopy exact sequence. Now if $N$ is simply connected so is $S_{\bf n}$. Thus, by choosing a primitive Kähler class on $S_{\bf n}$, we can take $M^{2d+3}$ to be simply connected. Furthermore, we can choose the Kähler class on $S_{\bf n}$ such that its restriction to ${\mathbb C}{\mathbb P}^1$ is primitive. It then follows that $F=S^3$. ◻ # Yamazaki's Fiber Join {#Yamsect} Yamazaki [@Yam99] constructed his fiber join in the category of regular K-contact manifolds which as shown in [@BoTo20b] restricts to the Sasakian case in a natural way. We refer to ob.cit for details. Here we briefly recall that the fiber join is constructed by considering $d+1$ regular Sasaki manifolds $M_j$ over a smooth algebraic variety $N$ with $d+1$ Kähler forms $\omega_j$ on $N$ that are not necessarily distinct. One then constructs a smooth manifold $M=M_1\star_f\cdots\star_f M_{d+1}$ as the unit sphere in the complex vector bundle $E=\oplus_{j=1}^{d+1}L_j^*$ where $L_j$ denotes the complex line bundle on $N$ associated to $M_j$ such that $c_1(L_j)=[\omega_j]$ and $L_j^*$ is its dual. We shall refer to such a fiber join as a *Sasaki-Yamazaki fiber join*. Topologically, we have **Proposition 4**. *Let $M$ be a Sasaki-Yamazaki fiber join as described above. Then* 1. *$M$ is a $S^{2d+1}$ bundle over $N$ with a $d+1$ dimensional Sasaki cone. Moreover,* 2. *if $d\geq n$ then $M$ has the cohomology groups of the product $S^{2d+1}\times N$; whereas,* 3. *if $d<n$ then the Euler class of the bundle does not vanish, and the Betti numbers satisfy $b_{2d+2i}(M)=b_{2d+2i}(N)-1$ where $i=1,\ldots,n-d$.* *Proof.* That $M$ is an $S^{2d+1}$ bundle follows from the construction, and Theorem 3.4 in [@BoTo20b] shows that $M$ admits a $d+1$ dimensional family of Sasakian structures. When $d\geq n$ the Euler class of the bundle vanishes and the Leray-Serre spectral sequence collapses giving the product groups in the limit. However, if $d<n$ with $M$ having a Sasakian structure, the odd Betti numbers less than half the dimension must be even (cf. [@BG05]). Moreover, the odd Betti numbers of $N$ are also even, and the even Betti numbers are greater than zero. So if the Euler class vanishes the orientation class $\alpha$ of the sphere which lies in the $E^{0,2d+1}_2$ term of the spectral sequence would survive to infinity which would imply that the Betti number $b_{2d+1}$ is odd. This contradicts the fact that $M$ has a Sasakian structure since $2d+1<2n< \frac{1}{2}\dim~M$. Thus, the Euler class, which is represented by the differential $d_{2d+2}(\alpha)$, cannot vanish in this case. So the real class $d_{2d+2}(\alpha)\in E^{2d+2,0}_\infty$ is killed which reduces the $(2d+2)th$ Betti number by one. The other equalities follow from this and naturality of the differential. ◻ The Euler class of the bundle is $\omega^{d+1}$ where $\omega$ is an integral Kähler form on $N$. We want to determine the conditions under which a sphere bundle is a fiber join. It is convenient to think of this in terms of $G$-structures. An oriented $S^{2d+1}$-bundle over $N$ is an associated bundle to a principal bundle with group $SO(2d+2)$. **Proposition 5**. *An $S^{2d+1}$-bundle $S(E)$ over a smooth projective algebraic variety $N$ is of the form $S(\oplus_iL_i)$ if and only if the group of the corresponding principal bundle is the maximal torus ${\mathbb T}^{d+1}_{\mathbb C}$. Moreover, this is a Sasaki-Yamazaki fiber join if there is a choice of complex line bundles $L_i$ such that $c_1(L_i^*)$ is positive definite for all $i=1,\ldots,d+1$.* *Proof.* The only if part is clear. Conversely, let $M$ be the total space of the unit sphere bundle in a complex vector bundle $E$ over a smooth projective algebraic variety $N$. Assume that the structure group of $E$ reduces to a maximal torus ${\mathbb T}^{d+1}_{\mathbb C}$. Then $E$ is isomorphic to a sum of complex line bundles $\oplus_{i=1}^{d+1}L_i$. Assume further that the $L_i$ can be chosen such that $c_1(L_i^*)$ is positive definite for $i=1,\ldots,d+1$. But this gives precisely the fiber join of the corresponding $S^1$ bundles over $N$. ◻ Let $M$ be a Sasaki-Yamazaki fiber join. Then as discussed above $M$ is an $S^{2d+1}$ bundle over a smooth projective algebraic variety $N$ for some $d\geq 1$. The Sasakian structure on $M$ restricts to the standard weighted Sasakian structure on each fiber $S^{2d+1}$. When the weights are integers, it is convenient to describe this by the following commutative diagram of $S^1$ actions labelled by a weight vector ${\bf w}$: $$\label{bundiag} \xymatrix{ S^1_{\bf w}\ar[d]^{id} \ar[r] &S^{2d+1}\ar[d] \ar[r] &{\mathbb C}{\mathbb P}^d[{\bf w}] \ar[d] \\ S^1_{\bf w}\ar[r]&M_{\mathfrak w}\ar[r] \ar[d] &{\mathbb P}_{\bf w}(\oplus_{j=1}^{d+1}L^*_j)\ar[d] \\ &N \ar[r]^{id} &N.}$$ ## Quasi-regular Quotients when $d=1$ {#qrquotients} For the case $d=1$ and co-prime ${\bf w}= (w_1,w_2) \in ({\mathbb Z}^+)^2$, we want to understand ${\mathbb P}_{\bf w}(\oplus_{j=1}^{d+1}L^*_j)$ in the diagram [\[bundiag\]](#bundiag){reference-type="eqref" reference="bundiag"}. To this end we will follow the ideas in Section 3.6 of [@BoTo13]. Let $M_i^3 \rightarrow N$ denote the primitive principal $S^1$-bundle corresponding to the line bundle $L_i$. Here we assume that $N$ is a smooth projective algebraic manifold. So $c_1(L_i^*)$ equals some (negative) integer $d_i$ times a primitive cohomology class that in turns defines $M_i^3$. \[Recall that $L_i$ has to be a positive line bundle over $N$.\] Consider the $S^1 \times S^1 \times {\mathbb C}^*$ action ${\mathcal A}_{{\bf w}, L_1,L_2}$ on $M_1^3\times M_2^3 \times {\mathbb C}^2$ defined by $${\mathcal A}_{{\bf w}, L_1,L_2}(\lambda_1,\lambda_2,\tau)(x_1,u_1,x_2,u_2;z_1,z_2)=(x_1, \lambda_1 u_1, x_2, \lambda_2 u_2; \tau^{w_1} \lambda_1^{d_1} z_1, \tau^{w_2} \lambda_2^{d_2}z_2),$$ where $\lambda_1,\lambda_2,\tau\in {\mathbb C}^*$ and $|\lambda_i|=1$. Then ${\mathbb P}_{\bf w}(L^*_1\oplus L^*_2)$ should equal $$M_1^3\times M_2^3 \times {\mathbb C}^2/{\mathcal A}_{{\bf w}, L_1,L_2}(\lambda_1,\lambda_2,\tau).$$ Now, we also can define a $w_1w_2$-fold covering map $\tilde{h}_{\bf w}: M_1^3\times M_2^3 \times {\mathbb C}^2 \rightarrow M_1^3\times M_2^3 \times {\mathbb C}^2$ by $$\tilde{h}(x_1,u_1,x_2,u_2;z_1,z_2)= (x_1,u_1,x_2,u_2;z_1^{w_2},z_2^{w_1})$$ and this gives a commutative diagram $$\xymatrix{M_1^3\times M_2^3 \times {\mathbb C}^2 \ar[d]^{\tilde{h}_{\bf w}}& \xrightarrow{{\mathcal A}_{{\bf w}, L_1,L_2}\left(\lambda_1,\lambda_2,\tau\right)} & M_1^3\times M_2^3 \times {\mathbb C}^2 \ar[d]^{\tilde{h}_{\bf w}}\\ M_1^3\times M_2^3 \times {\mathbb C}^2& \xrightarrow{{\mathcal A}_{{\bf 1}, L_1^{w_2},L_2^{w_1}}\left(\lambda_1,\lambda_2,\tau^{w_1w_2}\right)}& M_1^3\times M_2^3 \times {\mathbb C}^2}$$ and so we have a fiber preserving biholomorphism $h_{\bf w}: {\mathbb P}_{\bf w}(L^*_1\oplus L^*_2) \rightarrow {\mathbb P}((L^*_1)^{w_2}\oplus (L^*_2)^{w_1})$ and we can write ${\mathbb P}_{\bf w}(L^*_1\oplus L^*_2)$ as the log pair $({\mathbb P}((L^*_1)^{w_2}\oplus (L^*_2)^{w_1}),\Delta_{\bf w})$, where $\Delta_{\bf w}= (1-1/w_1)D_1 + (1-1/w_2)D_2$ and $D_1,D_2$ are the zero and infinity sections, respectively, of the bundle ${\mathbb P}((L^*_1)^{w_2}\oplus (L^*_2)^{w_1}) \rightarrow N$. **Remark 6**. Note that if ${\bf w}= (1,1)$ this checks out with the usual regular quotient. If the principal bundles $M_1^3$ and $M_2^3$ are equal, we can choose $(w_1,w_2)=(d_1,d_2)/a$ with $a=-\gcd(|d_1|,|d_2|)$ to get that $(L^*_1)^{w_2}=(L^*_2)^{w_1}$ and so ${\mathbb P}((L^*_1)^{w_2}\oplus (L^*_2)^{w_1})$ is trivial and the quasi-regular quotient is a product as expected from Proposition 3.8 of [@BoTo20b] By utilizing the set-up in Section A.3 of [@BoTo20b] we can also determine the quasi-regular Kähler class (up to scale) in the case with $d=1$ and co-prime ${\bf w}= (w_1,w_2) \in ({\mathbb Z}^+)^2$ as above. Indeed, from (9) of [@BoTo20b] we have that $$\label{quasiKahlerClass} w_1 w_2 d\eta_{\bf w}= w_2(r_1^2d\eta_1 + 2(r_1dr_1\wedge (\eta_1+d\theta_1)))+w_1(r_2^2d\eta_2 + 2(r_2dr_2\wedge (\eta_2+d\theta_2))),$$ where $(r_j,\theta_j)$ denote the polar coordinates on the fiber of the line bundle $L^*_j$ (chosen via a Hermitian metric on the line bundle). As explained in Section A.3 of [@BoTo20b], we can say that $z_0:= \frac{1}{2}r_1^2$ and $z_\infty:= \frac{1}{2}r_2^2$ are the moment maps of the natural $S^1$ action on $L^*_1$ and $L^*_2$, respectively. On $2=z_0+z_\infty$, the function $z:= z_0-1=1-z_\infty$ descends to a fiberwise moment map (with range \[-1,1\]) for the induced $S^1$ action on ${\mathbb P}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}\oplus (L_1)^{w_2}\otimes (L^*_2)^{w_1}) \rightarrow N$. Using that $r_1^2=2(z+1)$, $r_2^2=2(1-z)$, $r_1\,dr_1=dz$, and $r_2\,dr_2=-dz$, we rewrite [\[quasiKahlerClass\]](#quasiKahlerClass){reference-type="eqref" reference="quasiKahlerClass"} to $$w_1 w_2 d\eta_{\bf w}= 2(w_2 d\eta_1+w_1 d\eta_2) + 2d(z\theta),$$ where $\theta := w_2(\eta_1+d\theta_1)-w_1(\eta_2+d\theta_2)$ is a connection form on $(L_1)^{w_2}\otimes (L^*_2)^{w_1}$. Now this descends to a Kähler form on ${\mathbb P}((L^*_1)^{w_2}\oplus (L^*_2)^{w_1})={\mathbb P}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}\oplus (L_1)^{w_2}\otimes (L^*_2)^{w_1}) \rightarrow N$ with Kähler class $2(2\pi(w_2[\omega_1] + w_1[\omega_2]) + \Xi)$ where $c_1(L_j)=[\omega_j]$ and $\Xi/(2\pi)$ is the Poincare dual of $(D_1+D_2)$. We can summarize our findings for $d=1$ in the following proposition. **Proposition 7**. *For $d=1$ and co-prime ${\bf w}= (w_1,w_2) \in ({\mathbb Z}^+)^2$, the quasi-regular quotient of $M_{\mathfrak w}$ with respect to $\xi_{\bf w}$ is the log pair $B_{{\mathfrak w}, {\bf w}}:=({\mathbb P}((L^*_1)^{w_2}\oplus (L^*_2)^{w_1}),\Delta_{\bf w})$, where $\Delta_{\bf w}= (1-1/w_1)D_1 + (1-1/w_2)D_2$ and $D_1,D_2$ are the zero and infinity sections, respectively, of the bundle ${\mathbb P}((L^*_1)^{w_2}\oplus (L^*_2)^{w_1}) \rightarrow N$. Moreover, up to scale, the induced transverse Kähler class on $B_{{\mathfrak w}, {\bf w}}$ is equal to $2\pi(w_2[\omega_1] + w_1[\omega_2]) + \Xi$ where $c_1(L_j)=[\omega_j]$ and $\Xi/(2\pi)$ is the Poincare dual of $(D_1+D_2)$.* **Remark 8**. We can do the following sanity check: If colinearity (see [@BoTo20b] for the definition) holds on top of the above assumptions, we have according to Proposition 15 of [@BoTo20b] that the fiber join is just a regular $S^3_{\tilde{{\bf w}}}$-join as in [@BoTo14a]. Here $\omega_i=b_i \omega_N$ for $[\omega_N]$ a primitive integer Kähler class. Connecting with the notation in [@BoTo14a] (setting $w_i$ from [@BoTo14a] equal to $\tilde{w}_i$) we have $l_1(\tilde{w}_1,\tilde{w}_2)=(b_1,b_2)$ and $l_2=1$. Now Proposition [Proposition 7](#d=1quasiregular){reference-type="ref" reference="d=1quasiregular"} is consistent with Theorem 3.8 of [@BoTo14a] (with $v_i=w_i$) saying that the quotient of $\xi_{\bf w}$ has $n= b_1w_2-b_2w_1$. Moreover, the transverse Kähler class is then $$2\pi (w_2[\omega_1]+w_1[\omega_2]) + \Xi = 2\pi (b_1w_2+b_2w_1) [\omega_N] +\Xi = \frac{1}{r} [\omega_{N_n}] + \Xi$$ with $r_a=\frac{b_1w_2-b_2w_1}{b_1w_2+b_2w_1}$ and $[\omega_{N_n}]:=2\pi n [\omega_N]=c_1((L_1)^{w_2}\otimes (L^*_2)^{w_1})$. This is consistent with (44) and (59) in [@BoTo14a]. ## The General $d$ Case {#gendsec} For the fiber join $M_{\mathfrak w}$ we have in particular that the complex manifold arising as the quotient of the regular Reeb vector field $\xi_1$ is equal to ${\mathbb P}\left(\oplus_{j=1}^{d+1} L^*_j\right) \rightarrow N$. Recall from [@BoTo20b] that this is an *admissible* projective bundle as defined in [@ACGT08] exactly when the following all hold true: 1. The base $N$ is a local product of Kähler manifolds $(N_a,\Omega_{N_a})$, $a \in {\mathcal A}\subset {\mathbb N}$, where ${\mathcal A}$ is a finite index set. This means that there exist simply connected Kähler manifolds $N_{a}$ of complex dimension $d_a$ such that $N$ is covered by $\prod_{a\in {\mathcal A}}N_a$. On each $N_a$ there is an $(1,1)$ form $\Omega_{N_a}$, which is a pull-back of a tensor (also denoted by $\Omega_{N_a}$) on $N$, such that $\Omega_{N_a}$ is a Kähler form of a constant scalar curvature Kähler (CSCK) metric $g_a$. 2. There exist $d_0, d_\infty \in {\mathbb N}\cup \{0\}$, with $d=d_0+d_\infty +1$, such that $E_0:= \oplus_{j=1}^{d_0+1} L^*_j$ and $E_\infty := \oplus_{j=d_0+2}^{d_0+d_\infty +2} L^*_j$ are both projectively flat hermitian holomorphic vector bundles. *This would, for example, be true if $L^*_j= L_0$ for $j=1,...,d_0+1$ and $L^*_j= L_\infty$ for $j=d_0+2,...,d_0+d_\infty+2$, where $L_0$ and $L_\infty$ are some holomorphic line bundles. That is, $E_0=L_0\otimes {\mathbb C}^{d_0+1}$ and $E_\infty = L_\infty\otimes{\mathbb C}^{d_\infty+1}$. More generally, $c_1(L^*_1) = \cdots = c_1(L^*_{d_0+1})$ and $c_1(L^*_{d_0+2})= \cdots = c_1(L^*_{d_0+d_\infty+2})$ would be sufficient.* 3. $\frac{c_1(E_\infty)}{d_\infty+1}-\frac{c_1(E_0)}{d_0+1}= \sum_{a\in {\mathcal A}} [\epsilon_a\Omega_{N_a}]$, where $\epsilon_a=\pm 1$. The Kähler cone of the total space of an admissible bundle ${\mathbb P}\left( E_0\oplus E_\infty\right) \rightarrow N$ has a subcone of so-called **admissible Kähler classes** (defined in Section 1.3 of [@ACGT08]). This subcone has dimension $|{\mathcal A}|+1$ and, in general, this is not the entire Kähler cone. However, by Remark 2 in [@ACGT08], if $b_2(N_a)=1$ for all $a\in {\mathcal A}$ and $b_1(N_a) \neq 0$ for at most one $a\in {\mathcal A}$, then the entire Kähler cone is indeed admissible. ## Admissibility As briefly discussed in [@BoTo20b], it is convenient to have refined notions of admissibility. **Definition 9**. Any fiber join $M_{\mathfrak w}$ where the quotient of the regular Reeb vector field $\xi_1$ is an admissible projective bundle will also be called **admissible**. If further the transverse Kähler class of the regular quotient is a pullback of an admissible Kähler class, then we call $M_{\mathfrak w}$ **strongly admissible**. **Remark 11**. Note that in Definition 4.1 of [@BoTo20b] we introduced the condition of being **super admissible**. There we required the entire Kähler cone of the regular admissible quotient to be admissible. Of course, if that is the case then in particular the transverse Kähler class of the regular quotient is a pullback of an admissible Kähler classes. Thus $M_{\mathfrak w}$ is strongly admissible if it is super admissible. In fact we have **Proposition 10**. *Generally the inclusions $${\rm super~ admissible} \subset {\rm strongly~ admissible} \subset {\rm admissible}$$ are proper.* The proof of this proposition is a consequence of either of the Examples [Example 1](#admnotstrong){reference-type="ref" reference="admnotstrong"} or [Example 2](#2ndex){reference-type="ref" reference="2ndex"} below. **Example 1**. Let $\Sigma_g$ be a Riemann surface of genus $g>1$ and let $\omega_{\Sigma_g}$ denote the unit area Kähler form of the constant scalar curvature Kähler metric on $\Sigma_g$. Now consider $N=\Sigma_g\times\Sigma_g$ (i.e. $N_1=N_2=\Sigma_g$) and let $\pi_a$ denote the projection from $N$ to the $a^{th}$ factor. Then $\gamma_a:=[\pi_a^*\omega_{\Sigma_g}]\in H^2(N,{\mathbb Z})$. Let $\delta \in H^2(N,{\mathbb Z})$ denote the Poincaré dual of the diagonal divisor in $N$ defined by the diagonal curve $\{(x,x)\,|\,x\in \Sigma_g\}$. Then from Theorem 3.1 of [@Shu] (which uses Nakai's criterion for ample divisors), we know that $l_s:=(s-1)(\gamma_1+\gamma_2)+\delta \in H^2(N,{\mathbb Z})$ is in the Kähler cone of $N$ if and only if $s>g$. Now we form a $d=1$ Yamazaki fiber join by choosing line bundles $L_1$ and $L_2$ over $N$ such that $c_1(L_1)=[\omega_1]=l_{g+2}$ and $c_1(L_2)=[\omega_2]=l_{g+1}$. In the above setting $L_1^*=E_0$ and $L_2^*=E_\infty$ and we easily see that the fiber join is indeed admissible with $c_1(E_\infty)-c_1(E_0)=c_1(L_1)-c_1(L_2)=l_{g+2}-l_{g+1}=\gamma_1+\gamma_2$. Specifically, the regular quotient equals the admissible bundle $$S_g:={\mathbb P}\left(L_1^*\oplus L_2^*\right) \rightarrow N={\mathbb P}\left(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}\oplus L_1\otimes L_2^*\right) \rightarrow N= {\mathbb P}\left(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}\oplus {\mathcal O}(1,1)\right) \rightarrow \Sigma_g\times\Sigma_g.$$ Note that with the above notation $[\Omega_{N_a}]=\gamma_a$ (and $\epsilon_a=+1$). On $S_g$, the admissible Kähler classes are up to scale of the form $$\frac{1}{x_1}[\Omega_{N_1}]+\frac{1}{x_2}[\Omega_{N_2}]+\Xi,$$ where $0<x_a<1$ (following Section 1.3 of [@ACGT08]). According to Proposition [Proposition 7](#d=1quasiregular){reference-type="ref" reference="d=1quasiregular"}, the regular transverse Kähler class is, up to scale, the pull-back of $2\pi([\omega_1] + [\omega_2]) + \Xi$. This equals $$2\pi(l_{g+2}+l_{g+1})+\Xi=2\pi((2g+1)(\gamma_1+\gamma_2)+2\delta)+\Xi=2\pi\bigl((2g+1)[\Omega_{N_1}]+(2g+1) [\Omega_{N_2}]+2\delta\bigr)+\Xi$$ which due to the "$2\delta$" bit is not an admissible Kähler class. Therefore, the fiber join is not strongly admissible. Furthermore, it is possible to chose the line bundles $L_1$ and $L_2$ so that the fiber join is strongly admissible (cf. Section 5), but it will never be super admissible due to the fact that the Kähler cone of $N$ consist of more than just product classes and thus there are non-admissible Kähler classes on the total space of the ${\mathbb C}{\mathbb P}^1$-bundle of the regular quotient. Hence, the inclusions in Proposition [Proposition 10](#properincl){reference-type="ref" reference="properincl"} are proper. **Example 2**. Another example of admissible but not strongly admissible is the following case. Let $N={\mathbb P}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}\oplus {\mathcal O}(1,-1)) \rightarrow {\mathbb C}{\mathbb P}^1\times{\mathbb C}{\mathbb P}^1$. Let $\Omega_{FS}$ denote the standard Fubini-Study Kähler form on ${\mathbb C}{\mathbb P}^1$, let $\pi_i$ denote the projection from $N$ to the $i^{th}$ factor in the product ${\mathbb C}{\mathbb P}^1\times{\mathbb C}{\mathbb P}^1$, and let $\chi$ denote the Poincaré dual of $2\pi(D^N_1+D^N_2)$, where $D^N_1$, $D^N_2$ are the zero and infinity sections of $N \rightarrow {\mathbb C}{\mathbb P}^1\times{\mathbb C}{\mathbb P}^1$. Now consider the two CSC Kähler forms $\omega_1$ and $\omega_2$ on $N$ with Kähler classes $$[\omega_1]=2\left(3[\pi_1^*\Omega_{FS}] +3[\pi_2^*\Omega_{FS}] +\frac{\chi}{2\pi}\right)$$ and $$[\omega_2]=2[\pi_1^*\Omega_{FS}] +2[\pi_2^*\Omega_{FS}] +\frac{\chi}{2\pi},$$ respectively. (See e.g. Theorem 9 in [@ACGT08] to confirm that $[\omega_1]$ and $[\omega_2]$ are indeed represented by CSC Kähler forms.) Now we form a $d=1$ Yamazaki fiber join by choosing line bundles $L_1$ and $L_2$ over $N$ such that $c_1(L_1)=[\omega_1]$ and $c_1(L_2)=[\omega_2]$. In the above setting $L_1^*=E_0$ and $L_2^*=E_\infty$ and we easily see that the fiber join is indeed admissible with $c_1(E_\infty)-c_1(E_0)=c_1(L_1)-c_1(L_2)=4[\pi_1^*\Omega_{FS}] +4[\pi_2^*\Omega_{FS}] +\frac{\chi}{2\pi}$. Specifically, the regular quotient equals the admissible bundle $$S:{\mathbb P}\left(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}\oplus L\right) \rightarrow N$$ such that $c_1(L)= [\Omega_{N}]:=4[\pi_1^*\Omega_{FS}] +4[\pi_2^*\Omega_{FS}] +\frac{\chi}{2\pi}$ and $\Omega_N$ is a CSC Kähler form on $N$. Note that $S$ is a so-called *stage four Bott manifold* given by the matrix $$A=\begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 1&-1&1&0\\ 5&3&2&1 \end{pmatrix}.$$ \[See e.g. Section 1 of [@BoCaTo17] for details.\] It is important to note that the CSC Kähler manifold $(N,\Omega_N)$ is irreducible in the sense that for (1) at the beginning of Subsection [3.2](#gendsec){reference-type="ref" reference="gendsec"}, ${\mathcal A}$ must be just $\{1\}$. Following Section 1.3. in [@ACGT08], we have that on $S$, the admissible Kähler classes are up to scale of the form $$\frac{2\pi}{x}[\Omega_{N}]+\Xi,$$ where $0<x<1$, $\Xi$ denote the Poincare dual of $2\pi(D_1+D_2)$, and $D_1,D_2$ are the zero and infinity sections, respectively, of the bundle ${\mathbb P}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}\oplus L) \rightarrow N$. According to Proposition [Proposition 7](#d=1quasiregular){reference-type="ref" reference="d=1quasiregular"}, the regular transverse Kähler class is, up to scale, the pull-back of $2\pi([\omega_1] + [\omega_2]) + \Xi$. This equals $$2\pi(8[\pi_1^*\Omega_{FS}] +8[\pi_2^*\Omega_{FS}]+3 \frac{\chi}{2\pi}) + \Xi$$ which cannot be written as (the rescale of) $\frac{2\pi}{x}[\Omega_{N}]+\Xi$ for any $0<x<1$. Thus this is not an admissible Kähler class and therefore the fiber join is not strongly admissible. ## The Main Theorems {#mainthm} For Theorems [Theorem 12](#1stroundexistence){reference-type="ref" reference="1stroundexistence"} and [Theorem 14](#2ndroundexistence){reference-type="ref" reference="2ndroundexistence"} below, we only need the strongly admissible condition. In [@BoTo20b] we used the above observations together with existence results in [@Gua95], [@Hwa94], and [@HwaSi02] (specifically, the slight generalization in the form of Propostion 11 of [@ACGT08]) to prove the following theorem: **Theorem 12** ([@BoTo20b]). *Let $M_{\mathfrak w}$ be a strongly admissible fiber join whose regular quotient is a ruled manifold of the form ${\mathbb P}(E_0\oplus E_\infty)\longrightarrow N$ where $E_0 ,E_\infty$ are projectively flat hermitian holomorphic vector bundles on $N$ of complex dimension $(d_0+1),(d_\infty+1)$ respectively, and $N$ is a local Kähler product of non-negative CSC metrics. Then the Sasaki cone of $M_{\mathfrak w}$ has an open set of extremal Sasaki metrics (up to isotopy).* Together with E. Legendre and H. Huang we recently obtained the following result on admissible Kähler manifolds: **Theorem 13** (Theorem 3.1 in [@BHLT22]). *Suppose $\Omega$ is a rational admissible Kähler class on the admissible manifold $N^{ad}={\mathbb P}(E_0 \oplus E_{\infty}) \longrightarrow N$, where $N$ is a compact Kähler manifold which is a local product of nonnegative CSCK metrics. Let $(M,{\oldmathcal S})$ be the Boothby-Wang constructed Sasaki manifold given by an appropriate rescale of $\Omega$. Then the corresponding Sasaki-Reeb cone will always have a (possibly irregular) CSC-ray (up to isotopy).* The proof of this theorem (Section 3.1 of [@BHLT22]) reveals that this CSC Sasaki metric lies in a 2-dimensional subcone of ${\mathfrak t}^+({\oldmathcal S})$ which is exhausted by extremal Sasaki metrics. Further, since this subcone is constructed via Killing potentials coming from a moment map induced by a fiber wise $S^1$-action on the admissible bundle, it is clear that this is also a subcone of ${\mathfrak t}^+_{sph}$ (and all of ${\mathfrak t}^+_{sph}$ when $d=1$). Recall from Section 2.2 of [@BoTo20b] that ${\mathfrak t}^+_{sph}$ is defined to be the natural $(d+1)$-subcone of the Sasaki-Reeb cone of $M_{\mathfrak w}$ coming from considering the standard Sasaki CR structure on $S^{2d+1}$. In light of all this, we can thus easily improve Theorem [Theorem 12](#1stroundexistence){reference-type="ref" reference="1stroundexistence"} to give Theorem [Theorem 1](#2ndroundexistenceintro){reference-type="ref" reference="2ndroundexistenceintro"} in the Introduction, namely **Theorem 14**. *Let $M_{\mathfrak w}$ be a strongly admissible Yamazaki fiber join whose regular quotient is a ruled manifold of the form ${\mathbb P}(E_0\oplus E_\infty)\longrightarrow N$ where $E_0 ,E_\infty$ are projectively flat hermitian holomorphic vector bundles on $N$ of complex dimension $(d_0+1),(d_\infty+1)$ respectively, and $N$ is a local Kähler product of non-negative CSC metrics. Then ${\mathfrak t}^+_{sph}$ has a $2$-dimensional subcone of extremal Sasaki metrics (up to isotopy) which contains at least one ray of CSC Sasaki metrics.* # Further Examples {#Exsect} In this section we work out the details of examples of fiber joins in dimensions 5 and 7. We consider only the case with $d=1$, i.e. $d_0=d_\infty=0$. So we have an $S^3$ bundle $M$, which we shall assume to be strongly admissible, over a smooth projective algebraic variety $N$. We begin with the simplest case, namely where $N$ is a Riemann surface, so the simplest fiber join is of dimension 5. Even in this case the geometry is quite involved. Note that the genus $g=0$ case is a straightforward special case of Theorem [Theorem 14](#2ndroundexistence){reference-type="ref" reference="2ndroundexistence"} whose Sasaki cone is strictly larger than ${\mathfrak t}^+_{sph}$; hence, we concentrate on $g\geq 1$. In this case the fiber ${\mathbb C}{\mathbb P}^d[{\bf w}]$ is the log pair $({\mathbb C}{\mathbb P}^1,\Delta_{\bf w})$ with branch divisors $$\Delta_{\bf w}=\bigl(1-\frac{1}{w_1}\bigr)D_1 +\bigl(1-\frac{1}{w_2}\bigr)D_2.$$ Here we have $c_1(L_\infty)-c_1(L_0)=\sum_a[\epsilon_a\Omega_a]$. In order to construct a non-colinear fiber join of this kind we must have the Picard number $\rho(N)\geq 2$. In this case we may see the rays determined by $\xi_{\bf w}$ explicitly as $CR$-twists of the regular quotient [@ApCa18]. Indeed, following the notation of Section 3 of [@BHLT22], on the regular quotient, $N^{ad} = {\mathbb P}\bigl({\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}} \oplus (L_0^*\otimes L_\infty)\bigr) \rightarrow N$, we have a moment map ${\mathfrak z}: N^{ad} \rightarrow [-1,1]$. A choice of $c\in (-1,1)$ creates a new Sasaki structure (with Reeb vector field $\xi_c$) on $M_{\mathfrak w}$ via the lift of $f=c{\mathfrak z}+1$ from $S$ to $M_{\mathfrak w}$. In turn, this lift may be identified with $c\,z+1$, where $z:= z_0-1=1-z_\infty$ is given in the discussion above Proposition [Proposition 7](#d=1quasiregular){reference-type="ref" reference="d=1quasiregular"}. In particular, $z_0$ and $z_\infty$ are the moment maps of the natural $S^1$ action on $L^*_1$ and $L^*_2$, respectively. Thus, the weighted combination, $w_1 z_0+w_2 z_\infty$, should define the Reeb vector field $\xi_{\bf w}$ and since $$w_1 z_0+w_2 z_1 = (w_1-w_2) z +(w_1+w_2)=(w_1+w_2)(\frac{w_1-w_2}{w_1+w_2} z +1),$$ we see that (up to scale) $\xi_{\bf w}$ corresponds to choosing $c=\frac{w_1-w_2}{w_1+w_2}$ in the $CR$-twist. ## $N=\Sigma_g$, a compact Riemann surface of genus $g\geq1$. It is well known that if $N=\Sigma_g$, a Riemann surface of genus $g$, then an odd dimensional sphere bundle $M$ over $N$ is diffeomorphic to the trivial bundle $S^{2d+1}\times \Sigma_g$ or the unique non-trivial bundle $S^{2d+1}\tilde{\times} \Sigma_g$ [@Stee51]. We will consider $d=1$ fiber joins over $N=\Sigma_g$. Since these are necessarily colinear, they have earlier been treated as $S^3_{\bf w}$ joins [@BoTo13], but not in the setting of Yamazaki fiber joins. Let $\omega_{\Sigma_g}$ denote the unit area Kähler form of the constant scalar curvature Kähler metric on $\Sigma_g$ and let $k_1>k_2>0$ be integers (the case $0<k_1< k_2$ is completely similar) and let $L_1,L_2$ be holomorphic line bundles over $\Sigma_g$ such that $c_1(L_i)=k_i [\omega_{\Sigma_g}]$. The corresponding $d=1$ Yamazaki fiber join, $M_{\bf k}=S(L_1^*\oplus L_2^*) \rightarrow \Sigma_g$ has regular quotient $S_{\bf n}= {\mathbb P}\bigl({\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}} \oplus {\mathcal O}(k_1-k_2)\bigr) \rightarrow \Sigma_g$ and regular transverse Kähler class equal (up to scale) to the admissible Kähler class $2\pi(k_1+k_2)[\omega_{\Sigma_g}] + \Xi$, which we can write as $\frac{1}{x}\left( 2\pi (k_1-k_2)[\omega_{\Sigma_g}] \right)+\Xi$ with $x=\frac{k_1-k_2}{k_1+k_2}$. \[See Remark [Remark 8](#colinearsanitycheck){reference-type="ref" reference="colinearsanitycheck"}.\] Note that since $g\geq 1$, we have that the Sasaki cone equals the $2$-dimensional cone ${\mathfrak t}^+_{sph}$ We now follow Section 3 of [@BHLT22]. On the regular quotient, $S_{\bf n}$, we have a moment map ${\mathfrak z}: S_{\bf n}\rightarrow [-1,1]$. A choice of $c\in (-1,1)$ creates a new Sasaki structure (with Reeb vector field $\xi_c=f\,\xi_{\mathbf 1}$) on $M_{\mathfrak w}$ via the lift of $f=c{\mathfrak z}+1$ from $S_{\bf n}$ to $M_{\mathfrak w}$. From the discussion in the beginning of Section [4](#Exsect){reference-type="ref" reference="Exsect"} we know that $c=\frac{k_1-k_2}{k_1+k_2}=x$ corresponds to the Reeb vector field of the $S^3_{\tilde{{\bf w}}}$ join, $M^5_{g,l,\tilde{{\bf w}}}=S^3_g\star_{l,1}S^3_{\tilde{{\bf w}}}$ from Section 3.2 of [@BoTo13] with $S^3_g$ being the Boothby-Wang constructed smooth Sasaki structure over $(\Sigma_g,[\omega_{\Sigma_g}])$, $l=\gcd(k_1,k_2)$, and $(\tilde{w_1},\tilde{w_2})=(\frac{k_1}{l},\frac{k_2}{l})$. Since this Reeb vector field is extremal by construction, we know a priori that the set of extremal Sasaki rays in the Sasaki cone, ${\mathfrak t}^+_{sph}$ is not empty. Proposition 3.10 of [@BHLT22] tells us that the Reeb vector field determined - up to homothety - by $c\in (-1,1)$ (as explained in the beginning of Section [4](#Exsect){reference-type="ref" reference="Exsect"}) is extremal (up to isotopy) if and only if $F_c({\mathfrak z})>0$, for $-1<{\mathfrak z}<1$, where the polynomial $F_c({\mathfrak z})$ is given as follows: Let $s=\frac{2(1-g)}{k_1-k_2}$, $x=\frac{k_1-k_2}{k_1+k_2}$ and define $$\begin{array}{ccl} \alpha_{r,-4} & = & \int_{-1}^1(ct+1)^{-4}t^r(1+xt)\,dt\\ \\ \alpha_{r,-5} & = & \int_{-1}^1(ct+1)^{-5}t^r(1+xt)\,dt\\ \\ \beta_{r,-3} & = & \int_{-1}^1(ct+1)^{-3}xst^r\,dt \\ \\ &+ & (-1)^r(1-c)^{-3}(1-x) +(1+c)^{-3}(1+x). \end{array}$$ Then, $$\label{wextrpol5mnf} F_c({\mathfrak z})=(c{\mathfrak z}+1)^3 \left[ \frac{2(1-x)}{(1-c)^3}({\mathfrak z}+1) + \int_{-1}^{\mathfrak z}Q(t)({\mathfrak z}-t)\,dt\right],$$ where $$Q(t) = \frac{2xs}{(ct+1)^3} - \frac{(A_1 t+A_2)(1+xt)}{(ct+1)^5}$$ and $A_1$ and $A_2$ are the unique solutions to the linear system $$\label{wextrpol5mnf2} \begin{array}{ccl} \alpha_{1,-5}A_1+\alpha_{0,-5}A_2&=& 2\beta_{0,-3}\\ \\ \alpha_{2,-5}A_1+\alpha_{1,-5}A_2&=& 2\beta_{1,-3}. \end{array}$$ Further, if the positivity of $F_c({\mathfrak z})$ is satisfied, then the extremal Sasaki structure is CSC exactly when $$\label{scsc5mnf} \alpha_{1,-4}\beta_{0,-3} - \alpha_{0,-4}\beta_{1,-3}=0$$ is satisfied. The left hand side of [\[scsc5mnf\]](#scsc5mnf){reference-type="eqref" reference="scsc5mnf"} equals $\frac{4h(c)}{3(1-c^2)^5}$, with polynomial $h(c)=x(sx-2) +(5+x^2-sx)c-x(6+s x)c^2-(1-sx-3x^2)c^3$ and $h(\pm 1)=\pm 4(1\mp x)^2$. Thus, since $h(c)$ is negative at $c=-1$ and positive at $c=1$, [\[scsc5mnf\]](#scsc5mnf){reference-type="eqref" reference="scsc5mnf"} always has at least one solution $c\in (-1,1)$. We calculate $F_c({\mathfrak z})$: $$F_c({\mathfrak z})=\frac{(k_1+k_2)^2(1-{\mathfrak z}^2)p({\mathfrak z})}{4((1 - c)^2 k_1^2 + (1 + c)^2 k_2^2 + 4 (1 - c^2) k_1 k_2)},$$ where $p({\mathfrak z})$ is a polynomial of degree $2$ whose coefficients depend on $k_1,k_2,g$ and $c$, but is more conveniently written as $$\begin{array}{ccl} p({\mathfrak z})&=& c^2 s x+3 c^2 x^2-c^2-2 c s x^2+3 c x^3-7 c x+s x^3-4 x^2+6\\ \\ &+& 2 x \left(3 c^2 x^2-c^2-4 c x-x^2+3\right){\mathfrak z}\\ \\ &+& (c-x) \left(-c s x+3 c x^2-c+s x^2-2 x\right){\mathfrak z}^2, \end{array}$$ where $s=\frac{2(1-g)}{k_1-k_2}$, $x=\frac{k_1-k_2}{k_1+k_2}$. Clearly $F_c({\mathfrak z})>0$ for all ${\mathfrak z}\in (-1,1)$ exactly when $p({\mathfrak z})>0$ for all ${\mathfrak z}\in (-1,1)$. We have arrived at **Proposition 15**. *Consider the $d=1$ fiber join $S^3\longrightarrow M_{\bf k}\longrightarrow\Sigma_g$ over a Riemann surface $\Sigma_g$ of genus $g\geq1$ with its natural Sasakian structure ${\oldmathcal S}_c$ as described above. Then ${\oldmathcal S}_c$ is extremal (up to isotopy) if and only if $p({\mathfrak z})>0$ for all ${\mathfrak z}\in (-1,1)$.* Note that $p(-1)=\frac{8k_2((1 - c)^2 k_1^2 + (1 + c)^2 k_2^2 + 4 (1 - c^2) k_1 k_2)}{(k_1+k_2)^3}$ and $p(1)=\frac{8k_1((1 - c)^2 k_1^2 + (1 + c)^2 k_2^2 + 4 (1 - c^2) k_1 k_2)}{(k_1+k_2)^3}>0$, thus $p(\pm 1)>1$, so we see right away that when $c=x$, $p({\mathfrak z})$ (which is now of degree one) is positive for $-1<{\mathfrak z}<1$. This confirms our expectation from above that $\xi_c$ is extremal when $c= \frac{k_1-k_2}{k_1+k_2}=x$. It is easy to check that for $g>\frac{31 k_1^2+14 k_1 k_2+k_1+k_2^2+k_2}{k_1+k_2}$ and $c=\frac{k_1}{k_1+k_2}$, $p(0)<0$. Thus we see that for any fixed choice of integers $k_1>k_2>0$, ${\mathfrak t}^+_{sph}$ is not exhausted by extremal rays when $g$ is very large. This is expected in light of Theorem 5.1 in [@BoTo13]. From [@BoTo13] we have the following results: 1. (Proposition 5.5 in [@BoTo13] combined with Theorem 3 in [@ApCaLe21]) There is a unique ray in ${\mathfrak t}^+_{sph}$ with a CSC Sasaki metric (up to isotopy). 2. (Proposition 5.10 in [@BoTo13]) If $g\leq 1+3k_2$ then every ray in ${\mathfrak t}^+_{sph}$ has an extremal Sasaki metric (up to isotopy). In particular, this is true whenever $g\leq 4$. Statement (1) means that [\[scsc5mnf\]](#scsc5mnf){reference-type="eqref" reference="scsc5mnf"} has a unique solution $c\in (-1,1)$ (i.e. the cubic $h(c)$ above has a unique real root $c\in (-1,1)$) and for this unique solution, $p({\mathfrak z})>0$ for all ${\mathfrak z}\in (-1,1)$. An easy way to see the uniqueness of the real root directly from the present setup is to make a change of variable[^3] $c=\phi(b)=\frac{1-b}{1+b}$ \[$\phi: (0,+\infty) \rightarrow (-1,1)$\] Then $h(c)$ transforms to $\tilde{h}(b)$, where $$\tilde{h}(b)= \frac{4}{(b+1)^3}\left( (1-x)^2+(1-x)(2+2x-sx)b -(1+x)(2(1-x)-sx)b^2-(1+x)^2b^3\right).$$ Since the polynomial coefficients of the cubic $$(1-x)^2+(1-x)(2+2x-sx)b -(1+x)(2(1-x)-sx)b^2-(1+x)^2b^3$$ change sign exactly once (recall $sx\leq 0$ and $0<x<1$), we have (using Descartes' rule of signs) exactly one positive root $b\in (0,+\infty)$ (corresponding to a unique root $c\in(-1,1)$ of $h(c)$). Then too see that this $c$ value (let us call it $\hat{c}$) satisfies that $p({\mathfrak z})>0$ for all ${\mathfrak z}\in (-1,1)$ we can first observe that since $h(x)=3x(1-x^2)^2\neq 0$, $\hat{c}\neq x$. With that settled we may (solve for $s$ in $h(\hat{c})=0$ and) write $s=\frac{3 \hat{c}^3 x^2-\hat{c}^3-6 \hat{c}^2 x+\hat{c} x^2+5 \hat{c}-2 x}{(1-\hat{c}^2) x (\hat{c}-x)}$. Substituting this into $p({\mathfrak z})$ (and using that $x=\frac{k_1-k_2}{k_1+k_2}$) gives us $p({\mathfrak z})=\frac{4 ((1 - \hat{c})^2 k_1^2 + (1 + \hat{c})^2 k_2^2 + 4 (1 - \hat{c}^2) k_1 k_2)(1 + \hat{c} {\mathfrak z}) (1 - \hat{c} x -\hat{c}{\mathfrak z}+x{\mathfrak z})}{(1- \hat{c}^2)(k_1+k_2)^2}$. Since $0<x<1$ and $-1<\hat{c}<1$, it easily follows that $p({\mathfrak z})>0$ for $-1<{\mathfrak z}<1$. Similarly, statement (2) is (re)verified if we show that for $g\leq 1+3k_2$, $p({\mathfrak z})>0$ for all $c,{\mathfrak z}\in (-1,1)$. This is done easily by writing $p({\mathfrak z})$ in a new variable $y$: ${\mathfrak z}=\psi(y)=\frac{1-y}{1+y}$ ($0<y<+\infty$) along with using the above transformation $c=\phi(b)=\frac{1-b}{1+b}$. After multiplying by $(1+b)^2(1+y)^2$, this results in a polynomial in the two variables $b,y>0$. The coefficients of this polynomial are all non-negative (with some strictly positive) precisely when $g\leq 1+3k_i$ for $i=1,2$. Since (we assumed without loss of generality that) $k_1>k_2$, this is manifested by $g\leq 1+3k_2$. **Example 3**. Assume now that $k_2=1$ and $g=5$ or $g=6$. Thus $g\leq 1+3k_2$ is false and Statement (2) cannot be applied. Nevertheless we shall see that positivity of $p({\mathfrak z})$ for $-1<{\mathfrak z}<1$ still holds for all $k_1>1$: With $g=5$, $k_2=1$, $c=\phi(b)=\frac{1-b}{1+b}$, and ${\mathfrak z}=\psi(y)=\frac{1-y}{1+y}$, $p({\mathfrak z})$ rewrites to $$\frac{32 \left(b^2 k_1^2(k_1-y+y^2)+3 b k_1^2 y+4 b k_1^2+4 b k_1 y^2+11 b k_1 y+(3 k_1-4) y+k_1+y^2\right)}{(b+1)^2 (k_1+1)^3 (y+1)^2}.$$ Since $k_1\geq 2$ and $y,b>0$, it is easy to see that this is always positive. With $g=6$, $k_2=1$, $c=\phi(b)=\frac{1-b}{1+b}$, and ${\mathfrak z}=\psi(y)=\frac{1-y}{1+y}$, $p({\mathfrak z})$ rewrites to $$\frac{32 \left(b^2 k_1^2(k_1-2y+y^2)+3 b k_1^2 y+4 b k_1^2+4 b k_1 y^2+13 b k_1 y+(3 k_1-5) y+k_1+y^2\right)}{(b+1)^2 (k_1+1)^3 (y+1)^2}$$ Since $k_1\geq 2$ and $y,b>0$, we see also in this case that this is always positive. On the other hand, for $k_2\geq 2$, then $1+3k_2\geq 7$ and since $7$ is larger than both $5$ and $6$, we already know from Statement (2) above that positivity of $p({\mathfrak z})$ for $-1<{\mathfrak z}<1$ holds. In conclusion, when $g\leq6$ we have that for all integers $k_1>k_2>0$, every ray in ${\mathfrak t}^+_{sph}$ has an extremal Sasaki metric (up to isotopy). This improves the result we had in [@BoTo13]. Finally notice that when $g=7$, $k_1=2$, $k_2=1$ and $c=-\frac{299}{301}$, we get that $p(-\frac{1}{5})=-\frac{7794656}{61155675}<0$ and so positivity of $p({\mathfrak z})$ fails in this case. The case $k_1=k_2$ and $g\leq 6$ was already handled in Example 5.11 of [@BoTo13] (recall that $(k_1,k_2)=l(\tilde{w_1},\tilde{w_2})$ in the $S^3_{\tilde{{\bf w}}}$ join, $M^5_{g,l,\tilde{{\bf w}}}=S^3_g\star_{l,1}S^3_{\tilde{{\bf w}}}$). Similarly to the example above we had that every ray in ${\mathfrak t}^+_{sph}$ has an extremal Sasaki metric (up to isotopy). We can thus state the following result. **Proposition 16**. *Let ${\bf k}=(k_1,k_2)$ with $k_1\geq k_2> 0$ being integers and consider the Yamazaki fiber join $M_{\bf k}$ as described above. For $1\leq g \leq 6$ or $1\leq g \leq 1+3k_2$ we have that the entire Sasaki cone is extremal (up to isotopy).* ## $N={\mathbb C}{\mathbb P}^1\times {\mathbb C}{\mathbb P}^1$ {#NNsect} Let $\Omega_i$ denote the standard area forms on the ith copy of ${\mathbb C}{\mathbb P}^1$. With slight abuse of notation, we denote the pull-back of their Kähler classes to $H^2(N,{\mathbb Z})$ by $[\Omega_1]$ and $[\Omega_2]$. The Kähler cone of $N$ then equals $span_{{\mathbb R}^+}\{[\Omega_1],[\Omega_2]\}$. Let $M_{\mathfrak w}$ be a $d=1$ Yamazaki fiber join formed from a choice of Kähler classes which are represented by Kähler forms $$\label{kmatrixeqn} \omega_j=k^1_j\Omega_1 +k^2_j\Omega_2, \qquad k^1_j,k^2_j\in {\mathbb Z}^+,$$ for $j=1,2$. The line bundles $L_1, L_2$ satisfy that $c_1(L_j)=[\omega_j] = k^1_j[\Omega_1] +k^2_j[\Omega_2]$. So the choices of Kähler forms is given by the $2$ by $2$ matrix $$\label{Kmatrix} K= \begin{pmatrix} k^1_1 & k^2_1 \\ \\ k^1_{2} & k^2_{2}, \end{pmatrix}$$ and the fiber join is non-colinear exactly when $det\,K\neq0$. Now the quotient complex manifold of $M_{\mathfrak w}$ arising from the regular Sasakian structure with Reeb vector field $\xi_{\mathbf 1}$ is equal to the following ${\mathbb C}{\mathbb P}^1$ bundle over ${\mathbb C}{\mathbb P}^1\times{\mathbb C}{\mathbb P}^1$: $$\label{KSprojeqn} \notag {\mathbb P}\bigl(L_1^*\oplus L_2^*) = {\mathbb P}\bigl({\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}} \oplus L_1\otimes L_2^*\bigr) = {\mathbb P}\bigl({\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}} \oplus {\mathcal O}(k^1_1-k^1_2,k^2_1-k^2_2)\bigr) \rightarrow {\mathbb C}{\mathbb P}^1\times{\mathbb C}{\mathbb P}^1.$$ We assume here that $k^i_1\neq k^i_2$ for $i=1,2$. If we don't make this assumption our regular quotient could be a product of ${\mathbb C}{\mathbb P}^1$ with a Hirzebruch surface. This is not a problem per se, but needs to be treated slightly differently, so we will avoid this here. Every Kähler class on ${\mathbb P}\bigl({\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}} \oplus L_1\otimes L_2^*\bigr)$ is admissible in the broader sense of the definition given in [@ACGT08]. Thus the fiber join is super admissible and therefore strongly admissible. This case is hence a special case of Theorem [Theorem 14](#2ndroundexistence){reference-type="ref" reference="2ndroundexistence"}, with ${\mathfrak t}^+_{sph}$ a proper subcone of the (unreduced) Sasaki cone. Nevertheless we shall study this example in details since it will illustrate two different approaches for locating CSC ray(s) in ${\mathfrak t}^+_{sph}$. At the end of the section we will also discuss which polarized Kähler manifolds $(S_{\bf n},[\omega])$ of the form $S_{\bf n}={\mathbb P}\bigl({\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}} \oplus {\mathcal O}(n_1,n_2)\bigr) \rightarrow {\mathbb C}{\mathbb P}^1\times{\mathbb C}{\mathbb P}^1$ appear as regular quotients of a Sasaki Yamazaki fiber join. For $n_1,n_2 \in {\mathbb Z}\setminus \{0\}$, a Kähler class on the complex manifold $S_{\bf n}= {\mathbb P}\bigl({\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}} \oplus {\mathcal O}(n_1,n_2)\bigr) \rightarrow {\mathbb C}{\mathbb P}^1\times{\mathbb C}{\mathbb P}^1$ is, up to scale, of the form $2\pi(\frac{n_1}{x_1}[\Omega_1] +\frac{n_2}{x_2}[\Omega_2]) + \Xi)$, where $0<|x_i|<1$ and $x_in_i>0$. As we saw in Section 5.3.3 of [@BoTo20b], as well as in Section [3.1](#qrquotients){reference-type="ref" reference="qrquotients"} of the current paper, here we can calculate the quotient Kähler class up to scale and all in all we get a smooth admissible Kähler manifold with admissible data $$\label{regadmdata} n_1=k^1_1-k^1_2,\quad n_2=k^2_1-k^2_2, \quad x_1=\frac{k^1_1-k^1_2}{k^1_1+k^1_2}, \quad x_2=\frac{k^2_1-k^2_2}{k^2_1+k^2_2}.$$ Indeed, more generally, using Proposition [Proposition 7](#d=1quasiregular){reference-type="ref" reference="d=1quasiregular"} we have that for co-prime ${\bf w}= (w_1,w_2) \in ({\mathbb Z}^+)^2$ the quasi-regular quotient of $M_{\mathfrak w}$ with respect to $\xi_{\bf w}$ is the log pair $B_{{\mathfrak w}, {\bf w}}:=({\mathbb P}({\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}} \oplus {\mathcal O}(w_2k^1_1-w_1k^1_2,w_2 k^2_1-w_1 k^2_2)),\Delta_{\bf w})$. Together with the quotient Kähler class (up to scale, also from Proposition [Proposition 7](#d=1quasiregular){reference-type="ref" reference="d=1quasiregular"}) this gives (assuming $w_2k^i_1-w_1k^i_2\neq 0$) admissible data $$\label{qregadmdata} n_1=w_2k^1_1-w_1k^1_2,\quad n_2=w_2 k^2_1-w_1 k^2_2, \quad x_1=\frac{w_2 k^1_1- w_1 k^1_2}{w_2 k^1_1+w_1k^1_2}, \quad x_2=\frac{w_2k^2_1-w_1k^2_2}{w_2k^2_1+w_1k^2_2}.$$ Note that if $w_2k^i_1-w_1k^i_2=0$ for one of (or both) $i=1,2$, we get a product of ${\mathbb C}{\mathbb P}^1$ with a so-called Hirzebruch orbifold. From the discussion above we can see the rays, given up to scale by $\xi_{\bf w}$, as $CR$-twists of the regular quotient [@ApCa18]. So choosing $c=\frac{w_1-w_2}{w_1+w_2}$ creates a new Sasaki structure via the lift of $f=c{\mathfrak z}+1$ from $S_{\bf n}$ to $M_{\mathfrak w}$. With this correspondence in mind we can take two different approaches when seeking out rays in ${\mathfrak t}^+_{sph}$ with constant scalar curvature. From the $CR$-twist point of view, the Reeb vector field $\xi_c$ given by the $CR$-twist has a constant scalar curvature Sasaki metric (up to isotopy) exactly when Equation (10) from [@BHLT22] holds. Applying this equation to the regular quotient with admissible data from [\[regadmdata\]](#regadmdata){reference-type="eqref" reference="regadmdata"} yields the equation $f_{CR}(c)=0$, where $$\label{cscEQinc} \begin{split} f_{CR}(c) & := 18 c \left(c^2-1\right)^2 {k^1_1} {k^2_1} {k^1_2} {k^2_2}+3 (c-1)^5 (k^1_1)^2 (k^2_1)^2+3 (c+1)^5 (k^1_2)^2 (k^2_2)^2\\ &+(c+1) (c-1)^4 k^1_1k^2_1\left(k^1_1 + k^2_1-3 k^2_1{k^1_2}-3 k^1_1 {k^2_2}\right)\\ &+(c+1)^2 (c-1)^3 \left( (k^2_1)^2 {k^1_2} +(k^1_1)^2 {k^2_2}-4 {k^1_1} {k^2_1} {k^1_2}-4 {k^1_1} {k^2_1} {k^2_2} \right) \\ &+(c+1)^3 (c-1)^2\left( {k^1_1} (k^2_2)^2 + {k^2_1} (k^1_2)^2-4 {k^2_1} {k^1_2} {k^2_2}-4 {k^1_1} {k^1_2} {k^2_2} \right) \\ &+(c+1)^4 (c-1)k^1_2k^2_2\left( k^1_2 + k^2_2-3 {k^1_1} k^2_2 -3 {k^2_1} k^1_2 \right)\\ \end{split}$$ If $c\in {\mathbb Q}\cap (-1,1)$, we can then set $c=\frac{w_1-w_2}{w_1+w_2}$ to get an equation in $(w_1,w_2)\in {\mathbb Z}^+\times {\mathbb Z}^+$ for $\xi_{\bf w}$ being CSC (up to isotopy): $$\label{cscfinal} \begin{split} 0&= -3 (k^1_2)^2 (k^2_2)^2 w_1^5 \\ &+ k^1_2 k^2_2 \left(k^1_2 + k^2_2 - 3 k^2_1 k^1_2 - 3 k^1_1 k^2_2\right)w_1^4 w_2 \\ &+\left(4 k^1_1 k^1_2 k^2_2 + 4 k^2_1 k^1_2 k^2_2 - 9 k^1_1 k^2_1 k^1_2 k^2_2 -k^2_1 (k^1_2)^2 -k^1_1 (k^2_2)^2\right) w_1^3 w_2^2\\ &+\left(9 k^1_1 k^2_1 k^1_2 k^2_2 + (k^2_1)^2 k^1_2 + (k^1_1)^2 k^2_2 - 4 k^1_1 k^2_1 k^2_2- 4 k^1_1 k^2_1 k^1_2 \right)w_1^2 w_2^3\\ &+ k^1_1 k^2_1 \left(3 k^2_1 k^1_2 + 3 k^1_1 k^2_2-k^1_1 - k^2_1\right)w_1 w_2^4 \\ &+ 3(k^1_1)^2 (k^2_1)^2 w_2^5. \end{split}$$ On the other hand, Proposition 4.13 of [@BoTo21] (with $m_0=w_1$, $m_\infty=w_2$, $r_1=x_1$, and $r_2=x_2$) tells us that the Kähler class given by $(x_1,x_2)$ on the log pair $(S_{\bf n},\Delta_{\bf w})$ has a constant scalar curvature Kähler metric when the following equation holds true: $$\label{cscEQinw} \begin{split} 0= & 9 (w_1 - w_2) n_1 n_2 - 6 (w_1 + w_2) n_1 n_2 (x_1 + x_2) + 6 (w_1 - w_2) n_1 n_2 x_1x_2\\ &+ 3n_2 (4 w_1 w_2 - n_1 (w_1 - w_2)) x_1^2 + 3n_1 (4 w_1 w_2 - n_2 (w_1 - w_2)) x_2^2 \\ & - (4 w_1 w_2 (n_1 + n_2) - 3 (w_1 - w_2) n_1 n_2) x_1^2 x_2^2. \end{split}$$ We can then use the data in [\[qregadmdata\]](#qregadmdata){reference-type="eqref" reference="qregadmdata"} above to get an equation for the existence of a constant scalar curvature Kähler metric in the Kähler class of the quasi-regular Kähler quotient of $\xi_{\bf w}$. As expected from the above discussion and the fact that a quasi-regular Sasaki structure has constant scalar curvature (up to isotopy) exactly when its Kähler quotient has a constant scalar curvature Kähler metric in its Kähler class, this gives an equation equivalent to [\[cscfinal\]](#cscfinal){reference-type="eqref" reference="cscfinal"}. Consider a given complex manifold $S_{\bf n}= {\mathbb P}\bigl({\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}} \oplus {\mathcal O}(n_1,n_2)\bigr) \rightarrow {\mathbb C}{\mathbb P}^1\times{\mathbb C}{\mathbb P}^1$. This will be the regular quotient of a $d=1$ Yamazaki fiber join given by $K$ for any matrix $K$ of the form $$K= \begin{pmatrix} n_1+k^1 & n_2+k^2 \\ \\ k^1 & k^2, \end{pmatrix}$$ where $k^i \in {\mathbb Z}$ such that $k^i>Max\{0,-n_i\}$. For a given choice of $k^1,k^2$, the quotient Kähler class is then determined, up to scale, by $x_1=\frac{n_1}{n_1+2k^1}$ and $x_2=\frac{n_2}{n_2+2k^2}$. This gives a criterion for which Kähler classes on $S_{\bf n}$ can show up as regular quotient Kähler classes of a $d=1$ Yamazaki fiber join. For example, if $n_1=1$ and $n_2=-1$, we have $x_1=\frac{1}{1+2k^1}$ and $x_2=\frac{-1}{-1+2k^2}$. Here $k^1 \in {\mathbb Z}^+$ and $k^2 \in {\mathbb Z}^+\setminus\{1\}$. The Koiso-Sakane KE class is given by $x_1=1/2$ and $x_2=-1/2$ and we see right away that this class is out of range. The other CSC classes on this manifold are given by $x_2=-x_1$ and $x_2=x_1-1$ (see e.g. Theorem 9 in [@ACGT08]). Now, $$\begin{array}{ccl} x_2 & = &-x_1\\ \\ &\iff& \\ \frac{-1}{-1+2k^2} & = & -\frac{1}{1+2k^1}\\ \\ &\iff& \\ k^2&=& k^1+1, \end{array}$$ which then gives us a one parameter family $(x_1,x_2) = (\frac{1}{1+2k^1},\frac{-1}{1+2k^1})$ , $k^1\in {\mathbb Z}^+$ of CSC Kähler classes that each are regular quotient Kähler classes of a $d=1$ Yamazaki fiber join. One the other hand, $$\begin{array}{ccl} x_2 & = &x_1-1\\ \\ &\iff& \\ \frac{-1}{-1+2k^2} & = & \frac{-2k^1}{1+2k^1}\\ \\ &\iff& \\ 1&=& -4k^1-4k^1 k^2, \end{array}$$ which has no solutions for $k^1 \in {\mathbb Z}^+$ and $k^2 \in {\mathbb Z}^+\setminus\{1\}$. Thus, none of the CSC Kähler classes from this family can be regular quotient Kähler classes of a $d=1$ Yamazaki fiber join. ## $N= \Sigma_{g_1}\times\Sigma_{g_2}$, a product of Riemann surfaces {#highergenusprod} We can generalize the example of Section [4.2](#NNsect){reference-type="ref" reference="NNsect"} to consider the case where $N=\Sigma_{g_1}\times\Sigma_{g_2}$ with $\Sigma_{g_i}$ each being compact Riemann surfaces of genus $g_i$, equipped with a standard CSC area form $\Omega_i$. Similarly to Section [4.2](#NNsect){reference-type="ref" reference="NNsect"}, each choice of matrix $K= \begin{pmatrix} k^1_1 & k^2_1 \\ \\ k^1_{2} & k^2_{2}. \end{pmatrix}$, consisting of positive integer entries $k^i_j$, yields a $d=1$ Yamazaki fiber join $M_{\mathfrak w}=S(L_1^*\oplus L_2^*)$ via the line bundles $L_1, L_2$ satisfying $c_1(L_j)=[\omega_j] = k^1_j[\Omega_1] +k^2_j[\Omega_2]$. We assume here that $k^i_1\neq k^i_2$ for $i=1,2$. The case that $M$ is the total space of a Sasakian fiber join with $N=\Sigma_{g_1}\times\Sigma_{g_2}$ was treated in Proposition 5.8 of [@BoTo20b]. When $d>1$ the spectral sequence of the fibration collapses, so the cohomology groups of $M$ are the cohomology groups of the product $S^{2d+1}\times \Sigma_{g_1}\times\Sigma_{g_2}$. When $d=1$ we have $$\label{HM} H^p(M^7,{\mathbb Z})= \begin{cases} {\mathbb Z}~&\text{if $p=0,7$} \\ {\mathbb Z}^{2g_1+2g_2} ~&\text{if $p=1,3,6$} \\ {\mathbb Z}^{4g_1g_2+2}~ &\text{if $p=2,5$} \\ {\mathbb Z}^{2g_1+2g_2} +{\mathbb Z}_e~ &\text{if $p=4$} \\ 0 &\text{otherwise} \end{cases}$$ where the image of the differential $d_4$ in $E^{4,0}_2$ is the Euler class of the bundle with $e=k^1_1k^2_2+k^2_1k^1_2$. In both cases with $g_1,g_2$ and $e$ fixed we know that $H^4(N,{\mathbb Z})={\mathbb Z}$, so it follows from a theorem of Pontrjagin [@Pon45] (see also [@Mas58; @DoWh59]) that the sphere bundles $M$ are classified by their 2nd and 4th Stiefel-Whitney classes $w_2,w_4$, and their Pontrjagin class $p_1(M)$. Similarly to Section [4.2](#NNsect){reference-type="ref" reference="NNsect"}, the quotient complex manifold of $M_{\mathfrak w}$ arising from the regular Sasakian structure with Reeb vector field $\xi_{\mathbf 1}$ is equal to the following ${\mathbb C}{\mathbb P}^1$ bundle over $\Sigma_{g_1}\times\Sigma_{g_2}$: $${\mathbb P}\bigl(L_1^*\oplus L_2^*) = {\mathbb P}\bigl({\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}} \oplus L_1\otimes L_2^*\bigr) = {\mathbb P}\bigl({\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}} \oplus {\mathcal O}(n_1,n_2)\bigr) \rightarrow \Sigma_{g_1}\times\Sigma_{g_2},$$ with $n_1=k^1_1-k^1_2$ and $n_2=k^2_1-k^2_2$. Further, the regular quotient Kähler class is, up to scale, equal to the admissible Kähler class $2\pi(\frac{n_1}{x_1}[\Omega_1] +\frac{n_2}{x_2}[\Omega_2]) + \Xi)$ where $x_1=\frac{k^1_1-k^1_2}{k^1_1+k^1_2}, \quad x_2=\frac{k^2_1-k^2_2}{k^2_1+k^2_2}$. When $g_i\geq 2$ for at least one of $i=1,2$, we cannot use Theorem [Theorem 14](#2ndroundexistence){reference-type="ref" reference="2ndroundexistence"} to get existence of extremal/CSC Sasaki metrics. Further, we know from the examples in Sections 3.3 and 3.4 of [@BHLT22] that the existence of CSC or even just extremal Sasaki metrics is by no means a given. More specifically, Proposition 3.10 of [@BHLT22] tells us that the Reeb vector field determined - up to homothety - by $c\in (-1,1)$ (as explained in the beginning of the section) is extremal (up to isotopy) if and only if $F_c({\mathfrak z})>0$, for $-1<{\mathfrak z}<1$, where the polynomial $F_c({\mathfrak z})$ is given as follows: Let $s_i=\frac{2(1-g_i)}{n_i}=\frac{2(1-g_i)}{k^i_1-k^i_2}$, $x_i=\frac{k^i_1-k^i_2}{k^i_1+k^i_2}$, and define $$\begin{array}{ccl} \alpha_{r,-5} & = & \int_{-1}^1(ct+1)^{-5}t^r(1+x_1t)(1+x_2t)\,dt\\ \\ \alpha_{r,-6} & = & \int_{-1}^1(ct+1)^{-6}t^r(1+x_1t)(1+x_2t)\,dt\\ \\ \beta_{r,-4} & = & \int_{-1}^1(ct+1)^{-4}t^r(x_1s_1(1+x_2 t)+x_2 s_2(1+x_1 t))\,dt \\ \\ &+ & (-1)^r(1-c)^{-4}(1-x_1)(1-x_2) +(1+c)^{-4}(1+x_1)(1+x_2). \end{array}$$ Then, $$\label{wextrpol7mnf} F_c({\mathfrak z})=(c{\mathfrak z}+1)^4 \left[ \frac{2(1-x_1)(1-x_2)}{(1-c)^4}({\mathfrak z}+1) + \int_{-1}^{\mathfrak z}Q(t)({\mathfrak z}-t)\,dt\right],$$ where $$Q(t) = \frac{2\left( x_1s_1(1+x_2t)+x_2s_2(1+x_1t)\right)}{(ct+1)^4} - \frac{(A_1 t+A_2)(1+x_1t)(1+x_2t)}{(ct+1)^6}$$ and $A_1$ and $A_2$ are the unique solutions to the linear system $$\label{wextrpol7mnf2} \begin{array}{ccl} \alpha_{1,-6}A_1+\alpha_{0,-6}A_2&=& 2\beta_{0,-4}\\ \\ \alpha_{2,-6}A_1+\alpha_{1,-6}A_2&=& 2\beta_{1,-4}. \end{array}$$ Further, if the positivity of $F_c({\mathfrak z})$ is satisfied, then the extremal Sasaki structure is CSC exactly when $$\label{scsc7mnf} \alpha_{1,-5}\beta_{0,-4} - \alpha_{0,-5}\beta_{1,-4}=0$$ is satisfied. A direct calculation shows that $\alpha_{1,-5}\beta_{0,-4} - \alpha_{0,-5}\beta_{1,-4}=\frac{4h(c)}{9(1-c^2)^7}$ where $h(c)$ is the polynomial given by $$\begin{aligned} h(c)&=(3 x_1 x_2( s_1 x_2 + s_2x_1)- s_1 x_1- s_2 x_2+3(3 x_1^2 x_2^2- x_1^2+2 x_1 x_2- x_2^2+1)) c^5\\ &+(s_1 x_1^2 +s_2 x_2^2- 3 (s_1+s_2) x_1^2 x_2^2- 4 (s_1+s_2) x_1 x_2-6 (x_1+x_2)(4x_1 x_2+1))c^4 \\ &+4(((s_1x_1+s_2x_2)- (s_1x_2+s_2x_1)) x_1 x_2 + s_1 x_1+ s_2 x_2+3 x_1 x_2(x_1 x_2+5) +6 (x_1^2+x_2^2))c^3\\ &+4((s_1+s_2) (x_1x_2+ 1) x_1 x_2- s_1 x_1^2- s_2 x_2^2-3(x_1+x_2)(2x_1x_2+3 ))c^2\\ &+((s_1 x_2+s_2 x_1)x_1 x_2- (s_1 x_1+s_2 x_2)(4x_1 x_2+3) +3 (x_1^2 x_2^2+ x_1^2+ x_2^2+10 x_1 x_2+7)) c\\ &+3 (s_1 x_1^2+ s_2 x_2^2) -(s_1+s_2) x_1^2 x_2^2-6 (x_1+ x_2) \end{aligned}$$ and $h(\pm 1)=\pm 24(1\mp x_1)^2(1\mp x_2)^2$. Thus, equation [\[scsc7mnf\]](#scsc7mnf){reference-type="eqref" reference="scsc7mnf"} always have a solution $c\in (-1,1)$. In the event that $g_1,g_2\leq 1$, this is predicted by (the proof of) Theorem [Theorem 13](#bhlt22){reference-type="ref" reference="bhlt22"} and in the event that $g_1,g_2\geq 1$ (where ${\mathfrak t}^+$ is $2$-dimensional) this is predicted by Corollary 1.7 of [@BHL17]. The $g_1=0$ and $g_2>1$ (or vice versa) case falls outside of these results. Of course, a solution to $h(c)=0$ only corresponds to an actual CSC ray if we also have that the positivity condition of $F_c({\mathfrak z})$ is satisfied. **Proposition 17**. *Let $M_{\mathfrak w}$ be a $d=1$ fiber join over $\Sigma_{g_1}\times \Sigma_{g_2}$ with its induced Sasakian structure. Then for all $g_1, g_2 \geq 1$, there exists a matrix $K=\tiny \begin{pmatrix} k^1_1 & k^2_1 \\ \\ k^1_{2} & k^2_{2}. \end{pmatrix}$ such that the entire Sasaki cone of $M_{\mathfrak w}$ is extremal and contains a CSC ray.* *Proof.* Without loss of generality, we assume that $g_2\geq g_1\geq 1$. First we note that since $g_1,g_2\geq 1$, the Sasaki cone is of dimension $2$. Thus, the proof will consist of showing that for all $g_2\geq g_1\geq 1$, $\exists$ a two-by-two matrix $K$ such that $\forall c\in (-1,1)$, $F_c({\mathfrak z})$ as defined in [\[wextrpol7mnf\]](#wextrpol7mnf){reference-type="eqref" reference="wextrpol7mnf"} is positive for $-1< {\mathfrak z}<1$. Once this is proven we already know from the above discussion that for this such a choice of $K$, [\[scsc7mnf\]](#scsc7mnf){reference-type="eqref" reference="scsc7mnf"} has a solution $c\in (-1,1)$. This $c$ will correspond to a CSC ray. If $g_1=g_2=1$, the result follows from Theorem [Theorem 14](#2ndroundexistence){reference-type="ref" reference="2ndroundexistence"}. Thus, we assume for the rest of the proof that $g_2>1$. Now, let $K=\tiny \begin{pmatrix} 10g_1& 100g_2 \\ \\ 2 g_1 & g_2. \end{pmatrix}$. Using [\[wextrpol7mnf\]](#wextrpol7mnf){reference-type="eqref" reference="wextrpol7mnf"}, we can calculate that $$F_c({\mathfrak z})=\frac{(1-{\mathfrak z}^2)p({\mathfrak z})}{1212g_1g_2 h_0(c)},$$ where $$h_0(c)=544829 - 1814364 c + 2225984 c^2 - 1185624 c^3 + 229199 c^4,$$ and $p({\mathfrak z})$ is a cubic given by $$p({\mathfrak z})=8g_1g_2h_1(c)+4h_2(c,g_1,g_2)(1+{\mathfrak z})+2h_3(c,g_1,g_2)(1+{\mathfrak z})^2+h_4(c,g_1,g_2)(1+{\mathfrak z})^3,$$ where $$\begin{array}{ccl} h_1(c)&=&h_0(c)\\ \\ h_2(c,g_1,g_2) &=& 6h_{21}(c)+h_{22}(c)(g_2-2)+\left(5h_{23}(c)+h_{24}(c)(g_2-2)\right)(g_1-2)\\ \\ h_3(c,g_1,g_2) &=& 2h_{31}(c)+2h_{32}(c)(g_2-2)+\left(h_{33}(c)+2h_{34}(c)(g_2-2)\right)(g_1-2)\\ \\ h_4(c,g_1,g_2) &=& 10h_{41}(c)+h_{42}(c)(g_2-2)+\left(2h_{43}(c)+h_{44}(c)(g_2-2)\right)(g_1-2)\end{array}$$ with $$\begin{array}{ccl} h_{21}(c) &=&1849633 - 3952908 c + 2583653 c^2 - 545438 c^3 + 68368 c^4\\ \\ h_{22}(c) &=&5029446 - 10073556 c + 5505031 c^2 - 421486 c^3 - 29519 c^4\\ \\ h_{23}(c) &=&1085299 - 2250304 c + 1327594 c^2 - 148704 c^3 - 11901 c^4\\ \\ h_{24}(c) &=&2453521 - 4733176 c + 2196021 c^2 + 235654 c^3 - 147064 c^4\\ \\ h_{31}(c) &=&173925883 - 629489348 c + 863749558 c^2 - 530449308 c^3 + 122385903 c^4\\ \\ h_{32}(c) &=&86771822 - 314540932 c + 432305747 c^2 - 265928422 c^3 + 61453077 c^4\\ \\ h_{33}(c) &=&169929491 - 609982556 c + 828678836 c^2 - 502956696 c^3 + 114452421 c^4\\ \\ h_{34}(c) &=&42386813 - 152393768 c + 207385193 c^2 - 126091058 c^3 + 28743168 c^4\\ \\ h_{41}(c) &=&72852912 - 233877440 c + 270006303 c^2 - 130233426 c^3 + 21229919 c^4\\ \\ h_{42}(c) &=&365166252 - 1171579852 c + 1351415507 c^2 - 650974422 c^3 + 105863967c^4\\ \\ h_{43}(c) &=&184191678 - 594750598 c + 693107613 c^2 - 339776268 c^3 + 57173843 c^4\\ \\ h_{44}(c) &=&184642524 - 595846924 c + 693799609 c^2 - 339679914 c^3 + 57031029 c^4. \end{array}$$ We also notice that $p(1)=4000g_1g_2 h_0(c)$. *Claim:* For all $c\in (-1,1)$, $h_0(c)>0$. Further, for all $c\in (-1,1)$, $i=2,3$, and $j=1,2,3,4$, $h_{ij}(c)>0$. From this claim it then follows that for $g_1,g_2>1$, all $c\in (-1,1)$, and $i=0,1,2,3$, $h_i(c)>0$. Thus, in this case, we have $p(\pm 1)>0$, $p'(-1)>0$, and $p''(-1)>0$. Since $p({\mathfrak z})$ is a cubic, a moment's thought tells us that $p({\mathfrak z})>0$ for $-1<{\mathfrak z}<1$. Finally, since the claim also tells us that $h_0(c)>0$ for $c\in (-1,1)$, we conclude that $F_c({\mathfrak z})$ is positive for all $c\in (-1,1)$ and ${\mathfrak z}\in (-1,1)$ as desired. The proof of the claim is a standard exercise: For example, one easily checks that $h_0(\pm 1)>0$, $h_0'(\pm 1)<0$. Further, since $h_0''(c)$ is a second order polynomial in $c$ with $h_0''(\pm 1)>0$, and $h_0'''(1)<0$, we know $h_0''(c)>0$ for $c\in (-1,1)$. Thus for $-1\leq c \leq 1$, $h_0(c)$, is a (concave up and) decreasing function that is positive at $c=\pm 1$. It therefore must be positive for all $c\in (-1,1)$, as desired. The argument for the claim concerning $h_{ij}(c)$ with $i=2,3$ and $j=1,2,3,4$ is completely similar. Finally, if $g_1=1$ (and $g_2>1$), we still have that $h_0(c)=h_1(c)>0$ for $c\in (-1,1)$. Further, note that $$\begin{array}{ccl} h_2(c,1,g_2) &=& \tilde{h}_{21}(c)+5\tilde{h}_{22}(c)(g_2-2)\\ \\ h_3(c,1,g_2) &=& 5\tilde{h}_{31}(c)+2\tilde{h}_{32}(c)(g_2-2), \end{array}$$ where $$\begin{array}{ccl} \tilde{h}_{21}(c) &=&5671303 - 12465928 c + 8863948 c^2 - 2529108 c^3 + 469713 c^4\\ \\ \tilde{h}_{22}(c) &=&515185 - 1068076 c + 661802 c^2 - 131428 c^3 + 23509 c^4\\ \\ \tilde{h}_{31}(c) &=&35584455 - 129799228 c + 179764056 c^2 - 111588384 c^3 + 26063877 c^4\\ \\ \tilde{h}_{32}(c) &=&44385009 - 162147164 c + 224920554 c^2 - 139837364 c^3 + 32709909 c^4 \end{array}$$ Now, in an exact similar way as above, we can prove that for all $c\in (-1,1)$, $i=2,3$, and $j=1,2$, $\tilde{h}_{ij}(c)>0$. Therefore we may still conclude that $p(\pm 1)>0$, $p'(-1)>0$, and $p''(-1)>0$ and the proof finishes as above. ◻ ## $N={\mathbb P}(E) \rightarrow \Sigma_g$, where $E\rightarrow \Sigma_g$ is a polystable rank $2$ holomorphic vector bundle over a compact Riemann surface of genus $g\geq 1$. Let $\Sigma_g$ be a compact Riemann surface and let $E\rightarrow \Sigma_g$ be a holomorphic vector bundle. The degree of $E$, is defined by $deg\,E=\int_{\Sigma_g} c_1(E)$. Then $E$ is *stable*(or *semistable*) in the sense of Mumford if for any proper coherent subsheaf $F$, $\frac{deg\, F}{rank\,F} < \frac{deg\, E}{rank\,E}$ (or $\frac{deg\, F}{rank\,F} \leq \frac{deg\, E}{rank\,E}$). Further, a semistable holomorphic vector bundle, $E$, is called *polystable* if it decomposes as a direct sum of stable holomorphic vector bundles, $E=F_1\oplus \cdots\oplus F_l$, such that that $\frac{deg\, F_i}{rank\,F_i} = \frac{deg\, E}{rank\,E}$, for $i=1,\dots,l$. (See e.g. [@Kobbook] for more details on this.) Assume $N={\mathbb P}(E) \stackrel{\pi}{\rightarrow} \Sigma_g$, where $E\rightarrow \Sigma_g$ is a polystable rank $2$ holomorphic vector bundle over a compact Riemann surface of genus $g\geq 1$. Note that the polystabilty of $E$ is independent of the choice of $E$ in ${\mathbb P}(E)$. Indeed, by the theorem of Narasimhan and Seshadri [@NaSe65], polystability of $E$ is equivalent to ${\mathbb P}(E) \stackrel{\pi}{\rightarrow} \Sigma_g$ admitting a flat projective unitary connection which in turn is equivalent to $N$ admitting a local product Kähler metric induced by constant scalar curvature Kähler metrics on $\Sigma_g$ and ${\mathbb C}{\mathbb P}^1$. We shall explain and explore the latter in more detail below. Likewise, the condition of whether $deg E$ is even ($E$ spin) or odd ($E$ is non-spin), is independent of the choice of $E$. Unless $E$ is decomposable, we must have that $Aut(N,J)$ is discrete ([@Mar71]). Let ${\mathbf v}=c_1(VP(E))\in H^2(N,{\mathbb Z})$ denoted the Chern class of the vertical line bundle and let ${\mathbf f}\in H^2(N,{\mathbb Z})$ denote the Poincaré dual of the fundamental class of a fiber of ${\mathbb P}(E) \rightarrow \Sigma_g$. From e.g. [@Fuj92] we know that if ${\mathbf h}\in H^2(N,{\mathbb Z})$ denote the Chern class of the ($E$-dependent) tautological line bundle on $N$, then $H^2(N,{\mathbb Z})={\mathbb Z}{\mathbf h} \oplus {\mathbb Z}{\mathbf f}$ and ${\mathbf v} = 2{\mathbf h}+(deg E){\mathbf f}$. Due to the fact that $N={\mathbb P}(E) \stackrel{\pi}{\rightarrow} \Sigma_g$ admits a flat projective unitary connection, we know that $N$ has a universal cover $\tilde{N} ={\mathbb C}{\mathbb P}^1\times\tilde{\Sigma_g}$ (where $\tilde{\Sigma_g}$ is the universal cover of $\Sigma_g$). Let $\Omega_1$ denote the standard Fubini-Study area form on ${\mathbb C}{\mathbb P}^1$ and let $\Omega_2$ denote a standard CSC area form on $\Sigma_g$. Now consider the projection $\pi_1: {\mathbb C}{\mathbb P}^1\times\tilde{\Sigma_g}\rightarrow {\mathbb C}{\mathbb P}^1$ to the first factor. Then $\pi_1^*(\Omega_1)$ descends to a closed $(1,1)$ form on $N$ representing the class ${\mathbf v}/2$ and $[\pi^*\Omega_2]={\mathbf f}$. If we (abuse notation slightly and) think of $q_1\Omega_1+q_2\Omega_2$ as a local product of CSC Kähler forms on $N$, then this represents the cohomology class $\frac{q_1}{2}{\mathbf v} + q_2{\mathbf f}=q_1{\mathbf h} +(\frac{q_1}{2}(deg E)+q_2){\mathbf f}$. If $deg E$ is even, this class is in $H^2(N,{\mathbb Z})$ (and hence can represent a holomorphic line bundle) precisely when $q_1,q_2\in {\mathbb Z}$. If $deg E$ is odd, then the class is in $H^2(N,{\mathbb Z})$ iff ($q_1$ is an even integer and $q_2\in {\mathbb Z}$) or ($q_1$ is an odd integer and $(q_2-1/2)\in {\mathbb Z}$). Note that a similar discussion appears in the proof of Theorem 4.6 of [@ACGT08b]. With this in mind, we can we can (yet again) generalize to consider the case where $N$ is as described above. We consider a matrix $K= \begin{pmatrix} k^1_1 & k^2_1 \\ \\ k^1_{2} & k^2_{2}. \end{pmatrix}$, consisting of entries $k^i_j$, such that: - If $deg E$ is even, $k^i_j \in {\mathbb Z}^+$ - If $deg E$ is odd, one of the following is true: - $k^1_j$ is an even positive integer and $k^2_j\in {\mathbb Z}^+$ - $k^1_j$ is an odd positive integer and $(k^2_j-1/2)\in {\mathbb Z}^+$. Such a choice of $K$ yields a $d=1$ Yamazaki fiber join $M_{\mathfrak w}=S(L_1^*\oplus L_2^*)$ via the line bundles $L_1, L_2$ satisfying $c_1(L_j)=[\omega_j] = k^1_j[\Omega_1] +k^2_j[\Omega_2]=k^1_j{\mathbf h} +(\frac{k^1_j}{2}(deg E)+k^2_j){\mathbf f}$. As before we assume that $k^i_1\neq k^i_2$ for $i=1,2$. As we know, the quotient complex manifold of $M_{\mathfrak w}$ arising from the regular Sasakian structure with Reeb vector field $\xi_{\mathbf 1}$ is equal to the following ${\mathbb C}{\mathbb P}^1$ bundle over $N$: ${\mathbb P}\bigl(L_1^*\oplus L_2^*) = {\mathbb P}\bigl({\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold 1$}}{\scalebox{1.16}{$\textstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold 1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}} \oplus L_1\otimes L_2^*\bigr),$ with $c_1( L_1\otimes L_2^*)=(k^1_1-k^1_2)[\Omega_1] +(k^2_1-k^2_2)[\Omega_2]=(k^1_1-k^1_2){\mathbf h} +(\frac{(k^1_1-k^1_2)}{2}(deg E)+(k^2_1-k^2_2)){\mathbf f}$. Similarly, as before, the regular quotient Kähler class is, up to scale, equal to the admissible Kähler class $2\pi(\frac{k^1_1-k^1_2}{x_1}[\Omega_1] +\frac{k^2_1-k^2_2}{x_2}[\Omega_2]) + \Xi)$ where $x_1=\frac{k^1_1-k^1_2}{k^1_1+k^1_2}, \quad x_2=\frac{k^2_1-k^2_2}{k^2_1+k^2_2}$. We can now adapt the set-up from Section [4.3](#highergenusprod){reference-type="ref" reference="highergenusprod"} with $s_1=\frac{2}{k^1_1-k^1_2}$ and $s_2=\frac{2(1-g)}{k^2_1-k^2_2}$. In particular, equation [\[scsc7mnf\]](#scsc7mnf){reference-type="eqref" reference="scsc7mnf"} continues to have some solution $c\in (-1,1)$ and we can calculate $F_c({\mathfrak z})$ using [\[wextrpol7mnf\]](#wextrpol7mnf){reference-type="eqref" reference="wextrpol7mnf"}. If a choice of $K$ satisfies that $F_c({\mathfrak z})$ is positive for all $c\in (-1,1)$ and ${\mathfrak z}\in (-1,1)$, then we will have a conclusion similar to the result in Proposition [Proposition 17](#2highergenus){reference-type="ref" reference="2highergenus"}. Indeed, we have the following proposition. **Proposition 18**. *Let $N={\mathbb P}(E) \stackrel{\pi}{\rightarrow} \Sigma_g$, where $E\rightarrow \Sigma_g$ is a polystable rank $2$ holomorphic vector bundle over a compact Riemann surface of genus $g\geq 1$. Let $K=\tiny \begin{pmatrix} k^1_1 & k^2_1 \\ \\ k^1_{2} & k^2_{2} \end{pmatrix} = \begin{pmatrix} 10 g & 100 g\\ \\ 2g & g \end{pmatrix}$ and let $M_{\mathfrak w}$ be the $d=1$ fiber join over $N$ as described above with its induced Sasakian structure. Then the entire subcone, ${\mathfrak t}^+_{sph}$, is extremal and contains a CSC ray.* *In particular, if $E$ is indecomposable, then the entire Sasaki cone of $M_{\mathfrak w}$ is extremal and contains a CSC ray.* *Proof.* First we notice that $k^1_j$ is even for $j=1,2$ and thus this choice of $K$ is allowed whether or not $E$ is spin. Second, we have that the set of rays in ${\mathfrak t}^+_{sph}$ is parametrized by $c\in (-1,1)$ in the same manner as in Section [4.3](#highergenusprod){reference-type="ref" reference="highergenusprod"}. Further, in the case where $E$ is indecomposable, $Aut(N,J)$ is discrete and thus the Sasaki cone is exactly ${\mathfrak t}^+_{sph}$. Therefore all we need to do to prove the proposition is to check that for this choice of $K$, the polynomial $F_c({\mathfrak z})$, defined by [\[wextrpol7mnf\]](#wextrpol7mnf){reference-type="eqref" reference="wextrpol7mnf"}, is positive for all $c\in (-1,1)$ and ${\mathfrak z}\in (-1,1)$. If $g=1$, we already know from (the proof of) Theorem 3.1 in [@ApMaTF18] that for any choice of $K$, $F_c({\mathfrak z})>0$ for all $c\in (-1,1)$ and ${\mathfrak z}\in (-1,1)$. Thus we will assume that $g>1$ for the rest of the proof. By direct calculations we get that $$F_c({\mathfrak z}) = \frac{(1-{\mathfrak z}^2)p({\mathfrak z})}{1212 g h_0(c)},$$ where $$h_0(c) = 544829 - 1814364 c + 2225984 c^2 - 1185624 c^3 + 229199 c^4$$ and $p({\mathfrak z})$ is a cubic in ${\mathfrak z}$ that we may write as $$\begin{array}{ccl} p({\mathfrak z}) & = & 8 g h_1(c) + \left(4h_{21}(c)+20h_{22}(c)(g-2)\right)({\mathfrak z}+1) + \left(2h_{31}(c)+4h_{32}(c)(g-2)\right)({\mathfrak z}+1)^2 \\ \\ &+& \left(h_{41}(c)+2h_{42}(c)(g-2)\right)({\mathfrak z}+1)^3, \end{array}$$ where $$\begin{array}{ccl} h_1(c)&=&h_0(c)\\ \\ h_{21}(c)&=&5793707 - 13073132 c + 9976937 c^2 - 3421902 c^3 + 734322 c^4\\ \\ h_{22}(c)&=&515185 - 1068076 c + 661802 c^2 - 131428 c^3 + 23509 c^4\\ \\ h_{31}(c)&=&181918667 - 668502932 c + 933891002 c^2 - 585434532 c^3 + 138252867 c^4\\ \\ h_{32}(c)&=&44385009 - 162147164 c + 224920554 c^2 - 139837364 c^3 + 32709909 c^4\\ \\ h_{41}(c)&=&356026968 - 1129159208 c + 1277664093 c^2 - 594396318 c^3 + 89753413 c^4\\ \\ h_{42}(c)&=&90261864 - 287866464 c + 328807949 c^2 - 155647254 c^3 + 24416469 c^4.\\ \end{array}$$ Note also that $p(1)=4000gh_0(c)$. Completely similar to the way the claim at the end of the proof of Proposition [Proposition 17](#2highergenus){reference-type="ref" reference="2highergenus"} is verified, we can now show that for all $c\in (-1,1)$, $h_0(c)>0$ and for all $c\in (-1,1)$, $i=2,3$, and $j=1,2$, $h_{ij}(c)>0$. This tells us that $p(\pm 1)>0$, $p'(-1)>0$, and $p''(-1)>0$. Since $p({\mathfrak z})$ is a cubic, we conclude that $p({\mathfrak z})>0$ for $-1<{\mathfrak z}<1$. Finally, since $h_0(c)>0$ for $c\in (-1,1)$, $F_c({\mathfrak z})$ is positive for all $c\in (-1,1)$ and ${\mathfrak z}\in (-1,1)$ as desired. ◻ **Remark 19**. Note that if we fix a matrix $K$ and calculate $F_c({\mathfrak z})$, then we can observe that $$\lim_{g\rightarrow +\infty} F_0(0)=-\infty.$$ Thus is it clear that for any choice of $K$ there exist values $g>1$ such that the corresponding Sasaki cone is NOT exhausted by extremal Sasaki metrics. 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MR 2001e:53094 [^1]: The authors were partially supported by grants from the Simons Foundation, CPB by (\#519432), and CWT-F by (\#422410) [^2]: The combinatorics studied in these references is that of the moment cone which is dual to the Sasaki cone ${\mathfrak t}^+({\mathcal D},J)$. [^3]: Note that $b$ is exactly what $c$ is in (51) of [@BoTo13]. This follows from our discussion in Section [3](#Yamsect){reference-type="ref" reference="Yamsect"}.

arxiv_math

--- abstract: | Stochastic optimization often involves calculating the expected value of a first-order max or min function, known as a first-order loss function. In this context, loss functions are frequently approximated using piecewise linear functions. Determining the approximation error and the number of breakpoints (segments) becomes a critical issue during this approximation. This is due to a trade-off: increasing the number of breakpoints reduces the error but also increases the computational complexity of the embedded model. As this trade-off is unclear in advance, preliminary experiments are often required to determine these values. The objective of this study is to approximate the trade-off between error and breakpoints in piecewise linearization for first-order loss functions. To achieve this goal, we derive an upper bound on the minimum number of breakpoints required to achieve a given absolute error. This upper bound can be easily precomputed once the approximation intervals and error are determined, and serves as a guideline for the trade-off between error and breakpoints. Furthermore, we propose efficient algorithms to obtain a piecewise linear approximation with a number of breakpoints below the derived upper bound. author: - Yotaro Takazawa bibliography: - cas-refs.bib title: Precomputable Trade-off Between Error and Breakpoints in Piecewise Linearization for First-Order Loss Functions --- stochastic programming,inventory, piecewise linear approximation # Introduction Stochastic optimization often involves the calculation of an expected value of a max (min) function consisting of a decision variable and a random variable. One of the most fundamental functions among them can be expressed as a univariate function $\ell_X: \mathbb{R} \rightarrow \mathbb{R}$, $$\label{expectation_function} \ell_X(s) = \mathbb{E}[\max(a_1s+b_1X+c_1, a_2s+b_2X+c_2)],$$ where $X$ is a given one-dimensional random variable and $a_i, b_i, c_i \in \mathbb{R}$ ($i \in \{1, 2\}$). This function is used in many areas, especially in inventory control. For example, let $X$ be an uncertain demand for some product $P$, and $s$ be the ordering quantity for $P$. Then $\mathbb{E}[\max(X-s, 0)]$ and $\mathbb{E}[\max(s- X, 0)]$ are considered as shortage and inventory costs in inventory control. These functions are known as first-order loss functions [@snyder2019fundamentals] and are used in inventory control and other applications [@rossi2014piecewise]. Thus, we call $\ell_X$ a general (first-order) loss function, as it is a generalization of them. As another example, $\mathbb{E}[ \min(s, X)]$ is regarded as the expected number of units of product P sold, which is often used in a profit function in newsvendor models. While a general first-order loss function is often embedded in mixed-integer linear programming (MILP) models, calculating these values is a challenging task for the following reasons. When a target random variable $X$ is continuous, these expectation functions are often nonlinear and thus difficult to embed in MILP. On the other hand, when $X$ is discrete, these expectation functions can be directly embedded in MILP. However, when $X$'s support has a large cardinality or is infinite, solving optimization problems becomes challenging. For the above reasons, loss functions are often approximated by a piecewise linear function, which is a tractable format for MILP. In a piecewise linear approximation, we choose $n$ points in ascending order, known as breakpoints. We approximate the loss function using linear segments connecting each breakpoint to its corresponding function value. Determining the appropriate parameters, such as the acceptable error and the number of breakpoints for piecewise linear functions, is a crucial aspect of this approach. While increasing the number of breakpoints can decrease the approximation error, it also increases the computational complexity of MILP. Therefore, we must set a careful balance when choosing these parameters. Often, the appropriate parameters are determined through preliminary and sometimes heavy numerical experiments, as the theoretical relationship between these parameters is not always clear. This study aims to understand the trade-off between error and breakpoints in a piecewise linear approximation for general first-order loss functions. To achieve this objective, we derive a tight upper bound for the minimum number of breakpoints needed to achieve a given error. We also propose an efficient method for constructing a piecewise linear function with a number of breakpoints below this upper bound. As a result, we enable the determination of appropriate levels of error and breakpoints, using this upper bound as a guideline. Specifically, we have obtained the following results in this study for a general loss function: 1. Given an approximation error $\epsilon > 0$, we propose algorithms to make a piecewise linear function such that the maximum absolute error is within $\epsilon$ and the breakpoints are bounded by $M\sqrt{\frac{W}{\epsilon}}$, where $M \ (\leq 1)$ is a parameter dependent on the setting and $W$ is the width of the approximation interval. Among the proposed algorithms, one guarantees minimality in the number of breakpoints, while the others, although not guaranteeing minimality, offer the same upper bound on the number of breakpoints. We also demonstrate that this upper bound on the number of breakpoints is tight from a certain perspective. 2. Through computational experiments, we compare the actual number of breakpoints generated by our proposed algorithms with the derived bounds across various distributions. In many cases, we find that the minimal number of breakpoints can be approximated by $\frac{1}{2\sqrt{2}}\sqrt{\frac{W}{\epsilon}}$. We review related approaches for piecewise linear approximation, specifically focusing on methods that provide theoretical guarantees in terms of approximation error or the number of breakpoints. These can be broadly divided into two categories based on whether they fix the error or the number of breakpoints. Note that the setting of our study falls under fixing the error. As the foundation of our research, we first introduce the important work by [@rossi2014piecewise], which focuses on minimizing error in piecewise linear approximation for first-order loss functions with a fixed number of breakpoints. Their method divides the domain into $n$ intervals and uses a new discrete random variable to calculate the conditional expectation for each interval. Based on this, various heuristics have been proposed for different distributions [@rossi2014piecewise-conf; @rossi2015piecewise]. This approach is widely used, especially in the field of inventory management [@tunc2018extended; @kilic2019heuristics; @gutierrez2023stochastic; @xiang2023mathematical]. While the method in [@rossi2014piecewise] focuses on guarantees for minimizing error, it does not fully explore the relationship between error and the number of breakpoints and is specifically tailored for normal distributions. In contrast, our research can determine the relationship between error and breakpoints a priori. Our study is applicable to both continuous and discrete distributions and can also be used for scenario reduction in scenario data. There is a history of research on minimizing error in piecewise linear approximation for convex functions with fixed breakpoints [@cox1971algorithm; @gavrilovic1975optimal; @imamoto2008recursive; @liu2021optimal], which serve as the basis for [@rossi2014piecewise]. Similar to [@rossi2014piecewise], most of these works focus solely on minimizing error without delving into the relationship between error and the number of breakpoints. The sole exception is the study by [@liu2021optimal], which provides the first trade-off between error and breakpoints. Although their study achieves an error analysis and trade-off similar to ours, it is difficult to apply to the loss function for the following two reasons: 1.Their trade-off analysis relies on derivative information, making it inapplicable to general loss functions composed of discrete distributions. 2.Their algorithm requires simplicity in the derivative form of the target function for error computation, making it challenging to directly apply to general loss functions. Additionally, most of the studies mentioned here require solving non-convex optimization problems, leaving the computational complexity unknown. In contrast, our research has the advantage of bounding the number of iterations of our algorithms by the number of breakpoints. Next, we review studies that aim to minimize the number of breakpoints given a specified allowable error, which, as stated in [@ngueveu2019piecewise], are much fewer in number compared to the other setting. [@ngueveu2019piecewise] proposed a method for minimizing breakpoints that can be applied to a specific MILP. While their results are substantially different from ours, their motivation to focus on the trade-off between error and the number of breakpoints is similar to ours. Here, we discuss the study by [@rebennack2015continuous], which is most closely related to our research. In [@rebennack2015continuous], they formulated a MILP optimization problem to minimize the number of breakpoints for general univariate functions, given an allowable error $\epsilon$. Additionally, they proposed a heuristic for determining an adjacent breakpoint such that the error within an interval does not exceed $\epsilon$ when a specific breakpoint is given. Even though the heuristic algorithm's approach is similar to ours, it does not provide bounds on the number of breakpoints, making our results non-trivial. Although it is slightly outside the scope, we discuss the research on the scenario generation approach, which is another popular method for dealing with a random variable $X$ in stochastic optimization. In the scenario generation approach, a simpler discrete random variable $Y$ is generated to approximate a random variable $X$, with each value of $Y$ referred to as a scenario. Sample average approximation, such as Monte Carlo Sampling, is one of the most widely used methods in scenario generation [@shapiro2003monte]. Recall that our proposed method approximates $X$ by a new discrete distribution $\tilde{X}$ using conditional expectations based on [@rossi2014piecewise]. Thus, it can also be viewed as a scenario generation approach that guarantees the number of generated scenarios and the error. Note that, in existing studies in scenario generation, the function incorporating $X$ (in our case, $\ell_X$) is not specified, and the error is evaluated using some form of distance, such as the Wasserstein distance, between the target random variable $X$ and the generated variable $Y$. This is significantly different from our study, which evaluates the absolute error between $\ell_X$ and $\ell_Y$ when the function is specified. For the main methods related to scenario generation, please refer to [@lohndorf2016empirical]. An overview and structure of this study can be found in the next section; please refer to it for details. # Research Setting and Outline In this section, we outline the research, discuss its structure, and introduce the notation used throughout the study. ## Scope and Setting The following box summarizes the settings of this study. Input: : 1. a half open interval $(a, b] \subseteq \mathbb{R}$ 2. a one-dimensional random variable $X$ 3. a univariate function $f_X: (a, b] \rightarrow \mathbb{R}; \ f_X(s) = \mathbb{E}[ \min (s, X)]$ Task: : 1. Create a new discrete random variable $\tilde{X}$ from $X$ such that $f_{\tilde{X}}$ is a piece-wise linear approximation of $f_X$, which is based on [@rossi2014piecewise]. 2. Analyze $f_{\tilde{X}}$ based on the following evaluation criteria. Evaluation: : 1. the absolute approximation error defined by $$e_{X, \tilde{X}} \coloneqq \max_{s \in (a,b]} |f_{\tilde{X}}(s) - f_X(s)|$$ 2. the number of the breakpoints of $\tilde{f}$ For the setting, the following two points need further clarification: 1. Target Function:\ We can show that a general loss function $\ell_X$ can be expressed as the sum of an affine function and $\mathbb{E}[\min(s, X)]$ (please refer to [8.1](#sec_reduction){reference-type="ref" reference="sec_reduction"}). Without loss of generality, this reduction allows us to focus on the piecewise linear approximation of a simple function $f_X(s) := \mathbb{E}[\min(s, X)]$. Thus, in the rest of paper, we focus on $f_X$. We use the half-open interval $(a, b]$ as the approximation domain in order to simplify the discussion when dealing with discrete distributions. 2. Method of Piecewise Linear Approximation:\ In this study, we employ the method proposed by [@rossi2014piecewise] for piecewise linear approximation. In their method, we transform the target random variable $X$ into a discrete random variable $\tilde{X}$ that takes on fewer possible values than $X$ as follows. First, we divide $\mathbb{R}$ into $n$ consecutive regions $\mathcal{I} = (I_1, \dots, I_n)$. We construct $\tilde{X}$ from $\mathcal{I}$, which takes the conditional expectation $\mathbb{E}[X \mid X \in I]$ with probability $P(X \in I)$ for each $I \in \mathcal{I}$. Then we consider the loss function $f_{\tilde{X}}$ obtained by replacing $X$ with $\tilde{X}$ in $f_X$, which can be shown to be a piecewise linear function with $n+1$ breakpoints. It implies that the piecewise linear function $f_{\tilde{X}}$ is uniquely determined once a partition $\mathcal{I}$ is fixed. ## Structure of the Paper The structure of the rest of this paper is as follows: In Section 3, we analyze the error $e_{X, \tilde{X}}$ under the given partition $\mathcal{I}$. In Section 4, we propose algorithms for generating a partition $\mathcal{I}$ with an error less than $\epsilon$ and derive upper bounds on the number of breakpoints for the induced piecewise linear functions. In Section 5, we derive lower bounds on the number of breakpoints based on the framework of our analysis. In Section 6, we compare the actual errors and the number of breakpoints with their theoretical values through numerical experiments and discuss the results. In Section 7, we give our conclusion. ## Notation and Assumptions We assume that all random variables treated in this paper are real-valued random variables with expected values. Random variables can be either continuous or discrete unless explicitly stated. For a random variable $X$, its distribution function is denoted by $p_X: \mathbb{R}\rightarrow [0,1]$. For a half-open interval $I=(a, b] \subseteq \mathbb{R}$, we define the part of expectation of $\mathbb{E}[X]$ as $$\mathbb{E}_{a,b}[X] := \left \{\begin{array}{ll} \int_a^b \ p_X(x) x \, dx & \text{if $X$ is continuous}, \\ \sum_{x \in (a, b] \cap S} P(X=x)x & \text{if $X$ is discrete}, \\ \end{array} \right .$$ where $S$ is the support of $X$. # Analysis of Approximation Error In this section, we introduce the piecewise linear method used in the study and evaluate the error in its approximation. ## Piecewise Linear Approximation Framework First, we introduce a piecewise linear approximation framework for the function $f_X(s) = \mathbb{E}[\min(s, X)]$ on some half interval $(a, b] \subseteq \mathbb{R}$, which is based on [@rossi2014piecewise]. Let $\mathcal{I}=(I_0, I_1, \dots I_n, I_{n+1})$ be a partition of $\mathbb{R}$ such that - $I_j = (a_j, b_j]$ for $j \in \{0, 1, \dots, n, n+1\}$ such that $b_j = a_{j+1}$ for $j \in \{0, 1, \dots, n\}$ - $a_0 = - \infty$, $b_0 = a$, $a_{n+1}=b$ and $b_{n+1} = \infty$. To simplify the discussion, we assume that for all $I \in \mathcal{I}$, $P(X \in I)$ is positive. We consider a new discrete random variable $\tilde{X}$ with $\mathcal{I}$ to approximate $X$ as follows. **Definition 1**. A discrete random variable $\tilde{X}$ is said to be induced by a random variable $X$ with $\mathcal{I}$ if $\tilde{X}$ takes $\mathbb{E}[X \mid X \in I_j ]$ with probability $P(X \in I_j)$ for $j \in \{0, 1, \dots, n, n+1\}$. With $\tilde{X}$ instead of $X$ of $f_X$, $f_{\tilde{X}}$ is written as $$\label{def_f_tilde} f_{\tilde{X}}(s) \coloneqq \mathbb{E}[ \min(s, \tilde{X}) ] = \sum_{j=0}^{n+1} (P(X \in I_j) \cdot \min (s, \mu_j)),$$ where for $j \in \{0, 1, \dots, n, n+1\}$ we define $$\mu_j \coloneqq \mathbb{E}[X \mid X \in I_j].$$ We easily show that $f_{\tilde{X}}$ is a continuous piecewise linear function as follows. **Proposition 2**. $f_{\tilde{X}}$ on $(a, b]$ is a continuous piecewise linear function on with $n$ breakpoints. *Proof.* Assume that $\mu_i < s \leq \mu_{i + 1}$ for some $i \in \{0, 1, \dots, n\}$. Then we have $$f_{\tilde{X}}(s) = \sum_{j = 0}^i (P(X \in I_j) \cdot \mu_j) + s \cdot \sum_{j = i+1}^{n+1} P(X \in I_j).$$ Therefore, $f_{\tilde{X}}$ is a continuous piecewise linear function, whose breakpoints are $\mu_1, \dots, \mu_n$. ◻ Based on the above results, the piecewise linear function $f_{\tilde{X}}$ is uniquely determined once a partition $\mathcal{I}$ of $(a, b]$ and the associated random variable $\tilde{X}$ have been specified. ## Analysis of Approximation Error Now, we evaluate the absolute error between the piecewise linear function $f_{\tilde{X}}$ and $f_X$ defined as $$\label{def:approximation_error} e_{X, \tilde{X}} = \max_{s \in (a, b]} |f_{\tilde{X}}(s) - f_X(s)|.$$ Since $(I_1, \cdots, I_n)$ is a partition of $(a, b]$, by taking the maximum of the errors in each region, $e_{X, \tilde{X}}$ is rewritten as $$\begin{aligned} e_{X, \tilde{X}} &= \max_{j \in \{1,\dots, n\}} \max_{s \in I_j}| f_{\tilde{X}}(s) - f_X(s)|\\ &= \max_{j \in \{1,\dots, n\}} \Delta_X(I_j), \end{aligned}$$ where we define $$\label{def_delta} \Delta_X(I_j) \coloneqq \max_{s \in I_j}| f_{\tilde{X}}(s) - f_X(s)| \quad (j \in \{1, \dots, n\}).$$ **Remark 3**. Due to strict concavity of $f_X$, $e_{X, \tilde{X}}$ is minimized when all $\Delta_X(I_j)$ have the same value [@imamoto2008recursive]. Utilizing this property, research led by [@rossi2014piecewise] and others employ an approach that solves a nonlinear optimization problem to make all $\Delta_X(I_j)$ equal. These analyses are only valid for the optimal partition. In our study, we evaluate the approximation error for any partition for $(a, b]$, which makes it possible to evaluate the piecewise linear approximation function that provides a good but non-optimal approximation. From here on, we will evaluate $\Delta_X(I_k)$ for fixed $k \in \{1, \dots n\}$. For $s \in (a, b]$, we transform $f_X$ as follows. $$\begin{aligned} f_X(s) &= \mathbb{E}[\min(s, X)] = \sum_{j=0}^{n+1}\mathbb{E}_{(a_j, b_j]}[\min (s, X)]\\ &= \sum_{j=0}^{n+1}P(X \in I_j) \mathbb{E}[ \min (s, X_{I_j})], \end{aligned}$$ where we define the conditional random variable when $X \in I_j$ as $$X_{I_j} \coloneqq X \mid X \in I_j \quad (j \in \{1, \dots, n\}).$$ When $t \in I_k$, for any $j \neq k$, we have $$\mathbb{E}[ \min(t, X_{I_j})] = \min(t, \mu_j)$$ because we see $$\mathbb{E}[ \min(t, X_{I_j})] = \begin{cases} \mu_j \quad (\leq b_j \leq b_k \leq t) & \text{if} \ j < k, \\ t \quad (\leq b_k \leq a_j \leq \mu_j) & \text{if} \ j > k. \end{cases}$$ Comparing $f_{\tilde{X}}(s)$ as defined in [\[def_f\_tilde\]](#def_f_tilde){reference-type="eqref" reference="def_f_tilde"} with $f_X$, each term corresponding to $j \neq k$ has the same value in both $f_X$ and $f_{\tilde{X}}$. Thus, by subtracting $f_X(t)$ from $f_{\tilde{X}}(t)$, we obtain the following. $$f_{\tilde{X}}(t) - f_X(t) = P(X \in I_k) (\min (t, \mu_k) - \min(t, \mathbb{E}[ \min (t, X_{I_k})] )) \geq 0,$$ where the last inequality is from Jensen's inequality since $f_{X_{I_k}}$ is proven to be concave in Lemma [Lemma 19](#lemma:concave){reference-type="ref" reference="lemma:concave"}. By substituting the above equation into the definition of $\Delta_{X}$ given by [\[def_delta\]](#def_delta){reference-type="eqref" reference="def_delta"}, we obtain the following lemma. **Lemma 4**. For each $j \in \{1, \cdots, n\}$, $$\label{eq:delta} \Delta_X (I_j) = P(X \in I_j) \cdot \max_{s \in I_j} ( \min(s, \mu_j) -\mathbb{E}[ \min(s, X_{I_j})] ).$$ We can show that the maximum value in $\Delta_X(I_j)$ is attained at $s = \mathbb{E}[X \mid X \in I_j]$ and its value is calculated as follows, which is proved in the next subsection. The relationship is illustrated in Figure [1](#fig_1){reference-type="ref" reference="fig_1"}. {#fig_1 width="70%"} **Lemma 5**. For an interval $I_j = (a_j, b_j]$, $$\max_{s \in I_j} ( \min(s, \mu_j) -\mathbb{E}[ \min(s, X_{I_j})] ) = \mathbb{E}_{a,\mu_j}[\mu_j -X_{I_j}] \leq \frac{b_j-a_j}{4}.$$ From the above lemma, we have the following theorem, where the error in each interval can be analytically computed and is bounded by a value proportional to the product of the length of each region and the probability within that region. **Theorem 6**. For a interval $I_j = (a_j, b_j] \ (j \in \{1, \dots,n\})$, $$\Delta_X(I_j) = \mathbb{E}_{a_j,\mu_j}[\mu_j -X] \leq P(X \in I_j) \cdot \frac{b_j -a_j}{4}.$$ *Proof.* From the inequality in Lemma [Lemma 5](#lemma:one-break){reference-type="ref" reference="lemma:one-break"} and Lemma [Lemma 4](#lemma:delta){reference-type="ref" reference="lemma:delta"}, we have $$\Delta_{X}(I_j) \leq P(X\in I_j) \cdot \frac{b_j -a_j}{4}.$$ Next, we show the equality. From the equality in Lemma [Lemma 5](#lemma:one-break){reference-type="ref" reference="lemma:one-break"} and Lemma [Lemma 4](#lemma:delta){reference-type="ref" reference="lemma:delta"}, we have $$\Delta_{X}(I_j) = P(X \in I_j) \cdot \mathbb{E}_{a_j, \mu_j}[\mu_j - X_{I_j}].$$ Since $$\mathbb{E}_{a_j, \mu_j}[\mu_j - X_{I_j}] = \frac{\mathbb{E}_{a_j, \mu_j}[\mu_j - X]}{P(X \in I_j)},$$ we obtain $$\Delta_X(I_j) = \mathbb{E}_{a_j, \mu_j}[\mu - X].$$ ◻ We finally derive the following result. **Corollary 7**. $$e_{X, \tilde{X}} = \max_{j \in \{1, \dots n\} } \mathbb{E}_{a_j,\mu_j}[\mu_j - X]\leq \max_{j \in \{1, \dots n\}} \left(P(X \in I_j) \cdot \frac{b_j - a_j}{4}\right).$$ ## Proof of Lemma [Lemma 5](#lemma:one-break){reference-type="ref" reference="lemma:one-break"}: Approximation Error of Piecewise Linear Approximation with One Breakpoint {#proof-of-lemma-lemmaone-break-approximation-error-of-piecewise-linear-approximation-with-one-breakpoint} In this subsection, we provide a proof of Lemma [Lemma 5](#lemma:one-break){reference-type="ref" reference="lemma:one-break"}. Let $Y$ be a real-valued random variable whose support is a subset of $(a, b] \subseteq \mathbb{R}$. Let $\mu$ be the mean value of $Y$, that is, $\mu := \mathbb{E}[Y]$. Consider the piecewise linear approximation function to $f_Y(s) = \mathbb{E}[\min(s, Y)]$, whose breakpoint is only the mean value $\mathbb{E}[Y]$. Then, this function is $\min(s, \mathbb{E}[Y])$ and we define its approximation error at a point $s \in (a, b]$ as $$\delta_Y (s) \coloneqq |\min(s, \mathbb{E}[Y]) - f_Y(s)|.$$ To show the claim of Lemma [Lemma 5](#lemma:one-break){reference-type="ref" reference="lemma:one-break"}, it suffices to show that the maximum value of $\delta_Y(s)$ is equal to $\mathbb{E}_{a,\mu}[\mu -Y]$ and it is bounded by $\frac{b-a}{4}$. The following lemma shows that $\delta_Y(s)$ attains its maximum value at $s=\mu$ and its maximum value is $\mathbb{E}_{a,\mu}[\mu -Y]$. The relationship is illustrated in Figure [2](#fig_2){reference-type="ref" reference="fig_2"}. {#fig_2 width="70%"} **Lemma 8**. $$\max_{s \in (a,b]} \delta_Y(s) = \delta_Y(\mu) = \mathbb{E}_{a,\mu}[\mu -Y]$$ *Proof.* Assume $s \in (a, b]$. Then, $f_Y$ is written as $$f_Y(s) = \mathbb{E}_{a,s}[Y] + \mathbb{E}_{s,b}[s].$$ Thus, if $s \in (a, \mu]$, from $\min(s, \mu)=s$, we see that $$\begin{aligned} \delta_Y(s) &= s - (\mathbb{E}_{a,s}[Y] + \mathbb{E}_{s,b}[s])\\ &=\mathbb{E}_{a,s}[s] + \mathbb{E}_{s,b}[s] - \mathbb{E}_{a,s}[Y] - \mathbb{E}_{s,b}[s]\\ &= \mathbb{E}_{a,s}[s-Y]. \end{aligned}$$ Similarly, if $s \in (\mu, b]$, from $\min(s, \mu) = \mu$, we see that $$\begin{aligned} \delta_Y(s) &= \mu - (\mathbb{E}_{a,s}[Y] + \mathbb{E}_{s,b}[s])\\ &=\mathbb{E}_{a,s}[Y] + \mathbb{E}_{s,b}[Y] - \mathbb{E}_{a,s}[Y] - \mathbb{E}_{s,b}[s]\\ &= \mathbb{E}_{s,b}[Y-s]. \end{aligned}$$ Hence, we obtain $$\delta_Y(s) = \left \{\begin{array}{ll} \mathbb{E}_{a,s}[s-Y] & \text{if} \ s \in (a, \mu], \\ \mathbb{E}_{s,b}[Y-s] & \text{if} \ s \in (\mu, b]. \\ \end{array} \right .$$ For $s \in (a, \mu)$ and $s +\epsilon \in (a, \mu]$ such that $\epsilon >0$, we see that $$\begin{aligned} \mathbb{E}_{a, s + \epsilon}[s + \epsilon - Y] &= \mathbb{E}_{a, s}[s + \epsilon - Y] + \mathbb{E}_{s, s+\epsilon}[s + \epsilon - Y]\\ &= \mathbb{E}_{a, s}[s - Y] + \mathbb{E}_{a, s}[\epsilon] + \mathbb{E}_{s, s+\epsilon}[s + \epsilon - Y]\\ &\geq \mathbb{E}_{a, s}[s - Y]. \end{aligned}$$ Thus, $\delta_Y(s)$ is increasing for $s \in (a, \mu]$, and decreasing for $s \in (\mu, b]$ as well. Also, it is continuous since $f_{Y}$ and $f_{\tilde{Y}}$ are continuous from Lemma [Lemma 19](#lemma:concave){reference-type="ref" reference="lemma:concave"}. Therefore, it attains its maximum value at $s=\mu$. ◻ We finally obtain the upper bound as follows. **Theorem 9**. $$\max_{s \in (a, b]} \delta_Y(s) \leq \frac{b-a}{4}.$$ *Proof.* Since $\max_{s \in (a, b]}\delta_Y(s) = \delta_Y(\mu)$ from Lemma [Lemma 8](#lemma:delta_max){reference-type="ref" reference="lemma:delta_max"}, we have $$\begin{aligned} \delta_Y(\mu) &= \mathbb{E}_{a, \mu}[\mu - Y] \notag \\ &= P(a < Y \leq \mu) \mathbb{E}[\mu - Y | a < Y \leq \mu] \notag \\ &= P_A (\mu -\mu_A) \label{eq:last}, \end{aligned}$$ where $P_A = P(Y \leq \mu)$, $\mu_A = \mathbb{E}[Y \mid Y \leq \mu]$. Define $P_B = P( \mu < Y \leq b)$ and $\mu_B = \mathbb{E}[Y \mid \mu < Y \leq b]$, where we assume that $P_B > 0$ since $\delta_Y(\mu) = 0$ holds when $P_B = 0$. Then, we have the following relation: $$\begin{aligned} P_A + P_B &= 1, \\ P_A \mu_A + P_B \mu_B &= \mu. \end{aligned}$$ Thus, we have $P_A = \frac{\mu_B - \mu}{\mu_B - \mu_A}$. Therefore, substituting $P_A$ into [\[eq:last\]](#eq:last){reference-type="eqref" reference="eq:last"}, we obtain the following equation: $$\delta_Y(\mu) = \frac{(\mu - \mu_A)(\mu_B - \mu)}{\mu_B - \mu_A}.$$ Moreover, we obtain $$\frac{(\mu - \mu_A)(\mu_B - \mu)}{\mu_B - \mu_A} \leq \max_{\mu' \in [\mu_A, \mu_B]} \frac{(\mu' - \mu_A)(\mu_B - \mu')}{\mu_B - \mu_A} = \frac{\mu_B-\mu_A}{4},$$ where the last equation is from that the maximum value is attained when $\mu' = \frac{\mu_B - \mu_A}{2}$. Since $\mu_B - \mu_A \leq b-a$ holds, we finally get $\delta_Y(\mu) \leq \frac{b-a}{4}$. ◻ Under certain assumptions, we can approximate $\delta_Y(\mu) \approx \frac{b-a}{8}$, which is useful in the design of algorithms in the next section. **Remark 10**. Let $Z$ be a random variable having the probability density function $f: [a, b] \rightarrow \mathbb{R}$ and assume $f$ is absolutely continuous on $[a,b]$ and its first derivative $f'$ belongs to the Lebesgue space $L_\infty [a,b]$. From Theorem 2 in [@barnett2000some], we have $$\begin{aligned} \left|\mathbb{E}[Z] - \frac{a + b}{2}\right| \leq \frac{(b-a)^3}{12} \| f'\|_{\infty}. \end{aligned}$$ Thus, when the first derivative of the probability function in $(a, b]$ is small enough, in the proof above, we can approximate the error as $$\delta_Y(\mu) = \frac{(\mu - \mu_A)(\mu_B - \mu)}{\mu_B - \mu_A} \approx \frac{b-a}{8},$$ where $$\mu \approx \frac{a+b}{2}, \ \mu_A \approx \frac{a + \mu}{2} \ \text{and}\ \mu_B \approx \frac{\mu+b}{2}.$$ Under this assumption, we can also approximate $\Delta_{X}(I_j)$ as follows: $$\label{eq:approx} \Delta_X(I_j) \approx P(X \in I_j) \cdot \frac{b_j - a_j}{8}.$$ # Partition Algorithms In this section, we propose a partition algorithm that guarantees a bounded number of breakpoints while keeping the error below $\epsilon$. Our algorithm is shown in Algorithm [\[partition-algorithm\]](#partition-algorithm){reference-type="ref" reference="partition-algorithm"} and based on the results of Section 3.  \ - error bound function, $B \in \{B_{\text{exact}}, B_{1/4}, B_{1/8} \}$ defined by [\[def_B\]](#def_B){reference-type="eqref" reference="def_B"}\ - support of the random variable, S $\subseteq \mathbb{R}$\ - target interval $(a, b] \subseteq \mathbb{R}$\ - acceptable error $\epsilon > 0$\ a partition of $\mathbb{R}$, $I=(I_0, I_1, \dots, I_n, I_{n+1})$, such that $(I_1, \dots, I_n)$ is also a partition of $(a,b]$ $a_1 \leftarrow a$ $j \leftarrow 1$ $\displaystyle b_j \leftarrow \max_{y \in (a_j, b) \cap S} y \ \text{s.t.} \ B(a_j, y) \leq \epsilon$ [\[find_b\]]{#find_b label="find_b"} $a_{j+1} \leftarrow b_j$ $I_j \leftarrow (a_j, b_j]$ $j \leftarrow j+1$ $b_j \leftarrow b$ $I_j \leftarrow (a_j, b_j]$ $n \leftarrow j$ $I_0 \leftarrow (-\infty, a]$ $I_{n+1} \leftarrow (b, -\infty)$ $\mathcal{I}=(I_0, I_1, \dots, I_n, I_{n+1})$ The input to the Partition Algorithm consists of a tolerance $\epsilon > 0$, an interval $(a, b]$, and a bound function $B(x, y)$ that roughly represents the error in the piecewise linear approximation over the interval $(x, y]$. $B$ is chosen in $\{B_{\text{exact}}, B_{1/4}, B_{1/8}\}$ defined as $$\begin{aligned} \begin{split} B_{\text{exact}}(x, y) & \coloneqq \mathbb{E}_{x, \mu}[\mu - X] \ \ (\mu = \mathbb{E}[X \mid X \in (x, y]]) \\ B_{1/4}(x, y) &\coloneqq P(X \in (x, y]) \cdot \frac{y-x}{4}\\ B_{1/8}(x, y) &\coloneqq P(X \in (x, y]) \cdot \frac{y-x}{8}, \end{split} \label{def_B} \end{aligned}$$ where $B_{\text{exact}}$ and $B_{1/4}$ are derived from the equality and inequality of Theorem [Theorem 6](#theorem:delta){reference-type="ref" reference="theorem:delta"}, respectively and $B_{1/8}$ is derived from Remark [Remark 10](#remark:uniform){reference-type="ref" reference="remark:uniform"}. For simplicity, we refer to the Partition algorithm using the bound function $B$ as Algorithm $B$. The output is a partition of $\mathbb{R}$. The idea behind the algorithm is quite simple. We start with $a_1 = a$, and in the $j$-th iteration, with $a_j$ already determined, we choose $b_j$ to be the largest value such that an error function $B(a_j, b_j)$ does not exceed $\epsilon$. In this study, we assume that we can find $b_j$ exactly. The validity of this assumption and an actual method for finding $b_j$ will be discussed later. Our algorithms bear a resemblance to the heuristic algorithms presented by [@rebennack2015continuous], in which the next maximum breakpoint is selected to ensure that the error does not exceed $\epsilon$. The key distinction is that, in our algorithms, we determine the breakpoints indirectly by defining the intervals. From Theorem [Theorem 6](#theorem:delta){reference-type="ref" reference="theorem:delta"}, we see the following relation $$\Delta_X((x,y]) = B_{\text{exact}}(x, y) \leq 2B_{1/8}(x, y) = B_{1/4}(x, y).$$ Therefore, we have the following result about approximation errors from Colorollary [Corollary 7](#theorem:approx){reference-type="ref" reference="theorem:approx"}. **Theorem 11**. Let $\tilde{X}$ be the discrete random variable induced by piecewise linear approximation with the output $\mathcal{I}$ of Algorithm [\[partition-algorithm\]](#partition-algorithm){reference-type="ref" reference="partition-algorithm"}. Depending on the setting of $B$, the following results are obtained: $$\begin{aligned} {2} e_{X, \tilde{X}} &\leq \epsilon &\quad \text{ if } \quad & B = B_{\text{{exact}}} \text{ or } B=B_{1/4}, \\ e_{X, \tilde{X}} &\leq 2\epsilon &\quad \text{ if } \quad & B=B_{1/8}. \end{aligned}$$ The remaining issue is how large the upper bound of the number of breakpoints for each algorithm is. To summarize, the results obtained in this study can be compiled in the following table (Table [1](#tab:breakpoints){reference-type="ref" reference="tab:breakpoints"}). ---------------------------------- ------------- -------------------------------------------------- -------------------------------------------------- Algorithm error breakpoints continuous discrete $B_{\text{exact}}$ and $B_{1/4}$ $\epsilon$ $\frac{1}{2} \sqrt{\frac{b-a}{\epsilon}}$ $\sqrt{\frac{b-a}{\epsilon}}$ $B_{1/8}$ $2\epsilon$ $\frac{1}{2\sqrt{2}}\sqrt{\frac{b-a}{\epsilon}}$ $\frac{1}{\sqrt{2}} \sqrt{\frac{b-a}{\epsilon}}$ ---------------------------------- ------------- -------------------------------------------------- -------------------------------------------------- : Summary of Properties for Each Algorithm As can be seen from Table [1](#tab:breakpoints){reference-type="ref" reference="tab:breakpoints"}, the bounds on the number of breakpoints differ depending on whether the target random variable is continuous or discrete. Furthermore, $B_{\text{exact}}$ and $B_{1/4}$ yield equivalent results in terms of theoretical guarantees. Apart from the above differences, each algorithm has its own unique characteristics. We can show that Algorithm $B_{\text{exact}}$ is guaranteed to have the minimum number of breakpoints. However, for continuous random variables, the calculation of $B_{\text{exact}}$ involves numerical integration for conditional expectations, making its implementation costly and potentially introducing numerical errors. Algorithm $B_{1/8}$ theoretically could have an error up to $2\epsilon$, but in practice, it behaves almost identically to $B_{\text{exact}}$ (see the section on numerical experiments for details). Finally, we outline the remaining structure of this section. In Section 4.1, we discuss the specific implementation methods for finding $b_j$. In Section 4.2, we prove the optimality of Algorithm $B_{\text{exact}}$. Lastly, in Section 4.3, we provide bounds on the number of breakpoints for each algorithm. ## Implementation of Finding Next Point We discuss the actual implementation methods for finding $b_j$ even though we assume that $b_j$ is determined exactly in line [\[find_b\]](#find_b){reference-type="ref" reference="find_b"} of Algorithm 1, Unless otherwise specified, $B$ can be any of [\[def_B\]](#def_B){reference-type="eqref" reference="def_B"}, where we assume $B(x, x) = 0$ for any $x$. Here, we describe the implementation method for finding $b_j$ as performed in line [\[find_b\]](#find_b){reference-type="ref" reference="find_b"} of the algorithm. First, for $x \in (a, b)$, we define the following function $L_x \colon (x, b] \rightarrow \mathbb{R}$: $$L_x(y) = B(x, y) - \epsilon.$$ At the time of the $j$-th iteration in line [\[find_b\]](#find_b){reference-type="ref" reference="find_b"}, $L_{a_j}(a_j) = -\epsilon < 0$ and $L_{a_j}(b) > 0$ hold. We first consider the case where $X$ is continuous. For simplicity, we assume that the support of $X$ is $\mathbb{R}$. We assume that $L_{a_j}$ is continuous and strictly monotonically increasing. This is an assumption that holds for standard distributions. At this time, there exists only one $y$ such that $L_{a_j}(y) = 0$, that is, $B(a_j, y) = \epsilon$, within $(a_j, b)$. This can be realized by calling $B$ $O(\log (\frac{b-a}{d}))$ times using binary search for a small allowable error $d > 0$. Next, we consider the case where $X$ is discrete. Let $Z = \{x_1, \dots, x_K\}$ where $x_1 < \dots < x_K$ be the intersection of the support $S$ of $X$ and $(a, b]$. We assume that $a_j = x_{k[j]} \in Z$. We can find the largest $y$ from $\{x_{k[j]+1}, \dots, x_{K-1}\}$ satisfying $L_{a_j}(y) \leq \epsilon$ by $O(\log(b-a))$ calls to $B$. Finally, we note that special handling is needed for $B_{1/4}$ and $B_{1/8}$ when $\epsilon$ is extremely small. In such cases, it is possible that no $y$ satisfying $B(a_j, b_j) \leq \epsilon$ exists within $\{x_{k[j]+1}, \dots, x_{K-1}\}$. In this situation, we set $b_j = x_{k[j] + 1}$. Although this would make $B(a_j, b_j) > \epsilon$, the actual error on $(a_j, b_j]$ can still be bounded below $\epsilon$ as the exact error $B_{\text{exact}}(a_j,b_j)=0$. ## Optimality In this section, we show the optimality of Algorithm $B_{\text{exact}}$. Specifically, we show that the output of this algorithm is an optimal solution to the following optimization problem as follows: for given $\epsilon > 0$ and $(a, b] \subseteq \mathbb{R}$: $$\begin{aligned} \label{opt_breakpoint} \begin{array}{lll} \text{minimize} & n \\ \text{subject to} & \Delta_X(I_j) \leq \epsilon \quad \forall i \in \{1, \dots, n\}, \\ \end{array} \end{aligned}$$ where $n \in \mathbb{N}$ and a partition $\mathcal{I}=(I_0, \dots, I_{n+1})$ are decision variables. **Theorem 12**. Let $\mathcal{I} = (I_0,I_1, \dots, I_n, I_{n+1})$ be the output of Algorithm $B_{\text{exact}}$. Then, $\mathcal{I}$ is an optimal solution of [\[opt_breakpoint\]](#opt_breakpoint){reference-type="eqref" reference="opt_breakpoint"}. *Proof.* Let $\mathcal{I} = (I_0,I_1, \dots, I_n, I_{n+1})$ be the output of Algorithm [\[partition-algorithm\]](#partition-algorithm){reference-type="ref" reference="partition-algorithm"}, where we use the notations $I_j = (a_j, b_j]$ for $j \in \{1, \dots, n\}$. We derive a contradiction by assuming that there is a feasible solution $\mathcal{I}^* = (I^*_0, \dots, I_{m+1}^*)$ such that $m < n$, where we also use the notation $I_j^* = (a_j^*, b_j^*)$. Without loss of generality, we assume that $a_j^*, b_j^*$ are chosen in the support $S$ of $X$. We prove for any $j \in \{1, \dots, m\}$, $b_j^* \leq b_j$ by induction. When $j=1$, we see that $a_1=a_1^* = a$, $\Delta(a, b_1) \leq \epsilon$ and $\Delta(a, b_1^*) \leq \epsilon$. Since $b_1$ is the maximum in $S \cap (a, b]$ such that $\Delta(a, b_1) \leq \epsilon$, we have $b_1^* \leq b_1$. Now, for $j \in \{2, \dots, m\}$, we assume that $b_{j-1} \leq b_{j-1}^*$. First, we consider the case when $b_j^* \leq a_j$. In this case $b_j^* \leq a_j < b_j$ holds. Second, we assume that $a_j \leq b_j^*$. From the assumption of induction, we have $$a_j^* = b_{j-1}^*\leq b_{j-1} = a_j.$$ Therefore, $a_j^* \leq a_j \leq b_j^*$ holds. From this relation, then it leads $$\Delta_X(a_j, b_j^*) \leq \Delta_X(a_j^*, b_j^*) \leq \epsilon.$$ where we use Lemma [Lemma 20](#lemma:lower_delta){reference-type="ref" reference="lemma:lower_delta"}. When $b_j^* > b_j$, it contradicts that $b_j$ is maximal in $(a_j, b] \cap S$ such that $\Delta_X(a_j,b_j) \leq \epsilon$. Thus, we have $b_j^* \leq b_j$ for all $j \in \{1, \dots, m\}$. When $j=m$ in the result above, we have the following contradiction $$b = b_m^* \leq b_m < b_n = b.$$ Hence, for any feasible solution $\mathcal{I}^* = (I^*_0, \dots, I_{m+1}^*)$, $m \geq n$ holds, which implies $\mathcal{I}$ is optimal. ◻ ## Upper Bounds of Breakpoints In this section, we provide bounds on the number of output intervals for each algorithm. Note that we give proof of the bounds only for $B_{1/4}$ and $B_{1/8}$ since the number of the breakpoints with Algorithm $B_{\text{exact}}$ is less than or equal to that of Algorithm $B_{1/4}$ from Theorem [Theorem 12](#theorem:opt){reference-type="ref" reference="theorem:opt"}. First, we derive bounds for the case when $X$ is continuous. **Theorem 13**. Assume that $X$ is continuous. Let $\mathcal{I}=(I_0, I_1, \dots I_n, I_{n+1})$ be the outputs of Algorithm [\[partition-algorithm\]](#partition-algorithm){reference-type="ref" reference="partition-algorithm"}. Let $\tilde{X}$ be the discrete random variable induced by piecewise linear approximation with $\mathcal{I}$. Depending on the setting of $B$, the following inequalities holds: $$\begin{aligned} {2} n &\leq \frac{1+P}{4} \sqrt{\frac{b - a}{\epsilon}} + 1 \leq \frac{1}{2} \sqrt{\frac{b - a}{\epsilon}} + 1&\quad \text{ if } \quad & B \in \{B_{\text{{exact}}}, B_{1/4}\}, \\ n &\leq \frac{1+P}{4\sqrt{2}} \sqrt{\frac{b - a}{\epsilon }} + 1 \leq \frac{1}{2\sqrt{2}} \sqrt{\frac{b - a}{\epsilon }} + 1&\quad \text{ if } \quad & B=B_{1/8}, \end{aligned}$$ where $P = P(X \in (a, b])$. *Proof.* Let $\mathcal{I} = (I_0,I_1, \dots, I_n, I_{n+1})$ be the output of Algorithm [\[partition-algorithm\]](#partition-algorithm){reference-type="ref" reference="partition-algorithm"}, where we use the notations $I_j = (a_j, b_j]$ for $j \in \{1, \dots, n\}$. Let $B \in \{ B_{1/4}, B_{1/8}\}$ and $$M = \begin{cases} 1 & \text{if} \quad B = B_{1/4}, \\ 2 & \text{if} \quad B=B_{1/8}. \end{cases}$$ For $j \in \{1, \dots, n\}$, define $p_j = P(X \in I_j)$ and $r_j = \frac{b_j -a_j}{b-a}$. From $\sum_{j=1}^{n-1} p_j \leq P$ and $\sum_{j=1}^{n-1}r_j \leq 1$, we have $$\sum_{j=1}^{n-1} (p_j + r_j) \leq 1+P.$$ Now, we will get an upper bound of $n$ by estimating a lower bound of $p_j + r_j$. For any $j \in \{1, \dots, n-1\}$, since $B(a_j, b_j) = \epsilon$ holds in Line [\[find_b\]](#find_b){reference-type="ref" reference="find_b"} in Partition Algorithm, we have $$\epsilon = P(X \in I_j) \frac{b_j-a_j}{4M} = \frac{p_jr_j(b-a)}{4M}.$$ Thus, we have $$p_jr_j =\frac{4M\epsilon}{b-a}.$$ From the relationship of the geometric mean, we obtain $$p_j + r_j \geq 2\sqrt{p_jr_j} = 4 \sqrt{\frac{M\epsilon}{b-a} }.$$ By summing $j$ from $1$ to $n-1$, we obtain the following expression: $$4(n-1) \sqrt{\frac{M\epsilon}{b -a}} \leq \sum_{j=1}^{n-1}(p_j + r_j) \leq 1+P$$ Therefore, $$n \leq \frac{1+P}{4\sqrt{M}}\sqrt{\frac{b - a}{\epsilon}} + 1.$$ ◻ Next, we derive bounds for the case when $X$ is discrete. **Theorem 14**. Assume that $X$ is discrete. Let $\mathcal{I}=(I_0, I_1, \dots I_n, I_{n+1})$ be the outputs of Algorithm [\[partition-algorithm\]](#partition-algorithm){reference-type="ref" reference="partition-algorithm"}. Let $\tilde{X}$ be the discrete random variable induced by piecewise linear approximation with $\mathcal{I}$. Depending on the setting of $B$, the following inequalities holds: $$\begin{aligned} {2} n &\leq \frac{1+P}{2} \sqrt{\frac{b - a}{\epsilon}} + 1 \leq \sqrt{\frac{b - a}{\epsilon}} + 1&\quad \text{ if } \quad & B = B_{\text{{exact}}} \text{ or } B=B_{1/4}, \\ n &\leq \frac{1+P}{2\sqrt{2}} \sqrt{\frac{b - a}{\epsilon }} + 1 \leq \frac{1}{\sqrt{2}} \sqrt{\frac{b - a}{\epsilon }} + 1&\quad \text{ if } \quad & B=B_{1/8}, \end{aligned}$$ where $P = P(X \in (a, b])$. *Proof.* Recall that the endpoints of the interval $I_j = (a_j, b_j]$, $a_j$ and $b_j$, are chosen from within the support of $X$, denoted as $\{x_1, \dots, x_K\}$. In the $j$-th iteration, the index selected is represented by $k[j]$ such that $x_{k[j]} = b_j$. Define $p_j = P(X \in I_j)+ P(X=x_{k[j] + 1})$ and $r_j = \frac{b_j -a_j}{b -a} + \frac{ x_{k[j] + 1} - x_{k[j]}}{b-a}$ for $j \in \{1, \dots, n-1\}$. From $\sum_{j=1}^{n-1} p_j \leq 2P$ and $\sum_{j=1}^{n-1}r_j \leq 2$, we have $$\sum_{j=1}^{n-1} (p_j + r_j ) \leq 2(1+P).$$ Now, we will get an upper bound of $n$ by estimating a lower bound of $p_j + r_j$. For any $j \in \{1, \dots, n-1\}$, since $k[j]$ is the maximum index in $\{x_{k[j-1]}, \dots, x_{K-1}\}$ such that $B(a_j, x_{k[j]}) \leq \epsilon$, we have $$B(a_j, x_{k[j] + 1})= (P(X \in I_j) + P(X=x_{k[j] + 1}))\frac{x_{k[j] + 1} -a_j}{4M} > \epsilon.$$ From $b_j = x_{k[j]}$ and the definitions of $p_j$ and $r_j$, we have $$\begin{aligned} \epsilon &< (P(X \in I_j) + P(X=x_{k[j] + 1})) \frac{b_j -a_j + x_{k[j]+1} - x_{k[j]}}{4M} \\ &= p_j r_j \frac{b-a}{4M}. \end{aligned}$$ Thus, we have $$\frac{4M\epsilon}{b-a} < p_jr_j.$$ Following the almost same procedure as in the continuous case, we obtain the following result. $$n \leq \frac{1+P}{2\sqrt{M}} \sqrt{\frac{b - a}{\epsilon}} + 1.$$ ◻ # Lower bounds of the breakpoints In this section, we derive a lower bound on the number of breakpoints to achieve a given absolute error $\epsilon > 0$ when we use a piecewise linear approximation based on the approach in Section 2. Table [2](#tab:result){reference-type="ref" reference="tab:result"} summarizes the results from Section 3 and this section. From these results, we see that the upper bounds of the number of breakpoints in our proposed algorithm are close to the lower bounds. Continuous Discrete ------------------------------------------------ ----------------------------------- ----------------------------------- Lower Bound $\frac{\sqrt{2}}{4} \approx 0.36$ $\frac{\sqrt{2}}{2} \approx 0.71$ Upper Bound of $B_{\text{exact}}$ or $B_{1/4}$ $\frac{1}{2}$ 1 : Coefficients of the Upper and Lower Bounds in Terms of $\sqrt{\frac{b-a}{\epsilon}}$ We first introduce a common setting in discrete or continuous cases. For any given $a$, $b$, $\epsilon$ such that $a < b$ and $\epsilon > 0$, consider a necessary number of intervals, whose approximation error is within $\epsilon$. Define $$\label{def:N} N \coloneqq \left\lceil \frac{\sqrt{2}}{2} \sqrt{\frac{W}{\epsilon}} \right\rceil + 1 > \frac{\sqrt{2}}{2} \sqrt{\frac{W}{\epsilon}} ,$$ where $W = b - a$. Define the following $N$ equally spaced points in $(a, b]$: $$x_k = a + \frac{W}{N}k \quad (k = 1, \dots N).$$ Roughly speaking, we consider a random variable that takes the value $x_k$ with probability $1/N$ in both the discrete and continuous cases. ## Discrete Case Let $X$ be the discrete random variable, whose support is $\{x_1, \dots, x_{N}\}$ and probability function is $$p_{X}(x) = \left \{\begin{array}{ll} \frac{1}{N} & \text{if} \ x = x_k \ (k=1,\dots,N), \\ 0 & \text{otherwise.} \\ \end{array} \right .$$ The distribution of $X$ is illustrated in Figure [3](#fig_3){reference-type="ref" reference="fig_3"}. {#fig_3 width="60%"} **Lemma 15**. For any $I=(a', b'] \subseteq (a, b]$, $$P(X \in I) > \frac{1}{N} \Rightarrow \Delta_{X}(I) > \epsilon.$$ *Proof.* Let $I \subseteq (a, b]$ be an interval such that $|I \cap \{x_1, \dots, x_N\}| \geq 2$. Without loss of generality, from Lemma [Lemma 20](#lemma:lower_delta){reference-type="ref" reference="lemma:lower_delta"}, for some small positive $o > 0$ and $k \in \{1, \dots, N-1\}$, we assume $I=(x_{k}-o, x_{k+1}]$. From Lemma [Lemma 4](#lemma:delta){reference-type="ref" reference="lemma:delta"}, by defining $\mu = \mathbb{E}[X \mid X \in I] = \frac{x_k + x_{k+1}}{2}$, we have $$\Delta_{X}(I) = \mathbb{E}_{x_k-o, \mu}[\mu-X] = \mathbb{E}_{x_k,\mu}[\mu- X]$$ where the last equality holds since $X$ can only take the value $x_k$ with probability $1/N$ between $x_k - o$ and $\mu$. Finally we have $$\mathbb{E}_{x_k, \mu}[\mu-X] = \left(\frac{x_k + x_{k+1}}{2} - x_k \right) \cdot \frac{1}{N} = \frac{W}{2N^2} > \epsilon,$$ where we use $x_{k+1} - x_{k} = \frac{W}{N}$ and the last inequality is from [\[def:N\]](#def:N){reference-type="eqref" reference="def:N"}. ◻ **Theorem 16**. Let $\tilde{X}_L$ be a discrete random variable induced by piecewise linear approximation with $\mathcal{I}$ of the random variable $X$. Then, $$e_{X, \tilde{X}}\leq \epsilon \Rightarrow | \mathcal{I} | > N.$$ *Proof.* Let $\tilde{X}$ be a discrete random variable induced by piecewise linear approximation with $\mathcal{I}$ of a random variable $X$ such that $\max_{s \in (a, b]} |f_{\tilde{X}}(s) -f_X(s)| \leq \epsilon$. For any $I \in \mathcal{I}$, from $\Delta_X(I) \leq \epsilon$ and Lemma [Lemma 15](#lemma:low_discrete){reference-type="ref" reference="lemma:low_discrete"}, $$1 = \sum_{I \in \mathcal{I}} P(X \in I) < \frac{|\mathcal{I}|}{N}.$$ ◻ ## Continuous Case For some small $w > 0$, let $X$ be the continuous random variable, whose support is $S \coloneqq \cup_{k=1}^{N} (x_k -w, x_k]$ and probability function is $$p_X(x) = \left \{\begin{array}{ll} \frac{1}{Nw} & \text{if} \ x \in S, \\ 0 & \text{otherwise.} \\ \end{array} \right .$$ The distribution of $X$ is illustrated in Figure [4](#fig_4){reference-type="ref" reference="fig_4"}. {#fig_4 width="60%"} **Lemma 17**. For $I=(a', b'] \subseteq (a, b]$, $$P(X \in I) \geq \frac{2}{N} \Rightarrow \Delta_{X}(I) > \epsilon.$$ *Proof.* From Lemma [Lemma 20](#lemma:lower_delta){reference-type="ref" reference="lemma:lower_delta"}, without loss of generality, we assume that $I=(x_{k}-w, x_{k+1}]$, where $P(X \in I) = \frac{2}{N}$. From Lemma [Lemma 4](#lemma:delta){reference-type="ref" reference="lemma:delta"}, we have $$\Delta_{X}(I) = \mathbb{E}_{x_k-w, \mu}[\mu-X],$$ where $\mu = \mathbb{E}[X \mid X \in I] = \frac{x_k + x_{k+1}}{2} - \frac{w}{2}$. For each term, we have $$\begin{aligned} \mathbb{E}_{x_k -w, \mu}[\mu] &=\frac{1}{Nw} \cdot w \cdot \mu = \frac{\mu}{N}\\ \mathbb{E}_{x_k -w, \mu}[X]&= \mathbb{E}[X \mid X \in (x_k-w, \mu]] \cdot P(X\in (x_k-w, \mu]) \\ &= \left(x_k - \frac{w}{2} \right) \cdot \frac{w}{Nw}. \end{aligned}$$ Finally, we get $$\begin{aligned} \mathbb{E}_{x_k-w, \mu}[\mu-X] &= \left( \frac{x_k + x_{k+1}}{2} - \frac{w}{2}- x_k + \frac{w}{2} \right) \cdot \frac{1}{N}\\ &= \frac{W}{2N^2} > \epsilon \end{aligned}$$ where we use $x_{k+1} - x_{k} = \frac{W}{N}$ and the last inequality is from the definition of $N$ in [\[def:N\]](#def:N){reference-type="eqref" reference="def:N"}. ◻ **Theorem 18**. Let $\tilde{X}$ be a discrete random variable induced by piecewise linear approximation with $\mathcal{I}$ of the random variable $X$. Then, $$e_{X, \tilde{X}} \leq \epsilon \Rightarrow | \mathcal{I} | > \frac{N}{2}.$$ *Proof.* Let $\tilde{X}$ be a discrete random variable induced by piecewise linear approximation with $\mathcal{I}$ of a random variable $X$ such that $\max_{s \in (a, b]} |f_{\tilde{X}}(s) -f_X(s)| \leq \epsilon$. For any $I \in \mathcal{I}$, from $\Delta_X(I) \leq \epsilon$ and Lemma [Lemma 17](#lemma:low_cont){reference-type="ref" reference="lemma:low_cont"}, $$1 = \sum_{I \in \mathcal{I}} P(X \in I) < \frac{2|\mathcal{I}|}{N}.$$ ◻ # Numerical Experiments In this section, through numerical experiments, we compare the actual number of breakpoints generated by our proposed algorithms with the derived bounds in Section 4 across various distributions[^1]. ## Implementation We implemented our algorithms using Python 3.9 on a Macbook Pro laptop equipped with an Apple M1 Max CPU. We implemented all procedures required for the algorithms and generating distributions using the free scientific computing library, SciPy version 1.9.1 [@2020SciPy-NMeth]. For distribution generation, we utilized the distribution classes available in `scipy.stats`. We performed the calculation of conditional expectations for continuous distributions using the numerical integration function `quad` in `scipy.integrate`, with a numerical absolute tolerance set to $10^{-8}$. Furthermore, we conducted the process of finding the $b_j$ that satisfies $B(a_j, b_j) - \epsilon = 0$ in continuous distributions using the `find_root` function in `scipy.optimize`, through binary search, with an absolute tolerance related to $b_j$ also set to $10^{-8}$. ## Target Distributions Table [3](#tab:dist){reference-type="ref" reference="tab:dist"} summarizes the distributions that are the subject of focus in this study. The `instance` column lists the instance names, while the `distribution` column specifies the distribution names. In the `instance` column, the prefix C or D signifies a continuous or discrete distribution, respectively. In the `parameter` column, we show the parameters set for each instance. We utilized representative parameters for each distribution; however, for discrete distributions, we set the mean to be approximately 100. Additionally, we represent the approximation intervals in the columns for $a$ and $b$. For this study, we set them to be about $\pm 3$ times the standard deviation from the mean. instance distribution parameter $a$ $b$ ---------- ------------------- --------------------- ------- ------- C-N1 Normal $\mu=0, \sigma=1$ -3.0 3.0 C-N2 Normal $\mu=0, \sigma=5$ -15.0 15.0 C-Exp Exponential $\lambda=1$ 0.0 4.0 C-Uni Uniform $a=0, b=1$ 0.0 1.0 C-Bet Beta $\alpha=2, \beta=5$ 0.0 0.8 C-Gam Gamma $k=2, \theta=1$ 0.0 6.2 C-Chi Chi-Squared $k=3$ 0.0 10.3 C-Stu Student's t $\nu=10$ -3.4 3.4 C-Log Logistic $\mu=0, s=1$ -5.4 5.4 C-Lgn Lognormal $\mu=0, \sigma=1$ 0.0 8.1 D-Bin Binomial $n=200, p=0.5$ 78.0 121.0 D-Poi Poisson $\lambda=100$ 70.0 130.0 D-Geo Geometric $p=0.01$ 1.0 398.0 D-Neg Negative Binomial $r=100, p=0.5$ 57.0 142.0 : Target Distributions and Intervals ## Results and Discussion ---------- ------------ ------------------------- ----------- ----------- ------------ ------------ --------------------------------------- ----------- ----------- instance $\epsilon$ the number of intervals error (=$e_{X, \tilde{X}}/ \epsilon$) $B_{\text{exact}}$ $B_{1/8}$ $B_{1/4}$ $UB_{1/8}$ $UB_{1/4}$ $B_{\text{exact}}$ $B_{1/4}$ $B_{1/8}$ C-N1 0.100 3 3 4 3 4 1.000 0.486 0.949 0.050 4 4 6 4 6 1.000 0.495 0.973 0.010 8 8 11 9 13 1.000 0.499 0.996 C-N2 0.100 6 6 8 7 9 1.000 0.498 0.991 0.050 8 8 11 9 13 1.000 0.499 0.996 0.010 17 18 25 20 28 1.000 0.500 0.999 C-Exp 0.100 2 3 3 3 4 1.000 0.490 0.955 0.050 3 3 4 4 5 1.000 0.496 0.981 0.010 7 7 9 8 10 1.000 0.499 0.997 C-Uni 0.100 2 2 2 2 2 1.000 0.500 1.000 0.050 2 2 3 2 3 1.000 0.500 1.000 0.010 4 4 5 4 6 1.000 0.500 1.000 C-Bet 0.100 1 1 2 1 2 0.641 0.418 0.641 0.050 2 2 2 2 2 1.000 0.425 0.837 0.010 3 3 5 4 5 1.000 0.495 0.976 C-Gam 0.100 3 3 4 3 4 1.000 0.491 0.939 0.050 4 4 6 4 6 1.000 0.496 0.983 0.010 8 9 12 9 13 1.000 0.499 0.997 C-Chi 0.100 4 4 5 4 6 1.000 0.495 0.974 0.050 5 5 7 6 8 1.000 0.497 0.991 0.010 11 11 15 12 16 1.000 0.499 0.998 C-Stu 0.100 3 3 4 3 5 1.000 0.483 0.947 0.050 4 4 6 5 6 1.000 0.494 0.967 0.010 8 9 12 10 13 1.000 0.499 0.995 C-Log 0.100 4 4 5 4 6 1.000 0.491 0.961 0.050 5 5 7 6 8 1.000 0.496 0.983 0.010 11 11 15 12 17 1.000 0.499 0.997 C-Lgn 0.100 3 3 4 4 5 1.000 0.480 0.892 0.050 4 4 6 5 7 1.000 0.492 0.960 0.010 8 9 12 10 15 1.000 0.498 0.993 D-Bin 0.100 7 7 12 15 21 0.979 0.445 0.979 0.050 11 12 17 21 30 0.922 0.497 0.889 0.010 27 27 33 47 66 0.970 0.452 0.931 D-Poi 0.100 9 9 13 18 25 0.978 0.440 0.969 0.050 12 13 19 25 35 0.938 0.437 0.880 0.010 33 34 42 55 78 0.995 0.433 0.995 D-Geo 0.100 20 20 29 44 63 0.996 0.497 0.996 0.050 29 29 41 63 88 0.995 0.500 0.995 0.010 66 68 98 139 197 0.983 0.496 0.995 D-Neg 0.100 10 10 15 21 30 0.986 0.496 0.992 0.050 15 15 22 30 42 0.992 0.451 0.992 0.010 41 42 55 66 93 0.961 0.470 0.948 ---------- ------------ ------------------------- ----------- ----------- ------------ ------------ --------------------------------------- ----------- ----------- : Number of Intervals and Errors ### Details of Table Fields Table [4](#tab:all_results){reference-type="ref" reference="tab:all_results"} presents the results obtained in this study. We conducted experiments by varying the allowable error for each instance as $\epsilon \in \{0.1, 0.05, 0.01\}$ (as shown in the column $\epsilon$). We conducted experiments on Algorithm $B \in \{B_{\text{exact}}, B_{1/4}, B_{1/8}\}$ and compared the following 2 items: `Number of Interval Column`: : We compared the number of intervals dividing the interval $(a,b]$ output by each algorithm to the upper bound obtained in this study. These are displayed in the `intervals` column. The $B \in \{B_{\text{exact}}, B_{1/4}, B_{1/8}\}$ columns indicate the actual number of intervals output by each algorithm $B$. The columns for $B \in \{B_{\text{exact}}, B_{1/4}, B_{1/8}\}$ indicate the actual number of intervals output by each algorithm $B$. More precisely, this refers to $n$ in Algorithm 1. The $UB \in \{ UB_{1/4}, UB_{1/8}\}$ columns indicate the upper bounds of the breakpoints, which are rounded to integers, shown in Table [1](#tab:breakpoints){reference-type="ref" reference="tab:breakpoints"} in Section 3. Here, $UB_{1/4}$ and $UB_{1/8}$ are the upper bounds corresponding to Algorithm $B_{1/4} \ (B_{\text{exact}})$ and $B_{1/8}$, respectively. `Error Column:` : We conducted a comparison between the input $\varepsilon$ and the error $e_{X, \tilde{X}}$ associated with $\tilde{X}$ obtained from the partition output by each algorithm. We analytically calculated $e_{X, \tilde{X}}$ based on Corollary [Corollary 7](#theorem:approx){reference-type="ref" reference="theorem:approx"}. Although the results are not included in the paper, we confirmed that the error calculated analytically based on Corollary [Corollary 7](#theorem:approx){reference-type="ref" reference="theorem:approx"} matched the error calculated by SciPy's optimization library. In the table, the error column lists the ratio obtained by dividing the actual error by $\epsilon$. ### Results 1. Difference between Algorithms: Although not shown in the table, the computation time for each algorithm was less than 0.2 seconds. First, we discuss the number of intervals. For any distribution, the number of intervals output by $B_{\text{exact}}$ and $B_{1/8}$ differed by only 1-2. The number of intervals for $B_{1/4}$ was approximately 40% greater in each instance compared to $B_{\text{exact}}$ or $B_{1/8}$. Next, we discuss the error. For $B_{\text{exact}}$, the error values for continuous distributions were all 1.0. This is because, except for the last interval, we selected intervals where the error was precisely $\epsilon$. On the other hand, in the case of discrete distributions, it was not possible to select a point where the error was exactly $\epsilon$, resulting in slightly smaller values. The errors for $B_{1/4}$ and $B_{1/8}$ were slightly less than 0.5 and 1 for most distributions, respectively. 2. Difference between Actual Values and Upper Bounds: In continuous distributions, we found that the difference between the actual measured values of intervals and the upper bounds was small. Specifically, the difference between the values of $B_{\text{exact}}$ or $B_{1/8}$ and $UB_{1/8}$, as well as $B_{1/4}$ and $UB_{1/4}$, was generally 2 or less. In the case of discrete distributions, the upper bounds were approximately twice the actual measurements. ### Discussion We observe that many values in the error column of Algorithm $B_{1/8}$ are close to 1. This suggests that, in many cases, the approximation formula for [\[eq:approx\]](#eq:approx){reference-type="eqref" reference="eq:approx"} can be valid and a almost tight upper bound of the approximation error. Furthermore, considering that the number of breakpoints output by $B_{1/8}$ hardly differs from $B_{\text{exact}}$, we conclude that $B_{1/8}$ is an algorithm that outputs near-optimal solutions. Regarding continuous distributions, the obtained upper bounds $UB_{1/4}$ and $UB_{1/8}$ can be regarded as good approximations to the output values of the corresponding algorithms. In other words, $UB$ can be considered as an expression representing the trade-off between error and the number of breakpoints. Therefore, it is conceivable for users to refer to this trade-off when setting an appropriate acceptable error. On the other hand, for discrete distributions, we confirm a discrepancy of about twice between the upper bound of intervals and the algorithm output. We speculate that this is because, under the worst-case analysis due to the nature of discrete distributions within the proof of Theorem [Theorem 14](#theorem_discrete){reference-type="ref" reference="theorem_discrete"}, the upper bound becomes twice as large compared to continuous distributions. However, looking at the experimental results, the number of intervals output by the algorithm is close to $\frac{1}{2 \sqrt{2}} \sqrt{\frac{b-a}{\epsilon}}$ which is the upper bound obtained for continuous distributions. Thus, for the discrete case, $\frac{1}{2 \sqrt{2}} \sqrt{\frac{b-a}{\epsilon}}$ may also be useful as a practical guideline for the minimum number of intervals for given error $\epsilon$. Note that this upper bound may not always align with the actual output. If we prepare the number of intervals for the number of possible values of the discrete random variable, the error at that time can become 0. Thus, even when the number of possible values for the random variable is small, if $b-a$ is large, the actual number of intervals can be much smaller than the bound. Lastly, the following table summarizes these observations. algorithm actual error actual intervals ---------------------------------- ---------------- ---------------------------------------------------- $B_{\text{exact}}$ and $B_{1/8}$ $\epsilon$ $\frac{1}{2 \sqrt{2}} \sqrt{\frac{b-a}{\epsilon}}$ $B_{1/4}$ $0.5 \epsilon$ $\frac{1}{2}\sqrt{\frac{b-a}{\epsilon}}$ : Overview of Experimental Results for Allowable Error $\epsilon$ # Conclusions This study investigated the trade-off between error and the number of breakpoints when performing a piecewise linear approximation of $f_X(s)=\mathbb{E}[\min(s, X)]$. As a result, when conducting piecewise linear approximation based on the method proposed by [@rossi2014piecewise], we obtained $\frac{1}{2\sqrt{2}}\sqrt{\frac{W}{\epsilon}}$ as an approximately upper bound for the minimum number of breakpoints required to achieve an error less than $\epsilon$ through theoretical analysis and numerical experiments, where $W$ is the width of the approximation interval. Subsequently, we also proposed efficient algorithms to obtain a piecewise linear approximation with a number of breakpoints close to this bound. These results provide a guideline for the error and number of breakpoints to consider when we use piecewise linear approximation for a general first-order loss function. # Acknowledgment {#acknowledgment .unnumbered} This work was partially supported by JSPS KAKENHI Grant Numbers JP21K14368. # Proofs **Lemma 19**. $f_X$ is concave on $\mathbb{R}$. *Proof.* For any $\alpha \in [0, 1]$ and $s, t \in \mathbb{R}$, $$\begin{aligned} f_X(\alpha s + (1-\alpha)t) &= \mathbb{E}[\min(\alpha s + (1-\alpha)t, X)] \\ &\geq \mathbb{E}[ \alpha \min (s, X) + (1-\alpha) \min (t, X) ]\\ &= \alpha \mathbb{E}[\min(s, X)] + (1-\alpha) \mathbb{E}[ \min (t, X) ], \end{aligned}$$ where the inequality is from that $c(s) = \min(s, X)$ is concave. ◻ **Lemma 20**. Let $X$ be a random variable. Let $I=(a, b]$ and $I' = (a', b']$ such that $P(X \in I) >0$ and $I \subseteq I'$. Then, $$\Delta_X (I') \geq \Delta_X(I).$$ *Proof.* Let $I=(a, b]$. Define $I^L=(a', b]$ and $I^R = (a, b']$ for any $a' \leq a$ and $b' \geq b$. It suffices to show that $\Delta_X (I^L) \geq \Delta_X (I)$ and $\Delta_X (I^R) \geq \Delta_X (I)$. First, we will show that $\Delta_X (I^L) \geq \Delta_X (I)$. From Theorem [Theorem 6](#theorem:delta){reference-type="ref" reference="theorem:delta"}, $$\Delta_X (I^L) - \Delta_X (I) = \mathbb{E}_{a, \mu_L}[\mu_L - X] - \mathbb{E}_{a, \mu_I}[\mu_I - X],$$ where $\mu_L = \mathbb{E}[X \mid X \in I^L]$ and $\mu_I = \mathbb{E}[X \mid X \in I]$. Then, we get $$\begin{aligned} & \mathbb{E}_{a, \mu_L}[\mu_L - X] - \mathbb{E}_{a, \mu_I}[\mu_I - X]\\ &= \mathbb{E}_{a, \mu_I}[\mu_L - X] + \mathbb{E}_{\mu_I, \mu_L}[\mu_L - X] - \mathbb{E}_{a, \mu_I}[\mu_I - X]\\ &= \mathbb{E}_{a, \mu_I}[\mu_L - \mu_I] + \mathbb{E}_{\mu_I, \mu_L}[\mu_L - X] \geq 0. \end{aligned}$$ Thus, we have $\Delta_X (I^L) \geq \Delta_X (I)$. We can also show $\Delta_X (I^R) \geq \Delta_X (I)$ similarly. Therefore we have $\Delta_X (I') \geq \Delta_X(I)$. ◻ ## Reduction of General Loss Function to Min Function {#sec_reduction} We show that the general loss function can be expressed as a sum of $f_X$ and an affine function. **Lemma 21**. For any $A_1, A_2, B_1, B_2 \in \mathbb{R}$, $$\min(A_1 + B_1, A_2 + B_2) = A_2 + B_1 + \min(A_1 - A_2, B_2 - B_1)$$ *Proof.* For any $A, B \in \mathbb{R}$, the following equation holds. $$\min (A, B) = A - \min(A-B, 0).$$ By setting $A=A_1+A_2$ and $B = A_2 + B_2$, we have $$\begin{aligned} & \min(A_1 + B_1, A_2 +B_2) & \\ &= A_1 +B_1 - \min(A_1 + B_1 - (A_2 + B_2), 0) & (\because A=A_1+A_2, B = A_2 + B_2)\\ &= A_1 + B_1 - \min((A_1 - A_2) - (B_2 - B_1), 0) &\\ &= A_1 +B_1 - (A_1 - A_2 - \min(A_1- A_2, B_2 - B_1)& (\because A = A_1-A_2, B=B_2-B_1)\\ &= A_2 + B_1 + \min(A_1-A_2, B_2 - B_1).& \end{aligned}$$ ◻ **Theorem 22**. For a random variable $X$, define $\ell_X: \mathbb{R} \rightarrow \mathbb{R}$ as, $$\ell_X(s) = \mathbb{E}[ \min(a_1 s + b_1 X + c_1, a_2s+b_2X + c_2)]$$ where $a_i, b_i, c_i \in \mathbb{R} \ (i=0, 1, 2)$ such that $a_1 \neq a_2$ and $b_1 \neq b_2$. Then, $\ell_X$ can be written as $$\ell_X(s) = As' + B \mathbb{E}[Y] + C + \mathbb{E}[\min(s', Y)]$$ where,$Y=(b_2-b_1)X$ $s'= (a_1-a_2)s + c_1 - c_2$, $A=\frac{a_2}{a_1-a_2}$, $B=\frac{b_1}{b_2-b_1}$, and $C=\frac{a_1c_2- a_2c_1}{a_1-a_2}$. *Proof.* From the lemma above, we have $$\begin{aligned} & \min(a_1 s + b_1 X + c_1, a_2s+b_2X + c_2)\\ &= \min(\underbrace{a_1 s + c_1}_{A_1} +\underbrace{b_1X}_{B_1}, \underbrace{a_2s+c_2}_{A_2} + \underbrace{b_2X}_{B_2})\\ &= \min(A_1 + B_1, A_2 + B_2)\\ &= A_2 + B_1 + \min(A_1 - A_2, B_2 - B_1)\\ &= a_2s+c_2 + b_1X + \min((a_1-a_2)s + c_1 - c_2, (b_2-b_1)X)\\ &= As' + BY + C + \min(Y, s'). \end{aligned}$$ Thus, $$\ell_X(s) = As' + B\mathbb{E}[Y] + C + \mathbb{E}[\min(s', Y)].$$ ◻ [^1]: We have released the proposed algorithm as a Python package at <https://github.com/takazawa/piecewise-linearization-first-order-loss>, where the experimental code is also included.

arxiv_math

--- abstract: | We say that a hypergraph $\mathcal{H}$ contains a graph $H$ as a trace if there exists some set $S\subset V(\mathcal{H})$ such that $\mathcal{H}|_S=\{h\cap S: h\in E(\mathcal{H})\}$ contains a subhypergraph isomorphic to $H$. We study the largest number of hyperedges in uniform hypergraphs avoiding some graph $F$ as trace. In particular, we determine this number in the case $F=K_3$ and any uniformity, resolving a special case of a conjecture of Mubayi and Zhao, and we improve a bound given by Luo and Spiro in the case $F=C_4$ and uniformity 3. author: - "Dániel Gerbner[^1]" title: On forbidding graphs as traces of hypergraphs --- # Introduction A fundamental theorem in extremal Combinatorics is due to Turán [@T] and determines the largest number of edges in $n$-vertex $K_k$-free graphs (the case $k=3$ was proved earlier by Mantel [@man]). More generally, given a family ${\mathcal F}$ of graphs, $\ensuremath{\mathrm{ex}}(n,{\mathcal F})$ denotes the largest number of edges in $n$-vertex graphs that do not contain any member of ${\mathcal F}$ as a (not necessarily induced) subgraph, and if there is one forbidden subgraph, we use the simpler notation $\ensuremath{\mathrm{ex}}(n,F)$ instead of $\ensuremath{\mathrm{ex}}(n,\{F\})$. The Erdős-Stone-Simonovits theorem [@ES1966; @ES1946] determines the asymptotics of $\ensuremath{\mathrm{ex}}(n,{\mathcal F})$ in the case ${\mathcal F}$ does not contain any bipartite graphs. The bipartite case is much less understood and is the subject of extensive research, see [@fusi] for a survey. There is a natural analogue of this problem for hypergraphs and was already asked by Turán. Given a family ${\mathcal F}$ of hypergraphs, we denote be $\ensuremath{\mathrm{ex}}_r(n,{\mathcal F})$ the largest number of edges in an $r$-uniform hypergraph that does not contain any member of ${\mathcal F}$. This problem is much more complicated, for example we still do not know the asymptotics in the next obvious question, when the complete 4-vertex 3-uniform hypergraph is forbidden. A relatively recent line of research is to consider *graph-based hypergraps*. This is an informal common name of hypergraph classes that are obtained from graphs by enlarging their edges according to some set of rules. Extremal results concerning such hypergraphs were collected in Section 5.2. in [@gp]. The most studied graph-based hypergraps are the following. The *expansion* $F^{(r)+}$ of $F$ is obtained by adding $r-2$ new vertices to each edge such that each new vertex is added to only one edge, see [@mubver] for a survey on expansions. A *Berge copy* of $F$ is obtained by adding $r-2$ new vertices to each edge arbitrarily. The new vertices may be already in $F$ or not, and they may be added to any number of hyperedges. More precisely, we say that a hypergraph ${\mathcal F}$ is a Berge copy of $F$ if there is a bijection $f$ between the edges of $F$ and the hyperedges of ${\mathcal F}$ such that for each edge $e$ we have $e\subset f(e)$. Observe that there can be several non-isomorphic $r$-uniform Berge copies of $F$, the expansion being one of them. We denote by Berge-$F$ the family of Berge copies of $F$. Each Berge copy of $F$ is defined by a graph copy of $F$ on a subset of the vertices, we call that the *core* of the Berge-$F$. Berge hypergraphs were defined (generalizing the notion of hypergraph cycles due to Berge [@Be87]) by Gerbner and Palmer [@gp1]. Here we study a third type of graph based hypergraphs. We denote by $\ensuremath{\mathrm{Tr}}(F)$ the family of Berge copies of $F$ where the vertices added to the edges of $F$ are each outside $V(F)$. In other words, the *trace* of these Berge copies is $F$ on $V(F)$, i.e., $f(e)\cap V(F)=e$ for each edge $e$ of $F$. These were called *induced Berge* in [@fl]. The maximum number of hyperedges in hypergraphs with some forbidden traces have long been studied. For example, the celebrated Sauer Lemma [@sau; @she; @vc] deals with the case ${\mathcal H}$ does not contain the power set of a $t$-element set as a trace. The Turán problem for these graph-based hypergraphs is closely related to the so-called *generalized Turán problems*. Given two graphs $H$ and $G$, we denote by ${\mathcal N}(H,G)$ the number of copies of $H$ contained in $G$. Given an integer $n$ and graphs $H$ and $F$, we let $\ensuremath{\mathrm{ex}}(n,H,F)=\max\{{\mathcal N}(H,G): \text{ $G$ is an $n$-vertex $F$-free graph}\}$. After several sporadic results, the systematic study of this function was initiated by Alon and Shikhelman [@ALS2016]. It is easy to see that if we take the vertex sets of $r$-cliques in an $F$-free graph as hyperedges, the resulting graph is Berge-$F$-free. Therefore, we have $\ensuremath{\mathrm{ex}}(n,K_r,F)\le \ensuremath{\mathrm{ex}}_r(n,\text{Berge-}F)\le\ensuremath{\mathrm{ex}}_r(n,\ensuremath{\mathrm{Tr}}(F))\le\ensuremath{\mathrm{ex}}_r(n,F^{(r)+})$, where the second and third inequality follows from $F^{(r)+}\in \ensuremath{\mathrm{Tr}}(F)\subset \text{Berge-}F$. Stronger connection was established for these cases: $\ensuremath{\mathrm{ex}}_r(n,\text{Berge-}F)\le \ensuremath{\mathrm{ex}}(n,K_r,F)+\ensuremath{\mathrm{ex}}(n,F)$ [@gp2], $\ensuremath{\mathrm{ex}}_r(n,\ensuremath{\mathrm{Tr}}(F))=\Theta(\max_{s\le k}\ensuremath{\mathrm{ex}}(n,K_s,F))$ [@fl] and $\ensuremath{\mathrm{ex}}_r(n,F^{(r)+})=\ensuremath{\mathrm{ex}}(n,K_r,F)+O(n^{r-1})$ [@gerbner]. The first to study forbidden graphs as traces were Mubayi and Zhao [@muzh]. They studied the case $F=K_k$ (in fact they considered the more general case when complete hypergraphs are forbidden as traces). They observed that in the case $r<k$ we have $\ensuremath{\mathrm{ex}}_r(n,\ensuremath{\mathrm{Tr}}(K_k))=\ensuremath{\mathrm{ex}}_r(n,F^{(r)+})$ for sufficiently large $n$, and the exact value of that was determined by Pikhurko [@pikhu]. In the case $r\ge k$, Mubayi and Zhao conjectured that the extremal construction for sufficiently large $n$ is the following. We take a $K_k$-free graph on $n-r+k-1$ vertices with $\ensuremath{\mathrm{ex}}(n,K_{k-1},K_k)$ copies of $K_{k-1}$. Note that this is the so-called *Turán graph* by a theorem of Zykov [@zykov]. Then we take a set $U$ of $r-k+1$ new vertices, and pick as hyperedges the union of $U$ with the vertex set of any $(r-1)$-clique of the Turán graph. Mubayi and Zhao proved this conjecture asymptotically if $k=3$, i.e., they showed that $\ensuremath{\mathrm{ex}}_r(n,\ensuremath{\mathrm{Tr}}(K_3))=n^2/2+o(n^2)$. They proved this exactly if $r=3$. Our main result is the proof of the $k=3$ case of their conjecture for any $r$. **Theorem 1**. *For sufficiently large $n$ we have $ex_r(n,\ensuremath{\mathrm{Tr}}(K_3))=\lfloor(n-r+2)^2/4\rfloor$.* We can extend the asymptotic bound to book graphs in the 3-uniform case. The *book graph* $B_t$ consists of $t$ triangles sharing an edge, i.e., $B_t=K_{1,1,t}$. **Proposition 2**. *$\ensuremath{\mathrm{ex}}_3(n,\ensuremath{\mathrm{Tr}}(B_t))=(1+o(1))n^2/4$.* Other specific graphs $F$ such that $\ensuremath{\mathrm{ex}}_r(n,\ensuremath{\mathrm{Tr}}(F))$ has been studied include stars [@fl; @qg] and $K_{2,t}$ [@lusp; @qg]. In particular, Luo and Spiro [@lusp] showed $n^{3/2}/2+o(n^{3/2})\le \ensuremath{\mathrm{ex}}_3(n,\ensuremath{\mathrm{Tr}}(C_4))\le 5 n^{3/2}/6+o(n^{3/2})$. We improve the constant factor in the main term of the upper bound. **Theorem 3**. *$\ensuremath{\mathrm{ex}}_3(n,\ensuremath{\mathrm{Tr}}(C_4))\le 3 n^{3/2}/4+o(n^{3/2})$.* Finally, we show a connection of $\ensuremath{\mathrm{ex}}_r(\ensuremath{\mathrm{Tr}}(n,F))$ and generalized supersaturation. Given a graph $F$ and a positive integer $m$, the supersaturation problem deals with the minimum number of copies of $F$ in $n$-vertex graphs with at least $m$ edges. In the generalized version, we are also given a graph $H$ and the $n$-vertex graphs contain at least $m$ copies of $H$. Such problems were studied in [@cnr; @gnv; @hapa]. Note that this is equivalent to studying the most number of copies of $H$ when we are given an upper bound on the number of copies of $F$. **Proposition 4**. *There is an $n$-vertex graph $G$ with $O(n^{|V(F)|-1})$ copies of $F$ such that $\ensuremath{\mathrm{ex}}_r(n,\ensuremath{\mathrm{Tr}}(F))\le {\mathcal N}(K_r,G)$.* Obviously this connection to generalized Turán problems is less useful than the previously mentioned ones, partly because there are less results on generalized supersaturation problems. In the non-degenerate case when $\chi(F)>r$, a special case of a result of Halfpap and Palmer [@hapa] states that ${\mathcal N}(F,G)=o(n^{|V(F)|})$ implies ${\mathcal N}(K_r,G)\le (1+o(1))\ensuremath{\mathrm{ex}}(n,K_r,F)$. Therefore, with Proposition [Proposition 4](#propi){reference-type="ref" reference="propi"} we obtain the (already known) asymptotics of $\ensuremath{\mathrm{ex}}(n,\ensuremath{\mathrm{Tr}}(F))$ in this case. We could not find any application of Proposition [Proposition 4](#propi){reference-type="ref" reference="propi"} that gives new results, although it plays a small role in the proof of Proposition [Proposition 2](#book){reference-type="ref" reference="book"}. # Proofs The *shadow graph* of a hypergraph ${\mathcal H}$ has vertex set $V({\mathcal H})$ and $uv$ is an edge if and only if there is a hyperedge in ${\mathcal H}$ containing both $u$ and $v$. If ${\mathcal H}$ contains a $\ensuremath{\mathrm{Tr}}(F)$ (or any Berge-$F$), then the core of that is a copy of $F$ in the shadow graph. The converse is not true, for example if ${\mathcal H}$ is $r$-uniform, then the shadow graph contains copies of $K_r$, even if ${\mathcal H}$ is Berge-$K_r$-free. Note that in the shadow graph of a $\ensuremath{\mathrm{Tr}}(F)$-free hypergraph, each copy of $F$ has to contain an edge $uv$ with the property that each hyperedge containing $uv$ contains another vertex from that copy of $F$. In multiple proofs below, we will pick such an edge for each copy of $F$ and call it the *special edge* of that copy of $F$. We can show the following generalization of Proposition [Proposition 4](#propi){reference-type="ref" reference="propi"}. **Proposition 5**. *If ${\mathcal H}$ is $\ensuremath{\mathrm{Tr}}(F)$-free, then the shadow graph $G$ of ${\mathcal H}$ contains $O(n^{|V(F)|-1})$ copies of $F$.* Note that hyperedges of ${\mathcal H}$ create distinct copies of $K_r$ in $G$, thus $|E({\mathcal H})|\le {\mathcal N}(K_r,G)$, hence the above proposition implies Proposition [Proposition 4](#propi){reference-type="ref" reference="propi"}. *Proof.* Let us fix an edge $uv$ and count the number of copies of $F$ that have $uv$ as special edge. Let $h$ be a hyperedge containing $u$ and $v$. Then each copy of $F$ containing $uv$ contains also at least one of the other $r-2$ vertices of $h$. Therefore, we can count the copies of $F$ containing $uv$ by picking a non-empty subset of the other vertices of $h$ ($O(1)$ ways), then picking the rest of the vertices of $F$ ($O(n^{|V(F)|-3})$ ways) and then picking a copy of $F$ on those $|V(F)|$ vertices ($O(1)$ ways). As there are $O(n^2)$ special edges, the proof is complete. ◻ Let us continue with the proof of Theorem [Theorem 3](#4cyc){reference-type="ref" reference="4cyc"}. We will use the following lemma from [@lusp]. We say that a subset of a hyperedge is *light* if exactly one hyperedge contains it, and heavy otherwise. **Lemma 6** (Luo, Spiro [@lusp]). *Let ${\mathcal H}$ be a $\ensuremath{\mathrm{Tr}}(C_4)$-free 3-uniform hypergraph. Let ${\mathcal H}_2$ denote the subhypergraph of ${\mathcal H}$ consisting of the hyperedges that do not contain light edges. Then every edge of the shadow graph is in at most two hyperedges of ${\mathcal H}_2$.* We restate Theorem [Theorem 3](#4cyc){reference-type="ref" reference="4cyc"} here for convenience. **Theorem 1**. *$\ensuremath{\mathrm{ex}}_3(n,\ensuremath{\mathrm{Tr}}(C_4))\le 3 n^{3/2}/4+o(n^{3/2})$.* We denote hyperedges consisting of vertices $u,v,w$ by $uvw$. Note that there are 4-uniform hyperedges in the proof below, but we do not use the analogous notation, because we use that for 4-cycles. We let $uvwx$ denote the 4-cycle with edges $uv,vw,wx,xu$. *Proof.* Let ${\mathcal H}$ be a 3-uniform $Tr(C_4)$-free hypergraph. Let ${\mathcal H}_1$ be the subhypergraph consisting of the hyperedges that contain a light edge. Let us pick a light edge from each hyperedge of ${\mathcal H}_1$ and let $G_1$ denote the resulting graph. Then $G_1$ is $C_4$-free, thus $|E({\mathcal H}_1)|=|E(G_1)|\le \ensuremath{\mathrm{ex}}(n,C_4)=(1+o(1))n^{3/2}/2$. Let ${\mathcal H}_2$ denote the rest of the hyperedges of ${\mathcal H}$ and $G_2$ denote the shadow graph of ${\mathcal H}_2$. Then $G_2$ may contain copies of $C_4$, but each copy $uvwx$ of $C_4$ contains an edge $uv$ such that each hyperedge in ${\mathcal H}$ that contains $u$ and $v$ also contains $w$ or $x$. We say that $uv$ is a *special edge* for this $C_4$. In particular, $u$ and $v$ are contained together in at most two hyperedges of ${\mathcal H}$. This also implies that each $C_4$ in $G_2$ contains an edge $uv$ that is contained only in $C_4$'s in the same four vertices. Either we have a $K_4$ on those vertices, or $uv$ is contained in exactly one $C_4$. Let us consider a $K_4$ in $G_2$ with special edge $uv$ and other vertices $w,x$. Then $uvw$ and $uvx$ are in ${\mathcal H}_2$, otherwise one of their subedges would be light and not in $G_2$. We move the hyperedges $uvw$ and $uvx$ to ${\mathcal H}_1$ and the edge $uv$ to $G_1$. We repeat this as long as there is a $K_4$. We denote the hypergraph we obtain this way from ${\mathcal H}_1$ by ${\mathcal H}_1'$ and the graph we obtain from $G_1$ by $G_1'$. Note that a $K_4$ may contain multiple special edges, but we do this with only one special edge (since by removing that one edge, we already destroy the $K_4$). **Claim 7**. *$G_1'$ is $C_4$-free.* *Proof of Claim.* First we show that each hyperedge of ${\mathcal H}$ contains at most one edge of $G_1'$. This holds for $G_1$ since the edges are light and we picked only one from each hyperedge. The hyperedges of ${\mathcal H}_1$ do not contain any of the edges $uv$ in $E(G_1')\setminus E(G_1)$, since the only hyperedges of ${\mathcal H}$ that contain $uv$ are $uvw$ and $uvx$. Consider now the additional hyperedges added to ${\mathcal H}_1$. Recall that we add $uvw$ and $uvx$ at the same time. Both the hyperedges $uvw$ and $uvx$ contain only heavy edges, thus no edge of $G_1$. Finally, we need to show that no hyperedge contains two of the special edges added to $G_1$, say, $uv$ and $uy$. Assume otherwise, then without loss of generality, $uv$ is in $uvw$ and $uvx$, $y=x$ and $uy$ is added to $G_1'$ after $uv$. Then $uy$ is the special edge of a $K_4$ containing each vertex of $uvx$, hence $uv$ is in that $K_4$. But at that point $uv$ was already removed, a contradiction. We obtained that no two edges of $G_1'$ are contained in the same hyperedge of ${\mathcal H}$. But the special edge of a $C_4$ would be contained in such a hyperedge, a contradiction. ◻ Let us return to the proof of the theorem. **Claim 8**. *$|E(G_2)|\le (1+o(1))n^{3/2}/2$.* *Proof.* We will show that there are $O(n^{3/2})$ copies of $C_4$ in $G_2$. This implies the bound on the number of edges together with well-known supersaturation results [@ersi] (in fact the bound $o(n^2)$ would suffice). Recall that each hyperedge of ${\mathcal H}$ containing a special edge of a $C_4$ has that its remaining vertex is also in that $C_4$. As these special edges are heavy, it means that there are two hyperedges containing that edge, thus they together contain each vertex of the $C_4$. It implies that each edge is the special edge of at most two 4-cycles. As each $C_4$ contains a special edge, it is enough to show that there are $O(n^{3/2})$ special edges. A special edge from each $K_4$ were moved to $G_1'$, their number is at most $|E(G_1')|\le \ensuremath{\mathrm{ex}}(n,C_4)=(1+o(1))n^{3/2}/2$. Since each $K_4$ contains 6 edges, there are at most 6 times $(1+o(1))n^{3/2}/2$ special edges inside 4-cliques. The rest of the special edges are each contained in a hyperedge of ${\mathcal H}_1$. Since those hyperedges contain at most two edges of $G_2$, the number of these special edges is at most $2|E({\mathcal H}_1)|=2|E(G_1)|\le 2\ensuremath{\mathrm{ex}}(n,C_4)=(1+o(1))n^{3/2}$. ◻ Let $x=|E(G_1)'\setminus E(G_1)|$. The number of hyperedges in ${\mathcal H}_1'$ is at most the number of edges in $G_1$ plus twice the number of new edges in $G_1'$., i.e., $|E({\mathcal H}_1')|\le (1+o(1))n^{3/2}/2+x$. We also have $|E({\mathcal H}_1)|=|E(G_1)|\le (1+o(1))n^{3/2}/2-x$. Let ${\mathcal H}_2'$ denote the hyperedges of ${\mathcal H}$ that are not in ${\mathcal H}_1'$, thus ${\mathcal H}_2'$ is a subhypergraph of ${\mathcal H}_2$. Let $G_2'$ denote the graph of we obtain from $G_2$ by deleting the edges we moved to $G_1'$. By construction, $G_2'$ is $K_4$-free and we have that $|E(G_2')|\le |E(G_2)|-x\le (1+o(1))n^{3/2}/2-x$. Recall that by Lemma [Lemma 6](#lemls){reference-type="ref" reference="lemls"}, each edge of $G_2$ is in at most two hyperedges of ${\mathcal H}_2$. Let $y$ denote the number of edges in $G_2'$ that are in exactly two hyperedges of ${\mathcal H}_2$. Then those two hyperedges create a $C_4$ with a chord. That $C_4$ has a special edge $uv$, and $uv$ is contained in only one hyperedge of ${\mathcal H}_2$ (otherwise the shadow of those two hyperedges would create a $K_4$ and $uv$ would be moved to $G_1'$). As $uv$ is the special edge of only one 4-clique, we have that there are at least $y$ edges in $G_2'$ that are each in only one hyperedge of ${\mathcal H}_2'$. This implies that $y\le |E(G_2')|/2\le (1+o(1))n^{3/2}/4-x/2$. Then in $G_2$, we have at most $(1+o(1))n^{3/2}/4+x/2$ edges that are contained in exactly two hyperedges of ${\mathcal H}_2$. This shows that the total number of edge-hyperedge incidences between $G_2$ and ${\mathcal H}_2$ is at most $3(1+o(1))n^{3/2}/4+x/2$, hence $|E({\mathcal H}_2)|\le (1+o(1))n^{3/2}/4+x/6$. Combining the upper bounds we obtained on $|E({\mathcal H}_1)|$ and $|E({\mathcal H}_2)|$, we obtain that $|E({\mathcal H})|\le 3(1+o(1))n^{3/2}/4$. ◻ Let us continue with the proof of Theorem [Theorem 1](#tria){reference-type="ref" reference="tria"}, which we restate here for convenience. **Theorem 2**. *$ex_r(n,\ensuremath{\mathrm{Tr}}(K_3))=\lfloor(n-r+2)^2/4\rfloor$ for sufficiently large $n$.* We will use a result of Brouwer [@br], which states that if a triangle-free $n$-vertex graph is not bipartite, then it has at most $\lfloor n^2/4\rfloor-\lfloor n/2\rfloor+1$ edges, provided $n\ge 5$ (note that Brouwer dealt with $K_k$-free graphs in general). *Proof.* The lower bound is given by taking a complete bipartite graph on $n-r+2$ vertices, and adding the same $r-2$ new vertices to each edge. Let ${\mathcal H}$ be an $r$-uniform $\ensuremath{\mathrm{Tr}}(K_3)$-free hypergraph. First we show that there are at most two heavy $(r-1)$-subsets of any hyperedge $h$ of ${\mathcal H}$. If there are at least three such $(r-1)$-edges, they belong to three other hyperedges and each avoids a vertex of $h$, this gives three vertices and it is easy to see that the three hyperedges form a $\ensuremath{\mathrm{Tr}}(K_3)$ on those three vertices. **Claim 9**. *Each hyperedge $h$ of ${\mathcal H}$ contains a light 2-edge. Moreover, $h$ either contains at least two light 2-edges or contains two heavy $(r-1)$-sets.* *Proof.* Assume first that there are two heavy $(r-1)$-sets $A$ and $B$ in $h$, with $a\in h\setminus A$ and $b\in h \setminus B$. Let $h_A$ resp. $h_B$ be hyperedges distinct from $h$ that contain $A$ (resp. $B$). Assume that $\{a,b\}$ is heavy. Then it is contained in a hyperedge $h'$ distinct from $h$. Then $h'$ contains a vertex outside $h$, thus $h'$ contains at most $r-3$ vertices from $A\cap B$, hence avoids a vertex $c$. Then $h_A,h_B$ and $h'$ form a $\ensuremath{\mathrm{Tr}}(K_3)$ on $a,b,c$. Assume now that there is one heavy $(r-1)$-set $A$ in $h$, let $a\in h\setminus A$ and $h_A\neq h$ be a hyperedge of ${\mathcal H}$ containing $A$. Then a hyperedge $h_b$ contains $a$ and and a vertex $b\in h$, otherwise we found $r-1\ge 2$ light edges. We pick a hyperedge with the largest intersection with $A$. Clearly $h_b$ avoids at least two vertices $c$ and $d$ of $A$. Let $h_c\neq h$ be a hyperedge containing $b$ and $c$ (if exists), then it avoids a vertex $b'$ of $A$. Then we have a $\ensuremath{\mathrm{Tr}}(K_3)$ with vertices $a,b',c$ and hyperedges $h_A, h_b,h_c$, a contradiction showing that $h_c$ does not exist, i.e., $\{b,c\}$ is light. The same argument shows that $\{b,d\}$ is light. Assume now that $A$ is a largest heavy subset of $h$ with $|A|<r-1$, $h_A\neq h$ contains $A$ and $a,a'\in h\setminus A$. Let $h_a$ be a hyperedge that contains $a$ and a vertex $b\in A$. We pick among such hyperedges one with the largest intersection with $A$. Then $h_a$ avoids a vertex $c\in A$. Let $h_c$ be a hyperedge that contains $a$ and $c$, then it avoids a vertex $b'\in A\cap h_c$. Then there is a $\ensuremath{\mathrm{Tr}}(K_3)$ with vertices $a,b',c$ and hyperedges $h_A,h_a,h_c$, a contradiction showing that $\{a,c\}$ is light. The same argument shows that $\{a',c'\}$ is light for some $c'\in A$. ◻ Let us return to the proof of the theorem. We consider the following graph. For each hyperedge with two heavy $(r-1)$-subsets, we color the light subedge *green*, and color the other subedges *yellow*. For the other hyperedges, we color two light subedges *brown*. First observe that the graph $G_1$ consisting of the green and brown edges is triangle-free. Indeed, if a triangle contains edges from three different hyperedges, those form a $Tr(K_3)$. Otherwise there are two brown edges $uv$ and $uw$ from the same hyperedge. But then $vw$ is from the same hyperedge as well, and since each edge of $G_1$ is light, we picked three edges from that hyperedge, a contradiction. Note that this implies the asymptotically sharp upper bound $\lfloor n^2/4\rfloor$. More precisely, let $b$ denote the number of brown edges and $g$ denote the number of green edges. Then $|E({\mathcal H})|\le \frac{b}{2}+g\le b+g=|E(G_1)|\le \lfloor n^2/4\rfloor$ by Mantel's theorem. This also shows that we are done if there are at least $2(\lfloor n^2/4\rfloor-\lfloor (n-r+2)^2/4\rfloor)$ brown edges in $G_1$. Hence we can assume that there are at most $nr$ brown edges. We are done if the number of green edges is at most $\lfloor (n-r+2)^2/4\rfloor-nr\ge n(n-6r)/4$. We claim that there is no rainbow triangle, i.e., a triangle with a green, a brown and a yellow edge, and similarly there is no triangle with two green edges and a yellow edge. Indeed, if $uvw$ is a triangle and $uv$ is yellow, then there is no hyperedge of ${\mathcal H}$ containing $u,v,w$, as then we would pick two green edges or a green and a brown edge from that hyperedge, a contradiction. Therefore, $uw$ is in a single hyperedge $h_1$, $vw$ is in another single hyperedge $h_2$ and $uv$ is contained in a hyperedge $h_3$, which is different from $h_1$ and $h_2$ since $h_3$ does not contain $w$. Then $h_1,h_2,h_3$ form a $\ensuremath{\mathrm{Tr}}(K_3)$, a contradiction. Note that there may be a triangle with two brown edges and a yellow edge, and by definition there are triangles with a green and edge and two yellow edges. Let $G$ denote the graph of the green edges. For every vertex $v$, we simply denote its degree in $G$ by $d(v)$, and if we talk about degrees without specifying the host graph in the rest of the proof, we mean the degree in $G$. If $d(u)+d(v)\le n-6r$ for every edge $uv$, we can follow the argument of Mubayi and Zhao in the case $r=3$, which in turn follows Mantel's proof of his theorem. We have $$\frac{4|E(G)^2|}{n}=\frac{\left(\sum_{v\in V(G)}d(v)\right)^2}{n}\le \sum_{v\in V(G)}d(v)^2=\sum_{uv\in E(G)}(d(u)+d(v))\le(n-6r)|E(G)|,$$ thus $|E(G)|\le\frac{n(n-6r)}{4}$, completing the proof. Assume now that for the green edge $uv$ we have $d(u)+d(v)\ge n-6r$. There is a set $R$ of $r-2$ vertices such that each vertex $w\in R$ is connected by yellow edges to $u$ and $v$. This implies that none of the neighbors of $u$ or $v$ is adjacent to $w$ in $G$ (nor in $G_1$). Also the elements of $R$ are not adjacent to $w$ in $G$ or in $G_1$. Therefore, $d(w)\le 5r$, moreover, $d_{G_1}(w)\le 5r$. Here we use that $u$ and $v$ do not have common neighbors because of the triangle-free property. Now observe that the vertices in $R$ are incident to at most $5r(r-2)$ brown or green edges in total. Let $U=V({\mathcal H})\setminus R$, then we are done if $G_1[U]$ has at most $\lfloor (n-r+2)^2/4\rfloor-5r(r-2)$ edges. Assume that $G_1[U]$ has more than that many edges. As $G_1[U]$ is triangle-free, we have that $G_1[U]$ is bipartite, using the result of Brouwer. Moreover, let $U_1$ and $U_2$ be the two parts, then clearly $|U_1|,|U_2|\ge n/3$. There are at most $5r(r-2)$ edges missing between the two parts $U_1$ and $U_2$ of $G_1[U]$. This implies that every vertex in $U_1$ has degree at least $|U_2|-5r(r-2)$ and every vertex in $U_2$ has degree at least $|U_1|-5r(r-2)$. Assume that there is a green edge $uv$ in $G$ such that the set of $r-2$ vertices in the hyperedge containing $uv$ is not exactly $R$. Then we have at least $r-1$ vertices of degree at most $5r$. Therefore, the total number of edges in $G_1$ is at most $\ensuremath{\mathrm{ex}}(n-r+1,K_3)+5r(r-1)<\lfloor (n-r+2)^2/4\rfloor$ and we are done. We obtained that for the green edges, the same $r-2$ vertices extend them to hyperedges. In particular, there are no brown or green edges inside $R$. Assume that there is a brown edge $vw$ in $G_1$ with $v\in U_1$ and $w\in R$. Then $v$ is not incident to any green edges, as a green edge $uw$ would result in a rainbow triangle with vertices $u,v,w$. Let $b'$ denote the number of brown edges inside $U_1$ and $b''$ denote the number of brown edges incident to $R$. Since $v$ is incident to at least $n/3-5r(r-2)$ edges inside $U_1$, we have $b'\ge n/3-5r(r-2)$ and $b''\le 5r(r-2)$, thus $b/2\le b'$. We have $|E({\mathcal H})|\le \frac{b}{2}+g\le b'+g\le |E(G_1[U])|\le \lfloor (n-r+2)^2/4\rfloor$ and we are done. Finally, if there are no brown edges incident to $R$, then we have $|E(G_1)|= |E(G_1[U])|\le \lfloor (n-r+2)^2/4\rfloor$ and we are done. ◻ We finish the paper by proving Proposition [Proposition 2](#book){reference-type="ref" reference="book"}. Recall that it states $\ensuremath{\mathrm{ex}}_3(n,\ensuremath{\mathrm{Tr}}(B_t))=(1+o(1))n^2/4$. *Proof.* The lower bound is given by taking a complete bipartite graph on $n-1$ vertices, and adding the same new vertex to each edge. Let ${\mathcal H}$ be a $\ensuremath{\mathrm{Tr}}(B_t)$-free hypergraph and let ${\mathcal H}_1$ be the subhypergraph consisting of the hyperedges having a light subedge. Let $G_1$ be a graph obtained by taking a light subedge for each hyperedge in ${\mathcal H}_1$. It is easy to see that $G_1$ is $B_t$-free, hence $|E({\mathcal H}_1)|=|E(G_1)|\le \ensuremath{\mathrm{ex}}(n,B_t)=(1+o(1))n^2/4$ by the Erdős-Simonovits theorem (in fact $\ensuremath{\mathrm{ex}}(n,B_t)=\lfloor n^2/4\rfloor$ for sufficiently large $n$ by a theorem of Simonovits [@S]). Let ${\mathcal H}_2$ denote the rest of the hyperedges of ${\mathcal H}$ and $G_2$ denote the shadow graph of ${\mathcal H}$. Then $G_2$ contains $O(n^{t+1})$ copies of $B_2$ by Proposition [Proposition 5](#propi2){reference-type="ref" reference="propi2"}. Hence by the removal lemma [@efr] there is a set $A$ of $o(n^2)$ edges in $G_2$ such that each copy of $B_2$ contains an edge from $A$. We claim that each edge of $G_2$ is in at most $3t-3$ hyperedges of ${\mathcal H}_2$. We use a lemma of Luo and Spiro [@lusp], who showed the analogous statement in the case of forbidden $\ensuremath{\mathrm{Tr}}(K_{2,t})$. More precisely, they showed that if an edge $xy$ is in at least $3t-2$ hyperedges of ${\mathcal H}_2$, then there is a $\ensuremath{\mathrm{Tr}}(K_{2,t})$ in ${\mathcal H}$ such that $x$ and $y$ are the vertices in the smaller part of the core $K_{2,t}$. To find a $\ensuremath{\mathrm{Tr}}(K_{2,t})$ in ${\mathcal H}$, we need to find a hyperedge containing $x,y$ such that the third vertex of that hyperedge is not in this core $K_{2,t}$. This is doable if $t>1$, since we can pick any of the $2t-2$ hyperedges containing $x,y$ and avoiding the other $t$ vertices of the core $K_{2,t}$. Now we are ready to count the number of triangles in $G_2$. There are at most $(3t-3)|A|=o(n^2)$ triangles containing an edge in $A$. The rest of $G_2$ is $B_t$-free, thus contains $o(n^2)$ triangles by a result of Alon and Shikhelman [@ALS2016]. As each hyperedge of ${\mathcal H}_2$ creates a triangle in $G_2$, we have $|E({\mathcal H}_2)|=o(n^2)$, completing the proof. ◻ **Funding**: Research supported by the National Research, Development and Innovation Office - NKFIH under the grants FK 132060 and KKP-133819. 99 N. Alon, C. Shikhelman. Many $T$ copies in $H$-free graphs. *Journal of Combinatorial Theory, Series B*, **121**, 146--172, 2016. C. Berge, Hypergraphes: combinatoire des ensembles finis, *Gauthier-Villars*, 1987. A. Brouwer. Some lotto numbers from an extension of Turán's theorem. *Afdeling Zuivere Wiskunde \[Department of Pure Mathematics\]*, **152**, 1981. J. Cutler, J.D. Nir, A.J. Radcliffe, (2022). Supersaturation for subgraph counts. Graphs and Combinatorics, 38(3), 65. P. Erdős, P. Frankl, V. Rödl, The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent Graphs Combin., 2 (1986), 113--121. P. Erdős, M. Simonovits. A limit theorem in graph theory. *Studia Sci. Math. Hungar.* **1**, (1966), 51--57. P. Erdős, M. Simonovits. Supersaturated graphs and hypergraphs ,Combinatorica 3 (1983), 181--192. P. Erdős, A. H. Stone. On the structure of linear graphs. *Bulletin of the American Mathematical Society* **52**, (1946), 1087--1091. Z. Füredi, R. Luo. Induced Turán problems and traces of hypergraphs. *European Journal of Combinatorics*, 103692, 2023. Z.Füredi, M. Simonovits, (2013). The history of degenerate (bipartite) extremal graph problems. In Erdős centennial (pp. 169-264). Berlin, Heidelberg: Springer Berlin Heidelberg. D. Gerbner, On non-degenerate Turán problems for expansions, arXiv preprint arXiv:2309.01857 D. Gerbner, Z.L. Nagy, M. Vizer, (2022). Unified approach to the generalized Turán problem and supersaturation. Discrete Mathematics, 345(3), 112743. D. Gerbner, C. Palmer, Extremal Results for Berge hypergraphs. *SIAM Journal on Discrete Mathematics*, **31**, 2314--2327, 2017. D. Gerbner, C. Palmer. Counting copies of a fixed subgraph in $F$-free graphs. *European Journal of Combinatorics* **82**, 103001, 2019. D. Gerbner, B. Patkós, Extremal Finite Set Theory, 1st Edition, CRC Press, 2018. A. Halfpap, C. Palmer, (2021). On supersaturation and stability for generalized Turán problems. Journal of Graph Theory, 97(2), 232-240. R. Luo, S. Spiro, (2021). Forbidding $K_{2,t}$ Traces in Triple Systems. The Electronic Journal of Combinatorics, P2.4. W. Mantel, Problem28, Wiskundige Opgaven 10 (1907), 60--61. D. Mubayi, J. Verstraëte. A survey of Turán problems for expansions. *Recent Trends in Combinatorics*, 117--143, 2016. D. Mubayi, Y. Zhao, (2007). Forbidding complete hypergraphs as traces. Graphs and Combinatorics, 23(6), 667-679. O. Pikhurko. Exact computation of the hypergraph Turán function for expanded complete 2-graphs, *Journal of Combinatorial Theory, Series B*, **103**(2) 220--225, 2013. B. Qian, G. Ge, (2022). A note on the Turán number for the traces of hypergraphs. arXiv preprint arXiv:2206.05884. N.Sauer, On the density of families of sets, J. Combinatorial Theory Ser.A 13(1972), 145--147. S. Shelah, A combinatorial problem; stability and order for models and theories in infinitary languages, Pacifc J.Math. 41 (1972), 247--261. M. Simonovits, A method for solving extremal problems in graph theory, stability problems. In Theory of Graphs (Proc. Colloq., Tihany, 1966), pp. 279--319. 1968. P. Turán. Egy gráfelméleti szélsőértékfeladatról. *Mat. Fiz. Lapok*, **48**, 436--452, 1941. V.N. Vapnik, A.Ya. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Theory Probab. Appl. 16 (1971), 264--280. A. A. Zykov. On some properties of linear complexes. *Matematicheskii Sbornik*,**66**(2), 163--188, 1949. [^1]: Alfréd Rényi Institute of Mathematics, HUN-REN, E-mail: `gerbner@renyi.hu.`

arxiv_math

--- abstract: | Due to the ongoing electrification of transport in combination with limited power grid capacities, efficient ways to schedule electric vehicles (EVs) are needed for intraday operation of, for example, large parking lots. Common approaches like model predictive control repeatedly solve a corresponding offline problem. In this work, we present and analyze the Flow-based Offline Charging Scheduler ([Focs]{.smallcaps}), an offline algorithm to derive an optimal EV charging schedule for a fleet of EVs that minimizes an increasing, convex and differentiable function of the corresponding aggregated power profile. To this end, we relate EV charging to mathematical speed scaling models with job-specific speed limits. We prove our algorithm to be optimal. Furthermore, we derive necessary and sufficient conditions for any EV charging profile to be optimal. author: - | \ bibliography: - literature_references_and_summaries.bib title: | Relating Electric Vehicle Charging to Speed Scaling with Job-Specific Speed Limits\ [^1] --- algorithm, electric vehicle, scheduling, speed scaling # Introduction Due to the on-going electrification of transport in combination with limited power grid capacities, efficient ways to schedule the charging of electric vehicles (EVs) are needed for intraday operation of, for example, large parking lots. Many approaches, based on e.g., model predictive control [@2021VanKriekingePeakShavingCostMinMPCUniandBiDirectionalEVs] or valley-filling algorithms [@2018SchootUiterkampFillLevelPredictioninOnlineValleyFillingAlgsforEVcharging] repeatedly solve a corresponding offline problem, or solve it once at the beginning to derive a target function for intraday operation. To account for the limited grid capacity, one natural objective in EV scheduling problems is to minimize the $\ell_2$-norm of the aggregated power profile of for example a parking lot hosting multiple EVs. EV scheduling problems with this objective naturally reduce to (processor) speed scaling problems with job-specific speed limits. In speed scaling, tasks are scheduled on a processor within their respective availability such that a (typically increasing, convex and differentiable) function of the processing speed is minimized. One such function may be the $\ell_2$-norm. Speed scaling problems without speed limits are well-studied, with the YDS algorithm being one of the core approaches [@1995YaoSchedulingModelforReducedCPUEnergyYDSalg]. Already before YDS, Vizing, Komzakova and Tarchenko studied the same problem and came up with a similar solution as early as 1981 [@1981VizingSovjetYDS]. An extension of YDS considering global speed limits per time interval is given in [@2017AntoniadisContSpeedScalingWithVariability]. Another variant considers job-specific speed functions and uses a maximum flow formulation to find an optimal solution for both a single-processor and multi-processor setup [@2017ShiouraMachineSpeedScalingbyAdaptingMethodsforConvexOptimizationwithSubmodularConstraints]. However, to the best of our knowledge, the use case with job-specific speed limits, as is relevant to EV scheduling with EV-specific maximum charging powers, has not yet been studied. In this work, we present and analyze a novel offline algorithm to derive an optimal charging schedule for a fleet of EVs, minimizing an increasing, convex and differentiable function of the aggregated power profile. Furthermore, we derive necessary and sufficient conditions for solutions to the problem to be optimal. The remainder of the paper is organized as follows. Section [2](#sec:model){reference-type="ref" reference="sec:model"} describes the extended speed scaling model. After that, in Section [3](#sec:kkt){reference-type="ref" reference="sec:kkt"} we derive sufficient and necessary optimality conditions of a feasible solution of the speed scaling model for EV scheduling. Section [4](#sec:offlineAlg){reference-type="ref" reference="sec:offlineAlg"} presents the main result of this paper: the *Flow-based Offline Charging Scheduler* ([Focs]{.smallcaps}) which is an offline algorithm that, using maximum flows, computes an optimal solution for the given problem. We analyze that algorithm in Section [5](#sec:algAnalysis){reference-type="ref" reference="sec:algAnalysis"} and provide a mathematical proof of optimality. Finally, Section [6](#sec:conclusion){reference-type="ref" reference="sec:conclusion"} presents the conclusion of the paper. # Speed scaling model {#sec:model} In this section, we describe the considered EV scheduling problem. To this end, we extend the mathematical program discussed in [@2007BansalSpeedScalingManageEnergyAndTemperature]. Note that we slightly change phrasing and notation to fit the application to EVs. We model EVs as jobs $1, ..., n$. Each job $j$ corresponds to a pending charging session (or EV) with associated energy requirement $e_{j}$, arrival time $a_j$, and departure time $d_j$. In addition, we consider job-specific power limits $p_j^{\max}$ which form an extension to the traditional speed scaling model. We denote the set of all jobs as $[n]$. Next, we discretize the time horizon $[ \min_j a_j, \max_j d_j]$ into atomic intervals by using all arrival and departure times as breakpoints. The resulting (ordered) sequence of breakpoints we denote as $t_1, \dots, t_{m+1}$, corresponding to $m \leq 2n -1$ atomic intervals $I_i = [t_i, t_{i+1}]$ for $i = 1, ..., m$. Hereby, the length of interval $I_i$ is denoted as $|I_{i}|$. Furthermore, we introduce operator $L$ for the combined length of a set of atomic intervals, i.e., if $I$ is a set of atomic intervals, then $L(I) = \sum_{I_i\in I}|I_i|$. We denote the set of all atomic intervals by $\mathcal{I} = \{I_i| i=1, \dots, m\}$. To relate jobs to atomic intervals, denote by $J(i)$ the jobs that are available in interval $I_i$, i.e., $$J(i) =\{ j | (a_j \leq t_i) \land (t_{i+1} \leq d_j)\}.$$ Similarly, $J^{-1}(j)$ is defined as the set of indices $i$ of the intervals $I_i$ where $j$ is available. Furthermore, we consider a problem instance feasible if $$\begin{aligned} \nonumber e_j \leq p_j^{\max}\sum_{i\in J^{-1}(j)} |I_i| \hspace{10pt} \forall j \in [n].\end{aligned}$$ Finally, as atomic intervals are in general not unit-sized, we introduce maximum energy limits $e_{i,j}^{\max} = p_j^{\max} |I_{i}|$ per job $j$ and interval $I_i$. As decision variables, let $e_{i,j}$ be the energy that EV $j$ charges in $I_i$. A corresponding schedule is considered feasible if jobs are scheduled only between their arrival and departure time, with at most their maximum power, and charge their entire energy requirement $e_j$. Furthermore, preemption is allowed, i.e., a job can be suspended to be continued at a later interval. In this work, we do not consider V2G applications, therefore requiring that $e_{i,j}\geq 0$. Summarizing, the EV model is constrained by [\[eq:MIP\]]{#eq:MIP label="eq:MIP"} $$\begin{aligned} \sum_{i\in J^{-1}(j)} e_{i,j} &\geq e_{j} &\forall j\in[n] \label{eq:MIPnoENS}\\ e_{i,j} &\geq 0& \forall j \in [n], i\in J^{-1}(j) \hspace{3pt}\label{eq:MIPnonnegLoad}\\ e_{i,j} &\leq e_{i,j}^{\max} & \forall j \in [n], i\in J^{-1}(j). \label{eq:MIPspeedlimit}\end{aligned}$$ Note that those constraints are the same as those used by Bansal, Kimbrel and Pruhs [@2007BansalSpeedScalingManageEnergyAndTemperature] (up to notation), extended by Inequality ([\[eq:MIPspeedlimit\]](#eq:MIPspeedlimit){reference-type="ref" reference="eq:MIPspeedlimit"}) which models the job-specific speed limits. From a grid perspective, the aggregated power level resulting from an EV schedule is of interest. For a given schedule, the average aggregated power level $p_i$ in atomic interval $I_i$ is given by $$\begin{aligned} p_i = \frac{\sum_{j\in J(i)} e_{i,j}}{|I_i|}. \nonumber\end{aligned}$$ The two most frequently considered objectives for the overall grid usage are the $\ell_{\infty}$ and $\ell_{2}$-norms of the aggregated power. Hereby, the $\ell_{\infty}$-norm is of interest for cases with bounded grid capacity. In this work we focus on the overall impact of the charging on the grid, which is often modelled by the $\ell_2$-norm. Minimizing this norm results in what we refer to as the flattest possible aggregated power profile. This objective is similar to the objective function $\sum_{i=0}^m \left( p_i \right)^{\alpha}$ referred to in speed-scaling literature, for suitable $\alpha > 0$. In this work, we consider a generalization of this to minimizing any objective function $$\begin{aligned} F\left( p_1, \dots, p_m\right), \label{eq:MIPobjective} \nonumber\end{aligned}$$ that is 1) convex, 2) differentiable and 3) has the property that increasing any value $p_i$ to $p_{i'} > p_i$ will increase the value of the objective function. For convenience, we call a function $F$ that satisfies the last property *increasing*. # Optimality conditions {#sec:kkt} In the following section, we derive KKT conditions for the convex program presented in Section [2](#sec:model){reference-type="ref" reference="sec:model"} and interpret the resulting necessary and sufficient conditions for corresponding optimal solutions. To this end, we build upon earlier results by Bansal, Kimbrel and Pruhs [@2007BansalSpeedScalingManageEnergyAndTemperature]. The KKT conditions for a convex program $$\begin{aligned} \min \text{ } & \phi(x) &\nonumber\\ \text{s.t. } & \displaystyle \psi_{k}(x) \leq 0 & k = 1,\dots ,N \nonumber\end{aligned}$$ with differentiable functions $\psi_k$ are expressed using the KKT multipliers $\lambda_k$ associated with $\psi_k$. These necessary and sufficient conditions for optimality of solutions $x$ and $\lambda$ [@2004BoydVandenbergheConvexOptimization] are $$\begin{aligned} \psi_k(x) &\leq 0 & k = 1,\dots , N \\ \lambda_k &\geq 0 & k = 1,\dots , N \label{eq:KKTlambda}\\ \lambda_k \psi_k(x) &= 0 & k = 1,\dots , N \label{eq:KKTcomplementarySlackness}\\ \nabla \phi(x) + \sum_{k=1}^N\lambda_k \nabla \psi(x) &= 0. \label{eq:KKTgradientCondition}\end{aligned}$$ Applying this to our formulation of speed scaling with individual speed constraints ([\[eq:MIP\]](#eq:MIP){reference-type="ref" reference="eq:MIP"}), we introduce dual variables, denoted by $\delta_j$ for ([\[eq:MIPnoENS\]](#eq:MIPnoENS){reference-type="ref" reference="eq:MIPnoENS"}), $\gamma_{i,j}$ for ([\[eq:MIPnonnegLoad\]](#eq:MIPnonnegLoad){reference-type="ref" reference="eq:MIPnonnegLoad"}), and $\zeta_{i,j}$ for ([\[eq:MIPspeedlimit\]](#eq:MIPspeedlimit){reference-type="ref" reference="eq:MIPspeedlimit"}). Applying KKT condition ([\[eq:KKTgradientCondition\]](#eq:KKTgradientCondition){reference-type="ref" reference="eq:KKTgradientCondition"}) to the problem, we get $$\begin{aligned} 0 &= \nabla F\left(\sum_{j=1}^n p_{1,j}, \dots, \sum_{j=1}^n p_{m,j}\right) \\ &+ \sum_{j=1}^n \delta_j \nabla \left(e_j - \sum_{i \in J^{-1}(j)} e_{i,j}\right) \\ &- \sum_{i=1}^m \sum_{j \in J(i)} \gamma_{i,j} \nabla e_{i,j} \\ &- \sum_{i=1}^m \sum_{j \in J(i)} \zeta_{i,j} \nabla \left(e_{i,j}^{\max} - e_{i,j}\right).\end{aligned}$$ Note that the component of this gradient that corresponds to the partial derivative with respect to $e_{i,j}$ is $$\begin{aligned} \label{eq:nablacomponent} 0 = \frac{\partial F }{\partial e_{i,j}} -\delta_j -\gamma_{i,j} + \zeta_{i,j}.\end{aligned}$$ We analyze the condition in the components corresponding to partial derivatives with respect to $e_{i,j}$, where job $j\in J(i)$. We consider three cases in our analysis. First, consider $0 < e_{i,j} < e_{i,j}^{max}$. In this case job $j$ charges in interval $i$, but not at full power. Complementary slackness (compare ([\[eq:KKTcomplementarySlackness\]](#eq:KKTcomplementarySlackness){reference-type="ref" reference="eq:KKTcomplementarySlackness"})), now implies that $e_{i,j}\gamma_{i,j} = 0$ and $(e_{i,j} - e_{i,j}^{\max}) \zeta_{i,j} = 0$. In the considered case, this implies that $\gamma_{i,j} = \zeta_{i,j} = 0$. Therefore, ([\[eq:nablacomponent\]](#eq:nablacomponent){reference-type="ref" reference="eq:nablacomponent"}) simplifies to $$\begin{aligned} & 0 = -\delta_j + \frac{\partial F}{\partial e_{i,j}} \nonumber\\ \iff & \delta_j = \frac{\partial F}{\partial e_{i,j}}. \label{eq:KKTanalysisDelta}\end{aligned}$$ This derivation shows that we can interpret the dual variable $\delta_j$ as the derivative of the intensity function $F$ with respect to $e_{i,j}$. Since $\delta_j$ does not depend on $i$ and $F$ is convex and increasing, the aggregated power needs to be the same for any atomic interval $i'$ where job $j$ charges at a rate strictly between 0 and its power limit. Next, consider the case where $0 = e_{i,j} < e_{i,j}^{\max}$. Complementary slackness gives $\zeta_{i,j} = 0$, leaving us with $$\begin{aligned} & 0 = -\delta_j + \frac{\partial F}{\partial e_{i,j}} - \gamma_{i,j} \nonumber\\ \iff & \gamma_{i,j} = -\delta_j + \frac{\partial F}{\partial e_{i,j}}. \label{eq:KKTanalysisGamma}\end{aligned}$$ Using non-negativity of $\gamma_{i,j}$ (see ([\[eq:KKTlambda\]](#eq:KKTlambda){reference-type="ref" reference="eq:KKTlambda"})), it follows that $\frac{\partial F}{\partial e_{i,j}} \geq \delta_j$. As shown above, $\delta_j$ is independent of $i$ and characterizes $\frac{\partial F}{\partial e_{i',j}}$ for intervals $i'$ where $0<e_{i',j}<e_{i,j}^{\max}$. Given that $F$ is convex and increasing, and that $\frac{\partial F}{\partial e_{i,j}} \geq \frac{\partial F}{\partial e_{i',j}}$ for such $i$ and $i'$, we conclude that the power during interval $I_i$ where by assumption job $j$ does not charge, is at least as high as during intervals where job $j$ does charge at a (positive) power below its maximum. Lastly, consider the case where $0 < e_{i,j} = e_{i,j}^{\max}$. Complementary slackness gives us $\gamma_{i,j} = 0$, leaving us with $$\begin{aligned} & 0 = -\delta_j + \frac{\partial F}{\partial e_{i,j}} + \zeta_{i,j} \nonumber\\ \iff & \zeta_{i,j} = \delta_j - \frac{\partial F}{\partial e_{i,j}}. \label{eq:KKTanalysisZeta}\end{aligned}$$ We apply similar reasoning as in the previous case while carefully considering that the signs in the right hand sides of ([\[eq:KKTanalysisGamma\]](#eq:KKTanalysisGamma){reference-type="ref" reference="eq:KKTanalysisGamma"}) and ([\[eq:KKTanalysisZeta\]](#eq:KKTanalysisZeta){reference-type="ref" reference="eq:KKTanalysisZeta"}) are reversed. Based on this, we conclude that the power in any interval $I_i$ where job $j$ is executed at maximum speed, is at most as high as during intervals where $j$ is available and is charged at a (positive) power below its maximum, or is available and not charged at all. From the above analysis, the following necessary and sufficient conditions for a schedule to be optimal follow: 1. The aggregated power in all intervals where $j$ charges but does not reach its power-limit is the same. 2. The aggregated power in intervals where $j$ could, but does not charge is at least as high as where $j$ is actually charging. 3. The aggregated power in intervals where $j$ charges at maximum power is smaller or equal than in intervals where $j$ is charging and does not reach its limit. The first two conditions are similar to those derived by Bansal, Kimbrel and Pruhs, whereas the last is associated with the addition of job-specific speed limits. In Section [5.2](#sec:optimality){reference-type="ref" reference="sec:optimality"} we show that the output of [Focs]{.smallcaps}, the algorithm introduced in Section [4.2](#sec:alg){reference-type="ref" reference="sec:alg"}, is a feasible schedule that satisfies said conditions. Assuming we determine such a schedule, we can solve the system ([\[eq:KKTanalysisDelta\]](#eq:KKTanalysisDelta){reference-type="ref" reference="eq:KKTanalysisDelta"}), ([\[eq:KKTanalysisGamma\]](#eq:KKTanalysisGamma){reference-type="ref" reference="eq:KKTanalysisGamma"}) and ([\[eq:KKTanalysisZeta\]](#eq:KKTanalysisZeta){reference-type="ref" reference="eq:KKTanalysisZeta"}), proving optimality of the derived primal solution. # Offline algorithm using flows {#sec:offlineAlg} In this section, we present an iterative offline algorithm to determine an optimal EV charging schedule minimizing an increasing, convex and differentiable function of the aggregated power profile, given the individual arrival and departure times, energy requirements and maximum charging power of the EVs. The algorithm, similarly to YDS, uses the notion of critical intervals. These intervals are exactly those intervals that in an optimal solution require the highest aggregated power. Formally, these intervals are defined as follows. **Definition 1** (Critical intervals). *An atomic interval $I_i$ is *critical* if for any optimal schedule its aggregated power level $p_i \geq p_{i'}$ for any $i'\in [m]$ with $i'\neq i$.* Note that there may be multiple critical (atomic) intervals. Furthermore, one major difference with critical intervals as defined for YDS is that jobs do not have to fall entirely within a (set of) critical interval(s) to be scheduled to charge there. This difference with YDS follows from the job-specific power limits. The flat profile that YDS assigns to what they call a critical interval is not necessarily feasible in the setting with power limits, see e.g., Fig. [2](#fig:YDSinfeasible){reference-type="ref" reference="fig:YDSinfeasible"}. $j$ $a_j$ $d_j$ $e_j$ $p_j^{\max }$ ----- ------- ------- ------- --------------- 1 0 2 2 1 2 1 2 2 2 {#fig:YDSinfeasible} {#fig:YDSinfeasible} Compared to YDS, determining critical intervals and their power level is more involved. In the algorithm presented in Section [4.2](#sec:alg){reference-type="ref" reference="sec:alg"}, determining critical intervals requires the computation of multiple maximum flows. To be able to construct the flows and to keep track of the developments over the iterations of the proposed algorithm, we follow the speed-scaling notation introduced in Section [2](#sec:model){reference-type="ref" reference="sec:model"} and introduce some additional notation. ## Flow formulation {#sec:flow} We initialize a network $G = (V,D)$ as follows (see also Fig. [3](#fig:flowSchematic){reference-type="ref" reference="fig:flowSchematic"}). The vertex set V consists of source and sink vertices $s$ and $t$, as well as two sets of vertices representing job vertices and atomic interval vertices respectively, i.e., $V = \{s,t\}\cup[n]\cup\{I_i|i\in [m]\}$. Furthermore, the edge set $D$ consists of the union of the following three sets: $$\begin{aligned} D_s &= \{(s,j) | j\in [n]\} \\ D_0 &= \{(j,I_i) | j\in [n], i\in J^{-1}(j)\} \\ D_t &= \{(I_i,t) | i\in [m]\}\end{aligned}$$ with respective edge capacities $$\begin{aligned} c_{u,v} = \begin{cases} e_v & \text{if } u = s, \ v \in [n] \\ e_{i,u}^{max} & \text{if } u \in [n], \ v = I_i,\ i\in J^{-1}(u)\\ g_{r,k}(u) & \text{if } u \in [m], \ v = t \end{cases}.\end{aligned}$$ Note that the function $g_{r,k}$ is not defined yet. The algorithm works with rounds $r$, each of which executes iterations $k$. Intuitively, $g_{r,k}$ is a lower bound on the flatness of the aggregated power profile. It varies over the execution of the algorithm, and is discussed in more detail in Section [4.2](#sec:alg){reference-type="ref" reference="sec:alg"}. {#fig:flowSchematic} Given a flow $f$ in network $G$, we denote the flow value as $|f|$ and call an edge $(u,v)$ *saturated* if $f(u,v) = c_{u,v}$. Note that a flow in $G$ corresponds to an EV schedule. Here, an EV $j$ charges $f(j,I_i)$ in interval $I_i$ and the capacities on edges in $D_0$ model the job-specific power limits. Furthermore, any flow for which the edges in $D_s$ are saturated corresponds to a feasible EV charging schedule and the flow through $D_t$ models the aggregated energy in the atomic intervals of the charging schedule corresponding to $f$. By normalizing for the length of each atomic interval, we can deduce the aggregated power profile. Based on this correspondence, we may use the network structure to not only derive a feasible, but an optimal schedule for objective function $F(p_1, \dots, p_m)$. ## Algorithm formulation {#sec:alg} In the following, we use network $G$ defined in Section [4.1](#sec:flow){reference-type="ref" reference="sec:flow"} to derive an iterative algorithm that gives an optimal schedule and power profile for (aggregated) EV charging with an increasing, convex and differentiable objective function $F(p_1, \dots, p_m)$. Before going into detail, we provide some intuition and a rough overview of the workings of the algorithm. Intuitively, the edge set $D_s$ can be interpreted as the energy demand vector of the EVs. For any feasible EV schedule, those demands have to be met. The flow through edge set $D_0$, on the other hand, is what we are trying to determine: the EV schedule. For any interval node $I_i$, the incoming flow corresponds to the load scheduled in that interval. In particular, $f(j,I_i)$ is the energy charged to EV $j$ in interval $I_i$. Whereas the capacities of edges in $D_s$ and $D_0$ are determined by the instance, edge capacities of edges in $D_t$ are not. However, the flow through $D_t$ directly corresponds to the value of the objective function. Therefore, the algorithm presented in this section defines edge capacities for $D_t$ such that they are a lower bound on the highest aggregated power contributing to the objective function, i.e., a lower bound on the outgoing flow of nodes $I_i$ where $I_i$ is a critical interval. If given those capacities, we find a maximum flow that saturates all edges in $D_s$, we have found a feasible solution, and use this to determine the partial schedule for any critical interval $I_i$. This partial schedule corresponds to the incoming flow at each such node $I_i$. Else, we adapt the lower bound and repeat the process until we do find such a maximum flow and partial schedule. ![Schematic overview of [Focs]{.smallcaps}.](202309Arxiv-figure3.pdf){#fig:algSchematic} In Fig [4](#fig:algSchematic){reference-type="ref" reference="fig:algSchematic"} we provide a rough outline of an algorithm that exploits the bottleneck function of the critical intervals. In both the algorithm formulation and analysis, we distinguish between iterations and rounds of the algorithm. In Fig. [4](#fig:algSchematic){reference-type="ref" reference="fig:algSchematic"}, a new round starts every time that Algorithm [\[alg:round\]](#alg:round){reference-type="ref" reference="alg:round"} is called. To determine a (set of) critical interval(s) (see Definition [Definition 1](#def:crit){reference-type="ref" reference="def:crit"}), we may require multiple iterations in which we adapt the lower limit. Given the dynamic nature of this lower limit, we denote it as $g_{r,k}$ where $r$ and $k$ denote the current round and iteration respectively. At the end of a round, we have determined a (set of) critical interval(s). We determine the schedule for those intervals to be the incoming flow at the corresponding interval nodes. For non-critical intervals, there is no schedule yet. Their schedules will be determined in the next rounds. In that fashion, we will construct a schedule for the entire instance. To keep track of what has yet to be scheduled, we introduce the notion of *active intervals*. At the beginning of a round, interval $I_i$ is active if it has not yet been scheduled (i.e., has not yet been critical) in previous rounds. Let $\mathcal{I}_a$ be the set of active intervals, which we initialize to be all atomic intervals, i.e., $\mathcal{I}_a = \mathcal{I}$. The first iteration of the first round goes as follows. Given that we have to schedule a certain amount of energy and that the objective function is increasing, the most optimistic lower bound on the aggregated power is a constant profile over all intervals. Therefore, we initialize the capacities of the edges in $D_t$ by $$\begin{aligned} g_{1,1}(i) = \frac{\sum_{j=1}^n e_j}{L\left(\mathcal{I}_a\right)}|I_i| \ \forall i \in \mathcal{I}_a,\ \nonumber\end{aligned}$$ which is the aggregated energy charged in atomic interval $I_i$ given that in all intervals the same aggregated charging power is used, and all energy requirements are met. In that way, the edge capacities of $D_t$ act as lower bounds to the highest aggregated power level. They are dynamic and will be increased over iterations. Given the capacities, we determine a maximum flow $f_{1,1}$ for this instance. If the flow value $|f_{1,1}|$ of $f_{1,1}$ is $\sum_{j=1}^n e_{j}$, we have found a feasible schedule, and all active intervals are critical. If not, then there is at least one non-saturated edge $(I_i,t)$ with $I_i$ an active interval. We call the intervals corresponding to such edges *subcritical*. Note that those intervals will not be critical in this round. We therefore temporarily remove them from the set of active intervals and add them to what we call the collection of *parked intervals* $\mathcal{I}_p$. At the beginning of each round, this collection is initialized to be empty. This is the end of the first iteration. From here, we structurally increase the edge capacities of edges in $D_t$ and again compute a maximum flow until all edges in $D_s$ are saturated, and we found a feasible EV schedule. To this end, first note that after the first iteration, $$\begin{aligned} \sum_{j=1}^n c_{s,j} - |f_{1,1}| = \sum_{j=1}^n e_j - |f_{1,1}| > 0 \nonumber\end{aligned}$$ if there were subcritical intervals. In particular, this means that there are jobs $j$ for which additional charge still needs to be scheduled. Among the interval-vertices, the only candidates for additional flow are those vertices $I_i$ for which edge $(I_i,t)$ was saturated in $f_{1,1}$, i.e., the remaining active intervals. Keeping the objective in mind, we therefore proportionally increase the capacities at the remaining active intervals to $$\begin{aligned} g_{1,2}(i) = g_{1,1}(i) + \frac{\sum_{j=1}^n e_j - |f_{1,1}|}{L\left(\mathcal{I}_a\right)}|I_i| \ \forall i \in \mathcal{I}_a. \nonumber\end{aligned}$$ We repeat this process until we find a flow with flow value $\sum_{j=1}^n e_{j}$. Then we have found a feasible EV schedule for which the maximum aggregated power is minimal. Say this happens after $K_1$ iterations. The remaining active intervals in that iteration make up the set of critical intervals in the corresponding round. In Fig. [4](#fig:algSchematic){reference-type="ref" reference="fig:algSchematic"}, this is the first time we leave the box of Algorithm [\[alg:round\]](#alg:round){reference-type="ref" reference="alg:round"} and move on to fix parts of the schedule we are computing. We generalize the steps discussed thus far to any round $r$ and iteration $k$ with $1\leq k <K_r-1$ where $K_r$ is the number of iterations in round $r$. Then $$\begin{aligned} g_{r,1}(i) &= \frac{\sum_{j=1}^n c_{s,j}}{L\left(\mathcal{I}_a\right)}|I_i| \ &\forall i \in \mathcal{I}_a \\ g_{r,k+1}(i) &= g_{r,k}(i) + \frac{\sum_{j=1}^n c_{s,j} - |f_{r,k}|}{L\left(\mathcal{I}_a\right)}|I_i| \ &\forall i \in \mathcal{I}_a\end{aligned}$$ given that flow $f_{r,k}$ is the maximum flow in round $r$ and iteration $k$, and that between iterations active intervals and flow networks are updated. We end the round when we find a maximum flow with flow value $\sum_{j=1}^n c_{s,j}$. After each round $r$, we fix the part of the schedule associated with the critical interval(s) (top right box in Fig. [4](#fig:algSchematic){reference-type="ref" reference="fig:algSchematic"}) to correspond to the flow incoming at the respective (critical) interval nodes, and reduce the remainder of the problem by constructing a new network $G_{r+1}$ (bottom left box in Fig. [4](#fig:algSchematic){reference-type="ref" reference="fig:algSchematic"}) as follows. First, we exploit the acyclic topology of the network to define a flow $f_r|_{I_r^*}$ of the determined maximum flow $f_r$, where $I_r^* = \{i \in [m] | I_i \text{\ is critical in round\ }r \}$ is the set of indices of critical intervals and $$\begin{aligned} f_r|_{I_r^*}(I_i,t) &= \begin{cases} f(I_i,t) & \text{if } i \in I_r^* \\ 0 & \text{otherwise} \end{cases} \\ f_{r}|_{I_r^*}(j,I_i) &= \begin{cases} f(j,I_i) & \text{if } i \in I_r^* \\ 0 & \text{otherwise} \end{cases} \\ f_r|_{I_r^*}(s,j) &= \sum_{i\in J^{-1}(j)} f_r|_{I_r^*}(j,I_i).\end{aligned}$$ Note that this definition backpropagates flow from the sink to the source. Intuitively, $f_r|_{I_r^*}$ denotes the flow that goes through critical intervals. In the YDS-sense, $f_r|_{I_r^*}(s,j)$ is the critical load of job $j$ in round $r$. Now, $G_{r+1}$ is the network obtained by removing edges $(I_i,t)$ with $i\in I^*_r$ from $G_r$, and updating edge capacities to be $c_{u,v} - f_r|_{I_r^*}(u,v)$. From here, we start the next round of the algorithm and initialize a new flow $f_{r+1,1}$. The optimal flow output by the algorithm will be $f = \sum_r f_r|_{I_r^*}$. Implicitly, we use that augmenting paths in future rounds will not reshuffle the already determined subschedule induced by the critical intervals. We will come back to that in Lemma [Lemma 2](#lemma:Isolation){reference-type="ref" reference="lemma:Isolation"}. For more information about augmenting paths, and their relation to maximum flows, please refer to e.g., [@1972EdmondsKarpMaxFlowAlgs]. The algorithm to derive a feasible schedule within a round is summarized in Algorithm [\[alg:round\]](#alg:round){reference-type="ref" reference="alg:round"}. This algorithm is then embedded in the global algorithm (Algorithm [\[alg:alg\]](#alg:alg){reference-type="ref" reference="alg:alg"}) described in this section, outputting a flow $f$ corresponding to an optimal EV charging schedule. We refer to this algorithm as *Flow-based Offline Charging Scheduler* ([Focs]{.smallcaps}). To illustrate, Fig. [12](#fig:flowExWithProfile){reference-type="ref" reference="fig:flowExWithProfile"} displays both the flow and aggregated power profile of an example instance over the rounds and iterations of the algorithm. Here, the first three flows display $f_{r,k}$, whereas the last flow is the optimal flow $f$. In the respective power profiles corresponding to the flow-induced schedules, shaded intervals are parked, and solid green intervals are critical. In general, maximum flows are not unique. To illustrate that, the first flow is deliberately chosen such that the flow through $(1,I_1)$ differs from that through $(1,I_3)$. Note how the optimal power profile in the bottom graph is a sum of the green components at the end of each round of the algorithm. Note that Step [\[alg:alg:repeat\]](#alg:alg:repeat){reference-type="ref" reference="alg:alg:repeat"} in Algorithm [\[alg:alg\]](#alg:alg){reference-type="ref" reference="alg:alg"} can be reformulated as a recursion by calling [Focs]{.smallcaps}($G_r$). $G_r, \ r, \ \mathcal{I}_a$\ $\text{feasible flow } f_r, \ \text{critical sets } I_r^*$\ $\mathcal{I}_p = \varnothing, \ e^r = \sum_{j=1}^n c_{s,j}, \ k = 0 \ $ [\[alg:rep:defineG\]]{#alg:rep:defineG label="alg:rep:defineG"} $c_{I_i,t} = g_{r,k}(i) \ \forall i \in \mathcal{I}_a$ Determine a maximum flow $f_{r,k}$ $\mathcal{I}_p = \mathcal{I}_p \cup \{i\in \mathcal{I}_a | i \text{ subcritical in } f_{r,k}\}$ $\mathcal{I}_a = \mathcal{I}_a \setminus \mathcal{I}_p$ $f_r = f_{r,k},\ I_r^* = \mathcal{I}_a$ $k = k + 1$ and repeat from Step [\[alg:rep:defineG\]](#alg:rep:defineG){reference-type="ref" reference="alg:rep:defineG"} $G$ $\text{optimal flow } f$ $\mathcal{I}_a = \mathcal{I}, \ \mathcal{I}_p = \varnothing, \ r = 0, \ G_r = G , \ f$ [\[alg:alg:callRep\]]{#alg:alg:callRep label="alg:alg:callRep"} $f_r$, $I_r^* =$ [Round]{.smallcaps}($G_r,r,\mathcal{I}_a$) $\mathcal{I}_a = \mathcal{I}_a \setminus I_r^*$ $f = f + f_r|_{I_r^*}$ $G_{r+1} = G_r$ with capacities reduced by $f_r|_{I_r^*}$ and vertices $(I_i,t)$ removed for $i\in I_r^*$ $r = r + 1$ $f$ Repeat from Step [\[alg:alg:callRep\]](#alg:alg:callRep){reference-type="ref" reference="alg:alg:callRep"} [\[alg:alg:repeat\]]{#alg:alg:repeat label="alg:alg:repeat"} $j$ $a_j$ $d_j$ $e_j$ $p_j^{\max }$ ----- ------- ------- ------- --------------- 1 0 3 2 2 2 1 2 2 2 \ ![Intermediate states of [Focs]{.smallcaps} for an example instance, tracked over rounds and iterations.](202309Arxiv-figure4.pdf "fig:"){#fig:flowExWithProfile} ![Intermediate states of [Focs]{.smallcaps} for an example instance, tracked over rounds and iterations.](202309Arxiv-figure5.pdf "fig:"){#fig:flowExWithProfile}\ ![Intermediate states of [Focs]{.smallcaps} for an example instance, tracked over rounds and iterations.](202309Arxiv-figure6.pdf "fig:"){#fig:flowExWithProfile} ![Intermediate states of [Focs]{.smallcaps} for an example instance, tracked over rounds and iterations.](202309Arxiv-figure7.pdf "fig:"){#fig:flowExWithProfile}\ ![Intermediate states of [Focs]{.smallcaps} for an example instance, tracked over rounds and iterations.](202309Arxiv-figure8.pdf "fig:"){#fig:flowExWithProfile} ![Intermediate states of [Focs]{.smallcaps} for an example instance, tracked over rounds and iterations.](202309Arxiv-figure9.pdf "fig:"){#fig:flowExWithProfile}\ ![Intermediate states of [Focs]{.smallcaps} for an example instance, tracked over rounds and iterations.](202309Arxiv-figure10.pdf "fig:"){#fig:flowExWithProfile} ![Intermediate states of [Focs]{.smallcaps} for an example instance, tracked over rounds and iterations.](202309Arxiv-figure11.pdf "fig:"){#fig:flowExWithProfile} # Algorithm analysis {#sec:algAnalysis} In this section, we analyze [Focs]{.smallcaps}, the algorithm presented above. In particular, Section [5.1](#sec:properties){reference-type="ref" reference="sec:properties"} shortly discusses its complexity and properties, after which we prove its optimality in Section [5.2](#sec:optimality){reference-type="ref" reference="sec:optimality"}. ## Properties and complexity {#sec:properties} In this section, we discuss some properties and lemmas that apply to the flow model and algorithm. In particular, we establish some building blocks that enable us to prove optimality of the algorithm in Section [5.2](#sec:optimality){reference-type="ref" reference="sec:optimality"}. First, we note that feasibility of the algorithm output follows by design. At the end of any round, the resulting schedule is indeed feasible, or else there was no feasible schedule for the given instance. Secondly, we consider the complexity of the algorithm. In each iteration, we determine at least one subcritical interval, or else are left with only critical intervals. The number $m$ of atomic intervals is bounded by the number of charging jobs $j$, implying that $m \leq 2n$. Therefore, there are at most $2n - 1$ iterations within each round. Note that there are efficient algorithms available to solve maximum flow problems, e.g., [@1956FordFulkersonMaxFlows; @1972EdmondsKarpMaxFlowAlgs; @1970DinitzMaxFlows; @1974KarzanovMaxFlowsPreflows]. Furthermore, a comprehensive overview of traditional polynomial time maximum flow algorithms is given in [@1988GoldbergMaxFlows]. If we denote their complexity as $\mu$, we find that [Round]{.smallcaps} has a complexity of $\mathcal{O}(n \mu)$. Similarly, in every round we find at least one critical atomic interval. Therefore, there are at most $2n-1$ rounds before the algorithm terminates. This implies that the run time of [Focs]{.smallcaps} is bounded by $\mathcal{O}(n^2\mu)$. Note that this may be reduced further by exploiting the underlying structure of EV charging schedules, and by considering the decrease in network size over the rounds of the algorithm. Furthermore, there are maximum flow algorithms that are cubic in the number of nodes [@1988GoldbergMaxFlows]. The largest flow network that is considered in [Focs]{.smallcaps} (the network in the initial round) has $n+m+2 \leq 3n +2$ nodes. Therefore, a rough upper bound of the complexity of maximum flows in [Focs]{.smallcaps} is given by $\mu \leq n^3$. Next, we extend on the concept of work-transferability as described in [@2017AntoniadisContSpeedScalingWithVariability] to fit job-specific power limits. Formally: **Definition 2** (Work-transferability). *For a given schedule and atomic intervals $I_i$ and $I_{i'}$, the *work-transferable* relation $i\rightarrow i'$ holds if there exists a job $j\in J(i) \cap J(i')$ such that $e_{i,j} > 0$ and $e_{i',j} < e_{i',j}^{\max}$. Furthermore, let $\twoheadrightarrow$ be the transitive closure of $\rightarrow$.* Intuitively, if we have work-transferability from one atomic interval $i$ to another atomic interval $i'$, then we can transfer some work that was scheduled during $i$ to $i'$. In EV charging terms, we can advance or delay some charging from one point in time to another. Applying the concept to flows, we can make the following statement. **Lemma 1** (Work-transferability in flows). *For a given schedule and atomic intervals $I_i$ and $I_{i'}$, we have $i\rightarrow i'$ if and only if there exists a path $(I_i,j,I_{i'})$ in the residual graph corresponding to the schedule, where $j\in [n]$. Similarly, $i\twoheadrightarrow i''$ if and only if in the residual network corresponding to the schedule there exists an ($I_i$-$I_{i''}$)-path through interval and job vertices only.* *Proof of Lemma [Lemma 1](#lemma:WT){reference-type="ref" reference="lemma:WT"}:* We show only the first statement as the extension follows naturally using unions of paths. Assume that $i\rightarrow i'$. Then there exists a job $j$ such that $j\in J(i) \cap J(i')$ such that $e_{i,j} > 0$ and $e_{i',j} < e_{i',j}^{\max}$. The former implies that edge $(I_i,j)$ exists in the residual graph. As $c_{j,I_{i'}} = e_{i',j} < e_{i',j}^{\max}$, edge $(j,I_{i'})$ is in the residual graph. This proves existence of path $(I_i,j,I_{i'})$ in the residual graph. For the opposite direction, assume the existence of a path $(I_{i},j,I_{i'})$. Since $j\in [n]$, we know the edge capacity $c_{j,I_{i'}}$ in the original network to be $e_{i',j}^{\max}$. The existence of the edge in the residual graph gives that for the flow going through this edge which is defined by the schedule to be $e_{i',j}$, we have $e_{i',j} < e_{i,j}^{\max}$. Furthermore, existence of edge $(I_i,j)$ in the residual graph indicates positive flow through $(j,I_i)$ in the original network, implying $e_{i,j} > 0$. From the presence of both edges, it follows that $j\in J(i) \cap J(i')$, proving that $i\rightarrow i'$. $\blacksquare$ Fig. [13](#fig:wtFlow){reference-type="ref" reference="fig:wtFlow"} illustrates the concept of work-transferability. Here, dashed edges are those that are not in the original network, but might be present in the residual network. Lemma [Lemma 1](#lemma:WT){reference-type="ref" reference="lemma:WT"} translates work-transferability to the existence of (in the figure) red paths in the residual network. Next, we consider two lemmas that have a more direct relation to the algorithm. {#fig:wtFlow} **Lemma 2** (Isolation of critical intervals). *If $I_i$ is a critical interval in round $r$ and if the round consists of multiple iterations whereby $I_{i'}$ was subcritical in one of those iterations, then there is no work-transferable relation between $i$ and $i'$ in the schedule corresponding to the flow at the end of round $r$, i.e., $i\not \twoheadrightarrow i'$ with respect to flow $f_r$.* *Proof of Lemma [Lemma 2](#lemma:Isolation){reference-type="ref" reference="lemma:Isolation"}:* We prove the lemma by constructing an augmenting path (see Fig. [14](#fig:isolationProof){reference-type="ref" reference="fig:isolationProof"}). Assume in round $r$ interval $I_{i'}$ was parked in iteration $k$ and let $f_{r,k}$ be the flow at the end of iteration $k$. Since $I_{i'}$ is subcritical, we have $|f_{r,k}| < \sum_{j=1}^n e_j$, implying that for the next iteration the lower bound $g_{r,k}$ will be increased to $$\begin{aligned} g_{r,k+1}(i'') &= g_{r,k}(i'') + \frac{\sum_{j=1}^n c_{s,j} - |f_{r,k}|}{L\left(\mathcal{I}_a\right)}|I_{i''}| \ &\forall i'' \in \mathcal{I}_a.\end{aligned}$$ By criticality of $I_i$, the interval is active at the end of the iteration, implying $g_{r,k+1}(i) > g_{r,k}(i)$. Furthermore, criticality implies that there is no iteration in this round where $I_i$ is subcritical. Combing those facts, the flow through $(I_i,t)$ increases in iteration $k+1$ compared to iteration $k$. This is only possible if there is a job $j$ such that $(s,j)$ is not saturated and there exists a ($j$-$I_i$)-path $P$ in the residual graph. Furthermore, note that since $I_{i'}$ is being parked in iteration $k$, edge $(I_{i'},t)$ is not saturated and therefore exists in the residual graph. Now, assume $i\twoheadrightarrow i'$. By Lemma [Lemma 1](#lemma:WT){reference-type="ref" reference="lemma:WT"}, there exists an ($I_i$-$I_{i'}$)-path $P'$ that passes only through job and interval vertices. This implies that $P'' = (s,P,P',t)$ exists in the residual graph and contains an ($s$-$t$)-path, proving existence of an augmenting path in $f_{r,k}$. This contradicts maximality of the flow, implying $i \not \twoheadrightarrow i'$. $\blacksquare$ Intuitively, this lemma says that we cannot push any charging from (high power) critical intervals to (low power) subcritical intervals. This is in line with the notion of critical intervals as introduced for the YDS algorithm, and will be a key element in the optimality proof in Section [5.2](#sec:optimality){reference-type="ref" reference="sec:optimality"}. Furthermore, this particular lemma justifies that we fix the schedule of critical intervals at the end of each round. {reference-type="ref" reference="lemma:Isolation"}.](202309Arxiv-figure13.pdf){#fig:isolationProof} For the next lemma, we first introduce the notion of ranks. **Definition 3** (Rank). *Given an atomic interval $I_i$, its rank $r(i)$ is defined such that $I_i$ was critical in the $r(i)^{\text{th}}$ round, i.e., $i\in I_{r(i)}^*$.* **Lemma 3** (Monotonicity). *Given the schedule corresponding to output flow $f$ of algorithm [Focs]{.smallcaps} and atomic intervals $I_i$ and $I_{i'}$ where $r(i) < r(i')$, the aggregated power in $I_i$ is strictly larger than in $I_{i'}$: $$\begin{aligned} p_i > p_{i'}. \nonumber \end{aligned}$$* *Proof of Lemma [Lemma 3](#lemma:Monotonicity){reference-type="ref" reference="lemma:Monotonicity"}:* We prove the lemma by contradiction. In this proof, we consider flows at the end of rounds. Let interval $I_i$ be the lowest ranked interval such that its aggregated power level in the flow outputted by the algorithm is larger than that in intervals with rank $r(i) - 1$. By the algorithm not changing schedules at critical intervals, this already occurs at round $r(i)$ itself. Given that $I_i$ was subcritical in the previous round, the power level for $I_i$ increased. In particular, there is a job $j$ for which the power during interval $I_i$ increased compared to the previous round. Furthermore, we know that the flow through $(j,I_i)$ is positive in round $r(i)$, implying that edge $(I_i,j)$ is in the residual graph. However, applying Lemma [Lemma 2](#lemma:Isolation){reference-type="ref" reference="lemma:Isolation"} to the previous round, the amount of energy charged to job $j$ remains the same. Therefore, there is an interval $I_{i'}$ for which the flow through $(j,I_{i'})$ decreased compared to the previous round. As a consequence, the flow in round $r(i)$ does not saturate the edge, implying that edge $(j,I_{i'})$ is in the residual graph. Combining these findings, the path $(I_i,j,I_{i'})$ is in the residual graph, implying $i\rightarrow i'$, and thus contradicting Lemma [Lemma 2](#lemma:Isolation){reference-type="ref" reference="lemma:Isolation"}. $\blacksquare$ The lemma shows that the average aggregated power of atomic intervals is decreasing in their rank. Therefore, critical intervals as determined using the method presented in this paper share the monotonicity property known for YDS for the speed scaling problem without speed limits. We first find those intervals with the highest intensity, and then iteratively determine the next power levels. We also note that, similarly to YDS, if the objective function is strictly convex, the power profile outputted by [Focs]{.smallcaps} is unique. However, this does not necessarily apply to the schedule. ## Optimality proof {#sec:optimality} In this section, we prove that Algorithm [\[alg:alg\]](#alg:alg){reference-type="ref" reference="alg:alg"} as described in Section [4.2](#sec:alg){reference-type="ref" reference="sec:alg"} is optimal for any increasing, convex and differentiable objective function $F(p_1, \dots, p_m)$. To this end, we first prove some auxiliary lemmas that show compliance with the sufficient conditions derived in Section [3](#sec:kkt){reference-type="ref" reference="sec:kkt"}. **Lemma 4**. *The output of the algorithm complies with [Kkt1]{.smallcaps}.* *Proof of Lemma [Lemma 4](#lemma:kkt1){reference-type="ref" reference="lemma:kkt1"}:* If in the final output of the algorithm there are two distinct atomic intervals $I_i$ and $I_{i'}$ such that for job $j$ $0 < \frac{e_{i,j}}{|I_{i}|} = \frac{e_{i,j}}{|I_{i}|} < p_{j}^{max}$, then by defintion of worktransferability we have $i\rightarrow i'$ and $i'\rightarrow i$. By Lemma [Lemma 2](#lemma:Isolation){reference-type="ref" reference="lemma:Isolation"} and the strict monotonicity in Lemma [Lemma 3](#lemma:Monotonicity){reference-type="ref" reference="lemma:Monotonicity"}, this implies that the aggregated power in both intervals is the same. $\blacksquare$ **Lemma 5**. *The output of the algorithm complies with [Kkt2]{.smallcaps}.* *Proof of Lemma [Lemma 5](#lemma:kkt2){reference-type="ref" reference="lemma:kkt2"}:* Let $i\in J^{-1}(j)$ be such that $e_{i,j} = 0$ in the output of the algorithm. Assume that there is an interval $I_{i'}$ with $i\neq i'$ and $i'\in J^{-1}(j)$ for which the aggregated power in $I_{i'}$ is strictly greater than in $I_{i}$, i.e., $\frac{\sum_{j=1}^n e_{i,j}}{|I_{i}|} < \frac{\sum_{j=1}^n e_{i',j}}{|I_{i'}|}$. By Lemma [Lemma 3](#lemma:Monotonicity){reference-type="ref" reference="lemma:Monotonicity"} we have $r(i') < r(i)$, implying by Lemma [Lemma 2](#lemma:Isolation){reference-type="ref" reference="lemma:Isolation"} that $i' \not \twoheadrightarrow i$. Applying the definition of work-transferability, it follows that $e_{i',j} = 0$, proving compliance with [Kkt2]{.smallcaps}. $\blacksquare$ **Lemma 6**. *The output of the algorithm complies with [Kkt3]{.smallcaps}.* *Proof of Lemma [Lemma 6](#lemma:kkt3){reference-type="ref" reference="lemma:kkt3"}:* Let job $j$ run at maximum speed in $I_i$ in the schedule found by [Focs]{.smallcaps}. Assume that there is an interval $I_{i'}$ with $i\neq i'$ and $i'\in J^{-1}(j)$, such that the aggregated power in $I_i$ is strictly greater than in $I_{i'}$. By Lemma [Lemma 3](#lemma:Monotonicity){reference-type="ref" reference="lemma:Monotonicity"}, we know that $r(i)<r(i')$. Therefore, by Lemma [Lemma 2](#lemma:Isolation){reference-type="ref" reference="lemma:Isolation"}, there is no work-transferable relation between $i$ and $i'$ ($i\not \twoheadrightarrow i'$). From the definition of work-transferability it now follows directly that $e_{i',j} \geq e_{i',j}^{\max}$, proving compliance with [Kkt3]{.smallcaps}. $\blacksquare$ Combining all discussed above, we conclude optimality of the algorithm output. **Theorem 7** (Optimality). *For any feasible input instance, the schedule produced by the algorithm is an optimal solution minimizing any convex, increasing and differentiable objective function of the aggregated output powers.* *Proof of Theorem [Theorem 7](#thm:optimality){reference-type="ref" reference="thm:optimality"}:* The proof follows directly from the [Kkt]{.smallcaps} conditions derived in Section [3](#sec:kkt){reference-type="ref" reference="sec:kkt"}, the inherent feasibility of the output and Lemmas [Lemma 4](#lemma:kkt1){reference-type="ref" reference="lemma:kkt1"}--[Lemma 6](#lemma:kkt3){reference-type="ref" reference="lemma:kkt3"}. $\blacksquare$ # Conclusion {#sec:conclusion} In this work, we consider an EV scheduling problem with as objective to minimize an increasing, convex and differentiable function of the aggregated power profile. To this end, we relate EV charging to speed scaling with job-specific speed limits and derive sufficient and necessary optimality conditions. Furthermore, we present an offline algorithm that determines an optimal schedule in $\mathcal{O}(n^2 \mu)$ time where $\mu$ is the run time of an efficient maximum flow algorithm. We argue that this run time can further be reduced by exploiting the underlying problem structure of the EV scheduling problem. Lastly, we provide a mathematical proof of the optimality of the output of the algorithm. Future work may investigate an online variant of the presented algorithm, as well as additional problem constraints such as global power limits. Furthermore, numerical experiments are of interest, especially their integration with control strategies such as model predictive control or fill-level algorithms. Lastly, given that optimal schedules are not necessarily unique, scheduling rules resulting in a robust output should be explored. [^1]: This research is conducted within the SmoothEMS met GridShield project subsidized by the Dutch ministries of EZK and BZK (MOOI32005).

arxiv_math

--- abstract: | We consider the coupled system of the Landau--Lifshitz--Gilbert equation and the conservation of linear momentum law to describe magnetic processes in ferromagnetic materials including magnetoelastic effects in the small-strain regime. For this nonlinear system of time-dependent partial differential equations, we present a decoupled integrator based on first-order finite elements in space and an implicit one-step method in time. We prove unconditional convergence of the sequence of discrete approximations towards a weak solution of the system as the mesh size and the time-step size go to zero. Compared to previous numerical works on this problem, for our method, we prove a discrete energy law that mimics that of the continuous problem and, passing to the limit, yields an energy inequality satisfied by weak solutions. Moreover, our method does not employ a nodal projection to impose the unit length constraint on the discrete magnetisation, so that the stability of the method does not require weakly acute meshes. Furthermore, our integrator and its analysis hold for a more general setting, including body forces and traction, as well as a more general representation of the magnetostrain. Numerical experiments underpin the theory and showcase the applicability of the scheme for the simulation of the dynamical processes involving magnetoelastic materials at submicrometer length scales. address: Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom author: - Hywel Normington - Michele Ruggeri bibliography: - ref.bib title: | A decoupled, convergent and fully linear algorithm for the Landau--Lifshitz--Gilbert equation\ with magnetoelastic effects --- # Introduction Magnetoelastic (or magnetostrictive) materials are smart materials characterised by a strong interplay between their mechanical and magnetic properties [@brown1966]. On the one hand, they change shape when subject to applied magnetic fields (direct magnetostrictive effect), and on the other, they undergo a change in their magnetic state when subject to externally applied mechanical stresses (inverse magnetostrictive effect). Because of these properties, magnetoelastic materials currently find use in many technological applications requiring a magnetomechanical transducer, e.g. actuators or sensors [@pasquale2003]. In this work, we design and analyse a fully discrete numerical scheme for a coupled nonlinear system of partial differential equations (PDEs) modelling the dynamics of magnetisation and displacement in magnetoelastic materials in the small-strain regime. The system consists of the Landau--Lifshitz--Gilbert (LLG) equation for the magnetisation and the conservation of linear momentum law for the displacement (see [\[eq:newton\]](#eq:newton){reference-type="eqref" reference="eq:newton"}--[\[eq:llg\]](#eq:llg){reference-type="eqref" reference="eq:llg"} below). The two equations are nonlinearly coupled to each other: One of the contributions to the effective field appearing in the LLG equation depends on the mechanical stress in the body (and thus on the displacement) and there is a magnetisation-dependent contribution to the strain (usually referred to as the magnetostrain) in the conservation of momentum law. One additional difficulty is represented by a nonconvex pointwise constraint on the magnetisation, which is a vector field of constant unit length. Several versions of this PDE system have been used for physical investigations of magnetoelastic materials; see e.g. [@shu2004micromagnetic; @Mballa2014; @bhbvfs2014; @pwhcn2015; @rj2021; @renuka2021solution; @dw2023]. As far as the mathematical literature is concerned, we refer to [@visintin1985landau; @cef2011], in which existence of weak solutions has been established, and to a series of works by L. Baňas and coauthors [@bs2005; @banas2005a; @bs2006; @banas2008; @bppr2013] in numerical analysis. In [@bs2005; @banas2005a; @bs2006; @banas2008], the focus is on finite element methods for the approximation of strong solutions. More recently, [@bppr2013] extended the tangent plane scheme proposed in [@alouges2008a] for the LLG equation to this PDE system. The integrator, based on first-order finite elements in space and on an implicit first-order time-stepping method in time, decouples the system and only requires the solution of two linear systems per time-step. Under the assumption that all meshes used for the spatial discretisation are weakly acute (needed to guarantee the stability of the nodal projection used to impose the unit length constraint on the magnetisation [@bartels2005; @alouges2008a]), the authors proved unconditional convergence of the finite element approximations towards a weak solution of the problem. In this work, we generalise the PDE system considered in [@visintin1985landau; @cef2011; @bppr2013] by including volume and surfaces forces, as well as a more general expression for the magnetostrain [@federico2019tensor], which allows the description of a larger class of magnetoelastic materials. For this generalised system, we propose an integrator which resembles the one in [@bppr2013] (same finite element approximation spaces, same time discretisation method, same decoupled approach). However, following [@bartels2016; @abert_spin-polarized_2014] (and differently from [@bppr2013]), we remove the nodal projection from the update of the magnetisation (but we keep it in the discretisation of the elastic contributions). By doing this, we can avoid the requirement of weakly acute meshes at the expense of not maintaining the unit length constraint on the magnetisation at the vertices of the meshes. However, like in [@bartels2016; @abert_spin-polarized_2014], we can uniformly control the violation of the constraint by the time-step size. Despite the strong nonlinearity of the problem, the resulting integrator is fully linear (in the sense that it involves only linear operations like solving linear systems and updating the approximations using a linear time-stepping). For this generalised and modified integrator, we show unconditional well-posedness, a discrete energy law satisfied by the approximations, unconditional stability, and unconditional convergence of the approximations towards a weak solution of the problem (here, the adjective 'unconditional' refers to the fact that the analysis does not require any restrictive coupling condition between the time and spatial discretisation parameters). Moreover, assuming a (very restrictive, but artificial) Courant--Friedrichs--Lewy (CFL) condition on the time-step size and the spatial mesh size, we can pass the discrete energy law to the limit and show that the weak solution towards which our finite element approximation is converging satisfies an energy inequality. Summarising, the contribution of the present paper over the existing literature (and, in particular, over [@visintin1985landau; @cef2011; @bppr2013]) is threefold: - We consider a more general setting than in [@visintin1985landau; @cef2011; @bppr2013] (including volume/surface forces and a more general magnetoelastic contribution). Since our convergence proof is constructive, a byproduct of our analysis is a proof of existence of weak solutions for a more general model of magnetoelastic materials in the small-strain regime. - Our integrator is energetically 'mindful', in the sense that our approximations satisfy a discrete energy law which resembles the one satisfied by solutions of the continuous problem (cf. Proposition [Proposition 6](#prop:energy){reference-type="ref" reference="prop:energy"} below). Under a CFL condition on the discretisation parameters, we can pass the result to the limit and obtain an energy inequality for weak solutions. This aspect was not considered in [@bppr2013], where only boundedness of energy was proven. - The spatial meshes used by our integrator are assumed to be only shape-regular (and do not need to be weakly acute as in [@bppr2013]). This allows for the use of general mesh generators. Moreover, the discrete variational problems appearing in our integrator are standard and therefore easy to implement in standard finite element packages. For example, in the numerical experiments included in this work, we use Netgen/NGSolve [@netgen]. The remainder of this work is organised as follows: In Section [2](#sec:model){reference-type="ref" reference="sec:model"}, we present the PDE system we are interested in; In Section [3](#sec:ingredients){reference-type="ref" reference="sec:ingredients"}, we introduce the 'ingredients' that are necessary for the definition of our numerical scheme and for its analysis; In Section [4](#sec:main){reference-type="ref" reference="sec:main"}, we present our numerical integrator (Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"}) and state the main results of the work; Section [5](#sec:numerics){reference-type="ref" reference="sec:numerics"} is devoted to numerical experiments. In Section [6](#sec:proofs){reference-type="ref" reference="sec:proofs"}, we collect the proofs of all results. For the convenience of the reader, we conclude the paper with two appendices, Appendix [7](#sec:linear_algebra){reference-type="ref" reference="sec:linear_algebra"}, in which we collect several linear algebra definitions and results used throughout the work, and Appendix [8](#sec:physics){reference-type="ref" reference="sec:physics"}, in which we show how to pass from the fully dimensional model considered in the physics literature to the dimensionless setting we study. # Model problem {#sec:model} Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain representing the volume occupied by a ferromagnetic body. We assume the boundary $\partial\Omega$ is split into two disjoint relatively open parts $\Gamma_D$ (of positive measure) and $\Gamma_N$, i.e. $\partial\Omega = \overline{\Gamma}_D \cup \overline{\Gamma}_N$ and $\Gamma_D \cap \Gamma_N = \emptyset$. Let $T>0$ denote some final time. The magnetomechanical state of the material is described by two vector fields: the displacement $\boldsymbol{u}: \Omega \times (0,T) \to \mathbb{R}^3$ and the magnetisation $\boldsymbol{m}: \Omega \times (0,T) \to \mathbb{S}^2$. The total strain $\boldsymbol{\varepsilon}$ is made up of the elastic strain $\boldsymbol{\varepsilon}_{\mathrm{el}}$ and the magnetisation-dependent generally incompatible (in the sense that it does not satisfy the Saint-Venant compatibility conditions [@amrouche2006saint; @Mballa2014]) magnetostrain $\boldsymbol{\varepsilon}_{\mathrm{m}}$, i.e. $\boldsymbol{\varepsilon}= \boldsymbol{\varepsilon}_{\mathrm{el}}+ \boldsymbol{\varepsilon}_{\mathrm{m}}$. The total strain is given by $$\boldsymbol{\varepsilon}(\boldsymbol{u}) = \frac{1}{2}\left(\boldsymbol{\nabla}\boldsymbol{u}+ \boldsymbol{\nabla}\boldsymbol{u}^{\top}\right)$$ (strain-displacement relation). Following [@federico2019tensor], we consider the expression $$\label{eq:magnetostrain} \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}) = \mathbb{Z}: (\boldsymbol{m}\otimes \boldsymbol{m}),$$ where $\mathbb{Z}\in \mathbb{R}^{3^4}$ is a fourth-order tensor, which we assume to be minorly symmetric (i.e. $\mathbb{Z}_{ij\ell m} = \mathbb{Z}_{ji\ell m} = \mathbb{Z}_{ijm \ell}$ for all $i,j,\ell,m = 1,2,3$, cf. Appendix [7](#sec:linear_algebra){reference-type="ref" reference="sec:linear_algebra"}). It follows that $$\boldsymbol{\varepsilon}_{\mathrm{el}}(\boldsymbol{u},\boldsymbol{m}) = \boldsymbol{\varepsilon}(\boldsymbol{u}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}).$$ The elastic part of the strain compensates for the magnetic part to make the total strain compatible [@Mballa2014]. The elastic strain is related to the stress tensor $\boldsymbol{\sigma}$ by Hooke's law $$\boldsymbol{\sigma}(\boldsymbol{u},\boldsymbol{m}) = \mathbb{C}: \boldsymbol{\varepsilon}_{\mathrm{el}}(\boldsymbol{u},\boldsymbol{m}),$$ where $\mathbb{C}\in \mathbb{R}^{3^4}$ is the fourth-order, fully symmetric (i.e. $\mathbb{C}_{ij\ell m} = \mathbb{C}_{\ell mij} = \mathbb{C}_{ji\ell m} = \mathbb{C}_{ij m \ell}$ for all $i,j,\ell,m = 1,2,3$, cf. Appendix [7](#sec:linear_algebra){reference-type="ref" reference="sec:linear_algebra"}), positive definite stiffness tensor. The elastic energy reads as $$\mathcal{E}_{\mathrm{el}}[\boldsymbol{u},\boldsymbol{m}] = \frac{1}{2} \int_{\Omega}[\boldsymbol{\varepsilon}(\boldsymbol{u}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m})] : \{ \mathbb{C}: [\boldsymbol{\varepsilon}(\boldsymbol{u}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m})] \} - \int_{\Omega}\boldsymbol{f}\cdot\boldsymbol{u} - \int_{\Gamma_{N}}\boldsymbol{g}\cdot\boldsymbol{u},$$ where the last two terms model the work done by a volume force $\boldsymbol{f}: \Omega \to \mathbb{R}^3$ and a surface force $\boldsymbol{g}: \Gamma_N \to \mathbb{R}^3$ (traction), both assumed to be constant in time. The magnetic energy, for simplicity assumed to comprise only the Heisenberg exchange contribution, is given by $$\label{eq:energy_mag} \mathcal{E}_{\mathrm{m}}[\boldsymbol{m}] = \frac{1}{2}\int_{\Omega}|\boldsymbol{\nabla}\boldsymbol{m}|^2.$$ The total free energy of the system is defined as the sum of the magnetic and elastic energies, i.e. $$\label{eqn:definition_of_energy} \begin{split} &\mathcal{E}[\boldsymbol{u},\boldsymbol{m}] = \mathcal{E}_{\mathrm{m}}[\boldsymbol{m}] + \mathcal{E}_{\mathrm{el}}[\boldsymbol{u},\boldsymbol{m}] \\ & \quad = \frac{1}{2}\int_{\Omega}|\boldsymbol{\nabla}\boldsymbol{m}|^2 + \frac{1}{2} \int_{\Omega}[\boldsymbol{\varepsilon}(\boldsymbol{u}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m})] : \{ \mathbb{C}: [\boldsymbol{\varepsilon}(\boldsymbol{u}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m})] \} - \int_{\Omega}\boldsymbol{f}\cdot\boldsymbol{u} - \int_{\Gamma_{N}}\boldsymbol{g}\cdot\boldsymbol{u}. \end{split}$$ The dynamics of $\boldsymbol{u}$ and $\boldsymbol{m}$ is governed by the coupled system of the conservation of (linear) momentum law and the LLG equation $$\begin{aligned} {2} \label{eq:newton} \partial_{tt}\boldsymbol{u} & = \nabla \cdot \boldsymbol{\sigma}(\boldsymbol{u},\boldsymbol{m}) + \boldsymbol{f} && \text{in } \Omega \times (0,T),\\ \label{eq:llg} \partial_t \boldsymbol{m} & = - \boldsymbol{m}\times \boldsymbol{h}_{\mathrm{eff}}[\boldsymbol{u},\boldsymbol{m}] + \alpha \, \boldsymbol{m}\times \partial_t \boldsymbol{m} &\quad& \text{in } \Omega \times (0,T),\end{aligned}$$ supplemented with the initial and boundary conditions [\[eq:ibc\]]{#eq:ibc label="eq:ibc"} $$\begin{aligned} {2} \label{eq:ic_u} \boldsymbol{u}(0) &= \boldsymbol{u}^0 &\quad&\text{in } \Omega,\\ \label{eq:ic_ut} \partial_{t}\boldsymbol{u}(0) &= \dot\boldsymbol{u}^0 &&\text{in } \Omega,\\ \label{eq:ic_m} \boldsymbol{m}(0) &= \boldsymbol{m}^0 &&\text{in } \Omega,\\ \label{eq:bc_u_d} \boldsymbol{u}&= \boldsymbol{0} &&\text{on } \Gamma_D \times (0, T),\\ \label{eq:bc_u_n} \boldsymbol{\sigma}\boldsymbol{n}&= \boldsymbol{g}&& \text{on } \Gamma_{N} \times (0, T),\\ \label{eq:bc_m} \partial_{\boldsymbol{n}} \boldsymbol{m}&= \boldsymbol{0} && \text{on } \partial\Omega \times (0, T), \end{aligned}$$ where $\boldsymbol{u}^0, \dot\boldsymbol{u}^0 : \Omega \to \mathbb{R}^3$ and $\boldsymbol{m}^0 : \Omega \to \mathbb{S}^2$ are suitable initial data, while $\boldsymbol{n}:\partial\Omega \to \mathbb{S}^2$ denotes the outward-pointing unit normal vector to $\partial\Omega$. In [\[eq:llg\]](#eq:llg){reference-type="eqref" reference="eq:llg"}, $\alpha>0$ denotes the Gilbert damping parameter, whereas the effective field $\boldsymbol{h}_{\mathrm{eff}}[\boldsymbol{u},\boldsymbol{m}]$ is the variational derivative of the free energy with respect to the magnetisation, i.e. $$\boldsymbol{h}_{\mathrm{eff}}[\boldsymbol{u},\boldsymbol{m}] = -\frac{\delta \mathcal{E}[\boldsymbol{u},\boldsymbol{m}]}{\delta \boldsymbol{m}} = \boldsymbol{\Delta}\boldsymbol{m}+ \boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u},\boldsymbol{m}],$$ where the elastic field reads as $$\label{eqn:MagnetostrictiveEffectiveField} \boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u},\boldsymbol{m}] = 2 \, [\mathbb{Z}^\top : \boldsymbol{\sigma}(\boldsymbol{u},\boldsymbol{m})] \boldsymbol{m} = 2 \, (\mathbb{Z}^\top : \{\mathbb{C}: [\boldsymbol{\varepsilon}(\boldsymbol{u}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m})] \}) \boldsymbol{m},$$ with $\mathbb{Z}^\top$ being the transpose of $\mathbb{Z}$ (cf. Appendix [7](#sec:linear_algebra){reference-type="ref" reference="sec:linear_algebra"}). Note that [\[eq:newton\]](#eq:newton){reference-type="eqref" reference="eq:newton"} can be rewritten as $$\partial_{tt}\boldsymbol{u}= -\frac{\delta \mathcal{E}[\boldsymbol{u},\boldsymbol{m}]}{\delta \boldsymbol{u}}.$$ A simple formal calculation reveals that sufficiently smooth solutions to [\[eq:newton\]](#eq:newton){reference-type="eqref" reference="eq:newton"}--[\[eq:ibc\]](#eq:ibc){reference-type="eqref" reference="eq:ibc"} satisfy the energy law $$\label{eq:energy_law} \frac{\mathrm{d}}{\mathrm{d}t} \left( \mathcal{E}[\boldsymbol{u}(t),\boldsymbol{m}(t)] + \frac{1}{2}\left\lVert \partial_{t}\boldsymbol{u}(t) \right\rVert_{}^2 \right) = - \alpha \left\lVert \partial_t \boldsymbol{m}(t) \right\rVert_{}^2 \le 0,$$ i.e. the sum of the total energy [\[eqn:definition_of_energy\]](#eqn:definition_of_energy){reference-type="eqref" reference="eqn:definition_of_energy"} (which can be understood as a potential energy) and the kinetic energy $\left\lVert \partial_{t}\boldsymbol{u} \right\rVert_{}^2/2$ decays over time, with the decay being modulated by $\alpha$. For the data of the problem, we assume that $\mathbb{C}\in \boldsymbol{L}^{\infty}(\Omega)$ is uniformly positive definite, i.e. there exists $C_0>0$ such that $$\label{eq:tensor_coercivity} \boldsymbol{A}:(\mathbb{C}: \boldsymbol{A}) \ge C_0 \left\lVert \boldsymbol{A} \right\rVert_{}^2 \quad \text{for all } \boldsymbol{A}\in \mathbb{R}^{3 \times 3},$$ $\mathbb{Z}\in \boldsymbol{L}^{\infty}(\Omega)$, $\boldsymbol{f}\in\boldsymbol{L}^2(\Omega)$, $\boldsymbol{g}\in\boldsymbol{L}^2(\Gamma_{N})$, $\boldsymbol{u}^0\in\boldsymbol{H}^1(\Omega)$, $\dot\boldsymbol{u}^0\in\boldsymbol{L}^2(\Omega)$, and $\boldsymbol{m}^0\in\boldsymbol{H}^1(\Omega;\mathbb{S}^2)$. In the following definition, we state the notion of a weak solution of the initial boundary value problem [\[eq:newton\]](#eq:newton){reference-type="eqref" reference="eq:newton"}--[\[eq:ibc\]](#eq:ibc){reference-type="eqref" reference="eq:ibc"}; see [@cef2011]. Hereafter, we shall denote $L^2$-integrals in space over some domain $D$ with $\langle \cdot,\cdot \rangle_{D}$, omitting the subscript if $D = \Omega$. Moreover, we denote by $\Omega_T$ the space-time cylinder $\Omega \times (0,T)$. **Definition 1**. We say that a pair $(\boldsymbol{u},\boldsymbol{m}) :\Omega_T\to\mathbb{R}^3 \times \mathbb{R}^3$ is a weak solution to the initial boundary value problem [\[eq:newton\]](#eq:newton){reference-type="eqref" reference="eq:newton"}--[\[eq:ibc\]](#eq:ibc){reference-type="eqref" reference="eq:ibc"} if the following conditions hold: (i) $\boldsymbol{u}\in L^\infty(0,T;\boldsymbol{H}^1_D(\Omega))$ with $\partial_{t}\boldsymbol{u}\in L^\infty(0,T;\boldsymbol{L}^2(\Omega))$ and $\boldsymbol{m}\in L^\infty(0,T;\boldsymbol{H}^1(\Omega;\mathbb{S}^2))$ with $\partial_t \boldsymbol{m}\in L^2(0,T;\boldsymbol{L}^2(\Omega))$; (ii) for all $\boldsymbol{\xi}\in \boldsymbol{C}_{c}^{\infty}([0,T);\boldsymbol{C}^{\infty}(\overline{\Omega}))$ and $\boldsymbol{\varphi}\in \boldsymbol{C}^{\infty}(\overline{\Omega_T})$, we have $$\begin{aligned} \label{eq:weak_u} \begin{split} & -\int_{0}^{T}\langle \partial_{t}\boldsymbol{u}(t),\partial_{t}\boldsymbol{\xi}(t) \rangle_{}\mathrm{d}t + \int_{0}^{T} \langle \mathbb{C} : [\boldsymbol{\varepsilon}(\boldsymbol{u}(t)) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}(t))],\boldsymbol{\varepsilon}(\boldsymbol{\xi}(t)) \rangle_{} \mathrm{d}t \\ & \qquad = \int_{0}^{T}\langle \boldsymbol{f},\boldsymbol{\xi}(t) \rangle_{} \mathrm{d}t + \int_{0}^{T}\langle \boldsymbol{g},\boldsymbol{\xi}(t) \rangle_{\Gamma_{N}}\mathrm{d}t + \langle \dot{\boldsymbol{u}}^{0},\boldsymbol{\xi}(0) \rangle_{}, \end{split}\\ \label{eq:weak_m} \begin{split} & \int_{0}^{T}\langle \partial_t \boldsymbol{m}(t),\boldsymbol{\varphi}(t) \rangle_{}\mathrm{d}t - \alpha \int_{0}^{T}\langle \boldsymbol{m}(t)\times\partial_t \boldsymbol{m}(t),\boldsymbol{\varphi}(t) \rangle_{}\mathrm{d}t\\ & \qquad = \int_{0}^{T}\langle \boldsymbol{m}(t)\times\boldsymbol{\nabla}\boldsymbol{m}(t),\boldsymbol{\nabla}\boldsymbol{\varphi}(t) \rangle_{}\mathrm{d}t -\int_{0}^{T}\langle \boldsymbol{m}(t)\times\boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u}(t),\boldsymbol{m}(t)],\boldsymbol{\varphi}(t) \rangle_{}\mathrm{d}t; \end{split} \end{aligned}$$ (iii) the initial conditions $\boldsymbol{u}(0)=\boldsymbol{u}^0$ and $\boldsymbol{m}(0)=\boldsymbol{m}^0$ hold in the sense of traces; (iv) for almost all $t' \in (0,T)$, it holds that $$\label{eqn:EnergyLawDefinition} \mathcal{E}[\boldsymbol{u}(t'),\boldsymbol{m}(t')] + \frac{1}{2}\left\lVert \partial_t \boldsymbol{u}(t') \right\rVert_{}^2 + \alpha \int_{0}^{t'}\left\lVert \partial_t \boldsymbol{m}(t) \right\rVert_{}^2 \mathrm{d}t \le \mathcal{E}[\boldsymbol{u}^0,\boldsymbol{m}^0] + \frac{1}{2}\left\lVert \dot{\boldsymbol{u}}^{0} \right\rVert_{}^2.$$ Equations [\[eq:weak_u\]](#eq:weak_u){reference-type="eqref" reference="eq:weak_u"} and [\[eq:weak_m\]](#eq:weak_m){reference-type="eqref" reference="eq:weak_m"} are space-time variational formulations of [\[eq:newton\]](#eq:newton){reference-type="eqref" reference="eq:newton"} and [\[eq:llg\]](#eq:llg){reference-type="eqref" reference="eq:llg"}, respectively. The initial condition [\[eq:ic_ut\]](#eq:ic_ut){reference-type="eqref" reference="eq:ic_ut"} and the boundary conditions [\[eq:bc_u\_n\]](#eq:bc_u_n){reference-type="eqref" reference="eq:bc_u_n"} and [\[eq:bc_m\]](#eq:bc_m){reference-type="eqref" reference="eq:bc_m"} are imposed as natural boundary conditions in the variational formulations; The initial conditions [\[eq:ic_u\]](#eq:ic_u){reference-type="eqref" reference="eq:ic_u"} and [\[eq:ic_m\]](#eq:ic_m){reference-type="eqref" reference="eq:ic_m"} are imposed in the sense of traces in (iii); The Dirichlet boundary condition [\[eq:bc_u\_d\]](#eq:bc_u_d){reference-type="eqref" reference="eq:bc_u_d"} is imposed as essential boundary condition. Equation [\[eqn:EnergyLawDefinition\]](#eqn:EnergyLawDefinition){reference-type="eqref" reference="eqn:EnergyLawDefinition"} is the weak counterpart of the energy law [\[eq:energy_law\]](#eq:energy_law){reference-type="eqref" reference="eq:energy_law"} satisfied by strong solutions. **Remark 2**. Formula [\[eq:magnetostrain\]](#eq:magnetostrain){reference-type="eqref" reference="eq:magnetostrain"} is the general expression of the magnetostrain for anisotropic ferromagnets [@federico2019tensor] and covers the typical forms of the magnetostrain found in literature. These usually assume that the magnetostrain is *isochoric* [@hubert1998magnetic Section 3.2.6] (i.e. it has zero trace). Importantly, formula [\[eq:magnetostrain\]](#eq:magnetostrain){reference-type="eqref" reference="eq:magnetostrain"} covers the common *cubic* case, considered e.g. in [@james1998magnetostriction; @shu2004micromagnetic; @Mballa2014; @rj2021; @renuka2021solution] and given by $$\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}) = \frac{3}{2} \Bigg\{ \lambda_{100} \left( \boldsymbol{m}\otimes\boldsymbol{m}- \frac{I}{3}\right) + (\lambda_{111} - \lambda_{100}) \sum_{\substack{i,j=1 \\ i \neq j}}^3 (\boldsymbol{m}\cdot\boldsymbol{e}_i^\mathrm{c})(\boldsymbol{m}\cdot\boldsymbol{e}_j^\mathrm{c}) (\boldsymbol{e}_i^\mathrm{c} \otimes \boldsymbol{e}_j^\mathrm{c}) \Bigg\},$$ where $I \in \mathbb{R}^{3 \times 3}$ denotes the $3$-by-$3$ identity matrix, $\lambda_{100},\lambda_{111} \in \mathbb{R}$ are material constants, and $\{\boldsymbol{e}_1^\mathrm{c}, \boldsymbol{e}_2^\mathrm{c}, \boldsymbol{e}_3^\mathrm{c}\}$ denotes an orthonormal set yielding the crystal basis. When $\lambda_{100}=\lambda_{111}$, the latter reduces to the so-called *isotropic* case $$\label{eqn:magnetostrain_isotropic} \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}) = \frac{3}{2} \lambda_{\textnormal{100}} \left( \boldsymbol{m}\otimes\boldsymbol{m}- \frac{I}{3}\right),$$ considered e.g. in [@bhbvfs2014; @pwhcn2015; @dw2023]. For further details regarding specific crystal classes and their magnetostrain representation, we refer to [@federico2019tensor]. **Remark 3**. For the sake of simplicity (and since the focus of this work is on the design of a numerical method for the coupled system [\[eq:newton\]](#eq:newton){reference-type="eqref" reference="eq:newton"}--[\[eq:llg\]](#eq:llg){reference-type="eqref" reference="eq:llg"}), we neglect from the magnetic energy [\[eq:energy_mag\]](#eq:energy_mag){reference-type="eqref" reference="eq:energy_mag"} all lower-order contributions (magnetocrystalline anisotropy, Zeeman energy, magnetostatic energy, Dzyaloshinskii---Moriya interaction). However, we note that their numerical integration is well understood; see e.g. [@bffgpprs2014; @dpprs2019; @hpprss2019]. # Preliminaries {#sec:ingredients} In this section, we collect some notation and preliminary results that will be necessary to introduce and analyse the fully discrete algorithm we propose to approximate solutions to the initial boundary value problem [\[eq:newton\]](#eq:newton){reference-type="eqref" reference="eq:newton"}--[\[eq:ibc\]](#eq:ibc){reference-type="eqref" reference="eq:ibc"}. Hereafter, as customary in numerical analysis, given $A,B \in \mathbb{R}$, we shall write $A \lesssim B$ if there exists a constant $c>0$, clear from the context and always independent of the discretisation parameters, such that $A \le c \, B$. ## Time discretisation Let $0=t_{0}<t_{1}<\dots<t_{N}=T$ be a uniform partition of the time interval into $N$ uniform intervals with constant time-step size $k = T/N$, i.e. $t_i = ik$ for all $i=0,\dots,N$. Given values $\{\phi^i\}_{0 \le i \le N}$ and $\dot\phi^0$, we define the discrete time derivatives by $$\begin{aligned} \mathrm{d}_t\phi^i &:= \begin{cases} \dot\phi^0, & \text{if } i=0,\\ \displaystyle \frac{\phi^i - \phi^{i-1}}{k}, & \text{if } 1 \le i \le N, \end{cases} \\ \label{eq:second_derivative} \mathrm{d}_t^2 \phi^{i+1} &:= \frac{\mathrm{d}_t\phi^{i+1} - \mathrm{d}_t\phi^i}{k} = \begin{cases} \displaystyle \frac{\phi^1 - \phi^0 - k \dot\phi^0}{k^2}, & \text{if } i=0,\\ \displaystyle \frac{\phi^{i+1} - 2\phi^i+\phi^{i-1}}{k^2}, & \text{if } 1 \le 1 \le N-1. \end{cases}\end{aligned}$$ Moreover, we define the time reconstructions $\phi_{k}$, $\phi^-_{k}$, $\phi^+_{k}$, $\dot\phi_{k}$, $\dot\phi^-_{k}$, $\dot\phi^+_{k}$, defined, for all $0 \le i \le N-1$ and $t \in [t_i,t_{i+1})$, by [\[eq:reconstructions\]]{#eq:reconstructions label="eq:reconstructions"} $$\begin{gathered} \phi_{k}(t) := \frac{t-t_i}{k}\phi^{i+1} + \frac{t_{i+1} - t}{k}\phi^i, \quad \phi_{k}^-(t) := \phi^i, \quad \phi_{k}^+(t) := \phi^{i+1}, \\ \dot\phi_{k}(t) := \mathrm{d}_t\phi^i + (t - t_i)\mathrm{d}_t^2\phi^{i+1}, \quad \dot\phi_{k}^-(t) := \mathrm{d}_t\phi^i, \quad \dot\phi_{k}^+(t) := \mathrm{d}_t\phi^{i+1}.\end{gathered}$$ Note that $\partial_t \phi_k (t) = \dot\phi^+_{k}(t) = d_t \phi^{i+1}$ for all $t \in [t_i,t_{i+1})$. ## Space discretisation Let $\Omega$ be a polyhedral domain. Let $\{\mathcal{T}_h\}_{h>0}$ be a shape-regular family of meshes of $\Omega$ into tetrahedra, where $h=\max_{K\in\mathcal{T}_{h}}h_K$ denotes the mesh size of $\mathcal{T}_{h}$ and $h_K = \mathop{\mathrm{diam}}K$ for all $K \in \mathcal{T}_h$. We denote by $\mathcal{N}_{h}$ the set of nodes in the triangulation $\mathcal{T}_{h}$. For all $K \in \mathcal{T}_h$, we denote by $\mathcal{P}_{1}(K)$ the space of polynomials of degree at most 1 over $K$. We denote by $\mathcal{S}^1(\mathcal{T}_{h})$ the space of piecewise affine and globally continuous functions from $\Omega$ to $\mathbb{R}$, i.e. $$\mathcal{S}^1(\mathcal{T}_{h}) = \{\phi_{h}\in C(\overline{\Omega}):\phi_{h}\vert_{K}\in \mathcal{P}_{1}(K) \text{ for all } K\in\mathcal{T}_{h}\} \subset H^1(\Omega).$$ We denote by $\mathcal{I}_{h}:C(\overline{\Omega})\to \mathcal{S}^1(\mathcal{T}_{h})$ the nodal interpolant satisfying $\mathcal{I}_{h}[\phi](z) = \phi(z)$ for each $z\in\mathcal{N}_{h}$, where $\phi$ is a continuous function. Moreover, we consider the space $\mathcal{S}^{1}_{D}(\mathcal{T}_{h}) = \mathcal{S}^{1}(\mathcal{T}_{h})\cap H_{D}^{1}(\Omega)$, where homogeneous Dirichlet boundary conditions on $\Gamma_{D}$ are imposed explicitly. Since the unknowns of the problem in which we are interested are vector fields, we consider the vector-valued finite element space $\mathcal{S}^1(\mathcal{T}_{h})^3$ and use the same notation adopted in the scalar case to denote the vector-valued nodal interpolant $\mathcal{I}_{h}:\boldsymbol{C}(\overline{\Omega})\to \mathcal{S}^1(\mathcal{T}_{h})^3$. For all $0 \le i \le N$, the approximate displacement at time $t_i$, $\boldsymbol{u}_h^i \approx \boldsymbol{u}(t_i)$, will be sought in the finite element space $\mathcal{S}^1_D(\mathcal{T}_{h})^3$, whereas the approximate magnetisation, $\boldsymbol{m}_h^i \approx \boldsymbol{m}(t_i)$, will be sought in the set $$\label{eq:discrete_magnetisation} \boldsymbol{\mathcal{M}}_{h,\delta} = \big\{\boldsymbol{\phi}_{h}\in \mathcal{S}^1(\mathcal{T}_{h})^3: \left\lvert \boldsymbol{\phi}_{h}(z) \right\rvert\geq 1\text{ for all } z\in\mathcal{N}_{h} \text{ and } \left\lVert \mathcal{I}_{h}\left[|\boldsymbol{\phi}_h|^2\right] - 1 \right\rVert_{L^1(\Omega)}\leq \delta \big\}$$ for some $\delta>0$. Note that discrete magnetisations in $\boldsymbol{\mathcal{M}}_{h,\delta}$ generally do not satisfy the unit length constraint, not even at the vertices of the mesh, but the error is controlled in the $L^1$-sense by $\delta$. For the case $\delta=0$, we obtain the set $$\boldsymbol{\mathcal{M}}_{h,0} = \{\boldsymbol{\phi}_{h}\in \mathcal{S}^1(\mathcal{T}_{h})^3:\left\lvert \boldsymbol{\phi}_{h}(z) \right\rvert=1\text{ for all } z\in\mathcal{N}_{h}\},$$ in which the constraint holds at the vertices of the mesh. We define the nodal projection operator $\Pi_{h}:\boldsymbol{\mathcal{M}}_{h,\delta}\to\boldsymbol{\mathcal{M}}_{h,0}$ by $\Pi_{h}\boldsymbol{\phi}_{h}(z) = \boldsymbol{\phi}_{h}(z)/|\boldsymbol{\phi}_{h}(z)|$ for all $z\in\mathcal{N}_{h}$ and $\boldsymbol{\phi}_{h}\in\boldsymbol{\mathcal{M}}_{h,\delta}$. Another important property of solutions to the LLG equation is the orthogonality $\partial_t \boldsymbol{m}\cdot\boldsymbol{m}= 0$. To realise it at the discrete level, given an approximation $\boldsymbol{m}_h^i \approx \boldsymbol{m}(t_i)$ in $\boldsymbol{\mathcal{M}}_{h,\delta}$, we consider the discrete tangent space $$\boldsymbol{\mathcal{K}}_{h}[\boldsymbol{m}_{h}^i] = \{\boldsymbol{\psi}_{h}\in \mathcal{S}^1(\mathcal{T}_{h})^3: \boldsymbol{m}_{h}^i(z)\cdot\boldsymbol{\psi}_{h}(z) = 0 \text{ for all } z\in\mathcal{N}_{h}\},$$ where approximations $\boldsymbol{v}_h^i \approx \partial_t \boldsymbol{m}(t_i)$ will be sought. Note that the desired orthogonality property is imposed only at the vertices of the mesh. To conclude, we recall the definition of mass-lumped $L^2$-product $\langle \cdot,\cdot \rangle_{h}$, i.e. $$\label{eq:mass-lumping} \langle \boldsymbol{\psi},\boldsymbol{\phi} \rangle_{h} = \int_\Omega \mathcal{I}_h[\boldsymbol{\psi}\cdot \boldsymbol{\phi}] \quad \text{for all } \boldsymbol{\psi}, \boldsymbol{\phi}\in \boldsymbol{C}^0(\overline{\Omega}),$$ which is a scalar product on $\mathcal{S}^1(\mathcal{T}_h)^3$. # Algorithm and main results {#sec:main} In the following algorithm, we state the fully discrete numerical scheme we propose to approximate solutions to the initial boundary value problem [\[eq:newton\]](#eq:newton){reference-type="eqref" reference="eq:newton"}--[\[eq:ibc\]](#eq:ibc){reference-type="eqref" reference="eq:ibc"}. **Algorithm 4** (decoupled algorithm for the LLG equation with magnetostriction). [Discretisation parameters:]{.ul} Mesh size $h>0$, time-step size $k>0$, $\theta\in(1/2,1]$.\ [Input:]{.ul} Approximate initial conditions $\boldsymbol{m}_{h}^{0} \in \boldsymbol{\mathcal{M}}_{h,0}$, $\boldsymbol{u}_{h}^{0} \in \mathcal{S}^1_D(\mathcal{T}_h)^3$, $\dot\boldsymbol{u}_{h}^{0} \in \mathcal{S}^1(\mathcal{T}_h)^3$.\ [Loop:]{.ul} For all integers $0 \le i \le N-1$, iterate (i)--(iii): (i) Compute $\boldsymbol{v}_{h}^{i}\in\boldsymbol{\mathcal{K}}_{h}[\boldsymbol{m}_{h}^{i}]$ such that, for all $\boldsymbol{\phi}_{h}\in\boldsymbol{\mathcal{K}}_{h}[\boldsymbol{m}_{h}^{i}]$, it holds that $$\begin{gathered} \label{alg1:magnetisation_update} \alpha \langle \boldsymbol{v}_{h}^{i},\boldsymbol{\phi}_{h} \rangle_{h} + \langle \boldsymbol{m}_{h}^{i}\times \boldsymbol{v}_{h}^{i},\boldsymbol{\phi}_{h} \rangle_{h} + \theta k \langle \boldsymbol{\nabla}\boldsymbol{v}_{h}^{i},\boldsymbol{\nabla}\boldsymbol{\phi}_{h} \rangle_{} \\ = -\langle \boldsymbol{\nabla}\boldsymbol{m}_{h}^{i},\boldsymbol{\nabla}\boldsymbol{\phi}_{h} \rangle_{} + \langle \boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u}_{h}^{i},\Pi_h\boldsymbol{m}_{h}^{i}],\boldsymbol{\phi}_{h} \rangle_{}.\end{gathered}$$ (ii) Define $$\label{eq:alg_update_m} \boldsymbol{m}_{h}^{i+1} := \boldsymbol{m}_{h}^{i} + k \boldsymbol{v}_{h}^{i} \in \mathcal{S}^1(\mathcal{T}_h)^3.$$ (iii) Compute $\boldsymbol{u}_{h}^{i+1}\in\mathcal{S}^{1}_{D}(\mathcal{T}_{h})^3$ such that, for all $\boldsymbol{\psi}_{h}\in\mathcal{S}^{1}_{D}(\mathcal{T}_{h})^3$, it holds that $$\begin{gathered} \label{alg1:displacement_update} \langle \mathrm{d}_t^2 \boldsymbol{u}_{h}^{i+1},\boldsymbol{\psi}_h \rangle_{} + \langle \mathbb{C}:\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}),\boldsymbol{\varepsilon}(\boldsymbol{\psi}_{h}) \rangle_{} \\ = \langle \mathbb{C}:\boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1}),\boldsymbol{\varepsilon}(\boldsymbol{\psi}_{h}) \rangle_{} + \langle \boldsymbol{f},\boldsymbol{\psi}_{h} \rangle_{} + \langle \boldsymbol{g},\boldsymbol{\psi}_{h} \rangle_{\Gamma_{N}}.\end{gathered}$$ [Output:]{.ul} Approximations $\{ (\boldsymbol{u}_h^i,\boldsymbol{m}_h^i) \}_{0 \le i \le N}$. Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} resembles the decoupled algorithm proposed in [@bppr2013]. The discrete initial data $\boldsymbol{m}_{h}^{0} \in \boldsymbol{\mathcal{M}}_{h,0}$, $\boldsymbol{u}_{h}^{0} \in \mathcal{S}^1_D(\mathcal{T}_h)^3$ and $\dot\boldsymbol{u}_{h}^{0} \in \mathcal{S}^1(\mathcal{T}_h)^3$ denote suitable approximations of the initial conditions $\boldsymbol{m}^0$, $\boldsymbol{u}^0$ and $\dot\boldsymbol{u}^0$, respectively. For every time-step, given current approximations of the magnetisation and the displacement, we compute the new magnetisation first, and then the updated displacement using this. Specifically, to compute the new magnetisation, we use the tangent plane scheme [@aj2006; @bkp2008; @alouges2008a]: In step (i), given $\boldsymbol{u}_{h}^i$ and $\boldsymbol{m}_{h}^i$, we compute an approximation $\boldsymbol{v}_{h}^{i} \approx \partial_t \boldsymbol{m}(t_i)$ residing in the discrete tangent space $\boldsymbol{\mathcal{K}}_{h}[\boldsymbol{m}_{h}^{i}]$. The variational problem [\[alg1:magnetisation_update\]](#alg1:magnetisation_update){reference-type="eqref" reference="alg1:magnetisation_update"} solved by $\boldsymbol{v}_{h}^{i}$ is a discretisation of the equivalent formulation of the LLG equation $$\label{eq:equivalent_llg} \alpha \, \partial_t \boldsymbol{m}+ \boldsymbol{m}\times \partial_t \boldsymbol{m}= \boldsymbol{h}_{\mathrm{eff}}[\boldsymbol{u},\boldsymbol{m}] - (\boldsymbol{h}_{\mathrm{eff}}[\boldsymbol{u},\boldsymbol{m}]\cdot\boldsymbol{m})\boldsymbol{m},$$ which can be obtained from [\[eq:llg\]](#eq:llg){reference-type="eqref" reference="eq:llg"} via simple algebraic manipulations; cf. [@aj2006]. Looking at [\[alg1:magnetisation_update\]](#alg1:magnetisation_update){reference-type="eqref" reference="alg1:magnetisation_update"}, we note that the discrete variational formulation of the left-hand side of [\[eq:equivalent_llg\]](#eq:equivalent_llg){reference-type="eqref" reference="eq:equivalent_llg"} makes use of the mass-lumped $L^2$-product [\[eq:mass-lumping\]](#eq:mass-lumping){reference-type="eqref" reference="eq:mass-lumping"}. The two terms constituting the effective field $\boldsymbol{h}_{\mathrm{eff}}[\boldsymbol{u},\boldsymbol{m}]$ are treated differently: The exchange contribution is treated implicitly and therefore contributes to the left-hand side of [\[alg1:magnetisation_update\]](#alg1:magnetisation_update){reference-type="eqref" reference="alg1:magnetisation_update"}. The 'degree of implicitness' is modulated by the parameter $\theta \in (1/2,1]$. The elastic field is treated explicitly. In step (ii), with $\boldsymbol{v}_h^i$ at hand, we compute the new magnetisation $\boldsymbol{m}_{h}^{i+1}$ using a first-order time-stepping; cf. [\[eq:alg_update_m\]](#eq:alg_update_m){reference-type="eqref" reference="eq:alg_update_m"}. Differently from the seminal papers on the tangent plane schemes [@aj2006; @bkp2008; @alouges2008a] and from [@bppr2013], we follow the approach of [@bartels2016; @abert_spin-polarized_2014] and in our update we do not use the nodal projection. In particular, it holds that $\mathrm{d}_t\boldsymbol{m}_h^{i+1} = \boldsymbol{v}_h^i$. Finally, in step (iii), we compute the new displacement $\boldsymbol{u}_{h}^{i+1}$ using a standard finite element discretisation of [\[eq:newton\]](#eq:newton){reference-type="eqref" reference="eq:newton"}. We use the backward Euler method in time (the second time derivative in [\[eq:newton\]](#eq:newton){reference-type="eqref" reference="eq:newton"} is approximated using the different quotient [\[eq:second_derivative\]](#eq:second_derivative){reference-type="eqref" reference="eq:second_derivative"}). In Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"}, we apply the nodal projection to all approximate magnetisations arising from the elastic energy, i.e. in the elastic field on the right-hand side of [\[alg1:magnetisation_update\]](#alg1:magnetisation_update){reference-type="eqref" reference="alg1:magnetisation_update"} and in the magnetostrain term on the right-hand side of [\[alg1:displacement_update\]](#alg1:displacement_update){reference-type="eqref" reference="alg1:displacement_update"}, whereas the nodal projection is omitted from the magnetisation in the exchange field on the right-hand side of [\[alg1:magnetisation_update\]](#alg1:magnetisation_update){reference-type="eqref" reference="alg1:magnetisation_update"}, the cross product on the left hand side of [\[alg1:magnetisation_update\]](#alg1:magnetisation_update){reference-type="eqref" reference="alg1:magnetisation_update"}, and from the update [\[eq:alg_update_m\]](#eq:alg_update_m){reference-type="eqref" reference="eq:alg_update_m"}. Notably, despite the nonlinearity of the LLG equation and its nonlinear coupling with the conservation of momentum law, Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} is *fully linear* and only requires the solution of two linear systems per time-step. In the following proposition, we show the well-posedness of Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"}. The proof, based on standard arguments, is postponed to Section [6.1](#sec:proofs_wellposedness){reference-type="ref" reference="sec:proofs_wellposedness"}. **Proposition 5**. Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} is well defined for every $\theta\in(1/2,1]$, i.e. for every integer $0 \le i \le N-1$, there exists a unique $(\boldsymbol{v}_{h}^{i},\boldsymbol{m}_{h}^{i+1},\boldsymbol{u}_{h}^{i+1}) \in \boldsymbol{\mathcal{K}}_{h}[\boldsymbol{m}_{h}^{i}] \times \mathcal{S}^{1}(\mathcal{T}_{h})^3 \times \mathcal{S}^{1}_{D}(\mathcal{T}_{h})^3$ satisfying [\[alg1:magnetisation_update\]](#alg1:magnetisation_update){reference-type="eqref" reference="alg1:magnetisation_update"}--[\[alg1:displacement_update\]](#alg1:displacement_update){reference-type="eqref" reference="alg1:displacement_update"}. In the following proposition, we establish a discrete counterpart of the energy law [\[eq:energy_law\]](#eq:energy_law){reference-type="eqref" reference="eq:energy_law"} satisfied by smooth solutions of the continuous problem (see also [\[eqn:EnergyLawDefinition\]](#eqn:EnergyLawDefinition){reference-type="eqref" reference="eqn:EnergyLawDefinition"} for the corresponding property for weak solutions). Its proof is postponed to Section [6.2](#sec:proofs_energy_law){reference-type="ref" reference="sec:proofs_energy_law"}. **Proposition 6**. For every integer $0 \le i \le N-1$, the iterates of Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} satisfy the discrete energy law $$\label{eq:discrete_energy_law} \mathcal{E}[\boldsymbol{u}_h^{i+1},\boldsymbol{m}_h^{i+1}] + \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_h^{i+1} \right\rVert_{}^2 - \mathcal{E}[\boldsymbol{u}_h^i,\boldsymbol{m}_h^i] - \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_h^i \right\rVert_{}^2 = - \alpha k \left\lVert \boldsymbol{v}_h^i \right\rVert_{h}^2 - D_{h,k}^i - E_{h,k}^i,$$ where $D_{h,k}^i$ and $E_{h,k}^i$ are given by $$\begin{gathered} \label{eq:dissipation} D_{h,k}^i = k^2 (\theta-1/2) \left\lVert \boldsymbol{\nabla}\boldsymbol{v}_h^i \right\rVert_{}^2 + \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_h^{i+1} - \mathrm{d}_t\boldsymbol{u}_h^i \right\rVert_{}^2 \\ + \frac{1}{2} \left\lVert [\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1})-\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})] - [\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^i)-\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^i)] \right\rVert_{\mathbb{C}}^2 \ge 0\end{gathered}$$ and $$\label{eq:error_energy} \begin{split} E_{h,k}^i & = k^2 \langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{v}_{h}^{i}) \rangle_{} \\ & \quad + 2k \langle \mathbb{C}:\{[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})] - [\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i})] \},\mathbb{Z}(\boldsymbol{m}_{h}^{i} \otimes \boldsymbol{v}_{h}^{i}) \rangle_{} \\ & \quad + 2k\langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i})],\mathbb{Z}[(\boldsymbol{m}_h^i - \Pi_h \boldsymbol{m}_h^i) \otimes \boldsymbol{v}_{h}^{i}] \rangle_{}\\ & \quad + \langle \mathbb{C}:[\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{}. \end{split}$$ respectively. In [\[eq:dissipation\]](#eq:dissipation){reference-type="eqref" reference="eq:dissipation"}, we use the norm $\left\lVert \cdot \right\rVert_{\mathbb{C}}^2 = \langle \mathbb{C}:(\cdot),\cdot \rangle_{}$ for matrix-valued functions in $L^2(\Omega)^{3 \times 3}$. Thanks to our assumptions on $\mathbb{C}$ (cf. [\[eq:tensor_coercivity\]](#eq:tensor_coercivity){reference-type="eqref" reference="eq:tensor_coercivity"}), this norm is equivalent to the standard $L^2$-norm. Looking at the right-hand side of [\[eq:discrete_energy_law\]](#eq:discrete_energy_law){reference-type="eqref" reference="eq:discrete_energy_law"}, we see that the inherent $\alpha$-modulated energy dissipation of the model (cf. [\[eq:energy_law\]](#eq:energy_law){reference-type="eqref" reference="eq:energy_law"}) is spoiled by two terms: - the artificial damping $D_{h,k}^i$, arising from the implicit treatment of the exchange contribution of the effective field in [\[alg1:magnetisation_update\]](#alg1:magnetisation_update){reference-type="eqref" reference="alg1:magnetisation_update"} (the first term) and the use of the backward Euler method in [\[alg1:displacement_update\]](#alg1:displacement_update){reference-type="eqref" reference="alg1:displacement_update"} (the last two terms), - the error $E_{h,k}^i$ due to linearisation (the first term) decoupling (the second term), and use of the nodal projection to impose the unit length constraint on the magnetisations appearing in the elasticity terms (the third and fourth terms). **Remark 7**. Our argument to show Proposition [Proposition 6](#prop:energy){reference-type="ref" reference="prop:energy"} for Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} can be transferred to the algorithm of [@bppr2013], hence a by-product of our analysis is a discrete energy law for that algorithm. Due to the use of the nodal projection in [@bppr2013], the counterpart of [\[eq:discrete_energy_law\]](#eq:discrete_energy_law){reference-type="eqref" reference="eq:discrete_energy_law"} is only an inequality (not an identity), its proof requires to assume that the mesh is weakly acute, and the error term $E_{h,k}^i$ does not include the last two terms in [\[eq:error_energy\]](#eq:error_energy){reference-type="eqref" reference="eq:error_energy"}. Now, we discuss the stability and the convergence of Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"}. To this end, we consider the following convergence assumption on the approximate initial conditions: $$\label{eq:convergence_initial_data} \boldsymbol{u}_h^0 \to \boldsymbol{u}^0 \text{ in } \boldsymbol{H}^1(\Omega), \ \ \dot\boldsymbol{u}_h^0 \to \dot\boldsymbol{u}^0 \text{ in } \boldsymbol{L}^2(\Omega), \ \ \text{and} \ \ \boldsymbol{m}_h^0 \to \boldsymbol{m}^0 \text{ in } \boldsymbol{H}^1(\Omega), \ \ \text{ as } h \to 0.$$ Firstly, we can show that Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} is unconditionally stable and that the error in the unit length constraint can be controlled by the time-step size. **Proposition 8**. Suppose that assumption [\[eq:convergence_initial_data\]](#eq:convergence_initial_data){reference-type="eqref" reference="eq:convergence_initial_data"} is satisfied. There exists a threshold $k_0>0$ such that, if $k < k_0$, for every integer $1 \le j \le N$, the iterates of Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} satisfy $$\begin{gathered} \label{eq:stability} \left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^j \right\rVert_{}^2 + \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{j}) \right\rVert_{}^2 + \sum_{i=0}^{j-1}\left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} - \mathrm{d}_t\boldsymbol{u}_{h}^i \right\rVert_{}^2 + \sum_{i=0}^{j-1}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1})-\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \right\rVert_{}^2 \\ + \left\lVert \boldsymbol{m}_{h}^{j} \right\rVert_{\boldsymbol{H}^1(\Omega)}^2 + k \sum_{i=0}^{j-1} \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{}^2 + \left(\theta - \frac{1}{2}\right) k^2\sum_{i=0}^{j-1} \left\lVert \boldsymbol{\nabla}\boldsymbol{v}_{h}^{i} \right\rVert_{}^2 \leq C\end{gathered}$$ and $$\label{eq:general_constraint} \big\lVert \mathcal{I}_h\big[\left\lvert \boldsymbol{m}_h^j \right\rvert^2\big]-1 \big\rVert_{L^1(\Omega)} \le C k.$$ The threshold $k_0>0$ and the constant $C>0$ depend only on the shape-regularity parameter of $\mathcal{T}_h$, the problem data $\alpha$, $T$, $\Omega$, $\mathbb{C}$, $\mathbb{Z}$, $\boldsymbol{f}$ and $\boldsymbol{g}$, and the uniform bounds of the energy of the approximate initial data guaranteed by [\[eq:convergence_initial_data\]](#eq:convergence_initial_data){reference-type="eqref" reference="eq:convergence_initial_data"}. For the proof of the result, we refer to Section [6.3](#sec:proofs_stability){reference-type="ref" reference="sec:proofs_stability"}. Note that [\[eq:general_constraint\]](#eq:general_constraint){reference-type="eqref" reference="eq:general_constraint"} implies that, if the time-step size is sufficiently small, the approximate magnetisations generated by the algorithm belong to the set $\boldsymbol{\mathcal{M}}_{h,\delta}$ from [\[eq:discrete_magnetisation\]](#eq:discrete_magnetisation){reference-type="eqref" reference="eq:discrete_magnetisation"} with $\delta = Ck$. With the approximations generated by Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"}, we can construct the piecewise affine time reconstructions $\boldsymbol{u}_{hk} : (0,T) \to \mathcal{S}^1(\mathcal{T}_h)^3$ and $\boldsymbol{m}_{hk} : (0,T) \to \mathcal{S}^1(\mathcal{T}_h)^3$; see [\[eq:reconstructions\]](#eq:reconstructions){reference-type="eqref" reference="eq:reconstructions"}. In the following theorem, we show that the sequences $\{\boldsymbol{u}_{hk}\}$ and $\{\boldsymbol{m}_{hk}\}$ converge in a suitable sense towards a weak solution of the initial boundary value problem [\[eq:newton\]](#eq:newton){reference-type="eqref" reference="eq:newton"}--[\[eq:ibc\]](#eq:ibc){reference-type="eqref" reference="eq:ibc"} as $h$, $k$ go to $0$. Its proof is postponed to Sections [6.4](#sec:proofs_convergence){reference-type="ref" reference="sec:proofs_convergence"}--[6.5](#sec:proofs_energy_inequality){reference-type="ref" reference="sec:proofs_energy_inequality"}. **Theorem 9**. Suppose that assumption [\[eq:convergence_initial_data\]](#eq:convergence_initial_data){reference-type="eqref" reference="eq:convergence_initial_data"} is satisfied. (i) There exist a weak solution $(\boldsymbol{u},\boldsymbol{m})$ of [\[eq:newton\]](#eq:newton){reference-type="eqref" reference="eq:newton"}--[\[eq:ibc\]](#eq:ibc){reference-type="eqref" reference="eq:ibc"} in the sense of Definition [Definition 1](#def:weak){reference-type="ref" reference="def:weak"}(i)--(iii) and a (nonrelabeled) subsequence of $\{ (\boldsymbol{u}_{hk},\boldsymbol{m}_{hk}) \}$ which converges towards $(\boldsymbol{u},\boldsymbol{m})$ as $h,k \to 0$. In particular, as $h,k \to 0$, it holds that $\boldsymbol{u}_{hk} \overset{\ast}{\rightharpoonup}\boldsymbol{u}$ in $L^{\infty}(0,T; \boldsymbol{H}^1_D(\Omega))$, $\partial_t\boldsymbol{u}_{hk} \overset{\ast}{\rightharpoonup}\boldsymbol{u}$ in $L^{\infty}(0,T; \boldsymbol{L}^2(\Omega))$, $\boldsymbol{m}_{hk} \overset{\ast}{\rightharpoonup}\boldsymbol{m}$ in $L^{\infty}(0,T; \boldsymbol{H}^1(\Omega ;\mathbb{S}^2))$, and $\partial_t\boldsymbol{m}_{hk} \rightharpoonup\boldsymbol{m}$ in $\boldsymbol{L}^2(\Omega_T)$. (ii) If the discretisation parameters additionally satisfy the CFL condition $k=o(h^9)$, the weak solution from part (i) satisfies the energy inequality [\[eqn:EnergyLawDefinition\]](#eqn:EnergyLawDefinition){reference-type="eqref" reference="eqn:EnergyLawDefinition"} from Definition [Definition 1](#def:weak){reference-type="ref" reference="def:weak"}(iv). The proof of Theorem [Theorem 9](#thm:convergence){reference-type="ref" reference="thm:convergence"} is constructive and provides also a proof of existence of weak solutions. We recall that, due to the non-convex nature of the problem, uniqueness of weak solutions cannot be expected (cf. the explicit proof of non-uniqueness of weak solutions to the pure LLG equation in [@as1992]). Moreover, if $\theta \in [0,1/2]$, then Theorem [Theorem 9](#thm:convergence){reference-type="ref" reference="thm:convergence"} still holds, but with an additional CFL condition for part (i), i.e. $k=o(h^2)$ if $\theta \in [0,1/2)$ and $k=o(h)$ if $\theta = 1/2$; see [@alouges2008a]. **Remark 10**. The application of the nodal projection to all approximate magnetisations arising from the elastic energy is responsible for two of the error terms in [\[eq:error_energy\]](#eq:error_energy){reference-type="eqref" reference="eq:error_energy"} and for the severe CFL condition in Theorem [Theorem 9](#thm:convergence){reference-type="ref" reference="thm:convergence"}(ii) (cf. the analysis in Section [6.5](#sec:proofs_energy_inequality){reference-type="ref" reference="sec:proofs_energy_inequality"} below), so one would be tempted to completely remove it. However, we believe that a fully projection-free approach would not lead to an unconditionally stable method. In particular, the use of the nodal projection on the outermost magnetisation in the elastic field (cf. [\[eqn:MagnetostrictiveEffectiveField\]](#eqn:MagnetostrictiveEffectiveField){reference-type="eqref" reference="eqn:MagnetostrictiveEffectiveField"}) is non-negotiable as the total strain $\boldsymbol{\varepsilon}(\boldsymbol{u})$ is only in $\boldsymbol{L}^2(\Omega)$. For a stable method, it would be sufficient to take only one projection, not two, within the magnetostrain as this would yield the estimate $\left\lVert \mathbb{Z}: (\Pi_{h}\boldsymbol{m}_{h} \otimes \boldsymbol{m}_{h}) \right\rVert_{} \lesssim \left\lVert \boldsymbol{m}_{h} \right\rVert_{}$, which would allow for the stability estimate of Proposition [Proposition 8](#prop:stability){reference-type="ref" reference="prop:stability"}. However, we prefer not to use this approach as it would introduce some 'unnatural' non-symmetry. **Remark 11**. The proof of the energy inequality typically requires extra assumptions to be proven. In [@bffgpprs2014 Appendix A], in the case of the LLG equation (with full effective field), its proof requires higher regularity and stronger convergence assumptions on the applied field and general contribution terms. In [@dpprs2019 Theorem 3.2], in the case of the coupled system of the LLG equation and the eddy current equation, a similar situation arises with a CFL condition $k=o(h^{3/2})$. The very severe CFL condition in Theorem [Theorem 9](#thm:convergence){reference-type="ref" reference="thm:convergence"}(ii) is an artifact of the analysis and is due to the nonlinearity of the coupling and the fact that our proof requires explicit estimates of the error associated with the use of the nodal projection in the elastic terms. In particular, we need to estimate this error in a norm that is stronger than the $L^1$-norm, which leads to a reduced convergence rate with respect to the time-step size $k$ (see Lemma [Lemma 17](#lem:projection_error_Lp){reference-type="ref" reference="lem:projection_error_Lp"} below). This, combined with the fact that we need inverse estimates to obtain quantities we are able to control, leads to the CFL condition. For more details, we refer to the proof of the result in Section [6.5](#sec:proofs_energy_inequality){reference-type="ref" reference="sec:proofs_energy_inequality"} below. However, we stress that this restriction does not show up within the numerics (see, in particular, the experiment Section [5.3.3](#sec:numerics_cfl){reference-type="ref" reference="sec:numerics_cfl"}). # Numerical experiments {#sec:numerics} In this section, to show the applicability of our algorithm, we present a collection of numerical experiments. The implementation of Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} was written using the Netgen/NGSolve package [@netgen] using version 6.2.2302. The solution of the constrained linear system [\[alg1:magnetisation_update\]](#alg1:magnetisation_update){reference-type="eqref" reference="alg1:magnetisation_update"} in Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} is based on the null-space method given in [@ramage2013preconditioned; @kraus2019iterative]. The resulting system is solved using GMRES with an incomplete LU decomposition preconditioner, with the previous linear update $\boldsymbol{v}_{h}^{i-1}$ as a starting guess. The elastic equation [\[alg1:displacement_update\]](#alg1:displacement_update){reference-type="eqref" reference="alg1:displacement_update"} is solved using a Jacobi preconditioned conjugate gradient method. All computations were made on an i5-9500 CPU with 16GB of installed memory. ## Material parameters In the upcoming numerical experiments, we use material parameters estimated for (which we shall call FeCoSiB) from [@dw2022]. For the mass density and the Gilbert damping parameter (needed in our model, but not in [@dw2022]), we take the values used in [@kohmoto1985mass] and [@hu2023temperature], respectively. The resulting exchange length is $\ell_{\textnormal{ex}}= \sqrt{2A/(\mu_0 M_s^2)} \approx 3\cdot 10^{-9}$ m. The stiffness tensor $\mathbb{C}$ is assumed to be isotropic and acts on symmetric matrices $\boldsymbol{\varepsilon}$ (the only type required) as $$\mathbb{C}:\boldsymbol{\varepsilon}= 2\mu \, \boldsymbol{\varepsilon}+ \lambda \mathop{\mathrm{tr}}(\boldsymbol{\varepsilon}) \,I,$$ where $\mu$ and $\lambda$ are referred to as Lamé constants (for FeCoSiB after non-dimensionalisation we have $\mu\approx 6.89$ and $\lambda \approx 21.96$). For the magnetostrain, we consider the expression in [\[eqn:magnetostrain_isotropic\]](#eqn:magnetostrain_isotropic){reference-type="eqref" reference="eqn:magnetostrain_isotropic"}. In some experiments, the magnetic energy [\[eq:energy_mag\]](#eq:energy_mag){reference-type="eqref" reference="eq:energy_mag"} will be supplemented with the term $-\langle \boldsymbol{h}_{\mathrm{ext}},\boldsymbol{m} \rangle_{}$ (Zeeman energy), modelling the interaction of the magnetisation with an applied external field $\boldsymbol{h}_{\mathrm{ext}}$. For the sake of reproducibility, the values used are reported in Table [1](#tab:valuetable){reference-type="ref" reference="tab:valuetable"} (we refer to Appendix [8](#sec:physics){reference-type="ref" reference="sec:physics"} for the relationship between the fully dimensional model and the dimensionless setting of this paper). Symbol Name Value ----------------- ---------------------------- --------------------------------------- $A$ Exchange constant $1.5\cdot 10^{-11}$ J m^-1^ $\alpha$ Gilbert damping parameter $0.005$ $\gamma$ Gyromagnetic ratio $1.761 \cdot 10^{11}$ rad s^-1^ T^-1^ $\mu_0$ Permeability of free space $1.25663706 \cdot 10^{-6}$ $M_s$ Saturation magnetisation $1.5 \cdot 10^{6}$ A m^-1^ $\lambda_{100}$ Saturation magnetostrain $30 \cdot 10^{-6}$ $\rho$ Density $7900$ kg m^-3^ $g$ Gravitational acceleration $9.81$ m s^-2^ $\mu$ First Lamé constant $172$ GPa $\lambda$ Second Lamé constant $54$ GPa : Estimated material parameters for FeCoSiB taken from [@dw2022; @kohmoto1985mass; @hu2023temperature]. ## Magnetoelastic coupling {#sec: experiments_physics} In this section, we present two numerical experiments aimed at showcasing the capability of Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} to simulate physical processes involving magnetoelastic materials. The simulation object is a bar of FeCoSiB, clamped at one end ($y=0$ plane), shown in Figure [1](#fig:xy_box_schematic){reference-type="ref" reference="fig:xy_box_schematic"}. The bar has a physical length of $20\ell_{\textnormal{ex}}$ and width/height of $6\ell_{\textnormal{ex}}$. The maximum mesh size is $h_{\max} \approx 0.9 \ell_{\textnormal{ex}}$ (thereby being below the exchange length). The initial magnetisation is uniformly in the $x$-direction $\boldsymbol{m}_{h}^{0} = (1,0,0)$, whereas we set zero initial displacement $\boldsymbol{u}_{h}^{0} = \boldsymbol{0}$ with zero initial velocity $\dot{\boldsymbol{u}}_{h}^{0} = \boldsymbol{0}$. Gravity is enabled and implemented as a volume force $\boldsymbol{f}= (0,0,-g)$, with a value of $-g = -2.97 \cdot 10^{-14}$ after non-dimensionalisation. If enabled, tractions (represented by a surface force $\boldsymbol{g}$ applied on $\Gamma_N$) and applied external fields $\boldsymbol{h}_{\mathrm{ext}}$ are applied along the $+y$ direction. Simulations are run for $1$ ns, using time-steps of size $2\cdot 10^{-12}$ s. This corresponds to a non-dimensional time length of $T \approx 330$ and time-step $k \approx 0.66$. {reference-type="ref" reference="sec: experiments_physics"}: View from above of the FeCoSiB bar of dimensions $(20\ell_{\textnormal{ex}},6\ell_{\textnormal{ex}},6\ell_{\textnormal{ex}})$.](Schematics/xy_box_schematic.pdf){#fig:xy_box_schematic height="2.5cm"} ### Direct magnetostrictive effect {#sec:applied_magnetic_field} In this experiment, we show that changes in the magnetisation yield changes in the mechanical state of the body. To this end, we neglect traction and apply a uniform applied external field $\boldsymbol{h}_{\mathrm{ext}}$ along the $+y$ direction with low values of $0,1\cdot 10^{-4},3\cdot 10^{-4},5\cdot 10^{-4},7\cdot 10^{-4}$, which corresponds to fields of strength $0$, $0.2$, $0.6$, $0.9$, $1.3$ mT. The fields are weak so that the dynamics is not too fast. \ \ We observe the magnetisation aligning with the applied external field as expected through a precession, yielding an effect on the displacement. The coupling is clearly visible in Figure [\[fig:appliedfieldfigure\]](#fig:appliedfieldfigure){reference-type="ref" reference="fig:appliedfieldfigure"}, where we plot the time evolution of the average magnetisation and displacement components, e.g. $\langle u_{x} \rangle = (1/|\Omega|)\int_{\Omega}u_{x}$. The applied field is pointing in the $y$-direction, so the $y$ and $z$ components begin to increase in magnitude as seen in Figures [\[zeeman_avg_y\_mag\]](#zeeman_avg_y_mag){reference-type="ref" reference="zeeman_avg_y_mag"} and [\[zeeman_avg_z\_mag\]](#zeeman_avg_z_mag){reference-type="ref" reference="zeeman_avg_z_mag"}, taking from the $x$ component. The displacement on the other hand mirrors the magnetisation in the $y$ and $z$ components, with the $x$ component increasing due to magnetostriction, and then changing slowly as the magnetisation changes. Moreover, we see that, with stronger applied magnetic fields, the average magnetisation in the $y$ direction increases, displacing the body in the same direction. In Figure [\[fig:totalenergyzeeman\]](#fig:totalenergyzeeman){reference-type="ref" reference="fig:totalenergyzeeman"}, we plot the time evolution of the energy for all considered applied external fields. For greater strength applied fields, the energy reaches a lower value at later times. Importantly, we always see the energy decreasing. ### Inverse magnetostrictive effect {#sec:applied_traction} In this experiment, we show that changes in the mechanical state of the body yield changes in the magnetisation. To this end, we disable the Zeeman field and apply a traction on the $y=20$ plane in the $+y$ direction. Specifically, we consider a surface force of the form $\boldsymbol{g} = (0,b,0)$ for $b \in \{0,1.28\cdot10^{-9},3.19\cdot10^{-9},6.38\cdot10^{-9},1.28\cdot10^{-8}\}$, which corresponds to forces of strength $0$, $10$, $25$, $50$, $100$ N m^-2^. \ \ The time evolution of the average displacement and magnetisation components is shown in Figure [\[fig:tractionfigure\]](#fig:tractionfigure){reference-type="ref" reference="fig:tractionfigure"}. When more traction is applied, the average displacement in the $y$ direction increases. The $z$ component of the magnetisation in Figure [\[traction_avg_z\_mag\]](#traction_avg_z_mag){reference-type="ref" reference="traction_avg_z_mag"} is the most interesting, as it decreases more strongly due to stronger tractions. ## Properties of Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} {#properties-of-algorithm-algorithm} In this section, we present three experiments to numerically investigate the properties of Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"}. For all of them, the computational domain will be a cube with edge length equal to $6\ell_{\textnormal{ex}}$. ### $\theta$-dependence {#sec:theta_dependence} In this experiment, we investigate the effect on numerical simulations of the parameter $\theta \in (1/2,1]$, which controls the 'degree of implicitness' in the treatment of the exchange contribution in [\[alg1:magnetisation_update\]](#alg1:magnetisation_update){reference-type="eqref" reference="alg1:magnetisation_update"}. We use material parameters for FeCoSiB (cf. Table [1](#tab:valuetable){reference-type="ref" reference="tab:valuetable"}) except for the Gilbert damping parameter, for which we use the smaller value $\alpha=0.001$. The initial condition for the magnetisation is a 'hot' magnetic state, i.e. the values at the vertices of the mesh (which in this experiment has mesh size $h_{\max}\approx 3 \ell_{\textnormal{ex}}$) are assigned randomly to the magnetisation before being normalised. The displacement and its time derivative are initialised by zero. We run the simulation for $1\cdot 10^{-11}$ s using a time step size of $1\cdot 10^{-15}$ s and different values of $\theta \in \{ 0.50000005 , 0.505 , 0.6 , 0.7 , 0.8 , 0.9 , 1 \}$. The energy-decreasing behaviour can be seen in Figure [\[fig:thetatesting\]](#fig:thetatesting){reference-type="ref" reference="fig:thetatesting"}, with considerably more energy loss associated with greater $\theta$ values. So changing the $\theta$-implicitness parameter away from $1/2$ can yield considerable amounts of artificial numerical damping, which can be particularly bad in certain situations (e.g. in the case of long-time simulations). ### Unit length constraint violation {#sec:numerics_constraint} An essential property of the LLG equation at constant temperature is the unit length constraint on the magnetisation. Hence, an essential feature of any approximation algorithm must be the capability to achieve the unit length constraint. For Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"}, this property is the subject of Proposition [Proposition 8](#prop:stability){reference-type="ref" reference="prop:stability"}, particularly [\[eq:general_constraint\]](#eq:general_constraint){reference-type="eqref" reference="eq:general_constraint"}, i.e. $$\big\lVert \mathcal{I}_h\big[\left\lvert \boldsymbol{m}_h^j \right\rvert^2\big]-1 \big\rVert_{L^1(\Omega)} \le C k,$$ which shows that the unit length constraint is violated at most linearly in time (if measured in the $L^1$-norm). To see this numerically, we again consider a hot magnet as in Section [5.3.1](#sec:theta_dependence){reference-type="ref" reference="sec:theta_dependence"}, a particularly bad case with plenty of rotation by the magnetisation (note that the constant $C>0$ in [\[eq:general_constraint\]](#eq:general_constraint){reference-type="eqref" reference="eq:general_constraint"} depends, among other things, upon the energy of the initial magnetisation and is large for a random configuration), and use various time-steps with $\theta = 0.50000005$. In Figure [\[fig:ConstraintViolation\]](#fig:ConstraintViolation){reference-type="ref" reference="fig:ConstraintViolation"}, we plot the constraint violation (measured as the left-hand side of [\[eq:general_constraint\]](#eq:general_constraint){reference-type="eqref" reference="eq:general_constraint"}) at the finale iterate $\boldsymbol{m}_{h}^{N}$ of Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} against the time-step size $k$. We observe that the error decays linearly in $k$ as predicted by [\[eq:general_constraint\]](#eq:general_constraint){reference-type="eqref" reference="eq:general_constraint"}. The constraint violation is of the order $10^2$ for $k$ on the order of $10^{-3/2}$ due to the hot initial state, as the magnetisation at a node may need to rotate several times. Note that these simulations are run for only $0.01$ ns as we are only interested in verifying the constraint violation inequalities. In Figure [\[fig:Linfty\]](#fig:Linfty){reference-type="ref" reference="fig:Linfty"}, we plot the $L^\infty$-norm of the magnetisation at the final iterate against the time-step size $k$. We note that while we can control the integral violation with [\[eq:general_constraint\]](#eq:general_constraint){reference-type="eqref" reference="eq:general_constraint"}, in our projection-free algorithm we cannot directly control the maximum norm $\left\lVert \boldsymbol{m}_{h}^{j} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}$, which with a projection would be $1$ for each $j$. We see that the nodal maximum numerically tends to $1$ as desired, but the decay is not linear. Using similar methods to those to prove [\[eq:general_constraint\]](#eq:general_constraint){reference-type="eqref" reference="eq:general_constraint"} (see Lemma [Lemma 13](#lem:Magnetisation_Estimate){reference-type="ref" reference="lem:Magnetisation_Estimate"} below) and classical inverse estimates [@bartels2015 Lemma 3.5], one can show that $$\begin{split} \left\lVert \boldsymbol{m}_{h}^{j} \right\rVert_{\boldsymbol{L}^{\infty}}^2 - 1 &= \max_{z\in\mathcal{N}_{h}}|\boldsymbol{m}_{h}^{j}(z)|^2 - 1 \leq k^2 \sum_{i=0}^{j-1}\max_{z\in\mathcal{N}_{h}}|\boldsymbol{v}_{h}^{i}(z)|^2\\ &= k^2 \sum_{i=0}^{j-1}\left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 \lesssim h_{\min}^{-3}k^2 \sum_{i=0}^{j-1}\left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{\boldsymbol{L}^{2}(\Omega)}^2 \lesssim h_{\min}^{-3}k, \end{split}$$ thus the desired convergence $\left\lVert \boldsymbol{m}_{h}^{j} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)} \to 1$ as $h,k\to 0$ can be obtained assuming the CFL condition $k = o(h^3)$. ### Energy law robustness {#sec:numerics_cfl} In this experiment, we investigate the robustness of the evolution of the energy of the approximations generated by Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} with respect to the discretisation parameters. We consider a similar setup to the one used in Section [5.3.1](#sec:theta_dependence){reference-type="ref" reference="sec:theta_dependence"}. Specifically, we keep $\theta = 0.50000005$ and $\alpha=0.001$, but we add a Zeeman field $\boldsymbol{h}_{\mathrm{ext}} = (0.001, 0, 0)=(1.9, 0, 0)mT$ to encourage the system to approach the same final state. To give consistency between mesh refinements we change from a purely random initial state to the following initial condition for the magnetisation, $$\boldsymbol{m}^{0}(x,y,z) = \frac{1}{\sqrt{5}}(2,\sin(x+y+z),\cos(x+y+z)) \quad \text{for all } (x,y,z)\in\Omega.$$ It is easily shown that the initial condition satisfies $\left\lVert \boldsymbol{\nabla}\boldsymbol{m}^{0} \right\rVert_{}^2/2 = 64.8$ and $\left\lvert \boldsymbol{m}^0 \right\rvert=1$ in $\Omega$. NGSolve interpolates the initial condition onto the mesh via an Oswald-type interpolation [@oswald1993], applying an $L^2$-projection and then averaging for conformity, thus to enforce the condition $\boldsymbol{m}_{h}^{0}\in\boldsymbol{\mathcal{M}}_{h,0}$ we apply the nodal projection to the result of this interpolation. We then ran the simulation for $T \approx 3.32$ with combinations of $k=0.01$, $0.005$, $0.0025$, $0.00125$, $0.000625$ as time-step size and $h = 1.59$, $1.09$, $0.84$, $0.45$ as mesh size. As can be observed in Figure [\[fig:CFLGraph\]](#fig:CFLGraph){reference-type="ref" reference="fig:CFLGraph"}, the energy decay (and thus stability of the algorithm) occurs for all mesh sizes and time-steps. The initial energy is different for each due to the differing underlying mesh, and the interpolation process mentioned above (which is also different for each mesh), however the initial energies approach the actual energy. The different energy progressions are clustered into the four groups with similar energy decay when the time-step is the same. When the time-step size is smaller, the energy decay is slower, likely due to the error term in the discrete energy law (cf. the term $E_{h,k}^i$ in [\[eq:discrete_energy_law\]](#eq:discrete_energy_law){reference-type="eqref" reference="eq:discrete_energy_law"}). With no error term present, the dissipation would always reduce with lower time-step sizes. These results show that the algorithm behaves energetically well for all combinations of mesh and time-step size considered, including the worst case scenario for a CFL condition (when the finest mesh with $h=0.45$ and the largest time-step size $k=0.01$ are used). Clearly, this is not a mathematical proof that the restrictive CFL condition we need to show Theorem [Theorem 9](#thm:convergence){reference-type="ref" reference="thm:convergence"}(ii) is not needed, however our numerical experiments seem to corroborate this claim. # Proofs {#sec:proofs} In this section, we collect the proofs of the results presented in Section [4](#sec:main){reference-type="ref" reference="sec:main"}. For the convenience of the reader, we start with recalling some well-known results that will be used multiple times throughout the upcoming analysis. The norm $\left\lVert \cdot \right\rVert_{h}$ induced by the mass-lumped $L^2$-product [\[eq:mass-lumping\]](#eq:mass-lumping){reference-type="eqref" reference="eq:mass-lumping"} satisfies the norm equivalence $$\label{eq:h-scalar-product_equivalence} \left\lVert \boldsymbol{\phi}_h \right\rVert_{} \leq \left\lVert \boldsymbol{\phi}_h \right\rVert_{h} \leq \sqrt{5} \, \left\lVert \boldsymbol{\phi}_h \right\rVert_{} \quad \text{for all } \boldsymbol{\phi}_h \in \mathcal{S}^1(\mathcal{T}_h)^3,$$ and we have the error estimate $$\label{eq:h-scalar-product} \lvert \langle \boldsymbol{\phi}_h,\boldsymbol{\psi}_h \rangle_{} - \langle \boldsymbol{\phi}_h,\boldsymbol{\psi}_h \rangle_{h} \rvert \le C h^2 \left\lVert \boldsymbol{\nabla}\boldsymbol{\phi}_h \right\rVert_{} \left\lVert \boldsymbol{\nabla}\boldsymbol{\psi}_h \right\rVert_{} \quad \text{for all } \boldsymbol{\phi}_h, \boldsymbol{\psi}_h \in \mathcal{S}^1(\mathcal{T}_h)^3,$$ (cf. [@bartels2015 Lemma 3.9]). For all $K \in \mathcal{T}_h$ and $1 \le r,p \le \infty$, we have the local inverse estimate $$\label{eq:inverse_estimate} \left\lVert \boldsymbol{\phi}_h \right\rVert_{\boldsymbol{L}^p(K)} \le C h_K^{3(r-p)/(pr)} \left\lVert \boldsymbol{\phi}_h \right\rVert_{\boldsymbol{L}^r(K)} \quad \text{for all } \boldsymbol{\phi}_h \in \mathcal{S}^1(\mathcal{T}_h)^3$$ (see, e.g. [@bartels2015 Lemma 3.5]). For all $1 \le p < \infty$, the $L^p$-norm of functions in $\mathcal{S}^1(\mathcal{T}_h)^3$ is equivalent with the $\ell^p$-norm of the vector collecting their nodal values, weighted by the local mesh size, i.e. $$\label{eq:Lp_equivalence} C^{-1} \left\lVert \boldsymbol{\phi}_h \right\rVert_{\boldsymbol{L}^p(\Omega)} \leq \left( \sum_{z \in \mathcal{N}_h} h_z^3 \left\lvert \boldsymbol{\phi}_h(z) \right\rvert^p\right)^{1/p} \leq C \left\lVert \boldsymbol{\phi}_h \right\rVert_{\boldsymbol{L}^p(\Omega)} \quad \text{for all } \boldsymbol{\phi}_h \in \mathcal{S}^1(\mathcal{T}_h)^3,$$ where $h_z>0$ denotes the diameter of the node patch of $z \in \mathcal{N}_h$ (cf. [@bartels2015 Lemma 3.4]). If $p = \infty$, we have that $$\left\lVert \boldsymbol{\phi}_h \right\rVert_{\boldsymbol{L}^\infty(\Omega)} = \max_{z \in \mathcal{N}_h} \left\lvert \boldsymbol{\phi}_h(z) \right\rvert \quad \text{for all } \boldsymbol{\phi}_h \in \mathcal{S}^1(\mathcal{T}_h)^3.$$ Finally, the nodal projection is $H^1$-stable, i.e, it holds that $$\label{eq:projection_stability} \left\lVert \boldsymbol{\nabla}\Pi_h\phi_h \right\rVert_{} \le C \left\lVert \boldsymbol{\nabla}\phi_h \right\rVert_{} \quad \text{for all } \boldsymbol{\phi}\in \mathcal{S}^1(\mathcal{T}_h)^3 \text{ satisfying } \left\lvert \boldsymbol{\phi}(z) \right\rvert \ge 1 \text{ for all } z \in \mathcal{N}_h;$$ see [@bartels2016 Lemma 2.2]. We recall that [\[eq:projection_stability\]](#eq:projection_stability){reference-type="eqref" reference="eq:projection_stability"} holds with $C=1$ if all non-diagonal entries of the stiffness matrix are non-positive (cf. [@bartels2015 Proposition 3.2]). This assumption, which is satisfied under very restrictive geometric conditions on the mesh in three dimensions, is *not* required by the upcoming analysis. In all these inequalities, the constant $C>0$ (not the same at each occurrence) depends only on the shape-regularity of $\mathcal{T}_h$. ## Well-posedness {#sec:proofs_wellposedness} We start by showing an estimate of the $L^2$-norm of the discrete elastic field. **Lemma 12**. For all $\boldsymbol{u}_h,\boldsymbol{m}_h \in \mathcal{S}^1(\mathcal{T}_h)^3$ with $\left\lvert \boldsymbol{m}_h(z) \right\rvert\ge 1$ for all $z \in \mathcal{N}_h$, it holds that $$\label{eq:boundedness_elastic_field} \left\lVert \boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u}_{h},\Pi_h\boldsymbol{m}_{h}] \right\rVert_{}^2 \le 8 \left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 \left\lVert \mathbb{C} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 \left(\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}) \right\rVert_{}^2 + \left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 |\Omega| \right).$$ *Proof.* Using the expression of the discrete elastic field, we have $$\begin{split} & \left\lVert \boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u}_{h},\Pi_h\boldsymbol{m}_{h}] \right\rVert_{}^2 \\ & \quad \stackrel{\eqref{eqn:MagnetostrictiveEffectiveField}}{=} \left\lVert 2 (\mathbb{Z}^{\top}: \{ \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h})]\} ) \Pi_{h} \boldsymbol{m}_{h} \right\rVert_{}^2 \\ & \quad \stackrel{\phantom{\eqref{eqn:MagnetostrictiveEffectiveField}}}{\le} 4 \left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 \left\lVert \mathbb{C} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}) - \mathbb{Z}:(\Pi_h\boldsymbol{m}_{h} \otimes \Pi_h\boldsymbol{m}_{h}) \right\rVert_{}^2 \left\lVert \Pi_{h} \boldsymbol{m}_{h} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 \\ & \quad \stackrel{\phantom{\eqref{eqn:MagnetostrictiveEffectiveField}}}{\le} 8 \left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 \left\lVert \mathbb{C} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 \left(\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}) \right\rVert_{}^2 + \left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 |\Omega| \right), \end{split}$$ where we have used the boundedness of the fourth-order tensors and $\Pi_{h} \boldsymbol{m}_{h}$. ◻ We can now show the well-posedness of Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"}. *Proof of Proposition [Proposition 5](#prop:wellposedness){reference-type="ref" reference="prop:wellposedness"}.* The proof is basically identical to the one given in [@bppr2013] for the algorithm proposed therein. We restate it here including other terms. For the magnetisation term, define the family of bilinear form $a^{i}_{1}(\cdot,\cdot):\boldsymbol{\mathcal{K}}_{h}[\boldsymbol{m}_{h}^{i}]\times \boldsymbol{\mathcal{K}}_{h}[\boldsymbol{m}_{h}^{i}]\to\mathbb{R}$ for $i=0,\dots,N-1$, by $$a^{i}_{1}(\boldsymbol{\phi}_h,\boldsymbol{\psi}_h) := \alpha\langle \boldsymbol{\phi}_h,\boldsymbol{\psi}_h \rangle_{h} + \theta k \langle \boldsymbol{\nabla}\boldsymbol{\phi}_h,\boldsymbol{\nabla}\boldsymbol{\psi}_h \rangle_{} + \langle \boldsymbol{m}_{h}^{i}\times \boldsymbol{\phi}_h,\boldsymbol{\psi}_h \rangle_{}$$ and the family of linear (and bounded by Lemma [Lemma 12](#lem:boundedness_elastic_field){reference-type="ref" reference="lem:boundedness_elastic_field"}) functionals $L_{1}^{i}$ for $i=0,\ldots,N-1$ by $$L_{1}^{i}(\boldsymbol{\phi}_h) := -\langle \boldsymbol{\nabla}\boldsymbol{m}_{h}^{i},\boldsymbol{\nabla}\boldsymbol{\phi}_h \rangle_{} + \langle \boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u}_{h}^{i},\Pi_h\boldsymbol{m}_{h}^{i}],\boldsymbol{\phi}_h \rangle_{}.$$ Then [\[alg1:magnetisation_update\]](#alg1:magnetisation_update){reference-type="eqref" reference="alg1:magnetisation_update"} can be rewritten as $a^{i}_{1}(\boldsymbol{v}_h^i,\boldsymbol{\psi}_h) = L_{1}^{i}(\boldsymbol{\psi}_h)$ for all $\boldsymbol{\psi}_h\in\boldsymbol{\mathcal{K}}_{h}[\boldsymbol{m}_{h}^{i}]$. We can see that $a_{1}^{i}(\cdot,\cdot)$ is positive definite (in $\boldsymbol{L}^2(\Omega)$ and $\boldsymbol{H}^1(\Omega)$), as letting $\boldsymbol{\phi}_h=\boldsymbol{\psi}_h$ eliminates the final term, leaving a combination of the $L^2$-norm and $H^1$-seminorm. It follows by the finite dimensionality that [\[alg1:magnetisation_update\]](#alg1:magnetisation_update){reference-type="eqref" reference="alg1:magnetisation_update"} has a unique solution $\boldsymbol{v}_{h}^{i}\in \boldsymbol{\mathcal{K}}_{h}[\boldsymbol{m}_{h}^{i}]$. For the displacement term, define the bilinear form $a_{2}:\mathcal{S}^{1}_{D}(\mathcal{T}_{h})^3\times \mathcal{S}^{1}_{D}(\mathcal{T}_{h})^3\to\mathbb{R}$ by $$a_{2}(\boldsymbol{\phi}_h,\boldsymbol{\psi}_h) := \langle \boldsymbol{\phi}_h,\boldsymbol{\psi}_h \rangle_{} + k^2\langle \mathbb{C}:\boldsymbol{\varepsilon}(\boldsymbol{\phi}_h),\boldsymbol{\varepsilon}(\boldsymbol{\psi}_h) \rangle_{}.$$ As $\mathbb{C}$ is positive definite by assumption, applying Korn's inequality (see, e.g. [@brenner2008mathematical  Theorem 11.2.6]) yields positive definiteness of $a_{2}(\cdot,\cdot)$ in $\boldsymbol{H}^1(\Omega)$. Furthermore, defining the family of linear functionals $$L^{i}_{2}(\boldsymbol{\psi}_h) := k^2 \langle \mathbb{C}:\boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1}),\boldsymbol{\varepsilon}(\boldsymbol{\psi}_h) \rangle_{} + k\langle \mathrm{d}_t\boldsymbol{u}_{h}^{i},\boldsymbol{\psi}_h \rangle_{} + \langle \boldsymbol{u}_{h}^{i},\boldsymbol{\psi}_h \rangle_{} + k^2 \langle \boldsymbol{f},\boldsymbol{\psi}_h \rangle_{} + k^2 \langle \boldsymbol{g},\boldsymbol{\psi}_h \rangle_{\Gamma_{N}},$$ we have that [\[alg1:displacement_update\]](#alg1:displacement_update){reference-type="eqref" reference="alg1:displacement_update"} is equivalent to $a_{2}(\boldsymbol{u}_h^{i+1},\boldsymbol{\psi}_h) = L^{i}_{2}(\boldsymbol{\psi}_h)$ for all $\boldsymbol{\psi}_h\in\mathcal{S}^{1}_{D}(\mathcal{T}_{h})^3$, for each $i=0,\ldots,N-1$. Again exploiting the finite dimension, we have existence and uniqueness of a solution $\boldsymbol{u}_{h}^{i+1} \in \mathcal{S}^{1}_{D}(\mathcal{T}_{h})^3$ to [\[alg1:displacement_update\]](#alg1:displacement_update){reference-type="eqref" reference="alg1:displacement_update"}. ◻ ## Discrete energy law {#sec:proofs_energy_law} We now prove the discrete energy law satisfied by the iterates of Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"}. *Proof of Proposition [Proposition 6](#prop:energy){reference-type="ref" reference="prop:energy"}.* Let $0 \le i \le N-1$ be an arbitrary integer. Choosing the test function $\boldsymbol{\phi}_h = \boldsymbol{v}_{h}^{i} \in \boldsymbol{\mathcal{K}}_h[\boldsymbol{m}_{h}^{i}]$ in [\[alg1:magnetisation_update\]](#alg1:magnetisation_update){reference-type="eqref" reference="alg1:magnetisation_update"}, we obtain $$\alpha \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{h}^2 + \theta k\left\lVert \boldsymbol{\nabla}\boldsymbol{v}_{h}^{i} \right\rVert_{}^2 = - \langle \boldsymbol{\nabla}\boldsymbol{m}_{h}^{i},\boldsymbol{\nabla}\boldsymbol{v}_{h}^{i} \rangle_{} + \langle \boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u}_{h}^{i},\Pi_h\boldsymbol{m}_{h}^{i}],\boldsymbol{v}_{h}^{i} \rangle_{}.$$ Moreover, we have $$\frac{1}{2}\left\lVert \boldsymbol{\nabla}\boldsymbol{m}_{h}^{i+1} \right\rVert_{}^2 = \frac{1}{2}\left\lVert \boldsymbol{\nabla}\boldsymbol{m}_{h}^{i} \right\rVert_{}^2 + k\langle \boldsymbol{\nabla}\boldsymbol{m}_{h}^{i},\boldsymbol{\nabla}\boldsymbol{v}_{h}^{i} \rangle_{} +\frac{k^2}{2}\left\lVert \boldsymbol{\nabla}\boldsymbol{v}_{h}^{i} \right\rVert_{}^2.$$ Combining the two above equations, we obtain $$\label{eqn:MagneticNextStepEnergyRelation} \mathcal{E}_{\mathrm{m}}[\boldsymbol{m}_{h}^{i+1}] - \mathcal{E}_{\mathrm{m}}[\boldsymbol{m}_{h}^i] = - \alpha k \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{h}^2 - k^2(\theta - 1/2) \left\lVert \boldsymbol{\nabla}\boldsymbol{v}_{h}^{i} \right\rVert_{}^2 + k\langle \boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u}_{h}^{i},\Pi_h\boldsymbol{m}_{h}^{i}],\boldsymbol{v}_{h}^{i} \rangle_{}.$$ Choosing the test function $\boldsymbol{\psi}_h = \boldsymbol{u}_h^{i+1} - \boldsymbol{u}_h^{i} = k \, \mathrm{d}_t\boldsymbol{u}_h^{i+1}$ in [\[alg1:displacement_update\]](#alg1:displacement_update){reference-type="eqref" reference="alg1:displacement_update"} yields $$\begin{gathered} \langle \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} - \mathrm{d}_t\boldsymbol{u}_{h}^{i},\mathrm{d}_t\boldsymbol{u}_h^{i+1} \rangle_{} + \langle \mathbb{C}[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{} \\ = \langle \boldsymbol{f},\boldsymbol{u}_h^{i+1} - \boldsymbol{u}_h^{i} \rangle_{} + \langle \boldsymbol{g},\boldsymbol{u}_h^{i+1} - \boldsymbol{u}_h^{i} \rangle_{\Gamma_{N}}.\end{gathered}$$ Using Lemma [Lemma 25](#lem:abel){reference-type="ref" reference="lem:abel"}, the first term on the left-hand side can be reformulated as $$\langle \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} - \mathrm{d}_t\boldsymbol{u}_{h}^{i},\mathrm{d}_t\boldsymbol{u}_h^{i+1} \rangle_{} = \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} \right\rVert_{}^2 - \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^i \right\rVert_{}^2 + \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} - \mathrm{d}_t\boldsymbol{u}_{h}^i \right\rVert_{}^2$$ which yields $$\begin{gathered} \label{eqn:AfterAbelFirstSummation} \frac{1}{2}\left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} \right\rVert_{}^2-\frac{1}{2}\left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^i \right\rVert_{}^2+\frac{1}{2}\left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} - \mathrm{d}_t\boldsymbol{u}_{h}^i \right\rVert_{}^2 \\ + \langle \mathbb{C}[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{} \\ = \langle \boldsymbol{f},\boldsymbol{u}_h^{i+1} - \boldsymbol{u}_h^{i} \rangle_{} + \langle \boldsymbol{g},\boldsymbol{u}_h^{i+1} - \boldsymbol{u}_h^{i} \rangle_{\Gamma_{N}}.\end{gathered}$$ Similarly, we have $$\begin{split} &\langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{} \\ & \quad \stackrel{\phantom{\eqref{eq:abel}}}{=} \langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{}\\ & \qquad\quad + \langle \mathbb{C}:[\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{} \\ & \quad \stackrel{\phantom{\eqref{eq:abel}}}{=} \langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})],[\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})] - [\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^i)] \rangle_{}\\ & \qquad\quad + \langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_h^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_h^{i}) \rangle_{} \\ & \qquad\quad + \langle \mathbb{C}:[\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{} \\ & \quad \stackrel{\eqref{eq:abel}}{=} \frac{1}{2} \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1}) \right\rVert_{\mathbb{C}}^2 - \frac{1}{2} \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^i) \right\rVert_{\mathbb{C}}^2 \\ & \qquad\quad + \frac{1}{2} \left\lVert [\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})] - [\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^i)] \right\rVert_{\mathbb{C}}^2\\ & \qquad\quad + \langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_h^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_h^{i}) \rangle_{}\\ & \qquad\quad + \langle \mathbb{C}:[\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{}, \end{split}$$ respectively. Altogether, we thus obtain $$\label{eqn:ElasticNextStepEnergyRelation} \begin{split} & \mathcal{E}_{\mathrm{el}}[\boldsymbol{u}_{h}^{i+1},\boldsymbol{m}_{h}^{i+1}] + \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} \right\rVert_{}^2 - \mathcal{E}_{\mathrm{el}}[\boldsymbol{u}_{h}^i,\boldsymbol{m}_{h}^i] - \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^i \right\rVert_{}^2 \\ & \quad = - \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} - \mathrm{d}_t\boldsymbol{u}_{h}^i \right\rVert_{}^2 - \frac{1}{2} \left\lVert [\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})] - [\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^i)] \right\rVert_{\mathbb{C}}^2 \\ & \qquad - \langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_h^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_h^{i}) \rangle_{}\\ & \qquad - \langle \mathbb{C}:[\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{}. \end{split}$$ Combining [\[eqn:MagneticNextStepEnergyRelation\]](#eqn:MagneticNextStepEnergyRelation){reference-type="eqref" reference="eqn:MagneticNextStepEnergyRelation"} and [\[eqn:ElasticNextStepEnergyRelation\]](#eqn:ElasticNextStepEnergyRelation){reference-type="eqref" reference="eqn:ElasticNextStepEnergyRelation"} yields $$\begin{split} & \mathcal{E}[\boldsymbol{u}_{h}^{i+1},\boldsymbol{m}_{h}^{i+1}] + \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} \right\rVert_{}^2 - \mathcal{E}[\boldsymbol{u}_{h}^i,\boldsymbol{m}_{h}^i] - \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^i \right\rVert_{}^2 \\ & \quad = - \alpha k \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{h}^2 - k^2(\theta - 1/2) \left\lVert \boldsymbol{\nabla}\boldsymbol{v}_{h}^{i} \right\rVert_{}^2 + k\langle \boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u}_{h}^{i},\Pi_h\boldsymbol{m}_{h}^{i}],\boldsymbol{v}_{h}^{i} \rangle_{} \\ & \qquad - \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} - \mathrm{d}_t\boldsymbol{u}_{h}^i \right\rVert_{}^2 - \frac{1}{2} \left\lVert [\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})] - [\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^i)] \right\rVert_{\mathbb{C}}^2 \\ & \qquad - \langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_h^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_h^{i}) \rangle_{} \\ & \qquad - \langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{}\\ & \quad = - \alpha k \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{h}^2 - D_{h,k}^i - E_{h,k}^i, \end{split}$$ where, in the last identity, we have used the expression of $D_{h,k}^i$ in [\[eq:dissipation\]](#eq:dissipation){reference-type="eqref" reference="eq:dissipation"} and we have defined $$\begin{aligned} E_{h,k}^i &:= \langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_h^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_h^{i}) \rangle_{}\\ & \quad - k\langle \boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u}_{h}^{i},\Pi_h\boldsymbol{m}_{h}^{i}],\boldsymbol{v}_{h}^{i} \rangle_{}\\ & \quad + \langle \mathbb{C}:[\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{}\end{aligned}$$ To conclude the proof of [\[eq:discrete_energy_law\]](#eq:discrete_energy_law){reference-type="eqref" reference="eq:discrete_energy_law"}, it remains to show that the latter coincides with [\[eq:error_energy\]](#eq:error_energy){reference-type="eqref" reference="eq:error_energy"}. To this end, using the expression of the elastic field and Lemma [Lemma 24](#lem:tensor){reference-type="ref" reference="lem:tensor"}, we obtain $$\begin{split} k\langle \boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u}_{h}^{i},\Pi_h\boldsymbol{m}_{h}^{i}],\boldsymbol{v}_{h}^{i} \rangle_{} & \,\stackrel{\eqref{eqn:MagnetostrictiveEffectiveField}}{=} 2k\langle \mathbb{Z}^{\top}\mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i})]\Pi_h \boldsymbol{m}_h^i,\boldsymbol{v}_{h}^{i} \rangle_{} \\ & \stackrel{\eqref{eq:tensor_identity}}{=} 2k\langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i})],\mathbb{Z}(\Pi_h \boldsymbol{m}_h^i \otimes \boldsymbol{v}_{h}^{i}) \rangle_{}. \end{split}$$ Moreover, from [\[eq:alg_update_m\]](#eq:alg_update_m){reference-type="eqref" reference="eq:alg_update_m"} and the minor symmetry of $\mathbb{Z}$, we get the expansion $$\label{eq:expansion_magnetostrain} \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1}) = \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i}) + 2k \, \mathbb{Z}(\boldsymbol{m}_{h}^{i} \otimes \boldsymbol{v}_{h}^{i}) + k^2 \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{v}_{h}^{i}).$$ Altogether, it follows that $$\begin{split} E_{h,k}^i & = k^2 \langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{v}_{h}^{i}) \rangle_{} \\ & \quad + 2k \langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})],\mathbb{Z}(\boldsymbol{m}_{h}^{i} \otimes \boldsymbol{v}_{h}^{i}) \rangle_{} \\ & \quad - 2k\langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i})],\mathbb{Z}(\Pi_h \boldsymbol{m}_h^i \otimes \boldsymbol{v}_{h}^{i}) \rangle_{} \\ & \quad + \langle \mathbb{C}:[\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{}\\ & = k^2 \langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{v}_{h}^{i}) \rangle_{} \\ & \quad + 2k \langle \mathbb{C}:\{[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1})] - [\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i})] \},\mathbb{Z}(\boldsymbol{m}_{h}^{i} \otimes \boldsymbol{v}_{h}^{i}) \rangle_{} \\ & \quad + 2k\langle \mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i})],\mathbb{Z}[(\boldsymbol{m}_h^i - \Pi_h \boldsymbol{m}_h^i) \otimes \boldsymbol{v}_{h}^{i}] \rangle_{}\\ & \quad + \langle \mathbb{C}:[\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1})],\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{}. \end{split}$$ This shows [\[eq:error_energy\]](#eq:error_energy){reference-type="eqref" reference="eq:error_energy"} and concludes the proof. ◻ ## Stability {#sec:proofs_stability} We now prove Proposition [Proposition 8](#prop:stability){reference-type="ref" reference="prop:stability"} showing unconditional stability of Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} and an estimate of the violation of the unit length constraint. For the sake of clarity, we split the proof into several lemmas. An immediate consequence of the projection-free update [\[alg1:magnetisation_update\]](#alg1:magnetisation_update){reference-type="eqref" reference="alg1:magnetisation_update"} is the following $L^2$-bound for the approximate magnetisations. **Lemma 13**. For every integer $1 \le j \le N$, it holds that $$\begin{aligned} \label{alg2:m_bound_estimate} \left\lVert \boldsymbol{m}_{h}^j \right\rVert_{}^2 &\le C_1 \left( 1 + k^2 \sum_{i=0}^{j-1}\left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{}^2 \right),\\ \label{eq:auxiliary_constraint} \big\lVert \mathcal{I}_h\big[\left\lvert \boldsymbol{m}_h^j \right\rvert^2\big]-1 \big\rVert_{L^1(\Omega)} &\le C_2 k^2 \sum_{i=0}^{j-1}\left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{}^2.\end{aligned}$$ where $C_1,C_2>0$ are constants depending the shape-regularity parameter of $\mathcal{T}_h$ ($C_1$ depends also on $\left\lvert \Omega \right\rvert$). *Proof.* We follow [@bartels2016]. Starting from [\[eq:alg_update_m\]](#eq:alg_update_m){reference-type="eqref" reference="eq:alg_update_m"} and noting that $\boldsymbol{v}_{h}^{i}\in\boldsymbol{\mathcal{K}}_{h}[\boldsymbol{m}_{h}^{i}]$, we have for each $z\in\mathcal{N}_{h}$ that for every $0 \le i \le j-1$ $$\left\lvert \boldsymbol{m}_{h}^{i+1}(z) \right\rvert^2 = \left\lvert \boldsymbol{m}_{h}^{i}(z) \right\rvert^2 + k^2\left\lvert \boldsymbol{v}_{h}^{i}(z) \right\rvert^2.$$ Inductively, starting with $|\boldsymbol{m}_{h}^{0}(z)| = 1$, we deduce that $$|\boldsymbol{m}_{h}^{j}(z)|^2= 1+ k^2\sum_{i=0}^{j-1}|\boldsymbol{v}_{h}^{i}(z)|^2.$$ Then, noting that $\left\lVert 1 \right\rVert_{}=\left\lvert \Omega \right\rvert^{1/2}$ and using [\[eq:Lp_equivalence\]](#eq:Lp_equivalence){reference-type="eqref" reference="eq:Lp_equivalence"} yields [\[alg2:m_bound_estimate\]](#alg2:m_bound_estimate){reference-type="eqref" reference="alg2:m_bound_estimate"} (for a suitable constant $C_1>0$ we do not explicitly compute). The same argument shows [\[eq:auxiliary_constraint\]](#eq:auxiliary_constraint){reference-type="eqref" reference="eq:auxiliary_constraint"}. ◻ We also have the following estimate of all quantities involving the magnetisation. **Lemma 14**. For every integer $1 \le j \le N$, it holds that $$\begin{gathered} \label{eqn:MagneticStability} \left\lVert \boldsymbol{\nabla}\boldsymbol{m}_{h}^{j} \right\rVert_{}^2 + k \sum_{i=0}^{j-1} \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{}^2 + \left(\theta - \frac{1}{2}\right) k^2\sum_{i=0}^{j-1} \left\lVert \boldsymbol{\nabla}\boldsymbol{v}_{h}^{i} \right\rVert_{}^2 \\ \leq C_3 \left[ \left\lVert \boldsymbol{\nabla}\boldsymbol{m}_{h}^0 \right\rVert_{}^2 + k \sum_{i=0}^{j-1}\left( 1 + \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \right\rVert_{}^2 \right) \right],\end{gathered}$$ where $C_3>0$ depends only on $\alpha$, $\left\lvert \Omega \right\rvert$, $\left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}$, and $\left\lVert \mathbb{C} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}$. *Proof.* Let $1 \le j \le N$ be an integer. Starting from [\[eqn:MagneticNextStepEnergyRelation\]](#eqn:MagneticNextStepEnergyRelation){reference-type="eqref" reference="eqn:MagneticNextStepEnergyRelation"} (cf. the proof of Proposition [Proposition 6](#prop:energy){reference-type="ref" reference="prop:energy"}), we sum up from $0$ to $j-1$ to obtain $$\begin{gathered} \frac{1}{2}\left\lVert \boldsymbol{\nabla}\boldsymbol{m}_{h}^{j} \right\rVert_{}^2 + \alpha k\sum_{i=0}^{j-1} \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{h}^2 + \left(\theta - \frac{1}{2}\right) k^2\sum_{i=0}^{j-1} \left\lVert \boldsymbol{\nabla}\boldsymbol{v}_{h}^{i} \right\rVert_{}^2 \\ = \frac{1}{2}\left\lVert \boldsymbol{\nabla}\boldsymbol{m}_{h}^0 \right\rVert_{}^2 + k\sum_{i=0}^{j-1} \langle \boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u}_{h}^{i},\Pi_h\boldsymbol{m}_{h}^{i}],\boldsymbol{v}_{h}^{i} \rangle_{}.\end{gathered}$$ Using Lemma [Lemma 12](#lem:boundedness_elastic_field){reference-type="ref" reference="lem:boundedness_elastic_field"}, we can estimate the term involving the elastic field for some $\nu >0$ by $$\begin{split} \left\lvert \langle \boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u}_{h}^{i},\Pi_h\boldsymbol{m}_{h}^{i}],\boldsymbol{v}_{h}^{i} \rangle_{} \right\rvert & \stackrel{\phantom{\eqref{eq:boundedness_elastic_field}}}{\leq} \frac{1}{4\nu}\left\lVert \boldsymbol{h}_{\mathrm{m}}[\boldsymbol{u}_{h}^{i},\Pi_h\boldsymbol{m}_{h}^{i}] \right\rVert_{}^2 + \nu\left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{}^2\\ & \stackrel{\eqref{eq:boundedness_elastic_field}}{\leq} \frac{2}{\nu} \left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 \left\lVert \mathbb{C} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 \left(\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \right\rVert_{}^2 + \left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 |\Omega| \right) + \nu\left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{}^2. \end{split}$$ Then we get $$\begin{gathered} \frac{1}{2}\left\lVert \boldsymbol{\nabla}\boldsymbol{m}_{h}^{j} \right\rVert_{}^2 + \alpha k\sum_{i=0}^{j-1} \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{h}^2 + \left(\theta - \frac{1}{2}\right) k^2\sum_{i=0}^{j-1} \left\lVert \boldsymbol{\nabla}\boldsymbol{v}_{h}^{i} \right\rVert_{}^2 \le \frac{1}{2}\left\lVert \boldsymbol{\nabla}\boldsymbol{m}_{h}^0 \right\rVert_{}^2 \\ + \frac{2k}{\nu}\sum_{i=0}^{j-1} \left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 \left\lVert \mathbb{C} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 \left(\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \right\rVert_{}^2 + \left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}^2 |\Omega| \right) + \nu k\sum_{i=0}^{j-1} \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{}^2.\end{gathered}$$ Using [\[eq:h-scalar-product_equivalence\]](#eq:h-scalar-product_equivalence){reference-type="eqref" reference="eq:h-scalar-product_equivalence"} and choosing $\nu = \alpha/2$ yields [\[eqn:MagneticStability\]](#eqn:MagneticStability){reference-type="eqref" reference="eqn:MagneticStability"} (for a suitable constant $C_3>0$ which we do not compute explicitly). ◻ In the following lemma, we show that the magnetostrain is Lipschitz continuous with respect to the magnetisation (the use of the nodal projection is exploited here). **Lemma 15**. For all $\boldsymbol{m}_{h,1} , \boldsymbol{m}_{h,2} \in \mathcal{S}^1(\mathcal{T}_h)^3$ satisfying $\left\lvert \boldsymbol{m}_{h,\ell}(z) \right\rvert \ge 1$ for all $\ell=1,2$ and $z \in \mathcal{N}_h$, it holds that $$\label{eq:lipschitz_magnetostrain} \left\lVert \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h,1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h,2}) \right\rVert_{} \le C_{\mathrm{m}} \left\lVert \boldsymbol{m}_{h,1} - \boldsymbol{m}_{h,2} \right\rVert_{},$$ where $C_{\mathrm{m}}>0$ depends only on $\left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}$ and the shape-regularity parameter of $\mathcal{T}_h$. *Proof.* Straightforward calculations exploiting the boundedness guaranteed by the nodal projection, i.e. $\left\lVert \Pi_h\boldsymbol{m}_{h,1} \right\rVert_{\boldsymbol{L}^\infty(\Omega)} = \left\lVert \Pi_h\boldsymbol{m}_{h,2} \right\rVert_{\boldsymbol{L}^\infty(\Omega)} = 1$, show that $$\left\lVert \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h,1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h,2}) \right\rVert_{} \lesssim \left\lVert \Pi_h\boldsymbol{m}_{h,1} - \Pi_h\boldsymbol{m}_{h,2} \right\rVert_{},$$ where the hidden constant depends on $\left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)}$. From the norm equivalence in [@bartels2015 Lemma 3.4]) and the fact that the projection onto the sphere is non-expanding (i.e. Lipschitz continuous with constant 1), it follows that $$\left\lVert \Pi_h\boldsymbol{m}_{h,1} - \Pi_h\boldsymbol{m}_{h,2} \right\rVert_{} \lesssim \left\lVert \boldsymbol{m}_{h,1} - \boldsymbol{m}_{h,2} \right\rVert_{},$$ where the hidden constant depends on the shape-regularity of the mesh. Combining the above two estimates yields the desired result, where $C_{\mathrm{m}}>0$ is the product of the two constants hidden above. ◻ **Lemma 16**. For every integer $1 \le j \le N$, the following estimate holds $$\begin{gathered} \label{eq:StrainBoundedness} \left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^j \right\rVert_{}^2 + \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{j}) \right\rVert_{}^2 + \sum_{i=0}^{j-1}\left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} - \mathrm{d}_t\boldsymbol{u}_{h}^i \right\rVert_{}^2 + \sum_{i=0}^{j-1}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1})-\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \right\rVert_{}^2 \\ \le C_4 \left[ 1 + \left\lVert \dot\boldsymbol{u}_{h}^0 \right\rVert_{}^2 + \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{0}) \right\rVert_{}^2 + \left\lVert \boldsymbol{\nabla}\boldsymbol{m}_{h}^0 \right\rVert_{}^2 + k \sum_{i=0}^{j-1}\left( 1 + \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \right\rVert_{}^2 \right) \right],\end{gathered}$$ where $C_4>0$ depends only on the shape-regularity parameter of $\mathcal{T}_h$ and the problem data $\alpha$, $\Omega$, $\mathbb{C}$, $\mathbb{Z}$, $\boldsymbol{f}$ and $\boldsymbol{g}$. *Proof.* Let $1 \le j \le N$ be an integer. Starting from [\[eqn:AfterAbelFirstSummation\]](#eqn:AfterAbelFirstSummation){reference-type="eqref" reference="eqn:AfterAbelFirstSummation"} (cf. the proof of Proposition [Proposition 6](#prop:energy){reference-type="ref" reference="prop:energy"}), summing up from $0$ to $j-1$, we have $$\begin{gathered} \frac{1}{2}\left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{j} \right\rVert_{}^2 -\frac{1}{2}\left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^0 \right\rVert_{}^2 +\frac{1}{2}\sum_{i=0}^{j-1}\left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} - \mathrm{d}_t\boldsymbol{u}_{h}^i \right\rVert_{}^2 + \sum_{i=0}^{j-1} \langle \mathbb{C}:\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}),\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{} \\ = \sum_{i=0}^{j-1} \langle \mathbb{C}:\boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1}),\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{} \\ +\langle \boldsymbol{f},\boldsymbol{u}_h^{j} \rangle_{} -\langle \boldsymbol{f},\boldsymbol{u}_h^{0} \rangle_{} + \langle \boldsymbol{g},\boldsymbol{u}_h^{j} \rangle_{\Gamma_{N}} -\langle \boldsymbol{g},\boldsymbol{u}_h^{0} \rangle_{\Gamma_{N}}.\end{gathered}$$ Applying Lemma [Lemma 25](#lem:abel){reference-type="ref" reference="lem:abel"} to the last term on the left-hand side and rearranging we have $$\begin{gathered} \frac{1}{2}\left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^j \right\rVert_{}^2 + \frac{1}{2}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{j}) \right\rVert_{\mathbb{C}}^2 + \frac{1}{2}\sum_{i=0}^{j-1}\left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} - \mathrm{d}_t\boldsymbol{u}_{h}^i \right\rVert_{}^2 + \frac{1}{2} \sum_{i=0}^{j-1}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1})-\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \right\rVert_{\mathbb{C}}^2 \\ = \frac{1}{2}\left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^0 \right\rVert_{}^2 + \frac{1}{2}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{0}) \right\rVert_{\mathbb{C}}^2 + \sum_{i=0}^{j-1}\langle \mathbb{C}:\boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1}),\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{}\\ + \langle \boldsymbol{f},\boldsymbol{u}_h^{j} \rangle_{} - \langle \boldsymbol{f},\boldsymbol{u}_h^{0} \rangle_{} + \langle \boldsymbol{g},\boldsymbol{u}_h^{j} \rangle_{\Gamma_{N}} - \langle \boldsymbol{g},\boldsymbol{u}_h^{0} \rangle_{\Gamma_{N}}.\end{gathered}$$ The term involving the magnetostrain can be estimated as $$\begin{split} & \sum_{i=0}^{j-1}\langle \mathbb{C}:\boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_h\boldsymbol{m}_{h}^{i+1}),\boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i}) \rangle_{} \\ & \ = \langle \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_{h}\boldsymbol{m}_{h}^{j}),\mathbb{C}:\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{j}) \rangle_{} - \langle \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_{h}\boldsymbol{m}_{h}^{1}),\mathbb{C}:\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{0}) \rangle_{} - k \sum_{i=1}^{j-1} \langle \mathrm{d}_t\boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_{h}\boldsymbol{m}_{h}^{i+1}),\mathbb{C}:\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \rangle_{} \\ & \ \leq \left\lVert \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_{h}\boldsymbol{m}_{h}^{j}) \right\rVert_{}^2 + \frac{1}{4}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{j}) \right\rVert_{\mathbb{C}}^2 + \frac{1}{2}\left\lVert \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_{h}\boldsymbol{m}_{h}^{1}) \right\rVert_{\mathbb{C}}^2 + \frac{1}{2}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{0}) \right\rVert_{\mathbb{C}}^2 \\ & \quad + \frac{k}{2}\sum_{i=1}^{j-1}\left\lVert \mathrm{d}_t\boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_{h}\boldsymbol{m}_{h}^{i+1}) \right\rVert_{}^2 + \frac{k}{2}\sum_{i=1}^{j-1}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \right\rVert_{\mathbb{C}}^2 \\ & \ \leq \frac{3}{2} \left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^\infty(\Omega)}^2 \left\lvert \Omega \right\rvert + \frac{1}{4}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{j}) \right\rVert_{\mathbb{C}}^2 + \frac{1}{2}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{0}) \right\rVert_{\mathbb{C}}^2 \\ & \quad + \frac{k}{2}\sum_{i=1}^{j-1}\left\lVert \mathrm{d}_t\boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_{h}\boldsymbol{m}_{h}^{i+1}) \right\rVert_{}^2 + \frac{k}{2}\sum_{i=1}^{j-1}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \right\rVert_{\mathbb{C}}^2. \end{split}$$ Using Lemma [Lemma 15](#lem:lipschitz_magnetostrain){reference-type="ref" reference="lem:lipschitz_magnetostrain"}, we get $$\begin{split} \left\lVert \mathrm{d}_t\boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_{h}\boldsymbol{m}_{h}^{i+1}) \right\rVert_{} &\stackrel{\phantom{\eqref{eq:lipschitz_magnetostrain}}}{=} \frac{1}{k}\left\lVert \boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_{h}\boldsymbol{m}_{h}^{i+1})-\boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_{h}\boldsymbol{m}_{h}^{i}) \right\rVert_{} \\ &\stackrel{\eqref{eq:lipschitz_magnetostrain}}{\le} \frac{1}{k}C_{\mathrm{m}} \left\lVert \boldsymbol{m}_{h}^{i+1}-\boldsymbol{m}_{h}^{i} \right\rVert_{} = C_{\mathrm{m}} \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{}. \end{split}$$ It follows that $$\frac{k}{2}\sum_{i=1}^{j-1}\left\lVert \mathrm{d}_t\boldsymbol{\varepsilon}_{\mathrm{m}}(\Pi_{h}\boldsymbol{m}_{h}^{i+1}) \right\rVert_{}^2 \leq \frac{C_{\mathrm{m}}^2 \, k}{2}\sum_{i=1}^{j-1}\left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{}^2.$$ Moreover, for every $\delta >0$ we have that $$\begin{aligned} \begin{split} \left\lvert \langle \boldsymbol{f},\boldsymbol{u}_h^j \rangle_{} + \langle \boldsymbol{g},\boldsymbol{u}_h^j \rangle_{\Gamma_{N}} \right\rvert & \leq C_{\mathrm{KPC}} (\left\lVert \boldsymbol{f} \right\rVert_{} + \left\lVert \boldsymbol{g} \right\rVert_{\Gamma_{N}}) \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_h^j) \right\rVert_{\mathbb{C}} \\ &\leq \frac{C_{\mathrm{KPC}}^2}{4\delta}(\left\lVert \boldsymbol{f} \right\rVert_{} + \left\lVert \boldsymbol{g} \right\rVert_{\Gamma_{N}})^2+ \delta\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_h^j) \right\rVert_{\mathbb{C}}^2, \end{split}\end{aligned}$$ where $C_{\mathrm{KPC}}>0$ is a constant depending only on $\left\lvert \Omega \right\rvert$ and $\mathbb{C}$ (a combination of the continuity constant of the trace operator $\boldsymbol{H}^1(\Omega) \to \boldsymbol{L}^2(\Gamma_N)$, the constants appearing in Poincaré's and Korn's inequalities, and the equivalence constant in the norm equivalence $\left\lVert \cdot \right\rVert_{} \simeq \left\lVert \cdot \right\rVert_{\mathbb{C}}$). Overall, choosing $\delta = 1/8$ and recalling that $\mathrm{d}_t\boldsymbol{u}_{h}^0 = \dot\boldsymbol{u}_{h}^0$, we obtain $$\begin{split} & \frac{1}{2}\left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^j \right\rVert_{}^2 + \frac{1}{8}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{j}) \right\rVert_{\mathbb{C}}^2 + \frac{1}{2}\sum_{i=0}^{j-1}\left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} - \mathrm{d}_t\boldsymbol{u}_{h}^i \right\rVert_{}^2 + \frac{1}{2} \sum_{i=0}^{j-1}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1})-\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \right\rVert_{\mathbb{C}}^2 \\ & \quad\le \frac{3}{2} \left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^\infty(\Omega)}^2 \left\lvert \Omega \right\rvert + 2 C_{\mathrm{KPC}}^2(\left\lVert \boldsymbol{f} \right\rVert_{} + \left\lVert \boldsymbol{g} \right\rVert_{\Gamma_{N}})^2 + \frac{1}{2}\left\lVert \dot\boldsymbol{u}_{h}^0 \right\rVert_{}^2 \\ & \qquad + \left[1 + C_{\mathrm{KPC}} \left(\left\lVert \boldsymbol{f} \right\rVert_{} + \left\lVert \boldsymbol{g} \right\rVert_{\Gamma_{N}}\right)\right] \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{0}) \right\rVert_{\mathbb{C}}^2 + \frac{k}{2}\sum_{i=1}^{j-1}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \right\rVert_{\mathbb{C}}^2 + \frac{C_{\mathrm{m}}^2 \, k}{2}\sum_{i=1}^{j-1}\left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{}^2. \end{split}$$ Applying Lemma [Lemma 14](#lem:StabilityOfMagnetisation){reference-type="ref" reference="lem:StabilityOfMagnetisation"} to estimate the last term on the right-hand side, we obtain [\[eq:StrainBoundedness\]](#eq:StrainBoundedness){reference-type="eqref" reference="eq:StrainBoundedness"} (for a suitable constant $C_4>0$ which we do not compute explicitly). ◻ We are now in a position to prove Proposition [Proposition 8](#prop:stability){reference-type="ref" reference="prop:stability"}. *Proof of Proposition [Proposition 8](#prop:stability){reference-type="ref" reference="prop:stability"}.* We apply Lemmas [Lemma 13](#lem:Magnetisation_Estimate){reference-type="ref" reference="lem:Magnetisation_Estimate"}, [Lemma 14](#lem:StabilityOfMagnetisation){reference-type="ref" reference="lem:StabilityOfMagnetisation"} and [Lemma 16](#lem:StrainBoundedness){reference-type="ref" reference="lem:StrainBoundedness"}. Combining [\[alg2:m_bound_estimate\]](#alg2:m_bound_estimate){reference-type="eqref" reference="alg2:m_bound_estimate"}, [\[eqn:MagneticStability\]](#eqn:MagneticStability){reference-type="eqref" reference="eqn:MagneticStability"} and [\[eq:StrainBoundedness\]](#eq:StrainBoundedness){reference-type="eqref" reference="eq:StrainBoundedness"}, we obtain $$\begin{split} & \left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^j \right\rVert_{}^2 + \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{j}) \right\rVert_{}^2 + \sum_{i=0}^{j-1}\left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} - \mathrm{d}_t\boldsymbol{u}_{h}^i \right\rVert_{}^2 + \sum_{i=0}^{j-1}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1})-\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \right\rVert_{}^2 \\ & \qquad + \left\lVert \boldsymbol{m}_{h}^{j} \right\rVert_{\boldsymbol{H}^1(\Omega)}^2 + (1-C_1 k) k \sum_{i=0}^{j-1} \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{}^2 + \left(\theta - \frac{1}{2}\right) k^2\sum_{i=0}^{j-1} \left\lVert \boldsymbol{\nabla}\boldsymbol{v}_{h}^{i} \right\rVert_{}^2 \\ & \qquad\qquad \leq C_1 + C_4 + (C_3 + C_4) \left\lVert \boldsymbol{\nabla}\boldsymbol{m}_{h}^0 \right\rVert_{}^2 + C_4 \left\lVert \dot\boldsymbol{u}_{h}^0 \right\rVert_{}^2 + C_4 \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{0}) \right\rVert_{}^2 \\ & \qquad\qquad\qquad + (C_3 + C_4)k \sum_{i=0}^{j-1}\left( 1 + \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \right\rVert_{}^2 \right) \\ & \qquad\qquad \leq C_1 + C_4 + (C_3 + C_4) \left\lVert \boldsymbol{\nabla}\boldsymbol{m}_{h}^0 \right\rVert_{}^2 + C_4 \left\lVert \dot\boldsymbol{u}_{h}^0 \right\rVert_{}^2 + C_4 \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{0}) \right\rVert_{}^2 \\ & \qquad\qquad\qquad + (C_3 + C_4)T + (C_3 + C_4)k \sum_{i=0}^{j-1}\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i}) \right\rVert_{}^2, \end{split}$$ where in the last estimate we have used that $kj \le T$. If the time-step size $k$ is sufficiently small, the coefficients in front of all terms on the left-hand side are strictly positive. Given the boundedness of the approximate initial data guaranteed by assumption [\[eq:convergence_initial_data\]](#eq:convergence_initial_data){reference-type="eqref" reference="eq:convergence_initial_data"}, the desired stability estimate [\[eq:stability\]](#eq:stability){reference-type="eqref" reference="eq:stability"} then follows from the discrete Grönwall lemma; see e.g. [@thomee2006 Lemma 10.5]. Finally, [\[eq:general_constraint\]](#eq:general_constraint){reference-type="eqref" reference="eq:general_constraint"} follows from [\[eq:stability\]](#eq:stability){reference-type="eqref" reference="eq:stability"} and [\[eq:auxiliary_constraint\]](#eq:auxiliary_constraint){reference-type="eqref" reference="eq:auxiliary_constraint"}. This concludes the proof. ◻ ## Convergence {#sec:proofs_convergence} The proof of convergence of Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} (Theorem [Theorem 9](#thm:convergence){reference-type="ref" reference="thm:convergence"}(i)) follows the standard argument to prove existence of weak solutions for parabolic equations (uniform boundedness of Galerkin approximations, extraction of subsequences with suitable convergence properties, identification of the limit with a weak solution of the problem; see, e.g., [@evans2010 Section 7.1]) and thus has the same structure as the one which proves the convergence of [@bppr2013 Algorithm 4.1]. Therefore, in the upcoming analysis, we will provide only a sketch of the steps of the proof that can be found in [@bppr2013]. However, we will present in detail the (non-obvious) steps that we have to perform to cope with the partial omission of the nodal projection (for which we borrow ideas from [@abert_spin-polarized_2014; @bartels2016]) and to prove our novel energy estimate. We start the proof with showing the following lemma, which provides an estimate of the $L^p$-norm ($p \ge 1$) of the difference between the approximate magnetisations generated by Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} and their nodal projections. **Lemma 17**. Let $p \in [1,\infty)$. For all integers $1 \le j \le N$, it holds that $$\label{eq:projection_error_Lp} \left\lVert \boldsymbol{m}_h^j - \Pi_h \boldsymbol{m}_h^j \right\rVert_{\boldsymbol{L}^p(\Omega)} \le C \frac{T^{1-1/p}}{2} k^{1+1/p} \sum_{i=0}^{j-1} \left\lVert \boldsymbol{v}_h^i \right\rVert_{\boldsymbol{L}^{2p}(\Omega)}^{2},$$ where $C>0$ depends only on the shape-regularity of $\mathcal{T}_h$. *Proof.* Let $1 \le j \le N$ be an integer. For all $z \in \mathcal{N}_h$, we have that $$\begin{split} \left\lvert \boldsymbol{m}_h^j(z) - \Pi_h \boldsymbol{m}_h^j(z) \right\rvert & = \left\lvert \boldsymbol{m}_h^j(z) - \frac{\boldsymbol{m}_h^j(z)}{\left\lvert \boldsymbol{m}_h^j(z) \right\rvert} \right\rvert = \left\lvert \boldsymbol{m}_h^j(z) \right\rvert - 1 \\ & = \frac{\left\lvert \boldsymbol{m}_h^j(z) \right\rvert^2 - 1}{\left\lvert \boldsymbol{m}_h^j(z) \right\rvert + 1} \le \frac{1}{2} \left( \left\lvert \boldsymbol{m}_h^j(z) \right\rvert^2 - 1 \right) = \frac{k^2}{2} \sum_{i=0}^{j-1} \left\lvert \boldsymbol{v}_h^i(z) \right\rvert^2. \end{split}$$ If $p=1$, the norm equivalence [\[eq:Lp_equivalence\]](#eq:Lp_equivalence){reference-type="eqref" reference="eq:Lp_equivalence"} immediately yields [\[eq:projection_error_Lp\]](#eq:projection_error_Lp){reference-type="eqref" reference="eq:projection_error_Lp"}. If $p>1$, applying [\[eq:Lp_equivalence\]](#eq:Lp_equivalence){reference-type="eqref" reference="eq:Lp_equivalence"} twice and using the convexity of $x^p$ for $x>0$ as well as $jk \le T$, we obtain $$\begin{split} \left\lVert \boldsymbol{m}_h^j - \Pi_h \boldsymbol{m}_h^j \right\rVert_{\boldsymbol{L}^p(\Omega)}^p &\lesssim \sum_{z \in \mathcal{N}_h} h_z^3 \left\lvert \boldsymbol{m}_h^j(z) - \Pi_h \boldsymbol{m}_h^j(z) \right\rvert^p \le \sum_{z \in \mathcal{N}_h} h_z^3 \left(\frac{k^2}{2} \sum_{i=0}^{j-1} \left\lvert \boldsymbol{v}_h^i(z) \right\rvert^2 \right)^p \\ & \le \sum_{z \in \mathcal{N}_h} h_z^3 \frac{k^{2p}}{2^p} j^{p-1} \sum_{i=0}^{j-1} \left\lvert \boldsymbol{v}_h^i(z) \right\rvert^{2p} \le \sum_{z \in \mathcal{N}_h} h_z^3 \frac{k^{p+1}}{2^p} T^{p-1} \sum_{i=0}^{j-1} \left\lvert \boldsymbol{v}_h^i(z) \right\rvert^{2p} \\ & \lesssim \frac{k^{p+1}}{2^p} T^{p-1} \sum_{i=0}^{j-1} \left\lVert \boldsymbol{v}_h^i \right\rVert_{\boldsymbol{L}^{2p}(\Omega)}^{2p}, \end{split}$$ where the hidden constants depend only on the shape-regularity of $\mathcal{T}_h$. Then, [\[eq:projection_error_Lp\]](#eq:projection_error_Lp){reference-type="eqref" reference="eq:projection_error_Lp"} for $p>1$ follows from the inequality $\left\lVert \cdot \right\rVert_{\ell^p} \le \left\lVert \cdot \right\rVert_{\ell^1}$ satisfied by the $p$-norms in finite dimensions. This concludes the proof. ◻ Now, let $\{ \boldsymbol{m}_{hk}\}$, $\{ \boldsymbol{m}_{hk}^{\pm} \}$, $\{ \boldsymbol{v}_{hk}^- \}$, $\{ \boldsymbol{u}_{hk}\}$, $\{ \boldsymbol{u}_{hk}^\pm \}$, $\{ \dot\boldsymbol{u}_{hk}\}$, $\{ \dot\boldsymbol{u}_{hk}^\pm \}$ be the time reconstructions defined according to [\[eq:reconstructions\]](#eq:reconstructions){reference-type="eqref" reference="eq:reconstructions"} using the approximations $\{ (\boldsymbol{u}_h^i,\boldsymbol{m}_h^i) \}_{0 \le i \le N}$ generated by Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"}. In the following lemma, we show that the uniform stability established in Proposition [Proposition 8](#prop:stability){reference-type="ref" reference="prop:stability"} allows us to extract convergent subsequences from the sequences of time reconstructions. **Lemma 18**. Under the assumptions of Theorem [Theorem 9](#thm:convergence){reference-type="ref" reference="thm:convergence"}(i), there exist $\boldsymbol{u}\in L^\infty(0,T;\boldsymbol{H}^1_D(\Omega))$ with $\partial_{t}\boldsymbol{u}\in L^\infty(0,T;\boldsymbol{L}^2(\Omega))$ and $\boldsymbol{m}\in L^\infty(0,T;\boldsymbol{H}^1(\Omega;\mathbb{S}^2))$ with $\partial_t \boldsymbol{m}\in L^2(0,T;\boldsymbol{L}^2(\Omega))$ such that, upon extraction of (non-relabeled) subsequences, we have the following convergence results: [\[eq:convergences\]]{#eq:convergences label="eq:convergences"} $$\begin{aligned} \label{eq:conv_weakH1_u} \boldsymbol{u}_{hk} \rightharpoonup\boldsymbol{u}\quad &\text{in } \boldsymbol{H}^1(\Omega_T), \\ \label{eq:conv_weakstarLinftyH1_u} \boldsymbol{u}_{hk}, \boldsymbol{u}_{hk}^{\pm} \overset{\ast}{\rightharpoonup}\boldsymbol{u}\quad &\text{in } L^{\infty}(0,T;\boldsymbol{H}^1(\Omega)), \\ \label{eq:conv_weakL2H1_u} \boldsymbol{u}_{hk}, \boldsymbol{u}_{hk}^{\pm} \rightharpoonup\boldsymbol{u}\quad &\text{in } L^2(0,T;\boldsymbol{H}^1(\Omega)), \\ \label{eq:conv_strongL2_u} \boldsymbol{u}_{hk}, \boldsymbol{u}_{hk}^{\pm} \to \boldsymbol{u}\quad &\text{in } \boldsymbol{L}^2(\Omega_T), \\ \label{eq:conv_weakLinftyL2_ut} \dot\boldsymbol{u}_{hk}, \dot\boldsymbol{u}_{hk}^{\pm} \overset{\ast}{\rightharpoonup}\partial_{t}\boldsymbol{u}\quad &\text{in } L^\infty(0,T;\boldsymbol{L}^2(\Omega)), \\ \label{eq:conv_weakL2_ut} \dot\boldsymbol{u}_{hk}, \dot\boldsymbol{u}_{hk}^{\pm} \rightharpoonup\partial_{t}\boldsymbol{u}\quad &\text{in } \boldsymbol{L}^2(\Omega_T), \\ \label{eq:conv_weakH1} \boldsymbol{m}_{hk} \rightharpoonup\boldsymbol{m}\quad &\text{in } \boldsymbol{H}^1(\Omega_T), \\ \label{eq:conv_strongHs} \boldsymbol{m}_{hk} \to \boldsymbol{m}\quad &\text{in } \boldsymbol{H}^s(\Omega_T) \text{ for all } s \in (0,1), \\ \label{eq:conv_weakstarLinftyH1} \boldsymbol{m}_{hk}, \boldsymbol{m}_{hk}^{\pm} \overset{\ast}{\rightharpoonup}\boldsymbol{m}\quad &\text{in } L^{\infty}(0,T;\boldsymbol{H}^1(\Omega)), \\ \label{eq:conv_weakL2H1} \boldsymbol{m}_{hk}, \boldsymbol{m}_{hk}^{\pm} \rightharpoonup\boldsymbol{m}\quad &\text{in } L^2(0,T;\boldsymbol{H}^1(\Omega)), \\ \label{eq:conv_strongL2Hs} \boldsymbol{m}_{hk}, \boldsymbol{m}_{hk}^{\pm} \to \boldsymbol{m}\quad &\text{in } L^2(0,T;\boldsymbol{H}^s(\Omega)) \text{ for all } s \in (0,1),\\ \label{eq:conv_strongL2} \boldsymbol{m}_{hk}, \boldsymbol{m}_{hk}^{\pm} \to \boldsymbol{m}\quad &\text{in } \boldsymbol{L}^2(\Omega_T), \\ \label{eq:conv_pointwise} \boldsymbol{m}_{hk}, \boldsymbol{m}_{hk}^{\pm} \to \boldsymbol{m}\quad &\text{pointwise a.e.\ in } \Omega_T, \\ \label{eq:conv_weakL2_v} \boldsymbol{v}_{hk}^- \rightharpoonup\partial_t \boldsymbol{m}\quad &\text{in } \boldsymbol{L}^2(\Omega_T),\end{aligned}$$ as $h,k \to 0$. *Proof.* Using the boundedness expressed in Proposition [Proposition 8](#prop:stability){reference-type="ref" reference="prop:stability"}, we can successively extract weakly(-star) convergent subsequences (non-relabeled, with possibly different limits) from $\{\boldsymbol{u}_{hk}\}$ and $\{\boldsymbol{u}_{hk}^{\pm}\}$, from $\{\dot{\boldsymbol{u}}_{hk}\}$ and $\{\dot{\boldsymbol{u}}_{hk}^{\pm}\}$, from $\{\boldsymbol{m}_{hk}\}$ and $\{\boldsymbol{m}_{hk}^{\pm}\}$, and from $\{\boldsymbol{v}_{hk}^{-}\}$. Let $\boldsymbol{u}\in\boldsymbol{H}^1(\Omega_T)$ satisfy the weak convergence [\[eq:conv_weakH1_u\]](#eq:conv_weakH1_u){reference-type="eqref" reference="eq:conv_weakH1_u"}. Owing to the continuous inclusions $\boldsymbol{H}^1(\Omega_T) \subset L^2(0,T;\boldsymbol{H}^1(\Omega)) \subset \boldsymbol{L}^2(\Omega_T)$ and the compact inclusion $\boldsymbol{H}^1(\Omega_T) \Subset \boldsymbol{L}^2(\Omega_T)$, we obtain convergences [\[eq:conv_weakL2H1_u\]](#eq:conv_weakL2H1_u){reference-type="eqref" reference="eq:conv_weakL2H1_u"} and [\[eq:conv_strongL2_u\]](#eq:conv_strongL2_u){reference-type="eqref" reference="eq:conv_strongL2_u"}. Moreover, from the continuous inclusion $L^{\infty}(0,T;\boldsymbol{H}^1(\Omega)) \subset L^2(0,T;\boldsymbol{H}^1(\Omega))$, we can identify the weak-star limit of $\{\boldsymbol{u}_{hk}\}$ in $L^\infty(0,T;\boldsymbol{H}^1(\Omega))$ with the weak limit in $L^2(0,T;\boldsymbol{H}^1(\Omega))$, which shows [\[eq:conv_weakstarLinftyH1_u\]](#eq:conv_weakstarLinftyH1_u){reference-type="eqref" reference="eq:conv_weakstarLinftyH1_u"} for $\{\boldsymbol{u}_{hk}\}$. Let $\boldsymbol{m}\in\boldsymbol{H}^1(\Omega_T)$ satisfy the weak convergence [\[eq:conv_weakH1\]](#eq:conv_weakH1){reference-type="eqref" reference="eq:conv_weakH1"}. Arguing as above and using a well-known result for convergence in $L^p$-spaces, we obtain convergences [\[eq:conv_weakL2H1\]](#eq:conv_weakL2H1){reference-type="eqref" reference="eq:conv_weakL2H1"}, [\[eq:conv_strongL2\]](#eq:conv_strongL2){reference-type="eqref" reference="eq:conv_strongL2"} and (upon extraction of a further subsequence) [\[eq:conv_pointwise\]](#eq:conv_pointwise){reference-type="eqref" reference="eq:conv_pointwise"} for $\{\boldsymbol{m}_{hk}\}$. The continuous inclusion $L^{\infty}(0,T;\boldsymbol{H}^1(\Omega)) \subset L^2(0,T;\boldsymbol{H}^1(\Omega))$, shows [\[eq:conv_weakstarLinftyH1\]](#eq:conv_weakstarLinftyH1){reference-type="eqref" reference="eq:conv_weakstarLinftyH1"} for $\{\boldsymbol{m}_{hk}\}$. Let $0 < s < 1$ be arbitrary. Since $\boldsymbol{H}^s(\Omega_T) = [\boldsymbol{L}^2(\Omega_T),\boldsymbol{H}^1(\Omega_T)]_s$ and $L^2(0,T;\boldsymbol{H}^s(\Omega)) = [\boldsymbol{L}^2(\Omega_T),L^2(0,T;\boldsymbol{H}^1(\Omega))]_s$, well-known results from interpolation theory (see, e.g., [@bl1976 Theorem 6.4.5 and Theorem 3.8.1] and [@bl1976 Theorem 5.1.2]) yield the compact embedding $\boldsymbol{H}^1(\Omega_T) \Subset \boldsymbol{H}^s(\Omega_T)$ and the continuous inclusion $\boldsymbol{H}^s(\Omega_T) \subset L^2(0,T;\boldsymbol{H}^s(\Omega))$. These in turn show convergences [\[eq:conv_strongHs\]](#eq:conv_strongHs){reference-type="eqref" reference="eq:conv_strongHs"} and [\[eq:conv_strongL2Hs\]](#eq:conv_strongL2Hs){reference-type="eqref" reference="eq:conv_strongL2Hs"} for $\{\boldsymbol{m}_{hk}\}$. Furthermore, [\[eq:conv_weakL2_v\]](#eq:conv_weakL2_v){reference-type="eqref" reference="eq:conv_weakL2_v"} follows directly from $\partial_t \boldsymbol{m}_{hk}=\boldsymbol{v}_{hk}^{-}$. Overall, this shows the convergence results [\[eq:conv_weakH1_u\]](#eq:conv_weakH1_u){reference-type="eqref" reference="eq:conv_weakH1_u"}--[\[eq:conv_strongL2_u\]](#eq:conv_strongL2_u){reference-type="eqref" reference="eq:conv_strongL2_u"} and [\[eq:conv_weakH1\]](#eq:conv_weakH1){reference-type="eqref" reference="eq:conv_weakH1"}--[\[eq:conv_weakL2_v\]](#eq:conv_weakL2_v){reference-type="eqref" reference="eq:conv_weakL2_v"} for the sequences $\{\boldsymbol{u}_{hk}\}$, $\{\boldsymbol{m}_{hk}\}$ and $\{\boldsymbol{v}_{hk}^-\}$. Using the same argument, one can obtain the same results for $\{ \boldsymbol{u}_{hk}^{\pm} \}$ and $\{ \boldsymbol{m}_{hk}^{\pm} \}$. Since the quantity $$\label{eq:quantity_to_identify_limits} \sum_{i=0}^{j-1} \left\lVert \boldsymbol{m}_{h}^{i+1}- \boldsymbol{m}_{h}^{i} \right\rVert_{}^2 + \sum_{i=0}^{j-1}\left\lVert \boldsymbol{u}_{h}^{i+1} - \boldsymbol{u}_{h}^{i} \right\rVert_{}^2 + \sum_{i=0}^{j-1} \left\lVert \mathrm{d}_t\boldsymbol{u}_{h}^{i+1} - \mathrm{d}_t\boldsymbol{u}_{h}^{i} \right\rVert_{}$$ is uniformly bounded, arguing as in [@bppr2013 Lemma 5.7] we can show that the limits of $\{ \boldsymbol{u}_{hk} \}$ and $\{ \boldsymbol{u}_{hk}^{\pm} \}$ (resp. $\{ \boldsymbol{m}_{hk} \}$ and $\{ \boldsymbol{m}_{hk}^{\pm} \}$) coincide. The continuous inclusion $L^{\infty}(0,T;\boldsymbol{L}^2(\Omega)) \subset L^2(0,T;\boldsymbol{L}^2(\Omega)) = \boldsymbol{L}^2(\Omega_T)$, the boundedness of the third term in [\[eq:quantity_to_identify_limits\]](#eq:quantity_to_identify_limits){reference-type="eqref" reference="eq:quantity_to_identify_limits"} and the identity $\partial_{t}\boldsymbol{u}_{hk} = \dot\boldsymbol{u}_{hk}^+$ imply [\[eq:conv_weakLinftyL2_ut\]](#eq:conv_weakLinftyL2_ut){reference-type="eqref" reference="eq:conv_weakLinftyL2_ut"}--[\[eq:conv_weakL2_ut\]](#eq:conv_weakL2_ut){reference-type="eqref" reference="eq:conv_weakL2_ut"}. Finally, the fact that $\boldsymbol{m}$ satisfies $\left\lvert \boldsymbol{m} \right\rvert=1$ a.e. in $\Omega$ follows from the available convergence results and [\[eq:general_constraint\]](#eq:general_constraint){reference-type="eqref" reference="eq:general_constraint"}. For the details, we refer to Step 3 of the proof of [@hpprss2019 Proposition 6]. This concludes the proof. ◻ Let $\{ \hat\boldsymbol{m}_{hk}^\pm\}$ be the piecewise constant time reconstructions defined using the projection of the approximate magnetisations, i.e, $\hat\boldsymbol{m}_{hk}^-(t) := \Pi_h \boldsymbol{m}_h^i$ and $\hat\boldsymbol{m}_{hk}^+(t) := \Pi_h \boldsymbol{m}_h^{i+1}$ for all $i=0,\dots,N_1$ and $t \in [t_i,t_{i+1})$ (cf. [\[eq:reconstructions\]](#eq:reconstructions){reference-type="eqref" reference="eq:reconstructions"}). In the following lemma, we establish further convergence results that will be needed to identify the limit $(\boldsymbol{u},\boldsymbol{m})$ constructed in Lemma [Lemma 18](#lem:convergent_subsequences){reference-type="ref" reference="lem:convergent_subsequences"} with a weak solution of [\[eq:newton\]](#eq:newton){reference-type="eqref" reference="eq:newton"}--[\[eq:ibc\]](#eq:ibc){reference-type="eqref" reference="eq:ibc"}. **Lemma 19** (auxiliary convergences). Under the assumptions of Theorem [Theorem 9](#thm:convergence){reference-type="ref" reference="thm:convergence"}(i), upon extraction of a further (non-relabeled) subsequence, we have the following convergence results: $$\begin{aligned} \label{eq:conv_mhat_LinftyH1} \hat\boldsymbol{m}_{hk}^\pm \overset{\ast}{\rightharpoonup}\boldsymbol{m}\quad &\text{in } L^\infty(0,T;\boldsymbol{H}^1(\Omega)), \\ \hat\boldsymbol{m}_{hk}^\pm \rightharpoonup\boldsymbol{m}\quad &\text{in } L^2(0,T;\boldsymbol{H}^1(\Omega)), \\ \label{eq:conv_mhat_L2} \hat\boldsymbol{m}_{hk}^\pm \to \boldsymbol{m}\quad &\text{in } \boldsymbol{L}^2(\Omega_T), \\ \label{eq:conv_m2_L2} \boldsymbol{m}_{hk}^\pm\otimes\boldsymbol{m}_{hk}^\pm \to \boldsymbol{m}\otimes\boldsymbol{m}\quad &\text{in } \boldsymbol{L}^2(\Omega_T), \\ \label{eq:conv_mhat2_L2} \hat\boldsymbol{m}_{hk}^\pm\otimes\hat\boldsymbol{m}_{hk}^\pm \to \boldsymbol{m}\otimes\boldsymbol{m}\quad &\text{in } \boldsymbol{L}^2(\Omega_T),\end{aligned}$$ as $h,k \to 0$. *Proof.* Firstly, we note that $\left\lVert \Pi_h \boldsymbol{m}_h^i \right\rVert_{\boldsymbol{L}^\infty(\Omega)} = 1$ and $\left\lVert \boldsymbol{\nabla}\Pi_h \boldsymbol{m}_h^i \right\rVert_{} \lesssim \left\lVert \boldsymbol{\nabla}\boldsymbol{m}_h^i \right\rVert_{} \lesssim 1$ for all $i=0,\dots,N$ (the estimate of the gradient follows from [\[eq:stability\]](#eq:stability){reference-type="eqref" reference="eq:stability"} and [\[eq:projection_stability\]](#eq:projection_stability){reference-type="eqref" reference="eq:projection_stability"}). We infer that the sequences $\{ \hat\boldsymbol{m}_{hk}^\pm \}$ are uniformly bounded in $L^\infty(0,T;\boldsymbol{H}^1(\Omega))$ and arguing as in the proof of Lemma [Lemma 18](#lem:convergent_subsequences){reference-type="ref" reference="lem:convergent_subsequences"}, we can extract subsequences satisfying the convergence properties in [\[eq:conv_mhat_LinftyH1\]](#eq:conv_mhat_LinftyH1){reference-type="eqref" reference="eq:conv_mhat_LinftyH1"}--[\[eq:conv_mhat_L2\]](#eq:conv_mhat_L2){reference-type="eqref" reference="eq:conv_mhat_L2"}. The fact that the limit is exactly the function $\boldsymbol{m}\in L^\infty(0,T;\boldsymbol{H}^1(\Omega;\mathbb{S}^2))$ constructed in Lemma [Lemma 18](#lem:convergent_subsequences){reference-type="ref" reference="lem:convergent_subsequences"} follows from Lemma [Lemma 17](#lem:projection_error_Lp){reference-type="ref" reference="lem:projection_error_Lp"} (applied $p=1$), which guarantee that $\hat\boldsymbol{m}_{hk}^{\pm} \to \boldsymbol{m}$ in $\boldsymbol{L}^1(\Omega_T)$, which in turn implies that the limit functions in $\boldsymbol{L}^2(\Omega_T)$, $L^2(0,T;\boldsymbol{H}^1(\Omega))$ and $L^\infty(0,T;\boldsymbol{H}^1(\Omega))$ must necessarily be the same. To show [\[eq:conv_m2_L2\]](#eq:conv_m2_L2){reference-type="eqref" reference="eq:conv_m2_L2"}--[\[eq:conv_mhat2_L2\]](#eq:conv_mhat2_L2){reference-type="eqref" reference="eq:conv_mhat2_L2"}, we note that for $\boldsymbol{x}, \boldsymbol{y} \in\mathbb{R}^3$ we have $$\boldsymbol{x}\otimes \boldsymbol{x} - \boldsymbol{y}\otimes \boldsymbol{y} = \frac{1}{2} [(\boldsymbol{x}+\boldsymbol{y})\otimes (\boldsymbol{x}-\boldsymbol{y}) + (\boldsymbol{x}-\boldsymbol{y})\otimes(\boldsymbol{x}+\boldsymbol{y})].$$ Let $3/4 \le s < 1$. Using the above identity and the continuous inclusion $\boldsymbol{H}^s(\Omega) \subset \boldsymbol{L}^4(\Omega)$ for all $s \ge 3/4$, for arbitrary $t \in (0,T)$, we have $$\begin{split} \left\lVert \boldsymbol{m}_{hk}^\pm(t)\otimes\boldsymbol{m}_{hk}^\pm(t) - \boldsymbol{m}(t)\otimes\boldsymbol{m}(t) \right\rVert_{} & \leq \left\lVert \boldsymbol{m}_{hk}^\pm(t) + \boldsymbol{m}(t) \right\rVert_{\boldsymbol{L}^4(\Omega)} \left\lVert \boldsymbol{m}(t) - \boldsymbol{m}_{hk}^\pm(t) \right\rVert_{\boldsymbol{L}^4(\Omega)} \\ & \lesssim \left\lVert \boldsymbol{m}_{hk}^\pm(t) + \boldsymbol{m}(t) \right\rVert_{\boldsymbol{H}^1(\Omega)} \left\lVert \boldsymbol{m}(t) - \boldsymbol{m}_{hk}^\pm(t) \right\rVert_{\boldsymbol{H}^{s}(\Omega)}. \end{split}$$ It follows that $$\begin{split} \left\lVert \boldsymbol{m}_{hk}^\pm\otimes\boldsymbol{m}_{hk}^\pm - \boldsymbol{m}\otimes\boldsymbol{m} \right\rVert_{\boldsymbol{L}^2(\Omega_T)} & \lesssim \left\lVert \boldsymbol{m}_{hk}^\pm + \boldsymbol{m} \right\rVert_{L^\infty(0,T;\boldsymbol{H}^1(\Omega))} \left\lVert \boldsymbol{m}- \boldsymbol{m}_{hk}^\pm \right\rVert_{L^2(0,T;\boldsymbol{H}^{s}(\Omega))}. \end{split}$$ Convergence [\[eq:conv_m2_L2\]](#eq:conv_m2_L2){reference-type="eqref" reference="eq:conv_m2_L2"} then follows from the uniform boundedness of both $\boldsymbol{m}_{hk}^\pm$ and $\boldsymbol{m}$ in $L^\infty(0,T;\boldsymbol{H}^1(\Omega))$ and the strong convergence [\[eq:conv_strongL2Hs\]](#eq:conv_strongL2Hs){reference-type="eqref" reference="eq:conv_strongL2Hs"} from Lemma [Lemma 18](#lem:convergent_subsequences){reference-type="ref" reference="lem:convergent_subsequences"}. The proof of [\[eq:conv_mhat2_L2\]](#eq:conv_mhat2_L2){reference-type="eqref" reference="eq:conv_mhat2_L2"} is identical (due to the use of the nodal projection, one can use the Hölder inequality $\left\lVert \cdot \right\rVert_{L^2} \le \left\lVert \cdot \right\rVert_{L^\infty}\left\lVert \cdot \right\rVert_{L^2}$). This concludes the proof. ◻ Now, we are in a position to prove Theorem [Theorem 9](#thm:convergence){reference-type="ref" reference="thm:convergence"}(i). *Proof of Theorem [Theorem 9](#thm:convergence){reference-type="ref" reference="thm:convergence"}(i).* We apply Lemma [Lemma 18](#lem:convergent_subsequences){reference-type="ref" reference="lem:convergent_subsequences"}, which yields $\boldsymbol{u}\in L^\infty(0,T;\boldsymbol{H}^1_D(\Omega))$ with $\partial_{t}\boldsymbol{u}\in L^\infty(0,T;\boldsymbol{L}^2(\Omega))$ and $\boldsymbol{m}\in L^\infty(0,T;\boldsymbol{H}^1(\Omega;\mathbb{S}^2))$ with $\partial_t \boldsymbol{m}\in L^2(0,T;\boldsymbol{L}^2(\Omega))$ as well as subsequences of $\{\boldsymbol{u}_{hk}\}$ and $\{\boldsymbol{m}_{hk}\}$ satisfying the desired convergence properties. This already shows that $\boldsymbol{u}$ and $\boldsymbol{m}$ satisfy property (i) of Definition [Definition 1](#def:weak){reference-type="ref" reference="def:weak"}. Property (iii) follows from the available convergence results, the continuity of the trace operator $\boldsymbol{H}^1(\Omega_T) \to \boldsymbol{H}^{1/2}(\Omega)$, and assumption [\[eq:convergence_initial_data\]](#eq:convergence_initial_data){reference-type="eqref" reference="eq:convergence_initial_data"} on the discrete initial data. To conclude the proof, it remains to show that property (ii) holds, i.e., that $\boldsymbol{u}$ and $\boldsymbol{m}$ satisfy the variational formulations [\[eq:weak_u\]](#eq:weak_u){reference-type="eqref" reference="eq:weak_u"} and [\[eq:weak_m\]](#eq:weak_m){reference-type="eqref" reference="eq:weak_m"}, respectively. The result follows from the convergence properties established in Lemmas [Lemma 18](#lem:convergent_subsequences){reference-type="ref" reference="lem:convergent_subsequences"}--[Lemma 19](#lem:convergent_subsequences_aux){reference-type="ref" reference="lem:convergent_subsequences_aux"}. We omit the details, because - the proof that $\boldsymbol{u}$ satisfies [\[eq:weak_u\]](#eq:weak_u){reference-type="eqref" reference="eq:weak_u"} is identical to the one presented in [@bppr2013 page 1378], which is a consequence of the fact that in the displacement update [\[alg1:displacement_update\]](#alg1:displacement_update){reference-type="eqref" reference="alg1:displacement_update"} we employ the nodal projection for the magnetisation appearing on the right-hand side (our generalised setting involving a more general expression for the magnetostrain, body forces and traction does not pose further mathematical challenges here). - the proof that $\boldsymbol{m}$ satisfies [\[eq:weak_m\]](#eq:weak_m){reference-type="eqref" reference="eq:weak_m"} can be obtained combining the argument of [@bppr2013 pages 1376--1378] (which show convergence of the method with nodal projection towards a variational formulation of the LLG equation with magnetoelastic term) with the one of [@hpprss2019 page 1363] (where the modifications due to the omission of the nodal projection are presented). This concludes the proof. ◻ ## Energy inequality {#sec:proofs_energy_inequality} In this section, we use the compact notation $\hat\boldsymbol{m}_h = \Pi_h\boldsymbol{m}_h$ to denote the nodal projection of a general magnetisation approximation $\boldsymbol{m}_h$. To start with, in the following proposition, we state a variant of Proposition [Proposition 6](#prop:energy){reference-type="ref" reference="prop:energy"} for the discrete energy $$\hat\mathcal{E}_h[\boldsymbol{u}_h,\boldsymbol{m}_h] = \frac{1}{2}\left\lVert \boldsymbol{\nabla}\boldsymbol{m}_h \right\rVert_{}^2 + \frac{1}{2} \left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_h) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_h) \right\rVert_{\mathbb{C}}^2 - \langle \boldsymbol{f},\boldsymbol{u}_h \rangle_{} - \langle \boldsymbol{g},\boldsymbol{u}_h \rangle_{\Gamma_{N}},$$ which is obtained from [\[eqn:definition_of_energy\]](#eqn:definition_of_energy){reference-type="eqref" reference="eqn:definition_of_energy"} by applying the nodal projection to the discrete magnetisation appearing in the elastic energy. We omit the proof since it is very similar to the one of Proposition [Proposition 6](#prop:energy){reference-type="ref" reference="prop:energy"}. **Proposition 20**. For every integer $0 \le i \le N-1$, the iterates of Algorithm [Algorithm 4](#algorithm){reference-type="ref" reference="algorithm"} satisfy the discrete energy law $$\label{eq:modified_discrete_energy_law} \hat\mathcal{E}_h[\boldsymbol{u}_h^{i+1},\boldsymbol{m}_h^{i+1}] + \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_h^{i+1} \right\rVert_{}^2 - \hat\mathcal{E}_h[\boldsymbol{u}_h^i,\boldsymbol{m}_h^i] - \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_h^i \right\rVert_{}^2 = - \alpha k \left\lVert \boldsymbol{v}_h^i \right\rVert_{h}^2 - \hat D_{h,k}^i - \hat E_{h,k}^i,$$ where $\hat D_{h,k}^i$ and $\hat E_{h,k}^i$ are given by $$\begin{gathered} \hat D_{h,k}^i := k^2 (\theta-1/2) \left\lVert \boldsymbol{\nabla}\boldsymbol{v}_h^i \right\rVert_{}^2 + \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_h^{i+1} - \mathrm{d}_t\boldsymbol{u}_h^i \right\rVert_{}^2 \\ + \frac{1}{2} \left\lVert [\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1})-\boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^{i+1})] - [\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^i)-\boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^i)] \right\rVert_{\mathbb{C}}^2 \ge 0\end{gathered}$$ and $$\begin{split} \hat E_{h,k}^i & := \sum_{\ell=1}^5 \hat E_{h,k,\ell}^i \\ & := \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^{i+1},\hat\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i),\boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^i) \rangle_{} \\ & \quad + \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i),\boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1}) \rangle_{} + \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i),\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^i) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^i) \rangle_{} \\ & \quad + 2k \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i),\mathbb{Z}:[(\hat\boldsymbol{m}_{h}^{i} - \boldsymbol{m}_{h}^{i}) \otimes \boldsymbol{v}_{h}^{i}] \rangle_{} + k^2 \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i),\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{v}_{h}^i) \rangle_{}. \end{split}$$ respectively. Now, we are in a position to prove Theorem [Theorem 9](#thm:convergence){reference-type="ref" reference="thm:convergence"}(ii). *Proof of Theorem [Theorem 9](#thm:convergence){reference-type="ref" reference="thm:convergence"}(ii).* Let $t' \in (0,T)$. Let $1 \le j \le N$ such that $t' \in (t_{j-1},t_j)$. Summing [\[eq:modified_discrete_energy_law\]](#eq:modified_discrete_energy_law){reference-type="eqref" reference="eq:modified_discrete_energy_law"} for $i=0,\dots,j-1$ yields $$\hat\mathcal{E}_h[\boldsymbol{u}_h^j,\boldsymbol{m}_h^j] + \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_h^j \right\rVert_{}^2 - \hat\mathcal{E}_h[\boldsymbol{u}_h^0,\boldsymbol{m}_h^0] - \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_h^0 \right\rVert_{}^2 + \alpha k \sum_{i=0}^{j-1}\left\lVert \boldsymbol{v}_h^i \right\rVert_{h}^2 + \sum_{i=0}^{j-1} \hat D_{h,k}^i = - \sum_{i=0}^{j-1} \hat E_{h,k}^i.$$ Using the Cauchy--Schwarz inequality, the weighted Young inequality, and Lemma [Lemma 15](#lem:lipschitz_magnetostrain){reference-type="ref" reference="lem:lipschitz_magnetostrain"}, we obtain the estimate $$\begin{split} \lvert E_{h,k,1}^i \rvert & = \lvert \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^{i+1},\hat\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i),\boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^i) \rangle_{} \rvert \\ & \le \left\lVert [\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^{i+1})] - [\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^i) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^i)] \right\rVert_{\mathbb{C}} \left\lVert \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^i) \right\rVert_{\mathbb{C}} \\ & \le \frac{1}{4} \left\lVert [\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^{i+1})] - [\boldsymbol{\varepsilon}(\boldsymbol{u}_{h}^i) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^i)] \right\rVert_{\mathbb{C}}^2 \\ & \quad + C_{\mathrm{m}}^2 \left\lVert \mathbb{C} \right\rVert_{\boldsymbol{L}^{\infty}(\Omega)} k^2 \left\lVert \boldsymbol{v}_{h}^i \right\rVert_{}^2. \end{split}$$ We now estimate $E_{h,k,4}^i$ (assuming $i \ge 1$, because $E_{h,k,4}^0=0$ as $\hat\boldsymbol{m}_{h}^0 = \boldsymbol{m}_{h}^0$ by assumption). Using the Cauchy--Schwarz inequality, the Hölder inequality (for $p=2/(1-2\varepsilon)$ and $p' = 2p/(p-2)$ with $0 < \varepsilon\ll 1/2$ arbitrary), Lemma [Lemma 17](#lem:projection_error_Lp){reference-type="ref" reference="lem:projection_error_Lp"}, and classical inverse estimates (see, e.g. [@bartels2015 Lemma 3.5]), we obtain $$\begin{split} \lvert E_{h,k,4}^i \rvert & = 2k \lvert \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i),\mathbb{Z}:[(\hat\boldsymbol{m}_{h}^{i} - \boldsymbol{m}_{h}^{i}) \otimes \boldsymbol{v}_{h}^{i}] \rangle_{} \rvert \\ & \le 2 \left\lVert \mathbb{Z} \right\rVert_{\boldsymbol{L}^\infty(\Omega)} k \left\lVert \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i) \right\rVert_{} \left\lVert \hat\boldsymbol{m}_{h}^{i} - \boldsymbol{m}_{h}^{i} \right\rVert_{\boldsymbol{L}^p(\Omega)}\left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{\boldsymbol{L}^{p'}(\Omega)} \\ & \lesssim k \left\lVert \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i) \right\rVert_{} k^{(p+1)/p} \left( \sum_{\ell=0}^{i-1} \left\lVert \boldsymbol{v}_{h}^{\ell} \right\rVert_{\boldsymbol{L}^{2p}(\Omega)}^2 \right) \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{\boldsymbol{L}^{p'}(\Omega)} \\ & \lesssim k^{2+1/p} \left\lVert \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i) \right\rVert_{} h_{\min}^{3(1-p)/p} \left( \sum_{\ell=0}^{i-1} \left\lVert \boldsymbol{v}_{h}^{\ell} \right\rVert_{}^2 \right) h_{\min}^{3(2-p')/(2p')} \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{} \\ & \lesssim h_{\min}^{-3 } k^{5/2-\varepsilon} \left\lVert \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i) \right\rVert_{} \left( \sum_{\ell=0}^{i-1} \left\lVert \boldsymbol{v}_{h}^{\ell} \right\rVert_{}^2 \right) \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{}. \end{split}$$ Similarly, we obtain $$\lvert E_{h,k,5}^i \rvert = \lvert k^2 \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i),\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{v}_{h}^i) \rangle_{} \rvert \lesssim h_{\min}^{-3/2} k^2 \left\lVert \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i) \right\rVert_{} \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{}^2.$$ Moreover, noting $\boldsymbol{m}_{h}^{0} = \hat{\boldsymbol{m}}_{h}^{0}$ we have that $$\begin{split} & \sum_{i=0}^{j-1} \left(\hat E_{h,k,2}^i + \hat E_{h,k,3}^i \right) \\ & \quad = \sum_{i=0}^{j-1} \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i),\boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^{i+1}) \rangle_{} + \sum_{i=0}^{j-1} \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i),\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^i) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^i) \rangle_{} \\ & \quad = \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^0,\hat\boldsymbol{m}_{h}^0),\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^0) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^0) \rangle_{} - \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^{j-1},\hat\boldsymbol{m}_{h}^{j-1}),\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^j) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^j) \rangle_{} \\ & \qquad + \sum_{i=0}^{j-2} \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^{i+1},\hat\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i),\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^i) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^i) \rangle_{}.\\ & \quad = - \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^{j-1},\hat\boldsymbol{m}_{h}^{j-1}),\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^j) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^j) \rangle_{} \\ & \qquad + \sum_{i=0}^{j-2} \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^{i+1},\hat\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i),\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^i) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^i) \rangle_{} \end{split}$$ Using inverse estimates, Lemma [Lemma 15](#lem:lipschitz_magnetostrain){reference-type="ref" reference="lem:lipschitz_magnetostrain"} and Lemma [Lemma 17](#lem:projection_error_Lp){reference-type="ref" reference="lem:projection_error_Lp"}, we obtain the estimate $$\begin{split} & \lvert \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^{i+1},\hat\boldsymbol{m}_{h}^{i+1}) - \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i),\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^i) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^i) \rangle_{} \rvert \\ & \quad \lesssim \left(\left\lVert \boldsymbol{\varepsilon}(\boldsymbol{u}_h^{i+1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_h^i) \right\rVert_{} + \left\lVert \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_h^{i+1}) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_h^i) \right\rVert_{} \right) \left\lVert \boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^i) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^i) \right\rVert_{} \\ & \quad \lesssim \left(h_{\min}^{-1} k \left\lVert \mathrm{d}_t\boldsymbol{u}_h^{i+1} \right\rVert_{} + k^2 \left\lVert \boldsymbol{v}_h^i \right\rVert_{}^2 \right) \left\lVert \boldsymbol{m}_h^i + \hat\boldsymbol{m}_h^i \right\rVert_{\boldsymbol{H}^1(\Omega)} k^{4/3} h_{\min}^{-2} \sum_{\ell=0}^{i-1} \left\lVert \boldsymbol{v}_h^\ell \right\rVert_{}^2. \end{split}$$ Altogether, omitting all non-negative dissipative terms and using the stability from Proposition [Proposition 8](#prop:stability){reference-type="ref" reference="prop:stability"}, we thus obtain $$\begin{split} & \hat\mathcal{E}_h[\boldsymbol{u}_h^j,\boldsymbol{m}_h^j] + \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_h^j \right\rVert_{}^2 - \hat\mathcal{E}_h[\boldsymbol{u}_h^0,\boldsymbol{m}_h^0] - \frac{1}{2} \left\lVert \mathrm{d}_t\boldsymbol{u}_h^0 \right\rVert_{}^2 + \alpha k \sum_{i=0}^{j-1}\left\lVert \boldsymbol{v}_h^i \right\rVert_{h}^2 \\ & \qquad\quad - \langle \boldsymbol{\sigma}(\boldsymbol{u}_{h}^{j-1},\hat\boldsymbol{m}_{h}^{j-1}),\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{h}^j) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{h}^j) \rangle_{} \\ & \quad \lesssim \sum_{i=0}^{j-1} \Big( k^2 \left\lVert \boldsymbol{v}_h^i \right\rVert_{}^2 + h_{\min}^{-3} k^{4/3} \left\lVert \mathrm{d}_t\boldsymbol{u}_h^{i+1} \right\rVert_{}\left\lVert \boldsymbol{m}_h^i + \hat\boldsymbol{m}_h^i \right\rVert_{\boldsymbol{H}^1(\Omega)} + h_{\min}^{-2} k^{7/3} \left\lVert \boldsymbol{v}_h^i \right\rVert_{}^2 \\ & \qquad\quad + h_{\min}^{-3} k^{3/2-\varepsilon} \left\lVert \boldsymbol{\sigma}(\boldsymbol{u}_{h}^i,\hat\boldsymbol{m}_{h}^i) \right\rVert_{}\left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{} + h_{\min}^{-3/2} k^2 \left\lVert \boldsymbol{v}_{h}^{i} \right\rVert_{}^2 \Big) \\ & \quad \lesssim k + h_{\min}^{-3} k^{1/3} + h_{\min}^{-2} k^{4/3} + h_{\min}^{-3} k^{1/2-\varepsilon} + h_{\min}^{-3/2} k. \end{split}$$ Using [\[eq:h-scalar-product_equivalence\]](#eq:h-scalar-product_equivalence){reference-type="eqref" reference="eq:h-scalar-product_equivalence"}, rewriting the above using the time reconstructions [\[eq:reconstructions\]](#eq:reconstructions){reference-type="eqref" reference="eq:reconstructions"} and integrating in time over an arbitrary measurable set $\mathfrak{T} \subset [0,T]$, we obtain $$\begin{split} & \int_{\mathfrak{T}} \left( \mathcal{E}[\boldsymbol{u}_{hk}^+(t'),\boldsymbol{m}_{hk}^+(t')] + \frac{1}{2} \left\lVert \dot\boldsymbol{u}_{hk}^+(t') \right\rVert_{}^2 - \hat\mathcal{E}_h[\boldsymbol{u}_{hk}^-(0),\boldsymbol{m}_{hk}^-(0)] - \frac{1}{2} \left\lVert \dot\boldsymbol{u}_{hk}^-(0) \right\rVert_{}^2 \right) \mathrm{d}t' \\ & \qquad + \int_{\mathfrak{T}} \left( \alpha \int_0^{t'} \left\lVert \boldsymbol{v}_{hk}^-(t) \right\rVert_{}^2 \mathrm{d}t \right) \mathrm{d}t' + \int_{\mathfrak{T}} \left( \alpha \int_{t'}^{t_j} \left\lVert \boldsymbol{v}_{hk}^-(t) \right\rVert_{}^2 \mathrm{d}t \right) \mathrm{d}t' \\ & \qquad - \int_{\mathfrak{T}} \langle \boldsymbol{\sigma}(\boldsymbol{u}_{hk}^-(t'),\hat\boldsymbol{m}_{hk}^-(t')),\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}_{hk}^+(t')) - \boldsymbol{\varepsilon}_{\mathrm{m}}(\hat\boldsymbol{m}_{hk}^+(t')) \rangle_{} \mathrm{d}t' \\ & \quad \lesssim k + h_{\min}^{-3} k^{1/3} + h_{\min}^{-2} k^{4/3} + h_{\min}^{-3} k^{1/2-\varepsilon} + h_{\min}^{-3/2} k. \end{split}$$ We now consider the limit of this inequality as $h,k \to 0$. The assumed CFL condition $k=o(h^9)$ implies that the right-hand side converges to $0$ in the limit as $h,k \to 0$. The last two terms on the left-hand side converge to $0$: the first one by no concentration of Lebesgue functions, the other thanks to the available convergence results (cf. the convergences guaranteed by Lemmas [Lemma 18](#lem:convergent_subsequences){reference-type="ref" reference="lem:convergent_subsequences"}--[Lemma 19](#lem:convergent_subsequences_aux){reference-type="ref" reference="lem:convergent_subsequences_aux"}). Weak lower semicontinuity guarantees $$\begin{gathered} \int_{\mathfrak{T}} \left( \mathcal{E}[\boldsymbol{u}(t'),\boldsymbol{m}(t')] + \frac{1}{2} \left\lVert \partial_t\boldsymbol{u}(t') \right\rVert_{}^2 + \alpha \int_0^{t'} \left\lVert \partial_t \boldsymbol{m}(t) \right\rVert_{}^2 \mathrm{d}t\right) \mathrm{d}t' \\ \le \liminf_{h,k \to 0} \int_{\mathfrak{T}} \left( \hat\mathcal{E}_h[\boldsymbol{u}_{hk}^+(t'),\boldsymbol{m}_{hk}^+(t')] + \frac{1}{2} \left\lVert \dot\boldsymbol{u}_{hk}^+(t') \right\rVert_{}^2 + \alpha \int_0^{t'} \left\lVert \boldsymbol{v}_{hk}^-(t) \right\rVert_{}^2 \mathrm{d}t\right) \mathrm{d}t'.\end{gathered}$$ Assumption [\[eq:convergence_initial_data\]](#eq:convergence_initial_data){reference-type="eqref" reference="eq:convergence_initial_data"} yields $$\lim_{h,k \to 0} \left( \hat\mathcal{E}_h[\boldsymbol{u}_{hk}^-(0),\boldsymbol{m}_{hk}^-(0)] + \frac{1}{2} \left\lVert \dot\boldsymbol{u}_{hk}^-(0) \right\rVert_{}^2 \right) = \mathcal{E}[\boldsymbol{u}^0,\boldsymbol{m}^0] + \frac{1}{2} \left\lVert \dot\boldsymbol{u}^0 \right\rVert_{}^2.$$ Since $\mathfrak{T} \subset [0,T]$ was arbitrary, this shows that the energy inequality [\[eqn:EnergyLawDefinition\]](#eqn:EnergyLawDefinition){reference-type="eqref" reference="eqn:EnergyLawDefinition"} holds a.e. in $(0,T)$ and concludes the proof. ◻ # Acknowledgements {#acknowledgements .unnumbered} MR is a member of the 'Gruppo Nazionale per il Calcolo Scientifico (GNCS)' of the Italian 'Istituto Nazionale di Alta Matematica (INdAM)'. The authors thank Martin Kružík (Institute of Information Theory and Automation, Czech Academy of Sciences) for several interesting discussions on magnetoelasticity. The support of the Royal Society (grant IES\\R2\\222118) is thankfully acknowledged. # Linear algebra definitions and identities {#sec:linear_algebra} In this section, for the convenience of the reader, we collect some definitions and vector/matrix/tensor identities from linear algebra that are used throughout the work. **Definition 21**. Let $\mathbb{A}\in \mathbb{R}^{3^4}$ be a fourth-order tensor (4-tensor) with components $\mathbb{A}_{ij\ell m}$, where $i,j,\ell,m = 1,2,3$. We say that 1. $\mathbb{A}$ is minorly symmetric if $\mathbb{A}_{ij\ell m} = \mathbb{A}_{ji\ell m}=\mathbb{A}_{ij m \ell}$, 2. $\mathbb{A}$ is majorly symmetric if $\mathbb{A}_{ij\ell m} = \mathbb{A}_{\ell mji}$, 3. and $\mathbb{A}$ is (fully) symmetric if the aforementioned two conditions hold together. The transpose of $\mathbb{A}$ is the 4-tensor $\mathbb{A}^\top \in \mathbb{R}^{3^4}$ given by $(\mathbb{A}^\top)_{ij\ell m} = \mathbb{A}_{\ell mji}$. In particular, $\mathbb{A}$ is majorly symmetric if $\mathbb{A}^\top = \mathbb{A}$. **Remark 22**. In three dimensions, 4-tensors have $3^4=81$ components. Minorly symmetric 4-tensors have 36 independent components, majorly symmetric 4-tensors have 45 independent components, and fully symmetric 4-tensors have 21 independent components. Throughout this work the stiffness tensor $\mathbb{C}$ is assumed to be fully symmetric, whereas the magnetostriction tensor $\mathbb{Z}$ is assumed to be only minorly symmetric. In the numerical experiments of Section [5](#sec:numerics){reference-type="ref" reference="sec:numerics"}, we consider the isotropic case, in which $\mathbb{C}$ and $\mathbb{Z}$ have only two (the so-called Lamé constants) and one (the so-called saturation magnetostriction) independent components, respectively. In the following definition, we recall some operations between tensors. **Definition 23**. Let $\mathbb{A},\mathbb{B} \in \mathbb{R}^{3^4}$ be 4-tensors, let $\boldsymbol\nu,\boldsymbol\mu \in \mathbb{R}^{3 \times 3}$ be 2-tensors (matrices), and let $\boldsymbol{m},\boldsymbol{w}\in \mathbb{R}^3$ be 1-tensors (vectors). - We denote the double contraction between $\mathbb{A}$ and $\mathbb{B}$ as the 4-tensor $\mathbb{A}:\mathbb{B} \in \mathbb{R}^{3^4}$ given by $$(\mathbb{A}:\mathbb{B})_{ij\ell m} = \sum_{p,q}\mathbb{A}_{ijpq}\mathbb{B}_{pq\ell m}.$$ - We denote the double contraction between $\mathbb{A}$ and $\boldsymbol\nu$ as the 2-tensor $\mathbb{A}: \boldsymbol\nu \in \mathbb{R}^{3 \times 3}$ given by $$(\mathbb{A}: \boldsymbol\nu)_{ij} = \sum_{\ell,m}\mathbb{A}_{ij\ell m}\nu_{\ell m}.$$ - We denote the Frobenius product of $\boldsymbol\mu$ and $\boldsymbol\nu$ as the scalar $\boldsymbol\mu:\boldsymbol\nu \in \mathbb{R}$ given by $$\boldsymbol\mu : \boldsymbol\nu = \sum_{i,j}\mu_{ij} \nu_{ij}.$$ - We denote the tensor product of $\boldsymbol{m}$ and $\boldsymbol{w}$ as the 2-tensor $\boldsymbol{m}\otimes\boldsymbol{w}\in \mathbb{R}^{3\times 3}$ given by $$(\boldsymbol{m}\otimes\boldsymbol{w})_{ij} = m_{i}w_{j}.$$ The following result is useful for manipulation of the magnetostrain terms. **Lemma 24**. Let $\mathbb{Z}\in \mathbb{R}^{3^4}$ be a minorly symmetric 4-tensor, let $\boldsymbol{\sigma}\in \mathbb{R}^{3 \times 3}$ be a symmetric 2-tensor, and let $\boldsymbol{m},\boldsymbol{w}\in\mathbb{R}^3$ be two 1-tensors. We have the identity $$\label{eq:tensor_identity} [(\mathbb{Z}^\top:\boldsymbol{\sigma})\boldsymbol{w}]\cdot\boldsymbol{m}=[\mathbb{Z}^\top:\boldsymbol{\sigma})\boldsymbol{m}]\cdot\boldsymbol{w}= \boldsymbol{\sigma}:[\mathbb{Z}:(\boldsymbol{m}\otimes\boldsymbol{w})].$$ *Proof.* We have by the minor symmetry of $\mathbb{Z}$ that $$\begin{aligned} (\mathbb{Z}^\top:\boldsymbol{\sigma}) \boldsymbol{m}\cdot \boldsymbol{w} &= \sum_{i,j,\ell,m}\mathbb{Z}_{ijk\ell}^\top\sigma_{\ell m}m_{j}w_{i}\\ &= \sum_{i,j,\ell,m}\mathbb{Z}_{jik\ell}^\top\sigma_{\ell m}m_{i}w_{j}\quad\textnormal{by relabelling}\\ &= \sum_{i,j,\ell,m} \mathbb{Z}_{ijk\ell}^\top\sigma_{\ell m}w_{j}m_{i}\quad\textnormal{via minor symmetry}\\ &= (\mathbb{Z}^\top:\boldsymbol{\sigma}) \boldsymbol{w}\cdot \boldsymbol{m}.\end{aligned}$$ Furthermore, we have that $$\begin{aligned} (\mathbb{Z}^\top:\boldsymbol{\sigma}) \boldsymbol{m}\cdot \boldsymbol{w} &= \sum_{i,j,\ell,m} \mathbb{Z}_{ij\ell m}^\top\sigma_{\ell m}m_{j}w_{i} = \sum_{i,j,\ell,m} \sigma_{\ell m}\mathbb{Z}_{ij\ell m}^\top m_{j}w_{i}\\ &= \sum_{i,j,\ell,m} \sigma_{\ell m}\mathbb{Z}_{\ell mji}m_{j}w_{i}\quad\textnormal{via minor symmetry}\\ &= \sum_{\ell,m} \sigma_{\ell m}[\mathbb{Z}:(\boldsymbol{m}\otimes\boldsymbol{w})]_{\ell m} = \boldsymbol{\sigma}:[\mathbb{Z}:(\boldsymbol{m}\otimes\boldsymbol{w})].\end{aligned}$$ This show both identities in [\[eq:tensor_identity\]](#eq:tensor_identity){reference-type="eqref" reference="eq:tensor_identity"}. ◻ The following identity is useful to show the stability of numerical schemes. **Lemma 25**. Let $\{\nu_i\}$ be a sequence in an inner product space with inner product $\langle \cdot,\cdot \rangle_{}$ and associated norm $\left\lVert \cdot \right\rVert_{}$. We have the identity $$\label{eq:abel} \langle \nu_{i+1} - \nu_{i},\nu_{i+1} \rangle_{} = \frac{1}{2}\left\lVert \nu_{i+1} \right\rVert_{}^2 - \frac{1}{2}\left\lVert \nu_{i} \right\rVert_{}^2 + \frac{1}{2}\left\lVert \nu_{i+1} - \nu_{i} \right\rVert_{}^2.$$ # Nondimensionalisation {#sec:physics} Let $\Omega \subset \mathbb{R}^3$ denote the volume occupied by a ferromagnetic body (with the spatial variable $x \in \Omega$ measured in meter). Consider the magnetisation $\boldsymbol{M}$ (measured in A/m), which satisfies the length constraint $\left\lvert \boldsymbol{M} \right\rvert = M_s$, where the saturation magnetisation $M_s>0$ is also measured in A/m, and the displacement $\boldsymbol{U}$ (measured in m). We denote by $\boldsymbol{\varepsilon}(\boldsymbol{U})$ the total strain given by $$\boldsymbol{\varepsilon}(\boldsymbol{U}) = (\boldsymbol{\nabla}\boldsymbol{U}+ \boldsymbol{\nabla}\boldsymbol{U}^\top)/2$$ and by $\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{M})$ the magnetostrain given by $$\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{M})=\mathbb{Z}:(\boldsymbol{M}\otimes \boldsymbol{M}/M_s^2),$$ where $\mathbb{Z}$ is a dimensionless fourth-order tensor. The total energy of the system (measured in J) is given by $$\begin{gathered} \mathcal{E}[\boldsymbol{U},\boldsymbol{M}] = \frac{A}{M_s^2}\int_{\Omega}|\boldsymbol{\nabla}\boldsymbol{M}|^2 - \mu_0 \int_{\Omega} \boldsymbol{H}_{\mathrm{ext}} \cdot \boldsymbol{M} \\ + \frac{1}{2}\int_{\Omega}[\boldsymbol{\varepsilon}(\boldsymbol{U}) -\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{M})]:\{\boldsymbol{C}:[\boldsymbol{\varepsilon}(\boldsymbol{U}) -\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{M})]\} - \int_{\Omega}\boldsymbol{F}\cdot\boldsymbol{U} - \int_{\Gamma_N}\boldsymbol{G}\cdot\boldsymbol{U},\end{gathered}$$ where $A$ is the exchange constant (measured in J m^-1^), $\mu_0$ is the permeability of free space (measured in N A^-2^), $\boldsymbol{H}_{\mathrm{ext}}$ is an applied external field (measured in A/m), $\boldsymbol{C}$ is the fourth-order stiffness tensor (measured in N m^-2^), $\boldsymbol{F}$ is a body force (measured in N m^-3^), and $\boldsymbol{G}$ is a surface force (measured in N m^-2^). The dynamics of $\boldsymbol{M}$ is described by the LLG equation: $$\partial_t \boldsymbol{M} = -\gamma \mu_{0} \, \boldsymbol{M} \times \boldsymbol{H}_{\mathrm{eff}}[\boldsymbol{U},\boldsymbol{M}] + \frac{\alpha}{M_s}\boldsymbol{M} \times \partial_t \boldsymbol{M},$$ where $\gamma$ is the gyromagnetic ratio (measured in rad s^-1^T^-1^), $\alpha>0$ is the dimensionless Gilbert damping parameter, and the effective field $\boldsymbol{H}_{\mathrm{eff}}$ (measured in A/m) reads as $$\boldsymbol{H}_{\mathrm{eff}}[\boldsymbol{U},\boldsymbol{M}] = - \frac{1}{\mu_0} \frac{\delta\mathcal{E}[\boldsymbol{U},\boldsymbol{M}]}{\delta\boldsymbol{M}} = \frac{2A}{\mu_0 M_s^2}\boldsymbol\Delta \boldsymbol{M} + \boldsymbol{H}_{\mathrm{ext}} + \frac{2}{\mu_0 M_s^2}[\mathbb{Z}^\top :\boldsymbol\Sigma(\boldsymbol{U},\boldsymbol{M})]\boldsymbol{M},$$ where $\boldsymbol{\Sigma}(\boldsymbol{U},\boldsymbol{M}) = \boldsymbol{C}:[\boldsymbol{\varepsilon}(\boldsymbol{U}) -\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{M})]$ is the stress (measured in N m^-2^). The LLG equation is coupled with the conservation of momentum equation satisfied by the displacement: $$\rho \, \partial_{tt} \boldsymbol{U}= \boldsymbol{\nabla}\cdot \boldsymbol{\Sigma}(\boldsymbol{U},\boldsymbol{M}) + \boldsymbol{F},$$ where $\rho$ is the mass density (measured in kg m^-3^). Let $\boldsymbol{m}= \boldsymbol{M} / M_s$ denote the normalised magnetisation. We define the exchange length $\ell_{\textnormal{ex}}^2 = 2A / \mu_0 M_s^2$ (measured in m) and use it to rescale the spatial variable and the displacement according to $x' = x/\ell_{\textnormal{ex}}$ and $\boldsymbol{u}= \boldsymbol{U}/ \ell_{\textnormal{ex}}$, respectively. Additionally we introduce the dimensionless domain $\Omega'=\Omega/\ell_{\textnormal{ex}}$, the dimensionless time $t' = \gamma \mu_0 M_s t$, the dimensionless coupling parameter $\kappa = \rho \ell_{\textnormal{ex}}^2\gamma^2 \mu_0$, as well as the dimensionless differential operators $\boldsymbol{\nabla}=\boldsymbol{\nabla}'/\ell_{\textnormal{ex}}$ and $\boldsymbol\Delta=\boldsymbol\Delta'/\ell_{\textnormal{ex}}^2$. Further we define the dimensionless energy as $$\begin{split} \mathcal{E}'[\boldsymbol{u},\boldsymbol{m}] & = \frac{\mathcal{E}[\ell_{\textnormal{ex}}\boldsymbol{U},M_s \boldsymbol{m}]}{\mu_0 M_s^2\ell_{\textnormal{ex}}^3} \\ & = \frac{1}{2}\int_{\Omega'}|\boldsymbol{\nabla}' \boldsymbol{m}|^2 - \int_{\Omega'} \boldsymbol{h}_{\mathrm{ext}} \cdot\boldsymbol{m}\\ & \qquad + \frac{\kappa}{2}\int_{\Omega'}[\boldsymbol{\varepsilon}(\boldsymbol{u}) - \boldsymbol{\varepsilon}_{\mathrm{m}}'(\boldsymbol{m})]: \{\mathbb{C}:[\boldsymbol{\varepsilon}(\boldsymbol{u}) -\boldsymbol{\varepsilon}_{\mathrm{m}}'(\boldsymbol{m})] - \kappa\int_{\Omega'}\boldsymbol{f}\cdot\boldsymbol{u} - \kappa\int_{\Gamma_N'}\boldsymbol{g}\cdot\boldsymbol{u}, \end{split}$$ where $\boldsymbol{h}_{\mathrm{ext}} = \boldsymbol{H}_{\mathrm{ext}} / M_s$, $\boldsymbol{\varepsilon}_{\mathrm{m}}(\boldsymbol{m}) = \mathbb{Z}:(\boldsymbol{m}\otimes\boldsymbol{m})$, $\boldsymbol\Sigma = \kappa \mu_0 M_s^2\boldsymbol{\sigma}$, $\boldsymbol{G} = \kappa \mu_0 M_s^2\boldsymbol{g}$, $\boldsymbol{F}= \kappa(\mu_0 M_s^2/\ell_{\textnormal{ex}})\boldsymbol{f}$, and $\boldsymbol{C}= \kappa \mu_0 M_s^2\mathbb{C}$ (all dimensionless). The dimensionless effective field, defined by $\boldsymbol{h}_{\mathrm{eff}}[\boldsymbol{u},\boldsymbol{m}] = \boldsymbol{H}_{\mathrm{eff}}[\ell_{\textnormal{ex}}\boldsymbol{u},M_s\boldsymbol{m}]/M_s$, satisfies the relation $$\boldsymbol{h}_{\mathrm{eff}}[\boldsymbol{u},\boldsymbol{m}] = -\frac{\delta \mathcal{E}'[\boldsymbol{u},\boldsymbol{m}]}{\delta \boldsymbol{m}} = \boldsymbol\Delta'\boldsymbol{m}+ 2\kappa[\mathbb{Z}^\top :\boldsymbol{\sigma}(\boldsymbol{u},\boldsymbol{m})]\boldsymbol{m}+ \boldsymbol{h}_{\mathrm{ext}}.$$ With all these definitions, we retrieve the coupled system $$\begin{aligned} \partial_{t't'} \boldsymbol{u}&= \boldsymbol{\nabla}\cdot \boldsymbol{\sigma}(\boldsymbol{u},\boldsymbol{m}) + \boldsymbol{f},\\ \partial_{t'} \boldsymbol{m}&= -\boldsymbol{m}\times \boldsymbol{h}_{\textnormal{eff}}[\boldsymbol{u},\boldsymbol{m}] + \alpha \, \boldsymbol{m}\times \partial_{t'} \boldsymbol{m}.\end{aligned}$$ Altogether, we thus obtain the dimensionless model problem discussed throughout this work. Note that, to simplify the notation, in Sections [2](#sec:model){reference-type="ref" reference="sec:model"}--[6](#sec:proofs){reference-type="ref" reference="sec:proofs"} we omit all 'primes' from the dimensionless quantities, we assume $\kappa=1$, and we neglect the applied external field (unless otherwise mentioned).

arxiv_math

--- abstract: | We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral equation when one surface is close to another or to obtain values at grid points. We replace the singular kernel with a regularized version having a length parameter $\delta$ in order to control discretization error. Analysis near the singularity leads to an expression for the error due to regularization which has terms with unknown coefficients multiplying known quantities. By computing the integral with three choices of $\delta$ we can solve for an extrapolated value that has regularization error reduced to $O(\delta^5)$. In examples with $\delta/h$ constant and moderate resolution we observe total error about $O(h^5)$. For convergence as $h \to 0$ we can choose $\delta$ proportional to $h^q$ with $q < 1$ to ensure the discretization error is dominated by the regularization error. With $q = 4/5$ we find errors about $O(h^4)$. For harmonic potentials we extend the approach to a version with $O(\delta^7)$ regularization; it typically has smaller errors but the order of accuracy is less predictable. author: - "J. Thomas Beale[^1]" - "Svetlana Tlupova[^2]" bibliography: - Bibtrap.bib title: | Extrapolated Regularization of\ Nearly Singular Integrals on Surfaces --- **Keywords:** boundary integral method, nearly singular integral, layer potential, Stokes flow\ **Mathematics Subject Classifications:** 65R20, 65D30, 31B10, 76D07 # Introduction The evaluation of singular or nearly singular surface integrals, on or near the surface, requires special care. Here we are concerned with single and double layer integrals for harmonic potentials or for Stokes flow. One of several possible approaches is to regularize the singular kernel in order to control the discretization error. A natural choice is to replace the $1/r$ singularity in the single layer potential with $\mathop{\mathrm{erf}}(r/\delta)/r$, where $\mathop{\mathrm{erf}}$ is the error function and $\delta$ is a numerical parameter setting the length scale of the regularization. This replacement introduces an additional error due to smoothing. For the singular case, evaluating at points on the surface, we can modify the choice of regularization so that the new error is $O(\delta^5)$; see [@b04; @byw; @tbjcp]. The nearly singular case, evaluation at points near the surface, could be needed e.g. when surfaces are close together or to find values at grid points. For this case, in the previous work, we used analysis near the singularity to derive corrections which leave a remaining error of $O(\delta^3)$. It does not seem practical to extend the corrections to higher order. In the present work we show by local analysis that the simpler regularization can be used with extrapolation, rather than corrections, to improve the error to $O(\delta^5)$ in the nearly singular case. For ${\mathbf y}$ near the surface, at signed distance $b$, if $\mathcal{S}$ is the single layer potential with some density function and $\mathcal{S}_\delta$ is the regularized integral, we show that $$\label{front} \mathcal{S}_\delta({\mathbf y}) = \mathcal{S}({\mathbf y}) + C_1\delta I_0(b/\delta) + C_2\delta^3 I_2(b/\delta) + O(\delta^5)$$ where $I_0$ and $I_2$ are certain integrals, known explicitly, and $C_1$, $C_2$ are coefficients which depend on ${\mathbf y}$, $b$, the surface, and the density function. We can regard $\mathcal{S}$, $C_1$, $C_2$ as unknowns at one point ${\mathbf y}$. Our strategy is to calculate the regularized integrals $\mathcal{S}_\delta$ for three different choices of $\delta$ and then solve for $\mathcal{S}$, within $O(\delta^5$), from the system of three equations. We treat the double layer potential in a similar way, as well as the single and double layer integrals for Stokes flow. For the harmonic potentials we extend the approach to a method with $O(\delta^7)$ regularization error; it requires four choices of $\delta$ rather than three. To compute the integrals we use a quadrature rule for surface integrals for which the quadrature points are points where the surface intersects lines in a three-dimensional grid and the weights are determined by the normal vector to the surface. In our examples with moderate resolution we take $\delta/h$ constant where $h$ is the grid spacing. With the fifth order method we observe errors about $O(h^5)$. For convergence as $h \to 0$ we need to increase $\delta/h$ to ensure that the discretization error is dominated by the regularization error. To do this we choose $\delta$ proportional to $h^q$, e.g. with $q = 4/5$, resulting in an error about $O(h^4)$. With the seventh order regularization we typically see smaller errors in both versions but the order in $h$ is less predictable. Considerable work has been devoted to the computation of singular integrals such as layer potentials. Only a portion of this work has concerned nearly singular integrals on surfaces. Often values close to the surface are obtained by extrapolating from values further away [@ying-biros-zorin-06], sometimes as part of the quadrature by expansion method [@baggetorn; @klinttorn; @klockner-barnett-greengard-oneil-13; @zorincplx; @siegel-tornberg-18]. In [@steinbarn] sources are placed on the opposite side of the surface to produce a kernel independent method. With the singularity subtraction technique [@helsing-13] a most singular part is evaluated analytically leaving a more regular remainder. In [@nitsche], for the nearly singular axisymmetric case, the error in computing the most singular part provides a correction. In [@perez] an approximation to the density function is used to reduce the singularity. Regularization has been used extensively to model Stokes flow in biology [@cortez2d; @cortez3d]; see also [@tbjcp]. While the choice of numerical method depends on context, the present approach is simple and direct. The work required is similar to that for a surface integral with smooth integrand, except that three (or four) related integrals must be computed rather than one. No special gridding or separate treatment of the singularity is needed. The surface must be moderately smooth, without corners or edges. Geometric information about the surface is not needed other than normal vectors; further geometry was needed for the corrections of [@b04; @byw; @tbjcp] and in some other methods. It would be enough for the surface to be known through values of a level set function at grid points nearby. For efficiency fast summation methods suitable for regularized kernels [@yingradial; @rbfstokes; @wang-krasny-tlupova-20] could be used. The approach here is general enough that it should apply to other singular kernels; however, a limitation is discussed at the end of the next section. Results are described more specifically in Sect. 2. The analysis leading to [\[front\]](#front){reference-type="eqref" reference="front"} is carried out in Sect. 3. In Sect. 4 we discuss the quadrature rule and the discretization error. In Sect. 5 we present numerical examples which illustrate the behavior of the method. # Summary of results For a single layer potential $$\label{sgllayer} \mathcal{S}({\mathbf y}) = \int_\Gamma G({\mathbf x}- {\mathbf y})f({\mathbf x})\,dS({\mathbf x})\,, \quad G({\mathbf r}) = -\frac{1}{4\pi|{\mathbf r}|}$$ on a closed surface $\Gamma$, with given density function $f$, we define the regularized version $$\label{sglreg} \mathcal{S}_\delta({\mathbf y}) = \int_\Gamma G_\delta({\mathbf x}- {\mathbf y})f({\mathbf x})\,dS({\mathbf x})\,, \quad G_\delta({\mathbf r}) = G({\mathbf r})s_1(|{\mathbf r}|/\delta)$$ with $$\label{sglshape} s_1(r) = \mathop{\mathrm{erf}}(r) = \frac{2}{\sqrt{\pi}}\int_0^r e^{-s^2}\,ds$$ Then $G_\delta$ is smooth, with $G_\delta(0) = -1/(2\pi^{3/2}\delta)$, and $\mathop{\mathrm{erf}}(r/\delta) \to 1$ rapidly as $r/\delta$ increases. Typically $\mathcal{S}_\delta- \mathcal{S}= O(\delta)$. If ${\mathbf y}$ is near the surface, then ${\mathbf y}= {\mathbf x}_0 + b{\mathbf n}$, where ${\mathbf x}_0$ is the closest point on $\Gamma$, ${\mathbf n}$ is the outward normal vector at ${\mathbf x}_0$, and $b$ is the signed distance. From a series expansion for ${\mathbf x}$ near ${\mathbf x}_0$ and $b$ near $0$ we show in Sect. 3 that $$\label{basic} \mathcal{S}({\mathbf y}) + C_1\delta I_0(\lambda) + C_2\delta^3 I_2(\lambda) = \mathcal{S}_\delta({\mathbf y}) + O(\delta^5)$$ where $\lambda= b/\delta$; $C_1$, $C_2$ are unknown coefficients; and $I_0$ and $I_2$ are integrals occurring in the derivation that are found to be $$I_0(\lambda) = e^{-\lambda^2}/\sqrt{\pi} - |\lambda|\mathop{\mathrm{erfc}}|\lambda|$$ $$I_2(\lambda) = \frac23\left( (\frac12 - \lambda^2) e^{-\lambda^2}/\sqrt{\pi} + |\lambda|^3 \mathop{\mathrm{erfc}}|\lambda| \right)$$ Here $\mathop{\mathrm{erfc}}= 1 - \mathop{\mathrm{erf}}$. To obtain an accurate value of $\mathcal{S}$, we calculate the regularized integrals $\mathcal{S}_\delta$ for three different choices of $\delta$, with the same grid size $h$, resulting in a system of three equations with three unknowns. We can then solve for the exact integral $\mathcal{S}$ within error $\delta^5$. We typically choose $\delta_i = \rho_i h$ with $\rho_i = 2, 3, 4$ or $3, 4, 5$. To improve the conditioning we write three versions of [\[basic\]](#basic){reference-type="eqref" reference="basic"} in terms of $\rho$ rather than $\delta$, $$\label{form5} \mathcal{S}({\mathbf y}) + c_1\rho_i I_0(\lambda_i) + c_2\rho_i^3 I_2(\lambda_i) = \mathcal{S}_{\delta_i}({\mathbf y}) + O(\delta^5) \,,\quad i = 1,2,3$$ with $\lambda_i = b/\delta_i$. It is important that $c_1, c_2$ do not depend on $\delta$ or $\lambda$. We solve this $3\times 3$ system for $S$. The $i$th row is $[1\;,\rho_iI_0(\lambda_i)\;\,\rho_i^3I_2(\lambda_i)]$; the entries depend only on $\lambda_i$ as well as $\rho_i$. The value obtained for $\mathcal{S}$ has the form $$\label{comb} \mathcal{S}({\mathbf y}) = \sum_{i=1}^3 a_i(\lambda_i)S_{\delta_i}({\mathbf y})$$ For each $\lambda$, $a_1 + a_2 + a_3 = 1$. At $\lambda= 0$, $a_1 = 14/3$, $a_2 = -16/3$, $a_3 = 5/3$ provided $\rho_i = 2, 3, 4$. As $\lambda$ increases, the coefficients approach $(1,0,0)$, allowing a gradual transition to the region far enough from $\Gamma$ to omit the regularization. To ensure the smoothing error is dominant as $h \to 0$ we may choose $\delta = \rho h^q$ with $q < 1$, rather than $q = 1$, to obtain convergence $O(h^{5q})$; see Sect. 4. For the double layer potential $$\label{dbllayer} \mathcal{D}({\mathbf y}) = \int_\Gamma \frac{\partial G({\mathbf x}-{\mathbf y})}{\partial{\mathbf n}({\mathbf x})} g({\mathbf x})\,dS({\mathbf x})$$ the treatment is similar after a subtraction. Using Green's identities we rewrite [\[dbllayer\]](#dbllayer){reference-type="eqref" reference="dbllayer"} as $$\label{dblsub} \mathcal{D}({\mathbf y}) = \int_\Gamma \frac{\partial G({\mathbf x}-{\mathbf y})}{\partial{\mathbf n}({\mathbf x})} [g({\mathbf x}) - g({\mathbf x}_0)]\,dS({\mathbf x}) + \chi({\mathbf y})g({\mathbf x}_0)$$ where again ${\mathbf x}_0$ is the closest point on $\Gamma$ and $\chi = 1$ for ${\mathbf y}$ inside, $\chi = 0$ for ${\mathbf y}$ outside, and $\chi = \frac12$ on $\Gamma$. To regularize we replace $\nabla G$ with the gradient of the smooth function $G_\delta$, obtaining $$\label{gradreg} \nabla G_\delta({\mathbf r}) = \nabla G({\mathbf r})s_2(|{\mathbf r}|/\delta) = \frac{{\mathbf r}}{4\pi|{\mathbf r}|^3}s_2(|{\mathbf r}|/\delta)$$ with $$\label{dblshape} s_2(r) = \mathop{\mathrm{erf}}(r) - (2/\sqrt{\pi})re^{-r^2}$$ Thus $$\label{dblreg} \mathcal{D}_\delta({\mathbf y}) = \int_\Gamma \frac{{\mathbf r}\cdot{\mathbf n}({\mathbf x})}{4\pi|{\mathbf r}|^3}s_2(|{\mathbf r}|/\delta) [g({\mathbf x}) - g({\mathbf x}_0)]\,dS({\mathbf x}) + \chi({\mathbf y})g({\mathbf x}_0)\,, \quad {\mathbf r}= {\mathbf x}- {\mathbf y}$$ The expansion for $\mathcal{D}_\delta- \mathcal{D}$ near ${\mathbf x}_0$ is somewhat different but coincidentally leads to the same relation as in [\[form5\]](#form5){reference-type="eqref" reference="form5"} with $\mathcal{S}$ and $\mathcal{S}_\delta$ replaced by $\mathcal{D}$ and $\mathcal{D}_\delta$. Thus we can solve for $\mathcal{D}$ to $O(\delta^5)$ in the same way as for $\mathcal{S}$. There is a straightforward extension to a method with $O(\delta^7)$ regularization error. In equation [\[form5\]](#form5){reference-type="eqref" reference="form5"} there is now an additional term $C_3\delta^5 I_4$. There are four unknowns, so that four choices of $\delta$ are needed. Otherwise this version is similar to the original one. On the other hand, we could use only two choices of $\delta$, omitting the $\delta^3$ term in [\[form5\]](#form5){reference-type="eqref" reference="form5"}, obtaining a version with error $O(\delta^3)$. The special case of evaluation at points ${\mathbf y}$ on the surface is important because it is used to solve integral equations for problems such as the Dirichlet or Neumann problem for harmonic functions. We could use the procedure described with $b = 0$ and $\lambda= 0$. However in this case we can modify the regularization to obtain $O(\delta^5)$ error more directly [@b04; @byw]. For the single layer integral, in place of [\[sglreg\]](#sglreg){reference-type="eqref" reference="sglreg"} we use $$\label{sglon} G_\delta({\mathbf r}) = -\frac{s_1^\sharp(|{\mathbf r}|/\delta)}{4\pi|{\mathbf r}|}\,, \quad s_1^\sharp(r) = \mathop{\mathrm{erf}}(r) + \frac{2}{3\sqrt{\pi}}(5r - 2r^3)e^{-r^2}$$ For the double layer we use [\[dblreg\]](#dblreg){reference-type="eqref" reference="dblreg"} with $\chi = \frac12$ and [\[dblshape\]](#dblshape){reference-type="eqref" reference="dblshape"} replaced by $$\label{dblonshape} s_2^\sharp(r) = \mathop{\mathrm{erf}}(r) - \frac{2}{\sqrt{\pi}} \left(r - \frac{2r^3}{3}\right)e^{-r^2}$$ We typically use $\delta= 3h$ with these formulas for evaluation on the surface [@byw; @tbjcp]. They were derived by imposing conditions to eliminate the leading error [@b04], and the error can be checked using the analysis in the next section. Formulas with $O(\delta^7)$ error could be produced with the same approach. The equations of Stokes flow represent the motion of incompressible fluid in the limit of zero Reynolds number; e.g. see [@pozbook]. In the simplest form they are $$\label{stokeseqns} \Delta{\mathbf u}- \nabla p = 0\,,\quad \nabla\cdot{\mathbf u}= 0$$ where ${\mathbf u}$ is the fluid velocity and $p$ is the pressure. The primary fundamental solutions for the velocity are the Stokeslet and stresslet, $$\begin{aligned} \label{Stokeslet} S_{ij}(\mathbf{y,x}) &= \frac{\delta_{ij}}{|\mathbf{y} - \mathbf{x}|} + \frac{(y_i - x_i )(y_j - x_j)}{|\mathbf{y} - \mathbf{x}|^3} \\[6pt] \label{stresslet} T_{ijk} (\mathbf{y,x}) &= -\frac{6(y_i - x_i)(y_j - x_j)(y_k - x_k)}{|\mathbf{y} - \mathbf{x}|^5}\end{aligned}$$ where $\delta_{ij}$ is the Kronecker delta and $i,j,k = 1,2,3$. They are the kernels for the single and double layer integrals $$\begin{aligned} \label{SingleLayer} u_i(\mathbf{y}) &= \frac{1}{8\pi}\int_\Gamma S_{ij}(\mathbf{y,x}) f_j(\mathbf{x}) dS(\mathbf{x}) \\[6pt] \label{DoubleLayer} v_i(\mathbf{y}) &= \frac{1}{8\pi}\int_\Gamma T_{ijk} (\mathbf{y,x}) q_j(\mathbf{x}) n_k(\mathbf{x})dS(\mathbf{x})\end{aligned}$$ where $f_j$ and $q_j$ are components of vector quantities ${\mathbf f}$ and ${\mathbf q}$ on the surface and $n_k$ is a component of the normal vector ${\mathbf n}$. A subtraction can be used in both cases; e.g., see [@pozbook], Sect. 6.4. With ${\mathbf x}_0$ as before we rewrite [\[SingleLayer\]](#SingleLayer){reference-type="eqref" reference="SingleLayer"} as $$\label{stosglsub} u_i(\mathbf{y}) = \frac{1}{8\pi}\int_\Gamma S_{ij}(\mathbf{y,x}) [f_j({\mathbf x}) - f_k({\mathbf x}_0)n_k({\mathbf x}_0)n_j({\mathbf x})]\,dS(\mathbf{x})$$ The subtracted form of [\[DoubleLayer\]](#DoubleLayer){reference-type="eqref" reference="DoubleLayer"} is $$\label{stodblsub} v_i(\mathbf{y}) = \frac{1}{8\pi}\int_\Gamma T_{ijk} (\mathbf{y,x}) [q_j(\mathbf{x }) - q_j(\mathbf{x}_0)] n_k(\mathbf{x}) dS(\mathbf{x}) + \chi (\mathbf{y}) q_i(\mathbf{x }_0)$$ To compute [\[stosglsub\]](#stosglsub){reference-type="eqref" reference="stosglsub"} we replace $S_{ij}$ with the regularized version $$\label{Sreg} S_{ij}^\delta(\mathbf{y,x}) = \frac{\delta_{ij}}{r}s_1(r/\delta) + \frac{(y_i - x_i)(y_j - x_j)}{r^3}s_2(r/\delta)\,, \quad r = |{\mathbf y}- {\mathbf x}|$$ with $s_1$ and $s_2$ as in [\[sglshape\]](#sglshape){reference-type="eqref" reference="sglshape"},[\[dblshape\]](#dblshape){reference-type="eqref" reference="dblshape"}, resulting in a smooth kernel. For the Stokes double layer integral we need to rewrite the kernel for a reason described below. For ${\mathbf y}$ near the surface we have ${\mathbf y}= {\mathbf x}_0 + b{\mathbf n}_0$ with ${\mathbf x}_0 \in \Gamma$ and ${\mathbf n}_0 = {\mathbf n}({\mathbf x}_0)$. In $T_{ijk}$ we substitute $y_i - x_i = bn_i - {\hat x}_i$ where $n_i$ and ${\hat x}_i$ are the $i$th components of ${\mathbf n}_0$ and ${\hat {\mathbf x}}= {\mathbf x}- {\mathbf x}_0$. With $r = |{\mathbf y}- {\mathbf x}|$, where possible, we replace $b^2/r^2$ with $1 - (r^2 - b^2)/r^2$ to eliminate factors in the numerator which are nonzero at ${\mathbf x}_0$. We obtain $$\label{Tsplit} T_{ijk} = T_{ijk}^{(1)} + T_{ijk}^{(2)} = - 6\left(\frac{t_{ijk}^{(1)}}{r^3} + \frac{t_{ijk}^{(2)} - (r^2 - b^2)t_{ijk}^{(1)}}{r^5}\right)$$ where $$t_{ijk}^{(1)} = bn_in_jn_k - ({\hat x}_in_jn_k + n_i{\hat x}_jn_k + n_in_j{\hat x}_k)$$ $$t_{ijk}^{(2)} = b({\hat x}_i{\hat x}_jn_k + {\hat x}_in_j{\hat x}_k + n_i{\hat x}_j{\hat x}_k) - {\hat x}_i{\hat x}_j{\hat x}_k$$ and we substitute $r^2 - b^2 = |{\hat {\mathbf x}}|^2 - 2b{\hat {\mathbf x}}\cdot{\mathbf n}_0$. We compute [\[stodblsub\]](#stodblsub){reference-type="eqref" reference="stodblsub"} with $T_{ijk}$ replaced with the regularized version of [\[Tsplit\]](#Tsplit){reference-type="eqref" reference="Tsplit"} $$\label{Treg} T_{ijk}^\delta= T_{ijk}^{(1)}s_2(r/\delta) + T_{ijk}^{(2)}s_3(r/\delta)$$ where $$\label{s3} s_3(r) = \mathop{\mathrm{erf}}(r) - \frac{2}{\sqrt{\pi}}\left(\frac23 r^3 + r\right)e^{-r^2}$$ For both Stokes integrals, calculated in the manner described, we find in Sect. 3 that the error has a form equivalent to [\[form5\]](#form5){reference-type="eqref" reference="form5"}, and we extrapolate with three choices of $\delta$ as before. Again for the special case of evaluation on the surface we can obtain an $O(\delta^5)$ regularization directly. Formulas were given in [@tbjcp] and an improved formula for the stresslet case was given in [@novel]. It appears this method would not be successful if applied directly to the double layer potential or the Stokeslet integral without the subtraction. There would be a term in the integrand proportional to $1/r^3$. The equation [\[basic\]](#basic){reference-type="eqref" reference="basic"} for the regularization error would then have an additional term which, to first approximation, does not change as $\delta$ is varied. As a result the extrapolated value of the integral becomes unstable as $b \to 0$; i.e., the coefficients in the linear combination replacing [\[comb\]](#comb){reference-type="eqref" reference="comb"} become large as $b \to 0$. A similar consideration motivates the expression for $T_{ijk}$ above. For other kernels general techniques to reduce the singularity could be used if necessary, e.g. [@perez]. # Local analysis near the singularity We derive an expansion for the error due to regularizing a singular integral, when evaluated at a point ${\mathbf y}$ near the surface $\Gamma$. The expression obtained leads to the formula [\[basic\]](#basic){reference-type="eqref" reference="basic"} and the extrapolation strategy used here. The first few terms of the expansion were used in [@b04; @byw; @tbjcp] to find corrections to $O(\delta^3)$. We begin with the single layer potential [\[sgllayer\]](#sgllayer){reference-type="eqref" reference="sgllayer"}. The error $\epsilon$ is the difference between [\[sglreg\]](#sglreg){reference-type="eqref" reference="sglreg"} and [\[sgllayer\]](#sgllayer){reference-type="eqref" reference="sgllayer"}. Given ${\mathbf y}$ near $\Gamma$, we assume for convenience that the closest point on $\Gamma$ is ${\mathbf x}= 0$. Then ${\mathbf y}= b{\mathbf n}_0$, where ${\mathbf n}_0$ is the outward normal at ${\mathbf x}= 0$ and $b$ is the signed distance from the surface. We choose coordinates $\alpha= (\alpha_1,\alpha_2)$ on $\Gamma$ near ${\mathbf x}= 0$ so that ${\mathbf x}(0) = 0$, the metric tensor $g_{ij} = \delta_{ij}$ at $\alpha= 0$, and the second derivatives ${\mathbf x}_{ij}$ are normal at $\alpha= 0$. E.g., if the tangent plane at ${\mathbf x}= 0$ is $\{x_3 = 0\}$, we could use $(\alpha_1,\alpha_2) = (x_1,x_2)$. Since the error in the integral is negligible for ${\mathbf x}$ away from $0$ we can assume the density $f$ is zero outside this coordinate patch, regard it as a function of $\alpha$, and write the regularization error as $$\epsilon= \int [G_\delta({\mathbf x}(\alpha) - {\mathbf y}) - G({\mathbf x}- {\mathbf y})]f(\alpha)\,dS(\alpha)$$ Then $$\epsilon= \frac{1}{4\pi}\int\frac{\mathop{\mathrm{erfc}}(r/\delta)}{r}f(\alpha)\,dS(\alpha) \,, \qquad r = |{\mathbf x}(\alpha) - {\mathbf y}|$$ We can expand ${\mathbf x}$ near $0$ as $${\mathbf x}(\alpha) = {\mathbf T}_1(0)\alpha_1 + {\mathbf T}_2(0)\alpha_2 + \sum_{2\leq |\nu|\leq 4} c_\nu \alpha^\nu D^\nu{\mathbf x}(0) + O(|\alpha|^5)$$ Here ${\mathbf T}_i = \partial{\mathbf x}/\partial\alpha_i$, the tangent vector at ${\mathbf x}(\alpha)$, and we use multi-index notation: $\nu = (\nu_1,\nu_2)$, $\alpha^\nu = \alpha_1^{\nu_1}\alpha_2^{\nu_2}$, $D^\nu$ is mixed partial derivative of order $(\nu_1,\nu_2)$, and $|\nu| = \nu_1 + \nu_2$. We will use the notation $c_\nu$ for generic constants whose value will not be needed. We first get an expression for $r^2$. We start with $$|{\mathbf x}(\alpha)|^2 = \alpha_1^2 + \alpha_2^2 + \sum_{|\nu|=4,5} c_\nu\alpha^\nu + O(|\alpha|^6)$$ There is no term with $|\nu| = 3$ since the first and second order terms in ${\mathbf x}$ are orthogonal. Also $${\mathbf x}(\alpha)\cdot {\mathbf n}_0 = \sum_{2\leq\nu\leq 4}c'_\nu\alpha^\nu + O(|\alpha|^5)$$ Then $$r^2 = |x(\alpha) - b{\mathbf n}_0|^2 = |\alpha|^2 + b^2 + \sum_{|\nu|=4,5} c_\nu\alpha^\nu + b\sum_{2\leq|\nu|\leq 4} c'_\nu\alpha^\nu + O(|\alpha|^6 + |b||\alpha|^5)$$ We will make a change of variables $\alpha= (\alpha_1,\alpha_2) \to \xi = (\xi_1,\xi_2)$ defined by $$|\xi|^2 + b^2 = r^2\,,\quad \xi/|\xi| = \alpha/|\alpha|$$ This allows us to write the error as $$\label{epsxib} \epsilon= \frac{1}{4\pi}\int \frac{\mathop{\mathrm{erfc}}(\sqrt{|\xi|^2 + b^2}/\delta)}{\sqrt{|\xi|^2 + b^2}} w(\xi,b)\,d\xi$$ where $$\label{wsgl} w(\xi,b) = f(\alpha)\left|\frac{\partial\alpha}{\partial\xi}\right| |T_1\times T_2|$$ The mapping $\xi = \xi(\alpha)$ is close to the identity but it is not smooth at $\alpha= 0$, so that we cannot write $w$ directly in a power series in $(\xi,b)$. We will see that $w$ is a sum of terms of the form $b^m \xi^\nu /|\xi|^{2p}$ with $|\nu| \geq 2p$, and such a term makes a contribution to the error $\epsilon$ of order $\delta^{m+|\nu|-2p+1}$. To obtain a suitable series we need to express $\alpha$ as a function of $\xi$. From above we have $$\label{xi2al2} |\xi|^2/|\alpha|^2 = 1 + \sum_{|\nu|=4,5} c_\nu\alpha^\nu/|\alpha|^2 + b\sum_{2\leq|\nu|\leq 4} c'_\nu\alpha^\nu/|\alpha|^2 + O(|(\alpha,b)|^4)$$ Here $O(|(\alpha,b)|^4)$ means $O(|\alpha|^4 + b^4)$. With $u = \alpha/|\alpha| = \xi/|\xi|$, we can substitute $\alpha^\nu = u^\nu|\alpha|^{|\nu|}$ in [\[xi2al2\]](#xi2al2){reference-type="eqref" reference="xi2al2"}. We then regard [\[xi2al2\]](#xi2al2){reference-type="eqref" reference="xi2al2"} as a power series in $|\alpha|$ in which the coefficient of $|\alpha|^n$ is a polynomial in $b$ and $u$ with terms $u^\nu$ such that $|\nu| - n$ is even. It is important that this form is preserved by multiplication, and that the $k$th term in a product series depends only on the first $k$ terms in the factors. Using the power series for $(1+x)^{-1/2}$ we can write a similar expression for $|\alpha|/|\xi|$ with terms as in [\[xi2al2\]](#xi2al2){reference-type="eqref" reference="xi2al2"} and their products. The coefficient of $|\alpha|^n$ is a polynomial in $b$ and $u$, again with terms $u^\nu$, $|\nu| - n$ even; the same is true for powers of $|\alpha|/|\xi|$. We will invert the function $|\alpha| \to |\xi|$ using the Lagrange Inversion Theorem [@ww; @krantz]; the theorem is usually stated for analytic functions, but for $C^N$ functions it can be applied to the Taylor polynomial. We conclude from the theorem that $|\alpha|$ can be expressed as a power series in $|\xi|$, with remainder, such that the coefficient of $|\xi|^n$ is proportional to the coefficient of $|\alpha|^{n-1}$ in the series for $(|\alpha|/|\xi|)^n$ as a function of $|\alpha|$. This quantity has factors $u^\nu$ with $|\nu| - (n-1)$ even. We now divide this expression for $|\alpha|$ by $|\xi|$ so that the earlier parity is restored. Finally we rewrite $u^\nu|\xi|^n$ as $\xi^\nu|\xi|^{n-|\nu|}$, and in summary we have shown that $$\frac{|\alpha|}{|\xi|} = \sum c_{m \nu p} b^m \frac{\xi^\nu}{|\xi|^{2p}} + O(|(\xi,b)|^4)$$ where $m \geq 0$, $|\nu| \geq 2p$, and $m + |\nu| - 2p \leq 3$. With $\alpha_j = (|\alpha|/|\xi|)\xi_j$ we get a similar expression for $\alpha$ as a function of $\xi$. The function $f(\alpha)$ and the factor $|T_1\times T_2|$ in $w$ have series in $\alpha$ which can be converted to $\xi$. The Jacobian is $$\left| \frac{\partial\alpha}{\partial\xi} \right| = \mu^2 + \mu\xi\frac{\partial\mu}{\partial|\xi|}\,, \quad \mu = \frac{|\alpha|}{|\xi|}$$ It has terms of the same type as those in $|\alpha|/|\xi|$. The Jacobian has leading term $1$ and is bounded but not smooth as $\xi \to 0$. We conclude that $w$ has the expression $$\label{wsglexpand} w(\xi,b) = \sum c_{m \nu p} b^m \frac{\xi^\nu}{|\xi|^{2p}} + R(\xi,b)$$ where $m\geq 0$, $|\nu| \geq 2p$, $m + |\nu| - 2p \leq 3$, and $R(\xi,b) = O(|(\xi,b)|^4)$. To find the contribution $\epsilon_{m\nu p}$ to the error [\[epsxib\]](#epsxib){reference-type="eqref" reference="epsxib"} from a term in [\[wsglexpand\]](#wsglexpand){reference-type="eqref" reference="wsglexpand"} with a particular $(m,\nu,p)$ we will integrate in polar coordinates. The angular integral is zero by symmetry unless $\nu_1$, $\nu_2$ are both even. Let $n = |\nu| - 2p$, the degree of $\xi$. With the restriction $m + n \leq 3$ the possible nonzero terms have $n = 0$ and $0 \leq m \leq 3$ or $n = 2$ with $m = 0,1$. To carry out the integration, we rescale variables to $\xi = \delta\zeta$, $b = \delta\lambda$, and write $\zeta$ in polar coordinates. With $s = |\zeta|$ we obtain $$\epsilon_{m\nu p} = c_{m\nu p} b^m \delta^{n+1} I_n(\lambda) = c_{m\nu p} \lambda^m \delta^{m+ n+1} I_n(\lambda)$$ where $$I_n(\lambda) = \int_0^\infty \frac{\mathop{\mathrm{erfc}}(\sqrt{s^2 + \lambda^2})} {\sqrt{s^2 + \lambda^2}} s^{n+1}\,ds$$ In a similar way we see that the remainder $R$ leads to an error which is $O(\delta^5)$. In summary we can express the error as $$\label{epsform} \epsilon= \delta p_0(b)I_0(\lambda) + \delta^3 p_2(b)I_2(\lambda) + O(\delta^5)$$ where $p_0, p_2$ are polynomials in $b$ with $\deg p_0 = 3$, $\deg p_2 = 1$. They depend only on the surface and $b$, not $\delta$ or $\lambda$. For fixed $b$ and $h$ they are unknown coefficients. To normalize the equation we set $\delta= \rho h$ and rewrite it as $$\label{formnorm} \epsilon= c_1\rho I_0(\lambda) + c_2\rho^3 I_2(\lambda) + O(\delta^5)$$ This conclusion is equivalent to [\[form5\]](#form5){reference-type="eqref" reference="form5"}, which we use with three choices of $\delta$ to solve for the single layer potential within $O(\delta^5)$. For the double layer potential, in view of [\[dblreg\]](#dblreg){reference-type="eqref" reference="dblreg"} and [\[dblsub\]](#dblsub){reference-type="eqref" reference="dblsub"}, we can write the error from regularizing as $$\epsilon= \frac{1}{4\pi}\int \phi(r/\delta) \frac{({\mathbf x}(\alpha)-{\mathbf y})\cdot {\mathbf n}(\alpha)}{r^3}(g(\alpha)-g(0))\,dS(\alpha)$$ where $$\phi(r) = - \mathop{\mathrm{erfc}}(r) - (2/\sqrt{\pi})re^{-r^2}$$ and after changing from $\alpha$ to $\xi$, $$\epsilon= \frac{1}{4\pi}\int \frac{\phi(\sqrt{|\xi|^2 + b^2}/\delta)}{(|\xi|^2 + b^2)^{3/2} } w(\xi,b)\,d\xi$$ where now $$w(\xi,b) = [({\mathbf x}-{\mathbf y})\cdot {\mathbf n}][g(\alpha) - g(0)] \left|\frac{\partial\alpha}{\partial\xi}\right| |T_1\times T_2|$$ We find $$({\mathbf x}-{\mathbf y})\cdot {\mathbf n}= -b + O(|(\xi,b)|^2)$$ and note $g(\alpha) - g(0) = O(|\xi|)$. Thus each term in $w$ now has at least two additional factors. We expand $w$ as in [\[wsglexpand\]](#wsglexpand){reference-type="eqref" reference="wsglexpand"} but now include terms with $m + n \leq 5$, where again $n = |\nu| - 2p$. The term $(m,\nu,p)$ now contributes an error of order $\delta^{m+n-1}$, rather than $\delta^{m+n+1}$ as before. From the last remark, each nonzero term must have $m \geq 1$ and $n \geq 1$ or $m = 0$ and $n \geq 3$. By symmetry a term that contributes a nonzero error must have $m \geq 1$ and $n \geq 2$ or $m = 0$ and $n \geq 4$. The possible terms with $m+n \leq 5$ are $(m,2)$ with $m = 1,2,3$ and $(m,4)$ with $m = 0,1$. Rescaling the integrals we find $$\label{formdbl} \epsilon= \delta p_0(b)J_0(\lambda) + \delta^3 p_2(b)J_2(\lambda) + O(\delta^5)$$ with $\deg p_0 = 3$, $\deg p_2 = 1$, and $$J_n(\lambda) = \int_0^\infty \frac{\phi(\sqrt{s^2+\lambda^2})}{(s^2+\lambda^2)^{3/2}}s^{n+3}\,ds$$ In fact $$J_0 = - 2I_0\,, \qquad J_2 = -4I_2$$ so that [\[formdbl\]](#formdbl){reference-type="eqref" reference="formdbl"} is equivalent to [\[epsform\]](#epsform){reference-type="eqref" reference="epsform"}, and we can solve for the double layer as in [\[form5\]](#form5){reference-type="eqref" reference="form5"}. The expansions can be carried further in the same manner. For the single layer integral we can refine the error expression [\[epsform\]](#epsform){reference-type="eqref" reference="epsform"} to $$\epsilon= \delta p_0(b)I_0(\lambda) + \delta^3 p_2(b)I_2(\lambda) + \delta^5 p_4(b)I_4(\lambda) + O(\delta^7)$$ For the double layer [\[formdbl\]](#formdbl){reference-type="eqref" reference="formdbl"} is replaced by $$\epsilon= \delta p_0(b)J_0(\lambda) + \delta^3 p_2(b)J_2(\lambda) + \delta^5 p_4(b)J_4(\lambda) + O(\delta^7)$$ Each of these expressions leads to a system of four equations in four unknowns, using four different choices of $\delta$. In fact $J_4 = -6I_4$, so that again we may use the same equations for both cases. For the Stokes single layer integral, calculated in the form [\[stosglsub\]](#stosglsub){reference-type="eqref" reference="stosglsub"}, [\[Sreg\]](#Sreg){reference-type="eqref" reference="Sreg"}, the first term is equivalent to the single layer potential [\[sgllayer\]](#sgllayer){reference-type="eqref" reference="sgllayer"}. The second term resembles the double layer [\[dbllayer\]](#dbllayer){reference-type="eqref" reference="dbllayer"}. We note the integrand has a factor ${\tilde {\mathbf f}}({\mathbf x})\cdot({\mathbf y}- {\mathbf x})$ with ${\tilde {\mathbf f}}({\mathbf x}) = {\mathbf f}({\mathbf x}) - ({\mathbf f}({\mathbf x}_0)\cdot{\mathbf n}({\mathbf x}_0)){\mathbf n}({\mathbf x})$. Thus ${\tilde {\mathbf f}}\cdot{\mathbf n}= 0$ at ${\mathbf x}= {\mathbf x}_0$, and since ${\mathbf y}- {\mathbf x}= b{\mathbf n}({\mathbf x}_0) + O(\xi)$, the numerator of the integrand is $O(\xi)$. The discussion above for the double layer now applies to this second term, leading to the same expression for the error. For the Stokes double layer integral, with the subtraction [\[stodblsub\]](#stodblsub){reference-type="eqref" reference="stodblsub"} and the kernel rewritten as in [\[Treg\]](#Treg){reference-type="eqref" reference="Treg"}, the first term is again like the harmonic double layer. For the second term, regularized with $s_3$, the numerator in the expansion will have terms $O(\xi^3)$ or higher. By symmetry the terms that contribute nonzero error have $O(\xi^4)$ or higher. We get an expansion for the error in the second term in the form $$\epsilon= \delta p_0(b)K_0(\lambda) + \delta^3 p_2(b)K_2(\lambda) + O(\delta^5)$$ with $$K_n(\lambda) = \int_0^\infty \frac{\phi_3(\sqrt{s^2+\lambda^2})}{(s^2+\lambda^2)^{5/2}}s^{n+5}\,ds$$ and $\phi_3 = 1 - s_3$. We find that $K_0 = (8/3)I_0$ and $K_2 = 8I_2$, so that once again we can use [\[form5\]](#form5){reference-type="eqref" reference="form5"} for extrapolation. # Surface quadrature and the discretization error We use a quadrature rule for surface integrals introduced in [@wilson-10] and used in [@b04; @byw; @tbjcp]. We cover the surface with a three-dimensional grid with spacing $h$. The quadrature points have the form ${\mathbf x}= (ih,jh,x_3)$, i.e., points on the surface $\Gamma$ whose projections on the $(x_1,x_2)$ plane are grid points, and similarly for the other two directions. We only use points for which the component of the normal vector in the distinguished direction is no smaller than $\cos\theta$ for a chosen angle $\theta$. In our case we take $\theta = 70^o$. The weights are determined by a partition of unity $\psi_1, \psi_2, \psi_3$ on the unit sphere; it is applied to the normal vector at each point. We define three sets of quadrature points $\Gamma_1, \Gamma_2, \Gamma_3$ as $$\Gamma_3 = \{{\mathbf x}= (ih,jh,x_3) \in \Gamma : |n_3({\mathbf x})| \geq \cos\theta\}$$ where $n_3$ means the third component of the normal vector, and similarly for $\Gamma_1,\Gamma_2$. To construct the partition of unity we start with the bump function $$b(r) = \exp(a r^2/(r^2-1))\,, \quad |r|<1\,; \qquad b(r) = 0\,, \quad |r| \geq 1$$ Here $a$ is a parameter. For a unit vector ${\mathbf n}= (n_1,n_2,n_3)$ we define $$\beta_i({\mathbf n}) = b(\cos^{-1}|n_i|)/\theta) \,, \quad \psi_i({\mathbf n}) = \beta_i({\mathbf n})/\left(\sum_{j=1}^3 \beta_j({\mathbf n})\right)$$ The quadrature rule for a surface integral with integrand $f$ is $$\int_\Gamma f({\mathbf x})\,dS({\mathbf x}) \,\approx\, \sum_{i=1}^3 \sum_{{\mathbf x}\in \Gamma_i} f({\mathbf x})w_i({\mathbf x}) \,, \qquad w_i({\mathbf x}) = \frac{\psi_i({\mathbf n}({\mathbf x}))}{|n_i({\mathbf x})|} \,h^2$$ It has high order accuracy as allowed by the smoothness of the surface and the integrand. In earlier work we chose the parameter $a$ to be $1$. Here we use $a = 2$. We have found from error estimates in [@neglect], discussed below, as well as numerical experiments, that the discretization error is controlled better with this choice. We do not recommend using $a > 2$ because of increased derivatives. The full error in this method consists of the regularization error plus the discretization error; symbolically $$\textstyle\sum_\delta\,-\, \textstyle{\int}\,=\, \left( \textstyle{\int}_\delta\,-\, \textstyle{\int}\right) \,+\, \left(\textstyle\sum_\delta\,-\, \textstyle{\int}_\delta\right)$$ For either the single layer potential [\[sgllayer\]](#sgllayer){reference-type="eqref" reference="sgllayer"} or the double layer [\[dblsub\]](#dblsub){reference-type="eqref" reference="dblsub"} the discretization error can be written as $$\label{derr} c_1 h \,+\, C_2 h^2 \exp(-c_0 \delta^2/h^2) \,+\, O(\delta^5) \,, \quad c_1 = c_1(\delta/h)$$ which at first appears inaccurate. Formulas for the first term were given in [@b04; @byw], based on approximating the surface locally as a plane. They can be used as corrections. Estimates for these formulas were given in [@neglect]. With the parameter choices here, in particular with $\delta/h \geq 2$, it was shown that $$c_1^{(S)} \leq 2.1\cdot 10^{-7}\max{|f|}\,, \quad c_1^{(D)} \leq 8.3\cdot 10^{-7}\max{|\nabla g|}$$ for the single and double layer respectively, and they decrease rapidly as $\delta/h$ increases. Here $\nabla g$ means the tangential gradient. The $h^2$ term in [\[derr\]](#derr){reference-type="eqref" reference="derr"} evidently decreases rapidly as $\delta/h$ increases, as does $c_1$. With $\theta = 70^o$, $c_0 \approx 1.15$; see [@byw], Sect. 3.4. However $C_2$ depends on the surface and integrand and could be large. With moderate resolution we expect that the discretization error is controlled by the regularization. If desired the formulas for $c_1h$ in [@byw] could be used as corrections with the present method; they are infinite series, but only the first few terms are significant. To ensure that the regularization error dominates the discretization error for small $h$ we can choose $\delta$ proportional to $h^q$, with $q < 1$, so that $\delta/h$ increases as $h \to 0$. # Numerical examples We present examples computing single and double layer integrals at grid points within $O(h)$ of a surface, for harmonic potentials and for Stokes flow. The points are selected from the three-dimensional grid with spacing $h$ which determines the quadrature points on the surface, as described in Sect. 4. With the fifth order regularization the results are in general agreement with the theoretical predictions. With moderate resolution and $\delta/h$ constant the errors are about $O(h^5)$. With $\delta$ proportional to $h^{4/5}$ the error is about $O(h^4)$. For the harmonic potentials we also test the seventh order method; the errors are typically smaller but the order of accuracy is less predictable. It is likely that the discretization error is relatively more significant with the smaller errors of the seventh order case. We report maximum errors and $L^2$ errors, defined as $$\|e\|_{L^2} = \left( \sum_{{\mathbf y}} |e(y)|^2 / N \right)^{1/2}$$ where $e(y)$ is the error at ${\mathbf y}$ and $N$ is the number of points. We present absolute errors; for comparison we give approximate norms of the exact solution. **Harmonic Potentials.** We begin with known solutions on the unit sphere. We test the single and double layer separately. We compute the integrals at grid points first within distance $h$ and then on shells at increasing distance. In the latter case we also find values computed without regularization. We then compute known harmonic functions on three other surfaces which combine single and double layers. The single and double layer potentials, [\[sgllayer\]](#sgllayer){reference-type="eqref" reference="sgllayer"} and [\[dbllayer\]](#dbllayer){reference-type="eqref" reference="dbllayer"}, are harmonic inside and outside the surface $\Gamma$. They are characterized by the jump conditions $$= 0\,,\quad [\partial\mathcal{S}({\mathbf x})/\partial{\mathbf n}] = f({\mathbf x})$$ $$= -g({\mathbf x})\,,\quad [\partial\mathcal{D}({\mathbf x})/\partial{\mathbf n}] = 0$$ where $[\cdot]$ means the value outside $\Gamma$ minus the value inside. For the unit sphere we use the spherical harmonic function $$f({\mathbf x}) = 1.75(x_1 - 2x_2)(7.5x_3^2 - 1.5)\,, \quad |{\mathbf x}| = 1$$ for both the single and double layer integrals. The functions $$u_-({\mathbf y}) = r^3f({\mathbf y}/r)\,,\quad u_+({\mathbf y}) = r^{-4}f({\mathbf y}/r)\,, \quad\; r = |{\mathbf y}|$$ are both harmonic. We define $\mathcal{S}({\mathbf y})$ by [\[sgllayer\]](#sgllayer){reference-type="eqref" reference="sgllayer"} and $\mathcal{D}({\mathbf y})$ by [\[dbllayer\]](#dbllayer){reference-type="eqref" reference="dbllayer"} with $g = f$. They are determined by the jump conditions, $$\mathcal{S}({\mathbf y}) = - (1/7)u_-({\mathbf y})\,, \quad |{\mathbf y}|<1\,; \quad\; \mathcal{S}({\mathbf y}) = - (1/7)u_+({\mathbf y})\,, \quad |{\mathbf y}|>1$$ $$\mathcal{D}({\mathbf y}) = (4/7)u_-({\mathbf y})\,, \quad |{\mathbf y}|<1\,; \quad\; \mathcal{D}({\mathbf y}) = - (3/7)u_+({\mathbf y})\,, \quad |{\mathbf y}|>1$$ We present errors for the single and double layer potentials at grid points at various distances from the sphere. We begin with the single layer. We compute the integral as in [\[sglreg\]](#sglreg){reference-type="eqref" reference="sglreg"} and extrapolate as in [\[form5\]](#form5){reference-type="eqref" reference="form5"}. Near the sphere the maximum of $|\mathcal{S}|$ is about $1.15$ and the $L^2$ norm is about $.50$. Table 1 shows the $L^2$ and maximum errors for grid points within distance $h$ of the sphere, using fifth or seventh order extrapolation. For the fifth order we take $\delta/h = 2,3,4$ as previously described, and for the seventh order we take $\delta/h = 2,3,4,5$. The expected order of accuracy is evident in the fifth order case; the seventh order method has somewhat smaller errors but does not have a discernible order of accuracy, probably because the discretization error is significant. In subsequent tables we display the errors at nearby grid points at distance between $mh$ and $(m+1)h$ from the sphere, both inside and outside, for $m = 1,2,3$. We compute the integral with no regularization as well as the fifth and seventh order methods. Table 2 shows errors for $m=1$ and Table 3 for $m = 2$ and $3$. The values without regularization in Table 2 appear to be about $O(h)$ accurate. The fifth order method again has the expected order of accuracy at least for $m = 1$ but becomes less steady with distance. The errors become smaller overall as the distance increases. Beyond $4h$ the error without regularization is quite small, suggesting that we can discontinue the regularization for points at least $4h$ from the surface. ------- -------------- ---------- ------------ --------- -------------- ---------- ------------ --------- $1/h$ 5th order 7th order  $L^2$ err    ratio    max err   ratio    $L^2$ err    ratio    max err    ratio  32 7.60e-6 2.83e-5 2.43e-7 3.57e-6 64 2.39e-7 31.8 8.84e-7 32.0 5.20e-9 46.7 8.27e-8 43.2 128 7.48e-9 31.9 2.98e-8 29.7 5.20e-10 10.0 8.30e-9 10.0 ------- -------------- ---------- ------------ --------- -------------- ---------- ------------ --------- : Errors for the single layer potential on the unit sphere, at grid points within distance $h$, computed with the 5th and 7th order regularization. ------- --------------------- ------------ -------------- ------------ -------------- ------------ $1/h$  no regularization   5th order    7th order   $L^2$ err    max err    $L^2$ err    max err    $L^2$ err    max err   32 2.26e-5 2.38e-4 1.89e-6 8.37e-6 1.20e-7 1.74-6 64 1.18e-5 1.21e-4 5.94e-8 2.43e-7 1.73e-9 2.48e-8 128 5.63e-6 5.75e-5 1.86e-9 8.95e-9 1.82e-10 3.67e-9 ------- --------------------- ------------ -------------- ------------ -------------- ------------ : Errors for the single layer potential on the unit sphere, evaluated at distance between $h$ and $2h$, without regularization and with the 5th and 7th order methods. ------- -------------------------------- ----------- ----------- -------------------------------- ----------- ----------- $1/h$  $2h <\,$ distance $\, < 3h$    $3h <\,$ distance $\, < 4h$   no reg'n 5th order 7th order no reg'n 5th order 7th order 32 5.04e-7 4.13e-7 6.28e-8 4.51e-8 8.64e-8 4.16e-8 64 1.77e-7 1.18e-8 6.10e-10 2.90e-9 1.74e-9 2.35e-10 128 8.89e-8 3.77e-10 5.43e-11 1.24e-9 5.83e-11 1.29e-11 ------- -------------------------------- ----------- ----------- -------------------------------- ----------- ----------- : $L^2$ errors in the single layer potential on the unit sphere, evaluated at distance between $2h$ and $3h$ or $3h$ and $4h$. In Tables 4,5,6 we present results of the same type for the double layer potential, computed as in [\[dblreg\]](#dblreg){reference-type="eqref" reference="dblreg"}. They are similar in behavior to those for the single layer. The maximum of $|\mathcal{D}|$ is about $4.6$ and $\|\mathcal{D}\|_{L^2} \approx 1.8$. ------- -------------- ---------- ------------ ---------- -------------- ---------- ------------ ---------- $1/h$ 5th order 7th order  $L^2$ err    ratio    max err    ratio    $L^2$ err    ratio    max err    ratio   32 2.25e-5 8.13e-5 3.88e-7 5.22e-6 64 7.17e-7 31.4 2.61e-6 31.1 1.31e-8 29.7 2.79e-7 18.7 128 2.25e-8 31.9 8.29e-8 31.5 1.62e-9 8.1 3.57e-8 7.8 ------- -------------- ---------- ------------ ---------- -------------- ---------- ------------ ---------- : Errors for the double layer potential on the unit sphere, at grid points within distance $h$, computed with the 5th and 7th order regularization. ------- --------------------- ------------ -------------- ------------ -------------- ------------ $1/h$  no regularization   5th order    7th order   $L^2$ err    max err    $L^2$ err    max err    $L^2$ err    max err   32 2.78e-4 2.70e-3 5.46e-6 2.17e-5 3.76e-7 5.05e-6 64 1.39e-4 1.45e-3 1.79e-7 7.11e-7 1.27e-8 2.43e-7 128 6.87e-5 7.27e-4 5.77e-9 2.53e-8 1.66e-9 3.18e-8 ------- --------------------- ------------ -------------- ------------ -------------- ------------ : Errors for the double layer potential on the unit sphere, evaluated at distance between $h$ and $2h$, without regularization and with the 5th and 7th order methods. ------- -------------------------------- ----------- ----------- -------------------------------- ----------- ----------- $1/h$  $2h <\,$ distance $\, < 3h$    $3h <\,$ distance $\, < 4h$   no reg'n 5th order 7th order no reg'n 5th order 7th order 32 7.99e-6 1.08e-6 2.31e-7 2.75e-7 1.48e-7 1.36e-7 64 3.31e-6 3.62e-8 6.45e-9 6.76e-8 6.14e-9 2.52e-9 128 1.58e-6 1.37e-9 8.32e-10 2.97e-8 3.04e-10 2.60e-10 ------- -------------------------------- ----------- ----------- -------------------------------- ----------- ----------- : $L^2$ errors in the double layer potential on the unit sphere, evaluated at distance between $2h$ and $3h$ or $3h$ and $4h$. For the remaining tests on other surfaces we use a procedure as in [@byw] which allows us to have known solutions with an arbitrary surface $\Gamma$. This provides a test of the single and double layer combined, rather than separately. We choose harmonic functions $u_+$ outside and $u_-$ inside. We set $f = -[u]$ and $g = [\partial u/\partial n]$, the jumps across $\Gamma$ as above. Then assuming $u_+$ decays at infinity, $u({\mathbf y}) = \mathcal{S}({\mathbf y}) + \mathcal{D}({\mathbf y})$ on both sides, where $\mathcal{S}$ and $\mathcal{D}$ are defined in [\[sgllayer\]](#sgllayer){reference-type="eqref" reference="sgllayer"}, [\[dbllayer\]](#dbllayer){reference-type="eqref" reference="dbllayer"}. We choose $$u_-({\mathbf y}) = (\sin{y_1} + \sin{y_2})\exp{y_3}\,, \quad u_+({\mathbf y}) = 0$$ In these tests we again use $\delta/h = 2,3,4$ with the fifth order method and $\delta/h = 2,3,4,5$ with seventh order. We also choose $\delta$ proportional to $h^{4/5}$ with the fifth order method and $h^{4/7}$ with the seventh order method, so that the predicted order of error is $O(h^4)$. We choose constants so that $\delta$ agrees with the earlier choice at $1/h = 64$. Our first surface with this procedure is a rotated ellipsoid $$\frac{z_1^2}{a^2} + \frac{z_2^2}{b^2} + \frac{z_3^2}{c^2} = 1$$ where $a = 1$, $b = .8$, $c = .6$ and ${\mathbf z}= M{\mathbf x}$, where $M$ is the orthogonal matrix $$M = (1/\sqrt{6})\,[\sqrt{2}\quad 0\quad -2;\; \sqrt{2}\quad \sqrt{3}\quad 1;\; \sqrt{2}\quad -\sqrt{3}\quad 1]$$ We present results in two tables. In Table 7 we evaluate at all grid points within distance $h$ with both regularizations. Table 8 has values at points $y$ within distance $h$ in the first octant, i.e., those with $y_1, y_2, y_3 \geq 0$. The accuracy of the fifth order version is close to the prediction; the seventh order version has smaller errors in Table 8 and perhaps approximates the predicted order $O(h^4)$ but not clearly so. For the first table the $L^2$ norm of the exact solution is about $.5$ and the maximum about 1.7. For the second, within the first octant, they are about .76 and 1.4. ------- -------------- ---------- ------------ --------- -------------- ---------- ------------ --------- $1/h$ 5th order 7th order  $L^2$ err    ratio    max err   ratio    $L^2$ err    ratio    max err    ratio  32 1.47e-5 1.88e-4 2.60e-6 3.28e-5 64 5.02e-7 29.3 7.10e-6 26.5 3.01e-8 86.4 7.08e-7 46.4 128 1.61e-8 31.2 2.40e-7 29.6 5.26e-10 57.3 1.53e-8 46.4 ------- -------------- ---------- ------------ --------- -------------- ---------- ------------ --------- : Errors for the single and double layers on a rotated ellipsoid at grid points within distance $h$, with the 5th order and 7th order methods, $\delta$ proportional to $h$. ------- ------------------------------------------- ---------- ------------ --------- ------------------------------------------- ---------- ------------ --------- $1/h$ $5$th order, $\delta$ prop'l to $h^{4/5}$ $7$th order, $\delta$ prop'l to $h^{4/7}$  $L^2$ err    ratio    max err   ratio    $L^2$ err    ratio    max err    ratio  32 7.10e-6 4.29e-5 2.52e-6 2.07e-5 64 4.98e-7 14.2 2.95e-6 14.5 1.86e-8 135 1.86e-7 111 128 3.35e-8 14.9 1.88e-7 15.7 1.00e-9 18.6 6.40e-9 29 256 2.21e-9 15.2 1.22e-8 15.4 6.98e-11 14.4 4.21e-10 15.2 ------- ------------------------------------------- ---------- ------------ --------- ------------------------------------------- ---------- ------------ --------- : Errors for the rotated ellipsoid, at grid points within distance $h$ in the first octant; $5$th and $7$th order methods with $\delta$ chosen to correspond to $O(h^4)$ accuracy. The next example is a surface obtained by revolving a Cassini oval about the $x_3$ axis, $$(x_1^2 + x_2^2 + x_3^2 + a^2)^2 - 4a^2(x_1^2 + x_2^2) = b^2$$ with $a = .65$ and $b = .7$. The final surface represents a molecule with four atoms, $$\label{molesurf} \sum_{i=1}^4 \exp(-|{\mathbf x}- {\mathbf x}_k|^2/r^2) = c$$ with $r = .5$, $c = .6$, and ${\mathbf x}_k$ given by $$(\sqrt{3}/3,0,-\sqrt{6}/12)\,,\; (-\sqrt{3}/6,\pm .5,-\sqrt{6}/12)\,,\; (0,0,\sqrt{6}/4)$$ We compute the solution for grid points in the first octant as before for the ellipsoid, with $\delta$ related to $h$ in the same way. We present errors with fifth or seventh order regularization, with $\delta$ proportional to $h$ or fractional. The results, reported in Tables 9 and 10, are generally similar to those for the rotated ellipsoid. For both surfaces we see roughly the predicted orders of accuracy in the fifth order case. For seventh order the errors are smaller, but the accuracy in the fractional case is somewhat less than fourth order in $h$. For the Cassini surface the $L^2$ norm for the exact values is about $.78$ and the maximum is about $1.45$. For the molecular surface they are about $.57$ and $1.0$. ------- ------------------ ------------ ------------------------ ------------ ------------------ ------------ ------------------------ ------------ $1/h$ 5th order 7th order $\delta= \rho h$ $\delta= \rho h^{4/5}$ $\delta= \rho h$ $\delta= \rho h^{4/7}$  $L^2$ err    max err    $L^2$ err    max err    $L^2$ err    max err    $L^2$ err    max err   32 5.16e-5 5.44e-4 2.75e-5 3.00e-4 1.98e-5 2.24e-4 8.06e-6 4.47e-5 64 2.46e-6 3.98e-5 2.46e-6 3.98e-5 4.82e-7 9.99e-6 4.82e-7 9.99e-6 128 8.64e-8 1.57e-6 1.79e-7 3.07e-6 5.59e-9 1.22e-7 4.18e-8 8.67e-7 256 2.81e-9 5.15e-8 1.23e-8 2.05e-7 1.14e-10 1.34e-9 3.39e-9 7.44e-8 ------- ------------------ ------------ ------------------------ ------------ ------------------ ------------ ------------------------ ------------ : Errors for the Cassini oval surface, at grid points within distance $h$ in the first octant; $5$th and $7$th order method with $\delta$ proportional to $h$ or corresponding to $h^4$. ------- ------------------ ------------ ------------------------ ------------ ------------------ ------------ ------------------------ ------------ $1/h$ 5th order 7th order $\delta= \rho h$ $\delta= \rho h^{4/5}$ $\delta= \rho h$ $\delta= \rho h^{4/7}$  $L^2$ err    max err    $L^2$ err    max err    $L^2$ err    max err    $L^2$ err    max err   32 5.80e-5 2.51e-4 3.09e-5 1.81e-4 2.10e-5 1.45e-4 1.58e-5 1.17e-4 64 2.40e-6 1.33e-5 2.40e-6 1.33e-5 4.24e-7 3.92e-6 4.24e-7 3.92e-6 128 8.40e-8 4.98e-7 1.75e-7 9.77e-7 5.57e-9 7.86e-8 3.92e-8 3.03e-7 256 2.73e-9 1.82e-8 1.20e-8 7.15e-8 1.55e-10 2.51e-9 3.22e-9 2.60e-8 ------- ------------------ ------------ ------------------------ ------------ ------------------ ------------ ------------------------ ------------ : Errors for the molecular surface, at grid points within distance $h$ in the first octant; $5$th and $7$th order method with $\delta$ proportional to $h$ or corresponding to $h^4$. **Stokes Flow.** We present examples of three types. First we calculate the velocity near a translating spheroid in Stokes flow, given as a single layer integral. We then compute a standard identity for the double layer integral. Finally we compute a velocity that combines single and double layer integrals on an arbitrary surface, as in the examples above with harmonic potentials. We have increased $\rho$ to $(3,4,5)$ to make the order of accuracy more evident, even though errors are typically smaller with $(2,3,4)$. In each case we report errors at grid points within distance $h$ of the surface. In our first example we compare the single layer or Stokeslet integral with an exact solution. We compute the Stokes flow around a prolate spheroid $$\label{spheroid} x_1^2 + 4x_2^2 + 4x_3^2 = 1$$ with semi-axes $1$, $.5$, $.5$, translating with velocity $(1,0,0)$. The fluid velocity is determined by the integral [\[SingleLayer\]](#SingleLayer){reference-type="eqref" reference="SingleLayer"} from the surface traction ${\mathbf f}$. Formulas for the solution are given in [@chwang; @liron; @tbjcp]. The surface traction is $${\mathbf f}({\mathbf x}) = (f_1({\mathbf x}),0,0)\,,\quad f_1({\mathbf x}) = \frac{F_0}{\sqrt{1 - 3x_1^2/4}}$$ where $F_0$ is a constant. We compute the fluid velocity ${\mathbf u}$ as in [\[stosglsub\]](#stosglsub){reference-type="eqref" reference="stosglsub"},[\[Sreg\]](#Sreg){reference-type="eqref" reference="Sreg"} and extrapolate as before. Results are presented in Table 11. The exact solution has maximum amplitude $1$ and $L^2$ norm about $1$. ------- -------------------------------------- ---------- ------------ ---------- ------------------------------------------------------- ---------- ------------ ---------- $1/h$ $\delta=\rho h, \quad \rho =(3,4,5)$ $\delta=\rho h^{4/5}, \quad \rho =(3,4,5)/(64)^{1/5}$  $L^2$ err    ratio    max err    ratio    $L^2$ err    ratio    max err    ratio   32 3.35e-4 3.27e-3 1.85e-4 1.99e-3 64 1.65e-5 20.3 2.03e-4 16.1 1.65e-5 11.2 2.03e-4 9.8 128 6.02e-7 27.4 8.09e-6 25.1 1.23e-6 13.4 1.57e-5 12.9 256 1.95e-8 30.9 2.71e-7 29.9 8.34e-8 14.7 1.06e-6 14.8 ------- -------------------------------------- ---------- ------------ ---------- ------------------------------------------------------- ---------- ------------ ---------- : Errors for the Stokes single layer on a prolate spheroid, at grid points within distance $h$ outside the spheroid. Next we test the double layer integral [\[DoubleLayer\]](#DoubleLayer){reference-type="eqref" reference="DoubleLayer"} using the identity (2.3.19) from [@pozbook] $$\label{DL_identity} \frac{1}{8\pi} \epsilon_{jlm} \int_\Gamma x_m T_{ijk} (\mathbf{x_0,x}) n_k(\mathbf{x})dS(\mathbf{x}) = \chi (\mathbf{x}_0) \epsilon_{ilm} x_{0,m}$$ where $\chi$ = 1, 1/2, 0 when $\mathbf{x}_0$ is inside, on, and outside the boundary. We set $l=1$ and define $q_j(\mathbf{x}) = \epsilon_{j1m}x_m = (0,-x_3,x_2)$. We compute the integral according to [\[stodblsub\]](#stodblsub){reference-type="eqref" reference="stodblsub"}, [\[Tsplit\]](#Tsplit){reference-type="eqref" reference="Tsplit"}, [\[Treg\]](#Treg){reference-type="eqref" reference="Treg"} and extrapolate. We report errors for a sphere and for the spheroid [\[spheroid\]](#spheroid){reference-type="eqref" reference="spheroid"} in Tables 12 and 13. For the sphere the maximum value is $1$ and the $L^2$ norm is about $.57$. For the spheroid the maximum is $\approx .5$ and the $L^2$ norm is $\approx .3$. ------- -------------------------------------- ---------- ------------ ---------- ------------------------------------------------------- ---------- ------------ ---------- $1/h$ $\delta=\rho h, \quad \rho =(3,4,5)$ $\delta=\rho h^{4/5}, \quad \rho =(3,4,5)/(64)^{1/5}$  $L^2$ err    ratio    max err    ratio    $L^2$ err    ratio    max err    ratio   16 2.86e-4 5.14e-4 6.54e-5 1.49e-4 32 8.81e-6 32.5 1.63e-5 31.5 4.20e-6 15.6 9.05e-6 16.5 64 2.76e-7 31.9 5.07e-7 32.1 2.76e-7 15.2 5.07e-7 17.8 128 8.61e-9 32.0 1.57e-8 32.2 1.80e-8 15.3 3.10e-8 16.4 ------- -------------------------------------- ---------- ------------ ---------- ------------------------------------------------------- ---------- ------------ ---------- : Error for the Stokes double layer on the unit sphere, at grid points within distance $h$ on either side of the sphere. ------- -------------------------------------- ---------- ------------ ---------- ------------------------------------------------------- ---------- ------------ ---------- $1/h$ $\delta=\rho h, \quad \rho =(3,4,5)$ $\delta=\rho h^{4/5}, \quad \rho =(3,4,5)/(64)^{1/5}$  $L^2$ err    ratio    max err    ratio    $L^2$ err    ratio    max err    ratio   32 3.24e-4 1.51e-3 1.67e-4 8.95e-4 64 1.32e-5 24.5 8.07e-5 18.7 1.32e-5 12.6 8.07e-5 11.1 128 4.50e-7 29.3 3.11e-6 26.0 9.31e-7 14.2 6.08e-6 13.3 256 1.44e-8 31.3 1.04e-7 29.9 6.22e-8 15.0 4.10e-7 14.8 ------- -------------------------------------- ---------- ------------ ---------- ------------------------------------------------------- ---------- ------------ ---------- : Errors for the Stokes double layer on a prolate spheroid, at grid points within distance $h$ on either side of the spheroid. In order to test integrals on general surfaces we again use a formula combining the single and double layer integrals. If ${\mathbf u}$ is the velocity of Stokes flow outside and inside a surface $\Gamma$, with suitable decay at infinity, then $$\label{Sum_SLDL} u_i(\mathbf{y}) = -\frac{1}{8\pi}\int_\Gamma S_{ij} (\mathbf{y,x}) [f]_j(\mathbf{x}) dS(\mathbf{x}) - \frac{1}{8\pi}\int_\Gamma T_{ijk} (\mathbf{y,x}) [u]_j(\mathbf{x}) n_k(\mathbf{x})dS(\mathbf{x})$$ Here $[f] = f^+ - f^-$ is the jump in surface force, outside minus inside, and $[u]$ is the jump in velocity. The surface force is the normal stress, $f^\pm = \sigma^\pm \cdot{\mathbf n}$, where ${\mathbf n}$ the outward normal. The jump conditions are derived e.g. in [@pozbook]. As a test problem we take the inside velocity to be the Stokeslet due to a point force singularity of strength $\mathbf{b} = (4\pi,0,0)$, placed at $\mathbf{y}_0 = (2,0,0)$. The velocity is $$\label{Stokeslet_point} u^-_i(\mathbf{y}) = \frac{1}{8\pi} S_{ij}b_j = \frac{1}{8\pi} \Big( \frac{\delta_{ij}}{r} + \frac{\hat{y}_i \hat{y}_j}{r^3} \Big)b_j$$ and the stress tensor is $$\label{Stress_point} \sigma^-_{ik}(\mathbf{y}) = \frac{1}{8\pi} T_{ijk} b_j = \frac{-6}{8\pi}\frac{\hat{y}_i \hat{y}_j \hat{y}_k}{r^5} b_j$$ where $\hat{\mathbf{y}} = \mathbf{y}-\mathbf{y}_0$, $r=|\hat{\mathbf{y}}|$. We choose the outside velocity and stress to be zero. We compute the two integrals in the same manner as above. We present results for three surfaces: the unit sphere, Table 14; an ellipsoid with semi-axes $1$, $.8$, $.6$, Table 15; and the molecular surface [\[molesurf\]](#molesurf){reference-type="eqref" reference="molesurf"}, Table 16. For the first two surfaces, the errors are at all grid points within $h$, but for the molecular surface the points are in the first octant only. For the sphere or ellipsoid the maximum velocity magnitude is $\approx 1$ and the $L^2$ norms are $\approx .35$ and $.37$, respectively. For the molecular surface they are $\approx .9$ and $\approx .4$. ------- -------------------------------------- ---------- ------------ ---------- ------------------------------------------------------- ---------- ------------ ---------- $1/h$ $\delta=\rho h, \quad \rho =(3,4,5)$ $\delta=\rho h^{4/5}, \quad \rho =(3,4,5)/(64)^{1/5}$  $L^2$ err    ratio    max err    ratio    $L^2$ err    ratio    max err    ratio   32 4.94e-5 9.00e-4 2.51e-5 4.87e-4 64 1.86e-6 26.5 3.64e-5 24.7 1.86e-6 13.4 3.64e-5 13.3 128 6.10e-8 30.6 1.23e-6 29.7 1.27e-7 14.7 2.43e-6 15.0 ------- -------------------------------------- ---------- ------------ ---------- ------------------------------------------------------- ---------- ------------ ---------- : Errors for the Stokes single and double layers on the unit sphere, at grid points within distance $h$ on either side of the sphere. ------- ------------------------------------- ---------- ------------ ---------- ------------------------------------------------------- ---------- ------------ ---------- $1/h$ $\delta=\rho h, \quad \rho=(3,4,5)$ $\delta=\rho h^{4/5}, \quad \rho =(3,4,5)/(64)^{1/5}$  $L^2$ err    ratio    max err    ratio    $L^2$ err    ratio    max err    ratio   32 1.80e-4 2.43e-3 9.68e-5 1.41e-3 64 8.43e-6 21.4 1.26e-4 19.2 8.43e-6 11.5 1.26e-4 11.1 128 2.97e-7 28.4 4.63e-6 27.3 6.12e-7 13.8 9.09e-6 13.9 ------- ------------------------------------- ---------- ------------ ---------- ------------------------------------------------------- ---------- ------------ ---------- : Errors for the Stokes single and double layers on an ellipsoid, at grid points within distance $h$ on either side of the ellipsoid. ------- -------------------------------------- ---------- ------------ ---------- ------------------------------------------------------- ---------- ------------ ---------- $1/h$ $\delta=\rho h, \quad \rho =(3,4,5)$ $\delta=\rho h^{4/5}, \quad \rho =(3,4,5)/(64)^{1/5}$  $L^2$ err    ratio    max err    ratio    $L^2$ err    ratio    max err    ratio   32 2.89e-4 1.39e-3 1.54e-4 7.53e-4 64 1.49e-5 19.4 9.08e-5 15.3 1.49e-5 10.4 9.08e-5 8.3 128 6.01e-7 24.7 4.17e-6 21.8 1.21e-6 12.2 7.95e-6 11.4 256 2.04e-8 29.4 1.47e-7 28.3 8.67e-8 14.0 5.77e-7 13.8 ------- -------------------------------------- ---------- ------------ ---------- ------------------------------------------------------- ---------- ------------ ---------- : Errors for the Stokes single and double layers on the four-atom molecular surface, at grid points in the first octant within distance $h$ on either side of the molecule. # Declarations {#declarations .unnumbered} ## Conflict of interest {#conflict-of-interest .unnumbered} The authors declare no competing interests. # Acknowledgment {#acknowledgment .unnumbered} The work of ST was supported by the National Science Foundation grant DMS-2012371. [^1]: Department of Mathematics, Duke University, Durham, NC, 27708 USA beale\@math.duke.edu [^2]: Department of Mathematics, Farmingdale State College, SUNY, Farmingdale, NY 11735, USA tlupovs\@farmingdale.edu

arxiv_math

--- author: - - - - bibliography: - sn-bibliography.bib title: Reduction of a biochemical network mathematical model by means of approximating activators and inhibitors as perfect inverse relationships. --- Models of biochemical networks are usually presented as connected graphs where vertices indicate proteins and edges are drawn to indicate activation or inhibition relationships. These diagrams are useful for drawing qualitative conclusions from the identification of topological features, for example positive and negative feedback loops. These topological features are usually identified under the presumption that activation and inhibition are inverse relationships. The conclusions are often drawn without quantitative analysis, instead relying on rules of thumb. We investigate the extent to which a model needs to prescribe inhibition and activation as true inverses before models behave idiosyncratically; quantitatively dissimilar to networks with similar typologies formed by swapping inhibitors as the inverse of activators. The purpose of the study is to determine under what circumstances rudimentary qualitative assessment of network structure can provide reliable conclusions as to the quantitative behaviour of the network and how appropriate is it to treat activator and inhibitor relationships as opposite in nature. # Introduction {#sec1} Whilst there are different approaches to mathematically model biochemical networks, a commonly used model formalism describes continuous changes in concentration of molecular species over time using ordinary differential equations (ODEs), commonly known as reaction rate equations (RREs) [@jiang2022identification]. ODE models are studied to better understand generic biochemical network topologies and motifs as well as dynamics of well-defined pathways in specific organisms [@vaseghi2001signal; @swameye2003identification]. RREs are almost always non-linear as they describe protein interactions [@rao2014model]. Reaction rates can vary greatly, giving rise to different time scales. Elaborate integrated interactions give rise to complex nonlinear behaviours, such as ultra-sensitivity [@angeli2004detection], bi-stability [@rombouts2021dynamic], and oscillation [@russo2009equilibria]. Numerous studies of RREs have revealed general network properties that give rise to these nonlinear behaviours and, at the same time, give high level understanding for how biochemical networks encode cellular scale responses to stimuli. The general network properties are qualitative in nature, identified by investigating only the network topology, and are now often considered without substantial accompanying quantitative mathematical analysis. Some of these network-scale properties that can be qualitatively associated with complex behaviours include feedback loops, feed-forward loops, cross-talk, compartmentalisation, and noise [@schwartz1999interactions]. Whilst it may be easy to identify a macro-feature responsible for certain behaviour, it is not always easy to *predict* the behaviour simply because the macro-feature is present without quantitative analysis. Analysing the topological structure of the network through its interactive diagram is often a focal point for understanding the system's driving mechanisms and qualitative behaviour [@navlakha2014topological; @glass1975classification; @glass1972co]. A common qualitative technique involves systematically processing the causal links described by the diagram's edges [@gilbert2006petri]. For example, if a node $i$ inhibits node $j$ which inhibits node $k$, it is qualitatively assumed that the two inhibitions counteract each other in such a way that this may be seen as node $i$ activates a 'reverse' of node $j$ which then activates node $k$; in both cases, node $i$ indirectly activates node $k$. This qualitative equivalency is shown in Figure [1](#fig:similarity){reference-type="ref" reference="fig:similarity"}. Typically, this equivalency is assumed irrespective of the specific mathematical representation of the model. However, the similarities between these two interpretations can vary depending on the quantitative description of activators and inhibitors; in particular, how closely these relationships are opposites/inverses of each other. The ability to discern when two pathways will result in 'similar' behaviour is highly useful. In this simple example, we may be able to scientifically agree on a single oriented model (say the one with only activators), which characterises all "qualitatively similar" networks. In the absence of any standard form, we define the oriented model as a model which contains all activation interactions along predefined linear pathways embedded within the network. In the case where activators and inhibitors can be interchanged (exactly or approximately), the oriented model characterises a broad collection of models whereby activation and inhibition are able to be used interchangeably as opposites; thereby making it easier to identify similar networks as well as the nature of features such as feedback, feed forward and cross-talk. The pertinent question is 'What properties allow for the exact or approximate reduction of a biochemical mathematical model to its oriented model?'. We shall focus our attention predominantly on a property of a biochemical model which we call 'model bias' (the tendency for an interaction to be conferred directly effecting activation or deactivation of the downstream node) and secondly on a property which we call 'model symmetry' (the tendency for an interaction to be conferred directly as a result of elevated or lowered amounts of the upstream node). We find that for 'symmetric models' (where equal changes above and below a rest state in the upstream node confer equal and opposite changes in the downstream node), model bias plays a central role in determining the approximate tractability of a network to an oriented form. We investigate how the network connections together with model bias determine the suitability of this oriented form simplification through a numerical study. The manuscript is structured in the follow way. In Section [2](#sec:Frame){reference-type="ref" reference="sec:Frame"}, we mathematically define the framework for orientation of a biochemical network. We also define an unbiased symmetric model; a standard for which exact equivalence between a model and its oriented form can be shown. In Section [3](#sec:Numerics){reference-type="ref" reference="sec:Numerics"}, we explore increases in intrinsic bias in a model and how this mainly into divergence in its oriented form and in particular how this divergence is regulated specifically by network/pathway properties. We study systematically increasing complexity from simple chain-like pathways, to an example of an integrated pathway with cross-talk. Our studies provide new insights into when reduction of a network with an oriented form *may* be appropriate and when caution should be taken. {#fig:similarity width="30%"} # Model Framework {#sec:Frame} In the context of systems biology, the terms 'pathway' and 'network' are sometimes used interchangeably. A 'pathway' typically refers to a relatively small, well-defined set of entities and relationships, such as a specific signal transduction pathway [@alberts2002general]. As the name implies, it should be clear in a pathway where the signal 'begins' and where it 'ends'. On the other hand, a 'network' frequently refers to a larger, less constrained set of entities and relationships [@azeloglu2015signaling]. A network, on the other hand, is often a set of pathways connected through cross-talk and feedback mechanisms. We will define both a pathway and a network (the latter consisting as a collection of the former) as a directed graph where each node $x_i$, $i=1,\ldots,N$, is represented by a scalar state variable (with the same label), and each edge determines distinct terms on the RHS of the dynamical system describing the rates of change in these variables. Importantly, we use the term 'state' rather than concentration. Non-dimensionalisation of the active protein concentrations associated with each node followed by shifting allows us define each state variable on the range $x_i\in[-1,1]$. Here 0 represents half of the saturation concentration and -1 and 1 represents no protein and maximum protein respectively. We choose this framework because we consider that absence (and not just presence) of a protein can cause a response in downstream nodes. The general mathematical model we investigate consists of the differential equations ([\[rreswing\]](#rreswing){reference-type="ref" reference="rreswing"}); $$\label{rreswing} \frac{\mathrm{d} \mathbf{x}}{ \mathrm{d} t} = \boldsymbol{\psi}(\mathbf{x}),$$ describing the evolution of all $N$ state variables $\mathbf{x} = (x_1,x_2,\ldots, x_N)^{\dagger}$. Here $\psi_i = \varrho_i^+ - \varrho_i^-$ and $$\begin{aligned} \varrho_i^+ &= \sum_{\mathcal{E}\in \mathcal{E}_i} r_{\mathcal{E}}^+(x_i,y_{\mathcal{E}};\mathcal{T}_{\mathcal{E}}) , \quad \text{and} \label{rplus} \\ \varrho_i^- &= \sum_{\mathcal{E}\in \mathcal{E}_i} r_{\mathcal{E}}^-(x_i,y_{\mathcal{E}};\mathcal{T}_{\mathcal{E}}) , \label{rminus}\end{aligned}$$ where each of the sums are taken over the set of edges $\mathcal{E}_i$ that point towards the node $x_i$, $r_{\mathcal{E}}^\pm$ are functions that completely depend on the model and the type of edge $\mathcal{T}_{\mathcal{E}}\in\{ \mathcal{A},\mathcal{I}\}$ (activator $\mathcal{A}$ or inhibitor $\mathcal{I}$). These are functions of the node $y_{\mathcal{E}}$ from which each edge $\mathcal{E}$ originates. Finally, some edges in a network may not come explicitly from another node in the network. These edges represent sources or external stimuli to the system and must point towards a node in the network and for these edges it is assumed in the context of ([\[rplus\]](#rplus){reference-type="ref" reference="rplus"}) and ([\[rminus\]](#rminus){reference-type="ref" reference="rminus"}) that $y_{\mathcal{E}} = 0$ such that $y_{\mathcal{E}}$ and $y_{\mathcal{E}}^*$ are non-zero and balanced. We use the superscript asterisk to represent a 'flip' in a state variable; negating it algebraically and representing it in diagrams by changing nodes between solid and hollow style. An edge $\mathcal{E}$ is associated with the qualitative description of an activator or inhibitor. Activation is achieved in one of two ways, either $r_{\mathcal{E}}^+$ relatively increases with $y_{\mathcal{E}}$ and/or $r_{\mathcal{E}}^-$ relatively decreases with $y_{\mathcal{E}}$. Inhibition is associated with opposite conditions. In particular, for a given activation edge $\mathcal{E}$ connecting node $y$ to $x$, we require by definition that $$\label{condact}\frac{\partial r_{\mathcal{E}}^+(x,y;\mathcal{A})}{\partial y} \geq \frac{\partial r_{\mathcal{E}}^-(x,y;\mathcal{A})}{\partial y}$$ everywhere in the state space determined by $x$ and $y$. On the other hand, we require $$\label{condin}\frac{\partial r_{\mathcal{E}}^+(x,y;\mathcal{I})}{\partial y} \leq \frac{\partial r_{\mathcal{E}}^-(x,y;\mathcal{I})}{\partial y}$$ everywhere if an edge is to be an inhibition edge. ## Model Bias and Symmetry The conditions ([\[condact\]](#condact){reference-type="ref" reference="condact"}) and ([\[condin\]](#condin){reference-type="ref" reference="condin"}) can be achieved by adjusting either side of the inequalities. Treating activation as the opposite of inhibition requires that condition ([\[condin\]](#condin){reference-type="ref" reference="condin"}) is the same as ([\[condin\]](#condin){reference-type="ref" reference="condin"}) after swapping $+$ and $-$ then swapping the direction of the inequality. We will show it is necessary and sufficient for activation and inhibition to equal and opposite in this respect if both sides of the inequalities are equal and opposite in magnitude (unbiased) and this magnitude is the same for each condition (symmetric). We define these properties mathematically as follows. **Definition 1**. *An edge $\mathcal{E}$ of type $\mathcal{T}\in\{ \mathcal{A},\mathcal{I}\}$ is *unbiased* if it defines the function pair $\{ r_\mathcal{E}^+,r_\mathcal{E}^-\}$ in the model ([\[rreswing\]](#rreswing){reference-type="ref" reference="rreswing"}) and ([\[rplus\]](#rplus){reference-type="ref" reference="rplus"})-([\[rminus\]](#rminus){reference-type="ref" reference="rminus"}) such that, for all $(x,y)\in [-1,1]\times[-1,1]$, $$r_{\mathcal{E}}^{+}(x,y;\mathcal{T}) = r_{\mathcal{E}}^{-}(-x,-y;\mathcal{T}).$$ The edge is considered to be *positively biased* if $r_{\mathcal{E}}^{+}(x,y;\mathcal{T}) > r_{\mathcal{E}}^{-}(-x,-y;\mathcal{T})$ and *negatively biased* if $r_{\mathcal{E}}^{+}(x,y;\mathcal{T}) < r_{\mathcal{E}}^{-}(-x,-y;\mathcal{T})$ for all $(x,y)\in [-1,1]\times[-1,1]$.* **Definition 2**. *A model or biochemical network is *positively* or *negatively* biased or *unbiased* if all edges in the model or network are positively or negatively biased or unbiased, respectively.* **Definition 3**. *An edge $\mathcal{E}$ is *symmetric* if, in the context of the model ([\[rreswing\]](#rreswing){reference-type="ref" reference="rreswing"}) and ([\[rplus\]](#rplus){reference-type="ref" reference="rplus"})-([\[rminus\]](#rminus){reference-type="ref" reference="rminus"}), the functions $r_\mathcal{E}^\pm$ for activation and inhibition are defined such that $$r_{\mathcal{E}}^{\pm}(x,y;\mathcal{A}) = r_{\mathcal{E}}^{\pm}(x,-y;\mathcal{I}),$$ for all $(x,y)\in [-1,1]\times[-1,1]$. The edge is considered to be *activator weighted* if $r_{\mathcal{E}}^{\pm}(x,y;\mathcal{A}) > r_{\mathcal{E}}^{\pm}(x,-y;\mathcal{I})$ and *inhibitor weighted* if $r_{\mathcal{E}}^{\pm}(x,y;\mathcal{A}) < r_{\mathcal{E}}^{\pm}(x,-y;\mathcal{I})$ for all $(x,y)\in [-1,1]\times[-1,1]$.* **Definition 4**. *A model or biochemical network is *activator* or *inhibitor* weighted or *symmetric* if all edges in the model or network are activator or inhibitor weighted or symmetric, respectively.* In practise, each node of a biochemical network graph/diagram often represents a protein concentration. Concentrations can either be increased/activated (increasing the state variable for the node) or decreased/deactivated (decreasing the state variable for the node). To better explain the mechanisms which determine bias and asymmetry (weighting) in a model, it is useful to add detail to each node and consider an 'active' (solid dots) and 'inactive' (hollow dots) component. Of course, these are just labels as we assume that information may equally use the active and inactive components and that these labels just determine the positive and negative direction of the node state variable. A protein $x_i$ can be activated or deactivated and the mathematical model encodes this in the terms $\varrho_i^+$ and $\varrho_i^-$ respectively. Showing generic nodes $x$ and $y$ where $y$ affects $x$ through an edge, it is useful to visualise the mechanistic manner with which bias and weight/asymmetry are manifested in models. We catalogue the different extreme cases for both activator and inhibitor in Figure [2](#fig:definitions){reference-type="ref" reference="fig:definitions"}. ![Mechanistic diagrams for asymmetric and biased activator (a) and inhibitor (b) model edges. Each node $y$ (upstream) and $x$ (downstream) is represented as an actively switching chemical species between 'active' (solid dot) and 'inactive' (hollow dot) forms. The switching from active to inactive form is represented in the model for each node by $\varrho^-$ (Equation ([\[rminus\]](#rminus){reference-type="ref" reference="rminus"})) and from inactive to active by $\varrho^+$ (Equation ([\[rplus\]](#rplus){reference-type="ref" reference="rplus"})) which are influenced by the network edges. The dotted connectors indicate how biased (negative or positive) and weighted (activator or inhibitor) models for the network edges mechanistically confer activation or inhibition from node $y$ to node $x$.](Figures/definitions_4.pdf){#fig:definitions width="100%"} Bias and weighting just indicate deviation away from the unbiased symmetric case as it is difficult to assign a sensible generalised quantitative definition to these qualities. We focus primarily on bias in this manuscript as issues with symmetry can be somewhat mitigated by ensuring that the amplitude $r_{\mathcal{E}}$ is unchanged when comparing activators and inhibitors. It is possible to get a sense of increasing and decreasing bias by plotting the nullcline in the ($x$-$y$) state space that corresponds to $r_{\mathcal{E}}^+(x,y) = r_{\mathcal{E}}^-(x,y)$ (the steady state in $x$ caused by a state variable $y$ in the absence of other edges). We show these nullclines for a symmetric model edge of activator (Fig [3](#fig:dynamics_act){reference-type="ref" reference="fig:dynamics_act"}) and inhibitor (Fig [4](#fig:dynamics_inh){reference-type="ref" reference="fig:dynamics_inh"}) type respectively. The unbiased case can be seen clearly in the solid black curves. Increasing negative bias is shown by the dashed blue curves whereas increasing positive bias is shown by the dashed red curves. Increasing bias shifts the nullcline further to the left or right respectively when compared to the unbiased case in black. Furthermore, the unbiased case necessarily has the feature that the nullclines can be rotated around $(0,0)$ an angle of $\pi$ without changing the nullcline. ![[\[fig:dynamics_act\]]{#fig:dynamics_act label="fig:dynamics_act"}](Figures/activation.pdf){#fig:dynamics_act width="\\textwidth"} ![[\[fig:dynamics_inh\]]{#fig:dynamics_inh label="fig:dynamics_inh"}](Figures/inhibition.pdf){#fig:dynamics_inh width="\\textwidth"} ## Oriented pathways and pathway similarity Consider the example of the simple linear pathway shown in Figure [1](#fig:similarity){reference-type="ref" reference="fig:similarity"}. If it is justified that both representations behave the same, we can orient the pathway by a standard 'oriented' form. We shall denote the *oriented form* of the pathway as that portrayed on the right of Figure [1](#fig:similarity){reference-type="ref" reference="fig:similarity"}; that is, where all relationships are denoted as activators. If we are unable to treat activators and inhibitors as opposites (because the model is biased or assymetric) then it is still qualitatively feasible that the oriented and unoriented pathways of Figure [1](#fig:similarity){reference-type="ref" reference="fig:similarity"} (and others) behave 'similarly' but not exactly the same. To what degree bias results in dissimilar/divergent oriented and unoriented pathways is the focus of this paper. Here we define how to find the oriented form of a pathway and a network. **Definition 5**. *A (linear) *pathway* of length $N\geq 1$ is a set of ordered nodes $\{x_i\}_{i=1}^{N}$ connected by $N-1$ edges $\{\mathcal{E}_i\}_{i=1}^{N-1}$ where $\mathcal{E}_i$ connects $x_i$ to $x_{i+1}$. The pathway must have a prescribed *input* node $x_1$ and *output* node $x_N$. All non-output nodes are *internal* nodes.* **Definition 6**. *A *network* is a set of one or more pathways possibly connected by cross-talk and/or connected to external edges (edges not associated with a pathway/external stimuli).* **Definition 7**. *Two networks are *equivalent* (or *similar*) if all constituent pathway output nodes have identical (or similar) response outputs.* **Definition 8**. *An *oriented* pathway is one which is equivalent or similar to any given pathway but contains all activation edges. An *oriented* network is one that contains all oriented pathways. A pathway or network is called *orientable* if it has an oriented form, which is equivalent.* These definitions bring us to a key result expressed in the following lemma and generalised in the proceeding theorem. **Lemma 1**. *All unbiased and symmetric pathways are orientable.* *Proof.* To prove Lemma [Lemma 1](#thm1){reference-type="ref" reference="thm1"}, we first demonstrate an interesting property of unbiased and symmetric pathway/network models ([\[rreswing\]](#rreswing){reference-type="ref" reference="rreswing"}). Consider any node $x_i$ which may have any number of edges pointed towards it $\mathcal{E}_i^{(\text{in})}$ and out of it $\mathcal{E}_i^{(\text{out})}$. We have $$\begin{aligned} \dot{x}_i &= \sum_{\mathcal{E}_{ki}\in \mathcal{E}_{i}^{(\text{in})}} r^{+}_{\mathcal{E}_{ki}}(x_i,x_k;\mathcal{T}_{ki}) - r^{-}_{\mathcal{E}_{ki}}(x_i,x_k;\mathcal{T}_{ki}), \quad \text{and} \\ \dot{x}_{j} &= r^+_{\mathcal{E}_{ij}}(x_{j},x_{i};\mathcal{T}_{ij}) - r^-_{\mathcal{E}_{ij}}(x_{j},x_{i};\mathcal{T}_{ij})+ \psi_{ij}', \quad \text{for each } \mathcal{E}_{ij}\in \mathcal{E}_i^{(\text{out})},\end{aligned}$$ where $\mathcal{E}_{ki}$ is an edge of type $\mathcal{T}_{ki}$ that points from a node $x_k$ to $x_i$ and $\psi_{ij}'$ are due to edges that affect a node $x_j$ but are not connected to the node $x_i$. Here we have all terms which include the node $x_i$ in the model. We shall now perform a *flip* of the node $x_i$ by changing its state variable to its complements. That is, everywhere, we shall write these equations replacing $x_i$ for $-x_i$. $$\begin{aligned} \dot{x}^{\text{(flip i)}}_i &= \sum_{\mathcal{E}_{ki}\in \mathcal{E}_i^{(\text{in})}} - r^+_{\mathcal{E}_{ki}}(-x_i,x_k;\mathcal{T}_{ki}) + r^-_{\mathcal{E}_{ki}}(-x_i,x_k;\mathcal{T}_{ki}), \quad \text{and} \\ \dot{x}_j^{\text{(flip i)}} &= r^+_{\mathcal{E}_{ij}}(x_j,-x_i;\mathcal{T}_{ij}) - r^-_{\mathcal{E}_{ij}}(x_j,-x_i;\mathcal{T}_{ij})+ \psi_{ij}', \quad \text{for each } \mathcal{E}_{ij}\in \mathcal{E}_i^{(\text{out})}\end{aligned}$$ Consider now coupling this flipping of node $x_i$ operation with the operation of also performing a *flip* in the type of all edges in $\mathcal{E}_i^{(\text{in})}$ or $\mathcal{E}_i^{(\text{out})}$. That is, we change respective types for all edges connected to node $i$ $\mathcal{T}\rightarrow \mathcal{T}^*$ (which represents interchanging $\mathcal{A}\rightarrow \mathcal{I}$ and $\mathcal{I}\rightarrow \mathcal{A}$). We notice that if the model is symmetric, $\dot{x}_j^{\text{(flip i)}}$ reverts back to $\dot{x}_j$ after implementation of Definition [Definition 3](#symedge){reference-type="ref" reference="symedge"}. On the other hand, since the model is also unbiased, after implementation of Definition [Definition 1](#biasedge){reference-type="ref" reference="biasedge"}, $\dot{x}^{\text{(flip i)}}_i$ becomes $-\dot{x}_i$ as expected after flipping both the node $x_i$ and all connected edges. Therefore, if a model is unbiased and symmetric then flipping any node $x_i$ along with all of its connecting (in and out) edges from activation to inhibition and vice versa will leave all other nodes in the model completely unchanged. We continue this proof as a special case of Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"}. ◻ **Theorem 2**. *All unbiased and symmetric networks are orientable.* *Proof.* Lemma [Lemma 1](#thm1){reference-type="ref" reference="thm1"}, and by trivial extension Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"}, can be proven by induction. We prove this by showing orientability of a single pathway noting that a network of pathways can be approached in the same way. The induction base case for a pathway is where $N=2$ (a pathway with a single edge). The pathway consists of the output node $x_i$ and input node, $x_j$ as shown in Fig. [5](#fig:pf1){reference-type="ref" reference="fig:pf1"}. If the edge connecting $x_j$ to $x_i$ is an activation, then it is already in oriented form. If not, then, as we have shown, it is possible to *flip* the node $x_j$ along with all edges connected to $x_j$ without changing the output in the manner just described. The subsequent oriented pathway is therefore equivalent to the original, and thus a pathway of length $N=2$ is orientable if the model is symmetric and unbiased. It is important here that in performing the flipping of the node $x_j$ and the edge that the node $x_i$ is left unchanged (despite the fact that $x_j$ has become $-x_j$) because $x_j$ is an internal node whilst $x_i$ is the output node that needs to be identical if equivalence is to be achieved between the oriented and unoriented pathways. ![[\[fig:pf1\]]{#fig:pf1 label="fig:pf1"} A figure showing the equivalence in (a) the signalling pathway with $N=2$ and a single inhibition edge with (b) its oriented form. A hollow node is used to indicate that it is the complement (negative/flipped) of the original state variable for this node. The output nodes are in a dashed box to indicate that these nodes are unchanged between the unoriented and oriented pathways making these pathways equivalent.](Figures/pf1.pdf){#fig:pf1 width="30%"} We now assume that any unbiased symmetric pathway of length $k\geq 2$ is orientable. Consider a pathway of length $N=k+1$ with ordered nodes $\{x_i \}_{i=1}^{k+1}$. This pathway contains within it a pathway of length $k$ with input node $x_2$ and output node $x_{k+1}$. Since the pathway of length $k$ is orientable, we may perform any necessary flips of internal nodes and edges to generate the oriented form of the pathway from node $x_2$ to $x_{k+1}$. At this point, we are now permitted to flip node $x_1$ and all connected edges to place the full pathway into oriented form if the edge from $x_1$ to $x_2$ is an inhibition edge (otherwise the full pathway is already in oriented form). Thus, if a pathway of length $k$ is orientable then so is a pathway of length $k+1$. Application of this process to all pathways in a network proves Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"}. ◻ Inspired by the proof of Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"}, the follow algorithm is used to orient any given pathway. $x_i \leftarrow x_N$\ The steps outlined in Algorithm [\[algorithm\]](#algorithm){reference-type="ref" reference="algorithm"} are shown in Figure [6](#fig:algo1){reference-type="ref" reference="fig:algo1"}. ![An example of how to convert a pathway into its oriented form using the steps outlined in Algorithm [\[algorithm\]](#algorithm){reference-type="ref" reference="algorithm"}](Figures/algorithm_1.pdf){#fig:algo1 width="90%"} Generalisation to orienting a full network is done using Algorithm [\[algorithm2\]](#algorithm2){reference-type="ref" reference="algorithm2"}. Identify all constituent signalling pathways comprising the network.\ # Numerical Results and Discussion {#sec:Numerics} Evident by the prevalence of the idea of protein 'activation' and 'deactivation', most biochemical network models have bias. We conduct a numerical exploration of positive and negative bias and the robustness of the oriented form approximation. We focus first on single pathways and then explore the effect of feedback and network cross-talks. As expected, making the networks more complicated makes the pathways more idiosyncratic (diverges away from the behaviour of a common oriented form). However, for a given network, the oriented form can be much more robust if the bias is positive or negative depending on the situation. Furthermore, we find that for simple networks robustness is strong as bias is increased but beyond a critical bias robustness is lost rapidly and the oriented form does not represent the unoriented network; in some cases even behaving in an opposite way. ## Test model {#testmodel} We use a caricature test model so that we can increase or decrease bias explicitly. The model is parameterised by three parameters: $\alpha$, $\beta$, and $\phi$. The parameter $\alpha$ describes the amplitude (timescale), $\beta$ prescribes variable nonlinearity, and $\phi$ is a proxy for the model's bias and is chosen independently for each edge. This model is unlikely to represent a real biological system. That being said, by the manipulation of parameters $\alpha$, $\beta$ and $\phi$ there is a lot of flexibility in the model functions $r_\mathcal{E}^\pm(x,y;\mathcal{T}_\mathcal{E})$. In this sense, the test model allows for generic observations in the role of bias in the approximation that inhibition is the opposite of activation in the context of increasingly complex networks. The rates $r_{\mathcal{E}}^+$ and $r_{\mathcal{E}}^-$ are defined as follows: For activation, &r\_\^+(x,y;) = ( )(1+y) F(x;,)[\[eq:twoway_act_mod1\]]{#eq:twoway_act_mod1 label="eq:twoway_act_mod1"}\ &r\_\^-(x,y;) = ( )(1-y)F(-x;,)[\[eq:twoway_act_mod2\]]{#eq:twoway_act_mod2 label="eq:twoway_act_mod2"}. For inhibition, &r\_\^+(x,y;) = ( )(1-y) F(x;,)[\[eq:twoway_act_mod3\]]{#eq:twoway_act_mod3 label="eq:twoway_act_mod3"}\ &r\_\^-(x,y;) = ( )(1+y)F(-x;,)[\[eq:twoway_act_mod4\]]{#eq:twoway_act_mod4 label="eq:twoway_act_mod4"}. where, $$F(x;\alpha,\beta) = \frac{\alpha\beta(1-x)}{2\beta-(1+x)}.$$ Here $\phi = 0$ indicates no bias and $\phi>0$ ($\phi<0$) indicates positive (negative) bias. The function $F$ is based on Michaelis-Menten enzyme kinetics but translated to our state variable framework (where state variables vary between -1 and 1). Our study uses MATLAB's ODE45 solver to run simulations. Specifically, where not stated, for each bias value $\phi$, we test a total of 150 diverse sets of $\alpha$ and $\beta$ values to better isolate behaviour attributable to bias and network topology rather than from non-linearity and timescale. ## Linear pathways ### Unbiased linear pathways We begin by investigating single linear pathways of up to 5 nodes activated at the input node. Node 1 is the input node and Node 5 is the output node. In Fig. [7](#fig:comparepathway){reference-type="ref" reference="fig:comparepathway"} we compare the time evolution of each node for a sample pathway against its oriented form obtained by Algorithm [\[algorithm\]](#algorithm){reference-type="ref" reference="algorithm"}. In this particular case, we demonstrate numerically Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"} by setting the bias $\phi = 0$. The observation here is that the outputs (orange) are identical between pathway and oriented pathway whilst all flipping of internal nodes that is done in the orienting process correctly negates the time evolution of the node at all times. It is clear that when there is no (or very little) bias in the symmetric model, observing two inhibitions in series is quantitatively identical (or very similar) to two activators. This is because, at least qualitatively, it is understood that inhibiting an inhibitor is akin to a double negative; resulting in a positive. Understanding this equivalence (and when it is appropriate to assume it) makes it easier to qualitatively assess the pathway (and network) behaviour from generic topological relationships -- as is common with biochemical network models. {reference-type="ref" reference="testmodel"}) is used to model the pathway edges. In this model, each edge of the network is associated with two parameters, $\alpha_{ij}$ and $\beta_{ij}$ where $i$ is the source and $j$ is the destination of the connector between two nodes. The parameter values for $\alpha$ and $\beta$ were chosen arbitrarily as $\alpha = \{\alpha_{12},\alpha_{23},\alpha_{34},\alpha_{45}\}= \{0.5,1,2,1.3\}$ and $\beta = \{\beta_{12},\beta_{23},\beta_{34},\beta_{45}\}= \{10,5,15,18\}$. A boxes on the right side each node displays the state variables of each node from $t=0$ to $t=100$. All states are initialised in the neutral position of $0$. The pathway in (b) is the oriented representation of the signalling pathway shown in (a). The hollow nodes represent nodes that are flipped in the orienting process described by Algorithm [\[algorithm\]](#algorithm){reference-type="ref" reference="algorithm"}. Since this process results in a flipped $x_1$ the input edge should be interpreted as an inhibition which is indicated in blue. In choosing $\phi = 0$ (no bias) and noting that $x_5$ is the same for (a) and (b) we have verified the result of Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"}.](Figures/compare_pathway.pdf){#fig:comparepathway width="71%"} ### Biased linear pathways We now shift our investigation to the robustness of the oriented pathway reduction of single pathways with bias. To compare a pathway/network against its oriented form we measure the similarity between the two by focusing on two metrics. In both metrics we initialise the pathway and its oriented form in the neutral position (all node states equal to zero). We then run the model for both oriented and unoriented forms numerically until steady state. The first metric comparing the unoriented to oriented descriptions is the difference in the long term behaviour. We have restricted our study to non-oscillatory systems which limit to fixed steady states over time. We define the steady state difference to be $$\label{deltass} \delta_{ss} = \left< \delta - \bar{\delta}\right>_{\alpha,\beta} = \left<\lim_{t\rightarrow \infty} x_N(t) - \lim_{t\rightarrow \infty} \bar{x}_N(t)\right>_{\alpha,\beta} ,$$ where $\delta$ and $\bar{\delta}$ are the unoriented and oriented steady states of the output node $x_N$, respectively. The bar notation here corresponds to the oriented form. The brackets $\left<\cdot \right>_{\alpha,\beta}$ indicate averaging over many combinations of parameter sets $\alpha$ and $\beta$ describing the edges in the model. A second metric is defined to measure the difference in the transient aspects of the response of the unoriented and oriented cases. It is constructed by first normalising each output node by its steady state and then integrating the difference between the unoriented and oriented transient output values. The resultant error is positive (negative) if the unoriented form approaches steady state faster (slower) than the oriented form. Very large values in this metric may indicate the presence of some temporal behaviour not present in the other form (for example, rebounding behaviour). This is especially the case if one of the forms responds to the input and returns close to the neutral state over time. $$\delta_\tau = \left<\int_0^\infty \left( \frac{x_N(t)}{\delta } - \frac{\bar{x}_N(t)}{\bar{\delta}} \right) \ \mathrm{d}t \right>_{\alpha,\beta}$$ Behavioural differences between unoriented and oriented pathways can also lead to small errors $\delta_\tau$. The purpose of $\delta_\tau$ is to get insight into response times of unoriented and oriented pathways under simple conditions. This metric is less useful in the case where responses are sufficiently different in nature. Fig. [8](#fig:delta_ss){reference-type="ref" reference="fig:delta_ss"} displays the steady-state error $\delta_{ss}$ for all possible linear pathways of length $N=5$. Each chart in the figure is a plot of $\delta_{ss}$ versus $\phi$ in the model. That is, left of centre represented increasing negative bias and right of centre represents increasing positive bias. The pathways in the figure are shown at the top of each chart and immediately below these pathways are the oriented forms (showing specifically which nodes are flipped in the orienting process). The charts themselves are strategically organised into columns based on the total number of nodes that are flipped in order to create the oriented form (one node on the left, two nodes in the centre, three and four nodes on the right). From top to bottom, we order the charts such that in the orienting process flips that are generally more upstream are at the top whereas downstream flips are generally down the bottom. We also plot each chart in red if, after the orienting process, the input stimulus does not change type from activation but in blue if the input stimulus changes from activation to inhibition. Setting the charts out like that in Fig. [8](#fig:delta_ss){reference-type="ref" reference="fig:delta_ss"} shows a number of interesting and nontrivial properties for linear pathways: 1. Locally around the unbiased case $\phi = 0$ we see good agreement in steady states between the unoriented and oriented pathways. 2. Agreement with the oriented pathway is very robust if the model is *negatively* biased the oriented form requires a flip in the input stimulus from activation to inhibition. If there is no flip in the input stimulus then agreement is very robust if the model is *positively* biased. 3. As bias is increased, agreement with the oriented pathway remains robust up until some critical level of bias at which stage agreement rapidly decreases until under some conditions the worst cases reach $\delta_{ss} \approx \pm 2$ indicating that the oriented and unoriented forms of the pathway limit towards extreme opposite outputs. 4. The critical bias, before disagreement with the oriented form rapidly increases, is closer to the unbiased case -- that is, the oriented form is less robust -- if the orienting process flips over nodes in higher quantities *or* further downstream. Pathways which satisfy these criterion also have more extreme disagreements with their oriented form. These observations were also consistent with results found for pathways of lengths $N=3$ and $N=4$ (not shown here). ![This figure displays the steady-state error $\delta_{ss}$ of different signalling pathways of length $N=5$ as defined by Equation ([\[deltass\]](#deltass){reference-type="ref" reference="deltass"}). The model used is the test model in Section [3.1](#testmodel){reference-type="ref" reference="testmodel"}. The error is plotted in charts as functions of the bias parameter $\phi$ and the pathway with its oriented form are shown above each chart. The colors of the plots correspond to flipping (blue) or no flipping (red) of the input stimulus as a result of the orienting process. The columns separate the number of flipped nodes in the orienting process whilst each column is ordered from top to bottom where these flips occur more upstream or downstream respectively.](Figures/mean_dss_pathway.pdf){#fig:delta_ss width=".8\\linewidth"} It is also interesting to observe that disagreement between the oriented and unoriented forms as a result of bias compounds as the number of flips increases. This can be seen in Fig. [9](#fig:all_in_one){reference-type="ref" reference="fig:all_in_one"} where the sum of all errors for each of the four single flip pathways (left column) in Fig. [8](#fig:delta_ss){reference-type="ref" reference="fig:delta_ss"} are plotted against the steady state error associated with the pathway which requires all four flips to orient (bottom right chart in Fig. [8](#fig:delta_ss){reference-type="ref" reference="fig:delta_ss"}). The later error eclipses the sum of the former errors however in both cases robustness remains fairly strong for a significant interval around $\phi = 0$ (the unbiased case). {#fig:all_in_one width=".6\\linewidth"} Using the same charting order used in Fig. [8](#fig:delta_ss){reference-type="ref" reference="fig:delta_ss"}, we chart the errors $\delta_\tau$ in Fig. [10](#fig:pathway_dtau){reference-type="ref" reference="fig:pathway_dtau"}. The plots of $\delta_\tau$ show similar behaviour under changes in bias to $\delta_{ss}$. This is especially the case for pathways requiring few flips in orienting. In this case, oriented and unoriented pathways which tend towards different steady states also take different rates in getting there. Pathways requiring many flips produce more complex behaviour over time and $\delta_\tau$ is less insightful. ![This figure displays the steady-state error $\delta_{ss}$ of different signalling pathways of length $N=5$ as defined by Equation ([\[deltass\]](#deltass){reference-type="ref" reference="deltass"}). The model used is the test model in Section [3.1](#testmodel){reference-type="ref" reference="testmodel"}. The error is plotted in charts as functions of the bias parameter $\phi$ and the pathway with its oriented form are shown above each chart. The colors of the plots correspond to flipping (blue) or no flipping (red) of the input stimulus as a result of the orienting process. The chart arrangement is the same as that of Fig. [8](#fig:delta_ss){reference-type="ref" reference="fig:delta_ss"}.](Figures/mean_dtau_pathway_v2.pdf){#fig:pathway_dtau width=".9\\linewidth"} In the following section we shift our attention to the affect of bias on the oriented form on networks. We begin with simple pathways with feedback before looking at an example network with significant integration of multiple pathways. ## Networks ### Unbiased pathways with feedback Investigation of linear pathways in the previous section shows that care should be taken when assuming that inhibitors act as diametrically opposite activators under biased conditions in the case where there this assumption is taken in multiple instances (where there are many flips to get to the oriented form). We now attempt to investigate the validity of this assumption under increasing complexity and in particular in the case of feedback. In this section, we will fix a linear pathway of length $N=5$ consisting of 3 inhibitions and ending in a single activation. The orienting procedure then requires a flip of node 1 and 3; in the process flipping the input stimulus from activator to inhibitor. We then add a single feedback in the form of an inhibition to this pathway. Fig. [11](#fig:comparefeedback){reference-type="ref" reference="fig:comparefeedback"}, like Fig. [7](#fig:comparepathway){reference-type="ref" reference="fig:comparepathway"}, compares the evolution of each node in this pathway under unbiased conditions in using the test model in Section [3.1](#testmodel){reference-type="ref" reference="testmodel"} with an inhibition feedback from the output node 5 to node 3. The purpose is to validate Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"}; noting that output nodes are exactly the same between oriented and unoriented pathways. {reference-type="ref" reference="testmodel"} under unbiased conditions ($\phi = 0$). The parameters and formatting used is the same as that in Fig. [7](#fig:comparepathway){reference-type="ref" reference="fig:comparepathway"}. The figure shows equivalency in the two networks and validates Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"}.](Figures/feedback_compare.pdf){#fig:comparefeedback width="80%"} Figs [12](#fig:dss_feedback){reference-type="ref" reference="fig:dss_feedback"} and [13](#fig:dtau_feedback){reference-type="ref" reference="fig:dtau_feedback"} show an array of pathways of length $N=5$. These pathways all require two flips at nodes 1 and 3 to orient. The figures show how the errors $\delta_{ss}$ (Fig. [12](#fig:dss_feedback){reference-type="ref" reference="fig:dss_feedback"} showing error in steady state) and $\delta_\tau$ (Fig. [13](#fig:dtau_feedback){reference-type="ref" reference="fig:dtau_feedback"} describing discrepancy in temporal behaviour) are influenced by bias. In each figure, the array consists of the same pathway subject to all combinations of inhibition feedback. The first column consists of feedback a distance of a single node, the second column, a feedback a distance of two nodes, and in the last column a feedback is a distance of three and four nodes. The top charts display feedback which is introduced further downstream than the charts at the bottom. The pathways are oriented the same in all cases but the process leaves the feedback sometimes as an activator (red) and sometimes as an inhibitor (blue) depending on its location in the pathway. The charts are plotted in these colours respectively to highlight this feature across the chart which ultimately determines if the feedback is a negative or positive feedback. {#fig:dss_feedback width="80%"} {reference-type="ref" reference="fig:dss_feedback"}, as determined by varying $\phi$ values under an external stimulus. The y-axis represents $\delta_{\tau}$, while the x-axis shows $\phi$ values ranging from -1 to 1. The figure arrangement is the same as that of Fig. [12](#fig:dss_feedback){reference-type="ref" reference="fig:dss_feedback"}.](Figures/Feedback_dtau_v3.pdf){#fig:dtau_feedback width="80%"} The following key observations are found across our numerical investigation and exemplified in Figs [12](#fig:dss_feedback){reference-type="ref" reference="fig:dss_feedback"} and [13](#fig:dtau_feedback){reference-type="ref" reference="fig:dtau_feedback"}. 1. Orientability is more robust with the addition of negative (or positive) bias than the alternative if the oriented pathway is inhibited (or activated, respectively). This was found previously in the case of a simple linear pathway but this phenomena seems to be also valid with the introduction of complexity such as feedback (and later we will show this to be also the case for a significantly more complex network). Feedback tends to even amplify the significance of negative versus positive bias, especially in the case of the steady state (and excepting for our next observation). 2. Orientability (in the *steady state* but not the temporal behaviour) even close to the unbiased case is compromised in the specific case where (a) there is negative feedback (blue curves in Fig [12](#fig:dss_feedback){reference-type="ref" reference="fig:dss_feedback"}) *and* (b) when the feedback loop includes at least one node which is flipped in the orienting process. 3. Orientability (in the *temporal behaviour* but not the steady state) even close to the unbiased case is compromised under positive feedback (red curves in Fig. [13](#fig:dtau_feedback){reference-type="ref" reference="fig:dtau_feedback"}) but not under negative feedback (red curves). This compromise is still more pronounced in the case of positive bias due to the inhibition stimulation of the pathway. ## Network complexity and pathway cross-talk As it is difficult to do a systematic numerical study of complex network topologies and pathway crosstalk we will instead only look anecdotally at a realistic network and in particular the EGFR/HER2 signalling network model published by Yamaguchi *et al.* in 2014 [@yamaguchi2014signaling]. Using Algorithm [\[algorithm2\]](#algorithm2){reference-type="ref" reference="algorithm2"} to orient a general network requires first to assign nodes to pathways (as defined explicitly in this manuscript not necessarily in a general biological sense) based on key flows of information. Each node must belong to a pathway and each pathway has to have an input node and an output node with edges directed from input to output. Of course, this leaves the choice of pathways an open problem with many solutions. We attempt to do this in such a way that makes the most amount of sense biologically and favoring input nodes as nodes with external stimulus and aligning as much as possible to conventional pathway families. The unoriented network diagram and its oriented form are shown in Fig. [14](#fig:network){reference-type="ref" reference="fig:network"}(a) and (b) respectively. Pathway edges are represented in black. There are six mathematical pathways in the network and some can be collectively grouped into four biological pathways. To satisfy our mathematical definition, a single node (representing the protein PTEN) which acts as an intermediary between the biological pathways $P_3$ and $P_4$ is technically labelled as its own pathway. The red and blue colours indicate activation and inhibition cross-talks and/or external stimulus, respectively. The diagram includes various coupled biological pathways, such as Wnt/$\beta$-catenin ($P_1$), EGFR family ($P_2$), Notch family ($P_3$), and TNF-R pathway ($P_4$). The two output nodes labelled $x_1$ and $x_2$ are shown in orange and we do not demand similarity in any of the other nodes. We choose these two nodes as the output as this model has been constructed to focus on the EGFR pathway and how it is affected by the other signalling pathways [@yamaguchi2014signaling]. Immediately we highlight the power of the oriented form. In ensuring that all pathways are oriented, we can see the effect of the stimulii on this network. In the oriented form all stimulii except the stimulii of $P_4$ are contributing to activation of the output via the direct pathways. Furthermore, it is more easy to see the effect of the cross talk interactions. All cross-talk in the oriented form is positively stimulatory. That is, they all contribute to positive feedback or feed forward through cross-talk. The exception, of course, is the cross-talk with $P_4$ which is inhibited by its stimulus and through single inhibitory cross-talk with other pathways also acts as a positive stimulant. It is satisfying therefore that a network which is actively stimulated and contains positive feedback behaves (despite its added complexity) as a simple pathway in regards to the bias in the model. We can see this explicitly in the plots of $\delta_{ss}$ in Fig. [\[fig:dss_network_twoway\]](#fig:dss_network_twoway){reference-type="ref" reference="fig:dss_network_twoway"}. In particular, we notice that error $\delta_{ss}$ is greatly restricted to negative bias which has been an observation derived from our simpler tests for activated oriented pathways with positive feedback. ![A network model of EGFR/HER2 and its integration with other pathways as described in [@yamaguchi2014signaling] in both its unoriented form (a) and oriented form (b). The dashed boxes represent the four underlying interacting biological pathways. These are $P_1$ Wnt/$\beta$-catenin, $P_2$ EGFR, $P_3$ Notch and $P_4$ TNF-R. On the other hand, individual columns (shown with black edges) represent the six pathways as we define them mathematically in this manuscript. The red and blue arrows indicate activation and inhibition cross-talks and stimulii, respectively. We focus on the outputs of the model in [@yamaguchi2014signaling] which are indicated in orange and labelled $x_1$ and $x_2$.](Figures/network_4.pdf){#fig:network width="100%"} {#fig:dss_x4_twoway width=".95\\linewidth"} {#fig:dss_x8_twoway width=".95\\linewidth"} We can also visualise (more explicitly than simply plotting $\delta_\tau$ for this system) the discrepancy in the temporal evolution of the nodes $x_1$ and $x_2$ by plotting $x_1(t)$ and $x_2(t)$ explicitly for $\phi = 0$ and $\phi = \pm 0.5$ (representative of positive and negative bias respectively). This is done in the set of charts in Fig. [17](#fig:x4_and_x8){reference-type="ref" reference="fig:x4_and_x8"}. The first row of the figure shows the non-oriented output of $x_1$ (purple), while the second row shows the oriented network output. The third and fourth rows display the non-oriented and oriented network outputs for $x_2$ (blue) respectively. We observe perfect agreement for the unbiased model (centre column) as expected and very good agreement for the positively biased model as a result of positive stimulus and positive feedback throughout the oriented network (right column). Furthermore, as expected, the negatively biased model exhibits significant discrepancies between the oriented and unoriented forms in line with the observations of $\delta_{ss}$ in Fig. [\[fig:dss_network_twoway\]](#fig:dss_network_twoway){reference-type="ref" reference="fig:dss_network_twoway"} and consistent with observations of simpler pathways. {reference-type="ref" reference="fig:network"}. The simulation output is averaged over many combinations of parameters $\alpha$ and $\beta$ for each edge and shown in three columns for $\phi =-0.5$ (left), $\phi = 0$ (centre), and $\phi = 0.5$ (right). The first and third rows display the non-oriented output of $x_1$ and $x_2$, while the second and the fourth row show the respective oriented network outputs.](Figures/network_output_x4_x8.pdf){#fig:x4_and_x8 width="80%"} # Discussion and conclusion {#sec:conclusion} This manuscript is concerned with the appropriateness of reducing the complexity of a biochemical network/pathway by assuming that inhibition and activation relationships behave as perfect inverses of each other. Such reduction is common as a method of identification of topological properties and possible qualitative behaviours but this reduction is only appropriate under certain conditions explored in this manuscript. We define an oriented form as a standard/simplest way of reducing a pathway/network. We also prove that if a model is so-called 'unbiased' and 'symmetric' then reduction to the oriented form by successive exchanging of activations and inhibitions (keeping track of all operations by 'flipping' the sign of the relevant nodes) produces a model indistinguishable (from the perspective of measured output) from the unoriented form. We focus our study on the nature of (positive and negative) bias in the formation of error between an unoriented form and its reduced oriented form (both in the case of differing steady states and differing dynamic properties). We restrict ourself to symmetric models only. Our investigation is computational. Pathways in their oriented form have more easily identifiable structure and our investigation leads us to propose the following general conjectures for symmetrically modelled pathways and networks. 1. For many systems bias can be added to a model without generating significant errors in the orienting process. When errors appear they do so rapidly and suddenly and therefore care should be exercised when reducing a network qualitatively in this way. 2. When the oriented network is *externally stimulated* then the reduction to the oriented form produces significant errors if the model is negatively biased. If it is *externally inhibited* errors are formed if the model is positively biased. 3. Errors are compounded as more approximations (replacing activators for inhibitors and flipping a node to compensate) are taken. Furthermore, if these approximations are made further downstream, the errors are expected to be larger. 4. The former statements extend to pathways with feedback. However, error in the steady state is observed for both negative and positive bias models if the feedback is negative (and the loop includes a flipped node) and error in the temporal behaviour is observed for both negative and positive bias models if the feedback is positive. 5. The conclusions seem to be consistent if these general properties can be identified in the oriented form of more complex network models. This study has many limitations and suggests very strongly more rigorous work that should be done. We have opted not to look into detail at symmetry in this paper. This is partly due to the length of the paper but also because bias seems more common/significant in mathematical models in the literature. Furthermore, we do not have an objective definition to measure the extent of bias that makes sense. We have a measure of bias for the model used in this paper $\phi$ but this measure is arbitrary. It is therefore important that the scale of $\phi$ not be given too much emphasis. We also investigate one single model. We attempted to create this model with symmetry but also with the kinds of nonlinear relationships common in biochemical network models. It remains as future work to investigate the generalisability and analysis of the conjectures posed in this manuscript to a much more broad class of model. The observations found in this study form a framework with which to assess biochemical networks and determine if qualitative reductions are appropriate or if typologies are likely to be more idiosyncratic based on the specific quantitative model used to simulate it.

arxiv_math

** Inverse problem for fractional order subdiffusion equation **\ **Marjona Shakarova$^{1}$**\ *shakarova2104\@gmail.com\ * * $^{1}$ Institute of Mathematics, Academy of Science of Uzbekistan* **Abstract**: The study examines the inverse problem of finding the appropriate right-hand side for the subdiffusion equation with the Caputo fractional derivative in a Hilbert space represented by $H$. The right-hand side of the equation has the form $g(t)f$ and an element $f\in H$ is unknown. If the sign of $g(t)$ is a constant, then the existence and uniqueness of the solution is proved. When $g(t)$ changes sign, then in some cases, the existence and uniqueness of the solution is proved, in other cases, we found the necessary and sufficient condition for a solution to exist. Obviously, we need an extra condition to solve this inverse problem. We take the additional condition in the form $\int\limits_0^Tu(t)dt=\psi$. Here $\psi$ is a given element, of $H$. : Primary 35R11; Secondary 34A12.\ *Key words*: subdiffusion equation, inverse problem, the Caputo derivative, Fourier method. # Introduction Suppose that $H$ is a separable Hilbert space with the scalar product $(\cdot, \cdot)$, and let $A$ be an operator on $H$, with a domain of definition $D(A)$, satisfying the following conditions: 1\) $A=A^*$, where $A^*$ denotes the adjoint operator of $A$, 2\) $(Ah,h)\geq C(h,h)$, $h \in D(A)$, for some $C>0$. Assume that $A$ has a complete system of orthonormal eigenfunctions ${v_k}$ in $H$ and a countable set of positive eigenvalues $\lambda_k$. It is assumed that the eigenvalues are ordered such that $0<\lambda_1\leq\lambda_2\leq \cdots\rightarrow +\infty$. Let $C((a,b);H)$ stand for a set of continuous functions $u(t)$ of $t\in (a,b)$ with values in $H$. $D_t^\rho y(t)$ is the Caputo fractional derivative defined as (see, [@Pskhu]): $$D_t^\rho y(t)=\frac{Y(t)}{\Gamma (1-\rho)}, \quad Y(t)=\int\limits_0^t \frac{\frac{d}{d\xi}y(\xi)}{(t-\xi)^{\rho}}d\xi, \quad t>0,$$ where $\Gamma(\rho)$ is Euler's gamma function. We note that the fractional derivative and the regular classical derivative of the first order are equivalent if $\rho=1$: $D_t h(t)= \frac{d}{dt} h(t)$. **Problem**. We study the inverse problem of finding functions $\{u(t), f\}$ that satisfy the following subdiffusion equation $$\label{prob1} D_t^\rho u(t) + Au(t) =g(t)f,\quad \rho\in(0,1],\quad t\in (0, T],$$ with the initial $$\label{in.c} u(0)=\varphi,$$ and the additional conditions $$\label{ad} \int\limits_0^T u(t)dt=\psi.$$ Here $g(t)\in C[0,T]$ is a given function and $\varphi,\psi\in H$ are known elements. The solution of the inverse problem will involve examining the Cauchy problem for different types of differential equations. In this context, when we refer to the solution of the problem, we specifically mean the classical solution. This implies that all the derivatives and functions involved in the equation are assumed to be continuous with respect to the variable $t$. As an example, present the definition of the solution of the inverse problem ([\[prob1\]](#prob1){reference-type="ref" reference="prob1"})-([\[ad\]](#ad){reference-type="ref" reference="ad"}). **Definition 1**. *A pair of functions $\{u(t), f\}$ with the properties $D_t^\rho u(t), Au(t)\in C((0,T]; H)$, $u(t)\in C([0,T];H)$, $f\in H$ satisfying conditions ([\[prob1\]](#prob1){reference-type="ref" reference="prob1"})-([\[ad\]](#ad){reference-type="ref" reference="ad"}) is called **the solution** of the inverse problem.* Recently, inverse problems related to integer or fractional order differential equations have received more attention among researchers. Most research on source function determination focuses on specific processes such as $F = g(t)f(x)$, where either $g (t)$ or $f(x)$ is unknown. Inverse problems of finding the function $g(t)$ have been studied, for example, in [@Hand1]-[@Ash2]). When $f(x)$ is unknown and $g(t)\equiv1$, the inverse problems have been studied by many authors (see [@Fur]-[@4]). In this work, we focus on the problem of determining the function $f(x)$, when $g(t)\not\equiv1$. Similar problems for the diffusion equation are studied in the well-known monographs of S.Kabanikhin [@Kab1] and the papers [@Pr]-[@FN3]. As for the subdiffusion equation, such inverse problems are studied in papers [@MS]-[@AshM2]. Let us mention some of the results obtained for the diffusion and subdiffusion equations. We briefly note some known results on inverse problems for the diffusion equation. A.I. Prilepko and A.B. Kostin [@Pr] presented the elliptic part of the diffusion equation as a second-order differential expression. The authors consider both a non-self-adjoint and a self-adjoint elliptic part. They established a criterion of uniqueness of the generalized solution of the inverse problem when elliptic part is self-adjoint. Note, that here the additional condition is taken in an integral form. Unlike to the paper [@Pr], in papers [@Sab], [@Sab2] the problem of finding the function $f(x)$ for the diffusion equation was studied using the additional condition $u(x,t_0)=\psi$. Some authors set the additional condition as $t_0=T$ (see, e.g. [@Orl], [@Tix] for classical diffusion equations and for subdiffusion equations see [@MS], [@MS1]). An inverse problem similar to ([\[prob1\]](#prob1){reference-type="ref" reference="prob1"})-([\[ad\]](#ad){reference-type="ref" reference="ad"}) for various operators $A$ and with the Caputo and Riemann-Liouville derivatives are considered in [@FN]-[@FN2], and in [@FN] the fractional derivative is taken in the sense of Caputo and in [@FN2] in the sense of Riemann-Liouville. In [@FN], the criteria for the uniqueness of the solution of the inverse problem are found. And in work [@FN2] the question of the correctness of the inverse problem by operator methods was studied. In the paper [@AshM2] of the researchers analyzed subdiffusion equation with the Caputo derivative in which the Laplace operator forms the elliptic part. This paper focused on forward and inverse problems for the subdiffusion equation. The authors of the study proved the uniqueness and existence of the solution of the inverse problem, if the function $g(t)$ preserves its sign. Moreover, if the function $g(t)$ changes sign, a necessary and sufficient condition for the existence of a classical solution was found, and all solutions of the inverse problem were constructed using the classical Fourier method. It should be noted that all the findings presented in this paper for the case where $g(t)$ changes its sign are also new for the classical diffusion equation. Finally, we will use some original ideas from this work to solve our inverse problem.      We introduce the power of operator $A$ with domain $$D(A^\tau)=\{h\in H: \sum\limits_{k=1}^\infty \lambda_k^{2\tau} |h_k|^2 < \infty\},$$ acting in $H$ according to the rule: $$A^\tau h= \sum\limits_{k=1}^\infty \lambda_k^\tau h_k v_k.$$ Here $\tau$ is an arbitrary real number and $h_k=(h,v_k)$ are the Fourier coefficients of a element $h \in H$. For elements $h,g \in D(A^\tau)$ we introduce the scalar product: $$(h,g)_\tau=\sum\limits_{k=1}^\infty \lambda_k^{2\tau} h_k \overline {g_k} = (A^\tau h,A^\tau g)$$ and together with this norm $D(A^\tau)$ turns into a Hilbert space. # Preliminaries The problem of finding the function $u(t)$ satisfying subdiffusion equation ([\[prob1\]](#prob1){reference-type="ref" reference="prob1"}) with initial condition ([\[in.c\]](#in.c){reference-type="ref" reference="in.c"}) is also called *the forward problem*. The forward problem is well-studied in the literature, and the existence and uniqueness of the solution have been proved in various works, including [@AshM2], [@AshM]. These works provide important theoretical foundations for studying the inverse problem. We mention the solution of the forward problem to solve the inverse problem ([\[prob1\]](#prob1){reference-type="ref" reference="prob1"})-([\[ad\]](#ad){reference-type="ref" reference="ad"}) we are studying: $$\label{fp} u(t)=\sum\limits_{k=1}^{\infty}\left[\varphi_k E_{\rho,1}(-\lambda_k t^\rho)+ f_k\int\limits_0^t (t-\eta)^{\rho-1} E_{\rho, \rho} (-\lambda_k (t-\eta)^\rho) g(\eta)d\eta\right]v_k,$$ where $\varphi_k$, $f_k$ are the Fourier coefficients of functions $\varphi$, $f$, respectively and $$E_{\rho, \mu}(z)= \sum\limits_{n=0}^\infty \frac{z^n}{\Gamma(\rho n+\mu)} \quad 0 < \rho < 1, \quad z,\mu\in \mathbb{C}$$ is called the Mittag-Leffler function with two-parameters (see, [@Dzh66], p. 133). To find the unknowns $\{u(t), f\}$ of inverse problem ([\[prob1\]](#prob1){reference-type="ref" reference="prob1"})-([\[ad\]](#ad){reference-type="ref" reference="ad"}), we apply additional condition ([\[ad\]](#ad){reference-type="ref" reference="ad"}) to equality ([\[fp\]](#fp){reference-type="ref" reference="fp"}). Then obtain the following equality: $$\sum\limits_{k=1}^{\infty}\left[\varphi_k \int\limits_0^T E_{\rho,1}(-\lambda_k t^\rho)dt+ f_k \int\limits_0^T\int\limits_0^t (t-\eta)^{\rho-1} E_{\rho, \rho}(-\lambda_k(t-\eta)^\rho) g(\eta) d\eta dt \right]v_k=\psi.$$ Now we introduce the following lemmas: **Lemma 2**. *Let $\rho> 0$, then the following equality is hold: $$\int\limits_0^TE_{\rho,1}(-\lambda_k t^\rho)dt =TE_{\rho,2}(-\lambda_k T^\rho).$$* *Proof.* The proof of this lemma follows from the following equality (see, [@Gor], formula (4.4.4), p. 61): $$\label{MLI} \int\limits_0^t \eta^{\beta-1}E_{\rho,\beta}(\lambda\eta^\rho)d\eta=t^{\beta} E_{\rho,\beta+1}(\lambda t^\rho), \quad \rho>0, \quad\beta > 0, \quad \lambda \in C,$$ ◻ **Lemma 3**. *Let $\rho> 0$, then $$\int\limits_0^T\int\limits_0^{t} (t-\eta)^{\rho-1} E_{\rho, \rho} (-\lambda_k (t-\eta)^\rho ) g(\eta) d\eta dt = \int\limits_0^T g(\eta) (T-\eta)^{\rho} E_{\rho, \rho+1} (-\lambda_k (T-\eta)^\rho ) d\eta.$$* **Proof.* By calculating the double integral, we obtain the following equality: $$\label{in2} \int\limits_0^T\int\limits_0^{t} (t-\eta)^{\rho-1} E_{\rho, \rho} (-\lambda_k (t-\eta)^\rho ) g(\eta) d\eta dt$$ $$=\int\limits_0^T g(\eta)d\eta \int\limits_\eta^{T} (t-\eta)^{\rho-1} E_{\rho, \rho} (-\lambda_k (t-\eta)^\rho ) dt=\int\limits_0^T g(\eta)d\eta \int\limits_0^{T-\eta} s^{\rho-1} E_{\rho, \rho} (-\lambda_k s^\rho ) ds$$ $$=\int\limits_0^T g(\eta)d\eta \int\limits_0^{T-\eta} s^{\rho-1} E_{\rho, \rho} (-\lambda_k s^\rho ) ds.$$ Due to equality ([Lemma 2](#lem1){reference-type="ref" reference="lem1"}), ([\[in2\]](#in2){reference-type="ref" reference="in2"}) is equal to the following integral: $$\int\limits_0^T g(\eta) (T-\eta)^{\rho} E_{\rho, \rho+1} (-\lambda_k (T-\eta)^\rho ) d\eta.$$ ◻* According to Lemma [Lemma 2](#lem1){reference-type="ref" reference="lem1"} and Lemma [Lemma 3](#lem2){reference-type="ref" reference="lem2"}, we have the following equality: $$\sum\limits_{k=1}^{\infty}\left[\varphi_k TE_{\rho,2}(-\lambda_k T^\rho )dt+ f_k \int\limits_0^T (t-\eta)^{\rho} E_{\rho, \rho+1} (-\lambda_k (t-\eta)^\rho ) g(\eta) d\eta dt \right ]v_k=\psi.$$ If we expand the function $\psi$ into the Fourier series according to the system $\{ v_k\}$ and equate the Fourier coefficients, then we have the following equality: $$\label{EqFor_fk1} f_k p_{k,\rho}(T)= \psi_k - \varphi_k TE_{\rho,2}(-\lambda_k T^\rho).$$ where $$p_{k,\rho}(T)=\int\limits_0^T g(\eta) (T-\eta)^{\rho} E_{\rho, \rho+1} (-\lambda_k (T-\eta)^\rho ) d\eta.$$ According to the idea of the authors of [@AshM2], we divide $\mathbb{N}$ into two sets, i.e. $N=B_\rho \cup B_{0,\rho}$. Here, $\mathbb{N}$ represents the set of all natural numbers. The sets $B_\rho$ and $B_{0,\rho}$ are defined as follows: 1\) If the function $p_{k,\rho}(T)\neq 0$, then $k\in B_\rho$, 2\) Alternatively, if the function $p_{k,\rho}(T)=0$, then $k\in B_{0,\rho}$. It is obvious, if $g(t)$ is a sign-preserving function, then $p_{k,\rho}(T)\neq 0$. Therefore, in this case the set $B_{0,\rho}$ is empty and $B_\rho=\mathbb{N}$. Equation ([\[EqFor_fk1\]](#EqFor_fk1){reference-type="ref" reference="EqFor_fk1"}) provides us with a means to determine $f_k$. It can be observed that the criterion for the uniqueness of the solution to the inverse problem associated with the diffusion and subdiffusion equations can be expressed as follows: $$p_{k,\rho}(T)\neq 0.$$ According to this criterion, for the solution to be unique, it is necessary that the expression $p_{k,\rho}(T)$ does not equal zero. To establish two-sided estimates for $p_{k,\rho}(T)$, let's consider the case where the function $g(t)$ does not change sign. In this case, the set $B_{0,\rho}$ is empty. Then the following lemma holds. **Lemma 4**. *Let $\rho\in (0,1]$, $g(t)\in C[0,T]$ and $g(t)\neq 0$, $t\in [0,T]$. Then there are constants $C_0,C_1>0$, depending on $T$, such that for all $k$: $$\frac{C_0}{\lambda_k}\leq |p_{k,\rho}(T)|\leq\frac{C_1}{\lambda_k}.$$* *Proof.* Let $\rho=1$. By integrating by parts and the mean value theorem, we obtain $$p_{k,1}(T)=\frac{1}{\lambda_k}\int\limits_0^{T} (1-e^{-\lambda_k s}) g(T-s)ds =$$ $$=\frac{g(\xi_k)}{\lambda_k} \bigg[{T}-\frac{1}{\lambda_k} (1- e^{-\lambda_k T})\bigg], \quad \xi_k\in [0, T].$$ By virtue of the Weierstrass theorem, we have $|g(t)|\geq g_0=const >0$. Then we can establish the lower and upper bounds as follows: $$\frac{g_0c_0}{\lambda_k}\leq|p_{k,1}(T)|\leq\frac{\max\limits_{0\leq\xi \leq T}|g(\xi)|T}{\lambda_k}.$$ Let $\rho \in (0,1)$. Apply the mean value theorem and equality ([\[MLI\]](#MLI){reference-type="ref" reference="MLI"}) to obtain $$|p_{k,\rho}(T)| =\bigg|\int\limits _0^{T} \eta^{\rho} E_{\rho, \rho+1} (-\lambda_k \eta^\rho)g(T-\eta)d\eta\bigg|=$$ $$=|g(\xi_k)| T^{\rho+1} E_{\rho, \rho+2} (-\lambda_k T^\rho ) , \quad \xi_k\in[0,T].$$ Therefore, using the asymptotic estimate of the Mittag-Leffler function (see, [@Dzh66], p. 134) $$\label{MLA} E_{\rho, \mu}(-t)=\frac{t^{-1}}{\Gamma(\mu-\rho)}+O(t^{-2})$$ and the estimate $|g(t)|\geq g_0$ one has $$|p_{k,\rho}(T)|={|g(\xi_k)|{T^{\rho+1}}}\bigg(({T^\rho\lambda_k})^{-1}+O({\lambda_kT^{\rho}})^{-2}\bigg)\geq \frac{C_0}{\lambda_k}.$$ Finally, according to the estimate of the Mittag-Leffler function (see, [@Dzh66], p. 136) $$\label{Ml} |E_{\rho, \mu}(-t)|\leq \frac{C}{1+t}, \quad t\geq0$$ (where constant $C$ does not depend on $t$ and $\mu$), we have $$|p_{k,\rho}(T)|\leq C\frac{|g(\xi_k)|T^{\rho+1}}{1+\lambda_k T^\rho} \leq C\frac{ \max\limits_{0\leq\xi \leq T}|g(\xi)|T}{\lambda_k}\leq \frac{C_1}{\lambda_k}.$$ ◻ Now consider the case when $g(t)$ changes sign. Then the function $p_{k,\rho}(T)$ can become zero, and as a result, the set $B_{0,\rho}$ may turn out to be non-empty. In the case where the sign of $g(t)$ is a variable function, we will present the following lemma. **Lemma 5**. *Let $\rho\in (0,1]$, $g(t)\in C^1[0, T]$ and $g(0)\neq 0$. Then there exist numbers $m_0>0$ and $k_0$ such that, for all $T\leq m_0$ and $k\geq k_0$, the following estimates hold: $$\label{estimateSub} \frac{C_0}{\lambda_k}\leq |p_{k,\rho}(T)|\leq\frac{C_1}{\lambda_k}.$$ where constants $C_0$ and $C_1>0$ depend on $m_0$ and $k_0$.* *Proof.* Let $\rho=1$. By integrating by parts and the mean value theorem, we get $$p_{k,1}(T)=\frac{1}{\lambda_k}\int\limits_0^{T} (1-e^{-\lambda_k s}) g(T-s)ds =\frac{1}{\lambda_k}\bigg[g(T-s)(s+\frac{e^{-\lambda_k s}}{\lambda_k})\bigg|^{T}_0$$ $$+\int\limits_0^{T} (s+\frac{e^{-\lambda_k s}}{\lambda_k}) g'(T-s)ds\bigg]$$ $$=\frac{g(0)}{\lambda_k} \bigg(T+\frac{e^{-\lambda_k T}}{\lambda_k}\bigg)-\frac{g(T)}{\lambda_k^2} + \frac{g'(\xi_k)}{\lambda_k}\big[\frac{T^2}{2}-\frac{1}{\lambda_k^2}(1-e^{-\lambda_k T} )\big], \quad \xi_k\in [0, T].$$ since $k\geq k_0$ $$|p_{k,1}(T)|\geq\bigg|\frac{g(0)}{\lambda_k}T-\frac{g(T)}{\lambda_k^2}\bigg|.$$ If $g(0)\neq 0$, then for large $k$ we can conclude that there exists a constant $C_0$ such that the lower bound in the estimate holds. To establish the upper estimate, we utilize the boundedness of the function $g(t)$. Let $\rho\in (0,1)$. Using equality ([\[MLI\]](#MLI){reference-type="ref" reference="MLI"}) we integrate by parts, then apply the mean value theorem. Then we have $$p_{k,\rho}(T)=\int\limits_0^{T}g(T-s) s^{\rho} E_{\rho, \rho+1} (-\lambda_k s^\rho ) ds=\int\limits_0^{T}g(T-s) d\big[ s^{\rho+1} E_{\rho, \rho+2} (-\lambda_k s^\rho ) \big] =$$ $$=g(T-s) s^{\rho+1} E_{\rho, \rho+2} (-\lambda_k s^\rho )\bigg|^{T}_0+\int\limits_0^{T}g'(T-s) s^{\rho+1} E_{\rho, \rho+2} (-\lambda_k s^\rho )ds=$$ $$=g(0)\, T^{\rho+1} \,E_{\rho, \rho+2} (-\lambda_k T^\rho)+ g'(\xi_k) \int\limits_0^{T} s^{\rho+1} E_{\rho, \rho+2} (-\lambda_k s^\rho )ds, \quad \xi_k\in [0, T].$$ For the last integral formula ([\[MLI\]](#MLI){reference-type="ref" reference="MLI"}) implies $$\int\limits_0^{T} s^{\rho+1} E_{\rho, \rho+2} (-\lambda_k s^\rho )ds=T^{\rho+2} E_{\rho, \rho+3}(-\lambda_k T^\rho).$$ Apply the asymptotic estimate ([\[MLA\]](#MLA){reference-type="ref" reference="MLA"}) to get $$p_{k,\rho}(T)=\frac{g(0)T}{\lambda_k} +\frac{g'(\xi_k)}{\lambda_k} T^2 + O\bigg(\frac{1}{(\lambda_k T^\rho)^2}\bigg).$$ If $g(0)\neq 0$, we can infer that for sufficiently small $T$ and sufficiently large $k$, the required lower estimate holds. Additionally, this implies the required upper bound as well. ◻ **Corollary 6**. *If conditions of Lemma [Lemma 5](#lemmaSub){reference-type="ref" reference="lemmaSub"} are satisfied, then estimate ([\[estimateSub\]](#estimateSub){reference-type="ref" reference="estimateSub"}) holds for suffuciently small $T$ and $k\in B_{\rho}$.* **Corollary 7**. *If conditions of Lemma [Lemma 5](#lemmaSub){reference-type="ref" reference="lemmaSub"} are satisfied and $T$ is sufficiently small, then set $B_{0,\rho}$ has a finite number elements.* **Remark 8**. *In the paper [@AshM2], a lemma similar to the above lemma was proved for the diffusion and subdiffusion equations. In this paper $g(t_0)\neq 0$ and $g(0)\neq 0$ were for $\rho=1$ and $\rho\in (0,1)$, respectively. In this paper, in cases where $\rho=1$ and $\rho\in(0.1)$, conditions $g (t_0) \neq 0$ and $g(0) \neq 0$ for function $g(t)$ were found, respectively. However, in our lemma, for the diffusion and subdiffusion equations, for function $g(t)$ one has the same condition, i.e. $g(0)\neq 0$.* # The solution of problem ([\[prob1\]](#prob1){reference-type="ref" reference="prob1"})-([\[ad\]](#ad){reference-type="ref" reference="ad"}) If $g(t)$ is a sign-preserving function, then the following theorem holds. **Theorem 9**. *Let $\rho\in (0,1]$, $\varphi \in H$, $\psi \in D(A)$, $g(t)\in C[0,T]$ and $g(t)\neq 0$, $t\in [0,T]$. Then there exists a unique solution of the inverse problem ([\[prob1\]](#prob1){reference-type="ref" reference="prob1"})-([\[ad\]](#ad){reference-type="ref" reference="ad"}): $$f=\sum\limits_{k=1}^\infty \frac{1}{p_{k,\rho}(T)}\left[ \psi_k-\varphi_k TE_{\rho,2}(-\lambda_k T^\rho)\right]v_k,$$ $$u(t)=\sum\limits_{k=1}^\infty \left[\varphi_k E_{\rho,1}(-\lambda_kt^\rho)+ \frac{p_{k,\rho}(t)}{p_{k,\rho}(T)}\left[ \psi_k-\varphi_k TE_{\rho,2}(-\lambda_k T^\rho)\right]\right]v_k.$$* Now we form the following corresponding result for the case when sign of function $g(t)$ has changed. **Theorem 10**. *Let $\rho\in (0,1]$, $\varphi \in H$, $\psi \in D(A)$, $g(t)\in C^1[0,T]$. Further, we will assume that the conditions of Lemma [Lemma 5](#lemmaSub){reference-type="ref" reference="lemmaSub"} are satisfied and $T$ is sufficiently small.* *1) If set $B_{0,\rho}$ is empty, for all $k$, then there exists a unique solution of the inverse problem ([\[prob1\]](#prob1){reference-type="ref" reference="prob1"})-([\[ad\]](#ad){reference-type="ref" reference="ad"}): $$f=\sum\limits_{k=1}^\infty \frac{1}{p_{k,\rho}(T)}\left[ \psi_k-\varphi_k TE_{\rho,2}(-\lambda_k T^\rho)\right]v_k,$$ $$u(t)=\sum\limits_{k=1}^\infty \left[\varphi_k E_{\rho,1}(-\lambda_kt^\rho)+ \frac{p_{k,\rho}(t)}{p_{k,\rho}(T)}\left[ \psi_k-\varphi_k TE_{\rho,2}(-\lambda_k T^\rho)\right]\right]v_k.$$ 2) If set $B_{0,\rho}$ is not empty, then for the existence of a solution to the inverse problem, it is necessary and sufficient that the following conditions $$\label{ortogonal} \psi_k=\varphi_k T E_{\rho,2}(-\lambda_kT^\rho),\quad k\in B_{0,\rho}$$ be satisfied. In this case, the solution to the problem ([\[prob1\]](#prob1){reference-type="ref" reference="prob1"})-([\[ad\]](#ad){reference-type="ref" reference="ad"}) exists, but is not unique: $$\label{f_2} f=\sum\limits_{k\in B_{\rho}} \frac{1}{p_k(T)}\left[ \psi_k-\varphi_k TE_{\rho,2}(-\lambda_k T^\rho)\right]v_k+\sum\limits_{k \in B_{0,\rho}} f_k v_k,$$ $$\label{u_2} u(t)=\sum\limits_{k=1}^\infty\big[\varphi_k E_{\rho,1}(-\lambda_kt^\rho)+f_k\big]v_k,$$ where $f_k$, $k\in B_{0,\rho}$, are arbitrary real numbers.* As mentioned earlier, Theorem [Theorem 9](#thmNotChange){reference-type="ref" reference="thmNotChange"} for the diffusion equation ($\rho=1$) with the additional condition $u(x,t_0)=\psi$ has only been proven in the cases where $\Omega$ is an interval on $\mathbb{R}$ (see, [@Sab]) or a rectangle in $\mathbb{R}^2$ (see, [@Sab2]). The inverse problem ([\[prob1\]](#prob1){reference-type="ref" reference="prob1"})-([\[in.c\]](#in.c){reference-type="ref" reference="in.c"}) with the same additional condition, considering both the cases when the function $g(t)$ changes sign and when it does not change sign, has been addressed in the work of Ashurov et al. (see, [@AshM2]). However, the theorems we have presented above, for both the diffusion and subdiffusion equations, involve an integral additional condition ([\[ad\]](#ad){reference-type="ref" reference="ad"}). It is worth noting that these theorems are also novel for diffusion equations. Besides, we must also note that, unlike the paper [@AshM2], in the theorems we have proven, the condition is given not to point $t_0$, but to the boundary of the domain i.e $T$. **Proof of Theorem [Theorem 9](#thmNotChange){reference-type="ref" reference="thmNotChange"}.** Since $p_{k,\rho}(T)\neq 0$ for all $k \in \mathbb{N}$, then we get the following equations from ([\[EqFor_fk1\]](#EqFor_fk1){reference-type="ref" reference="EqFor_fk1"}): $$f_k=\frac{1}{p_{k,\rho}(T)}\left[ \psi_k-\varphi_kTE_{\rho,2}(-\lambda_k T^\rho)\right].$$ From these $f_k$ are Fourier coefficients of the unknown $f$, has the form: $$\label{inv10} f=\sum\limits_{k=1}^\infty \frac{1}{p_{k,\rho}(T)}\left[ \psi_k-\varphi_kTE_{\rho,2}(-\lambda_k T^\rho)\right]v_k.$$ Let us prove the uniformly convergence of this series. Let $F_j$ be the partial sum of series ([\[inv10\]](#inv10){reference-type="ref" reference="inv10"}): $$F_j=\sum\limits_{k=1}^j \frac{1}{p_{k,\rho}(T)}\left[ \psi_k-\varphi_kTE_{\rho,2}(-\lambda_k T^\rho)\right]v_k= F_{j,1}+F_{j,2}.$$ Then we show that every series $F_{j,1}$ and $F_{j,2}$ are absolutely and uniformly convergent. First we estimate of the series $F_{j,1}$. For this, applying Parseval's equality, we arrive at: $$||F_{j,1}||^2=\bigg|\bigg|\sum\limits_{k=1}^j \frac{\psi_k}{p_{k,\rho}(T)}v_k\bigg|\bigg|^2 \leq \sum\limits_{k=1}^j \frac{1}{|p_{k,\rho}(T)|^2} |\psi_k|^2\leq C\sum\limits_{k=1}^j\lambda_k^2|\psi_k|^2 = C ||\psi||^2_1.$$ Now, we estimate of the series $F_{j,2}$. According to Parseval's equality and estimate ([\[Ml\]](#Ml){reference-type="ref" reference="Ml"}), we have: $$||F_{j,2}||^2=\bigg|\bigg|\sum\limits_{k=1}^j \frac{\varphi_kTE_{\rho,2}(-\lambda_k T^\rho)}{p_{k,\rho}(T)}v_k\bigg|\bigg|^2 \leq \sum\limits_{k=1}^j \left|\frac{TE_{\rho,2}(-\lambda_k T^\rho)}{p_{k,\rho}(T)}\right|^2 |\varphi_k |^2\leq C ||\varphi||^2.$$ Thus, if $\varphi \in H$, $\psi \in D(A)$, then from estimates of $F_{i,j}$ we obtain $f \in H$. If $f\in H$ is known function, then we obtained the following equality for function $u(t)$: $$\label{inv11} u(t)=\sum\limits_{k=1}^\infty \left[\varphi_k E_{\rho,1}(-\lambda_kt^\rho)+ \frac{p_{k,\rho}(t)}{p_{k,\rho}(T)}\left[ \psi_k-\varphi_k TE_{\rho,2}(-\lambda_k T^\rho)\right]\right]v_k.$$ From this equality, we have the following form for Fourier coefficients $u_k(t)$ of function $u(t)$: $$u_k(t)=\varphi_k E_{\rho,1}(-\lambda_kt^\rho)+ \frac{p_{k,\rho}(t)}{p_{k,\rho}(T)}\left[ \psi_k-\varphi_k TE_{\rho,2}(-\lambda_k T^\rho)\right].$$ Now we need to show that function $u(t)$ is a solution of inverse problem ([\[prob1\]](#prob1){reference-type="ref" reference="prob1"})-([\[ad\]](#ad){reference-type="ref" reference="ad"}). Fulfillment of the conditions of Definition [Definition 1](#def1){reference-type="ref" reference="def1"} for function $u(t)$, defined by the series ([\[inv11\]](#inv11){reference-type="ref" reference="inv11"}) is proved in exactly the same way as the solution of the forward problem ([\[prob1\]](#prob1){reference-type="ref" reference="prob1"}). As we noted above, the solution to the forward problem was proved in papers [@AshM2], [@AshM]. The uniqueness of the solution was proved in paper [@AshM2]. Therefore, we briefly cite the proof of the uniqueness. To prove the uniqueness of the solution, assume the opposite, that is, there are two different solutions $\{u_1, f_1\}$ and $\{u_2, f_2\}$ satisfying the inverse problem ([\[prob1\]](#prob1){reference-type="ref" reference="prob1"} )-([\[ad\]](#ad){reference-type="ref" reference="ad"}). We must show that $u\equiv u_1-u_2 \equiv 0$, $f\equiv f_1-f_2\equiv 0$. For $\{u, f\}$ we have the following problem:: $$\label{prob20} \left\{ \begin{aligned} & D_t^\rho u(t)+A u(t) =g(t)f,\quad t\in (0,T],\\ &u(0)=0, \\ &\int\limits_0^T u(t)dt=0.\\ \end{aligned} \right.$$ We take any solution $\{u,f\}$ and define $u_k=(u,v_k)$ and $f_k=(f,v_k)$. Then, due to the self-adjointness of the operator $A$, we obtain $$D_t^\rho u_k(t)= (D_t^\rho u, v_k)= -(A u, v_k)+f_k g(t)=-( u,A v_k)+f_k g(t)=-\lambda_k u_k(t)+f_k g(t).$$ Therefore, for $u_k$ we obtain the Cauchy problem $$D_t^\rho u_k(t)+\lambda_k u_k(t) =f_kg(t),\quad t>0,\quad u_k(0)=0,$$ and the additional condition $$\int\limits_0^T u_k(t)dt=0.$$ If $f_k$ is known, then the unique solution of the Cauchy problem has the form $$u_k(t)= f_k\int\limits_0^t \eta^{\rho-1} E_{\rho, \rho} (-\lambda_k \eta^\rho) g(t-\eta) d\eta.$$ Apply the additional condition to get $$\int\limits_0^T u_k(t)dt= f_k\int\limits_0^T g(\eta) (T-\eta)^{\rho} E_{\rho, \rho+1} (-\lambda_k (T-\eta)^\rho ) d\eta=f_kp_{k,\rho}(T)=0.$$ Since $p_{k,\rho}(T) \neq 0$ for all $k \in \mathbb{N}$, then due to completeness of the set of eigenfunctions $\{v_k\}$ in $H$, we finally have $f\equiv 0$ and $u(t)\equiv0$. $\Box$   We will now proceed with the proof of Theorem [Theorem 10](#thm2){reference-type="ref" reference="thm2"}. **Proof of Theorem [Theorem 10](#thm2){reference-type="ref" reference="thm2"}.** We will consider the proof of the theorem for cases where the set $B_{0,\rho}$ is empty and non-empty. When $p_{k,\rho}(T)\neq 0$ for all $k$, we can prove the existence and uniqueness of the solution of functions ${\{u(t),f\}}$ in the same way as in Theorem [Theorem 9](#thmNotChange){reference-type="ref" reference="thmNotChange"}. Next, we consider the case where $B_{0,\rho}$ is not an empty set. If $k\in B_{\rho}$, we can use Lemma [Lemma 5](#lemmaSub){reference-type="ref" reference="lemmaSub"} to prove the first part of equalities ([\[f_2\]](#f_2){reference-type="ref" reference="f_2"})-([\[u_2\]](#u_2){reference-type="ref" reference="u_2"}) in the same way as the existence of a solution was proved in Theorem [Theorem 9](#thmNotChange){reference-type="ref" reference="thmNotChange"}. However, when $k\in B_{0,\rho}$, the solution of equation ([\[EqFor_fk1\]](#EqFor_fk1){reference-type="ref" reference="EqFor_fk1"}) with respect to $f_k$ exists if and only if the extra conditions ([\[ortogonal\]](#ortogonal){reference-type="ref" reference="ortogonal"}) are satisfied. The solution of equation ([\[EqFor_fk1\]](#EqFor_fk1){reference-type="ref" reference="EqFor_fk1"}) in this case can be arbitrary numbers $f_k$. Instead of condition ([\[ortogonal\]](#ortogonal){reference-type="ref" reference="ortogonal"}), according to $0<E_{\rho,2} (-t)<1$, (see [@Gor], p. 47) we can use the orthogonality conditions which are easy to verify: $$\varphi_k=(\varphi, v_k)=0, \quad \psi_k=(\psi, v_k)=0, \quad k\in B_{0,\rho}.$$ $\Box$   # Acknowledgements {#acknowledgements .unnumbered} The author is grateful to R.R. Ashurov for discussions of these results. The author acknowledges financial support from the Ministry of Innovative Development of the Republic of Uzbekistan, Grant No F-FA-2021-424. 99 Y. Liu, Z. Li, M. Yamamoto. Inverse problems of determining sources of the fractional partial differential equations, Handbook of Fractional Calculus with Appl. J.A.T. Marchado Ed. De Gruyter. 2, 411-430 (2019). K. Sakamoto, M. Yamamoto. Initial value boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. **382**, 426-447 (2011). R. Ashurov, M. Shakarova. Time-dependent source identification problem for fractional Schrödinger type equations, Lobachevskii Journal of Mathematics. **42**:3, 517-525 (2022). R. Ashurov, M. Shakarova. Time-dependent source identification problem for a fractional Schrodinger equation with the Riemann-Liouville derivative, https: arxiv.org/abs/2205.03407. 6 may 2022. K. Furati, O. Iyiola, M. Kirane. An inverse problem for a generalized fractional diffusion, Applied Mathematics and Computation. **249**, 24-31 (2014). M. Kirane, A. Malik. Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time, Applied Mathematics and Computation. **218**, 163-170 (2011). M. Kirane, B. Samet, B. Torebek. Determination of an unknown source term and the temperature distribution for the subdiffusion equation at the initial and final data, Electronic Journal of Differential Equations. **217**, 1-13 (2017). Z. Li, Y. Liu, M. Yamamoto. Initial-boundary value problem for multi-term time-fractional diffusion equation with positive constant coefficients, Applied Mathematica and Computation. **257**, 381-397 (2015). S. Malik, S. Aziz. An inverse source problem for a two parameter anomalous diffusion equation with nonlocal boundary conditions, Computers and Mathematics with applications. 3, 7-19 (2017). M. Ruzhansky, N. Tokmagambetov, B.T. Torebek. Inverse source problems for positive operators, I: Hypoelliptic diffusion and subdiffusion equations, J. Inverse Ill-Possed Probl. **27**, 891-911 (2019). R.R. Ashurov, A.T. Mukhiddinova. Inverse Problem of Determining the Heat Source Density for the Subdiffusion Equation, Differential Equations. **56**:12, 1550-1563 (2020). S.I. Kabanikhin. 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Zam. **56**:2, 99-113 (1994). V.E. Fedorov, A. V. Urazaeva. An inverse problem for linear Sobolev type equations, Journal of Inverse and Ill-Posed Problems, 2004, V. 121, pp. 387-395. M. Slodichka, Uniqueness for an inverse source problem of determining a space-dependent source in a non-autonomous time-fractional diffusion equation, Frac.Calculus and Appl. Anal., 2020. V. 23, N 6, pp. 1703-1711. DOI:10.1515/fca-2020-0084. M. Slodichka, K. Sishskova, V. Bockstal. Uniqueness for an inverse source problem of determining a space dependent source in a time-fractional diffusion equation, Appl. Math. Letters, 2019, V. 91, pp. 15-21. V.E. Fedorov, A.V. Nagumanova. Inverse problem for evolutionary equation with the Gerasimov--Caputo fractional derivative in the sectorial case, The Bulletin of Irkutsk State University. Series Mathematics, 2019, V. 28, pp. 123-137. V.E. Fedorov, R.R. Nazhimov. Inverse problems for a class of degenerate evolution equations with Riemann -- Liouville derivative, Frac.Cal., 2019, V. 22, pp. 271--286. R.R. Ashurov, M.D. Shakarova. Inverse problem for the subdiffusion equation with fractional Caputo derivative, 16 november 2022, https://doi.org/10.48550/arXiv.2211.00081. A.V. Pskhu, Fractional Differential Equations. Moscow: NAUKA. 2005 \[in Russian\]. R. Ashurov, A. Mukhiddinova. Initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary elliptic differential operator, Lobachevskii Journal of Mathematics. **42**:3, 517-525 (2021). M. Dzherbashian M \[=Djrbashian\]. Integral Transforms and Representation of Functions in the Complex Domain, Moscow: NAUKA. 1966 (in Russian). R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogozin. Mittag-Leffler Functions, Related Topics and Applications, Springer. Berlin-Heidelberg, Germany, 2014, doi: 10.1007/978-3-662-61550-8.

arxiv_math

--- abstract: | In this paper, the two-player leader-follower game with private inputs for feedback Stackelberg strategy is considered. In particular, the follower shares its measurement information with the leader except its historical control inputs while the leader shares none of the historical control inputs and the measurement information with the follower. The private inputs of the leader and the follower lead to the main obstacle, which causes the fact that the estimation gain and the control gain are related with each other, resulting that the forward and backward Riccati equations are coupled and making the calculation complicated. By introducing a kind of novel observers through the information structure for the follower and the leader, respectively, a kind of new observer-feedback Stacklberg strategy is designed. Accordingly, the above-mentioned obstacle is also avoided. Moreover, it is found that the cost functions under the presented observer-feedback Stackelberg strategy are asymptotically optimal to the cost functions under the optimal feedback Stackelberg strategy with the feedback form of the state. Finally, a numerical example is given to show the efficiency of this paper. author: - "Yue Sun, Hongdan Li and Huanshui Zhang\\* [^1] [^2]" title: Private Inputs for Leader-Follower Game with Feedback Stackelberg Strategy --- feedback Stackelberg strategy, private inputs, observers, asymptotic optimality. # Introduction In the traditional control model, centralized control is a basic concept and has been extensively studied from time-invariant system to time-variant system and system with time-delay [@Anderson1971; @Rami2001; @HSZhang2015]. However, with the development of wireless sensor network and artificial intelligence, the centralized control will no longer be applicable due to the fact that the achievable bandwidth would be limited by long delays induced by the communication between the centralized controller [@Bauer2013]. The task of effectively controlling multiple decision-makers systems in the absence of communication channels is increasingly an interesting and challenging control problem. Correspondingly, the decentralized control of large-scale systems arises accordingly, which has widespread implementation in electrical power distribution networks, cloud environments, multi-agent systems, reinforcement learning and so on [@Blaabjerg2006; @Hoogenkamp2022; @Ha2004; @Gorges2019], where decisions are made by multiple different decision-makers who have access to different information. Decentralized control can be traced back to 1970s [@Witsenhausen1968; @Davison1973; @Davison1976]. The optimization of decentralized control can be divided into two categories. The first category is the decentralized control for multi-controllers with one associated cost function [@Yoshikawa1975; @Swigart2010; @Liang2023]. Nayyar studied decentralized stochastic control with partial history observations and control inputs sharing in [@Nayyar2013] by using the common information approach and the $n$-step delayed sharing information structure was investigated in [@Nayyar2011]. [@Liang2021] focused on decentralized control in networked control system with asymmetric information by solving the forward and backward coupled Riccati equations through forward iteration, where the historical control inputs was shared unilaterally compared with the information structure shared with each other in [@Nayyar2013; @Nayyar2011]. [@BWang2019] designed decentralized strategies for mean-field system, which was further shown to have asymptotic robust social optimality. The other category is the decentralized control for game theory [@Pachter2017; @Sun2022; @Zhi2018]. Two-criteria LQG decision problems with one-step delay observation sharing pattern for stochastic discrete-time system in Stackelberg strategy and Nash equilibrium strategy were considered in [@Basar1978] and [@George1982], respectively. Necessary conditions for an optimal Stackelberg strategy with output feedback form were given in [@Suzumura1993] with incomplete information of the controllers. [@Klompstra2000] investigated feedback risk-sensitive Nash equilibrium solutions for two-player nonzero-sum games with complete state observation and shared historical control inputs. Static output feedback incentive Stackelberg game with markov jump for linear stochastic systems was taken into consideration in [@Mukaidani2016] and a numerical algorithm was further proposed which guaranteed local convergence. Noting that the information structure in the decentralized control systems mentioned above has the following feature, that is, all or part of historical control inputs of the controllers are shared with the other controllers. However, the case, where the controllers have its own private control inputs, has not been addressed in decentralized control system, which has applications in a personalized healthcare setting, in the states of a virtual keyboard user (e.g., Google GBoard users) and in the social robot for second language education of children [@Chowdhury2021]. It should be noted that the information structure where the control information are unavailable to the other decision makers will cause the estimation gain depends on the control gain and vice versa, which means the forward and backward Riccati equations are coupled, and make the calculation more complicated. Motivated by [@Juan2023], which focused on the LQ optimal control problem of linear systems with private input and measurement information by using a kind of novel observers to overcome the obstacle, in this paper, we are concerned with the feedback Stackelberg strategy for two-player game with private control inputs. In particular, the follower shares its measurement information to the leader, while the leader doesn't share any information to the follower due to the hierarchical relationship and the historical control inputs for the follower and the leader are both private, which is the main obstacle in this paper. To overcome the problem, firstly, the novel observers based on the information structure of each controller are proposed. Accordingly, a new kind of observer-feedback Stackelberg strategy for the follower and the leader is designed. Finally, it proved that the associated cost functions for the follower and the leader under the proposed observer-feedback Stackelberg strategy are asymptotically optimal as compared with the cost functions under the optimal feedback Stackelberg strategy with the feedback form of the state obtained in [@Castanon1976]. The outline of this paper is given as follows. The problem formulation is given in Section II. The observers and the observer-feedback Stackelberg strategy with private inputs are designed in Section III. The asymptotical optimal analysis is shown in Section IV. Numerical examples are presented in Section V. Conclusion is given in Section VI. *Notations*: $\mathbb{R}^n$ represents the space of all real $n$-dimensional vectors. $A'$ means the transpose of the matrix $A$. A symmetric matrix $A>0$ (or $A\geq 0$) represents that the matrix $A$ is positive definite (or positive semi-definite). $\|x\|$ denotes the Euclidean norm of vector $x$, i.e., $\|x\|^2=x'x$. $\|A\|$ denotes the Euclidean norm of matrix $A$, i.e., $\|A\|=\sqrt{\lambda_{max}(A'A)}$. $\lambda(A)$ represents the eigenvalues of the matrix $A$ and $\lambda_{max}(A)$ represents the largest eigenvalues of the matrix $A$. $I$ is an identity matrix with compatible dimension. $0$ in block matrix represents a zero matrix with appropriate dimensions. # Problem Formulation Consider a two-player leader-follower game described as: $$\begin{aligned} % \nonumber to remove numbering (before each equation) x(k+1) &\hspace{-0.8em}=&\hspace{-0.8em} Ax(k)+B_1u_1(k)+B_2u_2(k),\label{1-1}\\ y_1(k)&\hspace{-0.8em}=&\hspace{-0.8em}H_1x(k),\label{1-2}\\ y_2(k)&\hspace{-0.8em}=&\hspace{-0.8em}H_2x(k),\label{1-3}\end{aligned}$$ where $x(k)\in \mathbb{R}^n$ is the state with initial value $x(0)$. $u_1(k)\in \mathbb{R}^{m_1}$ and $u_2(k)\in \mathbb{R}^{m_2}$ are the two control inputs of the follower and the leader, respectively. $y_i(k)\in \mathbb{R}^{s_i}$ is the measurement information. $A$, $B_i$ and $H_i$ ($i=1, 2$) are constant matrices with compatible dimensions. The associated cost functions for the follower and the leader are given by $$\begin{aligned} % \nonumber to remove numbering (before each equation) J_1&\hspace{-0.8em}=&\hspace{-0.8em} \sum\limits^{\infty}_{k=0}[x'(k)Q_1x(k)+u'_1(k)R_{11}u_1(k)\nonumber\\ &\hspace{-0.8em} &\hspace{-0.8em} +u'_2(k)R_{12}u_2(k)],\label{2-1}\\ J_2&\hspace{-0.8em}=&\hspace{-0.8em} \sum\limits^{\infty}_{k=0}[x'(k)Q_2x(k)+u'_1(k)R_{21}u_1(k)\nonumber\\ &\hspace{-0.8em} &\hspace{-0.8em}+u'_2(k)R_{22}u_2(k)],\label{2-2}\end{aligned}$$ where the weight matrices are such that $Q_i\geq0$, $R_{ij}\geq0$ ($i\neq j$) and $R_{ii}>0$ ($i, j= 1, 2$) with compatible dimensions. Feedback Stackelberg strategy with different information structure for controllers had been considered since 1970s in [@Castanon1976], where the information structure satisfied that the controller shared all or part of historical inputs to the other. To the best of our knowledge, there has been no efficiency technique to deal with the case of private inputs for controllers. The difficultly lies in the unavailability of other controllers' historical control inputs, which leads to the fact that the estimation gain depends on the control gain and makes the forward and backward Riccati equations coupled. In this paper, our goal is that by designing the novel observers based on the measurements and private inputs for the follower and the leader, respectively, we will show the proposed observer-feedback Stackelberg strategy is asymptotic optimal to the deterministic case in [@Castanon1976]. Mathematically, by denoting $$\begin{aligned} % \nonumber to remove numbering (before each equation) Y_i(k)&\hspace{-0.8em}=&\hspace{-0.8em} \{y_i(0), ..., y_i(k)\}, \nonumber\\ U_i(k-1)&\hspace{-0.8em}=&\hspace{-0.8em} \{u_i(0), ..., u_i(k-1)\},\nonumber\\ {F}_1(k)&\hspace{-0.8em}=&\hspace{-0.8em}\{Y_1(k), U_1(k-1)\}, \label{sy-1}\\ {F}_2(k)&\hspace{-0.8em}=&\hspace{-0.8em}\{Y_1(k), Y_2(k), U_2(k-1)\},\label{sy-2}\end{aligned}$$ we will design the observer-feedback Stackelberg strategy based on the information $\mathcal{F}_i(k)$, where $u_i(k)$ is $\mathcal{F}_i(k)$-casual for $i= 1, 2$ in this paper. The following assumptions will be used in this paper. **Assumption 1**. *System $(A, B)$ is stabilizable with $B=\left[\hspace{-0.4em} \begin{array}{cc} B_1 & B_2 \\ \end{array} \hspace{-0.4em}\right]$ and system $(A, Q_i)$ ($i=1, 2$) is observable.* By denoting the admissible controls sets $\mathcal{U}_i$ (i=1, 2) for the feedback Stackelberg strategy of the follower and the leader: $$\begin{aligned} \label{3} % \nonumber to remove numbering (before each equation) \mathcal{U}_1&\hspace{-0.8em}=\{&\hspace{-0.8em}u_1: \Omega\times[0, N]\times \mathbb{R}^n \times U_2 \longrightarrow U_1 \},\nonumber\\ \mathcal{U}_2&\hspace{-0.8em}=\{&\hspace{-0.8em} u_2: \Omega\times[0, N]\times \mathbb{R}^n \longrightarrow U_2\},\end{aligned}$$ where $U_1$ and $U_2$ represent the strategy for the follower and the leader, respectively, the definition of the feedback Stackelberg strategy [@Bensoussan2015] is given. **Definition 1**. *$(u^*_1(k), u^*_2(k))\in \mathcal{U}_1\times \mathcal{U}_2$ is the optimal feedback Stackelberg strategy, if there holds that: $$\begin{aligned} \label{4} % \nonumber to remove numbering (before each equation) J_1(u^{*}_1(k, u^*_2(k)), u^*_2(k))&\hspace{-0.8em}\leq&\hspace{-0.8em} J_1(u_1(k, u^*_2(k)), u^*_2(k)), \forall u_1\in \mathcal{U}_1,\nonumber\\ J_2(u^{*}_1(k, u^*_2(k)), u^*_2(k))&\hspace{-0.8em}\leq &\hspace{-0.8em}J_2(u^{*}_1(k, u_2(k)), u_2(k)), \forall u_2\in \mathcal{U}_2.\nonumber\end{aligned}$$* Firstly, the optimal feedback Stackelberg strategy in deterministic case with perfect information structure is given, that is, the information structure of the follower and the leader both satisfy $$\begin{aligned} % \nonumber to remove numbering (before each equation) Y_k=\{x(0), ..., x(k), u_i(0), ..., u_i(k-1), \quad i=1,2\}.\end{aligned}$$ **Lemma 1**. *Under Assumption [Assumption 1](#assum1){reference-type="ref" reference="assum1"}, the optimal feedback Stackelberg strategy with the information structure for the follower and the leader satisfying $Y_k$, is given by $$\begin{aligned} % \nonumber to remove numbering (before each equation) u_1(k) &\hspace{-0.8em}=&\hspace{-0.8em} K_1x(k),\label{5-1}\\ u_2(k) &\hspace{-0.8em}=&\hspace{-0.8em} K_2x(k), \label{5-2}\end{aligned}$$ where the feedback gain matrices $K_1$ and $K_2$ satisfy $$\begin{aligned} % \nonumber to remove numbering (before each equation) K_1 &\hspace{-0.8em}=&\hspace{-0.8em} -\Gamma^{-1}_{1}Y_{1},\label{6-1}\\ K_2 &\hspace{-0.8em}=&\hspace{-0.8em} -\Gamma^{-1}_{2}Y_{2},\label{6-2}\end{aligned}$$ with $$\begin{aligned} % \nonumber to remove numbering (before each equation) \Gamma_{1} &\hspace{-0.8em}=&\hspace{-0.8em} R_{11}+B'_1P_{1}B_1,\label{7-1} \\ \Gamma_{2} &\hspace{-0.8em}=&\hspace{-0.8em}R_{22}+B'_2M'_{1}P_{2}M_{1}B_2+B'_2S'R_{21}SB_2,\label{7-2}\\ M_{1} &\hspace{-0.8em}=&\hspace{-0.8em}I-B_1S,\label{7-3} \quad S=\Gamma^{-1}_{1}B'_1P_{1},\label{7-4}\\ Y_{1} &\hspace{-0.8em}=&\hspace{-0.8em} B'_1P_{1}A+B'_1P_{1}B_2K_2,\label{7-5}\\ Y_{2} &\hspace{-0.8em}=&\hspace{-0.8em}B'_2M'_{1}P_{2}M_{1}A+B'_2S'R_{21}SA,\label{7-6}\end{aligned}$$ where $P_1$ and $P_2$ satisfy the following two-coupled algebraic Riccati equations: $$\begin{aligned} % \nonumber to remove numbering (before each equation) P_1&\hspace{-0.8em}=&\hspace{-0.8em}Q_1+(A+B_2K_2)'P_{1}(A+B_2K_2)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-Y'_{1}\Gamma^{-1}_{1}Y_{1}+K'_2R_{12}K_2, \label{8-1} \\ P_2&\hspace{-0.8em}=&\hspace{-0.8em}Q_2+A'M'_{1}P_{2}M_{1}A+A'S'R_{21}SA\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em} -Y'_{2}\Gamma^{-1}_{2}Y_{2}. \label{8-2}\end{aligned}$$* *The optimal cost functions for feedback Stackelberg strategy are such that $$\begin{aligned} % \nonumber to remove numbering (before each equation) J^*_1 &\hspace{-0.8em}=&\hspace{-0.8em}x'(0)P_1x(0),\label{9-1}\\ J^*_2 &\hspace{-0.8em}=&\hspace{-0.8em}x'(0)P_2x(0).\label{9-2}\end{aligned}$$* *Proof.* The optimal feedback Stackelberg strategy for deterministic case with perfect information structure for the follower and the leader in finite-time horizon has been shown in (18)-(28) with $\theta(t)=\Pi_1(t)=\Pi_2(t)=0$ in [@Castanon1976]. By using the results in Theorem 2 in [@HSZhang2015], the results obtained in [@Castanon1976] can be extended into infinite horizon, i.e., (18)-(28) in [@Castanon1976] are convergent to the algebraic equations obtained in ([\[6-1\]](#6-1){reference-type="ref" reference="6-1"})-([\[6-2\]](#6-2){reference-type="ref" reference="6-2"}) and ([\[8-1\]](#8-1){reference-type="ref" reference="8-1"})-([\[8-2\]](#8-2){reference-type="ref" reference="8-2"}) in Lemma [Lemma 1](#lem1){reference-type="ref" reference="lem1"} of this paper by using the monotonic boundedness theorem. This completes the proof. ◻ **Remark 1**. *$P_1>0$ and $P_2>0$ in ([\[8-1\]](#8-1){reference-type="ref" reference="8-1"})-([\[8-2\]](#8-2){reference-type="ref" reference="8-2"}) can be shown accordingly by using Theorem 2 in [@HSZhang2015], which guaranteed the invertibility of $\Gamma_1$ and $\Gamma_2$.* **Remark 2**. *Compared with [@Castanon1976], where the historical control inputs of the follower and the leader are shared with each other, the historical control inputs of this paper are private, leading to the main obstacle.* # The observer-feedback Stackelberg strategy Based on the discussion above, we are in position to consider the leader-follower game with private inputs, i.e., ${u}_i(k)$ is ${F}_i(k)$-casual. **Remark 3**. *As pointed out in [@Liang2021], the information structure in decentralized control, where one of the controllers (C1) doesn't share the historical control inputs to the other controller (C2) while C2 shares its historical control inputs with C1, is a challenge problem due to the control gain and estimator gain are coupled. The difficulty with private inputs for the follower and the leader is even more complicated due to the unavailability of the historical control inputs of each controller.* Considering the private inputs of the follower and the leader, the observers $\hat{x}_i(k)$ ($i=1, 2$) are designed as follows: $$\begin{aligned} % \nonumber to remove numbering (before each equation) \hat{x}_1(k+1)&\hspace{-0.8em}=&\hspace{-0.8em}A\hat{x}_1(k)+B_1u^{\star}_1(k)+B_2K_2\hat{x}_1(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}+L_1[y_1(k)-H_1\hat{x}_1(k)],\label{11-1}\\ \hat{x}_2(k+1)&\hspace{-0.8em}=&\hspace{-0.8em}A\hat{x}_2(k)+B_1K_1\hat{x}_2(k)+B_2u^{\star}_2(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}+L_2[y_2(k)-H_2\hat{x}_2(k)],\label{11-2}\end{aligned}$$ where the observer gain matrices $L_1$ and $L_2$ are chosen to make the observers stable. Accordingly, the observer-feedback Stackelberg strategy is designed as follows: $$\begin{aligned} % \nonumber to remove numbering (before each equation) u^{\star}_1(k) &\hspace{-0.8em}=&\hspace{-0.8em} K_1\hat{x}_1(k),\label{10-1}\\ u^{\star}_2(k) &\hspace{-0.8em}=&\hspace{-0.8em} K_2\hat{x}_2(k), \label{10-2}\end{aligned}$$ where $K_1$ and $K_2$ are given in ([\[6-1\]](#6-1){reference-type="ref" reference="6-1"})-([\[6-2\]](#6-2){reference-type="ref" reference="6-2"}), respectively. For convenience of future discussion, some symbols will be given beforehand. $$\begin{aligned} % \nonumber to remove numbering (before each equation) \mathcal{A} &\hspace{-0.8em}=&\hspace{-0.8em} \left[\hspace{-0.3em} \begin{array}{cc} A+B_2K_2-L_1H_1 &\hspace{-1em} -B_2K_2 \\ -B_1K_1 &\hspace{-1em} A+B_1K_1-L_2H_2 \\ \end{array} \hspace{-0.3em}\right],\label{sy-11}\\ \mathcal{B} &\hspace{-0.8em}=&\hspace{-0.8em}\left[\hspace{-0.3em} \begin{array}{cc} -B_1K_1 & -B_2K_2 \\ \end{array} \hspace{-0.3em}\right]\nonumber\\ &\hspace{-0.8em}=&\hspace{-0.8em}\left[\hspace{-0.3em} \begin{array}{cc} B_1S(A+B_2K_2) & -B_2K_2 \\ \end{array} \hspace{-0.3em}\right],\nonumber\\ \bar{A}&\hspace{-0.8em}=&\hspace{-0.8em}\left[\hspace{-0.3em} \begin{array}{cc} A+B_1K_1+B_2K_2 & \mathcal{B} \\ 0 & \mathcal{A} \\ \end{array} \hspace{-0.3em}\right],\nonumber\\ \tilde{x}(k)&\hspace{-0.8em}=&\hspace{-0.8em}\left[\hspace{-0.3em} \begin{array}{cc} \tilde{x}'_1(k) & \tilde{x}'_2(k) \\ \end{array} \hspace{-0.3em}\right]',\nonumber\\ \tilde{x}_i(k)&\hspace{-0.8em}=&\hspace{-0.8em} x(k)-\hat{x}_i(k), \quad i=1, 2.\nonumber\end{aligned}$$ Subsequently, the stability of the observers $\hat{x}_i(k)$ ($i=1, 2$) and the stability of the closed-loop system ([\[1-1\]](#1-1){reference-type="ref" reference="1-1"}) under the designed observer-feedback Stackelberg strategy ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}) are shown, respectively. **Theorem 1**. *If there exist optional gain matrices $L_1$ and $L_2$ such that the matrix $\mathcal{A}$ is stable, then, the observers $\hat{x}_i(k)$ for $i=1, 2$ are stable with the controllers of the follower and the leader satisfying ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}), i.e., there holds $$\begin{aligned} \label{sy-1} % \nonumber to remove numbering (before each equation) \lim_{k\rightarrow \infty}\|x(k)-\hat{x}_i(k)\|=0.\end{aligned}$$* *Proof.* By substituting the observer-feedback controllers ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}) into ([\[1-1\]](#1-1){reference-type="ref" reference="1-1"}), then $x(k+1)$ is recalculated as: $$\begin{aligned} \label{sy-2} % \nonumber to remove numbering (before each equation) x(k+1)&\hspace{-0.8em}=&\hspace{-0.8em} Ax(k)+B_1K_1\hat{x}_1(k)+B_2K_2\hat{x}_(k)\nonumber\\ &\hspace{-0.8em}=&\hspace{-0.8em}[A+B_1K_1+B_2K_2]x(k)-B_1K_1\tilde{x}_1(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-B_2K_2\tilde{x}_2(k).\end{aligned}$$ Accordingly, by adding ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}) into the observers ([\[11-1\]](#11-1){reference-type="ref" reference="11-1"})-([\[11-2\]](#11-2){reference-type="ref" reference="11-2"}) and combining with ([\[sy-2\]](#sy-2){reference-type="ref" reference="sy-2"}), the derivation of $\tilde{x}_i(k)$ for $i=1, 2$ are given as $$\begin{aligned} % \nonumber to remove numbering (before each equation) \tilde{x}_1(k+1)&\hspace{-0.8em}=&\hspace{-0.8em}(A+B_2K_2-L_1H_1)\tilde{x}_1(k)-B_2K_2\tilde{x}_2(k),\\ \tilde{x}_2(k+1)&\hspace{-0.8em}=&\hspace{-0.8em}(A+B_1K_1-L_2H_2)\tilde{x}_1(k)-B_1K_1\tilde{x}_1(k),\end{aligned}$$ that is $$\begin{aligned} \label{12-1} % \nonumber to remove numbering (before each equation) \tilde{x}(k+1)&\hspace{-0.8em}=&\hspace{-0.8em}\mathcal{A}\tilde{x}(k).\end{aligned}$$ Subsequently, if there exist matrices $L_1$ and $L_2$ making $\mathcal{A}$ stable, then, the stability of the matrix $\mathcal{A}$ means that $$\begin{aligned} % \nonumber to remove numbering (before each equation) \lim_{k\rightarrow \infty} \tilde{x}(k)=0,\end{aligned}$$ i.e., ([\[sy-1\]](#sy-1){reference-type="ref" reference="sy-1"}) is established. That is to say, the observers $\hat{x}_i(k)$ are stable under ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}). The proof is completed. ◻ **Remark 4**. *Noting that in Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"} the key point lies in that how to select $L_i$ ($i=1, 2$) so that the eigenvalues of the matrix $\mathcal{A}$ are within the unit circle. The following analysis gives an method to find $L_i$.* *According to the Lyapunov stability criterion, i.e., $\mathcal{A}$ is stable if and only if for any positive definite matrix $Q$, $\mathcal{A}'P\mathcal{A}-P =-Q$ admits a solution such that $P>0$. Thus, if there exists a $P>0$ such that $$\begin{aligned} % \nonumber to remove numbering (before each equation) \mathcal{A}'P\mathcal{A}-P<0,\end{aligned}$$ then $\mathcal{A}$ is stable. Following from the elementary row transformation, one has $$\begin{aligned} % \nonumber to remove numbering (before each equation) &\hspace{-0.8em}&\hspace{-0.8em}\left( \begin{array}{cc} I & I \\ 0 & I \\ \end{array} \right)\left( \begin{array}{cc} I & 0 \\ 0 & \mathcal{A}' \\ \end{array} \right)\left( \begin{array}{cc} -P & \mathcal{A}'P \\ P\mathcal{A} & -P \\ \end{array} \right)\left( \begin{array}{cc} I & 0 \\ 0 & \mathcal{A} \\ \end{array} \right)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}\times \left( \begin{array}{cc} I & 0 \\ I & I \\ \end{array} \right)=\left( \begin{array}{cc} \mathcal{A}'P\mathcal{A}-P & 0 \\ 0 & -\mathcal{A}'P\mathcal{A} \\ \end{array} \right)<0,\end{aligned}$$ that is, $\mathcal{A}'P\mathcal{A}-P<0$ is equivalent to the following matrix inequality $$\begin{aligned} \label{0908-1} % \nonumber to remove numbering (before each equation) \left( \begin{array}{cc} -P & \mathcal{A}'P \\ P\mathcal{A} & -P \\ \end{array} \right)<0.\end{aligned}$$ Noting that $\mathcal{A}$ is related with $L_i$, in order to use the linear matrix inequality (LMI) Toolbox in Matlab to find $L_i$, ([\[0908-1\]](#0908-1){reference-type="ref" reference="0908-1"}) will be transmit into a LMI form. Let $$\begin{aligned} % \nonumber to remove numbering (before each equation) P=\left( \begin{array}{cc} P & 0 \\ 0 & P \\ \end{array} \right), \quad \tilde{W}=\left( \begin{array}{cc} W_1 & 0 \\ 0 & W_2 \\ \end{array} \right),\end{aligned}$$ and rewrite $\mathcal{A}$ in ([\[sy-11\]](#sy-11){reference-type="ref" reference="sy-11"}) as $\mathcal{A}=\tilde{A}-\tilde{L}\tilde{H}$, where $$\begin{aligned} % \nonumber to remove numbering (before each equation) \mathcal{A} &\hspace{-0.8em}=&\hspace{-0.8em}\left( \begin{array}{cc} A+B_2K_2 &\hspace{-1em} -B_2K_2 \\ -B_1K_1 &\hspace{-1em} A+B_1K_1 \\ \end{array} \right),\\ \tilde{L}&\hspace{-0.8em}=&\hspace{-0.8em}\left( \begin{array}{cc} L_1 & 0 \\ 0 & L_2 \\ \end{array} \right),\quad \tilde{H}=\left( \begin{array}{cc} H_1 & 0 \\ 0 & H_2 \\ \end{array} \right).\end{aligned}$$ To this end, we have $$\begin{aligned} % \nonumber to remove numbering (before each equation) P\mathcal{A}=P\tilde{A}-P\tilde{L}\tilde{H}=P\tilde{A}-\tilde{W}\tilde{H},\end{aligned}$$ with $\tilde{W}=P\tilde{L}$. Based on the discussion above, it concludes that $\mathcal{A}$ is stable if there exists a $P>0$ such that the following LMI: $$\begin{aligned} \label{0908-2} % \nonumber to remove numbering (before each equation) \left( \begin{array}{cc} -P & (P\tilde{A}-\tilde{W}\tilde{H})'\\ P\tilde{A}-\tilde{W}\tilde{H} & -P \\ \end{array} \right)<0.\end{aligned}$$* *In this way, by using the LMI Toolbox in Matlab, $L_i$ can be found according, which stabilizes $\mathcal{A}$ where $L_i=P^{-1}W_i$.* Under the observer-feedback controllers ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}), the stability of ([\[1-1\]](#1-1){reference-type="ref" reference="1-1"}) is given. **Theorem 2**. *Under Assumption [Assumption 1](#assum1){reference-type="ref" reference="assum1"} and if there exists $L_i$ stabilizing $\mathcal{A}$, then the closed-loop system ([\[1-1\]](#1-1){reference-type="ref" reference="1-1"}) is stable with the observer-feedback controllers ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}).* *Proof.* According to ([\[sy-2\]](#sy-2){reference-type="ref" reference="sy-2"}), the closed-loop system ([\[1-1\]](#1-1){reference-type="ref" reference="1-1"}) is reformulated as $$\begin{aligned} \label{sy-3} % \nonumber to remove numbering (before each equation) x(k+1)&\hspace{-0.8em}=&\hspace{-0.8em} [A+B_1K_1+B_2K_2]x(k)+\mathcal{B}\tilde{x}(k).\end{aligned}$$ Together with ([\[12-1\]](#12-1){reference-type="ref" reference="12-1"}), we have $$\begin{aligned} \label{sy-4} % \nonumber to remove numbering (before each equation) \left[ \begin{array}{c} x(k+1) \\ \tilde{x}(k+1) \\ \end{array} \right]&\hspace{-0.8em}=&\hspace{-0.8em}\bar{A}\left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right].\end{aligned}$$ The stability of $A+B_1K_1+B_2K_2$ is guaranteed by the stabilizability of $(A, B)$ and the observability of $(A, Q_i)$ for $i=1, 2$. Following from Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"}, $\mathcal{A}$ is stabilized by selecting appropriate gain matrices $L_1$ and $L_2$. Subsequently, the stability of the closed-loop system ([\[1-1\]](#1-1){reference-type="ref" reference="1-1"}) is derived. This completes the proof. ◻ # The asymptotical optimal analysis The stability of the state and the observers, i.e., $x(k)$ and $\hat{x}_i$ for $i=1, 2$ has been shown in Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"} and Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"} under the observer-feedback controllers ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}). To shown the rationality of the design of the observer-feedback controllers ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}), the asymptotical optimal analysis relating with the cost functions under ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}) is given. To this end, denote the cost functions for the follower and the leader satisfying $$\begin{aligned} % \nonumber to remove numbering (before each equation) J_1(s, M)&\hspace{-0.8em}=&\hspace{-0.8em} \sum\limits^{M}_{k=s}[x'(k)Q_1x(k)+u'_1(k)R_{11}u_1(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}+u'_2(k)R_{12}u_2(k)],\label{13-1}\\ J_2(s, M)&\hspace{-0.8em}=&\hspace{-0.8em} \sum\limits^{M}_{k=s}[x'(k)Q_2x(k)+u'_1(k)R_{21}u_1(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}+u'_2(k)R_{22}u_2(k)].\label{13-2}\end{aligned}$$ Now, we are in position to show that the observer-feedback Stackelberg strategy ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}) is asymptotical optimal to the optimal feedback Stackelberg strategy presented in Lemma [Lemma 1](#lem1){reference-type="ref" reference="lem1"}. **Theorem 3**. *Under Assumption [Assumption 1](#assum1){reference-type="ref" reference="assum1"}, the corresponding cost functions ([\[13-1\]](#13-1){reference-type="ref" reference="13-1"})-([\[13-2\]](#13-2){reference-type="ref" reference="13-2"}) under the observer-feedback Stackelberg strategy ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}) with $L_i$ ($i=1, 2$) selected from Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"} are given by $$\begin{aligned} % \nonumber to remove numbering (before each equation) J^{\star}_1(s, \infty) &\hspace{-0.8em}=&\hspace{-0.8em} x'(s)P_1x(s)\nonumber\\ &\hspace{-0.8em} &\hspace{-0.8em}+\sum\limits^{\infty}_{k=s} \left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right]'\left[ \begin{array}{cc} 0 & T_1 \\ T'_1 & S_1\\ \end{array} \right] \left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right],\label{14-1}\\ J^{\star}_2(s, \infty) &\hspace{-0.8em}=&\hspace{-0.8em} x'(s)P_2x(s)\nonumber\\ &\hspace{-0.8em} &\hspace{-0.8em}+\sum\limits^{\infty}_{k=s} \left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right]'\left[ \begin{array}{cc} 0 & T_2\\ T'_2 & S_2\\ \end{array} \right] \left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right],\label{14-2}\end{aligned}$$ where $$\begin{aligned} % \nonumber to remove numbering (before each equation) S_1 &\hspace{-0.8em}=&\hspace{-0.8em} \mathcal{B}'P_1 \mathcal{B}-\left[ \begin{array}{cc} K'_1R_{11}K_1 & 0 \\ 0 & K'_2R_{12}K_2 \\ \end{array} \right],\\ S_2 &\hspace{-0.8em}=&\hspace{-0.8em} \mathcal{B}'P_2 \mathcal{B}-\left[ \begin{array}{cc} K'_1R_{21}K_1 & 0 \\ 0 & K'_2R_{22}K_2 \\ \end{array} \right],\\ T_1&\hspace{-0.8em}=&\hspace{-0.8em}(A+B_2K_2)'M'_1P_1 \mathcal{B},\\ T_2&\hspace{-0.8em}=&\hspace{-0.8em}(A+B_2K_2)'M'_1P_2 \mathcal{B}.\end{aligned}$$ Moreover, the differences, which are denoted as $\delta J_1(s, \infty)$ and $\delta J_2(s, \infty)$, between ([\[14-1\]](#14-1){reference-type="ref" reference="14-1"})-([\[14-2\]](#14-2){reference-type="ref" reference="14-2"}) and the optimal cost functions ([\[9-1\]](#9-1){reference-type="ref" reference="9-1"})-([\[9-2\]](#9-2){reference-type="ref" reference="9-2"}) obtained in Lemma [Lemma 1](#lem1){reference-type="ref" reference="lem1"} under the optimal feedback Stackelberg strategy are such that $$\begin{aligned} % \nonumber to remove numbering (before each equation) \delta J_1(s, \infty) &\hspace{-0.8em}=&\hspace{-0.8em} J^{\star}_1(s, \infty)-J^*_1(s, \infty)\nonumber\\ &\hspace{-0.8em}=&\hspace{-0.8em}\sum\limits^{\infty}_{k=s} \left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right]'\left[ \begin{array}{cc} 0 & T_1\\ T'_1 & S_1\\ \end{array} \right] \left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right],\label{15-1}\\ \delta J_2(s, \infty) &\hspace{-0.8em}=&\hspace{-0.8em} J^{\star}_2(s, \infty)-J^*_2(s, \infty)\nonumber\\ &\hspace{-0.8em}=&\hspace{-0.8em}\sum\limits^{\infty}_{k=s} \left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right]'\left[ \begin{array}{cc} 0 & T_2\\ T'_2 & S_2\\ \end{array} \right] \left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right].\label{15-2}\end{aligned}$$* *Proof.* The proof will be divided into two parts. The first part is to consider the cost function of the follower under the observer-feedback controllers ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}). Following from ([\[sy-2\]](#sy-2){reference-type="ref" reference="sy-2"}), system ([\[1-1\]](#1-1){reference-type="ref" reference="1-1"}) it can be rewritten as $$\begin{aligned} \label{16} % \nonumber to remove numbering (before each equation) x(k+1)&\hspace{-0.8em}=&\hspace{-0.8em} [A+B_1K_1+B_2K_2]x(k)-B_1K_1\tilde{x}_1(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-B_2K_2\tilde{x}_2(k)\nonumber\\ &\hspace{-0.8em}=&\hspace{-0.8em}(I-B_1S)(A+B_2K_2)x(k)+\mathcal{B}\tilde{x}(k),\end{aligned}$$ where $K_1$ in ([\[6-1\]](#6-1){reference-type="ref" reference="6-1"}) have been used in the derivation of the last equality. Firstly, we will prove $J^{\star}_1(s, \infty)$ satisfies ([\[14-1\]](#14-1){reference-type="ref" reference="14-1"}). Combing ([\[16\]](#16){reference-type="ref" reference="16"}) with ([\[8-1\]](#8-1){reference-type="ref" reference="8-1"}), one has $$\begin{aligned} \label{17} % \nonumber to remove numbering (before each equation) &\hspace{-0.8em}&\hspace{-0.8em} x'(k)P_1x(k)-x(k+1)'P_1x(k+1)\nonumber\\ =&\hspace{-0.8em}&\hspace{-0.8em}x'(k)[P_1-(A+B_2K_2)'(I-B_1S)'P_1(I-B_1S)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}\times(A+B_2K_2)]x(k)-x'(k)(A+B_2K_2)'M'_1P_1\mathcal{B}\tilde{x}(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-\tilde{x}'(k)\mathcal{B}'P_1M_1(A+B_2K_2)x(k) -\tilde{x}'(k)\mathcal{B}'P_1\mathcal{B}\tilde{x}(k)\nonumber\\ =&\hspace{-0.8em}&\hspace{-0.8em}x'(k)[Q_1+K'_2R_{12}K_2-(A+B_2K_2)'P_1B_1\Gamma^{-1}_{1}B'_{1}P_1\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}\times (A+B_2K_2)+(A+B_2K_2)'P_1B_1S(A+B_2K_2)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}+(A+B_2K_2)'S'B'_{1}P_1(A+B_2K_2)-(A+B_2K_2)'S'\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}\times B'_{1}P_1B_1S(A+B_2K_2)]x(k)-x'(k)(A+B_2K_2)'M'_1\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}\times P_1\mathcal{B}\tilde{x}(k)-\tilde{x}'(k)\mathcal{B}'P_1M_1(A+B_2K_2)x(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-\tilde{x}'(k)\mathcal{B}'P_1\mathcal{B}\tilde{x}(k)\nonumber\\ =&\hspace{-0.8em}&\hspace{-0.8em}x'(k)[Q_1+K'_2R_{12}K_2+K'_1(R_{11}+B'_{1}P_1B_1)K_1\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-K'_1B'_{1}P_1B_1K_1]x(k)-x'(k)(A+B_2K_2)'M'_1P_1\mathcal{B}\tilde{x}(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-\tilde{x}'(k)\mathcal{B}'P_1M_1(A+B_2K_2)x(k) -\tilde{x}'(k)\mathcal{B}'P_1\mathcal{B}\tilde{x}(k)\nonumber\\ =&\hspace{-0.8em}&\hspace{-0.8em}x'(k)[Q_1+K'_1R_{11}K_1+K'_2R_{12}K_2]x(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-x'(k)(A+B_2K_2)'M'_1P_1\mathcal{B}\tilde{x}(k)-\tilde{x}'(k)\mathcal{B}'P_1M_1\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}\times(A+B_2K_2)x(k)-\tilde{x}'(k)\mathcal{B}'P_1\mathcal{B}\tilde{x}(k).\end{aligned}$$ Substituting ([\[17\]](#17){reference-type="ref" reference="17"}) from $k=s$ to $k=M$ on both sides, we have $$\begin{aligned} \label{18} % \nonumber to remove numbering (before each equation) &\hspace{-0.8em}&\hspace{-0.8em} x'(s)P_1x(s)-x'(M+1)P_1x(M+1)\nonumber\\ =&\hspace{-0.8em}&\hspace{-0.8em} J_1(s, M)+\sum\limits^{M}_{k=s}\tilde{x}'(k)\left[ \begin{array}{cc} K'_1R_{11}K_1 & 0 \\ 0 & K'_2R_{12}K_2 \\ \end{array} \right]\tilde{x}(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-\sum\limits^{M}_{k=s}\left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right]'\left[ \begin{array}{cc} 0 & T_1\\ T'_1 & \mathcal{B}'P_1\mathcal{B}\\ \end{array} \right] \left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right].\end{aligned}$$ According to Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"}, the stability of ([\[1-1\]](#1-1){reference-type="ref" reference="1-1"}) means that $$\begin{aligned} % \nonumber to remove numbering (before each equation) \lim_{M\rightarrow \infty} x'(M+1)P_1x(M+1)=0.\end{aligned}$$ Thus, following from ([\[18\]](#18){reference-type="ref" reference="18"}) and letting $M\rightarrow \infty$, ([\[14-1\]](#14-1){reference-type="ref" reference="14-1"}) can be obtained exactly. The second part is to consider the cost function of the leader under the observer-feedback controllers ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}), that is, we will show that $J^{\star}_2(s, \infty)$ satisfies ([\[14-2\]](#14-2){reference-type="ref" reference="14-2"}). Following from ([\[16\]](#16){reference-type="ref" reference="16"}), it derives $$\begin{aligned} \label{19-1} % \nonumber to remove numbering (before each equation) &\hspace{-0.8em}&\hspace{-0.8em} x'(k)P_2x(k)-x(k+1)'P_2x(k+1)\nonumber\\ =&\hspace{-0.8em}&\hspace{-0.8em}x'(k)[P_2-(A+B_2K_2)'M'_1P_2M_1(A+B_2K_2)]x(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-x'(k)(A+B_2K_2)'M'_1P_2\mathcal{B}\tilde{x}(k)-\tilde{x}'(k)\mathcal{B}'P_2M_1(A\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}+B_2K_2)x(k) -\tilde{x}'(k)\mathcal{B}'P_2\mathcal{B}\tilde{x}(k)\nonumber\\ =&\hspace{-0.8em}&\hspace{-0.8em}x'(k)[Q_2+A'S'R_{21}SA-Y'_2\Gamma^{-1}_2Y_2\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-A'M'_1P_2M_1B_2K_2-K'_2B'_2M'_1P_2M_1A\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-K'_2B'_2M'_1P_2M_1B_2K_2]x(k)-\tilde{x}'(k)\mathcal{B}'P_2\mathcal{B}\tilde{x}(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-x'(k) (A+B_2K_2)'M'_1P_2\mathcal{B}\tilde{x}(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-\tilde{x}'(k)\mathcal{B}'P_2M_1(A+B_2K_2)x(k),\end{aligned}$$ where the algebraic Riccati equation ([\[8-2\]](#8-2){reference-type="ref" reference="8-2"}) has been used in the derivation of the last equality. For further optimization, we make the following derivation: $$\begin{aligned} \label{19} % \nonumber to remove numbering (before each equation) &\hspace{-0.8em}&\hspace{-0.8em} x'(k)P_2x(k)-x(k+1)'P_2x(k+1)\nonumber\\ =&\hspace{-0.8em}&\hspace{-0.8em}x'(k)[Q_2+K'_1R_{21}K_1+K'_2R_{22}K_2]x(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}+x'(k)[-(A+B_2K_2)'S'R_{21}S(A+B_2K_2)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-K'_2R_{22}K_2+A'S'R_{21}SA-Y'_2\Gamma^{-1}_2Y_2\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-A'M'_1P_2M_1B_2K_2-K'_2B'_2M'_1P_2M_1A\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-K'_2B'_2M'_1P_2M_1B_2K_2]x(k)-x'(k)(A+B_2K_2)'\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}\times M'_1P_2\mathcal{B}\tilde{x}(k)-\tilde{x}'(k)\mathcal{B}'P_2M_1(A+B_2K_2)x(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-\tilde{x}'(k)\mathcal{B}'P_2\mathcal{B}\tilde{x}(k)\nonumber\\ =&\hspace{-0.8em}&\hspace{-0.8em}x'(k)[Q_2+K'_1R_{21}K_1+K'_2R_{22}K_2]x(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-x'(k)(A+B_2K_2)'M'_1P_2\mathcal{B}\tilde{x}(k)-\tilde{x}'(k)\mathcal{B}'P_2M_1\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}\times(A+B_2K_2)x(k)-\tilde{x}'(k)\mathcal{B}'P_2\mathcal{B}\tilde{x}(k).\end{aligned}$$ Substituting ([\[19\]](#19){reference-type="ref" reference="19"}) from $k=s$ to $k=M$ on both sides, one has $$\begin{aligned} \label{20} % \nonumber to remove numbering (before each equation) &\hspace{-0.8em}&\hspace{-0.8em} x'(s)P_2x(s)-x'(M+1)P_2x(M+1)\nonumber\\ =&\hspace{-0.8em}&\hspace{-0.8em} J_2(s, M)+\sum\limits^{M}_{k=s}\tilde{x}'(k)\left[ \begin{array}{cc} K'_1R_{21}K_1 & 0 \\ 0 & K'_2R_{22}K_2 \\ \end{array} \right]\tilde{x}(k)\nonumber\\ &\hspace{-0.8em}&\hspace{-0.8em}-\sum\limits^{M}_{k=s}\left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right]'\left[ \begin{array}{cc} 0 & T_2\\ T'_2 & \mathcal{B}'P_1\mathcal{B}\\ \end{array} \right] \left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right].\end{aligned}$$ Due to $\lim_{M\rightarrow \infty} x'(M+1)P_2x(M+1)=0$, ([\[14-2\]](#14-2){reference-type="ref" reference="14-2"}) can be immediately obtained by letting $M\rightarrow \infty$ in ([\[20\]](#20){reference-type="ref" reference="20"}). Moreover, together with Lemma [Lemma 1](#lem1){reference-type="ref" reference="lem1"}, the optimal cost functions of ([\[13-1\]](#13-1){reference-type="ref" reference="13-1"})-([\[13-2\]](#13-2){reference-type="ref" reference="13-2"}) under the optimal feedback Stackelberg strategy are given by $$\begin{aligned} % \nonumber to remove numbering (before each equation) J^*_1(s, \infty)&\hspace{-0.8em}=&\hspace{-0.8em} x'(s)P_1x(s),\label{sy-5-1}\\ J^*_2(s, \infty)&\hspace{-0.8em}=&\hspace{-0.8em} x'(s)P_2x(s).\label{sy-5-2}\end{aligned}$$ Together with ([\[14-1\]](#14-1){reference-type="ref" reference="14-1"})-([\[14-2\]](#14-2){reference-type="ref" reference="14-2"}), $\delta J_1(s, \infty)$ and $\delta J_2(s, \infty)$ in ([\[15-1\]](#15-1){reference-type="ref" reference="15-1"})-([\[15-2\]](#15-2){reference-type="ref" reference="15-2"}) are obtained. This completes the proof. ◻ Finally, we will show the asymptotical optimal property under the observer-feedback Stackelberg strategy ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}). **Theorem 4**. *Under the condition of Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"}, the optimal cost functions ([\[14-1\]](#14-1){reference-type="ref" reference="14-1"})-([\[14-2\]](#14-2){reference-type="ref" reference="14-2"}) under the observer-feedback Stackelberg strategy ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}) are asymptotical optimal to the optimal cost functions ([\[sy-5-1\]](#sy-5-1){reference-type="ref" reference="sy-5-1"})-([\[sy-5-2\]](#sy-5-2){reference-type="ref" reference="sy-5-2"}) under the optimal feedback Stackelberg strategy ([\[5-1\]](#5-1){reference-type="ref" reference="5-1"})-([\[5-2\]](#5-2){reference-type="ref" reference="5-2"}), that is to say, for any $\varepsilon >0$, there exists a sufficiency large integer $N$ for $i=1, 2$ such that $$\begin{aligned} % \nonumber to remove numbering (before each equation) \delta J_i(N, \infty)< \varepsilon.\end{aligned}$$* *Proof.* Following from Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"}, there exists a stable matrix $\bar{A}$. Thus, by [@Rami2001], there exist constants $0< \lambda <1$ and $c>0$ such that $$\begin{aligned} \label{sy-6} % \nonumber to remove numbering (before each equation) \Big\|\left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right] \Big\|\leq c \lambda^k \Big\|\left[ \begin{array}{c} x(0) \\ \tilde{x}(0) \\ \end{array} \right]\Big\|.\end{aligned}$$ In this way, one has $$\begin{aligned} \label{sy-7} % \nonumber to remove numbering (before each equation) \delta J_i(N, \infty)&\hspace{-0.8em}=&\hspace{-0.8em}\sum\limits^{\infty}_{k=s} \left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right]'\left[ \begin{array}{cc} 0 & T_i\\ T'_i & S_i\\ \end{array} \right] \left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right]\nonumber\\ \leq&\hspace{-0.8em}&\hspace{-0.8em}\sum\limits^{\infty}_{k=s}\Big\|\left[ \begin{array}{cc} 0 & T_i\\ T'_i & S_i\\ \end{array} \right]\Big\|\Big\|\left[ \begin{array}{c} x(k) \\ \tilde{x}(k) \\ \end{array} \right]\Big\|^2\nonumber\\ \leq&\hspace{-0.8em}&\hspace{-0.8em}\sum\limits^{\infty}_{k=s}\lambda^{2k}\cdot c^2\Big\|\left[ \begin{array}{cc} 0 & T_i\\ T'_i & S_i\\ \end{array} \right]\Big\|\Big\|\left[ \begin{array}{c} x(0) \\ \tilde{x}(0) \\ \end{array} \right]\Big\|^2\nonumber\\ <&\hspace{-0.8em}&\hspace{-0.8em} \frac{\lambda^{2s}}{1-\lambda^2} \cdot c^2\Big\|\left[ \begin{array}{cc} 0 & T_i\\ T'_i & S_i\\ \end{array} \right]\Big\|\Big\|\left[ \begin{array}{c} x(0) \\ \tilde{x}(0) \\ \end{array} \right]\Big\|^2 \nonumber\\ \doteq &\hspace{-0.8em}&\hspace{-0.8em} \bar{c} \lambda^{2s}.\end{aligned}$$ Since $0< \lambda <1$, thus there exists a sufficiency large integer $N$ such that for any $\varepsilon >0$, satisfying $$\begin{aligned} % \nonumber to remove numbering (before each equation) \lambda^{2N}<\frac{1}{\bar{c}+1} \varepsilon.\end{aligned}$$ Combing with ([\[sy-7\]](#sy-7){reference-type="ref" reference="sy-7"}), one has $$\begin{aligned} \label{sy-8} % \nonumber to remove numbering (before each equation) \delta J_i(N, \infty) < \frac{\bar{c}}{\bar{c}+1} \varepsilon< \varepsilon.\end{aligned}$$ That is to say, the cost functions ([\[14-1\]](#14-1){reference-type="ref" reference="14-1"})-([\[14-2\]](#14-2){reference-type="ref" reference="14-2"}) under the observer-feedback Stackelberg strategy ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}) are asymptotical optimal to the cost functions ([\[sy-5-1\]](#sy-5-1){reference-type="ref" reference="sy-5-1"})-([\[sy-5-2\]](#sy-5-2){reference-type="ref" reference="sy-5-2"}) under the optimal feedback Stackelberg strategy ([\[5-1\]](#5-1){reference-type="ref" reference="5-1"})-([\[5-2\]](#5-2){reference-type="ref" reference="5-2"}) when the integer $N$ is large enough. The proof is now completed. ◻ # Numerical Examples To show the validity of the results in Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"} to Theorem [Theorem 4](#thm4){reference-type="ref" reference="thm4"}, the following example is presented. Consider system ([\[1-1\]](#1-1){reference-type="ref" reference="1-1"})-([\[1-3\]](#1-3){reference-type="ref" reference="1-3"}) with $$\begin{aligned} % \nonumber to remove numbering (before each equation) A &\hspace{-0.8em}=&\hspace{-0.8em} \left[ \begin{array}{cc} 1 & -0.7 \\ 1 & -0.3 \\ \end{array} \right], \quad B_1=\left[ \begin{array}{c} -5 \\ -1 \\ \end{array} \right], \\ B_2&\hspace{-0.8em}=&\hspace{-0.8em}\left[ \begin{array}{c} 0 \\ 1 \\ \end{array} \right], \quad H_1=\left[ \begin{array}{cc} 1 & 0 \\ \end{array} \right], \quad H_2=\left[ \begin{array}{cc} 0 & 1 \\ \end{array} \right],\end{aligned}$$ and the associated cost functions ([\[2-1\]](#2-1){reference-type="ref" reference="2-1"})-([\[2-2\]](#2-2){reference-type="ref" reference="2-2"}) with $$\begin{aligned} % \nonumber to remove numbering (before each equation) Q_1 &\hspace{-0.8em}=&\hspace{-0.8em} \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right], \quad Q_2 = \left[ \begin{array}{cc} 2 & 0 \\ 0 & 1 \\ \end{array} \right],\\ R_{11}&\hspace{-0.8em}=&\hspace{-0.8em}1, \quad R_{11}=2,\quad R_{21}=0,\quad R_{22}=1.\end{aligned}$$ By decoupled solving the algebraic Riccati equations ([\[8-1\]](#8-1){reference-type="ref" reference="8-1"})-([\[8-2\]](#8-2){reference-type="ref" reference="8-2"}), the feedback gains in ([\[6-1\]](#6-1){reference-type="ref" reference="6-1"})-([\[6-2\]](#6-2){reference-type="ref" reference="6-2"}) are respectively calculated as $$\begin{aligned} % \nonumber to remove numbering (before each equation) K_1 &\hspace{-0.8em}=&\hspace{-0.8em} \left[ \begin{array}{cc} 0.2028 & -0.1374 \\ \end{array} \right],\\ K_2 &\hspace{-0.8em}=&\hspace{-0.8em} \left[ \begin{array}{cc} -0.4005 & 0.0791 \\ \end{array} \right].\end{aligned}$$ By using the LMI Toolbox in Matlab, $L_i$ ($i=1, 2$) are calculated as $$\begin{aligned} % \nonumber to remove numbering (before each equation) L_1=\left[ \begin{array}{c} 1.2364 \\ 0.4246 \\ \end{array} \right], \quad L_2=\left[ \begin{array}{c} 0.0039 \\ 0.1925 \\ \end{array} \right],\end{aligned}$$ while the four eigenvalues of matrix $\mathcal{A}$ are calculated as: $$\begin{aligned} % \nonumber to remove numbering (before each equation) \lambda_1(\mathcal{A}) &\hspace{-0.8em}=&\hspace{-0.8em}0.1949, \quad \lambda_2(\mathcal{A})=0.6791,\\ \lambda_3(\mathcal{A}) &\hspace{-0.8em}=&\hspace{-0.8em}\lambda_4(\mathcal{A})=0.7317,\end{aligned}$$ which means that $\mathcal{A}$ in ([\[sy-11\]](#sy-11){reference-type="ref" reference="sy-11"}) is sable. In this way, following from Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"}, the state error estimation $\tilde{x}(k)$ in ([\[12-1\]](#12-1){reference-type="ref" reference="12-1"}) is stable, which is shown in Fig. [1](#Fig1){reference-type="ref" reference="Fig1"}, where data 1 to data 4 represent the four components of vector $\tilde{x}(k)\doteq \left[ \begin{array}{cccc} \tilde{x}_{11}(k) &\hspace{-0.5em} \tilde{x}_{21}(k) &\hspace{-0.5em} \tilde{x}_{31}(k) &\hspace{-0.5em} \tilde{x}_{41}(k) \\ \end{array} \right]'$. Moreover, under the observer-feedback Stackelberg strategy ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}), the state $x(k)$ in ([\[1-1\]](#1-1){reference-type="ref" reference="1-1"}) is also stable which can be seen in Fig. [2](#Fig2){reference-type="ref" reference="Fig2"}, where data 1 and data 2 represent the two components of $x(k)\doteq \left[ \begin{array}{cc} x_{11}(k) &\hspace{-0.5em} x_{21}(k)\\ \end{array} \right]'$. Finally, by analyzing Fig. [1](#Fig1){reference-type="ref" reference="Fig1"} and Fig. [2](#Fig2){reference-type="ref" reference="Fig2"} and selecting $N=30$ in Theorem [Theorem 4](#thm4){reference-type="ref" reference="thm4"}, the asymptotical optimal property of the cost functions ([\[14-1\]](#14-1){reference-type="ref" reference="14-1"})-([\[14-2\]](#14-2){reference-type="ref" reference="14-2"}) under the observer-feedback Stackelberg strategy ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}) is guaranteed. ![Trajectory of $\tilde{x}(k)$ in ([\[12-1\]](#12-1){reference-type="ref" reference="12-1"}) under the observer-feedback Stackelberg strategy ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}).](tildex.jpg){#Fig1 width="3.1in" height="2.4in"} ![Trajectory of $x(k)$ in ([\[1-1\]](#1-1){reference-type="ref" reference="1-1"}) under the observer-feedback Stackelberg strategy ([\[10-1\]](#10-1){reference-type="ref" reference="10-1"})-([\[10-2\]](#10-2){reference-type="ref" reference="10-2"}).](x.jpg){#Fig2 width="3.1in" height="2.4in"} # Conclusion In this paper, we have considered the feedback Stackelberg strategy for two-player leader-follower game with private inputs, where the follower only shares its measurement informaiton with the leader, while none of the historical control inputs and measurement information of the leader are shared with the follower due to the hierarchical relationship. The unavaliable access of the historical inputs for both controllers causes the main difficulty. The obstacle is overcome by designing the observers based on the informaiton structure and the observer-feedback Stackelberg strategy. Moreover, we have shown that the cost functions under the proposed observer-feedback Stackelberg strategy are asymptotical optimal to the cost functions under the optimal feedback Stackelberg strategy. 00 D. Anderson and B. Moore, "Linear optimal control", *Prentice-Hall, Englewood Cliffs, NJ*, 1971. M. Rami, X. Chen, J. Moore and X. Zhou. "Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ Controls", *IEEE Transactions on Automatic*, 46(3): 428-440, 2001. H. S. Zhang, L. Li, J. J. Xu and M. Y. Fu, "Linear quadratic regulation and stabilization of discrete-time systems with delay and multiplicative noise", *IEEE Transactions on Automatic Control*, 60(10): 2599-2613, 2015. N. W. Bauer, M. Donkers, N. van de Wouw and W. Heemels, "Decentralized observer-based control via networked communication", *Automatica*, 49: 2074-2086, 2013. F. Blaabjerg, R. Teodorescu, M. Liserre and A. V. Timbus, "Overview of control and grid synchronization for distributed power generation systems", *IEEE Transactions on Industrial Electronics*, 53: 1398-1409, 2006. B. Hoogenkamp, S. Farshidi, R. Y. Xin, Z. Shi, P. Chen and Z. M. Zhao, "A decentralized service control framework for decentralized applications in cloud environments", *Service-Oriented and Cloud Computation*, 13226: 65-73, 2022. Q. P. Ha, and H. Trinh, "Observer-based control of multi-agent systems under decentralized information structure", *International Journal of Systems Science*, 35(12): 719-728, 2004. D. Görges, "Distributed adaptive linear quadratic control using distributed reinforcement learning", *IFAC-PapersOnLine*, 52(11): 218-223, 2019. H. Witsenhausen, "A counterexample in stochastic optimum control", *SIAM Journal on Control and Optimization*, 6(1): 131-147, 1968. E. Davison, N. Rau and F. Palmay, "The optimal decentralized control of a power system consisting of a number of interconnected synchronous machines", *International Journal of Control*, 18(6): 1313-1328, 1973. E. Davison, "The robust decentralized control of a general servomechanism problem", *IEEE Transactions on Automatic Control*, AC-21: 14-24, 1976. T. Yoshikawa, "Dynamic programming approach to decentralized stochastic control problem", *IEEE Transactions on Automatic Control*, 20(6): 796-797, 1975. J. Swigart and S. Lall, "An explicit state-space solution for a deffcentralized two-player optimal linear-quadratic regulator", *American Control Conference*, 6385-6390, 2010. X. Liang, J. J Xu, H. X. Wang and H. S. Zhang, "Decentralized output-feedback control with asymmetric one-step delayed information", *IEEE Transactions on Automatic Control*, doi: 10.1109/TAC.2023.3250161, 2023. A. Nayyar, A. Mahajan and T. Teneketzis, "Decentralized stochastic control with partial history sharing: A common information approach", *IEEE Transactions on Automatic Control*, 58(7): 1644-1658, 2013. A. Nayyar, A. Mahajan and T. Teneketzis, "Optimal control strategies in delayed sharing information structures", *IEEE Transactions on Automatic Control*, 56(7): 1606-1620, 2011. X. Liang, Q. Q. Qi, H. S. Zhang and L. H. Xie, "Decentralized control for networked control systems with asymmetric information", *IEEE Transactions on Automatic Control*, 67(4): 2067-2083, 2021. B. C. Wang, X. Yu and H. L. Dong, "Social optima in linear quadratic mean field control withunmodeled dynamics and multiplicative noise" *Aisan Journal of Control*, 23(3): 1572-1582, 2019. T. Başar, "Two-criteria LQG decision problems with one-step delay observation sharing pattern", *Information and Control*, 38: 21-50, 1978. P. George, "On the linear-quadratic-gaussian Nash game with one-step delay observation sharing pattern", *IEEE Transactions on Automatic Control*, 27: 1065-1071, 1982. F. Suzumura and K. Mizukami, "Closed-loop strategy for Stackelberg game problem with incomplete information structures" *IFAC 12th Triennial World Congress*, Australia, 413-418, 1993. M. B. Klompstra, "Nash equilibria in risk-sensitive dynamic games", *IEEE Transactions on Automatic Control*, 45(7): 1397-1401, 2000. M. Pachter, "LQG dynamic games with a control-sharing information pattern", *Dynamic Games and Applications*, 7: 289-322, 2017. Y. Sun, J. J. Xu and H. S. Zhang, "Feedback Nash equilibrium with packet dropouts in networked control systems", *IEEE Transactions on Circuits and Systems II: Express Briefs*, 70(3): 1024-1028, 2022. Z. P. Li, M. Y. Fu, H. S. Zhang and Z. Z. Wu, "Mean field stochastic linear quadratic games for continuum-parameterized multi-agent systems", *Journal of the Franklin Institute*, 355: 5240-5255, 2018. H. Mukaidani, H. Xu and V. Dragan, "Static output-feedback incentive Stackelberg game for discrete-time markov jump linear stochastic systems with external disturbance", *IEEE Control Systems Letters*, 2(4): 701-706, 2016. S. R. Chowdhury, X. Y. Zhou and N. Shroff, "Adaptive control of differentially private linear quadratic systems", *IEEE International Symposium on Information Theory*, 485-490, 2021. J. J. Xu and H. S. Zhang, "Decentralized control of linear systems with private input and measurement information", *arXiv:2305.14921*, 1-6, 2023. D. Castanon and M. Athans, "On stochastic dynamic Stackelberg strategies", *Automatica*, 12: 177-183, 1976. A. Bensoussan, S. Chen and S. P. Sethi, "The maximum principle for global solutions of stochastic Stackelberg differential games", *SIAM Journal of Control and Optimization*, 53(4): 1965-1981, 2015. [^1]: This work was supported by the National Natural Science Foundation of China under Grants 61821004 and the Natural Science Foundation of Shandong Province (ZR2021ZD14, ZR2021JQ24), and Science and Technology Project of Qingdao West Coast New Area (2019-32, 2020-20, 2020-1-4), High-level Talent Team Project of Qingdao West Coast New Area (RCTD-JC-2019-05), Key Research and Development Program of Shandong Province (2020CXGC01208). \*Corresponding author. [^2]: Y. Sun is with the School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China (e-mail: sunyue9603\@163.com). H. Li and H. Zhang are with College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, Shandong 266590, China (e-mail: hszhang\@sdu.edu.cn; lhd200908\@163.com).

arxiv_math

--- abstract: | A *$\frac{1}{k}$-majority $l$-edge-colouring* of a graph $G$ is a colouring of its edges with $l$ colours such that for every colour $i$ and each vertex $v$ of $G$, at most $\frac{1}{k}$'th of the edges incident with $v$ have colour $i$. We conjecture that for every integer $k\geq 2$, each graph with minimum degree $\delta\geq k^2$ is $\frac{1}{k}$-majority $(k+1)$-edge-colourable and observe that such result would be best possible. This was already known to hold for $k=2$. We support the conjecture by proving it with $2k^2$ instead of $k^2$, which confirms the right order of magnitude of the conjectured optimal lower bound for $\delta$. We at the same time improve the previously known bound of order $k^3\log k$, based on a straightforward probabilistic approach. As this technique seems not applicable towards any further improvement, we use a more direct non-random approach. We also strengthen our result, in particular substituting $2k^2$ by $(\frac{7}{4}+o(1))k^2$. Finally, we provide the proof of the conjecture itself for $k\leq 4$ and completely solve an analogous problem for the family of bipartite graphs. address: AGH University, al. A. Mickiewicza 30, 30-059 Krakow, Poland author: - Paweł Pękała - Jakub Przybyło title: On generalised majority edge-colourings of graphs --- majority colouring ,edge majority colouring ,$\frac{1}{k}$-majority edge-colouring # Introduction A *majority colouring* of a digraph $D$ is a colouring of the vertices of $D$ such that for every vertex of $D$ at most half the out-neighbours of $v$ receive the same colour as $v$. This notion was first considered by Kreutzer, Oum, Seymour, van der Zypen and Wood [@digraph1], who in particular proved that every digraph has a majority $4$-colouring, and conjectured that $3$ colours should always suffice. They also posed several other related problems, addressed in a few consecutive papers. In particular, in [@digraph2] Anholcer, Bosek and Grytczuk extended the result above to a list setting. Further, in [@digraph3; @Knox-Samal] the authors studied the problem of $\frac{1}{k}$-majority colourings of digraphs, that is such colourings of the vertices of a digraph that each vertex receives the same colour as at most $\frac{1}{k}$'th of its out-neighbours, which is a natural generalisation, proposed already in [@digraph1]. Girão, Kittipassorn and Popielarz [@digraph3], and independently Knox and Šámal [@Knox-Samal] proved that for each $k\geq 2$, every digraph is $\frac{1}{k}$-majority $2k$-colourable, while there are digraphs requiring no less than $2k-1$ colours. Further results and extensions can be found e.g. in [@Szabo-majority; @digraph2; @MajorityGeneralOur]. It is worth mentioning that optimal results concerning the natural correspondents of the notions above, but in the environment of graphs follow by the argument of Lovász [@Lovasz-majority], printed already in 1966, see also [@MajorityGeneralOur] for further comments and results. A *majority edge-colouring* of a graph $G$ is a colouring of the edges of $G$ such that for each vertex $v$ of $G$, at most half of the edges incident with $v$ have the same colour. More generally, for an integer $k\geq 2$, a *$\frac{1}{k}$-majority edge-colouring* of $G$ is a colouring of its edges such that for every colour $i$ and each vertex $v$ of $G$ at most $\frac{1}{k}$'th of the edges incident with $v$ have colour $i$. One of characteristic features of these notions, introduced recently by Bock, Kalinowski, Pardey, Pilśniak, Rautenbach and Woźniak [@majority23], is that unlike in the case of vertex-colourings, such edge-colourings do not exist for all graphs. In particular, for every $k\geq 2$, graphs with vertices of degree $1$ do not admit a $\frac{1}{k}$-majority edge-colouring with any number of colours. In [@majority23] it was however proven that every graph $G$ of minimum degree $\delta\geq 2$ has a majority $4$-edge-colouring. On the other hand, the minimum degree of a graph may have significant influence on the number of colours sufficient to provide such colourings, and examining this problem seems to be the primal issue in this area. The main result of [@majority23] solves this problem for $k=2$. **Theorem 1** ([@majority23]). *Every graph $G$ of minimum degree $\delta\geq 4$ has a majority $3$-edge-colouring.* This result is twofold best possible. Firstly, $4$ cannot be decreased, as exemplify e.g. cubic graphs with chromatic index $4$. Secondly, $2$ colours are not sufficient e.g. for any graph containing an odd degree vertex. The main motivation of our research is thus the quest for best possible extension of Theorem [Theorem 1](#majority23-res2){reference-type="ref" reference="majority23-res2"} towards all $k\geq 3$. Note first that for any fixed $k\geq 2$, no minimum degree constraint can guarantee the existence of a $\frac{1}{k}$-majority edge-colouring with at most $k$ colours -- it is enough to consider graphs containing vertices of (arbitrarily large) degrees not divisible by $k$. We thus must admit (at least) $k+1$ colours, and in fact the authors of [@majority23] showed that this number of colours is sufficient (and hence, optimal) within our quest. **Theorem 2** ([@majority23]). *For every integer $k\geq 2$, there exists $\delta_k$ such that each graph $G$ of minimum degree $\delta\geq \delta_k$ has a $\frac{1}{k}$-majority $(k+1)$-edge-colouring.* For $k=2$ this follows by Theorem [Theorem 1](#majority23-res2){reference-type="ref" reference="majority23-res2"}, while in the remaining cases it was proven by a fairly standard application of the Lovász Local Lemma, which admitted to get the result above with $\delta_k=\Omega(k^3 \log k)$. We believe that much smaller values of $\delta_k$ should allow obtaining the same result. Our main objective concerns finding the optimal value of $\delta_k$ and can be formulated as follows. **Problem 3**. *For every integer $k\geq 2$, find the least value $\delta_k^{\rm opt}$ such that each graph $G$ of minimum degree $\delta\geq \delta_k^{\rm opt}$ has a $\frac{1}{k}$-majority $(k+1)$-edge-colouring.* It seems one needs to use a non-probabilistic argument in order to completely solve this problem, we discuss this issue at the end of the paper. In the next section we present the main tool we shall use within our approach, and show how it allows to solve the problem in the environment of bipartite graphs. In Section [3](#SectionGeneralGraphs){reference-type="ref" reference="SectionGeneralGraphs"} we shall in turn provide order-wise tight estimations for $\delta_k^{\rm opt}$ in the general case and formulate our main conjecture. In the following section we also confirm that the conjecture holds for $k\leq 4$. The last section is devoted to a short discussion concerning our results and further perspectives. # Bipartite graphs Let us begin with a simple observation implying the lower bound for $\delta_k^{\rm opt}$, also within the family of bipartite graphs. **Observation 4**. *For every $k\geq 2$ there exist bipartite graphs $G$ with $\delta(G)\geq k^2-k-1$ which are not $\frac{1}{k}$-majority $(k+1)$-edge-colourable.* *Proof.* Let $G$ be a graph containing a vertex $v$ of degree $k^2-k-1$ and suppose $G$ has a $\frac{1}{k}$-majority $(k+1)$-edge-colouring. Then at most $$\left\lfloor\frac{k^2-k-1}{k}\right\rfloor=k-2$$ edges incident with $v$ may have the same colour, and hence at most $$(k+1)(k-2) = k^2-k-2 < d(v)$$ edges incident with $v$ can be coloured, a contradiction. Therefore, in particular the complete bipartite graph $K_{k^2-k,k^2-k}$ with a single edge removed is an example of a bipartite graph which is not $\frac{1}{k}$-majority $(k+1)$-edge-colourable and has minimum degree $k^2-k-1$. ◻ We thus must require that $\delta(G)\geq k(k-1)$ in order to have a chance to show that such assumption guarantees that $G$ is $\frac{1}{k}$-majority edge-colourable with $k+1$ colours. In [@majority23] it was actually already proven that $k+2$ colours are sufficient in such a case. We shall show that in fact in the case of bipartite graphs, $k+1$ colours always suffice, which, in view of Observation [Observation 4](#ObservationLowerBipartite){reference-type="ref" reference="ObservationLowerBipartite"}, yields an optimal result. **Theorem 5**. *For every integer $k \geq 2$, if a bipartite graph $G$ has minimum degree at least $k(k-1)$, then $G$ has a $\frac{1}{k}$-majority $(k+1)$-edge-colouring.* In order to prove Theorem [Theorem 5](#bipartiteTheorem){reference-type="ref" reference="bipartiteTheorem"} we shall make use of Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"} below. This was essentially proven by Alon and Wei [@alon]. We present a slight, yet very useful refinement of their result. We also include its full proof for the sake of completeness. We say a cycle is *odd* if it has odd length. Moreover, cycles $C_1,\ldots,C_t$ in a graph $G$ are called *independent* if for any $i\neq j$ there is no vertex of $C_i$ adjacent to a vertex of $C_j$ in $G$. (Note that such cycles are in particular pairwise distinct.) **Lemma 6**. *Let $G = (V,E)$ be a graph, and let $z:E\rightarrow [0,1]$ be a weight function assigning to each edge $e\in E$ a real weight $z(e)$ in $[0,1]$. Then there is a function $x: E\rightarrow \{0,1\}$ assigning to each edge an integer value in $\{0,1\}$ such that* - *$\sum\limits_{e\ni v} z(e) -1 < \sum\limits_{e\ni v} x(e) \leq \sum\limits_{e\ni v} z(e) +1$ for every $v\in V$;* - *if $\sum\limits_{e\ni u} x(e) < \sum\limits_{e\ni u} z(e)$ and $\sum\limits_{e\ni v} x(e) < \sum\limits_{e\ni v} z(e)$ for some $uv\in E$, then $x(uv)=1$;* - *each vertex $v$ with $\sum\limits_{e\ni v} x(e)=\sum\limits_{e\ni v} z(e)+1$ belongs to an odd cycle $C_v$ with $\sum\limits_{e\ni u} z(e)\in\mathbb{Z}$ for every vertex $u$ of $C_v$, and moreover all such cycles are independent in $G$.* *Proof.* If $z(e) \in \{0,1\}$ for any edge $e$ of $G$, then fix $x(e) = z(e)$. Let $G'$ be a subgraph of $G$ obtained by removing all edges of $G$ for which $z(e)$ is an integer. Consider an incidence matrix $M$ of $G'$. For every edge $e\in E(G')$ let $y_e$ be its corresponding column in $M$. Note that if $G'$ contains a closed walk of even length whose every two consecutive edges are distinct from each other, then the columns of $M$ are linearly dependent over reals, i.e. there exist real numbers $\alpha_e$, $e\in E(G')$, such that $$\sum_{e\in E(G')} \alpha_e y_e = \overline{0}$$ and at least one $\alpha_e$ is nonzero. Note that for any nonzero real number $c$, if we modify all of the values $x(e)$ by $c\alpha_e$, then the sum $\sum_{e\ni v} x(e)$ remains the same for all vertices $v$, but the value of $x(e)$ shall change for at least one edge $e$. Therefore we can choose the value of the coefficient $c$ so that all modified values of $x(e)$ remain in $[0,1]$, but at least one of them gets an integer. Remove all integer valued edges from the graph $G'$ and continue with the same procedure until $G'$ does not contain an even closed walk with no repeated consecutive edges. Note every component of the resulting graph $G'$ contains at most one cycle, which has to be odd. Observe that at this point for all vertices $v$ of $G$ we have $$\sum_{e\ni v} x(e) = \sum_{e\ni v} z(e)$$ and for all edges $e$ of $G'$ the value of $x(e)$ is in the open interval $(0,1)$. Hence, after modifying each of these values to an integer in $\{0,1\}$, for all vertices $v$ with $d_{G'}(v)\leq 1$ we shall have $$\sum\limits_{e\ni v} z(e) -1 < \sum\limits_{e\ni v} x(e) < \sum\limits_{e\ni v} z(e) +1 \text{.}$$ Thus, in the following step we shall focus on modifying the values of $x(e)$ for the edges $e$ of $G'$ in such a way that $$\sum_{e\ni v} x(e) = \sum_{e\ni v} z(e)$$ for all vertices $v$ with $d_{G'}(v)\geq 2$. Suppose $G'$ has a component containing both a vertex of degree $1$ and a vertex of degree at least $2$. Consider the system of linear equations $$\sum_{e\ni v} x(e) = \sum_{e\ni v} z(e)$$ for all vertices $v$ of the component which have degrees at least $2$, where the variables are $x(e)$ for all edges $e$ of this component. The number of variables in this system is greater than the number of equations, hence its solution set is infinite. We choose a solution such that all values of $x(e)$ remain in the interval $[0,1]$ and at least one of them is an integer. We then remove integer valued edges from $G'$. We repeat this procedure until all components of $G'$ are odd cycles or isolated edges. For all isolated edges $e$ of $G'$ we can then simply set $x(e)=1$. The remaining edges $e$ with non-integer values of $x(e)$ induce disjoint odd cycles in $G$. By previous arguments all vertices $v$ of such cycles satisfy $$\sum_{e\ni v} x(e) = \sum_{e\ni v} z(e) \text{.}$$ We call an odd cycle *bad* if $x(e)=1/2$ for all its edges $e$. Note $\sum_{e\ni v} z(e)\in \mathbb{Z}$ for every vertex $v$ of any bad cycle in $G'$. We shall show that we may modify $G'$ so that all its bad cycles are independent in $G$. Suppose there are bad (odd) cycles $C_1,C_2$ in $G'$ joined by an edge, say $e_0$ in $G$. Let $H$ be the subgraph of $G$ induced by $e_0$ and the edges of $C_1,C_2$. Then it is straightforward to notice that there exist $\alpha_e\in\{-1,1\}$, $e\in C_1\cup C_2$, such that $$\alpha_{e_0}y_{e_0} + \sum_{e\in C_1\cup C_2}\alpha_ey_e = \overline{0}$$ where $\alpha_{e_0}=2$. (It suffices to alternately set $\alpha_e$ to $-1$ and $1$ along both of the cycles, starting from a vertex of $e_0$.) Since $x(e)=1/2$ for $e\in C_1\cup C_2$ and $x(e_0)\in\{0,1\}$, by adding $\alpha_e/2$ or $-\alpha_e/2$ (depending on $x(e_0)$) to all $x(e)$, $e\in E(H)$ we shall thus not change the sum $\sum_{e\ni v} x(e)$ at any vertex $v$ while all edges of $H$ shall get integer valued. We shall thus remove all these edges from $G'$. We repeat this procedure until all bad cycles of $G'$ are independent in $G$. For each cycle $C$ of $G'$ we then proceed as follows. If $C$ is bad, we denote any of its vertices as $v$. Then for all edges $e$ of $C$ we round the value of $x(e)$ to the nearest integer with the additional restriction that if for the two edges $e',e''$ incident to any given vertex $u$ in $C$ we had $x(e') = 1/2 = x(e'')$, then one of these values must be rounded down to $0$ and the other one rounded up to $1$ if $u\neq v$, while we round both values up to $1$ for $u=v$. As a result we obtain a function $x$ such that $\sum\limits_{e\ni v} x(e) = \sum\limits_{e\ni v} z(e) +1$ and $$\sum\limits_{e\ni u} z(e) -1 < \sum\limits_{e\ni u} x(e) < \sum\limits_{e\ni u} z(e) +1$$ for all the remaining vertices $u$ of $C$ (other than $v$). Thus, since bad cycles of $G'$ were independent in $G$, we obtain a function $x: E\rightarrow \{0,1\}$ satisfying (i) and (iii). Finally, we shall show that we can further modify the function $x$ so that it also satisfies (ii). Assume (ii) is not satisfied and there exists an edge $uv \in E$ with $x(uv)=0$ such that $\sum\limits_{e\ni u} x(e) < \sum\limits_{e\ni u} z(e)$ and $\sum\limits_{e\ni v} x(e) < \sum\limits_{e\ni v} z(e)$. If we modify the value of $x(uv)$ to $1$, then $\sum\limits_{e\ni u} x(e) < \sum\limits_{e\ni u} z(e) +1$ and $\sum\limits_{e\ni v} x(e) < \sum\limits_{e\ni v} z(e) +1$, hence (i) and (iii) are still satisfied, but there are less edges that contradict (ii). Therefore, we can construct in this manner a function $x: E\rightarrow \{0,1\}$ satisfying all three conditions (i) -- (iii). ◻ *Proof of Theorem [Theorem 5](#bipartiteTheorem){reference-type="ref" reference="bipartiteTheorem"}.* Let $G=(V,E)$ be a bipartite graph with $\delta(G)\geq k(k-1)$, $k\geq 2$. Set $\overline{G}_{k+1} = G$. For $i=k+1,k,\ldots,2$ let further $G_i$ be a subgraph of $\overline{G}_{i}$ obtained via applying to it Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"} with a constant weight function $z_i(e)=\frac{1}{i}$ and setting $E(G_i)=\{e\in E(\overline{G}_{i}): x_i(e)=1\}$ where $x_i:E(\overline{G}_{i})\to\{0,1\}$ is a function resulting from the lemma; let also $\overline{G}_{i-1} = (V(G), E(\overline{G}_i) \setminus E(G_i))$. Finally, set $G_1 = \overline{G}_1$. We shall prove that for every $i \in \{1,\dotsc,k+1\}$ and each vertex $v\in V$ $$\label{BipartiteDegreIneq} d_{G_i}(v) \leq \left\lfloor \frac{d_G(v)}{k} \right\rfloor,$$ and thus the edges of $G_1,\ldots,G_{k+1}$ partition $E$ to $k+1$ colour classes inducing a $\frac{1}{k}$-majority $(k+1)$-edge-colouring of $G$. In other words, the colouring $c:E\to\{1,\ldots,k+1\}$ can be defined by setting $c(e)=i$ iff $e\in E(G_i)$. Let $v$ be an arbitrary vertex of $G$ and let $$d_G(v) = (k+1) l + j$$ where $j\in \{0,\dotsc,k\}$. Since $d_G(v) \geq \delta(G) \geq k(k-1)$ we have that $l\geq k-1$ or $l=k-2$ and $j\geq 2$. Hence $$\label{l+1-Estimation} \left\lfloor \frac{d_G(v)}{k} \right\rfloor = \left\lfloor \frac{(k+1) l + j}{k} \right\rfloor = l+ \left\lfloor \frac{l + j}{k} \right\rfloor \geq l+1$$ unless $l=k-1$ and $j=0$. Since $G$ is bipartite, none of $G_i$ contains odd cycles. Thus, Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"} (iii), exploited to construct each $G_i$, guarantees the following to hold. **Claim 1**. *If $d_{\overline{G}_{t}}(v) = td$ for some $t\in\{1,\ldots,k+1\}$ and $d\in\mathbb{Z}$, hence $\sum\limits_{e\ni v} z_t(e)=d$, then $d_{G_t}(v) = d$.* This almost immediately yields ([\[BipartiteDegreIneq\]](#BipartiteDegreIneq){reference-type="ref" reference="BipartiteDegreIneq"}) in the case when $d_G(v)$ is divisible by $k+1$. It is sufficient to apply $k$ times Claim [Claim 1](#Claim1-bipartite){reference-type="ref" reference="Claim1-bipartite"} to obtain the following. **Claim 2**. *If $j=0$, i.e. $d_G(v) = (k+1)l$, then $d_{G_i}(v) = l$ for all $i$.* Then however, we have that for every $i \in \{1,\dotsc,k+1\}$, $$d_{G_i}(v) = l = \left\lfloor \frac{kl}{k} \right\rfloor \leq \left\lfloor \frac{d_G(v)}{k} \right\rfloor \text{,}$$ and thus ([\[BipartiteDegreIneq\]](#BipartiteDegreIneq){reference-type="ref" reference="BipartiteDegreIneq"}) follows. It remains to prove ([\[BipartiteDegreIneq\]](#BipartiteDegreIneq){reference-type="ref" reference="BipartiteDegreIneq"}) in the case when $j\neq 0$. By ([\[l+1-Estimation\]](#l+1-Estimation){reference-type="ref" reference="l+1-Estimation"}) it is sufficient to show that $d_{G_i}(v)\leq l+1$ for every $i$. This is implied by $k$ times repeated application of the following claim. **Claim 3**. *If $d_G(v) = (k+1)l + j$ and $j\neq 0$, then for every $t \in \{2,\dotsc,k+1\}$, if $d_{\overline{G}_t}(v) \in \{ tl, tl+1, \dotsc, t(l+1) \}$, then $d_{G_t}(v) \in \{l,l+1\}$ and $d_{\overline{G}_{t-1}}(v) \in \{(t-1)l,(t-1)l+1,\dotsc,(t-1)(l+1)\}$.* *Proof.* Note that analogously as above, if $d_{\overline{G}_t}(v) = tl$ (respectively, $t(l+1)$), then by Claim [Claim 1](#Claim1-bipartite){reference-type="ref" reference="Claim1-bipartite"}, $d_{G_t}(v) = l$ (respectively, $l+1$) and $d_{\overline{G}_{t-1}}(v) = (t-1)l$ (respectively, $(t-1)(l+1)$). In the remaining cases, $d_{G_t}(v) \in \{l,l+1\}$ by Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"} (i), and thus $d_{\overline{G}_{t-1}}(v) \in \{(t-1)l,(t-1)l+1,\dotsc,(t-1)(l+1)\}$. ◻  ◻ # General graphs {#SectionGeneralGraphs} The main obstacle in obtaining a similar result as in Theorem [Theorem 5](#bipartiteTheorem){reference-type="ref" reference="bipartiteTheorem"} for the general case, not only for bipartite graphs, is that we cannot show Claim [Claim 1](#Claim1-bipartite){reference-type="ref" reference="Claim1-bipartite"} to hold any more if a graph has odd cycles, as the proof of this fact relied on Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"} (iii). Actually, this inconvenience shall have much further reaching consequences than we initially suspected, and shall (most likely) disallow us obtaining sharp results in most of the general cases. Before we discuss our upper bounds, let us however first demonstrate that it is after all not that surprising we could not solve this apparent sole obstacle on the way towards extending Theorem [Theorem 5](#bipartiteTheorem){reference-type="ref" reference="bipartiteTheorem"} to all graphs, as the upper bound in it does not hold any more in general. **Observation 7**. *For every $k\geq 2$ there exists a graph $G$ with $\delta(G)\geq k^2-1$ which is not $\frac{1}{k}$-majority $(k+1)$-edge-colourable.* *Proof.* For every fixed $k\geq 2$ we construct a graph $G$ as follows. We first take the complete graph on $k^2+1$ vertices and remove the edges of any fixed Hamilton cycle from it. Then we add a new vertex to the constructed graph and connect it with single edges with all the remaining vertices. Note that the obtained graph $G$ has $k^2+1$ vertices of degree $k^2-1$ and a single vertex of degree $k^2+1$. Suppose there is a $\frac{1}{k}$-majority $(k+1)$-edge-colouring of $G$. Since $k^2-1=(k+1)(k-1)$ and at most $$\left\lfloor\frac{k^2-1}{k}\right\rfloor = k-1$$ edges incident to every vertex $v$ of degree $k^2-1$ can be coloured with the same colour, then for every such vertex $v$ and for each of the $k+1$ colours, exactly $k-1$ edges incident to such $v$ are coloured with this colour. Analogously, for the only vertex $u$ of $G$ with degree $k^2+1 = (k+1)(k-1)+2$, at most $$\left\lfloor\frac{k^2+1}{k}\right\rfloor = k$$ edges incident to the vertex $u$ can be coloured with the same colour, and hence exactly $k$ edges incident to the vertex $u$ must be coloured with some colour, say $\alpha$. Consider the subgraph of $G$ induced by the edges of $G$ coloured with $\alpha$. The sum of degrees in this subgraph equals $$(k^2+1)(k-1)+k,$$ which is always an odd number, a contradiction. ◻ Observation [Observation 7](#ExampleGeneral){reference-type="ref" reference="ExampleGeneral"} thus implies that $\delta_k^{\rm opt}\geq k^2$ for every $k\geq 2$. We conjecture that in fact $\delta_k^{\rm opt} = k^2$ for each $k\geq 2$, which holds for $k=2$ by Theorem [Theorem 1](#majority23-res2){reference-type="ref" reference="majority23-res2"}. **Conjecture 8**. *For every integer $k\geq 2$, if a graph $G$ has minimum degree $\delta\geq k^2$, then $G$ is $\frac{1}{k}$-majority $(k+1)$-edge-colourable.* By means of Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"} we shall now show that $\delta_k^{\rm opt}$ is at most twice the conjectured value. The proof of this fact shall also be an indispensable ingredient of the further slight improvement included in Theorem [Theorem 11](#thm2-GenMajEdg){reference-type="ref" reference="thm2-GenMajEdg"}. **Theorem 9**. *For every integer $k\geq 2$, if a graph $G$ has minimum degree $\delta\geq 2k^2$, then $G$ is $\frac{1}{k}$-majority $(k+1)$-edge-colourable.* *Proof.* Let $G=(V,E)$ be a graph with minimum degree $\delta\geq 2k^2$, for some fixed integer $k\geq 2$. Analogously as within the proof of Theorem [Theorem 5](#bipartiteTheorem){reference-type="ref" reference="bipartiteTheorem"}, we shall use Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"} in order to choose $k$ consecutive subgraphs $G_1,\dotsc,G_k$ of $G$ and colour the edges of each $G_i$ with colour $i$. The remaining edges shall be coloured with colour $k+1$. Set $\overline{G}_0 = G$. For $i = 1,\dotsc,k$ let $G_i$ be a subgraph of $\overline{G}_{i-1}$ induced by the edges $e$ with $x_i(e)=1$ where $x_i$ is a function resulting from applying Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"} to $\overline{G}_{i-1}$ with a constant weight function $z_i:E(\overline{G}_{i-1})\rightarrow \{\alpha_i\}$, where the value of $\alpha_i \in [0,1]$ shall be specified later; we also set $\overline{G}_i = (V(G), E(\overline{G}_{i-1}) \setminus E(G_i))$. Let $G_{k+1} = \overline{G}_k$. We shall show we may choose $\alpha_i$ for $i \in \{ 1, \dotsc, k\}$ so that for every vertex $v$ of $G$ and each $j \in \{1,\dotsc,k+1\}$, $$\label{dGjineqdG} d_{G_j}(v) \leq \frac{d_G(v)}{k}.$$ By Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"} (i), for every $v\in V$: $$\label{dG1inequalities} \alpha_1 d_G(v) -1 < \sum\limits_{e\ni v} x_1(e) = d_{G_1}(v) \leq \alpha_1 d_G(v) +1 \text{.}$$ In order to satisfy ([\[dGjineqdG\]](#dGjineqdG){reference-type="ref" reference="dGjineqdG"}) it is thus necessary and sufficient to choose $\alpha_1$ so that $\alpha_1 d_G(v) +1 \leq \frac{d_G(v)}{k}$ for all vertices $v$ of $G$, that is $\alpha_1 \leq \frac{1}{k} - \frac{1}{d_G(v)}$. Since the function $f(n) = \frac{1}{k}-\frac{1}{n}$ is increasing for $n>0$, we shall achieve our goal by setting $$\label{Alpha1setting} \alpha_1 = \frac{\frac{\delta}{k}-1}{\delta} \text{.}$$ Consequently, a vertex $v$ of degree $\delta$, which in some sense are the most restrictive ones, may in theory end up with $d_{G_1}(v)$ arbitrarily close to $\frac{\delta}{k}-2$, cf. ([\[dG1inequalities\]](#dG1inequalities){reference-type="ref" reference="dG1inequalities"}) and ([\[Alpha1setting\]](#Alpha1setting){reference-type="ref" reference="Alpha1setting"}). Hence for $i \in \{ 1, \dotsc, k\}$ we in general set: $$\label{Alphaisetting} \alpha_i = \frac{\frac{\delta}{k} -1}{\delta-(i-1)(\frac{\delta}{k}-2)} \text{.}$$ We shall now formally prove that such choices of $\alpha_i$ guarantee ([\[dGjineqdG\]](#dGjineqdG){reference-type="ref" reference="dGjineqdG"}) to hold for all $j$ and $v$. Let $v$ be an arbitrarily chosen vertex of $G$. There exists $\beta \geq 1$ such that $d_G(v) = \beta \delta$. We shall precisely show that for every $i \in \{1,\dotsc,k\}$, $$\label{bound1} d_{\overline{G}_i}(v) \leq \beta \left( \delta - i \left( \frac{\delta}{k} - 2 \right) \right)$$ and $$\label{bound2} d_{G_i}(v) \leq \frac{\beta \delta}{k} = \frac{d_G(v)}{k} \text{.}$$ We proceed by induction with respect to $i$. Since $$\alpha_1 d_G(v) = \frac{\frac{\delta}{k}-1}{\delta} \cdot \beta \delta = \beta \left( \frac{\delta}{k}-1 \right)$$ and $\beta \geq 1$, by ([\[dG1inequalities\]](#dG1inequalities){reference-type="ref" reference="dG1inequalities"}) we obtain $$\beta \left( \frac{\delta}{k}-2 \right) < d_{G_1}(v) \leq \beta \frac{\delta}{k} \text{,}$$ so [\[bound1\]](#bound1){reference-type="eqref" reference="bound1"} and [\[bound2\]](#bound2){reference-type="eqref" reference="bound2"} hold for $i=1$, which yields the base case of induction. For an induction step, assume that $d_{\overline{G}_{i-1}} (v) = D \leq \beta \left( \delta - (i-1) \left( \frac{\delta}{k} - 2 \right) \right)$ for some $i\leq k$. Note that by Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"} (i), $$d_{G_i}(v) > \alpha_i D -1 \text{,}$$ and thus, by ([\[Alphaisetting\]](#Alphaisetting){reference-type="ref" reference="Alphaisetting"}): $$\begin{aligned} d_{\overline{G}_i}(v) &< D - (\alpha_i D -1) = (1 - \alpha_i) D + 1 \\ &\leq (1 - \alpha_i) \beta \left( \delta - (i-1) \left( \frac{\delta}{k} - 2 \right) \right) + 1 \\ &= \beta \left( \delta - (i-1) \left( \frac{\delta}{k} - 2 \right) - \left(\frac{\delta}{k} -1 \right) \right) +1\\ &\leq \beta \left(\delta - (i-1) \left( \frac{\delta}{k} - 2 \right) - \left( \frac{\delta}{k} - 1 \right) \right) + \beta \\ &= \beta \left(\delta - i \left( \frac{\delta}{k} - 2 \right) \right) \text{,} \end{aligned}$$ cf. [\[bound1\]](#bound1){reference-type="eqref" reference="bound1"}. On the other hand, by ([\[dG1inequalities\]](#dG1inequalities){reference-type="ref" reference="dG1inequalities"}) and ([\[Alphaisetting\]](#Alphaisetting){reference-type="ref" reference="Alphaisetting"}): $$\begin{aligned} d_{G_i}(v) &\leq \alpha_i D + 1 \leq \alpha_i \beta \left( \delta - (i-1) \left( \frac{\delta}{k} - 2 \right) \right) +1 \\ &= \beta \left( \frac{\delta}{k} - 1 \right) +1 \leq \beta \left( \frac{\delta}{k} - 1 \right) + \beta = \frac{\beta \delta}{k} = \frac{d_G(v)}{k} \text{,} \end{aligned}$$ hence [\[bound2\]](#bound2){reference-type="eqref" reference="bound2"} and [\[bound1\]](#bound1){reference-type="eqref" reference="bound1"} hold (where [\[bound2\]](#bound2){reference-type="eqref" reference="bound2"} implies ([\[dGjineqdG\]](#dGjineqdG){reference-type="ref" reference="dGjineqdG"})). Finally, observe that by [\[bound1\]](#bound1){reference-type="eqref" reference="bound1"} we have: $$d_{G_{k+1}}(v) = d_{\overline{G}_{k}}(v) \leq \beta \left( \delta - k \left( \frac{\delta}{k} - 2 \right) \right) = 2 \beta k \text{.}$$ Since $\delta \geq 2k^2$, we obtain $d_{G_{k+1}}(v) \leq \frac{\beta\delta}{k} = \frac{d_G(v)}{k}$, which concludes the proof of Theorem [Theorem 9](#thm1-GenMajEdg){reference-type="ref" reference="thm1-GenMajEdg"}. ◻ Note that the bound on the minimum degree of $G$ was only required to bound $d_{G_{k+1}}$ in the proof of Theorem [Theorem 9](#thm1-GenMajEdg){reference-type="ref" reference="thm1-GenMajEdg"}. This remark allows us to use almost entire reasoning above within the proof of Theorem [Theorem 11](#thm2-GenMajEdg){reference-type="ref" reference="thm2-GenMajEdg"} below, which improves the general lower bound for $\delta_k^{\rm opt}$. This refinement exploits the following straightforward and direct consequence of Euler's Theorem (through adding a single auxiliary vertex to a graph, if necessary). Details of its proof can be found e.g. in [@majority23; @Przybylo22standard] and most likely in many other papers. **Observation 10**. *Let $G$ be a connected graph.* 1. *If $G$ has an even number of edges or $G$ contains vertices of odd degree, then $G$ has a $2$-edge-colouring such that for every vertex $u$ of $G$, at most $\left\lceil \frac{d_G(u)}{2} \right\rceil$ of the edges incident with $u$ have the same colour.* 2. *If $G$ has an odd number of edges, all vertices of $G$ have even degree and $u_G$ is any vertex of $G$, then $G$ has a $2$-edge-colouring such that for every vertex $u$ of $G$ distinct from $u_G$, exactly $\frac{d_G(u)}{2}$ of the edges incident with $u$ have the same colour, and at most $\frac{d_G(u_G)}{2}+1$ of the edges incident with $u_G$ have the same colour.* In what follows, a *bad vertex* shall mean a vertex of $G$ which was chosen as the vertex $u_G$ while applying Observation [Observation 10](#euler){reference-type="ref" reference="euler"} above, that is the vertex with exactly $\frac{d_G(v)}{2}+1$ incident edges coloured the same in one of the two colours. **Theorem 11**. *Let $k = 2^n + m - 1\geq 2$ where $n$ is a positive integer and $m$ is a nonnegative integer less than $2^n$. If $G$ is a graph with minimum degree $\delta\geq \frac{3}{2}k^2+\frac{1}{2}km+\frac{1}{2}k$, then $G$ has a $\frac{1}{k}$-majority $(k+1)$-edge-colouring.* *Proof.* We start by partially colouring the graph $G=(V,E)$ with $m$ colours, choosing $G_1,\ldots,G_m$, corresponding to colours $1,\ldots,m$, the same way as in the proof of Theorem [Theorem 9](#thm1-GenMajEdg){reference-type="ref" reference="thm1-GenMajEdg"}. By [\[bound2\]](#bound2){reference-type="eqref" reference="bound2"} these colours satisfy the *majority rule*, that is for every vertex $u$, at most $d_G(u)/k$ edges incident with $u$ are coloured with any of the colours in $\{1,\ldots,m\}$. Let $H$ be a subgraph of $G$ induced by the uncoloured edges. Notice that $H=\overline{G}_m$ (using the notation from the proof of Theorem [Theorem 9](#thm1-GenMajEdg){reference-type="ref" reference="thm1-GenMajEdg"}), and thus by [\[bound1\]](#bound1){reference-type="eqref" reference="bound1"}, if $v$ is a vertex of $G$ such that $d_G(v)=\beta \delta$, then $$\label{dHvInequ} d_H(v) \leq \beta \left( \delta - m \left( \frac{\delta}{k} - 2 \right) \right) \text{.}$$ We shall colour the edges of $H$ with new $2^n$ colours, namely the elements of the set $\{0,1\}^n$, hence we shall be colouring these edges with binary vectors of length $n$. For any vector $w\in\{0,1\}^n$ and $0\leq j\leq n$ we denote by $[w]_j$ the *prefix* of length $j$ of $w$, that is the vector in $\{0,1\}^j$ formed of $j$ first consecutive coordinates of $w$ (where $[w]_j=\emptyset$ for $j=0$). Initially (in step $0$) we associate the vector $c_e=(0,\ldots,0)$ to every edge $e$ of $H$. The vectors $c_e$, $e\in E(H)$, shall be modified one coordinate after another in $n$ steps. In certain situations we shall however be finally fixing all the remaining coordinates of some of these vectors at once -- the corresponding edges shall be called *determined*. In what follows, $c_e$ shall always refer to the current value of the colour (vector) associated with an edge $e$. Suppose for a given $i\in\{1,\ldots,n\}$ we have completed step $i-1$ of our construction, hence each $c_e$ has the first $i-1$ coordinates finally fixed (or all, for selected, determined, $e\in E(H)$), and we are about to perform step $i$. We proceed as follows. For each possible prefix $p\in\{0,1\}^{i-1}$ we denote by $H_p$ the subgraph induced in $H$ by all not yet determined edges $e$ with $[c_e]_{i-1}=p$ (after step $i-1$). In each such $H_p$ we consider all components one after another. Let $H'$ be such a component for any fixed $p\in\{0,1\}^{i-1}$. Let us give an advance notice to the fact that at most one vertex of $H'$ shall be chosen to be so-called *special for $H_{p'}$*, where $p'$ is the extensions of $p$ with $1$ added to its end (i.e. $p'\in\{0,1\}^i$, $[p']_{i-1}=p$ and $p'(i)=1$), according to the rule specified below. 1. If for each vertex $v$ of $H'$ there exists a prefix $q$ of $p$ (possibly $q=p$) such that $v$ is special for $H_q$, then for every edge $e$ of $H'$ we fix as $0$ all the remaining (starting from the $i$'th one) coordinates of $c_e$, hence all edges of $H'$ become determined. 2. Otherwise, we use Observation [Observation 10](#euler){reference-type="ref" reference="euler"} to temporarily colour the edges of $H'$ blue and red. For each edge $e$ of $H'$ we fix the $i$'th coordinate of $c_e$ as $0$ if $e$ is blue, and $1$ otherwise. Moreover, if we are forced to create a bad vertex $u_{H'}$ (with $(d_{H'}(u_{H'})/2)+1$ incident edges of the same colour), then we choose it so that it was not special for $H_q$ for any prefix $q$ of $p$ and assign colours blue and red so that $u_{H'}$ is incident with exactly $(d_{H'}(u_{H'})/2)+1$ red edges; we then also choose $u_{H'}$ to be special for $H_{p'}$ where $p'$ is the extensions of $p$ with $1$ added to its end. After going through all $n$ steps described above, we complete a $(k+1)$-edge-colouring of the graph $G$. It remains to show that all new colours satisfy the $\frac{1}{k}$-majority rule. Consider a vertex $v\in V$ and any fixed colour $\alpha\in\{0,1\}^n$. Let us denote by $G_\alpha$ the subgraph induced in $G$ (in fact in $H$) by all the edges coloured with $\alpha$. Suppose first that $v$ is incident with some edge $e\in E(H)$ which was coloured (determined) with a colour $\alpha$ according to Rule (a) above, i.e. at certain iteration $i$, the edge $e$ belonged to a component $H'$ of some $H_p$ with all vertices being special for some $H_q$ where $q$ is a prefix of $H_p$. Note however that by our construction, $H'$ must have been a (connected) subgraph of every such $H_q$, and thus for each such $H_q$ at most one vertex of $H'$ might have been chosen to be special for $H_q$. Consequently, as $p$ has no more than $n$ distinct prefixes, $H'$ must have contained at most $n$ vertices. Hence for each its vertex, in particular $v$, $$d_{G_\alpha}(v) < n \leq k \leq\frac{d_G(v)}{k}.$$ Assume in turn that every edge $e\in E(G_\alpha)$ incident with $v$ was coloured by means of Rule (b) exclusively. This rule was thus utilised $n$ times in order to settle all edges incident with $v$ coloured $\alpha$, each time via application of Observation [Observation 10](#euler){reference-type="ref" reference="euler"} to a component $H'$ (containing $v$) of some $H_p$, where $p$ is a prefix of $\alpha$. Suppose $v$ had degree $d$ in such $H'$, say in iteration $i$. If $v$ was chosen to be special for $H_p$, which could happen only once during $n$ steps of our construction (for prefixes $p$ of the fixed $\alpha$), then at most $\frac{d}{2}+1$ edges incident with $v$ got their colours' prefixes fixed as $p$ after step $i$; otherwise the number of such edges is bounded above by $\frac{d}{2} + \frac{1}{2}$, cf. Observation [Observation 10](#euler){reference-type="ref" reference="euler"}. Only such edges retained the chance to belong to $G_\alpha$. In order to estimate the maximum number of edges incident with $v$ which eventually could be coloured $\alpha$ let us thus consider two following functions. Let $f(d) = \frac{d}{2} + \frac{1}{2}$ and $g(d) = \frac{d}{2} + 1$. By the observations above, $d_{G_\alpha}(v)$ is bounded above by the maximum of $f^n(d)$ and $f^i(g(f^j(d)))$ for all natural numbers $i$ and $j$ such that $i+j = n-1$ where $d=d_H(v)$. Since $f(d)<g(d)$ for all $d$, the value of $f^i(g(f^j(d)))$ is greater that $f^n(d)$ for all $i$ and $j$ satisfying $i+j=n-1$. We shall prove the following upper bound. **Claim 4**. *The inequality $f^i(g(f^j(d))) \leq \frac{d-1}{2^n} + \frac{3}{2}$ holds for all $d$ and all natural numbers $i$ and $j$ such that $i+j=n-1$.* *Proof of Claim [Claim 4](#obs6){reference-type="ref" reference="obs6"}.* We begin by proving that $$f^i(d) = \frac{d-1}{2^i} + 1$$ holds for any nonnegative integer $i$. We proceed by induction with respect to $i$. Clearly $f^0(d) = d = \frac{d-1}{2^0} + 1$. Assume that $f^j(d) = \frac{d-1}{2^j} + 1$ holds for all $j<i$. Thus, $$f^i(d) = f(f^{i-1}(d)) = f\left( \frac{d-1}{2^{i-1}} + 1 \right) = \frac{\frac{d-1}{2^{i-1}} + 1}{2} + \frac{1}{2} = \frac{d-1}{2^{i}} + 1 \text{.}$$ Finally, for any fixed $i$ and $j$ such that $i+j=n-1$, we have $$\begin{aligned} f^i(g(f^j(d))) &= f^i \left( g \left( \frac{d-1}{2^j} + 1 \right)\right) = f^i \left( \frac{\frac{d-1}{2^j} + 1}{2} + 1 \right) \\ &= f^i \left( \frac{d-1}{2^{j+1}} + \frac{3}{2} \right) = \frac{\frac{d-1}{2^{j+1}} + \frac{3}{2}-1}{2^i} + 1 = \frac{d-1}{2^n} + \frac{1}{2^{i+1}} + 1 \text{.} \end{aligned}$$ The value of $f^i(g(f^j(d)))$ is greatest when $i=0$, and thus $f^i(g(f^j(d))) \leq \frac{d-1}{2^n} + \frac{3}{2}$. ◻ By Claim [Claim 4](#obs6){reference-type="ref" reference="obs6"} and discussion above we obtain that $$\label{dGAlphaIneq} d_{G_\alpha}(v)\leq \frac{d_H(v)-1}{2^n} + \frac{3}{2}.$$ It remains to show that if $d_G(v)=\beta\delta$, then $d_{G_\alpha}(v)\leq \frac{\beta\delta}{k}$. Recall that by [\[dHvInequ\]](#dHvInequ){reference-type="eqref" reference="dHvInequ"}, $d_H(v) \leq \beta \left( \delta - m \left( \frac{\delta}{k} - 2 \right) \right)$. This combined with [\[dGAlphaIneq\]](#dGAlphaIneq){reference-type="eqref" reference="dGAlphaIneq"} yield the following, where we make use of the facts that $k = 2^n + m - 1$, $\delta \geq \frac{3}{2}k^2+\frac{1}{2}km+\frac{1}{2}k$ and $\beta \geq 1$: $$\begin{aligned} d_{G_\alpha}(v) &\leq \frac{\beta \left( \delta - m \left( \frac{\delta}{k} - 2 \right) \right) - 1}{2^n} + \frac{3}{2} \\ &= \frac{\beta \delta \left( 1-\frac{m}{k} \right) + 2\beta m - 1}{2^n} + \frac{3}{2} \\ &= \frac{\beta \delta \left( \frac{2^n - 1}{k} \right) + 2\beta m - 1}{2^n} + \frac{3}{2} \\ &= \frac{\beta \delta}{k} + \frac{2\beta m - 1 - \frac{\beta \delta}{k}}{2^n} + \frac{3}{2} \\ &\leq \frac{\beta \delta}{k} + \frac{2\beta m - 1 - \beta \left( \frac{3}{2}k + \frac{1}{2}m + \frac{1}{2} \right)}{2^n} + \frac{3}{2} \\ &= \frac{\beta \delta}{k} + \frac{\beta \left( - \frac{3}{2}k + \frac{3}{2}m - \frac{1}{2} \right) - 1}{2^n} + \frac{3}{2} \\ &= \frac{\beta \delta}{k} + \frac{\beta \left( - \frac{3}{2} \left( 2^n - 1 \right) - \frac{1}{2} \right) - 1}{2^n} + \frac{3}{2} \\ &= \frac{\beta \delta}{k} - \frac{3}{2}\beta + \frac{\beta - 1}{2^n} + \frac{3}{2} \\ &= \frac{\beta \delta}{k} + (1-\beta) \left( \frac{3}{2} - \frac{1}{2^n} \right) \\ &\leq \frac{\beta \delta}{k} = \frac{d_G(v)}{k}. \end{aligned}$$ This concludes the proof of Theorem [Theorem 11](#thm2-GenMajEdg){reference-type="ref" reference="thm2-GenMajEdg"}. ◻ Note that the formula for $k$ used in Theorem [Theorem 11](#thm2-GenMajEdg){reference-type="ref" reference="thm2-GenMajEdg"} implies that $m$ is always bounded above by $\frac{k}{2}$, and thus we immediately obtain the following corollary. **Corollary 12**. *For every integer $k\geq 2$, if a graph $G$ has minimum degree $\delta\geq \frac{7}{4}k^2+\frac{1}{2}k$, then $G$ has a $\frac{1}{k}$-majority $(k+1)$-edge-colouring.* # Confirmation of Conjecture [Conjecture 8](#EdgeMajconj){reference-type="ref" reference="EdgeMajconj"} for initial values of $k$ {#SectionSmallCases} The main result of [@majority23] confirms Conjecture [Conjecture 8](#EdgeMajconj){reference-type="ref" reference="EdgeMajconj"} for $k=2$. In this section we shall extend this result towards two following values of $k$. To achieve this we shall use the two observations below. **Observation 13**. *Let $k\geq 2$ be an integer. If every graph with minimum degree $\delta \geq k^2$ and maximum degree $\Delta < 2k^2$ has a $\frac{1}{k}$-majority $(k+1)$-edge-colouring, then every graph with minimum degree at least $k^2$ has a $\frac{1}{k}$-majority $(k+1)$-edge-colouring.* **Proof.* Let $G$ be an arbitrary graph with minimum degree at least $k^2$. If the maximum degree of $G$ is less than $2k^2$ then by assumption it has a $\frac{1}{k}$-majority $(k+1)$-edge-colouring. Otherwise, let $v$ be a vertex of $G$ such that $d_G(v) \geq 2k^2$. There exist unique integers $n$ and $d$ such that $d_G(v) = n k^2 + d$ and $k^2 \leq d < 2k^2$. Partition the neighbourhood of $v$ into $n+1$ disjoint sets $N_0,\dotsc,N_n$ such that $|N_0| = d$ and $|N_i|=k^2$ for $i\geq 1$. Let $H$ be a graph such that $V(H) = V(G)\setminus\{v\} \cup \{v_0,v_1,\dotsc,v_n\}$ and $E(H) = E(G-v)\cup \bigcup\limits_{i=0}^{n} \{uv_i : u\in N_i\}$. Note that this operation yields a natural bijection between the edges of $G$ and the edges of $H$. Let $\overline{G}$ be a graph constructed from $G$ by applying the above operation to all vertices of $G$ with degree at least $2k^2$. By construction, the maximum degree of $\overline{G}$ is less than $2k^2$, hence $\overline{G}$ has a $\frac{1}{k}$-majority $(k+1)$-edge-colouring, which yields a $(k+1)$-edge-colouring of $G$. It remains to prove that this is also a $\frac{1}{k}$-majority edge-colouring of $G$.* *Let $v$ be a vertex of $G$ with degree $d_G(v) = n k^2 + d$ where $k^2 \leq d < 2k^2$ (with $n$ possibly equal $0$). The number of edges adjacent to $v$ coloured with the same colour is bounded above by $$n \left\lfloor \frac{k^2}{k} \right\rfloor + \left\lfloor \frac{d}{k} \right\rfloor = nk + \left\lfloor \frac{d}{k} \right\rfloor = \left\lfloor \frac{nk^2 + d}{k} \right\rfloor = \left\lfloor \frac{d_G(v)}{k} \right\rfloor \text{,}$$ hence the colouring of $G$ is indeed a $\frac{1}{k}$-majority $(k+1)$-edge-colouring. ◻* **Observation 14**. *For every integer $k\geq 2$, let $S_k$ be the set of all integers $i$ between $k^2$ and $2k^2$ such that $i \equiv k-1 \pmod k$. Let $\mathcal{G}_k$ be the set of all graphs for which the degrees of all vertices are in the set $S_k$. If every graph in $\mathcal{G}_k$ has a $\frac{1}{k}$-majority $(k+1)$-edge-colouring, then every graph with minimum degree at least $k^2$ has a $\frac{1}{k}$-majority $(k+1)$-edge-colouring.* **Proof.* By Observation [Observation 13](#limit_deg_1){reference-type="ref" reference="limit_deg_1"} it is sufficient to consider graphs with maximum degree less than $2k^2$. Let $G$ be an arbitrary graph with minimum degree $\delta \geq k^2$ and maximum degree $\Delta<2k^2$. If all vertices of $G$ have degrees in the set $S_k$, then by assumption $G$ has a $\frac{1}{k}$-majority $(k+1)$-edge-colouring. Otherwise, let $H$ be a graph constructed from $G$ by taking two copies of $G$ and joining by edges the vertices of $G$ which do not have degrees in the set $S_k$ with their corresponding counterparts in the second copy of $G$. Note that $G$ is a subgraph of $H$. Moreover, for every vertex $v$ of $G$, either $d_H(v)=d_G(v) \in S_k$ or $d_G(v) \equiv i \pmod k$ and $d_H(v) \equiv i+1 \pmod k$ (and the same holds for the vertices in the second copy of $G$). We repeat this operation until all vertices of the obtained graph $\overline{G}$ have degrees in the set $S_k$. Note the degree of each vertex $v$ of $G$ increased by at most $k-1$, and more importantly, $$\label{dHvdGvFloorsTheSame} \left\lfloor \frac{d_H(v)}{k} \right\rfloor = \left\lfloor \frac{d_G(v)}{k} \right\rfloor.$$ Since the maximum degree of $G$ is at most $2k^2-1 \equiv k-1 \pmod k$, the maximum degree of $\overline{G}$ is also less than $2k^2$, hence $\overline{G}\in \mathcal{G}_k$. By our assumption, there is a $\frac{1}{k}$-majority $(k+1)$-edge-colouring $c$ of $\overline{G}$. Since $G$ is a subgraph of $\overline{G}$, by [\[dHvdGvFloorsTheSame\]](#dHvdGvFloorsTheSame){reference-type="eqref" reference="dHvdGvFloorsTheSame"}, the colouring $c$ restricted to the edges of $G$ yields a $\frac{1}{k}$-majority $(k+1)$-edge-colouring of $G$. ◻* Observations [Observation 13](#limit_deg_1){reference-type="ref" reference="limit_deg_1"} and [Observation 14](#limit_deg_2){reference-type="ref" reference="limit_deg_2"} allow us to narrow down the set of graphs we need to consider in order to prove Conjecture [Conjecture 8](#EdgeMajconj){reference-type="ref" reference="EdgeMajconj"}. To start with, we exemplify their usefulness by reproving Theorem [Theorem 1](#majority23-res2){reference-type="ref" reference="majority23-res2"}, whose proof provided in [@majority23] is rather lengthy. Tools and observations introduced above yield a short and straightforward argument. *Proof of Theorem [Theorem 1](#majority23-res2){reference-type="ref" reference="majority23-res2"}.* Let $G$ be an arbitrary graph with minimum degree $\delta\geq 4$. By Observation [Observation 14](#limit_deg_2){reference-type="ref" reference="limit_deg_2"}, we can assume that the degrees of all vertices of $G$ are in the set $S_2 = \{5,7\}$. Let $D_2$ be the set of vertices of degree $5$, and $D_3$ the set of vertices of degree $7$. Vertices in $D_2$ can have at most $2$ incident edges in the same colour, and vertices of $D_3$ can have at most $3$ such edges. We shall construct a majority $3$-edge-colouring of $G$ in two steps. First, use Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"} with a weight function assigning $1/3$ to every edge of $G$ to colour some edges of $G$ with one of the three colours (similarly as in the proof of Theorem [Theorem 9](#thm1-GenMajEdg){reference-type="ref" reference="thm1-GenMajEdg"}). Vertices in $D_2$ have $1$ or $2$ edges coloured, and vertices in $D_3$ -- $2$ or $3$. Let $H$ be the graph induced by uncoloured edges of $G$. Vertices in $D_2$ have degrees $3$ or $4$ in $H$, and vertices in $D_3$ have degrees $4$ or $5$. Finally, use Observation [Observation 10](#euler){reference-type="ref" reference="euler"} to colour the edges of $H$ with the remaining two colours. Note that every component of $H$ either has vertices of odd degree or all of its vertices have degree $4$ and thus such component has an even number of edges. Hence, by Observation [Observation 10](#euler){reference-type="ref" reference="euler"}, at most $\left\lceil \frac{d_H(u)}{2} \right\rceil$ of the edges incident with any given vertex $u$ shall get the same colour, which satisfies the majority rule for the graph $G$. ◻ **Theorem 15**. *Every graph with minimum degree $\delta\geq 9$ has a $\frac{1}{3}$-majority 4-edge-colouring.* **Proof.* Let $G$ be a graph with minimum degree $\delta\geq 9$. By Observation [Observation 14](#limit_deg_2){reference-type="ref" reference="limit_deg_2"}, we can assume that the degrees of all vertices of $G$ are in the set $S_3 = \{11,14,17\}$. Similarly as before, let $D_3$ be the set of vertices of degree $11$ (which can have at most $3$ incident edges with the same colour), $D_4$ be the set of vertices of degree $14$ (allowing $4$ incident monochromatic edges), and $D_5$ -- vertices of degree $17$ (allowing $5$ incident edges with the same colour). Let $G'$ be a graph constructed from the graph $G$ by removing all components which have all vertices of degree $14$ and an odd number of edges. Hence, each component of $G'$ has an even number of edges or contains vertices of odd degree. Colour the edges of $G'$ using Observation [Observation 10](#euler){reference-type="ref" reference="euler"} with colours blue and red. The number of incident edges with the same colour shall equal $5$ or $6$ for vertices in $D_3$, $7$ for vertices in $D_4$, and $8$ or $9$ for vertices in $D_5$.* *We shall show that in fact we can choose such a $2$-edge-colouring of $G'$ complying with Observation [Observation 10](#euler){reference-type="ref" reference="euler"} that neither of the two colours induces a component with an odd number of edges and all vertices of degree $6$. Assume this is not the case and consider a $2$-edge-colouring of $G'$ consistent with Observation [Observation 10](#euler){reference-type="ref" reference="euler"} with the least number of such bad components. Without loss of generality we can assume that there exists such a bad component, say $H$ in the graph induced by the blue edges. Notice that $H$ is in particular Eulerian, and thus it is $2$-edge-connected. Clearly, all the vertices in $H$ are in the set $D_3$ and have exactly 5 red incident edges (in $G'$). Let $v$ be an arbitrary vertex of $H$, and let $u_1, u_2$ be any two distinct neighbours of $v$ in $H$. Consider a component in the subgraph of $G'$ induced by the red edges such that $v$ is in this component, denote it $H'$. If $u_1$ is not in the same (red) component as $v$, then we can recolour the edge $u_1v$ with red colour. In such a case, $H$ shall no longer have exclusively vertices of degree $6$, and no new $6$-regular monochromatic component shall be created, since at least one other than $u_1$ vertex in $H'$ needs to have odd degree. We proceed similarly if $u_1$ is in the same red component as $v$, but $u_2$ is not. If both $u_1$ and $u_2$ are in the same red component as $v$, then we can recolour the edge $u_1v$ to red colour. Then, neither $H$ nor $H'$ shall be a $6$-regular components. Hence, in each case, the number of monochromatic components with an odd number of edges and all vertices of degree $6$ can be decreased, which is in contradiction with the assumption that our colouring had the least possible number of bad components.* *As a result, both in the graph induced by the red edges and in the graph induced by the blue edges each component has an even number of edges or contains vertices of odd degree or contains a vertex of degree $8$. Hence, we can again use Observation [Observation 10](#euler){reference-type="ref" reference="euler"} (separately for graphs induced by both of the colours), choosing a vertex of degree $8$ as the bad vertex if necessary. The $4$-edge-colouring obtained this way satisfies the $\frac{1}{3}$-majority rule for the graph $G'$.* *Finally, consider components of $G$ with all vertices of degree $14$ and an odd number of edges. Using Observation [Observation 10](#euler){reference-type="ref" reference="euler"} we obtain a $2$-edge-colouring of such components with colours red and blue, such that in the subgraph generated by the edges of one of the colours all vertices shall have degree $7$, except one vertex of degree $6$ or $8$. In either case, there shall be a vertex of odd degree in each of the obtained monochromatic components, hence using again Observation [Observation 10](#euler){reference-type="ref" reference="euler"} (and merging the result with the colouring of $G'$) yields a $\frac{1}{3}$-majority $4$-edge-colouring of $G$. ◻* **Theorem 16**. *Every graph with minimum degree $\delta\geq 16$ has a $\frac{1}{4}$-majority $5$-edge-colouring.* **Proof.* Let $G$ be a graph with minimum degree $\delta\geq 16$. By Observation [Observation 14](#limit_deg_2){reference-type="ref" reference="limit_deg_2"}, we can assume that the degrees of all vertices of $G$ are in the set $S_4 = \{19,23,27,31\}$. Let $D_4$ be the set of vertices of degree $19$ (which can have at most $4$ incident edges with the same colour), $D_5$ be the set of vertices of degree $23$ (allowing $5$ incident monochromatic edges), $D_6$ be the set of vertices of degree $27$ (allowing $6$ incident edges with the same colour), and $D_7$ -- vertices of degree $31$ (allowing $7$ incident monochromatic edges). First, use Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"} with a weight function assigning $1/5$ to all edges of $G$ to colour some edges of $G$ with colour 1. As a result, every vertex $v$ of $G$ has either $\lfloor \frac{d(v)}{5} \rfloor$ or $\lceil \frac{d(v)}{5} \rceil$ incident edges coloured $1$. As none of the vertices has degree divisible by $5$, these two values are distinct, and by Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"} (ii), every edge $uv$ with exactly $\lfloor \frac{d(u)}{5} \rfloor$ edges incident with $u$ coloured $1$ and exactly $\lfloor \frac{d(v)}{5} \rfloor$ edges incident with $v$ coloured $1$ must be coloured $1$ as well.* *Let $H$ be the subgraph of $G$ induced by the uncoloured edges. Vertices in $D_4$ have degrees in $\{15,16\}$ in $H$, vertices in $D_5$ -- degrees in $\{18,19\}$, vertices in $D_6$ -- degrees in $\{21,22\}$, and vertices in $D_7$ -- degrees in $\{24,25\}$. Using Observation [Observation 10](#euler){reference-type="ref" reference="euler"} divide the graph $H$ into two subgraphs $H_1$ (coloured blue) and $H_2$ (coloured red), choosing vertices of degree $18$ or $22$ as the bad vertices, if necessary. In the components of the graphs $H_1$ and $H_2$ we have the following situation: vertices in $D_4$ have degrees in $\{7,8\}$, vertices in $D_5$ have degrees in $\{9,10\}$ (and possibly a single vertex has degree $8$), vertices in $D_6$ have degrees in $\{10,11\}$ (and possibly a single vertex has degree $12$), and vertices in $D_7$ have degrees in $\{12,13\}$. Notice that if a vertex in $D_6$ has degree $12$, then in the graph $H$, it had to be in a component with no vertex of odd degree, hence in the graphs $H_1$ and $H_2$, this vertex cannot be in the same component as any vertex of degree $10$.* *We shall show that we can recolour the graph $H$ (retaining conditions mentioned above) in such a way that neither $H_1$ nor $H_2$ contains a component with an odd number of edges whose every vertex either has degree $10$ and belongs to $D_5$ or has degree $8$ and belongs to $D_4$. Assume this is not possible and consider a colouring with the least number of such components. Let $\overline{H}$ be one of these components. Since the number of edges of $\overline{H}$ is odd, at least one of the vertices in $\overline{H}$ must have degree $10$. Let $v$ be a vertex of degree $10$ in $\overline{H}$ and let $u_1$ and $u_2$ be two distinct neighbours of $v$ in $\overline{H}$. If $v$ is the only vertex of degree $10$ in $\overline{H}$ and $d_H(v)=18$, then $v$ is a bad vertex and for all the remaining vertices $u$ of $\overline{H}$ we must have $d_H(u)=16=\lceil \frac{d(u)}{5} \rceil$, and two such vertices must be adjacent in $\overline{H}$ (hence also in $H$), which is impossible by the last remark of the first paragraph of the proof. We may thus assume that $d_H(v)=19=\lceil \frac{d(v)}{5} \rceil$, and hence, again by the last remark of the first paragraph of the proof, $u_1$ and $u_2$ must be vertices of degree $8$ in $\overline{H}$ and $15$ in $H$. Thus, proceeding analogously as in the proof of Theorem [Theorem 15](#small3){reference-type="ref" reference="small3"} we can obtain a colouring of $H$ with a smaller number of bad components, a contradiction.* *Hence, each component of $H_1$ and $H_2$ contains vertices of odd degree or has an even number of edges or contains a vertex of degree $10$ belonging to $D_6$ or a vertex of degree $12$ belonging to $D_7$. Thus, using Observation [Observation 10](#euler){reference-type="ref" reference="euler"} (with one of the mentioned vertices being chosen as the bad vertex, if necessary) we can obtain a $4$-edge-colouring of $H$, which completes a $\frac{1}{4}$-majority $5$-edge-colouring of $G$. ◻* # Concluding remarks Theorem [Theorem 9](#thm1-GenMajEdg){reference-type="ref" reference="thm1-GenMajEdg"} and the construction in Observation [Observation 7](#ExampleGeneral){reference-type="ref" reference="ExampleGeneral"} imply we managed to settle the order of magnitude of our main objective: $\delta_k^{\rm opt}$ and approximate it within a multiplicative factor of $2$. Our Conjecture [Conjecture 8](#EdgeMajconj){reference-type="ref" reference="EdgeMajconj"} clearly conveys we expect that $2$ is a redundant factor in our $2k^2$ upper bound for $\delta_k^{\rm opt}$. In fact, Corollary [Corollary 12](#7_4_Corollary){reference-type="ref" reference="7_4_Corollary"} shows that the leading factor in this bound should not be larger than $\frac{7}{4}k^2$. Moreover, Theorem [Theorem 11](#thm2-GenMajEdg){reference-type="ref" reference="thm2-GenMajEdg"} also implies that there is e.g. an infinite sequence of values of $k$ for which $\delta_k^{\rm opt}\leq (\frac{3}{2}+o(1))k^2$. On the other hand, even though we were able to confirm the conjecture for several initial values of $k$ in Section [4](#SectionSmallCases){reference-type="ref" reference="SectionSmallCases"}, we are not entirely convinced that the postulated quantity of $\delta_k^{\rm opt}$ has to be precisely correct for all $k$. One may possibly come up with some more sophisticated construction than the one in Observation [Observation 7](#ExampleGeneral){reference-type="ref" reference="ExampleGeneral"}, and this seems an interesting direction to be more thoroughly investigated. However, we would not expect the lower bound stemming from such a potential construction to exceed $k^2$ by far. In any case we strongly expect an upper bound of the form $(1+o(1))k^2$ to be valid for $\delta_k^{\rm opt}$. Recall that Observation [Observation 7](#ExampleGeneral){reference-type="ref" reference="ExampleGeneral"} implies we cannot directly extend to all graphs our Theorem [Theorem 5](#bipartiteTheorem){reference-type="ref" reference="bipartiteTheorem"}, yielding an optimal solution for the family of bipartite graphs. However, as mentioned, the main obstacle on the way towards obtaining some form of such an extension was lack of a valid in the general setting correspondent of Claim [Claim 1](#Claim1-bipartite){reference-type="ref" reference="Claim1-bipartite"} from the proof of Theorem [Theorem 5](#bipartiteTheorem){reference-type="ref" reference="bipartiteTheorem"}, where in some sense it allowed us to control and 'capture' degrees of consecutively constructed subgraphs of a given bipartite graph within a reasonably narrow interval. In fact, in pursuit of such a correspondent we came up with our refinement of the lemma of Alon and Wei [@alon], that is Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"}. Even though some aspects of this slight improvement were useful and crucial in the case of bipartite graphs, we did not use it in full measure, while at the same time it was not strong enough to provide a result we expect in a general case. Nevertheless, we decided to include in our paper this slightly excessive form of Lemma [Lemma 6](#mod_alon){reference-type="ref" reference="mod_alon"}, as a suggestion for possible further development of this tool, which might hopefully lead to solving Conjecture [Conjecture 8](#EdgeMajconj){reference-type="ref" reference="EdgeMajconj"}, or at least help closing the current gap. Let us also mention we believe that even solving our Problem [Problem 3](#GeneralizedMajorityProblem){reference-type="ref" reference="GeneralizedMajorityProblem"} for the first open case of $k=5$ (and maybe some consecutive initial ones) seems interesting by itself, and may furthermore shed light on a possible approach to attack Conjecture [Conjecture 8](#EdgeMajconj){reference-type="ref" reference="EdgeMajconj"} in its entirety. Finally, let us remark why we believe the probabilistic approach seems difficult to be (directly) utilised while trying to prove Conjecture [Conjecture 8](#EdgeMajconj){reference-type="ref" reference="EdgeMajconj"}. It stems from the fact that if a graph has a vertex $v$ with degree (close to) $k^2$, then while colouring its edges randomly with $k+1$ colours we expect every colour to appear roughly $d(v)/(k+1)>k-1$ times (and in fact some colours must appear at least this many times) around $v$, while we admit at most $\lfloor d(v)/k\rfloor \leq k$ appearances of each colour. Thus in a way we admit an error of at most $1$ in frequency of appearing of each colour, which does not seem achievable via probabilistic approach, as e.g. typical concentration tools require admitting an error "slightly" larger than $\sqrt{(d(v)/k)}$ (which is enough as long as $d(v)$ is of magnitude roughly $k^3\log k$). This is also why we reckon that our, rather naive in nature, approach is surprisingly efficient. 9 Noga Alon and Fan Wei. Irregular subgraphs. *Combin. Probab. Comput.* **32** (2023), no. 2, 269-283. doi:https://doi.org/10.1017/S0963548322000220 M. Anastos, A. Lamaison, R. Steiner, T. Szabó, Majority colorings of sparse digraphs, Electron. J. Combin. 28(2) (2021) \# P2.31. doi:10.37236/10067 Marcin Anholcer, Bartłomiej Bosek and Jarosław Grytczuk. Majority Choosability of digraphs. *Electron. J. Combin.* **24** (2017), no. 3, Paper No. 3.57, 5 pp. doi:https://doi.org/10.37236/6923 M. Anholcer, B. Bosek, J. Grytczuk, G. Gutowski, J. Przybyło, M. Zając, Mrs. Correct and Majority Colorings, arXiv:2207.09739. Felix Bock, Rafał Kalinowski, Johannes Pardey, Monika Pilśniak, Dieter Rautenbach and Mariusz Woźniak. Majority Edge-Colorings of Graphs. *Electron. J. Combin.* **30** (2023), no. 1, Paper No. 1.42, 8 pp. doi:https://doi.org/10.37236/11291 António Girão, Teeradej Kittipassorn and Kamil Popielarz. Generalized Majority Colourings of Digraphs. *Combin. Probab. Comput.* **26** (2017), no. 6, 850-855. doi:https://doi.org/10.1017/S096354831700044X F. Knox, R. Šámal, Linear bound for majority colourings of digraphs, Electron. J. Combin. 25(3) (2018) P3.29. doi:10.37236/6762 Stephan Kreutzer, Sang-il Oum, Paul Seymour, Dominic van der Zypen and David R. Wood. Majority Colourings of Digraphs. *Electron. J. Combin.* **24** (2017), no. 2, Paper No. 2.25, 9pp. doi: 10.37236/6410 L. Lovász, On decomposition of graphs, Studia Scientiarum Mathematicarum Hungarica I(1-2) (1966) 237--238. J. Przybyło, On the standard $(2,2)$-Conjecture, European J. Comb. 94 (2021) 103305. doi:10.1016/j.ejc.2020.103305

arxiv_math

--- abstract: | We continue the analysis of the Modular Isomorphism Problem for $2$-generated $p$-groups with cyclic derived subgroup, $p>2$, started in [@GLdRS2022]. We show that if $G$ belongs to this class of groups, then the isomorphism type of the quotients $G/(G')^{p^3}$ and $G/\gamma_3(G)^p$ are determined by its modular group algebra. In fact, we obtain a more general but technical result, expressed in terms of the classification [@OsnelDiegoAngel]. We also show that for groups in this class of order at most $p^{11}$, the Modular Isomorphism Problem has positive answer. Finally, we describe some families of groups of order $p^{12}$ whose group algebras over the field with $p$ elements cannot be distinguished with the techniques available to us. address: - Departamento de Matemáticas, Universidad de Murcia, Spain - Departamento de Matemáticas, Universidad de Murcia, Spain author: - Diego Garcı́a-Lucas - Ángel del Río bibliography: - MIP.bib title: | On the Modular Isomorphism Problem for\ 2-generated groups with cyclic derived subgroup --- The class of $2$-generated finite $p$-groups with cyclic derived subgroup, despite its apparent simplicity, has proven to be a rich class of $p$-groups, specially regarding the Modular Isomorphism Problem: the only known indecomposable groups to fail to satisfy the statement of this problem are $2$-groups that belong to this class (see [@GarciaMargolisdelRio]), while for $p>2$, the situation being quite different, the problem is still to be decided. Our main result settles the Modular Isomorphism Problem in the positive for groups of this class under additional constraints on the size of the initial terms of the lower central series: **Theorem 1**. *Let $p$ be an odd prime, let $k$ be the field with $p$ elements and let $G$ be a $2$-generated finite $p$-group with cyclic derived subgroup. If $kG\cong k H$ for some group $H$, then* 1. *[\[theorem1.1\]]{#theorem1.1 label="theorem1.1"}$G/\gamma_3(G)^p \cong H/\gamma_3(H)^p$ and* 2. *[\[theorem1.2\]]{#theorem1.2 label="theorem1.2"}$G/(G')^{p^3}\cong H/(H')^{p^3}$.* This result fails for $p=2$ because the counter-example in [@GarciaMargolisdelRio] is formed by groups with derived subgroup of order $4$. The proof of is based upon a more technical result in terms of the invariants described in [@OsnelDiegoAngel], that resumes the work started in [@GLdRS2022]. Namely, with the notation in , we prove the following theorem. **Theorem 2**. *Let $p$ be an odd prime, let $k$ be the field with $p$ elements and let $G$ be a $2$-generated finite $p$-group with cyclic derived subgroup and $$\textup{inv}(G)=(p,m,n_1,n_2,o_1,o_2,o'_1,o'_2,u_1^G,u_2^G).$$ If $k G\cong k H$ for some group $H$, then $H$ is also a $2$-generated finite $p$-group with cyclic derived subgroup and $$\textup{inv}(H)=(p,m,n_1,n_2,o_1,o_2,o'_1,o'_2,u_1^H,u_2^H)$$ such that $$u_2^G\equiv u_2^H \mod p,$$ and one of the following holds:* 1. *[\[theorem2.1\]]{#theorem2.1 label="theorem2.1"} $u_1^G\equiv u_1^H\mod p$.* 2. *[\[theorem2.2\]]{#theorem2.2 label="theorem2.2"} $o_1o_2>0$, $n_1+o_1'=n_2+o_2'$ and at least one of the following conditions fails:* - *$u_2^G\equiv u_2^H \equiv 1 \mod p^{o_1+1-o_2}$,* - *$n_2+o'_2=2m-o_1$.* Observe that if $G$ and $H$ are as in the previous theorems, then $G\cong H$ if and only if $\textup{inv}(G)=\textup{inv}(H)$, so is another step towards a solution of the Modular Isomorphism Problem for our target class of groups. As an application we obtain a positive answer for the Modular Isomorphism Problem for $2$-generated $p$-groups with $p>2$ having cyclic derived subgroup and order at most $p^{11}$. Moreover, for groups of order $p^{12}$ we also obtain a positive solution except for $p-2$ families of containing $p$ groups each. The paper is organized as follows. In we establish the notation and prove some general auxiliary results. In the remainder of the paper**[,]{style="color: blue"}** $p$ is an odd prime and all the groups are $2$-generated finite $p$-groups with cyclic derived subgroup. In we recall the classification of such groups from [@OsnelDiegoAngel] and establish some basic facts for these groups and their group algebras. In we prove Theorems [Theorem 1](#theorem1){reference-type="ref" reference="theorem1"} and [Theorem 2](#theorem2){reference-type="ref" reference="theorem2"}. Finally, in we prove the mentioned results about groups of small order. # Preliminaries {#SectionPreliminaries} Throughout the paper, $p$ denotes an odd prime number, $k$ is the field with $p$ elements, $G$ is a finite $p$-group and $N$ is a normal subgroup of $G$. The group algebra of $G$ over $k$ is denoted by $kG$ and its *augmentation ideal* is denoted by $\mathrm{I}(G)$. It is a classical result that $\mathrm{I}(G)$ is also the Jacobson ideal of $kG$. If $C$ is a subset of $G$ then $\hat C = \sum_{c\in C} c\in kG$. It is well known that the center $\mathrm{Z}(kG)$ is the $k$-span of the class sums $\hat C$ with $C$ running on the set $\mathop{\mathrm{Cl}}(G)$ of conjugacy classes of $G$. The rest of group theoretical notation is mostly standard: $[g,h]=g^{-1}h^{-1}gh$ for $g,h\in G$, $|G|$ denotes the order of $G$, $\mathrm{Z}(G)$ its center, $\{\gamma_i(G)\}_{i\geq 1}$ its lower central series and $G'=\gamma_2(G)$ its commutator subgroup. For $n\geq 1$, we denote by $C_n$ the cyclic group of order $n$. Moreover, if $g\in G$ and $X\subseteq G$ then $|g|$ denotes the order of $g$ and ${\rm C}_G(X)$ the centralizer of $X$ in $G$. For a subgroup $A$ of $G$, we denote $A^n=\left\langle a^n:x\in A \right\rangle$. If $A$ is normal cyclic subgroup of $G$, then $\mathrm{I}(A^{p^n})=\mathrm{I}(A)^{p^n}$ and hence $(\mathrm{I}(A)kG)^{p^n}=\mathrm{I}(A)^{p^n}kG=\mathrm{I}(A^{p^n})kG$. We take the following the following notation from [@OsnelDiegoAngel] for integers $s,t$ and $n$ with $n \ge 0$: $$\mathcal{S}\left(s\mid n\right) = \sum_{i=0}^{n-1} s^i.$$ We will use the following elementary lemma. **Lemma 1**. *If $G$ is a finite $p$-group with cyclic derived subgroup and $p>2$, then every conjugacy class of $G$ is a coset modulo a subgroup of $G'$.* *Proof.* Let $C$ be a conjugacy class of $G$, let $g\in C$ and $H=\{[x,g^{-1}] : x\in G\}$. Then $C=Hg$ and hence it is enough to prove that $H$ is a subgroup of $G'$. As $G'$ is cyclic and $H\subseteq G'$, it is enough to prove that if $h\in H$ then $h^i\in H$ for every non-negative integer $i$. Let $h=[x,g^{-1}]$ with $x\in G$. Then $h^x=h^r$ for some integer $r$ with $r\equiv 1 \mod p$. Therefore, using [@OsnelDiegoAngel Lemma 2.1], we have $[x^i,g^{-1}] = x^{-i} (x^i)^{g^{-1}} = x^{-i} (x^{g^{-1}})^i = x^{-i} (xh)^i=h^{\mathcal{S}\left(r\mid i\right)}$. This proves that $H$ contains all the elements of the form $h^{\mathcal{S}\left(r\mid i\right)}$ with $i\ge 0$. By [@GLdRS2022 Lemma 2.2] we deduce that $H$ contains $h^i$ for every non-negative integer. ◻ Let $n$ be a positive integer. We set $$\Omega_n(G)=\left\langle g\in G : g^{p^n}=1 \right\rangle \quad \text{and} \quad %\mho_n(G)=\GEN{g^{p^n} : g \in G}. %$$ %We also denote %$$ \Omega_n(G:N)=\left\langle g\in G: g^{p^{n}}\in N \right\rangle.$$ Observe that $\Omega_n(G:N)$ is the only subgroup of $G$ containing $N$ such that $$\Omega_n(G:N)/N=\Omega_n(G/N).$$ The next lemma collects some well-known results about the Modular Isomorphism Problem which will be used throughout the paper. **Lemma 2**. *The Modular Isomorphism Problem has a positive solution for $G$ if one of the following holds:* 1. *[\[MIPAbelian\]]{#MIPAbelian label="MIPAbelian"} $G$ is abelian [@Deskins1956].* 2. *[\[Metacyclic\]]{#Metacyclic label="Metacyclic"} $G$ is metacyclic [@BaginskiMetacyclic; @San96].* 3. *[\[2GenClass2\]]{#2GenClass2 label="2GenClass2"} $G$ is $2$-generated of class $2$ [@BdR20].* ## The Jennings series We denote ${\mathrm{D}}_n(G)$ the $n$-th term of the *Jennings series* of $G$, i.e. $${\mathrm{D}}_{n}(G) = \{g\in G : g-1\in \mathrm{I}(G)^n\}= \prod_{ip^j\ge n} \gamma_i(G)^{p^j}.$$ It is straightforward (see [@GL2022 Lemma 4.10]) that $$\label{UsefiN} G\cap (1+\mathrm{I}( G)^n+\mathrm{I}(N)kG)= {\mathrm{D}}_n(G)N.$$ Each quotient ${\mathrm{D}}_n(G)/{\mathrm{D}}_{n+1}(G)$ is elementary abelian and, if $t$ is the smallest non-negative integer with ${\mathrm{D}}_{t+1}(G)=1$, then a *Jennings set* of $G$ is a subset $\{g_{11},\dots,g_{1d_1},g_{21},\dots,g_{2d_2},\dots|g_{t1},\dots,g_{td_t}\}$ of $G$ such that $g_{i1}{\mathrm{D}}_{i+1}(G), \dots, g_{id_i}{\mathrm{D}}_{i+1}(G)$ is a basis of ${\mathrm{D}}_n(G)/{\mathrm{D}}_{n+1}(G)$ for each $i$. Observe that $|G|=p^{\sum_{i=1}^t d_i}$. If $x_1,\dots,x_n$ are the elements of a Jennings set of $G$, in some order, then $$\mathscr B = \{(x_1-1)^{e_1}\cdots (x_n-1)^{e_n} : 0\le e_i \le p-1 \text{ and }\sum_{i=1}^n e_i >0\}$$ is a basis of $\mathrm{I}(G)$, called a *Jennings basis* of $\mathrm{I}(G)$ associated to the given Jennings set. We denote $\mathscr B^n = \mathscr B \cap \mathrm{I}(G)^n$, which is a basis of $\mathrm{I}(G)^n$. **Lemma 3**. *There is a Jennings set $\mathscr S$ of $G$ such that $N\cap \mathscr S$ is a Jennings set of $N$.* *Proof.* We argue by induction on $|N|$. If $|N|=1$, then there is nothing to prove. Now suppose that the result holds for normal subgroups of order $p^n$, and assume that $N$ has order $p^{n+1}$. Since $G$ is a $p$-group, the center of $G$ intersects $N$ non-trivially, so we can choose a subgroup $L\subseteq N\cap \mathrm{Z}(G)$ of order $p$. By the induction hypothesis, we can choose a Jennings set $\bar {\mathscr S}$ of $G/L$ such that $\bar {\mathscr S}\cap (N/L)$ is a Jennings set of $N/L$. Let $\mathscr S$ be a set of representatives of the elements of $\bar {\mathscr S}$ in $G$. Clearly, the representatives of elements in $N/L$ are in $N$. For some $i$ we have that $L \subseteq D_i(G)$ but $L \not \subseteq D_{i+1}(G)$, and for some $j$, that $L\subseteq {\mathrm{D}}_j (N)$ but $L\not\subseteq {\mathrm{D}}_{j+1}(N)$. Observe that $\mathscr S$ is almost a Jennings basis of $G$ except it does not contain representatives of a basis of ${\mathrm{D}}_i(G)/{\mathrm{D}}_{i+1}(G)$, only of a maximal linear subspace which is a direct complement of $L$. Similarly, $\mathscr S\cap N$ is almost a Jennings basis of $N$ except it does not contain representatives of a basis of ${\mathrm{D}}_j(N)/{\mathrm{D}}_{j+1}(N)$, only of a maximal linear subspace which is a direct complement of $L$. Hence it suffices to take the Jennings set $\mathscr S\cup \{l\}$, where $l$ is a generator of $L$. ◻ The following equality is [@Usefi2008 Theorem A] and its symmetric analogue: $$\label{Usefi} {\mathrm{D}}_{n+1}(N)=G\cap (1+\mathrm{I}(N)^n\mathrm{I}(G))=G\cap (1+\mathrm{I}(G)\mathrm{I}(N)^n).$$ It can be generalized as follows. **Lemma 4**. *If $n$ and $m$ are positive integers, then $$(1+\mathrm{I}(G)^n+\mathrm{I}(N)^{m }\mathrm{I}(G))\cap G = {\mathrm{D}}_n(G){\mathrm{D}}_{m+1}(N)=(1+\mathrm{I}(G)^n+\mathrm{I}(G)\mathrm{I}(N)^{m })\cap G .$$* *Proof.* We prove only the first identity, the second being analogous. Since $(1+\mathrm{I}(G)^n)\cap G={\mathrm{D}}_{n}(G)$ and $(1+\mathrm{I}(N)^m\mathrm{I}(G))\cap G \supseteq (1+\mathrm{I}(N)^{m+1}) \cap G= {\mathrm{D}}_{m+1}(N)$, the right-to-left inclusion is clear. Thus it suffices to prove the converse. Taking quotients modulo ${\mathrm{D}}_n(G){\mathrm{D}}_{m+1}(N)$, it is enough to prove that $$\label{UsefiFrattini:AngelLaVaAQuitar} {\mathrm{D}}_n(G){\mathrm{D}}_{m+1}(N)=1\qquad \text{implies}\qquad (1+\mathrm{I}(G)^n+\mathrm{I}(N)^{m }\mathrm{I}(G))\cap G=1.$$ By , there is a Jennings set $\mathscr S$ of $G$ such that $N\cap \mathscr S$ is a Jennings set of $N$. Ordering the elements of $\mathscr S$ so that those in $N$ are placed first we obtain a Jennings basis $\mathscr B$ of $\mathrm{I}(G)$ associated to $\mathscr S$ containing a Jennings basis $\mathscr B_0$ of $\mathrm{I}(N)$ associated to $N\cap \mathscr S$. Recall that the set $\mathscr B^n= \mathscr B\cap \mathrm{I}(G)^n$ is a basis of $\mathrm{I}(G)^n$. Moreover, the set $\mathscr B_0^m=\mathscr B\cap \mathrm{I}(N)^m\mathrm{I}(G)$ is a basis of $\mathrm{I}(N)^m\mathrm{I}(G)$, and coincides with the set of elements of $\mathscr B$ of the form $xy$ with $x\in \mathscr B_0\cap \mathrm{I}(N)^m$ and $y\in \mathrm{I}(G)$. Then the following implication is clear: if $y\in \mathscr B$ occurs in the support in the basis $\mathscr B$ of an element $x\in \mathrm{I}(G)^n+\mathrm{I}(N)^m\mathrm{I}(G)$, then $y\in \mathscr B^n\cup \mathscr B_0^m$. Moreover, it is clear $(1+\mathscr B^n)\cap G\subseteq (1+\mathrm{I}(G)^n)\cap G={\mathrm{D}}_n(G)$ and $(1+\mathscr B_0^m)\cap G\subseteq (1+\mathrm{I}(N)^m\mathrm{I}(G))\cap G= {\mathrm{D}}_{m+1}(N)$ by [\[Usefi\]](#Usefi){reference-type="eqref" reference="Usefi"}. Thus $(1+\mathscr B^n\cup \mathscr B_0^m)\cap G \subseteq {\mathrm{D}}_n(G) {\mathrm{D}}_{m+1}(N)$. We prove [\[UsefiFrattini:AngelLaVaAQuitar\]](#UsefiFrattini:AngelLaVaAQuitar){reference-type="eqref" reference="UsefiFrattini:AngelLaVaAQuitar"} by induction on $m$. Suppose first that $m=1$ and that ${\mathrm{D}}_n(G){\mathrm{D}}_2(N)=1$, so [\[UsefiN\]](#UsefiN){reference-type="eqref" reference="UsefiN"} yields $$(1+\mathrm{I}(G)^n+\mathrm{I}(N)^{ }\mathrm{I}(G))\cap G \subseteq (1+\mathrm{I}(G)^n+\mathrm{I}(N)kG )={\mathrm{D}}_n(G)N=N.$$ So, if $1\neq g\in (1+\mathrm{I}(G)^n+\mathrm{I}(N)^{ }\mathrm{I}(G))\cap G$, then $g\in N$. Since $N$ is elementary abelian, $g-1\in \mathrm{I}(N)\setminus \mathrm{I}(N)^2$. Thus the support of $g-1$ in the basis $\mathscr B_0$ contains an element of the form $h-1$, with $1\neq h\in N$. Then, by the two previous paragraphs, $h\in (1+ \mathscr B^n\cup \mathscr B _0^1)\cap G\subseteq {\mathrm{D}}_n(G){\mathrm{D}}_{2}(N)=1$, a contradiction. For $m> 1$, the induction step is similar. Suppose that ${\mathrm{D}}_n(G){\mathrm{D}}_{m+1}(N)=1$, so ${\mathrm{D}}_m(N)$ is elementary abelian. Take $$1\neq g\in (1+\mathrm{I}(G)^n+\mathrm{I}(N)^{ m }\mathrm{I}(G))\cap G \subseteq (1+\mathrm{I}(G)^n+\mathrm{I}(N)^{m-1}\mathrm{I}(G) )={\mathrm{D}}_n(G){\mathrm{D}}_{m }(N)={\mathrm{D}}_m(N).$$ Since $\mathscr B_0\cap \mathrm{I}({\mathrm{D}}_{m }(N))$ is a Jennings basis of $\mathrm{I}({\mathrm{D}}_m(N))$ and $g-1\in \mathrm{I}({\mathrm{D}}_m(N))\setminus \mathrm{I}({\mathrm{D}}_m(N))^2$, we have that the support of $g-1$ in this basis (and hence in the basis $\mathscr B$) contains an element of the form $h-1$, with $1\neq h\in {\mathrm{D}}_m(N)$. However, $h\in (1+\mathscr B^n\cup \mathscr B_0^m)\subseteq {\mathrm{D}}_n(G){\mathrm{D}}_{m+1}(N)=1$, a contradiction. ◻ ## The relative lower central series The *lower central series of $N$ relative to $G$* is the series defined recursively by $$\gamma_1^G(N)=G \quad \text{and} \quad \gamma_{n+1}^G(N)=[ \gamma_n^G(N),N ].$$ We consider also the sequence of ideals of $kG$ defined recursively by setting $$\mathrm{J}^{1}(N,G) = \mathrm{I}(N)\mathrm{I}(G) \quad \text{and} \quad \mathrm{J}^{+1}(N,G)=\mathrm{I}(N)\mathrm{J}^{i}(N,G) +\mathrm{J}^{i}(N,G) \mathrm{I}(N).$$ This can be also defined with a closed formulae: $$\label{JClosed} \mathrm{J}^{n}(N,G)=\mathrm{I}(N)^n\mathrm{I}(G)+\sum_{i=1}^{n-1} \mathrm{I}(N)^{n-i}\mathrm{I}(G)\mathrm{I}(N)^{i}.$$ From $\mathrm{I}(N)kG=kG\mathrm{I}(N)$ and [\[JClosed\]](#JClosed){reference-type="eqref" reference="JClosed"} it easily follows that $$\label{IJISandwich} \mathrm{I}(N)^n\mathrm{I}(G)\subseteq \mathrm{J}^{n}(N,G)\subseteq \mathrm{I}(N)^n kG.$$ **Lemma 5**. *The following is a well defined map: $$\Lambda_N^n=\Lambda^n_{N,G}:\frac{\mathrm{I}(N)kG }{\mathrm{I}(N)\mathrm{I}(G)} \longrightarrow \frac{\mathrm{I}(N)^{p^{n}}kG}{\mathrm{J}^{p^{n}}(N,G)}, \qquad x+\mathrm{I}(N)\mathrm{I}(G)\mapsto x^{p^n}+\mathrm{J}^{p^{n}}(N,G).$$* *Proof.* Let $x\in \mathrm{I}(N)kG$ and $y\in \mathrm{I}(N)\mathrm{I}(G)$. Then $(x+y)^{p^n}-x^{p^n} = \sum_i a_i$ where each $a_i$ is a product of $p$ elements of $\{x,y\}$ with at least one equal to $y$. Hence each $a_i\in I_1\dots I_{p^n}$, where each $I_i$ is either $\mathrm{I}(N)kG$ or $\mathrm{I}(N) \mathrm{I}(G)$, and at least one of the $I_i$'s is of the second type. Since $\mathrm{I}(N) \mathrm{I}(G)\subseteq \mathrm{I}(N)kG$, $I_1\dots I_{p^n}\subseteq I(N)^{p^n-j} I(G) I(N)^j$ for some $0\leq j\leq p^n$, and hence, by [\[JClosed\]](#JClosed){reference-type="eqref" reference="JClosed"}, $I_1\dots I_{p^n}\subseteq \mathrm{J}^{p^{n}}(N,G)$. Therefore $(x+y)^{p^n}-x^{p^n} \in \mathrm{J}^{p^{n}}(N,G)$, so $\Lambda^n_N$ is well defined. ◻ The ambient group $G$ will be always clear from the context so we just write $\Lambda^n_N$. In particular, $$\Lambda_G^n: \frac{\mathrm{I}(G)}{\mathrm{I}(G)^2}\to \frac{\mathrm{I}(G)^{p^n}}{\mathrm{I}(G)^{p^{n}+1}}$$ is the usual map used in the *kernel size* computations (see [@Passman1965p4]). The first statement of the next lemma is just a slight modification of a well-known identity (see [@San89 Lemma 2.2]), while the second one is inspired, together with the definition of the ideals $\mathrm{J}^{i}(N,G)$, by the first section of [@BC88]. For the convenience of the reader we include a proof. **Lemma 6**. *Let $L$ and $N$ be normal subgroups of $G$. Then the following equations hold $$\begin{aligned} \label{eq71} \mathrm{I}(L) \mathrm{I}(N)kG+\mathrm{I}(N) \mathrm{I}(L)kG &=& \mathrm{I}([L,N])kG +\mathrm{I}(N)\mathrm{I}(L)kG, \\ \label{eq72} \mathrm{J}^{n}(N,G)&=& \sum_{i=1}^{n } \mathrm{I}(N)^{n+1-i} \mathrm{I}(\gamma_i^G(N))kG .\end{aligned}$$* *Proof.* Since the terms at both sides of [\[eq71\]](#eq71){reference-type="eqref" reference="eq71"} are two-sided ideals of $kG$, the equation follows from $$(g-1)(h-1)=hg([g,h]-1)+(h-1)(g-1)\qquad \text{for }g,h\in G.$$ In order to prove [\[eq72\]](#eq72){reference-type="eqref" reference="eq72"} we proceed by induction on $n$. For $n=1$ there is nothing to prove, and the following chain of equations $$\begin{aligned} \mathrm{J}^{n+1}(N,G) &=&\mathrm{J}^{n}(N,G)\mathrm{I}(N) + \mathrm{I}(N) \mathrm{J}^{n}(N,G)\\ & =& \sum_{i=1}^n \mathrm{I}(N)^{n+1-i} \mathrm{I}(\gamma_i^G(N))kG \mathrm{I}(N) + \mathrm{I}(N)\sum_{i=1}^n \mathrm{I}(N)^{n+1-i} \mathrm{I}(\gamma_i^G(N))kG \\ &= & \sum_{i=1}^n \mathrm{I}(N)^{n+1-i} \left[ \mathrm{I}(\gamma_i^G(N)) \mathrm{I}(N)kG + \mathrm{I}(N) \mathrm{I}(\gamma_i^G(N))kG\right] \\ \text{(by \eqref{eq71} with $L=\gamma_i^G(N)$)} &=& \sum_{i=1}^n \mathrm{I}(N)^{n+1-i} \left(\mathrm{I}(\gamma_{i+1}^G(N))kG + \mathrm{I}(N) \mathrm{I}(\gamma_i^G(N))kG \right) \\ &=& \sum_{i=1}^{n+1} \mathrm{I}(N)^{n+2-i} \mathrm{I}(\gamma_i^G(N))kG \end{aligned}$$ completes the induction argument. ◻ **Lemma 7**. *Let $N$ be a normal subgroup of $G$.* 1. *If $\gamma_i^G(N)\subseteq {\mathrm{D}}_i(N)$ for every $i\ge 2$ then for every $n\ge 1$ we have $\mathrm{J}^{n}(N,G)= \mathrm{I}(N)^n\mathrm{I}(G)$.* 2. *If $[G,N]\subseteq N^p$ then $\gamma_i^G(N)\subseteq {\mathrm{D}}_i(N)$ for every $i\ge 2$.* *Proof.* (1) Suppose that $\gamma_i^G(N)\subseteq {\mathrm{D}}_i(N)$ for $i\ge 2$. Since ${\mathrm{D}}_i(N) \subseteq 1+\mathrm{I}(N)^i$, it follows that if $i\ge 2$ then $\mathrm{I}(\gamma_i^G(N))\subseteq \mathrm{I}(N)^i$ and hence, using [\[eq72\]](#eq72){reference-type="eqref" reference="eq72"} we have $$\mathrm{J}^{s}(N,G)=\mathrm{I}(N)^s\mathrm{I}(G) + \sum_{i=2}^s \mathrm{I}(N)^{s+i-1}\mathrm{I}(\gamma_i^G(N))kG\subseteq \mathrm{I}(N)^s\mathrm{I}(G) + \mathrm{I}(N)^{s+1}kG \subseteq \mathrm{I}(N)^s\mathrm{I}(G).$$ This, together with [\[IJISandwich\]](#IJISandwich){reference-type="eqref" reference="IJISandwich"}, completes the proof. \(2\) Suppose that $[G,N]\subseteq N^p$. Then $\gamma_2^G(N)=[G,N]\subseteq N^p\subseteq {\mathrm{D}}_2(N)$. Then arguing by induction on $i$, for every $i\ge 3$ we obtain $\gamma_i^G(N)=[\gamma_{i-1}^G(N),N]\subseteq [{\mathrm{D}}_{i-1}(N),{\mathrm{D}}_1(N)] \subseteq {\mathrm{D}}_i(N)$, because $({\mathrm{D}}_i(N))_i$ is an $N_p$-series. ◻ ## Canonical subquotients and maps Let $\mathcal{G}$ be a class of groups. Roughly speaking, we say that a certain assignation defined on $\mathcal{G}$ is canonical if it "depends only on the isomorphism type of $kG$ as $k$-algebra". More precisely, suppose that for each $G$ in $\mathcal{G}$ we have associated a subquotient $U_G$ of $k G$ as $k$-space. We say that $G\mapsto U_G$ is canonical in $\mathcal{G}$ if every isomorphism $k$-algebras $\psi:kG\to kH$, with $G$ and $H$ in $\mathcal{G}$, induces an isomorphism $\tilde{\psi}:U_G\mapsto U_H$ in the natural way. If $(G\mapsto U_G^{(x)})_{x\in X}$ is a family of canonical subquotients in $\mathcal{G}$ then we also say that $G\mapsto \prod_{x\in X} U_G^{(x)}$ is canonical in $\mathcal{G}$. In this case every isomorphism $\psi:kG \to kH$ with $G$ and $H$ in $\mathcal{G}$ induces an isomorphism $\prod_{x\in X} U_G^{(x)}\to \prod_{x\in X} U_H^{(x)}$ in the natural way. **Lemma 8**. *The following assignations are canonical in the class of $p$-groups:* - *$G\mapsto \mathrm{I}(\Omega_n(G:G'))kG$.* - *$G\mapsto \mathrm{I}(\Omega_n(G:\mathrm{Z}(G)G'))kG$.* *Proof.* See [@GLdRS2022 Proposition 2.3(1) and Lemma 3.6]. ◻ **Lemma 9**. *[@GLdRS2022 Theorem 4.2(1)][\[lemma:canonicalsubgroups2\]]{#lemma:canonicalsubgroups2 label="lemma:canonicalsubgroups2"} The assignation $G\mapsto \mathrm{I}({\rm C}_G(G'))kG$ is canonical in the class of $p$-groups with cyclic derived subgroup and $p$ odd.* We note that, if $G\mapsto \mathrm{I}(N_G)kG$ is canonical in $\mathcal{G}$, where $N_G$ is a normal subgroup of $G$, then an easy induction on $n$ shows that $G\mapsto \mathrm{J}^{n}(N_G,G))$ is canonical in $\mathcal{G}$ too. Now suppose that for each $G$ in $\mathcal{G}$ we have associated a map $f_G: U_G \to V_G$, with $U$ and $V$ products of canonical subquotients in $\mathcal{G}$. We say that $G\mapsto f_G$ is *canonical* if for every isomorphism $\psi :kG\to k H$ the following square is commutative $$\xymatrix{ U_G \ar[d]_-{\tilde \psi} \ar[r]^-{f_G} & V_G \ar[d]^-{\tilde \psi} \\ U_H \ar[r]_-{f_H} & V_H }$$ For example, the assignation $G\mapsto \Lambda_G^n$ described above is canonical in the class of finite $p$-groups, and so is $G\mapsto \Delta_G$, where $\Delta_G$ is the natural projection: $$\Delta_G:\frac{\mathrm{I}(G')kG}{\mathrm{I}(G')\mathrm{I}(G)} \longrightarrow \frac{\mathrm{I}(G')kG+\mathrm{I}(G)^3}{\mathrm{I}(G)^3}, \quad x+ \mathrm{I}(G')\mathrm{I}(G) \mapsto x+ \mathrm{I}(G)^3.$$ Observe that $\Delta_G$ is well defined homomorphism of $k$-algebras because $\mathrm{I}(G')\subseteq \mathrm{I}(G)^2$. In order to simplify notation, instead of writing "$G\mapsto A_G$ is canonical" we just write "$A_G$ is canonical", where $A_G$ is either a product of subquotients or a map between canonical products of subquotients. For mnemonic purposes we use variations of the symbols $\Lambda^n$ and $\Upsilon^n$ for maps of the kind $x\mapsto x^{p^n}$. Moreover we will encounter a number of projection maps of the kind $x+I\mapsto x+J$ for ideals $I\subseteq J$, for which we use variations of the symbols $\Delta, \zeta$ and $\nu$, with the hope they help the reader to recall the domain: $\Delta$ refers to derived subgroup, $\zeta$ to center and $\nu$ to some normal subgroup $N$. Other projection maps are denoted with variations of $\pi$ and $\eta$. # 2-generated finite $p$-groups with cyclic derived subgroup {#SectionApp2Gen} The non-abelian 2-generated finite $p$-groups with cyclic derived subgroup have been classified in [@OsnelDiegoAngel] in terms of numerical invariants. For the reader's convenience, we include in the following theorem a simplification of this classification for the case $p>2$. **Theorem 10** ([@OsnelDiegoAngel]). *For a list of non-negative integers $I=(p,m,n_1,n_2,o_1,o_2,o'_1,o'_2,u_1,u_2)$ where $p>2$ is a prime number, let $\mathcal{G}_I$ be the group defined by $$\mathcal{G}_I=\left\langle b_1,b_2,a=[b_2,b_1] \mid a^{p^m}=1, a^{b_i}=a^{r_i}, b_i^{p^{n_i}}=a^{u_ip^{m-o'_i}} \right\rangle,$$ where $$\label{Erres} r_1=1+p^{m-o_1} \quad \text{and} \quad r_2 = \begin{cases} 1+p^{m-o_2}, & \text{if } o_2>o_1; \\ r_1^{p^{o_1-o_2}}, & \text{otherwise}. \end{cases}$$ Then $I\mapsto [\mathcal{G}_I]$, where $[\mathcal{G}_I]$ denotes the isomorphism class of $\mathcal{G}_I$, defines a bijection between the set of lists of integers $(p,m,n_1,n_2, o_1,o_2,o'_1,o'_2,u_1,u_2)$ satisfying conditions [\[1\]](#1){reference-type="ref" reference="1"}-[\[6\]](#6){reference-type="ref" reference="6"}, and the isomorphism classes of $2$-generated non-abelian groups of odd prime-power order with cyclic derived subgroup.* 1. *[\[1\]]{#1 label="1"} $p$ is prime and $n_1\geq n_2 \ge 1$.* 2. *[\[2\]]{#2 label="2"} $0\le o_i<\min(m,n_i)$, $0\le o'_i \le m-o_i$ and $p\nmid u_i$ for $i=1,2$.* 3. *[\[3\]]{#3 label="3"} One of the following conditions holds:* 1. *$o_1=0$ and $o'_1\le o'_2\le o'_1+o_2+n_1-n_2$.* 2. *$o_2=0<o_1$, $n_2<n_1$ and $o'_1+\min(0,n_1-n_ 2 -o_1)\le o'_2\le o'_1+n_1-n_2$.* 3. *$0<o_2< o_1<o_2+n_1-n_2$ and $o'_1\le o'_2\le o'_1+n_1-n_2$.* 4. *[\[4\]]{#4 label="4"} $o_2+o_1'\leq m\le n_1$ and one of the following conditions hold:* 1. *$o_1+o'_2\le m \le n_2$.* 2. *$2m-o_1-o'_2=n_2<m$ and $u_2\equiv 1 \mod p^{m-n_2}$.* 5. *[\[5\]]{#5 label="5"}$1\le u_1 \leq p^{a_1}$, where $a_1=\min(o'_1,o_2+\min(n_1-n_2+o'_1-o'_2,0)).$* 6. *[\[6\]]{#6 label="6"}One of the following conditions holds:* 1. *$1\le u_2 \leq p^{a_2}$.* 2. *$o_1o_2\neq 0$, $n_1-n_2+o_1'-o_2'=0<a_1$, $1+p^{a_2}\leq u_2 \leq 2p^{a_2}$, and $u_1 \equiv 1\mod p$;* *where $$a_2= \begin{cases} 0, &\text{if } o_1=0; \\ \min(o_1,o'_2,o'_2-o'_1+\max(0,o_1+n_2-n_1)), & \text{if } o_2=0<o_1; \\ \min(o_1-o_2,o'_2-o'_1), & \text{otherwise.} \end{cases}$$* For every non-abelian 2-generated finite $p$-group $\Gamma$ with cyclic derived subgroup and $p$ odd, let $\textup{inv}(\Gamma)$ denote the unique list satisfying the conditions of the previous theorem such that $\Gamma$ is isomorphic to $\mathcal{G}_{\textup{inv}(\Gamma)}$. An explicit description of $\textup{inv}(\Gamma)$ can be found in [@OsnelDiegoAngel] and also in [@GLdRS2022]. In these references the list $\textup{inv}(\Gamma)$ has two additional entries $\sigma_1$ and $\sigma_2$ which for $p>2$ always equal 1, so we drop them. In this section $\Gamma$ is a $2$-generated finite $p$-group with cyclic derived subgroup, and we set $$\textup{inv}(\Gamma)=(p,m,n_1,n_2,o_1,o_2,o'_1,o'_2,u_1,u_2).$$ Hence $\Gamma$ is given by the following presentation $$\Gamma=\left\langle b_1,b_2 \mid a=[b_2,b_1], a^{b_i}=a^{r_i}, b_i^{p^{n_i}}=a^{u_i p^{m-o'_i}} \right\rangle,$$ where $r_1$ and $r_2$ are as in [\[Erres\]](#Erres){reference-type="eqref" reference="Erres"}. By [@GLdRS2022 Lemma 3.5], $$\label{gamma} \gamma_n({\Gamma}) = \left\langle a^{p^{(n-2)(m-\max(o_1,o_2))}} \right\rangle, \text{ for } n\ge 2.$$ In particular $[\Gamma,\Gamma']=\gamma_3(\Gamma)\subseteq \left\langle a^p \right\rangle=(\Gamma')^p$, and hence, by , $$\mathrm{J}^{n}(\Gamma',\Gamma)=\mathrm{I}(\Gamma')^n\mathrm{I}(\Gamma) \text{ for every } n\ge 1.$$ By [@GLdRS2022 Lemma 2.2], there is a unique integer $\delta$ satisfying $$\label{eq:CenteredCongruence} 1\le \delta \le p^{o_1} \quad \text{and} \quad\mathcal{S}\left(r_2\mid \delta p^{m-o_1}\right)\equiv -p^{m-o_1}\mod p^m.$$ Moreover, $p\nmid \delta$. By [@GLdRS2022 Lemma 3.7] $$\label{eq:centerGens} \mathrm{Z}(\Gamma)=\left\langle b_1^{p^m}, b_2^{p^m}, c \right\rangle, \quad \text{where }c=\begin{cases} b_1^{\delta p^{m-o_2}}a, & \text{if }o_1=0; \\ b_1^{-\delta p^{m-o_2} } b_2^{\delta p^{ m-o_1}} a, & \text{otherwise}. \end{cases}$$ Observe that $$\label{nMenorni} n< n_i \quad \text{implies} \quad b_i^{p^n}\not\in {\mathrm{D}}_{p^n+1}(\Gamma)\Gamma', \text{ for } i=1,2.$$ Furthermore, for every $n\ge 0$, $$\label{Mpn} {\mathrm{D}}_{p^n}({\Gamma})={\Gamma}^{p^n}.$$ To prove this t suffices to show that $ip^j\ge p^n$ implies $\gamma_i({\Gamma})^{p^j}\subseteq \Gamma^{p^n}$. This is clear if $j\ge n$. Otherwise, $j<n$, $i\ge 2$ and $i-2\ge p^{n-j}-2\ge n-j$, since $p\ge 3$. Using [\[gamma\]](#gamma){reference-type="eqref" reference="gamma"} we obtain that $\gamma_i({\Gamma})^{p^j}=\left\langle a^{p^{j+(i-2)(m-\max(o_1,o_2))}} \right\rangle \subseteq \left\langle a^{p^n} \right\rangle \subseteq {\Gamma}^{p^n}$. Thus [\[Mpn\]](#Mpn){reference-type="eqref" reference="Mpn"} follows. Moreover, $$\label{eq:n1=m} n_1=m \qquad \text{implies} \qquad o_1o_2=0.$$ To see this, observe that if $o_1o_2>0$ and $n_1=m$ then $m>n_2$ by condition [\[3\]](#3){reference-type="ref" reference="3"}, so $n_2=2m-o_1-o_2'$ by condition [\[4\]](#4){reference-type="ref" reference="4"}. Thus, by conditions [\[2\]](#2){reference-type="ref" reference="2"} and [\[3\]](#3){reference-type="ref" reference="3"}, $o_1-o_2< n_1-n_2=o_1+o_2'- m \leq o_1-o_2$, a contradiction. In the rest of this section we assume the following: $$\label{Assumptions} o_1\ne o_2, \quad 0<\max(o'_1,o'_2)<m \quad \text{and} \quad n_2\ge 2.$$ In the next section we will see that this is the only case of interest, as if any of these conditions fails, then the Modular Isomorphism Problem has a positive solution for $\Gamma$. Observe that if $n<m-1$ then $\mathrm{I}(\Gamma')^{p^n}k\Gamma/\mathrm{I}(\Gamma')^{p^n}\mathrm{I}(\Gamma)$ is a one-dimensional $k$-space generated by the class of $a^{p^n}-1$. Moreover the image of $\Delta_\Gamma$ is spanned by $a-1+ \mathrm{I}(\Gamma)^3$. As $p$ is odd, $\Gamma^p = {\mathrm{D}}_3(\Gamma)$, and as $\max(o'_1,o'_2)<m$, $a\not\in \Gamma^p$. Thus $a-1\not\in \mathrm{I}(\Gamma)^3$. Then, we have the following **Lemma 11**. *$\Delta_\Gamma$ is an isomorphism.* **Lemma 12**. *$\hat C\in\mathrm{I}(\Gamma)^{(p-1)p^m}$ for each non-central conjugacy class $C$ of $\Gamma$.* *Proof.* By hypothesis $o_i'>0$ for some $i\in \{1,2\}$. In that case $m \leq n_i+o_i'-1$, by condition [\[4\]](#4){reference-type="ref" reference="4"}. Thus, it is enough to show that if $o'_i>0$, then $\hat C\in \mathrm{I}(\Gamma)^{(p-1)p^{n_i+o_i'-1}}$. If $x$ is an indeterminate over $k$ and $n\ge 1$ then we have $$\sum_{i=1}^{p^n-1} x^i = \frac{x^{p^n}-1}{x-1} = (x-1)^{p^n-1}.$$ Hence, using for each $C\in \mathop{\mathrm{Cl}}(\Gamma)$ such that $|C|>1$, and $g\in C$, there exists $0\leq n<m$ such that $$\begin{aligned} \hat C= \sum_{i=0}^{p^{m-n}-1} a^{ip^n}g= (a^{p^n}-1)^{p^{m-n}-1} g&=&(a^{p^n}-1)^{(p-1)p^{m-n-1} } (a^{p^n}-1)^{ p^ {m-n-1}-1}g \\ & =&(a^{p^{m-1}}-1)^{(p-1)} (a^{p^n}-1)^{p^{m-n-1}-1}g, \end{aligned}$$ and this element belongs to $\mathrm{I}(\Gamma)^{ (p-1)p^{n_i+o_i'-1} }$, as the hypothesis $o'_i>0$ implies $$a^{p^{m-1}}=b_i^{p^{n_i+o_i'-1}}\in {\mathrm{D}}_{p^{n_i+o_i'-1}}(\Gamma).$$ ◻ In the remainder of the section we consider a series of subquotients of $k\Gamma$ and maps which, by construction, are canonical in the class of 2-generated finite $p$-groups with cyclic derived subgroup satisfying [\[Assumptions\]](#Assumptions){reference-type="eqref" reference="Assumptions"}, and will play a central rôle in the proof of our main results. Recall from [@Sandling85 Lemma 6.10] that $$\label{eq:center} \mathrm{Z}(\mathrm{I}(\Gamma))=\mathrm{I}(\mathrm{Z}(\Gamma))\oplus \left( \bigoplus_{C\in \mathop{\mathrm{Cl}}(\Gamma),|C|>1} k \hat C \right).$$ Observe that as $o_i<m$ for $i=1,2$, $c\in {\mathrm{D}}_2(\Gamma)$, where $c$ is as in [\[eq:centerGens\]](#eq:centerGens){reference-type="eqref" reference="eq:centerGens"}, hence $c-1\in \mathrm{I}(\Gamma)^2$. Then and [\[eq:center\]](#eq:center){reference-type="eqref" reference="eq:center"} yield $$\begin{aligned} \label{eq:killcc} \begin{split} \frac{\mathrm{Z}(I( \Gamma))+ \mathrm{I}(\Gamma)^{p^m}}{\mathrm{I}(\Gamma)^{p^m}} &= \frac{\mathrm{I}(\mathrm{Z}(\Gamma))+ \mathrm{I}(\Gamma)^{p^m}}{\mathrm{I}(\Gamma)^{p^m}} \\ &= \frac{ k(c-1)+ k(c-1)^2+\dots + k(c-1)^{\frac{p^{m}-1}{2}} +\mathrm{I}(\Gamma)^{p^{m}}}{\mathrm{I}(\Gamma)^{p^{m}}}. \end{split}\end{aligned}$$ Hence, $$\frac{ \mathrm{Z}( \mathrm{I}(\Gamma))+\mathrm{I}(\Gamma)^{3 }}{\mathrm{I}(\Gamma)^{3}}=\frac{k(a-1)+\mathrm{I}(\Gamma)^{3 }}{\mathrm{I}(\Gamma)^{3 }}$$ since $c-a\in \mathrm{I}(\Gamma)^3$, and, for $o=\max(o_1,o_2)$, $$\label{ZIGpm} \frac{ \mathrm{Z}( \mathrm{I}(\Gamma))+\mathrm{I}(\Gamma)^{p^{m-o}+1} +\mathrm{I}(\Gamma')k\Gamma}{\mathrm{I}(\Gamma)^{p^{m-o}+1}+\mathrm{I}(\Gamma')k\Gamma}= \begin{cases} \frac{ k(b_1^{ p^{m-o_2}}-1)+\mathrm{I}(\Gamma)^{p^{m-o_2}+1} +\mathrm{I}(\Gamma')k\Gamma}{\mathrm{I}(\Gamma)^{p^{m-o_2}+1}+\mathrm{I}(\Gamma')k\Gamma}, & \text{if }o_1=0; \\ \frac{ k(b_2^{ p^{m-o_1}}-1)+\mathrm{I}(\Gamma)^{p^{m-o_1}+1} +\mathrm{I}(\Gamma')k\Gamma}{\mathrm{I}(\Gamma)^{p^{m-o_1}+1}+\mathrm{I}(\Gamma')k\Gamma}, &\text{if }o_1\ne 0. \end{cases}$$ This subquotient of $k \Gamma$ is one-dimensional by [\[nMenorni\]](#nMenorni){reference-type="eqref" reference="nMenorni"} and [@GL2022 Lemma 4.10]. Then we consider the canonical maps $$\zeta_\Gamma^1: \frac{ \mathrm{Z}( \mathrm{I}(\Gamma))+\mathrm{I}(\Gamma)^{p^{m}}}{\mathrm{I}(\Gamma)^{p^{m}}} \to \frac{ \mathrm{Z}( \mathrm{I}(\Gamma))+\mathrm{I}(\Gamma)^{3 }}{\mathrm{I}(\Gamma)^{3 }}, \ w+\mathrm{I}(\Gamma)^{p^m}\mapsto w+ \mathrm{I}(\Gamma)^3,$$ and $$\zeta_\Gamma^2: \frac{ \mathrm{Z}( \mathrm{I}(\Gamma))+\mathrm{I}(\Gamma)^{p^{m}}}{\mathrm{I}(\Gamma)^{p^{m}}} \to \frac{ \mathrm{Z}( \mathrm{I}(\Gamma))+\mathrm{I}(\Gamma)^{p^{m-o}+1} +\mathrm{I}(\Gamma')k\Gamma}{\mathrm{I}(\Gamma)^{p^{m-o}+1}+\mathrm{I}(\Gamma')k\Gamma}, \ w+\mathrm{I}(\Gamma)^{p^{m}}\mapsto w+\mathrm{I}(\Gamma)^{p^{m-o}+1}+\mathrm{I}(\Gamma')k\Gamma.$$ It is immediate that for $x_1,\dots,x_{(p^m-1)/2}\in k$, $$\zeta_\Gamma^1\left(\sum_{i=1}^{\frac{p^m-1}{2}} x_i(c-1)^i+\mathrm{I}(\Gamma)^{p^m}\right)= x_1(a-1)+\mathrm{I}(\Gamma)^3$$ and $$\zeta_\Gamma^2\left(\sum_{i=1}^{\frac{p^m-1}{2}} x_i(c-1)^i+\mathrm{I}(\Gamma)^{p^m}\right)=\begin{cases} x_1 (b_1^{p^{m-o_2}}-1)+\mathrm{I}(\Gamma)^{p^{m-o}+1}+\mathrm{I}(\Gamma')k\Gamma, &\text{if }o_1=0;\\ x_1\delta(b_2^{p^{m-o_1}}-1)+\mathrm{I}(\Gamma)^{p^{m-o}+1}+\mathrm{I}(\Gamma')k\Gamma, & \text{if }o_1\ne 0. \end{cases}$$ The first implies that $\mbox{\rm Im }(\zeta^1_\Gamma)=\mbox{\rm Im }(\Delta_\Gamma)$. For each $n\geq 1$ let $$\mathcal{C}_\Gamma=\frac{\mathrm{I}(C_\Gamma(\Gamma'))k\Gamma+\mathrm{I}(\Gamma)^2}{\mathrm{I}(\Gamma)^2}= \begin{cases} \frac{k(b_1-1)+\mathrm{I}(\Gamma)^2}{\mathrm{I}(\Gamma)^2}, &\text{if }o_1=0; \\ \frac{k(b_2-1)+\mathrm{I}(\Gamma)^2}{\mathrm{I}(\Gamma)^2}, & \text{if } o_1\ne 0. \end{cases}$$ Then $$\label{ImagenLambda} \Lambda_{\Gamma}^{n}(\mathcal{C}_\Gamma)= \begin{cases}\frac{k(b_1-1)^{p^{n}}+ \mathrm{I}(\Gamma)^{p^{n}+1}}{\mathrm{I}(\Gamma)^{p^{n}+1}}, &\text{if }o_1=0; \\ \frac{k(b_2-1)^{p^n}+ \mathrm{I}(\Gamma)^{p^n+1}}{\mathrm{I}(\Gamma)^{p^n+1}},&\text{if }o_1\ne 0. \end{cases}$$ Let $\tilde \Lambda_{\Gamma}^{n}:\mathcal{C}_\Gamma\to \Lambda^n_\Gamma(\mathcal{C}_\Gamma)$ be the restriction of $\Lambda_{\Gamma}^{n}$ to $\mathcal{C}_\Gamma$. By [\[nMenorni\]](#nMenorni){reference-type="eqref" reference="nMenorni"}, $$\label{LambdaTildeIso} \text{if either $o_1=0$ and $n<n_1$ or $o_1\ne 0$ and $n<n_2$, then $\tilde\Lambda^n_\Gamma$ is an isomorphism.}$$ Observe that $m-o<n_i$ for $i=1,2$. Indeed, if $m-o\ge n_i$ then, as $o>0$ and $o'_2<m$, by condition [\[Assumptions\]](#Assumptions){reference-type="eqref" reference="Assumptions"}, $i=2$ and $n_2=2m-o_1-o'_2> m-o_1\ge m-o$, a contradiction. Thus $\tilde\Lambda^{m-o}_\Gamma$ is an isomorphism and hence $\Lambda^{m-o}_\Gamma(\mathcal{C}_\Gamma)$ is one-dimensional. Therefore we have isomorphisms $$\begin{aligned} \label{CG} \mathcal C_\Gamma \stackrel{\tilde \Lambda^{m-o}_\Gamma}{\longrightarrow} \Lambda^{m-o}_\Gamma(\mathcal{C}_\Gamma) \stackrel{\pi_\Gamma}{\longrightarrow} \frac{ \mathrm{Z}( \mathrm{I}(\Gamma))+\mathrm{I}(\Gamma)^{p^{m-o}+1} +\mathrm{I}(\Gamma')k\Gamma}{\mathrm{I}(\Gamma)^{p^{m-o}+1}+\mathrm{I}(\Gamma')k\Gamma} \end{aligned}$$ where $\pi_\Gamma$ is another natural projection, i.e. $\pi_\Gamma\left(x+\mathrm{I}(\Gamma)^{p^{m-o}+1}\right) = x+\mathrm{I}(\Gamma)^{p^{m-o}+1}+\mathrm{I}(\Gamma')k\Gamma$. # Proof of the main results {#SectionProofs} Recall that $p$ is an odd prime integer and $k$ the field with $p$ elements. For the remainder of the paper, we fix the following notation. Let $G$ denote a $2$-generated finite $p$-group with cyclic derived subgroup, let $H$ denote another group and let $\psi:kG\rightarrow kH$ be an isomorphism of $k$-algebras. By [@GLdRS2022 Theorem C], $H$ is $2$-generated with cyclic derived subgroup, and $\textup{inv}(G)$ and $\textup{inv}(H)$ coincide in all but the last entries. So we may write $$\textup{inv}(G)=(p,m,n_1,n_2,o_1,o_2,o'_1,o'_2,u_1^G,u_2^G) \quad \text{and} \quad \textup{inv}(H)=(p,m,n_1,n_2,o_1,o_2,o'_1,o'_2,u_1^H,u_2^H).$$ To give a positive answer to the Modular Isomorphism Problem in this case we should prove that $G\cong H$, or equivalently that $u_i^G=u_i^H$ for $i=1,2$. Unfortunately, we are only able to prove the statement of , namely that $u_2^G\equiv u_2^H \mod p$ and, under some extra assumptions, that $u_1^G\equiv u_1^H \mod p$. By statements [\[Metacyclic\]](#Metacyclic){reference-type="eqref" reference="Metacyclic"} and [\[2GenClass2\]](#2GenClass2){reference-type="eqref" reference="2GenClass2"} of , we may assume that the groups $G$ and $H$ are not metacyclic, and both are of class at least 3. The first is equivalent to $\max(o_1,o_2)>0$ and the second is equivalent to $\max(o'_1,o'_2)<m$. In particular, $m\ge 2$. Moreover $n_2\geq 2$, as otherwise $n_2<m$ and condition [\[4\]](#4){reference-type="ref" reference="4"} yields $1=n_2=2m-o_1-o_2'$, but this last quantity is strictly greater than $1$ because $\max(o_1,o_2')<m$, by condition [\[2\]](#2){reference-type="ref" reference="2"} and since $\Gamma$ is not metacyclic. We also have that $o_1\neq o_2$ by condition [\[3\]](#3){reference-type="ref" reference="3"}. Finally, if $o_i'=0$ for some $i\in \{1,2\}$, then $u_i^G=1=u_i^H$ by conditions [\[5\]](#5){reference-type="ref" reference="5"} and [\[6\]](#6){reference-type="ref" reference="6"}; therefore we can assume that $\max(o_1',o_2')>0$. Thus the conditions in [\[Assumptions\]](#Assumptions){reference-type="eqref" reference="Assumptions"} hold, so we can freely use the statements of the previous section. In order to deal with $G$ and $H$ simultaneously, in the remainder of the paper ${\Gamma}$ denotes a 2-generated finite $p$-group with cyclic derived subgroup such that $$\textup{inv}({\Gamma})=(p,m,n_1,n_2,o_1,o_2,o'_1,o'_2,u_1^{\Gamma},u_2^{\Gamma}).$$ ## Proof of Recall that $o=\max(o_1,o_2)$. We let $$N_{\Gamma}= \begin{cases} \Omega_{m-o-1}({\Gamma}:Z({\Gamma}){\Gamma}'), & \text{if either } o_1=0 \text{ or } o_2=0 \text{ and } o'_1\ge o'_2;\\ \Omega_{n_2-1}({\Gamma}:{\Gamma}'), & \text{otherwise} \end{cases}$$ and $$\mathcal{N}_{\Gamma}=\frac{ \mathrm{I}(N_{\Gamma})k{\Gamma}\cap \mathrm{I}({\Gamma})^p}{\mathrm{I}(N_{\Gamma}) \mathrm{I}({\Gamma})}.$$ By , the subquotients $\mathrm{I}(N_{\Gamma})k{\Gamma}$, $\mathrm{J}^{n}(N_{\Gamma},{\Gamma})$ and $\mathcal{N}_{\Gamma}$ are canonical. Moreover, $$\label{NGG} N_{\Gamma}=\left\langle a,d,e \right\rangle, \quad \text{ where } \quad (d,e) = \begin{cases} (b_1^p,b_2^{p^{o_2+1}}), & \text{if } o_1=0; \\ (b_2^p,b_1^{p^{o_1+1}}), & \text{if } o_2=0 \text{ and } o'_1\ge o'_2; \\ (b_2^p,b_1^{p^{n_1-n_2+1}}), & \text{otherwise}; \end{cases}$$ and $\mathcal{N}_{\Gamma}$ is spanned by the classes of $d-1$ and $e-1$. **Lemma 13**. *For every $n\ge 0$, $\mathrm{J}^{n}(N_{\Gamma},{\Gamma})=\mathrm{I}(N_{\Gamma})^n\mathrm{I}({\Gamma})$.* *Proof.* Suppose first that either $o_1=0$ or $o_2=0$ and $o'_1\ge o'_2$. Then $\gamma_1^{\Gamma}(N_{\Gamma})={\Gamma}$, $\gamma_2^{\Gamma}(N_{\Gamma})= ({\Gamma}')^p$, and $\gamma_i^{\Gamma}(N_{\Gamma})=1$ for $i\geq 3$. Since ${\Gamma}'\subseteq N_{\Gamma}$ and , it follows that $$\mathrm{I}(N_{\Gamma})^{n-1} \mathrm{I}(({\Gamma}')^p)k{\Gamma} \subseteq \mathrm{I}(N_{\Gamma})^{n-1+p} k{\Gamma}\subseteq \mathrm{I}(N_{\Gamma})^{n}\mathrm{I}({\Gamma}).$$ Then the desired equality follows from [\[eq72\]](#eq72){reference-type="eqref" reference="eq72"}. Suppose that $o_1\ne 0$ and either $o_2\ne 0$ or $o'_1<o'_2$. Then again $\gamma_1^{\Gamma}(N_{\Gamma})={\Gamma}$, $\gamma_2^{\Gamma}(N_{\Gamma})= ({\Gamma}')^p$ and $\mathrm{I}(N_{\Gamma})^{n-1} \mathrm{I}(({\Gamma}')^p)k{\Gamma} \subseteq \mathrm{I}(N_{\Gamma})^{n}\mathrm{I}({\Gamma})$. For $i\ge 3$, an easy induction argument, using the description of $N_{\Gamma}$ in [\[NGG\]](#NGG){reference-type="eqref" reference="NGG"}, shows that $\gamma_i^{\Gamma}(N_{\Gamma}) = ({\Gamma}')^{p^{1+(i-2)k}}$, where $k=n_1-n_2+1+m-o_1$ if $o_2=0$, and $k=1+m-o_2$ otherwise. Either way $k\ge 2$ and hence $$\mathrm{I}(N_{\Gamma})^{n+1-i} \mathrm{I}(\gamma_i^{\Gamma}(N_{\Gamma}))k{\Gamma} \subseteq \mathrm{I}(N_{\Gamma})^{n+1-i} \mathrm{I}(({\Gamma}')^{p^{1+(i-2)k}})k{\Gamma} \subseteq \mathrm{I}(N_{\Gamma})^{n+1-i+p^{1+(i-2)k}}k{\Gamma}\subseteq \mathrm{I}(N_{\Gamma})^n\mathrm{I}({\Gamma}).$$ Then again [\[eq72\]](#eq72){reference-type="eqref" reference="eq72"} yields the desired equality. ◻ Denote $$\ell=\begin{cases} n_1+o'_1-2, & \text{if } o_1=0; \\ n_2+o'_2-2, & \text{otherwise}.\end{cases}$$ Combining and [\[Usefi\]](#Usefi){reference-type="eqref" reference="Usefi"} and using regularity it is easy to obtain $$\label{JUsefi} {\Gamma}\cap (1+\mathrm{J}^{p^{\ell}}(N_{\Gamma},{\Gamma})) = 1.$$ The next lemma covers most cases of . **Lemma 14**. *The following hold:* 1. *$u_2^G\equiv u_2^H \mod p$.* 2. *If $o_1o_2=0$ then $u_1^G\equiv u_1^H \mod p$.* *Proof.* Let $t\in \{1,2\}$ with $t=2$ in case $o_1o_2\ne 0$, and let $s$ be the other element of $\{1,2\}$, i.e. $\{s,t\}=\{1,2\}$. We have to prove that $u_t^G\equiv u_t^H\mod p$. If $a_t=0$ then $u_t^G=u_t^G=1$, so we assume that $a_t\ne 0$. In particular, $o'_t>0$ and $o_s>0$. Therefore $$t=\begin{cases} 1, & \text{if } o_1=0;\\ 2, & \text{otherwise}. \end{cases}$$ So, $\ell=n_t+o'_t-2$. If $t=1$ then $n_1+o'_1+o_2>n_2+o'_2$, by condition [\[5\]](#5){reference-type="ref" reference="5"}, as $a_1>0$. If $t=2$ and $o_1'\ge o_2'$ then, by condition [\[6\]](#6){reference-type="ref" reference="6"}, $o'_2-o'_1\le 0<a_2\le o'_2-o'_1+\max(0,o_1+n_2-n_1)$ and hence $n_1+o_1'< n_2+o_2'+o_1$ and $o_2=0$. We claim that for $x,y\in k$ $$\label{Lamdaell} \Lambda^{\ell}_{N_{\Gamma}}(x(d-1)+y(e-1)+\mathrm{I}(N_{\Gamma})\mathrm{I}({\Gamma})) = x u_t^{\Gamma}(a^{p^{m-1}}-1)+\mathrm{J}^{p^{\ell}}(N_{\Gamma},{\Gamma}).$$ Indeed, if $t=1$ then $o_1=0$, $o'_1>0$, $n_1+o'_1+o_2>n_2+o'_2$, $\ell=n_1+o'_1-2$, $d=b_1^p$ and $e=b_2^{p^{o_2+1}}$. Thus $$\begin{aligned} \Lambda^{\ell}_{N_{\Gamma}}(x(d-1)+y(e-1)+\mathrm{I}(N_{\Gamma})\mathrm{I}({\Gamma})) &=& x(b_1^{p^{n_1+o'_1-1}}-1) + y(b_2^{p^{n_1+o'_1+o_2-1}}-1)+\mathrm{J}^{p^{\ell}}(N_{\Gamma},{\Gamma}) \\ &=& xu_1^{\Gamma}(a^{p^{m-1}-1}-1)+\mathrm{J}^{p^{\ell}}(N_{\Gamma},{\Gamma}).\end{aligned}$$ Suppose that $t=2$. Then $o'_2>0$, $o_1>0$ and $\ell=n_2+o'_2-2$. If $o'_2\le o'_1$ then $o_2=0$ and $n_2+o'_2+o_1> n_1+o'_1$, and [\[Lamdaell\]](#Lamdaell){reference-type="eqref" reference="Lamdaell"} follows as in the previous case. If $o'_2>o'_1$ then $$\begin{aligned} \Lambda^{\ell}_{N_{\Gamma}}(x(d-1)+y(e-1)+\mathrm{I}(N_{\Gamma})\mathrm{I}({\Gamma})) &=& x(b_2^{p^{n_2+o'_2-1}}-1) + y(b_1^{p^{n_1+o'_2-1}}-1)+\mathrm{J}^{p^{\ell}}(N_{\Gamma},{\Gamma}) \\ &=& xu_2^{\Gamma}(a^{p^{m-1}-1}-1)+\mathrm{J}^{p^{\ell}}(N_{\Gamma},{\Gamma}).\end{aligned}$$ This finishes the proof of [\[Lamdaell\]](#Lamdaell){reference-type="eqref" reference="Lamdaell"}. By [\[JUsefi\]](#JUsefi){reference-type="eqref" reference="JUsefi"} and [\[Lamdaell\]](#Lamdaell){reference-type="eqref" reference="Lamdaell"}, $\Lambda^{\ell}_{N_{\Gamma}}(\mathcal{N}_{\Gamma})$ is one dimensional spanned by the class of $a^{p^{m-1}}-1$. Moreover, as $o'_t>0$, $a^{u_t^{\Gamma}p^{m-1}} = d^{p^\ell} \in N_{\Gamma}^{p^{\ell}}$ and hence the natural projection defines an isomorphism $$\Delta'_{\Gamma}:\frac{\mathrm{I}({\Gamma}')^{p^{m-1}} k{\Gamma}}{\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma})}\to \Lambda_{N_{\Gamma}}^\ell(\mathcal{N}_{\Gamma}).$$ Using [\[NGG\]](#NGG){reference-type="eqref" reference="NGG"} and [\[ImagenLambda\]](#ImagenLambda){reference-type="eqref" reference="ImagenLambda"} it is easy to see that the natural projections $$\eta_{\Gamma}:\mathcal N_{\Gamma}\to \frac{\mathrm{I}(N_{\Gamma})k{\Gamma}+\mathrm{I}({\Gamma})^{p+1}} { \mathrm{I}({\Gamma}')k{\Gamma}+\mathrm{I}({\Gamma})^{p+1}} \quad \text{and} \quad \Lambda_{\Gamma}^1(\mathcal C_{\Gamma}) \to \frac{\mathrm{I}(N_{\Gamma})k{\Gamma}+\mathrm{I}({\Gamma})^{p+1}} { \mathrm{I}({\Gamma}')k{\Gamma}+\mathrm{I}({\Gamma})^{p+1}}$$ make sense, their images coincide and the second map is injective. Thus the natural projection induces an isomorphism $\Lambda_{\Gamma}^1(\mathcal C_{\Gamma})\to \eta_{\Gamma}(\mathcal N_{\Gamma})$. On the other hand, by [\[LambdaTildeIso\]](#LambdaTildeIso){reference-type="eqref" reference="LambdaTildeIso"}, $\tilde\Lambda^1_{\Gamma}:\mathcal{C}_{\Gamma}\to \Lambda^1_{\Gamma}(\mathcal{C}_{\Gamma})$ is an isomorphism. Composing these isomorphisms we obtain an isomorphism $$\hat\Lambda^1_{\Gamma}:\mathcal{C}_{\Gamma}\to \mbox{\rm Im }(\eta_{\Gamma}), \quad w+\mathrm{I}({\Gamma})^2\mapsto w^p+\mathrm{I}({\Gamma}')k{\Gamma}+\mathrm{I}({\Gamma})^{p+1}.$$ This provides another canonical map $$\nu_{\Gamma}= (\hat\Lambda^1_{\Gamma})^{-1}\circ \eta_{\Gamma}: \mathcal{N}_{\Gamma}\to \mathcal{C}_{\Gamma},\quad w+\mathrm{I}(N_\Gamma)\mathrm{I}(\Gamma)\mapsto (\hat\Lambda^1_{\Gamma})^{-1}(w+\mathrm{I}({\Gamma}')k{\Gamma}+\mathrm{I}({\Gamma})^{p+1}).$$ Define the linear map $$\mu_{\Gamma}:\mathcal{C}_{\Gamma}\to \frac{\mathrm{I}({\Gamma}')^{p^{m-1}} k{\Gamma}}{\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma})}$$ sending the class of $x(b_t-1)$ to the class of $xu_t^{\Gamma}(a^{p^{m-1}}-1)$. A straightforward calculation shows that the following diagram commutes. $$\xymatrix{ \mathcal{N}_{\Gamma}\ar[d]_-{\nu_{\Gamma}} \ar[rr]^-{(\Delta'_{\Gamma})^{-1}\circ\Lambda_{N_{\Gamma}}^{p^{\ell}}}& & \frac{\mathrm{I}({\Gamma}')^{p^{m-1}} k{\Gamma}}{\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma})} \\ \mathcal{C}_{\Gamma}\ar[rru]_-{\mu_{\Gamma}} & & }$$ As the vertical map is surjective, $\mu_{\Gamma}$ is the unique map making the previous commutative. Then $\mu_{\Gamma}$ is canonical, since the other maps in the diagram are so. Consider the following equation where $X$ stands for an element of $k$. $$\label{diagram1} X\cdot \left(\Lambda_{{\Gamma}'}^{p^{m-1}}\circ \Delta_{\Gamma}^{-1}\circ\zeta_{\Gamma}^1\right) = \mu_\Gamma\circ (\tilde \Lambda_{{\Gamma}}^{p^ {m-o}})^{-1}\circ\pi_{\Gamma}^{-1}\circ \zeta_{\Gamma}^2.$$ Here, given a map $f$ with codomain in a vector space over $k$ and $x\in k$, $x\cdot f$ denotes the map given by $(x\cdot f)(w)=xf(w)$, for each $w$ in the domain of $f$. The unique solution for equation [\[diagram1\]](#diagram1){reference-type="eqref" reference="diagram1"} is $X=\delta u_t^{\Gamma}1_k$. Since all the maps involved are canonical, the solution when ${\Gamma}=G$ coincides with the solution when ${\Gamma}=H$. Furthermore, $p\nmid \delta$ and thus $u_t^G \equiv u^H_t \bmod p$, as desired. ◻ Most of the remaining cases of are covered by the next lemma. **Lemma 15**. *If $n_1+o_1'\neq n_2+o_2'$, then $u_1^G\equiv u_1^H\mod p$.* *Proof.* By we may assume that $o_1o_2\ne 0$. Hence condition [\[3\]](#3){reference-type="ref" reference="3"} and the hypothesis imply $n_1+o_1'> n_2+o_2'$. As in the proof of we may assume that $a_1>0$ and hence $o_1'>0$. Consider the subgroup $$M_{\Gamma}=\Omega_{n_2-m+o_1}({\Gamma}:{\Gamma}')=\left\langle b_1^{p^{n_1-n_2+m-o_1}}, b_2^{p^{m-o_1}}, a \right\rangle.$$ Recall that $c=b_1^{-\delta p^{m-o_2}} b_2^{\delta p^{m-o_1}}a$ and $\frac{\mathrm{Z}(\mathrm{I}({\Gamma}))+ \mathrm{I}({\Gamma})^{p^m}}{\mathrm{I}({\Gamma})^{p^m}}$ is spanned by the classes of $c-1,(c-1)^2,\dots,(c-1)^{\frac{p^m-1}{2}}$. The natural projection $$\zeta_{\Gamma}^3: \frac{\mathrm{Z}(\mathrm{I}({\Gamma}))+ \mathrm{I}({\Gamma})^{p^m}}{\mathrm{I}({\Gamma})^{p^m}} \to \frac{\mathrm{Z}(\mathrm{I}({\Gamma}))+ \mathrm{I}({\Gamma})^{p^{m-o_2}+1}+\mathrm{I}(M_{\Gamma})k{\Gamma}}{\mathrm{I}({\Gamma})^{p^{m-o_2}+1}+\mathrm{I}(M_{\Gamma})k{\Gamma}}$$ maps the class of $x(c-1)+ y(c-1)^2+ \dots$ to the class of $-x\delta (b_1^{p^{m-o_2}}-1)$, which is non-zero if $x\neq 0$ because $n_1-n_2+m-o_1>m-o_2$. So $\mbox{\rm Im }(\zeta_{\Gamma}^3)$ is $1$-dimensional. Now consider the composition $$\hat{\Lambda}^{m-o_2}_{\Gamma}: \frac{\mathrm{I}({\Gamma})}{\mathrm{I}({\Gamma})^2} \stackrel{\Lambda^{m-o_2}_{\Gamma}}{\longrightarrow} \frac{\mathrm{I}({\Gamma})^{p^{m-o_2}}}{\mathrm{I}({\Gamma})^{p^{m-o_2}+1}} \longrightarrow \frac{\mathrm{I}({\Gamma})^{p^{m-o_2}}+\mathrm{I}(M_{\Gamma})k{\Gamma}}{\mathrm{I}({\Gamma})^{p^{m-o_2}+1}+\mathrm{I}(M_{\Gamma})k{\Gamma}}$$ where the second map is the natural projection. It maps $x(b_1-1)+y(b_2-1)$ to $x(b_1^{p^{m-o_2}}-1)$, so $\mbox{\rm Im }(\hat{\Lambda}^{m-o_1}_{\Gamma})=\mbox{\rm Im }(\zeta^3_{\Gamma})$. The image of $\Lambda_{\Gamma}^{n_1+o'_1-1}$ is the subspace of $\mathrm{I}({\Gamma})^{p^{n_1+o'_1-1}}/\mathrm{I}({\Gamma})^{p^{n_1+o'_1-1}+1}$ spanned by the class of $a^{p^{m-1}}-1$. It coincides with the image of the natural projection $$\frac{\mathrm{I}({\Gamma}')^{p^{m-1}} k{\Gamma}}{\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma})} \to \frac{\mathrm{I}({\Gamma})^{p^{n_1+o'_1-1}}}{\mathrm{I}({\Gamma})^{p^{n_1+o'_1-1}+1}}.$$ Thus this natural projection yields an isomorphism $\tilde\Delta_{\Gamma}:\frac{\mathrm{I}({\Gamma}')^{p^{m-1}} k{\Gamma}}{\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma})}\to \mbox{\rm Im }(\Lambda_{\Gamma}^{n_1+o'_1-1})$. Let $\mu_{\Gamma}:\mbox{\rm Im }(\zeta^3_{\Gamma})\to \frac{\mathrm{I}({\Gamma}')^{p^{m-1}} k{\Gamma}}{\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma})}$ be the map that sends the class of $x (b_1^{p^{m-o_1}}-1)$ to the class of $xu_1(a^{p^{m-1}}-1)$. Then it is easy to see that the following diagram commutes $$\xymatrix{ \frac{\mathrm{I}({\Gamma})}{\mathrm{I}({\Gamma})^2} \ar[rr]^-{ \tilde \Delta_{\Gamma}^{-1}\circ\Lambda_{{\Gamma}}^{ n_1+o_1'-1 }} \ar[d]_-{\hat{\Lambda}^{m-o_2}_{\Gamma}}& & \frac{\mathrm{I}({\Gamma}')^{p^{m-1}} k{\Gamma}}{\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma})} \\ \mbox{\rm Im }( \zeta_{\Gamma}^3) \ar[rru]_-{ \mu_{\Gamma}} & & }$$ As the vertical map is surjective, $\mu_{\Gamma}$ is the unique map making the previous commutative, so $\mu_{\Gamma}$ is canonical. Then $-\delta u_1^{\Gamma} 1_k$ is the unique solution of the equation $$X \cdot (\Lambda_{{\Gamma}'}^{p^{m-1}} \circ \Delta_{\Gamma}^{-1}\circ\zeta_{\Gamma}^1)= \mu_{\Gamma}\circ \zeta_{\Gamma}^3.$$ Arguing as at the end of the proof of we conclude that $u_1^G\equiv u_1^H\mod p$. ◻ The proof of fails if $n_1+o_1'=n_2+o_2'$, because in that case $\ker(\hat{\Lambda}^{m-o_2}_{\Gamma})\not\subseteq \ker(\Delta_{\Gamma}^{-1}\circ \Lambda_{{\Gamma}}^{n_1+o_1'-1})$, and hence there is no map $\mu_{\Gamma}$ such that $\mu_{\Gamma}\circ \hat{\Lambda}^{m-o_2}_{\Gamma}= \Delta_{\Gamma}^{-1}\circ \Lambda_{{\Gamma}}^{n_1+o_1'-1}$. However, some special subcases can be handled with slight modifications of the previous arguments. For a non-negative integer $n$ define the map $$\begin{aligned} \Upsilon^n_{\Gamma}:\frac{\mathrm{Z}(\mathrm{I}({\Gamma}))+\mathrm{I}({\Gamma})^{p^m}}{\mathrm{I}({\Gamma})^{p^m}} & \longrightarrow & \frac{\mathrm{Z}(\mathrm{I}({\Gamma}))+\mathrm{I}({\Gamma})^{p^{n+m}}+\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma})}{\mathrm{I}({\Gamma})^{p^{n+m}}+\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma})} \\ w+\mathrm{I}({\Gamma})^{p^m} & \mapsto & w^{p^n}+\mathrm{I}({\Gamma})^{p^{n+m}}+\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma}). \end{aligned}$$ It is well defined because the elements of $\mathrm{Z}(\mathrm{I}({\Gamma}))$ are central. **Lemma 16**. *If $o_1o_2>0$, $n_1+o_1'=n_2+o_2'=2m-o_1$ and $u_2^G\equiv u_2^H \equiv 1 \mod p^{o_1+1-o_2}$, then $u_1^G\equiv u_1^H \mod p$.* *Proof.* As in previous proofs we may assume that $a_1\ne 0$ and hence $0<o'_1$. As $o_1o_2>0$ implies $n_1>n_2$, necessarily $1\le o_1'<o_2'$. Recall that $\mathrm{Z}({\Gamma})=\left\langle b_1^{p^m},b_2^{p^m},c \right\rangle$, where $c =b_1^{-\delta p^{m-o_2}}b_2^{\delta p^{m-o_1}}a$. We claim that $$\label{Congruenciadelta} (\delta u_2^{\Gamma}+1)p^{m+o_2-o_1-1} \equiv 0 \bmod p^m.$$ To prove this, it suffices to show that $\delta\equiv -1 \mod p^{o_1+1-o_2}$. As $v_p(r_2-1)=m-o_2$, $m-o_1\ge 1 = m+1-o_2-v_p(r_2)$. Hence [@OsnelDiegoAngel Lemma A.2] yields $\mathcal{S}\left(r_2\mid \delta p^{m-o_1}\right)\equiv \delta p^{m-o_1} \mod p^{m+1-o_2}$. Thus [\[eq:CenteredCongruence\]](#eq:CenteredCongruence){reference-type="eqref" reference="eq:CenteredCongruence"} implies that $\delta \equiv -1 \mod p^{o_1+1-o_2}$. This proves [\[Congruenciadelta\]](#Congruenciadelta){reference-type="eqref" reference="Congruenciadelta"}. Next we claim that $$\label{powerofc} c^{p^{ n_1+o_1'-1 -m+o_2}}=a^{-\delta u_1^{\Gamma}p^{m-1}}.$$ Indeed, first observe that condition implies $$\label{Desigualdad} n_1+o_1'-1=2m-o_1-1 \ge 2m -o_2 +n_2-n_1 = 2m-o_2-o_2'+o_1' \geq 2m-o_2-o_2' \ge m-o_2.$$ Thus the exponent in the left side of [\[powerofc\]](#powerofc){reference-type="eqref" reference="powerofc"} is a positive integer. Observe that $$\left\langle b_1^{p^{m-o_2}}, b_2^{p^{m-o_1}},a \right\rangle$$ is a regular group with derived subgroup $\left\langle a^{p^{2m-o_1-o_2}} \right\rangle$. As $m-o_2'+2m-o_1-o_2= 3m-o_1-o_2-o_2'\geq 2m -o_1>m$ (since $o_2+o_2'\leq m$), we derive that $$c^{p^{m-o_2'}}= b_1^{-\delta p^{2m-o_2-o_2'}} b_2^{\delta p^{2m-o_1-o_2'}} a^{p^{m-o_2'}}= b_1^{-\delta p^{2m-o_2-o_2'}} b_2^{\delta p^{n_2}} a^{p^{m-o_2'}}= b_1^{-\delta p^{2m-o_2-o_2'}} a^{(\delta u_2^{\Gamma}+1)p^{m-o_2'}}.$$ As $b_1^{p^{2m-o_2-o_2'}}\in \mathrm{Z}({\Gamma})$ and recalling [\[Desigualdad\]](#Desigualdad){reference-type="eqref" reference="Desigualdad"} we get $$\begin{aligned} c^{p^{ n_1+o_1'-1 -m+o_2}} &= (c^{p^{m-o_2'}})^{p^{n_1+o_1'-1-(2m-o_2-o_2')}}= b_1^{-\delta p^{n_1+o_1'-1}} a^{ (\delta u_2^{\Gamma}+1) p^{n_1+o_1'-1-m+o_2}} \\ &=a^{-\delta u_1^{\Gamma}p^{m-1}}a^{(\delta u_2^{\Gamma}+1) p^{m+o_2-o_1-1}} = a^{-\delta u_1^{\Gamma}p^{m-1}},\end{aligned}$$ where the last equality follows from [\[Congruenciadelta\]](#Congruenciadelta){reference-type="eqref" reference="Congruenciadelta"}. This proves [\[powerofc\]](#powerofc){reference-type="eqref" reference="powerofc"}. Using [\[powerofc\]](#powerofc){reference-type="eqref" reference="powerofc"} we obtain that $\Upsilon^{n_1+o'_1+o_2-m-1}_{\Gamma}$ maps the class of $\sum_{i=1}^{\frac{p-1}{2}} x_i(c-1)^i$, with $x_i\in k$, to the class of $-x_1\delta u_1^{\Gamma}(a^{p^{m-1}}-1)$. If $x_1\ne 0$, then the latter is not the class zero, by . Then the natural projection defines an isomorphism $\pi_{\Gamma}:\frac{\mathrm{I}({\Gamma}')^{p^{m-1}} k{\Gamma}}{\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma})}\to \mbox{\rm Im }(\Upsilon^{n_1+o'_1+o_2-m-1}_{\Gamma})$. So we have a canonical map $$\pi_{\Gamma}^{-1}\circ \Upsilon^{n_1+o'_1+o_2-m-1}_{\Gamma}:\frac{\mathrm{Z}(\mathrm{I}({\Gamma}))+\mathrm{I}({\Gamma})^{p^m}}{\mathrm{I}({\Gamma})^{p^m}} \to \frac{\mathrm{I}({\Gamma}')^{p^{m-1}} k{\Gamma}}{\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma})},$$ mapping the class of $\sum_{i=1}^{\frac{p-1}{2}} x_i(c-1)^i$ to the class of $x_1(-\delta u_1^{\Gamma}) (a^{p^{m-1}}-1)$. But we also have the canonical map $$\Lambda_{{\Gamma}'}^{m-1} \circ \Delta_{\Gamma}^{-1}\circ \zeta_{\Gamma}^1: \frac{\mathrm{Z}(\mathrm{I}({\Gamma}))+\mathrm{I}({\Gamma})^{p^m}}{\mathrm{I}({\Gamma})^{p^m}} \to \frac{\mathrm{I}({\Gamma}')^{p^{m-1}}k{\Gamma}}{\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma})}$$ that maps the class of $\sum_{i=1}^{\frac{p-1}{2}} x_i(c-1)^i$ to the class of $x_1(a^{p^{m-1}}-1)$. Thus the unique element $x\in k$ such that $\pi_{\Gamma}^{-1}\circ \Upsilon^{n_1+o'_1+o_2-m-1}_{\Gamma}=x\cdot (\Lambda_{{\Gamma}'}^{m-1} \circ \Delta_{\Gamma}^{-1}\circ \zeta_{\Gamma}^1)$ is $-\delta u_1^\Gamma 1_k$. Since all the maps are canonical, this has to be the same for ${\Gamma}=G$ and ${\Gamma}=H$. Hence $u_1^G\equiv u_1^H \mod p$. ◻ follows at once from Lemmas [Lemma 14](#MIPUesCaso1){reference-type="ref" reference="MIPUesCaso1"}, [Lemma 15](#MIPUeso1o2>0){reference-type="ref" reference="MIPUeso1o2>0"} and [Lemma 16](#MIPUesSpecial){reference-type="ref" reference="MIPUesSpecial"}. ## Proof of {#SubsectionTheoremA} Since $\psi(\mathrm{I}(G')kG)=\mathrm{I}(H')kH$, we have that $\psi(\mathrm{I}((G')^{p^n})kG)=\mathrm{I}((H')^{p^n})kH$ for each $n\geq 1$. Hence $\psi$ induces isomorphims $\psi_n: k(G/(G')^{p^n})\to k(H/(H')^{p^n})$. We first proof [\[theorem1.1\]](#theorem1.1){reference-type="eqref" reference="theorem1.1"}. By [\[gamma\]](#gamma){reference-type="eqref" reference="gamma"}, $\gamma_3(G)=(G')^{p^{m-\max(o_1,o_2)}}$. Hence, $\psi_{m-\max(o_1,o_2)+1}$ is an isomorphism $k(G/\gamma_3(G)^p)\cong k(H/\gamma_3(H)^p)$. Hence we can assume that $|\gamma_3(G)|=|\gamma_3(H)|=p$, so necessarily $\max(o_1,o_2)=1$. This means that $\{o_1,o_2\}=\{0,1\}$, by condition [\[3\]](#3){reference-type="ref" reference="3"}. Thus $a_1\leq o_2$ and $a_2\leq o_1$. Then $1\leq u_i^{\Gamma}<p$ for $i\in\{1,2\}$ and ${\Gamma}\in \{G,H\}$ by conditions [\[5\]](#5){reference-type="ref" reference="5"} and [\[6\]](#6){reference-type="ref" reference="6"}. Therefore $u_1^G=u_1^H$ and $u_2^G=u_2^H$ by , and the result follows. This proves [\[theorem1.1\]](#theorem1.1){reference-type="eqref" reference="theorem1.1"}. To prove [\[theorem1.2\]](#theorem1.2){reference-type="eqref" reference="theorem1.2"} we need one more result, which allows us to recover $u_i^{\Gamma}$ modulo a higher power of $p$ in very special situations (see ). For that, we define $$q^{\Gamma}=\min\{n\geq 0 : \Omega_1({\Gamma}') \cap {\mathrm{D}}_{p^n}({\Gamma})=1\}.$$ We claim that $$\label{lemma:descriptionq} q^{\Gamma}=\begin{cases} m, & \text{if } o'_1=o'_2=0; \\ n_2+o_2',& \text{if }0=o_1'<o'_2; \\ \max(n_1+o_1',n_2+o_2'), &\text{if } o'_1>0. \end{cases}$$ Indeed, first recall that $n_1\geq m$ by condition [\[4\]](#4){reference-type="ref" reference="4"}. Moreover, $n_2+o_2'\geq m$, since otherwise, by the same condition, $n_2=2m-o_1-o_2'$, so $m< 2m-o_1 =n_2+o_2'\leq m$, a contradiction. Clearly, $m\le q^{\Gamma}$, since $1\ne a^{p^{m-1}}\in \Omega_1({\Gamma}')\cap {\mathrm{D}}_{p^{m-1}}({\Gamma})$. Moreover, using regularity and [\[Mpn\]](#Mpn){reference-type="eqref" reference="Mpn"} we derive that if $n\ge m$, then ${\mathrm{D}}_{p^n}({\Gamma})=\left\langle b_1^{p^n},b_2^{p^n} \right\rangle$. If $o'_1=o'_2=0$, then ${\mathrm{D}}_{p^m}({\Gamma})\cap \Omega_1({\Gamma}')=1$, so $q^{\Gamma}=m$. Suppose that $0=o_1'<o_2'$. Then $a^{p^{m-1}}\in {\mathrm{D}}_{p^{n_2+o_2'-1}}({\Gamma})$, but ${\mathrm{D}}_{p^{n_2+o_2'}}({\Gamma})=\left\langle b_1^{p^{n_2+o_2'}} \right\rangle$, which does not intersect with ${\Gamma}'$. Thus $q^{\Gamma}=n_2+o'_2$. Finally suppose that $o_1'>0$. Then $a^{p^{m-1}}\in {\mathrm{D}}_{p^{\max(n_1+o_1',n_2+o_2')-1}}({\Gamma})$ because if $n_2+o_2'> n_1+o_1'$ then $o_2'>0$ since $n_1\ge n_2$. As ${\mathrm{D}}_{p^{\max(n_1+o_1',n_2+o_2')}}({\Gamma})=1$, we conclude that $q^\Gamma=\max(n_1+o'_1,n_2+o'_2)$. This finishes the proof of [\[lemma:descriptionq\]](#lemma:descriptionq){reference-type="eqref" reference="lemma:descriptionq"}. **Lemma 17**. *Let $t$ be a positive integer such that $t\leq 2m-1-q^G$.* 1. *[\[MIPUesHigherPower1\]]{#MIPUesHigherPower1 label="MIPUesHigherPower1"} Suppose that $o_1=0$ and $n_1=2m-o_2-o_1'$. If $u_1^G\equiv u_1^H \equiv -1 \mod p^{t}$, then $u_1^G\equiv u_1^H \mod p^{t +1}$.* 2. *[\[MIPUesHigherPower2\]]{#MIPUesHigherPower2 label="MIPUesHigherPower2"}* *Suppose that $o_2=0$ and $n_2=2m-o_1-o_2'$. If $u_2^G\equiv u_2^H \equiv 1 \mod p^{t}$, then $u_2^G\equiv u_2^H\mod p^{t+1}$.* *Proof.* Suppose first that the hypotheses of [\[MIPUesHigherPower1\]](#MIPUesHigherPower1){reference-type="eqref" reference="MIPUesHigherPower1"} hold. If $a_1\le t$ then $u_1^G=u_2^H=-1+p^{a_1}$. Thus we may assume that $t<a_1$ and in particular $t<o'_1$. Then $q^{\Gamma}=\max(n_1+o'_1,n_2+o'_2)$. Write $u_1^{\Gamma}=-1+ v_1^{\Gamma}p^{t}$. Recall that $\mathrm{Z}({\Gamma})= \left\langle b_1^{p^m}, b_2^{p^m}, c= b_1^{p^{m-o_2}}a \right\rangle$, by [\[eq:centerGens\]](#eq:centerGens){reference-type="eqref" reference="eq:centerGens"}. As $o_1=0$, $[b_1,a]=1$ and hence $$(b_1^{p^{m-o_2}}a)^{p^{m-o_1'}}=b_1^{p^{n_1}}a^{p^{m-o_1'}}=a^{(u_1^{\Gamma}+1) p^{m-o_1'}}=a^{v_1^{\Gamma}p^{m-o_1'+t}}.$$ Therefore $$(b_1^{p^{m-o_2}}a)^{p^{m-t-1}}=((b_1^{ p^{m-o_2}}a)^{p^{m-o_1'}})^{p^{o_1'-t-1}}= a^{v_1^{\Gamma}p^{m-1}}.$$ Then $\Upsilon^{m-t-1}_{\Gamma}$ maps the class of $x(c-1)+y(c-1)^2+ \dots$ to the class of $x v_1^{\Gamma}(a^{p^{m-1}}-1)$. Observe that $a^{p^{m-1}}\not\in {\mathrm{D}}_{2m-t-1}({\Gamma})$ since $2m-t-1\geq q^{\Gamma}$. Hence $(a^{p^{m-1}}-1)\not\in \mathrm{I}({\Gamma})^{p^{2m-t-1}}+\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma})$, by . Thus $\mbox{\rm Im }(\Upsilon^{m-t-1}_{\Gamma})$ has dimension $1$, and the natural projection $$\omega_{\Gamma}: \frac{\mathrm{I}({\Gamma}')^{p^{m-1}} k{\Gamma}}{\mathrm{I}({\Gamma}')^{p^{m-1}}\mathrm{I}({\Gamma})} \to \mbox{\rm Im }(\Upsilon^{m-t-1}_{\Gamma})$$ is an isomorphism. If $x\in k$, then $$(\omega_{\Gamma})^{-1}\circ \Upsilon^{m-t-1}_{\Gamma}= x\cdot (\Lambda_{{\Gamma}'}^{m-1} \circ \Delta_{\Gamma}^{-1}\circ \zeta_{\Gamma}^1)$$ if and only if $x= v_1^{\Gamma}\cdot 1_k$. As this holds both for ${\Gamma}=G$ and for ${\Gamma}=H$ and all the maps are canonical, we conclude that $v_1^G\equiv v_1^H \mod p$, so $u_1^G\equiv u_1^H\mod p^{t+1}$. This finishes the proof of [\[MIPUesHigherPower1\]](#MIPUesHigherPower1){reference-type="eqref" reference="MIPUesHigherPower1"}. Under the assumptions of [\[MIPUesHigherPower2\]](#MIPUesHigherPower2){reference-type="eqref" reference="MIPUesHigherPower2"}, the congruence in [\[eq:CenteredCongruence\]](#eq:CenteredCongruence){reference-type="eqref" reference="eq:CenteredCongruence"} yields $\delta\equiv -1 \mod p^{o_1}$, and hence $\mathrm{Z}({\Gamma})=\left\langle b_1^{p^m}, b_2^{p^m}, c=b_2^{-p^{m-o_1}} a \right\rangle$. Then setting $u_2^{\Gamma}=1+v_2^{\Gamma}p^t$ and arguing as above we obtain $(b_2^{-p^{m-o_1}}a)^{p^{m-t^{\Gamma}-1}}= a^{-v_2^{\Gamma}p^{m-1}}$. The rest of the proof is completely analogous to the previous case. ◻ **Lemma 18**. *If $n_2\leq 2$, then $G\cong H$.* *Proof.* Recall that we are assuming that [\[Assumptions\]](#Assumptions){reference-type="eqref" reference="Assumptions"} holds, so $m\ge 2$ and we may assume that $n_2=2$. If $m=2$ then $|\gamma_3({\Gamma})|=p$, and hence the result follows from [\[theorem1.1\]](#theorem1.1){reference-type="eqref" reference="theorem1.1"}. Thus we assume $m\geq 3$. Then $n_2<m$, and by condition [\[4\]](#4){reference-type="ref" reference="4"}, $2=n_2=2m-o_1-o_2'$ and $u_2^{\Gamma}\equiv 1 \mod p^{m-2}$. Then $2(m-1)=o_1+o_2'$. Since $o_1<m$ by condition [\[2\]](#2){reference-type="ref" reference="2"}, and $o_2'<m$ by [\[Assumptions\]](#Assumptions){reference-type="eqref" reference="Assumptions"}, we derive that $o_1=o_2'=m-1$. As $o_i+o_i'\leq m$ by condition [\[2\]](#2){reference-type="ref" reference="2"}, also $o_1'\leq 1$ and $o_2\leq 1$. Therefore $1\leq u_1^{\Gamma}\leq p$. Then implies that $u_1^G=u_1^H$ or condition [\[theorem2.2\]](#theorem2.2){reference-type="eqref" reference="theorem2.2"} in the theorem holds. In the latter case $o_1o_2>0$ and $n_1+o_1'=n_2+o_2'=m+1$. The former implies $n_1>m$ by [\[eq:n1=m\]](#eq:n1=m){reference-type="eqref" reference="eq:n1=m"}. Therefore $o_1'=0$, so $u_1^G=1=u_1^H$. Observe that $1\leq u_2^{\Gamma}\leq p^{a_2}$, for otherwise, $o_2=1$ and $n_1+o'_1-m-1 = n_1-n_2+o'_1-o'_2=0<a_1\le o'_1\le 1$ by condition [\[6\]](#6){reference-type="ref" reference="6"}, so $n_1=m$ and $o_1o_2>0$, in contradiction with [\[eq:n1=m\]](#eq:n1=m){reference-type="eqref" reference="eq:n1=m"}. If $o_2=1$, then $a_2\leq o_1-o_2=m-2$, so $1\leq u_2^{\Gamma}\leq p^{m-2}$ and hence $u_2^G=u_2^H$. Thus we assume $o_2=0$. Suppose that $o_1'=0$. Then by [\[lemma:descriptionq\]](#lemma:descriptionq){reference-type="eqref" reference="lemma:descriptionq"} $q^{\Gamma}=n_2+o_2'=m+1$. Thus $m-2\leq 2m-1-q^{\Gamma}=m-2$. Therefore [\[MIPUesHigherPower2\]](#MIPUesHigherPower2){reference-type="eqref" reference="MIPUesHigherPower2"} with $t=m-2$ yields that $u_2^G\equiv u_2^H\mod p^{m-1}$, i.e., $u_2^G=u_2^H$. Now suppose that $o_1'=1$. Since $n_1\geq m$ by condition [\[4\]](#4){reference-type="ref" reference="4"}, $q^{\Gamma}=n_1+1$. If $n_1>m$, then by condition [\[6\]](#6){reference-type="ref" reference="6"} $a_2=o_2'-o_ 1+\max(0,m+1-n_1)=m-2$ and $1\leq u_2^{\Gamma}\leq p^{m-2}$, so $u_2^G=u_2^H$. Hence we assume $n_1=m$. Then $q^{\Gamma}=m+1$ and $m-2\leq 2m-1-q^{\Gamma}=m-2$. Thus, again [\[MIPUesHigherPower2\]](#MIPUesHigherPower2){reference-type="eqref" reference="MIPUesHigherPower2"} with $t=m-2$ yields $u_2^G\equiv u_2^H\mod p^{m-1}$, i.e., $u_2^G=u_2^H$. ◻ Observe that is equivalent to the following proposition which may be of interest by itself. We do not know whether the hypothesis $p>2$ is needed. **Proposition 19**. *Let $G$ be a $2$-generated finite $p$-group with cyclic derived subgroup. Suppose that $p>2$ and $(G/G')^{p^2}$ is cyclic. If $kG\cong kH$ for some group $H$ then $G\cong H$.* We are finally ready to prove [\[theorem1.2\]](#theorem1.2){reference-type="eqref" reference="theorem1.2"}. Via the isomorphism $\psi_3$ introduced at the beginning of , we can assume that $(G')^{p^3}=1=(H')^{p^3}$, i.e., $m\le 3$. If $n_2\leq 2$, then the result follows from , so we assume $3\leq n_2$. If $|\gamma_3(G)|\leq p$, then the result follows from [\[theorem1.1\]](#theorem1.1){reference-type="eqref" reference="theorem1.1"}. Thus we assume $|\gamma_3(G)|=|\gamma_3(H)|=p^2$, so $m=3$. Then $\gamma_3(G)=(G')^{p^{m-\max(o_1,o_2)}}$, by [\[gamma\]](#gamma){reference-type="eqref" reference="gamma"}, which implies that $\max(o_1,o_2)=2$. By condition [\[3\]](#3){reference-type="ref" reference="3"}, we have three possibilities: $0=o_1<o_2=2$, $0=o_2<o_1=2$ and $1=o_2<o_1=2$. Suppose that $0=o_1<o_2=2$. Then $u_2^G=1=u_2^H$, by condition [\[6\]](#6){reference-type="ref" reference="6"}. Since $m=3$ and $o_2+o_1'\leq m$ by condition [\[4\]](#4){reference-type="ref" reference="4"}, we have that $o_2'\leq 1$, so $1\leq u_1^{\Gamma}\leq p$ for ${\Gamma}\in \{G,H\}$. Thus $u_1^G=u_1^H$, by . Suppose that $0=o_2<o_1=2$. Then $u_1^G=1=u_1^H$ by condition [\[5\]](#5){reference-type="ref" reference="5"}. Recall that $m=3\leq n_2$. Then $o_2'+2=o_2'+o_1\leq m=3$ by condition [\[4\]](#4){reference-type="ref" reference="4"}, so $o_2'\leq 1$. Hence $1\leq u_2^{\Gamma}\leq p$ for ${\Gamma}\in\{G; H\}$, by condition [\[6\]](#6){reference-type="ref" reference="6"}. Thus $u_2^G=u_2^H$ by . Finally suppose that $1=o_2<o_1=2$. By condition [\[2\]](#2){reference-type="ref" reference="2"}, $o_1'\leq 1$, and since $n_2\geq m$, by condition [\[4\]](#4){reference-type="ref" reference="4"}, $o_2' \leq 1$. Then $1\leq u_1^{\Gamma}\leq p$. Observe that neither condition [\[theorem2.2\]](#theorem2.2){reference-type="eqref" reference="theorem2.2"} in nor condition [\[6\]](#6){reference-type="ref" reference="6"}(b) holds since, by condition [\[3\]](#3){reference-type="ref" reference="3"}, in any of these cases $1=o_1-o_2<n_1-n_2=o_2'-o_1' \leq 1$, a contradiction. Therefore $1\leq u_2^{\Gamma}\leq p$ and, by , we derive that $u_1^G=u_1^H$ and $u_2^G=u_2^H$. # Applications to groups of small order {#SectionApplications} Recall that $p$ is an odd prime and $k$ is the field with $p$ elements. We first solve the Modular Isomorphism Problem for our target groups when their order is at most $p^{11}$. **Proposition 20**. *Let $G$ be a $2$-generated $p$-group with cyclic derived subgroup such that $|G|\leq p^{11}$. If $kG\cong kH$ for some group $H$, then $G\cong H$.* *Proof.* We may assume that $G$ is neither metacyclic nor of class at most $2$. Thus conditions [\[Assumptions\]](#Assumptions){reference-type="eqref" reference="Assumptions"} are satisfied and hence we can use all the results in previous sections. Let $G$ and $H$ be a $2$-generated $p$-groups ($p>2$) with cyclic derived subgroup of order at most $p^{11}$, with $kG\cong kH$ and the usual notation $\textup{inv}({\Gamma})=(p,m,n_1,n_2,o_1,o_2,o_1',o_2',u_1^{\Gamma},u_2^{\Gamma})$ for ${\Gamma}\in\{G,H\}$. If $m\leq 3$, the result follows from [\[theorem1.1\]](#theorem1.1){reference-type="eqref" reference="theorem1.1"}. Thus we assume $m>3$. Then $n_1\geq m>3$ by condition [\[4\]](#4){reference-type="ref" reference="4"}. We can assume that $n_2\geq 3$ by . Thus $|{\Gamma}|=p^{n_1+n_2+m}=p^{11}$. Therefore $n_2=3$ and $m=n_1=4$. As $n_2<m$, by condition [\[4\]](#4){reference-type="ref" reference="4"} $u_2^{\Gamma}\equiv 1 \mod p$ and $8-o_1-o_2'=2m-o_1-o_2'=n_2=3$, so $o_1+o_2'=5$. Since $o_1<m$ and $o_2'<m$ because ${\Gamma}$ is not metacyclic, we derive that $\{o_1,o_2'\}=\{3,2\}$. Then, by [\[eq:n1=m\]](#eq:n1=m){reference-type="eqref" reference="eq:n1=m"}, $o_2=0$. Thus $u_1^G=1=u_1^H$. It also follows that $a_2\leq 2$, so $1\leq u_2^{\Gamma}\leq p^2$. Since $2\leq o_1$, by condition [\[4\]](#4){reference-type="ref" reference="4"}, $o_1'\leq 2$. Thus $q^{\Gamma}\leq \max(n_1+o_1',n_2+o_2') \leq 6$. Write $t=1$, so $t=1= 2m-1-q^{\Gamma}$. Therefore, by [\[MIPUesHigherPower2\]](#MIPUesHigherPower2){reference-type="eqref" reference="MIPUesHigherPower2"}, $u_2^G\equiv u_2^H\mod p^2$. Thus $u_2^G=u_2^H$. ◻ For groups of order $p^{12}$, we can solve the Modular Isomorphism Problem except for $p-2$ families of groups of size $p$ each one: **Proposition 21**. *Let $G$ be a $2$-generated finite $p$ group with cyclic derived subgroup and $|G|\leq p^{12}$. If $kG\cong kH$ for some group $H$, then one of the following holds:* 1. *$G\cong H$.* 2. *There exist $i\in \{1,\dots,p-2\}$ and $u_1^G,u_1^H \in \{ i+jp: 0\leq j \leq p-1 \}$ such that $$\begin{aligned} \textup{inv}(G)&=(p,4,4,4,0,2,2,2,u_1^G,1) \text{ and} \\ \textup{inv}(H)&=(p,4,4,4,0,2,2,2,u_1^H,1). \end{aligned}$$* *Proof.* By and , we may assume that $|G'|>p^3$ and $|G|=p^{12}$. Moreover we can assume that neither $G$ nor $H$ is metacyclic nor of class at most $2$. With the notation of , $\textup{inv}(G)$ equals $\textup{inv}(H)$ except the last two entries, $(u_1^{\Gamma}, u_2^{\Gamma})$, where ${\Gamma}\in\{G,H\}$. Moreover, we can assume $n_2\geq 3$ by . Then either $n_2=3$, $m=4$ and $n_1=5$, or $n_2=m=n_1=4$. [Suppose that $m=4$, $n_1=5$ and $n_2=3$.]{.ul} By condition [\[4\]](#4){reference-type="ref" reference="4"}, $u_2^{\Gamma}\equiv 1 \mod p$ and $3=n_2=2m-o_1-o_2'$, so $5=o_1+o_2'$, and hence $\{o_1,o_2'\}=\{2,3\}$ because $o_1<m$ and $o_2'<m$. Suppose that $o_2'=3$. Then $o_1=2$, $o_2\leq m-o_2'=1$ and $o_1' \leq m-o_1=2$. Assume that $o_2=1$. Then $a_2\leq o_1-o_2=1$. If condition [\[6\]](#6){reference-type="ref" reference="6"}(b) does not hold, then yields $u_2^G=u_2^H$. Thus suppose this condition holds. Then $u_1^{\Gamma}\equiv 1 \bmod p$ and $5+o_1'=n_1+o_1'=n_2+o_2'=6$, so $o_1'=1$. Hence $1\leq u_1^{\Gamma}\leq p$, and we get $u_1^G=u_1^H=1$. Moreover $a_2=\min(o_1-o_2,o_2'-o_1')=1$. Thus $u_2^{\Gamma}\in\{1,1+p\}$. Summarizing, after exchanging $G$ and $H$, if necessary, $$\begin{aligned} \textup{inv}(G)&=(p,4,5,3,2,1,1,3,1, 1) ; \\ \textup{inv}(H)&= (p,4,5,3,2,1,1,3,1, 1+p) .\end{aligned}$$ But then a straightforward computation, using [\[eq:centerGens\]](#eq:centerGens){reference-type="eqref" reference="eq:centerGens"}, shows that $\mathrm{Z}(G)$ has exponent $p^2$ while the exponent of $\mathrm{Z}(H)$ is $p^3$, in contradiction with a result of Ward [@Ward] (see [@Pas77 Lemma 2.7]). Now assume $o_2=0$. Then $a_1=0$, so $u_1^G=1=u_1^H$. Observe that $a_2=\min (o_1, 3-o_1') \leq 2$. Moreover $3-o_1'=o_2'-o_1'\leq n_1-n_2=2$, so $1\leq o_1'$. If $o_1'=1$ then $q^{\Gamma}= 6$, and setting $t=1= 2m-1-q^{\Gamma}$, [\[MIPUesHigherPower2\]](#MIPUesHigherPower2){reference-type="eqref" reference="MIPUesHigherPower2"} yields $u_2^G=u_2^H$. Otherwise, i.e. if $o_1'\geq 2$, then $a_2\leq 1$, and $u_2^G=u_2^H$ by . Now suppose that $o_2'=2$. Then $o_1=3$, $o_2\leq m-o_2'=2$ and $o_1'\leq m-o_1=1$. We claim that $u_1^G=u_1^H$. Indeed, if $o_1'=0$ then $u_1^G=1=u_1^H$, and if $o_1'=1$ then condition [\[theorem2.2\]](#theorem2.2){reference-type="eqref" reference="theorem2.2"} of does not hold, and hence that theorem yields the claim. Moreover $a_2\leq o_2'=2$ and if condition [\[6\]](#6){reference-type="ref" reference="6"}(b) holds, then $o_2>0$ and $o_1'\ge a_1>0$ so that $a_2\le 1$. Thus $1\leq u_2^{\Gamma}\leq 2p< p^2$. Observe that $q^{\Gamma}=\max(5+o_1',5)\leq 6$. Then set $t=1\leq 2m-1-q^{\Gamma}$, and [\[MIPUesHigherPower2\]](#MIPUesHigherPower2){reference-type="eqref" reference="MIPUesHigherPower2"} yields that $u_2^G=u_2^H$. [Finally, suppose that $m=n_1=n_2=4$.]{.ul} By condition [\[3\]](#3){reference-type="ref" reference="3"} we have that $o_1=0$. Then $u_2^G=1=u_2^H$. Moreover $a_1=\min(o_1',o_2+o_1'-o_2')$. If $a_1\leq 1$, then $u_1^G=u_1^H$ by . Thus we assume $a_1\geq 2$, i.e., $2\leq o_1'\leq 3$ and $2\leq o_1'+o_2-o_2'$. If $o_2\leq 1$ then $|\gamma_3({\Gamma})|\leq p$, and $G\cong H$ by [\[theorem1.1\]](#theorem1.1){reference-type="eqref" reference="theorem1.1"}. Thus we suppose $o_2\geq 2$. Since $o_1'+o_2\leq m=4$, we derive that $o_1'=o_2=2$. Hence $a_1=o_1'=2$, and $o_2'\geq o_1'=2$, by condition [\[3\]](#3){reference-type="ref" reference="3"}. Since $o_2+o_2'\leq m =4$, necessarily $o_2'=2$. Hence we have that $$\textup{inv}({\Gamma})=(p,4,4,4,0,2,2,2,u_1^{\Gamma},1)$$ with $1\leq u_1^{\Gamma}\leq p^2$. Moreover, by [\[theorem1.2\]](#theorem1.2){reference-type="eqref" reference="theorem1.2"}, we have that $u_1^G \equiv u_1^H \mod p$. Hence there is an integer $1\leq i\leq p-1$ such that $u_1^{\Gamma}=i+j^{\Gamma}p$, for some integers $0 \leq j^G, j^H\leq p-1$. Finally, assume $i=p-1$, so $u_1^{\Gamma}\equiv -1 \mod p$. Since $q^{\Gamma}=6$, setting $t=1=2m-1-q^{\Gamma}$, [\[MIPUesHigherPower1\]](#MIPUesHigherPower1){reference-type="eqref" reference="MIPUesHigherPower1"} yields that $u_1^G=u_1^H$. ◻ **Remark 22**. *Observe that shows that $kG\cong kH$ implies $u_1^G\equiv u_1^H\mod p$ in almost all situations. A pair of groups $G$ and $H$ of minimal size with $u_1^G\not \equiv u_1^H \mod p$ and not covered by this theorem (i.e., such that it is still open whether they have isomorphic group algebras or not) consists in groups of order $3^{17}$ and $$\begin{aligned} \textup{inv}(G)=(3, 5, 7, 5, 1, 1, 2, 1, 1, 3, 1, 2); \\ \textup{inv}(H)=(3,5, 7, 5, 1, 1, 2, 1, 1, 3, 2, 2). \end{aligned}$$* **Acknowledgements**: We are grateful to Mima Stanojkovski, with whom we started the study of the Modular Isomorphism Problem for this class of groups, for useful comments and discussions on early drafts of this paper. , if not folklore, was written by Sofia Brenner and the first author for another project: we are grateful to her for allowing us to include it here.

arxiv_math

--- abstract: | The classical Galton--Watson process works with a fixed probability of fission at each time step. One of the generalizations is that the probabilities depend on time. We consider one of the most complex and interesting cases when we do not know the exact probabilities of fission at each time step - these probabilities are random variables themselves. The limit distributions of the number of descendants are described in terms of generalized integral and differential functional equations of the Schröder type. There are no more analogs of Karlin-McGregor functions, which were very helpful in the analysis of the asymptotic behavior of limit distributions for the classical case. We propose some approximate asymptotic methods. Even simple cases of random families with one or two members lead to nice asymptotics involving, perhaps, open problems related to special constants. The most exciting is Example 2 announced on [this site](https://math.stackexchange.com/questions/4748129/asymptotics-of-sequence-of-rational-numbers). Finally, the phenomenon of why the oscillations in the main asymptotic term usual for the classical case become rare in the case of random environments is discussed. address: "Mathematical Institute for Machine Learning and Data Science, KU Eichstätt--Ingolstadt, Germany; email: akucenko\\@gmail.com" author: - Anton A. Kutsenko title: Generalized Schröder-type functional equations for Galton--Watson processes in random environments --- Galton--Watson process, asymptotic behavior, Schröder-type functional equations, Poincaré-type functional equations, Karlin-McGregor functions # Introduction {#sec0} A simple Galton--Watson process, see, e.g. [@H], is defined by $$\label{001} X_{t+1}=\sum_{j=1}^{X_t}\xi_{j,t},\ \ \ X_0=1,\ \ \ t\in{\mathbb N}\cup\{0\},$$ where all $\xi_{j,t}$ are independent and identically-distributed natural number-valued random variables with the probability-generating function $$\label{002} P(z):=\mathbb{E}z^{\xi}=p_0+p_1z+p_2z^2+p_3z^3+....$$ All $p_j$ are non-negative and their sum equals $P(1)=1$. We focus on the so-called supercritical case $p_0=0$ and $0<p_1<1$. In this case, one may define the limit distribution $$\label{003} \varphi_n:=\lim_{t\to+\infty}\frac{\mathbb{P}(X_t=n)}{\mathbb{P}(X_t=1)}$$ describing the so-called relative limit densities $\varphi_n$, see [@H]. These values are coefficients of the analytic function $$\label{004} \Phi(z)=\varphi_1z+\varphi_2z^2+\varphi_3z^3+....$$ The function $\Phi$ satisfies the Scröder-type functional equation $$\label{005} \Phi(P(z))=p_1\Phi(z),\ \ \ \Phi(0)=0,\ \ \ \Phi'(0)=1,$$ it is defined at least for $|z|<1$. A Galton--Watson process in a random environment is characterized by the fact that $\xi_{j,t}$ are still independent but they are identically distributed for fixed $t$ only. The simplest case is when $P(z)$, see [([\[002\]](#002){reference-type="ref" reference="002"})]{.roman}, is fixed for each time $t$, but it depends on $t$ deterministically, see, e.g., [@J]. We consider a more complex case when the dependence on $t$ is random. This means that $P(z)$ is again not fixed and, at each time step $t$, it is chosen randomly among $$\label{006} P_r(z)=p_{1r}z+p_{2r}z^2+p_{3r}z^3+...,$$ where $r\in R$ is a random parameter having a distribution measure $\mu$ defined on some probabilistic space $R$. Note that some other generalizations, e.g., when $(p_n(t))_{n=0}^{+\infty}$ are independent identically distributed random vectors, are considered in, e.g., [@VD], see also references therein. For the case [([\[006\]](#006){reference-type="ref" reference="006"})]{.roman}, we can still define the limit distribution [([\[003\]](#003){reference-type="ref" reference="003"})]{.roman} and the corresponding analytic function [([\[004\]](#004){reference-type="ref" reference="004"})]{.roman}. The difference between simple and random cases is that $\Phi$ satisfies the generalized Schröder-type functional equation $$\label{007} \int_R\Phi(P_r(z))\mu(dr)=\Phi(z)\int_Rp_{1r}\mu(dr),\ \ \ \Phi(0)=0,\ \ \ \Phi'(0)=1.$$ **Remark on the derivation of [([\[007\]](#007){reference-type="ref" reference="007"})]{.roman}.** By the analogy with the simple case, we have $$\begin{gathered} \label{008a} \Phi_t(z):=\frac{\mathbb{E}z^{X_t}}{\mathbb{P}(X_t=1)}=\frac{\int_{R^t}P_{r_t}\circ...\circ P_{r_1}(z)\mu(dr_1)...\mu(dr_t)}{\int_{R^t}p_{1r_t}...p_{1r_1}\mu(dr_1)...\mu(dr_t)}=\\ \frac{\int_{R^t}P_{r_t}\circ...\circ P_{r_1}(z)\mu(dr_1)...\mu(dr_t)}{(\int_{R}p_{1r}\mu(dr))^t}=\frac{\int_R\Phi_{t-1}(P_{r_1}(z))\mu(dr_1)}{\int_Rp_{1r}\mu(dr)},\ \ \Phi_t(0)=0,\ \ (\Phi_t)'(0)=1.\end{gathered}$$ Taking the limit in [([\[008a\]](#008a){reference-type="ref" reference="008a"})]{.roman} for $t\to+\infty$, we arrive at [([\[007\]](#007){reference-type="ref" reference="007"})]{.roman}. height6pt width5.5pt depth0pt Differentiating [([\[007\]](#007){reference-type="ref" reference="007"})]{.roman} at $z=0$ and using Faà di Bruno's formula for the composition of two functions, we derive explicit recurrence expressions for the Tailor coefficients $\varphi_n$ of $\Phi(z)$, see [([\[003\]](#003){reference-type="ref" reference="003"})]{.roman} and [([\[004\]](#004){reference-type="ref" reference="004"})]{.roman}, $$\label{008} \varphi_n=\biggl(\int_R(p_{1r}-p_{1r}^{n})\mu(dr)\biggr)^{-1}\sum_{k=1}^{n-1}\varphi_{k}\int_RB_{n,k}(p_{1r},2p_{2r},...,(n-k+1)!p_{n-k+1,r})\mu(dr),$$ where $B_{n,k}$ are Bell polynomials $$\label{Bell1} B_{n,k}(x_1,x_2,...,x_{n-k+1}):=\sum\frac{n!}{j_1!j_2!...j_{n-k+1}!}\biggl(\frac{x_1}{1!}\biggr)^{j_1}\biggl(\frac{x_2}{2!}\biggr)^{j_2}...\biggl(\frac{x_{n-k+1}}{(n-k+1)!}\biggr)^{j_{n-k+1}},$$ and the sum for Bell polynomials is taken over all sequences $j_r$ of non-negative integers such that $$\label{Bell2} j_1+j_2+...+j_{n-k+1}=k,\ \ \ j_1+2j_2+3j_3+...+(n-k+1)j_{n-k+1}=n.$$ The natural assumption is that all the integrals exist in [([\[008\]](#008){reference-type="ref" reference="008"})]{.roman}. Usually, it is enough to assume that all the moments exist. In particular, polynomials $P_r(z)$ of bounded degrees, which is commonly the practice case, are admissible in [([\[008\]](#008){reference-type="ref" reference="008"})]{.roman}. In some cases, especially when $P_r$ depends on $r$ linearly, it is convenient to rewrite [([\[007\]](#007){reference-type="ref" reference="007"})]{.roman} in a differential form introducing $F(z)=\int_0^z\Phi(\zeta)d\zeta$. We consider this case in the Example section. Integral and differential extensions of the Schröder-type functional equations are much more complex than the original form, where $P_r$ does not depend on $r$. In particular, it is hard to determine the asymptotic behavior of $\varphi_n$ for large $n\to+\infty$. One of the reasons is that there are no more direct analogs of Karlin--McGregor functions defined in [@KM1] and [@KM2] for the classical case. Let us recall some facts about this function. For the case $P_r\equiv P$, it is possible to define $$\label{009} \Pi(z):=\lim_{t\to+\infty}\underbrace{P\circ...\circ P}_{t}(1-E^{-t}z),$$ where $E=P'(1)$ is the expectation $E=p_1+2p_2+3p_3+...$. The function $\Pi(z)$ is entire when $P$ is entire, and it satisfies the Poincaré-type functional equation $$\label{010} P(\Pi(z))=\Pi(Ez),\ \ \ \Pi(0)=1,\ \ \ \Pi'(0)=-1.$$ Note that this function is related to the probability density function for the so-called *martingal limit*, which is another limit distribution for branching processes apart from [([\[003\]](#003){reference-type="ref" reference="003"})]{.roman}: $$\label{011} p(x):=\lim_{t\to+\infty}E^t\mathbb{P}(X_t=[xE^t])=\frac1{2\pi\mathbf{i}}\int_{\mathbf{i}{\mathbb R}}\Pi(z)e^{zx}dz,$$ where the square brackets denote the integer part of a number, see, e.g. [@D1]. Combining $\Phi$ and $\Pi$, and using their functional equations, we can define the $1$-periodic function $$\label{012} K(z):=\Phi(\Pi(E^z))p_1^{-z}.$$ This function is very efficient in the analysis of asymptotic series for both limit distributions: [([\[003\]](#003){reference-type="ref" reference="003"})]{.roman} and [([\[011\]](#011){reference-type="ref" reference="011"})]{.roman}. Using ideas presented above [([\[008\]](#008){reference-type="ref" reference="008"})]{.roman}, we can define an analog of [([\[009\]](#009){reference-type="ref" reference="009"})]{.roman}: $$\label{013} \Pi(z):=\lim_{t\to+\infty}\int_{R^t}\underbrace{P_{r_1}\circ...\circ P_{r_t}}_{t}(1-E^{-t}z)\mu(dr_t)...\mu(dr_1),\ \ \ \Pi(0)=1,\ \ \ \Pi'(0)=-1,$$ where the average expectation $E$ is given by $$\label{014} E:=\int_R(P_r)'(1)\mu(dr).$$ Defining some analog of [([\[011\]](#011){reference-type="ref" reference="011"})]{.roman} is still possible. However, there are no appropriate variants of [([\[010\]](#010){reference-type="ref" reference="010"})]{.roman}, making the definition of $1$-periodic analogs of Karlin--McGregor functions [([\[012\]](#012){reference-type="ref" reference="012"})]{.roman} almost impossible. This fact significantly complicates the asymptotic analysis of the coefficients $\varphi_n$ when $n\to+\infty$. However, a few methods allow us to obtain good approximations of $\varphi_n$. Often, the main contribution to the asymptotic behavior of $\varphi_n$ comes from the singularity of $\Phi(z)$ at $z=1$. Substituting *ansatz* $\Phi(z)\approx A(1-z)^{\alpha}$ into the main equation [([\[007\]](#007){reference-type="ref" reference="007"})]{.roman}, we obtain the equation for $\alpha$: $$\label{015} \int_{R}((P_r)(1)')^{\alpha}\mu(dr)=\int_Rp_{1r}\mu(dr).$$ Usually, [([\[015\]](#015){reference-type="ref" reference="015"})]{.roman} has infinitely many solutions $\alpha$, which are zeros of the corresponding (entire) function. The largest contribution comes from $\alpha$ with the smallest real parts, because $$\label{016} \Phi(z)\approx A(1-z)^{\alpha}=A\sum_{n=0}^{+\infty}(-1)^n\binom{\alpha}{n} z^n,$$ and using the asymptotic for generalized binomial coefficients $$\label{017} \binom{\alpha}{n}\simeq\frac{(-1)^n}{\Gamma(-\alpha)n^{\alpha+1}}+\frac{(-1)^n\alpha(\alpha+1)}{2\Gamma(-\alpha)n^{\alpha+2}}+...,$$ we obtain the final approximation $$\label{018} \varphi_n\approx \sum C_{\alpha}n^{-\alpha-1},$$ where the sum is taken over the solutions of [([\[015\]](#015){reference-type="ref" reference="015"})]{.roman} with small real parts. Because $P_r'(1)\geqslant 1$ and $p_{1r}$ are non-negative, the real parts of solutions $\alpha$ are bounded from below. Note that if $\alpha=\alpha_1+\mathbf{i}\alpha_2$ is complex then the corresponding term in [([\[018\]](#018){reference-type="ref" reference="018"})]{.roman} gives the long-phase logarithmic oscillation $C_{\alpha}n^{-\alpha_1-1}\exp(-\mathbf{i}\alpha_2\ln n)$ (plus the corresponding terms related to the complex conjugate $\overline{\alpha}$) similar to that in the asymptotic of the classical Galton--Watson process with one generating function, see, e.g., [@K] and [@DG]. However, in the mixed case of many generating functions [([\[007\]](#007){reference-type="ref" reference="007"})]{.roman}, the derivation of complete asymptotic series for power coefficients of $\Phi(z)$ is, generally speaking, an open problem. Summarizing the above, we can formulate the general rough idea of finding approximations: Substitute $$\label{019} \varphi_n\simeq\sum_{\alpha\in\{Solutions\ of\ \textrm{(\ref{015})}\},\ j\in{\mathbb N}}\frac{C_{\alpha,j}}{n^{\alpha+j}}$$ into [([\[008\]](#008){reference-type="ref" reference="008"})]{.roman} directly or [([\[016\]](#016){reference-type="ref" reference="016"})]{.roman} into [([\[007\]](#007){reference-type="ref" reference="007"})]{.roman} and use [([\[017\]](#017){reference-type="ref" reference="017"})]{.roman} with some *a priori* and/or empirical additional information to obtain relations for coefficients $C_{\alpha,j}$. However, in many cases, it does not work properly and more sophisticated tricks are required. We will treat the explained approximate method in the examples below. Moreover, in some cases, we take more approximate terms for $\Phi(z)$ at $z=1$ and other critical points of the unit circle. We consider four examples. For the simplest linear fractional generating functions, which correspond to the power-law distribution, [([\[007\]](#007){reference-type="ref" reference="007"})]{.roman} can be solved explicitly. This is the complete analog of the similar case for the classical Galton--Watson process. The next two examples work with families of one or two particles. At each time step the generating function [([\[006\]](#006){reference-type="ref" reference="006"})]{.roman} is simply $P_r(z)=rz+(1-r)z^2$, where $r$ is distributed uniformly in $(1/2,1)$ or in $(0,1)$. This means that at each time step each particle generates one new particle or stays alone. The generation probabilities are any from $(1/2,1)$ or $(0,1)$ uniformly. Such types of problems are natural in situations when we do not know the exact probabilities of generation (or fission) at each time step but expect them with some probabilities. Two cases with distribution intervals $(1/2,1)$ and $(0,1)$ differ significantly. Relative limit densities [([\[003\]](#003){reference-type="ref" reference="003"})]{.roman} for both cases satisfy power laws with different exponents. In contrast to the classical case, see, e.g., [@K], it seems that there are no long-phase oscillations for the first approximation term. These long-phase oscillations may appear in the next asymptotic terms. However, in contrast to the half-interval $(1/2,1)$, for the full interval $(0,1)$ there are short-phase oscillations with bounded periods. In the last example, we consider the uncertainty produced by only two polynomials. This case is closest to the classical point of one polynomial and shows the long-phase logarithmic oscillations usual for classical cases. As mentioned in, e.g., [@DIL; @DMZ; @CG], such short- and long-phase oscillations can be important in applications in physics and biology. Let us discuss oscillations a little bit more. Due to [([\[018\]](#018){reference-type="ref" reference="018"})]{.roman} and the arguments below, they appear in some asymptotic terms for $\varphi_n$. However, for the classical Galton--Watson process, the oscillations appear for the main asymptotic term already, because $\alpha=(\ln P'(0)+2\pi\mathbf{i}m)/\ln P'(1)$, $m\in{\mathbb Z}$ are the only roots of [([\[015\]](#015){reference-type="ref" reference="015"})]{.roman}. Each complex $\alpha$ has the same real part as the primary real $\alpha$ and contributes to the main asymptotic term with some long-phase oscillations. In the generalized case, when we have more than one generating function $P_r(z)$, there is still one primary real solution $\alpha$ of [([\[015\]](#015){reference-type="ref" reference="015"})]{.roman}. The existence of complex solutions $\alpha$ of [([\[015\]](#015){reference-type="ref" reference="015"})]{.roman} with the same real part as the primary one requires that there is some real $B$ such that $P_r(1)'=B^{n_r}$ for some integer $n_r$ and almost all $r$. This condition describes some one-dimensional "algebraic\" curve in ${\mathbb R}^R$ with the parameter-argument $B$. The union of all such curves over all integer vectors $(n_r)$ is dense but has a "zero measure\" in most acceptable models. Let us briefly discuss short-phase oscillations considered in Example 2 below. They usually come from other critical points of the unit circle, not equal to $z=1$. Substituting such $z_0\ne1$ ($|z_0|=1$) into the argument of $P_r$ in [([\[015\]](#015){reference-type="ref" reference="015"})]{.roman}, we obtain the equation for the corresponding powers $\alpha$. However, since the real part of $P_r(z_0)$ with complex $z_0$ is less than $P_r(1)$, see [([\[002\]](#002){reference-type="ref" reference="002"})]{.roman}, the real part of a solution $\alpha$ of [([\[015\]](#015){reference-type="ref" reference="015"})]{.roman} should be larger than the minimal real solution $\alpha$ corresponding to $P_r(1)$. This is because the RHS of [([\[015\]](#015){reference-type="ref" reference="015"})]{.roman} is the same for both cases corresponding to $P_r(1)$ and $P_r(z_0)$. Hence, short-phase oscillations appear not in the main asymptotic term. Thus, oscillations in the main term of the asymptotics of the number of descendants, which is mostly common in the classical branching processes with fixed offspring distributions, become rare for generalized cases taken in a random environment. Two examples related to this discussion are considered at the beginning (Example 1) and end (Example 3) of the paper. # Examples ## Example 0. In rare cases, [([\[007\]](#007){reference-type="ref" reference="007"})]{.roman} can be solved explicitly. As in the classical case of the Galton--Watson process, if we assume $P_r(z)=(1-r)z/(1-rz)$ but with a varying distribution of $r$ with a measure $\mu(dr)$ then $\Phi(z)=z/(1-z)$ is the solution of [([\[007\]](#007){reference-type="ref" reference="007"})]{.roman}. ## Example 1. The case $P_r(z)=rz+(1-r)z^2$, with uniform $r\in(1/2,1)$. Equation [([\[007\]](#007){reference-type="ref" reference="007"})]{.roman} is $$\label{100} \int_{\frac12}^{1}\Phi(rz+(1-r)z^2)dr=\frac38\Phi(z).$$ Taking $F(z)=\int_0^z\Phi(\zeta)d\zeta$, [([\[100\]](#100){reference-type="ref" reference="100"})]{.roman} can be rewritten as $$\label{101} \frac{F(z)-F(\frac{z+z^2}2)}{z-z^2}=\frac38F'(z),\ \ F(0)=F'(0)=0,\ \ F''(0)=1.$$ Tailor coefficients, see [([\[004\]](#004){reference-type="ref" reference="004"})]{.roman}, can be determined from [([\[101\]](#101){reference-type="ref" reference="101"})]{.roman} after some transformations: $$\label{102} \sum_{n=1}^{+\infty}\frac{\varphi_nz^n}{2(n+1)}\sum_{j=0}^n\biggl(\frac{z+z^2}2\biggr)^j=\frac38\sum_{n=1}^{+\infty}\varphi_nz^n,\ \ \ \varphi_1=1,$$ which leads to $$\label{103} \frac38\varphi_n=\frac{\varphi_n}{2(n+1)}\biggl(\frac{1}{2^0}\binom{0}{0}+...+\frac1{2^n}\binom{n}{0}\biggr)+\frac{\varphi_{n-1}}{2n}\biggl(\frac1{2^1}\binom{1}{1}+...+\frac1{2^{n-1}}\binom{n-1}{1}\biggr)+....$$ Identity [([\[103\]](#103){reference-type="ref" reference="103"})]{.roman} gives the recurrence relation $$\label{104} \biggl(\frac38-\frac{1-2^{-n-1}}{n+1}\biggr)\varphi_n=\frac{\varphi_{n-1}}{2n}c_{n,1}+\frac{\varphi_{n-2}}{2(n-1)}c_{n,2}+...+\frac{\varphi_{n-k}}{2(n-k+1)}c_{n,k},\ \ \ \varphi_1=1,$$ where $n\geqslant 2$ and $k$ is the maximal number such that $2k\leqslant n$. The coefficients $c_{n,j}$ are given by $$\label{105} c_{n,j}=\frac1{2^{j}}\binom{j}{j}+...+\frac1{2^{n-j}}\binom{n-j}{j}=c_{n-1,j}+\frac1{2^{n-j}}\binom{n-j}{j},$$ for all $n\geqslant 2$ and $2j\leqslant n$. Formulas [([\[104\]](#104){reference-type="ref" reference="104"})]{.roman} and [([\[105\]](#105){reference-type="ref" reference="105"})]{.roman} are used for the numerical implementations presented below. They show the dominant power-law behavior of $\varphi_n$. It is possible to obtain a good approximation using the following non-rigorous reasoning. The domain of definition for $F(z)$ contains the intersection of the Julia sets related to the quadratic polynomials $rz+(1-r)z^2$ for $r\in(1/2,1)$. Thus, there is only one critical point for $F(z)$ in $\{|z|<1\}$. It is $z=1$. Substituting an assumption $F(z)\approx A(1-z)^{\alpha}$ into [([\[101\]](#101){reference-type="ref" reference="101"})]{.roman}, we obtain equation for $\alpha$, the analog of [([\[017\]](#017){reference-type="ref" reference="017"})]{.roman}: $$\label{106} 1-\biggl(\frac32\biggr)^{\alpha}=-\frac38\alpha.$$ The root of [([\[106\]](#106){reference-type="ref" reference="106"})]{.roman} with the minimal real part is $\alpha=-0.3904295156631794...$. This is a unique root with the negative real part. The corresponding term $A(1-z)^{\alpha}$ makes the greatest contribution to the growth of the coefficients $\varphi_n$. We have $$\label{107} \Phi(z)\approx-A\alpha(1-z)^{\alpha-1}=-A\alpha\sum_{n=0}^{+\infty}(-1)^n\binom{\alpha-1}{n}z^n.$$ Using asymptotic expansion for the extended binomial coefficients [([\[017\]](#017){reference-type="ref" reference="017"})]{.roman} in [([\[107\]](#107){reference-type="ref" reference="107"})]{.roman}, we obtain a good approximation $$\label{109} \varphi_n\approx C n^{-\alpha},$$ with some constant $C$, numerical approximation for which is $$\label{110} C=1.2232199....$$ Taking more terms $F(z)\approx A(1-z)^{\alpha}+B(1-z)^{1+\alpha}$ and using the same argument as above we can improve a little bit [([\[109\]](#109){reference-type="ref" reference="109"})]{.roman}, namely $$\label{111} \varphi_n\approx C \biggl(n^{-\alpha}-\frac{(3\alpha^2+11\alpha+2)\alpha}{2(6+9\alpha)}n^{-\alpha-1}\biggr).$$ The second term $n^{-\alpha-1}$ is dominant in comparison with other terms related to the roots of [([\[106\]](#106){reference-type="ref" reference="106"})]{.roman} with positive real parts since these real parts are larger than $1+\alpha$. However, the contribution of the root $\alpha=0$ is unclear to me. It seems that there is no term $\ln(1-z)$ in the asymptotic of $F(z)$ at $z=1$, but some more complex terms related to $\alpha=0$ may appear. A comparison between exact values $\varphi_n$, computed by [([\[104\]](#104){reference-type="ref" reference="104"})]{.roman} and [([\[105\]](#105){reference-type="ref" reference="105"})]{.roman}, and approximation terms, see [([\[111\]](#111){reference-type="ref" reference="111"})]{.roman}, is presented in Fig. [\[fig1\]](#fig1){reference-type="ref" reference="fig1"}. Two terms in [([\[111\]](#111){reference-type="ref" reference="111"})]{.roman} already give a good approximation of $\varphi_n$ as it is seen in Figs. [1](#fig1a){reference-type="ref" reference="fig1a"} and [2](#fig1b){reference-type="ref" reference="fig1b"}. Double precision computations are not enough for good accuracy and lead to numerical errors appearing in Fig. [3](#fig1c){reference-type="ref" reference="fig1c"}. Nevertheless, some attenuated oscillations coming from complex roots of [([\[106\]](#106){reference-type="ref" reference="106"})]{.roman}, and, maybe, some bounded non-attenuated impacts of the root $\alpha=0$ are expected. The contribution of root $\alpha=0$ looks intriguing, but in reality, it is much simpler. After I studied the next Example 2 in detail, I recalculated Fig. [3](#fig1c){reference-type="ref" reference="fig1c"} with double-double precision and with a more precise constant $$\label{110dd} C=1.223219951386792...,$$ see [([\[110\]](#110){reference-type="ref" reference="110"})]{.roman}. The result is seen in Fig. [4](#fig1cdd){reference-type="ref" reference="fig1cdd"}. I think further comments are unnecessary. Let us only note that obtaining the most accurate values of the constants is necessary for asymptotics of this type. In the next Example 2, special attention is paid to this aspect. {#fig1a width="\\linewidth"} {#fig1b width="\\linewidth"} {#fig1c width="\\linewidth"} {reference-type="ref" reference="fig1c"} with double-double precision and more precise $C$, see [([\[110dd\]](#110dd){reference-type="ref" reference="110dd"})]{.roman}.](original3dd.png){#fig1cdd width="0.99\\linewidth"} Before we continue, let us emulate the power law [([\[109\]](#109){reference-type="ref" reference="109"})]{.roman} by using one polynomial only $P(z)=pz+(1-p)z^2$ with fixed $p$. We can use the results of the last Example 3 with $p=a=b$, see [([\[126\]](#126){reference-type="ref" reference="126"})]{.roman}. In our case, $$\label{1110} p=0.6791281732038788538781...$$ leads to $$\label{1111} \alpha=-1.3904295156631794...,\ \ \ C=1.20166998031....$$ The power $\alpha$ is the same as in [([\[109\]](#109){reference-type="ref" reference="109"})]{.roman}, but $C$ is a little bit smaller, see [([\[110dd\]](#110dd){reference-type="ref" reference="110dd"})]{.roman}. However, the other asymptotic terms differ significantly in these two examples. In the example with the fixed offspring distribution, the oscillations appear even in the leading term, see Fig. [\[fig11\]](#fig11){reference-type="ref" reference="fig11"} and compare Fig. [4](#fig1cdd){reference-type="ref" reference="fig1cdd"} with Fig. [7](#fig11c){reference-type="ref" reference="fig11c"}, as discussed at the end of the Introduction section. Just like in the last example, we broke tradition and made all the calculations with double-double precision, bypassing double precision. {#fig11a width="\\linewidth"} {#fig11b width="\\linewidth"} {#fig11c width="\\linewidth"} ## Example 2. The case $P_r(z)=rz+(1-r)z^2$, with uniform $r\in(0,1)$. Applying the same arguments as in the previous example, we obtain the analog of [([\[101\]](#101){reference-type="ref" reference="101"})]{.roman} for the integral $F(z)=\int_0^z\Phi(\zeta)d\zeta$, namely $$\label{112} \frac{F(z)-F(z^2)}{z-z^2}=\frac12F'(z),\ \ F(0)=F'(0)=0,\ \ F''(0)=1,$$ which gives the analog of [([\[103\]](#103){reference-type="ref" reference="103"})]{.roman}: $$\label{113} %\ca \frac{\varphi_{2n}}{2}=\frac{\varphi_{2n}}{2n+1}+\frac{\varphi_{2n-1}}{2n}+...+\frac{\varphi_{n}}{n+1},\ \ \ \frac{\varphi_{2n+1}}{2}=\frac{\varphi_{2n+1}}{2n+2}+\frac{\varphi_{2n}}{2n+1}+...+\frac{\varphi_{n+1}}{n+2}. %\ac$$ RHSs in [([\[113\]](#113){reference-type="ref" reference="113"})]{.roman} are almost identical to each other. This fact allows us to simplify [([\[113\]](#113){reference-type="ref" reference="113"})]{.roman} easily $$\label{114} \varphi_n=\frac{n+1}{n-1}\varphi_{n-1},\ \ n\ {\rm mod}\ 2=0;\ \ \ \varphi_n=\frac{n+1}{n-1}\varphi_{n-1}-\frac{4}{n-1}\varphi_{\frac{n-1}2},\ \ n\ {\rm mod}\ 2=1,$$ with the initial state $\varphi_1=1$. Introducing $\psi_n=\varphi_n/n$, we can rewrite [([\[114\]](#114){reference-type="ref" reference="114"})]{.roman} as $$\label{115} \psi_n=\biggl(1+\frac1n\biggr)\psi_{n-1},\ \ n\ {\rm mod}\ 2=0;\ \ \ \psi_n=\biggl(1+\frac1n\biggr)\psi_{n-1}-\frac{2}{n}\psi_{\frac{n-1}2},\ \ n\ {\rm mod}\ 2=1,$$ with the initial state $\psi_1=1$. By induction, one can check $$\label{115a} \psi_n=\frac{n+1}2\biggl(1+\sum_k\prod_{1\leqslant r\leqslant k,\ 1<a_{j_r}\leqslant n,\ 2<\frac{a_{j_r}}{a_{j_{r+1}}},\ a_{j_r}\ {\rm mod}\ 2=1}\frac{-1}{a_{j_r}}\biggr).$$ There are a few elements of this sequence $$\psi_2=\frac32,\ \ \psi_3=\frac42\biggl(1-\frac13\biggr),\ \ \psi_4=\frac52\biggl(1-\frac13\biggr),\ \ \psi_5=\frac62\biggl(1-\frac15-\frac13\biggr),\ \ \psi_6=\frac72\biggl(1-\frac15-\frac13\biggr),$$ $$\psi_7=\frac82\biggl(1-\frac17-\frac15-\frac13+\frac1{7\cdot3}\biggr),\ \ \psi_8=\frac92\biggl(1-\frac17-\frac15-\frac13+\frac1{7\cdot3}\biggr),\ \ \psi_{15}=\frac{16}2\biggl(1-\frac1{15}-\frac1{13}-\frac1{11}-\frac19-$$ $$\frac17-\frac15-\frac13+\frac1{15\cdot7}+\frac1{15\cdot5}+\frac1{13\cdot5}+\frac1{11\cdot5}+\frac1{15\cdot3}+\frac1{13\cdot3}+\frac1{11\cdot3}+\frac1{9\cdot3}+\frac1{7\cdot3}-\frac1{15\cdot7\cdot3}\biggr).$$ The analog of [([\[017\]](#017){reference-type="ref" reference="017"})]{.roman} and, respectively [([\[106\]](#106){reference-type="ref" reference="106"})]{.roman}, for [([\[112\]](#112){reference-type="ref" reference="112"})]{.roman} is $$\label{116} 1-2^{\alpha}=-\frac12\alpha.$$ The root of [([\[116\]](#116){reference-type="ref" reference="116"})]{.roman} with the minimal real part is $\alpha=-1$. Thus, the corresponding approximation of $F(z)\approx A(1-z)^{-1}$ leads to the linear growth of $\varphi_n\approx An$, which is in good agreement with numerical computations. However, in this case $z=1$ is not a unique singularity for $F(z)$ in $\{z:\ |z|\leqslant 1\}$. One of the reasons is that the Julia set for $P_0(z)=z^2$ is the unit disk. Analyzing [([\[112\]](#112){reference-type="ref" reference="112"})]{.roman}, it is seen that $e^{\frac{2\pi\mathbf{i}k}{2^n}}$, $k,n\geqslant 0$ are singularities for $F(z)$ giving different impacts to the power series coefficients. Let us extract the impact of two major singularities $z=1$ and $z=-1$. Substituting $$\label{117} F(z)\approx A(1-z)^{-1}+B\ln(1-z)+C\ln(1+z)$$ into [([\[112\]](#112){reference-type="ref" reference="112"})]{.roman} and eliminating two main terms at $z=1$ and one main term at $z=-1$, we obtain $$\label{118} B=\frac{A}{4\ln 2-2},\ \ \ C=\frac A2.$$ For the constant $A$, we have the limit $A=\lim_{n\to\infty}\psi_n$. Researcher Tian Vlašić from *math.stackexchange.com* note that the knowledge system WolframAlpha shows $$\label{119} A=\frac1{2-2\ln2},$$ see the discussion at [this site](https://math.stackexchange.com/questions/4748129/asymptotics-of-sequence-of-rational-numbers). At the moment, I cannot prove [([\[119\]](#119){reference-type="ref" reference="119"})]{.roman}, but this agrees very well with numerical data. It is useful to note that the constant $1/(1-\ln2)$ appears at least in one somewhat similar but inhomogeneous equation, see [@BS]. Also, equations similar to [([\[114\]](#114){reference-type="ref" reference="114"})]{.roman} but with constant coefficients were considered in, e.g., [@EHOP]. The corresponding asymptotic includes factors with the logarithm of natural numbers in the denominator. Finally, substituting [([\[119\]](#119){reference-type="ref" reference="119"})]{.roman} and [([\[118\]](#118){reference-type="ref" reference="118"})]{.roman} into [([\[117\]](#117){reference-type="ref" reference="117"})]{.roman} and using $\Phi(z)=F'(z)$, we obtain very good approximations $$\label{120} \varphi_n\approx A(n+1)-B+C(-1)^n=\frac{n}{2(1-\ln2)}+\frac{4\ln2-3}{4(1-\ln2)(2\ln2-1)}+\frac{(-1)^n}{4(1-\ln2)}.$$ A comparison between exact values $\varphi_n$, computed by [([\[114\]](#114){reference-type="ref" reference="114"})]{.roman}, and approximation terms, see [([\[120\]](#120){reference-type="ref" reference="120"})]{.roman}, is presented in Fig. [\[fig2\]](#fig2){reference-type="ref" reference="fig2"}. Three terms in [([\[120\]](#120){reference-type="ref" reference="120"})]{.roman} already give a good approximation of $\varphi_n$ as it is seen in Figs. [8](#fig2a){reference-type="ref" reference="fig2a"} and [9](#fig2b){reference-type="ref" reference="fig2b"}. Note that, the values at even and odd $n$ are different in Fig. [9](#fig2b){reference-type="ref" reference="fig2b"}. Again, double precision computations are not enough for good accuracy and lead to some small numerical errors appearing in Fig. [10](#fig2c){reference-type="ref" reference="fig2c"}. Nevertheless, some attenuated factors coming from complex singularities of $F(z)$ are seen in Fig. [10](#fig2c){reference-type="ref" reference="fig2c"}. The singularities $z=\pm\mathbf{i}$ give the largest impact to the remainder. Along with the next asymptotic terms related to $z=\pm1$, they lead to oscillations of period $4$ - values of the remainder at $n$, $n+1$, $n+2$, and $n+3$ differ significantly. The corresponding exact terms can be obtained by the analogy with the use of [([\[117\]](#117){reference-type="ref" reference="117"})]{.roman}. On the other hand, analyzing numerical results, one can directly substitute four next asymptotic terms of the order $n^{-1}$ into [([\[114\]](#114){reference-type="ref" reference="114"})]{.roman}, and use the symmetry among them - two unique values and two others with the different sign - to obtain the next more accurate approximation $$\label{120a} \varphi_n=\frac{1}{2(1-\ln2)}\biggl(n+\frac{4\ln2-3}{4\ln2-2}+\frac{(-1)^n}{2}-\frac{(-1)^{n}(1-\ln2)}{(2\ln2-1)n}+\frac{\cos\frac{\pi n}2-\sin\frac{\pi n}2}{n}+...\biggr).$$ It seems that the remainder in [([\[120a\]](#120a){reference-type="ref" reference="120a"})]{.roman} is of the order $n^{-2}$. Continuing the process with the next asymptotic terms, we hopefully can reach the long-phase oscillations related to the complex roots $\alpha$ of [([\[116\]](#116){reference-type="ref" reference="116"})]{.roman}. In this case, the asymptotic for $\varphi_n$ will contain both: short-phase and long-phase oscillations. Since [([\[114\]](#114){reference-type="ref" reference="114"})]{.roman} is much simpler than [([\[104\]](#104){reference-type="ref" reference="104"})]{.roman}, we can perform computations with double-double precision in a moderate time. For the computations with a standard double precision we use [Embarcadero Delphi CE](https://www.embarcadero.com/ru/products/delphi/starter) with a highly optimized parallel library [MtxVec](https://www.dewresearch.com/products/mtxvec/mtxvec-for-delphi-c-builder). For very precise computations with double-double precision, we use an additional library [NesLib](https://github.com/neslib/Neslib.MultiPrecision). While this precision is enough for the accurate numerical implementation of [([\[114\]](#114){reference-type="ref" reference="114"})]{.roman}, this computation is not vectorized. Anyway, Fig. [11](#fig2cdd){reference-type="ref" reference="fig2cdd"} is an improved version of Fig. [10](#fig2c){reference-type="ref" reference="fig2c"}. The double-double precision is also enough to estimate how good the next approximation [([\[120a\]](#120a){reference-type="ref" reference="120a"})]{.roman} is, see Fig. [12](#fig3cdd){reference-type="ref" reference="fig3cdd"}. It seems that two limit values among eight in Fig. [12](#fig3cdd){reference-type="ref" reference="fig3cdd"} coincide with some others - that is why there are only six distinct values, while the short-phase period is $8$. Moreover, the discussed long-phase oscillations are more or less visible now. Let us extract them explicitly. The values of $8$-periodic asymptotic term $\rho_n/n^2(2-2\ln2)$ multiplied by the second power of $n$ can be computed in the same way as the previous terms - by using [([\[114\]](#114){reference-type="ref" reference="114"})]{.roman} and the empirical fact that $\rho$ has a zero average. Skipping these straightforward computations, we write the final values $$\rho_{8n}=\frac{11\ln2-9}{4\ln2-2},\ \ \ \rho_{8n+1}=\frac{19-31\ln2}{4\ln2-2},\ \ \ \rho_{8n+2}=\frac{27-45\ln2}{4\ln2-2},\ \ \ \rho_{8n+3}=\frac{\ln2-5}{4\ln2-2},$$ $$\rho_{8n+4}=\frac{11\ln2-9}{4\ln2-2},\ \ \ \rho_{8n+5}=\frac{33\ln2-13}{4\ln2-2},\ \ \ \rho_{8n+6}=\frac{19\ln2-5}{4\ln2-2},\ \ \ \rho_{8n+7}=\frac{\ln2-5}{4\ln2-2}.$$ In principle, we can continue to compute the next asymptotic terms related to the attenuated factors $n^{-j}$, where $j>2$ is an integer. However, there are complex zeros of [([\[116\]](#116){reference-type="ref" reference="116"})]{.roman} $$\alpha=2.545364930374021...\pm10.75397517526887...\mathbf{i}$$ with a non-integer real part less than $3$. This is the smallest real part of zeros lying in the right half plane $\mathop{\mathrm{Re}}\nolimits\alpha>0$. According to the discussion around [([\[017\]](#017){reference-type="ref" reference="017"})]{.roman} and [([\[018\]](#018){reference-type="ref" reference="018"})]{.roman}, these zeros contribute to a long-phase oscillation. A double-double precision still allows us to compute this contribution explicitly, see Fig. [13](#fig4cdd){reference-type="ref" reference="fig4cdd"} {#fig2a width="\\linewidth"} {#fig2b width="\\linewidth"} {#fig2c width="\\linewidth"} {reference-type="ref" reference="fig2c"} with using double-double precision instead of double precision.](Ftest.png){#fig2cdd width="0.9\\linewidth"} ![A few next asymptotic terms to [([\[120\]](#120){reference-type="ref" reference="120"})]{.roman} are added, see [([\[120a\]](#120a){reference-type="ref" reference="120a"})]{.roman}.](testFF1.png){#fig3cdd width="0.9\\linewidth"} {reference-type="ref" reference="fig3cdd"} are added. Red points are $\exp(2\pi j/10.75397517526887)$ for $j=21,...,27$.](testFF2.png){#fig4cdd width="0.85\\linewidth"} Let us make a few remarks regarding the computation of $F(z)$. Introducing new function $$\label{n000} H(z)=\frac{(1-z)F(z)}{z^2},$$ equation [([\[112\]](#112){reference-type="ref" reference="112"})]{.roman} can be rewritten in the integral form $$\label{n001} H(z)=\frac1{2(1-z)}-\frac2{1-z}\int_0^z\frac{\zeta}{1+\zeta} H(\zeta^2)d\zeta,\ \ \ H(0)=\frac12.$$ For $|z|<1$, since $|z^2|<|z|$ (when $z\ne0$), one can use the Picard-Lindelöf iterations $$\label{n002} H_0\equiv\frac12,\ \ \ H_{n+1}(z)=\frac1{2(1-z)}-\frac2{1-z}\int_0^z\frac{\zeta }{1+\zeta}H_n(\zeta^2)d\zeta$$ to obtain the converging approximations $H_n(z)\to H(z)$. It also gives some new sequences converging to $H(1)$, for which we believe $H(1)=1/2(1-\ln2)$. On the other hand, if $H(1)$ is bounded as it follows from [([\[117\]](#117){reference-type="ref" reference="117"})]{.roman} and [([\[n000\]](#n000){reference-type="ref" reference="n000"})]{.roman}, then [([\[n001\]](#n001){reference-type="ref" reference="n001"})]{.roman} leads to $$\label{n003} \int_0^1\frac{\zeta }{1+\zeta}H(\zeta^2)d\zeta=\frac14,$$ and, hence, $$\label{n004} H(z)=\frac1{1-z}\int_z^1\frac{2\zeta }{1+\zeta}H(\zeta^2)d\zeta.$$ Again, starting with $H_0\equiv H(1)$ in [([\[n004\]](#n004){reference-type="ref" reference="n004"})]{.roman}, the Picard-Lindelöf iterations give converging approximations for $H(z)$, and some new sequence for $H(1)$ if we take into account the fact that $H(0)=\frac12$. In other words, is it true that $$\label{n005} \lim_{n\to+\infty}\int_0^1\int_{z_n^2}^1...\int_{z_3^2}^1\int_{z_2^2}^1\frac{2z_1}{1+z_1}\prod_{k=2}^n\frac{2z_k}{(1+z_k)(1-z_k^2)}dz_1dz_2...dz_n=1-\ln2?%\frac{1-\ln2}2?$$ This is an equivalent problem to [([\[119\]](#119){reference-type="ref" reference="119"})]{.roman}. In fact, for the iterations, instead of $H_0\equiv 1$, one can use any function such that $H_0(1)=1$ and $H_0(z)$ is continuous at $z=1$. As an example, the result of iterations for two different $H_0$ is given in Fig. [\[fig5iter\]](#fig5iter){reference-type="ref" reference="fig5iter"}. There are no visible differences between Figs. [14](#fig5iter1){reference-type="ref" reference="fig5iter1"} and [15](#fig5iter2){reference-type="ref" reference="fig5iter2"}. Hence, there is an idea that probably can help to solve the problem about the constant [([\[119\]](#119){reference-type="ref" reference="119"})]{.roman}: try to find a function $H_0(z)$ continuous at $z=1$ such that $H_0(1)\ne0$ and integrals $$\label{n006} ...\int_{z_4^2}^1\frac{2z_3 }{1+z_3}\cdot\frac1{1-z_3^2}\int_{z_3^2}^1\frac{2z_2 }{1+z_2}\cdot\frac1{1-z_2^2}\int_{z_2^2}^1\frac{2z_1 }{1+z_1}H_0(z_1^2)dz_1dz_2dz_3...$$ can be computed explicitly. ![Initial function $H_0\equiv 1$ is constant in the whole interval $[0,1]$.](test.val.png){#fig5iter1 width="\\textwidth"}   ![Initial function $H_0(z)=0$ for $z\in[0,0.5)$, and $H_0(z)=2-z$ for $z\in(0.5,1]$.](test1.val.png){#fig5iter2 width="\\textwidth"} ## Example 3. Let us consider the case of two polynomials $P_1(z)=az+(1-a)z^2$ and $P_2(z)=bz+(1-b)z^2$, $a,b\in(0,1)$, which appear with equal probability at each time step. Equation [([\[007\]](#007){reference-type="ref" reference="007"})]{.roman} becomes $$\label{121} \Phi(az+(1-a)z^2)+\Phi(bz+(1-b)z^2)=(a+b)\Phi(z).$$ By analogy with the previous examples, we obtain the explicit recurrence expression for $\varphi_n$: $$\begin{gathered} \label{121a} (a+b-a^n-b^n)\varphi_n=(a^{n-2}(1-a)^1+b^{n-2}(1-b)^1)\binom{n-1}{1}\varphi_{n-1}+\\ (a^{n-4}(1-a)^2+b^{n-4}(1-b)^2)\binom{n-2}{2}\varphi_{n-2}+(a^{n-6}(1-a)^3+b^{n-6}(1-b)^3)\binom{n-3}{3}\varphi_{n-3}+...,\end{gathered}$$ where the sum breaks when the lower index becomes strictly greater than the upper index in the binomial coefficient. We substitute an *ansatz* $$\label{122} \Phi(z)\approx A(1-z)^{\alpha}+B(1-z)^{\alpha+1}$$ into [([\[121\]](#121){reference-type="ref" reference="121"})]{.roman} to obtain the analog of [([\[015\]](#015){reference-type="ref" reference="015"})]{.roman} for $\alpha$: $$\label{123} (2-a)^{\alpha}+(2-b)^{\alpha}=a+b,$$ and another one connecting $A$ and $B$: $$\label{124} A(2-a)^{\alpha}(-\alpha)\frac{1-a}{2-a}+B(2-a)^{\alpha+1}+A(2-b)^{\alpha}(-\alpha)\frac{1-b}{2-b}+B(2-b)^{\alpha+1}=B(a+b),$$ which leads to $$\label{125} B=A\frac{-\alpha(2-a)^{\alpha-1}(1-a)-\alpha(2-b)^{\alpha-1}(1-b)}{a+b-(2-a)^{\alpha+1}-(2-b)^{\alpha+1}}.$$ Taking into account arguments [([\[016\]](#016){reference-type="ref" reference="016"})]{.roman}, [([\[017\]](#017){reference-type="ref" reference="017"})]{.roman}, along with [([\[122\]](#122){reference-type="ref" reference="122"})]{.roman}, [([\[123\]](#123){reference-type="ref" reference="123"})]{.roman}, and [([\[125\]](#125){reference-type="ref" reference="125"})]{.roman}, we obtain $$\label{126} \varphi_n\approx C\biggl(\frac1{n^{\alpha+1}}+\frac{(2-a)^{\alpha-1}(a-a^2)+(2-b)^{\alpha-1}(b-b^2)}{(2-a)^{\alpha}(a-1)+(2-b)^{\alpha}(b-1)}\cdot\frac{\alpha(\alpha+1)}{2n^{\alpha+2}}\biggr),$$ where $C=A/\Gamma(-\alpha)$. Now, the concrete values $$\label{127} a=\frac{7}{16},\ \ \ b=\frac{3}{4}$$ are chosen such that [([\[123\]](#123){reference-type="ref" reference="123"})]{.roman} can be solved explicitly $$\label{128} \alpha=\log_{\frac54}\frac{-1\pm\sqrt{\frac{23}{4}}}{2}.$$ For the parameter $C$ in [([\[126\]](#126){reference-type="ref" reference="126"})]{.roman}, numerical computations give $$\label{129} C\approx1.3355247475.$$ There is one real $\alpha$ in [([\[128\]](#128){reference-type="ref" reference="128"})]{.roman} with the minimal real part, chosen for the approximation [([\[126\]](#126){reference-type="ref" reference="126"})]{.roman}, and infinitely many complex $\alpha$ with the same minimal real part, and other complex values. As it is seen in Fig. [\[fig3\]](#fig3){reference-type="ref" reference="fig3"}, two terms in [([\[126\]](#126){reference-type="ref" reference="126"})]{.roman} give a good approximation of $\varphi_n$. Again, double precision does not provide accurate results in Fig. [18](#fig3c){reference-type="ref" reference="fig3c"}, but acceptable. The corresponding double-double precision corrections are given in Fig. [19](#fig3cdd1){reference-type="ref" reference="fig3cdd1"} The long-phase logarithmic oscillations coming from the complex $\alpha$, discussed above, will be dominant in comparison with the second term for very large $n$. This completely resembles the situation with the classical Galton--Watson process described in, e.g., [@K]. The difference is that in the classical case of one polynomial, we have a complete asymptotic series for $\varphi_n$ and corresponding fast algorithms for the computation of the asymptotic terms. In the considered mixed case of two polynomials, there are no direct analogs of Karlin--McGregor functions and, at the moment, I do not know fast algorithms for the computation of coefficients in [([\[019\]](#019){reference-type="ref" reference="019"})]{.roman} - only in some cases and for some of them it is sometimes possible to give explicit formulas, as in the examples above. {#fig3a width="\\linewidth"} {#fig3b width="\\linewidth"} {#fig3c width="\\linewidth"} {reference-type="ref" reference="fig3c"} with double-double precision.](u4test3dd.png){#fig3cdd1 width="0.99\\linewidth"} Now, let us change a little bit the probability in the first polynomial and take $$\label{130} a=\frac{1}{2},\ \ \ b=\frac{3}{4},$$ see [([\[127\]](#127){reference-type="ref" reference="127"})]{.roman}. In this case, [([\[123\]](#123){reference-type="ref" reference="123"})]{.roman} cannot be solved explicitly. It has one zero with a minimal real part $$\label{131} \alpha=-1.526066812384411....$$ The corresponding factor in the main asymptotic term $\varphi_n=Cn^{-\alpha-1}+...$ is approximately $$\label{132} C\approx 1.28574621970439$$ All other zeros of [([\[123\]](#123){reference-type="ref" reference="123"})]{.roman} are complex and lie in a strip. Some of the zeros are arbitrarily close to $\alpha$ mentioned in [([\[131\]](#131){reference-type="ref" reference="131"})]{.roman}. They make a big contribution to the asymptotic of $\varphi_n$, but for very large $n$, since their factors $C_{\alpha}$ are tiny. For the moderate values of $n$, other $\alpha$ contribute more. We plot the corresponding approximations in Fig. [\[fig4\]](#fig4){reference-type="ref" reference="fig4"}. Breaking tradition, we performed all the calculations with double-double precision. In contrast to Fig. [18](#fig3c){reference-type="ref" reference="fig3c"}, the oscillation in Fig. [22](#fig4c){reference-type="ref" reference="fig4c"} has a smaller order of amplification than the main asymptotic term presented in Fig. [20](#fig4a){reference-type="ref" reference="fig4a"}. In addition, if you are interested in learning about the distribution of zeros in equations of type [([\[015\]](#015){reference-type="ref" reference="015"})]{.roman} with a finite discrete $(R,\mu)$, you can see, e.g., [@HW] and references therein. {#fig4a width="\\linewidth"} {#fig4b width="\\linewidth"} {#fig4c width="\\linewidth"} # Conclusion Galton--Watson processes in random environments, when probabilities of fission at each time step are random variables themselves, are considered. The corresponding generalized Schröder type functional equation describing the relative limit densities of descendants $\varphi_n$ is derived. For the classical Galton--Watson process with fixed probabilities of fission, the corresponding asymptotic is $\varphi_n=\sum_{\alpha,j}C_{\alpha,j}n^{-\alpha-j}$, where all the values $\alpha$ have the same real part and linearly distributed imaginary parts. Moreover, there are fast numerical procedures for the computation of $C_{\alpha,j}$. In contrast to the classical case, in the generalized asymptotic the real parts of $\alpha$ may differ, and the distribution of both imaginary and real parts can be irregular. The corresponding examples are considered. At the moment, I do not know fast algorithms for the computation of $C_{\alpha,j}$ in the generalized cases. These cases are the sources of many interesting analytic problems related to special functions and constants. One of them is considered in Example 2 announced also on [this site](https://math.stackexchange.com/questions/4748129/asymptotics-of-sequence-of-rational-numbers). # Acknowledgements {#acknowledgements .unnumbered} This paper is a contribution to the project M3 of the Collaborative Research Centre TRR 181 \"Energy Transfer in Atmosphere and Ocean\" funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer 274762653. I would like to thank users of *math.stackexchange.com* for very useful discussions. 9 T. E. Harris, "Branching processes\", *Ann. Math. Statist.*, **41**, 474-494, 1948. P. Jagers, "Galton--Watson process in varying environments\". *J. Appl. Probab.*, **11**, 174--178, 1974. V. A. Vatutin and E. E. D'yakonova, "Galton--Watson branching processes in a random environment I: Limit Theorems\". *Theory Probab. its Appl.* , **48**, 314-336, 2004. S. Karlin and J. McGregor, "Embeddability of discrete-time branching processes into continuous-time branching processes\". *Trans. Amer. Math. Soc.*, **132**, 115-136, 1968. S. Karlin and J. McGregor, "Embedding iterates of analytic functions with two fixed points into continuous groups\". *Trans. Amer. Math. Soc.*, **132**, 137-145, 1968. M. S. Dubuc. "La densité de la loi-limite d'un processus en cascade expansif\". *Z. Wahrsch. Verw. Gebiete*, **19**, 281-290, 1971. A. A. Kutsenko, "Approximation of the number of descendants in branching processes\". *J. Stat. Phys.*, **190**, 68, 2023. B. Derrida, C. Itzykson, and J. M. Luck, "Oscillatory critical amplitudes in hierarchical models\". *Comm. Math. Phys.*, **94**, 115--132, 1984. B. Derrida, S. C. Manrubia, D. H. Zanette. "Distribution of repetitions of ancestors in genealogical trees\". *Physica A*, **281**, 1-16, 2000. B. Derrida and G. Giacomin, "Log-periodic critical amplitudes: A perturbative approach\". *J. Stat. Phys.*, **154**, 286--304, 2014. O. Costin and G. Giacomin, "Oscillatory critical amplitudes in hierarchical models and the Harris function of branching processes\". *J. Stat. Phys.*, **150**, 471--486, 2013. G. G. Brown and B. O. Shubert, "On random binary trees\". *Math. Oper. Res.*, **9**, 43-64, 1984. P. Erdös, A. Hildebrand, A. Odlyzko, P. Pudaite and B. Reznick, "The asymptotic behavior of a family of sequences\". *Pac. J. Math.*, **126**, 227-241, 1987. J. Heittokangas and Z. T. Wen, "The asymptotic number of zeros of exponential sums in critical strips\". *Monatsh. Math.*, **194**, 261--273, 2021.

arxiv_math

--- author: - "Mya Davis[^1]Carl Hammarsten[^2]Siddarth Menon[^3]Maria Pasaylo[^4]Dane Sheridan[^5]" bibliography: - references.bib title: Classifying Tractable Instances of the Generalized Cable-Trench Problem --- ## Abstract {#abstract .unnumbered} Given a graph $G$ rooted at a vertex $r$ and weight functions, $\gamma, \tau: E(G)\to \mathbb{R}$, the generalized cable-trench problem (CTP) is to find a single spanning tree that simultaneously minimizes the sum of the total edge cost with respect to $\tau$ and the single-source shortest paths cost with respect to $\gamma$ [@Vasko1; @Vasko2]. Although this problem is provably $NP$-complete in the general case, we examine certain tractable instances involving various graph constructions of trees and cycles, along with quantities associated to edges and vertices that arise out of these constructions. We show that given a path decomposition oracle, for graphs in which all cycles are edge disjoint, there exists a fast method to determine the cabl-trench tree. Further, we examine properties of graphs which contribute to the general intractability of the CTP and present some open questions in this direction. # Introduction We begin with a connected graph $G=(V, E)$, where $V$ is the set of vertices and $E$ is the set of edges. Each edge is represented by pairs of the vertices they adjoin along with a weight function $\gamma: E\to \mathbb{R}$. The *minimum spanning tree* problem and the *single-source shortest paths* problem are problems in the study of combinatorial algorithms with efficient and well-studied solutions. The *minimum spanning tree* problem is the problem of finding a connected acyclic graph $T = (V', E')$ such that $V'=V$ and $E'\subseteq E$ (referred to as a spanning tree) such that the sum of the weight of each edge in $E'$ is minimized over all possible spanning trees. The problem is efficiently solved by Prim's algorithm, which builds the tree by placing vertices in a priority queue and adding edges between vertices already inside the tree and vertices adjacent to vertices inside the tree. The *single-source shortest paths* problem specifies a distinguished vertex, called the root $r$, and asks to find a spanning tree $T$ for which the distance of a root path from $r$ to every vertex $w\in V$ with respect to weight function $\gamma$ is minimized for each $w$ (by convention, the length of the path from $r$ to $r$ is $0$). The single-source shortest paths problem is efficiently solved by Dijkstra's algorithm. Similarly, vertices are placed in a priority queue and vertices are added to the shortest paths tree as the distance of the corresponding root path is updated. The *single constraint cable-trench problem* (SCTP) asks for a spanning tree $T$ that minimizes some linear combination of the cost of a minimum spanning tree $W_{MST}(T)$ and the cost of the single-source shortest paths tree $W_{SPT}(T)$ with respect to a root $r$. The cable-trench cost $cost_{SCTP}(T)$ is the linear combination with real coefficients $\alpha$ and $\beta$ defined by the following equation: $$cost_{SCTP}(T) = \alpha W_{MST}(T) + \beta W_{SPT}(T)$$ This problem was shown to be $NP$-complete by noting that given any vertices $s,t$, finding a minimum spanning tree for which the $s-t$ path length is minimal is $NP$-hard [@Vasko1]. Consequently, the cable-trench problem which minimizes the length of all $r-w$ paths is certainly $NP$-complete. In this case however, there are reasonably efficient and effective algorithms that approximate optimal solutions depending on the ratio $\frac{\alpha}{\beta}$. In this article, we focus on the *generalized cable-trench problem* which is a variation of the cable-trench problem in which there are two independent weights assigned to each edge. Essentially, there are functions, $\gamma: E\to \mathbb{R}$, $\tau: E\to \mathbb{R}$, and the problem is to find a spanning tree which minimizes the sum of the weights of the single-source shortest paths tree with respect to $\gamma$ and the minimum spanning tree with respect to $\tau$. Colloquially, $\gamma$ is called the cable cost function and $\tau$ is called the trench cost function. Because edge weights can always be normalized with respect to $\alpha$ and $\beta$, we may omit the coefficients entirely. Subsequently, we aim to minimize the generalized cable-trench cost $cost(T)$ function: $$cost(T) = W_{MST}(T, \tau) + W_{SPT}(T, \gamma)$$ It immediately follows that the generalized cable-trench problem is also $NP$-complete because of the complexity of the single constraint variation. Vasko et al. [@Vasko2] formulated the mixed integer linear program for the cable-trench problem as follows: Minimize: $$\sum\sum\gamma_{ij}x_{ij}+\sum\sum\tau_{ij}y_{ij}$$ subject to $$\begin{aligned} &\sum x_{1j}=n-1 && i=1\\ &\sum x_{ij}-\sum x_{ki} =-1 && i=2, 3, \dots\\ &\sum y_{ij}=n-1 && \text{all edges }i<j\\ &(n-1)y_{ij}-x_{ij}-x_{ji}\geq 0 && \text{all edges }i<j\\ &x_{ij} \geq 0 && \forall i,j\\ &y_{ij} \in \{0, 1\} && \forall i,j.\end{aligned}$$ Here, $x_{ij}$ denotes the number of cables between vertices $i,j$, and $y_{ij}$ is 1 if and only if a trench is dug between vertices $i,j$. The first set of constraints ensures that exactly $n-1$ cables leave the root. The second set ensures that a root path terminates at every root vertex $i$ (i.e. every vertex has a cable laid *to it* from the root). The third set of constraints ensures that $n-1$ trenches are dug. The fourth set of constraints ensure that cables are laid where trenches are dug, and the last two constraints guarantee positivity and integrality of the linear programming LP variables. ## Heuristic algorithms and inapproximability Significant research has been done into heuristic algorithms that can approximate optimal spanning tree solutions to the cable-trench problem. In the general case, given the similarity of Dijkstra's solution to the single-source shortest paths problem and Prim's solution to the minimum spanning tree problem, a natural idea is to modify the algorithm such that vertices are added to a spanning tree with priority relative to the cable-trench cost that is added to the tree. Vasko et al. [@Vasko2] analyzed such a modified algorithm in the case of large cable-trench networks and found that it reasonably approximated efficient solutions. In [@Khullar], Khuller, Raghavachari, and Young describe an algorithm CREW-PRAM which computes a spanning tree with a continuous tradeoff between minimum spanning tree cost and single-source shortest paths tree cost. For a given $\lambda > 0$, the algorithm approximates a minimum spanning tree up to a factor of $1+\sqrt{2}\lambda$ and a single-source shortest paths tree up to a factor of $1+\frac{\sqrt{2}}{\lambda}$. Further work done by Benedito, Pedrosa, and Rosado [@Benedito] show that there exists an $\epsilon > 0$ for which approximating an optimal solution up to a factor of $1+\epsilon$ is NP-hard. # Definitions **Definition 1**. Recall that a **spanning tree** of a graph $G = (V(G), E(G))$ is a connected, acyclic graph $T = (V(T), E(T))$ such that $V(T) = V(G)$ and $E(T)\subseteq E(G)$. **Definition 2**. Given a path $P$, we define the **trench length** $\tau(E)$ as the weighted size of $P$ with respect to the edge-weight function $\tau: E(P)\to \mathbb{R}$. $$\tau(P) = \sum_{e\in P}\tau(e)$$ **Definition 3**. Given a path $P$, we define the **cabling length** $L(P)$ as the weighted length of a path $P$ with respect to the edge-weight function $\gamma: E(P)\to \mathbb{R}$. $$L(P) = \sum_{e\in P}\gamma(e)$$ **Definition 4**. Given a path $P$, we define the **cabling cost** $C(P)$ as the $\gamma$-weight of all subpaths along path $P$ with respect to some orientation of edges on $P: (e_1, \dots, e_n)$. $$C(P) = \sum_{i = 1}^{n} \sum_{j = 1}^{i}\gamma(e_j)$$ **Lemma 5**. *Given paths $A,B$, we can define the cabling cost of the path concatenation $AB$ as follows: $$C(AB) = C(A) + L(A)|B| + C(B)$$* *Proof.* Assume path $AB$ comes with oriented edge list $(e_1, \dots , e_n)$ where path $A = (e_1, \dots , e_k)$ and path $B = (e_{k+1}, \dots , e_n)$. The costs of cabling path $A$ and path $B$ are $$C(A) = \sum_{i = 1}^{k} \sum_{j = 1}^{i}\gamma(e_j)$$ $$C(B) = \sum_{i = k+1}^{n} \sum_{j = k+1}^{i}\gamma(e_j)$$ and lengths of paths $A$ and $B$ are $$L(A) = \sum_{i = 1}^k\gamma(e_i)$$ $$L(B) = \sum_{i = k+1}^n\gamma(e_i)$$ By observation, $$L(AB) = L(A) + L(B)$$ Then, $$\begin{aligned} C(AB) &= \sum_{i = 1}^{n} \sum_{j = 1}^{i}\gamma(e_j)\\ &= \sum_{i = 1}^{k} \sum_{j = 1}^{i}\gamma(e_j) + (n-k)\sum_{i = 1}^{k}\gamma(e_j) + \sum_{i = k+ 1}^{n} \sum_{j = k+1}^{i}\gamma(e_j)\\ &= C(A) + |B|L(A) + C(B)\\ &= C(A) + L(A)|B| + C(B)\end{aligned}$$ ◻ Essentially, we account for the cabling cost of $A$ in the first term $C(A)$. Every cable that starts at $A$ and ends in $B$ must pay the entire cabling cost of $A$, after which it is cabled normally, and is accounted for by the cabling cost of $B$ in the last term $C(B)$. We thus think of the $L(A)|B|$ term as **pre-cabling** path $B$ through $A$, which yields the recursive structure shown in Figure [1](#fig:precabling){reference-type="ref" reference="fig:precabling"}. [\[fig:precabling\]]{#fig:precabling label="fig:precabling"} {#fig:precabling width="300pt"} Informally, we use both **cable-trench solution** and **cable-trench tree** to refer to the minimum cost spanning tree with respect to edge weight functions $\gamma$ and $\tau$. Lastly, we define the graph operation central to our study of the cable-trench problem: **Definition 6**. Given graphs $G, H$, and vertices $v_G$ on $G$ and $v_H$ on $H$, we define the **wedge** $G\wedge H$ (omitting the vertex information for brevity) as the graph formed by combining $G, H$ such that vertices $v_G, v_H$ are identified as the same vertex. # Results on Wedging [\[section3\]]{#section3 label="section3"} Let $G$ be a tree. The optimal cable-trench tree is the minimum cost spanning tree of $G$. Since $G$ is already a tree, then the minimum cost cable-trench tree for $G$ is $G$ itself. We can easily compute the cable-trench tree for the wedge of two graphs $G, H$ through a shared root of these two graphs with known cable-trench solution: **Proposition 7**. *[\[wedge_trees_at_roots\]]{#wedge_trees_at_roots label="wedge_trees_at_roots"} Given graphs $G, H$ rooted at $r_G, r_H$, with known spanning trees $T_G$, $T_H$ that minimize total cable-trench cost, then the cable-trench tree for $G\wedge H$ formed by identifying $r_G, r_H$ is the tree formed from wedging $T_G \wedge T_H$ by identifying $r_G, r_H$.* *Proof.* Since $G\wedge H$ is edge disjoint and its root is the same as $G, H$, the cost of a spanning tree $T$ can be decomposed as the sum of the costs of $G$'s spanning tree $T_G$ and $H$'s spanning tree $T_H$: $$cost(T) = cost(T_G)+cost(T_H)$$ The cost of each subtree $T_G, T_H$ is independent of each other. In order to minimize the total sum, we minimize each subtree. Thus, the weight of the minimum cost spanning tree is $cost(T_G)+cost(T_H)$. Therefore, the cable-trench tree for $G\wedge H$ is $T_G \wedge T_H$. ◻ To further extend the complexity of graphs for which the cable-trench solution is tractably computable, we can consider the case in a graph with known cable-trench solution wedged at the root onto a cycle $G$ at a vertex $v\in V(G)$, where $v \neq r$. [\[fig:CycleTerms\]]{#fig:CycleTerms label="fig:CycleTerms"} {#fig:CycleTerms width="200pt"} Given the diagram for cycle $G$ in Figure [2](#fig:CycleTerms){reference-type="ref" reference="fig:CycleTerms"}, we define the following components of $G$ for an edge $e$: - $T$ is the spanning tree of $G$ rooted at $r$ that does not contain edge $e$. - $v$ is the wedge vertex on $G$ which is identified with the root of $H$ in the graph $G\wedge H$. We enforce $v \neq r$. In the case where $v = r$, refer to Proposition [\[wedge_trees_at_roots\]](#wedge_trees_at_roots){reference-type="ref" reference="wedge_trees_at_roots"}. - $P$ is the path from $r$ to $e$ that includes edge $e$. $Q$ is the path from $e$ to $v$ that does not include edge $e$. $R$ is the path from $r$ to $v$ that does not include $e$. Note that all three paths $P, Q, R$ are edge disjoint, and that $P \cup Q \cup R = G$. - All paths are assumed to originate from the root. When the direction of a path is ambiguous, notation $Q^+$ indicates the path is oriented clockwise, and $Q^-$ indicates the path is oriented counter-clockwise. For notation purposes, if an edge has a subscript, the corresponding components in the decomposition in Figure [2](#fig:CycleTerms){reference-type="ref" reference="fig:CycleTerms"} associated to this edge all share the same subscript (i.e. $e_i$ specifies spanning tree $T_i$, and splits $G$ into paths $P_i, Q_i$, and $R_i$. When comparing two trees $T_1$ and $T_2$, we denote the path $Q_1 \cup Q_2$ by $\tilde{Q}$. Note, the orientation for $Q_1$ and $Q_2$ must be the same for $\tilde{Q}$ to be an oriented path. So, if we assume $e_1$ is on the lower path and $e_2$ is on the upper path, then the clockwise orientation from $e_2$ to $e_1$ on $\tilde{Q}$ arises as $\tilde{Q}^+ = Q_2^+ \cup Q_1^+$. Similarly, the counter-clockwise orientation from $e_1$ to $e_2$ arises as $\tilde{Q}^- = Q_1^- \cup Q_2^-$. See Figure [3](#fig:multiple_wedges){reference-type="ref" reference="fig:multiple_wedges"} for an example labeling. The following results holds for wedging $H$ with known cable-trench solution onto cycle $G$: **Lemma 8**. *Let $G$ be an $n$-cycle and $H$ be a graph with known cable-trench tree $T_H$. Consider the cable-trench tree $T$ of $H \wedge G$ formed by identifying vertices $v$, $r_H$. The subgraph of $T$ induced by $H$ is precisely $T_H$.* *Proof.* Assume $H$ has an cable-trench solution $T_H$ and $G$ is an n-cycle. Since $G \wedge T_H$ is edge disjoint and $T_H$ is wedged onto $G$ at $r_H$, the cable-trench solution of $T_H$ is independent of $G$ and $G$ is dependent on the number of edges in $T_H$. Thus, the cable-trench tree of $H$ remains the same. ◻ **Lemma 9**. *Given the cable-trench tree $T_1$ that deletes $e_1$ as shown in cycle $G$ in Figure [2](#fig:CycleTerms){reference-type="ref" reference="fig:CycleTerms"}, for any $e_2 \neq e_1$ on the path $P_1 \cup Q_1$, the $cost(T_{2} \wedge H) \geq cost(T_{1} \wedge H)$* *Proof.* Suppose that $cost(T_2 \wedge H) < cost(T_1\wedge H)$. Given that deleting $e_1$ was the optimal edge to delete internal to $G$, we know $cost(T_2) \geq cost(T_1)$. In order to satisfy our assumption, the cost of wedging $H$ must be higher when wedging onto $T_{1}$ compared to $T_{2}$. However, observe that $R_1=R_2$ because $e_1$ and $e_2$ are on the same path. Thus, we refer to it generically as $R$, and the additional cost of wedging $H$ at $v$ in both cases is $L(R)|H|$. Therefore, the $cost(T_{2} \wedge H) \geq cost(T_1 \wedge H)$ which contradicts our initial assumption. ◻ Before using these lemmas to present an algorithm for computing the cable-trench solution for wedging a graph $H$ with known cable-trench solution onto a cycle graph $G$, we define the subroutine **CycleCTP** that takes a cycle graph $G$ as an input and returns the edge $e_1$ for which $T_1$ is the cable-trench solution for $G$ via a brute force minimization over all $n$ spanning trees of $G$. Consider the following algorithm that takes the following as input: a cycle $G$, a vertex $v\in G$, and a cable-trench solution $T_H$ for graph $H$ with root $r_H$ to be wedged onto $v$. Note that computing the cost of a given tree can be done in $O(n)$ time. $e_1 \gets \textbf{CycleCTP}(G)$ $\min \gets cost(T_1 \wedge T_H)$ $\text{min} \gets cost(T_2 \wedge T_H)$ $\text{min}$ Notice by Lemma [Lemma 9](#lem:BetterAlpha){reference-type="ref" reference="lem:BetterAlpha"}, we must only loop over edges on path $R_1$, thus performing fewer computations than a naive brute force approach. Based on the given cost function and algorithm as defined in Algorithm [\[alg:cap\]](#alg:cap){reference-type="ref" reference="alg:cap"}, we present the following proposition as a proof of correctness. **Proposition 10**. *Assume $G$ is a cycle rooted at $r$ with cable-trench solution $T_1$ that excludes edge $e_1$. Assume $H$ is an arbitrary graph with known cable-trench solution $T_H$. If we wedge $H$ at its root onto $G$ at a non-root vertex $v$, the cable-trench solution for $G \wedge H$ is $T_1 \wedge T_H$ as long as for all edges $e_2 \in V(G)$ are not on path $R_1$, we have:* *$$0 \geq \tau(e_2) - \tau(e_1) + L(P_2) - L(P_1) + (L(P_2)-L(P_1))|\tilde{Q}| + C(\tilde{Q}^+) - C(\tilde{Q}^-) + (L(R_1) - L(R_2))|H|$$* *Proof.* Following the convention of Figure [2](#fig:CycleTerms){reference-type="ref" reference="fig:CycleTerms"}, the total cable-trench cost of the spanning tree $T_1$ in $G$ is: $$\tau(E(G)) - \tau(e_1) + C(P_1^-) - L(P_1) + C(R_1^+) + L(R_1)|Q_1| + C(Q_1^+)$$ So, when we consider the spanning tree $T_1 \wedge T_H$ in $G \wedge H$ the resulting total cable-trench cost may be computed as: $$\tau(E(G)) - \tau(e_1) + C(P_1^-) - L(P_1) + C(R_1^+) + L(R_1)|Q_1| + C(Q_1^+) + L(R_1)|H| + cost(T_H)$$ Here, the $L(R_1)|H|$ term arises as pre-cabling all of the cables that are internal to $T_H$ backward from the root of $H$ to the root of $G$. By Lemmas [Lemma 8](#lem:WedgeTreeRemains){reference-type="ref" reference="lem:WedgeTreeRemains"} and [Lemma 9](#lem:BetterAlpha){reference-type="ref" reference="lem:BetterAlpha"}, we know only an edge $e_2$ which is not on $R_1$ could possibly result in a lower cost cable-trench tree $T_2 \wedge T_H$. Similarly, we compute the total cable-trench cost of $T_2 \wedge T_H$ as follows: $$\tau(E(G)) - \tau(e_2) + C(P_2^+) - L(P_2) + C(R_2^-) + L(R_2)|Q_2| + C(Q_2^-) + L(R_2)|H| + cost(T_H)$$ If the cable-trench cost of excluding edge $e_1$ were to be cheaper than when excluding edge $e_2$, the difference of the above total cable-trench costs must be non-positive. Hence, after some direct cancellations and rearranging of terms, we have the following inequality: $$\begin{aligned} 0 \geq\ &\tau(e_2) - \tau(e_1) \\ +\ &L(P_2) - L(P_1)\\ +\ &[C(R_1^+) - C(P_2^+)] - [C(R_2^-) - C(P_1^-)]\\ +\ &L(R_1)|Q_1| - L(R_2)|Q_2|\\ +\ &C(Q_1^+) - C(Q_2^-)\\ +\ &L(R_1)|H| - L(R_2)|H|\end{aligned}$$ Now, observe that $R_1^+$ is the concatenation of $P_2^+$ and $Q_2^+$. That is, $R_1^+ = P_2^+Q_2^+$. Similarly, $R_2^- = P_1^-Q_1^-$. Hence, by Lemma [Lemma 5](#lem:ConcatenatedPreCabling){reference-type="ref" reference="lem:ConcatenatedPreCabling"} we have: $$C(R_1^+) - C(P_2^+)=L(P_2)|Q_2| + C(Q_2^+)$$ $$C(R_2^-) - C(P_1^-)=L(P_1)|Q_1| + C(Q_1^-)$$ And so, after plugging these in and more rearranging of terms, the inequality becomes: $$\begin{aligned} 0 \geq\ &\tau(e_2) - \tau(e_1) + \\ +\ &L(P_2) - L(P_1) \\ +\ &[L(P_2)|Q_2| + L(R_1)|Q_1|] - [L(P_1)|Q_1| + L(R_2)|Q_2|] \\ +\ &C(Q_2^+) - C(Q_1^-)\\ +\ &C(Q_1^+) - C(Q_2^-) \\ +\ &L(R_1)|H| - L(R_2)|H|\end{aligned}$$ Now, by the additivity of $L$, we have that $L(R_1) = L(P_2) + L(Q_2)$. So, recalling that we use $\tilde{Q} = Q_1 \cup Q_2$ (as sets of edges), it follows that: $$\begin{aligned} L(P_1)|Q_1| + L(R_2)|Q_2| &= L(P_1)|Q_1| + L(P_1)|Q_2| + L(Q_1)|Q_2| \\ &= L(P_1)|\tilde{Q}| + L(Q_1)|Q_2|\end{aligned}$$ Similarly, we also get: $$L(P_2)|Q_2| + L(R_1)|Q_1| = L(P_2)|\tilde{Q}| + L(Q_2)|Q_1|$$ And hence the inequality becomes: $$\begin{aligned} 0 \geq\ &\tau(e_2) - \tau(e_1)\\ +\ &L(P_2) - L(P_1)\\ +\ &L(P_2)|\tilde{Q}| - L(P_1)|\tilde{Q}| \\ +\ &[C(Q_2^+) + L(Q_2)|Q_1| + C(Q_1^+)]\\ +\ &[C(Q_1^-) + L(Q_1)|Q_2| + C(Q_2^-)]\\ +\ &L(R_1)|H| - L(R_2)|H|\end{aligned}$$ Finally, noting that our orientation convention gives $\tilde{Q}^+ = Q_2^+Q_1^+$ and $\tilde{Q}^- = Q_1^-Q_2^-$, we can use Lemma [Lemma 5](#lem:ConcatenatedPreCabling){reference-type="ref" reference="lem:ConcatenatedPreCabling"} again to get: $$C(\tilde{Q}^+) = C(Q_2^+) + L(Q_2)|Q_1| + C(Q_1^+)$$ $$C(\tilde{Q}^-) = C(Q_1^-) + L(Q_1)|Q_2| + C(Q_2^-)$$ So our inequality becomes: $$\begin{aligned} 0 \geq\ &\tau(e_2) - \tau(e_1)\\ +\ &L(P_2) - L(P_1) \\ +\ &L(P_2)|\tilde{Q}| - L(P_1)|\tilde{Q}|\\ +\ &C(\tilde{Q}^+) - C(\tilde{Q}^-) \\ +\ &L(R_1)|H| - L(R_2)|H|\end{aligned}$$ ◻ **Remark:** The inequality can be interpreted as being divided into pairs of terms that represent the difference in cost between $T_2$ and $T_1$ of each pair: - $\tau(e_2) - \tau(e_1)$ represents the difference in trench length. $T_2$ does not have to 'dig the trench' through $e_2$, and $T_1$ does not have to dig the trench through $e_1$. - In both $T_1$ and $T_2$, paths $P_1$ and $P_2$ have to be cabled, and they are both cabled from the same direction (originating from root $r$). However, in $T_1$, the last cable along edge $e_1$ does not need to be laid, as the edge is excluded from the tree. Similarly, $T_2$ does not include the last edge on its path. Therefore, while the cost of cabling $P_1$ and $P_2$ cancel, $L(P_2) - L(P_1)$ remains as the costs arising from including edges $e_1$ and $e_2$ on paths $P_1$ and $P_2$ respectively. - The term $(L(P_2) - L(P_1))|\tilde{Q}|$ represents the difference in the cost of pre-cabling $\tilde{Q}$. In $T_1$, we lay cables to $\tilde{Q}$ with respect to one orientation and thus pre-cable it through $P_2$. While in $T_2$, we lay cables to $\tilde{Q}$ with respect to the opposite orientation and thus pre-cable it through $P_1$. - $C(\tilde{Q}^+) - C(\tilde{Q}^-)$ accounts for the difference in cabling region $\tilde{Q}$ with respect to the two different possible orientations arising from $T_2$ and $T_1$. - Finally, $L(R_1)|H| - L(R_2)|H|$ represents the difference in the pre-cabling costs for graph $H$ through either $R_1$ or $R_2$. Note that the cost of the continuation of these cables internal to $H$ cancels out as they are the same in either case of the cable-trench solution restricted to $G$ being $T_1$ or $T_2$. Observe that our particular choice of $v$ only really impacts the last element of the inequality. The remainder of the inequality is largely dependent on our choices of $e_1, e_2$. As such, if we were to generalize the above argument to wedging multiple graphs onto a cycle at multiple points, we would expect the majority of the contribution to come from this $\tilde{Q}$ path in between our choices of $e_1$ and $e_2$. Now, consider a case where we wedge multiple graphs with cable-trench solution at distinct vertices $v_1,v_2, ..., v_k$ of $n$-cycle G where $v_i \neq r$ for all $i$. [\[fig:multiple_wedges\]]{#fig:multiple_wedges label="fig:multiple_wedges"} {#fig:multiple_wedges width="200pt"} In the general case, let $\mathcal{H} = \{H_1, \dots, H_k\}$ be a set of graphs for which each graph $H_i$ has known cable-trench solution. Let $\mathcal{V} = \{v_1, \dots, v_k\}$ be a set of vertices, where $v_i$ is the vertex on $G$ which is identified with the root of $H_i$ in the graph $G\wedge H_i$. We define the set $H(\tilde{Q})\subseteq \mathcal{V}$ as the set of vertices on path $\tilde{Q}$ that are contained in $\mathcal{V}$. Adapting notation conventions specified earlier, for an edge $e \in G$, denote by $R^{(v)}$ as the path from root $r$ to $v$ that does not include $e$. **Theorem 11**. *Assume $G$ with root $r$ has cable-trench solution $T_1$ for some $e_1$. We wedge graphs $H_{v_1}, H_{v_2}, \dots, H_{v_n}$ at distinct vertices $v_1, v_2, \dots, v_n$. $H_i$ has cable-trench solution $S_i$. The minimum cable-trench tree for $G\wedge H_{v_1} \wedge H_{v_2} \wedge \dots \wedge H_{v_n}$ is $T_1 \wedge S_1 \wedge S_2 \wedge \dots \wedge S_n$ if for all edges $e_2 \neq e_1$: $$\begin{aligned} \label{eq:multi_wedge_ineq} 0 \geq \tau(e_2) - \tau(e_1) + &L(P_2) - L(P_1) + (L(P_2)-L(P_1))|\tilde{Q}| + C(\tilde{Q}^+) - C(\tilde{Q}^-)\\ &+ \sum_{v\in H(\tilde{Q})} (L(R_{2}^{(v)}) - L(R_{1}^{(v)}))|H_v| \nonumber \end{aligned}$$* *Proof.* The details of the proof are essentially parallel to the proof of Proposition [Proposition 10](#prop:inequality){reference-type="ref" reference="prop:inequality"}. Recall the decomposition of $G$ specified by $e_1$ and our arbitrary choice of $e_2\neq e_1$. In these terms, construct the cost (internal to $G$) for the cable-trench tree deleting $e_1$ is: $$cost(T_1)=\tau(E(G))-\tau(e_1) + C(P_1) - L(P_1) + C(P_2) + L(P_2)|Q| + C(Q^+)$$ The analogous cost for the cable-trench tree deleting $e_2$ is: $$cost(T_2)=\tau(E(G))-\tau(e_2) + C(P_2) - L(P_2) + C(P_1) + L(P_1)|Q| + C(Q^-)$$ The cost of each tree $T_1$ and $T_2$, internal to just $G$, is done analogously to Proposition [Proposition 10](#prop:inequality){reference-type="ref" reference="prop:inequality"}, with the only difference being the contribution of the costs of trees $S_1, \dots, S_n$. We will not switch to the tree deleting $e_2$ if $cost(T_1 \wedge S_1\wedge \dots \wedge S_n) \leq cost(T_2 \wedge S_1\wedge \dots \wedge S_n)$, as doing so will increase the cost of our cable-trench tree. When we take the difference of these two quantities, observe that any graph $H_i$ for which $v_i$ is not on path $\tilde{Q}$ is cabled the same way, and therefore its cable-trench cost will cancel in the difference $cost(T_1\wedge S_1\wedge\dots \wedge S_n) - cost(T_2\wedge S_1\wedge \dots \wedge S_n)$. The only meaningful contributions arise from vertices $v_i \in \tilde{Q}$. Thus, the contribution of these graphs to the difference is precisely: $$\sum_{v\in H(\tilde{Q})} (L(R_{2}^{(v)}) - L(R_{1}^{(v)}))|H_v|$$ As for each graph $H_i$ such that $v_i\in \tilde{Q}$, we must take the difference between pre-cabling $|H_i|$ cables along path $R_2^{(v_i)}$ versus along path $R_1^{(v_i)}$. The same algebraic manipulations as in the proof of Proposition [Proposition 10](#prop:inequality){reference-type="ref" reference="prop:inequality"} yield that $0 \leq cost(T_1) - cost(T_2)$ if and only if the desired inequality ([\[eq:multi_wedge_ineq\]](#eq:multi_wedge_ineq){reference-type="ref" reference="eq:multi_wedge_ineq"}) holds, in which case the minimum cable-trench tree is $T_1 \wedge S_1 \wedge S_2 \wedge \dots \wedge S_n$ ◻ In each case, finding the particular edge $e_2$ for which deleting $e_2$ would yield a cheaper total cost of the cable-trench tree involves (in the worst case) verifying the inequality over all possible values of $e_2\in R_1$. This is captured by the loop condition in Algorithm [\[alg:cap\]](#alg:cap){reference-type="ref" reference="alg:cap"}. # The Strength Index of a Graph [\[section4\]]{#section4 label="section4"} Notice, when wedging a graph $H$ onto $G$, the only extra cost within $G$ is the cost to cable from the root to the wedge vertex $v$. **Lemma 12**. *If $L(R_2) > L(R_1)$, then there is no $e_2$ such that $cost(T_{2}) < cost(T_{1})$.* *Proof.* By assumption, $T_{1}$ is the cable-trench solution internal to the graph $G$, the following inequality must hold: $$\begin{aligned} 0 \leq \tau(e_1) - \tau(e_2) + &C(R_2) - C(R_1) + (C(P_2) - L(P_2)) - (C(P_1) - L(P_1))\\ &+ L(R_2)|Q_2| - L(R_1)|Q_1| + C(Q_2) - C(Q_1) \end{aligned}$$ Suppose there exists an edge $e_2$ satisfying the condition in the statement of Lemma [Lemma 12](#lem:Myas){reference-type="ref" reference="lem:Myas"}. Then, once we wedge graph $H$ onto $G$, if $cost(T_{2}) < cost(T_{1})$, then from Proposition [Proposition 10](#prop:inequality){reference-type="ref" reference="prop:inequality"}: $$\begin{aligned} 0 > \tau(e_1) - \tau(e_2) &+ C(R_2) - C(R_1) + (C(P_2) - L(P_2)) - (C(P_1) - L(P_1)) + (L(R_2)-L(R_1))|\tilde{Q}|\\ &+ C(Q_2) - C(Q_1) + (L(R_2) - L(R_1))|H| \end{aligned}$$ Thus, $L(R_2) - L(R_1) < 0$ must be true if $cost(T_2) < cost(T_1)$ will be true for a given $e_2$. ◻ In essence, if $L(R_2) > L(R_1)$, then no matter how large the graph $H$ is, the cable-trench solution internal to $G$ will never change. Intuitively, switching to $T_2$ would mean that graph $H$ is cabled through path $R_2$. If $L(R_2) > L(R_1)$, the cost of adding $H$ increases with no savings to the spanning tree internal to $G$. We can expand this into an idea of the *strength* of a cycle $G$. **Definition 13**. The **strength** of an edge-vertex pair is denoted $\sigma(v,e)$. The vertex $v$ is the wedge vertex and the edge $e \in R_1$ is the edge which we will compare against $e_1$. The value of $\sigma(v,e)$ is the size of the vertex set for the largest graph $H$ that can be wedged onto $G$ such that the cable-trench solution remains $T_1 \wedge T_H$. As shown in Lemma [Lemma 12](#lem:Myas){reference-type="ref" reference="lem:Myas"}, if $L(R_2) > L(R_1)$, then $H$ can be any size, so we define $\sigma(v,e) = \infty$ for the $v,e$ pair. **Definition 14**. The **vertex strength** of a wedge vertex $v$ is $\sigma(v) = \min\{\sigma(v,e): e \in R\}$. The vertex strength is essentially the size of the largest tree that can be wedged onto $G$ such that $e_1$ remains the best edge to exclude in a cable-trench solution for the composite graph. **Definition 15**. The **breaking edge** is the edge $e$ on $R_1$ with $\sigma(v,e) = \sigma(v)$. The breaking edge is called as such since it is the first edge to "break" when enough weight is added onto $H$. **Lemma 16**. *Given cycle $G$ with cable-trench solution $T_1$ obtained via deletion of edge $e_1$. If the wedge vertex $v$ is chosen so that $L(R_2) < L(R_1)$, then there exists a breaking edge $e_2$ on the path $R_1$ $($the path from the root to the the wedge vertex not through $e_1)$.* *Proof.* For arbitrary graph $H$ with internal cable-trench solution $T_H$, consider the inequality comparing $T_1 \wedge T_H$ and $T_2 \wedge T_H$: $$\begin{aligned} 0 \leq \tau(e_1) - \tau(e_2) &+C(R_2) - C(R_1) + (C(P_2) - L(P_2)) - (C(P_1) - L(P_1)) + (L(R_2)-L(R_1))|\tilde{Q}|\\ &+ C(Q_2) - C(Q_1) + (L(R_2) - L(R_1))|H| \end{aligned}$$ This inequality is the case when $cost(T_2 \wedge T_H) \leq cost(T_1 \wedge T_H)$ All of the terms except the last are fixed for our choice of $e_1$ and $e_2$. When $L(R_2) < L(R_1)$, then $(L(R_2) - L(R_1))|H|$ is negative, so eventually for a large enough $|H|$: $$0 > \tau(e_1) - \tau(e_2) + L(P_1) - L(P_2) + (L(P_1)-L(P_2))|\tilde{Q}| + C(Q^-) - C(Q^+) + (L(R_2) - L(R_1))|H|$$ and thus $T_2 \wedge T_H$ is the better cable-trench solution. ◻ **Theorem 17**. *Once a breaking edge $e_1$ is determined, then there will never be a distinct edge $e_2$ on $P_1\cup Q_1$ such that $cost(T_{2} \wedge H) < cost(T_{1} \wedge H)$* *Proof.* Follows immediately from Lemma [Lemma 9](#lem:BetterAlpha){reference-type="ref" reference="lem:BetterAlpha"}. ◻ In the broader algorithmic context, the strength indices and breaking edge of a graph $G$ are internal properties of $G$, and thus for a fixed graph $G$, can be precomputed. When wedging graph $H$ onto $G$, the spanning tree computation reduces to a simple check of whether or not $|H|$ exceeds the strength of the graph, and if so, we can immediately determine the optimal spanning tree of the wedge via the breaking edge. In essence, these internal graph properties offer more savings in Algorithm [\[alg:cap\]](#alg:cap){reference-type="ref" reference="alg:cap"} as the cost computation is drastically simplified. # The Theta Graph To demonstrate the *local-to-global* notion that we established in Section [\[section3\]](#section3){reference-type="ref" reference="section3"}, we will broaden the classes of graphs for which we can efficiently solve the generalized cable-trench problem. We will consider a variant of a cycle graph which we will refer to as a $\theta$ graph. Such a graph consists of two vertices (a root vertex $r$ and another vertex $v$) connected through 3 edge-disjoint paths (which, for now, we refer to as $X_1, X_2, X_3$. Observe that in such a graph, any spanning tree is uniquely specified removing an edge from two distinct paths, i.e. $(e_i, e_j)$ where $e_i \in X_i$ and $e_j\in X_j$ for $i\neq j$. Figure [4](#fig:ThetaTerms){reference-type="ref" reference="fig:ThetaTerms"} demonstrates the analogous decomposition for $\theta$-graphs as we had in Figure [2](#fig:CycleTerms){reference-type="ref" reference="fig:CycleTerms"}. Suppose, without loss of generality, that we are certain to exclude an edge $e_3$ on path $P_3\cup Q_3$. By Proposition [\[wedge_trees_at_roots\]](#wedge_trees_at_roots){reference-type="ref" reference="wedge_trees_at_roots"}, we can essentially ignore $P_3$ in the computation that follows. We can simply determine the cable-trench solution for the cycle $R_{1,3} \cup R_{2,3}$, and determine whether $|Q_3|$ exceeds its strength. Iterating this procedure over all $e_3\in P_3\cup Q_3$ and repeating the procedure for (again without loss of generality) $e_2\in P_2\cup Q_2$, then taking the minimum over all computed costs yields the best solution for this class of graphs. Further, we can attempt to extend the notions of strength to this class of graphs to extend our local-to-global principle further. ## $\theta$-analogs of the strength index and breaking edges Without loss of generality, for some $\theta$-graph $G$, suppose that the internal cable-trench solution excludes some fixed edges $e_1\in P_1\cup Q_1$ and $e_2\in P_2\cup Q_2$. **Definition 18**. For a vertex $v$ and an edge $e_3 \in R_{1,2}$, define the **first edge strength** $\sigma_1(v, e_3)$ is the largest $h$ such that there exists a graph $H$ wedged onto $G$ at $v$ with $|H|=h$ for which $cost(T_{1,2}\wedge H) < cost(T_{1,3}\wedge H)$ and $cost(T_{1,2}\wedge H) < cost(T_{2,3}\wedge H)$ both hold. **Definition 19**. For a vertex $v$ and edge $e_3 \in R_{1,2}$, define the **second edge strength** $\sigma_2(v, e_3)$ is the largest $h$ such that there exists a graph $H$ wedged onto $G$ at $v$ with $|H|=h$ for which exactly *one* of $cost(T_{1,2}\wedge H) < cost(T_{1,3}\wedge H)$ and $cost(T_{1,2}\wedge H) < cost(T_{2,3}\wedge H)$ holds. Note that $\sigma_1(v, e) \leq \sigma_2(v, e)$. We use this to define the strengths of a vertex $v$ by minimizing over all edges $e \in R_{1,2}$, hence $\sigma_1(v) = \min_{e\in R_{1,2}}\sigma_1(v, e)$ and $\sigma_2(v) = \min_{e\in R_{1,2}}\sigma_2(v, e)$. With these in hand, we obtain a generalization of Lemma [Lemma 12](#lem:Myas){reference-type="ref" reference="lem:Myas"}. **Lemma 20**. *As in Lemma [Lemma 12](#lem:Myas){reference-type="ref" reference="lem:Myas"}:* 1. *If both $L(R_{2,3}) > L(R_{1,2})$ and $L(R_{1,3}) > L(R_{1,2})$, then $\sigma_1(v) = \sigma_2(v) = \infty$.* 2. *If exactly one of $L(R_{2,3}) > L(R_{1,2})$ or $L(R_{1,3}) > L(R_{1,2})$ holds, then $\sigma_2(v) = \infty$ and $\sigma_1(v) < \infty$.* *Proof.* We first present a proof of (1). We partition our $\theta$ graph into the following path segments, fixing some choices of edges $e_1 \in R_{2,3}$, $e_2 \in R_{1,3}$ and $e_3\in R_{1,2}$. [\[fig:ThetaTerms\]]{#fig:ThetaTerms label="fig:ThetaTerms"} {#fig:ThetaTerms width="200pt"} Without loss of generality, we assume that the cable-trench solution internal to this graph removes (fixed) edges $e_1, e_2$. Again, without loss of generality, we will show that $L(R_{1,3}) > L(R_{1,2})$ implies that there exists no edge $e_3 \in R_{1,2}$ that $cost(T_{2,3}\wedge H)$ is cheaper than $cost(T_{1, 2}\wedge H)$ (a similar argument will show that if $L(R_{2,3}) > L(R_{1,2})$ then there is no $e_3\in R_{1,2}$ such that $cost(T_{1,3}\wedge H)$ is cheaper than $cost(T_{1,2}\wedge H)$. The total cost of $T_{1,2} \wedge H$ can be computed as: $$\begin{aligned} (\tau(E(G))-\tau(e_1)-\tau(e_2)) &+ C(P_1)+C(P_2) + C(R_{1,2})+L(R_{1,2})(|H|+|Q_1|+|Q_2|)\\ &+ C(Q_1^-) + C(Q_2^-) + cost(H)\end{aligned}$$ We can further decompose $C(R_{1,2})$ as follows: $$C(R_{1,2}) = C(P_3) + L(P_3) + C(e_3) + (L(P_3) + L(e_3))|Q_3| + C(Q_3^+)$$ The total cost of $T_{1,3}\wedge H$ can similarly be computed as: $$\begin{aligned} (\tau(E(G))-\tau(e_1)-\tau(e_3)) &+ C(P_1)+C(P_3) + C(R_{1,3})+L(R_{1,3})(|H|+|Q_1|+|Q_3|)\\ &+ C(Q_1^-) + C(Q_3^-) + cost(H)\end{aligned}$$ We can further decompose $C(R_{1,3})$ as follows: $$C(R_{1,3}) = C(P_2) + L(P_2) + C(e_2) + (L(P_2) + L(e_2))|Q_2| + C(Q_2^+)$$ If $H$ is sufficiently large, such that $T_{1,3}$ is eventually cheaper in the above expression, then we expect the difference of the two expressions to be positive (i.e. $cost(T_{1,2}\wedge H) - cost(T_{1,3}\wedge H) > 0$). After cancelling out terms in both expressions, this assumption implies that for arbitrary $|H|$: $$\begin{aligned} \label{theta_ineq} &(\tau(e_3)-\tau(e_2))\\ &+ L(P_3)+L(e_3) + L(P_3)|Q_3|+L(e_3)|Q_3| +L(R_{1,2})(|H|+|Q_1| + |Q_2|)\\ &-L(P_2)-L(e_2) - L(P_2)|Q_2| -L(e_2)|Q_2| - L(R_{1,3})(|H|+|Q_1| + |Q_3|) \\ &+ C(Q_2^-) - C(Q_2^+) + C(Q_3^+)-C(Q_3^-) > 0\end{aligned}$$ Again, as in Proposition [Proposition 10](#prop:inequality){reference-type="ref" reference="prop:inequality"}, (2) accounts for the difference in trench lengths, (3) and (4) account for the difference in contributions from paths $R_{1,3}$ and $R_{1,2}$, and (5) accounts for the disoriented regions $Q_2$ and $Q_3$. We are assuming that $T_{1,2}$ is the internal cable-trench solution for $G$ (i.e. the cable-trench solution when $|H|=0$). Therefore, when $|H|=0$, we have that the difference is actually $\leq 0$. Thus, if the inequality is satisfied, we *must* have that: $$L(R_{1,2})|H| - L(R_{1,3})|H|>0$$ Since $|H|>0$, in order for there to exist a suitable $e_3$, we must have that $L(R_{1,2}) > L(R_{1,3})$. If not, then no such edge exists. If we apply the same reasoning to the tree $T_{1,3}$, we get that in order for there to exist a suitable $e_3$, we must have that $L(R_{1,2}) > L(R_{2,3})$. Thus, if $L(R_{2,3}) > L(R_{1,2})$ and $L(R_{1,3}) > L(R_{1,2})$, for all $|H|$, there is no cheaper tree. Thus, $\sigma_1(v) = \sigma_2(v) = \infty$. In order to prove the second case of the lemma, we simply note that we can apply the above rationale to exactly one of the pairs $R_{2,3}, R_{1,2}$ or $R_{1,3}, R_{1,2}$. Without loss of generality say that $L(R_{2,3}) > L(R_{1,2})$ and $L(R_{1,3})\leq L(R_{1,2})$. Then, for some choice of $e_3 \in R_{1,2}$ we can just solve for $|H|$ to find a graph where $T_{1,3}$ is less costly than $T_{1,2}$, and then minimize over all $e_3\in R_{1,2}$. ◻ **Lemma 21**. *[\[lem: better_alpha_theta\]]{#lem: better_alpha_theta label="lem: better_alpha_theta"} Let $G$ be a $\theta$ graph with disjoint $r,v$ paths $R_{2,3}$, $R_{1,3}$ and $R_{1,2}$. Without loss of generality, suppose that the cable-trench solution excludes edges $e_1\in R_{2,3}$ and $e_2\in R_{1,3}$. Then for all graphs $H$ wedged onto $G$ at $v$, if the cable-trench solution of $G\wedge H$ excludes edges $e_1'\in R_{2,3}$ and $e_2'\in R_{1,3}$, then $e_1'= e_1$ and $e_2' = e_2$.* *Proof.* The proof is analogous to the proof of Lemma [Lemma 9](#lem:BetterAlpha){reference-type="ref" reference="lem:BetterAlpha"}. Given that the cable-trench solution for $G\wedge H$ excludes edges $e_1' \in R_{2,3}$ and $e_2'\in R_{1,3}$, the contribution of wedging graph $H$ at vertex $v$ is $$\label{cont_H} L(R_{1,2})|H| + cost(H)$$ as root paths from $R$ to the vertices in $H$ must be cabled through $R_{1,2}$. We observe that in any spanning tree which excludes an edge on both $R_{2,3}$ and $R_{1,3}$ (thus, routes cables to $H$ in ([\[cont_H\]](#cont_H){reference-type="ref" reference="cont_H"}) through $R_{1,2}$) the contribution of $H$ is the same. Thus, the choices of $e_1'$ and $e_2'$ must be the optimal internal cable-trench solution to $G$, as the contribution of $H$ is equal for all pairs of edges in $R_{2,3}\times R_{1,3}$. By assumption, $e_1 \in R_{2,3}$ and $e_2 \in R_{1,3}$ are edges removed in the cable-trench solution of $G$, and therefore are optimal to remove internal to $G$. Therefore, we must have that $e_1' = e_1$ and $e_2' = e_2$. ◻ Note that Lemma [\[lem: better_alpha_theta\]](#lem: better_alpha_theta){reference-type="ref" reference="lem: better_alpha_theta"} makes no claims on the specific edges excluded in a cable-trench solution of $G\wedge H$ *if* the paths ($R_{2,3}, R_{1,3},$ or $R_{1,2}$) of the excluded edges differ from the paths of excluded edges from $G$. In essence, the fact that $\theta$ graphs admit tractable solutions is a direct consequence of the results of Sections [\[section3\]](#section3){reference-type="ref" reference="section3"} and [\[section4\]](#section4){reference-type="ref" reference="section4"}. By showing that similar constructions exist for $\theta$ graphs, we expect that computing the cable-trench solution for a graph in which two vertices are connected by 4 edge-disjoint paths should be tractable as well, by a similar argument as presented at the beginning of the section. The strength discussion enables us to further extend the class of graphs for which we can quickly compute the cable-trench solution. # Intractability of CTP in General Graphs The generalization of the strength precomputation to $\theta$ graphs demonstrates that the 'inductive' use of strength in order to increase the class of graphs that can be extended. Although, the increased complexity presents itself in the increasingly expensive precomputation of strength statistics about the graph. Further, the class of graphs attainable using the above methods is quite delicate; the methods described above would not clearly generalize when moving the wedge vertex $v$ such that it lies entirely on one of the three paths, $R_{2,3}$, $R_{1,3}$, or $R_{2,3}$, in the $\theta$ graph. We return to our discussion of graphs for which we can compute the cable-trench solution quickly, and will refer to such graphs as **tractable**. We also describe a class of graphs, given an oracle that provides some structural information about the graph, admit an efficient scheme for solving the generalized cable-trench problem. We will call such graphs **tractable with an oracle**. ## Tractability in cactus graphs As shown in Section [\[section3\]](#section3){reference-type="ref" reference="section3"}, given a collection of cycles and trees, we can iteratively construct a graph $G$ by wedging cycles and trees onto it and, in parallel, keep track of the cable-trench solution in polynomial time. Such a construction ensures a graph in which all cycles are edge-disjoint (i.e. each edge is in at most one cycle). These graphs are called *cactus graphs*. Keeping consistent with existing literature, a **decomposition** of a graph $G$ is defined as a collection of edge-disjoint subgraphs $G_1, G_2, \dots, G_m$ such that the edge sets $E(G_1), \dots, E(G_m)$ form a partition of the edge set of $G$. We say that this is a **path decomposition** if each subgraph is either a path or a cycle. Every graph clearly admits a path decomposition, and the following theorem of Lovász [@Lovasz] characterizes the size of such a decomposition. **Theorem 22**. *A graph $G$ on $n$ vertices (not necessarily connected) can be decomposed into $\lfloor n/2\rfloor$ paths and cycles.* As an algorithm, finding such a decomposition is closely related to the pathwidth metric on graphs. Although computing the pathwidth of arbitrary graphs is NP-complete, there exist fixed-parameter tractable algorithms (i.e. if the pathwidth is known to be bounded above, there exist tractable algorithms to compute it)[@Bodlaender2] [@Bodlaender] . Further, the decomposition itself is quite delicate, as we require a sort of *maximal* decomposition, one in which every pair of component intersects at most at one vertex. Such a decomposition must exist for a cactus graph essentially by definition of a cactus graph. Given an oracle that finds such a path decomposition for cactus graphs, we obtain the following corollary as a consequence of Theorem [Theorem 11](#thm:Wedging){reference-type="ref" reference="thm:Wedging"}. **Corollary 23**. *Cactus graphs are tractable with an oracle.* {#fig:example width="250pt"} [\[fig:example\]]{#fig:example label="fig:example"} **Example 24**. Consider the graph in Figure [5](#fig:example){reference-type="ref" reference="fig:example"}. If we were to wedge cycle $A$ to tree $B$ rooted at $r_B$, we could determine the cable-trench solution. At this point, we could wedge it to $C$ at $r_C$, and continue doing this until we wedged the graph $A\wedge B\wedge C\wedge D \wedge E$ onto $F$ at point $r_F$. At each step we know the optimal weight spanning tree, which allows us to wedge it to another tree or cycle. **Observation 25**. Let $T$ be a spanning tree for a cactus graph $G$. Given a subset of the vertices $V'$, if we look at the induced subgraph $G' \subseteq G$, and take the intersection $T' = G' \cap T$, we note that $T'$ is necessarily a spanning tree of $G'$. Essentially, our knowledge of spanning trees of individual components of $G$ determine the total number of spanning trees of the entire graph. ## Identifying multiple vertices in graph constructions We have seen that for cycles, trees, restricted cases of $\Theta$ graphs, and graphs constructible by wedging the previous graphs under specific conditions, the cable-trench solution can be found quickly. In general, there is a brute-force computation over all spanning trees via Kirchhoff's result on spanning trees that enables a fast search over a reasonably small set of spanning trees. A natural next step is to consider the point at which finding the cable-trench solution becomes computationally difficult. Of course, a necessary condition for a family of graphs to be *intractable* is for the family to admit an exponential number of spanning trees. The example we will use is the $2\times m$ grid graph ($G[2,m]$). By a simple inductive argument, it is clear that the number of spanning trees of $G[2,m]$ grid graph is $\Omega(2^n)$. Interestingly, we note that there exist spanning trees $T$ of the $2\times (m+1)$ grid such that the induced subgraph $T'\subset T$ restricted to the $2\times m$ grid is not a tree. This graph does not satisfy the property established in Example [Example 24](#ex_cactus){reference-type="ref" reference="ex_cactus"}, and as a consequence, an algorithm dependent on decomposing $G[2, m]$ should not be expected to consistently find the cable-trench solution. As iteratively wedging graphs together at chosen roots preserves the tractability of the graph, and any arbitrary graph admits a path decomposition, we must have a connection (or a class of connections) between graph components for which the graph is no longer tractable. We conjecture that families of graphs with exponentially many spanning trees, and those whose construction necessitates identifying paths and cycles at multiple points, do not easily admit tractable solution schemes. [^1]: Michigan Technological University [^2]: DeSales University [^3]: University of California, Berkeley [^4]: University of Florida [^5]: University of California, Los Angeles

arxiv_math

--- abstract: | We consider a Bayesian approach for the recovery of the thermal diffusivity for the heat equation when the initial temperature map of the system is unknown. This is a semiparametric inverse problem as the diffusivity is a one-dimensional parameter while the initial condition is infinite-dimensional. We assume that we have access to noisy observations of the system at the initial time $0$ and a final time $T$. We put a Gaussian process prior on the initial condition, and prove a Bernstein-von Mises (BvM) theorem for the marginal posterior of the thermal diffusivity provided that the prior regularity satisfies a restriction determined by the smoothness of the initial condition. We also investigate the behaviour of the marginal posterior for different prior regularities numerically, indicating that the BvM result is not valid if the prior regularity does not satisfy the restriction. address: - | VU Amsterdam\ - | VU Amsterdam\ author: - - bibliography: - references.bib title: Semiparametric Bernstein-von Mises for a Parameter in the Heat Equation --- # Introduction In nonparametric statistical inverse problems the goal is to recover a function $f$ from a noisy observation of a transformed version $K\!f$ of the function, where $K$ is some known operator, typically with unbounded inverse. Many methods have been proposed and studied for this problem, including frequentist regularization methods and nonparametric Bayes approaches. The focus in the theoretical literature is mostly on the setting that the operator $K$ is known and the function $f$ is the unknown object of interest that needs to be recovered from the data. In several important applications however, the operator $K = K_\theta$ depends on an unknown, Euclidean parameter $\theta$, and it is that parameter which is actually the main object of interest. An example from biology occurs in the paper [@gao], which deals with the modelling of biochemical interaction networks. In that case $K= K_\theta$ is the solution operator of a linear ordinary differential equation describing the time evolution of gene expression levels. The function $f$ describes how the activity of a so-called transcription factor changes over time and the parameter vector $\theta$ describes important aspects of the chemical reaction that is modelled, like the basal transcription rate and the rate of decay of mRNA. Another setting in which problems of this type arise naturally is in cosmology. In general relativity for instance, Einstein's equations describe how the large-scale structure of the universe evolves from small energy density fluctuations in the early universe. These equations are partial differential equations that depend on a number of important cosmological constants that cosmologists want to learn. Given the early state $f$ of the universe, general relativity gives "forward" predictions $K_\theta f$ of certain observable quantities. By comparing these predictions with (noisy) observations, inference can be made about the parameters $\theta$. In such applications the operator $K_\theta$ is typically an operator giving an appropriate solution to Einstein's equations. A particular example of this approach occurs in weak lensing, which is the state-of-the-art method used in recent and future cosmological surveys, see for instance [@lensing]. In that case the parameter vector $\theta$ of interest includes important constants like the matter density of the universe and the matter inhomogeneity. The natural Bayesian approach to this type of statistical inverse problems with parameter-dependent operators, often followed in practice, starts with endowing the pair $(\theta, f)$ with a prior distribution. The particular data generating mechanism gives rise to a likelihood and together the prior and the likelihood yield a posterior distribution for the unknown pair $(\theta, f)$. The marginal posterior for $\theta$ can then be used to make inference about $\theta$. This paper is motivated by the fact that this type of Bayesian methodology is commonly used in inverse problems with parameter-dependent operators, yet there are no readily applicable theoretical results giving insight into their fundamental performance. To obtain some first insight we study these matters in this paper in the context of a semiparametric inverse problem that is interesting yet relatively tractable mathematically. We consider the one-dimensional heat equation which describes the evolution of the temperature in a thin metal rod as a function of position and time. The dynamics of the system up to a final time $T> 0$ are governed by the partial differential equation $$\begin{aligned} \label{eq:heatEquation} {\frac{\partial }{\partial t}} u(x,t) = \theta \frac{\partial^2}{\partial x^2}u(x,t), \text{ \text{ }} u(x,0)=f(x), \text{ \text{ }} u(0,t)=u(1,t)=0,\end{aligned}$$ where $u$ defined on $[0,1]\times[0,T]$ is the temperature function and $f\in L^2[0,1]$ represents the initial condition of the temperature in the system, which is assumed to satisfy the same boundary conditions $f(0)=f(1)=0$. The parameter $\theta > 0$ is the thermal diffusivity constant of the metal and is our main object of interest. If $\theta$ is known, then inferring the initial function $f$ from noisy observations of the function $u(\cdot, T)$ at the final time $T$ is a well-known statistical inverse problem, see for instance [@Bissantz2008], [@Cavalier2008; @Cavalier2011], [@Golubev1999], [@Knapik2013], [@Mair1994], [@Mair1996], [@Stuart2010]. It is a severely ill-posed problem, since the solution operator that maps $f$ to $u(\cdot, T)$ takes functions in $L^2[0,1]$ to super smooth functions. Indeed, if we denote by $f_k$ the coefficients of $f$ with respect to the sine basis functions $e_k(x)=\sqrt{2}\sin(k\pi x)$, then the solution of [\[eq:heatEquation\]](#eq:heatEquation){reference-type="eqref" reference="eq:heatEquation"} can be expressed as $$\begin{aligned} \label{eq: sol} u(x,t)=\sqrt{2}\sum_{k=1}^{\infty}f_k e^{-\theta\pi^2 t k^2}\sin(k\pi x).\end{aligned}$$ As a consequence, the optimal, minimax rate for estimators that recover $f$ from a noisy observation of $u(0,T)$ is known to be of the order $(\log n)^{-\beta/2}$ (with respect to $L^2$-loss), where $\beta$ is the (Sobolev) regularity of $f$ and $n$ is the signal-to-noise ratio, or sample size. Various frequentist and Bayesian methods achieve this rate, see for instance [@Mair1994; @Mair1996; @Golubev1999; @Bissantz2008; @Knapik2013]. In this paper our interest is not in recovering the initial temperature distribution $f$ however, but rather in learning the diffusivity parameter $\theta$ from data, while still assuming that $f$ is unknown as well. Just observing the solution of the equation at final time $T$ is then not sufficient to identify the parameter $\theta$, as can be seen from [\[eq: sol\]](#eq: sol){reference-type="eqref" reference="eq: sol"}. Instead we assume that we have noisy observations of the system [\[eq:heatEquation\]](#eq:heatEquation){reference-type="eqref" reference="eq:heatEquation"} at times $0$ and $T$. More precisely, if we denote by $K_\theta: L^2[0,1]\to L^2[0,1]$ the solution operator $$\label{eq: k} K_\theta f = \sum_{k=1}^{\infty}\langle f, e_k \rangle e^{-\theta\pi^2 T k^2}e_k,$$ we assume that we have observations $X^{(1)} = (X^{(1)}_t: t \in [0,1])$ and\ $X^{(2)} = (X^{(2)}_t: t \in [0,1])$ satisfying $$\begin{aligned} \label{eq: sde1} dX^{(1)}_t & = f(t)\,dt + \frac1{\sqrt{n}}\,dW^{(1)}_t,\\ \label{eq: sde2} dX^{(2)}_t & = K_\theta f(t)\,dt + \frac1{\sqrt{n}}\,dW^{(2)}_t, \end{aligned}$$ where $W^{(1)}$ and $W^{(2)}$ are independent Brownian motions and $n$ is the signal-to-noise ratio (Expanding in the basis $e_k$ shows that this is equivalent to observing noisy versions of the coefficients of $f$ and $u(\cdot,T)$ relative to the basis $e_k$). We take a Bayesian approach and endow the pair $(\theta, f)$ with a prior distribution by putting a Gaussian process prior on $f$ and an independent prior with a positive, continuous Lebesgue density on $\theta$. We present results that describe the behaviour of the marginal posterior for the parameter $\theta$ of interest as $n\to \infty$. Our main result is Theorem [Theorem 1](#thm: main){reference-type="ref" reference="thm: main"}, a semiparametric Bernstein-von Mises (BvM) theorem for the marginal posterior of $\theta$ when putting a Gaussian process prior on $f$. It gives conditions under which the marginal posterior behaves asymptotically like a normal distribution centered at an efficient estimator of $\theta$, with a variance equal to the inverse of the efficient Fisher information. Such a result guarantees in particular validity of uncertainty quantification, in the sense that that credible intervals for the parameter $\theta$ obtained from the marginal posterior are also asymptotic frequentist confidence intervals in that case, see for instance [@castillo]. As the examples provided in Section [3](#sec: ex){reference-type="ref" reference="sec: ex"} show, the essential condition for BvM to hold is that the regularity of the prior on $f$ is not much greater than the regularity of the true underlying $f$. We complement this in Section [4](#sec:simulations){reference-type="ref" reference="sec:simulations"} with numerical experiments. These confirm our theoretical results and indicate that if the prior on $f$ is too smooth relative to the underlying truth then the posterior is biased, leading to unreliable inference about the parameter $\theta$. We restrict our analysis to the particular problem [\[eq:heatEquation\]](#eq:heatEquation){reference-type="eqref" reference="eq:heatEquation"}--[\[eq: sde2\]](#eq: sde2){reference-type="eqref" reference="eq: sde2"} in this first paper on this topic, but there are certainly more general underlying mechanisms and hence potential for generalization and extension. Investigation of our proofs shows that part of the arguments would go through for other operators $K_\theta$ as well without many changes. On the other hand, the dependence of $K_\theta$ on $\theta$ plays a crucial role and results like Lemma [Lemma 6](#lem: metrics){reference-type="ref" reference="lem: metrics"} ahead will have to be derived separately for each particular operator $K_\theta$. For other examples this may be much more challenging. We present the semiparametric setting and our main Theorem [Theorem 1](#thm: main){reference-type="ref" reference="thm: main"} in Section [2](#sec:result){reference-type="ref" reference="sec:result"} and give examples of Gaussian process priors for which it can be applied in Section [3](#sec: ex){reference-type="ref" reference="sec: ex"}. In Section [4](#sec:simulations){reference-type="ref" reference="sec:simulations"}, we experimentally assess the result of the theorem by providing a Metropolis Hastings algorithm for sampling from an approximation of the marginal posterior for $\theta$. The proof of Theorem [Theorem 1](#thm: main){reference-type="ref" reference="thm: main"} is given in Section [5](#sec: proof){reference-type="ref" reference="sec: proof"}. ## Notation On the interval $[0,1]$, the space of square integrable functions is denoted by $L^2[0,1]$ and the space of Hölder continuous functions of order $\alpha$ by $C^\alpha[0,1]$. For functions $f,g\in L^2[0,1]$, $\|f\|^2:=\int_0^1 f(x)^2\text{d}x$ is the $L^2$ norm of $f$ and $\langle f, g \rangle = \int_0^1 f(x)g(x)\text{d}x$ is the $L^2$ inner product of $f$ and $g$. Note that when writing $\|A\|$ for an operator $A$, no confusion is possible and we naturally mean the operator norm of $A$. For two numbers $a$ and $b$, we denote by $a\wedge b$ the minimum of $a$ and $b$. Similarly $a\vee b$ denotes the maximum of $a$ and $b$. For two sequences $a_n$ and $b_n$, we mean by $a_n\asymp b_n$ that $|a_n/b_n|$ is bounded away from zero and infinity as $n\to\infty$. By $a_n \lesssim b_n$, we mean that $a_n/b_n$ is bounded. For two probability measures $P$ and $Q$ defined on the same probability space $(\Omega,\mathcal{F}), \|P-Q\|_{\text{TV}}$ is the total variational distance between $P$ and $Q$ defined by $\|P-Q\|_{\text{TV}}:=2\sup_{F\in\mathcal{F}} |P(A)-Q(A)|$. Finally, for a metric space $(A, d)$ and $\varepsilon> 0$ we denote by $N(\varepsilon, A, d)$ the minimum number of balls of radius $\varepsilon$ needed to cover $A$. # General result {#sec:result} We assume that the pair $(\theta, f)$ belongs to $\Theta \times L^2[0,1]$, where $\Theta = (a,b)$ for some $0< a<b<\infty$. We endow the unknown pair with a prior $\Pi$ of the form $\Pi = \pi_\theta \times \pi_f$, where $\pi_\theta$ has a continuous Lebesgue density that is bounded away from $0$ and $\infty$ on $\Theta$ and $\pi_f$ is a centered Gaussian process prior on $L^2[0,1]$, that is, the law of a centered Gaussian process with sample paths that belong to $L^2[0,1]$. Girsanov's theorem (see for instance [@KS]) gives an expression for the likelihood, $$\begin{aligned} \log \frac{dP_{\theta, f}}{dP_{0}} = n\int_0^1 f(t)\,dX^{(1)}_t + n \int_0^1 K_\theta f(t)\,dX^{(2)}_t -\frac{n}2\|f\|^2 -\frac{n}2\|K_\theta f\|^2,\end{aligned}$$ where $P_{\theta, f}$ is the law of the pair $(X^{(1)}, X^{(2)})$ defined by [\[eq: sde1\]](#eq: sde1){reference-type="eqref" reference="eq: sde1"}--[\[eq: sde2\]](#eq: sde2){reference-type="eqref" reference="eq: sde2"} and the dominating measure $P_0$ is the law of $n^{-1/2}(W^{(1)}, W^{(2)})$. Combined with the prior this results in a posterior distribution $\Pi(\cdot \,|\,X)$ for the pair $(\theta, f)$. Here we denote by $X = (X^{(1)}, X^{(2)})$ all available data. We are in particular interested in the marginal posterior $B \mapsto \Pi(\theta \in B\,|\,X)$ of the thermal diffusivity parameter. Since we have a real-valued parameter $\theta$ of interest and an infinite-dimensional nuisance parameter $f$, the behaviour of this marginal posterior is determined by the semiparametric structure of the model. In many texts semiparametric concepts are mainly treated in the context of i.i.d. models, see for instance [@AadsBoek], but of course they can be extended to non-i.i.d. models, including the one we study in this paper. The paper [@McNeney] considers a quite general setting for instance. In Section [5](#sec: proof){reference-type="ref" reference="sec: proof"} ahead we show that our model fits into the semiparametric framework of [@castillo]. To describe our semiparametric Bernstein-von Mises theorem with as little technicalities as possible, we first recall that if $f$ is known (so that we only have to consider the observations $X^{(2)}$), the model is called differentiable in quadratic mean at $\theta$ if there exists a $g_\theta$ such that $$\frac{K_{\theta+h} f - K_{\theta} f}{h} \to g_\theta$$ in $L^2[0,1]$ as $h \to 0$, and that we then have local asymptotic normality (LAN) (see [@ih]). We note that this is the case in our model with $g_\theta = \dot K_\theta f$ if we define $\dot K_\theta: L^2[0,1] \to L^2[0,1]$ by $$\label{eq: kdot} \dot K_\theta f = - \pi^2 T\sum_{k=1}^{\infty}\langle f, e_k \rangle k^2e^{-\theta\pi^2 T k^2}e_k.$$ The notation is meant to indicate that $\dot K_\theta$ is simply the derivative of $K_\theta$ with respect to $\theta$. The Fisher information at $\theta$ in the model with known $f$ is then given by $I_\theta = \|\dot K_\theta f\|^2$. To understand what happens in the semiparametric case that $f$ is unknown, consider a one-dimensional submodel of the form $t \mapsto (\theta + t/\sqrt n, f +tg/\sqrt n)$, for $g \in L^2[0,1]$. Then realizing that $$\frac{K_{\theta+ah} (f+hg) - K_{\theta} f}{h} \to a\dot K_\theta f + K_\theta g$$ in $L^2[0,1]$ as $h \to 0$ we see using Girsanov's theorem that for the submodel we have the LAN expansion $$\label{eq: lan} \begin{split} \log \frac{dP_{\theta+t/\sqrt n, f+tg/\sqrt n}}{dP_{\theta, f}} & = t\int g\,dW^{(1)}+ t\int (\dot K_\theta f + K_\theta g)\,dW^{(2)}\\ & \quad -\frac{1}2t^2\|g\|^2 -\frac{1}2t^2\|\dot K_\theta f + K_\theta g\|^2 + o_{P_{\theta, f}}(1) \end{split}$$ as $n \to \infty$, with $W^{(1)}$ and $W^{(2)}$ independent $P_{\theta, f}$-Brownian motions. Hence, the Fisher information in the submodel at $t=0$ is given by $\|g\|^2 + \|\dot K_\theta f + K_\theta g\|^2$. Using the fact that $K_\theta$ is self-adjoint it is not hard to see that this is minimal for $g = -\gamma$, where $$\gamma = \gamma_{\theta, f} = (I+K^2_\theta)^{-1}K_\theta\dot K_\theta f.$$ We therefore call $\gamma$ the least favourable direction in our model. The Fisher information at $t=0$ in the least favourable submodel is given by $$\tilde I_{\theta, f} = \|\dot K_\theta f\|^2 - \langle K_\theta\dot K_\theta f, (I+ K^2_\theta)^{-1}K_\theta\dot K_\theta f \rangle.$$ The latter quantity is then the efficient Fisher information at $(\theta, f)$. Note that it is strictly positive and strictly smaller than the parametric Fisher information if $f \not = 0$, so there is loss of information due to the fact that $f$ is unknown. Our main general result asserts that the semiparametric BvM theorem holds for the marginal posterior distribution of the parameter $\theta$ if $\pi_f$ is a "good enough" Gaussian prior on the function $f$. Here "good enough" means that if $\pi_f$ is used as a prior to make inference about $f$ in the model [\[eq: sde1\]](#eq: sde1){reference-type="eqref" reference="eq: sde1"}, then the corresponding posterior contracts around the true $f_0$ (with respect to the $L^2$-norm) at a rate $\varepsilon_n$ that is of smaller order than $n^{-1/4}$. We recall that the rate $\varepsilon_n$ is determined by the so-called the concentration function $\varphi_{f_0}$ associated to the centered Gaussian prior $\pi_f$ on $L^2[0,1]$, which is defined by $$\varphi_{f_0}(\varepsilon) = \inf_{h \in \mathbb{H}:\|h-f_0\| \le \varepsilon}\|h\|^2_\mathbb{H}- \log\pi_f(f: \|f\| \le \varepsilon),$$ where $\mathbb{H}$ is the reproducing kernel Hilbert space (RKHS) of $\pi_f$ (see [@rkhs]). If $f_0$ is the true function in the model [\[eq: sde1\]](#eq: sde1){reference-type="eqref" reference="eq: sde1"} and $\varepsilon_n$ is a sequence of positive numbers such that $n\varepsilon^2_n \to \infty$ and the concentration inequality $\varphi_{f_0}(\varepsilon_n) \le n\varepsilon^2_n$ holds, then the posterior corresponding to the prior $\pi_f$ contracts around $f_0$ with respect to the $L^2$-norm at the rate $\varepsilon_n$ (see Theorem 3.4 of [@vdVvZ08]). The theorem below says that semiparametric Bernstein-von Mises holds for the marginal posterior of $\theta$ if the concentration inequality $\varphi_{f_0}(\varepsilon_n) \le n\varepsilon^2_n$ holds for a rate $\varepsilon_n$ that is of smaller order than $n^{-1/4}$ and, in addition, the least favourable direction $\gamma_{\theta_0, f_0}$ can be approximated by elements of the RKHS of $\pi_f$ in the same way as the true function $f_0$. The latter condition is expressed by imposing that the concentration inequality holds for $\gamma_{\theta_0, f_0}$ as well, that is, we also have $\varphi_{\gamma_{\theta_0, f_0}}(\varepsilon_n) \le n\varepsilon^2_n$. In principle the statement of the theorem can be extended to allow for approximation of $\gamma_{\theta_0, f_0}$ at a slower rate $\rho_n$, adding the condition that $\sqrt n \varepsilon_n\rho_n \to 0$. However, as we will discuss in the next section, the mapping properties of the solution operator $K_{\theta_0}$ imply that $\gamma_{\theta_0, f_0}$ is very smooth, and in particular smoother than the true function $f_0$. This means that the condition on the least favourable direction as phrased in the theorem is automatically fulfilled for many Gaussian process priors with a rich enough RKHS. In fact, we will see that for many priors we have that $\gamma_{\theta_0, f_0}$ actually belongs to the RKHS itself, so that the condition is trivially satisfied. What remains is the condition $n\varepsilon_n^4 \to 0$, which typically gives a restriction on the relation between the regularity of the true $f_0$ and the regularity of the prior $\pi_f$, as we will illustrate in the examples in Section [3](#sec: ex){reference-type="ref" reference="sec: ex"}. Recall that by $\|\cdot\|_{\text{TV}}$ we denote the total variation distance between probability measures. We denote the true parameter pair by $(\theta_0, f_0)$. **Theorem 1**. *Let $f_0 \not = 0$ and suppose there exists a sequence $\varepsilon_n \downarrow 0$ such that $n \varepsilon^2_n \gtrsim \log n$ and $n\varepsilon^4_n \to 0$ and we have the concentration inequalities $\varphi_{f_0}(\varepsilon_n) \le n\varepsilon^2_n$ and $\varphi_{\gamma_{\theta_0, f_0}}(\varepsilon_n) \le n\varepsilon^2_n$. Then $$\begin{aligned} \Big\|\Pi(\theta \in \cdot \,|\,X) - N\Big(\theta_0 + \frac{1}{\sqrt{n}} \Delta_{\theta_0, f_0}, \frac{1}{n}\tilde I^{-1}_{\theta_0, f_0}\Big)\Big\|_{\text{TV}}\to 0 \end{aligned}$$ in $P_{\theta_0, f_0}$-probability as $n\to\infty$, where $\Delta_{\theta_0, f_0 } \sim N(0, \tilde I^{-1}_{\theta_0, f_0})$.* The proof of the theorem is given in Section [5](#sec: proof){reference-type="ref" reference="sec: proof"}. In the next section we first investigate what the result looks like for a number of concrete Gaussian process priors $\pi_f$. # Results for specific Gaussian process priors {#sec: ex} In this section we consider various natural and commonly used choices of Gaussian process priors $\pi_f$. These priors all have a hyperparameter than can be viewed as describing a form of "smoothness", or "regularity" of the prior. We investigate in particular for which combinations of prior regularity and regularity of the true function $f_0$ we have that the Bernstein-von Mises result of Theorem [Theorem 1](#thm: main){reference-type="ref" reference="thm: main"} holds. To verify the condition involving the least favourable direction we note that by the explicit expressions for the operators $K_\theta$ and $\dot K_\theta$ it is given by $$\gamma_{\theta_0, f_0} = - \pi^2 T\sum_{k=1}^{\infty} \frac{k^2 \langle f_0, e_k \rangle e^{-2\theta_0\pi^2 T k^2}} {1+e^{-2\theta_0\pi^2 T k^2}} e_k.$$ The fact that the coefficients of $\gamma_{\theta_0, f_0}$ with respect to the sine basis functions $e_k$ are decreasing exponentially fast implies that the function is very smooth in various senses. It is for instance infinitely often continuously differentiable. As a result $\gamma_{\theta_0, f_0}$ belongs to the RKHS of many Gaussian process priors that are natural choices for $\pi_f$ in the present setting. *Example 2* (Series prior). Since the operator $K_\theta$ diagonalizes on the sine basis $e_k$, it is natural to consider a prior on $f$ with a covariance that diagonalizes on this basis as well. In this example we therefore consider the prior $\pi_f$ defined as the distribution of the random series $$W = \sum_{k=1}^\infty \sigma_kZ_ke_k,$$ where the $e_k$ are the sine basis functions introduced above, the $Z_k$ are independent standard normal variables and $\sigma_k$ is a sequence of standard deviations that satisfies $\sum \sigma^2_k < \infty$, ensuring that the random series defines a random element in $L^2[0,1]$. We will in particular consider the choice $\sigma_k \asymp k^{-1/2-\alpha}$ for $\alpha> 0$. This yields a prior on $f$ which (almost) has regularity $\alpha$ in Sobolev-type sense with respect to the sine basis. Indeed, for all $s < \alpha$ we have that $\sum k^{2s}\langle W, e_k \rangle^2 < \infty$ almost surely. We assume that $f_0$ has regularity $\beta$ in the same sense, that is, $\|f_0\|^2_\beta = \sum k^{2\beta}\langle f_0, e_k \rangle^2 < \infty$. (We note that since we consider the sine basis here, this notion of regularity is closely related to, but not exactly the same as the usual Sobolev regularity.) Let $\mathbb{H}$ be the reproducing kernel Hilbert space of the process $W$. According to Theorem 4.2 of [@rkhs] it is given by $$\mathbb{H}= \Big\{h= \sum c_k e_k: \|h\|^2_\mathbb{H}= \sum \frac{c^2_k}{\sigma^2_k} < \infty\Big\}.$$ By Corollary 4.3 of [@dunker] we have that $-\log\pi_f(\|f\| \le \varepsilon) \lesssim\varepsilon^{-1/\alpha}$. For $K \ge 1$, consider $h = \sum_{k \le K}\langle f_0, e_k \rangle e_k \in \mathbb{H}$. We have $\|h - f_0\| \le K^{-\beta}\|f_0\|_{\beta}$ and $\|h\|^2_\mathbb{H}= \sum_{k \le K} \langle f_0, e_k \rangle^2/\sigma^2_k \lesssim (1\vee K^{1+2(\alpha - \beta)})\|f_0\|^2_{\beta}$. Given $\varepsilon> 0$ we take $K \asymp (\|f_0\|_{\beta}/\varepsilon)^{1/\beta}$ to find that $$\varphi_{f_0}(\varepsilon) \lesssim(\varepsilon^{-\frac{1+2(\alpha - \beta)}{\beta}} \vee 1) + \varepsilon^{-\frac1\alpha}.$$ It follows that for $\varepsilon_n = cn^{-\alpha \wedge \beta/(1+2\alpha)}$, with $c > 0$ large enough, we have the inequality $\varphi_{f_0}(\varepsilon_n) \le n\varepsilon^2_n$. We clearly have $n \varepsilon^2_n \gtrsim \log n$ and the condition $n\varepsilon^4_n \to 0$ translates into the requirement that $1/2 < \alpha < 2\beta - 1/2$. The fact that the coefficients of $\gamma_{\theta_0, f_0}$ with respect to the sine basis decay exponentially fast implies that $\gamma_{\theta_0, f_0} \in \mathbb{H}$, irrespective of the values of $\alpha$ and $\beta$. Hence, the condition on $\gamma_{\theta_0, f_0}$ is trivially satisfied. We conclude that the Bernstein-von Mises result holds in this case if $1/2 < \alpha < 2\beta - 1/2$. The next examples show that it is in fact not necessary to use a prior on $f$ that is compatible with the operator $K_\theta$ as in the preceding case. Commonly used Gaussian process priors work just as well and give similar results. Only the appropriate type of regularity to describe the result is slightly different in each case. *Example 3* (Integrated Brownian motion). Define the $k$-fold integration operators $I_{0+}^k$ by $I_{0+}^0 f = f$ and $I^{k}_{0+}f(t) = \int_0^t I^{k-1}_{0+}f(s)\,ds$ for $k \in \mathbb{N}$. Now fix $k \in \mathbb{N}_0$ and let $\pi_f$ be the law of the centered Gaussian process $$W_t = \sum_{i=0}^{k} t^iZ_i + I_{0+}^k B(t), \qquad t \in [0,1],$$ where $Z_0, \ldots, Z_{k}$ are independent standard normal variables and $B$ is a standard Brownian motion independent of the $Z_i$. (The independent polynomial is added to the $k$-fold integrated Brownian motion because the process itself and its $k$ derivatives would otherwise all vanish at $0$, which is undesirable.) The well-known properties of Brownian motion imply that the process $W$ (almost) has regularity $\alpha = k+1/2$, in the sense that its sample paths almost surely belong to $C^s[0,1]$ for every $s < \alpha$. We assume that $f_0 \in C^\beta[0,1]$ for $\beta > 0$. It is known that in this situation the concentration inequality $\varphi_{f_0}(\varepsilon) \le n\varepsilon^2_n$ holds again for $\varepsilon_n = cn^{-\alpha \wedge \beta/(1+2\alpha)}$, with $c > 0$ large enough, see Section 11.4.1 of [@vaartghosal]. The RKHS of $W$ is the (usual) $L^2$-Sobolev space of regularity $\alpha+1/2 = k+1$, consisting of the functions on $[0,1]$ that are $k$ times differentiable, and with $k$th derivative $f^{(k)}$ that is absolutely continuous with derivative $f^{(k+1)}$ in $L^2[0,1]$ (see Lemma 11.29 of [@vaartghosal]). Hence, since $\gamma_{\theta_0, f_0}$ is infinitely often continuously differentiable it belongs to this RKHS and the condition on $\gamma_{\theta_0, f_0}$ is trivially satisfied again. We conclude that also in this case we have that the Bernstein-von Mises result holds if $1/2 < \alpha < 2\beta - 1/2$. It is straightforward to extend this example from multiply integrated Brownian motion to the more general Riemann-Liouville process, which also covers fractional integrals of Brownian motion. See Section 11.4.2 of [@vaartghosal]. *Example 4* (Matérn process). In this example we let $\pi_f$ be the law of a (one-dimensional) Matérn process with parameter $\alpha > 0$. This is a centered, stationary Gaussian process $W$ with spectral measure $\mu_\alpha(d\lambda) = (1+\lambda^2)^{-1/2-\alpha}d\lambda$, that is, $$\mathbb{E}W_s W_t = \int_\mathbb{R}\frac{e^{i\lambda(t-s)}}{(1+\lambda^2)^{1/2+\alpha}}\,d\lambda.$$ (The stationary Ornstein-Uhlenbeck process is a particular example, corresponding to $\alpha = 1/2$.) It can be shown that the sample paths of the Matérn process almost surely belong to $C^s[0,1]$ for all $s < \alpha$. We assume that for $\beta > 0$ we have $f_0 \in C^\beta[0,1] \cap H^\beta[0,1]$, where $H^\beta[0,1]$ is the Sobolev space defined as the space of restrictions to $[0,1]$ of functions $f \in L^2(\mathbb{R})$ with Fourier transform $\hat f$ that satisfies $\int (1+\lambda^2)^\beta |\hat f(\lambda)|^2\,d\lambda < \infty$. By Lemmas 11.36 and 11.37 of [@vaartghosal] we have $\varphi_{f_0}(\varepsilon_n) \le n\varepsilon^2_n$ for $\varepsilon_n = cn^{-\alpha \wedge \beta/(1+2\alpha)}$, with $c > 0$ large enough. The RKHS of $W$ is the space of (restrictions to $[0,1]$ of) real parts of functions $h$ that can be written as $h(t) = \int e^{i\lambda t}\psi(\lambda)\mu_\alpha(d\lambda)$ for some $\psi \in L^2(\mu_\alpha)$, see Lemma 11.35 of [@vaartghosal]. The smoothness on $[0,1]$ implies that $\gamma_{\theta_0, f_0}$ can be extended to a compactly supported $C^\infty$-function on the whole line. The Fourier transform $\hat \gamma_{\theta_0, f_0}$ of this extension then has the property that $|\lambda|^p |\hat \gamma_{\theta_0, f_0}(\lambda)| \to 0$ for every $p > 0$ as $|\lambda| \to \infty$. By Fourier inversion the extended function can be written as $$\gamma_{\theta_0, f_0}(t) = \int e^{i\lambda t} \hat \gamma_{\theta_0, f_0}(\lambda)\,d\lambda = \int e^{i\lambda t}\psi(\lambda)\mu_\alpha(d\lambda),$$ where $\psi(\lambda) = (1+\lambda^2)^{1/2+\alpha}\hat \gamma_{\theta_0, f_0}(\lambda)$. By the observation just made about the tails of $\hat \gamma_{\theta_0, f_0}$ we have that $\psi \in L^2(\mu_\alpha)$, hence $\gamma_{\theta_0, f_0}$ belongs to the RKHS of $W$. So as in the preceding examples we have that the condition on $\gamma_{\theta_0, f_0}$ is trivially satisfied and the Bernstein-von Mises result holds if $1/2 < \alpha < 2\beta - 1/2$. The least favourable direction generally does not belong to the RKHS of the squared exponential process. It can can be suitably approximated by elements of the RKHS of this prior however, so the Bernstein-von Mises result also holds if the (rescaled) squared exponential is used as prior on $f$. We note that in this example the prior depends on the signal-to-noise ratio $n$ through the length scale of the Gaussian process. Although it is not made explicit in the statement of Theorem [Theorem 1](#thm: main){reference-type="ref" reference="thm: main"}, inspection of the proofs shows that it remains valid for $n$-dependent priors. *Example 5* (Squared exponential process). Let $\pi_f$ be the law of the squared exponential process $W$ with length scale $l = l_{n, \alpha} = n^{-1/(1+2\alpha)}$ for $\alpha > 0$, so $$\mathbb{E}W_s W_t = e^{-(t-s)^2/l^2}.$$ The sample paths of the squared exponential process are analytic functions, but still in view of the results of [@scaling] it makes sense to think of the rescaled process $W$ as "essentially" having regularity $\alpha$. We assume that $f_0 \in C^\beta[0,1]$ for $\beta > 0$. By Lemma 2.2 and Theorem 2.4 of [@scaling] the inequality $\varphi_{f_0}(\varepsilon_n) \le n\varepsilon^2_n$ holds in this case for $\varepsilon_n = c n^{-\alpha \wedge \beta/(1+2\alpha)}\log n$, with $c > 0$ large enough. Note that also in this case we have $n\varepsilon_n^4 \to 0$ if and only if $1/2 < \alpha < 2\beta - 1/2$, the extra logarithmic factor in the rate $\varepsilon_n$ has no influence on this condition. The spectral measure of the stationary process $W$ is given by $\mu(d\lambda) = \mu_{n,\alpha}(d\lambda) = l(2\sqrt\pi)^{-1}e^{-l^2\lambda^2/4}\,d\lambda$ and as in the preceding example the RKHS $\mathbb{H}$ of $W$ is the space of (restrictions to $[0,1]$ of) real parts of functions $h$ that can be written as $h(t) = \int e^{i\lambda t}\psi(\lambda)\mu(d\lambda)$ for some $\psi \in L^2(\mu)$, and the RKHS norm of such an element of $\mathbb{H}$ is given by $\|h\|_\mathbb{H}= \|\psi\|_{L^2(\mu)}$ (see Lemma 11.35 of [@vaartghosal]). In the preceding example we noted that $\gamma_{\theta_0, f_0}$ extends to a compactly support $C^\infty$-function on $\mathbb{R}$ that can be written as $\gamma_{\theta_0, f_0}(t) = \int e^{i\lambda t} \hat \gamma_{\theta_0, f_0}(\lambda)\,d\lambda$, where the Fourier transform $\hat \gamma_{\theta_0, f_0}(\lambda)$ has tails that decay faster than any polynomial. Now for $K > 0$, let $$h(t) = \int_{-K}^K e^{i\lambda t} \hat \gamma_{\theta_0, f_0}(\lambda)\,d\lambda.$$ Then for every $p> 0$ we have $\|h-\gamma_{\theta_0, f_0}\|^2 \lesssim\int 1_{|\lambda| > K}|\lambda|^{-1-2p}\,d\lambda \lesssim K^{-2p}$. Moreover we have $h(t) = \int e^{i\lambda t}\psi(\lambda)\mu(d\lambda)$, where $$\psi(\lambda) = \frac{2\sqrt\pi}{l}1_{|\lambda| \le K} e^{\frac{l^2\lambda^2}{4}}\hat \gamma_{\theta_0, f_0}(\lambda).$$ It follows that $\|h\|^2_\mathbb{H}= \|\psi\|^2_{L^2(\mu)} \lesssim l^{-1} e^{\frac{l^2K^2}{4}}$. Taking and $K = 1/l$ and $p > \alpha \wedge \beta$ we obtain $$\inf_{h \in \mathbb{H}:\|h-\gamma_{\theta_0, f_0}\| \le \varepsilon_n}\|h\|^2_\mathbb{H}\lesssim n^{1/(1+2\alpha)} \le n\varepsilon^2_n$$ and hence, after enlarging the constant $c$ if necessary, $\varphi_{\gamma_{\theta_0, f_0}}(\varepsilon_n) \le n\varepsilon^2_n$. So also with this prior $\pi_f$ the Bernstein-von Mises result holds if $1/2 < \alpha < 2\beta - 1/2$. In fact by the results of [@scaling] this example generalizes to any process $W_{t/l}$, where $W$ is a centered stationary Gaussian process with a spectral measure $\mu$ that satisfies $\int e^{\delta|\lambda|}\,\mu(d\lambda) < \infty$ for some $\delta > 0$. So we see that many common choices of Gaussian process priors for $\pi_f$ lead to the Bernstein-von Mises result for the marginal posterior of $\theta$. Although the precise appropriate notion of smoothness differs from case to case, the condition is every time that the regularity $\alpha$ of the prior and the regularity $\beta$ of the true $f_0$ should satisfy the constraints $1/2 < \alpha < 2\beta - 1/2$. In other words it is not necessary that the prior $\pi_f$ is tuned optimally, leading to an optimal contraction rate for the marginal posterior of $f$. It is fine if the prior is undersmoothing, and a limited degree of oversmoothing is allowed as well. This phenomenon has been observed before in several other instances of the semiparametric Bernstein-von Mises theorem, see for instance [@castillo], [@rene], [@ismaeljudith]. # Posterior simulations {#sec:simulations} In this section, we present the results of our simulations to illustrate the application of Theorem [Theorem 1](#thm: main){reference-type="ref" reference="thm: main"}. We begin by reformulating the model to use the series representation for the prior, which is more convenient for sampling using Monte-Carlo Markov Chain (MCMC) methods. We select a ground truth $(\theta_0,f_0)$ and implement a Metropolis-Hastings (MH) algorithm to sample from the marginal posterior distribution of $\theta$. We repeat this process for different prior regularities $\alpha$ and examine the resulting marginal posterior distribution of $\theta$. We also provide trace plots to assess the convergence of our algorithm. For a given pair of parameters, the signal-in-white noise model we consider is equivalent to observing the following samples for all $k\in\mathbb{N}$: $$\begin{aligned} X^{(1)}_k = f_k + \frac{1}{\sqrt{n}}\zeta^1_k, \\ X^{(2)}_k = e^{-ck^2\theta}f_k + \frac{1}{\sqrt{n}}\zeta^2_k,\end{aligned}$$ where the $\zeta^1_k$ and $\zeta^2_k$ are i.i.d. standard Gaussians. In practice, we compute an approximation of the posterior by considering only the first $m$ observations. This yields the following expression for the approximate likelihood: $$\begin{aligned} \label{eq:likelihood_cut} \ell(\theta,f\mid X)= \dfrac{n^{K}}{\sqrt{2\pi}^{2K}}\prod_{k\leq m} \exp\left(-\dfrac{n}{2}(X^{(1)}_k-f_k)^2 - \dfrac{n}{2}(X^{(2)}_k- e^{-ck^2\theta}f_k)^2\right).\end{aligned}$$ We put a uniform prior on $\theta$ and consider the prior series representation of the GP prior on $f$ already mentioned in Example [Example 2](#seriesprior){reference-type="ref" reference="seriesprior"}. $$\begin{aligned} \pi^\sigma_f\sim \sum_{k=1}^{\infty} \sigma_k\nu_ke_k,\end{aligned}$$ where the $\nu_k$ are i.i.d $N(0,1)$ random variables and where $\sigma_k = (k+1)^{-1/2-\alpha}$ where $\alpha>0$ is the chosen regularity of the prior. It follows that the prior is distributed as follows $$\begin{aligned} (\theta,f_1,\cdots,f_m) \sim U(\Theta)\times \prod_{k=1}^m N\left(0,(k+1)^{-1-2\alpha}\right).\end{aligned}$$ Furthermore, by integrating $\Pi(\theta,f)\ell(\theta,f|X)$ with respect to to the $f_k$'s, we obtain that the marginal posterior for $\theta, \Pi(\theta\mid X)$ is proportional to the following quantity: $$\begin{aligned} \label{eq:post_theta} \prod_{k\leq m}(n+ne^{-ck^2\theta}+(k+1)^{2\alpha+1})^{-1/2} \exp\left(\dfrac{(nX^{(1)}_k+nX^{(2)}_ke^{-ck^2\theta})^2}{2(n+ne^{-ck^2\theta}+(k+1)^{2\alpha+1})}\right).\end{aligned}$$ Our MH algorithm will use [\[eq:post_theta\]](#eq:post_theta){reference-type="eqref" reference="eq:post_theta"} as a target distribution. For the proposal step, we sample from a normal distribution centered at the previous proposal value with standard deviation $\sigma_\theta$. The details of our sampling procedure are described in Algorithm [\[alg:MH\]](#alg:MH){reference-type="ref" reference="alg:MH"}. We fix $T=1$, let $\theta_0=0.01$ and consider the function $f_0$ defined as follows: $$\begin{aligned} f_0(x)=\sum_{k=1}^\infty k^{-2}e_k(x).\end{aligned}$$ We have that $f$ is Sobolev smooth of order $\beta = 3/2$. Our BvM theorem is guaranteed for $0.5< \alpha < 2.5$. We fix $m=100$ and generate data vectors $X^{(1)}$ and $X^{(2)}$ with signal-to-noise ratio $n=10^5$. We run Algorithm [\[alg:MH\]](#alg:MH){reference-type="ref" reference="alg:MH"} for $10^5$ iterations with a burn in period of 1000. For a correctly chosen $\alpha$ in the zone ($\alpha=1$), we run Algorithm [\[alg:MH\]](#alg:MH){reference-type="ref" reference="alg:MH"} on different datasets and observe that the marginal posterior of $\theta$ indeed seems to satisfy the BvM (Figure [1](#fig:goodAlpha){reference-type="ref" reference="fig:goodAlpha"}). We also plot the resulting marginal posterior distributions for $\theta$ for different values of $\alpha$ outside of the zone prescribed by the BvM theorem in Figure [2](#fig:ALPHAS_outisde){reference-type="ref" reference="fig:ALPHAS_outisde"}, that is for $\alpha>2\beta-1/2$. As anticipated, we observe the appearance of a bias which becomes larger when $\alpha$ increases. This process was repeated with different datasets to confirm the observations. ![Approximations of the marginal posterior of $\theta$ obtained when running algorithm [\[alg:MH\]](#alg:MH){reference-type="ref" reference="alg:MH"} on three different datasets with $\alpha=1$. The red line marks the true value of $\theta$ while the blue curve is the theoretical limiting distribution in Theorem [Theorem 1](#thm: main){reference-type="ref" reference="thm: main"}. That is, a normal distribution centered at an efficient estimator (here the posterior mean) with variance the efficient Fisher information.](images/bvm_valid.png){#fig:goodAlpha width="\\textwidth"} {reference-type="ref" reference="thm: main"} The approximations are realized for the same dataset.](images/bias_outside_zone.png){#fig:ALPHAS_outisde width="\\textwidth"} Overall, the presented figures validate the predictions of our semiparametric BvM. They also exhibit the positive relationship between the prior regularity $\alpha$ and the magnitude of a bias in the marginal posterior distributions which can be observed for $\alpha$ outside the BvM zone. To verify the convergence of our sampling algorithm, we present two of the trace plots of the chains of $\theta$ obtained during the sampling process for different values of $\alpha$ and on different datasets. These trace plots showcase no particular behavior, and no burn in period seems to really be required to obtain convergence. This contributes to the evidence that our algorithm is indeed converging and sampling from the posterior distribution. {#fig:traceplots width="80%"} # Proof of Theorem [Theorem 1](#thm: main){reference-type="ref" reference="thm: main"} {#sec: proof} The proof relies on application of Theorem 2 of [@castillo]. To set the stage we endow $L = \mathbb{R}\times L^2[0,1]$ with the inner product $$\langle (a_1,g_1), (a_1, g_2) \rangle_L = \langle g_1,g_2 \rangle+ \langle K_{\theta_0}g_1+a_1\dot{K}_{\theta_0}f_0, K_{\theta_0}g_2+a_2\dot{K}_{\theta_0}f_0 \rangle,$$ with corresponding norm given by $\|a, g\|^2_L = \langle (a,g), (a, g) \rangle_L$. The LAN expansion [\[eq: lan\]](#eq: lan){reference-type="eqref" reference="eq: lan"} then reads $$\log \frac{dP_{\theta_0+a/\sqrt n, f+g/\sqrt n}}{dP_{\theta_0, f_0}} = W(a,g) - \frac12\|a,g\|^2_L +o_{P_{\theta_0, f_0}}(1),$$ where $$W(a,g) = \int g\,dW^{(1)} + \int (a\dot K_{\theta_0} f_0 + K_{\theta_0} g)\,dW^{(2)},$$ with $W^{(1)}$ and $W^{(2)}$ independent $P_{\theta_0, f_0}$-Brownian motions. Note that $W$ is the isonormal process on the Hilbert space $L$. Also observe that for the least favourable direction $\gamma = \gamma_{\theta_0, f_0}$ it holds that $(0, \gamma) \in L$ is the orthogonal projection in $L$ of the point $(1,0)$ on the subspace $\{0\}\times L^2[0,1]$. Moreover, we have $\|1,0\|^2_L = \|\dot K_{\theta_0} f_0\|^2 = I_{\theta_0, f_0}$ and $\|0,\gamma\|^2_L = \langle K_{\theta_0}\dot K_{\theta_0} f_0, (I+ K^2_{\theta_0})^{-1}K_{\theta_0}\dot K_{\theta_0} f_0 \rangle$, so that $\tilde I_{\theta_0, f_0} = \|1,0\|^2_L - \|0,\gamma\|^2_L = \|1,-\gamma\|^2_L$. (Note that any $(a, g) \in L$ can be written as $(a,g) = a(1, -\gamma) + (0, g+a\gamma)$, where the terms are orthogonal in $L$. In particular, by Pythagoras, $\|a,g\|^2_L = a^2 \tilde I_{\theta_0, f_0} + \|0, g+a\gamma\|^2_L$.) Let $$R_n(\theta, f) = \log \frac{dP_{\theta, f}}{dP_{\theta_0, f_0}} - \Big(\sqrt n W(\theta-\theta_0 ,f-f_0) - \frac n2\|\theta-\theta_0 ,f-f_0\|^2_L\Big)$$ be the remainder in the LAN expansion. By Theorem 2 of [@castillo] it suffices to show that there exist $\rho_n \to 0$, and $\gamma_n \in \mathbb{H}$ satisfying $\|\gamma_n-\gamma\| \to 0$ and $\|\gamma_n\|_\mathbb{H}^2\le 2n\rho^2_n$, and $\varepsilon_n \to 0$ and $\mathcal{F}_n \subset L^2[0,1]$ such that with $\mathcal{F}_n(\theta) = \mathcal{F}_n + (\theta-\theta_0)\gamma_n$, the following conditions are satisfied: 1. Posterior contraction: for some $c > 0$ we have $\log \pi_f(\|f-f_0\| \le c \varepsilon_n) \lesssim n\varepsilon^2_n$ and $$\begin{aligned} \label{eq: c1} \Pi((\theta, f) \in \Theta\times \mathcal{F}_n: \|\theta-\theta_0, f-f_0\|_L \le \varepsilon_n\,|\,X) & \to 1,\\ \label{eq: c2} \inf_{|\theta-\theta_0|{\tilde I^{1/2}_{\theta_0, f_0}}\le \varepsilon_n} \Pi^{\theta =\theta_0}(f \in \mathcal{F}_n(\theta): \|0, f-f_0\|_L \le \varepsilon_n/2\,|\,X) & \to 1 \end{aligned}$$ in $P_{\theta_0, f_0}$-probability, where $\Pi^{\theta =\theta_0}$ is the posterior in the model in which $\theta=\theta_0$ is fixed. 2. Uniform LAN: $$\sup_{(\theta, f) \in \Theta\times \mathcal{F}_n \atop \|(\theta, f) - (\theta_0, f_0)\|_L \le 2\varepsilon_n} \frac{R_n(\theta, f) - R_n(\theta_0, f-(\theta-\theta_0)\gamma_n)}{1+n(\theta-\theta_0)^2} \to 0$$ in $P_{\theta_0, f_0}$-probability. 3. Approximation of the least favourable direction: $\sqrt n\varepsilon_n\rho_n \to 0$. The assumption $\varphi_{\gamma_{\theta_0, f_0}}(\varepsilon_n) \le n\varepsilon^2_n$ implies that there exist $\gamma_n \in \mathbb{H}$ such that $\|\gamma_n - \gamma_{\theta_0, f_0}\| \le \varepsilon_n$ and $\|\gamma_n\|^2_\mathbb{H}\le n\varepsilon^2_n$. We see that with this sequence $\gamma_n$ and $\rho_n = \|\gamma_n\|_\mathbb{H}/\sqrt{2n}$ we have $\rho_n \lesssim\varepsilon_n$ and $\sqrt n\varepsilon_n\rho_n \lesssim\sqrt n \varepsilon_n^2$, hence since $n \varepsilon^4_n \to 0$ condition (iii) above is satisfied. In the sections ahead we will prove that the posterior contraction and uniform LAN conditions (ii) and (iii) are satisfied for this sequence $\gamma_n$ as well. The following lemma provides bounds for various norms and distances that we encounter in the proof. For $\eta \ge 0$ we denote the Sobolev-type space of regularity $\eta$ with respect to the sine basis by $S^\eta[0,1]$. This is the space of functions $f$ for which $\|f\|^2_{S^\eta} = \sum k^{2\eta} \langle f, e_k \rangle^2 < \infty$. The operator norm of a bounded linear operator $T: L^2[0,1] \to S^\eta[0,1]$ is denoted by $\|T\|_{L^2 \to S^\eta} = \sup_{\|f\| \le 1} \|Tf\|_{S^\eta}$. **Lemma 6**. *For all $f_1, f_2 \in \mathcal{F}$ and $\theta_1, \theta_2, \theta \in \Theta$ and $\eta \ge 0$ we have* 1. *$\|K_\theta\| \le 1$, $\|\dot K_\theta\| \le \frac1{2\theta}$,* 2. *$$\|{K_\theta - K_{\theta_0}}\|_{L^2 \to S^\eta} \le \frac{|\theta - \theta_0|}{\pi^\eta T^{\eta/2}}\Big(\frac{2+\eta}{2e(\theta_0\wedge\theta)}\Big)^{\frac{2+\eta}{2}},$$* 3. *$$\|K_\theta - K_{\theta_0} -(\theta-\theta_0)\dot K_{\theta_0}\|_{L^2 \to S^\eta} \le \frac{(\theta - \theta_0)^2}{2\pi^\eta T^{\eta/2}}\Big(\frac{4+\eta}{2e\theta_0}\Big)^{\frac{4+\eta}{2}},$$* 4. *$\|K_{\theta_1} f_1 - K_{\theta_2}f_2\| \le \|f_1-f_2\| + \frac\pi2\sqrt{\frac{ T}{\theta_1 \wedge \theta_2}} |\theta_1-\theta_2| (\|f_1\| \wedge \|f_2\|),$* 5. *$\|(K_{\theta} - K_{\theta_0})f_0\| \ge T e^{-(\theta \vee \theta_0)\pi^2 T k^2}|\langle f_0, e_k \rangle|\, |\theta-\theta_0|$ for all $k \ge 1$.* ***Proof**.* (i). Follows from the explicit expressions for the operators and the inequality $\sup_{x \ge 0} x^2e^{-x} = 4e^{-2}\le 1$. (ii). We have $$\begin{aligned} \|(K_\theta & - K_{\theta_0})f\|^2_{S^\eta}\\ & = \sum k^{2\eta}\langle f, e_k \rangle^2(e^{-\theta \pi^2Tk^2}- e^{-\theta_0 \pi^2Tk^2})^2\\ & \le \sum k^{2\eta}\langle f, e_k \rangle^2 e^{-2(\theta_0\wedge\theta) \pi^2Tk^2}((\theta-\theta_0) \pi^2Tk^2)^2\\ & = \frac{(\theta-\theta_0)^2}{2^{2+\eta}(\theta_0\wedge\theta)^{2+\eta}\pi^{2\eta}T^\eta} \sum \langle f, e_k \rangle^2 e^{-2(\theta_0\wedge\theta) \pi^2Tk^2} (2(\theta_0\wedge\theta)\pi^2Tk^2)^{2+\eta}.\end{aligned}$$ Now use the fact that $\sup_{x \ge 0} e^{-x} x^{2+\eta} = e^{-(2+\eta)}(2+\eta)^{2+\eta}$. (iii). We have $$\begin{aligned} \|(K_\theta - K_{\theta_0} -(\theta-\theta_0)\dot K_{\theta_0})f\|^2_{S^\eta} & = \sum k^{2\eta}\langle f, e_k \rangle^2 e^{-2\theta_0\pi^2Tk^2}h^2((\theta- \theta_0)\pi^2 T k^2), \end{aligned}$$ where $h(x) = e^{-x} -1 + x$. The proof is completed as the proof of (ii), using that $|h(x)| \le x^2/2$ for $x \ge 0$ and $\sup_{x \ge 0} e^{-x} x^{4+\eta} = e^{-(4+\eta)}(4+\eta)^{4+\eta}$. (iv). By the triangle inequality and (i) we have $\|K_{\theta_1} f_1 - K_{\theta_2}f_2\| \le \|f_1-f_2\| + \|(K_{\theta_1} - K_{\theta_2})f_2\|$. By [\[eq: k\]](#eq: k){reference-type="eqref" reference="eq: k"} and the inequalities $|e^{-x}-e^{-y}| \le |x-y| e^{-x\wedge y}$ and $\sup_{x \ge 0} xe^{-x} =e^{-1}\le 1/2$, $$\|(K_{\theta_1} - K_{\theta_2})f_2\|^2 \le \frac{\pi^2 T}{4(\theta_1 \wedge \theta_2)} (\theta_1-\theta_2)^2 \|f_2\|^2.$$ Since the same can be done with the roles of $f_1$ and $f_2$ reversed, we arrive at the desired inequality. (v). For every $k \ge 1$ we obviously have $$\begin{aligned} \|(K_{\theta} - K_{\theta_0})f_0\|^2 & = \sum (e^{-\theta \pi^2 T j^2}-e^{-\theta_0 \pi^2 T j^2})^2\langle f_0, e_j \rangle^2 \\ & \ge (e^{-\theta \pi^2 T k^2}-e^{-\theta_0 \pi^2 T k^2})^2\langle f_0, e_k \rangle^2.\end{aligned}$$ The inequality then follows from the elementary fact that $|e^{-x}-e^{-y}| \ge |x-y| e^{-x\vee y}$. ◻ ## Posterior contraction condition In this section we consider conditions [\[eq: c1\]](#eq: c1){reference-type="eqref" reference="eq: c1"}--[\[eq: c2\]](#eq: c2){reference-type="eqref" reference="eq: c2"}. Items (i) and (v) of Lemma [Lemma 6](#lem: metrics){reference-type="ref" reference="lem: metrics"} and the triangle inequality imply that if $f_0 \not = 0$, there exists a constant $C> 0$ such that $$\|\theta-\theta_0, f-f_0\|_L \le C d((\theta, f), (\theta_0, f_0)),$$ where the metric $d$ is defined by $d^2((\theta_1, f_1), (\theta_2, f_2)) = \|f_1-f_2\|^2 + \|K_{\theta_1} f_1 - K_{\theta_2}f_2\|^2$. A straightforward adaptation of Theorem 8.31 of [@vaartghosal] then implies that [\[eq: c1\]](#eq: c1){reference-type="eqref" reference="eq: c1"} holds for a constant times $\varepsilon_n$ if for some $c_0, c_1, c_2 > 0$ we have $$\begin{aligned} \label{eq: cc1} -\log \Pi((\theta, f): d((\theta, f), (\theta_0, f_0)) \le c_0\varepsilon_n) & \le c_1 n\varepsilon^2_n\\ \label{eq: cc2} \log N(\varepsilon_n, \Theta \times \mathcal{F}_n, d) & \le c_2n\varepsilon^2_n,\\ \label{eq: cc3} \Pi((\theta, f) \not \in \Theta\times\mathcal{F}_n) & \le e^{-65c_1n\varepsilon^2_n}.\end{aligned}$$ By part (iv) of Lemma [Lemma 6](#lem: metrics){reference-type="ref" reference="lem: metrics"} and the fact that $\Theta$ is bounded away from $0$ we have $d((\theta, f), (\theta_0, f_0)) \lesssim\|f-f_0\| + |\theta-\theta_0|$. Since the prior density for $\theta$ is bounded away from $0$, a sufficient condition for [\[eq: cc1\]](#eq: cc1){reference-type="eqref" reference="eq: cc1"} is therefore that $$-\log \pi_f(\|f-f_0\| \le 2\varepsilon_n) \lesssim n\varepsilon^2_n.$$ It is well known that this is equivalent to the concentration inequality $\varphi_{f_0}(\varepsilon_n) \le n\varepsilon^2_n$, see Lemma 5.3 of [@rkhs]. For the sieves $\mathcal{F}_n$ we take $$\mathcal{F}_n= \varepsilon_nL^2_1 + M\sqrt n \varepsilon_n\mathbb{H}_1$$ for appropriate $M > 0$, where $\mathbb{H}_1$ and $L^2_1$ are the unit balls of the RKHS $\mathbb{H}$ and $L^2[0,1]$, respectively. Theorem 3.1 of [@vdVvZ08] implies that [\[eq: cc3\]](#eq: cc3){reference-type="eqref" reference="eq: cc3"} is fulfilled if $M$ is chosen large enough. It remains to check the entropy condition [\[eq: cc2\]](#eq: cc2){reference-type="eqref" reference="eq: cc2"}. Since $\mathbb{H}$ is continuously embedded in $L^2[0,1]$, we have that $\mathcal{F}_n \subset c'M\sqrt n \varepsilon_n L^2_1$ for some $c' > 0$. Hence, by Lemma [Lemma 6](#lem: metrics){reference-type="ref" reference="lem: metrics"}.(iv), we have $$d((\theta_1, f_1), (\theta_2, f_2)) \lesssim\|f_1-f_2\| + \sqrt n \varepsilon_n |\theta - \theta_0|$$ for all $\theta_1, \theta_2 \in \Theta$ and $f_1, f_2 \in \mathcal{F}_n$. It follows that $$\log N(\varepsilon_n, \Theta \times \mathcal{F}_n, d) \lesssim\log N(\varepsilon_n, \Theta, \sqrt n \varepsilon_n|\cdot|) + \log N(\varepsilon_n, \mathcal{F}_n, \|\cdot\|).$$ The first term is of order $\log n$ and the second one is bounded by a constant times $n\varepsilon^2_n$ by Theorem 3.1 of [@vdVvZ08]. Hence, the assumption that $n\varepsilon^2_n \gtrsim \log n$ implies that the left-hand side of the display is bounded by a constant times $n\varepsilon^2_n$. We conclude that [\[eq: cc1\]](#eq: cc1){reference-type="eqref" reference="eq: cc1"}--[\[eq: cc3\]](#eq: cc3){reference-type="eqref" reference="eq: cc3"} are indeed fulfilled with $\mathcal{F}_n$ as above, so that condition [\[eq: c1\]](#eq: c1){reference-type="eqref" reference="eq: c1"} is satisfied. The proof that [\[eq: c2\]](#eq: c2){reference-type="eqref" reference="eq: c2"} is satisfied is very similar, but now we need to bound the entropy of $\mathcal{F}_n(\theta) = \mathcal{F}_n + (\theta-\theta_0)\gamma_n$ and the remaining mass $\pi_f(f \not \in \mathcal{F}_n(\theta))$, uniformly in $\theta$. For the entropy we note that if $\{f_i\}$ is an $\varepsilon$-net for $\mathcal{F}_n$ and $\{\theta_j\}$ is an $\delta$-net for $\Theta$, then $\{f_i + (\theta_j-\theta_0)\gamma_n\}$ is an $(\varepsilon+ \delta\|\gamma_n\|)$-net for $\mathcal{F}_n(\theta)$. Hence, by setting $\delta = \varepsilon/\|\gamma_n\|$ we get $$\log N(2\varepsilon, \mathcal{F}_n(\theta), \|\cdot\|) \le \log N(\varepsilon, \mathcal{F}_n, \|\cdot\|) + \log\frac{\text{diam}(\Theta)\|\gamma_n\|}{\varepsilon}.$$ Since we have $\gamma_n \to \gamma$ and $n\varepsilon^2_n \gtrsim \log n$, it follows that $\log N(2\varepsilon_n, \mathcal{F}_n(\theta), \|\cdot\|) \lesssim n\varepsilon^2_n$. Finally, for the remaining mass, we note that we can write $\mathcal{F}_n(\theta) = (\varepsilon_nL^2_1 + (\theta-\theta_0)\gamma_n) + M\sqrt n \varepsilon_n\mathbb{H}_1$. Hence by the Borell-Sudakov-Tsirelson inequality, $$\pi_f(f \not \in \mathcal{F}_n(\theta)) \le 1-\Phi(\Phi^{-1}(\pi_f(\|f - (\theta - \theta_0)\gamma_n\| \le \varepsilon_n)) + M\sqrt n \varepsilon_n ).$$ By Lemma 5.2 of [@rkhs], $$\pi_f(\|f - (\theta - \theta_0)\gamma_n\| \le \varepsilon_n) \ge e^{-\frac12\|(\theta - \theta_0)\gamma_n\|^2_\mathbb{H}} \pi_f(\|f\| \le \varepsilon_n) \ge e^{-K \text{diam}^2(\Theta)n\varepsilon^2_n}$$ for some $K > 0$. Since $\Phi^{-1}(x) \sim - \sqrt{2\log(1/x)}$ for $x \to 0$, it follows that for given $C > 0$, it holds that $\pi_f(f \not \in \mathcal{F}_n(\theta)) \le e^{-Cn\varepsilon^2_n}$, provided $M$ is chosen large enough. The proof that condition [\[eq: c2\]](#eq: c2){reference-type="eqref" reference="eq: c2"} holds is now easily completed. ## Uniform LAN condition {#subsec:LAN} Straightforward algebra gives $$\begin{aligned} R_n(\theta, f) & = \sqrt n \int (K_\theta-K_{\theta_0}) (f-f_0)\,dW^{(2)} \\ & \quad + \sqrt n \int \Big((K_\theta-K_{\theta_0})f_0-(\theta-\theta_0)\dot K_{\theta_0} f_0 \Big)\,dW^{(2)}\\ & \quad -\frac{n}2\Big\|(K_\theta-K_{\theta_0}) (f-f_0) + \Big((K_\theta-K_{\theta_0})f_0-(\theta-\theta_0)\dot K_{\theta_0} f_0 \Big)\Big\|^2 \\ & \quad - n\langle (K_\theta-K_{\theta_0})f_0-(\theta-\theta_0)\dot K_{\theta_0} f_0, K_{\theta_0}(f-f_0) + (\theta-\theta_0)\dot K_{\theta_0}f_0 \rangle \\ & \quad - n\langle (K_\theta-K_{\theta_0}) (f-f_0), (\theta-\theta_0)\dot K_{\theta_0}f_0 \rangle\\ & \quad - n\langle (K_\theta-K_{\theta_0}) (f-f_0), K_{\theta_0}(f-f_0) \rangle. \end{aligned}$$ In particular we have $R(\theta_0, f) = 0$ for all $f$, so to verify the uniform LAN condition we need to show that $R_n(\theta, f) = o(1+n(\theta-\theta_0)^2)$, uniformly over the set $V_n = \{(\theta, f) \in \Theta\times \mathcal{F}_n: \|\theta -\theta_0, f-f_0\|_L \le 2\varepsilon_n\}$. Note that for $(\theta, f) \in V_n$ we have $\|f-f_0\| \lesssim\varepsilon_n$ and $|\theta-\theta_0| \lesssim\varepsilon_n$ and $\|f\| \lesssim 1+\varepsilon_n \lesssim 1$. We also repeatedly use the simple fact that $|x| \le 1 + x^2$, which implies that $\sqrt n/(1+n(\theta-\theta_0)^2) \le 1/|\theta-\theta_0|$ and $\sqrt n |\theta-\theta_0| \le 1+n(\theta-\theta_0)^2$. To deal with the first stochastic term, let $$\mathcal{G}= \Big\{(\theta-\theta_0)^{-1} (K_\theta-K_{\theta_0}) (f-f_0): (\theta, f) \in V_n\Big\}.$$ We have $0 \in \mathcal{G}$ and by Lemma [Lemma 6](#lem: metrics){reference-type="ref" reference="lem: metrics"}.(ii) (applied with $\eta=0$) the $L^2$-diameter of $\mathcal{G}$ is bounded by a constant times $\varepsilon_n$. Part (ii) of Lemma [Lemma 6](#lem: metrics){reference-type="ref" reference="lem: metrics"} also shows that for every $\eta > 0$, the set $\mathcal{G}$ belongs to a ball $B^\eta_{R\varepsilon_n}$ of radius $R\varepsilon_n$ in the Sobolev-type space $S^\eta[0,1]$ for some $R > 0$. By Dudley's theorem it follows that $$\begin{aligned} \mathbb{E}\sup_{g \in \mathcal{G}} \Big|\int g\,dW^{(2)}\Big| & \le \mathbb{E}\sup_{g_1, g_2, \in \mathcal{G}\atop \|g_1-g_2\| \le c\varepsilon_n} \Big|\int (g_1-g_2)\,dW^{(2)}\Big| \\ & \lesssim\int_0^{c\varepsilon_n} \sqrt{\log N(\varepsilon, B^\eta_{R\varepsilon_n}, \|\cdot\|) }\,d\varepsilon\\ & \lesssim\int_0^{c\varepsilon_n} \Big(\frac{\varepsilon_n R}{\varepsilon}\Big)^{1/(2\eta)}\,d\varepsilon= \varepsilon_n \int_0^c \Big(\frac{ R}{x}\Big)^{1/(2\eta)}\,dx. \end{aligned}$$ For $\eta > 1/2$ the last integral is finite. Hence, we have $$\sup_{\theta, f \in V_n} \frac{\sqrt n}{1+n(\theta-\theta_0)^2} \Big|\int (K_\theta-K_{\theta_0}) (f-f_0)\,dW^{(2)} \Big|\le \sup_{g \in \mathcal{G}} \Big|\int g\,dW^{(2)}\Big| \to 0$$ in probability. For the second stochastic integral, consider $$\mathcal{G}' = \{ (\theta-\theta_0)^{-1}((K_\theta-K_{\theta_0})f_0-(\theta-\theta_0)\dot K_{\theta_0} f_0): (\theta, f_0) \in V_n \}.$$ By Lemma [Lemma 6](#lem: metrics){reference-type="ref" reference="lem: metrics"}.(iii) (applied with $\eta=0$) we have $0 \in \mathcal{G}'$ and the $L^2$-diameter of $\mathcal{G}'$ is bounded by a constant times $\varepsilon_n$. Part (iii) of Lemma [Lemma 6](#lem: metrics){reference-type="ref" reference="lem: metrics"} also implies that for every $\eta > 0$, the set $\mathcal{G}$ belongs to a ball $B^\eta_{R\varepsilon_n}$ of radius $R\varepsilon_n$ in the Sobolev-type space $S^\eta[0,1]$ for some $R > 0$. The uniform bound for the second stochastic integral is then obtained using Dudley's theorem again. It remains to consider the deterministic terms in the remainder. Using parts (ii) and (iii) of Lemma [Lemma 6](#lem: metrics){reference-type="ref" reference="lem: metrics"} (applied with $\eta = 0$) and Cauchy-Schwarz it is straightforward to verify that the first three deterministic terms are $o(1+n(\theta-\theta_0)^2)$, uniformly over $V_n$. By Cauchy-Schwarz and part (ii) of Lemma [Lemma 6](#lem: metrics){reference-type="ref" reference="lem: metrics"} the last deterministic term is bounded by a constant times $n |\theta-\theta_0| \varepsilon_n^2 = \sqrt n |\theta-\theta_0| \sqrt n\varepsilon_n^2$. This is $o(\sqrt n|\theta-\theta_0|) = o(1+n(\theta-\theta_0)^2)$ under the assumption that $\sqrt n\varepsilon^2_n \to 0$. The authors would like to thank Frank van der Meulen and Aad van der Vaart for their helpful assistance and comments.

arxiv_math

--- abstract: | We study a distance graph $\Gamma_n$ that is isomorphic to the $1$-skeleton of an $n$-dimensional unit hypercube. We show that every measurable set of positive upper Banach density in the plane contains all sufficiently large dilates of $\Gamma_n$. This provides the first examples of distance graphs other than the trees for which a dimensionally sharp embedding in positive density sets is known. address: Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia author: - Vjekoslav Kovač - Bruno Predojević bibliography: - cubicalgraphs.bib title: Large dilates of hypercube graphs in the plane --- # Introduction One line of investigation in geometric measure theory was initiated with a question posed by Székely [@Sze83] (and popularized by Erdős [@Erd83:open]), who asked if a planar set of positive upper density (see Section [2](#sec:notation){reference-type="ref" reference="sec:notation"} for definitions of the usual densities) realizes all sufficiently large distances between pairs of its points. This question has been answered affirmatively by Furstenberg, Katznelson, and Weiss [@FKW90:dist], and independently also by Falconer and Marstrand [@FM86:dist] and by Bourgain [@B86:roth]. Each of the three proofs turned out quite influential; for instance the approach of Falconer and Marstrand [@FM86:dist] has been generalized to "very dense" sets in $\mathbb{R}^d$ [@FKY22]. Bourgain's approach [@B86:roth] made the greatest impact and it triggered a series of results studying more general rigid configurations in $\mathbb{R}^d$ [@LM16:prod; @LM19:hypergraphs; @K20:anisotrop; @DS22], not necessarily with respect to the Euclidean distance [@Kol04; @CMP15:roth; @DKR18; @DK21; @DK22]. Results of this type are often shown in sufficiently large dimensions only, as higher dimensions add "degrees of freedom" and make it easier to identify the desired pattern. The "slicing argument" rigorously justifies this reasoning, so the results are sometimes formulated only in the minimal number of dimensions $d$ in which they are known to hold. However, it is usually still unresolved what this minimal dimension $d$ needed to identify sufficiently large dilates of a given configuration in every set of positive upper density actually is. A large class of "flexible" configurations was covered by Lyall and Magyar [@LM20] in their study of embeddings of the so-called distance graphs which they define as finite connected graphs whose set of vertices is contained in some Euclidean space. Furthermore, they define a distance graph $\Gamma$ to be $k$-degenerate, for some positive integer $k$, if every induced subgraph of $\Gamma$ contains a vertex of degree at most $k$. The smallest such $k$ is known as the *degeneracy* of $\Gamma$. Their main result, [@LM20 Theorem 2], requires the dimensional threshold $d\geqslant k+1$ for every $k$-degenerate distance graph they discussed (which they called proper distance graphs). In other words, for a certain subclass of $k$-degenerate distance graphs $\Gamma$ they showed that every set $A\subseteq\mathbb{R}^{k+1}$ of positive upper density contains an isometric copy of the dilate $\lambda\cdot\Gamma$ for all sufficiently large numbers $\lambda>0$. One can easily check that a distance graph is a tree if and only if it is $1$-degenerate, so it follows from their result that trees can be embedded in the minimal possible dimension $d=2$. (An alternative approach to trees can be found in [@K20:anisotrop Section 5].) However it is natural to ask if there are other distance graphs, besides trees, for which such a dimensionally optimal embedding is possible. Similarly, one can ask whether there exist distance graphs of arbitrarily large degeneracy whose large dilates can still be embedded in the plane $\mathbb{R}^2$. The motivation for this paper lies in providing examples of such graphs. We are interested in the distance graph $\Gamma_n$ that is (isomorphic to) a $1$-skeleton of an $n$-dimensional unit hypercube. Notice that $\Gamma_n$ is $n$-degenerate. Unfolding the more general terminology from [@LM20], one says that a set $A\subseteq\mathbb{R}^2$ *contains an isometric copy of* $\lambda\cdot\Gamma_n$ for some number $\lambda>0$ if there exist a point $x\in\mathbb{R}^2$ and vectors $y_1,\ldots,y_n\in\mathbb{R}^2$ of Euclidean length $\lambda$ such that $$\label{eq:allpoints} x + r_1 y_1 + \cdots + r_n y_n \in A$$ for all $2^n$ tuples $(r_1,\ldots,r_n)\in\{0,1\}^n$ and that all $2^n$ points in [\[eq:allpoints\]](#eq:allpoints){reference-type="eqref" reference="eq:allpoints"} are mutually distinct; see Figure [1](#fig:graph1){reference-type="ref" reference="fig:graph1"}. Here is the precise formulation of our first result. {#fig:graph1 width="0.5\\linewidth"} **Theorem 1**. *For a positive integer $n$ and a measurable set of positive upper Banach density $A\subseteq\mathbb{R}^2$ there exists a number $\lambda_0(A,n)>0$ such that for every number $\lambda\geqslant\lambda_0(A,n)$ the set $A$ contains an isometric copy of the distance graph $\lambda\cdot\Gamma_n$.* There is some history of variants of Theorem [Theorem 1](#thm:large){reference-type="ref" reference="thm:large"} if one is content with embedding $\lambda\cdot\Gamma_n$ in higher-dimensional Euclidean spaces. - Groundbreaking work by Lyall and Magyar [@LM16:prod] developed a possible approach to product configurations $\Delta_1\times\Delta_2$ and, in particular, showed that sufficiently large dilates of rigid squares can be found in positive density subsets of $\mathbb{R}^4$; see [@LM16:prod Theorem 1.2]. - Higher-dimensional hypercubes were a bit more difficult and the same result for rigid $n$-hypercubes when $n\geqslant 3$ follows from the studies of more general configurations, done independently by Durcik and Kovač in $\mathbb{R}^{5n}$ [@DK21 Theorem 1] and by Lyall and Magyar in $\mathbb{R}^{2n}$ [@LM19:hypergraphs Theorem 1.1(i)]. - The most general rigid configuration for which a result like Theorem [Theorem 1](#thm:large){reference-type="ref" reference="thm:large"} is known is due to Lyall and Magyar [@LM19:hypergraphs Theorem 1.2(i)]; it embeds an $n$-fold product $\Delta_1\times\cdots\times\Delta_n$ of vertex-sets of non-degenerate simplices in positive density subsets of $\mathbb{R}^{\mathop{\textup{card}}(\Delta_1)+\cdots+\mathop{\textup{card}}(\Delta_n)}$. - Additional improvements are possible if one gives up the rigidity of the hypercube and starts "flexing" its $1$-skeleton. The aforementioned work by Lyall and Magyar [@LM20] recognizes $\Gamma_n$ as a proper $n$-degenerate distance graph, so it has already been known prior to this paper that a positive upper Banach density subset of $\mathbb{R}^{n+1}$ contains an isometric copy of $\lambda\cdot\Gamma_n$ at all sufficiently large scales $\lambda$ [@LM20 Theorem 2(i)]. - Theorem [Theorem 1](#thm:large){reference-type="ref" reference="thm:large"} above embeds $\lambda\cdot\Gamma_n$ in $\mathbb{R}^2$, which is clearly dimensionally optimal. Already Székely's question (which is the case $n=1$) obviously has a negative answer in $\mathbb{R}$: just consider $A=[-1/10,1/10]+\mathbb{Z}$. There has also been interest in "compact versions" (in the language of Bourgain [@B86:roth]) of Euclidean density theorems. The following weaker but quantitative variant of Theorem [Theorem 1](#thm:large){reference-type="ref" reference="thm:large"} will follow the same line of proof. Let $|A|$ denote the Lebesgue measure of $A\subseteq\mathbb{R}^2$. **Theorem 2**. *For a positive integer $n$ there exists a positive constant $C(n)$ with the following property: for every $0<\delta\leqslant 1/2$ and every measurable set $A\subseteq[0,1]^2$ satisfying $|A|\geqslant\delta$ there is an interval $I(A,n)\subseteq(0,1]$ of length at least $\exp(-\delta^{-C(n)})$ such that for every $\lambda\in I(A,n)$ the set $A$ contains an isometric copy of the distance graph $\lambda\cdot\Gamma_n$.* Weaker variants of Theorem [Theorem 2](#thm:interval){reference-type="ref" reference="thm:interval"} in higher dimensions can also be deduced from the existing literature in numerous ways. - Lyall and Magyar [@LM20 Theorem 2(ii)] showed a general quantitative result of this type for proper distance graphs, but there $\lambda\cdot\Gamma_n$ needs to be embedded in $[0,1]^{n+1}$. - For completely rigid hypercubes of side-length $\lambda$, Lyall and Magyar [@LM19:hypergraphs Theorem 1.1(ii)] showed an analogous result in $[0,1]^{2n}$, but with a tower-exponential bound on the length of the interval $I(A,n)$. Then they proceeded to study multiple products of vertex-sets of simplices $\Delta_1\times\cdots\times\Delta_n$ [@LM19:hypergraphs Theorem 1.2(ii)]. - Durcik and Kovač [@DK22 Theorem 3] sharpened the last result for hypercubes in $[0,1]^{2n}$ to the same type of bound for the length of $I(A,n)$ as in Theorem [Theorem 2](#thm:interval){reference-type="ref" reference="thm:interval"}, namely a single exponential in a negative power of $\delta$. - Durcik and Stipčić [@DS22 Theorem 1] extended the aforementioned quantitatively reasonable result from [@DK22] to multiple products of vertex-sets of simplices, simultaneously generalizing [@LM19:hypergraphs Theorem 1.2(ii)] and [@DK22 Theorem 3]. - Theorem [Theorem 2](#thm:interval){reference-type="ref" reference="thm:interval"} is also clearly dimensionally optimal for embeddings of $\Gamma_n$: in $[0,1]$ one can simply consider $A=[0,\varepsilon]\cup[3\varepsilon,4\varepsilon]\cup[6\varepsilon,7\varepsilon]\cup\cdots$, which has density about $1/3$, while $\varepsilon>0$ can be arbitrarily small. {#fig:graph2 width="0.43\\linewidth"} Certain subgraphs of $\Gamma_n$ (or their minor modifications) have appeared in the work of Fitzpatrick, Iosevich, McDonald, and Wyman [@FIMW21 Section 4] (see the collection of adjoined rhombi in Figure [2](#fig:graph2){reference-type="ref" reference="fig:graph2"}) albeit in the context of configurations in two-dimensional vector spaces over finite fields. This fact provided us with another source of motivation for formulating Theorems [Theorem 1](#thm:large){reference-type="ref" reference="thm:large"} and [Theorem 2](#thm:interval){reference-type="ref" reference="thm:interval"}. **Remark 3**. All of the aforementioned results in the literature relevant to embeddings of large dilates of $\Gamma_n$ also apply to the $1$-skeleton of a rectangular box with edge lengths $c_1,\ldots,c_n\in(0,\infty)$, viewed as a distance graph. This is also the case with our Theorems [Theorem 1](#thm:large){reference-type="ref" reference="thm:large"} and [Theorem 2](#thm:interval){reference-type="ref" reference="thm:interval"}, which we formulated in the case $c_1=\cdots=c_n=1$ only, not to overwhelm the notation. For instance, purely cosmetic changes in the proof of the first theorem can also show that the $1$-skeleton of a rectangular box with edge lengths $a_j=\lambda c_j$, $j=1,\ldots,n$, can be embedded in a positive upper Banach density set $A\subseteq\mathbb{R}^2$ for all sufficiently large numbers $\lambda$ depending on $A$; see Figure [3](#fig:graph3){reference-type="ref" reference="fig:graph3"}. In the other direction, each prescribed edge length of the distance graph that we are embedding clearly needs to be sufficiently large, as the case $a_1=1$ (or, in fact, and other fixed number) is easily prohibited by constructing a set $A$ of positive upper density that avoids unit distances between its points. Quite interestingly, a certain coloring constructed in [@Kov23color] to answer a seemingly unrelated question of Erdős and Graham [@EP \#189] (also see [@EG79 p. 331] and [@GB15 p. 56]) can be modified to show that there exists a positive upper density planar set $A$ in which we cannot embed any $1$-skeleton of a rectangular box with sides $a_j$ satisfying $$\label{eq:productrelation} a_1 a_2 \cdots a_n=1;$$ the details are given in [@Kov23color]. The fact that even a single relation between edge lengths, like [\[eq:productrelation\]](#eq:productrelation){reference-type="eqref" reference="eq:productrelation"}, is problematic for embeddings in $\mathbb{R}^2$ adds to the subtleties of Theorems [Theorem 1](#thm:large){reference-type="ref" reference="thm:large"} and [Theorem 2](#thm:interval){reference-type="ref" reference="thm:interval"}. {#fig:graph3 width="0.5\\linewidth"} A crucial ingredient in the proofs of Theorems [Theorem 1](#thm:large){reference-type="ref" reference="thm:large"} and [Theorem 2](#thm:interval){reference-type="ref" reference="thm:interval"} is a bound for the multilinear singular integral forms associated with the hypercube graphs. We will include a self-contained treatment of those analytical objects to the extent needed for the geometric application in this paper, but let us also briefly comment on them in higher generality. The singular integral form that will appear in Section [4](#sec:error){reference-type="ref" reference="sec:error"} (see [\[eq:theform\]](#eq:theform){reference-type="eqref" reference="eq:theform"}) is a particular case of a *singular Brascamp--Lieb form*, namely $$\label{eq:BLgeneral} \Lambda(f_1,\ldots,f_m) = \mathop{\textup{p.v.}}\int_{\mathbb{R}^D} \Big( \prod_{j=1}^{m} f_j (\Pi_j\mathbf{x}) \Big) K(\Pi\mathbf{x}) \,\textup{d}\mathbf{x},$$ where $\Pi_1,\ldots,\Pi_m,\Pi$ are surjective linear maps from $\mathbb{R}^D$ to lower-dimensional Euclidean spaces, while $K$ is a smooth Calderón-Zygmund kernel. The term *singular Brascamp--Lieb inequality* was first used by Durcik and Thiele [@DT20] for any $\textup{L}^p$ estimate for the form [\[eq:BLgeneral\]](#eq:BLgeneral){reference-type="eqref" reference="eq:BLgeneral"}. These inequalities are significantly more difficult than their non-singular counterparts (with kernel $K$ omitted) and the literature is very far from their complete theory; see the survey paper [@DT21survey] by Durcik and Thiele. No estimates are known already for seemingly simple instances, such as the so-called triangular Hilbert transform [@KTZ15]. However, the particular "cubical" case, when there are $2^n$ maps $\Pi_j$ and they are projections of the form $$\mathbb{R}^{2n}\to\mathbb{R}^n,\quad (x_1^0, \ldots, x_n^0, x_1^1,\ldots, x_n^1) \mapsto (x_1^{r_1}, \ldots, x_n^{r_n})$$ for $(r_1,\ldots,r_n)\in\{0,1\}^n$, has seen much progress over the last ten years [@Kov12; @Kov11; @Dur15; @Dur17; @DT20] (even before they were called the singular Brascamp--Lieb forms). The current state-of-the-art paper is the one by Durcik, Slavı́ková, and Thiele [@DST22]. Particular instances of the singular cubical forms have already found applications in geometric measure theory [@DKR18; @DK21; @DK22; @DS22], probability [@KS15; @KS20], and ergodic theory [@Kov16; @DKST19]. The form [\[eq:theform\]](#eq:theform){reference-type="eqref" reference="eq:theform"} needed in Section [4](#sec:error){reference-type="ref" reference="sec:error"} is not of the cubical type, but in the case $d=1$ it can be reduced to the studied cubical form by composing the functions $f_j$ with certain skew-projections, while the cases $d\geqslant 2$ would be quite analogous. Thus, we could have essentially just invoked the result of [@DST22] in the second half of Section [4](#sec:error){reference-type="ref" reference="sec:error"}. However, in order to make the proofs of Theorems [Theorem 1](#thm:large){reference-type="ref" reference="thm:large"} and [Theorem 2](#thm:interval){reference-type="ref" reference="thm:interval"} more elementary, we prefer to give a short self-contained proof of the particular estimate that we need, namely, [\[eq:singspec\]](#eq:singspec){reference-type="eqref" reference="eq:singspec"} below. Proofs of Theorems [Theorem 1](#thm:large){reference-type="ref" reference="thm:large"} and [Theorem 2](#thm:interval){reference-type="ref" reference="thm:interval"} will respectively follow the approaches from [@K20:anisotrop Section 4] and [@DK22 Section 7]. However, some technicalities need to be done differently for a flexible configuration and in the plane. The most notable differences are the appearance of the Gowers' uniformity norms $\|\cdot\|_{\textup{U}^n}$ in the study of the so-called structured part of the counting form in Section [3](#sec:structured){reference-type="ref" reference="sec:structured"} (and not when studying the so-called uniform part in Section [5](#sec:uniform){reference-type="ref" reference="sec:uniform"}), and the appearance of the aforementioned hypercube graph singular integral forms in Section [4](#sec:error){reference-type="ref" reference="sec:error"}. Density theorems in geometric measure theory have gained a lot of attention in the literature over the last ten years. The reader can consult [@Kovac:survey] for a very brief survey of the very recent developments. # Notation, basic definitions, and counting forms {#sec:notation} We will write $A\lesssim_P B$ and $B\gtrsim_P A$ if the inequality $A\leqslant C_P B$ holds for some unimportant finite constant $C_P$ depending only on a set of parameters $P$. We also write $A \sim_P B$ if both $A\lesssim_P B$ and $B\gtrsim_P A$ hold. Sometimes it is understood that the constant $C_P$ also depends on a few paramaters that are not explicitly listed in the set $P$, and which will be fixed throughout the proof. In our case this will be the pattern dimension $n$. We will write $\mathbbm{1}_{S}$ for the indicator function (i.e., the characteristic function) of a set $S\subseteq\mathbb{R}^2$. The imaginary unit will be denoted by $\mathbbm{i}$. The Euclidean norm of a vector $v$ will be written as $|v|$, while the standard inner product of $v$ and $w$ will be denoted $v\cdot w$. We will write $\textup{B}_r(x)$ for the Euclidean ball of radius $r>0$ centered at $x\in\mathbb{R}^2$. The *upper Banach density* of a measurable set $A$ in the Euclidean plane $\mathbb{R}^2$ is defined to be $$\overline{\delta}(A) := \limsup_{R\to\infty} \sup_{x\in\mathbb{R}^2} \frac{|A\cap(x+[0,R]^2)|}{R^2},$$ where $|B|$ denotes the two-dimensional Lebesgue measure of $B$. It is clearly greater than or equal to the more common (*centered*) *upper density* of a measurable $A\subseteq\mathbb{R}^2$, given by $$\limsup_{R\to\infty} \frac{|A\cap[-R/2,R/2]^2|}{R^2}.$$ Note that $\overline{\delta}(A)$ is a more flexible quantity, in the sense that it also detects accumulated "mass" on arbitrary translates of large squares $[-R/2,R/2]^2$. Fix an integer $n\geqslant 1$. Let $\mathcal{B}(\mathbb{R}^D)$ denote the Borel sigma algebra on $\mathbb{R}^D$. We write $\sigma$ for the normalized spherical measure on $\mathbb{S}^{1}\subset\mathbb{R}^2$. Let $\mathbbm{g}$ be the *standard Gaussian* on $\mathbb{R}^2$, $\mathbbm{h}^{(i)}$ its partial derivatives, and $\mathbbm{k}$ its Laplacian, i.e., $$\mathbbm{g}(x) := e^{-\pi|x|^2},\quad \mathbbm{h}^{(i)} := \partial_i \mathbbm{g}, \ i=1,2,\quad \mathbbm{k} := \Delta \mathbbm{g}.$$ Using the usual normalization for the Fourier transform, $$\widehat{f}(\xi) := \int_{\mathbb{R}^2} f(x) e^{-2\pi \mathbbm{i} x\cdot\xi} \,\textup{d}x,$$ we easily get $$\label{eq:FtofGauss} \widehat{\mathbbm{g}}(\xi) = e^{-\pi|\xi|^2}, \quad \widehat{\mathbbm{h}^{(i)}}(\xi) = 2\pi \mathbbm{i}\xi_i e^{-\pi|\xi|^2}, \quad \widehat{\mathbbm{k}}(\xi) = -4\pi^2 |\xi|^2 e^{-\pi|\xi|^2}.$$ In general, we denote by $\tau_\lambda$ and $f_\lambda$ dilates of a measure $\tau$ on $\mathcal{B}(\mathbb{R}^D)$ and an integrable function $f$ on $\mathbb{R}^D$ by a factor $\lambda>0$, respectively defined as $$\tau_{\lambda}(E) := \tau\Big(\frac{1}{\lambda}E\Big), \quad f(x) := \frac{1}{\lambda^D} f\Big(\frac{1}{\lambda}x\Big).$$ Their convolution is, on the other hand, defined as a function $\tau\ast f$ on $\mathbb{R}^D$ given by $$(\tau\ast f)(x) := \int_{\mathbb{R}^D} f(x-y) \,\textup{d}\tau(y).$$ Using [\[eq:FtofGauss\]](#eq:FtofGauss){reference-type="eqref" reference="eq:FtofGauss"} and the basic identity for the Fourier transform of a convolution, we easily obtain $$\begin{aligned} \sum_{l=1}^2 \mathbbm{h}^{(l)}_{\alpha} \ast \mathbbm{h}^{(l)}_{\beta} &=\frac{\alpha \beta}{\alpha^2 + \beta ^2} \mathbbm{k}_{\sqrt{\alpha^2 + \beta^2}} , \label{conv par der Gauss} \\ \mathbbm{k}_{\alpha} \ast \mathbbm{g}_{\beta} &=\frac{\alpha^2}{\alpha^2 + \beta^2} \mathbbm{k}_{\sqrt{\alpha^2 + \beta^2}} \label{conv Gauss Laplace}\end{aligned}$$ for some $\alpha, \beta >0$. Dilates of $\mathbbm{g}$ and $\mathbbm{k}$ satisfy the (re-parameterized) *heat equation*, $$\label{eq:heateq} \frac{\partial}{\partial t} \mathbbm{g}_t(x) = \frac{1}{2\pi t} \mathbbm{k}_t(x)$$ for $(t,x)\in(0,\infty)\times\mathbb{R}^2$, which is easily seen by straightforward differentiation. Let $f\colon\mathbb{R}^2\to[0,1]$ be a compactly supported measurable function. It is convenient to denote $$\mathcal{F}_n(x;y_1,\ldots,y_n) := \prod_{(r_1,\ldots,r_n)\in\{0,1\}^n} f(x + r_1 y_1 + \cdots + r_n y_n)$$ for $x,y_1,\ldots,y_n\in\mathbb{R}^2$. For instance, in the case $n=2$ we have $$\mathcal{F}_2(x;y_1,y_2) = f(x) f(x+y_1) f(x+y_2) f(x+y_1+y_2).$$ Throughout this paper we will always use this notation in the special case when $f$ is equal to an indicator function of some set. These expressions are well-known from additive combinatorics, because their integrals are powers of the *Gowers uniformity norms* $\|\cdot\|_{\textup{U}^n}$, namely, $$\label{eq:Gowersdef} \int_{(\mathbb{R}^2)^{n+1}} \mathcal{F}_n(x;y_1,\ldots,y_n) \,\textup{d}x \,\textup{d}y_1 \cdots \textup{d}y_n = \|f\|_{\textup{U}^n(\mathbb{R}^2)}^{2^n}.$$ Note that $\mathcal{F}_n(x;y_1,\ldots,y_n)$ is symmetric in $y_1,\ldots,y_n$ and that it satisfies the recurrence relation: $$\label{eq:rekurzija} \mathcal{F}_n(x;y_1,\ldots,y_n) = \mathcal{F}_{n-1}(x;y_1,\ldots,y_{n-1}) \mathcal{F}_{n-1}(x+y_n;y_1,\ldots,y_{n-1}).$$ The well-known *Gowers--Cauchy--Schwarz inequality* [@Gow01; @HK05; @ET12] on the group $\mathbb{R}^2$ reads $$\begin{aligned} \bigg| \int_{(\mathbb{R}^2)^{n+1}} \prod_{(r_1,\ldots,r_n)\in\{0,1\}^n} \mathcal{C}^{r_1+\cdots+r_n} f_{r_1,\ldots,r_n}(x + r_1 y_1 + \cdots + r_n y_n) \,\textup{d}x \,\textup{d}y_1 \cdots \textup{d}y_n \bigg| & \\ \leqslant\prod_{(r_1,\ldots,r_n)\in\{0,1\}^n} \|f_{r_1,\ldots,r_n}\|_{\textup{U}^n(\mathbb{R}^2)} & .\end{aligned}$$ for bounded, compactly supported, measurable, complex functions $f_{r_1,\ldots,r_n}$, where $\mathcal{C}$ denotes the operator of complex conjugation (so that $\mathcal{C}^2$ is the identity). It is merely a consequence of many applications of the Cauchy--Schwarz inequality. Moreover, if $f$ is nonnegative, bounded, measurable, and supported on a cube $Q\subset\mathbb{R}^d$, then we have $$\label{eq:GCScorr} \|f\|_{\textup{U}^n(\mathbb{R}^2)} \gtrsim_{n} |Q|^{-1+(n+1)/2^n} \int_Q f .$$ To see this, we simply apply the Gowers--Cauchy--Schwarz inequality when $f_{0,\ldots,0}=f$ and all other $f_{r_1,\ldots,r_n}$ are equal to the characteristic function of the cube $\widetilde{Q}$ obtained by dilating $Q$ from its center by the factor of $n+1$. Namely, let $Q_0$ be the cube congruent to $Q$, but translated so that its center coincides with the origin. If $x\in Q$ and $y_1,\ldots,y_n\in Q_0$, then $x + r_1 y_1 + \cdots + r_n y_n\in\widetilde{Q}$ for any choice of $r_1,\ldots,r_n\in\{0,1\}$. Let us finally define an appropriate *counting form* by $$\mathcal{N}^{0}_{\lambda}(f) := \int_{\mathbb{R}^{2}} \int_{(\mathbb{R}^2)^n} \mathcal{F}_n(x;y_1,\ldots,y_n) \,\textup{d}\sigma_{\lambda}^{\otimes n}(y_1,\ldots,y_n) \,\textup{d}x$$ for every $\lambda>0$. Moreover, we will also need the smoothed out version of $\mathcal{N}_{\lambda}^{0}$, defined as $$\mathcal{N}^{\varepsilon}_{\lambda}(f) := \int_{(\mathbb{R}^2)^{n+1}} \mathcal{F}_n(x;y_1,\ldots,y_n) \prod_{k=1}^n (\sigma_{\lambda} \ast \mathbbm{g}_{\varepsilon \lambda})(y_k) \,\textup{d}y_1 \cdots \textup{d}y_n \,\textup{d}x$$ for $\lambda>0$ and $0<\varepsilon\leqslant 1$. Note that we have $$\label{eq:Nconvergence} \lim_{\varepsilon\to0+} \mathcal{N}_{\lambda}^{\varepsilon}(f) = \mathcal{N}_{\lambda}^{0}(f).$$ This can be verified by first observing that $$\mathcal{G}(y_1,\ldots,y_n) := \int_{\mathbb{R}^2} \mathcal{F}_n(x;y_1,\ldots,y_n) \,\textup{d}x$$ is a continuous function on $(\mathbb{R}^2)^n$, merely by the continuity of translation operators on $\textup{L}^1(\mathbb{R}^2)$. Then [\[eq:Nconvergence\]](#eq:Nconvergence){reference-type="eqref" reference="eq:Nconvergence"} follows simply by rewriting $$\begin{aligned} \mathcal{N}^{\varepsilon}_{\lambda}(f) & = \int_{(\mathbb{R}^2)^n} (\mathcal{G}\ast\mathbbm{g}^{\otimes n}_{\varepsilon\lambda})(y_1,\ldots,y_n)\,\textup{d}\sigma_{\lambda}^{\otimes n}(y_1,\ldots,y_n), \\ \mathcal{N}^{0}_{\lambda}(f) & = \int_{(\mathbb{R}^2)^n} \mathcal{G}(y_1,\ldots,y_n)\,\textup{d}\sigma_{\lambda}^{\otimes n}(y_1,\ldots,y_n) \end{aligned}$$ and using the pointwise limit $\lim_{\varepsilon\to0+}\mathcal{G}\ast\mathbbm{g}^{\otimes n}_{\varepsilon\lambda}=\mathcal{G}$ and the dominated convergence theorem with respect to the measure $\textup{d}\sigma_{\lambda}^{\otimes n}(y_1,\ldots,y_n)$. Using the scheme of approach inaugurated by Cook, Magyar, and Pramanik [@CMP15:roth], the main idea is to decompose $$\label{eq:decomposition} \mathcal{N}^{0}_{\lambda}(f) = \mathcal{N}^{1}_{\lambda}(f) + \big(\mathcal{N}^{\varepsilon}_{\lambda}(f) - \mathcal{N}^{1}_{\lambda}(f)\big) + \big(\mathcal{N}^{0}_{\lambda}(f) - \mathcal{N}^{\varepsilon}_{\lambda}(f)\big)$$ for an appropriately chosen $\varepsilon\in(0,1]$. The three summands on the right hand side of [\[eq:decomposition\]](#eq:decomposition){reference-type="eqref" reference="eq:decomposition"} are respectively called the *structured part*, the *error part*, and the *uniform part*; see the "philosophical" discussion in [@K20:anisotrop]. Proofs of Theorems [Theorem 1](#thm:large){reference-type="ref" reference="thm:large"} and [Theorem 2](#thm:interval){reference-type="ref" reference="thm:interval"} proceed by controlling the structured part from below and bounding the other two parts from above, modulo the fact that pigeonholing in the scales $\lambda$ is used in the discussion of the error part. **Remark 4**. It is not immediately clear that the positivity of the counting form $\mathcal{N}_{\lambda}^{0}(f)$ implies the existence of a nondegenerate hypercube configuration. In principle, some of the verticies may overlap. However, such degenerate cases are negligible. To see this, let $S,T\subseteq \{1, \ldots, n\}$, $S\neq T$, for some natural number $n$. Define $$H_{S,T} := \bigg\{(y_1, \ldots , y_n) \in (\mathbb{R}^2)^n \, : \, \sum_{i\in S}y_i = \sum_{i\in T}y_i \bigg\}.$$ Finally set $$H=\bigcup_{\substack{S,T\subseteq \{1, \ldots, n\}\\S\neq T}} H_{S,T}.$$ For arbitrary fixed $S,T$ as above consider the *degenerate counting form* $$\begin{aligned} \mathcal{N}^{0,S,T}_{\lambda}(f) :&= \int_{\mathbb{R}^{2}} \int_{H_{S,T}} \mathcal{F}_n(x;y_1,\ldots,y_n) \,\textup{d}\sigma_{\lambda}^{\otimes n}(y_1,\ldots,y_n) \,\textup{d}x.\end{aligned}$$ There exist $1\leqslant j\leqslant n$ and $s_1,s_2,\ldots,s_{j-1}\in\{-1,0,1\}$ (all depending on $S,T$) such that the hyperplane $H_{S,T}$ can be written as $$H_{S,T} = \{(y_1, \ldots , y_n) \in (\mathbb{R}^2)^n \, : \, y_j = s_1 y_1 + s_2 y_2 + \cdots + s_{j-1} y_{j-1}\},$$ so, consequently, $$\begin{aligned} \mathcal{N}^{0,S,T}_{\lambda}(f) = \int_{\mathbb{R}^{2}} \int_{(\mathbb{R}^2)^{j-1}} \int_{\{s_1 y_1 + s_2 y_2 + \cdots + s_{j-1} y_{j-1}\}} \int_{(\mathbb{R}^2)^{n-j}} \mathcal{F}_n(x;y_1,\ldots,y_n) & \\ \textup{d}\sigma_{\lambda}^{\otimes (n-j)}(y_{j+1},\ldots,y_n) \,\textup{d}\sigma_{\lambda}(y_j) \,\textup{d}\sigma_{\lambda}^{\otimes (j-1)}(y_1,\ldots,y_{j-1}) \,\textup{d}x & = 0,\end{aligned}$$ since the integral over the singleton $\{s_1 y_1 + s_2 y_2 + \cdots + s_{j-1} y_{j-1}\}$ with respect to the measure $\sigma_{\lambda}$ certainly equals $0$. Therefore, $$\begin{aligned} \int_{\mathbb{R}^{2}} \int_{(\mathbb{R}^2)^n\setminus H} \mathcal{F}_n(x;y_1,\ldots,y_n) \,\textup{d}\sigma_{\lambda}^{\otimes n}(y_1,\ldots,y_n) \,\textup{d}x & \\ \geqslant\mathcal{N}^{0}_{\lambda}(f) - \sum_{\substack{S,T\subseteq \{1, \ldots, n\}\\S\neq T}}\mathcal{N}^{0,S,T}_{\lambda}(f) = \mathcal{N}^{0}_{\lambda}(f) & .\end{aligned}$$ Once we show that $\mathcal{N}^{0}_{\lambda}(\mathbbm{1}_A)>0$, it will follow that there exist $x\in\mathbb{R}^2$ and $(y_1,\ldots,y_n)\in(\mathbb{R}^2)^n\setminus H$ such that $\mathcal{F}_n(x;y_1,\ldots,y_n)>0$, which means that all $2^n$ points [\[eq:allpoints\]](#eq:allpoints){reference-type="eqref" reference="eq:allpoints"} really belong to $A$ and are mutually distinct. We conclude that $\mathcal{N}^{0}_{\lambda}(\mathbbm{1}_A)>0$ is sufficient to guarantee that $A$ contains an isometric copy of $\lambda\cdot\Gamma_n$. Now we are ready to begin the proofs of Theorems [Theorem 1](#thm:large){reference-type="ref" reference="thm:large"} and [Theorem 2](#thm:interval){reference-type="ref" reference="thm:interval"}, which occupy the rest of the paper. As we have already mentioned, the strategy is to control separately each of the three terms from the decomposition [\[eq:decomposition\]](#eq:decomposition){reference-type="eqref" reference="eq:decomposition"}. # The structured part {#sec:structured} **Proposition 5**. *For numbers $R\geqslant\lambda>0$ and a measurable set $B\subseteq[0,R]^2$ we have $$\label{eq:structured} \mathcal{N}^{1}_{\lambda}(\mathbbm{1}_B) \geqslant c_{\textup{str}} \Big(\frac{|B|}{R^2}\Big)^{2^n} R^{2},$$ where $c_{\textup{str}}\in(0,\infty)$ is a constant that depends only on $n$.* *Proof.* Recall that $$\mathcal{N}^{1}_{\lambda}(\mathbbm{1}_B) = \int_{(\mathbb{R}^2)^{n+1}} \mathcal{F}_n(x;y_1,\ldots,y_n) \prod_{k=1}^n (\sigma_{\lambda} \ast \mathbbm{g}_{\lambda})(y_k) \,\textup{d}y_1 \cdots \textup{d}y_n \,\textup{d}x .$$ On the right hand side we use an easy pointwise estimate, $$\sigma\ast\mathbbm{g} \gtrsim \mathbbm{1}_{\textup{B}_2(0)} \geqslant\mathbbm{1}_{[-1,1]^2};$$ see [@K20:anisotrop Section 3.1] for a slightly more general formulation. That way we obtain $$\mathcal{N}^{1}_{\lambda}(\mathbbm{1}_B) \gtrsim_n \lambda^{-2n} \int_{(\mathbb{R}^2)^{n+1}} \mathcal{F}_n(x;y_1,\ldots,y_n) \prod_{k=1}^n \mathbbm{1}_{[-\lambda,\lambda]^2}(y_k) \,\textup{d}y_1 \cdots \textup{d}y_n \,\textup{d}x.$$ We partition $[0,R]^2$ into squares of side length $\lambda$. More precisely, we consider the set $$\mathcal{Q}:=\bigg\{ [(k-1)\lambda,k\lambda) \times [(l-1)\lambda,l\lambda) \, : \, k,l\in\Big\{1,2, \dots, \Big\lceil\frac{R}{\lambda}\Big\rceil\Big\} \bigg\},$$ which covers $[0,R]^2$ completely. Notice that $|\mathcal{Q}|=\lceil R/\lambda\rceil^2\leqslant(2R/\lambda)^2$ and that each square $Q \in \mathcal{Q}$ has area $\lambda^2$. We impose further restrictions on the above integral by requiring that all the $2^n$ vertices of the degenerate paralelotope [\[eq:allpoints\]](#eq:allpoints){reference-type="eqref" reference="eq:allpoints"} lie within the same element of $\mathcal{Q}$, that is, due to the fact that the elements of $\mathcal{Q}$ are mutually disjoint, we can write $$\begin{aligned} \mathcal{N}^{1}_{\lambda}(\mathbbm{1}_B) \gtrsim_n \lambda^{-2n} \sum_{Q\in\mathcal{Q}}\int_{(\mathbb{R}^2)^{n+1}} \prod_{(r_1,\dots,r_n)\in\{0,1\}^n}\mathbbm{1}_{B\cap Q}(x+r_1y_1+\cdots+r_ny_n) & \\ \prod_{k=1}^n \mathbbm{1}_{[-\lambda,\lambda]^2}(y_k) \,\textup{d}y_1 \cdots \textup{d}y_n \,\textup{d}x & .\end{aligned}$$ Also note that if $x+r_1y_1+\cdots+r_ny_n$ all lie within the same square of side-length $\lambda$, it trivially follows that all $y_k$ belong to $[-\lambda,\lambda]^2$. Therefore, the last display becomes simply $$\begin{aligned} \mathcal{N}^{1}_{\lambda}(\mathbbm{1}_B) \gtrsim_n \lambda^{-2n} \sum_{Q\in\mathcal{Q}}\int_{(\mathbb{R}^2)^{n+1}} \prod_{(r_1,\dots,r_n)\in\{0,1\}^n} & \mathbbm{1}_{B\cap Q}(x+r_1y_1+\cdots+r_ny_n) \,\textup{d}y_1 \cdots \textup{d}y_n \,\textup{d}x,\end{aligned}$$ i.e., by the definition of the Gowers norms [\[eq:Gowersdef\]](#eq:Gowersdef){reference-type="eqref" reference="eq:Gowersdef"}, $$\mathcal{N}^{1}_{\lambda}(\mathbbm{1}_B) \gtrsim_n \lambda^{-2n} \sum_{Q\in\mathcal{Q}} \|\mathbbm{1}_{B\cap Q}\|_{\textup{U}^n(\mathbb{R}^d)}^{2^n}.$$ Using [\[eq:GCScorr\]](#eq:GCScorr){reference-type="eqref" reference="eq:GCScorr"} with $f=\mathbbm{1}_{B\cap Q}$ followed by Jensen's inequality, we obtain $$\begin{aligned} \mathcal{N}^{1}_{\lambda}(\mathbbm{1}_B) & \gtrsim_n \lambda^{-2n} \sum_{Q\in\mathcal{Q}} |Q|^{-2^n+n+1} \Big(\int_{Q} \mathbbm{1}_{B\cap Q}\Big)^{2^n} = \lambda^{-2^{n+1}+2} \sum_{Q\in\mathcal{Q}} |B\cap Q|^{2^n} \\ & \geqslant\lambda^{-2^{n+1}+2} |\mathcal{Q}|^{-2^n+1} \Big(\sum_{Q\in\mathcal{Q}} |B\cap Q| \Big)^{2^n} = \lambda^{-2^{n+1}+2} |\mathcal{Q}|^{-2^n+1} |B|^{2^n} \\ & \gtrsim_n \lambda^{-2^{n+1}+2} \Big(\frac{R}{\lambda}\Big)^{-2^{n+1}+2} |B|^{2^n} = \Big(\frac{|B|}{R^2}\Big)^{2^n} R^{2}.\end{aligned}$$ This completes the proof of [\[eq:structured\]](#eq:structured){reference-type="eqref" reference="eq:structured"}. ◻ # The error part {#sec:error} **Proposition 6**. *For every positive integer $J$, every "scales" $0<\lambda_1<\cdots<\lambda_J$ satisfying $\lambda_{j+1}\geqslant 2\lambda_j$ for $j=1,\ldots,J-1$, every number $R\geqslant 2\lambda_J$, every measurable set $B\subseteq[0,R]^{2}$, and every $\varepsilon\in(0,1]$ we have the estimate: $$\label{eq:error} \sum_{j=1}^{J} \big|\mathcal{N}^{\varepsilon}_{\lambda_j}(\mathbbm{1}_B)-\mathcal{N}^{1}_{\lambda_j}(\mathbbm{1}_B)\big| \leqslant C_{\textup{err}}\varepsilon^{-3n}\log\Big(\frac{1}{\varepsilon}\Big) R^{2},$$ where $C_{\textup{err}}\in(0,\infty)$ is some constant that depends only on $n$.* *Proof.* Write $f=\mathbbm{1}_{B}$. Differentiating the product and using the heat equation [\[eq:heateq\]](#eq:heateq){reference-type="eqref" reference="eq:heateq"} we get $$\label{eq:productrule} \frac{\partial}{\partial t} \prod_{k=1}^n(\sigma_{\lambda} \ast \mathbbm{g}_{t\lambda})(y_k) = \frac{1}{2\pi t} \sum_{m=1}^{n} (\sigma_{\lambda} \ast \mathbbm{k}_{t\lambda})(y_m) \Big( \prod_{\substack{1\leqslant k\leqslant n\\ k\neq m}} (\sigma_{\lambda} \ast \mathbbm{g}_{t\lambda})(y_k) \Big).$$ Therefore, for $0<\alpha<\beta\leqslant 1$ we can write $$\mathcal{N}^{\alpha}_{\lambda}(f)-\mathcal{N}^{\beta}_{\lambda}(f) = \sum_{m=1}^{n} \mathcal{L}_{\lambda}^{\alpha,\beta,m}(f),$$ where $$\begin{aligned} \mathcal{L}_{\lambda}^{\alpha,\beta,m}(f) := -\frac{1}{2\pi} \int_{\alpha}^{\beta} \int_{(\mathbb{R}^2)^{n+1}} &\mathcal{F}_n(x;y_1,\ldots,y_n) \\ & (\sigma_{\lambda} \ast \mathbbm{k}_{t \lambda})(y_m) \prod_{\substack{1\leqslant k\leqslant n\\ k\neq m}}(\sigma_{\lambda} \ast \mathbbm{g}_{t\lambda})(y_k) \,\textup{d}y_1 \cdots \textup{d}y_n \,\textup{d}x \,\frac{\textup{d}t}{t}.\end{aligned}$$ That way we obtain the bound $$\label{error to bound} \sum_{j=1}^{J} \big|\mathcal{N}^{\varepsilon}_{\lambda_j}(\mathbbm{1}_B)-\mathcal{N}^{1}_{\lambda_j}(\mathbbm{1}_B)\big| \lesssim \sum_{m=1}^{n} \sum_{j=1}^{J} \big|\mathcal{L}_{\lambda_j}^{\varepsilon,1,m}(\mathbbm{1}_B) \big| .$$ Due to the symmetry of the variables $y_1,\dots, y_n$ in the previous expressions, it is sufficient to work with the last summand (i.e., the one for $m=n$) on the right hand side of [\[error to bound\]](#error to bound){reference-type="eqref" reference="error to bound"}. Let $\theta= 10^{-1}e^{-1}$ and observe that, for any $\lambda, t >0$, $$\int_{\theta t \lambda}^{e \theta t \lambda} \frac{\textup{d}s}{s}=1 .$$ For $j \in \lbrace 1,2, \ldots J\rbrace$, $t \in [\varepsilon,1]$, and $s \in [\theta t \lambda_j, e \theta t \lambda_j]$, set $$r=r(j,t,s):=\sqrt{(t \lambda_j)^2-2s^2}.$$ A simple calculation shows that $r\sim s \sim t\lambda_j$. Using the Gaussian convolution identities [\[conv par der Gauss\]](#conv par der Gauss){reference-type="eqref" reference="conv par der Gauss"} and [\[conv Gauss Laplace\]](#conv Gauss Laplace){reference-type="eqref" reference="conv Gauss Laplace"}, we can rewrite $$\label{eq:erraux1} \mathbbm {k}_{t\lambda_j}=\frac{(t\lambda_j)^2}{s^2}\sum_{l=1}^2 \mathbbm{h}^{(l)}_{s} \ast \mathbbm{h}^{(l)}_{s} \ast \mathbbm{g}_{r}.$$ The following Gaussian inequalities appear as [@K20:anisotrop (4.5)] and [@K20:anisotrop (4.6)], respectively: $$\begin{aligned} (\sigma_{\lambda_j}\ast\mathbbm{g}_{t\lambda_j})(x) & \lesssim \varepsilon^{-3} \int_1^{\infty} \mathbbm{g}_{s\gamma}(x) \,\frac{\textup{d}\gamma}{\gamma^2}, \label{eq:erraux2} \\ (\sigma_{\lambda_j}\ast\mathbbm{g}_{r})(x) & \lesssim \varepsilon^{-3} \int_1^{\infty} \mathbbm{g}_{s\gamma}(x) \,\frac{\textup{d}\gamma}{\gamma^2}. \label{eq:erraux3}\end{aligned}$$ Using the identity [\[eq:erraux1\]](#eq:erraux1){reference-type="eqref" reference="eq:erraux1"} and expanding out the convolution, using the inequalities [\[eq:erraux2\]](#eq:erraux2){reference-type="eqref" reference="eq:erraux2"} and [\[eq:erraux3\]](#eq:erraux3){reference-type="eqref" reference="eq:erraux3"}, recalling the recursive identity [\[eq:rekurzija\]](#eq:rekurzija){reference-type="eqref" reference="eq:rekurzija"} and the fact that $\mathbbm{h}^{(l)}_s$ is odd, and introducing the change of variables $u=y_n+x$, we obtain that $$\sum_{j=1}^{J} \big|\mathcal{L}_{\lambda_j}^{\varepsilon,1,n}(\mathbbm{1}_B) \big|$$ is at most $$\begin{aligned} \varepsilon^{-3n}\sum_{l=1}^2 \sum _{j=1}^J \int_{[1,\infty)^n}\int_{\varepsilon}^1\int_{\theta t \lambda_j}^{e \theta t \lambda_j}\int_{(\mathbb{R}^2)^{n+1}} &\Big| \int_{\mathbb{R}^2}\mathcal{F}_{n-1}(u;y_1,\ldots,y_{n-1}) \mathbbm{h}^{(l)}_{s}(u-z_1) \,\textup{d}u \Big|\\ &\Big| \int_{\mathbb{R}^2} \mathcal{F}_{n-1}(x;y_1,\ldots,y_{n-1}) \mathbbm{h}^{(l)}_{s}(x-z_2) \,\textup{d}x \Big| \\ & \mathbbm{g}_{s \gamma_n} (z_1 -z_2) \prod_{1\leqslant k\leqslant n-1} \mathbbm{g}_{s \gamma_k}(y_k) \\ & \textup{d}z_1\,\textup{d}z_2\,\textup{d}y_1 \cdots \textup{d}y_{n-1} \,\frac{\textup{d}s}{s} \,\frac{\textup{d}t}{t}\,\frac{\textup{d}\gamma_1}{\gamma_1^2} \cdots\,\frac{\textup{d}\gamma_n}{\gamma_n^2} .\end{aligned}$$ The expression above can be treated in the same way as a similar expression in [@K20:anisotrop Subsection 4.2], namely, we apply the Cauchy--Schwarz inequality to separate the product of functions $\mathcal{F}_{n-1}$ and then we notice that after a change of variables we can multiply the two square roots to obtain $$\begin{aligned} \varepsilon^{-3n} \sum_{l=1}^2 \sum _{j=1}^J \int_{[1,\infty)^n}\int_{\varepsilon}^1\int_{\theta t \lambda_j}^{e \theta t \lambda_j}\int_{(\mathbb{R}^2)^{n+1}} \Big(& \int_{\mathbb{R}^2} \mathcal{F}_{n-1}(u;y_1,\ldots,y_{n-1}) \mathbbm{h}^{(l)}_{s}(u-z_1)\,\textup{d}u \Big)^2 \\ &\mathbbm{g}_{s \gamma_n} (z_1 -z_2) \prod_{1\leqslant k\leqslant n-1} \mathbbm{g}_{s \gamma_k}(y_k) \\ &\textup{d}z_1\,\textup{d}z_2\,\textup{d}y_1 \cdots \textup{d}y_{n-1} \,\frac{\textup{d}s}{s} \,\frac{\textup{d}t}{t}\,\frac{\textup{d}\gamma_1}{\gamma_1^2} \cdots\,\frac{\textup{d}\gamma_n}{\gamma_n^2} .\end{aligned}$$ Using the translation invariance of the Lebesgue measure, we integrate out the Gaussian dilated by $s \gamma_n$ in the variable $z_2$ and then we integrate out $1/\gamma_n^{2}$. Next, we expand the square term inside the integral, and finally we undo the previously introduced change of variables and use the recursive identity [\[eq:rekurzija\]](#eq:rekurzija){reference-type="eqref" reference="eq:rekurzija"}. This will allow us to notice a convolution of the partial derivatives of the Gaussian, which we can rewrite as a dilated Gaussian Laplacian $\mathbbm{k}$ using identity [\[conv par der Gauss\]](#conv par der Gauss){reference-type="eqref" reference="conv par der Gauss"}. Finally, we are in a position to eliminate the scales $\lambda_j$ out of the consideration. We observe that due to the fact that the scales $\lambda_j$ grow for at least a factor of $2$, for a fixed $t \in [\varepsilon, 1]$, each $s\in(0,\infty )$ belongs to at most two of the intervals $[\theta t \lambda_j, e \theta t \lambda_j]$, therefore, we can bound the previous by $$\begin{aligned} -\varepsilon^{-3n} \int_{[1,\infty)^{n-1}}\int_{\varepsilon}^1\int_{0}^{\infty}\int_{(\mathbb{R}^2)^{n+1}} & \mathcal{F}_{n}(x;y_1,\ldots,y_{n}) \mathbbm{k}_{\sqrt{2}s}(y_n) \\ & \prod_{1\leqslant k\leqslant n-1}\mathbbm{g}_{s \gamma_k}(y_k) \,\textup{d}x\,\textup{d}y_1 \cdots \textup{d}y_{n} \,\frac{\textup{d}s}{s} \,\frac{\textup{d}t}{t}\,\frac{\textup{d}\gamma_1}{\gamma_1^2} \cdots\,\frac{\textup{d}\gamma_{n-1}}{\gamma_{n-1}^2} . \end{aligned}$$ Integrating with respect to $t$ further yields the factor of $\log(1/\varepsilon)$. To complete the proof, we need to bound an integral form similar to [@K20:anisotrop (2.11)], which is precisely the aforementioned case of the singular Brascamp--Lieb form [\[eq:BLgeneral\]](#eq:BLgeneral){reference-type="eqref" reference="eq:BLgeneral"} mentioned in the introduction. For $n \in \mathbb{N}$, $m \in \{1, \ldots, n \}$, scales $\gamma_1,\ldots, \gamma_n \in ( 0 ,\infty )$ and a compactly supported $f \in \textup{L}^{2^n}(\mathbb{R}^2)$, let $$\begin{aligned} \Theta_{\gamma_1, \ldots, \gamma_n}^{n,m}(f):=-\int_0^{\infty}\int_{(\mathbb{R}^2)^{n+1}} & \mathcal{F}_n(x;y_1,\ldots, y_n) \mathbbm{k}_{s\gamma_m}(y_m) \nonumber \\ & \Big(\prod_{\substack{1 \leqslant k \leqslant n \\ k \neq m }}\mathbbm{g}_{s\gamma_k}(y_k)\Big) \,\textup{d}x\,\textup{d}y_1\cdots\textup{d}y_n \, \frac{\textup{d}s}{s}. \label{eq:theform}\end{aligned}$$ Using the convolution identity [\[conv par der Gauss\]](#conv par der Gauss){reference-type="eqref" reference="conv par der Gauss"} it follows that $$\mathbbm{k}_{s\gamma_m}=2\sum_{l=1}^2\mathbbm{h}_{s\gamma_m/\sqrt{2}}^{(l)} \ast \mathbbm{h}_{s\gamma_m/\sqrt{2}}^{(l)} .$$ For fixed $y_1, \dots, y_n$ and $m \in \{1, \dots, n \}$, let $$F_m(u):=\mathcal{F}_{n-1}(u;y_1,\dots,y_{m-1},y_{m+1},\dots,y_n).$$ Using a recursive relation similar to [\[eq:rekurzija\]](#eq:rekurzija){reference-type="eqref" reference="eq:rekurzija"}, making the change of variables $u=x+y_n$ and using the above identity, we can write $$\begin{aligned} \Theta_{\gamma_1, \ldots ,\gamma_n}^{n,m}(f)=-2\sum_{l=1}^2\int_0^{\infty}\int_{(\mathbb{R}^2)^{n-1}} &\left\langle F_m \ast\mathbbm{h}_{s\gamma_m/\sqrt{2}}^{(l)} \ast \mathbbm{h}_{s\gamma_m/\sqrt{2}}^{(l)}, F_m\right\rangle_{\textup{L}^2(\mathbb{R}^2)}\\ &\Big(\prod_{\substack{1 \leqslant k \leqslant n \\ k \neq m }}\mathbbm{g}_{s\gamma_k}(y_k)\Big) \,\textup{d}y_1\cdots\,\textup{d}y_{m-1}\,\textup{d}y_{m+1}\cdots\textup{d}y_n \, \frac{\textup{d}s}{s}.\end{aligned}$$ A simple calculation shows that $$\left\Vert F_m \ast\mathbbm{h}_{s\gamma_m/\sqrt{2}}^{(l)}\right\Vert_{\textup{L}^2(\mathbb{R}^2)}^2=- \left\langle F_m \ast\mathbbm{h}_{s\gamma_m/\sqrt{2}}^{(l)} \ast \mathbbm{h}_{s\gamma_m/\sqrt{2}}^{(l)}, F_m\right\rangle_{\textup{L}^2(\mathbb{R}^2)} ,$$ from which follows that $\Theta_{\gamma_1, \ldots, \gamma_n}^{n,m}(f)$ is well defined and nonnegative. Using the product formula [\[eq:productrule\]](#eq:productrule){reference-type="eqref" reference="eq:productrule"} again and the fundamental theorem of calculus, we obtain $$\begin{aligned} \sum_{m=1}^n\Theta_{\gamma_1, \ldots, \gamma_n}^{n,m}(f) &= 2\pi\lim_{\alpha \to 0^+}\int_{(\mathbb{R}^2)^{n+1}}\mathcal{F}_n(x;y_1,\ldots, y_n)\int_0^{\infty} \Big(\prod_{k=1}^n\mathbbm{g}_{\alpha\gamma_k}(y_k)\Big) \,\textup{d}s\,\textup{d}x\,\textup{d}y_1\cdots\textup{d}y_n \\ & \quad -2\pi\lim_{\beta \to \infty}\int_{(\mathbb{R}^2)^{n+1}}\mathcal{F}_n(x;y_1,\ldots, y_n)\int_0^{\infty} \Big(\prod_{k=1}^n\mathbbm{g}_{\beta\gamma_k}(y_k)\Big) \,\textup{d}s\,\textup{d}x\,\textup{d}y_1\cdots\textup{d}y_n .\end{aligned}$$ Interpreting the above limits as distributions, we find that the second limit is equal to $0$, while the first is equal to $$\int_{\mathbb{R}^2}f(x)^{2^n}\,\textup{d}x=\Vert f \Vert_{\textup{L}^{2^n}(\mathbb{R}^2)}^{2^n}.$$ In particular, $$\label{eq:singspec} 0\leqslant\Theta_{\gamma_1, \ldots, \gamma_n}^{n,m}(f) \leqslant 2\pi \Vert f \Vert_{\textup{L}^{2^n}(\mathbb{R}^2)}^{2^n}$$ for any choice of the parameters. Using this and summing in $m$ we finally get that [\[error to bound\]](#error to bound){reference-type="eqref" reference="error to bound"} is bounded by $$\begin{aligned} &\varepsilon^{-3n}\log\Big(\frac{1}{\varepsilon}\Big) \int_{[1,\infty)^{n-1}}\sum_{m=1}^n\Theta_{\gamma_1,\ldots,\gamma_{n-1},\sqrt{2}}^{n,m}(f) \frac{\textup{d}\gamma_1}{\gamma_1^2} \cdots\,\frac{\textup{d}\gamma_{n-1}}{\gamma_{n-1}^2} \\ &\lesssim\varepsilon^{-3n}\log\Big(\frac{1}{\varepsilon}\Big) \Vert f \Vert_{\textup{L}^{2^n}(\mathbb{R}^2)}^{2^n} \leqslant\varepsilon^{-3n}\log\Big(\frac{1}{\varepsilon}\Big)R^{2}.\end{aligned}$$ This finalizes the proof of [\[eq:error\]](#eq:error){reference-type="eqref" reference="eq:error"}. ◻ # The uniform part {#sec:uniform} **Proposition 7**. *For real numbers $\lambda>0$, $\varepsilon\in (0,1]$, and a measurable set $B\subseteq[0,R]^2$ we have $$\label{eq:uniform} \big|\mathcal{N}^{0}_{\lambda}(\mathbbm{1}_B)-\mathcal{N}^{\varepsilon}_{\lambda}(\mathbbm{1}_B)\big| \leqslant C_{\textup{uni}} \varepsilon^{1/2} R^2,$$ where $C_{\textup{uni}}\in(0,\infty)$ is some constant that depends only on $n$.* *Proof.* Again write $f=\mathbbm{1}_B$. Motivated by the limit [\[eq:Nconvergence\]](#eq:Nconvergence){reference-type="eqref" reference="eq:Nconvergence"}, instead of bounding $\big | \mathcal{N}^{0}_{\lambda}(\mathbbm{1}_B)-\mathcal{N}^{\varepsilon}_{\lambda}(\mathbbm{1}_B)\big|$, we rather focus on controlling $\big |\mathcal{N}^{\theta}_{\lambda}(\mathbbm{1}_B)-\mathcal{N}^{\varepsilon}_{\lambda}(\mathbbm{1}_B)\big|$ for every $0<\theta<\varepsilon\leqslant 1$. This allows us to reuse the approach from the previous section, which has already proven itself convenient. We begin by bounding $\big| \mathcal{L}_{\lambda}^{\theta,\varepsilon,m}(f)\big|$ for a fixed $m \in \{1,\ldots,n\}$. Recall $$\begin{aligned} \mathcal{L}_{\lambda}^{\theta,\varepsilon,m}(f)=\frac{-1}{2\pi}\int_\theta^{\varepsilon}\int_{(\mathbb{R}^2)^{n+1}} &\mathcal{F}_{n}(x;y_1,\ldots,y_n)\\ & (\sigma_{\lambda}\ast \mathbbm{k}_{\lambda t})(y_m)\prod_{\substack{1\leqslant k \leqslant n \\ k\neq m}}(\sigma_{\lambda}\ast \mathbbm{g}_{t\lambda})(y_k) \,\textup{d}x\,\textup{d}y_1\cdots\textup{d}y_n \frac{\textup{d}t}{t} .\end{aligned}$$ Let us focus our attention only on the integrals in the variables $x$ and $y_m$. Expanding the previous display using [\[eq:rekurzija\]](#eq:rekurzija){reference-type="eqref" reference="eq:rekurzija"}, $$\begin{aligned} \int_{(\mathbb{R}^2)^2} \mathcal{F}_{n-1}(x+y_m;y_1,\ldots, y_{m-1},y_{m+1}, \ldots,y_{n}) \mathcal{F}_{n-1}(x;y_1,\ldots, y_{m-1},y_{m+1}, \ldots,y_{n}) & \\ \sigma_{\lambda} \ast \mathbbm{k}_{t \lambda}((x+y_m)-x) \,\textup{d}y_m\,\textup{d}x & .\end{aligned}$$ Using (by now standard) change of variables $u=x+y_m$ we notice a triple convolution $$\int_{\mathbb{R}^2} \big (F_m \ast \sigma_{\lambda} \ast \mathbbm{k}_{t \lambda}\big)(u) \overline{F_m(u)} \,\textup{d}u.$$ Applying Plancherel's theorem and the basic identity for the Fourier transform of a convolution, we can write the previous integral as $$\int_{\mathbb{R}^2} \widehat{F_m}(\xi)\widehat{\sigma_{\lambda}}(\xi)\widehat{\mathbbm{k}_{t \lambda}}(\xi) \overline{\widehat{F_m}(\xi)} \,\textup{d}\xi.$$ Using the decay of the Fourier transform of the spherical measure and the basic identity for the Fourier transform of the Laplacian, we get the following bound $$\begin{aligned} \big | \widehat{\sigma}(\lambda \xi) \widehat{\mathbbm{k}}(t \lambda \xi)\big | &\leqslant | \lambda |^{-1/2} | \xi |^{-1/2} t^2 \lambda^2 |\xi|^2 e^{-2\pi i |t\xi|^2} \lesssim t^{1/2} (t|\xi|)^{3/2}e^{-2\pi i |t\xi|^2}\\ &\leqslant t^{1/2} \sup_{u \in [0,\infty)}u^{3/2}e^{-2\pi i u^2} \lesssim t^{1/2}.\end{aligned}$$ Combining these results and once more using Plancherel's identity, we obtain the bound $$\begin{aligned} &\big| \mathcal{L}_{\lambda}^{\theta,\varepsilon,m}(f) \big| \lesssim \int_{\theta}^{\varepsilon}\int_{(\mathbb{R}^2)^{n}} t^{1/2} \Big |\widehat{ F_m}(\xi)\Big |^2 \prod_{\substack{1\leqslant k \leqslant n \\ k\neq m}}\sigma_{\lambda}\ast \mathbbm{g}_{t\lambda}(y_k) \,\textup{d}\xi\,\textup{d}y_1\cdots\textup{d}y_{m-1}\,\textup{d}y_{m+1}\cdots\textup{d}y_n \frac{\textup{d}t}{t} \\ &=\int_{\theta}^{\varepsilon}\int_{(\mathbb{R}^2)^{n-1}} t^{1/2} \left\Vert \widehat{F_m}(\xi)\right\Vert_{\textup{L}^2(\mathbb{R}^2)}^2 \prod_{\substack{1\leqslant k \leqslant n \\ k\neq m}}\sigma_{\lambda}\ast \mathbbm{g}_{t\lambda}(y_k) \,\textup{d}y_1\cdots\textup{d}y_{m-1}\,\textup{d}y_{m+1}\cdots\textup{d}y_n \frac{\textup{d}t}{t} \\ &=\int_{\theta}^{\varepsilon}\int_{(\mathbb{R}^2)^{n-1}} t^{1/2} \left\Vert F_m(\xi)\right\Vert_{\textup{L}^2(\mathbb{R}^2)}^2 \prod_{\substack{1\leqslant k \leqslant n \\ k\neq m}}\sigma_{\lambda}\ast \mathbbm{g}_{t\lambda}(y_k) \,\textup{d}y_1\cdots\textup{d}y_{m-1}\,\textup{d}y_{m+1}\cdots\textup{d}y_n \frac{\textup{d}t}{t} .\end{aligned}$$ Recall that $f=\mathbbm{1}_{B}$ and $B\subseteq [0,R]^2$, use the trivial bound on $\left\Vert F_m(\xi)\right\Vert_{\textup{L}^2(\mathbb{R}^2)}^2$, and integrate in all remaining variables except $t$ to obtain $$\begin{aligned} \big| \mathcal{L}_{\lambda}^{\theta,\varepsilon,m}(f) \big| &\lesssim \int_{\theta}^{\varepsilon}\int_{(\mathbb{R}^2)^{n-1}} t^{1/2} R^2 \prod_{\substack{1\leqslant k \leqslant n \\ k\neq m}}(\sigma_{\lambda}\ast \mathbbm{g}_{t\lambda})(y_k) \,\textup{d}y_1\cdots\textup{d}y_{m-1}\,\textup{d}y_{m+1}\cdots\textup{d}y_n \frac{\textup{d}t}{t}\\ &\lesssim \int_{\theta}^{\varepsilon} t^{-1/2} R^2 \,\textup{d}t \lesssim R^2(\varepsilon^{1/2}-\theta^{1/2}).\end{aligned}$$ Hence, it follows that $$\begin{aligned} \big|\mathcal{N}^{\theta}_{\lambda}(\mathbbm{1}_B)-\mathcal{N}^{\varepsilon}_{\lambda}(\mathbbm{1}_B)\big| \leqslant \sum_{m=1}^n\big| \mathcal{L}_{\lambda}^{\theta,\varepsilon,m}(f) \big| &\lesssim R^2(\varepsilon^{1/2}-\theta^{1/2})\,\end{aligned}$$ Finally, letting $\theta$ go to $0$ from the right and using [\[eq:Nconvergence\]](#eq:Nconvergence){reference-type="eqref" reference="eq:Nconvergence"}, we obtain [\[eq:uniform\]](#eq:uniform){reference-type="eqref" reference="eq:uniform"}, as desired. ◻ # Combination of the estimates To prove Theorem [Theorem 1](#thm:large){reference-type="ref" reference="thm:large"}, we argue by contradiction, i.e., assume the statement does not hold. Then, there exists a strictly increasing sequence of scales $(\lambda_j)_{j \in \mathbb{N}}$ such that $A$ does not contain an isometric copy of $\lambda_j \cdot \Gamma_n$, for all $j\in\mathbb{N}$. By passing to a subsequence, we can assume that $\lambda_{j+1}>2\lambda_{j}$ for all $j \in \mathbb{N}$. Choose $\varepsilon\in (0 ,1 ]$ small enough so that $$\label{eq:combaux1} C_{\textup{uni}}\varepsilon^{1/2} <\frac{c_{\textup{str}}}{3} \Big(\frac{\overline{\delta}(A)}{2}\Big)^{2^n}.$$ Next choose $J \in \mathbb{N}$ large enough so that $$\label{eq:combaux2} J^{-1}C_{\textup{err}}\varepsilon^{-3n}\log\Big(\frac{1}{\varepsilon}\Big) < \frac{c_{\textup{str}}}{3} \Big(\frac{\overline{\delta}(A)}{2}\Big)^{2^n}.$$ Finally, take $R\geqslant 1$ such that $$\sup_{x\in\mathbb{R}^2} \frac{|A\cap(x+[0,R]^2)|}{R^2}>\frac{\overline{\delta}(A)}{2}$$ and also make sure that it is large enough, i.e., $R>2\lambda_J$. It follows that there exists some $x\in \mathbb{R}^2$ such that the set $$B:=(A-x)\cap[0,R]^2$$ satisfies $$\label{eq:combaux3} |B| >\frac{\overline{\delta}(A)}{2} R^2.$$ We claim that there has to exists an index $j\in\{1,\ldots,J\}$ such that $$\label{eq:pigeon_error} \big|\mathcal{N}^{\varepsilon}_{\lambda_j}(\mathbbm{1}_B)-\mathcal{N}^{1}_{\lambda_j}(\mathbbm{1}_B)\big| \leqslant J^{-1}C_{\textup{err}}\varepsilon^{-3n}\log\Big(\frac{1}{\varepsilon}\Big) R^{2}.$$ If that was not the case, then simply by summing over all $j$, we would obtain a contradiction with [\[eq:error\]](#eq:error){reference-type="eqref" reference="eq:error"}. Combining the bounds [\[eq:structured\]](#eq:structured){reference-type="eqref" reference="eq:structured"}, [\[eq:uniform\]](#eq:uniform){reference-type="eqref" reference="eq:uniform"}, [\[eq:combaux1\]](#eq:combaux1){reference-type="eqref" reference="eq:combaux1"}, [\[eq:combaux2\]](#eq:combaux2){reference-type="eqref" reference="eq:combaux2"}, [\[eq:combaux3\]](#eq:combaux3){reference-type="eqref" reference="eq:combaux3"}, and [\[eq:pigeon_error\]](#eq:pigeon_error){reference-type="eqref" reference="eq:pigeon_error"}, and applying the decomposition [\[eq:decomposition\]](#eq:decomposition){reference-type="eqref" reference="eq:decomposition"} with $\lambda=\lambda_j$, we obtain $$\mathcal{N}_{\lambda}^0(\mathbbm{1}_B)>\frac{c_{\textup{str}}}{3}\Big(\frac{\overline{\delta}(A)}{2}\Big)^{2^n}R^2>0 .$$ Therefore, by Remark [Remark 4](#rem:nondegenerate){reference-type="ref" reference="rem:nondegenerate"}, the set $B$ contains an isometric copy of $\lambda_j \cdot \Gamma_n$. Since $B$ is obtained simply as a subset of a translate of $A$, it follows that $A$ also contains an isometric copy of $\lambda_j \cdot \Gamma_n$, but this contradicts our choice of the sequence $(\lambda_j)_{j\in\mathbb{N}}$. To prove Theorem [Theorem 2](#thm:interval){reference-type="ref" reference="thm:interval"}, consider the intervals $I_j = [2^{-2j},2^{-2j+1})$, where $j\in \mathbb{N}$, and suppose that for each of these intervals we choose some $\lambda_j\in I_j$ such that $A$ does not contain an isometric copy of $\lambda_j\cdot \Gamma_n$. Notice that $\lambda_{j+1}\leqslant\lambda_j/2$ for all $j\in \mathbb{N}$. We choose $\varepsilon >0$ such that $$C_{\textup{uni}}\varepsilon^{\frac{1}{2}}<\frac{c_{\textup{str}}}{3}\delta^{2^n}$$ and choose $J$ large enough so that $$J^{-1}C_{\textup{err}}\varepsilon^{-3n}\log\Big(\frac{1}{\varepsilon}\Big) < \frac{c_{\textup{str}}}{3}\delta^{2^n} .$$ Clearly, such $J$ can be chosen somewhat sparingly, to satisfy $$J \leqslant C \delta^{-(3n+1)2^{n+1}}$$ for a constant $C$ depending on $c_{\textup{str}},C_{\textup{err}},C_{\textup{uni}}$ and thus actually only depending on $n$. Because at this point we are considering only finitely many scales $\lambda_j$, $1\leqslant j\leqslant J$, simply by relabeling them, we can view them as an increasing finite sequence which allows us to reuse the pigeonhole argument from the proof of Theorem [Theorem 1](#thm:large){reference-type="ref" reference="thm:large"}, that is, we conclude that for some $j\in \{1,\ldots, J\}$ we have a bound which formally looks identical to [\[eq:pigeon_error\]](#eq:pigeon_error){reference-type="eqref" reference="eq:pigeon_error"}, only $B$ is replaced with $A$. From the decomposition [\[eq:decomposition\]](#eq:decomposition){reference-type="eqref" reference="eq:decomposition"} we conclude $\mathcal{N}_{\lambda_j}^0(\mathbbm{1}_A)>0$ for some index $j$. By Remark [Remark 4](#rem:nondegenerate){reference-type="ref" reference="rem:nondegenerate"}, there exists an isometric copy of $\lambda_j\cdot\Gamma_n$ in the set $A$, which contradicts our choices of the numbers $(\lambda_j)_{j\in\mathbb{N}}$. Thus, there exists an index $1\leqslant j\leqslant J$ such that $A$ contains an isometric copy of $\lambda\cdot\Gamma_n$ for every scale $\lambda\in I_j$; set $$I(A, n):=I_j$$ for one such value of $j$. An easy calculation shows that the length of this interval, $$|I(A, n)| = 2^{-2j} \geqslant 2^{-2J},$$ satisfies the requirement from the statement of Theorem [Theorem 2](#thm:interval){reference-type="ref" reference="thm:interval"}. # Acknowledgment {#acknowledgment .unnumbered} B. P. is supported by the *Croatian Science Foundation*.

arxiv_math

--- abstract: | We study the tracking controllability of the heat equation that, as we shall see, through transmutation, is intimately related to similar properties of the wave equation. More precisely, we seek for controls that, acting on a part of the boundary of the domain where the heat or wave process evolves, aims to assure that the trace on the complementary set tracks a given trajectory. We identify the dual observability problem, which consists on estimating the boundary sources, localized on a given subset of the boundary, out of boundary measurements on the complementary subset. Classical unique continuation and smoothing properties of the heat equation allow proving approximate tracking controllability properties and the smoothness of the class of trackable trajectories. We develop a new transmutation method which allows to transfer results on the sidewise controllability of the wave equation onto the tracking controllability of the heat one. This allows to achieve some estimates on the cost of control. The paper is complemented with the discussion of some possible variants of these results and a list of open problems. author: - "Jon Asier Bárcena-Petisco, Enrique Zuazua [^1] [^2] [^3]" bibliography: - Nodalheat.bib title: Tracking controllability of heat and wave equations --- tracking controllability, sidewise controllability, heat equation, wave equation, transmutation # Introduction The initial motivation of this problem is to analyze the tracking or sidewise controllability problem for the heat equation: $$\label{con:heat} \begin{cases} y_{t}-\Delta y=0 & \mbox{ in } (0,T)\times\Omega,\\ y=v1_\gamma & \mbox{ on } (0,T)\times\partial\Omega,\\ y(0)=y_0 & \mbox{ on }\Omega, \end{cases}$$ when $\Omega\subset\mathbb R^d$ is a given open bounded domain, $T>0$ a given time horizon, $\gamma\subset\partial\Omega$ a subset of the boundary, $v$ the control and $y_0$ the initial value. Hereafter, we denote by $1_\gamma$ the characteristic function of the set $\gamma$ of the boundary where the source term acts. The *(sidewise) tracking controllability problem* is formulated mathematically in the following way: given $\tilde \gamma\subset \partial\Omega$ (usually, but not necessarily, $\tilde \gamma \subset\partial\Omega\setminus\gamma$), and $w\in L^2((0,T)\times\tilde\gamma)$, to find a control $v\in L^2((0,T)\times\gamma)$ such that: $$\label{eq:parny} \partial_\nu y=w \ \ \ \ \mbox{ on } (0,T)\times\tilde\gamma,$$ where $\nu$ denotes the normal vector to $\partial\Omega$ pointing outwards. In other words, we seek to control the flux on $(0,T)\times\tilde \gamma$ by acting on $(0,T)\times \gamma$. When such a control $v$ exists, so that [\[eq:parny\]](#eq:parny){reference-type="eqref" reference="eq:parny"} is satisfied, the target $w$ is said to be *reachable*. Obviously, this kind of problems is not exclusive of the heat equation. One could formulate similar ones for other models like the wave equation. And, actually, one of the main results of this paper will show that the tracking controllability of the heat and wave equation can be related through a suitable subordination or transmutation principle. In the particular one-dimensional case the reachable space has been analyzed in the context of motion planning in the pioneering work [@laroche2000motion], by using power series representation methods. Other works on 1d parabolic equations in which boundary traces are controlled include [@dunbar2003motion], [@lynch2002flatness], and [@schorkhuber2013flatness]. In the multi-dimensional setting the only known results are only valid for cylinders (see [@martin2016null] and [@martin2016reachable]), where separation of variables can be employed, reducing the problem to the $d=1$ case. In this paper, first, in Section [2](#sec:gendom){reference-type="ref" reference="sec:gendom"}, by duality, we transform the tracking controllability problem on its dual observability one, which consists on identifying sources on a part of the boundary of the domain out of measurements on the control domain. This observability problem differs from classical ones on the fact that, normally, the initial data of the system are the objects to be identified. Duality, together with the Holmgren's Uniqueness Theorem, allows to prove easily the approximate tracking controllability property, i.e. the fact that [\[eq:parny\]](#eq:parny){reference-type="eqref" reference="eq:parny"} can be achieved for all target up to an arbitrarily small $\varepsilon$ error. Second, in Section [3](#sec:trasm){reference-type="ref" reference="sec:trasm"}, using a new transmutation formula, inspired on the classical Kannai transform [@kannai1977off], we show that the tracking controllability of the heat equation is subordinated to the analog property of the wave equation. The tracking controllability of the wave equation has been mainly analyzed for $d=1$, first in [@li2010exact] and [@li2016exact], with constructive methods; and then, in [@sarac2021sidewise], with a duality approach that inspires this paper. A third contribution of this paper, in Section [4](#sec:segcyl){reference-type="ref" reference="sec:segcyl"}, concerns the quantification of the cost of approximate controllability for the heat equation. This is done by carefully analyzing power series representations, a method that, as mentioned above, has already been used to tackle the tracking control of the $1d$ heat equation. The results of this paper can be extended to many situations: the control may act on Neumann boundary conditions and aim at the Dirichlet trace, the heat equation may have space-dependent coefficients, etc. Section [5](#sec:opprob){reference-type="ref" reference="sec:opprob"} is devoted to present some of these variants and some interesting and challenging open problems. # Framework for tracking controllability {#sec:gendom} In this section we study the framework for tracking controllability in an abstract setting and then apply it to study the controllability of the heat equation. We recall that the framework for the wave equation was already introduced in [@sarac2021sidewise]. ## An abstract setting Let us consider the Hilbert spaces $Y$, $U$ and $W$ endowed with the scalar products $\langle\cdot,\cdot\rangle_Y$, $\langle\cdot,\cdot\rangle_U$ and $\langle\cdot,\cdot\rangle_W$ respectively. Let $A:D(A)\to Y$, $B:U\to Y$ and $E:D(A)\to W$ be linear operators. We can consider the tracking controllability problem in the abstract setting: $$\label{con:abssys} \begin{cases} y_{t}=Ay+Bu,\\ y(0)=y_0,& \end{cases}$$ for the target: $$\label{eq:Ew} Ey(t)=w(t) \mbox{ on } (\tau,T),$$ for some $\tau\geq0$. As it is a classical in control problems, we consider the dual problem of [\[con:abssys\]](#con:abssys){reference-type="eqref" reference="con:abssys"}-[\[eq:Ew\]](#eq:Ew){reference-type="eqref" reference="eq:Ew"}, which reads as follows: $$\label{eq:adjabstract} \begin{cases} -p_t=A^*p+E^*f,\\ p(T)=0. \end{cases}$$ **Proposition 1** (Dual notion of approx. contr.). *Let $Y$, $U$ and $W$ Hilbert spaces and $A:D(A)\to Y$, $B:U\to Y$ and $E:D(A)\to W$ be linear operators. Then, for all $w\in L^2(0,T;W)$ and $\varepsilon>0$ there is a control $f\in L^2(0,T;U)$ such that the solution of [\[con:abssys\]](#con:abssys){reference-type="eqref" reference="con:abssys"} satisfies: $$\label{eq:approxheat} \|Ey-w\|_{L^2(0,T;W)}<\varepsilon,$$ if and only if for all $f\in L^2(0,T;U)\setminus \{0\}$ the solution $p_f$ of [\[eq:adjabstract\]](#eq:adjabstract){reference-type="eqref" reference="eq:adjabstract"} satisfy that: $$\label{eq:pfnotnull} B^*p_f\neq 0.$$ In fact, if [\[eq:pfnotnull\]](#eq:pfnotnull){reference-type="eqref" reference="eq:pfnotnull"} is satisfied for all $f\in L^2(0,T;U)\setminus \{0\}$ and if $y_0=0$, we obtain [\[eq:approxheat\]](#eq:approxheat){reference-type="eqref" reference="eq:approxheat"} applying the control $v=B^*p_f$, where $f$ is the minimizer of: $$\label{eq:defJf} \begin{split} J(f)=\frac{1}{2}\|B^*p_f\|^2_{L^2(0,T;U)} &- \int_0^T\langle f,w \rangle_W dt \\&+\varepsilon\|f\|_{L^2(0,T;W)}. \end{split}$$* Proposition [Proposition 1](#prop:absdualapprx){reference-type="ref" reference="prop:absdualapprx"} is based on the Hilbert Uniqueness Method, which is explained for instance in [@lions1988controlabilite], and more recently, in [@coron2007control Section 2.3]. *Proof.* Let us first suppose that [\[eq:pfnotnull\]](#eq:pfnotnull){reference-type="eqref" reference="eq:pfnotnull"} is satisfied for all $f\in L^2(0,T;U)\setminus\{0\}$. By linearity, it suffices to prove the approximate controllability for $y_0=0$. From [\[eq:pfnotnull\]](#eq:pfnotnull){reference-type="eqref" reference="eq:pfnotnull"}, $J$ is strictly convex, continuous and coercive, so $J$ has a unique minimizer $\tilde f\in L^2(0,T;U)$. Thus, we find for all $f\in L^2(0,T;U)$ and $\delta\neq0$ that: $$\label{eq:defpf} \begin{split} &\delta\int_{0}^T\langle B^*p_{\tilde f},B^* p_f\rangle_U dt -\delta \int_{0}^T\langle f,w\rangle_Wdt \\&+\varepsilon(\|\tilde f+\delta f\|_{L^2(0,T;W)}-\|\tilde f\|_{L^2(0,T;W)})+O_{\delta\to0}(\delta^2) \\&=J(\tilde f+\delta f)-J(\tilde f)\geq0. \end{split}$$ Moreover, if $y$ is the solution of [\[con:abssys\]](#con:abssys){reference-type="eqref" reference="con:abssys"} with $y_0=0$ and $v=B^*p_{\tilde f}$, then: $$\label{eq:dualcompint} \begin{split} 0&=\int_0^T\langle y_t-Ay-BB^*p_{\tilde f}, p_f\rangle_Y dt \\&=\int_0^T \langle y,E^*f\rangle_Y -\int_0^T \langle BB^*p_{\tilde f},p_f\rangle_Y dt %-\int_0^T \langle \partial_\nu y fdxdt\\& %+\int_0^T\int_\gamma \partial_\nu p_{\tilde f}\partial_\nu p_f dxdt - %\int_0^T\int_\Omega y(\partial_tp_f+\Delta p_f)dxdt\\&= %-\int_0^T\int_{\tilde\gamma} \partial_\nu y fdxdt %+\int_0^T\int_\gamma \partial_\nu p_{\tilde f}\partial_\nu p_f dxdt, \end{split}$$ which implies that: $$\label{eq:bdeq} \int_0^T\langle B^*p_{\tilde f},B^* p_f\rangle_U dt= \int_0^T\langle Ey,f\rangle_W dt.$$ Thus, considering [\[eq:defpf\]](#eq:defpf){reference-type="eqref" reference="eq:defpf"}-[\[eq:bdeq\]](#eq:bdeq){reference-type="eqref" reference="eq:bdeq"}, the solution of [\[con:abssys\]](#con:abssys){reference-type="eqref" reference="con:abssys"} with control $v=B^*p_{\tilde f}$ satisfies: $$\label{eq:deltafunceps} \begin{split} &\delta\int_0^T\langle Ey-w,f\rangle_W dt+O(\delta^2) \\&\geq -\varepsilon(\|\tilde f+\delta f\|_{L^2(0,T;W)}-\|\tilde f\|_{L^2(0,T;W)}) \\&\geq -\varepsilon|\delta| \|f\|_{L^2(0,T;W)}. \end{split}$$ Taking $\delta\to 0^+$ and $\delta\to0^-$ we obtain from [\[eq:deltafunceps\]](#eq:deltafunceps){reference-type="eqref" reference="eq:deltafunceps"} that: $$\left|\int_0^T\langle Ey-w,f\rangle_Wdt \right| \leq\varepsilon\|f\|_{L^2(0,T;W)},$$ for all $f\in L^2(0,T;W)$, which implies [\[eq:approxheat\]](#eq:approxheat){reference-type="eqref" reference="eq:approxheat"}. Reciprocally, if $B^*p_f=0$ for some $f\neq0$, considering [\[eq:dualcompint\]](#eq:dualcompint){reference-type="eqref" reference="eq:dualcompint"}, $E y$ is orthogonal to $f$ for all $v\in L^2(0,T;U)$, so under that hypothesis the system [\[con:abssys\]](#con:abssys){reference-type="eqref" reference="con:abssys"} is not approximately controllable. ◻ In a similar way, based on the Hilbert Uniqueness Method, we can obtain the duality result for exact controllability: **Proposition 2** (Dual notion of exact contr.). *Let $Y$, $U$ and $W$ Hilbert spaces and $A:D(A)\to Y$, $B:U\to Y$ and $E:D(A)\to W$ be linear operators. Then, for all $w\in L^2(0,T;W)$ and $\varepsilon>0$ there is a control $f\in L^2(0,T;U)\setminus \{0\}$ such that the solution of [\[con:abssys\]](#con:abssys){reference-type="eqref" reference="con:abssys"} satisfies: $$\label{eq:exactheat} Ey=w,$$ if and only if $$\label{eq:ratioabs} \sup_{f\in L^2(0,T;U)\setminus \{0\}} \frac{\|f\|_{L^2(0,T;W)}} {\|B^*p_f\|_{L^2(0,T;U)}}<+\infty,$$ for $p$ the solution of [\[eq:adjabstract\]](#eq:adjabstract){reference-type="eqref" reference="eq:adjabstract"}. In fact, if [\[eq:ratioabs\]](#eq:ratioabs){reference-type="eqref" reference="eq:ratioabs"} is satisfied, we obtain [\[eq:exactheat\]](#eq:exactheat){reference-type="eqref" reference="eq:exactheat"} applying the control $v=\partial_\nu p_f$, where $f$ is the minimizer of: $$\label{eq:defJfexact} J(f)=\frac{1}{2}\|B^*p_f\|^2_{L^2(0,T;U)} - \int_0^T\langle f,w \rangle_W dt.$$* **Remark 3**. *The existence of Carleman inequalities could allows us to prove some kind of exact controllability for the wave equation, an existence which remains as a challenging open problem.* ## Tracking of the heat equation {#sec:obsprob} A consequence of Proposition [Proposition 1](#prop:absdualapprx){reference-type="ref" reference="prop:absdualapprx"} is the approximate controllability of the heat equation: **Proposition 4** (Approx. contr.). *Let $\Omega$ be an analytic domain, $\gamma\subset \partial\Omega$ be relatively open and non-empty, and $\tilde\gamma\subset\subset\partial\Omega\setminus\gamma$. Then, for all $w\in L^2((0,T)\times\tilde\gamma)$ and $\varepsilon>0$ there is a control $v\in L^2((0,T)\times \gamma)$ such that the solution of [\[con:heat\]](#con:heat){reference-type="eqref" reference="con:heat"} satisfies [\[eq:approxheat\]](#eq:approxheat){reference-type="eqref" reference="eq:approxheat"}.* *Proof.* The dual problem of [\[con:heat\]](#con:heat){reference-type="eqref" reference="con:heat"}-[\[eq:parny\]](#eq:parny){reference-type="eqref" reference="eq:parny"} reads as follows: $$\label{adj:heat} \begin{cases} -p_{t}-\Delta p=0 & \mbox{ in } (0,T)\times\Omega,\\ p=f1_{\tilde\gamma} & \mbox{ on } (0,T)\times\partial\Omega,\\ p(T)=0 & \mbox{ on }\Omega. \end{cases}$$ By Proposition [Proposition 2](#prop:dualapprx){reference-type="ref" reference="prop:dualapprx"}, it suffices to prove that $\partial_\nu p_f=0$ on $\gamma$ implies that $f=0$. This result follows from Holmgren's Uniqueness Theorem (see, for instance, [@ladyvzenskaja1988linear]), which implies that the solution of the heat equations are analytic, and by Hopf's Lemma (which states that if $p=0$ on a part of the boundary and $p>0$ on a neighbourhood of the boundary, then $\partial_\nu p$ on that part of the boundary). In fact, if $p$ and $\partial_\nu p$ are both null on $\gamma$, all the derivatives of $p$ derivatives are null on $\gamma$. Otherwise, their is $\bar x\in\gamma$ such that $p$ is strictly positive or negative in an open $\tilde \Omega\subset\Omega$ such that $\bar x\in\partial\tilde\Omega$, which by Hopf's lemma implies that $\partial_\nu p(\bar x)\neq 0$. Finally, the result follows from analyticity. ◻ **Remark 5** (Regularity of the trackable space). *Due to the regularizing effect of the heat equation, we cannot expect that the trackable space contains irregular traces if $\tilde\gamma\subset\subset\partial\Omega\setminus\gamma$. Indeed, with a classic bootstrapping argument as in [@doubova2002controllability Lemma 2.5], we can show that the reachable space must be regular (notably, if $\Omega$ is a $C^\infty$ domain, the trace must be $C^\infty$).* # Transmutation for tracking controllability {#sec:trasm} In this section we relate the tracking controllability of the heat equation and of the wave equation by using a variant of the Kannai transform (see [@kannai1977off], [@miller2004geometric] and [@miller2006control]). We recall that the Kannai consists on averaging the solutions of the wave equation with the kernel: $$\label{def:k} k(t,s):= \frac{e^{-s^2/(4t)}}{\sqrt{4\pi t}}.$$ We recall that $k$ is the fundamental solution of the heat equation; i.e. it satisfies: $$\label{eq:kernel} \partial_t k=\partial_{ss} k;\ \ \ k(0,s)=\delta_0(s).$$ As we show in Section [3.1](#sec:Kannai){reference-type="ref" reference="sec:Kannai"}, the Kannai transform links the heat and wave equations. We prove in this section that irregular traces independent of time cannot be reached. Notably, let us consider the following control problem for the wave equation: $$\label{eq:trackwaved} \begin{cases} z_{tt}-\Delta z=0 & \mbox{ in }\mathbb R\times \Omega,\\ z(t,\cdot)= g1_\gamma & \mbox{ on } \mathbb R\times \partial\Omega,\\ z(0,\cdot)=z_0& \mbox{ on }\Omega,\\ z_t(0,\cdot)=z_1& \mbox{ on }\Omega, \end{cases}$$ Here, $\Omega\subset\mathbb R^d$ is a $C^2$ domain, $g$ is the control and $(z_0,z_1)\in L^2(\Omega)\times H^{-1}(\Omega)$ the initial states. For this section we define the functional space: $$%\label{eq:defcalER} \begin{split} \mathcal E(\mathbb R; H):= &\left\{g\in L^\infty_{\operatorname{loc}}(\mathbb R;H): \forall\delta>0\ \exists C_\delta>0: \right. \\& \left. \|g(t)\|_{H}\leq C_\delta e^{\delta t^2} \ \forall t\in\mathbb R\right\}, \end{split}$$ for a given Hilbert space $H$. The main result that we prove is the following: **Theorem 6** (Untrackable traces for the wave eq.). *Let $\Omega$ be a $C^2$ domain, $\gamma\subset\partial\Omega$, and $\tilde\gamma\subset\subset \partial\Omega \setminus\gamma$. Let $w\in L^\infty(\tilde\gamma)\setminus C^0(\tilde\gamma)$. Then, for any $\tau>0$ there is no $g\in\mathcal E(\mathbb R; L^2(\gamma))$, $z_0\in L^2(\Omega)$ and $z_1\in H^{-1}(\Omega)$ such that the solution $z$ of [\[eq:trackwaved\]](#eq:trackwaved){reference-type="eqref" reference="eq:trackwaved"} satisfies: $$\partial_\nu z(t,x)=w(x) \mbox{ on } (\tau, \infty)\times\tilde\gamma$$* This is done with the Kannai transform. Indeed, this is the first time that the Kannai transform is used to obtain negative results about tracking controllability. For that, we consider targets that present discontinuities on the space variable. This means that our result apply to all the dimensions except dimension 1. ## The Kannai transform {#sec:Kannai} The importance of the Kannai transform is reflected by following link between the control problems [\[con:heat\]](#con:heat){reference-type="eqref" reference="con:heat"} and [\[eq:trackwaved\]](#eq:trackwaved){reference-type="eqref" reference="eq:trackwaved"}: **Lemma 7** (Kannai transform). *Let $\Omega$ be a $C^2$ domain, $\gamma\subset\partial\Omega$, $g\in\mathcal E(\mathbb R, L^2(\gamma))$, $z_0\in L^2(\Omega)$, $z_1\in H^{-1}(\Omega)$ and $z$ the solution of [\[eq:trackwaved\]](#eq:trackwaved){reference-type="eqref" reference="eq:trackwaved"}. Then, $$\label{eq:ytransf} y(t,x)=\int_{-\infty}^{\infty}k(t,s)z(s,x)ds,$$ is a solution of [\[con:heat\]](#con:heat){reference-type="eqref" reference="con:heat"} for $T=\infty$, $y_0=z_0$, $$v(t,x):=\int_{-\infty}^{\infty}k(t,s)g(s,x)ds,$$ and it satisfies: $$\partial_\nu y(t,x)=\int_{-\infty}^{\infty}k(t,s)\partial_\nu z(s,x)ds.$$* *Proof.* It is trivial that the function $y$ given by [\[eq:ytransf\]](#eq:ytransf){reference-type="eqref" reference="eq:ytransf"} satisfy the boundary conditions of [\[con:heat\]](#con:heat){reference-type="eqref" reference="con:heat"}. Moreover, it satisfies the initial condition because of [\[eq:kernel\]](#eq:kernel){reference-type="eqref" reference="eq:kernel"}. Finally, it is a solution of the heat equation. Indeed, if $g\in\mathcal D(\mathbb R\setminus\{0\}\times\gamma)$, then for all $t\in(0,T)$ and $x\in(0,L)$ the following equality holds: $$\begin{split} y_t&=\int_{-\infty}^{\infty}k_t(t,s)z(s,x)ds =\int_{-\infty}^{\infty}k_{ss}(t,s)z(s,x)ds \\&=\int_{-\infty}^{\infty}k(t,s)z_{ss}(s,x)ds =\int_{-\infty}^{\infty}k(t,s)\Delta z(s,x)ds \\&=\Delta\left(\int_{-\infty}^{\infty} k(t,s)z(s,x)ds\right) =\Delta y. \end{split}$$ We have used [\[eq:kernel\]](#eq:kernel){reference-type="eqref" reference="eq:kernel"} in the second equality. Moreover, the integration by parts on the third equality is rigorous because $k$ decays when $s\to\infty$. Finally, for any $g\in \mathcal E(\mathbb R;L^2(\gamma))$ we can prove that $\int_{-\infty}^{\infty}k_{ss}(t,s)z(s,x)ds= \Delta\int_{-\infty}^{\infty} k(t,s)z(s,x)ds$ with a density argument, since for all $t>0$ the function $e^{-s^2/(4t)}z$ decays quadratic exponentially when $s\to\infty$, so $y_t=\Delta y$. ◻ ## Proof of Theorem [Theorem 6](#tm:notwavecon){reference-type="ref" reference="tm:notwavecon"} {#proof-of-theorem-tmnotwavecon} Let us now prove Theorem [Theorem 6](#tm:notwavecon){reference-type="ref" reference="tm:notwavecon"}. For that, we first prove the following result: **Lemma 8**. *Let $\Omega$ be a $C^2$ domain, $\gamma\subset\partial\Omega$, and $\tilde\gamma\subset\subset \partial\Omega \setminus\gamma$ be relatively open. Let $w\in L^\infty(\tilde\gamma)$ such that $w\not\in C^0(\tilde\gamma)$. Then, for any $\tau>0$ there is no $g\in\mathcal E(\mathbb R; L^2(\gamma))$, $z_0\in L^2(\Omega)$ and $z_1\in H^{-1}(\Omega)$ such that the solution $z$ of [\[eq:trackwaved\]](#eq:trackwaved){reference-type="eqref" reference="eq:trackwaved"} satisfies: $$\label{eq:falseres} \begin{cases} \partial_\nu z(t,x)=w_1 & \mbox{ on }\left((-\infty,-\tau)\cup (\tau, \infty)\right)\times\tilde\gamma_1,\\ \partial_\nu z(t,x)=-w_2 & \mbox{ on }\left((-\infty,-\tau)\cup (\tau,\infty)\right)\times\tilde\gamma_2. \end{cases}$$* *Proof.* We are going to prove Lemma [Lemma 8](#lm:notwavecon){reference-type="ref" reference="lm:notwavecon"} by reductio ad absurdum. Let us consider the function: $$y(t,x)=\int_{-\infty}^{\infty}k(t,s)z(s,x)ds,$$ for $k$ given by [\[def:k\]](#def:k){reference-type="eqref" reference="def:k"}. By Lemma [Lemma 7](#lm:Kannaitrans){reference-type="ref" reference="lm:Kannaitrans"}, $y$ is a solution of [\[con:heat\]](#con:heat){reference-type="eqref" reference="con:heat"} for $T=\infty$, $$v(t,x):=\int_{-\infty}^{\infty}k(t,s)g(s,x)ds$$ and it satisfies: $$\label{eq:parnheatvar} \partial_\nu y(t,x)= \int_{-\infty}^{\infty}k(t,s)\partial_\nu z(s,x)ds \mbox{ on }\mathbb R\times\tilde\gamma.$$ Because of the regularizing effect of the heat equation, $\partial_\nu u\in C^0([0,T]\times\tilde\gamma)$. However, the following limit holds on $L^\infty(\tilde\gamma)$: $$\label{lim:parnheatvar} \begin{split} \displaystyle\lim_{t\to\infty}&\int_{-\infty}^{\infty}k(t,s)\partial_\nu z(s,x)ds \\&= \displaystyle\lim_{t\to\infty}\int_{-\infty}^{\infty}\frac{e^{-s^2}}{\sqrt\pi}\partial_\nu z\left(s\sqrt{4t},x\right)ds=w_1.\\ \end{split}$$ Since $C^0(\tilde\gamma)$ is close in $L^\infty(\tilde\gamma)$, we arrive at a contradiction. ◻ From Lemma [Lemma 8](#lm:notwavecon){reference-type="ref" reference="lm:notwavecon"} we deduce the proof of Theorem [Theorem 6](#tm:notwavecon){reference-type="ref" reference="tm:notwavecon"}: *Proof.* The proof of Theorem [Theorem 6](#tm:notwavecon){reference-type="ref" reference="tm:notwavecon"} is based on a symmetry reasoning. In fact, if such a solution $z$ exists we define: $$\tilde z:= \begin{cases} z(t,x)&t\geq0,\\ z(-t,x)&t<0. \end{cases}$$ Clearly, $\tilde z$ satisfies [\[eq:trackwaved\]](#eq:trackwaved){reference-type="eqref" reference="eq:trackwaved"} for: $$\tilde g(t)=g(t)1_{t\geq0}+g(-t)1_{t<0}$$ and [\[eq:falseres\]](#eq:falseres){reference-type="eqref" reference="eq:falseres"}, which contradicts the results of Theorem [Theorem 6](#tm:notwavecon){reference-type="ref" reference="tm:notwavecon"}. ◻ **Remark 9** (Implications of Theorem [Theorem 6](#tm:notwavecon){reference-type="ref" reference="tm:notwavecon"}). *Theorem [Theorem 6](#tm:notwavecon){reference-type="ref" reference="tm:notwavecon"} has one of the following implications: either the trackable space on the interval $(\tau,\infty)$ does not contain some discontinuous functions or the cost for getting those traces is exponentially large in the sense that there is $\delta>0$ such that for all $C>0$ there is a sequence $t_n$ such that $\|g\|_{L^2(\gamma)}\geq C e^{\delta t^2_n}$. Determining which assertion holds, though, is an open problem.* # Controls for the 1d heat equation {#sec:segcyl} In this section we study the tracking controllability of the 1d heat equation by using the flatness approach and duality. Notably, we study the solutions of: $$\label{con:heat0} \begin{cases} y_{t}-\partial_{xx} y=0 & \mbox{ in } (0,T)\times(0,L),\\ y(\cdot,0)=0 &\mbox{ on } (0,T),\\ y(\cdot,L)=v& \mbox{ on } (0,T),\\ y(0)=0 & \mbox{ on }(0,L). \end{cases}$$ First, we recall how to compute explicitly the controls so that the solution of [\[con:heat0\]](#con:heat0){reference-type="eqref" reference="con:heat0"} satisfies: $$\label{tarjet:heat0} \partial_x y(\cdot,0)=w \ \ \ \ \mbox{ on } (0,T),$$ where $w$ is a flat function. Then, we use those controls to get an upper bound on the cost for approximate tracking controllability. Notably, we derive an upper bound for the minimum cost of the control which acts on $(0,T)\times\{L\}$ so that: $$\label{tarjet:heatapprox0} \|\partial_x y(\cdot,0)-w\|_{C^0([0,T])}\leq\varepsilon,$$ with the controls obtained explicitly with the flatness method. In particular, we prove the following result: **Theorem 10** (Cost for approx. contr.). *Let $L>0$, $T>0$ and $w\in W^{1,\infty}(0,T)$ a function satisfying $w(0)=0$. Then, for all $s\in (0,1)$ there is a constant $C>0$ such that for all $\varepsilon>0$ we can obtain a control $v_\varepsilon$ satisfying: $$\label{est:costveps} \begin{split} &\|v_\varepsilon\|_{C^0([0,T])}\\&\leq C \exp\left[ C\left(\frac{\|w\|_{W^{1,\infty}(0,T)}} {\varepsilon}\right)^{1/s}\right]\|w\|_{W^{1,\infty}(0,T)} \end{split}$$ and such that the solution of [\[con:heat0\]](#con:heat0){reference-type="eqref" reference="con:heat0"} satisfies [\[tarjet:heatapprox0\]](#tarjet:heatapprox0){reference-type="eqref" reference="tarjet:heatapprox0"}.* The proof of Theorem [Theorem 10](#tm:approxcost){reference-type="ref" reference="tm:approxcost"} consists on approximating the targets with Gevrey function by convolution with cut-off functions. The requirement of being $w$ more regular than the space in which the norm is taken is common on parabolic control theory (see, for instance, [@fernandez2000cost]). In this paper we use this to approximate the target with an element of the trackable space in a uniform way with the $C^0$ norm. Before proving Theorem [Theorem 10](#tm:approxcost){reference-type="ref" reference="tm:approxcost"} we first recall some results of Gevrey functions, and then perform some technical estimates on an auxiliary function. ## Previous results In this section we recall the known controllability results to flat outputs in one-dimensional domains and the existence of Gevrey cut-off functions. By definition $w$ is a Gevrey function of order $s$ if and only if $w\in C^\infty([0,T])$ and satisfies for some $C,R>0$: $$|w^{(i)}(t)|\leq C\frac{(i!)^s}{R^i}, \ \ \forall t\in[0,T], \forall i\geq0.$$ Here we consider $w\in[1,2)$. If $s=1$, then the Gevrey function is analytic. **Lemma 11** (Controls for flat targets). *Let $s\in [0,2)$, $L>0$, $T>0$ and $w\in C^\infty([0,T])$ be a Gevrey function of order $s$ satisfying $w^{(i)}(0)=0$ for all $i\in\mathbb N$. Then, there is $v$ a Gevrey function of order $s$ such that the solution of [\[con:heat0\]](#con:heat0){reference-type="eqref" reference="con:heat0"} satisfies [\[tarjet:heat0\]](#tarjet:heat0){reference-type="eqref" reference="tarjet:heat0"}.* The proof of Lemma [Lemma 11](#lm:trackGevrey){reference-type="ref" reference="lm:trackGevrey"} is mainly contained in [@laroche2000motion]. We recall that they consider the controls: $$%\label{def:flatlinev} v(t)= \sum_{i\geq0}\frac{L^{2i+1}}{(2i+1)!}w^{(i)}(t).$$ and that the solution of [\[con:heat0\]](#con:heat0){reference-type="eqref" reference="con:heat0"} are given by: $$\label{def:solyline} y(t,x)=\sum_{i\geq0}\frac{x^{2i+1}}{(2i+1)!}w^{(i)}(t).$$ Next, we recall the existence of cut-off functions that belong to a Gevrey space: **Lemma 12** (Gevrey cut-off functions). *Let $s>1$. There is a cut-off function $\xi$ supported in $[0,1]$ of Gevrey order $s$ and satisfying $\int_0^1\xi(t)dt=1$.* We may construct the function in Lemma [Lemma 12](#lm:defxi){reference-type="ref" reference="lm:defxi"} by considering that for all $s >0$ the function $\exp\left( \frac{-1}{((1-t)t)^s }\right)1_{(0,1)}$ is of order $1+1/s$. This was proved first in [@ramis1978devissage], and an English version of the proof can be consulted in [@schorkhuber2013flatness Lemma 4]. Their proofs are based on Cauchy integral formula and on Stirling formula. ## Upper bounds for auxiliary functions In order to quantify the cost we introduce the auxiliary functions: $$\label{def:Gsx} \mathcal G_s (x):= \sum_{i\geq 0}\frac{x^i}{(i!)^s}.$$ We remark that these functions have an exponential bound: **Lemma 13** (Upper bounds for $\mathcal G_s$). *Let $s >0$. Then, there is $C>0$ depending on $s$ such that: $$\label{est:bdGtheta} \mathcal G_s(x)\leq C\exp\left(Cx^\frac{1}{s}\right), \ \ \ \forall x\geq0.$$* Lemma [Lemma 13](#lm:bdexp){reference-type="ref" reference="lm:bdexp"} is proved by estimating $\partial_x[\ln(\mathcal G_s(x))]$ with Stirling's formula and a splitting of the lower and the higher order terms of the sum: *Proof.* In order to prove [\[est:bdGtheta\]](#est:bdGtheta){reference-type="eqref" reference="est:bdGtheta"} it suffices to prove that there is $C>0$ such that: $$\label{est:Gthetaprim} \mathcal G_s'(x)\leq Cx^{\frac{1-s}{s}}\mathcal G_s(x), \ \ \ \forall x\geq 1.$$ For that purpose, we remark that: $$\mathcal G_s '(x)=\sum_{i\geq1}i^{1-s }\frac{x^{i-1}}{((i-1)!)^s } =\sum_{i\geq0}(i+1)^{1-s }\frac{x^i}{(i!)^s }.$$ In order to prove [\[est:Gthetaprim\]](#est:Gthetaprim){reference-type="eqref" reference="est:Gthetaprim"} we split the terms into $i< 2x^{1/s}e$ and $i\geq 2x^{1/s}e$. On the one hand, if $i\geq 2x^{1/s}e$, from Stirling's formula we get that: $$\label{est:xiitheta} \frac{x^i}{(i!)^s}\leq C \frac{x^ie^{is}}{i^{is}} =C\left(\frac{x^{1/s}e}{i}\right)^{is}\leq C2^{-is}.$$ Thus, from [\[est:xiitheta\]](#est:xiitheta){reference-type="eqref" reference="est:xiitheta"} we obtain for all $x\geq1$ that: $$\label{est:sumupterm} \sum_{i\geq 2x^{1/s}e}(i+1)^{1-s}\frac{x^i}{(i!)^s} \leq C\leq Cx^{\frac{1-s}{s}}\mathcal G_s(x).$$ On the other hand, we find for all $x\geq1$ that: $$\label{est:sumdownterms} \begin{split} \sum_{i< 2x^{1/s}e}(i+1)^{1-s}\frac{x^i}{(i!)^s} &\leq \sum_{i<2x^{1/s}e}(4e)^{1-s} x^{\frac{1-s}{s}}\frac{x^i}{(i!)^s}\\&\leq C x^{\frac{1-s}{s}}\mathcal G_s(x). \end{split}$$ Therefore, [\[est:Gthetaprim\]](#est:Gthetaprim){reference-type="eqref" reference="est:Gthetaprim"} follows from [\[est:sumupterm\]](#est:sumupterm){reference-type="eqref" reference="est:sumupterm"} and [\[est:sumdownterms\]](#est:sumdownterms){reference-type="eqref" reference="est:sumdownterms"}. ◻ ## Conclusion of the proof of Theorem [Theorem 10](#tm:approxcost){reference-type="ref" reference="tm:approxcost"} {#conclusion-of-the-proof-of-theorem-tmapproxcost} We now have all the ingredients to compute the upper bound of the cost of approximate controllability. The proof of Theorem [Theorem 10](#tm:approxcost){reference-type="ref" reference="tm:approxcost"} is divided in 2 steps: first we approximate the target $w$ by convolution with the cut-off function given in Lemma [Lemma 12](#lm:defxi){reference-type="ref" reference="lm:defxi"}, and secondly we apply the control given in Lemma [Lemma 11](#lm:trackGevrey){reference-type="ref" reference="lm:trackGevrey"} and compute estimates on it with Lemma [Lemma 13](#lm:bdexp){reference-type="ref" reference="lm:bdexp"}. *Proof.* *Step 1: approximating the normal trace.* Let $s\in (0,1)$ and $w\in W^{1,\infty}(0,T)$ a function satisfying $w(0)=0$. By linearity, we can suppose that: $$\label{eq:wnorm} \|w\|_{W^{1,\infty}(0,T)}=1.$$ We define $\xi_\varepsilon:= \varepsilon^{-1}\xi(x\varepsilon^{-1})$, for $\xi$ the Gevrey function of order $2-s$ given in Lemma [Lemma 12](#lm:defxi){reference-type="ref" reference="lm:defxi"} and: $$\label{def:weps} w_\varepsilon:= \tilde w\ast\xi_\varepsilon= \int_{t-\varepsilon}^t\tilde w(t')\xi_\varepsilon(t-t')dt',$$ for $\tilde w$ the prolongation of $w$ by $0$ to $\mathbb R^-$. Since $\xi$ is supported in $[0,1]$, $w_\varepsilon=0$ in $(-\infty,0]$, so $w_\varepsilon$ annihilates at $t=0$. Moreover, from $w(0)=0$ we get that: $$|w_\varepsilon(t) -w(t)| \leq \sup_{t'\in (0,\varepsilon)}|\tilde w(t-t')-w(t)| \leq \varepsilon\|w\|_{W^{1,\infty}(0,T)}.$$ Thus, from [\[eq:wnorm\]](#eq:wnorm){reference-type="eqref" reference="eq:wnorm"} we obtain that: $$\label{eq:approxwC0} \|w_\varepsilon-w\|_{C^0([0,T])}\leq \varepsilon.$$ Since $\xi$ is a Gevrey function of order $2-s$ we can show easily that $w_\varepsilon$ is a Gevrey function of order $2-s$. In fact, considering [\[eq:wnorm\]](#eq:wnorm){reference-type="eqref" reference="eq:wnorm"} and [\[def:weps\]](#def:weps){reference-type="eqref" reference="def:weps"} we get that: $$\label{est:wepsN} \|w_\varepsilon\|_{C^i([0,T])}\leq \varepsilon^{-i}\|\xi\|_{C^i([0,1])}, \ \ \ \forall i\in\mathbb N.$$ Thus, from the assumption that $\xi$ is a Gevrey function of order $2-s$ and [\[est:wepsN\]](#est:wepsN){reference-type="eqref" reference="est:wepsN"} we deduce that: $$\label{est:wepsNcomplete} \|w_\varepsilon\|_{C^i([0,T])}\leq \left(\frac{C}{\varepsilon}\right)^{i}(i!)^{2-s}, \ \ \ \forall i\in\mathbb N.$$ *Step 2: estimation of the control.* From Lemma [Lemma 11](#lm:trackGevrey){reference-type="ref" reference="lm:trackGevrey"} we obtain that $\partial_xy(\cdot,0)=w_\varepsilon$ if we apply the control: $$%\label{def:flatlinev} v_\varepsilon(t)= \sum_{i\geq0}\frac{L^{2i+1}}{(2i+1)!}w_\varepsilon^{(i)}(t).$$ In particular, from [\[est:wepsNcomplete\]](#est:wepsNcomplete){reference-type="eqref" reference="est:wepsNcomplete"} we find that: $$\|v_\varepsilon\|_{C^0([0,T])}\leq \sum_{i\geq0}\left( \frac{C}{\sqrt\varepsilon}\right)^{2i} \frac{(i!)^{2-s}}{(2i+1)!},$$ for $C$ a constant independent of $i$. Next, we consider that: $$\label{est:facti} \frac{(i!)^{2-s}}{(2i+1)!}\leq \frac{1}{(i!)^{s}},$$ since: $$\frac{(2i)!}{(i!)^2}={2i \choose i}>1.$$ Thus, $$\begin{split} \|v_\varepsilon\|_{C^0([0,T])}&\leq \sum_{i\geq0}\left( \frac{C}{\sqrt\varepsilon}\right)^{2i}\frac{(i!)^{2-s}}{(2i+1)!} \\&\leq \displaystyle\sum_{i\geq0}\left(\frac{C}{\sqrt\varepsilon}\right)^{2i}\frac{1}{(i!)^{s}}\\&\leq \displaystyle\sum_{i\geq0} \left(\frac{C}{\varepsilon}\right)^{i}\frac{1}{(i!)^{s}}=\mathcal G_s(C\varepsilon^{-1}). \end{split}$$ Hence, we obtain [\[est:costveps\]](#est:costveps){reference-type="eqref" reference="est:costveps"} from [\[est:bdGtheta\]](#est:bdGtheta){reference-type="eqref" reference="est:bdGtheta"}. ◻ # Open problems {#sec:opprob} First of all, we would like to remark that our technique can clearly work when we have other boundary conditions, such as Neumann or Robin. The method and results in this paper lead to some interesting open problems and could be extended in various directions (in addition to the ones proposed in Remark [Remark 9](#rk:costcon){reference-type="ref" reference="rk:costcon"}) that we briefly describe: - **Multi-dimensional domains.** It is an open problem to get more precise information about controls for multi-dimensional domains, both for the heat and wave equations. The ideal scenario is to have explicit formulas; though it is more realistic to search for quantitative estimates for the cost of approximate controllability. In fact, generalizing the flatness method for the heat equation and characteristic or energy methods for the wave equation in non-cylindrical domains seem challenging tasks. - **Optimality on the cost of approximate controllability.** One relevant open problem is whether we can sharpen the bounds given in Theorem [Theorem 10](#tm:approxcost){reference-type="ref" reference="tm:approxcost"} to $\exp(C\varepsilon^{-1})$ or $\exp(C\varepsilon^{-1/2})$, which is more in line with the known bounds for classical approximate controllability of parabolic equations. Indeed, the cost of getting at $\varepsilon$ distance in the $L^2$ norm to a function $y^T\in H^2(0,L)\cap H_0^1(0,L)$ in the heat equation with constant coefficients is bounded by $\exp(C\varepsilon^{-1/2})$ (see [@fernandez2000cost]). In addition, a cost of $\exp(C\varepsilon^{-1})$ has been obtained for more general heat equations (see [@fernandez2000cost], [@phung2004note] and [@boutaayamou2020cost]), for the semi-linear heat equation (see [@yan2009cost]), for the Ginzburg-Landau equation (see [@aramua2017cost]) and for the hypoelliptic heat equation (see [@laurent2017tunneling]). Their proofs are based on an observability inequality with appropriate weights for the null controllability, so we cannot replicate them for the control problem [\[con:heat\]](#con:heat){reference-type="eqref" reference="con:heat"}. - **Observability estimates for the heat equation.** The obtention of observability inequalities for system [\[adj:heat\]](#adj:heat){reference-type="eqref" reference="adj:heat"} remains open. In fact, it is interesting to see if we can obtain an observability inequality from Lemma [Lemma 11](#lm:trackGevrey){reference-type="ref" reference="lm:trackGevrey"}. The main difficulty is that Gevrey functions do not form a Banach space and its dual is the space of ultra-distributions (see, for example, [@dasgupta2014gevrey]). - **Tracking plus null control.** It is an interesting problem to determine if we may take the heat equation to equilibrium in addition to controlling the trace. It is clear that for all $\delta>0$ we can control the trace on $(0,T-\delta)$ and then obtain $y(T,\cdot)=0$, as the heat equation is controllable at arbitrary small times (see, for instance, [@lebeau1995controle] and [@lafuncion]). However, because of continuity, this is not true when aiming both at the trace on $(0,T)$ and at the final state. However, it is a relevant problem to determine if this is true with proper compatibility conditions. - **Other parabolic equations.** A possible extension is to determine both theoretically and numerically if analog properties are satisfied by more complex parabolic equations like the heat equation with time and space dependent parameters, Stokes equation and Stefan equation, or if there is any remarkable difference. - **Sidewise controllability of the wave equation.** Sidewise observability estimates for the wave equation are also an open problem. This includes establishing observability inequalities under suitable concavity and microlocal geometric conditions on the support of the source and the measurement set, for sources fulfilling suitable pseudo-differential conditions. Some breakthroughs are done in [@Dehman2024boundary]. # Conclusions {#sec:concl} In this paper we have studied the tracking controllability for the heat equation and its relation with the sidewise controllability of the wave equation. First, duality allows to prove approximate controllability results in multi-dimensional domains. Second, the transmutation method provides a way for obtaining controls for the heat equation, which are useful for numerical purposes, and a way for proving that irregular traces on the wave equation require controls with a norm increasing exponentially. Third, revisiting the flatness approach we obtain the cost of approximate controllability for the 1d heat equation and, combining this with duality methods, we control the trajectories on two interior points. In the future, efforts should be devoted to the understanding of controllability in multi-dimensional domains and of more complex parabolic and hyperbolic equations: finding in that complex setting a method that can provide explicit formulas or quantitative estimates for the controls remains an interesting while challenging issue. Also, efforts should be devoted to obtaining Carleman inequalities for proving the tracking null controllability of the heat equation. # References {#references .unnumbered} Jon Asier Bárcena-Petisco Jon Asier Bárcena-Petisco recived in 2016 a Bachellor's degree in Mathematics from the University of the Basque Country, being the first of his promotion. A year later, he obtained a Master's degree in Mathematics from Sorbonne University. Afterwards, in 2020 he obtained his PhD at the LJLL (Sorbonne University), where he did a thesis on Control Theory under the supervision of Sergio Guerrero, obtaining the prize Vicent Caselles, granted to the best 6 PhD thesis in Mathematics by Spaniards or made in a Spanish university. In the academic year 2020-2021, he completed his postdoctoral training in the Autonomous University of Madrid, within the ERC Dycon. Currently, he is an Assistant Professor at the Department of Mathematics of the University of the Basque Country in the field of Applied Mathematics. He currently has 9 published papers and the Spanish national habilitation necessary for obtaining a permanent position in a Spanish public university. Enrique Zuazua Enrique Zuazua received the Licenciatura degree in mathematics from the University of Basque Country, Leioa, Spain, in 1984, and the dual Ph.D. degree in mathematics from the University of the Basque Country in 1987 and the Université Pierre et Marie Curie, Paris, France, in 1988.,He holds, since 2019, the Chair for Dynamics, Control and Numerics - Alexander von Humboldt Professorship, at the Department of Data Science, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany, and part-time appointments at Universidad Autónoma de Madrid, Madrid, Spain, and the Fundación Deusto, Bilbao, Spain. He is a member of Jakiunde, Gipuzkoa, Spain, the Basque Academy of Sciences, Letters and Humanities and of the Academia of Europaea, London, U.K., and cooperates with the University of Sichuan, Chengdu, China, and Sherpa.ai, Bilbao, Spain. His fields of expertise in the broad area of applied mathematics cover topics related to partial differential equations, systems control and numerical analysis, and machine learning. He was the first Manager for the area of Mathematics of the Spanish National Research Plan from 1999 to 2002, the Founding Scientific Director of the Basque Centre for Applied Mathematics from 2008 to 2012, and, in 2016, he launched the Chair of Computational Mathematics at the Deusto Foundation, both in Bilbao, Spain. Since 2021, he has been the speaker of the FAU Research Center for Mathematics of Data, Erlangen, Germany. He also develops an intense dissemination agenda, gathered at https://cmc.deusto.eus/enzuazua/.,Dr. Zuazua has been awarded the Euskadi (Basque Country) Prize for Science and Technology 2006 and the National Julio Rey Pastor Prize 2007 in Mathematics and Information and Communication Technology and the Advanced Grants of the European research Council (ERC) NUMERIWAVES in 2010, DYCON in 2016 and CoDeFeL in 2023. [^1]: The work of J.A.B.P. is funded by the Basque Government, under grant IT1615-22; and by the Spanish Government's Ministry of Science, Innovation and Universities (MICINN), under grant PID2021-126813NB-I00. The work of E.Z. is partially funded by the Alexander von Humboldt-Professorship program, the COST Action grant CA18232, "Mathematical models for interacting dynamics on networks" (MAT-DYN-NET), the ModConFlex Marie Curie Action, the Transregio 154 Project ''Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks'' of the German DFG, the grants PID2020-112617GB-C22 and TED2021-131390B-I00 of MINECO (Spain), and by the Madrid Goverment - UAM Agreement for the Excellence of the University Research staff in the context of the V PRICIT (V Regional Plan of Scientific Research and Technological Innovation) [^2]: Department of Mathematics, University of the Basque Country, 48080, Bilbao Spain (e-mail: jonasier.barcena\@ehu.eus). [^3]: \[1\] Chair for Dynamics, Control, Machine Learning, and Numerics (Alexander von Humboldt-Professorship), Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany. \[2\] Chair of Computational Mathematics, Fundación Deusto, Avenida de las Universidades 24, 48007 Bilbao, Basque Country, Spain. \[3\] Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain. (e-mail: enrique.zuazua\@fau.de).

arxiv_math

--- abstract: | We study a new flocking model which has the versatility to capture the physically realistic qualitative behavior of the Motsch-Tadmor model, while also retaining the entropy law, which lends to a similar 1D global well-posedness analysis to the Cucker-Smale model. This is an improvement to the situation in the Cucker-Smale case, which may display the physically unrealistic behavior that large flocks overpower the dynamics of small, far away flocks; and it is an improvement in the situation in the Motsch-Tadmor case, where 1D global well-posedness is not known. The new model was proposed in [@shvydkoy2022environmental] and has a similar structure to the Cucker-Smale and Motsch-Tadmor hydrodynamic systems, but with a new feature: the communication strength is not fixed, but evolves in time according to its own transport equation along the Favre-filtered velocity field. This transport of the communication strength is precisely what preserves the entropy law. A variety of phenomenological behavior can be obtained from various choices of the initial communication strength, including the aforementioned Motsch-Tadmor-like behavior. We develop the general well-posedness theory for the new model and study the long time behavior-- including alignment, strong flocking in 1D, and entropy estimates to estimate the distribution of the limiting flock, all of which extend the classical results of the Cucker-Smale case. In addition, we provide numerical evidence to show the similar qualitative behavior between the new model and the Motsch-Tadmor model for a particular choice of the initial communication strength. address: - 851 S Morgan St, M/C 249, Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607 - 851 S Morgan St, M/C 249, Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607 author: - Roman Shvydkoy - Trevor Teolis title: Well-posedness and long time behavior of the Euler Alignment System with adaptive communication strength --- [^1] # Introduction ## Brief background and motivation The pressureless Euler Alignment system based on the classical Cucker-Smale model is given by $$\begin{aligned} \label{EAS_CS} \tag{CS} \begin{cases} \partial_t \rho + \nabla \cdot (\textbf{u}\rho) = 0 \\ \partial_t \textbf{u}+ \textbf{u}\cdot \nabla \textbf{u}= (\textbf{u}\rho)_{\phi} - \textbf{u}\rho_{\phi} \end{cases}\end{aligned}$$ where we use the shorthand notation $f_{\phi} := f \ast \phi$ for convolutions, see [@CS2007a; @HT2008]. Here, $\phi\in C^1$ is a smooth non-negative radially decreasing communication kernel, $\rho, \textbf{u}$ are density and velocity of the flock, respectively, and the environment in question is either $\Omega= \mathbb{T}^n$ or $\mathbb{R}^n$ (although our results for the $\mathrm{s}$-model will be stated only for the Torus $\mathbb{T}^n$). Analysis and relevance to applications of [\[EAS_CS\]](#EAS_CS){reference-type="eqref" reference="EAS_CS"} has been the subject of many studies in recent years, see [@ABFHKPPS; @FK2019; @HeT2017; @Sbook; @TT2014] and references therein. In particular, flocking in the classical sense of uniformly bounded radius and exponential alignment $$\label{e:flocking} \sup_{t\geqslant 0} (\mathop{\mathrm{diam}}(\mathop{\mathrm{supp}}\rho )) < \infty , \quad \sup_{x \in\mathop{\mathrm{supp}}\rho} |\textbf{u}(t,x) - \textbf{u}_\infty|\leqslant C_0 e^{- \delta t}$$ holds under "heavy-tail\" condition on the kernel, [@TT2014] $$\label{ } \int_0^\infty \phi(r) \, \mbox{d}r= \infty,$$ by direct analogy with the agent-based result of Cucker and Smale [@CS2007a; @CS2007b; @HL2009]. Here, the limiting velocity $\textbf{u}_\infty$ is determined by the initial momentum, which is conserved. The alignment force in the system is mildly diffusive as seen for instance from the energy balance law $$\label{e:enlaw} \frac{\mbox{d\,\,}}{\mbox{d}t}\frac{1}{2} \int_\Omega\rho |\textbf{u}|^2 \, \mbox{d}x= - \int_{\Omega\times \Omega} \phi(x-y) |\textbf{u}(x) - \textbf{u}(y)|^2 \rho(x)\rho(y) \, \mbox{d}y\, \mbox{d}x.$$ Therefore the regularity theory for [\[EAS_CS\]](#EAS_CS){reference-type="eqref" reference="EAS_CS"} in the smooth communication case runs somewhat parallel to hyperbolic conservation laws; the difference being that ther is room for regularization to effect the force. For instance, in 1D, Carrillo, Choi, Tadmor and Tan [@CCTT2016] establish an exact threshold regularity criterion in terms of the so called "e-quantity\" $$\label{e:e} e = \partial_x u + \rho_\phi, \qquad \partial_t e + \partial_x (u e) = 0.$$ The solution with smooth initial condition remains smooth if and only if $e_0 \geqslant 0$. In multi-D partial results are found in [@HKK2014; @HKK2015; @HeT2017; @TT2014] and the book [@Sbook] presents a general continuation criterion in the spirit of Grassin [@Grassin99], Poupaud [@Poupaud99]: as long as $$\label{e:BKMdiv} \int_0^{T_0} \inf_{x\in \Omega} \nabla\cdot {\bf u}(t,x) \, \mbox{d}t> - \infty$$ the solution can be continued smoothly beyond $T_0$. Phenomenologically the Euler alignment system performs well when the flock is mono-scale. However, in heterogeneous formations, when two remote clusters of largely diverse size appear, the dynamics according to [\[EAS_CS\]](#EAS_CS){reference-type="eqref" reference="EAS_CS"} yields pathological results. The large cluster hijacks evolution of the smaller cluster removing any fine features of the latter. Motsch and Tadmor argue in [@MT2011; @MT2014] that rebalancing the averaging operation in the alignment force cures such issues. They proposed the following modification $$\begin{aligned} \label{EAS_MT} \tag{MT} \begin{cases} \partial_t \rho + \nabla \cdot (\textbf{u}\rho) = 0 \\ \partial_t \textbf{u}+ \textbf{u}\cdot \nabla \textbf{u}= \frac{1}{\rho_{\phi}} \big( (\textbf{u}\rho)_{\phi} - \textbf{u}\rho_{\phi} \big). \end{cases}\end{aligned}$$ The model has the exact same flocking behavior [\[e:flocking\]](#e:flocking){reference-type="eqref" reference="e:flocking"} under heavy-tail kernel, but progress in well-posedness theory of the system has been stalled even in 1D due to the lack of the energy law [\[e:enlaw\]](#e:enlaw){reference-type="eqref" reference="e:enlaw"} or the e-quantity [\[e:e\]](#e:e){reference-type="eqref" reference="e:e"}. Therefore the need for a model with qualitative features similar to those of Motsch-Tadmor but better analytical properties has become a pressing problem. A model with the potential to achieve these features has been proposed in [@shvydkoy2022environmental] in the context of Environmental Averaging models, but has not received anymore attention there. The goal of this paper is to show that the model proposed there, which we call the adaptive strength model or $\mathrm{s}$-model for short, does indeed possess the desirable qualitative and analytic properties. We describe the $\mathrm{s}$-model next. ## Environmental Averaging Models and the $\mathrm{s}$-model Despite their differences both systems [\[EAS_CS\]](#EAS_CS){reference-type="eqref" reference="EAS_CS"} and [\[EAS_MT\]](#EAS_MT){reference-type="eqref" reference="EAS_MT"} share similar structure of the alignment force. It can be written as $$\label{e:F} F = \mathrm{s}_\rho ( \left[ \textbf{u} \right]_\rho - \textbf{u}),$$ where $\left[ \textbf{u} \right]_\rho$ is the averaging component, and $\mathrm{s}_\rho \geqslant 0$ is the communication strength. In both cases $\left[ \textbf{u} \right]_\rho= \frac{(\textbf{u}\rho)_\phi}{\rho_\phi}$, which is also known in turbulence literature as the Favre filtration, see [@Favre]. The difference comes only in prescription of the communication strength. In the Cucker-Smale case $\mathrm{s}_\rho = \rho_\phi$, while in the Motsch-Tadmor case $\mathrm{s}_\rho \equiv 1$. Many other examples encountered in the literature, including multi-flocks and multi-species, share the same structure and fall under the category of so called 'environmental averaging models'. The general theory of such models has been developed in [@shvydkoy2022environmental]. The alignment characteristics and well-posedness are determined by a strength function $\mathrm{s}_{\rho}$ and the weighted averaging operator $\mathrm{s}_\rho [\cdot]_{\rho}$. It is observed in [@shvydkoy2022environmental] that the main reason why the e-equation [\[e:e\]](#e:e){reference-type="eqref" reference="e:e"} holds in the Cucker-Smale case is because, for this model, the strength function $\rho_{\phi}$ happens to evolve according to its own transport equation along the Favre-averaged field: $$\label{ } \partial_t \rho_\phi + \partial_x ( \rho_\phi \left[ \textbf{u} \right]_\rho ) = 0.$$ So, a new model was proposed where instead of prescribing communication strength $\mathrm{s}_\rho$ a priori, one lets it adapt to the environment through transport along the averaged field $$\label{ } \partial_t \mathrm{s}+ \partial_x ( \mathrm{s} \left[ \textbf{u} \right]_\rho) = 0, \hspace{5mm} \mathrm{s}\geqslant 0$$ As such, the adaptive strength $\mathrm{s}$ becomes another unknown and may not explicitly depend on the density. The resulting full model, which we call $\mathrm{s}$-model for short, reads $$\label{s_model} \tag{SM} \begin{cases} \partial_t \rho + \nabla \cdot (\textbf{u}\rho) = 0 \\ \partial_t \mathrm{s}+ \nabla \cdot (\mathrm{s} \left[ \textbf{u} \right]_{\rho}) = 0, \hspace{15mm} \mathrm{s}\geqslant 0 \\ \partial_t \textbf{u}+ \textbf{u}\cdot \nabla \textbf{u}= \mathrm{s}( \left[ \textbf{u} \right]_{\rho} - \textbf{u}), \end{cases}$$ Now, regardless of the particular averaging used, the model always admits a conserved quantity in 1D similar to [\[e:e\]](#e:e){reference-type="eqref" reference="e:e"}, which lands it more amenable well-posedness analysis. Indeed, if we differentiate the velocity equation in $x$, we get $$\begin{aligned} \partial_t \partial_x u + \partial_x u (\partial_x u + \mathrm{s}) + u(\partial_x^2 u + \partial_x \mathrm{s}) = \partial_x (\mathrm{s}[u]_{\rho}) .\end{aligned}$$ If $\mathrm{s}$ satisfies the transport equation in [\[s_model\]](#s_model){reference-type="eqref" reference="s_model"}, then it becomes $$\begin{aligned} \partial_t (\partial_x u + \mathrm{s}) + \partial_x u (\partial_x u + \mathrm{s}) + u(\partial_x^2 u + \partial_x \mathrm{s}) = 0\end{aligned}$$ which is the desired conservation law (a.k.a. the entropy law): $$\begin{aligned} \label{conservation_law} e = \partial_x u + \mathrm{s}, \hspace{5mm} \partial_t e + \partial_x (ue) = 0.\end{aligned}$$ In the Cucker-Smale theory, this extra conservation law holds the key to 1D global well-posedeness, strong flocking, and entropy estimates on the limiting distribution of the flock. We affirm in this paper that these results can be extended to the case of the $\mathrm{s}$-model (albeit with the specific case of the Favre averaging, which is the most relevant to the Cucker-Smale and Motsch-Tadmor models, see [1.4](#intro:main_results){reference-type="ref" reference="intro:main_results"} for further justification for working with the case of the Favre averaging). *Remark 1*. The $e$-quantity is sometimes referred to as an entropy because, in 1D, it's magnitude provides a measure of distance of the limiting flock from the uniform distribution. For the Cucker-Smale model, the particular result was proved in [@LS-entropy] and is extended to the $\mathrm{s}$-model with Favre averaging here in Theorem [\[thm:entropy_estimate_intro\]](#thm:entropy_estimate_intro){reference-type="ref" reference="thm:entropy_estimate_intro"}. We note that any attempt to develop well-posedness and alignment analysis of [\[s_model\]](#s_model){reference-type="eqref" reference="s_model"} for most general averaging operators $[\textbf{u}]_{\rho}$ necessitates many technical assumptions and leads to an obscure exposition. So, to avoid such technicalities and to keep our analysis close to the Cucker-Smale and Motsch-Tadmor cases, we limit ourselves to the Favre-based models, setting $\textbf{u}_F := (\textbf{u}\rho)_{\phi} / \rho_{\phi}$ to be our fixed averaging protocol. That is, with regard to the local and 1D global well-posedness results, the choice of $\textbf{u}_F$ is merely convenient and the results can be extended to general averaging protocols $[\textbf{u}]_{\rho}$. Notably, however, the small data and long-time behavior results depend on the explicit structure of $\textbf{u}_F$ and therefore these results may not be extendable to general averaging protocols. Luckily, choosing the specific averaging $\textbf{u}_F$ over general averaging operators is a small sacrifice. Indeed, even with the $\mathrm{s}$-model with the specific Favre averaging has versatility to capture the Motsch-Tadmor-like behavior while retaining the nice analytic properties of the Cucker-Smale model (owing to the conservation law [\[conservation_law\]](#conservation_law){reference-type="eqref" reference="conservation_law"}). We will from here on refer to the $\mathrm{s}$-model with Favre averaging as just the '$\mathrm{s}$-model', unless it is stated otherwise. To rewrite the $\mathrm{s}$-model in a simpler form and more explicit form, we introduce the new variable $$\mathrm{w}:= \frac{\mathrm{s}}{\rho_{\phi}}.$$ As $\rho_{\phi}$ and $\mathrm{s}$ satisfy the same continuity equation, $\mathrm{w}$ satisfies the pure transport equation along the characteristics of $\textbf{u}_F$ $$\begin{aligned} \partial_t \mathrm{w}+ \textbf{u}_F \cdot \nabla \mathrm{w}= 0.\end{aligned}$$ We will refer to it as the \"weight\" in order to distinguish it from the strength in the $\mathrm{s}$-model. The $\mathrm{s}$-model can now be written in a way that eliminates division by $\rho_\phi$ in the alignment force: $$\begin{aligned} \label{EAS_WM} \tag{WM} \begin{cases} \partial_t \rho + \nabla \cdot (\textbf{u}\rho) = 0 \\ \partial_t \mathrm{w}+ \textbf{u}_F \cdot \nabla \mathrm{w}= 0 \\ \partial_t \textbf{u}+ \textbf{u}\cdot \nabla \textbf{u}= \mathrm{w}((\textbf{u}\rho)_{\phi} - \textbf{u}\rho_{\phi}). \end{cases}\end{aligned}$$ Setting $\mathrm{w}=1$ we recover the CS case, while setting $\mathrm{w}_0 = 1/({\rho_0})_{\phi}$ at least initially we recover the Motsch-Tadmor data. In the latter case, as it evolves the strength will deviate from the Motsch-Tadmor strength. The question arises as to whether this new strength still retains the same balancing properties as the original Motsch-Tadmor model. In Section [8](#description_of_numerics){reference-type="ref" reference="description_of_numerics"} we present numerical evidence that it is indeed the case -- small flocks are not overly influenced by large far away flocks. ## Assumptions and Notation Before stating the main results, we will describe notational conventions and outline our main assumptions, which will hold throughout unless stated otherwise. The domain is assumed to be the torus (i.e. $\Omega = \mathbb{T}^n$). We will indicate if and how our results can be extended to the open space $\mathbb{R}^n$. The kernel $\phi$ is a smooth, positive, radial, decreasing function; the density and weight are non-negative functions (i.e. $\rho, \mathrm{w}\geqslant 0$). We will use $\partial^k_x$ and $\partial^k_t$ to denote the $k^{th}$ partial derivative with respect to time and space, respectively. In multi-D, we will at times use $\partial$ to denote an arbitrary spatial derivative (i.e. the partial derivative in an arbitrary coordinate direction). Since the kernel $\phi$ is a radial function, we will denote the derivatives by $\phi'$, $\phi''$, etc. We will use $c$'s to denote lower bounds. For instance, $c_0$, $c_1$, and $c_2$, will refer to lower bounds on the $\rho$, $\phi$, and $e$, respectively. To abbreviate maximum and minimum values of a function, we write $f_+ := \sup_{x \in \mathbb{T}^n} f(x)$ and $f_- = \inf_{x \in \mathbb{T}^n} f(x)$. As mentioned in the introduction, we abbreviate convolutions by $f_{\phi} := f \ast \phi$. Regarding function spaces, $H^m := H^m(\mathbb{T}^n)$ is the Sobolev space of functions whose first $m$ derivates lie in $L^2(\mathbb{T}^n)$. We will denote by $L^1_+$ the space of positive $L^1(\mathbb{T}^n)$ functions. Finally, $C_w([0,T]; X)$ denotes the space of weakly continuous functions with values in $X$ on the time interval $[0,T]$. ## The scope and main results {#intro:main_results} Let us now state the main results. It will be more convenient to develop regularity theory for the $\mathrm{s}$-model in the form [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"} treating $\mathrm{w}$ as an unknown. In section [4](#LWP){reference-type="ref" reference="LWP"}, local existence and continuation is proved in higher regularity Sobolev classes via a Banach Fixed Point argument for a viscous regularization of [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"}; the full result is then obtained by compactness arguments. Energy estimates are also established that give rise to to the continuation criterion. In particular, **Theorem 2**. *(Local well-posedness) [\[lwp\]]{#lwp label="lwp"} Given initial data $(\rho_0, \mathrm{w}_0, \textbf{u}_0) \in (H^k \cap L_+^1) \times H^l \times H^m$ with $l \geqslant m \geqslant k+1 \geqslant n/2 + 3$, and suppose $(\rho_0)_{\phi} \geqslant c_0 > 0$. Then there exists a time $T > 0$ and a unique solution $(\rho, \mathrm{w}, \textbf{u}) \in C_w([0,T]; (H^k \cap L_+^1) \times H^l \times H^m)$ to [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"} satisfying the initial condition and $\inf_{[0,T]} \rho_{\phi} >0$.* *Moreover, if $$\label{cont_criterion} \int_0^{T}\left( \|\nabla \textbf{u}\|_{\infty} + \Big\|\frac{1}{\rho_{\phi}} \Big\|_{\infty}^{l+1} \right) \, \mbox{d}s< \infty$$ then the solution can be continued beyond time $T$.* *Remark 3*. Provided, $\phi \geqslant c_1 > 0$, Theorem [\[lwp\]](#lwp){reference-type="ref" reference="lwp"} also holds in $\mathbb{R}^n$. The proof relies on a lower bound on $\rho_{\phi}$, which necessitates a lower bound on $\phi$ when $\rho \in L^1(\mathbb{R}^n)$. The continuation crieterion can be used to obtain a small data result. See Theorem [Theorem 14](#small_data){reference-type="ref" reference="small_data"} for the complete statement. **Theorem 4**. *(Global well-posedness for small initial data in multi-D) [\[thm:small_data_intro\]]{#thm:small_data_intro label="thm:small_data_intro"} Assume the kernel $\phi$ is bounded below, $\phi \geqslant c_1 > 0$. If in addition, the initial data satisfies the assumptions of Theorem [\[lwp\]](#lwp){reference-type="ref" reference="lwp"} and the initial velocity and initial variation of the velocity is small enough, then there is a unique solution $(\rho, \mathrm{w}, \textbf{u}) \in C_w([0,T]; (H^k \cap L^1_+) \times H^l \times H^m)$ to [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"} existing globally in time.* In 1D, having the additional conservation law [\[conservation_law\]](#conservation_law){reference-type="eqref" reference="conservation_law"} helps establish control over $\partial_x u$ first and then over decay rate of $\rho_\phi$ to achieve the same threshold criterion for global well-posedness as in the classical Cucker-Smale case. In fact, we extend this result to multi-dimensional unidirectional flocks introduced in [@LS-uni1] $$\label{e:uniintro} \textbf{u}(x,t) = u(x,t) \textbf{d}, \hspace{8mm} \textbf{d}\in \mathbb{S}^{n-1}, \hspace{2mm} \textbf{u}: \mathbb{T}^n \times \mathbb{R}^+ \to \mathbb{R}.$$ The kew feature of these solutions is possession of the same conservation law [\[conservation_law\]](#conservation_law){reference-type="eqref" reference="conservation_law"} $$e = \nabla u \cdot \textbf{d}+ \mathrm{s}$$ although in this case it does not give control over the full gradient of ${\bf u}$. In Section [7](#UniGWP){reference-type="ref" reference="UniGWP"}, we present a bootstrap argument that establishes full control provided the weight $\mathrm{w}$ is bounded above and below. See Theorem [Theorem 17](#t:GWP_1D){reference-type="ref" reference="t:GWP_1D"} and Theorem [Theorem 21](#t:uni){reference-type="ref" reference="t:uni"} for the complete statement of the 1D and multi-D cases, respectively. **Theorem 5**. *(Global well-posedness for unidirectional flocks) The unidirectional flock of the form [\[e:uniintro\]](#e:uniintro){reference-type="eqref" reference="e:uniintro"} starting from a smooth initial data gives rise to a unique global solution provided $e_0 \geqslant 0$.* Turning to long time behavior, we note that there is exponential $L^{\infty}$-based alignment when the kernel is bounded below, see Theorem [\[thm:linf_alignment\]](#thm:linf_alignment){reference-type="ref" reference="thm:linf_alignment"}. The proof is analogous to the Cucker-Smale case given by Ha and Liu in [@HL2009] so we don't include it as a main result; but it is an important one as it shows that the new model retains strong alignment characterstics. Additionally, $L^{\infty}$-based alignment will be used for the small data and strong flocking results. In Section [3.2](#local_alignment){reference-type="ref" reference="local_alignment"}, for local communication kernels, we show conditional alignment of the velocity in the $L^2$ sense (as opposed to the unconditional $L^2$-based alignment result in the Cucker-Smale case). Let $V_2(t) = \frac{1}{2} \int |\textbf{u}(t,x) - \bar{\textbf{u}}(t)|^2 \rho(t,x) dx$, where $\bar{\textbf{u}}(t) = \frac{1}{M} \int_{\mathbb{T}^n} \textbf{u}(t,x) \rho(t,x) dx$ is the average momentum at time $t$ and $M = \int_{\mathbb{T}^n} \rho_0(x) dx$ is the mass of the flock. **Theorem 6**. *(Alignment in $L^2$ under local communication). [\[thm:l2_alignment_intro\]]{#thm:l2_alignment_intro label="thm:l2_alignment_intro"} For smooth initial data and local kernels $\phi(x,y) \geqslant c_1 \mathbbm{1}_{|x-y|<r_0}$, there exists $c_1' := c_1'(r_0, c_1) > 0$ such that if the solution satisfies $$\begin{aligned} 0 < \frac{\mathrm{w}_+ - \mathrm{w}_-}{\mathrm{w}_-} \leqslant\frac{c_1'}{M \|\phi\|_{\infty}} \frac{\rho_-^2(t)}{\rho_+(t)}, \hspace{5mm} t \geqslant 0 \end{aligned}$$ Then there is exponential $V_2$-based alignment. In other words, there exists a constant $\delta > 0$ such that $$\begin{aligned} V_2(t) \leqslant V_2(0) e^{-\delta t} \end{aligned}$$* *Remark 7*. Non-integrability of $\frac{\rho_-^2}{\rho_+}$ is the key for alignment under local kernels. This was first observed by Tadmor in [@Tadmor-notices] in the case of the Cucker-Smale model. However, when $\mathrm{w}$ is non-constant, the contraint is a (more stringent) uniform lower bound on $\frac{\rho_-^2}{\rho_+}$ and thus non-integrability is automatic. Notably, $V_2$-based alignment in the Cucker-Smale case with the relaxed non-integrability assumption is not necessarily exponential. With alignment and a threshold condition for 1D global well-posedness in hand, the question arises as to whether the density converges to a limiting distribution (a.k.a strong flocking). In Section [6.1](#sec:strong_flocking){reference-type="ref" reference="sec:strong_flocking"}, we affirm this is the case in 1D, provided the entropy $e_0$ and the kernel $\phi$ both bounded away from zero. **Theorem 8**. *(Strong Flocking in 1D) [\[thm:strong_flocking_intro\]]{#thm:strong_flocking_intro label="thm:strong_flocking_intro"} Assume the kernel is bounded below, $\phi \geqslant c_1 > 0$ and $e_0 = \partial_x u_0 + \mathrm{w}_0 (\rho_0)_{\phi} \geqslant c_2 > 0$. for some constants $c_1$, $c_2$. If in addition, the initial data satisfies the assumptions of Theorem [\[lwp\]](#lwp){reference-type="ref" reference="lwp"}, then there exists a global in time solution with a limiting velocity $u_{\infty}$. In particular, there exists $\delta > 0$ such that $$\begin{aligned} |u - u_{\infty}| + |\partial_x u| + |\partial_x^2 u| \leqslant e^{-\delta t} \end{aligned}$$ As a consequence, there exists a limiting density distribution $\rho_{\infty}$ such that $$\begin{aligned} |\rho(t, \cdot) - \rho_{\infty}(\cdot)| \leqslant e^{-\delta t} \end{aligned}$$* It is not known, even in the Cucker-Smale case, what the limiting distribution $\rho_{\infty}$ looks like. However, the $L^1$ distance from the uniform distibrution $\bar{\rho} = \frac{M}{2\pi}$ can be estimated using relative entropy estimates. In Section [6.2](#sec:entropy){reference-type="ref" reference="sec:entropy"}, we establish the following theorem, which is an extension of the result by Leslie and Shvydkoy for the Cucker-Smale case established in [@LS-entropy]. **Theorem 9**. *(Deviation of limiting flock from uniform distribution) [\[thm:entropy_estimate_intro\]]{#thm:entropy_estimate_intro label="thm:entropy_estimate_intro"} Suppose the initial data is smooth, $\phi$ satisfies $\phi(r) \geqslant\mathds{1}_{r \leqslant r_0}$, and $e_0 = \partial_x(u_0) + \mathrm{w}_0 (\rho_0)_{\phi} \geqslant 0$. Let $\tilde{e} = \partial_x u + \mathcal{L}_{\phi} \rho$, where $\mathcal{L}_{\phi} \rho = \mathrm{w}(x) \int_{\mathbb{T}} (\rho(y) - \rho(x)) \phi(x-y) dy$ and $\tilde{q} = \frac{\tilde{e}}{\rho}$. If $\rho$ remains bounded and $\|\tilde{q}\|_{\infty} \leqslant\tilde{Q} < \mathrm{w}_+ \|\phi\|_{L^1}$ for some constant $\tilde{Q}$, then there exists a constant $c := c(r_0)$ such that $$\begin{aligned} \limsup_{t \to \infty} \|\rho(t) - \bar{\rho} \|_{L^1} &\leqslant\Big( \tilde{Q} + \|\phi\|_{\infty} (\mathrm{w}_+ - \mathrm{w}_-) \Big) \frac{M \mathrm{w}_+ \|\phi\|_{\infty}}{c(\mathrm{w}_+ \|\phi\|_{L^1} - \tilde{Q})} \\ \end{aligned}$$* *Remark 10*. In the Cucker-Smale case $\mathrm{w}= 1$, the quantity $q = e/\rho$ is transported. Consequently, $\tilde{Q} = \|q_0\|_{\infty}$ and we recover the result of Leslie and Shvydkoy in [@LS-entropy]. They observed that the bounding expression is linear in $\|q_0\|_{\infty}$ for small values of $\|q_0\|_{\infty}$, showing that small initial values for $q_0$ lead to close to uniform distributions of the flock for large times. In the case presented here, where $\mathrm{w}$ is not necessarily equal to $1$, we pick up an additional term with linear dependence on $(\mathrm{w}_+ - \mathrm{w}_-)$. In particular, to obtain close to uniform distrubtions of the flock for large times, one requires both smallness $\tilde{Q}$ and smallness of $(\mathrm{w}_+ - \mathrm{w}_-)$. See Remark [Remark 19](#rmk:bd_on_q_tilde){reference-type="ref" reference="rmk:bd_on_q_tilde"} for smallness conditions on $\tilde{Q}$. ## Outline of Paper The paper will be organized as follows. In section [2](#Properties){reference-type="ref" reference="Properties"}, we will discuss the basic properties of the $\mathrm{s}$-model-- namely, the maximum principle and the lack of both momentum conservation and an energy law--and compare these properties to that of the Cucker-Smale and Motsch-Tadmor models. In section [3](#alignment){reference-type="ref" reference="alignment"}, we record the $L^{\infty}$-based exponential alignment result and establish the conditional $L^2$-based alignment result under a local communication kernel, Theorem [\[thm:l2_alignment_intro\]](#thm:l2_alignment_intro){reference-type="ref" reference="thm:l2_alignment_intro"}. In section [4](#LWP){reference-type="ref" reference="LWP"}, local in time well-posedness along with the continuation criterion is established, Theorem [\[lwp\]](#lwp){reference-type="ref" reference="lwp"}. In section [5](#GWP_1D){reference-type="ref" reference="GWP_1D"}, 1D global well-posedness is established under the threshold criterion: $e_0 = \partial_x u_0 + \mathrm{w}_0(\rho_0)_{\phi} \geqslant 0$, i.e. the 1D version of Theorem [Theorem 5](#thm:UniGWP){reference-type="ref" reference="thm:UniGWP"}. In section [6](#sec:limiting_density_profile){reference-type="ref" reference="sec:limiting_density_profile"}, we establish, in 1D, strong flocking and estimate the deviation of the limiting density from the uniform distribution via relative entropy estimates, Theorem [\[thm:strong_flocking_intro\]](#thm:strong_flocking_intro){reference-type="ref" reference="thm:strong_flocking_intro"} and Theorem [\[thm:entropy_estimate_intro\]](#thm:entropy_estimate_intro){reference-type="ref" reference="thm:entropy_estimate_intro"}. In section [7](#UniGWP){reference-type="ref" reference="UniGWP"}, the 1D global well-posedness argument is extended to uni-directional flocks in multiple dimensions, i.e. the full version of Theorem [Theorem 5](#thm:UniGWP){reference-type="ref" reference="thm:UniGWP"}. In Section [8](#description_of_numerics){reference-type="ref" reference="description_of_numerics"}, we provide a comparison of numerical solutions to the $\mathrm{s}$-model, Motsch-Tadmor model, and Cucker-Smale model and a description of the numerical method. The numerics illustrate that when $\mathrm{w}_0 = 1/\rho_{\phi}$, the $\mathrm{s}$-model displays similar qualitative behavior to the Motsch-Tadmor model, see tables [2](#plots:solns_CS){reference-type="ref" reference="plots:solns_CS"}, [3](#plots:solns_WM_MT){reference-type="ref" reference="plots:solns_WM_MT"}, [4](#plots:solns_MT){reference-type="ref" reference="plots:solns_MT"}. Convergence plots as the mesh coarsity approach zero are also included in order to validify the numerical method, see table [5](#plots:vary_k_and_h){reference-type="ref" reference="plots:vary_k_and_h"}. Lastly, the Appendix contains some of the technical estimates used throughout the paper. # Properties of $\mathrm{s}$-model {#Properties} The alignment force in [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"} has a similar structure to the alignment force in [\[EAS_CS\]](#EAS_CS){reference-type="eqref" reference="EAS_CS"} and thus inherits similar features. For instance, it inherits the maximum principle for the velocity (i.e. $\|\textbf{u}\|_{\infty} \leqslant\|\textbf{u}_0\|_{\infty}$) which is crucial for alignment. Nonetheless, there are some key differences. The presence of the weight $\mathrm{w}$ destroys the symmetry of the communication strength, as in the Motsch-Tadmor case. That is to say, the force exerted by particle 'x' on particle 'y' may not be equal to the force exerted by particle 'y' on particle 'x'. As a consequence, there is no conservation of momentum nor is there a dissapative energy law. Let us illustrate how the weight $\mathrm{w}$ obstructs these laws. The momemtum $P$ and the energy $\mathcal{E}$ are given by $$\begin{aligned} P = \int_{\mathbb{T}} \rho \textbf{u}dx, \hspace{5mm} \mathcal{E}= \frac{1}{2} \int_{\mathbb{T}} |\textbf{u}|^2 \rho dx \end{aligned}$$ For the momentum, we have $$\begin{aligned} \frac{d}{dt} P &= - \int_{\mathbb{T}} \textbf{u}\nabla \cdot (\textbf{u}\rho) dx + \int -\rho \textbf{u}\cdot \nabla \textbf{u}+ \rho \mathrm{w}((\textbf{u}\rho)_{\phi} - \textbf{u}\rho_{\phi}) \hspace{1mm} dx \\ &= - \int_{\mathbb{T}} div(\rho |\textbf{u}|^2) + \int \rho \mathrm{w}((\textbf{u}\rho)_{\phi} - \textbf{u}\rho_{\phi}) \hspace{1mm} dx \\ &= \int_{\mathbb{T}^2} \mathrm{w}(x) (\textbf{u}(y) - \textbf{u}(x)) \phi(x-y) \rho(x) \rho(y) \hspace{1mm} dy dx \\\end{aligned}$$ Symmetrizing, $$\begin{aligned} &= -\int_{\mathbb{T}^2} \mathrm{w}(y) (\textbf{u}(y) - \textbf{u}(x)) \phi(x-y) \rho(x) \rho(y) \hspace{1mm} dy dx \\\end{aligned}$$ and averaging the last two lines, we obtain $$\begin{aligned} \label{momemtum_time_deriv} \frac{d}{dt} P &= -\frac{1}{2} \int_{\mathbb{T}^2} (\textbf{u}(x) - \textbf{u}(y)) (\mathrm{w}(x) - \mathrm{w}(y)) \phi(x-y) \rho(x) \rho(y) \hspace{1mm} dy dx \end{aligned}$$ With the presence of a non-constant $\mathrm{w}$, we cannot conclude that this is equal to zero. A similar computation shows that the same problem persists for the energy. $$\begin{aligned} \frac{d}{dt} \mathcal{E} &= \int_{\mathbb{T}} \rho \textbf{u}\mathrm{w}\cdot (\phi * (\rho \textbf{u}) - \textbf{u}\phi * \rho) \hspace{1mm} dx \\ &= \int_{\mathbb{T}} \int_{\mathbb{T}} \textbf{u}(x) \mathrm{w}(x) \cdot (\textbf{u}(y) - \textbf{u}(x)) \phi(x-y) \rho(x) \rho(y) \hspace{1mm} dy dx \\ &= -\int_{\mathbb{T}} \int_{\mathbb{T}} \textbf{u}(y) \mathrm{w}(y) \cdot (\textbf{u}(y) - \textbf{u}(x)) \phi(x-y) \rho(x) \rho(y) \hspace{1mm} dy dx \\\end{aligned}$$ Averaging the last two lines, we get $$\begin{aligned} &= -\frac{1}{2} \int_{\mathbb{T}} \int_{\mathbb{T}} \big( \textbf{u}(x) \mathrm{w}(x) - \textbf{u}(y) \mathrm{w}(y) \big) \cdot \big(\textbf{u}(y) - \textbf{u}(x) \big) \phi(x-y) \rho(x) \rho(y) \hspace{1mm} dy dx \\\end{aligned}$$ The presence of the weight $\mathrm{w}$ prevents us from guarenteeing pure dissipation. However, we can still decompose the law into a dissapative term and an extra term. We have $$\begin{aligned} \label{energy_time_deriv} \frac{d}{dt} \mathcal{E}&= -\frac{1}{2} \int_{\Omega^2} \mathrm{w}(x) |\textbf{u}(x)-\textbf{u}(y)|^2 \phi(x-y) \rho(x) \rho(y) dx dy \\ &-\frac{1}{2} \int_{\Omega} \textbf{u}(y) (\textbf{u}(x) - \textbf{u}(y)) (\mathrm{w}(x) - \mathrm{w}(y)) \phi(x-y) \rho(x) \rho(y) dx dy \nonumber\end{aligned}$$ At first, the lack of a dissapative energy law looks to pose an obstruction for alignment since energy decay is connected to $L^2$-based alignment. However, we are placated by the following two facts: (i) The maximum principle still holds for the velocity equation. In particular, $\|\textbf{u}\|_{\infty} \leqslant\|\textbf{u}_0\|_{\infty}$. As a result, the $L^{\infty}$-based alignment results can be established. See Section [3.1](#linf_alignment){reference-type="ref" reference="linf_alignment"} for the precise statement. (ii) A dissapative energy law can be recovered provided there are constraints on $\mathrm{w}$ and $\rho$, which allow the second term to absorbed into the dissapative term. With an energy law at hand, we recover a conditional alignment result in the case of local kernels. The details are discussed in Section [3.2](#local_alignment){reference-type="ref" reference="local_alignment"}. We summarize the differences between [\[EAS_CS\]](#EAS_CS){reference-type="eqref" reference="EAS_CS"}, [\[EAS_MT\]](#EAS_MT){reference-type="eqref" reference="EAS_MT"}, and [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"} in table [\[table:compare_models\]](#table:compare_models){reference-type="ref" reference="table:compare_models"} below. Entropy Law Performs well in Heterogeneous Formations Conservation of Momentum Energy Dissapation ---- ------------- ------------------------------------------- -------------------------- -------------------- CS MT WM [\[table:compare_models\]]{#table:compare_models label="table:compare_models"} # Alignment ## Alignment in $L^{\infty}$ {#linf_alignment} The alignment $L^{\infty}$-based alignment result follows a similar Lyapunov-based approach to the Cucker-Smale case, given by Ha and Liu in [@HL2009]. The presence of the weight does not introduce any difficulties as long as it's bounded away from zero. Indeed, the Lagrangian formulation of the velocity equation of [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"} is given by $$\begin{cases} \dot{X}(t, \alpha) = \textbf{V}(t, \alpha) \\ \dot{\textbf{V}}(\alpha, t) = \mathrm{w}(t, X(t,\alpha)) \int_{\mathbb{T}^n} \phi(X(t, \alpha) - X(t, \beta)) (\textbf{V}(t,\alpha) - \textbf{V}(t, \beta)) \rho_0(\beta) d\beta \end{cases}$$ We now state the alignment theorem without proof. Let $\mathcal{A}(t) = \max_{\alpha, \beta \in \mathbb{T}^n} |\textbf{V}(\alpha,t) - \textbf{V}(\beta,t)|$. **Theorem 11**. *(Alignment and Flocking) [\[thm:linf_alignment\]]{#thm:linf_alignment label="thm:linf_alignment"} If $\phi$ is bounded below, $\phi \geqslant c_1 > 0$, then $$\begin{aligned} \mathcal{A}(t) \leqslant\mathcal{A}_0 e^{-\mathrm{w}_- M c_1 t} \end{aligned}$$* *Remark 12*. When $\Omega = \mathbb{R}^n$, the Theorem [\[thm:linf_alignment\]](#thm:linf_alignment){reference-type="ref" reference="thm:linf_alignment"} also holds. In addition, the diameter $\mathcal{D}(t) = \max_{\alpha, \beta \in \mathbb{T}^n} |X(\alpha,t) - X(\beta,t)|$ of the flock remains bounded. Flocking then reduces to showing global well-posedness. ## Alignment in $L^2$ {#local_alignment} We now turn to the case of local kernels and the proof of Theorem [\[thm:l2_alignment_intro\]](#thm:l2_alignment_intro){reference-type="ref" reference="thm:l2_alignment_intro"}. For local kernels, the communication strength vanishes for agents that are more than a distance $r_0$ apart. In other words, $\phi(x,y) \geqslant c_1 \mathbbm{1}_{|x-y|<r_0}$. In this case, the $L^{\infty}$-based arguments in Section [3.1](#linf_alignment){reference-type="ref" reference="linf_alignment"} fail due to the lack of a lower bound on the alignment force. However, there is alignment in $L^2$ provided there is energy dissapation. Dissapation can be achieved provided $0 < \frac{\mathrm{w}_+ - \mathrm{w}_-}{\mathrm{w}_-} \leqslant\frac{c_1 c_1'}{M \|\phi\|_{\infty}} \frac{\rho_-^2}{\rho_+}$, as shown in the following proof. *Proof of Theorem [\[thm:l2_alignment_intro\]](#thm:l2_alignment_intro){reference-type="ref" reference="thm:l2_alignment_intro"}.* Let $\mathcal{E}(t)$ and $P(t)$ be the energy and momentum as defined in Section [2](#Properties){reference-type="ref" reference="Properties"}. Observe that $V_2 = \mathcal{E}(t) - \frac{1}{2M} P(t)^2$. Using this along with the momentum and energy equations [\[momemtum_time_deriv\]](#momemtum_time_deriv){reference-type="eqref" reference="momemtum_time_deriv"} and [\[energy_time_deriv\]](#energy_time_deriv){reference-type="eqref" reference="energy_time_deriv"}, we obtain $$\begin{aligned} \frac{d}{dt} V_2 &= \frac{d}{dt} \mathcal{E}- \frac{1}{M} P \frac{d}{dt} P \\ &= -\frac{1}{2} \int_{\mathbb{T}^{2n}} \mathrm{w}(x) |\textbf{u}(x)-\textbf{u}(y)|^2 \phi(x-y) \rho(x)\rho(y) dx dy \\ &- \frac{1}{2} \int_{\mathbb{T}^{2n}} \textbf{u}(y) (\textbf{u}(x)-\textbf{u}(y)) (\mathrm{w}(x)-\mathrm{w}(y)) \phi(x-y) \rho(x) \rho(y) dx dy \\ &+ \frac{1}{2M} \int_{\mathbb{T}^n} \textbf{u}(x) \rho(x) dx \int_{\mathbb{T}^{2n}} (\textbf{u}(x)-\textbf{u}(y)) (\mathrm{w}(x)-\mathrm{w}(y)) \phi(x-y) \rho(x)\rho(y) dx dy \\ \end{aligned}$$ With the goal relating it back to $V_2$, we write it as $$\begin{aligned} &= -\frac{1}{2} \int_{\mathbb{T}^{2n}} \mathrm{w}(x) |\textbf{u}(x)-\textbf{u}(y)|^2 \phi(x-y) \rho(x)\rho(y) dx dy \\ &- \frac{1}{2} \int_{\mathbb{T}^{2n}} (\textbf{u}(y) - \bar{\textbf{u}}) (\textbf{u}(x) - \bar{\textbf{u}} - (\textbf{u}(y) - \bar{\textbf{u}})) (\mathrm{w}(x)-\mathrm{w}(y)) \phi(x-y) \rho(x) \rho(y) dx dy \\ &:= -I_1 + I_2 \end{aligned}$$ The dissapation term $I_1$ can be bounded from below by $$\begin{aligned} I_1 &\geqslant c_0 \mathrm{w}_- \rho_-^2 \int_{|x-y| \leqslant r_0} |\textbf{u}(x) - \textbf{u}(y)|^2 dx dy \\ \end{aligned}$$ Let $Ave(\textbf{u}) = \frac{1}{(2\pi)^n} \int_{\mathbb{T}^n} \textbf{u}(x) dx$. Using Lemma 2.1 of [@LS-entropy] we obtain for some constant $c_1' := c_1'(r_0, c_1)$, $$\begin{aligned} &\geqslant c_1' \mathrm{w}_- \rho_-^2 \frac{1}{2} \int_{\mathbb{T}^n} |\textbf{u}(x) - Ave(\textbf{u})|^2 dx \\ \end{aligned}$$ And from the idendity $\int_{\mathbb{T}^n} |\textbf{u}(x) - Ave(\textbf{u})|^2 \rho(x) dx = \int_{\mathbb{T}^n} |\textbf{u}(x) - \bar{\textbf{u}}|^2 \rho(x) dx + \int_{\mathbb{T}^n} |Ave(\textbf{u}) - \bar{\textbf{u}}|^2 \rho(x) dx$, $$\begin{aligned} &\geqslant c_1' \mathrm{w}_- \frac{\rho_-^2}{\rho_+} \frac{1}{2} \int_{\mathbb{T}^n} |\textbf{u}(x) - \bar{\textbf{u}}|^2 \rho(x) dx \\ &= c_1' \mathrm{w}_- \frac{\rho_-^2}{\rho_+} V_2 \\ \end{aligned}$$ In $I_2$, the first term vanishes due to symmetrization. So we are left with $$\begin{aligned} |I_2| &\leqslant\frac{1}{2} \int_{\mathbb{T}^{2n}} |\textbf{u}(y) - \bar{\textbf{u}}|^2 |\mathrm{w}(x)-\mathrm{w}(y)| \phi(x-y) \rho(x) \rho(y) dx dy \\ &\leqslant\frac{1}{2} (\mathrm{w}_+ - \mathrm{w}_-) \|\phi\|_{\infty} \int_{\mathbb{T}^2} |\textbf{u}(y) - \bar{\textbf{u}}|^2 \rho(x) \rho(y) dx dy \\ &= \frac{M}{2} (\mathrm{w}_+ - \mathrm{w}_-) \|\phi\|_{\infty} V_2 \\ \end{aligned}$$ To absorb $I_2$ into the dissapative term $I_1$, we require that $$\begin{aligned} \frac{M}{2} (\mathrm{w}_+ - \mathrm{w}_-) \|\phi\|_{\infty} \leqslant\frac{1}{2} c_1' \mathrm{w}_- \frac{\rho_-^2}{\rho_+} \end{aligned}$$ which is equivalent to the contraint given in the theorem. Under this constraint, we have $$\begin{aligned} \frac{d}{dt} V_2 \leqslant- \frac{1}{2} c_1' \mathrm{w}_- \frac{\rho_-^2}{\rho_+} V_2 \end{aligned}$$ Owing to the uniform lower bound on $\frac{\rho_-^2}{\rho_+}$, we obtain exponential decay of $V_2$. ◻ # Local Well-Posedness {#LWP} In this section, we will prove Theorem [\[lwp\]](#lwp){reference-type="ref" reference="lwp"} and establish global well-posedness for small initial data, Theorem [\[thm:small_data_intro\]](#thm:small_data_intro){reference-type="ref" reference="thm:small_data_intro"}. ## Local well-posedness for a viscous regularization The strategy is to obtain a local well-posedness result for a viscous regularizion of [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"} given by $$\begin{aligned} \label{regularized_EAS_WM} \tag{WM'} \begin{cases} \partial_t \rho + \nabla \cdot (\textbf{u}\rho) = \epsilon \Delta \rho \\ \partial_t \mathrm{w}+ \textbf{u}_F \cdot \nabla \mathrm{w}= \epsilon \Delta \mathrm{w}\\ \partial_t \textbf{u}+ \textbf{u}\cdot \nabla \textbf{u}= \mathrm{w}((\textbf{u}\rho)_{\phi} - \textbf{u}\rho_{\phi}) + \epsilon \Delta \textbf{u} \end{cases}\end{aligned}$$ Then we will show that the time of existence for the regularized equation [\[regularized_EAS_WM\]](#regularized_EAS_WM){reference-type="eqref" reference="regularized_EAS_WM"} does not depend on $\epsilon$ and moreover that there's a subsequence of solutions converging to solutions to [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"}. The proof of local well-posedness of [\[regularized_EAS_WM\]](#regularized_EAS_WM){reference-type="eqref" reference="regularized_EAS_WM"} uses the standard Banach Fixed Point argument. Let $Z(t,x) = (\rho(t,x), \mathrm{w}(t,x), \textbf{u}(t,x))$ and let $\mathcal{N}(Z)$ denote the non-linear terms in [\[regularized_EAS_WM\]](#regularized_EAS_WM){reference-type="eqref" reference="regularized_EAS_WM"}. Define the map $$\begin{aligned} \label{duhamel_formula} \mathcal{F}[Z](t) = e^{\epsilon t \Delta} Z_0 + \int_0^t e^{\epsilon (t-s) \Delta} \mathcal{N}(Z(s)) ds \end{aligned}$$ Let $r = c_0 / (2 \|\phi\|_{L^{\infty}})$. We will show that there exists a small time $T$ so that this map is a contraction mapping on $C([0,T); B_{r}(Z_0))$ where $B_{r}(Z_0)$ denotes the the ball of radius $r$ in $X = (H^k \cap L^1_+) \times H^l \times H^m$ centered at $Z_0$. The choice of the $r$ guarentees that $$\begin{aligned} \|\rho_{\phi}\|_{\infty} \geqslant c_0 - \|(\rho - \rho_0)_{\phi}\|_{\infty} \geqslant c_0 - \|\rho - \rho_0\|_{L^1} \|{\phi}\|_{L^{\infty}} \geqslant c_0/2 \end{aligned}$$ so the lower bound $\|\rho_{\phi}\|_{\infty} \geqslant c_0/2$ automatically holds. Invariance and contractivity of the map $\mathcal{F}$ will be obtained from estimates on $\|\int_0^t e^{\epsilon(t-s)\Delta} \mathcal{N}(Z(s)) ds\|_X$. In the estimates below, we will use the following notation. If $U$ and $V$ are quantities, then $U \lesssim V$ is equivalent to $U \leqslant C(n, c_0, l, M, \phi, \epsilon) V$, where $M$ is the mass. Importantly, the constant $C$ does not depend on the time $T$. The non-linear estimates on derivatives from the Appendix [10](#appendix){reference-type="ref" reference="appendix"} will be used in order to estimate Sobolev norms of the Favre Filtration. We now proceed to show invariance of the map $\mathcal{F}$ by showing that $$\begin{aligned} \|\mathcal{F}(Z(s)) - Z_0\|_X \leqslant \Big\| e^{\epsilon t \Delta} Z_0 - Z_0 \Big\|_X + \Big\| \int_0^t e^{\epsilon (t-s) \Delta} \mathcal{N}(Z(s)) ds \Big\|_X \leqslant 1\end{aligned}$$ provided $T$ is small. The first term can be made small by the continuity property of the heat semigroup. Regarding the second term, the $\rho$-equation has been estimated in [@Sbook]. For the $\mathrm{w}$ equation, we use the Analyticity property of the heat equation and $l \geqslant n/2+2$ to get $$\begin{aligned} \Big\| \partial^l &\int_0^t e^{\epsilon(t-s) \Delta} \textbf{u}_F \cdot \nabla \mathrm{w}ds \Big\|_2 \leqslant\int_0^t \frac{1}{\sqrt{\epsilon(t-s)}} \big\| \partial^{l-1} ( \textbf{u}_F \cdot \nabla \mathrm{w}) \big\|_2 ds \\ &\leqslant\frac{T^{1/2}}{\epsilon^{1/2}} \Big( \|\nabla \mathrm{w}\|_{\infty} \|\textbf{u}_F\|_{\dot{H}^{l-1}} + \| \mathrm{w}\|_{\dot{H}^l} \|\textbf{u}_F\|_{\infty} \Big) \\\end{aligned}$$ Using [\[favre_estimate\]](#favre_estimate){reference-type="ref" reference="favre_estimate"} to estimate $\|\textbf{u}_F\|_{\dot{H}^{l-1}}$, we get $$\begin{aligned} &\lesssim \frac{T^{1/2}}{\epsilon^{1/2}} \Big( \|\nabla \mathrm{w}\|_{\infty} \Big( \Big\| \frac{1}{\rho_{\phi}} \Big\|_{\infty} \|(\textbf{u}\rho)_{\phi} \|_{H^{l-1}} + \Big\| \frac{1}{\rho_{\phi}} \Big\|_{\infty}^{l} \| \rho_{\phi} \|_{H^{l-1}} \|(\textbf{u}\rho)_{\phi} \|_{\infty} \Big) + \| \mathrm{w}\|_{\dot{H}^{l}} \|\textbf{u}\|_{\infty} \Big) \\ &\lesssim \frac{T^{1/2}}{\epsilon^{1/2}} \Big( \|\nabla \mathrm{w}\|_{\infty} \|\textbf{u}\|_{\infty} + \| \mathrm{w}\|_{\dot{H}^{l}} \|\textbf{u}\|_{\infty} \Big) \\ &\lesssim \frac{T^{1/2}}{\epsilon^{1/2}} \| \mathrm{w}\|_{\dot{H}^{l}} \|\textbf{u}\|_{\infty} \\\end{aligned}$$ This quantity is small for small $T$. We only needed $l \geqslant n/2 + 1$ here, but the energy estimates later will impose a more strict conditon on the exponent. For the velocity equation, the transport term is estimated in [@Sbook]. It remains to estimate the alignment term. $$\begin{aligned} \Big\| \partial^m &\int_0^t e^{\epsilon(t-s) \Delta} \mathrm{w}\big( (\textbf{u}\rho)_{\phi} - \textbf{u}\rho_{\phi} \big) ds \Big\|_2 \leqslant\int_0^t \frac{1}{\sqrt{\epsilon(t-s)}} \big\| \partial^{m-1} \big( \mathrm{w}\big( (\textbf{u}\rho)_{\phi} - \textbf{u}\rho_{\phi} \big) \big\|_2 ds \\ &\leqslant\frac{T^{1/2}}{\epsilon^{1/2}} \Big( \| \mathrm{w}\|_{\infty} \|(\textbf{u}\rho)_{\phi} - \textbf{u}\rho_{\phi}\|_{\dot{H}^{m-1}} + \| \mathrm{w}\|_{\dot{H}^{m-1}} \|(\textbf{u}\rho)_{\phi} - \textbf{u}\rho_{\phi}\|_{\infty} \Big) \\ &\lesssim \frac{T^{1/2}}{\epsilon^{1/2}} \Big( \| \mathrm{w}\|_{\infty} \|\textbf{u}\|_{\dot{H}^{m-1}} + \| \mathrm{w}\|_{\dot{H}^{m-1}} \|\textbf{u}\|_{\infty} \Big) \\ \end{aligned}$$ This expression is small for small $T$. To show that $\mathcal{F}$ is a contraction, we will to show for $Z_1, Z_2 \in C([0,T]; B_r(Z_0))$ that $$\begin{aligned} \| \mathcal{F}(Z_1) - \mathcal{F}(Z_2) \|_{C([0,T]; B_{r}(Z_0))} \leqslant\alpha \|Z_1 - Z_2\|_{C([0,T]; B_{r}(Z_0))}\end{aligned}$$ for some $0 < \alpha < 1$ where $\|f\|_{C([0,T]; B_{r}(Z_0))} = \sup_{0 \leqslant s \leqslant T} \|f(s)\|_X$. In the following contractivity estimates, $C = C(n, c_0, l, M_1, M_2, \epsilon, \phi)$ is a constant where $M_1$, $M_2$ dentoes the mass of the two respective solutions. The constant $C$ may change in each line; and $[\textbf{u}_i]_{\rho_j} := (\textbf{u}_i \rho_j)_{\phi}/(\rho_j)_{\phi}$ denotes the Favre averaging associated to $\textbf{u}_i$, $\rho_j$. For the $\mathrm{w}$-equation, using the algebra property of the $H^l$ norm, we obtain $$\begin{aligned} \Big\| \partial^l &\int_0^t e^{\epsilon(t-s) \Delta} \Big( [\textbf{u}_1]_{\rho_1} \cdot \nabla \mathrm{w}_1 - [\textbf{u}_2]_{\rho_2} \cdot \nabla \mathrm{w}_2 \Big) ds \Big\|_2 \\ &\lesssim \Big\| \partial^l \int_0^t e^{\epsilon(t-s) \Delta} \Big( \big( [\textbf{u}_1]_{\rho_1} - [\textbf{u}_1]_{\rho_2} \big) \cdot \nabla \mathrm{w}_1 + [\textbf{u}_1 - \textbf{u}_2]_{\rho_2} \cdot \ \nabla \mathrm{w}_1 + [\textbf{u}_2]_{\rho_2} \cdot \nabla \big(\mathrm{w}_1 - \mathrm{w}_2 \big) \Big) ds \Big\|_2 \\ &\lesssim \frac{T^{1/2}}{\epsilon^{1/2}} \Big\| \big( [\textbf{u}_1]_{\rho_1} - [\textbf{u}_1]_{\rho_2} \big) \cdot \nabla \mathrm{w}_1 + [\textbf{u}_1 - \textbf{u}_2]_{\rho_2} \cdot \ \nabla \mathrm{w}_1 + [\textbf{u}_2]_{\rho_2} \cdot \nabla \big(\mathrm{w}_1 - \mathrm{w}_2 \big) \Big\|_{\dot{H}^{l-1}} \\ &\lesssim \frac{T^{1/2}}{\epsilon^{1/2}} \Big( \| [\textbf{u}_1]_{\rho_1} - [\textbf{u}_1]_{\rho_2} \|_{\dot{H}^{l-1}} \| \mathrm{w}_1 \|_{\dot{H}^{l}} \ + \| [\textbf{u}_1 - \textbf{u}_2]_{\rho_2} \|_{\dot{H}^{l-1}} \cdot \| \mathrm{w}_1\|_{\dot{H}^{l}} \\ &+ \| [\textbf{u}_2]_{\rho_2}\|_{\dot{H}^{l-1}} \| \mathrm{w}_1 - \mathrm{w}_2 \|_{\dot{H}^{l}} \Big) \\\end{aligned}$$ By [\[favre_estimate\]](#favre_estimate){reference-type="ref" reference="favre_estimate"}, we have $$\begin{aligned} \|[\textbf{u}_2]_{\rho_2}\|_{\dot{H}^{l-1}} &\leqslant\Big\| \frac{1}{{\rho_2}_{\phi}} \Big\|_{\infty} \|(\textbf{u}_2 \rho_2)_{\phi} \|_{H^{l-1}} + \Big\| \frac{1}{{\rho_2}_{\phi}} \Big\|_{\infty}^{l} \| {\rho_2}_{\phi} \|_{H^{l-1}} \|(\textbf{u}_2 \rho_2)_{\phi} \|_{\infty} \\ &\leqslant C \|\textbf{u}_2\|_{\infty} \end{aligned}$$ And by [\[contractivity_estimate\]](#contractivity_estimate){reference-type="ref" reference="contractivity_estimate"}, we have $$\begin{aligned} \|[\textbf{u}_1]_{\rho_1} - [\textbf{u}_1]_{\rho_2}\|_{\dot{H}^{l}} &\leqslant\Big\| \frac{1}{{\rho_1}_{\phi} {\rho_2}_{\phi}} \Big\|_{\infty} \| (\textbf{u}_1 \rho_1)_{\phi} ({\rho_2})_{\phi} - (\textbf{u}_1 \rho_2)_{\phi} ({\rho_1})_{\phi} \|_{H^l} \\ &+ \Big\| \frac{1}{{\rho_1}_{\phi} {\rho_2}_{\phi}} \Big\|_{\infty}^{l+1} \| {\rho_1}_{\phi} {\rho_2}_{\phi} \|_{H^l} \| (\textbf{u}_1 \rho_1)_{\phi} ({\rho_2})_{\phi} - (\textbf{u}_1 \rho_2)_{\phi} ({\rho_1})_{\phi} \|_{\infty} \\ &\leqslant C \| (\textbf{u}_1 \rho_1)_{\phi} ({\rho_2})_{\phi} - (\textbf{u}_1 \rho_2)_{\phi} ({\rho_1})_{\phi} \|_{H^l} \\ &= C \| ({\textbf{u}_1 \rho_1- \textbf{u}_1\rho_2})_{\phi} ({\rho_2})_{\phi} + (\textbf{u}_1 {\rho_2})_{\phi} ({\rho_2 - \rho_1})_{\phi} \|_{H^l} \\ &\leqslant C \|\textbf{u}_1\|_{\infty} \| Z_1 - Z_2 \|_X \\\end{aligned}$$ All together, we obtain $$\begin{aligned} \Big\| \partial^l &\int_0^t e^{\epsilon(t-s) \Delta} \Big[ [\textbf{u}_1]_{\rho_1} \cdot \nabla \mathrm{w}_1 - [\textbf{u}_2]_{\rho_2} \cdot \nabla \mathrm{w}_2 \Big] ds \Big\|_2 \\ &\leqslant C \frac{T^{1/2}}{\epsilon^{1/2}} \Big[ \| Z_1\|_X \|Z_1 - Z_2\|_{X} + \|Z_2\|_X \| Z_1 - Z_2 \|_X \Big] \\ &\leqslant C \frac{T^{1/2}}{\epsilon^{1/2}} \|Z_0 + r\|_X \| Z_1 - Z_2 \|_X \\\end{aligned}$$ The contractivity for the $\rho$ and $\textbf{u}$-equations can be estimated similarly. This shows $F$ is a contraction mapping for some small enough time of existence, $T$. ## Time of existence is independent of $\epsilon$ and energy estimates {#energy_estimates} We will now show that $T$ does not depend on $\epsilon$ by obtaining $\epsilon$-independent energy estimates on the norm of the solution $$\begin{aligned} Y_{m,l,k} = \|\textbf{u}\|_{H^m}^2 + \| \mathrm{w}\|_{H^l}^2 + \|\rho\|_{H^k}^2 + \|\rho\|_1^2\end{aligned}$$ The mass $\|\rho\|_1$ is conserved so it remains to control the other terms. The $\epsilon$-independent energy estimate for the $\rho$ equation is given in [@Sbook]. We record it here. Provided $m \geqslant k+1$, $$\begin{aligned} \frac{d}{dt} \|\rho\|_{\dot{H}^k}^2 \leqslant C (\|\nabla \textbf{u}\|_{\infty} + \|\nabla \rho\|_{\infty} + \|\rho\|_{\infty}) Y_{m,l,k}\end{aligned}$$ For the purpose of obtaining a continuation criterion in Section [4.4](#sec_cont_criterion){reference-type="ref" reference="sec_cont_criterion"}, we will explicitly include the dependence of the energy estimates on $\|1/\rho_{\phi}\|_{\infty}$ instead of absorbing it into the implied constant. Testing the $\mathrm{w}$-equation with $\partial^{2l} \mathrm{w}$, we get $$\begin{aligned} \frac{d}{dt} \| \mathrm{w}\|_{\dot{H}^l}^2 = \int_{\mathbb{T}^n} \textbf{u}_F \cdot (\nabla \mathrm{w}) \partial^{2l} \mathrm{w}dx - \epsilon \| \mathrm{w}\|_{\dot{H}^{l+1}}^2\end{aligned}$$ Integrating by parts and using the commutator estimate and [\[favre_estimate\]](#favre_estimate){reference-type="ref" reference="favre_estimate"}, we get $$\begin{aligned} \frac{d}{dt} \| \mathrm{w}\|_{\dot{H}^l}^2 &= \frac{1}{2} \int_{\mathbb{T}^n} \nabla \cdot \textbf{u}_F |\partial^l \mathrm{w}|^2 dx - \int_{\mathbb{T}^n} \Big( \partial^l ( \textbf{u}_F \cdot \nabla \mathrm{w}) - \textbf{u}_F \cdot \nabla \partial^l \mathrm{w}\Big) \partial^l \mathrm{w}dx - \epsilon \| \mathrm{w}\|_{\dot{H}^{l+1}}^2 \\ &\lesssim \|\nabla \textbf{u}_F\|_{\infty} \| \mathrm{w}\|_{\dot{H}^l}^2 + \big( \|\textbf{u}_F\|_{\dot{H}^l} \|\nabla \mathrm{w}\|_{\infty} + \|\nabla \textbf{u}_F\|_{\infty} \| \mathrm{w}\|_{\dot{H}^l} \big) \| \mathrm{w}\|_{\dot{H}^l} \\ &\lesssim \Big\| \frac{1}{\rho_{\phi}} \Big\|_{\infty}^2 \|\textbf{u}\|_{\infty} \| \mathrm{w}\|_{\dot{H}^l}^2 + \Big( \Big\| \frac{1}{\rho_{\phi}} \Big\|_{\infty}^{l+1} \|\textbf{u}\|_{\infty} \|\nabla \mathrm{w}\|_{\infty} + \Big\| \frac{1}{\rho_{\phi}} \Big\|_{\infty}^2 \|\textbf{u}\|_{\infty} \Big) \| \mathrm{w}\|_{\dot{H}^l} \\ &\lesssim \Big\| \frac{1}{\rho_{\phi}} \Big\|_{\infty}^{l+1} \|\textbf{u}\|_{\infty} \| \mathrm{w}\|_{\dot{H}^l}^2 \end{aligned}$$ Testing the velocity equation with $\partial^{2m} \textbf{u}$, and using the commutator estimate, we get $$\begin{aligned} \int_{\mathbb{T}^n} \partial^m (\textbf{u}\cdot \nabla \textbf{u}) \partial^m \textbf{u}dx \lesssim \|\nabla \textbf{u}\|_{\infty} \|\textbf{u}\|_{\dot{H}^m}^2\end{aligned}$$ For the alignment terms, we use the product estimate to get $$\begin{aligned} \int_{\mathbb{T}^n} \partial^m \Big( \mathrm{w}( (\textbf{u}\rho)_{\phi} - \rho_{\phi} ) \Big) \partial^m \textbf{u}dx &\leqslant\big(\| \mathrm{w}\|_{\infty} \|(\textbf{u}\rho)_{\phi} - \rho_{\phi}\|_{\dot{H}^m} + \| \mathrm{w}\|_{\dot{H}^m} \|(\textbf{u}\rho)_{\phi} - \rho_{\phi} \|_{\infty} \big) \|\textbf{u}\|_{\dot{H}^m} \\ &\lesssim (\| \mathrm{w}\|_{\infty} + \| \mathrm{w}\|_{\dot{H}^m}) \|\textbf{u}\|_{\infty} \|\textbf{u}\|_{\dot{H}^m} \end{aligned}$$ Provided $l \geqslant m$, the energy estimate for the velocity equation becomes, $$\begin{aligned} \frac{d}{dt} \|\textbf{u}\|_{\dot{H}^m}^2 \leqslant(\|\nabla \textbf{u}\|_{\infty} + \| \mathrm{w}\|_{\infty} \|\textbf{u}\|_{\infty} + \|\textbf{u}\|_{\infty}) Y_{m,l,k}\end{aligned}$$ Combining the energy estimates, we get an $\epsilon$-independent estimate on the norm $Y_{m,l,k}$. $$\begin{aligned} \frac{d}{dt} Y_{m,l,k} &\leqslant C \Big(\|\nabla \textbf{u}\|_{\infty} + \|\nabla \rho\|_{\infty} + \|\rho\|_{\infty} + \Big\| \frac{1}{\rho_{\phi}} \Big\|_{\infty}^{l+1} \|\textbf{u}\|_{\infty} + (1 + \|\mathrm{w}\|_{\infty}) \| \textbf{u}\|_{\infty} \Big) Y_{m,l,k} \end{aligned}$$ Recall that $\rho_{\phi} \geqslant c_0/2$ up to some $\epsilon$-independent time $T'$ and that $\mathrm{w}$ is bounded uniformly in time. So, for $l \geqslant m \geqslant k+1 > n/2 + 2$, we obtain $$\begin{aligned} \frac{d}{dt} Y_{m,l,k} &\lesssim Y_{m,l, k}^{2}, \hspace{5mm} \text{ for } t < T'\end{aligned}$$ This gives a bound on $Y_{m,l,k}$ up to some positive time $T>0$. Due to the $\epsilon$-independence of these energy estimates, the local solutions to [\[regularized_EAS_WM\]](#regularized_EAS_WM){reference-type="eqref" reference="regularized_EAS_WM"} exist on the common time interval $[0,T]$ independent of $\epsilon$. We will conclude by taking $\epsilon \to 0$ in [\[regularized_EAS_WM\]](#regularized_EAS_WM){reference-type="eqref" reference="regularized_EAS_WM"} and using compactness properties to obtain a local in time solution to [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"}. *Remark 13*. The full energy estimate including the dissapative terms show that for some $\epsilon$-independent constant $C$, $$\begin{aligned} \epsilon \int_0^T \big( \|\rho\|_{H^{k+1}}^2 + \| \mathrm{w}\|_{H^{l+1}}^2 + \|\textbf{u}\|_{H^{m+1}}^2 \big) ds \leqslant C \end{aligned}$$ ## Viscous solutions to [\[regularized_EAS_WM\]](#regularized_EAS_WM){reference-type="eqref" reference="regularized_EAS_WM"} approach solutions to [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"} The following estimate on the time derivative will yield the necessary compactness properties. We will denote the solution to [\[regularized_EAS_WM\]](#regularized_EAS_WM){reference-type="eqref" reference="regularized_EAS_WM"} by $Z^{\epsilon}$ instead of just $Z$ in order to emphasize the dependence of the solution on $\epsilon$. From squaring the time derivatives in [\[regularized_EAS_WM\]](#regularized_EAS_WM){reference-type="eqref" reference="regularized_EAS_WM"}, we see that $$\begin{aligned} \|\partial_t Z^{\epsilon}\|_{L^2}^2 \lesssim \|Z^{\epsilon}\|_X^2 + \epsilon \|Z^{\epsilon}\|_{H^2}^2\end{aligned}$$ From this inequality and Remark [Remark 13](#est_on_dissapative_terms){reference-type="ref" reference="est_on_dissapative_terms"}, we obtain $\partial_t Z^{\epsilon} \in L^2([0,T]; L^2 \times L^2 \times L^2)$. Of course we also have $Z^{\epsilon} \in L^{\infty}([0,T]; X)$ by local well-posedness. Applying the Aubin-Lions Lemma with $Y_0 = H^{k} \times H^{l} \times H^{m}$, $Y = H^{k-1} \times H^{l-1} \times H^{m-1}$, and $Y_1 = L^2 \times L^2 \times L^2$, we obtain a convergent subsequence $Z^{\epsilon} \to Z^*$ in $C([0,T]; Y)$ for some $Z^*$. That $Z^* = Z$, the solution to [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"}, can be seen by taking the limit as $\epsilon \to 0$ in the Duhamel formula [\[duhamel_formula\]](#duhamel_formula){reference-type="eqref" reference="duhamel_formula"}. Indeed, since $l \geqslant m \geqslant k+1 \geqslant n/2 + 3$, we have the pointwise convergence $\mathcal{N}(Z^{\epsilon}) \to \mathcal{N}(Z^*)$ so by the dominated convergence theorem and the continuity property of the heat semigroup, we conclude that $$\begin{aligned} \mathcal{F}[Z^*](t) = Z^*_0 + \int_0^t \mathcal{N}(Z^*(s)) ds \end{aligned}$$ which is the solution to [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"}. Further, since $Z^* = Z \in C([0,T]; Y)$ and $Y$ is dense in $H^{-k} \times H^{-l} \times H^{-m}$, the solution is weakly continuous, i.e. $Z \in C_w([0,T]; X)$. This concludes the local in time existence and uniqueness part of Theorem [\[lwp\]](#lwp){reference-type="ref" reference="lwp"}. ## Conditions for Continuation of the Solution {#sec_cont_criterion} Let us now establish the continuation criterion [\[cont_criterion\]](#cont_criterion){reference-type="eqref" reference="cont_criterion"}, which we will use to prove conditional global existence for unidirection flocks in Sections [5](#GWP_1D){reference-type="ref" reference="GWP_1D"} and [7](#UniGWP){reference-type="ref" reference="UniGWP"}. Recall the relevant energy estimates from Section [4.2](#energy_estimates){reference-type="ref" reference="energy_estimates"}. $$\begin{aligned} \frac{d}{dt} &\|\rho\|_{\dot{H}^k}^2 \leqslant C (\|\nabla \textbf{u}\|_{\infty} + \|\nabla \rho\|_{\infty} + \|\rho\|_{\infty}) Y_{m,l,k} \label{rho_energy_estimate} \\ \frac{d}{dt} &\| \mathrm{w}\|_{\dot{H}^l}^2 \lesssim \Big\| \frac{1}{\rho_{\phi}} \Big\|_{\infty}^{l+1} \|\textbf{u}\|_{\infty} \| \mathrm{w}\|_{\dot{H}^l}^2 \label{w_energy_estimate} \\ \frac{d}{dt} &\|\textbf{u}\|_{\dot{H}^m}^2 \lesssim \|\nabla \textbf{u}\|_{\infty} \|\textbf{u}\|_{\dot{H}^m}^2 + (\| \mathrm{w}\|_{\infty} \|\textbf{u}\|_{\infty} + \|\textbf{u}\|_{\infty} \| \mathrm{w}\|_{\dot{H}^m}) \|\textbf{u}\|_{\dot{H}^m} \label{u_energy_estimate}\end{aligned}$$ A sufficient continuation criterion is then given by $$\begin{aligned} \int_0^{T} \Big(\|\nabla \textbf{u}\|_{\infty} + \|\rho\|_{\infty} + \|\nabla \rho\|_{\infty} + \Big\|\frac{1}{\rho_{\phi}} \Big\|_{\infty}^{l+1} \Big) ds < \infty\end{aligned}$$ That is, the $X$-norm of the solution will not blow up for finite times provided this holds. We can simplify this criterion to [\[cont_criterion\]](#cont_criterion){reference-type="eqref" reference="cont_criterion"} by showing that $\|\nabla \textbf{u}\|_{\infty}$ and $\| 1/\rho_{\phi} \|_{\infty}$ control $\|\rho\|_{\infty}$ and $\|\nabla \rho\|_{\infty}$. Indeed, assume that [\[cont_criterion\]](#cont_criterion){reference-type="eqref" reference="cont_criterion"} holds. Then solving the continuity equation along characteristics, $\dot{X}(t,\alpha) = \textbf{u}(t, X(t,\alpha))$, we get $$\begin{aligned} \rho(X(t, \alpha)) = \rho(0, \alpha) \exp \Big\{ -\int_0^T (\nabla \cdot \textbf{u}) (s, X(s,\alpha)) ds \Big\}\end{aligned}$$ From the integrability of $\|\nabla \textbf{u}\|_{\infty}$ in [\[cont_criterion\]](#cont_criterion){reference-type="eqref" reference="cont_criterion"}, we obtain that $\|\rho\|_{\infty}$ is bounded. Further, boundedness of $\nabla \rho$ can be obtained similarly by differentiating the continuity equation. The equation for an arbitrary partial derivative $\partial \rho$ is given by $$\begin{aligned} \partial_t (\partial \rho) + \textbf{u}\cdot \nabla \partial \rho + \partial \textbf{u}\cdot \nabla \rho + (\nabla \cdot \textbf{u}) \partial \rho + (\nabla \cdot \partial \textbf{u}) \rho = 0\end{aligned}$$ Then $$\begin{aligned} \frac{d}{dt}\|\partial \rho\|_{\infty} \leqslant\| \frac{d}{dt} \partial \rho \|_{\infty} \leqslant\| \nabla \textbf{u}\|_{\infty} \| \nabla \rho \|_{\infty} + \|\nabla \textbf{u}\|_{\infty} \|\partial \rho\|_{\infty} + \| \nabla^2 \textbf{u}\|_{\infty} \|\rho\|_{\infty} \end{aligned}$$ Summing over the partials, we get $$\begin{aligned} \frac{d}{dt}\|\nabla \rho\|_{\infty} \lesssim \| \nabla \textbf{u}\|_{\infty} \| \nabla \rho \|_{\infty} + \| \nabla^2 \textbf{u}\|_{\infty} \|\rho\|_{\infty} \end{aligned}$$ To conclude by Gronwall, we need to bound $\| \nabla^2 \textbf{u}\|_{\infty}$. Observe that $\| \nabla^2 \textbf{u}\|_{\infty} \leqslant\|\textbf{u}\|_{H^m}$ since $m > n/2 + 2$ so it suffices to bound $\|\textbf{u}\|_{H^m}$. From the energy estimate [\[w_energy_estimate\]](#w_energy_estimate){reference-type="eqref" reference="w_energy_estimate"} on the $\mathrm{w}$-equation, [\[cont_criterion\]](#cont_criterion){reference-type="eqref" reference="cont_criterion"} implies that $\| \mathrm{w}\|_{H^l}$ is bounded for finite times, and in turn the energy estimate [\[u_energy_estimate\]](#u_energy_estimate){reference-type="eqref" reference="u_energy_estimate"} on the $\textbf{u}$-equation implies that $\|\textbf{u}\|_{H^m}$ is bounded for finite times. ## Small Initial Data With the continuation criterion in hand, we will prove Theorem [\[thm:small_data_intro\]](#thm:small_data_intro){reference-type="ref" reference="thm:small_data_intro"}. The precise statement is given below in Theorem [Theorem 14](#small_data){reference-type="ref" reference="small_data"}. Provided the kernel is bounded below and the initial variations of $\textbf{u}$ are sufficiently small, control of $\|\nabla \textbf{u}\|_{\infty}$ can be established in any dimension. The small variation of $\textbf{u}$ allows the quadratic term in the equation for $\partial \textbf{u}$ to be absorbed into the dissapative term. Intuitively speaking, the strength of the alignment force will overpower the Burger's transport term. Letting $\mathcal{A}(t) = \max_{x,y \in \mathbb{T}} |\textbf{u}(x) - \textbf{u}(y)|$, we state the result. **Theorem 14**. *Assume the kernel is bounded below, $\phi \geqslant c_1 > 0$. If the intitial data satisfies the conditions of Theorem [\[lwp\]](#lwp){reference-type="ref" reference="lwp"} and the following smallness conditions: $$\begin{aligned} &\mathcal{A}_0 < \epsilon^2, \hspace{5mm} \|\textbf{u}_0\|_{\infty} < \epsilon, \hspace{5mm} \epsilon < \frac{c_1 \mathrm{w}_- M}{2 + \eta M \|\phi\|_{\infty} + \mathrm{w}_+ M \|\nabla \phi\|_{\infty}}, \\ &\eta = \|\nabla \mathrm{w}_0\|_{\infty} \exp \Big\{ \frac{2 \|\phi\|_{\infty} \|\phi'\|_{\infty}}{M (\mathrm{w}_-) c_1^3} \mathcal{A}_0 \Big\} \end{aligned}$$ then there is a unique solution $(\rho, \mathrm{w}, \textbf{u}) \in C_w([0,T]; (H^k \cap L^1_+) \times H^l \times H^m)$ to [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"} existing globally in time such that $$\begin{aligned} \| \nabla \textbf{u}\|_{\infty} < 2 \epsilon, \hspace{5mm} t > 0 \end{aligned}$$* *Remark 15*. The essential ingredients for the proof are exponential alignment in $L^{\infty}$ and a bound from below on $\rho_{\phi}$, from which we can obtain a lower bound on the dissaptive term in the equation for $\partial \textbf{u}$ and a uniform bound on $\|\nabla \mathrm{w}\|_{\infty}$, which in turn controls another term in the equation for $\partial \textbf{u}$. The quantity $\eta$ above denotes this uniform bound on $\|\nabla \mathrm{w}\|_{\infty}$. For general kernels, $\|\nabla \mathrm{w}\|_{\infty}$ may not be bounded. *Remark 16*. The conclusion $\|\nabla \textbf{u}\|_{\infty} < 2 \epsilon$ for $t > 0$ can be bootstrapped to obtain exponential decay of $\|\nabla \textbf{u}\|_{\infty}$. *Proof.* By assumption, $\rho_{\phi} \geqslant c_1 M > 0$. Then the continuation criterion [\[cont_criterion\]](#cont_criterion){reference-type="eqref" reference="cont_criterion"} reduces to control of $\|\nabla \textbf{u}\|_{\infty}$. The equation for an arbitrary partial derivative $\partial \textbf{u}$ is given by $$\label{eqn:partial_u} \partial_t (\partial \textbf{u}) + \textbf{u}\cdot \nabla \partial \textbf{u}= \partial \mathrm{w}( (\textbf{u}\rho)_{\phi} - \textbf{u}\rho_{\phi} ) + \mathrm{w}( (\textbf{u}\rho)_{\phi'} - \textbf{u}\rho_{\phi'} ) - \nabla \textbf{u}\cdot \partial \textbf{u}- \mathrm{w}\rho_{\phi} \partial \textbf{u}$$ Then $$\frac{d}{dt} \|\partial \textbf{u}\|_{\infty} \leqslant\big( \| \partial \mathrm{w}\|_{\infty} M \|\phi\|_{\infty} + \mathrm{w}_+ M \|\nabla \phi\|_{\infty} \big) \mathcal{A}(t) + \big( \|\nabla \textbf{u}\|_{\infty} - c_1 \mathrm{w}_- M \big) \|\partial \textbf{u}\|_{\infty}$$ The equation for $\partial \mathrm{w}$ is given by $$\partial_t (\partial \mathrm{w}) + \textbf{u}_F \cdot \nabla \partial \mathrm{w}= \partial \textbf{u}_F \cdot \nabla \mathrm{w}$$ Therefore, in order to bound $\|\partial \mathrm{w}\|_{\infty}$, the exponential decay of $\|\partial \textbf{u}_F\|_{\infty}$ is sufficient. Due to $\rho_{\phi} \geqslant c_1 M$ and exponential alignment (Theorem [\[thm:linf_alignment\]](#thm:linf_alignment){reference-type="ref" reference="thm:linf_alignment"}), we have $$\begin{aligned} \label{ineq:decay_u_F} |\partial \textbf{u}_F| &= \Big| \frac{\rho_{\phi} (\textbf{u}\rho)_{\phi'} - \rho_{\phi'} (\textbf{u}\rho)_{\phi} }{\rho_{\phi}^2} \Big| = \Big| \frac{\rho_{\phi} \big( (\textbf{u}\rho)_{\phi'} - \textbf{u}\rho_{\phi'} \big) + \rho_{\phi'} \big( (\textbf{u}\rho)_{\phi} - \textbf{u}\rho_{\phi} \big) }{\rho_{\phi}^2} \Big| \\ &\leqslant\frac{2 \|\phi\|_{\infty} \|\phi'\|_{\infty}}{c_1^2} \mathcal{A}(t) \leqslant\frac{2 \|\phi\|_{\infty} \|\phi'\|_{\infty}}{c_1^2} \mathcal{A}_0 e^{-\mathrm{w}_- Mc_1t} \nonumber \end{aligned}$$ As a result, integrating the $\partial \mathrm{w}$ equation, we obtain $$\begin{aligned} \label{eqn:bd_on_w_x} \|\nabla \mathrm{w}\|_{\infty} \leqslant\|\nabla \mathrm{w}_0\|_{\infty} \exp \Big\{ \frac{2 \|\phi\|_{\infty} \|\phi'\|_{\infty}}{M (\mathrm{w}_-) c_1^3} \mathcal{A}_0 \Big\} := \eta \end{aligned}$$ Now, given that $\mathcal{A}_0 < \epsilon^2$ (and therefore by alignment, $\mathcal{A}(t) \leqslant\epsilon^2$) and $\|\nabla \textbf{u}_0\|_{\infty} < \epsilon$, let $[0,T)$ be the maximal interval of existence and let $[0, t^*)$ be the the maximal time interval on the interval existence for which $\|\nabla \textbf{u}\|_{\infty} < 2 \epsilon$. Let $a = \eta M \|\phi\|_{\infty} + \mathrm{w}_+ M \|\nabla \phi\|_{\infty}$ and $b = c_1 \mathrm{w}_- M$. Then $$\frac{d}{dt} \|\partial \textbf{u}\|_{\infty} \leqslant a \epsilon^2 - (b - 2\epsilon) \|\partial \textbf{u}\|_{\infty}, \hspace{5mm} t < t^*$$ Integrating, we obtain $$\|\partial \textbf{u}\|_{\infty} \leqslant\|\partial \textbf{u}_0\|_{\infty} + \frac{a \epsilon^2}{b-2\epsilon} \leqslant\epsilon + \frac{a \epsilon^2}{b-2\epsilon}$$ Fix $0 < \gamma < 1$. Provided $\epsilon < b(2\gamma - 1)/(4\gamma - 2 + a)$, $\|\partial \textbf{u}\|_{\infty} < 2 \gamma \epsilon < 2 \epsilon$ for all $t < t^*$. Thus, for small enough $\epsilon$, $t^* = T$ and by the continuation criterion, the solution can be continued beyond $T$, contradicting that it is the maximal time of existence. Thus $T = \infty$. This argument holds for all $\gamma < 1$ so Theorem [Theorem 14](#small_data){reference-type="ref" reference="small_data"} follows. ◻ # Global Well Posedness in 1D {#GWP_1D} In this section, we prove Theorem [Theorem 5](#thm:UniGWP){reference-type="ref" reference="thm:UniGWP"} in the 1D case first in order to illustrate the core of the argument before proceeding to the multi-D case, which is proved in Section [7](#UniGWP){reference-type="ref" reference="UniGWP"}. We will establish a threshold condition for global well-posedness in 1D in a similar manner to Carrillo, Choi, Tadmor and Tan in [@CCTT2016]. The precise statement is as follows. **Theorem 17**. *Suppose the initial data is smooth. Then we have* (i) *if $e_0 = \partial_x u_0 + \mathrm{w}_0 (\rho_0)_{\phi} \geqslant 0$, then $e$ remains positive for all $t>0$ and there's a unique solution $(\rho, \mathrm{w}, u) \in C_w([0,T]; (H^k \cap L^1_+) \times H^l \times H^m)$ to [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"} existing globally in time and satisfying the initial data.* (ii) *if $e_0 = \partial_x u_0 + \mathrm{w}_0 (\rho_0)_{\phi} < 0$, then $e$ approaches $-\infty$ in finite time and there is finite time blow-up of the solution.* *Proof.* By design, the entropy, $e = \partial_x u + \mathrm{w}\rho_{\phi}$, in 1D is conserved. We have $$\begin{aligned} \partial_t e + \partial_x(u e) = 0 \end{aligned}$$ Written along characteristics, we have the ODE $$\begin{aligned} \dot{e} = e(\mathrm{w}\rho_{\phi} -e) \end{aligned}$$ Provided $0 < \mathrm{w}_0 < \infty$, the logistic ODE on $e$ implies (i) if $e_0 \geqslant 0$ then $e(t) > 0$ and it remains bounded. (ii) if $e_0 < 0$ then $\dot{e} \leqslant-e^2$ so $e$ blows up. Therefore we have the threshold condition: if $e_0 < 0$, the solution blows up; but if $e_0 \geqslant 0$, then $\partial_x u$ remains bounded by some constant $C > 0$. The continuity equation implies that $$\begin{aligned} \rho(t,x) \geqslant\rho_0 e^{-\int_0^t \|\partial_x u\|_{\infty} ds} = \rho_0(x) e^{-Ct} \end{aligned}$$ This is enough to conlude global well-posedness via the continuation criterion [\[cont_criterion\]](#cont_criterion){reference-type="eqref" reference="cont_criterion"}. However, the lower bound can be improved to be of order $1/(1+t)$. Let us include the argument for the sake of optimality. It may be relevant to future flocking results. Observe that $e$ and $\rho$ satisfy the same transport equation and a result, the quantity $\frac{e}{\rho}$ is transported. $$\partial_t \big( \frac{e}{\rho} \big) + u \partial_x \big(\frac{e}{\rho} \big) = 0$$ In particular, letting $C = \frac{e_0}{\rho_0}$, we have $e \leqslant C \rho$. Writing the $\rho$-equation along characteristics, we get $$\begin{aligned} \dot{\rho} &= -\partial_x u \rho = (\mathrm{w}\rho_{\phi} - e) \rho \geqslant-C \rho^2 \\ \end{aligned}$$ Consequently, $\dot{(1/\rho)} \leqslant C$ and integrating we obtain $1/\rho \leqslant 1/\rho_0 + C t$. It total, we conclude that $1/\rho_{\phi}$ is bounded on any finite time interval and, by the continuation criterion, any local solution for which $e_0 \geqslant 0$ can be extended to any time interval. ◻ # Limiting Density Profile {#sec:limiting_density_profile} With the conditions for alignment and global well-posedness in 1D at hand, two natural questions arise. Is there a limiting density distribution for the flock? If so, what does the limiting density profile look like? In the 1D Cucker-Smale case with a heavy tail kernel and $e_0 > 0$, the former is answered in the affirmative in [@Sbook]; the latter is answered partially by establishing an estimate on its deviation from the uniform distribution in [@LS-entropy]. We extend these results to the $W$-model. In particular, we prove Theorem [\[thm:strong_flocking_intro\]](#thm:strong_flocking_intro){reference-type="ref" reference="thm:strong_flocking_intro"} and Theorem [\[thm:entropy_estimate_intro\]](#thm:entropy_estimate_intro){reference-type="ref" reference="thm:entropy_estimate_intro"}. ## Strong Flocking in 1D {#sec:strong_flocking} The solution flocks strongly if there is alignment of the velocities as well as convergence of the density $\rho$ to a limiting distribution $\rho_{\infty}$. This can be established in 1D provided $e_0 > 0$ and there is exponential alignment of the velocities, which acording to Theorem [\[thm:linf_alignment\]](#thm:linf_alignment){reference-type="ref" reference="thm:linf_alignment"}, necessitates the bound from below $\phi \geqslant c_1$ (in particular, $\rho_{\phi} \geqslant c_1 M$). The strict positivity of $e_0$ guarentees dissapation in the $\partial_x u$ equation, which is crucial for strong flocking. Indeed, let $u_{\infty}$ denote the limiting velocity. If there were a limiting density profile, it must be that the time derivative of the density in the moving frame with coordinates $x' = x - u_{\infty}t$, $t' = t$ is approaching zero sufficiently fast. Examining the equation for the density in the moving frame, $$\begin{aligned} \label{eq:rho_moving_frame} \partial_{t'} \rho + (u - u_{\infty}) \partial_{x'} \rho + (\partial_{x'} u) \rho = 0\end{aligned}$$ we see that boundedness of $\rho$ and $\partial_x \rho$ along with sufficiently fast decay of $\partial_x u$ is sufficient for strong flocking. *Remark 18*. In the case of small data, Theorem [Theorem 14](#small_data){reference-type="ref" reference="small_data"}, the smallness of the initial variation of $u$ led to a dissaptive term in the equation for $\partial u$. Here, we replace the small data assumption with $e_0 > 0$, which, in 1D, also leads to a dissapitive term in the equation for $\partial u$. *Proof of Theorem [\[thm:strong_flocking_intro\]](#thm:strong_flocking_intro){reference-type="ref" reference="thm:strong_flocking_intro"}.* First, note that if $e_0 \geqslant c_2 > 0$, then it remains bounded from below. Indeed, along characteristics, $\dot{e} = e(\mathrm{w}\rho_{\phi} - e)$ is non-negative whenever $0 \leqslant e \leqslant\mathrm{w}\rho_{\phi}$. Therefore, $e \geqslant\min\{ c_2, \mathrm{w}\rho_{\phi} \} := c$. Let $E(t)$ denote a generic exponentially decaying quanitity, which may vary from line to line. From the equation for $\partial_x u$ and $e > c > 0$, we have $$\begin{aligned} \frac{d}{dt} \|\partial_x u\| \leqslant\big( \mathrm{w}_+ + \| \partial_x \mathrm{w}\|_{\infty} \big) E(t) - c \|\partial_x u\|_{\infty} \end{aligned}$$ The exponential decay of $u_F$ (given that $\phi$ is bounded below away from zero) was shown in estimate[\[ineq:decay_u\_F\]](#ineq:decay_u_F){reference-type="eqref" reference="ineq:decay_u_F"}. As a result, $\|\partial_x \mathrm{w}\|$ is bounded and $\|\partial_x u\|_{\infty}$ is exponentially decaying. Turning to the second derivative, the equation for $\partial^2_x u$ is given by $$\begin{aligned} \partial_t (\partial_x^2 u) + u \cdot \partial^3_x u &= \partial_x^2 \mathrm{w}( (u\rho)_{\phi} - u \rho_{\phi} ) + 2 \partial_x \mathrm{w}( (u\rho)_{\phi'} - u \rho_{\phi'} ) \\ &+ \mathrm{w}( (u\rho)_{\phi''} - u \rho_{\phi''} ) - 2\partial_x(\mathrm{w}\rho_{\phi})) \partial_x u - 2 (\partial_x u) \partial_x^2 u - e \partial^2_x u \end{aligned}$$ To control the first term, we will show that $\|\partial^2_x \mathrm{w}\|_{\infty}$ is bounded. The equation for $\partial^2_x \mathrm{w}$ is given by $$(\partial_t + u_F \partial_x)(\partial^2_x \mathrm{w}) = - 2(\partial_x u_F) \partial^2_x \mathrm{w}+ (\partial_x^2 u_F) \partial_x \mathrm{w}$$ We claim that $\partial^2_x u_F$ is exponentially decaying. Indeed, $$\begin{aligned} \partial^2_x u_F &= \frac{(u \rho)_{\phi''}} {\rho_{\phi}} - \frac{\rho_{\phi''} (u\rho)_{\phi}} {\rho_{\phi}^2} - 2 \frac{\rho_{\phi'} (u\rho)_{\phi'}} {\rho_{\phi}^2} + 2\frac{\rho_{\phi'}^2 (u\rho)_{\phi}} {\rho_{\phi}^3} \\ &:= A_1 - A_2 - B_1 + B_2 \end{aligned}$$ By the exponential alignment [\[thm:linf_alignment\]](#thm:linf_alignment){reference-type="eqref" reference="thm:linf_alignment"}, we have $$\begin{aligned} A_1 - A_2 = \frac{ \rho_{\phi} ((u \rho)_{\phi''} - u \rho_{\phi''}) - \rho_{\phi''} ((u\rho)_{\phi} - u\rho_{\phi})} {\rho_{\phi}^2} \leqslant E(t) \\ \end{aligned}$$ Similarly, $$\begin{aligned} B_2 - B_1 = 2 \frac{\rho_{\phi'}^2 ((u\rho)_{\phi} - u\rho_{\phi}) - \rho_{\phi} \rho_{\phi'} ((u\rho)_{\phi'} - u\rho_{\phi'}) } {\rho_{\phi}^3} \leqslant E(t) \end{aligned}$$ In total, we gather that $$\begin{aligned} \frac{d}{dt} \| \partial^2_x \mathrm{w}\|_{\infty} \leqslant E(t) + E(t) \|\partial^2_x w\|_{\infty} \end{aligned}$$ In particular, $\|\partial^2_x \mathrm{w}\|_{\infty}$ is bounded. Returning to $\partial_x^2 u$, we obtain $$\begin{aligned} \frac{d}{dt} \| \partial_x^2 u \|_{\infty} \leqslant E(t) + (E(t) - c) \|\partial^2_x u\|_{\infty} \end{aligned}$$ Thus $\|\partial^2_x u\|_{\infty}$ is exponentially decaying. With exponential decay of $\|\partial_x u\|_{\infty}$ and $\|\partial^2_x u\|_{\infty}$ in hand, boundedness of $\rho$ and $\partial_x \rho$ follows. The former follows from the continuity equation and latter from $$\begin{aligned} \partial_t (\partial_x \rho) = -u \partial^2_x \rho - 2 (\partial_x u) (\partial_x \rho) - (\partial^2_x u) \rho \end{aligned}$$ so that $$\begin{aligned} \partial_t \|\partial_x \rho\|_{\infty} \leqslant E(t) + E(t) \|\partial_x \rho\|_{\infty} \end{aligned}$$ Integrating gives a uniform bound on $\|\partial_x \rho\|_{\infty}$. Now, from the density equation in the moving frame [\[eq:rho_moving_frame\]](#eq:rho_moving_frame){reference-type="eqref" reference="eq:rho_moving_frame"} along with exponential alignment, we have $$\begin{aligned} \partial_t \rho = E(t) \end{aligned}$$ In particular, $\rho(t, x)$ is Cauchy in time, uniformly in $x$, so there exists a limiting function $\rho_{\infty}(x)$. Moreover, the exponential decay of $\partial_t \rho$ implies exponential convergence of $\rho(t,x)$ to $\rho_{\infty}(x)$ in $L^{\infty}$. ◻ ## Relative Entropy Estimate and Distribution of Limiting Flock in 1D {#sec:entropy} In this section, we prove Theorem [\[thm:entropy_estimate_intro\]](#thm:entropy_estimate_intro){reference-type="ref" reference="thm:entropy_estimate_intro"}. We aim to estimate the $L^1$ distance of the limiting flock to the uniform distribution, $\bar{\rho} = M/2\pi$: $$\begin{aligned} \limsup_{t \to \infty} \|\rho(t) - \bar{\rho} \|_{L^1} &\leqslant\Big( \tilde{Q} + \|\phi\|_{\infty} (\mathrm{w}_+ - \mathrm{w}_-) \Big) \frac{M \mathrm{w}_+ \|\phi\|_{\infty}}{c(\mathrm{w}_+ \|\phi\|_{L^1} - \tilde{Q})} \\\end{aligned}$$ Recall that $\tilde{e} = \partial_x u + \mathcal{L}_{\phi} \rho$, where $\mathcal{L}_{\phi} \rho = \mathrm{w}(x) \int_{\mathbb{T}} (\rho(y) - \rho(x)) \phi(x-y) dy$ and $\tilde{q} = \frac{\tilde{e}}{\rho}$; and it is assumed that $\tilde{q} \leqslant\tilde{Q}$ for some constant $\tilde{Q}$. We remark on the conditions for such a constant $\tilde{Q}$ to exist. *Remark 19*. The boundedness of $\rho$ and $\|\tilde{q}\|_{\infty} \leqslant\tilde{Q} < \mathrm{w}_+ \|\phi\|_{L^1}$ for some constant $\tilde{Q}$ is satisfied when $e = \partial_x u + \mathrm{w}\rho_{\phi}$, the kernel $\phi$, and the weight $\mathrm{w}$ are bounded away from zero, i.e. $\phi \geqslant c_1 > 0$, $e_0 \geqslant c_2 > 0$, $\mathrm{w}\geqslant\mathrm{w}_- > 0$ and there is a smallness assumption on the derivative of the initial weight and/or the initial variation of the velocity. Indeed, if $e_0 > 0$ then it remains bounded away from zero for all time. Since $\tilde{q}$ satisfies $\partial_t \tilde{q} + u \partial_x \tilde{q} = \partial_x \mathrm{w}(u - u_F)$ and the kernel is bounded away from zero, we have exponential alignment and, as a result, $\|\partial_x \mathrm{w}\|_{\infty}$ remains bounded by $\|\partial_x \mathrm{w}_0\|_{\infty} \exp \Big\{ \frac{2 \|\phi\|_{\infty} \|\phi'\|_{\infty}}{M (\mathrm{w}_-) c_1^3} \mathcal{A}_0 \Big\}$, where $\mathcal{A}(t) = \max_{(x,y) \in \mathbb{T}^2} |u(t,x) - u(t,y)|$. This was shown in [\[eqn:bd_on_w\_x\]](#eqn:bd_on_w_x){reference-type="eqref" reference="eqn:bd_on_w_x"}. As a result, $$\begin{aligned} \|\tilde{q}\|_{\infty} \leqslant\| \partial_x \mathrm{w}\|_{\infty} \frac{\mathcal{A}_0}{ M c_1 \mathrm{w}_-} \leqslant\|\partial_x \mathrm{w}_0\|_{\infty} \exp \Big\{ \frac{2 \|\phi\|_{\infty} \|\phi'\|_{\infty}}{M (\mathrm{w}_-) c_1^3} \mathcal{A}_0 \Big\} \frac{\mathcal{A}_0}{ M c_1 \mathrm{w}_-} \end{aligned}$$ We see that small values of $\|\partial_x \mathrm{w}_0\|_{\infty}$ or $\mathcal{A}_0$ or $\frac{1}{\mathrm{w}_-}$ are sufficient to achieve $\|\tilde{q}\|_{\infty} < \mathrm{w}_+ \|\phi\|_{L^1}$. The proof relies on an estimate of the *relative* entropy $\mathcal{H}= \int_{\mathbb{T}} \rho \log \frac{\rho}{\bar{\rho}}$ in order to achieve the desired estimate. This is to be distinguished from the entropy $e$. *Proof of Theorem [\[thm:entropy_estimate_intro\]](#thm:entropy_estimate_intro){reference-type="ref" reference="thm:entropy_estimate_intro"}.* By the Csiszar-Kullback inequality [\[lma:ck_inequality\]](#lma:ck_inequality){reference-type="ref" reference="lma:ck_inequality"}, to control $\|\rho - \bar{\rho}\|_1$, it suffices to control $\mathcal{H}= \int_{\mathbb{T}} \rho \log \frac{\rho}{\bar{\rho}} dx$, We have $$\begin{aligned} \partial_t (\rho \log \rho) &= -[u (\rho \log \rho)]' - \rho u' = -[u (\rho \log \rho)]' - \rho \tilde{e} - \rho \mathcal{L}\rho \end{aligned}$$ So that, $$\begin{aligned} \frac{d \mathcal{H}}{dt} &= \frac{d}{dt} \int_{\mathbb{T}} \rho \log \rho dx \\ &= \int_{\mathbb{T}} (\rho - \bar{\rho}) \tilde{e} dx - \int_{\mathbb{T}^2} \rho(x) \mathrm{w}(x) (\rho(y) - \rho(x)) \phi(x-y) dy dx \\ &= -\int_{\mathbb{T}} (\rho - \bar{\rho}) \tilde{e} dx - \bar{\rho} \int_{\mathbb{T}} \tilde{e} dx + \int_{\mathbb{T}^2} \rho(x) \mathrm{w}(x) (\rho(y) - \rho(x)) \phi(x-y) dy dx \\ &:= I_1 + I_2 + I_3 \end{aligned}$$ Symmetrizing the $I_3$ term, we obtain $$\begin{aligned} I_3 &= -\frac{1}{2} \int_{\mathbb{T}^2} \mathrm{w}(x) |\rho(x) - \rho(y)|^2 \phi(x-y) dy dx - \frac{1}{2} \int_{\mathbb{T}^2} \rho(y) (\mathrm{w}(x) - \mathrm{w}(y)) (\rho(x) - \rho(y)) \phi(x-y) dy dx \\ &\leqslant-\frac{1}{2} \mathrm{w}_- \int_{|x-y| \leqslant r_0} |\rho(y) - \rho(x)|^2 dy dx + \frac{1}{2} \|\rho\|_{\infty} \|\phi\|_{\infty} (\mathrm{w}_+ - \mathrm{w}_-) \int_{\mathbb{T}^2} |\rho(x) - \bar{\rho} - (\rho(y) - \bar{\rho})| dy dx \\ &\leqslant-\frac{1}{2} \mathrm{w}_- \int_{|x-y| \leqslant r_0} |\rho(y) - \rho(x)|^2 dy dx + 2\pi \|\rho\|_{\infty} \|\phi\|_{\infty} (\mathrm{w}_+ - \mathrm{w}_-) \|\rho - \bar{\rho}\|_{L^1} \\ \end{aligned}$$ By Lemma 2.1 of [@LS-entropy], we have, for some positive constant $c := c(r_0)$, $$\begin{aligned} \leqslant-c \mathrm{w}_- \|\rho - \bar{\rho}\|_{L^2}^2 + 2\pi \|\rho\|_{\infty} \|\phi\|_{\infty} (\mathrm{w}_+ - \mathrm{w}_-) \|\rho - \bar{\rho}\|_{L^1} \\ \end{aligned}$$ In the remainder, $c$ may change from line to line, but it will remain solely dependent on $r_0$. Symmetrizing $I_2$, we have $$\begin{aligned} I_2 &= \bar{\rho} \int_{\mathbb{T}^2} \mathrm{w}(x) (\rho(y) - \rho(x)) \phi(x-y) dy dx \\ &= \frac{1}{2} \bar{\rho} \int_{\mathbb{T}^2} (\mathrm{w}(y) - \mathrm{w}(x)) (\rho(y) - \rho(x)) \phi(x-y) dy dx \\ \end{aligned}$$ Using $\bar{\rho} \leqslant\|\rho\|_{\infty}$, we obtain the same estimate as the non-dissapative term in $I_3$. $$\begin{aligned} |I_2| \leqslant 2\pi \|\rho\|_{\infty} (\mathrm{w}_+ - \mathrm{w}_-) \|\phi\|_{\infty} \|\rho - \bar{\rho}\|_{L^1} \end{aligned}$$ For $I_1$, we have $$\begin{aligned} |I_1| &= \big| \int_{\mathbb{T}} (\rho - \bar{\rho}) \rho \tilde{q} dx \big| \\ &= \|\rho\|_{\infty} \|\tilde{q}\|_{\infty} \|\rho - \bar{\rho}\|_{L^1} \end{aligned}$$ Combining these estimates with the the Csiszar-Kullback inequality, we obtain $$\begin{aligned} \frac{d \mathcal{H}}{dt} &\leqslant\Big( \|\rho(t)\|_{\infty} \|\tilde{q}(t)\|_{\infty} + 4\pi \|\rho(t)\|_{\infty} \|\phi\|_{\infty} (\mathrm{w}_+ - \mathrm{w}_-) \Big) \|\rho(t) - \bar{\rho}\|_{L^1} - c \mathrm{w}_- \|\rho(t) - \bar{\rho}\|_{L^2}^2 \\ &\leqslant\Big( \|\rho(t)\|_{\infty} \|\tilde{q}(t)\|_{\infty} + 4\pi \|\rho(t)\|_{\infty} \|\phi\|_{\infty} (\mathrm{w}_+ - \mathrm{w}_-) \Big) \sqrt{4 \pi \bar{\rho} \mathcal{H}(t)} - c \mathrm{w}_- \bar{\rho} \mathcal{H}(t) \end{aligned}$$ Letting $Y = \sqrt{\mathcal{H}}$, we obtain via Gronwall, $$\begin{aligned} Y(t) &\leqslant Y_0 e^{-c \mathrm{w}_- \bar{\rho}t} + \sqrt{\pi \bar{\rho}} \int_0^t \|\rho(s)\|_{\infty} \|\tilde{q}(s)\|_{\infty} e^{-c \bar{\rho}(t-s)} ds \\ &+ 4\pi \sqrt{\pi \bar{\rho}} \|\phi\|_{\infty} (\mathrm{w}_+ - \mathrm{w}_-) \int_0^t \|\rho(s)\|_{\infty} e^{-c \bar{\rho}(t-s)} ds \\ \end{aligned}$$ To relate it back to $\|\rho - \bar{\rho}\|_{L^1}$, we multiply both sides of the inequality by $\sqrt{4\pi \bar{\rho}}$ and apply the Csizar-Kullback inequality again. Combining this with the following elementary fact: for a bounded function $a(t)$ and a constant $b$, $\limsup_{t \to \infty} \int_0^t a(s) e^{b(t-s)} ds \leqslant\frac{1}{b} \limsup_{t \to \infty} a(t)$, we obtain (since $\rho$ is bounded by assumption) $$\begin{aligned} \limsup_{t \to \infty} \|\rho(t) - \bar{\rho}\|_{L^1} &\leqslant\frac{1}{c} \limsup_{t \to \infty} \|\rho(t)\|_{\infty} \|\tilde{q}(t)\|_{\infty} \\ &+ \frac{1}{c} \|\phi\|_{\infty} (\mathrm{w}_+ - \mathrm{w}_-) \limsup_{t \to \infty} \|\rho(t)\|_{\infty} \\ \end{aligned}$$ Applying Lemma [Lemma 20](#lma:bd_on_rho){reference-type="ref" reference="lma:bd_on_rho"} to bound $\limsup_{t \to \infty} \|\rho(t)\|_{\infty}$ gives the result. ◻ **Lemma 20**. *If there exists a constant $\tilde{Q}$ such that $\|\tilde{q}\|_{\infty} \leqslant\tilde{Q} < \mathrm{w}_+ \|\phi\|_{L^1}$, then $$\begin{aligned} \limsup_{t \to \infty} \|\rho(t)\|_{\infty} \leqslant\frac{M \mathrm{w}_+ \|\phi\|_{\infty}}{\mathrm{w}_+ \|\phi\|_{L^1} - \|\tilde{q}\|_{\infty}} \end{aligned}$$* *Proof.* Let $x_+$ denote the maximizer of $\rho(t)$. That is, $\rho_+(t) = \rho(t, x_+)$. From the continuity equation, we have $$\begin{aligned} \frac{d}{dt} \rho_+(t) &= -\rho_+(t) \partial_x u(t, x_+) = -\rho_+(t) (\tilde{e} - \mathcal{L}_{\phi} \rho) \\ &= -\rho_+(t)^2 \tilde{q}(t, x_+) + \rho_+(t) \mathrm{w}(t, x_+) \int_{\mathbb{T}} \phi(x_+ - y) (\rho(t,y) - \rho_+(t)) dy \\ &\leqslant(\tilde{Q} - \mathrm{w}_+ \|\phi\|_{L^1}) \rho_+(t)^2 + M \mathrm{w}_+ \|\phi\|_{\infty} \rho_+(t) \\ &= (\mathrm{w}_+ \|\phi\|_{L^1} - \tilde{Q}) \rho_+(t) \bigg[ \frac{M \mathrm{w}_+ \|\phi\|_{\infty}}{\mathrm{w}_+ \|\phi\|_{L^1} - \tilde{Q}} - \rho_+(t) \bigg] \\ \end{aligned}$$ Observe that if $\dot{X}(t) \leqslant A X(t) [B - X(t)]$ where $A,B > 0$ are constants and $X(t) > 0$, then $$\begin{aligned} X(t) \leqslant\frac{BX(0)} {X(0) + (B - X(0)) \exp(-ABt)} \end{aligned}$$ Applying this differential inequality and taking $t \to \infty$ gives the result. ◻ # Unidirectional Flocks {#UniGWP} In this section, we prove Theorem [Theorem 5](#thm:UniGWP){reference-type="ref" reference="thm:UniGWP"} in full. As in the 1D case, Theorem [Theorem 17](#t:GWP_1D){reference-type="ref" reference="t:GWP_1D"}, the $e$-quantity is used to control the gradient of the velocity. The difference here is that the full gradient needs to be controlled. Let us recall the definition of unidirectional flocks [\[e:uniintro\]](#e:uniintro){reference-type="eqref" reference="e:uniintro"}. A flock is unidirectional if it has the form $$\textbf{u}(x,t) = u(x,t) \textbf{d}, \hspace{8mm} \textbf{d}\in \mathbb{S}^{n-1}, \hspace{2mm} \textbf{u}: \mathbb{T}^n \times \mathbb{R}^+ \to \mathbb{R}$$ for all time $t$. The precise statement to be proved is as follows. **Theorem 21**. *Given smooth initial data with $\textbf{u}_0$ unidirectional in the direction $\textbf{d}$,* (i) *if $e_0 = \nabla u_0 \cdot \textbf{d}+ \mathrm{w}_0 (\rho_0)_{\phi} \geqslant 0$, then there's a unique solution $(\rho, \mathrm{w}, \textbf{u}) \in C_w([0,T]; (H^k \cap L^1_+) \times H^l \times H^m)$ to [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"} existing globally in time and satisfying the initial data.* (ii) *if $e_0 = \nabla u_0 \cdot \textbf{d}+ \mathrm{w}_0 (\rho_0)_{\phi} < 0$, then there is finite time blow-up of the solution.* *Proof.* By the maximum principle applied to each direction, the solution $\textbf{u}$ remains unidirectional for all time. By rotation invariance of [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"}, we can assume WLOG that $\textbf{d}= x_1$. The velocity then takes the form $\textbf{u}= (u(x,t), 0, \dots, 0)$ and the system [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"} can be written $$\label{SM_uni} \begin{cases} \partial_t \rho + \partial_1 (u\rho) = 0 \\ \partial_t \mathrm{w}+ [u]_{\rho} \partial_1 \mathrm{w}= 0\\ \partial_t u + u \partial_1 u = \mathrm{w}((u\rho)_{\phi} - u\rho_{\phi}) \end{cases}$$ The entropy equation is given by $$e = \partial_1 u + \mathrm{w}\rho_{\phi}, \hspace{5mm} \partial_t e + \partial_1(ue) = 0$$ Written along characteristics, we recover the same ODE from the 1D case. $$\begin{aligned} \dot{e} = e(\mathrm{w}\rho_{\phi}-e) \end{aligned}$$ As a result, we obtain the same threshold condition. (i) if $e_0 \geqslant 0$ then $e(t) > 0$ bounded. (ii) if $e_0 < 0$ then $\dot{e} \leqslant-e^2$ so $e$ blows up. Considering the case $e_0 \geqslant 0$, the bound $1/\rho_{\phi} \leqslant 1/\rho_0 + C_1 t$ follows a similar argument to the 1D case. Turning to $\nabla u$, we write the equation for a generic partial derivative $\partial u$. $$\begin{aligned} (\partial_t + u \partial_1) \partial u &= -(\partial_1 u)(\partial u) + \partial \big( \mathrm{w}((u\rho)_{\phi} - u \rho_{\phi}) \big) \\ &= -e (\partial u) + \partial \mathrm{w}((u\rho)_{\phi} - u \rho_{\phi}) + \mathrm{w}((u\rho)_{\phi'} - u \rho_{\phi'}) \\ \end{aligned}$$ Multiplying by $\partial u$ and taking the supremum over the support of $\rho$ and using that $e \geqslant 0$, we get $$\begin{aligned} \frac{d}{dt} \|\partial u\|_{L^{\infty}} &\leqslant\|\partial \mathrm{w}\|_{\infty} \|(u\rho)_{\phi} + u \rho_{\phi}\|_{\infty} + \|\mathrm{w}((u\rho)_{\phi'} - u \rho_{\phi'}) \|_{\infty} \\ &\lesssim \|u\|_{\infty} (\mathrm{w}_+ + \|\partial \mathrm{w}\|_{\infty}) \end{aligned}$$ It remains to bound $\|\partial \mathrm{w}\|_{\infty}$. We have $$\begin{aligned} (\partial_t + u_F \partial_1) \partial \mathrm{w}+ (\partial u_F) (\partial_1 \mathrm{w}) = 0 \end{aligned}$$ In the case that $\partial = \partial_1$, solving along characteristics, we have $$\begin{aligned} |\partial_1 \mathrm{w}| &= \big| (\partial_1 \mathrm{w}_0) \exp \Big\{ - \int_0^t \partial_1 u_F ds \Big\} \big| \\ &\leqslant|\partial_1 \mathrm{w}_0| \exp \Big\{ \int_0^t C \Big\| \frac{1}{\rho_{\phi}} \Big\|_{\infty}^2 \|u\|_{\infty} ds \Big\} \\ &\leqslant|\partial_1 \mathrm{w}_0| \exp \Big\{ \int_0^t C t^2 \|u\|_{\infty} ds \Big\} \end{aligned}$$ For an arbitrary partial derivative along characteristics, we obtain $$\begin{aligned} |\partial \mathrm{w}| &= \big| \partial \mathrm{w}_0 \int_0^t (\partial u_F) (\partial_1 \mathrm{w}) ds \big| \\ &\leqslant|\partial \mathrm{w}_0| \int_0^t Ct^2 \|u\|_{\infty} |\partial_1 \mathrm{w}_0| \exp \Big\{ \int_0^s C t^2 \|u\|_{\infty} dr \Big\} ds \end{aligned}$$ which is bounded for finite times. Moreover $\|\partial u\|_{\infty}$ is bounded for finite times. The continuation criterion implies that a local solution can be extended to any finite time interval. ◻ # Numerical Simulation Plots and Description {#description_of_numerics} To illustrate that the $\mathrm{s}$-model with Motsch-Tadmor initial data ($\mathrm{w}_0 = (1/\rho_0)_{\phi}$) possesses similar qualitative features to the Motsch-Tadmor model, we provide numerical solution plots for these two cases and for the Cucker-Smale case for comparison. The general scheme is a linearized Finite Element discretization in space with conforming elements and semi-implicit first order finite differences in time. The stability and error analysis for this numerical scheme is not known. However, we provide evidence of error convergence in [8.4](#appdx:convg_experiment){reference-type="ref" reference="appdx:convg_experiment"} by plotting the error between a known solution and the numerical solution as the mesh parameters go to zero. Before writing the variational problem, let us describe the discretization of the torus and the numerical solution spaces. ## Local polynomial vector spaces and variational problem The discretization of the torus is a uniform partition of the interval $[0,1]$ into $M$ pieces of size $h = 1/M$, where the point $0$ is indentified with the point $1$. We will refer each subinterval of size $h$ as an element. The numerical solutions will lie in discrete finite element-based vector spaces with local 3rd and 2nd order local polynomial basis functions, which we denote $P_3$ and $P_2$, respectively. We provide details of the construction of the basis functions for $P_3$; the construction of the basis functions for $P_2$ is analgous. There are $3M + 1$ nodes (for $P_2$, there are $2M + 1$ nodes) placed uniformly over the unit interval. For each node, there will be a corresponding basis function (which is a piecewise 3rd order polynomial) with support only in nearby elements whose value is equal to $1$ at the given node and $0$ at nearby nodes. To describe the basis functions associated to each node, it is convenient to describe the basis functions whose support intersects a given element. To that end, suppose the given element is $[0,1]$ (the basis functions with support in element $[0,1]$ can easily be adapted to any element by shifting and scaling). The four nodes, placed at positions $0$, $1/3$, $2/3$, and $1$, correspond to four basis functions, which on the interval $[0,1]$, have the form $\psi_k(x) = a_{k0} + a_{k1} x + a_{k2} x^2 + a_{k3} x^3$, $k = 0$ to $3$. The coefficients are chosen so that the $\psi_k(x)$ is equal to $1$ at the node $k/3$ and $0$ at the other three nodes. In particular, the coefficients are given by the solution to the matrix equation, $$\begin{aligned} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 1/3 & 1/3^2 & 1/3^3 \\ 1 & 2/3 & (2/3)^2 & (2/3)^3 \\ 1 & 1 & 1 & 1 \\ \end{bmatrix} \begin{bmatrix} a_{00} & a_{01} & a_{02} & a_{03} \\ a_{10} & a_{11} & a_{12} & a_{13} \\ a_{20} & a_{21} & a_{22} & a_{23} \\ a_{30} & a_{31} & a_{32} & a_{33} \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \end{aligned}$$ Now let us describe how the entire basis functions are formed from these $\psi_k$. Let $v_k^*$, $k = 0$ to $3$, be the basis funciton associated to node at position $k/3$. We will choose $v_k^*$ so that they are continuous at the boundary of the element (however, there is no continuity of the derivatives of the boundary of the element). In particular, the $v_k^*$ are constructed using $\psi_k$ as follows. We simply choose $v_1^* = \psi_1 \mathcal{X}_{[0,1]}$ and $v_2^* = \psi_2 \mathcal{X}_{[0,1]}$. Continuity at the boundary of the element holds by the construction of $\psi_1$, $\psi_2$. The functions $\psi_0$ and $\psi_3$ are equal to $1$ at the boundary of the element. To retain continuity at the boundary, let $\psi_k'$, $\psi_k''$ denote the basis functions with support in the left adjacent element $[-1,0]$ and the right adjacent element $[1,2]$, respectively (that is, $\psi_k'(x) = \psi_k(x+1)$ and $\psi_k''(x) = \psi_k(x-1)$). Then $v_0^* = \psi_3' \mathcal{X}{[-1,0]} + \psi_0 \mathcal{X}{[0,1]}$ and $v_3^* = \psi_3 \mathcal{X}{[0,1]} + \psi_0'' \mathcal{X}{[1,2]}$. A general basis function $v_k$ on the mesh with $M$ elements of size $h$ is obtained by shifting and scaling the $v_k^*$'s. For instance, consider the $i^{th}$ element $[\frac{i}{M}, \frac{i+1}{M}]$, $0 \leqslant i \leqslant M-1$, on a mesh of $M$ elements. The four basis functions associated to the nodes on this element are given by $v_k^i = v_k^*(M(x - i/M))$, $k = 0$ to $3$. The trial and test function spaces for $P_3$ are both equal and we denote them by $\{ v_k \}_{k=1}^{3M+1}$, as there are $3M+1$ nodes. Similarly, the trial and test function spaces for $P_2$ are both equal and follow a similar construction. We denote the trial and test functions for $P_2$ by $\{ q_k \}_{k=1}^{2M+1}$. Now let $(\rho^n, \mathrm{w}^n, u^n) \in (P_3, P_3, P_2)$ be the numerical solution at the $n^{th}$ time step. Then for some coefficients $b_k^n$, $(b^n_k)'$, and $c_k^n$ and $(v_k, q_k) \in (P_3, P_2)$. $$\begin{aligned} \rho^n(x) = \sum_{k=0}^{3M + 1} b_k^n v_k, \hspace{5mm} \mathrm{w}^n(x) = \sum_{k=0}^{3M + 1} (b^n_k)' v_k, \hspace{5mm} u^n(x) = \sum_{k=0}^{2M + 1} c_k^n q_k \end{aligned}$$ Let $V_I, Q_I$ be the interpolant operators on $P_3$ and $P_2$, respectively. Given intial data $(\rho_0, \mathrm{w}_0, u_0)$, we set $(\rho^0, \mathrm{w}^0, u^0) = (V_I \rho_0, V_I \mathrm{w}_0, Q_I u_0)$; and we set $\phi_h = V_I \phi$. To obtain the solutions at the next time step, we solve the following variational problem. For all test functions $(v, q) \in (P_3, P_2)$, $$\begin{aligned} \label{eqn:numerical_scheme} \begin{cases} \frac{1}{k} \langle\rho^{n+1} - \rho^n, v \rangle- \langle\rho^{n+1} u^n, \frac{d}{dx} v \rangle= 0 \\ \frac{1}{k} \langle\mathrm{w}^{n+1} - \mathrm{w}^n, v \rangle+ \langle(\frac{d}{dx} \mathrm{w}^{n+1}) \frac{(u^n \rho^n)_{\phi_h}}{\rho^n_{\phi_h}}, v \rangle= 0 \\ \frac{1}{k} \langle u^{n+1} - u^n, q \rangle+ \langle u^{n+1} \frac{d}{dx} u^n, q \rangle= \langle\mathrm{w}^n (u^n \rho^n)_{\phi_h}, q \rangle- \langle\mathrm{w}^n u^{n+1} \rho^n_{\phi_h}, q \rangle \end{cases} \end{aligned}$$ For the original Motsch-Tadmor model, the weight is set to $1/(\rho^n)_{\phi}$. The variational problem is given by $$\begin{aligned} \label{eqn:numerical_scheme_MT} \begin{cases} \frac{1}{k} \langle\rho^{n+1} - \rho^n, v \rangle- \langle\rho^{n+1} u^n, \frac{d}{dx} v \rangle= 0 \\ \frac{1}{k} \langle u^{n+1} - u^n, q \rangle+ \langle u^{n+1} \frac{d}{dx} u^n, q \rangle= \langle\frac{1}{(\rho^n)_{\phi}} (u^n \rho^n)_{\phi_h}, q \rangle- \langle\frac{1}{(\rho^n)_{\phi}} u^{n+1} \rho^n_{\phi_h}, q \rangle \end{cases} \end{aligned}$$ The term $\langle\rho^{n+1} u^n, \frac{d}{dx} v \rangle$ in both variational forms is obtained via integration by parts. The spaces $P_3$, $P_2$ and the numerical solutions to [\[eqn:numerical_scheme\]](#eqn:numerical_scheme){reference-type="eqref" reference="eqn:numerical_scheme"} were computed with the aid of the FENICS software library [@LoggWells2010] [@LoggEtal_10_2012]. *Remark 22*. The choice $(P_3, P_3, P_2)$ was chosen purely heuristically in order to resemble the inf-sup stability condition for the Stokes equation, however, it has not been proven that these spaces satsify the inf-sup condition for the system [\[eqn:numerical_scheme\]](#eqn:numerical_scheme){reference-type="eqref" reference="eqn:numerical_scheme"}. ## Comparison of Cucker-smale and $\mathrm{s}$-model with Motsch-Tadmor initial data {#appdx:comparison_CS_WM} Numerical solution plots for the solution to [\[eqn:numerical_scheme\]](#eqn:numerical_scheme){reference-type="eqref" reference="eqn:numerical_scheme"} in the Cucker-Smale case ($\mathrm{w}_0 = 1$) and for the $\mathrm{s}$-model with Motsch-Tadmor initial data ($\mathrm{w}_0 = 1/(\rho_0)_{\phi}$) are given in tables [2](#plots:solns_CS){reference-type="ref" reference="plots:solns_CS"} and [3](#plots:solns_WM_MT){reference-type="ref" reference="plots:solns_WM_MT"}. The numerical solution plots for the solution to [\[eqn:numerical_scheme_MT\]](#eqn:numerical_scheme_MT){reference-type="eqref" reference="eqn:numerical_scheme_MT"}, for the original Motsch-Tadmor model, is given in table [4](#plots:solns_MT){reference-type="ref" reference="plots:solns_MT"}. The parameters for all three cases are given in [8.3](#appdx:parameters){reference-type="ref" reference="appdx:parameters"}. In the Cucker-Smale case, there is rapid decay of the velocity of the small flock (i.e. rapid alignment to the large flock's velocity) and, as a result, less movement in the density of the small flock. Conversely, in the case of $\mathrm{s}$-model with Motsch-Tadmor initial data and for the original Mostch-Tadmor model, the velocity decays at a slower rate so there is more movement in the density of small flock. The point we are highlighting here is that the dynamics of the small flock in the Cucker-Smale case gets overpowered by the large flock, while in the case of the $\mathrm{s}$-model with Motsch-Tadmor initial data (and, of course, in the original Motsch-Tadmor model as stipulated in [@MT2011]), it does not. *Remark 23*. The $\mathrm{s}$-model and Motsch-Tadmor cases appear to be identical in the plots. This is due to the fact that the large flock remains almost stationary and, as a result, $\rho_{\phi}$ is almost stationary. Even though the differences of the solutions are not perceptable, the Motsch-Tadmor model does not have global well-posedness analysis, unlike the $\mathrm{s}$-model as we demonstrated across the paper. ## Parameters of the model {#appdx:parameters} In the numerical experiments shown described in [8.2](#appdx:comparison_CS_WM){reference-type="ref" reference="appdx:comparison_CS_WM"}, the time step $k = 2 / 40$ and the number of mesh elements is $M = 100$ (i.e. $h = 1/100$). The number of time steps is $40$ (i.e. the total time elapsed is $2$). The initial density comprises of a small mass flock and a large mass flock. The initial velocity gives the small flock a negative velocity and the large flock a zero velocity. Explicitly, initial density, initial velocity, and kernel are given by 1. $$\rho_0(x) = \begin{cases} \frac{1}{2} \exp \Big\{-\frac{1}{1 - (10*(x-0.25)^2)} \Big\} \hspace{8mm} \text{if } 0.15 < x < 0.35 \\ 50 \exp \Big\{-\frac{1}{1 - (10*(x-0.75)^2)} \Big\} \hspace{8mm} \text{if } 0.65 < x < 0.85 \\ 0 \hspace{8mm} \text{otherwise} \end{cases}$$ 2. $$u_0(x) = \begin{cases} \frac{1}{12 \pi} \cos(10 \pi (x - 0.15)) - \frac{1}{12 \pi} \hspace{8mm} \text{if } 0.15 < x < 0.35 \\ 0 \hspace{8mm} \text{otherwise} \end{cases}$$ 3. $$\phi(x) = \frac{1}{(1+x^2)^{1/2}}$$ The constant of $1/(12\pi)$ was chosen to guarentee that $e_0 = \partial_x u_0 + \mathrm{w}_0 (\rho_0)_{\phi} > 0$ (so the solution will not blow up, see Section [5](#GWP_1D){reference-type="ref" reference="GWP_1D"} for details on this threshold condition). In the Cucker-Smale simulation $\mathrm{w}_0 = 1$ (and therefore remains $1$ for all time). In the case of Motsch-Tadmor initial data, $\mathrm{w}_0 = 1/(\rho_0)_{\phi}$. Plots of the kernel and initial data are given in table [1](#plots:kernel_and_initial_data){reference-type="ref" reference="plots:kernel_and_initial_data"}. -------------------------------------------------------------------- ---------------------------------------------------------------------- {width="0.39\\linewidth"} {width="0.39\\linewidth"} {width="0.39\\linewidth"} {width="0.39\\linewidth"} -------------------------------------------------------------------- ---------------------------------------------------------------------- : The kernel $\phi$, initial density $\rho_0$, initial weight for the case of Motsch Tadmor initial data $\mathrm{w}_0 = 1/(\rho_0)_{\phi}$ and intial velocity $u_0$. Note that these graphs are at different scales. [\[plots:kernel_and_initial_data\]]{#plots:kernel_and_initial_data label="plots:kernel_and_initial_data"} --------------------------------------------------------------------------- ----------------------------------------------------------- {width="0.47\\linewidth"} {width="0.47\\linewidth"} --------------------------------------------------------------------------- ----------------------------------------------------------- : The computed solution densities zoomed into the small flock and the computed velocities for the Cucker-Smale case [\[plots:solns_CS\]]{#plots:solns_CS label="plots:solns_CS"} ------------------------------------------------------------------------------- --------------------------------------------------------------- {width="0.47\\linewidth"} {width="0.47\\linewidth"} ------------------------------------------------------------------------------- --------------------------------------------------------------- : The computed solution densities zoomed into the small flock and the computed velocities for the $\mathrm{s}$-model with Motsch-Tadmor initial data. [\[plots:solns_WM_MT\]]{#plots:solns_WM_MT label="plots:solns_WM_MT"} --------------------------------------------------------------------------- ----------------------------------------------------------- {width="0.47\\linewidth"} {width="0.47\\linewidth"} --------------------------------------------------------------------------- ----------------------------------------------------------- : The computed solution densities zoomed into the small flock and the computed velocities for the Motsch-Tadmor case. [\[plots:solns_MT\]]{#plots:solns_MT label="plots:solns_MT"} ## Numerical Convergence Experiment {#appdx:convg_experiment} The well-posedness and error analysis of the numerical scheme is not analyzed here. Instead, we provide evidence of convergence to a true solution as $k, h \to 0$. Observe that $\rho(t,x) = 1 + \sin(t)$, $\mathrm{w}(t,x) = \sin(t) + \frac{1}{2\pi} (2 + \sin(2\pi x))$, and $u(t,x) = \sin(t) + \frac{1}{2\pi} \sin(2\pi x)$ is a solution to the $\mathrm{s}$-model system with a forcing given by $$\begin{aligned} \label{eqn:wm_with_forcing} \begin{cases} \partial_t \rho + \nabla \cdot (u\rho) = \cos(t) + \sin(t) \cos(2 \pi x) + \cos(2 \pi x) \\ \partial_t \mathrm{w}+ u_F \cdot \nabla \mathrm{w}= \cos(t) + \sin(t) \cos(2 \pi x) \\ \partial_t u + u \cdot \nabla u = \mathrm{w}((u \rho)_{\phi} - u \rho_{\phi}) + \cos(t) + \sin(t) \cos(2 \pi x) + \frac{1}{4 \pi} \sin(4 \pi x) + \big( \sin(t) + \frac{1}{\pi} + \\\frac{1}{2 \pi} \sin(2 \pi x) \big) \big( \frac{1}{2 \pi} \sin(2 \pi x) + \frac{1}{2 \pi} \sin(t) \sin(2 \pi x) \big) \end{cases} \end{aligned}$$ The corresponding variational problem with a forcing $f = (f_1, f_2, f_3)$ is given by $$\begin{aligned} \label{eqn:numerical_scheme_with_forcing} \begin{cases} \frac{1}{k} \langle\rho^{n+1} - \rho^n, v \rangle- \langle\rho^{n+1} u^n, \frac{d}{dx} v \rangle= \langle f_1, v \rangle\\ \frac{1}{k} \langle\mathrm{w}^{n+1} - \mathrm{w}^n, v \rangle+ \langle(\frac{d}{dx} \mathrm{w}^{n+1}) \frac{(u^n \rho^n)_{\phi_h}}{\rho^n_{\phi_h}}, v \rangle= \langle f_2, v \rangle\\ \frac{1}{k} \langle u^{n+1} - u^n, q \rangle+ \langle u^{n+1} \frac{d}{dx} u^n, q \rangle= \langle\mathrm{w}^n (u^n \rho^n)_{\phi_h}, q \rangle- \langle\mathrm{w}^n u^{n+1} \rho^n_{\phi_h}, q \rangle+ \langle f_3, q \rangle \end{cases} \end{aligned}$$ We will provide evidence that the numerical solution to [\[eqn:numerical_scheme_with_forcing\]](#eqn:numerical_scheme_with_forcing){reference-type="eqref" reference="eqn:numerical_scheme_with_forcing"}, with forcing $f$ equal to the right hand side of [\[eqn:wm_with_forcing\]](#eqn:wm_with_forcing){reference-type="eqref" reference="eqn:wm_with_forcing"}, converges to the solution to [\[eqn:wm_with_forcing\]](#eqn:wm_with_forcing){reference-type="eqref" reference="eqn:wm_with_forcing"}, which in turn provides evidence that the original numerical scheme [\[eqn:numerical_scheme\]](#eqn:numerical_scheme){reference-type="eqref" reference="eqn:numerical_scheme"} is a (conditionally) stable and convergent scheme. Convergence is measured in the $H^1$ norm. To distinguish the numerical solutions, let $(\rho_h, \mathrm{w}_h, u_h)$ denote the solutions to [\[eqn:numerical_scheme\]](#eqn:numerical_scheme){reference-type="eqref" reference="eqn:numerical_scheme"}. For a given numerical solution, we compute, at a specified time $T$, the $L^2$ error and the $L^2$ error of the gradient seperately (since we expect faster convergence rates for the $L^2$ error). Here, we assume that $T$ coincides with one of the discrete times (i.e. for instance, $\rho_h(T, x) = \rho^n(x)$ for some $n$). We denote the errors for a given mesh $h,k$ by $E_{h,k}^0, E_{h,k}^1$, respectively. $$\begin{aligned} &E_{h,k}^0(T) = \|\rho(T, \cdot) - \rho_h(T, \cdot)\|_{L^2}^2 + \|\mathrm{w}(T, \cdot) - \mathrm{w}_h(T, \cdot)\|_{L^2}^2 + \|u(T, \cdot) - u_h(T, \cdot)\|_{L^2}^2, \\ &E_{h,k}^1(T) = \|\partial_x \rho(T, \cdot) - \partial_x \rho_h(T, \cdot) \|_{L^2}^2 + \|\partial_x \mathrm{w}(T, \cdot) - \partial_x \mathrm{w}_h(T, \cdot) \|_{L^2}^2 + \| \partial_x u(T, \cdot) - \partial_x u_h(T, \cdot) \|_{L^2}^2 \end{aligned}$$ To illustrate convergence of the numerical scheme in $H^1$, we perform the following test. Fix $T = 0.5$. Vary the spatial and temporal mesh size simaltaneously, $h_i = \frac{1}{2^i}$ with $2 \leqslant i \leqslant 7$, $k_i = h_i/4$. For each $(h,k)$, the errors $E_{h,k}^0(T)$ and $E_{h,k}^1(T)$ are computed and the loglog graph of the errors with respect to $h$ (with the understanding that $k = h/2$) is shown in Table [5](#plots:vary_k_and_h){reference-type="ref" reference="plots:vary_k_and_h"}. *Remark 24*. In the numerical experiments for [\[eqn:numerical_scheme_with_forcing\]](#eqn:numerical_scheme_with_forcing){reference-type="eqref" reference="eqn:numerical_scheme_with_forcing"}, a Courant-Friedrichs-Lewy condition for the mesh parameters $h,k$ was observed. In some cases, when $k \approx h$, numerical instability was observed. We found, experimentally, that $k = h/4$ suppressed these instabilities. Although, we did not find that such a condition was necessary for the unforced equations, [\[eqn:numerical_scheme\]](#eqn:numerical_scheme){reference-type="eqref" reference="eqn:numerical_scheme"} and [\[eqn:numerical_scheme_MT\]](#eqn:numerical_scheme_MT){reference-type="eqref" reference="eqn:numerical_scheme_MT"}. --------------------------------------------------------------------------- -------------------------------------------------------------------------------- {width="0.47\\linewidth"} {width="0.47\\linewidth"} --------------------------------------------------------------------------- -------------------------------------------------------------------------------- : Loglog plot of the $L^2$ error and $L^2$ error of the gradient with resepct to $h$, where $k = h/4$, i.e. for fixed $T = 0.5$, the mesh sizes are $(h,k) = (\frac{1}{4}, \frac{1}{16}), (\frac{1}{8}, \frac{1}{32}), (\frac{1}{16}, \frac{1}{64}), (\frac{1}{32}, \frac{1}{128}), (\frac{1}{64}, \frac{1}{256})$. [\[plots:vary_k\_and_h\]]{#plots:vary_k_and_h label="plots:vary_k_and_h"} # Conclusion and Future Work We have extended many important classical results about the Cucker-Smale model to the more versatile $\mathrm{s}$-model with adaptive strength and Favre averaging protocol [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"}. In order to gain versatility of behavior, it sacrificed the conservation of momentum and the energy law. Nonetheless, we showed that it still retains many of the desirable analytical qualities of the Cucker-Smale model-- namely alignment, local well-posedness, a threshold condition for global well-posedness in 1D, existence for small data and uni-directional flocks, strong flocking, and relative entropy estimates on the limiting flock. Although such results were obtained for the $\mathrm{s}$-model with Favre averaging, conceivably an extension to the general environmental averaging model in the sense of [@Sbook] is possible. Indeed, with the appropriate assumptions on the averaging $[\textbf{u}]_{\rho}$, one expects to obtain alignment results, local well-posedness, and a threshold condition for global well-posedness for uni-directional flocks. However, the small data, strong flocking, and entropy estimates depends on the exponential decay of the derivatives of the Favre averaged velocity $\textbf{u}_F$, which incidentally, depends on the algebraic structure of the Favre averaging. Such results will therefore not be able to be easily extended to more general averaging operators $[\textbf{u}]_{\rho}$ in the $\mathrm{s}$-model [\[s_model\]](#s_model){reference-type="eqref" reference="s_model"}. While not as general as [\[s_model\]](#s_model){reference-type="eqref" reference="s_model"}, [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"} provides an important unification of the Cucker-Smale and Motsch-Tadmor models into one which retains many of the desirable analytic and qualitative features of both. Unlike Cucker-Smale and Motsch-Tadmor, it was conceived at the hydrodynamic level and lacks a discrete and kinetic description. It may therefore be of general interest to put [\[EAS_WM\]](#EAS_WM){reference-type="eqref" reference="EAS_WM"} on more firm physical and theoretical grounds by researching its discrete and kinetic counterparts from which it arises. # Appendix The invariance and contractivity estimates on the map $\mathcal{F}$ for the local well-posedness argument in Section [4](#LWP){reference-type="ref" reference="LWP"} use the analyticity property of the heat semigroup and non-linear estimates on the derivatives. For the non-linear estimates on derivatives, the Faa di Bruno Formula and the Gagliardo-Nirenburg inequality are used to obtain estimates on $\|[\textbf{u}_1]_{\rho_1} - [\textbf{u}_1]_{\rho_2} \|_{H^l}$ and $\| \textbf{u}_F \|_{H^l}$, where $\textbf{u}_F = (\textbf{u}\rho)_{\phi} / \rho_{\phi}$ is the Favre-Filtration. Due to the presence of $\rho_{\phi}$ in the denominator of the Favre-filtration, an estimate on the sobolev norm of the reciprical is necessary. The estimate [\[recip_rho_estimate\]](#recip_rho_estimate){reference-type="eqref" reference="recip_rho_estimate"} on the Sobolev norm of $1/f$ is a specific case of Lemma 2.5 of [@lear_reynolds_shvydkoy2020local], which estimates the fractional Sobolev norm for the purpose of showing local well-posedness of topological models. However, we record a simplified version here in order to highlight the dependence of the estimates on $\|1/\rho\|_{\infty}$ and to avoid the unnecessary dependence on $\nabla \rho$. We also record the Csizar-Kullback inequality used in the entropy estimates in Section [6.2](#sec:entropy){reference-type="ref" reference="sec:entropy"}. ## Classical Lemmas We record the Gagliardo-Nirenberg inequality, Csiszar-Kullback inequality, and analyticity proprerty of the heat semi-group. (Gagliardo-Nirenburg Inequality) [\[lma:GN_inequality\]]{#lma:GN_inequality label="lma:GN_inequality"} Assume the domain is $\mathbb{T}^n$ or $\mathbb{R}^n$. If $1 \leqslant q \leqslant\infty$, $0 \neq j < m$ integers, $1 \leqslant r \leqslant\infty$, $p \geqslant 1$ and $0 \leqslant\theta < 1$ such that $$\begin{aligned} \frac{1}{p} = \frac{j}{n} + \theta \Big(\frac{1}{r} - \frac{m}{n} \Big) + \frac{1-\theta}{q}, \hspace{5mm} \frac{j}{m} \leqslant\theta < 1 \end{aligned}$$ then there's a constant $C := C(j,m,n,q,r,\theta)$ such that $$\begin{aligned} \|D^j f\|_{L^p} \leqslant C \|D^m f\|_{L^r}^{\theta} \|f\|_{L^{q}}^{1-\theta} \end{aligned}$$ for any $f \in L^q \cap L^2 \cap W^{m,r}$. For our purposes, we set $r = 2$ and place the smallest power on the $W^{m,r}(\mathbb{R}^n)$ norm, i.e. $\theta = j/m$. We obtain $$\begin{aligned} \|f\|_{W^{j,p}} \leqslant C \|f\|_{H^m}^{j/m} \|f\|_{L^q}^{1-j/m} \end{aligned}$$ where $$\begin{aligned} \frac{1}{p} = \frac{j}{2m} + \frac{1}{q} \Big(1 - \frac{j}{m} \Big) \end{aligned}$$ (Csiszar-Kullback Inequality) [\[lma:ck_inequality\]]{#lma:ck_inequality label="lma:ck_inequality"} The entropy is given by $\mathcal{H}= \int_{\mathbb{T}} \rho \log \frac{\rho}{\bar{\rho}} dx$ where $\bar{\rho} = \frac{1}{2\pi} \int_{\mathbb{T}} \rho(x)dx$. Then $$\begin{aligned} \frac{1}{4\pi} \|\rho - \bar{\rho}\|_{L^1}^2 \leqslant\bar{\rho} \mathcal{H}\leqslant\|\rho - \bar{\rho}\|_{L^2}^2 \end{aligned}$$ (Analyticity Property of Heat Semigroup) For all $\epsilon, t > 0$, there exists a constant $C > 0$ independent of $f$, $\epsilon$, $t$ such that $$\begin{aligned} \|\partial e^{\epsilon t \Delta} f \|_p \leqslant\frac{C}{\sqrt{\epsilon t}} \|f\|_p \hspace{10mm} 1 \leqslant p \leqslant\infty \end{aligned}$$ ## Non-linear estimates on derivatives To establish the aforementioned non-linear estimates on derivatives, we first record the Fa di Bruno Formula. For a more general version of [\[recip_rho_estimate\]](#recip_rho_estimate){reference-type="ref" reference="recip_rho_estimate"} on a fractional Sobolev space, we refer to Lemma 2.5 of [@lear_reynolds_shvydkoy2020local]. (Faa di Bruno Formula) Let $f^{(i)}$ denote the $i^{th}$ partial derivative of $f$. $$\begin{aligned} \partial^P h(g) = \sum_{\textbf{j}} \frac{P!}{j_1!1!^{j_1} j_2!2!^{j_2} \dots j_P!P!^{j_P}} h^{(j_1 + \dots + j_p)}(g) \prod_{i=1}^{|\textbf{j}|} g^{(k_i)} \end{aligned}$$ where the sum is over all $P$-tuples of non-negative integers $\textbf{j} = (j_1, \dots, j_P)$ satisfying $$\begin{aligned} 1*j_1 + 2*j_2 + 3*j_3 + \dots + P*j_P = P \end{aligned}$$ and $$\begin{aligned} k_1 + k_2 + \dots + k_{|\textbf{j}|} = P \end{aligned}$$ ($H^l$ estimate on $1/f$) [\[recip_rho_estimate\]]{#recip_rho_estimate label="recip_rho_estimate"} Assume the domain is $\mathbb{T}^n$ or $\mathbb{R}^n$. For $f \in H^l$, there's a constant $C := C(l,n)$ such that $$\begin{aligned} \Big\| \partial^l\Big( \frac{1}{f} \Big) \Big\|_2 \leqslant C \Big\| \frac{1}{f} \Big\|_{\infty}^{l+1} \| f \|_{H^l} \end{aligned}$$ and $$\begin{aligned} \Big\| \partial^l\Big( \frac{\nabla f}{f} \Big) \Big\|_2 \leqslant C \Big( \|f\|_{H^{l+1}} \Big\|\frac{1}{f} \Big\|_{\infty} + \|\nabla f\|_{\infty} \Big\| \frac{1}{f} \Big\|_{\infty}^{l+1} \| f \|_{H^l} \Big) \\ \end{aligned}$$  *Proof.* Using $h(x) = \frac{1}{x}$ and $g(x) = f(x)$ in Faa di Bruno's formula and Holder's Inequality, we have for some constant $C' := C'(l)$, which may change from line to line, $$\begin{aligned} \Big\| \partial^l\Big( \frac{1}{f} \Big) \Big\|_2 &= \Big\| \sum_{\textbf{j}} \frac{l!}{j_1!1!^{j_1} j_2!2!^{j_2} \dots j_l!l!^{j_l}} \frac{(-1)^{j_1 + \dots + j_l}}{f^{j_1 + \dots + j_l + 1}} \prod_{i=1}^{|\textbf{j}|} f^{(k_i)} \Big\|_2 \\ &\leqslant C' \Big\| \frac{1}{f} \Big\|_{\infty}^{l+1} \prod_{i=1}^{|\textbf{j}|} \| (\partial^{k_i} f) \|_{L^{p_i}} \\ &\leqslant C' \Big\| \frac{1}{f} \Big\|_{\infty}^{l+1} \prod_{i=1}^{|\textbf{j}|} \| f \|_{W^{k_i, p_i}} \\ \end{aligned}$$ where $\sum_{i=1}^{|\textbf{j}|} k_i = l$ and $\sum_{i=1}^{|\textbf{j}|} 1/p_i = 1/2$. Choosing $\frac{1}{p_i} = \frac{k_i}{2l}$ and $q=\infty$ in the Gagliardo-Nirenberg inequality [\[lma:GN_inequality\]](#lma:GN_inequality){reference-type="ref" reference="lma:GN_inequality"}, we obtain for some constant $C := C(l, n)$, which may change from line to line $$\begin{aligned} &\leqslant C \Big\| \frac{1}{f} \Big\|_{\infty}^{l+1} \prod_{i=1}^{|\textbf{j}|} \| f \|_{H^l}^{k_i/l} = C \Big\| \frac{1}{f} \Big\|_{\infty}^{l+1} \| f \|_{H^l}\\ \end{aligned}$$ Using this and the product estimate, we can estimate for $f \in H^{l+1}$, $$\begin{aligned} \Big\| \partial^l\Big( \frac{\nabla f}{f} \Big) \Big\|_2 &\leqslant C \Big( \|f\|_{H^{l+1}} \Big\|\frac{1}{f} \Big\|_{\infty} + \|\nabla f\|_{\infty} \Big\|\frac{1}{f} \Big\|_{H^l} \Big) \\ &\leqslant C \Big( \|f\|_{H^{l+1}} \Big\|\frac{1}{f} \Big\|_{\infty} + \|\nabla f\|_{\infty} \Big\| \frac{1}{f} \Big\|_{\infty}^{l+1} \| f \|_{H^l} \Big) \\ \end{aligned}$$ ◻ ($H^l$ Contractivity estimate on Favre Filtration) [\[contractivity_estimate\]]{#contractivity_estimate label="contractivity_estimate"} Let $\textbf{u}_F = (\textbf{u}\rho)_{\phi} / \rho_{\phi}$. Then there exists a contant $C := C(l,n)$ such that $$\begin{aligned} \| [\textbf{u}_1]_{\rho_1} - [\textbf{u}_1]_{\rho_2} \|_{H^{l}} &\leqslant\Big\| \frac{1}{{\rho_1}_{\phi} {\rho_2}_{\phi}} \Big\|_{\infty} \| (\textbf{u}_1 \rho_1)_{\phi} ({\rho_2})_{\phi} - (\textbf{u}_1 \rho_2)_{\phi} ({\rho_1})_{\phi} \|_{H^{l}} \\ &+ C \Big\| \frac{1}{{\rho_1}_{\phi} {\rho_2}_{\phi}} \Big\|_{\infty}^{l+1} \| {\rho_1}_{\phi} {\rho_2}_{\phi} \|_{H^{l}} \| (\textbf{u}_1 \rho_1)_{\phi} ({\rho_2})_{\phi} - (\textbf{u}_1 \rho_2)_{\phi} ({\rho_1})_{\phi} \|_{\infty} \end{aligned}$$ *Proof.* Using the commutator estimate, we obtain $$\begin{aligned} \| [\textbf{u}_1]_{\rho_1} - [\textbf{u}_1]_{\rho_2} \|_{H^l} &= \Big\| \frac{ (\textbf{u}_1 \rho_1)_{\phi} ({\rho_2})_{\phi} - (\textbf{u}_1 \rho_2)_{\phi} ({\rho_1})_{\phi}} { {\rho_1}_{\phi} {\rho_2}_{\phi} } \Big\|_{H^l} \\ &\leqslant\Big\| \frac{1}{{\rho_1}_{\phi} {\rho_2}_{\phi}} \Big\|_{\infty} \| (\textbf{u}_1 \rho_1)_{\phi} ({\rho_2})_{\phi} - (\textbf{u}_1 \rho_2)_{\phi} ({\rho_1})_{\phi} \|_{H^l} \\ &+ \Big\| \frac{1}{{\rho_1}_{\phi} {\rho_2}_{\phi}} \Big\|_{H^l} \| (\textbf{u}_1 \rho_1)_{\phi} ({\rho_2})_{\phi} - (\textbf{u}_1 \rho_2)_{\phi} ({\rho_1})_{\phi} \|_{\infty} \\ \end{aligned}$$ Then [\[recip_rho_estimate\]](#recip_rho_estimate){reference-type="ref" reference="recip_rho_estimate"} applied to $1/({\rho_1}_{\phi} {\rho_2}_{\phi})$ gives $$\begin{aligned} \Big\| \frac{1}{{\rho_1}_{\phi} {\rho_2}_{\phi}} \Big\|_{H^l} \leqslant C \Big\| \frac{1}{{\rho_1}_{\phi} {\rho_2}_{\phi}} \Big\|_{\infty}^{l+1} \| {\rho_1}_{\phi} {\rho_2}_{\phi} \|_{H^l} \end{aligned}$$ and the desired inequality follows. ◻ ($H^l$ estimate on Favre Filtration) [\[favre_estimate\]]{#favre_estimate label="favre_estimate"} Let $\textbf{u}_F = (\textbf{u}\rho)_{\phi} / \rho_{\phi}$. Then there's a constant $C := C(l,m)$ $$\| \textbf{u}_F \|_{H^l} \leqslant\Big\| \frac{1}{\rho_{\phi}} \Big\|_{\infty} \|(\textbf{u}\rho)_{\phi} \|_{H^l} + C \Big\| \frac{1}{\rho_{\phi}} \Big\|_{\infty}^{l+1} \| \rho_{\phi} \|_{H^l} \|(\textbf{u}\rho)_{\phi} \|_{\infty}$$ *Proof.* Apply the commutator estimate and the Gagliardo-Nirenberg inequality as in [\[contractivity_estimate\]](#contractivity_estimate){reference-type="ref" reference="contractivity_estimate"}. ◻ 10 G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato, and J. Soler. Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives. , 29(10):1901--2005, 2019. J. A. Carrillo, Y.-P. Choi, E. Tadmor, and C. Tan. Critical thresholds in 1D Euler equations with non-local forces. , 26(1):185--206, 2016. F. Cucker and S. Smale. Emergent behavior in flocks. , 52(5):852--862, 2007. F. Cucker and S. Smale. On the mathematics of emergence. , 2(1):197--227, 2007. A. Favre. Turbulence: Space-time statistical properties and behavior in supersonic flows. , 26(10):2851--2863, 1983. A. Figalli and M.-J. Kang. A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignment. , 12(3):843--866, 2019. M. Grassin. Existence of global smooth solutions to Euler equations for an isentropic perfect gas. In *Hyperbolic problems: theory, numerics, applications, Vol. I (Zürich, 1998)*, volume 129 of *Internat. Ser. Numer. Math.*, pages 395--400. Birkhäuser, Basel, 1999. S.-Y. Ha, M.-J. Kang, and B. Kwon. A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid. , 24(11):2311--2359, 2014. S.-Y. Ha, M.-J. Kang, and B. Kwon. Emergent dynamics for the hydrodynamic Cucker-Smale system in a moving domain. , 47(5):3813--3831, 2015. S.-Y. Ha and J.-G. Liu. A simple proof of the Cucker-Smale flocking dynamics and mean-field limit. , 7(2):297--325, 2009. S.-Y. Ha and E. Tadmor. From particle to kinetic and hydrodynamic descriptions of flocking. , 1(3):415--435, 2008. S. He and E. Tadmor. Global regularity of two-dimensional flocking hydrodynamics. , 355(7):795--805, 2017. D. Lear and R. Shvydkoy. Existence and stability of unidirectional flocks in hydrodynamic Euler Alignment systems. , 15(1):175--196, 2019. https://arxiv.org/abs/1911.10661. T. M. Leslie and R. Shvydkoy. On the structure of limiting flocks in hydrodynamic Euler Alignment models. , 29(13):2419--2431, 2019. A. Logg and G. N. Wells. automated finite element computing. , 37, 2010. A. Logg, G. N. Wells, and J. Hake. a C++/Python finite element library. In A. Logg, K.-A. Mardal, and G. N. Wells, editors, *Automated Solution of Differential Equations by the Finite Element Method*, volume 84 of *Lecture Notes in Computational Science and Engineering*, chapter 10. Springer, 2012. S. Motsch and E. Tadmor. A new model for self-organized dynamics and its flocking behavior. , 144(5):923--947, 2011. S. Motsch and E. Tadmor. Heterophilious dynamics enhances consensus. , 56(4):577--621, 2014. F. Poupaud. Global smooth solutions of some quasi-linear hyperbolic systems with large data. , 8(4):649--659, 1999. David N Reynolds and Roman Shvydkoy. Local well-posedness of the topological euler alignment models of collective behavior. , 33(10):5176, 2020. Roman Shvydkoy. . Nečas Center Series. Birkhäuser/Springer, Cham, \[2021\] ©2021. Roman Shvydkoy. Environmental averaging. , 2022. E. Tadmor and C. Tan. Critical thresholds in flocking hydrodynamics with non-local alignment. , 372(2028):20130401, 22, 2014. Eitan Tadmor. On the mathematics of swarming: emergent behavior in alignment dynamics. , 68(4):493--503, 2021. [^1]: **Acknowledgment.** This work was supported in part by NSF grant DMS-2107956.

arxiv_math

--- abstract: | Numerical semigroups with multiplicity $m$ are parameterized by integer points in a polyhedral cone $C_m$, according to Kunz. For the toric ideal of any such semigroup, the main result here constructs a free resolution whose overall structure is identical for all semigroups parametrized by the relative interior of a fixed face of $C_m$. This resolution is minimal when the semigroup is maximal embedding dimension, which is the case parametrized by the interior of $C_m$ itself. This resolution is employed to show uniformity of Betti numbers for all toric ideals defined by semigroups parametrized by points interior to a single face of $C_m$. address: - | Mathematics Department\ University of Kentucky\ Lexington, KY 40506 - | School of Mathematics\ University of Minnesota\ Minneapolis, MN 55455 - | Department of Mathematics\ Duke University\ Durham, NC 27708 - | Mathematics Department\ San Diego State University\ San Diego, CA 92182 - | Mathematics Department\ University of Wisconsin Madison\ Madison, WI 53706 author: - Benjamin Braun - Tara Gomes - Ezra Miller - Christopher O'Neill - Aleksandra Sobieska date: 3 October 2023 title: | Minimal free resolutions of numerical\ semigroup algebras via Apéry specialization --- [^1] # Introduction {#sec:intro} Given a numerical semigroup $S$, the corresponding semigroup algebra has a defining toric ideal $I_S$. While the study of algebraic invariants of $I_S$ falls within the broader study of toric ideals, the family of toric ideals for numerical semigroups forms a rich and interesting area of study that often affords more refined general results than are known or possible for the general toric setting. The subject of this paper is free resolutions and Betti numbers for $I_S$. For general toric ideals, there is a substantial literature on their resolutions and Betti numbers. In 1998, Peeva and Sturmfels described minimal free resolutions for generic lattice ideals [@genericlattice]. More recently, Tchernev gave an explicit recursive algorithm for canonical minimal resolutions of toric rings [@minresrecursive]. Further, Li, Miller, and Ordog construct a canonical minimal free resolution of an arbitrary positively graded lattice ideal with a closed-form combinatorial description of the differential in characteristic $0$ and all but finitely many positive characteristics [@minreslattice]. However, these constructions are all quite general, and one would hope that in the special case of numerical semigroups, explicit resolutions of $I_S$ more directly tied to the combinatorics of $S$ are possible. Free resolutions of $I_S$ for a numerical semigroup $S$ are known in some special cases. The surveys [@nssyzygysurvey; @nsbettisurvey] include most results concerning special families. We outline a few here. If $S$ is maximal embedding dimension (MED) and $I_S$ is determinantal (that is, generated by the minors of a matrix), then $I_S$ is resolved by the Eagon--Northcott complex [@gotoencomplex]. This accounts for some, but not all, MED numerical semigroups; a characterization of determinantal MED numerical semigroups is given in [@determinentalmed]. If $S$ is generated by an arithmetic sequence, then $I_S$ is minimally resolved by a variant of the Eagon--Northcott complex [@nsfreeresarith]. If $S$ is obtained as a gluing of two numerical semigroups $T$ and $T'$, then a minimal free resolution of $I_S$ can be obtained from the minimal free resolutions of $I_T$ and $I_{T'}$ via a mapping cone construction [@nsfreeresgluing]; this includes the case where $I_S$ is complete intersection. If $S$ has at most 3 generators, then a minimal free resolution is known, and in particular $\beta_0(I_S) \in \{2, 3\}$. Numerous families of 4-generated numerical semigroups have been investigated, in part to illustrate some extremal Betti number behavior; see the survey [@nsbettisurvey] for more detail. The present work is motivated by recent papers that examine a family of convex rational polyhedra $C_m$ called *Kunz cones*, one for each integer $m \ge 2$, for which each numerical semigroup $S$ with multiplicity $m$---that is, with $m = \min(S \setminus \{0\})$---corresponds to an integer point of $C_m$. When these were first introduced in [@kunz], it was shown that the point in $C_m$ corresponding to $S$ lies in the interior of $C_m$ if and only if $S$ is MED. Most subsequent on this topic papers have employed lattice point techniques to examine enumerative questions [@alhajjarkunz; @kaplancounting; @kunzcoords]. More recently, it was shown that two numerical semigroups $S$ and $T$ correspond to points (relative) interior to the same face of $C_m$ if and only if certain subsets of their divisibility posets coincide [@wilfmultiplicity; @kunzfaces1]. It was then proven in [@kunzfaces3] that if $S$ and $T$ lie interior to the same face of $C_m$, then $\beta_0(I_S) = \beta_0(I_T)$, and the authors conjectured that in this case, $\beta_d(I_S) = \beta_d(I_T)$ for every $d$. Our main result is a proof of this conjecture by constructing explicit free resolutions of all numerical semigroup ideals. These resolutions are minimal when the semigroup has maximal embedding dimension and are minimalized uniformly for all numerical semigroups lying interior to the same face of the Kunz cone $C_m$. More precisely, our contributions are as follows. 1. We introduce the Apéry toric ideal $J_S$ of $S$, an analogue of $I_S$ that lies in a ring with $m$ variables instead of a ring with one variable per minimal generator of $S$. A generating set for $J_S$ can be obtained by concatenating any generating set for $I_S$ and a regular sequence with one element for each additional variable. 2. For any positive integer $m\geq 2$, we construct a free resolution of $J_S$ called the *Apéry resolution*. The rank of the $d$-th free module depends only on $m$ and $d$, and the positions of the nonzero entries of the matrices representing the boundary maps depend only on $m$. An example of this for $m=4$ is given in Figure [\[fig:m4res\]](#fig:m4res){reference-type="ref" reference="fig:m4res"}, where the values $b_{i,j}$ depend on $S$. 3. When $S$ corresponds to a point interior to $C_m$, i.e., when $S$ is MED and thus $J_S = I_S$, the Apéry resolution is a minimal free resolution of $I_S$. 4. [\[i:step4\]]{#i:step4 label="i:step4"} For any numerical semigroups $S$ and $T$ corresponding to point interior to the same face $F$ of $C_m$, we prove there exists a uniform method for modifying the Apéry resolutions of $J_S$ and $J_T$ to minimal free resolutions in such a way that the resulting ranks of the free modules and the positions of the nonzero entries of the matrices representing the boundary maps depend only on $m$, $F$, and $d$. 5. As a result, we conclude that for any $S$ and $T$ whose points lie interior to the same face $F$ of $C_m$, the Betti numbers of $I_S$ and $I_T$ are identical. $%$ \begin{array}{c@{\:}c@{\:}c@{\:}c@{\:}c} & \begin{blockarray}{rcccccc} \\ \\ & \scriptstyle\textbf{1,1} & \scriptstyle\textbf{2,2} & \scriptstyle\textbf{3,3} & \scriptstyle\textbf{2,1} & \scriptstyle\textbf{3,1} & \scriptstyle\textbf{3,2} \\ \begin{block}{r@{\,\,}[*{6}{@{\,}l}]} \scriptstyle\varnothing& x_1^2 - x_2 y^{b_{11}} & x_2^2 - y^{b_{22}} & x_3^2 - x_2 y^{b_{33}} & x_1x_2 - x_3 y^{b_{12}} & x_1x_3 - y^{b_{13}} & x_2x_3 - x_1 y^{b_{23}} \\ \end{block} \end{blockarray} & \\[-1em] 0 \leftarrow R & \mathord\leftarrow \mkern-6mu \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill \mkern-6mu \mathord-& \end{array}$ $%$ \begin{array}{c@{\:}c@{\:}c@{\:}c@{\:}c@{}c} & \begin{blockarray}{@{}r*{8}{@{}c}} \\ \\ & \scriptstyle\textbf{1,12} & \scriptstyle\textbf{1,13} & \scriptstyle\textbf{2,12} & \scriptstyle\textbf{2,23} & \scriptstyle\textbf{3,13} & \scriptstyle\textbf{3,23} & \scriptstyle\textbf{2,13} & \scriptstyle\textbf{3,12} \\ \begin{block}{@{}l@{\,\,\,}[*{8}{@{}l}]} \scriptstyle\textbf{1,1} & -x_2 & -x_3 & \phantom{-}& \phantom{-} & \phantom{-}& \phantom{-}&\,\phantom{-}y^{b_{23}}& \phantom{-}y^{b_{23}} \\ \scriptstyle\textbf{2,2} & -y^{b_{11}} & \phantom{-}& \,\phantom{-}x_1 & -x_3 & \phantom{-}& \phantom{-}y^{b_{33}} & \phantom{-}& \phantom{-} \\ \scriptstyle\textbf{3,3} & \phantom{-}& \phantom{-}& \phantom{-}& \phantom{-} & \phantom{-}x_1 & \phantom{-}x_2 &\,-y^{b_{12}}& \phantom{-} \\ \scriptstyle\textbf{2,1} & \phantom{-}x_1 & \phantom{-}& \,-x_2 & \phantom{-}y^{b_{23}} & \phantom{-}y^{b_{33}} & \phantom{-}&\,-x_3 & \phantom{-} \\ \scriptstyle\textbf{3,1} & \phantom{-}y^{b_{12}} & \phantom{-}x_1 & \phantom{-}& \phantom{-} & -x_3 & -y^{b_{23}} & \phantom{-}& -x_2 \\ \scriptstyle\textbf{3,2} & \phantom{-}& -y^{b_{11}} &\,-y^{b_{12}}& \phantom{-}x_2 & \phantom{-}& -x_3 &\,\phantom{-}x_1 & \phantom{-}x_1 \\ \end{block} \end{blockarray} && \begin{blockarray}{@{}r*{3}{@{}c}} & \scriptstyle\textbf{1,[3]} & \scriptstyle\textbf{2,[3]} & \scriptstyle\textbf{3,[3]} \\ \begin{block}{@{}l@{\,\,}[*{3}{@{\,}l}]} \scriptstyle\textbf{1,12} & \phantom{-}x_3 & -y^{b_{23}} & \\ \scriptstyle\textbf{1,13} & -x_2 & & \phantom{-}y^{b_{23}}\\ \scriptstyle\textbf{2,12} & & \phantom{-}x_3 & -y^{b_{33}}\\ \scriptstyle\textbf{2,23} & -y^{b_{11}} & \phantom{-}x_1 & \\ \scriptstyle\textbf{3,13} & \phantom{-}y^{b_{12}} & & -x_2 \\ \scriptstyle\textbf{3,23} & & -y^{b_{12}} & \phantom{-}x_1 \\ \scriptstyle\textbf{2,13} & \phantom{-}x_1 & -x_2 & \\ \scriptstyle\textbf{3,12} & -x_1 & \phantom{-}& \phantom{-}x_3 \\ \end{block} \end{blockarray} & \\[-1em] R^6 & \mathord\leftarrow \mkern-6mu \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill \mkern-6mu \mathord-& \!R^8 & \mathord\leftarrow \mkern-6mu \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill \mkern-6mu \mathord-& \!R^3 & \leftarrow 0 \end{array}$ The term "specialization" in the titles of the paper and Section [4](#sec:specialization){reference-type="ref" reference="sec:specialization"} refers to passage from the interior of the Kunz cone to a face, which entails some facet inequalities to become equalities. Consequently, some exponents on $y$ variables (as in Figure [\[fig:m4res\]](#fig:m4res){reference-type="ref" reference="fig:m4res"}) pass from positive to $0$, which results in the specialization that sets $y = 1$. Further substitutions among the $x$ variables---extraneous ones are set equal to monomials in the others---combine in Step [\[i:step4\]](#i:step4){reference-type="ref" reference="i:step4"} with row and column operations to produce minimal free resolutions from the original Apéry resolution. The remainder of this paper is structured as follows. Section [2](#sec:background){reference-type="ref" reference="sec:background"} reviews basic properties of numerical semigroups and Kunz cones, and we define the modules and maps used in the Apéry resolution. Section [3](#sec:resolution){reference-type="ref" reference="sec:resolution"} proves that the Apéry resolution is indeed a resolution, and establish the minimality of this resolution when $S$ is MED. Section [4](#sec:specialization){reference-type="ref" reference="sec:specialization"} describes how to modify the Apéry resolution in a uniform way for all numerical semigroups in the interior of a fixed face of $C_m$ to obtain a minimal resolution. Further research directions are outlined in Section [5](#sec:openquesitons){reference-type="ref" reference="sec:openquesitons"}. # Kunz polyhedra and Apéry resolutions {#sec:background} ## Semigroups and toric ideals {#b:toric-ideals} A *numerical semigroup* is a subsemigroup of $(\mathbb{Z}_{\ge 0}, +)$ that contains $0$ and has finite complement. Throughout this work, fix a numerical semigroup $S \subset \mathbb{Z}_{\ge 0}$ with *multiplicity* $$%$$ $ \mathsf m(S) = \min(S \setminus \{0\}) = m$$ and write $$\begin{aligned} %\end{align*} \mathop{\mathrm{Ap}}(S) &= \{n \in S : n - m \notin S\} \\ &= \{0, a_1, \ldots, a_{m-1}\}\end{aligned}$$ for the *Apéry set* consisting of the minimal element of $S$ from each equivalence class modulo $m$, where each $a_i$ satisfies $a_i \equiv i \bmod m$. For convenience, define $a_0 = m$; this convention plays an important role in our later formulas. In particular, $$%$$ $ S = \langle m, a_1, \ldots, a_{m-1}\rangle=\langle a_0, a_1, \ldots, a_{m-1}\rangle,$$ though this generating set need not be the unique minimal generating set $\mathcal A(S)$ of $S$, such as when $a_i + a_j = a_{i+j}$ for some $i, j$, where indices are summed modulo $m$. The semigroup $S$ has *maximum embedding dimension (MED)* if $\mathcal A(S) = \{a_0, \ldots, a_{m-1}\}$. **Example 1**. The semigroup $S=\langle 4,9,11,14\rangle$ has multiplicity $\mathsf m(S) = 4$ and Apéry set $\mathop{\mathrm{Ap}}(S)=\{0,9,14,11\}$. The semigroup $T=\langle 4,13,23\rangle$ has multiplicity $\mathsf m(T) = 4$ and $\mathop{\mathrm{Ap}}(T)=\{0,13,26,23\}$. Note that $a_1 + a_1 = a_2$ in $T$, and thus the Apéry set is not a minimal generating set. Let $R = \Bbbk[x_0, x_1, \ldots, x_{m-1}]$ with the natural grading by $\mathbb{Z}$ via $\deg(x_i) = a_i$ and set $y = x_0$. The *Apéry toric ideal* of $S$ is the kernel $J_S = \ker(\varphi)$ of the homomorphism $$\begin{aligned} %\end{align*} \varphi:R & \longrightarrow \Bbbk[t] \\ x_i & \longmapsto t^{a_i}\end{aligned}$$ and the *defining toric ideal* of $S$ is $$%$$ $ I_S = J_S \cap \Bbbk[x_i : a_i \in \mathcal A(S)].$$ For every $1\leq i,j\leq m-1$ define $$\label{eq:cij}%\end{equation} c_{i,j} = \tfrac{1}{m}(a_i + a_j - a_{i+j}) \ge 0$$ and $$%$$ $ b_{i,j} = \begin{cases} c_{i,j} & \text{if } i + j \ne m \\ c_{i,j} + 1 & \text{if } i + j = m. \end{cases}$$ In particular, $c_{i,j} = 0$ if and only if $a_i + a_j = a_{i+j}$; this is impossible if $i + j = m$, since $m$ is the multiplicity, so $b_{i,j} = 0$ if and only if $c_{i,j} = 0$. It is known that $$\label{eq:medbinomials}%\end{equation} J_S = \langle x_ix_j - y^{c_{i,j}}x_{i+j} : 1 \le i \le j \le m-1\rangle,$$ though it also follows from Lemma [Lemma 10](#l:idealgens){reference-type="ref" reference="l:idealgens"} here. **Example 2**. The semigroup $S=\langle 4,9,11,14\rangle$ has $(a_1,a_2,a_3)=(9,14,11)$ and $$%$$ $ J_S = I_S = \langle x_1^2-yx_2, x_1x_2-y^3x_3,x_1x_3-y^4y,x_2^2-y^6y,x_2x_3-x_1y^4,x_3^2-x_2y^2\rangle.$$ The terms here are written in a way that emphasizes the convention $a_0 = m$ and $x_0 = y$, such as to produce the binomial $x_1x_3 - y^4x_0 = x_1x_3 - y^4y = x_1x_3 - y^5$. The semigroup $T=\langle 4,13,23\rangle$ has $(a_1,a_2,a_3)=(13,26,23)$ and Apéry ideal $$\begin{aligned} %\end{align*} J_T &= \langle x_1^2-x_2, x_1x_2-y^4x_3, x_1x_3-y^9, x_2^2-y^{13}, x_2x_3-x_1y^9, x_3^2-x_2y^5\rangle \\ &= \langle x_1^2-x_2, x_1^3-y^4x_3, x_1x_3-y^9, x_3^2-x_1^2y^5\rangle\end{aligned}$$ and defining toric ideal $$%$$ $ I_T = \langle x_1^3-y^4x_3, x_1x_3-y^9, x_3^2-x_1^2y^5\rangle = J_T \cap \Bbbk[y,x_1,x_3].$$ ## Kunz cone {#b:Kunz-cone} This subsection describes the Kunz cone and its relationship to the values $b_{i,j}$. Letting $\mathop{\mathrm{Ap}}(S)=\{0,a_1,\ldots,a_{m-1}\}$ with $a_i \equiv i \bmod m$ for each $i$ as in Section [2.1](#b:toric-ideals){reference-type="ref" reference="b:toric-ideals"}, the *Apéry coordinate vector* of $S$ with respect to $m$ is the tuple $(a_1, \ldots, a_{m-1})$. The following set of linear inequalities exactly characterizes the set of Apéry coordinate vectors for numerical semigroups of multiplicity $m$ [@kunzfaces1; @kunz]. **Definition 3**. For each $m\geq 2$, the *Kunz cone* $C_m \subseteq \mathbb{R}_{\ge 0}^{m-1}$ has facet inequalities $$%$$ $ z_i + z_j \geq z_{i+j} \quad \text{for} \quad 1 \le i \le j \le m-1 \quad \text{with} \quad i+j \ne m,$$ where addition of subscripts is modulo $m$. **Lemma 4**. *If $S$ is a numerical semigroup of multiplicity $m$, then $b_{i,j} = 0$ if and only if $a_i+a_j = a_{i+j}$. Hence the Apéry coordinate vector of $S$ lies on the boundary of $C_m$ if and only if $b_{i,j} = 0$ for some $i,j$.* *Proof.* Follows from the definitions, using ([\[eq:cij\]](#eq:cij){reference-type="ref" reference="eq:cij"}) for the claim about $b_{i,j}$. ◻ The lemma has the following consequence [@kunz]. **Proposition 5**. *A vector $z = (z_1, \ldots, z_{m-1}) \in \mathbb{Z}_{\ge 1}^{m-1}$ with $z_i \equiv i \bmod m$ for all $i$ lies in $C_m$ if and only if $z$ is the Apéry coordinate vector of a numerical semigroup $S$. Moreover, $z$ is in the interior of $C_m$ if and only if $S$ has maximal embedding dimension.* **Example 6**. The cone $C_4 \subseteq \mathbb{R}_{\ge 0}^3$ is defined by the inequalities $$%$$ $ z_2 + z_3 \ge z_1, \qquad z_1 + z_2 \ge z_3, \qquad 2z_1 \ge z_2, \qquad \text{and} \qquad 2z_3 \ge z_2,$$ and has extremal rays generated by $(1,0,1)$, $(1,2,3)$, $(1,2,1)$, and $(3,2,1)$. All positive-dimensional faces of $C_4$ contain numerical semigroups (in the sense of Proposition [Proposition 5](#prop:kunzinteriormed){reference-type="ref" reference="prop:kunzinteriormed"}) except the rays through $(1,0,1)$ and $(1,2,1)$. Numerical semigroups on the rays through $(1,2,3)$ and $(3,2,1)$ have embedding dimension 2, and numerical semigroups in the relative interior of the facets $z_1 + z_2 = z_3$ and $z_2 + z_3 = z_1$ are complete intersections [@kunzfaces2]. In particular, a minimal free resolution for the defining toric ideal of any semigroup in these $4$ faces is known. The numerical semigroup $S$ from Example [Example 2](#ex:semigrouptoric){reference-type="ref" reference="ex:semigrouptoric"} corresponds to the point $(9,14,11)$ in the relative interior of $C_4$, while $T$ corresponds to the point $(13,26,23)$ in the relative interior of the facet $2z_1 = z_2$. A minimal free resolution for $J_S$ is obtained by substituting the appropriate values for $b_{i,j}$ in the free resolution in Figure [\[fig:m4res\]](#fig:m4res){reference-type="ref" reference="fig:m4res"}, while a minimal free resolution for $J_T$ is obtained via analogous substitution into Figure [\[fig:m4spec\]](#fig:m4spec){reference-type="ref" reference="fig:m4spec"} (at the end of Section [4](#sec:specialization){reference-type="ref" reference="sec:specialization"}). This leaves the facet $2z_3 = z_2$, and, courtesy of the action of $\mathbb{Z}_4^*$ on $C_4$, free resolutions for semigroups in this face can be obtained from the ones exhibited in Figure [\[fig:m4spec\]](#fig:m4spec){reference-type="ref" reference="fig:m4spec"} by interchanging 1's and 3's in every subscript. ## Modules and maps for the Apéry resolution {#b:modules} For any numerical semigroup $S$ of multiplicity $m$, this subsection defines the free modules and linear maps between them that form the *Apéry resolution* $$%$$ $ \mathcal F_\bullet: 0 \longleftarrow R \longleftarrow F_1 \longleftarrow F_2 \longleftarrow \cdots$$ of $J_S$. Theorem [Theorem 12](#t:medresolution){reference-type="ref" reference="t:medresolution"} shows that it is a resolution. Of particular note is that the ranks of its modules and the locations of the nonzero coefficients in the matrices representing its linear maps depend only on $m$, not on the actual values of $\mathop{\mathrm{Ap}}(S)$. Theorem [Theorem 12](#t:medresolution){reference-type="ref" reference="t:medresolution"} and Corollary [Corollary 13](#c:minimal-for-med){reference-type="ref" reference="c:minimal-for-med"} show that this resolution is minimal if and only if $S$ has maximal embedding dimension, i.e., corresponds to a point interior to $C_m$. Theorem [Theorem 19](#t:specializationindependent){reference-type="ref" reference="t:specializationindependent"} shows that when $S$ lies on the boundary of $C_m$, a minimal resolution of $J_S$ can be obtained from the Apéry resolution in a manner that is uniform for all semigroups in the interior of a fixed face of $C_m$, parametrized by the $b_{i,j}$. ### Modules {#ss:modules} For $d=0,1,\ldots,m-1$, define $F_d$ to be the free module over $R$ with formal basis elements $$%$$ $ \left\{e_{i,A} : i \in [m-1], \, A \subset [m-1], |A| = d, i \ge \min(A)\right\},$$ where $\deg(e_{i,A}) = a_i + \sum_{j \in A} a_j$. Since every pair $(i,A)$ such that $|A|=d$ and $i<\min(A)$ corresponds to a $(d+1)$-element subset $\{i\}\cup A$ of $[m-1]$, it is immediate that $$\begin{aligned} %\end{align*} \mathop{\mathrm{rank}}F_d &= (m-1)\binom{m-1}{d} - \binom{m-1}{d+1} = d\binom{m}{d+1} \, .\end{aligned}$$ **Example 7**. For $m=3$, $F_0 = Re_\varnothing$, where $Re_\varnothing= \{re_\varnothing: r\in R\}$. Similarly, $$\begin{aligned} %\end{align*} F_1 &= Re_{1,\{1\}}+Re_{2,\{2\}}+Re_{2,\{1\}} \\ &= \{\alpha e_{1,\{1\}}+\beta e_{2,\{2\}}+\gamma e_{2,\{1\}}: \alpha,\beta,\gamma \in R\} \end{aligned}$$ with $\deg(e_{1,\{1\}})= a_1+a_1$, $\deg(e_{2,\{2\}})=a_2+a_2$, and $\deg(e_{2,\{1\}})=a_2+a_1$. Note that $\mathop{\mathrm{rank}}F_1 = 1\cdot\binom{3}{1+1}$. Finally, $$%$$ $ F_2 = Re_{1,12}+Re_{2,12}$$ with $\deg(e_{1,12}) = a_1+a_1+a_2$ and $\deg(e_{2,12}) = a_2+a_1+a_2$. ### Maps {#ss:maps} A few notational conventions help to define the boundary maps between the $F_d$. For $A \subseteq [m-1]$, set $$%$$ $ \mathop{\mathrm{sign}}(j,A)=(-1)^t \qquad \text{for} \qquad j\in A=\{\ell_0<\ell_1<\cdots <\ell_t=j <\cdots <\ell_r\}.$$ For convenience, set $e_{0,A} = 0$, and for $i \in [m-1]$ with $i < \min(A)$, define $$\label{eq:quotientsub}%\end{equation} e_{i,A} = \sum_{j \in A} \mathop{\mathrm{sign}}(j,A) e_{j,A \cup i \setminus j}\, .$$ As a consequence of this definition of $e_{i,A}$, for each $B \subseteq [m-1]$, $$\label{eq:quotientrel}%\end{equation} \sum_{i \in B} \mathop{\mathrm{sign}}(i,B) e_{i,B \setminus i} = 0.$$ With these conventions in hand, and considering $i+j$ modulo $m$ in subscripts as usual, define the map $\partial_d: F_d \to F_{d-1}$ by $$\label{eq:definepartial}%\end{equation} e_{i,A} \mapsto \sum_{j \in A} \mathop{\mathrm{sign}}(j, A) (x_j e_{i,A \setminus j} - y^{b_{i,j}} e_{i+j,A \setminus j})$$ with the exception of $d = 1$, in which case $$%$$ $ \partial_1(e_{i,j}) = x_ix_j - y^{c_{i,j}} x_{i+j}\, .$$ **Example 8**. Figure [\[fig:m3res\]](#fig:m3res){reference-type="ref" reference="fig:m3res"} shows the modules and maps for the case $m=3$. Note that by definition the bases for the modules are indexed by $(i,A)$ pairs, and these are used to label the rows and columns of the matrices representing the maps. Consider the term $\partial_2(e_{1,12})$, which by definition is $$\begin{aligned} %\end{align*} \partial_2(e_{1,12}) & = \left(x_1e_{1,2}-y^{b_{1,1}}e_{2,2}\right) - \left(x_2e_{1,1}-y^{b_{1,2}}e_{0,1}\right) \\ & = x_1e_{2,1} - y^{b_{1,1}} e_{2,2} - x_2e_{1,1}\end{aligned}$$ where the relation $e_{1,2} - e_{2,1} = 0$ is used. This illustrates how the relation [\[eq:quotientrel\]](#eq:quotientrel){reference-type="eqref" reference="eq:quotientrel"} ensures that $\partial_d$ is well defined. **Example 9**. Figure [\[fig:m4res\]](#fig:m4res){reference-type="ref" reference="fig:m4res"} shows the modules and maps for the case $m=4$. The ranks of the modules and the general structure of the maps are independent of the values of $\mathop{\mathrm{Ap}}(S)$ except that the exponents on the $y$-variables in the matrices. The Apéry resolution of $I_S$ for $S$ introduced in Example [Example 1](#ex:semigroup){reference-type="ref" reference="ex:semigroup"} is given in the upper portion of Figure [\[fig:determinantalEN\]](#fig:determinantalEN){reference-type="ref" reference="fig:determinantalEN"} (toward the end of Section [3](#sec:resolution){reference-type="ref" reference="sec:resolution"}). $%$ \begin{array}{c@{\:}c@{\:}c@{\:}c@{\:}c} & \begin{blockarray}{rccc} \\ \\ & \scriptstyle\textbf{1,1} & \scriptstyle\textbf{2,2} & \scriptstyle\textbf{2,1} \\ \begin{block}{r@{\,\,\,}[lll]} \scriptstyle\varnothing& x_1^2 - x_2 y^{b_{11}} & x_2^2 - x_1 y^{b_{22}} & x_1x_2 - y^{b_{12}} \\ \end{block} \end{blockarray} && \begin{blockarray}{rc@{\,\,}c} & \scriptstyle\textbf{1,12} & \scriptstyle\textbf{2,12} \\ \begin{block}{r@{\,\,\,}[l@{\,\,}l]} \scriptstyle\textbf{1,1} & -x_2 & \phantom{-}y^{b_{22}} \\ \scriptstyle\textbf{2,2} & -y^{b_{11}} & \phantom{-}x_1 \\ \scriptstyle\textbf{2,1} & \phantom{-}x_1 & -x_2 \\ \end{block} \end{blockarray} & \\[-1em] 0 \leftarrow R & \mathord\leftarrow \mkern-6mu \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill \mkern-6mu \mathord-& R^3 & \mathord\leftarrow \mkern-6mu \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill \mkern-6mu \mathord-& R^2 \leftarrow 0 \end{array}$ # Maximal embedding dimension numerical semigroups {#sec:resolution} This section proves the Apéry resolution to be indeed a resolution of $J_S$. The core of the proof is Schreyer's theorem, which identifies a Gröbner basis (under a carefully chosen term order) for the syzygy module of a Gröbner basis (see [@clo2] for a thorough overview of Schreyer's theorem). Lemma [Lemma 11](#l:quotient){reference-type="ref" reference="l:quotient"} verifies important subtleties about the boundary maps $\partial_d$: they are consistent with the definition of $e_{i,A}$ in [\[eq:quotientsub\]](#eq:quotientsub){reference-type="eqref" reference="eq:quotientsub"} and [\[eq:quotientrel\]](#eq:quotientrel){reference-type="eqref" reference="eq:quotientrel"} when $i < \min A$, and substituting [\[eq:quotientsub\]](#eq:quotientsub){reference-type="eqref" reference="eq:quotientsub"} into the definition of $\partial_d$ still yields matrix entries that are monomials. **Lemma 10**. *The generating set [\[eq:medbinomials\]](#eq:medbinomials){reference-type="eqref" reference="eq:medbinomials"} is a Gröbner basis for $J_S$ under any term order $\preceq$ on $R$ for which ${\mathbf x}^{\mathbf a}y^r \succ {\mathbf x}^{\mathbf b}y^s$ whenever $a_1 + \cdots + a_{m-1} > b_1 + \cdots + b_{m-1}$, where ${\mathbf x}^{\mathbf a}$ and ${\mathbf x}^{\mathbf b}$ are monomials in $x_1,\dots,x_{m-1}$.* *Proof.* Since $J_S$ is generated by binomials, it suffices to consider binomials when computing initial ideals. The key observation is that in any graded degree, exactly one monomial in $R$ has the form $x_i y^a$ with $a \in \mathbb{Z}_{\ge 0}$, since the graded degrees of the variables $x_i$ are distinct modulo $m$. Hence the larger term under $\preceq$ in any nonzero binomial from $J_S$ is divisible by $x_ix_j = \mathop{\mathrm{In}}_\preceq(x_ix_j - y^{c_{i,j}} x_{i+j})$ for some $i, j \in [m-1]$. As such, $$%$$ $ \mathop{\mathrm{In}}_\preceq(J_S) = \langle x_ix_j : 1 \le i, j \le m-1\rangle,$$ and thus the generating set in [\[eq:medbinomials\]](#eq:medbinomials){reference-type="eqref" reference="eq:medbinomials"} is a Gröbner basis for $J_S$. ◻ The next aim is to establish that applying $\partial$ to $e_{i,A}$ when $i<\min(A)$ using the expression given in [\[eq:definepartial\]](#eq:definepartial){reference-type="eqref" reference="eq:definepartial"} is consistent with [\[eq:quotientrel\]](#eq:quotientrel){reference-type="eqref" reference="eq:quotientrel"}; this is needed when considering the result of applying $\partial$ repeatedly. Further, careful analysis of the use of [\[eq:quotientrel\]](#eq:quotientrel){reference-type="eqref" reference="eq:quotientrel"} is required when $i+j<\min(A\setminus j)$ in [\[eq:definepartial\]](#eq:definepartial){reference-type="eqref" reference="eq:definepartial"}. These issues are addressed in the following lemma. **Lemma 11**. *The maps $\partial_d$ respect [\[eq:quotientrel\]](#eq:quotientrel){reference-type="eqref" reference="eq:quotientrel"}, and for $d > 1$, the entries of each $\partial_d$ are monomials. Furthermore, $\mathcal F_\bullet$ is a complex.* *Proof.* If $d = 1$, then [\[eq:quotientsub\]](#eq:quotientsub){reference-type="eqref" reference="eq:quotientsub"} yields $e_{i,j} = e_{j,i}$, and the first claim is immediate. If $d > 1$, then for each $B \subseteq [m-1]$ with $|B| = d+1$, $$%$$ $ \sum_{i \in B} \mathop{\mathrm{sign}}(i,B) \partial e_{i,B \setminus i} = \sum_{i \in B} \mathop{\mathrm{sign}}(i,B) \sum_{j \in B \setminus i} \mathop{\mathrm{sign}}(j, B \setminus i) (x_j e_{i,B \setminus ij} - y^{b_{i,j}} e_{i+j,B \setminus ij}),$$ wherein the coefficient of $y^{b_{i,j}} e_{i+j,B \setminus ij}$ for distinct $i, j \in B$ equals $$%$$ $ \mathop{\mathrm{sign}}(i,B) \mathop{\mathrm{sign}}(j,B \setminus i) + \mathop{\mathrm{sign}}(j,B) \mathop{\mathrm{sign}}(i,B \setminus j) = 0$$ and the remaining terms yield $$\begin{aligned} %\end{align*} \sum_{i \in B} \mathop{\mathrm{sign}}(i,B) \partial e_{i,B \setminus i} &= \sum_{i \in B} \mathop{\mathrm{sign}}(i,B) \sum_{j \in B \setminus i} \mathop{\mathrm{sign}}(j, B \setminus i) x_j e_{i,B \setminus ij} \\ % &= \sum_{\{i,j\} \subseteq B} \sign(i,B)\sign(j,B \setminus % i)x_je_{i,B \setminus ij} + \sign(j,B)\sign(i,B \setminus % j)x_ie_{j,B \setminus ij} % \\ &= \sum_{j \in B} x_j \sum_{i \in B \setminus j} \mathop{\mathrm{sign}}(i,B)\mathop{\mathrm{sign}}(j,B \setminus i) e_{i,B \setminus ij} \\ &= -\sum_{j \in B} \mathop{\mathrm{sign}}(j,B) x_j \sum_{i \in B \setminus j} \mathop{\mathrm{sign}}(i,B \setminus j) e_{i,B \setminus ij} \\ &= -\sum_{j \in B} \mathop{\mathrm{sign}}(j,B) x_j \cdot 0 \\ &= 0.\end{aligned}$$ Now proceed to the second claim that each $\partial_d$ is a matrix whose entries are monomials. Call $e_{i,A}$ *squarefree* if $i \notin A$. Note that every term in [\[eq:quotientrel\]](#eq:quotientrel){reference-type="eqref" reference="eq:quotientrel"} is squarefree, and no two equalities of the form [\[eq:quotientrel\]](#eq:quotientrel){reference-type="eqref" reference="eq:quotientrel"} share any terms. As such, it suffices to ensure that no two squarefree terms in $\partial e_{i,A}$ lie in the same equality in [\[eq:quotientrel\]](#eq:quotientrel){reference-type="eqref" reference="eq:quotientrel"}. To this end, fix $j, k \in A$. If $e_{i, A \setminus j}$ and $e_{i, A \setminus k}$ lie in the same equality in [\[eq:quotientrel\]](#eq:quotientrel){reference-type="eqref" reference="eq:quotientrel"}, then $(A \setminus j) \cup i = (A \setminus k) \cup i$ and thus $j = k$. If $e_{i+j, A \setminus j}$ and $e_{i+k, A \setminus k}$ lie in the same equality in [\[eq:quotientrel\]](#eq:quotientrel){reference-type="eqref" reference="eq:quotientrel"}, then $i + j \notin A$ but $i + j \in (A \cup \{i + k\}) \setminus k$, so necessarily $i + j = i + k$ and thus $j = k$. Lastly, if $e_{i, A \setminus j}$ and $e_{i+k, A \setminus k}$ lie in the same equality in [\[eq:quotientrel\]](#eq:quotientrel){reference-type="eqref" reference="eq:quotientrel"}, then $i+k \notin A$ but $i+k \in (A \setminus j) \cup i$, so $i + k = i$, which is impossible. It remains to prove $\mathcal F_\bullet$ is a complex. First, suppose $A = \{j,k\}$ with $j < k$ and $i \ge j$. If $i + j$, $i + k$, and $i + j + k$ are all nonzero, then $$\begin{aligned} %\end{align*} \partial^2 e_{i,A} &= x_j \partial e_{i,k} - x_k \partial e_{i,j} - y^{b_{i,j}} \partial e_{i+j,k} + y^{b_{i,k}} \partial e_{i+k,j} \\ &= x_j (x_ix_k - y^{c_{i,k}}x_{i+k}) - x_k (x_ix_j - y^{c_{i,j}}x_{i+j}) \\&\phantom{={}} - y^{b_{i,j}} (x_{i+j}x_k - y^{c_{i+j,k}}x_{i+j+k}) + y^{b_{i,k}} (x_{i+k}x_j - y^{c_{i+k,j}}x_{i+j+k}) \\ &= x_{i+j}x_k (y^{c_{i,j}} - y^{b_{i,j}}) + x_{i+k}x_j (y^{c_{i,k}} - y^{b_{i,k}}) \\&\phantom{={}} + x_{i+j+k} (y^{b_{i,j}} y^{c_{i+j,k}} - y^{b_{i,k}} y^{c_{i+k,j}}) = 0\end{aligned}$$ by homogeneity of $\partial$. In the event $i + j = 0$, or $i + k = 0$, or $i + j + k = 0$, replacing $x_0$ with zero as appropriate in the above algebra yields the desired equality. For all remaining cases, $|A| > 2$, and in the expansion of $$\begin{aligned} %\end{align*} \partial^2 e_{i,A} &= \sum_{j \in A} \mathop{\mathrm{sign}}(j, A) x_j \partial e_{i,A \setminus j} - \sum_{j \in A} \mathop{\mathrm{sign}}(j, A) y^{b_{i,j}} \partial e_{i+j,A \setminus j} \\ &= \sum_{j \in A} \mathop{\mathrm{sign}}(j, A) x_j \bigg( \sum_{k \in A \setminus j} \mathop{\mathrm{sign}}(k, A \setminus j) (x_k e_{i,A \setminus jk} - y^{b_{i,k}} e_{i+k,A \setminus jk}) \bigg) \\* & \phantom{={}} - \sum_{j \in A} \mathop{\mathrm{sign}}(j, A) y^{b_{i,j}} \bigg( \sum_{k \in A \setminus j} \mathop{\mathrm{sign}}(k, A \setminus j) (x_k e_{i+j,A \setminus jk} - y^{b_{i+j,k}} e_{i+j+k,A \setminus jk}) \bigg),\end{aligned}$$ the terms $x_j x_k e_{i,A \setminus jk}$, $x_j y^{b_{i,k}}e_{i+k,A \setminus jk}$, and $y^{b_{i,j} + b_{i+j,k}} e_{i+j+k, A \setminus jk}$ each have coefficient $$%$$ $ \mathop{\mathrm{sign}}(j,A)\mathop{\mathrm{sign}}(k,A \setminus j) + \mathop{\mathrm{sign}}(k,A)\mathop{\mathrm{sign}}(j, A \setminus k) = 0$$ for any distinct $j, k \in A$. ◻ **Theorem 12**. *The complex $\mathcal F_\bullet$ is a resolution.* *Proof.* Proceed by induction on $d$ to show that the columns of the matrices for $\partial_d$ form a Gröbner basis for $\ker\partial_{d-1}$. The case $d = 1$ is handled by Lemma [Lemma 10](#l:idealgens){reference-type="ref" reference="l:idealgens"}, so suppose $d = 2$. Let $\preceq'$ denote the partial order on $F_1$ given by $x^\beta e_{k,\ell} \preceq' x^\alpha e_{i,j}$ whenever $$%$$ $ \mathop{\mathrm{In}}_\preceq(\partial_1(x^\beta e_{k,b})) \prec \mathop{\mathrm{In}}_\preceq(\partial_1(x^\alpha e_{i,a})),$$ or when equality holds above and $a< b$, or if equality holds above, $a = b$, and $i < k$. Note $x^\beta e_{j,\ell} \preceq' x^\alpha e_{i,k}$ whenever $x^\alpha$ has higher total degree in $x_1, \ldots, x_{m-1}$ than $x^\beta$, so $$%$$ $ \mathop{\mathrm{In}}_{\preceq'}(\partial_2(e_{i,jk})) = x_k e_{i,j} \qquad \text{where} \qquad j < k \text{ and } i \ge j.$$ Schreyer's theorem [@clo2 Chapter 5, (3.3)] implies the elements $$\label{eq:schreyergb}%\end{equation} s_{i,a; \, k,b} = \frac{L}{x_ix_a} e_{i,a} - \frac{L}{x_kx_b} e_{k,b} - \sum_{\ell \ge c \ge 1} f_{\ell,c} e_{\ell,c} \qquad \text{for} \qquad i \ge a, \, k \ge b$$ form a Gröbner basis for $\ker(\partial_1)$ under $\preceq'$, where $L = \mathop{\mathrm{lcm}}(x_ix_a, x_kx_b)$ and the $f_{\ell,c}$ are coefficients obtained from polynomial long division when dividing $S(\partial_1(e_{i,a}), \partial_1(e_{k,b}))$ by [\[eq:medbinomials\]](#eq:medbinomials){reference-type="eqref" reference="eq:medbinomials"}. In particular, we claim $$%$$ $ \mathop{\mathrm{In}}_{\preceq'}(\ker(\partial_1)) = \langle x_k e_{i,j} \mid j < k \text{ and } i \ge j\rangle$$ is generated by initial terms of the columns of $\partial_2$. Indeed, by construction $\mathop{\mathrm{In}}_{\preceq'}(s_{i,a; \, k,b})$ must be one of the first two terms in [\[eq:schreyergb\]](#eq:schreyergb){reference-type="eqref" reference="eq:schreyergb"} and $\partial_1(\frac{L}{x_ix_a} e_{i,a}) =\partial_1( \frac{L}{x_kx_b} e_{k,b} )$, so without loss of generality say $e_{k,b} \prec' e_{i,a}.$ Then either $a<b \leq k$ and $x_k$ or $x_b$ appear as a coefficient of $e_{i,a}$, or $a=b$, $a\leq i <k$ and $x_k$ appears as a coefficient of $e_{i,a}$, so $\mathop{\mathrm{In}}_{\preceq'}(s_{i,a; \, k,b})$ is divisible by the initial term of some column of $\partial_2$. This implies that $\mathop{\mathrm{In}}_{\preceq'}(\mathop{\mathrm{im}}(\partial_2)) = \mathop{\mathrm{In}}_{\preceq'}(\ker(\partial_1))$, which, together with $\mathop{\mathrm{im}}(\partial_2) \subseteq \ker(\partial_1)$, implies $\mathop{\mathrm{im}}(\partial_2) = \ker(\partial_1)$ and the columns of $\partial_2$ form a Gröbner basis under $\preceq'$. Lastly, suppose $d > 2$, let $\preceq$ denote the term order on $F_{d-2}$ obtained inductively, and let $\preceq'$ denote the term order on $F_{d-1}$ so that $x^\beta e_{j,B}\preceq' x^\alpha e_{i,A}$ whenever $$%$$ $ \mathop{\mathrm{In}}_\preceq(x^\beta \partial_{d-1}(e_{j,B})) \prec \mathop{\mathrm{In}}_\preceq(x^\alpha \partial_{d-1}(e_{i,A})),$$ or if equality holds above and $A$ precedes $B$ lexicographically, or if equality holds above, $A = B$, and $i < j$. One readily obtains $$%$$ $ \mathop{\mathrm{In}}_{\preceq'}(\partial_d(e_{i,A})) = x_j e_{i,A \setminus j} \qquad \text{with} \qquad j = \max(A)$$ after checking the following: - $x^\alpha e_{k,B} \preceq' x^\beta e_{\ell,C}$ whenever $x^\beta$ has higher total degree in $x_1, \ldots, x_{m-1}$ than $x^\alpha$; - $x_k \mathop{\mathrm{In}}_\preceq(\partial_{d-1}(e_{i,A \setminus k})) = x_\ell \mathop{\mathrm{In}}_\preceq(\partial_{d-1}(e_{i,A \setminus \ell}))$ for all $k, \ell \in A$; and - the substitution [\[eq:quotientsub\]](#eq:quotientsub){reference-type="eqref" reference="eq:quotientsub"} need only be made if $\min(A) \le i < \min(A \setminus \min(A))$, in which case $A \setminus j$ lexicographically precedes the second subscript of every summand in [\[eq:quotientrel\]](#eq:quotientrel){reference-type="eqref" reference="eq:quotientrel"}. The equality $S(\partial_{d-1}(e_{i,A}), \partial_{d-1}(e_{j,B})) = 0$ holds due to initial terms having distinct basis vectors unless $i = j$, $A = C \cup \{\gamma\}$, and $B = C \cup \{\delta\}$ for some $\delta, \gamma \in [m-1]$ and some nonempty $C \subseteq [m-1]$ with $\delta, \gamma > \max(C)$. As such, Schreyer's theorem yields $$%$$ $ \mathop{\mathrm{In}}_{\preceq'}(\ker(\partial_{d-1})) = \langle x_\delta e_{i, C} : i, \delta \in [m-1], \, C \subseteq [m-1], \, \delta > \max(C)\rangle,$$ and since $x_\delta e_{i, C} = \mathop{\mathrm{In}}_{\preceq'}(\partial_d(e_{i,C \cup \delta}))$ for each $i$, $\delta$, and $C$, the columns of $\partial_d$ form a Gröbner basis for $\ker(\partial_{d-1})$. The proof is completed by observing that induction also ensures none of the intial terms in question involve $e_{i,A}$ with $i < \min(A)$. ◻ **Corollary 13**. *The resolution $\mathcal F_\bullet$ is minimal if and only if $S$ is MED.* *Proof.* A resolution is minimal if and only if the matrices for $\partial_d$ contain no nonzero constant entries. The only entries that depend on $a_1, \ldots, a_{m-1}$ are powers of $y$, and their exponents $b_{i,j}$ are all strictly positive precisely when $S$ is MED. ◻ **Remark 14**. MED semigroups whose associated toric ideal is determinantal are exactly those semigroups where $a_1, a_2, \ldots, a_{m-1}$ form an arithmetic sequence (not necessarily in that order) [@gotoencomplex; @determinentalmed]. In this case, $I_S$ is resolved by the Eagon--Northcott complex [@eagonnorthcott]; a detailed treatment on the Eagon--Northcott resolution can be found in [@Eis05 Appendix A2H]. The strict requirements on an MED semigroup to make its associated toric ideal determinantal mean that such semigroups form only a small proportion of all numerical semigroups: in the Kunz cone, these semigroups lie in the union of a finite set of affine $2$-planes, whose union cannot be the whole cone. Although relatively few toric ideals of MED semigroup ideals are minimally resolved by Eagon--Northcott complexes, the occasional overlap does mean that all toric ideals for MED numerical semigroups share Betti numbers with the Eagon--Northcott resolution of a $2 \times m$ matrix, despite the impossibility of using the Eagon--Northcott construction to resolve most such toric ideals. Even in the case where the ideal is determinantal, the Apéry resolution differs from the Eagon--Northcott resolution. As an example, consider the numerical semigroup $S = \langle 4, 9, 10, 11 \rangle$, whose defining toric ideal $I_S$ is generated by the $2 \times 2$ minors of $$%$$ $ \begin{bmatrix} x_1 & x_2 & x_3 & y^3 \\ y^2 & x_1 & x_2 & x_3 \end{bmatrix}.$$ The key difference is the presentation of the generators of $I_S$. Namely, the generators as provided in [\[eq:medbinomials\]](#eq:medbinomials){reference-type="eqref" reference="eq:medbinomials"} are of the form $x_ix_j - x_{i+j}y^{b_{i,j}}$, while those given by determinants may have the form $x_{i}x_j - x_{i+1}x_{j-1}$. Figure [\[fig:determinantalEN\]](#fig:determinantalEN){reference-type="ref" reference="fig:determinantalEN"} shows the Apéry resolution and the Eagon--Northcott resolution of $I_S$, with basis elements in the Eagon--Northcott resolution ordered to mimic the Apéry resolution. It is worth noting that in the $m=3$ case, $a_1$ and $a_2$ trivially form an arithmetic sequence, and in fact the Apéry resolution and the Eagon--Northcott resolution coincide. $%$ \begin{array}{c@{\:}c@{\:}c@{\:}c@{\:}c} & \begin{blockarray}{cccccc} \\ \\ \begin{block}{[*{6}{@{\,\,}l}]} x_1^2 - x_2 y^2 & x_2^2 - y^5 & x_3^2 - x_2 y^3 & x_1x_2 - x_3 y^2 & x_1x_3 - y^5 & x_2x_3 - x_1 y^3 \\ \end{block} \end{blockarray} & \\[-1em] 0 \leftarrow R & \mathord\leftarrow \mkern-6mu \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill \mkern-6mu \mathord-& \end{array}$ $%$ \begin{array}{c@{\:}c@{\:}c@{\:}c@{\:}c@{}c} & \begin{blockarray}{@{}*{8}{@{\,\,}c}} \\ \\ \begin{block}{@{}[*{8}{@{\,\,}l}]} -x_2 & -x_3 & \phantom{-}& \phantom{-} & \phantom{-}& \phantom{-}& \phantom{-}y^3 & \phantom{-}y^3 \\ -y^2 & \phantom{-}& \phantom{-}x_1 & -x_3 & \phantom{-}& \phantom{-}y^3 & \phantom{-}& \phantom{-} \\ \phantom{-}& \phantom{-}& \phantom{-}& \phantom{-} & \phantom{-}x_1 & \phantom{-}x_2 & -y^2 & \phantom{-} \\ \phantom{-}x_1 & \phantom{-}& -x_2 & \phantom{-}y^3 & \phantom{-}y^3 & \phantom{-}& -x_3 & \phantom{-} \\ \phantom{-}y^2 & \phantom{-}x_1 & \phantom{-}& \phantom{-} & -x_3 & -y^3 & \phantom{-}& -x_2 \\ \phantom{-}& -y^2 & -y^2 & \phantom{-}x_2 & \phantom{-}& -x_3 & \phantom{-}x_1 & \phantom{-}x_1 \\ \end{block} \end{blockarray} && \begin{blockarray}{@{}*{3}{@{\,\,}c}} \begin{block}{@{}[*{3}{@{\,\,}l}]} \phantom{-}x_3 & -y^3 & \\ -x_2 & & \phantom{-}y^3 \\ & \phantom{-}x_3 & -y^3 \\ -y^2 & \phantom{-}x_1 & \\ \phantom{-}y^2 & & -x_2 \\ & -y^2 & \phantom{-}x_1 \\ \phantom{-}x_1 & -x_2 & \\ -x_1 & \phantom{-}& \phantom{-}x_3 \\ \end{block} \end{blockarray} & \\[-1em] R^6 & \mathord\leftarrow \mkern-6mu \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill \mkern-6mu \mathord-& R^8 & \mathord\leftarrow \mkern-6mu \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill \mkern-6mu \mathord-& R^3 & \leftarrow 0 \end{array}$ $%$ \begin{array}{c@{\:}c@{\:}c@{\:}c@{\:}c} & \begin{blockarray}{cccccc} \begin{block}{[*{6}{@{\,\,}l}]} x_1^2 - x_2 y^2 & x_2^2 - x_1x_3 & x_3^2 - x_2 y^3 & x_1x_2 - x_3 y^2 & x_1x_3 - y^5 & x_2x_3 - x_1 y^3 \\ \end{block} \end{blockarray} & \\[-1em] 0 \leftarrow R & \mathord\leftarrow \mkern-6mu \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill \mkern-6mu \mathord-& \end{array}$ $%$ \begin{array}{c@{\:}c@{\:}c@{\:}c@{\:}c@{}c} & \begin{blockarray}{*{8}{@{\,\,}c}} \\ \\ \begin{block}{[*{8}{@{\,\,}l}]} \phantom{-}x_2 &\phantom{-}x_3 &\phantom{-}x_3 &\phantom{-}&\phantom{-}&\phantom{-}&\phantom{-}&\phantom{-}y^3 \\ \phantom{-}y^2 &\phantom{-}&\phantom{-}x_1 &\phantom{-}x_3 &\phantom{-}&\phantom{-}y^3 &\phantom{-}&\phantom{-} \\ \phantom{-}&\phantom{-}&\phantom{-}&\phantom{-}x_1 &\phantom{-}x_1 &\phantom{-}x_2 &\phantom{-}y^2 &\phantom{-} \\ -x_1 &\phantom{-}&-x_2 &\phantom{-}&\phantom{-}y^3 &\phantom{-}&\phantom{-}x_3 &\phantom{-} \\ \phantom{-}&-x_1 &\phantom{-}&\phantom{-}&-x_3 &\phantom{-}&-x_2 &-x_2 \\ \phantom{-}&\phantom{-}y^2 &\phantom{-}&-x_2 &\phantom{-}&-x_3 &\phantom{-}&\phantom{-}x_1 \\ \end{block} \end{blockarray} && \begin{blockarray}{*{3}{@{\,\,}c}} \begin{block}{[*{3}{@{\,\,}l}]} -x_3 & -y^3 & \\ \phantom{-}x_2 & \phantom{-}x_3 & \phantom{-}\\ \phantom{-}& -x_3 & -y^3 & \\ \phantom{-}y^2 & \phantom{-}x_1 & \\ \phantom{-}& -x_1 & -x_2 \\ \phantom{-}& \phantom{-}y^2 & \phantom{-}x_1 & \\ -x_1 & -x_2 & \\ \phantom{-}& \phantom{-}x_2 & \phantom{-}x_3 \\ \end{block} \end{blockarray} & \\[-1em] R^6 & \mathord\leftarrow \mkern-6mu \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill \mkern-6mu \mathord-& R^8 & \mathord\leftarrow \mkern-6mu \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill \mkern-6mu \mathord-& R^3 & \leftarrow 0 \end{array}$ **Remark 15**. When $S$ is MED, quotienting the Apéry resolution of $I_S$ by the ideal $\langle y\rangle$ yields a minimal resolution of the ideal $\langle x_1, \ldots, x_{m-1}\rangle^2$ over the ambient polynomial ring $\Bbbk[x_1, \ldots, x_{m-1}]$. This ideal is known to be resolved by the Eagon--Northcott complex on the $2 \times m$ matrix $$%$$ $ \begin{bmatrix} x_1 & x_2 & \cdots & x_{m-2} & x_{m-1} & 0\\ 0 & x_ 1 &x_2 & x_3 & \cdots & x_{m-1} \end{bmatrix}.$$ Thus, in the MED case, the Eagon--Northcott complex "sits inside" the Apery resolution; indeed, it is the result of an artinian reduction of $R/I_S$. # Specialization for arbitrary numerical semigroups {#sec:specialization} The Apéry resolution can be thought of as a family of free resolutions, one for the Apéry ideal $J_S$ of each numerical semigroup $S$ with multiplicity $m$, that is parametrized by the values $b_{i,j}$. Given a numerical semigroup $S$, a free resolution of $J_S$ is obtained by simply computing the values $b_{i,j}$ from the Apéry set of $S$ and substituting them into the Apéry resolution. By Corollary [Corollary 13](#c:minimal-for-med){reference-type="ref" reference="c:minimal-for-med"}, restricting to semigroups $S$ in the interior of $C_m$, the Apéry resolutions form a parametrized family of minimal free resolutions. The main result of this section is Theorem [Theorem 19](#t:specializationindependent){reference-type="ref" reference="t:specializationindependent"}, which implies that for each face $F$ of $C_m$, there exists a family of minimal free resolutions, one for the Apéry ideal $J_S$ of each numerical semigroup $S$ indexed by the interior of $F$, that is analogously parametrized by the positive $b_{i,j}$. Figure [\[fig:m4spec\]](#fig:m4spec){reference-type="ref" reference="fig:m4spec"} depicts one such resolution for the $z_2 = 2z_1$ facet of $C_4$. Our proof of Theorem [Theorem 19](#t:specializationindependent){reference-type="ref" reference="t:specializationindependent"} is nonconstructive: it carefully argues that there exists a change of basis for the Apéry resolution, depending only on $F$, that yields the desired minimal free resolution of $J_S$ as a summand. Together with Proposition [Proposition 16](#p:freeresextravars){reference-type="ref" reference="p:freeresextravars"}, which gives the algebraic relationship between minimal resolutions of $J_S$ and $I_S$, the Betti numbers of $J_S$ and $I_S$ can be recovered from $F$ (Theorem [Theorem 18](#t:ungradedbetti){reference-type="ref" reference="t:ungradedbetti"} and Corollary [Corollary 20](#c:gradedbetti){reference-type="ref" reference="c:gradedbetti"}). **Proposition 16**. *A minimal free resolution of $J_S$ can be obtained as the tensor product of a minimal free resolution of $I_S$ with a Koszul complex.* *Proof.* Non-minimality of $m, a_1, \ldots, a_{m-1}$ as generators for $S$ is reflected in $J_S$ by binomial generators without $y$. More specifically, if $a_i + a_j = a_{i+j}$, then $b_{i,j}=0$ and the binomial $x_i x_j - x_{i+j}$ appears in $J_S$. Let $\mathcal{A}(S) = \{m,a_{i_1}, a_{i_2}, \ldots, a_{i_r}\}$ be the elements $a_i$ that minimally generate $S$. Though $I_S$ naturally lives in $\Bbbk[y,x_{i_1},x_{i_2}, \ldots, x_{i_r}]$, consider it as an ideal in $R$ via the natural inclusion map. For each nonzero $w \in \mathop{\mathrm{Ap}}(S) - \mathcal{A}(S)$, pick one of the binomials $f_w = x_w - x_u x_v$. These binomials form a regular sequence on $R$, so the ideal $I_W$ generated by the $f_w$ is resolved by a Koszul complex $\mathcal{K}_\bullet$. Writing $\mathcal{G}_\bullet$ for a minimal free resolution of $I_S$, the only nontrivial homology of $\mathcal{G}_\bullet \otimes_R \mathcal{K}_\bullet$ occurs in homological degree $0$ and is isomorphic to $H_0(\mathcal{G}_\bullet) \otimes H_0(\mathcal{K}_\bullet) = R/I_S \otimes_R R/I_W = R/J_S$, where the last equality is because the $f_w$ form a regular sequence over $R/I_S$. Therefore $\mathcal{G}_\bullet \otimes \mathcal{K}_\bullet$ is a minimal free resolution of $R/J_S$. ◻ **Example 17**. The underlying structure as a tensor of two resolutions is readily seen in Figure [\[fig:specialization\]](#fig:specialization){reference-type="ref" reference="fig:specialization"}, which resolves $J_S$ for $\mathop{\mathrm{Ap}}(S)=\{4,a_1,2a_1,a_3\}$. This example was obtained by computing the Apéry resolution for $J_S$ and then trimming away any constant entries using row and column operations as described in Theorem [Theorem 19](#t:specializationindependent){reference-type="ref" reference="t:specializationindependent"}. **Theorem 18**. *Let $S$ and $T$ be numerical semigroups corresponding to points interior to the same face $F$ of the Kunz cone $\mathcal{C}_m$. The Apéry ideals of $S$ and $T$ share the same Betti numbers, as do the defining toric ideals of $S$ and $T$. In particular, $\beta_d(J_S) = \beta_d(J_T)$ and $\beta_d(I_S) = \beta_d(I_T)$ for all $d \geq 0$.* *Proof.* Let $\mathcal F_\bullet$ and $\mathcal F_\bullet'$ be the Apéry resolutions of $J_S$ and $J_T$ respectively. In the case that $S$ and $T$ are both MED, so the face $F$ is the entirety of $C_m$, both resolutions are minimal by Corollary [Corollary 13](#c:minimal-for-med){reference-type="ref" reference="c:minimal-for-med"} and have the same modules at each homological degree, so $\beta_d(J_S) = \beta_d(J_T)$ holds immediately. If $\mathcal F_\bullet$ and $\mathcal F_\bullet'$ are not minimal, then the resolutions have $\pm 1$ entries in identical places in their resolutions, once again because $S$ and $T$ lie interior to the same face $F$ and thus have the same $b_{i,j} = 0$, meaning that the same entries $\pm y^{b_{i,j}}$ become $\pm 1$. Because the Betti numbers of any positively graded ideal $I$ equal the dimensions of the graded vector spaces $\mathop{\mathrm{Tor}}_\bullet(I,\Bbbk)$, consider $\mathcal F_\bullet \otimes_\Bbbk\Bbbk$ and $\mathcal F_\bullet' \otimes_\Bbbk\Bbbk$. The differentials in these complexes are identical: they are matrices of $0$s and $\pm 1$s with units in matching places. Therefore, their kernels and images are the same at each homological degree, so $$%$$ $ \beta_d(J_S) = \dim \mathop{\mathrm{Tor}}_\Bbbk(\mathcal F_\bullet,\Bbbk) = \dim \mathop{\mathrm{Tor}}_\Bbbk(\mathcal F_\bullet',\Bbbk) = \beta_d(J_T).$$ Next consider $I_S$ and $I_T$. By Proposition [Proposition 16](#p:freeresextravars){reference-type="ref" reference="p:freeresextravars"}, $\mathcal F_\bullet = (\mathcal G_\bullet \otimes \mathcal K_\bullet)$ and $\mathcal F_\bullet' = (\mathcal G_\bullet' \otimes \mathcal K_\bullet)$, where $\mathcal G_\bullet$ and $\mathcal G_\bullet'$ are minimal free resolutions of $I_S$ and $I_T$, respectively, and $\mathcal K_\bullet$ is the Koszul resolution on the extraneous binomials. Tensoring with $\mathcal K_\bullet$ exerts the same invertible change on the Betti numbers of $\mathcal G_\bullet$ and $\mathcal G_\bullet'$. More specifically, let $$%$$ $ g_S(t) = \sum\limits_{i=0}^p \beta_i(I_S) t^i \qquad \text{and} \qquad f_S(t) = \sum\limits_{i=0}^q \beta_i(J_S) t^i$$ be the generating functions for the Betti numbers of $\mathcal G_\bullet$ and $\mathcal F_\bullet$ respectively. Since $\mathcal K_\bullet$ is a Koszul resolution of $r = m - |\mathcal A(S)|$ elements, $f_S(t) = (1+t)^r g_S(t)$. Thus, $$%$$ $ (1+t)^r g_S(t) = f_S(t) = f_T(t) = (1+t)^r g_T(t)$$ $g_S(t) = g_T(t)$, meaning $\beta_i(I_S) = \beta_i (I_T)$ for all $i \geq 0$. ◻ **Theorem 19**. *Consider the set $$%$$ $ \mathcal M = \{x_i : 1 \le i \le m-1\} \cup \{y^{b_{i,j}} : 1 \le i, j \le m - 1\}$$ of formal symbols appearing as matrix entries in Apéry resolutions. (Lemma [Lemma 11](#l:quotient){reference-type="ref" reference="l:quotient"} ensures every nonzero matrix entry is accounted for in $\mathcal M$). Fix a face $F$ of $C_m$. There is a sequence of matrices, whose entries are $\Bbbk$-linear combinations of formal products of elements of $\mathcal M$, with the following property: for each numerical semigroup $S$ indexed by the relative interior of $F$, substituting $R$-variables and the values $b_{i,j}$ for $S$ into the entries of each matrix yields boundary maps for a graded minimal free resolution of $J_S$.* *Proof.* Fix a numerical semigroup $S$ with multiplicity $m$. Let $$%$$ $ \mathcal N_S = \{ x_i : 1 \le i \le m-1 \} \cup \{ y^{b_{i,j}} : 1 \le i, j \le m - 1 \text{ and } b_{i,j} > 0 \} \subseteq \mathcal M$$ denote the set of elements of $\mathcal M$ corresponding to positive-degree monomials in $R$ under the grading by $S$. If $S$ is MED, then $\mathcal M = \mathcal N_S$; otherwise they are distinct. By [@Eis95 Theorem 20.2] (see also [@cca Exercises 1.10 and 1.11]), the matrices in any free resolution for $J_S$ can, via a sequence of row and column operations that preserve homogeneity, be turned into block diagonal matrices with $2$ blocks: (i) a matrix with no nonzero constant entries and at least one nonzero entry in each row and column; and (ii) a matrix with no nonconstant entries and at most one nonzero entry in each row and column. After doing this, restricting to each block (i) yields a minimal free resolution for $J_S$. One way to select the aforementioned row and column operations is as follows. Begin with the matrices $M_i$ for the maps $\partial_i$ for the Apéry resolution, and perform the following for each $i = 1, 2, \ldots, m-1$, assuming that, as a result of prior operations, any column of $M_i$ with a nonzero constant entry has no other nonzero entries. - First use nonzero constant entries of $M_i$ to clear all other entries in their respective rows. If $i=1$, then no such rows exist. Fix a row $R$ of $M_i$ with a nonzero constant entry $c$, say in column $C_1$ with corresponding row $R_1$ in the matrix $M_{i+1}$. For each nonzero entry $f$ in $R$, say in a column $C_2 \ne C_1$ with corresponding row $R_2$ in $M_{i+1}$, subtract $c^{-1} f \cdot C_1$ from $C_2$ and add $c^{-1} f \cdot R_2$ to $R_1$. Once this is done, $c$ will be the only nonzero entry in $R$, and in fact $c$ will be the only nonzero entry in row $R$ and column $C_1$. Moreover, since $M_iM_{i+1} = 0$, the row $R_1$ in $M_{i+1}$ only has entries $0$. - Next use nonzero constant entries of $M_{i+1}$ to clear all other entries in their respective columns. Fix a column $C$ of $M_{i+1}$ with a nonzero constant entry $c$, say in row $R_1$ with corresponding column $C_1$ in the matrix $M_i$. For each nonzero entry $f$ in $C$, say in a row $R_2 \ne R_1$ with corresponding column $C_2$ in $M_i$, subtract $c^{-1} f \cdot R_1$ from $R_2$ and add $c^{-1} f \cdot C_2$ to $C_1$. Once this is done, $c$ will be the only nonzero entry in column $C$, so since $M_iM_{i+1} = 0$, the column $C_1$ now only has entries $0$. Moreover, all changes to $M_i$ only affect the (now all 0) column $C_1$, so $M_i$ still has the property that every nonzero constant entry is the only nonzero entry in its row and column. Once the above operations are completed for each $i$, the rows and columns may be permuted to obtain the desired blocks. The key observation is that in the above sequence of row and column operations, the entry $f$ is an existing matrix entry. As such, after each row or column operation, every matrix entry $g$ can be written as a $\Bbbk$-linear combination of products of (possibly constant) elements of $\mathcal M$. Moreover, if $g$ is a nonzero constant, then $g$ has degree $0$ under the grading by $S$, so by homogeneity, the aforementioned expression for $g$ cannot contain any monomials in $\mathcal N_S$, since it must be a $\Bbbk$-linear combination of products of degree-$0$ elements of $\mathcal M$. Now, fix a numerical semigroup $T$ in the same face $F$ of the Kunz cone $C_m$ as $S$. The sets $\mathcal M \setminus \mathcal N_S$ and $\mathcal M \setminus \mathcal N_T$ each contain $y^{b_{i,j}}$ whenever $F$ is contained in the facet $z_i + z_j = z_{i+j}$, and thus $\mathcal N_S = \mathcal N_T$. As a consequence of the preceding paragraph, applying an identical sequence of row and column operations to the Apéry resolution for $J_T$ yields nonzero constant entries in precisely the same locations at each step of the process. This completes the proof. ◻ Theorem [Theorem 19](#t:specializationindependent){reference-type="ref" reference="t:specializationindependent"} yields the following graded refinement of Theorem [Theorem 18](#t:ungradedbetti){reference-type="ref" reference="t:ungradedbetti"}. **Corollary 20**. *For each $i \in \{0, \ldots, m-1\}$, writing $[i]_m = i + m\mathbb{Z}$, $$%$$ $ \sum_{b \in [i]_m} \beta_{d,b}(J_S) = \sum_{b \in [i]_m} \beta_{d,b}(J_{S'}).$$ The same relationship holds between the Betti numbers of $I_S$ and $I_{S'}$.* *Proof.* Apply Theorem [Theorem 19](#t:specializationindependent){reference-type="ref" reference="t:specializationindependent"} for the first claim, and subsequently apply Proposition [Proposition 16](#p:freeresextravars){reference-type="ref" reference="p:freeresextravars"} for the final claim. ◻ [\[fig:specialization\]]{#fig:specialization label="fig:specialization"} $%$ \begin{array}{c@{\:}c@{\:}c@{\:}c@{\:}c} & \begin{blockarray}{rcccc} & \scriptstyle\textbf{1,1} & \scriptstyle\textbf{3,3} & \scriptstyle\textbf{2,1} & \scriptstyle\textbf{3,1} \\ \begin{block}{r@{\,\,}[@{\,\,}l@{\,}|*{3}{@{\,\,}l}]} \scriptstyle\textbf{000} & x_1^2 - x_2 & x_3^2 - x_1^2 y^{b_{33}} & x_1^3 - x_3 y^{b_{12}} & x_1x_3 - y^{b_{13}} \\ \end{block} \end{blockarray} & \\[-1em] 0 \leftarrow R & \mathord\leftarrow \mkern-6mu \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill \mkern-6mu \mathord-& \end{array}$ $%$ \begin{array}{c@{\:}c@{\:}c@{\:}c@{\:}c@{}c} & \begin{blockarray}{@{}r*{5}{@{\,\,}c}} \\ & \scriptstyle\textbf{3,23} & \scriptstyle\textbf{2,12} & \scriptstyle\textbf{2,13} & \scriptstyle\textbf{3,13} & \scriptstyle\textbf{3,12} \\ \begin{block}{@{}l@{\,\,}[*{3}{@{\,\,}l}|*{2}{@{\,\,}l}@{\,}]} \scriptstyle\textbf{1,1} & \phantom{-}x_3^2 - x_1^2 y^{b_{33}} & \phantom{-}x_1^3 - x_3 y^{b_{12}} & \phantom{-}x_1x_3 - y^{b_{13}} & \phantom{-}& \phantom{-}\\ \cline{2-6} \scriptstyle\textbf{3,3} & -(x_1^2 - x_2) & \phantom{-}& \phantom{-}& \phantom{-}x_1 & -y^{b_{12}} \\ \scriptstyle\textbf{2,1} & \phantom{-}& -(x_1^2 - x_2) & \phantom{-}& \phantom{-}y^{b_{33}} & -x_3 \\ \scriptstyle\textbf{3,1} & \phantom{-}& \phantom{-}& -(x_1^2 - x_2) & -x_3 & \phantom{-}x_1^2 \\ \end{block} \end{blockarray} && \begin{blockarray}{@{}r*{3}{@{\,\,}c}} & \scriptstyle\textbf{3,[3]} & \scriptstyle\textbf{2,[3]} \\ \begin{block}{@{}l@{\,\,}[*{3}{@{\,\,}l}@{\,}]} \scriptstyle\textbf{3,23} & \phantom{-}x_1 & -y^{b_{12}} \\ \scriptstyle\textbf{2,12} & \phantom{-}y^{b_{33}} & -x_3 \\ \scriptstyle\textbf{2,13} & -x_3 & \phantom{-}x_1^2 \\ \cline{2-3} \scriptstyle\textbf{3,13} & \phantom{-}x_1^2 - x_2 & \phantom{-}\\ \scriptstyle\textbf{3,12} & \phantom{-}& \phantom{-}x_1^2 - x_2 \\ \end{block} \end{blockarray} & \\[-1em] R^4 & \mathord\leftarrow \mkern-6mu \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill \mkern-6mu \mathord-& R^5 & \mathord\leftarrow \mkern-6mu \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill \mkern-6mu \mathord-& R^2 & \leftarrow 0 % \\ % \begin{array}{@{\hspace{-2ex}}c@{\hspace{-2ex}}} % \s 1|100 \\ % \s 3|001 \\ % \s 2|100 \\ % \s 3|100 \\ % \s 2|010 \\ % \s 3|010 \\ % \\ % \\ % \end{array} % && % \begin{array}{@{\hspace{-2ex}}c@{\hspace{-2ex}}} % \s 1|110 \\ % \s 1|101 \\ % \s 2|110 \\ % \s 2|011 \\ % \s 3|101 \\ % \s 3|011 \\ % \s 2|101 \\ % \s 3|110 \\ % \end{array} % && % \begin{array}{@{\hspace{-2ex}}c@{\hspace{-2ex}}} % \s 1|111 \\ % \s 2|111 \\ % \s 3|111 \\ % \\ % \\ % \\ % \\ % \\ % \end{array} \end{array}$ # Open questions {#sec:openquesitons} Several of the open questions presented here relate to the defining toric ideal $I_S$. One of the main results of [@kunzfaces1] identifies a finite poset corresponding to each face $F$ of $C_m$, called the *Kunz poset* of $F$. If a point interior to $F$ indexes a numerical semigroup $S$, then this poset coincides with the divisibility poset of $\mathop{\mathrm{Ap}}(S)$. In [@kunzfaces3], the Kunz poset of $F$ is used to obtain a parametrized family of minimal binomial generating sets, one for the defining toric ideal $I_S$ of each numerical semigroup $S$ in $F$. The last three binomials in the first matrix in Figure [\[fig:specialization\]](#fig:specialization){reference-type="ref" reference="fig:specialization"} constitute one such example for the relevant facet $F$ of $C_4$. This provides a natural candidate for the first matrix in the resolution conjectured as follows. **Conjecture 21**. *For each face $F$ of $C_m$, there exists a parametrized family of minimal resolutions, one for the defining toric ideal $I_S$ of each numerical semigroup $S$ indexed by the interior of $F$, akin to those obtained in Theorem [Theorem 19](#t:specializationindependent){reference-type="ref" reference="t:specializationindependent"} for Apéry ideals.* Unlike the proof of Theorem [Theorem 19](#t:specializationindependent){reference-type="ref" reference="t:specializationindependent"}, the proofs in [@kunzfaces3] are constructive, utilizing the Kunz poset structure of the face $F$ containing $S$. Intuitively, the set of factorizations of elements of $\mathop{\mathrm{Ap}}(S)$ (a set which depends only on $F$) forms the staircase of a monomial ideal $M$. If each element of $\mathop{\mathrm{Ap}}(S)$ factors uniquely, then $I_S$ has exactly one binomial generator for each minimal monomial generator of $M$. If some elements of $\mathop{\mathrm{Ap}}(S)$ have multiple factorizations, then graph-theoretic methods can be used to partition some of the minimal generators of $M$ into blocks (called *outer Betti elements*) and construct one minimal binomial generator of $I_S$ for each block. We refer the reader to [@kunzfaces3 Section 5] for the full construction; additional examples can be found in [@minprescard]. The non-constructive nature of the proof of Theorem [Theorem 19](#t:specializationindependent){reference-type="ref" reference="t:specializationindependent"}, along with the constructive nature of the proofs in [@kunzfaces3], motivates the following. **Problem 22**. Find an explicit combinatorial (e.g., poset-theoretic) construction of the matrices obtained in Theorem [Theorem 19](#t:specializationindependent){reference-type="ref" reference="t:specializationindependent"} and conjectured in Conjecture [Conjecture 21](#conj:parametrizedfreeres){reference-type="ref" reference="conj:parametrizedfreeres"}. Following Remark [Remark 15](#r:artinianreduction){reference-type="ref" reference="r:artinianreduction"}, substituting $y = 0$ into the resolution in Figure [\[fig:specialization\]](#fig:specialization){reference-type="ref" reference="fig:specialization"} yields a minimal free resolution for the ideal $\langle x_1^2 - x_2, x_1^3, x_1x_3, x_3^2\rangle$. More generally, for any numerical semigroup $S$, the construction in [@kunzfaces3] produces a generating set for $I_S$ with the property that substituting $y = 0$ yields an artinian binomial ideal depending only on the face $F$ of $C_m$ containing $S$. Each binomial in this generating set results from first syzygies whose graded degrees lie in $\mathop{\mathrm{Ap}}(S)$, while each monomial generator results from one of the outer Betti elements of the Kunz poset of $F$. It is thus natural to suspect the artinian reduction in Remark [Remark 15](#r:artinianreduction){reference-type="ref" reference="r:artinianreduction"} reflects a more general phenomenon. **Conjecture 23**. *Substituting $y = 0$ into the minimal resolution for $J_S$ obtained in Theorem [Theorem 19](#t:specializationindependent){reference-type="ref" reference="t:specializationindependent"} (or, analogously, the one for $I_S$ from Conjecture [Conjecture 21](#conj:parametrizedfreeres){reference-type="ref" reference="conj:parametrizedfreeres"}) yields a minimal free resolution of an artinian binomial ideal depending only on the face $F$ containing $S$.* There is a long history of topological formulas for Betti numbers of graded ideals (e.g., Hochster's formula for squarefree monomial ideals [@Hoc77] (see [@cca Chapter 1]), squarefree divisor complexes for toric ideals [@squarefreedivisorcomplex], or the use of poset homology for computing Poincaré series of semigroup algebras [@shellmonoid]). The following is thus a natural problem. **Problem 24**. Given a face $F$ of $C_m$, find a topological formula for extracting the value in the equation in Corollary [Corollary 20](#c:gradedbetti){reference-type="ref" reference="c:gradedbetti"} from the Kunz poset of $F$. As mentioned in Example [Example 6](#ex:kunzm4){reference-type="ref" reference="ex:kunzm4"}, the ray $(1,2,1)$ of $C_4$ contains positive integer points, but none correspond to a numerical semigroup under Proposition [Proposition 5](#prop:kunzinteriormed){reference-type="ref" reference="prop:kunzinteriormed"}. Indeed, the first and third coordinates of any such point must be equal, but any point $(a_1, a_2, a_3) \in C_4$ corresponding to a numerical semigroup must have $a_i \equiv i \bmod 4$ for each $i$. However, the construction in [@kunzfaces1] still associates a poset to this ray, and naively following the construction in [@kunzfaces3] for a binomial generating set with $y = 0$ as in Conjecture [Conjecture 23](#conj:artinianreduction){reference-type="ref" reference="conj:artinianreduction"} yields the artinian binomial ideal $\langle x_1^2 - x_3^2, x_1^3, x_1x_3, x_3^3\rangle$. This motivates the following. **Problem 25**. Extend the correspondance in Proposition [Proposition 5](#prop:kunzinteriormed){reference-type="ref" reference="prop:kunzinteriormed"} to a family of lattice ideals that includes the defining toric ideals of numerical semigroups but reaches points in faces of $C_m$ that do not contain numerical semigroups. Our final conjecture is supported by computational evidence for small $m$. 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arxiv_math

--- abstract: | In 1971, Ray and Singer proposed an analytic equivalent of a classical topological invariant, the $R$-torsion. This Ray-Singer torsion has had many ramifications in mathematics and physics. I will describe the background, the Ray-Singer papers and some subsequent work. address: | Department of Mathematics\ University of California, Berkeley\ Berkeley, CA 94720-3840\ USA author: - John Lott date: September 11, 2023 title: The Ray-Singer Torsion --- [^1] # Introduction {#sect1} As a bridge between two worlds, it's always interesting to find analytic equivalents for topological invariants. An outstanding example is the Hodge theorem, relating the real cohomology of a smooth compact manifold to the harmonic differential forms, after equipping the manifold with a Riemannian metric. In a 1971 paper [@Ray-Singer; @(1971)], Daniel Ray and Isadore Singer of MIT proposed an analytic equivalent of a classical topological invariant called the $R$-torsion. In so doing, they introduced a remarkable notion of a regularized determinant of a Laplace-type operator. The Ray-Singer work turned out to have contact with many areas of mathematics and physics, in ways that would have been hard for Ray and Singer to predict. In a second paper in 1973, they extended their methods to the holomorphic setting [@Ray-Singer; @(1973)]. After the Ray-Singer papers appeared, it was probably not clear how the Ray-Singer torsion fit into the wider framework of geometry and topology. This was clarified in later decades by looking at families of manifolds, rather than individual manifolds. In Section [2](#sect2){reference-type="ref" reference="sect2"}, I'll review some of the work that lead up to the Ray-Singer papers. Section [3](#sect3){reference-type="ref" reference="sect3"} describes the content of the Ray-Singer papers. Section [4](#sect4){reference-type="ref" reference="sect4"} has selected further developments. To give a coherent narrative, I haven't tried to be comprehensive. To restrict the length of this article, most of the work described is from the $20^{th}$ century. There are excellent expositions of the Ray-Singer work, such as Werner Müller's article in the Notices [@Muller; @(2022)]. I thank Jeff Cheeger and Dan Freed for their comments on an earlier version of this article. # Topological precedents {#sect2} There were both topological and analytical results that inspired the Ray-Singer work. Although I was a thesis student of Singer, I never asked him about the motivations for the Ray-Singer papers, so some of what follows are my surmises. Section [2.1](#sect2.1){reference-type="ref" reference="sect2.1"} has a description of lens spaces. Section [2.2](#sect2.2){reference-type="ref" reference="sect2.2"} has the definition of $R$-torsion and Section [2.3](#sect2.3){reference-type="ref" reference="sect2.3"} gives some of its properties. Some applications of the $R$-torsion are given in Section [2.4](#sect2.4){reference-type="ref" reference="sect2.4"}, namely the homeomorphism classification of lens spaces and the disproof of the Hauptvermutung for simplicial complexes. Section [2.5](#sect2.5){reference-type="ref" reference="sect2.5"} mentions the role of Arnold Shapiro. A reference for the material in this section is [@Milnor; @(1966)]. ## Lens spaces {#sect2.1} A basic problem in topology is to understand the homeomorphism types of manifolds. This is already challenging in three dimensions. There's a class of three dimensional manifolds called *lens spaces* which, although easy to define, have interesting topological properties. Let $p > q$ be relatively prime positive integers. Writing $S^3 = \{(z_1, z_2) \in {\mathbb C}^2 \: : \: |z_1|^2 + |z_2|^2 = 1 \}$, there's a free ${\mathbb Z}_p$-action on $S^3$ whose generator sends $(z_1, z_2)$ to $\left( e^{\frac{2 \pi i}{p}} z_1, e^{\frac{2 \pi iq}{p}} z_2 \right)$. The lens space $L(p,q)$ is the quotient of $S^3$ by this ${\mathbb Z}_p$-action. The word "lens" comes from a way of picturing a fundamental domain for the action [@Thurston; @(1997) Example 1.4.6]. There are analogous higher dimensional lens spaces in which ${\mathbb Z}_p$ acts on an odd dimensional sphere. When is $L(p,q)$ homeomorphic to $L(p^\prime, q^\prime)$? Considering fundamental groups, $p$ must be equal to $p^\prime$. One gets some further information from homotopy theory. However, it turns out that $L(7,1)$ is homotopy equivalent to $L(7,2)$, so standard algebraic topology won't help to decide whether they are homeomorphic. ## $R$-torsion {#sect2.2} The $R$-torsion was developed by Kurt Reidemeister [@Reidemeister; @(1935)], Wolfgang Franz [@Franz; @(1935)] and Georges de Rham [@de; @Rham; @(1936)] to understand when lens spaces are combinatorially equivalent. Franz did his Habilitation degree under the supervision of Reidemeister.[^2] Reidemeister's work was about three dimensional manifolds and had the aim of classifying the lens spaces up to combinatorial equivalence. De Rham and Franz extended Reidemeister's work to higher dimension. Franz dealt with arbitrary finite simplicial complexes. To establish some notation, let $V$ be a finite dimensional real vector space. If ${\bf v} = (v_1, \ldots v_n)$ and ${\bf w} = (w_1, \ldots w_n)$ are two bases of $V$ then we write $[{\bf w}/{\bf v}]$ for $| \det T|$, where $T$ is the change-of-basis matrix from ${\bf v}$ to ${\bf w}$, i.e. $w_i = \sum_j T_{ij} v_j$. Now let $C$ be a finite chain complex $$\label{2.1} C_N \stackrel{\partial}{\longrightarrow} C_{N-1} \stackrel{\partial}{\longrightarrow} \ldots C_1 \stackrel{\partial}{\longrightarrow} C_{0}$$ of finite dimensional real vector spaces. As usual, we write $Z_q \subset C_q$ and $B_q \subset C_q$ for the kernel and image of $\partial$, respectively, and put $H_q = Z_q/B_q$, so that we have short exact sequences $$\label{2.2} 0 \longrightarrow Z_q \longrightarrow C_q \stackrel{\partial}{\longrightarrow} B_{q-1} \longrightarrow 0$$ and $$\label{2.3} 0 \longrightarrow B_q \longrightarrow Z_q \longrightarrow H_q \longrightarrow 0.$$ Suppose that ${\bf c}_q$ is a preferred basis for $C_q$ and ${\bf h}_q$ is a preferred basis for $H_q$. Choose an auxiliary basis ${\bf b}_q$ for $B_q$. Choose elements $\tilde{\bf b}_{q-1} \subset C_q$ so that $\partial \tilde{\bf b}_{q-1} = {\bf b}_{q-1}$ and choose elements $\tilde{\bf h}_q \subset Z_q$ that project to ${\bf h}_q$. Then $({\bf b}_q, \tilde{\bf h}_q, \tilde{\bf b}_{q-1})$ is a basis for $C_q$. Since $[({\bf b}_q, \tilde{\bf h}_q, \tilde{\bf b}_{q-1})/{\bf c}_q]$ only depends on ${\bf b}_q$, ${\bf h}_q$, ${\bf b}_{q-1}$ and ${\bf c}_q$, we write it as $[({\bf b}_q, {\bf h}_q, {\bf b}_{q-1})/{\bf c}_q]$. **Definition 1**. *The torsion of $C$ is the positive real number $\tau(C)$ given by $$\label{2.5} \log \tau(C) = \sum_{q=0}^N (-1)^q \log [({\bf b}_q, {\bf h}_q, {\bf b}_{q-1})/{\bf c}_q].$$* It's not hard to see that $\tau(C)$ is independent of the choices of the ${\bf b}_q$ bases. And it just depends on the ${\bf c}_q$ and ${\bf h}_q$ bases through their induced volume forms on $C_q$ and $H_q$, respectively. If $K$ is a connected finite simplicial complex then its real chain groups have preferred bases and we can talk about the torsion of the chain complex, provided that we are given a preferred basis of the real homology groups. In order to compare two different simplicial complexes using numerical invariants, it turns out to be useful to have *acyclic* complexes, i.e. ones with vanishing homology, so that no choice of basis is needed. This can often be achieved by *local systems* on $K$. For us, a local system can be thought of as a flat Euclidean vector bundle on $K$ or, more algebraically, as arising from a homomorphism $\rho : \pi_1(K, k_0) \rightarrow O(n)$. Letting $\widetilde{K}$ denote the universal cover of $K$, the twisted chain groups are given by $C_q(K, \rho) = C_q(\widetilde{K}) \otimes_{{\mathbb R}\pi_1(K,k_0)} {\mathbb R}^n$, where an element $\sum_g a_g g$ of ${\mathbb R}\pi_1(K,k_0)$ acts on $v \in {\mathbb R}^n$ by $(\sum_g a_g g) \cdot v = \sum_g a_g \: \rho(g) v$. Choosing a fundamental domain of $K$ in $\widetilde{K}$, the natural basis of $C_q({K})$ and the standard basis of ${\mathbb R}^n$ combine to give a preferred basis of $C_q(K, \rho)$. **Definition 1**. *Given the finite simplicial complex $K$ and a representation $\rho$ so that $C(K, \rho)$ has vanishing homology, the $R$-torsion is given by $\tau_K(\rho) = \tau(C(K, \rho))$.* One can show that $\tau_K(\rho)$ is independent of the choice of fundamental domain in $\widetilde{K}$. ## Properties of the $R$-torsion {#sect2.3} The main point of the construction of $\tau_K(\rho)$ is that it is *invariant under subdivison*. In particular, if two triangulated manifolds can be subdivided to become combinatorially equivalent then they have the same values for the $R$-torsion. It follows that the $R$-torsion is a PL (piecewise linear) invariant of PL manifolds. In particular, it is a diffeomorphism invariant of smooth manifolds. (Some of the early literature doesn't distinguish clearly between PL homeomorphism and topological homeomorphism.) Now $\tau_K$ is not a homotopy invariant of $K$. However, it is invariant under a more restricted notion of homotopy equivalence, called *simple homotopy equivalence* [@Cohen; @(1973)]. As to whether two homeomorphic complexes have the same torsion, this will be true if the complexes are necessarily simple homotopy equivalent. Such was proven for PL manifolds by Kirby and Siebenmann [@Kirby-Siebenmann; @(1977)] and more generally for CW complexes by Chapman [@Chapman; @(1974)]. Chapman's proof involved a foray into "manifolds" locally modelled on Hilbert cubes. In some ways, the $R$-torsion is a cousin of the Euler characteristic. For example, the Euler characteristic of an odd dimensional closed (= compact boundaryless) manifold vanishes, while the $R$-torsion of an even dimensional (oriented) closed manifold vanishes. There is a product formula: If $K$ and $K^\prime$ are finite complexes, and $\rho : \pi_1(K^\prime, k_0^\prime) \rightarrow O(n)$ is a homomorphism, let $\widehat{\rho}$ be the composite homomorphism $\pi_1(K \times K^\prime, (k_0,k_0^\prime)) \rightarrow \pi_1(K^\prime, k_0^\prime) \rightarrow O(n)$. Assuming that $C(K^\prime, \rho)$ is acyclic, one finds that $$\label{2.7} T_{K \times K^\prime}(\widehat{\rho}) = \chi(K) T_{K^\prime}({\rho}).$$ The relation between $R$-torsion and Euler characteristic will be clarified in Section [5.4](#sect5.4){reference-type="ref" reference="sect5.4"}. ## Applications of the $R$-torsion {#sect2.4} Returning to lens spaces, we can triangulate $L(7,1)$ and $L(7,2)$. Running through the homomorphisms from ${\mathbb Z}_7$ to $O(2)$, one finds that the possible numerical values of the $R$-torsion differ for $L(7,1)$ and $L(7,2)$. Hence they cannot be homeomorphic. (Without using [@Chapman; @(1974)] or [@Kirby-Siebenmann; @(1977)], the $R$-torsion computations imply that $L(7,1)$ and $L(7,2)$ are not PL homeomorphic, and in three dimensions topological manifolds have unique PL structures [@Moise; @(1952)].) More generally, one sees that $L(p,q)$ is homeomorphic to $L(p, q^\prime)$ if $q^\prime \equiv \pm q \mod p$ (coming from the involution $z_2 \rightarrow \overline{z_2}$) or $\pm qq^\prime \equiv 1 \mod p$ (coming from the additional involution $(z_1, z_2) \rightarrow (z_2, z_1)$). The $R$-torsion shows that this is the only way that $L(p,q)$ can be *homeomorphic* to $L(p, q^\prime)$. In contrast, $L(p,q)$ is *homotopy equivalent* to $L(p, q^\prime)$ if and only if $\pm q q^\prime$ is a quadratic residue mod $p$. There are similar statements for higher dimensional lens spaces. More generally, consider *spherical space forms*, meaning quotients $M = S^r/\Gamma$ where $\Gamma$ is a finite subgroup of $O(r+1)$ that acts freely on $S^r$. De Rham showed that spherical space forms of a given dimension can be classified up to isometry by their fundamental groups and $R$-torsions [@de; @Rham; @(1950)]. As a consequence, two spherical space forms are homeomorphic if and only if they are isometric. A classical application of the $R$-torsion was to disprove the *Hauptvermutung* (or "main conjecture") for simplical complexes. The motivation went back to the beginnings of homology theory. The homology groups of a finite simplicial complex were first defined combinatorially using simplicial homology. While simplicial homology had many nice features, it was not at all clear whether *homeomorphic* simplicial complexes had isomorphic simplicial homology. One approach to this was to try to prove the Hauptvermuting, saying that homeomorphic simplicial complexes have combinatorially equivalent subdivisions. Using the subdivision invariance of simplicial homology, one would then conclude that simplicial homology is homeomorphism invariant. The homeomorphism invariance of simplicial homology was proven instead using the simplicial approximation theorem, but the validity of the Hauptvermutung stayed open until 1961 when John Milnor gave a counterexample [@Milnor; @(1961)]. Let $\sigma^r$ denote the $r$-simplex. For $j \in \{1,2\}$, let $L_j$ be a triangulation of $L(7,j)$. Consider the product $L_j \times \sigma^r$ and let $X_j$ denote the result of coning off its boundary $L_j \times \partial \sigma^r$. If $r \ge 3$ then $X_1$ is homeomorphic to $X_2$. The reason is that from an argument of Barry Mazur, $L_1 \times {\mathbb R}^r$ is homeomorphic to $L_2 \times {\mathbb R}^r$, and $X_j$ is homeomorphic to the one point compactification of $L_j \times {\mathbb R}^r$. On the combinatorial side, $X_1$ and $X_2$ are simply connected, so one can't use the $R$-torsion directly. Milnor instead used a variant of the $R$-torsion. Let $Y_j$ be the one point compactification of the universal cover of $L_j \times {\mathbb R}^r$, with $y_{j,0}$ being the added point. It inherits a cellular structure on which ${\mathbb Z}_7$ acts, freely off of $y_{j,0}$, with quotient $X_j$. Using a representation $\rho : {\mathbb Z}_7 \rightarrow O(2)$, the torsion of the relative chain complex $C_*(Y_j, y_{j,0}) \otimes_{{\mathbb R}{\mathbb Z}_7} {\mathbb R}^2$ and its subdivision invariance, Milnor showed that $X_1$ and $X_2$ do not have combinatorially equivalent subdivisions. ## Arnold Shapiro {#sect2.5} In the first Ray-Singer paper, one sees, "We raise the question as to how to describe this manifold invariant in analytic terms. Arnold Shapiro once suggested that there might be a formula for the torsion in terms of the Laplacian $\Delta$ acting on differential forms." Arnold Shapiro was a topologist who was born in 1921 and died in 1962, nine years before the first Ray-Singer paper appeared. Shapiro was a professor at Brandeis and was apparently the first person to come up with an explicit way to turn the sphere inside out. He is known for his paper with Atiyah and Bott on Clifford modules and K-theory, which appeared two years after his death. Referring to the period 1955-1957, Raoul Bott wrote, "During that time, and largely at Princeton, I met Serre, Thom, Hirzebruch, Atiyah, Singer, Milnor, Borel, Harish-Chandra, James, Adams,\... I could go on and on. But these people, together with Kodaira and Spencer, and my more or less 'personal remedial tutor', Arnold Shapiro, were the ones I had the most mathematical contact with.\" Regarding Shapiro's suggestion, there is a way to write the $R$-torsion in terms of *combinatorial* Laplacians. In reference to the chain complex ([\[2.1\]](#2.1){reference-type="ref" reference="2.1"}), the preferred basis of $C_q$ defines an inner product on $C_q$ for which the basis elements are orthonormal. Let $\partial^* : C_q \rightarrow C_{q+1}$ be the adjoint operator to $\partial$. Define the combinatorial Laplacian $\triangle^{(c)}_q : C_q \rightarrow C_q$ by $\triangle^{(c)} = - \left( \partial^* \partial + \partial \partial^* \right)$, a nonpositive self-adjoint operator. If the chain complex $C$ is acyclic then each $\triangle^{(c)}_q$ is invertible and [@Ray-Singer; @(1971) Proposition 1.7] states that $$\label{2.8} \log \tau(C) = \frac12 \sum_{q=0}^N (-1)^{q+1} q \log \det( - \triangle^{(c)}_q).$$ # Analytical precedents {#sect3} This section describes some of the analytical work leading up to the Ray-Singer papers. In Section [3.1](#sect3.1){reference-type="ref" reference="sect3.1"}, I recall the relation between the Riemann zeta function and heat conduction on a circle. Section [3.2](#sect3.2){reference-type="ref" reference="sect3.2"} describes the work of Minakshisundaram and Pleijel on defining the zeta function of the Laplacian on a compact Riemannian manifold. Section [3.3](#sect3.3){reference-type="ref" reference="sect3.3"} is about the subsequent work by McKean and Singer. A reference for the material in this section is [@Berline-Getzler-Vergne; @(2004) Chapter 2]. Following Shapiro's suggestion to write the $R$-torsion in terms of differential form Laplacians, and in view of ([\[2.8\]](#2.8){reference-type="ref" reference="2.8"}), Ray and Singer faced the task of making sense of the determinant of a self-adjoint operator on a Hilbert space. The relevant operators had discrete spectrum but the product of the eigenvalues was not convergent. Instead it was typically a product such as $1 \cdot 2 \cdot 3 \cdot \ldots$. There is a well known way to take the *sum* of such numbers, giving the semiserious formula $1 + 2 + 3 + \ldots = - \frac{1}{12}$. The meaning of this formula is in terms of the Riemann zeta function $$\label{3.1} \zeta(s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \ldots.$$ Formally, $1 + 2 + 3 + \ldots = \zeta(-1)$. On the other hand, we know that the zeta function can be meromorphically continued from $\Re(s) > 1$ to the complex plane and its value at $s = -1$ is rigorously $- \frac{1}{12}$. Ray and Singer showed how to give a meaning to determinants of Laplacians on manifolds, using analytic continuation. ## Riemann zeta function and heat conduction {#sect3.1} Let us first recall the relationship between the Riemann zeta function and heat conduction on a circle. Let $S^1(L)$ denote a circle of length $L$. The heat equation on the circle, with initial condition $u_0$, is $$\label{3.2} \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} , \: \: \: \: \: \: u(0,x) = u_0(x).$$ Here $u_0$ is a function on $S^1(L)$ or, equivalently, an $L$-periodic function on ${\mathbb R}$. The time-$t$ solution can be written as $u_t(x) = \int_{S^1(L)} K(t,x,y) \: u_0(y) \: dy$ where $K(t,x,y)$ is the heat kernel or, in operator terms, as $u_t = e^{t \partial_x^2} u_0$. Just as the trace of a matrix can be written as a sum of diagonal entries, we can write $$\label{3.3} \operatorname{Tr}\left( e^{t \partial_x^2} \right) = \int_{S^1(L)} K(t,x,x) \: dx.$$ The eigenvalues of $\partial_x^2$ are zero, with multiplicity one, and $\left\{- \left( \frac{2 \pi j}{L} \right)^2 \right\}_{j=1}^\infty$, each with multiplicity two. Hence $$\label{3.4} \operatorname{Tr}\left( e^{t \partial_x^2} \right) = 1 + 2 \sum_{j=1}^\infty e^{- t \left( \frac{2 \pi j}{L} \right)^2}.$$ Using the formula $\lambda^{-s} = \frac{1}{\Gamma(s)} \int_0^\infty t^{s-1} e^{- \lambda t} dt$ for positive $s$ and $\lambda$, we find that $$\label{3.5} 2 \left( \frac{L}{2\pi} \right)^{2s} \zeta(2s) = \frac{1}{\Gamma(s)} \int_0^\infty t^{s-1} \left( \operatorname{Tr}\left( e^{t \partial_x^2} \right) - 1 \right) dt$$ provided that $\Re(s) > \frac12$. We could describe the meromorphic continuation of ([\[3.5\]](#3.5){reference-type="ref" reference="3.5"}) using what is known about the Riemann zeta function but to give a direct description, we choose $\epsilon > 0$ and write the right-hand side as a sum of two terms, the first coming from the $t$-integration between $0$ and $\epsilon$, and the second coming from the $t$-integration between $\epsilon$ and $\infty$. Since $\operatorname{Tr}\left( e^{t \partial_x^2} \right) - 1$ decays exponentially fast in $t$, the second term is analytic in $s$. To handle the first term, there is an explicit formula $$\label{3.6} K(t,x,y) = \frac{1}{\sqrt{4 \pi t}} \sum_{k \in {\mathbb Z}} e^{- \frac{|x-y-kL|^2}{4t}}$$ for $x, y \in [0, L)$, coming from the heat kernel $\frac{1}{\sqrt{4 \pi t}} e^{- \frac{|x-y|^2}{4t}}$ on ${\mathbb R}$ and thinking of $x$ as receiving heat from sources $\{y+kL\}_{k \in {\mathbb Z}}$ in ${\mathbb R}$. Then the first term is $$\begin{aligned} \label{3.7} & \frac{1}{\Gamma(s)} \int_0^\epsilon t^{s-1} \left( \operatorname{Tr}\left( e^{t \partial_x^2} \right) - 1 \right) dt = \frac{1}{\Gamma(s)} \int_0^\epsilon t^{s-1} \left( \int_{S^1(L)} K(t,x,x) \: dx - 1 \right) dt = \\ & \frac{1}{\Gamma(s)} \int_0^\epsilon t^{s-1} \left( \frac{1}{\sqrt{4 \pi t}} L \sum_{k \in {\mathbb Z}} e^{- \frac{L^2 k^2}{4t}} - 1 \right) dt. \notag\end{aligned}$$ The only possible singularities come from the $k=0$ term, i.e. $\frac{1}{\Gamma(s)} \int_0^\epsilon t^{s-1} \frac{L}{\sqrt{4 \pi t}} \: dt$, which for large $s$ equals $\frac{L}{\sqrt{4 \pi}} \frac{\epsilon^{s-\frac12}}{\Gamma(s)} \frac{1}{s-\frac12}$. Hence the only singularity is a simple pole at $s = \frac12$ with residue $\frac{L}{2\pi}$. There is a similar story when the circle is replaced with a flat torus of dimension $N$. The Riemann zeta function is replaced by the Epstein zeta function. The analogous expression to ([\[3.5\]](#3.5){reference-type="ref" reference="3.5"}) acquires a simple pole at $\frac{N}{2}$ with a residue equal to $\frac{1}{(4\pi)^{\frac{N}{2}} \Gamma(\frac{N}{2})}$ times the volume of the torus. ## Minakshisundaram-Pleijel {#sect3.2} Going from circles to manifolds, Subbaramiah Minakshisundaram, from India, and Åke Pleijel, from Norway, defined the zeta function of the Laplacian on a compact Riemannian manifold [@Minakshisundaram-Pleijel; @(1949)]. The collaboration happened when they were visiting the Institute for Advanced Study during the 1947-1948 academic year. They had each worked on related problems for domains in the plane but neither of them had used Riemannian manifolds before. If $\triangle$ denotes the (nonpositive) Laplace operator then heat conduction on a closed Riemannian manifold $M$ satisfies the equation $$\label{3.8} \frac{\partial u}{\partial t} = \triangle u , \: \: \: \: \: \: u(0,x) = u_0(x).$$ The solution can be written $u_t(x) = \int_{M} K(t,x,y) \: u_0(y) \: \operatorname{dvol}(y)$, where $K(t,x,y)$ is the heat kernel, or in operator terms as $u(t) = e^{t \triangle} u_0$. The method of Minakshisundaram-Pleijel was to first write an approximate solution to $K(t,x,y)$ in normal coordinates around a point $y \in M$. Their approximate solution or "parametrix" was of the form $$\label{3.9} H(t,x,y) = (4 \pi t)^{- \frac{N}{2}} e^{- \frac{r^2}{4t}} \left( U_0 + U_1 t + \ldots + U_n t^n \right).$$ Here $N = \dim(M)$, $r$ is the distance from $x$ to $y$, each $U_j$ is a function of $x$ and $y$, and $n$ is a parameter. The $U_j$'s were computed recursively by requiring that $H$ satisfy the PDE ([\[3.8\]](#3.8){reference-type="ref" reference="3.8"}) to leading order. This gave a formula for $U_j$ in terms of $U_{j-1}$; the starting point was $U_0 = 1$. Minakshisundaram and Pleijel were able to estimate the error when approximating $K$ by $H$. (More precisely, they passed to Green's functions.) They then considered the expression $$\label{3.10} \zeta_{x,y}(s) = \frac{1}{\Gamma(s)} \int_0^\infty t^{s-1} \left( K(t,x,y) - 1 \right) \: dt.$$ If $\{\lambda_j\}_{j=1}^\infty$ are the nonzero eigenvalues of $\triangle$ then the corresponding zeta function is defined to be $$\label{3.11} \zeta_{\triangle}(s) = \sum_j (- \lambda_j)^{-s} = \int_M \zeta_{x,x}(s) \: \operatorname{dvol}(x).$$ The main result of Minakshisundaram and Pleijel was the following. **Theorem 1**. *[@Minakshisundaram-Pleijel; @(1949)] If $x \neq y$ then $\zeta_{x,y}(s)$ extends to an analytic function of $s$. If $x = y$ then $\zeta_{x,x}(s)$ extends to a meromorphic function of $s$. If $N$ is odd then $\zeta_{x,x}(s)$ has simple poles at $\frac{N}{2} - j$ for $j = 0, 1, 2, \ldots$, while if $N$ is even then $\zeta_{x,x}(s)$ has simple poles at $\frac{N}{2}, \frac{N}{2} - 1, \ldots, 1$. If $N$ is odd then $\zeta_{x,x}(s)$ vanishes at nonpositive integers.* *There is a similar statement for $\zeta_{\triangle}(s)$.* The statements about the locations of the poles of $\zeta_{x,x}(s)$ and the values at nonpositive integers can be seen by plugging the parametrix ([\[3.9\]](#3.9){reference-type="ref" reference="3.9"}) into ([\[3.10\]](#3.10){reference-type="ref" reference="3.10"}) and integrating from $0$ to $\epsilon$. If $N$ is even then the values of $\zeta_{x,x}(s)$ at nonpositive integers can be computed this way. In particular, they are expressions in the Riemannian metric and its derivatives, up to a certain order, at $x$. ## McKean-Singer {#sect3.3} A 1967 paper by Henry McKean and Singer extended the Minakshisundaram-Pleijel work in several ways [@McKean-Singer; @(1967)]. First, the coefficients $U_1(x,x)$ and $U_2(x,x)$ in ([\[3.9\]](#3.9){reference-type="ref" reference="3.9"}) were computed. They found that $U_1(x,x) = \frac{R(x)}{6}$, where $R$ is the scalar curvature, and $U_2(x,x)$ is the sum of a quadratic expression in the curvature tensor and a multiple of $\triangle R$. Second, McKean and Singer considered Laplacians on manifolds with boundary, with Dirichlet or Neumann boundary conditions. They constructed a parametrix and showed that there is again an asymptotic expansion $$\label{3.13} \operatorname{Tr}\left( e^{t \triangle} \right) = \int_M K(t,x,x) \: \operatorname{dvol}(x) \sim (4 \pi t)^{- \frac{N}{2}} (c_0 + c_1 t^{\frac12} + c_2 t + \ldots)$$ for small $t$, but now with half-integer powers. Third, and what is most relevant for the Ray-Singer papers, they discussed such asymptotic expansions when the function Laplacian is replaced by the Hodge Laplacian on differential forms. Their motivation for this discussion came from the possibility of proving the Chern-Gauss-Bonnet theorem using heat equation methods, a possibility that was later realized. # The Ray-Singer papers {#sect4} The Ray-Singer papers were joint works between Daniel Ray and Isadore Singer. Ray was a faculty member at MIT from 1957 to 1979. His specialties were stochastic processes and spectral problems. I had Ray as a teacher for undergraduate analysis. Section [4.1](#sect4.1){reference-type="ref" reference="sect4.1"} describes the first Ray-Singer paper and Section [4.2](#sect4.2){reference-type="ref" reference="sect4.2"} describes the second Ray-Singer paper. ## The first Ray-Singer paper {#sect4.1} Recall equation ([\[2.8\]](#2.8){reference-type="ref" reference="2.8"}), giving the $R$-torsion in terms of combinatorial Laplacians, and Shapiro's suggestion that there may be a formula for the torsion in terms of the Laplacian acting on differential forms. Whereas the determinants in ([\[2.8\]](#2.8){reference-type="ref" reference="2.8"}) are of finite dimensional operators, it is not clear what the determinant of an (infinite dimensional) differential form Laplacian should mean. It was for this purpose that Ray and Singer introduced *zeta function regularized determinants*. To motivate this notion, consider the calculus formula $$\label{4.1} \log \lambda = - \frac{d}{ds} \Big|_{s=0} e^{-s \log(\lambda)} = - \frac{d}{ds} \Big|_{s=0} \lambda^{-s}$$ for $\lambda$ positive. Extending this to matrices, if $M$ is a positive definite square matrix then using the spectral theorem, $\operatorname{Tr}\left( M^{-s} \right)$ is an analytic function of $s$ and $$\label{4.2} \log \det(M) = - \frac{d}{ds} \Big|_{s=0} \operatorname{Tr}\left( M^{-s} \right).$$ The idea of Ray and Singer was to use ([\[4.2\]](#4.2){reference-type="ref" reference="4.2"}) to *define* the determinant of suitable infinite dimensional operators. To specify the relevant operators, let $W$ be a connected closed orientable Riemannian manifold and let $E$ be a flat orthogonal vector bundle on $W$ or, equivalently, a homomorphism $\rho : \pi_1(W, w_0) \rightarrow O(n)$. Let $\Omega^q(W, E)$ denote the smooth $q$-forms on $W$ with value in $E$. The Riemannian metric on $W$, along with the standard inner product on ${\mathbb R}^n$, gives an $L^2$-inner product on $\Omega^q(W, E)$. The exterior derivative $d : \Omega^*(W, E) \rightarrow \Omega^{*+1}(W, E)$ has a formal adjoint $\delta : \Omega^*(W, E) \rightarrow \Omega^{*-1}(W, E)$. The differential form Laplacian is defined to be $\triangle = - (\delta d + d \delta)$. Let $\triangle_q$ denote the restriction of $\triangle$ to $\Omega^q(W, E)$. Let us initially suppose that $\triangle_q$ is negative definite for all $q$. By the Hodge theorem, this is equivalent to saying that $\operatorname{H}^q(W, E) = 0$ for all $q$. Putting $\zeta_{q,\rho}(s) = \operatorname{Tr}\left( (- \triangle_q)^{-s} \right)$, we can *define* $\det (\triangle_q)$ by $$\label{4.3} \log \det(\triangle_q) = - \frac{d}{ds} \Big|_{s=0} \zeta_{q,\rho}(s).$$ The key point is that $\zeta_{q,\rho}(s)$ is analytic near $s = 0$, so the definition makes sense. If we think of descending from $s$ large, where $\zeta_{q,\rho}(s)$ makes conventional sense, then we encounter poles in $\zeta_{q,\rho}(s)$ at $s = \frac{N}{2}, \frac{N}{2} - 1$, etc. So $\log \det(\triangle_q)$ can only be computed after traversing all of these poles. **Definition 1**. *Suppose that the flat vector bundle $E$ is acyclic, i.e. that $\operatorname{H}^*(W, E) = 0$. The analytic torsion is the positive real number $T_W(\rho)$ such that $$\label{4.5} \log T_W(\rho ) = \frac12 \sum_{q=0}^N (-1)^q q \zeta^\prime_{q,\rho}(0).$$* Note the similarity with ([\[2.8\]](#2.8){reference-type="ref" reference="2.8"}). Using Hodge duality, one finds that $T_W(\rho) = 1$ if $W$ is even dimensional. The main result of the Ray-Singer paper, that $T_W(\rho)$ is a diffeomorphism invariant of $W$, follows from the next theorem. **Theorem 1**. *[@Ray-Singer; @(1971)] $T_W(\rho)$ is independent of the Riemannian metric on $W$.* *Proof.* We may assume that $W$ is odd dimensional. Suppose that $g_0$ and $g_1$ are two Riemannian metrics on $W$. Putting $g(u) = u g_1 + (1-u) g_0$ and computing the analytic torsion with respect to $g(u)$, it suffices to show that $\frac{d}{du} \log T_W(\rho) = 0$. In terms of the Hodge duality operator $\star$, one can write $\delta = \pm \star^{-1} \circ d \circ \star$. Putting $\alpha = \star^{-1} \frac{d \star}{du}$, one has $\frac{d\delta}{du} = [\delta, \alpha]$. Then $\frac{d\triangle}{du} = -[\delta, \alpha]d - d [\delta, \alpha]$. Now $\frac{d}{du} \operatorname{Tr}(- \triangle_q)^{-s} = s \operatorname{Tr}\left( \frac{d \triangle_q}{du} (- \triangle_q)^{-s-1} \right)$; this is justified when $\Re(s)$ is large enough and then extends by analytic continuation. After some rearrangement, one finds $$\label{4.7} \frac{d}{du} \log T_W(\rho) = \frac12 \frac{d}{ds} \Bigg|_{s=0} \sum_{q=0}^N (-1)^{q+1} s \operatorname{Tr} \left( \alpha (- \triangle_q)^{-s} \right).$$ The key term is the $(- \triangle_q)^{-s}$ appearing on the right-hand side of ([\[4.7\]](#4.7){reference-type="ref" reference="4.7"}). Theorem [Theorem 1](#3.12){reference-type="ref" reference="3.12"}, or more precisely its extension to differential forms, implies that the expression $$\label{4.8} \operatorname{Tr} \left( \alpha (- \triangle_q)^{-s} \right) = \int_M \operatorname{tr}\left( \alpha(x) \: \zeta_{x,x}(s) \right) \: \operatorname{dvol}(x)$$ vanishes at $s=0$. Then the additional factor of $s$ in ([\[4.7\]](#4.7){reference-type="ref" reference="4.7"}) gives $\frac{d}{du} \log T_W(\rho) = 0$. ◻ Ray and Singer gave evidence that the analytic torsion equals the $R$-torsion. For example, they showed that the analog of ([\[2.7\]](#2.7){reference-type="ref" reference="2.7"}) holds for the analytic torsion. In some ways, it is convenient to remove the acyclicity assumption that $\operatorname{H}^*(W, E) = 0$. If $\operatorname{H}^*(W, E)$ is nonzero then Ray and Singer defined $T_W(\rho)$ as in ([\[4.5\]](#4.5){reference-type="ref" reference="4.5"}), where the zero eigenvalues of $\triangle_q$ are removed when constructing $\zeta_{q,\rho}(s)$. That is, $$\label{4.9} \zeta_{q,\rho}(s) = \frac{1}{\Gamma(s)} \int_0^\infty t^{s-1} \operatorname{Tr}\left( e^{t \triangle_q} - P_{\operatorname{Ker}(\triangle_q)} \right) \: dt,$$ where $P_{\operatorname{Ker}(\triangle_q)}$ denotes orthogonal projection onto the (finite dimensional) kernel of $\triangle_q$. On the $R$-torsion side, the Hodge isomorphism gives an $L^2$-inner product on $\operatorname{H}^q(W, E)$, and hence an orthonormal basis. Fixing this inner product, Ray and Singer showed that the $R$-torsion $\tau_{K}({\rho})$ is unchanged upon a subdivision of a triangulation $K$ of $W$. If the Riemannian metric varies, they computed the variations of $T_W(\rho)$ and $\tau_{K}({\rho})$. The variations were generally nonzero but just involved the change of volume form on $\operatorname{H}^*(W, E)$. In particular, they derived that $T_W(\rho)/\tau_{K}({\rho})$ is metric independent, even in the nonacyclic case. Much of the Ray-Singer paper dealt with the technically more challenging case when $W$ is allowed to have boundary. Without entering into the details, suffice it to say that most of their results for the closed case extend to the case of nonempty boundary. ## The second Ray-Singer paper {#sect4.2} In a sequel, Ray and Singer defined an analytic torsion in the holomorphic setting [@Ray-Singer; @(1973)]. In this case there was no classical counterpart. The analogy was that a smooth manifold goes to a complex manifold, a Riemannian metric goes to a Hermitian metric and the Hodge Laplacian goes to the $\overline{\partial}$-Laplacian. Suppose now that $W$ is a compact connected complex manifold of complex dimension $N$. Let $\rho : \pi_1(W, w_0) \rightarrow U(n)$ be a homomorphism, with corresponding flat holomorphic vector bundle $L$ on $W$. For each $p$, there is a complex $$\label{4.10} \ldots \stackrel{\overline{\partial}}{\longrightarrow} \Omega^{p,q}(W, L) \stackrel{\overline{\partial}}{\longrightarrow} \Omega^{p,q+1}(W, L) \stackrel{\overline{\partial}}{\longrightarrow} \ldots$$ and corresponding Laplacian $\triangle_p = - \left( \overline{\partial}^* \overline{\partial} + \overline{\partial} \overline{\partial}^* \right)$. Let $\triangle_{p,q}$ denote the restriction of $\triangle_p$ to $\Omega^{p,q}(W, L)$. Put $\zeta_{p,q,\rho}(s) = \operatorname{Tr}\left( (- \triangle^\prime_{p,q})^{-s} \right)$, where the $\prime$ on $\triangle^\prime_{p,q}$ indicates that zero eigenvalues are neglected. **Definition 1**. *Given $p$, the $\overline{\partial}$-torsion $T_p(W, \rho)$ is the positive real number such that $$\label{4.12} \log T_p(W,\rho) = \frac12 \sum_{q=0}^N (-1)^q q \zeta^\prime_{p,q,\rho}(0).$$* Regarding the dependence of $T_p(W,\rho)$ on the Hermitian metric, the proof of Theorem [Theorem 1](#4.6){reference-type="ref" reference="4.6"} goes through except for the last step. Because the real dimension of $W$ is even, $\operatorname{Tr} \left( \alpha (- \triangle_q)^{-s} \right)$ need not vanish at $s=0$. However, Ray and Singer deduced the following statement. **Theorem 1**. *[@Ray-Singer; @(1973)] Let $\rho_1$ and $\rho_2$ be two homomorphisms from $\pi_1(W, w_0)$ to $U(n)$, with corresponding flat vector bundles $L_1$ and $L_2$, respectively. Given $p$, suppose that $\operatorname{H}^{p,q}(W, L_j) = 0$ for all $q \in [0, N]$ and for $j \in \{1,2\}$. Then $T_p(W,\rho_1)/T_p(W,\rho_2)$ is independent of the Hermitian metric on $W$.* The reason that the theorem holds is that if $h(u)$ is a one-parameter family of Hermitian metrics on $W$ then the argument for Theorem [Theorem 1](#4.6){reference-type="ref" reference="4.6"} shows that $\frac{d}{du} \log T_p(W,\rho_j)$ is an integral over $W$ whose integrand just depends on the local geometry. In taking the difference $\frac{d}{du} \left( \log T_p(W,\rho_1) -\log T_p(W,\rho_2) \right)$ the integrand cancels out. If $N=1$ then one can remove the assumption that $\operatorname{H}^{p,q}(W, L_j) = 0$. Hence under the assumptions of Theorem [Theorem 1](#4.13){reference-type="ref" reference="4.13"}, the ratio $T_p(W,\rho_1)/T_p(W,\rho_2)$ is an invariant of the complex manifold $W$. Ray and Singer computed it explicitly for Riemann surfaces. In the case of genus one, i.e. tori, the answer was in terms of theta functions. In the case of genus greater than one, they used the Selberg trace formula to express the answer in terms of Selberg zeta functions. # Further developments {#sect5} After the first Ray-Singer paper, an outstanding problem was to show that the analytic torsion equals the $R$-torsion. Section [5.1](#sect5.1){reference-type="ref" reference="sect5.1"} describes the proofs of this by Cheeger and Müller, along with the subsequent proof by Bismut and Zhang. A further understanding of the Ray-Singer torsion came from looking at *families*. Section [5.2](#sect5.2){reference-type="ref" reference="sect5.2"} explains how Quillen used the $\overline{\partial}$-torsion of the second Ray-Singer paper in the setting of a family of $\overline{\partial}$-operators on a complex vector bundle over a Riemann surface. Section [5.3](#sect5.3){reference-type="ref" reference="sect5.3"} has the extension to higher dimension, due to Bismut-Gillet-Soulé, along with their construction of a holomorphic torsion form. Section [5.4](#sect5.4){reference-type="ref" reference="sect5.4"} describes the analytic torsion form of a smooth fiber bundle, due to Bismut and me. ## Cheeger-Müller theorem {#sect5.1} In their first paper, Ray and Singer showed that the $R$-torsion $\tau_K(\rho)$ and the analytic torsion $T_W(\rho)$ have formal similarities. Furthermore, Ray showed by explicit computation that they coincide for lens spaces [@Ray; @(1970)]. This naturally lead to the problem of showing that $T_W(\rho) = \tau_K(\rho)$ when $K$ is a triangulation of $W$. There are various proofs of this, each being technically involved. In order to show that $T_W(\rho)$ and $\tau_K(\rho)$ are the same, a first approach might be to use the similarity between ([\[2.8\]](#2.8){reference-type="ref" reference="2.8"}) and ([Definition 1](#4.11){reference-type="ref" reference="4.11"}), take finer and finer triangulations of $W$, and hope that ([\[2.8\]](#2.8){reference-type="ref" reference="2.8"}) approaches ([Definition 1](#4.11){reference-type="ref" reference="4.11"}). This approach cannot work directly since ([Definition 1](#4.11){reference-type="ref" reference="4.11"}) requires an analytic continuation beyond singularities in the $s$-plane, whereas one does not see any such singularities in ([\[2.8\]](#2.8){reference-type="ref" reference="2.8"}). Rather than trying to show directly that $T_W(\rho)/\tau_K(\rho) = 1$ Ray and Singer proposed to first take two representations $\rho_1, \rho_2 : \pi_1(W, w_0) \rightarrow O(n)$ and show that the difference $$\label{5.1} \log \left( \frac{T_W(\rho_1)}{\tau_K(\rho_1)} \right) - \log \left( \frac{T_W(\rho_2)}{\tau_K(\rho_2)} \right) =\log \left( \frac{T_W(\rho_1)}{T_W(\rho_2)} \right) - \log \left( \frac{\tau_K(\rho_1)}{\tau_K(\rho_2)} \right)$$ vanishes. Since the residues of the (simple) poles of the zeta function are the integrals of local expressions on $W$, the difference $\zeta_{q,\rho_1}(s)-\zeta_{q,\rho_2}(s)$ is analytic in $s$. Because of this, $\log (T_W(\rho_1)/T_W(\rho_2))$ should have better approximation properties than $\log T_W(\rho_1)$ or $\log T_W(\rho_2)$ individually. Ray and Singer's idea was to put a Morse function $f$ on $W$, look at its sublevel sets $W_u = f^{-1}(-\infty, u]$, compute the various torsions on $W_u$ (with appropriate boundary conditions) and analyze how the expression in ([\[5.1\]](#5.1){reference-type="ref" reference="5.1"}) depends on $u$. Away from the critical values of $f$, it should be constant in $u$. One would then want to analyze how it changes when one passes through a critical value. In this approach it is important to remove the acyclicity condition. The equality of $T_W(\rho)$ and $\tau_K(\rho)$ was proven independently by Jeff Cheeger [@Cheeger; @(1979)] and Werner Müller [@Muller; @(1978)]. As in the Ray-Singer idea, the proofs involved analyzing how $T_W(\rho)/\tau_K(\rho)$ varies under topological change. How this was implemented differed from what Ray and Singer had in mind. One common idea to the Cheeger and Müller papers was to use the fact that the torsions are defined independent of orientation, and the torsion of $W \cup W$ is twice the torsion of $W$. As $W \cup W$ is the boundary of $[0,1] \times W$, there is a sequence of surgeries that transform $W \cup W$ to $S^N$. If one can keep track of how the torsions change under surgery then one can reduce to checking the equality on $S^N$. Müller's first step was to use approximations of the differential form Laplacian by the combinatorial Laplacian to show that ([\[5.1\]](#5.1){reference-type="ref" reference="5.1"}) vanishes. Hence $T_W(\rho)/\tau_K(\rho)$ was independent of the representation and it sufficed to work with the trivial representation $\rho$. Next, suppose that one has a presurgery manifold $W_1$ and a postsurgery manifold $W_2$. The surgery amounts to removing a copy of $S^k \times D^{N-k}$ and adding a copy of $D^{k+1} \times S^{N-k-1}$. One can assume that the Riemannian metric is standard on those pieces. Let $W_3$ be the double $S^k \times S^{N-k}$ of $S^k \times D^{N-k}$, and let $W_4$ be the double $S^{k+1} \times S^{N-k-1}$ of $D^{k+1} \times S^{N-k-1}$. In the combination $\zeta^{W_1}_{q,\rho}(s)-\zeta^{W_2}_{q,\rho}(s) - \frac12 \zeta^{W_3}_{q,\rho}(s) + \frac12 \zeta^{W_4}_{q,\rho}(s)$ the singularities cancelled out, and so one obtained an analytic function of $s$. Müller showed that this combination can be approximated by the analogous expression in combinatorial Laplacians, obtaining in the end that $T_{W_1}(\rho)/\tau_{K_1}(\rho) = T_{W_2}(\rho)/\tau_{K_2}(\rho)$. This finally reduced to checking the ratio for $W = S^N$, where it followed from Ray's calculations. Cheeger's approach to the surgery was to keep careful track of the eigenvalues under a conical degeneration. Let $W_1(u)$ denote the result of removing $S^k \times D^{N-k}(u)$ from $W_1$, where $D^{N-k}(u)$ denotes a disk of radius $u$. Cheeger imposed absolute boundary conditions on $W_1(u)$ and made a refined analysis of how the heat kernel for the differential form Laplacian behaved as $u \rightarrow 0$. A model space for this analysis was the product $S^k \times A_{u,1}^{N-k}$, where $A_{u,1}^{N-k}$ denotes the annulus in ${\mathbb R}^{N-k}$ with outer radius one and inner radius $u$. With the symmetric analysis for $W_2$, he was able to show that $T_{W_1}(\rho)/\tau_{K_1}(\rho) = T_{W_2}(\rho)/\tau_{K_2}(\rho)$, provided that $1 \le k \le n-2$. Finally, replacing $W$ by $W \times S^6$, he was able to prove that $T_{W}(\rho)/\tau_{K}(\rho) = 1$. Besides proving the equality of the torsions, Cheeger's methods lead to his later work on the spectral geometry of singular Riemannian spaces [@Cheeger; @(1983)]. Jean-Michel Bismut and Weiping Zhang gave an alternative proof of the Cheeger-Müller theorem using a Morse function $f$ on $W$ [@Bismut-Zhang; @(1992)]. The idea was to do a Witten deformation, meaning that one replaces $d$ by $e^{-Tf} \circ d \circ e^{Tf}$ and $\delta$ by $e^{Tf} \circ \delta \circ e^{-Tf}$. As in the proof of Theorem [Theorem 1](#4.6){reference-type="ref" reference="4.6"}, the ensuing analytic torsion is independent of $T$. As $T \rightarrow \infty$, most of the eigenvalues of $\triangle$ go to minus infinity. The ones that stay bounded have eigenfunctions that give the (finite dimensional) Witten complex computing $\operatorname{H}^*(W, E)$. By means of this limit, using generic Morse functions $f$ (i.e. $\nabla f$ satisfies the Smale transversality conditions) Bismut and Zhang were able to prove that $T_W(\rho) = \tau_K(\rho)$ without performing surgery on $W$. Finally, an approach using a gluing formula for the analytic torsion was given by Simeon Vishik [@Vishik; @(1995)]. ## Determinant line bundle {#sect5.2} In 1985, Daniel Quillen published a four page paper that gave a new understanding of the $\overline{\partial}$-torsion [@Quillen; @(1985)]. Quillen's paper had one reference, the second Ray-Singer paper. As a first step, Quillen applied the definition of the $\overline{\partial}$-torsion not just for a flat unitary bundle, but rather for a general holomorphic bundle equipped with a Hermitian inner product. Since the analytic torsion of the first Ray-Singer paper was defined using a $d$-flat vector bundle, it is natural in the holomorphic setting to replace this by a $\overline{\partial}$-flat vector bundle, i.e. a holomorphic vector bundle $E$. We write the corresponding $\overline{\partial}$-torsion as $T_W(E)$, taking $p=0$. Quillen considered a compact Riemann surface $W$ and a smooth complex vector bundle $E$ on $W$. A holomorphic structure on $E$ corresponds to a choice of $\overline{\partial}$-operator $\overline{\partial} : \Omega^{0,0}(W, E) \rightarrow \Omega^{0,1}(W, E)$; the local holomorphic sections $s$ of $E$ correspond to the local solutions of $\overline{\partial} s = 0$. Rather than looking at a single holomorphic structure on $E$, Quillen looked at the family ${\mathcal A}$ of *all* such structures. It has a natural complex structure. In this setting, there is a holomorphic line bundle $Det$ on ${\mathcal A}$ called the *determinant line bundle*. Given a holomorphic structure $a \in {\mathcal A}$ on $E$, the fiber of $Det$ over $a$ is $$\label{5.2} Det_a = \left( \Lambda^{max} \operatorname{H}^0(W, E) \right)^* \otimes \Lambda^{max} \operatorname{H}^1(W, E),$$ where $\Lambda^{max}$ denotes the highest exterior power. Suppose that we want to put an inner product on $Det$. Given a Riemannian metric on $W$ and a Hermitian inner product on $E$, for each $a \in {\mathcal A}$ the Hodge isomorphism gives an $L^2$ inner product $\langle \cdot, \cdot \rangle_{L^2,a}$ on $Det_a$. Unfortunately, this inner product need not be continuous in $a$. The issue is that while $Det_a$ is smooth in $a$, the individual factors $\operatorname{H}^0(W, E)$ and $\operatorname{H}^1(W, E)$ can abruptly jump in dimension. Quillen's insight was that this lack of continuity can be corrected using the $\overline{\partial}$-torsion. The *Quillen metric* on $Det_a$ is defined by $\langle \cdot, \cdot \rangle_{Q,a} = T_W(E)^2 \langle \cdot, \cdot \rangle_{L^2,a}$. It gives rise to a *smooth* inner product on $Det$. An inner product on a holomorphic line bundle induces a preferred connection on the line bundle. One can then talk about the curvature of the connection. In the case of the determinant line bundle, computing the curvature essentially amounts to computing $\partial \overline{\partial} \log T_W(E)$, a $2$-form on ${\mathcal A}$. Quillen found a formula for the curvature involving the integral of a *local* expression on $W$, in contrast to the nonlocal nature of $T_W(E)$, If $\operatorname{H}^0(W, E)$ and $\operatorname{H}^1(W, E)$ both happen to vanish then the *determinant line* $Det_a$ can be identified with ${\mathbb C}$, with a canonical element $1 \in Det_a$. In this setting, the Quillen norm of $1$ is $T_W(E)$. Although it may look as if we haven't achieved anything new, the framework of determinant line bundles is useful as one can sometimes use holomorphic methods to *compute* $T_W(E)$ [@Bost-Nelson; @(1986)]. ## Holomorphic torsion form {#sect5.3} Quillen's work was extended to higher dimensions by Jean-Michel Bismut, Henri Gillet and Christophe Soulé [@BGS; @(1988)]. Furthermore, they found that the Ray-Singer torsion $T_W(E)$ entered into a differential form version of the Riemann-Roch-Grothendieck (RRG) theorem. The setup of [@BGS; @(1988)] was a family of complex structures on a compact manifold $Z$, parametrized by a complex manifold $B$. That is, one has a holomorphic fiber bundle $\pi : M \rightarrow B$ whose fibers are diffeomorphic to $Z$. They also assumed that the fibers carry Kähler metrics that form a Kähler fibration, in the sense that the Kähler forms on the fibers are the restrictions of a closed $(1,1)$-form on $M$. Let $E$ be a holomorphic vector bundle on $M$, equipped with a Hermitian inner product $h^E$. There is an induced Chern connection on $E$. Put $Z_b = \pi^{-1}(b)$. The determinant line bundle $Det$ is a holomorphic line bundle on $B$ whose fiber over $b \in B$ is $$\begin{aligned} \label{5.3} Det_b = & \left( \Lambda^{max} \operatorname{H}^0 \left( Z_b, E \big|_{Z_b} \right) \right)^* \otimes \Lambda^{max} \operatorname{H}^1 \left( Z_b, E \big|_{Z_b} \right) \otimes \\ & \left( \Lambda^{max} \operatorname{H}^2 \left( Z_b, E \big|_{Z_b} \right) \right)^* \otimes \Lambda^{max} \operatorname{H}^3 \left( Z_b, E \big|_{Z_b} \right) \otimes \ldots \notag\end{aligned}$$ If $\langle \cdot, \cdot \rangle_{L^2,b}$ denotes the $L^2$-inner product on $Det_b$ then the Quillen metric $\langle \cdot, \cdot \rangle_{Q}$ on $Det$ is defined by $\langle \cdot, \cdot \rangle_{Q,b} = T\left( Z_b, E \big|_{Z_b} \right)^2 \langle \cdot, \cdot \rangle_{L^2,b}$. Put $TZ = \operatorname{Ker}(d\pi)$, a holomorphic vector bundle on $M$. The Kähler fibration gives a connection on $TZ$. **Theorem 1**. *[@BGS; @(1988)] The curvature $2$-form associated to the Quillen metric is $2 \pi i$ times the $2$-form component of $\int_Z \operatorname{Td}(TZ, g^{TZ}) \wedge \operatorname{ch}(E, h^E)$.* Here $\int_Z$ is integration over the fiber, $\operatorname{Td}$ is the Todd form and $\operatorname{ch}$ is the Chern character form. The normalization is such that $\operatorname{Td}$ and $\operatorname{ch}$ represent rational cohomology classes. The validity of the theorem was previously known on the level of cohomology. The point is that an explicit $2$-form representative arises geometrically as the curvature of $Det$, when the latter is equipped with the Quillen metric. On the level of cohomology, the expression $\int_Z \operatorname{Td}(TZ) \: \cup \: \operatorname{ch}(E)$ is the right-hand side of the RRG theorem. To state the theorem, let us make the additional assumption that for each $q$, the dimension of $\operatorname{H}^q \left( Z_b, E \big|_{Z_b} \right)$ is constant in $b$. Then the vector spaces $\left\{ \operatorname{H}^q \left( Z_b, E \big|_{Z_b} \right) \right\}_{b \in B}$ fit together to form a holomorphic vector bundle $\underline{\operatorname{H}}^q$ on $B$. In this setting the RRG theorem says that $$\label{5.5} \sum_{q=0}^{\dim_{\mathbb C}(Z)} (-1)^q \operatorname{ch}(\underline{\operatorname{H}}^q) = \int_Z \operatorname{Td}(TZ) \cup \operatorname{ch}(E),$$ in $\operatorname{H}^{even}(B; {\mathbb R})$. The vector bundle $\underline{\operatorname{H}}^q$ acquires an $L^2$-inner product $h^{\underline{\operatorname{H}}^q}$ and corresponding connection. One can ask whether ([\[5.5\]](#5.5){reference-type="ref" reference="5.5"}) becomes an equality of differential forms. This is not the case, but the discrepancy can be described using the *holomorphic torsion form*. **Theorem 1**. *[@BGS; @(1988)] There is a canonical form ${\mathcal T} \in \Omega^{even}(B)$, depending on the above geometric data, so that $$\label{5.7} \sqrt{-1} \partial \overline{\partial} {\mathcal T} = \int_Z \operatorname{Td}\left( TZ, \nabla^{TZ} \right) \wedge \operatorname{ch}(E, h^E) - \sum_{q=0}^{\dim_{\mathbb C}(Z)} (-1)^q \operatorname{ch}\left( \underline{\operatorname{H}}^q, h^{\underline{\operatorname{H}}^q} \right)$$ in $\Omega^{even}(B)$. The degree-zero component ${\mathcal T}_{[0]} \in C^\infty(B)$ of ${\mathcal T}$ is related to the $\overline{\partial}$-torsion by ${\mathcal T}_{[0]}(b) = \frac{1}{\pi} \log T\left( Z_b, E \big|_{Z_b} \right)$.* Because of Theorem [Theorem 1](#5.6){reference-type="ref" reference="5.6"}, the form ${\mathcal T}$ can be called the *holomorphic torsion form*. It can be considered to be a *transgression* of the RRG theorem, on the level of differential forms. As a remark, the formalism of determinant line bundles and Quillen metrics extends to smooth families of Dirac-type operators [@Berline-Getzler-Vergne; @(2004) Chapters 9.7 and 10.6], [@Bismut-Freed; @(1986); @Bismut-Freed; @(1987)]. The definition of the Quillen metric again involves a product of regularized determinants, although in general it cannot be identified with the Ray-Singer torsion. ## Analytic torsion form {#sect5.4} As described in the previous section, the $\overline{\partial}$-torsion is the $0$-form component of a torsion form that represents a transgression of the RRG formula. To come full circle, one can ask if there's a similar interpretation of the original Ray-Singer torsion. It turns out that there is, as was shown by Bismut and me. In order to see this interpretation, it was necessary to extend the definition of the Ray-Singer torsion $T_W(\rho)$ beyond the case of flat orthogonal or unitary vector bundles. Let $E$ be a flat complex vector bundle on $W$. Suppose that $E$ is equipped with a Hermitian inner product $h^E$, not necessarily covariantly constant. Then one can still use the formula ([\[4.5\]](#4.5){reference-type="ref" reference="4.5"}) to define the Ray-Singer torsion $T_W(E)$. Considering the role that volume forms play in the $R$-torsion, a natural extension of the Ray-Singer work was to assume that $E$ has unimodular holonomy, i.e. taking values in $\{ A \in \operatorname{GL}(n, {\mathbb C}) \: : \: |\det A| = 1\}$. In this case Müller proved the extension of the Cheeger-Müller theorem [@Muller; @(1993)]. This had later application to the growth of torsion in the cohomology of locally symmetric spaces, as described in [@Muller; @(2022)]. Going beyond this, Bismut and Zhang considered arbitrary flat complex vector bundles $E$ [@Bismut-Zhang; @(1992)]. The topological meaning of $T_W(E)$ was not so clear in this case but Bismut and Zhang proved "anomaly" formulas showing how $T_W(E)$ depends on $g_W$ and $h^E$. Using the analogy that $\overline{\partial}$-flat vector bundles, i.e. holomorphic bundles, are like $d$-flat vector bundles, i.e. flat vector bundles, the first question was whether there is a analog of the RRG theorem for flat vector bundles. It turns out that there is. To state it, let us define certain characteristic classes of *flat* vector bundles. Let $W$ be a smooth manifold and let $E$ be a flat complex vector bundle on $W$. Let $h^E$ be a Hermitian inner product on $E$ (not necesarily covariantly constant). With respect to a local covariantly constant basis of $E$, we can think of $h^E$ locally as a Hermitian matrix valued function on $B$. Put $\omega(E, h^E) = (h^E)^{-1} dh^E$, a globally defined $\operatorname{End}(E)$-valued $1$-form on $B$. If $k$ is a positive odd integer, put $$\label{5.8} c_k(E, h^E) = (2 \pi i)^{- \frac{k-1}{2}} 2^{-k} \operatorname{tr}\left( (\omega(E, h^E))^k \right).$$ Then $c_k(E, h^E)$ is closed and its de Rham cohomology class $c_k(E)$ is independent of $h^E$. To give some idea of what $c_k(E)$ measures, it vanishes if $E$ has unitary holonomy. And $c_1(E)$ vanishes if the holonomy is unimodular. Now let $\pi : M \rightarrow B$ be a fiber bundle with closed fibers $Z_b = \pi^{-1}(b)$. Let $E$ be a flat complex vector bundle on $M$ and let $\underline{\operatorname{H}}^q$ be the flat complex vector bundle on $B$ whose fiber over $b \in B$ is $\operatorname{H}^q \left( Z_b, E \Big|_{Z_b} \right)$. Let $TZ = \operatorname{Ker}(d\pi)$ be the vertical tangent bundle, a vector bundle on $M$, and let $o(TZ)$ be its orientation bundle, a flat ${\mathbb R}$-bundle on $M$. Let $e(TZ) \in \operatorname{H}^{\dim(Z)}(M; o(TZ))$ be the Euler class of $TZ$. The following is an analog of the RRG theorem, for flat vector bundles. **Theorem 1**. *[@Bismut-Lott; @(1995)] For any positive odd number $k$, $$\label{5.10} \sum_{q=0}^{\dim (Z)} (-1)^q c_k(\underline{\operatorname{H}}^q) = \int_Z e(TZ) \cup c_k(E)$$ in $\operatorname{H}^k(B; {\mathbb R})$.* To explain where the torsion comes in, equip the fiber bundle with a horizontal distribution $T^HM$ and a vertical Riemannian metric $g^{TZ}$. Also equip $E$ with a Hermitian inner product. Then the vector bundle $\underline{\operatorname{H}}^q$ acquires an $L^2$-inner product $h^{\underline{\operatorname{H}}^q}$. The next result states the existence of "higher" analytic torsion forms that realize ([\[5.10\]](#5.10){reference-type="ref" reference="5.10"}) on the level of differential forms. **Theorem 1**. *[@Bismut-Lott; @(1995)] For any positive odd number $k$, there is an explicit $(k-1)$-form ${\mathcal T}_{k-1}$, depending on the geometric data, so that $$\label{5.12} d{\mathcal T}_{k-1} = \int_Z e \left( TZ, \nabla^{TZ} \right) \wedge c_k(E, h^E) - \sum_{q=0}^{\dim (Z)} (-1)^q c_k \left( \underline{\operatorname{H}}^q, h^{\underline{\operatorname{H}}^q} \right)$$ in $\Omega^k(B; {\mathbb R})$. When $k = 1$, the function ${\mathcal T}_0 \in C^\infty(B)$ is such that ${\mathcal T}_0(b)$ is the negative of the logarithm of the Ray-Singer torsion $T_{Z_b}( E \Big|_{Z_b})$.* When $k=1$, equation ([\[5.12\]](#5.12){reference-type="ref" reference="5.12"}) recovers the anomaly formula of Bismut and Zhang. If $\dim(Z)$ is odd and $\underline{\operatorname{H}}^q = 0$ for all $q$ then ([\[5.12\]](#5.12){reference-type="ref" reference="5.12"}) implies that ${\mathcal T}_{k-1}$ is closed, and hence has a de Rham representative $[{\mathcal T}_{k-1}] \in \operatorname{H}^{k-1}(B, {\mathbb R})$. It turns out that $[{\mathcal T}_{k-1}]$ is independent of the choices of $T^H M$, $g^TZ$ and $h^E$, i.e. just depends on the smooth fiber bundle $\pi : M \rightarrow B$ and the flat complex vector bundle $E \rightarrow M$. On the other hand, there are "higher" versions of the $R$-torsion, which are also invariants of smooth fiber bundles [@Dwyer-Weiss-Williams; @(2003); @Igusa; @(2002)]. The question then arises if there is a higher Cheeger-Müller theorem. The latest on this is [@Puchol-Zhang-Zhu; @(2021)]. 99 F. Bachmann, H. Behnke and W. Franz, "In memoriam Kurt Reidemeister", Math. Ann. 199, p. 1-11 (1972) N. Berline, E. Getzler and M. Vergne, [Heat kernels and Dirac operators]{.ul}, $2^{nd}$ edition, Springer, New York (2004) J.-M. Bismut and D. Freed, "The analysis of elliptic families. I. Metrics and connections on determinant bundles", Comm. Math. Phys. 106, p. 159-176 (1986) J.-M. Bismut and D. Freed, "The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem", Comm. Math. Phys. 107, p. 103-163 (1986) J.-M. Bismut, Henri Gillet and Christophe Soulé, "Analytic torsion and holomorphic determinant bundles I, II, III", Comm. Math. Phys. 115, p. 49-351 (1988) J.-M. Bismut and J. Lott, "Flat vector bundles, direct images and higher real analytic torsion", J. Amer. Math. Soc. 8, p. 291-363 (1995) J.-M. Bismut and W. Zhang, "An extension of a theorem by Cheeger and Müller", Astérisque 205 (1992) J.-B. Bost and P. Nelson, "Spin-$\frac12$ bosonization on compact surfaces", Phys. Rev. Lett. 57, p. 795-798 (1986) T. Chapman, "Topological invariance of Whitehead torsion", Am. J. Math. 96, p. 488-497 (1974) J. Cheeger, "Analytic torsion and the heat equation", Ann. of Math. 109, p. 259-322 (1979) J. Cheeger, "Spectral geometry of singular Riemannian spaces", J. Diff. Geom. 18, p. 575-657 (1983) M. Cohen, [A course in simple homotopy theory]{.ul}, Springer, New York (1973) W. Dwyer, M. Weiss and B. Williams, "A parametrized index theorem for the algebraic K-theory Euler class", Acta Math. 190, p. 1-104 (2003) W. Franz, "Über die Torsion einer Überdeckung\", J. Reine Angew. Math. 173, p. 245--254 (1935) K. Igusa, [Higher Franz-Reidemeister torsion]{.ul}, AMS/IP Studies in Advanced Mathematics 31, International Press, Somerville (2002) R. Kirby and L. Siebenmann, [Foundational Essays on Topological Manifolds, Smoothings and Triangulations]{.ul}, Annals of Math. Studies 88, Princeton University Press, Princeton (1977) H. McKean and I. Singer, "Curvature and the eigenvalues of the Laplacian", J. Diff. Geom. 1, p. 43-69 (1967) J. Milnor, "Two complexes which are homeomorphic but combinatorially distinct", Ann. Math. 74, p. 575-590 (1961) J. Milnor, "Whitehead torsion", Bull. Amer. Math. Soc. 72, p. 358-426 (1966) S. Minakshisundaram and Å. Pleijel, "Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds", Canad. J. Math. 1, p. 242--256 (1949) E. Moise, \"Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung\", Ann. Math. 56, p. 96-114 (1962) W. Müller, "Analytic torsion and R-torsion of Riemannian manifolds", Adv. Math.28, p. 233-305 (1978) W. Müller, "Analytic torsion and R-torsion for unimodular representations", J. Amer. Math. Soc.6, p. 721-753 (1993) W. Müller, "Analytic torsion", Notices of the AMS, 69, p. 1557-1559 (2022) M. Puchol, Y. Zhang and J. Zhu, "A comparison between the Bismut-Lott torsion and the Igusa-Klein torsion", preprint, https://arxiv.org/abs/2105.11985 (2021) D. Quillen, "Determinants of Cauchy-Riemann operators on Riemann surfaces", Funct. Anal. Appl. 19, p. 31-34 (1985) D. Ray, "Reidemeister torsion and the Laplacian on lens spaces", Adv. Math. 4, p. 109-126 (1970) D. Ray and I. Singer, "R-torsion and the Laplacian on Riemannian manifolds", Adv. Math. 7, p. 145-210 (1971) D. Ray and I. Singer, "Analytic torsion for complex manifolds", Ann. Math 98, p. 154-177 (1973) K. Reidemeister, "Homotopieringe und Linsenräume", Abh. Math. Sem. Univ. Hamburg 11, p.102--109 (1935) G. de Rham, "Sur les nouveaux invariants topologiques de M. Reidemeister, Rec. Math. Moscou, n. Ser. 1, p. 737-742 (1936) G. de Rham, "Complexes à automorphismes et homéomorphie différentiable", Ann. Inst. Fourier 2, p. 51-67 (1950) W. Thurston, [Three-dimensional geometry and topology]{.ul}, Princeton University Press, Princeton (1997) S. Vishik, "Generalized Ray-Singer conjecture. I. A manifold with a smooth boundary", Comm. Math. Phys.167, p. 1-102 (1995) [^1]: [^2]: Reidemeister was removed from his professorship at the University of Königsberg in 1933, as retaliation for his anti-Nazi statements. He learned of his dismissal by reading about it in the local newspaper [@BBF; @(1972)]. He got a position at the University of Marburg in 1934, where he remained until 1955. During the war Franz worked in the Wehrmacht's codebreaking group. He held a position at the Goethe University Frankfurt between 1946 and 1974.

arxiv_math

--- abstract: | We study germs of hypersurfaces $(Y,0)\subset (\mathbb{C}^{n+1},0)$ that can be described as the image of $\mathscr{A}$-finite mappings $f:(X,S)\rightarrow (\mathbb{C}^{n+1},0)$ defined on an [icis]{.smallcaps} $(X,S)$ of dimension $n$. We extend the definition of the Jacobian module given by Fernández de Bobadilla, Nuño-Ballesteros and Peñafort-Sanchis when $X=\mathbb{C}^n$, which controls the image Milnor number $\mu_I(X,f)$. We apply these results to prove the case $n=2$ of the generalised Mond conjecture, which states that $\mu_I(X,f)\geq \textup{codim}_{\mathscr{A}_e}(X,f)$, with equality if $(Y,0)$ is weighted homogeneous. address: - Departament de Matemàtiques, Universitat de València, Campus de Burjassot, 46100 Burjassot SPAIN - Departament de Matemàtiques, Universitat de València, Campus de Burjassot, 46100 Burjassot SPAIN. Departamento de Matemática, Universidade Federal da Paraíba CEP 58051-900, João Pessoa - PB, BRAZIL author: - Alberto Fernández-Hernández, Juan J. Nuño-Ballesteros bibliography: - referencearticle2.bib title: Disentangling mappings defined on ICIS --- # Introduction The main objects that are studied along this paper are hypersurface singularities $(Y,0)\subset (\mathbb{C}^{n+1},0)$. It is well known that if a hypersurface germ has isolated singularity, then one can define two different invariants to encode information of the hypersurface, namely the Milnor and Tjurina numbers, $\mu$ and $\tau$, respectively. The first of them, $\mu$, has a more topological flavour, since it counts the number of spheres in the homotopy type of a Milnor fibre of $Y$. On the other hand, the Tjurina number has a much more rigid nature, and counts the number of parameters of a versal deformation of the hypersurface. An inspection of the algebraic formulas that define them makes clear that $\mu \geq \tau$, with equality in the case that $(Y,0)$ is weighted homogeneous. In [@vanishingcycles], D. Mond extended the definition of $\mu$ to a particular kind of hypersurfaces which could be described as the image of a mapping $f:(\mathbb{C}^n,S)\rightarrow (\mathbb{C}^{n+1},0)$ that has isolated instability, or, equivalently, that has finite codimension. In this case, $f$ is the normalisation mapping of the hypersurface $(Y,0)$. He then introduced the concept of *image Milnor number* $\mu_I(f)$ of the mapping $f$, which is well defined if the dimensions $(n,n+1)$ are Mather's nice dimensions, or if $f$ has corank one, as will be commented on in the following lines. This new invariant played the role of the Milnor number for this kind of hypersurfaces that do not have isolated singularity anymore. Moreover, the *codimension* of $f$, denoted by $\textup{codim}_{\mathscr{A}_e}(f)$, is an analytical invariant that measures the minimum number of parameters required to fully deform $f$ through versal unfoldings, in the same way the Tjurina number does. In terms of this invariant, one says that a mapping is $\mathscr{A}$*-finite* provided it has finite codimension, or that is *stable* whenever it has codimension 0. Although the invariants are written and described in terms of the mapping $f$, they can be seen to depend only on the analytic class of the image $(Y,0)$, and hence $\mu_I(f)$ and $\textup{codim}_{\mathscr{A}_e}(f)$ are the natural invariants that correspond to $\mu$ and $\tau$ in this different context. The question is therefore natural: **Conjecture 1** (Mond conjecture). *Let $f:(\mathbb{C}^n,S)\rightarrow (\mathbb{C}^{n+1},0)$ be an $\mathscr{A}$-finite mapping, where $(n,n+1)$ are Mather's nice dimensions. Then, $\mu_I(f)\geq \textup{codim}_{\mathscr{A}_e}(f)$, with equality if $f$ is weighted homogeneous.* The only known cases are $n=1,2$, and it is still an open problem if $n\geq 3$. This conjecture is therefore the extension of the $\mu\geq \tau$ statement for hypersurfaces that admit an $\mathscr{A}$-finite normalisation mapping with normal space $(\mathbb{C}^n,S)$. Nevertheless, the amount of hypersurfaces satisfying this property is limited. In particular, one would have interest in studying this statement if the normal space of $(Y,0)$ is singular. More precisely, in this article we study the case of hypersurfaces whose normalisation is $\mathscr{A}$-finite and whose normal space is an isolated complete intersection singularity ([icis]{.smallcaps}). In [@Mond-Montaldi], D. Mond and J. Montaldi settled the deformation theory of mappings $f:(X,S)\rightarrow (\mathbb{C}^p,0)$ with $(X,S)$ being an [icis]{.smallcaps} of dimension $n$, and they took the first steps to study an extension of the Mond conjecture in this general setting. It turns out that the notions of $\textup{codim}_{\mathscr{A}_e}(X,f)$ and $\mu_I(X,f)$ can be defined in this wider context, as Mond and Montaldi showed. The former invariant, namely the codimension of $(X,f)$, equals the minimum number of parameters of a versal unfolding of $(X,f)$, where now unfoldings are allowed to deform both the mapping $f$ and the space $X$. On the other hand, the definition of $\mu_I(X,f)$ is given in the same way as it is done for mappings with smooth source. In order to properly define it, one requires the existence of *stabilisations*, which are deformations $f_t$ of $f$ with the property that $f_t$ is stable for every $t\neq 0$ small enough. These particular deformations do only exist provided $(n,n+1)$ are nice dimensions or provided $f$ has corank one. In these terms, it can be checked that the images $Y_t=\textup{im}\,f_t$, which are called *disentanglements* of $Y$, all have the homotopy type of a wedge of $n$-spheres. The number of such spheres is therefore defined as the image Milnor number $\mu_I(X,f)$. This settles the framework to study in this general case whether $\mu_I(X,f)\geq \textup{codim}_{\mathscr{A}_e}(X,f)$, with equality in the weighted homogeneous case. This question is what we refer to as the generalised Mond conjecture. The only case that was known before this article is studied in [@juanjo-henrique] by D. Henrique and J.J. Nuño-Ballesteros, and provides a postive answer for the generalised Mond conjecture if $n=1$ and $(X,S)\subset (\mathbb{C}^2,0)$ is a plane curve. Our main contribution in this article is that we prove the generalised Mond conjecture in the case that $n=2$ in its whole generality. Therefore, we show that the $\mu\geq \tau$ statement can be extended to surfaces $(Y,0)$ in $(\mathbb{C}^3,0)$ with $\mathscr{A}$-finite normalisation mapping on an [icis]{.smallcaps}. Formally, the main theorem of this paper is the following result: **Theorem 2**. *Let $f:(X,S)\rightarrow (\mathbb{C}^3,0)$ be an $\mathscr{A}$-finite mapping where $(X,S)$ is an [icis]{.smallcaps} of dimension $2$ and with image $(Y,0)$. Then, $$\mu_I(X,f)\geq \textup{codim}_{\mathscr{A}_e}(X,f),$$ with equality if $(Y,0)$ is weighted homogeneous.* In order to prove it, we adapt the construction of the module $M(g)$ and its relative version for unfoldings $M_\textup{rel}(G)$ that were defined in [@BobadillaNunoPenafort] by J. Fernández de Bobadilla, J. J. Nuño-Ballesteros and G. Peñafort-Sanchis. We extend to this general framework their main result, which states that the length of $M(g)$ equals $\mu_I(X,f)$ if and only if $M_\textup{rel}(G)$ is a Cohen-Macaulay module, and that, if each of these cases hold, the generalised Mond conjecture holds for the mapping $(X,f)$. # Mappings on [icis]{.smallcaps} Singularities of smooth mappings between manifolds is a classical subject in Singularity Theory. The infinitesimal methods were developed by Thom and Mather in the late sixties and by this reason it is known as Thom-Mather theory. We refer to the book [@juanjo] for a modern presentation of the theory, which also includes the extension to holomorphic germs between complex manifolds. In [@Mond-Montaldi] Mond and Montaldi extended the Thom-Mather theory of singularities of mappings $f\colon (X,S)\to(\mathbb{C}^p,0)$ defined on an [icis]{.smallcaps} $(X,S)$. The crucial point here is that they consider deformations not only of the mapping $f$, but also of the [icis]{.smallcaps} $(X,S)$. Here we summarise some of the basic definitions and properties, in order to make this paper more self-contained. For a more detailed account we refer to the original paper by Mond and Montaldi [@Mond-Montaldi]. Along this section we work with holomorphic map germs $f\colon (X,S)\to(\mathbb{C}^p,0)$, where $(X,S)$ is an [icis]{.smallcaps} of dimension $n$. It is usual to denote such a map germs by a pair $(X,f)$, although sometimes we may omit the base set of the germ if it does not provide relevant information or it is clear from the context. The two first definitions declare the type of equivalences and deformations we are dealing with in this theory. **Definition 3**. Two holomorphic map germs $f,g:(X,S)\rightarrow (\mathbb{C}^p,0)$ are called $\mathscr{A}$*-equivalent* if we have a commutative diagram $$\begin{tikzcd} (X,S) \arrow[r, "f" ]\arrow[d,"\varphi"]& (\mathbb{C}^p,0)\arrow[d, " \psi"] \\ (X,S) \arrow[r, "g" ]& (\mathbb{C}^p,0) \end{tikzcd}$$ where the columns are biholomorphisms. **Definition 4**. An *unfolding of the pair* $\left(X,f\right)$ is a map germ $F\colon(\mathcal{X},S')\rightarrow \left(\mathbb{C}^p\times \mathbb{C}^r,0\right)$ together with a flat projection $\pi\colon(\mathcal{X},S')\rightarrow (\mathbb{C}^r,0)$ and an isomorphism $j\colon(X,S)\to\big(\pi^{-1}(0),S'\big)$ such that the following diagram commutes $$\begin{tikzcd}[column sep=tiny] &(X,S) \arrow[dr, "f\times\left\{0\right\}" ] \arrow[dl, "j" ' ]& \\ (\pi^{-1}(0),S') \arrow[d,hook] & &\big(\mathbb{C}^p\times\left\{0\right\}, 0\big) \arrow[d,hook]\\ (\mathcal{X},S')\arrow[rr, "F"] \arrow[dr,"\pi" '] && (\mathbb{C}^p\times \mathbb{C}^r,0) \arrow[dl,"\pi_2"]\\ &(\mathbb{C}^r,0)& \end{tikzcd} ,$$ where $\pi_2:\mathbb{C}^p\times \mathbb{C}^r\rightarrow \mathbb{C}^r$ is the Cartesian projection. In Definition [Definition 4](#unfolding){reference-type="ref" reference="unfolding"}, $\mathbb{C}^r$ is called the *parameter space of the unfolding*. It is common to denote the unfolding by $(\mathcal{X},\pi,F,j)$. For each parameter $u\in\mathbb{C}^r$ in a neighbourhood of the origin, we have a mapping $f_u\colon X_u\rightarrow \mathbb{C}^p$, where $X_u:= \pi^{-1}(u)$, denoted also by $(X_u,f_u)$. **Definition 5**. Two unfoldings $(\mathcal{X},\pi,F,j)$ and $(\mathcal{X}',\pi',F',j')$ over $\mathbb{C}^r$ are *isomorphic* if the following diagram commutes: $$\begin{tikzcd}[column sep=0.4cm] & (\mathcal{X},j(S)) \arrow[dd," \Phi"'] \arrow[rr, "F"]\arrow[dr, "\pi"]& & (\mathbb{C}^p\times \mathbb{C}^r,0)\arrow[dd," \Psi"] \arrow[dl, "\pi_2"']\\ (X,S) \arrow[ur, "j"] \arrow[dr, "j'"'] & & (\mathbb{C}^r,0) & \\ & (\mathcal{X}',j'(S)) \arrow[rr, "F'"] \arrow[ur, "\pi'"]& & (\mathbb{C}^p\times \mathbb{C}^r,0)\arrow[ul, "\pi_2"'] \end{tikzcd},$$ where $\Phi$ and $\Psi$ are biholomorphisms and also $\Psi$ is an unfolding of the identity over $\mathbb{C}^d$. If $(\mathcal{X},\pi,F,j)$ is an unfolding of $(X,f)$ over $(\mathbb{C}^r,0)$, a germ $\rho: (\mathbb{C}^{s},0) \rightarrow (\mathbb{C}^r,0)$ induces and unfolding $(\mathcal{X}_\rho,\pi_\rho,F_\rho,j_\rho)$ of $(X,f)$ by a *base change* or, in other words, by the fibre product of $F$ and $\text{id}_{\mathbb{C}^p}\times \rho$: $$\begin{tikzcd} \mathcal{X}_\rho:= \mathcal{X}\times_{\mathbb{C}^p\times \mathbb{C}^s}\left(\mathbb{C}^p\times \mathbb{C}^s\right)\arrow[r,"F_\rho"]\arrow[d]&\mathbb{C}^p\times \mathbb{C}^s\arrow[d,"\text{id}_{\mathbb{C}^p}\times \rho"]\\ \mathcal{X}\arrow[r,"F"]&\mathbb{C}^p\times \mathbb{C}^r \end{tikzcd},$$ where we omit the points of the germs for simplicity. The unfolding $(\mathcal{X},\pi,F,j)$ is *versal* if every other unfolding, for example $(\mathcal{X}',\pi',F',j')$, is isomorphic to an unfolding induced from the former by a base change, $(\mathcal{X}_\rho, \pi_\rho, F_\rho, j_\rho)$. A versal unfolding is called *miniversal* if it has a parameter space with minimal dimension. **Definition 6**. A germ $(X,f)$ is *stable* if any unfolding is *trivial*, that is, isomorphic to the constant unfolding $(X\times\mathbb{C}^r,\pi_2,f\times\textnormal{id}_{\mathbb{C}^r},i)$. We say that $(X,f)$ has *isolated instability* if there exists a representative $f\colon X\to\mathbb{C}^p$ such that the restriction $f\colon X\setminus f^{-1}(0)\to \mathbb{C}^p\setminus\{0\}$ has only stable singularities. A *stabilisation* of $(X,f)$ is a 1-parameter unfolding $(\mathcal{X},\pi,F,j)$ with the property that for any small enough $s\in\mathbb{C}\setminus\{0\}$, $(X_s,f_s)$ has only stable singularities. Such a mapping $(X_s,f_s)$, with $s\ne0$, is called a *stable perturbation* of $(X,f)$. A crucial fact is that any germ $(X,f)$ with isolated instability admits a stabilisation, provided that $(n,p)$ are nice dimensions in the sense of Mather or $f$ has only kernel rank one singularities (that is, $f$ admits an extension whose differential has kernel rank $\le 1$ everywhere). A proof in the case $X=\mathbb{C}^n$ can be found in [@juanjo] and the extension to the case of mappings on [icis]{.smallcaps} appears in [@roberto]. Next, we recall the notion of $\mathscr{A}_e$-codimension of a germ $(X,f)$. In order to do this, we introduce the following notation: - $\mathscr{O}_p$ is the local ring of holomorphic functions $(\mathbb{C}^p,0)\to\mathbb{C}$, - $\mathscr{O}_{X,S}$ is the (semi-)local ring of holomorphic functions $(X,S)\to\mathbb{C}$, - $f^*:\mathscr{O}_p\to\mathscr{O}_{X,S}$ is the induced ring morphism $f^*(h)=h\circ f$, - $\theta_{\mathbb{C}^p,0}$ is the $\mathscr{O}_p$-module of germs of vector fields on $(\mathbb{C}^p,0)$, - $\theta_{X,S}$ is the $\mathscr{O}_{X,S}$-module of germs of vector fields on $(X,S)$, - $\theta(f)$ is the module of vector fields along $f$, - $\omega f:\theta_{\mathbb{C}^p,0}\rightarrow \theta(f)$ is the mapping $\omega(\eta)=\eta\circ f$, - $tf:\theta_{X,S}\rightarrow \theta(f)$ is mapping $tf(\xi)=d\tilde f\circ\xi$, for some analytic extension $\tilde f$ of $f$. **Definition 7**. The *$\mathscr{A}_e$-codimension* of $(X,f)$ is defined as $$\textup{codim}_{\mathscr{A}_e}(X,f)=\dim_\mathbb{C}\frac{\theta(f)}{{tf(\theta_{X,S})+\omega f(\theta_p)}}+\sum_{x\in S} \tau(X,x),$$ where $\tau(X,x)$ is the Tjurina number of $(X,x)$. When $\textup{codim}_{\mathscr{A}_e}(X,f)<\infty$, the germ $(X,f)$ is called $\mathscr{A}$-finite. The versality theorem holds also for mappings on [icis]{.smallcaps}, as the reader can find in [@Mond-Montaldi]: $(X,f)$ is $\mathscr{A}$-finite if and only if it admits a versal unfolding and in case it is $\mathscr{A}$-finite, then $\textup{codim}_{\mathscr{A}_e}(X,f)$ is equal to the number of parameters in a miniversal unfolding. As a consequence, $(X,f)$ is stable if and only if $(X,S)$ is smooth and $f$ is stable in the usual sense (see [@juanjo]). Another important issue with $\mathscr{A}$-finiteness is the extension of the Mather-Gaffney geometric criterion for mappings on [icis]{.smallcaps}: $(X,f)$ is $\mathscr{A}$-finite if and only if it has isolated instability (see [@roberto]). Finally, we will recall the definition of image Milnor number in the case $p=n+1$. The original definition when $X=\mathbb{C}^n$ is due to Mond (see [@vanishingcycles; @juanjo]), but it can be adapted quite easily to mappings on [icis]{.smallcaps} (see [@roberto]). In the case $p\le n$, the analogous invariant is called the discriminant Milnor number, considered for the first time by Damon and Mond in [@Damon-Mond] in the case $X=\mathbb{C}^n$ and extended to mappings on [icis]{.smallcaps} by Mond and Montaldi in [@Mond-Montaldi]. The definition is clearly inspired in the classical Milnor number and is motivated by the following theorem. We denote by $B_\epsilon$ the closed ball in $\mathbb{C}^{n+1}$ of radius $\epsilon>0$ centered at the origin. We assume $f\colon (X,S)\to(\mathbb{C}^{n+1},0)$ is $\mathscr{A}$-finite and that either $(n,n+1)$ are nice dimensions of Mather or $f$ has only corank one singularities. We take a stabilisation of $(X,f)$ with stable perturbation $(X_s,f_s)$. **Theorem 8**. *[@vanishingcycles; @roberto] For all $\epsilon,\eta$, with $0<\eta\ll\epsilon\ll 1$ and for all $s\in\mathbb{C}$, with $0<|s|<\eta$, $f_s(X_s)\cap B_\epsilon$ has the homotopy type of a bouquet of spheres of dimension $n$. Moreover, the number of such spheres, denoted by $\mu_I(X,f)$, is independent of the choice of the stabilisation, the parameter $s$ and the numbers $\epsilon,\eta$.* **Definition 9**. With the notation of Theorem [Theorem 8](#disent){reference-type="ref" reference="disent"}, $f_s(X_s)\cap B_\epsilon$ is called the *disentanglement* and $\mu_I(X,f)$ is called the *image Milnor number* of $(X,f)$. The proof of Theorem [Theorem 8](#disent){reference-type="ref" reference="disent"} is based on arguments by Lê [@Le] and by Siersma [@Siersma]. In fact, the original formulation in [@Siersma] gives a recipe of how to compute $\mu_I(X,f)$ in a more algebraic way: **Theorem 10**. *[@Siersma][\[Siersma\]]{#Siersma label="Siersma"} With the notation of Theorem [Theorem 8](#disent){reference-type="ref" reference="disent"}, let $G\colon(\mathbb{C}^{n+1}\times\mathbb{C},0)\to(\mathbb{C},0)$ be a reduced equation of the image of the stabilistation. Then, $$\mu_I(X,f)=\sum_{y\in B_\epsilon \setminus f_s(X_s)} \mu (g_s; y),$$ where $g_s(y)=G(y,s)$ and $\mu (g_s; y)$ is the Milnor number of $g_s$ at $y$.* *Remark 11*. In order to compute the image Milnor number $\mu_I(X,f)$, sometimes is more convenient to consider a stable unfolding (i.e., an unfolding which is stable as a germ) instead of a stabilisation. In such a case, the bifurcation set $\mathcal B$ is the set of parameters $u\in\mathbb{C}^r$ in a neighbourhood of the origin, such that $f_u\colon X_u\to \mathbb{C}^{n+1}$ has only stable singularities. When $(n,n+1)$ are nice dimensions or when $f$ has only corank one singularities, $\mathcal B$ is a proper closed analytic subset germ in a neighbourhood of $0$ in $\mathbb{C}^r$ (see [@juanjo] or [@roberto]). A stabilisation can be constructed easily by taking a line $L\subset\mathbb{C}^r$ with the property that $L\cap \mathcal B=\{0\}$. If $u\notin\mathcal B$, then $f_u(X_u)\cap B_\epsilon$ has the homotopy type of a bouquet of $n$-spheres and the number of such spheres is $\mu_I(X,f)$. Analogously, if $G\colon(\mathbb{C}^{n+1}\times\mathbb{C}^r,0)\to(\mathbb{C},0)$ is a reduced equation of the image of the unfolding, then $$\mu_I(X,f)=\sum_{y\in B_\epsilon \setminus f_u(X_u)} \mu (g_u; y),$$ where $g_u(y)=G(y,u)$ and $\mu (g_u; y)$ is the Milnor number of $g_u$ at $y$. # The generalised version of the Jacobian module $M(g)$ {#section3} Let $f:(X,S)\rightarrow (\mathbb{C}^{n+1},0)$ be an $\mathscr{A}$-finite mapping defined on an [icis]{.smallcaps} $(X,S)\subset (\mathbb{C}^{n+k},S)$ of dimension $n$. As $f$ is finite, its image $(Y,0)$ is a hypersurface of $(\mathbb{C}^{n+1},0)$, and hence it can be described by a reduced equation $g\in \mathscr{O}_{n+1}$. Let $h:(\mathbb{C}^{n+k},S)\rightarrow (\mathbb{C}^k,0)$ be a mapping so that $(X,S)=h^{-1}(0)$. Consider an analytic extension $\tilde{f}:(\mathbb{C}^{n+k},S)\rightarrow (\mathbb{C}^{n+1},0)$ of $f$, and write $\hat{f}=(\Tilde{f},h):(\mathbb{C}^{n+k},S)\rightarrow (\mathbb{C}^{n+1+k},0)$. It is therefore clear that the restriction of $\hat{f}$ to $(X,S)$ is precisely the mapping $(f,0)$. Hence, $\hat{f}$ is a finite mapping, since $\hat{f}^{-1}(0)=f^{-1}(0)=S$. Moreover, the diagram commutes, where $i$ is the inclusion and $j$ is the natural immersion. In particular, $\hat{f}$ is an unfolding of $f$ deforming both the mapping and the domain. After taking representatives, the induced deformations of $f$ are the mappings $\hat{f}_t:X_t\subset \mathbb{C}^{n+k}\rightarrow \mathbb{C}^{n+1}$ defined as $\hat{f}_t(x)=\tilde{f}(x,t)$, where $X_t=h^{-1}(t)$ for $t\in \mathbb{C}^k$ small enough. The key idea is that $\hat{f}$ is the simplest unfolding of $f$ with smooth source. Hence, the definition of the module for the mapping $f$ will be performed through a specialisation of the one from $\hat{f}$. Since $\hat{f}$ is a finite mapping, its image $(\hat{Y},0)$ is a hypersurface of $(\mathbb{C}^{n+1+k},0)$. Furthermore, the restriction of $\hat{f}$ to its image $(\mathbb{C}^{n+k},S)\rightarrow (\hat{Y},0)$ is the normalisation mapping of $(\hat{Y},0)$, and hence it induces a monomorphism of rings $\mathscr{O}_{\hat{Y},0}\hookrightarrow \mathscr{O}_{n+k}$ that lets us consider $\mathscr{O}_{\hat{Y},0}$ to be a subring of $\mathscr{O}_{n+k}$. In this case, the diagram $$\begin{tikzcd} \mathscr{O}_{n+1+k} \arrow[r, "\hat{f}^*"] \arrow[dr,"\pi",two heads] & \mathscr{O}_{n+k} \\ & \mathscr{O}_{\hat{Y},0}\arrow[u,hook] \end{tikzcd}$$ commutes, where $\pi$ is the epimorphism associated to the natural inclusion of $(\hat{Y},0)$ in $(\mathbb{C}^{n+1+k},0)$. We consider both $\mathscr{O}_{\hat{Y},0}$ and $\mathscr{O}_{n+k}$ to be $\mathscr{O}_{n+1+k}$-modules via the corresponding morphisms $\pi$ and $\hat{f}^*$, respectively. Let us consider $\hat{g} \in \mathscr{O}_{n+1+k}$ to be a reduced equation of $(\hat{Y},0)$ in such a way that $\hat{g} \circ j =g$. The following result from R. Piene [@Piene] relates the conductor ideal of $\hat{f}$, given by $$C(\hat{f})= \{ h \in \mathscr{O}_{\hat{Y},0} : h\cdot \mathscr{O}_{n+k} \subset \mathscr{O}_{\hat{Y},0}\},$$ with the partial derivatives of $\hat{g}$ and with the minors of the Jacobian matrix $d\hat{f}$. **Theorem 12** ([@Piene]). *There exists a unique $\lambda \in \mathscr{O}_{n+k}$ such that, for every $l\in \{1, \ldots, n+1+k\}$, $$\partial_l \hat{g}\circ \hat{f}=(-1)^l\cdot \lambda \cdot\det (d\hat{f}_1, \ldots, d\hat{f}_{l-1}, d\hat{f}_{l+1}, \ldots, d\hat{f}_{n+1+k}),$$ where $\partial_l \hat{g}$ denotes the partial derivative of $\hat{g}$ with respect to the $l$-th variable. Furthermore, the ideal $C(\hat{f})$ is principal, and generated by the element $\lambda$.* Let us denote by $J(\hat{g})$ to the Jacobian ideal of $\hat{g}$, which is generated by the partial derivatives $\partial_l \hat{g}$. This result therefore shows that $J(\hat{g})\cdot \mathscr{O}_{n+k}\subset C(\hat{f})$, where $J(\hat{g})\cdot \mathscr{O}_{n+k} = \hat{f}^*(J(\hat{g}))$. On the other hand, since $\hat{f}$ is a finite mapping with degree 1 onto its image, an application of Theorem 3.4 of [@Mond-Pellikaan] of D. Mond and R. Pellikaan shows that $\mathscr{F}_1(\hat{f}) \cdot \mathscr{O}_{n+k} = C(\hat{f})$, where $\mathscr{F}_1(\hat{f})$ denotes the first Fitting ideal of $\mathscr{O}_{n+k}$ as an $\mathscr{O}_{n+1+k}$-module via $\hat{f}^*$. Let us denote by $(y,z)$ to the coordinates in $(\mathbb{C}^{n+1+k},0)$, with $y\in \mathbb{C}^{n+1}$ and $z\in \mathbb{C}^k$, and consider the Jacobian ideal $J_y(\hat{g} ) = \langle \partial \hat{g}/\partial y_1, \ldots, \partial \hat{g}/\partial y_{n+1}\rangle$ generated by the derivatives with respect to the variables $y=(y_1, \ldots, y_{n+1})$. **Definition 13**. The restriction of $\hat{f}^*$ to $\mathscr{F}_1(\hat{f})$ induces an epimorphism of $\mathscr{O}_{n+1+k}$-modules $$\dfrac{\mathscr{F}_1(\hat{f})}{J_y(\hat{g})}\rightarrow \dfrac{C(\hat{f})}{J(\hat{g})\cdot \mathscr{O}_{n+k}}.$$ We define $N(\hat{g})$ to be the $\mathscr{O}_{n+1+k}$-module given by the kernel of this morphism, and define $$M(g)=N(\hat{g})\otimes\dfrac{\mathscr{O}_{n+1+k}}{\mathfrak{m}_k\cdot \mathscr{O}_{n+1+k}},$$ where $\mathfrak{m}_k=(z_1, \ldots, z_k)$ is generated by the parameters of the unfolding $\hat{f}$ of $f$. Note that $M(g)$ has a natural $\mathscr{O}_{n+1}$-module structure inherited from the tensor product. Hence, $M(g)$ is defined by taking into account that $\hat{f}$ is an unfolding of $f$, and following the spirit of [@BobadillaNunoPenafort] in the case of mappings with smooth source. Furthermore, it is relevant to notice that this module $M(g)$ coincides with the given in the smooth case just by taking $\hat{f}=f$ and $k=0$. *Remark 14*. Since $\hat{f}$ is defined in a smooth source, it lies naturally in the context of [@BobadillaNunoPenafort]. However, the module $N(\hat{g})$ that we have defined is not exactly the same as the module that is defined in [@BobadillaNunoPenafort]. Indeed, the source of the morphism that defines $N(\hat{g})$ has the Jacobian ideal $J_y(\hat{g})$ where only partial derivatives with respect to the parameters $y_1, \ldots, y_{n+1}$ are taken into consideration, while the module in the target does have all the partial derivatives. Although this may seem to be somewhat whimsical, this will be shown to be exactly what is needed for the module to specialise properly. This is due to the fact that, in general, $J(\hat{g})\cdot \mathscr{O}_{n+k}\neq J_y(\hat{g})\cdot \mathscr{O}_{n+k}$, in contrast with what happens in the smooth case. This important fact is what infuences this definition for $M(g)$, and what makes that another possible definition may not specialise properly. *Remark 15*. The module $N(\hat{g})$ is determined by the short exact sequence $$0\rightarrow N(\hat{g}) \rightarrow \dfrac{\mathscr{F}_1(\hat{f})}{J_y(\hat{g})}\rightarrow \dfrac{C(\hat{f})}{J(\hat{g})\cdot \mathscr{O}_{n+k}}\rightarrow 0.$$ **Proposition 16**. *The following formula holds: $$N(\hat{g})=\dfrac{(\hat{f}^*)^{-1} (J(\hat{g})\cdot \mathscr{O}_{n+k})}{J_y(\hat{g})}.$$* *Proof.* By definition, it is clear that $$N(\hat{g})=\dfrac{(\hat{f}^*)^{-1} (J(\hat{g})\cdot \mathscr{O}_{n+k})\cap \mathscr{F}_1(\hat{f})}{J_y(\hat{g})}.$$ Since $J(\hat{g})\cdot \mathscr{O}_{n+k}\subset C(\hat{f})$, then $(\hat{f}^*)^{-1} (J(\hat{g})\cdot \mathscr{O}_{n+k})\subset \mathscr{F}_1(\hat{f})$, and the formula follows. ◻ # A relative version for the generalised Jacobian module In this section, a relative version of the module for unfoldings of $\hat{f}$ is defined. For the sake of simplicity, we consider unfoldings $F:(\mathbb{C}^{n+k+r},S\times 0)\rightarrow (\mathbb{C}^{n+1+k+r},0)$ of the mapping $\hat{f}$ instead of general unfoldings of $f$ with possibly non-smooth source. Let $G\in \mathscr{O}_{n+1+k+r}$ be an equation for the image of $F$ such that $G(y,z,0)=\hat{g}(y,z)$, where $(y,z,u)$ are the coordinates of $(\mathbb{C}^{n+1+k+r},0)$, with $y\in \mathbb{C}^{n+1}, z\in \mathbb{C}^k$ and $u\in \mathbb{C}^r$. We then have that the diagram $$\begin{tikzcd} {(\mathbb{C}^{n+k+r},S\times 0)} & {(\mathbb{C}^{n+1+k+r},0)} \\ {(\mathbb{C}^{n+k},S)} & {(\mathbb{C}^{n+1+k},0)} & {(\mathbb{C},0)} \\ {(X,S)} & {(\mathbb{C}^{n+1},0)} \arrow["{\hat{f}}", from=2-1, to=2-2] \arrow["f", from=3-1, to=3-2] \arrow["i", hook, from=3-1, to=2-1] \arrow["j"', hook, from=3-2, to=2-2] \arrow["{\hat{i}}", hook, from=2-1, to=1-1] \arrow["{\hat{j}}"', hook, from=2-2, to=1-2] \arrow["F", from=1-1, to=1-2] \arrow["g", from=3-2, to=2-3] \arrow["{\hat{g}}", from=2-2, to=2-3] \arrow["G", from=1-2, to=2-3] \end{tikzcd}$$ commutes, where $\hat{i}, \hat{j}$ are the natural immersions. Let us denote by $$J_y(G ) = \Big\langle \dfrac{\partial G}{\partial y_1}, \ldots, \dfrac{\partial G}{\partial y_{n+1}}\Big\rangle, \,\, J_z(G ) = \Big\langle \dfrac{\partial G}{\partial z_1}, \ldots, \dfrac{\partial G}{\partial z_k}\Big\rangle,$$ and $J_{y,z}(G)=J_y(G)+J_z(G)$. We therefore define $M_\textup{rel}(G)$ as the kernel of the module epimorphism $$\dfrac{\mathscr{F}_1(F)}{J_y (G)}\rightarrow \dfrac{C(F)}{J_{y,z}(G)\cdot \mathscr{O}_{n+k+r}}.$$ Hence, $M_\textup{rel}(G)$ fits into the short exact sequence $$0\rightarrow M_\textup{rel}(G) \rightarrow \dfrac{\mathscr{F}_1(F)}{J_y (G)}\rightarrow \dfrac{C(F)}{J_{y,z}(G)\cdot \mathscr{O}_{n+k+r}}\rightarrow 0.$$ Furthermore, it is straightforward to verify that **Proposition 17**. *The following formula holds: $$M_\textup{rel}(G)=\dfrac{(F^*)^{-1} (J_{y,z}(G)\cdot \mathscr{O}_{n+k+r})}{J_y(G)}.$$* In contrast with the case of $M(g)$, we have that $J_{y,z}(G)\cdot \mathscr{O}_{n+k+r}=J(G)\cdot \mathscr{O}_{n+k+r}$ in virtue of lemma 4.5 of [@BobadillaNunoPenafort]. Hence, they can be interchanged indistinctively in this formula. Notice, in addition, that the relative module in [@BobadillaNunoPenafort] is denoted by $M_y(G)$ with the intention to highlight that the module is relative with respect to the parameters $y=(y_1, \ldots, y_{n+1})$. Since this interpretation in the case of mappings with non-smooth source is less clear, due to the fact that one has both parameters $y,z$, the authors have decided to consider the notation $M_\textup{rel}(G)$. In what follows, we show that the given definition of the $\mathscr{O}_{n+1+k+r}$-module $M_\textup{rel}(G)$ properly specialises to $M(g)$ seen as an $\mathscr{O}_{n+1}$-module. Before giving a proof of this result, let us comment on the specialisation method that should be considered. *Remark 18*. It is relevant to notice that the specialisation process that is performed in [@BobadillaNunoPenafort] for mappings with smooth source restricts the relative module $M_y(G)=M_\textup{rel}(G)$ to be an $\mathscr{O}_r$-module, and then $M_\textup{rel}(G)$ is tensored with $\mathscr{O}_r/\mathfrak{m}_r$. Hence, one naturally obtains a $\mathbb{C}$-vector space that is isomorphic to $M(g)$ (see Theorem 4.6 of [@BobadillaNunoPenafort]). However, this specialisation process ignores the fact that $M(g)$ is an $\mathscr{O}_{n+1}$-module. In order to take this into account, one should perform the specialisation process in a slightly different way. Since $M_\textup{rel}(G)$ is an $\mathscr{O}_{n+1+r}$-module, the tensor product $$M_\textup{rel}(G)\otimes \dfrac{\mathscr{O}_{n+1+r}}{\mathfrak{m}_r\cdot \mathscr{O}_{n+1+r}}$$ provides naturally an $\mathscr{O}_{n+1}$-module, due to the fact that $\mathscr{O}_{n+1+r}/(\mathfrak{m}_r\cdot \mathscr{O}_{n+1+r})\cong \mathscr{O}_{n+1}$, where $\mathfrak{m}_r=(u_1, \ldots , u_r)$ is the maximal ideal of $\mathscr{O}_r$ generated by the parameters of the unfolding $F$. In fact, this specialisation process both captures the spirit of forcing the parameters of the unfolding to be equal to 0 and keeps the natural $\mathscr{O}_{n+1}$-module structure of $M(g)$, as it can be easily verified with minor modifications in the proofs of section 4 of [@BobadillaNunoPenafort]. Hence, it can be easily checked that the module $M_\textup{rel}(G)$ of [@BobadillaNunoPenafort] in the smooth case satisfies that $$M_\textup{rel}(G)\otimes \dfrac{\mathscr{O}_{n+1+r}}{\mathfrak{m}_r\cdot \mathscr{O}_{n+1+r}} \cong M(g),$$ where $\cong$ denotes isomorphism of $\mathscr{O}_{n+1}$-modules. Although this is not relevant in the smooth case, we have already seen in the definition [Definition 13](#def:module){reference-type="ref" reference="def:module"} that specialisations should be taken through this method in order to preserve the module structure. Taking this into account, we are now able to check that the module $M_\textup{rel}(G)$ in this setting properly specialises to $M(g)$: **Theorem 19**. *If $n\geq 2$, then $$M_\textup{rel}(G) \otimes \dfrac{\mathscr{O}_{n+1+k+r}}{\mathfrak{m}_{k+r}\cdot \mathscr{O}_{n+1+k+r}}\cong M(g)$$ as $\mathscr{O}_{n+1}$-modules.* *Proof.* The same proof of the smooth case (Theorem 4.6 of [@BobadillaNunoPenafort] and the previous remark) shows that $$M_\textup{rel}(G)\otimes \dfrac{\mathscr{O}_{n+1+k+r}}{\mathfrak{m}_r\cdot \mathscr{O}_{n+1+k+r}}\cong N(\hat{g}),$$ since this first specialisation restricts the parameters that were added in the unfolding $F$ of $\hat{f}$, both with smooth domain. Indeed, the only difference is that the definition of $N(\hat{g})$ has a term $J_y(\hat{g})$, but it holds that $\hat{j}(J_y(G))=J_y(\hat{g})$ and, as in the smooth case, $\hat{j}(J_{y,z}(G))=J(\hat{g})$. Hence, $$\begin{aligned} M_\textup{rel}(G) \otimes \dfrac{\mathscr{O}_{n+1+k+r}}{\mathfrak{m}_{k+r}\cdot \mathscr{O}_{n+1+k+r}} &= \left( M_\textup{rel}(G) \otimes \dfrac{\mathscr{O}_{n+1+k+r}}{\mathfrak{m}_{r}\cdot \mathscr{O}_{n+1+k+r}}\right)\otimes \dfrac{\mathscr{O}_{n+1+k}}{\mathfrak{m}_{k}\cdot \mathscr{O}_{n+1+k}} \\ &= N(\hat{g})\otimes \dfrac{\mathscr{O}_{n+1+k}}{\mathfrak{m}_{k}\cdot \mathscr{O}_{n+1+k}} =M(g),\end{aligned}$$ where the last equality holds by definition of $M(g)$. ◻ *Remark 20*. As we have commented on before in Remark [Remark 18](#remark:specialisation){reference-type="ref" reference="remark:specialisation"}, the specialisation performed is different than the one appearing in [@BobadillaNunoPenafort] for mappings with smooth source. Although this different approach lets us transfer the module structure, it is important for applications to analyse the analogue process. It turns out that, when we restrict $M_\textup{rel}(G)$ to be an $\mathscr{O}_{k+r}$-module via the natural inclusion $\mathscr{O}_{k+r}\rightarrow \mathscr{O}_{n+1+k+r}$ induced by the projection $\mathbb{C}^{n+1}\times \mathbb{C}^{k+r}\rightarrow \mathbb{C}^{k+r}$, then it is straightforward to verify that $$M_\textup{rel}(G)\otimes \dfrac{\mathscr{O}_{k+r}}{\mathfrak{m}_{k+r}}\cong M(g)$$ as $\mathbb{C}$-vector spaces, just by noticing that $$\begin{aligned} M_\textup{rel}(G)\otimes \dfrac{\mathscr{O}_{k+r}}{\mathfrak{m}_{k+r}}&\cong \dfrac{M_\textup{rel}(G)}{\mathfrak{m}_{k+r}\cdot M_\textup{rel}(G)} = \dfrac{M_\textup{rel}(G)}{(\mathfrak{m}_{k+r}\cdot \mathscr{O}_{n+1+k+r})\cdot M_\textup{rel}(G)} \cong \\ &\cong M_\textup{rel}(G) \otimes \dfrac{\mathscr{O}_{n+1+k+r}}{\mathfrak{m}_{k+r}\cdot \mathscr{O}_{n+1+k+r}} \cong M(g).\end{aligned}$$ In a nutshell, the specialisation process can be performed either restricting first $M_\textup{rel}(G)$ to be an $\mathscr{O}_{k+r}$-module, or rather keeping its whole $\mathscr{O}_{n+1+k+r}$-module structure. The obtained modules are, respectively, $$M_\textup{rel}(G)\otimes (\mathscr{O}_{n+1+k+r}/\mathfrak{m}_{k+r}\cdot \mathscr{O}_{n+1+k+r}) \text{ and } M_\textup{rel}(G)\otimes (\mathscr{O}_{k+r}/\mathfrak{m}_{k+r}).$$ In both scenarios one recovers $M(g)$ with some structure: in the former case, the result is an $\mathscr{O}_{n+1}$-module isomorphic to $M(g)$, and, in the latter, a $\mathbb{C}$-module isomorphic to $M(g)$. Hence, the specialisation procces that one should perform depends on whether one needs to keep the module structure or not. For most of the cases, the only information that needs to be keeped is the complex dimension as a vector space, and hence both methods are valid. Lastly, an important result regarding the form of the module $M_\textup{rel}(G)$ when $F$ is stable is the following: **Proposition 21**. *Let $F$ be a stable unfolding of $\hat{f}$ and $G$ an equation such that $G\in J(G)$. Then, $$M_\textup{rel}(G)= \dfrac{J(G)}{J_y(G)}.$$* *Proof.* Since $F$ is stable and $G\in J(G)$, the smooth version of the module $M(G)$ from [@BobadillaNunoPenafort] satisfies that $M(G)=0$. In this case, the formula of proposition 5.1 of this article yields that $$M(G)=\dfrac{(F^*)^{-1}(J(G)\cdot \mathscr{O}_{n+k+r})}{J(G)},$$ and hence $(F^*)^{-1}(J(G)\cdot \mathscr{O}_{n+k+r})=J(G)$. Furthermore, in the smooth case, it follows that $J(G)\cdot \mathscr{O}_{n+k+r}=J_{y,z}(G)\cdot \mathscr{O}_{n+k+r}$. Hence, proposition [Proposition 17](#prop:formulaMrelG){reference-type="ref" reference="prop:formulaMrelG"} yields $$M_\textup{rel}(G)=\dfrac{(F^*)^{-1} (J_{y,z}(G)\cdot \mathscr{O}_{n+k+r})}{J_y(G)}=\dfrac{J(G)}{J_y(G)},$$ and the claim follows. ◻ Note that, since $\hat{f}$ is a finite mapping, the existence of a stable unfolding $F$ of $\hat{f}$ is granted. Furthermore, as it is commented in [@BobadillaNunoPenafort], there is always a stable unfolding $F$ which admits an equation $G$ with the property that $G\in J(G)$. Such an equation is referred to as a *good defining equation*. Indeed, if $F(u,x)=(u,f_u(x))$ is a stable unfolding of $\hat{f}$, then let $F'$ be the $1$-parameter stable unfolding of $F$ given by $F'(t,u,x)=(t,u,f_u(x))$. Let $G=0$ be a reduced equation of the image of $F$ and consider $G'(t,u,y)=e^t G(u,y)$. It therefore follows that $G'=0$ is a reduced equation defining the image of $F'$ with the property that $G'=\partial_t G'\in J(G')$. # Relation between $\dim_\mathbb{C}M(g)$ and $\textup{codim}_{\mathscr{A}_e}(X,f)$ Let $F$ be a stable unfolding of $\hat{f}$, and consider an equation $G$ of the image of $F$, namely, $(Z,0)$, that satisfies $G\in J(G)$. Then, the last result of the previous section showed that $M_\textup{rel}(G) =J(G)/J_y(G)$. Let us relate this with the codimension of $(X,f)$, which can be determined through the formula $$\textup{codim}_{\mathscr{A}_e}(X,f)=\dim_\mathbb{C}\dfrac{\theta(i)}{ti(\theta_{n+1})+i^*(\textup{Derlog}\,Z)},$$ where $i:(\mathbb{C}^{n+1},0)\rightarrow (\mathbb{C}^{n+1+k+r},0)$ denotes the natural immersion (see [@Mond-Montaldi] for more details). Recall that $\textup{Derlog}\,Z = \{\xi \in \theta_{n+1+k+r}: \xi(G)=\lambda (G)\}$, and $\textup{Derlog}\,G = \{\xi \in \theta_{n+1+k+r}: \xi(G)=0\}$. Notice that $G\in J(G)$, so that $G=\sum_{s} a_s \partial_s G$, where $\partial_s G$ denotes the partial derivatives with respect to all the variables in $(y,z,u)\in \mathbb{C}^{n+1+k+r}$. Hence, the vector field $\epsilon = \sum_s a_s\partial_s$ satisfies that $\epsilon (G)=G$, where $\partial_s$ denotes the coordinate vector field associated with the $s$-th coordinate, where $s\in \{1, \ldots, n+1+k+r\}$. Furthermore, $\textup{Derlog}\,Z = \textup{Derlog}\,G \oplus \langle \epsilon \rangle$. We therefore have that $$\textup{codim}_{\mathscr{A}_e}(X,f)=\dim_\mathbb{C}\dfrac{\theta(i)}{ti(\theta_{n+1})+i^*(\textup{Derlog}\,G)+i^*(\epsilon)}.$$ Notice that the evaluation mapping $\text{ev}:\theta_{n+1+k+r}\rightarrow J(G)$ given by $\xi \mapsto \xi (G)$ is a surjective mapping with kernel $\textup{Derlog}\,G$. Hence, it induces an isomorphism $$\dfrac{\theta_{n+1+k+r}}{\textup{Derlog}\,G} \cong J(G).$$ Thus, $$\dfrac{\theta_{n+1+k+r}}{\langle \frac{\partial}{\partial y_1}, \ldots, \frac{\partial}{\partial y_{n+1}} \rangle +\textup{Derlog}\,G} \cong \dfrac{J(G)}{J_y (G)}=M_\textup{rel}(G).$$ Tensoring with $\mathscr{O}_{k+r}/\mathfrak{m}_{k+r}$ yields that $$\dfrac{\theta(i)}{ti(\theta_{n+1}) +i^*\textup{Derlog}\,G} \cong M(g).$$ Now, notice that the evaluation map acting on $\epsilon$ gives $\epsilon (G)=G$, and hence $i^*(\text{ev}(\epsilon))=i^*(G)=g$. Therefore, if $K(g)=(J(g)+(g))/J(g)$, then $$0 \rightarrow K(g) \rightarrow \dfrac{\theta(i)}{ti(\theta_{n+1}) +i^*\textup{Derlog}\,G} \rightarrow \dfrac{\theta(i)}{ti(\theta_{n+1}) +i^*\textup{Derlog}\,G +i^*(\epsilon)}\rightarrow 0$$ is a short exact sequence. Indeed, the evaluation map satisfies that $\text{ev}(ti(\theta_{n+1}))=J(g)$ and that $\text{ev}(i^*\textup{Derlog}\,G)=0$. Hence, the evaluation map yields an isomorphism $$\dfrac{ti(\theta_{n+1}) +i^*\textup{Derlog}\,G+i^*(\epsilon)}{ti(\theta_{n+1}) +i^*\textup{Derlog}\,G }\cong \dfrac{J(g)+(g)}{J(g)}=K(g).$$ After taking lengths in the exact sequence, and taking into account the previous assertions, it follows that **Theorem 22**. *In the above conditions, $$\dim_\mathbb{C}M(g) = \dim_\mathbb{C}K(g)+\textup{codim}_{\mathscr{A}_e}(X,f).$$ In particular, $\dim_\mathbb{C}M(g)\geq \textup{codim}_{\mathscr{A}_e}(X,f)$, with equality in case that $g$ is weighted homogeneous.* Furthermore, this formula shows that $\dim_\mathbb{C}M(g)$ only depends on the isomorphism class of $g$, and neither depends on the mapping $f$ nor on the chosen extension $\hat{f}$. When either $(n,n+1)$ are nice dimensions or in corank one, all stable singularities are weighted homogeneous (see [@juanjo]). This gives the following direct consequence of Theorem [Theorem 22](#relation){reference-type="ref" reference="relation"}: **Corollary 23**. *Assume that either $(n,n+1)$ are nice dimensions or $(X,f)$ has corank one. Then, $M(g)=0$ if and only if $(X,f)$ is stable.* # The Jacobian module and the Mond conjecture {#section:5_3} This section is the centerpiece of this paper. We present one of the main results we aim to establish, namely, a formula for the image Milnor number expressed in terms of the Samuel multiplicity of the module $M_\textup{rel}(G)$. Additionally, we prove that the generalised Mond conjecture holds provided the module $M_\textup{rel}(G)$ is Cohen-Macaulay. This theorem is the key ingredient to show the main result of the article [@BobadillaNunoPenafort], which is Theorem 6.1, and whose proof can be easily adapted to this new setting: **Theorem 24**. *Assume that either $(n,n+1)$ are nice dimensions or $(X,f)$ has corank one. Let $F$ be a stable unfolding of $\hat{f}$. With the notations of Section [3](#section3){reference-type="ref" reference="section3"}, $$\mu_I(X,f)=e(\mathfrak{m}_{k+r},M_\textup{rel}(G)),$$ that is, $\mu_I(X,f)$ equals the Samuel multiplicity of the $\mathscr{O}_{k+r}$-module $M_\textup{rel}(G)$ with respect to $\mathfrak{m}_{k+r}$.* *Proof.* Take a representative of $F$ and let $w\in \mathbb{C}^k \times \mathbb{C}^r$ be a generic value. The conservation of multiplicity implies that $$e\Big(\mathfrak{m}_{k+r}, M_\textup{rel}(G)\Big)=\sum_{p\in B_\epsilon} e\Big( \mathfrak{m}_{k+r,w}, M_\textup{rel}(G)_{(p,w)} \Big).$$ In order to compute the previous multiplicity, let us take first the points $p\in Y_w\cap B_\epsilon$, where $Y_w$ is the image of $f_w:X_w\rightarrow \mathbb{C}^{n+1}$. Since the module $M_\textup{rel}(G)$ specialises to $M(g)$, it follows that $$M_\textup{rel}(G)_{(p,w)}\otimes \dfrac{\mathscr{O}_{k+r,w}}{\mathfrak{m}_{k+r,w}}\cong M(g_w)_p$$ as $\mathbb{C}$-vector spaces in virtue of Remark [Remark 20](#remark:specialisation2){reference-type="ref" reference="remark:specialisation2"}. If $F$ is a stable unfolding, it follows that, provided $w$ is generic, $f_w$ is a stable mapping, and hence $M(g_w)_p=0$ for each $p\in Y_w$, by Corollary [Corollary 23](#nice){reference-type="ref" reference="nice"}. Hence, the points $p\in Y_w \cap B_\epsilon$ do not contribute to the term $e(\mathfrak{m}_{k+r}, M_\textup{rel}(G))$. On the other hand, if $p\in B_\epsilon / Y_w$, then $$M_\textup{rel}(G)_{(p,w)} = \dfrac{\mathscr{O}_{\mathbb{C}^{k+r}\times B_\epsilon, (p,w)}}{J_y (G)}$$ is a module with dimension $\geq k+r$. Indeed, this follows from the exact sequence that defines $M_\textup{rel}(G)$, since the localisation of $C(F)/(J_{y,z}(G)\cdot \mathscr{O}_{n+k+r})$ is zero if $p\notin Y_w$, and $\mathscr{F}_1(F)$ localises to $\mathscr{O}_{\mathbb{C}^{k+r}\times B_\epsilon, (p,w)}$. Moreover, $$M_\textup{rel}(G)_{(p,w)}\otimes \dfrac{\mathscr{O}_{k+r,w}}{\mathfrak{m}_{k+r,w}} \cong \dfrac{\mathscr{O}_{B_\epsilon, p}}{J(g_w)}$$ is a module with dimension $0$, since it has finite length due to the fact that $g_u$ has isolated singularity. This implies that the dimension of $M_\textup{rel}(G)$ is $\leq k+r$. Hence, $M_\textup{rel}(G)_{(p,u)}$ is a complete intersection ring, and in particular a Cohen-Macaulay $\mathscr{O}_{k+r}$-module of dimension $k+r$. Hence, $$\begin{aligned} e\Big( \mathfrak{m}_{k+r,w}, M_\textup{rel}(G)_{(p,w)} \Big) &= \dim_\mathbb{C}\left( M_\textup{rel}(G)_{(p,w)} \otimes \dfrac{\mathscr{O}_{k+r,w}}{\mathfrak{m}_{k+r,w}} \right) \\&= \dim_\mathbb{C}\dfrac{\mathscr{O}_{B_\epsilon, p}}{J(g_w)} = \mu(g_w, p). \end{aligned}$$ By Siersma's Theorem [\[Siersma\]](#Siersma){reference-type="ref" reference="Siersma"}, it follows that $\sum_{p\in B_\epsilon / Y_w} \mu(g_w,p) = \mu_I(X,f)$. ◻ Once this result has been proven, the following theorem is now an immediate consequence: **Theorem 25**. *In the above conditions, the following statements are equivalent and imply the generalised Mond conjecture for $(X,f)$:* 1. *$\dim_\mathbb{C}M(g)=\mu_I(X,f)$,* 2. *$M_\textup{rel}(G)$ is a Cohen-Macaulay $\mathscr{O}_{k+r}$-module of dimension $k+r$.* *Furthermore, if $g$ is weighted homogeneous and satisfies the generalised Mond conjecture, then the above assertions hold.* *Proof.* Recall that the Samuel multiplicity of an $R$-module $M$ is generally smaller than the length of $M/(\mathfrak{m}\cdot M)$, with equality if and only if $M$ is a Cohen-Macaulay module with the same dimension as $R$. Hence, $M_\textup{rel}(G)$ is Cohen-Macaulay of dimension $k+r$ if and only if $$\begin{aligned} \mu_I(X,f)&=e\Big(\mathfrak{m}_{k+r}, M_\textup{rel}(G)\Big) = \dim_\mathbb{C}\dfrac{M_\textup{rel}(G)}{\mathfrak{m}_{k+r} M_\textup{rel}(G)}\\&= \dim_\mathbb{C}M_\textup{rel}(G)\otimes \dfrac{\mathscr{O}_{k+r}}{\mathfrak{m}_{k+r}}=\dim_\mathbb{C}M(g). \end{aligned}$$ Therefore, if one of the items hold, it then follows that $$\mu_I(X,f)=\dim_{\mathbb{C}} M(g) = \dim_\mathbb{C}K(g) + \textup{codim}_{\mathscr{A}_e}(X,f) \geq \textup{codim}_{\mathscr{A}_e}(X,f),$$ and hence, the generalised Mond conjecture holds for $(X,f)$. Moreover, if $g$ is weighted homogeneous, then $K(g)=0$. Thus, if the generalised Mond conjecture holds for $(X,f)$, then $$\mu_I(X,f)=\textup{codim}_{\mathscr{A}_e}(X,f)=\dim_\mathbb{C}M(g)-\dim_\mathbb{C}K(g)=\dim_\mathbb{C}M(g).$$ Hence, the above assertions hold. ◻ *Remark 26*. Recent work by Nuño-Ballesteros and Giménez Conejero [@roberto2] has shown that the requirement for $M_\textup{rel}(G)$ to be $k+r$-dimensional can be eliminated from the second condition. Thus, we have that $\dim_\mathbb{C}M(g)=\mu_I(X,f)$ if and only if $M_\textup{rel}(G)$ is a Cohen-Macaulay module. Indeed, if the dimension of $M_\textup{rel}(G)$ is strictly less than $k+r$, then $\mu_I(X,f)$ is zero, which, according to [@roberto2], implies that $(X,f)$ is a stable map-germ, and hence $\textup{codim}_{\mathscr{A}_e}(X,f)=0$. This result shows that in order to establish the validity of the Mond conjecture for $f$, it suffices to check the Cohen-Macaulay property of the module $M_\textup{rel}(G)$. While it has been observed that this module is Cohen-Macaulay in every computed example, it remains an open question to provide a rigorous proof of this statement. # Proof of the generalised Mond conjecture for *n = 2* {#section:5_4} In this section, we achieve our main objective of the article, which is to show the validity of the generalised Mond conjecture for $n=2$ by employing the results from the previous section. Building on the main result of the previous section, to establish the generalised Mond conjecture for mappings $f:(X,0)\rightarrow (\mathbb{C}^3,0)$ where $(X,0)$ is a 2-dimensional [icis,]{.smallcaps} it suffices to check that $M_\textup{rel}(G)$ is a Cohen-Macaulay module of dimension $k+r$. Notice first that the dimension of $M_\textup{rel}(G)$ is $\leq k+r$, due to the fact that $$\dim_\mathbb{C}M_\textup{rel}(G)\otimes \dfrac{\mathscr{O}_{n+1+k+r}}{\mathfrak{m}_{k+r}\cdot \mathscr{O}_{n+1+k+r}} =\dim_\mathbb{C}M(g) <+\infty.$$ Therefore, it is enough to show that $\textup{depth}\,M_\textup{rel}(G) \geq k+r$. To achieve this, we apply the depth lemma to the exact sequence $$0\rightarrow M_\textup{rel}(G) \rightarrow \dfrac{\mathscr{F}_1(F)}{J_y(G)}\rightarrow \dfrac{C(F)}{J_{y,z}(G)\cdot \mathscr{O}_{n+k+r}} \rightarrow 0$$ to obtain that $$\textup{depth}\,M_\textup{rel}(G)\geq \min \left\{ \textup{depth}\,\dfrac{\mathscr{F}_1(F)}{J_y (G)}, \textup{depth}\,\dfrac{C(F)}{J_{y,z} (G)\cdot \mathscr{O}_{n+k+r}} + 1 \right\}.$$ Hence, both terms of the minimum have to be shown to be greater than or equal to $k+r$. For the module $C(F)/J_{y,z} (G)\cdot \mathscr{O}_{n+k+r}$, it has been checked in Remark 3.9 of [@BobadillaNunoPenafort] that this module is Cohen-Macaulay of dimension $n+k+r-2$ provided $n\geq 2$, due to the fact that it is isomorphic to the determinantal ring $\mathscr{O}_{n+k+r}/R(F)$, where $R(F)$ denotes the ramification ideal of $F$. In particular, it follows that $$\textup{depth}\,\dfrac{C(F)}{J_y (G)\cdot \mathscr{O}_{n+k+r}} + 1 = n+k+r-1 \geq k+r.$$ Therefore, it is enough to verify that $\textup{depth}\,\mathscr{F}_1(F)/J_y (G) \geq k+r$. Notice that, up to this point, the assumption $n=2$ has not been used yet. In general, the module $\mathscr{F}_1(F)/J_y (G)$ has dimension $$\dim \dfrac{\mathscr{F}_1(F)}{J_y (G)} = \max \left\{\dim M_\textup{rel}(G), \dim \dfrac{C(F)}{J_y (G)\cdot \mathscr{O}_{n+k+r,0}} \right\} =n+k+r-2,$$ due to the fact that $\dim M_\textup{rel}(G) \leq k+r \leq n+k+r-2$ provided $n\geq 2$. Therefore, it is not expected that $M_\textup{rel}(G)$ will be Cohen-Macaulay for every $n\geq 2$. The only case in which we can expect this is when $n=2$, since, in this case, its dimension is precisely $k+r$. To verify this claim, we make use of Pellikaan's Theorem: **Theorem 27** (Pellikaan, Section 3 of [@Pellikaan]). *If $J\subset F\subset R$ are ideals of $R$ where $J$ is generated by $m$ elements, $\textup{grade}\,(F/J)\geq m$ and $\textup{pd}\, (R/F)=2$, then $F/J$ is a perfect module and grade$\,(F/J)=m$.* This result plays a crucial role to show that the module is indeed Cohen-Macaulay, as it is proven in the following proposition: **Theorem 28**. *If $n=2$, then $\mathscr{F}_1(F)/J_y (G)$ is a Cohen-Macaulay module.* *Proof.* We follow the notation of the previous result, taking $R=\mathscr{O}_{n+1+k+r}$, $F=\mathscr{F}_1(F)$ and $J=J_y (G)$. It follows from the proof of Theorem 3.4 of [@Mond-Pellikaan] that $R/F=\mathscr{O}_{n+1+k+r}/\mathscr{F}_1(F)$ is a determinantal ring of dimension $n+k+r-1$, and hence Cohen-Macaulay. Hence, by the Auslander-Buchsbaum formula, $$\begin{aligned} \textup{pd}\, (R/F) &= \textup{depth}\,R - \textup{depth}\,R/F = \dim R - \dim R/F \\&= n+1+k+r - (n+k+r-1) = 2. \end{aligned}$$ Moreover, $J=J_y (G)$ is clearly generated by $n+1=3$ elements, namely the partial derivatives of $G$ with respect to the variables $y_1, \ldots, y_{n+1}$. Furthermore, $$\begin{aligned} \textup{grade}\,(F/J) &= \textup{depth}\,(\textup{Ann}\,(F/J), \mathscr{O}_{n+1+r})= \textup{ht}\,\textup{Ann}\,(F/J) = \\ &= \dim \mathscr{O}_{n+1+r}-\dim F/J= n+1+r-(n+r-2)=3.\end{aligned}$$ Thus, Pellikaan's Theorem states that $F/J$ is a perfect $\mathscr{O}_{n+1+k+r}$-module. Furthermore, since $\mathscr{O}_{n+1+k+r}$ is a local Cohen-Macaulay ring, then $F/J$ is Cohen-Macaulay. ◻ Now the main result of this article follows easily as an application of the results presented in the previous section. *Proof of Theorem [Theorem 2](#thrm:main){reference-type="ref" reference="thrm:main"}.* Theorem [Theorem 28](#theorem:n=2){reference-type="ref" reference="theorem:n=2"} implies that $M_\textup{rel}(G)$ is a Cohen-Macaulay module. Hence, Theorem [Theorem 25](#theorem:MC){reference-type="ref" reference="theorem:MC"} implies that the generalised Mond conjecture holds for $(X,f)$. ◻ In this setting, the image Milnor number therefore satisfies that $\mu_I(X,f)=\dim_\mathbb{C}M(g)$. This gives an operative method to compute this number, as the following example shows: **Example 29**. Let $(X,0)\subset (\mathbb{C}^3,0)$ be the hypersurface defined by $x^3+y^3-z^2=0$ and let $f:(X,0)\rightarrow (\mathbb{C}^3,0)$ be the $\mathscr{A}$-finite mapping $f(x,y,z)=(x,y,z^3+xz+y^2)$. In this case, $\hat{f}(x,y,z)=(x,y,z^3+xz+y^2, x^3+y^3-z^2)$ turns out to be a stable mapping. With [Singular]{.smallcaps} [@DGPS], one easily obtains that $\textup{codim}_{\mathscr{A}_e}(X,f)=6$ and $\mu_I(X,f)=\dim_\mathbb{C}M(g)=6$.

arxiv_math

--- abstract: | We prove comparison principle for viscosity solutions of a Hamilton--Jacobi--Bellman equation in a strong coupling regime considering a stationary and a time-dependent version of the equation. We consider a Hamiltonian that has a representation as the supremum of a difference of two functions: an internal Hamiltonian depending on a control variable and a function interpreted as a cost of applying the controls. Our major innovation lies in the use of a cost function that can be discontinuous, unbounded and depending on momenta, enabling us to address previously unexplored scenarios such as cases arising from the theory of large deviations and homogenisation. For completeness, we also state the existence of viscosity solutions and we verify the assumptions for an example arising from biochemistry. **Keywords***: Hamilton--Jacobi--Bellman equations, comparison principle, viscosity solutions, optimal control theory, Large deviations;* author: - "Serena Della Corte[^1] Richard C. Kraaij[^2]" bibliography: - KraaijBib.bib title: Well-posedness of a Hamilton-Jacobi-Bellman equation in the strong coupling regime --- # Introduction In the present work we study well--posedness of the Hamilton--Jacobi--Bellman equation on a subset $E\subseteq{{\mathbb R}}^d$, $$\begin{aligned} \label{eq:Hamilton-Jacobi-Bellman} u(x)-\lambda \mathcal{H}(x,\nabla u(x))=h(x),\end{aligned}$$ where $\lambda$ is a positive constant and $h$ is a continuous bounded function, and for the time--dependent version $$\begin{gathered} \label{eq:time-dep-HJB} \begin{cases} \partial_t u(x,t) - \mathcal{H}(x,\nabla_x u(t,x)) = 0, & \text{if $t>0$,}\\ u(0,x)=u_0(x) & \text{if $t=0$.} \end{cases}\end{gathered}$$ In the entire work we consider a Hamiltonian of the type $$\label{def:hamiltonian} \mathcal{H}(x,p)=\sup_{\theta\in\Theta}\left[\Lambda(x,p,\theta) - \mathcal{I}(x,p,\theta)\right].$$ The main goal of this work is to prove the *comparison principle* for viscosity solutions of the above equations [\[eq:Hamilton-Jacobi-Bellman\]](#eq:Hamilton-Jacobi-Bellman){reference-type="eqref" reference="eq:Hamilton-Jacobi-Bellman"} and [\[eq:time-dep-HJB\]](#eq:time-dep-HJB){reference-type="eqref" reference="eq:time-dep-HJB"}, implying also the uniqueness of solutions. Comparison principle for viscosity solutions has been largely studied in the past years with an increasingly complex Hamiltonian. Above all, we mention [@BaCD97] and [@DLLe11] for a proof of comparison principle for equations arising from *optimal control* problems and [@BuDuGa18], [@KuPo17] for Hamiltonians coming from the theory of large deviations for Markov processes. In these settings, the standard assumptions used to obtain the comparison principle are usually either the *modulus continuity* of $\mathcal{H}$ i.e. $$\vert \mathcal{H}(x,p)- \mathcal{H}(y,p)\vert \leq \omega(|x-y|(1+|p|)),$$ or uniformly coercivity of $\mathcal{H}$, that is $$\sup_{x\in K} \mathcal{H}(x,p) \to \infty \qquad \text{if $|p|\to \infty$.}$$ In the case in which $\mathcal{H}$ is in a variational representation as in [\[def:hamiltonian\]](#def:hamiltonian){reference-type="eqref" reference="def:hamiltonian"}, the above assumptions can be derived from conditions on $\Lambda$ and $\mathcal{I}$ such that coercivity or pseudo-coercivity of $\Lambda$ and regularity and boundedness of the cost function $\mathcal{I}$. However, there is a wide class of examples violating the above assumptions. In particular, this is the case of Hamiltonians arising in the study of systems with multiple time--scales (see for example [@BuDuGa18]). More recently, in [@KrSc21] the authors prove well--posedness for viscosity solutions of a general Hamilton--Jacobi--Bellman equation that can be applied in many of the above contexts. It is proved comparison principle under more generic and weaker assumptions then the common ones explained above. To be more precised, the authors in [@KrSc21] prove for the first time comparison principle for an Hamilton--Jacobi--Bellman equation with Hamiltonian of the type [\[def:hamiltonian\]](#def:hamiltonian){reference-type="eqref" reference="def:hamiltonian"} with $\Lambda$ that can be non coercive, non pseudo--coercive and non Lipschitz and $\mathcal{I}$ that can be unbounded and discontinuous, but not depending on momenta $p$. Our work can then be seen as an extension of the above mentioned work as we introduce a cost function $\mathcal{I}$ depending on momenta $p$. The introduction of the momenta in the function $\mathcal{I}$ makes on one hand the setting even more general including examples arising from problems in homogenisation theory that could not be treated before, and on the other hand the Hamiltonian more difficult to treat as it takes into account contributions from both parts $\Lambda$ and $\mathcal{I}$. For this reason, it is necessary a change of the starting assumptions based on the difference $\Lambda - \mathcal{I}$ and not on the two separate functions. In section [3](#section:clarify-assumptions){reference-type="ref" reference="section:clarify-assumptions"} we explain in more details how our work includes all the examples previously covered in [@BaCD97] and [@KrSc21]; in Section [6](#section:verification-for-examples-of-Hamiltonians){reference-type="ref" reference="section:verification-for-examples-of-Hamiltonians"} we give an extra example extending one of the examples presented in [@Po18]. In the following we present a concise overview of our strategy, without delving deep into specific details. Proving comparison principle one usually wants to bound the difference between subsolution and supersolution $\sup_E u_1- u_2$ by using a doubling variables procedure and typically ends up with an estimate of the following type $$\begin{aligned} \sup_E(u_1-u_2) \leq &\lambda \liminf_{\varepsilon\to 0}\liminf_{\alpha\to\infty} \left[\mathcal{H}\left(x_{\alpha,\varepsilon}, \mathrm{d}_x \frac{\alpha}{2}d^2(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon})\right) - \mathcal{H}\left(y_{\alpha,\varepsilon}, - \mathrm{d}_y \frac{\alpha}{2}d^2(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon})\right)\right] \\ &+ \sup_{E}(h_1 - h_2). \label{eq:intro:estimate}\end{aligned}$$ Therefore, the aim is usually to bound the difference of Hamiltonians in two sequences of points, $x_{\alpha,\varepsilon}$ and $y_{\alpha,\varepsilon}$, obtained as optimizers in the doubling variables procedure, and corresponding momenta $p^1_{\alpha,\varepsilon}= \mathrm{d}_x \frac{\alpha}{2}d^2(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon})= - \mathrm{d}_y \frac{\alpha}{2}d^2(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon}) = p^2_{\alpha,\varepsilon}$. Unlike the approach taken in [@KrSc21] and in [@BaCD97], where $\Lambda$ and $\mathcal{I}$ were worked on independently, in the strong coupling regime, where $\mathcal{I}$ depends on $p$, we need to consider their difference as a single function. Indeed, we give assumptions on $\Lambda-\mathcal{I}$. Our main assumptions are as follows: - Firstly, we rely on the *continuity estimate* of $\Lambda - \mathcal{I}$ that is morally the comparison principle for $\Lambda - \mathcal{I}$ for fixed $\theta$. Indeed, it enables us to control the difference of $\mathcal{H}$ in [\[eq:intro:estimate\]](#eq:intro:estimate){reference-type="eqref" reference="eq:intro:estimate"} by managing the difference of $\left(\Lambda(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon})-\mathcal{I}(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon})\right) - \left(\Lambda(y_{\alpha,\varepsilon},p_{\alpha,\varepsilon}^2,\theta_{\alpha,\varepsilon})-\mathcal{I}(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon})\right)$ for $\theta_{\alpha,\varepsilon}$ optimizing $\mathcal{H}$ and well chosen as explained in the next point. See Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} [\[item:assumption:regularity:continuity_estimate\]](#item:assumption:regularity:continuity_estimate){reference-type="ref" reference="item:assumption:regularity:continuity_estimate"} for the rigorous notions. - In order to get comparison for the Hamilton--Jacobi-Bellman equation in terms of $\mathcal{H}$ by the continuity estimate for $\Lambda-\mathcal{I}$, we also need to control the $\theta_{\alpha,\varepsilon}$. For this reason, we assume the compactness of the level sets of $\mathcal{I}- \Lambda$. Using this assumption we are indeed able to prove that the above sequence $\theta_{\alpha,\varepsilon}$ is relatively compact, i.e. [\[item:def:continuity_estimate:3\]](#item:def:continuity_estimate:3){reference-type="ref" reference="item:def:continuity_estimate:3"}. This assumption is made rigorous in Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"}. - We assume $\Gamma$--convergence for $\mathcal{I}-\Lambda$ to prove regularity of $\mathcal{H}$. This assumption is typically true for the most treated examples, e.g. when $\Lambda$ and $\mathcal{I}$ are continuous or when $\mathcal{I}$ arises as a Donsker--Varadhan functional (see [@DoVa75a]). See Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} [\[item:assumption:Gamma-convergence\]](#item:assumption:Gamma-convergence){reference-type="ref" reference="item:assumption:Gamma-convergence"}. To complete the well-posedness study of the Hamilton--Jacobi--Bellman equations [\[eq:Hamilton-Jacobi-Bellman\]](#eq:Hamilton-Jacobi-Bellman){reference-type="eqref" reference="eq:Hamilton-Jacobi-Bellman"} and [\[eq:time-dep-HJB\]](#eq:time-dep-HJB){reference-type="eqref" reference="eq:time-dep-HJB"}, we also state the existence of viscosity solutions for equation [\[eq:Hamilton-Jacobi-Bellman\]](#eq:Hamilton-Jacobi-Bellman){reference-type="eqref" reference="eq:Hamilton-Jacobi-Bellman"} in Theorem [Theorem 10](#theorem:existence_of_viscosity_solution){reference-type="ref" reference="theorem:existence_of_viscosity_solution"} and we mention of our ongoing work towards a new improved proof of the existence of solutions for [\[eq:time-dep-HJB\]](#eq:time-dep-HJB){reference-type="eqref" reference="eq:time-dep-HJB"}. To prove Theorem [Theorem 10](#theorem:existence_of_viscosity_solution){reference-type="ref" reference="theorem:existence_of_viscosity_solution"}, the main ingredient is to establish the existence of the differential inclusion in terms of $\partial_p \mathcal{H}$. Moreover, we need to make sure that the solutions to the differential inclusions remains inside our set $E$. To this aim, we add Assumption [Assumption 18](#assumption:Hamiltonian_vector_field){reference-type="ref" reference="assumption:Hamiltonian_vector_field"}. Our work is then structured as follows: In Section [2](#section:results){reference-type="ref" reference="section:results"} we firstly give some preliminaries and an overview of the general setting and secondly we state our main results, Theorems [Theorem 8](#theorem:comparison_principle_variational){reference-type="ref" reference="theorem:comparison_principle_variational"} and [Theorem 10](#theorem:existence_of_viscosity_solution){reference-type="ref" reference="theorem:existence_of_viscosity_solution"}, and the assumptions needed to prove them. In Section [3](#section:clarify-assumptions){reference-type="ref" reference="section:clarify-assumptions"} we give a list of examples showing that our method is able to include examples treated in previous works as well as new examples that were previously beyond the scope of their applicability. Then, we prove continuity of the Hamiltonian in Section [4](#section:regularity-of-H){reference-type="ref" reference="section:regularity-of-H"}. In Section [5](#section:comparison_principle){reference-type="ref" reference="section:comparison_principle"} we give the proof of comparison principle and we state the existence of solutions. Finally, in Section [6](#section:verification-for-examples-of-Hamiltonians){reference-type="ref" reference="section:verification-for-examples-of-Hamiltonians"} we treat an example to show that our assumptions are well-posed. **Acknowledgement.** The authors are supported by The Netherlands Organisation for Scientific Research (NWO), grant number 613.009.148 . # General setting and main results {#section:results} In this section, we firstly give some notions and definitions used throughout all the paper. Then, we proceed with the assumptions needed for the statement of our main results given in Section [2.2](#section:results:HJ-of-Perron-Frobenius-type){reference-type="ref" reference="section:results:HJ-of-Perron-Frobenius-type"}. ## Preliminaries {#section:preliminaries} For a Polish space $\mathcal{X}$ we denote by $C(\mathcal{X})$ and $C_b(\mathcal{X})$ the spaces of continuous and bounded continuous functions respectively. We denote $C_l(\mathcal{X})$ and $C_u(\mathcal{X})$ the spaces of lower bounded continuous and upper bounded continuous functions respectively. If $\mathcal{X}\subseteq \mathbb{R}^d$ then we denote by $C_c^\infty(\mathcal{X})$ the space of smooth functions that vanish outside a compact set. We denote by $C_{cc}^\infty(\mathcal{X})$ the set of smooth functions that are constant outside of a compact set in $\mathcal{X}$, and by $\mathcal{P}(\mathcal{X})$ the space of probability measures on $\mathcal{X}$. We equip $\mathcal{P}(\mathcal{X})$ with the weak topology induced by convergence of integrals against bounded continuous functions. Throughout the paper, $E$ will be the set on which we base our Hamilton-Jacobi equations. We assume that $E$ is a subset of $\mathbb{R}^d$ that is a Polish space which is contained in the $\mathbb{R}^d$ closure of its $\mathbb{R}^d$ interior. This ensures that gradients of functions are determined by their values on $E$. Note that we do not necessarily assume that $E$ is open. We assume that the space of controls $\Theta$ is Polish. We next introduce *viscosity solutions* for an Hamilton-Jacobi-Bellman equation $f-\lambda Af=h$ and for the time-dependent version $f_t - A f = 0$. **Definition 1** (Viscosity solutions for the stationary equation). Let $A_\dagger : \mathcal{D}(A_\dagger) \subseteq C_l(E) \to C_b(E)$ be an operator with domain $\mathcal{D}(A_\dagger)$, $\lambda > 0$ and $h_\dagger \in C_b(E)$. Consider the Hamilton-Jacobi equation $$f - \lambda A_\dagger f = h_\dagger. \label{eqn:differential_equation}$$ We say that $u$ is a *(viscosity) subsolution* of equation [\[eqn:differential_equation\]](#eqn:differential_equation){reference-type="eqref" reference="eqn:differential_equation"} if $u$ is bounded from above, upper semi-continuous and if, for every $f \in \mathcal{D}(A)$ there exists a sequence $x_n \in E$ such that $$\begin{gathered} \lim_{n \uparrow \infty} u(x_n) - f(x_n) = \sup_x u(x) - f(x), \\ \limsup_{n \uparrow \infty} u(x_n) - \lambda A_\dagger f(x_n) - h_\dagger(x_n) \leq 0. \end{gathered}$$ Let $A_\ddagger: \mathcal{D}(A_\ddagger) \subseteq C_u (E) \to C_b (E)$ be an operator with domain $\mathcal{D}(A_\ddagger)$, $\lambda > 0$ and $h_\ddagger \in C_b(E)$. Consider the Hamilton-Jacobi equation $$f - \lambda A_\ddagger f = h_\ddagger. \label{eqn:differential_equation-ddagger}$$ We say that $v$ is a *(viscosity) supersolution* of equation [\[eqn:differential_equation-ddagger\]](#eqn:differential_equation-ddagger){reference-type="eqref" reference="eqn:differential_equation-ddagger"} if $v$ is bounded from below, lower semi-continuous and if, for every $f \in \mathcal{D}(A)$ there exists a sequence $x_n \in E$ such that $$\begin{gathered} \lim_{n \uparrow \infty} v(x_n) - f(x_n) = \inf_x v(x) - f(x), \\ \liminf_{n \uparrow \infty} v(x_n) - \lambda A_\ddagger f(x_n) - h_\ddagger(x_n) \geq 0. \end{gathered}$$ We say that $u$ is a *(viscosity) solution* of the set of equations [\[eqn:differential_equation\]](#eqn:differential_equation){reference-type="eqref" reference="eqn:differential_equation"} and [\[eqn:differential_equation-ddagger\]](#eqn:differential_equation-ddagger){reference-type="eqref" reference="eqn:differential_equation-ddagger"}, if it is both a subsolution of [\[eqn:differential_equation\]](#eqn:differential_equation){reference-type="eqref" reference="eqn:differential_equation"} and a supersolution of [\[eqn:differential_equation-ddagger\]](#eqn:differential_equation-ddagger){reference-type="eqref" reference="eqn:differential_equation-ddagger"}. **Definition 2** (Viscosity solutions for the time-dependent equation). Let $A_\dagger : \mathcal{D}(A_\dagger) \subseteq C_l(E) \to C_b(E)$ be an operator with domain $\mathcal{D}(A_\dagger)$. Consider the Hamilton-Jacobi equation with the initial value, $$\begin{gathered} \begin{cases} \partial_t u(t,x) - A_\dagger u(t,\cdot)(x) = 0, & \text{if } t > 0, \\ u(0,x) = u_0(x) & \text{if } t = 0, \end{cases} \label{eqn:HJ_def_subsolution} \\ \end{gathered}$$ Let $T>0$, $f\in D(A_\dagger)$ and $g\in C^1([0,T])$ and let $F_\dagger(x,t): E\times [0,T] \to {{\mathbb R}}$ be the function $$F_\dagger (x,t) = \begin{cases} \partial_t g(t) - A_\dagger f(x) & \text{if $t>0$} \\ \left[\partial_t g(t) - A_\dagger f(x) \right] \wedge \left[u(t,x)-u_0(x) \right] & \text{if $t=0$.} \end{cases}$$ We say that $u$ is a *(viscosity) subsolution* for [\[eqn:HJ_def_subsolution\]](#eqn:HJ_def_subsolution){reference-type="eqref" reference="eqn:HJ_def_subsolution"} if for any $T > 0$ any $f \in D(A_\dagger)$ and any $g\in C^1([0,T])$ there exists a sequence $(t_n,x_n) \in [0,T] \times E$ such that $$\begin{aligned} & \lim_{n \uparrow \infty} u(t_n,x_n) - f(x_n) -g(t_n) = \sup_{t\in[0,T],x} u(t,x) - f(x) - g(t), \\ & \limsup_{n \uparrow \infty} F_\dagger (x_n,t_n) \leq 0. \end{aligned}$$ Let $A_\ddagger : \mathcal{D}(A_\ddagger) \subseteq C_u(E) \to C_b(E)$ be an operator with domain $\mathcal{D}(A_\ddagger)$. Consider the Hamilton-Jacobi equation with the initial value, $$\begin{gathered} \begin{cases} \partial_t u(t,x) - A_\ddagger u(t,\cdot)(x) = 0, & \text{if } t > 0, \\ u(0,x) = u_0(x) & \text{if } t = 0, \end{cases} \label{eqn:HJ_def_supersolution} \\ \end{gathered}$$ Let $T>0$, $f\in D(A_\ddagger)$ and $g\in C^1([0,T])$ and let $F_\ddagger(x,t): E\times [0,T] \to {{\mathbb R}}$ be the function $$F_\ddagger (x,t) = \begin{cases} \partial_t g(t) - A_\dagger f(x) & \text{if $t>0$} \\ \left[\partial_t g(t) - A_\dagger f(x) \right] \vee \left[u(t,x)-u_0(x) \right] & \text{if $t=0$.} \end{cases}$$ We say that $v$ is a viscosity supersolution for [\[eqn:HJ_def_supersolution\]](#eqn:HJ_def_supersolution){reference-type="eqref" reference="eqn:HJ_def_supersolution"} if for any $T > 0$ any $f \in D(A_\ddagger)$ and $g\in C^1([0,T])$ there exists a sequence $(t_n,x_n) \in [0,T] \times E$ such that $$\begin{aligned} & \lim_{n \uparrow \infty} u(t_n,x_n) - f(x_n) -g(t_n) = \inf_{t\in[0,T],x} u(t,x) - f(x) - g(t), \\ & \liminf_{n\uparrow} F_\ddagger (x_n,t_n) \geq 0 \end{aligned}$$ We say that $u$ is a *(viscosity) solution* of the set of equations [\[eqn:HJ_def_subsolution\]](#eqn:HJ_def_subsolution){reference-type="eqref" reference="eqn:HJ_def_subsolution"} and [\[eqn:HJ_def_supersolution\]](#eqn:HJ_def_supersolution){reference-type="eqref" reference="eqn:HJ_def_supersolution"}, if it is both a subsolution of [\[eqn:HJ_def_subsolution\]](#eqn:HJ_def_subsolution){reference-type="eqref" reference="eqn:HJ_def_subsolution"} and a supersolution of [\[eqn:HJ_def_supersolution\]](#eqn:HJ_def_supersolution){reference-type="eqref" reference="eqn:HJ_def_supersolution"}. *Remark 3*. Consider the definition of subsolutions for $f-\lambda A f= h$. Suppose that the testfunction $f \in \mathcal{D}(A)$ has compact sublevel sets, then instead of working with a sequence $x_n$, there exists $x_0 \in E$ such that $$\begin{gathered} u(x_0) - f(x_0) = \sup_x u(x) - f(x), \\ u(x_0) - \lambda A f(x_0) - h(x_0) \leq 0. \end{gathered}$$ A similar simplification holds in the case of supersolutions and in the case of the time-dependent equation $\partial_t f - A f = 0$. *Remark 4*. For an explanatory text on the notion of viscosity solutions and fields of applications, we refer to [@CIL92]. *Remark 5*. At present, we refrain from working with unbounded viscosity solutions as we use the upper bound on subsolutions and the lower bound on supersolutions in the proof of Theorem [Theorem 8](#theorem:comparison_principle_variational){reference-type="ref" reference="theorem:comparison_principle_variational"}. We, however, believe that the methods presented in this paper can be generalized if $u$ and $v$ grow slower than the containment function $\Upsilon$ that will be defined below in Definition [Definition 14](#def:containment-new){reference-type="ref" reference="def:containment-new"}. We give now the definition of *comparison principle* leading to a uniqueness notion as stated in Remark [Remark 7](#remark:uniqueness){reference-type="ref" reference="remark:uniqueness"}. **Definition 6** (Comparison Principle). For two operators $A_\dagger, A_\ddagger \subseteq C(E) \times C(E)$, we say that the comparison principle holds if for any viscosity subsolution $u$ of $f - \lambda A_\dagger f = h_1$ (resp. $\partial_t f - A_\dagger f = 0$) and viscosity supersolution $v$ of $f - \lambda A_\ddagger f = h_2$ (resp. $\partial_t f - A_\ddagger f = 0$), $\sup_x \left\{u(x) - v(x)\right\} \leq \sup_x \{h_1 (x) - h_2 (x)\}$ (resp. $\sup_{t\in [0,T],x} \left\{u(t,x) - v(t,x)\right\} \leq \sup_x \{ u(0,x)- v(0,x)\}$ for all $T>0$ ) holds on $E$. *Remark 7* (Uniqueness). If $u$ and $v$ are two viscosity solutions of [\[eqn:differential_equation\]](#eqn:differential_equation){reference-type="eqref" reference="eqn:differential_equation"} or [\[eqn:HJ_def_subsolution\]](#eqn:HJ_def_subsolution){reference-type="eqref" reference="eqn:HJ_def_subsolution"}, then we have $u\leq v$ and $v\leq u$ by the comparison principle, and, hence, uniqueness of solutions. ## Main results: comparison and existence {#section:results:HJ-of-Perron-Frobenius-type} In this section, we state our main results: the comparison principle, that is Theorem [Theorem 8](#theorem:comparison_principle_variational){reference-type="ref" reference="theorem:comparison_principle_variational"}, and existence of solutions in Theorem [Theorem 10](#theorem:existence_of_viscosity_solution){reference-type="ref" reference="theorem:existence_of_viscosity_solution"}. Consider the variational Hamiltonian $\mathcal{H}: E \times \mathbb{R}^d \rightarrow \mathbb{R}$ given by $$\label{eq:results:variational_hamiltonian} \mathcal{H}(x,p) = \sup_{\theta \in \Theta}\left[\Lambda(x,p,\theta) - \mathcal{I}(x,p,\theta)\right].$$ The precise assumptions on the maps $\Lambda$ and $\mathcal{I}$ are formulated in Section [2.3](#section:assumptions){reference-type="ref" reference="section:assumptions"}. **Theorem 8** (Comparison principle). *Consider the map $\mathcal{H}: E \times \mathbb{R}^d \rightarrow \mathbb{R}$ as in [\[eq:results:variational_hamiltonian\]](#eq:results:variational_hamiltonian){reference-type="eqref" reference="eq:results:variational_hamiltonian"}. Suppose that Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} is satisfied. Define the operator $\mathbf{H}f(x) := \mathcal{H}(x,\nabla f(x))$ with domain $\mathcal{D}(\mathbf{H}) = C_{cc}^\infty(E)$. Then:* (a) *[\[item:theorem-comparison\]]{#item:theorem-comparison label="item:theorem-comparison"} For any $h \in C_b(E)$ and $\lambda > 0$, the comparison principle holds for $$\label{eq:results:HJ-eq} f - \lambda \, \mathbf{H}f = h.$$* (b) *For any $f_0 \in C_b(E)$, the comparison principle holds for $$\begin{gathered} \begin{cases} \partial_t f(t,x) - \mathbf{H}f(t,\cdot)(x) = 0, & \text{if $t>0$}\\ f(0,x)=f_0(x) &\text{if $t=0$.} \end{cases} \end{gathered}$$* *Remark 9* (Domain). The comparison principle holds with any domain that satisfies $C_{cc}^\infty(E)\subseteq \mathcal{D}(\mathbf{H})\subseteq C^1_b(E)$. We state it with $C^\infty_{cc}(E)$ to connect it with the existence result of Theorem [Theorem 10](#theorem:existence_of_viscosity_solution){reference-type="ref" reference="theorem:existence_of_viscosity_solution"}, where we need to work with test functions whose gradients have compact support. Consider the Legendre dual $\mathcal{L}: E \times \mathbb{R}^d \rightarrow [0,\infty]$ of the Hamiltonian, $$\mathcal{L}(x,v) := \sup_{p\in\mathbb{R}^d} \left[\left \langle p,v \right \rangle - \mathcal{H}(x,p)\right],$$ and denote the collection of absolutely continuous paths in $E$ by $\mathcal{A}\mathcal{C}$. **Theorem 10** (Existence of viscosity solution). *Consider $\mathcal{H}: E \times \mathbb{R}^d \rightarrow \mathbb{R}$ as in [\[eq:results:variational_hamiltonian\]](#eq:results:variational_hamiltonian){reference-type="eqref" reference="eq:results:variational_hamiltonian"}. Suppose that Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} is satisfied for $\Lambda$ and $\mathcal{I}$, and that $\mathcal{H}$ satisfies Assumption [Assumption 18](#assumption:Hamiltonian_vector_field){reference-type="ref" reference="assumption:Hamiltonian_vector_field"}. For each $\lambda > 0$, let $R(\lambda)$ be the operator $$\label{resolvent} R(\lambda) h(x) = \sup_{\substack{\gamma \in \mathcal{A}\mathcal{C}\\ \gamma(0) = x}} \int_0^\infty \lambda^{-1} e^{-\lambda^{-1}t} \left[h(\gamma(t)) - \int_0^t \mathcal{L}(\gamma(s),\dot{\gamma}(s))\right] \, \mathrm{d}t.$$ Then $R(\lambda)h$ is the unique viscosity solution to $f - \lambda \mathbf{H}f = h$.* *Remark 11*. The form of the solution is typical, see for example Section III.2 in [@BaCD97]. It is the value function obtained by an optimization problem with exponentially discounted cost. The difficulty of the proof of Theorem [Theorem 10](#theorem:existence_of_viscosity_solution){reference-type="ref" reference="theorem:existence_of_viscosity_solution"} lies in treating the irregular form of $\mathcal{H}$. *Remark 12*. We mention that in Theorem [Theorem 10](#theorem:existence_of_viscosity_solution){reference-type="ref" reference="theorem:existence_of_viscosity_solution"} we only state the existence of viscosity solution for the stationary equation $f - \lambda \mathbf{H}f = h$. In our work [@DeCoKr23], we will prove the existence also for the evolutionary equation. ## Assumptions {#section:assumptions} In this section, we formulate and comment on the assumptions imposed on the Hamiltonian defined in the previous section. We start with the *continuity estimate*. We will apply the definition below for $\mathcal{G}= \Lambda-\mathcal{I}$. **Definition 13** (Continuity estimate). Let $\mathcal{G}: E \times \mathbb{R}^d\times\Theta \rightarrow \mathbb{R}$, $(x,p,\theta)\mapsto \mathcal{G}(x,p,\theta)$ be a function. Suppose that for each $\varepsilon > 0$, there is a sequence of positive real numbers $\alpha \rightarrow \infty$. Suppose that for each $\varepsilon$ and $\alpha$ we have variables $(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon})$ in $E^2$ and variables $\theta_{\alpha,\varepsilon}$ in $\Theta$. We say that this collection is *fundamental* for $\mathcal{G}$ with if: 1. [\[item:def:continuity_estimate:1\]]{#item:def:continuity_estimate:1 label="item:def:continuity_estimate:1"} For each $\varepsilon$, there are compact sets $K_\varepsilon \subseteq E$ and $\widehat{K}_\varepsilon\subseteq\Theta$ such that for all $\alpha$ we have $x_{\alpha,\varepsilon},y_{\alpha,\varepsilon} \in K_\varepsilon$ and $\theta_{\alpha,\varepsilon}\in\widehat{K}_\varepsilon$. 2. [\[item:def:continuity_estimate:2\]]{#item:def:continuity_estimate:2 label="item:def:continuity_estimate:2"} For each $\varepsilon > 0$, we have $\lim_{\alpha \rightarrow \infty} \frac{\alpha}{2} d^2(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon}) = 0$. For any limit point $(x_\varepsilon,y_\varepsilon)$ of $(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon})$, we have $\mathrm{d}(x_{\varepsilon},y_{\varepsilon}) = 0$. 3. [\[item:def:continuity_estimate:3\]]{#item:def:continuity_estimate:3 label="item:def:continuity_estimate:3"} We have for all $\varepsilon > 0$ $$\begin{aligned} & \sup_{\alpha} \mathcal{G}\left(y_{\alpha,\varepsilon}, - \frac{\alpha}{2} \mathrm{d}_y d^2 (x_{\alpha,\varepsilon},\cdot)(y_{\alpha,\varepsilon}),\theta_{\alpha,\varepsilon}\right) < \infty, \label{eqn:control_on_Gbasic_sup} \\ & \inf_\alpha \mathcal{G}\left(x_{\alpha,\varepsilon}, \frac{\alpha}{2} \mathrm{d}_x d^2(\cdot,y_{\alpha,\varepsilon})(x_{\alpha,\varepsilon}),\theta_{\alpha,\varepsilon}\right) > - \infty. \label{eqn:control_on_Gbasic_inf} \end{aligned}$$ [\[itemize:funamental_inequality_control_upper_bound\]]{#itemize:funamental_inequality_control_upper_bound label="itemize:funamental_inequality_control_upper_bound"} We say that $\mathcal{G}$ satisfies the *continuity estimate* if for every fundamental collection of variables we have for each $\varepsilon > 0$ that $$\label{equation:Xi_negative_liminf} \liminf_{\alpha \rightarrow \infty} \mathcal{G}\left(x_{\alpha,\varepsilon}, \alpha \mathrm{d}_x\frac{1}{2}d^2(\cdot,y_{\alpha,\varepsilon})(x_{\alpha,\varepsilon}),\theta_{\alpha,\varepsilon}\right) - \mathcal{G}\left(y_{\alpha,\varepsilon}, - \alpha \mathrm{d}_y\frac{1}{2}d^2(x_{\alpha,\varepsilon},\cdot)(y_{\alpha,\varepsilon}),\theta_{\alpha,\varepsilon}\right) \leq 0.$$ The continuity estimate is indeed exactly the estimate that one would perform when proving the comparison principle for the Hamilton-Jacobi equation in terms of the Hamiltonian [\[eq:results:variational_hamiltonian\]](#eq:results:variational_hamiltonian){reference-type="eqref" reference="eq:results:variational_hamiltonian"} (disregarding the supremum over $\theta$). Indeed, in standard proofs of comparison principle one usually wants to control the difference of Hamiltonians calculated in particular collections of points. Typically, the control on $(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon})$ that is assumed in [\[item:def:continuity_estimate:1\]](#item:def:continuity_estimate:1){reference-type="ref" reference="item:def:continuity_estimate:1"} and [\[item:def:continuity_estimate:2\]](#item:def:continuity_estimate:2){reference-type="ref" reference="item:def:continuity_estimate:2"} is obtained from choosing $(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon})$ as optimizers in the doubling of variables procedure (see Lemma [Lemma 31](#lemma:doubling_lemma){reference-type="ref" reference="lemma:doubling_lemma"}), and the control that is assumed in [\[item:def:continuity_estimate:3\]](#item:def:continuity_estimate:3){reference-type="ref" reference="item:def:continuity_estimate:3"} is obtained by using the viscosity sub- and supersolution properties in the proof of the comparison principle. The required restriction to compact sets in Lemma [Lemma 31](#lemma:doubling_lemma){reference-type="ref" reference="lemma:doubling_lemma"} is obtained by including in the test functions a *containment function*. **Definition 14** (Containment function). We say that a function $\Upsilon: E\to [0,\infty]$ is a *containment function* for $\mathcal{H}$ if $\Upsilon\in C^1(E)$ and there exists a constant $c_\Upsilon$ such that - For every $c\geq 0$, the set $\{x\, | \, \Upsilon(x)\leq c\}$ is compact; - $\sup_\theta \sup_x \left(\Lambda(x,\nabla\Upsilon(x), \theta)- \mathcal{I}(x, \nabla \Upsilon(x), \theta)\right)\leq c_\Upsilon.$ For two constants $M_1, M_2$ and a compact set $K\subseteq E$ we write $$\begin{aligned} \label{eq:def_levelsets} \Theta_{M_1,M_2,K} := \bigcup_{x, y\in K} \bigcup_{\alpha > 1} \biggl\{\theta\in \Theta \, | \, \mathcal{I}\left(x,\partial_x \frac{\alpha}{2} d^2 (x,y),\theta\right) &-\Lambda\left(x,\partial_x \frac{\alpha}{2} d^2 (x,y),\theta\right)\leq M_1 , \\ \Lambda\left(y, \partial_y \frac{- \alpha}{2} d^2 (x,y), \theta\right) &- \mathcal{I}\left(y,\partial_y \frac{-\alpha}{2} d^2 (x,y),\theta\right)\leq M_2\biggr\}.\end{aligned}$$ To prove the main results we will also make use of the continuity of $\mathcal{H}$. The continuity of $\mathcal{H}$ is proved in Proposition [Proposition 25](#prop:reg-of-H-and-L:continuity){reference-type="ref" reference="prop:reg-of-H-and-L:continuity"} by making use of the following notion of convergence for the function $\mathcal{I}-\Lambda$. **Definition 15** ($\Gamma$--convergence). Let $J:E\times{{\mathbb R}}^d\times \Theta\to {{\mathbb R}}\cup\{\infty\}$. We say that $J$ is $\Gamma$--convergent in terms of $(x,p)$, if 1. If $x_n\to x$ in $E$, $p_n\to p$ in ${{\mathbb R}}^d$ and $\theta_n\to \theta$ then $\liminf_n J(x_n,p_n,\theta_n)\geq J(x,p,\theta)$, 2. For $x_n\to x$ and $p_n\to p$ and for all $\theta\in\Theta$ there are $\theta_n$ such that $\theta_n\to \theta$ and $\limsup_n J(x_n,p_n,\theta_n)\leq J(x,p,\theta)$. We will consider the following assumption. **Assumption 16**. The functions $\mathcal{H}$ and $\Lambda-\mathcal{I}$ verify the following properties. (I) [\[item:assumption:convexity\]]{#item:assumption:convexity label="item:assumption:convexity"}The map $p\mapsto\mathcal{H}(x,p)$ is convex and $\mathcal{H}(x,0)=0$ for every $x\in E$. (II) [\[item:assumption:regularity:boundness\]]{#item:assumption:regularity:boundness label="item:assumption:regularity:boundness"}The function $\theta\mapsto\Lambda(x,p,\theta)-\mathcal{I}(x,p,\theta)$ is bounded from above for every $x,p$. (III) [\[item:assumption:compact_containment\]]{#item:assumption:compact_containment label="item:assumption:compact_containment"}There exists a containment function $\Upsilon:E\to[0,\infty]$ in the sense of Definition [Definition 14](#def:containment-new){reference-type="ref" reference="def:containment-new"}. (IV) [\[item:assumption:Gamma-convergence\]]{#item:assumption:Gamma-convergence label="item:assumption:Gamma-convergence"} The function $\mathcal{I}-\Lambda$ is $\Gamma$--convergent in terms of $(x,p)$. (V) [\[item:assumption:compact-sublevelsets\]]{#item:assumption:compact-sublevelsets label="item:assumption:compact-sublevelsets"} $\forall M_1, M_2 \in {{\mathbb R}}$ and $\forall K$ compact, the set $\Theta_{M_1,M_2,K}$ is relatively compact. (VI) [\[item:assumption:statinary_meas\]]{#item:assumption:statinary_meas label="item:assumption:statinary_meas"} $\forall x \in E$ , $\forall p \in {{\mathbb R}}^d$ and for all small neighborhood of $x$ and $p$, $U_x$ $V_p$, there exists a continuous function $g:U_x\times V_p \to {{\mathbb R}}$ such that the set $\phi_g(y,q)=\{\theta \in \Theta \, | \, \mathcal{I}(y,q,\theta) - \Lambda (y,q,\theta) \leq g(y,q)\}$ is non-empty and compact $\forall \, y\in U_x, q \in V_p$. (VII) [\[item:assumption:regularity:continuity_estimate\]]{#item:assumption:regularity:continuity_estimate label="item:assumption:regularity:continuity_estimate"}The function $\Lambda - \mathcal{I}$ verifies the continuity estimate in the sense of Definition [Definition 13](#def:results:continuity_estimate){reference-type="ref" reference="def:results:continuity_estimate"}. To establish the existence of viscosity solutions, we will impose one additional assumption. For a general convex functional $p \mapsto \Phi(p)$ we denote $$\label{eqn:subdifferential} \partial_p \Phi(p_0) := \left\{ \xi \in \mathbb{R}^d \,:\, \Phi(p) \geq \Phi(p_0) + \xi \cdot (p-p_0) \quad (\forall p \in \mathbb{R}^d) \right\}.$$ **Definition 17**. The tangent cone (sometimes also called *Bouligand cotangent cone*) to $E$ in $\mathbb{R}^d$ at $x$ is $$T_E(x) := \left\{z \in \mathbb{R}^d \, \middle| \, \liminf_{\lambda \downarrow 0} \frac{d(x + \lambda z, E)}{\lambda} = 0\right\}.$$ **Assumption 18**. One of the following conditions hold: (a) the set $E$ is open; (b) the set $E$ is closed and convex and the map $p\mapsto \mathcal{H}(x,p)$ is such that $\partial_p \mathcal{H}(x,p) \cap T_E(x) \neq \emptyset$ for all $x \in E$, $p \in \mathbb{R}^d$. The above assumption implies that the solutions of the differential inclusion in terms of $\partial_p \mathcal{H}(x,p)$ remain inside $E$. # Basic examples to clarify the assumptions and comparison with previous works {#section:clarify-assumptions} Although in Section [6](#section:verification-for-examples-of-Hamiltonians){reference-type="ref" reference="section:verification-for-examples-of-Hamiltonians"} we present a fully new example beyond present results, in this section, we give four examples. The first two are those treated in the previous works [@BaCD97] [@KrSc21], showing that our results includes these examples. To emphasize the fact that we can address more examples then the previous works, we illustrate a third example that does not fall into the list of examples considered in [@KrSc21] but that can be studied with our results. The final example clarifies Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"}. ## The classical control Hamiltonian The following first example is a classical example that was firstly given in [@BaCD97]. **Example 19**. *Consider the Hamiltonian $$\mathcal{H}(x,p) = \sup_{a\in A} \left\{ - f(x,a) \cdot p - l(x,a)\right\},$$ with $A$ a compact set, $f$ a Lipschitz continuous function and $l$ a non-negative continuous function such that $$\sup_{a\in A}\vert l(x,a) - l(y,a)\vert \leq w ( \vert x-y \vert),$$ with $w$ a modulus of continuity. Then, $\mathcal{H}$ verifies Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"}.* *Proof sketch..* We show in the following that the example verifies our assumptions. (I) $\mathcal{H}(x,p)$ is convex in $p$ and $\mathcal{H}(x,0)=\sup_{a\in A} \left\{ - l(x,a)\right\}= 0$. (II) $a\mapsto \Lambda(x,p,a) - \mathcal{I}(x,a)$ is bounded since $A$ is compact and $\Lambda - \mathcal{I}$ is continuous. (III) $\Upsilon(x) = \frac{1}{2} \log (1 + \vert x \vert ^2)$ is a containment function, in the sense of Definition [Definition 14](#def:containment-new){reference-type="ref" reference="def:containment-new"}, since $f$ is Lipschitz. (IV) $\mathcal{I}- \Lambda$ is continuous and hence $\Gamma$- convergent in $(x,p)$. (V) For every $M_1,M_2 \in {{\mathbb R}}$ and $K$ compact, the closure of $\Theta_{M_1,M_2,K}$ is a closed subset of $A$ and hence it is compact. (VI) Since $\mathcal{I}(x,p,\cdot) - \Lambda(x,p,\cdot)$ is a continuous function on $A$ compact, there exists $M\geq 0$ such that $\mathcal{I}(x,p,a) - \Lambda(x,p,a)\leq M$, for every $a\in A$. (VII) The function $\Lambda - \mathcal{I}$ verifies trivially the continuity estimate due to the Lipschitz property of $f$ and the modulus continuity of $l$.  ◻ We can then conclude that our result extends [@BaCD97 Theorem 3.1]. In the same way it is possible to show that our result extends also the time--dependent case [@BaCD97 Theorem 3.7]. ## Hamiltonians with discontinuous and x dependent cost function As explained in the introduction, cases in which $\mathcal{I}$ is not bounded or not continuous and $\Lambda$ is not Lipschitz or not coercive are not covered by [@BaCD97]. In [@KrSc21], the authors instead treat these cases, but keeping the cost function $\mathcal{I}$ independent of $p$. We would like to emphasize that while many of our assumptions are implied by the assumptions in [@KrSc21], it is not straightforward and clear whether our fifth assumption can be derived from the assumptions in [@KrSc21] (in particular by their assumption ($\mathcal{I}3$)). The inclusion of momentum in the cost function adds an element of complexity, making it challenging to generalize the previous assumption that does not account for momentum. Nevertheless, our approach includes all the examples addressed in [@KrSc21] and more. In particular, we show in the following two examples of Hamiltonians with a discontinuous and $x$ dependent cost function $\mathcal{I}$. The first one was treated in Proposition 5.9 in [@KrSc21] and in Example [Example 20](#example:x_dep_cost-function_workingforkraaijsch){reference-type="ref" reference="example:x_dep_cost-function_workingforkraaijsch"} we show that it also matches our assumptions. The second example is a case of Hamiltonian that can not be addressed by [@KrSc21]. But in Example [Example 21](#example:x_dep_cost-function_workingforus){reference-type="ref" reference="example:x_dep_cost-function_workingforus"} we show that it does fit our assumptions and hence that we can cover more cases then previous works. **Example 20**. *We consider $E={{\mathbb R}}^d$, $F= \{1, \dots, J\}$ and $\Theta=\mathcal{P}(F)$ and $$\begin{aligned} &\Lambda(x,p,\theta) = \sum_{i\in F} \left[\langle a(x,i) p , p \rangle + \langle b(x,i), p \rangle \right]\theta_i \\ &\mathcal{I}(x,\theta) = \sup_{w\in {{\mathbb R}}^J} \sum_{ij} r(i,j,x) \theta_i \left [ 1 - e^{w_j - w_i}\right],\end{aligned}$$ where $a: E \times F \to {{\mathbb R}}^{d\times d}$, $b: E\times F \to {{\mathbb R}}^d$, $r: F^2\times E \to [0,\infty)$ and $\theta_i = \theta(\{i \})$. Then, Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} holds.* *Proof Sketch..* (I) Convexity of $\mathcal{H}$ in $p$ and the fact that $\mathcal{H}(x,0)=0$ is shown in [@KrSc21] Proposition 5.11. (II) $\theta \mapsto \Lambda(x,p,\theta) - \mathcal{I}(x,p,\theta)$ is bounded since $\Lambda$ is bounded and continuous and $\mathcal{I}$ has compact sublevel sets in $\Theta$ as shown in Proposition 5.9 of [@KrSc21]. (III) The existence of a containment function $\Upsilon$ is shown in Proposition 5.11 of [@KrSc21]. Indeed, in [@KrSc21] the authors show that there exists a function $\Upsilon$ that has compact sublevel sets and such that $\Lambda(x,\nabla \Upsilon(x),\theta)$ is bounded from above. By observing that $\Lambda(x,\nabla \Upsilon(x), \theta) - \mathcal{I}(x,\theta)$ is bounded above by $\Lambda(x,\nabla \Upsilon(x),\theta)$, it becomes evident that $\Upsilon$ is also a containment function in the sense of Definition [Definition 14](#def:containment-new){reference-type="ref" reference="def:containment-new"}. (IV) $\mathcal{I}- \Lambda$ is $\Gamma$-convergent as $\mathcal{I}$ is $\Gamma$-convergent (see Proposition 5.9 in [@KrSc21]) and $\Lambda$ is continuous. (V) For every $M_1,M_2\in {{\mathbb R}}$ and every $K$ compact, the closure of $\Theta_{M_1,M_2,K}$ is a closed subset of $\Theta$ and, hence, compact. (VI) In Proposition 5.9 of [@KrSc21] it is shown that there exists a $\theta^0(x,p)$ such that $\mathcal{I}(x,p,\theta^0(x,p))=0$. Hence, taking $g(x,p)=-\Lambda(x,p,\theta^0(x,p))$, it follows that the set $\phi_{g}(x,p)$ is not empty. (VII) $\Lambda - \mathcal{I}$ verifies the continuity estimate since $\Lambda$ verifies the continuity estimate as shown in Proposition 5.11 in [@KrSc21] and $\mathcal{I}$ is equicontinuous by Proposition 5.9 in [@KrSc21].  ◻ The following example is a case of Hamiltonian with an unbounded internal Hamiltonian $\Lambda$ in terms of $\theta$. This leads to a situation where Assumption $(\Lambda 4)$ in [@KrSc21] fails. In the following we prove the validity of our Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"}. The key distinction in our approach, enabling the success of this example while it remained unaddressed in [@KrSc21], lies in our treatment of $\Lambda$ and $\mathcal{I}$ in an integrated whole $\Lambda - \mathcal{I}$ rather than separately and with two different types of assumptions. In this case, while $\Lambda$ is not bounded in terms of $\theta$, the composite $\Lambda - \mathcal{I}$ is bounded. Consequently, in contrast to Assumption $(\Lambda 4)$ in [@KrSc21], which solely considers the boundedness of $\Lambda$ and subsequently fails in this scenario, our Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} [\[item:assumption:regularity:boundness\]](#item:assumption:regularity:boundness){reference-type="ref" reference="item:assumption:regularity:boundness"} instead requires the boundedness of $\Lambda - \mathcal{I}$ and consequently holds. **Example 21**. *Consider $E={{\mathbb R}}^d$ and $\Theta = \mathcal{P}({{\mathbb R}})$ and $$\begin{aligned} &\Lambda(x,p,\theta) = \int_{{\mathbb R}}- x^3 p + \frac{1}{2}(1+|z|) p^2 \, \theta(dz) \\ &\mathcal{I}(x,\theta) = - \inf_{\phi \in C^2({{\mathbb R}})} \int_{{\mathbb R}}\frac{L_x \phi(z)}{\phi(z)} \, \theta(dz), \end{aligned}$$ where $$L_x \phi(z) := - (z - x) \phi ' (z) + \frac{1}{2}\Delta \phi(z) .$$ Then, Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} holds.* *Proof sketch..* In the following we prove Assumptions [\[item:assumption:compact_containment\]](#item:assumption:compact_containment){reference-type="ref" reference="item:assumption:compact_containment"} and [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"}. The proof of the other assumptions are or trivial or similar to the proof for Example [Example 20](#example:x_dep_cost-function_workingforkraaijsch){reference-type="ref" reference="example:x_dep_cost-function_workingforkraaijsch"}. We firstly prove that $$\label{eq:costfunction_bound} \mathcal{I}(x,\theta) \geq -\frac{1}{2} (1+ x^2) + \frac{1}{2}\int_{{\mathbb R}}(x-z)^2 \theta(dz),$$ then, we will proceed with the proof of Assumptions [\[item:assumption:compact_containment\]](#item:assumption:compact_containment){reference-type="ref" reference="item:assumption:compact_containment"} and [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"}. *Proof of [\[eq:costfunction_bound\]](#eq:costfunction_bound){reference-type="eqref" reference="eq:costfunction_bound"}.* Let $\tilde{\phi}(z) = e^{1+\frac{1}{2} z^2}$ and $\xi_n(r) = \frac{n r}{n+r}$ and let $\phi_n = \xi_n \circ \tilde{\phi}$ and call $J(x,\theta)$ the function at the right-hand side of [\[eq:costfunction_bound\]](#eq:costfunction_bound){reference-type="eqref" reference="eq:costfunction_bound"}. Note that $$J(x,\theta) = - \int_{{\mathbb R}}\frac{L_x \tilde{\phi}}{\tilde{\phi}}.$$ Moreover, $$\inf_g \int_{{\mathbb R}}\frac{L_x g}{g} \leq \liminf_n \int_{{\mathbb R}}\frac{L_x \phi_n}{\phi_n}.$$ Consequently, we only need to prove that $$\liminf_n \int_{{\mathbb R}}\frac{L_x \phi_n}{\phi_n} \leq \int_{{\mathbb R}}\frac{L_x \tilde{\phi}}{\tilde{\phi}}.$$ By definition of $L_x$, we have $$\begin{aligned} \label{eq:generator_testfun} L_x \phi_n(z) = & - (z-x) \left(\frac{n^2}{(n+e^{1+1/2 z^2})^2} \right) \tilde{\phi}'(z) + \frac{1}{2} \left(\frac{n^2}{(n+e^{1+1/2 z^2})^2} \right) \tilde{\phi}''(z) \\ & + \frac{1}{2}\left[\frac{-2n^2}{(n+ e^{1+1/2 z^2})^3} z^2 e^{2(1+1/2 z^2)} \right].\end{aligned}$$ Dividing [\[eq:generator_testfun\]](#eq:generator_testfun){reference-type="eqref" reference="eq:generator_testfun"} by $\phi_n$, $$\frac{L_x\phi_n}{\phi_n} = \left(\frac{n^2}{(n+e^{1+1/2 z^2})^2} \right) \frac{L_x \tilde{\phi}}{\phi_n} - \left(\frac{n^2}{(n+e^{1+1/2 z^2})^3} \right) z^2 e^{2(1+1/2 z^2)}.$$ Moreover, $$\begin{aligned} \liminf_n \int_{{\mathbb R}}\frac{L_x \phi_n}{\phi_n} &= \liminf_n \int_{{\mathbb R}}\underbrace{\left(\frac{n^2}{(n+e^{1+1/2 z^2})^2} \right) \frac{L_x \tilde{\phi}}{\phi_n}}_{\downarrow \frac{L_x \tilde{\phi}}{\tilde{\phi}}} + \liminf_n \int_{{\mathbb R}}\underbrace{-\left(\frac{n^2}{(n+e^{1+1/2 z^2})^3} \right) z^2 e^{2(1+1/2 z^2)}}_{\leq 0} \\ &\leq \int_{{\mathbb R}}\frac{L_x \tilde{\phi}}{\tilde{\phi}},\end{aligned}$$ where in the last inequality we used that $\xi_n (r)$ converges to $r$ for $n \to \infty$. We can conclude that $$\begin{aligned} \mathcal{I}(x,\theta) \geq \int_{{\mathbb R}}-(z-x) z + \frac{1}{2} (1+z^2) = \int_{{\mathbb R}}- \frac{1}{2}(1+x^2) + \frac{1}{2}(x - z)^2 \, \theta(dz),\end{aligned}$$ establishing [\[eq:costfunction_bound\]](#eq:costfunction_bound){reference-type="eqref" reference="eq:costfunction_bound"}. *Proof of Assumption [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"}*. We want to prove that $\Theta_{M_1,M_2,K}$ defined in [\[eq:def_levelsets\]](#eq:def_levelsets){reference-type="eqref" reference="eq:def_levelsets"} is relatively compact for all $M_1,M_2$ constants and $K$ compact. To this aim, note that, by definition of $\Theta_{M_1,M_2,K}$ ( [\[eq:def_levelsets\]](#eq:def_levelsets){reference-type="eqref" reference="eq:def_levelsets"} at page ) and by [\[eq:costfunction_bound\]](#eq:costfunction_bound){reference-type="eqref" reference="eq:costfunction_bound"}, we have $$\begin{aligned} \label{eq:subseteq_levelset} \Theta_{M_1,M_2,K} &\subseteq \bigcup_{x,y\in K} \bigcup_{\alpha>1} \left\{ \theta \, | \, \mathcal{I}(x,\theta) - \Lambda(x,\partial_x\frac{\alpha}{2} d^2(x,y),\theta) < M_1\right\} \\ &\subseteq \bigcup_{x,y\in K} \bigcup_{\alpha>1} \left\{ \theta \, | \, J(x,\theta) - \Lambda(x,\partial_x\frac{\alpha}{2} d^2(x,y),\theta) < M_1\right\} := \Theta^*_{M_1,K}.\end{aligned}$$ Moreover, since $J(x,\theta)$ has compact sublevel sets in $\theta$ for $x$ in a compact $K$, and considering the quadratic nature of $J$ together with the linearity of $\Lambda$ in $z$, it follows that the sublevel sets of $J - \Lambda$ are also compact in $\theta$. Consequently, this implies that the set $\Theta^*_{M_1,K}$ is compact. Then, by [\[eq:subseteq_levelset\]](#eq:subseteq_levelset){reference-type="eqref" reference="eq:subseteq_levelset"}, we can conclude that $\Theta_{M_1,M_2,K}$ is relatively compact. *Proof of Assumption [\[item:assumption:compact_containment\]](#item:assumption:compact_containment){reference-type="ref" reference="item:assumption:compact_containment"}.* We prove that $\Upsilon(x) = \frac{1}{2} x^2$ is a containment function. To do so, note that by [\[eq:costfunction_bound\]](#eq:costfunction_bound){reference-type="eqref" reference="eq:costfunction_bound"} we have $$\begin{aligned} \Lambda(x,\nabla \Upsilon(x),\theta) - \mathcal{I}(x,\theta) &\leq - x^4 + \frac{1}{2}\int_{{\mathbb R}}(1+|z|) x^2 \, \theta(dz) + \frac{1}{2} (1+ x^2) - \frac{1}{2}\int_{{\mathbb R}}(x-z)^2 \, \theta(dz) \\ &= x^2 - x^4 + \frac{1}{2} + \frac{1}{2} \int_{{\mathbb R}}|z| x^2 -\frac{1}{2} (x-z)^2 \theta(dz)\\ &= x^2 - x^4 + \frac{1}{2} + \int_{{\mathbb R}}\frac{1}{2}|z| x^2 -\frac{1}{2} x^2 - \frac{1}{2} z^2 +xz \, \theta(dz)\\ &= \frac{1}{2}x^2 - x^4 + \frac{1}{2} + \int_{{\mathbb R}}\frac{1}{2}|z| x^2 - \frac{1}{2} z^2 +\left(x\sqrt{2}\right)\left(\frac{z}{\sqrt{2}}\right) \, \theta(dz) \\ & \leq \frac{1}{2} x^2 - x^4 + \frac{1}{2} + \int_{{\mathbb R}}\frac{1}{4} z^2 + \frac{1}{4} x^4 - \frac{1}{2} z^2 + x^2 + \frac{1}{4}z^4 \, \theta(dz)\\ &= - \frac{3}{4} x^4 + \frac{3}{2}x^2 + \frac{1}{2},\end{aligned}$$ where, in the last inequality we used the rule $ab \leq \frac{1}{2}a^2 +\frac{1}{2}b^2$ for $\frac{1}{2}|z|x^2$ and $xz$. Note that the last line is bounded from above for all $x$ and $\theta$. This concludes the proof. ◻ ## Hamiltonian with x and p dependent cost function Although [@KrSc21] effectively covers a greater number of cases than [@BaCD97], as explained in the previous subsection, it is unable to address scenarios where the cost function depends on momenta $p$. For instance, in [@DecoKr] and [@Po18], it is proved comparison principle with two different \"ad hoc\" proofs involving coercivity or Lipschitz estimates and optimization problems, for an Hamilton--Jacobi--Bellman equation with an Hamiltonian of the type as in [\[def:hamiltonian\]](#def:hamiltonian){reference-type="eqref" reference="def:hamiltonian"}. These examples does not fall into the cases of [@KrSc21] due to the presence of $p$ in $\mathcal{I}$. Indeed, assumptions as $(\mathcal{I}5)$ and $(\mathcal{I}4)$ in [@KrSc21] are not satisfied and quite challenging to modify to incorporate the momenta $p$. Our assumptions are instead satisfied. We also want to mention that in [@FK06] it is possible to find some examples of Hamiltonians with cost function depending on $p$. We confidently assert that our results can cover all these examples. We can conclude, then, that our work can be used for a large class of examples including examples that fall into previously treated theories, as the two examples above, as well as those that have been addressed by means of \"ad hoc\" proofs, as the examples in [@DecoKr] and [@Po18], and cases left unexplored as the one that we study in Section [6](#section:verification-for-examples-of-Hamiltonians){reference-type="ref" reference="section:verification-for-examples-of-Hamiltonians"}. ## Examples to clarify Assumption [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"} In this subsection, we want to clarify our Assumption [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"} using two straightforward examples. In the introduction we explained that the first three assumptions are standard in the proof of comparison principle and that the $\Gamma$ - convergence and assumption [\[item:assumption:statinary_meas\]](#item:assumption:statinary_meas){reference-type="ref" reference="item:assumption:statinary_meas"} is needed to prove continuity of the Hamiltonian. We show now that Assumption [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"} is a key assumption. Indeed, the continuity estimate of $\Lambda - \mathcal{I}$ is morally equivalent to the comparison principle for $\Lambda - \mathcal{I}$ for well chosen $\theta$. Thus, one is enable to control the difference of Hamiltonians, and hence to prove comparison principle for $\mathcal{H}$ by controlling the difference of $\Lambda- \mathcal{I}$, if it is also possible to control the variable $\theta$, i.e., if assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"} holds. In the first example we will show how assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"} may not hold and subsequently lead to the failure of the comparison principle. In the second example, we show how to get Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"} and, hence, comparison. **Example 22**. *Consider $E= {{\mathbb R}}^2$, $\Theta = {{\mathbb R}}$ and for $x=(x_1,x_2)\in E$ and $p=(p_1,p_2) \in {{\mathbb R}}^2$ consider $$\Lambda(x,p,\theta)= \frac{1}{2} a(x) p_1 ^2 + b(x,\theta) p_1,$$ and $$\mathcal{I}(p,\theta) = |p_2 - \theta|.$$ Then, Assumption [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"} fails.* *Proof sketch..* In a standard comparison principle proof, one usually finds two sequences of points $x_{\alpha,\varepsilon}$ and $y_{\alpha,\varepsilon}$, optimizers in a doubling variable procedure, and the corresponding momenta $p_{\alpha,\varepsilon}$. Assuming that, for some $\theta_{\alpha,\varepsilon}$ and all fixed small $\varepsilon>0$, $$\liminf_{\alpha\to\infty}( \Lambda - \mathcal{I})( x_{\alpha,\varepsilon}, p_{\alpha,\varepsilon}, \theta_{\alpha,\varepsilon}) - ( \Lambda - \mathcal{I})( y_{\alpha,\varepsilon}, p_{\alpha,\varepsilon}, \theta_{\alpha,\varepsilon}) \leq 0,$$ i.e., that morally comparison holds for $\Lambda - \mathcal{I}$ along $\theta_{\alpha,\varepsilon}$, one wants then use this bound to prove comparison for $\mathcal{H}$. However, for $p_1\neq p_2$, Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"} fails since it is not possible to control the $\theta_{\alpha,\varepsilon}$. Indeed, using the subsolution and supersolution properties, one is able to show that $$\begin{aligned} \label{example:subsolution} \sup_\alpha (\mathcal{I}- \Lambda)(x_{\alpha,\varepsilon}, p_{\alpha,\varepsilon}, \theta_{\alpha,\varepsilon}) < \infty,\end{aligned}$$ $$\label{example:supersolution} \sup_\alpha (\Lambda - \mathcal{I}) (y_{\alpha,\varepsilon}, p_{\alpha,\varepsilon}, \theta_{\alpha,\varepsilon}) < \infty.$$ However, with the above bounds, in this case, we can not conclude any control on the the sequence $\theta_{\alpha,\varepsilon}$. Indeed, if for example $p_{2,\varepsilon,\alpha}$ blows up, $\theta_{\alpha,\varepsilon}$ consequentially does the same in order to have [\[example:subsolution\]](#example:subsolution){reference-type="eqref" reference="example:subsolution"}. Using this observation, we can conclude that comparison principle in this case fails. ◻ In the following example we show how to make Assumption [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"} work. **Example 23**. *We consider $E={{\mathbb R}}$, $\Theta = {{\mathbb R}}$ and for $x\in E$ and $p\in {{\mathbb R}}$ consider $$\Lambda(x,p,\theta)=\frac{1}{2} a(x) p^2 + b(x,\theta)p,$$ and $$\mathcal{I}(p,\theta) = |p-\theta|.$$ Assumption [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"} holds.* *Proof sketch..* The Hamiltonian now is coercive as $\Lambda$ controls the order of $p$. This means that $$\lim_{|p| \to \infty} \Lambda(x,p,\theta) - \mathcal{I}(x,p,\theta) = \infty.$$ In this case, then, by coercivity and [\[example:supersolution\]](#example:supersolution){reference-type="eqref" reference="example:supersolution"}, $p_{\alpha,\varepsilon}$ can not blow up. Using this fact and the bound [\[example:subsolution\]](#example:subsolution){reference-type="eqref" reference="example:subsolution"}, we can conclude that $\theta_{\alpha,\varepsilon}$ can not go to infinity as well. In this case assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"} holds and comparison as well. ◻ *Remark 24*. Note that the Hamiltonian in Examples [Example 22](#example:assumptionV_notworking){reference-type="ref" reference="example:assumptionV_notworking"} and [Example 23](#example:assumptionV_working){reference-type="ref" reference="example:assumptionV_working"} is not convex in $p$. For the sake of clarity, we used this example as it is easy to see the role of Assumption [\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"}. Replacing the above $\mathcal{I}$ with the Donsker-Varadhan rate function for the Ornstein--Uhlenbeck diffusion process centered in $p_2$, one does get a convex Hamiltonian for which the above argument applies. # Regularity of the Hamiltonian {#section:regularity-of-H} We start showing that the Hamiltonian is continuous. This is the content of the following proposition. **Proposition 25** (Continuity of the Hamiltonian). *Let $\mathcal{H} : E \times \mathbb{R}^d\to \mathbb{R}$ be the Hamiltonian defined in [\[eq:results:variational_hamiltonian\]](#eq:results:variational_hamiltonian){reference-type="eqref" reference="eq:results:variational_hamiltonian"}, and suppose that Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} is satisfied. Then the map $(x,p) \mapsto \mathcal{H}(x,p)$ is continuous and the Lagrangian $(x,v) \mapsto \mathcal{L}(x,v) := \sup_{p} \left \langle p,v \right \rangle - \mathcal{H}(x,p)$ is lower semi-continuous.* We will use the following technical result to establish upper semi-continuity of $\mathcal{H}$. **Lemma 26** (Lemma 17.30 in [@AlBo06]). *Let $\mathcal{X}$ and $\mathcal{Y}$ be two Polish spaces. Let $\phi : \mathcal{X}\rightarrow \mathcal{K}(\mathcal{Y})$, where $\mathcal{K}(\mathcal{Y})$ is the space of non-empty compact subsets of $\mathcal{Y}$. Suppose that $\phi$ is upper hemi-continuous, that is if $x_n \rightarrow x$ and $y_n \rightarrow y$ and $y_n \in \phi(x_n)$, then $y \in \phi(x)$.* *Let $f : \text{Graph} (\phi) \rightarrow \mathbb{R}$ be upper semi-continuous. Then the map $m(x) = \sup_{y \in \phi(x)} f(x,y)$ is upper semi-continuous.* *Proof of Proposition [Proposition 25](#prop:reg-of-H-and-L:continuity){reference-type="ref" reference="prop:reg-of-H-and-L:continuity"}.* We start by establishing the upper semi-continuity arguing on the basis of Lemma [Lemma 26](#lemma:upper_semi_continuity_abstract){reference-type="ref" reference="lemma:upper_semi_continuity_abstract"}. Firstly, note that $f(x,p,\theta)=\Lambda(x,p,\theta)-\mathcal{I}(x,p,\theta)$ is upper semi-continuous since, by Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} [\[item:assumption:Gamma-convergence\]](#item:assumption:Gamma-convergence){reference-type="ref" reference="item:assumption:Gamma-convergence"}, $-f(x,p,\theta)= \mathcal{I}(x,p,\theta)-\Lambda(x,p,\theta)$ is lower semi-continuous. Moreover, by Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} [\[item:assumption:statinary_meas\]](#item:assumption:statinary_meas){reference-type="ref" reference="item:assumption:statinary_meas"}, for all small neighborhood of $x$ and $p$ there exists a continuous function $g$ in these neighborhoods such that there exists $\theta^0(x,p)$ for which $\mathcal{I}(x,p,\theta^0(x,p)) - \Lambda(x,p,\theta^0(x,p)) \leq g(x,p)$. Hence, we can write the supremum over $\theta \in \Theta$ as the supremum over $\theta \in \phi(x,p)$ where $$\phi_g(x,p)=\overline{\left\{ \theta \in \Theta \, \vert \, \mathcal{I}(x,p,\theta)-\Lambda(x,p,\theta)\leq g(x,p) \right\}}.$$ $\phi_g(x,p)$ is non empty, since $\theta^0(x,p) \in \phi_g(x,p)$, and it is compact for Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} [\[item:assumption:statinary_meas\]](#item:assumption:statinary_meas){reference-type="ref" reference="item:assumption:statinary_meas"}. We are left to show that $\phi$ is upper hemi-continuous. Thus, let $(x_n,p_n) \rightarrow (x,p)$ and $\theta_n \rightarrow \theta$ with $\theta_n \in \phi_g(x_n,p_n)$. We establish that $\theta \in \phi_g(x,p)$. By the definition of $\phi_g(x_n,p_n)$, $\mathcal{I}(x_n,p_n,\theta_n)- \Lambda(x_n, p_n, \theta_n)\leq g(x,p)$. Then, by the lower semi-continuity of $\mathcal{I}-\Lambda$, we can write $$\begin{aligned} \mathcal{I}(x,p,\theta) - \Lambda(x,p,\theta) &\leq \liminf_n \mathcal{I}(x_n,p_n,\theta_n) - \Lambda(x_n,p_n,\theta_n) \leq \liminf_{n} g(x_n,p_n)=g(x,p), \end{aligned}$$ which implies indeed that $\theta \in \phi_g(x,p)$. Thus, upper semi-continuity follows by an application of Lemma [Lemma 26](#lemma:upper_semi_continuity_abstract){reference-type="ref" reference="lemma:upper_semi_continuity_abstract"}. We prove now the lower semi--continuity of $\mathcal{H}$. Precisely, we want to show that if $(x_n,p_n)\rightarrow (x,p)$ then $\liminf_n \mathcal{H}(x_n,p_n)\geq \mathcal{H}(x,p)$. Let $\theta\in\Theta$ be such that $\mathcal{H}(x,p)=\Lambda(x,p,\theta)-\mathcal{I}(x,p,\theta)$. By Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"}[\[item:assumption:Gamma-convergence\]](#item:assumption:Gamma-convergence){reference-type="ref" reference="item:assumption:Gamma-convergence"}, the function $\mathcal{I}-\Lambda$ is $\Gamma$--convergent in the sense of Definition [Definition 15](#def:Gamma-convergence){reference-type="ref" reference="def:Gamma-convergence"}. That means that there exist $\theta_n$ converging to $\theta$ such that $\limsup_n \mathcal{I}(x_n,p_n,\theta_n)- \Lambda(x_n,p_n,\theta_n) \leq \mathcal{I}(x,p,\theta)-\Lambda(x,p,\theta)$. Therefore, $$\begin{aligned} \liminf_n \mathcal{H}(x_n,p_n)&\geq \liminf_n \left[\Lambda(x_n,p_n,\theta_n) - \mathcal{I}(x_n,p_n,\theta_n)\right]\\ &= -\limsup_n \left[\mathcal{I}(x_n,p_n,\theta_n)-\Lambda(x_n,p_n,\theta_n) \right]\geq - \left[\mathcal{I}(x,p,\theta)- \Lambda(x,p,\theta) \right]\\ &=\mathcal{H}(x,p),\end{aligned}$$ establishing the lower semi-continuity of $\mathcal{H}$ and, hence, the continuity. Moreover, since the Lagrangian $\mathcal{L}$ is the Legendre transform of $\mathcal{H}$, it is lower semi-continuous. ◻ # Proofs of the main theorems {#section:comparison_principle} In this section we establish Theorem [Theorem 8](#theorem:comparison_principle_variational){reference-type="ref" reference="theorem:comparison_principle_variational"} and Theorem [Theorem 10](#theorem:existence_of_viscosity_solution){reference-type="ref" reference="theorem:existence_of_viscosity_solution"}. To prove the comparison principle for $f-\lambda \mathbf{H}f= h$ and $\partial_t f - \mathbf{H}f = 0$, we relate them to a set of Hamilton--Jacobi--Bellman equation with Hamiltonians constructed from $\mathbf{H}$. To do this, we introduce two operators $H_\dagger$ and $H_\ddagger$ that will be respectively an upper and lower bound for $\mathbf{H}$. The two new Hamiltonians, defined in Subsection [5.1](#subsection:definition_of_Hamiltonians){reference-type="ref" reference="subsection:definition_of_Hamiltonians"}, are constructed in terms of a containment function $\Upsilon$ that allows us to restrict our analysis to a compact set. Schematically, we will establish the diagram in Figure [\[SF:fig:CP-diagram-in-proof-of-CP\]](#SF:fig:CP-diagram-in-proof-of-CP){reference-type="ref" reference="SF:fig:CP-diagram-in-proof-of-CP"}. The arrows will be established in Subsection [5.1](#subsection:definition_of_Hamiltonians){reference-type="ref" reference="subsection:definition_of_Hamiltonians"}. Finally, we will establish the comparison principle for $H_\dagger$ and $H_\ddagger$ in Subsection [5.2](#subsection:proof_of_comparison_principle){reference-type="ref" reference="subsection:proof_of_comparison_principle"}. The combination of these two results imply the comparison principle for $\mathbf{H}$ as shown in the following. *Proof of Theorem [Theorem 8](#theorem:comparison_principle_variational){reference-type="ref" reference="theorem:comparison_principle_variational"}.* We only prove the first item. The proof for the time-dependent case follows the same lines. Fix $h_1,h_2 \in C_b(E)$ and $\lambda > 0$. Let $u,v$ be a viscosity sub- and supersolution to $f - \lambda \mathbf{H}f = h_1$ and $f - \lambda \mathbf{H}f = h_2$ respectively. By Lemma [Lemma 29](#lemma:viscosity_solutions_compactify2){reference-type="ref" reference="lemma:viscosity_solutions_compactify2"} proven in Section [5.1](#subsection:definition_of_Hamiltonians){reference-type="ref" reference="subsection:definition_of_Hamiltonians"}, $u$ and $v$ are a sub- and supersolution to $f - \lambda H_\dagger f = h_1$ and $f - \lambda H_\ddagger f = h_2$ respectively. Thus $\sup_E u - v \leq \sup_E h_1 - h_2$ by Proposition [Proposition 30](#prop:CP){reference-type="ref" reference="prop:CP"} of Section [5.2](#subsection:proof_of_comparison_principle){reference-type="ref" reference="subsection:proof_of_comparison_principle"}. ◻ ## Auxiliary operators {#subsection:definition_of_Hamiltonians} In this section, we repeat the definition of $\mathbf{H}$, and introduce the operators $H_\dagger$ and $H_\ddagger$. **Definition 27**. The operator $\mathbf{H}\subseteq C_b^1(E) \times C_b(E)$ has domain $\mathcal{D}(\mathbf{H}) = C_{cc}^\infty(E)$ and satisfies $\mathbf{H}f(x) = \mathcal{H}(x, \nabla f(x))$, where $\mathcal{H}$ is the map $$\mathcal{H}(x,p) = \sup_{\theta \in \Theta}\left[\Lambda(x,p,\theta) - \mathcal{I}(x,p,\theta)\right].$$ We proceed by introducing $H_\dagger$ and $H_\ddagger$ serving as natural upper and lower bounds to $\mathbf{H}$. Recall Assumption [\[item:assumption:compact_containment\]](#item:assumption:compact_containment){reference-type="ref" reference="item:assumption:compact_containment"} and the constant $C_\Upsilon := \sup_{\theta}\sup_x \Lambda(x,\nabla \Upsilon(x),\theta)- \mathcal{I}(x,\nabla \Upsilon(x),\theta)$ therein. Denote by $C_\ell^\infty(E)$ the set of smooth functions on $E$ that have a lower bound and by $C_u^\infty(E)$ the set of smooth functions on $E$ that have an upper bound. The definitions below are motivated by the convexity of the map $p \mapsto \mathcal{H}(x,p)$. **Definition 28** (The operators $H_\dagger$ and $H_\ddagger$). For $f \in C_\ell^\infty(E)$ and $\varepsilon \in (0,1)$ set $$\begin{gathered} f^\varepsilon_\dagger := (1-\varepsilon) f + \varepsilon \Upsilon \\ H_{\dagger,f}^\varepsilon(x) := (1-\varepsilon) \mathcal{H}(x,\nabla f(x)) + \varepsilon C_\Upsilon. \end{gathered}$$ and set $$H_\dagger := \left\{(f^\varepsilon_\dagger,H_{\dagger,f}^\varepsilon) \, \middle| \, f \in C_\ell^\infty(E), \varepsilon \in (0,1) \right\}.$$ For $f \in C_u^\infty(E)$ and $\varepsilon \in (0,1)$ set $$\begin{gathered} f^\varepsilon_\ddagger := (1+\varepsilon) f - \varepsilon \Upsilon \\ H_{\ddagger,f}^\varepsilon(x) := (1+\varepsilon) \mathcal{H}(x,\nabla f(x)) - \varepsilon C_\Upsilon. \end{gathered}$$ and set $$H_\ddagger := \left\{(f^\varepsilon_\ddagger,H_{\ddagger,f}^\varepsilon) \, \middle| \, f \in C_u^\infty(E), \varepsilon \in (0,1) \right\}.$$ The operator $\mathbf{H}$ is related to $H_\dagger, H_\ddagger$ by the following Lemma whose proof is standard and can be found for example in [@KrSc21]. We include it for completeness. **Lemma 29**. *Fix $\lambda > 0$ and $h \in C_b(E)$.* (a) *Every subsolution to $f - \lambda \mathbf{H}f = h$ is also a subsolution to $f - \lambda H_\dagger f = h$.* (b) *Every supersolution to $f - \lambda \mathbf{H}f = h$ is also a supersolution to $f-\lambda H_\ddagger f=~h$.* (c) *Every subsolution to $\partial_t f - \mathbf{H}f = 0$ is also a subsolution to $\partial_t f - H_\dagger f = 0$.* (d) *Every supersolution to $\partial_t f - \mathbf{H}f = 0$ is also a supersolution to $\partial_t f - H_\ddagger f=0$.* *Proof.* We only prove (a) as the other claims can be carried out analogously. Fix $\lambda > 0$ and $h \in C_b(E)$. Let $u$ be a subsolution to $f - \lambda \mathbf{H}f = h$. We prove it is also a subsolution to $f - \lambda H_\dagger f = h$. Fix $\varepsilon > 0$ and $f\in C_\ell^\infty(E)$ and let $(f^\varepsilon_\dagger,H^\varepsilon_{\dagger,f}) \in H_\dagger$ as in Definition [Definition 28](#definiton:HdaggerHddagger){reference-type="ref" reference="definiton:HdaggerHddagger"}. We will prove that there are $x_n\in E$ such that $$\begin{gathered} \lim_{n\to\infty}\left(u-f_\dagger^\varepsilon\right)(x_n) = \sup_{x\in E}\left(u(x)-f_\dagger^\varepsilon(x) \right),\label{eqn:proof_lemma_conditions_for_subsolution_first}\\ \limsup_{n\to\infty} \left[u(x_n)-\lambda H_{\dagger,f}^\varepsilon(x_n) - h(x_n)\right]\leq 0.\label{eqn:proof_lemma_conditions_for_subsolution_second} \end{gathered}$$ As the function $\left[u -(1-\varepsilon)f\right]$ is bounded from above and $\varepsilon \Upsilon$ has compact sublevel-sets, the sequence $x_n$ along which the first limit is attained can be assumed to lie in the compact set $$K := \left\{x \, | \, \Upsilon(x) \leq \varepsilon^{-1} \sup_x \left(u(x) - (1-\varepsilon)f(x) \right)\right\}.$$ Set $M = \varepsilon^{-1} \sup_x \left(u(x) - (1-\varepsilon)f(x) \right)$. Let $\gamma : \mathbb{R}\rightarrow \mathbb{R}$ be a smooth increasing function such that $$\gamma(r) = \begin{cases} r & \text{if } r \leq M, \\ M + 1 & \text{if } r \geq M+2. \end{cases}$$ Denote by $f_\varepsilon$ the function on $E$ defined by $$f_\varepsilon(x) := \gamma\left((1-\varepsilon)f(x) + \varepsilon \Upsilon(x) \right).$$ By construction $f_\varepsilon$ is smooth and constant outside of a compact set and thus lies in $\mathcal{D}(H) = C_{cc}^\infty(E)$. As $u$ is a viscosity subsolution for $f - \lambda Hf = h$ there exists a sequence $x_n \in K \subseteq E$ (by our choice of $K$) with $$\begin{gathered} \lim_n \left(u-f_\varepsilon\right)(x_n) = \sup_x \left(u(x)-f_\varepsilon(x)\right), \label{eqn:visc_subsol_sup} \\ \limsup_n \left[u(x_n) - \lambda \mathbf{H} f_\varepsilon(x_n) - h(x_n)\right] \leq 0. \label{eqn:visc_subsol_upperbound} \end{gathered}$$ As $f_\varepsilon$ equals $f_\dagger^\varepsilon$ on $K$, we have from [\[eqn:visc_subsol_sup\]](#eqn:visc_subsol_sup){reference-type="eqref" reference="eqn:visc_subsol_sup"} that also $$\lim_n \left(u-f_\dagger^\varepsilon\right)(x_n) = \sup_{x\in E}\left(u(x)-f_\dagger^\varepsilon(x)\right),$$ establishing [\[eqn:proof_lemma_conditions_for_subsolution_first\]](#eqn:proof_lemma_conditions_for_subsolution_first){reference-type="eqref" reference="eqn:proof_lemma_conditions_for_subsolution_first"}. Convexity of $p \mapsto \mathcal{H}(x,p)$ yields for arbitrary points $x\in K$ the estimate $$\begin{aligned} \mathbf{H} f_\varepsilon(x) &= \mathcal{H}(x,\nabla f_\varepsilon(x)) \\ & \leq (1-\varepsilon) \mathcal{H}(x,\nabla f(x)) + \varepsilon \mathcal{H}(x,\nabla \Upsilon(x)) \\ &\leq (1-\varepsilon) \mathcal{H}(x,\nabla f(x)) + \varepsilon C_\Upsilon = H^\varepsilon_{\dagger,f}(x). \end{aligned}$$ Combining this inequality with [\[eqn:visc_subsol_upperbound\]](#eqn:visc_subsol_upperbound){reference-type="eqref" reference="eqn:visc_subsol_upperbound"} yields $$\limsup_n \left[u(x_n) - \lambda H^\varepsilon_{\dagger,f}(x_n) - h(x_n)\right] \leq \limsup_n \left[u(x_n) - \lambda \mathbf{H} f_\varepsilon(x_n) - h(x_n)\right] \leq 0,$$ establishing [\[eqn:proof_lemma_conditions_for_subsolution_second\]](#eqn:proof_lemma_conditions_for_subsolution_second){reference-type="eqref" reference="eqn:proof_lemma_conditions_for_subsolution_second"}. This concludes the proof. ◻ ## The comparison principle {#subsection:proof_of_comparison_principle} In the following we prove the comparison principle for the operators $H_\dagger$ and $H_\ddagger$. **Proposition 30**. *Fix $\lambda > 0$ and $h_1,h_2 \in C_b(E)$. The following holds:* (a) *Let $u$ be a viscosity subsolution to $f - \lambda H_\dagger f = h_1$ and let $v$ be a viscosity supersolution to $f - \lambda H_\ddagger f = h_2$. Then we have $\sup_x u(x) - v(x) \leq \sup_x h_1(x) - h_2(x)$.* (b) *Let $u$ be a viscosity subsolution to $\partial_t f - H_\dagger f = 0$ and let $v$ be a viscosity supersolution to $\partial_t f - H_\ddagger f = 0$. Then we have $\sup_{t\in [0,T], x} u(x,t) - v(x,t) \leq \sup_x{ u(x,0) - v(x,0)}$ for all $T>0$.* The strategy of the proof is the same for both equations. In both cases, the aim is to prove that it is possible to bound the difference of the Hamiltonians, in well-chosen sequences of points, by using the continuity estimate for $\Lambda - \mathcal{I}$. This is the content of Proposition [Proposition 33](#prop:continuity_estimate){reference-type="ref" reference="prop:continuity_estimate"}. With this aim, we use two variants of a classical estimate, that was proven e.g. in [@CIL92 Proposition 3.7], given respectively in Lemma [Lemma 31](#lemma:doubling_lemma){reference-type="ref" reference="lemma:doubling_lemma"} for the stationary equation and Lemma [Lemma 32](#lemma:quadrupling_lemma){reference-type="ref" reference="lemma:quadrupling_lemma"} for the evolutionary case. **Lemma 31**. *Let $u$ be bounded and upper semi-continuous, let $v$ be bounded and lower semi-continuous and let $\Upsilon$ be a containment function.* *Fix $\varepsilon > 0$. For every $\alpha >0$ there exists $(x_\alpha,y_\alpha)=(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon}) \in E \times E$ such that $$\begin{gathered} \frac{u(x_{\alpha})}{1-\varepsilon} - \frac{v(y_{\alpha})}{1+\varepsilon} - \frac{\alpha}{2} d^2(x_{\alpha},y_{\alpha}) - \frac{\varepsilon}{1-\varepsilon}\Upsilon(x_{\alpha}) -\frac{\varepsilon}{1+\varepsilon}\Upsilon(y_{\alpha}) \\ = \sup_{x,y \in E} \left\{\frac{u(x)}{1-\varepsilon} - \frac{v(y)}{1+\varepsilon} - \frac{\alpha }{2}d^2(x,y) - \frac{\varepsilon}{1-\varepsilon}\Upsilon(x) - \frac{\varepsilon}{1+\varepsilon}\Upsilon(y)\right\}. \end{gathered}$$ Additionally, for every $\varepsilon > 0$ we have that* (a) *The set $\{x_{\alpha}, y_{\alpha} \, | \, \alpha > 0\}$ is relatively compact in $E$.* (b) *All limit points of $\{(x_{\alpha},y_{\alpha})\}_{\alpha > 0}$ as $\alpha \rightarrow \infty$ are of the form $(z,z)$ and for these limit points we have $\frac{u(z)}{1-\varepsilon} - \frac{v(z)}{1+\varepsilon} - \frac{2\varepsilon}{1-\varepsilon^2}\Upsilon(z) = \sup_{x \in E} \left\{\frac{u(x)}{1-\varepsilon} - \frac{v(x)}{1+\varepsilon} - \frac{2\varepsilon}{1-\varepsilon^2}\Upsilon(x) \right\}$.* (c) *We have $$\lim_{\alpha \rightarrow \infty} \alpha d^2(x_{\alpha},y_{\alpha}) = 0.$$* **Lemma 32**. *Let $u$ be bounded and upper semi-continuous, let $v$ be bounded and lower semi-continuous and let $\Upsilon$ be a containment function.* *Fix $\varepsilon > 0$, $\beta>0$ and $T>0$. For every $\alpha >0$ and $\gamma >0$ there exists $$(x_{\alpha,\gamma},t_{\alpha,\gamma}, y_{\alpha,\gamma}, s_{\alpha,\gamma}) = (x_{\alpha,\gamma,\varepsilon,\beta}, t_{\alpha,\gamma,\varepsilon,\beta}, y_{\alpha,\gamma,\varepsilon,\beta}, s_{\alpha,\gamma,\varepsilon,\beta})\in E\times [0,T], \times E \times [0,T]$$ such that $$\begin{gathered} \frac{u(t_{\alpha,\gamma}, x_{\alpha,\gamma})}{1-\varepsilon} - \frac{v(s_{\alpha,\gamma},y_{\alpha,\gamma})}{1+\varepsilon} - \frac{\alpha}{2}d^2(x_{\alpha,\gamma},y_{\alpha,\gamma}) - \frac{\gamma}{2}(s_{\alpha,\gamma}-t_{\alpha,\gamma})^2 - \frac{\varepsilon}{1-\varepsilon} \Upsilon(x_{\alpha,\gamma}) - \frac{\varepsilon}{1+\varepsilon}\Upsilon(y_{\alpha,\gamma}) \\ - \frac{\beta}{2} (t_{\alpha,\gamma}+s_{\alpha,\gamma}) + \beta T \\ = \sup_{s,t \in [0,T],x,y} \biggl\{\frac{u(t,x)}{1-\varepsilon} - \frac{v(s,y)}{1+\varepsilon} - \frac{\alpha}{2}d^2(x,y) - \frac{\alpha}{2}(s-t)^2 - \frac{\varepsilon}{1-\varepsilon} \Upsilon(x) - \frac{\varepsilon}{1+\varepsilon}\Upsilon(y) - \frac{\beta}{2} (t+s) + \beta T \biggr\}. \end{gathered}$$ Additionally, for every $\varepsilon > 0$ and $\beta>0$ we have that* (a) *For any $\gamma>0$,* (i) *the set $\{x_{\alpha,\gamma}, y_{\alpha,\gamma} \, | \, \alpha > 0\}$ is relatively compact in $E$,* (ii) *we have $$\lim_{\alpha \rightarrow \infty} \alpha d^2(x_{\alpha,\gamma},y_{\alpha,\gamma}) = 0,$$* (iii) *all limit points of $\{(x_{\alpha,\gamma},y_{\alpha,\gamma},t_{\alpha,\gamma},s_{\alpha,\gamma})\}_{\alpha > 0}$ as $\alpha \rightarrow \infty$ are of the form $(z_{\gamma},z_{\gamma},t_\gamma,s_\gamma)$.* (b) *Let $(z_\gamma,z_\gamma,t_\gamma,s_\gamma)$ be a limit point as in (a)(iii). Then* (i) *the set $\{z_\gamma \, | \, \gamma >0\}$ is relatively compact in $E$,* (ii) *all limit point of $\{(z_\gamma,z_\gamma,t_\gamma,s_\gamma)\}_{\gamma>0}$ as $\gamma \rightarrow \infty$ are of the form $(z,z,w,w)$ and for these limit points we have $$\frac{u(w,z)}{1-\varepsilon} - \frac{v(w,z)}{1+\varepsilon} - \frac{2\varepsilon}{1-\varepsilon^2}\Upsilon(z) - \beta ( T - w) = \sup_{x \in E, t\in[0,T]} \left\{\frac{u(t,x)}{1-\varepsilon} - \frac{v(t,x)}{1+\varepsilon} -\frac{2\varepsilon}{1-\varepsilon^2}\Upsilon (x) - \beta (T - t) \right\}.$$* In the following proposition we prove the continuity estimate for $\mathcal{H}$ by using the continuity estimate of $\Lambda - \mathcal{I}$. **Proposition 33**. *Consider $(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon})$ found in Lemma [Lemma 31](#lemma:doubling_lemma){reference-type="ref" reference="lemma:doubling_lemma"} or Lemma [Lemma 32](#lemma:quadrupling_lemma){reference-type="ref" reference="lemma:quadrupling_lemma"} (for which we fix $\gamma$ and $\beta$) and denote $p^1_{\alpha,\varepsilon} := \alpha \mathrm{d}_x \frac{1}{2}d^2(\cdot,y_{\alpha,\varepsilon})(x_{\alpha,\varepsilon})$ and $p^2_{\alpha,\varepsilon} := - \alpha \mathrm{d}_y \frac{1}{2}d^2(x_{\alpha,\varepsilon},\cdot)(y_{\alpha,\varepsilon})$. Suppose that $$\label{subsolution_bound} \inf_\alpha \mathcal{H}(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon}) > - \infty$$ and $$\label{supersolution_bound} \sup_\alpha \mathcal{H}(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon}) < \infty.$$* *Then, for all $\varepsilon>0$ there exists a sequence $\alpha(\varepsilon)\to \infty$, such that $$\label{eqn:difference_hamiltonian} \liminf_{\varepsilon\to 0}\liminf_{\alpha\to\infty}\mathcal{H}(x_\alpha,p^1_{\alpha,\varepsilon})- \mathcal{H}(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon})\leq 0.$$* *Proof.* We only prove the statement for $(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon})$ found in Lemma [Lemma 31](#lemma:doubling_lemma){reference-type="ref" reference="lemma:doubling_lemma"}. The proof in the context of Lemma [Lemma 32](#lemma:quadrupling_lemma){reference-type="ref" reference="lemma:quadrupling_lemma"} is analogous. The proof is given in two steps. We sketch the steps, before giving full proof. [*Step 1*]{.ul}: We will show that there are controls $\theta_{\alpha,\varepsilon}$ such that $$\label{eqn:choice_control} \mathcal{H}(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon}) = \Lambda(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}) - \mathcal{I}(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}).$$ As a consequence we have $$\begin{gathered} \label{eqn:basic_decomposition_Hamiltonian_difference1} \mathcal{H}(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon})- \mathcal{H}(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon}) \leq \Lambda(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon})- \Lambda(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon})\\ +\mathcal{I}(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon})- \mathcal{I}(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}). \end{gathered}$$ For establishing [\[eqn:difference_hamiltonian\]](#eqn:difference_hamiltonian){reference-type="eqref" reference="eqn:difference_hamiltonian"}, it is sufficient to bound the differences in [\[eqn:basic_decomposition_Hamiltonian_difference1\]](#eqn:basic_decomposition_Hamiltonian_difference1){reference-type="eqref" reference="eqn:basic_decomposition_Hamiltonian_difference1"} by using Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"}[\[item:assumption:regularity:continuity_estimate\]](#item:assumption:regularity:continuity_estimate){reference-type="ref" reference="item:assumption:regularity:continuity_estimate"}. [*Step 2*]{.ul}: We verify the conditions to apply the continuity estimate, Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"} [\[item:assumption:regularity:continuity_estimate\]](#item:assumption:regularity:continuity_estimate){reference-type="ref" reference="item:assumption:regularity:continuity_estimate"} which then concludes the proof. [*Proof of Step 1*]{.ul}: Recall that $\mathcal{H}(x,p)$ is given by $$\mathcal{H}(x,p) = \sup_{\theta \in \Theta}\left[\Lambda(x,p,\theta) - \mathcal{I}(x,p,\theta) \right].$$ Since $\Lambda(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\cdot) - \mathcal{I}(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\cdot) : \Theta \to \mathbb{R}$ is upper semi-continuous and bounded by [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"}[\[item:assumption:Gamma-convergence\]](#item:assumption:Gamma-convergence){reference-type="ref" reference="item:assumption:Gamma-convergence"} and [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"}[\[item:assumption:regularity:boundness\]](#item:assumption:regularity:boundness){reference-type="ref" reference="item:assumption:regularity:boundness"}, there exists an optimizer $\theta_{\alpha,\varepsilon} \in\Theta$ such that $$\label{eqn:choice_of_optimal_measure} \mathcal{H}(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon}) = \Lambda(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}) - \mathcal{I}(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}) .$$ Choosing the same point in the supremum of the second term $\mathcal{H}(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon})$, we obtain for all $\varepsilon > 0$ and $\alpha > 0$ the estimate $$\begin{gathered} \label{eqn:basic_decomposition_Hamiltonian_difference} \mathcal{H}(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon})- \mathcal{H}(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon}) \leq \Lambda(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon})- \Lambda(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon})\\ + \mathcal{I}(y_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon})- \mathcal{I}(x_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}) . \end{gathered}$$ [*Proof of Step 2*]{.ul}: We will construct for each $\varepsilon > 0$ a sequence $\alpha = \alpha(\varepsilon) \rightarrow \infty$ such that the collection $(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon})$ is fundamental for $\Lambda-\mathcal{I}$ in the sense of Definition [Definition 13](#def:results:continuity_estimate){reference-type="ref" reference="def:results:continuity_estimate"}. We thus need to verify for each $\varepsilon > 0$ (i) [\[item:fundamental_liminf\]]{#item:fundamental_liminf label="item:fundamental_liminf"} $$\inf_\alpha \Lambda(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}) - \mathcal{I}(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}) > - \infty,\label{eq:proof-CP:Vx-unif-bound-below_first}$$ (ii) [\[item:fundamental_limsup\]]{#item:fundamental_limsup label="item:fundamental_limsup"} $$\sup_{\alpha}\Lambda(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon})-\mathcal{I}(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}) < \infty$$ (iii) [\[item:fundamental_compactcontrols\]]{#item:fundamental_compactcontrols label="item:fundamental_compactcontrols"} The set of controls $\theta_{\alpha,\varepsilon}$ is relatively compact. We will first establish [\[item:fundamental_liminf\]](#item:fundamental_liminf){reference-type="ref" reference="item:fundamental_liminf"} and [\[item:fundamental_limsup\]](#item:fundamental_limsup){reference-type="ref" reference="item:fundamental_limsup"} for all $\alpha$. Then, [\[item:fundamental_compactcontrols\]](#item:fundamental_compactcontrols){reference-type="ref" reference="item:fundamental_compactcontrols"} will follow from [\[item:fundamental_liminf\]](#item:fundamental_liminf){reference-type="ref" reference="item:fundamental_liminf"} and [\[item:fundamental_limsup\]](#item:fundamental_limsup){reference-type="ref" reference="item:fundamental_limsup"} and Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"}[\[item:assumption:compact-sublevelsets\]](#item:assumption:compact-sublevelsets){reference-type="ref" reference="item:assumption:compact-sublevelsets"}. By [\[subsolution_bound\]](#subsolution_bound){reference-type="eqref" reference="subsolution_bound"} and [\[eqn:choice_of_optimal_measure\]](#eqn:choice_of_optimal_measure){reference-type="eqref" reference="eqn:choice_of_optimal_measure"}, $$- \infty < \inf_\alpha \mathcal{H}(x_{\alpha,\varepsilon}, p^1_{\alpha,\varepsilon}) = \inf_\alpha \Lambda(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}) - \mathcal{I}(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon})$$ establishing [\[item:fundamental_liminf\]](#item:fundamental_liminf){reference-type="ref" reference="item:fundamental_liminf"}. By [\[supersolution_bound\]](#supersolution_bound){reference-type="eqref" reference="supersolution_bound"}, $$\sup_\alpha \Lambda(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}) - \mathcal{I}(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}) < \sup_\alpha \mathcal{H}(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon}) < \infty$$ implying [\[item:fundamental_limsup\]](#item:fundamental_limsup){reference-type="ref" reference="item:fundamental_limsup"}. ◻ *Proof of Proposition [Proposition 30](#prop:CP){reference-type="ref" reference="prop:CP"}.* *Proof of (a).* Fix $\lambda >0$ and $h_1,h_2 \in C_b(E)$. Let $u$ be a viscosity subsolution and $v$ be a viscosity supersolution of $f - \lambda H_\dagger f = h_1$ and $f - \lambda H_\ddagger f = h_2$ respectively. For any $\varepsilon > 0$ and any $\alpha > 0$, define the map $\Phi_{\alpha,\varepsilon}: E \times E \to \mathbb{R}$ by $$\Phi_{\alpha,\varepsilon}(x,y) := \frac{u(x)}{1-\varepsilon} - \frac{v(y)}{1+\varepsilon} - \frac{\alpha}{2} d^2(x,y) - \frac{\varepsilon}{1-\varepsilon} \Upsilon(x) - \frac{\varepsilon}{1+\varepsilon}\Upsilon(y).$$ Let $\varepsilon > 0$. By Lemma [Lemma 31](#lemma:doubling_lemma){reference-type="ref" reference="lemma:doubling_lemma"}, there is a compact set $K_\varepsilon \subseteq E$ and there exist points $x_{\alpha,\varepsilon},y_{\alpha,\varepsilon} \in K_\varepsilon$ such that $$\label{eqn:comparison_optimizers} \Phi_{\alpha,\varepsilon}(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon}) = \sup_{x,y \in E} \Phi_{\alpha,\varepsilon}(x,y),$$ and $$\label{eq:proof-CP:Psi-xy-converge} \lim_{\alpha \to \infty} \frac{\alpha }{2}d^2(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon}) = 0.$$ For all $\alpha$ it follows that $$\begin{aligned} \label{eq:proof-CP:general-bound-u1u2} \sup_E (u - v) & = \lim_{\varepsilon\to 0} \sup_{x\in E} \frac{u(x)}{1-\varepsilon} - \frac{v(x)}{1+\varepsilon}\\ &\leq \liminf_{\varepsilon\to 0} \sup_{x,y \in E} \frac{u(x)}{1-\varepsilon} - \frac{v(y)}{1+\varepsilon} - \frac{\alpha}{2} d^2(x,y) - \frac{\varepsilon}{1-\varepsilon} \Upsilon (x) - \frac{\varepsilon}{1+\varepsilon} \Upsilon (y) \\ & = \liminf_{\varepsilon\to 0} \frac{u(x_{\alpha,\varepsilon})}{1-\varepsilon} - \frac{v(y_{\alpha,\varepsilon})}{1+\varepsilon} - \frac{\alpha}{2} d^2(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon}) - \frac{\varepsilon}{1-\varepsilon}\Upsilon(x_{\alpha,\varepsilon}) -\frac{\varepsilon}{1+\varepsilon}\Upsilon(y_{\alpha,\varepsilon})\\ &\leq \liminf_{\varepsilon \to 0} \left[ \frac{u(x_{\alpha,\varepsilon})}{1-\varepsilon} - \frac{v(y_{\alpha,\varepsilon})}{1+\varepsilon}\right]. \end{aligned}$$ At this point, we want to use the sub- and supersolution properties of $u$ and $v$. Define the test functions $\varphi^{\varepsilon,\alpha}_1 \in \mathcal{D}(H_\dagger), \varphi^{\varepsilon,\alpha}_2 \in \mathcal{D}(H_\ddagger)$ by $$\begin{aligned} \varphi^{\varepsilon,\alpha}_1(x) & := (1-\varepsilon) \left[\frac{v(y_{\alpha,\varepsilon})}{1+\varepsilon} + \frac{\alpha}{2} d^2(x,y_{\alpha,\varepsilon}) + \frac{\varepsilon}{1-\varepsilon}\Upsilon(x) + \frac{\varepsilon}{1+\varepsilon}\Upsilon(y_{\alpha,\varepsilon})\right] \\ & \hspace{5cm} + (1-\varepsilon)(x-x_{\alpha,\varepsilon})^2, \\ \varphi^{\varepsilon,\alpha}_2(y) & := (1+\varepsilon)\left[\frac{u_1(x_{\alpha,\varepsilon})}{1-\varepsilon} - \frac{\alpha}{2} d^2(x_{\alpha,\varepsilon},y) - \frac{\varepsilon}{1-\varepsilon}\Upsilon(x_{\alpha,\varepsilon}) - \frac{\varepsilon}{1+\varepsilon}\Upsilon(y)\right] \\ & \hspace{5cm} - (1+\varepsilon) (y-y_{\alpha,\varepsilon})^2. \end{aligned}$$ Using [\[eqn:comparison_optimizers\]](#eqn:comparison_optimizers){reference-type="eqref" reference="eqn:comparison_optimizers"}, we find that $u - \varphi^{\varepsilon,\alpha}_1$ attains its supremum at $x = x_{\alpha,\varepsilon}$, and thus $$\sup_E (u-\varphi^{\varepsilon,\alpha}_1) = (u-\varphi^{\varepsilon,\alpha}_1)(x_{\alpha,\varepsilon}).$$ Denote $p_{\alpha,\varepsilon}^1 := \alpha \mathrm{d}_x \frac{1}{2}d^2(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon})$. By our addition of the penalization $(x-x_{\alpha,\varepsilon})^2$ to the test function, the point $x_{\alpha,\varepsilon}$ is in fact the unique optimizer, and we obtain from the subsolution inequality that $$\label{eq:proof-CP:subsol-ineq} u(x_{\alpha,\varepsilon}) - \lambda \left[ (1-\varepsilon) \mathcal{H}\left(x_{\alpha,\varepsilon}, p_{\alpha,\varepsilon}^1 \right) + \varepsilon C_\Upsilon\right] \leq h_1(x_{\alpha,\varepsilon}).$$ With a similar argument for $u_2$ and $\varphi^{\varepsilon,\alpha}_2$, we obtain by the supersolution inequality that $$\label{eq:proof-CP:supersol-ineq} v(y_{\alpha,\varepsilon}) - \lambda \left[(1+\varepsilon)\mathcal{H}\left(y_{\alpha,\varepsilon}, p_{\alpha,\varepsilon}^2 \right) - \varepsilon C_\Upsilon\right] \geq h_2(y_{\alpha,\varepsilon}),$$ where $p_{\alpha,\varepsilon}^2 := -\alpha \mathrm{d}_y \frac{1}{2} d^2 (x_{\alpha,\varepsilon},y_{\alpha,\varepsilon})$. With that, estimating further in [\[eq:proof-CP:general-bound-u1u2\]](#eq:proof-CP:general-bound-u1u2){reference-type="eqref" reference="eq:proof-CP:general-bound-u1u2"} leads to $$\begin{gathered} \sup_E(u-v) \leq \liminf_{\varepsilon\to 0}\liminf_{\alpha \to \infty} \bigg[\frac{h_1(x_{\alpha,\varepsilon})}{1-\varepsilon} - \frac{h_2(y_{\alpha,\varepsilon})}{1+\varepsilon} + \frac{\varepsilon}{1-\varepsilon} C_\Upsilon \\ + \frac{\varepsilon}{1+\varepsilon} C_\Upsilon + \lambda \left[\mathcal{H}(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon}) - \mathcal{H}(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon})\right]\bigg]. \end{gathered}$$ Note that, by the subsolution inequality [\[eq:proof-CP:subsol-ineq\]](#eq:proof-CP:subsol-ineq){reference-type="eqref" reference="eq:proof-CP:subsol-ineq"}, $$\begin{aligned} - \infty < \frac{1}{\lambda} \inf_E\left(u - h_1\right) & \leq (1-\varepsilon) \mathcal{H}(x_{\alpha,\varepsilon},p^1_{\alpha,\varepsilon}) + \varepsilon C_{\Upsilon}, \end{aligned}$$ and by the supersolution inequality [\[eq:proof-CP:supersol-ineq\]](#eq:proof-CP:supersol-ineq){reference-type="eqref" reference="eq:proof-CP:supersol-ineq"}, $$(1+\varepsilon)\mathcal{H}\left(y_{\alpha,\varepsilon},p^2_{\alpha,\varepsilon}\right) - \varepsilon C_\Upsilon \leq \frac{1}{\lambda} \sup_E (v-h_2) < \infty.$$ Thus, comparison principle follows from Proposition [Proposition 33](#prop:continuity_estimate){reference-type="ref" reference="prop:continuity_estimate"}. *Proof of (b).* Let $u$ be a subsolution for $H_\dagger$, and $v$ a supersolution for $H_\ddagger$. Let $T > 0$ be fixed. For any $\beta > 0$, we have $$\sup_{t \in [0,T],x} u(t,x) - v(t,x) \leq \sup_{t \in [0,T],x} u(t,x) - v(t,x) -\beta t + \beta T$$ We next incorporate our Lyapunov type functions $$\sup_{t \in [0,T],x} u(t,x) - v(t,x) -\beta t + \beta T = \lim_{\varepsilon \downarrow 0} \sup_{t \in [0,T],x} \frac{u(t,x)}{1-\varepsilon} - \frac{v(t,x)}{1+\varepsilon} - \frac{2\varepsilon}{1-\varepsilon^2}\Upsilon(x) -\beta t + \beta T.$$ Thus, for any $\varepsilon > 0$, $\alpha,\gamma > 0$, we have $$\begin{gathered} \label{eq:quadruplication} \sup_{t \in [0,T], x} \frac{u(t,x)}{1-\varepsilon} - \frac{v(t,x)}{1+\varepsilon} - \frac{2\epsilon}{1-\epsilon^2}\Upsilon(x) -\beta t + \beta T \leq \sup_{s,t \in [0,T],x,y} \frac{u(t,x)}{1-\varepsilon} - \frac{v(s,y)}{1+\varepsilon} - \frac{\alpha}{2}d^2(x,y) - \frac{\gamma}{2}(s-t)^2 \\ - \frac{\varepsilon}{1-\varepsilon} \Upsilon(x) - \frac{\varepsilon}{1+\varepsilon}\Upsilon(y) - \frac{\beta}{2} (t+s) + \beta T.\end{gathered}$$ By Lemma [Lemma 32](#lemma:quadrupling_lemma){reference-type="ref" reference="lemma:quadrupling_lemma"}, there are $(x_{\alpha,\gamma},t_{\alpha,\gamma},y_{\alpha,\gamma},s_{\alpha,\gamma}) = (x_{\varepsilon,\beta,\alpha,\gamma},t_{\varepsilon,\beta,\alpha,\gamma},y_{\varepsilon,\beta,\alpha,\gamma},s_{\varepsilon,\beta,\alpha,\gamma})$ optimizing the supremum on the right-hand side and such that $$\begin{aligned} \frac{\alpha}{2}d^2(x_{\alpha,\gamma},y_{\alpha,\gamma}) \to 0 \quad \text{as $\alpha\to\infty$}.\end{aligned}$$ First assume $t_{\alpha,\gamma},s_{\alpha,\gamma} > 0$. We will aim for a contradiction. Define the functions $f^\dagger_{\varepsilon,\beta,\alpha}(x)\in D(H_\dagger)$ and $f^\ddagger_{\varepsilon,\beta,\alpha}(y)\in D(H_\ddagger)$ by $$\begin{aligned} f^\dagger(x) & = (1-\varepsilon)\left[\frac{v(s_{\alpha,\gamma},y_{\alpha,\gamma})}{1+\varepsilon} +\frac{\alpha}{2}d^2(x,y_{\alpha,\gamma}) +\frac{\varepsilon}{1+\varepsilon}\Upsilon(y_{\alpha,\gamma})- \beta T \right] +\varepsilon\Upsilon(x),\\ f^\ddagger(y) & = (1+\varepsilon)\left[ \frac{u(t_{\alpha,\gamma},x_{\alpha,\gamma})}{1-\varepsilon} -\frac{\alpha}{2}d^2(x_{\alpha,\gamma},y) - \frac{\varepsilon}{1-\varepsilon}\Upsilon(x_{\alpha,\gamma}) +\beta T\right] - \varepsilon\Upsilon(y).\end{aligned}$$ Observe that $$\begin{aligned} u(t_{\alpha,\gamma},x_{\alpha,\gamma})-f^\dagger(x_{\alpha,\gamma}) - h_1(t_{\alpha,\gamma}) & = \sup_{t\in[0,T],x} u(t,x)- f^\dagger(x)- h_1(t),\\ v(s_{\alpha,\gamma},y_{\alpha,\gamma})-f^\ddagger(y_{\alpha,\gamma})- h_2(s_{\alpha,\gamma}) & = \inf_{t\in [0,T],y} v(s,y) - f^\ddagger(y) - h_2(s),\end{aligned}$$ where $h_1(t)= (1-\varepsilon)(\frac{\beta}{2}(t+s_{\alpha,\gamma}) + \frac{\gamma}{2}(t-s_{\alpha,\gamma})^2)$ and $h_2(s)= (1+\varepsilon)(-\frac{\beta}{2}(t_{\alpha,\gamma}+s) - \frac{\gamma}{2}(t_{\alpha,\gamma}- s)^2)$. Then, using the sub and supersolution inequalities, dividing them by $(1-\varepsilon)$ and $(1+\varepsilon)$ respectively, we find $$\begin{aligned} & \gamma (t_{\alpha,\gamma} - s_{\alpha,\gamma}) + \frac{\beta}{2} - \mathcal{H}(x_{\alpha,\gamma}, \alpha \mathrm{d}_x \frac{1}{2} d^2(\cdot,y_{\alpha,\gamma})(x_{\alpha,\gamma})) - \frac{\varepsilon}{1-\varepsilon}c_\Upsilon \leq 0, \\ & \gamma (t_{\alpha,\gamma} - s_{\alpha,\gamma}) - \frac{\beta}{2} - \mathcal{H}(y_{\alpha,\gamma}, - \alpha \mathrm{d}_y \frac{1}{2} d^2(x_{\alpha,\gamma},\cdot)(y_{\alpha,\gamma})) + \frac{\varepsilon}{1+\varepsilon}c_\Upsilon \geq 0.\end{aligned}$$ Combining the two equations yields $$\beta \leq \mathcal{H}(x_{\alpha,\gamma}, \alpha \mathrm{d}_x \frac{1}{2} d^2(\cdot,y_{\alpha,\gamma})(x_{\alpha,\gamma})) - \mathcal{H}(y_{\alpha,\gamma}, - \alpha \mathrm{d}_y \frac{1}{2} d^2(x_{\alpha,\gamma},\cdot)(y_{\alpha,\gamma})) + \frac{2\varepsilon}{1-\varepsilon^2} c_\Upsilon$$ sending $\alpha \rightarrow \infty$ and $\varepsilon \rightarrow 0$, and using Proposition [Proposition 33](#prop:continuity_estimate){reference-type="ref" reference="prop:continuity_estimate"}, we get a contradiction for small $\varepsilon$ as $\beta > 0$. So it holds that for small $\varepsilon$, large $\alpha$ and all $\gamma>0$, we have $t_{\alpha,\gamma} = 0$ or $s_{\alpha,\gamma} = 0$. Proceeding from equation [\[eq:quadruplication\]](#eq:quadruplication){reference-type="eqref" reference="eq:quadruplication"}, we get $$\begin{aligned} \sup_{t \in [0,T],x} u(t,x) - v(t,x) & \leq \sup_{t \in [0,T],x} u(t,x) - v(t,x) -\beta t + \beta T \\ & \leq \lim_{\varepsilon \downarrow 0} \sup_{s,t \in [0,T],x,y} \frac{u(t,x)}{1-\varepsilon} - \frac{v(s,y)}{1+\varepsilon} - \frac{\alpha}{2}d^2(x,y) - \frac{\gamma}{2}(s-t)^2 \\ & \qquad - \frac{\varepsilon}{1-\varepsilon} \Upsilon(x) - \frac{\varepsilon}{1+\varepsilon}\Upsilon(y) - \frac{\beta}{2} (t+s) + \beta T \\ & \leq \lim_{\varepsilon \downarrow 0} \lim_{\gamma \rightarrow \infty} \lim_{\alpha \rightarrow \infty} \frac{u(t_{\alpha,\gamma},x_{\alpha,\gamma})}{1-\varepsilon} - \frac{v(s_{\alpha,\gamma},y_{\alpha,\gamma})}{1+\varepsilon}\\ &\qquad +\frac{\varepsilon}{1-\varepsilon} \Upsilon(x_{\alpha,\gamma}) - \frac{\varepsilon}{1+\varepsilon}\Upsilon(y_{\alpha,\gamma}) - \frac{\beta}{2} (t_{\alpha,\gamma}+s_{\alpha,\gamma}) + \beta T \\ & \leq \sup_x u(0,x) - v(0,x) + \beta T\end{aligned}$$ where we used that $u$ is upper semi-continuous, $v$ is lower semi-continuous, and Lemma [Lemma 32](#lemma:quadrupling_lemma){reference-type="ref" reference="lemma:quadrupling_lemma"} (b)(ii) and the fact that $t_{\alpha,\gamma}=0$ or $s_{\alpha,\gamma} = 0$. As $\beta > 0$ was arbitrary, we conclude. ◻ ## Existence of viscosity solutions {#sec:existence} In this subsection, we will prove Theorem [Theorem 10](#theorem:existence_of_viscosity_solution){reference-type="ref" reference="theorem:existence_of_viscosity_solution"}. In other words, we show that for $h\in C_b(E)$ and $\lambda>0$, the function $R(\lambda)h$ given by $$R(\lambda) h(x) = \sup_{\substack{\gamma \in \mathcal{A}\mathcal{C} \\ \gamma(0) = x}} \int_0^\infty \lambda^{-1} e^{-\lambda^{-1}t} \left[h(\gamma(t)) - \int_0^t \mathcal{L}(\gamma(s),\dot{\gamma}(s))\right] \, \mathrm{d}t$$ is indeed a viscosity solution to $f - \lambda \mathbf{H}f = h$. *Proof of Theorem [Theorem 10](#theorem:existence_of_viscosity_solution){reference-type="ref" reference="theorem:existence_of_viscosity_solution"}.* The result follows from Theorem 2.8 in [@KrSchl20] where the authors show that $\partial_p \mathcal{H}(x,p) \cap T_E(x) \neq \emptyset$ implies the existence of solutions. ◻ We want to mention that in our ongoing work [@DeCoKr23] we prove the existence of solutions for the evolutionary equation as well. # Example: Hamiltonians arising from biochemistry {#section:verification-for-examples-of-Hamiltonians} The purpose of this section is to showcase that the method introduced in this paper is versatile enough to capture interesting examples that could not be treated before. The Hamiltonian that we consider arises from the study of systems biology. More precisely, it plays a crucial role in the study, with a *large deviations* approach, of multi--scale Markov processes modelling chemical reactions. Over the past few decades, there has been significant research conducted on multi-scale chemical reactions networks (see for example [@KaKu13] and [@BaKuPoRe06]). They are usually described by the use of continuous-time Markov chains with generators of the following form $$\label{eq:multiple-time-Markov-process-generator} A f(z) = \sum_{\gamma \in \Gamma} r(z,\gamma)\left[f(z+\gamma) - f(z) \right],$$ with $z$ in the space $\mathbb{N}^J$, with $J$ the set of chemical species, and $r\in C^1 (\mathbb{N}^J\times \Gamma)$ is a non-negative smooth function. In the above, the state $z$ is a vector whose components describe the number of molecules of a chemical species, $r$ is the transition rate of the reaction, $\Gamma$ is the set representing all reactions. In particular, every $\gamma \in \Gamma$ describes a reaction in the sense that its $i$--th component represents the number of molecules of the $i$--th species that is used (if the component is negative) or obtained (if the component is positive) in the reaction. Motivated by a huge class of examples arising from biochemistry in which two dominant time--scales occur (see for instance Example [Example 35](#ex:mm){reference-type="ref" reference="ex:mm"}), we will consider a two - scale process $Z=(X,Y)$ in the space $E^0 = \mathbb{N}^l \times \mathbb{N}^m$. The amount of molecules of the first type is an order of magnitude greater then the amount of the second type. For this reason and to be able to study the limit behaviour of the process, we consider the scaled species $X_N=X/N$ and $Y_N=Y$ in the space $E^0_N = \left(\frac{1}{N} \mathbb{N}\right)^l \times \mathbb{N}^m$. The time-scale separation between the slow process $X_N$ and the fast process $Y_N$ is then $N$ and the generator of the rescaled process $Z_N=(X_N,Y_N)$ is given by $$\label{eq:generator_rescaled} A_N f(x,y) = N \sum_{\gamma=(\gamma_x,\gamma_y) \in \Gamma} r(x,y,\gamma) [f(x+N^{-1}\gamma_x , y + \gamma_y)],$$ for $f \in D(A_N) \subseteq C (E_N^0)$. In order to gain a more comprehensive understanding of the system and simplify the limit procedure that will follow, we divide the generator into three distinct parts. These parts will specifically describe reactions occurring on the macroscopic, microscopic, and a combination of the two scales, respectively. The generator of $(X_N, Y_N)$ is then, $$\begin{aligned} \label{eq:multiple-time: generator_extended} A_N f(x,y) = & N \sum_{\gamma=(\gamma_x,\gamma_y)\in \Gamma_1} r (x,y,\gamma)\left[f(x+N^{-1}\gamma_x,y) - f(x,y) \right] + \\ & N \sum_{\gamma=(\gamma_x,\gamma_y)\in \Gamma_2} r (x,y,\gamma)\left[f(x+N^{-1}\gamma_x,y + \gamma_y) - f(x,y) \right] + \\ & N \sum_{\gamma=(\gamma_x,\gamma_y)\in \Gamma_3} r (x,y,\gamma)\left[f(x,y + \gamma_y) - f(x,y) \right],\end{aligned}$$ where we write $$\begin{aligned} &\Gamma_1 = \left\{ \gamma=(\gamma_x,\gamma_y)\in \mathbb{Z}^l \times \mathbb{Z}^m \, : \, \gamma_{y_i}=0 \, \forall i \in \{1,\dots,m\} \right\} ;\\ &\Gamma_2 = \left\{ \gamma =(\gamma_x,\gamma_y)\in \mathbb{Z}^l \times \mathbb{Z}^m \, : \, \exists i\in\{1,\dots,l\} \,, j \in \{1,\dots, m\} \, | \, \gamma_{x_i}\neq 0, \gamma_{y_j}\neq 0 \right\} ;\\ &\Gamma_3 = \left\{ \gamma =(\gamma_x,\gamma_y)\in \mathbb{Z}^l \times \mathbb{Z}^m \, : \, \gamma_{x_i}=0 \, \forall i \in \{1,\dots,l\} \right\}.\end{aligned}$$ The model is subjected to the following assumption. **Assumption 34**. The molecules that are part of the fast process are subjected to a conservation law. More precisely, there exists a constant $M>0$ such that $$\sum_{i=1}^m Y_{i} = M, \qquad \text{and} \qquad \sum_{i=1}^m \gamma_{y,i} = 0 \qquad \forall \gamma \in \Gamma_2 \cup \Gamma_3.$$ Assumption [Assumption 34](#assumption:conservation){reference-type="ref" reference="assumption:conservation"} allows us to restrict the set of values of $Y_N$ to the set $$F_M = \{ n \in \mathbb{N}^m \, : \, \sum_{i= 1}^m n_i = M \},$$ and, hence, we consider for our analysis of $Z_N$ the set $$E_N = \left(\frac{1}{N} \mathbb{N}\right)^l \times F_M.$$ To show an example, we describe in the following the model for enzyme kinetics with an inflow of the substrate, also called Michaelis--Menten model, studied in [@Po18]. **Example 35**. *Consider four types of molecules. Namely, $S$, $E$, $ES$ and $P$ representing respectively the substrate, enzyme, enzyme--substrate complex and the product. The following four reactions occur.* (1) *$\emptyset \xhookrightarrow[]{k_0} S$* (2) *$E + S \xhookrightarrow[]{k_1} ES$* (3) *$ES \xhookrightarrow[]{k_2} E + S$* (4) *$ES \xhookrightarrow[]{k_3} P +E$.* *Let $X^1, X^2, Y^1, Y^2$ represent the amount of $S, P, E, ES$ respectively.* *In real-world physical scenarios, the quantities of enzyme molecules and enzyme-substrate complexes are typically small when compared to the number of substrate and product molecules. Consequently, it is reasonable to assume that $X^1$ and $X^2$ are an order of magnitude greater then $Y^1$ and $Y^2$. In this way, we lead to the scaled amounts represented by a slow process $X_N = (X^1/ N, X^2/N)$ and a fast process $Y_N=(Y^1, Y^2)$.* *The generator of the two--scales process $Z_N = (X_N,Y_N)$ is as in [\[eq:multiple-time: generator_extended\]](#eq:multiple-time: generator_extended){reference-type="eqref" reference="eq:multiple-time: generator_extended"}, with the following rates, each describing one of the reactions above.* (1) *$r(x,y,(1,0, 0, 0))= k_0 \in {{\mathbb R}}_+;$* (2) *$r(x,y,(-1, 0 , -1, 1)) = k_1 x_1 y_1 \qquad k_1, \in {{\mathbb R}}_+;$* (3) *$r(x,y,(1,0,1,-1)) = k_2 y_2 \qquad k_2 \in {{\mathbb R}}_+;$* (4) *$r(x,y,(0,1,1,-1)) = k_3 y_2 \qquad k_3\in {{\mathbb R}}_+$.* The above model falls into the class of examples described by [\[eq:multiple-time: generator_extended\]](#eq:multiple-time: generator_extended){reference-type="eqref" reference="eq:multiple-time: generator_extended"}. Moreover, we will observe in Subsection [6.2](#subsection:example_Hamiltonian){reference-type="ref" reference="subsection:example_Hamiltonian"} that our approach proves to be suitable and applicable to a bigger class of scenarios. We are interested in the limit behaviour of the two component $E_N$-valued Markov process $(X_N, Y_N)$. In the limit regime, the fast component $Y_N$ converges to equilibrium and the slow component $X_N$ converges to a deterministic limit. To characterize the speed of convergence, we are interested in the large deviation behavior for the slow process. In particular, we want to characterize the function $I:C_E [0,\infty) \to [0,\infty]$ (that will depend on the generator of the fast process), with $C_E [0,\infty)$ the space of all continuous path $x: [0,\infty)\to E$, with $E= [0,\infty)^l$, such that $$\mathbb{P}\left[X_N \approx x \right] \sim e^{-N I(x)}.$$ This means that $X_N$ has a limit path $\tilde{x} \in C_{E}$ and this limit is the unique minimizer of the function $I$. Moreover, for any path $x\neq \tilde{x}$ such that $I(x)>0$, the probability that $X_N$ is close to $x$ is exponentially small. Hence, characterizing the function $I$ means to characterize the limit behaviour of $X_N$. In Chapter 8 of [@FK06], it is shown that the function $I$ is characterized by the unique solution of the Hamilton-Jacobi equation $$f-\lambda \mathbf{H}f = h \qquad (\text{or $\partial_t f - \mathbf{H}f = 0$}),$$ with $\mathbf{H}f = \mathcal{H}(f,\nabla f)$ and $\mathcal{H}: E \times {{\mathbb R}}^l \to {{\mathbb R}}$ of the type $$\label{hamiltonian_example} \mathcal{H}(x,p)=\sup_{\theta \in \Theta} \left[ \int_{F_M} V(y;x,p) \, d\theta(y) + \inf_{\varphi \in C^2(F_M)} \int_{F_M} e^{-\varphi}L_{x,p}e^{\varphi}\,d\theta \right].$$ We refer to the Subsection [6.2](#subsection:example_Hamiltonian){reference-type="ref" reference="subsection:example_Hamiltonian"} for a detailed explanation on how to construct the Hamiltonian using a limiting procedure built on [\[eq:multiple-time: generator_extended\]](#eq:multiple-time: generator_extended){reference-type="eqref" reference="eq:multiple-time: generator_extended"}. In this context, $\Theta = \mathcal{P}(F_M)$, that is the probability measures over the set $F_M$, and $$\begin{aligned} V(y;x,p) &= \sum_{\gamma=(\gamma_x,\gamma_y)\in\Gamma_1} r(x,y,\gamma) (e^{\langle p,\gamma_x\rangle}-1) + \sum_{\gamma=(\gamma_x,\gamma_y)\in\Gamma_2} r(x,y,\gamma) (e^{\langle p, \gamma_x\rangle}-1),\\ L_{x,p}f(x,y) &= \sum_{\gamma=(\gamma_x,\gamma_y)\in\Gamma_2} r(x,y,\gamma) e^{\langle p, \gamma_x\rangle} [ f(x,y+\gamma_y)- f(x,y)]\\ &+ \sum_{\gamma=(\gamma_x,\gamma_y)\in\Gamma_3} r(x,y,\gamma) [f(x,y+\gamma_y) - f(x,y)],\end{aligned}$$ where $r \in C^1(E \times \Gamma_i)$ are non negative smooth functions. Hence, in this context the well-posedness of the Hamilton-Jacobi equation plays a central role. In Subsection [6.1](#subsection:example_comparison){reference-type="ref" reference="subsection:example_comparison"}, we prove comparison principle for the equations $f - \lambda \mathbf{H}f = h$ and $\partial_t f - \mathbf{H}f = 0$, with $\mathbf{H}f = \mathcal{H}(x,\nabla f(x))$ and $\mathcal{H}$ as in [\[hamiltonian_example\]](#hamiltonian_example){reference-type="eqref" reference="hamiltonian_example"}, and existence of solutions for the stationary equation. In Subsection [6.2](#subsection:example_Hamiltonian){reference-type="ref" reference="subsection:example_Hamiltonian"}, we outline how the above Hamiltonian can be obtained from a limit procedure and an eigenvalue problem arising from the multi--scale Markov process described above. For these aims, we need two additional assumptions. **Assumption 36**. The matrix $(R_x)_{y_1,y_2}= \sum_{k\in\{2,3\}} \sum_{\gamma \in \Gamma_k : y_2 = y_1 + \gamma_y} r(x,y_1,\gamma)$ is irreducible for every $x$. **Assumption 37**. For all $z$ and $\gamma$ there exist continuous functions $\phi^{z,\gamma}_1,\phi^{z,\gamma}_2: [0,\infty)^l \to {{\mathbb R}}$ such that $r(x, z, \gamma) = \phi^{z,\gamma}_1(x)\phi^{z,\gamma}_2(x)$ and such that 1. $\inf \phi_2^{z,\gamma}>0$; 2. if $\langle \gamma_x , x-y \rangle >0$ then $\phi_1^{z,\gamma} (x) < \phi_1 ^{z,\gamma}(y)$; 3. if $\gamma_{x_i} < 0$ and $x_i = 0$ then $\phi_1(x) = 0$. Assumption [Assumption 36](#assumption:irreducibility){reference-type="ref" reference="assumption:irreducibility"} guarantees the existence of an eigenvalue of $V_{x,p} + L_{x,p}$ (i.e. step [\[item: FengKurtz_eigenvalue\]](#item: FengKurtz_eigenvalue){reference-type="ref" reference="item: FengKurtz_eigenvalue"} of Subsection [6.2](#subsection:example_Hamiltonian){reference-type="ref" reference="subsection:example_Hamiltonian"}). Assumption [Assumption 37](#assumption:inner_product){reference-type="ref" reference="assumption:inner_product"} restricts the possible rates and hence Hamiltonians of the type [\[hamiltonian_example\]](#hamiltonian_example){reference-type="eqref" reference="hamiltonian_example"} that we are able to treat with our theorems. First of all it removes cases of type $\mathcal{H}(x,p)=x (e^p -1)$, with $x\in [0,\infty)$, for which it is well known that comparison principle fails (see e.g. Example E in [@ShWe05]). Moreover, it is straightforward to show Assumption [Assumption 37](#assumption:inner_product){reference-type="ref" reference="assumption:inner_product"} in many cases in one dimension, e.g. $\mathcal{H}(x,p)=x(e^{-p}-1)$, $\mathcal{H}(x,p)=x e^{\beta x} (e^{-p} - 1)$ in $[0,\infty)$. It is also straightforward to verify the above assumptions for the two dimensional Example [Example 35](#ex:mm){reference-type="ref" reference="ex:mm"}. Regarding higher dimensional cases, the assumption excludes examples in which there is an interaction between two molecules of the slow process. These cases, indeed, produce rates of the type $r_{z,\gamma}(x) = x_i x_j$ with $i,j \in \{1,\dots, l\}$ or such that $\langle \gamma_x, x- y \rangle > 0$ and not equal to $\phi_1(y) - \phi_1(x)$, for which Assumption [Assumption 37](#assumption:inner_product){reference-type="ref" reference="assumption:inner_product"} does not necessary hold. However, to our knowledge, examples of this type are largely unexplored. Cases in higher dimension for which the above assumption holds are e.g. Hamiltonians of the type $\mathcal{H}(x,p) = x_1 (1+ x_2) (e^{p_1} -1)$. ## Comparison principle and existence of viscosity solutions {#subsection:example_comparison} **Theorem 38** (Comparison principle). *Consider $\mathcal{H}(x,p)$ as in [\[hamiltonian_example\]](#hamiltonian_example){reference-type="eqref" reference="hamiltonian_example"}. Suppose Assumption [Assumption 36](#assumption:irreducibility){reference-type="ref" reference="assumption:irreducibility"} and Assumption [Assumption 37](#assumption:inner_product){reference-type="ref" reference="assumption:inner_product"}. Then comparison principles for $f-\lambda \mathcal{H}(x, \nabla f(x)) = h$ and $\partial_t f - \mathcal{H}(x,\nabla f(x)) = 0$ hold.* *Proof.* To prove the comparison principle we firstly mention that $\mathcal{H}(x,p)$ is of the form [\[eq:results:variational_hamiltonian\]](#eq:results:variational_hamiltonian){reference-type="eqref" reference="eq:results:variational_hamiltonian"} with $\Theta = \mathcal{P}(F_M)$ and $$\label{ex:Lambda} \Lambda(x,p,\theta)= \int_{F_M} V(y;x,p) \, d\theta(y),$$ and $$\begin{aligned} \label{ex:I} \mathcal{I}(x,p,\theta)& = - \inf_{u\in C^2(F_M), \inf u>0} \int_{F_M} \frac{ L_{x,p} u } {u}\,d\theta= - \inf_{\varphi \in C^2(F_M)} \int_{F_M} e^{-\varphi} L_{x,p}e^{\varphi}\,d\theta. \end{aligned}$$ We can then apply Theorem [Theorem 8](#theorem:comparison_principle_variational){reference-type="ref" reference="theorem:comparison_principle_variational"} to show the comparison principle. In the following we verify Assumption [Assumption 16](#assumption:regularity:Lambda-I){reference-type="ref" reference="assumption:regularity:Lambda-I"}. (I) The function $p\mapsto \mathcal{H}(x,p)$ is convex. Moreover, note that $V_{x,0}=0$. Hence, $$\mathcal{H}(x,0)= - \inf_\theta \mathcal{I}(x,p,\theta) = 0,$$ being $\mathcal{I}\geq 0$ (see [@DoVa75a]) and there exists a measure $\theta^0_{x,p}$ such that $\mathcal{I}(x,p,\theta^0_{x,p})=0$, due to Assumption [Assumption 36](#assumption:irreducibility){reference-type="ref" reference="assumption:irreducibility"} (see [@Kl14] Theorem 17.51). (II) $\theta \mapsto \Lambda(x,p,\theta)-\mathcal{I}(x,p,\theta)$ is bounded for every $p$ and $x$ being a continuous function over a compact set. (III) $\Upsilon(x)= \frac{1}{2} \log(1+ \sum_{i=1}^l x_i^2)$ is a containment function since the functions $$(r(x,y,\gamma))\left(e^{\frac{x}{x^2+1}\gamma_x}-1\right),$$ $$e^{-\varphi}r(x,y,\gamma)e^{\frac{x}{1+x^2}\gamma_x}e^\varphi$$ and $$e^{-\varphi}r(x,y,\gamma)e^{\varphi}$$ are bounded for every $\gamma\in\Gamma$, every $\varphi \in C^2(F_M)$ and every $(x,y)\in E\times F_M$. (IV) Let $(x,p)\in E \times{{\mathbb R}}^l$. The function $V(\cdot,x,p)$ is continuous and hence $(x,p,\theta)\mapsto\Lambda(x,p,\theta)$ is continuous. Moreover, $\mathcal{I}$ is lower semicontinuous, as the supremum over continuous functions. Then, $\mathcal{I}- \Lambda$ is lower semicontinuous and the first property of Definition [Definition 15](#def:Gamma-convergence){reference-type="ref" reference="def:Gamma-convergence"} follows. We prove now that if $x_n \to x$ and $p_n \to p$ and for all $\theta \in \Theta$, there are $\theta_n$ such that $\theta_n \to \theta$ and $$\label{eq:gammaconv} \limsup_n \mathcal{I}(x_n, p_n, \theta_n) \leq \mathcal{I}(x,p,\theta).$$ Then, the $\Gamma$-- convergence of $\mathcal{I}- \Lambda$ will follow from [\[eq:gammaconv\]](#eq:gammaconv){reference-type="eqref" reference="eq:gammaconv"} and continuity of $\Lambda$. For any $m \in \mathbb{N}$, there exists $\varphi_m \in C^2(F_M)$ such that $$\mathcal{I}(x,p,\theta) \leq \int_{F_M} e^{-\varphi_m} L_{x,p} e^{\varphi_m} \, d\theta + \frac{1}{m}.$$ Then, taking into account the continuity of $L_{x,p}$ and choosing $\theta_n = \theta$ for every $n$, we get $$\limsup_n \mathcal{I}(x_n,p_n,\theta_n) \leq \int_{F_M} e^{-\varphi_m} L_{x,p} e^{\varphi_m} \, d\theta + \frac{1}{m}.$$ By letting $m$ to infinity we obtain [\[eq:gammaconv\]](#eq:gammaconv){reference-type="eqref" reference="eq:gammaconv"}. (V) As $\Theta$ is compact, any closed subset of $\Theta$ is compact. (VI) As explained above, there exists a measure $\theta^0_{x,p}$ such that $\mathcal{I}(x,p,\theta^0_{x,p})=0$. Then, $\mathcal{I}(x,p,\theta^0_{x,p}) - \Lambda(x,p,\theta^0_{x,p}) \leq - \Lambda(x,p,\theta^0_{x,p})$. Taking $g(x,p)= - \Lambda(x,p,\theta^0_{x,p} )$, $\phi_{g}(x,p)$ is not empty, as $\theta_{x,p}^0 \in \phi_{g}(x,p)$. (VII) Let $(x_{\alpha,\varepsilon},y_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon})$ be a fundamental sequence as in Definition [Definition 13](#def:results:continuity_estimate){reference-type="ref" reference="def:results:continuity_estimate"}. Set $p_{\alpha,\varepsilon} = \alpha (x_{\alpha,\varepsilon} - y_{\alpha,\varepsilon})$. We aim to show $$\begin{aligned} & \liminf_{\alpha \rightarrow \infty} (\Lambda - \mathcal{I})\left(x_{\alpha,\varepsilon}, p_{\alpha,\varepsilon}, \theta_{\alpha,\varepsilon}\right) - (\Lambda-\mathcal{I})\left(y_{\alpha,\varepsilon},p_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}\right)\leq 0. \end{aligned}$$ By the definition of $\Lambda$ and $\mathcal{I}$ in [\[hamiltonian_example\]](#hamiltonian_example){reference-type="eqref" reference="hamiltonian_example"}, the difference above is of the type $$\begin{aligned} \label{eq:fundamentalseq} &\int_{F_M} \sum_{\gamma\in\Gamma_1 \cup \Gamma_2} \left(r(x_{\alpha,\varepsilon},z,\gamma)-r(y_{\alpha,\varepsilon},z,\gamma)\right)\left(e^{\langle p_{\alpha,\varepsilon}, \gamma_x \rangle}-1 \right)\, d\theta +\\ & \inf_{\varphi \in C^2(F_M)} \int_{F_M} e^{-\varphi} \left(\sum_{\gamma\in\Gamma_2} r(x_{\alpha,\varepsilon},z,\gamma)-r(y_{\alpha,\varepsilon},z,\gamma)\right) e^{\langle p_{\alpha,\varepsilon}, \gamma_x \rangle} e^{\varphi}+\\ & e^{-\varphi}\left(\sum_{\gamma\in \Gamma_3} r(x_{\alpha,\varepsilon},z,\gamma) - r(y_{\alpha,\varepsilon},z,\gamma)\right) e^{\varphi}\, d\theta. \end{aligned}$$ Note that if $r(x,y,\gamma)$ is constant in $x$, the difference above is zero. Hence, we only take into account the parameters $\gamma$ such that $r$ depends on $x$. Moreover, by the upper bound [\[item:def:continuity_estimate:3\]](#item:def:continuity_estimate:3){reference-type="ref" reference="item:def:continuity_estimate:3"} in Definition [Definition 13](#def:results:continuity_estimate){reference-type="ref" reference="def:results:continuity_estimate"}, we find that there is some $\alpha(\varepsilon)$ such that $$\label{eq:boundsupersolution} \sup_{\alpha\geq \alpha(\varepsilon)} \Lambda(y_{\alpha,\varepsilon}, p_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}) - \mathcal{I}(y_{\alpha,\varepsilon}, p_{\alpha,\varepsilon},\theta_{\alpha,\varepsilon}) <\infty.$$ If $\lim_{\alpha} r(y_{\alpha,\varepsilon},z,\gamma) > 0$ for all $\gamma$, we can conclude by the bound [\[eq:boundsupersolution\]](#eq:boundsupersolution){reference-type="eqref" reference="eq:boundsupersolution"} that $e^{\langle p_{\alpha,\varepsilon}, \gamma_x \rangle}$ is bounded. Then, by property [\[item:def:continuity_estimate:2\]](#item:def:continuity_estimate:2){reference-type="ref" reference="item:def:continuity_estimate:2"} of Definition [Definition 13](#def:results:continuity_estimate){reference-type="ref" reference="def:results:continuity_estimate"} and continuity of the rates, [\[eq:fundamentalseq\]](#eq:fundamentalseq){reference-type="eqref" reference="eq:fundamentalseq"} converges to $0$ for $\alpha \to \infty$. Consider now all terms $\gamma$ such that $r(y_{\alpha,\varepsilon},z,\gamma) \to 0$ as $\alpha \to \infty$. Firstly note that, by Assumption [Assumption 37](#assumption:inner_product){reference-type="ref" reference="assumption:inner_product"}, [\[eq:fundamentalseq\]](#eq:fundamentalseq){reference-type="eqref" reference="eq:fundamentalseq"} equal to $$\begin{aligned} \label{eq:fundamentalseq_two} &\int_{F_M} \sum_{\gamma\in\Gamma_1 \cup \Gamma_2} \left(\phi^{z,\gamma}_1 (x_{\alpha,\varepsilon})\phi^{z,\gamma}_2(x_{\alpha,\varepsilon}) - \phi^{z,\gamma}_1 (y_{\alpha,\varepsilon})\phi^{z,\gamma}_2(y_{\alpha,\varepsilon})\right) \left(e^{\langle p_{\alpha,\varepsilon}, \gamma_x \rangle}-1 \right)\, d\theta +\\ & \inf_{\varphi \in C^2(F_M)} \int_{F_M} e^{-\varphi} \left(\sum_{\gamma\in\Gamma_2} \left(\phi^{z,\gamma}_1 (x_{\alpha,\varepsilon})\phi^{z,\gamma}_2(x_{\alpha,\varepsilon}) - \phi^{z,\gamma}_1 (y_{\alpha,\varepsilon})\phi^{z,\gamma}_2(y_{\alpha,\varepsilon})\right) e^{\langle p_{\alpha,\varepsilon}, \gamma_x \rangle} \right)e^{\varphi}+\\ & e^{-\varphi}\left(\sum_{\gamma\in \Gamma_3} r(x_{\alpha,\varepsilon},z,\gamma) - r(y_{\alpha,\varepsilon},z,\gamma)\right) e^{\varphi}\, d\theta.\end{aligned}$$ The last line converges to $0$ for $\alpha \to \infty$ by the continuity of the rates. If $\langle p_{\alpha,\varepsilon} , \gamma_x \rangle <0$, $e^{\langle p_{\alpha,\varepsilon} , \gamma_x \rangle}$ is bounded and the first two lines also converge to $0$ by continuity of the rates. Consider the case $\langle p_{\alpha,\varepsilon} , \gamma_x \rangle >0$. Then, by Assumption [Assumption 37](#assumption:inner_product){reference-type="ref" reference="assumption:inner_product"}, $\phi_1(y_{\alpha,\varepsilon}) > \phi_1(x_{\alpha,\varepsilon}) \geq 0$ and $\phi_2 (y_{\alpha,\varepsilon}) > 0$. Then, we can write the first two lines of [\[eq:fundamentalseq_two\]](#eq:fundamentalseq_two){reference-type="eqref" reference="eq:fundamentalseq_two"} as $$\begin{aligned} &\int_{F_M} \sum_{\gamma\in\Gamma_1 \cup \Gamma_2} \underbrace{\left(\frac{\phi^{z,\gamma}_1(x_{\alpha,\varepsilon}) \phi^{z,\gamma}_2 (x_{\alpha,\varepsilon})}{\phi^{z,\gamma}_1(y_{\alpha,\varepsilon}) \phi^{z,\gamma}_2 (y_{\alpha,\varepsilon})} - 1 \right)}_{(1)}\underbrace{\left( \phi^{z,\gamma}_1(y_{\alpha,\varepsilon}) \phi^{z,\gamma}_2 (y_{\alpha,\varepsilon}) \right) \left(e^{\langle p_{\alpha,\varepsilon}, \gamma_x \rangle}-1 \right)}_{(2)}\, d\theta +\\ & \inf_{\varphi \in C^2(F_M)} \int_{F_M} e^{-\varphi} \left(\sum_{\gamma\in\Gamma_2} \underbrace{\left(\frac{\phi_1^{z,\gamma}(x_{\alpha,\varepsilon}) \phi^{z,\gamma}_2 (x_{\alpha,\varepsilon})}{\phi_1^{z,\gamma}(y_{\alpha,\varepsilon}) \phi^{z,\gamma}_2 (y_{\alpha,\varepsilon})} - 1 \right)}_{(3)}\underbrace{\left( \phi^{z,\gamma}_1(y_{\alpha,\varepsilon})\phi^{z,\gamma}_2(y_{\alpha,\varepsilon}) \right) e^{\langle p_{\alpha,\varepsilon}, \gamma_x \rangle}}_{(4)} \right)e^{\varphi}\end{aligned}$$ For $\alpha \to \infty$, by Assumption [Assumption 37](#assumption:inner_product){reference-type="ref" reference="assumption:inner_product"} (2), $(1)$ and $(3)$ are negative and $(2)$ and $(4)$ are positive and bounded by [\[eq:boundsupersolution\]](#eq:boundsupersolution){reference-type="eqref" reference="eq:boundsupersolution"}. Then, for $\alpha$ big the second and third lines of [\[eq:fundamentalseq\]](#eq:fundamentalseq){reference-type="eqref" reference="eq:fundamentalseq"} are bounded above from zero and this concludes the proof.  ◻ **Proposition 39** (Existence of viscosity solutions). *Consider $\mathcal{H}(x,p)$ as in [\[hamiltonian_example\]](#hamiltonian_example){reference-type="eqref" reference="hamiltonian_example"}. Suppose Assumptions [Assumption 36](#assumption:irreducibility){reference-type="ref" reference="assumption:irreducibility"} and [Assumption 37](#assumption:inner_product){reference-type="ref" reference="assumption:inner_product"}. Then, the function $R(\lambda)h$ defined in [\[resolvent\]](#resolvent){reference-type="eqref" reference="resolvent"} is the unique viscosity solution to $f-\lambda \mathbf{H}f = h$.* *Proof.* We show that $\mathcal{H}(x,p)$ in [\[hamiltonian_example\]](#hamiltonian_example){reference-type="eqref" reference="hamiltonian_example"} verifies Assumption [Assumption 18](#assumption:Hamiltonian_vector_field){reference-type="ref" reference="assumption:Hamiltonian_vector_field"}. Then, the result follows from Theorem [Theorem 10](#theorem:existence_of_viscosity_solution){reference-type="ref" reference="theorem:existence_of_viscosity_solution"}. We need to show that $\partial_p \mathcal{H}(x,p) \cap T_E(x) \neq \emptyset$ for all $x\in E$. We prove this in two steps: 1. We firstly show that for an Hamiltonian of the type [\[hamiltonian_example\]](#hamiltonian_example){reference-type="eqref" reference="hamiltonian_example"}, $\partial_p \left[\int_{F_M} V_{x,p} + e^{-\varphi} L_{x,p} e^{\varphi} \, d\theta \right] \subseteq \partial_p \mathcal{H}(x,p)$; 2. Secondly, we show that $\partial_p \left[\int_{F_M} V_{x,p} + e^{-\varphi} L_{x,p} e^{\varphi} \, d\theta \right] \cap T_E(x) \neq \emptyset$. *Proof of step 1.* We call $\Psi_{x,\varphi,\theta} (p) = \int_{F_M} V_{x,p} + e^{-\varphi} L_{x,p} e^{\varphi} \, d\theta$. Firstly, recall that for a general convex function $p \mapsto \Phi(p)$ we denote $$\partial_p \Phi (p_0) = \{ \xi \in {{\mathbb R}}^l \, : \, \Phi(p) \geq \Phi(p_0) + (p-p_0)\cdot \xi \quad \forall p\in {{\mathbb R}}^l\}.$$ Let $\xi \in \partial_p \Psi_{x,\varphi,\theta} (p)$ and $q \in {{\mathbb R}}^l$. We call $\varphi_q$ the optimal map for $\mathcal{H}(x,q)$ and $\theta_p$ the optimal measure for $\mathcal{H}(x,p)$. Then we have $$\begin{aligned} \mathcal{H}(x,q) & \geq \int_{F_M} V_{x,q} + e^{-\varphi_q}L_{x,q} e^{\varphi_q} \, d\theta_p\\ &\geq \int_{F_M} V_{x,p} + e^{-\varphi_q}L_{x,p} e^{\varphi_q} \, d\theta_p + (q-p)\cdot \xi \\ &\geq \inf_\varphi \int_{F_M} V_{x,p} + e^{-\varphi}L_{x,p} e^{\varphi} \, d\theta_p + (q-p)\cdot \xi \\ &= \mathcal{H}(x,p) + (q-p) \cdot \xi\end{aligned}$$ showing that $\xi \in \partial_p \mathcal{H}(x,p)$ and hence that $\partial_p \Psi_{x,\varphi,\theta}(p)\subseteq \partial_p \mathcal{H}(x,p)$. *Proof of step 2.* We claim that $\partial_p \Psi_{x,\theta,\varphi}(p) \cap T_E(x)$ is not empty for every $x\in E$. Indeed, note that $$T_{[0,\infty)^l}(x) = \Pi_{i=1}^l T_i,$$ with $$T_i = \begin{cases} {{\mathbb R}}\qquad &\text{if $x_i \neq 0$,}\\ [0,\infty) \qquad &\text{if $x_i = 0$.} \end{cases}$$ Then, if $x_i \neq 0$, $\partial_{p_i} \Psi _{x_i, \varphi, \theta} (p) \subseteq T_i$ trivially. If $x_i = 0$, by assumption [Assumption 37](#assumption:inner_product){reference-type="ref" reference="assumption:inner_product"}(3), if $\gamma_{x_i} <0$, $r(x,y,\gamma) = 0$. Hence, we can conclude that $\partial_{p_i} \Psi_{x_i, \varphi, \theta} (p) \geq 0$. And this conclude the proof. ◻ ## Construction of the Hamiltonian with a Large deviations approach {#subsection:example_Hamiltonian} In this subsection we show how the Hamiltonian in [\[hamiltonian_example\]](#hamiltonian_example){reference-type="eqref" reference="hamiltonian_example"} can be obtained from the study of the multi-scale Markov process arising in biological systems described above. In a large deviations context the aim is to characterize the function $I:C_E [0,\infty) \to [0,\infty]$, with $C_E [0,\infty)$ the space of all continuous path $x: [0,\infty) \to E$, such that $$\mathbb{P}\left[X_N \approx x \right] \sim e^{-N I(x)}.$$ In Chapter 8 of [@FK06], it is shown that the function $I$ is characterized by the unique solution of the Hamilton-Jacobi equation $$f-\lambda \mathbf{H}f = h,$$ with $\mathbf{H}f(x) = \mathcal{H}(x,\nabla f(x))$ found in three steps: 1. Given the *generator* $A_N$ of the process $(X_N,I_N)$, define the *non linear generator* $H_N f = \frac{1}{N}e^{-Nf}A_N e^{Nf}$ for $f$ such that $e^{Nf}\in D(A_N)$; 2. Given $f\in C(E)$ and $h \in C(E\times F_M)$ define $f_N(x,y) = f(x) + N^{-1}h(x,y)$ and find $H_{h}$ such that $\lim_{N\to \infty} H_N f_N(x,y) = H_{h}(x,\nabla f(x), y)$; 3. [\[item: FengKurtz_eigenvalue\]]{#item: FengKurtz_eigenvalue label="item: FengKurtz_eigenvalue"}For every $x$, find $h_x$ such that $H_{h_x}$ does not depend on $y$. Calling $p= \nabla f(x)$, this is equivalent to solve an eigenvalue problem : for all $x\in E$ and $p\in {{\mathbb R}}^l$, there exists $\mathcal{H}(x,p)$ and $\bar{h}$ such that $H (x,p,y) \bar{h}(x,y) = \mathcal{H}(x,p) \bar{h}(x,y)$. In this subsection, we show that in the analysis of the limit behaviour of the process $(X_N, Y_N)$, the three steps above give the Hamiltonian [\[hamiltonian_example\]](#hamiltonian_example){reference-type="eqref" reference="hamiltonian_example"} studied in the previous subsection. In [@Po18], the author studies the limit behaviour of a similar example following the method developed by Feng and Kurtz described in the steps above. However, in contrast with what it has been done in the above cited work, with our approach it is not necessary to find and calculate an explicit expression of the eigenvalue $\mathcal{H}$ found in step [\[item: FengKurtz_eigenvalue\]](#item: FengKurtz_eigenvalue){reference-type="ref" reference="item: FengKurtz_eigenvalue"}. Therefore, our method presents an effective solution that can be applied to similar scenarios in which the eigenvalue problem can not be easily solved. We have the following result. **Proposition 40** (Markov processes on multiple time-scale). *Consider the Markov process $(X_N,Y_N)$ having the operator [\[eq:generator_rescaled\]](#eq:generator_rescaled){reference-type="eqref" reference="eq:generator_rescaled"} as generator. Suppose Assumptions [Assumption 36](#assumption:irreducibility){reference-type="ref" reference="assumption:irreducibility"} and [Assumption 37](#assumption:inner_product){reference-type="ref" reference="assumption:inner_product"}. Then, the following hold:* 1. *Let $f\in C(E)$ and $h\in C(E\times F_M)$. Define $f_N (x,y)= f(x) + N^{-1} h(x,y)$ and $H_N f =\frac{1}{N} e^{-Nf}A_N e^{Nf}$ provide $e^{Nf} \in D(A_N)$. Then, $$\lim_N H_N f_N(x,y) = V(y; x, \nabla_xf(x))+ e^{-h(x,y)}L_{x,\nabla_xf(x)}e^{h(x,y)},$$ with $$V(y;x,p) = \sum_{\gamma=(\gamma_x,\gamma_y)\in\Gamma_1} r(x,y,\gamma) (e^{\langle p, \gamma_x \rangle}-1) + \sum_{\gamma=(\gamma_x,\gamma_y)\in\Gamma_2} r(x,y,\gamma) (e^{\langle p , \gamma_x\rangle}-1),$$ and $$\begin{aligned} L_{x,p}f(x,y) = &\sum_{\gamma=(\gamma_x,\gamma_y)\in\Gamma_2} r(x,y,\gamma) e^{\langle p, \gamma_x\rangle} [ f(x,y+\gamma_y)- f(x,y)]\\ &+ \sum_{\gamma=(\gamma_x,\gamma_y)\in\Gamma_3} r(x,y,\gamma) [f(x,y+\gamma_y) - f(x,y)]. \end{aligned}$$* 2. *There exists a unique constant $\mathcal{H}(x,p)$ and a unique function $g(x,y)$ solving the eigenvalue problem $(V(y;x,p) + L_{x,p})g(x,y)=\mathcal{H}(x,p)g(x,y)$.* 3. *Given the map $\mathbf{H}f = \mathcal{H}(x,\nabla f(x))$, the Hamilton--Jacobi equation $f- \lambda \mathbf{H}f= h$ verifies the comparison principle.* *Proof sketch..* Recalling the generator $A_N$ in [\[eq:multiple-time: generator_extended\]](#eq:multiple-time: generator_extended){reference-type="eqref" reference="eq:multiple-time: generator_extended"}, note that the exponential generator $H_N$ acting on the test functions $f_N$ are $$\begin{aligned} H_Nf_N(x,y)=& \sum_{\gamma \in \Gamma_1} r(x,y,\gamma) \left[ e^{N\left(f\left(x+\frac{1}{N}\gamma_x\right) - f(x)\right)+h\left(x+\frac{1}{N}\gamma_x,y\right)- h(x,y)}-1\right]\\ &+ \sum_{\gamma\in\Gamma_2} r(x,y,\gamma) \left[e^{N\left(f\left(x+\frac{1}{N}\gamma_x\right)- f(x)\right)+h\left(x+\frac{1}{N}\gamma_x,y+\gamma_y\right)-h(x,y)}-1\right]\\ &+\sum_{\gamma\in\Gamma_3} r (x,y,\gamma) \left[e^{h(x,y+\gamma_y)-h(x,y)}-1\right].\end{aligned}$$ As a consequence, its limit is $$\begin{aligned} \lim_N H_N f_N (x,y) = &\sum_{\gamma\in\Gamma_1} r(x,y,\gamma) [e^{\langle \nabla f(x), \gamma_x\rangle} - 1] + \sum_{\gamma\in\Gamma_2} r(x,y,\gamma) [e^{\langle \nabla f(x), \gamma_x\rangle} e^{h(x,y+\gamma_y)- h(x,y)}-1]\\ & \sum_{\gamma\in\Gamma_3} r(x,y,\gamma) [ e^ {h(x,y+\gamma_y)- h(x,y)}-1] \end{aligned}$$ and the first claim is proven. The second point follows from Assumption [Assumption 36](#assumption:irreducibility){reference-type="ref" reference="assumption:irreducibility"} and Perron -- Frobenius Theorem. The third point follows from the fact that, being the eigenvalue of $V_{x,p} + L_{x,p}$, $\mathcal{H}(x,p)$ is the Hamiltonian [\[hamiltonian_example\]](#hamiltonian_example){reference-type="eqref" reference="hamiltonian_example"} and from Theorem [Theorem 38](#theorem:comparison_example){reference-type="ref" reference="theorem:comparison_example"} (see [@DoVa75a] for more details about the representation of the eigenvalue). ◻ [^1]: Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands. *E-mail address*: s.dellacorte\@tudelft.nl [^2]: Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands. *E-mail address*: r.c.kraaij\@tudelft.nl

arxiv_math

--- abstract: | This paper presents a decentralized volt-var control (VVC) and network reconfiguration strategy to address the challenges arising from the growing integration of distributed energy resources, particularly photovoltaic (PV) generation units, in active distribution networks. To reconcile control measures with different time resolutions and empower local control centers to handle intermittency locally, the proposed approach leverages a two-stage distributionally robust optimization; decisions on slow-responding control measures and set points that link neighboring subnetworks are made in advance while considering all plausible distributions of uncertain PV outputs. We present a decomposition algorithm with an acceleration scheme for solving the proposed model. Numerical experiments on the IEEE 123 bus distribution system are given to demonstrate its outstanding out-of-sample performance and computational efficiency, which suggests that the proposed method can effectively localize uncertainty via risk-informed proactive timely decisions. author: - "Geunyeong Byeon , Kibaek Kim  [^1] [^2] [^3]" bibliography: - ref.bib title: Distributionally Robust Decentralized Volt-Var Control with Network Reconfiguration --- Distributionally robust, active distribution networks, multi-timescale, network partition, decentralized, probabilistic forecast, coordinated voltage control # Introduction energy resources (DERs), particularly photovoltaic (PV) systems, are increasingly causing uncertain and intermittent influxes of electric power at the edge of distribution grids. Traditional operating practices for distribution systems, which monitor and control a small subset of network components on a slow timescale, are inadequate to handle instant voltage fluctuations and overvoltage problems caused by the volatile bidirectional power flow, which requires active controls based on optimal power flow (OPF) [@jin2020local]. Because of the limitations, distribution operators often limit the capacity of PV systems, known as hosting capacity [@ismael2019state]. One of the key reliability requirements that limit PV integration is a national standard for voltage regulation, such as ANSI C84.1, which demands that voltages be maintained within safe limits across the grid. In order to regulate voltages, on-load tap changers (OLTCs) and switchable capacitor banks (CBs) have been traditionally used. OLTCs alter the voltage on the secondary winding, while CBs provide reactive power near demand nodes. Recently, network reconfiguration via line switching and reactive power support by smart inverters have emerged as promising control measures for handling the DER-related issues [@zhou2021three]; the new IEEE 1547-2018 standard even stipulates the reactive power support capability of smart inverters for DER integration. Recent studies have developed decentralized approaches for effective operations of such legacy and emerging control devices in the distribution grid (e.g., [@antoniadou2017distributed; @li2018decentralized; @li2019distributed; @feng2017decentralized]). These approaches aim to balance between centralized and distributed methods by assuming a hierarchical control system that consists of local control centers (LCCs) and a central coordinator (CC). Specifically, the CC makes the slow timescale decisions on OLTCs, capacitors, and/or line switches and also determines some set-points or policies for coordinating the LCCs, each of which independently makes operational decisions on local smart inverters at a much higher time resolution to respond to the intermittent voltage changes. In order to define the LCCs' governing regions, network partitioning methods are proposed in [@li2018decentralized; @li2019distributed], and a componentwise decomposition is employed in [@feng2017decentralized]. In addition, the CC and/or LCCs are often modeled as optimization problems [@li2019distributed] or a machine learning architecture/reinforcement learning (RL) agents for fast control and coordination [@sun2021optimal; @sun2021two; @cao2021deep; @liu2021robust]. However, none of the literature considers network reconfiguration, and RL-based approaches face challenges such as scalability, safety, and robustness, as discussed in [@chen2022reinforcement]. Uncertain intermittency of PV generation further complicates the distribution systems control. A risk-aware OPF-based decentralized control was proposed in [@li2019distributed] based on a two-stage adaptive robust optimization (TSARO) problem that considers an interval estimate of uncertain PV outputs and demand. The solutions to TSARO can be conservative, however, as it ignores the likelihood of each outcome [@ben2002robust]. Alternatively, two-stage distributionally robust optimization (TSDRO) provides a more flexible approach for making risk-informed decisions for the situations when probability distributions of uncertain factors are hard to specify because of a limited number of samples or nonstationarity. TSDRO hedges against a worst-case probability distribution within a set of plausible probability distributions, namely, an ambiguity set. Several TSDRO models are proposed for dispatch and/or network reconfiguration in [@zhou2020linear; @zheng2020adaptive; @zhou2021three; @chen2021fast; @liu2021data; @zhai2022distributionally; @maghami2023two], but none of the work accounts for decentralization. In [@huang2020distributionally], decentralized dispatch is considered in a TSDRO setting, where an ambiguity set is constructed on a finite sample space. In this paper we propose a TSDRO problem for the decentralized VVC and reconfiguration of distribution systems with uncertain PV outputs. The ambiguity set is constructed with all the probability distributions that are within a certain Wasserstein distance from a reference distribution \[22\], namely the Wasserstein ambiguity set, to account for the uncertain PV outputs. Figure [\[fig:communication-network\]](#fig:communication-network){reference-type="ref" reference="fig:communication-network"} illustrates the proposed TSDRO approach; in the first stage, the CC decides on slow-responding control measures and determines the set points for variables linking neighboring subnetworks while considering the Wasserstein ambiguity set; and in the second stage, each LCC operates its region by using PV inverters while meeting the set points. TSDRO over a Wasserstein ambiguity set is particularly a good choice for this application since it is robust even with a limited number of samples and can hedge against potential bias in the probabilistic forecast [@li2020review]. To the best of our knowledge, no previous work investigated distributionally robust decentralized VVC and reconfiguration over a Wasserstein ambiguity set. The key contributions of this paper are threefold. First, it introduces a new formulation of the TSDRO model that effectively coordinates multiple LCCs and various control measures in the presence of intermittent PV generation. Notably, it presents a concise formulation for reconfiguring radial networks. Second, it proposes a scheme that enhances the solution approach developed in [@byeon2022two] for solving the TSDRO model. Unlike the heuristic solution presented in [@li2019distributed] for their TSARO model, the solution approach provides an exact solution to the TSDRO model. Third, the paper presents numerical results from a case study that demonstrate the potential advantages of the TSDRO model over two-stage robust optimization (TSRO) and the sample average approximation (SAA) of two-stage stochastic optimization (TSSO) in terms of out-of-sample performance. Specifically, the proposed method, even with a limited sample size, ensures reliable load shedding and power import in a majority of scenarios via an efficient and effective utilization of PVs. The rest of this paper is organized as follows. Section [2](#sec:preliminaries){reference-type="ref" reference="sec:preliminaries"} presents notations and preliminaries, and Section [3](#sec:centralized){reference-type="ref" reference="sec:centralized"} formalizes the decentralized control problem. Section [4](#sec:solution){reference-type="ref" reference="sec:solution"} briefly reviews a solution method, and Section [5](#sec:case-study){reference-type="ref" reference="sec:case-study"} analyzes the out-of-sample performance of the model on a test system. Section [6](#sec:conclusion){reference-type="ref" reference="sec:conclusion"} concludes the paper. # Notations and Preliminaries {#sec:preliminaries} The parameters and variables are summarized in Tables [1](#table:param){reference-type="ref" reference="table:param"} and [2](#table:var){reference-type="ref" reference="table:var"}, respectively. We denote sets with calligraphic letters (e.g., $\mathcal{N}$ and $\mathcal{E}$) and use capital letters to denote matrices, unless otherwise stated. We denote random numbers or vectors with the tilde and their realizations without the tilde. For a set $\mathcal{A}$ and a bus $i \in \mathcal{N}$, we let $\mathcal{A}(i)$ denote the subset of $\mathcal{A}$ that is associated with $i$ (e.g., $\mathcal{K}(i)$ is the set of distributed generators located at $i$). For an integer $n$, $[n]$ denotes a set $\{1,\cdots, n\}$. Sets with subscript $i$ denote its subset associated with subregion $i$; for example, $\mathcal{S}_i$ denotes the set of shunt capacitors located in subregion $i$. We assume that the uncertain PV output levels $\tilde \xi_i$ associated with subregion $i$ are independent of $\tilde \xi_j$ for $j \neq i$. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Notation Description ----------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------- $\mathcal{G} = (\mathcal{N},\mathcal{E})$ undirected graph representing the distribution grid $\quad \mathcal{N}$ set of buses, indexed by $j \in \{1, \cdots, |\mathcal{N}|\}$, where $1$ denotes the substation bus $\quad \mathcal{E}$ set of lines, indexed by $\{(j,k) \in \mathcal{N} \times \mathcal{N}: j< k\}$ $\mathcal{E}^u \subseteq \mathcal{E}$ set of switchable lines $\mathcal{K} = \mathcal{K}^D \cup \mathcal{K}^{PV}$ set of energy resources, where $\mathcal{K}^{D}$ and $\mathcal{K}^{PV}$ denote the set of dispatchable units and $k$ residential PVs, respectively $\overline g_{j}^p + {\bf i} \overline g_j^q$ upper bound on power output of $j \in \mathcal{K}^D$ $\underline g_{j}^p + {\bf i} \underline g_j^q$ lower bound on power output of $j \in \mathcal{K}^D$ $\overline p_l$ installed capacity of $l \in \mathcal{K}^{PV}$ $\mathcal{S}$ set of shunt capacitors $\overline q_k$ reactive power rating of $k \in \mathcal{S}$ $\overline v_j, \underline v_j$ upper and lower limits of voltage magnitude squared at $j \in \mathcal{N}$ $d^p_j + {\bf i} d^q_j$ load on $j \in \mathcal{N}$ $\overline\ell_{jk}$ limit on current magnitude squared passing through $(j,k) \in \mathcal{E}$ $r_{jk} + {\bf i} x_{jk}$ complex impedance of $(j,k) \in \mathcal{E}$ $\beta_v, \beta_{shed}, \beta_{slack}$ weights on objective terms for substation voltage magnitude, and consensus violation, respectively Network decomposition $\mathcal{L}$ set of LCCs $\mathcal{N}_i, \mathcal{E}_i$ set of buses and lines of the subnetwork governed by LCC $i \in \mathcal{L}$, respectively $\mathcal{C} \subseteq \mathcal{E}$ set of lines connecting each pair of neighboring subnetworks $k_i$ number of PVs in the subregion $i \in \mathcal{L}$ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Parameters --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Notation Description ------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Continuous variables $v_{j}$ voltage magnitude squared at $j \in \mathcal{N}$ $p_{jk} + {\bf i} q_{jk}$ complex power flow toward $(j,k) \in \mathcal{E}$ at $j$ $\ell_{jk}$ current magnitude squared on $(j,k) \in \mathcal{E}$ $g^p_{j} + {\bf i} g^q_{j}$ power generation of $j \in \mathcal{K}$ ${\bf i} q^c_{k}$ reactive power generated by $k \in \mathcal{S}$ $f_{jk}$ artificial flow between $(j,k) \in \mathcal{E}$ ${\theta}^p_{j} + {\bf i} {\theta}^q_{j}$ amount of load shed at $j \in \mathcal{N}$ $\kappa_{jk}, \iota_{jk}, o_{jk}$ auxiliary variables for converting [\[eq:2nd:power-flow:power:conv\]](#eq:2nd:power-flow:power:conv){reference-type="eqref" reference="eq:2nd:power-flow:power:conv"} into 3D Lorentz cone constraints Binary variables $u_{jk}$ 1 if $(j,k) \in \mathcal{E}^u$ is open (i.e., its end buses are not adjacent), 0 otherwise $w_{k}$ 1 if $k\in \mathcal{S}$ is on, 0 otherwise $s_{jk}$ adjacency of $j ,k \in \mathcal{N}$ $s_{0j}$ 1 if $j \in \mathcal{N}$ receives a nontrivial artificial flow from the dummy node, 0 otherwise Random variables $\tilde \xi_{i}=(\tilde \xi_{il})_{l\in[k_i]}$ random output level of PVs in subregion $i$ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Variables In this paper we pose the control problem as a TSDRO problem that features two types of decisions: a first-stage CC's action $x$ that is made before the uncertain PV output levels $(\tilde \xi_i)_{i \in \mathcal{L}}$ are realized and a second-stage LCC's action $y_i$ that is made to support $x$ after observing a realization $\xi_i$ of $\tilde \xi_i$ for all $i \in \mathcal{L}$. For each $i \in \mathcal{L}$, we let $\mathcal{P}_i$ represent a set of plausible probability distributions of $\tilde \xi_i$, which is referred to as an *ambiguity set*. The aim of TSDRO is to find $x$ that hedges against a worst-case probability distributions of $\tilde \xi_i$'s among those in $\mathcal{P}_i$'s. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Notation Description --------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------- $\Xi_i := [0, 1]^{k_i}$ support of $\tilde \xi_i$ for $i \in \mathcal{L}$ $n_i$ sample cardinality (equiv. number of scenarios) of $\tilde \xi_i$ $\zeta_{i1}, \cdots, \zeta_{in_i} \in \Xi_i$ sample (equiv. scenarios) of $\tilde \xi_i$ $P_{ij}$ probability associated with scenario $\zeta_{ij}$ $\hat{\mathbb{P}}_i:= \sum_{j \in [n_i]} P_{ij}\delta_{\zeta_{ij}}$ discrete reference distribution of $\tilde \xi_i$, where $\delta_{\zeta_{ij}}$ denotes Dirac measure concentrated on $\{\zeta_{ij}\}$ $p$ scalar from $[1,\infty]$ that defines the $l_p$-norm over $\Xi_i$ for all $i \in \mathcal{L}$ $\epsilon_i$ Wasserstein ball radius for $i \in \mathcal{L}$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Parameters for Wasserstein ambiguity set Specifically, for each $i \in \mathcal{L}$, we use a *Wasserstein ambiguity set*, parameters of which are given in Table [3](#table:param:W){reference-type="ref" reference="table:param:W"}. The Wasserstein ambiguity set is composed of probability distributions that are within $\epsilon_i$-distance of a reference distribution $\hat{\mathbb{P}}_i$ with regard to a Wasserstein metric $d(\cdot, \cdot)$, for some scalar $\epsilon_i > 0$. Let ${\mathcal{P}_i}(\Xi_i)$ denote the collection of all probability distributions $\mathbb{Q}$ on $\Xi_i$ with a finite first moment. A *Wasserstein distance* between two probability distributions $\mathbb{Q}_1$ and $\mathbb{Q}_2$ in $\mathcal{P}_i(\Xi_i)$ is defined as $$d(\mathbb{Q}_1 ,\mathbb{Q}_2):=\inf _{\gamma \in \Gamma (\mathbb{Q}_1 ,\mathbb{Q}_2 )}\left\{\int _{\Xi_i\times \Xi_i}\|\xi_1- \xi_2\|_p\, \mathrm{d}\gamma (\xi_1,\xi_2)\right\},$$ where $\Gamma (\mathbb{Q}_1 ,\mathbb{Q}_2 )$ denotes the collection of all probability distributions with marginals $\mathbb{Q}_1$ and $\mathbb{Q}_2$, respectively, and $\|\cdot\|_p$ is an $l_p$-norm in $\mathbb{R}^{k_i}$ for some $p \in [1,\infty]$. In this paper we use a discrete distribution $\hat{\mathbb{P}}_i$ as the reference (e.g., the empirical distribution or a probabilistic forecast given as a histogram). Formally, the Wasserstein ambiguity set is defined as $$\mathcal{P}_i := \{\mathbb{Q} \in \mathcal{P}_i(\Xi_i): d(\mathbb{Q}, \hat{\mathbb{P}}_i) \le \epsilon_i\}. % \label{eq:ambiguity-set}$$ # Distributionally Robust Reconfiguration and Decentralized Control {#sec:centralized} We present a TSDRO formulation for distributionally robust decentralized VVC and reconfiguration under uncertain PV generation. On a rolling basis, the CC solves [\[prob\]](#prob){reference-type="eqref" reference="prob"} to decide on the first-stage decision $x$ that contains the consensus decision $\mu$ for coordinating LCCs (e.g., voltages and power flows on coupling lines), together with the slow-responding control decisions $\sigma$ (e.g., OLTC operations and switches for lines and capacitor banks). The decision is made while considering the worst-case expected real-time system cost $\sup_{\mathbb{P}_i \in \mathcal{P}_i} \mathbb{E}_{\tilde\xi_i \sim \mathbb{P}_i}[Z_i(x,\tilde\xi_i)]$ for each $i \in \mathcal{L}$, where $Z_i(x,\xi_i)$ computes the real-time cost of subregion $i \in \mathcal{L}$ for given first-stage decision $x$ and a realization $\xi_i$ of $\tilde \xi_i$: $$\begin{aligned} \min_{x} \ & c^T x + \sum_{i \in \mathcal{L}} \sup_{\mathbb P_i \in \mathcal{P}_i} \mathbb{E}_{\tilde\xi_i \sim \mathbb{P}_i} [ Z_i(x, \tilde \xi_i)]\\ \mbox{s.t.} \ & x = (\sigma, \mu),\\ \ & \sigma = (u, v_1, w) \in \mathcal{A},\label{prob:A}\\ \ & \mu = (v_k, v_l, \ell_{kl}, p_{kl}, q_{kl})_{(k,l) \in \mathcal{C}} \in \mathcal{B}, % \ & u \in \mb B^{|\mc E_u|}, v_1 \in \mb R, w \in \mb B^{|\mc S|}, \end{aligned}$$[\[prob\]]{#prob label="prob"} where $c^T x := \beta_v v_1 + \sum_{(i,j) \in \mathcal C} r_{ij}\ell_{ij}$ and $\mathcal{A}$ and $\mathcal{B}$ respectively denote the feasible regions of $\sigma$ and $\mu$. The detailed definitions of $\mathcal{A}$, $\mathcal{B}$, and $Z_i$ are discussed in the following sections. The inclusion of the objective term $\beta_v v_1$ aims to achieve conservation voltage reduction (CVR) to some extent when dealing with voltage-dependent loads. To be more accurate, one can include the sum of voltage magnitudes squared over all voltage-dependent loads as in [@farivar2012optimal]. **Remark 1** (Extendability). *For simplicity, we model the OLTC decision, $v_1$, as a continuous variable. However, one can readily introduce additional binary variables and constraints for the tap position to [\[prob\]](#prob){reference-type="eqref" reference="prob"} as in [@li2017distributed].* **Remark 2** (Rolling decision horizon). *Problem [\[prob\]](#prob){reference-type="eqref" reference="prob"} is designed to be solved on a rolling basis. As illustrated in Figure [1](#fig:decision-rolling-horizon){reference-type="ref" reference="fig:decision-rolling-horizon"}, on a rolling basis, the CC is given a probabilistic forecast $\hat{\mathbb{P}}_i$ of $\tilde \xi_i$ for upcoming time periods. With $\{\hat{\mathbb{P}}_i\}_{i \in \mathcal{L}}$ updated, the CC solves [\[prob\]](#prob){reference-type="eqref" reference="prob"} to obtain $x$ to be in effect in the next period, say, $x^{t}$; then, $x^{t}$ can be used in the following time period to enable autonomous operations of LCCs. More specifically, at time $t$, each LCC $i$ observes a realization $\xi^{t}_i$ of $\tilde \xi^t_i$ and operates its region by solving $Z_i(x^{t}, \xi^{t}_i)$ independently.* {#fig:decision-rolling-horizon width="0.9\\linewidth"} ## Compact radiality constraints $\mathcal{A}$ {#sec:model:first-stage} To define $\mathcal{A}$, we first establish the following constraints on line switching: [\[eq:1st:switch\]]{#eq:1st:switch label="eq:1st:switch"} $$\begin{aligned} &s_{jk} = 1, \ \forall (j,k) \in \mathcal{E} \setminus \mathcal{E}_u, \label{eq:1st:switch:line-without-switch}\\ &s_{jk} = 1 - u_{jk}, \ \forall (j,k) \in \mathcal{E}_u. \label{eq:1st:switch:edge-on-off}\end{aligned}$$ Note that, by [\[eq:1st:switch\]](#eq:1st:switch){reference-type="eqref" reference="eq:1st:switch"}, $s_{jk}$ indicates whether buses $j$ and $k$ are adjacent or not. Therefore, $s$ represents the topology of the distribution grid in effect, which we denote by $\mathcal{G}' = (\mathcal{N}, \mathcal{E}')$ with $\mathcal{E}' :=\{(j,k) \in \mathcal{E}: s_{jk} = 1\}$. {#fig:network-flow width="0.3\\linewidth"} Radial topology is often required for the operations of distribution grids [@lei2020radiality]. We propose a new compact formulation that guarantees network radiality. Consider a network flow problem defined on an extended graph of $\mathcal{G}'$ with a dummy node, indexed by $0$, and a set of lines connecting the dummy node to some buses in $\mathcal{N}$; the dummy node is the supply node that supplies $|\mathcal{N}|$ units of flow, and each node in $\mathcal{N}$ demands a single unit of flow. We define the following set of constraints to formulate the network flow: [\[eq:1st:network-flow\]]{#eq:1st:network-flow label="eq:1st:network-flow"} $$\begin{aligned} &0 \le f_{0j} \le |\mathcal{N}| s_{0j}, \ \forall j \in \mathcal{N}, \label{eq:1st:network-flow:1}\\ &-|\mathcal{N}| s_{jk} \le f_{jk} \le |\mathcal{N}| s_{jk}, \ \forall (j,k) \in \mathcal{E}, \label{eq:1st:network-flow:2}\\ &f_{0j} + \sum_{(k,j) \in \mathcal{E}} f_{kj} - \sum_{(j,k) \in \mathcal{E}} f_{jk} = 1, \ \forall j \in \mathcal{N},\label{eq:1st:network-flow:demand}\\ &\sum_{j \in \mathcal{N}} f_{0j} = |\mathcal{N}|.\label{eq:1st:network-flow:supply} \end{aligned}$$ By [\[eq:1st:network-flow:1\]](#eq:1st:network-flow:1){reference-type="eqref" reference="eq:1st:network-flow:1"}--[\[eq:1st:network-flow:2\]](#eq:1st:network-flow:2){reference-type="eqref" reference="eq:1st:network-flow:2"}, nonzero flow $f_{jk}$ is allowed between adjacent buses $(j,k)$ only. Equations [\[eq:1st:network-flow:demand\]](#eq:1st:network-flow:demand){reference-type="eqref" reference="eq:1st:network-flow:demand"}--[\[eq:1st:network-flow:supply\]](#eq:1st:network-flow:supply){reference-type="eqref" reference="eq:1st:network-flow:supply"} respectively formulate the flow balance equation for the demand and supply nodes. Then, the following constraint ensures the radial topology as stated in Proposition [\[prop:radiality\]](#prop:radiality){reference-type="ref" reference="prop:radiality"}, the proof of which is given in Appendix [7](#appendix:radiality){reference-type="ref" reference="appendix:radiality"}: $$\sum_{(j,k) \in \mathcal{E}} s_{jk} \le |\mathcal{N}| - \sum_{j\in \mathcal{N}} s_{0j}.\label{eq:1st:tree}$$ **Proposition 1**. *Let $\mathcal{G}'=(\mathcal{N}, \mathcal{E}')$ be the graph with $\mathcal{E}' \subseteq \mathcal{E}$ defined as $\{(j,k) \in \mathcal{E}: s_{jk} = 1\}$. Then, [\[eq:1st:network-flow\]](#eq:1st:network-flow){reference-type="eqref" reference="eq:1st:network-flow"} and [\[eq:1st:tree\]](#eq:1st:tree){reference-type="eqref" reference="eq:1st:tree"} ensure the radiality of $\mathcal{G}'$. [\[prop:radiality\]]{#prop:radiality label="prop:radiality"}* To summarize, the feasible set $\mathcal{A}$ is given as $$\begin{aligned} \mathcal{A} := & \left\{(u, v_1, w) \in \mathbb{B}^{|\mathcal{E}_u|}\times \mathbb{R} \times \mathbb{B}^{|\mathcal{S}|}: \eqref{eq:1st:switch}-\eqref{eq:1st:tree}\right\},\end{aligned}$$ where $s$ and $f$ are used as auxiliary binary and continuous variables, respectively. **Remark 3**. *In [@lei2020radiality], an extended formulation is proposed for ensuring radial topology, which is a tighter formulation but requires $2|\mathcal{N}| \cdot |\mathcal{E}|$ continuous variables, $2|\mathcal{E}|$ binary variables, and $|\mathcal{N}|^2 + 2|\mathcal{N}| \cdot |\mathcal{E}| - |\mathcal{L}| - |\mathcal{E}| +1$ constraints, which can be huge for large-scale networks. On the other hand, the proposed formulation adds at most $|\mathcal{N}|+|\mathcal{E}|$ continuous variables, $|\mathcal{N}|$ binary variables, and $2|\mathcal{N}|+|\mathcal{E}|+2$ constraints.* ## Decentralized operations cost $Z_i$ {#sec:model:second-stage} Given $\hat x$ and $\xi_i$, each LCC $i \in \mathcal{L}$ is operated to minimize generation cost, transmission loss, and load shed by utilizing local smart PV inverters. The LCC operations problem can be formulated as a second-order cone programming (SOCP) problem that takes the SOCP relaxation of the power flow physics, as proposed in [@farivar2012optimal], together with constraints on line and CB switching. The optimal decentralized operations cost $Z_i(\hat x, \xi_i)$ given the first-stage variables $\hat x$ and $\xi_i$ is computed by solving the following SOCP problem: where $C_j(\cdot)$ denotes a linear cost function for each dispatchable unit $j \in \mathcal{K}_i^D$. Equations [\[eq:2nd:power-flow:power:conv\]](#eq:2nd:power-flow:power:conv){reference-type="eqref" reference="eq:2nd:power-flow:power:conv"}, [\[eq:2nd:power-flow:Ohm:conv\]](#eq:2nd:power-flow:Ohm:conv){reference-type="eqref" reference="eq:2nd:power-flow:Ohm:conv"}, [\[eq:2nd:power-flow:balance:p\]](#eq:2nd:power-flow:balance:p){reference-type="eqref" reference="eq:2nd:power-flow:balance:p"}, [\[eq:2nd:power-flow:balance:q\]](#eq:2nd:power-flow:balance:q){reference-type="eqref" reference="eq:2nd:power-flow:balance:q"}, and [\[eq:2nd:power-flow:bound:volt\]](#eq:2nd:power-flow:bound:volt){reference-type="eqref" reference="eq:2nd:power-flow:bound:volt"} formulate the SOCP relaxation of OPF proposed in [@farivar2012optimal]. In order to prevent overloading, [\[eq:2nd:bound:current:active\]](#eq:2nd:bound:current:active){reference-type="eqref" reference="eq:2nd:bound:current:active"} is added, which puts a limit on the magnitude of current passing through each line. In order to model line switching, [\[eq:2nd:power-flow:Ohm:conv\]](#eq:2nd:power-flow:Ohm:conv){reference-type="eqref" reference="eq:2nd:power-flow:Ohm:conv"}--[\[eq:2nd:bound:current:active\]](#eq:2nd:bound:current:active){reference-type="eqref" reference="eq:2nd:bound:current:active"} are replaced with [\[eq:2nd:power-flow:Ohm:inactive:MC1\]](#eq:2nd:power-flow:Ohm:inactive:MC1){reference-type="eqref" reference="eq:2nd:power-flow:Ohm:inactive:MC1"}--[\[eq:2nd:bound:current:inactive\]](#eq:2nd:bound:current:inactive){reference-type="eqref" reference="eq:2nd:bound:current:inactive"} for $(j,k) \in \mathcal{E}^u_i$; note that [\[eq:2nd:power-flow:Ohm:inactive:MC1\]](#eq:2nd:power-flow:Ohm:inactive:MC1){reference-type="eqref" reference="eq:2nd:power-flow:Ohm:inactive:MC1"}--[\[eq:2nd:power-flow:Ohm:inactive:MC4\]](#eq:2nd:power-flow:Ohm:inactive:MC4){reference-type="eqref" reference="eq:2nd:power-flow:Ohm:inactive:MC4"} are equivalent to $v^d_{jk} = (1-u_{jk})(v_j - v_{k})$ since $v^d_{jk} = v_j - v_k$ when $u_{jk} = 0$ and $v^d_{jk} = 0$ otherwise. Therefore, when the switchable line $(j,k)$ is inactive, that is, $u_{jk}=1$, $v_{jk}^d$ and $\ell_{jk}$ become zeros, and thus $p_{jk}$ and $q_{jk}$ become zeros by [\[eq:2nd:power-flow:power:conv\]](#eq:2nd:power-flow:power:conv){reference-type="eqref" reference="eq:2nd:power-flow:power:conv"}. It is only when the line is active, namely, $u_{jk} = 0$, that [\[eq:2nd:power-flow:power:conv\]](#eq:2nd:power-flow:power:conv){reference-type="eqref" reference="eq:2nd:power-flow:power:conv"} and [\[eq:2nd:bound:current:active\]](#eq:2nd:bound:current:active){reference-type="eqref" reference="eq:2nd:bound:current:active"} are imposed for the line. In addition, for the load located at bus $j \in \mathcal{N}_i$, [\[eq:2nd:bound:loadshed\]](#eq:2nd:bound:loadshed){reference-type="eqref" reference="eq:2nd:bound:loadshed"} bounds the amount of load shed by the load, and [\[eq:2nd:gen-bound-dispatchable\]](#eq:2nd:gen-bound-dispatchable){reference-type="eqref" reference="eq:2nd:gen-bound-dispatchable"} enforces generation bounds for dispatchable generators. Equation [\[eq:2nd:shunt-capacity\]](#eq:2nd:shunt-capacity){reference-type="eqref" reference="eq:2nd:shunt-capacity"} models the reactive power support of switchable CBs, which is equivalent to $q_{k}^c = \overline q_k v_j w_k$. Equation [\[eq:2nd:gen-bound-PV\]](#eq:2nd:gen-bound-PV){reference-type="eqref" reference="eq:2nd:gen-bound-PV"} constrains the AC power outputs of each PV $l$ within its capacity, where $\xi_{il}$ represents the ratio of the DC power generated to the installed capacity, which ranges from 0 to 1, as in [@li2019distributed; @farivar2012optimal]. Equation [\[eq:2nd:consensus\]](#eq:2nd:consensus){reference-type="eqref" reference="eq:2nd:consensus"} represents the consensus constraint that fixes $x_i$ to be the corresponding values of $\hat x$ with a proper definition of a matrix $A_i$. **Remark 4**. *Instead of enforcing [\[eq:2nd:consensus\]](#eq:2nd:consensus){reference-type="eqref" reference="eq:2nd:consensus"} strictly as a constraint, we penalize any of its violations with some large penalty $\beta_{slack}$ in the objective of $Z_i(x,\tilde \xi_i)$. This soft enforcement guarantees relatively complete recourse, which the solution procedure can utilize. [\[rema:consensus\]]{#rema:consensus label="rema:consensus"}* ## Feasible region $\mathcal{B}$ of consensus variables $\mu$ The CC decides on $\mu$ that links adjacent subregions while making sure they obey the physical constraints: $$\begin{aligned} \mathcal{B} := & \left\{\mu= (v_k, v_l, \ell_{kl}, p_{kl}, q_{kl})_{(k,l) \in \mathcal{C}}: \right.\forall (k,l) \in \mathcal{C},\nonumber\\ & \qquad v_k, v_l, \ell_{kl}, p_{kl}, q_{kl} \in \mathbb{R}, \ \eqref{eq:2nd:power-flow:power:conv},\\ & \qquad \begin{cases} \mbox{Eqs. } \eqref{eq:2nd:power-flow:Ohm:conv},\eqref{eq:2nd:bound:current:active} & \mbox{ if } (k,l) \notin \mathcal{E}_u,\\ \mbox{Eqs. } \eqref{eq:2nd:power-flow:Ohm:inactive:MC1}-\eqref{eq:2nd:bound:current:inactive} & \mbox{ o.w.} \end{cases}\\ &\qquad \left.\underline v_k \le v_k \le \overline v_k, \underline v_l \le v_l \le \overline v_l \right\}. % &\qquad\left.\eqref{eq:lorentz:linear}, \mbox{ linearized } \eqref{eq:lorentz} \mbox{ by } \eqref{eq:2nd:Lorentz:linear}, \ \eqref{eq:2nd:linear:switch}\right\}.\end{aligned}$$ # Solution Approach {#sec:solution} It is shown in [@byeon2022two] that [\[prob\]](#prob){reference-type="eqref" reference="prob"} can be posed as follows: $$\begin{aligned} \min_{x \in \mathcal{X}, \lambda_i \ge 0, i \in \mathcal{L}} & c^T x + \sum_{i\in \mathcal{L}} \epsilon_i \lambda_i + \sum_{j \in [n_i]} P_{ij} t_{ij}\\ \mbox{s.t.} \ & t_{ij} \ge g_{ij}(x,\lambda_i), \forall i \in \mathcal{L}, j \in [n_i],\label{eq:extended:epigraph}\end{aligned}$$[\[prob:extended\]]{#prob:extended label="prob:extended"} where $g_{ij}(x,\lambda_i)$ is the optimal objective value of $$\begin{aligned} \max \ & h_i(x)^T \pi + b_i^T \psi_{i} - \lambda_i \|B_{ij} z_{i} -\zeta_{ij}\|\\ \mbox{s.t.} \ &z_{i} \in \mathcal{Z}_i, \pi \in \Pi_i, \psi_{i} \in \mathcal{MC}(\pi,z_{i}).\end{aligned}$$ [\[prob:sub\]]{#prob:sub label="prob:sub"} From the relatively complete recourse in Remark [\[rema:consensus\]](#rema:consensus){reference-type="ref" reference="rema:consensus"}, it is guaranteed that $g_{ij}(x,\lambda_i)$ is always bounded for any $x$ and $\lambda_i$. Therefore, Algorithm [\[algo\]](#algo){reference-type="ref" reference="algo"} can be employed to solve the problem, in which ($M^0$) denotes the problem obtained by relaxing [\[eq:extended:epigraph\]](#eq:extended:epigraph){reference-type="eqref" reference="eq:extended:epigraph"} from [\[prob:extended\]](#prob:extended){reference-type="eqref" reference="prob:extended"} [@byeon2022two]. $\texttt{k} \gets 0$; $(M) \gets$($M^0$); $LB \gets -\infty$; $UB \gets \infty$ Solve ($M$) $v^\texttt{k}, (x^\texttt{k}, \lambda^\texttt{k}, t^\texttt{k}) \gets$ the optimal objective value and solution of ($M$) Solve [\[prob:sub\]](#prob:sub){reference-type="eqref" reference="prob:sub"} with $x,\lambda_i$ fixed as $x^\texttt{k}, \lambda^\texttt{k}_i$ $g_{ij}^{\texttt{k}} \gets$ the optimal objective value of [\[prob:sub\]](#prob:sub){reference-type="eqref" reference="prob:sub"} $(\pi^\texttt{k}, \psi_{i}^\texttt{k}, z_{i}^\texttt{k}) \gets$ an optimal solution of [\[prob:sub\]](#prob:sub){reference-type="eqref" reference="prob:sub"}; $\omega^\texttt{k} \gets B_{ij} z_{i}^\texttt{k}$ Add $t_{ij} \ge (h_i(x) + T_i \omega^\texttt{k})^T\pi^\texttt{k} - \lambda \|\omega^\texttt{k} - \zeta_{ij}\|$ to ($M$) $UB \gets \min\{UB, c^T x^\texttt{k} + \sum_{i\in\mathcal{L}} \epsilon_i \lambda_i^\texttt{k} + \sum_{j \in [n_i]} P_{ij} g_{ij}^{\texttt{k}}\}$ $LB \gets v^\texttt{k}$; $\texttt{k}\gets \texttt{k}+1$ To accelerate the algorithm, in line 7 of Algorithm [\[algo\]](#algo){reference-type="ref" reference="algo"}, instead of solving [\[prob:sub\]](#prob:sub){reference-type="eqref" reference="prob:sub"} from scratch, we first solve it with $B_{ij}z_i$ fixed at $\zeta_{ij}$ and then resolve it with $B_{ij}z_i$ fixed at $0$ to alternatively add a suboptimal cut if it cuts off the current candidate solution. # Numerical Results {#sec:case-study} In our numerical experiments we observed that SOCP solvers suffer from numerical instability and result in inconsistent solutions. Therefore, throughout the experiments reported in this paper, we use a linear asymptotic relaxation of the second-stage SOCP problem $Z_i(x, \xi_i)$, denoted by $\hat Z_i(x, \xi_i)$, for the experiment. Note, however, that the implications derived from the LP relaxation should align with the SOCP problem as the relaxation becomes tightened. The details of the relaxation are given in Appendix [8](#appendix:linear-relaxation){reference-type="ref" reference="appendix:linear-relaxation"}. In this section we compare the performance of the proposed TSDRO model with that of TSRO and SAA of the TSSO models as follows: - `opt`: [\[prob\]](#prob){reference-type="eqref" reference="prob"} with an optimal choice of $\epsilon$; - `hm`: [\[prob\]](#prob){reference-type="eqref" reference="prob"} with $\epsilon$ chosen via a holdout method, in which we spare 20% of the data for validation; - `saa`: $\min_{x \in \mathcal{X}} \ c^T x + \sum_{i \in \mathcal{L}} \mathbb{E}_{\hat{\mathbb{P}}_i} [\hat Z_i(x, \tilde \xi_i)]$; and - `ro`: $\min_{x \in \mathcal{X}} \ c^T x + \sum_{i \in \mathcal{L}} \max_{\xi_i \in \Xi_i} \hat Z_i(x,\xi_i)$. {#fig:test-system width="0.8\\linewidth"} To this end we use a modified IEEE 123 bus system with 8 PVs with a maximum capacity of 0.05 p.u, as illustrated in Figure [3](#fig:test-system){reference-type="ref" reference="fig:test-system"}. See Appendix [9](#appendix:test-system){reference-type="ref" reference="appendix:test-system"} for more details. The second stage aims to minimize the total import from the transmission grid, that is, $C_i(g_i^p) = g_i^p$ for $i \in \mathcal{K}^D$. The objective weights are set as $\beta_v = 0.01, \beta_{shed} = 10^2, \beta_{slack} = 10^5$ to prioritize consensus. We set the upper bound $\bar \pi$ of the dual variable associated with [\[eq:second-stage-constraint:xi\]](#eq:second-stage-constraint:xi){reference-type="eqref" reference="eq:second-stage-constraint:xi"} as $10^6$. We use $l_1$-norm to define the Wasserstein ball (i.e., $p=1$), and thus the algorithm explained in Section [4](#sec:solution){reference-type="ref" reference="sec:solution"} is exact. We assume 5 local control centers coupled via $\mathcal C=\{$sw2, sw3, sw4, sw5, sw7, sw8$\}$. The subregions have total loads of around $0.044 +{\bf i}0.022, 0.032+{\bf i}0.017, 0.044+{\bf i}0.027, 0.064+{\bf i}0.035$, and $0.018+{\bf i}0.009$ p.u., respectively. For each subregion $i$, we assume that the true distribution of $\tilde \xi_i$ for the next time periods, denoted by $\mathbb{Q}_i$, is a truncated multivariate normal distribution [@doubleday2020probabilistic] with a mean of 0.8 and a covariance matrix having 0.1's on its diagonal and 0.001's on its off-diagonal entries so that PVs in the same region are slightly correlated. For the numerical experiments, the Wasserstein radii $\{\epsilon_i\}_{i \in \mathcal{L}}$ and the number of available scenarios $\{n_i\}_{i \in \mathcal{L}}$ are varied homogeneously across the subregions, so we let $\epsilon$ and $n$ denote the common radius and cardinality. For each $n \in \{5, 10, \cdots, 30\}$, we generate 50 training datasets, each of which consists of $n$ potential scenarios of $(\tilde \xi_1, \cdots, \tilde \xi_{|\mathcal{L}|})$ sampled from $\mathbb{Q}_1 \times \cdots \times \mathbb{Q}_{|\mathcal{L}|}$. Each training dataset is used to construct $\{\hat{\mathbb{P}}_i\}_{i \in \mathcal{L}}$ for simulation. For testing, we also sample $10^3$ out-of-sample scenarios from $\mathbb{Q}_1 \times \cdots \times \mathbb{Q}_{|\mathcal{L}|}$. The data files are available in json format on <https://github.com/gbyeon/DROControl-dataset.git>. All experiments were executed on a Dell PowerEdge R650 server, with 56 cores and 512GB of RAM. The implementation is in Julia and uses IBM CPLEX 12.10. To accelerate Algorithm [\[algo\]](#algo){reference-type="ref" reference="algo"}, we add the cuts in a lazy manner to ($M$) during its branch-and-bound process using a callback function and parallelize the cut generation procedure in Lines 6-13. ## Expected total system cost We first compare the expected total system cost of those four models, estimated based on the $10^3$ testing scenarios, over 50 independent simulation runs. To be specific, let $\hat x$ denote a solution obtained by one of the models. Then its expected system cost is computed by $c^T \hat x + \sum_{i \in \mathcal{L}} \sum_{j=1}^{10^3} \frac{1}{10^3}\hat Z_i(\hat x, \hat\zeta_{ij})$, where $\{\hat\zeta_{ij}\}_{j=1,\cdots, 10^3}$ denotes the set of testing scenarios of subnetwork $i$. To first compare `opt`, `saa`, and `ro`, Figure [\[fig:OOS_reliability\]](#fig:OOS_reliability){reference-type="ref" reference="fig:OOS_reliability"} visualizes the 20th and 80th percentiles (shaded areas) and the means (solid lines with markers) of the expected total system cost of the solution of [\[prob\]](#prob){reference-type="eqref" reference="prob"} with respect to varying $\epsilon$, denoted by $\tilde J_{n}(\epsilon)$. Note that as $\epsilon \rightarrow 0$, the solution of [\[prob\]](#prob){reference-type="eqref" reference="prob"} becomes the solution of `saa`; and when $\epsilon \ge 10$, the solution of [\[prob\]](#prob){reference-type="eqref" reference="prob"} becomes the solution of `ro` since the Wasserstein distance is bounded from above by the distance between two extreme scenarios $\|\boldsymbol 1 - \boldsymbol 0\|_1$, where $\boldsymbol 1$ and $\boldsymbol 0$ are vectors of ones and zeros with the dimensionality of $\max_{i \in \mathcal{L}}k_i=3$. In Figure [\[fig:OOS_reliability\]](#fig:OOS_reliability){reference-type="ref" reference="fig:OOS_reliability"} we observe that the total cost improves up to a certain point of $\epsilon$ and then declines for all of the simulations. This empirically shows the superiority of `opt` over `saa` and `ro`, as `opt` chooses $\tilde\epsilon$ that minimizes $\tilde J(\epsilon)$. Since `hm` chooses different $\epsilon$ for each training dataset based on its performance on a validation dataset, the expected total system cost of `hm` is not visualized in Figure [\[fig:OOS_reliability\]](#fig:OOS_reliability){reference-type="ref" reference="fig:OOS_reliability"}. To compare all of those four models, Figure [4](#fig:OOS_hm_opt){reference-type="ref" reference="fig:OOS_hm_opt"} plots the estimated total cost with respect to the training sample size $n$. The figure highlights the sample efficiency of the DRO method. As expected, the total costs of `saa` and `hm` highly depend on the sample size $n$; one can see from the figure that the cost improves as $n$ increases. However, since `hm` considers ambiguity in the sample, despite the suboptimal choice of $\epsilon$, its total cost is much lower than that of `saa`. On the other hand, `ro` remains the same since it does not take advantage of samples. Note that `opt` gives the minimum expected cost, even with a limited sample size. As expected, the performance of those three data-driven solutions `opt`, `ro`, and `saa` becomes similar as $n$ increases. {#fig:OOS_hm_opt width="0.5\\linewidth"} ## Impacts on load shedding {#sec:result:loadshed} To see the out-of-sample performance of the four models on important system statistics, we first compare the expected networkwide load shed. Figure [\[fig:loadshed_by_methods\]](#fig:loadshed_by_methods){reference-type="ref" reference="fig:loadshed_by_methods"} visualizes the amount of networkwide load shed for each of the $10^3$ testing scenarios. For each scenario, the figure plots a red dot for the median and grey bars for the 10th and 90th percentiles of the results obtained over the 50 simulation runs. As Figure [\[a\]](#a){reference-type="ref" reference="a"} indicates, with the limited sample size `saa` can result in significant load shedding for many scenarios, and the results vary a lot per simulation, indicating its vulnerability to the limited sample size. With the suboptimal choice of $\epsilon$, the results of `hm` also fluctuate per simulation, but the red dots corresponding to the median values are much more concentrated in the lower region compared with those of `saa`. On the other hand, with the optimal choice of $\epsilon$, the results of `opt` do not vary much over different simulation runs; and, under many scenarios, there is no load shed. Lastly, `ro` has a zero load shedding for all of the testing scenarios. Figure [\[b\]](#b){reference-type="ref" reference="b"} shows how the results of the data-driven methods, `saa`, `hm`, `opt`, improve as we have more data. `opt` achieves zero load shedding for many of the testing scenarios, and `hm` and `saa` get much closer to `opt`. Since `ro` is not affected by the samples, it remains to have zero load shed for $n=30$. ## Impacts on power imports and PV utilization To show how much efficiency is sacrificed to prevent load shedding in those four models, we analyze PV utilization with respect to the amount of power imported from the main grid, that of utilized power outputs of PVs, and the power factor of PV outputs. Figure [\[fig:pgimport\]](#fig:pgimport){reference-type="ref" reference="fig:pgimport"} displays the violin plots showing the distribution of the amount of real power imported at the substation over the testing scenarios averaged for the 50 simulations. The thickness of each graph represents the density of the corresponding value. Again `ro` shows the same behavior independently of $n$, since it does not use sample data; also, its import amount is almost the total real power demand, which is 0.202. This implies that `ro` uses PV power outputs only for reactive power support, which explains why it has zero load shedding for all the scenarios. On the other hand, for $n=5$, `saa` and `hm` are often too realistic and import no more than 50% of the total demand most of the time or even export some of the power to the main grid, resulting in a high load shed in many testing scenarios. However, `opt` imports a reliable amount that mostly covers 50% of the demand while maintaining the level of load shed significantly lower than those of `saa` and `hm`. For $n=30$, the three data-driven methods, `saa`, `hm`, and `opt`, become similar. {#fig:util_N5 width="0.85\\linewidth"} {#fig:pf_N5 width="0.85\\linewidth"} Figures [5](#fig:util_N5){reference-type="ref" reference="fig:util_N5"} and [6](#fig:pf_N5){reference-type="ref" reference="fig:pf_N5"} analyze how each method utilizes PVs; the former plots the distribution of PV utilization and the latter depicts that of the power factor. As shown in Figure [5](#fig:util_N5){reference-type="ref" reference="fig:util_N5"}, `saa` uses PVs the most while `ro` does not utilize PVs at buses 48 and 300 at all. Moreover, `ro` use PVs mostly only for reactive power support, as indicated by Figure [6](#fig:pf_N5){reference-type="ref" reference="fig:pf_N5"}. On the other hand, `saa` utilizes PVs mostly for real power generation as it is too optimistic, while `hm` and `opt` do so in a more balanced way. ## Computation times As the number of subproblems increases with the same cardinality $n$ grows, we use a varying number of cores per different $n$; 14 cores for $n=5$, 28 cores for $n=15$, 56 cores for $n=30$. Table [\[table:comp-time\]](#table:comp-time){reference-type="ref" reference="table:comp-time"} summarizes the computation time in seconds for `opt` for the 50 simulation runs. Note that the solution time can be further reduced by utilizing more cores. R0.1 R0.2R0.2R0.2 $n$ & 10th & median & 90th\ & 197.05 & 446.88 & 1338.50\ 15 & 122.88& 417.89& 1212.63\ 30 & 135.44& 285.32& 750.97\ # Conclusions {#sec:conclusion} We proposed a decentralized control method for active distribution networks that effectively coordinates local control centers and control measures on varying timescales. This coordination is achieved by determining risk-aware here-and-now decisions centrally, which serve as set points for coupling variables and slow-responding control decisions. The here-and-now decisions are informed by a set of plausible distributions of uncertain PV outputs that are close enough to a reference distribution, such as a probabilistic forecast, in a Wasserstein distance sense. Numerical studies on the IEEE 123 bus system demonstrate the outstanding out-of-sample performance of the proposed approach; the proposed method maintained reliable load shed under a majority of testing scenarios while making the most of PVs in a balanced manner even with a limited sample size. It is demonstrated that the proposed DRO model has the potential to achieve (1) a great sample efficiency; (2) a high PV utilization level while avoiding load shed; (3) proactive coordination of LCCs and slow-responding control measures, and (4) a successful uncertainty localization. This suggests that the proposed approach can be strengthened by incorporating more flexible units such as storage systems. Future research will be devoted to modeling unbalanced multiphase systems, such as the one proposed in [@byeon2022linear], as well as incorporating more control devices such as storage systems. Additionally, exploring the impact of heterogeneous Wasserstein radii over subregions may be worthwhile. # Acknowledgment {#acknowledgment .unnumbered} The authors thank Dr. Dae-Hyun Choi for their invaluable comments on this paper. # Proof of Proposition [\[prop:radiality\]](#prop:radiality){reference-type="ref" reference="prop:radiality"} {#appendix:radiality} Let $c$ denote the number of connected components of $\mathcal{G}'$. Note that since the dummy node is the only supply node, in order for each node in $\mathcal{G}'$ to receive a unit of flow, at least one active edge must connect $0$ to each connected component. Therefore, $\sum_{i \in \mathcal{N}} s_{0i} \ge c$ by [\[eq:1st:network-flow\]](#eq:1st:network-flow){reference-type="eqref" reference="eq:1st:network-flow"}, and thus $\sum_{(i,j) \in \mathcal{E}'} s_{ij} = \sum_{(i,j) \in \mathcal{E}} s_{ij} \le |\mathcal{N}| - \sum_{i\in \mathcal{N}} s_{0i} \le |\mathcal{N}| - c$ by [\[eq:1st:tree\]](#eq:1st:tree){reference-type="eqref" reference="eq:1st:tree"}. Let $\mathcal{N}(k)$ and $\mathcal{E}'(k)$ denote the set of nodes and lines in the $k$th connected component of $\mathcal{G}'$ for $k \in [c]$ (i.e., $\bigcup_{k\in [c]} \mathcal{N}(k) = \mathcal{N}$, $\bigcup_{k\in [c]} \mathcal{E}'(k) = \mathcal{E}'$, and $\mathcal{N}(k_1) \bigcap \mathcal{N}(k_2) = \emptyset$, $\mathcal{E}'(k_1) \bigcap \mathcal{E}'(k_2) = \emptyset$ for any $k_1 \neq k_2$). By the connectivity of each component $k$, it follows that $\sum_{(i,j) \in \mathcal{E}'(k)} s_{ij} \ge |\mathcal{N}(k)|-1$, which implies $\sum_{(i,j) \in \mathcal{E}'} s_{ij} \ge |\mathcal{N}|-c$. Therefore, it follows that $\sum_{(i,j) \in \mathcal{E}'} s_{ij} = |\mathcal{N}|-c$, which implies $\sum_{(i,j) \in \mathcal{E}'(k)} s_{ij} = |\mathcal{N}(k)|-1$ for each $k \in [c]$. Therefore, each connected component is a tree, and the desired result follows. 0◻ # A Linear Relaxation $\hat Z_i(\hat x,\xi_i)$ of $Z_i(\hat x,\xi_i)$ {#appendix:linear-relaxation} {#fig:linear-approximation width="0.85\\linewidth"} In the problem of $Z_i(\hat x,\xi_i)$, nonlinearity appears in [\[eq:2nd:power-flow:power:conv\]](#eq:2nd:power-flow:power:conv){reference-type="eqref" reference="eq:2nd:power-flow:power:conv"} and [\[eq:2nd:gen-bound-PV\]](#eq:2nd:gen-bound-PV){reference-type="eqref" reference="eq:2nd:gen-bound-PV"}. Note that each of these constraints can be posed as a collection of 3-dimensional Lorentz cones $\{(x,t) \in \mathbb R^2 \times \mathbb R : \sqrt{x_1^2+x_2^2} \le t \}$ and linear constraints with some auxiliary variables as follows: $$\begin{aligned} \eqref{eq:2nd:power-flow:power:conv} \Leftrightarrow \ & \kappa_{jk} = \frac{\ell_{jk}+v_{j}}{2}, \ o_{jk} = \frac{\ell_{jk}-v_{j}}{2} \label{eq:lorentz:linear}\\ & \iota_{jk} \ge \sqrt{q_{jk}^2 + o_{jk}^2}, \kappa_{jk} \ge \sqrt{p_{jk}^2 + \iota_{jk}^2}. \label{eq:lorentz}\\ % \end{align} % \end{subequations} % \begin{subequations} % \begin{align} \eqref{eq:2nd:gen-bound-PV} \Leftrightarrow \ & \kappa_l \le \overline p_{l}\xi_{il}, \label{eq:second-stage-constraint:xi}\\ & \kappa_{l} \ge \sqrt{(g^p_{l})^2 + (g^q_{l})^2}. % \mbox{Equation } \eqref{eq:2nd:PV-on-off} \Leftrightarrow \ & \iota_{kt} = \overline C_{k} s_{kt}, \ \forall t \in \mc T,\\ % & \iota_{kt} \ge \sqrt{(g^p_{kt})^2 + (g^q_{kt})^2}, \end{aligned}$$ We relax each of the 3-dimensional Lorentz cone constraints with a set of $m$ linear constraints using its supporting hyperplanes at $(\hat x_1, \hat x_2, \hat t) = \left(\hat t\cos(\frac{2\pi j}{m}), \hat t\sin(\frac{2\pi j}{m}), \hat t \right)$ for $j\in [m]$ for some $m \ge 4$: $$\cos\left(2\pi j/m\right) x_1 + \sin\left(2\pi j/m\right) x_2 \le t, \ \forall j \in [m]. \label{eq:2nd:Lorentz:linear}$$ Figure [7](#fig:linear-approximation){reference-type="ref" reference="fig:linear-approximation"} illustrates the linear relaxation and indicates that the relaxation is confined in the following relaxed Lorentz cone constraint: $$(1+\epsilon) t \ge \sqrt{x_1^2+ x_2^2},$$ where $\epsilon = \frac{1}{\cos(\frac{\pi}{m})} - 1$. As $m$ goes to infinity, $\epsilon$ approaches zero, becoming tight to the original constraint. **Remark 5**. *Ben-Tal and Nemirovski [@ben2001polyhedral] proposed a linear relaxation that approximates the 3D Lorentz cone constraint with a polynomially increasing number of additional variables and constraints. It is shown that with $2(v+1)$ additional variables and $3(v+2)$ constraints for some integer $v$, $\epsilon = \frac{1}{\cos(\frac{\pi}{2^{v+1}})}-1$ relaxation is achieved. Although the growth in problem size is at a much slower pace in the relaxation proposed in [@ben2001polyhedral], [\[eq:2nd:Lorentz:linear\]](#eq:2nd:Lorentz:linear){reference-type="eqref" reference="eq:2nd:Lorentz:linear"} requires fewer variables and constraints up to $\epsilon = 1\%$ and is simpler; thus, we use [\[eq:2nd:Lorentz:linear\]](#eq:2nd:Lorentz:linear){reference-type="eqref" reference="eq:2nd:Lorentz:linear"} with $\epsilon = 1\%$ (i.e., $m = 23$). One can use the linear relaxation in [@ben2001polyhedral] if higher accuracy is desired.* The relaxation for [\[eq:2nd:power-flow:power:conv\]](#eq:2nd:power-flow:power:conv){reference-type="eqref" reference="eq:2nd:power-flow:power:conv"} no longer guarantees $p_{jk}$ and $q_{jk}$ to be zero when $\ell_{jk} = 0$, that is, when $u_{jk} = 1$. Therefore, we add the following constraints for $(j,k) \in \mathcal{E}^u_i$: $$\begin{aligned} -\overline s_{jk} \left(1-u_{jk}\right)\le p_{jk} \le \overline s_{jk} \left(1-u_{jk}\right),\\ -\overline s_{jk} \left(1-u_{jk}\right)\le q_{jk} \le \overline s_{jk} \left(1-u_{jk}\right), \end{aligned}$$ [\[eq:2nd:linear:switch\]]{#eq:2nd:linear:switch label="eq:2nd:linear:switch"} where $\overline s_{jk}$ is the apparent power limit on line $(j,k)$. # Test System Description {#appendix:test-system} The test system has 8 line switches named $\{$sw1, sw2, $\cdots$, sw8$\}$ and 4 CBs at buses 83, 88, 90, and 92. The substation regulator is treated as an OLTC, in which the CC decides its voltage magnitude squared, $v_1$. Other regulators are removed, and the load transformers are modeled as lines with equivalent impedance. Since the original data is in three phases, we modify the data to get a single-phase test system as follows: (i) loads are averaged over phases, and (ii) the line impedance $r_{ij} + {\bf i}x_{ij}$ is obtained by the maximum diagonal element of its impedance matrix. Only the voltage source serves as a dispatchable generator. The voltage bounds are set to be \[0.8 p.u., 1.2 p.u.\]. [^1]: G. Byeon is with the School of Computing and Augmented Intelligence, Arizona State University, Tempe, AZ, USA (contact: geunyeong.byeon\@asu.edu) [^2]: K. Kim is with the Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, IL, USA (contact: kimk\@anl.gov). [^3]: This material is based upon work supported by the U.S. Department of Energy, Office of Science, under contract number DE-AC02-06CH11357.

arxiv_math

--- abstract: | We provide novel lower bounds on the Betti numbers of Vietoris--Rips complexes of hypercube graphs of all dimensions, and at all scales. In more detail, let $Q_n$ be the vertex set of $2^n$ vertices in the $n$-dimensional hypercube graph, equipped with the shortest path metric. Let $\mathrm{VR}(Q_n;r)$ be its Vietoris--Rips complex at scale parameter $r \ge 0$, which has $Q_n$ as its vertex set, and all subsets of diameter at most $r$ as its simplices. For integers $r<r'$ the inclusion $\mathrm{VR}(Q_n;r)\hookrightarrow\mathrm{VR}(Q_n;r')$ is nullhomotopic, meaning no persistent homology bars have length longer than one, and we therefore focus attention on the individual spaces $\mathrm{VR}(Q_n;r)$. We provide lower bounds on the ranks of homology groups of $\mathrm{VR}(Q_n;r)$. For example, using cross-polytopal generators, we prove that the rank of $H_{2^r-1}( \mathrm{VR}(Q_{n};r))$ is at least $2^{n-(r+1)}\binom{n}{r+1}$. We also prove a version of *homology propagation*: if $q\ge 1$ and if $p$ is the smallest integer for which $\mathrm{rank}H_q(\mathrm{VR}(Q_p;r))\neq 0$, then $\mathrm{rank}H_{q}( \mathrm{VR}(Q_{n};r)) \ge \sum_{i=p}^n 2^{i-p} \binom{i-1}{p-1} \cdot \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r))$ for all $n \ge p$. When $r\le 3$, this result and variants thereof provide tight lower bounds on the rank of $H_{q}(\mathrm{VR}(Q_{n};r))$ for all $n$, and for each $r \ge 4$ we produce novel lower bounds on the ranks of homology groups. Furthermore, we show that for each $r\ge 2$, the homology groups of $\mathrm{VR}(Q_{n};r)$ for $n \ge 2r+1$ contain propagated homology not induced by the initial cross-polytopal generators. author: - Henry Adams - Žiga Virk bibliography: - VRHypercube-LowerBoundHomology.bib title: Lower bounds on the homology of Vietoris--Rips complexes of hypercube graphs --- [^1] # Introduction Let $Q_n$ be the vertex set of the hypercube graph, equipped with the shortest path metric. In other words, $Q_n$ can be thought of the set of all $2^n$ binary strings of $0$'s and $1$'s equipped with the Hamming distance, or alternatively, as the set $\{0,1\}^n\subseteq \mathbb{R}^n$ equipped with the $\ell^1$ metric. In this paper, we study the topology of the *Vietoris--Rips simplicial complexes* of $Q_n$. Given a metric space $X$ and a scale $r\ge 0$, the Vietoris--Rips simplicial complex $\mathrm{VR}(X;r)$ has $X$ as its vertex set, and a finite subset $\sigma \subseteq X$ as a simplex if and only if the diameter of $\sigma$ is at most $r$. Originally introduced for use in algebraic topology [@Vietoris27] and geometric group theory [@bridson2011metric; @Gromov], Vietoris--Rips complexes are now a commonly-used tool in applied and computational topology in order to approximate the shape of a dataset [@Carlsson2009; @EdelsbrunnerHarer]. Important results include the fact that nearby metric spaces give nearby Vietoris--Rips persistent homology barcodes [@ChazalDeSilvaOudot2014; @chazal2009gromov], that Vietoris--Rips complexes can be used to recover the homotopy types of manifolds [@Hausmann1995; @Latschev2001; @virk2021rips; @majhi2023demystifying], and that Vietoris--Rips persistent homology barcodes can be efficiently computed [@bauer2021ripser]. Nevertheless, not much is known about Vietoris--Rips complexes of manifolds or of simple graphs at large scale parameters, unless the manifold is the circle [@AA-VRS1], unless the graph is a cycle graph [@Adamaszek2013; @AAFPP-J], or unless one restricts attention to $1$-dimensional homology [@virk20201; @gasparovic2018complete]. Let $\mathrm{VR}(Q_n;r)$ be the Vietoris--Rips complex of the vertex set of the $n$-dimensional hypercube at scale parameter $r$. The homotopy types of $\mathrm{VR}(Q_n;r)$ are known for $r\le 3$ (and otherwise mostly unknown); see Table [1](#table:homotopy-types){reference-type="ref" reference="table:homotopy-types"}. For $r=0$, $\mathrm{VR}(Q_n;0)$ is the disjoint union of $2^n$ vertices, and hence homotopy equivalent to a wedge sum $(2^n-1)$-fold wedge sum of zero-dimensional spheres. For $r=1$, $\mathrm{VR}(Q_n;1)$ is a connected graph (the hypercube graph), which by a simple Euler characteristic computation is homotopy equivalent to a $((n-2)2^{n-1}+1)$-fold wedge sum of circles. For $r=2$, Adams and Adamaszek [@adamaszek2022vietoris] prove that $\mathrm{VR}(Q_n;2)$ is homotopy equivalent to a wedge sum of $3$-dimensional spheres; see Theorem [Theorem 4](#ThmAA){reference-type="ref" reference="ThmAA"} for a precise statement which also counts the number of $3$-spheres. For $r=3$, Shukla proved in [@shukla2022vietoris Theorem A] that for $n\ge 5$, the $q$-dimensional homology of $\mathrm{VR}(Q_n;3)$ is nontrivial if and only if $q=7$ or $q=4$. The study of $r=3$ was furthered by Feng [@feng2023homotopy] (based on work by Feng and Nukula [@feng2023vietoris]), who proved that $\mathrm{VR}(Q_n;3)$ is always homotopy equivalent to a wedge sum of $7$-spheres and $4$-spheres; see Theorem [Theorem 5](#ThmZiqin){reference-type="ref" reference="ThmZiqin"} for a precise statement which also counts the number of spheres of each dimension. When $r=n-1$, $\mathrm{VR}(Q_n;n-1)$ is isomorphic to the boundary of the cross-polytope with $2^n$ vertices, and hence is homeomorphic to a sphere of dimension $2^{n-1}-1$. For $r\ge n$, the space space $\mathrm{VR}(Q_n;n)$ is a complete simplex, and hence contractible. However, there is an entire infinite "triangle" of parameters, namely $r\ge 4$ and $r\le n-2$, for which essentially nothing is known about the homotopy types of $\mathrm{VR}(Q_n;r)$. $_n \backslash ^r$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ -------------------- ---------------------- ----------------------- ------------------------ ------------------------------------------------- ------------------------------------------ ----------------------------------------- ----------------------------------------- ----------------------------------------- $1$ $S^0$ $*$ $*$ $*$ $*$ $*$ $*$ $*$ $2$ $\bigvee^3 S^0$ $S^1$ $*$ $*$ $*$ $*$ $*$ $*$ $3$ $\bigvee^7 S^0$ $\bigvee^5 S^1$ $S^3$ $*$ $*$ $*$ $*$ $*$ $4$ $\bigvee^{15} S^0$ $\bigvee^{17} S^1$ $\bigvee^9 S^3$ $S^7$ $*$ $*$ $*$ $*$ $5$ $\bigvee^{31} S^0$ $\bigvee^{49} S^1$ $\bigvee^{49} S^3$ $\bigvee^{10} S^7 \vee S^4$ $S^{15}$ $*$ $*$ $*$ $6$ $\bigvee^{63} S^0$ $\bigvee^{129} S^1$ $\bigvee^{209} S^3$ $\bigvee^{60} S^7 \vee \bigvee^{11} S^4$ $\textcolor{blue}{\beta_{15}\ge 12}$ $S^{31}$ $*$ $*$ $7$ $\bigvee^{127} S^0$ $\bigvee^{321} S^1$ $\bigvee^{769} S^3$ $\bigvee^{280} S^7 \vee \bigvee^{71} S^4$ $\textcolor{blue}{\beta_{15}\ge 84}$ $\textcolor{blue}{\beta_{31}\ge 14}$ $S^{63}$ $*$ $8$ $\bigvee^{255} S^0$ $\bigvee^{769} S^1$ $\bigvee^{2561} S^3$ $\bigvee^{1120} S^7 \vee \bigvee^{351} S^4$ $\textcolor{blue}{\beta_{15}\ge 448}$ $\textcolor{blue}{\beta_{31}\ge 112}$ $\textcolor{blue}{\beta_{63}\ge 16}$ $S^{127}$ $9$ $\bigvee^{511} S^0$ $\bigvee^{1793} S^1$ $\bigvee^{7937} S^3$ $\bigvee^{4032} S^7 \vee \bigvee^{1471} S^4$ $\textcolor{blue}{\beta_{15}\ge 2016}$ $\textcolor{blue}{\beta_{31}\ge 672}$ $\textcolor{blue}{\beta_{63}\ge 144}$ $\textcolor{blue}{\beta_{127}\ge 18}$ $10$ $\bigvee^{1023} S^0$ $\bigvee^{4097} S^1$ $\bigvee^{23297} S^3$ $\bigvee^{13440} S^7 \vee \bigvee^{5503} S^4$ $\textcolor{blue}{\beta_{15}\ge 8064}$ $\textcolor{blue}{\beta_{31}\ge 3360}$ $\textcolor{blue}{\beta_{63}\ge 960}$ $\textcolor{blue}{\beta_{127}\ge 180}$ $11$ $\bigvee^{2047} S^0$ $\bigvee^{9217} S^1$ $\bigvee^{65537} S^3$ $\bigvee^{42240} S^7 \vee \bigvee^{18943} S^4$ $\textcolor{blue}{\beta_{15}\ge 29568}$ $\textcolor{blue}{\beta_{31}\ge 14784}$ $\textcolor{blue}{\beta_{63}\ge 5280}$ $\textcolor{blue}{\beta_{127}\ge 1320}$ $12$ $\bigvee^{4095 $\bigvee^{20481} S^1$ $\bigvee^{178177} S^3$ $\bigvee^{126720} S^7 \vee \bigvee^{61183} S^4$ $\textcolor{blue}{\beta_{15}\ge 101376}$ $\textcolor{blue}{\beta_{31}\ge 59136}$ $\textcolor{blue}{\beta_{63}\ge 25344}$ $\textcolor{blue}{\beta_{127}\ge 7920}$ } S^0$ : Black entries are the known homotopy types of $\mathrm{VR}(Q_n;r)$; blue entries are sample novel lower bounds on the Betti numbers of $\mathrm{VR}(Q_n;r)$ based on Theorem [Theorem 9](#ThmMain1){reference-type="ref" reference="ThmMain1"}. (See Table [2](#table:homotopy-types2){reference-type="ref" reference="table:homotopy-types2"} for improved lower bounds for the $r=4$ column.) In this paper, instead of focusing on a single value of $r$, we provide novel lower bounds on the ranks of homology groups of $\mathrm{VR}(Q_n;r)$ for all values of $r$. Some of these lower bounds are shown in blue in Table [1](#table:homotopy-types){reference-type="ref" reference="table:homotopy-types"}: using cross-polytopal generators, in Theorem [Theorem 9](#ThmMain1){reference-type="ref" reference="ThmMain1"} we prove $\mathrm{rank}H_{2^r-1}( \mathrm{VR}(Q_{n};r)) \geq 2^{n-(r+1)}\binom{n}{r+1}$ (although we show in Section [7](#sec:geometric){reference-type="ref" reference="sec:geometric"} that for $r\ge 2$ and $n > 2r$ this does not constitute the entire reduced homology in all dimensions). This is the first result showing that the topology of $\mathrm{VR}(Q_n;r)$ is nontrivial for *all* values of $r\le n-1$. Furthermore, we often show that $\mathrm{VR}(Q_n;r)$ is far from being contractible, with the rank of $(2^r-1)$-dimensional homology tending to infinity exponentially fast as a function of $n$ (with $n$ increasing and with $r$ fixed). Our general strategy, which we refer to as *homology propagation*, is as follows. Let $q\ge 1$. Suppose that one can show that the $q$-dimensional homology group $H_q(\mathrm{VR}(Q_{p};r))$ is nonzero (for example, using a homology computation on a computer, or alternatively a theoretical result such as the mentioned cross-polytopal elements or the geometric generators of Section [7](#sec:geometric){reference-type="ref" reference="sec:geometric"}). Then we provide lower bounds on the ranks of the homology groups $H_q(\mathrm{VR}(Q_{n};r))$ for all $n\ge p$. In particular, in Theorem [Theorem 18](#ThmMain4){reference-type="ref" reference="ThmMain4"} we prove that if $p\ge 1$ is the smallest integer for which $H_q(\mathrm{VR}(Q_{p};r))\neq 0$, then $$\mathrm{rank}H_{q}(\mathrm{VR}(Q_{p};r)) \geq \sum_{i=p}^n 2^{i-p} \binom{i-1}{p-1} \cdot \mathrm{rank}H_{q}(\mathrm{VR}(Q_{p};r)).$$ Thus, a homology computation for a low-dimensional hypercube $Q_p$ has consequences for the homology of $\mathrm{VR}(Q_{n};r)$ for all $n\ge p$. See Table [2](#table:homotopy-types2){reference-type="ref" reference="table:homotopy-types2"} for some consequences of this result and of related results. As we explain in Section [6.4](#SubsComparison){reference-type="ref" reference="SubsComparison"}, when $r\le 3$ our results are known to provide tight lower bounds on all Betti numbers of $\mathrm{VR}(Q_{n};r)$. We take this as partial evidence that our novel results on the Betti numbers of $\mathrm{VR}(Q_{n};r)$ for $r\ge 4$ are likely to be good lower bounds, though we do not know how close they are to being tight as no upper bounds are known. Indeed, the main "upper bound" we know of on the Betti numbers of $\mathrm{VR}(Q_{n};r)$ is a triviality result for 2-dimensional homology: Carlsson and Filippenko [@carlsson2020persistent] prove that $H_2(\mathrm{VR}(Q_n;r))=0$ for all $n$ and $r$. For integers $r<r'$, we prove via a simple argument that the inclusion $\mathrm{VR}(Q_n;r)\hookrightarrow\mathrm{VR}(Q_n;r')$ is nullhomotopic. Therefore, there are no persistent homology bars of length longer than one, and all homological information about the filtration $\mathrm{VR}(Q_n;\bullet)$ is determined by $\mathrm{VR}(Q_n;r)$ for individual integer values of $r$. Though we have stated our results for $Q_n=\{0,1\}^n$ equipped with the $\ell^1$ metric, we remark that these results hold for any $\ell^p$ metric with $1\le p < \infty$. Indeed for $x,y\in Q_n$, the $i$-th coordinates of $x$ and $y$ differ by either $0$ or $1$ for each $1\le i\le n$, and hence $\mathrm{VR}((Q_n,\ell^p);r)=\mathrm{VR}((Q_n,\ell^1);r^p)$. So, our results can be translated into any $\ell^p$ metric by a simple reparametrization of scale. We expect that some of our work could be transferred over to provide results for Čech complexes of hypercube graphs, as studied in [@adams2022v], though we do not pursue that direction here. We begin with some preliminaries in Section [2](#sec:preliminaries){reference-type="ref" reference="sec:preliminaries"}. In Section [3](#sec:contractions-ph){reference-type="ref" reference="sec:contractions-ph"} we review contractions, and we prove that $\mathrm{VR}(Q_n;\bullet)$ has no persistent homology bars of length longer than one. In Section [4](#sec:maximal){reference-type="ref" reference="sec:maximal"} we use cross-polytopal generators to prove $\mathrm{rank}H_{2^r-1}( \mathrm{VR}(Q_{n};r)) \geq 2^{n-(r+1)}\binom{n}{r+1}$. We introduce concentrations in Section [5](#sec:concentrations){reference-type="ref" reference="sec:concentrations"}, which we use to prove our more general forms of homology propagation in Section [6](#sec:contractions-homology){reference-type="ref" reference="sec:contractions-homology"}. In Section [7](#sec:geometric){reference-type="ref" reference="sec:geometric"} we prove the existence of novel lower-dimensional homology generators, and we conclude with some open questions in Section [8](#sec:conclusion){reference-type="ref" reference="sec:conclusion"}. # Preliminaries and geometry of hypercubes. {#sec:preliminaries} ## Homology All homology groups will be considered with coefficients in $\mathbb{Z}$ or in a field. The rank of a finitely generated abelian group is the cardinality of a maximal linearly independent subset. We let $\beta_q$ denote the $q$-th *Betti number* of a space, i.e., the rank of the $q$-dimensional homology group. ## Hypercubes Hypercubes are among the simplest examples of product spaces. **Definition 1**. Given $n\in \{1,2,\ldots\}$, the *hypercube graph* $Q_n$ is the metric space $\{0,1\}^n$, equipped with the $\ell^1$ metric. In particular, the elements of the space are $n$-tuples $(a_1, a_2, \ldots, a_n)=(a_i)_{i\in [n]}$ with $a_i\in \{0,1\}$, and the $\ell^1$ distance is defined as $$d \big((a_1, a_2, \ldots, a_n),(b_1, b_2, \ldots, b_n)\big)= \sum_{i=1}^n |a_i-b_i|.$$ In other words, the distance between two $n$-tuples is the number of coordinates in which they differ. For $a\in Q_n$, its antipodal point $\bar a$ is given as $\bar a = (1,1,\ldots, 1)-a$. In particular, $\bar a$ is the furthest point in $Q_n$ from $a$ and thus shares no coordinate with $a$. Observe that $d(a,\bar a)=n$. ## Vietoris--Rips complexes A Vietoris--Rips complex is a way to "thicken" a metric space, as we describe via the definitions below. **Definition 2**. Given a metric space $X$ and a finite subset $A \subseteq X$, the *diameter of $A$* is $$\mathrm{diam}(A) = \max_{a,b\in A} d(a,b).$$ The *local diameter* of $A$ at a point $a\in A$ equals $$\mathrm{localDiam}(A,a) = \max_{b\in A} d(a,b).$$ **Definition 3**. Given $r \geq 0$ and a metric space $X$ the *Vietoris--Rips complex* $\mathrm{VR}(X;r)$ is the simplicial complex with vertex set $X$, and with a finite subset $\sigma \subseteq X$ being a simplex whenever $\mathrm{diam}(\sigma) \leq r$. This is the *closed* Vietoris--Rips complex, since we are using the convention $\le$ instead of $<$. But, since the metric spaces $Q_n$ are finite, all of our results have analogues if one instead considers the *open* Vietoris--Rips complex that uses the $<$ convention. In [@adamaszek2022vietoris] it was proven that $\mathrm{VR}(Q_n;2)$ is homotopy equivalent to a wedge sum of $3$-dimensional spheres: **Theorem 4** (Theorem 1 of [@adamaszek2022vietoris]). *For $n\ge 3$, we have the homotopy equivalence $$\mathrm{VR}(Q_n;2) \simeq \bigvee_{c_n} S^3, \textrm{ where } c_n= \sum_{0 \leq j < i < n}(j+1)(2^{n-2}- 2^{i-1}).$$* See [@saleh2023vietoris] for some relationships between this result and generating functions. In [@feng2023homotopy], it was proven that $\mathrm{VR}(Q_n;3)$ is always homotopy equivalent to a wedge sum of $7$-spheres and $4$-spheres: **Theorem 5** (Theorem 24 of [@feng2023homotopy]). *For $n\ge 5$, we have the homotopy equivalence $$\mathrm{VR}(Q_n;3) \simeq \bigvee_{2^{n-4}\binom{n}{4}} S^7 \ \vee \bigvee_{\sum_{i=4}^{n-1} 2^{i-4} \binom{i}{4}} S^4.$$* ## Embeddings of hypercubes {#ssec:embeddings} For $k$ a positive integer, let $[k]=\{1,2,\ldots, k\}$. Given $p \in [n-1]$ there are many isometric copies of $Q_p$ in $Q_n$. For any subset $S \subseteq [n]$ of cardinality $p$ we can isometrically embed $Q_p$ in $Q_n$, using set $S$ as its variable coordinates, and leaving the rest of the entries fixed. In more detail, we define an isometric embedding $\iota_S^b \colon Q_p \hookrightarrow Q_n$ associated to a subset $S=\{s_1,s_2, \ldots, s_p\} \subseteq [n]$ of coordinates and an offset $(b_i)_{i\in [n]\setminus S} \in \{0,1\}^{n-|S|}$, maps $(a_i)_{i\in [p]}$ to $(a'_i)_{i\in [n]}$ with: - $a'_{s_i}=a_i$ for $i\in [p]$, and - $a'_i=b_i$ otherwise. Given a fixed set $S$, there are $2^{n-p}$ such embeddings $\iota_S$, each associated to a different offset $b$. Let $\pi_S \colon Q_n \to Q_p$ be the map projecting onto the coordinates in $S$. Then $\pi_s \circ \iota_S = id_{Q_p}$ for any map $\iota_S$ (i.e., for any choice of an offset $b$). Given an offset $(b_i)_{i\in [n]\setminus S}$, let $Q_p^b$ denote the image of $\iota_S^b$ corresponding to the offset $b$, and let $\pi_S^b \colon Q_n \to Q_p^b$ be defined as $\iota_S^b \circ \pi_S$. Given $B \subseteq Q_n$ its *cubic hull* $\mathrm{cHull}(B)$ is the smallest isometric copy of a cube (i.e., the image of $Q_{p'}$ via some map $\iota$) containing $B$. For our purposes we will only consider isometric embeddings $Q_p \hookrightarrow Q_n$ (also denoted by $Q_p \leq Q_n$) that retain the order of coordinates, although any permutation of coordinates of $Q_p$ results in a non-constant isometry of $Q_p$ and thus a different isometric embedding into $Q_n$. With this convention of retaining the coordinate order, there are $\binom{n}{p} 2^{n-p}$ isometric embeddings $\iota \colon Q_p \hookrightarrow Q_n$ and $\binom{n}{p}$ projections $\pi \colon Q_n \to Q_p$. # Contractions and the persistent homology of hypercubes {#sec:contractions-ph} In this section we prove the following results. First, fix the scale $r\ge 0$, and let $p\le n$. An isometric embedding $Q_p \hookrightarrow Q_n$ gives an inclusion $\mathrm{VR}(Q_p;r)\hookrightarrow \mathrm{VR}(Q_n;r)$, which is injective on homology in all dimensions. Alternatively, fix dimension $n$, and consider integer scale parameters $r < r'$. The inclusion $\mathrm{VR}(Q_n;r)\hookrightarrow \mathrm{VR}(Q_n;r')$ is nullhomotopic, and hence the filtration $\mathrm{VR}(Q_n;\bullet)$ has no persistent homology bars of length longer than one. These results follow from the properties of contractions, which we introduce now. ## Contractions {#ssec:contractions} A map $f \colon X \to A$ from a metric space $(X,d)$ onto a closed subspace $A \subseteq X$ is a *contraction* if $f|_{A}=id_A$ and if $d(f(x),f(y)) \leq d(x,y)$ for all $x,y\in X$. Our interest in contractions stems from the fact that if a contraction $X \to A$ exists, then the homology of a Vietoris--Rips complex of $A$ maps injectively into the homology of the corresponding Vietoris--Rips complex of $X$: **Proposition 6** ([@virk2022contractions]). *If $f\colon X \to A$ is a contraction, then the embedding $A \hookrightarrow X$ induces injections on homology $H_q(\mathrm{VR}(A;r))\to H_q(\mathrm{VR}(X;r))$ for all integers $q\ge 0$ and scales $r\ge 0$.* We prove that the projections from a higher-dimensional cube to a lower-dimensional cube in Section [2.4](#ssec:embeddings){reference-type="ref" reference="ssec:embeddings"} are contractions: **Lemma 7**. *Given fixed $p\in[n-1]$, set $S\subseteq [n]$ of cardinality $p$, and offset $b$ as in Section [2.4](#ssec:embeddings){reference-type="ref" reference="ssec:embeddings"}, the following hold:* 1. *Maps $\pi_S$ and $\pi_S^b$ are contractions.* 2. *For each $x\in Q_p^b$ and $y\in Q_n$, we have $d(x,y) = d(x, \pi_S^b(y)) + d(\pi_S^b(y),y)$.* 3. *For each offset $b'$:* 1. *For each $x,y\in Q_p^{b'}$, we have $d(x,y)=d(\pi_p^b(x),\pi_p^b(y))$.* 2. *For each $x\in Q_p^{b'}$ and $y\notin Q_p^{b'}$, we have $d(x,y) -1 \geq d(\pi_p^b(x),\pi_p^b(y)) \geq d(x,y) - (n-p)$.* *Proof.* For $x,y\in Q_n$, the distance $d(x,y)$ is the number of components in which $x$ and $y$ differ. On the other hand, $d(\pi_S^b(x), \pi_S^b(y))$ is the number of components from $S$ in which $x$ and $y$ differ (and the same holds for map $\pi_S$ instead of $\pi_S^b$ as well). Thus $d(x,y) \leq d(\pi_S^b(x), \pi_S^b(y))$ with: - $d(x,y) = d(\pi_S^b(x), \pi_S^b(y))$ if $x,y\in Q_p^{b'}$ as their coordinates outside $S$ agree, yielding (1) and (3)(a), and - (3)(b) follows from the fact that for each $x\in Q_p^{b'}, y\notin Q_p^{b'}$, the number of coordinates outside of $S$ on which $x$ and $y$ disagree is at least $1$ (due to $x,y$ not being in the same $Q_p^*$) and at most $n-p$ (which is the cardinality of $[n]\setminus S$). Item (2) follows from the observation that: - $d(x, \pi_S^b(y))$ is the number of components from $S$ in which $x$ and $y$ differ, and - $d(\pi_S^b(y),y)$ is the number of components from $[n]\setminus S$ in which $x$ and $y$ differ, as $x\in Q_p^b$.  ◻ Since each of the projections $\pi \colon Q_n \to Q_p$ is a contraction by Lemma [Lemma 7](#Lemma1){reference-type="ref" reference="Lemma1"}, Proposition [Proposition 6](#PropContrEmbed){reference-type="ref" reference="PropContrEmbed"} then implies that each of the embeddings $Q_p \hookrightarrow Q_n$ induces an injective map on homology $H_q(\mathrm{VR}(Q_p;r)) \to H_q(\mathrm{VR}(Q_n;r))$ for all dimensions $q$. ## Persistent homology of hypercubes {#ssec:ph} The emphasis in modern topology is often on persistent homology arising from the Vietoris--Rips filtration. However, in the setting of Vietoris--Rips complexes of hypercubes, persistent homology does not provide any more information beyond the homology groups at fixed scale parameters. Indeed, the following proposition implies that for any integers $r < r'$, the inclusion $\mathrm{VR}(Q_n;r) \hookrightarrow \mathrm{VR}(Q_n;r+1)$ induces a map that is trivial on homology. **Proposition 8**. *For any positive integers $n$ and $r$, the natural inclusion $\mathrm{VR}(Q_n;r) \hookrightarrow \mathrm{VR}(Q_n;r+1)$ is homotopically trivial.* *Proof.* We first claim that the inclusion $\mathrm{VR}(Q_n;r) \hookrightarrow \mathrm{VR}(Q_n;r+1)$ is homotopic to the projection $\pi_{[n-1]}\colon \mathrm{VR}(Q_n;r) \to \mathrm{VR}(Q_{n-1};r)$ in $\mathrm{VR}(Q_n;r+1)$. In order to prove the claim we will show that the two maps are contiguous in $\mathrm{VR}(Q_n;r+1)$ (i.e., for each simplex $\sigma \in \mathrm{VR}(Q_n;r)$ the union $\sigma \cup \pi_{[n-1]}(\sigma)$ is contained in a simplex of $\mathrm{VR}(Q_n;r+1)$), which implies that the two maps are homotopic. Let $\sigma \in \mathrm{VR}(Q_n;r)$. By definition $\mathrm{diam}(\sigma)\leq r$. As $\pi_{[n-1]}(\sigma)$ is obtained by dropping the final coordinate we also have $\mathrm{diam}(\pi_{[n-1]}(\sigma))\leq r$. Taking $x\in \sigma$ and $y\in \pi_{[n-1]}(\sigma)$, i.e. $y=\pi_{[n-1]}(y')$ for some $y'\in\sigma$, we see that $$d(x,y)\leq d(x,y')+d(y',y) \leq r + 1$$ as $d(y,y')\leq 1$. This $\sigma \cup \pi_{[n-1]}(\sigma) \in \mathrm{VR}(Q_n;r+1)$, and the claim is proved. We proceed inductively, proving that each projection $\pi_{[k]}\colon \mathrm{VR}(Q_n;r) \to \mathrm{VR}(Q_k;r)$ is homotopic to the projection $\pi_{[k-1]}\colon \mathrm{VR}(Q_n;r) \to \mathrm{VR}(Q_{k-1};r)$ in $\mathrm{VR}(Q_n;r+1)$, by the same argument as above. As a result, the embedding $\mathrm{VR}(Q_n;r) \hookrightarrow \mathrm{VR}(Q_n;r+1)$ is homotopic to the projection $\pi_{\{1\}}\colon \mathrm{VR}(Q_n;r) \to \mathrm{VR}(Q_1;r)$. Since $\mathrm{VR}(Q_1;r)$ is clearly contractible, this completes the proof. ◻ # Homology bounds via cross-polytopes and maximal simplices {#sec:maximal} Fix a scale $r\ge 2$, and consider an isometric embedding $\iota \colon Q_{r+1} \hookrightarrow Q_n$ for $n\ge r+1$. The aim of this section is to prove not only that the induced map $\mathrm{VR}(Q_{r+1};r)\hookrightarrow \vr{Q_n;r}$ is injective on $(2^r-1)$-dimensional homology, but also that different (ordered) embeddings $\iota$ produce independent homology generators. Let us explain this in detail. We first observe that $\mathrm{VR}(Q_{r+1};r)$ is homeorphic to a $(2^r-1)$-dimensional sphere, i.e. $\mathrm{VR}(Q_{r+1};r) \cong S^{2^r-1}$. The reason is that each vertex $x \in Q_{r+1}$ is connected by an edge in $\mathrm{VR}(Q_{r+1};r)$ to every vertex of $Q_{r+1}$ *except* for $\bar x$, the antipodal vertex. Therefore, after taking the clique complex of this set of edges, we see that $\mathrm{VR}(Q_{r+1};r)$ is isomorphic (as simplicial complexes) to the boundary of the cross-polytope with $2^{r+1}$ vertices. This cross-polytope is a $2^r$-dimensional ball in $2^r$-dimensional Euclidean space, and therefore its boundary is a sphere of dimension $2^r-1$. In particular, $\mathrm{rank}H_{2^r-1}( \mathrm{VR}(Q_{r+1};r))=1$. Since $\mathrm{VR}(Q_{r+1};r)$ is the boundary of a cross-polytope, there is a convenient $(2^r -1)$-dimensional cycle $\gamma$ generating $H_{2^r-1}( \mathrm{VR}(Q_{r+1};r))$. Define the set of maximal antipode-free simplices as $$\mathcal{A}_r = \{Y \subseteq Q_{r+1} \mid x \in Y \Leftrightarrow \bar x \notin Y\}.$$ The cycle $\gamma$ is defined as the sum of appropriately oriented elements of $\mathcal{A}_r$. The space $Q_{r+1}$ consists of $2^{r+1}$ points, which can be partitioned into $2^r$ pairs of mutually antipodal points. If a subset of $Q_{r+1}$ contains exactly one point from each such pair, it is of cardinality $2^r$. Thus $\mathcal{A}_r$ consists of sets of cardinality $2^r$. Given $x\in Q_{r+1}$, the only element of $Q_{r+1}$ which disagrees with $x$ on all $r+1$ coordinates is $\bar x$. As a result each element of $\mathcal{A}_r$ is of diameter at most $r$ and thus a simplex of $\mathrm{VR}(Q_{r+1};r)$. Observe also that any element of $\mathcal{A}_r$ is a maximal simplex of $\mathrm{VR}(Q_{r+1};r)$: adding any point to such a simplex would mean the presence of an antipodal pair, and so the diameter would thus grow to $r+1$. As explained above, the embeddings $\iota \colon Q_{r+1} \hookrightarrow Q_n$ induce injections on homology by Lemma [Lemma 7](#Lemma1){reference-type="ref" reference="Lemma1"} and Proposition [Proposition 6](#PropContrEmbed){reference-type="ref" reference="PropContrEmbed"}. The fact that these embeddings give independent homology generators is formalized in the following statement, which is also the main result of this section. **Theorem 9**. *For $r \geq 2$, $$\mathrm{rank}H_{2^r-1}( \mathrm{VR}(Q_{n};r)) \geq 2^{n-(r+1)}\binom{n}{r+1}.$$* The proof will be provided at the conclusion of the section. Recall that $2^{n-(r+1)}\binom{n}{r+1}$ is the number of different (ordered) embeddings $\iota \colon Q_{r+1} \hookrightarrow Q_n$. We will use maximal simplices and the pairing between homology and cohomology in order to prove that these $2^{n-(r+1)}\binom{n}{r+1}$ different embeddings provide independent cross-polytopal generators for homology. **Proposition 10**. *Suppose $K$ is a simplicial complex and $\sigma$ is a maximal simplex of dimension $p$ in $K$. If there is a $p$-cycle $\alpha$ in $K$ in which $\sigma$ appears with a non-trivial coefficient $\lambda$, then any representative $p$-cycle of $[\alpha]$ also contains $\sigma$ with the same coefficient $\lambda$.* *Proof.* As $\sigma$ is maximal, the $p$-cochain mapping $\sigma$ to $1$ and all other $p$-simplices to $0$ is a $p$-cocycle denoted by $\omega_\sigma$. Utilizing the cap product we see that for each representative $\alpha'$ of $[\alpha]$, the cap product $[\omega_\sigma] \frown [\alpha'] = \lambda$ is the coefficient of $\sigma$ in $\alpha'$. ◻ **Remark 11**. Proposition [Proposition 10](#PropCap){reference-type="ref" reference="PropCap"} could also be proved directly. If $\alpha$ and $\alpha'$ are homologous $p$-cycles, then their difference is a boundary of a $(p+1)$-chain. The later cannot contain $\sigma$ since $\sigma$ is maximal; hence the coefficients of $\sigma$ in $\alpha$ and in $\alpha'$ coincide. We emphasize the cohomological proof because we will use the cochain $\omega_\sigma$ again. We next focus on the construction of maximal simplices of $\mathrm{VR}(Q_{r+1};r)$ which are furthermore also maximal simplices in $\mathrm{VR}(Q_{n};r)$. The following is a simple criterion identifying such a simplex as a maximal simplex in $\mathrm{VR}(Q_{n};r)$; see Figure [1](#fig:maximalSimplices){reference-type="ref" reference="fig:maximalSimplices"}. (We recall that the *local diameter* of $\sigma\subseteq Q_n$ at a point $w\in \sigma$ is defined as $\mathrm{localDiam}(\sigma,w) = \max_{z\in \sigma} d(w,z)$.) {reference-type="ref" reference="PropReduction"}, and also Lemma [Lemma 13](#LemmaREvenAndOdd){reference-type="ref" reference="LemmaREvenAndOdd"} when $r$ is even. An inclusion of $\sigma$ in $Q_4$ also gives a maximal simplex $\iota_S^b(\sigma)\in\mathrm{VR}(Q_4;2)$. *(Right)* Subcube $Q_4$ with a maximal simplex $\sigma\times\{0,1\}\in\mathrm{VR}(Q_4;3)$ drawn in blue, illustrating Lemma [Lemma 13](#LemmaREvenAndOdd){reference-type="ref" reference="LemmaREvenAndOdd"} when $r$ is odd. ](maximalSimplices.pdf){#fig:maximalSimplices width="3in"} **Proposition 12**. *Let $n\ge r+1$, let $S \subseteq [n]$, and let $b$ be an associated offset. Let $\sigma \subseteq Q_{r+1}^b$, and suppose $\sigma \in \mathcal{A}_r$ as a subset of $Q_{r+1}$. If $\mathrm{localDiam}(\sigma,w) = r$ for all $w \in \sigma$, then $\sigma$ is a maximal simplex in $\mathrm{VR}(Q_{n};r)$.* *Proof.* Assume a point $x\in Q_n\setminus \sigma$ is added to $\sigma$. We will show that this increases the diameter of $\sigma$ beyond $r$, by repeatedly using Lemma [Lemma 7](#Lemma1){reference-type="ref" reference="Lemma1"}. - If $\pi_S^b(x) \notin \sigma$ then $\overline{\pi_S^b(x)} \in \sigma$ and thus $$d(x,\overline{\pi_S^b(x)} ) = d(x,\pi_S^b(x)) + d(\pi_S^b(x),\overline{\pi_S^b(x)}) \geq 0 + (r+1) =r+1.$$ - If $\pi_S^b(x) \in \sigma$ then $d(x,\pi_S^b(x)) \geq 1$, and also the local diameter assumption implies there exists $y\in \sigma$ with $d(\pi_S^b(x),y)=r$. Thus $$d(x,y ) = d(x,\pi_S^b(x)) + d(\pi_S^b(x),y) \geq 1 + r.$$ Hence $\sigma$ is maximal in $\mathrm{VR}(Q_n;r)$. ◻ We now construct maximal simplices $\sigma$ in $\mathrm{VR}(Q_{r+1};r)$ that, by Proposition [Proposition 12](#PropReduction){reference-type="ref" reference="PropReduction"}, will remain maximal in $\mathrm{VR}(Q _n;r)$. We recall that the *cubic hull* $\mathrm{cHull}(\sigma)$ is the smallest isometric copy of a cube containing $\sigma$. That the convex hull of $\sigma$ is all of $Q_{r+1}$ will later be used to give the independence of homology generators in the proof of Theorem [Theorem 9](#ThmMain1){reference-type="ref" reference="ThmMain1"}. **Lemma 13**. *If $r \geq 2$, then there exists a maximal simplex $\sigma \subseteq Q_{r+1}$ from $\mathcal{A}_r$ with $\mathrm{localDiam}(\sigma,y)=r$ for all $y \in \sigma$, and with $\mathrm{cHull}(\sigma)=Q_{r+1}$.* *Proof.* We first prove the case when $r\ge 2$ is odd (before afterwards handling the case when $r\ge 3$ is odd). Define $\sigma$ as the collection of vertices in $Q_{r+1}$ whose coordinates contain an even number of values $1$. As $r+1$ is odd, this means $x\in \sigma$ iff $\bar x \notin \sigma$, so $\sigma \in \mathcal{A}_r$. We proceed by determining the local diameter. Let $y\in \sigma$ and define $y'$ by taking $\bar y$ and flipping one of its coordinates. Then $y' \in \sigma$ as it has an even number of ones, and it disagrees with $y$ on all coordinates except the flipped one, hence $d(y,y')=r$. So $\mathrm{localDiam}(\sigma,y) = r$ for all $y\in\sigma$. It remains to show that $\mathrm{cHull}(\sigma)=Q_{r+1}$. If $\mathrm{cHull}(\sigma) \subsetneq Q_{r+1}$, there would be a single coordinate shared by all the points of $\sigma$. However, as $r \geq 2$ we can prescribe any single coordinate as we please, and then fill in the rest of the coordinates to obtain a vertex of $\sigma$: - if the chosen coordinate was $1$, fill another coordinate as $1$ and the rest as $0$; - if the chosen coordinate was $0$, fill all other coordinates as $0$. Next, we handle the case when $r\ge 3$ is odd. Let $\tau$ be the the maximal simplex in $Q_r$ obtained in the proof of the even case. Define $$\sigma = \tau \times \{0,1\}\subseteq Q_{r+1}.$$ Formally speaking, $\sigma = \iota_{[r]}^{(0)}(Q_r) \cup \iota_{[r]}^{(1)}(Q_r) = Q_r^{(0)} \cup Q_r^{(1)}$, with the associated index set being $S=[r]$. We first prove $\sigma \in \mathcal{A}_r$. A point $x\in \sigma$ is of the form $x=y \times \{i\}$ with $y\in \tau, i\in \{0,1\}$. As $\bar x = \bar y \times \{1-i\}$ and $y\in \mathcal{A}_{r-1}$, we see that $x\in \sigma$ iff $\bar x \notin \sigma$. We proceed by determining the local diameter. Take $x=y \times \{i\} \in \sigma$. As $\mathrm{localDiam}(\tau,y)=r-1$, there exists $y'\in \tau$ with $d(y,y')=r-1$. But then $y' \times \{1-i\} \in \sigma$ and $d\left(y \times \{i\},y' \times \{1-i\}\right)=r$. It remains to show that $\mathrm{cHull}(\sigma)=Q_{r+1}$. Similarly as in the proof of the even case, this follows from the fact that as $r \geq 3$ we can prescribe any single coordinate as we please, and then fill in the rest of the coordinates to obtain a vertex of $\sigma$: - the last coordinate can be chosen freely by the construction of $\sigma$; - any of the first $r$ coordinates can be choosen freely by the construction and by the even case.  ◻ We are now in position to prove the main result of this section, Theorem [Theorem 9](#ThmMain1){reference-type="ref" reference="ThmMain1"}, which states that $\mathrm{rank}H_{2^r-1}( \mathrm{VR}(Q_{n};r)) \geq 2^{n-(r+1)}\binom{n}{r+1}$. *Proof of Theorem [Theorem 9](#ThmMain1){reference-type="ref" reference="ThmMain1"}.* For notational convenience, let $k = 2^{n-(r+1)}\binom{n}{r+1}$. There are $k$ isometric copies of $Q_{r+1}$ in $Q_n$ obtained via embeddings $\iota$, which we enumerate as $C_1, C_2, \ldots, C_{k}$. For each $i$: 1. Let $\sigma_i$ be the maximal simplex in $C_i$ obtained from Proposition [Proposition 12](#PropReduction){reference-type="ref" reference="PropReduction"} and Lemma [Lemma 13](#LemmaREvenAndOdd){reference-type="ref" reference="LemmaREvenAndOdd"}. 2. Let $[\alpha_i]$ be the mentioned cross-polytopal generator of $H_{2^r -1}(\mathrm{VR}(C_i;r))$, and recall the coefficient of $\sigma_i$ in $\alpha_i$ is $1$. 3. Let $\omega_i$ be the $(2^r - 1)$-cochain on $Q_n$ mapping $\sigma_i$ to $1$ and the rest of the $(2^r-1)$-dimensional simplices to $0$. As $\sigma_i$ is a maximal simplex in $Q_n$, the cochains $\omega_i$ are cocycles. 4. Note that $\sigma_i$ is not contained as a term in $\alpha_j$ for any $i \neq j$. Indeed, if that was the case, $\sigma_i$ would be contained in the lower-dimensional cube $C_i \cap C_j$, contradicting the conclusion $\mathrm{cHull}(\sigma)=Q_{r+1}$ from Lemma [Lemma 13](#LemmaREvenAndOdd){reference-type="ref" reference="LemmaREvenAndOdd"}. As a result, $[\omega_i] \frown [\alpha_i]=1$ and $[\omega_i] \frown [\alpha_j]=0$ for $i \neq j$. It remains to prove that homology classes $[\alpha_i]$ in $H_{2^r-1}( \mathrm{VR}(Q_{n};r))$ (via the natural inclusion) are linearly independent. If $\sum_{i=1}^{k} \lambda_i [\alpha_i]=0$ for some $\lambda_i \in \mathbb{Z}$, then applying $[\omega_j]$ via the cap product we obtain $\lambda_j=0$ by (4) above. Hence the rank of $H_{2^r-1}( \mathrm{VR}(Q_{n};r))$ is at least $k=2^{n-(r+1)}\binom{n}{r+1}$. ◻ # More contractions: concentrations {#sec:concentrations} Up to now the only contractions that we have utilized are the projections $\pi_S$. In order to establish additional homology bounds we need to employ a new kind of contractions called concentrations. The idea of a such maps in low-dimensional settings is shown in Figures [2](#Fig1){reference-type="ref" reference="Fig1"} and [3](#Fig3){reference-type="ref" reference="Fig3"}. We proceed with an explanation of the general case. ![Two contractions on $Q_2$: a projection $\pi_{[1]}$ and a concentration map.](Fig1.pdf){#Fig1 width="4in"} Let $n > k$ be positive integers. Choose $a=(a_{k+1}, a_{k+2}, \ldots, a_n) \in \{0,1\}^{n-k}$ and let $C$ denote the copy of $Q_k$ identified as $Q_k^a$, i.e., $$C= \{0,1\}^k \times \{a_{k+1}\} \times \{a_{k+2}\} \times \ldots \times \{a_{n}\}.$$ Working towards a contraction we define a *concentration map $f\colon Q_n \to C$ of codimension $n-k$* by the following rule: 1. $f|_C = \textrm{Id}_C$, and 2. for $x=(x_1, x_2, \ldots, x_n)\in Q_n \setminus C$ we define $$f(x)=(x_1, x_2, \ldots, x_{k-1}, 1, a_{k+1}, a_{k+2}, \ldots, a_n).$$ In particular, we concentrate the $k$-th coordinate of $Q_n \setminus C$ to $1$ (although we might as well have used $0$). Permuting the coordinates of $Q_n$ generates other concentration maps. In order to discuss the properties of concentration functions it suffices to consider the concentrations defined as $f$ above. {#Fig3 width="4in"} **Proposition 14**. *Let $f$ be the concentration map as defined above.* 1. *Map $f$ is a contraction.* 2. *The $n-k+1$ many $k$-dimensional cubes $Q_{k}\subseteq Q_n$ containing the $(k-1)$-dimensional cube $$\{0,1\}^{k-1} \times \{0\} \times \{a_{k+1}\} \times \{a_{k+2}\} \times \ldots \times \{a_{n}\}$$ are mapped onto $C$ isometrically by $f$.* 3. *Map $f$ maps all other $k$-dimensional cubes $Q_{k}\subseteq Q_n$ onto cubes of dimension less than $k$.* *Proof.* (i) We verify the claim by a case analysis: For $x,y\in C$ we have $$d(f(x),f(y))=d(x,y)$$ as $f$ is the identity on $C$. For $x,y\notin C$ the quantity $d(x,y)$ is the number of coordinates in which $x$ and $y$ differ, while $d(f(x),f(y))$ is the number of coordinates amongst the first $k-1$ in which $x$ and $y$ differ. Thus $d(f(x),f(y)) \leq d(x,y)$. Let $x\in C, y\notin C$. Then $d(x,y)$ is the sum of the following two numbers: - The number of coordinates amongst the first $k$ coordinates in which $x$ and $y$ differ. - The number of coordinates amongst the last $n-k$ coordinates in which $x$ and $y$ differ. Note that this quantity is at least $1$ as $y \notin C$. On the other side, $d(f(x),f(y))$ is less than or equal to the sum of the following two numbers: - The number of coordinates amongst the first $k-1$ coordinates in which $x$ and $y$ differ. - The number $1$ if the $k$-th coordinate of $x$ does not equal $1$. Together we obtain $d(f(x),f(y)) \leq d(x,y)$. This covers all possible cases. We conclude that $f$ is a contraction. \(ii\) The $n-k+1$ copies of $Q_k$ in question are the ones of the form $$\{0,1\}^{k-1} \times \{0\} \times \{a_{k+1}\} \times \{a_{k+2}\} \times \ldots \times \{a_{p-1}\} \times \{0,1\} \times \{a_{p+1}\} \times \ldots \times \{a_{n}\}$$ for $p\in \{k, k+1, \ldots, n\}$. The case $p=k$ shows that $C$ is one of these copies. Note that the part $$\{0,1\}^{k-1} \times \{0\} \times \{a_{k+1}\} \times \{a_{k+2}\} \times \ldots \times \{a_{p-1}\} \times \{a_p\} \times \{a_{p+1}\} \times \ldots \times \{a_{n}\}$$ is contained in $C$ and thus $f$ is identity on it. On the other hand, the part $$\{0,1\}^{k-1} \times \{0\} \times \{a_{k+1}\} \times \{a_{k+2}\} \times \ldots \times \{a_{p-1}\} \times \{1- a_p\} \times \{a_{p+1}\} \times \ldots \times \{a_{n}\}$$ gets mapped to $$\{0,1\}^{k-1} \times \{1\} \times \{a_{k+1}\} \times \{a_{k+2}\} \times \ldots \times \{a_{p-1}\} \times \{a_p\} \times \{a_{p+1}\} \times \ldots \times \{a_{n}\}$$ by retaining the first $k-1$ coordinates. Together these two parts form $C$. \(iii\) Any $k$-dimensional cube $Q_{k}\subseteq Q_n$ not mentioned in (2) has one of the first $k-1$ coordinates (say, the $p$-th coordinate) constant. Thus the same holds for its image via $f$ and consequently, its image is contained in the corresponding $Q_{k-1}\subseteq Q_n$, i.e., the one having the $p$-th coordinate constant, and having the last $n-k$ coordinates prescribed as in $C$. ◻ # Homology bounds via contractions {#sec:contractions-homology} {reference-type="ref" reference="ThmMain2"} than the linear increase in Theorems [Theorem 17](#ThmMain3){reference-type="ref" reference="ThmMain3"} and [Theorem 18](#ThmMain4){reference-type="ref" reference="ThmMain4"}. ](Fig4.pdf){#Fig4 width="4in"} In Section [4](#sec:maximal){reference-type="ref" reference="sec:maximal"} we showed how the appearance of cross-polytopal homology classes in certain dimensions of the Vietoris--Rips complexes of cubes generate independent homology elements in Vietoris--Rips complexes of higher-dimensional cubes. Applying Proposition [Proposition 6](#PropContrEmbed){reference-type="ref" reference="PropContrEmbed"} to the canonical projections implies that the homology of each smaller subcube embeds. However, the independence of homology classes arising from various subcubes was proved using maximal simplices; this argument depended heavily on the fact that convenient (cross-polytopal) homology representatives were available to us. In this section we aim to provide an analogous result for homology in any dimension, without prior knowledge of homology generators. **Example 15**. As a motivating example, consider the graph $\mathrm{VR}(Q_3;1)$. The cube $Q_3$ contains six subcubes $Q_2$ and each $\mathrm{VR}(Q_2;1)\simeq S^1$ has the first Betti number equal to $1$. However, the first Betti number of $Q_3$ is not $6$ but rather $5$, demonstrating that homology classes of various subcubes might in general interfere, i.e., not be independent. In Theorem [Theorem 9](#ThmMain1){reference-type="ref" reference="ThmMain1"} we proved there is no such interference in a specific case (involving cross-polytopes). In the more general case of this section, we prove lower bounds even when independence may not hold. **The general setting of this section**: Fix $r, q\in \{1,2,\ldots\}$. Let $p\ge 1$ be the smallest integer for which $H_q(\mathrm{VR}(Q_p;r))\neq 0$. We implicitly assume that such a $p$ exists, i.e., that $H_q(\mathrm{VR}(Q_n;r))$ is non-trivial for some $n$. The cube $Q_n$ contains $2^{n-p} \binom{n}{p}$ canonical $Q_p$ subcubes. We aim to estimate the rank of the homomorphism $$\bigoplus_{{2^{n-p} \binom{n}{p}}} H_q(\mathrm{VR}(Q_p;r)) \to H_q(\mathrm{VR}(Q_n;r))$$ induced by the map $$\coprod_{2^{n-p} \binom{n}{p}} Q_p \to Q_n,$$ consisting of the natural inclusions of all of the $Q_p$ subcubes. ## Homology bounds via projections We will start with the simplest argument to demonstrate how contractions, in this case projections, may be used to lower bound the homology. **Theorem 16**. *Let $q\ge 1$. If $p$ is the smallest integer for which $H_q(\mathrm{VR}(Q_p;r))\neq 0$, then for $n \geq p$, $$\mathrm{rank}H_{q}( \mathrm{VR}(Q_{n};r)) \geq \binom{n}{p} \cdot \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r)).$$* *Proof.* Let the $Q_p^*$ denote the $\binom{n}{p}2^{n-p}$ different $p$-dimensional subcubes of $Q_n$ (say as $*$ varies from $1$ to $\binom{n}{p}2^{n-p}$). For a subset $S \subseteq [n]$ of cardinality $p$, the projection $\pi_S \colon Q_n \to Q_p$ is an isometry on $2^{n-p}$ of these subcubes $Q_p^*$ (the ones having exactly the coordinates $S$ as the free coordinates). For each such $S$ choose one of these cubes and designate it as $Q_{p,S}$, thus marking $\binom{n}{p}$ copies of $Q_p \subseteq Q_n$. For each $S$ let $a_{1,S}, a_{2,S}, \ldots, a_{\mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r)),S}$ denote a largest linearly independent collection in $H_{q}( \mathrm{VR}(Q_{p,S};r))$. We claim that the collection $\{a_{i,S}\}$ of cardinality $\binom{n}{p}\cdot \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r))$ is linearly independent in $H_{q}( \mathrm{VR}(Q_{n};r))$. Assume $$\sum_{i,S}\lambda_{i,S} \cdot a_{i,S}=0$$ for some coefficients $\lambda_{i,S}$. Fix a subset $S' \subseteq [n]$ of cardinality $p$, and to the equality above apply the map on $H_q$ induced by the projection $\pi_{S'}\colon Q_n \to Q_p$. As $p$ is the minimal dimension of a cube in which $H_q$ is non-trivial on Vietoris--Rips complexes, and as $\pi_{S'}$ maps all of the $Q_p^*$ to a smaller-dimensional cube except for the ones with exactly the coordinates in $S'$ as the free coordinates, we obtain $(\pi_{S'})_*(a_{i,S})=0$ for all $S \neq S'$, where $*$ denotes the induced map on homology. As $\pi_{S'}|_{Q_{p,S'}}\to Q_p$ is a bijection on the corresponding Vietoris--Rips complexes, and induces an isomorphism homology, we have reduced our equation to $$\sum_{i}\lambda_{i,S'} \cdot a_{i,S'}=0.$$ Consequently, $\lambda_{i,S'}=0$ for all $i$, as $\{a_{i,S'}\}$ forms a linearly independent collection in $H_{q}( \mathrm{VR}(Q_{p};r))$ by definition. As $S'$ was arbitrary we conclude $\lambda_{i,S}=0$ for all $i$ and $S$, and thus the claim holds. ◻ ## Codimension $1$ homology bounds via concentrations The following result states that in codimension $1$ (i.e., when we increase the dimension of the cube by $1$ from $p$ to $p+1$), all but at most one of the subcubes induce independent inclusions on homology. Example [Example 15](#Ex1){reference-type="ref" reference="Ex1"} shows that all subcubes need not induce independent inclusions of homology (and we see that Theorem [Theorem 17](#ThmMain3){reference-type="ref" reference="ThmMain3"} is tight in the case of Example [Example 15](#Ex1){reference-type="ref" reference="Ex1"}). **Theorem 17**. *Let $q\ge 1$. If $p$ is the smallest integer for which $H_q(\mathrm{VR}(Q_p;r))\neq 0$, then $$\mathrm{rank}H_{q}( \mathrm{VR}(Q_{p+1};r)) \geq %(2(p+1)-1) (2p+1) \cdot \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r)).$$* *Proof.* The cube $Q_{p+1}$ contains $2p+2$ subcubes $Q_p$, which we enumerate as $Q_{p,1}, Q_{p,2}, \ldots, %Q_{p,2(p+1)} Q_{p,2p+2}$. For each $1 \le j \le %2(p+1)-1 2p+1$, let $\{a_{i,j} \mid 1\le i \le \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r))\}$ denote a largest linearly independent collection in $H_{q}( \mathrm{VR}(Q_{p,j};r))$. We claim that the collection $$\left\{a_{i,j}\mid 1\le i \le \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r)), 1\le j\le 2p+1\right\}$$ of cardinality $(2p+1)\cdot \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r))$ is linearly independent. Assume $$\label{Eq2} \sum_{i,j}\lambda_{i,j} \cdot a_{i,j}=0$$ for some coefficients $\lambda_{i,j}$ with $1\le i \le \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r))$ and $1\le j\le 2p+1$. Note that there are no representatives from $Q_{p,2p+2}$. Choose a subcube $Q_{p-1}\subseteq Q_{p,2p+2}$. It is the intersection of $Q_{p,2p+2}$ and another $p$-dimensional cube of the form $Q_{p,*}$, say $Q_{p,1}$. We apply the concentration map $f$ corresponding to these choices using Proposition [Proposition 14](#PropConcentration){reference-type="ref" reference="PropConcentration"}: 1. $f$ is bijective on exactly two $p$-cubes: $Q_{p,2p+2}$ and $Q_{p,1}$. 2. $f$ maps all other $p$-cubes to cubes of dimension less than $p$. By the choice of $p$ (as the first dimension of a cube in which nontrivial $p$-dimensional homology appears in its Vietoris--Rips complex), we get $f_*(a_{i,j})=0$ for all $i>1$. 3. As a result, after applying the induced map $f_*$ on homology, Equation [\[Eq2\]](#Eq2){reference-type="eqref" reference="Eq2"} simplifies to $$\sum_{j}\lambda_{j} \cdot a_{1,j}=0.$$ Consequently, $\lambda_{1,j}=0$ for all $i$, as $\{a_{1,j}\}$ forms a linearly independent collection in $H_{q}( \mathrm{VR}(Q_{p,1};r))$ by definition. We keep repeating the procedure of the previous paragraph: - Choose a subcube $Q_{p,j}$, whose corresponding coefficients $\lambda_{i,j}$ have been determined to be zero, and choose a neighboring subcube $Q_{p,j'}$ (i.e., a subcube with a common $(p-1)$-dimensional cube), whose corresponding coefficients $\lambda_{i,j'}$ have not yet been determined to be zero. - Apply the concentration map corresponding to these two $p$-dimensional cubes to deduce that the coefficients $\lambda_{i,j'}$ also equal zero. Any cube $Q_{p,j'}$ can be reached from $Q_{p,2p+2}$ by an appropriate sequence of cubes (i.e., $Q_{p,2p+2}$, a $(p-1)$-dimensional subcube thereof, an enclosing $p$-dimensional cube, a $(p-1)$-dimensional subcube thereof, $\ldots, Q_{p,j'}$). Therefore, we can eventually deduce that $\lambda_{i,j}=0$ for all $i$ and $j$. Hence the rank bound holds due to the setup of Equation [\[Eq2\]](#Eq2){reference-type="eqref" reference="Eq2"}. ◻ ## General homology bounds via concentrations {#ssec:gen-via-concentrations} In this subsection we will generalize the argument of Theorem [Theorem 17](#ThmMain3){reference-type="ref" reference="ThmMain3"} to deduce a lower bound on homology on all subsequent larger (not just codimension one) cubes. The core idea is the following. In the previous subsection we were able to "connect" $p$-dimensional cubes by concentrations. Each chosen concentration was bijective on exactly two adjacent cubes of the form $Q_p$ sharing a common $(p-1)$-dimensional cube; see item (i) in the proof of Theorem [Theorem 17](#ThmMain3){reference-type="ref" reference="ThmMain3"}. If the coefficients of Equation [\[Eq2\]](#Eq2){reference-type="eqref" reference="Eq2"} corresponding to one of the two copies of $Q_p$ were known to be trivial, then the homological version of the said concentration transformed Equation [\[Eq2\]](#Eq2){reference-type="eqref" reference="Eq2"} so that only the coefficients corresponding to the other of the two cubes $Q_p$ were retained; see item (ii). These coefficients were then deduced to be trivial by their definition as the resulting equation contained only terms arising from a single $Q_p$, see item (iii). In this subsection we generalize this argument to all higher-dimensional cubes. Instead of concentrations isolating $2$ adjacent copies of $Q_p$ (as happens in codimension $1$), the concentrations will in general isolate $n-p+1$ (i.e., codimension plus one; see Proposition [Proposition 14](#PropConcentration){reference-type="ref" reference="PropConcentration"}(ii)) copies of $Q_p$. The main technical question is thus to determine how many of the subcubes are independent in the above sense. **Theorem 18**. *Let $q\ge 1$. If $p$ is the smallest integer for which $H_q(\mathrm{VR}(Q_p;r))\neq 0$, then for $n \geq p$, $$\mathrm{rank}H_{q}( \mathrm{VR}(Q_{n};r)) \geq \sum_{i=p}^n 2^{i-p} \binom{i-1}{p-1} \cdot \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r)).$$* In particular cases, Theorem [Theorem 18](#ThmMain4){reference-type="ref" reference="ThmMain4"} reduces to the following. For $n=p$ we get the tautology that $\mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r))$ is at least as large as itself. For $n=p+1$ we recover Theorem [Theorem 17](#ThmMain3){reference-type="ref" reference="ThmMain3"}: $$\begin{aligned} \mathrm{rank}H_{q}( \mathrm{VR}(Q_{n};r)) &\geq \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r))+ 2 \binom{p}{p-1} \cdot \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r)) \\ &= (2p+1)\cdot \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r)).\end{aligned}$$ {reference-type="ref" reference="ThmMain4"}. In each step the thick dashed lines represent copies of $Q_p$ in $M_n$ yielding new independent homology classes within the Vietoris-Rips complex of $Q_n$, in addition to the established independent homology classes (thick solid lines) arising from certain copies of $Q_p$, denoted by $F_n$, within the front face $Q_{n-1}^1$. The multiplicative factor in the theorem is the total number of the thick edges, both dashed and solid ones.](Fig2.pdf){#Fig5 width="4in"} *Proof of Theorem [Theorem 18](#ThmMain4){reference-type="ref" reference="ThmMain4"}.* The cube $Q_n$ consists of two disjoint copies of $Q_{n-1}$; see Figure [5](#Fig5){reference-type="ref" reference="Fig5"}: - the rear one with the last coordinate $0$, denoted by $Q_{n-1}^0$, and - the front one with the last coordinate $1$, denoted by $Q_{n-1}^1$. We partition the $Q_p$ subcubes of $Q_n$ into three classes: - The ones contained in the rear $Q_{n-1}^0$ where vertices have last coordinate $0$, denoted by $R_n$. - The ones contained in the front $Q_{n-1}^1$ where vertices have last coordinate $1$, denoted by $F_n$. - The ones contained in the middle passage between them, denoted by $M_n$. Each such $Q_p$ in $M_n$ is of the form $D \times \{0,1\}$, where $D \subseteq Q_{n-1}$ is a copy of $Q_{p-1}$. We will prove that the following $Q_p$ subcubes of $Q_n$ induce independent embeddings on homology $H_q$ of Vietoris--Rips complexes: the elements of $M_n$ (dashed cubes in Figure [5](#Fig5){reference-type="ref" reference="Fig5"}) and the elements of $F_n$ that have inductively been shown to include independent embeddings on homology $H_q$ of Vietoris--Rips complexes on $Q^1_{n-1}$ (bold cubes in Figure [5](#Fig5){reference-type="ref" reference="Fig5"}). The initial cases of the inductive process have been discussed in the paragraph before the proof. For $n=p+1$ this is Theorem [Theorem 17](#ThmMain3){reference-type="ref" reference="ThmMain3"}. The cardinality of $M_n$ is $2^{(n-1)-(p-1)} \binom{n-1}{p-1} = 2^{n-p} \binom{n-1}{p-1}$, which is the number of $Q_{p-1}$ subcubes in $Q_{n-1}^0$. Each such subcube has the last coordinate constantly $0$. Taking a union with a copy of the same $Q_{p-1}$ subcube with the last coordinates changed to $1$, we obtain a $Q_p$ subcube in $M_n$. It is apparent that all elements of $M_n$ arise this way. Let us enumerate the elements of $M_n$ as $Q_{p,j}^M$ with $1\le j \le 2^{n-p} \binom{n-1}{p-1}$. For each such $j$ let $\{a_{i,j}\mid 1\le i \le \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r))\}$ denote a largest linearly independent collection in $H_{q}( \mathrm{VR}(Q_{p,j}^M;r))$. The cardinality of the copies of $Q_p$ in $F_n$ that have inductively been shown to include independent embeddings on homology $H_q$ of Vietoris--Rips complexes on $Q^1_{n-1}$ equals $\sum_{i=p}^{n-1} 2^{i-p} \binom{i-1}{p-1}$, by inductive assumption. Let us enumerate them by $Q_{p,j}^F$ with $1\le j\le \sum_{i=p}^{n-1} 2^{i-p} \binom{i-1}{p-1}$. For each such $j$ let $\{b_{i,j} \mid 1\le i\le \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r))\}$ denote a largest linearly independent collection in $H_{q}( \mathrm{VR}(Q_{p,j}^F;r))$. Assuming the equality $$\label{Eq3a} \sum_{i,j}\lambda_{i,j} \cdot a_{i,j} + \sum_{i,j}\mu_{i,j} \cdot b_{i,j}=0$$ in $H_{q}( \mathrm{VR}(Q_{n};r))$ for some coefficients $\lambda_{i,j}$, $\mu_{i,j}$, we claim that all coefficients equal zero. This will prove the theorem as the number of involved terms equals $\sum_{i=p}^n 2^{i-p} \binom{i-1}{p-1} \cdot \mathrm{rank}H_{q}( \mathrm{VR}(Q_{p};r))$. We will first prove that the coefficients $\lambda_{i,j}$ are all zero. Fix some $1\le j \le 2^{n-p} \binom{n-1}{p-1}$, and let $D$ be the copy of $Q_{p-1}$ in $Q_{n-1}^0$ so that $C\coloneqq D \times \{0,1\}$ is equal to $Q_{p,j}^M$. Let $f$ be any concentration $Q_n \to C:=D \times \{0,1\}$. (For example, in Figure [4](#Fig4){reference-type="ref" reference="Fig4"} one can visualize $D$ as the solid round vertex, and $C$ as the edge between the two solid vertices.) By Proposition [Proposition 14](#PropConcentration){reference-type="ref" reference="PropConcentration"}: - $f$ maps any $Q_p$ subcube of $Q_n$ that contains $D$ bijectively onto $C=Q_{p,j}^M$. All such subcubes except for $C$ are contained in $R_n$. - $f$ maps all of the other $Q_p$ subcubes of $Q_n$ to lower-dimensional subcubes. These two observations imply that applying the induced map $f_*$ on homology to Equation [\[Eq3a\]](#Eq3a){reference-type="eqref" reference="Eq3a"}, we obtain $\sum_{i}\lambda_{i,j} \cdot a_{i,j}=0.$ By the choice of $\{a_{i,j}\}_i$ as an independent collection of homology classes for $H_{q}(\mathrm{VR}(Q_{p,j}^M;r))$, we obtain $\lambda_{i,j}=0$ for all $i$. Since this can be done for any $1\le j \le 2^{n-p} \binom{n-1}{p-1}$, we have $\lambda_{i,j}=0$ for all $i$ and $j$. We have thus reduced Equation [\[Eq3a\]](#Eq3a){reference-type="eqref" reference="Eq3a"} to $\sum_{i,j}\mu_{i,j} \cdot b_{i,j}=0.$ Let $\pi_S \colon Q_n \to Q_{n-1}$ be the projection that forgets the last coordinate of each vector (explicitly, $S=[n-1]\subseteq [n]$). Note that the restrictions of $\pi_S$ to $Q_n^0$ and to $Q_n^1$ are bijections. Hence, after applying the induced map $(\pi_S)_*$ on homology, the inductive definition of the $b_{i,j}$ implies that $\mu_{i,j}=0$ for all $i$ and $j$. ◻ ## Comparison with known results {#SubsComparison} In this subsection we demonstrate that our lower bounds agree with actual ranks of homology in many known cases. In particular, for $r=1,2,3$, if we assume that we know the homotopy types or Betti numbers for the first few cases ($n\le r+1$ or $n \le r+2$), then we show that our lower bounds on the Betti numbers of $\mathrm{VR}(Q_n;r)$ are tight (i.e. optimal) for all $n$ and for all dimensions of homology. For $r=4$ we explain the best lower bounds we know on the Betti numbers of $\mathrm{VR}(Q_n;4)$, which are based on homology computations by Ziqin Feng. Since the homotopy types of $\mathrm{VR}(Q_n;4)$ are unknown for $n\ge 6$, we do not know if these bounds are tight. For a summary see Table [2](#table:homotopy-types2){reference-type="ref" reference="table:homotopy-types2"}. ### The case $\mathbf{r=1}$ Assuming the obvious homeomorphism $\mathrm{VR}(Q_2;1) \cong S^1$, the lower bound of Theorem [Theorem 18](#ThmMain4){reference-type="ref" reference="ThmMain4"} with $p=2$ gives $$\mathrm{rank}H_1(\mathrm{VR}(Q_n;1)) \ge \sum_{i=2}^n 2^{i-2} \binom{i-1}{1} =\sum_{i=2}^n (i-1) 2^{i-2} =n 2^{n-1} - 2^n + 1,$$ where the last step is explained in Appendix [9.1](#app:1){reference-type="ref" reference="app:1"}. This inequality is actually an equality, as one can see via an Euler characteristic computation. Indeed, $\mathrm{VR}(Q_n;1)$ has $2^n$ vertices and $n2^{n-1}$ edges, and so the Euler characteristic is $2^n - n2^{n-1}$. As $\mathrm{VR}(Q_n;1)$ is connected, the rank of $H_1(\mathrm{VR}(Q_n;1))$ equals $n 2^{n-1} - 2^n + 1$. See [@carlsson2020persistent Proposition 4.12] for a related computation. ### The case $\mathbf{r=2}$ We know that the embedding of each individual subcube induces an injection on homology. Our results provide the lower bounds on the rank of the map on homology induced by the inclusion of *all* subcubes $Q_p$ (where $p$ is the dimension of the first appearance of $q$-dimensional homology $H_q$). The upper bound for homology obtained in this way is $2^{n-p}\binom{n}{p} \mathrm{rank}H_q(\mathrm{VR}(Q_p;r))$, where the the multiplicative constant is the number of all $Q_p$ subcubes on $Q_n$. These possible generators are all independent in the case of cross-polytopal generators (Theorem [Theorem 9](#ThmMain1){reference-type="ref" reference="ThmMain1"}). In case this bound is exceeded we can thus deduce that certain new homology classes appear that are *not* generated by the embeddings of $Q_p$ subcubes. $_n \backslash ^r$ $1$ $1$ $2$ $2$ -------------------- -------------------------------------------------------- ---------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------ $1$ $*$ $*$ $*$ $*$ $2$ $\textcolor{violet}{\mathbf{S^1}}$ $\beta_{1}\geq \textcolor{violet}{1}$ $*$ $*$ $3$ $\bigvee^5 S^1$ $\beta_{1}\geq \textcolor{violet}{ 5}$ $\textcolor{red}{\mathbf{S^3}}$ $\beta_{3}\geq \textcolor{red}{1}$ $4$ $\bigvee^{17} S^1$ $\beta_{1}\geq \textcolor{violet}{17}$ $\bigvee^8 S^3 \vee \textcolor{blue}{\mathbf{S^3}}$ $\beta_{3}\geq \textcolor{red}{ 8} + \textcolor{blue}{1}$ $5$ $\bigvee^{49} S^1$ $\beta_{1}\geq \textcolor{violet}{49}$ $\bigvee^{49} S^3$ $\beta_{3}\geq \textcolor{red}{ 40} + \textcolor{blue}{9}$ $6$ $\bigvee^{129} S^1$ $\beta_{1}\geq \textcolor{violet}{129}$ $\bigvee^{209} S^3$ $\beta_{3}\geq \textcolor{red}{ 160} + \textcolor{blue}{49}$ $7$ $\bigvee^{321} S^1$ $\beta_{1}\geq \textcolor{violet}{321}$ $\bigvee^{769} S^3$ $\beta_{3}\geq \textcolor{red}{ 560} + \textcolor{blue}{209}$ $8$ $\bigvee^{769} S^1$ $\beta_{1}\geq \textcolor{violet}{769}$ $\bigvee^{2561} S^3$ $\beta_{3}\geq \textcolor{red}{ 1792} + \textcolor{blue}{769}$ $9$ $\bigvee^{1793} S^1$ $\beta_{1}\geq \textcolor{violet}{1793}$ $\bigvee^{7937} S^3$ $\beta_{3}\geq \textcolor{red}{ 5376} + \textcolor{blue}{2561}$ $10$ $\bigvee^{4097} S^1$ $\beta_{1}\ge \textcolor{violet}{4097}$ $\bigvee^{23297} S^3$ $\beta_{3}\geq \textcolor{red}{ 15360} + \textcolor{blue}{7937}$ $11$ $\bigvee^{9217} S^1$ $\beta_{1}\ge \textcolor{violet}{9217}$ $\bigvee^{65537} S^3$ $\beta_{3}\geq \textcolor{red}{ 42240} + \textcolor{blue}{23297}$ $12$ $\bigvee^{20481} S^1$ $\beta_{1}\ge \textcolor{violet}{20482}$ $\bigvee^{178177} S^3$ $\beta_{3}\geq \textcolor{red}{ 112640} + \textcolor{blue}{65537}$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $_n \backslash ^r$ $3$ $3$ $4$ $4$ $1$ $*$ $*$ $*$ $*$ $2$ $*$ $*$ $*$ $*$ $3$ $*$ $*$ $*$ $*$ $4$ $\textcolor{red}{\mathbf{S^7}}$ $\beta_{7}\geq \textcolor{red}{1}$ $*$ $*$ $5$ $\bigvee^{10} S^7 \vee \textcolor{blue}{\mathbf{S^4}}$ $\beta_{7}\geq \textcolor{red}{10}, \beta_{4}\geq \textcolor{blue}{1}$ $\textcolor{red}{\mathbf{S^{15}}}$ $\beta_{15}\geq \textcolor{red}{1}$ $6$ $\bigvee^{60} S^7 \vee \bigvee^{11} S^4$ $\beta_{7}\geq \textcolor{red}{60}, \beta_{4}\geq \textcolor{blue}{11}$ $\beta_{15}=\textcolor{red}{12}+\textcolor{gray}{\mathbf{2}}, \beta_7=\textcolor{brown}{\mathbf{239}}$ $\beta_{15}\ge\textcolor{red}{12}+\textcolor{gray}{2}, \beta_7\ge\textcolor{brown}{239}$ $7$ $\bigvee^{280} S^7 \vee \bigvee^{71} S^4$ $\beta_{7}\geq \textcolor{red}{280}, \beta_{4}\geq \textcolor{blue}{71}$ $?$ $\beta_{15}\ge\textcolor{red}{84}+\textcolor{gray}{26}, \beta_7\ge\textcolor{brown}{3107}$ $8$ $\bigvee^{1120} S^7 \vee \bigvee^{351} S^4$ $\beta_{7}\geq \textcolor{red}{1120}, \beta_{4}\geq \textcolor{blue}{351}$ $?$ $\beta_{15}\ge\textcolor{red}{448}+\textcolor{gray}{194}, \beta_7\ge\textcolor{brown}{23183}$ $9$ $\bigvee^{4032} S^7 \vee \bigvee^{1471} S^4$ $\beta_{7}\geq \textcolor{red}{4032}, \beta_{4}\geq \textcolor{blue}{1471}$ $?$ $\beta_{15}\ge\textcolor{red}{2016}+\textcolor{gray}{1090}, \beta_7\ge\textcolor{brown}{130255}$ $10$ $\bigvee^{13440} S^7 \vee \bigvee^{5503} S^4$ $\beta_{7}\geq \textcolor{red}{13440}, \beta_{4}\geq \textcolor{blue}{5503}$ $?$ $\beta_{15}\ge\textcolor{red}{8064}+\textcolor{gray}{5122}, \beta_7\ge\textcolor{brown}{612079}$ $11$ $\bigvee^{42240} S^7 \vee \bigvee^{18943} S^4$ $\beta_{7}\geq \textcolor{red}{42240}, \beta_{4}\geq \textcolor{blue}{18943}$ $?$ $\beta_{15}\ge\textcolor{red}{29568}+\textcolor{gray}{21250}, \beta_7\ge\textcolor{brown}{2539375}$ $12$ $\bigvee^{126720} S^7 \vee \bigvee^{61183} S^4$ $\beta_{7}\geq \textcolor{red}{126720}, \beta_{4}\geq \textcolor{blue}{61183}$ $?$ $\beta_{15}\ge\textcolor{red}{101376}+\textcolor{gray}{80386}, \beta_7\ge\textcolor{brown}{9606127}$ : For each of $r=1,2,3,4$, the pair of columns represent the comparison between *(left)* known homotopy types and homology groups of $\mathrm{VR}(Q_n;r)$ with *(right)* the bounds arising from our results. The bold red spheres are the initial cross-polytopal spheres that induce the red lower bounds on Betti numbers due to Theorem [Theorem 9](#ThmMain1){reference-type="ref" reference="ThmMain1"}. For $r=1,2,3$, the bold blue and violet spheres induce the blue and violet lower bounds on Betti numbers due to Theorems [Theorem 18](#ThmMain4){reference-type="ref" reference="ThmMain4"} and [Theorem 19](#ThmMain5){reference-type="ref" reference="ThmMain5"}. Observe that the total lower bounds match the known Betti numbers for $r=1,2,3$. The homology computations for $\mathrm{VR}(Q_6;4)$ by Ziqin Feng induce the lower bounds on Betti numbers of $\mathrm{VR}(Q_n;4)$ for $n\ge 6$ by Theorems [Theorem 18](#ThmMain4){reference-type="ref" reference="ThmMain4"} and [Theorem 19](#ThmMain5){reference-type="ref" reference="ThmMain5"}. Theorem [Theorem 20](#ThmGeom){reference-type="ref" reference="ThmGeom"}(**iii**) states that in each column $r\ge 2$, we have at least one homology class (such as the features in blue) that is not induced from a red cross-polytopal sphere. In the case $r=2$, $H_{3}( \mathrm{VR}(Q_{3};2))$ has rank one and is generated by a cross-polytopal element. Even though $Q_4$ has only $2 \binom{4}{1}=8$ subcubes $Q_3$, we have $\mathrm{rank}H_{3}( \mathrm{VR}(Q_{4};2))=9>8$. This indicates the appearance of a homology class $\alpha$ not generated by embedded homologies of $Q_3$ subcubes; see Section [7](#sec:geometric){reference-type="ref" reference="sec:geometric"} for a description of this "geometric" generator. This new homology class contributes to the homology of higher-dimensional cubes in the same way as the homology described by Theorem [Theorem 18](#ThmMain4){reference-type="ref" reference="ThmMain4"}. We formalize this in the next subsection; see Theorem [Theorem 19](#ThmMain5){reference-type="ref" reference="ThmMain5"}. Together, this cross-polytopal generator and this geometric generator $\alpha$ explain all of the homology when $r=2$: 1. The cross-polytopal elements provide $2^{n-3} \binom{n}{3}$ independent generators for $H_{3}( \mathrm{VR}(Q_{n};2))$ (Theorem [Theorem 9](#ThmMain1){reference-type="ref" reference="ThmMain1"}). 2. The non-cross-polytopal element $\alpha$ provides $\sum_{i=4}^n 2^{i-4} \binom{i-1}{3}=\sum_{i=3}^{n-1} 2^{i-3} \binom{i}{3}$ more independent generators for $H_{3}( \mathrm{VR}(Q_{n};2))$ (Theorem [Theorem 19](#ThmMain5){reference-type="ref" reference="ThmMain5"}). Thus, the combined lower bound says that the rank of $H_{3}( \mathrm{VR}(Q_{n};2))$ is at least $$\begin{aligned} &2^{n-3} \binom{n}{3} + \sum_{i=3}^{n-1} 2^{i-3} \binom{i}{3} \\ =& \sum_{i=3}^{n} 2^{i-3} \binom{i}{3} \\ =& 2^{n-2}\binom{n}{3} - \sum_{i=1}^{n-2} 2^{i-1}\binom{i+1}{2} && \text{as explained in Appendix~\ref{app:2}}\\ =& \sum_{i=1}^{n-2} \left(2^{n-2}-2^{i-1}\right)\binom{i+1}{2} && \text{since }\binom{n}{3}=\sum_{i=1}^{n-2}\binom{i+1}{2} \\ =& \sum_{i=1}^{n-1} \left(2^{n-2}-2^{i-1}\right)\binom{i+1}{2} && \text{since }2^{n-2}-2^{(n-1)-1}=0 \\ =& \sum_{0 \leq j < i < n}(j+1)(2^{n-2}- 2^{i-1}) && \text{since }\binom{i+1}{2}=\sum_{j=0}^{i-1}(j+1) \\ \eqqcolon&\ c_n.\end{aligned}$$ Since $\mathrm{VR}(Q_n;2) \simeq \bigvee_{c_n} S^3$ (Theorem [Theorem 4](#ThmAA){reference-type="ref" reference="ThmAA"}), this combined lower bound explains all of the homology when $r=2$. ### The case $\mathbf{r=3}$ Using only the cross-polytopal generator for $\mathrm{VR}(Q_4;3) \cong S^7$ and also the homotopy equivalence $\mathrm{VR}(Q_5;3) \simeq \bigvee^{10} S^7 \bigvee S^4$, we obtain the following lower bounds bounds on homology when $r=3$: 1. The cross-polytopal elements provide $2^{n-4} \binom{n}{4}$ independent generators for $H_{7}( \mathrm{VR}(Q_{n};3))$ (Theorem [Theorem 9](#ThmMain1){reference-type="ref" reference="ThmMain1"}). 2. The non-cross-polytopal four-dimensional homology element appearing at $n=5$ provides\ $\sum_{i=5}^n 2^{i-5} \binom{i-1}{4} = \sum_{i=4}^{n-1} 2^{i-4} \binom{i}{4}$ independent generators for $H_{4}( \mathrm{VR}(Q_{n};3))$ (Theorem [Theorem 18](#ThmMain4){reference-type="ref" reference="ThmMain4"}). The total lower bound equals the actual rank of all homology of $\mathrm{VR}(Q_{n};3)$, due to Theorem [Theorem 5](#ThmZiqin){reference-type="ref" reference="ThmZiqin"} which says $$\mathrm{VR}(Q_n;3) \simeq \bigvee_{2^{n-4}\binom{n}{4}} S^7 \ \vee \bigvee_{\sum_{i=4}^{n-1} 2^{i-4} \binom{i}{4}} S^4.$$ ### The case $\mathbf{r=4}$ {#SubSubsComparison:4} Ziqin Feng at Auburn University has computed the homology of $\mathrm{VR}(Q_6;4)$. To do so, he used the Easley Cluster at Auburn University (a system for high-performance and parallel computing), about 180 GB of memory, and the Ripser software package [@bauer2021ripser]. His computations show that $$H_q(\mathrm{VR}(Q_6;4);\mathbb{Z}/2) \cong \begin{cases} 0 & \text{if }1\le q\le 6\\ (\mathbb{Z}/2)^{239} & \text{if }q=7\\ 0 & \text{if }8\le q\le 14\\ (\mathbb{Z}/2)^{14} & \text{if }q=15. \end{cases}$$ This computation is shown in the first $r=4$ column in Table [2](#table:homotopy-types2){reference-type="ref" reference="table:homotopy-types2"}, and the consequences implied by this computation and by our results are shown in the second $r=4$ column in that table. ### The case of general $\mathbf{r}$ In Section [7](#sec:geometric){reference-type="ref" reference="sec:geometric"} we prove that in each column of Tables [1](#table:homotopy-types){reference-type="ref" reference="table:homotopy-types"} or [2](#table:homotopy-types2){reference-type="ref" reference="table:homotopy-types2"} with $r \geq 2$, a new homology generator appears in $\mathrm{VR}(Q_n;r)$ with $n\ge r+1$, i.e., below the diagonal entry $n=r+1$ where the cross-polytopal generator appears. Examples of these new homology generators are in the items (b) above for $r=2$ and $3$. ## Propagation of non-initial homology For positive integers $m < n$ let $$\Psi_{m,n} \colon \coprod_{2^{n-m} \binom{n}{m}} Q_m \to Q_n$$ denote the natural inclusion of all the $Q_m$ subcubes of $Q_n$. Given a positive integer $q$, let the homomorphism $$(\Psi_{m,n})_* \colon \bigoplus_{{2^{n-m} \binom{n}{m}}} H_q(\mathrm{VR}(Q_m;r)) \to H_q(\mathrm{VR}(Q_n;r))$$ be the induced map on homology. Our previous results have provided lower bounds on the rank of maps $(\Psi_{p,n})_*$, where $p$ is the smallest parameter for which $H_q(\mathrm{VR}(Q_p;r))$ is non-trivial. Our next result explains how an analogous result also holds for other maps $(\Psi_{m,n})_*$. **Theorem 19**. *Let $q\ge 1$. Let $$\mathcal{R}_m = \mathrm{rank}\Big(H_q(\mathrm{VR}(Q_m;r)) \ \big/ \ \mathrm{im}(\Psi_{m-1,m})_*\Big).$$ Then for $n \geq m \geq p$, $$\mathrm{rank}\Big(H_q(\mathrm{VR}(Q_n;r)) \ \big/ \ \mathrm{im}(\Psi_{m-1,n})_*\Big) \geq \sum_{i=m}^n 2^{i-m} \binom{i-1}{m-1} \cdot \mathcal{R}_m.$$* Note that $(\Psi_{p-1,p})_*=0$ by the definition of $p$, and so we recover Theorem [Theorem 18](#ThmMain4){reference-type="ref" reference="ThmMain4"} by setting $m=p$ in Theorem [Theorem 19](#ThmMain5){reference-type="ref" reference="ThmMain5"}. *Proof.* The proof proceeds by induction on $m$. We will actually prove $$\mathrm{rank}(\rho_{m,n}) \geq \sum_{i=m}^n 2^{i-m} \binom{i-1}{m-1} \cdot \mathcal{R}_m, \quad \text{where} \quad \rho_{m,n}=\mathrm{im}(\Psi_{m,n})_* \ \big/ \ \mathrm{im}(\Psi_{m-1,n})_*$$ The base case of the induction at $m=p$ is Theorem [Theorem 18](#ThmMain4){reference-type="ref" reference="ThmMain4"}, since in this case $\mathrm{im}(\Psi_{p-1,p})_*=0$ as $H_q(\mathrm{VR}(Q_{p-1};r))=0$. It remains to show the inductive step. Our proof is essentially the same as the proof of Theorem [Theorem 18](#ThmMain4){reference-type="ref" reference="ThmMain4"} applied to $\rho_{m,n}$ instead of to $H_q(\mathrm{VR}(Q_n;r))$. The cube $Q_n$ consists of two disjoint copies of $Q_{n-1}$; see Figure [5](#Fig5){reference-type="ref" reference="Fig5"}: - the rear one with the last coordinate $0$, denoted by $Q_{n-1}^0$, and - the front one with the last coordinate $1$, denoted by $Q_{n-1}^1$. We partition the $Q_m$ subcubes of $Q_n$ into three classes: - The ones contained in the rear $Q_{n-1}^0$ where vertices have last coordinate $0$, denoted by $R_n$. - The ones contained in the front $Q_{n-1}^1$ where vertices have last coordinate $1$, denoted by $F_n$. - The ones contained in the middle passage between them, denoted by $M_n$. Each such $Q_m$ in $M_n$ is of the form $D \times \{0,1\}$, where $D \subseteq Q_{n-1}$ is a copy of $Q_{m-1}$. We will prove that the following $Q_m$ subcubes of $Q_n$ induce independent homology in $\rho_{m,n}$: the elements of $M_n$ (dashed cubes in Figure [5](#Fig5){reference-type="ref" reference="Fig5"}) and the elements of $F_n$ that have inductively been shown to include independent embeddings in $H_{q}(\mathrm{VR}(Q^1_{n-1};r)) /\mathrm{im}(\Psi_{m-1,n-1})_*$ (bold cubes in Figure [5](#Fig5){reference-type="ref" reference="Fig5"}). The base case of the inductive process is Theorem [Theorem 18](#ThmMain4){reference-type="ref" reference="ThmMain4"}. The cardinality of $M_n$ is $2^{(n-1)-(m-1)} \binom{n-1}{m-1} = 2^{n-m} \binom{n-1}{m-1}$, which is the number of $Q_{m-1}$ subcubes in $Q_{n-1}^0$. Each such subcube has the last coordinate constantly $0$. Taking a union with a copy of the same $Q_{m-1}$ subcube with the last coordinates changed to $1$, we obtain a $Q_m$ subcube in $M_n$. It is apparent that all elements of $M_n$ arise this way. Let us enumerate the elements of $M_n$ as $Q_{m,j}^M$ with $1\le j \le 2^{n-m} \binom{n-1}{m-1}$. For each such $j$ let $\{a_{i,j}\mid 1\le i \le \mathcal{R}_m\}$ denote a largest linearly independent collection in $H_{q}( \mathrm{VR}(Q_{m,j}^M;r))/\mathrm{im}(\Psi_{m-1,m})_*$. The cardinality of the copies of $Q_m$ in $F_n$ that have inductively been shown to include independent embeddings in $H_{q}(\mathrm{VR}(Q^1_{n-1};r))/\mathrm{im}(\Psi_{m-1,n-1})_*$ equals $\sum_{i=m}^{n-1} 2^{i-m} \binom{i-1}{m-1}$, by inductive assumption. Let us enumerate them by $Q_{m,j}^F$ with $1\le j\le \sum_{i=m}^{n-1} 2^{i-m} \binom{i-1}{m-1}$. For each such $j$ let $\{b_{i,j} \mid 1\le i\le \mathcal{R}_m\}$ denote a largest linearly independent collection in $H_{q}( \mathrm{VR}(Q_{m,j}^F;r))/\mathrm{im}(\Psi_{m-1,m})_*$. Assuming the equality $$\label{Eq3b} \sum_{i,j}\lambda_{i,j} \cdot a_{i,j} + \sum_{i,j}\mu_{i,j} \cdot b_{i,j}=0$$ in $\rho_{m,n}$ for some coefficients $\lambda_{i,j}$, $\mu_{i,j}$, we claim that all coefficients equal zero. This will prove the theorem as the number of involved terms equals $\sum_{i=m}^n 2^{i-m} \binom{i-1}{m-1} \cdot \mathcal{R}_m$. We will first prove that the coefficients $\lambda_{i,j}$ are all zero. Fix some $1\le j \le 2^{n-m} \binom{n-1}{m-1}$, and let $D$ be the copy of $Q_{m-1}$ in $Q_{n-1}^0$ so that $C\coloneqq D \times \{0,1\}$ is equal to $Q_{m,j}^M$. Let $f$ be any concentration $Q_n \to C:=D \times \{0,1\}$. (For example, in Figure [4](#Fig4){reference-type="ref" reference="Fig4"} one can visualize $D$ as the solid round vertex, and $C$ as the edge between the two solid vertices.) By Proposition [Proposition 14](#PropConcentration){reference-type="ref" reference="PropConcentration"}: - $f$ maps any $Q_m$ subcube of $Q_n$ that contains $D$ bijectively onto $C=Q_{m,j}^M$. All such subcubes except for $C$ are contained in $R_n$. - $f$ maps all of the other $Q_m$ subcubes of $Q_n$ to lower-dimensional subcubes, and hence the map $$H_q (\mathrm{VR}(Q_m;r))\ \big/ \ \mathrm{im}(\Psi_{m-1,m})_* \to H_q(\mathrm{VR}(C;r))\ \big/ \ \mathrm{im}(\Psi_{m-1,m})_*$$ induced by $f|_C$ is trivial. These two observations imply that applying the induced map $f_*$ on homology to Equation [\[Eq3b\]](#Eq3b){reference-type="eqref" reference="Eq3b"}, we obtain $\sum_{i}\lambda_{i,j} \cdot a_{i,j}=0.$ By the choice of $\{a_{i,j}\}_i$ as an independent collection of homology classes, we obtain $\lambda_{i,j}=0$ for all $i$. Since this can be done for any $1\le j \le 2^{n-p} \binom{n-1}{p-1}$, we have $\lambda_{i,j}=0$ for all $i$ and $j$. We have thus reduced Equation [\[Eq3b\]](#Eq3b){reference-type="eqref" reference="Eq3b"} to $\sum_{i,j}\mu_{i,j} \cdot b_{i,j}=0.$ Let $\pi_S \colon Q_n \to Q_{n-1}$ be the projection that forgets the last coordinate of each vector (explicitly, $S=[n-1]\subseteq [n]$). Note that the restrictions of $\pi_S$ to $Q_n^0$ and to $Q_n^1$ are bijections. Hence, after applying the induced map $(\pi_S)_*$ on homology, the inductive definition of the $b_{i,j}$ as being independent in $H_{q}(\mathrm{VR}(Q^1_{n-1};r))/\mathrm{im}(\Psi_{m-1,n-1})_*$ implies that $\mu_{i,j}=0$ for all $i$ and $j$. ◻ # Geometric generators {#sec:geometric} For $r\ge 0$, let $f_n\colon \mathrm{VR}(Q_n;r)\to[0,1]^n$ be the map defined by sending each vertex of $Q_n$ to the corresponding point in $\{0,1\}^n\subseteq [0,1]^n$, and then by extending linearly to simplices. Let $n(r)\in\{0,1,\ldots\}$ be the smallest integer $n$ such that $f_n\colon \mathrm{VR}(Q_n;r)\to[0,1]^n$ is not surjective. The fact that this is well-defined follows from Lemma [Lemma 24](#LemGeomGen){reference-type="ref" reference="LemGeomGen"}, which proves that $n(r) \leq 2r+1$. The main theorem in this section is the following. **Theorem 20**. *For all $r\ge 2$,* i : *There exists some $m \leq n(r)$ such that $\pi_{m-1}(\mathrm{VR}(Q_m;r))\neq 0$;* ii : *There exists some $k \leq m$ such that $H_{k-1}(\mathrm{VR}(Q_m;r))\neq 0$;* iii : *Not all of the above non-trivial homotopy group (resp. homology group) is generated by the initial cross-polytopal spheres, i.e., the image of the induced map $(\Psi_{r+1,m})_*$ of Vietoris-Rips complexes $\mathrm{VR}(Q_{r+1};r) \hookrightarrow \mathrm{VR}(Q_m;r)$ at scale $r$ is not all of $\pi_{m-1}(\mathrm{VR}(Q_m;r))$ (resp. $H_{k-1}(\mathrm{VR}(Q_m;r))$).* Statement **ii** above is true for homology taken with any choice of coefficients. An important consequence of this theorem is the following. Together, statements **i**--**iii** imply that for each $r\ge 2$, there is a new topological feature in $\mathrm{VR}(Q_m;r)$ that is not induced from an inclusion $\mathrm{VR}(Q_{r+1};r)\hookrightarrow\mathrm{VR}(Q_m;r)$. In Table [2](#table:homotopy-types2){reference-type="ref" reference="table:homotopy-types2"}, these appear as the new blue $S^3$ feature in $\mathrm{VR}(Q_4;2)\simeq \vee^9 S^3$, and as the new blue $S^4$ feature in $\mathrm{VR}(Q_5;3)\simeq \vee^{10} S^7 \vee S^4$. In the following example, when $r=2$, we see that we can take $k=m=n(r)$. However, we do not know if this is the case in general. **Example 21**. Fix $r=2$. Note that $n(2)=4$ is the smallest integer $n$ such that $f_n\colon \mathrm{VR}(Q_n;2)\to[0,1]^n$ is not surjective. We will use this to show $\pi_3(\mathrm{VR}(Q_4;2))\neq 0$. The following five tetrahedra form a triangulation of $[0,1]^3$ (see Section 5.3 and in particular Figure 4 of  [@VargasRosarioThesis]). $$\begin{aligned} \tau_1 &= \{(0,1,1),(1,1,0),(1,0,1),(0,0,0)\} \\ \tau_2 &= \{(0,0,0),(1,0,0),(1,1,0),(1,0,1)\} \\ \tau_3 &= \{(0,1,1),(1,1,0),(0,0,0),(0,1,0)\} \\ \tau_4 &= \{(1,1,1),(0,1,1),(1,1,0),(1,0,1)\} \\ \tau_5 &= \{(0,0,0),(0,1,1),(0,0,1),(1,0,1)\}\end{aligned}$$ Furthermore, each tetrahedron $\tau_i$ has diameter at most $2$. Since $[0,1]^3=\cup_{i=1}^5 |\tau_i|$, and since each $\tau_i$ is a simplex in $\mathrm{VR}(Q_3;2)$, we define a surjective map $h\colon [0,1]^3 \to \mathrm{VR}(Q_3;2)$ by letting $h(x)=\sum_{v\in \sigma(x)}\lambda_v v$, where $\sigma(x)$ is the unique simplex in the triangulation $[0,1]^3=\cup_{i=1}^5 |\tau_i|$ that contains $x$ in its interior.[^2] We note that the map $h$ is continuous. Note that $\partial([0,1]^4)$ is composed of $8$ faces, where each face is a 3-dimensional cube $[0,1]^3$. By piecing together $8$ copies of the map $h$ in a continuous way, we obtain a map $g\colon \partial([0,1]^4) \to \mathrm{VR}(Q_4;2)$. The map $f_4\colon \mathrm{VR}(Q_4;2)\to[0,1]^4$ is not surjective, although its image does contain $\partial([0,1]^4)$, which follows since $f_3\colon \mathrm{VR}(Q_3;2)\to[0,1]^3$ is surjective. Therefore, there exists a point $v\in \mathrm{int}([0,1]^4)\setminus \mathrm{im}(f_4)$. We thus obtain a composition $$\partial([0,1]^4) \xrightarrow{g} \mathrm{VR}(Q_4;2) \xrightarrow{f_4} [0,1]^4\setminus\{v\} \xrightarrow{\pi_v} \partial([0,1]^4),$$ where $\pi_v \colon [0,1]^4\setminus\{v\} \to \partial([0,1]^4)$ is the radial projection away from the point $v\in \mathrm{int}([0,1]^4)$. Note that the composition $\pi_v \circ f_4 \circ g$ is equal to the identity map on $\partial([0,1]^4)$, and that we have a homeomorphism $\partial([0,1]^4) \cong S^3$. Since this map $\pi_v \circ f_4 \circ g$ factors through $\mathrm{VR}(Q_4;2)$, it follows that $\pi_3(\mathrm{VR}(Q_4;2))\neq 0$. The proof of Theorem [Theorem 20](#ThmGeom){reference-type="ref" reference="ThmGeom"} follows this strategy. **Example 22**. Fix $r=3$. By [@feng2023homotopy] we have $\mathrm{VR}(Q_5;3) \simeq \bigvee^{10} S^7 \vee S^4$. Theorem [Theorem 20](#ThmGeom){reference-type="ref" reference="ThmGeom"} is satisfied with $m=k=5$. **Non-Example 23**. Fix $r=4$. By Ziqin Feng's homology computations in Section [6.4.4](#SubSubsComparison:4){reference-type="ref" reference="SubSubsComparison:4"} we have $H_7(\mathrm{VR}(Q_6;4);\mathbb{Z}/2)\cong (\mathbb{Z}/2)^{239}$, $H_{15}(\mathrm{VR}(Q_6;4);\mathbb{Z}/2)\cong (\mathbb{Z}/2)^{14}$, and $H_{q}(\mathrm{VR}(Q_6;4);\mathbb{Z}/2)=0$ for $1\le q\le 6$ and $8 \le q\le 14$. Since the dimensions $7$ and $15$ of nontrivial homology are not smaller than $6$, neither of these nontrivial homology groups fits the description in Theorem [Theorem 20](#ThmGeom){reference-type="ref" reference="ThmGeom"}(**ii**). This shows that $m>6$ when $r=4$. In other words, Theorem [Theorem 20](#ThmGeom){reference-type="ref" reference="ThmGeom"} guarantees that a new topological feature not yet present in the $r=4$ row of Table [2](#table:homotopy-types2){reference-type="ref" reference="table:homotopy-types2"} will appear in some $\mathrm{VR}(Q_m;4)$ with $m>6$. We now build up towards the proof of Theorem [Theorem 20](#ThmGeom){reference-type="ref" reference="ThmGeom"}. We will use the following notation: - $C_n = [0,1]^n$ denotes the $n$-dimensional cube. - $k$-skeleton: $C_n^{(k)}= \cup \{k\text{-dimensional subcubes of } C_n \}= \cup \{\mathrm{Conv}Q_k \mid Q_k \leq Q_n\}$. - $\partial C_n = C_n^{(n-1)} \cong S^{n-1}$. We will use the following lemma in the proof of Theorem [Theorem 20](#ThmGeom){reference-type="ref" reference="ThmGeom"}. **Lemma 24**. *For each $r\ge 2$ the following hold:* 1. *$n(r) \leq 2r+1$.* 2. *The image of $f_{n(r)}$ contains $\partial C_{n(r)-1}$.* 3. *For each $r\ge 2$ and for each $n$, $\mathrm{VR}(Q_n;r)$ is connected and simply connected.* *Proof.* (1) Choose a simplex $\sigma \in \mathrm{VR}(Q_{n(r)-1};r)$ whose image via $f_{n(r)-1}$ contains the center $z=(\frac{1}{2}, \frac{1}{2}, \ldots, \frac{1}{2})$ of the cube. Without loss of generality (due to the symmetry) we may assume $\sigma$ contains the origin $(0, 0, \ldots, 0)$. As $||z||_{\ell_1}= \frac{n(r)-1}{2}$, $\sigma$ must contain a vertex of $\ell_1$-norm at least $\frac{n(r)-1}{2}$. On the other hand, the $\ell_1$-norm of each vertex of $\sigma$ is at most $r$, due to the inclusion of the origin in $\sigma$. Thus $\frac{n(r)-1}{2} \leq r$, which implies (1). \(2\) Holds since $f_{n(r)-1}$ is not surjective. \(3\) Choose $r>0$. The complex $\mathrm{VR}(Q_n;r)$ is obviously connected. Let $\alpha$ be a based simplicial loop in $\mathrm{VR}(Q_n;r)$ and let $[v,w]$ be an edge, which is a part of $\alpha$. Setting $v=v_0$ and $w=v_r$, we can replace $[v,w]$ by a homotopic (rel {v,w}) concatenation of edges $[v_0,v_1] * [v_1, v_2]* \ldots* [v_{r-1},v_r]$, whose pairwise distances are at most $1$. In particular, since $v$ and $w$ differ in at most $r$ coordinates, we can choose $v_i$ inductively so that $v_i$ differs from $v_{i+1}$ in at most one coordinate. Thus $\mathrm{diam}\{v_0, v_1, \ldots, v_r\} \leq r$ which means that the defined vertices $v_i$ form a simplex contained in $\mathrm{VR}(Q_n;r)$. As any simplex is contractible, $[v,w]$ is homotopic (rel {v,w}) to the concatenation of edges $[v_0,v_1] * [v_1, v_2]* \ldots* [v_{r-1},v_r]$. Replacing each edge of $\alpha$ in this manner we obtain a based homotopic simplicial loop $\beta$, such that the endpoints of all the edges are at distance at most $1$. The loop $\beta$ is thus contained $\mathrm{VR}(Q_n;2)$, which is simply connected by [@adamaszek2022vietoris], and contained in $\mathrm{VR}(Q_n;r)$. As a result, $\beta$ and $\alpha$ are contractible loops. As $\alpha$ was arbitrary, this means $\mathrm{VR}(Q_n;r)$ is simply connected. ◻ *Proof of Theorem [Theorem 20](#ThmGeom){reference-type="ref" reference="ThmGeom"}.* **i**: Fix $r\ge 2$. In order to show **i**, it is equivalent to assume $\pi_{n-1}(\mathrm{VR}(Q_n;r))$ is trivial for all $n \in \{1,2, \ldots, n(r)-1\}$, and then prove that $\pi_{N-1}(\mathrm{VR}(Q_{N};r))\neq 0$ for $N\coloneqq n(r)$. This is what we will do. First, we define $\varphi\colon \partial C_{N} \to \mathrm{VR}(Q_{N};r)$ by induction on the skeleta of $C_{N}$. For the base case, the 0-skeleton, define $\varphi|_{C_{N}^{(0)}}$ as the identity on $Q_{N}$, mapping a point of $Q_{N}$ to the corresponding vertex in $\mathrm{VR}(Q_{N};r)$. Now, assume that $$\varphi\colon C_{N}^{(j)} \to \mathrm{VR}(Q_{N};r)$$ has been defined for some $j \in \{0,1,\ldots, N-2\}$ in a **subcube-preserving** manner,. i.e., $$\label{eq:subcube-preserving} \forall i \leq j, \ \forall Q_i \leq Q_{N}: \ \varphi(\mathrm{Conv}Q_i) \subseteq \mathrm{VR}(Q_i;r).$$ Before defining $\varphi$ on all of $C_N^{(j+1)}$, we first explain how to define $\varphi$ on a single $(j+1)$-dimensional cube. Fix $Q_{j+1} < Q_{N}$, and let $C_{j+1} = \mathrm{Conv}Q_{j+1}$. Define $\varphi|_{C_{j+1}} \colon C_{j+1} \to \mathrm{VR}(Q_{j+1};r)$ as follows: - By [\[eq:subcube-preserving\]](#eq:subcube-preserving){reference-type="eqref" reference="eq:subcube-preserving"} above we have $$\varphi({\partial C_{j+1}}) \subseteq \cup_{Q_j \leq Q_{j+1}} \mathrm{VR}(Q_j;r) \leq \mathrm{VR}(Q_{j+1};r).$$ - By the assumption at the beginning of the proof, $\varphi|_{\partial C_{j+1}} \colon \partial C_{j+1} \to \mathrm{VR}(Q_{j+1};r)$ is contractible and can thus be extended over $C_{j+1}$. In particular, the subcube-preserving condition $\varphi({ C_{j+1}}) \subseteq \mathrm{VR}(Q_{j+1};r)$ holds. Defining $\varphi$ on $\mathrm{Conv}Q_j$ for each $Q_j \leq Q_{j+1}$ we obtain a continuous subcube-preserving map $\varphi$ defined on $\partial C_{N}^{(j+1)}$. This concludes the inductive step, and thus we obtain a subcube-preserving map $$\varphi\colon \partial C_{N} \to \mathrm{VR}(Q_{N};r).$$ Next, we next show that $\varphi$ is not contractible. Choose $z\in C_{N} \setminus \partial C_{N}$ such that $z$ is not contained in the image of $f_{N}$. Let $\nu \colon C_{N} \setminus \{z\} \to \partial C_{N}$ be the radial projection map, which is a retraction. Define $\psi = \nu \circ f_{N} \circ \varphi\colon \partial C_{N} \to \partial C_{N}$ as the composition of maps $$\partial C_{N} \stackrel{\varphi}{\to} \mathrm{VR}(Q_{N};r)\stackrel{f_{N}}{\to} C_{N} \setminus \{z\} \stackrel{\nu}{\to} \partial C_{N},$$ and note it is a map between topological $(N-1)$-spheres. Observe that $\psi$ is subcube-preserving, i.e., $\forall Q_j < Q_{N}: \psi(\mathrm{Conv}Q_j) \subseteq \mathrm{Conv}Q_j$. Map $\psi \colon \partial C_{N} \to \partial C_{N}$ is homotopic to the identity, as is demonstrated by the linear homotopy $$H \colon \partial C_{N} \times I \to \partial C_{N}, \qquad H(x,t) = (1-t)\varphi(x) + t x,$$ which is well-defined by the subcube-preserving property. As $\psi$ is homotopically non-trivial, so is $\varphi$. Since $\mathrm{VR}(Q_{N};r)$ is path connected, this implies $\pi_{N} (\mathrm{VR}(Q_{N};r))$ is non-trivial regardless of which basepoint is used, giving **i**. **ii**: By the Hurewicz theorem and Lemma [Lemma 24](#LemGeomGen){reference-type="ref" reference="LemGeomGen"}(3), the first non-trivial homotopy group of $\mathrm{VR}(Q_m;r)$ is isomorphic to the corresponding homology group with integer coefficients in the same dimension. By **i**, the mentioned dimension is at most $m-1$. **iii**: Let $m \in \{1,2, n(r)\}$ be the parameter from the proof of **i** that satisfies $\pi_{m-1}\mathrm{VR}(Q_{m_r};r)\neq 0$. For $r>2$ we claim that $m-1 < 2^r-1$. Indeed, $m-1 \le n(r)-1 \leq 2r < 2^r-1$ by Lemma [Lemma 24](#LemGeomGen){reference-type="ref" reference="LemGeomGen"}(2). So for $r>2$, the dimension ($m-1$ or lower) of the homotopy and homology groups in **i** and **ii** is lower than the dimension $2^r-1$ of the $\Psi_{r+1}^{m_r}$ induced invariants, giving **iii**. Finally, in the case $r=2$, we have $m=2$ and $\mathrm{VR}(Q_4;2) \simeq \vee^9 S^3$ by [@adamaszek2022vietoris]. Since $Q_4$ contains only $8$ copies of $Q_3$, the image of $\Psi_3^4$ induced map on $\pi_3$ is of rank at most $8$; thus the claim **iii** follows also for $r=2$. ◻ # Conclusion and open questions {#sec:conclusion} We conclude with a description of some open questions. We remind the reader of questions from [@adamaszek2022vietoris], which ask if $\mathrm{VR}(Q_n;r)$ is always a wedge of spheres, what the homology groups and homotopy types of $\mathrm{VR}(Q_n;r)$ are for $3\le r\le n-2$, and if $\mathrm{VR}(Q_n;r)$ collapses to its $(2^r-1)$-skeleton (which would imply that the homology groups $H_q(\mathrm{VR}(Q_n;r))$ are zero for $q\geq 2^r$). Below we pose some further questions. The first four questions are related to the geometric generators in Section [7](#sec:geometric){reference-type="ref" reference="sec:geometric"}. Understanding the answers to any of them would provide further information about parameters of Theorem [Theorem 20](#ThmGeom){reference-type="ref" reference="ThmGeom"}. **Question 25**. Recall that $n(r)\in\{0,1,\ldots\}$ is the smallest integer $n$ such that $f_n\colon \mathrm{VR}(Q_n;r)\to[0,1]^n$ is not surjective. What are the values of $n(r)$ as a function of $r$? **Question 26**. If $f_n\colon \mathrm{VR}(Q_n;r)\to[0,1]^n$ is not surjective, then is it necessarily the case that the center $(\frac{1}{2}, \frac{1}{2}, \ldots, \frac{1}{2})$ is not in the image of $f_n$? **Question 27**. If $f_n\colon \mathrm{VR}(Q_n;r)\to[0,1]^n$ is surjective, then does their exist a triangulation of $[0,1]^n$ by simplices of diameter at most $r$? **Question 28**. The following question is based on a StackExchange post [@StackExchangeBalanced]. A subset $B\subseteq\{0,1\}^n$ is *balanced* if $\frac{1}{|B|}\sum_{b\in B} b = (\frac{1}{2},\ldots,\frac{1}{2})\in\mathbb{R}^n$. For example, the tetrahedron $\tau_1$ in Example [Example 21](#ex:geom-r=2){reference-type="ref" reference="ex:geom-r=2"} is a set of four vertices that forms a balanced subset. If $(\frac{1}{2}, \frac{1}{2}, \ldots, \frac{1}{2})$ is in the image of $f_n\colon \mathrm{VR}(Q_n;r)\to[0,1]^n$, then does there necessarily exist a balanced subset of $\{0,1\}^n$ of diameter at most $r$? One of the reasons we ask this question is that the answers to the StackExchange post [@StackExchangeBalanced] place constraints on the smallest diameter for a balanced subset of the $n$-dimensional cube. The remaining questions are more general. **Question 29**. In Section [4](#sec:maximal){reference-type="ref" reference="sec:maximal"} we described cross-polytopal homology generators. In Section [7](#sec:geometric){reference-type="ref" reference="sec:geometric"} we described geometric homology generators. In Non-Example [Non-Example 23](#nonexample-feng){reference-type="ref" reference="nonexample-feng"} we described homology generators, due to computations by Ziqin Feng, that are neither cross-polytopal in the sense of Section [4](#sec:maximal){reference-type="ref" reference="sec:maximal"} (arising from an isometric embedding $Q_{r+1}\hookrightarrow Q_n$) nor geometric in the sense of Section [7](#sec:geometric){reference-type="ref" reference="sec:geometric"}. What other types of homology generators are there for $H_q(\mathrm{VR}(Q_n;r))$? **Question 30**. Our main results show how the homology (and persistent homology) of $\mathrm{VR}(Q_p;r)$ for $1\le p \le m$ place lower bounds on the Betti numbers of $\mathrm{VR}(Q_n;r)$ for all $n\ge m$. For every $r\ge 1$, is there some integer $m(r)$ such that our induced lower bounds are tight for all $n\ge m(r)$ and for all homology dimensions? **Question 31**. The group of symmetries of the $n$-dimensional cube is the hyperoctahedral group. How does this group act on the homology $H_q(\mathrm{VR}(Q_n;r))$? **Question 32**. What homology propogation results can be proven for Čech complexes of hypercube graphs, as studied in [@adams2022v]? # Acknowledgements {#acknowledgements .unnumbered} We would like to thank the Institute of Science and Technology Austria (ISTA) for hosting research visits, and we would like to thank John Bush, Ziqin Feng, Michael Moy, Samir Shukla, Anurag Singh, Daniel Vargas-Rosario, and Hubert Wagner for helpful conversations. The second author was supported by Slovenian Research Agency grants No. N1-0114, J1-4001, and P1-0292. We acknowledge the Auburn University Easley Cluster for support of this work, via computations carried out by Ziqin Feng. # Proofs of numerical identities This appendix contains two short proofs of numerical identities, both of which are used in Section [6.4](#SubsComparison){reference-type="ref" reference="SubsComparison"} in the cases $r=1$ and $r=2$. ## First proof {#app:1} Let $S_n=\sum_{i=2}^n (i-1) 2^{i-2}$; we claim $S_n = n2^{n-1}-2^n+1$. Indeed, note that $$\begin{aligned} S_n=2 S_n - S_n&=0 &&+2 &&+2\cdot2^2 &&+3\cdot 2^3 &&+\ \ \ \ \ \ldots &&+(n-2)\cdot 2^{n-2}&&+(n-1)\cdot 2^{n-1}\\ &\ \ \ -1 &&-2\cdot 2 &&-3\cdot2^2 &&-4\cdot 2^3 &&-\ \ \ \ \ \ldots &&-(n-1)\cdot 2^{n-2}\\ &=-1 &&-2 &&-2^2 &&-2^3 &&-\ \ \ \ \ \ldots &&-2^{n-2}&&+(n-1)\cdot 2^{n-1},\end{aligned}$$ which gives $S_n = -(2^{n-1}-1)+(n-1)2^{n-1} = n2^{n-1}-2^n+1$. ## Second proof {#app:2} We prove that $\sum_{i=3}^{n} 2^{i-3} \binom{i}{3} = 2^{n-2}\binom{n}{3} - \sum_{i=1}^{n-2} 2^{i-1}\binom{i+1}{2}$ by induction on $n$. For the base case $n=3$, note that both sides equal $1$. For the inductive step, note that if we assume the formula is true for $n-1$, then as desired we get $$\begin{aligned} \sum_{i=3}^{n} 2^{i-3} \binom{i}{3} &= 2^{n-3} \binom{n}{3} + \left(2^{n-3}\binom{n-1}{3} - \sum_{i=1}^{n-3} 2^{i-1} \binom{i+1}{2}\right) \\ &= 2^{n-3} \binom{n}{3} + 2^{n-3}\binom{n-1}{3} + 2^{n-3}\binom{n-1}{2}- \sum_{i=1}^{n-2} 2^{i-1} \binom{i+1}{2} \\ &= 2^{n-3} \binom{n}{3} + 2^{n-3}\binom{n}{3}- \sum_{i=1}^{n-2} 2^{i-1} \binom{i+1}{2} \\ &= 2^{n-2}\binom{n}{3} - \sum_{i=1}^{n-2} 2^{i-1}\binom{i+1}{2}.\end{aligned}$$\ \ [^1]: *MSC codes.* 05E45, 55N31, 55U10 [^2]: By convention, the interior of a vertex is that vertex.

arxiv_math

--- author: - Ravil Bildanov, Ilya Gorshkov title: On 3-generated axial algebras of Jordan type half --- *Abstract: Axial algebras of Jordan type $\eta$ are a special type of commutative non-associative algebras. They are generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-\eta)$, where $\eta$ is a fixed value that is not equal to $0$ or $1$. These algebras have restrictive multiplication rules that generalize the Pierce decomposition for idempotents in Jordan algebras. Axial algebras of Jordan type $1/2$ we will call Jordan type half algebras. In fact, Jordan algebras generated by idempotents are an Jordan type half algebras.* An universal $3$-generated Jordan type half algebra as an algebra with $4$ parameters was construsted by I. Gorshkov and A. Staroletov. Depending on the value of the parameter, the universal algebra may contain a non-trivial form radical. In the present paper, all semisimple $3$-generated Jordan type half algebras have been described. MSC code: 20D60 Keywords: axial algebras, Jordan type algebras. # Introduction {#introduction .unnumbered} Axial algebras of Jordan type $\eta$ were introduced by Hall, Rehren and Shpektorov [@hrs] within the framework of the general theory of axial algebras. These algebras are commutative non-associative algebras over a field $\mathbb{F}$, generated by a set of special idempotents known as primitive axes. While Jordan algebras generated by primitive idempotents are an example of Jordan type half algebras, not all algebras of this type are Jordan algebras. The Matsuo algebras constructed from the group of 3-transpositions are an example of such algebras. It was proved in [@hrs] (with a correction in [@hss]) that, for $\eta\neq 1/2$, algebras of Jordan type $\eta$ are the Matsuo algebras or their factor algebras. Therefore, the case $\eta=1/2$ is special for algebras of Jordan type and for this $\eta$ they are called algebras of Jordan type half. The class of Matsuo algebras was introduced by Matsuo [@m] and later generalized in [@hrs]. Jordan type half algebras are not exhausted by Matsuo algebras and their factors. Moreover, the factor algebras of Matsuo algebras do not contain all Jordan algebras generated by primitive idempotents. So, for example, the $27$-dimensional Albert algebra is generated by $4$ primitive idempotents and hence it is a Jordan type half algebra but not a Matsuo algebra [@kms]. An universal 3-generated algebra $A(\alpha, \beta,\gamma,\phi)$ of Jordan type half was constructed in [@gs], where $\alpha, \beta,\gamma,\phi$ are parameters. There it is proved that if $(\alpha+\beta+\gamma-2\phi-1)(\alpha\beta\gamma-\phi^2)\neq 0$ and $\phi^2-\alpha\beta\gamma$ is a square in $\mathbb{F}$, then $A(\alpha, \beta,\gamma,\phi)$ is isomorphic to the matrix algebra $M^+_3(\mathbb{F})$ of $3\times 3$ matrices with Jordan multiplication. Otherwise, the algebra $A(\alpha, \beta,\gamma,\phi)$ is not simple. A Frobenius form $(\ ,\ )$ on $A$ is a nonzero symmetric bilinear form, which associates with multiplication in $A$ i.e. $\forall a,b,c \in A$ we have $(ab,c)=(ac,b)$. Hall, Rehren and Shpectorov [@hrs] showed that in Jordan type algebras there exists a unique Frobenius form with the property $(a,a)=1$ for any primitive axis $a$. Let $A$ be an algebra with a Frobenius form $(\ ,\ )$. The radical of the form $(\ ,\ )$ is an ideal $R(A)$ generated by elements $x$ such that $(x,a)=(a,x)=0$ for any element $a\in A$. The purpose of this article is to describe all $3$-generated algebras of Jordan type half with trivial radical. # Preliminary results We consider commutative (but usually non-associative) algebras over a ground field $\mathbb{F}$ with characteristic not two. For the definitions and preliminary results, we will almost always follow [@hrs] and [@gg]. Let's denote as $L \langle X \rangle$ the linear span of the set $X$ over $F$, as $\langle\langle X \rangle\rangle$ the algebra generated by the set $X$, as $\{ a,b,c \}$ the set which contains $a,b,c$. **Definition 1**. *Given an element $a \in A$ and $\lambda \in \mathbb{F}$, we introduce the subspace\ $A_{\lambda} (a)=\{u \in A \ | \ au=\lambda u\}$.* Obviously, $A_{\lambda} (a)$ is an eigenspace of operator $ad_a: x \rightarrow ax$, associated with $\lambda \in \mathbb{F}$. **Definition 2**. *An idempotent $a \in A$ is primitive if $\dim(A_1(a)) = 1$.* **Definition 3**. *An algebra $A$ is an algebra of Jordan type half, if $A$ is generated by the set of primitive idempotents $X$ such that for every $x \in X$ we have a decomposition $A=A_0(x) \oplus A_1(x) \oplus A_{1/2}(x)$ with following fusion (multiplication) rules: $$A_0(a)A_{1/2}(a) \subseteq A_{1/2}(a), A_1(a)A_{1/2}(a) \subseteq A_{1/2}(a), A_0(a)A_1(a) \subseteq \{0\}$$ $$A_0^2(a) \subseteq A_0(a), A_1^2(a) \subseteq A_1(a), A_{1/2}^2(a) \subseteq A_0 \oplus A_1$$.* Let us introduce some classes of simple Jordan algebras. **Definition 4**. *Denote by $M_n^{+}(\mathbb{F})$ the matrix algebra $M_n(\mathbb{F})$ with Jordan product\ $A \circ B = \frac{1}{2}(AB+BA)$.* **Definition 5**. *If $\mathbb{F}$ has an involution $\theta \in Aut(\mathbb{F})$, then denote by\ $H_n^{+}(\mathbb{F})=\{ A \in M_n^{+}(\mathbb{F}) \ | \ \theta(A)^T=A \}$ the Hermitian Jordan algebra.* **Definition 6**. *We define an algebra $JForm_n(\mathbb{F})$ on $\mathbb{F}\oplus V$ over an arbitrary field $\mathbb{F}$ and vector space $V$ of dimension $n$ over $\mathbb{F}$ with bilinear form $\phi$ with the product $$(a \oplus \mathbf{v}) \bullet (b \oplus \mathbf{w}) = (ab + \phi(\mathbf{v}, \mathbf{w})) \oplus (a\mathbf{w} + b\mathbf{v}), \ where \ a,b \in \mathbb{F} \ and \ \mathbf{v}, \mathbf{w} \in V.$$* It is well known that $M_n^{+}(\mathbb{F})$, $H_n^{+}(\mathbb{F})$, $JForm_n(\mathbb{F})$ are simple Jordan algebras generated by primitive idempotents, so they are algebras of Jordan type half. **Lemma 1**. *(Theorem 4.1. [@hss]) Every algebra of Jordan type $\eta$ admits a unique Frobenius form which satisfies the property $(a,a)=1$ for all axes $a \in X$.* **Lemma 2**. *(Proposition 2.7. [@hrs]) The radical of Frobenius form $R(A)$ coincides with the largest ideal of $A$ containing no axes from $A$.* **Lemma 3**. *Let $A$ be an algebra of Jordan type $\eta$. Then for all $a,b \in A$ and their images $\Bar{a}, \Bar{b} \in A\slash R(A)$ $(a,b)=(\Bar{a}, \Bar{b})$.* *Proof.* Let $a=\Bar{a}+r_a, b=\Bar{b}+r_b$, where $\Bar{a}, \Bar{b} \in A\slash R(A), r_a, r_b \in R(A)$. Then $(a,b)=(\Bar{a}+r_a, \Bar{b}+r_b)=(\Bar{a}, \Bar{b})+(\Bar{a}, r_b)+(\Bar{b}, r_a)+(r_a, r_b)=(\Bar{a}, \Bar{b})$. ◻ **Lemma 4**. *(Lemma 1. [@gg]) Let $A$ be a finitely generated algebra of Jordan type half, $a,b$ are axes, $\alpha=(a,b)$. Then we have the following equalities:* 1. *$a_0^2(b)=(1-\alpha)a_0(b)$* 2. *$a_{1/2}^2(b)=\alpha a_0(b)+(\alpha-\alpha^2)a$* 3. *$a_0(b)a_{1/2}(b)=\frac{1}{2}(1-\alpha)a_{1/2}(b)$* **Lemma 5**. *Let $A=\langle\langle a,b\rangle\rangle$ be a $2$-generated algebra of Jordan type half. Then it is one of following:* 1. *$\dim(A) = 1, (a,b)=1, a=b$* 2. *$\dim(A) = 2, (a,b)=0, A \cong \mathbb{F} \oplus \mathbb{F}$* 3. *$\dim(A) = 2, (a,b)=1, \dim(R(A))=1$* 4. *$\dim(A) = 3, (a,b)=0, \dim(R(A))=1$, $A/R(A) \cong \mathbb{F} \oplus \mathbb{F}$* 5. *$\dim(A) = 3, (a,b)=1, \dim(R(A))=2$* 6. *$\dim(A) = 3, (a,b) \neq 0,1$ and $A$ is a Matsuo algebra.* *Proof.* The assertion of the lemma is a simple consequence of Proposition 1 [@hss] ◻ **Lemma 6**. *(Corollary 1.[@gg]) Let $A$ be a $2$-generated algebra of Jordan type half with generating axes $a,b$. Denote $\alpha=(a,b)$. Then we have* 1. *$a(ab)=\frac{1}{2}(\alpha a+ab)$* 2. *$(ab)b=\frac{1}{2}(\alpha b+ab)$* 3. *$(ab)(ab)=\frac{\alpha}{4}(a+b+2ab)$* **Lemma 7**. *(Main theorem [@gs])[\[mainGS\]]{#mainGS label="mainGS"} There is a $3$-generated $9$-dimensional algebra $A(\alpha,\beta,\gamma,\phi)$ such that each $3$-generated algebra of Jordan type half is a factor algebra of this algebra for suitable values of parameters.* Let $A=\langle\langle a,b,c \rangle\rangle, \dim(A)=9, \alpha=(a,b), \beta=(b,c), \gamma=(a,c), \phi=(ab,c)$. We put in Table 1 all possible conditions for $\alpha,\beta,\gamma,\phi$ for $A(\alpha,\beta,\gamma,\phi)$ to be non-simple. We also put in this table from [@gs] basis of the radical $R(A)$ and dimension of the factor algebra $A \slash R(A)$ in each case. $\begin{tabu}[h!]{|c|c|c|c|c|c|c|} \hline \text{Number}&\text{Condition} & dim(A/R(A)) & \text{Basis of radical} \\ \hline 1&\psi=\alpha=\beta=\gamma=1 & 1 & \begin{tabu}{@{}c@{}} b-a, c-a, ab-a, bc-a, ac-a, \\ a(bc)-a, b(ac)-a, c(ab)-a \end{tabu} \\ \hline %case 4 2&\psi=\alpha=\beta=0, \gamma=1 & 2 & \begin{tabu}{@{}c@{}} c-a, ab, bc, ac-a, \\ a(bc), b(ac), c(ab) \end{tabu} \\ \hline %case 3 3&\psi=\alpha=\beta=\gamma=0 & 3 & \begin{tabu}{@{}c@{}} ab, bc, ac, a(bc), b(ac), c(ab) \end{tabu} \\ \hline %case 5 4&\begin{tabu}{@{}c@{}} \psi=\alpha=0, \beta, \gamma\neq0, \\ \beta+\gamma=1 \end{tabu} & 3 & \begin{tabu}{@{}c@{}} ab, \cfrac{1}{2}\gamma a-\cfrac{1}{2}\beta b-\cfrac{1}{2}c+bc, \\ -\cfrac{1}{2}\gamma a+\cfrac{1}{2}\beta b-\cfrac{1}{2}c+ac, \\ \cfrac{1}{4}\gamma a+\cfrac{1}{4}\beta b-\cfrac{1}{4}c+a(bc), \\ \cfrac{1}{4}\gamma a+\cfrac{1}{4}\beta b-\cfrac{1}{4}c+b(ac), c(ab) \end{tabu} \\ \hline %case 2 5&\begin{tabu}{@{}c@{}} \alpha\beta\gamma=\psi^2, \psi\neq0,\alpha\neq1, \\ \alpha+\beta+\gamma=2\psi+1 \end{tabu} & 3 & \begin{tabu}{@{}c@{}} \alpha(\beta-1)a+\alpha(\gamma-1)b+\alpha(1-\alpha)c+(2\alpha-2\psi)ab, \\ (\alpha\beta-\alpha\psi)b+(\psi-\alpha\beta)ab+(\alpha^2-\alpha)bc, \\ (\alpha\gamma-\alpha\psi)a+(\psi-\alpha\gamma)ab+(\alpha^2-\alpha)ac, \\ (\alpha\psi-\alpha^2\beta)a+(\alpha+\psi-\alpha^2-\alpha\gamma)ab+2\alpha(\alpha-1)a(bc), \\ \alpha(\psi-\alpha\gamma)b+(\alpha+\psi-\alpha^2-\alpha\beta)ab+2\alpha(\alpha-1)b(ac), \\ (\psi-\alpha\beta)a+(\psi-\alpha\gamma)b+(1-\alpha)ab+2(\alpha-1)c(ab) \end{tabu} \\ \hline %case 7$`<!-- -->`{=html}6$&\psi=\alpha=\beta=0, \gamma\neq0,1 & 4 & \begin{tabu}{@{}c@{}} ab, bc, ac, a(bc), b(ac), c(ab) \end{tabu} \\ \hline %case 8 7&\begin{tabu}{@{}c@{}} \psi^2\neq\alpha\beta\gamma, \\ \alpha+\beta+\gamma=2\psi+1, \\ \alpha\neq1 \end{tabu} & 4 & \begin{tabu}{@{}c@{}} \frac{1}{2}(\beta-1)a+\frac{1}{2}(\beta-\alpha)b+\frac{1}{2}(1-\alpha)c+(1-\beta)ab+(\alpha-1)bc, \\ \frac{1}{2}(\gamma-\alpha)a+\frac{1}{2}(\gamma-1)b+\frac{1}{2}(1-\alpha)c+(1-\gamma)ab+(\alpha-1)ac, \\ (2\psi-2\alpha\beta+\beta-1)a+(\gamma-1)b+(1-\alpha)c+(4-2\alpha-2\gamma)ab+(4\alpha-4)a(bc), \\ (\beta-1)a+(2\psi-2\alpha\gamma+\gamma-1)b+(1-\alpha)c+(4-2\alpha-2\beta)ab+(4\alpha-4)b(ac), \\ (\psi-\alpha)a+(\psi-\alpha)b+\alpha(1-\alpha)c+(2-\beta-\gamma)ab+(2\alpha-2)c(ab) \end{tabu} \\ \hline %case 6 8&\begin{tabu}{@{}c@{}} \psi=\alpha=0, \beta,\gamma\neq0, \\ \beta+\gamma\neq1 \end{tabu} & 6 & \begin{tabu}{@{}c@{}} ab, b(ac)-a(bc), cab \end{tabu} \\ \hline %case 1 9&\begin{tabu}{@{}c@{}} \alpha\beta\gamma=\psi^2, \psi\neq0, \\ \alpha+\beta+\gamma\neq2\psi+1 \end{tabu} & 6 & \begin{tabu}{@{}c@{}} -\beta\gamma ab-\alpha\beta ac+2\psi a(bc),\\ -\beta\gamma ab-\alpha\gamma bc+2\psi b(ac), \\ -\alpha\gamma bc-\alpha\beta ac+2\psi c(ab) \end{tabu} \\ \hline \end{tabu}$ # Factors In this section we will describe $3$-generated algebra of Jordan types half with trivial radical. We will prove the following theorem. **Theorem 1**. *Let $A$ be a $3$-generated algebra of Jordan type half with trivial radical over a ground field $\mathbb{F}$ with characteristic not equal to two or three. Then A is isomorphic to one of the following algebras:* 1. *$\mathbb{F}^n, n \in \{ 1, 2, 3 \}$* 2. *$JForm_3(\mathbb{F})$* 3. *$\mathbb{F} \oplus JForm_3(\mathbb{F})$* 4. *$M^{+}_2(\mathbb{F})$* 5. *$H^{+}_3(\mathbb{F})$* 6. *$M^{+}_3(\mathbb{F})$* It follows from Lemma [\[mainGS\]](#mainGS){reference-type="ref" reference="mainGS"} that we must describe the factors of the algebra $A(\alpha, \beta, \gamma, \psi)$ by the radical. We will use the description of the algebra $A(\alpha, \beta, \gamma, \psi)$ in Theorem 2 from [@gs]. Follow by [@gs], denote $\alpha=(a,b), \beta=(b,c), \gamma=(a,c), \phi=(ab,c)$. In [@gs], the dimensions and bases of the radicals of the algebra $A(\alpha, \beta, \gamma, \psi)$ are described (see Table $6$). Table $6$ of [@gs] we write in present text, see Table $1$. Denote by $A_i$ the universal $9$-dimensional algebra $A(\alpha_i, \beta_i, \gamma_i, \psi_i)$ with parameters and numeration from Table $1$, by $R_i$ the radical of this algebra and by $S_i$ the factor algebra by radical. We begin from two trivial propositions for $1$-dimensional and $2$-dimensional algebras. It is obvious that such algebras aren't generated by three linear independent axes. **Proposition 1**. *Let $A$ be a $1$-dimensional algebra of Jordan type half with trivial radical. Then $A \cong S_1$.* *Proof.* It is easy to see that $S_1 \cong \mathbb{F}$. We have $A$ is 1-dimensional, so $\dim L\langle a,b,c\rangle=1$ and $a=b=c$. Hence $A \cong \mathbb{F} \cong S_1$. ◻ **Proposition 2**. *Let $A$ be a $2$-dimensional $3$-generated algebra of Jordan type half with trivial radical. Then $A \cong \mathbb{F} \oplus \mathbb{F}\cong S_2$.* *Proof.* By Lemma [Lemma 5](#twogen){reference-type="ref" reference="twogen"}, there is only one $2$-dimensional algebra of Jordan type half with trivial radical, so $A \cong \mathbb{F} \oplus \mathbb{F} \cong S_2$. ◻ **Proposition 3**. *Let $A$ be $3$-dimensional $3$-generated algebra of Jordan type half with trivial radical. Then $A$ is isomorphic to one of the algebras $S_3, S_5$.* *Proof.* Assume that the algebra $A$ generated by $a$ and $b$. From Lemma [Lemma 5](#twogen){reference-type="ref" reference="twogen"} it follows that there is only one $3$-dimensional $2$-generated Jordan type half algebra with trivial radical. In this case, as the axis $c$ we can choose any axis of the algebra $A$. Put $c=a^{\tau_b}=a-4ab+4\alpha b$. We have $\beta=\alpha,\ \gamma=(1-2\alpha)^2$ and $\psi=\alpha(2\alpha-1)$. Therefore $\alpha\beta\gamma=\psi^2,\ \psi\neq0,\alpha\neq1,\ \alpha+\beta+\gamma=2\psi+1$. In particular in this case $A\simeq S_5$. Assume that the algebra $A$ is not generated by $2$ axes. Therefore, based on the dimension of $A$, we conclude that $A$ is the linear span of the axes $a$, $b$, and $c$. Assume that $ab \notin L\langle a,b \rangle$. Hence $\dim \langle\langle a, b \rangle\rangle = 3$. Therefore $c \in L\langle a,b,ab \rangle=A$; the contradiction. By the same way, we can show that if $ac \in L\langle a,c \rangle$ and $bc \in L \langle b, c \rangle$. In particular, we have $\dim(\langle\langle a,b\rangle\rangle)=\dim(\langle\langle a,c\rangle\rangle)=\dim(\langle\langle c,b\rangle\rangle)=2$. From Lemma [Lemma 5](#twogen){reference-type="ref" reference="twogen"} it follows that $\{(a,b),(a,c),(b,c)\}\subseteq \{0,1\}$. Moreover, if $(a,b)=0$, then $\langle\langle a,b\rangle\rangle\simeq \mathbb{F}\oplus\mathbb{F}$. Therefore, if $(a,b)=(a,c)=(b,c)=0$ then $A\simeq \mathbb{F}\oplus\mathbb{F}\oplus\mathbb{F}$ and $\psi=0$. In this case the Gram matrix of the algebra $A$ is the identity matrix and hence the radical of $A$ is trivial. In We conclude that in this case $A\simeq S_3$. Assume that $(a,c)\neq0$. We have $(a,c)=1$. In this case, $R(\langle\langle a,c\rangle\rangle)$ is not trivial and contains the element $a-c$. Assume that $(a,b)=(b,c)=0$. In this case we have $(a-c,b)=0$. Consequently $a-c\in R(A)$; the contradiction. Therefore, without loss of generality, we can assume that $(b,c)=1$. If $(a,b)=1$ then $(a-c,b)=0$ and consequently $a-c\in R(A)$; the contradiction. Therefore $(a,b)=0$. From the description of two generated algebras we have $ab=0$, $a= c+a_h$, $b=c+ b_h$ where $a_h,b_h\in A_{1/2}(c)$. Therefore $0=ab=(c+a_h)(c+b_h)=c+1/2(a_h+b_h)+a_hb_h$, where $c+a_hb_h\in A_{0+1}(c)$ and $a_h+b_h\in A_{1/2}(c)$. Therefore $a_h+b_h=0$. In particular $b=a^{\tau_c}$ and $\dim(A)=2$. ◻ **Proposition 4**. *Algebras $S_4$ and $S_5$ are isomorphic.* *Proof.* It follows from Lemma [Lemma 5](#twogen){reference-type="ref" reference="twogen"} that the subalgebra of the algebra $S_4$ generated by the axes $a$ and $b$ has dimension $3$. Hence $S_5$ is generated by $2$ axes and is isomorphic to $S_3$. ◻ **Proposition 5**. *Let $A$ be $4$-dimensional $3$-generated Jordan type half algebra with trivial radical. Then one of the assertions hold:* 1. *$A \simeq S_6 \simeq \mathbb{F} \oplus JForm_3(\mathbb{F})$* 2. *$A \simeq S_7 \simeq M^{+}_2(\mathbb{F})$* *Proof.* The algebra $M^{+}_2(\mathbb{F})$ is a simple Jordan algebra. Algebra $\mathbb{F} \oplus JForm_3(\mathbb{F})$ includes non-trivial ideals. Therefore $M^{+}_2(\mathbb{F})\not\simeq \mathbb{F} \oplus JForm_3(\mathbb{F})$. Hence, to prove this proposition, it suffices to show that $S_6 \simeq \mathbb{F} \oplus JForm_3(\mathbb{F})$ and $S_7 \simeq M^{+}_2(\mathbb{F})$. ◻ **Lemma 8**. *$S_6$ is isomorphic to $\mathbb{F} \oplus JForm_3(\mathbb{F})$.* *Proof.* Let $\langle\langle a,b,c\rangle\rangle\simeq S_6$. We have $(a,c)\not\in\{0,1\}$. Therefore $\langle\langle a,c\rangle\rangle$ is isomorphic to $JForm_3(\mathbb{F})$. From Table 1 it follows that radical of $A(0,0,\gamma,0)$ contains $ab$ and $bc$. Therefore $ab=bc=0$ and $S_6\simeq \langle\langle a,c\rangle\rangle\oplus \langle\langle b\rangle\rangle\simeq \mathbb{F} \oplus JForm_3(\mathbb{F})$. ◻ **Lemma 9**. *$S_7$ is isomorphic to $M^{+}_2(\mathbb{F})$.* *Proof.* Let $$A = \left( \begin{array}{cc} 1 & 0 \\ \lambda_a & 0 \\ \end{array} \right) B = \left( \begin{array}{ccc} 0 & 1 \\ 0 & \lambda_b \\ \end{array} \right) C = \left( \begin{array}{ccc} -2\lambda_b\lambda_c & 2\lambda_c(1+2\lambda_b\lambda_c) \\ -\lambda_b & 1+2\lambda_b\lambda_c \\ \end{array} \right)$$ Where $\lambda_a,\lambda_b, \lambda_c\in \mathbb{F}\setminus \{0\}$. Consider the following map $f: S_7 \rightarrow M^{+}_2(\mathbb{F}), f(a)=A, f(b)=B, f(c)=C$. Using calculations, we show that multiplication table for $f(\langle\langle a,b,c \rangle\rangle)$ [^1] coincides with multiplication table for $S_7$. Hence, $f$ is an isomorphism. All calculations are straightforward and can be done by hand [^2]. ◻ **Proposition 6**. *Let $A$ be $6$-dimensional $3$-generated algebra of Jordan type half with trivial radical. Then $A \simeq S_8 \simeq S_9\simeq H^{+}_3(\mathbb{F})$.* For proof the isomorphism between $S_8$ and $S_9$, we will proof the isomorphism of both algebras to $H^{+}_3(\mathbb{F})$. **Lemma 10**. *$S_8 \cong H^{+}_3(\mathbb{F})$* *Proof.* Consider the following matrices in $H^{+}_3(\mathbb{F})$ and map $f: S_8 \rightarrow H^{+}_3(\mathbb{F})$, $f(a)=A, f(b)=B, f(c)=C$, $\lambda_a, \lambda_b, \lambda_c \in \mathbb{F}\setminus\{0\}$ are the invariant by $\theta$ parameters which are defined later from conditions to $\alpha,\beta,\gamma$ and $\psi$. $$A = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right) B = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & \frac{1+\sqrt{1-4\lambda_b^2}}{2} & \lambda_b \\ 0 & \lambda_b & \frac{1-\sqrt{1-4\lambda_b^2}}{2} \\ \end{array} \right) C = \left( \begin{array}{ccc} \frac{1+\sqrt{1-4\lambda_c^2}}{2} & 0 & \lambda_c \\ 0 & 0 & 0 \\ \lambda_c & 0 & \frac{1-\sqrt{1-4\lambda_c^2}}{2} \\ \end{array} \right)$$ Below we show that the mapping $f$ is an isomorphism between the algebras $S_8$ and $H^+_3(\mathbb{F})$. All calculations are straightforward and can be done by hand [^3]. It is easy to see that $A^2=A, B^2=B, C^2=C$. Symmetric involution matrices $e_{12}, e_{13}, e_{23}$ from $H^{+}_3(\mathbb{F})$ are in $\langle\langle A,B,C \rangle\rangle$, hence $\langle\langle A,B,C \rangle\rangle=H^{+}_3(\mathbb{F})$. A map $(\ ,\ ): H^{+}_3(\mathbb{F})^2 \rightarrow \mathbb{F}$ such that $(X,Y)=tr(XY)=tr(X \circ Y)$, where $X,Y\in H^{+}_3(\mathbb{F})$, is a symmetric bilinear form on $H^{+}_3(\mathbb{F})$. This form associates with the product $\circ$. Clearly, we have $tr(A \circ A)=tr(B \circ B)=tr(C \circ C)=1$.\ Furthermore, we see that $tr(A \circ B)=0, tr(B \circ C)=\frac{1}{4}(1-\sqrt{1-4\lambda_b^2})(1-\sqrt{1-4\lambda_c^2})=\beta, tr(A \circ C)=\frac{1+\sqrt{1-4\lambda_c^2}}{2}=\gamma$ and $tr(A \circ (B \circ C))=tr(B \circ (A \circ C))=tr(C \circ (A \circ B))=0$. So $\lambda_b=\pm\sqrt{\dfrac{-\beta(1+\beta+\gamma)}{(\gamma-1)^2}}, \lambda_c=\pm\sqrt{-\gamma(\gamma-1)}$ and we can take any solution of this equation. Take basis $a, b, c, b \cdot c, a \cdot c, a \cdot (b \cdot c)$ for $S_8$. Multiplication table for $f(\langle\langle a,b,c \rangle\rangle)$ [^4] coincides with multiplication table for $S_8$. We have $A, B, C, B\cdot C, A\cdot C, A \cdot (B \cdot C)$ is a basis of the algebra $H^3_+(\mathbb{F})$ and hence the kernel of $f$ is trivial. Thus $f$ is an isomorphism of the algebras $S_8$ and $H^+_3(\mathbb{F})$. ◻ **Lemma 11**. *$S_9 \cong H^{+}_3(\mathbb{F})$* *Proof.* Let $f: S_9 \rightarrow H^{+}_3(\mathbb{F})$, $f(a)=A, f(b)=B, f(c)=C$, and $\lambda_a, \lambda_b, \lambda_c \in \mathbb{F}$ are the invariant by $\theta$ parameters which are defined later from conditions to $\alpha,\beta,\gamma$ and $\psi$. $$A = \left( \begin{array}{ccc} \frac{1+\sqrt{1-4\lambda_b^2}}{2} & \lambda_a & 0 \\ \lambda_a & \frac{1+\sqrt{1-4\lambda_b^2}}{2} & 0 \\ 0 & 0 & 0 \\ \end{array} \right), B = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & \frac{1+\sqrt{1-4\lambda_b^2}}{2} & \lambda_b \\ 0 & \lambda_b & \frac{1-\sqrt{1-4\lambda_b^2}}{2} \\ \end{array} \right),$$ $$C = \left( \begin{array}{ccc} \frac{1+\sqrt{1-4\lambda_c^2}}{2} & 0 & \lambda_c \\ 0 & 0 & 0 \\ \lambda_c & 0 & \frac{1-\sqrt{1-4\lambda_c^2}}{2} \\ \end{array} \right)$$ Below we show that the mapping $f$ is an isomorphism between the algebras $S_8$ and $H^+_3(\mathbb{F})$. All calculations are straightforward and can be done by hand[^5]. It is easy to see that $A^2=A, B^2=B, C^2=C$. Symmetric involution matrices $e_{12}, e_{13}, e_{23}$ from $H^{+}_3(\mathbb{F})$ are in $\langle\langle A,B,C \rangle\rangle$, hence $\langle\langle A,B,C \rangle\rangle=H^{+}_3(\mathbb{F})$. A map $(\ ,\ ): H^{+}_3(\mathbb{F})^2 \rightarrow \mathbb{F}$ such that $(X,Y)=tr(XY)=tr(X \circ Y)$, where $X,Y\in H^{+}_3(\mathbb{F})$, is a symmetric bilinear form on $H^{+}_3(\mathbb{F})$. It is easy to see that this form associates with the product $\circ$. Clearly, we have $tr(A \circ A)=tr(B \circ B)=tr(C \circ C)=1$.\ Furthermore, we see that $tr(A \circ B)=\frac{1}{4}(1-\sqrt{1-4\lambda_a^2})(1-\sqrt{1-4\lambda_b^2})=\alpha, tr(B \circ C)=\frac{1}{4}(1-\sqrt{1-4\lambda_b^2})(1-\sqrt{1-4\lambda_c^2})=\beta, tr(A \circ C)=\frac{1}{4}(1-\sqrt{1-4\lambda_a^2})(1-\sqrt{1-4\lambda_c^2})=\gamma$ and $tr(A \circ (B \circ C))=tr(B \circ (A \circ C))=tr(C \circ (A \circ B))=\lambda_a\lambda_b\lambda_c=\psi$. So, we have equations to $\lambda_a, \lambda_b, \lambda_c$. We can obtain roots of these expressions by Wolfram Mathematica, but the roots are too long to show it here. We can easily check in Wolfram Mathematica that $\psi^2=\alpha\beta\gamma$ and $2\psi A \circ (B \circ C)-\beta\gamma (A \circ B)-\alpha\beta (A \circ C)=2\psi B \circ (A \circ C)-\beta\gamma (A \circ B)-\alpha\gamma (B \circ C)=2\psi C \circ (A \circ B)-\alpha\gamma(B \circ C)-\alpha\beta(A \circ C)=0$. Take basis $A, B, C, A \cdot B, B \cdot C, A \cdot C$ for $S_8$. Multiplication table for $f(\langle\langle a,b,c \rangle\rangle)$ [^6] coincides with multiplication table for $S_9$. ◻ # Acknowlengements The work was supported by the grant of the President of the Russian Federation for young scientists (MD-1264.2022.1.1). # Tables Here are multiplication tables for $S_7$, $S_8$, $S_9$. \| c \|\| c \| c \| c \| c \| $\ast$ & $a$ & $b$ & $c$ & $ab$\ $a$ & $a$ & \* & \* & \*\ $b$ & $ab$ & $b$ & \* & \*\ $c$ & \@c@$\frac{1}{2}((\gamma-\alpha)a+(\gamma-1)b$\ $+(1-\alpha)c+2(-\gamma+1)ab)$ & \@c@$\frac{1}{2}((\beta-1)a+(\beta-\alpha)b$\ $+(1-\alpha)c+2(-\beta+1)ab)$ & $c$ & \*\ $ab$ & $\frac{1}{2}(a\alpha+ab)$ & $\frac{1}{2}(b\alpha+ab)$ & \@c@$(\psi-\alpha)a+(\psi-\alpha)b$\ $+(\alpha-\alpha^2)c+(2-\beta-\gamma)ab$ & $\frac{1}{4}\alpha(a+b+2ab)$\ \| c \|\| c \| c \| c \| c \| c \| c \| $\ast$ & $a$ & $b$ & $c$ & $ab$ & $bc$ & $ac$\ $a$ & $a$ & \* & \* & \* & \* & \*\ $b$ & $ab$ & $b$ & \* & \* & \* & \*\ $c$ & $ac$ & $bc$ & $c$ & \* & \* & \*\ $ab$ & $\frac{1}{2}(\alpha a+ab)$ & $\frac{1}{2}(\alpha b+ab)$ & $-(\frac{\psi}{2\beta}bc+\frac{\psi}{2\gamma}ac)$ & $\frac{1}{4}\alpha(a+b+2ab)$ & \* & \*\ $bc$ & $-(\frac{\psi}{2\alpha}ab+\frac{\psi}{2\gamma}ac)$ & $\frac{1}{2}(\beta b+bc)$ & $\frac{1}{2}(\beta c+bc)$ & \@c@$\frac{\psi}{4}a+\dfrac{2\beta^2-\psi}{8\beta}ab-\frac{\alpha\beta}{8\psi}ac$\ $+ \dfrac{\psi^2+2\alpha\beta\psi-\alpha\beta^2}{8\beta\psi}bc$ & $\frac{1}{4}\beta(b+c+2bc)$ & \*\ $ac$ & $\frac{1}{2}(\gamma a+ac)$ & $-\frac{\psi}{2\alpha}ab-\frac{\psi}{2\beta}bc$ & $\frac{1}{2}(\gamma c+ac)$ & \@c@$\frac{\psi}{4}b+\dfrac{2\beta^2-\psi}{8\beta}ab-\dfrac{\alpha\beta}{8\psi}ac$\ $+\dfrac{\alpha\beta^2+\psi^2+2\alpha\beta\psi}{8\beta\psi}bc$ & \@c@$\frac{\psi}{4}c+\dfrac{\alpha\psi-2\beta\psi}{8\alpha\beta}ab$\ $+ \dfrac{\alpha^2\beta^2+2\psi^3-\alpha\psi^2}{8\alpha\beta\psi}bc$\ $+ \dfrac{2\beta\psi-\alpha\beta}{8\psi}ac$ & $\frac{1}{4}\gamma(a+c+2ac)$\ \|c\|\|c\|c\|c\|c\|c\|c\| $\ast$ & $a$ & $b$ & $c$ & $bc$ & $ac$ & $a(bc)$\ $a$ & $a$ & \* & \* & \* & \* & \*\ $b$ & $ab$ & $b$ & \* & \* & \* & \*\ $c$ & $ac$ & $bc$ & $c$ & \* & \* & \*\ $bc$ & $a(bc)$ & $\frac{1}{2}(b\beta+bc)$ & $\frac{1}{2}(c\beta+bc)$ & $\frac{1}{4}(\beta(b+c+2bc))$ & \* & \*\ $ac$ & $\frac{1}{2}(a\gamma+ac)$ & $a(bc)$ & $\frac{1}{2}(c\gamma+ac)$ & $\frac{1}{4}(\gamma bc + \beta ac + 2a(bc))$ & $\frac{1}{4}\gamma(a+c+2ac)$ & \*\ $a(bc)$ & $0$ & $\frac{1}{4}(\beta ac+2a(bc))$ & $\frac{1}{4}(\gamma bc + \beta ac)$ & $\frac{\beta\gamma}{8}b+\frac{\beta}{8}ac+\frac{\beta}{4}a(bc)$ & $\frac{\beta\gamma}{8}a+\frac{\gamma}{8}bc+\frac{\gamma}{4}a(bc)$ & $\frac{\beta\gamma}{16}a+\frac{\beta\gamma}{16}b$\ 99 *I. Gorshkov, A. Staroletov*, On primitive 3-generated axial algebras of Jordan type. J. Algebra, 563:74--99, 2020. *J. I. Hall, F. Rehren, and S. Shpectorov*, Primitive axial algebras of Jordan type. J. Algebra, 437:79--115, 2015. *J. I. Hall, Y.Segev, and S. Shpectorov*, On primitive axial algebras of Jordan type. Bull. Inst. Math. Acad. Sin. (N.S.), 13(4):397--409, 2018. *I. Gorshkov, V. Gubarev*, Quasi-definite axial algebras of Jordan type half. Preprint, ArXiv.org, https://arxiv.org/abs/2204.02578 *S.M.S. Khasraw, J. McInroy, S. Shpectorov*, On the structure of axial algebras. Trans. Amer. Math. Soc. 373:2135-2156, 2020. *M. Miyamoto*, Griess algebras and conformal vectors in vertex operator algebras, J. Algebra 179(2):523-548, 1996. Ravil Bildanov, [Sobolev Institute of Mathematics, Novosibirsk, Russia;]{.smallcaps}\ [Novosibirsk State University, Novosibirsk, Russia;]{.smallcaps}\ *E-mail address:* `ravilbildanov@gmail.com ` Ilya Gorshkov, [Sobolev Institute of Mathematics, Novosibirsk, Russia;]{.smallcaps}\ [Novosibirsk State Technical University, Novosibirsk, Russia;]{.smallcaps}\ *E-mail address:* `ilygor8@gmail.com` [^1]: Calculations for multiplication table in $S_7$ can be found in [https://github.com/RavilBildanov/3gen-axial-algebras/blob/main/S8 multiplication table.nb](https://github.com/RavilBildanov/3gen-axial-algebras/blob/main/S8 multiplication table.nb), see paragraph Tables. [^2]: One can find our calculations here: [https://github.com/RavilBildanov/3gen-axial-algebras/blob/main/M2+ (S7).nb](https://github.com/RavilBildanov/3gen-axial-algebras/blob/main/M2+ (S7).nb) [^3]: One can find our calculations here: [https://github.com/RavilBildanov/3gen-axial-algebras/blob/main/H3+ (S8).nb](https://github.com/RavilBildanov/3gen-axial-algebras/blob/main/H3+ (S8).nb) [^4]: Calculations for multiplication table in $S_8$ can be found in [https://github.com/RavilBildanov/3gen-axial-algebras/blob/main/S8 multiplication table.nb](https://github.com/RavilBildanov/3gen-axial-algebras/blob/main/S8 multiplication table.nb), see paragraph Tables. [^5]: One can find our calculations here: [https://github.com/RavilBildanov/3gen-axial-algebras/blob/main/H3+ (S9).nb](https://github.com/RavilBildanov/3gen-axial-algebras/blob/main/H3+ (S9).nb) [^6]: Calculations for multiplication table in $S_9$ can be found in [https://github.com/RavilBildanov/3gen-axial-algebras/blob/main/S9 multiplication table.nb](https://github.com/RavilBildanov/3gen-axial-algebras/blob/main/S9 multiplication table.nb)

arxiv_math

--- author: - Xiaoxiao Peng - Shijie Zhou - Wei Lin - Xuerong Mao bibliography: - bibliography.bib title: Invariance Principles for $G$-Brownian-Motion-Driven Stochastic Differential Equations and Their Applications to $G$-Stochastic Control --- # abstract The *G*-Brownian-motion-driven stochastic differential equations (*G*-SDEs) as well as the *G*-expectation, which were seminally proposed by Peng and his colleagues, have been extensively applied to describing a particular kind of uncertainty arising in real-world systems modeling. Mathematically depicting long-time and limit behaviors of the solution produced by *G*-SDEs is beneficial to understanding the mechanisms of system's evolution. Here, we develop a new *G*-semimartingale convergence theorem and further establish a new invariance principle for investigating the long-time behaviors emergent in *G*-SDEs. We also validate the uniqueness and the global existence of the solution of *G*-SDEs whose vector fields are only locally Lipschitzian with a linear upper bound. To demonstrate the broad applicability of our analytically established results, we investigate its application to achieving *G*-stochastic control in a few representative dynamical systems. # Introduction Long-time and limit behaviors of the solutions generated by stochastic differential equations (SDEs) have received growing attention because such behaviors usually correspond to particular functions in real-world systems [@gevers1991continuous; @Mao-5; @Mao-6; @florchinger1995lyapunov; @brockwell1999stability]. Interesting physical or/and biological phenomena have been systematically investigated, including asymptotic behaviors of random matrices in quantum physics [@mendelson2014singular], stochastic resonance [@benzi1983theory], stochastic homogeneity [@DabrockHofmanova-520], stochastic stabilization or synchronization [@b33; @b34; @li2019robust; @li2011output], and random-temporal-structure-induced emergence [@b38; @b39; @b40; @b41]. Also developed were stochastic versions of invariance principle, which originated from LaSalle's invariance principle [@Lasalle-1; @Lasalle-2] for deterministic systems and then has been extended successfully to study the SDEs [@Mao-7; @zhou2023generalized], the stochastic differential delayed equations (SDDEs) [@XuerongMao-8; @Mao-9], the stochastic functional differential equations (SFDEs) [@Mao-10; @YiQi-11] and even the discrete stochastic dynamical systems [@zhou2022generalized]. These versions of invariance principles are often used to elucidate the asymptotic behaviors, such as stability, boundedness, and invariance in some chaotic attractors, emergent in random systems. In addition to the traditional frameworks of randomness and stochasticity, measuring uncertainties of randomness is another important issue in those areas replete with fluctuations and risks of high level, such as economics [@Knight-12]. A seminal framework by means of sublinear expectation was fundamentally built by Peng and his colleagues to quantify such uncertainties [@Peng-13] and then extended broadly in line with the modern probability theory. Indeed, the framework has been put forward to investigating the $G$-Brownian-motion-driven stochastic differential equations ($G$-SDEs), which thus provides a model to describe the randomness with uncertainties in evolutionary dynamics. Also systematically investigated was the well-posedness of $G$-SDEs [@Gao-14; @Peng-13] and stochastic functional differential equations ($G$-SFDEs) [@YongRen-15; @FaizullahFaiz-18]. Furthermore, although the stability of $G$-SDEs has been widely investigated [@LiLin-16; @RenYin-17], rigorously delicate descriptions of stability, boundedness, control and even invariance property in dynamical attractors using $G$-SDEs are still lacking. In this article, we, therefore, intends to fill in this gap through novelly developing an invariance principle for $G$-SDEs and investigate its applicability to the stochastic control, especially in the case that the noise is uncertain. As such, this invariance principle can render the analytical investigations of dynamics produced by $G$-SDEs much clearer and more complete. In order to develop this new principle, we need to establish a new version of $G$-semimartingale convergence theorem, nontrivially generalizing the classical semimartingale convergence theorem developed in [@LiptserShiryayev-24]. The remaining of this article is organized as follows. Section [3](#pre){reference-type="ref" reference="pre"} introduces some basic concepts and provides some preliminary theorems of sublinear expectations. Section [4](#Gsemi){reference-type="ref" reference="Gsemi"} rigorously proves the $G$-semimartingale convergence theorem as follows. **THEOREM 1**. *Assume $A^{1}$ and $A^{2}$ are two non-decreasing process with initial value 0, $A^{1}(t)$ is a continuous process and $\mathbb{\hat{E}}[A^{1}(+\infty)]<+\infty$. Assume that $Z$ is a non-negative $G$-semimartingale satisfying $\mathbb{\hat{E}}[Z^{+}({0})]<\infty$ with the form as $Z({t})=Z({0})+A^{1}({t})-A^{2}(t)+M({t}), \ t\geq 0,$ where $M({t})$ is a continuous $G$-supermartingale with initial value 0. $M({t})\in L_{G}^{1}(\Omega_{t})$ for every $t\geq 0$. Then, we have that $A^{2}(+\infty)<+\infty$, $\lim_{t\rightarrow +\infty}Z({t})$ finitely exists, and that $\lim_{t\rightarrow +\infty}M({t})$ finitely exists quasi-surely.* Here, we sketch the proof of the above convergence theorem as follows. By extending the space of random variables, we generalize Fatou's Lemma on the $G$-conditional expectation. Combining with the uppercrossing inequality, we derive the $G$-martingale convergence theorem for a continuous process and then establish the essential $G$-semimartingale convergence theorem. Also in this section, we present the other more applicable versions of the $G$-semimartingale convergence theorem. With all these preparations, Section [5](#secGSDE){reference-type="ref" reference="secGSDE"} presents our main result, the *invariance principle* for the $G$-SDEs, and validates it using the established $G$-semimartingale convergence theorem. Here, we show this principle as follows. **THEOREM 2**. *With those conditions and assumptions listed in Section [5](#secGSDE){reference-type="ref" reference="secGSDE"}, we suppose that there exists a function $V \in C^{2,1}(\mathbb{R}^{d}\times \mathbb{R}_{+}; \mathbb{R}_{+})$, a function $\gamma \in L^{1}(\mathbb{R}_{+}; \mathbb{R}_{+})$ and a continuous function $\eta: \mathbb{R}^{d} \rightarrow \mathbb{R}_{+}$ such that $\lim_{\vert x\vert \rightarrow \infty} \inf_{0\leq t <+\infty} V({\bm x},t)=\infty$ and $\mathcal{L}V({\bm x},t)\leq \gamma(t)-\eta({\bm x})$, where the diffusive operator $\mathcal{L}V=V_{t}+V_{x_{i}}f^{i}+G\Big((V_{x_{k}}(h^{kij}+h^{kji})+V_{x_{k}x_{l}} g^{ki}g^{lj})_{i,j=1}^{n}\Big )$ where Einstein's notations are applied here. Then, we have that $\lim_{t \rightarrow +\infty}V({\bm x}(t),t)$ finitely exists quasi-surely and that $\lim_{t \rightarrow +\infty}\eta({\bm x}(t))=0$ quasi-surely. Moreover, we have $\lim_{t \rightarrow +\infty}d(\bm x(t), {\rm Ker}(\eta))=0.$ Here, $\bm{x}(t)$ is the solution of the $G$-SDEs which read $${\rm d}{\bm x}(t)={\bm f}({\bm x}(t),t){\rm d}t+{\bm g}({\bm x}(t),t){\rm d}{\bm B}(t)+{\bm h}({\bm x}(t),t){\rm d}\langle {\bm B}\rangle (t).$$* The proof of such theorem, though inspired by [@Mao-7], is rather different. By $G$-Itô's formula, we write out the function in a form of the $G$-semimartingale and then apply the corresponding convergence theorem. By estimating the calculus of $\eta$ based on the uppercrossing stopping time, we show that all trajectories converge to the kernel of the function $\eta$ quasi-surely. Still in this section, we further present several generalized versions of invariance principle. All these build up a solid foundation for Section [6](#example){reference-type="ref" reference="example"}, where we use the *G*-stochastic control to stabilize representative complex dynamics, demonstrating the broad applicability of our analytically-established results. Finally, Section [7](#discussion){reference-type="ref" reference="discussion"} provides some discussion and concluding remarks. # Preliminaries {#pre} In this section, we present some frequently used definitions and results of sublinear expectation theory, which will be useful for our following investigations. For more details, we refer to [@LaurentDenis-25; @Peng-13; @Peng-26; @Lipeng-27]. To begin with, we let $\Omega$ be a given set, and $\mathcal{H}$ be the space of all real-valued functions defined on $\Omega$. Denote by ${C}_{l,{\rm Lip}}(\mathbb{R}^{d })$ the space of all locally Lipschitz-continuous functions on $\mathbb{R}^{d }$. And, for any function $\varphi \in {C}_{l,{\rm Lip}}(\mathbb{R}^{d })$, if $x_{i}(\omega) \in \mathcal{H}$ for all $i=1,2,\cdots,d$, then $\varphi(x_{1}(\omega), \cdots, x_{d}(\omega))\in \mathcal{H}$. Next, we provide some basic concepts on the sublinear expectation. **Definition 3** (Sublinear Expectation [@Peng-13]). *A functional $\mathbb{{E}}[\cdot]$ is said to be a sublinear expectation on $\mathcal{H}$ if it satisfies: (1) $\mathbb{E}[c]=c, \ {\rm for} \ {\rm any}~c \in \mathbb{R}$, (2) $\mathbb{E}[X] \leq \mathbb{E}[Y], \ {\rm for} \ {\rm any}~X\leq Y$, (3) $\mathbb{E}[X+Y]\leq \mathbb{E}[X]+\mathbb{E}[Y]$, and (4) $\mathbb{E}[\lambda X]=\lambda\mathbb{E}[X], \ {\rm for} \ {\rm any} \lambda \geq 0$.* **Definition 4** ($G$-Function [@Peng-13]). *A function $G:\mathbb{R}^{d} \times \mathbb{S}^{d} \to \mathbb{R}$ is said to be sublinear and monotone if it satisfies $(1)~G({\bm p}+\bar{\bm p}, {\bm A}+\bar{{\bm A}}) \leq G({\bm p}, {\bm A})+G(\bar{\bm p}, \bar{\bm A})$, $(2)~G({\bm p}, {\bm A}) \leq G({\bm p}, \bar{\bm A}), \text{if}~{\bm A} \leq \bar{\bm A}$, and $(3)~G(\lambda {\bm p}, \lambda {\bm A})=\lambda G({\bm p}, {\bm A}),~\forall~\lambda \geq 0$.* *Here, $\mathbb{S}^{d}$ denotes the space of $d \times d$ symmetric matrices. And ${\bm A} \leq \bar{\bm A}$ implies the nonnegativity of the symmetric matrix $\bar{\bm A}-{\bm A}$.* In the following, we assume the function $G$ defined in Definition [Definition 4](#def2){reference-type="ref" reference="def2"} is independent of the vector $p$. It is worthwhile to mention that, when $d=1$, $G$ is reduced to the form $G(r)=\frac{1}{2}(r^{+}\overline{\sigma}^{2}-r^{-}\underline{\sigma}^{2})$ for some [non-negative]{style="color: black"} $\underline{\sigma} \leq \overline{\sigma}$. Here $r^+$ and $r^-$ correspond to the non-negative and the non-positive parts of $r$, respectively. Moreover, if a symmetric $G$-Brownian motion satisfies ${\mathbb{\hat{E}}}[{\bm A}{\bm B}(t),{\bm B}(t)]=2G(\bm {A})t$ with $G({\bm A})=\frac{1}{2}{\mathbb{\hat{E}}}[{\bm A}{\bm B}(1),{\bm B}(1)]$, then $G$ is said to be a $G$-function related to the symmetric $G$-Brownian motion ${\bm B}$. Here, the definition of $G$-Brownian motion, as well as $G$-conditional expectation, can be found in [@Peng-13]. Moreover, it is necessary to introduce some definitions on some spaces of functions and measures. Here, we denote, respectively, by $\bullet$ $\mathscr{F}_{t}:$ The completion of $\sigma({\bm B}(s):s\leq t)$, $\bullet$ $\mathscr{B}(\Omega):$ The Borel $\sigma$-algebra on $\Omega$, $\bullet$ $L^{0}(\Omega)$: The space of all $\mathscr{B}(\Omega)$-measurable functions, $\bullet$ $L_{G}^{p}(\Omega):$ The completion of the space ${\rm Lip}(\Omega)$ under the norm $\|\cdot\|_{L_{G}^{p}}:=(\mathbb{\hat{E}}[\vert \cdot\vert ^{p}])^{\frac{1}{p}}$, $\bullet$ ${\rm Lip}\left(\Omega_{t}\right)$: $\{\varphi({\bm B}({t_{1}}), {\bm B}({t_{2}})-{\bm B}({t_{1}}), \cdots,$ ${\bm B}({t_{k}})-{\bm B}({t_{k-1}})):\varphi \in {C}_{l,{\rm Lip}}(\mathbb{R}^{m \times k }), ~0\leq t_{1}<\cdots<t_{k}\leq t\}$, $\bullet$ $L_{G}^{p}(\Omega_{t}):$ $L_{G}^{p}(\Omega) \cap {\rm Lip}\left(\Omega_{t}\right)$, $\bullet$ $\mathcal{M}:$ The set of all probability measure defined on $\Omega$, $\bullet$ $E_{Q}[\cdot]$: The expectation under the traditional probability measure $Q$, $\bullet$ [$\mathcal{P}(t, Q)$:=]{style="color: black"} $\left\{R \in \mathcal{M}: E_{Q}[X]=E_{R}[X], \forall X \in {\rm Lip}\left(\Omega_{t}\right)\right\}$, $\bullet$ $\mathcal{Q}:=\{Q\in \mathcal{M}:E_{Q}[X]\leq {\mathbb{\hat{E}}}[X], \forall X \in L_{G}^{1}\left(\Omega\right)\}$, and $\bullet$ [$\mathcal{L}^{0}(\Omega)$:=]{style="color: black"} {$X \in L^{0}(\Omega)$: $E_{Q}[X]$ exists for any $Q \in \mathcal{Q}$}. From Theorem 1.2.1 in [@Peng-26], it follows that the sublinear expectation satisfies $\mathbb{\hat{E}}[X]=\sup_{Q \in \mathcal{Q}}E_{Q}[X]$ for each $X \in$ Lip($\Omega$). Thus, the definition of $\mathbb{\hat{E}}[\cdot]$ can be extended to $\mathcal{L}^0(\Omega)$. In addition, for the $G$-conditional expectation defined above, it can be represented by means of the probability space. **THEOREM 5** ([@HuPeng-29]). *For each $Q \in \mathcal{Q}$ and $X \in L_{G}^{1}(\Omega)$, ${\mathbb{\hat{E}}}_{t}[X]=\mathop{\mathrm{ess\,sup}}_{R \in \mathcal{P}(t, Q)}{ }^{Q} E_{R}\left[X \mid \mathscr{F}_{t}\right], \ \ Q\text{-}a.s..$ Here, if  $Y=\mathop{\mathrm{ess\,sup}}_{R \in \mathcal{P}(t, Q)}{ }^{Q} E_{R}\left[X \mid \mathscr{F}_{t}\right]$, it means that for every $R \in \mathcal{P}(t, Q)$, $E_{R}\left[X \mid \mathscr{F}_{t}\right] \leq Y, Q\text{-}a.s..$ Moreover, if $E_{R}\left[X \mid \mathscr{F}_{t}\right] \leq Z$ [for each $R \in \mathcal{P}(t, Q)$]{style="color: black"}, $Q\text{-}a.s.$, then we must have $Y\leq Z, Q\text{-}a.s..$* For introducing $G$-Itô's calculus, we define $M_{G}^{p}([0, T])$, a space of random process, and the $G$-Itô's calculus on it (refer to [@Peng-13] for details). Moreover, the quadratic variation is defined in the same manner as that in normal stochastic analysis. However, the range of the quadratic variation here is much different. **Lemma 6** ([@Peng-13]). *For an $m$-dimensional $G$-Brownian motion ${\bm B}$, there exists a bounded, convex and closed set $\Gamma \in \mathbb{S}_{+}^{m}$ such that $\langle {\bm B} \rangle({t}) \in t\Gamma:=\{t \gamma: \gamma \in \Gamma\}$, where ${\mathbb{S}_{+}^{n}}$ represents the space of all positive symmetric matrices. Also, $\langle {\bm B} \rangle({t})$ and $\langle {\bm B} \rangle({t+s})- \langle {\bm B} \rangle({s})$ are identically distributed.* **Remark 7**. *In what follows, denote by $\bar{\gamma}:= \max_{\gamma \in \Gamma} (\vert \gamma\vert _{F}\vee \vert \gamma\vert_{2} )$ where $\vert \cdot\vert _{F}$ and $\vert \cdot\vert_{2}$, respectively, are the Frobenius norm [@belitskii2013matrix] and 2-norm for the matrix. Then, it follows from Lemma [Lemma 6](#lemmalemma){reference-type="ref" reference="lemmalemma"} that $\vert \langle {\bm B} \rangle({t})\vert_{F} \vee \vert \langle {\bm B} \rangle({t})\vert_{2} \leq \bar{\gamma}t$. Especially when $m=1$, we have $\bar{\gamma}=\overline{\sigma}^{2}$. Also, the largest eigenvalue of a matrix is denoted by $\lambda_{\rm max}(\cdot)$.* There are some very useful inequalities for our investigation in this article. Combining the results of Sections 3.3-3.5 in [@Peng-13], Lemma [Lemma 6](#lemmalemma){reference-type="ref" reference="lemmalemma"}, and Remark [Remark 7](#lemmaqua){reference-type="ref" reference="lemmaqua"}, we give the conclusions as follows. **THEOREM 8**. *For any $\eta(t), ~\gamma(t)\in M_{G}^{2}[0, T]$, we have $\mathbb{\hat{E}}\left(\int_{0}^{T} \eta(t) \mathrm{~d} B_{i}(t)\right)=0$ and $\mathbb{\hat{E}}\left(\int_{0}^{T} \eta(t) \mathrm{~d} B_{i}(t)\int_{0}^{T} \gamma(t) \mathrm{~d} B_{j}(t)\right) =\mathbb{\hat{E}}\left(\int_{0}^{T} \eta(t)\gamma(t) \mathrm{~d}\langle B_{i},B_{j}\rangle(t)\right) \leq \bar{\gamma} \cdot \mathbb{\hat{E}}\left(\int_{0}^{T} \vert \eta(t)\gamma(t)\vert \mathrm{~d} t\right).$* Now we introduce the Choquet capacity and some related propositions. **Definition 9** (Choquet Capacity, [@Peng-13]). *For $\mathcal{A} \in \mathscr{B}(\Omega)$, define by $c(\mathcal{A}):=\sup_{Q \in \mathcal{Q}}Q[\mathcal{A}]=\mathbb{\hat{E}}[1_{\mathcal{A}}].$ A property is called *valid quasi-surely* if this property is valid on the set $\Omega \backslash \mathcal{A}$ with $c(\mathcal{A})=0$.* **Proposition 10** (Monotone Convergence Theorem, [@LaurentDenis-25; @Peng-26]). *If $X({n}) \uparrow X$, $\{X(n)\}\subset \mathcal{L}^{0}(\Omega)$, $X({n})$ is nonnegative, then $\mathbb{\hat{E}}[X({n})]\uparrow \mathbb{\hat{E}}[X]$.* **THEOREM 11** ([@Li-31]). *Assume that $\{M(n) \}$ is a $G$-supermartingale, satisfying $\sup_{n}\mathbb{\hat{E}}[M^{-}(n)]<+\infty$. Then, $\lim_{n\rightarrow \infty}M(n)$ exists, which is finite quasi-surely. Here, the definition of $G$-martingale can be found in [@Peng-13].* # $G$-Semimartingale Convergence Theorem {#Gsemi} In the literature, the semimartingale convergence theorem mainly describes the asymptotic property of the semimartingale, which is a random variable comprising a martingale and a process with bounded variation. Inspired by this well-established and broadly-applied convergence theorem, we are to establish a $G$-semimartingale convergence theorem and its variant. It will be shown that the $G$-semimartingale convergence theorem is based crucially on Doob's $G$-martingale convergence theorem. In fact, to our best knowledge, the continuous version of Doob's $G$-martingale convergence theorem has not yet been established until the result presented [as follows]{style="color: black"}. **Proposition 12** ($G$-Martingale Convergence Theorem, A Continuous Version). *Assume that $\{M({t}):t\in[0,+\infty)\}$ is a right- or left-continuous $G$-supermartingale, and $M(t)\in L_{G}^{1}(\Omega_{t})$. Moreover, assume that $\mathbb{\hat{E}}[\sup_{t\geq 0}M^{-}({t})]<+\infty$. Then, $M({t})$ converges finitely to $M(+{\infty})\in L_{G}^{1^{*}_{*}}(\Omega)$ quasi-surely. Moreover, $\mathbb{\hat{E}}_{t}[M({+\infty})]\leq M({t})$. [Here, the definition of $L_{G}^{1^{*}_{*}}(\Omega)$ is provided in Definition [Definition 34](#remark1){reference-type="ref" reference="remark1"} of Appendix [8.1](#appendix1){reference-type="ref" reference="appendix1"}]{style="color: black"}.* [The proof of this proposition is tedious and tangential to the main focus of this article. To enhance the readability, we include the proof into Appendix [8.1](#appendix1){reference-type="ref" reference="appendix1"}.]{style="color: black"} Now, with this preparation, we establish the following $G$-semimartingale convergence theorem. **THEOREM 13** ($G$-Semimartingale Convergence Theorem). *Assume that $A^{1}$ and $A^{2}$ are two non-decreasing processes with initial value $0$, and that $A^{1}(t)$ is a continuous process with $\mathbb{\hat{E}}[A^{1}(+\infty)]<+\infty$. Also, assume that $Z$ is a non-negative $G$-semimartingale satisfying $\mathbb{\hat{E}}[Z^{+}({0})]<\infty$ with the form $Z({t})=Z({0})+A^{1}({t})-A^{2}(t)+M({t}), \ t\geq 0,$ where $M({t})$ is a continuous $G$-supermartingale with initial value $0$ and $M({t})\in L_{G}^{1}(\Omega_{t})$ for every $t\geq 0$. Then, we have that $A^{2}(+\infty)<+\infty$, $\lim_{t\rightarrow +\infty}Z({t})$ finitely exists and $\lim_{t\rightarrow +\infty}M({t})$ finitely exists quasi-surely.* Notice that $M({t})=Z({t})-Z({0})-A^{1}(t)+A^{2}(t)\geq -Z({0})-A^{1}(+\infty).$ Then, $\sup_{t\geq 0}M^{-}({t}) \leq Z^{+}(0)+A^{1}(+\infty)$. By Proposition [Proposition 12](#convergence){reference-type="ref" reference="convergence"}, we have $\lim_{t \rightarrow \infty}M({t})$ finitely exists quasi-surely. Because $A^2({t})=Z({0})+A^{1}({t})+M({t})-Z({t})\leq Z({0})+A^{1}({t})+M({t})$ and $Z({t})=Z({0})+A^{1}({t})-A^{2}({t})+M({t}),$ their limits also exist quasi-surely. It is mentioned that this $G$-semimartingale convergence theorem can only deal with the case where the limit of $A^{1}({t})$ is supposed to be finite under the sublinear expectation. We now give its variant, the $G$-semimartingale convergence theorem with the $\mathbb{F}$-stopping time. It can deal with the case where the condition on the finite limit of $A^{1}({t})$ in Theorem [THEOREM 13](#semimartinagel){reference-type="ref" reference="semimartinagel"} is removed. The tradeoff however requires more conditions for the $G$-martingale $M$. **THEOREM 14** ($G$-Semimartingale Convergence Theorem with Stopping Time). *Assume that $A^{1}$ and $A^{2}$ are two non-decreasing processes both with initial value $0$, and that $A^{1}({t})$ is a continuous adapted process. Also assume that $Z$ is a non-negative adapted process satisfying $\mathbb{\hat{E}}[\vert Z({0})\vert ]<\infty$ with the form $Z({t})=Z({0})+A^{1}({t})-A^{2}({t})+M({t}), \ \ t\geq 0,$ where $M({t})$ is a continuous process with initial value $0$. Furthermore, assume that there exists a series of $\mathbb{F}$-stopping times $\tau_{N}$ satisfying $\{\tau_{N}\rightarrow +\infty\}$ quasi-surely such that, for any $Q \in \mathcal{Q}$, ${E}_{Q}[M({t\wedge\tau_{N}})\vert \mathscr{F}_{s}]=M({s\wedge\tau_{N}})$. Then, we have quasi-surely $$\begin{array}{l} \displaystyle\left\{\omega:A^{1}(+\infty)<+\infty\right\}\subset \left\{\omega:\lim_{t\rightarrow +\infty}Z({t}) \mbox{ finitely exists} \right\} \\ \displaystyle ~~~~\cap \left\{\omega:A^{2}(+\infty)<+\infty\right\}\cap \left\{\omega:\lim_{t\rightarrow +\infty}M({t}) \ \mbox{finitely exists}\right\}. \end{array}$$ Here, $\mathcal{A} \subset \mathcal{B}$ quasi-surely means that $c(\mathcal{A}\backslash \mathcal{B})=0$, where $c$ is the Choquet capacity provided in Definition [Definition 9](#defcho){reference-type="ref" reference="defcho"}.* Denote by $\mathcal{A}=\Omega \backslash \Big(\{\omega:\lim_{t\rightarrow +\infty}Z({t}) \ \mbox{finitely exists} \}\cap \{\omega:A^{2}(+\infty)<+\infty\}\cap\{\omega:\lim_{t\rightarrow +\infty}M({t}) \ \mbox{finitely exists} \} \Big)$. For every $Q \in \mathcal{Q}$, we have $E_{Q}[\vert Z({0})\vert ]\leq \mathbb{\hat{E}}[\vert Z({0})\vert ]$. By the $G$-semimartingale convergence theorem for the normal probability space [@LiptserShiryayev-24], we have $Q(\mathcal{A})=0$. By the arbitrariness of the $Q$'s choice, we obtain that $c(\mathcal{A})=\sup_{Q \in \mathcal{Q}}Q(\mathcal{A})=0$, which therefore completes the proof. # Invariance Principle in Sublinear Expectation {#secGSDE} Now, we consider a $d$-dimensional $G$-stochastic differential equation which reads $$\label{GSDE} {\rm d}{\bm x}(t)={\bm f}({\bm x}(t),t){\rm d}t+{\bm g}({\bm x}(t),t){\rm d}{\bm B}(t)+{\bm h}({\bm x}(t),t){\rm d}\langle {\bm B}\rangle (t),$$ where the initial value $x(0)=x_{0}$. Furthermore, we denote, respectively, by $\vert \bm A\vert _{2}:=\sqrt{{\rm tr}(\bm A^{\top}\bm A)}~~\mbox{ and} ~~\vert \bm A\vert :=\vert \bm A\vert_{F}=\sqrt{\sum_{i,j=1}^{n}a_{ij}^{2}}$ different norms of a given matrix $\bm{A}$. All functions ${\bm f}:\mathbb{R}^{d}\times \mathbb{R}_{+}\rightarrow \mathbb{R}^{d}$, ${\bm g}:\mathbb{R}^{d}\times \mathbb{R}_{+} \rightarrow \mathbb{R}^{d\times m}$, and ${\bm h}:\mathbb{R}^{d}\times \mathbb{R}_{+} \rightarrow \mathbb{R}^{d\times m \times m}$ are supposed to be continuous. In addition, $h^{kij}=h^{kji}$, and $f^{i}({\bm x},\cdot)$, $g^{ij}({\bm x},\cdot)$ and $h^{kij}({\bm x},\cdot) \in M_{G}^{2}[0,T]$ for every $T>0$. We need the following assumptions. **Assumption 15**. *For any $N \in \mathbb{N}$, there exists a number $C_{N}$ such that $\vert {\bm f}({\bm x},t)-f({\bm y},t)\vert +\vert {\bm g}({\bm x},t)-{\bm g}({\bm y},t)\vert +\vert {\bm h}({\bm x},t)-{\bm h}({\bm y},t)\vert \leq C_{N}\vert {\bm x}-{\bm y}\vert$ for all $\vert {\bm x}\vert \wedge \vert {\bm y}\vert \leq N$. Here, $\vert {\bm h}\vert$ still represents the norm for $\bm h$ of $d\times m\times m$ dimensions.* **Assumption 16**. *There exists a number $C_{l}$ such that $\vert {\bm f}({\bm x},t)\vert +\vert {\bm g}({\bm x},t)\vert +\vert {\bm h}({\bm x},t)\vert \leq C_{l}(1+\vert {\bm x}\vert ),$ for all $({\bm x},t)\in \mathbb{R}^{d}\times \mathbb{R}_{+}$.* [Underlying these assumptions as prerequisites, the solutions of Eq. [\[GSDE\]](#GSDE){reference-type="eqref" reference="GSDE"} are well-posed from a certain perspective as follows.]{style="color: black"} **Proposition 17**. *If Assumption [Assumption 15](#asump1){reference-type="ref" reference="asump1"} holds, there is a global unique solution in a quasi-sure sense on $[0,\tau_{\infty})$, where $\tau_{\infty}=\lim_{n\rightarrow +\infty}\tau_{N}, \ \tau_{N}:=\inf \{t\geq 0:\vert {\bm x(t)}\vert \geq N\}.$For given $N>0$, there exists $\bm x^{N}\in M_{G}^{2}[0,T]$ with $T>0$ such that $\bm x=\bm x^{N}$ on $[0,\tau_{N})$. Additionally, for ${\bm A}=(a^{ij}):\mathbb{R}^{d}\times \mathbb{R}_{+} \rightarrow \mathbb{R}^{d\times m}$ with $a^{ij}({\bm x},\cdot)\in M_{G}^{1}[0,T]$ and $T>0$, we have ${\bm M}(t)=\int_{0}^{t\wedge \tau_{N}}{\bm A}({\bm x}(s),s) {\rm d}{\bm B}(s)$ is $Q$-martingale for each $Q \in \mathcal{Q}$. If Assumption [Assumption 16](#asump2){reference-type="ref" reference="asump2"} holds, we have $\tau_{\infty}=+\infty$ quasi-surely.* **Remark 18**. *[ The proof of Proposition [Proposition 17](#exist){reference-type="ref" reference="exist"} is similar to those presented in Refs. [@Mao-6; @LiLin-16], which we omit here. It is worth mentioning that $\bm x(\cdot)$, the solution to Eq. [\[GSDE\]](#GSDE){reference-type="eqref" reference="GSDE"}, does not belong to $M_{G}^{2}([0,T];\mathbb{R}^{d})$. Actually, $\bm x(\cdot\wedge \tau_N)\in M_{*}^{2}([0,T];\mathbb{R}^{d})$ for each $N>0$, which implies that our solution is locally integrable. In particular, if $\tau_\infty =+\infty$, we have $\bm x(\cdot) \in M_{w}^{2}([0,T];\mathbb{R}^{d})$ and it is globally integrable on $[0,+\infty)$ now. Here, both $M_{*}^{2}([0,T];\mathbb{R}^{d})$ and $M_{w}^{2}([0,T];\mathbb{R}^{d})$ are expanded integrand space defined in Chapter 8 of Ref. [@Peng-13] satisfying $M_{G}^{2}([0,T];\mathbb{R}^{d})\subset M_{*}^{2}([0,T];\mathbb{R}^{d})\subset M_{w}^{2}([0,T];\mathbb{R}^{d})$. ]{style="color: black"}* Next, we introduce $G$-Itô's formula which is useful in the following discussions. **THEOREM 19** ($G$-Itô's formula [@Lipeng-27]). *Let $V \in C^{2,1}(\mathbb{R}^{d}\times \mathbb{R}_{+}; \mathbb{R}_{+})$. For the $d$-dimensional $G$-stochastic differential equations ${\rm d}{\bm x}(t)={\bm f}(t){\rm d}t+{\bm g}(t){\rm d}{\bm B}(t)+{\bm h}(t){\rm d}\langle {\bm B}\rangle (t)$ with the initial value ${\bm x}( 0)={\bm x}_{0}$. Moreover, ${\bm f}:\mathbb{R}_{+}\rightarrow \mathbb{R}^{d}$, ${\bm g}: \mathbb{R}_{+} \rightarrow \mathbb{R}^{d\times m}$, and ${\bm h}:\mathbb{R}_{+} \rightarrow \mathbb{R}^{d\times m^2}$ with $f^{i}(\cdot),~g^{ij}(\cdot) \in M_{G}^{1}[0,T]$, $h^{kij}(\cdot) \in M_{G}^{2}[0,T]$ for every $T>0$. Then, $V({\bm x}(t),t)=V({\bm x}_{0},0)+\int_{0}^{t} V_{t}({\bm x}(s),s){\rm d}s+\int_{0}^{t} V_{x_{i}}({\bm x}(s),s)f^{i}(s){\rm d}s + \int_{0}^{t} V_{x_{i}}({\bm x}(s),s)g^{ij}(s){\rm d}B_{j}(s) + \int_{0}^{t} V_{x_{k}}({\bm x}(s),s)h^{kij}(s){\rm d}\langle B_{i}, B_{j} \rangle(s) +\int_{0}^{t}\frac{1}{2} V_{x_{k}x_{l}}({\bm x}(s),s)g^{ki}(s)g^{lj}(s){\rm d}\langle B_{i}, B_{j} \rangle(s).$* [Actually, $G$-Itô's formula presented above could be applicable to $M_{*}^{2}([0,T];\mathbb{R}^{d})$ and $M_{w}^{2}([0,T];\mathbb{R}^{d})$ according to Theorem 5.4 established in [@Lipeng-27]. By virture of $G$-Itô's formula,]{style="color: black"} Assumption [Assumption 16](#asump2){reference-type="ref" reference="asump2"} used above can be replaced. To present this result, we introduce the notation as $\mathcal{L}V:=V_{t}+V_{x_{i}}f^{i}+G\Big((V_{x_{k}}(h^{kij}+h^{kji})+V_{x_{k}x_{l}} g^{ki}g^{lj})_{i,j=1}^{n}\Big),$ where the function $V \in C^{2,1}(\mathbb{R}^{d}\times \mathbb{R}_{+}; \mathbb{R}_{+})$. As such, we obtain the following result. **Proposition 20**. *Suppose that Assumption [Assumption 15](#asump1){reference-type="ref" reference="asump1"} holds and that there exists a function $\gamma \in L^{1}(\mathbb{R}_{+}; \mathbb{R}_{+})$ such that $\mathcal{L}V({\bm x},t)\leq \gamma(t).$ Moreover, $V$ satisfies $$\label{conditionofV2} \lim_{\vert x\vert \rightarrow \infty} \inf_{0\leq t <+\infty} V({\bm x},t)=+\infty.$$ Then, $\tau_{\infty}$, as defined in Proposition [Proposition 17](#exist){reference-type="ref" reference="exist"}, satisfies $\tau_{\infty}=+\infty$ quasi-surely.* For simplicity of expression, we still include the proof of Proposition [Proposition 20](#exist2){reference-type="ref" reference="exist2"} in Appendix [8.2](#appendix3){reference-type="ref" reference="appendix3"}, where the following proposition is needed. **Proposition 21** ([@LiLin-16]). *Let $M(t)=\int_{0}^{t} \kappa_{ij}(s){\rm d}\langle B_{i},B_{j} \rangle (s)-\int_{0}^{t} 2G(\bm \kappa){\rm d}s,$ where ${\bm \kappa} \in M_{G}^{1}([0,T];\mathbb{S}^{n})$. Then, we have $M(t)\leq 0$ quasi-surely. Particularly $\mathbb{\hat{E}}[M(t)]\leq 0$.* [In addition, we present the following $G$-stochastic Barbalat's lemma that will be used later, and its proof is provided in Appendix [8.3](#appendix_barbalat){reference-type="ref" reference="appendix_barbalat"}]{style="color: black"}. **Lemma 22**. *[Suppose that Assumption [Assumption 15](#asump1){reference-type="ref" reference="asump1"} holds and $\tau_{\infty}=+\infty$ quasi-surely, where $\tau_\infty$ is defined in Proposition [Proposition 17](#exist){reference-type="ref" reference="exist"}. Also suppose that the solution to Eq. [\[GSDE\]](#GSDE){reference-type="eqref" reference="GSDE"} satisfies $\sup_{t\in \mathbb{R}^{+}}\vert \bm x(t) \vert<+\infty \ q.s.$. Besides, there exists $\eta\in C(\mathbb{R}^{d}; \mathbb{R}_{+})$ such that $$\label{integralbound} \int_{0}^{+\infty}\eta(\bm x(t)){\rm d}t<+\infty, \ \ q.s..$$ Then, we have $\lim_{t\rightarrow +\infty}\eta(\bm x(t))=0$ quasi-surely.]{style="color: black"}* Now, with the following assumption, we state our main theorem. **Assumption 23**. *For each $N>0$, $t \in \mathbb{R}_{+}$ and all $\vert \bm x\vert \leq N$, there exists a number $K_{N}>0$ such that $\vert {\bm f}({\bm x},t)\vert +\vert {\bm g}({\bm x},t)\vert +\vert {\bm h}({\bm x},t)\vert \leq K_{N}.$* **THEOREM 24**. *Suppose that Assumptions [Assumption 15](#asump1){reference-type="ref" reference="asump1"} and [Assumption 23](#asump3){reference-type="ref" reference="asump3"} hold. Also suppose that there exist three functions $V \in C^{2,1}(\mathbb{R}^{d}\times \mathbb{R}_{+}; \mathbb{R}_{+})$, $\gamma \in L^{1}(\mathbb{R}_{+}; \mathbb{R}_{+})$ and $\eta\in C(\mathbb{R}^{d}; \mathbb{R}_{+})$ such that $({\rm UB})$ $\lim_{\vert x\vert \rightarrow \infty} \inf_{0\leq t <+\infty} V({\bm x},t)=\infty$ and $\mathcal{L}V({\bm x},t)\leq \gamma(t)-\eta({\bm x})$. Then, we have that $\lim_{t \rightarrow +\infty}V({\bm x}(t),t)$ finitely exists quasi-surely and that $$\label{limitofeta} \lim_{t \rightarrow +\infty}\eta({\bm x}(t))=0 \ \ q.s..$$ Moreover, $\lim_{t \rightarrow +\infty}d(\bm x, {\rm Ker}(\eta))=0$, where $d(\bm x, {\rm Ker}(\eta)):= \inf_{{\bm y}\in {\rm Ker}(\eta)}\vert \bm x- \bm y \vert$.* Using Proposition [Proposition 20](#exist2){reference-type="ref" reference="exist2"}, the $G$-SDEs satisfying the conditions assumed in this theorem have a global solution on $[0,+\infty)$ with a property that $\mathcal{L}V({\bm x},t)\leq \gamma(t)-\eta({\bm x})\leq \gamma(t).$ By $G$-Itô's formula in Theorem [THEOREM 19](#GIto){reference-type="ref" reference="GIto"}, Proposition [Proposition 17](#exist){reference-type="ref" reference="exist"} and Remark [Remark 18](#rk4.4){reference-type="ref" reference="rk4.4"}, we have $$\begin{aligned} &&V({\bm x}(t\wedge \tau_{N}),t\wedge \tau_{N})=V({\bm x}_{0},0)+\int_{0}^{t\wedge \tau_{N}} V_{t}({\bm x}(s),s){\rm d}s\\ &&+\int_{0}^{t\wedge \tau_{N}} V_{x_{i}}({\bm x}(s),s)f^{i}({\bm x}(s),s){\rm d}s + \int_{0}^{t\wedge \tau_{N}} V_{x_{i}}({\bm x}(s),s)g^{ij}({\bm x}(s),s){\rm d}B_{j}(s)\\ &&+ \int_{0}^{t\wedge \tau_{N}} V_{x_{k}}({\bm x}(s),s)h^{kij}({\bm x}(s),s){\rm d}\langle B_{i}, B_{j} \rangle(s) +\int_{0}^{t\wedge \tau_{N}}\frac{1}{2} V_{x_{k}x_{l}}({\bm x}(s),s)g^{ki}({\bm x}(s),s)\\ &&g^{lj}({\bm x}(s),s){\rm d}\langle B_{i}, B_{j} \rangle(s),\end{aligned}$$ where $\tau_{N}:=\inf \{t\geq 0:\vert {\bm x(t)}\vert \geq N\}$. Letting $N \rightarrow+\infty$ and setting $\bm \kappa=(\kappa_{ij})_{i,j=1}^{m}$ for every $t\geq 0$ where ${\kappa}_{ij}=V_{x_{k}}(h^{kij}+h^{kji})+V_{x_{k}x_{l}}g^{ki}g^{lj},$ we get that $\tau_{N}$ tends to $+\infty$ by Proposition [Proposition 17](#exist){reference-type="ref" reference="exist"} and $$\begin{aligned} V({\bm x}(t),t)&=&V({\bm x}_{0},0)+\int_{0}^{t} V_{t}({\bm x}(s),s){\rm d}s+\int_{0}^{t} V_{x_{i}}({\bm x}(s),s)f^{i}({\bm x}(s),s){\rm d}s \\ &&+ \int_{0}^{t} V_{x_{i}}({\bm x}(s),s)g^{ij}({\bm x}(s),s){\rm d}B_{j}(s)+\int_{0}^{t} \frac{1}{2}\kappa_{ij}({\bm x}(s),s) {\rm d}\langle B_{i},B_{j} \rangle(s). \end{aligned}$$ Thus, if we set $$V({\bm x}(t),t)=V({\bm x}_{0},0) + \int_{0}^{t} \gamma(s){\rm d}s-A_{2}(t) +\int_{0}^{t} V_{x_{i}}({\bm x}(s),s)g^{ij}({\bm x}(s),s){\rm d}B_{j}(s),$$ then $A_{2}(0)=0$. Besides, according to Proposition [Proposition 21](#prop5){reference-type="ref" reference="prop5"}, for every $0\leq t_{1}<t_{2}<+\infty$, we have $$\begin{aligned} && A_{2}(t_{2})-A_{2}(t_{1})=\int_{t_{1}}^{t_{2}} \gamma(s){\rm d}s-\int_{t_{1}}^{t_{2}} V_{x_{i}}({\bm x}(s),s)f^{i}({\bm x}(s),s){\rm d}s\\ && -\int_{t_{1}}^{t_{2}} V_{t}({\bm x}(s),s){\rm d}s-\int_{t_{1}}^{t_{2}} \frac{1}{2}\kappa_{ij}({\bm x}(s),s) {\rm d}\langle B_{i},B_{j} \rangle (s)\\ &\geq& \int_{t_{1}}^{t_{2}} \gamma(s){\rm d}s-\int_{t_{1}}^{t_{2}} V_{t}({\bm x}(s),s){\rm d}s\\ &&-\int_{t_{1}}^{t_{2}} V_{x_{i}}({\bm x}(s),s)f^{i}({\bm x}(s),s){\rm d}s-\int_{t_{1}}^{t_{2}} G(\eta ({\bm x}(s),s)) {\rm d}s\\ &=&\int_{t_{1}}^{t_{2}} \gamma(s){\rm d}s- \int_{t_{1}}^{t_{2}} \mathcal{L} V({\bm x}(s),s){\rm d}s \geq\int_{t_{1}}^{t_{2}} \eta(s){\rm d}s \geq 0\end{aligned}$$ which implies that $A_{2}(t)$ is a non-decreasing process. Using Proposition [Proposition 17](#exist){reference-type="ref" reference="exist"}, we obtain that $\int_{0}^{t\wedge \tau_{N}} V_{x_{i}}({\bm x}(s),s)g^{ij}({\bm x}(s),s){\rm d}B_{j}(s)$ is a $Q$-martingale for every $Q \in \mathcal{Q}$. Noticing $\int_{0}^{+ \infty} \gamma(s){\rm d}s<+\infty$ and according to Proposition [THEOREM 14](#semimartinage2){reference-type="ref" reference="semimartinage2"}, we have a set ${\Omega_{0}}\subset \Omega$ such that $c({\Omega}\backslash\Omega_{0})=0$. Then, we have that, for all $\omega \in \Omega_{0}$, $\lim_{n \rightarrow +\infty}A_{2}(t) \ {\rm finitely} \ {\rm exists}$ and $\lim_{n \rightarrow +\infty}V({\bm x}(t),t)$ finitely exists. Thus, on ${\Omega_0}$, $\int_{0}^{+\infty}\eta(\bm x(t)){\rm d}t<+\infty.$ From the finite existence of the limit of $V$, we obtain that, on $\Omega_0$, $\sup_{t \geq 0}V({\bm x}(t;\omega),t)<+\infty$. Hence, from the above-assumed condition (UB), it follows that there exists $K(\omega)$ such that $\sup_{t\geq 0}\vert {\bm x}(t;\omega)\vert \leq K(\omega)$. [According to Lemma [Lemma 22](#theorembarbalat){reference-type="ref" reference="theorembarbalat"}, we obtain $\lim_{t\rightarrow +\infty} \eta(\bm x(t))=0$ quasi-surely.]{style="color: black"} For every $\omega$ satisfying $\lim_{t\rightarrow +\infty}\eta(\bm x(t;\omega))=0$ and $\sup_{t \in \mathbb{R}_{+}}\vert \bm x(t;\omega)\vert <+\infty$, there exists $\bm y(\omega)$ and a sequence $\{t_{i}\}$ having $\lim_{i\rightarrow +\infty}\bm x(t_{i};\omega)=\bm y(\omega)$. So, $\lim_{i\rightarrow +\infty}\eta(\bm x(t_{i};\omega))=\bm \eta(\bm y(\omega))=0$ and ${\rm Ker}(\eta)\neq \emptyset$. If $\limsup_{t\rightarrow +\infty} d(\bm x(t;\omega),{\rm ker}(\eta))$ is positive, there exist a sequence $\{t_{i}\}$ such that $d(\bm x(t_{i};\omega),{\rm ker}(\eta))\geq \epsilon,$ for some $\epsilon>0$. This implies $\eta(\bm y)>0$, which is a contradiction. **Remark 25**. *[Here, our conclusions nontrivially extend the corresponding results obtained for the traditional SDEs. Particularly, the significant differences do exist. First, in terms of the conclusions, we are able to induce relevant results even when the system randomness itself is uncertain, greatly surpassing the applicability scope of existing Brownian motion-driven stochastic systems. From a technical standpoint, our generalized stochastic differential equation (i.e., G-SDE) cannot measure the occurrence probability of events from the perspective of traditional probability measures, but the capacities instead. Second, the construction of the monotone functions in our semi-martingales differs significantly from the invariance principles in the traditional stochastic analysis. ]{style="color: black"}* Next, we present another version of invariance principle, where $\eta$ is a function with respect to the function $V$. **THEOREM 26**. *Suppose that Assumption [Assumption 15](#asump1){reference-type="ref" reference="asump1"} holds, and that there exist three functions $V \in C^{2,1}(\mathbb{R}^{d}\times \mathbb{R}_{+}; \mathbb{R}_{+})$, $\gamma \in L^{1}(\mathbb{R}_{+}; \mathbb{R}_{+})$ and $\eta\in C(\mathbb{R}_{+} ;\mathbb{R}_{+})$ such that $\mathcal{L}V({\bm x},t)\leq \gamma(t)-\eta(V({\bm x},t))$ for all $({\bm x},t)\in \mathbb{R}^{d}\times \mathbb{R}_{+}$. Then, we obtain that $\lim_{t \rightarrow +\infty}V({\bm x}(t),t)$ finitely exists quasi-surely and $\lim_{t \rightarrow +\infty}\eta(V({\bm x}(t),t))=0 \ q.s..$ Moreover, $\lim_{t \rightarrow +\infty}d( V(\bm x(t),t), {\rm Ker}(\eta))=0.$* Analogously, the $G$-SDEs have a global solution on $[0,+\infty)$ according to Proposition [Proposition 20](#exist2){reference-type="ref" reference="exist2"}. By the arguments akin to those for validating Theorem [THEOREM 24](#lasalle1){reference-type="ref" reference="lasalle1"}, we obtain $V({\bm x}(t),t) = V({\bm x}_{0},0) + \int_{0}^{t} \gamma(s){\rm d}s-A_{2}(t) +\int_{0}^{t} V_{x_{i}}({\bm x}(s),s)g^{ij}({\bm x}(s),s){\rm d}B_{j}(s),$ where $A_{2}(0)=0$ and for every $0\leq t_{1}<t_{2}<+\infty$, $$\begin{aligned} &&A_{2}(t_{2})-A_{2}(t_{1})=\int_{t_{1}}^{t_{2}} \gamma(s){\rm d}s-\int_{t_{1}}^{t_{2}} V_{x_{i}}({\bm x}(s),s)f^{i}({\bm x}(s),s){\rm d}s\\ &&-\int_{t_{1}}^{t_{2}} V_{t}({\bm x}(s),s){\rm d}s-\int_{t_{1}}^{t_{2}} \frac{1}{2}\kappa_{ij}({\bm x}(s),s) {\rm d}\langle B_{i},B_{j} \rangle(s) \\\end{aligned}$$ $$\begin{aligned} &\geq& \int_{t_{1}}^{t_{2}} \gamma(s){\rm d}s-\int_{t_{1}}^{t_{2}} V_{x_{i}}({\bm x}(s),s)f^{i}({\bm x}(s),s){\rm d}s\\ &&-\int_{t_{1}}^{t_{2}} V_{t}({\bm x}(s),s){\rm d}s-\int_{t_{1}}^{t_{2}} G(\eta ({\bm x}(s),s)) {\rm d}s\\ &=&\int_{t_{1}}^{t_{2}} \gamma(s){\rm d}s- \int_{t_{1}}^{t_{2}} \mathcal{L} V({\bm x}(s),s){\rm d}s \geq\int_{t_{1}}^{t_{2}} \eta(V({\bm x}(s),s)){\rm d}s \geq 0.\end{aligned}$$ Hence, by the $G$-semimartingale Convergence Theorem [THEOREM 14](#semimartinage2){reference-type="ref" reference="semimartinage2"}, there exists $\bar{\Omega}\subset \Omega$ such that $c({\Omega}\backslash \bar{\Omega})=0$. Furthermore, we have that, on $\bar{\Omega}$, $$\int_{0}^{\infty} \eta(V({\bm x}(t),t)) {\rm d}t<+\infty~~\mbox{and}~~ \lim_{n \rightarrow +\infty}V({\bm x}(t),t) \ {\rm finitely} \ {\rm exists} .$$ Now, we claim that, for every $\omega\in \bar{\Omega}$, we have $\lim_{t \rightarrow +\infty}\eta(V({\bm x}(t;\omega),t))=0$. We validate the claim by contradiction. If this is not the case, then we have a sequence $\{t_{k}\}$ with $t_{k+1}-t_{k}>1$ and $\epsilon>0$, such that $\eta(V({\bm x}(t_{k};\omega),t_{k}))>\epsilon$. Assume $\sup_{t\geq 0}V({\bm x}(t;\omega),t)\leq K(\omega)$. Hence, there exists $\delta_{1}$ such that $\vert \eta(x)-\eta(y)\vert \leq \frac{\epsilon}{2}$ for $0\leq x,y\leq K(\omega)$ and $\vert x-y\vert \leq\delta_{1}$. As $\lim_{t\rightarrow +\infty}V({\bm x}(t;\omega),t)$ finitely exists and $V({\bm x}(t;\omega),t)$ is continuous about $t$, we can easily check that it is uniformly continuous on $\mathbb{R}^+$. Thus, there exists $\delta_{2}<1$ such that $\vert V({\bm x}(t;\omega),t)-V({\bm x}(s;\omega),s)\vert < \delta_{1}, \ \ \vert t-s\vert < \delta_{2}.$ Consequently, for $t_{k}\leq t < t_{k}+\delta_{2}$, we have $\eta(V({\bm x}(t;\omega),t))\geq \eta(V({\bm x}(t_{k};\omega),t_{k}))-\vert \eta(V({\bm x}(t_{k};\omega),t_{k})) -\eta(V({\bm x}(t;\omega),t))\vert\geq \frac{\epsilon}{2}.$ Therefore, $+\infty>\int_{0}^{\infty} \eta(V({\bm x}(t),t)) {\rm d}t\geq \sum_{k=1}^{+\infty}\int_{t_k}^{t_k+\delta_{2}} \eta(V({\bm x}(t),t)) {\rm d}t \geq \sum_{k=1}^{+\infty} \frac{\epsilon \delta_{2}}{2}=+\infty,$ which indicates a contradiction. Finally, the arguments for proving $\lim_{t \rightarrow +\infty}d( V(\bm x(t),t), {\rm Ker}(\eta))=0$ are the same as those for validating the last conclusion in Theorem [THEOREM 24](#lasalle1){reference-type="ref" reference="lasalle1"}. **Remark 27**. *A set $\mathcal{A} \in \mathscr{B}(\Omega)$ is said to be invariant if $c\big(\{\exists t\geq 0, ~x(t;{\bm x}_{0})\notin \mathcal{A} \}\big)=0,$ for every ${\bm x}_{0} \in \mathcal{A}$. Actually, if we suppose some conditions to be valid only in the invariant set $\mathcal{A}$ for Theorems $\ref{lasalle1}$ and $\ref{lasalle2}$, the conclusions in these theorems still sustain.* Finally, we present two corollaries which can be obtained directly form the invariance principles established above. These results are related to the stability or the exponential stability of the solution $\bm x(t)$. **Corollary 28**. *Let Assumption [Assumption 15](#asump1){reference-type="ref" reference="asump1"} hold. Assume further that there exists a function $V \in C^{2,1}(\mathbb{R}^{d}\times \mathbb{R}_{+}; \mathbb{R}_{+})$ such that $$\label{cor1} \mu_{1}(\vert {\bm x}\vert )\leq V({\bm x},t)\leq \mu_{2}(\vert {\bm x}\vert ), \ \ \mathcal{L}V({\bm x},t)\leq -\mu_{3}(\vert {\bm x}\vert ),$$ where $\mu_{1}$, $\mu_{2}$ and $\mu_{3}$ are three strictly increasing functions in $[0,+\infty)$ with the initial value $0$ and $\mu_{1}(r), \mu_{2}(r) \rightarrow +\infty$ as $r \rightarrow +\infty$. Then, we have $\lim_{t \rightarrow +\infty}\vert {\bm x}(t)\vert =0 ~q.s.$.* From the condition assumed in [\[cor1\]](#cor1){reference-type="eqref" reference="cor1"}, it follows that $\mu_{2}^{-1}(V({\bm x},t))\leq \vert {\bm x}\vert$, which implies $\mathcal{L}V({\bm x},t)\leq -\mu_{3}(\mu_{2}^{-1}(V({\bm x},t))).$ According to Theorem [THEOREM 26](#lasalle2){reference-type="ref" reference="lasalle2"}, we have $\lim_{t \rightarrow \infty}\mu_{3}(\mu_{2}^{-1}(V({\bm x}(t),t)))=0 ~ q.s.$, which implies $\lim_{t \rightarrow \infty}V({\bm x}(t),t)=0 ~ q.s.$. Therefore, we have $\lim_{t \rightarrow \infty}\mu_{1}(\vert {\bm x}(t)\vert )=0 ~ q.s.$, which finally gives $\lim_{t \rightarrow \infty} \vert {\bm x}(t)\vert =0 ~ q.s.$. **Corollary 29**. *Let Assumption [Assumption 15](#asump1){reference-type="ref" reference="asump1"} hold. Assume further that there exist two functions: $V \in C^{2,1}(\mathbb{R}^{d}\times \mathbb{R}_{+}; \mathbb{R}_{+})$ and $\gamma \in L^{1}(\mathbb{R}_{+};\mathbb{R}_{+} )$, such that ${\rm e}^{\lambda t}\vert {\bm x}\vert ^{p}\leq V({\bm x}(t),t) \ \ \ {\rm and} \ \ \ \mathcal{L}V({\bm x},t)\leq \gamma(t),$ where $\lambda$ and $p$ are positive numbers. Then, we have $\varlimsup_{t \rightarrow +\infty}\frac{1}{t} {\rm log}\vert {\bm x}(t)\vert \leq -\frac{\lambda}{p} \ \ \ q.s..$* Set $\eta=0$ in Theorem [THEOREM 26](#lasalle2){reference-type="ref" reference="lasalle2"}. Then, $\lim_{t \rightarrow +\infty}V({\bm x}(t),t)$ finitely exists quasi-surely. Further use the condition that ${\rm e}^{\lambda t}\vert {\bm x}\vert ^{p}\leq V({\bm x}(t),t)$. The proof is therefore complete. # Illustrative Examples: Applying $G$-invariance principle to achieving $G$-stochastic control {#example} In this section, we use several representative examples to illustrate the applicability of our analytical results to realizing $G$-stochastic control of the unstable dynamical systems. **Example 30**. *Consider a linear (complex network) system ${\rm d}{\bm x}(t)={\bm A}{\bm x}(t){\rm d}t$. Here, $\bm{A}=[11,5,2;5,11,2;2,2,14]$. Then, it is easy to check that ${\lambda}_{\rm max}({\bm A})=18$ and the system is unstable. Now, for a $G$-Brownian motion where $\underline{\sigma}^2=3.5$ and $\overline{\sigma}^2=4$, we choose ${\bm D}={\bm I}_{3}$ and $\bm C = [-19,11,2;11,-19,2;2,2,-10]$ to $G$-stochastically control the linear system as ${\bm x}(t)={\bm x}_{0}+\int_{0}^{t} {\bm A}{\bm x}(s){\rm d}s + \int_{0}^{t} {\bm D}{\bm x}(s) {\rm d}{\bm B}(s)+\int_{0}^{t} {\bm C}{\bm x}(s) {\rm d}\langle {\bm B}\rangle(s).$ Choosing $V({\bm x}):=\vert {\bm x}\vert ^2$ yields: $\mathcal{L}V({\bm x})=2{\bm x}^{\top}{\bm A}{\bm x}+G(2{\bm x}^{\top}{\bm D}^{\top}{\bm D}{\bm x}+4{\bm x}^{\top}{\bm C}{\bm x}).$ As ${\lambda}_{\rm max}({\bm C})=-6$, we easily derive that $\mathcal{L}V({\bm x})\leq -2.5\vert {\bm x}\vert ^2$. This, according to Corollary [Corollary 28](#cor4.8){reference-type="ref" reference="cor4.8"}, ensures the asymptotic stability of the controlled system in a quasi-sure sense.* *Moreover, if we set $V({\bm x},t)={\rm e}^{\lambda t}\vert {\bm x}\vert ^2$, we obtain that $\mathcal{L}V({\bm x},t)= \mathcal{L}V({\bm x})=\left[{\bm x}^{\top}(2{\bm A}+\lambda {\bm I}_{d}){\bm x}+G(2{\bm x}^{\top}{\bm D}^{\top}{\bm D}{\bm x}+4{\bm x}^{\top}{\bm C}{\bm x})\right]{\rm e}^{\lambda t},$ which, using the parameters $\underline{\sigma}^2=3.5$ and $\overline{\sigma}^2=4$, yields $\mathcal{L}V({\bm x},t)\leq (\lambda-1.5)\vert {\bm x}\vert ^{2}$. If we set $\lambda \leq 1.5$, using Corollary [Corollary 29](#cor2){reference-type="ref" reference="cor2"} gives $\varlimsup_{t \rightarrow +\infty}\frac{1}{t} {\rm log}\vert {\bm x}(t)\vert \leq -0.75 \ q.s.$. This clearly illustrates the exponential stability of the controlled system.* **Example 31**. *Consider an autonomous system, which reads ${\rm d}{\bm x}(t)={\bm f}({\bm x}(t)){\rm d}t$. Here, ${\bm f}$ satisfies Assumption [Assumption 15](#asump1){reference-type="ref" reference="asump1"} and $\bm f(\bm 0)=\bm 0$. Moreover, $\bm f$ satisfies one-sided Lipschitz condition, i.e., there exists a number $L>0$ such that $\langle \bm x,\bm f(\bm x) \rangle \leq L \vert \bm x \vert^{2}.$ There are many systems, not globally Lipschitzian, only satisfying this one-sided Lipschitz condition. For instance, both $f(x)=x-x^3$ and the Lorenz system with $\bm{f}(\bm x)=[ \sigma x_{2}-\sigma x_{1}, \rho x_{1}-x_{3}x_{1}-x_{2}, x_{1}x_{2}-\beta x_{3} ]^{\top}$ satisfy the one-sided Lipschitz condition. Now, we apply the $G$-stochastic control to the original dynamics, which yields ${\rm d}{\bm x}(t)={\bm f}({\bm x}(t)){\rm d}t+k\sum_{j=1}^{m}{\bm x}(t){\rm d}{B}_{j}(t)$ with $k>\left(-{L}/{c_{-1}}\right)^{{1/2}}$ with $c_{-1}:=G\left((-1)_{i,j=1}^{m}\right)$. Here, $(-1)_{i,j=1}^{m}$ corresponds to an $m \times m$ matrix with all elements are $-1$. Then, the controlled system becomes stochastically stable, whose proof is included in Appendix [8.4](#app4){reference-type="ref" reference="app4"}. Take the three-dimensional Lorenz system for example. We are able to use a one-dimensional $G$-Brownian motion to render the controlled system stable quasi-surely, if we set $m=1$, $c_{-1}=G(-1)=-\frac{1}{2}\underline{\sigma}^{2}$, $L\leq \frac{1}{2}(\sigma +\rho)$, and $k>(\sigma +\rho)^{1/2}\underline{\sigma}^{-1}$.* **Example 32**. *Consider an oscillating system ${\rm d}{\bm x}(t)=\bm C \bm f(\bm x(t)){\rm d}t$, where $\bm C = [1,1,4;5,-1,4;8,1,0]$ and $\bm f(\bm x)=[-x_{1},\arctan (x_{2}), \tanh (x_{3})]^{\top}$. Now, we consider the $G$-stochastically controlled system as ${\rm d}{\bm x}(t)=\bm C \bm f(\bm x(t)){\rm d}t+ \bm g(\bm x(t)){\rm d}{\bm B}(t)$, where ${\bm B}$ is a two-dimensional, independent and identically distributed $G$-Brownian motion with $\bar{\sigma}^{2}=50$ and $\underline{\sigma}^{2}=40$, and $\bm g(\bm x)=[\bm A_{1}\bm x, \bm A_{2}\bm x]$ in which ${\bm A_{1}}=[1,0.5,0;0,1,0;0,0,1]$ and ${\bm A_{2}}=[1,0,0;0,1,0.5;0,0,1]$. Additionally, the $G$-function of $\bm B$ satisfies $\bm G(\bm M)=\sum_{j=1}^{2}G_{j}(a_{jj})$, where $\bm M=(m_{ij})_{i,j=1}^{2}$ is a two-dimensional matrix, and $G_{j}$ is the $G$-function related to the one-dimensional $G$-Brownian motion $B_{j}$. Set $V(\bm x)=\vert \bm x \vert^{\alpha}$ for some $\alpha>0$. By Appendix [8.5](#appendix2){reference-type="ref" reference="appendix2"}, $\mathbb{R}^{3}\backslash \{\bm 0\}$ is an invariant set of the system. It follows that, on $\mathbb{R}^{3}\backslash \{\bm 0\}$, $$\begin{aligned} \mathcal{L}V(\bm x) &=&\alpha \vert \bm x\vert^{\alpha-2}\big[-x_{1}^{2}+x_{1}\arctan (x_{2})+4x_{1}\tanh (x_{3})-5x_{1}x_{2}-x_{2}\arctan (x_{2})\\ & &+4x_{2}\tanh(x_{3})-8x_{3}x_{1}+x_{3}\arctan (x_{2})\big]\\ & & +\alpha \vert \bm x\vert^{\alpha-4} G\left(\vert \bm x\vert^{2}\bm g^{\top}\bm g+(\alpha-2)\bm g^{\top}\bm x \bm x^{\top}\bm g\right)\\ &\leq &\alpha \vert \bm x\vert^{\alpha-2}(-x_{1}^{2}+6\vert x_{1}x_{2}\vert + 12\vert x_{1}x_{3}\vert + 5\vert x_{2}x_{3}\vert) \\ &&+\sum_{j=1}^{2}\alpha \vert \bm x\vert^{\alpha-4}G_{j}\left(\vert \bm x\vert^{2}\vert \bm A_{j}\bm x\vert^{2}+ (\alpha-2)(\bm x^{\top} \bm A_{j}\bm x)^{2}\right).\end{aligned}$$ Notice that $(\bm x^{\top}\bm A_{j}\bm x)^{2}\geq \frac{1}{2} \vert \bm x \vert^{2}\vert \bm A_{j}\bm x \vert^{2}+\frac{1}{8} \vert \bm x \vert^{4}$ and $\bm x^{\top}\bm A_{j}\bm x \leq \frac{5}{4} \vert \bm x \vert^{2}$ for $j=1,2,$ and set $\alpha = \frac{2}{25}$. Then, we obtain $\mathcal{L}V(\bm x)\leq \frac{17}{25} \vert \bm x\vert^{\frac{2}{25}}+\sum_{j=1}^{2}\frac{2}{25}\vert \bm x\vert^{-\frac{98}{25}}G_{j}\Big(\frac{2}{25}(\bm x^{\top} \bm A_{j}\bm x)^{2}-\frac{1}{4} \vert \bm x\vert^{4}\Big )\leq -\frac{3}{25}\vert \bm x\vert^{\frac{2}{25}}$. Setting $\eta$ in Theorem [THEOREM 24](#lasalle1){reference-type="ref" reference="lasalle1"} as $\eta(\bm x)=\frac{3}{25}\vert \bm x\vert^{\frac{2}{25}}$ guarantees the quasi-sure stability of the above controlled system.* In Appendix [8.6](#appendix_numerical){reference-type="ref" reference="appendix_numerical"}, we further provide a few numerical evidences for illustrating the above examples. It is emphasized that those numerically-presented results do not represent all the exact solution produced by the $G$-SDEs, but only provide some evidences partially supporting the analytical results obtained in the above examples. The numerical scheme used there is not complete, so it awaits further development for rigorously approximating the solution of $G$-SDEs. # Conclusion {#discussion} In this article, we have developed several invariance principles for the stochastic differential equations driven by the $G$-Brownian motions. Our work is basically inspired by the seminal works from two directions: one is from the stability theory of the traditional SDEs [@Mao-7] and the other is from the fundamentally-innovative works on the sublinear expectation [@Peng-13]. Our contributions include not only the establishment of the $G$-semimartingale convergence theorem and its variants for the sublinear expectation, but also the establishment of several invariance principles and their applications in investigating the long-term behaviors of $G$-SDEs. Indeed, we anticipate that our analytical results can be beneficial to understanding and solving the problems associated with uncertain randomness in dynamical systems. As for the future research directions, the assumption on the linear growth and the locally Lipschitz conditions can be further weakened through restricting the discussion for the operator $\mathcal{L}$ in some specific space. Also, further development of the invariance principles for the $G$-SDDEs and the $G$-SFDEs could be promoted. More practically, complete scheme for rigorously approximating the solution produced by the $G$-SFDEs deserves deep investigation. # Appendix ## Proof of Proposition [Proposition 12](#convergence){reference-type="ref" reference="convergence"} {#appendix1} [First, we establish Fatou's lemma for the $G$-conditional expectation, which is a prerequisite for our proposition to be demonstrated]{style="color: black"}. **Lemma 33** (Fatou's Lemma for $G$-conditional Expectation). *$\{X(n)\}\in L_{G}^{1}(\Omega)$ are a series of random vectors, and there exists a random variable $M$ such that $\mathbb{\hat{E}}[\vert M\vert ]<+\infty$ and $X(n)\geq M$ for any $n>0$. Then, $\mathbb{\hat{E}}_{t}\left[\varliminf_{n\rightarrow \infty}X(n)\right]\leq \varliminf_{n\rightarrow \infty}\mathbb{\hat{E}}_{t}[X(n)].$* In order to present the proof for this lemma, we need to extend the space of random variables and make some necessary preparations. **Definition 34** ([@HuPeng-29]). *Introduce some extended spaces of random variables as follows:* *$\begin{array}{l} \mathcal{L}_{G}^{{1}^{*}}(\Omega):=\Big\{X\in L^{0}(\Omega): \exists X({n})\in {L}_{G}^{1}(\Omega) \ {\rm such} \ {\rm that} \ X({n}) \downarrow X \Big\}, \\ {L}_{G}^{{1}^{*}}(\Omega):=\Big\{X\in L^{0}(\Omega): \mathbb{\hat{E}}[\vert X\vert ]<+\infty, \ \ X \in \mathcal{L}_{G}^{{1}^{*}}(\Omega) \Big\},\\ \mathcal{L}_{G}^{{1}^{*}_{*}}(\Omega):= \Big\{X\in L^{0}(\Omega): \exists X({n})\in L_{G}^{1^{*}}(\Omega) \ {\rm such} \ {\rm that} \ X({n}) \uparrow X \Big\},\\ {L}_{G}^{{1}^{*}_{*}}(\Omega):=\Big\{X\in L^{0}(\Omega): \mathbb{\hat{E}}[ \vert X\vert ]<+\infty, \ \ X \in \mathcal{L}_{G}^{{1}^{*}_{*}}(\Omega) \Big\}. \end{array}$* *Then, we extend the $G$-conditional expectation on $\mathcal{L}_{G}^{{1}^{*}_{*}} (\Omega)$. Directly, we have ${L}_{G}^{{1}^{*}}(\Omega) \subset \mathcal{L}_{G}^{{1}^{*}}(\Omega) \subset \mathcal{L}_{G}^{{1}^{*}_{*}}(\Omega)$ and ${L}_{G}^{{1}^{*}}(\Omega) \subset {L}_{G}^{{1}^{*}_{*}}(\Omega) \subset \mathcal{L}_{G}^{{1}^{*}_{*}}(\Omega)$.* **Lemma 35** ([@HuPeng-29]). *Suppose that $\{X(n)\}\subset L_{G}^{1^{*}_{*}}(\Omega)$ is a series of non-decreasing random variables. Denote by $X:=\lim_{n \to \infty}X(n)$. Then, we have quasi-surely $\lim_{n \to \infty}\mathbb{\hat{E}}_{t}[X(n)]=\mathbb{\hat{E}}_{t}[X].$* **Lemma 36**. *If $X, Y \in L_{G}^{1}(\Omega)$, then $X\wedge Y \in L_{G}^{1}(\Omega)$ $($resp. $X\vee Y \in L_{G}^{1}(\Omega))$.* As $X, Y \in L_{G}^{1}(\Omega)$, there exists $\{X_{n}\}$ and $\{Y_{n}\}$ contained in ${\rm Lip}(\Omega)$ such that $\mathbb{\hat{E}}[\vert X(n)-X\vert ]\rightarrow 0$ and $\mathbb{\hat{E}}[\vert Y(n)-Y\vert ]\rightarrow 0$. For $\varphi$, $\psi \in C_{l,{\rm Lip}}(\Omega)$, we have $\varphi \wedge \psi=\frac{\varphi+\psi-\vert \varphi-\psi\vert }{2}\in C_{l,{\rm Lip}}(\Omega).$ Thus, $X(n)\wedge Y(n) \in {\rm Lip}(\Omega)$. So we derive $\mathbb{\hat{E}}[\vert X\wedge Y-X(n)\wedge Y(n)\vert ]\leq \mathbb{\hat{E}}[\vert X-X(n)\vert ]+\mathbb{\hat{E}}[\vert Y-Y(n)\vert ]\rightarrow 0,$ which implies $X\wedge Y \in L_{G}^{1}(\Omega)$. The case that $X\vee Y \in L_{G}^{1}(\Omega)$ is analogous. **Lemma 37**. *If $X(n)\in L_{G}^{1}(\Omega)$ and $X(n)$ converges to $X$, and there exists a random variable $M$ such that $\mathbb{\hat{E}}[\vert M\vert ]<+\infty$ and $X(n)\geq M$ for any $n>0$. Then, $X\in \mathcal{L}_{G}^{{1}^{*}_{*}}(\Omega).$* For any $m,n>0$, by Lemma [Lemma 36](#smalllemma1){reference-type="ref" reference="smalllemma1"}, we obtain that $\inf_{n \leq k\leq m} X(k) \in L_{G}^{1}(\Omega)$. Then, from Definition [Definition 34](#remark1){reference-type="ref" reference="remark1"}, it follows that $\inf_{k\geq n} X(k) \in \mathcal{L}_{G}^{{1}^{*}}(\Omega)$. Also, by the fact that $M\leq\inf_{k\geq n} X(k)\leq X(n)$, we have $\left\vert \inf_{k\geq n} X(k)\right\vert \leq \vert X(n)\vert +\vert M\vert .$ Thus, $\mathbb{\hat{E}}[\vert \inf_{k\geq n} X(k)\vert ]\leq +\infty$ and $\inf_{k\geq n} X(k) \in{L}_{G}^{{1}^{*}}(\Omega)$ using Definition [Definition 34](#remark1){reference-type="ref" reference="remark1"}. As $X=\lim_{n\rightarrow +\infty}\inf_{k\geq n} X(k)$, we immediately obtain the conclusion using Definition [Definition 34](#remark1){reference-type="ref" reference="remark1"}. Set $Y(n):=\inf_{k\geq n}\mathbb{\hat{E}}_{t}[X(k)]$. Using the arguments analogous to those performed in Lemma [Lemma 37](#lemma3){reference-type="ref" reference="lemma3"}, we get $Y(n) \in {L}_{G}^{1^{*}}(\Omega)$. According to Lemma [Lemma 35](#lemma2.1){reference-type="ref" reference="lemma2.1"}, we obtain $\lim_{n \rightarrow \infty}\mathbb{\hat{E}}_{t}[Y(n)]=\mathbb{\hat{E}}_{t}[\lim_{n \rightarrow \infty}Y(n)]$. Because of $Y(n)\leq X(n)$, we derive $\mathbb{\hat{E}}_{t}[Y(n)]\leq\mathbb{\hat{E}}_{t}[X(n)]$ and $\lim_{n \rightarrow \infty}\mathbb{\hat{E}}_{t}[Y(n)]\leq \varliminf_{n\rightarrow \infty}\mathbb{\hat{E}}_{t}[X(n)],$ which implies $\mathbb{\hat{E}}_{t}[\varliminf_{n\rightarrow \infty}X(n)]\leq \varliminf_{n\rightarrow \infty}\mathbb{\hat{E}}_{t}[X(n)]$ we expect. Now, we are in a position to prove the $G$-martingale convergence theorem step-by-step using the uppercrossing inequality. **Definition 38**. *A random time $\tau: \Omega\rightarrow [0,+\infty)$ is called an $\mathbb{F}$-stopping time, if $\{\tau \leq t\}\in \mathscr{F}_{t}$ for every $t\geq 0$.* **Definition 39**. *For a finite subset $F\subset [0,+\infty)$, the interval $[\alpha,\beta]$ and the process $M=\{M(t)\}$ with $M(t)\in L_{G}^{1}(\Omega)$, we define the a series of $\mathbb{F}$-stopping times recursively by: $$\begin{array}{l} \tau_{1}(\omega)=\min \left\{t \in F ; M(t;\omega) \leq \alpha\right\},~ \sigma_{j}(\omega)=\min \left\{t \in F ; t \geq \tau_{j}(\omega), \ \ M(t;\omega)\geq\beta\right\},\\ \tau_{j+1}(\omega)=\min \left\{t \in F ; t \geq \sigma_{j}(\omega), \ \ M(t;\omega)\leq\alpha\right\}. \end{array}$$ And the minimum of an empty set is defined as $+\infty$. Let $U_{F}(\alpha, \beta ; M(\omega))$ be the largest number $j$ such that $\sigma_{j}(\omega)<+\infty$. For any general set $I\subset [0,+\infty)$, we define $U_{I}(\alpha, \beta ; M(\omega))=\sup \left\{U_{F}(\alpha, \beta ; M(\omega)) ; F \subseteq I, \ F \ {\text is} \text { finite}\right\}.$* **Proposition 40** (Upcrossing Inequality, A Discrete Version, [@Li-31]). *Assume that $\{-M(n):n=1,2,\cdots,N\}$ is a $G$-supermartingale. If $M(n)\in L_{G}^{1}(\Omega_{n})$, then we have $\mathbb{\hat{E}}[U_{\{1,2,\cdots,N\}}(\alpha, \beta ; M(\omega))]\leq \frac{\mathbb{\hat{E}}[(M(N)-\alpha)^{+}]}{\beta-\alpha}.$* **Lemma 41** (Uppercrossing Inequality, A Continuous Version). *Assume that $\{M(t):t\in[0,+\infty)\}$ is a right- or left-continuous function and $\{-M(t):t\in[0,+\infty)\}$ is a $G$-supermartingale. If $M(t)\in L_{G}^{1}(\Omega_{t})$, then we have that, for any integer $n>0$, $\mathbb{\hat{E}}[U_{[0,n]}(\alpha, \beta ; M(\omega))]\leq \frac{\mathbb{\hat{E}}[(M({n})-\alpha)^{+}]}{\beta-\alpha}.$* Define $A_{j}:=\cup_{1\leq k\leq j}\{ni/k: i=0,1,\cdots, k\}$. Then, the monotone convergence theorem (Theorem [Proposition 10](#monotone){reference-type="ref" reference="monotone"}), together with Definition [Definition 38](#upcrossingd){reference-type="ref" reference="upcrossingd"} and Proposition [Proposition 40](#upcrossing){reference-type="ref" reference="upcrossing"}, immediately yields: $\mathbb{\hat{E}} \left[U_{[0,n]\cap \mathbb{Q}}(\alpha, \beta ; M(\omega))\right]=\lim_{j\rightarrow +\infty} \mathbb{\hat{E}}[U_{A_{j}}(\alpha, \beta ; M(\omega))]\leq \frac{\mathbb{\hat{E}}[(M({n})-\alpha)^{+}]}{\beta-\alpha}.$ Thus, for any sufficiently small $\epsilon>0$, as $M$ is right- or left-continuous, $\mathbb{\hat{E}}\left[U_{[0,n]}(\alpha, \beta ; M(\omega))\right]\leq \mathbb{\hat{E}}\left[U_{[0,n]\cap \mathbb{Q}}(\alpha+\epsilon, \beta-\epsilon ; M(\omega))\right]\leq \frac{\mathbb{\hat{E}}[(M({n})-\alpha)^{+}]}{\beta-\alpha-2\epsilon},$ which validates the conclusion as required due to the arbitrariness of $\epsilon$'s selection. From Lemma [Lemma 41](#upcrossing2){reference-type="ref" reference="upcrossing2"} and Proposition [Proposition 10](#monotone){reference-type="ref" reference="monotone"}, it follows that $$\begin{aligned} & & \mathbb{\hat{E}}[U_{[0,+\infty)}(\alpha, \beta ; -M(\omega))]=\lim_{n\rightarrow +\infty}\mathbb{\hat{E}}[U_{[0,n]}(\alpha, \beta ; -M(\omega))] \leq \sup_{n\in \mathbb{N}}\frac{\mathbb{\hat{E}}[(-M({n})-\alpha)^{+}]}{\beta-\alpha} \leq \\ & & \frac{\sup_{t\geq 0}\mathbb{\hat{E}}[(-M)^{+}({t})]+\vert \alpha\vert }{\beta-\alpha} =\frac{\sup_{t\geq 0}\mathbb{\hat{E}}[M^{-}({t})]+\vert \alpha\vert }{\beta-\alpha} \leq \frac{\mathbb{\hat{E}}[\sup_{t\geq 0}M^{-}({t})]+\vert \alpha\vert }{\beta-\alpha} <+\infty.\end{aligned}$$ So $U_{[0,+\infty)}(\alpha, \beta ; -M(\omega))<+\infty$ quasi-surely. Denote by $A_{\alpha,\beta}:=\big\{U_{[0,+\infty)}(\alpha, \beta ; -M(\omega))=+\infty\big\}.$ Since $\{\omega:-M(t;\omega) \ {\rm does} \ {\rm not} \ {\rm converge}\}\subset \cup_{\alpha,\beta\in \mathbb{Q}}A_{\alpha,\beta}$, $-M(t)$ converges quasi-surely to some $-M(+{\infty})$. Here, $M(+{\infty})$ can be $+\infty$ or $-\infty$. By the fact that $M(t)\geq \inf_{t\geq 0}-M^{-}(t)=-\sup_{t\geq 0}M^{-}(t)$ and Lemma [Lemma 37](#lemma3){reference-type="ref" reference="lemma3"}, we have $M(+{\infty})\in \mathcal{L}_{G}^{{1}^{*}_{*}}(\Omega)$. And by Lemma [Lemma 33](#lemma2.2){reference-type="ref" reference="lemma2.2"}, we further have $$\begin{aligned} \mathbb{\hat{E}}[\vert M(+{\infty})\vert ]&&\leq \varliminf_{n \rightarrow \infty} {\mathbb{\hat{E}}}\left[\left\vert M({n})\right\vert \right]< 2\mathbb{\hat{E}}\left[\sup_{n\in \mathbb{N}}M^{-}({n})\right]+\varliminf_{n \rightarrow \infty}\mathbb{\hat{E}}[M({n})]\\ &&\leq 2\mathbb{\hat{E}}\left[\sup_{t\geq 0}M^{-}(t)\right]+\mathbb{\hat{E}}[M({1})]<\infty.\end{aligned}$$ Thus, $M({+\infty})$, finite quasi-surely, belongs to ${L}_{G}^{{1}^{*}_{*}}(\Omega)$. Finally, by virtue of Lemma [Lemma 33](#lemma2.2){reference-type="ref" reference="lemma2.2"}, we have $\mathbb{\hat{E}}_{t}[M(+{\infty})]\leq \varliminf _{k\rightarrow +\infty}\mathbb{\hat{E}}_{t}[M({t_{k}})]\leq M({t}),$ which completes the proof. ## Proof of Proposition [Proposition 20](#exist2){reference-type="ref" reference="exist2"} {#appendix3} From Propositions [THEOREM 19](#GIto){reference-type="ref" reference="GIto"} and [Proposition 17](#exist){reference-type="ref" reference="exist"}, it follows that $$\begin{aligned} &&V({\bm x}(t\wedge \tau_{N}),t\wedge \tau_{N})=V({\bm x}_{0},0)+\int_{0}^{t\wedge \tau_{N}} V_{t}({\bm x}(s),s){\rm d}s\\ &&+\int_{0}^{t\wedge \tau_{N}} V_{x_{i}}(x(s),s)f^{i}({\bm x}(s),s){\rm d}s + \int_{0}^{t\wedge \tau_{N}} V_{x_{i}}({\bm x}(s),s)g^{ij}({\bm x}(s),s){\rm d}B_{j}(s)\\ &&+ \int_{0}^{t\wedge \tau_{N}} V_{x_{k}}({\bm x}(s),s)h^{kij}({\bm x}(s),s){\rm d}\langle B_{i}, B_{j} \rangle(s)\\ &&+\int_{0}^{t\wedge \tau_{N}}\frac{1}{2} V_{x_{k}x_{l}}({\bm x}(s),s)g^{ki}({\bm x}(s),s) g^{lj}({\bm x}(s),s){\rm d}\langle B_{i}, B_{j} \rangle(s).\end{aligned}$$ Set ${\bm \eta}=(\kappa_{ij})\in M_{G}^{1}([0,T]; \mathbb{S}^{m})$, where ${ \eta}_{ij}=V_{x_{k}}(h^{kij}+h^{kji})+V_{x_{k}x_{l}}g^{ki}g^{lj}$. Using Proposition [Proposition 21](#prop5){reference-type="ref" reference="prop5"} leads us to the calculations as follows: $$\begin{aligned} &&V({\bm x}(t\wedge \tau_{N}),t\wedge \tau_{N})=V({\bm x}_{0},0)+\int_{0}^{t\wedge \tau_{N}} V_{t}({\bm x}(s),s){\rm d}s\\ &&+\int_{0}^{t\wedge \tau_{N}} V_{x_{i}}({\bm x}(s),s)f^{i}({\bm x}(s),s){\rm d}s + \int_{0}^{t\wedge \tau_{N}} V_{x_{i}}({\bm x}(s),s)g^{ij}({\bm x}(s),s){\rm d}B_{j}(s)\\ &&+\int_{0}^{t\wedge \tau_{N}} \frac{1}{2}\kappa_{ij}({\bm x}(s),s) {\rm d}\langle B_{i},B_{j} \rangle \\ &\leq&V({\bm x}_{0},0)+\int_{0}^{t\wedge \tau_{N}} V_{t}({\bm x}(s),s){\rm d}s+\int_{0}^{t\wedge \tau_{N}} G(\bm \eta) {\rm d}s\\ &&+\int_{0}^{t\wedge \tau_{N}} V_{x_{i}}({\bm x}(s),s)f^{i}({\bm x}(s),s){\rm d}s + \int_{0}^{t\wedge \tau_{N}} V_{x_{i}}({\bm x}(s),s)g^{ij}({\bm x}(s),s){\rm d}B_{j}(s) \\ &=&V({\bm x}_{0},0)+\int_{0}^{t\wedge \tau_{N}} \mathcal{L} V({\bm x}(s),s){\rm d}s + \int_{0}^{t\wedge \tau_{N}} V_{x_{i}}({\bm x}(s),s)g^{ij}({\bm x}(s),s){\rm d}B_{j}(s)\\ &\leq &V({\bm x}_{0},0)+\int_{0}^{+\infty} \gamma(t){\rm d}t + \int_{0}^{t\wedge \tau_{N}} V_{x_{i}}({\bm x}(s),s)g^{ij}({\bm x}(s),s){\rm d}B_{j}(s)\end{aligned}$$ Then, $\mathbb{\hat{E}}[\vert V({\bm x}(t\wedge \tau_{N}),t\wedge \tau_{N})\vert ]\leq \vert V({\bm x}_{0},0)\vert + \int_{0}^{+\infty} \gamma(t){\rm d}t:=K<+\infty,$ which implies $$\begin{aligned} \label{inequ} \infty&>&K\geq \mathbb{\hat{E}}[\vert V({\bm x}(t\wedge \tau_{N}),t\wedge \tau_{N})\vert ]\geq \mathbb{\hat{E}}[\mu(\vert {\bm x}(t\wedge \tau_{N})\vert)]\geq \nonumber\\ &\geq& \mu(N)c(\tau_{N}\leq t)\geq \mu(N)c(\tau_{\infty}\leq t)\end{aligned}$$ where $\mu(r):=\inf_{\vert \bm x\vert \geq r, t\geq 0}V(\bm x,t)$ and $\lim_{r\rightarrow+\infty}\mu(r)=+\infty$ because of the condition assumed in [\[conditionofV2\]](#conditionofV2){reference-type="eqref" reference="conditionofV2"}. Now, letting $N\rightarrow+\infty$ in [\[inequ\]](#inequ){reference-type="eqref" reference="inequ"} yields $c(\tau_{\infty}\leq t)=0$ for any $t$. Finally, further letting $t\rightarrow+\infty$ gives $c(\tau_{\infty}\leq +\infty)=0$, which completes the proof. ## Proof of Lemma [Lemma 22](#theorembarbalat){reference-type="ref" reference="theorembarbalat"} {#appendix_barbalat} [To prove Lemma [Lemma 22](#theorembarbalat){reference-type="ref" reference="theorembarbalat"}, we first establish the inequality as follows.]{style="color: black"} **Lemma 42**. *For $A_{ij}(t)\in M_{G}^{2}[0, T]$, denote by ${\bm A}(t)=(a_{ij}(t))_{d\times m}$. Then, we have $\mathbb{\hat{E}}\left\vert \int_{0}^{T} {\bm A}(t) \mathrm{~d} {\bm B}(t)\right \vert ^2 \leq d\bar{\gamma} \ \mathbb{\hat{E}}\int_{0}^{T} \vert {\bm A}(t)\vert ^{2} \mathrm{~d} t.$* For simplicity of expression, we apply Einstein's notations [@einstein1922general] in the following arguments and throughout if they are necessary. From Theorem [THEOREM 8](#BDG){reference-type="ref" reference="BDG"} and Remark [Remark 7](#lemmaqua){reference-type="ref" reference="lemmaqua"}, it follows that $$\begin{aligned} &&\mathbb{\hat{E}}\left\vert \int_{0}^{T} {\bm A}(t) \mathrm{~d} {\bm B}(t) \right\vert ^2 =\mathbb{\hat{E}} \left (\int_{0}^{T}a_{ij}(t) \mathrm{~d} B_{j}(t) \int_{0}^{T}a_{ik}(t) \mathrm{~d} B_{k}(t) \right )\\ &=& \mathbb{\hat{E}} \int_{0}^{T}a_{ij}(t)a_{ik}(t) \mathrm{~d} \langle B_{j}, B_{k}\rangle(t) =\mathbb{\hat{E}} \int_{0}^{T} {\rm trace}({\bm A}(t)\mathrm{~d} \langle {\bm B} \rangle(t) {\bm A}^{\top}(t) )\\ &\leq & d\cdot \mathbb{\hat{E}} \int_{0}^{T} \lambda_{\rm max}(A(t)\mathrm{~d} \langle B \rangle(t) A^{\top}(t) ) = d\cdot\mathbb{\hat{E}} \int_{0}^{T} \vert {\bm A}(t)\mathrm{~d} \langle {\bm B} \rangle(t) {\bm A}^{\top}(t) \vert_{2} \\ &=& d\cdot\mathbb{\hat{E}} \int_{0}^{T} \vert {\bm A}(t)\vert_{2} ^{2} \mathrm{~d} \vert \langle {\bm B} \rangle\vert_{2} (t) \leq d\bar{\gamma} \cdot \mathbb{\hat{E}} \int_{0}^{T} \vert {\bm A}(t)\vert_{2} ^{2} \mathrm{~d} t \leq d\bar{\gamma} \cdot \mathbb{\hat{E}} \int_{0}^{T} \vert {\bm A}(t)\vert^{2} \mathrm{~d} t.\end{aligned}$$ The proof is therefore completed. Now, we need to prove the lemma using contradiction. If this is not true, then there exists $Q \in \mathcal{Q}$ such that $Q \left(\left\{\omega :\liminf_{t \rightarrow +\infty} \eta(\bm x(t;\omega))>0\right\} \right)>0.$ Thus, there exists $\epsilon>0$ such that $Q(\Omega_{1})\geq 2\epsilon$ with $\Omega_{1}=\left\{\omega\in \Omega_{0}: \liminf_{t \rightarrow +\infty} \eta(\bm x(t))>2\epsilon\right\}.$ Since $\Omega_{1}=\cup_{n=1}^{+\infty} \left( \Omega_{1} \cap \left\{\omega: \sup_{t \geq 0}\vert \bm x(t;\omega)\vert <n \right\}\right),$ there exists a number $N>0$ such that $Q(\Omega_{2})\geq \epsilon$ in which $\Omega_{2}= \Omega_{1} \cap \left\{\omega: \sup_{t \geq 0}\vert \bm x(t;\omega)\vert <N \right\}$. Now, we define the $\mathbb{F}$-stopping times as $$\begin{array}{l} \sigma_{1}(\omega):=\inf\{t: \eta(\bm x(t;\omega))\geq 2\epsilon\}, ~~ \sigma_{2i}(\omega):=\inf\{t: \eta(\bm x(t;\omega))\leq \epsilon, ~ t\geq \sigma_{2i-1}(\omega)\}, \\ \sigma_{2i+1}(\omega):=\inf\{t: \eta(\bm x(t;\omega))\geq 2\epsilon, ~ t\geq \sigma_{2i}(\omega)\}, ~~ \tau_{N}(\omega):=\inf\{t: \vert \bm x(t;\omega)\vert\geq N\}. \end{array}$$ For all $\omega \in \Omega_{2}$, $\tau_{N}(\omega)=+\infty$ and $\sigma_{i}(\omega)<+\infty$ for all $i>0$ using the formula [\[integralbound\]](#integralbound){reference-type="eqref" reference="integralbound"} and the definition of $\Omega_1$. By virtue of Proposition [Proposition 17](#exist){reference-type="ref" reference="exist"}, ${\bm M}(t)=\int_{0}^{t\wedge \tau_{N}}{\bm g}({\bm x}(s),s) {\rm d}{\bm B}(s)$ is a $Q$-martingale for each $Q \in \mathcal{Q}$. Hence, using Assumption [Assumption 15](#asump1){reference-type="ref" reference="asump1"}, Lemma [Lemma 42](#theoreminequality){reference-type="ref" reference="theoreminequality"}, Hölder's inequality, and Doob's martingale inequality in traditional stochastic analysis, we obtain that for each $T>0$, $$\begin{aligned} && E_{Q}[1_{\{\tau_{N}\wedge \sigma_{2i-1}<+\infty\}}\sup_{0 \leq t \leq T}\vert \bm x(\tau_{N}\wedge (\sigma_{2i-1}+t))-\bm x(\tau_{N}\wedge \sigma_{2i-1})\vert ^{2}] \\ &\leq& 3E_{Q}\left [1_{\{\tau_{N}\wedge \sigma_{2i-1}<+\infty\}}\sup_{0 \leq t \leq T}\left \vert \int_{\tau_{N}\wedge \sigma_{2i-1}}^{\tau_{N}\wedge (\sigma_{2i-1}+t)} \bm f(\bm x(s),s) {\rm d}s \right \vert ^{2}\right ] \\ & &+3E_{Q}\left [1_{\{\tau_{N}\wedge \sigma_{2i-1}<+\infty\}}\sup_{0 \leq t \leq T}\left \vert \int_{\tau_{N}\wedge \sigma_{2i-1}}^{\tau_{N}\wedge (\sigma_{2i-1}+t)} \bm g(\bm x(s),s) {\rm d}\bm B(s) \right \vert ^{2}\right ] \\ & &+3E_{Q}\left [1_{\{\tau_{N}\wedge \sigma_{2i-1}<+\infty\}}\sup_{0 \leq t \leq T}\left \vert \int_{\tau_{N}\wedge \sigma_{2i-1}}^{\tau_{N}\wedge (\sigma_{2i-1}+t)} \bm h(\bm x(s),s) {\rm d}\langle \bm B \rangle(s) \right \vert ^{2}\right ] \\ &\leq& 3TE_{Q}\left [1_{\{\tau_{N}\wedge \sigma_{2i-1}<+\infty\}}\sup_{0 \leq t \leq T}\int_{\tau_{N}\wedge \sigma_{2i-1}}^{\tau_{N}\wedge (\sigma_{2i-1}+t)} \vert \bm f(\bm x(s),s)\vert ^{2} {\rm d}s \right ] \\ & &+12E_{Q}\left [1_{\{\tau_{N}\wedge \sigma_{2i-1}<+\infty\}}\left \vert \int_{\tau_{N}\wedge \sigma_{2i-1}}^{\tau_{N}\wedge (\sigma_{2i-1}+T)} \bm g(\bm x(s),s) {\rm d} \bm B(s) \right \vert \right ] \\ & &+3T\bar{\gamma}^{2}m^2E_{Q}\left [1_{\{\tau_{N}\wedge \sigma_{2i-1}<+\infty\}}\sup_{0 \leq t \leq T}\int_{\tau_{N}\wedge \sigma_{2i-1}}^{\tau_{N}\wedge (\sigma_{2i-1}+t)} \vert \bm h(\bm x(s),s)\vert ^{2} {\rm d}s \right ]\end{aligned}$$ $$\begin{aligned} &\leq& 3T\mathbb{\hat{E}}\left [1_{\{\tau_{N}\wedge \sigma_{2i-1}<+\infty\}}\int_{\tau_{N}\wedge \sigma_{2i-1}}^{\tau_{N}\wedge (\sigma_{2i-1}+T)} \vert \bm f(\bm x(s),s)\vert ^{2} {\rm d}s \right ]\\ & &+12 d \bar{\gamma}\mathbb{\hat{E}}\left [1_{\{\tau_{N}\wedge \sigma_{2i-1}<+\infty\}}\int_{\tau_{N}\wedge \sigma_{2i-1}}^{\tau_{N}\wedge (\sigma_{2i-1}+T)} \vert \bm g(\bm x(s),s)\vert ^{2} {\rm d}s \right ] \\ & &+3T\bar{\gamma}^{2}m^2\mathbb{\hat{E}}\left [1_{\{\tau_{N}\wedge \sigma_{2i-1}<+\infty\}}\int_{\tau_{N}\wedge \sigma_{2i-1}}^{\tau_{N}\wedge (\sigma_{2i-1}+T)} \vert \bm h(\bm x(s),s)\vert ^{2} {\rm d}s \right ]\\ &\leq& 3K_{N}^{2} T(T+4d\bar{\gamma}+T\bar{\gamma}^{2}m^2).\end{aligned}$$ As $\eta$ is continuous, there exists a number $\delta>0$ such that, for every $\bm x,\bm y \in B(N)$ and $\vert \bm x-\bm y\vert \leq \delta$, $\vert \eta(\bm x)-\eta(\bm y)\vert <\epsilon$. We select sufficiently small $T>0$ such that ${ 3K_{N}^{2}T(T+4d\bar{\gamma}+T\bar{\gamma}^{2}m^2)}/{\delta^2}<\frac{\epsilon}{2}.$ Thus, we have $Q\left(1_{\{\tau_{N}\wedge \sigma_{2i-1}<+\infty\}}\sup_{0 \leq t \leq T}\vert \bm x(\tau_{N}\wedge (\sigma_{2i-1}+t))-\bm x(\tau_{N}\wedge \sigma_{2i-1})\vert \geq \delta \right)\leq { 3K_{N}^{2}T(T+4d\bar{\gamma}+T\bar{\gamma}^{2}m^2)}/{\delta^2}<\frac{\epsilon}{2}$. Hence, we have $Q(\{\sigma_{2i-1}<+\infty, \tau_{N}=+\infty\}\cap \{\sup_{0\leq t\leq T}\vert \bm x(\sigma_{2i-1}+t)-\bm x(\sigma_{2i-1})\vert \geq \delta\}) \leq \frac{\epsilon}{2}.$ By the definition and the property of $\Omega_{2}$, we conclude that $Q\left(\{\sigma_{2i-1}<+\infty, \tau_{N}=+\infty\}\cap\left\{\sup_{0\leq t\leq T}\vert \bm x(\sigma_{2i-1}+t)-\bm x(\sigma_{2i-1})\vert < \delta\right\}\right) \geq \epsilon-\frac{\epsilon}{2}=\frac{\epsilon}{2},$ which further implies that $$\begin{aligned} &&Q\left(\{\sigma_{2i-1}<+\infty, \tau_{N}=+\infty\}\cap\left\{\sup_{0\leq t\leq T}\vert \eta(\bm x(\sigma_{2i-1}+t))-\eta(\bm x(\sigma_{2i-1}))\vert < \epsilon\right\}\right)\\ && \geq Q\left(\{\sigma_{2i-1}<+\infty, \tau_{N}=+\infty\} \cap\left\{\sup_{0\leq t\leq T}\vert \bm x(\sigma_{2i-1}+t)-\bm x(\sigma_{2i-1})\vert < \delta\right\}\right) \geq \frac{\epsilon}{2}. \end{aligned}$$ Define $\tilde{\Omega}_{i}:=\left \{ \sup_{0\leq t \leq T} \vert \eta(\bm x(\sigma_{2i-1}+t))-\eta(\bm x (\sigma_{2i-1}))\vert < \epsilon \right \}.$ Then, on $\tilde{\Omega}_{i}\cap \{\sigma_{2i-1}<+\infty\}$, we have $\sigma_{2i}-\sigma_{2i-1}\geq T$. By [\[limitofeta\]](#limitofeta){reference-type="eqref" reference="limitofeta"}, if $\sigma_{2i-1}<+\infty$, then $\sigma_{2i}<+\infty$ quasi-surely. Thus, $$\begin{aligned} +\infty &>& \mathbb{\hat{E}} \int_{0}^{+\infty} \eta(\bm x(t)){\rm d}t \geq E_{Q} \int_{0}^{+\infty} \eta(\bm x(t)){\rm d}t\\ &\geq &\sum_{i=1}^{+\infty} E_{Q}\left [1_{\{\tau_{N}=+\infty, \sigma_{2i-1}<+\infty,\sigma_{2i}<+\infty\}} \int_{\sigma_{2i-1}}^{\sigma_{2i}} \eta(\bm x(t)){\rm d}t\right ]\\ &\geq& \epsilon \sum_{i=1}^{+\infty} E_{Q}\left [1_{\{\tau_{N}=+\infty, \sigma_{2i-1}<+\infty\}} (\sigma_{2i}-\sigma_{2i-1})\right ]\\ &\geq& \epsilon \sum_{i=1}^{+\infty} E_{Q}\left [1_{\{\tau_{N}=+\infty, \sigma_{2i-1}<+\infty\}\cap \tilde{\Omega}_{i}} (\sigma_{2i}-\sigma_{2i-1})\right ]\\ &\geq& \epsilon T \sum_{i=1}^{+\infty} Q(\{\tau_{N}=+\infty, \sigma_{2i-1}<+\infty\}\cap \tilde{\Omega}_{i}) \geq \epsilon T \sum_{i=1}^{+\infty} \frac{\epsilon}{2} =+\infty,\end{aligned}$$ which indicates a contradiction. Consequently, we get $\lim_{t \rightarrow +\infty}\eta(\bm x(t))=0$ quasi-surely. ## Dynamic Stability in Example [Example 31](#exstability){reference-type="ref" reference="exstability"} {#app4} Here, we validate the quasi-sure stability of the considered equations in Example [Example 31](#exstability){reference-type="ref" reference="exstability"}. To this end, we set $V(\bm x):=\vert \bm x\vert^{\alpha}$ for some given $0<\alpha<1$, which yields $\mathcal{L}V(\bm x)= \alpha \vert \bm x \vert^{\alpha-2} \langle \bm x, \bm f (\bm x)\rangle+G\left(\left(k^2(\alpha-1)\alpha \vert \bm x \vert^{\alpha}\right)_{i,j=1}^{m}\right),$ where $\left(k^2(\alpha-1)\alpha \vert \bm x \vert^{\alpha}\right)_{i,j=1}^{m}$ stands for an $m\times m$ matrix such that all elements are $k^2(\alpha-1)\alpha \vert \bm x \vert^{\alpha}$. As $c_{-1}:=(-1)_{i,j=1}^{m}$ is a non-positive symmetric matrix with eigenvalues 0 and $-m$, we have $c_{-1}<0$. Set $0<\alpha<1+\frac{L}{k^2c_{-1}}<1$, we obtain that $\mathcal{L}V(\bm x)= \alpha \vert \bm x \vert^{\alpha-2} \langle \bm x, \bm f (\bm x)\rangle+k^2c_{-1}(1-\alpha)\alpha \vert \bm x \vert^{\alpha} \leq\alpha \vert \bm x \vert^{\alpha}(L+k^2c_{-1}(1-\alpha)).$ Set $\eta({\bm x}):=\alpha \vert \bm x \vert^{\alpha}(L+k^2c_{-1}(1-\alpha))<0$. Hence, in light of Proposition [Proposition 20](#exist2){reference-type="ref" reference="exist2"} and Theorem [THEOREM 24](#lasalle1){reference-type="ref" reference="lasalle1"}, if we could confirm a *statement* that the system in Example $\ref{exstability}$ does not reach $\bm 0$ before it explodes, $V(\bm x)$ with $\alpha<1$ and along any trajectory apart from $\bm 0$ is differentiable to the second order, so that the quasi-sure convergence of $\bm x$ is guaranteed to $\bm 0$, the kernel of $\eta$. To make confirm the statement, we first introduce the following result. **Proposition 43**. *Let $M(t)=\int_{0}^{t} \kappa_{ij}(s){\rm d}\langle B_{i},B_{j} \rangle (s)+\int_{0}^{t} 2G(-\bm \eta){\rm d}s,$ where ${\bm \eta} \in M_{G}^{1}([0,T];\mathbb{S}^{m})$. Then, we have $M(t)\geq 0$ quasi-surely. Particularly, $\mathbb{\hat{E}}[M(t)]\geq 0$.* The proof of the above proposition is akin to the proof for Proposition [Proposition 21](#prop5){reference-type="ref" reference="prop5"}, which is omitted here. Now, we make the final confirmation. We set $\tau_{N}:=\inf \{t\geq 0:\vert {\bm x(t)}\vert \geq N\}$ and $\xi_{\epsilon}=\inf \{t\geq 0:\vert {\bm x(t)}\vert \leq \epsilon\}$ for $\epsilon,N>0$, and select $V(\bm x)=\log\vert \bm x \vert$. Then, using the formula presented in Theorem [THEOREM 19](#GIto){reference-type="ref" reference="GIto"} and Proposition [Proposition 17](#exist){reference-type="ref" reference="exist"}, we get $$\begin{aligned} \log \vert \bm x(t\wedge \tau_{N} \wedge \xi_{\epsilon}) \vert&=&\log \vert \bm x_{0} \vert +\int_{0}^{t\wedge \tau_{N} \wedge \xi_{\epsilon}}\frac{\langle \bm x(s), \bm f(\bm x(s)) \rangle}{\vert \bm x \vert^2}\rm d s\\ &&+\sum_{j=1}^{n}\int_{0}^{t\wedge \tau_{N} \wedge \xi_{\epsilon}}k{\rm d} B_{j}(s)-\sum_{i,j=1}^{n}\int_{0}^{t\wedge \tau_{N} \wedge \xi_{\epsilon}}\frac{1}{2}k^2 \langle B_{i},B_{j} \rangle(s)\end{aligned}$$ Noticing the local Lipschitz property of $\bm f$ gives $\vert \langle \bm x, \bm f(\bm x) \rangle\vert \leq \vert \bm x \vert \vert \bm f(\bm x) \vert\leq K_{N}\vert \bm x \vert^{2}$ on $[0,\tau_{N})$. Set $c_{1}:=G((1)_{i,j=1}^{m})>0$. Then, by Proposition [Proposition 43](#prop5ex){reference-type="ref" reference="prop5ex"}, we have $\mathbb{\hat{E}}[\log \vert \bm x(t\wedge \tau_{N} \wedge \xi_{\epsilon}) \vert\geq \mathbb{\hat{E}}[\log \vert \bm x_{0} \vert]-\int_{0}^{t\wedge \tau_{N} \wedge \xi_{\epsilon}}(K_{N}+k^2c_{1}){\rm d}s] \geq \mathbb{\hat{E}}[\log \vert \bm x_{0}\vert]-(K_{N}+k^2c_{1})t.$ On the other hand, $\mathbb{\hat{E}}[\log \vert \bm x(t\wedge \tau_{N} \wedge \xi_{\epsilon}) \vert ] \leq c(\xi_{\epsilon}< t\wedge \tau_{N} )\log\epsilon + c(\xi_{\epsilon}\geq t\wedge \tau_{N} )\log N \leq c(\xi_{\epsilon}< t\wedge \tau_{N} )\log\epsilon + \log N.$ Hence, we obtain $\mathbb{\hat{E}}[\log \vert \bm x_{0}\vert]-(K_{N}+k^2c_{1})t \leq c(\xi_{\epsilon}< t\wedge \tau_{N} )\log\epsilon + \log N.$ First, letting $\epsilon \rightarrow 0$ results in $c(\xi_{0}< t\wedge \tau_{N} )=0$. Then, letting both $t$ and $N \rightarrow +\infty$ yields $c(\xi_{0}< \tau_{\infty} )=0$, which confirms the above statement and finally completes the proof. ## Invariant Set Associated with Autonomous $G$-SDEs {#appendix2} **THEOREM 44**. *We consider the following autonomous $G$-SDEs: $$\label{equationapp1} {\rm d}{\bm x}(t)={\bm f}({\bm x}(t)){\rm d}t+{\bm g}({\bm x}(t)){\rm d}{\bm B}(t)+{\bm h}({\bm x}(t)){\rm d}\langle {\bm B} \rangle (t),$$ where ${\bm f}:\mathbb{R}^{d}\rightarrow \mathbb{R}^{d}$, ${\bm g}:\mathbb{R}^{d} \rightarrow \mathbb{R}^{d\times m}$, ${\bm h}:\mathbb{R}^{d} \rightarrow \mathbb{R}^{d\times m^2}$, and ${\bm f}({\bm a})={\bm g}({\bm a})={\bm h}({\bm a})={\bm 0}$. Clearly, ${\bm f},{\bm g}$ and ${\bm h}$ are all globally Lipschitzian. Then, we have that, for all ${\bm x}_{0}\neq a$, $c\big(\{\omega: \exists~t>0, ~~\bm{x}(t,\omega;{\bm x}_{0})={\bm a}\} \big)=0,$ which indicates that the trajectory does not approach ${\bm a}$ quasi-surely in a finite time.* We know that the $G$-SDEs [\[equationapp1\]](#equationapp1){reference-type="eqref" reference="equationapp1"} have a unique solution on $M_{G}[0,T]$ for every $T>0$ according to [@Peng-13]. First, we need to perform the proof for the situation of $\bm a={\bm 0}$. Now set $\mathcal{A}:=\{\omega: {\bm x}(t,\omega)={\bm 0} \ {\rm for} \ {\rm some} \ t \in [0,+\infty)\}.$ If $c(\mathcal{A})>0$, then there exists a number $T>0$ such that $c(\mathcal{A}_{T})>0$ where $\mathcal{A}_{T}:=\{\omega: {\bm x}(t,\omega)={\bm 0} \ {\rm for} \ {\rm some} \ t \in [0,T]\},$ which is due to the fact that $\mathcal{A}=\cup_{T=1}^{+\infty}\mathcal{A}_{T}$. Next, introduce the stopping time $\tau_{\epsilon}:=\inf\{t \in [0,+\infty): \vert {\bm x}(t,\omega)\vert \leq \epsilon \} .$ Set $V({\bm x}):=1/\vert {\bm x}\vert =(\vert {\bm x}\vert ^{2})^{-\frac{1}{2}}$. Then, we perform the calculations using $G$-Itô's formula, obtaining that $$\begin{aligned} && V({\bm x}(T\wedge \tau_{\epsilon}))=V({\bm x}_{0})+\int_{0}^{T\wedge \tau_{\epsilon}} V_{x_{i}}({\bm x}(s))f^{i}({\bm x}(s)){\rm d}s \\ &&+ \int_{0}^{T\wedge \tau_{\epsilon}} V_{x_{i}}({\bm x}(s))g^{ij}({\bm x}(s)){\rm d}{ B}_{j}(s) +\int_{0}^{T\wedge \tau_{\epsilon}} \frac{1}{2}\kappa_{ij}({\bm x}(s)) {\rm d}\langle B_{i},B_{j} \rangle(s) \\ &=&V({\bm x}_{0})-\int_{0}^{T\wedge \tau_{\epsilon}} \frac{\langle {\bm x}(s),f({\bm x}(s))\rangle}{\vert {\bm x}\vert ^{3}}{\rm d}s -\int_{0}^{T\wedge \tau_{\epsilon}} \frac{x_{i}(s)g^{ij}({\bm x}(s))}{\vert {\bm x}\vert ^{3}}{\rm d}B_{j}(s)\\ &&+\int_{0}^{T\wedge \tau_{\epsilon}} \Bigg [-\frac{g^{\mu{i}}({\bm x}(s))g^{\mu{j}}({\bm x}(s))}{2\vert {\bm x}\vert ^{3}} +\frac{3}{2\vert {\bm x}\vert ^{5}}x_{\mu}x_{v} g^{\mu{i}}({\bm x}(s))g^{\nu{j}}({\bm x}(s))\\ &&-\frac{x_{v}h^{vij}({\bm x}(s))}{\vert {\bm x}\vert ^{3}}\Bigg ]{\rm d}\langle B_{i},B_{j} \rangle (s) \leq V({\bm x}_{0})+\int_{0}^{T\wedge \tau_{\epsilon}}\Bigg [\frac{\vert {\bm f}({\bm x})\vert }{\vert {\bm x}\vert ^{2}} +\frac{d\bar{\gamma} \vert {\bm g}({\bm x})\vert ^{2}}{2\vert {\bm x}\vert ^{3}}+\frac{3\bar{\gamma}\vert {\bm g}({\bm x})\vert ^{2}}{2\vert {\bm x}\vert ^3}\\ &&+\frac{\vert {\bm h}({\bm x})\vert \bar{\gamma}}{\vert {\bm x}\vert ^{2}}\Bigg ]{\rm d}s + \int_{0}^{T\wedge \tau_{\epsilon}} V_{x_{i}}({\bm x}(s))g^{ij}({\bm x}(s)){\rm d}B_{j}(s),\end{aligned}$$ where $\kappa_{ij}=V_{x_{k}}(h^{kij}+h^{kji})+V_{x_{k}x_{l}}g^{ki}g^{lj}$ and Einstein's notations are applied here. Let $\rho({\bm x}):=\frac{\vert {\bm f}({\bm x})\vert }{\vert {\bm x}\vert } +\frac{(d+3)\bar{\gamma} \vert {\bm g}({\bm x})\vert ^{2}}{2\vert {\bm x}\vert ^{2}}+\frac{\vert {\bm h}({\bm x})\vert \bar{\gamma}}{\vert {\bm x}\vert }.$ Then, there exists a number $K>0$ such that $\rho ({\bm x})\leq K<+\infty$ because ${\bm f}$, ${\bm g}$ and ${\bm h}$ are globally Lipschitzian as mentioned above. Hence, it follows that $$\begin{aligned} && V({\bm x}(T\wedge \tau_{\epsilon})) \leq V({\bm x}_{0})+ \int_{0}^{T\wedge \tau_{\epsilon}} V({\bm x}(s)) \rho({\bm x}(s)){\rm d}s+ \int_{0}^{T\wedge \tau_{\epsilon}} V_{x_{i}}({\bm x}(s))g^{ij}({\bm x}(s)){\rm d}B_{j}(s)\\ & & =V({\bm x}_{0})+ \int_{0}^{T} V({\bm x}(s)) \rho ({\bm x}(s))1_{[0,\tau_{\epsilon}]}{\rm d}s + \int_{0}^{T} V_{x_{i}}({\bm x}(s))g^{ij}({\bm x}(s))1_{[0,\tau_{\epsilon}]}{\rm d}B_{j}(s)\\ && \leq V({\bm x}_{0})+K\int_{0}^{T} V({\bm x}(s)) 1_{[0,\tau_{\epsilon}]} {\rm d}s + \int_{0}^{T} V_{x_{i}}({\bm x}(s))g^{ij}({\bm x}(s)) 1_{[0,\tau_{\epsilon}]}{\rm d}B_{j}(s),\end{aligned}$$ which implies that $\mathbb{\hat{E}}[V({\bm x}(T\wedge \tau_{\epsilon}))] \leq \mathbb{\hat{E}}[V({\bm x}_{0})]+ K \mathbb{\hat{E}}\int_{0}^{T} V({\bm x}(s)) 1_{[0,\tau_{\epsilon}]} {\rm d}s\\ \leq \mathbb{\hat{E}}[V({\bm x}_{0})]+ K \int_{0}^{T}\mathbb{\hat{E}}[ V({\bm x}(s\wedge \tau_{\epsilon}))]{\rm d}s.$ Now, using Gronwall's inequality, we have $\mathbb{\hat{E}}\left [\frac{1}{\vert {\bm x}(T\wedge \tau_{\epsilon})\vert }\right ]\leq \mathbb{\hat{E}}[V({\bm x}_{0})]{\rm e}^{KT}.$ From the definition of $\tau_{\epsilon}$ and also from the continuity of ${\bm x}(t)$, it follows that $\vert {\bm x}(T\wedge \tau_{\epsilon})\vert =\epsilon$ on $\mathcal{A}_{T}$. Thus, $c(\mathcal{A}_{T})= \epsilon \mathbb{\hat{E}}\left [\frac{1}{\vert {\bm x}(T\wedge \tau_{\epsilon})\vert }1_{\mathcal{A}_{T}} \right ] \leq \epsilon \mathbb{\hat{E}}[V({\bm x}_{0})]{\rm e}^{KT},$ which is valid for every $\epsilon>0$. Therefore, we immediately obtain $c(\mathcal{A}_{T})=0$, which is a contradiction. For the general situation of $\bm{a}$, we set ${\bm y}(t):={\bm x}(t)-{\bm a}$. Then, $\bm{y}(t)$ satisfies the $G$-SDEs: ${\rm d}{\bm y}(t)={\bm f}({\bm y}(t)+\bm a){\rm d}t+{\bm g}({\bm y}(t)+{\bm a}){\rm d}{\bm B}(t)+{\bm h}({\bm y}(t)+{\bm a}){\rm d}\langle {\bm B} \rangle (t).$ Consequently, we know that ${\bm y}(t)$ never approaches ${\bm 0}$ quasi-surely, i.e., ${\bm x}(t)$ never approaches ${\bm a}$ quasi-surely. Therefore, the proof is complete. rew ## Numerical evidences {#appendix_numerical} Here, we describe the numerical scheme that we use for partially illustrating the analytical results obtained in the main text. Actually, we do not provide a complete simulation for the solutions of $G$-SDEs but only simulate the corresponding SDEs under a group of probability measures. A rigorous and complete scheme for simulating the solution of $G$-SDEs still awaits further investigations. To this end, we first suppose $\bm W(t)$ to be a standard $m$-dimensional Brownian motion on the probability space $(\Omega, \mathcal{B}(\Omega), P)$. Also suppose that $\Theta$ is a bounded, closed and convex subset of $\mathbb{R}^{m\times m}$, where $\Theta=[\underline{\sigma}, \overline{\sigma}]$ for $m=1$. In addition, $\mathcal{\tilde{Q}}:=\Big\{P_{\bm \theta} \in \mathcal{M}:P_{\bm \theta} \ {\rm is} \ {\rm the} \ {\rm law} \ {\rm of} \ {\rm process} \ \int_{0}^{t}{\bm \theta}(s){\rm d}{\bm W}(s)\ {\rm for~} \forall \ t\geq 0, {\bm \theta} \in \mathscr{A}_{0,\infty}^{\Theta}\Big\} \subset \mathcal{Q},$ where $\mathscr{A}_{0,\infty}^{\Theta}$ denotes the collection of all $\Theta$-valued $\mathscr{F}$ adapted function in $[0,+\infty)$. According to Remark 15 in Ref. [@HuPeng-29], the capacity satisfies $c(\mathcal{A})=\sup_{Q \in \mathcal{\tilde{Q}}}P[\mathcal{A}]$ for any $\mathcal{A}\in \mathscr{B}(\Omega)$, so we can check whether an event is correct quasi-surely on the probability measures space $\mathcal{\tilde{Q}}$. Thus, we make our numerical simulations on a finite subset of $\mathcal{\tilde{Q}}$ repeatedly as follows and use the case where $\langle B_{i}, B_{j}\rangle=0$ for each $i \neq j$ and all $B_{i}$ are identically distributed. For the time interval $[0, T]$, we introduce a uniform time partition $0=t_0<t_1<\cdots<t_N=T$ with $\Delta t:=t_{n+1}-t_n=T/N$. We use the following Euler-Maruyama scheme, as proposed in [@maruyama1954transition], to investigate the solution of the SDEs correspondingly from the $G$-SDEs in [\[GSDE\]](#GSDE){reference-type="eqref" reference="GSDE"}: $$\label{Euler} {\bm X}({n+1})={\bm X}(n)+{\bm f}({\bm X}(n),t_{n}){\rm \Delta}t+{\bm g}({\bm X}(n),t_{n}){\Delta\bm B}(t_{n})+{\bm h}({\bm X}(n),t_{n})\Delta\langle {\bm B}\rangle (t_{n})$$ with $\bm{X}(0)={\bm x}_0$ and $n=0,1,\cdots,N-1$. [Here, $\Delta{B}_i(t_{n}) \sim \mathcal{N}(0,\sigma_{i,n}^{2} \Delta t)$ and $\Delta \langle B_{i} \rangle (t_n)=\sigma_{i,n}^{2}{\Delta t}$ with $\sigma_{i,n}\in [\underline{\sigma}, \overline{\sigma}]$ and $i=0,1,\cdots,m$.]{style="color: black"} [In order to investigate the dynamics of the corresponding SDEs on the probability measures space $\mathcal{\tilde{Q}}$, the covariance $\{\sigma_{i,n}\}_{1\leq i\leq m, 1\leq n\leq N}$ should be taken from all the element of the set $[\underline{\sigma}, \overline{\sigma}]^{ m \times N}$. To do this numerically, we introduce a uniform interval partition $\underline{\sigma}=\sigma_{0}<\sigma_{1}<\cdots<\sigma_{k}=\overline{\sigma}$ with $\Delta\sigma=\sigma_{i+1}-\sigma_{i}=(\overline{\sigma}-\underline{\sigma})/k$. Denote by $\Sigma_{jl}:=\{ \sigma_i | j\leq i\leq l \}$, where $1\leq j\leq l \leq k$. For any given tuple $(j,l)$, we choose an element $(\mu_{in})_{1\leq i\leq m, 1\leq n\leq N}\in\Sigma_{jl}^{m\times N}$, set $\sigma_{i,n}=\mu_{in}$ for all $1\leq i\leq m, 1\leq n\leq N$, and then approximate the dynamics of the SDEs correspondingly from [\[GSDE\]](#GSDE){reference-type="eqref" reference="GSDE"} using the scheme specified in [\[Euler\]](#Euler){reference-type="eqref" reference="Euler"}, which enables us to numerically produce a large number of simulating trials. ]{style="color: black"} In Figure [\[fig1\]](#fig1){reference-type="ref" reference="fig1"}, we show the numerical results, respectively, for Examples [Example 30](#example1){reference-type="ref" reference="example1"}-[Example 32](#example3){reference-type="ref" reference="example3"}.

arxiv_math

--- author: - Sean Fancher - Prashant K. Purohit - Eleni Katifori bibliography: - refs.bib title: An efficient spectral method for the dynamic behavior of truss structures --- # Abstract {#abstract .unnumbered} Truss structures at macro-scale are common in a number of engineering applications and are now being increasingly used at the micro-scale to construct metamaterials. In analyzing the properties of a given truss structure, it is often necessary to understand how stress waves propagate through the system and/or its dynamic modes under time dependent loading so as to allow for maximally efficient use of space and material. This can be a computationally challenging task for particularly large or complex structures, with current methods requiring fine spatial discretization or evaluations of sizable matrices. Here we present a spectral method to compute the dynamics of trusses inspired by results from fluid flow networks. Our model accounts for the full dynamics of linearly elastic truss elements via a network Laplacian; a matrix object which couples the motions of the structure joints. We show that this method is equivalent to the continuum limit of linear finite element methods as well as capable of reproducing natural frequencies and modes determined by more complex and computationally costlier methods. # Introduction {#sec:intro} Trusses have been a mainstay of structural engineering for a substantial amount of human history, with applications including bridges, buildings, airplanes, and spacecraft. While these applications at the scale of several tens of meters are well established, truss metamaterials with microstructure in the range of a few microns are of intense research interest currently due to their easy manufacturability by 3D printing and other techniques. Wave propagation through truss metamaterials [@mueller2019energy; @glaesener2021viscoelastic], is at the forefront of this research since porous metamaterials can provide resistance to impact while being lightweight and inexpensive. More generally, wave propagation through metamaterials has been of interest for some time since they can be designed to allow only certain wavelengths/frequencies to propagate, thus enabling applications in acoustic absorbers and transmitters [@wang2015locally; @krushynska2018accordion]. Similar techniques have also been applied to structures comprised of polymer networks for the purpose of understanding response to propagating loads and defect particles [@zhang2015novel; @lu2022double]. In all these applications the central question of concern is to determine the dynamic behavior of the structure. Furthermore, current imaging techniques for measurement of local strains have become so sophisticated that it is possible to observe wavefronts propagating through individual elements at the microscale [@branch2017controlling]. These experimental developments must be matched with computational techniques that can provide similarly detailed information about the passage of wavefronts. The dynamic behavior of truss structures is most commonly studied using the finite element method. Each member of the structure is discretized into appropriate truss elements, then stiffness and mass matrices are assembled, then the equations of motion (including external forces) are written in matrix form starting from Newton's laws. The natural frequencies and mode shapes (which are critical in structural design and in understanding the dispersion relations of metamaterials) are computed by solving an eigenvalue problem using the stiffness and mass matrices [@cook2007concepts]. The frequencies obtained depend on the mass and stiffness matrices; they vary depending on how fine the discretization is, whether the consisent or lumped mass matrix (or a combination) is used, and on the shape functions used to assemble the stiffness and mass matrices. Often, very fine discretization is required to get an accurate estimate of the natural frequencies and the computational costs can be prohibitive if the strcture has a large number of degrees of freedom. Methods that retain the full element dynamics have also been developed via solutions in Fourier space [@pao1999dynamic] or integral operators [@polz2019wave]. These produce accurate solutions for large structures but require more sophisticated mathematics and analysis algorithms. While it is possible to directly simulate the temporal response of a truss structure by monitoring the propagation of reactive wavefronts [@howard1998analysis; @messner2015wave; @trainiti2016wave], this can become computationally costly as more wavefronts are produced via scattering. In this article, we present a new method for determining the dynamic behavior of a given truss structure based on our work on fluid flow networks [@fancher2022mechanical]. We retain a full description of the element dynamics much like the methodology of [@howard1998analysis; @pao1999dynamic], but we maintain a focus on the motions of the structure nodes rather than stress within the elements. By assuming linear stress and displacement dynamics, we can express the propagation of stress waves in terms of the node motions in Fourier space and develop a linear relation between the displacements of and forces acting on the nodes. This allows our resulting matrix objects to be smaller in size and thus more computationally efficient to analyze in comparison to those produced via element stress calculations. We then provide an analysis of two example structures to directly compare our results to those obtain by other methods. # Methods {#sec:methods} R0.4 {width="38%"} ## Single Rod Dynamics {#sec:SRD} We begin by considering a solid, cylindrical rod of length $L$ and cross sectional area $A$. As the rod undergoes tension and compression in response to externally applied forces, each infinitesimally thin segment is displaced from its rest position by an amount $u(z,t)$, where $z$ is the distance from the $z=0$ end of the rod in its unperturbed configuration and $t$ is time. Specifically, $u(z,t)>0$ implies the segment has shifted in the direction of increasing $z$ and $u(z,t)<0$ implies a shift in the direction of decreasing $z$, as depicted in Fig. [\[Fig:cartoons\]](#Fig:cartoons){reference-type="ref" reference="Fig:cartoons"}A. From this displacement field we can derive two other critically important fields; the velocity field, $v(z,t)$, and strain field, $\epsilon (z,t)$. Defining the sign of $v$ and $\epsilon$ to denote movement in the direction of $z$ and tensile expansion of the material respectively allows for the relations $$v\left(z,t\right) = \frac{\partial u}{\partial t}, \quad\quad\quad \epsilon\left(z,t\right) = \frac{\partial u}{\partial z}. \label{vepdef}$$ Under the assumption that the material is linearly elastic, the stress field, $\sigma(z,t)$, can be expressed simply as $\sigma(z,t)=E\epsilon(z,t)$, where $E$ is the Young's modulus. This forces the sign of $\sigma$ to be such that $\sigma>0$ represents tensile stress while $\sigma<0$ represents compressive stress. These definitions of $v$, $\epsilon$, and $\sigma$ immediately allow for the identity $$\frac{\partial v}{\partial z}-\frac{1}{E}\frac{\partial\sigma}{\partial t} = \frac{\partial}{\partial z}\left(\frac{\partial u}{\partial t}\right)-\frac{\partial}{\partial t}\left(\frac{\partial u}{\partial z}\right) = 0. \label{masscon}$$ We now consider the forces acting on an infinitesimally thin segment of the rod, such as that of width $\delta z$ shown in Fig. [\[Fig:cartoons\]](#Fig:cartoons){reference-type="ref" reference="Fig:cartoons"}A. Following our sign convention for the stress field, Newton's second law takes the form $$\frac{\partial\sigma}{\partial z}-\rho\frac{\partial v}{\partial t} = 0. \label{forcebalance}$$ where $\rho$ is the material density and the $\delta z\to 0$ has been taken. Together, Eqs. [\[masscon\]](#masscon){reference-type="ref" reference="masscon"} and [\[forcebalance\]](#forcebalance){reference-type="ref" reference="forcebalance"} represent the dynamic coupling between the stress and velocity fields of the rod [@howard1998analysis; @pao1999dynamic]. Additionally, we can express these in terms of the displacement field to transform Eq. [\[forcebalance\]](#forcebalance){reference-type="ref" reference="forcebalance"} into $$E\frac{\partial^{2}u}{\partial z^{2}}-\rho\frac{\partial^{2}u}{\partial t^{2}} = 0, \label{dispwave}$$ thus producing the simple wave equation wherein waves can propagate through the displacement field at speed $c=\sqrt{E/\rho}$. One important implication of Eq. [\[dispwave\]](#dispwave){reference-type="ref" reference="dispwave"} is that $u(z,t)$ can be expanded into its Fourier modes via $$u\left(z,t\right) = \int d\omega\>\left(F\left(\omega\right)e^{i\omega\left(t-\frac{z}{c}\right)}+B\left(\omega\right)e^{i\omega\left(t+\frac{z}{c}\right)}\right), \label{uFint}$$ where $F(\omega)$ and $B(\omega)$ are the forward and backward wave amplitudes respectively and the integral being over all real $\omega$ is implied. This in turn allows for the velocity and stress fields to be expressed as $$v\left(z,t\right) = \frac{\partial u}{\partial t} = \int d\omega\>i\omega\left(F\left(\omega\right)e^{i\omega\left(t-\frac{z}{c}\right)}+B\left(\omega\right)e^{i\omega\left(t+\frac{z}{c}\right)}\right), \label{vFint}$$ $$\sigma\left(z,t\right) = E\frac{\partial u}{\partial z} = \Gamma\int d\omega\>i\omega\left(-F\left(\omega\right)e^{i\omega\left(t-\frac{z}{c}\right)}+B\left(\omega\right)e^{i\omega\left(t+\frac{z}{c}\right)}\right), \label{sFint}$$ [\[vsFint\]]{#vsFint label="vsFint"} where $\Gamma=E/c=\sqrt{E\rho}$ is the material impedance. ## Rod and Joint Network Construction {#sec:RJNC} We now consider a network comprised of rods obeying the dynamic equations outlined thus far connected via a series of joints such as the one depicted in Fig. [\[Fig:cartoons\]](#Fig:cartoons){reference-type="ref" reference="Fig:cartoons"}B. Here we will use the index $\mu$ to denote a particular joint and the index $\nu$ to denote a joint connected to $\mu$ through one of the network rods. In this way, $\nu\in\mathcal{N}_{\mu}$, where $\mathcal{N}_{\mu}$ is the set of all joints connected to joint $\mu$ through a single rod. The rods themselves and their various fields will be labelled with a two component index, $\mu\nu$, comprised of the two joints the rod connects. Specifically, $u_{\mu\nu}(z,t)$ is the displacement field of rod $\mu\nu$ with the $z=0$ end of the rod being at joint $\mu$. Similar notation also applies to other fields as well as rod specific parameters such as the length, $L_{\mu\nu}$, and cross sectional area, $A_{\mu\nu}$. Exchanging the index order thus also reverses the directionality of the rod, leading to the relations $$u_{\mu\nu}\left(z,t\right) = -u_{\nu\mu}\left(L_{\mu\nu}-z,t\right), \label{uexchange}$$ $$v_{\mu\nu}\left(z,t\right) = -v_{\nu\mu}\left(L_{\mu\nu}-z,t\right), \label{vexchange}$$ $$\sigma_{\mu\nu}\left(z,t\right) = \sigma_{\nu\mu}\left(L_{\mu\nu}-z,t\right). \label{sexchange}$$ [\[uvsexchange\]]{#uvsexchange label="uvsexchange"} We can also apply the Fourier expansion used in Eq. [\[uFint\]](#uFint){reference-type="ref" reference="uFint"} alongside this notation to obtain the index exchange laws for $F_{\mu\nu}(\omega)$ and $B_{\mu\nu}(\omega)$. Letting $\tau_{\mu\nu}=L_{\mu\nu}/c_{\mu\nu}$ allows for these to be expressed as $$F_{\mu\nu}\left(\omega\right) = -B_{\nu\mu}\left(\omega\right)e^{i\omega\tau_{\mu\nu}}, \quad\quad\quad B_{\mu\nu}\left(\omega\right) = -F_{\nu\mu}\left(\omega\right)e^{-i\omega\tau_{\mu\nu}}. \label{FBexchange}$$ Finally, we can Fourier transform Eq. [\[vsFint\]](#vsFint){reference-type="ref" reference="vsFint"} in time and combine the result with Eq. [\[FBexchange\]](#FBexchange){reference-type="ref" reference="FBexchange"} to produce the relations $$\tilde{v}_{\mu\nu}\left(0,\omega\right) = i\omega\left(F_{\mu\nu}\left(\omega\right)+B_{\mu\nu}\left(\omega\right)\right), \label{v0FT}$$ $$\tilde{\sigma}_{\mu\nu}\left(0,\omega\right) = i\omega\Gamma_{\mu\nu}\left(-F_{\mu\nu}\left(\omega\right)+B_{\mu\nu}\left(\omega\right)\right), \label{s0FT}$$ [\[vsFT\]]{#vsFT label="vsFT"} $$\tilde{v}_{\mu\nu}\left(0,\omega\right) = -\tilde{v}_{\nu\mu}\left(0,\omega\right)\cos\left(\omega\tau_{\mu\nu}\right)-i\Gamma_{\mu\nu}^{-1}\tilde{\sigma}_{\nu\mu}\left(0,\omega\right)\sin\left(\omega\tau_{\mu\nu}\right), \label{vFTexchange}$$ $$\tilde{\sigma}_{\mu\nu}\left(0,\omega\right) = \tilde{\sigma}_{\nu\mu}\left(0,\omega\right)\cos\left(\omega\tau_{\mu\nu}\right)+i\Gamma_{\mu\nu}\tilde{v}_{\nu\mu}\left(0,\omega\right)\sin\left(\omega\tau_{\mu\nu}\right). \label{sFTexchange}$$ [\[vsFTexchange\]]{#vsFTexchange label="vsFTexchange"} From here we assign a $D_{\mu}$-dimensional coordinate system, denoted as $\mathbb{R}_{\mu}^{D}$, to the $\mu$th joint such that the origin is located at the rest location of the joint and the set of rods connected to that joint span $\mathbb{R}_{\mu}^{D}$. We can then define a unit vector, $\hat{e}_{\mu\nu}\in\mathbb{R}_{\mu}^{D}$, to rod $\mu\nu$ with equivalent directionality, thus implying that $\hat{e}_{\mu\nu}$ points from joint $\mu$ to joint $\nu$. Of note is that since the opposing vector $\hat{e}_{\nu\mu}$ exists in $\mathbb{R}_{\nu}^{D}$, there is no implicit index exchange relation between $\hat{e}_{\mu\nu}$ and $\hat{e}_{\nu\mu}$ without first defining the relation between $\mathbb{R}_{\mu}^{D}$ and $\mathbb{R}_{\nu}^{D}$. However, if $\hat{e}_{\mu\nu}$ and $\hat{e}_{\nu\mu}$ are expressed in the global coordinate system, denoted $\mathbb{R}_{g}^{D}$, then they must of course point in opposing directions. This is particularly easy to the achieve if $\text{dim}(\mathbb{R}_{\mu}^{D})=\text{dim}(\mathbb{R}_{g}^{D})$ for all $\mu$. In this case, we can define an "aligned coordinate set\" in which $\vec{x}_{\mu}=\vec{x}_{g}-\vec{r}_{\mu}$; where $\vec{x}_{\mu}$ is a position vector in $\mathbb{R}_{\mu}^{D}$, $\vec{x}_{g}$ is the same position in $\mathbb{R}_{g}^{D}$, and $\vec{r}_{\mu}$ is the position of the $\mu$th joint in $\mathbb{R}_{g}^{D}$. With the coordinate systems defined, we next investigate the dynamics of the joints by assuming that each joint is massless and incapable of carrying force. Newton's second law applied to joint $\mu$ then takes the form $$\vec{P}_{\mu}\left(t\right)+\sum_{\nu\in\mathcal{N}_{\mu}}\hat{e}_{\mu\nu}A_{\mu\nu}\sigma_{\mu\nu}\left(0,t\right) = \vec{0}, \label{jointforce}$$ where $\vec{P}_{\mu}\left(t\right)$ is the force being applied to the joint by some entity external to the system, again expressed within $\mathbb{R}_{\mu}^{D}$. Finally, the joint itself moves within this coordinate system with a velocity given by the vector $\vec{w}_{\mu}(t)$. Enforcing that the $z=0$ end of rod $\mu\nu$ must have a velocity equivalent to the projection of $\vec{w}_{\mu}$ in the direction of $\hat{e}_{\mu\nu}$ yields the condition $$\vec{w}_{\mu}\cdot\hat{e}_{\mu\nu} = v_{\mu\nu}\left(0,t\right) \quad\forall\quad \nu\in\mathcal{N}_{\mu}. \label{jointvelcon}$$ Eqs. [\[jointforce\]](#jointforce){reference-type="ref" reference="jointforce"} and [\[jointvelcon\]](#jointvelcon){reference-type="ref" reference="jointvelcon"} provide the connectivity laws of the network and define how the stress and velocity fields of different rods interact [@howard1998analysis; @pao1999dynamic]. Their form can be somewhat simplified by introducing the matrix $\overset\Leftrightarrow{e}_{\mu}$, defined to be a $D_{\mu}\times|\mathcal{N}_{\mu}|$ matrix in which each column is a distinct $\hat{e}_{\mu\nu}$. The joint stress and velocity vectors, $\vec{\sigma}_{\mu}(t)$ and $\vec{v}_{\mu}(t)$, can then be defined as column vectors of length $|\mathcal{N}_{\mu}|$ such that their respective $j$th components give $\sigma_{\mu\nu}(0,t)$ and $v_{\mu\nu}(0,t)$ for the same $\nu$ as was used to generate the $j$th column of $\overset\Leftrightarrow{e}_{\mu}$. Similarly, we construct $\overset\Leftrightarrow{A}_{\mu}$ as a diagonal matrix of size $|\mathcal{N}_{\mu}|\times|\mathcal{N}_{\mu}|$ whose $j$th diagonal entry is $A_{\mu\nu}$. Finally, treating $\vec{P}_{\mu}\left(t\right)$ and $\vec{w}_{\mu}$ as a column vectors allows Eqs. [\[jointforce\]](#jointforce){reference-type="ref" reference="jointforce"} and [\[jointvelcon\]](#jointvelcon){reference-type="ref" reference="jointvelcon"} to be expressed as $$\overset\Leftrightarrow{e}_{\mu}\overset\Leftrightarrow{A}_{\mu}\vec{\sigma}_{\mu} = -\vec{P}_{\mu}, \label{jointforcemat}$$ $$\left(\vec{w}_{\mu}\right)^{\text{T}}\overset\Leftrightarrow{e}_{\mu} = \left(\vec{v}_{\mu}\right)^{\text{T}}. \label{jointvelconmat}$$ [\[jointmateqs\]]{#jointmateqs label="jointmateqs"} Of note is that due to our construction of $\mathbb{R}_{\mu}^{D}$ we have $D_{\mu}\le |\mathcal{N}_{\mu}|$. In the specific case of equality, $\overset\Leftrightarrow{e}_{\mu}$ must be an invertible square matrix due to the set of vectors $\{\hat{e}_{\mu\nu}\> |\>\nu\in\mathcal{N}_{\mu}\}$ spanning the local coordinate system. This causes Eqs. [\[jointforcemat\]](#jointforcemat){reference-type="ref" reference="jointforcemat"} and [\[jointvelconmat\]](#jointvelconmat){reference-type="ref" reference="jointvelconmat"} to become a bijective linear relation between the rod dynamics, $\vec{\sigma}_{\mu}$ and $\vec{v}_{\mu}$, and the joint dynamics, $\vec{P}_{\mu}$ and $\vec{w}_{\mu}$. Thus, the rods are all effectively uncoupled from each other when $D_{\mu}=|\mathcal{N}_{\mu}|$. This is because for any given stress and velocity field within a single rod there must exist an applied force and joint velocity such that Eqs. [\[jointforcemat\]](#jointforcemat){reference-type="ref" reference="jointforcemat"} and [\[jointvelconmat\]](#jointvelconmat){reference-type="ref" reference="jointvelconmat"} are satisfied without inducing any additional dynamics in any other rod. ## The Laplacian Block Matrix {#sec:MML} We now seek to construct a global relation between the motions of every joint in the network. We achieve this by first noting the similarities between the element velocity and stress fields with the flow velocity and pressure of a fluid in a compliant vessel, then following a methodology previously developed for calculating the pressure distribution in such a fluid flow network [@fancher2022mechanical]. To begin, we combine Eqs. [\[vexchange\]](#vexchange){reference-type="ref" reference="vexchange"} and [\[v0FT\]](#v0FT){reference-type="ref" reference="v0FT"} to produce $$\tilde{v}_{\nu\mu}\left(0,\omega\right) = -\tilde{v}_{\mu\nu}\left(L_{\mu\nu},\omega\right) = -i\omega\left(F_{\mu\nu}\left(\omega\right)e^{-i\omega\tau_{\mu\nu}}+B_{\mu\nu}\left(\omega\right)e^{i\omega\tau_{\mu\nu}}\right). \label{vLtoFB}$$ Solving $F_{\mu\nu}$ and $B_{\mu\nu}$ yields $$F_{\mu\nu}\left(\omega\right) = \frac{\tilde{v}_{\mu\nu}\left(0,\omega\right)+\tilde{v}_{\nu\mu}\left(0,\omega\right)e^{-i\omega\tau_{\mu\nu}}}{i\omega\left(1-e^{-2i\omega\tau_{\mu\nu}}\right)} = \frac{\tilde{\vec{w}}_{\mu}^{\text{T}}\left(\omega\right)\hat{e}_{\mu\nu}+\tilde{\vec{w}}_{\nu}^{\text{T}}\left(\omega\right)\hat{e}_{\nu\mu}e^{-i\omega\tau_{\mu\nu}}}{i\omega\left(1-e^{-2i\omega\tau_{\mu\nu}}\right)}, \label{Ftov0L}$$ $$B_{\mu\nu}\left(\omega\right) = \frac{\tilde{v}_{\mu\nu}\left(0,\omega\right)+\tilde{v}_{\nu\mu}\left(0,\omega\right)e^{i\omega\tau_{\mu\nu}}}{i\omega\left(1-e^{2i\omega\tau_{\mu\nu}}\right)} = \frac{\tilde{\vec{w}}_{\mu}^{\text{T}}\left(\omega\right)\hat{e}_{\mu\nu}+\tilde{\vec{w}}_{\nu}^{\text{T}}\left(\omega\right)\hat{e}_{\nu\mu}e^{i\omega\tau_{\mu\nu}}}{i\omega\left(1-e^{2i\omega\tau_{\mu\nu}}\right)}, \label{Btov0L}$$ [\[FBtov0L\]]{#FBtov0L label="FBtov0L"} where we have replaced $\tilde{v}_{\mu\nu}$ and $\tilde{v}_{\nu\mu}$ with the Fourier transformed velocities of joints $\mu$ and $\nu$ respectively as per Eq. [\[jointvelcon\]](#jointvelcon){reference-type="ref" reference="jointvelcon"}. From here we use Eq. [\[s0FT\]](#s0FT){reference-type="ref" reference="s0FT"} to express the Fourier transformed stress as $$\begin{aligned} \tilde{\sigma}_{\mu\nu}\left(0,\omega\right) &= i\omega\Gamma_{\mu\nu}\left(-F_{\mu\nu}\left(\omega\right)+B_{\mu\nu}\left(\omega\right)\right) \nonumber\\ &= \Gamma_{\mu\nu}\left(-\frac{\tilde{\vec{w}}_{\mu}^{\text{T}}\left(\omega\right)\hat{e}_{\mu\nu}+\tilde{\vec{w}}_{\nu}^{\text{T}}\left(\omega\right)\hat{e}_{\nu\mu}e^{-i\omega\tau_{\mu\nu}}}{1-e^{-2i\omega\tau_{\mu\nu}}}+\frac{\tilde{\vec{w}}_{\mu}^{\text{T}}\left(\omega\right)\hat{e}_{\mu\nu}+\tilde{\vec{w}}_{\nu}^{\text{T}}\left(\omega\right)\hat{e}_{\nu\mu}e^{i\omega\tau_{\mu\nu}}}{1-e^{2i\omega\tau_{\mu\nu}}}\right) \nonumber\\ &= \frac{i\Gamma_{\mu\nu}}{\sin\left(\omega\tau_{\mu\nu}\right)}\left(\hat{e}_{\mu\nu}^{\text{T}}\tilde{\vec{w}}_{\mu}\left(\omega\right)\cos\left(\omega\tau_{\mu\nu}\right)+\hat{e}_{\nu\mu}^{\text{T}}\tilde{\vec{w}}_{\nu}\left(\omega\right)\right). \label{s0towmn}\end{aligned}$$ Inserting Eq. [\[s0towmn\]](#s0towmn){reference-type="ref" reference="s0towmn"} into the Fourier transform of Eq. [\[jointforce\]](#jointforce){reference-type="ref" reference="jointforce"} and introducing the parameter $\Lambda_{\mu\nu}=A_{\mu\nu}\Gamma_{\mu\nu}$ then yields $$-\tilde{\vec{P}}_{\mu}\left(\omega\right) = \sum_{\nu\in\mathcal{N}_{\mu}}\hat{e}_{\mu\nu}A_{\mu\nu}\tilde{\sigma}_{\mu\nu}\left(0,\omega\right) = \sum_{\nu\in\mathcal{N}_{\mu}}\hat{e}_{\mu\nu}\frac{i\Lambda_{\mu\nu}}{\sin\left(\omega\tau_{\mu\nu}\right)}\left(\hat{e}_{\mu\nu}^{\text{T}}\tilde{\vec{w}}_{\mu}\left(\omega\right)\cos\left(\omega\tau_{\mu\nu}\right)+\hat{e}_{\nu\mu}^{\text{T}}\tilde{\vec{w}}_{\nu}\left(\omega\right)\right). \label{s0sumtowmn}$$ Based on Eq. [\[s0sumtowmn\]](#s0sumtowmn){reference-type="ref" reference="s0sumtowmn"} we can construct the block vectors $\mathbf{\vec{W}}$ and $\mathbf{\vec{P}}$ as well as the block matrix $\mathbf{\overset\Leftrightarrow{D}}$. These are objects whose individual components are themselves vectors and matrices. Specifically, $\mathbf{\vec{W}}$ and $\mathbf{\vec{P}}$ have $J$ components each, where $J$ is the number of joints in the network, with the $\mu$th components being the vectors $\tilde{\vec{w}}_{\mu}$ and $\tilde{\vec{P}}_{\mu}$ respectively, each expressed in terms of $\mathbb{R}_{\mu}^{D}$. Similarly, $\mathbf{\overset\Leftrightarrow{D}}$, denoted here as the network Laplacian, has a $J\times J$ structure with components defined as $$\mathbf{\overset\Leftrightarrow{D}}_{\mu\nu} = \begin{cases} \sum_{\gamma\in\mathcal{N}_{\mu}}\Lambda_{\mu\gamma}\omega\cot\left(\omega\tau_{\mu\nu}\right)\hat{e}_{\mu\gamma}\hat{e}_{\mu\gamma}^{\text{T}} & \mu=\nu \\ \Lambda_{\mu\nu}\omega\csc\left(\omega\tau_{\mu\nu}\right)\hat{e}_{\mu\nu}\hat{e}_{\nu\mu}^{\text{T}} & \nu\in\mathcal{N}_{\mu} \\ \overset\Leftrightarrow{0} & \text{otherwise} \end{cases}, \label{Ddef}$$ thus making the $\mu\nu$ component of $\mathbf{\overset\Leftrightarrow{D}}$ a $D_{\mu}\times D_{\nu}$ matrix that transforms a vector in $\mathbb{R}_{\nu}^{D}$ into one in $\mathbb{R}_{\mu}^{D}$. Given these definitions, Eq. [\[s0sumtowmn\]](#s0sumtowmn){reference-type="ref" reference="s0sumtowmn"} clearly dictates $$-\frac{1}{i\omega}\mathbf{\overset\Leftrightarrow{D}}\mathbf{\vec{W}} = -\mathbf{\vec{P}} \quad\quad\quad \implies \quad\quad\quad \mathbf{\overset\Leftrightarrow{D}}\mathbf{\vec{U}} = \mathbf{\vec{P}}. \label{DWPrel}$$ Eq. [\[DWPrel\]](#DWPrel){reference-type="ref" reference="DWPrel"} provides a direct relation between the force applied to the system and its dynamics. The first equality generates a linear transformation between the joint velocities and forces. The second equality generates a similar relation in terms of $\mathbf{\vec{U}}$, the block vector of Fourier transformed joint displacements, and is obtained from the first by noting that $\mathbf{\vec{W}}=i\omega\mathbf{\vec{U}}$ since velocity is the time derivative of displacement. Of note is that the network Laplacian derived here is similar in function to the global scattering matrix of the system [@pao1999dynamic], but importantly represents a relation over the joints rather than elements. Thus, the network Laplacian is typically smaller in size and more computationally manageable. # Results {#sec:results} ## Single rod resonance {#sec:SRR} R0.5 {width="48%"} Several important system properties can be derived from the explicit relations derived here. Of particular importance are the restrictions placed on $\mathbf{\vec{U}}$, $\mathbf{\vec{W}}$, and $\mathbf{\vec{P}}$ as the driving frequency changes. At frequencies where $\mathbf{\overset\Leftrightarrow{D}}$ is finite and invertible, Eq. [\[DWPrel\]](#DWPrel){reference-type="ref" reference="DWPrel"} represents the only restriction, thus allowing finite joint dynamics to be obtained from any finite forcing. However, if $\mathbf{\overset\Leftrightarrow{D}}$ is finite but noninvertible, then the space of allowable forcings is reduced to the image of $\mathbf{\overset\Leftrightarrow{D}}$. The frequencies wherein this occurs are noted as the natural frequencies while the sets of joint dynamics that cause $\mathbf{\vec{U}}$ and $\mathbf{\vec{W}}$ to exist within the nonempty null space of $\mathbf{\overset\Leftrightarrow{D}}$ are the natural modes. Being within the null space, these modes necessarily exist with no forces applied to any joint and thus represent all possible force-free dynamics of the system. There also exist frequencies at which components of $\mathbf{\overset\Leftrightarrow{D}}$ diverge due to the cotangent and cosecant functions utilized in Eq. [\[Ddef\]](#Ddef){reference-type="ref" reference="Ddef"}. Specifically, from Eq. [\[s0sumtowmn\]](#s0sumtowmn){reference-type="ref" reference="s0sumtowmn"} it is clear that this occurs whenever $\omega\tau_{\mu\nu}=n\pi$ for any rod $\mu\nu$ and $n\in\mathbb{Z}$. Thus, in order to retain a finite $\tilde{\vec{P}}_{\mu}(\omega)$, the velocities of joints $\mu$ and $\nu$ must satisfy $$\left(-1\right)^{n}\hat{e}_{\mu\nu}^{\text{T}}\tilde{\vec{w}}_{\mu}\left(\frac{n\pi}{\tau_{\mu\nu}}\right)+\hat{e}_{\nu\mu}^{\text{T}}\tilde{\vec{w}}_{\nu}\left(\frac{n\pi}{\tau_{\mu\nu}}\right) = 0. \label{wmnrescon}$$ This restricts the allowable joint dynamics such that the ends of the rod must move in phase ($\tilde{v}_{\mu\nu}(0,n\pi/\tau_{\mu\nu})=\tilde{v}_{\mu\nu}(L_{\mu\nu},n\pi/\tau_{\mu\nu})$) when $n$ is even and out of phase ($\tilde{v}_{\mu\nu}(0,n\pi/\tau_{\mu\nu})=-\tilde{v}_{\mu\nu}(L_{\mu\nu},n\pi/\tau_{\mu\nu})$) when $n$ is odd. If $\tilde{\vec{w}}_{\mu}$ and $\tilde{\vec{w}}_{\nu}$ are also differentiable at $\omega\tau_{\mu\nu}=n\pi$ then we can use L'Hopital's rule to obtain $$\lim_{\omega\to n\pi/\tau_{\mu\nu}}\frac{\hat{e}_{\mu\nu}^{\text{T}}\tilde{\vec{w}}_{\mu}\left(\omega\right)\cos\left(\omega\tau_{\mu\nu}\right)+\hat{e}_{\nu\mu}^{\text{T}}\tilde{\vec{w}}_{\nu}\left(\omega\right)}{\sin\left(\omega\tau_{\mu\nu}\right)} = \frac{1}{\tau_{\mu\nu}}\left.\left(\hat{e}_{\mu\nu}^{\text{T}}\frac{\partial\tilde{\vec{w}}_{\mu}}{\partial\omega}+\left(-1\right)^{n}\hat{e}_{\nu\mu}^{\text{T}}\frac{\partial\tilde{\vec{w}}_{\nu}}{\partial\omega}\right)\right|_{\omega=\frac{n\pi}{\tau_{\mu\nu}}}. \label{wmnlimcon}$$ The nonvanishing nature of Eq. [\[wmnlimcon\]](#wmnlimcon){reference-type="ref" reference="wmnlimcon"} implies that there can be nonzero forces applied to these nodes even at these divergent frequencies. Additionally, Eq. [\[wmnlimcon\]](#wmnlimcon){reference-type="ref" reference="wmnlimcon"} is seen to be symmetric with respect to exchanging the order of $\mu$ and $\nu$ when $n$ is even and antisymmetric when $n$ is odd. However, this term is multiplied into $\hat{e}_{\mu\nu}$ when contributing to $\tilde{\vec{P}}_{\mu}$ and into $\hat{e}_{\nu\mu}$ when contributing to $\tilde{\vec{P}}_{\nu}$. This induces a relative phase inversion between these forces since $\hat{e}_{\mu\nu}$ and $\hat{e}_{\nu\mu}$ point in opposing directions when expressed in terms of $\mathbb{R}_{g}^{D}$. Thus, in contrast to the relative phases of the rod end velocities, the contributions of the rod dynamics to the joint forces must be out of phase when $n$ is even and in phase when $n$ is odd. These phase conditions for different parities of $n$ are reflected in Fig. [\[Fig:resonance\]](#Fig:resonance){reference-type="ref" reference="Fig:resonance"}. ## Joint wave dispersion {#sec:JWD} Shifting focus from an individual rod to an individual joint, one direct result that can be obtained from the theory presented here is the well known phenomenon of wave dispersion. By constructing the forward and backward wave amplitude vectors at joint $\mu$ as $\vec{F}_{\mu}(\omega)$ and $\vec{B}_{\mu}(\omega)$ such that their respective $j$th components are $F_{\mu\nu}(\omega)$ and $B_{\mu\nu}(\omega)$ with $\nu$ being the same index used to construct the $j$th column of $\overset\Leftrightarrow{e}_{\mu}$, exactly the same indexing scheme as was used for $\vec{\sigma}_{\mu}(t)$ and $\vec{v}_{\mu}(t)$, we can utilize the Fourier transform of Eq. [\[jointmateqs\]](#jointmateqs){reference-type="ref" reference="jointmateqs"} to express $\vec{F}_{\mu}(\omega)$, the set of outgoing wave amplitudes, as a function of $\vec{B}_{\mu}(\omega)$, the set of incoming wave amplitudes. This can be compactly expressed by first defining the matrix $\overset\Leftrightarrow{\Lambda}_{\mu}$ in exactly the same manner as $\overset\Leftrightarrow{A}_{\mu}$ but with the diagonal terms being the respective $\Lambda_{\mu\nu}$. The Fourier transform of Eq. [\[jointforcemat\]](#jointforcemat){reference-type="ref" reference="jointforcemat"} can then be written as $$\begin{aligned} -\tilde{\vec{P}}_{\mu}\left(\omega\right) &= \overset\Leftrightarrow{e}_{\mu}\overset\Leftrightarrow{A}_{\mu}\tilde{\vec{\sigma}}_{\mu}\left(\omega\right) = \overset\Leftrightarrow{e}_{\mu}\overset\Leftrightarrow{\Lambda}_{\mu}\cdot i\omega\left(\vec{B}_{\mu}\left(\omega\right)-\vec{F}_{\mu}\left(\omega\right)\right) = \overset\Leftrightarrow{e}_{\mu}\overset\Leftrightarrow{\Lambda}_{\mu}\left(2i\omega\vec{B}_{\mu}\left(\omega\right)-\tilde{\vec{v}}_{\mu}\left(\omega\right)\right) \nonumber\\ &= \overset\Leftrightarrow{e}_{\mu}\overset\Leftrightarrow{\Lambda}_{\mu}\left(2i\omega\vec{B}_{\mu}\left(\omega\right)-\overset\Leftrightarrow{e}_{\mu}^{\text{T}}\tilde{\vec{w}}_{\mu}\left(\omega\right)\right), \label{jointforcematFT}\end{aligned}$$ where we have used the Fourier transform of Eq. [\[jointvelconmat\]](#jointvelconmat){reference-type="ref" reference="jointvelconmat"} in the final equality. Eq. [\[jointforcematFT\]](#jointforcematFT){reference-type="ref" reference="jointforcematFT"} can then be solved for $\tilde{\vec{w}}_{\mu}(\omega)$, which can be used to produce $$\vec{F}_{\mu}\left(\omega\right) = \frac{1}{i\omega}\tilde{\vec{v}}_{\mu}\left(\omega\right)-\vec{B}_{\mu}\left(\omega\right) = \frac{1}{i\omega}\overset\Leftrightarrow{e}_{\mu}^{\text{T}}\tilde{\vec{w}}_{\mu}\left(\omega\right)-\vec{B}_{\mu}\left(\omega\right) = \overset\Leftrightarrow{T}_{\mu}\vec{B}_{\mu}\left(\omega\right)+\frac{1}{i\omega}\overset\Leftrightarrow{e}_{\mu}^{\text{T}}\left(\overset\Leftrightarrow{e}_{\mu}\overset\Leftrightarrow{\Lambda}_{\mu}\overset\Leftrightarrow{e}_{\mu}^{\text{T}}\right)^{-1}\tilde{\vec{P}}_{\mu}\left(\omega\right), \label{jointFampsol}$$ where $$\overset\Leftrightarrow{T}_{\mu} = 2\overset\Leftrightarrow{e}_{\mu}^{\text{T}}\left(\overset\Leftrightarrow{e}_{\mu}\overset\Leftrightarrow{\Lambda}_{\mu}\overset\Leftrightarrow{e}_{\mu}^{\text{T}}\right)^{-1}\overset\Leftrightarrow{e}_{\mu}\overset\Leftrightarrow{\Lambda}_{\mu}-\overset\Leftrightarrow{I}_{\mu}, \label{Tdef}$$ is the transmission matrix and $\overset\Leftrightarrow{I}_{\mu}$ is the $|\mathcal{N}_{\mu}|\times|\mathcal{N}_{\mu}|$ identity matrix. Eq. [\[jointFampsol\]](#jointFampsol){reference-type="ref" reference="jointFampsol"} represents the method by which one may directly calculate the amplitude of all outgoing waves from the amplitudes of all incoming waves and the force applied to the joint. It is also functionally equivalent to other representations of the same wave scattering phenomenon [@howard1998analysis], but is here presented in a compacted notation. Of note is that when $D_{\mu}=|\mathcal{N}_{\mu}|$, $\overset\Leftrightarrow{e}_{\mu}$ is invertible and $\overset\Leftrightarrow{T}_{\mu}$ simply reduces to $\overset\Leftrightarrow{I}_{\mu}$. In this case, the wave amplitudes of each element are completely decoupled from each other so that no energy can pass through the joint except for that which is supplied by any external forcing. The transmission matrix can be used in a variety of ways. The scattering of wavefronts can be easily tracked by propagating them through the rods of a structure, using Eq. [\[jointFampsol\]](#jointFampsol){reference-type="ref" reference="jointFampsol"} to calculate how they transmit through and reflect off of the joints, and repeating for the desired amount of simulation time. The natural frequencies of the system can even be determined by also enforcing a matching condition throughout this process (as we explore in Sec. [3.4](#sec:Square){reference-type="ref" reference="sec:Square"} and explicitly calculate in Appendix [5](#app:sswd){reference-type="ref" reference="app:sswd"}). This method of wavefront tracking is functionally equivalent to the reverberation matrix method explored in [@pao1999dynamic]; wherein a global scattering matrix is constructed as a diagonal block matrix whose component matrices are the transmission matrices. The reverberation matrix is then constructed by matching the backward waves with their corresponding index exchanged forward waves and solving Eq. [\[jointFampsol\]](#jointFampsol){reference-type="ref" reference="jointFampsol"} for the applied forces. Just like the network Laplacian developed here, the frequencies at which this reverberation matrix becomes singular are the natural frequencies of the system and correspond to the existence of force free oscillatory modes. Importantly, this construction requires separating the forward and backward traveling waves, thus causing the reverberation matrix to represent a double cover over the set of structure rods. This is in contrast to the network Laplacian, which represents a single cover over the set of joints, thus making the network Laplacian generally smaller and more computationally efficient than the reverberation matrix. ## Connection to stiffness and mass matrices {#sec:SMM} Our theory can also be used to analyze the global properties of a structure. Eq. [\[DWPrel\]](#DWPrel){reference-type="ref" reference="DWPrel"} in particular can be interpreted as a dynamic extension of the static stiffness matrix method. In this method, $\mathbf{\overset\Leftrightarrow{K}}$ is the static stiffness matrix, $\mathbf{\vec{U}}$ is the displacement vector whose elements give the static displacement from equilibrium of each joint, and $\mathbf{\vec{Q}}$ is the vector of externally applied forces that maintain this out of equilibrium positioning. The stiffness matrix can be defined by treating each rod as a Hookian spring of stiffness $k_{\mu\nu}=A_{\mu\nu}E_{\mu\nu}/L_{\mu\nu}=\Lambda_{\mu\nu}/\tau_{\mu\nu}$ so that the force it exerts at joint $\mu$ is given by $-k_{\mu\nu}\hat{e}_{\mu\nu}(\hat{e}_{\mu\nu}^{\text{T}}\vec{u}_{\mu}-\hat{e}_{\nu\mu}^{\text{T}}\vec{u}_{\nu})$, where $\vec{u}_{\mu}\in\mathbb{R}_{\mu}^{D}$ is the displacement of the $\mu$th joint in its own local coordinate system. Summing this effect over all rods and using our construction of $\mathbf{\overset\Leftrightarrow{D}}$ given by Eq. [\[Ddef\]](#Ddef){reference-type="ref" reference="Ddef"} automatically provides the relation $$\mathbf{\overset\Leftrightarrow{K}} = \lim_{\omega\to 0}\mathbf{\overset\Leftrightarrow{D}}\left(\omega\right). \label{KDrel}$$ From here, we can use the fact that $\mathbf{\vec{U}}$ and $\mathbf{\vec{Q}}$ represent static quantities to express their Fourier transforms as simply the vectors themselves multiplied by a $\delta$-function. This allows the second form of Eq. [\[DWPrel\]](#DWPrel){reference-type="ref" reference="DWPrel"} to be expressed as $$\mathbf{\vec{Q}}\delta\left(\omega\right) = \mathbf{\vec{P}} = \mathbf{\overset\Leftrightarrow{D}}\mathbf{\vec{U}}\delta\left(\omega\right) = \left(\mathbf{\overset\Leftrightarrow{K}}\mathbf{\vec{U}}\right)\delta\left(\omega\right). \label{Pstatic}$$ Equating the prefactors of the $\delta$-functions on either side of Eq. [\[Pstatic\]](#Pstatic){reference-type="ref" reference="Pstatic"} yields the condition $\mathbf{\vec{Q}}=\mathbf{\overset\Leftrightarrow{K}}\mathbf{\vec{U}}$, which is precisely a statement of static equilibrium written in matrix form. We can further explore the limiting behavior of Eq. [\[DWPrel\]](#DWPrel){reference-type="ref" reference="DWPrel"} by defining the consistent mass matrix of the system as $$\mathbf{\overset\Leftrightarrow{M}}_{\mu\nu} = \begin{cases} \sum_{\gamma\in\mathcal{N}_{\mu}}\int_{0}^{L_{\mu\gamma}}dz\>A_{\mu\gamma}\rho_{\mu\gamma}\left(N\left(\frac{z}{L_{\mu\gamma}}\right)\right)^{2}\hat{e}_{\mu\gamma}\hat{e}_{\mu\gamma}^{\text{T}} & \mu=\nu \\ -\int_{0}^{L_{\mu\nu}}dz\>A_{\mu\nu}\rho_{\mu\nu}N\left(\frac{z}{L_{\mu\nu}}\right)\left(1-N\left(\frac{z}{L_{\mu\nu}}\right)\right)\hat{e}_{\mu\nu}\hat{e}_{\nu\mu}^{\text{T}} & \nu\in\mathcal{N}_{\mu} \\ \overset\Leftrightarrow{0} & \text{otherwise} \end{cases}, \label{MCdef}$$ where $N(x):\lbrack 0,1\rbrack\to\lbrack 0,1\rbrack$ is the shape function of the rod. Note the sign negation in the off diagonal terms which takes into account the opposing directionalities of $\hat{e}_{\mu\nu}$ and $\hat{e}_{\nu\mu}$ when expressed in terms of $\mathbb{R}_{g}^{D}$. Using the standard linear shape function ($N(x)=1-x$) allows the integrals to be easily performed while also yielding the relation $$\mathbf{\overset\Leftrightarrow{M}} = -\frac{1}{2}\lim_{\omega\to 0}\left(\frac{\partial^{2}}{\partial\omega^{2}}\left(\mathbf{\overset\Leftrightarrow{D}}\right)\right) \quad\quad\implies\quad\quad \mathbf{\overset\Leftrightarrow{D}} = \mathbf{\overset\Leftrightarrow{K}}-\omega^{2}\mathbf{\overset\Leftrightarrow{M}}+\mathcal{O}\left(\omega^{4}\right), \label{MDrel}$$ where $\mathbf{\overset\Leftrightarrow{D}}$ has been approximated by its Taylor expansion to second order. This expansion shows that the practice [@cook2007concepts] of finding the natural frequencies of the system by observing where $\det(\mathbf{\overset\Leftrightarrow{K}}-\omega^{2}\mathbf{\overset\Leftrightarrow{M}})=0$ is merely a second order approximation of defining the natural frequencies by where $\det(\mathbf{\overset\Leftrightarrow{D}})=0$. ## Square structure with a cross bar {#sec:Square} {width="50%"} We now seek to apply the theoretical framework developed to two specific examples. To show that our theory produces results that are equivalent to previously established methods, we first consider a two dimensional structure comprised of four joints arranged in a square connected by five total rods, as depicted in Fig. [\[Fig:squaredisp\]](#Fig:squaredisp){reference-type="ref" reference="Fig:squaredisp"}. We choose a global coordinate and label system such that the joint 1 is located at $(0,0)$, joint 2 at $(L,0)$, joint 3 at $(0,L)$, and joint 4 at $(L,L)$. Four of the five rods form the square outline of the structure and can thus be denoted by the connectivity labels 12, 13, 24, and 34. The fifth rod, labeled 23, creates a cross bar of length $\sqrt{2}L$. Finally, we invoke an aligned coordinate set to define the local coordinate system of each joint. Precisely how this system responds to external forces can be determined from the use of scattering wavefronts. Fig. [\[Fig:squaredisp\]](#Fig:squaredisp){reference-type="ref" reference="Fig:squaredisp"} shows a time lapse of a compression wave generated by a sudden impact moving through the structure. Eq. [\[jointFampsol\]](#jointFampsol){reference-type="ref" reference="jointFampsol"} dictates how the initial wave interacts with joint 2 to become two separate transmitted waves and one reflected wave (see Appendix [5](#app:sswd){reference-type="ref" reference="app:sswd"} for explicit derivation of transmission matrices). This process repeats itself every time one of the waves reaches joint 2 or 3, steadily creating a continuously increasing number of waves. Fig. [\[Fig:squaredisp\]](#Fig:squaredisp){reference-type="ref" reference="Fig:squaredisp"} also highlights how waves merely reflect off joints 1 and 4 rather than scattering through them due to the fact that $D_{\mu}=|\mathcal{N}_{\mu}|$ at these joints. As discussed in Sec. [3.2](#sec:JWD){reference-type="ref" reference="sec:JWD"}, this causes the rods attached to them to be effectively uncoupled and incapable of passing energy to one another. If the wave is oscillatory with a well defined, nonzero frequency rather than a single pulse, then the forcing conditions become more restricted. For most frequencies, oscillatory forcing at one or more joints is necessary to maintain a well defined steady state as opposed to a divergently growing number of wavefronts. The natural frequencies of the system are an exception in that waves at these frequencies can propagate through the structure in a steady state that does not require any external forces to be applied. By examining how waves traverse each rod via Eq. [\[FBexchange\]](#FBexchange){reference-type="ref" reference="FBexchange"} and scatter through joints 2 and 3 via Eq. [\[jointFampsol\]](#jointFampsol){reference-type="ref" reference="jointFampsol"} we can calculate the relation between the forward and backward travelling wave amplitudes in rod 23 from either the scattering at joint 2 or that at joint 3 (see Appendix [5](#app:sswd){reference-type="ref" reference="app:sswd"}). Requiring these amplitudes to match yields the condition $$\Lambda_{23}^{-2}-\frac{1}{2}\eta_{23}\left(\omega\right)\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)+\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right)-\frac{1}{4}\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)\right)\left(\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right) = 0, \label{squarenatfreq}$$ where $\eta_{\mu\nu}(\omega)=\Lambda_{\mu\nu}^{-1}\cot(\omega\tau_{\mu\nu})$. Any frequency which satisfies Eq. [\[squarenatfreq\]](#squarenatfreq){reference-type="ref" reference="squarenatfreq"} is a natural frequency of the system. With the theory developed here, we can rederive Eq. [\[squarenatfreq\]](#squarenatfreq){reference-type="ref" reference="squarenatfreq"} via the methodology presented in Sec. [3.3](#sec:SMM){reference-type="ref" reference="sec:SMM"}. The network Laplacian can be constructed rather simply as a $4\times4$ block matrix in which each component is a $2\times2$ matrix. The specific properties of these component matrices allows for the inverse and determinant of the network Laplacian to be readily calculated (see Appendix [6](#app:ssli){reference-type="ref" reference="app:ssli"}). Requiring that this determinant vanish at some nonzero frequency then produces exactly the same condition as that given by Eq. [\[squarenatfreq\]](#squarenatfreq){reference-type="ref" reference="squarenatfreq"}. For more complex structures with network Laplacians too large to analytically invert, numerical methods can be used to find the frequencies which satisfy $\text{det}(\mathbf{\overset\Leftrightarrow{D}})=0$. {width="50%"} Importantly, the natural frequencies obtained from these methods are not the same as those that can be obtained by using a mass matrix approach. As discussed in Sec. [3.3](#sec:SMM){reference-type="ref" reference="sec:SMM"}, traditionally one would define a mass matrix, $\mathbf{\overset\Leftrightarrow{M}}$, then determine the natural frequencies by observing where $\det(\mathbf{\overset\Leftrightarrow{K}}-\omega^{2}\mathbf{\overset\Leftrightarrow{M}})=0$. Eq. [\[MCdef\]](#MCdef){reference-type="ref" reference="MCdef"} gives one possible definition of $\mathbf{\overset\Leftrightarrow{M}}$, denoted as the consistent mass matrix. Another even simpler definition is known as the lumped mass matrix in which $N(x)=\Theta(1/2-x)$, with $\Theta(x)$ being the Heaviside step function, thus causing $\mathbf{\overset\Leftrightarrow{M}}$ to become diagonal with $\mathbf{\overset\Leftrightarrow{M}}_{\mu\mu}=(1/2)m_{\mu}\overset\Leftrightarrow{I}_{\mu}$. Here $m_{\mu}$ is the total mass of all rods connected to joint $\mu$. This lumped mass matrix simply compresses the entire system into the joints by splitting the mass of each rod evenly between its connected joints. Either of these mass matrices can be used to calculate a set of natural frequencies, though these will generally not be equivalent due to the distinct way each handles the mass of the various system elements. The theory presented here is also distinct from this approach as the network Laplacian was developed without any approximation of the distribution of mass within the rods. We thus expect the natural frequencies calculated from the condition $\det(\mathbf{\overset\Leftrightarrow{D}})=0$ to represent a continuum limit of those derived from either mass matrix. To show this, we considered the effects of dividing each bar of the square structure depicted in Fig. [\[Fig:squaredisp\]](#Fig:squaredisp){reference-type="ref" reference="Fig:squaredisp"} into a number of elements. Specifically, for $n$ divisions we introduce $n-1$ evenly spaced new joints in each rod to transform it into $n$ smaller identical rods. We then recalculate the five lowest nonzero natural frequencies using all four methods (the lumped mass matrix, consistent mass matrix, network Laplacian, and reverberation matrix) by steadily increasing $\omega$ and observing where the determinant of each matrix vanishes. Of course, the spectrum of the network Laplacian and reverberation matrix are completely equivalent, but the larger reverberation method ultimately requires more computation time to explore. Fig. [\[Fig:squarediv\]](#Fig:squarediv){reference-type="ref" reference="Fig:squarediv"}A shows the results of this process and highlights two critical aspects: 1) the natural frequencies of the network Laplacian (and equivalently the reverberation matrix) are invariant to these divisions and 2) the natural frequencies of both mass matrix methods steadily converge to those of the network Laplacian as the structure becomes more finely divided. The first of these is a consequence of our aforementioned treatment of the mass distribution of each rod without approximation. Dividing the rods and adding new joints does not affect how any waves traverse the structure in our model. The second explicitly shows how our model acts as a continuum limit of the mass matrix method. The network Laplacian produces natural frequencies equivalent to the mass matrices in the limit of infinite subdivision of the system elements. In the particular case considered here in which all rods are made of identical material and have identical cross sections, the network Laplacian and reverberation matrix also find a natrual mode at $\omega=\pi c/L$ that the mass matrices do not. This is due to this particular value of $\omega$ being a resonant frequency of each rod except for the cross bar, thus requiring the more careful handling explored in Sec. [3.1](#sec:SRR){reference-type="ref" reference="sec:SRR"}. We also compare the overall computation times required to obtain the spectra plotted in Fig. [\[Fig:squarediv\]](#Fig:squarediv){reference-type="ref" reference="Fig:squarediv"}A as performed on an AMD Ryzen 5 1500x processor with Numpy's linear algebra package used to calculate the determinants. Fig. [\[Fig:squarediv\]](#Fig:squarediv){reference-type="ref" reference="Fig:squarediv"}B shows how these computation times increase for the mass matrix methods as the system becomes more finely divided. As mentioned previously, the natural frequencies and modes determined by the network Laplacian and reverberation matrix are invariant to such divisions, so only the computation time of the divisionless case was measured. We see from this data that the smaller network Laplacian is indeed more computationally efficient than the larger reverberation matrix and that both are substantially more efficient than either mass matrix method when the system rods are subdivided into more than 3 pieces. ## 7 Truss Bridge {#sec:7TB} The next structure we will examine is a small bridge. Specifically, we consider a system with seven rods connected through five joints. The joints are numbered 1 through 5 and exist at locations $\vec{j}_{1}=\lbrack -L,0\rbrack$, $\vec{j}_{2}=\lbrack -L/2,L\sqrt{3}/2\rbrack$, $\vec{j}_{3}=\lbrack 0,0\rbrack$, $\vec{j}_{4}=\lbrack L/2,L\sqrt{3}/2\rbrack$, and $\vec{j}_{5}=\lbrack L,0\rbrack$. The seven rods have connectivity labels 12, 13, 23, 24, 34, 35, and 45. For the analysis considered here, we will assume each rod has the same cross section and is made of the same material so that $\Lambda_{\mu\nu}$ and $\tau_{\mu\nu}$ are constant across all rods. This construction creates a parallelogram frame trisected into equilateral triangles in which all rods are identical. We will use this system to showcase how the theory presented here can determine not only the natural frequencies of a structure but natural modes as well. Additional constraints will also be imposed by forcing joints 1 and 5 to have zero displacement and velocity at all times, thus anchoring the structure in place. Finally, we will assume that joints 2, 3, and 4 have no external forces imposed on them, but allow forcing at joints 1 and 5 so as to maintain the anchoring constraints. To analyze this structure, we begin by noting that Eq. [\[DWPrel\]](#DWPrel){reference-type="ref" reference="DWPrel"} can be expressed in the form (see Appendix [7](#app:7TB){reference-type="ref" reference="app:7TB"} for express form of components) $$\begin{bmatrix} \mathbf{\overset\Leftrightarrow{D}}_{11} & \mathbf{\overset\Leftrightarrow{D}}_{12} & \mathbf{\overset\Leftrightarrow{D}}_{13} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} \\ \mathbf{\overset\Leftrightarrow{D}}_{12} & \mathbf{\overset\Leftrightarrow{D}}_{22} & \mathbf{\overset\Leftrightarrow{D}}_{23} & \mathbf{\overset\Leftrightarrow{D}}_{24} & \overset\Leftrightarrow{0} \\ \mathbf{\overset\Leftrightarrow{D}}_{13} & \mathbf{\overset\Leftrightarrow{D}}_{23} & \mathbf{\overset\Leftrightarrow{D}}_{33} & \mathbf{\overset\Leftrightarrow{D}}_{34} & \mathbf{\overset\Leftrightarrow{D}}_{35} \\ \overset\Leftrightarrow{0} & \mathbf{\overset\Leftrightarrow{D}}_{24} & \mathbf{\overset\Leftrightarrow{D}}_{34} & \mathbf{\overset\Leftrightarrow{D}}_{44} & \mathbf{\overset\Leftrightarrow{D}}_{45} \\ \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} & \mathbf{\overset\Leftrightarrow{D}}_{35} & \mathbf{\overset\Leftrightarrow{D}}_{45} & \mathbf{\overset\Leftrightarrow{D}}_{55} \end{bmatrix}\begin{bmatrix} \vec{0} \\ \tilde{\vec{u}}_{2} \\ \tilde{\vec{u}}_{3} \\ \tilde{\vec{u}}_{4} \\ \vec{0} \end{bmatrix} = \begin{bmatrix} \tilde{\vec{P}}_{1} \\ \vec{0} \\ \vec{0} \\ \vec{0} \\ \tilde{\vec{P}}_{5} \end{bmatrix}. \label{DWPformTB}$$ Since there is nonzero forcing at joints 1 and 5 we cannot simply find the frequencies which cause the network Laplacian determinant to vanish as we did in the previous section. However, the anchoring of joints 1 and 5 does allow us to examine the subsystem of joints 2, 3, and 4. Extracting only the second, third, and fourth rows of Eq. [\[DWPformTB\]](#DWPformTB){reference-type="ref" reference="DWPformTB"} produces the simpler equation $$\begin{bmatrix} \mathbf{\overset\Leftrightarrow{D}}_{22} & \mathbf{\overset\Leftrightarrow{D}}_{23} & \mathbf{\overset\Leftrightarrow{D}}_{24} \\ \mathbf{\overset\Leftrightarrow{D}}_{23} & \mathbf{\overset\Leftrightarrow{D}}_{33} & \mathbf{\overset\Leftrightarrow{D}}_{34} \\ \mathbf{\overset\Leftrightarrow{D}}_{24} & \mathbf{\overset\Leftrightarrow{D}}_{34} & \mathbf{\overset\Leftrightarrow{D}}_{44} \\ \end{bmatrix}\begin{bmatrix} \tilde{\vec{u}}_{2} \\ \tilde{\vec{u}}_{3} \\ \tilde{\vec{u}}_{4} \end{bmatrix} = \begin{bmatrix} \vec{0} \\ \vec{0} \\ \vec{0} \end{bmatrix}. \label{DWPformTB_small}$$ Eq. [\[DWPformTB_small\]](#DWPformTB_small){reference-type="ref" reference="DWPformTB_small"} does represent a null space problem that can be solved in much the same way as was done in the previous section, thus producing the frequency condition (see Appendix [7](#app:7TB){reference-type="ref" reference="app:7TB"}) $$\frac{27}{64}\left(5\cos^{2}\left(\omega\tau\right)-5\cos\left(\omega\tau\right)+1\right)\left(3\cos^{2}\left(\omega\tau\right)-\cos\left(\omega\tau\right)-1\right)\left(3\cos\left(\omega\tau\right)+1\right)\left(\cos\left(\omega\tau\right)+1\right) = 0. \label{7TBfreqcon}$$ {#Fig:bridgemodes width="95%"} Eq. [\[7TBfreqcon\]](#7TBfreqcon){reference-type="ref" reference="7TBfreqcon"} is a sixth degree polynomial in $\cos(\omega\tau)$ in which all six roots are real and exist in the range $\lbrack-1,1\rbrack$. This allows for six possible displacement modes to exist, each of which has its own set of applied forces that can be easily calculate via the first and fifth rows of Eq. [\[DWPformTB\]](#DWPformTB){reference-type="ref" reference="DWPformTB"} once the displacements themselves are known. These modes are shown in Fig. [1](#Fig:bridgemodes){reference-type="ref" reference="Fig:bridgemodes"} and calculated explicitly in Appendix [7](#app:7TB){reference-type="ref" reference="app:7TB"}. Despite the relative simplicity of our model, we see that these modes are nearly identical to those determined in [@lahe2019exact] by nonlinear means which include shear stress and transverse motion. # Discussion {#sec:Disc} In this work we have derived a new approach for analyzing the dynamic behavior of truss structures. The linear element dynamics given by Eqs. [\[masscon\]](#masscon){reference-type="ref" reference="masscon"} and [\[forcebalance\]](#forcebalance){reference-type="ref" reference="forcebalance"} allowed us to adapt the methodologies of our previous study in fluid flow networks [@fancher2022mechanical] and construct an effective network Laplacian that directly relates the motion of joints to their externally applied forces. The frequencies at which this network Laplacian matrix becomes singular are then taken to be the natural frequencies of the structure while elements of the null space at these frequencies represent the natural modes. By examining two simple example structures we have shown that these natural features as calculated here agree exceptionally well with those obtained by more traditional methods such as high resolution finite element analysis. The key strength of the network Laplacian derived here is its relative compactness. The mesh refinement necessary to perform high resolution finite element analysis incurs large computational costs, but such additional work is completely unnecessary with the network Laplacian. Other models, such as the reverberation matrix, have achieved this same feat by also fully solving for the dynamics of each element of the structure in Fourier space, but these have typically been expressed over the set of elements rather than the smaller set of joints [@howard1998analysis; @pao1999dynamic]. Comparatively, the methods developed in this work utilize notably smaller matrices and, in turn, less computation time for analyzing larger, more complicated structures. We expect these computational advantages to persist as this model is developed further to overcome some of its current shortcomings. Importantly, in solving for the element dynamics we have neglected transverse motion and shear stress. More realistic material responses such as viscoelastic effects have also been excluded [@glaesener2021viscoelastic; @ananthapadmanabhan2020numerical]. These can be introduced into our model in such a way as to retain the linear nature of the system, thus providing fruitful directions for future works. More arduous issues stem from effects that are inherently nonlinear. For any structure of dimensionality 2 or greater, a proper accounting of large deformations requires geometric nonlinearities that greatly complicate the network connectivity [@saka1992optimum]. The elements can also be subject to nonlinear material response such as strain hardening [@shi2015geometric] or nodal density dependent effects [@he2017mechanical]. These nonlinearities disrupt the algebraic nature of the Fourier space solutions and thus represent significant obstacles that the methods used here cannot immediately surpass. However, this method can still be adapted to compute dynamic small perturbations of truss structures around large static deformations. In this case, the matrices will become a function of the state of (static) strain of the bars making up the truss. Finally, the model as we have developed it here is applicable to nonperiodic transient dynamics in the time domain via careful transformation of the Fourier space solutions. This style of response analysis has already been explored in other works [@pao1999dynamic; @polz2019wave], including our own mathematically similar modelling of fluid flow networks [@fancher2022mechanical], but we anticipate the network Laplacian to provide measurably more efficient computational algorithms. Due to this, we also anticipate that with the appropriate extensions, such as the aforementioned linear shear stress and viscoelasticity, the network Laplacian method will have broad application to fields such as optimization in metamaterials and transient dynamics in finite element methods. **Acknowledgements** This reasearch was funded by the ARO MURI grant 10085212, the University of Pennsylvania Materials Research Science and Engineering Center (MRSEC) through Grant No. DMR-1720530 and the Simons Foundation through Grant No. 568888. PKP acknowledges support for this work through a seed grant from Penn's Materials Science and Engineering Center (MRSEC) grant DMR-1720530. # Appendices {#appendices .unnumbered} # Square structure: wave dispersion condition {#app:sswd} Here we calculate the natural frequencies of the square structure studied in Sec. [3.4](#sec:Square){reference-type="ref" reference="sec:Square"} by examining the dispersion of wave amplitudes and applying a matching condition. To begin, we note that given the structure as it is shown in Fig. [\[Fig:squaredisp\]](#Fig:squaredisp){reference-type="ref" reference="Fig:squaredisp"}, joints 1 and 4 in particular satisfy the condition that $D_{\mu}=|\mathcal{N}_{\mu}|$ so as to allow $\overset\Leftrightarrow{T}_{\mu}$ to reduce to $\overset\Leftrightarrow{I}_{\mu}$. Given this and the fact we are considering the force free case, Eq. [\[jointFampsol\]](#jointFampsol){reference-type="ref" reference="jointFampsol"} dictates $$F_{12}\left(\omega\right) = B_{12}\left(\omega\right), \quad\quad F_{13}\left(\omega\right) = B_{13}\left(\omega\right), \quad\quad F_{42}\left(\omega\right) = B_{42}\left(\omega\right), \quad\quad F_{43}\left(\omega\right) = B_{43}\left(\omega\right). \label{sswd:FB14rels}$$ We next consider the transmission matrices for joints 2 and 3. We will define the global coordinate system as oriented relative to Fig. [\[Fig:squaredisp\]](#Fig:squaredisp){reference-type="ref" reference="Fig:squaredisp"} with $\hat{x}$ pointing from joint 1 to 2 and $\hat{y}$ pointing from joint 1 to 3. Aligning all local coordinates with this global system allows Eq. [\[Tdef\]](#Tdef){reference-type="ref" reference="Tdef"} to produce $$\begin{aligned} \overset\Leftrightarrow{T}_{2} &= 2\begin{bmatrix} -1 & 0 \\ 0 & 1 \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix}\left(\begin{bmatrix} -1 & 0 & -\frac{1}{\sqrt{2}} \\ 0 & 1 & \frac{1}{\sqrt{2}} \end{bmatrix}\begin{bmatrix} \Lambda_{12} & 0 & 0 \\ 0 & \Lambda_{24} & 0 \\ 0 & 0 & \Lambda_{23} \end{bmatrix}\begin{bmatrix} -1 & 0 \\ 0 & 1 \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix}\right)^{-1}\begin{bmatrix} -1 & 0 & -\frac{1}{\sqrt{2}} \\ 0 & 1 & \frac{1}{\sqrt{2}} \end{bmatrix}\begin{bmatrix} \Lambda_{12} & 0 & 0 \\ 0 & \Lambda_{24} & 0 \\ 0 & 0 & \Lambda_{23} \end{bmatrix} \nonumber\\ &\quad -\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \nonumber\\ &= \begin{bmatrix} \Lambda_{12}\Lambda_{24}+\frac{1}{2}\Lambda_{23}\left(\Lambda_{12}-\Lambda_{24}\right) & -\Lambda_{24}\Lambda_{23} & \Lambda_{24}\Lambda_{23}\sqrt{2} \\ -\Lambda_{12}\Lambda_{23} & \Lambda_{12}\Lambda_{24}+\frac{1}{2}\Lambda_{23}\left(\Lambda_{24}-\Lambda_{12}\right) & \Lambda_{12}\Lambda_{23}\sqrt{2} \\ \Lambda_{12}\Lambda_{24}\sqrt{2} & \Lambda_{12}\Lambda_{24}\sqrt{2} & \frac{1}{2}\Lambda_{23}\left(\Lambda_{12}+\Lambda_{24}\right)-\Lambda_{12}\Lambda_{24} \end{bmatrix} \nonumber\\ &\quad \cdot\left(\Lambda_{12}\Lambda_{24}+\frac{1}{2}\Lambda_{23}\left(\Lambda_{12}+\Lambda_{24}\right)\right)^{-1}, \label{sswd:T2}\end{aligned}$$ $$\begin{aligned} \overset\Leftrightarrow{T}_{3} &= 2\begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}\left(\begin{bmatrix} 1 & 0 & \frac{1}{\sqrt{2}} \\ 0 & -1 & -\frac{1}{\sqrt{2}} \end{bmatrix}\begin{bmatrix} \Lambda_{34} & 0 & 0 \\ 0 & \Lambda_{13} & 0 \\ 0 & 0 & \Lambda_{23} \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}\right)^{-1}\begin{bmatrix} 1 & 0 & \frac{1}{\sqrt{2}} \\ 0 & -1 & -\frac{1}{\sqrt{2}} \end{bmatrix}\begin{bmatrix} \Lambda_{34} & 0 & 0 \\ 0 & \Lambda_{13} & 0 \\ 0 & 0 & \Lambda_{23} \end{bmatrix} \nonumber\\ &\quad -\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \nonumber\\ &= \begin{bmatrix} \Lambda_{34}\Lambda_{13}+\frac{1}{2}\Lambda_{23}\left(\Lambda_{34}-\Lambda_{13}\right) & -\Lambda_{13}\Lambda_{23} & \Lambda_{13}\Lambda_{23}\sqrt{2} \\ -\Lambda_{34}\Lambda_{23} & \Lambda_{34}\Lambda_{13}+\frac{1}{2}\Lambda_{23}\left(\Lambda_{13}-\Lambda_{34}\right) & \Lambda_{34}\Lambda_{23}\sqrt{2} \\ \Lambda_{34}\Lambda_{13}\sqrt{2} & \Lambda_{34}\Lambda_{13}\sqrt{2} & \frac{1}{2}\Lambda_{23}\left(\Lambda_{34}+\Lambda_{13}\right)-\Lambda_{34}\Lambda_{13} \end{bmatrix} \nonumber\\ &\quad \cdot\left(\Lambda_{34}\Lambda_{13}+\frac{1}{2}\Lambda_{23}\left(\Lambda_{34}+\Lambda_{13}\right)\right)^{-1}. \label{sswd:T3}\end{aligned}$$ [\[sswd:T23\]]{#sswd:T23 label="sswd:T23"} We can use the index exchange relations given by Eq. [\[FBexchange\]](#FBexchange){reference-type="ref" reference="FBexchange"} along with the equalities established in Eq. [\[sswd:FB14rels\]](#sswd:FB14rels){reference-type="ref" reference="sswd:FB14rels"} to express Eq. [\[jointFampsol\]](#jointFampsol){reference-type="ref" reference="jointFampsol"} at joints 2 and 3 as $$\begin{bmatrix} F_{21}\left(\omega\right) \\ F_{24}\left(\omega\right) \\ F_{23}\left(\omega\right) \end{bmatrix} = \overset\Leftrightarrow{T}_{2}\begin{bmatrix} B_{21}\left(\omega\right) \\ B_{24}\left(\omega\right) \\ B_{23}\left(\omega\right) \end{bmatrix} \quad\quad\implies\quad\quad \begin{bmatrix} -F_{12}\left(\omega\right)e^{i\omega\tau_{12}} \\ -F_{42}\left(\omega\right)e^{i\omega\tau_{24}} \\ -B_{32}\left(\omega\right)e^{i\omega\tau_{23}} \end{bmatrix} = \overset\Leftrightarrow{T}_{2}\begin{bmatrix} -F_{12}\left(\omega\right)e^{-i\omega\tau_{12}} \\ -F_{42}\left(\omega\right)e^{-i\omega\tau_{24}} \\ B_{23}\left(\omega\right) \end{bmatrix}. \label{sswd:joint2amps}$$ $$\begin{bmatrix} F_{34}\left(\omega\right) \\ F_{31}\left(\omega\right) \\ F_{32}\left(\omega\right) \end{bmatrix} = \overset\Leftrightarrow{T}_{3}\begin{bmatrix} B_{34}\left(\omega\right) \\ B_{31}\left(\omega\right) \\ B_{32}\left(\omega\right) \end{bmatrix} \quad\quad\implies\quad\quad \begin{bmatrix} -F_{43}\left(\omega\right)e^{i\omega\tau_{34}} \\ -F_{13}\left(\omega\right)e^{i\omega\tau_{13}} \\ -B_{23}\left(\omega\right)e^{i\omega\tau_{23}} \end{bmatrix} = \overset\Leftrightarrow{T}_{3}\begin{bmatrix} -F_{43}\left(\omega\right)e^{-i\omega\tau_{34}} \\ -F_{13}\left(\omega\right)e^{-i\omega\tau_{13}} \\ B_{32}\left(\omega\right) \end{bmatrix}. \label{sswd:joint3amps}$$ [\[sswd:joint23amps\]]{#sswd:joint23amps label="sswd:joint23amps"} Eq. [\[sswd:joint23amps\]](#sswd:joint23amps){reference-type="ref" reference="sswd:joint23amps"} represents a total of 6 equations with 6 unknown variables; $F_{12}(\omega)$, $F_{42}(\omega)$, $F_{43}(\omega)$, $F_{13}(\omega)$, $B_{23}(\omega)$, and $B_{32}(\omega)$. However, there is clearly not one unique solution to these equations as simply multiplying each variable by the same constant will also produce a solution. Here we will derive the frequency condition necessary to resolve this issue. We now begin to condense out some of these variables by manipulating the various pieces of Eq. [\[sswd:joint23amps\]](#sswd:joint23amps){reference-type="ref" reference="sswd:joint23amps"} into usable relations. First, we multiply the top entry of Eq. [\[sswd:joint2amps\]](#sswd:joint2amps){reference-type="ref" reference="sswd:joint2amps"} by $\Lambda_{12}$ and subtract from it the middle entry multiplied by $\Lambda_{24}$. A similar process can also be performed on the first two components of Eq. [\[sswd:joint3amps\]](#sswd:joint3amps){reference-type="ref" reference="sswd:joint3amps"} with $\Lambda_{34}$ and $\Lambda_{13}$. These manipulations eliminate $B_{23}(\omega)$ and $B_{32}(\omega)$ to produce $$\Lambda_{12}F_{12}\left(\omega\right)\sin\left(\omega\tau_{12}\right) = \Lambda_{24}F_{42}\left(\omega\right)\sin\left(\omega\tau_{24}\right), \quad\quad\quad \Lambda_{34}F_{43}\left(\omega\right)\sin\left(\omega\tau_{34}\right) = \Lambda_{13}F_{13}\left(\omega\right)\sin\left(\omega\tau_{13}\right). \label{sswd:F23rels}$$ We can immediately implement Eq. [\[sswd:F23rels\]](#sswd:F23rels){reference-type="ref" reference="sswd:F23rels"} by adding the first two components of Eq. [\[sswd:joint2amps\]](#sswd:joint2amps){reference-type="ref" reference="sswd:joint2amps"} without any extra factors of $\Lambda_{12}$ or $\Lambda_{24}$ and solving for $B_{23}(\omega)$ in terms of $F_{12}(\omega)$. Additionally solving for $B_{32}(\omega)$ via a similar process with Eq. [\[sswd:joint3amps\]](#sswd:joint3amps){reference-type="ref" reference="sswd:joint3amps"} then yields $$\begin{aligned} B_{23}\left(\omega\right) &= -\frac{\Lambda_{12}F_{12}\left(\omega\right)\sin\left(\omega\tau_{12}\right)}{\Lambda_{23}\sqrt{2}}\left(2i+\Lambda_{23}\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)\right)\right) \nonumber\\ &= -\frac{\Lambda_{24}F_{42}\left(\omega\right)\sin\left(\omega\tau_{24}\right)}{\Lambda_{23}\sqrt{2}}\left(2i+\Lambda_{23}\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)\right)\right), \label{sswd:BF23rels}\end{aligned}$$ $$\begin{aligned} B_{32}\left(\omega\right) &= -\frac{\Lambda_{34}F_{43}\left(\omega\right)\sin\left(\omega\tau_{34}\right)}{\Lambda_{23}\sqrt{2}}\left(2i+\Lambda_{23}\left(\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right)\right) \nonumber\\ &= -\frac{\Lambda_{13}F_{13}\left(\omega\right)\sin\left(\omega\tau_{13}\right)}{\Lambda_{23}\sqrt{2}}\left(2i+\Lambda_{23}\left(\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right)\right), \label{sswd:BF32rels}\end{aligned}$$ [\[sswd:BFrels\]]{#sswd:BFrels label="sswd:BFrels"} where we have introduced the notation $\eta_{\mu\nu}(\omega)=\Lambda_{\mu\nu}^{-1}\cot(\omega\tau_{\mu\nu})$. Eq. [\[sswd:BFrels\]](#sswd:BFrels){reference-type="ref" reference="sswd:BFrels"} then allows us to express the third components of both parts of Eq. [\[sswd:joint23amps\]](#sswd:joint23amps){reference-type="ref" reference="sswd:joint23amps"} as $$\begin{aligned} &B_{23}\left(\omega\right)\left(\frac{2\Lambda_{12}\Lambda_{24}\Lambda_{23}\left(\Lambda_{12}^{-1}\csc\left(\omega\tau_{12}\right)e^{-i\omega\tau_{12}}+\Lambda_{24}^{-1}\csc\left(\omega\tau_{24}\right)e^{-i\omega\tau_{24}}\right)}{\left(2i+\Lambda_{23}\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)\right)\right)\left(\Lambda_{12}\Lambda_{24}+\frac{1}{2}\Lambda_{23}\left(\Lambda_{12}+\Lambda_{24}\right)\right)}+\frac{\frac{1}{2}\Lambda_{23}\left(\Lambda_{12}+\Lambda_{24}\right)-\Lambda_{12}\Lambda_{24}}{\Lambda_{12}\Lambda_{24}+\frac{1}{2}\Lambda_{23}\left(\Lambda_{12}+\Lambda_{24}\right)}\right) \nonumber\\ &= B_{23}\left(\omega\right)\frac{\Lambda_{23}\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)\right)-2i}{\Lambda_{23}\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)\right)+2i} = -B_{32}\left(\omega\right)e^{i\omega\tau_{23}}, \label{sswd:B23con}\end{aligned}$$ $$\begin{aligned} &B_{32}\left(\omega\right)\left(\frac{2\Lambda_{34}\Lambda_{13}\Lambda_{23}\left(\Lambda_{34}^{-1}\csc\left(\omega\tau_{34}\right)e^{-i\omega\tau_{34}}+\Lambda_{13}^{-1}\csc\left(\omega\tau_{13}\right)e^{-i\omega\tau_{13}}\right)}{\left(2i+\Lambda_{23}\left(\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right)\right)\left(\Lambda_{34}\Lambda_{13}+\frac{1}{2}\Lambda_{23}\left(\Lambda_{34}+\Lambda_{13}\right)\right)}+\frac{\frac{1}{2}\Lambda_{23}\left(\Lambda_{34}+\Lambda_{13}\right)-\Lambda_{34}\Lambda_{13}}{\Lambda_{34}\Lambda_{13}+\frac{1}{2}\Lambda_{23}\left(\Lambda_{34}+\Lambda_{13}\right)}\right) \nonumber\\ &= B_{32}\left(\omega\right)\frac{\Lambda_{23}\left(\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right)-2i}{\Lambda_{23}\left(\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right)+2i} = -B_{23}\left(\omega\right)e^{i\omega\tau_{23}} \label{sswd:B32con}\end{aligned}$$ [\[sswd:Bcons\]]{#sswd:Bcons label="sswd:Bcons"} The only way for Eqs. [\[sswd:B23con\]](#sswd:B23con){reference-type="ref" reference="sswd:B23con"} and [\[sswd:B32con\]](#sswd:B32con){reference-type="ref" reference="sswd:B32con"} to both be true is for $$\begin{aligned} &e^{2i\omega\tau_{23}} = \frac{\Lambda_{23}\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)\right)-2i}{\Lambda_{23}\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)\right)+2i} \frac{\Lambda_{23}\left(\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right)-2i}{\Lambda_{23}\left(\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right)+2i} \nonumber\\ &\implies\quad \Lambda_{23}^{-2}-\frac{1}{2}\eta_{23}\left(\omega\right)\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)+\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right)-\frac{1}{4}\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)\right)\left(\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right) = 0 \label{sswd:omegacon}\end{aligned}$$ to also be true. Thus, frequencies which satisfy Eq. [\[sswd:omegacon\]](#sswd:omegacon){reference-type="ref" reference="sswd:omegacon"} are natural frequencies of the structure capable of force-free oscillatory modes. # Square structure: Laplacian inversion {#app:ssli} We now seek to derive the same natural frequency condition for the square structure studied in Sec. [3.4](#sec:Square){reference-type="ref" reference="sec:Square"} by calculating the determinant of the network Laplacian; which for this particular structure has components $$\mathbf{\overset\Leftrightarrow{D}}_{11} = \begin{bmatrix} \Lambda_{12}\omega\cot\left(\omega\tau_{12}\right) & 0 \\ 0 & \Lambda_{13}\omega\cot\left(\omega\tau_{13}\right) \end{bmatrix}, \label{ssli:D11}$$ $$\mathbf{\overset\Leftrightarrow{D}}_{22} = \begin{bmatrix} \Lambda_{12}\omega\cot\left(\omega\tau_{12}\right)+\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right) & -\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right) \\ -\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right) & \Lambda_{24}\omega\cot\left(\omega\tau_{24}\right)+\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right) \end{bmatrix}, \label{ssli:D22}$$ $$\mathbf{\overset\Leftrightarrow{D}}_{33} = \begin{bmatrix} \Lambda_{34}\omega\cot\left(\omega\tau_{34}\right)+\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right) & -\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right) \\ -\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right) & \Lambda_{13}\omega\cot\left(\omega\tau_{13}\right)+\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right) \end{bmatrix}, \label{ssli:D33}$$ $$\mathbf{\overset\Leftrightarrow{D}}_{44} = \begin{bmatrix} \Lambda_{34}\omega\cot\left(\omega\tau_{34}\right) & 0 \\ 0 & \Lambda_{24}\omega\cot\left(\omega\tau_{24}\right) \end{bmatrix}, \label{ssli:D44}$$ $$\mathbf{\overset\Leftrightarrow{D}}_{12} = \mathbf{\overset\Leftrightarrow{D}}_{21} = \begin{bmatrix} -\Lambda_{12}\omega\csc\left(\omega\tau_{12}\right) & 0 \\ 0 & 0 \end{bmatrix}, \label{ssli:D12}$$ $$\mathbf{\overset\Leftrightarrow{D}}_{24} = \mathbf{\overset\Leftrightarrow{D}}_{42} = \begin{bmatrix} 0 & 0 \\ 0 & -\Lambda_{24}\omega\csc\left(\omega\tau_{24}\right) \end{bmatrix}, \label{ssli:D24}$$ $$\mathbf{\overset\Leftrightarrow{D}}_{34} = \mathbf{\overset\Leftrightarrow{D}}_{43} = \begin{bmatrix} -\Lambda_{34}\omega\csc\left(\omega\tau_{34}\right) & 0 \\ 0 & 0 \end{bmatrix}, \label{ssli:D34}$$ $$\mathbf{\overset\Leftrightarrow{D}}_{13} = \mathbf{\overset\Leftrightarrow{D}}_{31} = \begin{bmatrix} 0 & 0 \\ 0 & -\Lambda_{13}\omega\csc\left(\omega\tau_{13}\right) \end{bmatrix}, \label{ssli:D13}$$ $$\mathbf{\overset\Leftrightarrow{D}}_{23} = \mathbf{\overset\Leftrightarrow{D}}_{32} = \begin{bmatrix} -\frac{1}{2}\Lambda_{23}\omega\csc\left(\omega\tau_{23}\right) & \frac{1}{2}\Lambda_{23}\omega\csc\left(\omega\tau_{23}\right) \\ \frac{1}{2}\Lambda_{23}\omega\csc\left(\omega\tau_{23}\right) & -\frac{1}{2}\Lambda_{23}\omega\csc\left(\omega\tau_{23}\right) \end{bmatrix}, \label{ssli:D23}$$ $$\mathbf{\overset\Leftrightarrow{D}}_{14} = \mathbf{\overset\Leftrightarrow{D}}_{41} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}. \label{ssli:D14}$$ [\[ssli:Dcoms\]]{#ssli:Dcoms label="ssli:Dcoms"} From here we can find the determinant of $\mathbf{\overset\Leftrightarrow{D}}$ by first finding its inverse. To do so, let us define the additional matrices $$\begin{aligned} \overset\Leftrightarrow{U}_{2} &= \mathbf{\overset\Leftrightarrow{D}}_{22}-\mathbf{\overset\Leftrightarrow{D}}_{12}\mathbf{\overset\Leftrightarrow{D}}_{11}^{-1}\mathbf{\overset\Leftrightarrow{D}}_{12}-\mathbf{\overset\Leftrightarrow{D}}_{24}\mathbf{\overset\Leftrightarrow{D}}_{44}^{-1}\mathbf{\overset\Leftrightarrow{D}}_{24} \nonumber\\ &= \begin{bmatrix} \frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right)-\Lambda_{12}\omega\tan\left(\omega\tau_{12}\right) & -\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right) \\ -\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right) & \frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right)-\Lambda_{24}\omega\tan\left(\omega\tau_{24}\right) \end{bmatrix}, \label{ssli:U2}\end{aligned}$$ $$\begin{aligned} \overset\Leftrightarrow{U}_{3} &= \mathbf{\overset\Leftrightarrow{D}}_{33}-\mathbf{\overset\Leftrightarrow{D}}_{13}\mathbf{\overset\Leftrightarrow{D}}_{11}^{-1}\mathbf{\overset\Leftrightarrow{D}}_{13}-\mathbf{\overset\Leftrightarrow{D}}_{34}\mathbf{\overset\Leftrightarrow{D}}_{44}^{-1}\mathbf{\overset\Leftrightarrow{D}}_{34} \nonumber\\ &= \begin{bmatrix} \frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right)-\Lambda_{34}\omega\tan\left(\omega\tau_{34}\right) & -\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right) \\ -\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right) & \frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right)-\Lambda_{13}\omega\tan\left(\omega\tau_{13}\right) \end{bmatrix}, \label{ssli:U3}\end{aligned}$$ $$\begin{aligned} \overset\Leftrightarrow{V}_{2} &= \overset\Leftrightarrow{U}_{2}-\mathbf{\overset\Leftrightarrow{D}}_{23}\overset\Leftrightarrow{U}_{3}^{-1}\mathbf{\overset\Leftrightarrow{D}}_{23} \nonumber\\ &= \begin{bmatrix} \frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right)\nu_{3}\left(\omega\right)-\Lambda_{12}\omega\tan\left(\omega\tau_{12}\right) & -\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right)\nu_{3}\left(\omega\right) \\ -\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right)\nu_{3}\left(\omega\right) & \frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right)\nu_{3}\left(\omega\right)-\Lambda_{24}\omega\tan\left(\omega\tau_{24}\right) \end{bmatrix}, \label{ssli:V2}\end{aligned}$$ $$\begin{aligned} \overset\Leftrightarrow{V}_{3} &= \overset\Leftrightarrow{U}_{3}-\mathbf{\overset\Leftrightarrow{D}}_{23}\overset\Leftrightarrow{U}_{2}^{-1}\mathbf{\overset\Leftrightarrow{D}}_{23} \nonumber\\ &= \begin{bmatrix} \frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right)\nu_{2}\left(\omega\right)-\Lambda_{34}\omega\tan\left(\omega\tau_{34}\right) & -\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right)\nu_{2}\left(\omega\right) \\ -\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right)\nu_{2}\left(\omega\right) & \frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right)\nu_{2}\left(\omega\right)-\Lambda_{13}\omega\tan\left(\omega\tau_{13}\right) \end{bmatrix}, \label{ssli:V3}\end{aligned}$$ [\[ssli:UVdefs\]]{#ssli:UVdefs label="ssli:UVdefs"} where $$\nu_{2}\left(\omega\right) = \frac{1+\frac{1}{2}\Lambda_{23}\tan\left(\omega\tau_{23}\right)\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)\right)}{1-\frac{1}{2}\Lambda_{23}\cot\left(\omega\tau_{23}\right)\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)\right)}, \label{ssli:nu2}$$ $$\nu_{3}\left(\omega\right) = \frac{1+\frac{1}{2}\Lambda_{23}\tan\left(\omega\tau_{23}\right)\left(\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right)}{1-\frac{1}{2}\Lambda_{23}\cot\left(\omega\tau_{23}\right)\left(\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right)}. \label{ssli:nu3}$$ [\[ssli:nudefs\]]{#ssli:nudefs label="ssli:nudefs"} Additionally, we note that $$\mathbf{\overset\Leftrightarrow{D}}_{12}\mathbf{\overset\Leftrightarrow{D}}_{11}^{-1}\mathbf{\overset\Leftrightarrow{D}}_{13} = \mathbf{\overset\Leftrightarrow{D}}_{13}\mathbf{\overset\Leftrightarrow{D}}_{11}^{-1}\mathbf{\overset\Leftrightarrow{D}}_{12} = \mathbf{\overset\Leftrightarrow{D}}_{24}\mathbf{\overset\Leftrightarrow{D}}_{44}^{-1}\mathbf{\overset\Leftrightarrow{D}}_{34} = \mathbf{\overset\Leftrightarrow{D}}_{34}\mathbf{\overset\Leftrightarrow{D}}_{44}^{-1}\mathbf{\overset\Leftrightarrow{D}}_{24} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}. \label{ssli:Dzeros}$$ These allow us to more easily construct the block matrices $$\mathbf{\overset\Leftrightarrow{E}} = \begin{bmatrix} \overset\Leftrightarrow{I}_{2} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} \\ -\mathbf{\overset\Leftrightarrow{D}}_{12}\mathbf{\overset\Leftrightarrow{D}}_{11}^{-1} & \overset\Leftrightarrow{I}_{2} & \overset\Leftrightarrow{0} & -\mathbf{\overset\Leftrightarrow{D}}_{24}\mathbf{\overset\Leftrightarrow{D}}_{44}^{-1} \\ -\mathbf{\overset\Leftrightarrow{D}}_{13}\mathbf{\overset\Leftrightarrow{D}}_{11}^{-1} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{I}_{2} & -\mathbf{\overset\Leftrightarrow{D}}_{34}\mathbf{\overset\Leftrightarrow{D}}_{44}^{-1} \\ \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{I}_{2} \end{bmatrix}, \label{ssli:E}$$ $$\mathbf{\overset\Leftrightarrow{F}} = \begin{bmatrix} \overset\Leftrightarrow{I}_{2} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} \\ \overset\Leftrightarrow{0} & \overset\Leftrightarrow{I}_{2} & -\mathbf{\overset\Leftrightarrow{D}}_{23}\overset\Leftrightarrow{U}_{3}^{-1} & \overset\Leftrightarrow{0} \\ \overset\Leftrightarrow{0} & -\mathbf{\overset\Leftrightarrow{D}}_{23}\overset\Leftrightarrow{U}_{2}^{-1} & \overset\Leftrightarrow{I}_{2} & \overset\Leftrightarrow{0} \\ \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{I}_{2} \end{bmatrix}, \label{ssli:F}$$ $$\mathbf{\overset\Leftrightarrow{G}} = \begin{bmatrix} \overset\Leftrightarrow{I}_{2} & -\mathbf{\overset\Leftrightarrow{D}}_{12}\overset\Leftrightarrow{V}_{2}^{-1} & -\mathbf{\overset\Leftrightarrow{D}}_{13}\overset\Leftrightarrow{V}_{3}^{-1} & \overset\Leftrightarrow{0} \\ \overset\Leftrightarrow{0} & \overset\Leftrightarrow{I}_{2} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} \\ \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{I}_{2} & \overset\Leftrightarrow{0} \\ \overset\Leftrightarrow{0} & -\mathbf{\overset\Leftrightarrow{D}}_{24}\overset\Leftrightarrow{V}_{2}^{-1} & -\mathbf{\overset\Leftrightarrow{D}}_{34}\overset\Leftrightarrow{V}_{3}^{-1} & \overset\Leftrightarrow{I}_{2} \end{bmatrix}, \label{ssli:G}$$ $$\mathbf{\overset\Leftrightarrow{H}} = \begin{bmatrix} \mathbf{\overset\Leftrightarrow{D}}_{11}^{-1} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} \\ \overset\Leftrightarrow{0} & \overset\Leftrightarrow{V}_{2}^{-1} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} \\ \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{V}_{3}^{-1} & \overset\Leftrightarrow{0} \\ \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} & \mathbf{\overset\Leftrightarrow{D}}_{44}^{-1} \end{bmatrix}. \label{ssli:H}$$ [\[ssli:EFGHdefs\]]{#ssli:EFGHdefs label="ssli:EFGHdefs"} Each of these matrices represents a distinct step in the process of inverting $\mathbf{\overset\Leftrightarrow{D}}$, thus allowing us to write $$\mathbf{\overset\Leftrightarrow{D}}^{-1} = \mathbf{\overset\Leftrightarrow{H}}\mathbf{\overset\Leftrightarrow{G}}\mathbf{\overset\Leftrightarrow{F}}\mathbf{\overset\Leftrightarrow{E}}. \label{ssli:Dinvsol}$$ Eq. [\[ssli:Dinvsol\]](#ssli:Dinvsol){reference-type="ref" reference="ssli:Dinvsol"} is particularly helpful due to each of the component matrices having relatively simple determinants, thus providing the solution $$\begin{aligned} &\text{det}\left(\mathbf{\overset\Leftrightarrow{D}}\right) = \left(\text{det}\left(\mathbf{\overset\Leftrightarrow{H}}\right)\text{det}\left(\mathbf{\overset\Leftrightarrow{G}}\right)\text{det}\left(\mathbf{\overset\Leftrightarrow{F}}\right)\text{det}\left(\mathbf{\overset\Leftrightarrow{E}}\right)\right)^{-1} = \frac{\text{det}\left(\mathbf{\overset\Leftrightarrow{D}}_{11}\right)\text{det}\left(\overset\Leftrightarrow{V}_{2}\right)\text{det}\left(\overset\Leftrightarrow{V}_{3}\right)\text{det}\left(\mathbf{\overset\Leftrightarrow{D}}_{44}\right)}{\text{det}\left(\overset\Leftrightarrow{I}_{2}-\mathbf{\overset\Leftrightarrow{D}}_{23}\overset\Leftrightarrow{U}_{2}^{-1}\mathbf{\overset\Leftrightarrow{D}}_{23}\overset\Leftrightarrow{U}_{3}^{-1}\right)} \nonumber\\ &= \text{det}\left(\mathbf{\overset\Leftrightarrow{D}}_{11}\right)\text{det}\left(\overset\Leftrightarrow{V}_{2}\right)\text{det}\left(\overset\Leftrightarrow{U}_{3}\right)\text{det}\left(\mathbf{\overset\Leftrightarrow{D}}_{44}\right) \nonumber\\ &= \left(\Lambda_{12}\Lambda_{13}\omega^{2}\cot\left(\omega\tau_{12}\right)\cot\left(\omega\tau_{13}\right)\right)\left(\Lambda_{24}\Lambda_{34}\omega^{2}\cot\left(\omega\tau_{24}\right)\cot\left(\omega\tau_{34}\right)\right) \nonumber\\ &\quad \cdot\left(\Lambda_{12}\Lambda_{24}\omega^{2}\tan\left(\omega\tau_{12}\right)\tan\left(\omega\tau_{24}\right)-\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right)\nu_{3}\left(\omega\right)\left(\Lambda_{12}\omega\tan\left(\omega\tau_{12}\right)+\Lambda_{24}\omega\tan\left(\omega\tau_{24}\right)\right)\right) \nonumber\\ &\quad \cdot\left(\Lambda_{34}\Lambda_{13}\omega^{2}\tan\left(\omega\tau_{34}\right)\tan\left(\omega\tau_{13}\right)-\frac{1}{2}\Lambda_{23}\omega\cot\left(\omega\tau_{23}\right)\left(\Lambda_{34}\omega\tan\left(\omega\tau_{34}\right)+\Lambda_{13}\omega\tan\left(\omega\tau_{13}\right)\right)\right) \nonumber\\ &= \omega^{8}\left(\Lambda_{12}\Lambda_{24}\Lambda_{34}\Lambda_{13}\Lambda_{23}\right)^{2} \nonumber\\ &\quad \cdot\left(\Lambda_{23}^{-2}-\frac{1}{2}\eta_{23}\left(\omega\right)\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)+\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right)-\frac{1}{4}\left(\eta_{12}\left(\omega\right)+\eta_{24}\left(\omega\right)\right)\left(\eta_{34}\left(\omega\right)+\eta_{13}\left(\omega\right)\right)\right). \label{ssli:Ddet}\end{aligned}$$ Thus, we see that the condition $\text{det}(\mathbf{\overset\Leftrightarrow{D}})=0$ is identical to that provided by Eq. [\[sswd:omegacon\]](#sswd:omegacon){reference-type="ref" reference="sswd:omegacon"} for nonzero frequencies. # 7 Truss Bridge: natural frequencies and modes {#app:7TB} Here, we solve for the allowable vibration modes of the 7 truss bridge discussed in Sec. [3.5](#sec:7TB){reference-type="ref" reference="sec:7TB"}. We again utilized an aligned coordinate set with respect to the given joint positions. Firstly, we note that due to each truss having the same value of $\Lambda$ and $\tau$, the components of the network Laplacian can be written as $$\mathbf{\overset\Leftrightarrow{D}}_{\mu\nu}\left(\omega\right) = \Lambda\omega\csc\left(\omega\tau\right)\mathbf{\overset\Leftrightarrow{C}}_{\mu\nu}\left(\omega\right), \label{ATB:DCrel}$$ where $$\mathbf{\overset\Leftrightarrow{C}}_{11}\left(\omega\right) = \cos\left(\omega\tau\right)\begin{bmatrix} \frac{5}{4} & \frac{\sqrt{3}}{4} \\ \frac{\sqrt{3}}{4} & \frac{3}{4} \end{bmatrix}, \label{ATB:C11}$$ $$\mathbf{\overset\Leftrightarrow{C}}_{22}\left(\omega\right) = \cos\left(\omega\tau\right)\begin{bmatrix} \frac{3}{2} & 0 \\ 0 & \frac{3}{2} \end{bmatrix}, \label{ATB:C22}$$ $$\mathbf{\overset\Leftrightarrow{C}}_{33}\left(\omega\right) = \cos\left(\omega\tau\right)\begin{bmatrix} \frac{5}{2} & 0 \\ 0 & \frac{3}{2} \end{bmatrix}, \label{ATB:C33}$$ $$\mathbf{\overset\Leftrightarrow{C}}_{44}\left(\omega\right) = \cos\left(\omega\tau\right)\begin{bmatrix} \frac{3}{2} & 0 \\ 0 & \frac{3}{2} \end{bmatrix}, \label{ATB:C44}$$ $$\mathbf{\overset\Leftrightarrow{C}}_{55}\left(\omega\right) = \cos\left(\omega\tau\right)\begin{bmatrix} \frac{5}{4} & -\frac{\sqrt{3}}{4} \\ -\frac{\sqrt{3}}{4} & \frac{3}{4} \end{bmatrix}, \label{ATB:C55}$$ $$\mathbf{\overset\Leftrightarrow{C}}_{12}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{21}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{34}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{43}\left(\omega\right) = -\begin{bmatrix} \frac{1}{4} & \frac{\sqrt{3}}{4} \\ \frac{\sqrt{3}}{4} & \frac{3}{4} \end{bmatrix}, \label{ATB:C1234}$$ $$\mathbf{\overset\Leftrightarrow{C}}_{13}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{31}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{24}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{42}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{35}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{53}\left(\omega\right) = -\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \label{ATB:C132435}$$ $$\mathbf{\overset\Leftrightarrow{C}}_{23}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{32}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{45}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{54}\left(\omega\right) = -\begin{bmatrix} \frac{1}{4} & -\frac{\sqrt{3}}{4} \\ -\frac{\sqrt{3}}{4} & \frac{3}{4} \end{bmatrix}, \label{ATB:C2345}$$ $$\mathbf{\overset\Leftrightarrow{C}}_{14}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{41}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{15}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{51}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{25}\left(\omega\right) = \mathbf{\overset\Leftrightarrow{C}}_{51}\left(\omega\right) = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}. \label{ATB:C141525}$$ [\[ATB:C\]]{#ATB:C label="ATB:C"} With this setup, we can solve Eq. [\[DWPformTB_small\]](#DWPformTB_small){reference-type="ref" reference="DWPformTB_small"} via a series of elementary row operations represented by the block matrices $$\mathbf{\overset\Leftrightarrow{E}} = \begin{bmatrix} \overset\Leftrightarrow{I}_{2} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} \\ -\mathbf{\overset\Leftrightarrow{C}}_{23}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1} & \overset\Leftrightarrow{I}_{2} & \overset\Leftrightarrow{0} \\ -\mathbf{\overset\Leftrightarrow{C}}_{24}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{I}_{2} \end{bmatrix}, \label{ATB:E}$$ $$\mathbf{\overset\Leftrightarrow{F}} = \begin{bmatrix} \overset\Leftrightarrow{I}_{2} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} \\ \overset\Leftrightarrow{0} & \overset\Leftrightarrow{I}_{2} & \overset\Leftrightarrow{0} \\ \overset\Leftrightarrow{0} & -\left(\mathbf{\overset\Leftrightarrow{C}}_{34}-\mathbf{\overset\Leftrightarrow{C}}_{24}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\mathbf{\overset\Leftrightarrow{C}}_{23}\right)\left(\mathbf{\overset\Leftrightarrow{C}}_{33}-\mathbf{\overset\Leftrightarrow{C}}_{23}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\mathbf{\overset\Leftrightarrow{C}}_{23}\right)^{-1} & \overset\Leftrightarrow{I}_{2} \end{bmatrix}. \label{ATB:F}$$ [\[ATB:EF\]]{#ATB:EF label="ATB:EF"} These definitions along allow for Eq. [\[DWPformTB_small\]](#DWPformTB_small){reference-type="ref" reference="DWPformTB_small"} to be transformed into the equality $$\begin{bmatrix} \vec{0} \\ \vec{0} \\ \vec{0} \end{bmatrix} = \mathbf{\overset\Leftrightarrow{F}}\mathbf{\overset\Leftrightarrow{E}}\mathbf{\overset\Leftrightarrow{C}}\begin{bmatrix} \tilde{\vec{u}}_{2} \\ \tilde{\vec{u}}_{3} \\ \tilde{\vec{u}}_{4} \end{bmatrix} = \begin{bmatrix} \mathbf{\overset\Leftrightarrow{C}}_{22} & \mathbf{\overset\Leftrightarrow{C}}_{23} & \mathbf{\overset\Leftrightarrow{C}}_{24} \\ \overset\Leftrightarrow{0} & \mathbf{\overset\Leftrightarrow{C}}_{33}-\mathbf{\overset\Leftrightarrow{C}}_{23}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\mathbf{\overset\Leftrightarrow{C}}_{23} & \mathbf{\overset\Leftrightarrow{C}}_{34}-\mathbf{\overset\Leftrightarrow{C}}_{23}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\mathbf{\overset\Leftrightarrow{C}}_{24} \\ \overset\Leftrightarrow{0} & \overset\Leftrightarrow{0} & \overset\Leftrightarrow{U}_{4} \end{bmatrix}\begin{bmatrix} \tilde{\vec{u}}_{2} \\ \tilde{\vec{u}}_{3} \\ \tilde{\vec{u}}_{4} \end{bmatrix}, \label{ATB:CWPdiag}$$ where the prefactor of $\Lambda\omega\csc(\omega\tau)$ has been temporarily ignored and $$\overset\Leftrightarrow{U}_{4} = \mathbf{\overset\Leftrightarrow{C}}_{44}-\mathbf{\overset\Leftrightarrow{C}}_{24}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\mathbf{\overset\Leftrightarrow{C}}_{24}-\left(\mathbf{\overset\Leftrightarrow{C}}_{34}-\mathbf{\overset\Leftrightarrow{C}}_{24}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\mathbf{\overset\Leftrightarrow{C}}_{23}\right)\left(\mathbf{\overset\Leftrightarrow{C}}_{33}-\mathbf{\overset\Leftrightarrow{C}}_{23}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\mathbf{\overset\Leftrightarrow{C}}_{23}\right)^{-1}\left(\mathbf{\overset\Leftrightarrow{C}}_{34}-\mathbf{\overset\Leftrightarrow{C}}_{23}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\mathbf{\overset\Leftrightarrow{C}}_{24}\right). \label{ATB:U4def}$$ The known forms of the various matrices given in Eq. [\[ATB:CWPdiag\]](#ATB:CWPdiag){reference-type="ref" reference="ATB:CWPdiag"}, along with denoting $\cos(\omega\tau)$ as simply $c$ for the sake of simplifying notation, can then be used to produce $$\mathbf{\overset\Leftrightarrow{C}}_{33}-\mathbf{\overset\Leftrightarrow{C}}_{23}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\mathbf{\overset\Leftrightarrow{C}}_{23} = c\begin{bmatrix} \frac{5}{2} & 0 \\ 0 & \frac{3}{2} \end{bmatrix}-\frac{2}{3c}\begin{bmatrix} \frac{1}{4} & -\frac{\sqrt{3}}{4} \\ -\frac{\sqrt{3}}{4} & \frac{3}{4} \end{bmatrix}, \label{ATB:C33mod}$$ $$\mathbf{\overset\Leftrightarrow{C}}_{34}-\mathbf{\overset\Leftrightarrow{C}}_{23}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\mathbf{\overset\Leftrightarrow{C}}_{24} = -\begin{bmatrix} \frac{1}{4} & \frac{\sqrt{3}}{4} \\ \frac{\sqrt{3}}{4} & \frac{3}{4} \end{bmatrix}-\frac{2}{3c}\begin{bmatrix} \frac{1}{4} & 0 \\ -\frac{\sqrt{3}}{4} & 0 \end{bmatrix}, \label{ATB:C34mod}$$ $$\mathbf{\overset\Leftrightarrow{C}}_{34}-\mathbf{\overset\Leftrightarrow{C}}_{24}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\mathbf{\overset\Leftrightarrow{C}}_{23} = -\begin{bmatrix} \frac{1}{4} & \frac{\sqrt{3}}{4} \\ \frac{\sqrt{3}}{4} & \frac{3}{4} \end{bmatrix}-\frac{2}{3c}\begin{bmatrix} \frac{1}{4} & -\frac{\sqrt{3}}{4} \\ 0 & 0 \end{bmatrix}, \label{ATB:C34modT}$$ $$\begin{aligned} &\overset\Leftrightarrow{U}_{4} = c\begin{bmatrix} \frac{3}{2} & 0 \\ 0 & \frac{3}{2} \end{bmatrix}-\frac{2}{3c}\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}-\frac{1}{\left(\frac{5}{2}c-\frac{1}{6c}\right)\left(\frac{3}{2}c-\frac{1}{2c}\right)-\frac{1}{12c^{2}}}\left(\frac{9c}{4}\begin{bmatrix} \frac{1}{4} & \frac{\sqrt{3}}{4} \\ \frac{\sqrt{3}}{4} & \frac{3}{4} \end{bmatrix}-\begin{bmatrix} \frac{1}{2} & \frac{\sqrt{3}}{4} \\ \frac{\sqrt{3}}{4} & 0 \end{bmatrix}+\frac{1}{2c}\begin{bmatrix} \frac{1}{4} & -\frac{\sqrt{3}}{4} \\ -\frac{\sqrt{3}}{4} & -\frac{3}{4} \end{bmatrix}\right) \nonumber\\ &= \frac{1}{c\left(\frac{15}{4}c^{2}-\frac{3}{2}\right)}\left(\frac{45c^{4}}{8}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}-\frac{9c^{2}}{4}\begin{bmatrix} \frac{85}{36} & \frac{\sqrt{3}}{4} \\ \frac{\sqrt{3}}{4} & \frac{7}{4} \end{bmatrix}+c\begin{bmatrix} \frac{1}{2} & \frac{\sqrt{3}}{4} \\ \frac{\sqrt{3}}{4} & 0 \end{bmatrix}+\frac{1}{2}\begin{bmatrix} \frac{7}{4} & \frac{\sqrt{3}}{4} \\ \frac{\sqrt{3}}{4} & \frac{3}{4} \end{bmatrix}\right). \label{ATB:U4sol}\end{aligned}$$ [\[ATB:submatsols\]]{#ATB:submatsols label="ATB:submatsols"} When the matrix given by Eq. [\[ATB:U4sol\]](#ATB:U4sol){reference-type="ref" reference="ATB:U4sol"} has a determinant of 0, then there exists a possible $\tilde{\vec{u}}_{4}$ that can satisfy Eq. [\[ATB:CWPdiag\]](#ATB:CWPdiag){reference-type="ref" reference="ATB:CWPdiag"}. This determinant takes the form of a polynomial function in $c$ that can be reduced via $$\begin{aligned} \text{det}\left(\overset\Leftrightarrow{U}_{4}\right) &= \frac{1}{c^{2}\left(\frac{15}{4}c^{2}-\frac{3}{2}\right)^{2}}\left(\left(\frac{45}{8}c^{4}-\frac{85}{16}c^{2}+\frac{1}{2}c+\frac{7}{8}\right)\left(\frac{45}{8}c^{4}-\frac{63}{16}c^{2}+\frac{3}{8}\right)-\frac{3}{16}\left(-\frac{9}{4}c^{2}+c+\frac{1}{2}\right)^{2}\right) \nonumber\\ &= \frac{1}{\frac{9}{4}c^{2}\left(\frac{15}{4}c^{2}-\frac{3}{2}\right)}\cdot\frac{27}{64}\left(5c^{2}-5c+1\right)\left(3c^{2}-c-1\right)\left(3c+1\right)\left(c+1\right). \label{ATB:U4det}\end{aligned}$$ A significant amount of information can be extracted from the explicit form of Eq. [\[ATB:U4det\]](#ATB:U4det){reference-type="ref" reference="ATB:U4det"}. Firstly, when calculating the determinant of the entire matrix given in Eq. [\[ATB:CWPdiag\]](#ATB:CWPdiag){reference-type="ref" reference="ATB:CWPdiag"}, we need only multiply Eq. [\[ATB:U4det\]](#ATB:U4det){reference-type="ref" reference="ATB:U4det"} by the determinants of $\mathbf{\overset\Leftrightarrow{C}}_{22}$ and $\mathbf{\overset\Leftrightarrow{C}}_{33}-\mathbf{\overset\Leftrightarrow{C}}_{23}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\mathbf{\overset\Leftrightarrow{C}}_{23}$. This process merely cancels the polynomial in the denominator of Eq. [\[ATB:U4det\]](#ATB:U4det){reference-type="ref" reference="ATB:U4det"}, leaving only the polynomial in the numerator. Thus, the zeros of this polynomial will determine the allowable frequencies for the entire system. Fortunately, this polynomial separates cleanly into the product of two quadratic and two linear polynomials. The roots can thus be easily calculated to be $(5\pm\sqrt{5})/10$, $(1\pm\sqrt{13})/6$, $-1/3$, and $-1$. The first five of these roots all lie in the open interval $(-1,1)$, which in turn means there exist real frequencies that achieve these roots without causing the overall factor of $\csc(\omega\tau)$ extracted in Eq. [\[ATB:DCrel\]](#ATB:DCrel){reference-type="ref" reference="ATB:DCrel"} to diverge. The sixth root can be satisfied when $\omega\tau$ is an odd multiple of $\pi$, but this causes $\csc(\omega\tau)$ to diverge, thus requiring noticeably more work to find an appropriate solution. From here the solution method is straightforward. For each allowable value of $\cos(\omega\tau)$ we can use Eq. [\[ATB:U4sol\]](#ATB:U4sol){reference-type="ref" reference="ATB:U4sol"} to determine the explicit form of $\overset\Leftrightarrow{U}_{4}$ and calculate its null vector, $\tilde{\vec{u}}_{4}$. With that, we can use Eqs. [\[DWPformTB\]](#DWPformTB){reference-type="ref" reference="DWPformTB"} and [\[ATB:CWPdiag\]](#ATB:CWPdiag){reference-type="ref" reference="ATB:CWPdiag"} to readily solve for each other relevant vector as $$\tilde{\vec{u}}_{3} = -\left(\mathbf{\overset\Leftrightarrow{C}}_{33}-\mathbf{\overset\Leftrightarrow{C}}_{23}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\mathbf{\overset\Leftrightarrow{C}}_{23}\right)^{-1}\left(\mathbf{\overset\Leftrightarrow{C}}_{34}-\mathbf{\overset\Leftrightarrow{C}}_{23}\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\mathbf{\overset\Leftrightarrow{C}}_{24}\right)\tilde{\vec{u}}_{4}, \label{ATB:u3sol}$$ $$\tilde{\vec{u}}_{2} = -\mathbf{\overset\Leftrightarrow{C}}_{22}^{-1}\left(\mathbf{\overset\Leftrightarrow{C}}_{23}\tilde{\vec{u}}_{3}+\mathbf{\overset\Leftrightarrow{C}}_{24}\tilde{\vec{u}}_{4}\right), \label{ATB:u2sol}$$ $$\tilde{\vec{P}}_{1} = \Lambda\omega\csc\left(\omega\tau\right)\left(\mathbf{\overset\Leftrightarrow{C}}_{12}\tilde{\vec{u}}_{2}+\mathbf{\overset\Leftrightarrow{C}}_{13}\tilde{\vec{u}}_{3}\right), \label{ATB:P1sol}$$ $$\tilde{\vec{P}}_{5} = \Lambda\omega\csc\left(\omega\tau\right)\left(\mathbf{\overset\Leftrightarrow{C}}_{35}\tilde{\vec{u}}_{3}+\mathbf{\overset\Leftrightarrow{C}}_{45}\tilde{\vec{u}}_{4}\right). \label{ATB:P5sol}$$ [\[ATB:u32P15sol\]]{#ATB:u32P15sol label="ATB:u32P15sol"} This process yields the results $\displaystyle \cos\left(\omega\tau\right) = \frac{1}{6}\left(1+\sqrt{13}\right)$ $\displaystyle \cos\left(\omega\tau\right) = \frac{1}{10}\left(5+\sqrt{5}\right)$ $\displaystyle \cos\left(\omega\tau\right) = -\frac{1}{3}$ ------------------------------------- ------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------ $\displaystyle \tilde{\vec{u}}_{2}$ $\displaystyle \frac{1}{\sqrt{\frac{215}{972}-\frac{59}{972}\sqrt{13}}}\begin{bmatrix} $\displaystyle \frac{1}{\sqrt{\frac{9}{20}-\frac{7}{60}\sqrt{5}}}\begin{bmatrix} $\displaystyle \frac{1}{2\sqrt{7}}\begin{bmatrix} \frac{17}{72}-\frac{5}{72}\sqrt{13} \\ -\frac{3}{8}-\frac{1}{40}\sqrt{5} \\ 1 \\ \frac{\sqrt{3}}{4}\left(\frac{31}{54}-\frac{7}{54}\sqrt{13}\right) -\frac{\sqrt{3}}{4}\left(\frac{5}{6}-\frac{13}{30}\sqrt{5}\right) -3\sqrt{3} \end{bmatrix}$ \end{bmatrix}$ \end{bmatrix}$ $\displaystyle \tilde{\vec{u}}_{3}$ $\displaystyle \frac{1}{\sqrt{\frac{215}{972}-\frac{59}{972}\sqrt{13}}}\begin{bmatrix} $\displaystyle \frac{1}{\sqrt{\frac{9}{20}-\frac{7}{60}\sqrt{5}}}\begin{bmatrix} $\displaystyle \frac{1}{2\sqrt{7}}\begin{bmatrix} 0 \\ \frac{3}{10}-\frac{1}{5}\sqrt{5} \\ -6 \\ -\frac{\sqrt{3}}{4}\left(\frac{10}{27}-\frac{4}{27}\sqrt{13}\right) 0 0 \end{bmatrix}$ \end{bmatrix}$ \end{bmatrix}$ $\displaystyle \tilde{\vec{u}}_{4}$ $\displaystyle \frac{1}{\sqrt{\frac{215}{972}-\frac{59}{972}\sqrt{13}}}\begin{bmatrix} $\displaystyle \frac{1}{\sqrt{\frac{9}{20}-\frac{7}{60}\sqrt{5}}}\begin{bmatrix} $\displaystyle \frac{1}{2\sqrt{7}}\begin{bmatrix} -\frac{17}{72}+\frac{5}{72}\sqrt{13} \\ -\frac{3}{8}-\frac{1}{40}\sqrt{5} \\ 1 \\ \frac{\sqrt{3}}{4}\left(\frac{31}{54}-\frac{7}{54}\sqrt{13}\right) \frac{\sqrt{3}}{4}\left(\frac{5}{6}-\frac{13}{30}\sqrt{5}\right) 3\sqrt{3} \end{bmatrix}$ \end{bmatrix}$ \end{bmatrix}$ $\displaystyle \tilde{\vec{P}}_{1}$ $\displaystyle \frac{\Lambda\omega\csc\left(\omega\tau\right)}{\sqrt{\frac{215}{972}-\frac{59}{972}\sqrt{13}}}\begin{bmatrix} $\displaystyle \frac{\Lambda\omega\csc\left(\omega\tau\right)}{\sqrt{\frac{9}{20}-\frac{7}{60}\sqrt{5}}}\begin{bmatrix} $\displaystyle \frac{\Lambda\omega\csc\left(\omega\tau\right)}{2\sqrt{7}}\begin{bmatrix} -\frac{1}{6}+\frac{1}{24}\sqrt{13} \\ -\frac{1}{20}+\frac{1}{8}\sqrt{5} \\ 12 \\ -\frac{\sqrt{3}}{4}\left(\frac{2}{3}-\frac{1}{6}\sqrt{13}\right) \frac{\sqrt{3}}{4}\left(1-\frac{3}{10}\sqrt{5}\right) 3\sqrt{3} \end{bmatrix}$ \end{bmatrix}$ \end{bmatrix}$ $\displaystyle \tilde{\vec{P}}_{5}$ $\displaystyle \frac{\Lambda\omega\csc\left(\omega\tau\right)}{\sqrt{\frac{215}{972}-\frac{59}{972}\sqrt{13}}}\begin{bmatrix} $\displaystyle \frac{\Lambda\omega\csc\left(\omega\tau\right)}{\sqrt{\frac{9}{20}-\frac{7}{60}\sqrt{5}}}\begin{bmatrix} $\displaystyle \frac{\Lambda\omega\csc\left(\omega\tau\right)}{2\sqrt{7}}\begin{bmatrix} +\frac{1}{6}-\frac{1}{24}\sqrt{13} \\ -\frac{1}{20}+\frac{1}{8}\sqrt{5} \\ 12 \\ -\frac{\sqrt{3}}{4}\left(\frac{2}{3}-\frac{1}{6}\sqrt{13}\right) -\frac{\sqrt{3}}{4}\left(1-\frac{3}{10}\sqrt{5}\right) -3\sqrt{3} \end{bmatrix}$ \end{bmatrix}$ \end{bmatrix}$ $\displaystyle \cos\left(\omega\tau\right) = \frac{1}{6}\left(1-\sqrt{13}\right)$ $\displaystyle \cos\left(\omega\tau\right) = \frac{1}{10}\left(5-\sqrt{5}\right)$ $\displaystyle \cos\left(\omega\tau\right) = -1$ $\displaystyle \tilde{\vec{u}}_{2}$ $\displaystyle \frac{1}{\sqrt{\frac{215}{972}+\frac{59}{972}\sqrt{13}}}\begin{bmatrix} $\displaystyle \frac{1}{\sqrt{\frac{9}{20}+\frac{7}{60}\sqrt{5}}}\begin{bmatrix} $\displaystyle \frac{1}{2\sqrt{3}}\begin{bmatrix} \frac{17}{72}+\frac{5}{72}\sqrt{13} \\ -\frac{3}{8}+\frac{1}{40}\sqrt{5} \\ -3 \\ \frac{\sqrt{3}}{4}\left(\frac{31}{54}+\frac{7}{54}\sqrt{13}\right) -\frac{\sqrt{3}}{4}\left(\frac{5}{6}+\frac{13}{30}\sqrt{5}\right) \sqrt{3} \end{bmatrix}$ \end{bmatrix}$ \end{bmatrix}$ $\displaystyle \tilde{\vec{u}}_{3}$ $\displaystyle \frac{1}{\sqrt{\frac{215}{972}+\frac{59}{972}\sqrt{13}}}\begin{bmatrix} $\displaystyle \frac{1}{\sqrt{\frac{9}{20}+\frac{7}{60}\sqrt{5}}}\begin{bmatrix} $\displaystyle \frac{1}{2\sqrt{3}}\begin{bmatrix} 0 \\ \frac{3}{8}+\frac{1}{5}\sqrt{5} \\ 0 \\ -\frac{\sqrt{3}}{4}\left(\frac{10}{27}+\frac{4}{27}\sqrt{13}\right) 0 -2\sqrt{3} \end{bmatrix}$ \end{bmatrix}$ \end{bmatrix}$ $\displaystyle \tilde{\vec{u}}_{4}$ $\displaystyle \frac{1}{\sqrt{\frac{215}{972}+\frac{59}{972}\sqrt{13}}}\begin{bmatrix} $\displaystyle \frac{1}{\sqrt{\frac{9}{20}+\frac{7}{60}\sqrt{5}}}\begin{bmatrix} $\displaystyle \frac{1}{2\sqrt{3}}\begin{bmatrix} -\frac{17}{72}-\frac{5}{72}\sqrt{13} \\ -\frac{3}{8}+\frac{1}{40}\sqrt{5} \\ 3 \\ \frac{\sqrt{3}}{4}\left(\frac{31}{54}+\frac{7}{54}\sqrt{13}\right) \frac{\sqrt{3}}{4}\left(\frac{5}{6}+\frac{13}{30}\sqrt{5}\right) \sqrt{3} \end{bmatrix}$ \end{bmatrix}$ \end{bmatrix}$ $\displaystyle \tilde{\vec{P}}_{1}$ $\displaystyle \frac{\Lambda\omega\csc\left(\omega\tau\right)}{\sqrt{\frac{215}{972}+\frac{59}{972}\sqrt{13}}}\begin{bmatrix} $\displaystyle \frac{\Lambda\omega\csc\left(\omega\tau\right)}{\sqrt{\frac{9}{20}+\frac{7}{60}\sqrt{5}}}\begin{bmatrix} $\displaystyle \begin{bmatrix} -\frac{1}{6}-\frac{1}{24}\sqrt{13} \\ -\frac{1}{20}-\frac{1}{8}\sqrt{5} \\ 0 \\ -\frac{\sqrt{3}}{4}\left(\frac{2}{3}+\frac{1}{6}\sqrt{13}\right) \frac{\sqrt{3}}{4}\left(1+\frac{3}{10}\sqrt{5}\right) 0 \end{bmatrix}$ \end{bmatrix}$ \end{bmatrix}$ $\displaystyle \tilde{\vec{P}}_{5}$ $\displaystyle \frac{\Lambda\omega\csc\left(\omega\tau\right)}{\sqrt{\frac{215}{972}+\frac{59}{972}\sqrt{13}}}\begin{bmatrix} $\displaystyle \frac{\Lambda\omega\csc\left(\omega\tau\right)}{\sqrt{\frac{9}{20}+\frac{7}{60}\sqrt{5}}}\begin{bmatrix} $\displaystyle \begin{bmatrix} \frac{1}{6}+\frac{1}{24}\sqrt{13} \\ -\frac{1}{20}-\frac{1}{8}\sqrt{5} \\ 0 \\ -\frac{\sqrt{3}}{4}\left(\frac{2}{3}+\frac{1}{6}\sqrt{13}\right) -\frac{\sqrt{3}}{4}\left(1+\frac{3}{10}\sqrt{5}\right) 0 \end{bmatrix}$ \end{bmatrix}$ \end{bmatrix}$ The $\cos(\omega\tau)=-1$ case is particularly interesting. The $\tilde{\vec{u}}_{\mu}$ solutions presented here satisfy Eq. [\[DWPformTB_small\]](#DWPformTB_small){reference-type="ref" reference="DWPformTB_small"} without issue when $\cos(\omega\tau)=-1$. Additionally, inserting them into Eq. [\[DWPformTB\]](#DWPformTB){reference-type="ref" reference="DWPformTB"} while ignoring the divergent $\csc(\omega\tau)$ term yields vanishing forces at joints 1 and 5. Since $\csc(\omega\tau)$ diverges as $-(\omega\tau-m\pi)^{-1}$, where $m$ is an odd integer, when $\cos(\omega\tau)$ approaches -1, L'Hopital's rule dictates that the first derivatives with respect to $\omega$ of the velocities must also obey the same solution set given above so as to satisfy Eq. [\[DWPformTB_small\]](#DWPformTB_small){reference-type="ref" reference="DWPformTB_small"}. This in turn means that only the second order and higher terms of the velocities can contribute to the forces applied to joints 1 and 5, causing said forces to only have terms of first order and higher in $(\omega\tau-m\pi)$. Thus, these solutions represents a force free oscillation mode of the system that still satisfies the conditions that joints 1 and 5 are immobile.

arxiv_math

--- abstract: | In this short note, we provide the well-posedness for an hyperbolic-hyperbolic-elliptic system of PDEs describing the motion of collision free-plasma in magnetic fields. The proof combines a pointwise estimate together with a bootstrap type of argument for the elliptic part of the system. address: - Departamento de Análisis Matemático y Instituto de Matemáticas y Aplicaciones (IMAULL), Universidad de La Laguna C/. Astrofísico Francisco Sánchez s/n, 38200 - La Laguna, Spain. - Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria. Avda. Los Castros s/n, Santander, Spain. author: - Diego Alonso-Orán - Rafael Granero-Belinchón title: Well-posedness for an hyperbolic-hyperbolic-elliptic system describing cold plasmas --- # Introduction and main result The motion of a cold plasma in a magnetic field consisting of singly-charged particles can be described by the following system of PDEs [@berezin1964theory; @gardner1960similarity] [\[eq:1\]]{#eq:1 label="eq:1"} $$\begin{aligned} n_t+(un)_x&=0,\\ u_t+uu_x+\frac{BB_x}{n}&=0,\\ B-n-\left(\frac{B_x}{n}\right)_x&=0,\end{aligned}$$ where $n,u$ and $B$ are the ionic density, the ionic velocity and the magnetic field, respectively. Moreover, it has also been used as a simplified model to describe the motion of collission-free two fluid model where the electron inertial, charge separation and displacement current are neglected and the Poisson equation ([\[eq:1\]](#eq:1){reference-type="ref" reference="eq:1"}c) is initially satisfied, [@berezin1964theory; @kakutani1968reductive]. In ([\[eq:1\]](#eq:1){reference-type="ref" reference="eq:1"}) the spatial domain $\Omega$ is either $\Omega=\mathbb{R}$ or $\Omega=\mathbb{S}^1$ (*i.e.* $x\in\mathbb{R}$ or $x\in [-\pi,\pi]$ with periodic boundary conditions) and the time variable satisfies $t\in [0,T]$ for certain $0<T\leq\infty$. The corresponding initial-value problem consists of the system [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"} along with initial conditions $$\label{initialdata} n(x,0)=n_0(x),\;u(x,0)=u_0(x),$$ which are assumed to be smooth enough for the purposes of the work. System [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"} was introduced by Gardner & Morikawa [@gardner1960similarity]. Furthermore, Gardner & Morikawa formally showed that the solutions of [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"} converge to solutions of the Korteveg-de Vries equation (see also the paper by Su & Gardner [@su1969korteweg]). Berezin & Karpman extended this formal limit to the case where the wave propagates at angles of certain size with respect to the magnetic field [@berezin1964theory]. Later on, Kakutani, Ono, Taniuti & Wei [@kakutani1968reductive] removed the hypothesis on the angle. This formal KdV limit was recently justified by Pu & Li [@pu2019kdv]. Very recently in [@AloDurGra], by means of a multi-scale expansion (cf. [@AloranWaves; @CGSW18]), the authors derived three asymptotic models of [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"} and studied several analytical properties of the models: the existence of conserved quantities, the Hamiltonian structure, the well-posedness and the formation of singularities in finite time. More precisely, for the uni-directional model which resembles the well-known Fornberg-Whitham equation (cf. [@Fornberg-Whitham-78]), the authors showed that wave-breaking occurs, that is, the formation of an infinite slope in the solution. In [@ChenYang], a new sufficient condition on the initial data is given which exhibits wave breaking extending the previous work [@AloDurGra]. To the best of the author's knowledge, although system [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"} has been introduced more than 50 year's ago, the well-posedness of the system has not been studied elsewhere. The result of this work is to give a positive answer to the previous problem and the main theorem reads as follows **Theorem 1**. *Let $n_0(x)>0$, $n_0(x)-1\in H^2$ and $u_0(x)\in H^3$. Then, there exists a $T>0$ and unique solution of [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"} such that $$(n-1,u)\in C([0,T],H^2\times H^3).$$* ## Notation {#notation .unnumbered} For $1\leq p\leq\infty$, let $L^{p}=L^{p}(\mathbb R)$ be the usual normed space of $L^{p}$-functions on $\mathbb R$ with $||\cdot ||_{p}$ as the associated norm. For $s\in\mathbb R$, the inhomogeneous Sobolev space $H^{s}=H^s(\mathbb R)$ is defined as $$\begin{aligned} H^s(\mathbb R)\triangleq\left\{f\in L^2(\mathbb R):\|f\|_{H^s(\mathbb R)}^2=\int_\mathbb R(1+\xi^2)^s|\widehat{f}(\xi)|^2<+\infty\right\}, \end{aligned}$$ with norm $$\left\| f \right\|_{H^s}=\left\| f \right\|_{L^2}^2+\left\| f \right\|_{\dot{H}^s}.$$ Moreover, throughout the paper $C = C(\cdot)$ will denote a positive constant that may depend on fixed parameters and $x \lesssim y$ ($x \gtrsim y$) means that $x\le C y$ ($x\ge C y$) holds for some $C$. # Proof of Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"} {#sec:proof:thm} The proof follows the classical a priori estimates approach which combines the derivation of useful a priori energy estimates and the use of a suitable approximation procedure via mollifiers (see for instance [@AloDurGra]). First, we write system [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"} in the new variables $n=1+\eta, B=1+b$. Then system [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"} becomes [\[eq:2\]]{#eq:2 label="eq:2"} $$\begin{aligned} \eta_t+(u\eta)_x+u_{x}&=0,\\ u_t+uu_x+\frac{(1+b)b_x}{1+\eta}&=0,\\ b-\eta-\left(\frac{b_x}{1+\eta}\right)_x&=0.\end{aligned}$$ We are going to find the appropriate energy estimates for the following energy $$\label{energy} \mathcal{E}(t)=\left\| \eta(t) \right\|^{2}_{H^{2}}+\left\| u(t) \right\|^{2}_{H^{3}}+\displaystyle\max_{x\in \mathbb{R}}\frac{1}{1+\eta(x,t)}.$$ In order to estimate the last term in the energy $\mathcal{E}(t)$, we need to derive a pointwise estimate. To that purpose, following [@Cordoba-Cordoba-04] and defining $$\begin{aligned} m(t)=\displaystyle\min_{x\in\mathbb R}\eta(x,t)=\eta(\underline{x}_{t},t), \mbox{ for } t>0,\end{aligned}$$ it is easy to check that $m(t)$ is a Lipschitz functions and one has the following bound $$\left|m(t)-m(s)\right|\leq \max_{y,z}|\partial_t \eta(y,z)||t-s|.$$ From Rademacher's theorem it holds that $m(t)$ is differentiable in $t$ almost everywhere and furthermore $$\label{adg19} m'(t)=\partial_t \eta(\underline{x}_{t},t) \text{ a.e.}$$ Then, using ([\[eq:1\]](#eq:1){reference-type="ref" reference="eq:1"}a) and noticing that $n_{x}(\underline{x}_{t},t)=$ we readily see that $$\begin{aligned} \label{expresion} m'(t)=-u_{x}(\underline{x}_{t},t)m(t)-u_{x}(\underline{x}_{t},t)=-u_{x}(\underline{x}_{t},t)(1+m(t))\end{aligned}$$ Moreover, since by assumption $m(0)>-1$ we also have that $$\label{pointwise} m(t)>-1, \quad \mbox{ for } 0<t\ll 1.$$ We remark that this is not a monotonicity statement relying on a sign condition for $u_{x}(\underline{x}_{t},t)$, but just a small in time argument. Hence, following the argument in [@CGO] and using [\[expresion\]](#expresion){reference-type="eqref" reference="expresion"} we find that $$\begin{aligned} \label{pointwise2} \frac{d}{dt} \left(\displaystyle\max_{x\in \mathbb{R}}\frac{1}{1+\eta(x,t)}\right)=-\frac{\partial_{t}\eta(\underline{x}_{t},t)}{(1+m(t))^2}=\frac{u_{x}(\underline{x}_{t},t)}{1+m(t)}\leq C (\mathcal{E}(t))^2.\end{aligned}$$ The lower order $L^{2}$ norm of $\eta$ is bounded by $$\begin{aligned} \label{L2:eta} \frac{1}{2}\frac{d}{dt}\left\| \eta \right\|_{L^{2}}^{2}&\lesssim \left\| \eta \right\|_{L^{2}}^{2}\left\| u_{x} \right\|_{L^\infty}+\left\| \eta \right\|_{L^{2}}\left\| u_{x} \right\|_{L^{2}}\end{aligned}$$ Similarly, we find that $$\label{L2:u} \frac{1}{2}\frac{d}{dt}\left\| u \right\|_{L^{2}}^{2}\lesssim \left( 1+\left\| b \right\|_{L^{\infty}}\right)\left\| \frac{b_{x}}{1+\eta} \right\|_{L^{2}}\left\| u \right\|_{L^{2}}$$ Testing equation ([\[eq:2\]](#eq:2){reference-type="ref" reference="eq:2"}a) and ([\[eq:2\]](#eq:2){reference-type="ref" reference="eq:2"}b) with $\partial_{x}^{4}\eta$ and $\partial_{x}^{6}u$ respectively, and integrating by parts we have that $$\begin{aligned} \frac{1}{2}\frac{d}{dt}\left\| \partial_{x}^{2}\eta \right\|_{L^{2}}^{2}&\lesssim \left\| \eta \right\|_{H^{2}}^{2}\left\| u \right\|_{H^{3}}+\left\| \eta \right\|_{H^{2}}\left\| u \right\|_{H^{3}} \label{H2:eta}, \\ \frac{1}{2}\frac{d}{dt}\left\| \partial_{x}^{3}u \right\|_{L^{2}}^{2}& \lesssim \left\| u \right\|_{H^{3}}^{3}+\left( 1+\left\| b \right\|_{L^{\infty}}\right)\left\| \frac{b_{x}}{1+\eta} \right\|_{H^{3}}\left\| u \right\|_{H^{3}}. \label{H3:u}\end{aligned}$$ Therefore, combining [\[L2:eta\]](#L2:eta){reference-type="eqref" reference="L2:eta"}-[\[H3:u\]](#H3:u){reference-type="eqref" reference="H3:u"} and using Sobolev embedding and Young's inequality that $$\begin{aligned} \label{estimate:energy1} \frac{1}{2}\frac{d}{dt}\left(\left\| \eta \right\|_{H^{2}}^{2}+\left\| u \right\|_{H^{3}}^{2}\right)&\lesssim \left\| \eta \right\|_{H^{2}}^{3}+\left\| u \right\|_{H^{3}}^{3} + \left( 1+\left\| b \right\|_{H^{1}}\right)^{2}\left\| \frac{b_{x}}{1+\eta} \right\|^{2}_{H^{3}}+\left\| u \right\|_{H^{3}}^{2}. \end{aligned}$$ Moreover, using ([\[eq:2\]](#eq:2){reference-type="ref" reference="eq:2"}c) we find that $$\begin{aligned} \left\| \frac{b_{x}}{1+\eta} \right\|^{2}_{\dot{H}^{3}}=\int_{\mathbb{R}} \left| \left(\frac{b_{x}}{1+\eta}\right)_{xxx} \right|^{2} dx=\int_{\mathbb{R}} \left(\frac{b_{x}}{1+\eta}\right)_{xxx}\left(b-\eta\right)_{xx} \ dx\leq \left\| \frac{b_{x}}{1+\eta} \right\|_{\dot{H}^{3}}\left( \left\| \eta \right\|_{H^{2}}+\left\| b \right\|_{H^{2}}\right).\end{aligned}$$ Therefore, we find that $$\left\| \frac{b_{x}}{1+\eta} \right\|_{H^{3}} \leq \left\| \eta \right\|_{H^{2}}+\left\| b \right\|_{H^{2}}.$$ Plugging the previous estimate in [\[estimate:energy1\]](#estimate:energy1){reference-type="eqref" reference="estimate:energy1"} we infer that $$\begin{aligned} \label{est:faltab} \frac{1}{2}\frac{d}{dt}\left(\left\| \eta \right\|_{H^{2}}^{2}+\left\| u \right\|_{H^{3}}^{2}\right)&\lesssim \left\| \eta \right\|_{H^{2}}^{3}+\left\| u \right\|_{H^{3}}^{3} + \left( 1+\left\| b \right\|_{H^{1}}\right)^{2}(\left\| \eta \right\|_{H^{2}}+\left\| b \right\|_{H^{2}})+\left\| u \right\|_{H^{3}}^{2} \nonumber \\ &\lesssim 1+ \left\| \eta \right\|_{H^{2}}^{3}+\left\| u \right\|_{H^{3}}^{3} +\left\| b \right\|_{H^{2}}^{3}.\end{aligned}$$ To close the energy estimate, we need to compute $\left\| b \right\|_{H^{2}}^{3}$. To that purpose, we first find using the elliptic equation ([\[eq:2\]](#eq:2){reference-type="ref" reference="eq:2"}c) and integrating by parts that $$\left\| b \right\|_{L^{2}}^{2}=\int_{\mathbb{R}} \eta b \ dx + \int_{\mathbb{R}} \left(\frac{b_{x}}{1+\eta}\right)_{x}b \ dx=\int_{\mathbb{R}} \eta b \ dx - \int_{\mathbb{R}} \frac{b_{x}^{2}}{1+\eta} \ dx$$ Therefore, using the pointwise estimate [\[pointwise\]](#pointwise){reference-type="eqref" reference="pointwise"} we find that the last term $$-\int_{\mathbb{R}} \frac{b_{x}^{2}}{1+\eta} \ dx \leq 0,$$ and hence Young's inequality yields $$\label{estimate0} \left\| b \right\|^{2}_{L^{2}}\leq \left\| \eta \right\|^{2}_{L^{2}}$$ To compute the higher-order norm, let us first write $$\label{estimate1} \left\| b_{x} \right\|^{2}_{L^2}=\int_{\mathbb{R}} \frac{1+\eta}{1+\eta}(b_{x})^2 \ dx=-\int_{\mathbb{R}} \frac{b_{x}}{1+\eta} (1+\eta)_{x}b \ dx-\int_{\mathbb{R}} \left(\frac{b_{x}}{1+\eta}\right)_{x} (1+\eta) \ b \ dx=I_1+I_2.$$ Using Hölders and Young's inequality we readily see that $$\label{estimate2} |I_{1}|\leq \left\| b_{x} \right\|_{L^{2}}\left\| \eta_{x} \right\|_{L^{\infty}}\left\| b \right\|_{L^{2}}\left\| \frac{1}{1+\eta} \right\|_{L^{\infty}} \leq \frac{1}{2}\left\| b_{x} \right\|_{L^{2}}^{2}+ C\left\| \eta \right\|^{2}_{H^{2}}\left\| b \right\|^{2}_{L^{2}}\left\| \frac{1}{1+\eta} \right\|^{2}_{L^{\infty}}.$$ On the other hand, using once again the elliptic equation ([\[eq:2\]](#eq:2){reference-type="ref" reference="eq:2"}c) we find that $$\label{estimate3} I_{2}=\int_{\mathbb{R}}(\eta-b)(1+\eta)b \ dx =\int_{\mathbb{R}} \left(\eta b+\eta^2 b-b^{2}(1+\eta) \right) \ dx \leq \left\| b \right\|_{L^{2}}\left\| \eta \right\|_{L^2}+ \left\| b \right\|_{L^{2}}\left\| \eta \right\|_{L^2}\left\| \eta \right\|_{L^{\infty}}.$$ Therefore, collecting [\[estimate1\]](#estimate1){reference-type="eqref" reference="estimate1"} -[\[estimate3\]](#estimate3){reference-type="eqref" reference="estimate3"} we infer that $$\label{estimate4} \left\| b_{x} \right\|^{2}_{L^2}\leq \frac{1}{2}\left\| b_{x} \right\|_{L^{2}}^{2}+ C\left\| \eta \right\|^{2}_{H^{2}}\left\| b \right\|^{2}_{L^{2}}\left\| \frac{1}{1+\eta} \right\|^{2}_{L^{\infty}}+\left\| b \right\|_{L^{2}}\left\| \eta \right\|_{L^2}+ \left\| b \right\|_{L^{2}}\left\| \eta \right\|_{L^2}\left\| \eta \right\|_{L^{\infty}}$$ and hence using [\[estimate0\]](#estimate0){reference-type="eqref" reference="estimate0"} we conclude that $$\begin{aligned} \label{estimate5} \left\| b_{x} \right\|^{2}_{L^2}&\lesssim \left\| \eta \right\|^{2}_{H^{2}}\left\| \eta \right\|^{2}_{L^{2}}\left\| \frac{1}{1+\eta} \right\|^{2}_{L^{\infty}}+\left\| \eta \right\|_{L^{2}}^{2}+ \left\| \eta \right\|_{L^{2}}^{2}\left\| \eta \right\|_{L^{\infty}}.\end{aligned}$$ We iterate the previous idea, to provide an estimate $\left\| b_{xx} \right\|_{L^{2}}$. To that purpose, we write $$\label{estimate6} \left\| b_{xx} \right\|^{2}_{L^2}=-\int_{\mathbb{R}}\frac{1+\eta}{1+\eta}b_{xxx} b_{x} \ dx=\int_{\mathbb{R}}(1+\eta)b_{xx} \left(\frac{b_{x}}{1+\eta}\right)_{x} \ dx+\int_{\mathbb{R}}(1+\eta)_{x}b_{xx} \frac{b_{x}}{1+\eta} \ dx=J_1+J_2.$$ Using the elliptic equation ([\[eq:2\]](#eq:2){reference-type="ref" reference="eq:2"}c), we have that $$\begin{aligned} J_{1}=\int_{\mathbb{R}}(1+\eta)b_{xx} (b-\eta) \ dx &\leq \left\| b_{xx} \right\|_{L^2} \left\| (1+\eta)(b-\eta) \right\|_{L^{2}} \nonumber \\ &\leq \frac{1}{2\epsilon}\left\| b_{xx} \right\|^{2}_{L^{2}} + C_{\epsilon}\left(1+\left\| \eta \right\|_{H^{2}}^{4}+\left\| b \right\|_{L^{2}}^{4} \right)\end{aligned}$$ where in the second inequality we have used the Sobolev embedding and Young's ineqality. Similarly, $$\begin{aligned} J_{2} &\leq \left\| b_{xx} \right\|_{L^2} \left\| (1+\eta)_{x}\frac{b_{x}}{1+\eta} \right\|_{L^{2}} \leq \frac{1}{2\epsilon}\left\| b_{xx} \right\|^{2}_{L^{2}} + C_{\epsilon}\left(1+\left\| \eta \right\|_{H^{2}}^{8}+\left\| \frac{1}{1+\eta} \right\|_{L^{\infty}}^{8} + \left\| b_{x} \right\|_{L^{2}}^{8} \right).\end{aligned}$$ Therefore taking $\epsilon\ll 1$ (for instance $\epsilon=1/4$), we find that $$\label{estimateforbxx} \frac{1}{2}\left\| b_{xx} \right\|_L^{2}\leq C\left(1+\left\| \eta \right\|_{H^{2}}^{8}+\left\| \frac{1}{1+\eta} \right\|_{L^{\infty}}^{8} + \left\| b_{x} \right\|_{L^{2}}^{8} \right).$$ Hence, estimate [\[estimateforbxx\]](#estimateforbxx){reference-type="eqref" reference="estimateforbxx"} combined with the previous estimates for $\left\| b_{x} \right\|_{L^2}$ given in [\[estimate5\]](#estimate5){reference-type="eqref" reference="estimate5"} and $\left\| b \right\|_{L^2}$ in [\[estimate0\]](#estimate0){reference-type="eqref" reference="estimate0"} we conclude that $$\left\| b \right\|^{3}_{H^{2}}\leq C \left(1+\mathcal{E}(t)\right)^{p},$$ for some $C>0$ and $p>2$ large enough. The precise power of $p$ can be computed though it is not essential to provide a local-in-time solution. Hence, plugging the previous estimate into [\[est:faltab\]](#est:faltab){reference-type="eqref" reference="est:faltab"} and taking into account [\[pointwise2\]](#pointwise2){reference-type="eqref" reference="pointwise2"} we conclude that $$\label{final:est} \frac{d}{dt}\mathcal{E}(t)\leq C \left(1+\mathcal{E}(t)\right)^{p}$$ for some $C>0$ and $p>2$ large enough which ensures a local time of existence $T^{\star}>0$ such that $$\mathcal{E}(t)\leq 4 \mathcal{E}(0), \quad \mbox{ for } 0\leq t \leq T^{\star}.$$ In order to construct the solution, we first define the approximate problems using mollifiers, which reads [\[eq:regularized\]]{#eq:regularized label="eq:regularized"} $$\begin{aligned} \eta^{\epsilon}_t+\mathcal{J}_{\epsilon}(\mathcal{J}_{\epsilon}u\mathcal{J}_{\epsilon}\eta)_x+\mathcal{J}_{\epsilon}\mathcal{J}_{\epsilon}u^{\epsilon}_{x}&=0,\\ u^{\epsilon}_t+\mathcal{J}_{\epsilon}\left(\mathcal{J}_{\epsilon}u\mathcal{J}_{\epsilon}u_x\right)+\frac{(1+b)b_x}{1+\eta^{\epsilon}}&=0,\\ b-\eta^{\epsilon}-\left(\frac{b_x}{1+\eta^{\epsilon}}\right)_x&=0.\end{aligned}$$ Repeating the previous estimates we find a time of existence $T^{\star}>0$ for the sequence of regularized problems. Using compactness arguments and passing to the limit we conclude the proof of existence. The time continuity for the solution is obtained by classical arguments. On the one hand, the differential equation [\[final:est\]](#final:est){reference-type="eqref" reference="final:est"} gives the strong right continuity at $t=0$. Using the change of variables $\hat{t}=-t$, we get the strong left continuity at $t=0$, which combined show the continuity in time of the solution. # Acknowledgments {#acknowledgments .unnumbered} D.A-O is supported by the Spanish MINECO through Juan de la Cierva fellowship FJC2020-046032-I. R.G-B is supported by the project "Mathematical Analysis of Fluids and Applications\" Grant PID2019-109348GA-I00 funded by MCIN/AEI/ 10.13039/501100011033 and acronym "MAFyA\". This publication is part of the project PID2019-109348GA-I00 funded by MCIN/ AEI /10.13039/501100011033. This publication is also supported by a 2021 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation. The BBVA Foundation accepts no responsibility for the opinions, statements, and contents included in the project and/or the results thereof, which are entirely the responsibility of the authors. D.A-O and R. G-B are also supported by the project "Análisis Matemático Aplicado y Ecuaciones Diferenciales\" Grant PID2022-141187NB-I00 funded by MCIN/ AEIand acronym "AMAED\". 10 url urlprefix href D. Alonso-Orán. Asymptotic shallow models arising in magnetohydrodynamics, , 3, 371--398 (2021). D. Alonso-Orán, A. Durán, R. Granero-Belinchón. Derivation and well-posedness for asymptotic models of cold plasmas.  , (2023). Y. A. Berezin, V. Karpman. Theory of nonstationary finite-amplitude waves in a low-density plasma, 19 1265--1271 (1964). A. Cheng, R. Granero-Belinchón, S. Shkoller, J. Wilkening. Rigorous Asymptotic Models of Water Waves, , 1, 71--130, (2019). 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arxiv_math

--- abstract: | Octonion algebras are certain algebras with a multiplicative quadratic form. In [@OA], Alsaody and Gille show that, for octonion algebras over unital commutative rings, there is an equivalence between isotopes and isometric quadratic forms. The contravariant equivalence from unital commutative rings to affine schemes leads us to a question: can the equivalence of isometry and isotopy be generalized to octonion algebras over a (not necessarily affine) scheme? We present the basic definitions and properties of octonion algebras, both over rings and over schemes. Then we show that an isotope of an octonion algebra $\mathcal{C}$ over a scheme is isomorphic to a twist by an $\mathbf{Aut}(\mathcal{C})$--torsor. We conclude the paper by giving an affirmative answer to our question. author: - "V. Hildebrandsson[^1]" bibliography: - ref.bib title: "Octonion algebras over schemes and the equivalence of isotopes and isometric quadratic forms[^2]" --- # Introduction The octonions $\mathbb{O}$ are a normed division algebra over $\mathbb{R}$ of dimension 8, discovered in 1843 by John Graves. Based on these, the definition of a octonion algebras over any field was given by Leonard Dickson in 1927, and this was generalized once more to octonion algebras over rings by Loos--Petersson--Racine in 2008 [@LPR]. An important part of these definitions is the assumption that the algebra has an associated quadratic form. In the case over a field, an octonion algebra is completely determined by its quadratic form [@octalgfield Claim 2.3]. Over rings however, this is not the case, proved by Gille in [@Gille_2014]. The main result of Alsaody and Gille gives a construction of the non--isomorphic octonion algebras with isometric quadratic forms. By the contravariant equivalence of the category of rings and the category of affine schemes, we expect this construction to generalize to octonion algebras over affine schemes. However, will the construction generalize to octonion algebras over general schemes? Composition algebras over locally ringed spaces were first defined by Petersson in [@HP]. Since a locally ringed space $(X,\mathcal{O}_X)$ comes with a sheaf of rings $\mathcal{O}_X$, the idea is to define the composition algebra as a pair $(\mathscr{C},N)$ where $\mathscr{C}$ is an $\mathcal{O}_X$--algebra that is locally a composition algebra in the ring sense. Composition algebras only exist in ranks 1, 2, 4, 8, and a composition algebra of rank 8 is called an octonion algebra. The structure of the article is as follows. In Section [2](#BG){reference-type="ref" reference="BG"}, we recall background of sheaf theory over general sites, the fppf site, and basic notions of octonion algebras over rings. We also present the main result of Alsaody and Gille. In Section [3](#OoS){reference-type="ref" reference="OoS"}, we give the corresponding definitions for octonion algebras over schemes. In Section [4](#TII){reference-type="ref" reference="TII"}, we prove several analogous results of those proved in [@OA §4 and §6]. The main results are (1) twisting an octonion algebra $\mathcal{C}$ over a scheme by an $\mathbf{Aut}(\mathcal{C})$--torsor corresponds naturally to an isotope of $\mathcal{C}$, and (2) two octonion algebras over schemes are isotopic if and only if their quadratic forms are isometric. ## Acknowledgments {#acknowledgments .unnumbered} I want to thank Susanne Pumplün for enlightening me of the work of Petersson. I also want to thank Philippe Gille for our discussion which helped improve the thesis. Lastly, I am grateful to Seidon Alsaody who introduced me to this problem and who helped me immensely, both throughout the master thesis project and when rewriting it into this article. # Background {#BG} In this section, we will present some background on sheaf theory on general sites and the site $(\mathbf{Sch}/X)_\text{fppf}$. We will assume knowledge in standard sheaf theory (i.e. over the Zariski site) and in basic schemes theory. We will also present theory of octonion algebras over rings and the main result of Alsaody and Gille. For more on sheaf theory on general sites, see [@stacks §7]. For more on the fppf site, see [@stacks §34.7]. For more on octonion algebras over rings and proofs of results in Section [2.2](#OoR){reference-type="ref" reference="OoR"}, see [@OA §2 and §6]. Throughout, if not stated otherwise, $R$ is a commutative and unital ring and $X$ is a scheme. By an *$R$--ring* we mean a unital, commutative and associative $R$--algebra. We denote the category of schemes by $\mathbf{Sch}$. A *$X$--scheme* is a scheme $Y$ together with a morphism of schemes $Y\to X$. A $X$--morphism $Y\to Z$ is a morphism of schemes such that the diagram $$\begin{tikzcd} Y\arrow[rr]\arrow[dr]&&Z\arrow[dl]\\ &X \end{tikzcd}$$ commutes. We denote the category of $X$--schemes by $\mathbf{Sch}/X$. By the Yoneda lemma, we can identify a scheme $Y\in\mathbf{Sch}$ (or $X$--scheme $Y\in\mathbf{Sch}/X$) with its representable functor $$h_Y:\mathbf{Sch}\to\mathbf{Set}\quad(\text{or }h_Y:\mathbf{Sch}/X\to\mathbf{Set}).$$ We use the notation $Y(T):=h_Y(T)$. A *group scheme* (or $X$--group scheme) is a scheme (or $X$--scheme) $G$ and a factorization of its representable functor $h_G$ through the forgetful functor of groups to sets. ## Sites and sheaves In order to avoid set theoretical problems, we work over a fixed Grothendieck universe. A *site* is a category $\mathscr{C}$ and a class $\mathbf{Cov}(\mathscr{C})$ of families of morphisms with fixed target $\{U_i\to U\}_{i\in I}$, called coverings of $\mathscr{C}$, satisfying certain conditions, see [@stacks Definition 7.6.2]. We will denote the site by its category $\mathscr{C}$. **Example 1** (Fppf site). An *fppf covering* of $X$ is a family of morphisms with fixed target in $\mathbf{Sch}$, $\{p_i:X_i\to X\}_{i\in I}$, such that $X=\bigcup_{i\in I}p_i(X_i)$ and each $p_i$ is flat and of locally finite presentation. These fppf coverings makes $\mathbf{Sch}$ into a site [@stacks Lemma 34.7.3], denoted by $\mathbf{Sch}_\text{fppf}$. We make $\mathbf{Sch}/X$ into a site by defining $$(\mathbf{Sch}/X)_\text{fppf}:=(\mathbf{Sch}_\text{fppf})/X.$$ In other words, a covering of $Y\in\mathbf{Sch}/X$ is an fppf covering $\{Y_i\to Y\}_{i\in I}$, where $Y_i\in\mathbf{Sch}/X$, such that the diagram $$\begin{tikzcd} Y_i\arrow[rr]\arrow[dr]&&Y\arrow[dl]\\ &X& \end{tikzcd}$$ commutes for all $i\in I$. Let $\mathscr{C}$ be a category. A *presheaf* $\mathcal{F}$ on $\mathscr{C}$ is a contravariant functor $\mathcal{F}:\mathscr{C}\to\mathbf{Set}$. If $\mathscr{C}$ is a site, a presheaf $\mathcal{F}$ is a *sheaf* if for every covering $\{U_i\to U\}_{i\in I}\in\mathbf{Cov}(\mathscr{C})$ the following holds: 1. if two sections $s,t\in\mathcal{F}(U)$ are such that $\forall i\in I$ $s|_{U_i}=t|_{U_i}$ then $s=t$, 2. if $\forall i\in I$ $s_i\in\mathcal{F}(U_i)$ are such that $\forall j\in I$ $s_i|_{U_i\times_UU_j}=s_j|_{U_i\times_UU_j}$, then there exists $s\in\mathcal{F}(U)$ such that $\forall i\in I$ $s|_{U_i}=s_i$. Let $\mathcal{F}$ and $\mathcal{G}$ be two presheaves. A *morphism of presheaves* is a natural transformation $\eta:\mathcal{F}\to\mathcal{G}$. If $\mathcal{F}$ and $\mathcal{G}$ are sheaves, then $\eta$ is a *morphism of sheaves*. The categories of presheaves and sheaves are denoted by $\mathbf{PSh}(\mathscr{C})$ and $\mathbf{Sh}(\mathscr{C})$ respectively. Let $\mathcal{F}$ be a presheaf on the site $\mathscr{C}$. A sheafification functor $L:\mathbf{PSh}(\mathscr{C})\to\mathbf{Sh}(\mathscr{C})$ is a left adjoint to the inclusion $\mathbf{Sh}(\mathscr{C})\hookrightarrow\mathbf{PSh}(\mathscr{C})$. The *sheafification of $\mathcal{F}$* is then $\mathcal{F}':=L(\mathcal{F})$ together with the canonical map $\mathcal{F}\to\mathcal{F}'$. For its construction, see [@stacks §7.10]. Let $\mathscr{C}$ be a site and $\mathcal{G}$ be a sheaf of groups on $\mathscr{C}$. A *$\mathcal{G}$--torsor* is a sheaf of sets $\mathcal{F}$ on $\mathscr{C}$ endowed with a $\mathcal{G}$--action $\mathcal{G}\times\mathcal{F}\to\mathcal{F}$ such that 1. whenever $\mathcal{F}(U)$ is non-empty the action $\mathcal{G}(U)\times\mathcal{F}(U)\to\mathcal{F}(U)$ is transitive, 2. for every $U\in\mathscr{C}$ there exists a covering $\{U_i\to U\}_{i\in I}\in\mathbf{Cov}(\mathscr{C})$ such that $\mathcal{F}(U_i)$ is non-empty for all $i\in I$. A morphism of $\mathcal{G}$--torsors is a morphism of sheaves compatible with the $\mathcal{G}$--action. ## Octonion algebras over rings {#OoR} Let $A=(A,\ast)$ be an algebra. It is *alternative* if for each $a,b\in A$ $$a(ab)=(aa)b\text{ and }(ba)a=b(aa).$$ For each $a\in A$ we have the linear maps $L_a:x\mapsto a\ast x$ and $R_a:x\mapsto x\ast a$. If $A$ is alternative then $L_aR_a=R_aL_a$ for all $a\in A$, and we denote this map $B_a$. Two $R$--algebras $(A,\ast_A)$ and $(B,\ast_B)$ are *isotopic* if there exist invertible linear maps $f_i:A\to B$, $i=1,2,3$, such that $$f_1(x\ast_Ay)=f_2(x)\ast_Bf_3(y)$$ for all $x,y\in A$. **Remark 2**. If $A$ and $B$ are isotopic, then $A$ and $B$ are isomorphic as $R$--modules, but not necessarily as $R$--algebras. However, $f_1:A\to B$ is an isomorphism of algebras $A$ and $(B,\ast'_B)$, where $$x\ast'_By=f_2f_1^{-1}(x)\ast_Bf_3f_1^{-1}(y).$$ An algebra $(B,\ast_B')$ is a *principal isotope* of $(B,\ast_B)$ if there exist invertible linear maps $g,h:B\to B$ such that $x\ast'y=g(x)\ast h(y)$ for all $x,y\in B$. We denote $(B,\ast_B')=:B_{g,h}$. The algebras $B$ and $B_{g,h}$ are obviously isotopic. It follows from remark [Remark 2](#anm3.2){reference-type="ref" reference="anm3.2"} that algebras $A$ and $B$ are isotopic if and only if $A$ is isomorphic to a principal isotope of $B$. Let $(A,\ast)$ be an $R$--algebra and $g,h:A\to A$ invertible maps. $A_{g,h}$ is unital if there exists $e\in A$ such that for all $x\in A$ $$g(x)\ast h(e)=g(e)\ast h(x)=x.$$ Equivalently, $$g^{-1}=R_{h(e)}\text{ and }h^{-1}=L_{g(e)}. \label{eq1}\tag{$\ast$}$$ If $A$ is a unital, alternative algebra, this implies that $g(e)$ and $h(e)$ are invertible elements (the inverse of an element is well defined in an alternative algebra). Let $a=g(e)^{-1}$ and $b=h(e)^{-1}$. Then ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"}) is equivalent to $$g=R_a \text{ and }h=L_b.$$ Also, for any $a,b\in A^\ast$, $A_{R_a,L_b}$ is unital with unity $(ab)^{-1}$. Here $A^\ast$ denotes the invertible elements of $A$. **Proposition 3** (Alsaody and Gille, Proposition 2.1). *Let $A$ be a unital alternative algebra over $R$ and let $A'$ be isotopic to $A$. Then $A'$ is unital if and only if $A'\simeq A_{R_a,L_b}$ for some $a,b\in A^\ast$.* **Remark 4**. 1. We denote $A_{R_a,L_b}=:A^{a,b}$. 2. If $\phi:R\to S$ is a ring morphism, we can pullback an $R$--algebra $A$ to an $S$--algebra $A_S:=A\otimes_RS$. For $a,b\in A^\ast$, we have $(A^{a,b})_S=(A_S)^{a_S,b_S}$. We denote this algebra by $A^{a,b}_S$. A quadratic form on an $R$--module $M$ is a map $q:M\to R$ such that $$q(rm)=r^2q(m),\quad r\in R, m\in M,$$ and $$b_q(m,n):=q(m+n)-q(m)-q(n),\quad m,n\in M$$ is bilinear. It is multiplicative if $$q(mn)=q(m)q(n), \quad m,n\in M,$$ and regular if the morphism $$\begin{aligned} \alpha:M&\to\text{Hom}(M,R)\\ m&\mapsto b_q(m,\_) \end{aligned}$$ is an isomorphism. If $M$ is finitely generated and projective, the rank of $M$ at $p\in\mathbf{Spec}(R)$ is the rank of the free $R_p$--module $M_p$. It is a locally constant function on $\mathbf{Spec}(R)$. Now, an *octonion algebra over $R$* is an $R$--algebra whose underlying module is projective of constant rank 8, and which is endowed with a regular multiplicative quadratic form. It is unital and alternative. **Example 5**. Over any ring $R$, we have the *Zorn algebra* $$\mathbf{Zorn}(R)= \begin{pmatrix} R&R^3\\ R^3&R \end{pmatrix}$$ with multiplication defined by $$\begin{pmatrix} a&u\\ u'&a' \end{pmatrix} \begin{pmatrix} b&v\\ v'&b' \end{pmatrix}= \begin{pmatrix} ab+u^tv'&av+b'u-u'\times v'\\ a'v'+bu'+u\times v&(u')^tv+a'b' \end{pmatrix}$$ where $u\times v$ is the vector cross product and $u^tv$ is the scalar product. With the quadratic form given by the determinant $$\det\begin{pmatrix} a&u\\ u'&a' \end{pmatrix} =aa'-u^tu',$$ $\mathbf{Zorn}(R)$ becomes an $R$--octonion algebra. **Remark 6**. 1. If $(C,q)$ is an octonion algebra, we have the equality $C^\ast=\{x\in C|q(x)\in R^\ast\}$. 2. The trace of a quadratic form $q$ is the map $tr:M\to R, m\mapsto b_q(1,m)$. Then we define the involution map by $\kappa:x\mapsto\overline{x}:=tr(x)\cdot1_C-x$. 3. For any octonion algebra $(C,q)$ over $R$, we have the *Hamilton--Cayley equation* $q(c)1=c\overline{c}$, for any $c\in C$ [@HP]. For any octonion algebra $C$ over $R$ we have the octonion sphere $\mathbf{S}_C$, which is the $R$--scheme $$\mathbf{S}_C(S)=\{c\in C_S|q_{C_S}(c)=1\}.$$ We can now state the result of Alsaody and Gille. **Theorem 7** (Alsaody and Gille, Corollary 6.7). *Let $C$ and $C'$ be octonion algebras over $R$. The quadratic forms $q_C$ and $q_{C'}$ are isometric if and only if there exist $a,b\in\mathbf{S}_C(R)$ such that $C'\simeq C^{a,b}$.* # Octonion algebras over schemes {#OoS} In this section, we will present the definition of an octonion algebra over a scheme and some of its constructions. These definitions will be generalizations of the ones defined in subsection [2.2](#OoR){reference-type="ref" reference="OoR"}. **Definition 8** (Octonion algebra over a scheme). An *octonion algebra* over $X$ is a tuple $(\mathcal{C},\mathcal{Q})$ consisting of 1. a sheaf $\mathcal{C}$ of $\mathcal{O}_X$-algebras such that for all open affine $U\subset X$, the underlying module of the $\mathcal{O}_{X}(U)$-algebra $\mathcal{C}(U)$ is projective with constant rank 8, 2. a morphism of sheaves $\mathcal{Q}:\mathcal{C}\to\mathcal{O}_X$ such that for all open affine $U\subset X$, $\mathcal{Q}(U)$ is a multiplicative, regular, quadratic form on the $\mathcal{O}_X(U)$--algebra $\mathcal{C}(U)$. We call $\mathcal{Q}$ the *quadratic form on $\mathcal{C}$*. **Remark 9**. In the case where $X=\mathbf{Spec}(R)$ is an affine scheme, $\mathcal{O}_X(X)=R$, so for an octonion algebra $\mathcal{C}$ over $X$, $\mathcal{C}(X)$ is an $R$-octonion algebra (as defined in subsection [2.2](#OoR){reference-type="ref" reference="OoR"}). Note that defining the stalks and induced stalk maps $(\mathcal{C}_x,\mathcal{Q}_x)$ to be octonion algebras over $\mathcal{O}_{X,x}$, as defined in [@HP §1.6 and §1.7], is another natural way to define the octonion algebra over a scheme. In the following proposition and corollary, we see that the definitions are equivalent. **Proposition 10**. *Let $\mathcal{Q}:\mathcal{C}\to\mathcal{O}_X$ be a natural transformation. Then, for some open affine set $U\subset X$, $\mathcal{Q}(U)$ is a regular, multiplicative, and quadratic form if and only if $\mathcal{Q}_x$ is a regular, multiplicative, and quadratic form for all $x\in U$.* *Proof.* We only show the multiplicative part, regularity and being quadratic is similar. The induced stalk map $\mathcal{Q}_x$ at $x\in U$ can be defined in two steps: $$\begin{aligned} 1.&\quad\Tilde{\mathcal{Q}}_x:(U,s)\mapsto(U,\mathcal{Q}(U)(s))\\ 2.&\quad \mathcal{Q}_x:\overline{(U,s)}\mapsto\overline{\Tilde{\mathcal{Q}}_x(U,s)}. \end{aligned}$$ Assume $\mathcal{Q}(U)$ is a regular, multiplicative, and quadratic form. Then, by the definition of $\mathcal{Q}_x$ for $x\in U$, it follows that $\mathcal{Q}_x$ is aswell. Now, assume for all $x\in U$ that $\mathcal{Q}_x$ is multiplicative. Then $\Tilde{\mathcal{Q}}_x$ must be multiplicative. For $a,b\in\mathcal{C}(U)$, we have $$\begin{aligned} (U,\mathcal{Q}(U)(ab))&=\Tilde{\mathcal{Q}}_x((U,ab))\\ &=\Tilde{\mathcal{Q}}_x((U,a)(U,b))\\ &=\Tilde{\mathcal{Q}}_x((U,a))\Tilde{\mathcal{Q}}_x((U,b))\\ &=(U,\mathcal{Q}(U)(a))(U,\mathcal{Q}(U)(b))\\ &=(U,\mathcal{Q}(U)(a)\mathcal{Q}(U)(b)) \end{aligned}$$ so $\mathcal{Q}(U)$ must also be multiplicative. The proof of regularity and being quadratic is similar. ◻ **Corollary 11**. *For some open affine set $U\subset X$, $(\mathcal{C}(U),\mathcal{Q}(U))$ is an octonion algebra over $\mathcal{O}_X(U)$ if and only if $(\mathcal{C}_x,\mathcal{Q}_x)$ is an octonion algebra over $\mathcal{O}_{X,x}$ for all $x\in U$.* *Proof.* The rank of $\mathcal{C}(U)$ at $x$ is by definition the rank of $\mathcal{C}_x$, and $\mathcal{C}(U)$ is projective if and only if $\mathcal{C}_x$ is projective for all $x\in U$ [@CA Exercise 4.11b]. Then the statement follows from proposition [Proposition 10](#prop3.3){reference-type="ref" reference="prop3.3"}. ◻ Let $(\mathcal{C},\mathcal{Q})$ and $(\mathcal{C}',\mathcal{Q}')$ be two octonion algebras over $X$. A *morphism* $\varphi:(\mathcal{C},\mathcal{Q})\to(\mathcal{C}',\mathcal{Q}')$ is a natural transformation $\varphi:\mathcal{C}\to\mathcal{C}'$ such that $\varphi(U)$ is an algebra morphism for all open sets $U\subset X$. **Remark 12**. If we have an isomorphism $\varphi:(\mathcal{C}',\mathcal{Q}')\to(\mathcal{C},\mathcal{Q})$ of octonion algebras over $X$, we also have an isomorphism of octonion algebras $(\mathcal{C}(U),\mathcal{Q}(U))\simeq(\mathcal{C}'(U),\mathcal{Q}'(U))$ over the ring $\mathcal{O}_{X}(U)$. It follows that $\mathcal{Q}(U)$ and $\mathcal{Q}'(U)$ are isometric, i.e. $$\mathcal{Q}'(U)=\mathcal{Q}(U)\circ\varphi(U)=(\mathcal{Q}\circ\varphi)(U).$$ Since this holds for all open affine $U\subset X$, and $X$ has an open affine cover, it follows that $\mathcal{Q}'=\mathcal{Q}\circ\varphi$, i.e. $\mathcal{Q}$ and $\mathcal{Q}'$ are isometric. We denote that two quadratic forms are isometric by $\mathcal{Q}\sim\mathcal{Q}'$. Let $(\mathcal{C},\mathcal{Q})$ be an octonion algebra over $X$. For an octonion algebra over a ring, the form induces a bilinear form, a trace, and a natural involution. The same goes for the quadratic form $\mathcal{Q}$. The bilinear form is the morphism of sheaves $\mathcal{B_Q}:\mathcal{C}\times\mathcal{C}\to\mathcal{O}_X$ defined for each open set $U\subset X$ by $$\mathcal{B_Q}(U):(c,c')\mapsto\mathcal{Q}(U)(c+c')-\mathcal{Q}(U)(c)-\mathcal{Q}(U)(c').$$ The trace is then given by the morphism of sheaves $\text{tr}_\mathcal{C}:\mathcal{C}\to\mathcal{O}_X$ defined for each open set $U\subset X$ by $$\text{tr}_\mathcal{C}(U):c\mapsto\mathcal{B_Q}(U)(1,c)$$ and the natural involution is the morphism of sheaves $\kappa_\mathcal{C}:\mathcal{C}\to\mathcal{C}$ defined for each open set $U\subset X$ by $$\kappa_\mathcal{C}(U):c\mapsto\text{tr}_\mathcal{C}(U)(c)\cdot1_{\mathcal{C}(U)}-c=:\overline{c}$$ Now, let $a\in\mathcal{C}(X)$ be a global section. Let $L_a:\mathcal{C}\to\mathcal{C}$ and $R_a:\mathcal{C}\to\mathcal{C}$ be the morphisms of sheaves defined for any open $U\subset X$ by $$\begin{aligned} L_a(U):x\mapsto\text{res}_{X,U}(a)\cdot x,\quad R_a(U):x\mapsto x\cdot\text{res}_{X,U}(a).\end{aligned}$$ Also, let $B_a:=R_aL_a$. The *octonion unit sphere* is the $X$--scheme defined by $$\begin{aligned} \mathbf{S}_\mathcal{Q_C}:\mathbf{Sch}/X&\to\mathbf{Set}\\ Y&\mapsto\{c\in \mathcal{C}_Y(Y)|\mathcal{Q}_{\mathcal{C}_Y}(Y)(c)=1\}.\end{aligned}$$ **Definition 13** (Isotope). Let $\mathcal{C}$ be an octonion algebra over $X$. For each pair $a,b\in\mathbf{S}_{\mathcal{Q_C}}(X)$, we have an *isotope of* $\mathcal{C}$, denoted $\mathcal{C}^{a,b}$, which is the sheaf of $\mathcal{O}_X$-algebras such that, for any open $U\subset X$, $(\mathcal{C}^{a,b}(U),\ast)$ is the $\mathcal{O}_X(U)$-algebra with multiplication $$x\ast y:=R_a(U)(x)\cdot L_b(U)(y).$$ **Remark 14**. Using the quadratic form $\mathcal{Q}$ of $\mathcal{C}$, the isotope $\mathcal{C}^{a,b}$ becomes an octonion algebra over $X$. Also, it follows that $(\mathcal{C}^{a,b}(U),\mathcal{Q}(U))$ is an isotope of $(\mathcal{C}(U),\mathcal{Q}(U))$ as $\mathcal{O}_X(U)$--octonion algebras, for all open affine $U\subset X$. By this and remark [Remark 9](#anm4.2){reference-type="ref" reference="anm4.2"}, we see that the definitions of an octonion algebra over a scheme and of an isotope are natural generalizations of those over rings, as we wanted. Now, let $Y$ be an $X$--scheme. We make an octonion algebra $\mathcal{C}$ over $X$ into an octonion algebra $\mathcal{C}_Y$ over $Y$. Consider the presheaf $$V\mapsto (f^{-1}\mathcal{C})(V)\otimes_{(f^{-1}\mathcal{O}_X)(V)}\mathcal{O}_Y(V),$$ and let $\mathcal{C}_Y$ be its sheafification (with respect to the Zariski site). Then $\mathcal{C}_Y$, together with $\mathcal{Q}_Y:=f^{-1}\mathcal{Q}\otimes id$, where $f^{-1}\mathcal{Q}$ is the inverse sheaf functor acting on the sheaf morphism $\mathcal{Q}$, is an octonion algebra over $Y$ [@HP Proposition 1.7]. (The motivation is the construction in remark [Remark 4](#anm3.5){reference-type="ref" reference="anm3.5"}. If $X=\mathbf{Spec}(R)$ and $Y=\mathbf{Spec}(S)$, we get exactly $\mathcal{C}_Y(Y)=C(X)_S$.) **Remark 15**. 1. Let $U\subset X$ be an open set. For a section $a\in\mathcal{C}(U)$ of an octonion algebra $\mathcal{C}$ over $X$, we have, from the definition of $\mathcal{C}_Y$, the associated section $a_Y:=[a]\otimes1\in\mathcal{C}_Y(f^{-1}(U))$, where $[a]$ is the equivalence class of $a$ in $(f^{-1}\mathcal{C})(f^{-1}(U))$. In particular, if $a\in\mathcal{C}(X)$ is a global section, $a_Y\in\mathcal{C}_Y(Y)$ is a global section. 2. Let $V\subset Y$ be an open set. In both $(\mathcal{C}_Y)^{a_Y,b_Y}(V)$ and $(\mathcal{C}^{a,b})_Y(V)$, the elements are of the form $c_Y$, for some $c\in \mathcal{C}(U)$, $fV\subset U\subset X$, where $U$ is open. Elements $c_Y,d_Y\in(\mathcal{C}_Y)^{a_Y,b_Y}$ are multiplied as $$c_Y\ast_1d_Y=\big(c_Y\cdot\text{res}_{Y,V}(a_Y)\big)\big(\text{res}_{Y,V}(b_Y)\cdot d_Y\big),$$ while $c_Y,d_Y\in(\mathcal{C}^{a,b})_Y$ are multiplied as $$c_Y\ast_2d_Y=\big[((c\cdot\text{res}_{X,U}(a))(\text{res}_{X,U}(b)\cdot d))\big]\otimes 1.$$ However, $$\begin{aligned} c_Y\ast_2d_Y&=\big[((c\cdot\text{res}_{X,U}(a))(\text{res}_{X,U}(b)\cdot d))\big]\otimes 1\\ &=\big[(c\cdot\text{res}_{X,U}(a))\big]\otimes1\cdot\big[(\text{res}_{X,U}(b)\cdot d)\big]\otimes1\\ &=\big([c]\otimes1\cdot[\text{res}_{X,U}(a)]\otimes1\big)\cdot\big([\text{res}_{X,U}(b)]\otimes1\cdot[d]\otimes1\big)\\ &=\big(c_Y\cdot\text{res}_{Y,V}(a_Y)\big)\big(\text{res}_{Y,V}(b_Y)\cdot d_Y\big)\\ &=c_Y\ast_1d_Y, \end{aligned}$$ by naturality of the restriction maps. So $(\mathcal{C}_Y)^{a_Y,b_Y}=(\mathcal{C}^{a,b})_Y$, and we denote this sheaf by $\mathcal{C}_Y^{a,b}$. **Example 16**. Recall that over $\mathbb{Z}$, we have the Zorn algebra $\mathbf{Zorn}(\mathbb{Z})$. By the contravariant equivalence of $\mathbf{Ring}$ and $\mathbf{Aff}$, $\mathbf{Zorn}(\mathbb{Z})$ defines an octonion algebra over $\mathbf{Spec}(\mathbb{Z})$ as well. Let $Y$ be a scheme. By the pullback construction, and the fact that any scheme is a $\mathbb{Z}$--scheme, $\mathbf{Zorn}(\mathbb{Z})_Y$ exists and is an octonion algebra over $Y$. # Twists, isotopes, and isometries {#TII} In this section, we will prove the generalizations of four results in [@OA] (theorems 4.1, 4.6, and 6.6, and corollary 6.7, corresponding to theorems 4.3, 4.7, and 4.12, and corollary 4.13 in this article). ## The twisted octonion algebra Let us start with the affine case. We want to define the $R$--group schemes $\mathbf{Aut}(\mathcal{C})$, $\mathbf{SO}(q_C)$, and $\mathbf{RT}(C)$. Let $(C,q_C)$ be an octonion $R$--algebra. Denote by $\mathbf{Aut}(C)$ the automorphism $R$--group scheme of $C$, defined, for any $R$--ring $S$ by $$\mathbf{Aut}(C)(S):=\mathbf{Aut}(C_S),$$ which is a closed subscheme of the $R$--group scheme $\mathbf{GL}(C)$. By $\mathbf{SO}(q_C)$ we denote the special orthogonal $R$--group scheme associated to $q_C$ [@calmès Section 4.3]. The embedding $\mathbf{Aut}(C)\to\mathbf{GL}(C)$ induces a closed immersion $\mathbf{Aut}(C)\to\mathbf{SO}(q_C)$. Define $\mathbf{RT}(C)$ as the closed $R$--subscheme of $\mathbf{SO}(q_C)^3$ such that for any $R$--ring $S$ we have $$\mathbf{RT}(C)(S)=\left\{(t_1,t_2,t_3)\in\mathbf{SO}(q_C)^3|t_{1}(xy)=\overline{t_{2}(\overline{x})}\cdot\overline{t_{3}(\overline{y})}\text{ for any }x,y\in C_S\right\}.$$ Now we can continue with octonion algebras over general schemes. Let $X$ be a scheme. From here onwards, unless otherwise stated, we denote an $X$--scheme $Y\overset{f}{\to}X$ by its scheme $Y$. Let $(\mathcal{C},\mathcal{Q_C})$ be an octonion algebra over $X$. Let $\mathbf{Aut}(\mathcal{C})$ be the affine group scheme over $X$ such that $$\mathbf{Aut}(\mathcal{C})(Y):=\mathbf{Aut}(\mathcal{C}_Y).$$ The $X$--group scheme $\mathbf{O}(\mathcal{Q}_\mathcal{C})$ is defined by $$\begin{aligned} \mathbf{O}(\mathcal{Q}_\mathcal{C})(Y):=\{\varphi:\mathcal{C}_Y\to\mathcal{C}_Y|&\varphi_Z:(\mathcal{C}_Y)_Z\to(\mathcal{C}_Y)_Z\text{ is linear, invertible,}\\ &\text{and }\mathcal{Q}_{(\mathcal{C}_Y)_Z}\circ\varphi=\mathcal{Q}_{(\mathcal{C}_Y)_Z}\text{ for any $Y$–scheme }Z\}\end{aligned}$$ [@calmès Section 4.1] and, similar to the affine case, $\mathbf{SO}(\mathcal{Q}_\mathcal{C})$ is the special orthogonal $X$--group scheme [@calmès Section 4.3]. Recall that for $c\in\mathcal{C}(U)$, we have the natural involution $\kappa_\mathcal{C}(U)(c)=\text{tr}_\mathcal{C}(U)(c)-c$. We then define $\mathbf{RT}(\mathcal{C})$ as the closed $X$--subscheme of $\mathbf{SO}(\mathcal{Q}_\mathcal{C})^3$ defined by $$\begin{aligned} \mathbf{RT}(\mathcal{C})(Y):=\{ (t_1,t_2,t_3)\in\mathbf{SO}&(\mathcal{Q}_\mathcal{C})(Y)^3\big|\text{ for any $Y$–scheme }Z\\ &t_{1Z}(Z)(xy)=(\kappa_{(\mathcal{C}_Y)_Z} t_{2Z}\kappa_{(\mathcal{C}_Y)_Z})(Z)(x)\cdot(\kappa_{(\mathcal{C}_Y)_Z} t_{3Z}\kappa_{(\mathcal{C}_Y)_Z})(Z)(y)\}.\end{aligned}$$ where $t_{iZ}$, $i=1,2,3$, are induced by the $Y$--scheme structure morphism $Z\to Y$. **Remark 17**. These schemes can equivalently be defined in with a global--local point of view: $$\mathbf{O}(\mathcal{Q}_\mathcal{C})(Y)=\{\varphi:\mathcal{C}_Y\to\mathcal{C}_Y|\text{ for all open affine }V\subset Y; \varphi(V)\in\mathbf{O}(\mathcal{Q}_{\mathcal{C}_Y}(V))\},$$ $$\mathbf{SO}(\mathcal{Q}_\mathcal{C})(Y)=\{\varphi\in\mathbf{O}(\mathcal{Q}_\mathcal{C})(Y)|\text{ for all open affine }V\subset Y; \varphi(V)\in\mathbf{SO}(\mathcal{Q}_{\mathcal{C}_Y}(V))\},$$ $$\begin{aligned} \mathbf{RT}(\mathcal{C})(Y)=\{ (t_1,t_2,t_3)\in\mathbf{SO}(\mathcal{Q}_\mathcal{C})(Y)^3|&\text{ for all open affine }V\subset Y\\ &(t_1(V),t_2(V),t_3(V))\in\mathbf{RT}(\mathcal{C}_Y(V))(\mathcal{O}_Y(V))\}, \end{aligned}$$ From [@OA], we know many properties of these group schemes over rings ($\simeq$ affine schemes). We will see that this perspective will prove useful. **Example 18**. We see that, for any affine open $U\subset X$, $a\in\mathcal{C}(X)$, $x,y\in\mathcal{C}(U)$, the triple $(B_a,R_{\overline{a}},L_{\overline{a}})$ satisfies $$\begin{aligned} B_a(U)(xy)&=R_a(U)L_a(U)(xy)\\ &=\text{res}_{X,U}(a)\cdot ((xy)\cdot\text{res}_{X,U}(a))\\ &=(\overline{\text{res}_{X,U}(\overline{a})\cdot\overline{x}})\cdot(\overline{\overline{y}\cdot\text{res}_{X,U}(\overline{a})})\\ &=R_{\overline{a}}(U)(x)L_{\overline{a}}(U)(y), \end{aligned}$$ where the third equality follows from the fourth Moufang identity. Hence $(B_a,R_{\overline{a}},L_{\overline{a}})\in\mathbf{RT}(\mathcal{C})(U)$. We have a natural action of $\mathbf{RT}(\mathcal{C})$ on $\mathbf{S}_\mathcal{Q_C}\times_X\mathbf{S}_\mathcal{Q_C}=:\mathbf{S}_\mathcal{Q_C}^2$ by $$(t_1,t_2,t_3).(u,v)=(t_3(Y)(u),t_2(Y)(v))$$ for any $(t_1,t_2,t_3)\in\mathbf{RT}(\mathcal{C})(Y)$ and any $u,v\in\mathbf{S}_\mathcal{Q_C}(Y)$. We also have a map $\Pi:\mathbf{RT}(\mathcal{C})\to\mathbf{S}_\mathcal{Q_C}^2$, defined for any $\mathbf{t}:=(t_1,t_2,t_3)\in\mathbf{RT}(\mathcal{C})(Y)$ by $$\Pi_Y: \mathbf{t}\mapsto(t_3(Y)(1),t_2(Y)(1)).$$ **Theorem 19**. 1. *The stabilizer $\text{\emph{Stab}}_{\mathbf{RT}(\mathcal{C})}(1,1)$ is $i(\mathbf{Aut}(\mathcal{C}))$, where $i:\mathbf{Aut}(\mathcal{C})\hookrightarrow\mathbf{RT}(\mathcal{C})$ is the natural transformation $i_Y:t\mapsto(t,t,t)$.* 2. *The fppf quotient sheaf $\mathbf{RT}(\mathcal{C})/\mathbf{Aut}(\mathcal{C})$ is representable by an $X$--scheme and the map $$\mathbf{RT}(\mathcal{C})/\mathbf{Aut}(\mathcal{C})\to\mathbf{S}_\mathcal{Q_C}^2,$$ induced by $\Pi$, is an $X$--isomorphism.* *Proof.* 1. First we have to show that $i$ is well-defined. Take $t\in\mathbf{SO}(\mathcal{Q}_\mathcal{C})(Y)$ and open affine $V\subset Y$. Then $t(V)$ is an automorphism if and only if $$(t(V),t(V),t(V))\in\mathbf{RT}(\mathcal{C}_Y(V))(\mathcal{O}_Y(V))$$ [@OA Lemma 3.5]. Hence $t\in\mathbf{Aut}(\mathcal{C})(Y)$ if and only if $(t,t,t)\in\mathbf{RT}(\mathcal{C})(Y)$. Now, $(t_1(V),t_2(V),t_3(V))\in\mathbf{RT}(\mathcal{C}_Y(V))(\mathcal{O}_Y(V))$ satisfies $$t_3(V)(1)=t_2(V)(1)=1$$ if and only if $$(t_1(V),t_2(V),t_3(V))\in i_V\big(\mathbf{Aut}(\mathcal{C}_Y(V))\big)$$ [@OA Proposition 3.8]. Since it holds for all open affine $V\subset Y$, and $Y$ can be covered by open affines, we have that Stab$_{\mathbf{RT}(\mathcal{C})}(1,1)=i(\mathbf{Aut}(\mathcal{C}))$. 2. We first want to show that $\mathbf{Aut}(\mathcal{C})$ is flat over $X$. Let $h:\mathbf{Aut}(\mathcal{C})\to X$ be the canonical map, and $X=\bigcup_iU_i$ an affine cover, $U_i=\mathbf{Spec}(R_i)$ (note that, by [@stacks Lemma 34.7.2], this cover is also an fppf covering). Since $\mathbf{Aut}(\mathcal{C})$ is an affine $X$-scheme, $V_i:=h^{-1}(U_i)$ is an affine open set. $\mathbf{Aut}(\mathcal{C})|_{U_i}$ is an $R_i$--scheme, and, by definition, $V_i$ is an open set of $\mathbf{Aut}(\mathcal{C})|_{U_i}$. Then $\mathcal{O}_{\mathbf{Aut}(\mathcal{C})|_{U_i}}(V_i)$ is a flat $R_i$--module [@LPR Corollary 4.12]. But $\mathcal{O}_{\mathbf{Aut}(\mathcal{C})|_{U_i}}(V_i)=\mathcal{O}_{\mathbf{Aut}(\mathcal{C})}(V_i)$, so $h$ is a flat morphism [@stacks Lemma 29.25.3]. Then $\mathbf{Aut}(\mathcal{C})$ is flat over $X$. It follows that $\mathbf{RT}(\mathcal{C})/\mathbf{Aut}(\mathcal{C})$ is representable by an $X$--scheme, and the induced map is a monomorphism [@SGA Expose XVI Theorem 2.2]. Let $$\begin{aligned} P:\mathbf{Sch}/X&\to\mathbf{Set}\\ Z&\mapsto\mathbf{RT}(\mathcal{C})(Z)/\mathbf{Aut}(\mathcal{C})(Z) \end{aligned}$$ be a presheaf. Then $\widetilde{P}:=\mathbf{RT}(\mathcal{C})/\mathbf{Aut}(\mathcal{C})$ is its the (fppf) sheafification of $P$. Let $\{U_i\}_{i\in I}$ be an affine cover of $X$ and consider the presheaf $$\begin{aligned} Q_i:\mathbf{Sch}/U_i&\to\mathbf{Set}\\ Z&\mapsto\mathbf{RT}(\mathcal{C})|_{U_i}(Z)/\mathbf{Aut}(\mathcal{C})|_{U_i}(Z) \end{aligned}$$ Then $\widetilde{Q_i}:=\mathbf{RT}(\mathcal{C})|_{U_i}/\mathbf{Aut}(\mathcal{C})|_{U_i}$ is its (fppf) sheafification. For any $Z\in\mathbf{Sch}/U_i\subset\mathbf{Sch}/X$, the presheaves $P$ and $Q_i$ coincide: $$P(Z)=\mathbf{RT}(\mathcal{C})(Z)/\mathbf{Aut}(\mathcal{C})(Z)=\mathbf{RT}(\mathcal{C})|_{U_i}(Z)/\mathbf{Aut}(\mathcal{C})|_{U_i}(Z)=Q_i(Z).$$ Then both $\widetilde{Q_i}$ and $\Big(\mathbf{RT}(\mathcal{C})/\mathbf{Aut}(\mathcal{C})\Big)\Big|_{U_i}$ satisfy the universal property of sheafification of $P|_{U_i}$. It follows from uniqueness that $$\Big(\mathbf{RT}(\mathcal{C})/\mathbf{Aut}(\mathcal{C})\Big)\Big|_{U_i}\simeq\widetilde{Q_i}=\mathbf{RT}(\mathcal{C})|_{U_i}/\mathbf{Aut}(\mathcal{C})|_{U_i}.$$ Note also that, for any $Z\in\mathbf{Sch}/U_i\subset\mathbf{Sch}/X$, we also have the equivalence $$\begin{aligned} (\mathbf{S}_\mathcal{Q_C}|_{U_i}\times_{U_i}\mathbf{S}_\mathcal{Q_C}|_{U_i})(Z)&=\mathbf{S}_\mathcal{Q_C}|_{U_i}(Z)\times_{U_i}\mathbf{S}_\mathcal{Q_C}|_{U_i}(Z)\\ &= \mathbf{S}_\mathcal{Q_C}(Z)\times_X\mathbf{S}_\mathcal{Q_C}(Z)\\ &=(\mathbf{S}_\mathcal{Q_C}\times_X\mathbf{S}_\mathcal{Q_C})(Z)\\ &=(\mathbf{S}_\mathcal{Q_C}\times_X\mathbf{S}_\mathcal{Q_C})|_{U_i}(Z). \end{aligned}$$ Now, restricting the induced monomorphism we get $$\mathbf{RT}(\mathcal{C})|_{U_i}/\mathbf{Aut}(\mathcal{C})|_{U_i}\to\big(\mathbf{S}_\mathcal{Q_C}|_{U_i}\big)^2,$$ which is an $U_i$--isomorphism [@OA Theorem 4.1(2)]. It follows that $$\mathbf{RT}(\mathcal{C})/\mathbf{Aut}(\mathcal{C})\overset{\sim}{\to}\mathbf{S}_\mathcal{Q_C}^2$$ is an $X$--isomorphism.  ◻ Let $X=\bigcup_iU_i$ be an affine cover and $\mathbf{G}:=\mathbf{Aut}(\mathcal{C})$. It follows from theorem [Theorem 19](#thm4.10){reference-type="ref" reference="thm4.10"} and [@GA Corollary III.4.1.8] that $\Pi^{-1}|_{U_i}$ is a $\mathbf{G}|_{U_i}$--torsor in the fppf topology. Then $\Pi^{-1}$ is a $\mathbf{G}$--torsor over $\mathbf{S}_\mathcal{Q_C}^2$ in the fppf topology. We have the additive $X$--group scheme $\mathbf{W}(\mathcal{C})$ defined by $$\mathbf{W}(\mathcal{C})(Y)=\mathcal{C}_Y(Y).$$ Given $a,b\in\mathbf{S}_\mathcal{Q_C}(Y)$, consider the $\mathbf{G}$--torsor $\mathbf{E}^{a,b}:=\Pi^{-1}(a,b)$ over $X$. Let the presheaf $\mathbf{P}^{a,b}(\mathcal{C})$ of $X$--schemes be definied by $$\mathbf{P}^{a,b}(\mathcal{C})(Y)=(\Pi^{-1}_Y(a_Y,b_Y)\times \mathcal{C}_Y(Y))/\sim$$ where $$(\mathbf{t},x)\sim(\mathbf{t}',x')\iff\exists g\in\mathbf{G}(Y):(\mathbf{t}g,x)=(\mathbf{t}',g(Y)(x')),$$ and let $\mathbf{E}^{a,b}\wedge^\mathbf{G} \mathbf{W}(\mathcal{C})$ be its sheafification. **Remark 20**. Let $\Pi_Y(\mathbf{t})=\Pi_Y(\mathbf{t}')$. Then $$\begin{aligned} t_3(Y)(1)&=t_3'(Y)(1)\iff t_3^{-1}(Y)t_3'(Y)(1)=1\iff(t_3^{-1}t_3')(Y)(1)=1\\ t_2(Y)(1)&=t_2'(Y)(1)\iff t_2^{-1}(Y)t_2'(Y)(1)=1\iff(t_2^{-1}t_2')(Y)(1)=1 \end{aligned}$$ which implies that $\mathbf{t}^{-1}\mathbf{t}'\in\mathbf{Stab}_{\mathbf{RT}(\mathcal{C})}(1,1)(Y)=i_Y(\mathbf{G}(Y))$. The presheaf $\mathbf{P}^{a,b}(\mathcal{C})$ is in fact a presheaf of algebras; the linear structure on $\mathbf{P}^{a,b}(\mathcal{C})(Y)$ is induced by $$(\mathbf{t},x)+(\mathbf{t}',x')=(\mathbf{t},x+(t_1^{-1}t_1')(Y)(x'))$$ and, for $s\in\mathcal{O}_Y(Y)$, $$s(\mathbf{t},x)=(\mathbf{t},sx),$$ and the multiplication $$(\mathbf{t},x)(\mathbf{t}',x')=(\mathbf{t},x(t_1^{-1}t_1')(Y)(x')).$$ This makes $\mathbf{P}^{a,b}(\mathcal{C})(Y)$ into an $\mathcal{O}_Y(Y)$--algebra. The unity is the class of $(\mathbf{t},1)$, indepentent of choice of $\mathbf{t}\in\Pi^{-1}_Y(a_Y,b_Y)$ by remark [Remark 20](#anm4.10){reference-type="ref" reference="anm4.10"}. From this and the fact that Cayley--Hamilton equation holds on $\mathcal{Q}_Y(V)$ for any open $V\subset Y$, it follows that $\mathbf{P}^{a,b}(\mathcal{C})(Y)$ has the same quadratic form as $\mathcal{C}_Y$ [@OA p.885]. So $\mathbf{P}^{a,b}(\mathcal{C})(Y)$ is an $\mathcal{O}_Y(Y)$--octonion algebra. **Lemma 21**. *Let $a,b\in\mathbf{S}_\mathcal{Q_C}(X)$ and $(t_1,t_2,t_3)\in\mathbf{RT}(\mathcal{C})(Y)$. Then $t_1(Y)$ is an isomorphism $\mathcal{C}_Y(Y)\to\mathcal{C}_Y^{a,b}(Y)$ if and only if $t_2(Y)(1)=\eta b_Y$ and $t_3(Y)(1)=\eta a_Y$ for some $\eta\in\boldsymbol{\boldsymbol{\mu}}_2(Y)$.* *Proof.* By definition, $\mathcal{C}_Y(Y)$ and $\mathcal{C}_Y^{a,b}(Y)$ are $\mathcal{O}_Y(Y)$--octonion algebras, and $t_1(Y)$ an algebra homomorphism between them. The statement then follows from [@OA Proposition 4.5]. ◻ **Lemma 22**. *Let $a,b\in\mathbf{S}_\mathcal{Q_C}(X)$. For each $Y\in\mathbf{Sch}/X$ such that $\Pi^{-1}_Y(a_Y,b_Y)\neq\varnothing$, the map $$\begin{aligned} \Omega_Y^{a,b}:\mathbf{P}^{a,b}(\mathcal{C})(Y)&\to\mathcal{C}_Y^{a,b}(Y)\\ (\mathbf{t},x)&\mapsto t_1(Y)(x)\end{aligned}$$ is an algebra isomorphism.* *Proof.* The proof is similar to that of [@OA Theorem 4.6]. The map $\Omega^{a,b}_Y$ is well--defined for any $X$--scheme $Y$ since if $(\mathbf{t},x)\sim(\mathbf{t}',x')$ then $x'=g(Y)^{-1}(x)$ and $\mathbf{t}'=\mathbf{t}g$, for some $g\in\mathbf{G}(Y)$, so $$t_1'(Y)(x')=(t_1g)(Y)(g(Y)^{-1}(x))=t_1(Y)g(Y)g(Y)^{-1}(x)=t_1(Y)(x).$$ $\Omega^{a,b}(Y)$ is linear since, $$\Omega^{a,b}_Y(s(\mathbf{t},x))=\Omega^{a,b}_Y((\mathbf{t},sx))=t_1(Y)(sx)=st_1(Y)(x)=s\Omega^{a,b}_Y(\mathbf{t},x),$$ for $s\in\mathcal{O}_Y(Y)$, and $$\begin{aligned} \Omega^{a,b}_Y((\mathbf{t},x)+(\mathbf{t}',x'))&=\Omega^{a,b}_Y(\mathbf{t},x+(t_1^{-1}t_1')(Y)(x'))\\ &=t_1(Y)(x+(t_1^{-1}t_1')(Y)(x'))\\ &=t_1(Y)(x)+t_1(Y)((t_1^{-1}t_1')(Y)(x'))\\ &=t_1(Y)(x)+t_1'(Y)(x')\\ &=\Omega^{a,b}_Y(\mathbf{t},x)+\Omega^{a,b}_Y(\mathbf{t}',x'). \end{aligned}$$ We have $$\Omega^{a,b}_Y((\mathbf{t},x)(\mathbf{t}',x'))=\Omega^{a,b}_Y(\mathbf{t},x(t_1^{-1}t_1')(Y)(x'))=t_1(Y)(x(t_1^{-1}t_1')(Y)(x')).$$ Since $\mathbf{t}\in\Pi^{-1}_Y(a_Y,b_Y)$, it follows from lemma [Lemma 21](#prop4.11){reference-type="ref" reference="prop4.11"} that this is equal to $$\begin{aligned} t_1(Y)(x)\ast_{a_Y,b_Y}t_1(Y)((t_1^{-1}t_1')(Y)(x'))&=t_1(Y)(x)\ast_{a_Y,b_Y}t_1'(Y)(x')\\ &=\Omega^{a,b}_Y(\mathbf{t},x)\ast_{a_Y,b_Y}\Omega^{a,b}_Y(\mathbf{t}',x') \end{aligned}$$ If $\Omega^{a,b}_Y(\mathbf{t},x)=t_1(Y)(x)=0$ then $x=0$ since $t_1(Y)$ is invertible. Then $$\ker\Omega^{a,b}_Y=\{(\mathbf{t},0)|\mathbf{t}\in\Pi^{-1}_Y(a_y,b_Y)\}.$$ By definition, for any $(\mathbf{t},0),(\mathbf{t}',0)\in\ker\Omega^{a,b}_Y$, $\Pi_Y(\mathbf{t})=\Pi_Y(\mathbf{t}')$. From remark [Remark 20](#anm4.10){reference-type="ref" reference="anm4.10"}, it follows that $\mathbf{t}^{-1}\mathbf{t}'\in i_Y(\mathbf{G}(Y))$. Then $$(\mathbf{t}(\mathbf{t}^{-1}\mathbf{t}'),(\mathbf{t}^{-1}\mathbf{t}')(Y)(0))=(\mathbf{t}',0)$$ so $(\mathbf{t},0)\sim(\mathbf{t}',0)$. Then $\ker\Omega^{a,b}_Y=0$ and $\Omega^{a,b}_Y$ is injective. Given $x\in\mathcal{C}^{a,b}_Y(Y)$ then any $\mathbf{t}\in\Pi^{-1}_Y(a,b)\neq\varnothing$ satisfy $\Omega^{a,b}_Y(\mathbf{t},t^{-1}_1(Y)(x))=x$, so $\Omega^{a,b}_Y$ is surjective. ◻ Now we can show the link between twisted octonion algebras and isotopes. **Theorem 23**. *Consider the algebra isomorphisms $\Omega^{a,b}_Y$, defined in lemma [Lemma 22](#lemma4.12){reference-type="ref" reference="lemma4.12"}. The induced map $$\widehat{\Omega}^{a,b}:\mathbf{E}^{a,b}\wedge^\mathbf{G} \mathbf{W}(\mathcal{C})\overset{\sim}{\to}\mathbf{W}(\mathcal{C}^{a,b}).$$ is a natural isomorphism of fppf-sheaves of algebras.* **Remark 24**. Naturality means that for any morphism $g:Z\to Y$ of $X$--schemes, the diagram $$\begin{tikzcd} \mathbf{P}^{a,b}(\mathcal{C})(Z)\arrow[r,"\Omega^{a,b}_Z"]&\mathcal{C}^{a,b}_Z(Z)\\ \mathbf{P}^{a,b}(\mathcal{C})(Y)\arrow[r,"\Omega^{a,b}_Y"]\arrow[u,"\widetilde{g}:=\mathbf{P}^{a,b}(\mathcal{C})(g)"]&\mathcal{C}^{a,b}_Y(Y)\arrow[u,swap,"\widehat{g}:=\mathcal{C}^{a,b}_Y(g)"] \end{tikzcd}$$ commutes. The maps are well-defined by remark [Remark 15](#anm4.3){reference-type="ref" reference="anm4.3"}. *Proof.* We first prove the isomorphism. It follows from the universal property of sheafification that if the map $\Omega_Y^{a,b}$ on $\mathbf{P}^{a,b}(\mathcal{C})(Y)$ is an algebra isomorphism then $\widehat{\Omega}^{a,b}_Y$ is also an algebra isomorphism. Since $\Omega^{a,b}$ is globally defined, it suffices to check an open cover of $Y$. The (fppf) sheaf $\mathbf{E}^{a,b}\wedge^\mathbf{G}\mathbf{W}(\mathcal{C})$ is a $\mathbf{G}$--torsor so for every $X$--scheme $Y$ there exists an fppf covering $\{Y_i\to Y\}_{i\in I}$ such that $\mathbf{P}^{a,b}(\mathcal{C})(Y_i)\neq\varnothing$. Hence we only need to check $Y$ such that $\mathbf{P}^{a,b}(\mathcal{C})(Y)\neq\varnothing$. Let $Y$ be such a scheme. It follows from lemma [Lemma 22](#lemma4.12){reference-type="ref" reference="lemma4.12"} that $\Omega^{a,b}_Y$ is an algebra isomorphism. Now we prove naturality. It suffices to check locally. For any open affine $U\subset X$ and any morphism $h:W\to V$ of $U$--schemes, the diagram $$\begin{tikzcd} \mathbf{P}^{a,b}(\mathcal{C})|_U(W)\arrow[r,"\Omega^{a,b}_W"]&(\mathcal{C}|_U)^{a,b}_W(W)\\ \mathbf{P}^{a,b}(\mathcal{C})|_U(V)\arrow[r,"\Omega^{a,b}_V"]\arrow[u,"\widetilde{h}"]&(\mathcal{C}|_U)^{a,b}_V(V)\arrow[u,"\widehat{h}"] \end{tikzcd}$$ commutes [@OA Theorem 4.6]. ◻ ## The equivalence of isotopes and isometries Consider the site $(\mathbf{Sch}/X)_\text{fppf}$. If $\mathbf{G}$ is a sheaf of groups in $(\mathbf{Sch}/X)_\text{fppf}$, we denote the set of isomorphism classes of $\mathbf{G}$--torsors, called the *first cohomology*, by $H^1_\text{fppf}(X,\mathbf{G})$ (see [@Gille §2.2]). **Lemma 25**. *Let $(\mathcal{C},\mathcal{Q_C})$ be an octonion algebra over $X$. We have bijections of pointed sets $$\begin{aligned} H^1_\text{fppf}(X,\mathbf{Aut}(\mathcal{C}))&\simeq\{C'|C'\simeq\mathcal{C}\text{ fppf locally}\}\\ H^1_\text{fppf}(X,\mathbf{O}(\mathcal{Q_C}))&\simeq\{(\mathcal{M},\mathcal{Q_M})|\mathcal{Q_M}\sim\mathcal{Q_C}\text{ fppf locally}\} \end{aligned}$$ with base points $\mathcal{C}$ and $(\mathcal{C},\mathcal{Q_C})$ respectively, which preserves the base points. (Here $C'$ are octonion algebras over $X$ and $(\mathcal{M},\mathcal{Q_M})$ are $\mathcal{O}_X$--module sheaves with a quadratic form.)* *Proof.* We have mutually inverse functions $$\begin{aligned} H^1_\text{fppf}(X,\mathbf{Aut}(\mathcal{C}))&\leftrightarrow\{C'|C'\simeq\mathcal{C}\text{ fppf locally}\}\\ E&\mapsto E\wedge^{\mathbf{Aut}(\mathcal{C})}\mathcal{C}\\ \mathbf{Isom}(\mathcal{C},C')&\mapsfrom C' \end{aligned}$$ where $E\wedge^{\mathbf{Aut}(\mathcal{C})}\mathcal{C}$ is the sheafification of $$\mathbf{E}(\mathcal{C})(Y)=(E(Y)\times \mathcal{C}_Y(Y))/\sim$$ where $$(u,x)\sim(u',x')\iff\exists g\in\mathbf{Aut}(\mathcal{C})(Y):(ug,x)=(u',g(Y)(x')),$$ and $\mathbf{Isom}(\mathcal{C},C')$ is the set of isomorphisms from $\mathcal{C}$ to $C$'. The other bijection is similar. ◻ Recall the scheme $\boldsymbol{\mu}_2$. We can embed this into $\mathbf{RT}(\mathcal{C})$ by $$\begin{aligned} \boldsymbol{\mu}_2(Y)&\hookrightarrow\mathbf{RT}(\mathcal{C})(Y)\\ \eta&\mapsto(1,\eta,\eta)\end{aligned}$$ We also have an embedding into $\mathbf{S}_\mathcal{Q_C}^2$: $$\begin{aligned} \boldsymbol{\mu}_2(Y)&\hookrightarrow\mathbf{S}_\mathcal{Q_C}^2(Y)\\ \eta&\mapsto(\eta\cdot1_\mathcal{C},\eta\cdot1_\mathcal{C})\end{aligned}$$ We can now mod out $\boldsymbol{\mu}_2$ from the mapping $\Pi$, and get $$\Pi_+:\mathbf{RT}(\mathcal{C})/\boldsymbol{\mu}_2\to\mathbf{S}_\mathcal{Q_C}^2/\boldsymbol{\mu}_2$$ Let $\{U_i\}$ be an affine cover of $X$ and $f_1:\mathbf{RT}(\mathcal{C})\to\mathbf{SO}(\mathcal{Q_C})$ be the projection onto the first coordinate. The map $f_1$ induces an $U_i$--isomorphism $$\big(\mathbf{RT}(\mathcal{C})/\boldsymbol{\mu}_2\big)\big|_{U_i}\simeq\mathbf{RT}(\mathcal{C})|_{U_i}\big/\boldsymbol{\mu}_2|_{U_i}\overset{\sim}{\longrightarrow}\mathbf{SO}(\mathcal{Q_C})|_{U_i}$$ [@OA p. 893], where the first isomorphism comes from a similar argument as for the quotient scheme in theorem [Theorem 19](#thm4.10){reference-type="ref" reference="thm4.10"}. It follows that we have an $X$--isomorphism $\mathbf{RT}(\mathcal{C})/\boldsymbol{\mu}_2\overset{\sim}{\longrightarrow}\mathbf{SO}(\mathcal{Q_C})$, so we get a commutative diagram $$\begin{tikzcd} \mathbf{RT}(\mathcal{C})\arrow[r,"f_1"]\arrow[d,"\Pi"]&\mathbf{SO}(\mathcal{Q_C})\arrow[d,"\Pi_+"]\\ \mathbf{S}_\mathcal{Q_C}^2\arrow[r,"\rho"]&\mathbf{S}_\mathcal{Q_C}^2/\boldsymbol{\mu}_2 \end{tikzcd} \label{cartdiag}\tag{$\dag$}$$ where $\Pi$ and $\Pi_+$ are $\mathbf{Aut}(\mathcal{C})$--torsors. **Lemma 26**. *The diagram $($[\[cartdiag\]](#cartdiag){reference-type="ref" reference="cartdiag"}$)$ is a Cartesian square.* *Proof.* Let $\{U_i\}_{i\in I}$ be an affine open cover of $X$. For each $i\in I$, the restricted diagram $$\begin{tikzcd} \mathbf{RT}(\mathcal{C})|_{U_i}\arrow[r,"f_1|_{U_i}"]\arrow[d,"\Pi|_{U_i}"]&\mathbf{SO}(\mathcal{Q_C})|_{U_i}\arrow[d,"\Pi_+|_{U_i}"]\\ \mathbf{S}_\mathcal{Q_C}^2|_{U_i}\arrow[r,"\rho|_{U_i}"]&(\mathbf{S}_\mathcal{Q_C}^2/\boldsymbol{\mu}_2)|_{U_i} \end{tikzcd}$$ is a Cartesian square [@OA p. 893]. Let $Y\in\mathbf{Sch}/X$, and let $Y\overset{g}{\to}\mathbf{SO}(\mathcal{Q_C})$ and $Y\overset{h}{\to}\mathbf{S}_\mathcal{Q_C}^2$ be $X$--morphisms such that $\Pi_+g=\rho h$. Then $$(\Pi_+|_{U_i})(g|_{U_i})=(\Pi_+g)|_{U_i}=(\rho h)|_{U_i}=(\rho|_{U_i})(h|_{U_i}),$$ so for all $i\in I$ there exists a morphism $\lambda_i:Y|_{U_i}\to\mathbf{RT}(\mathcal{C})|_{U_i}$ such that $$g|_{U_i}=f_1|_{U_i}\lambda_i\text{ and } h|_{U_i}=\Pi|_{U_i}\lambda_i.$$ On the intersection $U_i\cap U_j$, both $\lambda_i$ and $\lambda_j$ make the restricted diagram commute. By the uniqueness in a Cartesian square, we have $$\lambda_i|_{U_i\cap U_j}=\lambda_j|_{U_i\cap U_j}.$$ Then there exists a $\lambda:Y\to\mathbf{RT}(\mathcal{C})$ such that $\lambda|_{U_i}=\lambda_i$ [@stacks Lemma 6.33.1]. In particular, this $\lambda$ satisfies $$g=f_1\lambda\text{ and }h=\Pi\lambda$$ and we are done. ◻ We have two projections $p_i:\mathbf{S}_\mathcal{Q_C}^2/\boldsymbol{\mu}_2\to\mathbf{S}_\mathcal{Q_C}/\boldsymbol{\mu}_2$, $i=1,2$, defined by projection onto the first and second coordinate respectively. We get two actions of $\mathbf{RT}(\mathcal{C})$ on $\mathbf{S}_\mathcal{Q_C}$: $$\big((t_1,t_2,t_3),u\big)\mapsto t_3(u),\quad\text{and }\quad\big((t_1,t_2,t_3),v\big)\mapsto t_2(v).$$ This induces two actions of $\mathbf{RT}(\mathcal{C})/\boldsymbol{\mu}_2\simeq\mathbf{SO}(\mathcal{Q_C})$ on $\mathbf{S}_\mathcal{Q_C}/\boldsymbol{\mu}_2$, which we denote by $(g,x)\mapsto g\bullet_ix$, $i=1,2$. **Lemma 27**. *Both actions of $\mathbf{SO}(\mathcal{Q_C})(X)$ on $(\mathbf{S}_{\mathcal{Q_C}}/\boldsymbol{\mu}_2)(X)$ are transitive.* *Proof.* First we will prove that the orbit map $\mathbf{SO}(\mathcal{Q_C})\to\mathbf{S}_\mathcal{Q_C}/\boldsymbol{\mu}_2$, $g\mapsto g\bullet_j[1]$ admits a splitting for each $j=1,2$. Let $j=2$. We have a map $\mathbf{S}_\mathcal{Q_C}\to\mathbf{SO}(\mathcal{Q_C})$ defined by $$\begin{aligned} \mathbf{S}_\mathcal{Q_C}(Y)&\to\mathbf{SO}(\mathcal{Q_C})(Y)\\ a&\mapsto B_a \end{aligned}$$ The map is $\boldsymbol{\mu}_2$--invariant, since for any $X$--scheme $Y$ and any $\eta\in\boldsymbol{\mu}_2(Y)$ $$B_{\eta a}(Y)(x)=\eta a\cdot x\cdot\eta a=\eta^2\cdot a\cdot x\cdot a=a\cdot x\cdot a=B_a(V)(x),$$ so it induces a map $h:\mathbf{S}_\mathcal{Q_C}/\boldsymbol{\mu}_2\to\mathbf{SO}(\mathcal{Q_C})$. Let $x\in(\mathbf{S}_\mathcal{Q_C}/\boldsymbol{\mu}_2)(X)$ and $\{U_i\to X\}$ be an fppf covering. Then for every $i$, $x_{U_i}$ lifts to some $a_i\in\mathbf{S}_\mathcal{Q_C}(U_i)$. Then $h(X)(x)_{U_i}=B_{a_i}(U_i)$, which lifts to $(B_{a_i},R_{\overline{a_i}},L_{\overline{a_i}})=:\mathbf{t}\in\mathbf{RT}(\mathcal{C})(U_i)$ (see example [Example 18](#ex5.2){reference-type="ref" reference="ex5.2"}). We have $$h(X)(x)_{U_i}\bullet_2x_{U_i}=p_1(\mathbf{t}.x_{U_i})=R_{\overline{a_i}}(U_i)[a_i]=[R_{\overline{a_i}}(U_i)(a_i)]=[1],$$ and since it holds for each $U_i$ it follows that $$h(X)(x)^{-1}\bullet_2[1]=x.$$ So $x\mapsto h(X)(x)^{-1}$ is a section of the orbit map. The case $j=1$ is done similarly, but with the first projection $p_1$. Now, let $x,y\in \mathbf{S}_\mathcal{Q_C}/\boldsymbol{\mu}_2$. We have $h(X)(y)^{-1}\cdot h(X)(x)\in\mathbf{SO}(\mathcal{Q_C})(X)$, so $$(h(X)(y)^{-1}\cdot h(X)(x))\bullet_jx=h(X)(y)^{-1}\bullet_j(h(X)(x)\bullet_j x)=h(X)(y)^{-1}\bullet_j[1]=y.$$ Hence the actions are transitive. ◻ Now we have everything we need to state and prove our last theorem. **Theorem 28**. *Let $\mathbf{G}=\mathbf{Aut}(\mathcal{C})$ and let $$\begin{aligned} S_1&=\ker\big(H^1_\text{fppf}(X,\mathbf{G})\to H^1_\text{fppf}(X,\mathbf{RT}(\mathcal{C}))\big),\\ S_2&=\ker\big(H^1_\text{fppf}(X,\mathbf{G})\to H^1_\text{fppf}(X,\mathbf{O}(\mathcal{Q}_\mathcal{C}))\big), \end{aligned}$$ where the maps are induced by $$\begin{aligned} \mathbf{Aut}(\mathcal{C})&\to\mathbf{RT}(\mathcal{C}),\qquad\qquad\mathbf{Aut}(\mathcal{C})\to\mathbf{O}(\mathcal{Q}_\mathcal{C})\\ \varphi&\mapsto(\varphi,\varphi,\varphi)\qquad\qquad\qquad\psi\mapsto\psi \end{aligned}$$ Then $S_1=S_2$.* **Remark 29**. The maps are well--defined by remark [Remark 12](#remark3.5){reference-type="ref" reference="remark3.5"} and theorem [Theorem 19](#thm4.10){reference-type="ref" reference="thm4.10"}(1). *Proof of theorem [Theorem 28](#thm4.17){reference-type="ref" reference="thm4.17"}.* Parts of this proof is inspired by the proof of theorem 6.6 in [@OA]. Notice that $S_1$ classifies the octonion algebras over $X$ that are isomorphic to $\mathcal{C}^{a,b}$. We have homomorphisms $$\begin{aligned} i_1:\mathbf{Aut}(\mathcal{C})&\to\mathbf{RT}(\mathcal{C}),\qquad\qquad\qquad i_2:\mathbf{RT}(\mathcal{C})\to\mathbf{SO}(\mathcal{Q}_\mathcal{C})\\ \varphi&\mapsto(\varphi,\varphi,\varphi)\qquad\qquad\qquad(\psi_{1},\psi_{2},\psi_{3})\mapsto\psi_{1}\\ i_3:\mathbf{SO}(\mathcal{Q}_\mathcal{C})&\to\mathbf{O}(\mathcal{Q}_\mathcal{C}),\qquad\quad i_4:\mathbf{Aut}(\mathcal{C})\to\mathbf{O}(\mathcal{Q}_\mathcal{C})\\ \rho&\mapsto\rho\qquad\qquad\qquad\qquad\qquad\varphi\mapsto\varphi\\ \end{aligned}$$ We see that $i_4=i_3\circ i_2\circ i_1$. $H^1(X,\_)$ is a covariant functor so $i_4^\ast=i_3^\ast\circ i_2^\ast\circ i_1^\ast$, where $i_j^\ast:=H^1_\text{fppf}(X,i_j)$. From this it follows that $S_1\subseteq S_2$. Let $[C']\in S_2$. Then there exists $\varphi:\mathcal{C}\to C'$ such that $\varphi\in\mathbf{O}(\mathcal{Q_C})$. For any affine connected subset $U\subset X$, either $\varphi|_U\in\mathbf{SO}(\mathcal{Q_C})$ or $\varphi|_U\in\mathbf{O}(\mathcal{Q_C})\setminus\mathbf{SO}(\mathcal{Q_C})$ [@OA Theorem 6.6]. Let $X_i$ be the connected components of $X$, and $\{U_{ij}\}$ an affine connected cover of $X_i$. If for some $j_1,j_2$ we have $U_{ij_1}\cap U_{ij_2}\neq\varnothing$ and $$\varphi|_{U_{ij_1}}\in\mathbf{SO}(\mathcal{Q_C}),\text{ and }\varphi|_{U_{ij_2}}\in\mathbf{O}(\mathcal{Q_C})\setminus\mathbf{SO}(\mathcal{Q_C}),$$ then there exists an open affine $V\subset U_{ij_1}\cap U_{ij_2}$ such that $$\varphi|_V\in\mathbf{SO}(\mathcal{Q_C}),\text{ and }\varphi|_V\in\mathbf{O}(\mathcal{Q_C})\setminus\mathbf{SO}(\mathcal{Q_C}),$$ a contradiction. Hence, for every $i$, either $$\varphi|_{U_{ij}}\in\mathbf{SO}(\mathcal{Q_C})\quad\forall j$$ or $$\varphi|_{U_{ij}}\in\mathbf{O}(\mathcal{Q_C})\setminus\mathbf{SO}(\mathcal{Q_C})\quad\forall j.$$ Then there exists a morphism $\psi:\mathcal{C}\to C'$ such that $$\psi|_{U_{ij}}=\left\{ \begin{array}{cc} \varphi|_{U_{ij}},&\text{if }\varphi|_{U_{ij}}\in\mathbf{SO}(\mathcal{Q_C})\\ \kappa|_{U_{ij}}\circ\varphi|_{U_{ij}},&\text{if }\varphi|_{U_{ij}}\in\mathbf{O}(\mathcal{Q_C})\setminus\mathbf{SO}(\mathcal{Q_C}) \end{array} \right.$$ Then $\psi\in\mathbf{SO}(\mathcal{Q_C})$. There exists a bijection [@Gille Proposition 2.4.3] $$\begin{aligned} \phi:\mathbf{SO}(\mathcal{Q_C})(X)\backslash(\mathbf{S}_\mathcal{Q_C}^2/\boldsymbol{\mu}_2)(X)&\overset{\sim}{\longrightarrow}S_2\\ x&\longmapsto\Pi_+^{-1}(x)\wedge^\mathbf{G}\mathbf{W}(\mathcal{C}), \end{aligned}$$ where $\mathbf{SO}(\mathcal{Q_C})(X)\backslash(\mathbf{S}_\mathcal{Q_C}^2/\boldsymbol{\mu}_2)(X)$ is the set of orbits of the $\mathbf{SO}(\mathcal{Q_C})(X)$--action. Each orbit can be represented by an element in $(\mathbf{S}_\mathcal{Q_C}^2/\boldsymbol{\mu}_2)(X)$, so let $x\in(\mathbf{S}_\mathcal{Q_C}^2/\boldsymbol{\mu}_2)(X)$ such that $\phi(x)=[C']$ and let $x_1$ be its projection onto the first copy of $(\mathbf{S}_\mathcal{Q_C}/\boldsymbol{\mu}_2)(X)$. By Lemma [Lemma 27](#lemma4.17){reference-type="ref" reference="lemma4.17"}, we may assume $x_1=[1]\in(\mathbf{S}_\mathcal{Q_C}/\boldsymbol{\mu}_2)(X)$. Consider the commutative diagram $$\begin{tikzcd} \mathbf{S}_\mathcal{Q_C}^2\arrow[r]\arrow[d,"\varrho_1"]&\mathbf{S}_\mathcal{Q_C}\arrow[d,"\varrho_2"]\\ \mathbf{S}_\mathcal{Q_C}^2/\boldsymbol{\mu}_2\arrow[r]&\mathbf{S}_\mathcal{Q_C}/\boldsymbol{\mu}_2 \end{tikzcd}$$ where the horizontal maps are projections onto the first coordinate. Note that $\varrho_1$ and $\varrho_2$ define $\boldsymbol{\mu}_2$--torsors over $\mathbf{S}_\mathcal{Q_C}^2/\boldsymbol{\mu}_2$ and $\mathbf{S}_\mathcal{Q_C}/\boldsymbol{\mu}_2$ respectively. By the commutativity of the diagram, we get an isomorphism $\varrho_1^{-1}(x)\to\varrho_2^{-1}([1])$ of $\boldsymbol{\mu}_2$--torsors. Since $\varrho_2^{-1}(X)([1])=\boldsymbol{\mu}_2(X)\cdot1\neq\varnothing$, we have $\varrho_1^{-1}(x)\neq\varnothing$, so $x$ lifts to an element $(a,b)\in\mathbf{S}_\mathcal{Q_C}^2(X)$. By the Cartesian diagram $($[\[cartdiag\]](#cartdiag){reference-type="ref" reference="cartdiag"}$)$ we have a morphism $\Pi^{-1}(a,b)\overset{f_1}{\to}\Pi_+^{-1}(x)$ and, since $\Pi$ and $\Pi_+$ are $\mathbf{G}$--torsors, this morphism must be an isomorphism. Then we have an isomorphism of fppf--sheaves of algebras $$\mathbf{W}(\mathcal{C}^{a,b})\simeq\Pi^{-1}(a,b)\wedge^\mathbf{G}\mathbf{W}(\mathcal{C})\simeq\Pi_+^{-1}(x)\wedge^\mathbf{G}\mathbf{W}(\mathcal{C})\simeq\mathbf{W}(C').$$ Thus $[C']\in S_1$. ◻ **Corollary 30**. *Let $\mathcal{C}$ and $\mathcal{C}'$ be two octonion algebras over X. The quadratic forms $\mathcal{Q}_\mathcal{C}$ and $\mathcal{Q}_{\mathcal{C}'}$ are isometric if and only if there exists $a,b\in\mathbf{S}_\mathcal{Q_C}(X)$ such that $\mathcal{C}'$ is isomorphic to $\mathcal{C}^{a,b}$.* *Proof.* By theorem [Theorem 23](#thm4.7){reference-type="ref" reference="thm4.7"} and lemma [Lemma 25](#lemma4.9){reference-type="ref" reference="lemma4.9"}, the set $S_1$ classifies the octonion algebras over $X$ isomorphic to $\mathcal{C}^{a,b}$ for some $a,b\in\mathbf{S}_\mathcal{Q_C}(X)$. Also by lemma [Lemma 25](#lemma4.9){reference-type="ref" reference="lemma4.9"}, the set $S_2$ classifies octonion algebras whose norm is isometric to $\mathcal{Q_C}$. By the equality established in theorem [Theorem 28](#thm4.17){reference-type="ref" reference="thm4.17"}, the statement follows. ◻ [^1]: Department of Mathematics (MAI), Linköpings University, 581 83 Linköping, Sweden. Email address: victor.hildebrandsson\@gmail.com [^2]: This work was done while the author was a master student at the Department of Mathematics, Uppsala University. Supervisor was S. Alsaody.

arxiv_math

--- abstract: | The quasisymmetric generating function of the set of permutations whose inverses have a fixed descent set is known to be symmetric and Schur-positive. The corresponding representation of the symmetric group is called the descent representation. In this paper, we provide an extension of this result to colored permutation groups, where Gessel's fundamental quasisymmetric functions are replaced by Poirier's colored quasisymmetric functions. For this purpose, we introduce a colored analogue of zigzag shapes and prove that the representations associated with these shapes coincide with colored descent representations studied by Adin, Brenti and Roichman in the case of two colors and Bagno and Biagioli in the general case. Additionally, we provide a colored analogue of MaMahon's alternating formula which expresses ribbon Schur functions in the basis of complete homogeneous symmetric functions. author: - Vassilis Dionyssis Moustakas title: Descent representations and colored quasisymmetric functions --- # Introduction {#sec:intro} The basis of Schur functions forms one of the most interesting basis of the space of symmetric functions [@StaEC2 Chapter 7]. Schur functions appear in the representation theory of the symmetric group as characters of irreducible representations. A symmetric function is called Schur-positive if it is a linear combination of Schur functions with nonngegative coefficients. The problem of determining whether a given symmetric function is Schur-positive constitutes a major problem in algebraic combinatorics [@Sta00]. Adin and Roichman [@AR15] highlighted a connection between Schur-positivity of certain quasisymmetric generating functions and the existence of formulas which express the characters of interesting representations as weighted enumerations of nice combinatorial objects. Quasisymmetric functions are certain power series in infinitely many variables that generalize the notion of symmetric functions. They first appeared in the work of Stanley and were later defined and systematically studied by Gessel [@Ges84] (see also [@BM16]). An example of this connection of particular interest involves the quasisymmetric generating function of inverse descent classes of the symmetric group and the characters of Specht modules of zigzag shapes, often called descent representations (for all undefined terminology we refer to [2](#sec:pre){reference-type="ref" reference="sec:pre"}). Adin, Brenti and Roichman [@ABR05] studied descent representations by using the coinvariant algebra as a representation space and provided an extension to the hyperoctahedral group, which was later generalized to every complex reflection group by Bagno and Biagioli [@BB07]. Recently, Adin et al. [@AAER17] investigated an extension of the aforementioned connection to the hyperoctahedral setting, where Gessel's fundamental quasisymmetric functions were replaced by Poirier's signed quasisymmetric functions [@Poi98]. In particular, they proved [@AAER17 Proposition 5.5] that the signed quasisymmetric generating function of signed inverse descent classes is Schur-positive in the hyperoctahedral setting, but without explicitly specifying the corresponding characters. Motivated by the afore-mentioned result, in this paper, we aim to extend upon it in the case of colored permutation groups, a special class of complex reflection groups. In particular, we prove that the colored quasisymmetric generating function of inverse colored descent classes is Schur-positive in the colored setting and show that the corresponding characters are precisely the characters of colored descent representations studied by Bagno and Biagioli (see ). For this purpose, we suggest a colored analogue of Gessel's zigzag shape approach to descent representations. Furthermore, we provide a colored analogue of a well-known formula due to MacMahon, popularized by Gessel [@Ges84], which expresses the Frobenius image of colored descent representations, usually called ribbon Schur functions, as an alternating sum of complete homogeneous symmetric functions in the colored context (see ). The paper is structured as follows. discusses background on permutations, tableaux, compositions, zigzag diagrams, symmetric/quasisymmetric functions and descent representations. reviews the combinatorics of colored compositions, colored permutations and colored quasisymmetric functions. introduces and studies the notion of colored zigzag shapes and proves the main results of this paper, namely . # Preliminaries {#sec:pre} This section fixes notation and discusses background. Throughout this paper we assume familiarity with basic concepts in the theory of symmetric functions and representations of the symmetric group as presented, for example, in [@StaEC2 Chapter 7]. For a positive integer $n$, we write $[n]:=\{1,2,\dots,n\}$ and denote by $|S|$ the cardinality of a finite set $S$. ## Permutations, tableaux, compositions and zigzag diagrams {#subsec:permutation-preliminaries} A of a positive integer $n$ is a sequence $\alpha= (\alpha_1, \alpha_2, \dots, \alpha_k)$ of positive integers such that $\alpha_1 + \alpha_2 + \cdots + \alpha_k = n$. Compositions of $n$ are in one-to-one correspondence with subsets of $[n-1]$. In particular, let ${\rm S}_\alpha:= \{r_1,r_2, \dots, r_{k-1}\}$ be the set of partial sums $r_i := \alpha_1 + \alpha_2 + \cdots +\alpha_i$, for all $1 \le i \le k$. Conversely, given a subset $S = \{s_1 < s_2 < \cdots < s_k\} \subseteq [n-1]$, let ${\rm co}(S) = \left(s_1, s_2-s_1, \dots, s_k-s_{k-1}, n-s_k\right)$. The maps $\alpha\mapsto {\rm S}_\alpha$ and $S \mapsto {\rm co}(S)$ are bijections and mutual inverses. Sometimes, it will be convenient to work with subsets of $[n-1]$ which contain $n$. For this purpose, we will write $S^+ := S \cup\{n\}$. In this case, ${\rm S}^+_\alpha= \{r_1,r_2, \dots, r_k\}$ and the maps $\alpha\mapsto {\rm S}^+_\alpha$ and $S^+ \mapsto {\rm co}(S^+)$ remain bijections and mutual inverses. We make this (non-standard) convention because we will later need to keep track of the color of the last coordinate of a colored permutation (see ). The set of all compositions of $n$, written ${\rm Comp}(n)$, becomes a poset with the partial order of reverse refinement. The covering relations are given by $$(\alpha_1, \dots, \alpha_i + \alpha_{i+1}, \dots, \alpha_k) \prec (\alpha_1, \dots, \alpha_i,\alpha_{i+1}, \dots, \alpha_k).$$ The corresponding partial order on the set of all subsets of $[n-1]$ is inclusion of subsets. A of $n$, written $\lambda\vdash n$, is a composition $\lambda$ of $n$ whose parts appear in weakly decreasing order. A (also called border-strip, ribbon or skew hook) is a connected skew shape that does not contain a $2\times2$ square. Ribbons with $n$ cells are in one-to-one correspondence with compositions of $n$. Given $\alpha\in {\rm Comp}(n)$, let ${\rm Z}_\alpha$ be the ribbon with $n$ cells whose row lengths, when read from bottom to top, are the parts of $\alpha$. For example, for $n=9$ $$\alpha= (2,1,2,3,1) \quad \longmapsto \quad {\rm Z}_\alpha= \ytableausetup{centertableaux,smalltableaux}\ydiagram{4+1, 2+3, 1+2, 1+1, 2} \, .$$ Let $\mathfrak{S}_n$ be the symmetric group on $[n]$. We will think of permutations as words and write them in one-line notation $\pi = \pi_1\pi_2\cdots\pi_n$. The of $\pi \in \mathfrak{S}_n$ is defined by ${\rm Des}(\pi) := \{i \in [n-1]: \pi_i > \pi_{i+1}\}$. Also, let ${\rm co}(\pi) := {\rm co}({\rm Des}(\pi))$ be the of $\pi$. The descent composition of $\pi$ essentially records the lengths of increasing runs of $\pi$. For $\alpha\in {\rm Comp}(n)$, we define the $${\rm D}_\alpha:= \{\pi \in \mathfrak{S}_n : {\rm co}(\pi) = \alpha\}$$ and the corresponding $${\rm D}_\alpha^{-1} := \{\pi \in \mathfrak{S}_n : {\rm co}(\pi^{-1}) = \alpha\}.$$ Let ${\rm SYT}(\lambda/\mu)$ be the set of all standard Young tableaux of a skew shape $\lambda/\mu$. The of a standard Young tableau $Q\in{\rm SYT}(\lambda/\mu)$, written ${\rm Des}(Q)$, is the set of all $i \in [n-1]$ such that $i+1$ appears in a lower row than $i$ does in $Q$. Also, we write ${\rm co}(Q) := {\rm co}({\rm Des}(Q))$. It is well-known that permutations of $\mathfrak{S}_n$ are in one-to-one correspondence with standard Young tableaux of ribbon shape with $n$ cells. The following refinement of this fact explains the connection between (inverse) descent classes and tableaux of ribbon shape (see, for example, [@AR14 Propositions 3.5 and 10.12]). **Proposition 1**. *For every $\alpha\in {\rm Comp}(n)$, there exists a bijection ${\rm SYT}({\rm Z}_\alpha) \rightarrow{\rm D}_\alpha$ with $Q \mapsto \pi$ such that ${\rm Des}(Q) = {\rm Des}(\pi^{-1})$. In particular, the distribution of the descent set is the same over ${\rm D}_\alpha^{-1}$ and ${\rm SYT}({\rm Z}_\alpha)$.* The resulting permutation of is often called the of the standard Young tableau $Q$, and it is the word obtained by reading the cell entries of $Q$ in the northeast direction, starting from the southwestern corner. For example, for $n=4$ and $\alpha= (2,2)$ we have $$\ytableausetup{mathmode,centertableaux} \begin{ytableau} \none & {\color{darkcandyapplered}2} & 4 \\ 1 & 3 \end{ytableau} \mapsto 1324 \quad \begin{ytableau} \none & 2 & {\color{darkcandyapplered}3} \\ 1 & 4 \end{ytableau} \mapsto 1423 \quad \begin{ytableau} \none & {\color{darkcandyapplered}1} & 4 \\ 2 & 3 \end{ytableau} \mapsto 2314 \quad \begin{ytableau} \none & {\color{darkcandyapplered}1} & {\color{darkcandyapplered}3} \\ 2 & 4 \end{ytableau} \mapsto 2413 \quad \begin{ytableau} \none & 1 & {\color{darkcandyapplered}2} \\ 3 & 4 \end{ytableau} \mapsto 3412,$$ where colored entries represent the descents of the corresponding tableaux and $${\rm D}_\alpha^{-1} = \{13{\color{darkcandyapplered}\cdot}24, \, 134{\color{darkcandyapplered}\cdot}2, \, 3{\color{darkcandyapplered}\cdot}124, \, 3{\color{darkcandyapplered}\cdot}14{\color{darkcandyapplered}\cdot}2, \,34{\color{darkcandyapplered}\cdot}12\},$$ where colored dots represent the descents of the corresponding permutations. ## The characteristic map, quasisymmetric functions and descent representations {#subsec:qsym-desrep} Let ${\bm x}= (x_1, x_2, \dots)$ be a sequence of commuting indeterminates and consider the space ${\rm Sym}_n$ of homogeneous symmetric functions of degree $n$ in ${\bm x}$. The , written ${\rm ch}$, is a $\mathbb{C}$-linear isomorphism from the space of virtual $\mathfrak{S}_n$-representations to ${\rm Sym}_n$. The characteristic map sends the irreducible $\mathfrak{S}_n$-representations corresponding to $\lambda\vdash n$ to the $s_\lambda({\bm x})$ associated to $\lambda$ and in particular it maps non-virtual $\mathfrak{S}_n$-representations to Schur-positive symmetric functions. The associated to $\alpha\in {\rm Comp}(n)$ is defined by $$F_\alpha({\bm x}) := \sum_{\substack{1 \le i_1 \le i_2 \le \cdots \le i_n \\ j \in {\rm S}_\alpha\, \Rightarrow\, i_j < i_{j+1}}} x_{i_1}x_{i_2} \cdots x_{i_n}.$$ We recall the following well-known expansion [@StaEC2 Theorem 7.19.7] $$\label{eq:SchurF} s_{\lambda/\mu}({\bm x}) = \sum_{Q \in {\rm SYT}(\lambda/\mu)} F_{{\rm co}(Q)}({\bm x}),$$ for any skew shape $\lambda/\mu$. A subset $\mathcal{A}\subseteq \mathfrak{S}_n$ is called if the quasisymmetric generating function $$F(\mathcal{A};{\bm x}) := \sum_{\pi \in \mathcal{A}} F_{{\rm co}(\pi)}({\bm x})$$ is Schur-positive. In this case, it follows that ${\rm ch}(\varrho)({\bm x}) = F(\mathcal{A};{\bm x})$ for some non-virtual $\mathfrak{S}_n$-representation $\rho$ (see also [@AAER17 Corollary 3.3]) and we will say that $\mathcal{A}$ is Schur-positive for $\varrho$. The skew Schur function $r_\alpha({\bm x}) := s_{{\rm Z}_\alpha}({\bm x})$ is called the corresponding to $\alpha\in {\rm Comp}(n)$. The (virtual) $\mathfrak{S}_n$-representation $\varrho_\alpha$ such that ${\rm ch}(\varrho_\alpha)({\bm x}) = r_\alpha({\bm x})$ is called the of the symmetric group. We remark that this definition is not the standard way to define descent representations in the literature. For more information on descent representations from a combinatorial representation-theoretic point of view we refer to [@ABR05]. For example, descent representations are non-virtual $\mathfrak{S}_n$-representations, as the following proposition explains. Combining yields the following result of Gessel [@Ges84 Theorem 7] (see also [@StaEC2 Corollary 7.23.4]). **Proposition 2**. *For every $\alpha\in {\rm Comp}(n)$, $$\label{eq:RibbonSchur-to-Schur} r_\alpha({\bm x}) \ = \ F({\rm D}_\alpha^{-1}; {\bm x}) \ = \ \sum_{\lambda \vdash n} \, c_\lambda(\alpha) \, s_\lambda({\bm x}),$$ where $c_\lambda(\alpha)$ is the number of $Q \in {\rm SYT}(\lambda)$ such that ${\rm co}(Q) = \alpha$. In particular, inverse descent classes are Schur-positive for descent representations.* Descent representations in disguised form appear in Stanley's work [@Sta82] on group actions on posets. If $\chi_\alpha$ denotes the character of $\varrho_\alpha$, then [@Sta82 Theorem 4.3] is translated into the following alternating formula $$\label{eq:Stanley-desrep} \chi_\alpha= \sum_{\substack{\beta\in {\rm Comp}(n) \\ \beta\preceq \alpha}} (-1)^{\ell(\alpha) - \ell(\beta)} \, 1_\beta\uparrow_{\mathfrak{S}_\beta}^{\mathfrak{S}_n},$$ where - $\ell(\alpha)$ denotes the number of parts of $\alpha$, called length of $\alpha$ - $\mathfrak{S}_\alpha:= \mathfrak{S}_{\alpha_1}\times\mathfrak{S}_{\alpha_2}\times \cdots$ denotes the Young subgroup corresponding to $\alpha$ - $1_n$ (resp. $1_\alpha$) denotes the trivial $\mathfrak{S}_n$-character (resp. $\mathfrak{S}_\alpha$-character) - $\uparrow$ denotes induction of characters. Taking the Frobenius image, becomes $$\label{eq:ribbon-to-completehomogeneous} r_\alpha({\bm x}) = \sum_{\substack{\beta\in {\rm Comp}(n) \\ \beta\preceq \alpha}} (-1)^{\ell(\alpha) - \ell(\beta)} \, h_\beta({\bm x}),$$ where $h_\beta({\bm x})$ denotes the symmetric functions corresponding to $\beta$. As Gessel [@Ges84 page 293] points out, MacMahon was the first to study ribbon Schur functions by means of [\[eq:ribbon-to-completehomogeneous\]](#eq:ribbon-to-completehomogeneous){reference-type="ref" reference="eq:ribbon-to-completehomogeneous"}. In our running example, for $n=4$ and $\alpha= (2,2)$ $$r_\alpha({\bm x}) = 2F_{(2,2)}({\bm x}) + F_{(3,1)}({\bm x}) + F_{(1,3)}({\bm x}) + F_{(1,2,1)}({\bm x}) = s_{(2,2)}({\bm x}) + s_{(3,1)}({\bm x}),$$ since the tableaux of shape $(2,2)$ and $(3,1)$ and descent set $\{2\}$ are $$\begin{ytableau} 1 & {\color{darkcandyapplered}2} \\ 3 & 4 \end{ytableau} \quad \text{and} \quad \ytableausetup{mathmode,centertableaux} \begin{ytableau} 1 & {\color{darkcandyapplered}2} & 4 \\ 3 \end{ytableau} \qquad$$ respectively, which is also in agreement with $$r_\alpha({\bm x}) = h_{(2,2)}({\bm x}) - h_{(4)}({\bm x}).$$ # Combinatorics of colored objects {#sec:color} This section reviews the combinatorics of colored objects including colored permutations, colored compositions, $r$-partite tableaux, colored quasisymmetric functions and a colored analogue of the characteristic map. For the corresponding notions in the case of two colors we refer the reader to [@AAER17]. We fix a positive integer $r$ and view the elements of $\mathbb{Z}_r$, the cyclic group of order $r$, as colors $0,1,\dots,r-1$, totally ordered by the natural order inherited by the integers. Also, we will write $i^j$ instead of $(i,j)$ to represent colored integers, where $i$ is the underlying integer and $j$ is the color. ## Colored compositions and colored sets {#subsec:colored-comp-set} An of a positive integer $n$ is a pair $(\alpha,\epsilon)$ such that $\alpha\in {\rm Comp}(n)$ and $\epsilon\in \mathbb{Z}_r^{\ell(\alpha)}$ is a sequence of colors assigned to the parts of $\alpha$. An of $[n]$ is a pair $(S^+,\zeta)$ such that $S\subseteq [n-1]$ and $\zeta : S^+ \rightarrow\mathbb{Z}_r$ is a color map. For the examples, we will represent colored compositions (resp. sets) as ordered tuples (resp. sets) of colored integers. Colored compositions of $n$ are in one-to-one correspondence with colored subsets of $[n]$. The correspondence is given as follows: Given a colored composition $(\alpha,\epsilon)$, let $\sigma_{(\alpha,\epsilon)} := ({\rm S}^+(\alpha),\zeta)$ where $\zeta : {\rm S}^+(\alpha) \rightarrow\mathbb{Z}_r$ is defined by $\zeta(r_i) := \epsilon_i$. Conversely, given a colored subset $(S^+,\zeta)$ with $S^+ = \{s_1 < \cdots < s_k < s_{k+1}=n\}$, let ${\rm co}(S^+,\zeta) = ({\rm co}(S),\epsilon)$ where $\epsilon\in \mathbb{Z}_r^k$ is defined by letting $\epsilon_i = \zeta(s_i)$, for all $1 \le i \le k$. For example, for $n=10$ and $r=4$ $$\left( 2^{\color{mymagenta}0}, 2^{\color{mygreen}1} ,1^{\color{mygreen}1} , 1^{\color{myred}3}, 3^{\color{mygreen}1} , 1^{\color{myneworange}2}\right) \longleftrightarrow \left\{ 2^{\color{mymagenta}0}, 4^{\color{mygreen}1}, 5^{\color{mygreen}1}, 6^{\color{myred}3}, 9^{\color{mygreen}1}, 10^{\color{myneworange}2} \right\}.$$ Given a colored composition $(\alpha,\epsilon)$ of $n$, we can extend $\epsilon$ to a color vector $\tilde{\epsilon} \in \mathbb{Z}_r^n$ by letting $$\tilde{\epsilon} := (\underbrace{\epsilon_1, \epsilon_1, \dots, \epsilon_1}_{\text{$\alpha_1$ times}}, \underbrace{\epsilon_2, \epsilon_2, \dots, \epsilon_2}_{\text{$\alpha_2$ times}}, \dots, \underbrace{\epsilon_k, \epsilon_k, \dots, \epsilon_k}_{\text{$\alpha_k$ times}}).$$ Similarly, given a colored subset $(S^+,\zeta)$ of $[n]$ with $S^+ = \{s_1 < \cdots < s_k < s_{k+1}:=n\}$, we can extend the color map to a color vector $\tilde{\zeta} = (\tilde{\zeta}_1,\tilde{\zeta}_2, \dots, \tilde{\zeta}_n) \in \mathbb{Z}_r^n$, by letting $\tilde{\zeta}_j := \zeta(s_i)$ for all $s_{i-1} < j \le s_i$ where $s_0:=0$. The corresponding color vector of our running example is $$({\color{mymagenta}0}, {\color{mymagenta}0}, {\color{mygreen}1}, {\color{mygreen}1}, {\color{mygreen}1}, {\color{myred}3}, {\color{mygreen}1}, {\color{mygreen}1}, {\color{mygreen}1}, {\color{myneworange}2}).$$ The set of all $r$-colored compositions of $n$, written ${\rm Comp}(n,r)$, becomes a poset with the partial order of reverse refinement on consecutive parts of constant color. The covering relations are given by $$\left( (\dots, \alpha_i + \alpha_{i+1}, \dots), (\dots, \epsilon_i, \dots) \right) \prec \left( (\dots, \alpha_i, \alpha_{i+1}, \dots), (\dots, \epsilon_i, \epsilon_i, \dots) \right).$$ The corresponding partial order on $r$-colored subsets of $[n]$ is inclusion of subsets with the same color vector. Notice that these posets are not connected, since each color vector gives rise to a unique connected component (see, for example [@HP10 Figure 4]). ## Colored permutations and $r$-partite tableaux {#subsec:colored-perm-des} The wreath product $\mathbb{Z}_r\wr\mathfrak{S}_n$ is called the and we denote it by $\mathfrak{S}_{n,r}$. It consists of all pairs $(\pi,{\mathrm z})$, called , such that $\pi \in \mathfrak{S}_n$ is the underlying permutation and ${\mathrm z}= ({\mathrm z}_1, {\mathrm z}_2, \dots, {\mathrm z}_n) \in \mathbb{Z}_r^n$ is a color vector. When we consider specific examples, it will be convenient to write colored permutations in window notation, that is as words $\pi_1^{{\mathrm z}_1}\pi_2^{{\mathrm z}_2}\cdots\pi_n^{{\mathrm z}_n}$ on colored integers. The product in $\mathfrak{S}_{n,r}$ is given by the rule $$(\pi, {\mathrm z})(\tau, {\mathrm w}) = \left(\pi\tau, {\mathrm w}+ \tau({\mathrm z})\right)$$ where $\pi\tau$ is evaluated from right to left, $\tau({\mathrm z}) := ({\mathrm z}_{\tau_1},{\mathrm z}_{\tau_2}, \dots, {\mathrm z}_{\tau_n})$ and the addition is coordinatewise modulo $r$. The inverse (resp. conjugate) of $(\pi,{\mathrm z})$, written ${(\pi,{\mathrm z})}^{-1}$ (resp. $\overline{(\pi,{\mathrm z})}$) is the element $(\pi^{-1},-\pi^{-1}({\mathrm z}))$ (resp. $(\pi,-{\mathrm z})$). Colored permutation groups can be viewed as complex reflection groups (see, for example, [@BB07 Sections 1-2]). Therefore, $\mathfrak{S}_{n,r}$ can be realized as the group of all $n\times n$ matrices such that - the nonzero entries are $r$-th roots of unity, and - there is exactly one nonzero entry in every row and every column. For our purposes it is more convenient to view them as groups of colored permutations rather than groups of complex matrices. The case $r=2$ is of particular interest. In this case, it is often customary to write $\mathfrak{B}_n := \mathfrak{S}_{n,2}$ and identify colors $0$ and $1$ with signs $+$ and $-$, respectively. $\mathfrak{B}_n$ coincides with the , the symmetry group of the $n$-dimensional cube. The hyperoctahedral group is a real reflection group and its elements are called . Much of what is presented in this paper is motivated by Adin et al.'s work [@AAER17] on character formulas and descents for $\mathfrak{B}_n$. The of $(\pi,{\mathrm z}) \in \mathfrak{S}_{n,r}$, denoted by ${\rm sDes}(\pi,{\mathrm z})$, is the pair $(S^+,\zeta)$ where - $S$ consists of all $i \in [n-1]$ such that ${\mathrm z}_i \neq {\mathrm z}_{i+1}$ or ${\mathrm z}_i = {\mathrm z}_{i+1}$ and $i \in {\rm Des}(\pi)$ - $\zeta : S^+ \rightarrow\mathbb{Z}_r$ is the map defined by $\zeta(i) = {\mathrm z}_i$ for all $i \in S^+$. In words, the colored descent set records the ending positions of increasing runs of constant color together with their colors. Notice that the color vector of the colored descent set of $(\pi,{\mathrm z})$ is the same as ${\mathrm z}$. For example, for $n=10$ and $r=4$ $${\rm sDes}\left( 2^{\color{myred}3} 4^{\color{myred}3} 6^{\color{mygreen}1} 1^{\color{mygreen}1} 5^{\color{mygreen}1} {10}^{\color{myred}3} 3^{\color{mygreen}1} 7^{\color{mygreen}1} 9^{\color{mygreen}1}8^{\color{mymagenta}0}\right) = \{2^{\color{myred}3}, 3^{\color{mygreen}1}, 5^{\color{mygreen}1}, 6^{\color{myred}3}, 8^{\color{mygreen}1}, 9^{\color{mygreen}1}, 10^{\color{mymagenta}0}\}.$$ The $r$-colored composition which corresponds to the colored descent set ${\rm sDes}(\pi,{\mathrm z})$ is called of $(\pi,{\mathrm z})$ and is denoted by ${\rm co}(\pi,{\mathrm z})$. It records the lengths of increasing runs of constant color together with their colors. In our running example, we have $${\rm co}\left( 2^{\color{myred}3} 4^{\color{myred}3} 6^{\color{mygreen}1} 1^{\color{mygreen}1} 5^{\color{mygreen}1} {10}^{\color{myred}3} 3^{\color{mygreen}1} 7^{\color{mygreen}1} 9^{\color{mygreen}1}8^{\color{mymagenta}0}\right) = \left(2^{\color{myred}3}, 1^{\color{mygreen}1}, 2^{\color{mygreen}1}, 1^{\color{myred}3}, 1^{\color{mygreen}1}, 2^{\color{mygreen}1}, 1^{\color{mymagenta}0}\right).$$ For $(\alpha,\epsilon) \in {\rm Comp}(n,r)$, we define the $${\rm D}_{(\alpha,\epsilon)} := \{ (\pi,{\mathrm z}) \in \mathfrak{S}_{n,r} : {\rm co}(\pi,{\mathrm z}) = (\alpha,\epsilon)\}$$ and the corresponding $$\overline{{\rm D}}_{(\alpha,\epsilon)}^{-1} := \{ (\pi,{\mathrm z}) \in \mathfrak{S}_{n,r} : {\rm co}\left({\overline{(\pi,{\mathrm z})}}^{-1}\right) = (\alpha,\epsilon)\}.$$ For reasons that will become apparent in the sequel, instead of dealing with inverse descent classes it will be more convenient to deal with conjugate-inverse descent classes. Colored descent classes were introduced by Mantaci and Reutenauer [@MR95] who called them shape classes and used them to introduce and study a colored analogue of Solomon's descent algebra. We remark that in the hyperoctahedral case, where we have only two colors, there is no need to consider conjugate-inverse elements because $\mathfrak{B}_n$ is a real reflection group. An of $n$, written ${\bm \lambda}\vdash n$, is an $r$-tuple ${\bm \lambda}= (\lambda^{(0)}, \lambda^{(1)}, \dots, \lambda^{(r-1)})$ of (possibly empty) integer partitions of total sum $n$. For example, $${\bm \lambda}\ = \ \left( (2), (3,2,1), (1), (1) \right).$$ is a 4-partite partition of $10$. A of shape ${\bm \lambda}$ is an $r$-tuple ${\bm Q}= (Q^{(0)}, Q^{(1)}, \dots, Q^{(r-1)})$ of (possibly empty) tableaux, called parts, which are strictly increasing along rows and columns such that $Q^{(i)}$ has shape $\lambda^{(i)}$ and every element of $[n]$ appears exactly once as an entry of some $Q^{(i)}$. We denote by ${\rm SYT}({\bm \lambda})$ the set of all standard Young $r$-partite tableaux of shape ${\bm \lambda}$. To each $r$-partite tableau ${\bm Q}$, we associate a color vector ${\mathrm z}$, defined by letting ${\mathrm z}_i = j$, where $0 \le j \le r-1$ is such that $i \in Q^{(j)}$. For example, for $n=10$ and $r=4$ $${\bm Q} \ = \ \ytableausetup{mathmode} \left(\begin{ytableau} 1 & 9 \end{ytableau}\, , \ \begin{ytableau} 3 & 5 & 6 \\ 4 & 10 \\ 7 \end{ytableau}\, , \ \begin{ytableau} 2 \end{ytableau}\, , \ \begin{ytableau} 8 \end{ytableau} \right)$$ has color vector $${\mathrm z}= \left( {\color{mymagenta}0}, {\color{myneworange}2}, {\color{mygreen}1}, {\color{mygreen}1}, {\color{mygreen}1}, {\color{mygreen}1},{\color{mygreen}1}, {\color{myred}3}, {\color{mymagenta}0}, {\color{mygreen}1} \right)$$ The of an $r$-partite tableau ${\bm Q}$, denoted by ${\rm sDes}({\bm Q})$, is defined similarly to that for colored permutations. In this case, the colored descent set records the first element of a pair $(i, i+1)$ together with its color, such that $i$ and $i+1$ either belong to parts with different colors or they belong to the same part and $i$ is a descent of this part. In our running example, $${\rm sDes}({\bm Q}) = \left\{ 1^{\color{mymagenta}0}, 2^{\color{myneworange}2}, 3^{\color{mygreen}1}, 6^{\color{mygreen}1}, 7^{\color{mygreen}1}, 8^{\color{myred}3}, 9^{\color{mymagenta}0}, 10^{\color{mygreen}1} \right\}.$$ Also, we write ${\rm co}({\bm Q}) := {\rm co}({\rm sDes}({\bm Q}))$. ## Colored quasisymmetric functions and the characteristic map {#subsec:colored-qsym-ch} Consider $r$ copies ${\bm x}^{(0)}, {\bm x}^{(1)}, \dots, {\bm x}^{(r-1)}$ of ${\bm x}$, one for each color of $\mathbb{Z}_r$ and let ${\rm Sym}_n^{(r)}$ be the space of (homogeneous) formal power series of degree $n$ in ${\bm x}^{(0)}, {\bm x}^{(1)}, \dots, {\bm x}^{(r-1)}$ which are symmetric in each variable ${\bm x}^{(j)}$ separately. In particular, $${\rm Sym}_n^{(r)} = \bigoplus_{\substack{a_0, \dots, a_{r-1} \in \mathbb{N}\\ a_0 + \cdots + a_{r-1} = n}} \left( {\rm Sym}_{a_0}({\bm x}^{(0)}) \otimes \cdots \otimes {\rm Sym}_{a_{r-1}}({\bm x}^{(r-1)})\right).$$ Drawing parallel to the classical case, for an $r$-partite partition ${\bm \lambda}= (\lambda^{(0)}, \lambda^{(1)}, \dots, \lambda^{(r-1)})$, we define $$s_{\bm \lambda}:= s_{\lambda^{(0)}}({\bm x}^{(0)}) s_{\lambda^{(1)}}({\bm x}^{(1)}) \cdots s_{\lambda^{(r-1)}}({\bm x}^{(r-1)}).$$ The set $\{s_{\bm \lambda}: {\bm \lambda}\vdash n\}$ forms a basis for ${\rm Sym}_n^{(r)}$ which we call the . An element of ${\rm Sym}_n^{(r)}$ is called if all the coefficients in its expansion in the Schur basis are nonnegative. It is well-known that (complex) irreducible $\mathfrak{S}_{n,r}$-representations are indexed by $r$-partite partitions of $n$ (see, for example, [@BB07 Section 5]). Poirier [@Poi98] introduced a colored analogue of the characteristic map which we denote by ${\rm ch}^{(r)}$. This map is a $\mathbb{C}$-linear isomorphism from the space of virtual $\mathfrak{S}_{n,r}$-representations to ${\rm Sym}_n^{(r)}$ which sends the irreducible $\mathfrak{S}_{n,r}$-representation corresponding to ${\bm \lambda}\vdash n$ to $s_{\bm \lambda}$. In particular, it maps non-virtual $\mathfrak{S}_{n,r}$-representations to Schur-positive elements of ${\rm Sym}_n^{(r)}$. The () associated to $(\alpha,\epsilon) \in {\rm Comp}(n,r)$ is defined by $$\label{eq:colqsym-definition} F_{(\alpha,\epsilon)}^{(r)} := F_{(\alpha,\epsilon)}({\bm x}^{(0)}, \dots, {\bm x}^{(r-1)}) := \sum_{\substack{1 \le i_1 \le i_2 \le \cdots \le i_n \\ \epsilon_j \ge \epsilon_{j+1} \ \Rightarrow\ i_{r_j} < i_{r_{j+1}}}} x_{i_1}^{(\tilde{\epsilon_1})}x_{i_2}^{(\tilde{\epsilon_2})} \cdots x_{i_n}^{(\tilde{\epsilon_n})},$$ where the second restriction in the sum runs through all indices $1 \le j \le \ell(\alpha)-1$. For example, if $(m^n)$ denotes the vector (or sequence) of length $n$ and entries equal to $m$, then $$\begin{aligned} F_{\left((n), (k)\right)}^{(r)} &= \sum_{1 \le i_1 \le i_2 \le \cdots \le i_n} x_{i_1}^{(k)}x_{i_2}^{(k)} \cdots x_{i_n}^{(k)} = h_n({\bm x}^{(k)}) \\ F_{\left((1^n), (k^n)\right)}^{(r)} &= \sum_{1 \le i_1 < i_2 < \cdots < i_n} x_{i_1}^{(k)}x_{i_2}^{(k)} \cdots x_{i_n}^{(k)} = e_n({\bm x}^{(k)}),\end{aligned}$$ where $h_n$ (resp. $e_n$) denotes the $n$-th (resp. ) symmetric function. This colored analogue of Gessel's fundamental quasisymmetric function was introduced by Poirier [@Poi98] and has been studied by several people [@AAER17; @BaH08; @BeH06; @HP10; @Mou21]. It seems that this is particularly suitable when we consider colored permutation groups as wreath products. A different signed analogue of quasisymmetric functions was introduced by Chow [@Cho01] which has found applications when one considers the hyperoctahedral group as a Coxeter group (see, for example, [@BBJR20]). Steingrímsson [@Stei94 Definition 3.2] introduced a notion of descents for colored permutations which reduces to the classical one and using it we can provide an alternative (and more convenient) description for colored quasisymmetric functions. The of $(\pi,{\mathrm z}) \in \mathfrak{S}_{n,r}$ is defined by $${\rm Des}(\pi,{\mathrm z}) := \{i \in [n]: {\mathrm z}_i > {\mathrm z}_{i+1} \ \text{or} \ {\mathrm z}_i = {\mathrm z}_{i+1} \, \text{and} \, i \in {\rm Des}(\pi)\},$$ where $\pi_{n+1} := 0$ and ${\mathrm z}_{n+1} := 0$. In particular, $n \in {\rm Des}(\pi,{\mathrm z})$ if and only if ${\mathrm z}_n > 0$. With this in mind, for the colored descent composition of $(\pi,{\mathrm z})$ becomes $$\label{eq:colqsym-actual-definition} F_{(\pi,{\mathrm z})}^{(r)} \, := \, F_{{\rm co}(\pi,{\mathrm z})}^{(r)} \, = \sum_{\substack{1 \le i_1 \le i_2 \le \cdots \le i_n \\ j \in {\rm Des}(\pi,{\mathrm z}) \mathbin{\mathchoice{\hbox{\tikz{\draw[line width=0.6pt,line cap=round] (3pt,0) -- (0,6pt);}}}{\hbox{\tikz{\draw[line width=0.6pt,line cap=round] (3pt,0) -- (0,6pt);}}}{\hbox{\tikz{\draw[line width=0.45pt,line cap=round] (2pt,0) -- (0,4pt);}}}{\hbox{\tikz{\draw[line width=0.4pt,line cap=round] (1.5pt,0) -- (0,3pt);}}}}\{n\} \ \Rightarrow\ i_j < i_{j+1}}} x_{i_1}^{({\mathrm z}_1)}x_{i_2}^{({\mathrm z}_2)} \cdots x_{i_n}^{({\mathrm z}_n)}.$$ Adin et al. [@AAER17 Proposition 4.2] proved a signed analogue of , which can be trivially extended to the general case. **Proposition 3**. *For ${\bm \lambda}\vdash n$, $$\label{eq:SchurF-colored} s_{\bm \lambda}= \sum_{{\bm Q}\in {\rm SYT}({\bm \lambda})} F_{{\rm co}({\bm Q})}^{(r)}.$$* Finally, a subset $\mathcal{A}\subseteq \mathfrak{S}_{n,r}$ is called if the colored quasisymmetric generating function $$F^{(r)}(\mathcal{A}) := \sum_{(\pi,{\mathrm z}) \in \mathcal{A}} F_{(\pi,{\mathrm z})}^{(r)}$$ is a Schur-positive element of ${\rm Sym}_n^{(r)}$. In this case, it follows that ${\rm ch}^{(r)}(\varrho)({\bm x}) = F^{(r)}(\mathcal{A})$ for some non-virtual $\mathfrak{S}_{n,r}$-representation $\varrho$ (see also [@AAER17 Corollary 3.7]) and we will say that $\mathcal{A}$ is Schur-positive for $\varrho$. # Introducing colored zigzag shapes {#sec:zigzag-colored} This section introduces the notion of colored zigzag shapes and proves several properties which will be needed in the sequel. Following Bergeron and Hohlweg [@BeH06 Section 2.1] (see also [@HP10 Section 3.6]), the of a colored composition $(\alpha,\epsilon) \in {\rm Comp}(n,r)$ is the unique concatenation $(\alpha_{(1)},\epsilon_{(1)})(\alpha_{(2)},\epsilon_{(2)})\cdots(\alpha_{(m)},\epsilon_{(m)})$ of non-empty, monochromatic colored compositions $\alpha_{(i)}$ of color $\epsilon_{(i)}$ such that $\epsilon_{(i)} \neq \epsilon_{(i+1)}$ for all $1 \le i \le m-1$. For example, for $n=10$ and $r=4$ $$\left( 2^{\color{mymagenta}0}, 2^{\color{mygreen}1} ,1^{\color{mygreen}1} , 1^{\color{myred}3}, 3^{\color{mygreen}1} , 1^{\color{myneworange}2}\right) \ = \ {(2)}^{\color{mymagenta}0}{(2,1)}^{\color{mygreen}1}{(1)}^{\color{myred}3}{(3)}^{\color{mygreen}1}{(1)}^{\color{myneworange}2}.$$ Notice that each $\epsilon_{(i)}$ is a single color rather than a sequence of colors. **Definition 4**. An with $n$ cells is a pair $(Z,\epsilon)$, where $Z = (Z_1, \dots, Z_k)$ is a sequence of zigzag diagrams and $\epsilon= (\epsilon_1, \dots, \epsilon_k) \in \mathbb{Z}_r^k$ is a sequence of colors assigned to the parts of $Z$ such that $\epsilon_i \ne \epsilon_{i+1}$ for every $1 \le i \le k-1$. For example, there exist six 2-colored zigzag shapes with 2 cells $$\left(\,\ydiagram{2}\, ,{\color{mymagenta}0}\right), \ \left(\,\ydiagram{2}\, , {\color{mygreen}1}\right), \ \left(\left(\, \ydiagram{1}\, ,\ydiagram{1}\, \right), ({\color{mymagenta}0}, {\color{mygreen}1})\right), \ \left(\left(\, \ydiagram{1}\, ,\ydiagram{1}\, \right), ({\color{mygreen}1},{\color{mymagenta}0})\right), \ \left(\,\ydiagram{1,1}\, ,{\color{mymagenta}0}\right), \ \left(\,\ydiagram{1,1}\, ,{\color{mygreen}1}\right).$$ In general, as the following proposition suggests, the number of $r$-colored zigzag shapes with $n$ cells is equal to $r(r+1)^{n-1}$, the cardinality of ${\rm Comp}(n,r)$ (see [@HP10 Table 1]). **Proposition 5**. *The set of $r$-colored zigzag shapes with $n$ cells is in one-to-one correspondence with ${\rm Comp}(n,r)$ and therefore with the set of all $r$-colored subsets of $[n]$.* *Proof.* Given a colored composition of $n$ with rainbow decomposition $$(\alpha,\epsilon) = (\alpha_{(1)},\epsilon_{(1)})(\alpha_{(2)},\epsilon_{(2)})\cdots(\alpha_{(m)},\epsilon_{(m)})$$ we form the following colored zigag shape with $n$ cells $${\rm Z}_{(\alpha,\epsilon)} := \left( \left({\rm Z}_{\alpha_{(1)}}, {\rm Z}_{\alpha_{(2)}}, \dots, {\rm Z}_{\alpha_{(m)}}\right), \left(\epsilon_{(1)}, \epsilon_{(2)}, \dots, \epsilon_{(m)}\right) \right).$$ The map $(\alpha,\epsilon) \mapsto {\rm Z}_{(\alpha,\epsilon)}$ is the desired bijection. ◻ For example, the corresponding 4-colored zigzag shape with 10 cells to the 4-colored composition of our running example is $$\left( 2^{\color{mymagenta}0}, 2^{\color{mygreen}1} ,1^{\color{mygreen}1} , 1^{\color{myred}3}, 3^{\color{mygreen}1} , 1^{\color{myneworange}2}\right) \longleftrightarrow \left( \left( \, \ydiagram{2} \, , \, \ydiagram{1+1,2} \, , \, \ydiagram{1} \, , \, \ydiagram{3} \, , \, \ydiagram{1} \, \right), ({\color{mymagenta}0}, {\color{mygreen}1}, {\color{myred}3}, {\color{mygreen}1}, {\color{myneworange}2}) \right).$$ Now, to each colored zigzag shape we can associate an $r$-partite (skew) shape and consider standard Young $r$-partite tableaux of this shape. In particular, given an $r$-colored zigzag shape $(Z,\epsilon)$ we define the $r$-partite skew shape ${\bm \lambda}_{(Z,\epsilon)} := \left(Z^{(0)}, Z^{(1)}, \dots, Z^{(r-1)}\right)$, where $$Z^{(j)} := \bigoplus_{\substack{1 \le i \le k \\ \epsilon_i = j}} Z_i$$ for all $0 \le j \le r-1$. Here, the $\lambda\oplus \mu$ of two (skew) shapes $\lambda, \mu$ is the skew shape whose diagram is obtained by placing the diagram of $\lambda$ and $\mu$ in such a way that the upper-right vertex of $\lambda$ coincides with the lower-left vertex of $\mu$. In our running example, we have $$\left( \, \ydiagram{2} \, , \, \ydiagram{2+3,1+1,2} \, , \, \ydiagram{1} \, , \, \ydiagram{1} \, \right).$$ Notice that two different $r$-colored zigzag shapes can give rise to the same $r$-partite skew shape. The following proposition provides a colored analogue of [Proposition 1](#prop:ribbons-and-tableaux){reference-type="ref" reference="prop:ribbons-and-tableaux"}, which will be used in . **Proposition 6**. *For every $(\alpha,\epsilon) \in {\rm Comp}(n,r)$, there exists a bijection ${\rm D}_{(\alpha,\epsilon)} \rightarrow{\rm SYT}({\bm \lambda}_{{\rm Z}_{(\alpha,\epsilon)}})$ with $(\pi,{\mathrm z}) \mapsto {\bm Q}$ such that $${\rm sDes}({\bm Q}) = {\rm sDes}\left({\overline{(\pi,{\mathrm z})}}^{-1}\right).$$ In particular, the distribution of the colored descent set is the same over $\overline{{\rm D}}_{(\alpha,\epsilon)}^{-1}$ and ${\rm SYT}({\bm \lambda}_{{\rm Z}_{(\alpha,\epsilon)}})$.* To prepare for the proof, we remark that we can define the rainbow decomposition of any word (or sequence) of colored integers. In particular, the rainbow decomposition of a colored permutation $(\pi,{\mathrm z}) \in \mathfrak{S}_{n,r}$ is the unique concatenation $(\pi_{(1)},{\mathrm z}_{(1)})(\pi_{(2)},{\mathrm z}_{(2)})\cdots(\pi_{(m)},{\mathrm z}_{(m)})$ of non-empty, monochromatic permutations $\pi_{(i)}$ of color ${\mathrm z}_{(i)}$ such that ${\mathrm z}_{(i)} \neq {\mathrm z}_{(i+1)}$ for all $1 \le i \le m-1$. For example, for $n=10$ and $r=4$ $$2^{\color{mymagenta}0} 3^{\color{mymagenta}0} 7^{\color{mygreen}1} {10}^{\color{mygreen}1} 5^{\color{mygreen}1} 6^{\color{myred}3} 1^{\color{mygreen}1} 8^{\color{mygreen}1} 9^{\color{mygreen}1} 4^{\color{myneworange}2} \ = \ (23)^{\color{mymagenta}0} (7\,{10}\,5)^{\color{mygreen}1} (6)^{\color{myred}3} (189)^{\color{mygreen}1} (4)^{\color{myneworange}2}.$$ With this in mind, the colored descent composition of $(\pi,{\mathrm z})$ turns out to be the (unique) colored composition with the following rainbow decomposition $${\rm co}(\pi,{\mathrm z}) \ = \ ({\rm co}(\pi_{(1)}), {\mathrm z}_{(1)})({\rm co}(\pi_{(2)}), {\mathrm z}_{(2)})\cdots({\rm co}(\pi_{(m)}), {\mathrm z}_{(m)}).$$ *Proof of .* Let $(\alpha,\epsilon) \in {\rm Comp}(n,r)$ with rainbow decomposition $$(\alpha,\epsilon) = (\alpha_{(1)},\epsilon_{(1)})(\alpha_{(2)},\epsilon_{(2)})\cdots(\alpha_{(m)},\epsilon_{(m)}).$$ Given $(\pi,{\mathrm z}) \in {\rm D}_{(\alpha,\epsilon)}$, the discussion of the preceding paragraph implies that its rainbow decomposition satisfies ${\rm co}(\pi_{(i)}) = \alpha_{(i)}$ and ${\mathrm z}_{(i)} = \epsilon_{(i)}$, for all $1 \le i \le m$. Applying [Proposition 1](#prop:ribbons-and-tableaux){reference-type="ref" reference="prop:ribbons-and-tableaux"} yields a standard Young tableaux $Q_{(i)}$ of shape ${\rm Z}_{\alpha_{(i)}}$ corresponding to each $\pi_{(i)}$, for all $1 \le i \le m$. Now, define an $r$-partite tableau ${\bm Q}= (Q^{(0)}, Q^{(1)}, \dots, Q^{(r-1)})$ where $$Q^{(j)} = \bigoplus_{\substack{1 \le i \le m \\ \epsilon_{(i)} = j}} Q_{(i)}$$ for all $0 \le j \le r-1$. The shape of ${\bm Q}$ is ${\bm \lambda}_{{\rm Z}_{(\alpha,\epsilon)}}$, since $$\text{shape of $Q^{(j)}$} \ = \ \bigoplus_{\substack{1 \le i \le m \\ \epsilon_{(i)} = j}} {\rm Z}_{\alpha_{(i)}}.$$ The process can be reversed in a unique way and thus yielding the required bijection. For the second assertion, suppose $(\pi,{\mathrm z}) \mapsto {\bm Q}$. On the one hand, since $\overline{(\pi,{\mathrm z})}^{-1} = (\pi^{-1},\pi^{-1}({\mathrm z}))$ the color vector of ${\rm sDes}(\overline{(\pi,{\mathrm z})}^{-1})$ is equal to $\pi^{-1}({\mathrm z})$. On the other hand, the $i$-th entry of the color vector $\tilde{\zeta}$ of ${\rm sDes}({\bm Q})$ records the color of the part of ${\bm Q}$ in which $i$ belongs and therefore the way we defined ${\bm Q}$ implies $\tilde{\zeta} = \pi^{-1}({\mathrm z})$. These observations imply that ${\rm sDes}(\overline{(\pi,{\mathrm z})}^{-1})$ and ${\rm sDes}({\bm Q})$ have the same color vector and therefore they record the same changes of colors. It remains to examine what happens in the case of constant color. Suppose that the second component of ${\rm sDes}(\overline{(\pi,{\mathrm z})}^{-1})$ is $({\mathrm z}_{i_1}, {\mathrm z}_{i_2}, \dots, {\mathrm z}_{i_k})$, for some $1 \le i_1 < i_2 < \cdots < i_k \le n$. If ${\mathrm z}_{i_j} = {\mathrm z}_{i_{j+1}}$, then $i_j \in {\rm Des}(\pi^{-1})$ which implies that $i_j$ and $i_{j+1}$ belong to the same part $Q^{({\mathrm z}_{i_j})}$ of ${\bm Q}$ and that $i_j \in {\rm Des}(Q^{({\mathrm z}_{i_j})})$ which concludes the proof. ◻ **Example 7**. We illustrate the previous proof in a specific example for $n=10$ and $r=4$. Suppose $$(\alpha,\epsilon) \ = \ \left( 2^{\color{mymagenta}0}, 2^{\color{mygreen}1} ,1^{\color{mygreen}1} , 1^{\color{myred}3}, 3^{\color{mygreen}1} , 1^{\color{myneworange}2}\right) \ = \ (2^{\color{mymagenta}0})(2^{\color{mygreen}1} ,1^{\color{mygreen}1})(1^{\color{myred}3})(3^{\color{mygreen}1})(1^{\color{myneworange}2}).$$ As we have already computed, its corresponding colored zigzag shape is $${\rm Z}_{(\alpha,\epsilon)} \ = \ \left( \left( \, \ydiagram{2} \, , \, \ydiagram{1+1,2} \, , \, \ydiagram{1} \, , \, \ydiagram{3} \, , \, \ydiagram{1} \, \right), ({\color{mymagenta}0}, {\color{mygreen}1}, {\color{myred}3}, {\color{mygreen}1}, {\color{myneworange}2}) \right)$$ and thus it corresponds to the following 4-partite skew shape $${\bm \lambda}_{{\rm Z}_{(\alpha,\epsilon)}} \ = \ \left( \, \ydiagram{2} \, , \, \ydiagram{2+3,1+1,2} \, , \, \ydiagram{1} \, , \, \ydiagram{1} \, \right).$$ Now, we pick an element of ${\rm D}_{(\alpha,\epsilon)}$ $$(\pi,{\mathrm z}) \ = \ 2^{\color{mymagenta}0} 3^{\color{mymagenta}0} 7^{\color{mygreen}1} {10}^{\color{mygreen}1} 5^{\color{mygreen}1} 6^{\color{myred}3} 1^{\color{mygreen}1} 8^{\color{mygreen}1} 9^{\color{mygreen}1} 4^{\color{myneworange}2} \ = \ (23)^{\color{mymagenta}0} (7\,{10}\,5)^{\color{mygreen}1} (6)^{\color{myred}3} (189)^{\color{mygreen}1} (4)^{\color{myneworange}2}$$ and form the tableaux $$Q_{(1)} = \ytableausetup{mathmode} \begin{ytableau} 2 & 3 \end{ytableau} \, , \quad Q_{(2)} = \begin{ytableau} \none & 5 \\ 7 & 10 \end{ytableau} \, , \quad Q_{(3)} = \begin{ytableau} 6 \end{ytableau} \, , \quad Q_{(4)} = \begin{ytableau} 1 & 8 & 9 \end{ytableau} \, , \quad Q_{(5)} = \begin{ytableau} 4 \end{ytableau}$$ with corresponding colors $$\epsilon_{(1)} \ = \ {\color{mymagenta}0}, \quad \epsilon_{(2)} \ = \ {\color{mygreen}1}, \quad \epsilon_{(3)} \ = \ {\color{myred}3}, \quad \epsilon_{(4)} \ = \ {\color{mygreen}1}, \quad \epsilon_{(5)} \ = \ {\color{myneworange}2}.$$ Taking the direct sum of tableaux of the same color yields the following 4-partite tableau $${\bm Q}\ = \ \ytableausetup{mathmode} \left( \begin{ytableau} 2 & 3 \end{ytableau}\, , \ \begin{ytableau} \none & \none & 1 & 8 & 9 \\ \none & 5 \\ 7 & 10 \end{ytableau}\, , \ \begin{ytableau} 4 \end{ytableau}\, , \ \begin{ytableau} 6 \end{ytableau} \right)$$ with colored descent set $${\rm sDes}({\bm Q}) \ = \ \left\{ 1^{\color{mygreen}1}, 3^{\color{mymagenta}0}, 4^{\color{myneworange}2}, 5^{\color{mygreen}1}, 6^{\color{myred}3}, 9^{\color{mygreen}1}, 10^{\color{mygreen}1} \right\}$$ which coincides with the colored descent set of the conjugate-inverse of $(\pi,{\mathrm z})$ $$\overline{(\pi,{\mathrm z})}^{-1} \ = \ 7^{\color{mygreen}1} 1^{\color{mymagenta}0} 2^{\color{mymagenta}0} {10}^{\color{myneworange}2} 5^{\color{mygreen}1} 6^{\color{myred}3} 3^{\color{mygreen}1} 8^{\color{mygreen}1} 9^{\color{mygreen}1} 4^{\color{mygreen}1}.$$ # Character formulas for colored descent representations {#sec:char} This section studies colored descent representations in the context of colored zigzag shapes and proves the main results of this paper. In particular, proves that the colored quasisymmetric generating function of conjugate-inverse colored descent classes is Schur-positive and equals the Frobenius image of colored descent representations. provides an alternating formula for the latter in terms of complete homogeneous symmetric functions in the colored context. Bagno and Biagioli [@BB07 Section 8] studied colored descent representations using the coinvariant algebra as a representation space, extending the techniques of Adin, Brenti and Roichman [@ABR05]. We are going to define colored descent representations by means of colored zigzag shapes and prove that the two descriptions coincide by providing the decomposition into irreducible $\mathfrak{S}_{n,r}$-representations. **Definition 8**. Let $(\alpha,\epsilon)$ be an $r$-colored composition of $n$ with rainbow decomposition $(\alpha,\epsilon) = (\alpha_{(1)},\epsilon_{(1)})(\alpha_{(2)},\epsilon_{(2)})\cdots$ $(\alpha_{(m)},\epsilon_{(m)})$. The element $$r_{(\alpha,\epsilon)} := r_{(\alpha,\epsilon)}({\bm x}^{(0)},{\bm x}^{(1)}, \dots, {\bm x}^{(r-1)}) := r_{\alpha_{(1)}}({\bm x}^{(\epsilon_{(1)})})r_{\alpha_{(2)}}({\bm x}^{(\epsilon_{(2)})})\cdots r_{\alpha_{(m)}}({\bm x}^{(\epsilon_{(m)})})$$ of ${\rm Sym}_n^{(r)}$ is called the corresponding to $(\alpha,\epsilon)$ and the (virtual) $\mathfrak{S}_{n,r}$-representation $\varrho_{(\alpha,\epsilon)}$ such that $${\rm ch}^{(r)}(\varrho_{(\alpha,\epsilon)}) = r_{(\alpha,\epsilon)}$$ is called the corresponding to $(\alpha,\epsilon)$. For example, for $n=10$ and $r=4$ $$r_{(2^{\color{mymagenta}0}, 2^{\color{mygreen}1} ,1^{\color{mygreen}1} , 1^{\color{myred}3}, 3^{\color{mygreen}1} , 1^{\color{myneworange}2})} = r_{(2)}({\bm x}^{({\color{mymagenta}0})}) r_{(2,1)}({\bm x}^{({\color{mygreen}1})}) r_{(1)}({\bm x}^{({\color{myred}3})}) r_{(3)}({\bm x}^{({\color{mygreen}1})}) r_{(1)}({\bm x}^{({\color{myneworange}2})}).$$ The first part of following theorem shows that colored descent representations are actually non-virtual and coincide with the ones studied by Bagno and Biagioli [@BB07 Theorem 10.5], while the second part extends and complements Adin et al.'s [@AAER17 Proposition 5.5(i)] to general colored permutation groups. **Theorem 9**. *For every $(\alpha,\epsilon) \in {\rm Comp}(n,r)$, $$\label{eq:coloredRibbonSchur-to-Schur} r_{(\alpha,\epsilon)} \ = \ F^{(r)}(\overline{{\rm D}}_{(\alpha,\epsilon)}^{-1}) \ = \ \sum_{{\bm \lambda}\vdash n} c_{{\bm \lambda}}(\alpha,\epsilon) \, s_{{\bm \lambda}},$$ where $c_{{\bm \lambda}}(\alpha,\epsilon)$ is the number of ${\bm Q}\in {\rm SYT}({\bm \lambda})$ such that ${\rm co}({\bm Q}) = (\alpha,\epsilon)$. In particular, conjugate-inverse colored descent classes are Schur-positive for colored descent representations.* The proof of is essentially a colored version of that of . It is based on a colored analogue of the well-known , first considered by White [@Whi83] and further studied by Stanton and White [@SW85] (see also [@Sta82 Section 6] and [@AAER17 Section 5] for the case of two colors). It is a bijection from $\mathfrak{S}_{n,r}$ to the set of all pairs of standard Young $r$-partite tableaux of the same shape and size $n$. If $w \mapsto ({\bm P},{\bm Q})$ under this correspondence, then $$\begin{aligned} {\rm sDes}(w) &= {\rm sDes}({\bm Q}) \\ {\rm sDes}(\overline{w}^{-1}) &= {\rm sDes}({\bm P}).\end{aligned}$$ *Proof of .* The first equality of follows directly from . For the second equality, applying the colored analogue of the Robinson--Schensted correspondence yields $$F^{(r)}(\overline{{\rm D}}_{(\alpha,\epsilon)}^{-1}) = \sum_{{\bm \lambda}\vdash n} \, \sum_{\substack{{\bm P}, \hspace{1pt}{\bm Q}\, \in \, {\rm SYT}({\bm \lambda}) \\ {\rm co}({\bm P}) = (\alpha,\epsilon)}} F_{{\rm co}({\bm Q})}^{(r)}.$$ and the proof follows from . ◻ In our running example, we see that $$\begin{aligned} r_{(2^{\color{mymagenta}0}, 2^{\color{mygreen}1} ,1^{\color{mygreen}1} , 1^{\color{myred}3}, 3^{\color{mygreen}1} , 1^{\color{myneworange}2})} &= s_{2}({\bm x}^{({\color{mymagenta}0})}) s_{21}({\bm x}^{({\color{mygreen}1})}) s_{3}({\bm x}^{({\color{mygreen}1})}) s_{1}({\bm x}^{({\color{myneworange}2})}) s_{1}({\bm x}^{({\color{myred}3})}) \\ &= s_{2}({\bm x}^{({\color{mymagenta}0})}) \left((s_{321}({\bm x}^{({\color{mygreen}1})}) + s_{411}({\bm x}^{({\color{mygreen}1})}) + s_{42}({\bm x}^{({\color{mygreen}1})}) + s_{51}({\bm x}^{({\color{mygreen}1})})\right) s_{1}({\bm x}^{({\color{myneworange}2})}) s_{1}({\bm x}^{({\color{myred}3})}) \\ &= s_{(2,321,1,1)} + s_{(2,411,1,1)} + s_{(2,42,1,1)} + s_{(2,51,1,1)},\end{aligned}$$ where we omitted the parentheses and commas in (regular) partitions for ease of notation. There are many ways to make this computation, the most of which is to implement the [@StaEC2 Section 7.15]. Thus, the decomposition of the colored descent representation corresponding to $(2^{\color{mymagenta}0}, 2^{\color{mygreen}1} ,1^{\color{mygreen}1} , 1^{\color{myred}3}, 3^{\color{mygreen}1} , 1^{\color{myneworange}2})$ is the multiplicity free direct sum of the irreducible $\mathfrak{S}_{10,4}$-representations corresponding to the 4-partite partitions $(2,321,1,1), (2,411,1,1), (2,42,1,$ $1)$ and $(2,51,1,1)$ with corresponding 4-partite tableaux $$\begin{aligned} {2} \ytableausetup{mathmode} &\left( \begin{ytableau} 1 & 2 \end{ytableau}\, , \ \begin{ytableau} 3 & 4 & 9 \\ 5 & 8 \\ 7 \end{ytableau}\, , \ \begin{ytableau} 10 \end{ytableau}\, , \ \begin{ytableau} 6 \end{ytableau} \right), \quad %%%% &&\left( \begin{ytableau} 1 & 2 \end{ytableau}\, , \ \begin{ytableau} 3 & 4 & 8 & 9 \\ 5 \\ 7 \end{ytableau}\, , \ \begin{ytableau} 10 \end{ytableau}\, , \ \begin{ytableau} 6 \end{ytableau} \right), \\ %%%% &\left( \begin{ytableau} 1 & 2 \end{ytableau}\, , \ \begin{ytableau} 3 & 4 & 8 & 9 \\ 5 & 7 \end{ytableau}\, , \ \begin{ytableau} 10 \end{ytableau}\, , \ \begin{ytableau} 6 \end{ytableau} \right), \quad %%%% &&\left( \begin{ytableau} 1 & 2 \end{ytableau}\, , \ \begin{ytableau} 3 & 4 & 7 & 8 & 9 \\ 5 \end{ytableau}\, , \ \begin{ytableau} 10 \end{ytableau}\, , \ \begin{ytableau} 6 \end{ytableau} \right).\end{aligned}$$ We can express the colored ribbon Schur function as an alternating sum of elements of a basis of ${\rm Sym}_n^{(r)}$ which can be viewed as the colored analogue of the basis of complete homogeneous symmetric functions. For an $r$-partite partition ${\bm \lambda}= (\lambda^{(0)},\lambda^{(1)}, \dots, \lambda^{(r-1)})$, let $$h_{\bm \lambda}:= h_{\bm \lambda}({\bm x}^{(0)},{\bm x}^{(1)}, \dots, {\bm x}^{(r-1)}) := h_{\lambda^{(0)}}({\bm x}^{(0)})h_{\lambda^{(1)}}({\bm x}^{(1)})\cdots h_{\lambda^{(r-1)}}({\bm x}^{(r-1)}).$$ The set $\{h_{\bm \lambda}: {\bm \lambda}\vdash n\}$ forms a basis for ${\rm Sym}_n^{(r)}$. Similarly to the classical case, given $(\alpha,\epsilon) \in {\rm Comp}(n,r)$ we can form an $r$-partite partition ${\bm \lambda}_{(\alpha,\epsilon)}$ of $n$ by first splitting its entries into colored components and then rearranging the entries of each component in weakly decreasing order. We write $h_{(\alpha,\epsilon)}:=h_{{\bm \lambda}_{(\alpha,\epsilon)}}$. **Theorem 10**. *For every $(\alpha,\epsilon) \in {\rm Comp}(n,r)$, $$\label{eq:mainB} r_{(\alpha,\epsilon)} \ = \, \sum_{\substack{(\beta,\delta) \in {\rm Comp}(n,r) \\ (\beta,\delta) \preceq (\alpha,\epsilon)}} \, (-1)^{\ell(\alpha) - \ell(\beta)} \, h_{(\beta,\delta)}.$$* *Proof.* Let $(\alpha,\epsilon)$ be a colored composition of $n$ with rainbow decomposition $$(\alpha,\epsilon) = (\alpha_{(1)},\epsilon_{(1)})(\alpha_{(2)},\epsilon_{(2)})\cdots(\alpha_{(m)},\epsilon_{(m)}).$$ Expanding each term $r_{\alpha_{(i)}}({\bm x}^{(\epsilon_{(i)})})$ in the definition of the colored ribbon Schur function $r_{(\alpha,\epsilon)}$ according to yields $$\begin{aligned} r_{(\alpha,\epsilon)} &= r_{\alpha_{(1)}}({\bm x}^{(\epsilon_{(1)})})r_{\alpha_{(2)}}({\bm x}^{(\epsilon_{(2)})})\cdots r_{\alpha_{(m)}}({\bm x}^{(\epsilon_{(m)})}) \\ &= \prod_{1 \le i \le m} \sum_{\beta_{(i)} \preceq \, \alpha_{(i)}} \, (-1)^{\ell(\alpha_{(i)}) - \ell(\beta_{(i)})} h_{\beta_{(i)}}({\bm x}^{(\epsilon_{(i)})}) \\ &= \sum_{\substack{1 \le i \le m \\ \beta_{(i)} \preceq \, \alpha_{(i)}}} \, (-1)^{\ell(\alpha) - (\ell(\beta_{(1)}) + \cdots + \ell(\beta_{(m)}))} \, h_{\beta_{(1)}}({\bm x}^{(\epsilon_{(1)})})\cdots h_{\beta_{(m)}}({\bm x}^{(\epsilon_{(m)})}),\end{aligned}$$ since $\ell(\alpha) = \ell(\alpha_{(1)}) + \cdots + \ell(\alpha_{(m)})$. The proof follows by considering the colored composition with rainbow decomposition $(\beta,\delta) = (\beta_{(1)},\epsilon_{(1)})\cdots(\beta_{(m)},\epsilon_{(m)})$ and noticing that the conditions $\beta_{(i)} \preceq \alpha_{(i)}$ for all $1 \le i \le m$ are precisely equivalent to $(\beta,\delta) \preceq (\alpha,\epsilon)$ and that $$\begin{aligned} \ell(\beta) &= \ell(\beta_{(1)}) + \cdots + \ell(\beta_{(m)}) \\ h_{(\beta,\delta)} &= h_{\beta_{(1)}}({\bm x}^{(\epsilon_{(1)})})\cdots h_{\beta_{(m)}}({\bm x}^{(\epsilon_{(m)})}).\end{aligned}$$ ◻ In our running example, we have $$r_{(2^{\color{mymagenta}0}, 2^{\color{mygreen}1} ,1^{\color{mygreen}1} , 1^{\color{myred}3}, 3^{\color{mygreen}1} , 1^{\color{myneworange}2})} = h_{(2^{\color{mymagenta}0}, 2^{\color{mygreen}1} ,1^{\color{mygreen}1} , 1^{\color{myred}3}, 3^{\color{mygreen}1} , 1^{\color{myneworange}2})} - h_{(2^{\color{mymagenta}0}, 3^{\color{mygreen}1} , 1^{\color{myred}3}, 3^{\color{mygreen}1} , 1^{\color{myneworange}2})}$$ which is in agreement with the expansion in the Schur basis that we calculated above, since $$h_{321} - h_{33} = s_{321} + s_{411} + s_{42} + s_{51}.$$ Finally, let us describe the representation-theoretic version of . For this we need to introduce some notation. We fix a primitive $r$-th root of unity $\omega$. For all $0 \le j \le r-1$, let $\mathbb{1}_{n,j}$ be the irreducible $\mathfrak{S}_{n,r}$-representation corresponding to the $r$-partite partition having all parts empty, except for the part of color $j$ which is equal to $(n)$. Then, $$\mathbb{1}_{n,j}(\pi,\epsilon) = \omega^{j(\epsilon_1 + \epsilon_2 + \cdots + \epsilon_n)}$$ for all $(\pi,\epsilon) \in \mathfrak{S}_{n,r}$ (see, for example, [@BC12 Section 4]). For $(\alpha,\epsilon) \in {\rm Comp}(n,r)$ of length $k$, we define the following $\mathfrak{S}_{\alpha,r}$-representation $$\mathbb{1}_{(\alpha,\epsilon)} := \mathbb{1}_{\alpha_1,\epsilon_1} \otimes \mathbb{1}_{\alpha_2,\epsilon_2} \otimes \cdots \otimes \mathbb{1}_{\alpha_k,\epsilon_k},$$ where $\mathfrak{S}_{\alpha,r} := \mathfrak{S}_{\alpha_1,r}\times\mathfrak{S}_{\alpha_2,r}\times\cdots\times\mathfrak{S}_{\alpha_k,r}$ is embedded in $\mathfrak{S}_{n,r}$ in the obvious way. Since the colored characteristic map is a ring homomorphism, we have $$\label{eq:characteristicmap-shaperep} {\rm ch}^{(r)}\left(\mathbb{1}_{(\alpha,\epsilon)} \uparrow_{\mathfrak{S}_{\alpha,r}}^{\mathfrak{S}_{n,r}}\right) \ = \ h_{\alpha_1}({\bm x}^{(\epsilon_1)})h_{\alpha_2}({\bm x}^{(\epsilon_2)})\cdots h_{\alpha_k}({\bm x}^{(\epsilon_k)}),$$ where the second equality follows from basic properties of the colored characteristic map [@Poi98 Corollary 3] $${\rm ch}^{(r)}(\mathbb{1}_{\alpha_i,\epsilon_i}) = s_{(\alpha_i)}({\bm x}^{(\epsilon_i)}) = h_{\alpha_i}({\bm x}^{(\epsilon_i)}).$$ Since the characteristic map is actually a ring isomorphism [@Poi98 Theorem 2], applying it to yields the following. **Corollary 11**. *For all $(\alpha,\epsilon) \in {\rm Comp}(n,r)$, the character $\chi_{(\alpha,\epsilon)}$ of the colored descent representation corresponding to $(\alpha,\epsilon)$ satisfies $$\label{eq:mainB-rep} \chi_{(\alpha,\epsilon)} = \sum_{\substack{(\beta,\delta) \in {\rm Comp}(n,r) \\ (\beta,\delta) \preceq (\alpha,\epsilon)}} \, (-1)^{\ell(\alpha) - \ell(\beta)} \, \mathbb{1}_{(\beta,\delta)}\uparrow_{\mathfrak{S}_{\beta,r}}^{\mathfrak{S}_{n,r}}.$$* Schur functions can also be viewed as generating functions of certain $P$-partitions, where $P$ is a Schur labeled poset arising from the (possibly skew) Young diagram (see, for example, [@Ges84 Section 2] and [@StaEC2 Section 7.19]). 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arxiv_math

--- abstract: | In this article we exhibit new explicit families of congruences for the overpartition function, making effective the existence results given previously by Treneer. We give infinite families of congruences modulo $m$ for $m = 5, 7, 11$, and finite families for $m = 13, 17, 19$. address: - Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837 - Departamento de Matemática - FCEyN - UBA and IMAS - CONICET, Pabellón I, Ciudad Universitaria, Ciudad Autónoma de Buenos Aires (1428), Argentina - Escuela de Matemática, Universidad de Costa Rica, San José 11501, Costa Rica author: - Nathan C. Ryan, Nicolás Sirolli and Jean Carlos Villegas-Morales bibliography: - biblio.bib title: Explicit families of congruences for the overpartition function --- # Introduction Let $p(n)$ be the number of partitions of a positive integer $n$; that is, the number of ways $n$ can be written as a sum of non-increasing positive integers. Ramanujan [@ramanujan] proved congruences of the form: $$\begin{aligned} p(5n+4) & \equiv 0 \pmod{5},\\ p(7n+5) & \equiv 0 \pmod{7},\\ p(11n+6) &\equiv 0 \pmod{11},\end{aligned}$$ for every $n$. For decades it was difficult to find more congruences like these; nevertheless, Ono proved in [@ono] that for each prime $m \geq 5$ there exists an infinite family of congruences for the partition function modulo $m$: more precisely, he proved that a positive proportion of the primes $\ell$ are such that $$\begin{aligned} p\left(\frac{m\ell^3 n + 1}{24}\right) \equiv 0 \pmod m.\end{aligned}$$ for every $n$ coprime to $\ell$. The number of overpartitions $\overline{p}(n)$ of a positive integer $n$ is defined to be the number of ways in which $n$ can be written as a non-increasing sum of positive integers in which the first occurrence of a number may be overlined (see [@overpartitions]). The numbers of both partitions and overpartitions can be described in terms of eta-quotients; in particular, they are known to be coefficients of weakly holomorphic modular forms of half-integral weight, with integral coefficients. Treneer showed in [@treneer2] that Ono's existence results were valid, more generally, for the coefficients of such modular forms. In the particular case of the overpartition function, her results imply that for every prime $m \geq 5$, for sufficiently large $r$, $$\begin{aligned} \overline p\left(m^r\ell^3 n\right) \equiv 0 \pmod m.\end{aligned}$$ for every $n$ coprime to $\ell$. The main goal of this article is to show explicit instances of these (families of) congruences, as well as for certain variations similar to those considered by Ono for the partition function. Weaver devised a strategy in [@weaver] for making Ono's results explicit: she exhibited 76,065 new families of congruences for the partition function by finding congruences between its generating function and appropriate holomorphic modular forms, and then verifying a finite number of congruences for the partition function. Her computations were extended by Johansson [@johansson] who used efficient algorithms for computing the partition function to find more than $2.2 \cdot 10^{10}$ such families of congruences. Using Weaver's techniques along with the theory of Eisenstein series of half-integral weight from [@wang-pei], we were able to find *infinitely many* families of congruences for the overpartition function. Our first main results are the following two theorems. **Theorem 1**. *Let $m \in \{5,7,11\}$, and let $\ell$ be an odd prime such that $\ell^{m-4} \equiv -1 \pmod m$. Then $$\overline{p}\left(m \ell^3 n\right) \equiv 0 \pmod m$$ for every $n$ prime to $\ell$.* We remark that for $m=5$ the above result was given in [@treneer1 Prop. 1.4]. **Theorem 1**. *Let $m \in \{5,7,11\}$, and let $\ell$ be an odd prime such that $\ell^{m-4} \equiv -1 + \epsilon_{m,\ell}\, \ell^{\tfrac{m-5}2} \pmod m$, with $\epsilon_{m,\ell}\in \{\pm1\}$. Then $$\overline{p}\left(m \ell^2 n\right) \equiv 0 \pmod m$$ for every $n$ prime to $\ell$ such that $$\genfrac(){}{}{(-1)^{\tfrac{m-3}2}n}{\ell} = \epsilon_{m,\ell}.$$* For primes $m \geq 13$ the appearance of cusp forms in level $16$ and weight $\tfrac{m-2}2$ makes it more difficult to find infinitely many families of congruences. Using the results from [@barquero-etal] for efficiently computing the overpartition function, we obtained the following families of congruences. **Theorem 1**. *Let $m,\ell$ be primes as in Table [1](#tab:ml){reference-type="ref" reference="tab:ml"}. Then $$\overline{p}\left(m \ell^3 n\right) \equiv 0 \pmod m$$ for every $n$ prime to $\ell$.* $m$ $\ell$ ------ -------------------------------------------------------- $13$ $1811, 1871, 1949, 2207, 3301, 4001, 4079, 4289, 4931$ $17$ $2039, 2719, 3331, 4079$ $19$ $151, 1091, 2659, 3989$ : Congruences for primes $m \geq 13$. See Theorem [Theorem 1](#thm:fin0){reference-type="ref" reference="thm:fin0"}. **Theorem 1**. *Let $m,\ell$ be primes, and let $\epsilon_{m,\ell}\in \{\pm1\}$ be as in Table [2](#tab:ml_eps){reference-type="ref" reference="tab:ml_eps"}. Then $$\overline{p}\left(m \ell^2 n\right) \equiv 0 \pmod m$$ for every $n$ prime to $\ell$ such that $$\genfrac(){}{}{(-1)^{\tfrac{m-3}2}n}{\ell} = \epsilon_{m,\ell}.$$* $m$ $(\ell,\epsilon_{m,\ell})$ ------ ------------------------------------------------------------ $13$ $(431, 1), (2459, 1), (4513, 1), (4799, 1)$ $17$ $(167, 1), (541, 1), (911, -1), (1013, -1), (1153, 1)$, $(1867, 1), (1931, -1),(2543,-1), (2683, 1), (2887, 1)$, $(3019, -1), (3023, 1), (3329, 1), (4243, -1), (4651, -1)$ $19$ $(2207,-1)$ : Congruences for primes $m \geq 13$. See Theorem [Theorem 1](#thm:fineps){reference-type="ref" reference="thm:fineps"}. We point out that using different techniques, in [@rsst; @barquero-etal] the authors found (finite) families of congruences for the overpartition function modulo $m$ for $m = 3,5,7$; see also [@Chen] for $m = 5$, and [@Xia] for powers of $m=3$. Moreover, and independently from our work, in [@zheng] the author gives a proof of Theorem [Theorem 1](#thm:inf0){reference-type="ref" reference="thm:inf0"}, which uses eta-quotients instead of Eisenstein series. As far as we know, the results in this article give the first known congruences for $m > 11$. The rest of the paper is organized as follows. In the next section we give the necessary notation and preliminaries regarding half-integral weight modular forms and eta-quotients. In Section [3](#sect:eisen){reference-type="ref" reference="sect:eisen"} we state the results we need on Eisenstein series of half-integral weight and level 16. We conclude the article with the proofs of our main results in Section [4](#sect:proofs){reference-type="ref" reference="sect:proofs"}. # Preliminaries {#sect:prelims} ## Half-integral weight modular forms {#half-integral-weight-modular-forms .unnumbered} We refer the reader to [@wang-pei Sect. 5] for details on this subsection. Given a non zero integer $m$ we denote by $\chi_m$ the primitive Dirichlet character such that $\chi_m(a) = \genfrac(){}{}{m}{a}$ for every $a$ such that $(a,4m) = 1$. Given an odd integer $k\geq 3$, we denote $\lambda = \tfrac{k-1}2$. Furthermore, given a positive integer $m$ we denote $\omega_n = \chi_m$, with $m = (-1)^\lambda n$. Given $k$ as above, a positive integer $N$ divisible by $4$ and a character $\chi$ modulo $N$, we denote by $\mathcal{M}_{k/2}({N,\chi})$ the space of holomorphic modular forms of weight $k/2$, level $N$ and character $\chi$. We denote by $\mathcal{S}_{k/2}({N,\chi})$ and $\mathcal{E}_{k/2}({N,\chi})$ the subspace of cusp forms and the Eisenstein subspace, respectively. When $\chi$ is the trivial character, we omit it from the notation. We consider the following operators acting on half-integral weight modular forms. Let $g = \sum_{n \geq 0} a(n) q^n \in \mathcal{M}_{k/2}({N})$. - The Fricke involution $W(N)$, given by $$\begin{aligned} W(N) & : \mathcal{M}_{k/2}({N,\chi}) \to \mathcal{M}_{k/2}({N,\chi \chi_N}), \\ & (g\vert W(N))(z) = (Nz)^{-k/2} g(-1/Nz).\end{aligned}$$ We include here an extra factor of $N^{-k/2}$ not present in [@wang-pei]. - For a prime $\ell$, the Hecke operator $T(\ell^2)$, given by $$\begin{aligned} \nonumber T(\ell^2) & : \mathcal{M}_{k/2}({N,\chi}) \to \mathcal{M}_{k/2}({N,\chi}), \\ \label{eqn:hecke} g \vert T(\ell^2)& = \sum_{n \geq 0} \left(a(\ell^2 n) + \chi(\ell) \ell^{\lambda-1} \omega_n(\ell) a(n) %+ \chi(\ell) \kro{(-1)^\lambda n}{\ell} \ell^{\lambda-1} a(n) + \chi(\ell^2) \, \ell^{2\lambda-1}a(n/\ell^2)\right) q^n. %& g \Tls = \sum_{n \geq 0} b(n) q^n.\end{aligned}$$ - For an integer $m \geq 1$, the $V(m)$ operator, given by $$\begin{aligned} V(m) & : \mathcal{M}_{k/2}({N,\chi}) \to \mathcal{M}_{k/2}({mN,\chi \chi_m}), \\ & g \vert V(m) = \sum_{n \geq 0} a(n) q^{mn}. %& (g\vert V(m))(z) = g(mz).\end{aligned}$$ - For an integer $m \geq 1$, the $U(m)$ operator, given by $$\begin{aligned} U(m) & : \mathcal{M}_{k/2}({N,\chi}) \to \mathcal{M}_{k/2}({M,\chi \chi_m}),\\ & g \vert U(m)= \sum_{n \geq 0} a(mn) q^n,\end{aligned}$$ where $M$ is the smallest multiple of $N$ which is divisible by every prime dividing $m$, and such that the conductor of $\chi_m$ divides $M$. The latter two act as well on rings of formal power series. The following is the Sturm bound for general weights. Its proof follows from the integral weight case; see [@rsst Prop. 4.1]. **Proposition 1**. *Let $k \geq 3$ be an integer, and let $m$ be a prime. Suppose that $g=\sum_{n \geq 0} a(n) q^n \in \mathcal{M}_{k/2}({N}) \cap \mathbb{Z}\llbracket q\rrbracket$. Let $$%\label{eqn:sturm} n_0 = \left\lfloor\frac{k}{24} \cdot [\mathop{\mathrm{SL}}_2(\mathbb{Z}):\Gamma_0(N)]\right\rfloor.$$ If $a(n) \equiv 0 \pmod{m}$ for $1 \leq n \leq n_0$, then $g \equiv 0 \pmod{m\mathbb{Z}\llbracket q\rrbracket}$.* The result is also valid for proving equalities, namely when $m=0$. ## Eta-quotients {#eta-quotients .unnumbered} Let $\eta(z)$ denote the Dedekind eta function, which is given by $$\eta(z) = q^{\tfrac1{24}}\prod_{n=1}^\infty \left(1-q^n\right), \qquad q=e^{2\pi iz}.$$ Given a finite set $X=\{(\delta,r_{\delta})\} \subseteq \mathbb{Z}_{>0}\times\mathbb{Z}$, denote $s_X=\sum\delta r_{\delta}$. Assuming that $s_X \equiv 0 \pmod{24}$, the eta-quotient defined by $X$ is $$\label{eqn:etaq} \eta^X(z)=\prod_{X}\eta(\delta z)^{r_{\delta}} = q^{\tfrac{s_X}{24}} \prod_X \prod_{n=1}^\infty \left(1-q^{\delta n}\right)^{r_\delta}.$$ Note that $1/\eta^X$ is also an eta-quotient. Let $k = \sum_X r_\delta$, and let $N$ be the smallest multiple of every $\delta$, and of $4$ if $k$ is odd, such that $$N \sum_X \frac{r_\delta}{\delta} \equiv 0 \pmod{24},$$ Finally, letting $m'=\prod_X \delta^{r_\delta}$ we let $m = m'$ for even $k$, and $m = 2m'$ for odd $k$. Then (see [@gordon-hughes Thm. 3] and [@treneer3 Coro. 2.7]) we have the following result. **Proposition 1**. *With the notation as above, $\eta^X$ is a weakly holomorphic modular form of weight $k$, level $N$ and character $\chi_m$.* Thus, $\eta^X$ is holomorphic and nonzero in the upper half-plane, but it can have poles and zeros at the cusps. Furthermore, following [@ligozat], if $\gcd(a,c) = 1$, then the order of vanishing of $\eta^X$ at a cusp $s = a/c \in \mathbb{Q}\cup\{\infty\}$ is given by $$\label{eqn:ligozat} \mathop{\mathrm{ord}}_s\left(\eta^X\right) = \frac{N}{24 \gcd(c^2,N)} \, \sum_X \gcd(c,\delta)^2 \, \frac{r_\delta}{\delta}. %\frac{h_s}{24} \, \sum_X \gcd(c,\delta)^2 \, \frac{r_\delta}{\delta}.$$ In particular, $\mathop{\mathrm{ord}}_\infty(\eta^X) = 0$ whenever $s_X = 0$. Moreover, in this case by [\[eqn:etaq\]](#eqn:etaq){reference-type="eqref" reference="eqn:etaq"} we have that $\eta^X \in 1 + q\mathbb{Z}\llbracket q\rrbracket$. In particular, $\eta^X \in \mathbb{Z}\llbracket q\rrbracket^\times$. # Eisenstein spaces of half-integral weight and level 16 {#sect:eisen} Wang and Pei ([@wang-pei]) considered the Eisenstein spaces of half-integral weights, giving bases of eigenforms for these spaces in the case of level $4D$, with $D$ odd and squarefree. Relying on their definitions and results, we consider the case of level $16$. The main result of this section is the following. **Proposition 1**. *Let $\ell \geq 3$ be prime. Then $T(\ell^2)$ acts by multiplication by $\sigma_{k-1}(\ell)$ on $\mathcal{E}_{k/2}({16})$.* We also give in Proposition [Proposition 1](#prop:eis_coeffs){reference-type="ref" reference="prop:eis_coeffs"} exact formulas for the coefficients of the Eisenstein series, which are needed to prove the congruence in [\[eqn:g11\]](#eqn:g11){reference-type="eqref" reference="eqn:g11"}. Let $\Gamma_\infty = \left\{{\pm\left(\begin{smallmatrix} 1 & n \\ 0 & 1 \end{smallmatrix} \right)} \, : \, {n \in \mathbb{Z}}\right\} \leq \mathop{\mathrm{SL}}_2(\mathbb{Z})$. Let $k \geq 3$ be an odd integer. Denote $\lambda = \tfrac{k-1}2$. Let $N \in \{4,8\}$. For $\gamma \in \Gamma_0(N)$, let $j(\gamma,z)$ be the automorphy factor of weight $1/2$. For $k>3$ we denote $$\begin{aligned} E_{k,N}(z) & = \sum_{\gamma \in \Gamma_\infty \backslash \Gamma_0(N)} \frac{1}{j(\gamma,z)^k}, \\ %\frac{\chi(d_\gamma)}{j(\gamma,z)^k}, \\ E'_{k,N} & = \tfrac{2^k N^\lambda}{1-(-1)^{\lambda}i} \cdot E_{k,N} \vert W(N).\end{aligned}$$ For $k = 3$ we consider the difference $E_{3,N} - 2 \sqrt N \, E'_{3,N}$ defined by the formulas above, which, for simplicity, we will denote by $E_{3,N}$. We start by giving the Fourier expansions of these Eisenstein series, following [@wang-pei]. For this purpose we introduce the following notation, which will not used in other parts of the paper. For an even integer $v$ denote $$c_k^\pm(v) = \frac{1-2^{(2-k)v/2}}{1-2^{2-k}} \pm 2^{(2-k)v/2}.$$ Given a positive integer $n$, let $v_n = \mathop{\mathrm{val}}_2(n)$ and $n' = (-1)^\lambda n / 2^{v_n}$, and denote $$\begin{aligned} C_k(n) & = \begin{cases} c_k^-(v_n-1), & 2 \nmid v_n,\\ c_k^-(v_n) , & 2 \mid v_n,\, n' \equiv 3 \pmod 4, \\ c_k^+(v_n) + 2^{((2-k)v_n + (3-k))/2} \, \genfrac(){}{}{n'}{2}, & 2 \mid v_n,\, n' \equiv 1 \pmod 4, \end{cases} \\ \gamma_{k,4}(n) & = \begin{cases} C_k(n), & k > 3, \\ C_3(n) - 2 , & k = 3, %3(A_3(n) - 2) , & k = 3, N = 8, n \equiv 1,2 \pmod 4. \end{cases} \\ \gamma_{k,8}(n) & = \begin{cases} 0 , & (-1)^\lambda n \equiv 2,3 \pmod 4, \\ C_k(n) - 1, & (-1)^\lambda n \equiv 0,1 \pmod 4,\, k > 3, \\ C_3(n) - 2, & (-1)^\lambda n \equiv 0,1 \pmod 4,\, k = 3. %3(A_3(n) - 2) , & k = 3, N = 8, n \equiv 1,2 \pmod 4. \end{cases}\end{aligned}$$ Let $\omega$ denote a Dirichlet character of conductor $f$. Let $B_\lambda$ denote the $\lambda$-th Bernoulli polynomial. Then we consider the generalized Bernoulli number $$B_{\lambda,\omega} = f^{\lambda - 1} \sum_{a = 1}^{f} \omega(a) B_\lambda(\tfrac a{f})$$ Furthermore, letting $\mu$ denote the Möbius function, for each positive integer $n$ we denote $$\begin{aligned} \beta_{\lambda,\omega}(n) & = \sum_{a,b} \mu(a) \omega(a) a^{-\lambda} b^{-2\lambda+1}, \end{aligned}$$ where $a,b$ run over all positive odd integers such $(ab)^2 \mid n$. Recall that for each positive integer $m$ we consider the primitive Dirichlet character $\omega_m$ such that for $(a,4m) = 1$ we have $$\begin{aligned} \omega_m(a) = \genfrac(){}{}{(-1)^\lambda m}{a}.\end{aligned}$$ We denote by $f_m$ its conductor, and we remark that ${f_m/m}$ is the square of a rational number. We let $$\label{eqn:alpha} \alpha_{\lambda,m} = \frac { \sqrt{f_m/m} \, B_{\lambda,\omega_m} %\beta_{\lambda,\omega_n}(n) } { f_m^\lambda \, B_{2\lambda}} \frac{ 1-\omega_m(2) 2^{-\lambda} } { 1-2^{-2\lambda} }.$$ Finally, for each positive integer $n$ we denote $$\begin{aligned} \label{eqn:an} a_{k,N}(n) & = \alpha_{\lambda,n} \, \beta_{\lambda,\omega_n}(n) \, \gamma_{k,N}(n) \, n^\lambda, \\ %\nonumber a'_{k,N}(n) & = \alpha_{\lambda,nN} \, \beta_{\lambda,\omega_{nN}}(n) \, n^\lambda.\end{aligned}$$ **Proposition 1**. *For $N \in \{4,8\}$ and odd $k \geq 3$ we have $$\begin{aligned} E_{k,N} & = 1 + \sum_{n \geq 1} a_{k,N}(n) \, q^n, \quad k \geq 3,\\ E'_{k,N} & = \sum_{n \geq 1} a'_{k,N}(n) \, q^n, \quad k > 3. \end{aligned}$$* The proof is essentially given in [@wang-pei]; their formulas for the coefficients of these Eisenstein series involve values $L(\lambda, \omega_m)$ of $L$-series of quadratic characters at positive integers. The latter are well known; we use them in the result below. Given a positive integer $\lambda$ we denote $$\label{eqn:elambda} e_\lambda = \frac{ 2^{2+\lambda -k} \genfrac(){}{}{2\lambda+3}{2} \lambda! } { (1-2^{-2\lambda}) B_{2\lambda} \pi^\lambda }.$$ **Lemma 1**. *For every positive integer $m$ we have that $$\alpha_{\lambda,m} = e_\lambda \, \left(1-\omega_m(2) 2^{-\lambda}\right) \, L(\lambda,\omega_m) \, m^{-1/2}.$$ Moreover, $\mathop{\mathrm{sgn}}(\alpha_{\lambda,m}) = \genfrac(){}{}{2\lambda+1}{2}$.* *Proof.* From [@montgomery-vaughan p. 337] and [@montgomery-vaughan Thm. 9.17] we have that for every quadratic character $\omega$ with conductor $f$ and such that $\omega(1) = (-1)^\lambda$ we have that $$L(\lambda,\omega) = \frac{ \genfrac(){}{}{2\lambda+3}{2} 2^{\lambda -1} \pi^\lambda \sqrt f } { {\lambda!}f^\lambda } \, B_{\lambda,\omega},$$ from which the first claim follows. The second claim follows from the fact that for every such $\omega$ we have that $L(\lambda,\omega) > 0$; hence $$\mathop{\mathrm{sgn}}(\alpha_{\lambda,m}) = \mathop{\mathrm{sgn}}(e_\lambda) = \genfrac(){}{}{2\lambda+3}{2} \mathop{\mathrm{sgn}}(B_{2\lambda}) = \genfrac(){}{}{2\lambda+1}{2}. \qedhere$$ ◻ **Corollary 1**. *Let $n$ be a squarefree positive integer. Then $a'_{k,N}(n) \neq 0$. Furthermore, $a_{k,N}(n) = 0$ if and only if $\gamma_{k,N}(n) = 0$.* *Proof of Proposition [Proposition 1](#prop:eis_coeffs){reference-type="ref" reference="prop:eis_coeffs"}.* Using the well know formulas for $\zeta(2\lambda)$ and $\Gamma(\lambda + 1/2)$, and using that $$\frac{ (-i)^{\lambda+1/2} \left(1+(-1)^\lambda i\right) } { \sqrt 2 } = \genfrac(){}{}{2\lambda+1}{2},$$ we obtain that $$e_\lambda = \frac{ (-2\pi i)^{\lambda+1/2} \left(1+(-1)^\lambda i\right) } { 2^{2\lambda+1} \Gamma(\lambda+1/2) \zeta(2\lambda) (1-2^{-2\lambda}) } .$$ Then the result follows straightforwardly from Lemma [Lemma 1](#lem:alphael){reference-type="ref" reference="lem:alphael"} and the formulas [@wang-pei (2.30), (2.33), (2.35), (2.36) and (2.38)]. ◻ Proposition [Proposition 1](#prop:eis_coeffs){reference-type="ref" reference="prop:eis_coeffs"} shows that $E_{k,N}$ and $E'_{k,N}$, which a priori have their coefficients in a cyclotomic field ([@shimura Thm. 2.3]), actually have rational coefficients. The following results shows that, as in the integral weight case (see [\[eqn:eisenstein\]](#eqn:eisenstein){reference-type="eqref" reference="eqn:eisenstein"}), their denominators are controlled by $k$ and can be described in terms of Bernoulli numbers. Its proof will require the following result from Carlitz ([@carlitz Thm. 3]). **Lemma 1**. *Let $d$ be a fundamental discriminant, and let $\lambda$ be a positive integer.* 1. *If $d = -4$ and $\lambda$ is odd, then $2 B_{\lambda,\chi_d}/\lambda \in \mathbb{Z}$.* 2. *If $d = \pm p$, with $p>2$ an odd prime such that $2\lambda/(p-1)$ is an odd integer, then $p B_{\lambda,\chi_d} \in \mathbb{Z}$.* 3. *Otherwise, $B_{\lambda,\chi_d}/\lambda \in \mathbb{Z}$.* We denote by $S_\lambda$ the denominator of $(2^\lambda-1) B_\lambda / \lambda$; here we let $0 = 0/1$. **Proposition 1**. *For $N \in \{4,8\}$ and odd $k \geq 3$ we have $$\begin{aligned} E_{k,N} & \in 1 + \tfrac{ \lambda }{ 2^{\lambda-1} \left(2^{2\lambda}-1\right) B_{2\lambda} S_\lambda } \, \mathbb{Z}\llbracket q\rrbracket, \\ E'_{k,N} & \in \tfrac{ \lambda 2^\lambda }{ \left(2^{2\lambda}-1\right) B_{2\lambda} S_\lambda N^\lambda } \,\mathbb{Z}\llbracket q\rrbracket. \end{aligned}$$* *Proof.* We prove the claim for $E_{k,N}$; the proof for $E'_{k,N}$ follows by similar arguments. Let $n$ be a positive integer. Recalling that $f_n$ denotes the conductor of $\omega_n$, write $n = f_n q_n^2 = f'_n (q'_n)^2$ with $f'_n$ squarefree, so that $\sqrt{f_n / n} = 1 / q_n$ and $2q_n / q'_n \in \{1,2\}$. Moreover, let $w_n = \mathop{\mathrm{val}}_2(q'_n)$ and write $q'_n = 2^{w_n} q''_n$. Then letting $$\begin{aligned} r_n & = {q''_n}^{2\lambda-1}\, \beta_{\lambda,\omega_n}(n) , \\ s_n & = S_\lambda \left(2^\lambda-\omega_n(2)\right) B_{\lambda,\omega_n} / \lambda \\ t_n & = \left(2q_n/q''_n\right)^{2\lambda-1}\, \gamma_{k,N}(n) , \end{aligned}$$ and using [\[eqn:alpha\]](#eqn:alpha){reference-type="eqref" reference="eqn:alpha"}, according to [\[eqn:an\]](#eqn:an){reference-type="eqref" reference="eqn:an"} we can decompose $$a_{k,N}(n) = \frac{ \lambda \, } { 2^{\lambda-1} \, \left(2^{2\lambda}-1\right) B_{2\lambda} \, S_\lambda } \cdot r_n \, s_n \, t_n \,.$$ From the definition of $\beta_{\lambda,\omega}(n)$ it is easy to see that $r_n \in \mathbb{Z}$. Furthermore, by the definition of $\gamma_{k,N}(n)$, we have that $t_n \in \mathbb{Z}$. To prove the result it suffices then to show that $s_n \in \mathbb{Z}$. This is immediate when $\lambda$ is even and $n$ is a square, since in this case $s_n / S_\lambda = (2^\lambda-1) B_\lambda / \lambda$ (unless $n=1$, when they differ by a sign). Assume now that $\lambda$ is odd and $n$ is a square. Then $s_n / S_\lambda = 2 B_{\lambda,\omega_n} / \lambda$, hence the claim follows by part (a) of Lemma [Lemma 1](#lem:carlitz){reference-type="ref" reference="lem:carlitz"}. Finally, assume that $\lambda$ is odd or $n$ is not a square. In case (c) of Lemma [Lemma 1](#lem:carlitz){reference-type="ref" reference="lem:carlitz"}, the claim follows immediately. In case (b), let $p = f_n$. Noting that $\lambda \mid S_\lambda$, the claim follows from quadratic reciprocity and Euler's criterion, which give that $$\omega_n(2) = \genfrac(){}{}{2}{p} = \genfrac(){}{}{2}{p}^{ \frac{2\lambda}{p-1} } \equiv 2^\lambda \pmod p. \qedhere$$ ◻ **Proposition 1**. *Let $k \geq 3$ be odd. Then $$\dim \mathcal{E}_{k/2}({16}) = \begin{cases} 4, & k = 3,\\ 6, & k > 3. \end{cases}$$ Furthermore, $$\begin{gathered} \label{eqn:eis_gens} %\eisf{3/2}{16} & = %\label{eqn:ek2} \mathcal{E}_{k/2}({16}) = \\ \begin{cases} {\left\langle{ E_{3,4}, E_{3,4} \vert V(4), E_{3,8}, E_{3,4}\vert V(2) \vert U(2) }\right\rangle}, & k = 3,\\ {\left\langle{ E_{k,4}, E_{k,4}\vert V(4), E'_{k,4}, E'_{k,4}\vert V(4), E_{k,8}, E'_{k,8}\vert V(2) }\right\rangle}, & k > 3. \end{cases} \end{gathered}$$* *Proof.* The first claim follows from [@cohen-oesterle]. Let $N \in \{4,8\}$. In [@wang-pei Thm. 7.6] it is proved that $E_{k,N} \in \mathcal{E}_{k/2}({N})$. Considering the codomains of the operators $W(N),V(2),V(4),U(2)$ (see Section [2](#sect:prelims){reference-type="ref" reference="sect:prelims"}) we get that $\mathcal{E}_{k/2}({16})$ contains the subspace on the right hand side of [\[eqn:eis_gens\]](#eqn:eis_gens){reference-type="eqref" reference="eqn:eis_gens"}, for $k \geq 3$. We now prove that the generators on the right hand side of [\[eqn:eis_gens\]](#eqn:eis_gens){reference-type="eqref" reference="eqn:eis_gens"} are linearly independent, using the formulas for their coefficients given by Proposition [Proposition 1](#prop:eis_coeffs){reference-type="ref" reference="prop:eis_coeffs"}. Assume first that $k \equiv 5 \pmod 4$. Then $$\begin{aligned} E_{k,4} & = 1 + a_{k,4}(1) q + a_{k,4}(2) q^2+ a_{k,4}(3) q^3+ a_{k,4}(4) q^4+ a_{k,4}(5) q^5+ O(q^6), \\ E_{k,4} \vert V(4) & = 1 + a_{k,4}(1) q^4 + O(q^6), \\ E'_{k,4} & = a'_{k,4}(1) q + a'_{k,4}(2) q^2+ a'_{k,4}(3) q^3+ a'_{k,4}(4) q^4+ a'_{k,4}(5) q^5+ O(q^6), \\ E'_{k,4} \vert V(4) & = a'_{k,4}(1) q^4 + O(q^6), \\ E_{k,8} & = 1 + a_{k,8}(1) q + a_{k,8}(4) q^4+ a_{k,8}(5) q^5+ O(q^6), \\ E'_{k,8} \vert V(2) & = a'_{k,8}(1) q^2 + a'_{k,8}(2) q^4+ O(q^6). \end{aligned}$$ Then, since $a'_{k,4}(1) \, a'_{k,8}(1) \neq 0$ (see Corollary [Corollary 1](#coro:notzero){reference-type="ref" reference="coro:notzero"}), it suffices to prove that $$\begin{aligned} \left(\begin{matrix} a_{k,4}(1) & a_{k,4}(3) & a_{k,4}(5) \\ a'_{k,4}(1) & a'_{k,4}(3) & a'_{k,4}(5) \\ a_{k,8}(1) & 0 & a_{k,8}(5) \end{matrix} \right) \end{aligned}$$ is non-singular. We have that $\beta_{\lambda,\omega}(n) = 1$ for squarefree $n$. Furthermore, we have that $\gamma_{k,4}(1) > 0, \gamma_{k,4}(3) < 0, \gamma_{k,4}(5) >0$ and that $\gamma_{k,8}(1) > 0, \gamma_{k,8}(5) < 0$. Then by Lemma [Lemma 1](#lem:alphael){reference-type="ref" reference="lem:alphael"} the signs of the matrix above are given by $$\begin{aligned} \genfrac(){}{}{2\lambda+1}{2} \left(\begin{matrix} + & - & + \\ + & + & + \\ + & 0 & - \end{matrix} \right), \end{aligned}$$ hence its determinant is non-zero. The case $k \equiv 7 \pmod 4, k \neq 3$, can be proved similarly, using the $7$-th coefficient instead of the $5$-th coefficient in the matrix above. Finally, for $k = 3$ using Proposition [Proposition 1](#prop:eis_coeffs){reference-type="ref" reference="prop:eis_coeffs"} we get that $$\begin{aligned} E_{3,4} & = 1 + 6q + 12q^2 + 8 q^3 + O(q^4), \\ E_{3,4} \vert V(4) & = 1 + O(q^4), \\ E_{3,8} & = 1 + 8q^3 + O(q^4), \\ E_{3,4} \vert V(2) \vert U(2) & = 1 + 12q^2 + O(q^4), \end{aligned}$$ which completes the proof. ◻ *Proof of Proposition [Proposition 1](#prop:eis_eigen){reference-type="ref" reference="prop:eis_eigen"}.* Denote by $\mathcal{V}\subseteq \mathcal{E}_{k/2}({16})$ the $\sigma_{k-1}(\ell)$-eigenspace for $T(\ell^2)$. We claim first that $E_{k,4}, E_{k,8} \in \mathcal{V}$. For every $n$ we see easily from the definitions and Lemma [Lemma 1](#lem:alphael){reference-type="ref" reference="lem:alphael"} that $$\begin{aligned} \omega_{\ell^2 n} & = \omega_{n}, \\ \alpha_{\lambda,\ell^2 n} & = \ell^{-1} \, \alpha_{\lambda,n}, \\ \gamma_{k,N}\left(\ell^2 n\right) & = \gamma_{k,N}(n). \end{aligned}$$ Then the claim follows directly from [\[eqn:hecke\]](#eqn:hecke){reference-type="eqref" reference="eqn:hecke"}, using the equalities above and the transformation formulas for computing $\beta_{\lambda,\omega_{\ell^2 n}}(\ell^2 n)$ in terms of $\beta_{\lambda,\omega_n}(n)$ given in [@wang-pei p. 209]; we remark that though Wang and Pei are considering $k > 3$ and level $4D$ with $D$ odd and squarefree, these particular computations hold in our setting. The result follows, then, by noting that the remaining generators for $\mathcal{E}_{k/2}({16})$ given in Proposition [Proposition 1](#prop:eis_bases){reference-type="ref" reference="prop:eis_bases"} belong to $\mathcal{V}$, since by [@wang-pei Thm. 5.19] the Hecke operators $T(\ell^2)$ with $\ell \neq 2$ commute with the operators $W(N)$, and by $\eqref{eqn:hecke}$ they commute with $U(2),V(2),V(4)$. ◻ # Proofs {#sect:proofs} This section is devoted to give the proofs of our main results, namely Theorems [Theorem 1](#thm:inf0){reference-type="ref" reference="thm:inf0"}, [Theorem 1](#thm:infeps){reference-type="ref" reference="thm:infeps"}, [Theorem 1](#thm:fin0){reference-type="ref" reference="thm:fin0"} and [Theorem 1](#thm:fineps){reference-type="ref" reference="thm:fineps"}. We first state the following result for obtaining congruences for coefficients of (modulo $m$) eigenforms of half-integral weight used by [@ahlgrenono; @treneer2; @treneer1; @rsst], among others. **Proposition 1**. *Let $g = \sum_{n\geq 0} a(n) q^ n \in \mathcal{M}_{k/2}({N}) \cap \mathbb{Z}\llbracket q\rrbracket$, and let $\ell,m$ be primes such that $g \vert T(\ell^2) \equiv \lambda_{m,\ell}\, g \pmod{m\mathbb{Z}\llbracket q\rrbracket}$.* 1. *If $\lambda_{m,\ell}\equiv 0 \pmod m$, then $a(\ell^3 n) \equiv 0 \pmod m$ for every $n$ prime to $\ell$.* 2. *If there exists $\epsilon \in \{\pm1\}$ such that $$\lambda_{m,\ell} \equiv %\epsilon \, \ell^{\tfrac{k-1}2} \pmod m, \epsilon \, \ell^{\lambda} \pmod m,$$ then $a(\ell^2 n) \equiv 0 \pmod m$ for every $n$ prime to $\ell$ such that $\omega_n(\ell) = \epsilon$.* *Proof.* Both claims follow directly from [\[eqn:hecke\]](#eqn:hecke){reference-type="eqref" reference="eqn:hecke"}; for part (a), by replacing $n$ by $\ell n$, with $n$ prime to $\ell$. ◻ The goal of the following series of results is to prove that for prime $m$ the numbers $\overline{p}(mn)$ are congruent to the Fourier coefficients of a holomorphic modular form. We start with a preliminary result. **Lemma 1**. *Let $f$ and $g$ be power series, and let $m \geq 1$. Then $$(\left(f\vert V(m)\cdot g\right)\vert U(m)= f\cdot \left(g\vert U(m)\right).$$* *Proof.* Let $f=\sum_{n=0}^{\infty}a(n)q^n$ and $g=\sum_{n=0}^{\infty}b(n)q^n$. Denote $$\begin{aligned} \widetilde{a}(h) & = \begin{cases} a(n), &\text{if $h=nm$},\\ 0, &\text{otherwise}, \end{cases} %\qquad \text{and} \qquad \\ \widetilde{c}(h) & =\sum_{k=0}^{h}\widetilde{a}(k)b(h-k). \end{aligned}$$ Now, note that $$\begin{aligned} \widetilde{c}(hm)=\sum_{k=0}^{hm}\widetilde{a}(k)b(hm-k) =\sum_{k=0}^{h}a(k)b(hm-km). %=:c(h). \end{aligned}$$ Then we have $$\begin{aligned} ((f \vert V(m)) \cdot g)\vert U(m)&= \left( \left(\sum_{h=0}^{\infty}\widetilde{a}(h)q^{h}\right) \left(\sum_{n=0}^{\infty}b(n)q^n\right) \right)\vert U(m)\\ &=\left(\sum_{h=0}^{\infty}\widetilde{c}(h)q^{h}\right)\vert U(m) =\sum_{h=0}^{\infty}\widetilde{c}(hm)q^{h} \\ %& =\sum_{h=0}^{\infty}\widetilde{c}(hm)q^{h} \\ & =\sum_{h=0}^{\infty} \left(\sum_{k=0}^{h}a(k)b(mh-k)\right) q^h = f\cdot \left(g\vert U(m)\right). \qedhere \end{aligned}$$ ◻ **Lemma 1**. *Let $f$ be an eta-quotient. Then for every prime $m \geq 1$ we have that $$f\vert V(m)\equiv f^{m}\pmod{m\mathbb{Z}\llbracket q\rrbracket}.$$* *Proof.* Write $f$ as in [\[eqn:etaq\]](#eqn:etaq){reference-type="eqref" reference="eqn:etaq"}. Since both operators $V(m)$ and $g\mapsto g^m$ are multiplicative, it suffices to verify the congruence for every factor $g$ of $f$. For $g = q^{\tfrac{s_X}{24}}$ both operators clearly agree, and for $g = 1-q^{\delta n}$ the congruence follows from the fact that $(r+s)^m \equiv r^m + s^m \pmod{m\mathbb{Z}\llbracket q\rrbracket}$ for every $r,s \in \mathbb{Z}\llbracket q\rrbracket$. ◻ **Proposition 1**. *Let $f = \eta^X$ be an eta-quotient with $s_X = 0$, and let $F = 1/f$. Then for every prime $m \geq 1$ we have that $$F^{m^2-1} \vert U(m)\equiv F^m \cdot \left(f \vert U(m)\right) \pmod{m\mathbb{Z}\llbracket q\rrbracket}.$$* *Proof.* Applying Lemma [Lemma 1](#lem:Modm){reference-type="ref" reference="lem:Modm"} to the eta-quotient $F^m$ and using that, since $s_X = 0$, we have by [\[eqn:etaq\]](#eqn:etaq){reference-type="eqref" reference="eqn:etaq"} that $f \in \mathbb{Z}\llbracket q\rrbracket^\times$, we obtain that $$F^{m^{2}-1}\equiv F^{m}\vert V(m)\cdot f \pmod{m\mathbb{Z}\llbracket q\rrbracket}.$$ The result follows then by applying $U(m)$ to this congruence and using Lemma [Lemma 1](#lem:Power){reference-type="ref" reference="lem:Power"}. ◻ In what follows we consider the eta-quotient related to the generating function for $\overline{p}(n)$ (see [@overpartitions (1.1)]). Namely, we let $$\label{eqn:etaq_op} f = \frac{\eta(2z)}{\eta^2(z)} = \sum_{n\geq 0} \overline{p}(n)q^n,$$ and we denote $F = 1/f$. Note that $f$ (as well as $F$) satisfies the hypothesis of Proposition [Proposition 1](#prop:Um){reference-type="ref" reference="prop:Um"}. **Lemma 1**. *We have that $F \in \mathcal{M}_{1/2}({16})$. Furthermore, for every nonnegative integer $k$ such that $k\equiv 0 \pmod8$ we have that $F^k \in \mathcal{M}_{k/2}({2})$.* We remark that $f$ is not holomorphic: by [\[eqn:ligozat\]](#eqn:ligozat){reference-type="eqref" reference="eqn:ligozat"}, it has a simple pole at $s=0$. *Proof.* By Proposition [Proposition 1](#prop:eta_wh){reference-type="ref" reference="prop:eta_wh"} we have that $F$ is a weakly holomorphic modular form of level $16$ and weight $1/2$, with trivial character. Its possible singularities lie at the cusps $s$ for $\Gamma_0(16)$, namely $s \in \{0,1/8,1/4,1/2,3/4,\infty\}$. Then the claim follows from [\[eqn:ligozat\]](#eqn:ligozat){reference-type="eqref" reference="eqn:ligozat"}, which shows that the order of vanishing of $F$ at each such $s$ is nonnegative (moreover, it is positive only for $s=0$). The second claim follows from the holomorphicity of $F$ at the cusps $s$ for $\Gamma_0(2)$, namely $s \in \{0,\infty\}$, since by Proposition [Proposition 1](#prop:eta_wh){reference-type="ref" reference="prop:eta_wh"} we have that, for $k$ as above, $F^k$ has level $2$. ◻ As in Section [3](#sect:eisen){reference-type="ref" reference="sect:eisen"}, let $\Gamma_\infty = \left\{{\pm\left(\begin{smallmatrix} 1 & n \\ 0 & 1 \end{smallmatrix} \right)} \, : \, {n \in \mathbb{Z}}\right\}$. Let $k \geq 2$ be an even integer. We consider the Eisenstein series of integral weight $$\begin{aligned} E_{k}(z) & = \sum_{\gamma \in \Gamma_\infty \backslash \mathop{\mathrm{SL}}_2(\mathbb{Z})} \frac{1}{(c_\gamma z + d_\gamma)^k} \quad \in \mathcal{M}_{k}({1}),\\ D_2 & = 2 E_2 | V(2) - E_2 \quad \in \mathcal{M}_{2}({2}).\end{aligned}$$ Then $E_{k} \in 1 + q\mathbb{Z}\llbracket q\rrbracket$. Furthermore, $$\label{eqn:eisenstein} E_k = 1 - \frac{2k}{B_k} \sum_{n\geq1} \sigma_{k-1}(n) q^n.$$ The following result will not be used in our proofs, but explains the type of forms $h_m$ appearing in Table [4](#tab:hm){reference-type="ref" reference="tab:hm"} below (see also Remark [Remark 1](#rmk:conjecture){reference-type="ref" reference="rmk:conjecture"}). **Proposition 1**. *Let $D_2,E_4$ be as above. Then $\left\{{D_2^a E_4^b} \, : \, {2a+4b = k}\right\}$ is a basis for $\mathcal{M}_{k}({2})$.* *Proof.* Let $\Delta_2 = \eta^8(z) \eta(2z)^8$. Since [\[eqn:ligozat\]](#eqn:ligozat){reference-type="eqref" reference="eqn:ligozat"} implies that $\mathop{\mathrm{ord}}_0(\Delta_2) = \mathop{\mathrm{ord}}_\infty(\Delta_2) = 1$, by Proposition [Proposition 1](#prop:eta_wh){reference-type="ref" reference="prop:eta_wh"} we have that $\Delta_2 \in \mathcal{S}_{8}({2})$. Furthermore, since $\Delta_2$ does not vanish on the upper half-plane, the map $$\begin{aligned} \mathcal{M}_{k}({2}) & \to \mathcal{S}_{k+8}({2}),\\ g & \mapsto g\cdot \Delta_2 \end{aligned}$$ is an isomorphism. Denote by $\mathcal{V}_k$ the subspace of $\mathcal{M}_{k}({2})$ generated by $\left\{{D_2^a E_4^b} \, : \, {2a+4b = k}\right\}$. Using Proposition [Proposition 1](#prop:sturm){reference-type="ref" reference="prop:sturm"} and [\[eqn:eisenstein\]](#eqn:eisenstein){reference-type="eqref" reference="eqn:eisenstein"} we get that $$576\Delta_2 = 5 D_2^2 E_4 - E_4^2 - 4 D_2^4.$$ Hence $\Delta_2 \in \mathcal{V}_8$. Thus, to prove that $\mathcal{M}_{k}({2}) = \mathcal{V}_k$ it suffices to show that for every $f \in \mathcal{M}_{k+8}({2})$ there exists $g \in \mathcal{V}_{k+8}$ such that $f-g \in \mathcal{S}_{k+8}({2})$. For this purpose it suffices to prove that there exist $g_\infty,g_0 \in \mathcal{V}_{k+8}$ such that $g_\infty$ does not vanish at $\infty$, and $g_0$ vanishes at $\infty$ but not at $0$; equivalently, $g_0$ vanishes at $\infty$ but is not cuspidal. We can clearly let $g_\infty = D_2^a$ with $a = \tfrac{k+8}2$. In the case of $g_0$, it suffices to consider $k \in \{0,2,4,6\}$. Then using explicit bases for $\mathcal{S}_{k+8}({2})$ we see that we can let $g_0$ be as in Table [3](#tab:g0){reference-type="ref" reference="tab:g0"}. $k$ $g_0$ ----- --------------------- $0$ $D_2^4 - E_4^2$ $2$ $D_2^5 - D_2 E_4^2$ $4$ $D_2^6 - E_4^3$ $6$ $D_2^7 - D_2 E_4^3$ : Forms in $\mathcal{V}_{k+8}$ vanishing at $\infty$ but not at $0$. Used in the proof of Proposition [Proposition 1](#prop:Mk2){reference-type="ref" reference="prop:Mk2"}. Finally, the independence of the forms $D_2^a E_4^b$ follows using the formulas for $\dim(\mathcal{M}_{k}({2}))$ (see [@cohen-oesterle]). ◻ **Proposition 1**. *Let $5 \leq m \leq 19$ be a prime, and let $h_m$ be the corresponding form given in Table [4](#tab:hm){reference-type="ref" reference="tab:hm"}. Let $0 \leq k_m < 8$ be such that $k_m \equiv -m \pmod 8$ and let $g_m = F^{k_m} h_m$. Then $g_m \in \mathcal{M}_{\tfrac{m-2}2}({16}) \cap \mathbb{Z}\llbracket q\rrbracket$, and $$\label{eqn:fUmgm} f \vert U(m)\equiv g_m \pmod {m\mathbb{Z}\llbracket q\rrbracket}.$$* $m$ $h_m$ ------ -------------------------------- $5$ $1$ $7$ $D_2$ $11$ $D_2$ $13$ $E_4$ $17$ $13 D_{2}^{2} + 5 E_{4}$ $19$ $11 D_{2}^{3} + 9 D_{2} E_{4}$ : Holomorphic modular forms used in Proposition [Proposition 1](#prop:hm){reference-type="ref" reference="prop:hm"}. *Proof.* The first claim follows from Lemma [Lemma 1](#lem:F16){reference-type="ref" reference="lem:F16"}, since for every $m$, we have that $h_m \in \mathcal{M}_{\tfrac{m-k_m-2}{2}}({2})$. Since $F \in \mathbb{Z}\llbracket q\rrbracket^\times$, by Proposition [Proposition 1](#prop:Um){reference-type="ref" reference="prop:Um"} we have that [\[eqn:fUmgm\]](#eqn:fUmgm){reference-type="eqref" reference="eqn:fUmgm"} is equivalent to $$\label{eqn:FUm} F^{m^2-1} \vert U(m)\equiv F^{m+k_m} h_m \pmod{m\mathbb{Z}\llbracket q\rrbracket},$$ which, by Lemma [Lemma 1](#lem:F16){reference-type="ref" reference="lem:F16"}, is a congruence between integral weight modular forms. Over integral weights the operator $U(m)$ agrees, modulo $m\mathbb{Z}\llbracket q\rrbracket$, with the Hecke operator $T(m)$ (not to be confused with the Hecke operator [\[eqn:hecke\]](#eqn:hecke){reference-type="eqref" reference="eqn:hecke"} for half-integral weights). Since $E_{m-1} \equiv 1 \pmod{m\mathbb{Z}\llbracket q\rrbracket}$, we get then that [\[eqn:FUm\]](#eqn:FUm){reference-type="eqref" reference="eqn:FUm"} is equivalent to $$\label{eqn:FTm} F^{m^2-1} \vert T(m) \equiv F^{m+k_m} h_m \, E_{m-1}^{\tfrac{m-1}2}\pmod{m\mathbb{Z}\llbracket q\rrbracket},$$ which, by Lemma [Lemma 1](#lem:F16){reference-type="ref" reference="lem:F16"}, is a congruence between forms in $\mathcal{M}_{\tfrac{m^2-1}2}({2}) \cap \mathbb{Z}\llbracket q\rrbracket$. Thus, by Proposition [Proposition 1](#prop:sturm){reference-type="ref" reference="prop:sturm"} it suffices to prove that the $n$-th coefficients of both sides of [\[eqn:FTm\]](#eqn:FTm){reference-type="eqref" reference="eqn:FTm"} agree, modulo $m$, up to $n$ equal to $$\tfrac{m^2-1}{24} \cdot \left[\mathop{\mathrm{SL}}_2(\mathbb{Z}): \Gamma_0(2)\right] = \tfrac{m^2-1}8.$$ Moreover, reversing the reasoning above and denoting $h_m = \sum_{n \geq 0}a_m(n)q^n$, we get that it suffices to show that $$\overline{p}(n)^{k_m} \, \overline{p}(mn) \equiv a_m(n) \pmod m, \quad 1 \leq n \leq \tfrac{m^2-1}8,$$ which in each case can be proved by computing these numbers explicitly. ◻ *Remark 1*. In fact, using the techniques from the above proof and Proposition [Proposition 1](#prop:Mk2){reference-type="ref" reference="prop:Mk2"}, we have found forms $g_m$ as in Proposition [Proposition 1](#prop:gm_eigens){reference-type="ref" reference="prop:gm_eigens"} for every prime $m < 1000$. From here on, given primes $\ell,m$, we denote $$\lambda_{\ell,m} = \sigma_{m-4}(\ell) = 1 + \ell^{m-4},$$ the eigenvalue of $T(\ell^2)$ acting on $\mathcal{E}_{\tfrac{m-2}2}({16})$ (see Proposition [Proposition 1](#prop:eis_eigen){reference-type="ref" reference="prop:eis_eigen"}). **Proposition 1**. *Let $5 \leq m \leq 19$ be a prime, and let $g_m$ be the form given in Proposition [Proposition 1](#prop:congruences){reference-type="ref" reference="prop:congruences"}.* 1. *If $5 \leq m \leq 11$ then $g_m \vert T(\ell^2)\equiv \lambda_{m,\ell}g_m \pmod{m\mathbb{Z}\llbracket q\rrbracket}$ for every prime $\ell > 2$.* 2. *If $13 \leq m \leq 19$ then $g_m \vert T(\ell^2)\equiv \lambda_{m,\ell}g_m \pmod{m\mathbb{Z}\llbracket q\rrbracket}$ for every prime $\ell$ in Table [5](#tab:eigens){reference-type="ref" reference="tab:eigens"}.* $m$ $\ell$ ------ ------------------------------------------------------------------------------- $13$ $431, 1811, 1871, 1949, 2207, 2459, 3301, 4001, 4079, 4289, 4513, 4799, 4931$ $17$ $1999, 2207, 2243, 4759$ $19$ $151, 1091, 2207, 2659, 3989$ : Primes $\ell$ giving congruences modulo $m$. See Proposition [Proposition 1](#prop:gm_eigens){reference-type="ref" reference="prop:gm_eigens"}. *Proof.* To prove part (a) we can use Proposition [Proposition 1](#prop:eis_eigen){reference-type="ref" reference="prop:eis_eigen"}, once we verify that for $5 \leq m \leq 11$ we have that $g_m \in \mathcal{E}_{\tfrac{m-2}2}({16}) + m\mathbb{Z}\llbracket q\rrbracket$. The latter claim, in the case $5 \leq m \leq 7$ follows from the fact that $\mathcal{S}_{\tfrac{m-2}2}({16}) = \{0\}$. In the case $m = 11$, since for $N \in \{4,8\}$ by Proposition [Proposition 1](#prop:eis_denoms){reference-type="ref" reference="prop:eis_denoms"} we have that $$17 \cdot E_{9,N},\, 2^5 17 \cdot E'_{9,N} \quad \in \mathbb{Z}\llbracket q\rrbracket,$$ we can use Proposition [Proposition 1](#prop:sturm){reference-type="ref" reference="prop:sturm"} to obtain that $$\begin{gathered} \label{eqn:g11} g_{11}(z) \equiv 9 E_{9,4} + 4 E_{9,4} \vert V(4) + 7 E'_{9,4} + 4 E'_{9,4} \vert V(4) + 7 E'_{9,8} \vert V(2) \pmod{11\,\mathbb{Z}\llbracket q\rrbracket}. \end{gathered}$$ For proving part (b), by Proposition [Proposition 1](#prop:sturm){reference-type="ref" reference="prop:sturm"} it suffices to prove that the $n$-th coefficients of $g_m\vert T(\ell^2)$ and $\lambda_{m,\ell}\, g_m$ agree, modulo $m$, for $n$ up to $$\tfrac{m-2}{24} \cdot \left[\mathop{\mathrm{SL}}_2(\mathbb{Z}): \Gamma_0(16)\right] = m-2.$$ Moreover, by [\[eqn:fUmgm\]](#eqn:fUmgm){reference-type="eqref" reference="eqn:fUmgm"} it suffices to prove that $$\left(f\vert U(m)\vert T(\ell^2)\right)(n) \equiv \lambda_{m,\ell}\, (f\vert U(m))(n) \pmod m, \quad 1 \leq n \leq m-2,$$ which in each case can be proved by computing these numbers explicitly. ◻ *Remark 1*. The proof of Proposition [Proposition 1](#prop:congruences){reference-type="ref" reference="prop:congruences"} involves computing $\overline{p}(mn)$ and $a_m(n)$ modulo $m$ for small values of $n$. This can be accomplished easily by expanding the infinite product [\[eqn:etaq_op\]](#eqn:etaq_op){reference-type="eqref" reference="eqn:etaq_op"} defining $f$ (and $F$), and using [\[eqn:eisenstein\]](#eqn:eisenstein){reference-type="eqref" reference="eqn:eisenstein"}. The proof of Proposition [Proposition 1](#prop:gm_eigens){reference-type="ref" reference="prop:gm_eigens"} involves computing $\overline{p}(mn)$ modulo $m$ for large values of $n$ (e.g. $n = m(m-2)\ell^2$ with large $\ell$); in this case we resort to the efficient method provided by [@barquero-etal]. The proofs of our main results now follow easily. *Proof of Theorems [Theorem 1](#thm:inf0){reference-type="ref" reference="thm:inf0"} and [Theorem 1](#thm:fin0){reference-type="ref" reference="thm:fin0"}.* They follow using Proposition [Proposition 1](#prop:congruences){reference-type="ref" reference="prop:congruences"} (a) and Proposition [Proposition 1](#prop:gm_eigens){reference-type="ref" reference="prop:gm_eigens"}. ◻ *Proof of Theorems [Theorem 1](#thm:infeps){reference-type="ref" reference="thm:infeps"} and [Theorem 1](#thm:fineps){reference-type="ref" reference="thm:fineps"}.* They follow using Proposition [Proposition 1](#prop:congruences){reference-type="ref" reference="prop:congruences"} (b) and Proposition [Proposition 1](#prop:gm_eigens){reference-type="ref" reference="prop:gm_eigens"}; the eigenvalues $\lambda_{m,\ell}$ in Table [5](#tab:eigens){reference-type="ref" reference="tab:eigens"} satisfy the hypothesis of Proposition [Proposition 1](#prop:congruences){reference-type="ref" reference="prop:congruences"} (b), namely they are such that $$\lambda_{m,\ell}\equiv \epsilon_{m,\ell}\, \ell^{\tfrac{m-3}2}\pmod m,$$ where $\epsilon_{m,\ell}$ is as in Table [2](#tab:ml_eps){reference-type="ref" reference="tab:ml_eps"}. ◻ *Remark 1*. We found that $g_m$ is, modulo $m\mathbb{Z}\llbracket q\rrbracket$, an eigenfunction of $T(\ell^2)$ for more primes $\ell$ than those appearing in Table [5](#tab:eigens){reference-type="ref" reference="tab:eigens"}, but the eigenvalues are not useful for our purposes, since they do not satisfy any of the hypotheses of Proposition [Proposition 1](#prop:congruences){reference-type="ref" reference="prop:congruences"}. Moreover, the primes given are all the primes $\ell < 5000$ giving congruences. For $m = 23$ we found that $\ell = 5303, 8783$ yield eigenvalues, but they do not give congruences. For larger $m$ we have not been able to find eigenvalues.

arxiv_math

--- abstract: | In a recent paper [@Chen-Erchenko-Gogolev], Chen, Erchenko and Gogolev have proven that if a Riemannian manifold with boundary has hyperbolic geodesic trapped set, then it can be embedded into a compact manifold whose geodesic flow is Anosov. They have to introduce some assumptions that we discuss here. We explain how some can be removed, obtaining in particular a result applicable to all reasonable 3 dimensional examples. author: - Yannick Guedes Bonthonneau bibliography: - biblio.bib title: Extending a result of Chen, Erchenko and Gogolev. --- In all that follows, $(M,g)$ will denote a smooth compact Riemannian manifold with smooth boundary $\partial M$, of dimension $n$. The geodesic flow is defined on a subset of $S M \times\mathbb{R}$, and we will always assume that the set of points $v\in SM$ whose geodesic trajectory never encounters $\partial SM$ is a hyperbolic set. As in [@Chen-Erchenko-Gogolev], we consider **Problem 1**. *Can $M$ be isometrically embedded as an open set of a compact manifold $(N,g')$ without boundary, so that the geodesic flow of $N$ is Anosov?* Since metrics whose geodesic flow is Anosov form the $C^2$ interior of the metrics that do not have conjugate points [@Ruggiero-1991], certainly $M$ must not have any conjugate point, so we will make this assumption from now on. It is a standard assumption in the context of inverse problems for manifolds with boundary, that the boundary is strictly convex, i.e that the second fundamental form $$\label{eq:convex-boundary} II_{\partial M} > 0.$$ This ensures that the riemannian distance function for points that are close to each other on the boundary is realized by small geodesics that only touch the boundary at their endpoints. Thus, the set of points whose geodesic trajectory remains inside $M$ for all times does not meet $\partial M$. In practice, this simplifies many computations so we will assume [\[eq:convex-boundary\]](#eq:convex-boundary){reference-type="eqref" reference="eq:convex-boundary"} holds. **Definition 1**. *Let $M$ be a compact manifold with non empty boundary, with hyperbolic trapped set, no conjugate points, and strictly convex boundary. We say that it is *an Anosov manifold with boundary*.* Let us recall the result of [@Chen-Erchenko-Gogolev] **Theorem 1** (Theorem A, [@Chen-Erchenko-Gogolev]). *Let $M$ be an Anosov manifold with boundary, and furthermore assume that the boundary components of $M$ are topological spheres, then Problem [Problem 1](#probA){reference-type="ref" reference="probA"} has a positive solution.* Since the only connected compact manifold in one dimension is a sphere, this theorem in particular solves completely the case of surfaces. In higher dimension, the assumption that the boundary components are spheres seemed very stringent, the question being whether there exists any example where $M$ is not a topological ball. In a second version of their paper, the authors in [@Chen-Erchenko-Gogolev] claim that their argument also applies to the case $\partial M \simeq \mathbb{S}^1 \times \mathbb{S}^{n-2}$. In this paper, we will prove the following results. First, we forego the topology question, and observe that the metric extension of [@Chen-Erchenko-Gogolev] yields almost directly **Theorem 2**. *Let $M$ be an Anosov manifold with boundary. Then one can embed isometrically $M\subset N$ into a conformally compact complete $n$-manifold $N$ whose interior has uniformly hyperbolic geodesic flow, negative curvature at infinity, asymptotically constant, and no conjugate points. Also, $N$ is diffeomorphic to $M$.* Next, we consider the case of the boundary having a very simple topology, and obtain some partial results about the topology of $M$ **Theorem 3**. *Let $M$ be an Anosov manifold with boundary of dimension at least $3$* - *If at least one boundary component is diffeomorphic to a sphere, then $M$ is diffeomorphic to a ball, and has no trapped set.* - *If at least one boundary component is diffeomorphic to $\mathbb{S}^1\times \mathbb{S}^{n-2}$, then $M$ is diffeomorphic to a solid torus. It is a convex neighbourhood of a single closed geodesic.* As observed by the authors of [@Chen-Erchenko-Gogolev], in the second case, one can use the residual finitess of the $\pi_1$ of hyperbolic manifolds to embed $M$ in the finite cover of any given compact hyperbolic $n$-manifold. If one component of the boundary is a $\mathbb{S}^{n-p-1}$ bundle over a $p$-dimensional Anosov manifold, one could imagine that $M$ is diffeomorphic to the corresponding ball bundle ; except from some partial results, whether this is true or not has eluded the author. Finally, we show that for 3 manifolds, the problem can also be completely solved. **Theorem 4**. *Let $M$ be an oriented Anosov 3-manifold. Then Problem [Problem 1](#probA){reference-type="ref" reference="probA"} has a positive answer.* It seems that current knowledge of topology of Riemannian manifolds of dimension $\geq 4$ is not sufficient to settle the question in higher dimensions. **Structure of the arguments.** We will rely on a good part of the argument in [@Chen-Erchenko-Gogolev]. Let us recall its gist. They first construct an extension of the metric near the boundary, whose curvature is bounded above uniformly, but becomes arbitrarily negative at an arbitrarily small distance of the boundary, and constant at a fixed distance. The second part is to study the dynamics of Jacobi Fields, and prove that the passage of the geodesics through the patch of possibly positive curvature is compensated by the large negative curvature in the rest of the extension, so that the extension does not create conjugate points, and preserves the Axiom A property. The last part is to find a manifold of constant sectional curvature that can be glued to the extended manifold. We point out that in the arguments of the second part, it is not really important that the curvature becomes constant far from the boundary. What is crucial is that it is arbitrarily negative on the complement of an arbitrarily small patch near $\partial M$, and uniformly bounded above. If one were to build an extension satisfying only these two properties, the second part of the arguments would apply. Then if one glues to the extension a manifold with concave boundary and negative curvature, the rest of the arguments also apply. The remaining topological question is thus whether there exist concave manifold with negative curvature and prescribed metric near boundary components. **Organization of the paper.** We will first give a slightly different presentation of the construction of the metric extension, so that it applies to the most general case and prove Theorem [Theorem 2](#thm:conformally-compact){reference-type="ref" reference="thm:conformally-compact"}. Next, we will discuss some topological results about the general problem, and finish the proof of Theorem [Theorem 3](#thm:special-cases){reference-type="ref" reference="thm:special-cases"}. Finally, we will concentrate on the 3 dimensional case, and prove Theorem [Theorem 4](#thm:3D){reference-type="ref" reference="thm:3D"}. We happily thank C. Lecuire for providing the key argument for Theorem [Theorem 4](#thm:3D){reference-type="ref" reference="thm:3D"}. We also warmly thank B. Petri and F. Paulin for many explanations. The author of this paper is not an expert in topology of hyperbolic 3 manifold, or higher dimensional topology for that matter. If such an expert would take interest in these questions, maybe significant further progress could be made. # The extension problem {#sec:metric-extension} We will here present the constructive arguments of [@Chen-Erchenko-Gogolev] a little differently. We start by giving formulæ for the curvatures of a metric in coordinates adapted to an hypersurface. ## Study of the curvatures in a slice situation In this section, we consider $S$ a compact manifold endowed with a family of Riemannian metrics $(g_t)_t$, and a function $f$ of the parameter $t$, that we will assume to lie in an interval $I\subset \mathbb{R}$. Both are assumed to be smooth, and we are chiefly interested in the sectional curvatures of the metrics on $I\times S$ given by $$h= dt^2 + g_t,\qquad \tilde{h} = dt^2 + f(t)^2 g_t.$$ In what follows, will think of $g_t$ as fixed, and will let $f$ vary to obtain some interesting properties. Quantities related to $\tilde{h}$ will be denoted with a ${}^\sim$. In [@Chen-Erchenko-Gogolev], some very classical formulæ are recalled, but some computations have only been done in local coordinates using Christoffel coefficients; we will here try to give an intrinsic presentation, simplifying the proof of the first part of their statement. On the manifold $I\times S$, we denote by $T$ the vector field $\partial/\partial t$, and the letters $X,Y$ will denote vector fields tangent to $S$, that do not depend on $t$. Recall that the second fundamental form is $II =(1/2) \partial_t h$, and the Shape operator $A$ is defined by $$h(AX,Y) = II(X,Y).$$ Here, $$g_t(\tilde{A}X,Y) = \frac{f'}{f} g_t(X,Y) + \frac{1}{2}\partial_t g_t(X,Y),$$ so that $$\tilde{A} = \frac{f'}{f} + A.$$ We will assume that the slices $S_t =\{t\} \times S$ are strictly convex, i.e that $\tilde{A} > 0$. More precisely, we will assume that $A\geq 0$, and that $f'/f > 0$. We say that the slices are *uniformly* strictly convex if we have $\tilde{A}> C >0$ for some global constant $C>0$. By $\sigma_{U,V}$, we denote the plane generated by vectors $U,V$. **Proposition 1**. *We have the following formulæ for the sectional curvatures of $\tilde{h}$, where $a\neq 0$, and $X,Y$ are orthonormal for $g_t$ at the point of computation $$\begin{aligned} \tilde{K}(\sigma_{X,T}) &= -\frac{f''}{f} + K(\sigma_{X,T}) - 2 \frac{f'}{f} g_t(AX,X). \\ \tilde{K}(\sigma_{X,Y}) &= \frac{1}{f^2} K^{int}(\sigma_{X,Y}) + \left[ g_t(AX,Y)^2 - g_t(AX,X)g_t(AY,Y)\right] \\ &\qquad - \left(\frac{f'}{f}\right)^2 - 2 \frac{f'}{f}\left[ g_t(AX,X)+ g_t(AY,Y)\right]\\ \tilde{K}(\sigma_{X+aT,Y})&= \frac{ f(t)^2 \tilde{K}(\sigma_{X,Y}) + a^2 \tilde{K}(\sigma_{T,Y}) + 2a h( R(X,Y)Y,T)}{f(t)^2 + a^2} . \\\end{aligned}$$* *Proof.* This essentially follows from the Gauss and Codazzi equations. Let us start with the sectional curvature of the plane $\sigma_{X,T}$ generated by $T$ and $X\neq 0$. It is given by $$K(\sigma_{X,T}) = - \frac{g_t((A'+A^2)X,X)}{g_t(X,X)},$$ so that $$\tilde{K}(\sigma_{X,T}) = - \frac{f''}{f} + K(\sigma_{X,T}) - 2 \frac{f'}{f} \frac{g_t(AX,X)}{g_t(A,A)}.$$ Let us now turn to the curvature of the plane $\sigma_{X,Y}$ generated by $X$ and $Y$. The Gauss' equation gives $$K(\sigma_{X,Y}) = K^{int}(\sigma_{X,Y}) - \frac{ g_t(AX,X)g_t(AY,Y) - g_t(AX,Y)^2}{g_t(X,X)g_t(Y,Y) - g_t(X,Y)^2},$$ where $K^{int}$ is the sectional curvature of $g_t$. Since $f$ is constant on each slice, we find directly $$\tilde{K}^{int}(\sigma_{X,Y}) = \frac{1}{f^2} K^{int}(\sigma_{X,Y}).$$ We deduce that taking $X$ and $Y$ orthonormal (for $g_t$) at the point of interest, we get $$\begin{split} \tilde{K}(\sigma_{X,Y}) &= \frac{1}{f^2} K^{int}(\sigma_{X,Y}) - \left[ g_t(AX,X)g_t(AY,Y) - g_t(AX,Y)^2\right] \\ &\quad - \left( \frac{f'}{f}\right)^2 - 2 \frac{f'}{f}( g_t(AX,X) + g_t(AY,Y) ). \end{split}$$ Let us now turn to the curvature of the plane $\sigma_{T+aX,Y}$, generated by $X + aT$ and $Y$, for some $a\neq 0$. Without changing the plane (but changing $a$), we can assume that $X,Y$ are orthogonal (for $g_t$) at the point of interest. Then we have $$K(\sigma_{X+aT, Y}) = \frac{ h( R(X+aT, Y)Y, X+aT )}{(a^2+g_t(X,X))g_t(Y,Y)}.$$ This gives (using the symmetries of the curvature tensor) $$\begin{split} K(\sigma_{X+aT, Y}) =& \frac{1}{a^2+g_t(X,X)^2}\left(g_t(X,X)^2 K(\sigma_{X,Y}) + a^2 K(\sigma_{T,Y})\right) \\ & + \frac{2a}{(a^2+g_t(X,X)^2)g_t(Y,Y)}\langle R(X,Y)Y,T\rangle. \end{split}$$ It is this last mixed term in the RHS that was estimated using coordinates and Christoffel coefficients in [@Chen-Erchenko-Gogolev]. We are seeking to compute $$F(X,Y):= \langle \nabla_X \nabla_Y Y - \nabla_Y \nabla_X Y - \nabla_{[X,Y]} Y, T\rangle.$$ According to the Codazzi equation, we have $$F(X,Y)=\nabla_X^{int} II(Y,Y) - \nabla_Y^{int} II(X,Y).$$ Now, we observe that $$\tilde{F}(X,Y)= f(t)^2 F(X,Y).$$ This $f(t)^2$ term will be compensated by $\tilde{h}(Y,Y)$, so we conclude that $$\tilde{K}(\sigma_{X+aT,Y}) = \frac{ f(t)^2 \tilde{K}(\sigma_{X,Y}) + a^2 \tilde{K}(\sigma_{T,Y}) + 2 a F(X,Y) }{f(t)^2 + a^2}.$$ ◻ ## Extension of the metric We assume that we are given $(M,g)$ as in the introduction. Near the boundary, we can always add cylinders, to define $$N:= M \cup (\partial M)_x\times ]0, +\infty]_t,$$ which is a smooth manifold with boundary, diffeomorphic to $M$. Theorem [Theorem 2](#thm:conformally-compact){reference-type="ref" reference="thm:conformally-compact"} follows from the following **Lemma 1**. *We can build a metric $\tilde{h}$ on the interior of $N$ that extends the metric of $M$, and so that this metric is conformal to a smooth metric on $N$. Additionally, there exists a constant $C_0>0$ so that for $\kappa>0$ large enough, we can ensure that* 1. *In $\mathring{N}\setminus M$, the sets $\{t=t_0\}$ are equidistant sets from the boundary $\partial M$. They are uniformly strictly convex* 2. *The sectional curvature of $\tilde{h}$ tends to $-\kappa^2$, with equality for $t\geq 1$ if $\partial M$ admits a constant curvature metric.* 3. *The sectional curvature of $\tilde{h}$ is globally bounded above by $C_0$,* 4. *For $t\geq 1/\sqrt{\kappa}$, the sectional curvature of $\tilde{h}$ is less than $-\kappa^2/2$.* *Proof.* We will work near a fixed boundary component $P\subset \partial M$. Near $P$, we can write the metric in the form $$h=dt^2 + g_t(x,dx),$$ where $g_t$ is a family of metrics on $P$, $-\epsilon< t\leq 0$, and $t$ is the geodesic distance to $P$. We can extend the family $g_t$ smoothly to $]-\epsilon, +\infty[$, and since $\partial_t g_t > 0$ near $t = 0$, we can ensure that - $\partial_t g_t > \tfrac{1}{2}\partial_t g_t|_{t=0}$ for $t\in ]-\epsilon, 1/2]$ - $\partial_t g_t \geq 0$ for all $t$'s - $\partial_t g_t = 0$ for $t\geq 1$. Additionally, if we are given a metric $g'$ on $P$, we can ensure that $g_t = C g'$ for $t\geq 1$ and some constant $C>0$. When $P$ supports a metric of constant sectional curvature, this is the choice that we make. Next, we want to look for $\tilde{h} = dt^2 +f(t)^2 g_t(x,dx)$ in the form given by the previous §. We impose $f=1$ on $]-\epsilon, 0]$. Let us analyze the conditions of the theorem and their consequence for $f$. First, to ensure uniform strict convexity, it suffices to assume that $f''\geq 0$ globally, and that $f'/f>C>0$ on $[1/2, +\infty[$. Next, $f''\geq 0$ also ensures[^1] that $\tilde{K}\leq K$, so that (3) is satisfied with $C_0 \geq \sup K$. Let us now consider property (2). We observe that since for $t\geq 1$, $\partial_t g_t = 0$, we get the formulæ ($t\geq 1$) $$\begin{aligned} \tilde{K}(\sigma_{X,T}) &= -\frac{f''}{f} \\ \tilde{K}(\sigma_{X,Y}) &= \frac{1}{f^2}K^{int}(\sigma_{X,Y}) - \left(\frac{f'}{f}\right)^2\\ \tilde{K}(\sigma_{X+aT,Y})&= \frac{ f(t)^2 \tilde{K}(\sigma_{X,Y}) + a^2 \tilde{K}(\sigma_{Y,T})}{f(t)^2+ a^2}. \end{aligned}$$ This suggests to take for $t\geq 1$ $$\label{eq:f-away} f(t)= f_{cc}(t):= \begin{cases} \frac{C}{\kappa} \cosh(\kappa t) & \text{if $\inf K^{int}_{t\geq 1} = - C^2<0$,}\\ e^{\kappa t} &\text{if $K^{int}_{t\geq1} =0$,}\\ \frac{C}{\kappa} \sinh(\kappa t) & \text{if $K^{int}_{t\geq 1}\geq 0$ and $\sup K^{int}_{t\geq 1} = C^2 >0$.} \end{cases}$$ This ensures that the curvature tends to $-\kappa^2$ as $t\to+\infty$ (exponentially fast in $t$). When $g_{t\geq 1}$ has constant curvature, we have $\tilde{K}=-\kappa^2$ for $t\geq 1$. Now, for item (4), we observe that for $C_0$ large enough, $$\tilde{K}(\sigma_{X,T}) \leq - \frac{f''}{f} + C_0,\quad \tilde{K}(\sigma_{X,Y}) \leq \frac{C_0}{f^2} - (f'/f)^2,$$ $$\tilde{K}(\sigma_{X+aT,Y}) \leq \max(\tilde{K}(\sigma_{X,T}),\tilde{K}(\sigma_{X,Y}) ) + \frac{C_0}{f}.$$ Let $t_0\in(0,1)$ be such that the function $f_{cc}$ defined in [\[eq:f-away\]](#eq:f-away){reference-type="eqref" reference="eq:f-away"} satisfies $f_{cc}\geq 1$ for $t\geq t_0$. Then taking this suggestion for $f$, not only for $t\geq 1$ but $t\geq t_0$, we get $$\tilde{K} \leq 2C_0 - \kappa^2.$$ We observe that (for example), $t_0 = 1/\sqrt{\kappa}$ satisfies the condition for $\kappa$ large. We also observe that $f_{cc}''\geq 0$ for $t\geq 0$, and $f_{cc}'/f_{cc}> C>0$ for $t>1/2$, uniformly as $\kappa$ becomes large. It remains thus to define $f$ on the interval $[0,1/\sqrt{\kappa}]$ so that $f''\geq 0$, to obtain a smooth function. The only condition for this is that $(0,f(0))=(0,1)$ is strictly above the tangent to the graph of $f_{cc}$ at $t=t_0 =1/\sqrt{\kappa}$. Accordingly, we require that $$f_{cc}(t_0) - f_{cc}'(t_0) t_0 < 1.$$ We can then check case by case that this holds for $t_0 = 1/\sqrt{\kappa}$ and $\kappa$ large enough. Let us finally check the smooth conformal compactness. For this, it suffices to set $y = e^{-\kappa t}$, to write the metric in the cylinder in the form $$\tilde{h} = \frac{1}{\kappa^2 y^2}\left( dy^2 + m(y)g_{t\geq 1}(x,dx)\right).$$ Here, the function $m$ is a smooth function of $y^2$, and $y$ is a boundary defining function. ◻ # General observations about the topology of the problem Let us collect some general informations about universal covers of manifolds with hyperbolic geodesic flow. This is classical material. We will denote by $\pi:\widetilde{M}\to M$ the universal cover of $M$, and by $\widetilde{N}$ that of $N$. Since $M$ is geodesically convex, and does not have conjugate points, we can pick $x_0\in M$ and identify $\widetilde{M}$ with the set of $u\in T_x M$ such that $\exp_x(u)\in M\simeq \mathbb{R}^n$. By this construction, we identify $\widetilde{M}$ as a geodesically convex subset of $\widetilde{N}\simeq \mathbb{R}^n$. The fundamental group $\pi_1(M)$ is realized by isometries of $\widetilde{N}$, which preserve $\partial \widetilde{M}$. In particular, $\partial\widetilde{M}$ is a Galois cover of $\partial M$. The embedding $\imath : \partial M \hookrightarrow M$ induces a map $\imath_\ast : \pi_1(\partial M)\to \pi_1(M)$. If $P$ is a connected component of $\partial M$, and $\widetilde{P}$ a connected component of $\pi^{-1}(P)$, then we have $\pi_1(\widetilde{P}) = \{ \gamma \in \pi_1(P)\ |\ \imath_\ast(\gamma) = 0\}$, and $P = \widetilde{P}/\imath_\ast(\pi_1(P))$. The elements of $\pi_1(M)$ (except $1$) are represented by a special kind of isometries, called *loxodromic*. They have a continuous extension to (Hölder) homeomorphisms of $B^n = \widetilde{N}\cup\mathbb{S}^{n-1}$ (the *visual compactification* of $\widetilde{M}$). They have exactly two fixed points, which lie in $\mathbb{S}^{n-1}$. They preserve the corresponding geodesic, along which they are a translation. In particular, if two isometries $\gamma$, $\mu$ commute, there must exist another $\eta$, and $k,\ell\in\mathbb{N}$ so that $\gamma = \eta^k$, $\mu = \eta^\ell$. This implies that there is no copy of $\mathbb{Z}^2$ inside $\pi_1(M)$. Projecting $\partial \widetilde{M}$ along rays from $x_0$ on $\mathbb{S}^{n-1}$, we find that $\partial\widetilde{M}$ is (Hölder) homeomorphic to an open set of $\mathbb{S}^{n-1}$. Its complement $\Lambda$ is called the *limit set*. It is the set of endpoints of geodesics that remain inside in $M$ for all times. From these general facts, we deduce our Theorem [Theorem 3](#thm:special-cases){reference-type="ref" reference="thm:special-cases"}. *Proof.* Let us start by observing that the following are equivalent 1. At least one connected component of $\partial \widetilde{M}$ is a sphere 2. At least one connected component of $\partial \widetilde{M}$ is compact 3. $\partial\widetilde{M}\simeq \mathbb{S}^{n-1}$ 4. $\widetilde{M}\simeq B^n$ 5. $\pi_1(M) =\{1\}$ 6. $M$ is diffeomorphic to a closed ball. Certainly, (1) implies (2), and since $\partial \widetilde{M}$ is an open set of $\mathbb{S}^{n-1}$, if it is compact, it must the whole sphere, so (2) implies (3). Using rays starting from $x_0$, we can then build a diffeomorphism between $\widetilde{M}$ and $B^n$, so that (3) implies (4). In that case, every element of $\pi_1(M)$ preserves a compact set of $\mathbb{R}^n$, so must be trivial, and (4) implies (5). If (5) holds, then $M= \widetilde{M}$ must be compact, and using again rays from $x_0$, we find that $M$ is a closed ball. Finally, if $M$ is a closed ball, its boundary is a sphere. This takes care of item (1) of Theorem [Theorem 3](#thm:special-cases){reference-type="ref" reference="thm:special-cases"}. Now, we also have equivalence between (a) $\Lambda$ has exactly two points (b) The boundary of $M$ is diffeomorphic to a bundle: $\mathbb{S}^{n-2} \to \partial M \to \mathbb{S}^{1}$. (c) One connected component of $\partial M$ is diffeomorphic to a bundle $\mathbb{S}^{n-2} \to P \to \mathbb{S}^{1}$. (d) One connected component $P$ of $\partial M$ satisfies $\imath_\ast(\pi_1(P))\simeq \mathbb{Z}$. Certainly, (a) implies that $\pi_1(M)$ is generated by a non trivial loxodromic isometry $\gamma$. In particular the boundary of $M$ is the quotient of $\mathbb{R}\times \mathbb{S}^{n-2}$ by the group generated by $\gamma$. We can use Fermi coordinates along the geodesic preserved by $\gamma$ to write $\widetilde{M} \simeq \mathbb{R}\times \mathbb{R}^{n-1}$, a decomposition that is preserved by $\gamma$, so that $\widetilde{M}$ intersected with every slice ${t}\times \mathbb{R}^{n-1}$ is star shaped with smooth boundary (and thus diffeomorphic to a ball in $\mathbb{R}^{n-1}$). This implies that $M$ is a ball bundle above the circle, and its boundary a sphere bundle above the circle. This implies (b) (which implies (c)). Now assume (c) and let $P$ be the corresponding connected component, and $\widetilde{P}$ a connected component of $\pi^{-1}(P)$. From the arguments in the ball case above, we know that $\widetilde{P}$ cannot be compact. If $n>3$, this implies $\widetilde{P} = \mathbb{R}\times \mathbb{S}^{n-2}$, and $\imath(\pi_1(P)) \simeq \mathbb{Z}$ is generated by one non trivial isometry. In the case $n=3$, we have to consider that case $\widetilde{P}\simeq \mathbb{R}^2$. However, since $\pi_1(M)$ cannot contain $\mathbb{Z}^2$, this case is ruled out, and so it is the same as $n>3$, and we have (d). Now, assuming (d), let $\imath_\ast(\pi_1(P))=<\gamma>$. Then let $c(t)$ be the geodesic preserved by $\gamma$. We find that $\widetilde{P}$ is at bounded distance from $\{c(t) | {t\in\mathbb{R}}\}$. This implies that $\partial \widetilde{P}$ (seen as a subset of $\mathbb{S}^{n-1}$) can only contain the endpoints of $c(t)$, and so this implies (a). In the case that the bundle is trivial, this gives item (b) of Theorem [Theorem 3](#thm:special-cases){reference-type="ref" reference="thm:special-cases"}. ◻ We have identified two type of particularly simple boundaries. Let us discuss now some more partial results in this direction. **Lemma 2**. *Let us assume that a boundary component $P$ satisfies $\Gamma:=\imath(\pi_1(P))= \pi_1(\Sigma)$, where $\Sigma$ is a compact Anosov manifold of dimension $p\leq n-2$. Then $P$ is the only boundary component.* *Proof.* In that case, the visual boundary $\partial \Gamma$ is homeomorphic to a sphere $\mathbb{S}^{p-1}$, and since $\Gamma$ is a hyperbolic subgroup of $\pi_1(M)$, the limit set of $\Gamma$, which must be also $\partial \tilde{P}$, is also homeomorphic to $\mathbb{S}^{p-1}$. Now, since $\mathbb{S}^{n-1}\setminus\mathbb{S}^{p-1}$ must be connected, we deduce that $\mathbb{S}^{n-1}\setminus \mathbb{S}^{p-1} = \tilde{P}$. This means that $P$ is the whole boundary of $M$. ◻ If the embedding of $\mathbb{S}^{p-1}$ into $\mathbb{S}^{n-1}$ is the standard one, we get that $\tilde{P}\simeq \mathbb{S}^{n-p-1}\times\mathbb{R}^{p}$, and $M$ turns out to be tubular neighbourhood of $\Sigma$. However, there are more than one way to embed a sphere in another, except in the case that $n> 2p$. **Corollary 1**. *If $\Sigma$ is a surface, and $M$ has dimension at least $5$, then $M$ is diffeomorphic to a ball bundle over $\Sigma$.* In 4 dimensions, understanding exactly which $B_2$ bundles over a compact surface can be endowed with a complete convex hyperbolic structure is not completely solved. This question was investigated by several authors --- see [@gromov], [@convex-plumbing]. Let us close this section with a modicum of information regarding the general case for the possible shape of the boundary. We specialize to the case of $M$ having 4 dimensions, to be able to use the geometrization theorem. For this, we will rely on several results from the theory of the topology of 3 manifolds. For example, the reader can consult [@Freitas], in particular its §3. Since it is not our main focus, we will be very cursory. **Lemma 3**. *Let $M$ be an Anosov $4$-manifold with boundary, with some boundary component $P\neq \emptyset$. Assume that $P\subset M$ is $\pi_1$-injective. Then either $P=\mathbb{S}^3$ or it decomposes as a connected sum $$A_1 \# ... \# A_p \# B_1 \# \dots \# B_\ell,$$ where each $A_j$ is a copy of $\mathbb{S}^2\times \mathbb{S}^1$, and each $B_j$ is a compact hyperbolic 3 manifold.* *Proof.* According to the prime decomposition theorem, we can decompose $$P = P_1 \# \dots \# P_m,$$ where no $P_j$ can be decomposed as a non-trivial connected sum, and (unless $P= \mathbb{S}^3$) each $P_j$ is either $\mathbb{S}^1\times\mathbb{S}^2$ or is irreducible (i.e every embedding of $\mathbb{S}^2$ bounds a ball). Now, according to the geometrization theorem, we can decompose each $P_j$ as $$P_j = \cup Q_{j,\ell},$$ where each $Q_{j,\ell}$ is a manifold whose boundary components are torii, and whose interior admits a geometric structure, in the list of Thurston's 8 geometries. Also, each torus boundary is $\pi_1$ embedded. Now, since we assumed that $P$ is $\pi_1$-injective, and $\pi_1(M)$ cannot contain a $\mathbb{Z}^2$ subgroup, we deduce that there cannot exist any incompressible torus in the boundary, and, in particular, the decomposition of the $P_j$ must be trivial: each of them supports a Thurston geometry. Now, among the Thurston geometries, we observe that for either a compact Euclidean, Nil or Sol manifold, the fundamental group must be a semi-direct product $\mathbb{Z}^2 \times \mathbb{Z}$, which contains a $\mathbb{Z}^2$. In the $\mathbb{H}^2\times\mathbb{R}$ or $\widetilde{SL(2,\mathbb{R})}$ case, the fundamental group must contain a cyclic normal subgroup, i.e a $\mathbb{Z}$ center. Since we cannot have torsion, it must also contain a $\mathbb{Z}^2$, and this is also ruled out. Next, we observe that in the spherical case, elements of the $\pi_1$ must have finite order, which is not possible because there are no elliptic elements in $\pi_1(M)$. In particular, if $P_j$ has spherical geometry, $P_j = \mathbb{S}^3$. We deduce that each $P_j$ has geometry either $\mathbb{S}^2\times\mathbb{R}$, or $\mathbb{H}^3$, or is a sphere. ◻ # The case of 3-manifolds In this section, let us concentrate on the case that $M$ is 3 dimensional and oriented. Then the boundary components are compact oriented surfaces, so either a sphere, a torus or surfaces of genus $g>1$. As we have seen, if $M$ is neither a solid torus nor a ball, all the components must be of the latter variety. The authors of [@Chen-Erchenko-Gogolev] have already commented on how to embed the torus or the ball, so we will concentrate on the remaining case. As noted in §[1](#sec:metric-extension){reference-type="ref" reference="sec:metric-extension"} we can assume that the boundary components have constant curvature. **Theorem 5**. *Let $M$ be a 3-manifold whose boundary is strictly convex, with curvature $-\kappa^2$ constant near the boundary. Assume further that all connected components of the boundary are hyperbolic surfaces of genus $g>1$. Provided the hyperbolic metric is well chosen on the boundary, $M$ can be embedded into a compact manifold $N'$ without boundary, such that the curvature is $-\kappa^2$ in $N'\setminus M$.* *Proof.* We start by recalling this result (see Theorem 3.3 in [@Fujii]). **Lemma 4**. *For $g>1$, there exists $N_g$ a compact hyperbolic 3-manifold whose boundary is a totally geodesic surface $S_g$ of genus $g$.* (As explained in [@Fujii], not all hyperbolic structures on surfaces can be realized as totally geodesic boundaries, essentially because of Mostow rigidity). Let us come back to our problem. We will from now on assume that the boundary of $M$ is connected. If there are several components, one can work component wise. Let us hence denote $\Sigma=\partial M$, and endow $\Sigma$ with a hyperbolic metric $g_{\partial N_g}$ so that Lemma [Lemma 4](#lemma:Fujii){reference-type="ref" reference="lemma:Fujii"} applies, and we also have $\Sigma = \partial N_g$ as a totally geodesic boundary. Near the boundary of $(N_g, h_0)$, we have $$\label{eq:near-boundary-N_g} N_g \simeq \Sigma_x \times [0, \delta[_\tau,\quad h_0 = d\tau^2 + \cosh(\tau)^2 g_{\partial N_g}(x,dx).$$ Let us apply the construction of §[1](#sec:metric-extension){reference-type="ref" reference="sec:metric-extension"}. For $C_0>0$ large enough, we can ensure that $C_0 g_{\partial N_g}> g_{\partial M}$, where $g_{\partial M}$ is the metric on $\Sigma\simeq \partial M$. We can thus build an extension $M\subset N$ according to Lemma [Lemma 1](#lemma:extension){reference-type="ref" reference="lemma:extension"}. In $N\setminus M$, for $1\leq t\leq 2$, the metric takes the form (for some $\kappa>0$) $$dt^2 + \frac{1}{\kappa^2}\cosh(\kappa t)^2 g_{\partial N_g}(x,dx).$$ In the formula above we recognize the expression in coordinates of the metric $h_0/\kappa^2$ (here, $\tau = \kappa t$). If the local coordinates in [\[eq:near-boundary-N_g\]](#eq:near-boundary-N_g){reference-type="eqref" reference="eq:near-boundary-N_g"} extend as far as $\tau \leq 2\kappa$, we can thus glue $N_{\{t< 2\}}$ with $N_g\setminus \Sigma\times[0,2\kappa[$, and this will conclude the proof of Theorem [Theorem 5](#thm:compact-embedding-geometry){reference-type="ref" reference="thm:compact-embedding-geometry"} setting $$N' = N_{\{t<2\}}\cup (N_g\setminus \Sigma\times [0,2\kappa[).$$ The difficulty here is that for the dynamical arguments of [@Chen-Erchenko-Gogolev] to apply, we need to be able to take $\kappa$ arbitrarily large. We will thus be done if we can prove **Lemma 5**. *For any $\kappa>0$, we can choose $N_g$ such that a $2\kappa$ neighbourhood of $\partial N_g$ is diffeomorphic to $\partial N_g \times [0,2\kappa[$.* The argument of the proof was communicated by C. Lecuire. It relies on the following very fine statement from the topology of hyperbolic manifolds **Theorem 6** (Theorem 9.2, [@Agol2012]). *Fundamental groups of compact 3 dimensional hyperbolic manifolds are LERF (locally extended residually finite).* This means that whenever $H\subset \pi_1(M)$ is finitely generated, and $\gamma\in \pi_1(M)\setminus H$, there exists a finite index subgroup $\Gamma\subset\pi_1(M)$ such that $H\subset \Gamma$ and $\gamma\notin \Gamma$. Let us come back to *Proof of Lemma [Lemma 5](#lemma:good-N_g){reference-type="ref" reference="lemma:good-N_g"}.* We will build $N_g$ by induction, so we start by denoting $N_g^0$ the one provided by Lemma [Lemma 4](#lemma:Fujii){reference-type="ref" reference="lemma:Fujii"}. Let us now consider geodesics of $N_g^0$ with endpoints in its boundary. We will say that two such geodesics are boundary-homotopic, if there is a homotopy between them, so that at each time, the endpoints remain in the boundary. Working on the universal cover, it is elementary to observe that any such geodesic is either boundary homotopic to a point in the boundary, or to a geodesic that intersects the boundary orthogonally. Additionally, such a boundary-orthogonal geodesic uniquely minimizes the length among its boundary-homotopy class. Let us set $$\mathfrak{G}= \{ k a \ |\ k\in\mathbb{N},\ \text{some boundary-orthogonal geodesic has length $a$.} \}$$ Certainly, $\mathfrak{G}$ is a discrete set, and its only accumulation point is $+\infty$. We can enumerate it as $$\mathfrak{G}= \{ 2 \ell_j \ |\ j\geq 0\}.$$ With a bit more of elementary riemannian geometry, we find that $\ell_0$, the largest $\ell$ such that the $\ell$-neighbourhood of $\Sigma$ is product, is exactly half the length of a shortest boundary-orthogonal geodesic (there may be several non-homotopic orthogonal geodesics with the same length). Let us denote $m_0>0$ the number of boundary-orthogonal geodesic with length $2\ell_0$, and $\gamma_0$ one of them. Let us now consider the double $A_0 = DN_g^0$ of $N_g^0$ along $\Sigma$, and $D\gamma_0$ the double of $\gamma_0$, a closed geodesic of length $4\ell$. Using the LERF theorem, we find a finite index subgroup $\Gamma\subset \pi_1(A_0)$ so that $\pi_1(\Sigma) \subset \Gamma$, and $[D\gamma_0]\notin \Gamma$. Let us denote $B_0 = \mathbb{H}^3/\Gamma$ the corresponding cover of $A_0$. The geodesic $D\gamma_0$ is lifted to a closed geodesic $\gamma'_0$ of length at least $2 \times 2\ell$. Let us pick a lift $\Sigma\subset B_0$, intersecting $\gamma'_0$. Actually, $\gamma'_0$ only cuts $\Sigma$ twice. We cut $B_0$ along $\Sigma$, which cuts $\gamma'_0$ into two pieces. There are two sides to this cut, and we call "side A" the side corresponding to the longest piece of $\gamma'_0$. To the other side, we glue a copy of the original $N_g^0$. We obtain thus a manifold with $\Sigma$ as boundary, that we rename $N_g^{0,1}$. (Possibly, it may be non-connected, so we only keep the connected component that has a boundary). It is a cover of $N_g^0$. Let us now consider boundary-orthogonal geodesics of $N_g^{0,1}$. They cover boundary-orthogonal geodesics of $N_g^0$. On the other hand, we can lift those of $N_g^0$ uniquely to ones of $N_g^{0,1}$, so that there is a bijection between the two sets. Let us now consider the ones of length $2\ell_0$. We have ensured that the geodesic above $\gamma_0$ has length $\geq 4 \ell_0$, so that there are at most $m_0 - 1$ such geodesics. We can thus apply the same procedure to $N_g^{0,1}$, to obtain $N_g^{0,2}$, etc, until there are no boundary-orthogonal geodesics of length $2\ell_0$ left; we are sure this will end at the latest at the step $m_0$. We denote by $N_g^1$ the corresponding manifold. $N_g^1$ is a connected hyperbolic 3-manifold with boundary, whose boundary is a totally geodesic copy of $\Sigma$. Additionally, the lengths of its boundary-orthogonal geodesics are contained in $\mathfrak{G}$, and they are strictly larger than $2\ell_0$. So they must be at least $2\ell_1$. Proceeding by induction, we can thus construct a manifold whose boundary is a totally geodesic copy of $\Sigma$, and whose shortest boundary-orthogonal geodesic has length as large as desired. For example strictly larger than $4\kappa$. This implies that a $2\kappa$ neighbourhood of $\Sigma$ is product. ◻ This closes the proof of Theorem [Theorem 5](#thm:compact-embedding-geometry){reference-type="ref" reference="thm:compact-embedding-geometry"} ◻ Observe that there are examples of compact hyperbolic 4-manifolds whose $\pi_1$ is not LERF. One of the incarnations of the fact that our problem here becomes much more difficult when the dimension increases. Let us conclude the proof of the main theorem [Theorem 4](#thm:3D){reference-type="ref" reference="thm:3D"}. *Proof of Theorem [Theorem 4](#thm:3D){reference-type="ref" reference="thm:3D"}.* We have already embedded isometrically $M\subset N'$, so that the curvature of $N'$ satisfies 1. $K \leq C_0$ globally, 2. $K \leq - \kappa^2 /2$ on $N'\setminus V$, where $V$ is the $1/\sqrt{\kappa}$ neighbourhood of $M$. We have seen that $C_0$ is fixed, and $\kappa$ can be taken arbitrarily large. We have also ensured that the slices are uniformly strictly convex. In the arguments below, $C>0$ will denote a constant that does not depend on the choice $\kappa$, and that may change at every line. Except from this, we will use the notations for Jacobi fields introduced in [@Chen-Erchenko-Gogolev]. Observe that their $1/C_1$ corresponds to our $1/\sqrt{\kappa}$. The key technical tool is the comparison lemma 2.8 in [@Chen-Erchenko-Gogolev]. In particular, it tells us that in $N'\setminus V$, in the region $t>1/\sqrt{\kappa}$, where the curvature is below $-\kappa^2/2$, a Jacobi field with $\mu_J(0)> - Q$ satisfies $\mu_J(t)>(1-\epsilon)\kappa/\sqrt{2}$ for $t\gtrsim 1$, when $\kappa$ is large and $Q$ remains fixed. We will rely on their arguments of §8, trying to give some detail. Let us first sketch the proof of **Lemma 6**. *Let $v\in \partial_- SM$ and $J$ be a perpendicular Jacobi field along the corresponding geodesic. If $\mu_J(0)>Q_M$, then $J$ does not vanish in positive time.* *Proof.* According to Lemma 8.5 in [@Chen-Erchenko-Gogolev], the time travel in the region $t\in [0,1/\sqrt{\kappa}]$ is bounded above by $C/\sqrt{\kappa}$. If $J$ a perpendicular Jacobi field along a geodesic $\gamma_v$ starting at a point $v\in SN'$, let us assume that $v\in \partial SM$ is entering. Then according to Proposition 4.1 of [@Chen-Erchenko-Gogolev], if $\mu_J(0) > Q_M$, either $v\in\Gamma_-$ and we are done, or $\gamma\notin \Gamma_-$. Then as $\gamma_v$ leaves $SM$, $\mu_J(\ell(v)) > - Q_M$. Since the travel time in the collar is small and the curvature bounded above, we deduce that $\mu_J$ is at least $-2Q_M$ when $\gamma_v$ enters $\{t>1/\sqrt{\kappa}\}$. Now, in this region, the curvature is bounded above by $-\kappa^2/2$, so that applying the comparison lemma 2.8 of [@Chen-Erchenko-Gogolev], and taking into account that the travel time inside $t>1/\sqrt{\kappa}$ must be at least $1$, if $\kappa$ is large enough, $\mu_J$ must be at least $Q_M$ again when $\gamma_v$ enters again $SM$ (if it ever does). We can conclude by induction on the times. ◻ **Lemma 7**. *$N'$ has no conjugate points* *Proof.* Let us consider a nonzero Jacobi field $J$ that vanishes at some point above $N'\setminus M$. Then, using again the comparison lemma, we deduce that $\mu_J> Q_M$ as long as the geodesic remains in $SN'\setminus SM$, and thus cannot vanish again according to our lemma. Let us now consider a nonzero Jacobi field vanishing at some point. If the corresponding geodesic remains in $SM$, then we are done. Otherwise consider the first time that it exits $SM$. Then, using the argument of the proof of Lemma 8.11 of [@Chen-Erchenko-Gogolev], we must have $\mu_J> - Q_M$. Otherwise we could apply Proposition 4.1 in reversed time, and obtain a contradiction. Again by the smallness of the collar and the very negative curvature in $\{t>1/\sqrt{\kappa}\}$, if the geodesics come back again in $SM$, we must have $\mu_J> Q_M$ at the point of entry. Then proceeding by induction as above enables us to conclude. ◻ Finally, we have to prove that the geodesic flow is Anosov. For this it suffices, according to Eberlein's theorem, to prove that no nonzero Jacobi field can be globally bounded. Since $M$ has axiom A geodesic flow and the curvature is very negative in $\{t>1/\sqrt{\kappa}\}$, this is a given for any Jacobi field along a geodesic that remains either in $SM$ or $SN'\setminus SM$ for all times. Consider a geodesic $\gamma$ entering $SM$ at a point $v$, with a perpendicular Jacobi field $J$ satisfying $\mu_J> Q_M$ at that point. Then according to Proposition 4.1 of [@Chen-Erchenko-Gogolev], we have either $\|J\|\to+\infty$, or $\int_{SM} \mu_J > -C_0$ before $\gamma$ exits $SM$. Now, before the geodesic enters again (if it ever does) in $SM$, since the curvature is so negative, using the comparison lemma again, we must have $$\int_{SN\setminus SM} \mu_J\geq (1-\epsilon) \frac{\kappa}{\sqrt{2}} - \epsilon,$$ where $\epsilon$ tends to $0$ as $\kappa$ grows large. Here we have taken into account that the travel time must be at least $1$ above $N'\setminus M$. We also used Proposition 4.1, ensuring that $\mu_J>-Q_M$ entering $SN'\setminus SM$. Now, either the sequence of times when the geodesic changes component is finite, and we know directly by the comparison lemma and Proposition 4.1 that $|J|\to+\infty$, or the sequence is infinite. In that case, we observe that between two consecutive times the geodesic entered $SM$, we have $$\int \mu_J > \frac{1}{10}\kappa - C_0 > 1,$$ so that $\|J\|\to+\infty$ also in positive time. Let us now consider the case that as $\gamma$ enters $SM$, $\mu_J \leq Q_M$. Then, we reverse time, and see this as a geodesic entering $SN'\setminus SM$ with $\mu_J \geq - Q_M$. Then the same argument as above apply, only in negative time. ◻ [^1]: One needs that $f''\geq 0$, $f'\geq 0$ and $f\geq 1$. Since $f=1$ on $]-\epsilon, 0]$ and is smooth, it suffices to assume that $f''\geq 0$

arxiv_math

--- abstract: | In this paper, we propose a reduced-dimensional smoothed particle hydrodynamics (SPH) formulation for quasi-static and dynamic analyses of plate and shell structures undergoing finite deformation and large rotation. By exploiting Uflyand--Mindlin plate theory, the present surface-particle formulation is able to resolve the thin structures by using only one layer of particles at the mid-surface. To resolve the geometric non-linearity and capture finite deformation and large rotation, two reduced-dimensional linear-reproducing correction matrices are introduced, and weighted non-singularity conversions between the rotation angle and pseudo normal are formulated. A new non-isotropic Kelvin-Voigt damping is proposed especially for the both thin and moderately thick plate and shell structures to increase the numerical stability. In addition, a shear-scaled momentum-conserving hourglass control algorithm with an adaptive limiter is introduced to suppress the mismatches between the particle position and pseudo normal and those estimated with the deformation gradient. A comprehensive set of test problems, for which the analytical or numerical results from literature or those of the volume-particle SPH model are available for quantitative and qualitative comparison, are examined to demonstrate the accuracy and stability of the present method. address: - TUM School of Engineering and Design, Technical University of Munich, 85748 Garching, Germany - Huawei Technologies Munich Research Center, 80992 Munich, Germany author: - Dong Wu - Chi Zhang - Xiangyu Hu bibliography: - IEEEabrv.bib - mybibfile.bib title: An SPH formulation for general plate and shell structures with finite deformation and large rotation --- SPH ,Uflyand--Mindlin plate theory ,Finite deformations ,Geometric non-linearity ,Thin and moderately thick plate/shell structures ,Quasi-static and dynamic analyses ,Reduced-dimensional linear-reproducing correction matrices ,Non-singularity ,Non-isotropic damping ,Hourglass modes # Introduction For computational continuum dynamics, as alternatives to conventional mesh-based methods, e.g. finite element method (FEM) and finite volume method (FVM), meshless methods have flourished in the past decades [@belytschko1996meshless; @liu2005introduction; @liu2010smoothed; @zhang2022smoothed]. Smoothed particle hydrodynamics (SPH), initially developed by Lucy [@lucy1977numerical] and Gingold and Monaghan [@gingold1977smoothed] for astrophysical simulations, is one typical example. In SPH, the continuum is modeled by particles associated with physical properties such as mass and velocity, and the governing equations are discretized in the form of particle interactions using a Gaussian-like kernel function [@monaghan2005smoothed; @liu2010smoothed; @monaghan2012smoothed]. Since a significant number of physical system abstractions can be realized through particle interactions, SPH has been used to model multi-physical systems within a unified computational framework [@zhang2021sphinxsys], which is able to achieve seamless monolithic, strong and conservative coupling [@matthies2003partitioned; @matthies2006algorithms]. To achieve such a unified computational framework, it is crucial to discretize all relevant physics equations using effective and efficient SPH methods. In the case of plate and shell structures which are omnipresent thin structures in scientific and engineering fields such as shipbuilding [@cerik2019simulation; @peng2019meshfree], aerospace [@totaro2009optimal], and medical treatment [@laubrie2020new], etc., the traditional full-dimensional or volume-particle SPH method, is not computationally efficient [@cleary1998modelling]. Since there are well-developed and matured reduced-dimensional theories, such as Kirchhoff-Love [@love1887small] and Uflyand-Mindlin (or Mindlin-Reissner) [@uflyand1948wave; @mindlin1951influence; @elishakoff2017vibrations; @elishakoff2020handbook], for plate and shell structures based on mid-surface reconstruction, it is expected to develop the computationally much more efficient reduced-dimensional or surface-particle SPH method with a single-layer of particles only. The early meshless methods for plates and shells were based on Petrov or element-free Galerkin formulation [@krysl1996analysis; @li2000numerical; @rabczuk2007meshfree; @li2008numerical], or the reproducing kernel particle method [@donning1998meshless; @chen2006constrained; @peng2018thick]. As for SPH, Maurel and Combescure [@maurel2008sph] first developed a surface-particle SPH method for total Lagrangian quasi-static and dynamic analyses of moderately thick plates and shells based on the Uflyand-Mindlin theory and the assumption of small deformation. In their work, besides an artificial viscosity term to alleviate numerical instability issues, a stress point method is applied to temper hourglass or zero-energy modes which exhibit in the traditional SPH method using collocated particles for both deformation and stress. While being effective on preventing zero-energy modes, using stress points may faces several issues, such as how to locate or generate these points for complex geometries, complicated numerical algorithms and the compensation of computational efficiency [@ganzenmuller2015hourglass; @wu2023essentially]. Nevertheless, this method was later applied in large deformation analyses by Ming et al . [@ming2013robust] and dynamic damage-fracture analyses by Caleyron et al. [@caleyron2012dynamic]. Lin et al. [@lin2014efficient] developed a similar method for quasi-static analyses, but applied an artificial viscosity term based on membrane and shearing decomposition. Ming et al. [@ming2015smoothed] first considered finite deformation by taking all strain terms into account with the help of Gauss-Legendre quadrature for more accurately capturing of non-linear stress. In all of the surface-particle SPH methods mentioned above, the rotation angles of mid-surface are directly obtained from the pseudo normal in governing equations under the assumption of small rotation. In this work, we propose a collocated surface-particle SPH formulation for total Lagrangian quasi-static and dynamic analyses of general plate or shell structures, which may be thin or have moderate thickness, involving finite deformation or/and large rotation. First, to better resolve the geometric non-linearity induced by finite deformation and large rotation, two new reduced-dimensional correction matrices for linearly reproducing position and normal direction are introduced, and a weighted conversion algorithm, which achieves non-singularity under large rotation, is proposed. Second, a new non-isotropic Kelvin-Voigt damping base on Ref. [@zhang2022artificial] is proposed for achieve good numerical stability for both thin and moderately thick plate or shell structures. Third, in order to address hourglass modes using collocated particles only other than introducing extra stress points, drawing the inspiration from Refs. [@kondo2010suppressing; @ganzenmuller2015hourglass], a shear-scaled momentum-conserving formulation with an adaptive limiter is developed by mitigating the discrepancy between the actual particle position and pseudo normal and those estimated by the deformation gradient. A set of numerical examples involving quasi-static and dynamic analyses for both thin and moderately thick plate or shell structures are given. The results are compared with analytical, numerical solutions in literature or/and those obtained by the volume-particle SPH method to demonstrate the numerical accuracy and stability of the present method. The remainder of this manuscript is organized as follows. Section [2](#sec:governingeq){reference-type="ref" reference="sec:governingeq"} introduces the theoretical model of plates and shells, including the kinematics, constitutive relation, stress correction and conservation equations. The proposed surface-particle SPH formulation, including the reduced-dimensional linear-producing correction matrices, weighted conversion algorithm, non-isotropic damping and momentum-conserving hourglass control, is described in Section [3](#sec:SPH_method){reference-type="ref" reference="sec:SPH_method"}. Numerical examples are presented and discussed in Section [4](#sec:examples){reference-type="ref" reference="sec:examples"} and then concluding remarks are given in Section [5](#sec:conclusion){reference-type="ref" reference="sec:conclusion"}. For a better comparison and future opening for in-depth studies, all the computational codes of this work are released in the open-source repository of SPHinXsys [@zhang2020sphinxsys; @zhang2021sphinxsys] at `https://github.com/Xiangyu-Hu/SPHinXsys`. indicating the parameter $\left( \bullet \right)$ is defined at the initial configuration and global coordinate system indicating the parameter $\left( \bullet \right)$ is defined at the initial configuration and initial local coordinate system indicating the parameter $\left( \bullet \right)$ is defined at the current configuration and current local coordinate system global coordinate system initial local coordinate system current local coordinate system # Theoretical models {#sec:governingeq} We first introduce the theoretical mode of 3D plate, and then that of 3D shell in which material points may possess different initial normal directions leading to different initial local coordinate systems. After that, we briefly describe the 2D plate and shell models, which resolve the plane strain problem, as a simplification of the 3D counterparts. ## 3D plate model We consider the Uflyand--Mindlin plate theory [@uflyand1948wave; @mindlin1951influence] to account for transverse shear stress which is significant for moderately thick plates. The theory implies that the plate behavior can be represented by one layer of material points at its mid-surface, as shown in Figure [1](#figs:discretization_plate){reference-type="ref" reference="figs:discretization_plate"}. ### Kinematics We introduce $\bm{X} = \left( X, Y, Z \right)$ to represent the global coordinate system, and $\bm{\xi} = \left( \xi,\eta, \zeta \right)$ and $\bm{x} = \left( x, y, z \right)$, associated with so-called pseudo-normal vector $\bm{n}$, to denote the initial and current local coordinate systems, respectively. Note that the initial local coordinate system is same with the global one for plate. Each material point possesses five degrees of freedom, viz., three translations $\bm{u} = \left\{ u,v,w \right\}^{\operatorname{T}}$ and two rotations $\bm{\theta} = \left\{ \theta, \varphi \right\}^{\operatorname{T}}$ expressed in the global coordinates. Positive values of $\theta$ and $\varphi$ indicate that the plate is rotated anticlockwise around the coordinate axis when the axis points toward the observer and the coordinate system is right-handed. The two rotations are used to update the pseudo-normal $\bm{n} = \left\{ n_{1},n_{2},n_{3} \right\}^{\operatorname{T}}$ which is also defined in the global coordinate system and remains straight but is not necessarily perpendicular to the mid-surface, i.e., the pseudo normal may be different with the real normal $\bm{n}_r$, as shown in Figure [1](#figs:discretization_plate){reference-type="ref" reference="figs:discretization_plate"}. Note that $\bm{n}^0 = \left\{ 0, 0, 1 \right\}^{\operatorname{T}}$ denotes the pseudo-normal in the initial configuration with the superscript $\left( \bullet \right)^0$ denoting the initial configuration. {#figs:discretization_plate width="90%"} For a 3D plate, the position $\bm{r}$ of a material point at a distance $\chi$ away from the mid-surface along the pseudo normal $\bm n$ can be expressed as $$\bm{r}\left( \xi ,\eta ,\chi, t \right) = \bm{r}_m \left( \xi ,\eta, t \right) + \chi \bm{n} \left( \xi ,\eta, t \right), \quad \chi \in \left[- d/2, d/2 \right], \label{eq: position}$$ where $d$ is the thickness, $\bm{r}_m$ the position of the material point at the mid-surface with the subscript $\left( \bullet \right)_m$ denoting the mid-surface. Note that since the thickness is assumed to be constant during deformation and the pseudo normal $\bm n$ represents the plate thickness direction, the distance $\chi$ is always between $- d/2$ and $d/2$. Therefore, the displacement $\bm{u}$ of the material point can be determined by $$\bm{u} \left( \xi ,\eta ,\chi, t \right) = \bm{u}_m \left( \xi ,\eta, t \right) + \chi \Delta \bm{n} \left( \xi ,\eta, t \right),$$ where $\Delta \bm{n} = \bm{n} - \bm{n}^0$. representing the mid-surface parameter Then we can define the deformation gradient tensor as $$\label{deformation_tensor} \mathbb{F} = \nabla^0 \bm{r} = \nabla^0 \bm{u} + \mathbb{I} = \left( \bm{a}_1, \bm{a}_2, \bm{a}_3 \right),$$ where $\nabla^0 \equiv \partial / \partial \bm{\xi}$ is the gradient operator with respect to the initial configuration, $\mathbb{I}$ the identity matrix, and $\bm{a}_1$, $\bm{a}_2$, $\bm{a}_3$ are specified by $$\begin{cases} \bm{a}_1 = \bm{r}_{m,\xi} + \chi \bm{n}_{\xi} \\ \bm{a}_2 = \bm{r}_{m,\eta} + \chi \bm{n}_{\eta}\\ \bm{a}_3 = \bm{n} \end{cases}$$ with $\nabla^0 \bm{r}_{m} \equiv (\bm{r}_{m,\xi}, \bm{r}_{m,\eta})^{\operatorname{T}}$ and $\nabla^0 \bm{n} \equiv (\bm{n}_{\xi}, \bm{r}_{\eta})^{\operatorname{T}}$. The deformation gradient tensor can be decomposed into two components as $$\mathbb{F} = \mathbb{F}_m + \chi \mathbb{F}_n,$$ where $\mathbb{F}_m = \left( \bm{r}_{m,\xi}^{\operatorname{T}}, \bm{r}_{m,\eta}^{\operatorname{T}}, \bm{n}^{\operatorname{T}} \right)$ and $\mathbb{F}_n = \left( \bm{n}_{\xi}^{\operatorname{T}}, \bm{n}_{\eta}^{\operatorname{T}}, 0 \right)$. Furthermore, the real normal $\bm{n}_r$ is given as $$\bm{n}_r = \frac{\bm{r}_{m,\xi} \times \bm{r}_{m,\eta}}{\left|\bm{r}_{m,\xi} \times \bm{r}_{m,\eta}\right|}.$$ ### Constitutive relation {#sec:constitutive_relation} With the deformation gradient tensor $\mathbb{F}$, the Green-Lagrangian strain tensor $\mathbb{E}$ can be obtained as $$\label{Lagrangian-strain} \mathbb{E} = \frac{1}{2} \left(\mathbb{F}^{\operatorname{T}}\mathbb{F} - \mathbb{I}\right) = \frac{1}{2} \left(\mathbb{C} - \mathbb{I}\right),$$ where $\mathbb{C}$ is the right Cauchy deformation gradient tensor. The Eulerian Almansi strain can be converted from $\mathbb{E}$ as $$\label{Almansi_strain} {\color{white}\contour{black}{$\epsilon$}} = \mathbb{F}^{-\operatorname{T}} \cdot \mathbb{E} \cdot \mathbb{F}^{-1} = \frac{1}{2} \left(\mathbb{I} - \mathbb{F}^{-\operatorname{T}} \mathbb{F}^{-1}\right).$$ When the material is linear and isotropic, the Cauchy stress reads $$\label{constitutive_relation} \begin{split} {\color{white}\contour{black}{$\sigma$}} &= K \operatorname{tr}\left({\color{white}\contour{black}{$\epsilon$}} \right) \mathbb{I} + 2G\left( {\color{white}\contour{black}{$\epsilon$}} - \frac{1}{3} \operatorname{tr}\left({\color{white}\contour{black}{$\epsilon$}} \right) \mathbb{I} \right) \\ &= \lambda\operatorname{tr}\left( {\color{white}\contour{black}{$\epsilon$}} \right) \mathbb{I} + 2\mu {\color{white}\contour{black}{$\epsilon$}}, \\ \end{split}$$ where $\lambda$ and $\mu$ are the Lamé constants, $K = \lambda + 2\mu /3$ the bulk modulus and $G = \mu$ the shear modulus. The relationship between the two moduli is given by $$E = 2G \left( 1 + \nu \right) = 3K\left( 1 - 2\nu \right),$$ where $E$ denotes the Young's modulus and $\nu$ the Poisson's ratio. ### Stress correction {#sec:stress_correction} As the thickness is significantly less than the length and width of plate, the following boundary conditions hold when the plate is free from external forces on its surfaces where $\chi = \pm \frac{d}{2}$ or $z = \pm \frac{d}{2}$ $$\label{shear_stress_correction} \left. \sigma^l_{xz}\right|_{z = \pm \frac{d}{2}} = 0, \quad \left. \sigma^l_{yz}\right|_{z = \pm \frac{d}{2}} = 0,$$ $$\label{plane_stress} \left. \sigma^l_{zz}\right|_{z \in \left[- \frac{d}{2} , \frac{d}{2}\right]} = 0,$$ with the superscript $\left( \bullet \right)^l$ denoting the current local coordinates. Taking the boundary condition Eq. [\[plane_stress\]](#plane_stress){reference-type="eqref" reference="plane_stress"} and constitutive Eq. [\[constitutive_relation\]](#constitutive_relation){reference-type="eqref" reference="constitutive_relation"} into account, the following relation of strains holds [@donning1998meshless] $$\label{strain_relation} \bar \epsilon^l_{zz}=\frac{-\nu \left( \epsilon^l_{xx} + \epsilon^l_{yy} \right)}{1-\nu},$$ where the current local strain ${\color{white}\contour{black}{$\epsilon$}}^l$ is obtained by $${\color{white}\contour{black}{$\epsilon$}}^l = \mathbb{Q} {\color{white}\contour{black}{$\epsilon$}} \mathbb{Q}^{\operatorname{T}}.$$ Here, $\mathbb{Q}$ is the orthogonal transformation matrix from the global to current local coordinates. Following Batoz and Dhatt [@batoz1990modelisation], $\mathbb{Q}$ can be given as $$\label{3D_transformation_matrix} \mathbb{Q} = \begin{bmatrix} n_3 + \frac{(n_2)^2 }{1 + n_3 } & - \frac{n_1 n_2}{1 + n_3 } & - n_1 \\ - \frac{n_1 n_2 }{1 + n_3 } & n_3 + \frac{(n_1)^2 }{1 + n_3 }& -n_2 \\ n_1 & n_2 & n_3 \\ \end{bmatrix}.$$ To satisfy the boundary conditions of Eq. [\[shear_stress_correction\]](#shear_stress_correction){reference-type="eqref" reference="shear_stress_correction"}, the transverse shear stress should be corrected as [@wisniewski2010finite] $$\label{shear_correction} \bar\sigma^l_{xz} = \bar\sigma^l_{zx} = \kappa \sigma^l_{xz}, \quad \bar\sigma^l_{yz} = \bar\sigma^l_{zy} = \kappa \sigma^l_{yz},$$ where $\kappa$ denotes the shear correction factor which is typically set to $5/6$ for the rectangular section of the isotropic plate. Taking the corrected strain $\bar{{\color{white}\contour{black}{$\epsilon$}}}^l$ into constitutive Eq. [\[constitutive_relation\]](#constitutive_relation){reference-type="eqref" reference="constitutive_relation"} and then applying Eq. [\[shear_correction\]](#shear_correction){reference-type="eqref" reference="shear_correction"}, the corrected current local Cauchy stress $\bar{{\color{white}\contour{black}{$\sigma$}}}^l$ is obtained. ### Conservation equations The mass conservation equation can be written as $$\label{mass-conservation} \rho = J_m^{-1}\rho^0,$$ where $J_m = \det(\mathbb{F}_m)$, $\rho^0$ and $\rho$ represent the initial and current densities, respectively. The momentum conservation equation is $$\label{momentum-conservation1} \rho \ddot {\bm{u}}^l = \nabla \cdot \left(\bar{ {\color{white}\contour{black}{$\sigma$}}}^l\right)^{\operatorname{T}}$$ or $$\label{momentum-conservation2} \rho \begin{bmatrix} \ddot u^l \\ \ddot v^l \\ \ddot w^l \end{bmatrix} = \begin{bmatrix} \frac{\partial \bar \sigma_{xx}^l}{\partial x} + \frac{\partial \bar \sigma_{xy}^l}{\partial y} + \frac{\partial \bar \sigma_{xz}^l}{\partial z} \\ \frac{\partial \bar \sigma_{y x}^l}{\partial x} + \frac{\partial \bar \sigma_{yy}^l}{\partial y} + \frac{\partial \bar \sigma_{yz}^l}{\partial z} \\ \frac{\partial \bar \sigma_{z x}^l}{\partial x} + \frac{\partial \bar \sigma_{zy}^l}{\partial y} + \frac{\partial \bar \sigma_{zz}^l}{\partial z} \end{bmatrix}.$$ With Eqs. [\[shear_stress_correction\]](#shear_stress_correction){reference-type="eqref" reference="shear_stress_correction"} and [\[plane_stress\]](#plane_stress){reference-type="eqref" reference="plane_stress"}, we can integrate Eq. [\[momentum-conservation2\]](#momentum-conservation2){reference-type="eqref" reference="momentum-conservation2"} along $\chi$ or $z \in \left[- d/2, d/2 \right]$ as $$\label{momentum-conservation-integrated} d \rho \begin{bmatrix} \ddot u_m^l \\ \ddot v_m^l \\ \ddot w_m^l \end{bmatrix} = \begin{bmatrix} \frac{\partial N_{xx}^l}{\partial x} + \frac{\partial N_{xy}^l}{\partial y} \\ \frac{\partial N_{yx}^l}{\partial x} + \frac{\partial N_{yy}^l}{\partial y}\\ \frac{\partial N_{zx}^l}{\partial x} + \frac{\partial N_{zy}^l}{\partial y} \end{bmatrix},$$ where the stress resultant $\mathbb{N}^l$ is calculated by the Gauss--Legendre quadrature rule as $$\label{gaussian_quadrature_rule1} \mathbb{N}^l = \int_{-d/2}^{d/2} \bar {{\color{white}\contour{black}{$\sigma$}}}^l \left(z\right) dz = \sum\limits_{ip = 1}^N \bar {{\color{white}\contour{black}{$\sigma$}}}^l \left(z_{ip}\right) A_{ip}.$$ Here, $z_{ip}$ is the integral point, $A_{ip}$ the weight, and $N$ the number of the integral point. Since the quadrature rule is conducted to yield an exact result for polynomials of degree $2 N - 1$ or lower [@gil2007numerical], $N$ is determined by the applied constitutive relation. By multiplying both sides of Eq. [\[momentum-conservation1\]](#momentum-conservation1){reference-type="eqref" reference="momentum-conservation1"} by $z$ and integrating along $z \in \left[- d/2, d/2 \right]$, the angular momentum conservation equation can be obtained as $$\label{angular_momentum_conservation_integrated} \frac{d^3}{12} \rho \begin{bmatrix} \ddot n_1^l \\ \ddot n_2^l \\ \ddot n_3^l \end{bmatrix} = \begin{bmatrix} \frac{\partial M_{xx}^l}{\partial x} + \frac{\partial M_{xy}^l}{\partial y} \\ \frac{\partial M_{yx}^l}{\partial x} + \frac{\partial M_{yy}^l}{\partial y} \\ \frac{\partial M_{zx}^l}{\partial x} + \frac{\partial M_{zy}^l}{\partial y} \\ \end{bmatrix} + \begin{bmatrix} -N_{xz}^l \\ -N_{yz}^l \\ 0 \end{bmatrix},$$ where the moment resultant $\mathbb{M}^l$ is calculated as $$\label{gaussian_quadrature_rule2} \mathbb{M}^l = \int_{-d/2}^{d/2} z \bar {{\color{white}\contour{black}{$\sigma$}}}^l \left(z\right) dz = \sum\limits_{ip = 1}^N z_{ip} \bar {{\color{white}\contour{black}{$\sigma$}}}^l \left(z_{ip}\right) A_{ip}.$$ Note that $$\int_{-d/2}^{d/2} z \frac{\partial \bar \sigma_{xz}^l}{\partial z} dz = {\Big [}z \bar \sigma_{xz}^l {\Big ]}_{-d/2}^{d/2} - \int_{-d/2}^{d/2} \bar \sigma_{xz}^l dz = -N_{xz}^l.$$ Therefore, the two governing equations, including the evolution of mid-surface displacement and pseudo normal, respectively, for the 3D plate can be described as $$\begin{cases} d\rho \bm{\ddot u}_m^l = \nabla^l \cdot \left(\mathbb{N}^l\right) ^{\operatorname{T}} \\ \frac{d^3}{12}\rho \bm{\ddot n}^l = \nabla^l \cdot \left(\mathbb{M}^l\right)^{\operatorname{T}}+ {\bf{Q}}^l, \end{cases}$$ where $$\label{resultants} \mathbb{N}^l = \begin{bmatrix} N_{xx}^l & N_{xy}^l & 0 \\ N_{yx}^l & N_{yy}^l & 0 \\ N_{zx}^l & N_{zy}^l & 0 \\ \end{bmatrix}, \mathbb{M}^l = \begin{bmatrix} M_{xx}^l & M_{xy}^l & 0 \\ M_{yx}^l & M_{yy}^l & 0 \\ M_{zx}^l & M_{zy}^l & 0 \\ \end{bmatrix}, {\bf{Q}}^l = \begin{bmatrix} -N_{xz}^l \\ -N_{yz}^l \\ 0 \\ \end{bmatrix}.$$ In total Lagrangian formulation, the conservation equations above are converted into $$\label{plate-conservation-equation} \begin{cases} d\rho ^0 \bm{\ddot u}_m = \left(\mathbb{F}_m\right)^{\operatorname{-T}} \nabla^0 \cdot \left( J_m \mathbb{N}^{\operatorname{T}} \right) \\ \frac{d^3}{12}\rho^0 \bm{\ddot n} = \left(\mathbb{F}_m\right)^{\operatorname{-T}} \nabla^0 \cdot \left( J_m \mathbb{M}^{\operatorname{T}} \right) + J_m \mathbb{Q}^{\operatorname{T}}{\bf{Q}}^l, \end{cases}$$ where $\mathbb{N} = \mathbb{Q}^{\operatorname{T}} \mathbb{N}^l \mathbb{Q}$ and $\mathbb{M} = \mathbb{Q}^{\operatorname{T}} \mathbb{M}^l \mathbb{Q}$ are the stress and moment resultants, respectively, in global coordinates. ## 3D shell model Based on the 3D plate model, the 3D shell model is obtained by introducing the initial local coordinate system and the transformation matrix from the global to initial local coordinate system. ### Kinematics The kinematics of shell can be constructed in the initial local coordinates denoted with the superscript $\left( \bullet \right)^L$. Each material point possesses five degrees of freedom, viz., three translations $\bm{u}^L = \left\{ u^L, v^L, w^L \right\}^{\operatorname{T}}$ and two rotations $\bm{\theta}^L = \left\{ \theta^L, \varphi^L\right\}^{\operatorname{T}}$ as shown in Figure [2](#figs:discretization_shell){reference-type="ref" reference="figs:discretization_shell"}. The pseudo-normal vector is also presented in initial local coordinates by $\bm{n}^L = \left\{ n^L_1,n^L_2,n^L_3 \right\}^{\operatorname{T}}$, especially denoted by $\bm{n}^{0, L} = \left\{ 0, 0, 1 \right\}^{\operatorname{T}}$ in the initial local configuration. {#figs:discretization_shell width="90%"} The local position $\bm{r}^L$ of a material point can be expressed as $$\bm{r}^L\left( \xi ,\eta ,\chi, t \right) = \bm{r}^L_m \left( \xi ,\eta, t \right) + \chi \bm{n}^L \left( \xi ,\eta, t \right), \quad \chi \in \left[- d/2,d/2 \right].$$ The local displacement $\bm{u}^L$ can thus be obtained by $$\bm{u}^L \left( \xi ,\eta ,\chi, t \right) = \bm{u}^L_m \left( \xi ,\eta, t \right) + \chi \Delta \bm{n}^L \left( \xi ,\eta, t \right),$$ where $\Delta \bm{n}^L = \bm{n}^L - \bm{n}^{0, L}$. Similar to 3D plates, the local deformation gradient tensor of 3D shells can be defined as $$\label{shell_deformation_tensor} \mathbb{F}^L = \nabla^{0, L} \bm{r}^L + \nabla^{0, L} \bm{n}^L - \nabla^{0, L} \bm{n}^{0, L} = \left( {\bm{a}^L_1}, {\bm{a}^L_2}, {\bm{a}^L_3} \right),$$ where $\nabla^{0, L} \equiv \partial / \partial \bm{\xi}$ is the gradient operators defined in the initial local configuration, and $\bm{a}^L_1$, $\bm{a}^L_2$, $\bm{a}^L_3$ are detailed by $$\begin{cases} \bm{a}^L_1 = \bm{r}^L_{m,\xi} + \chi \bm{n}^L_{\xi} - \chi \bm{n}^{0, L}_{\xi} \\ \bm{a}^L_2 = \bm{r}^L_{m,\eta} + \chi \bm{n}^L_{\eta} - \chi \bm{n}^{0, L}_{\eta} \\ \bm{a}^L_3 = \bm{n}^L. \end{cases}$$ ### Stress correction and conservation equation With the local deformation gradient tensor $\mathbb{F}^L$, the local Eulerian Almansi strain ${\color{white}\contour{black}{$\epsilon$}}^L$ can be calculated by the Eq. [\[Almansi_strain\]](#Almansi_strain){reference-type="eqref" reference="Almansi_strain"}. After that, the current local ${\color{white}\contour{black}{$\epsilon$}}^l$ is obtained according to the coordinate transformation as $${\color{white}\contour{black}{$\epsilon$}}^l = \mathbb{Q} \left(\mathbb{Q}^0\right)^{\operatorname{T}} {\color{white}\contour{black}{$\epsilon$}}^L \mathbb{Q}^0 \mathbb{Q}^{\operatorname{T}},$$ where $\mathbb{Q}^0$, the orthogonal transformation matrix from the global to initial local coordinates, is calculated from Eq. [\[3D_transformation_matrix\]](#3D_transformation_matrix){reference-type="eqref" reference="3D_transformation_matrix"} while the current pseudo normal $\bm{n}$ is replaced by the initial one $\bm{n}^0$. And then the corrected strain $\bar{{\color{white}\contour{black}{$\epsilon$}}}^l$ is estimated by applying Eq. [\[strain_relation\]](#strain_relation){reference-type="eqref" reference="strain_relation"}. After getting the current local Cauchy stress ${\color{white}\contour{black}{$\sigma$}}^l$ by Eq. [\[constitutive_relation\]](#constitutive_relation){reference-type="eqref" reference="constitutive_relation"}, the corrected one $\bar{{\color{white}\contour{black}{$\sigma$}}}^l$ is obtained by Eq. [\[shear_correction\]](#shear_correction){reference-type="eqref" reference="shear_correction"}. Note that the total Lagrangian conservation equations of a 3D shell has the same form as Eqs. [\[plate-conservation-equation\]](#plate-conservation-equation){reference-type="eqref" reference="plate-conservation-equation"} with $\mathbb{F}_m = \left(\mathbb{Q}^0\right)^{\operatorname{T}} \mathbb{F}_m^L \mathbb{Q}^0$. ## 2D plate/shell model If a plate/shell is assumed to be a strip that is very long and has a finite width, and the transverse load is assumed to be uniform along the length, the analysis can be simplified at any cross section as a plane strain problem [@reddy2006theory]. The kinematics of 2D plate and shell can all be built in initial local coordinates, as the transformation matrix $\mathbb{Q}^0$ from the global to initial local coordinates for the plate is the unit matrix. The 2D model is in the global $X$-$Z$ plane, and each material point possesses three degrees of freedom, viz., two translations $\bm{u}^L = \left\{ u^L, w^L \right\}^{\operatorname{T}}$ and one rotation $\bm{\theta}^L = \left\{\varphi^L \right\}^{\operatorname{T}}$ expressed in the initial local coordinates. The pseudo-normal vector is presented in the initial local coordinates by $\bm{n}^L = \left\{ n^L_1, n^L_3 \right\}^{\operatorname{T}}$, especially denoted by $\bm{n}^{0, L} = \left\{ 0, 1 \right\}^{\operatorname{T}}$ in the initial local configuration. Similar to 3D model, the local position $\bm{r}^L$ of a material point can be expressed as $$\bm{r}^L\left( \xi ,\chi, t \right) = \bm{r}^L_m \left( \xi, t \right) + \chi \bm{n}^L \left( \xi, t \right), \quad \chi \in \left[- d/2, d/2 \right], \label{eq: shell_position}$$ the local displacement $\bm{u}^L$ can be evaluated as $$\bm{u}^L \left( \xi ,\chi, t \right) = \bm{u}^L_m \left( \xi, t \right) + \chi \Delta \bm{n}^L \left( \xi, t \right).$$ and the local deformation gradient tensor is written as $$\label{2D_shell_deformation_tensor} \mathbb{F}^L = \nabla^{0, L} \bm{r}^L + \nabla^{0, L} \bm{n}^L - \nabla^{0, L} \bm{n}^{0, L} = \left( {\bm{a}^L_1}, {\bm{a}^L_3} \right),$$ where $\bm{a}^L_1$ and $\bm{a}^L_3$ are given by $$\begin{cases} \bm{a}^L_1 = \bm{r}^L_{m,\xi} + \zeta \bm{n}^L_{\xi} - \zeta \bm{n}^{0, L}_{\xi} \\ \bm{a}^L_3 = \bm{n}^L. \end{cases}$$ The coordinate transformation matrix $\mathbb{Q}$ from global to current local coordinates is simplified from Eqs. [\[3D_transformation_matrix\]](#3D_transformation_matrix){reference-type="eqref" reference="3D_transformation_matrix"} as $$\label{2D_Q} \mathbb{Q} = \begin{bmatrix} n_3 & -n_1 \\ n_1 & n_3 \\ \end{bmatrix},$$ and the 2D transformation matrix $\mathbb{Q}^0$ from global to initial local coordinates can also calculated by Eq. [\[2D_Q\]](#2D_Q){reference-type="eqref" reference="2D_Q"} while the current pseudo normal $\bm{n}$ is replaced by the initial one $\bm{n}^0$. The corrected relation of strains is simplified from Eq. [\[strain_relation\]](#strain_relation){reference-type="eqref" reference="strain_relation"} as $$\label{2D_strain_relation} \bar\epsilon^l_{zz}=\frac{-\nu \epsilon^l_{xx} } {1-\nu}.$$ Finally, the 2D conservation equation is identical to 3D Eq. [\[plate-conservation-equation\]](#plate-conservation-equation){reference-type="eqref" reference="plate-conservation-equation"} with $$\mathbb{N}^l = \begin{bmatrix} N^l_{xx} & 0 \\ N^l_{z x}& 0 \\ \end{bmatrix}, \mathbb{M}^l = \begin{bmatrix} M^l_{xx} & 0 \\ M^l_{zx} & 0 \\ \end{bmatrix}, {\bf{Q}}^l = \begin{bmatrix} -N^l_{xz} \\ 0 \\ \end{bmatrix}.$$ # SPH method for plate and shell structures {#sec:SPH_method} In this section, we first introduce the reduced-dimensional SPH method, and detail the proposed formulations for plate and shell structures, including the discretization of conservation equations, non-singular conversion algorithm for the kinematics between rotation angles and pseudo normal, and the algorithms to increase numerical stability and alleviate hourglass modes. After that, the time-integration schemes are presented. ## Reduced-dimensional SPH method In full-dimensional SPH method, the smoothed field $f(\bm{r})$ is obtained as $$\label{eq:kenelintegral} f(\bm{r}) = \int_{\Omega} f(\bm{r}') W(\bm{r} - \bm{r}', h) d\bm{r}',$$ where $f(\bm{r}')$ is the original continuous field before smoothing, $\Omega$ the entire space and $W(\bm{r} - \bm{r}', h)$ a Gaussian-like kernel function with smoothing length $h$ denoting the compact support. By carrying out the integration of Eq. [\[eq:kenelintegral\]](#eq:kenelintegral){reference-type="eqref" reference="eq:kenelintegral"} along the thickness of the plate/shell structure, we can obtained the reduced-dimensional smoothed field by $$\label{eq:reducedkenelintegral} f \left( \mathbf r\right) \approx \int_{\widehat \Omega} f \left( \mathbf r' \right) \widehat W \left( \mathbf r-\mathbf r', h\right) d\mathbf r',$$ where $\widehat \Omega$ denotes the reduced space and $\widehat W(\bm{r} - \bm{r}', h)$ the reduced kernel function. Note that Eqs. ([\[eq:kenelintegral\]](#eq:kenelintegral){reference-type="ref" reference="eq:kenelintegral"}) and ([\[eq:reducedkenelintegral\]](#eq:reducedkenelintegral){reference-type="ref" reference="eq:reducedkenelintegral"}) have identical forms of formulation. A reduced-dimensional fifth-order Wendland kernel [@wendland1995piecewise] reads $$\widehat W (q, h) = \alpha \begin{cases} \left(1 + 2q \right) \left( 1 - q / 2\right)^4 & \text{if} \quad 0\le q\le2 \\ 0 & \text{otherwise} \end{cases},$$ where $q = \left| \mathbf r-\mathbf r'\right|/h$ and the constant $\alpha$ is equal to $\frac{3}{4 h}$ and $\frac{7}{4\pi h^2}$ for 2D and 3D problems, respectively. Also note that the reduced kernel function has identical form with the full-dimensional counterpart except different dimensional normalizing constant parameter, allowing the integration of unit can be satisfied in the reduced space. Due to the almost identical forms, in present work from here, we do not identify the full- and reduced-dimensional formulations unless explicitly mentioned. In the reduced-dimensional SPH method, similarly to the full-dimensional counterpart [@monaghan2005smoothed], the gradient of the variable field $f(\bm{r})$ at a surface particle $i$ can be approximated as $$\label{eq:gradsph} \begin{split} \nabla f_i & = \int_{\Omega} \nabla f (\bm{r}) W(\bm{r}_i - \bm{r}, h) d \bm{r} \\ & = - \int_{\Omega} f (\bm{r}) \nabla W(\bm{r}_i - \bm{r}, h) d \bm{r} \approx - \sum\limits_j f_j\nabla W_{ij}V_j , \end{split}$$ where $V$ is the reduced particle volume, i.e. length and area for 2D and 3D problems, respectively. Here, the summation is conducted over all the neighboring particles $j$ located at the support domain of the particle $i$, and $\nabla W_{ij} = -\frac{\partial W\left( \bm{r}_{ij}, h \right)}{\partial r_{ij}} \bm{e}_{ij}$ is the gradient of the kernel function with $\bm{r}_{ij} = \bm{r}_{i} - \bm{r}_{j}$ and $\bm{e}_{ij} = \bm{r}_{ij} / |\bm{r}_{ij}|$ denoting the unit vector pointing from particle $j$ to $i$. Equation [\[eq:gradsph\]](#eq:gradsph){reference-type="eqref" reference="eq:gradsph"} can be modified into a strong form as $$\label{eq:gradsph-strong} \nabla f_i = \nabla f_i - f_{i}\nabla 1 \approx \sum\limits_j f_{ij} \nabla W_{ij}V_j,$$ where $f_{ij} = f_{i} - f_{j}$ is the interparticle difference value. This strong-form derivative operator can be used to determine the local structure of a field, such as the deformation gradient tensor. And Eq. [\[eq:gradsph\]](#eq:gradsph){reference-type="eqref" reference="eq:gradsph"} can also be modified into a weak form as $$\label{eq:gradsph-weak} \nabla f_i = f_{i}\nabla 1 + \nabla f_i \approx - \sum\limits_j \left(f_{i} + f_{j}\right)\nabla W_{ij} V_j.$$ This weak-form derivative operator is applied here for solving the conservation equations. Thanks to its anti-symmetric feature, i.e., $\nabla W_{ij} = - \nabla W_{ji}$, the momentum conservation of the particle system is ensured [@monaghan2005smoothed]. Also note that, in the present work, the reduced-dimensional SPH method is employed for total Lagrangian formulation [@randles1996smoothed], such as Eq. [\[plate-conservation-equation\]](#plate-conservation-equation){reference-type="eqref" reference="plate-conservation-equation"}. Therefore, the smoothing kernel function and its derivatives are only evaluated once, also denoted with superscript $\left( \bullet \right)^0$ at the initial configuration, and kept unchanged during the simulation. ## First-order consistency corrections {#sec:consistency correction} For the full-dimensional SPH in total Lagrangian formulation, in order to remedy the 1st-order inconsistency which is caused by incomplete kernel support at domain boundary or with irregular particle distribution, the symmetric correction matrix $\mathbb{B}^{0}_i$ for each particle [@vignjevic2006sph; @liu2010smoothed] is introduced for each particle to satisfy the linear-reproducing condition $$\label{eq:consistency-condition} \left(\sum\limits_j {\bm{r}^{0}_{ij} \otimes \nabla^{0} W_{ij} V^{0}_j} \right) \mathbb{B}^{0}_i = \mathbb{I}.$$ Then the strong-form approximations of gradient Eq. [\[eq:gradsph-strong\]](#eq:gradsph-strong){reference-type="eqref" reference="eq:gradsph-strong"} is modified as $$\label{eq:gradsph-strong-corrected} \nabla^{0} f_i \approx \left( \sum\limits_j f_{ij} \nabla^{0} W_{ij}V^{0}_j \right) \mathbb{B}^{0}_i,$$ and the weak-form approximations of divergence Eq. [\[eq:gradsph-weak\]](#eq:gradsph-weak){reference-type="eqref" reference="eq:gradsph-weak"} as $$\label{eq:gradsph-weak-corrected} \nabla^{0} \cdot f_i \approx - \sum\limits_j \left(f_{i}\mathbb{B}^{0}_i + f_{j}\mathbb{B}^{0}_j\right)\nabla^{0} W_{ij} V^{0}_j.$$ In the reduced-dimensional SPH, we generalize the linear-reproducing condition as $$\label{eq:reduced-consistency-condition} \left[\mathbb{G}^{\operatorname{T}} \mathbb{Q}^{0}_i \left(\sum\limits_j {\bm{q}^{0}_{ij} \otimes \nabla^{0} W_{ij} V^{0}_j} \right) \left(\mathbb{Q}^{0}_i\right)^{\operatorname{T}} \mathbb{G} \right] \mathbb{B}^{0, L}_i = \mathbb{K}_i,$$ where $\bm{q}^{0}_{ij}$ is the initial inter-particle difference of a linear vector, $\mathbb{Q}^{0}_i$ is the transformation matrix from the global to initial local coordinates, and $\mathbb{G}$ is a reducing matrix, i.e., $$\mathbb{G} = \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix} ~\text{and}~ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ \end{bmatrix}$$ for 2D and 3D problems, respectively. It ensures that the corrections are carried out within the local reduced space. Similarly, the strong-form approximations of gradient Eq. [\[eq:gradsph-strong\]](#eq:gradsph-strong){reference-type="eqref" reference="eq:gradsph-strong"} is modified as $$\label{eq:shell-strong-corrected} \nabla^{0} f_i \approx \left( \sum\limits_j f_{ij} \nabla^{0} W_{ij}V^{0}_j \right)\widetilde{\mathbb{B}}^{0}_i,$$ where $\widetilde{\mathbb{B}}^{0}_i = \left(\mathbb{Q}^{0}_i\right)^{\operatorname{T}} \mathbb{G} \mathbb{B}^{0, L}_i \mathbb{G}^{\operatorname{T}} \mathbb{Q}^{0}_i$ and the weak-form approximations of divergence Eq. [\[eq:gradsph-weak\]](#eq:gradsph-weak){reference-type="eqref" reference="eq:gradsph-weak"} as $$\label{eq:shell-weak-corrected} \nabla^{0} \cdot f_i \approx - \sum\limits_j \left(f_{i}\widetilde{\mathbb{B}}^{0}_i + f_{j}\widetilde{\mathbb{B}}^{0}_j\right)\nabla^{0} W_{ij} V^{0}_j.$$ Here, we introduce the correction matrix $\widetilde{\mathbb{B}}^{0}_i = \widetilde{\mathbb{B}}^{0,\bm{r}}_i$, $\bm{q}^{0}_{ij} = \bm{r}_{ij}^0$ and $\mathbb{K}_i$ is the reduced identity matrix denoted as $$\mathbb{K}_i = \mathbb{K}^{\bm{r}} = \begin{bmatrix} 1 \end{bmatrix} ~\text{and}~ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ for 2D and 3D problems, respectively, to correct the position-based quantities. Similarly, we introduce the correction matrix $\widetilde{\mathbb{B}}^{0}_i = \widetilde{\mathbb{B}}^{0,\bm{n}}_i$, $\bm{q}^{0}_{ij} = \bm{n}_{ij}^0$ and $$\mathbb{K}_i = \mathbb{K}^{\bm{n}}_{i} = \begin{bmatrix} 1/R_i^L \end{bmatrix} ~\text{and}~ \begin{bmatrix} 1/R_{1,i}^L & 0 \\ 0 & 1/R_{2,i}^L \end{bmatrix},$$ where $R_i^L$, $R_{1,i}^L$ and $R_{2,i}^L$ are the curvature radii of particle $i$ for 2D and 3D problems, respectively, to correct rotation-based quantities. ## Discreization of conservation equations {#sec:reduced_dimensional_TLSPH} With two correction matrices obtained from Eq. [\[eq:reduced-consistency-condition\]](#eq:reduced-consistency-condition){reference-type="eqref" reference="eq:reduced-consistency-condition"} and following Eq. [\[eq:shell-weak-corrected\]](#eq:shell-weak-corrected){reference-type="eqref" reference="eq:shell-weak-corrected"}, the momentum equations [\[plate-conservation-equation\]](#plate-conservation-equation){reference-type="eqref" reference="plate-conservation-equation"} are discretized as $$\label{discrete_dynamic_equation1} d\rho _{i}^0 \bm{\ddot u}_{m, i} = \sum\limits_j \left( J_{m, i} \mathbb{N}_i \left(\mathbb{F}_{m, i} \right)^{\operatorname{-T}} \widetilde{\mathbb{B}}^{0, \bm{r}}_i + J_{m, j} \mathbb{N}_j \left(\mathbb{F}_{m, j} \right)^{\operatorname{-T}} \widetilde{\mathbb{B}}^{0, \bm{r}}_j \right) \nabla^0 W_{ij} V_j^0,$$ and $$\label{discrete_dynamic_equation2} \begin{split} \frac{d^3 } {12} \rho_i^0\bm{\ddot n }_i & = \sum\limits_j \left( J_{m, i} \mathbb{M}_i \left(\mathbb{F}_{m, i} \right)^{\operatorname{-T}} \widetilde{\mathbb{B}}^{0, \bm{n}}_i + J_{m,j} \mathbb{M}_j \left(\mathbb{F}_{m, j} \right)^{\operatorname{-T}} \widetilde{\mathbb{B}}^{0, \bm{n}}_j \right) \nabla^0 W_{ij} V_j^0 \\ & + {J_{m, i} \left(\mathbb{Q}_i^0\right)}^{\operatorname{T}} \bm{Q}^l_i. \end{split}$$ ## Kelvin--Voigt type damping Following Ref. [@zhang2022artificial], when calculating the current local Cauchy stress by using the constitutive Eq. [\[constitutive_relation\]](#constitutive_relation){reference-type="eqref" reference="constitutive_relation"}, an artificial damping stress ${\color{white}\contour{black}{$\sigma$}}_d^l$ based on the Kelvin-Voigt type damper is introduced here as $$\label{Cauchy_stress_damping} {\color{white}\contour{black}{$\sigma$}}_d^l = J_m^{-1} \mathbb{Q} \left(\mathbb{Q}^0\right)^{\operatorname{T}} \mathbb{F}^L \dot{\mathbb{E}}^L {\color{white}\contour{black}{$\gamma$}} \left(\mathbb{F}^L\right)^{\operatorname{T}} \mathbb{Q}^0 \mathbb{Q}^{\operatorname{T}},$$ where the numerical viscosity matrix $${\color{white}\contour{black}{$\gamma$}} = \begin{bmatrix} \rho c h / 2 & 0 \\ 0 & \rho c s / 2 \\ \end{bmatrix} ~\text{and}~ \begin{bmatrix} \rho c h / 2 & 0 & 0 \\ 0 & \rho c h / 2 & 0 \\ 0 & 0 & \rho c s / 2 \\ \end{bmatrix}$$ where $c = \sqrt {K/\rho}$ and $s = \min (h, d)$, for 2D and 3D problems, respectively. Note that, different from Ref. [@zhang2022artificial], where an isotropic numerical damping is applied, the present damping leads to a smaller out-of-plane contribution when $d \le h$, which makes it suitable for both thin and moderately thick plate and shell structures. The change rate of the Green-Lagrangian strain tensor is given as $$\dot{\mathbb{E}}^L = \frac{1}{2} \left[\left(\dot{\mathbb{F}}^L \right)^{\operatorname{T}} \mathbb{F}^L + \left(\mathbb{F}^L\right)^{\operatorname{T}} \dot{\mathbb{F}}^L \right].$$ Here, the change rate of the deformation gradient tensor of particle $i$ is $$\label{change_rate_deformation_gradient} {\mathbb{\dot F}}^L_i = \nabla^{0, L} \bm{\dot u}_i^L = \nabla^{0}\bm{\dot u}^L_{m, i} + \chi \nabla^{0}\bm{\dot n}^L_{i} ,$$ where $$\begin{cases} \nabla^{0}\bm{\dot u}^L_{m, i} = \mathbb{Q}_i^0\left(\sum\limits_j \bm{\dot u}_{m,ij} \otimes \nabla^0 W_{ij} V_j^0 \right) \widetilde{\mathbb{B}}_i^{0, \bm{r}} \left(\mathbb{Q}_i^0\right)^{\operatorname{T}} \\ \nabla^{0}\bm{\dot n}^L_{i} = \mathbb{Q}_i^0 \left(\sum\limits_j \bm{\dot n}_{ij} \otimes \nabla^0 W_{ij} V_j^0 \right) \widetilde{\mathbb{B}}_{i}^{0, \bm{n}} \left(\mathbb{Q}_i^0\right)^{\operatorname{T}} \end{cases}$$ are obtained following the consistency condition Eq. [\[eq:reduced-consistency-condition\]](#eq:reduced-consistency-condition){reference-type="eqref" reference="eq:reduced-consistency-condition"} and the strong-form correction Eq. [\[eq:shell-strong-corrected\]](#eq:shell-strong-corrected){reference-type="eqref" reference="eq:shell-strong-corrected"}. ## Hourglass control Inspired from Refs. [@kondo2010suppressing; @ganzenmuller2015hourglass] in full-dimensional SPH for total Lagrangian solid dynamics, we introduce a hourglass control algorithm here to alleviate the hourglass modes in the dynamics of plate and shell structures. First, we estimate the position of the inter-particle middle point linearly using the deformation gradient tensor from particles $i$ and $j$, respectively, as $$\label{predicted_position1} \bm{r}_{i + \frac{1}{2}} = \bm{r}_{m, i} - \frac{1}{2} \mathbb{F}_{m, i} \bm{r}_{m, ij}^0, \quad \bm{r}_{j - \frac{1}{2}} = \bm{r}_{m,j} + \frac{1}{2} \mathbb{F}_{m,j} \bm{r}_{m, ij}^0.$$ One can find that the inconsistency beyond linear estimation $\hat{\bm{r}}_{ij} = \bm{r}_{i + \frac{1}{2}} - \bm{r}_{j - \frac{1}{2}}$ is $$\hat{\bm{r}}_{ij} = \bm{r}_{m, ij} - \frac{1}{2} \left( \mathbb{F}_{m,i} + \mathbb{F}_{m,j} \right) \bm{r}_{m, ij}^0.$$ To suppress the position inconsistency $\hat{\bm{r}}_{ij}$, we introduce an extra correction term to the discrete momentum conservation Eq. [\[discrete_dynamic_equation1\]](#discrete_dynamic_equation1){reference-type="eqref" reference="discrete_dynamic_equation1"} as $$\label{artificial-stress} d \rho _{i}^0 \bm{\ddot u}^{cor}_{m,i} = \sum\limits_j \alpha G \beta_{ij} \gamma_{ij}^{\bm{r}} D \hat{\bm{r}}_{ij} \frac{\partial W\left( \bm{r}_{ij}^0, h \right)} {\partial r_{ij}^0} V_j^0 \\$$ where $\beta_{ij} = W^0_{ij} / W_{0}$ leads to less correction to further neighbors, $\gamma_{ij}^{\bm{r}} = \min \left(2\left|\hat{\bm{r}}_{ij}\right|/ \left|\bm{r}_{m, ij}\right|, 1\right)$ is an adaptive limiter for less correction on the domain where the inconsistency is less pronounced, D the dimension, and parameter $\alpha = 0.002$ according to the numerical experiment and remains constant throughout this work. Note that, since the inconsistency decreases with decreasing particle spacing, different from Refs. [@ganzenmuller2015hourglass], the present correction is purely numerical and vanishes with increasing resolution. Similarly, for the predicted pseudo normal, the difference of the intermediate point can be described as $$\hat{\bm{n}}_{ij} = \bm{n}_{ij} - \bm{n}_{ij}^0 - \frac{1}{2} \left(\mathbb{F}_{\bm{n},i} + \mathbb{F}_{\bm{n},j}\right) \bm{r}_{ij}^0.$$ Similar with Eq. [\[artificial-stress\]](#artificial-stress){reference-type="eqref" reference="artificial-stress"}, the extra correction term added to the discrete angular momentum conservation Eq. [\[discrete_dynamic_equation2\]](#discrete_dynamic_equation2){reference-type="eqref" reference="discrete_dynamic_equation2"} is $$\label{artificial-torque} \frac{d^{3}}{12}\rho _{i}^0 \bm{\ddot n}^{cor}_i = \sum\limits_j \alpha G d^2 \beta_{ij} \gamma_{ij}^{\bm{n}} D \hat{\bm{n}}_{ij} \frac{\partial W\left( \bm{r}_{ij}^0,h \right)}{\partial r_{ij}^0 } V_j^0,$$ where the adaptive limiter is $\gamma_{ij}^{\bm{n}} = \min \left(2 \left|\hat{\bm{n}}_{ij}\right| / \left|\bm{n}_{ij} - \bm{n}_{ij}^0\right|, 1\right)$. Note that, different with Refs. [@kondo2010suppressing; @ganzenmuller2015hourglass], the present correction force is introduced in particle pairwise pattern, implying momentum conservation [@monaghan2005smoothed]. Also note that, the correction force is scaled to the shear, rather than Young's, modulus, due to the fact that the hourglass modes are characterized by shear deformation [@wu2023essentially]. ## Conversion between rotations and pseudo normal Different from the mid-surface displacement, which can be numerically integrated directly from its evolution equation, the pseudo normal is not suitable for direct numerical integration since its unit magnitude may not be maintained strictly. Under the assumption of small rotation, one may have the simplified relation between the pseudo normal and rotations, i.e. $\bm{\ddot{\theta}} = (-\ddot{n}_2, \ddot{n}_1)$, so that one can obtain the rotation increment, and update rotation matrix $\mathbb{R}$ using Rodrigues formula [@betsch1998parametrization; @lin2014efficient] and finally the integrated pseudo normal by $\bm{n} = \mathbb{R}\bm{n}^0$. In the present work, the numerical integration of pseudo normal is carried out without the assumption of small rotation, that is, we strictly identify the rotations and pseudo normal by using the original evolution equation and obtain their conversion relations $\bm{\ddot{\theta}} = \bm{\ddot{\theta}}(\bm{\ddot{n}}, \bm{\dot{\theta}}, \bm{\theta})$. Different from using the rotation matrix $\mathbb{R}$ based on Rodrigues formula, we update the pseudo normal $\bm{n}^L$ with [@betsch1998parametrization; @wisniewski2010finite] $$\label{eq:updated-pseudo-normal} \bm{n}^L = \mathbb{R}^L_{\eta} \mathbb{R}^L_{\xi} \bm{n}^{0, L},$$ where $\mathbb{R}_{\xi}^L\equiv \mathbb{R}_{\xi}(\theta^{L})$ and $\mathbb{R}^L_{\eta} \equiv \mathbb{R}_{\eta}(\varphi^{L})$ are the local rotation matrices respected to the axes $\xi$ and $\eta$, respectively, or, equivalently, with the change rate $$\label{eq:pseudo-normal-rate} \bm{\dot{n}}^L = \dot{\mathbb{R}}^L_{\eta} \dot{\mathbb{R}}^L_{\xi},$$ where $\dot{\mathbb{R}}^L_{\xi}\equiv \mathbb{R}_{\xi}(\theta^{L}, \dot{\theta}^L)$ and $\dot{\mathbb{R}}^L_{\eta}\equiv \mathbb{R}_{\eta}(\varphi^{L}, \dot{\varphi}^L)$. Here, the rotations and their change rates are numerically integrated directly with the help of conversion relations. Specifically, for a 2D problem, $\mathbb{R}^L_{\xi}$ is a unit matrix, and $\mathbb{R}^L_{\eta}$ can be described as $$\mathbb{R}^L_{\eta} = \begin{bmatrix} \cos \varphi^L & \sin \varphi^L \\ - \sin \varphi^L & \cos \varphi^L \\ \end{bmatrix}.$$ Then, one has the relation as $$\label{2d-rotaion-pseudo-normal} \bm{n}^L = (\sin \varphi^L, \cos \varphi^L)^{\operatorname{T}},$$ its 1st-order time derivative corresponding Eq. [\[eq:pseudo-normal-rate\]](#eq:pseudo-normal-rate){reference-type="eqref" reference="eq:pseudo-normal-rate"} $$\label{2D_n_first_derivative} \dot {\bm{n}}^L = (\cos \varphi^L \cdot \dot \varphi^L, -\sin \varphi^L \cdot \dot \varphi^L)^{\operatorname{T}},$$ and 2nd-order derivative $$\label{2D_n_second_derivative} \ddot {\bm{n}}^L = (-\sin \varphi^L \cdot \left(\dot \varphi^L\right)^2 +\cos \varphi^L \cdot \ddot \varphi^L, -\cos \varphi^L \cdot \left(\dot \varphi^L\right)^2 -\sin \varphi^L \cdot \ddot \varphi^L)^{\operatorname{T}}.$$ Note that Eq. [\[2D_n\_second_derivative\]](#2D_n_second_derivative){reference-type="eqref" reference="2D_n_second_derivative"} suggests two theoretically equivalent conversion relations $$\label{2D_conversion_relation1} \ddot \varphi^L = \frac{\ddot n_1^L + \sin \varphi^L \cdot \left(\dot \varphi^L\right)^2} {\cos \varphi^L} \quad \text{and} \quad \ddot \varphi^L = \frac{\ddot n_2^L + \cos \varphi^L \cdot \left(\dot \varphi^L\right)^2} {-\sin \varphi^L}.$$ Although each of them can be used to obtain the rotation angle $\varphi^L$ and its change rate with direct numerical integration and hence the pseudo normal with Eq. [\[eq:updated-pseudo-normal\]](#eq:updated-pseudo-normal){reference-type="eqref" reference="eq:updated-pseudo-normal"}, there are singularities at large rotation angles $\varphi^L = 0.5 \pi + k \pi$ (1st relation) or $\varphi^L = k \pi$ (2nd relation) with $k=0, 1, 2, 3, ...$ [@simo1990stress; @betsch1998parametrization; @singla2004avoid]. In order to eliminate the singularities, we propose to uses both relations with a weighted average as $$\label{2D_conversion_relation3} \begin{split} \ddot \varphi^L & = \left(\cos \varphi^L\right)^2 \frac{\ddot n_1^L + \sin \varphi^L \cdot \left(\dot \varphi^L\right)^2} {\cos \varphi^L} + \left(\sin \varphi^L\right)^2 \frac{\ddot n_2^L + \cos \varphi^L \cdot \left(\dot \varphi^L\right)^2} {-\sin \varphi^L} \\ & = \cos \varphi^L \left(\ddot n_1^L + \sin \varphi^L \cdot \left(\dot \varphi^L\right)^2\right) - \sin \varphi^L \left(\ddot n_2^L + \cos \varphi^L \cdot \left(\dot \varphi^L\right)^2\right), \end{split}$$ which cancels both denominators. Note that the present formulation recovers $\ddot \varphi^L = \ddot n_1^L$ under the assumption of small rotation. As for 3D problems, the rotation matrices $\mathbb{R}^L_{\xi}$ and $\mathbb{R}^L_{\eta}$ are $$\mathbb{R}^L_\xi = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta^L & -\sin \theta^L \\ 0 & \sin \theta^L & \cos \theta^L \\ \end{bmatrix},$$ and $$\mathbb{R}^L_\eta = \begin{bmatrix} \cos \varphi^L & 0 & \sin \varphi^L \\ 0 & 1 & 0 \\ - \sin \varphi^L & 0 & \cos \varphi^L \\ \end{bmatrix}. %$$ Similarly, one has the relation between rotations and pseudo normal [@hughes1978consistent] $$\bm{n}^L = ( \cos \theta^L \sin \varphi^L, -\sin \theta^L, \cos \theta^L \cos \varphi^L)^{\operatorname{T}},$$ its 1st-order time derivatives corresponding Eq. [\[eq:pseudo-normal-rate\]](#eq:pseudo-normal-rate){reference-type="eqref" reference="eq:pseudo-normal-rate"} $$\label{3D_n_first_derivative} \begin{cases} \dot n^L_1 = -\sin \theta^L \sin \varphi^L \dot \theta^L + \cos \theta^L \cos \varphi^L \dot \varphi^L\\ \dot n^L_2 = -\cos \theta^L \dot \theta^L\\ \dot n^L_3 = -\sin \theta^L \cos \varphi^L \dot \theta^L - \cos \theta^L \sin \varphi^L \dot \varphi^L, \end{cases}$$ and 2nd-order derivatives $$\label{3D_n_second_derivative} \begin{cases} \begin{split} \ddot n^L_1 = & -\sin \theta^L \sin \varphi^L \ddot \theta^L - \cos \theta^L \sin \varphi^L ({\dot\theta}^L)^2 - 2 \sin \theta^L \cos\varphi^L \dot \theta^L \dot \varphi^L \\ & - \cos \theta^L \sin \varphi^L (\dot \varphi^L)^2 + \cos \theta^L \cos \varphi^L \ddot \varphi^L \end{split}\\ \ddot n^L_2 = \sin \theta^L (\dot \theta^L)^2 -\cos \theta^L \ddot \theta^L\\ \begin{split} \ddot n^L_3 &= -\sin \theta^L \cos \varphi^L \ddot \theta^L - \cos \theta^L \cos \varphi^L ({\dot\theta}^L)^2 + 2 \sin \theta^L \cos\varphi^L \dot \theta^L \dot \varphi^L \\ & - \cos \theta^L \cos \varphi^L (\dot \varphi^L)^2 - \cos \theta^L \sin \varphi^L \ddot \varphi^L. \end{split} \end{cases}$$ Note that, one can obtained 3 theoretically equivalent conversion relations, respectively, by 1st and 3rd expressions of Eq. [\[3D_n\_second_derivative\]](#3D_n_second_derivative){reference-type="eqref" reference="3D_n_second_derivative"} as $$\label{3D_n_second_derivative1} \begin{cases} \ddot \theta^L = -\left( \ddot n^L_3 \cos \varphi^L+\ddot n^L_1 \sin \varphi^L + \left(\dot \varphi^L \right)^2 \cos \theta^L + \left(\dot \theta^L \right)^2 \cos \theta^L \right) / \sin \theta^L\\ \ddot \varphi^L = \left( \ddot n^L_1 \cos \varphi^L - \ddot n^L_3 \sin \varphi^L + 2 \dot \varphi^L \dot \theta^L \sin \theta^L \right) / \cos \theta^L, \end{cases}$$ 1st and 2nd expressions $$\label{3D_n_second_derivative2} \begin{cases} \ddot \theta^L = \left( \sin \theta^L \left(\dot \theta^L \right)^2 - \ddot n^L_2 \right) / \cos \theta^L\\ \begin{split} \ddot \varphi^L &= ( \ddot n^L_1 \cos \theta^L + \left(\dot \varphi^L\right)^2 \cos^2 \theta^L \sin \varphi^L + \left(\dot \theta^L\right)^2 \sin \varphi^L - \ddot n^L_2 \sin \varphi^L \sin \theta^L \\ &+ 2 \dot \varphi^L \dot \theta^L \cos \varphi^L \cos\theta^L \sin \theta^L ) / \cos \varphi^L \cos^2 \theta^L, \end{split} \end{cases}$$ and 2nd and 3rd expressions $$\label{3D_n_second_derivative3} \begin{cases} \ddot \theta^L = \left( \sin \theta^L \left(\dot \theta^L \right)^2 - \ddot n^L_2 \right) / \cos \theta^L\\ \begin{split} \ddot \varphi^L &= -( \ddot n^L_3 \cos \theta^L + \left(\dot \varphi^L\right)^2 \cos \varphi^L \cos^2 \theta^L + \left(\dot \theta^L\right)^2 \cos \varphi^L - \ddot n^L_2 \cos \varphi^L \sin \theta^L \\ &- 2 \dot \varphi^L \dot \theta^L \cos \theta^L \sin \varphi^L \sin \theta^L ) / \sin \varphi^L \cos^2 \theta^L. \end{split} \end{cases}$$ Again, each of these conversion relations suffers singularities at large rotations similar to that of 2D formulations. To eliminate the singularities, we first apply the weighted average to the conversion between between $\ddot \theta^L$ and $\bm{\ddot n}^L$ with Eqs. [\[3D_n\_second_derivative1\]](#3D_n_second_derivative1){reference-type="eqref" reference="3D_n_second_derivative1"} and [\[3D_n\_second_derivative2\]](#3D_n_second_derivative2){reference-type="eqref" reference="3D_n_second_derivative2"} as $$\label{3D_conversion_relation1} \begin{split} \ddot \theta^L &= -\left( \ddot n^L_3 \cos \varphi^L+\ddot n^L_1 \sin \varphi^L + \left(\dot \varphi^L \right)^2 \cos \theta^L + \left(\dot \theta^L \right)^2 \cos \theta^L \right) \sin \theta^L \\ &+\left( \sin \theta^L \left(\dot \theta^L \right)^2 - \ddot n^L_2 \right) \cos \theta^L. \end{split}$$ Then, for the conversion relation between $\ddot \varphi^L$ and $\bm{\ddot n}^L$, according to Eq. [\[3D_n\_second_derivative1\]](#3D_n_second_derivative1){reference-type="eqref" reference="3D_n_second_derivative1"}, we can rewrite the relation as $$\label{3D_n_second_derivative4} \cos \theta^L = B / \ddot \varphi^L,$$ where $B$ denotes the numerator of the 2nd expression in Eq. [\[3D_n\_second_derivative1\]](#3D_n_second_derivative1){reference-type="eqref" reference="3D_n_second_derivative1"}. We further denote the numerators of the 2nd expressions in Eqs. [\[3D_n\_second_derivative2\]](#3D_n_second_derivative2){reference-type="eqref" reference="3D_n_second_derivative2"} and [\[3D_n\_second_derivative3\]](#3D_n_second_derivative3){reference-type="eqref" reference="3D_n_second_derivative3"}, respectively, as $B_1$ and $B_2$. Inserting Eq. [\[3D_n\_second_derivative4\]](#3D_n_second_derivative4){reference-type="eqref" reference="3D_n_second_derivative4"} into Eqs. [\[3D_n\_second_derivative2\]](#3D_n_second_derivative2){reference-type="eqref" reference="3D_n_second_derivative2"} and [\[3D_n\_second_derivative3\]](#3D_n_second_derivative3){reference-type="eqref" reference="3D_n_second_derivative3"}, we have $$\label{3D_n_second_derivative5} \begin{cases} \ddot \varphi^L = \left(B^2 \cos \varphi^L\right) / B_1\\ \ddot \varphi^L = \left(B^2 \sin \varphi^L\right) / B_2, \end{cases}$$ and obtain the weighted average of the conversion relation as $$\label{3D_conversion_relation2} \ddot \varphi^L = \frac{B_1 B^2 \cos \varphi^L + B_2 B^2 \sin \varphi^L} {B_1^2 + B_2^2}.$$ Again, one can easily find that the present relations recover $\bm{\ddot{\theta}}^L = (\ddot{\theta}^L, \ddot{\varphi}^L) = (-\ddot{n}_2^L, \ddot{n}_1^L)$ for small rotations. ## Time stepping For the time integration of plate and shell dynamics, we use the position-based Verlet scheme [@zhang2021multi]. At the beginning of each time step, the deformation gradient, particle position, rotation angles and pseudo normal are updated to the midpoint of the $n$-th time step as $$\label{eq:verlet-first-half-solid} \begin{cases} \mathbb{F}^{L, n + \frac{1}{2}} = \mathbb{F}^{L, n} + \frac{1}{2} \Delta t \dot{\mathbb{F}}^{L, n}\\ \bm{r}_m^{n + \frac{1}{2}} = \bm{r}_m^n + \frac{1}{2} \Delta t \bm{\dot u}_m^n\\ \bm{\theta}^{L, n + \frac{1}{2}} = \bm{\theta}^{L, n} + \frac{1}{2} \Delta t \bm{\dot \theta}^{L, n}\\ \bm{n}^{L, n + \frac{1}{2}} = \bm{n}^{L, n} + \frac{1}{2} \Delta t \bm{\dot n}^{L, n}. \end{cases}$$ After the stress correction and Gauss-Legendre quadrature Eqs. [\[gaussian_quadrature_rule1\]](#gaussian_quadrature_rule1){reference-type="eqref" reference="gaussian_quadrature_rule1"} and [\[gaussian_quadrature_rule2\]](#gaussian_quadrature_rule2){reference-type="eqref" reference="gaussian_quadrature_rule2"}, the conservation equations are solved to obtain the $\bm{\ddot u}_m^{n+1}$ and $\bm{\ddot n}^{n+1}$. After transforming $\bm{\ddot n}^{n+1}$ to $\bm{\ddot n}^{L, n+1}$, $\bm{\ddot \theta}^{L, n+1}$ is obtained through the conversion relation between the pseudo normal and rotation angle, i.e., Eq. [\[2D_conversion_relation3\]](#2D_conversion_relation3){reference-type="eqref" reference="2D_conversion_relation3"} for 2D problems and Eqs. [\[3D_conversion_relation1\]](#3D_conversion_relation1){reference-type="eqref" reference="3D_conversion_relation1"} and [\[3D_conversion_relation2\]](#3D_conversion_relation2){reference-type="eqref" reference="3D_conversion_relation2"} for 3D problems. At this point, the velocity and angular velocity are updated by $$\label{eq:verlet-first-mediate-solid} \begin{cases} \bm{\dot u}_m^{n + 1} = \bm{\dot u}_m^{n} + \Delta t \bm{\ddot u}_m^{n+1}\\ \bm{\dot \theta}^{L, n + 1} = \bm{\dot \theta}^{L, n} + \Delta t \bm{\ddot \theta}^{L, n+1}, \end{cases}$$ and the change rate of pseudo normal $\bm{\dot {n}}$ is updated by Eq. [\[2D_n\_first_derivative\]](#2D_n_first_derivative){reference-type="eqref" reference="2D_n_first_derivative"} or [\[3D_n\_first_derivative\]](#3D_n_first_derivative){reference-type="eqref" reference="3D_n_first_derivative"}. Finally, the change rate of the deformation gradient $\dot{\mathbb{F}}^{L, n + 1}$ is estimated by Eq. [\[change_rate_deformation_gradient\]](#change_rate_deformation_gradient){reference-type="eqref" reference="change_rate_deformation_gradient"}, and then the deformation gradient, density, particle position, rotation angles and pseudo normal are updated to the new time step with $$\label{eq:verlet-second-final-solid} \begin{cases} \mathbb{F}^{L, n + 1} = \mathbb{F}^{L, n + \frac{1}{2}} + \frac{1}{2} \Delta t \dot{\mathbb{F}}^{L, n + 1}\\ \rho^{n + 1} = \left(J_m^{n + 1} \right)^{-1} \rho^0 \\ \bm{r}_m^{n + 1} = \bm{r}_m^{n + \frac{1}{2}} + \frac{1}{2} \Delta t \bm{\dot u}_m^{n + 1}\\ \bm{\theta}^{L, n + 1} = \bm{\theta}^{L, n + \frac{1}{2}} + \frac{1}{2} \Delta t \bm{\dot \theta}^{L, n + 1}\\ \bm{n}^{L, n + 1} = \bm{\theta}^{L, n + \frac{1}{2}} + \frac{1}{2} \Delta t \bm{\dot n}^{L, n + 1}. \end{cases}$$ For the numerical stability, the time-step size $\Delta t$ is given by $$\label{eq:dt} \Delta t = \text{CFL}\min\left(\Delta t_1, \Delta t_2, \Delta t_3 \right),$$ where $$\label{time_step_size} \begin{cases} \Delta t_1 = \min\left(\frac{h}{c_v + |\bm{\dot u}_m|_{max}}, \sqrt{\frac{h}{|{\bm{\ddot u}_m}|_{max}}} \right)\\ \Delta t_2 = \min\left(\frac{1}{c_v + |\bm{\dot \theta}^L|_{max}}, \sqrt{\frac{1}{|{\bm{\ddot \theta}^L}|_{max}}} \right)\\ \Delta t_3 = h \left( \frac{\rho \left(1 - \nu^2 \right) / E} {2 + \left(\pi^2/12 \right) \left(1 - \nu \right) \left[ 1 + 1.5 \left(h/d \right)^2 \right] } \right)^{1/2}.\\ \end{cases}$$ Note that the time-step size $\Delta t_3$ follows the Refs. [@lin2014efficient; @tsui1971stability] and depends on the thickness and material properties, and the present Courant-Friedrichs-Lewy (CFL) number is set as 0.6. An overview of the complete solution procedure is presented in Algorithm [\[algorithm1\]](#algorithm1){reference-type="ref" reference="algorithm1"}. Setup parameters and initialize the simulation Construct the particle-neighbor list and compute the kernel values Compute the correction matrices $\widetilde{\mathbb{B}}^{0, \bm{r}}$ and $\widetilde{\mathbb{B}}^{0, \bm{n}}$ for each particle (Section [3.2](#sec:consistency correction){reference-type="ref" reference="sec:consistency correction"}) Compute the time-step size $\Delta t$ using Eq. [\[eq:dt\]](#eq:dt){reference-type="eqref" reference="eq:dt"} Update the deformation gradient tensor $\mathbb{F}^L$, particle position $\bm{r}_m$, rotation angle $\bm{\theta}^L$ and pseudo normal $\bm{n}$ for half time step $\Delta t/2$ Compute and correct the Cauchy stress ${\color{white}\contour{black}{$\sigma$}}^l$ (Sections [2.1.2](#sec:constitutive_relation){reference-type="ref" reference="sec:constitutive_relation"} and [2.1.3](#sec:stress_correction){reference-type="ref" reference="sec:stress_correction"}) Compute the resultants $\mathbb{N}^l$ and $\mathbb{M}^l$, and shear force $\bm{Q}^l$ (Eq. [\[resultants\]](#resultants){reference-type="eqref" reference="resultants"}) Compute the acceleration $\bm{\ddot u}_m$ (Eqs. [\[discrete_dynamic_equation1\]](#discrete_dynamic_equation1){reference-type="eqref" reference="discrete_dynamic_equation1"} and [\[artificial-stress\]](#artificial-stress){reference-type="eqref" reference="artificial-stress"}) and $\bm{\ddot n}$ (Eqs. [\[discrete_dynamic_equation2\]](#discrete_dynamic_equation2){reference-type="eqref" reference="discrete_dynamic_equation2"} and [\[artificial-torque\]](#artificial-torque){reference-type="eqref" reference="artificial-torque"}) Compute the angular acceleration $\bm{\ddot \theta}^L$ (Eq. [\[2D_conversion_relation3\]](#2D_conversion_relation3){reference-type="eqref" reference="2D_conversion_relation3"} for 2D problems, and Eqs. [\[3D_conversion_relation1\]](#3D_conversion_relation1){reference-type="eqref" reference="3D_conversion_relation1"} and [\[3D_conversion_relation2\]](#3D_conversion_relation2){reference-type="eqref" reference="3D_conversion_relation2"} for 3D problems) Update the velocity $\bm{\dot u}_m$ and angular velocity $\bm{\dot \theta}^L$ for a time step $\Delta t$ Compute the change rate of pseudo normal $\bm{\dot n}^L$ using Eq. [\[2D_n\_first_derivative\]](#2D_n_first_derivative){reference-type="eqref" reference="2D_n_first_derivative"} or [\[3D_n\_first_derivative\]](#3D_n_first_derivative){reference-type="eqref" reference="3D_n_first_derivative"} Compute the change rate of the deformation gradient tensor $\partial \mathbb{F}^L / \partial t$ (Eq. [\[change_rate_deformation_gradient\]](#change_rate_deformation_gradient){reference-type="eqref" reference="change_rate_deformation_gradient"}) Update the deformation gradient tensor $\mathbb{F}^L$, density $\rho$, particle position $\bm{r}_m$, rotation angle $\bm{\theta}^L$ and pseudo normal $\bm{n}$ for another half time step $\Delta t/ 2$ Terminate the simulation.  # Numerical examples {#sec:examples} To demonstrate the accuracy and stability of the proposed surface-particle SPH method (denoted as shell method), this section investigates a series of benchmark tests where analytical or numerical reference data from the literature or/and volume-particle SPH method (denoted as volume method) are available for qualitative and quantitative comparison. The smoothing length $h = 1.15~dp$, where $dp$ denotes the initial particle spacing, is employed in all the following simulations. ## 2D oscillating plate strip The first example involves a plate strip with initial uniform transverse velocity along the length with one edge fixed and the others free, which has previously been theoretically [@landau1986course] and numerically [@gray2001sph; @zhang2017generalized; @wu2023essentially] investigated in the literature. As shown in Figure [3](#figs:2D_plate_setup){reference-type="ref" reference="figs:2D_plate_setup"}(a), this plate strip is assumed to be infinitely long along the $y$-axis with a finite width $a = 0.2~\text{m}$ along the $x$-axis. To demonstrate that both thin and moderately thick structures can be simulated, this plate strip is modeled with the thicknesses $d = 0.01~\text{m}$ and $0.001~\text{m}$. The material properties are set as follows: density $\rho_0 = 1000.0 ~\text{kg} / \text{m}^3$, Young's modulus $E = 2.0~\text{MPa}$, and Poisson's ratio $\nu$ varies for different cases. Figure [3](#figs:2D_plate_setup){reference-type="ref" reference="figs:2D_plate_setup"}(b) shows the discrete model of the chosen cross-section with clamped edges at $x = 0$. {#figs:2D_plate_setup width="\\textwidth"} The transverse velocity $v_z$ is applied to the plate strip as $$v_z(x) = v_f c \frac{f(x)}{f(a)},$$ where $v_f$ is a constant that varies with different cases, and $$\begin{split} f(x) &= \left(\sin(ka) + \sinh(ka) \right) \left(\cos(kx) - \cosh(kx) \right) \\ & - \left(\cos(ka) + \cosh(ka) \right) \left(\sin(kx) - \sinh(kx) \right) \end{split}$$ with $k$ determined by $$\cos(ka) \cosh(ka) = -1$$ and $ka = 1.875$. The frequency $\omega$ of the oscillating plate strip is theoretically given by $$\omega ^2 = \frac{E d^2 k^4}{12 \rho \left(1 - \nu^2 \right)}.$$ Figure [4](#figs:2D_oscillating_plate_stress){reference-type="ref" reference="figs:2D_oscillating_plate_stress"} shows the particles with von Mises stress $\bar\sigma$ contour for the case of $d = 0.001~\text{m}$, $v_f = 0.01$, $\nu = 0.4$, and the initial particle spacing $dp = a / 40 = 0.005~\text{m}$. It should be noted that the present method predicts smooth deformation and stress fields without singularities for large rotations (more than $\pi$). {#figs:2D_oscillating_plate_stress width="\\textwidth"} Three different spatial resolutions, $a / dp =40$, $a / dp =80$, and $a / dp =160$, are tested in the convergence study. Figure [5](#figs:2D_oscillating_plate_convergence){reference-type="ref" reference="figs:2D_oscillating_plate_convergence"} shows the time history of vertical position $z$ of the strip endpoint with $d = 0.01~\text{m}$, $v_f = 0.025$ and $\nu = 0.4$. {#figs:2D_oscillating_plate_convergence width="\\textwidth"} It can be observed that typical 2nd-order convergence has been achieved. In addition, a long-term simulation is performed herein to demonstrate the numerical stability of the proposed formulation. For quantitative validation, $v_f$ $\nu$ $T_\text{Shell model}$ $T_\text{Theoretical}$ Error ------- ------- ------------------------ ------------------------ ------- 0.025 0.22 0.58137 0.54018 7.63% 0.05 0.22 0.57715 0.54018 6.92% 0.1 0.22 0.56801 0.54018 5.15%   0.025 0.30 0.56804 0.52824 7.53% 0.05 0.30 0.56308 0.52824 6.60% 0.1 0.30 0.55481 0.52824 5.03%   0.025 0.40 0.54447 0.50752 7.28% 0.05 0.40 0.53683 0.50752 5.78% 0.1 0.40 0.53252 0.50752 4.93% : 2D oscillating plate strip: Quantitative validation of the oscillation period for $a = 0.2~\text{m}$ and $d = 0.01~\text{m}$ with various $v_f$ and $\nu$. [\[tab:oscillating_plate_period1\]]{#tab:oscillating_plate_period1 label="tab:oscillating_plate_period1"} $v_f$ $\nu$ $T_\text{Shell model}$ $T_\text{Theoretical}$ Error -------- ------- ------------------------ ------------------------ ------- 0.0025 0.22 5.80249 5.40182 7.42% 0.005 0.22 5.75544 5.40182 6.55% 0.01 0.22 5.64181 5.40182 4.44%   0.0025 0.30 5.66756 5.28243 7.29% 0.005 0.30 5.61006 5.28243 6.20% 0.01 0.30 5.49156 5.28243 3.96%   0.0025 0.40 5.42826 5.07519 6.96% 0.005 0.40 5.34224 5.07519 5.26% 0.01 0.40 5.27522 5.07519 3.94% : 2D oscillating plate strip: Quantitative validation of the oscillation period for $a = 0.2~\text{m}$ and $d = 0.001~\text{m}$ with various $v_f$ and $\nu$. [\[tab:oscillating_plate_period2\]]{#tab:oscillating_plate_period2 label="tab:oscillating_plate_period2"} Tables [1](#tab:oscillating_plate_period1){reference-type="ref" reference="tab:oscillating_plate_period1"} and [2](#tab:oscillating_plate_period2){reference-type="ref" reference="tab:oscillating_plate_period2"} detail the oscillation period $T$ for a wide range of $v_f$ and $\nu$, obtained by the present method with the spatial particle resolution $a / dp =160$, when thickness $d = 0.01~\text{m}$ and $0.001~\text{m}$, respectively, and the comparison to theoretical solution obtained form small perturbation analysis. The differences, which are less than 8.00% for $\nu = 0.22$ and decrease to about 5.00% when the Poisson's ratio is increased to 0.4, are acceptable. ## 3D square plate {#3D_plate_steady} In this section, a 3D square plate under different types of boundary conditions is considered for quasi-steady analyses, as shown in Figure [6](#figs:3D_square_plate_setup){reference-type="ref" reference="figs:3D_square_plate_setup"}. {#figs:3D_square_plate_setup width="0.85 \\textwidth"} With side length $a = b =$ 254 mm and thickness $d =$ 25.4 mm, the plate material is defined with density $\rho_0 = 1600 ~\text{kg} / \text{m}^3$, Young's modulus $E = 53.7791 ~\text{GPa}$ and Poisson's ratio $\nu = 0.3$. Three types of boundary conditions denoted as SS0, SS1 and SS3 following Refs. [@reddy2006theory; @lin2014efficient] are implemented as - SS0: constrained mass center without constrained boundaries; - SS1: $u = w = \varphi = 0$ on edges parallel to $x$-axis and $v = w = \theta = 0$ on edges parallel to $y$-axis; - SS3: $u = v = w = 0$ on all edges. Note that, for the case of SS0, the outer square ring with width $d$ is imposed with negative pressure $q_{02}$. The uniformly distributed loads are parameterized by the loading factors $\bar P$ and $\bar P_1$ as $q_0 = \bar PE\left( {d/a} \right)^4$, $q_{01} = \bar P_1 E\left( {d/a} \right)^4$ and $q_{02} (2ad + 2bd + 4 d^2) = q_{01} a b$, so that the applied negative force along the $z$-axis prevents the center of mass from moving. For comprehensive validation, a convergence study of tests with SS0 is conducted, and the results are compared with those obtained by the volume method released in the SPHinXsys repository [@zhang2021sphinxsys]. Figure [7](#figs:3D_square_plate_comparison_volumn_stress){reference-type="ref" reference="figs:3D_square_plate_comparison_volumn_stress"} shows the particle distribution and stress fileds under the loading factor $\bar P_1 = 25$ with the spatial discretization $d/dp = 8$. {#figs:3D_square_plate_comparison_volumn_stress width="\\textwidth"} ![3D square plate: Load-deflection curves of tests with SS0 under three different spatial resolutions, and their comparison with those of the volume method [@zhang2021sphinxsys].](3D_square_plate_comparison_volumn.pdf){#figs:3D_square_plate_comparison_volumn width="\\textwidth"} Figure [8](#figs:3D_square_plate_comparison_volumn){reference-type="ref" reference="figs:3D_square_plate_comparison_volumn"} shows the non-dimensional deflection $\bar w_C = w_C /d$ and $\bar w_A = w_A / d$ probed at the central point $C$ and corner point $A$, respectively, obtained by both SPH shell and volume methods. It should be emphasized that there are only quite small differences between the results of the present reduced-dimensional and full-dimensional models. The particles colored by von Mises stress $\bar\sigma$ at the mid-surface for three spatial discretizations, $a/dp = 20$, $a/dp = 40$ and $a/dp = 80$, with the SS1 and SS3 boundary conditions under $\bar P = 200$ are shown in Figure [\[figs:3D_square_plate_SS1_SS3_stress\]](#figs:3D_square_plate_SS1_SS3_stress){reference-type="ref" reference="figs:3D_square_plate_SS1_SS3_stress"}. It can be observed that the regular particle distribution and smooth stress field are obtained. Also, both the deformation and von Mises stress $\bar \sigma$ exhibit good convergence properties with particle refinement. In order to demonstrate the accuracy of the present method, the non-dimensional deflections $\bar w_C$ for tests with SS1 and SS3 under various spatial resolutions are compared to those of the Ref. [@reddy2006theory]. As shown in Figs. [\[figs:3D_square_plate_SS1_SS3_comparison\]](#figs:3D_square_plate_SS1_SS3_comparison){reference-type="ref" reference="figs:3D_square_plate_SS1_SS3_comparison"}, the numerical results quickly converge to the reference solutions obtained by the finite element method (FEM) with increasing resolution. ## Dynamic response of a 3D square plate Following Ref. [@momenan2018new], the 3D square plate studied in Section [4.2](#3D_plate_steady){reference-type="ref" reference="3D_plate_steady"} is considered herein with the thickness $d = 12.7~\text{mm}$ and Young's modulus $E = 68.94 ~\text{GPa}$. The SS0 and SS3 boundary conditions are applied for dynamic analyses under a step loading of uniform normal pressure $q_{01} = q_0 = 2.068427 ~\text{MPa}$. For convergence study, three different spatial discretizations, i.e., $d/dp = 2$, $d/dp = 4$ and $d/dp = 8$, are considered. For quantitative validation, Figure [9](#figs:3D_plate_dynamic_solution_comparison_volumn){reference-type="ref" reference="figs:3D_plate_dynamic_solution_comparison_volumn"} shows the time history of the deflections $w_C$ probed at the central point $C$ and $w_A$ at the corner point $A$ with SS0 boundary condition and its comparison to the results obtained by the volume method. Also, Figure [10](#figs:3D_plate_dynamic_solution_comparison){reference-type="ref" reference="figs:3D_plate_dynamic_solution_comparison"} shows the time history of the deflection $w_C$ with the SS3 boundary condition and its comparison with that of Ref. [@momenan2018new]. In general, the present results are in good agreements with those obtained by the volume method and of Ref. [@momenan2018new]. {#figs:3D_plate_dynamic_solution_comparison_volumn width="\\textwidth"} {#figs:3D_plate_dynamic_solution_comparison width="0.5 \\textwidth"} ## 3D cantilevered plate Following Refs [@sze2002stabilized; @sze2004popular; @payette2014seven], the static response of a 3D cantilevered plate subjected to a distributed end shear load $f_0$ is considered. {#figs:3D_cantilevered_plate_setup width="0.5 \\textwidth"} As shown in Figure [11](#figs:3D_cantilevered_plate_setup){reference-type="ref" reference="figs:3D_cantilevered_plate_setup"}, the plate with length $a = 10~\text{m}$, width $b = 1~\text{m}$ and thickness $d = 0.1~\text{m}$ is clamped at $y = 0$, and has material parameters of density $\rho_0 = 1100 ~\text{kg} / \text{m}^3$, Young's modulus $E = 1.2 ~\text{MPa}$ and Poisson's ratio $\nu = 0.0$. The shear load is parameterized by a loading factor $\bar F$ as $f_0 = \bar F EI / a^2$ with the inertia moment $I = bd^3/12$. Three different resolutions, i.e., $b/dp = 5$, $b/dp = 7$ and $b/dp = 9$, are considered for convergence study. Figure [12](#figs:3D_cantilevered_plate_stress){reference-type="ref" reference="figs:3D_cantilevered_plate_stress"} shows the particles colored by the vertical displacement under different loading factor $\bar F$ at the spatial resolution $b / dp = 9$. A regular particle distribution and smooth vertical displacement field are noted. {#figs:3D_cantilevered_plate_stress width="50%"} ![ 3D cantilevered plate: Load-deflection curves with three various spatial discretizations, and their comparison with that of Payette et al. [@payette2014seven].](3D_cantilevered_plate_comparison.pdf){#figs:3D_cantilevered_plate_comparison width="0.5 \\textwidth"} Figure [13](#figs:3D_cantilevered_plate_comparison){reference-type="ref" reference="figs:3D_cantilevered_plate_comparison"} gives the displacement $u_c$ and $w_c$ of the point $C$, defined in Figure [11](#figs:3D_cantilevered_plate_setup){reference-type="ref" reference="figs:3D_cantilevered_plate_setup"}, as a function of the loading factor $\bar F$ and the initial particle spacing $dp$, and their comparison with those in Ref. [@payette2014seven]. It can be noted that the displacement is converging rapidly, again at about 2nd-order, with increasing resolution, demonstrating the accuracy of the present method. ## Scordelis-Lo roof As shown in Figure [14](#figs:3D_roof_setup){reference-type="ref" reference="figs:3D_roof_setup"}, the Scordelis-Lo roof with length $a = 50~\text{m}$, radius $r = 25~\text{m}$, thickness $d = 0.25~\text{m}$ and $\beta = 40^\circ$ is considered herein, and the material properties are density $\rho_0 = 36 ~\text{kg} / \text{m}^3$, Young's modulus $E = 432 ~\text{MPa}$ and zero Poisson's ratio. {#figs:3D_roof_setup width="0.5 \\textwidth"} The roof is supported at its ends by fixed diaphragms, i.e. the translations in $x$ and $z$ directions are constrained, and subjected to a gravity loading of $g = 10 ~\text{m} / \text{s}^2$. The FEM solution of the vertical displacement $w$ at the midpoint of the side edge converges to 0.3024 $\text{m}$ as reported in Refs. [@belytschko1985stress; @simo1989stress]. A sequentially refined resolutions of $b/dp = 15, 20, 25, 30~\text{and}~40$ with $b = 2 r \beta$ denoting the arc length of the roof end are considered to assess the convergence property of the present method. {#figs:3D_roof_convergence_stress width="\\textwidth"} {#figs:3D_roof_convergence width="0.5 \\textwidth"} Figure [15](#figs:3D_roof_convergence_stress){reference-type="ref" reference="figs:3D_roof_convergence_stress"} shows the particles colored with the von Mises stress $\bar \sigma$ of the mid-surface obtained at different resolutions. The regular particle distribution and smooth stress fields are noted. With increasing resolution, a clear convergence is exhibited. The profile of displacement $w$ with varying spatial resolution obtained by the present method is depicted in Figure [16](#figs:3D_roof_convergence){reference-type="ref" reference="figs:3D_roof_convergence"}. It can be noted that the result converges rapidly to $w = 0.2991~\text{m}$ when $b/dp = 40$ with 1.09% error compared to the solution of Refs. [@belytschko1985stress; @simo1989stress]. ## Pinched hemispherical shell We now consider a pinched hemispherical shell with an $18^\circ$ circular cutout at its pole following Refs. [@simo1990stressIV; @buechter1992shell; @jiang1994simple; @sze2004popular; @payette2014seven]. As shown in Figure [17](#figs:3D_pinched_hemisphere){reference-type="ref" reference="figs:3D_pinched_hemisphere"}(a), the hemispherical shell with the radius $r = 10.0~\text{m}$ and thickness $d = 0.04~\text{m}$ is loaded by four alternating radial point forces $\bm{F}$, prescribed along the equator at $90^\circ$ intervals. {#figs:3D_pinched_hemisphere width="\\textwidth"} A linear elastic material with the density $\rho_0 = 1100 ~\text{kg} / \text{m}^3$, Young's modulus $E = 68.25 ~\text{MPa}$ and Poisson's ratio $\nu = 0.3$ is applied. Figure [17](#figs:3D_pinched_hemisphere){reference-type="ref" reference="figs:3D_pinched_hemisphere"}(b-d) shows the distribution of von Mises stress $\bar \sigma$ at the mid-surface under varying magnitude of the point force $\bm{F}$ . The regular particle distribution is observed, although slight stress fluctuation is exhibited near the place where the point force is applied. For quantitative analysis and convergence study, the radial deflections $w_A$ and $w_B$ of monitoring points $A$ and $B$ as a function of the point force magnitude and resolution are compared with those of Ref. [@sze2004popular]. Three different spatial discretizations, i.e., $2 \pi r/dp = 80, 160~\text{and}~240$, are considered for convergence study. As shown in Figure [18](#figs:3D_hemisphere_comparison){reference-type="ref" reference="figs:3D_hemisphere_comparison"}, the results of present SPH shell model is quickly converging to those of Ref. [@sze2004popular]. ![Pinched hemispherical shell: Curves of radical displacements of points $A$ and $B$ as a function of the point force magnitude and spatial resolution, and their comparison with those of Sze et al. [@sze2004popular].](3D_pinched_hemisphere_comparison.pdf){#figs:3D_hemisphere_comparison width="\\textwidth"} ## Pulled-out cylindrical shell A more challenging benchmark test with large displacements is considered in this section following Refs. [@maurel2008sph; @jiang1994corotational]. As shown in Figure [19](#figs:3D_cylinder_setup){reference-type="ref" reference="figs:3D_cylinder_setup"}, a cylindrical shell with the radius $r = 5.0~\text{m}$, length $a = 10.35~\text{m}$ and thickness $d = 0.094~\text{m}$ is subjected to a pair of point forces $\bm{F}$ which are equal in magnitude and opposite in direction. {#figs:3D_cylinder_setup width="50%"} A linear elastic material with the density $\rho_0 = 1100 ~\text{kg} / \text{m}^3$, Young's modulus $E = 10.5 ~\text{MPa}$ and Poisson's ratio $\nu = 0.3125$ is applied. Figure [20](#figs:3D_cylinder_stress){reference-type="ref" reference="figs:3D_cylinder_stress"} shows the distribution of von Mises stress $\bar \sigma$ at the mid-surface under varying magnitude of the point force $\bm{F}$. The regular particle distribution and smooth stress fields, even close to the place where the point force is applied, are observed. {#figs:3D_cylinder_stress width="\\textwidth"} ![Pulled-out cylindrical shell: Curves of radical displacements of points $A$, $B$ and $C$ as a function of the point force magnitude and spatial resolution, and their comparison with those of Maurel and Combescure [@maurel2008sph] and Jiang et al. [@jiang1994corotational].](3D_cylinder_comparison.pdf){#figs:3D_cylinder_comparison width="\\textwidth"} For quantitative analysis and convergence study, the radial displacements $w_A$, $w_B$ and $w_C$ of monitoring points $A$, $B$ and $C$ as a function of the point force magnitude and resolution are compared with those of Ref. [@maurel2008sph; @jiang1994corotational]. Three different spatial discretizations, i.e., $b/dp = 80, 160~\text{and}~240$ with $b = 2 \pi r$ denoting the circumference length of the end, are considered for convergence study. As shown in Figure [21](#figs:3D_cylinder_comparison){reference-type="ref" reference="figs:3D_cylinder_comparison"}, the bifurcation point of the curve is accurately predicted, suggesting good accuracy and robustness of the present method. ## Pinched semi-cylindrical shell We further consider a pinched semi-cylindrical shell with finite deformation and rotation following Refs. [@stander1989assessment; @brank1995implementation; @sze2004popular; @arciniega2007tensor]. As shown in Figure [22](#figs:3D_pinched_cylinder){reference-type="ref" reference="figs:3D_pinched_cylinder"}(a), the semi-cylindrical shell with the radius $r = 1.016~\text{m}$, length $a = 3.048~\text{m}$ and thickness $d = 0.03~\text{m}$ is completely clamped at a circumferential periphery and experiences a pinching force at the center of free-hanging periphery. {#figs:3D_pinched_cylinder width="\\textwidth"} Along its longitudinal edges, the vertical direction and the rotation about the $y$-axis are constrained. The elastic material properties are density $\rho_0 = 1100 ~\text{kg} / \text{m}^3$, Young's modulus $E = 20.685 ~\text{MPa}$ and Poisson's ratio $\nu = 0.3$. Figure [22](#figs:3D_pinched_cylinder){reference-type="ref" reference="figs:3D_pinched_cylinder"}(b-d) shows the distribution of von Mises stress $\bar \sigma$ at the mid-surface under varying magnitude of the point force $\bm{F}$ . Noted that the present method features regular particle distribution and smooth stress fields, even close to the constrained edges and place where the point force is applied, without singularities for finite rotations (more than $0.5\pi$). For quantitative analysis and convergence study, the downward deflection $w_A$ of monitoring point $A$ as a function of the point force magnitude and resolution is compared with that of Ref. [@sze2004popular]. Three different spatial discretizations, i.e., $\pi r / dp = 20, 40~\text{and}~80$, are considered for convergence study. As shown in Figure [23](#figs:3D_pinched_cylinder_comparison){reference-type="ref" reference="figs:3D_pinched_cylinder_comparison"}, the result difference obtained by the present SPH shell method between different resolution rapidly decreases as the spatial refinement, and the results agree well with those of Ref. [@sze2004popular]. ![Pinched semi-cylindrical shell: Curves of radical displacements of point $A$ as a function of the point force magnitude and spatial resolution, and their comparison with those of Sze et al. [@sze2004popular].](3D_pinched_semi_cylinder_comparison.pdf){#figs:3D_pinched_cylinder_comparison width="50%"} # Concluding remarks {#sec:conclusion} In this paper, we present a reduced-dimensional SPH method for quasi-static and dynamic analyses of both thin and moderately thick plate and shell structures. By introducing two reduced-dimensional linear-reproducing correction matrices, the method reproduces linear gradients of the position and pseudo-normal. The finite deformation is taken into account by considering all terms of strain with the help of Gauss-Legendre quadrature along the thickness. To cope with large rotations, the method introduces weighted non-singularity conversion relation between the rotation angles and pseudo normal. A non-isotropic Kelvin-Voigt damping and a momentum-conserving hourglass control algorithm with a limiter are also proposed to increase numerical stability and to suppress hourglass modes. An extensive set of numerical examples have been investigated to demonstrate the accuracy and robustness of the present method. Note that, while the plate and shell structure considered here have moderate and high modulus, one extension of the present method is for soft thin structures, such as membranes. Another outlook, along with the multi-physical modeling within unified computational framework, is to develop the SPH method for the interaction between fluid and thin structures. # CRediT authorship contribution statement {#credit-authorship-contribution-statement .unnumbered} **Dong Wu:** Conceptualization, Methodology, Investigation, Visualization, Validation, Formal analysis, Writing - original draft, Writing - review & editing; **Chi Zhang:** Investigation, Writing - review & editing; **Xiangyu Hu:** Supervision, Methodology, Investigation, Writing - review & editing. # Declaration of competing interest {#declaration-of-competing-interest .unnumbered} The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. # Acknowledgment {#acknowledgment .unnumbered} D. Wu is partially supported by the China Scholarship Council (No. 201906130189). D. Wu, C. Zhang and X.Y. Hu would like to express their gratitude to the German Research Foundation (DFG) for its sponsorship of this research under grant number DFG HU1527/12-4. # References {#references .unnumbered}

arxiv_math

--- abstract: | By studying laminations of the unit disk, we can gain insight into the structure of Julia sets of polynomials and their dynamics in the complex plane. The polynomials of a given degree, $d$, have a parameter space. The hyperbolic components of such parameter spaces are in correspondence to rotational polygons, or classes of "rotational sets", which we study in this paper. By studying the count of such rotational sets, and therefore the underlying structure behind these rotational sets and polygons, we can gain insight into the interrelationship among hyperbolic components of the parameter space of these polynomials. These rotational sets are created by uniting rotational orbits, as we define in this paper. The number of such sets for a given degree $d$, rotation number $\frac pq$, and cardinality $k$ can be determined by analyzing the potential placements of pre-images of zero on the unit circle with respect to the rotational set under the $d$-tupling map. We obtain a closed-form formula for the count. Though this count is already known based upon some sophisticated results, our count is based upon elementary geometric and combinatorial principles, and provides an intuitive explanation. author: - John C. Mayer - Michael J. Moorman - Gabriel B. Quijano - Matthew C. Williams date: September 2023 title: Counting Rotational Sets for Laminations of the Unit Disk from First Principles --- # Introduction ## Motivation What are "rotational sets\" for laminations for the unit disk under the action of the angle $d$-tupling map, and why count them? Laminations are a topological and combinatorial model of the connected Julia sets of polynomials considered as functions of the complex numbers, modeled by the plane. Such models are used both to understand specific types of polynomials and their Julia sets, and to study the parameter spaces of polynomials. For example, the well-known Mandelbrot set [@Mandel] is the parameter space of quadratic polynomials of the form $P_c(z)= z^2+c$ with parameter $c$ with connected Julia set. The so-called hyperbolic components of that parameter space are of interest, including how they are connected to each other, how they are arranged in the Mandelbrot set, and how many components there are that are associated with attractive orbits (of the associated polynomials) of a given period, rotation number, and the like. These terms are defined below. Our research is concerned with polynomials of higher degree ($d>2$), about which much less is currently understood. Laminations are composed of *leaves* (chords of the unit circle) which form a closed collection of non-crossing segments that are forward and backward invariant under a natural extension of the *angle $d$-tupling map* (the angular part or argument of the complex power function $z\mapsto z^d$, where $z=r e^{2\pi it}$ and the argument is the exponent of $e$) on the unit circle. Leaves connecting points of a rotational set in circular order form polygons in the lamination. There is a correspondence between rotational polygons in laminations and fixed points of polynomials that have a non-zero infinitesimal rotation number (determined by the argument of the derivative of the polynomial at the fixed point). Such polygons are in correspondence to a fundamental class of hyperbolic components of the parameter space of degree $d$ polynomials with connected Julia set. For example, in the Mandelbrot set for the hyperbolic component marked star in Figure [3](#mandel){reference-type="ref" reference="mandel"}, all the Julia sets have a (repelling) fixed point (the marked point in the Julia set) which is represented in the lamination for that Julia set by a rotational triangle (marked star). The Julia set is actually the boundary of the shaded blue region, which contains all the points running off to infinity under iteration of the polynomial. The white regions in the lamination correspond to the black regions in the "filled-in\" Julia set. The filled-in Julia set consists of all points whose orbits under iteration of the polynomial are bounded. "Counting \... from First Principles\" in our title indicates that we will use the most fundamental geometric and combinatorial properties of the angle $d$-tupling map to make the count. By studying laminations in the abstract without reference to a particular polynomial or Julia set, we aim to reverse the process by which a Julia set leads to a lamination. By understanding what is possible for laminations, we can constrain what is possible for locally connected Julia sets. Our main result is Theorem [Theorem 21](#count){reference-type="ref" reference="count"}. For a preview of where that theorem takes us with Julia sets, skip ahead to view Figure [15](#rabbits){reference-type="ref" reference="rabbits"}. {#mandel width=".3\\textwidth"} {#mandel width=".3\\textwidth"} {#mandel width=".3\\textwidth"} ## Orbits, the Angle $d$-tupling Map, and Itineraries Most of the following definitions are adapted from [@Brittany:2023] and [@Blokh:2006]. Elementary proofs of some propositions are left to the reader. **Definition 1**. Let $f:X\to X$ be a function. By $f^q(x)$ we denote the composition $f(f(\dots f(x)\dots))$, where $f$ is composed with itself $q$ times. By convention, $f^0(x)=x$. **Definition 2**. Given a point $x\in X$, the *orbit* of $x$ is the set $\mathcal{O} = \{f^n(x)\}^{\infty}_{n=0}$. We denote the unit circle in the Cartesian plane by $S^1=\{(x,y)\mid x^2+y^2=1\}$. We will describe points in $S^1$ by their central angle and we will measure angles in revolutions rather than radians or degrees. A full circle is 1 revolution. We say we measure angles *mod 1*, such that angles which differ by a full revolution are the same. We now consider the specific function $\sigma_d$ in whose orbits we are interested. **Definition 3**. Let $d\in\mathbb{Z}$ with $d>1$. Define the *angle $d$-tupling map* $\sigma_d:S^1\to S^1$ by $\sigma_d(x)=dx \pmod 1$. We represent points on the circle coordinatized by $[0,1)$ by their base $d$ expansion. In base $d=2$, the notation $\_001$ denotes for us that the digits $001$ repeat infinitely which, as the reader can check, is the point $\frac13\in[0,1)$. However, in base $d=3$, $\frac13$ is the point $1\_0$, which in our notation means that the initial digit $1$ does not repeat, but the digit $0$ repeats. We can use the tools of symbolic dynamics, particularly the *forgetful shift*. Because points under $\sigma_d$ are multiplied by $d$ and are taken modulo $1$, the first digit of the base $d$ expansion becomes the integer part which goes away after taking the modulus. So we can quickly calculate the next point in an orbit by simply "forgetting" the leading digit (when written as a base $d$ expansion). For convenience, we will define $\mathbb{Z}_+$ to be $\{i\in\mathbb{Z}\mid i\geq0\}$. We denote the set of *pre-images* under $\sigma_d^q$ of a point $x$ by $\sigma_d^{-q}(x)=\{y\in S^1\mid \sigma_d^q(y)=x\}$. If there is a $q\in\mathbb{Z}_+$ such that $\sigma_d^q(x)=x$, then the orbit of $x$ is finite, and we say it is a *periodic orbit* and $x$ is a *periodic point*. If $q$ is least for which $\sigma_d^q(x)=x$, then we say $q$ is the *period* of the point (respectively, orbit). The set of points visited on those $q$ iterations make up the orbit $\mathcal O$ for that given point. If this $q$ exists, then for the set ${\mathcal O} = \{\sigma_d^n(x)\mid 0 \le n < q\}$ it is true that $\sigma_d:\mathcal O\to \mathcal O$ and $\sigma_d({\mathcal O})={\mathcal O}$. **Proposition 4**. *For a given degree $d$, the pre-images of $0$ are $$\sigma_d^{-1}(0)=\left\{0, \frac{1}{d}, \frac{2}{d},\dots,\frac{d-1}{d}\right\}$$ or when written in base $d$ expansion $$\sigma_d^{-1}(\_0)=\left\{\_0,1\_0,2\_0,\dots,(d-1)\_0\right\}$$* These pre-images serve as the border between neighboring intervals of length $\frac{1}{d}$ in $S^1$. **Definition 5**. Fix $d>1$. Define intervals $I_0=[0,\frac1d)$, and in general $I_j=[\frac{j}{d},\frac{j+1}{d})$ for $1\le j\le d-1$. Recall $0$ and $1$ are identical in $S^1$. Then $S^1$ is the disjoint union $\bigcup_{j=0}^{d-1} I_j$. For instance, the smallest non-zero pre-image of $0$, $\frac{1}{d}$ (equivalently, $1\_0$), is the border between $I_0$ and $I_1$. Note that each interval $I_j$ maps one-to-one in counterclockwise order from $\_0$ onto $S^1$. Consequently, within each $I_j$ there is a preimage of every $I_k$ consecutively in order, but of length $\frac{1}{d^2}$. With these tools in hand, we can now express orbits in a much more useful manner. ## Itineraries We have given the points in $S^1$ when considered under the map $\sigma_d$ their base $d$ expansion. This will allow us to describe the orbit of a point in the circle under $\sigma_d$ in terms the visits of its orbit to the distinguished intervals in Definition [Definition 5](#preimage0){reference-type="ref" reference="preimage0"}. The *itinerary* of a point $s\in S^1$ is the ordered list of its visits in its orbit to the distinguished intervals. The proof of the following Proposition is left to the reader. **Proposition 6**. *Under $\sigma_d$, the itinerary of a point is exactly its base $d$ expansion. Consequently, two points $s$ and $t$ in $S^1$ under $\sigma_d$ have the same itinerary if, and only if, $s=t$.* Now, the points of an orbit are defined by their relative placement to pre-images of zero. The utility in this is found in how orbits can be defined by the location of pre-images relative to the points of the orbit. This will be essential to counting rotational sets from first principles. {#fig:itinerary} Because any given itinerary for a point in a periodic orbit can be shifted to find the itineraries of every other point in that orbit, periodic orbits can be clearly referenced with the itinerary of any given point it contains. For consistency we will define orbits to have the same itinerary as their smallest (compared to $\_0$) point. **Definition 7**. Let ${\mathcal O}=\{0\leq x_1<x_2<x_3\dots<x_q<1\}$ be a periodic orbit. Define $Itin(\mathcal{O})=Itin(x_1)$. **Definition 8**. A *gap* is a complementary interval in $S^1 \setminus \mathcal{O}$. ## Spatial and Temporal Order of Orbits The count of rotational sets is fundamentally based upon the comparison of spatial and temporal orders of points in an orbit with reference to the pre-images of $\_0$. See Figure [5](#fig:temporal-spatial){reference-type="ref" reference="fig:temporal-spatial"}. **Definition 9**. *Spatial order* refers to the ordering of points in an orbit by value, from smallest to largest in $[0,1)$. In terms of a map onto $S^1$, this order is given by starting at $\_0$ and following our points counterclockwise. **Definition 10**. *Temporal order* orders the points in an orbit by starting with the lowest valued point in our orbit (or closest to $\_0$ spatially, measuring from $\_0$ counterclockwise) and applying $\sigma_d$ repeatedly. So, temporal ordering is based on how our itinerary is followed by repeated applications of $\sigma_d$. {#fig:temporal-spatial} # Rotational Sets Consider $\sigma_d:S^1\to S^1$ for a particular $d > 1$. **Definition 11**. Let ${\mathcal O}=\{x_1<x_2<x_3\dots<x_q\}$ be a periodic orbit under $\sigma_d$. If, and only if, there exists a $p \in \mathbb{Z}^+$ such that for all $i \in \{1,2,\dots q\}$, $\sigma_d(x_i)= x_{i+p\pmod q}$, then we say that $\mathcal{O}$ is a *rotational* periodic orbit with *rotation number* $\frac{p}{q}$ (in lowest terms). *Remark 12*. Note that the numerator $p$ of the rotation number $p/q$ of a rotational periodic orbit is sufficient to determine the temporal order of its points. Along with how many points of the orbit are in each interval, this is enough to determine its itinerary. It follows from Proposition [Proposition 6](#itin){reference-type="ref" reference="itin"} that if, and only if, two rotational periodic orbits have the same $p/q$ and each of their corresponding points are in the same intervals, then they have the same itinerary and are the same orbit. As for the practical generation of this itinerary, it can be found by reading off the interval of each point by starting with $x_1$ and "jumping" forwards $p$ points counter-clockwise along $S^1$, repeating until getting back to that initial point. Consider what $p$ is in the rotational orbit in Figure [5](#fig:temporal-spatial){reference-type="ref" reference="fig:temporal-spatial"}. A consequence of how the rotation number of an orbit can describe the forward orbits of its points is that it can also describe their pre-images. Since $\_0$ lies between $x_1$ and $x_q$, at least one pre-image of $\_0$ must lie between $x_{q-p} \in \sigma_d^{-1}(x_{q})$ and $x_{1+q-p} \in \sigma_d^{-1}(x_{1})$. We call such pre-images the *Principal Pre-image* of their respective orbits. **Definition 13**. Let the *principal preimage* for a rotational orbit $\mathcal{O}$ be a pre-image of $\_0$ lying between $x_{q-p} \in \sigma_d^{-1}(x_{q})$ and $x_{1+q-p} \in \sigma_d^{-1}(x_{1})$. The reader can verify that there must always be a principal preimage. ## Rotational Sets Containing Multiple Orbits Not only can points rotate together while maintaining order, but so too can multiple orbits together, forming a rotational set. **Definition 14**. Let ${\mathcal P}=\{x_i\mid 0\leq x_1<x_2<x_3<\dots<x_{qk}<1\}$ be a finite set in consecutive counterclockwise order in $S^1$. We say $\mathcal P$ is a *rotational set* containing $k$ orbits with rotation number $\frac{p}{q}$ for $\sigma_d$ if, and only if, 1. $\sigma_d({\mathcal{P}})={\mathcal P}$ 2. $x_i,x_{i+1},\dots,x_{i+k-1 \pmod {qk}}$ for $i \in [1,2,\dots qk]$ are in different orbits 3. $x_i$ moves to $x_{i + pk\pmod{qk}}$ **Definition 15**. Let $G_i = \{x_{(i-1)k + 1}, x_{(i-1)k + 2},\dots,x_{ik} \}$ where $i\in[1,2,\dots,q]$. We say $G_i$ is the $i$th *group* and $G_i$ is the set of the $i$th points spatially of each orbit. *Remark 16*. Items (2) and (3) of Definition 2.3 also show that $G_i$ moves together preserving spatial order to $G_{i + p\pmod{q}}$. **Definition 17**. Let a *principal preimage* for a rotational set ${\mathcal P}$ be the pre-image of $\_0$ that lies between $G_{q-p} \subset \sigma_d^{-1}(G_{q})$ and $G_{q-p+1} \subset \sigma_d^{-1}(G_{1})$. *Remark 18*. For the principal pre-image of a set, it must lie between $x_{(q-p)k} \in \sigma_d^{-1}(x_{qk})$ and $x_{1+(q-p)k} \in \sigma_d^{-1}(x_{1})$. $x_{1+(q-p)k}$ is the smallest point in group $G_{q-p+1}$ and $x_{(q-p)k}$ is the largest point in group $G_{q-p}$, therefore the principal pre-image lies between those two groups. {#fig:rotSet} In order to aid with the identification and counting of rotational sets, we need to differentiate gaps between points that are within groups from those outside groups. This distinction is necessary as pre-images that lie in the former are what differentiate orbits within a rotational set from each other. In other words, the pre-images that are within the groups are such that if they were removed, the orbits would no longer be different. **Definition 19**. Let *intra-group* gaps be gaps that lie within a group, or in other words are gaps that are in between two points from two different orbits that aren't the first and last orbits spatially in their group. **Definition 20**. Let *inter-group* gaps be gaps that lie outside a group (or between groups). These are all of the gaps that are not intra-group gaps. Note that with these definitions, the principal pre-image must always lie within an inter-group gap. See if you can build intuition for this by considering Figure [6](#fig:rotSet){reference-type="ref" reference="fig:rotSet"} through this new lens. # Algorithms and resulting Formulas ## Counting Rotational Sets Goldberg [@Goldberg:1992] counted the rotational orbits with a given rotation number $\frac{p}{q}$ for $\sigma_d$ and indicated that rotational sets containing multiple orbits with that rotation number could also be counted, providing an example count for $\sigma _3$, but no general formula. As a corollary to her characterization of rotational orbits in terms of their temporal and spatial placement with respect to the $d-1$ fixed points of $\sigma_d$, she showed that the maximal number of orbits in a rotational set for $\sigma_d$ was $d-1$. This also follows as a corollary to our main theorem, Theorem [Theorem 21](#count){reference-type="ref" reference="count"}. McMullen ([@McMullen:2010], Section 2) built upon Goldberg to provide a criterion for two orbits for $\sigma_d$ with the same rotation number to be compatible in one rotational set. Tan [@Tan:2019] used an algorithm based upon the Goldberg/McMullen criterion to count the number of rotational sets containing $k$ orbits for $\sigma_d$ with a given rotation number $\frac pq$. So, while the content of our main theorem is known, the proof here is new and more elementary. **Theorem 21** (Identifying and Counting Rotational Sets). *Consider the collection $\mathcal{B}$ of all rotational sets for a given degree $d$, rotation number $\frac{p}{q}$ in lowest terms, and number $k$ of distinct orbits per set. The cardinality of $\mathcal{B}$ is given by $$|\mathcal{B}| = \sum_{i=k-1}^l \left[ \binom{d+q-2}{d-2-i} \sum_{j=0}^{k-1} \left[ (-1)^j \binom{k-1}{j} \binom{q(k-1-j)}{i} \right] \right]$$* *where* *$$l = \begin{cases} q(k-1) & d-2 > q(k-1) \\ d-2 & otherwise \end{cases}$$* *Proof.* For any given rotational set $B$ within $\mathcal{B}$, there are $q$ points in each of the $k$ orbits within $B$. The count of each way to place pre-images between these neighboring points is the same as the count of rotational sets because the placement of pre-images dictates the digits for the itinerary of each point as can be seen in Remark [Remark 12](#itinToPreimage){reference-type="ref" reference="itinToPreimage"}. However, placing a pre-image between $\_0$ and the smallest point in the set is different from placing a pre-image between the largest point and $\_0$. The former would increase the digits in all the itineraries while the latter would not. Therefore, we need to count the number of ways to place pre-images within the gaps distinguished by $\_0$ and the points within $B$. Let the range of values from $0$ to $qk$ correspond with the gaps between neighboring points in the set of points that contain $\_0$ and the points within each orbit in $B$. $0$ corresponds with the gap between $\_0$ and the smallest point in $B$, $1$ with the gap between the smallest and second smallest, and so on. There are $d$ pre-images to place. The first is $\_0$, which is its own pre-image. The next is the principal pre-image, whose position is already determined (as can be seen in Remark [Remark 18](#principal){reference-type="ref" reference="principal"}). This leaves us with $d-2$ pre-images to place. Now, we must concern ourselves with where these pre-images can be placed to form a valid rotational set with $k$ orbits. **Lemma 22**. *Label gaps with their congruence class modulo $k$. Rotational sets in $\mathcal{B}$ must have at least one pre-image in each non-zero congruence class. Therefore, the cardinality of $\mathcal{B}$ is equivalent to that of $P$ when $P$ is defined to be the set of all sets composed of $d-2$ non-negative integers less than or equal to $qk$, such that each one contains at least one element from each non-zero congruence class modulo $k$.* *Proof.* In order to ensure differentiation between orbits, each orbit must be differentiated from its intra-group neighbors (the groups are differentiated by the principal pre-image and $\_0$). The first spatial orbit is not a neighbor with the last as they lie on opposite sides of any given group. So for each intra-group neighbor, an orbit must have at least one pre-image between one of its points and that neighbor's points. Therefore, this requirement can only be fulfilled by placing pre-images in intra-group gaps. Here is an example to provide clarity. The reader is invited to draw their own illustration for this example. For the first orbit (spatially) to be distinct from the second orbit, there must be a pre-image either in the gap between their smallest points respectively, second smallest, or any other pair of corresponding points. This particular restriction for the first and second orbits can be restated as the requirement for a pre-image to exist in a gap with label $n$ such that $n$ is in the congruence class of $1$ modulo $k$. This rule can be generalized for all intra-group gaps by separating them into congruence classes. For the $n$th and $(n+1)$th orbits to be differentiated, there must be at least one pre-image in the set of gaps with labels in $\{ x \in [0\dots qk] \mid x \in [n]_k\}$. Therefore, all non-zero congruence classes require at least one pre-image for the rotational set to be valid. The set of rotational sets, with the restrictions articulated above, will have the same cardinality as $P$ when defined as follows. $P$ is the set of all sets composed of $d-2$ non-negative integers less than or equal to $qk$, such that each one contains at least one element from each non-zero congruence class modulo $k$. ◻ We will now define sets that correspond with the choice of gaps in which to put pre-images. As of now, we are only placing one pre-image per gap even though more than one can be placed for a valid rotational set. This is done to make counting simpler later on. We will refer to the range of labels for gaps as $\lambda$, and define it as: $$\lambda = \{x \in \mathbb{Z}_{+} \mid x \leq qk\}$$ In other words, $\lambda$ is the set of non-negative integers less than or equal to $qk$. We will refer to the set of all labels for intra-group gaps as $\psi$ and define $\psi$ as $$\psi = \{x \in \lambda \mid x \ (\mathrm{mod}\ k) \neq 0 \}$$ There are multiple possible values for how many pre-images may be in intra-group gaps for any given rotational set. We will call the number of pre-images in intra-group gaps $i$. Define $T_i$ as the set of all sets that contain $i$ elements from $\psi$. We also know that $|T_i| = \binom{|\psi|}{i} = \binom{q(k-1)}{i}$ since $|\psi| = q(k-1)$ because there are $k-1$ non-zero congruence classes with $q$ integers in each. $$|T_i| = \binom{q(k-1)}{i}$$ **Lemma 23**. *The range for the possible number of pre-images in intra-group gaps varies from $k-1$ to $l$ where $$l = \begin{cases} q(k-1) & d-2 > q(k-1) \\ d-2 & otherwise \end{cases}$$* *Proof.* Only $k - 1$ pre-images are required for differentiation. The minimal value of $i$ is the minimal number of pre-images required for differentiation, $k - 1$. As for the maximum value of $i$, it can be limited by either the number of empty gaps, $q(k - 1)$, or the number of pre-images to place, $d-2$. Therefore, the maximal value of $i$ is $l$ where $$l = \begin{cases} q(k-1) & d-2 > q(k-1) \\ d-2 & otherwise \end{cases}$$ ◻ The sets in $T_i$ for $i \in [k-1, k, \dots ,l]$ that follow our requirement of differentiation are valid (yet may not be complete as these only correspond to $i$ pre-images of the $d-2$ that need to be placed). So we will define $\gamma_i$ as the subset of $T_i$ that contains all the sets where there is at least one element from each non-zero congruence class modulo $k$. **Lemma 24**. *The count of sets in $\mathcal{B}$ such that each $B \in \mathcal{B}$ has $i$ pre-images in its intra-group gaps is given by $$\sum_{j=0}^{k-1} (-1)^{j} \binom{k-1}{j} \binom{q(k-1-j)}{i}$$* *Proof.* Each set within $P$ that has $i$ elements in non-zero congruence classes corresponds with an element in $\gamma_i$. Similarly, each rotational set in $\mathcal{B}$ that has $i$ pre-images in intra-group gaps corresponds with an element in $\gamma_i$. For counting purposes, we found it easier to find $T_i \setminus \gamma_i$ than $\gamma_i$, so we will define $W_i=T_i \setminus \gamma_i$. We can identify and count the sets in $W_i$ (as opposed to the valid sets in $\gamma$) by noticing the equivalence of the problem with finding the union of sets. The goal is to find every possible way to leave at least one congruence class unfilled. Categorize each placement as belonging to a series of sets $(C_1, C_2,\dots , C_{k-1})$ where $C_j$ is defined as the set of elements of $T_i$ where the $j$th congruence class is not represented. With this definition, a singular placement can belong to multiple sets $C_j$, and we seek the union of each of these sets. In other words, $\bigcup\limits_{j=1}^{k-1} C_j = W_i$ because it contains every set where at least 1 non-zero congruence class is not represented. This is a well known problem, and the solution utilizes the inclusion-exclusion principle [@InEx] with the count given by $$|W_i| = \left|\bigcup\limits_{j=1}^{k-1} C_j\right| = \sum_{j=1}^{k-1} (-1)^{j+1} \binom{k-1}{j} \binom{q(k-1-j)}{i}$$ Since $\gamma_i$ is defined as the set difference of $T_i$ and $W_i$, $$|\gamma_i| = |T_i| - |W_i| = \binom{q(k-1)}{i} - \sum_{j=1}^{k-1} (-1)^{j+1} \binom{k-1}{j} \binom{q(k-1-j)}{i}$$which can be rewritten as $$\sum_{j=0}^{k-1} (-1)^{j} \binom{k-1}{j} \binom{q(k-1-j)}{i}$$ ◻ We have identified and counted all valid placements of intra-group pre-images. In order to finish constructing a rotational set, the remaining pre-images must be placed in between groups. For each of these placements in $\gamma$, the number of pre-images left to place naturally depends on the number of pre-images already placed. For any given set $g \in \gamma$ with cardinality $i$, there are $d-2-i$ elements from $\lambda$ (all the labels for the gaps) that need to be added to construct an element in $\mathcal{B}$. There are a few limitations on which gaps pre-images can be placed in, due to the fact that our prior count depends on certain gaps lacking pre-images and others having at least one. Therefore, we can place pre-images in intra-group gaps that already have pre-images or any given inter-group gap. In other words, the added $d-2-i$ elements must either be duplicates of elements already in $g$ which are also in $\psi$ or elements in $\lambda \setminus \psi$. There are $\binom{d+q-2}{d-2-i}$ ways to choose these integers, therefore there are these many elements in $P$ and corresponding orbits for any given set $g$. As argued before, $i$ (the cardinality of $g$) ranges from $k-1$ to $l$. Bringing all of this together, $|\mathcal{B}|$ and $|P|$ are equal to the sum of $\binom{d+q-2}{d-2-i}|\gamma|$ over possible values of $i$. $$|P| = |\mathcal{B}| = \sum_{i=k-1}^l \left[ \binom{d+q-2}{d-2-i} \sum_{j=0}^{k-1} \left[ (-1)^j \binom{k-1}{j} \binom{q(k-1-j)}{i} \right] \right]$$ ◻ ## Identifying Rotational Sets Containing a Given Orbit This count is certainly insightful, but in order to gain insight as to specific laminations that one may examine, it would be useful to be able to count and identify rotational sets that contain a given orbit. Here we state that this can be done, using a method quite similar to the method used in the proof of the previous counting theorem. **Theorem 25** (Identifying Maximal Rotational Sets that contain a given orbit). *Consider the orbit $\mathcal{O}$ with degree $d$ and rotation number $\frac{p}{q}$ in lowest terms. An exhaustive list of all the rotational sets that contain $\mathcal{O}$ can be found algorithmically, in a way inspired by the previous proof. The count of maximal rotational sets that contain $\mathcal{O}$ can be expressed as a closed-form formula in terms of the degree $d$, the rotation number $\frac{p}{q}$, and the digits of $\mathcal{O}$.* Similar to the count of all rotational sets under certain parameters being given through the valid placements of pre-images of $\_0$, the orbits that belong to rotational sets containing a given orbit $\mathcal{O}$ can be identified through a similar process. We leave this investigation to the reader. However, an algorithm for identifying such rotational sets containing an orbit $\mathcal{O}$ based on the proof strategy of the previous theorem can be accessed on GitHub [@Hugh:2023]. ## Examples In order to demonstrate the usefulness of the proposed algorithm, consider the rotational orbit $[\_012, \_120, \_201]$ under $\sigma_3$. Applying the algorithm, we determine that this orbit can be paired with either the orbit $[\_002, \_020, \_200]$ or the orbit $[\_112, \_121, \_211]$ (but not both since a rotational set for $\sigma_3$ contains at most two orbits). The rotational polygons formed by these three sets are in the first row of Figure [15](#rabbits){reference-type="ref" reference="rabbits"}. {#rabbits width=".33\\textwidth"} {#rabbits width=".33\\textwidth"} {#rabbits width=".33\\textwidth"}\ {#rabbits width=".33\\textwidth"} {#rabbits width=".33\\textwidth"} {#rabbits width=".33\\textwidth"}\ {#rabbits width=".33\\textwidth"} {#rabbits width=".33\\textwidth"} {#rabbits width=".33\\textwidth"} The second row of the figure shows the first few stages of the pullback lamination determined by the initial polygons and an appropriate choice of branches of the inverse of $\sigma_3$ [@LamBuilder:2021]. The corresponding Julia sets, found using Mathematica and FractalStream [@Brittany:2023], are displayed in the third row of the figure. Each polygon in the corresponding lamination represents a junction point in the Julia set. The central polygon in each case represents a fixed point of the polynomial for the corresponding Julia set. Note that the lamination represents reasonably well the geometry of the corresponding Julia set if you imagine the polygons shrunk to points. In this process it is important to note that we found the rotational polygons and laminations *before* we found the polynomials and their Julia sets. ## Future Questions If you find yourself interested in continuing this work, perhaps consider the following questions as starting points for areas of research: 1. The formula for the count provided in Theorem [Theorem 21](#count){reference-type="ref" reference="count"} is quite complicated. Tan ([@Tan:2019] Theorem 3.2) provides an equivalent count, though without a basis in first principles. How can our count be simplified from first principles? 2. Verify that the formula found by Tan and the closed-form formula in our Theorem 3.1 give the same count of rotational sets. 3. Consider the closed form formula given by Theorem [Theorem 21](#count){reference-type="ref" reference="count"}. The formula implies that for a fixed $d$ and $k$, you can express the formula as a polynomial equation in terms of $q$. What is the degree of the polynomial for a given $d$ and $k$? Is it possible to generate the coefficients for the polynomial for a given $d$ and $k$ in closed form? What can this teach us about the count of rotational sets for a fixed degree and rotational set size? 4. Consider the lattice [@Lattice] of rotational sets, partially ordered by subset inclusion, for a given degree $d$ and rotation number $\frac pq$, where a join [@Join] between sets $A$ and $B$ represents the union of $A$ and $B$. Given a degree $d$, what can we learn about the underlying structure of these lattices, and what can it teach us about rotational sets? 1 B. Burdette, C. Falcione, C. Hale, and J. Mayer. Unicritcal and maximally critical laminations. arXiv:2303.17668 \[math.DS\] (2023). A. Blokh, J. Malaugh, J. Mayer, L. Oversteegen, D. Parris. Rotational subsets of the circle under $z^d$. 153 (2006), 1540--1570. L. Goldberg. Fixed points of polynomial maps, Part 1, Rotation subsets of the circle. *Annales scientifiques de l'É.N.S. 4e série*, tome 25, no 6 (1992), p. 679--685. L. Goldberg and J. Milnor. Fixed points of polynomial maps, Part II. Fixed point portraits. , tome 26, no 1 (1993), p. 51--98. C. McMullen. Dynamics on the unit disk: Short geodesics and simple cycles. , 85 (2010) 723--749. Y. Tan. Counting rotational subsets of the circle $\mathbb{R}/\mathbb{Z}$ under the angle-multiplying map $t\mapsto dt$. arXiv:2207.03594v2 \[math.CO\] (2022). Github: MaxSetGeneratingAlgo. https://github.com/mjmoorman03/MaxSetGeneratingAlgo/tree/main. Falcione, C. Lamination Builder. https://csfalcione.github.io/lamination-builder/. Wikipedia: Inclusion-Exclusion Principle. https://en.wikipedia.org/wiki/Inclusion-exclusion_principle. Wikipedia: Join and Meet. https://en.wikipedia.org/wiki/Join_and_meet. Wikipedia: Lattice (order). https://en.wikipedia.org/wiki/Lattice\_(order). Wikipedia: Mandelbrot Set. https://en.wikipedia.org/wiki/Mandelbrot_set.

arxiv_math

--- abstract: | We study the concentration phenomenon of the Lane-Emden equation with vanishing potentials $$\begin{cases} -\Delta u_n=W_n(x)u_n^{p_n},\quad u_n>0,\quad\text{in}~\Omega,\\ u_n=0,\quad\text{on}~\partial\Omega,\\ \int_\Omega p_n W_n(x)u_n^{p_n}{\mathrm d}x\le C, \end{cases}$$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$, $W_n(x)\geq 0$ are bounded functions with zeros in $\Omega$, and $p_n\to\infty$ as $n\to\infty$. A typical example is $W_n(x)=|x|^{2\alpha}$ with $0\in\Omega$, i.e. the equation turns to be the well-known Hénon equation. The asymptotic behavior for $\alpha=0$ has been well studied in the literature. While for $\alpha>0$, the problem becomes much more complicated since a singular Liouville equation appears as a limit problem. In this paper, we study the case $\alpha>0$ and prove a quantization property (suppose $0$ is a concentration point) $$p_n|x|^{2\alpha}u_n(x)^{p_n-1+t}\to 8\pi e^{\frac{t}{2}}\sum_{i=1}^k\delta_{a_i}+8\pi(1+\alpha)e^{\frac{t}{2}}c^t\delta_0, \quad t=0,1,2,$$ for some $k\ge0$, $a_i\in\Omega\setminus\{0\}$ and some $c\ge1$. Moreover, for $\alpha\not\in{\mathbb N}$, we show that the blow up must be simple, i.e. $c=1$. As applications, we also obtain the complete asymptotic behavior of ground state solutions for the Hénon equation. address: - Department of Mathematical Sciences, Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China - Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, China author: - Zhijie Chen - Houwang Li title: Energy quantization of the two dimensional Lane-Emden equation with vanishing potentials --- # Introduction {#section-1} In the past years much attention has been paid to the blow-up analysis for solution sequences $u_n(x)$ of the Lane-Emden type equation $$\label{equ-1} \begin{cases} -\Delta u_n=W_n(x)|u_n|^{p_n-1}u_n,\quad\text{in}~\Omega,\\ u_n=0,\quad\text{on}~\partial\Omega,\\ \int_\Omega p_n W_n(x)|u_n|^{p_n}{\mathrm d}x\le C, \end{cases}$$ where $\Omega\subset\mathbb R^2$ is a smooth bounded domain, $W_n(x)\geq 0$ are bounded functions with zeros in $\Omega$, $p_n\to\infty$ as $n\to\infty$, and $C>0$ is a constant independent of $n$. As in the literature, $a\in\overline\Omega$ is called a blow-up point of $p_nu_n$ if there exists $\left\{x_n\right\}\subset\Omega$ such that $x_n\to a$ and $p_nu_n(x_n)\to\infty$. In this case, we also call this $a$ a blow-up point of $u_n$ for convenience. When $W_n(x)\equiv1$, [\[equ-1\]](#equ-1){reference-type="eqref" reference="equ-1"} turns to be the well known two demensional Lane-Emden equation $$\label{equ-2} \begin{cases} -\Delta u_n=|u_n|^{p_n-1}u_n,\quad\text{in}~\Omega,\\ u_n=0,\quad\text{on}~\partial\Omega,\\ \int_\Omega p_n|u_n|^{p_n}{\mathrm d}x\le C. \end{cases}$$ The asymptotic behaviors of *positive solutions* of [\[equ-2\]](#equ-2){reference-type="eqref" reference="equ-2"} have been well studied by various mathematicians in a series of papers [@LE-8; @LE-1; @asy2-4; @LE-5; @LE-6], and the main results can be summarized as follows: Let $u_n$ be a sequence of positive solutions of [\[equ-2\]](#equ-2){reference-type="eqref" reference="equ-2"}. Then there exists a finite set ${\mathcal S}=\left\{a_1,\cdots,a_k\right\}\subset\Omega$ consisting of blow-up points of $p_nu_n$ such that up to a subsequence, for a suitable $r_0>0$, $$\label{1-3} \sup_{B_{r_0}(a_i)} u_n(x)\to\sqrt{e}\quad\text{and}\quad \sup_{\overline{\Omega}\setminus \bigcup_{i=1}^k B_r(a_i)}p_nu_n(x)\le C_r, \quad\text{\it for any}~r>0,$$ $$\label{1-4} p_nu_n(x)^{p_n-1}\to 8\pi\sum_{i=1}^k \delta_{a_i}\quad\text{\it weakly in the sense of measures},$$ where $\delta_{a_i}$ is the Dirac measure at $a_i$, and $$B_r(a):=\{x\in\mathbb{R}^2\,:\, |x-a|<r\},\quad B_r:=B_r(0)$$ denote open balls. Furthermore, for each $1\leq i\leq k$, a suitable scaling of $u_n$ near $a_i$ converges in $\mathcal C_{loc}^2(\mathbb R^2)$ to an entire solution $U$ of the Liouville equation $$\label{Liouville-1} \begin{cases} -\Delta U=e^U\quad\text{in}~\mathbb R^2,\\ \int_{\mathbb R^2}e^U{\mathrm d}x<\infty. \end{cases}$$ On the other hand, the asymptotic behaviors of *nodal solutions* of [\[equ-2\]](#equ-2){reference-type="eqref" reference="equ-2"} are much more difficult to study and there are only some partial results; see [@LE-7; @LE-4; @LE-11]. In particular, comparing to positive solutions, new phenomena appear for nodal solutions. For example, Grossi, Grumiau and Pacella [@LE-11] studied *the least energy radial nodal solutions* in a ball, and proved that the limit profile of these nodal solutions looks like a superposition of two bubbles, one related to a regular limit problem [\[Liouville-1\]](#Liouville-1){reference-type="eqref" reference="Liouville-1"} and another one related to a singular limit problem $$\label{Liouville-2} \begin{cases} -\Delta U=e^U+H\delta_0,\quad\text{in}~\mathbb R^2,\\ \int_{\mathbb R^2}e^U{\mathrm d}x<\infty, \end{cases}$$ where $H$ is a suitable constant. More precisely, a suitable scaling of the positive parts $u_n^+=\max\{u_n, 0\}$ converges to a solution of the Liouville equation [\[Liouville-1\]](#Liouville-1){reference-type="eqref" reference="Liouville-1"}, while a suitable scaling of the negative parts $u_n^-=\min\{u_n,0\}$ converges to a singular solution of the singular Liouville equation [\[Liouville-2\]](#Liouville-2){reference-type="eqref" reference="Liouville-2"}. One purpose of this paper is to show that that for *positive solutions* of [\[equ-1\]](#equ-1){reference-type="eqref" reference="equ-1"}, if $W_n(x)$ vanishes (with finite order) at some points, then the singular Liouville equation [\[Liouville-2\]](#Liouville-2){reference-type="eqref" reference="Liouville-2"} appears again as a limit problem. This is a different feature comparing to positive solutions of the Lane-Emden equation [\[equ-2\]](#equ-2){reference-type="eqref" reference="equ-2"}. Our another interest of studying [\[equ-1\]](#equ-1){reference-type="eqref" reference="equ-1"} is originated from the Hénon equation $$\begin{cases} -\Delta u_n=|x|^{2\alpha}|u_n|^{p_n-1}u_n,\quad\text{in}~B_1,\\ u_n=0,\quad\text{on}~\partial B_1, \end{cases}$$ which was introduced by Hénon [@Henon-0] in the study of stellar clusters in radially symmetric settings in 1973. Here we consider more general potentials $W_n(x)$. Suppose $W_n(x)$ has the form $$\label{con-1} W_n(x)=\overline W_{n}(x)\prod_{i=1}^{m}|x-q_i|^{2\alpha_i},$$ where $m\ge1$, $\alpha_i>0$ and $\overline W_n$ satisfies $$\label{con-2} 0<\frac{1}{C}\le \overline W_{n}(x)\le C<\infty,\quad |\nabla \overline W_{n}(x)|\le C,\quad\text{for}~x\in\Omega,$$ for some positive constant $C$ independent of $n$. Denote the zero set of $W_n(x)$ by $$\label{Z} {\mathcal Z}:=\left\{x\in\Omega:~W_n(x)=0\right\}=\left\{q_1,\cdots,q_m\right\}.$$ We will see that the problem will become very subtle if $p_nu_n$ blows up at some points in ${\mathcal Z}$. ## Local problems We start from a Brézis-Merle type result. In [@MF-1] Brézis-Merle gave their famous alternative results for the Liouville problem $-\Delta u_n=V_n(x)e^{u_n}$ in $\Omega$. Later, Ren-Wei [@LE-5; @LE-6] developed their method to handle the least energy solutions of the Lane-Emden equation [\[equ-2\]](#equ-2){reference-type="eqref" reference="equ-2"}. Here we follow Ren-Wei's idea to prove the following Brézis-Merle type result. **Theorem 1**. *Suppose $p_n\to\infty$ and $u_n$ is a solution sequence of $$\label{equ-BM} \begin{cases} -\Delta u_n=V_n(x)u_n^{p_n},\quad u_n>0,\quad\text{in}~\Omega,\\ \int_\Omega p_n V_n(x)u_n^{p_n}{\mathrm d}x\le C. \end{cases}$$ Then under the condition that $$\label{con-BM} 0\le V_n(x)\le C,\quad |\nabla V_n(x)|\le C,\quad\text{for all}~x\in\Omega,$$ after passing to a subsequence (still called $u_n$), one of the following alternatives holds:* - *$u_n \to 0$ uniformly in $L_{loc}^\infty(\Omega)$ with $\|p_nu_n\|_{L^\infty(K)}\leq C_K$ for any compact subset $K\Subset\Omega$.* - *There exist a non-empty finite set $\Sigma=\left\{a_1,\cdots,a_k\right\}\subset\Omega$ and corresponding sequences $\left\{x_{n,i}\right\}_{n\in{\mathbb N}}$ in $\Omega$ for $i=1,\cdots,k$, such that $x_{n,i}\to a_i$ and $u_n(x_{n,i})\to\gamma_i\ge 1$ as $n\to\infty$. Moreover, $\|p_nu_n\|_{L^\infty(K)}\le C_K$ for any compact subset $K\Subset\Omega\setminus\Sigma$, and $$\label{lambda-i} p_nV_n(x)u_n(x)^{p_n-1}\to\sum_{i=1}^k\beta_{a_i}\delta_{a_i},\quad p_nV_n(x)u_n(x)^{p_n}\to\sum_{i=1}^k\lambda_{a_i}\delta_{a_i}$$ weakly in the sense of measures in $\Omega$ with $\beta_{a_i}\ge\frac{4\pi e}{\gamma_i}$ and $\lambda_{a_i}\ge4\pi e$.* Note that we need no boundary conditions on $u_n$ in Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"}. When the alternative (ii) holds, the set $\Sigma$ only consists of those blow-up points of $p_nu_n$ contained in $\Omega$, i.e. whether $p_nu_n$ blows up at some points of $\partial\Omega$ or not is unknown. After Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"}, a natural question arises: *When the alternative (ii) holds, can one compute the exact values of $\beta_{a_i}$, $\lambda_{a_i}$ for every $i$*? An easy situation is that $V_n(x)$ is bounded below away from zero near the blow-up point $a_i$, and we call this *a regular case*. **Theorem 2**. *Suppose $p_n\to\infty$, $r>0$ and $u_n$ is a solution sequence of $$\label{equ-regular} \begin{cases} -\Delta u_n=V_n(x)u_n^{p_n},\quad u_n>0,\quad\text{in}~ B_r,\\ \int_{B_r} p_n V_n(x)u_n^{p_n}{\mathrm d}x\le C, \end{cases}$$ with $0$ being the only blow-up point of $p_nu_n$ in $B_r$, i.e., $$\label{con-regular1} \max_{B_r}p_nu_n\to\infty\quad \text{and}\quad \max_{\overline B_r\setminus B_\delta} p_nu_n\le C_\delta,\quad\text{for any}~0<\delta<r.$$ Then under the condition that $$\label{con-regular2} 0<\frac{1}{C}\le V_n(x)\le C,\quad |\nabla V_n(x)|\le C,\quad\text{for}~x\in B_r,$$ after passing to a subsequence (still called $u_n$), it hold $\max_{B_r}u_n\to\sqrt e$ and $$\label{result-regular} p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+t}\to 8\pi e^{\frac{t}{2}}\delta_0, \quad t=0,1,2$$ weakly in the sense of measures.* Theorem [Theorem 2](#thm-regular){reference-type="ref" reference="thm-regular"} improves [@LE-1 Theorem 1.1] in the sense that $V_n\not\equiv1$ is allowed and no boundary condition $u_n=0$ is needed. The idea of proving Theorem [Theorem 2](#thm-regular){reference-type="ref" reference="thm-regular"} is similar to that of [@LE-1 Theorem 1.1], and for the reader's convenience we will give the proof in Section 3. First, by the blow-up analysis around a local maximum of $u_n$, we are led to a solution $U_0$ of the Liouville equation [\[Liouville-1\]](#Liouville-1){reference-type="eqref" reference="Liouville-1"}. The classical result of Chen-Li [@classification-1] characterizes all these solutions, which implies $\int_{\mathbb R^2}e^{U_0}=8\pi$. Then, by the local Pohozaev identity and the Green's representation formula, we get a decay estimate of $u_n$, which is used to apply the Dominate Convergence Theorem to get the convergence of energies, and hence get the desired results. Now a delicate situation is that $V_n(x)$ vanishes (with finite order) at a blow-up point $a_i$, and we call this *a singular case* since the blow-up around a local maximum of $u_n$ near $a_i$ will lead to the singular Liouville problem. Due to $\alpha\in{\mathbb N}$ or not, we have different results. **Theorem 3**. *Suppose $p_n\to\infty$, $r>0$, $\alpha>0$ and $u_n$ is a solution sequence of $$\label{equ-singular} \begin{cases} -\Delta u_n=|x|^{2\alpha}V_n(x)u_n^{p_n},\quad u_n>0,\quad\text{in}~ B_r,\\ \int_{B_r} p_n |x|^{2\alpha}V_n(x)u_n^{p_n}{\mathrm d}x\le C, \end{cases}$$ with $0$ being the only blow-up point of $p_nu_n$ in $B_r$, i.e., $$\label{con-singular1} \max_{B_r}p_nu_n\to\infty\quad \text{and}\quad \max_{\overline B_r\setminus B_\delta} p_nu_n\le C_\delta,\quad\text{for any}~0<\delta<r.$$ Then under the condition that $$\label{con-singular2} 0<\frac{1}{C}\le V_n(x)\le C,\quad |\nabla V_n(x)|\le C,\quad\text{for}~x\in B_r,$$ after passing to a subsequence (still called $u_n$), it holds that $\max_{B_r}u_n\to\gamma\ge\sqrt e$ and $$\label{result-singular1} p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+t}\to 8\pi(1+\alpha)e^{\frac{t}{2}}c^t\delta_0, \quad t=0,1,2,$$ weakly in the sense of measures for some $c\in[1,\gamma]$. Moreover, there holds $c=1$ and $\gamma=\sqrt e$ if $\alpha\not\in{\mathbb N}$.* *Remark 4*. It is interesting to compare Theorem [Theorem 3](#thm-singular){reference-type="ref" reference="thm-singular"} with the results for singular mean field problems. Suppose $u_n$ solves $$-\Delta u_n=|x|^{2\alpha}V_ne^{u_n}\quad\text{\it in }\;B_r,$$ and assume that $0$ is the only blow-up point. Then under the condition [\[con-singular2\]](#con-singular2){reference-type="eqref" reference="con-singular2"}, Tarantello [@SMF-1] proved that $|x|^{2\alpha}V_ne^{u_n}\to\beta_0\delta_0$ with $\beta_0\in 8\pi{\mathbb N}_{\geq 1}\cup\{8\pi(1+\alpha)+8\pi{\mathbb N}\}$. There are also explicit examples in [@SMF-1] to show that $\beta_0$ can take any value contained in $8\pi{\mathbb N}_{\geq 1}\cup\{8\pi(1+\alpha)+8\pi{\mathbb N}\}$. Here we get a quite surprising result, that is the energy must be $8\pi(1+\alpha)$, i.e., $$p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1}\to 8\pi(1+\alpha)\delta_0,$$ for any $\alpha>0$. The proof of Theorem [Theorem 3](#thm-singular){reference-type="ref" reference="thm-singular"} is much more complicated than that of Theorem [Theorem 2](#thm-regular){reference-type="ref" reference="thm-regular"}, and the main difficulty is the lack of the following condition $$\label{result-singular2} \sup_{B_{\varepsilon_0}}p_n|x|^{2+2\alpha}V_n(x)u_n(x)^{p_n-1}\le C, \quad\text{for some}~\varepsilon_0\in(0,r).$$ The proof consists of two main ingredients. First, we assume [\[result-singular2\]](#result-singular2){reference-type="eqref" reference="result-singular2"} holds, and then by a blow-up around a local maximum of $u_n$, we are led to a solution $U_\alpha$ of the singular Liouville problem $$\label{Liouville-3}\begin{cases} -\Delta U_\alpha=|x|^{2\alpha}e^{U_\alpha},\quad\text{in}~\mathbb R^2,\\ \int_{\mathbb R^2}|x|^{2\alpha}e^{U_\alpha}{\mathrm d}x<\infty. \end{cases}$$ Since $\Delta(\frac{1}{2\pi}\ln|x|)=\delta_0$, we see that [\[Liouville-2\]](#Liouville-2){reference-type="eqref" reference="Liouville-2"} and [\[Liouville-3\]](#Liouville-3){reference-type="eqref" reference="Liouville-3"} are equivalent in the sense that $U_\alpha$ is a solution of [\[Liouville-3\]](#Liouville-3){reference-type="eqref" reference="Liouville-3"} if and only if $U_\alpha+2\alpha\ln|x|$ is a solution of [\[Liouville-2\]](#Liouville-2){reference-type="eqref" reference="Liouville-2"} with $H=-4\pi\alpha$. A result of Prajapat-Tarantello [@classification-2] (see [@classification-1] for $\alpha=0$) characterizes all solutions of [\[Liouville-3\]](#Liouville-3){reference-type="eqref" reference="Liouville-3"}, from which we know that $$\int_{\mathbb R^2}|x|^{2\alpha}e^{U_\alpha}{\mathrm d}x=8\pi(1+\alpha).$$ Then by the local Pohozaev identity and the Green's representation formula, we get a decay estimate of $u_n$, and hence we obtain $\beta_t=8\pi(1+\alpha)e^{\frac{t}2}$ for $t=0,1,2$. Second, we assume [\[result-singular2\]](#result-singular2){reference-type="eqref" reference="result-singular2"} does not hold. Note that equation [\[equ-singular\]](#equ-singular){reference-type="eqref" reference="equ-singular"} is formally invariant under the transformation $$\label{tem-101} v_n(x)=r^{\alpha_n}u_n(rx),\quad\text{with}~\alpha_n=\frac{2+2\alpha}{p_n-1}.$$ Thanks to this transformation and inspired by [@SMF-1], we can construct a decomposition of $u_n$; see Proposition [Proposition 24](#decomposition){reference-type="ref" reference="decomposition"}. In this direction, we reduce the singular case to some regular cases. By accurate analysis, we show that there is no energy loss in neck domains. Then using Theorems [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"} and [Theorem 2](#thm-regular){reference-type="ref" reference="thm-regular"}, we compute the exact values of the correponding energies, which gives $$p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+t}\to 8\pi e^{\frac{t}{2}}(\sum_{i=1}^l N_ic_i^t)\delta_0,\quad t=0,1,2,$$ for some $l\ge1$, $c_i\ge1$, $N_i\in{\mathbb N}$ for $i=1,\cdots,l$. Then comparing these energies by Pohozaev identity, we get $l=1$ and $N_1=1+\alpha$. So that if $\alpha\not\in{\mathbb N}$, we get a contradiction, and then condition [\[result-singular2\]](#result-singular2){reference-type="eqref" reference="result-singular2"} holds, which gives Theorem [Theorem 3](#thm-singular){reference-type="ref" reference="thm-singular"} for $\alpha\not\in{\mathbb N}$. While for $\alpha\in{\mathbb N}$, the result is more complicated. For the mean field equation with integer singular sources $\alpha\in{\mathbb N}$, Kuo-Lin [@SMF-4] and Bartolucci-Tarantello [@SMF-5] showed the non-simple blow-up phenomena happens, i.e., condition [\[result-singular2\]](#result-singular2){reference-type="eqref" reference="result-singular2"} does not hold. We refer to [@Nonsimple-1; @Nonsimple-2] for more information of the non-simple blow-up. Hence it is an interesting problem to consider the blow-up phenomena of [\[equ-singular\]](#equ-singular){reference-type="eqref" reference="equ-singular"} with $\alpha\in{\mathbb N}$, and in a following paper, we would like to study this case. ## Boundary value problems Thanks to the above local properties, we are in position to study positive solutions of our initial problem [\[equ-1\]](#equ-1){reference-type="eqref" reference="equ-1"}, i.e. $$\label{equ-1-0} \begin{cases} -\Delta u_n=W_n(x)u_n^{p_n},\quad u_n>0,\quad\text{in}~\Omega,\\ u_n=0,\quad\text{on}~\partial\Omega,\\ \int_\Omega p_n W_n(x)u_n^{p_n}{\mathrm d}x\le C, \end{cases}$$ with $W_n(x)$ satisfying [\[con-1\]](#con-1){reference-type="eqref" reference="con-1"}-[\[con-2\]](#con-2){reference-type="eqref" reference="con-2"}. For a solution sequence $u_n$ of [\[equ-1-0\]](#equ-1-0){reference-type="eqref" reference="equ-1-0"}, we define the set ${\mathcal S}$ of blow-up points of $p_nu_n$ as $$\label{S} {\mathcal S}:=\left\{a\in\overline\Omega:~\exists \{x_n\}\subset\Omega,~x_n\to a,~p_nu_n(x_n)\to\infty\right\}.$$ By considering the maximum point of $u_n$, one can easily check ${\mathcal S}\neq\emptyset$; see Section 5. Then Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"} tells us that after passing to a subsequence, ${\mathcal S}\cap\Omega$ is an at most finite set. The problem is whether ${\mathcal S}\cap\partial\Omega=\emptyset$ or not. When $W_n(x)\equiv1$, by the moving plane method one can prove that $p_nu_n$ is uniformly bounded in a small neighbourhood of $\partial\Omega$, and hence there is no boundary blow-up. However, due to the appearance of $W_n(x)$, the moving plane method is not applicable anymore. Here we use the induction method developed in [@LE-7] for the Lane-Emden equation in a small neighbourhood of $\partial\Omega$, and get ${\mathcal S}\cap\partial\Omega=\emptyset$ by leading to a contradiction with $u_n|_{\partial\Omega}=0$. We point out that the induction method in [@LE-7] is inefficient at the places where $W_n(x)$ has zeros. Recall the zero set ${\mathcal Z}$ of $W_n$ defined in [\[Z\]](#Z){reference-type="eqref" reference="Z"}. Once we obtain ${\mathcal S}\subset\Omega$, we can apply Theorem [Theorem 2](#thm-regular){reference-type="ref" reference="thm-regular"} near any point $a\in{\mathcal S}\setminus{\mathcal Z}$, and apply Theorem [Theorem 3](#thm-singular){reference-type="ref" reference="thm-singular"} near any point $a\in{\mathcal S}\cap{\mathcal Z}$. Indeed, we obtain **Theorem 5**. *Let $u_n$ be a solution sequence of [\[equ-1-0\]](#equ-1-0){reference-type="eqref" reference="equ-1-0"}, and suppose $W_n(x)$ satisfies [\[con-1\]](#con-1){reference-type="eqref" reference="con-1"}-[\[con-2\]](#con-2){reference-type="eqref" reference="con-2"} with $\alpha_i>0$ for every $i=1,\cdots,m$. Then up to a subsequence, there exists a positive integer $k$ and different points $a_1,\cdots,a_k\in\Omega$ such that* - *The blow-up set ${\mathcal S}$ of $p_nu_n$ is given by ${\mathcal S}=\left\{a_1,\cdots,a_k\right\}$.* - *For small $r>0$, $\max_{B_r(a_i)}u_n\to \gamma_i\geq \sqrt{e}$ for all $1\leq i\leq k$. Furthermore, $\gamma_i=\sqrt{e}$ if $a_i\in{\mathcal S}\setminus{\mathcal Z}$.* - *For $t=0,1,2$, there holds $$\label{1234} p_n W_n(x)u_n(x)^{p_n-1+t}\to 8\pi e^{\frac{t}{2}} \Big(\sum_{a_i\in{\mathcal S}\setminus{\mathcal Z}}\delta_{a_i}+\sum_{a_j=q_{j'}\in{\mathcal S}\cap{\mathcal Z}}(1+\alpha_{j'})c_j^t\delta_{a_j} \Big),$$ weakly in the sense of measures for some $c_{j}\ge1$.* - *For $a_j=q_{j'}\in{\mathcal S}\cap {\mathcal Z}$, $\gamma_j=\sqrt{e}$ and $c_j=1$ if $\alpha_{j'}\not\in{\mathbb N}$.* *Remark 6*. - For $\alpha_i\not\in{\mathbb N}$ for all $i=1,\cdots,m$, the existence of blow-up solutions of [\[equ-1-0\]](#equ-1-0){reference-type="eqref" reference="equ-1-0"} satisfying [\[1234\]](#1234){reference-type="eqref" reference="1234"} has been constructed by Esposito-Pistoia-Wei [@Reduction-1] via the finite-dimensional reduction method. In particular, their result shows that $\mathcal S\cap \mathcal Z\neq \emptyset$ happens for some solutions. Therefore, in general we can not expect $\mathcal S\cap \mathcal Z=\emptyset$ in Theorem [Theorem 5](#thm-1){reference-type="ref" reference="thm-1"}. In Theorem [Theorem 5](#thm-1){reference-type="ref" reference="thm-1"}, we prove in another direction that if $\alpha_i\not\in{\mathbb N}$ for all $i=1,\cdots,m$, then any solution sequence $u_n$ with bounded energy must behave the multi-point blow-up phenomena, and at any point the blow-up is simple. - For mean field equation with non-quantized singularity, i.e., $\alpha\not\in{\mathbb N}$, the profile of blow-up solutions has been given in Bartolucci-Tarantello [@SMF-2] and Bartolucci-Chen-Lin-Tarantello [@SMF-3]. They showed that the solution sequences develop multi-point blow-up and at each point the blow-up is simple. Our results are similar to theirs but different. As one can see we have no $\max u_n\to\infty$ but instead $\max u_n\to\sqrt e$; the energy of each bubble is dependent on the local maximum of $u_n$, which makes the analysis very different. - For the Hénon equation $$\label{equ-henon} \begin{cases} -\Delta u_n=|x|^{2\alpha}u_n^{p_n},\quad u_n>0,\quad\text{in}~\Omega,\\ u_n=0,\quad\text{on}~\partial\Omega, \end{cases}$$and some more general equations, the uniform bounded energy condition $\int_\Omega p_n|x|^{2\alpha}u_n^{p_n}{\mathrm d}x\le C$ was proved to hold automatically in [@bound-1] for $\alpha>0$ and any simply connected domain $\Omega$ with $0\in\Omega$. As an application of Theorem [Theorem 5](#thm-1){reference-type="ref" reference="thm-1"}, we study the ground states of the Hénon equation [\[equ-henon\]](#equ-henon){reference-type="eqref" reference="equ-henon"} with $0\in\Omega$. Let $u_n(x)$ be a ground state (or called a least energy solution) of [\[equ-henon\]](#equ-henon){reference-type="eqref" reference="equ-henon"}, which by definition is a nontrivial solution of [\[equ-henon\]](#equ-henon){reference-type="eqref" reference="equ-henon"} such that the energy $\int_{\Omega}|x|^{2\alpha}|u_n|^{p_n+1}dx$ is smallest among all nontrivial solutions. It is standard to see that such ground state exists and is positive in $\Omega$ (up to a sign). We want to show that $0$ is not a blow-up point for the ground states. When $\alpha=0$, the complete asymptotic behavior of the ground states as $p_n\to \infty$ was obtained in [@LE-5; @LE-6; @LE-8], which says that the ground states behave as a single point blow-up (i.e. $k=1$ in [\[1-3\]](#1-3){reference-type="eqref" reference="1-3"}-[\[1-4\]](#1-4){reference-type="eqref" reference="1-4"}). For $\alpha>0$, Zhao [@Henon-1] proved some partial results for the ground state $u_n(x)$, which can be summarized as follows: - For $\alpha>0$, $$\label{tem-4-00}1\leq \liminf_{n\to\infty}\|u_n\|_{L^\infty(\Omega)}\leq \limsup_{n\to\infty}\|u_n\|_{L^\infty(\Omega)}\leq \sqrt{e},$$ $$\label{tem-4} \lim_{n\to\infty} p_n\int_{\Omega}|x|^{2\alpha}u_n^{p_n+1}{\mathrm d}x= 8\pi e.$$ - For $\alpha>e-1$, the ground state $u_n(x)$ behaves as at most two points blow-up, and $u_n(x)$ is not radially symmetric for $p_n$ large if $\Omega=B_r$ is an open ball. We want to improve these results and give a complete asymptotic behavior of the ground state $u_n(x)$ of the Hénon equation [\[equ-henon\]](#equ-henon){reference-type="eqref" reference="equ-henon"} for any $\alpha>0$. To state our result, we introduce some notations. Recall the Green function $G(x,y)$ of $-\Delta$ in $\Omega$ with the Dirichlet boundary condition: $$\left\{ \begin{aligned} &-\Delta_x G(x,y)=\delta_y \quad &\text{in}~\Omega,\\ &G(x,y)=0 \quad &\text{on}~\partial\Omega, \end{aligned} \right.$$ It has the following form $$G(x,y)=-\frac{1}{2\pi}\log|x-y|-H(x,y),\quad(x,y)\in\Omega\times\Omega,$$ where $H(x,y)$ is the regular part of $G(x,y)$. It is well known that $H$ is a smooth function in $\Omega\times\Omega$, both $G$ and $H$ are symmetric in $x$ and $y$. The Robin function of $\Omega$ is defined as $$\label{Robin} R(x):=H(x,x).$$ **Theorem 7**. *Let $0\in\Omega$, $\alpha>0$ and $u_n$ be a ground state of the Hénon equation [\[equ-henon\]](#equ-henon){reference-type="eqref" reference="equ-henon"}. Set $u_n(x_n)=\|u_n\|_{L^\infty(\Omega)}$. Then $u_n(x_n)\to\sqrt e$ and up to a subsequence, $x_n\to a\in\Omega\setminus\{0\}$, $$p_nu_n\to 8\pi\sqrt e G(x,a),\quad\text{in}~\mathcal C_{loc}^2(\overline{\Omega}\setminus\{a\}),$$ $$p_n|x|^{2\alpha}u_n(x)^{p_n-1+t}\to8\pi e^{\frac{t}{2}}\delta_a, \quad t=0,1,2,$$ $$\nabla \left(R(\cdot)-\frac{1}{4\pi}\log|\cdot|^{2\alpha}\right)(a)=0,$$ where $R(x)$ is the Robin function in [\[Robin\]](#Robin){reference-type="eqref" reference="Robin"}.* *Remark 8*. Theorem [Theorem 7](#thm-0){reference-type="ref" reference="thm-0"} improves those results in [@Henon-1]. It is also interesting to compare Theorem [Theorem 7](#thm-0){reference-type="ref" reference="thm-0"} with some other results in the literature. Consider the Hénon equation in general dimensions $$\begin{cases} -\Delta u_n=|x|^{2\alpha}u_n^{p_n},\quad u_n>0,\quad\text{in}~\Omega,\\ u_n=0,\quad\text{on}~\partial\Omega, \end{cases}$$ where $\Omega\subset\mathbb R^N$ is a smooth bounded domain. When $N\ge2$, the asymptotic behavior of ground states as $\alpha\to\infty$ was studied by Byeon-Wang in [@H-1; @H-2], where they proved that the ground states develop a boundary blow-up. In another direction, when $N\ge3$, $\alpha>0$ is fixed, $\Omega=B_1$ and $p_n\to\frac{N+2}{N-2}$, Cao-Peng [@H-3] showed that the ground states also develop a boundary blow-up. However, Theorem [Theorem 7](#thm-0){reference-type="ref" reference="thm-0"} shows that there is no boundary blow-up for planar domains. Especially, when $\Omega$ is the unit ball, we know that the ground state of the Hénon equation is not radially symmetric for $p_n$ large, since $x_n\to a\neq0$. The paper is organized as follows. In Section 2, we prove the Brézis-Merle type result Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"}. In Sections 3 and 4, we study respectively the regular case and the singular case, and then prove Theorems [Theorem 2](#thm-regular){reference-type="ref" reference="thm-regular"} and [Theorem 3](#thm-singular){reference-type="ref" reference="thm-singular"}. In Section 5, we study the boundary value problem and prove Theorem [Theorem 5](#thm-1){reference-type="ref" reference="thm-1"}. Finally in Section 6, we study the ground states of the Hénon equation. Throughout the paper, we denote by $C, C_0, C_1, \cdots$ to be positive constants independent of $n$ but may be different in different places. # The Brézis-Merle type result In this section, we follow Ren-Wei's idea [@LE-5; @LE-6] to prove Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"}. Let $u_n$ be a solution sequence of [\[equ-BM\]](#equ-BM){reference-type="eqref" reference="equ-BM"} and denote $$\label{label-11} \bar u_n:=p_nu_n \quad\text{and}\quad f_n:=p_nV_nu_n^{p_n}.$$ Then it holds $$\label{label-9} \begin{cases} -\Delta \bar u_n=f_n,\quad\text{in}~\Omega,\\ \bar u_n>0,\quad\text{in}~\Omega,\\ \bar u_n=0,\quad\text{on}~\partial\Omega. \end{cases}$$ Thanks to $\|f_n\|_{L^1(\Omega)}\le C$, we may assume that $$f_n\to\nu\quad\text{weakly in}~{\mathcal M}(\Omega)~\text{as}~n\to\infty,$$ where ${\mathcal M}(\Omega)$ is the space of Radon measures. Obviously $\nu(\Omega)<\infty$. For any $\delta>0$, we say a point $x_*\in\Omega$ to be *a $\delta$-regular point* with respect to $\nu$, if there exists $\varphi\in\mathcal C_0(\Omega)$ satisfying $0\le\varphi\le 1$, $\varphi\equiv1$ near $x_*$ such that $$\int_\Omega\varphi{\mathrm d}\nu<\frac{4\pi}{\tfrac{1}{e}+2\delta}.$$ Denote $$\Sigma_\nu(\delta):=\left\{x\in\Omega:~x~\text{is not a $\delta$-regular point w.r.t. $\nu$}~\right\}.$$ Before proceeding our discussion, we quote an $L^1$ estimates from [@MF-1]. **Lemma 9** ([@MF-1]). *Let $u$ be a solution of $$\begin{cases} -\Delta u=f\quad\text{in}~\Omega,\\ u=0\quad\text{on}~\partial\Omega, \end{cases}$$ where $\Omega$ is a smooth bounded domain in $\mathbb R^2$. Then for any $0<\varepsilon<4\pi$, we have $$\int_{\Omega}\exp\left(\frac{(4\pi-\varepsilon)|u(x)|}{\|f\|_{L^1(\Omega)}}\right){\mathrm d}x\le \frac{4\pi^2}{\varepsilon}(\mathrm{diam}~\Omega)^2.$$* Now we give an equivalent characterization of the set $\Sigma_\nu(\delta)$. **Lemma 10**. *For any $\delta>0$ and $x_*\in\Omega$, we have that $x_*\in\Sigma_\nu(\delta)$ if and only if for any $R>0$ such that $B_R(x_*)\subset\Omega$, it holds $\|\bar u_n\|_{L^\infty(B_R(x_*))}\to+\infty$ as $n\to\infty$. Consequently, $\Sigma_\nu(\delta)$ does not depend on the choice of $\delta$.* *Proof.* First, take $x_*\not\in\Sigma_\nu(\delta)$, we want to prove that there exists $R_0>0$ such that $\|\bar u_n\|_{L^\infty(B_{R_0}(x_*))}\le C$ as $n\to\infty$. Since $\|f_n\|_{L^1(\Omega)}\le C$, by applying the elliptic $L^p$ estimate with the duality argument (cf. [@p-estimate]) to [\[label-9\]](#label-9){reference-type="eqref" reference="label-9"}, one gets that $\bar u_n$ are uniformly bounded in $W^{1,s}(\Omega)$ for any $1\le s<2$. In particular, $$\label{dd} \|\bar u_n\|_{L^s(\Omega)}\leq C_s,\quad 1\leq s<2.$$ We claim that there exist small $R_0>0$ and $\delta_0>0$ such that $$\label{claim-1} \|f_n\|_{L^{1+\delta_0}(B_{2R_0}(x_*))}\le C,\quad\text{as}~n\to\infty.$$ Once [\[claim-1\]](#claim-1){reference-type="eqref" reference="claim-1"} is proved, we can apply the weak Harnack inequality ([@book-1 Theorem 8.17]) to obtain $$\|\bar u_n\|_{L^\infty(B_{R_0}(x_*))} \le C\left(\|\bar u_n\|_{L^{3/2}(B_{2R_0}(x_*))}+\|f_n\|_{L^{1+\delta_0}(B_{2R_0}(x_*))}\right)\le C.$$ Now we need to check the claim [\[claim-1\]](#claim-1){reference-type="eqref" reference="claim-1"}. Since $\frac{\log x}{x}\le\frac{1}{e}$ for any $x\in(0,+\infty)$, we obtain $$\log (p_n^{1/p_n}u_n(x))\le \frac{1}{e}p_n^{1/p_n}u_n(x),\quad \forall x.$$ Therefore, for any $x\in\Omega$ and $\delta>0$ $$f_n(x)=V_n(x)e^{p_n\log(p_n^{1/p_n}u_n(x))}\le Ce^{\frac{1}{e}p_n^{1+1/p_n}u_n(x)}\le Ce^{(\frac{1}{e}+\frac{\delta}{2})\bar u_n(x)},\quad\text{for $n$ large}.$$ Since $x_*\not\in\Sigma_\nu(\delta)$, i.e. $x_*$ is a $\delta$-regular point, it follows from the definition of $\delta$-regular points that there exists $R_1>0$ such that $B_{2R_1}(x_*)\subset \Omega$ and $$\int_{B_{2R_1}(x_*)}f_n<\frac{4\pi}{\tfrac{1}{e}+\delta}\quad\text{for $n$ large}.$$ Take $\bar u_n=\bar u_{n,1}+\bar u_{n,2}$ with $\bar u_{n,1}=0$ on the boundary $\partial B_{2R_1}(x_*)$ and $\bar u_{n,2}$ is harmonic in the ball $B_{2R_1}(x_*)$, i.e. $$\label{harmonic-0} \begin{cases} -\Delta \bar u_{n,1}=f_n\quad\text{in }\; B_{2R_1}(x_*),\\ \bar u_{n,1}=0\quad\text{on }\;\partial B_{2R_1}(x_*), \end{cases} \qquad \begin{cases} -\Delta \bar u_{n,2}=0\quad\text{in }\; B_{2R_1}(x_*),\\ \bar u_{n,2}=\bar u_n\quad\text{on }\;\partial B_{2R_1}(x_*). \end{cases}$$ By the maximum principle, $\bar u_{n,1}>0$ and $\bar u_{n,2}>0$ in $B_{2R_1}(x_*)$. Applying Lemma [Lemma 9](#label-7){reference-type="ref" reference="label-7"} to $\bar u_{n,1}$, we get $$\int_{B_{2R_1}(x_*)}\exp\left(\frac{\gamma\bar{u}_{n,1}(x)}{\|f_n\|_{L^1(B_{2R_1}(x_*))}}\right){\mathrm d}x \le C_\gamma,\quad\text{for any $\gamma\in(0,4\pi)$}.$$ Note that $0<\bar u_{n,2}< \bar u_n$ in $B_{2R_1}(x_*)$. Then by the mean value theorem for harmonic functions and [\[dd\]](#dd){reference-type="eqref" reference="dd"}, we obtain $$\|\bar u_{n,2}\|_{L^\infty(B_{R_1}(x_*))}\le C\|\bar u_{n,2}\|_{L^1(B_{2R_1}(x_*))}\le C\|\bar u_n\|_{L^1(B_{2R_1}(x_*))} \le C\|\bar u_{n}\|_{L^1(\Omega)}\le C.$$ Take $\delta_0>0$ such that $\gamma:=4\pi(1+\delta_0)\frac{1+\frac{\delta}{2}e}{1+\delta e}<4\pi$. Then using the above estimates, we conclude that for $n$ large, $$\begin{aligned} \int_{B_{R_1}(x_*)} f_n(x)^{1+\delta_0}{\mathrm d}x &\le \int_{B_{R_1}(x_*)}C\exp\left((1+\delta_0)(\tfrac{1}{e}+\tfrac{\delta}{2}) \bar u_n(x)\right) {\mathrm d}x\\ &\le C\int_{B_{R_1}(x_*)}\exp\left((1+\delta_0)(\tfrac{1}{e}+\tfrac{\delta}{2}) \bar u_{n,1}(x)\right) {\mathrm d}x\\ &\le C\int_{B_{2R_1}(x_*)}\exp\left((1+\delta_0)(\tfrac{1}{e}+\tfrac{\delta}{2}) \bar u_{n,1}(x)\right) {\mathrm d}x\\ &\le C\int_{B_{2R_1}(x_*)}\exp\left(4\pi(1+\delta_0)\frac{1+\tfrac{\delta}{2}e}{1+\delta e} \frac{\bar{u}_{n,1}(x)}{\|f_n\|_{L^1(B_{2R_1}(x_*))}}\right){\mathrm d}x\\ &=C\int_{B_{2R_1}(x_*)}\exp\left( \frac{\gamma\bar{u}_{n,1}(x)}{\|f_n\|_{L^1(B_{2R_1}(x_*))}}\right){\mathrm d}x\le C_{\gamma}. \end{aligned}$$ Thus by choosing $R_0=R_1/2$, we finish the proof of the claim [\[claim-1\]](#claim-1){reference-type="eqref" reference="claim-1"}. Finally, given any $x_*\in\Sigma_\nu(\delta)$, we claim that for any $R>0$, $\|\bar u_n\|_{L^\infty(B_R(x_*))}\to+\infty$ as $n\to\infty$. If not, then there exists $R_1>0$ such that up to a subsequence, $\|\bar u_n\|_{L^\infty(B_{R_1}(x_*))}\le C$ as $n\to\infty$. Consequently, $$\int_{B_{R_1}(x_*)}f_n=\int_{B_{R_1}(x_*)}p_nV_n(x)u_n^{p_n}\le Cp_n\int_{B_{R_1}(x_*)}\left(\frac{C}{p_n}\right)^{p_n} \to 0\quad\text{as $n\to\infty$.}$$ Thus by the definition of $\delta$-regular points, we obtain $x_*\not\in\Sigma_\nu(\delta)$, a contradiction. This finishes the proof. ◻ **Corollary 11**. *For any $\delta>0$, $\Sigma_\nu(\delta)\subset\Omega$ is an at most finite set.* *Proof.* Since $\nu(\left\{x_*\right\})\ge\frac{4\pi}{\tfrac{1}{e}+2\delta}$ for every $x_*\in\Sigma_\nu(\delta)$, it holds $$C\ge\nu(\Omega)\ge \frac{4\pi}{\tfrac{1}{e}+2\delta}\#\Sigma_\nu(\delta),$$ which implies $\#\Sigma_\nu(\delta)<\infty$. ◻ **Corollary 12**. *For any compact subset $K\Subset\Omega\setminus\Sigma_\nu(\delta)$, it holds $$\|u_n\|_{L^\infty(K)}\le \frac{C_K}{p_n},\quad\text{for $n$ large}.$$* *Proof.* Given any compact subsets $K\Subset\Omega\setminus\Sigma_\nu(\delta)$, for any $x\in K$, we have $x\notin \Sigma_\nu(\delta)$, then it follows from Lemma [Lemma 10](#equiv-1){reference-type="ref" reference="equiv-1"} that there exists $R_x>0$ such that $$\|\bar u_n\|_{L^\infty(B_{R_x}(x))}\le C_x,\quad\text{for $n$ large}.$$ From here and the finite covering theorem, we obtain $$\|\bar u_n\|_{L^\infty(K)}\le C,\quad\text{for $n$ large}.$$ This implies $$\|u_n\|_{L^\infty(K)}\le \frac{C_K}{p_n},\quad\text{for $n$ large}.$$ Thus the proof is complete. ◻ *Proof of Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"}.* If $\Sigma_\nu(\delta)=\emptyset$, then Corollary [Corollary 12](#label-10){reference-type="ref" reference="label-10"} implies that the alternative $(i)$ in Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"} holds. Thus we now suppose $\Sigma_\nu(\delta)\neq\emptyset$ and prove the alternative $(ii)$ in Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"} holds. Since $\Sigma_\nu(\delta)$ is a finite set and does not depend on the choice of $\delta$, we denote $$\label{label-12} \Sigma=\Sigma_\nu(\delta)=\left\{a_1,\cdots,a_k\right\}.$$ By Corollary [Corollary 12](#label-10){reference-type="ref" reference="label-10"}, we know that $\nu=\sum_{i=1}^k\lambda_{a_i}\delta_{a_i}$. Since $\nu(\left\{a_i\right\})\ge\frac{4\pi}{\tfrac{1}{e}+2\delta}$ for any $\delta>0$, we get $\lambda_{a_i}\ge4\pi e$, and hence $p_nV_n(x)u_n(x)^{p_n}\to\sum_{i=1}^k\lambda_{a_i}\delta_{a_i}$ weakly in the sense of measures in $\Omega$ with $\lambda_{a_i}\ge4\pi e$. Then by Hölder inequality, we get $p_nV_n(x)u_n(x)^{p_n-1}\to\sum_{i=1}^k\beta_{a_i}\delta_{a_i}$. Choose $r_0>0$ such that $$\label{label-44} B_{2r_0}(a_i)\subset\Omega\quad\text{and}\quad B_{2r_0}(a_i)\cap B_{2r_0}(a_j)=\emptyset,\quad\text{for}~i,j=1,\cdots,k,~i\neq j.$$ Define the local maximums $\gamma_{n,i}$ and the local maximum points $x_{n,i}$ of $u_n$ by $$\label{notation-1} \gamma_{n,i}=u_n(x_{n,i}):=\max_{B_{2r_0}(a_i)} u_n,\quad\text{for}~i=1,\cdots,k.$$ Recall the definition of $\bar u_n$ and $f_n$ in [\[label-11\]](#label-11){reference-type="eqref" reference="label-11"}, we have $-\Delta \bar u_n=f_n$ in $\Omega$. For any $i=1,\cdots,k$, it follows from [@book-1 Theorem 3.7] that $$\max_{B_{2r_0}(a_i)}\bar u_n\le C(\max_{\partial B_{2r_0}(a_i)}\bar u_n+\max_{B_{2r_0}(a_i)}f_n),$$ Since Corollary [Corollary 12](#label-10){reference-type="ref" reference="label-10"} implies $\max_{\partial B_{2r_0}(a_i)}\bar u_n\le C$ and Lemma [Lemma 10](#equiv-1){reference-type="ref" reference="equiv-1"} implies $\max_{B_{2r_0}(a_i)}\bar u_n\to+\infty$, we have $$\max_{B_{2r_0}(a_i)}f_n\to+\infty,$$ which, together with $\max_{B_{2r_0}(a_i)}f_n(x)\leq Cp_n \gamma_{n,i}^{p_n}$, yields that up to a subsequence, $\gamma_{n,i}\to\gamma_i\ge1$. By Corollary [Corollary 12](#label-10){reference-type="ref" reference="label-10"}, $u_n\to0$ in $L_{\operatorname{\rm loc}}^\infty(\overline B_{2r_0}(a_i)\setminus\left\{a_i\right\})$, so $x_{n,i}\to a_i$ as $n\to\infty$. Finally, it is easy to see $\beta_{a_i}\ge\frac{\lambda_{a_i}}{\gamma_i}\ge\frac{4\pi e}{\gamma_i}$. This completes the proof. ◻ # The regular case In this section, we prove Theorem [Theorem 2](#thm-regular){reference-type="ref" reference="thm-regular"}. Let $u_n$ be a solution sequence of [\[equ-regular\]](#equ-regular){reference-type="eqref" reference="equ-regular"}. Without loss of generality, we may assume the radius $r=1$. Suppose $0$ is the only blow-up point of $p_nu_n$ in $B_1$ and $V_n(x)$ satisfies [\[con-regular2\]](#con-regular2){reference-type="eqref" reference="con-regular2"}. Let $x_n$ be a maximum point of $u_n$ in $B_1$, i.e. $$u_n(x_n):=\max_{\overline B_1}u_n,$$ then [\[con-regular1\]](#con-regular1){reference-type="eqref" reference="con-regular1"} implies $p_nu_n(x_n)\to\infty$ and $x_n\to0$. Define the scaling parameter $\mu_n>0$ by $$\mu_n^{-2}:=p_nV_n(x_n)u_n(x_n)^{p_n-1},$$ and the scaling function by $$\label{55}v_n(x):=p_n\left(\frac{u_n(x_n+\mu_nx)}{u_n(x_n)}-1\right) \quad\text{for}~x\in D_n:=\frac{B_1-x_n}{\mu_n}.$$ It is easy to see that $v_n$ satisfies $$\label{label-1} \begin{cases} -\Delta v_n=\frac{V_n(x_n+\mu_nx)}{V_n(x_n)}\left(1+\frac{v_n}{p_n}\right)^{p_n}\quad\text{in}~D_n,\\ v_n(0)=0=\max_{D_n}v_n, \end{cases}$$ and $$\label{5-0}0<1+\frac{v_n(x)}{p_n}=\frac{u_n(x_n+\mu_nx)}{u_n(x_n)}\leq 1\quad \text{in }D_n.$$ **Lemma 13**. *After passing to a subsequence, it hold $\mu_n\to0$, $u_n(x_n)\to\gamma\in[1,\infty)$ and $v_n\to U_0(x)=-2\log(1+\frac{1}{8}|x|^2)$ in $\mathcal C_{loc}^2(\mathbb R^2)$.* *Proof.* Suppose $\mu_n\not\to0$, then up to subsequence we may assume $u_n(x_n)\le\left(\frac{C}{p_n}\right)^{\frac{1}{p_n-1}}$ for some constant $C>0$. Thus it holds $0\le -\Delta (p_nu_n)\le C$, which together with $\displaystyle\max_{\partial B_1}p_nu_n\le C$ implies $\displaystyle\max_{B_1}p_nu_n\le C$. This is a contradiction with that $0$ is a blow-up point of $p_nu_n$. So $\mu_n\to0$ and hence $u_n(x_n)\to\gamma\ge 1$. Now we prove $\gamma<\infty$. Recall [@LE-4 Proposition 2.7] that there is $C>0$ independent of $x\in\Omega$ and $p$ such that $$\|G(x,\cdot)\|_{L^{p}(\Omega)}^{p}\leq Cp^{p+1},\quad \text{for $p>1$ large}.$$ Then by the Green's representation formula and Hölder inequality, $$\begin{aligned} u_n(x_n) &=\int_{B_1} G(x_n,y)V_n(y)u_n(y)^{p_n}{\mathrm d}y-\int_{\partial B_1}\frac{\partial G(x_n,y)}{\partial\nu}u_n(y){\mathrm d}s_y \\ &\le C\|G(x_n,\cdot)\|_{L^{2p_n+1}(B_1)}\left(\int_{B_1}V_n(y)^{1+\frac{1}{2p_n}}u_n(y)^{p_n+\frac{1}{2}}\right)^{\frac{2p_n}{2p_n+1}}\\ &\qquad+\frac{C}{p_n}\int_{\partial B_1}\left\lvert\frac{\partial G(x_n,y)}{\partial\nu}\right\rvert{\mathrm d}s_y\\ &\le C(2p_n+1)^{\frac{2p_n+2}{2p_n+1}}u_n(x_n)^{\frac{p_n}{2p_n+1}}\left(\int_{B_1}V_n(y)u_n(y)^{p_n}\right)^{\frac{2p_n}{2p_n+1}}+\frac{C}{p_n}\\ &\le C(2p_n+1)^{\frac{2p_n+2}{2p_n+1}}u_n(x_n)^{\frac{p_n}{2p_n+1}}\left(\frac{C}{p_n}\right)^{\frac{2p_n}{2p_n+1}}+\frac{C}{p_n}\\ &\le C\left(u_n(x_n)^{\frac{p_n}{2p_n+1}}+\frac{1}{p_n}\right),\quad\text{for $p_n>1$ large enough},\end{aligned}$$ so $\limsup_{n\to\infty} u_n(x_n)\le C$, i.e., $\gamma<\infty$. For any $R>0$, $B_R\subset D_n$ for $n$ large. Like [\[harmonic-0\]](#harmonic-0){reference-type="eqref" reference="harmonic-0"} we let $$v_n=\varphi_n+\psi_n\quad\text{in}~B_R,$$ with $-\Delta \varphi_n=-\Delta v_n$ in $B_R$ and $\psi_n=v_n$ on $\partial B_R$. Thanks to [\[label-1\]](#label-1){reference-type="eqref" reference="label-1"}-[\[5-0\]](#5-0){reference-type="eqref" reference="5-0"}, we see that $|-\Delta v_n|\le C$ in $D_n$ for some constant $C>0$. Then by the standard elliptic theory, we obtain that $\varphi_n$ is uniformly bounded in $B_R$. Since $\psi_n=v_n-\varphi_n$, we know that $\psi_n$ is harmonic in $B_R$ and bounded from above. By the Harnack inequality, we see that if $\inf_{B_R}\psi_n\to-\infty$, then $\sup_{B_R}\psi_n\to-\infty$ as $n\to\infty$, which contradicts with $\psi_n(0)=-\varphi_n(0)\ge-C$. So $\psi_n$ and hence $v_n$ is uniformly bounded in $B_R$. After passing to a subsequence, the standard elliptic theory implies that $$v_n\to U_0 \quad\text{in}~\mathcal C_{loc}^2(\mathbb R^2)~\text{as}~n\to\infty,$$ and [\[label-1\]](#label-1){reference-type="eqref" reference="label-1"} implies $$\label{label-2} \begin{cases} -\Delta U_0=e^{U_0}\quad\text{in}~\mathbb R^2,\\ U_0(0)=0=\max_{\mathbb R^2}U_0. \end{cases}$$ Moreover, by Fatou's Lemma, $$\begin{aligned} \int_{\mathbb R^2}e^{U_0}{\mathrm d}x&\le \liminf_{n\to\infty}\int_{D_n}\frac{V_n(x_n+\mu_nx)}{V_n(x_n)}\Big(1+\frac{v_n}{p_n}\Big)^{p_n}{\mathrm d}x\\ &=\liminf_{n\to\infty}\frac{p_n}{u_n(x_n)}\int_{B_1}V_n(x)u_n(x)^{p_n}{\mathrm d}x\le C.\end{aligned}$$ Since $U_0(0)=0$, by the classification result due to Chen and Li [@classification-1] we obtain $$U_0(x)=-2\log \left(1+\frac{1}{8}|x|^2\right),$$ and $\int_{\mathbb R^2}e^{U_0}{\mathrm d}x=8\pi$. ◻ We introduce the local Pohozaev identity. **Lemma 14**. *Suppose $u$ satisfies $$\begin{cases} -\Delta u=V(x)u^p,\quad\text{in}~\Omega,\\ u>0,\quad\text{in}~\Omega,\end{cases}$$ then for any $y\in\mathbb R^2$ and any subset $\Omega'\subset\Omega$, it holds $$\label{pho-1} \begin{aligned} &\quad \frac{1}{p+1}\int_{\Omega'}\left(2V(x)+\left\langle\nabla V(x),x-y\right\rangle\right)u(x)^{p+1}{\mathrm d}x \\ &=\int_{\partial\Omega'}\left\langle\nabla u(x),\nu(x)\right\rangle\left\langle\nabla u(x),x-y\right\rangle-\frac{1}{2}|\nabla u(x)|^2\left\langle x-y,\nu(x)\right\rangle{\mathrm d}s_x \\ &\quad +\frac{1}{p+1}\int_{\partial\Omega'} V(x)u(x)^{p+1}\left\langle x-y,\nu(x)\right\rangle{\mathrm d}s_x, \end{aligned}$$ where $\nu(x)$ denotes the outer normal vector of $\partial \Omega'$ at $x$.* *Proof.* By direct computations, we have $$-\Delta u(x)\cdot\left\langle\nabla u(x),x-y\right\rangle=-\operatorname{div}\left(\nabla u(x)\left\langle\nabla u(x),x-y\right\rangle-\frac{1}{2}|\nabla u(x)|^2 (x-y)\right),$$ and $$\begin{aligned} V(x)u(x)^p\cdot\left\langle\nabla u(x),x-y\right\rangle&=\frac{1}{p+1}\operatorname{div}\left(V(x)u(x)^{p+1}(x-y)\right)\\ &\quad -\frac{1}{p+1}(2V(x)+\left\langle\nabla V(x),x-y\right\rangle)u(x)^{p+1}. \end{aligned}$$ Then multiplying $-\Delta u=V(x)u^p$ with $\left\langle\nabla u(x),x-y\right\rangle$, integrating on $\Omega'$ and using the divergence theorem, we obtain [\[pho-1\]](#pho-1){reference-type="eqref" reference="pho-1"}. ◻ By [\[con-regular1\]](#con-regular1){reference-type="eqref" reference="con-regular1"} we have that for any compact subset $K\Subset\overline B_1\setminus\{0\}$, $$\label{label-36} \|p_nu_n\|_{L^\infty(K)}\le C_K.$$ **Lemma 15**. *It holds $$p_nu_n(x)\to 8\pi\gamma G_1(x,0)+\psi(x),\quad\text{in}~\mathcal C_{loc}^2(\overline B_1\setminus\{0\})~\text{as}~n\to\infty.$$ where $\gamma$ is given in Lemma [Lemma 13](#converge-regular){reference-type="ref" reference="converge-regular"}, $\psi\in\mathcal C^2(\overline B_1)$ is a harmonic function, and $G_1(x,y)$ denotes the Green function of $-\Delta$ in $B_1$ with the Dirichlet boundary condition.* *Proof.* Like [\[harmonic-0\]](#harmonic-0){reference-type="eqref" reference="harmonic-0"} we set $u_n=\phi_n+\psi_n$ with $\phi_n=0$ on $\partial B_1$ and $\psi_n$ is harmonic in $B_1$. Since $\psi_n=u_n=O(\frac{1}{p_n})$ on $\partial B_1$, it follows from the standard elliptic theory that up to a subsequence, $p_n\psi_n\to\psi$ in $\mathcal C^2(\overline B_1)$. Since $p_n\psi_n$ is harmonic, so is $\psi$. Take $d\in(0,1)$ and any compact subset $K\Subset\overline B_1\setminus\{0\}$. Applying the Green's representation formula to $\phi_n$ and using [\[con-regular1\]](#con-regular1){reference-type="eqref" reference="con-regular1"}-[\[con-regular2\]](#con-regular2){reference-type="eqref" reference="con-regular2"}, we get that for any $x\in K$, $$\begin{aligned} p_n\phi_n(x) &=\int_{B_1} G_1(x,y)p_nV_n(y)u_n(y)^{p_n} {\mathrm d}y\\ &=\int_{B_d} G_1(x,y)p_nV_n(y)u_n(y)^{p_n} {\mathrm d}y+o_n(1)\int_{B_1\setminus B_d} G_1(x,y){\mathrm d}y\\ &\to\sigma_0 G_1(x,0),\quad\text{uniformly for $x\in K$ as}~n\to\infty, \end{aligned}$$ where $$\label{label-43} \sigma_0:=\lim_{d\to0}\lim_{n\to\infty}\int_{B_d}p_nV_n(x)u_n(x)^{p_n}{\mathrm d}x.$$ Again by the Green's representation formula, a similar argument implies $$\nabla_x(p_n\phi_n)(x)=\int_\Omega \nabla_x G_1(x,y)p_nV_n(y)u_n(y)^{p_n} {\mathrm d}y\to \sigma_0 \nabla_xG_1(x,0).$$ Thus $p_nu_n(x)\to \sigma_0 G_1(x,0)+\psi(x)$ in $\mathcal C_{loc}^1(\overline B_1\setminus\{0\})$. From here and $-\Delta (p_nu_n)=p_nV_n(x)u_n^{p_n}\to 0$ in $L_{\operatorname{\rm loc}}^\infty(\overline B_1\setminus\{0\})$ and $p_nu_n\to\psi$ in $\mathcal C^2(\partial B_1)$, it follows from the standard elliptic estimates that $$p_nu_n(x)\to \sigma_0 G_1(x,0)+\psi(x),\quad\text{in}~\mathcal C_{loc}^2(\overline B_1\setminus\{0\})~\text{as}~n\to\infty.$$ It remains to prove $\sigma_0=8\pi\gamma$. Since $$\begin{aligned} \int_{B_d}p_nV_n(x)u_n(x)^{p_n}{\mathrm d}x &=u_n(x_n)\int_{\frac{B_{d}-x_n}{\mu_n}}\frac{V_n(x_n+\mu_nx)}{V_n(x_n)}\left(1+\frac{v_n(x)}{p_n}\right)^{p_n}{\mathrm d}x\\ &\ge \gamma\int_{\mathbb R^2}e^{U_0}{\mathrm d}x+o_n(1)=8\pi\gamma+o_n(1), \end{aligned}$$ we get $\sigma_0\ge 8\pi\gamma$. On the other hand, applying the Pohozaev identity [\[pho-1\]](#pho-1){reference-type="eqref" reference="pho-1"} with $y=0$, $\Omega'=B_d$, $V=V_n$ and $u=u_n$, we obtain $$\label{label-46} \begin{aligned} &\quad \frac{p_n^2}{p_n+1}\int_{B_d}\left(2V_n(x)+\left\langle\nabla V_n(x),x\right\rangle\right)u_n(x)^{p_n+1}{\mathrm d}x \\ &=\int_{\partial B_d}\left\langle p_n\nabla u_n(x),\nu(x)\right\rangle\left\langle p_n\nabla u_n(x),x\right\rangle-\frac{1}{2}|p_n\nabla u_n(x)|^2\left\langle x,\nu(x)\right\rangle{\mathrm d}s_x \\ &\quad +\frac{p_n^2}{p_n+1}\int_{\partial B_d} V_n(x)u_n(x)^{p_n+1}\left\langle x,\nu(x)\right\rangle{\mathrm d}s_x. \end{aligned}$$ Note that for $x\in B_1\setminus\{0\}$, we have $$\label{label-45} p_n\nabla u_n(x)\to\sigma_0\nabla_x G(x,0)+\nabla\psi(x)=-\frac{\sigma_0}{2\pi}\frac{x}{|x|^2}+O(1).$$ Using [\[label-36\]](#label-36){reference-type="eqref" reference="label-36"} and [\[label-45\]](#label-45){reference-type="eqref" reference="label-45"}, we obtain (note $\nu(x)=\frac{x}{|x|}$ on $\partial B_d$) $$\lim_{n\to\infty}\text{RHS of \eqref{label-46}}=\frac{\sigma_0^2}{4\pi}+O(d).$$ From here and [\[label-46\]](#label-46){reference-type="eqref" reference="label-46"}, we conclude $$\label{label-48} \lim_{d\to 0}\lim_{n\to\infty}p_n\int_{B_d}\left(2V_n(x)+\left\langle\nabla V_n(x),x\right\rangle\right)u_n(x)^{p_n+1}{\mathrm d}x=\frac{\sigma_0^2}{4\pi}.$$ Since $V_n$ satisfies [\[con-regular2\]](#con-regular2){reference-type="eqref" reference="con-regular2"}, we have $$\left\lvert p_n\int_{B_d}\left\langle\nabla V_n(x),x\right\rangle u_n(x)^{p_n+1}{\mathrm d}x \right\rvert\le Cd\int_{B_d}p_nV_n(x)u_n(x)^{p_n}{\mathrm d}x\le C d,$$ which together with [\[label-48\]](#label-48){reference-type="eqref" reference="label-48"} and [\[label-43\]](#label-43){reference-type="eqref" reference="label-43"} implies $$\frac{\sigma_0^2}{8\pi}=\lim_{d\to0}\lim_{n\to\infty} p_n\int_{B_d}V_n(x)u_n(x)^{p_n+1}{\mathrm d}x\leq \lim_{n\to\infty}u_n(x_n) \sigma_0=\gamma\sigma_0,$$ so $\sigma_0\le 8\pi\gamma$. This proves $\sigma_0=8\pi\gamma$. ◻ For the scaling function $v_n$ defined in [\[55\]](#55){reference-type="eqref" reference="55"}, we need the following decay estimates, which will be used to apply the Dominated Convergence Theorem. **Lemma 16**. *For any $\eta\in(0,4)$, there exist small $r_\eta>0$, large $R_\eta>1$, $n_\eta>1$ and constant $C_\eta>0$ such that $$v_n(x)\le \eta\log\frac{1}{|x|}+C_\eta\quad \text{and} \quad \left\lvert v_n(x)\right\rvert\le C_\eta(1+\log|x|),$$ for any $2R_\eta\le |x|\le \frac{r_\eta}{\mu_n}$ and $n\ge n_\gamma$.* *Proof.* By Lemma [Lemma 13](#converge-regular){reference-type="ref" reference="converge-regular"} we have $$v_n(x)\to U_0(x)=-2\log\left(1+\frac{1}{8}|x|^2\right)\quad \text{in}~\mathcal C_{loc}^2(\mathbb R^2).$$ Moreover, Lemma [Lemma 15](#converge-regular2){reference-type="ref" reference="converge-regular2"} tells $\sigma_0= 8\pi\gamma$ with $\sigma_0$ defined by [\[label-43\]](#label-43){reference-type="eqref" reference="label-43"}. Applying the Green's representation formula, we have for any $x\in D_n$, $$\begin{aligned} u_n(x_n+\mu_nx) &=\int_{B_1} G_1(x_n+\mu_nx,y)V_n(y)u_n(y)^{p_n}{\mathrm d}y-\int_{\partial B_1} \frac{\partial G_1(x_n+\mu_nx,y)}{\partial\nu_y}u_n(y){\mathrm d}s_y\\ &=\frac{u_n(x_n)}{p_n}\int_{D_n} G_1(x_n+\mu_nx,x_n+\mu_nz) \frac{V_n(x_n+\mu_nz)}{V_n(x_n)} \left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &\quad -\int_{\partial B_1} \frac{\partial G_1(x_n+\mu_nx,y)}{\partial\nu_y}u_n(y){\mathrm d}s_y. \end{aligned}$$ Then it follows from [\[55\]](#55){reference-type="eqref" reference="55"} that $$\begin{aligned} v_n(x)&=-p_n+ \int_{D_n} G_1(x_n+\mu_nx,x_n+\mu_nz)\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &\quad -\frac{p_n}{u_n(x_n)}\int_{\partial B_1} \frac{\partial G_1(x_n+\mu_nx,y)}{\partial\nu_y}u_n(y){\mathrm d}s_y. \end{aligned}$$ Since $v_n(0)=0$ and $G_1(z,y)=-\frac{1}{2\pi}\log|z-y|-H_1(z,y)$, we have $$\begin{aligned} &\quad v_n(x)\\ &=v_n(x)-v_n(0)\\ &=\int_{D_n} \left[G_1(x_n+\mu_nx,x_n+\mu_nz)-G_1(x_n,x_n+\mu_nz)\right]\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &\quad -\frac{p_n}{u_n(x_n)}\int_{\partial B_1} \left(\frac{\partial G_1(x_n+\mu_nx,y)}{\partial\nu_y}-\frac{\partial G_1(x_n,y)}{\partial\nu_y}\right)u_n(y){\mathrm d}s_y\\ &=\frac{1}{2\pi}\int_{D_n} \log\frac{|z|}{|z-x|}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &\quad -\int_{D_n} \left[H_1(x_n+\mu_nx,x_n+\mu_nz)-H_1(x_n,x_n+\mu_nz)\right]\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &\quad -\frac{p_n}{u_n(x_n)}\int_{\partial B_1} \left(\frac{\partial G_1(x_n+\mu_nx,y)}{\partial\nu_y}-\frac{\partial G_1(x_n,y)}{\partial\nu_y}\right)u_n(y){\mathrm d}s_y\\ &=:\@slowromancap\romannumeral 1@(x)+\@slowromancap\romannumeral 2@(x)+\@slowromancap\romannumeral 3@(x).\end{aligned}$$ Since $H_1(x,y)$ is smooth in $B_1\times B_1$ and $\nabla G_1(x,y)$ is bounded for $|x-y|\ge c>0$, we have that for $|x|\le \frac{r_\eta}{\mu_n}$ with small $r_\eta<\frac{1}{2}$ to be chosen later, $$\begin{aligned} \@slowromancap\romannumeral 2@(x) &=O(1)\int_{D_n} \frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &=O(1)\frac{p_n}{u_n(x_n)}\int_{B_1} V_n(y)u_n(y)^{p_n}{\mathrm d}y=O(1), \end{aligned}$$ $$\begin{aligned} \@slowromancap\romannumeral 3@(x)=O(1)\int_{\partial B_1} \left\lvert\frac{\partial G_1(x_n+\mu_nx,y)}{\partial\nu_y}\right\rvert+\left\lvert\frac{\partial G_1(x_n,y)}{\partial\nu_y}\right\rvert{\mathrm d}y=O(1). \end{aligned}$$ For any fixed $\eta\in(0,4)$, let $\varepsilon=\frac{2\pi}{3}(4-\eta)>0$ and take $R_\eta>1$ large such that $\int_{B_{R_\eta}(0)}e^{U_0}>\int_{\mathbb R^2}e^{U_0}-\frac{\varepsilon}{2}=8\pi-\frac{\varepsilon}{2}$, where $U_0(z)=-2\log\left(1+\frac{1}{8}|z|^2\right)$. Then from $v_n\to U_0$ we get that for $n$ large, $$\label{label-54} \int_{B_{R_\eta}} \frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z \ge \int_{B_{R_\eta}}e^{U_0(z)}{\mathrm d}z-\frac{\varepsilon}{2}> 8\pi-\varepsilon.$$ From $\sigma_0=8\pi\gamma$, we see that $$\begin{aligned} &\lim_{r\to0}\lim_{n\to\infty}\int_{\left\{|z|\le\frac{2r}{\mu_n}\right\}}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ =&\lim_{r\to0}\lim_{n\to\infty}\frac{p_n}{u_n(x_n)}\int_{B_{2r}}V_n(y)u_n(y)^{p_n}dy=\frac{\sigma_0}{\gamma}= 8\pi.\end{aligned}$$ Thus we can choose $r_\eta\in (0,\frac12)$ small such that for $n$ large, $$\label{label-55} \int_{\left\{|z|\le\frac{2r_\eta}{\mu_n}\right\}}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\le 8\pi+\varepsilon.$$ By [\[label-54\]](#label-54){reference-type="eqref" reference="label-54"}-[\[label-55\]](#label-55){reference-type="eqref" reference="label-55"} we obtain $$\label{65} \int_{\left\{R_{\eta}\leq |z|\le\frac{2r_\eta}{\mu_n}\right\}}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\le 2\varepsilon.$$ Fix any $2R_\eta\le |x|\le \frac{r_\eta}{\mu_n}$, to compute the integral $\@slowromancap\romannumeral 1@(x)$, we divide $D_n$ into four domains $D_n=\sum_{i=1}^4D_{n,i}$ and divide the integral $\@slowromancap\romannumeral 1@(x)$ into four terms $\@slowromancap\romannumeral 1@(x)=\sum_{i=1}^4\@slowromancap\romannumeral 1@_{D_{n,i}}(x)$, where $D_{n,1}=\left\{z\in D_n:~|z|\le R_\eta\right\}$, $D_{n,2}=\left\{z\in D_n:~|z|\ge \frac{2r_\eta}{\mu_n}\right\}$ and $$D_{n,3}=\left\{z\in D_n:~R_\eta\le|z|\le\frac{2r_\eta}{\mu_n},~ |z|\le 2|z-x|\le 3|z|\right\},$$ $$D_{n,4}=\left\{z\in D_n:~R_\eta\le|z|\le\frac{2r_\eta}{\mu_n},~|z|\ge 2|z-x|~\text{or}~2|z-x|\ge 3|z|\right\}.$$ If $z\in D_{n,1}$, then $|x|\ge 2|z|$, $|x-z|\geq \frac12 |x|$ and hence $$\log\frac{|z|}{|z-x|}\le \log\frac{2R_\gamma}{|x|}\le 0.$$ From here and [\[label-54\]](#label-54){reference-type="eqref" reference="label-54"}, we have $$\begin{aligned} \label{label-60} \@slowromancap\romannumeral 1@_{D_{n,1}}(x)&\le \frac{1}{2\pi}\log\frac{2R_\eta}{|x|}\int_{D_{n,1}} \frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z \\ &\le (4-\frac{\varepsilon}{2\pi})\log\frac{1}{|x|}+C.\nonumber\end{aligned}$$ On the other hand, since $v_n(x)\le 0$ and $$0\ge\log\frac{1}{|z-x|}\ge\log\frac{2}{3|x|},\quad\text{for }z\in D_{n,1},$$ we get $$\begin{aligned} \label{label-61} \@slowromancap\romannumeral 1@_{D_{n,1}}(x) &\ge \frac{1}{2\pi}\log\frac{2}{3|x|}\int_{D_{n,1}} \frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z -C\int_{D_{n,1}}\left\lvert \log|z|\right\rvert{\mathrm d}z\nonumber\\ &\ge (4+\frac{\varepsilon}{2\pi})\log\frac{1}{|x|}-C.\end{aligned}$$ Note that $|z|\geq 2|x|$ for $z\in D_{n,2}$. Then it is easy to see that $$\log\frac{2}{3}\le\log\frac{|z|}{|z-x|}\le\log2,\quad\text{for }\;z\in D_{n,2}\cup D_{n,3},$$ which implies $$\begin{aligned} \label{label-62} \left\lvert\@slowromancap\romannumeral 1@_{D_{n,2}}(x)+\@slowromancap\romannumeral 1@_{D_{n,3}}(x)\right\rvert&\le C\int_{ D_n} \frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &= \frac{Cp_n}{u_n(x_n)}\int_{\Omega}V_n(y)u_n(y)^{p_n}{\mathrm d}y\le C.\nonumber\end{aligned}$$ Finally for $z\in D_{n,4}$, it holds $2\le |z|\le 2|x|$. Then we see from [\[65\]](#65){reference-type="eqref" reference="65"} that $$\label{label-56} \begin{aligned} 0&\le \frac{1}{2\pi}\int_{D_{n,4}} \log|z|\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &\le \frac{1}{2\pi}\log(2|x|)\int_{D_{n,4}}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &\le \frac{\varepsilon}{\pi}\log|x|+C, \end{aligned}$$ Furthermore, by $v_n(x)\le 0$, we get $$\label{label-57} \begin{aligned} 0&\le \frac{1}{2\pi}\int_{\left\{z\in D_{n,4}:~|z-x|\le 1\right\}} \log\frac{1}{|z-x|}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &\le C\int_{\left\{|z-x|\le 1\right\}}\log\frac{1}{|z-x|}{\mathrm d}x\le C. \end{aligned}$$ While for $z\in\left\{z\in D_{n,4}:~|z-x|\ge 1\right\}$, it holds $$\log\frac{1}{3|x|}\le \log\frac{1}{|z-x|}\le 0,$$ and hence $$\label{label-59} \begin{aligned} 0\ge&\frac{1}{2\pi}\int_{\left\{z\in D_{n,4}:~|z-x|\ge 1\right\}} \log\frac{1}{|z-x|}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &\ge \frac{1}{2\pi}\log\frac{1}{3|x|}\int_{D_{n,4}}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &\ge \frac{\varepsilon}{\pi}\log\frac{1}{3|x|}. \end{aligned}$$ Combining [\[label-56\]](#label-56){reference-type="eqref" reference="label-56"}-[\[label-59\]](#label-59){reference-type="eqref" reference="label-59"}, we get $$\label{label-63} \frac{\varepsilon}{\pi}\log\frac{1}{3|x|}\le \@slowromancap\romannumeral 1@_{D_{n,4}}(x)\le \frac{\varepsilon}{\pi}\log|x|+C.$$ By [\[label-60\]](#label-60){reference-type="eqref" reference="label-60"},[\[label-61\]](#label-61){reference-type="eqref" reference="label-61"},[\[label-62\]](#label-62){reference-type="eqref" reference="label-62"},[\[label-63\]](#label-63){reference-type="eqref" reference="label-63"} and $\@slowromancap\romannumeral 2@(x)=O(1)$, $\@slowromancap\romannumeral 3@(x)=O(1)$, we finally get $$\left\lvert v_n(x)\right\rvert\le C(1+\log|x|),$$ and $$v_n(x)\le \left[4-\frac{3\varepsilon}{2\pi}\right]\log\frac{1}{|x|}+C=\eta\log\frac{1}{|x|}+C,$$ for any $2R_\eta\le |x|\le \frac{r_\eta}{\mu_n}$ and some constant $C>0$. This completes the proof. ◻ *Remark 17*. For any $\eta\in(0,4)$, Lemma [Lemma 16](#decay-regular1){reference-type="ref" reference="decay-regular1"} implies $$\left(1+\frac{v_n(x)}{p_n}\right)^{p_n}=e^{p_n\log(1+\frac{v_n(x)}{p_n})}\le e^{v_n(x)} \le \frac{C_\eta}{|x|^\eta},\quad\text{\it for }\;2R_\eta\le |x|\le \frac{r_\eta}{\mu_n}.$$ Meanwhile, since $v_n\to U_0$ in $\mathcal C_{loc}^2(\mathbb R^2)$, we have $\left(1+\frac{v_n(x)}{p_n}\right)^{p_n}\le C$ for $|x|\le 2R_\eta$ and $n$ large. Therefore, $$0\le \left(1+\frac{v_n(x)}{p_n}\right)^{p_n}\le \frac{C_\eta}{1+|x|^\eta}, \quad\forall|x|\leq \frac{r_\eta}{\mu_n}.$$ Similarly, we have $$\left\lvert v_n(x)\right\rvert\le C_\eta\log\left(2+|x|\right), \quad\forall|x|\leq \frac{r_\eta}{\mu_n}.$$ As a direct application of the above decay estimates, we have **Lemma 18**. *It holds $\gamma=\sqrt e$.* *Proof.* Take $\eta=3$ in Lemma [Lemma 16](#decay-regular1){reference-type="ref" reference="decay-regular1"} and Remark [Remark 17](#decay-regular2){reference-type="ref" reference="decay-regular2"} and let $r=r_3/2$. Let $G_{r}(x,y)$ denote the Green function of $-\Delta$ in $B_{r}$ with the Dirichlet boundary condition. By the Green's representation formula, we have $$\begin{aligned} & u_n(x_n)=\int_{B_{r}} G_{r}(x_n,y)V_n(y)u_n(y)^{p_n}{\mathrm d}y-\int_{\partial B_{r}}\frac{\partial G_{r}(x_n,y)}{\partial\nu}u_n(y){\mathrm d}s_y \\ &=\frac{u_n(x_n)}{p_n}\int_{\frac{B_{r}-x_n}{\mu_n}} G_{r}(x_n,x_n+\mu_ny)\frac{V_n(x_n+\mu_ny)}{V_n(x_n)}\left(1+\frac{v_n(y)}{p_n}\right)^{p_n}{\mathrm d}y+O(\frac{1}{p_n}),\\ \end{aligned}$$ so $$\frac{1}{p_n}\int_{\frac{B_{r}-x_n}{\mu_n}} G_{r}(x_n,x_n+\mu_ny)\frac{V_n(x_n+\mu_ny)}{V_n(x_n)}\left(1+\frac{v_n(y)}{p_n}\right)^{p_n}{\mathrm d}y=1+O(\frac{1}{p_n}).$$ On the other hand, by Remark [Remark 17](#decay-regular2){reference-type="ref" reference="decay-regular2"}, for any $y\in \frac{B_{r}-x_n}{\mu_n}$, we have $|y|\le \frac{r_3}{\mu_n}$ for $n$ large and so $$\label{33-25} 0\le\left(1+\frac{v_n(y)}{p_n}\right)^{p_n}\le \frac{C}{1+|y|^3}.$$ Then by applying the Dominated Convergence Theorem, we get $$\lim_{n\to\infty}\int_{\frac{B_{r}-x_n}{\mu_n}} \frac{V_n(x_n+\mu_ny)}{V_n(x_n)}\left(1+\frac{v_n(y)}{p_n}\right)^{p_n}{\mathrm d}y=\int_{\mathbb R^2}e^{U_0}{\mathrm d}x=8\pi,$$ $$\begin{aligned} &\lim_{n\to\infty}\int_{\frac{B_{r}-x_n}{\mu_n}} \left(\frac{1}{2\pi}\log|y|+H_{r}(x_n,x_n+\mu_ny)\right)\frac{V_n(x_n+\mu_ny)}{V_n(x_n)}\left(1+\frac{v_n(y)}{p_n}\right)^{p_n}{\mathrm d}y\\ &\qquad=\int_{\mathbb R^2} \left(\frac{1}{2\pi}\log|y|+H_{r}(0,0)\right) e^{U_0(y)}{\mathrm d}y=C <\infty.\end{aligned}$$ From here, $G_{r}(x,y)=-\frac{1}{2\pi}\log|x-y|-H_{r}(x,y)$ and $\mu_n^{-2}=p_nV_n(x_n)u_n(x_n)^{p_n-1}$, we have $$\begin{aligned} &\quad 1+O(\frac{1}{p_n})\\ &= \frac{1}{p_n}\int_{\frac{B_{r}-x_n}{\mu_n}} G_{r}(x_n,x_n+\mu_ny)\frac{V_n(x_n+\mu_ny)}{V_n(x_n)}\left(1+\frac{v_n(y)}{p_n}\right)^{p_n}{\mathrm d}y\\ &=-\frac{1}{2\pi}\frac{\log\mu_n}{p_n}\int_{\frac{B_{r}-x_n}{\mu_n}} \frac{V_n(x_n+\mu_ny)}{V_n(x_n)}\left(1+\frac{v_n(y)}{p_n}\right)^{p_n}{\mathrm d}y+O(\frac1{p_n})\\ &=\frac{1}{4\pi}\left(\frac{\log p_n+\log V_n(x_n)}{p_n}+\frac{p_n-1}{p_n}\log u_n(x_n)\right)(8\pi+o_n(1))+o_n(1)\\ &=2\log \gamma+o_n(1). \end{aligned}$$ Thus $2\log \gamma=1$, i.e., $\gamma=\sqrt e$. ◻ Now we are ready to prove Theorem [Theorem 2](#thm-regular){reference-type="ref" reference="thm-regular"}. *Proof of Theorem [Theorem 2](#thm-regular){reference-type="ref" reference="thm-regular"}.* It has been proved in Lemma [Lemma 18](#tem-2){reference-type="ref" reference="tem-2"} that $\displaystyle\max_{\overline B_1}u_n\to \sqrt e$. Since $0$ is the only blow-up point of $p_nu_n$ in $B_1$, we see that $$p_nV_n(x)u_n(x)^{p_n-1+k}\to \beta_k\delta_0, \quad\text{for }k=0,1,2,$$ weakly in the sense of measures. For any small $r>0$, it follows from the Dominated Convergence Theorem that $$\begin{aligned} &\int_{B_r}p_nV_n(x)u_n(x)^{p_n-1+k}{\mathrm d}x\\ =&u_n(x_n)^{k}\int_{\frac{B_{r}-x_n}{\mu_n}}\frac{V_n(x_n+\mu_nx)}{V_n(x_n)}\left(1+\frac{v_n(x)}{p_n}\right)^{p_n-1+k}{\mathrm d}x \to \gamma^{k}\int_{\mathbb R^2}e^{U_0}{\mathrm d}x=8\pi e^{\frac{k}{2}},\end{aligned}$$ Thus $\beta_k=8\pi e^{\frac{k}{2}}$ for $k=0,1,2$. This completes the proof. ◻ # The singular case In this section, we prove Theorem [Theorem 3](#thm-singular){reference-type="ref" reference="thm-singular"}. Let $u_n$ be a solution sequence of [\[equ-singular\]](#equ-singular){reference-type="eqref" reference="equ-singular"}. Without loss of generality, we may assume the radius $r=1$. Suppose $0$ is the only blow-up point of $p_nu_n$ in $B_1$ and $V_n(x)$ satisfies [\[con-singular2\]](#con-singular2){reference-type="eqref" reference="con-singular2"}. Let $x_n$ be a maximum point of $u_n$ in $B_1$, i.e. $$\label{max-1} u_n(x_n)=\max_{\overline B_1}u_n,$$ then [\[con-singular1\]](#con-singular1){reference-type="eqref" reference="con-singular1"} implies $p_nu_n(x_n)\to\infty$ and $x_n\to0$. We claim that $$\label{4-42} \mu_n^{-2-2\alpha}:=p_nV_n(x_n)u_n(x_n)^{p_n-1}\to\infty.$$ Indeed, if $p_nV_n(x_n)u_n(x_n)^{p_n-1}\not\to\infty$, then up to a subsequence, we have $u_n(x_n)\le\left(\frac{C}{p_n}\right)^{\frac{1}{p_n-1}}$ for some constant $C>0$. Thus it holds $0\le -\Delta (p_nu_n)\le C$, which together with $\displaystyle\max_{\partial B_1}p_nu_n\le C$ implies $\displaystyle\max_{B_1}p_nu_n\le C$. This is a contradiction with that $0$ is a blow-up point of $p_nu_n$. So $p_nV_n(x_n)u_n(x_n)^{p_n-1}\to\infty$ and hence $\liminf_{n\to\infty}u_n(x_n)\ge 1$. Following the approach in Lemma [Lemma 13](#converge-regular){reference-type="ref" reference="converge-regular"}, we may assume that $$\label{max} u_n(x_n)=\max_{\overline B_1}u_n\to\gamma\in[1,\infty).$$ Up to a subsequence, we denote $$\begin{aligned} \label{tem-60-beta} \beta_k:=\lim_{n\to\infty} \int_{B_1}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+k}{\mathrm d}x,\quad k=0,1,2.\end{aligned}$$ Then $$\beta_2\leq \gamma \beta_1\leq \gamma^2\beta_0.$$ Furthermore, for any $0<d<1$, since [\[con-singular1\]](#con-singular1){reference-type="eqref" reference="con-singular1"} gives $\sup_{B_1\setminus B_{d}}p_nu_n\le C$, we have $$\lim_{n\to\infty} \int_{B_1\setminus B_d}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+k}{\mathrm d}x=0,$$ so $$\begin{aligned} \label{tem-60-beta1} \beta_k=\lim_{d\to 0}\lim_{n\to\infty} \int_{B_d}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+k}{\mathrm d}x,\quad k=0,1,2.\end{aligned}$$ **Lemma 19**. *We have $$\label{beta12}\beta_1^2=8\pi(1+\alpha)\beta_2,\quad \beta_1\leq 8\pi(1+\alpha)\gamma.$$* *Proof.* Let $G_1(x,y)$ denotes the Green function of $-\Delta$ in $B_1$ with the Dirichlet boundary condition. Exactly as in Lemma [Lemma 15](#converge-regular2){reference-type="ref" reference="converge-regular2"}, we get $$p_nu_n(x)\to \beta_1 G_1(x,0)+\psi(x),\quad\text{in}~\mathcal C_{loc}^2(\overline B_1\setminus\{0\})~\text{as}~n\to\infty,$$ where $\psi\in\mathcal C^2(\overline B_1)$ is a harmonic function. Consequently, for $x\in B_1\setminus\{0\}$, we have $$\label{tem-22} p_n\nabla u_n(x)\to\beta_1\nabla_x G(x,0)+\nabla\psi(x)=-\frac{\beta_1}{2\pi}\frac{x}{|x|^2}+O(1).$$ Applying the Pohozaev identity [\[pho-1\]](#pho-1){reference-type="eqref" reference="pho-1"} with $y=0$, $\Omega'=B_d$, $V=|x|^{2\alpha}V_n(x)$ and $u=u_n$, and by using [\[con-singular1\]](#con-singular1){reference-type="eqref" reference="con-singular1"} and [\[tem-22\]](#tem-22){reference-type="eqref" reference="tem-22"}, we obtain $$\label{tem-23} \begin{aligned} &\quad \lim_{n\to\infty}\frac{p_n^2}{p_n+1}\int_{B_d}|x|^{2\alpha}\left[(2+2\alpha)V_n(x)+\left\langle\nabla V_n(x),x\right\rangle\right]u_n(x)^{p_n+1}{\mathrm d}x \\ &=\frac{\beta_1^2}{4\pi}+O(d). \end{aligned}$$ Since $V_n$ satisfies [\[con-singular2\]](#con-singular2){reference-type="eqref" reference="con-singular2"}, we have $$\left\lvert p_n\int_{B_d}|x|^{2\alpha}\left\langle\nabla V_n(x),x\right\rangle u_n(x)^{p_n+1}{\mathrm d}x \right\rvert\le Cd\int_{B_d}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n}{\mathrm d}x\le C d.$$ From here and [\[tem-23\]](#tem-23){reference-type="eqref" reference="tem-23"}, we deduce $$\frac{\beta_1^2}{8\pi(1+\alpha)}=\lim_{d\to0}\lim_{n\to\infty}\int_{B_d}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n+1}{\mathrm d}x=\beta_2\le \gamma\beta_1,$$ namely [\[beta12\]](#beta12){reference-type="eqref" reference="beta12"} holds. ◻ Since $|x_n|\to0$ and $p_nV_n(x_n)u_n(x_n)^{p_n-1}\to\infty$, we need compare their convergence rates to analyse the values of $\beta_k$. ## A special case.  In this section, we assume that $$\label{assume-1} p_n|x_n|^{2+2\alpha}V_n(x_n)u_n(x_n)^{p_n-1}\le C.$$ Define the scaling function $$v_n(x):=p_n\left(\frac{u_n(x_n+\mu_nx)}{u_n(x_n)}-1\right) \quad\text{for}~x\in D_n:=\frac{B_1-x_n}{\mu_n}.$$ It is easy to see that $v_n$ satisfies $$\label{tem-20} \begin{cases} -\Delta v_n=\left\lvert x+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nx)}{V_n(x_n)}\left(1+\frac{v_n}{p_n}\right)^{p_n}\quad\text{in}~D_n,\\ v_n(0)=0=\max_{D_n}v_n. \end{cases}$$ Since [\[assume-1\]](#assume-1){reference-type="eqref" reference="assume-1"} implies $|\frac{x_n}{\mu_n}|\leq C$, up to a subsequence we have $\frac{x_n}{\mu_n}\to x_\infty$ for some $x_\infty\in\mathbb R^2$. Then by following the approach of Lemma [Lemma 13](#converge-regular){reference-type="ref" reference="converge-regular"}, we obtain $v_n\to U_\alpha$ in $\mathcal C_{loc}^2(\mathbb R^2)$, where $U_\alpha$ satisfies $$\begin{cases} -\Delta U_\alpha=|x+x_\infty|^{2\alpha}e^{U_\alpha}\quad\text{in}~\mathbb R^2,\\ U_\alpha(0)=0=\max_{\mathbb R^2}U_\alpha,\\ \int_{\mathbb R^2}e^{U_{\alpha}}{\mathrm d}x\le C. \end{cases}$$ By the classification result due to Prajapat and Tarantello [@classification-2], we obtain $$\label{tem-24} U_\alpha(z)=-2\log \left(1+\frac{1}{8(1+\alpha)^2}|(z+z_\infty)^{1+\alpha}-z_\infty^{1+\alpha}|^2\right),\quad z\in{\mathbb C},$$ where $z_\infty\in\mathbb{C}$ is the complex notation of $x_\infty$. Moreover, $$\int_{\mathbb R^2}|x+x_\infty|^{2\alpha}e^{U_\alpha}{\mathrm d}x=8\pi(1+\alpha).$$ **Lemma 20**. *Suppose [\[assume-1\]](#assume-1){reference-type="eqref" reference="assume-1"} holds, then $\beta_1=8\pi(1+\alpha)\gamma$.* *Proof.* By Fatou's Lemma, for any $d\in (0,1)$, $$\begin{aligned} &\quad \lim_{n\to\infty}\int_{B_d}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n}{\mathrm d}x\\ &=u_n(x_n)\int_{\frac{B_{d}-x_n}{\mu_n}}\left\lvert x+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nx)}{V_n(x_n)}\left(1+\frac{v_n(x)}{p_n}\right)^{p_n}{\mathrm d}x\\ &\ge \gamma\int_{\mathbb R^2}|x+x_\infty|^{2\alpha}e^{U_\alpha}{\mathrm d}x+o_n(1)=8\pi(1+\alpha)\gamma+o_n(1), \end{aligned}$$ we get $\beta_1\ge 8\pi(1+\alpha)\gamma$. Together with [\[beta12\]](#beta12){reference-type="eqref" reference="beta12"}, we obtain $\beta_1= 8\pi(1+\alpha)\gamma$. ◻ Since the following lemma is similar to Lemma [Lemma 16](#decay-regular1){reference-type="ref" reference="decay-regular1"}, we sketch the proof and only emphasize the different places. **Lemma 21**. *Suppose [\[assume-1\]](#assume-1){reference-type="eqref" reference="assume-1"} holds, then for any $\eta\in(0,4(1+\alpha))$, there exist small $r_\eta>0$, large $R_\eta>1$, $n_\eta>1$ and constant $C_\eta>0$ such that $$v_n(x)\le \eta\log\frac{1}{|x|}+C_\eta\quad \text{and} \quad \left\lvert v_n(x)\right\rvert\le C_\eta(1+\log|x|),$$ for any $2R_\eta\le |x|\le \frac{r_\eta}{\mu_n}$ and $n\ge n_\gamma$.* *Proof.* Exactly as in Lemma [Lemma 16](#decay-regular1){reference-type="ref" reference="decay-regular1"}, we have $$\label{tem-39}\begin{aligned} v_n(x) &=\frac{1}{2\pi}\int_{D_n} \log\frac{|z|}{|z-x|}\left\lvert z+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z+O(1)\\ &=:\@slowromancap\romannumeral 1@(x)+O(1) \end{aligned}$$ for $|x|\le \frac{r_\eta}{\mu_n}$ with small $r_\eta<\frac{1}{2}$ to be chosen later. For any fixed $\eta\in(0,4(1+\alpha))$, let $\varepsilon=\frac{2\pi}{3+4\alpha}[4(1+\alpha)-\eta]>0$ and take $R_\eta>0$ such that $$\int_{B_{R_\eta}}|z+x_\infty|^{2\alpha}e^{U_\alpha(z)}{\mathrm d}z>\int_{\mathbb R^2}|z+x_\infty|^{2\alpha}e^{U_\alpha(z)}{\mathrm d}z-\frac{\varepsilon}{2}=8\pi(1+\alpha)-\frac{\varepsilon}{2},$$ where $U_\alpha$ is given in [\[tem-24\]](#tem-24){reference-type="eqref" reference="tem-24"}. Then from $v_n\to U_\alpha$ we get that for $n$ large, $$\begin{aligned} \label{tem-26} & \int_{B_{R_\eta}} \left\lvert z+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z \\ \ge &\int_{B_{R_\eta}}|z+x_\infty|^{2\alpha}e^{U_\alpha}{\mathrm d}z-\frac{\varepsilon}{2}> 8\pi(1+\alpha)-\varepsilon. \nonumber\end{aligned}$$ From $\beta_1=8\pi(1+\alpha)\gamma$ with $\beta_1$ satisfying [\[tem-60-beta1\]](#tem-60-beta1){reference-type="eqref" reference="tem-60-beta1"}, we see that $$\lim_{r\to0}\lim_{n\to\infty}\int_{\left\{|z|\le\frac{2r}{\mu_n}\right\}}\left\lvert z+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z= 8\pi(1+\alpha).$$ Thus we can choose $r_\eta\in (0,\frac12)$ small such that for $n$ large, $$\label{tem-27} \int_{\left\{|z|\le\frac{2r_\eta}{\mu_n}\right\}}\left\lvert z+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\le 8\pi(1+\alpha)+\varepsilon,$$ and consequently, $$\label{tem-27-0} \int_{\left\{R_{\eta}\leq |z|\le\frac{2r_\eta}{\mu_n}\right\}}\left\lvert z+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\le 2\varepsilon.$$ Fix any $2R_\eta\le |x|\le \frac{r_\eta}{\mu_n}$, to compute the integral $\@slowromancap\romannumeral 1@(x)$, we divide $D_n$ into the same four domains $D_n=\sum_{i=1}^4D_{n,i}$ as in Lemma [Lemma 16](#decay-regular1){reference-type="ref" reference="decay-regular1"} and divide the integral $\@slowromancap\romannumeral 1@(x)$ into four terms $\@slowromancap\romannumeral 1@(x)=\sum_{i=1}^4\@slowromancap\romannumeral 1@_{D_{n,i}}(x)$. Then as in Lemma [Lemma 16](#decay-regular1){reference-type="ref" reference="decay-regular1"}, we obtain $$\label{tem-31} [4(1+\alpha)+\frac{\varepsilon}{2\pi}]\log\frac{1}{|x|}-C\le \@slowromancap\romannumeral 1@_{D_{n,1}}(x)\le [4(1+\alpha)-\frac{\varepsilon}{2\pi}]\log\frac{1}{|x|}+C,$$ $$\label{tem-32} \left\lvert\@slowromancap\romannumeral 1@_{D_{n,2}}(x)+\@slowromancap\romannumeral 1@_{D_{n,3}}(x)\right\rvert\le C.$$ Finally for $z\in D_{n,4}$, it holds $2\le |z|\le 2|x|$. Then it follows from [\[tem-27-0\]](#tem-27-0){reference-type="eqref" reference="tem-27-0"} that $$\label{tem-33} \begin{aligned} 0&\le \frac{1}{2\pi}\int_{D_{n,4}} \log|z|\left\lvert z+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &\le \frac{1}{2\pi}\log(2|x|)\int_{D_{n,4}}\left\lvert z+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &\le \frac{\varepsilon}{\pi}\log|x|+C. \end{aligned}$$ Furthermore, by $v_n(x)\le 0$, we get $$\begin{aligned} 0&\le \frac{1}{2\pi}\int_{\left\{z\in D_{n,4}:~|z-x|\le\frac{1}{|x|^{2\alpha}}\right\}} \log\frac{1}{|z-x|}\left\lvert z+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &\le C(1+|x|^{2\alpha})\int_{\left\{|z-x|\le\frac{1}{|x|^{2\alpha}}\right\}}\log\frac{1}{|z-x|}{\mathrm d}z\\ &\le C\frac{(1+|x|^{2\alpha})(1+\log|x|)}{|x|^{4\alpha}}\le C. \end{aligned}$$ While for $z\in\left\{z\in D_{n,4}:~|z-x|\ge\frac{1}{|x|^{2\alpha}}\right\}$, it holds $$\log\frac{1}{3|x|}\le \log\frac{1}{|z-x|}\le 2\alpha\log|x|,$$ and hence $$\begin{aligned} &~\frac{1}{2\pi}\int_{\left\{z\in D_{n,4}:~|z-x|\ge\frac{1}{|x|^{2\alpha}}\right\}} \log\frac{1}{|z-x|}\left\lvert z+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ &\le \frac{\alpha}{\pi}\log|x| \int_{D_4}\left\lvert z+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z \\ &\le \frac{2\alpha\varepsilon}{\pi}\log|x|, \end{aligned}$$ $$\begin{aligned} \label{tem-30-2} \frac{1}{2\pi}\int_{\left\{z\in D_{n,4}:~|z-x|\ge\frac{1}{|x|^{2\alpha}}\right\}}& \log\frac{1}{|z-x|}\left\lvert z+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\nonumber\\ \ge &\frac{1}{2\pi}\log \frac{1}{3|x|}\int_{D_{n,4}}\left\lvert z+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nz)}{V_n(x_n)}\left(1+\frac{v_n(z)}{p_n}\right)^{p_n}{\mathrm d}z\\ \ge &\frac{\varepsilon}{\pi}\log \frac{1}{3|x|}.\nonumber\end{aligned}$$ Combining [\[tem-33\]](#tem-33){reference-type="eqref" reference="tem-33"}-[\[tem-30-2\]](#tem-30-2){reference-type="eqref" reference="tem-30-2"}, we get $$\label{tem-40} \frac{\varepsilon}{\pi}\log\frac{1}{3|x|}\le \@slowromancap\romannumeral 1@_{D_{n,4}}(x)\le \frac{(1+2\alpha)\varepsilon}{\pi}\log|x|+C.$$ Finally, from [\[tem-39\]](#tem-39){reference-type="eqref" reference="tem-39"},[\[tem-31\]](#tem-31){reference-type="eqref" reference="tem-31"},[\[tem-32\]](#tem-32){reference-type="eqref" reference="tem-32"} and [\[tem-40\]](#tem-40){reference-type="eqref" reference="tem-40"}, we finally get $$\left\lvert v_n(x)\right\rvert\le C(1+\log|x|),$$ and $$v_n(x)\le \left[4(1+\alpha)-\frac{(3+4\alpha)\varepsilon}{2\pi}\right]\log\frac{1}{|x|}+C=\eta\log\frac{1}{|x|}+C,$$ for any $2R_\eta\le |x|\le \frac{r_\eta}{\mu_n}$ and some constant $C>0$. This completes the proof. ◻ *Remark 22*. Similarly as Remark [Remark 17](#decay-regular2){reference-type="ref" reference="decay-regular2"}, we have that for any $\eta\in(0,4(1+\alpha))$, there exists $C_\eta>0$ such that $$0\le \left(1+\frac{v_n(x)}{p_n}\right)^{p_n}\le \frac{C_\eta}{1+|x|^\eta}, \quad\forall|x|\leq \frac{r_{\eta}}{\mu_n},$$ $$\left\lvert v_n(x)\right\rvert\le C_\eta\log\left(2+|x|\right), \quad\forall|x|\leq \frac{r_{\eta}}{\mu_n}.$$ **Lemma 23**. *Suppose [\[assume-1\]](#assume-1){reference-type="eqref" reference="assume-1"} holds, then $\gamma=\sqrt e$ and $$p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+k}\to 8\pi(1+\alpha)e^{\frac{k}{2}}\delta_0,\quad\text{for }k=0,1,2$$ weakly in the sense of measures.* *Proof.* Take $\eta=3(1+\alpha)$ in Lemma [Lemma 21](#decay-singular1){reference-type="ref" reference="decay-singular1"} and Remark [Remark 22](#decay-singular2){reference-type="ref" reference="decay-singular2"}, and let $r=r_{3(1+\alpha)}/2$. Then the same argument as Lemma [Lemma 18](#tem-2){reference-type="ref" reference="tem-2"} implies $$\frac{1}{p_n}\int_{\frac{B_{r}-x_n}{\mu_n}} G_{r}(x_n,x_n+\mu_ny)\left\lvert y+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_ny)}{V_n(x_n)}\left(1+\frac{v_n(y)}{p_n}\right)^{p_n}{\mathrm d}y=1+O(\frac1{p_n}).$$ Since $|y|\leq \frac{r_{3(1+\alpha)}}{\mu_n}$ for $y\in \frac{B_{r}-x_n}{\mu_n}$, we see from Remark [Remark 22](#decay-singular2){reference-type="ref" reference="decay-singular2"} that $$0\le\left(1+\frac{v_n(y)}{p_n}\right)^{p_n}\le \frac{C}{1+|y|^{3(1+\alpha)}},\quad\text{\it for }\;y\in \frac{B_{r}-x_n}{\mu_n},$$ so it follows from the Dominated Convergence Theorem that $$\begin{aligned} &\lim_{n\to\infty}\int_{\frac{B_{r}-x_n}{\mu_n}} \left\lvert y+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_ny)}{V_n(x_n)}\left(1+\frac{v_n(y)}{p_n}\right)^{p_n}{\mathrm d}y\\ =&\int_{\mathbb R^2}|y+x_\infty|^{2\alpha}e^{U_\alpha}{\mathrm d}y=8\pi(1+\alpha), \end{aligned}$$ $$\begin{aligned} &\lim_{n\to\infty}\int_{\frac{B_{r}-x_n}{\mu_n}} \left(\frac{1}{2\pi}\log|y|+H_{r}(x_n,x_n+\mu_ny)\right)\left\lvert y+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_ny)}{V_n(x_n)}\left(1+\frac{v_n(y)}{p_n}\right)^{p_n}{\mathrm d}y\\ &\qquad=\int_{\mathbb R^2} \left(\frac{1}{2\pi}\log|y|+H_{r}(0,0)\right) |y+x_\infty|^{2\alpha}e^{U_\alpha(y)}{\mathrm d}y=C <\infty.\end{aligned}$$ From here, $G_{r}(x,y)=-\frac{1}{2\pi}\log|x-y|-H_{r}(x,y)$ and $\mu_n^{-2-2\alpha}=p_nV_n(x_n)u_n(x_n)^{p_n-1}$, we have $$\begin{aligned} &\quad 1+O(\frac1{p_n})\\ &=\frac{1}{p_n}\int_{\frac{B_{r}-x_n}{\mu_n}} G_{r}(x_n,x_n+\mu_ny)\left\lvert y+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_ny)}{V_n(x_n)}\left(1+\frac{v_n(y)}{p_n}\right)^{p_n}{\mathrm d}y\\ &=-\frac{1}{2\pi}\frac{\log\mu_n}{p_n}\int_{\frac{B_{r}-x_n}{\mu_n}} \left\lvert y+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_ny)}{V_n(x_n)}\left(1+\frac{v_n(y)}{p_n}\right)^{p_n}{\mathrm d}y+O(\frac{1}{p_n})\\ &=\frac{1}{4\pi(1+\alpha)}\left(\frac{\log p_n+\log V_n(x_n)}{p_n}+\frac{p_n-1}{p_n}\log u_n(x_n)\right)(8\pi(1+\alpha)+o_n(1))+o_n(1)\\ &=2\log \gamma+o_n(1).\end{aligned}$$ Thus $2\log \gamma=1$, i.e., $\gamma=\sqrt e$. Since $0$ is the only blow-up point of $p_nu_n$ in $B_1$, we see that $$p_nV_n(x)u_n(x)^{p_n-1+k}\to \beta_k\delta_0, \quad \text{for }k=0,1,2,$$ weakly in the sense of measures, where $\beta_k$ are given by [\[tem-60-beta\]](#tem-60-beta){reference-type="eqref" reference="tem-60-beta"}-[\[tem-60-beta1\]](#tem-60-beta1){reference-type="eqref" reference="tem-60-beta1"}. For any small $r>0$, again by the Dominated Convergence Theorem, we get $$\begin{aligned} &\quad\int_{B_r}p_nV_n(x)|x|^{2\alpha}u_n(x)^{p_n-1+k}{\mathrm d}x\\ &=u_n(x_n)^{k}\int_{\frac{B_{r}-x_n}{\mu_n}}\left\lvert x+\frac{x_n}{\mu_n}\right\rvert^{2\alpha}\frac{V_n(x_n+\mu_nx)}{V_n(x_n)}\left(1+\frac{v_n(x)}{p_n}\right)^{p_n-1+k}{\mathrm d}x\\ &\to \gamma^{k}\int_{\mathbb R^2}|x+x_\infty|^{2\alpha}e^{U_\alpha}{\mathrm d}x=8\pi (1+\alpha)e^{\frac{k}{2}}. \end{aligned}$$ Therefore, $\beta_k=8\pi(1+\alpha)e^{\frac{k}{2}}$, and the proof is complete. ◻ ## The general case. In this section, we do not assume the estimate [\[assume-1\]](#assume-1){reference-type="eqref" reference="assume-1"}, so the previous arguments in Section 4.1 do not work and different ideas are needed. We begin with the following decomposition result, whose proof is inspired by [@SMF-1 Proposition 1.4]. **Proposition 24**. *Let $u_n$ satisfy the assumptions of Theorem [Theorem 3](#thm-singular){reference-type="ref" reference="thm-singular"}. Then along a subsequence, one of the following alternatives holds.* - *Either there exists $\varepsilon_0\in(0,\frac{1}{2})$ such that $$\label{bound} \sup_{B_{2\varepsilon_0}}p_n|x|^{2+2\alpha}u_n(x)^{p_n-1}\le C,$$* - *or there exist $\varepsilon_0'\in(0,\frac{1}{2})$ and $l\geq 1$ sequences $\{z_{n,i}\}_{n\ge1}\subset B_1\setminus\{0\}$, $i=1,\cdots,l$, such that $$\label{tem-15} \lim_{n\to\infty} z_{n,i}=0,\quad \liminf_{n\to\infty}|z_{n,i}|^{\frac{2+2\alpha}{p_n-1}}u_n(z_{n,i})\geq 1,$$ $$\label{tem-15-0} \lim_{n\to\infty} p_n|z_{n,i}|^{2+2\alpha}u_n(z_{n,i})^{p_n-1}=\infty,$$ $$\label{tem-16} p_n|x|^{2+2\alpha}u_n(x)^{p_n-1}\le C,\;\text{for}~x\in\left\{y\in B_1:~|y|\le 2\varepsilon_0'|z_{n,1}|\;\text{or}\; |y|\ge \frac{1}{2\varepsilon_0'}|z_{n,l}|\right\},$$ and in case $l\ge 2$, then $\frac{|z_{n,i}|}{|z_{n,i+1}|}\to0$ as $n\to\infty$ and $$\label{tem-17} p_n|x|^{2+2\alpha}u_n(x)^{p_n-1}\le C,\;\text{for}~x\in\bigcup_{i=1}^{l-1}\left\{y\in B_1:~\frac{1}{2\varepsilon_0'}|z_{n,i}|\le |y|\le 2\varepsilon_0'|z_{n,i+1}|\right\}.$$* *Proof.* We devide the proof into several steps. **Step 1.** Assume there exists $\{z_n\}\subset B_1$ such that $p_n|z_n|^{2+2\alpha}u_n(z_n)^{p_n-1}\to\infty$, we prove that $\lim_{n\to\infty}z_n=0$, $\liminf_{n\to\infty}|z_{n}|^{\frac{2+2\alpha}{p_n-1}}u_n(z_{n})\geq 1$ and $$\label{tem-3} \limsup_{n\to\infty}\int_{B_{\delta|z_n|}(z_n)}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n}{\mathrm d}x\ge 4\pi e,\quad\text{for every}~\delta>0.$$ Indeed, $p_n|z_n|^{2+2\alpha}u_n(z_n)^{p_n-1}\to\infty$ implies $$\liminf_{n\to\infty}u_n(z_n)\ge1,\quad \liminf_{n\to\infty}|z_{n}|^{\frac{2+2\alpha}{p_n-1}}u_n(z_{n})\geq 1,$$ so $p_nu_n(z_n)\to \infty$. Since [\[con-singular1\]](#con-singular1){reference-type="eqref" reference="con-singular1"} says that $0$ is the only blow-up point of $p_nu_n$ in $B_1$, we obtain $z_n\to 0$. In particular, this argument shows that once [\[tem-15-0\]](#tem-15-0){reference-type="eqref" reference="tem-15-0"} holds, then [\[tem-15\]](#tem-15){reference-type="eqref" reference="tem-15"} holds. To prove [\[tem-3\]](#tem-3){reference-type="eqref" reference="tem-3"}, we let $$\label{al_n} v_n(x):=|z_n|^{\alpha_n}u_n(|z_n|x), \quad\text{with}~\alpha_n:=\frac{2+2\alpha}{p_n-1}.$$ Since $|z_n|\to0$ and $p_n|z_n|^{2+2\alpha}u_n(z_n)^{p_n-1}\to\infty$, we get $$1\ge|z_n|^{\alpha_n}\ge p_n^{-\frac{1}{p_n-1}}\frac{1}{u_n(z_n)}\ge\frac{1}{\gamma}+o(1),$$ where $\gamma$ is given in [\[max\]](#max){reference-type="eqref" reference="max"}. Then $$\int_{B_{\frac{1}{|z_n|}}}p_n|x|^{2\alpha}V_n(|z_n|x)v_n(x)^{p_n}{\mathrm d}x=|z_n|^{\alpha_n} \int_{B_1}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n}{\mathrm d}x\le C.$$ From [\[equ-singular\]](#equ-singular){reference-type="eqref" reference="equ-singular"}, we see that $$\label{4-32}\begin{cases} -\Delta v_n=|x|^{2\alpha}V_n(|z_n|x)v_n^{p_n},\quad\text{in}~B_{\frac{1}{|z_n|}},\\ p_nv_n(\frac{z_n}{|z_n|})\to\infty. \end{cases}$$ Take a subsequence so that $\frac{z_n}{|z_n|}$ converges to some point $x_0$ in the unit circle. Then $p_nv_n$ admits a blow-up point at $x_0$ and around it the function $W_n(x)=|x|^{2\alpha}V_n(|z_n|x)$ is uniformly bounded from above. Then by using Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"} for $v_n$ in any open bounded domain $\Omega$ containing $x_0$, we see that for any small $\delta>0$, $$\limsup_{n\to\infty}\int_{B_{\delta}(\frac{z_n}{|z_n|})}p_n|x|^{2\alpha}V_n(|z_n|x)v_n(x)^{p_n}{\mathrm d}x\ge 4\pi e.$$ A simple change of variables leads to $$\begin{aligned} &\limsup_{n\to\infty}\int_{B_{\delta |z_{n}|}(z_n)}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n}{\mathrm d}x\\ =&\limsup_{n\to\infty}|z_n|^{-\alpha_n}\int_{B_{\delta}(\frac{z_n}{|z_n|})}p_n|x|^{2\alpha}V_n(|z_n|x)v_n(x)^{p_n}{\mathrm d}x\geq 4\pi e,\end{aligned}$$ namely [\[tem-3\]](#tem-3){reference-type="eqref" reference="tem-3"} holds. This proves Step 1. **Step 2.** Suppose the alternative (i) does not hold for every $\varepsilon_0\in(0,\frac{1}{2})$, we prove that there exist $\varepsilon_0'\in(0,\frac{1}{2})$ and a sequence $\{z_{n,1}\}$ such that $$\label{tem-5} \lim_{n\to\infty} z_{n,1}=0,\quad \lim_{n\to\infty} p_n|z_{n,1}|^{2+2\alpha}u_n(z_{n,1})^{p_n-1}=\infty,$$ and $$\label{tem-6} p_n|x|^{2+2\alpha}u_n(x)^{p_n-1}\le C,\quad\text{for}~x\in\left\{y\in B_1:~|y|\le 2\varepsilon_0'|z_{n,1}|\right\}.$$ Indeed, since [\[bound\]](#bound){reference-type="eqref" reference="bound"} does not hold for every $\varepsilon_0\in(0,\frac{1}{2})$, up to a subsequence there is $z_n\in {B}_{1}$ such that $$p_n|z_n|^{2+2\alpha}u_n(z_n)^{p_n-1}=\sup_{B_{1/2}}p_n|x|^{2+2\alpha}u_n(x)^{p_n-1}\to\infty.$$ Then by Step 1 we have $z_n\to0$ and $$\limsup_{n\to\infty}\int_{B_{\delta|z_n|}(z_n)}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n}{\mathrm d}x\ge 4\pi e,\quad\text{for every}~\delta>0.$$ Let $v_n$ be defined by [\[al_n\]](#al_n){reference-type="eqref" reference="al_n"}, then $v_n$ satisfies [\[4-32\]](#4-32){reference-type="eqref" reference="4-32"}. There are the same alternatives for $v_n$. If there exists some $\varepsilon_0'\in(0,\frac{1}{2})$ such that $$\label{tem-7} \sup_{B_{2\varepsilon_0'}}p_n|x|^{2+2\alpha}v_n(x)^{p_n-1}\le C,$$ then $$\sup_{B_{2\varepsilon_0'|z_n|}}p_n|x|^{2+2\alpha}u_n(x)^{p_n-1}=\sup_{B_{2\varepsilon_0'}}p_n|x|^{2+2\alpha}v_n(x)^{p_n-1}\le C,$$ so setting $z_{n,1}=z_n$ we get [\[tem-5\]](#tem-5){reference-type="eqref" reference="tem-5"}-[\[tem-6\]](#tem-6){reference-type="eqref" reference="tem-6"} and we are done. Otherwise, for any $r\in (0,\frac12)$, $$\limsup_{n\to\infty}\sup_{B_{2r}}p_n|x|^{2+2\alpha}v_n(x)^{p_n-1}=\infty.$$Then up to a subsequence, there exists $r_n\to 0$ and $\bar z_n\in \overline{B}_{r_n}$ such that $$p_n|\bar z_n|^{2+2\alpha}v_n(\bar z_n)^{p_n-1}=\sup_{B_{r_n}}p_n|x|^{2+2\alpha}v_n(x)^{p_n-1}\to\infty,\quad\text{as }n\to\infty.$$ Let $\tilde z_n:=|z_n|\bar z_n$. Then $$\frac{|\tilde z_n|}{|z_n|}=|\bar z_n|\to0 \quad\text{and}\quad p_n|\tilde z_n|^{2+2\alpha}u_n(\tilde z_n)^{p_n-1}=p_n|\bar z_n|^{2+2\alpha}v_n(\bar z_n)^{p_n-1}\to\infty.$$ Consequently by Step 1, $$\displaystyle\limsup_{n\to\infty}\int_{B_{\delta|\tilde z_n|}(\tilde z_n)}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n}{\mathrm d}x\ge 4\pi e,\quad\text{for every}~\delta>0.$$ Furthermore, for each fixed $\delta\in (0,1)$, we have $$B_{\delta|z_n|}(z_n)\cap B_{\delta|\tilde z_n|}(\tilde z_n)=\emptyset\quad\text{ for $n$ large},$$ so $$\limsup_{n\to\infty}\int_{B_{\delta|z_n|}(z_n)\cup B_{\delta|\tilde z_n|}(\tilde z_n)}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n}{\mathrm d}x\ge 8\pi e.$$ Keep on repeating the alternatives above for the scaled new sequence (still called $v_n$) where in [\[al_n\]](#al_n){reference-type="eqref" reference="al_n"} we replace $z_n$ with the new sequence $\tilde z_n$, and so on. We see that, each time the scaled new sequence $v_n$ fails to verify [\[tem-7\]](#tem-7){reference-type="eqref" reference="tem-7"} for any $\varepsilon_0'\in (0,\frac12)$, we add a contribution of $4\pi e$ to the value $\int_{B_1}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n}{\mathrm d}x\le C$. So after finitely many steps we find a sequence $\{z_{n,1}\}$ and $\varepsilon_0'\in(0,\frac{1}{2})$ such that [\[tem-5\]](#tem-5){reference-type="eqref" reference="tem-5"}-[\[tem-6\]](#tem-6){reference-type="eqref" reference="tem-6"} hold. This proves Step 2. **Step 3.** Suppose the alternative (i) does not hold for every $\varepsilon_0\in(0,\frac{1}{2})$, we prove that the alternative (ii) holds. First, by Step 2, there are a sequence $\{z_{n,1}\}$ and $\varepsilon_0'\in(0,\frac{1}{2})$ such that [\[tem-5\]](#tem-5){reference-type="eqref" reference="tem-5"}-[\[tem-6\]](#tem-6){reference-type="eqref" reference="tem-6"} hold. If there exists $\varepsilon_1'\in(0,\varepsilon_0']$ such that $$\sup_{\frac{1}{2\varepsilon_1'}|z_{n,1}|\le |x|\le 1}p_n|x|^{2+2\alpha}u_n(x)^{p_n-1}\le C,$$ then by replacing $\varepsilon_0'$ with $\varepsilon_1'$, we see that the alternative (ii) holds with $l=1$. Otherwise, for any $\varepsilon\in (0,\varepsilon_0']$, $$\limsup_{n\to\infty}\sup_{\frac{1}{2\varepsilon}|z_{n,1}|\le |x|\le 1}p_n|x|^{2+2\alpha}v_n(x)^{p_n-1}=\infty.$$Then up to a subsequence, there exist $\varepsilon_n\to 0$ and $y_n$ satisfying $\frac{1}{2\varepsilon_n}|z_{n,1}|\le |y_n|\le 1$ such that $$p_n|y_n|^{2+2\alpha}u_n(y_n)^{p_n-1}=\sup_{\frac{1}{2\varepsilon_n}|z_{n,1}|\le |x|\le 1}p_n|x|^{2+2\alpha}u_n(x)^{p_n-1}\to\infty,\quad\text{as}~n\to\infty.$$ This implies $\frac{|z_{n,1}|}{|y_n|}\to 0$. Furthermore, by Step 1, we see that necessarily $$y_n\to0,\quad \limsup_{n\to\infty}\int_{B_{\delta|y_n|}(y_n)}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n}{\mathrm d}x\ge 4\pi e,$$ for every $\delta>0$. To obtain the second sequence $z_{n,2}$, for $\varepsilon\in(0,\frac{1}{2})$ we consider $$\label{tem-12} \sup_{\left\{ \frac{1}{2\varepsilon}|z_{n,1}|\le |x|\le 2\varepsilon|y_n| \right\}} p_n|x|^{2+2\alpha}u_n(x)^{p_n-1}.$$ If there is $\varepsilon\in(0,\frac{1}{2})$ such that [\[tem-12\]](#tem-12){reference-type="eqref" reference="tem-12"} is uniformly bounded for all $n$, we would simply take $z_{n,2}=y_n$, and adjust accordingly $\varepsilon_0'$ (for example, replace $\varepsilon_0'$ with $\min\{\varepsilon_0', \varepsilon\}$) in order to ensure [\[tem-6\]](#tem-6){reference-type="eqref" reference="tem-6"} with $i=1$. Otherwise, for any $\varepsilon\in (0,\frac12)$, $$\limsup_{n\to\infty}\sup_{\left\{ \frac{1}{2\varepsilon}|z_{n,1}|\le |x|\le 2\varepsilon|y_n| \right\}} p_n|x|^{2+2\alpha}u_n(x)^{p_n-1}=\infty.$$ Then up to a subsequence, there are $\varepsilon_n\to 0$ and $\bar{y}_n$ satisfying $\frac{1}{2\varepsilon_n}|z_{n,1}|\le |\bar{y}_n|\le 2\varepsilon_n|y_n|$ such that $$p_n|\bar y_n|^{2+2\alpha}u_n(\bar y_n)^{p_n-1}=\sup_{\frac{1}{2\varepsilon_n}|z_{n,1}|\le |x|\le 2\varepsilon_n|y_n|}p_n|x|^{2+2\alpha}u_n(x)^{p_n-1}\to\infty.$$ Therefore, we could replace $y_n$ with this new sequence $\bar y_n$ with the properties $\frac{|z_{n,1}|}{|\bar y_n|}\to0$, $\frac{|\bar y_n|}{|y_n|}\to0$, $$\bar y_n\to0,\quad \limsup_{n\to\infty}\int_{B_{\delta|\bar y_n|}(\bar y_n)}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n}{\mathrm d}x\ge 4\pi e,$$ and consider again whether [\[tem-12\]](#tem-12){reference-type="eqref" reference="tem-12"} is uniformly bounded for some $\varepsilon\in (0,\frac12)$. Note that, as above, each time we admit the existence of such a new sequences, we add a contribution of $4\pi e$ to the value $\int_{B_1}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n}{\mathrm d}x\le C$. So by repeating the same alternatives for any such new sequence, after finitely many steps we must arrive to one for which [\[tem-12\]](#tem-12){reference-type="eqref" reference="tem-12"} is uniformly bounded on $n\in{\mathbb N}$ for some $\varepsilon\in(0,\frac{1}{2})$. Such sequence defines $z_{n,2}$, and we can adjust $\varepsilon_0'\in(0,\frac{1}{2})$ accordingly in order to guarantee [\[tem-6\]](#tem-6){reference-type="eqref" reference="tem-6"} with $i=1$ and $$\sup_{\frac{1}{2\varepsilon_0'}|z_{n,1}|\le |x|\le 2\varepsilon_0'|z_{n,2}| } p_n|x|^{2+2\alpha}u_n(x)^{p_n-1}\leq C.$$ Finally we iterate the argument above by replacing $z_{n,1}$ with $z_{n,2}$. We are either able to check [\[tem-15\]](#tem-15){reference-type="eqref" reference="tem-15"}, [\[tem-15-0\]](#tem-15-0){reference-type="eqref" reference="tem-15-0"}, [\[tem-16\]](#tem-16){reference-type="eqref" reference="tem-16"} and [\[tem-17\]](#tem-17){reference-type="eqref" reference="tem-17"} for $l=2$ and so we are done, or obtain a third sequence $\{z_{n,3}\}$ for which we can verify [\[tem-15\]](#tem-15){reference-type="eqref" reference="tem-15"}, [\[tem-15-0\]](#tem-15-0){reference-type="eqref" reference="tem-15-0"} and [\[tem-17\]](#tem-17){reference-type="eqref" reference="tem-17"} for $i=1,2$. After finitely many steps we arrive to the desired conclusion, i.e. the alternative (ii) holds. ◻ To handle the alternative (ii) in Proposition [Proposition 24](#decomposition){reference-type="ref" reference="decomposition"}, we need to estimate the energies in neck domains. Let $\varepsilon_0'\in(0,\frac{1}{2})$ and $z_{n,i}$, $i=1,\cdots,l$, be given by the alternative (ii) in Proposition [Proposition 24](#decomposition){reference-type="ref" reference="decomposition"}. We define the subsets of $B_1$, $$\label{definition-PQ}\begin{aligned} Q_{n,i}&:=\left\{x\in B_1:~\varepsilon_0'|z_{n,i}|\le|x|\le \frac{1}{\varepsilon_0'}|z_{n,i}|\right\}, \quad \text{for }1\leq i\leq l,\\ P_{n,i}&:=\left\{x\in B_1:~\frac{1}{\varepsilon_0'}|z_{n,i-1}|\le|x|\le \varepsilon_0'|z_{n,i}|\right\}, \quad \text{for }1\leq i\leq l+1, \end{aligned}$$ where we set $|z_{n,0}|=0$ and $|z_{n,l+1}|=1$. Then for any $n\ge1$, $$B_1=\left(\bigcup_{i=1}^lQ_{n,i}\right)\cup\left(\bigcup_{j=1}^{l+1}P_{n,j}\right)\cup(B_1\setminus B_{\varepsilon_0'}).$$ We compute the integrals on each domain. Since [\[con-singular1\]](#con-singular1){reference-type="eqref" reference="con-singular1"} gives $$\label{max01}\sup_{B_1\setminus B_{\varepsilon_0'}}p_nu_n\le C,$$we immediately obtain $$\label{tem-56-0} \lim_{n\to\infty}\int_{B_1\setminus B_{\varepsilon_0'}}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+t}{\mathrm d}x=0,\quad t=0,1,2.$$ By [\[max\]](#max){reference-type="eqref" reference="max"} and [\[tem-15\]](#tem-15){reference-type="eqref" reference="tem-15"}, we have for $1\leq i\leq l$, $$1\geq |z_{n,i}|^{\alpha_n}\geq \frac{1+o_n(1)}{u_n(z_{n,i})}\geq \frac{1}{\gamma}+o_n(1),$$ so up to a subsequence we may assume $$\label{eq-ci} \lim_{n\to\infty} |z_{n,i}|^{-\alpha_n} =c_i \in [1, \gamma],\quad 1\leq i\leq l.$$ Clearly $$\label{eq-ci1} \gamma\geq c_1\geq c_2\cdots \geq c_l\geq 1.$$ **Lemma 25**. *Suppose the alternative (ii) in Proposition [Proposition 24](#decomposition){reference-type="ref" reference="decomposition"} holds, then there exists positive integers $N_i$, $i=1,\cdots,l$, such that for $t=0,1,2$, $$\label{energy-PQ}\begin{aligned} \lim_{n\to\infty}\int_{Q_{n,i}}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+t}{\mathrm d}x &=8\pi e^{\frac{t}{2}}c_i^tN_i,\quad \forall\, 1\leq i\leq l, \\ \lim_{n\to\infty}\int_{P_{n,i}}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+t}{\mathrm d}x &=0,\quad \forall\, 1\leq i\leq l+1. \end{aligned}$$* *Proof.* We devide the proof into several steps. **Step 1.** we consider the integral on $P_{n,i}$ for $2\leq i\leq l+1$. Fix any $2\leq i\leq l+1$, we claim $$\label{tem-50} \sup_{P_{n,i}}p_n u_n\le C,\quad\forall~n\ge1.$$ Assume by contradiction that [\[tem-50\]](#tem-50){reference-type="eqref" reference="tem-50"} does not hold, then up to a subsequence, we can take $y_n\in P_{n,i}$ such that $$\label{tem-51} p_nu_n(y_n)=\sup_{P_{n,i}}p_n u_n\to\infty.$$ Let $$w_{n}(x):=|y_n|^{\alpha_n}u_n(|y_n|x),\quad\text{with}~\alpha_n=\frac{2+2\alpha}{p_n-1}.$$ Denote $D_0=\{x\in\mathbb R^2:~\frac{1}{2}\le |x|\le 2\}$. It is easy to see that $-\Delta w_{n}=|x|^{2\alpha}V_n(|y_n|x)w_{n}^{p_n}$ in $D_0$. By [\[max\]](#max){reference-type="eqref" reference="max"}, [\[tem-15\]](#tem-15){reference-type="eqref" reference="tem-15"} and $\frac{1}{\varepsilon_0'}|z_{n,i-1}|\le|y_n|\le \varepsilon_0'|z_{n,i}|$, we get $$1\ge |y_n|^{\alpha_n}\ge\left(\frac{1}{\varepsilon_0'}|z_{n,i-1}|\right)^{\alpha_n}\geq \left(\frac{1}{\varepsilon_0'}\right)^{\alpha_n}\frac{1+o_n(1)}{u_n(z_{n,i-1})} \geq\frac{1}{\gamma}+o_n(1),$$ and hence $$\int_{D_0}p_n|x|^{2\alpha}V_n(|y_n|x)w_{n}(x)^{p_n}{\mathrm d}x=|y_n|^{\alpha_n}\int_{\frac{|y_n|}{2}\le |x|\le 2|y_n|}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n}{\mathrm d}x\le C.$$ Moreover, by [\[tem-16\]](#tem-16){reference-type="eqref" reference="tem-16"}-[\[tem-17\]](#tem-17){reference-type="eqref" reference="tem-17"}, we get $$\begin{aligned} \sup_{D_0}p_n|x|^{2\alpha+2}w_{n}(x)^{p_n-1}=\sup_{\frac{|y_n|}{2}\leq |x|\leq 2|y_n|}p_n|x|^{2\alpha+2}u_{n}(x)^{p_n-1}\\ \le \sup_{\frac{1}{2\varepsilon_0'}|z_{n,i-1}|\leq |x|\leq 2\varepsilon_0'|z_{n,i}|}p_n|x|^{2\alpha+2}u_{n}(x)^{p_n-1}\leq C.\end{aligned}$$ From here, and noting that $|x|^{2\alpha}V_n(|y_n|x)$ is uniformly bounded for $x\in D_0$, we can apply Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"} for $w_{n}$ in $D_0$, and conclude that the alternative (i) in Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"} holds, which implies $\sup_{|x|=1}p_nw_{n}(x)\le C$ for some $C>0$. Thus $$p_nu_n(y_n)=p_nw_n(\frac{y_n}{|y_n|})|y_n|^{-\alpha_n}\le C,$$ which is a contradiction with [\[tem-51\]](#tem-51){reference-type="eqref" reference="tem-51"}. So the claim [\[tem-50\]](#tem-50){reference-type="eqref" reference="tem-50"} holds. It follows that $$\label{tem-56} \lim_{n\to\infty}\int_{P_{n,i}}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+t}{\mathrm d}x=0,\quad t=0,1,2, \;\forall\, 2\leq i\leq l+1.$$ **Step 2.** we consider the integral on $P_{n,1}=\{x\in B_1:~|x|\le \varepsilon_0'|z_{n,1}|\}$. Let $$w_{n,1}(x):=|z_{n,1}|^{\alpha_n}u_n(|z_{n,1}|x),\quad\text{with}~\alpha_n=\frac{2+2\alpha}{p_n-1}.$$ Then it is easy to check that $w_{n,1}$ satisfies $$\begin{cases} -\Delta w_{n,1}=|x|^{2\alpha}V_n(|z_{n,1}|x)w_{n,1}^{p_n},\quad \text{in}~B_{(1+\delta)\varepsilon_0'},\\ \int_{B_{(1+\delta)\varepsilon_0'}}p_n|x|^{2\alpha}V_n(|z_{n,1}|x)w_{n,1}(x)^{p_n}{\mathrm d}x\le C. \end{cases}$$ where $\delta>0$ is a small constant. Applying Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"} for $w_{n,1}$ in $B_{(1+\delta)\varepsilon_0'}$, there are two possibilities. If the alternative (i) of Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"} holds, then we have $\sup_{\overline B_{\varepsilon_0'}}p_nw_{n,1}\le C$, and hence $$\label{max02}\sup_{P_{n,1}}p_nu_n=|z_{n,1}|^{-\alpha_n}\sup_{\overline B_{\varepsilon_0'}}p_nw_{n,1}\le C.$$It follows that $$\label{tem-57} \lim_{n\to\infty}\int_{P_{n,1}}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+t}{\mathrm d}x=0,\quad t=0,1,2.$$ If the alternative (ii) of Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"} holds, since [\[tem-16\]](#tem-16){reference-type="eqref" reference="tem-16"} implies $$\label{tem-54} \sup_{B_{(1+\delta)\varepsilon_0'}}p_n|x|^{2+2\alpha} w_{n,1}(x)^{p_n-1}\leq \sup_{B_{2\varepsilon_0'|z_{n,1}|}}p_n|x|^{2+2\alpha} u_{n}(x)^{p_n-1}\le C,$$ we see that $0$ is the only blow-up point of $p_nw_{n,1}$ in $B_{(1+\delta)\varepsilon_0'}$. Moreover, by [\[tem-54\]](#tem-54){reference-type="eqref" reference="tem-54"}, one can check that the assumption [\[assume-1\]](#assume-1){reference-type="eqref" reference="assume-1"} holds for $w_{n,1}$, so we can apply Lemma [Lemma 23](#tem-41){reference-type="ref" reference="tem-41"} to $w_{n,1}$ to conclude that $$\lim_{n\to\infty}\int_{B_{\varepsilon_0'}}p_n|x|^{2\alpha}V_n(|z_{n,1}|x)w_{n,1}(x)^{p_n-1+k}{\mathrm d}x=8\pi(1+\alpha)e^{\frac{k}{2}},\quad k=0,1,2,$$ $$\max_{B_{\varepsilon_0'}}w_{n,1}\to \sqrt{e}.$$ Backing to $u_n$ and using [\[eq-ci\]](#eq-ci){reference-type="eqref" reference="eq-ci"}, we get $$\label{tem-58} \begin{aligned} & \lim_{n\to\infty}\int_{P_{n,1}}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+k}{\mathrm d}x\\ =&\lim_{n\to\infty}|z_{n,1}|^{-k\alpha_n}\int_{B_{\varepsilon_0'}}p_n|x|^{2\alpha}V_n(|z_{n,1}|x)w_{n,1}(x)^{p_n-1+k}{\mathrm d}x\\ = &8\pi (1+\alpha) c_1^k e^{\frac{k}{2}}\geq 8\pi (1+\alpha) e^{\frac{k}{2}},\quad k=0,1,2, \end{aligned}$$ $$\label{max03} \max_{P_{n,1}}u_n=|z_{n,1}|^{-\alpha_n}\max_{B_{\varepsilon_0'}}w_{n,1}\to c_1\sqrt{e}.$$ We will show in Step 4 that [\[tem-58\]](#tem-58){reference-type="eqref" reference="tem-58"} can not hold, namely actually [\[tem-57\]](#tem-57){reference-type="eqref" reference="tem-57"} holds. **Step 3.** we consider the integral on $Q_{n,i}$ for $1\leq i\leq l$. Let $$w_{n,i}(x):=|z_{n,i}|^{\alpha_n}u_n(|z_{n,i}|x),\quad\text{with}~\alpha_n=\frac{2+2\alpha}{p_n-1}.$$ By [\[tem-15\]](#tem-15){reference-type="eqref" reference="tem-15"}, it is easy to check that $w_{n,i}$ satisfies $$\begin{cases} -\Delta w_{n,1}=|x|^{2\alpha}V_n(|z_{n,i}|x)w_{n,i}^{p_n},\quad \text{in}~D_0':=\left\{x\in\mathbb R^2:~\varepsilon_0'\le |x|\le \frac{1}{\varepsilon_0'}\right\},\\ \int_{D_0'}p_n|x|^{2\alpha}V_n(|z_{n,i}|x)w_{n,i}(x)^{p_n}{\mathrm d}x\le C,\\ p_nw_{n,i}(\frac{z_{n,i}}{|z_{n,i}|})\to\infty. \end{cases}$$ Note from [\[tem-16\]](#tem-16){reference-type="eqref" reference="tem-16"}-[\[tem-17\]](#tem-17){reference-type="eqref" reference="tem-17"} that $$\begin{aligned} \label{tem-55} &\sup_{D_0'\setminus\{2\varepsilon_0'\le |x|\le \frac{1}{2\varepsilon_0'}\} }p_n|x|^{2+2\alpha} w_{n,i}(x)^{p_n-1}\\ =&\sup_{\{\frac{|z_{n,i}|}{2\varepsilon_0'}\leq |x|\leq \frac{|z_{n,i}|}{\varepsilon_0'}\}\cup\{\varepsilon_0'|z_{n,i}|\leq |x|\leq 2\varepsilon_0'|z_{n,i}|\}}p_n|x|^{2+2\alpha} u_{n}(x)^{p_n-1}\le C.\nonumber \end{aligned}$$ Therefore by applying Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"}, the blow-up set $\Sigma_i$ of $w_{n,i}$ in $D_0'$ has the following property: $$\Sigma_i\neq\emptyset\;\text{is a finite set},\quad \lim_{n\to\infty}\frac{z_{n,i}}{|z_{n,i}|}\in\Sigma_i\subset\{2\varepsilon_0'\le |x|\le \frac{1}{2\varepsilon_0'}\}\Subset D_0'.$$ Hence, noting that $|x|^{2\alpha}V_n(|z_{n,i}|x)$ satisfies [\[con-regular2\]](#con-regular2){reference-type="eqref" reference="con-regular2"} for $x\in D_0'$, we are in position to apply Theorem [Theorem 2](#thm-regular){reference-type="ref" reference="thm-regular"} to $w_{n,i}$ around each point of $\Sigma_i$ and derive $$\lim_{n\to\infty}\int_{D_0'}p_n|x|^{2\alpha}V_n(|z_{n,i}|x)w_{n,i}(x)^{p_n-1+k}{\mathrm d}x=8\pi e^{\frac{k}{2}} N_i,\quad k=0,1,2,$$ $$\max_{D_0'}w_{n,i}\to \sqrt{e},$$ where $N_i=\#\Sigma_i\geq 1$. Backing to $u_n$ and using [\[eq-ci\]](#eq-ci){reference-type="eqref" reference="eq-ci"}, we get $$\label{4-56}\begin{aligned} &\lim_{n\to\infty}\int_{Q_{n,i}}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+k}{\mathrm d}x\\ =&\lim_{n\to\infty}|z_{n,i}|^{-k\alpha_n}\int_{D_0'}p_n|x|^{2\alpha}V_n(|z_{n,i}|x)w_{n,i}(x)^{p_n-1+k}{\mathrm d}x\\ = &8\pi c_i^{k} e^{\frac{k}{2}} N_i\geq 8\pi e^{\frac{k}{2}} N_i,\quad k=0,1,2, \end{aligned}$$ $$\label{max04}\max_{Q_{n,i}}u_n=|z_{n,i}|^{-\alpha_n}\max_{D_0'}w_{n,i}\to c_i\sqrt{e}.$$ **Step 4.** We claim that $\gamma=c_1\sqrt{e}\geq \sqrt{e}$, and [\[tem-58\]](#tem-58){reference-type="eqref" reference="tem-58"} can not hold in Step 2, so [\[tem-57\]](#tem-57){reference-type="eqref" reference="tem-57"} holds. Indeed, by [\[max01\]](#max01){reference-type="eqref" reference="max01"}, [\[tem-50\]](#tem-50){reference-type="eqref" reference="tem-50"}, [\[max02\]](#max02){reference-type="eqref" reference="max02"}, [\[max03\]](#max03){reference-type="eqref" reference="max03"}, [\[max04\]](#max04){reference-type="eqref" reference="max04"} and [\[eq-ci1\]](#eq-ci1){reference-type="eqref" reference="eq-ci1"}, we have $$\gamma=\lim_{n\to\infty}\max_{B_1}u_n=\lim_{n\to\infty}\max_{P_{n,1}\cup\cup_i Q_{n,i}} u_n=\max_{1\leq i\leq l} c_i\sqrt{e}=c_1\sqrt{e}.$$ Assume by contradiction that [\[tem-58\]](#tem-58){reference-type="eqref" reference="tem-58"} hold in Step 2. Recalling $\beta_k$ defined by [\[tem-60-beta\]](#tem-60-beta){reference-type="eqref" reference="tem-60-beta"}, we see from [\[tem-56-0\]](#tem-56-0){reference-type="eqref" reference="tem-56-0"}, [\[tem-56\]](#tem-56){reference-type="eqref" reference="tem-56"}, and [\[4-56\]](#4-56){reference-type="eqref" reference="4-56"} that $$\beta_1=8\pi (1+\alpha) c_1 e^{\frac{1}{2}}+\sum_{i=1}^l 8\pi c_i e^{\frac{1}{2}} N_i>8\pi (1+\alpha) c_1 e^{\frac{1}{2}}=8\pi (1+\alpha)\gamma,$$ a contradiction with [\[beta12\]](#beta12){reference-type="eqref" reference="beta12"} which says that $\beta_1\leq 8\pi(1+\alpha)\gamma$. Thus we finish the proof. ◻ **Lemma 26**. *Suppose the alternative (ii) in Proposition [Proposition 24](#decomposition){reference-type="ref" reference="decomposition"} holds, then there must be $l=1$, $N_1=1+\alpha$ and $$\label{tem-202} \lim_{n\to\infty}\int_{B_1}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+t}{\mathrm d}x =8\pi(1+\alpha) e^{\frac{t}{2}}c_1^t,\quad t=0,1,2.$$ In particular, $\alpha\in \mathbb{N}$ since $N_1$ is a positive integer.* *Proof.* We discuss two cases separately. **Case 1:** $l=1$. Recall the definition of $\beta_k$ in [\[tem-60-beta\]](#tem-60-beta){reference-type="eqref" reference="tem-60-beta"}, then [\[beta12\]](#beta12){reference-type="eqref" reference="beta12"} tells us that $\beta_1^2=8\pi(1+\alpha)\beta_2$. Since $l=1$, by [\[tem-56-0\]](#tem-56-0){reference-type="eqref" reference="tem-56-0"} and [\[energy-PQ\]](#energy-PQ){reference-type="eqref" reference="energy-PQ"}, we get $$\beta_k=8\pi e^{\frac{k}{2}}N_1c_1^k, \quad k=0,1,2,$$ so that we obtain $c_1^2N_1^2=(1+\alpha)c_1^2N_1$, which gives $N_1=1+\alpha$ and [\[tem-202\]](#tem-202){reference-type="eqref" reference="tem-202"}. **Case 2:** $l\ge2$. We first claim that $N_1=1+\alpha$. We define $r_n=|z_{n,2}|$ and $v_n(x)=r_n^{\alpha_n}u_n(r_nx)$ with $\alpha_n=\frac{2+2\alpha}{p_n-1}$. Then it is easy to see that $v_n$ satisfies $$-\Delta v_n=|x|^{2\alpha}\widetilde V_nv_n^{p_n},\quad\text{in}~B_{\varepsilon_0'},$$ where $\widetilde V_n(x)=V_n(r_nx)$. Let $$\begin{aligned} \label{tem-60-beta2} \widetilde\beta_k:=\lim_{n\to\infty} \int_{B_{\varepsilon_0'}}p_n|x|^{2\alpha}\widetilde V_nv_n(x)^{p_n-1+k}{\mathrm d}x,\quad k=0,1,2. \end{aligned}$$ By [\[energy-PQ\]](#energy-PQ){reference-type="eqref" reference="energy-PQ"}, we get $$\widetilde\beta_k =\lim_{n\to\infty}r_n^{k\alpha_n}\int_{P_{n,1}\cup Q_{n,1}\cup P_{n,2}}p_n|x|^{2\alpha}V_n(x)u_n(x)^{p_n-1+k}{\mathrm d}x =8\pi e^{\frac{k}{2}} N_1c_1^{k}c_2^{-k}.$$ Since $\sup_{P_{n,2}}p_n u_n\le C$, we obtain $$p_n v_n\le C, \quad\text{for any}~x\in{B_{\varepsilon_0'}\setminus B_{\frac{|z_{n,1}|}{\varepsilon_0'|z_{n,2}|}}}.$$ Thanks to $\frac{|z_{n,1}|}{|z_{n,2}|}\to0$, we know that $0$ is the only blow up point of $v_n$ in $B_{\varepsilon_0'}$. Now we are in the same situation of Lemma [Lemma 19](#tem-203){reference-type="ref" reference="tem-203"}, so that it holds $\widetilde\beta_1^2=8\pi(1+\alpha)\widetilde\beta_2$. It follows that $N_1=1+\alpha$. Now as in Case 1, by [\[tem-56-0\]](#tem-56-0){reference-type="eqref" reference="tem-56-0"} and [\[energy-PQ\]](#energy-PQ){reference-type="eqref" reference="energy-PQ"}, we get $$\beta_k=8\pi e^{\frac{k}{2}}\sum_{i=1}^lN_ic_i^k,\quad k=0,1,2.$$ Then by $\beta_1^2=8\pi(1+\alpha)\beta_2$, we get $(\sum_{i=1}^lN_ic_i)^2=N_1\sum_{i=1}^lN_ic_i^2$. Thanks to [\[eq-ci1\]](#eq-ci1){reference-type="eqref" reference="eq-ci1"}, we have $$N_1\sum_{i=2}^lN_ic_i^2=(\sum_{i=1}^lN_ic_i)^2-N_1^2c_1^2>2N_1c_1\sum_{i=2}^lN_ic_i\ge 2N_1\sum_{i=2}^lN_ic_i^2,$$ which is a contradiction, so $l=1$ and we finish the proof. ◻ *Proof of Theorem [Theorem 3](#thm-singular){reference-type="ref" reference="thm-singular"}.* If $\alpha\not\in{\mathbb N}$, from Proposition [Proposition 24](#decomposition){reference-type="ref" reference="decomposition"} and Lemma [Lemma 26](#key){reference-type="ref" reference="key"}, we know that [\[bound\]](#bound){reference-type="eqref" reference="bound"} holds, so that Lemma [Lemma 23](#tem-41){reference-type="ref" reference="tem-41"} gives the theorem. If $\alpha\in{\mathbb N}$, we don't know whether [\[bound\]](#bound){reference-type="eqref" reference="bound"} holds or not, but anyway this theorem follows from Lemma [Lemma 23](#tem-41){reference-type="ref" reference="tem-41"} and Lemma [Lemma 26](#key){reference-type="ref" reference="key"}. ◻ # The boundary value problem This section is devoted to the proof of Theorem [Theorem 5](#thm-1){reference-type="ref" reference="thm-1"}. Let $u_n$ be a solution sequence of [\[equ-1-0\]](#equ-1-0){reference-type="eqref" reference="equ-1-0"}, and $W_n(x)$ satisfy [\[con-1\]](#con-1){reference-type="eqref" reference="con-1"}-[\[con-2\]](#con-2){reference-type="eqref" reference="con-2"}. We denote $\|u\|_\infty=\|u\|_{L^\infty(\Omega)}$ for simplicity. Let $\varphi_1>0$, $\|\varphi_1\|_\infty=1$, be the eigenfunction of the first eigenvalue of $-\Delta$ in $\Omega$ with the Dirichlet boundary condition: $$\lambda_1(\Omega):=\inf_{u\in H_0^1(\Omega)}\frac{\int_\Omega|\nabla u|^2}{\int_\Omega u^2}>0.$$ Then $\varphi_1$ satisfies $-\Delta\varphi_1=\lambda_1(\Omega)\varphi_1$ and we have $$\int_\Omega (W_nu_n^{p_n-1}-\lambda_1(\Omega))u_n\varphi_1=\int_\Omega (-\varphi_1\Delta u_n+u_n\Delta\varphi_1 )=0.$$ So $(\|W_nu_n^{p_n-1}\|_\infty-\lambda_1(\Omega))\int_\Omega u_n\varphi_1\ge0$, which implies $$\|u_n\|_\infty\ge \left(\frac{\lambda_1(\Omega)}{\max_\Omega W_n}\right)^{\frac{1}{p_n-1}}\geq \left(\frac{\lambda_1(\Omega)}{C}\right)^{\frac{1}{p_n-1}}.$$ As a result, we obtain $\liminf_{n\to+\infty}\|u_n\|_\infty\ge1$, which yields that ${\mathcal S}\neq\emptyset$, where ${\mathcal S}$ is the set of blow-up points of $p_nu_n$ defined in [\[S\]](#S){reference-type="eqref" reference="S"}. Moreover, the same argument as Lemma [Lemma 13](#converge-regular){reference-type="ref" reference="converge-regular"} implies $\limsup_{n\to+\infty}\|u_n\|_\infty\le C$. Applying Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"} for $u_n$ in $\Omega$, we obtain a set $$\Sigma=\left\{a_1,\cdots,a_k\right\}\subset\Omega$$ satisfying the properties in Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"}, where we set $\Sigma=\emptyset$ and $k=0$ if the alternative (i) holds. Obviously, $\Sigma\subset{\mathcal S}$. Recall the zero set ${\mathcal Z}\subset\Omega$ of $W_n(x)$ defined in [\[Z\]](#Z){reference-type="eqref" reference="Z"}. We choose $r_0>0$ small such that $$B_{2r_0}(a)\subset\Omega\quad\text{and}\quad B_{2r_0}(a)\cap B_{2r_0}(a')=\emptyset,\quad\text{for}~a,a'\in\Sigma\cup{\mathcal Z},~a\neq a'.$$ When $\Sigma\neq\emptyset$, we define the local maximums $\gamma_{n,i}$ and the local maximum points $x_{n,i}$ of $u_n$ near $a_i\in\Sigma$ by $$\label{5--2} \gamma_{n,i}=u_n(x_{n,i}):=\max_{B_{2r_0}(a_i)} u_n,\quad\text{for}~i=1,\cdots,k.$$ Then the proof of Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"} shows that $x_{n,i}\to a_i$ and $\gamma_{n,i}\to \gamma_i\geq 1$. Now we aim to exclude the boundary concentration. Denote $$\Omega_\delta:=\left\{x\in\Omega:~d(x,\partial\Omega)\ge\delta\right\}.$$ Since ${\mathcal Z}\subset\Omega$ and $\Sigma\subset\Omega$, we take $\delta_0>0$ small such that $\Omega_{3\delta_0}$ is a compact subset satisfying $$\label{label-34} ({\mathcal Z}\cup\Sigma)\subset\Omega_{3\delta_0}\Subset\Omega.$$ Denote $$\widetilde{\Omega}:=\Omega\setminus\Omega_{2\delta_0}=\left\{x\in\Omega:~d(x,\partial\Omega)<2\delta_0\right\}.$$ Since Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"} says that $$\label{777}\text{\it $p_nu_n$ is uniformly bounded in $L_{loc}^{\infty}(\Omega\setminus\Sigma)$},$$we see from [\[label-34\]](#label-34){reference-type="eqref" reference="label-34"} that $$\label{label-20} 0\le p_nu_n(x)\le C,\quad\text{for}~x\in\partial\widetilde\Omega,$$ where $C>0$ is a constant. More precisely, $$\label{equ-1-2} \begin{cases} -\Delta u_n=W_n(x)u_n^{p_n},\quad u_n>0,\quad\text{in}~\widetilde\Omega,\\ u_n=0,\quad\text{on}~\partial\Omega\subset\partial\widetilde{\Omega},\\ 0<p_nu_n\leq C\quad\text{on}~\partial\widetilde{\Omega}\setminus\partial\Omega,\\ \|u_n\|_{L^\infty(\Omega)}\leq C,\\ \int_\Omega p_n W_n(x)u_n^{p_n}{\mathrm d}x\le C. \end{cases}$$ Furthermore, it follows from [\[con-1\]](#con-1){reference-type="eqref" reference="con-1"}-[\[con-2\]](#con-2){reference-type="eqref" reference="con-2"} and [\[label-34\]](#label-34){reference-type="eqref" reference="label-34"} that $$\label{con-2-0} 0<\frac{1}{C}\le W_{n}(x)\le C<\infty,\quad |\nabla W_{n}(x)|\le C,\quad\text{for}~x\in\widetilde\Omega.$$ **Proposition 27**. *There is no blow-up point in $\overline\Omega\setminus\Omega_{\delta_0}$ for $p_nu_n$.* To give the proof of Proposition [Proposition 27](#boundary){reference-type="ref" reference="boundary"}, we assume by contradiction that there exists a blow-up point in $\overline\Omega\setminus\Omega_{\delta_0}\subset\widetilde{\Omega}$, then $$\label{label-21} \limsup_{n\to\infty}\sup_{\widetilde\Omega} p_nu_n=\infty,$$ and the same argument as [\[4-42\]](#4-42){reference-type="eqref" reference="4-42"} implies $$\limsup_{n\to\infty}\sup_{\widetilde\Omega}p_nW_n(x)u_n(x)^{p_n-1}=\infty.$$ Thus up to a subsequence, there is a family of points $\{y_{n,1}\}$ such that $$\label{con-2-22} p_nW_n(y_{n,1})u_n(y_{n,1})^{p_n-1}\to\infty,\quad\text{as}~n\to\infty.$$ Now we suppose there exist $m\in{\mathbb N}\setminus\{0\}$ families of points $\{y_{n,i}\}_{n\ge1}$, $i=1,\cdots,m$, in $\widetilde\Omega$ such that $$\label{label-13} p_nW_n(y_{n,i})u_n(y_{n,i})^{p_n-1}\to\infty,\quad\text{as}~n\to\infty.$$ Define the parameters $\varepsilon_{n,i}>0$ by $$\label{laben-14} \varepsilon_{n,i}^{-2}:=p_nW_n(y_{n,i})u_n(y_{n,i})^{p_n-1},\quad\text{for}~i=1,\cdots,m,$$ then $$\label{5-13} \lim_{n\to\infty}\varepsilon_{n,i}=0,\quad\quad \liminf_{n\to\infty} u_n(y_{n,i})\ge 1.$$ Define $$\label{5-14} R_{n,m}(x):=\min_{i=1,\cdots,m}|x-y_{n,i}|,\quad\text{for}~x\in\widetilde\Omega.$$ As in [@LE-7], we introduce the following properties: - For any $1\leq i,j\leq m$ and $i\neq j$, $$\lim_{n\to\infty}\varepsilon_{n,i}=0,\quad\quad\lim_{n\to\infty}\frac{|y_{n,i}-y_{n,j}|}{\varepsilon_{n,i}}=\infty.$$ - For each $1\leq i\leq m$, for $x\in\widetilde\Omega_{n,i}:=\frac{\widetilde\Omega-y_{n,i}}{\varepsilon_{n,i}}$, $$\label{5--15} w_{n,i}(x):=p_n\left(\frac{u_n(y_{n,i}+\varepsilon_{n,i}x)}{u_n(y_{n,i})}-1\right)\to U_0(x)=-2\log\left(1+\frac{1}{8}|x|^2\right)$$ in $\mathcal C_{loc}^2(\mathbb R^2)$ as $n\to\infty$. - There exists $C>0$ independent of $n$ such that $$\sup_{x\in\widetilde{\Omega}}p_nR_{n,m}(x)^2W_n(x)u_n(x)^{p_n-1}\le C,\quad\forall\,n.$$ It is easy to see that once (${\mathcal P}_1^m$)-(${\mathcal P}_3^m$) hold for $m\in{\mathbb N}\setminus\{0\}$ families of points $\{y_{n,i}\}_{n\ge1}$, then we can not find an $m+1$ family of points $\{y_{n,m+1}\}_{n\ge1}$ such that (${\mathcal P}_1^{m+1}$) holds. **Lemma 28**. *There exists $l\in{\mathbb N}\setminus\{0\}$ families of points $\{y_{n,i}\}_{n\ge1}$ in $\widetilde\Omega$, $i=1,\cdots,l$, such that, after passing to a subsequencce, $({\mathcal P}_1^l)$-$({\mathcal P}_3^l)$ hold.* *Proof.* Thanks to [\[equ-1-2\]](#equ-1-2){reference-type="eqref" reference="equ-1-2"}, [\[con-2-0\]](#con-2-0){reference-type="eqref" reference="con-2-0"} and [\[con-2-22\]](#con-2-22){reference-type="eqref" reference="con-2-22"}, the proof of Lemma [Lemma 28](#label-31){reference-type="ref" reference="label-31"} is very similar to that of [@Druet Proposition 2.1] or [@LE-7 Proposition 2.2], so we omit it. ◻ Now we define the concentration set ${\mathcal T}$ in $\overline{\widetilde{\Omega}}$ by $$\label{label-33} {\mathcal T}:=\left\{a_{k+1},\cdots,a_{k+l}\right\}=\left\{\lim_{n\to\infty}y_{n,i},~i=1,\cdots,l\right\}\subset\overline{\widetilde \Omega},$$ with $y_{n,i}\to a_{k+i}$ as $n\to\infty$, where $y_{n,i}$ is given by Lemma [Lemma 28](#label-31){reference-type="ref" reference="label-31"}. **Lemma 29**. *We have ${\mathcal T}\subset\partial\Omega$ and $p_nu_n$ is uniformly bounded in $L_{loc}^{\infty}(\overline{\widetilde{\Omega}}\setminus {\mathcal T})$.* *Proof.* If there exists $a_{k+i}\in{\mathcal T}\cap\Omega$, then by [\[777\]](#777){reference-type="eqref" reference="777"} and [\[5-13\]](#5-13){reference-type="eqref" reference="5-13"} we have $a_{k+i}\in\Sigma$. However, by the choice of $\delta_0$ in [\[label-34\]](#label-34){reference-type="eqref" reference="label-34"}, it holds ${\mathcal T}\cap\Sigma=\emptyset$, a contradiction to $a_{k+i}\in{\mathcal T}\cap\Sigma$. Thus ${\mathcal T}\subset\partial\Omega$. Recalling [\[equ-1-2\]](#equ-1-2){reference-type="eqref" reference="equ-1-2"}, we set $p_nu_n=\phi_n+\psi_n$ with $$\begin{cases} -\Delta \phi_n=p_nW_n(x)u_n^{p_n},\quad\text{in}~\widetilde\Omega,\\ \phi_n=0\quad\text{on }\;\widetilde\Omega, \end{cases} \quad \begin{cases} -\Delta \psi_n=0\quad\text{in }\; \widetilde\Omega,\\ \bar \psi=p_nu_n\in [0,C]\quad\text{on }\;\partial\widetilde\Omega. \end{cases}$$ Then $\|\psi_n\|_{L^\infty(\widetilde\Omega)}\leq C$. We claim that $$\label{888} |\nabla \phi_n(x)|\leq \frac{C}{R_{n,l}(x)},\quad \forall x\in\widetilde{\Omega}.$$ To prove [\[888\]](#888){reference-type="eqref" reference="888"}, we fix any $x\in\widetilde{\Omega}$. Recalling [\[5-14\]](#5-14){reference-type="eqref" reference="5-14"}, we take $j_0$ such that $$R_{n,l}(x)=\min_{i=1,\cdots,m}|x-y_{n,i}|=|x-y_{n,j_0}|.$$ Let $\tilde{G}(x,y)$ be the Green function of $-\Delta$ in $\widetilde\Omega$ with the Dirichlet boundary condition. Then $$\begin{aligned} |\nabla\phi_n(x)|&=\left|\int_{\widetilde{\Omega}}\nabla_x\widetilde{G}(x,y)p_nW_n(y)u_n(y)^{p_n}dy\right|\\ &\leq \int_{\widetilde{\Omega}\cap \{|y-x|\geq\frac{R_{n,l}(x)}{2}\}} \frac{1}{|x-y|}p_nW_n(y)u_n(y)^{p_n}dy\\ &\quad+\int_{\widetilde{\Omega}\cap \{|y-x|\leq\frac{R_{n,l}(x)}{2}\}} \frac{1}{|x-y|}p_nW_n(y)u_n(y)^{p_n}dy=I_1+I_2.\end{aligned}$$ By [\[equ-1-2\]](#equ-1-2){reference-type="eqref" reference="equ-1-2"} we see that $I_1\leq \frac{C}{R_{n,l}(x)}$. For $|y-x|\leq \frac{R_{n,l}(x)}{2}=\frac{|x-y_{n,j_0}|}{2}$, we have $$|y-y_{n,i}|\geq |x-y_{n,i}|-|x-y|\geq \frac{R_{n,l}(x)}{2},\quad\forall i,$$ so $R_{n,l}(y)\geq \frac{R_{n,l}(x)}{2}$. Then by [\[equ-1-2\]](#equ-1-2){reference-type="eqref" reference="equ-1-2"} and (${\mathcal P}_3^l$), we have $$\begin{aligned} p_nW_n(y)u_n(y)^{p_n}\leq \frac{4C}{R_{n,l}(x)^2}p_n R_{n,l}(y)^2W_n(y)u_n(y)^{p_n-1}\leq \frac{C}{R_{n,l}(x)^2},\end{aligned}$$ and then $$I_2\leq \frac{C}{R_{n,l}(x)^2}\int_{|y-x|\leq\frac{R_{n,l}(x)}{2}} \frac{1}{|x-y|}dy\leq \frac{C}{R_{n,l}(x)}.$$ Therefore, [\[888\]](#888){reference-type="eqref" reference="888"} holds. From here and $\phi_n=0$ on $\partial\widetilde\Omega$, we see that $\phi_n$ is uniformly bounded in $L_{loc}^{\infty}(\overline{\widetilde{\Omega}}\setminus {\mathcal T})$ and so does $p_nu_n=\phi_n+\psi_n$. ◻ Recalling the set ${\mathcal S}$ of blow-up points defined by [\[S\]](#S){reference-type="eqref" reference="S"}, clearly we have $$\label{label-37} {\mathcal S}=\Sigma\cup{\mathcal T}=\left\{a_1,\cdots,a_{k+l}\right\}.$$ Recalling [\[5\--2\]](#5--2){reference-type="eqref" reference="5--2"}, [\[laben-14\]](#laben-14){reference-type="eqref" reference="laben-14"}, [\[5\--15\]](#5--15){reference-type="eqref" reference="5--15"} and Lemma [Lemma 28](#label-31){reference-type="ref" reference="label-31"}, we unify the notations by setting $$x_{n,k+i}:=y_{n,i},\quad \mu_{n,k+i}:=\varepsilon_{n,i},\quad v_{n,k+i}:=w_{n,i},\quad \gamma_{n,k+i}:=u_n(y_{n,i})$$ for $1\leq i\leq l$, and after passing to a subsequence, we assume $$\gamma_{n,i}\to \gamma_i\ge1,\quad\text{as}~n\to\infty, \quad\text{for }1\leq i\leq k+l.$$ By Theorem [Theorem 1](#thm-BM){reference-type="ref" reference="thm-BM"} and Lemma [Lemma 29](#lem-53){reference-type="ref" reference="lem-53"}, we conclude that $$\label{5-19}\|p_nu_n\|_{L^\infty(K)}\le C_K\quad\text{\it for any compact subset $K\in\overline\Omega\setminus{\mathcal S}$}.$$ More precisely, we have the following convergence result. **Lemma 30**. *There exists $\sigma_{a_i}\geq 8\pi$, $i=1,\cdots,k+l$, such that $$p_nu_n(x)\to \sum_{i=1}^{k+l}\sigma_{a_i} G(x,a_i),\quad\text{in}~\mathcal C_{loc}^2(\overline\Omega\setminus{\mathcal S})~\text{as}~n\to\infty.$$* *Proof.* Exactly as in Lemma [Lemma 15](#converge-regular2){reference-type="ref" reference="converge-regular2"}, we get $$p_nu_n(x)\to \sum_{i=1}^{k+l} \sigma_{a_i} G(x,a_i),\quad\text{in}~\mathcal C_{loc}^2(\overline{\Omega}\setminus {\mathcal S}),\quad\text{as}~n\to\infty,$$ where $$\sigma_{a_i}:=\lim_{d\to0}\lim_{n\to\infty}p_n\int_{B_d(a_i)\cap\Omega}W_n(x)u_n(x)^{p_n}{\mathrm d}x.$$ We show that $\sigma_{a_i}\geq 8\pi$. For $1\leq i\leq k$, by Theorems [Theorem 2](#thm-regular){reference-type="ref" reference="thm-regular"} and [Theorem 3](#thm-singular){reference-type="ref" reference="thm-singular"}, we see that $\sigma_{a_i}\geq 8\pi \sqrt{e}$. For $k+1\leq i\leq k+l$, since $x_{n,i}\to a_i$ as $n\to\infty$, then $B_{d/2}(x_{n,i})\subset B_d(a_i)$ for $n$ large, and hence $$\begin{aligned} p_n\int_{B_d(a_i)\cap\Omega}W_n(x)u_n(x)^{p_n}{\mathrm d}x &\ge p_n\int_{B_{d/2}(x_{n,i})\cap\Omega}W_n(x)u_n(x)^{p_n}{\mathrm d}x\\ &=\gamma_{n,i}\int_{B_{\frac{d}{2\mu_{n,i}}}\cap\Omega_{n,i}}\frac{W_n(x_{n,i}+\mu_{n,i}x)}{W_n(x_{n,i})}\left(1+\frac{v_{n,i}}{p_n}\right)^{p_n}{\mathrm d}x. \end{aligned}$$ Passing to the limit as $n\to\infty$, since $B_{\frac{d}{2\mu_{n,i}}}\cap\Omega_{n,i}\to\mathbb R^2$, thanks to $({\mathcal P}_2^l)$ we get $$\label{tem-100} \lim_{n\to\infty}p_n\int_{B_d(a_i)\cap\Omega}W_n(x)u_n(x)^{p_n}{\mathrm d}x\ge \gamma_i\int_{\mathbb R^2}e^{U_0}{\mathrm d}x\ge 8\pi\gamma_i,$$ which gives $\sigma_{a_i}\ge 8\pi$. ◻ Now we are ready to show that there is no boundary blow up. *Proof of Proposition [Proposition 27](#boundary){reference-type="ref" reference="boundary"}.* Assume by contradiction that there exists a concentration point in $\overline\Omega\setminus\Omega_{\delta_0}$. Then the above argument shows that the blow-up point ${\mathcal S}$ is given in [\[label-37\]](#label-37){reference-type="eqref" reference="label-37"}. Since ${\mathcal T}\neq\emptyset$ now, we take $a_i\in{\mathcal T}\subset\partial\Omega$ for some $k+1\leq i\leq k+l$. Choose $r>0$ such that ${\mathcal S}\cap B_r(a_i)=\{a_i\}$. Let $y_n=a_i+\rho_{n,d}\nu(a_i)$, where $\nu(x)$ is the outer normal vector of $\partial\Omega$ at the point $x\in\partial\Omega$, and $$\rho_{n,d}:=\frac{ \int_{\partial\Omega\cap B_d(a_i)}\left(\frac{\partial u_n(x)}{\partial\nu}\right)^2\left\langle x-a_i,\nu(x)\right\rangle{\mathrm d}s_x } { \int_{\partial\Omega\cap B_d(a_i)}\left(\frac{\partial u_n(x)}{\partial\nu}\right)^2\left\langle\nu(a_i),\nu(x)\right\rangle{\mathrm d}s_x }.$$ Recall the zero set ${\mathcal Z}$ of $W_n(x)$ defined in [\[Z\]](#Z){reference-type="eqref" reference="Z"}. Choose $0<d<\min\{r,\frac{d(a_i,{\mathcal Z})}{2}\}$ small enough such that $\frac{1}{2}\le \left\langle\nu(a_i),\nu(x)\right\rangle\le 1$ for $x\in \partial\Omega\cap B_d(a_i)$. With this choice of $d$, we have $$\label{label-40} |\rho_{n,d}|\le 2d.$$ Moreover, it is easy to see that the choice of $y_n$ implies $$\label{label-39} \int_{\partial\Omega\cap B_d(a_i)}\left(\frac{\partial u_n(x)}{\partial\nu}\right)^2\left\langle x-y_n,\nu(x)\right\rangle{\mathrm d}s_x=0.$$ Applying the local Pohozaev identity [\[pho-1\]](#pho-1){reference-type="eqref" reference="pho-1"} in the set $\Omega\cap B_d(a_i)$ with $u=u_n$, $V(x)=W_n(x)$ and $y=y_n$, using [\[label-39\]](#label-39){reference-type="eqref" reference="label-39"}, the boundary condition $u_n=0$ on $\partial\Omega$ (so that $\nabla u_n=-|\nabla u_n|\nu$ on $\partial\Omega$) we obtain $$\begin{aligned} \label{label-41} &\quad \frac{2p_n^2}{p_n+1}\int_{\Omega\cap B_d(a_i)}W_n(x)u_n(x)^{p_n+1}{\mathrm d}x\nonumber\\ &\quad+\frac{p_n^2}{p_n+1}\int_{\Omega\cap B_d(a_i)}\left\langle\nabla W_n(x),x-y_n\right\rangle u_n(x)^{p_n+1}{\mathrm d}x\nonumber\\ %&=\f{p_n^2}{2}\int_{\pa \Omega\cap B_d(a_i)}|\nabla p_nu_n|^2\abr{x-y_n,\nu}\rd s_x +\int_{\Omega\cap\pa B_d(a_i)} \abr{\nabla p_nu_n,\nu}\abr{\nabla p_nu_n,x-y_n} \rd s_x\\ %&\quad -\f{1}{2}\int_{\Omega\cap\pa B_d(a_i)} \abs{\nabla p_nu_n}^2\abr{x-y_n,\nu} \rd s_x +\f{p_n^2}{p_n+1}\int_{\Omega\cap\pa B_d(a_i)} |x|^{2\al}u_n(x)^{p_n+1}\abr{x-y_n,\nu} \rd s_x \\ &=\int_{\Omega\cap\partial B_d(a_i)} \left\langle p_n\nabla u_n,\nu\right\rangle\left\langle p_n\nabla u_n,x-y_n\right\rangle {\mathrm d}s_x\\ &\quad-\frac{1}{2}\int_{\Omega\cap\partial B_d(a_i)} \left\lvert p_n\nabla u_n\right\rvert^2\left\langle x-y_n,\nu\right\rangle {\mathrm d}s_x \nonumber\\ &\quad +\frac{p_n^2}{p_n+1}\int_{\Omega\cap\partial B_d(a_i)} W_n(x)u_n(x)^{p_n+1}\left\langle x-y_n,\nu\right\rangle {\mathrm d}s_x. \nonumber\end{aligned}$$ Next we estimate the second term in the left-hand side and all the three terms in the right-hand side. By the choice of $d$ and [\[label-40\]](#label-40){reference-type="eqref" reference="label-40"}, we have $W_n(x)\ge C>0$, $|\nabla W_n(x)|\le C$ and $|x-y_n|\le 3d$ for $x\in\Omega\cap B_d(a_i)$, so $$\begin{aligned} &\left\lvert\frac{p_n^2}{p_n+1}\int_{\Omega\cap B_d(a_i)}\left\langle\nabla W_n(x),x-y_n\right\rangle u_n(x)^{p_n+1}{\mathrm d}x\right\rvert\\ \le & Cd\int_{\Omega\cap B_d(a_i)}p_nW_n(x)u_n(x)^{p_n}{\mathrm d}x=O(d). \end{aligned}$$ By [\[5-19\]](#5-19){reference-type="eqref" reference="5-19"} we have $$\frac{p_n^2}{p_n+1}\left\lvert\int_{\Omega\cap\partial B_d(a_i)} W_n(x)u_n(x)^{p_n+1}\left\langle x-y_n,\nu\right\rangle {\mathrm d}s_x\right\rvert\le \frac{C_d^{p_n+1}}{p_n^{p_n}}\to 0\;\text{as }n\to\infty.$$ By [\[label-40\]](#label-40){reference-type="eqref" reference="label-40"} we may assume $y_n\to y_d$ with $|y_d-a_i|\leq 2d$. Recall Lemma [Lemma 30](#converge-2){reference-type="ref" reference="converge-2"} that $$p_nu_n(x)\to F(x):=\sum_{j=1}^{k+l}\sigma_{a_j}G(x,a_j),\quad\text{in }\mathcal C_{loc}^2(\overline\Omega\cap B_r(a_i)\setminus\{a_i\}).$$ Since $a_i\in\partial\Omega$ implies $G(x,a_i)\equiv 0$ for $x\neq a_i$, we have (see e.g. [@LE-1 (3.7)]) $$F(x)=O(1),\quad \nabla F(x)=O(1),\quad \text{\it for }x\in \overline{\Omega}\cap B_{r}(a_i)\setminus\{a_i\}.$$ From here and $0<d<r$, we obtain $$\begin{aligned} &\lim_{n\to\infty}\int_{\Omega\cap\partial B_d(a_i)} \left\langle p_n\nabla u_n,\nu\right\rangle\left\langle p_n\nabla u_n,x-y_n\right\rangle {\mathrm d}s_x\\ =&\int_{\Omega\cap\partial B_d(a_i)} \left\langle\nabla F,\nu\right\rangle\left\langle\nabla F,x-y_d\right\rangle {\mathrm d}s_x= O(1) \int_{\Omega\cap\partial B_d(a_i)}\left\lvert x-y_d\right\rvert {\mathrm d}s_x=O(d^2),\end{aligned}$$ and similarly $$\lim_{n\to\infty}\int_{\Omega\cap\partial B_d(a_i)} \left\lvert p_n\nabla u_n\right\rvert^2\left\langle x-y_n,\nu\right\rangle {\mathrm d}s_x=O(d^2).$$ Inserting these estimates into [\[label-41\]](#label-41){reference-type="eqref" reference="label-41"}, we finally obtain $$\lim_{d\to0}\lim_{n\to\infty}p_n\int_{\Omega\cap B_d(a_i)}W_n(x)u_n(x)^{p_n+1}{\mathrm d}x=0.$$ However, a similar argument as [\[tem-100\]](#tem-100){reference-type="eqref" reference="tem-100"} leads to $$\lim_{d\to0}\lim_{n\to\infty}p_n\int_{\Omega\cap B_d(a_i)}W_n(x)u_n(x)^{p_n+1}{\mathrm d}x\ge 8\pi\gamma_i^2>0,$$ which is a contradiction. This completes the proof. ◻ *Proof of Theorem [Theorem 5](#thm-1){reference-type="ref" reference="thm-1"}.* Since Proposition [Proposition 27](#boundary){reference-type="ref" reference="boundary"} tells us that ${\mathcal S}=\Sigma=\{a_1,\cdots,a_k\}\subset \Omega$, by using Theorem [Theorem 2](#thm-regular){reference-type="ref" reference="thm-regular"} for those $a_i\in {\mathcal S}\setminus{\mathcal Z}$ and Theorem [Theorem 3](#thm-singular){reference-type="ref" reference="thm-singular"} for those $a_i\in {\mathcal S}\cap{\mathcal Z}$, one can easily prove Theorem [Theorem 5](#thm-1){reference-type="ref" reference="thm-1"}. ◻ # The ground state of the Hénon equation This section is devoted to the proof of Theorem [Theorem 7](#thm-0){reference-type="ref" reference="thm-0"}. Let $u_n$ be a ground state of the Hénon equation [\[equ-henon\]](#equ-henon){reference-type="eqref" reference="equ-henon"}. Set $$u_n(x_n)=\max_{\Omega}u_n,$$ then [\[tem-4-00\]](#tem-4-00){reference-type="eqref" reference="tem-4-00"} implies $u_n(x_n)\to\gamma\in [1,\sqrt{e}]$, i.e. $p_nu_n(x_n)\to\infty$. Applying Theorem [Theorem 5](#thm-1){reference-type="ref" reference="thm-1"}, we see the existence of $k\in\mathbb{N}\setminus\{0\}$ and a set ${\mathcal S}=\left\{a_1,\cdots,a_k\right\}\subset\Omega$ consisting of blow-up points of $p_nu_n$ in $\overline{\Omega}$ such that $\max_{B_r(a_i)}u_n\to\gamma_i\geq \sqrt{e}$ for any small $r$ and $\|p_nu_n\|_{L^\infty(K)}\le C_K$ for any compact subsets $K\subset\overline\Omega\setminus{\mathcal S}$. In particular, $u_n(x_n)\to \sqrt{e}$. **Lemma 31**. *It holds ${\mathcal S}=\{a\}$ with $a=a_1\neq 0$. Consequently, $x_n\to a$ and $$\label{6--10} p_n|x|^{2\alpha}u_n^{p_n-1+k}\to8\pi e^{\frac{k}{2}}\delta_{a},\quad k=0,1,2$$ weakly in the sense of measures.* *Proof.* Assume by contradiction that $0\in{\mathcal S}$. By choosing $r>0$ small, we know that $0$ is the only blow-up point of $p_nu_n$ in $B_{r}$, i.e., $$\max_{B_r}p_nu_n\to\infty\quad \text{and}\quad \max_{\overline B_r\setminus B_\delta} p_nu_n\le C_\delta,\quad\text{for any}~0<\delta<r.$$ Applying Theorem [Theorem 3](#thm-singular){reference-type="ref" reference="thm-singular"}, up to a subsequence we obtain $$\int_{B_r} p_n|x|^{2\alpha}u_n^{p_n+1}{\mathrm d}x\to 8\pi(1+\alpha)ec^2\geq 8\pi(1+\alpha)e,$$ a contradiction with [\[tem-4\]](#tem-4){reference-type="eqref" reference="tem-4"}. This proves $0\not\in{\mathcal S}$. Then we can apply Theorem [Theorem 2](#thm-regular){reference-type="ref" reference="thm-regular"} around each point of ${\mathcal S}$ and obtain (note [\[tem-4\]](#tem-4){reference-type="eqref" reference="tem-4"}) $$8\pi e=\lim_{n\to\infty}\int_{\Omega}p_n|x|^{2\alpha}u_n^{p_n+1}{\mathrm d}x=8\pi e\times \#{\mathcal S}.$$ This implies $\#{\mathcal S}=1$, i.e. ${\mathcal S}=\{a\}$ with $a=a_1$ and so $x_n\to a$. Consequently, [\[6\--10\]](#6--10){reference-type="eqref" reference="6--10"} follow from Theorem [Theorem 2](#thm-regular){reference-type="ref" reference="thm-regular"}. ◻ We need the local Pohozaev identity. **Lemma 32**. *Suppose $u$ satisfies $$\begin{cases} -\Delta u=V(x)u^p,\quad\text{in}~\Omega,\\ u>0,\quad\text{in}~\Omega,\end{cases}$$ then for any $y\in\mathbb R^2$ and any subset $\Omega'\subset\Omega$, it holds $$\label{pho-2} \begin{aligned} &\quad \frac{1}{p+1}\int_{\Omega'}\partial_iV(x)u(x)^{p+1}{\mathrm d}x-\frac{1}{p+1}\int_{\partial\Omega'}V(x)u(x)^{p+1}\nu_i(x){\mathrm d}s_x \\ &=\int_{\partial\Omega'}\left\langle\nabla u(x),\nu(x)\right\rangle\partial_iu(x)-\frac{1}{2}|\nabla u(x)|^2\nu_i(x){\mathrm d}s_x,\quad i=1,2, \end{aligned}$$ where $\partial_i=\frac{\partial}{\partial x_i}$ and $\nu(x)=(\nu_1(x),\nu_2(x))$ is the outer normal vector of $\partial \Omega'$ at $x$.* *Proof.* By direct computations, for $i=1,2$, we have $$-\Delta u(x)\cdot\partial_iu(x)=-\operatorname{div}(\partial_iu(x)\nabla u(x) )+\frac{\partial_i|\nabla u(x)|^2}{2},$$ and $$V(x)u(x)^p\cdot\partial_iu(x)=\partial_i\left(\frac{V(x)u(x)^{p+1}}{p+1}\right)-\frac{\partial_iV(x)u(x)^{p+1}}{p+1}.$$ Then by multiplying $-\Delta u=V(x)u^p$ with $\partial_iu(x)$, integrating on $\Omega'$ and using the divergence theorem, we obtain [\[pho-2\]](#pho-2){reference-type="eqref" reference="pho-2"}. ◻ Now we can finish the proof of Theorem [Theorem 7](#thm-0){reference-type="ref" reference="thm-0"}. *Proof of Theorem [Theorem 7](#thm-0){reference-type="ref" reference="thm-0"}.* Thanks to Lemma [Lemma 31](#lem-6-1){reference-type="ref" reference="lem-6-1"}, we can apply Lemma [Lemma 30](#converge-2){reference-type="ref" reference="converge-2"} and Lemma [Lemma 31](#lem-6-1){reference-type="ref" reference="lem-6-1"} to obtain $$\label{6-66}p_nu_n\to 8\pi\sqrt e G(x,a),\quad\text{in}~\mathcal C_{loc}^2(\overline{\Omega}\setminus\{a\})~\text{as}~n\to\infty.$$ It remains to compute the location of the blow-up point $a$. Applying the Pohozaev identity [\[pho-2\]](#pho-2){reference-type="eqref" reference="pho-2"} with $y=0$, $\Omega'=B_d(a)$, $V=|x|^{2\alpha}$ and $u=u_n$, and by using $\max_{\partial B_d(a)}p_nu_n\le C_d$, we obtain $$\begin{aligned}\label{tem-09} &\int_{\partial B_d(a)}\left\langle p_n\nabla u_n,\nu\right\rangle p_n\partial_i u_n-\frac{1}{2}\left\lvert p_n\nabla u_n\right\rvert^2\nu_i{\mathrm d}\sigma_x\\ =&\frac{2\alpha p_n^2}{p_n+1}\int_{B_d(a)}|x|^{2\alpha-2}x_iu_n^{p_n+1}{\mathrm d}x+o_n(1),\quad i=1,2. \end{aligned}$$ Note from [\[6-66\]](#6-66){reference-type="eqref" reference="6-66"} that on $\partial B_d(a)$, $$\left\langle p_n\nabla u_n,\nu\right\rangle p_n\partial_i u_n-\frac{1}{2}\left\lvert p_n\nabla u_n\right\rvert^2\nu_i\to-64\pi^2e\left(\frac{(x-a)_i}{8\pi^2d^3}+\frac{1}{2\pi r}\partial_iH(x,a)+O(1)\right),$$ as $n\to\infty$. This means that $$\text{LHS of \eqref{tem-09}}=-\frac{64\pi^2e}{2\pi d}\int_{\partial B_d(a)}\partial_iH(x,a){\mathrm d}\sigma_x+O(r)+o_n(1).$$ On the other hand, recalling Remark [Remark 17](#decay-regular2){reference-type="ref" reference="decay-regular2"}, we use the Domainted Convergence Theorem to get (write $a=(a_1,a_2)$) $$\begin{aligned} &\quad\text{RHS of \eqref{tem-09}}\\ &=\frac{2\alpha p_n}{p_n+1}\frac{u_n(x_n)^2}{|x_n|^{2\alpha}}\int_{B_{\frac{d}{2\mu_n}}}\left(|x_n+\mu_ny|^{2\alpha-2}\right)(x_n+\mu_ny)_i\left(1+\frac{v_n}{p_n}\right)^{p_n+1}{\mathrm d}y+o_n(1)\\ &=2\alpha\frac{e}{|a|^{2\alpha}}\int_{\mathbb R^2}|a|^{2\alpha-2}a_ie^U{\mathrm d}y+o_n(1)=16\pi e\alpha\frac{a_i}{|a|^2}+o_n(1). \end{aligned}$$ Thus by letting $n\to\infty$ first and then $d\to0$ in [\[tem-09\]](#tem-09){reference-type="eqref" reference="tem-09"}, we obtain $\partial_iH(a,a)=\frac{\alpha a_i}{4\pi|a|^2}$ for $i=1,2$, which implies $$\nabla \left(R(\cdot)-\frac{1}{4\pi}\log|\cdot|^{2\alpha}\right)(a)=0.$$ This completes the proof. ◻ ## Acknowledgements {#acknowledgements .unnumbered} Z.Chen is supported by NSFC (No. 12222109, 12071240), and H.Li is supposed by the postdoctoral fundation of BIMSA. 10 Adimurthi; Grossi Massimo. 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arxiv_math

--- abstract: | Given a finite poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The induced saturation number of $\mathcal P$, denoted by $\text{sat}^*(n,\mathcal P)$, is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. In this paper we prove that the saturation number for any given poset grows at worst polynomially. More precisely, we show that $\text{sat}^*(n, \mathcal P)=O(n^c)$, where $c\leq|\mathcal{P}|^2/4+1$ is a constant depending on $\mathcal P$ only. We obtain this result by bounding the VC-dimension of our family. author: - | PAUL BASTIDE, CARLA GROENLAND,\ MARIA-ROMINA IVAN AND TOM JOHNSTON bibliography: - document.bib title: "**A POLYNOMIAL UPPER BOUND FOR POSET SATURATION**" --- # Introduction We say that a poset $(\mathcal Q,\preceq)$ contains an *induced copy* of a poset $(\mathcal P,\preceq')$ if there exists an injective order-preserving function $f:\mathcal P\rightarrow\mathcal Q$ such that $(f(\mathcal P),\preceq)$ is isomorphic to $(\mathcal P,\preceq')$. We denote by $2^{[n]}$ the power set of $[n]=\lbrace 1,2,\dots,n\rbrace$. We define the *$n$-hypercube*, denoted by $Q_n$ to be the poset formed by equipping $2^{[n]}$ with the partial order induced by inclusion. If $\mathcal P$ is a finite poset and $\mathcal F$ is a family of subsets of $[n]$, we say that $\mathcal F$ is $\mathcal P$-*saturated* if $\mathcal F$ does not contain an induced copy of $\mathcal P$, and for any $S\notin\mathcal F$, the family $\mathcal F\cup S$ contains an induced copy of $\mathcal P$. The smallest size of a $\mathcal P$-saturated family of subsets of $[n]$ is called the *induced saturated number*, and denoted by $\text{sat}^*(n,\mathcal P)$. It has been shown that the growth of $\text{sat}^*(n,\mathcal P)$ has a dichotomy. Keszegh, Lemons, Martin, Pálvölgyi and Patkós [@keszegh2021induced] proved that for any poset the induced saturated number is either bounded or at least $\log_2(n)$. They also conjectured that in fact sat$^*(n,\mathcal P)$ is either bounded, or at least $n+1$. Recently, Freschi, Piga, Sharifzadeh and Treglown [@freschi2022induced] improved this result by replacing $\log_2 (n)$ with $2\sqrt{n-2}$. There is no known poset $\mathcal P$ for which $\text{sat}^*(n,\mathcal P)=\omega(n)$, and it is in fact believed that for any poset, the saturation number is either constant or grows linearly. Whilst, as summarised above, some general lower bounds have been established, no non-trivial general upper bounds have yet been found. Given a general poset $\mathcal P$, what can we say about upper bounds on its saturation number? How fast can it grow? Is it possible to have an intricate partial relation that forces the saturation number to grow faster than any polynomial? The aim of this paper is to show that the answer is no: the saturation numbers have at worst polynomial growth. Our main result is the following. **Theorem 1**. *Let $\mathcal P$ be a finite poset, and let $|\mathcal P|$ denote the size of the poset. Then $\textnormal{sat}^*(n,\mathcal P)=O(n^c)$, where $c\leq|\mathcal P|^2/4+1$ is a constant depending on $\mathcal P$ only. [\[thm:main\]]{#thm:main label="thm:main"}* Induced and non-induced poset saturation numbers are a growing area of study in combinatorics. Saturation for posets was introduced by Gerbner, Keszegh, Lemons, Palmer, Pálvölgyi and Patkós [@gerbner2013saturating], although this was not for *induced* saturation. Induced poset saturation was first introduced in 2017 by Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan [@ferrara2017saturation]. We briefly summarise some of the recent developments below, and also refer the reader to the textbook of Gerbner and Patkós [@gerbner2019extremal] for a nice introduction to the area. Determining the saturation number, even for small posets, has proven to be a difficult question. The *exact* saturation number is known for only a precious few posets such as the $X$ and $Y$ posets [@freschi2022induced], chains with at most 6 sets [@morrison2014saturated], and the fork [@ferrara2017saturation]. The only class of large posets for which exact saturation numbers are known are the $k$-antichains, denoted by $\mathcal{A}_k$. It is easy to see that a collection of $k-1$ full chains (chains of order $n+1$) that intersect only at $\emptyset$ and $[n]$ form a $k$-antichain saturated family. Thus, for $n$ large enough, we certainly have $\text{sat}^*(n, \mathcal A_{k})\leq (k-1)(n-1)+2$. In the other direction, Martin, Smith and Walker [@martin2020improved] showed that for $k\geq4$ and $n$ large enough $\text{sat}^*(n,\mathcal A_{k})\geq \left(1-\frac{1}{\log_2(k-1)}\right)\frac{(k-1)n}{\log_2(k-1)}$. Recently, Bastide, Groenland, Jacob and Johnston [@bastide2022exact] showed that $\text{sat}^*(n,\mathcal A_k)=(k-1)n-\Theta(k\log k)$, and gave the exact value for $n$ sufficiently large compared to $k$. Other posets that have received special attention are the butterfly (Figure [\[fig:butterfly\]](#fig:butterfly){reference-type="ref" reference="fig:butterfly"}), which we denote by $\mathcal B$, and the diamond (Figure [\[fig:diamond\]](#fig:diamond){reference-type="ref" reference="fig:diamond"}), which we denote by $\mathcal D_2$. The butterfly poset is at least known to be linear, but the upper and lower bounds differ by a constant factor. Indeed, the best known lower bound is $\mathop{\mathrm{sat*}}(n, \mathcal{B}) \geq n +1$ as shown by Ivan in [@ivan2020saturation], while the best upper bound is currently $\mathop{\mathrm{sat*}}(n, \mathcal{B}) \leq 6n - 10$, as shown by Keszegh, Lemons, Martin, Pálvölgyi and Patkós in [@keszegh2021induced]. Even less is known about the diamond. Martin, Smith and Walker [@martin2020improved] proved that $\sqrt n\leq\text{sat}^*(n, \mathcal D_2)\leq n+1$. The lower bound was later improved by Ivan [@ivan2022minimal] and now stands at $\text{sat}^*(n, \mathcal D_2) \geq (2\sqrt2-o(1))\sqrt n$. Despite the simple structure of the diamond, whether its saturation number is linear is still unknown. The proof of Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"} uses the following two new key notions, 'cube-height' and 'cube-width'. For a poset $\mathcal{P}$, the 'cube-height' is the least $k$ such that, for some $n$, we can embed $\mathcal P$ into the first $k+1$ layers of $Q_n$, while the 'cube-width' is the smallest $n$ that makes such a 'small height' embedding possible. We give formal definitions and bounds on these two notions in Section 2. The cube-height and cube-width are designed to build a $\mathcal P$-saturated family with bounded VC-dimension. This is done in Section 3, where we prove Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"}. Our construction could be viewed as the result of a greedy algorithm where the sets are ordered according to size (and then arbitrarily within the layers), and an element is added to the family as long as it does not create a copy of $\mathcal{P}$ in the family. Greedy algorithms have been used before in studying poset saturation, most notably when a greedy colex algorithm gave the linear upper bound for the butterfly [@keszegh2021induced]. In particular, this implies that such 'layer-by-layer' greedy algorithms result in a saturated family of growth size $n^{|\mathcal{P}|^2}$ while having a near-linear time complexity of $O_\mathcal{P}(|Q_n| (\log|Q_n|)^{|\mathcal{P}|^3})$. This follows from the fact that for any family $\mathcal{F}$, it can be decided if it is $\mathcal{P}$-free in $O_\mathcal{P}(|\mathcal{F}|^{|\mathcal{P}|})$ time. We end the introduction by reminding the reader about the VC-dimension of a family of sets. We say that a family $\mathcal{F}$ of subsets of $[n]$ *shatters* a set $S \subseteq [n]$ if, for all $F\subseteq S$, there exists $A\in\mathcal{F}$ such that $A\cap S=F$. In other words, $\{A\cap S:A\in \mathcal{F}\}$ is the power set of $S$. The *VC-dimension* of $\mathcal{F}$ is the largest cardinality of a set shattered by $\mathcal{F}$. The size of a family $\mathcal F$ with bounded VC-dimension grows at worst polynomially, as shown by the following well-known result. **Lemma 2** (Sauer-Shelah lemma [@sauer1972density; @shelah1972combinatorial]). *If $\mathcal{F}\subseteq 2^{[n]}$ has VC-dimension $d$, then $|\mathcal{F}|\leq\displaystyle\sum_{i=0}^d \binom{n}{i}$. [\[lem:ss\]]{#lem:ss label="lem:ss"}* # Cube-height and cube-width In this section we discuss how to 'fit' a given poset $\mathcal P$ into a hypercube. We do this with the help of cube-height and cube-width, the two new quantities mentioned above, which we bound in terms of $|\mathcal{P}|$. Given two integers $h\leq w$, we denote by $\binom{[w]}{\leq h}$ the induced subposet of the hypercube $Q_w$ consisting of all the sets of size at most $h$, i.e the poset $Q_w$ restricted to the first $h + 1$ layers, $0,1,\dots,h$. **Definition 1**. *For a poset $\mathcal P$, we define the *cube-height* $h^*(\mathcal P)$ to be the minimum $h^* \in \mathbb{N}$ for which there exists $n \in \mathbb{N}$ such that $\binom{[n]}{\leq h^*}$ contains an induced copy of $\mathcal{P}$.* **Definition 2**. *For a poset $\mathcal P$, we define the *cube-width* $w^*(\mathcal P)$ to be the minimum $w^* \in \mathbb{N}$ such that there exists an induced copy of $\mathcal P$ in $\binom{[w^*]}{\leq h^*(\mathcal P)}$.* We stress that the two notions defined above are different from the usual height and width of $\mathcal P$, that is, from the size of the biggest chain and antichain, respectively. It is easy to see that the height of $\mathcal P$ is always at most $h^*(\mathcal P)+1$, and that equality can happen (e.g. for a chain), but that is not always the case. Indeed, if $\mathcal P$ is the butterfly poset (Figure [\[fig:butterfly\]](#fig:butterfly){reference-type="ref" reference="fig:butterfly"}), then the height of $\mathcal P$ is 2 and its cube-height is 3: in any hypercube, the first 3 layers are butterfly-free. Similarly, the width and the cube-width can be very different. For example, if $\mathcal P$ is a chain of size $k$, then its width is 1, but its cube-width is $k-1$. Cube-width is not even a monotone property. For example, the antichain of size $\binom{k}{k/2}$ has cube-height 1 and cube-width $\binom{k}{k/2}$, but adding a chain of length $k/2$ which is less than all elements of the antichain gives a poset with cube-height $k/2$ and cube-width $k$. It is important to remark that the cube-width is *not* the minimal $n$ for which the poset can be embedded in $Q_n$. Indeed, the cube-width of an antichain of size 20 is 20, but $Q_6$ contains an antichain of size 20, namely the middle layer. We now bound the cube-height and cube-width in terms of the size of the poset. **Lemma 3**. *For any poset $\mathcal P$, we have that $h^*(\mathcal P)\leq |\mathcal P|-1$. [\[lem:height\]]{#lem:height label="lem:height"}* We remark that the inequality in this lemma is tight, since a chain on $k$ elements has cube-height $k-1$. *Proof of Lemma [\[lem:height\]](#lem:height){reference-type="ref" reference="lem:height"}.* We prove that any poset $\mathcal{P}$ on $k$ elements embeds in $\binom{[n]}{\leq k-1}$ for all $n\geq k$ by induction on $k$. The base case $k=1$ is trivially true since the cube-height of a poset with 1 element is $0$. Let $k\geq 2$ and assume the claim is true for all posets of size less than $k$. Let $\mathcal P=(\{p_1,\dots,p_k\},\preceq)$, and suppose that $n\geq k$. We show that $\mathcal P$ appears as an induced poset in $\binom{[n]}{\leq k-1}$. Suppose first that $\mathcal{P}$ has a unique maximal element. After renumbering the elements as necessary, we may assume that $p_k$ is the unique maximal element of $\mathcal P$. In this case, using the induction hypothesis, we find sets $A_1,\dots, A_{k-1}\in \binom{[k-1]}{\leq k-2}$ such that they induce a copy of the poset $\mathcal{P}\setminus \{p_k\}$. Now let $A_k=[k-1]$ and observe that $A_1,\dots,A_k$ induce a copy of $\mathcal{P}$ in $\binom{[n]}{\leq k-1}$. Indeed, $A_i \subsetneq A_k$ for all $i\leq k-1$ since $A_k$ has size $k-1$, while $|A_i|\leq k-2$ for all $i\leq k$. Suppose now that $\mathcal{P}$ does not contain a unique maximal element. We construct the sets $A_1,\dots,A_k$ as follows: for any $i,j \in [k]$, $i\in A_j$ if and only if $p_i\preceq p_j$. We observe that all constructed sets are subsets of $[k]\subseteq [n]$ and the size of each $A_i$ is the number elements less than or equal to $p_i$ (including $p_i$), which is at most $k-1$ since $\mathcal{P}$ has no unique maximal element. It remains to argue that $\{A_1,\dots, A_k\}$ induces a copy of $\mathcal P$ in $Q_n$. If $p_i\not \preceq p_j$, then $A_i\not \subseteq A_j$ since $i\in A_i\setminus A_j$. On the other hand, if $p_i\preceq p_j$, then $\ell \in A_i$ implies $p_\ell\preceq p_i$, which implies $p_\ell\preceq p_j$ by transitivity. Therefore, $\ell \in A_j$, which shows that $A_i\subseteq A_j$, as required. ◻ Note that the proof above gives a simple algorithm for constructing an embedding. In the lemma above, we embedded $\mathcal P$ into $Q_{|\mathcal P|}$. This cannot be improved in general, as seen in the following example. Let $\mathcal P_t$ be the poset consisting of $t$ antichains $\mathcal A_1,\dots,\mathcal A_t$ of size $2$, where we further impose that any element of $\mathcal A_i$ is less than any element of $\mathcal A_j$ for all $i<j$. Now, $2t=|\mathcal P_t|$ and by induction on $t$ it follows that $\mathcal P_t$ does not embed into $Q_{2t-1}$. Indeed, since everything below one of these antichains is a subset of both of its elements, each $\mathcal A_i$ must use at least two new elements of the ground set. Thus, this poset can be embedded in $Q_{2t}$, but not in $Q_{2t-1}$. We say that a collection of sets $A_1,\dots,A_k\subseteq [n]$ forms an *optimal cube-height embedding of $\mathcal P$* if they are pairwise distinct and induce a copy of $\mathcal{P}$ in $\binom{[n]}{\leq h^*(\mathcal P)}$. By the definition of $h^*(\mathcal P)$ such an embedding exists, and its ground set is $A_1\cup \dots \cup A_k$, which has size at most $h^*(\mathcal{P})k$, thus we immediately get the following corollary. **Corollary 4**. *For any poset $\mathcal{P}$, $w^*(\mathcal{P})\leq h^*(\mathcal{P})|\mathcal{P}|\leq |\mathcal{P}|^2$.* This corollary can immediately be strengthened by noting that we only need to take the union over the maximal elements in the embedding, so $w^*(\mathcal{P})$ is bounded by $h^*(\mathcal{P})$ times the number of maximal elements. Since the number of maximal elements is bounded by the size of the largest antichain in $\mathcal{P}$, denoted by $w(\mathcal P)$, this gives the following bound $$w^*(\mathcal{P})\leq h^*(\mathcal{P})w(\mathcal{P}).$$ Whilst Corollary [Corollary 4](#lem:width_easy){reference-type="ref" reference="lem:width_easy"} is enough for us to prove that $\mathop{\mathrm{sat*}}(n, \mathcal{P}) = O(n^{|\mathcal{P}|^2 - 1})$, proving Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"} requires a stronger bound on $w^*(\mathcal{P})$, which is given by the following lemma. **Lemma 5**. *For any poset $\mathcal{P}$, we have that $w^*(\mathcal{P}) \leq |\mathcal{P}|^2/4 + 2$.* In order to prove Lemma [Lemma 5](#lem:width){reference-type="ref" reference="lem:width"} we will make use of Lemma 2.2 from [@freschi2022induced], which we state below for completeness. **Lemma 6** ([@freschi2022induced], Lemma 2.2). *Let $\mathcal{F} \subseteq 2^{[n]}$ be such that for every $i \in [n]$ there exist two elements $A, B \in \mathcal{F}$ such that $A \setminus B = \{i\}$. Then $|\mathcal{F}| \geq 2\sqrt{n - 2}$.* The next proposition tells us that any optimal cube-height embedding has the property stated in Lemma [Lemma 6](#lemma:treglown){reference-type="ref" reference="lemma:treglown"}. We remark that this property in itself may be of independent interest, as explained in the final section. **Proposition 7**. *Let $\mathcal P$ be a poset and let $A_1,\dots,A_k\in \binom{[w^*(\mathcal{P})]}{\leq h^*(\mathcal{P})}$ be distinct sets that induce a copy of $\mathcal P$. Then for all $a\in [w^*(\mathcal{P})]$, there exist $i,j\in [k]$ such that $A_i\setminus A_j =\{a\}$.* *Proof.* Suppose there exists $a \in [w^*(\mathcal{P})]$ such that there does not exist $i,j \in [k]$ with $A_i\setminus A_j =\{a\}$. By relabelling as necessary, we may assume that $a = w^*(\mathcal{P})$, which we denote by $w^*$ for clarity. We now replace $A_i$ by $A_i \setminus \{w^*\}$ for all $i\leq k$. This new family lives in $\binom{[w^*-1]}{\leq h^*(\mathcal P)}$ and we claim it still forms a copy of $\mathcal{P}$. First, notice that we do not decrease the size of the family: for that to happen there would have to be distinct $A_i, A_j$ such that $A_j = A_i \cup \{w^*\}$, but that would immediately imply $A_j \setminus A_i = \{w^*\}$, a contradiction. We are left to show that comparability and incomparability relations are preserved. Let $A_i, A_j$ be such that $A_i \subseteq A_j$. Then $A_i\setminus \{w^*\} \subseteq A_j \setminus \{w^*\}$, as required. Finally, let $A_i$ and $A_j$ be incomparable, and assume that $A_i\setminus \{w^*\} \subseteq A_j\setminus \{w^*\}$. This implies that $w^*\in A_i$ and $w^*\notin A_j$, and consequently $A_i\setminus A_j=\{w^*\}$, a contradiction. Therefore, $A_i\setminus\{w^*\}$ and $A_j\setminus\{w^*\}$ are incomparable, and we have indeed shown that the new family forms an induced copy of $\mathcal P$. However, this new family lives in $\binom{[w^*-1]}{\leq h^*(\mathcal P)}$, contradicting the definition of $w^*$. ◻ We are now ready to prove the stronger upper bound on $w^*(\mathcal P)$. *Proof of Lemma [Lemma 5](#lem:width){reference-type="ref" reference="lem:width"}.* Suppose $\mathcal F=\{A_1, \dots, A_k\}$ forms an optimal cube-height embedding of $\mathcal{P}$ in $Q_{w^*(\mathcal P)}$. By Proposition [Proposition 7](#prop:property){reference-type="ref" reference="prop:property"}, $\mathcal{F}\subseteq 2^{[w^*(\mathcal{P})]}$ is a family of sets such that, for every $a \in [w^*(\mathcal{P})]$, there exist two sets $A_i, A_j \in \mathcal{F}$ with $A_j \setminus A_i = \{a\}$. Lemma [Lemma 6](#lemma:treglown){reference-type="ref" reference="lemma:treglown"} then implies that $|\mathcal{P}| = |\mathcal{F}| \geq 2 \sqrt{w^*(\mathcal{P}) - 2}$, and rearranging $w^*({\mathcal P})\leq |\mathcal P|^2/4+2$. ◻ # Proof of the main result In this section we prove our main result, Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"}. Given a poset $\mathcal{P}$ and $n$ large enough, we will construct a $\mathcal P$-saturated family in $Q_n$ of size at most $2n^{w^*(\mathcal{P}) - 1}$ which, combined with the bound on the cube-width from the previous section, achieves the claimed result. *Proof of Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"}.* Let $h^*=h^*(\mathcal P)$, $w^*=w^*(\mathcal P)$, and assume $n\geq2 w^*$. Let $\mathcal F_0$ be the family consisting of the first $h^*$ layers, or in other words, all the elements of size at most $h^*-1$. By the definition of the cube-height, the family $\mathcal F_0$ does not contain an induced copy of $\mathcal P$. We now extend this family to a $\mathcal P$-saturated family in an arbitrary fashion. Let $\mathcal F$ be this resulting family. The crucial property of this family is the following. **Claim 8**. *The VC-dimension of $\mathcal{F}$ is less than $w^*$.* *Proof.* Suppose towards a contradiction that $\mathcal{F}$ shatters a set $S$ of size $w^*$. By definition this means that $\mathcal L=\{A\cap S: A\in \mathcal F\}$ is the power set of $S$, and it is isomorphic to $Q_{w^*}$. Since $w^*$ is the cube-width of $\mathcal P$, we can find a copy of $\mathcal P$ in $\mathcal L$ such that all sets have size at most $h^*$. For simplicity, we call this copy $\mathcal P$. Let $M_1,\dots, M_s$ be the maximal elements of $\mathcal P$, which are subsets of $S$ by construction, and let $\mathcal P'=\mathcal P\setminus\{M_1,\dots, M_s\}$. Since we have removed all the maximal elements of $\mathcal P$, the height of $\mathcal{P}'$ is less than that of $\mathcal{P}$ (i.e. $h^*(\mathcal P') \leq h^*(\mathcal P) - 1$), and $\mathcal{P}'$ is embedded in the first $h^*$ layers. Hence, the subposet $\mathcal P'$ is contained in $\mathcal F_0\subseteq \mathcal F$. Since each $M_i$ is a subset of $S$, we can find $A_i\in\mathcal F$ such that $A_i\cap S=M_i$ for all $i\leq s$. We now show that $\mathcal P'\cup\{A_1,\dots,A_s\}$ is an induced copy of $\mathcal P$ in $\mathcal F$, which will yield the desired contradiction. First, if $B\in\mathcal P'$ is incomparable to $M_i$, then $B$ is also incomparable to $A_i$. This is because if $B$ is a subset of $A_i$, then it is also a subset of $A_i\cap S=M_i$, a contradiction. Conversely, if $B\in\mathcal P'$ is a subset of $M_i=A_i\cap S$, then it is a subset of $A_i$ too. We also have that $A_i$ and $A_j$ are incomparable for $i\neq j$ as they are incomparable when restricted to $S$. Finally, $A_i$ can never be a subset of $B\in \mathcal{P}'$, since $A_i\cap S=M_i$ is not a subset of $B\cap S=B$. We conclude that $\mathcal P'\cup\{A_1,\dots,A_s\}$ is an induced copy of $\mathcal P$ in $\mathcal F$. This gives a contradiction, proving that the VC-dimension of $\mathcal{F}$ is strictly less than $w^*$, as desired. ◻ Combining Lemma [\[lem:ss\]](#lem:ss){reference-type="ref" reference="lem:ss"} and Claim [Claim 8](#claim:vc){reference-type="ref" reference="claim:vc"}, we conclude that, as $n\geq 2w^*$, $$\text{sat}^*(n,\mathcal P)\leq|\mathcal F|\leq \displaystyle\sum_{i=0}^{w^*-1} \binom{n}{i}\leq w^*\frac{n^{w^*-1}}{(w^*-1)!} \leq 2n^{w^*-1}.$$ Here we have used that $\frac{m}{(m-1)!}\leq2$ for all $m \in \mathbb{N}$, and that, since $n\geq 2w^*$, the largest binomial coefficient in the above sum is $\binom{n}{w^*-1}$. Finally, Lemma [Lemma 5](#lem:width){reference-type="ref" reference="lem:width"} tells us that $w^* \leq |\mathcal P|^2/4+2$, which proves Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"}. ◻ # Concluding remarks and further work A first very natural question is: how small can the cube-width be? The antichain shows that $w^*(\mathcal{P})$ may be as large as $|\mathcal{P}|$. However, for all the posets we have considered, the cube-width is always at most the size of the poset. We conjecture that this has to be true in general. **Conjecture 9**. *For any finite poset $\mathcal P$, $w^*(\mathcal P)\leq |\mathcal P|$. [\[conj:width\]]{#conj:width label="conj:width"}* Since we proved that $\mathop{\mathrm{sat*}}(n,\mathcal{P}) = O(n^{w^*(\mathcal{P}) - 1})$, Conjecture [\[conj:width\]](#conj:width){reference-type="ref" reference="conj:width"} would imply that $\text{sat}^*(n, \mathcal P)=O(n^{|\mathcal P| - 1})$. That upper bound seems the natural threshold for our VC dimension approach and indeed our construction may yield families of such a size (e.g. for the chain). To conclude the paper, we expand on perhaps one of the most surprising phenomenon we observed in our work. We say that a family $\mathcal{F}\subsetneq Q_n$ *separates* $[n]$ if for every $i \in [n]$ there exist two sets $A$ and $B$ in $\mathcal F$ such that $A\setminus B=\{i\}$. Freschi, Piga, Sharifzadeh and Treglown [@freschi2022induced] showed that if the saturation number of a poset $\mathcal{P}$ is unbounded, then any induced $\mathcal P$-saturated family separates $[n]$ (and therefore is of size $\Omega(\sqrt{n})$). On the other hand, in Proposition [Proposition 7](#prop:property){reference-type="ref" reference="prop:property"}, we proved that every optimal cube-height embedding separates its ground set. We also note that Keszegh, Lemons, Martin, Pálvölgyi and Patkós [@keszegh2021induced] arrived at their $\log_2 (n)$ lower bound via a weaker 'separability' property of $\mathcal P$-saturated families. This allowed them to build a complete graph on $n$ vertices covered by complete bipartite graphs, each of these corresponding to exactly one set in the family. Their lower bound then follows since $\log_2(n)$ is the biclique cover number for the complete graph on $n$ vertices. It seems that poset saturation and separability properties are in some sense deeply interlinked. In view of this, we feel that improvements towards Conjecture [\[conj:width\]](#conj:width){reference-type="ref" reference="conj:width"} may yield ideas for improvements on the general $\sqrt{n}$ lower bound, or vice versa. Paul Bastide, [LaBRI, Université de Bordeaux, Bordeaux, France]{.smallcaps} *Email address:* `paul.bastide@ens-rennes.fr` Carla Groenland, [Institute of Applied Mathematics, Technische Universiteit Delft (TU Delft), 2628 CD Delft, Netherlands.]{.smallcaps} *Email address:* `c.e.groenland@tudelft.nl` Maria-Romina Ivan, [Magdalene College, University of Cambridge, Cambridge, CB3 0AG, UK and Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK.]{.smallcaps} *Email address:* `mri25@dpmms.cam.ac.uk` Tom Johnston, [School of Mathematics, University of Bristol, Bristol, BS8 1UG, UK and Heilbronn Institute for Mathematical Research, Bristol, UK.]{.smallcaps} *Email address:* `tom.johnston@bristol.ac.uk`

arxiv_math

--- abstract: | Approximating differential operators defined on two-dimensional surfaces is an important problem that arises in many areas of science and engineering. Over the past ten years, localized meshfree methods based on generalized moving least squares (GMLS) and radial basis function finite differences (RBF-FD) have been shown to be effective for this task as they can give high orders of accuracy at low computational cost, and they can be applied to surfaces defined only by point clouds. However, there have yet to be any studies that perform a direct comparison of these methods for approximating surface differential operators (SDOs). The first purpose of this work is to fill that gap. For this comparison, we focus on an RBF-FD method based on polyharmonic spline kernels and polynomials (PHS+Poly) since they are most closely related to the GMLS method. Additionally, we use a relatively new technique for approximating SDOs with RBF-FD called the tangent plane method since it is simpler than previous techniques and natural to use with PHS+Poly RBF-FD. The second purpose of this work is to relate the tangent plane formulation of SDOs to the local coordinate formulation used in GMLS and to show that they are equivalent when the tangent space to the surface is known exactly. The final purpose is to use ideas from the GMLS SDO formulation to derive a new RBF-FD method for approximating the tangent space for a point cloud surface when it is unknown. For the numerical comparisons of the methods, we examine their convergence rates for approximating the surface gradient, divergence, and Laplacian as the point clouds are refined for various parameter choices. We also compare their efficiency in terms of accuracy per computational cost, both when including and excluding setup costs. author: - Andrew M. Jones - Peter A. Bosler - Paul A. Kuberry - Grady B. Wright bibliography: - biber.bib title: Generalized moving least squares vs. radial basis function finite difference methods for approximating surface derivatives --- PDEs on surfaces ,Meshfree ,Meshless ,RBF-FD ,GMLS ,Polyharmonic spline 65D05 ,65D25 ,65M06 ,65M75 ,65N06 ,65N75 ,41A05 ,41A10 ,41A15 # Introduction {#sec:intro} The problem of approximating differential operators defined on two dimensional surfaces embedded in $\mathbb{R}^3$ arises in many multiphysics models. For example, simulating atmospheric flows with Eulerian or Lagrangian numerical methods requires approximating the surface gradient, divergence, and Laplacian on the two-sphere [@vallis_2006; @AMJ:numericalweather07; @AMJ:Fornberg15; @AMJ:Bosler14]. Similar surface differential operators (SDOs) on more geometrically complex surfaces appear in models of ice sheet dynamics [@Gowan2021], biochemical signaling on cell membranes [@Liue2104191118], morphogenesis [@stoop2015curvature], texture synthesis [@Mikkelsen2020Bump], and sea-air hydrodynamics [@banaerjee2004surfdiv]. Localized meshfree methods based on generalized moving least squares (GMLS) and radial basis function finite differences (RBF-FD) have become increasingly popular over the last ten years for approximating SDOs and solving surface partial differential equations (PDEs); see, for example, [@mahadevan2022metrics; @LiangZhao13; @TraskKuberry20; @SUCHDE20192789; @GrossEtAl20] for GMLS and [@LSW2016; @FlyerLehtoBlaiseWrightStCyr2012; @FlyerWrightFornberg; @SHANKAR2014JSC; @PIRET2016; @SHANKAR2018722; @petras2018rbf; @ShawThesis; @GUNDERMAN2020109256; @Wendland2020; @Alvarez2021; @wright2022mgm] for RBF-FD. These methods can be applied to surfaces defined by point clouds, without having to form a triangulation of the surface like surface finite element methods [@dziuk_elliott_2013] or a level-set representation of the surface like embedded finite element methods [@BertalmioEtAl2001]. Additionally, for the special case of the sphere, RBF-FD has been shown to be highly competitive with element based methods in terms of accuracy per degree of freedom [@FlyerLehtoBlaiseWrightStCyr2012; @GUNDERMAN2020109256; @FlyerWrightFornberg]. While there is one study dedicated to comparing GMLS and RBF-FD for approximating functions and derivatives in $\mathbb{R}^2$ and $\mathbb{R}^3$ [@Bayona2019], there are no studies that compare them for approximating SDOs. The first purpose of the present work is to fill this gap. The RBF-FD methods referenced above use different approaches [for]{style="color: black"} approximating SDOs, while the GMLS methods essentially use the same approach based on (weighted) polynomial least squares. To keep the comparison to GMLS manageable, we will limit our focus to an RBF-FD method based on polyharmonic spline (PHS) kernels augmented with polynomials (or PHS+Poly) since they are most closely related to GMLS [@Bayona2019]. Additionally, these RBF-FD methods are becoming more and more prevalent as they can give high orders of accuracy that are controlled by the augmented polynomial degree [@bayona2017role] and they do not require choosing a shape parameter, which can be computationally intensive to do in an automated way. The techniques for formulating SDOs also vary significantly in the RBF-FD methods referenced above, while the formulations used in GMLS are similar, being based on local coordinates to the surface. In this work, we limit our focus to the so-called tangent plane formulation with RBF-FD, as it provides a more straightforward technique for incorporating polynomials in RBF-FD methods than [@Alvarez2021; @LSW2016; @FlyerLehtoBlaiseWrightStCyr2012; @SHANKAR2014JSC; @PIRET2016; @SHANKAR2018722; @petras2018rbf; @Wendland2020] and is related to the local coordinate formulation used in GMLS (see below). Additionally, the comparison in [@ShawThesis] of several RBF-FD methods for approximating the surface Laplacian (Laplace-Beltrami operator) revealed the tangent plane approach to be the most computationally efficient in terms of accuracy per computational cost. The tangent plane method was first introduced by Demanet [@DEMANET2006] for approximating the surface Laplacian using polynomial based approximations. Suchde & Kuhnert [@SUCHDE20192789] generalized this method to other SDOs using polynomial weighted least squares. Shaw [@ShawThesis] (see also [@wright2022mgm]) was the first to use this method for approximating the surface Laplacian with RBF-FD and Gunderman et. al. [@GUNDERMAN2020109256] independently developed the method for RBF-FD specialized to the surface gradient and divergence on the unit two-sphere. The second purpose of the present work is to analytically compare the local coordinate formulation of SDOs used in GMLS to the tangent plane formulation and to show that these formulations are in fact identical when the tangent space for the surface is known exactly for the given point cloud. When the tangent space is unknown, which is generally the case for surface represented by point clouds, it must be approximated. There has been little attention given in the literature on RBF-FD methods for how to do these approximations; commonly it is assumed that they are computed by some separate techniques (e.g., [@SHANKAR2014JSC; @LSW2016; @Wendland2020]). However, for GMLS, these approximations are incorporated directly in the methods (e.g., [@LiangZhao13; @TraskKuberry20; @GrossEtAl20]). The third purpose of this work is use the ideas from GMLS to develop a new RBF-FD technique for approximating the tangent space directly using PHS+Poly. By combining this with the tangent plane method, we arrive at the first comprehensive PHS+Poly RBF-FD framework for approximating SDOs on point cloud surfaces. The GMLS and RBF-FD methods both use weighted combinations of function values over a local stencil of points to approximate SDOs. They also feature a parameter $\ell$ for controlling the degree of polynomial precision of the formulas. For the numerical comparisons of the methods, we investigate how the size of the stencils and the polynomial degree effect the convergence rates of the methods for approximating SDOs under refinement. We focus on approximations of the surface gradient, divergence, and Laplacian operators on two topologically distinct surfaces, the unit two-sphere and the torus, which are representative of a broad range of application domains. In the case of the sphere, we also study the convergence rates of the methods for different point sets, including icosahedral points that are popular in applications. Finally, we investigate the efficiency of the methods in terms of their accuracy versus computational cost, both when including and excluding setup costs. Our numerical results demonstrate that RBF-FD and GMLS give similar convergence rates for the same choice of polynomial degree $\ell$, but overall RBF-FD results in lower errors. We also show that the often-reported super convergence of GMLS for the surface Laplacian only happens for highly structured, quasi-uniform point sets, and when the point sets are more general (but still possibly quasi-uniform), this convergence rate drops to the theoretical rate. Additionally, we find that the errors for RBF-FD can be further reduced with increasing stencil sizes, but that this does not generally hold for GMLS, and the errors can actually deteriorate. Finally, we find that when setup costs are included, GMLS has an advantage in terms of efficiency, but if these are neglected then RBF-FD is more efficient. The remainder of the paper is organized as follows. In Section [\[sec:background\]](#sec:background){reference-type="ref" reference="sec:background"}, we provide some background and notation on stencil-based approximations and on surface differential operators. We follow this with a detailed overview of the GMLS and RBF methods in Section [3](#sec:gmls){reference-type="ref" reference="sec:gmls"} and [4](#sec:rbffd_lap){reference-type="ref" reference="sec:rbffd_lap"}, respectively. In particular, Section [\[sec:tangent_plane\]](#sec:tangent_plane){reference-type="ref" reference="sec:tangent_plane"} shows the equivalence of the local coordinate and tangent plane formulations of some SDOs, and Section [\[sec:rbffd_tangent_space\]](#sec:rbffd_tangent_space){reference-type="ref" reference="sec:rbffd_tangent_space"} introduces an RBF-FD method for approximating the tangent space. A comparison of some theoretical properties of the two methods is given in Section [\[sec:theoretical\]](#sec:theoretical){reference-type="ref" reference="sec:theoretical"}, while extensive numerical comparisons are given in Section [6](#sec:results){reference-type="ref" reference="sec:results"}. We end with some concluding remarks in Section [7](#sec:remarks){reference-type="ref" reference="sec:remarks"}. # Background and notation[\[sec:background\]]{#sec:background label="sec:background"} ## Stencils {#sec:stencils} The RBF-FD and GMLS methods both discretize SDOs by weighted combinations of function values over a local *stencil* of points. This makes them similar to traditional finite-difference methods, but the lack of a grid, a tuple indexing scheme, and inherent awareness of neighboring points requires that some different notation and concepts be introduced. In this section we review the stencil notation that will be used in the subsequent sections. Let $\mathbf{X}=\{\mathbf{x}_i\}_{i=1}^N$ be a global set of points (point cloud) contained in some domain $\Omega$. A *stencil of $\mathbf{X}$* is a subset of $n \leq N$ nodes of $\mathbf{X}$ that are close (see discussion below for what this means) to some point $\mathbf{x}_{\rm c}\in\Omega$, which is called the *stencil center*. In this work, the stencil center is some point from $\mathbf{X}$, so that $\mathbf{x}_{\rm c} = \mathbf{x}_i$, for some $1\leq i \leq N$, and this point is always included in the stencil. We denote the subset of points making up the stencil with stencil center $\mathbf{x}_i$ as $\mathbf{X}^i$ and allow the number of points in the stencil to vary with $\mathbf{x}_i$. To keep track [of]{style="color: black"} which points in $\mathbf{X}^i$ belong to $\mathbf{X}$, we use *index set* notation and let $\sigma^i$ denote the set of indices of the $1 < n_i \leq N$ points from $\mathbf{X}$ that belong to $\mathbf{X}^i$. Using this notation, we write the elements of the stencil as $\mathbf{X}^i = \{\mathbf{x}_j\}_{j\in \sigma^i}$. We also use the convention that the indices are sorted by the distance the stencil points are from the stencil center $\mathbf{x}_i$, so that the first element of $\sigma^i$ is $i$. With the above notation, we can define a general stencil-based approximation method to a given (scalar) linear differential operator $\mathcal{L}$. Let $u$ be a scalar-valued function defined on $\Omega$ that is smooth enough so that $\mathcal{L}u$ is defined for all $\mathbf{x}\in\Omega$. The approximation to $\mathcal{L}u$ at any $\mathbf{x}_i\in \mathbf{X}$ is given as $$\label{eq:stencils} \mathcal{L}u|_{\mathbf{x}= \mathbf{x}_i} \approx \sum_{j\in\sigma^i} c_{ij}u(\mathbf{x}_j).$$ The weights $c_{ij}$ are determined by the method of approximation, which in this study will be either GMLS or RBF-FD. These weights can be assembled into a sparse $N\times N$ "stiffness" matrix, similar to mesh-based methods. Vector linear differential operators (e.g., the gradient) can be similarly defined where [\[eq:stencils\]](#eq:stencils){reference-type="eqref" reference="eq:stencils"} is used for each component and $\mathcal{L}$ is the scalar operator for that component. There are two main approaches used in the meshfree methods literature for determining the stencil points, one based on $k$-nearest neighbors (KNN) and one based on ball searches. These are illustrated in Figure [2](#fig:search_algos){reference-type="ref" reference="fig:search_algos"} for a scattered point set $\mathbf{X}$ in the plane. The approach that uses KNN is straightforward since it amounts to simply choosing the stencil $\mathbf{X}^i$ as the subset of $n_i$ points from $\mathbf{X}$ that are closest to $\mathbf{x}_i$. The approach that uses ball searches is a bit more involved, so we summarize it in Algorithm [\[alg:epsilon_ball\]](#alg:epsilon_ball){reference-type="ref" reference="alg:epsilon_ball"}. Both methods attempt to select points such that the stencil satisfies polynomial unisolvency conditions (see the discussion in Section [\[sec:metric_terms\]](#sec:metric_terms){reference-type="ref" reference="sec:metric_terms"}). In this work, we use the method in Algorithm [\[alg:epsilon_ball\]](#alg:epsilon_ball){reference-type="ref" reference="alg:epsilon_ball"} since - it is better for producing stencils with symmetries when $\mathbf{X}$ is regular, which can be beneficial for improving the accuracy of the approximations; - it is more natural to use with the weighting kernel inherent to GMLS; and - it produces stencils that are not biased in one direction when the spacing of the points in $X$ are anisotropic. To measure distance in the ball search, we use the standard Euclidean distance measured in $\mathbb{R}^3$ rather than distance on the surface since this is simple to compute for any surface. We also use a $k$-d tree to efficiently implement [the]{style="color: black"} method. Finally, the choice of parameters we use in Algorithm [\[alg:epsilon_ball\]](#alg:epsilon_ball){reference-type="ref" reference="alg:epsilon_ball"} are discussed in Section [\[sec:gmls_stencils_weights\]](#sec:gmls_stencils_weights){reference-type="ref" reference="sec:gmls_stencils_weights"}. **Input:** Point cloud $\mathbf{X}$; stencil center $\mathbf{x}_{\rm c}$; number initial stencil points $n$; radius factor $\tau \geq 1$ **Output:** Indices $\sigma^{\rm c}$ in $\mathbf{X}$ for the stencil center $\mathbf{x}_{\rm c}$ Find the $n$ nearest neighbors in $\mathbf{X}$ to $\mathbf{x}_c$, using the Euclidean distance Compute the max distance $h_{\max}$ between $\mathbf{x}_{\rm c}$ and its $n$ nearest neighbors Find the indices $\sigma^{\rm c}$ of the points in $\mathbf{X}$ contained in the ball of radius $\tau h_{\max}$ centered at $\mathbf{x}_{\rm c}$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- {#fig:search_algos width="40%"} {#fig:search_algos width="40%"} \(a\) KNN \(b\) Ball search ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ## Surface differential operators in local coordinates {#sec:local_coords} Here we review some differential geometry concepts that will be used in the subsequent sections. We refer the reader to the books [@Walker15; @ONeill2006; @koenderink1990solid] for a thorough discussion of these concepts and the derivations of what follows. We assume that $\mathcal{M}\subset\mathbb{R}^3$ is a regular surface and let $T_{\mathbf{x}}\mathcal{M}$ denote the set of all vectors in $\mathbb{R}^3$ that are tangent to $\mathcal{M}$ at $\mathbf{x}\in\mathcal{M}$ (i.e., the tangent space to $\mathcal{M}$ at $\mathbf{x}$). This assumption implies that for each point $\mathbf{x}\in\mathcal{M}$ there exists a local parameterization in $T_{\mathbf{x}}\mathcal{M}$ of a neighborhood (or patch) of $\mathcal{M}$ containing $\mathbf{x}$ of the form $$\begin{aligned} \mathbf{f}(\hat{x},\hat{y}) = (\hat{x},\hat{y},f(\hat{x},\hat{y})), \label{eq:monge}\end{aligned}$$ where $\hat{x}$, $\hat{y}$ are local coordinates for $T_{\mathbf{x}}\mathcal{M}$, and $f$ is a smooth function for the "height" of the surface patch over $T_{\mathbf{x}}\mathcal{M}$ [@koenderink1990solid]. This local parametric representation of a surface is called a Monge patch or Monge form [@ONeill2006] and is illustrated for a bumpy sphere surface in Figure [4](#fig:projection){reference-type="ref" reference="fig:projection"}. As we see below, it is particularly well suited for computing SDOs. Using the parameterization [\[eq:monge\]](#eq:monge){reference-type="eqref" reference="eq:monge"}, the local metric tensor $G$ about $\mathbf{x}$ for the surface is given as $$\begin{aligned} G = \begin{bmatrix} \partial_{\hat{x}} \mathbf{f}\cdot \partial_{\hat{x}} \mathbf{f}& \partial_{\hat{x}}\mathbf{f}\cdot \partial_{\hat{y}} \mathbf{f}\\ \partial_{\hat{y}} \mathbf{f}\cdot \partial_{\hat{x}} \mathbf{f}& \partial_{\hat{y}} \mathbf{f}\cdot \partial_{\hat{y}} \mathbf{f} \end{bmatrix}= \begin{bmatrix} 1 + (\partial_{\hat{x}} f)^2 & (\partial_{\hat{x}} f) (\partial_{\hat{y}}f) \\ (\partial_{\hat{x}} f) (\partial_{\hat{y}}f) & 1 + (\partial_{\hat{y}} f)^2 \end{bmatrix}. \label{eq:metric_tensor}\end{aligned}$$ Letting $g^{ij}$ denote the $(i,j)$ entry of $G^{-1}$, the surface gradient operator locally about $\mathbf{x}$ is given as $$\begin{aligned} \widehat{\nabla}_{\mathcal{M}} = (\partial_{\hat{x}} \mathbf{f}) \left(g^{11} \partial_{\hat{x}} + g^{12} \partial_{\hat{y}}\right) + (\partial_{\hat{y}} \mathbf{f}) \left(g^{21} \partial_{\hat{x}} + g^{22} \partial_{\hat{y}}\right). \label{eq:intrinsic_grad_rot}\end{aligned}$$ However, this is the surface gradient with respect to the horizontal $\hat{x}\hat{y}$-plane (see Figure [4](#fig:projection){reference-type="ref" reference="fig:projection"} (b)), and subsequently needs to be rotated so that it is with respect to $T_{\mathbf{x}}\mathcal{M}$ in its original configuration. If $\boldsymbol{\xi}^1$ and $\boldsymbol{\xi}^2$ are orthonormal vectors that span $T_{\mathbf{x}}\mathcal{M}$ and $\boldsymbol{\eta}$ is the unit outward normal to $\mathcal{M}$ at $\mathbf{x}$, then the surface gradient in the correct orientation is given as $$\begin{aligned} {\nabla}_{\mathcal{M}} = \underbrace{\begin{bmatrix} \boldsymbol{\xi}^1 & \boldsymbol{\xi}^2 & \boldsymbol{\eta}\end{bmatrix}}_{\displaystyle R} \widehat{\nabla}_{\mathcal{M}}. \label{eq:intrinsic_grad}\end{aligned}$$ Using this result, the surface divergence of a smooth vector $\mathbf{u}\in T_{\mathbf{x}}\mathcal{M}$ can be written as $$\begin{aligned} \nabla_{\mathcal{M}} \cdot \mathbf{u}= \left(g^{11} \partial_{\hat{x}} + g^{12} \partial_{\hat{y}}\right) (\partial_{\hat{x}} \mathbf{f})^T R^{T}\mathbf{u}+ \left(g^{21} \partial_{\hat{x}} + g^{22} \partial_{\hat{y}}\right)(\partial_{\hat{y}} \mathbf{f})^T R^{T}\mathbf{u} \label{eq:intrinsic_div}\end{aligned}$$ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Illustration of a Monge patch parameterization for a local neighborhood of a regular surface $\mathcal{M}$ in 3D. (a) Entire surface (in gray) together with the tangent plane (in cyan) for a point $\mathbf{x}_{\rm c}$ where the Monge patch is constructed (i.e., $T_{\mathbf{x}_{\rm c}}\mathcal{M}$); red spheres mark a global point cloud $\mathbf{X}$ on the surface. (b) Close-up view of the Monge patch parameterization, together with the points from a stencil $\mathbf{X}_{\rm c}$ (red spheres) formed from $\mathbf{X}$ and the projection of the stencil to the tangent plane (blue spheres); the stencil center $\mathbf{x}_{\rm c}$ is at the origin of the axes for the $\hat{x}\hat{y}$-plane and is marked with a violet sphere. [\[fig:projection\]]{#fig:projection label="fig:projection"}](figures/ProjectionIllustrationSurface1.pdf){#fig:projection width="40%"} ![Illustration of a Monge patch parameterization for a local neighborhood of a regular surface $\mathcal{M}$ in 3D. (a) Entire surface (in gray) together with the tangent plane (in cyan) for a point $\mathbf{x}_{\rm c}$ where the Monge patch is constructed (i.e., $T_{\mathbf{x}_{\rm c}}\mathcal{M}$); red spheres mark a global point cloud $\mathbf{X}$ on the surface. (b) Close-up view of the Monge patch parameterization, together with the points from a stencil $\mathbf{X}_{\rm c}$ (red spheres) formed from $\mathbf{X}$ and the projection of the stencil to the tangent plane (blue spheres); the stencil center $\mathbf{x}_{\rm c}$ is at the origin of the axes for the $\hat{x}\hat{y}$-plane and is marked with a violet sphere. [\[fig:projection\]]{#fig:projection label="fig:projection"}](figures/ProjectionIllustrationSurface2.pdf){#fig:projection width="40%"} \(a\) \(b\) --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The surface Laplacian operator locally about $\mathbf{x}$ is given as $$\begin{aligned} \Delta_{\mathcal{M}}= \frac{1}{\sqrt{|g|}} \biggl(& \partial_{\hat{x}}\left(\sqrt{|g|}g^{11}\partial_{\hat{x}}\right) + \partial_{\hat{x}}\left(\sqrt{|g|}g^{12}\partial_{\hat{y}}\right) + \\ & \partial_{\hat{y}}\left(\sqrt{|g|}g^{21}\partial_{\hat{x}}\right) + \partial_{\hat{y}}\left(\sqrt{|g|}g^{22}\partial_{\hat{y}} \right)\biggr), \end{aligned} \label{eq:intrinsic_lap}$$ where $|g|=\det(G)$. This operator is invariant to rotations of the surface in $\mathbb{R}^3$, so no subsequent modifications of [\[eq:intrinsic_lap\]](#eq:intrinsic_lap){reference-type="eqref" reference="eq:intrinsic_lap"} are necessary. # GMLS using local coordinates {#sec:gmls} The [formulation]{style="color: black"} of GMLS [on a manifold]{style="color: black"} was introduced by Liang & Zhao [@LiangZhao13] and further refined by Trask, Kuberry, and collaborators [@TraskKuberry20; @GrossEtAl20]. It uses local coordinates to approximate SDOs as defined in [\[eq:intrinsic_grad\]](#eq:intrinsic_grad){reference-type="eqref" reference="eq:intrinsic_grad"}--[\[eq:intrinsic_lap\]](#eq:intrinsic_lap){reference-type="eqref" reference="eq:intrinsic_lap"} and requires a method to also approximate the metric terms. Both approximations are computed for each $\mathbf{x}_i \in \mathbf{X}\subset \mathcal{M}$ using GMLS over a local stencil of points $\mathbf{X}^i\subset\mathbf{X}$. Below we give a brief overview of the method assuming that the tangent/normal vectors for the surface are known for each $\mathbf{x}_i\in\mathbf{X}$. We then discuss a method for approximating these that is used in the Compadre Toolkit [@AMJ:Compadre], which we use in the numerical experiments. We present the GMLS method through the lens of derivatives of MLS approximants as we feel this makes the analog to RBF-FD clearer, it is also closer to the description from [@LiangZhao13]. Other derivations of GMLS are based on weighted least squares approximants of general linear functionals given at some set of points, e.g. [@Wendland:2004; @Mirzaei11; @trask2017high]. However, both techniques produce the same result in the end [@Mirzaei11]. For a more thorough discussion of MLS approximants, see for example [@fasshauer2007meshfree ch. 22] and the references therein. ## Approximating the metric terms[\[sec:metric_terms\]]{#sec:metric_terms label="sec:metric_terms"} The metric terms are approximated from an MLS reconstruction of the Monge patch of $\mathcal{M}$ centered at each target point $\mathbf{x}_i$ using a local stencil of $n_i$ points $\mathbf{X}^i\subset\mathbf{X}$. This procedure is illustrated in Figure [4](#fig:projection){reference-type="ref" reference="fig:projection"} and can be described as follows. First, the stencil $\mathbf{X}^i$ is expressed in the form of [\[eq:monge\]](#eq:monge){reference-type="eqref" reference="eq:monge"} (i.e., $(\hat{x}_j,\hat{y}_j,f_j)$, $j\in\sigma^i$), where $(\hat{x}_j,\hat{y}_j)$ are the coordinates for the stencil points in $T_{\mathbf{x}_i}\mathcal{M}$, and $f_j = f(\hat{x}_j,\hat{y}_j)$ are samples of the surface as viewed from the $\hat{x}\hat{y}$-plane. These can be computed explicitly as $$\begin{aligned} \begin{bmatrix} \hat{x}_j \\ \hat{y}_j \\ f_j \end{bmatrix} = {\underbrace{\begin{bmatrix} \boldsymbol{\xi}_i^1 & \boldsymbol{\xi}_i^2 & \boldsymbol{\eta}_i\end{bmatrix}^T}_{\displaystyle R_i^T}} (\mathbf{x}_j - \mathbf{x}_i), \label{eq:projection}\end{aligned}$$ where $\boldsymbol{\xi}_i^{1}$ and $\boldsymbol{\xi}_i^{2}$ are orthonormal vectors that span $T_{\mathbf{x}_i}\mathcal{M}$ and $\boldsymbol{\eta}_i$ is the unit normal to $\mathcal{M}$ at $\mathbf{x}_i$. To simplify the notation that follows, we let $\mathbf{\hat{x}}_j=(\hat{x}_j,\hat{y}_j)$ and $\hat{\mathbf{X}}^i = \{\mathbf{\hat{x}}_j\}_{j\in\sigma^i}$ denote the projection of the stencil $\mathbf{X}^i$ to $T_{\mathbf{x}_i}\mathcal{M}$. Note that for convenience in what comes later we have shifted the coordinates so that the center of the projected stencil is $\mathbf{\hat{x}}_i = (0,0)$. In the second step, the approximate Monge patch at $\mathbf{x}_i$ is constructed from a MLS approximant of the data $(\mathbf{\hat{x}}_j,f_j)$, $j\in\sigma^i$, which can be written as $$\begin{aligned} q(\mathbf{\hat{x}}) = \sum_{k=1}^{L} b_k(\mathbf{\hat{x}}) p_k(\mathbf{\hat{x}}), \label{eq:mls_interp}\end{aligned}$$ where $\{p_1,\ldots,p_L\}$ is a basis for $\mathbb{P}^2_{\ell}$ (the space of bivariate polynomials of degree $\ell$) and $L = \text{dim}(\mathbb{P}^2_{\ell})=(\ell+1)(\ell+2)/2$ is the dimension of this space. The coefficients $b_k(\mathbf{\hat{x}})$ of the approximant are determined from the data according to the weighted least squares problem $$\begin{aligned} \underline{b}^{*}(\mathbf{\hat{x}}) = \mathop{\mathrm{argmin}}_{\underline{b}\in\mathbb{R}^L} \sum_{j\in\sigma^i} w_{\rho}(\mathbf{\hat{x}}_j,\mathbf{\hat{x}}) (q(\mathbf{\hat{x}}_j) - f_j)^2 = \mathop{\mathrm{argmin}}_{\underline{b}\in\mathbb{R}^L} \|W_{\rho}(\mathbf{\hat{x}})^{1/2}(P\underline{b}- \underline{f})\|_2^2, \label{eq:wls_fd_approx_system}\end{aligned}$$ where $w_{\rho}:\mathbb{R}^2\times\mathbb{R}^2\rightarrow\mathbb{R}^{\geq 0}$ is a weight kernel that depends on a support parameter $\rho$, $W_{\rho}(\mathbf{\hat{x}})$ is the $n_i\times n_i$ diagonal matrix $W_{\rho}(\mathbf{\hat{x}}) = \text{diag}(w_{\rho}(\mathbf{\hat{x}}_j,\mathbf{\hat{x}}))$, and $P$ is the $n_i\times L$ Vandermonde-type matrix $$\begin{aligned} P = \begin{bmatrix} p_1(\mathbf{\hat{x}}_j) & p_2(\mathbf{\hat{x}}_j) & \cdots & p_L(\mathbf{\hat{x}}_j) \end{bmatrix},\; j\in\sigma^i \label{eq:vandermonde}\end{aligned}$$ Here we use underlines to denote vectors (i.e., $\underline{b}$ and $\underline{f}$ denote vectors containing coefficients and data from [\[eq:wls_fd_approx_system\]](#eq:wls_fd_approx_system){reference-type="eqref" reference="eq:wls_fd_approx_system"}, respectively). Note that the coefficients $b_k$ depend on $\mathbf{\hat{x}}$ because the kernel $w_{\rho}$ depends on $\mathbf{\hat{x}}$ (this gives origin to the term "moving" in MLS). We discuss the selection of the stencils and weight[ing]{style="color: black"} kernel below, but for now it is assumed that $n_i > L$ and $\mathbf{X}^i$ is unisolvent on the space $\mathbb{P}^2_{\ell}$ (i.e., $P$ is full rank), so that [\[eq:wls_fd_approx_system\]](#eq:wls_fd_approx_system){reference-type="eqref" reference="eq:wls_fd_approx_system"} has a unique solution. The MLS approximant $q$ is used in place of $f$ in the Monge patch [\[eq:monge\]](#eq:monge){reference-type="eqref" reference="eq:monge"} and it is used to approximate the metric terms in [\[eq:intrinsic_grad\]](#eq:intrinsic_grad){reference-type="eqref" reference="eq:intrinsic_grad"}--[\[eq:intrinsic_lap\]](#eq:intrinsic_lap){reference-type="eqref" reference="eq:intrinsic_lap"}. To compute these terms, various derivatives need to be approximated at the projected stencil center $\mathbf{\hat{x}}_i$. Considering, for example, $\partial_{\hat{x}} q$, the approximation is computed as follows: $$\begin{aligned} \partial_{\hat{x}} q\bigr|_{\mathbf{\hat{x}}_i} \approx \sum_{k=1}^{L} b_k^{*}(\mathbf{\hat{x}}_i) \partial_{\hat{x}}(p_k(\mathbf{\hat{x}}))\bigr|_{\mathbf{\hat{x}}_i}, \label{eq:diff_mls_interp}\end{aligned}$$ where $b_k^{*}(\mathbf{\hat{x}}_i)$ come from [\[eq:wls_fd_approx_system\]](#eq:wls_fd_approx_system){reference-type="eqref" reference="eq:wls_fd_approx_system"} with $\mathbf{\hat{x}}=\mathbf{\hat{x}}_i$. Other derivatives of metric terms in [\[eq:intrinsic_grad\]](#eq:intrinsic_grad){reference-type="eqref" reference="eq:intrinsic_grad"}--[\[eq:intrinsic_lap\]](#eq:intrinsic_lap){reference-type="eqref" reference="eq:intrinsic_lap"} are approximated in a similar way to [\[eq:diff_mls_interp\]](#eq:diff_mls_interp){reference-type="eqref" reference="eq:diff_mls_interp"}. We note if the standard monomial basis is used for $\{p_1,\ldots,p_L\}$, then by centering the projected stencil in [\[eq:projection\]](#eq:projection){reference-type="eqref" reference="eq:projection"} about the origin, only one of the derivatives of $p_k$ in [\[eq:diff_mls_interp\]](#eq:diff_mls_interp){reference-type="eqref" reference="eq:diff_mls_interp"} is non-zero when evaluated at $\mathbf{\hat{x}}_i$. Note that [\[eq:diff_mls_interp\]](#eq:diff_mls_interp){reference-type="eqref" reference="eq:diff_mls_interp"} is only an approximation of $\partial_{\hat{x}}q$ because it does not include the contribution of $\partial_{\hat{x}}(b_k^{*}(\mathbf{\hat{x}}))\big|_{\mathbf{\hat{x}}_i}$. This approximation is referred to as a "diffuse derivative" in the literature and is equivalent to the GMLS formulation of approximating derivatives [@Mirzaei11]. The term "GMLS derivatives" is preferred over "diffuse derivatives" to describe [\[eq:diff_mls_interp\]](#eq:diff_mls_interp){reference-type="eqref" reference="eq:diff_mls_interp"}, since the approximation is not diffuse or uncertain and has the same order of accuracy as the approximations that include the derivatives of the weight kernels [@mirzaei2016error]. ## Approximating SDOs {#sec:gmls_sdos} The procedure for approximating any of the SDOs in [\[eq:intrinsic_grad\]](#eq:intrinsic_grad){reference-type="eqref" reference="eq:intrinsic_grad"}--[\[eq:intrinsic_lap\]](#eq:intrinsic_lap){reference-type="eqref" reference="eq:intrinsic_lap"} is similar to the one for approximating the metric terms, but for this task we are interested in computing stencil weights as in [\[eq:stencils\]](#eq:stencils){reference-type="eqref" reference="eq:stencils"} instead of the value of a derivative at a point. Since these SDOs involve computing various partial derivatives with respect to $\hat{x}$ and $\hat{y}$, we can use [\[eq:diff_mls_interp\]](#eq:diff_mls_interp){reference-type="eqref" reference="eq:diff_mls_interp"} as a starting point for generating these stencil weights. If $\{u_j\}_{j\in\sigma^i}$ are samples of a function $u$ over the projected stencil $\hat{\mathbf{X}}^i$, then we can again approximate $\partial_{\mathbf{\hat{x}}}u\bigr|_{\mathbf{\hat{x}}=\mathbf{\hat{x}}_i}$ using [\[eq:diff_mls_interp\]](#eq:diff_mls_interp){reference-type="eqref" reference="eq:diff_mls_interp"}, with $b_k^*(\mathbf{\hat{x}}_i)$ defined in terms of the samples of $u$. To write this in stencil form we note that [\[eq:diff_mls_interp\]](#eq:diff_mls_interp){reference-type="eqref" reference="eq:diff_mls_interp"} can be written using vector inner products as $$\begin{aligned} \partial_{\hat{x}} u\bigr|_{\mathbf{\hat{x}}_i} \approx \partial_{\hat{x}} q\bigr|_{\mathbf{\hat{x}}_i} \approx {\underbrace{\begin{bmatrix} \partial_{\hat{x}} p_1\bigr|_{\mathbf{\hat{x}}_i} & \cdots & \partial_{\hat{x}}p_L\bigr|_{\mathbf{\hat{x}}_i}\end{bmatrix}}_{\displaystyle(\partial_{\hat{x}}\underline{p}(\mathbf{\hat{x}}_i))^T}} \underline{b}^{*}(\mathbf{\hat{x}}_i) = {\underbrace{\begin{bmatrix} c_{1}^{i} & \cdots & c_{n^{i}}\end{bmatrix}}_{\displaystyle(\underline{c}^i_{\hat{x}})^T}} \underline{u}, \label{eq:wls_fd_approx}\end{aligned}$$ where we have substituted the solution of $\underline{b}^{*}(\mathbf{\hat{x}}_i)$ in [\[eq:wls_fd_approx_system\]](#eq:wls_fd_approx_system){reference-type="eqref" reference="eq:wls_fd_approx_system"} to obtain the term in the last equality. Using the normal equation solution for $\underline{b}^{*}(\mathbf{\hat{x}}_i)$, the stencil weights $\underline{c}^i_{\hat{x}}$ can be expressed as $$\begin{aligned} \underline{c}^i_{\hat{x}} = W_{\rho}(\mathbf{\hat{x}}_i)P(P^T W_{\rho}(\mathbf{\hat{x}}_i)P)^{-1} (\partial_{\hat{x}}\underline{p}(\mathbf{\hat{x}}_i)). %W_{\rho}(\vxh_i)^{1/2}P\uc^i_{\xh} = W_{\rho}(\vxh_i)^{1/2}(\partial_{\xh}\up(\vxh_i)). \label{eq:wls_fd_lap}\end{aligned}$$ This is typically computed using a QR factorization of $W_{\rho}(\mathbf{\hat{x}}_i)^{1/2}P$ to promote numerical stability. Stencil weights $\underline{c}^i_{\hat{y}}$, $\underline{c}^i_{\hat{x}\hat{x}}$, $\underline{c}^i_{\hat{x}\hat{y}}$, and $\underline{c}^i_{\hat{y}\hat{y}}$ for the other derivative operators appearing in [\[eq:intrinsic_grad\]](#eq:intrinsic_grad){reference-type="eqref" reference="eq:intrinsic_grad"}--[\[eq:intrinsic_lap\]](#eq:intrinsic_lap){reference-type="eqref" reference="eq:intrinsic_lap"} can be computed in a similar manner for each stencil $\hat{\mathbf{X}}^i$, $i=1,\dots,N$. These can then be combined together with the approximate metric terms to define the weights $\{c_{ij}\}$ in [\[eq:stencils\]](#eq:stencils){reference-type="eqref" reference="eq:stencils"} for any of the SDOs in [\[eq:intrinsic_grad\]](#eq:intrinsic_grad){reference-type="eqref" reference="eq:intrinsic_grad"}--[\[eq:intrinsic_lap\]](#eq:intrinsic_lap){reference-type="eqref" reference="eq:intrinsic_lap"}. ## Choosing the stencils and weight kernel[\[sec:gmls_stencils_weights\]]{#sec:gmls_stencils_weights label="sec:gmls_stencils_weights"} As discussed in Section [2.1](#sec:stencils){reference-type="ref" reference="sec:stencils"}, we use Algorithm [\[alg:epsilon_ball\]](#alg:epsilon_ball){reference-type="ref" reference="alg:epsilon_ball"} to choose the stencil weights. For the initial stencil size, we use $L=\text{dim}(\mathbb{P}^2_{\ell})$. The radius factor $\tau$ controls the size of the stencil, with larger $\tau$ resulting in larger stencils, and we experiment with this parameter in the numerical results section. There are many choices for the weight kernel $w_{\rho}$ in [\[eq:wls_fd_approx_system\]](#eq:wls_fd_approx_system){reference-type="eqref" reference="eq:wls_fd_approx_system"}. Typically, a single radial kernel is used to define $w_{\rho}$ as $w_{\rho}(\mathbf{x},\mathbf{y}) = w(\|\mathbf{x}-\mathbf{y}\|/\rho)$, where $\mathbf{x},\mathbf{y}\in\mathbb{R}^d$ and $\| \cdot \|$ is the standard Euclidean norm for $\mathbb{R}^d$. In this work, we use the same family of compactly supported radial kernels as [@TraskKuberry20; @GrossEtAl20] and implemented in [@AMJ:Compadre]: $$\begin{aligned} w_{\rho}(\mathbf{x},\mathbf{y}) = \left(1 - \frac{\|\mathbf{x}-\mathbf{y}\|}{\rho}\right)_{+}^{2m}, \label{eq:weight_kernel}\end{aligned}$$ where $m$ is a positive integer and $(\cdot)_{+}$ is the positive floor function. These $C^{0}$ kernels have support over the ball of radius $\rho$ centered at $\mathbf{y}$. While smoother kernels can be used such as Gaussians, splines, or Wendland kernels [@fasshauer2007meshfree], we have not observed any significant improvement in the accuracy of GMLS derivative approximations with smoother kernels. In general, proofs on how the choice of kernels effects the accuracy of GMLS approximations have yet to be found. Finally, we note that the support parameter $\rho$ is chosen on a per stencil basis and is set equal to $\tau h_{max}$ from Algorithm [\[alg:epsilon_ball\]](#alg:epsilon_ball){reference-type="ref" reference="alg:epsilon_ball"}. Picking an optimal value for $\tau$ to minimize the approximation error is a difficult problem. In general, the optimal value depends locally on the point set and the function (or its derivative) being approximated [@lipman2006error]. While there are some algorithms that attempt to approximate this value to minimize the local pointwise error (e.g., [@lipman2006error; @wang2008optimal]), they are computationally expensive. Typically, one chooses a single $\tau > 1$ such that the minimization problem [\[eq:wls_fd_approx_system\]](#eq:wls_fd_approx_system){reference-type="eqref" reference="eq:wls_fd_approx_system"} is well-posed (i.e., $P$ is full rank). This can be easily monitored for each stencil to adjust $\tau$ appropriately. ## Approximating the tangent space[\[sec:gmls_tangent_space\]]{#sec:gmls_tangent_space label="sec:gmls_tangent_space"} When the tangent space $T_{\mathbf{x}_i}\mathcal{M}$ is unknown, a coarse approximation to it can be computed for each stencil $\mathbf{X}^i$ using principal component analysis [@LiangZhao13]. In this method, one computes the eigenvectors of the covariance matrix $\overline{X}_i \overline{X}_i^T$, where $\overline{X}_i$ is the $3$-by-$n_i$ matrix formed from the stencil points $\mathbf{X}^i$ centered about their mean. The two dominant eigenvectors of this matrix are taken as a coarse approximation to $T_{\mathbf{x}_{i}}\mathcal{M}$ and the third is taken as a coarse approximation to the normal to $\mathcal{M}$ at $\mathbf{x}_i$; we denote these by $\tilde{\boldsymbol{\xi}}_{i}^1$, $\tilde{\boldsymbol{\xi}}_i^2$, and $\tilde{\boldsymbol{\eta}}_i$, respectively. Next, an approximate Monge patch parameterization is formed with respect to this approximate tangent space using MLS following the same procedure outlined at the beginning of Section [\[sec:metric_terms\]](#sec:metric_terms){reference-type="ref" reference="sec:metric_terms"}. This procedure is illustrated in Figure [6](#fig:tp_correction){reference-type="ref" reference="fig:tp_correction"} (a), where the coarse approximate tangent plane is given in yellow. A refined approximation to the true tangent plane and normal at the stencil center $\mathbf{x}_{\rm c}$ can be obtained by computing the tangent plane and normal to the MLS approximant of the Monge patch at $\mathbf{x}_{\rm c}$; this plane is given in cyan in Figure [6](#fig:tp_correction){reference-type="ref" reference="fig:tp_correction"} (a). Once this plane is computed, a new Monge patch parameterization with respect to this refined tangent plane approximation is formed, as illustrated in Figure [6](#fig:tp_correction){reference-type="ref" reference="fig:tp_correction"} (b). This procedure is repeated for each stencil $\mathbf{X}^i$ and the refined tangent space computed for each stencil is used in the procedure described in Section [\[sec:metric_terms\]](#sec:metric_terms){reference-type="ref" reference="sec:metric_terms"} for approximating the metric terms. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Illustration of the tangent plane correction method. (a) Monge patch parameterization for a local neighborhood of a regular surface $\mathcal{M}$ (in gray) in 3D using a coarse approximation to the tangent plane (in yellow) at the center of the stencil $\mathbf{x}_{\rm c}$ and the refined approximation to the tangent plane (in cyan). (b) Same as (a), but for the Monge patch with respect to the refined tangent plane. The red spheres denote the points from the stencil and the blue spheres mark the projection of the stencil to the (a) coarse and (b) refined tangent planes. The coarse and refined approximations to the tangent and normal vectors are given as $\boldsymbol{\xi}^{1}_{\rm c}$, $\boldsymbol{\xi}^{2}_{\rm c}$, and $\boldsymbol{\eta}_{\rm c}$, respectively, with tildes on these variables denoting the coarse approximation. [\[fig:tp_correction\]]{#fig:tp_correction label="fig:tp_correction"}](figures/TangentPlaneCorrection1.pdf){#fig:tp_correction width="40%"} ![Illustration of the tangent plane correction method. (a) Monge patch parameterization for a local neighborhood of a regular surface $\mathcal{M}$ (in gray) in 3D using a coarse approximation to the tangent plane (in yellow) at the center of the stencil $\mathbf{x}_{\rm c}$ and the refined approximation to the tangent plane (in cyan). (b) Same as (a), but for the Monge patch with respect to the refined tangent plane. The red spheres denote the points from the stencil and the blue spheres mark the projection of the stencil to the (a) coarse and (b) refined tangent planes. The coarse and refined approximations to the tangent and normal vectors are given as $\boldsymbol{\xi}^{1}_{\rm c}$, $\boldsymbol{\xi}^{2}_{\rm c}$, and $\boldsymbol{\eta}_{\rm c}$, respectively, with tildes on these variables denoting the coarse approximation. [\[fig:tp_correction\]]{#fig:tp_correction label="fig:tp_correction"}](figures/TangentPlaneCorrection2.pdf){#fig:tp_correction width="40%"} \(a\) \(b\) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- # RBF-FD using the tangent plane {#sec:rbffd_lap} As discussed in the introduction, there are several RBF-FD methods that have been developed over the past ten years for approximating SDOs. We use the one based on the tangent plane method for formulating SDOs and PHS+Poly interpolants for approximating the derivatives that appear in this formulation. The subsections below provide a detailed overview of these respective techniques. ## Tangent plane method[\[sec:tangent_plane\]]{#sec:tangent_plane label="sec:tangent_plane"} The tangent plane method similarly uses local coordinates for the surface in the tangent plane formed at each $\mathbf{x}_i\in \mathbf{X}$, but unlike the method from Section [\[sec:metric_terms\]](#sec:metric_terms){reference-type="ref" reference="sec:metric_terms"}, it does not use approximations to the metric terms. It instead approximates the SDOs at each $\mathbf{x}_i$ using the standard definitions for the derivatives in the tangent plane. So, using local coordinates [\[eq:monge\]](#eq:monge){reference-type="eqref" reference="eq:monge"} about $\mathbf{x}_i$, the surface gradient for the tangent plane method is taken as $$\begin{aligned} %{\nabla}_{\M}\bigr|_{\vx=\vx_i} = R_i \begin{bmatrix} \partial_{\xh}\bigr|_{\vxh_i} \\ \partial_{\yh}\bigr|_{\vxh_i} \\ 0 \end{bmatrix}, {\nabla}_{\mathcal{M}} = R_i \begin{bmatrix} \partial_{\hat{x}} \\ \partial_{\hat{y}} \\ 0 \end{bmatrix}, %\label{eq:intrinsic_grad} \label{eq:tp_grad}\end{aligned}$$ and the surface divergence of a smooth vector $\mathbf{u}\in T_{\mathbf{x}_i}\mathcal{M}$ is taken as $$\begin{aligned} %\nabla_{\M} \cdot \vu\bigr|_{\vx_i} = %\left(\begin{bmatrix} \partial_{\xh} & 0 & 0 \end{bmatrix}R_i^{T}\vu\right)\bigr|_{\vx_i} + %\left(\begin{bmatrix} 0 & \partial_{\yh} & 0 \end{bmatrix}R_i^{T}\vu\right)\bigr|_{\vy_i}, \nabla_{\mathcal{M}} \cdot \mathbf{u}= \begin{bmatrix} \partial_{\hat{x}} & \partial_{\hat{y}} & 0 \end{bmatrix}R_i^{T}\mathbf{u}, \label{eq:tp_div}\end{aligned}$$ where $R_i$ is the rotation matrix given in [\[eq:projection\]](#eq:projection){reference-type="eqref" reference="eq:projection"}. Similarly, the surface Laplacian in the tangent plane method is $$%\laps\bigr|_{\vx_i} = \partial_{\xh\xh}\bigr|_{\vxh_i} + \partial_{\yh\yh}\bigr|_{\vxh_i}. \Delta_{\mathcal{M}}= \partial_{\hat{x}\hat{x}} + \partial_{\hat{y}\hat{y}}. \label{eq:tp_lap}$$ We next show that if $T_{\mathbf{x}_i}\mathcal{M}$ is known exactly for each $\mathbf{x}_i\in \mathbf{X}$ and the point at which the SDOs are evaluated is $\mathbf{x}_i$, then the SDOs [\[eq:tp_grad\]](#eq:tp_grad){reference-type="eqref" reference="eq:tp_grad"}--[\[eq:tp_lap\]](#eq:tp_lap){reference-type="eqref" reference="eq:tp_lap"} are equivalent to the corresponding ones involving metric terms [\[eq:intrinsic_grad\]](#eq:intrinsic_grad){reference-type="eqref" reference="eq:intrinsic_grad"}--[\[eq:intrinsic_lap\]](#eq:intrinsic_lap){reference-type="eqref" reference="eq:intrinsic_lap"}. This was shown indirectly in [@DEMANET2006] for the surface Laplacian using the distributional definition of the surface Laplacian. Here we show the result follows explicitly for each surface differential operator [\[eq:intrinsic_grad\]](#eq:intrinsic_grad){reference-type="eqref" reference="eq:intrinsic_grad"}--[\[eq:intrinsic_lap\]](#eq:intrinsic_lap){reference-type="eqref" reference="eq:intrinsic_lap"} from the local coordinate formulation in Section [2.2](#sec:local_coords){reference-type="ref" reference="sec:local_coords"}. The first step is to note that the vectors $\partial_{\hat{x}} \mathbf{f}\bigr|_{\mathbf{\hat{x}}_i}$ and $\partial_{\hat{y}} \mathbf{f}\bigr|_{\mathbf{\hat{x}}_i}$ from the Monge parameterization [\[eq:monge\]](#eq:monge){reference-type="eqref" reference="eq:monge"} are tangential to the $\hat{x}\hat{y}$-plane and must therefore be orthogonal to the vector $\begin{bmatrix} 0 & 0 & 1 \end{bmatrix}$. This implies $\partial_{\hat{x}} f= \partial_{\hat{y}}f = 0$ at $\mathbf{\hat{x}}_i$, which means the metric tensor [\[eq:metric_tensor\]](#eq:metric_tensor){reference-type="eqref" reference="eq:metric_tensor"} reduces to the identity matrix when evaluated at $\mathbf{\hat{x}}_i$. Using this result in [\[eq:intrinsic_grad\]](#eq:intrinsic_grad){reference-type="eqref" reference="eq:intrinsic_grad"} for $\widehat{\nabla}_{\mathcal{M}}$ means that the surface gradient formula [\[eq:intrinsic_grad\]](#eq:intrinsic_grad){reference-type="eqref" reference="eq:intrinsic_grad"} is exactly [\[eq:tp_grad\]](#eq:tp_grad){reference-type="eqref" reference="eq:tp_grad"} when evaluated at $\mathbf{\hat{x}}_i$. The equivalence of the surface divergence formulas [\[eq:intrinsic_div\]](#eq:intrinsic_div){reference-type="eqref" reference="eq:intrinsic_div"} and [\[eq:tp_div\]](#eq:tp_div){reference-type="eqref" reference="eq:tp_div"} also follow immediately from this result. The steps for showing the equivalence of the surface Laplacian operator are more involved. To simplify the notation in showing this result, we denote partial derivatives of $f$ with subscripts. For the first step of this process, we substitute the explicit metric terms, $|g| = (1+f_{\hat{x}}^2)(1+f_{\hat{y}}^2) - (f_{\hat{x}} f_{\hat{y}})^2$, $g^{11} = (1+f_{\hat{y}})/|g|$, $g^{12}=g^{21}=-(f_{\hat{x}})(f_{\hat{y}})/|g|$, and $g^{22} = (1+f_{\hat{x}})/|g|$, into [\[eq:intrinsic_lap\]](#eq:intrinsic_lap){reference-type="eqref" reference="eq:intrinsic_lap"} and expand the derivatives. Next, we simplify to obtain the following formula: $$\begin{aligned} \begin{aligned} \Delta_{\mathcal{M}}= \frac{1}{\left(f_{\hat{x}}^2+f_{\hat{y}}^2+1\right)^{2}} \biggl( & \left(f_{\hat{y}} f_{\hat{x}\hat{y}} \left(1+2 f_{\hat{x}}^2+f_{\hat{y}}^2\right)-(f_{\hat{x}} f_{\hat{x}\hat{x}} + f_{\hat{y}}f_{\hat{x}\hat{y}})(1+f_{\hat{y}}^2) - f_{\hat{x}}f_{\hat{y}\hat{y}}(1 + f_{\hat{x}}^2)\right)\partial_{\hat{x}} + \\ & \left(f_{\hat{x}} f_{\hat{x}\hat{y}} \left(1+2 f_{\hat{y}}^2+f_{\hat{x}}^2\right)-(f_{\hat{y}} f_{\hat{y}\hat{y}} + f_{\hat{x}}f_{\hat{x}\hat{y}})(1 + f_{\hat{x}}^2) - f_{\hat{y}}f_{\hat{x}\hat{x}}(1+f_{\hat{y}}^2) \right)\partial_{\hat{y}}\biggr) + \\ & g^{11} \partial_{\hat{x}\hat{x}} + 2g^{12} \partial_{\hat{x}\hat{y}} + g^{22}\partial_{\hat{y}\hat{y}} \end{aligned}\end{aligned}$$ Using $f_{\hat{x}} = f_{\hat{y}} = g^{12} = 0$ and $g^{11}=g^{22}=1$ at $\mathbf{\hat{x}}_i$, this formula reduces to [\[eq:tp_lap\]](#eq:tp_lap){reference-type="eqref" reference="eq:tp_lap"}. ## Approximating the SDOs Since the tangent plane method does not require computing approximations to any metric terms, we only need to describe the RBF-FD method for approximating the derivatives that appear in [\[eq:tp_grad\]](#eq:tp_grad){reference-type="eqref" reference="eq:tp_grad"}--[\[eq:tp_lap\]](#eq:tp_lap){reference-type="eqref" reference="eq:tp_lap"}. We derive this method from derivatives of interpolants over the projected stencils for each point $\mathbf{x}_i\in X$ using the same notation as Section [3](#sec:gmls){reference-type="ref" reference="sec:gmls"} and we assume that the tangent space is known. A method for approximating the tangent space also using RBF-FD is discussed in Section [\[sec:rbffd_tangent_space\]](#sec:rbffd_tangent_space){reference-type="ref" reference="sec:rbffd_tangent_space"}. Let $\{u_j\}_{j\in\sigma^i}$ be samples of some function $u$ over the projected stencil $\hat{\mathbf{X}}^i = \{\mathbf{\hat{x}}_j\}_{j\in\sigma^i}$. The PHS+Poly interpolant to this data can be written $$\begin{aligned} %s(\vxh) = \sum_{j=1}^{n_i} a_j \|\vxh - \vxh_{\sigma^i_j}\|^{2\kappa+1} + \sum_{k=1}^{L} b_k p_k(\vxh), s(\mathbf{\hat{x}}) = \sum_{j=1}^{n_i} a_j \phi(\|\mathbf{\hat{x}}- \mathbf{\hat{x}}_{\sigma^i_j}\|) + \sum_{k=1}^{L} b_k p_k(\mathbf{\hat{x}}), \label{eq:phs_interp}\end{aligned}$$ where $\phi(r) = r^{2\kappa+1}$ is the PHS kernel of order $2\kappa+1$, $\kappa\in\mathbb{Z}^{\geq 0}$, $\sigma^i_j$ is the $j$th index in $\sigma^i$, $\|\cdot\|$ denotes the Euclidean norm, and $\{p_1,\ldots,p_L\}$ are a basis for $\mathbb{P}^2_{\ell}$. The expansion coefficients are determined by the $n_i$ interpolation conditions and $L$ additional moment conditions: $$\begin{aligned} s(\mathbf{\hat{x}}_j) = u_j,\; j\in\sigma^i \quad \text{and} \quad \sum_{j=1}^{n_i} a_j p_k(\mathbf{\hat{x}}_{\sigma^i_j}) = 0,\; k=1,\ldots,L. \label{eq:phs_interp_conditions}\end{aligned}$$ These conditions can be written as the following $(n_i+L)\times(n_i+L)$ linear system $$\begin{aligned} \begin{bmatrix} A & P \\ P^T & \mathbf{0} \end{bmatrix} \begin{bmatrix} \underline{a}\\ \underline{b} \end{bmatrix} = \begin{bmatrix} \underline{u}\\ \underline{0} \end{bmatrix}, \label{eq:rbf_fd_interp_system}\end{aligned}$$ where $A_{jk} = \|\mathbf{\hat{x}}_{\sigma_j^i} - \mathbf{\hat{x}}_{\sigma_k^i}\|^{2\kappa+1}$ ($j,k=1,\ldots,n_i$) and $P$ is the same Vandermonde-type matrix given in [\[eq:vandermonde\]](#eq:vandermonde){reference-type="eqref" reference="eq:vandermonde"}. The PHS parameter $\kappa$ controls the smoothness of the kernel and should be chosen such that $0\leq \kappa \leq \ell$. With this restriction on $\kappa$, it can be shown that $A$ is positive definite on the subspace of vectors in $\mathbb{R}^n$ satisfying the $L$ moment conditions in [\[eq:phs_interp_conditions\]](#eq:phs_interp_conditions){reference-type="eqref" reference="eq:phs_interp_conditions"} [@Wendland:2004]. Hence, if the stencil points $\mathbf{X}^i$ are such that $\text{rank}(P)=L$ (i.e., they are unisolvent on the space $\mathbb{P}^2_{\ell}$), then the system [\[eq:rbf_fd_interp_system\]](#eq:rbf_fd_interp_system){reference-type="eqref" reference="eq:rbf_fd_interp_system"} is non-singular and the PHS+Poly interpolant is well-posed. Note that this is the same restriction on $\mathbf{X}^i$ for the MLS problem [\[eq:wls_fd_approx_system\]](#eq:wls_fd_approx_system){reference-type="eqref" reference="eq:wls_fd_approx_system"} to have a unique solution. The stencil weights for approximating any of the derivatives appearing in the SDOs [\[eq:tp_grad\]](#eq:tp_grad){reference-type="eqref" reference="eq:tp_grad"}-[\[eq:tp_lap\]](#eq:tp_lap){reference-type="eqref" reference="eq:tp_lap"} can be obtained from differentiating the PHS+Poly interpolant [\[eq:phs_interp\]](#eq:phs_interp){reference-type="eqref" reference="eq:phs_interp"}. Without loss of generality, consider approximating the operator $\partial_{\hat{x}}$ over the stencil $\hat{\mathbf{X}}^i$. Using vector inner products as in [\[eq:wls_fd_approx\]](#eq:wls_fd_approx){reference-type="eqref" reference="eq:wls_fd_approx"}, the stencil weights for this operator are determined from the approximation $$\begin{aligned} %\partial_{\xh} u\bigr|_{\vxh_i} \approx \partial_{\xh} s\bigr|_{\vxh_i} %&= \sum_{j=1}^{n_i} a_j \partial_{\xh}\|\vxh - \vxh_{\sigma^i_j}\|^{2\kappa+1}\Bigr|_{\vxh_i} + \sum_{k=1}^{L} b_k \partial_{\xh}p_k(\vxh)\bigr|_{\vxh_i} \\ %& = \begin{bmatrix} \partial_{\xh}\us(\vxh_i) & \partial_{\xh}\up(\vxh_i) \end{bmatrix}^T\begin{bmatrix} \ua \\ \ub \end{bmatrix}. \partial_{\hat{x}} u\bigr|_{\mathbf{\hat{x}}_i} \approx \partial_{\hat{x}} s\bigr|_{\mathbf{\hat{x}}_i} &= \begin{bmatrix} \partial_{\hat{x}}\underline{\phi}(\mathbf{\hat{x}}_i) & \partial_{\hat{x}}\underline{p}(\mathbf{\hat{x}}_i) \end{bmatrix}^T\begin{bmatrix} \underline{a}\\ \underline{b}\end{bmatrix}.\end{aligned}$$ where $\partial_{\hat{x}}\underline{\phi}(\mathbf{\hat{x}}_i)$ and $\partial_{\hat{x}}\underline{p}(\mathbf{\hat{x}}_i)$ are vectors containing the entries $\partial_{\hat{x}}\|\mathbf{\hat{x}}- \mathbf{\hat{x}}_{\sigma^i_j}\|^{2\kappa+1}\bigr|_{\mathbf{\hat{x}}_i}$, $j=1,\ldots,n_i$, and $\partial_{\hat{x}}p_k(\mathbf{\hat{x}})\bigr|_{\mathbf{\hat{x}}_i}$, $k=1,\ldots,L$, respectively. Using [\[eq:rbf_fd_interp_system\]](#eq:rbf_fd_interp_system){reference-type="eqref" reference="eq:rbf_fd_interp_system"} in the preceding expression gives the stencil weights as the solution to the following linear system $$\begin{aligned} \begin{bmatrix} A & P \\ P^T & \mathbf{0} \end{bmatrix} \begin{bmatrix} \underline{c}_{\hat{x}}^i \\ \underline{\lambda} \end{bmatrix} = \begin{bmatrix} \partial_{\hat{x}}\underline{\phi}(\mathbf{\hat{x}}_i) \\ \partial_{\hat{x}}\underline{p}(\mathbf{\hat{x}}_i) \end{bmatrix}, \label{eq:rbf_fd_lap}\end{aligned}$$ where the entries in $\underline{\lambda}$ are not used as part of the weights. Note that this description is equivalent to applying $\partial_{\hat{x}}$ to the PHS+Poly cardinal basis functions defined over the stencil and evaluating them at $\mathbf{\hat{x}}_i$ [@AMJ:Fornberg15]. Stencil weights $\underline{c}^i_{\hat{y}}$, $\underline{c}^i_{\hat{x}\hat{x}}$, $\underline{c}^i_{\hat{x}\hat{y}}$, and $\underline{c}^i_{\hat{y}\hat{y}}$ for the other partial derivatives can be computed in an analogous way for each stencil $\hat{\mathbf{X}}^i$, $i=1,\dots,N$. These can then be combined together to define the weights $\{c_{ij}\}$ in [\[eq:stencils\]](#eq:stencils){reference-type="eqref" reference="eq:stencils"} for any of the SDOs in [\[eq:tp_grad\]](#eq:tp_grad){reference-type="eqref" reference="eq:tp_grad"}--[\[eq:tp_lap\]](#eq:tp_lap){reference-type="eqref" reference="eq:tp_lap"}. ## Choosing the stencils and PHS order[\[sec:rbffd_stencils_weights\]]{#sec:rbffd_stencils_weights label="sec:rbffd_stencils_weights"} Similar to GMLS, we use Algorithm [\[alg:epsilon_ball\]](#alg:epsilon_ball){reference-type="ref" reference="alg:epsilon_ball"} to choose the stencils and also use the same initial stencil size of $n=L$ for this algorithm. The parameter $\kappa$ used to determine the PHS order should be chosen with an upper bound of $\kappa \leq \ell$ (so that [\[eq:rbf_fd_lap\]](#eq:rbf_fd_lap){reference-type="eqref" reference="eq:rbf_fd_lap"} is well posed) and a lower bound such that the derivatives of the PHS kernels make sense for whatever operator the RBF-FD stencils are being used to approximate. In this work we use $\kappa=\ell$ as [we]{style="color: black"} have found [that]{style="color: black"} this choice works well for approximating various SDOs across a wide range of surfaces. Choosing $\kappa < \ell$ can be useful for improving the conditioning of the system [\[eq:rbf_fd_lap\]](#eq:rbf_fd_lap){reference-type="eqref" reference="eq:rbf_fd_lap"} and for reducing Runge Phenomenon-type edge effects in RBF-FD approximations near boundaries [@bayona2019role]. ## Approximating the tangent space[\[sec:rbffd_tangent_space\]]{#sec:rbffd_tangent_space label="sec:rbffd_tangent_space"} If $T_{\mathbf{x}_i}\mathcal{M}$ is unknown for any $\mathbf{x}_i\in \mathbf{X}$, then we use a similar procedure to the one discussed for GMLS in Section [\[sec:gmls_tangent_space\]](#sec:gmls_tangent_space){reference-type="ref" reference="sec:gmls_tangent_space"} (and illustrated in Figure [6](#fig:tp_correction){reference-type="ref" reference="fig:tp_correction"}) to approximate it. The difference for RBF-FD is that instead of using an MLS reconstruction of the Monge patch parameterization formed from the coarse tangent plane approximation at each $\mathbf{x}_i$, we use the PHS+Poly interpolant [\[eq:phs_interp\]](#eq:phs_interp){reference-type="eqref" reference="eq:phs_interp"} for the reconstruction. The refined approximation to the tangent plane at each $\mathbf{x}_i$ is then obtained from derivatives of the PHS+Poly interpolant of the Monge patch for stencil $\mathbf{X}^i$. We note that this approach is new amongst the different tangent plane methods, as previous approaches assumed the tangent space was computed by some other, possibly unrelated techniques, and not directly from the stencils (e.g., [@SUCHDE20192789; @ShawThesis; @wright2022mgm]). By combining this technique with the tangent plane method, we arrive at the first comprehensive PHS+Poly RBF-FD framework for approximating SDOs on point cloud surfaces. # Theoretical comparison of GMLS and RBF-FD[\[sec:theoretical\]]{#sec:theoretical label="sec:theoretical"} In this section, we make comparisons of the GMLS and RBF-FD methods in terms of some of their theoretical properties, including the different approaches in formulating SDOs, the parameters of the approximations, and the computational cost. One of the main differences between the GMLS and RBF-FD approaches is that the former uses the local coordinate method to formulate SDOs, while the latter uses the tangent plane method. As shown in Section [\[sec:tangent_plane\]](#sec:tangent_plane){reference-type="ref" reference="sec:tangent_plane"} these methods are equivalent if the tangent space for $\mathcal{M}$ is known for each $\mathbf{x}_i\in X$ and the SDOs are evaluated at the stencil center $\mathbf{x}_i$. However, the GMLS method does not take advantage of this and instead includes metric terms in the formulation. These metric terms are approximated with the same order of accuracy as the GMLS approximation of the derivatives (see below), so that these errors are asymptotically equivalent as the spacing of the points in the stencil goes to zero. When the tangent space is unknown, both methods again approximate it to the same order of accuracy as their respective approximations of the derivatives. The GMLS and RBF-FD methods each feature the parameter $\ell$, which controls the degree of the polynomials used in the approximation. For a given $\ell$, the formulas for either method are exact for all bivariate polynomials of degree $\ell$ in the tangent plane formed by the stencil center $\mathbf{x}_i$. Unsurprisingly, $\ell$ also effects the local accuracy of the formulas in the tangent plane with increasing $\ell$ giving higher orders of accuracy for smooth problems; see [@mirzaei2016error; @LiangZhao13] for a study of the accuracy of GMLS and [@davydov2019optimal; @bayona2019insight] for RBF-FD. The order of accuracy of both methods depends on the highest order derivative appearing in the SDOs, and is generally $\ell$ if th[e derivative order]{style="color: black"} is 1 and $\ell-1$ if th[e derivative]{style="color: black"} order is two. However, for certain quasi-uniform point clouds with symmetries, the order has been shown to be $\ell$ for GMLS applied to second order operators like the surface Laplacian [@LiangZhao13]. The computational cost of the methods can be split between the setup cost and the evaluation cost. The setup cost depends on $\ell$ and $n_i$ (which depends on $\tau$). For each stencil $\mathbf{X}^i$, the dominant setup cost of GMLS comes from solving the $n_i \times L$ system [\[eq:wls_fd_lap\]](#eq:wls_fd_lap){reference-type="eqref" reference="eq:wls_fd_lap"}, while the dominant cost for RBF-FD comes from solving the $(n_i + L)\times(n_i+L)$ system [\[eq:rbf_fd_lap\]](#eq:rbf_fd_lap){reference-type="eqref" reference="eq:rbf_fd_lap"}. We use QR factorization to solve the GMLS system and LU factorization to solve the RBF-FD system, which gives the following (to leading order): $$\begin{aligned} \text{Setup cost GMLS} \sim 2 \sum_{i=1}^{N} n_i L^2\quad\text{and}\quad \text{Setup cost RBF-FD} \sim \frac23 \sum_{i=1}^{N} (n_i + L)^3. \label{eq:setup_cost}\end{aligned}$$ The stencil sizes depend on $\ell$ and $\tau$, and for quasi-uniform point clouds $\mathbf{X}$, $n_i$ is typically some multiple $\gamma$ of $L$. In this case, the setup cost of RBF-FD is higher by approximately $(1+\gamma)^3/(3\gamma)$. We note that the setup procedures for both methods are an embarrassingly parallel process, as each set of stencil weights can be computed independently of every other set. The evaluation costs of both methods are the same and can be reduced to doing sparse matrix-vector products. So, for a scalar SDO like the surface Laplacian $$\begin{aligned} \text{evaluation cost GMLS \& RBF-FD:}\sim 2\sum_{i=1}^N n_i. \label{eq:runtime_cost}\end{aligned}$$ If the $\ell$ and $\tau$ parameters remain fixed so that size of the stencils remain fixed as $N$ increases, then both the setup and evaluation cost are linear in $N$. # Numerical comparison of GMLS and RBF-FD {#sec:results} We perform a number of numerical experiments comparing GMLS and RBF-FD for approximating the gradient, divergence, and Laplacian on two topologically distinct surfaces: the unit two sphere $\mathbb{S}^2$ and the torus defined implicitly as $$\begin{aligned} \mathbb{T}^2 = \left\{(x,y,z)\in\mathbb{R}^3\, \bigr|\, (1-\sqrt{x^2 + y^2})^2 + z^2 - 1/9 = 0\right\}. \label{eq:torus}\end{aligned}$$ For the experiments with the sphere, we consider two different node sets $\mathbf{X}$, icosahedral and Hammersley; see Figure [9](#fig:node_sets){reference-type="ref" reference="fig:node_sets"} (a) & (b) for examples. The first are highly structured, quasi-uniform points that are commonly used in numerical weather prediction [@AMJ:Bosler14; @AMJ:numericalweather07]. They have also been used in other studies on GMLS [@TraskKuberry20] and RBF-FD [@FlyerLehtoBlaiseWrightStCyr2012] methods on the sphere. Hammersely are low discrepancy point sequences commonly used in Monte-Carlo integration on the sphere [@Cui97equidistribution]. They are highly unstructured with some points that nearly overlap. For the experiments on the torus, we use Poisson disk points generated using the weighted sample elimination (WSE) algorithm [@Yuksel2015]. These points are also unstructured, but are quasi-uniform; see Figure [9](#fig:node_sets){reference-type="ref" reference="fig:node_sets"} (c) for an example. They have also previously been used in studies on GMLS and RBF-FD methods [@wright2022mgm]. Convergence results with other point sets can be found in the PhD thesis of the first author [@JonesThesis]. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![Examples from the three node sets considered in the numerical experiments: (a) $N=2562$, (b) $N=2048$, (c) $N=2038$.[\[fig:node_sets\]]{#fig:node_sets label="fig:node_sets"}](figures/SphereIcosPtsN2562.png){#fig:node_sets width="28%"} ![Examples from the three node sets considered in the numerical experiments: (a) $N=2562$, (b) $N=2048$, (c) $N=2038$.[\[fig:node_sets\]]{#fig:node_sets label="fig:node_sets"}](figures/SphereHammersleyPtsN2048.png){#fig:node_sets width="28%"} ![Examples from the three node sets considered in the numerical experiments: (a) $N=2562$, (b) $N=2048$, (c) $N=2038$.[\[fig:node_sets\]]{#fig:node_sets label="fig:node_sets"}](figures/TorusPoissonPtsN2038.png){#fig:node_sets width="33%"} (a) Icosahedral (b) Hammersley (c) Poisson disk ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Error estimates for GMLS and RBF-FD typically require the nodes to be quasi-uniform in the sense that the average spacing between the points $h$ (or more generally the mesh-norm) decreases like $h\sim N^{-1/2}$ [@fasshauer2007meshfree; @Wendland:2004]. As mentioned above, the icosahedral and Poisson disk node sets have this property and are thus well-suited for numerically testing convergence rates of GMLS and RBF-FD methods with increasing $N$ (i.e., convergence as the density of the sampling of the surfaces increases). Specifically, we experimentally examine the algebraic convergence rates $\beta$ versus $\sqrt{N}$, assuming the error behaves like $\mathcal{O}(N^{-\beta/2})$, and include results for polynomial degrees $\ell=2$, 4, and 6. The Hammersley node sets are well suited to testing how stable the methods are to stencils with badly placed points. Since these nodes have low discrepancy over the sphere, it also makes sense to test convergence in a similar manner to the other point sets. The exact values of $N$ used in the experiments for each of the node sets are as follows. For Icosahedral, $N=10242, 40962, 163842, 655362$, and for Hammersley and Poisson disk: $N=8153, 32615, 130463, 521855$. All RBF-FD results that follow were obtained from a Python implementation of the method that only utilizes the scientific computing libraries SciPy and NumPy. For the GMLS results, we use the software package Compadre [@AMJ:Compadre], which is implemented in C++ and uses the portable performance library Kokkos. ## Convergence comparison: Sphere {#subsec:sphere} We base all the convergence comparisons for the sphere on the following function consisting of a random linear combination of translates of $50$ Gaussians of different widths on the sphere: $$u(\mathbf{x}) = \sum_{j=1}^{50} d_{j} \exp(-\gamma_j \|\mathbf{x}- \mathbf{y}_{j}\|^2),\; \mathbf{x},\mathbf{y}_j\in\mathbb{S}^2, \label{eq:target_sphere}$$ where $\mathbf{y}_{j}$ are the centers and are randomly placed on the sphere, and $d_j$ & $\gamma_j$ are sampled from the normal distributions $\mathcal{N}(0,1)$ & $\mathcal{N}(15,4)$, respectively. This function has also been used in other studies on RBF-FD methods [@LSW2016]. We use samples of $u$ in the surface gradient tests and measure the error against the exact surface gradient, which can be computed using the Cartesian gradient $\nabla$ in $\mathbb{R}^3$ as $\nabla_\mathcal{M}u = \nabla u - \boldsymbol{\eta}(\boldsymbol{\eta}\cdot \nabla u)$, where $\boldsymbol{\eta}$ is the unit outward normal to $\mathbb{S}^2$ [@FuselierWright2013] (which is just $\mathbf{x}$). Applying this to [\[eq:target_sphere\]](#eq:target_sphere){reference-type="eqref" reference="eq:target_sphere"} gives $$\begin{aligned} \nabla_{\mathcal{M}} u = 2\sum_{j=1}^{50} d_{j} \gamma_j (\mathbf{y}_j - \mathbf{x}(\mathbf{x}\cdot\mathbf{y}_j))\exp(-\gamma_j \|\mathbf{x}- \mathbf{y}_{j}\|^2). \label{eq:grad_sphere}\end{aligned}$$ We use samples of this field in the surface divergence tests. Since $\nabla_{\mathcal{M}}\cdot\nabla_{\mathcal{M}} u = \Delta_{\mathcal{M}}u$, we compare the errors in this test against the exact surface Laplacian of $u$, which can be computed using the results of [@AMJ:Fornberg15] as $$\begin{aligned} \Delta_{\mathcal{M}}u = -\sum_{j=1}^{50} d_{j} \gamma_j (4 - \|\mathbf{x}- \mathbf{y}_{j}\|^2(2 + \gamma_j (4 - \|\mathbf{x}- \mathbf{y}_{j}\|^2)))\exp(-\gamma_j \|\mathbf{x}- \mathbf{y}_{j}\|^2).\end{aligned}$$ We also use this in the tests of the surface Laplacian using samples of $u$. For all these tests, we set radius factor $\tau$ in the stencil selection Algorithm [\[alg:epsilon_ball\]](#alg:epsilon_ball){reference-type="ref" reference="alg:epsilon_ball"} to 1.5, which gave good results for both RBF-FD and GMLS (see the next section for some results on the effects of increasing $\tau$). While the exact tangent space for the sphere is trivially determined, we approximate it in all the results using the methods discussed in the Section [\[sec:gmls_tangent_space\]](#sec:gmls_tangent_space){reference-type="ref" reference="sec:gmls_tangent_space"} for GMLS and Section [\[sec:rbffd_tangent_space\]](#sec:rbffd_tangent_space){reference-type="ref" reference="sec:rbffd_tangent_space"} for RBF-FD. These approximations are done with the same parameters for approximating the different SDOs to keep the asymptotic orders of accuracy comparable. Although not included here, we did experiments with the exact tangent space and obtained similar results to those presented here. -- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Convergence results for (a) surface gradient, (b) divergence, and (b) Laplacian on the sphere using icosahedral node sets. Errors are given in relative two-norms (first column) and max-norms (second column). Markers correspond to different $\ell$: filled markers are GMLS and open markers are RBF-FD. Dash-dotted lines without markers correspond to 2nd, 4th, and 6th order convergence with $1/\sqrt{N}$. $\beta$ are the measured order of accuracy computed using the lines of best fit to the last three reported errors. [\[fig:convergence_icos\]]{#fig:convergence_icos label="fig:convergence_icos"}](figures/errs_l2_grad_icos.pdf){#fig:convergence_icos width="40%"} ![Convergence results for (a) surface gradient, (b) divergence, and (b) Laplacian on the sphere using icosahedral node sets. Errors are given in relative two-norms (first column) and max-norms (second column). Markers correspond to different $\ell$: filled markers are GMLS and open markers are RBF-FD. Dash-dotted lines without markers correspond to 2nd, 4th, and 6th order convergence with $1/\sqrt{N}$. $\beta$ are the measured order of accuracy computed using the lines of best fit to the last three reported errors. [\[fig:convergence_icos\]]{#fig:convergence_icos label="fig:convergence_icos"}](figures/errs_linf_grad_icos.pdf){#fig:convergence_icos width="40%"} ![Convergence results for (a) surface gradient, (b) divergence, and (b) Laplacian on the sphere using icosahedral node sets. Errors are given in relative two-norms (first column) and max-norms (second column). Markers correspond to different $\ell$: filled markers are GMLS and open markers are RBF-FD. Dash-dotted lines without markers correspond to 2nd, 4th, and 6th order convergence with $1/\sqrt{N}$. $\beta$ are the measured order of accuracy computed using the lines of best fit to the last three reported errors. [\[fig:convergence_icos\]]{#fig:convergence_icos label="fig:convergence_icos"}](figures/errs_l2_divgrad_icos.pdf){#fig:convergence_icos width="40%"} ![Convergence results for (a) surface gradient, (b) divergence, and (b) Laplacian on the sphere using icosahedral node sets. Errors are given in relative two-norms (first column) and max-norms (second column). Markers correspond to different $\ell$: filled markers are GMLS and open markers are RBF-FD. Dash-dotted lines without markers correspond to 2nd, 4th, and 6th order convergence with $1/\sqrt{N}$. $\beta$ are the measured order of accuracy computed using the lines of best fit to the last three reported errors. [\[fig:convergence_icos\]]{#fig:convergence_icos label="fig:convergence_icos"}](figures/errs_linf_divgrad_icos.pdf){#fig:convergence_icos width="40%"} ![Convergence results for (a) surface gradient, (b) divergence, and (b) Laplacian on the sphere using icosahedral node sets. Errors are given in relative two-norms (first column) and max-norms (second column). Markers correspond to different $\ell$: filled markers are GMLS and open markers are RBF-FD. Dash-dotted lines without markers correspond to 2nd, 4th, and 6th order convergence with $1/\sqrt{N}$. $\beta$ are the measured order of accuracy computed using the lines of best fit to the last three reported errors. [\[fig:convergence_icos\]]{#fig:convergence_icos label="fig:convergence_icos"}](figures/errs_l2_lap_icos.pdf){#fig:convergence_icos width="40%"} ![Convergence results for (a) surface gradient, (b) divergence, and (b) Laplacian on the sphere using icosahedral node sets. Errors are given in relative two-norms (first column) and max-norms (second column). Markers correspond to different $\ell$: filled markers are GMLS and open markers are RBF-FD. Dash-dotted lines without markers correspond to 2nd, 4th, and 6th order convergence with $1/\sqrt{N}$. $\beta$ are the measured order of accuracy computed using the lines of best fit to the last three reported errors. [\[fig:convergence_icos\]]{#fig:convergence_icos label="fig:convergence_icos"}](figures/errs_linf_lap_icos.pdf){#fig:convergence_icos width="40%"} -- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Figures [15](#fig:convergence_icos){reference-type="ref" reference="fig:convergence_icos"} and [21](#fig:convergence_hammer){reference-type="ref" reference="fig:convergence_hammer"} display the convergence results for GMLS and RBF-FD as a function of $N$. Each figure is for a different point set type and contains the results for approximating the surface gradient, divergence, and Laplacian in both the relative two- and max-norms and for different polynomial degrees $\ell$. We see from all the results that the measured convergence rates for GMLS and RBF-FD are similar, but that RBF-FD gives lower errors for the same $N$ and $\ell$ for approximating the surface gradient and divergence. This is also true for the surface Laplacian when $\ell=4$ and $\ell=6$, but not for $\ell=2$. For this case, GMLS gives lower errors for the same $N$ on the icosahedral nodes and about the same error for the Hammersley nodes. We also see from Figure [21](#fig:convergence_hammer){reference-type="ref" reference="fig:convergence_hammer"} that both methods do not appear to be effected by stability issues associated with badly placed points in the stencils for the Hammersley nodes. The measured convergence rates in the two-norm for the surface gradient and divergence approximations are close to the expected rates of $\ell$ for both point sets. However, when looking at the convergence rates of the surface Laplacian, we see from Figure [15](#fig:convergence_icos){reference-type="ref" reference="fig:convergence_icos"} that the icosahedral nodes have higher rates than for the Hammersley nodes in Figure [21](#fig:convergence_hammer){reference-type="ref" reference="fig:convergence_hammer"}. These improved convergence rates have been referred to as superconvergence in the GMLS literature and rely on the point set being structured so that the stencils have certain symmetries [@LiangZhao13]. When these symmetries do not exist, as is the case for the Hammersley nodes, the convergence rates for the surface Laplacian more closely follow the expected rates of $\ell-1$. -- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ {reference-type="ref" reference="fig:convergence_icos"}, but for the Hammersley nodes on the sphere.[\[fig:convergence_hammer\]]{#fig:convergence_hammer label="fig:convergence_hammer"}](figures/errs_l2_grad_hammer.pdf){#fig:convergence_hammer width="40%"} {reference-type="ref" reference="fig:convergence_icos"}, but for the Hammersley nodes on the sphere.[\[fig:convergence_hammer\]]{#fig:convergence_hammer label="fig:convergence_hammer"}](figures/errs_linf_grad_hammer.pdf){#fig:convergence_hammer width="40%"} {reference-type="ref" reference="fig:convergence_icos"}, but for the Hammersley nodes on the sphere.[\[fig:convergence_hammer\]]{#fig:convergence_hammer label="fig:convergence_hammer"}](figures/errs_l2_divgrad_hammer.pdf){#fig:convergence_hammer width="40%"} {reference-type="ref" reference="fig:convergence_icos"}, but for the Hammersley nodes on the sphere.[\[fig:convergence_hammer\]]{#fig:convergence_hammer label="fig:convergence_hammer"}](figures/errs_linf_divgrad_hammer.pdf){#fig:convergence_hammer width="40%"} {reference-type="ref" reference="fig:convergence_icos"}, but for the Hammersley nodes on the sphere.[\[fig:convergence_hammer\]]{#fig:convergence_hammer label="fig:convergence_hammer"}](figures/errs_l2_lap_hammer.pdf){#fig:convergence_hammer width="40%"} {reference-type="ref" reference="fig:convergence_icos"}, but for the Hammersley nodes on the sphere.[\[fig:convergence_hammer\]]{#fig:convergence_hammer label="fig:convergence_hammer"}](figures/errs_linf_lap_hammer.pdf){#fig:convergence_hammer width="40%"} -- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ## Convergence comparison: Torus {#subsec:torus} -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- {reference-type="ref" reference="fig:convergence_icos"}, but for torus using Poisson disk points.[\[fig:convergence_torus_poisson\]]{#fig:convergence_torus_poisson label="fig:convergence_torus_poisson"}](figures/errs_l2_grad_torus_poisson.pdf){#fig:convergence_torus_poisson width="40%"} {reference-type="ref" reference="fig:convergence_icos"}, but for torus using Poisson disk points.[\[fig:convergence_torus_poisson\]]{#fig:convergence_torus_poisson label="fig:convergence_torus_poisson"}](figures/errs_linf_grad_torus_poisson.pdf){#fig:convergence_torus_poisson width="40%"} {reference-type="ref" reference="fig:convergence_icos"}, but for torus using Poisson disk points.[\[fig:convergence_torus_poisson\]]{#fig:convergence_torus_poisson label="fig:convergence_torus_poisson"}](figures/errs_l2_divgrad_torus_poisson.pdf){#fig:convergence_torus_poisson width="40%"} {reference-type="ref" reference="fig:convergence_icos"}, but for torus using Poisson disk points.[\[fig:convergence_torus_poisson\]]{#fig:convergence_torus_poisson label="fig:convergence_torus_poisson"}](figures/errs_linf_divgrad_torus_poisson.pdf){#fig:convergence_torus_poisson width="40%"} {reference-type="ref" reference="fig:convergence_icos"}, but for torus using Poisson disk points.[\[fig:convergence_torus_poisson\]]{#fig:convergence_torus_poisson label="fig:convergence_torus_poisson"}](figures/errs_l2_lap_torus_poisson.pdf){#fig:convergence_torus_poisson width="40%"} {reference-type="ref" reference="fig:convergence_icos"}, but for torus using Poisson disk points.[\[fig:convergence_torus_poisson\]]{#fig:convergence_torus_poisson label="fig:convergence_torus_poisson"}](figures/errs_linf_lap_torus_poisson.pdf){#fig:convergence_torus_poisson width="40%"} -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The convergence comparisons on the torus are based on the target function $$\label{eq:torus_forcing_exact} u(\mathbf{x}) = \displaystyle\frac{x}{8}(x^4 - 10x^2 y^2 + 5y^4)(r^2 - 60 z^2),\; \mathbf{x}\in \mathbb{T}^2,$$ where $r = \sqrt{x^2+y^2}$. This function has also been used in other studies of RBF methods for surfaces [@FuselierWright2013]. As with the sphere example, the surface gradient of $u$ can be computed as $\nabla_\mathcal{M}u = \nabla u - \boldsymbol{\eta}(\boldsymbol{\eta}\cdot \nabla u)$, where $\boldsymbol{\eta}$ is the unit outward normal to $\mathbb{T}^2$, which can be computed from the implicit equation [\[eq:torus\]](#eq:torus){reference-type="eqref" reference="eq:torus"}. The surface Laplacian of [\[eq:torus_forcing_exact\]](#eq:torus_forcing_exact){reference-type="eqref" reference="eq:torus_forcing_exact"} is given in [@FuselierWright2013] as $$\begin{aligned} \Delta_{\mathcal{M}}u(\mathbf{x}) = \displaystyle-\frac{3x}{8r^2}(x^4 - 10x^2 y^2 + 5y^4)(10248r^4 - 34335r^3 + 41359r^2 - 21320r + 4000),\; \mathbf{x}\in \mathbb{T}^2.\end{aligned}$$ Similar to the sphere, we use samples of $\nabla_{\mathcal{M}} u$ in the tests of the divergence and compare the results with $\Delta_{\mathcal{M}}u$ above. We first study the convergence rates with the stencil radius scaling $\tau=1.5$ and approximate the tangent space, as we did with the sphere tests. Figure [27](#fig:convergence_torus_poisson){reference-type="ref" reference="fig:convergence_torus_poisson"} displays the results for the surface gradient, divergence, and Laplacian. We see that errors for RBF-FD are again smaller than the errors for GMLS in almost all cases over the range of $N$ tested. However, GMLS has a slightly higher convergence rates in the case of the surface gradient and divergence, but not for the Laplacian. Both methods have convergence rates that are close to the expected rates of $\ell$ for these surface gradient and divergence and $\ell-1$ for the Laplacian. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Relative two-norm errors of the surface Laplacian approximations as the stencil radius parameter $\tau$ varies. Left figure shows errors for several different values of $\tau$ and a fixed $N=130463$. Right figure shows the convergence rates of the methods for different $\tau$ and a fixed $\ell=4$.[\[fig:tau_scaling\]]{#fig:tau_scaling label="fig:tau_scaling"}](figures/tau_scaling_N130463_torus.pdf){#fig:tau_scaling width="40%"} ![Relative two-norm errors of the surface Laplacian approximations as the stencil radius parameter $\tau$ varies. Left figure shows errors for several different values of $\tau$ and a fixed $N=130463$. Right figure shows the convergence rates of the methods for different $\tau$ and a fixed $\ell=4$.[\[fig:tau_scaling\]]{#fig:tau_scaling label="fig:tau_scaling"}](figures/convgtau_torus.pdf){#fig:tau_scaling width="40%"} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Next we investigate how the approximation properties of the two methods change when $\tau$ is increased, which results in larger stencil sizes. We focus on approximating the surface Laplacian as similar results were found for the other SDOs. In the left plot of Figure [29](#fig:tau_scaling){reference-type="ref" reference="fig:tau_scaling"}, we show the relative two-norm errors of the approximations for a fixed $N$ as $\tau$ varies from 1.5 to 2.5. We see that increasing $\tau$ has opposite effects on the two methods: the errors decrease for RBF-FD and increase with GMLS. We see similar results in the right plot of Figure [29](#fig:tau_scaling){reference-type="ref" reference="fig:tau_scaling"}, where we show the convergence of the methods with increasing $N$ for different fixed values of $\tau$ (and $\ell$ fixed at $4$). While the convergence rates do not appear to change with $\tau$, the overall errors decrease for RBF-FD and increase for GMLS. It should be noted that the errors eventually increase for GMLS as $\tau$ decreases to 1 (which has been observed in other studies) and picking an optimal $\tau$ in an automated way is challenging (e.g. [@lipman2006error; @wang2008optimal]). These results make sense when one considers the different types of approximations the methods are based on: RBF-FD is based on interpolation, while GMLS is based on least squares approximation. As the stencil sizes increase, RBF-FD has a larger approximation space consisting of more shifts of PHS kernels, which can reduce the errors [@davydov2019optimal]. However, GMLS has the same fixed approximation space of polynomials of degree $\ell$ regardless of the stencil size. Finally, we compare the errors when the exact and approximate tangent spaces are used in the two methods. We focus only on the surface Laplacian and for $\ell=4$ since similar results were obtained for the other operators and other $\ell$. Table [1](#tbl:tangent_space){reference-type="ref" reference="tbl:tangent_space"} shows the results for both methods. The approximate tangent spaces were computed using the methods from Sections [\[sec:gmls_tangent_space\]](#sec:gmls_tangent_space){reference-type="ref" reference="sec:gmls_tangent_space"} (GMLS) and [\[sec:rbffd_tangent_space\]](#sec:rbffd_tangent_space){reference-type="ref" reference="sec:rbffd_tangent_space"} (RBF-FD) also using the polynomial degree $\ell=4$. As discussed in Section 5, this choice is made so that the tangent spaces are approximated with the same asymptotic order of accuracy as the approximation of the metric terms with GMLS. We see from the table that the differences between using the exact or the approximate tangent spaces for approximating the surface Laplacian is minor. GMLS RBF-FD -------- ------------ ------------- ------------ ------------ $N$ Exact Approx. Exact Approx. 8153 4.7984e-04 4.8004e-04 1.3311e-04 1.3312e-04 32615 6.0457e-05 6.04654e-05 1.5321e-05 1.5322e-05 130463 7.5486e-06 7.5488e-06 1.8811e-06 1.8811e-06 521855 8.0158e-07 8.0159e-07 2.0177e-07 2.0176e-07 : Comparison of the relative $\ell_2$ errors for the surface Laplacian on the torus using the exact tangent space for the torus and approximations to it based on the methods from Sections [\[sec:gmls_tangent_space\]](#sec:gmls_tangent_space){reference-type="ref" reference="sec:gmls_tangent_space"} (GMLS) and [\[sec:rbffd_tangent_space\]](#sec:rbffd_tangent_space){reference-type="ref" reference="sec:rbffd_tangent_space"} (RBF-FD). In all cases, $\ell=4$ and the points are based on Poisson disk sampling.[\[tbl:tangent_space\]]{#tbl:tangent_space label="tbl:tangent_space"} ## Efficiency comparison The results in Section [6](#sec:results){reference-type="ref" reference="sec:results"} demonstrate that RBF-FD and GMLS have similar asymptotic convergence rates for the same $\ell$, but that RBF-FD can achieve lower errors for the same $N$ and stencil sizes. In this section, we consider which of the methods are more computationally efficient in terms of error per computational cost. We examine both the efficiency when the setup costs are included and when just the evaluation costs are included, as measured by [\[eq:setup_cost\]](#eq:setup_cost){reference-type="eqref" reference="eq:setup_cost"} and [\[eq:runtime_cost\]](#eq:runtime_cost){reference-type="eqref" reference="eq:runtime_cost"}, respectively. We limit this comparison to $\tau=1.5$, but note that it may be possible to tune this parameter to (marginally) optimize the efficiency of either method over this case. Figure [31](#fig:buildcost_figs){reference-type="ref" reference="fig:buildcost_figs"} displays the results of this examination for the case of computing the surface Laplacian on the torus discretized with Poisson disk sampling. Similar results were obtained for other SDOs and for the sphere, so we omit them. We see from the figure that GMLS is more efficient when the setup costs are included, but that RBF-FD is more efficient when only evaluation costs are included. For problems where the point sets are fixed and approximations to a SDO are required to be performed multiple times---as occurs when solving a time-dependent surface PDEs---the setup costs are not as important as the evaluation costs since they are amortized across all time-steps. In this scenario RBF-FD is the more efficient method. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Comparison of the computational efficiency of GMLS and RBF-FD for approximating the surface Laplacian in terms of accuracy per computational cost. (a) Shows the efficiency when considering the setup and evaluation costs as defined in [\[eq:setup_cost\]](#eq:setup_cost){reference-type="eqref" reference="eq:setup_cost"} and [\[eq:runtime_cost\]](#eq:runtime_cost){reference-type="eqref" reference="eq:runtime_cost"}, respectively, while (b) show the efficiency when only considering the evaluation cost. [\[fig:buildcost_figs\]]{#fig:buildcost_figs label="fig:buildcost_figs"}](figures/build_cost_v_err.pdf){#fig:buildcost_figs width="45%"} ![Comparison of the computational efficiency of GMLS and RBF-FD for approximating the surface Laplacian in terms of accuracy per computational cost. (a) Shows the efficiency when considering the setup and evaluation costs as defined in [\[eq:setup_cost\]](#eq:setup_cost){reference-type="eqref" reference="eq:setup_cost"} and [\[eq:runtime_cost\]](#eq:runtime_cost){reference-type="eqref" reference="eq:runtime_cost"}, respectively, while (b) show the efficiency when only considering the evaluation cost. [\[fig:buildcost_figs\]]{#fig:buildcost_figs label="fig:buildcost_figs"}](figures/evaluation_cost_v_err.pdf){#fig:buildcost_figs width="45%"} \(a\) Set-up + Evaluation \(b\) Evaluation -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- # Concluding remarks {#sec:remarks} We presented a thorough comparison of the GMLS and RBF-FD methods for approximating the three most common SDOs: the gradient, divergence, and Laplacian (Laplace-Beltrami). Our analysis of the two different formulations of SDOs used in the methods revealed that if the exact tangent space for the surface is used, these formulations are identical. We further derived a new RBF-FD method for approximating the tangent space of surfaces represented only by point clouds. Our numerical investigation of the methods showed that they appear to converge at similar rates when the same polynomial degree $\ell$ is used, but that RBF-FD generally gives lower errors for the same $N$ and $\ell$. We additionally examined the dependency of the stencil size on the methods (as measured by the $\tau$ parameter) and found that the errors produced by GMLS deteriorate as the stencil size increases. The errors for RBF-FD, contrastingly, appear to keep improving as the stencil size increases. However, we don't expect this trend to continue indefinitely, as eventually the tangent plane formulation breaks down when the stencil size becomes too large. Finally, we investigated the computational efficiency of the methods in terms of error versus computational cost and found GMLS to be more efficient when setup costs are included and RBF-FD to be more efficient when only considering evaluation costs. # Acknowledgements {#acknowledgements .unnumbered} #### Funding AMJ was partially supported by US NSF grant CCF-1717556. PAB & PAK were supported by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research (ASCR) Program and Biological and Environmental Research (BER) Program under a Scientific Discovery through Advanced Computing (SciDAC 4) BER partnership pilot project. PAK was additionally supported by the Laboratory Directed Research & Development (LDRD) program at Sandia National Laboratories and ASCR under Award Number DE-SC-0000230927. AMJ was also partially supported by the Climate Model Development and Validation (CMDV) program, funded by BER. Part of this work was conducted while AMJ was employed at the Computer Science Research Institute at Sandia National Laboratories. GBW was partially supported by U.S. NSF grants CCF-1717556 and DMS-1952674. # Statements and Declarations {#statements-and-declarations .unnumbered} Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC (NTESS), a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Administration (DOE/NNSA) under contract DE-NA0003525. This written work is authored by an employee of NTESS. The employee, not NTESS, owns the right, title and interest in and to the written work and is responsible for its contents. Any subjective views or opinions that might be expressed in the written work do not necessarily represent the views of the U.S. Government. The publisher acknowledges that the U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this written work or allow others to do so, for U.S. Government purposes. The DOE will provide public access to results of federally sponsored research in accordance with the DOE Public Access Plan. #### Competing Interests The authors have no relevant financial or non-financial interests to disclose. #### Author Contributions - AMJ: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing: Original draft preparation, Writing: Review & Editing. - PAB: Methodology, Resources, Writing: Review & Editing, Funding acquisition, Supervision - PAK: Resources, Software, Writing: Review & Editing - GBW: Methodology, Writing: Orignal draft preparation, Writing: Review & Editing, Funding acquisition, Supervision

arxiv_math

--- abstract: | A partition is finitary if all its members are finite. For a set $A$, $\mathscr{B}(A)$ denotes the set of all finitary partitions of $A$. It is shown consistent with $\mathsf{ZF}$ (without the axiom of choice) that there exist an infinite set $A$ and a surjection from $A$ onto $\mathscr{B}(A)$. On the other hand, we prove in $\mathsf{ZF}$ some theorems concerning $\mathscr{B}(A)$ for infinite sets $A$, among which are the following: 1. If there is a finitary partition of $A$ without singleton blocks, then there are no surjections from $A$ onto $\mathscr{B}(A)$ and no finite-to-one functions from $\mathscr{B}(A)$ to $A$. 2. For all $n\in\omega$, $|A^n|<|\mathscr{B}(A)|$. 3. $|\mathscr{B}(A)|\neq|\mathrm{seq}(A)|$, where $\mathrm{seq}(A)$ is the set of all finite sequences of elements of $A$. address: | School of Philosophy\ Wuhan University\ No. 299 Bayi Road\ Wuhan\ Hubei Province 430072\ People's Republic of China author: - Guozhen Shen title: Cantor's theorem may fail for finitary partitions --- # Introduction In 1891, Cantor [@Cantor1891] proved that, for all sets $A$, there are no surjections from $A$ onto $\mathscr{P}(A)$ (the power set of $A$). Under the axiom of choice, for infinite sets $A$, several sets related to $A$ have the same cardinality as $\mathscr{P}(A)$ or $A$; for example, $\mathcal{S}(A)$ (the set of all permutations of $A$) and $\mathrm{Part}(A)$ (the set of all partitions of $A$) have the same cardinality as $\mathscr{P}(A)$, and $A^2$, $\mathrm{fin}(A)$ (the set of all finite subsets of $A$), $\mathrm{seq}(A)$ (the set of all finite sequences of elements of $A$), and $\mathrm{seq}^{\text{1-1}}(A)$ (the set of all finite sequences without repetition of elements of $A$) have the same cardinality as $A$. However, without the axiom of choice, this is no longer the case. In 1924, Tarski [@Tarski1924] proved that the statement that $A^2$ has the same cardinality as $A$ for all infinite sets $A$ is in fact equivalent to the axiom of choice. Over the past century, various variations of Cantor's theorem have been investigated in $\mathsf{ZF}$ (the Zermelo--Fraenkel set theory without the axiom of choice), with $A$ or $\mathscr{P}(A)$ replaced by a set which has the same cardinality under the axiom of choice. Specker [@Specker1954] proves that, for all infinite sets $A$, there are no injections from $\mathscr{P}(A)$ into $A^2$. Halbeisen and Shelah [@HalbeisenShelah1994] prove that $|\mathrm{fin}(A)|<|\mathscr{P}(A)|$ and $|\mathrm{seq}^{\text{1-1}}(A)|\neq|\mathscr{P}(A)|\neq|\mathrm{seq}(A)|$. Forster [@Forster2003] proves that there are no finite-to-one functions from $\mathscr{P}(A)$ to $A$. Recently, Peng and Shen [@PengShen2022] prove that there are no surjections from $\omega\times A$ onto $\mathscr{P}(A)$, and Peng, Shen and Wu [@PengShenWu2022] prove that the existence of an infinite set $A$ and a surjection from $A^2$ onto $\mathscr{P}(A)$ is consistent with $\mathsf{ZF}$. The variations of Cantor's theorem with $\mathscr{P}(A)$ replaced by $\mathcal{S}(A)$ are investigated in [@DawsonHoward1976; @ShenYuan2020a; @ShenYuan2020b; @SonpanowVejjajiva2019]. For a set $A$, let $\mathscr{B}(A)$ be the set of all finitary partitions of $A$, where a partition is finitary if all its members are finite. We use the symbol $\mathscr{B}$ to denote this notion just because $|\mathscr{B}(n)|$ is the $n$-th Bell number. The axiom of choice implies that $\mathscr{B}(A)$ and $\mathscr{P}(A)$ have the same cardinality for infinite sets $A$, but each of "$|\mathscr{B}(A)|<|\mathscr{P}(A)|$", "$|\mathscr{P}(A)|<|\mathscr{B}(A)|$", and "$|\mathscr{B}(A)|$ and $|\mathscr{P}(A)|$ are incomparable" for some infinite set $A$ is consistent with $\mathsf{ZF}$. Recently, Phansamdaeng and Vejjajiva [@PhansamdaengVejjajiva2022] prove that $|\mathrm{fin}(A)|<|\mathscr{B}(A)|$ for all infinite sets $A$. In this paper, we further study the variations of Cantor's theorem with $\mathscr{P}(A)$ replaced by $\mathscr{B}(A)$. We prove that Cantor's theorem may fail for finitary partitions in the sense that the existence of an infinite set $A$ and a surjection from $A$ onto $\mathscr{B}(A)$ is consistent with $\mathsf{ZF}$. Nevertheless, we prove in $\mathsf{ZF}$ some theorems concerning $\mathscr{B}(A)$ for infinite sets $A$, among which are the following: 1. If there is a finitary partition of $A$ without singleton blocks, then there are no surjections from $A$ onto $\mathscr{B}(A)$ and no finite-to-one functions from $\mathscr{B}(A)$ to $A$. 2. For all $n\in\omega$, $|A^n|<|\mathscr{B}(A)|$. 3. $|\mathscr{B}(A)|\neq|\mathrm{seq}(A)|$. # Some notation and preliminary results Throughout this paper, we shall work in $\mathsf{ZF}$. In this section, we indicate briefly our use of some terminology and notation. For a function $f$, we use $\mathop{\mathrm{dom}}(f)$ for the domain of $f$, $\mathop{\mathrm{ran}}(f)$ for the range of $f$, $f[A]$ for the image of $A$ under $f$, $f^{-1}[A]$ for the inverse image of $A$ under $f$, and $f{\upharpoonright}A$ for the restriction of $f$ to $A$. For functions $f$ and $g$, we use $g\circ f$ for the composition of $g$ and $f$. We write $f:A\to B$ to express that $f$ is a function from $A$ to $B$, and $f:A\twoheadrightarrow B$ to express that $f$ is a function from $A$ *onto* $B$. For a set $A$, $|A|$ denotes the cardinality of $A$. **Definition 1**. Let $A,B$ be arbitrary sets. 1. $|A|=|B|$, or $A\approx B$, if there is a bijection between $A$ and $B$. 2. $|A|\leqslant|B|$, or $A\preccurlyeq B$, if there is an injection from $A$ into $B$. 3. $|A|\leqslant^\ast|B|$, or $A\preccurlyeq^\ast B$, if there is a surjection from a subset of $B$ onto $A$. 4. $|A|<|B|$ if $|A|\leqslant|B|$ and $|A|\neq|B|$. Clearly, if $A\preccurlyeq B$ then $A\preccurlyeq^\ast B$, and if $A\preccurlyeq^\ast B$ then $\mathscr{P}(A)\preccurlyeq\mathscr{P}(B)$. In the sequel, we shall frequently use expressions like "one can explicitly define" in our formulations, which is illustrated by the following example. **Theorem 2** (Cantor-Bernstein). *From injections $f:A\to B$ and $g:B\to A$, one can explicitly define a bijection $h:A\to B$.* *Proof.* See [@Halbeisen2017 Theorem 3.14]. ◻ Formally, Theorem [Theorem 2](#cbt){reference-type="ref" reference="cbt"} states that one can define a class function $H$ without free variables such that, whenever $f$ is an injection from $A$ into $B$ and $g$ is an injection from $B$ into $A$, $H(f,g)$ is defined and is a bijection between $A$ and $B$. Consequently, if $|A|\leqslant|B|$ and $|B|\leqslant|A|$, then $|A|=|B|$. **Definition 3**. Let $A$ be a set and let $f$ be a function. 1. $A$ is *Dedekind infinite* if $\omega\preccurlyeq A$; otherwise, $A$ is *Dedekind finite*. 2. $A$ is *power Dedekind infinite* if $\mathscr{P}(A)$ is Dedekind infinite; otherwise, $A$ is *power Dedekind finite*. 3. $f$ is (*Dedekind*) *finite-to-one* if for every $z\in\mathop{\mathrm{ran}}(f)$, $f^{-1}[\{z\}]$ is (Dedekind) finite. Clearly, if $f$ and $g$ are (Dedekind) finite-to-one functions, so is $g\circ f$ (cf. [@Shen2017 Fact 2.8]). It is well-known that $A$ is Dedekind infinite if and only if there exists a bijection between $A$ and a proper subset of $A$. For power Dedekind infinite sets, recall Kuratowski's celebrated theorem. **Theorem 4** (Kuratowski). *$A$ is power Dedekind infinite if and only if $\omega\preccurlyeq^\ast A$.* *Proof.* See [@Halbeisen2017 Proposition 5.4]. ◻ The following two facts are Corollaries 2.9 and 2.11 of [@Shen2017], respectively. **Fact 5**. *If $A$ is power Dedekind infinite and there exists a finite-to-one function from $A$ to $B$, then $B$ is power Dedekind infinite.* **Fact 6**. *If $A^n$ is power Dedekind infinite, so is $A$.* Let $P$ be a partition of $A$. We say that $P$ is *finitary* if all blocks of $P$ are finite, and write $\mathop{\mathrm{ns}}(P)$ for the set of non-singleton blocks of $P$. For $x\in A$, we write $[x]_P$ for the unique block of $P$ which contains $x$. The equivalence relation $\sim_P$ on $A$ induced by $P$ is defined by $$x\sim_Py\qquad\text{if and only if}\qquad[x]_P=[y]_P.$$ **Definition 7**. Let $A$ be an arbitrary set. 1. $\mathop{\mathrm{Part}}(A)$ is the set of all partitions of $A$. 2. $\mathop{\mathrm{Part_{fin}}}(A)=\{P\in\mathop{\mathrm{Part}}(A)\mid P\text{ is finite}\}$. 3. $\mathscr{B}(A)=\{P\in\mathop{\mathrm{Part}}(A)\mid P\text{ is finitary}\}$. 4. $\mathop{\mathrm{\mathscr{B}_{fin}}}(A)=\{P\in\mathscr{B}(A)\mid\mathop{\mathrm{ns}}(P)\text{ is finite}\}$. 5. $\mathop{\mathrm{fin}}(A)$ is the set of all finite subsets of $A$. 6. $\mathop{\mathrm{seq}}(A)=\{f\mid f\text{ is a function from an }n\in\omega\text{ to }A\}$. 7. $\mathop{\mathrm{seq^{1-1}}}(A)=\{f\mid f\text{ is an injection from an }n\in\omega\text{ into }A\}$. Below we list some basic relations between the cardinalities of these sets. We first note that $\mathop{\mathrm{fin}}(A)\preccurlyeq^\ast\mathop{\mathrm{seq^{1-1}}}(A)\preccurlyeq\mathop{\mathrm{seq}}(A)$. The next three facts are Facts 2.13, 2.16, and 2.17 of [@Shen2017], respectively. **Fact 8**. *If $A$ is infinite, then $\mathop{\mathrm{fin}}(A)$ and $\mathscr{P}(A)$ are power Dedekind infinite.* **Fact 9**. *$\mathop{\mathrm{seq^{1-1}}}(A)\preccurlyeq\mathop{\mathrm{fin}}(\mathop{\mathrm{fin}}(A))$.* **Fact 10**. *There is a finite-to-one function from $\mathop{\mathrm{fin}}(\mathop{\mathrm{fin}}(A))$ to $\mathop{\mathrm{fin}}(A)$.* The next three facts are Facts 2.19, 2.20, and Corollary 2.23 of [@ShenYuan2020a], respectively. **Fact 11**. *If $A$ is non-empty, then $\mathop{\mathrm{seq}}(A)$ is Dedekind infinite.* **Fact 12**. *If $A$ is Dedekind finite, then there is a Dedekind finite-to-one function from $\mathop{\mathrm{seq}}(A)$ to $\omega$.* **Fact 13**. *If $A$ is Dedekind infinite, then $\mathop{\mathrm{seq}}(A)\approx\mathop{\mathrm{seq^{1-1}}}(A)$.* **Fact 14**. *$\mathop{\mathrm{Part}}(A)\preccurlyeq\mathscr{P}(A^2)$.* *Proof.* The function that maps each partition $P$ of $A$ to $\sim_P$ is an injection from $\mathop{\mathrm{Part}}(A)$ into $\mathscr{P}(A^2)$. ◻ **Corollary 15**. *If $A$ is power Dedekind finite, then $\mathop{\mathrm{Part}}(A)$, and hence also $\mathop{\mathrm{Part_{fin}}}(A)$ and $\mathscr{B}(A)$, are Dedekind finite.* *Proof.* If $A$ is power Dedekind finite, so is $A^2$ by Fact [Fact 6](#sh02){reference-type="ref" reference="sh02"}, and thus $\mathscr{P}(A^2)$ is Dedekind finite, which implies that also $\mathop{\mathrm{Part}}(A)$, $\mathop{\mathrm{Part_{fin}}}(A)$ and $\mathscr{B}(A)$ are Dedekind finite by Fact [Fact 14](#sh09){reference-type="ref" reference="sh09"}. ◻ **Fact 16**. *$\mathop{\mathrm{\mathscr{B}_{fin}}}(A)\preccurlyeq\mathop{\mathrm{fin}}(\mathop{\mathrm{fin}}(A))$.* *Proof.* The function that maps each $P\in\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$ to $\mathop{\mathrm{ns}}(P)$ is an injection from $\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$ into $\mathop{\mathrm{fin}}(\mathop{\mathrm{fin}}(A))$. ◻ **Fact 17**. *$\mathop{\mathrm{\mathscr{B}_{fin}}}(A)\preccurlyeq\mathop{\mathrm{Part_{fin}}}(A)$.* *Proof.* If $A$ is finite, then $\mathop{\mathrm{\mathscr{B}_{fin}}}(A)=\mathop{\mathrm{Part_{fin}}}(A)$; otherwise, the function that maps each $P\in\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$ to $\mathop{\mathrm{ns}}(P)\cup\{\bigcup(P\setminus\mathop{\mathrm{ns}}(P))\}$ is an injection from $\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$ into $\mathop{\mathrm{Part_{fin}}}(A)$. ◻ **Fact 18**. *If $|A|\geqslant5$, then $\mathop{\mathrm{fin}}(A)\preccurlyeq\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$ and $\mathscr{P}(A)\preccurlyeq\mathop{\mathrm{Part_{fin}}}(A)$.* *Proof.* Let $E=\{a,b,c,d,e\}$ be a $5$-element subset of $A$. We define functions $f:\mathop{\mathrm{fin}}(A)\to\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$ and $g:\mathscr{P}(A)\to\mathop{\mathrm{Part_{fin}}}(A)$ by setting, for $B\in\mathop{\mathrm{fin}}(A)$ and $C\in\mathscr{P}(A)$, $$f(B)= \begin{cases} (\{B\}\cup[A\setminus B]^1)\setminus\{\varnothing\} & \text{if $B$ is not a singleton,}\\ \{\{a,x\},E\setminus\{a,x\}\}\cup[A\setminus(B\cup E)]^1 & \text{if $B=\{x\}$ for some $x\neq a$,}\\ \{\{a\},\{b,c\},\{d,e\}\}\cup[A\setminus E]^1 & \text{if $B=\{a\}$,} \end{cases}$$ and $$g(C)= \begin{cases} \{C,A\setminus C\}\setminus\{\varnothing\} & \text{if $a\notin C$,}\\ (\{C,A\setminus(C\cup E)\}\cup[E\setminus C]^1)\setminus\{\varnothing\} & \text{if $a\in C$, $|C\cap E|\leqslant3$,}\\ \{C\setminus\{a\},\{a\},E\setminus C,A\setminus(C\cup E)\}\setminus\{\varnothing\} & \text{if $a\in C$, $|C\cap E|=4$,}\\ \{C\setminus\{b,c,d,e\},\{b,c\},\{d,e\},A\setminus C\}\setminus\{\varnothing\} & \text{if $E\subseteq C$,} \end{cases}$$ where $[D]^1$ denotes the set of $1$-element subsets of $D$. It is easy to see that $f$ and $g$ are injective. ◻ The following corollary immediately follows from Facts [Fact 8](#sh03){reference-type="ref" reference="sh03"} and [Fact 18](#sh12){reference-type="ref" reference="sh12"}. **Corollary 19**. *If $A$ is infinite, then $\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$ and $\mathscr{B}(A)$ are power Dedekind infinite.* For infinite sets $A$, the relations between the cardinalities of $\mathop{\mathrm{fin}}(A)$, $\mathscr{P}(A)$, $\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$, $\mathscr{B}(A)$, $\mathop{\mathrm{Part_{fin}}}(A)$, $\mathop{\mathrm{Part}}(A)$, and $\mathscr{P}(A^2)$ can be visualized by the following diagram (where for two sets $X$ and $Y$, $X\longrightarrow Y$ means $|X|\leqslant|Y|$, $X \tikz[baseline=-\the\dimexpr\fontdimen 22\textfont 2\relax]{ \node[anchor=south,font=\scriptsize, inner ysep=1.5pt,outer xsep=2.2pt](x){\hspace*{4mm}}; \draw[shorten <=3.4pt,shorten >=3.4pt,dashed,->](x.south west)--(x.south east); } Y$ means $|X|<|Y|$, and $X \tikz[baseline=-\the\dimexpr\fontdimen 22\textfont 2\relax]{ \node[anchor=south,font=\scriptsize, inner ysep=1.5pt,outer xsep=2.2pt](x){\hspace*{4mm}}; \draw[shorten <=3.4pt,shorten >=3.4pt,dashed,-](x.south west)--(x.south east); } Y$ means $|X|\neq|Y|$). In the above diagram, the $\leqslant$-relations have already been established, and the inequalities $|\mathrm{fin}(A)|<|\mathscr{P}(A)|$ and $|\mathrm{fin}(A)|<|\mathscr{B}(A)|$ are proved in [@HalbeisenShelah1994 Theorem 3] and [@PhansamdaengVejjajiva2022 Theorem 3.7], respectively. The inequalities $|\mathscr{B}_{\mathrm{fin}}(A)|<|\mathrm{Part}_{\mathrm{fin}}(A)|$ and $|\mathscr{B}_{\mathrm{fin}}(A)|\neq|\mathscr{P}(A)|$ will be proved in Section [4](#sh00){reference-type="ref" reference="sh00"}. The other relations not indicted in the diagram cannot be proved in $\mathsf{ZF}$, as shown in the next section. # Permutation models and consistency results We refer the readers to [@Halbeisen2017 Chap. 8] or [@Jech1973 Chap. 4] for an introduction to the theory of permutation models. Permutation models are not models of $\mathsf{ZF}$; they are models of $\mathsf{ZFA}$ (the Zermelo--Fraenkel set theory with atoms). Nevertheless, they indirectly give, via the Jech--Sochor theorem (cf. [@Halbeisen2017 Theorem 17.2] or [@Jech1973 Theorem 6.1]), models of $\mathsf{ZF}$. Let $A$ be the set of atoms, let $\mathcal{G}$ be a group of permutations of $A$, and let $\mathfrak{F}$ be a normal filter on $\mathcal{G}$. We write $\mathop{\mathrm{sym_{\mathcal{G}}}}(x)$ for the set $\{\pi\in\mathcal{G}\mid\pi x=x\}$, where $\pi\in\mathcal{G}$ extends to a permutation of the universe by $$\pi x=\{\pi y\mid y\in x\}.$$ Then $x$ belongs to the permutation model $\mathcal{V}$ determined by $\mathcal{G}$ and $\mathfrak{F}$ if and only if $x\subseteq\mathcal{V}$ and $\mathop{\mathrm{sym_{\mathcal{G}}}}(x)\in\mathfrak{F}$. For each $E\subseteq A$, we write $\mathop{\mathrm{fix_{\mathcal{G}}}}(E)$ for the set $\{\pi\in\mathcal{G}\mid\forall a\in E(\pi a=a)\}$. Let $\mathcal{I}\subseteq\mathscr{P}(A)$ be a normal ideal and let $\mathfrak{F}$ be the normal filter on $\mathcal{G}$ generated by the subgroups $\{\mathop{\mathrm{fix_{\mathcal{G}}}}(E)\mid E\in\mathcal{I}\}$. Then $x$ belongs to the permutation model $\mathcal{V}$ determined by $\mathcal{G}$ and $\mathcal{I}$ if and only if $x\subseteq\mathcal{V}$ and there exists an $E\in\mathcal{I}$ such that $\mathop{\mathrm{fix_{\mathcal{G}}}}(E)\subseteq\mathop{\mathrm{sym_{\mathcal{G}}}}(x)$; that is, every $\pi\in\mathcal{G}$ fixing $E$ pointwise also fixes $x$. Such an $E$ is called a *support* of $x$. ## A model for $|\mathscr{B}(A)|\leqslant^\ast|A|$ and $|\mathscr{B}(A)|<|\mathscr{P}(A)|$ We construct a permutation model $\mathcal{V}_\mathscr{B}$ in which the set $A$ of atoms satisfies $|\mathscr{B}(A)|\leqslant^\ast|A|$ and $|\mathscr{B}(A)|<|\mathscr{P}(A)|$. The atoms are constructed by recursion as follows: 1. $A_0=\varnothing$ and $\mathcal{G}_0=\{\varnothing\}$ is the group of all permutations of $A_0$. 2. $A_{n+1}=A_n\cup\{(n,P,k)\mid P\in\mathscr{B}(A_n)\text{ and }k\in\omega\}$. 3. $\mathcal{G}_{n+1}$ is the group of permutations of $A_{n+1}$ consisting of all permutations $h$ for which there exists a $g\in\mathcal{G}_n$ such that - $g=h{\upharpoonright}A_n$; - for each $P\in\mathscr{B}(A_n)$, there exists a permutation $q$ of $\omega$ such that $h(n,P,k)=(n,\{g[D]\mid D\in P\},q(k))$ for all $k\in\omega$. Let $A=\bigcup_{n\in\omega}A_n$ be the set of atoms, let $\mathcal{G}$ be the group of permutations of $A$ consisting of all permutations $\pi$ such that $\pi{\upharpoonright}A_n\in\mathcal{G}_n$ for all $n\in\omega$, and let $\mathfrak{F}$ be the normal filter on $\mathcal{G}$ generated by the subgroups $\{\mathop{\mathrm{fix_{\mathcal{G}}}}(A_n)\mid n\in\omega\}$. The permutation model determined by $\mathcal{G}$ and $\mathfrak{F}$ is denoted by $\mathcal{V}_\mathscr{B}$. **Lemma 20**. *For every $P\in\mathscr{B}(A)$, $P\in\mathcal{V}_\mathscr{B}$ if and only if $\mathop{\mathrm{ns}}(P)\subseteq\mathscr{P}(A_m)$ for some $m\in\omega$.* *Proof.* Let $P\in\mathscr{B}(A)$. If $\mathop{\mathrm{ns}}(P)\subseteq\mathscr{P}(A_m)$ for some $m\in\omega$, then clearly $\mathop{\mathrm{fix_{\mathcal{G}}}}(A_m)\subseteq\mathop{\mathrm{sym_{\mathcal{G}}}}(P)$, which implies that $P\in\mathcal{V}_\mathscr{B}$. For the other direction, suppose $P\in\mathcal{V}_\mathscr{B}$ and let $m\in\omega$ be such that $\mathop{\mathrm{fix_{\mathcal{G}}}}(A_m)\subseteq\mathop{\mathrm{sym_{\mathcal{G}}}}(P)$; that is, every $\pi\in\mathcal{G}$ fixing $A_m$ pointwise also fixes $P$. We claim $\mathop{\mathrm{ns}}(P)\subseteq\mathscr{P}(A_m)$. Assume towards a contradiction that $x\sim_Py$ for some distinct $x,y$ such that one of $x$ and $y$ is not in $A_m$. Suppose that $x=(n,Q,k)$ and $y=(n',Q',k')$, and assume without loss of generality $n'\leqslant n$. Then $x\notin A_m$ and thus $m\leqslant n$. Let $l\in\omega$ be such that $(n,Q,l)\notin[y]_P$ and let $q$ be the transposition that swaps $k$ and $l$. Since $P$ is finitary, such an $l$ exists. Let $h$ be the permutation of $A_{n+1}$ such that $h$ fixes $A_n$ pointwise and for all $R\in\mathscr{B}(A_n)$ and all $j\in\omega$, $h(n,R,j)=(n,R,q(j))$ if $R=Q$, and $h(n,R,j)=(n,R,j)$ otherwise. Then $h\in\mathcal{G}_{n+1}$ fixes $A_{n+1}\setminus\{x,(n,Q,l)\}$ pointwise. Hence $h(y)=y$. Extend $h$ in a straightforward way to some $\pi\in\mathcal{G}$. Then $\pi\in\mathop{\mathrm{fix_{\mathcal{G}}}}(A_m\cup\{y\})$ and $\pi(x)=(n,Q,l)\notin[y]_P$. Thus $\pi$ moves $P$, which is a contradiction. ◻ **Lemma 21**. *In $\mathcal{V}_\mathscr{B}$, $|\mathscr{B}(A)|\leqslant^\ast|A|$ and $|\mathscr{B}(A)|<|\mathscr{P}(A)|$.* *Proof.* Let $\Phi$ be the function on $\{P\in\mathscr{B}(A)\mid\exists m\in\omega(\mathop{\mathrm{ns}}(P)\subseteq\mathscr{P}(A_m))\}$ defined by $$\Phi(u)=\{(n_P,P\cap\mathscr{P}(A_{n_P}),k)\mid k\in\omega\},$$ where $n_P$ is the least $m\in\omega$ such that $\mathop{\mathrm{ns}}(P)\subseteq\mathscr{P}(A_m)$. Clearly, $\Phi\in\mathcal{V}_\mathscr{B}$. In $\mathcal{V}_\mathscr{B}$, by Lemma [Lemma 20](#sh16){reference-type="ref" reference="sh16"}, $\Phi$ is an injection from $\mathscr{B}(A)$ into $\mathscr{P}(A)$, and the sets in the range of $\Phi$ are pairwise disjoint, which implies that $|\mathscr{B}(A)|\leqslant^\ast|A|$. Since $|\mathscr{P}(A)|\nleqslant^\ast|A|$ by Cantor's theorem, it follows that $|\mathscr{B}(A)|<|\mathscr{P}(A)|$. ◻ Now the next theorem immediately follows from Lemma [Lemma 21](#sh17){reference-type="ref" reference="sh17"} and the Jech--Sochor theorem. **Theorem 22**. *It is consistent with $\mathsf{ZF}$ that there exists an infinite set $A$ for which $|\mathscr{B}(A)|\leqslant^\ast|A|$ and $|\mathscr{B}(A)|<|\mathscr{P}(A)|$.* ## A model for $|\mathscr{P}(A)|<|\mathscr{B}(A)|<|\mathrm{Part}_{\mathrm{fin}}(A)|$ We show that the ordered Mostowski model $\mathcal{V}_\mathrm{M}$ (cf. [@Halbeisen2017 pp. 198--202] or [@Jech1973 §4.5]) is a model of this kind. Recall that the set $A$ of atoms carries an ordering $<_\mathrm{M}$ which is isomorphic to the ordering of the rational numbers, the permutation group $\mathcal{G}$ consists of all automorphisms of $\langle A,<_\mathrm{M}\rangle$, and $\mathcal{V}_\mathrm{M}$ is determined by $\mathcal{G}$ and finite supports. Clearly, the ordering $<_\mathrm{M}$ belongs to $\mathcal{V}_\mathrm{M}$ (cf. [@Halbeisen2017 Lemma 8.10]). In $\mathcal{V}_\mathrm{M}$, $A$ is infinite but power Dedekind finite (cf. [@Halbeisen2017 Lemma 8.13]), and thus, by Fact [Fact 6](#sh02){reference-type="ref" reference="sh02"} and Corollary [Corollary 15](#sh10){reference-type="ref" reference="sh10"}, $\mathscr{P}(A^2)$ and $\mathscr{B}(A)$ are Dedekind finite. **Lemma 23**. *In $\mathcal{V}_\mathrm{M}$, $\mathscr{B}(A)=\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$.* *Proof.* Let $P\in\mathcal{V}_\mathrm{M}$ be a finitary partition of $A$ and let $E$ be a finite support of $P$. We claim $\mathop{\mathrm{ns}}(P)\subseteq\mathscr{P}(E)$. Assume towards a contradiction that $x\sim_Py$ for some distinct $x,y$ such that $x\notin E$. Since $P$ is finitary, we can find a $\pi\in\mathop{\mathrm{fix_{\mathcal{G}}}}(E\cup\{y\})$ such that $\pi(x)\notin[y]_P$. Hence $\pi$ moves $P$, contradicting that $E$ is a support of $P$. Thus $\mathop{\mathrm{ns}}(P)\subseteq\mathscr{P}(E)$, so $P\in\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$. ◻ The next two lemmas are Lemmas 8.11(b) and 8.12 of [@Halbeisen2017], respectively. **Lemma 24**. *Every $x\in\mathcal{V}_\mathrm{M}$ has a least support.* **Lemma 25**. *If $E$ is an $n$-element subset of $A$, then $E$ supports exactly $2^{2n+1}$ subsets of $A$.* **Lemma 26**. *For each $n\in\omega$, let $B_n^\star$ be the number of partitions of $n$ without singleton blocks; that is, $$B_n^\star=|\{P\in\mathscr{B}(n)\mid\mathop{\mathrm{ns}}(P)=P\}|.$$ If $n\geqslant23$, then $2^{2n+2}<B_n^\star$.* *Proof.* For each $n\in\omega$, let $B_n$ be the $n$-th Bell number; that is, $B_n=|\mathscr{B}(n)|$. Recall Dobinski's formula (see, for example, [@Rota1964]): $$B_n=\frac{1}{e}\sum_{k=0}^\infty\frac{k^n}{k!}.$$ It is easy to see that $B_n=B_n^\star+B_{n+1}^\star$. Hence, for $n\geqslant23$, we have $$B_n^\star>\frac{B_{n-1}}{2}>\frac{8^{n-1}}{2e\cdot8!}=\frac{2^{n-5}}{2e\cdot8!}\cdot2^{2n+2} \geqslant\frac{2^{18}}{2e\cdot8!}\cdot2^{2n+2}>2^{2n+2}.\qedhere$$ ◻ **Lemma 27**. *In $\mathcal{V}_\mathrm{M}$, $|\mathscr{P}(A)|<|\mathscr{B}(A)|<|\mathrm{Part}_{\mathrm{fin}}(A)|<|\mathrm{Part}(A)|<|\mathscr{P}(A^2)|$.* *Proof.* In $\mathcal{V}_\mathrm{M}$, since $\mathscr{P}(A^2)$ is Dedekind finite, and the injections constructed in the proofs of Facts [Fact 14](#sh09){reference-type="ref" reference="sh09"} and [Fact 17](#sh14){reference-type="ref" reference="sh14"} are clearly not surjective, it follows that $|\mathscr{B}_{\mathrm{fin}}(A)|<|\mathrm{Part}_{\mathrm{fin}}(A)|<|\mathrm{Part}(A)|<|\mathscr{P}(A^2)|$. By Lemma [Lemma 23](#sh19){reference-type="ref" reference="sh19"}, it remains to show that $|\mathscr{P}(A)|<|\mathscr{B}(A)|$. For a finite subset $E$ of $A$, we can use $<_\mathrm{M}$ to define an ordering of the subsets of $A$ supported by $E$ and an ordering of the finitary partitions $P$ of $A$ with $\mathop{\mathrm{ns}}(P)\subseteq\mathscr{P}(E)$. Let $D=\{a_i\mid i<46\}$ be a $46$-element subset of $A$. In $\mathcal{V}_\mathrm{M}$, we define an injection $f$ from $\mathscr{P}(A)$ into $\mathscr{B}(A)$ as follows. Let $C\in\mathcal{V}_\mathrm{M}$ be a subset of $A$. By Lemma [Lemma 24](#sh20){reference-type="ref" reference="sh20"}, $C$ has a least support $E$. Suppose that $C$ is the $k$-th subset of $A$ with $E$ as its least support. Let $n=|D\bigtriangleup E|$, where $\bigtriangleup$ denotes the symmetric difference. Now, if $|E|\geqslant23$, define $f(C)$ to be the $k$-th finitary partition $P$ of $A$ with $\bigcup\mathop{\mathrm{ns}}(P)=E$; otherwise, define $f(C)$ to be the $(B_n^\star-k-1)$-th finitary partition $P$ of $A$ with $\bigcup\mathop{\mathrm{ns}}(P)=D\bigtriangleup E$. In the second case, $n\geqslant23$, and thus $B_n^\star-k-1>2^{2n+1}$ by Lemmas [Lemma 25](#sh21){reference-type="ref" reference="sh21"} and [Lemma 26](#sh22){reference-type="ref" reference="sh22"}. Thus $f$ is injective. Since $D$ is a finite support of $f$, it follows that $f\in\mathcal{V}_\mathrm{M}$. Finally, since $\mathscr{B}(A)$ is Dedekind finite and $f$ is not surjective, it follows that $|\mathscr{P}(A)|<|\mathscr{B}(A)|$. ◻ Now the next theorem immediately follows from Lemmas [Lemma 23](#sh19){reference-type="ref" reference="sh19"} and [Lemma 27](#sh23){reference-type="ref" reference="sh23"} and the Jech--Sochor theorem. **Theorem 28**. *It is consistent with $\mathsf{ZF}$ that there exists an infinite set $A$ for which $\mathscr{B}(A)=\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$ and $|\mathscr{P}(A)|<|\mathscr{B}(A)|<|\mathrm{Part}_{\mathrm{fin}}(A)|<|\mathrm{Part}(A)|<|\mathscr{P}(A^2)|$.* ## A model in which $\mathscr{B}(A)$ is incomparable with $\mathscr{P}(A)$ or $\mathop{\mathrm{Part_{fin}}}(A)$ We use a variation of the basic Fraenkel model (cf. [@Halbeisen2017 pp. 195--196] or [@Jech1973 §4.3]). Let $A$ be an uncountable set of atoms, let $\mathcal{G}$ be the group of all permutations of $A$, and let $\mathcal{V}_\mathrm{F}$ be the permutation model determined by $\mathcal{G}$ and countable supports. **Lemma 29**. *In $\mathcal{V}_\mathrm{F}$, $|\mathscr{B}(A)|\nleqslant|\mathrm{Part}_{\mathrm{fin}}(A)|$, $|\mathscr{B}_{\mathrm{fin}}(A)|\nleqslant|\mathscr{P}(A)|$, and $|\mathscr{P}(A)|\nleqslant|\mathscr{B}(A)|$.* *Proof.* (1) $|\mathscr{B}(A)|\nleqslant|\mathrm{Part}_{\mathrm{fin}}(A)|$. Assume towards a contradiction that in $\mathcal{V}_\mathrm{F}$ there is an injection $f$ from $\mathscr{B}(A)$ into $\mathop{\mathrm{Part_{fin}}}(A)$. Let $B$ be a countable support of $f$. Let $\{a_n\mid n\in\omega\}\subseteq A\setminus B$ with $a_i\neq a_j$ whenever $i\neq j$. Consider the finitary partition $P=\{\{a_{2n},a_{2n+1}\}\mid n\in\omega\}\cup[A\setminus\{a_n\mid n\in\omega\}]^1\in\mathcal{V}_\mathrm{F}$. Since $f(P)$ is a finite partition, there must be $i,j\in\omega$ with $i\neq j$ such that $a_{2i}$ and $a_{2j}$ are in the same block of $f(P)$. The transposition that swaps $a_{2i}$ and $a_{2i+1}$ fixes $P$, and thus also fixes $f(P)$, which implies that $a_{2i+1}$ and $a_{2j}$ are also in the same block of $f(P)$. Hence, the transposition that swaps $a_{2i+1}$ and $a_{2j}$ fixes $f(P)$, but it moves $P$, contradicting that $f$ is injective. \(2\) $|\mathscr{B}_{\mathrm{fin}}(A)|\nleqslant|\mathscr{P}(A)|$. Assume towards a contradiction that in $\mathcal{V}_\mathrm{F}$ there is an injection $g$ from $\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$ into $\mathscr{P}(A)$. Let $C$ be a countable support of $g$. Let $a_0,a_1,a_2,a_3$ be four distinct elements of $A\setminus C$. Consider the finitary partition $P=\{\{a_0,a_1\},\{a_2,a_3\}\}\cup[A\setminus\{a_0,a_1,a_2,a_3\}]^1\in\mathcal{V}_\mathrm{F}$. Clearly, for any $i,j<4$ with $i\neq j$, there is a $\pi\in\mathop{\mathrm{fix_{\mathcal{G}}}}(C)$ such that $\pi(P)=P$ and $\pi(a_i)=a_j$. Hence, $\{a_0,a_1,a_2,a_3\}\subseteq g(P)$ or $\{a_0,a_1,a_2,a_3\}\subseteq A\setminus g(P)$. Thus, the transposition that swaps $a_0$ and $a_2$ fixes $g(P)$, but it moves $P$, contradicting that $g$ is injective. \(3\) $|\mathscr{P}(A)|\nleqslant|\mathscr{B}(A)|$. Assume towards a contradiction that in $\mathcal{V}_\mathrm{F}$ there is an injection $h$ from $\mathscr{P}(A)$ into $\mathscr{B}(A)$. Let $D$ be a countable support of $h$. Let $C$ be a denumerable subset of $A\setminus D$. Take $x\in C$ and $y\in A\setminus(C\cup D)$. We claim that $\{x\}\in h(C)$. Assume not; since $h(C)$ is finitary, we can find a $z\in C$ such that $z\notin[x]_{h(C)}$, and then the transposition that swaps $x$ and $z$ would fix $C$ but move $h(C)$, which is a contradiction. Similarly, $\{y\}\in h(C)$. Hence, the transposition that swaps $x$ and $y$ fixes $h(C)$, but it moves $C$, contradicting that $h$ is injective. ◻ Now the next theorem immediately follows from Fact [Fact 18](#sh12){reference-type="ref" reference="sh12"}, Lemma [Lemma 29](#sh25){reference-type="ref" reference="sh25"}, and the Jech--Sochor theorem. **Theorem 30**. *It is consistent with $\mathsf{ZF}$ that there exists an infinite set $A$ such that* 1. *$|\mathscr{B}(A)|$ and $|\mathrm{Part}_{\mathrm{fin}}(A)|$ are incomparable;* 2. *$|\mathscr{B}(A)|$ and $|\mathscr{P}(A)|$ are incomparable;* 3. *$|\mathscr{B}_{\mathrm{fin}}(A)|$ and $|\mathscr{P}(A)|$ are incomparable.* As easily seen, for infinite well-orderable $A$, $|A|=|\mathrm{fin}(A)|=|\mathscr{B}_{\mathrm{fin}}(A)|$ and $|\mathscr{P}(A)|=|\mathscr{B}(A)|=|\mathrm{Part}_{\mathrm{fin}}(A)|=|\mathrm{Part}(A)|=|\mathscr{P}(A^2)|$. Therefore, Theorems [Theorem 22](#sh18){reference-type="ref" reference="sh18"}, [Theorem 28](#sh24){reference-type="ref" reference="sh24"}, and [Theorem 30](#sh26){reference-type="ref" reference="sh26"} show that Diagram [\[sh15\]](#sh15){reference-type="ref" reference="sh15"} is optimal in the sense that the $\leqslant$, $<$, or $\neq$ relations between the cardinalities of these sets not indicted in the diagram cannot be proved in $\mathsf{ZF}$. However, we do not know whether $|\mathrm{Part}_{\mathrm{fin}}(A)|<|\mathscr{B}(A)|$ for some infinite set $A$ is consistent with $\mathsf{ZF}$. # Theorems in $\mathsf{ZF}$ {#sh00} In this section, we prove in $\mathsf{ZF}$ some results concerning $\mathscr{B}(A)$, as well as the inequalities $|\mathscr{B}_{\mathrm{fin}}(A)|<|\mathrm{Part}_{\mathrm{fin}}(A)|$ and $|\mathscr{B}_{\mathrm{fin}}(A)|\neq|\mathscr{P}(A)|$ indicted in Diagram [\[sh15\]](#sh15){reference-type="ref" reference="sh15"}. Although Cantor's theorem may fail for $\mathscr{B}(A)$, it does hold under the existence of auxiliary functions. **Definition 31**. Let $f$ be a function on $A$. An *auxiliary function* for $f$ is a function $g$ defined on $\mathop{\mathrm{ran}}(f)$ such that, for all $z\in\mathop{\mathrm{ran}}(f)$, $g(z)$ is a finitary partition of $f^{-1}[\{z\}]$ with at least one non-singleton block. **Lemma 32**. *From a function $f:A\to\mathscr{B}(A)$ and an auxiliary function for $f$, one can explicitly define a finitary partition of $A$ not in $\mathop{\mathrm{ran}}(f)$.* *Proof.* Use Cantor's diagonal construction. Let $f$ be a function from $A$ to $\mathscr{B}(A)$ and let $g$ be an auxiliary function for $f$. Let $h$ be the function on $\mathop{\mathrm{ran}}(f)$ defined by $$h(P)= \begin{cases} \{\{z\}\mid f(z)=P\} & \text{if $x\sim_Py$ for some distinct $x,y\in f^{-1}[\{P\}]$,}\\ g(P) & \text{otherwise.} \end{cases}$$ Then $\bigcup_{P\in\mathop{\mathrm{ran}}(f)}h(P)$ is a finitary partition of $A$ not in $\mathop{\mathrm{ran}}(f)$. ◻ ## A general result We prove a general result which states that, if $\mathscr{B}(A)$ is Dedekind infinite, then there are no Dedekind finite-to-one functions from $\mathscr{B}(A)$ to $\mathop{\mathrm{fin}}(A)$. **Lemma 33**. *For any infinite ordinal $\alpha$, one can explicitly define an injection $f:\alpha\times\alpha\to\alpha$.* *Proof.* See [@Specker1954 2.1]. ◻ **Lemma 34**. *From an infinite ordinal $\alpha$, one can explicitly define an injection $f:\mathop{\mathrm{fin}}(\alpha)\to\alpha$.* *Proof.* See [@Halbeisen2017 Theorem 5.19]. ◻ **Lemma 35**. *From a finite-to-one function $f:\alpha\to A$, where $\alpha$ is an infinite ordinal, one can explicitly define an injection $g:\alpha\to A$.* *Proof.* See [@Shen2017 Lemma 3.3]. ◻ The main idea of the following proof is originally from [@HalbeisenShelah1994 Theorem 3]. **Lemma 36**. *From an injection $f:\alpha\to\mathop{\mathrm{fin}}(A)$, where $\alpha$ is an infinite ordinal, one can explicitly define an injection $h:\alpha\to\mathop{\mathrm{fin}}(A)$ such that the sets in $\mathop{\mathrm{ran}}(h)$ are pairwise disjoint and contain at least two elements.* *Proof.* Let $f$ be an injection from $\alpha$ into $\mathop{\mathrm{fin}}(A)$, where $\alpha$ is an infinite ordinal. Let $\sim$ be the equivalence relation on $A$ defined by $$x\sim y\quad\text{if and only if}\quad\forall\beta<\alpha(x\in f(\beta)\leftrightarrow y\in f(\beta)).$$ Clearly, for every $x\in\bigcup\mathop{\mathrm{ran}}(f)$, the equivalence class $[x]_\sim$ is finite. We want to show that there are $\alpha$ many such equivalence classes. In order to prove this, define a function $\Psi$ on $\bigcup\mathop{\mathrm{ran}}(f)$ by $$\Psi(x)=\bigl\{\gamma<\alpha\bigm|x\in f(\gamma)\text{ and } \textstyle\bigcap\{f(\beta)\mid\beta<\gamma\text{ and }x\in f(\beta)\}\nsubseteq f(\gamma)\bigr\}.$$ We claim that, for all $x,y\in\bigcup\mathop{\mathrm{ran}}(f)$, $$\label{sh32} x\sim y\quad\text{if and only if}\quad\Psi(x)=\Psi(y).$$ Clearly, if $x\sim y$ then $\Psi(x)=\Psi(y)$. For the other direction, assume towards a contradiction that $\Psi(x)=\Psi(y)$ but not $x\sim y$. Let $\delta<\alpha$ be the least ordinal such that $x\in f(\delta)$ is not equivalent to $y\in f(\delta)$. Without loss of generality, assume that $x\in f(\delta)$ but $y\notin f(\delta)$. Since $y\notin f(\delta)$, we have $\delta\notin\Psi(y)=\Psi(x)$, which implies that $\bigcap\{f(\beta)\mid\beta<\delta\text{ and }x\in f(\beta)\}\subseteq f(\delta)$. Since, for all $\beta<\delta$, $x\in f(\beta)$ if and only if $y\in f(\beta)$, it follows that $y\in\bigcap\{f(\beta)\mid\beta<\delta\text{ and }x\in f(\beta)\}\subseteq f(\delta)$, which is a contradiction. We also claim that, for all $x\in\bigcup\mathop{\mathrm{ran}}(f)$, $$\label{sh33} \Psi(x)\in\mathop{\mathrm{fin}}(\alpha).$$ Let $\xi$ be the least ordinal such that $x\in f(\xi)$. Then $\xi$ is the first element of $\Psi(x)$. For all $\gamma,\delta\in\Psi(x)$ with $\gamma<\delta$, if $f(\xi)\cap f(\gamma)=f(\xi)\cap f(\delta)$, then $\bigcap\{f(\beta)\mid\beta<\delta\text{ and }x\in f(\beta)\}\subseteq f(\xi)\cap f(\gamma)\subseteq f(\delta)$, contradicting that $\delta\in\Psi(x)$. Hence, the function that maps each $\gamma\in\Psi(x)$ to $f(\xi)\cap f(\gamma)$ is an injection from $\Psi(x)$ into $\mathscr{P}(f(\xi))$. Since $\mathscr{P}(f(\xi))$ is finite, it follows that $\Psi(x)$ is finite. Now, by Lemma [Lemma 34](#sh29){reference-type="ref" reference="sh29"}, we can explicitly define an injection $p:\mathop{\mathrm{fin}}(\alpha)\to\alpha$. By [\[sh33\]](#sh33){reference-type="eqref" reference="sh33"}, $\mathop{\mathrm{ran}}(\Psi)\subseteq\mathop{\mathrm{fin}}(\alpha)$. Let $R$ be the well-ordering of $\mathop{\mathrm{ran}}(\Psi)$ induced by $p$; that is, $R=\{(a,b)\mid a,b\in\mathop{\mathrm{ran}}(\Psi)\text{ and }p(a)<p(b)\}$. Let $\theta$ be the order type of $\langle\mathop{\mathrm{ran}}(\Psi),R\rangle$, and let $\Theta$ be the unique isomorphism of $\langle\mathop{\mathrm{ran}}(\Psi),R\rangle$ onto $\langle\theta,\in\rangle$. It is easy to see that $\theta$ is an infinite ordinal. Again, by Lemma [Lemma 34](#sh29){reference-type="ref" reference="sh29"}, we can explicitly define an injection $q:\mathop{\mathrm{fin}}(\theta)\to\theta$. By [\[sh32\]](#sh32){reference-type="eqref" reference="sh32"}, the function that maps each $\beta<\alpha$ to $\Psi[f(\beta)]$ is an injection from $\alpha$ into $\mathop{\mathrm{fin}}(\mathop{\mathrm{ran}}(\Psi))$. Let $g$ be the function on $\alpha$ defined by $$g(\beta)=\Theta^{-1}(q(\Theta[\Psi[f(\beta)]])).$$ $g$ is visualized by the following diagram: $$\begin{matrix} g: & \alpha & \to & \mathop{\mathrm{fin}}(\mathop{\mathrm{ran}}(\Psi)) & \to & \mathop{\mathrm{fin}}(\theta) & \to & \theta & \to & \mathop{\mathrm{ran}}(\Psi)\\ & \beta & \mapsto & \Psi[f(\beta)] & \mapsto & \Theta[\Psi[f(\beta)]] & \mapsto & q(\Theta[\Psi[f(\beta)]]) & \mapsto & g(\beta). \end{matrix}$$ Hence, $g$ is an injection from $\alpha$ into $\mathop{\mathrm{ran}}(\Psi)$. By Lemma [Lemma 33](#sh28){reference-type="ref" reference="sh28"}, we can explicitly define an injection $s:\alpha\times\alpha\to\alpha$. Then the function $h$ on $\alpha$ defined by $$h(\beta)=\Psi^{-1}[\{g(s(\beta,0)),g(s(\beta,1))\}]$$ is the required function. ◻ **Corollary 37**. *From an injection $f:\alpha\to\mathop{\mathrm{fin}}(A)$, where $\alpha$ is an infinite ordinal, one can explicitly define a surjection $g:A\twoheadrightarrow\alpha$ and an auxiliary function for $g$.* *Proof.* Let $f$ be an injection from $\alpha$ into $\mathop{\mathrm{fin}}(A)$, where $\alpha$ is an infinite ordinal. By Lemma [Lemma 36](#sh31){reference-type="ref" reference="sh31"}, we can explicitly define an injection $h:\alpha\to\mathop{\mathrm{fin}}(A)$ such that the sets in $\mathop{\mathrm{ran}}(h)$ are pairwise disjoint and contain at least two elements. Now, the function $g$ on $A$ defined by $$g(x)= \begin{cases} \text{the unique }\beta<\alpha\text{ for which }x\in h(\beta) & \text{if $x\in\bigcup\mathop{\mathrm{ran}}(h)$,}\\ 0 & \text{otherwise,} \end{cases}$$ is a surjection from $A$ onto $\alpha$, and the function $t$ on $\alpha$ defined by $$t(\beta)= \begin{cases} \{h(0)\}\cup\{\{x\}\mid x\in A\setminus\bigcup\mathop{\mathrm{ran}}(h)\} & \text{if $\beta=0$,}\\ \{h(\beta)\} & \text{otherwise,} \end{cases}$$ is an auxiliary function for $g$. ◻ Now we are ready to prove our main theorem. **Theorem 38**. *If $\mathscr{B}(A)$ is Dedekind infinite, then there are no Dedekind finite-to-one functions from $\mathscr{B}(A)$ to $\mathop{\mathrm{fin}}(A)$.* *Proof.* Assume towards a contradiction that $\mathscr{B}(A)$ is Dedekind infinite and there is a Dedekind finite-to-one function $\Phi:\mathscr{B}(A)\to\mathop{\mathrm{fin}}(A)$. Let $h$ be an injection from $\omega$ into $\mathscr{B}(A)$. In what follows, we get a contradiction by constructing by recursion an injection $H$ from the proper class of ordinals into the set $\mathscr{B}(A)$. For $n\in\omega$, take $H(n)=h(n)$. Now, we assume that $\alpha$ is an infinite ordinal and $H{\upharpoonright}\alpha$ is an injection from $\alpha$ into $\mathscr{B}(A)$. Then $\Phi\circ(H{\upharpoonright}\alpha)$ is a Dedekind finite-to-one function from $\alpha$ to $\mathop{\mathrm{fin}}(A)$. Since all Dedekind finite subsets of $\alpha$ are finite, $\Phi\circ(H{\upharpoonright}\alpha)$ is finite-to-one. By Lemma [Lemma 35](#sh30){reference-type="ref" reference="sh30"}, $\Phi\circ(H{\upharpoonright}\alpha)$ explicitly provides an injection $f:\alpha\to\mathop{\mathrm{fin}}(A)$. Therefore, by Corollary [Corollary 37](#sh34){reference-type="ref" reference="sh34"}, from $f$, we can explicitly define a surjection $g:A\twoheadrightarrow\alpha$ and an auxiliary function $t$ for $g$. Then $(H{\upharpoonright}\alpha)\circ g$ is a surjection from $A$ onto $H[\alpha]$ and $t\circ(H{\upharpoonright}\alpha)^{-1}$ is an auxiliary function for $(H{\upharpoonright}\alpha)\circ g$. Hence, it follows from Lemma [Lemma 32](#sh27){reference-type="ref" reference="sh27"} that we can explicitly define an $H(\alpha)\in\mathscr{B}(A)\setminus H[\alpha]$ from $H{\upharpoonright}\alpha$ (and $\Phi$). ◻ We draw some corollaries from the above theorem. The next two corollaries immediately follows from Theorem [Theorem 38](#sh35){reference-type="ref" reference="sh35"} and Facts [Fact 9](#sh04){reference-type="ref" reference="sh04"}, [Fact 10](#sh05){reference-type="ref" reference="sh05"}, and [Fact 16](#sh11){reference-type="ref" reference="sh11"}. **Corollary 39**. *If $\mathscr{B}(A)$ is Dedekind infinite, then there are no Dedekind finite-to-one functions from $\mathscr{B}(A)$ to $\mathop{\mathrm{seq^{1-1}}}(A)$.* **Corollary 40**. *If $\mathscr{B}(A)$ is Dedekind infinite, then there are no Dedekind finite-to-one functions from $\mathscr{B}(A)$ to $\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$, and thus $|\mathscr{B}_{\mathrm{fin}}(A)|<|\mathscr{B}(A)|$.* **Corollary 41**. *If $\mathscr{B}(A)\neq\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$, then $|\mathscr{B}_{\mathrm{fin}}(A)|<|\mathscr{B}(A)|$.* *Proof.* Suppose $\mathscr{B}(A)\neq\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$. Clearly, $|\mathscr{B}_{\mathrm{fin}}(A)|\leqslant|\mathscr{B}(A)|$. If $|\mathscr{B}(A)|=|\mathscr{B}_{\mathrm{fin}}(A)|$, since $\mathop{\mathrm{\mathscr{B}_{fin}}}(A)\subset\mathscr{B}(A)$, it follows that $\mathscr{B}(A)$ is Dedekind infinite, contradicting Corollary [Corollary 40](#sh37){reference-type="ref" reference="sh37"}. Hence, $|\mathscr{B}_{\mathrm{fin}}(A)|<|\mathscr{B}(A)|$. ◻ The following corollary is also proved in [@PhansamdaengVejjajiva2022 Theorem 3.7]. **Corollary 42**. *For all infinite sets $A$, $|\mathrm{fin}(A)|<|\mathscr{B}(A)|$.* *Proof.* By Fact [Fact 18](#sh12){reference-type="ref" reference="sh12"}, $|\mathrm{fin}(A)|\leqslant|\mathscr{B}_{\mathrm{fin}}(A)|\leqslant|\mathscr{B}(A)|$. If $|\mathscr{B}(A)|=|\mathrm{fin}(A)|$, since the injection constructed in the proof of Fact [Fact 18](#sh12){reference-type="ref" reference="sh12"} is not surjective, it follows that $\mathscr{B}(A)$ is Dedekind infinite, contradicting Theorem [Theorem 38](#sh35){reference-type="ref" reference="sh35"}. Thus, $|\mathrm{fin}(A)|<|\mathscr{B}(A)|$. ◻ ## $|\mathscr{B}(A)|\neq|\mathrm{seq}(A)|$ We need the following result, which is a Kuratowski-like theorem for $\mathscr{B}(A)$. **Theorem 43**. *For all sets $A$, the following are equivalent:* 1. *$\mathscr{B}(A)$ is Dedekind infinite;* 2. *$\mathop{\mathrm{ns}}(P)$ is power Dedekind infinite for some $P\in\mathscr{B}(A)$;* 3. *$\mathscr{P}(\omega)\preccurlyeq\mathscr{B}(A)$.* *Proof.* (i)${}\Rightarrow{}$(ii). Suppose that $\mathscr{B}(A)$ is Dedekind infinite. Assume towards a contradiction that $\mathop{\mathrm{ns}}(P)$ is power Dedekind finite for all $P\in\mathscr{B}(A)$. Let $\langle P_k\mid k\in\omega\rangle$ be a denumerable family of finitary partitions of $A$. We define by recursion a sequence $\langle Q_n\mid n\in\omega\rangle$ of finitary partitions of $A$ such that $\mathop{\mathrm{ns}}(Q_j)\neq\varnothing$ and $\bigcup\mathop{\mathrm{ns}}(Q_i)\cap\bigcup\mathop{\mathrm{ns}}(Q_j)=\varnothing$ whenever $i\neq j$ as follows. Let $n\in\omega$ and assume that $Q_i$ has already been defined for all $i<n$. Let $B=\bigcup_{i<n}\bigcup\mathop{\mathrm{ns}}(Q_i)$. By assumption, $\mathop{\mathrm{ns}}(Q_i)$ is power Dedekind finite, so is $\bigcup\mathop{\mathrm{ns}}(Q_i)$ by Fact [Fact 5](#sh01){reference-type="ref" reference="sh01"}. Hence, $B$ is power Dedekind finite. Consider the following two cases: [Case]{.smallcaps} 1. For a least $k\in\omega$, $x\sim_{P_k}y$ for some distinct $x,y\in A\setminus B$. Then we define $$Q_n=\{[z]_{P_k}\setminus B\mid z\in A\setminus B\}\cup\{\{z\}\mid z\in B\}.$$ It is easy to see that $Q_n$ is a finitary partition of $A$ such that $\mathop{\mathrm{ns}}(Q_n)\neq\varnothing$ and $\bigcup\mathop{\mathrm{ns}}(Q_i)\cap\bigcup\mathop{\mathrm{ns}}(Q_n)=\varnothing$ for all $i<n$. [Case]{.smallcaps} 2. Otherwise. Then, for all $k\in\omega$ and all $x\in\bigcup\mathop{\mathrm{ns}}(P_k)\setminus B$, $$\label{sh41} [x]_{P_k}\cap B\neq\varnothing\text{ and }[x]_{P_k}\setminus B=\{x\}.$$ Define by recursion a sequence $\langle k_m\mid m\in\omega\rangle$ of natural numbers as follows. Let $m\in\omega$ and assume that $k_j$ has been defined for $j<m$. By assumption, $\mathop{\mathrm{ns}}(P_{k_j})$ is power Dedekind finite, so is $\bigcup\mathop{\mathrm{ns}}(P_{k_j})$ by Fact [Fact 5](#sh01){reference-type="ref" reference="sh01"}. Therefore, $B\cup\bigcup_{j<m}\bigcup\mathop{\mathrm{ns}}(P_{k_j})$ is power Dedekind finite. Hence, there is a $k\in\omega$ such that $\mathop{\mathrm{ns}}(P_k)\nsubseteq\mathscr{P}(B\cup\bigcup_{j<m}\bigcup\mathop{\mathrm{ns}}(P_{k_j}))$. Define $k_m$ to be the least such $k$. Then $$\label{sh42} {\textstyle\bigcup}\mathop{\mathrm{ns}}(P_{k_m})\nsubseteq B\cup\bigcup_{j<m}{\textstyle\bigcup}\mathop{\mathrm{ns}}(P_{k_j}).$$ Let $f$ and $g$ be the functions on $\omega$ defined by $$\begin{aligned} f(m) & ={\textstyle\bigcup}\mathop{\mathrm{ns}}(P_{k_m})\setminus\bigl(B\cup\bigcup_{j<m}{\textstyle\bigcup}\mathop{\mathrm{ns}}(P_{k_j})\bigr)\\ g(m) & =\{[x]_{P_{k_m}}\cap B\mid x\in f(m)\}\end{aligned}$$ By [\[sh41\]](#sh41){reference-type="eqref" reference="sh41"}, for every $m\in\omega$, $g(m)$ is a finitary partition of a subset of $B$, and hence $\sim_{g(m)}$ is a subset of $\mathscr{P}(B^2)$. Since $\mathscr{P}(B^2)$ is Dedekind finite by Fact [Fact 6](#sh02){reference-type="ref" reference="sh02"}, there are least $l_0,l_1\in\omega$ with $l_0<l_1$ such that ${\sim_{g(l_0)}}={\sim_{g(l_1)}}$, and thus $g(l_0)=g(l_1)$. Let $$D=\bigl\{\{x,y\}\bigm|x\in f(l_0),y\in f(l_1)\text{ and }[x]_{P_{k_{l_0}}}\cap B=[y]_{P_{k_{l_1}}}\cap B\bigr\}.$$ Since $l_0<l_1$, $f(l_0)\cap f(l_1)=\varnothing$, and thus, by [\[sh41\]](#sh41){reference-type="eqref" reference="sh41"}, the sets in $D$ are pairwise disjoint. By [\[sh42\]](#sh42){reference-type="eqref" reference="sh42"}, $f(m)\neq\varnothing$ for all $m\in\omega$, and since $g(l_0)=g(l_1)$, it follows that $D\neq\varnothing$. Now, we define $$Q_n=D\cup\{\{z\}\mid z\in A\setminus\textstyle\bigcup D\}.$$ Then $Q_n$ is a finitary partition of $A$ such that $\mathop{\mathrm{ns}}(Q_n)=D\neq\varnothing$; since $B\cap\bigcup D=\varnothing$, it follows that $\bigcup\mathop{\mathrm{ns}}(Q_i)\cap\bigcup\mathop{\mathrm{ns}}(Q_n)=\varnothing$ for all $i<n$. Finally, $$Q=\bigcup_{n\in\omega}\mathop{\mathrm{ns}}(Q_n)\cup\bigl\{\{z\}\bigm|z\in A\setminus\bigcup_{n\in\omega}{\textstyle\bigcup}\mathop{\mathrm{ns}}(Q_n)\bigr\}$$ is a finitary partition of $A$ such that $\mathop{\mathrm{ns}}(Q)=\bigcup_{n\in\omega}\mathop{\mathrm{ns}}(Q_n)$ is power Dedekind infinite, which is a contradiction. (ii)${}\Rightarrow{}$(iii). Suppose that $P$ is a finitary partition of $A$ such that $\mathop{\mathrm{ns}}(P)$ is power Dedekind infinite. Let $p$ be a surjection from $\mathop{\mathrm{ns}}(P)$ onto $\omega$. Then the function $h$ on $\mathscr{P}(\omega)$ defined by $$h(u)=p^{-1}[u]\cup\{\{z\}\mid z\in A\setminus\textstyle\bigcup p^{-1}[u]\}$$ is an injection from $\mathscr{P}(\omega)$ into $\mathscr{B}(A)$. (iii)${}\Rightarrow{}$(i). Obviously. ◻ **Corollary 44**. *If $\mathscr{B}(A)$ is Dedekind infinite, then there are no Dedekind finite-to-one functions from $\mathscr{B}(A)$ to $\mathop{\mathrm{seq}}(A)$.* *Proof.* Assume towards a contradiction that $\mathscr{B}(A)$ is Dedekind infinite and there exists a Dedekind finite-to-one function from $\mathscr{B}(A)$ to $\mathop{\mathrm{seq}}(A)$. If $A$ is Dedekind infinite, then $\mathop{\mathrm{seq}}(A)\approx\mathop{\mathrm{seq^{1-1}}}(A)$ by Fact [Fact 13](#sh08){reference-type="ref" reference="sh08"}, contradicting Corollary [Corollary 39](#sh36){reference-type="ref" reference="sh36"}. Otherwise, by Fact [Fact 12](#sh07){reference-type="ref" reference="sh07"}, there is a Dedekind finite-to-one function from $\mathop{\mathrm{seq}}(A)$ to $\omega$. By Theorem [Theorem 43](#sh40){reference-type="ref" reference="sh40"}, $\mathscr{B}(\omega)\approx\mathscr{P}(\omega)\preccurlyeq\mathscr{B}(A)$, and therefore there is a Dedekind finite-to-one function from $\mathscr{B}(\omega)$ to $\omega$, contradicting again Corollary [Corollary 39](#sh36){reference-type="ref" reference="sh36"}. ◻ **Corollary 45**. *For all non-empty sets $A$, $|\mathscr{B}(A)|\neq|\mathrm{seq}(A)|$.* *Proof.* For all non-empty sets $A$, if $|\mathscr{B}(A)|=|\mathrm{seq}(A)|$, then it follows from Fact [Fact 11](#sh06){reference-type="ref" reference="sh06"} that $\mathscr{B}(A)$ is Dedekind infinite, contradicting Corollary [Corollary 44](#sh43){reference-type="ref" reference="sh43"}. ◻ We do not know whether $|\mathscr{B}(A)|\neq|\mathrm{seq}^{\text{1-1}}(A)|$ for all infinite sets $A$ is provable in $\mathsf{ZF}$. ## $|A^n|<|\mathscr{B}(A)|$ The main idea of the following proof is originally from [@Truss1973 Lemma(i)] (cf. also [@Shen2023b]). **Lemma 46**. *For all $n\in\omega$, if $|A|\geqslant 2n(2^{n+1}-1)$, then $|A^n|\leqslant|\mathscr{B}_{\mathrm{fin}}(A)|$.* *Proof.* Let $n\in\omega$ and let $A$ be a set with at least $2n(2^{n+1}-1)$ elements. Since $2^{i}=1+\sum_{k<i}2^k$, we can choose $2n(2^{n+1}-1)$ distinct elements of $A$ so that they are divided into $n+1$ sets $H_i$ ($i\leqslant n$) with $$H_i=\{a_{i,j}\mid j<2n\}\cup\{b_{i,x}\mid x\in H_k\text{ for some }k<i\}.$$ We construct an injection $f$ from $A^n$ into $\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$ as follows. Without loss of generality, assume that $A\cap\omega=\varnothing$. Let $s\in A^n$. Let $i_s$ be the least $i\leqslant n$ for which $\mathrm{ran}(s)\cap H_i=\varnothing$. There is such an $i$ because $|\mathrm{ran}(s)|\leqslant n$. Let $t_s$ be the function on $n$ defined by $$t_s(j)= \begin{cases} s(j) & \text{if $s(j)\neq s(k)$ for all $k<j$,}\\ \max\{k<j\mid s(j)=s(k)\} & \text{otherwise.} \end{cases}$$ Clearly, $t_s\in\mathop{\mathrm{seq^{1-1}}}(A\cup n)$. Let $u_s$ be the function on $n$ defined by $$u_s(j)= \begin{cases} a_{i_s,n+t_s(j)} & \text{if $t_s(j)\in n$,}\\ b_{i_s,t_s(j)} & \text{if $t_s(j)\in H_k$ for some $k<i_s$,}\\ t_s(j) & \text{otherwise.} \end{cases}$$ Then it is easy to see that $u_s\in\mathop{\mathrm{seq^{1-1}}}(A)$. Now, we define $$f(s)=\bigl\{\{a_{i_s,j},u_s(j)\}\bigm|j<n\bigr\}\cup\bigl\{\{z\}\bigm|z\in A\setminus(\{a_{i_s,j}\mid j<n\}\cup\mathop{\mathrm{ran}}(u_s))\bigr\}.$$ Clearly, $f(s)\in\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$. We prove that $f$ is injective by showing that $s$ is uniquely determined by $f(s)$ in the following way. First, $i_s$ is the least $i\leqslant n$ such that $H_i\cap\bigcup\mathop{\mathrm{ns}}(f(s))\neq\varnothing$. Second, $u_s$ is the function on $n$ such that $\{a_{i_s,j},u_s(j)\}\in f(s)$ for all $j<n$. Then, $t_s$ is the function on $n$ such that, for every $j<n$, either $t_s(j)$ is the unique element of $n$ for which $u_s(j)=a_{i_s,n+t_s(j)}$, or $t_s(j)$ is the unique element of $\bigcup_{k<i_s}H_k$ for which $u_s(j)=b_{i_s,t_s(j)}$, or $t_s(j)=u_s(j)\notin H_{i_s}$. Finally, $s$ is the function on $n$ recursively determined by $$s(j)= \begin{cases} t_s(j) & \text{if $t_s(j)\in A$,}\\ s(t_s(j)) & \text{otherwise.} \end{cases}$$ Hence, $f$ is an injection from $A^n$ into $\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$. ◻ **Corollary 47**. *For all $n\in\omega$ and all infinite sets $A$, $|A^n|<|\mathscr{B}(A)|$.* *Proof.* By Lemma [Lemma 46](#sh45){reference-type="ref" reference="sh45"}, $|A^n|\leqslant|\mathscr{B}_{\mathrm{fin}}(A)|\leqslant|\mathscr{B}(A)|$. If $|\mathscr{B}(A)|=|A^n|$, since the injection constructed in the proof of Lemma [Lemma 46](#sh45){reference-type="ref" reference="sh45"} is not surjective, $\mathscr{B}(A)$ is Dedekind infinite, contradicting Corollary [Corollary 44](#sh43){reference-type="ref" reference="sh43"}. Thus, $|A^n|<|\mathscr{B}(A)|$. ◻ **Lemma 48**. *If $A$ is Dedekind infinite, then $|\mathrm{seq}(A)|\leqslant|\mathscr{B}_{\mathrm{fin}}(A)|$.* *Proof.* Let $h$ be an injection from $\omega$ into $A$. By the proof of Lemma [Lemma 46](#sh45){reference-type="ref" reference="sh45"}, from $h{\upharpoonright}(2n(2^{n+1}-1))$, we can explicitly define an injection $$f_n:A^n\to\{P\in\mathscr{B}(A)\mid|\mathrm{ns}(P)|=n\}.$$ Then $\bigcup_{n\in\omega}f_n$ is an injection from $\mathop{\mathrm{seq}}(A)$ into $\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$. ◻ The following corollary immediately follows from Lemma [Lemma 48](#sh47){reference-type="ref" reference="sh47"} and Corollary [Corollary 45](#sh44){reference-type="ref" reference="sh44"}. **Corollary 49**. *For all Dedekind infinite sets $A$, $|\mathrm{seq}(A)|<|\mathscr{B}(A)|$.* ## A Cantor-like theorem for $\mathscr{B}(A)$ Under the assumption that there is a finitary partition of $A$ without singleton blocks, we show that Cantor's theorem holds for $\mathscr{B}(A)$. The key step of our proof is the following lemma. **Lemma 50**. *From a finitary partition $P$ of $A$ without singleton blocks and a surjection $f:A\twoheadrightarrow\alpha$, where $\alpha$ is an infinite ordinal, one can explicitly define a surjection $g:A\twoheadrightarrow\alpha$ and an auxiliary function for $g$.* *Proof.* Let $P$ be a finitary partition of $A$ without singleton blocks, and let $f$ be a surjection from $A$ onto $\alpha$, where $\alpha$ is an infinite ordinal. Let $$Q=\{f[E]\mid E\in P\}.$$ Clearly, $Q\subseteq\mathop{\mathrm{fin}}(\alpha)$. By Lemma [Lemma 34](#sh29){reference-type="ref" reference="sh29"}, we can explicitly define an injection $p:\mathrm{fin}(\alpha)\to\alpha$. Let $R$ be the well-ordering of $Q$ induced by $p$; that is, $R=\{(a,b)\mid a,b\in Q\text{ and }p(a)<p(b)\}$. Since $P$ is a partition of $A$, $Q$ is a cover of $\alpha$. Define a function $h$ from $\alpha$ to $Q$ by setting, for $\beta\in\alpha$, $$h(\beta)=\text{the $R$-least }c\in Q\text{ such that }\beta\in c.$$ Since $\beta\in h(\beta)$ and $h(\beta)$ is finite for all $\beta\in\alpha$, $h$ is a finite-to-one function from $\alpha$ to $Q$. Hence, by Lemma [Lemma 35](#sh30){reference-type="ref" reference="sh30"}, $h$ explicitly provides an injection from $\alpha$ into $Q$. Since $p{\upharpoonright}Q$ is an injection from $Q$ into $\alpha$, it follows from Theorem [Theorem 2](#cbt){reference-type="ref" reference="cbt"} that we can explicitly define a bijection $q$ between $Q$ and $\alpha$. Now, the function $g$ on $A$ defined by $$g(x)=q(f[[x]_P])$$ is a surjection from $A$ onto $\alpha$, and the function $t$ on $\alpha$ defined by $$t(\beta)=\{E\in P\mid q(f[E])=\beta\}$$ is an auxiliary function for $g$. ◻ **Theorem 51**. *For all infinite sets $A$, if there is a finitary partition of $A$ without singleton blocks, then there are no surjections from $A$ onto $\mathscr{B}(A)$.* *Proof.* Let $A$ be an infinite set and let $P$ be a finitary partition of $A$ without singleton blocks. Assume towards a contradiction that there is a surjection $\Phi:A\twoheadrightarrow\mathscr{B}(A)$. By Corollary [Corollary 19](#sh13){reference-type="ref" reference="sh13"}, $\mathscr{B}(A)$ is power Dedekind infinite, so is $A$. Since $\bigcup\mathop{\mathrm{ns}}(P)=A$ is power Dedekind infinite, so is $\mathop{\mathrm{ns}}(P)$ by Fact [Fact 5](#sh01){reference-type="ref" reference="sh01"}. Hence, by Theorem [Theorem 43](#sh40){reference-type="ref" reference="sh40"}, $\mathscr{B}(A)$ is Dedekind infinite. Let $h$ be an injection from $\omega$ into $\mathscr{B}(A)$. In what follows, we get a contradiction by constructing by recursion an injection $H$ from the proper class of ordinals into $\mathscr{B}(A)$. For $n\in\omega$, take $H(n)=h(n)$. Now, we assume that $\alpha$ is an infinite ordinal and $H{\upharpoonright}\alpha$ is an injection from $\alpha$ into $\mathscr{B}(A)$. Then $(H{\upharpoonright}\alpha)^{-1}\circ\Phi$ is a surjection from a subset of $A$ onto $\alpha$ and thus can be extended by zero to a surjection $f:A\twoheadrightarrow\alpha$. By Lemma [Lemma 50](#sh49){reference-type="ref" reference="sh49"}, from $P$ and $f$, we can explicitly define a surjection $g:A\twoheadrightarrow\alpha$ and an auxiliary function $t$ for $g$. Then $(H{\upharpoonright}\alpha)\circ g$ is a surjection from $A$ onto $H[\alpha]$ and $t\circ(H{\upharpoonright}\alpha)^{-1}$ is an auxiliary function for $(H{\upharpoonright}\alpha)\circ g$. Hence, it follows from Lemma [Lemma 32](#sh27){reference-type="ref" reference="sh27"} that we can explicitly define an $H(\alpha)\in\mathscr{B}(A)\setminus H[\alpha]$ from $H{\upharpoonright}\alpha$ (and $P,\Phi$). ◻ Under the same assumption, we also show that there are no finite-to-one functions from $\mathscr{B}(A)$ to $A^n$. **Theorem 52**. *For all infinite sets $A$, if there is a finitary partition of $A$ without singleton blocks, then there are no finite-to-one functions from $\mathscr{B}(A)$ to $A^n$ for every $n\in\omega$.* *Proof.* Let $A$ be an infinite set and let $P$ be a finitary partition of $A$ without singleton blocks. Assume towards a contradiction that there is a finite-to-one function from $\mathscr{B}(A)$ to $A^n$ for some $n\in\omega$. By Corollary [Corollary 19](#sh13){reference-type="ref" reference="sh13"}, $\mathscr{B}(A)$ is power Dedekind infinite, so is $A$ by Facts [Fact 5](#sh01){reference-type="ref" reference="sh01"} and [Fact 6](#sh02){reference-type="ref" reference="sh02"}. Since $\bigcup\mathop{\mathrm{ns}}(P)=A$ is power Dedekind infinite, so is $\mathop{\mathrm{ns}}(P)$ by Fact [Fact 5](#sh01){reference-type="ref" reference="sh01"}. Hence, by Theorem [Theorem 43](#sh40){reference-type="ref" reference="sh40"}, $\mathscr{B}(A)$ is Dedekind infinite, contradicting Corollary [Corollary 44](#sh43){reference-type="ref" reference="sh43"}. ◻ ## The inequalities $|\mathscr{B}_{\mathrm{fin}}(A)|<|\mathrm{Part}_{\mathrm{fin}}(A)|$ and $|\mathscr{B}_{\mathrm{fin}}(A)|\neq|\mathscr{P}(A)|$ **Lemma 53**. *If $A$ is power Dedekind infinite, then there are no finite-to-one functions from $\mathscr{P}(A)$ to $\mathop{\mathrm{fin}}(A)$.* *Proof.* See [@Shen2017 Corollary 3.7]. ◻ The next corollary immediately follows from Lemma [Lemma 53](#sh54){reference-type="ref" reference="sh54"} and Facts [Fact 10](#sh05){reference-type="ref" reference="sh05"} and [Fact 16](#sh11){reference-type="ref" reference="sh11"}. **Corollary 54**. *If $A$ is power Dedekind infinite, then there are no finite-to-one functions from $\mathscr{P}(A)$ to $\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$.* **Theorem 55**. *For all infinite sets $A$, $|\mathscr{B}_{\mathrm{fin}}(A)|<|\mathrm{Part}_{\mathrm{fin}}(A)|$.* *Proof.* Let $A$ be an infinite set. By Fact [Fact 17](#sh14){reference-type="ref" reference="sh14"}, $|\mathscr{B}_{\mathrm{fin}}(A)|\leqslant|\mathrm{Part}_{\mathrm{fin}}(A)|$. Assume $|\mathrm{Part}_{\mathrm{fin}}(A)|=|\mathscr{B}_{\mathrm{fin}}(A)|$. Since the injection constructed in the proof of Fact [Fact 17](#sh14){reference-type="ref" reference="sh14"} is not surjective, $\mathop{\mathrm{Part_{fin}}}(A)$ is Dedekind infinite, and thus $A$ is power Dedekind infinite by Corollary [Corollary 15](#sh10){reference-type="ref" reference="sh10"}. Now, by Fact [Fact 18](#sh12){reference-type="ref" reference="sh12"}, $|\mathscr{P}(A)|\leqslant|\mathrm{Part}_{\mathrm{fin}}(A)|=|\mathscr{B}_{\mathrm{fin}}(A)|$, contradicting Corollary [Corollary 54](#sh55){reference-type="ref" reference="sh55"}. Hence, $|\mathscr{B}_{\mathrm{fin}}(A)|<|\mathrm{Part}_{\mathrm{fin}}(A)|$. ◻ Finally, we prove $|\mathscr{B}_{\mathrm{fin}}(A)|\neq|\mathscr{P}(A)|$. For this, we need the following number-theoretic lemma. **Lemma 56**. *For each $n\in\omega$, let $B_n$ be the $n$-th Bell number; that is, $B_n=|\mathscr{B}(n)|$. Then $B_m$ is not a power of $2$ for all $m\geqslant3$.* *Proof.* Let $m\geqslant3$. Then $B_m>4$. It suffices to prove that $B_m$ is not divisible by $8$. By [@Lunnon1979 Theorem 6.4], $B_{n+24}\equiv B_n\pmod{8}$ for all $n\in\omega$. But $B_n$ modulo $8$ for $n$ from $0$ to $23$ are $$1,1,2,5,7,4,3,5,4,3,7,2,5,5,2,1,3,4,7,1,4,7,3,2.$$ Hence, $B_m$ is not divisible by $8$. ◻ **Lemma 57**. *If $\mathop{\mathrm{fin}}(\mathscr{P}(A))$ is Dedekind infinite, then $A$ is power Dedekind infinite.* *Proof.* See [@Shen2023a Theorem 3.2]. ◻ **Theorem 58**. *For all non-empty sets $A$, $|\mathscr{P}(A)|\neq|\mathscr{B}_{\mathrm{fin}}(A)|$.* *Proof.* If $A$ is a singleton, $|\mathscr{P}(A)|=2\neq1=|\mathscr{B}_{\mathrm{fin}}(A)|$. Suppose $|A|\geqslant2$, and fix two distinct elements $a,b$ of $A$. Assume toward a contradiction that there is a bijection $\Phi$ between $\mathscr{P}(A)$ and $\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$. We define by recursion an injection $f$ from $\omega$ into $\mathop{\mathrm{fin}}(\mathscr{P}(A))$ as follows. Take $f(0)=\{\{a\},\{b\}\}$. Let $n\in\omega$, and assume that $f(0),\dots,f(n)$ have been defined and are pairwise distinct elements of $\mathop{\mathrm{fin}}(\mathscr{P}(A))$. Let $\sim$ be the equivalence relation on $A$ defined by $$x\sim y\quad\text{if and only if}\quad\forall C\in f(0)\cup\dots\cup f(n)(x\in C\leftrightarrow y\in C).$$ Since $f(0),\dots,f(n)$ are finite, the quotient set $A/{\sim}$ is a finite partition of $A$. Let $k=|A/{\sim}|$ and let $U=\{\bigcup W\mid W\subseteq A/{\sim}\}$. Then $|U|=2^k$ and $$\label{sh58} f(0)\cup\dots\cup f(n)\subseteq U.$$ Since $\{a\},\{b\}\in A/{\sim}$, we have $k\geqslant2$. Let $D=\bigcup\{\bigcup\mathop{\mathrm{ns}}(P)\mid P\in\Phi[U]\}$. Since $U$ is finite and $\Phi[U]\subseteq\mathop{\mathrm{\mathscr{B}_{fin}}}(A)$, it follows that $D$ is finite. Let $m=|D|$ and let $E=\{P\in\mathop{\mathrm{\mathscr{B}_{fin}}}(A)\mid\bigcup\mathop{\mathrm{ns}}(P)\subseteq D\}$. Then $|E|=B_m$ and $\Phi[U]\subseteq E$. Hence, $2^k=|U|=|\Phi[U]|\leqslant|E|=B_m$. Since $k\geqslant2$, we have $m\geqslant3$, which implies that $B_m\neq2^k$ by Lemma [Lemma 56](#sh52){reference-type="ref" reference="sh52"}, and hence $\Phi[U]\subset E$. Now, we define $f(n+1)=\Phi^{-1}[E\setminus\Phi[U]]$. Then $f(n+1)$ is a non-void finite subset of $\mathscr{P}(A)$. By [\[sh58\]](#sh58){reference-type="eqref" reference="sh58"}, it follows that $f(n+1)$ is disjoint from each of $f(0),\dots,f(n)$, and thus is distinct from each of them. The existence of the above injection $f$ shows that $\mathop{\mathrm{fin}}(\mathscr{P}(A))$ is Dedekind infinite, which implies that, by Lemma [Lemma 57](#sh53){reference-type="ref" reference="sh53"}, $A$ is power Dedekind infinite, contradicting Corollary [Corollary 54](#sh55){reference-type="ref" reference="sh55"}. ◻ # Open questions We conclude the paper with the following five open questions. **Question 59**. Are the following statements consistent with $\mathsf{ZF}$? 1. There exist an infinite set $A$ and a finite-to-one function from $\mathscr{B}(A)$ to $A$. 2. There exists an infinite set $A$ for which $|\mathscr{B}(A)|<|\mathcal{S}_3(A)|$, where $\mathcal{S}_3(A)$ is the set of permutations of $A$ with exactly $3$ non-fixed points. 3. There exists an infinite set $A$ for which $|\mathscr{B}(A)|=|\mathrm{seq}^{\text{1-1}}(A)|$. 4. There exists an infinite set $A$ for which $|\mathrm{Part}_{\mathrm{fin}}(A)|<|\mathscr{B}(A)|$. 5. There exist an infinite set $A$ and a surjection from $A^2$ onto $\mathop{\mathrm{Part}}(A)$. ## Acknowledgements {#acknowledgements .unnumbered} I should like to give thanks to Professor Ira Gessel for providing a proof of Lemma [Lemma 56](#sh52){reference-type="ref" reference="sh52"}. The author was partially supported by National Natural Science Foundation of China grant number 12101466. 99 G. Cantor, *Über eine elementare Frage der Mannigfaltigkeitslehre*, Jahresber. Dtsch. Math.-Ver. 1 (1891), 75--78. J. W. Dawson, Jr. and P. E. Howard, *Factorials of infinite cardinals*, Fund. Math. 93 (1976), 185--195. T. Forster, *Finite-to-one maps*, J. Symb. Log. 68 (2003), 1251--1253. L. Halbeisen, *Combinatorial Set Theory: With a Gentle Introduction to Forcing*, 2nd ed., Springer Monogr. Math., Springer, Cham, 2017. L. Halbeisen and S. Shelah, *Consequences of arithmetic for set theory*, J. Symb. Log. 59 (1994), 30--40. T. Jech, *The Axiom of Choice*, Stud. Logic Found. Math. 75, North-Holland, Amsterdam, 1973. W. F. Lunnon, P. A. B. Pleasants and N. M. Stephens, *Arithmetic properties of Bell numbers to a composite modulus I*, Acta Arith. 39 (1979), 1--16. Y. Peng and G. Shen, *A generalized Cantor theorem in $\mathsf{ZF}$*, J. Symb. Log. (2022), to appear. Y. Peng, G. Shen and L. Wu, *A surjection from square onto power*, preprint (2022), arXiv:2207.13300. P. Phansamdaeng and P. Vejjajiva, *The cardinality of the partitions of a set in the absence of the axiom of choice*, Log. J. IGPL (2022), to appear. G. C. Rota, *The number of partitions of a set*, Amer. Math. Monthly 71 (1964), 498--504. G. Shen, *Generalizations of Cantor's theorem in $\mathsf{ZF}$*, Math. Log. Q. 63 (2017), 428--436. G. Shen, *The power set and the set of permutations with finitely many non-fixed points of a set*, Math. Log. Q. 69 (2023), 40--45. G. Shen, *On a cardinal inequality in $\mathsf{ZF}$*, Math. Log. Q. (2023), to appear. G. Shen and J. Yuan, *Factorials of infinite cardinals in $\mathsf{ZF}$ Part I: $\mathsf{ZF}$ results*, J. Symb. Log. 85 (2020), 224--243. G. Shen and J. Yuan, *Factorials of infinite cardinals in $\mathsf{ZF}$ Part II: Consistency results*, J. Symb. Log. 85 (2020), 244--270. N. Sonpanow and P. Vejjajiva, *Factorials and the finite sequences of sets*, Math. Log. Q. 65 (2019), 116--120. E. Specker, *Verallgemeinerte Kontinuumshypothese und Auswahlaxiom*, Arch. Math. 5 (1954), 332--337. A. Tarski, *Sur quelques théorèmes qui équivalent à l'axiome du choix*, Fund. Math. 5 (1924), 147--154. J. Truss, *The well-ordered and well-orderable subsets of a set*, Math. Log. Q. 19 (1973), 211--214.

arxiv_math

--- author: - Samantha Pilgrim bibliography: - mybibliography2.bib title: Topological Rigidity of the Dynamic Asymptotic Dimension --- #### Abstract: We show for a free action of a countable group $\Gamma$ on a compact metric space by homeomorphisms that the dynamic asymptotic dimension is either infinite or coincides with the asymptotic dimension of $\Gamma$. # Introduction The dynamic asymptotic dimension of a group action $\Gamma\curvearrowright X$ was first introduced in [@dasdimGWY]. Although known to be related to dimension theories for group actions which take into account both the asymptotic dimension of $\Gamma$ and the covering dimension of $X$ [@kerr2017dimension Theorem 5.14], there are many examples for which $\text{DAD}(\Gamma\curvearrowright X) = \text{asdim}\Gamma$. This was recently shown to always hold for finite dimensional actions on $0$-dimensional spaces (implied by a groupoid result in [@bonicke2023dynamic]); and for minimal actions by virtually cyclic groups [@amini2020dynamic]. Notwithstanding, a proof of this bound for general actions on higher dimensional spaces has until now been elusive. In this note, we will show that $\text{DAD}(\Gamma\curvearrowright X)\in \{\text{asdim}\Gamma, \infty\}$ for all free actions of countable groups by homeomorphisms on compact metric spaces. This sharp bound has been sought after for some time and is related to questions formulated by Willett and others (see [@warpedcones Section 8] for details). Moreover, the proof is elementary and relatively natural, especially considering certain arguments in large-scale geometry. While anticipated, this result is still somewhat surprising since the definition of $\text{DAD}$ involves open sets, and so one would expect it to be sensitive to the topology of $X$. In fact, our main theorem will imply it is equivalent to define the $\text{DAD}$ using borel or even arbitrary sets (at least for actions covered by the theorem), reminiscent of the work of [@Conley2020BorelAD]. The dynamic asymptotic dimension and related theories have found relevance in a number of interesting problems, including the Farrell-Jones conjecture on manifold topology [@dasdimGWY Section 4], calculations of $C^*$-algebra $K$-theory [@Guentner2016DynamicAD] [@bonicke2021dynamic], and the classifiability of $C^*$-algebras [@dasdimGWY section 8] (see also the introduction of [@https://doi.org/10.48550/arxiv.2201.03409]). Sharper bounds for the dimension in particular have implications for the homology of group actions [@bonicke2021dynamic Theorem 3.36], similar to how the homology of a manifold vanishes in degrees higher than its dimension. # Preliminaries The asymptotic dimension of a metric space was originally introduced by Gromov in [@Gromov1991AsymptoticIO]. There are many definitions, but we will use the one below. See [@Exel2015PartialDS Part I] for a more complete treatment of partial actions. [\[dad definition\]]{#dad definition label="dad definition"} [\[chain definition\]]{#chain definition label="chain definition"} [\[finite union lemma\]]{#finite union lemma label="finite union lemma"} *Proof.* Let $x_0, \ldots, x_n$ be an $F^{r_B}$-chain in $A\cup B$ with no repeated points. Suppose $x_j$ and $x_k$ are two consecutive points contained only in $A$. Then $x_{j+1}, \ldots, x_{k-1}$ is an $F^{r_B}$-chain in $B$ and therefore is $F^{R_B}$-bounded. We therefore have that $x_{j+1}$ and $x_{k-1}$ are in the same $F^{R_B}$-component, so $x_i$ and $x_j$ are in the same $F^{R_B + 2r_B}$-component of $A$. Since $R_B + 2r_B < r_A$, the points in the original chain which are contained only in $A$ therefore form an $F^{r_A}$-chain in $A$ and so (considered as a subset of $A$) is $F^{R_A}$-bounded. The original chain, considered as a set, is therefore $F^{2R_B + 2r_B + R_A}$-bounded. This implies the $F^{r_B}$-components of $A\cup B$ are $B_e^{2R_B + 2r_B + R_A}(C_F(\Gamma))$-bounded, which implies the result since $2R_B + 2r_B< 2r_A$. ◻ [\[basic lemma\]]{#basic lemma label="basic lemma"} 0◻ We will apply the following lemma several times in the proof of the main theorem, and it may be of general interest. [\[trick\]]{#trick label="trick"} *Proof.* For $x\in U$, let $Y^x$ be the set $$\Bigg\{\begin{array}{l|l} y\in U & \exists \text{ } f_1, \ldots, f_k\in F\text{, and } x\in U_j\text{ } (0\leq j\leq d) \text{ such that } \\ \text{ } & y = \theta_{f_k}\cdots \theta_{f_1}(x ) \text{ and } \theta_{f_{k_0}}\circ\cdots \circ \theta_{f_1}(x)\in U \text{ } \forall \text{ } 1\leq k_0\leq k \end{array}\Bigg\}$$ so $S^x\subset S\subset \Gamma$ is the unique smallest set such that $\theta_{S^x}(x) = Y^x$. Since $U$ is compact, there is $\epsilon>0$ such that $$\inf_{x\in U} \min_{y\in Y_x} \min_{\{f\in F\text{ } |\text{ } y\in D_f,\text{ } \theta_f(y)\notin U\}} d_X(U, \theta_f(y))>\epsilon$$ Let $\delta<\epsilon/2$ be such that $d(\theta_\gamma(x), \theta_\gamma(y))<\epsilon/2$ for all $\gamma\in S$ and $x, y\in D_\gamma$. Let $V^x = \cap_{\{s\in FS | x\notin D_s\}} D_s^c$ (this is an open set since each $D_s$ is closed). Then define $U^x = V^x\cap \text{Int}(B_x^{\delta}(X))$. It follows that we can replace $U$ by $\cup_x U^x$. ◻ [\[open or closed remark\]]{#open or closed remark label="open or closed remark"} [\[easy inequality\]]{#easy inequality label="easy inequality"} *Proof.* Since $\Gamma\curvearrowright X$ is free, we can identify $\Gamma$ with the orbit of any $x\in X$. Then a $(d, B_e^R(\Gamma), S)$-cover for $\Gamma\curvearrowright X$ gives rise to a $(d, R, \text{diam}(S))$-cover of $\Gamma$. In fact, this works if the cover of $X$ we start with uses Borel or even arbitrary sets. ◻ The reverse inequality is more difficult. # Sharp Bounds for the DAD When $X$ is a Cantor set, the proof that $\text{DAD}(\Gamma\curvearrowright X)\leq \text{asdim}\Gamma$ whenever $\text{DAD}(\Gamma\curvearrowright X)<\infty$ is fairly natural given the proof of [@boxspacesDT Proposition 3.1]. Being somewhat imprecise, the idea is to use the assumption that $\text{DAD}(\Gamma\curvearrowright X)<\infty$ to break $X$ into finitely-many pieces each with finite components at an appropriate scale. We can then cover each piece with disjoint clopen towers (see [@kerr2017dimension] for a definition), which are separated and each have dimension at most $\text{asdim}\Gamma$ at the given scale because the action restricted to a tower will resemble the structure of $\Gamma$ at that scale. Covers for these towers can be combined at no cost because they are separated, and the finite union lemma allows us to put the pieces back together. In higher dimensions, towers cannot be both open and disjoint unless they do not cover all of $X$. However, one can arrange things so that the remaining part of $X$ is contained in the boundaries of some sufficiently nice sets, and so is one dimension lower. This allows a pleasing argument using the inductive dimension. We need one more lemma, the proof of which also incidentally explains how to finish proving [\[open or closed remark\]](#open or closed remark){reference-type="ref" reference="open or closed remark"}. [\[helper lemma\]]{#helper lemma label="helper lemma"} *Proof.* Since $X$ is compact and $\mathcal{U}$ is open, the cover $\mathcal{U}$ has some lebesgue number $\lambda>0$. Let $U'_i$ be the $\frac{4\lambda}{5}$-interior of $U_i$, i.e. the largest open set such that the open ball of radius $\frac{4\lambda}{5}$ about each of its points is contained in $U_i$. Since $\lambda$ is the Lebesgue number of $\mathcal{U}$, the $U'_i$ still cover $X$. About each $x\in \overline{U'_i}$, there is a neighborhood $G^x_i$ of $x$ contained in the open ball of radius $\delta$ with $\partial G_i\leq m-1$. Since $\overline{U'_i}$ is compact, there is then a finite subcover $\{G^{j}_i\}$ of $\overline{U'_i}$ with $\dim_{ind}(\partial G_i^j)\leq m-1$ for all $i$ and $j$ and $\cup_j G^{j}_i\subset N_{\delta}(U'_i)$. Let $V_i^j = \overline{G}_i^j$. Then $\partial V_i^j = \overline{G}_i^j\setminus \text{int}(\overline{G}_i^j)\subset \overline{G}_i^j\setminus G = \partial G_i^j$ and so $\dim_{ind}(\partial V_i^j)\leq m-1$ as well. Now let $V_i = \cup_j V_i^j$, which is contained in the $\lambda/4$-neighborhood of $U'_i$ and hence the $\lambda/2$-interior of $U_i$. ◻ *Proof.* We will proceed by induction on the dimension of $Y$. Assume the theorem holds when $Y$ has dimension $\leq m-1$. Let $\mathcal{D}_\Gamma$ be the $\text{asdim}\Gamma$-dimensional control function for $\Gamma\curvearrowright\Gamma$ coming from [\[basic lemma\]](#basic lemma){reference-type="ref" reference="basic lemma"}. We can assume $\mathcal{D}_\Gamma(F)\supset F$ for all $F\in \mathcal{P}_{fs}(\Gamma)$. Fix $F_0\in \mathcal{P}_{fs}(\Gamma)$ and define $F_i$ for $i = 1, \ldots, d$ by the formula $F_{i+1} = \mathcal{D}_\Gamma(F_i)^3$. We will continue using $\gamma\cdot x$ only for the action $\Gamma\curvearrowright X$. Start with an open $(d, F_d, S)$-cover for $\Gamma\curvearrowright Y$ and use [\[helper lemma\]](#helper lemma){reference-type="ref" reference="helper lemma"} to obtain $\mathcal{V} = \{V_0, \ldots, V_d\}$, a closed $(d, F_d, S)$-cover for $\Gamma\curvearrowright Y$. Since $\partial(A\cup B)\subset \partial A\cup\partial B$, we have $\dim_{ind}(\partial_Y V_i)\leq \dim_{ind}(\cup_j \partial_Y V_i^j)\leq m-1$. Let $S_i^x$ be the subset of $S$ such that $S_i^x\cdot x = Y_i^x$ is the $F_d$-component of $x$ in $V_i$. Using [\[trick\]](#trick){reference-type="ref" reference="trick"}, we see that for each $x\in V_i$ there is an open (in $Y$) neighborhood $U_i^x$ about $x$ such that $S_i^{x}\cdot y$ contains the $F_d$ component of $y$ in $V_i\cup \big(\cup_x U_i^x\big) = \cup_x U_i^x$ for all $y\in U_i^x$. We can also ensure $U_i^x$ is small enough that the collection $\{s\cdot U_i^x | s\in S\}$ is disjoint (since the action is free). Since $\dim_{ind}(Y)\leq m$, there is an open neighborhood $W_i^x$ of $x$ contained in $U_i^x$ with $\dim_{ind}(\partial_{Y} W_i^x)\leq m-1$. Let $U_i = \cup_x U_i^x$. By our choice of $U_i^x$, $(S_i^x\cdot W_i^x)\cap U_i$ is equal to its own $F_d$-component in $U_i$. By passing to a finite subcover, we have a finite collection of open sets $\{W_i^{k}\}_{k=1}^{K_i}$ in $Y$ with $\dim_{ind}(\partial_Y W_i^k)\leq m-1$ such that $\cup_k W_i^k$ contains $V_i$. Moreover, since $(S_i^{x_k}\cdot W_i^k)\cap U_i$ is its own $F_d$-component in $U_i$, $(S_i^{x_k}\cdot W_i^k)\cap V_i$ is equal to its own $F_d$-component in $V_i$; that is, it cannot be that $f\cdot x\in V_i$ but $f\cdot x\notin (S_i^{x_k}\cdot W_i^k)\cap V_i$ for $f\in F_d$ and $x\in (S_i^{x_k}\cdot W_i^k)\cap V_i$. We will just write $S_i^k$ for the set $S_i^{x_k}$ from now on to simplify notation. Define $C_i^1 = W_i^1$ and $C_i^k = W_i^k\setminus \cup_{j=1}^{k-1} S_i^j\cdot C_i^j =W_i^k\setminus \cup_{j=1}^{k-1} S_i^{j}\cdot W_i^j$ for $2\leq k\leq K_i$. Notice that $\cup_k (S_{i}^k\cdot C_i^k)\supset V_i$. Moreover, since $S_i^k\cdot W_i^k\cap V_i$ is equal to its own $F_d$-component in $V_i$ (for all $k$), $C_i^k$ is disjoint from $S_i^{k'}\cdot W_i^{k'}$ for $k'<k$, $S_i^k$ is $F_d$-connected, and $F_d$ is symmetric, $S_i^k\cdot C_i^k\cap S_i^{k'}\cdot C_i^{k'}\cap V_i=\emptyset$ for $k'<k$ hence for any $k, k'$. In fact, $(S_i^k\cdot C_i^k)\cap V_i$ and $(S_i^{k'}\cdot C_i^k)\cap V_i$ are $F_d$-separated since they are contained in $V_i$, disjoint, and are are equal to their own $F_d$-components in $V_i$. In what follows all boundaries are taken with respect to the subspace topology on $Y$. Let $Z = \big(\cup_i \partial V_i\big)\cup \big(\cup_i \cup_k (S)^2\cdot \partial W_i^k\big)$ (by $(S)^2$ we mean $\{st : s, t\in S\}$). Recall that $\partial (A\cap B)\subset \partial A\cup\partial B$. Therefore, we have $$\partial((s\cdot C_i^k)\cap V_i) \subset \partial(s\cdot C_i^k)\cup \partial V_i$$ $$= s\cdot \partial C_i^k \cup \partial V_i= s\cdot \partial(W_i^k\setminus \cup_{j<k} S_i^j\cdot W_i^j) \cup \partial V_i$$ $$=s\cdot \partial\big(( \cdots (W_i^k\setminus S_i^1\cdot W_i^1)\setminus\cdots )\setminus S_i^{k-1}\cdot W_i^{k-1}\big) \cup \partial V_i\subset s\cdot \big( \cup_{j\leq k} \partial S_i^j\cdot W_i^j\big) \cup \partial V_i$$ $$\subset \big(\cup_{j\leq k} sS_i^j\cdot \partial W_i^j\big) \cup \partial V_i\subset \big(\cup_{j\leq k} (S)^2\cdot \partial W_i^j\big) \cup \partial V_i\subset Z$$ Since $Z$ is a finite union of spaces with inductive dimension at most $m-1$, $\dim_{ind}(Z)\leq m-1$. By the inductive hypothesis, we can find an $(\text{asdim}\Gamma, \mathcal{D}_\Gamma(F_d)^3, T)$-cover for $Z$ by open sets (open in the subspace topology on $Z$). Since $Z$ is closed and therefore compact, this cover has some lebesgue number $\mu$, and so we can replace the sets in the cover by the closures of their $\mu/2$-interiors to get a closed $(\text{asdim}\Gamma, \mathcal{D}_\Gamma(F_d)^3, T)$-cover for $Z$. Then by [\[trick\]](#trick){reference-type="ref" reference="trick"}, we can again replace this cover by a cover using sets which are open in $X$ and so cover an open neighborhood $N$ of $Z$ in $X$. Define $D_i^{k, s} = (s\cdot C_i^k \cap V_i)\setminus N$ for $s\in S_i^{k}$. Since $N$ is open (in $X$ in fact) and contains $Z$ (and so in particular, the boundary of each $s\cdot C_i^k\cap V_i$), each $D_i^{k, s}$ is closed in $Y$. For each $k$ and each $i$, there is a bijection $\{D_i^{k, s} : s\in S_i^k\}\leftrightarrow S_i^k\subset \Gamma$ which is equivariant for the obvious partial actions on each of these sets. We have $D_i^{k, s}\cap D_i^{k, s'}=\emptyset$ for $s\neq s'$. Notice also that $f\cdot D_i^{k, s}\subset D_i^{k, fs}\cup N\cup V_i^c$ for all $f\in F_d$ and $s\in S_i^k$ (where we interpret $D_i^{k, fs}$ as being empty if $fs\notin S_i^k$). We can therefore use the aforementioned bijection to form a closed $(\text{asdim}\Gamma, F_i, \mathcal{D}_\Gamma(F_i))$-cover for $A^k_i = \cup_{k, s} D_i^{k, s}$. By our construction of the sets $D_i^{k, s}$, $A_i^k$ and $A_i^{k'}$ are $F_d$-separated for $k\neq k'$, and so we can put their covers together at no cost to form a closed $(\text{asdim}\Gamma, F_i, \mathcal{D}_\Gamma(F_i))$-cover for $\cup_k A_i^k$. Then by [\[finite union lemma\]](#finite union lemma){reference-type="ref" reference="finite union lemma"}, our choice of $F_i$, and induction, we can put *these* covers together to get a closed $(\text{asdim}\Gamma, F_0, \mathcal{D}_\Gamma(F_d))$-cover for $A := \cup_i \cup_k A_i^k$. Notice that $A\cup N \supset Y$. Using [\[trick\]](#trick){reference-type="ref" reference="trick"} again, we have an open $(\text{asdim}\Gamma, F_0, \mathcal{D}_\Gamma(F_d))$-cover for an open neighborhood of $A$. By one more application of [\[finite union lemma\]](#finite union lemma){reference-type="ref" reference="finite union lemma"}, we get an open $(\text{asdim}\Gamma, F_0, T^3)$-cover for an open neighborhood of $Y$ so that we have shown $\text{DAD}(\Gamma\curvearrowright Y)\leq \text{asdim}\Gamma$. In the case where $m=0$, there is nothing more to do as $\partial W_i^k = \partial V_i = \emptyset$, and so $Z = \emptyset$. ◻ *Proof.* When $X$ is a normal space with countable base and so in particular when $X$ is a metric space, the inductive dimension coincides with the Lebesgue covering dimension. The corollary then follows immediately from the previous theorem and [\[easy inequality\]](#easy inequality){reference-type="ref" reference="easy inequality"}. ◻ *Proof.* We only need to know that $\text{DAD}(\Gamma\curvearrowright X)<\infty$, which follows from [@warpedcones Theorem 8.5]. ◻

arxiv_math

--- abstract: | In this paper, we summarize the work on the characterization of finite simple groups and the study on finite groups with "the set of element orders\" and "two orders\" (the order of group and the set of element orders). Some related topics, and the applications together with their generalizations are also discussed. address: Wujie Shi. School of Mathematics and Big Date, Chongqing University of Arts and Sciences, Chongqing, P.R.China, 402160; School of Mathematical Sciences, Soochow University, Suzhou, Jiangsu, P.R.China, 215006. author: - Wujie Shi title: Quantitative characterization of finite simple groups --- # Introduction Group theory is an important branch of mathematics, and it has important applications in mathematics, physics, chemistry and other fields. The classification theorem for the finite simple groups, which was completed in 2004, is one of the most important mathematical achievements of the 20th century. In terms of the number of participating researchers, the number of published papers, and the total length of the proof (more than 15,000 pages), it is unprecedented in the history of mathematics. The first paper classifying an infinite family of finite simple groups, starting from a hypothesis on the structure of certain proper subgroups, was published by Burnside in 1899. Recently, many significant topics in group representation theory have been successfully reduced to the cases of simple groups and quasi-simple groups, such as McKay Conjecture and Alperin Conjecture, and so on (See [@146]), we urgently need to understand and know more properties of simple groups. The famous researcher Solomon in the field of group theory pointed out that: "The ongoing project to publish a series of more than $12$ volumes presenting a complete proof of this theorem is expected to be completed by 2025.\" (See the email sent by R. Solomon to the author on November 2nd, 2020 or [@186]). In finite group, the order $|G|$ of group $G$ and its elements orders are two basic quantities. Let $G$ be a finite group and let $\pi_e(G)$ denote the set of element orders of $G$. The author of this article proposed the following conjecture in 1987: ****Conjecture** 1** (See [@165]). *Let $G$ be a finite group and $S$ a finite simple group. Then $G \cong S$ if and only if $(1)$ $\pi_e(G)= \pi_e(S)$; $(2)$ $|G|=|S|$, in other words, every finite simple group can be characterized by using only the order of the group and the orders of its elements (briefly, "two orders\").* In 1987, the author posed the above conjecture to Professor Thompson and received his encouragement and full affirmation. In his response, Thompson remarked: "Good luck with your conjecture about simple groups. I hope you continue to work on it\", "I like your arguments\", "This would certainly be a nice theorem\". At the same time, in two letters from Thompson dated 1987 and 1988 respectively, he posed the following problem and conjecture (see reference [@100 Problem 12.37-12.39]). ****Definition** 2**. *Let $G$ be a finite group, let $d$ be a positive integer and write $G(d)=\{x \in G | x^d = 1 \}$. We say that two groups $G_1$ and $G_2$ are of the same order type if and only if $|G_1(d)|=|G_2(d)|$ where $d=1$, $2, \cdots$.* **Thompson Problem (1987)** [^1] Suppose $G_1$ and $G_2$ are groups of the same order type. Suppose also that $G_1$ is solvable. Is it true that $G_2$ is also necessarily solvable? In Thompson's letter he pointed out that:"The problem arose initially in the study of algebraic number fields, and is of considerable interest.\" Let $G$ be a finite group, set $N(G)=\{n\in\mathbb{Z}^+ \mid G \text{ has conjugacy class } C \text{ with } |C| = n \}$, that is, it is the set of all lengths of the conjugacy classes of $G$. **Thompson Conjecture (1988)** [^2] Let $G$ and $M$ be two finite groups and $N(G)=N(M)$, if $M$ is a non-abelian simple group and the centre of $G$ is trivial, then $G$ and $M$ are isomorphic. This article provides an overview of the proof for Conjecture [**Conjecture** 1](#1.1){reference-type="ref" reference="1.1"}, Thompson Conjecture, the current status of solutions for Thompson Problem, as well as some problems related to quantities and groups, it involves their applications and generalizations, and also includes some unsolved problems. In fact, Conjecture [**Conjecture** 1](#1.1){reference-type="ref" reference="1.1"} originated from the author's master's thesis. The thesis studied finite groups whose elements with prime order except for the identity element (element prime order groups, $\mathop{\mathrm{EPO}}$-groups) (See [@184]), and finite groups whose elements with prime power orders (element prime power order groups, $\mathop{\mathrm{EPPO}}$-groups) (see [@185]). Moreover, the following result is obtained (See [@163]): ****Theorem** 3**. *[\[1.3\]]{#1.3 label="1.3"} Let $G$ be a finite group. Then $G\cong A_5$ if and only if $\pi_e(G)=\{1, 2, 3, 5\}$.* After learning about this result, Professor Xuefu Duan wrote in a letter to Professor Zhongmu Chen: "Very interesting, I will also think about it when I have time.\" The encouragement from both Professor Duan and Professor Chen prompted the author to characterize more simple groups by using "the set of element orders\". However, when dealing with simple groups (such as $A_6$) that could not be characterized only by the "set of element orders\", the natural thought is to add the "order of group.\" Consequently, the conjecture [**Conjecture** 1](#1.1){reference-type="ref" reference="1.1"} was posed. Article [@184] was published in Chinese. A similar article was published five years later in the *Proceedings of the American Mathematical Society* [@42], but the main theorem of that article contains some mistakes. In response, the author, together with coauthors, published article [@39] as a corrigendum to article [@42]. Initially published in Chinese as well, article [@185] gained recent attention, prompting the authors of [@185] to translate it into English and submited it on arXiv. [@149] generalized [@184] to investigate finite groups in which all elements, except those in a normal subgroup, have prime order. The recently published paper (arXiv:2203.02537v1) on arXiv by Lewis, titled "Groups having all elements of a normal subgroup with prime power order\" is a generalization of the reference [@185]. # The spectrum characterization of finite simple groups Before characterizing the alternating group $A_5$ only using the set of element orders, the author had used the number $|\pi(G)|$ of prime factors of $|G|$ and impose some restrictive conditions on "the set of element orders\" to characterize simple group ${\operatorname{PSL}}_2(7)$ (See [@160]) and some other simple groups (See [@159; @161; @162; @164]). Moreover, it is not difficult to deduce that the aforementioned constraints hold from the "set of element orders\". For instance, in [@160] the author proved the following result: ****Theorem** 4**. *Let $G$ be a finite group satisfying the following conditions:* - *$|G|$ contains at least three different prime factors, that is, $|\pi(G)|\geqslant 3$;* - *the order of every non-identity element in $G$ is either a power of $2$ or a prime different from $5$.* *Then G is isomorphic to ${\operatorname{PSL}}_2(7)$.* It is easy to deduce that the above conditions $(1)$ and $(2)$ hold from $\pi_e(G)=\{1, 2, 3, 4, 7\}$ ($=\pi_e({\operatorname{PSL}}_2(7))$). In [@163], the alternating group $A_5$ was characterized using the "set of element orders\" as a condition, and the proof only used elementary group theory concepts. Subsequently, in reference [@153], is also used an elementary method to prove that a finite group $G$ is isomorphic to $A_5$ if and only if $\pi_e(G)=\{1, p, q, r\}$, where $p$, $q$, and $r$ are distinct prime numbers. Due to the significance of the concept of "the set of element orders in a group\", it is referred to as the "spectrum\" in some subsequent articles. Mazurov (See [@129]) point out: "This result opened a wide way for investigations of recognizability of groups.\" Obviously, the set of element orders in a group $G$, denoted by $\pi_e(G)$, is a subset of the set of positive integers $\mathbb{Z}^+$. However, conversely, the following question is quite difficult: ****Problem** 5**. *What kind of sets of numbers can become a set of element orders $\pi_e(G)$ of a finite group $G$?* In October 1996, the author gave the following definition in a presentation of "Group Theory Seminar\" held at Princeton University: For a subset $\Gamma$ of the set of positive integers, the function $h$ can be defined as follows, $h(\Gamma)$ is the number of isomorphism classes of groups $G$ such that $\pi_e(G)= \Gamma$. ****Definition** 6** (See [@175]). *For a given group $G$, we have $h(\pi_e(G))\geqslant 1$. A group $G$ is called recognizable if $h(\pi_e(G))=1$. A group $G$ is called almost recognizable (or $k$-recognizable) if $h(\pi_e(G))=k$ is finite; otherwise $G$ is called unrecognizable.* Solvable groups are unrecognizable (see [@175 Theorem 4]). In particular, in [@67; @125; @136; @170; @177], the following result was obtained: ****Theorem** 7**. *If $G$ has a solvable minimal normal subgroup or $G$ is isomorphic to one of the following groups: $A_6$, $A_{10}$, $L_3(3)$, $U_3(3)$, $U_3(5)$, $U_3(7)$, $U_4(2)$, $U_5(2)$, $J_2$, $S_4(q)$ where $q\neq 3^{2m+1}$ and $m>0$, then $G$ is unrecognizable.* In April 2010, at the Ischia Group Theory Conference held in Italy, the author presented the following theorem for recognizable groups (See [@177]): ****Theorem** 8**. *Let $G$ be one of the following simple groups:* - *Alternating group $A_n$ where $n\neq 6$, $10$.* - *Sporadic simple group $S$ where $S\neq J_2$.* - *Simple group of Lie type:* - *$L_2(q)$ with $q\neq 9$; $L_3(2^m)$ for $m\geqslant 1$; $L_3(q)$ where $3<q\equiv 2(\bmod 5)$ and $(6,\frac{q-1}{2})=1$;* - *$U_3(2^m)$ where $m\geqslant 2$; $L_4(2^m)$ where $m\geqslant 1$; $U_4(2^m)$ where $m\geqslant 2$;* - *$Sz(2^{2m+1})$ with $m\geqslant 1$; $R(3^{2m+1})$ with $m\geqslant 1$; ${}^2F_4(2^{2m+1})$ where $m\geqslant 1$; $S_4(3^{2m+1})$ where $m\geqslant 0$;* - *$B_p(3)$ where $p>3$ is odd prime; $C_p(3)$ where $p$ is odd prime; $D_p(5)$ where $p$ is odd prime;* - *$D_n(q)$, where $q=2$, $3$ or $5$ for some $n$; $G_2(3^m)$;* - *${}^2F_4(2)'$, $L_3(7)$, $L_4(3)$; $L_n(2)$ where $n\geqslant 3$; $L_5(3)$, $U_3(9)$, $U_3(11)$, $U_4(3)$, $U_6(2)$, $G_2(3)$;* - *$G_2(4)$, $S_6(3)$, $O^+_8(2)$, $O^+_{10}(2)$, $F_4(2)$, ${}^3D_4(2)$, ${}^2E_6(2)$.* *Then $G$ is recognizable.* Now, the research on the recognizable groups mentioned above has already been summarized comprehensively, see [@67 Table 1-9], recognizable simple groups are those finite simple groups in which the function $h$ equal $1$ in these tables. Mazurov et al. relaxed the condition mentioned above by removing the restriction that the group is "finite\", proving a series of results (See [@70; @85; @114; @115; @116; @117; @118; @119; @126; @127; @128; @130; @131; @132; @133; @135; @208; @224; @225; @226; @227]). A typical result is as follows (See [@113]): ****Theorem** 9**. *If the spectrum of a group $G$ is equal to $\{1,2,3,4,7\}$ then $G\cong {\operatorname{PSL}}_2(7)$.* Note that if the group $G$ is assumed to be finite, then the conclusion of the above theorem appeared in the literature [@160] in 1984. Without the "finite\" condition, a paper with the same conclusion was published in 2007, after a span of 23 years. For the function $h$, it was conjectured in [@148] and [@100 Problem 12.84] that $h(\Gamma)=\{0,1,\infty\}$. In [@40; @122; @123; @138; @148; @173; @183; @192] they subsequently provided counterexamples to this conjecture and gave example with function $h$ equal to 2. Furthermore, the following question was put forward in [@173]: Let $\Gamma$ be an arbitrary subset of the natural numbers, does there exist a positive integer $k$ such that $h(\Gamma)=\{0, 1, 2,\cdots, k, \infty\}$? If the answer is positive, what is the value of such $k$? In [@214] Zavarnitsine gave an example as follows: For any $r>0$, $h(\pi_e(L_3(7^n)))=r+1$ where $n=3^r$, and these $r+1$ groups are all of the form $L_3(7^n)\langle \rho\rangle$, $\rho$ is a field automorphism of $L_3(7^n)$ ($n= 3^r$), $k= 0, 1, 2,\cdots r$. It is an example of function $h$ such that for any positive integer $k$, there exist $k$-recognizable groups. Observing example of group for which the function $h$ is equal to $2$ (See [@170 Theorem 5.4]), the author has raised the question of whether, for any $k$, there exist two finite $k$-recognizable groups that are section-free, in other words, they are not simple sections of each other (see [@177 Problem 5.2]). In [@66] Grechkoseeva pointed out the existence of such groups, which are $G_1 = L_{15}(2^{60}).3$ and $G_2 = L_{15}(2^{60}).5$ ([@100 Problem A:16.106]). Note that although $G_1$ and $G_2$ mentioned above are section-free, both of them have normal subgroup $L_{15}(2^{60})$. The spectrum of a finite group is a set of positive integers containing $1$. If the integer set $\pi_e(G)$ is partitioned into $1$, the set of prime number $\pi_e'(G)$ ($\pi_e'(G)=\pi(G)$) and the set of composite numbers $\pi''_e(G)$, then the following result holds: ****Theorem** 10** (See [@45]). *Suppose that $G$ is a finite group, then $|\pi_e'(G)|\leqslant |\pi_e''(G)|+3$. Moreover if $|\pi_e'(G)|=|\pi_e''(G)|+3$, then $G$ is simple and $G$ can be characterized by its spectrum. Furthermore, $G$ is one of the following simple groups.* 1. *$A_5$, $L_2(11)$, $L_2(13)$, $L_2(16)$, $L_3(4)$ and $J_1$.* 2. *$Sz(q)$, where $q=2^{2n+1}$ satisfies that each of $q-1$, $q-\sqrt{2q}+1$, and $q+\sqrt{2q} + 1$ is either a prime number or a product of two distinct prime numbers.* 3. *$L_2(2^n)$, where $n$ ($n\geqslant 5)$ is an odd prime and satisfies both $\frac{2^n+1}{3}$ is prime and $2^n-1$ is either a prime number or a product of two distinct prime numbers.* 4. *$L_2(3^n)$ where $n$ is an odd prime and satisfies both $\frac{3^n+1}{4}$ is a prime and $\frac{3^n-1}{2}$ is either prime or a product of two distinct prime numbers.* 5. *$L_2(5^n)$, where $n$ is an odd prime satisfying both $\frac{5^n-1}{4}$ and $\frac{5^n+1}{6}$ are primes.* 6. *$L_2(p)$, where $p$ is a prime greater than $13$ and one of the following holds:* 1. *$\frac{p-1}{4}$ and $\frac{p+1}{6}$ are primes.* 2. *$\frac{p-1}{6}$ and $\frac{p+1}{4}$ are primes.* ****Problem** 11**. *In the cases $(II)$--$(VI)$ of the aforementioned theorem, is the number of simple groups satisfying the respective conditions finite or infinite?* Let $\iota(G)$ denote the set of $pq$ ($p\neq q$) type numbers in the set $\pi_e(G)$. Then the following result holds: ****Theorem** 12** (See [@44]). *Let $G$ be a finite group, then $|\pi(G)|\leqslant |\iota(G)|+4$. If $|\pi(G)|= |\iota(G)|+4$ then $G$ is simple and $G$ can be characterized by its spectrum.* From Theorem [**Problem** 11](#2.8){reference-type="ref" reference="2.8"} (Theorem [**Theorem** 12](#2.9){reference-type="ref" reference="2.9"}) we have the number $|\pi(G)|$ of prime divisors of $|G|$ is constrained by $|\pi^{''}_e(G)|$ ($|\iota(G)|$). If the "order of a group\" is square-free, then the group is solvable (metacyclic group). With respect to this conclusion and the property of $A_5$ (Theorem [\[1.3\]](#1.3){reference-type="ref" reference="1.3"}), the author posed the following question: ****Problem** 13**. *Study on finite groups whose "element orders are square-free\".* Without attaching any additional conditions, what other properties can the spectrum of a finite group possess? That is the Problem [**Problem** 5](#2.2){reference-type="ref" reference="2.2"}: What kind of sets of numbers can become a set of element orders $\pi_e(G)$? After the publication of the literature [@160], Brandl wrote to the author and posed two questions, one of which is: If the orders of elements in a finite group $G$ are consecutive integers, that is, $\pi_e(G)= \{1, 2, 3,\cdots, n\}$, then what is the largest possible value of $n$ that can continue consecutively? Obviously, this question imposes a "consecutive \" condition on the spectrum of the group. In order to solve this question, the author considered prime graph components of finite groups and found the key reference [@201] to resolve this problem (Refer to [@83; @102] for the relevant literature and to [@103] for the most comprehensive description of the prime graph components). ****Definition** 14** (See [@201]). *Let $G$ be a finite group. The simple graph $\Gamma(G)$ of $G$ is given by $\pi_e(G)$ as follows: This graph has vertex set $\pi(G)$, and two vertices $p$ and $q$ are adjacent if and only if $pq\in \pi_e(G)$, denoted by $p\sim q$. Denote the connected components of the graph $\Gamma(G)$ by $t(G)$, and using $\pi_i=\pi_i(G)$ $(i=1,2,\cdots, t(G))$, we denote the $i$th connected component of $\Gamma(G)$. If the order of $G$ is even, denote the component containing $2$ by $\pi_1(G)$. This graph $\Gamma(G)$ is usually called the prime graph of group $G$. It was proposed by Grunberg and Kegel in 1975, hence it is also said to be the Grunberg-Kegel graph of group $G$ (abbreviated as GK(G)).* It is evident from the definition of the prime graph that $\pi_e(G)$ determines $\Gamma(G)$. However, conversely, the same $\Gamma(G)$ can correspond to different sets $\pi_e(G)$. Therefore, using $\Gamma(G)$ to study finite groups, especially finite simple groups, has become a new research topic. ****Theorem** 15** (See [@20]). *Let $G$ be a finite group. If the elements of spectrum of $G$ are consecutive integers, i.e., $\pi_e(G)=\{1,2,3,\cdots,n\}$ (such groups are called finite $\mathop{\mathrm{OC}}_n$-groups), then $n\leqslant 8$. Moreover, these groups have been classified as follows:* 1. *$n\leqslant 2$ and $G$ is an elementary abelian group.* 2. *$n=3$ and $G=[N]Q$ is a Frobenius group where either $N\cong Z^t_3$, $Q\cong Z_2$ or $N\cong Z^{2t}_2$, $Q\cong Z_3$.* 3. *$n=4$ and $G=[N]Q$ and one of the following holds:* 1. *$N$ has exponent $4$ and class $\leqslant 2$ and $Q\cong Z_3$.* 2. *$N\cong Z^{2t}_2$ and $Q\cong \Sigma_3$.* 3. *$N\cong Z^{2t}_3$ and $Q\cong Z_4$ or $Q_8$ and $G$ is a Frobenius group.* 4. *$n=5$ and $G\cong A_6$ or $G=[N]Q$, where $Q\cong A_5$ and $N$ is an elementary abelian $2$-group and a direct sum of natural ${\operatorname{SL}}(2,4)$-modules.* 5. *$n=6$ and $G$ is one of the following types:* 1. *$G=[P_5]Q$ is a Frobenius group, where $Q\cong[Z_3]Z_4$ or $Q\cong {\operatorname{SL}}(2,3)$ and $P_5\cong Z^{2t}_5$.* 2. *$G/O_2(G)\cong A_5$ and $O_2(G)$ is elementary abelian and a direct sum of natural and orthogonal ${\operatorname{SL}}(2,4)$-modules.* 3. *$G=\Sigma_5$ or $G\cong \Sigma_6$.* 4. *$n=7$ and $G\cong A_7$.* 5. *$n=8$ and $G=[{\operatorname{PSL}}(3,4)]\langle \beta\rangle$, $\beta$ is an unitary automorphism of ${\operatorname{PSL}}(3,4)$.* *where $[A]B$ denotes the split extension of its normal subgroup $A$ by a complement $B$.* ****Corollary** 16**. *Let $G$ be a finite group, then $\pi_e(G)=\{1,2,3,\cdots,7\}$ if and only if $G\cong A_7$.* For the infinite $\mathop{\mathrm{OC}}_7$ groups $G$, Mamontov and Jabara proved that it is locally finite (See [@120]), that is, any finite set of elements in $G$ generates a finite subgroup, thereby deducing that $G \cong A_7$ (See [@100 Problem A:19.80]). It is easy to see that $Z^{\infty}_2$ and $[Z^{\infty}_3]Z_2$ correspond to the infinite $\mathop{\mathrm{OC}}_2$ groups and $\mathop{\mathrm{OC}}_3$ groups, respectively. For $4\leqslant n\leqslant 6$, there also exist infinite $\mathop{\mathrm{OC}}_n$ groups. ****Problem** 17**. *What is the maximum value of $n$ for the infinite $\mathop{\mathrm{OC}}_n$ group? The infinite $\mathop{\mathrm{OC}}_n$ group is periodic, meaning that the orders of elements in the group are finite. Is it locally finite? This is a special case for Burnside's problem (See [@72]).* The literature [@108; @202] respectively discussed finite groups with orders of elements as consecutive odd integers, and finite groups with orders of elements as consecutive integers except for some primes. ****Problem** 18**. *By analogizing Theorem [**Theorem** 15](#2.12){reference-type="ref" reference="2.12"}, we investigate finite groups whose spectrum consist of segmented consecutive integers. For example, $\pi_e(A_5)=\{1, 2, 3, 5\}$ (See [@184]) and $\pi_e({\operatorname{PSL}}_2(7))=\{1, 2, 3, 4, 7\}$ (See [@160]) both represent finite simple groups with spectrum consisting of two segments of consecutive integers. Furthermore, for finite groups with segmented consecutiveness, determine their maximum length of consecutive integers. For instance, in the case of consecutive, which is an $\mathop{\mathrm{OC}}_n$ group, with a length of $8$.* Notice that the order of an element in a group is the order of cyclic subgroup. In [@51; @171] they provided finite groups with orders of abelian subgroups and orders of proper subgroups are consecutive integers, respectively. Furthermore, since the order of an element in a group is a conjugation invariant. In [@150] Qian investigated the finite groups with consecutive nonlinear character degrees. # Characterizing finite simple groups using the order of group and the set of element orders According to the classification theorem for finite simple groups (abbreviated CFSG), every finite simple group is isomorphic to one of the following (See [@54]): 1. the groups of prime order; 2. the alternating groups of degree at least $5$; 3. the simple classical groups; 4. the simple exceptional groups of Lie type; 5. the 26 sporadic groups. From 1987 to 2003, the author and his coauthors successively demonstrated Conjecture [**Conjecture** 1](#1.1){reference-type="ref" reference="1.1"} to be true for all finite simple groups except the families $B_n(q)$, $C_n(q)$, and $D_n(q)$ (where $n$ is even) (See [@32; @165; @172; @180; @181; @182; @205]). In 2009, this conjecture was also proved for the families $B_n(q)$, $C_n(q)$, and $D_n(q)$ (where $n$ is even) in [@195]. As a result, the aforementioned conjecture has been proved and established a theorem, which states that all finite simple groups can be determined by the "order of group\" and the "set of element orders\" (Simplified as the "two orders\"). For the $26$ sporadic simple groups, their "two orders\" are very clear, which can be studied only by using the prime graph components of finite groups in [@201]. In fact, these $26$ sporadic simple groups unless $J_2$ can all be characterized only by the "set of element orders\" (See Theorem [**Theorem** 8](#2.5){reference-type="ref" reference="2.5"}(2)). For alternating groups with degree greater than or equal to $5$, a series of lemmas are deduced in [@182] from the order of the group $A_n$, which is $\frac{n!}{2}$. In particular, the Stirling formula from number theory is used to prove the conclusion without using the prime graph components of finite groups in [@201]. In fact, all alternating groups of degree at least $5$, except $A_6$ and $A_{10}$, can be characterized only by using the "set of element orders\" (See Theorem [**Theorem** 8](#2.5){reference-type="ref" reference="2.5"}(1)). For the simple classical groups and exceptional simple groups of Lie type, the author first used the classification theorem for simple groups to solve a problem which was posed by Artin in 1955: Determining all finite simple groups with the order of Sylow subgroups greater than $|G|^{1/3}$ (See [@14]), i.e., the following two lemmas are proved (see [@168]): ****Lemma** 19**. *Let $G$ be a finite simple group and if $|G|=p^km$, where $p$ does not divide $m$, $p$ is an odd prime and $|G|<p^{3k}$, then $G$ is one of the following groups:* - *A simple group of Lie type in characteristic $p$;* - *$A_5$, $A_6$ and $A_9$;* - *$L_2(p-1)$ where $p$ is a Fermat prime, $L_2(8)$ and $U_5(2)$.* ****Lemma** 20**. *Let $G$ be a finite simple group and if $|G|=2^km$ with $m$ odd, and $|G|<2^{3k}$, then $G$ is one of the following:* - *A simple group of Lie type in characteristic $2$;* - *$L_2(r)$ where $r$ is Fermat prime or Mersenne prime;* - *$A_6$, $U_3(3)$, $A_9$, $M_{12}$, $U_3(4)$, $A_{10}$, $M_{22}$, $J_2$, $HS$, $M_{24}$, $Suz$, $Ru$, $Fi_{22}$, $Co_2$, $Co_1$ and $B$.* Above lemmas and the order of group narrow down the scope of the research to simple group of Lie type. Furthermore, upon deeper analysis, Conjecture [**Conjecture** 1](#1.1){reference-type="ref" reference="1.1"} has been proved to be true for all cases except groups $B_n(q)$, $C_n(q)$, and $D_n(q)$ (where $n$ is even). In fact, for exceptional simple groups of Lie type, Suzuki-Ree groups ${}^2B_2(2^{2n+1})$ ($n\geqslant 1$) (See [@169]), ${}^2G_2(3^{2n+1})$ ($n\geqslant 1$) (See [@21]), ${}^2F_4(2^{2n+1})$ ($n\geqslant 1$) (See [@46]), and $G_2(q)$ (See [@196]), $E_8(q)$ (See [@104]) and $F_4(2^m)$ (See [@31]), and others can be characterized by only using the "set of element orders\" (See [@67 Table 8]). For classical groups $B_n(q)$ and $C_n(q)$, which have the same order, does there exist some $n$ and prime power $q$ such that their spectra are also the same? Thereby constructing a counterexample to Conjecture [**Conjecture** 1](#1.1){reference-type="ref" reference="1.1"}. Shi (See [@176]) and Grechkoseeva (See [@65]) almost simultaneously considered this question and proved, using different methods, that for all $n$ and prime power $q$, $\pi_e(B_n(q))\neq \pi_e(Cn(q))$, thus such a counterexample does not exist. To finally prove Conjecture [**Conjecture** 1](#1.1){reference-type="ref" reference="1.1"}, in [@194], the nonabelian composition factors for these symplectic groups and orthogonal groups are given directly from the spectrum of the classical groups $B_n(q)$, $C_n(q)$, and $D_n(q)$ (where $n$ is even). Finally, Conjecture [**Conjecture** 1](#1.1){reference-type="ref" reference="1.1"} was proved in [@195] by using the three theorems of [@194]. Consequently, Conjecture [**Conjecture** 1](#1.1){reference-type="ref" reference="1.1"}, which characterizes finite simple groups by using "two orders\", has been proved and established as a theorem. Numbers and sets of numbers play an important role in mathematics. The concept of "two orders\" was introduced first by the author and has now been extensively studied in the field of group theory. Simple groups are complex, the concept of \"two orders\" is the simplest concept. The validation of Conjecture [**Conjecture** 1](#1.1){reference-type="ref" reference="1.1"} establishes a connection between the terms "simple\" and "complex\". Of course, its proof used the classification theorem for finite simple groups. All finite simple groups can be characterized by "two orders\", but it fails to characterize some groups with small orders. For example: dihedral group $D_8$ with $8$ elements and the quaternion group $Q_8$, obviously, $\pi_e(D_8)=\pi_e(Q_8)=\{1,2,4\}$ but $D_8$ and $Q_8$ are not isomorphism. We will give an application of Conjecture [**Conjecture** 1](#1.1){reference-type="ref" reference="1.1"} as following: Let $G$ be a finite group and let $B(G)$ be a Burnside ring of $G$. Yoshida posed the following open problem in [@211]: Let $G$ and $S$ be two finite groups, does $B(G)\cong B(H)$ implies $G\cong H$ (See [@211 Page 340, Proble 2])? It was proved in [@101 Corollary 5.2] that the spectrum of a finite group is determined by its Burnside ring, so we have the following application: ****Corollary** 21**. *Let $G$ be a finite group, $S$ a finite simple group. If $B(G)\cong B(S)$ then $G\cong S$.* Thus, simple groups are recognizable by Burnside ring. In other words, simple groups can be characterized using their Burnside ring. ****Problem** 22**. *Is it possible to prove Conjecture [**Conjecture** 1](#1.1){reference-type="ref" reference="1.1"} without using the classification theorem for finite simple groups?* For a few simple groups with small orders, it is possible to provide a proof without using the classification theorem for simple groups, for example $A_5$ (See [@163]). However, it seems impossible to provide a proof without using the classification theorem for all finite simple groups. ****Problem** 23**. *Weakening the condition of "two orders\", is it possible to provide a characterization for all finite simple groups? In other words, can they be characterized by the "group order\" and the "set of orders of certain elements\", or by the "orders of Hall subgroups in a group\" and the "set of orders of elements\" for all simple groups?* For a prime power $q=p^e$, we write $ch(q)=p$, and for a Lie type simple group $G$ defined over $GF(q)$, the characteristic of $G$ denoted by $ch(G)=p$. In [@92] Kantor and Seress proved the following theorem (See [@92 Theorem 1.2]): ****Theorem** 24**. *Let $G$ and $H$ be simple groups of Lie type of odd characteristic. If $m_i(G)= m_i(H)$ for $i=1$, $2$, $3$, then characteristics of $G$ and $H$ are same, where $m_1(H)$, $m_2(H)$ and $m_3(H)$ denote the largest, the second largest and the third largest element order of $G$, respectively.* This gives rise to the following question: Is it possible to characterize simple group using the "group order\" and the "set of orders of certain elements\"? In fact, the following theorem has been proved in [@75; @76; @107; @212]: ****Theorem** 25**. *Let $G$ be a group. $S$ is one of the following simple groups:* - *Sparodic simple groups (See [@76]);* - *$L_2(p)$ where $p\neq 7$ is prime (See [@107]);* - *Simple $K_4$-groups with type $L_2(q)$ where $q$ is a prime power (See [@75]);* - *Simple $K_5$-groups of type $L_3(p)$, where $p$ is a prime number and $(3,p-1)=1$ (See [@212]).* *Then $G\cong S$ if and only if $|G|=|S|$ and $m_1(G)=m_1(S)$, where $m_1(G)$ denotes the largest element order of $G$.* If we add the second largest element order or the third largest element order of group $G$, then it can also be used to characterize some simple groups (See [@74]). Let $G$ be a finite group, set Van($G$)=$\{g\in G|\text{ there exists } \chi \text{ such that } \chi(g)=0 \}$, where $\chi$ is an irreducible complex character of $G$, Vo($G$) denotes the set of element orders of Van($G$). Obviously, Vo($G$) is a subset of the set of element orders $\pi_e(G)$ of $G$. The following conjecture was posed in [@100 Problem 19.30]: ****Conjecture** 26**. *Let $G$ be a finite group and $S$ a finite simple group, then $G\cong S$ if and only if (1) Vo($G$)=Vo($S$); (2) $|G|=|S|$.* The conjecture mentioned above as a question was also put forward in Zhang Jinshan's PhD thesis. We refer to the literature [@158] and footnote [^3] for its recent studies. In fact, the group order, element order, the number of elements of the same order, the length of conjugacy class, and the degree of character, and others, are conjugation invariants. In addition, order of normalizer of a Sylow subgroup, order of maximal abelian (solvable) subgroup, the index of maximal subgroup, the number of Sylow subgroups, and so on, are important quantities in the study of groups. Study the structure of groups, especially finite simple groups, starting from these important quantities is a broad and meaningful subject. For example, study the structure of finite groups whose lengths of conjugacy classes are all prime powers. Qian[@151] and Qian, et al.[@154] defined codegree of the character and gave a connection between the element order and the degree of character. Next, we consider the characterization of simple groups by using "the order of Hall subgroup of a group\" and the "set of element orders\". Firstly, the "order ($|G|_2$) of Sylow $2$-subgroups\" of a group $G$ and the "set of element orders\" were used to characterize alternating and sporadic simple groups. We have the following theorems: ****Theorem** 27**. *Let $G$ be a finite group and $S$ an alternating simple group. Then $G\cong S$ if and only if (1) $|G|_2=|S|_2$; (2) $\pi_e(G)=\pi_e(S)$.* ****Theorem** 28**. *Let $G$ be a finite group and $S$ a sporadic simple group. Then $G\cong S$ if and only if (1) $|G|_2=|S|_2$; (2) $\pi_e(G)=\pi_e(S)$.* ****Lemma** 29** (See [@56]). *Suppose $G$ is a finite group with $\pi_e(G)=\pi_e(S)$, where $S$ is an alternating simple group $A_n$ and $n\geqslant 5$, $n\neq 6$, $10$. Then $G\cong S$.* In literature [@123; @148; @163] the authors earlier studied the characterization of alternating groups and symmetric groups using the set of element orders. The subsequent studies in [@56; @106; @213] the authors further investigated the characterization of alternating groups by using the set of element orders. It was pointed out in [@187] that $A_6$ and $A_{10}$ are unrecognizable. ****Lemma** 30** ([@20]). *Let $G$ be a finite group with $\pi_e(G)=\pi_e(A_6)=\{1,2,3,4,5\}$, then $G\cong A_6$ or $G=[N_1]Q$ where $Q\cong A_5$, $N_1$ is an elementary abelian $2$-group and a direct sum of natural ${\operatorname{SL}}(2,4)$-modules.* ****Lemma** 31** (See [@187]). *Let $G$ be a finite group with $\pi_e(G)=\pi_e(A_{10})=\{1,2,\cdots,$ $10,12,15,21\}$, then $G\cong A_{10}$, or $G=[A]C$ where $A$ is an Abelian $\{3,7\}$-group, $C=C_G(t)=[\langle t\rangle ]S_5$, where $t$ is an involution and $a^t=a^{-1}$, $a\in A$. And the Sylow $2$-subgroup of $C$ is a generalised quaternion group $Q_{16}$ with $16$ elements.* ****Lemma** 32** (See [@134; @170]). *Let $G$ be a finite group with $\pi_e(G)=\pi_e(M)$ where $M$ is a sporadic simple group distinct from $J_2$, then $G\cong M$.* It was proved in [@170] that except for $Co_2$ and $J_2$, the remaining $24$ sporadic simple groups can all be characterized by the set of element orders. In [@134] Mazurov and Shi proved that $Co_2$ can be characterized by using the "set of element orders\", however if $\pi_e(G)=\pi_e(J_2)=\{1,2,\cdots 8,10,12,15\}$, then $G\cong J_2$, $S_8$, or $G\cong [N]A_8$ is a split extension of a $2$-group $N$ with order $2^{6t}$ ($t=1,2,\cdots$) by $A_8$. Obviously, from the above arguments and Lemma [**Lemma** 29](#3.11){reference-type="ref" reference="3.11"}, we can deduce that Theorems [**Theorem** 27](#3.9){reference-type="ref" reference="3.9"} and [**Theorem** 28](#3.10){reference-type="ref" reference="3.10"} hold. ****Problem** 33**. *Let $G$ be a finite group and $S$ a finite simple group. If (1) $|G|_{\pi}=|S|_{\pi}$ where $|G|_{\pi}$ denotes the order of $\pi$-Hall subgroup of $G$, $\pi\neq \pi(G)$; (2) $\pi_e(G)=\pi_e(S)$, can we prove that $G\cong S$?* In 2007, Mazurov posed the following conjecture at International Algebraic Conference held in St. Petersburg: Let $G$ be a finite group, and let $L$ be an alternating group of sufficiently large degree or a simple group of Lie type of sufficiently large Lie rank. If $G$ is isospectral to $L$, then $G$ is an almost simple group with socle isomorphic to $L$. In fact, if $L$ is a finite simple group and $G$ is isospectral to $L$, then either $G$ is solvable, in which case $L$ is one of the following cases: $L_3(3)$, $U_3(3)$, and $S_4(3)$; or $G$ is an almost simple group with socle isomorphic to $L$ (See [@56 Theorem 2]). The conjecture proposed by Mazurov was finally proved to be valid in [@68]. By using these results and comparing $|G|_{\pi}$ and $|M|_{\pi}$, Question [**Problem** 33](#3.15){reference-type="ref" reference="3.15"} can be answered. # Thompson Conjecture and Thompson Problem ## The origin and early work of the Thompson Conjecture   As mentioned earlier, the author posed Conjecture [**Conjecture** 1](#1.1){reference-type="ref" reference="1.1"} to Thompson in 1987, and Thompson proposed the following conjecture in his response letter in 1988 (Thompson Conjecture (1988)): Let $G$ and $M$ be finite groups, set $N(G)=\{n\in\mathbb{Z}^+ \mid G \text{ has conjugacy class } C, |C| = n \}$. Suppose $M$ is a non-abelian simple group, and the center of $G$ is trivial. If $N(G)=N(M)$, then $G$ and $M$ are isomorphic. In [@33] Chen proved that if $M$ is a sporadic simpe group then Thompson conjecture is correct. Subsequently, he finished his PhD thesis titled "On Thompson Conjecture\". In [@34] the author based on the foundation of prime graph components ([@201]), a definition of the order components was provided as follows: ****Definition** 34**. *Let $G$ be a finite group, $\pi_1$, $\pi_2,\cdots, \pi_t$ are prime graph components of $G$, where $t=t(G)$ is the number of prime graph components of $\Gamma(G)$. Assume $|G|=m_1m_2\cdots m_t$ where $\pi_i=\pi(m_i)$. Then $m_1$, $m_2, \cdots, m_t$ are called order components of $G$. Set $OC(G)=\{m_1, m_2,\cdots, m_{t(G)}\}$, the set of order components of $G$.* Thompson Conjecture was proved to be ture for all sporadic simple groups in[@33; @34]. Its validity was also proved in [@35; @36] for finite group having at least $3$ prime graph components. For the case of $2$ prime graph components, we can use the same method to prove it. For instance, it was proved in [@37] that Thompson Conjecture is correct for the group ${}^3D_4(q)$. ## The case of groups with connected prime graph   In 2009, Vasil'ev proved that Thompson Conjecture is valid for the simple groups $A_{10}$ and $L_4(4)$ with connected prime graph. Subsequently, Ahanjideh proved that Thompson Conjecture is also correct for simple groups of Lie type $A_n(q)$ (See [@5]), $B_n(q)$ (See [@8]), $C_n(q)$ (See [@6]), $D_n(q)$ where $n\neq 4,8$ (See [@7]), ${}^2A_n(q)$ (See [@9]) and ${}^2D_n(q)$ (See [@10]) in a series of articles. For other exceptional simple groups of Lie type, Thompson Conjecture was established the validity in the literature [@62; @84; @97; @98; @206]. In [@13; @55; @57; @58; @59; @60; @110; @204] the authors proved that the Thompson Conjecture is true for alternating simple groups. Finally, Gorshkov (See [@61]) demonstrated the validity of the Thompson Conjecture for the remaining cases $D_4(q)$ and $D_8(q)$. Therefore, the Thompson Conjecture was completely proved and was became a theorem. ## Thompson Problem   Thompson problem posed by Thompson in a letter to the author in 1987. Let $G_1$ and $G_2$ be groups of the same order type (as defined in Definition [**Definition** 2](#1.2){reference-type="ref" reference="1.2"}). Suppose $G_1$ is solvable, is it true that $G_2$ is also necessarily solvable? For groups $G$ of even order, we can not determine the solvability of $G$ by using the order of $G$ (See [@50]). But we may judge it using the same order type of $G$ if the answer of Thompson Conjecture is positive. In particular, Thompson wrote in the letter: I have talked with several mathematicians concerning groups of the same order type. The problem arose initially in the study of algebraic number fields, and is of considerable interest. At the same letter, Thompson provided the following example of nonsolvable groups with the same order type in the letter: $$G_1=2^4:A_7, \hspace{.5cm} G_2=L_3(4):2_2.$$ Both of them are maximal subgroups of $M_{23}$, where "$:$\" denotes semidirect product, we refer to page 72 and page 23 of [@41] for the maximal subgroups of $M_{23}$ and the automorphism group of $L_3(4)$, respectively. The author gave a lecture titled "Thompson Problem and Thompson Conjecture\" at the international conference "Algebra and Mathematical Logic\" held in Russia in September 2011, commemorating the 100th anniversary of V. V. Morozov's birthday. During the conference, the author presented an overview of the aforementioned problem and conjecture. ## Order equation   Let $M_G(n)$ be the set of elements of $G$ with order $n$, the elements of $G$ can be partitioned into the elements of same order, that is, $G=\bigcup M_G(n)$ where $n\in \pi_e(G)$. Since every element of order $n$ must be contained in some cyclic subgroup of order $n$, we have $|M_G(n)|=V_n(G)\phi(n)$, where $V_n(G)$ is the number of cyclic subgroups of order $n$, $\phi(n)$ is Euler totient function. Hence we can obtain the following equation, called $order~equation$ of $G$, $|G|=\bigcup V_n(G)\phi(n)$, $n\in \pi_e(G)$. For convenience, we write order equation of $G$ as $Ord(G)$. Moreover, if two order equations of finite groups $G$ and $H$ are equal, then $\pi_e(G)=\pi_e(H)$ and $V_d(G)=V_d(F)$ for any $d\in \pi_e(G)$. It is easy to see that same order equation is equivalent to the same order type of Thompson Problem. Let $G$ and $H$ are two groups of same order type, it is not hard to prove that if $H$ is nilpotent then $G$ is also nilpotent and if $H$ is supersolvable then $G$ is solvable (See [@203]). Of course, if $H$ is a finite nonabelian simple group, then $G\cong H$ is nonsolvable because Conjecture [**Conjecture** 1](#1.1){reference-type="ref" reference="1.1"} has been proved. It was proved in [@156; @157] that if $H$ is solvable and the prime graph of $H$ is disconnected, then $G$ is solvable. The following results were proved in [@156; @157]: ****Lemma** 35**. *Let $G$ be a finite group and $H$ a Frobenius (resp. $2$-Frobenius) group. If $G$ and $H$ have same order equation, then $G$ is a Frobenius (resp. $2$-Frobenius) group. Moreover, if $H$ is solvable, then $G$ is also solvable.* From Lemma [**Lemma** 35](#4.2){reference-type="ref" reference="4.2"} we deduce the following theorem: ****Theorem** 36**. *Let $G$ and $H$ be any two finite groups of the same order type. If $H$ is solvable and its prime graph is disconnected, then $G$ is solvable.* Note that orders of two elements in a group are equal if they are conjugate. Consequently, "same order class\" is the union of several "conjugacy classes\". Conversely, two elements of the "same order\" might not be "conjugate\". The series of books *Unsolved Problems in Group Theory* records the following question: Suppose that, in a finite group $G$, each two elements of the same order are conjugate. Is then $|G|\leqslant 6$ (see [@100 Problem 7.48])? It was proved in [@49; @52; @191; @215] successively using the classification for finite simple groups (CFSG). ****Problem** 37**. *Without using classification for finite simple groups, can we prove that if each two elements of the same order are conjugate in finite group $G$, then $|G|\leqslant 6$?* Thompson Problem investigates the structure of groups from the "same order\" perspective, while Thompson Conjecture studies groups from the "conjugacy\" perspective. These two have both distinctions and connections. It is well-known that conjugacy classes play a significant role in the study of finite group structure. In [@30] it was introduced that the influence of the conjugacy class size or the number of conjugacy classes on finite group structure. In [@155], $A_5$ was characterized by using the number of elements of the same order, that is, the following theorem was proved: ****Theorem** 38**. *A group $G$ is isomorphic to $A_5$ if and only if the same order type of $G$ is $\{1,15,20,24\}$.* Now denote $nse(G)=\{m_k|k\in \pi_e(G)\}$, where $m_k$ denotes the number of elements of order $k$ in $G$, then the above theorem can be stated as follows: A group $G$ is isomorphic to $A_5$ if and only if $nse(G)=nse(A_5)$. For some alternating simple groups and linear simple groups, there is a lot of literature about characterizing these simple groups using $nse(G)$, see [@11; @15]. The following result was obtained in [@155]: ****Proposition** 39**. *Let $G$ be a group containing more than two elements. If the number of elements with the same order in $G$ is finite, and the maximal number is $s$, then $|G|\leqslant s(s^2-1)$.* Thompson Problem investigates that using "same order\" to judge the solvability of groups. The condition "same order\" cannot be changed to "conjugacy\" here. In fact, we have the following conclusion: ****Theorem** 40** (See [@145]). *There exist two finite groups $G$ and $H$ such that $N(G)=N(H)$, i.e., the set of their conjugacy class sizes are same, but $G$ is solvable and $H$ is nonsolvable.* A related work to Thompson Problem is the study of the solvability of groups using the number of elements of maximal order or the type of this number (See [@38; @48; @77; @87; @88; @89; @90; @91; @189; @207]). # Problems related to quantitative characterization ## Width of order and width of spectrum in finite groups   The fundamental theorem of arithmetic states that every integer $n$ can be represented as $n=p^aq^b\cdots r^c$, where $p$, $q,\cdots, r$ are distinct prime numbers. The prime factorization of numbers is one of the most important results in mathematics. In the following section, we will study the influence of the number of prime divisors on the structure of groups. Note that both the order $|G|$ of a finite group $G$ and the order $|g|$ of an element $g$ are numbers in $G$, let $|\pi(G)|$ and $|\pi(g)|$ denote the number of prime factors of $|G|$ and $|g|$, respectively. These are respectively called the width $\omega_0(G)$ of the order of finite group $G$ and the width of an element $g$. ****Definition** 41**. *Let $G$ be a finite group, we define $\omega_0(G)=|\pi(G)|$ the width of order of $G$ and $\omega_s(G)=max\{|\pi(g)|\big|g\in \pi_e(G)\}$ the width of spectrum of $G$.* For other sets of conjugation invariants of group $G$, the corresponding width can also be defined. In August 2020, at the online conference "Ural Workshop on Group Theory and Combinatorics\" held in Russia, the author gave the first presentation and introduced the aforementioned two definitions of width, and discussed the structure of finite groups with a small width, we refer to [@179] for some results about the "width of order\". This literature points out that groups with $\omega_0(G)=1$, finite $p$-groups, although the orders of these groups have only one prime factor, their structures are quite complex. Recent Chinese literature can be found in [@221; @222]. For the especially "simple\" condition, $\pi_e(G)=\{1,3\}$, but its structure is quite complex, and currently, there is no a detailed classification. When $\omega_0(G)=2$, that is, $|G|=p^aq^b$, it is a special class of solvable groups. In 1904, Burnside provided the first proof of its solvability using representation theory in [@25]. Later, in [@53] Goldschmidt provided a purely group theoretical proof for this theorem when both $p$ and $q$ are odd primes. Then, Bender (See [@16]) and Matsuyama (See [@121]), in 1972 and 1973 respectively, proved the remaining cases, thereby providing an abstract group proof for the $p^aq^b$ theorem. If $\omega_0(G)=3$, then $G$ can be nonsolvable group, and its chief factors are finite simple groups with $\omega_0(G)=3$. In [@80] Herzog gave all $8$ finite simple groups (simple $K_3$-groups) with $\omega_0(G)=3$, they are $A_5$, $L_2(7)$, $L_2(8)$, $A_6$, $L_2(17)$, $L_3(3)$, $U_3(3)$, and $U_4(2)$. Using classification for finite simple groups, simple groups with $\omega_0(G)=4$ (simple $K_4$ groups) were successively given in the papers [@82; @167; @200] [^4] . The further investigation of the simple group with $\omega_0(G)=4$ (simple $K_4$-group) has been discussed in [@24; @223], it is one of the following groups: 1. $A_n$ ($n=7,8,9,10$), $M_{11}$, $M_{12}$, $J_2$; $L_2(q)$ ($q=16,25,49,81,97,243,577$), $L_3(q)$ ($q=4,5,7,8,17$), $L_4(3)$; $O_5(q)$ ($q=4,5,7,9$), $O_7(2)$, $O^+_8(2)$, $G_2(3)$; $U_3(q)$ ($q=4,5,7,8,9$), $U_4(3)$, $U_5(2)$; ${}^3D_4(2)$, ${}^2F_4(2)'$, $Sz(8)$, $Sz(32)$; 2. $L_2(r)$, where $r$ is prime and satisfies the following equation: $r^2-1=2^a3^bu$ for $a\geqslant 1$, $b\geqslant 1$, $u>3$ is prime; 3. $L_2(2^m)$ and the following equations hold: $2^m-1=u$, $2^m+1=3t$ where $m\geqslant 1$, $u$ and $t$ are prime numbers and $t>3$; 4. $L_2(3^m)$ and the following equations hold: $3^m+1=4t$, $3^m-1=2u$ where $m\geqslant 1$, $u$ and $t$ are odd primes. Every group of order $p^aq^b$ is solvable, so the number of simple $K_2$ group is zero. However, the number of simple $K_3$ groups is $8$. The following is a conjecture about the number of simple $K_4$ groups. ****Conjecture** 42**. *There are infinite simple $K_4$ groups.* In particular, the author conjectured that there are infinite prime number $r$ such that $\omega_o(L_2(r))=4$. However, it is difficult to prove this result since it is a special case of the unsolved Dickson conjecture (See [@47]). In the following pairs of numbers $$(x,3x-2),\quad (x,2x+1),\quad (x,4x+1),\quad (x,6x-1),$$ $$(x,6x+1),\quad (x,2x-1),\quad (x,4x-1),\quad (x,8x-1),$$ if it can be proved that for prime number $x$ such that there are infinitely many prime pairs appearing in any of the aforementioned pairs, then the number of simple $K_4$ groups is infinite. Conversely, if for infinite prime numbers $x$, the second position of the above pairs can only appear finite prime numbers, can a contradiction be deduced from this? Simple groups with $\omega_0(G)=5$ and $\omega_0(G)=6$ were studied in [@86; @105]. The following results hold for the width $\omega_0(G)$ of order of $G$: $\omega_0(G)\leqslant |\pi^{''}_e(G)|+3$, and if $\omega_0(G)=|\pi^{''}_e(G)|+3$ then $G$ is simple where $\pi^{''}_e(G)$ denotes the set of composite numbers in the set $\pi_e(G)$ (See Theorem [**Theorem** 10](#2.7){reference-type="ref" reference="2.7"}); $\omega_0(G)\leqslant |\iota(G)|+4$, and if $\omega_0(G)=|\iota(G)|+4$ then $G$ is simple, where $\iota(G)$ denotes the set of $pq$ ($p\neq q$) type numbers in $\pi_e(G)$ (See Theorem [**Theorem** 12](#2.9){reference-type="ref" reference="2.9"}). Similar to Conjecture [**Conjecture** 42](#5.2){reference-type="ref" reference="5.2"}, there also arise the following number theory problems from the classification theorem for finite simple groups: ****Problem** 43**. *Are there infinite simple groups whose order being a square number?* ## Finite groups with a small spectrum width   A group $G$ is called prime power order group (briefly, $\mathop{\mathrm{EPPO}}$-group) if $\omega_s(G)=1$. Moreover, a group $G$ is called prime element group (briefly, $\mathop{\mathrm{EPO}}$-group) if its every nontrivial element has prime order. These groups were studied in [@184; @185]. For solvable $\mathop{\mathrm{EPPO}}$-groups $G$, Shi and Yang proved that $\omega_0(G)=2$ and investigated the structure of chief factor of $G$ (See [@185 Theorem 2.4]); Based on [@188], the structure of nonsolvable $\mathop{\mathrm{EPPO}}$-group was classified in [@185 Theorems 3.1 and 3.2]. In addition, it was proved in [@185 Theorem 2.1] that a solvable $\mathop{\mathrm{EPPO}}$-group $G$ is an $M$-group, that is, every irreducible complex representation of $G$ is a monomial representation. In [@185] the following result was proved for $\mathop{\mathrm{EPPO}}$-group (See [@185 Theorem 1.4]): ****Property** 44**. *Let $G$ be an $\mathop{\mathrm{EPPO}}$-group and $H$ be a subgroup of $G$ such that $(|H|, d)=1$ for a natural number $d>1$. Then $|H|$ divides the number of elements of order $d$ in $G$.* The following result shows that the aforementioned property is a characteristic property of $\mathop{\mathrm{EPPO}}$-groups. ****Theorem** 45** (See [@26]). *Let $G$ be a finite group and let $H<G$ such that $(|H|,d)=1$ where $1\neq d\in \pi_e(G)$. Then $G$ is a $\mathop{\mathrm{EPPO}}$-group if and only if $|H|$ divides the number of elements of order $d$ in $G$.* The study of finite $\mathop{\mathrm{EPPO}}$-groups was extended to the infinite case in [@79; @201]. ****Problem** 46**. *Study the structure of finite groups $G$ with $\omega_s(G)=2$ and $\omega_s(G)=3$.* In [@96] they studied finite groups $G$ with $\omega_s(G)=2$, that is, there exists $pq\in \pi_e(G)$ such that $pqr\not\in \pi_e(G)$ where $p$, $q$, $r$ are different primes. For the special cases, i.e., solvable $\{p,q,r\}$-groups, the following result holds (See [@96 Theorem C]): Let $G$ be a solvable $\{p,q,r\}$-group, $p$, $q$, and $r$ divide $|G|$. If $pqr\not\in \pi_e(G)$ (i.e., $\omega_s(G)\leqslant 2$), then $h(G)\leqslant 21$. Furthermore, if $|G|$ is odd, then $h(G)\leqslant 15$, where $h(G)$ is the Fitting height of $G$. Theorem $C$ of [@96] is a recently obtained result, it can be seen that study the structure of finite groups with $\omega_s(G)=2$ is a rather challenging problem. In [@152] Qian has recently provided the structure of solvable groups whose spectrum width is $3$. Note that, the classification of finite groups with $\omega_s(G)=1$ ($\mathop{\mathrm{EPPO}}$-group) was accomplished based on the work in [@188]. The classification for simple groups might be used to determine the structure of finite groups satisfying $\omega_s(G)=2$ and $\omega_s(G)=3$. ## Generalizations of related problems   Using spectrum to characterize the finite alternating group $A_5$ was a study conducted three to four decades ago. Here are several generalizations from this work. "Spectrum\" is the set of element orders, i.e., it is the set of orders of cyclic subgroups. We can consider the influence of the sets of numbers on group structures, such as: the set of orders of abelian subgroups, the set of orders of nilpotent subgroups, the set of orders of solvable subgroups (See [@19; @43]), the set of orders of the normalizer of Sylow subgroups (See [@18] ), etc. The order of element, the number of elements of the same order, the length of conjugacy classe, and the degree of character, and others, are conjugation invariants in $G$. Different sets of numbers can be constructed from these conjugation invariants. "Characterization\" can be "isomorphic\" or "homomorphic\". From the perspective of the "function $h$\", it can be $h(\Gamma)=1$ or $h(\Gamma)=k$ where $k$ is finite, otherwise, it is called unrecognizable. "Finite simple group\" can be a "simple group\" or a nonsolvable group, such as the automorphism groups of some simple groups, it can also be the direct product of simple groups, for example $Sz(2^7)\times Sz(2^7)$ (See [@124]) and $J_4\times J_4$ (See [@64]). "Spectrum\" represents the research conditions, "characterization\" represents the research methodology, and "finite simple groups\" represent the research subject. Many questions can be raised from various perspectives to conduct research on these three aspects. ## The quantity relationship between the width of order and the width of spectrum   In 1991, the author summarized the early research on characterizing simple groups by using the "order\" and introduced Thompson Problem and Thompson Conjecture in [@166] (see Sections 4.1 and 4.4 of this article). In Section 5 of [@166], the author posed the following question (see [@166 Question 4.3]): Given a positive integer $k$, denote by $|\pi(k)|$ the number of distinct prime divisors of $k$. Let $G$ be a finite group, $n=max\{|\pi(k)|\big | k\in \pi_e(G)\}$, does there exist a function $f$ such that $|\pi(G)|\leqslant f(n)$? In other words, for any finite group, is the width of its order bound by the width of its spectrum? If $\omega_s(G)=n=1$, then $G$ is a finite $\mathop{\mathrm{EPPO}}$-group, from [@185 Theorem 2.4, 3.1 and 3.2] we deduce that $\omega_0(G)=|\pi(G)|\leqslant 4$. For the above question, Zhang [@216] proved that if $G$ is solvable, then the width $\omega_0(G)$ of the order is bounded by a quadratic function of the spectrum width $\omega_s(G)$, that is, $$\omega_0(G)\leqslant \frac{\omega_s(G)(\omega_s(G)+3)}{2}.$$ For general group, $\omega_0(G)$ is bounded by a super-exponential function of $\omega_s(G)$. The above result was subsequently improved in the literature [@94; @95; @144]. Recently, the following theorem was proved in [@81]: ****Theorem** 47**. *Let $G$ be a finite group, then $\omega_0(G)<210\omega_s(G)^4$.* An analogous result regarding the order of a group and its co-degrees can be found in [@210]. It is easy to get that $\omega_s(G)\leqslant \omega_0(G)$ by the definitions of the order width and the spectrum width. The following result holds for all finite simple groups: ****Theorem** 48** (See [@179]). *Let $G$ be a finite simple group, then $\omega_s(G)<\omega_0(G)$.* Naturally, the author put forward a question to investigate the difference between two widths, denoted as $d=\omega_0(G)-\omega_s(G)$. For example, discuss the classification of finite simple groups with $d=1$. # Group, quantity and graph ## Group and quantity   Groups and quantities are closely connected. The group order can be used to determine some important properties of the group, such as the groups of order $15$ are cyclic, the groups of order $p^2$ are abelian, and the groups of order $135$ are nilpotent, and so on. For the solvability of groups, there are many well-known results were obtained in the research history of group theory, such as Feit-Thompson odd order theorem (See [@50]) and Thompson classification theorem for minimal simple group (See [@190]). From these, the following theorem can be obtained: ****Theorem** 49**. *Let $G$ be a finite group. If $(|G|,2)=1$ or $(|G|,15)=1$, then $G$ is solvable.* We refer to *Mathematical Reviews* MR0230809(37$\sharp$`<!-- -->`{=html}6367) for the case of $(|G|,15)=1$ mentioned above. In [@73], the numbers $2$ and $15$ in the theorem above are called "solvable coprime number\", meaning that a group whose order is coprime to $2$ or $15$ is necessarily a solvable group. Moreover, it is obtained in [@73] that all solvable coprime numbers must be multiples of either $2$ or $15$. Compared to the study of finite groups using the order of the group, we characterize finite simple groups using the set $\pi_e(G)$ of element orders. In addition, the set $\pi_e(G)$ of element orders can also be used to judge the solvability of $G$. ****Definition** 50**. *Let $G$ be a finite group and let $\pi_e(G)$ be the set of element orders in $G$. The set of numbers $T$ is called "set of intersection empty solvable\" if $\pi_e(G)\cap T=\varnothing$ implies that $G$ is a solvable group.* ****Theorem** 51** (See [@178]). *Let $G$ be a finite group and let $\pi_e(G)$ be the set of element orders in $G$. If $\pi_e(G)\cap T=\varnothing$ where $T=\{2\}$, $\{3,4\}$ or $\{3,5\}$, then $G$ is solvable. Furthermore, using the set of intersection empty solvable of $\pi_e(G)$ to judge $G$ is solvable or not, noly the above three cases.* ****Problem** 52**. *Let $G$ be a finite group. Is it possible to use $\pi_e(G)$ to provide sufficient conditions for $G$ to be cyclic, abelian, nilpotent, supersolvable, or an $M$-group, etc.?* Compared to the quantity problem of the "order of group\", some similar problems can also be posed by using the "element orders\". For example, Lagrange Theorem of finite group $G$ implies that the order of every subgroup of $G$ divides the order of $G$. However, for any divisor of $|G|$, a subgroup of that order may not necessarily exist. In other words, the converse of Lagrange's theorem does not hold. ****Definition** 53**. *A finite group $G$ is called $\mathop{\mathrm{CLT}}$ (the converse of Lagrange's theorem) group if and only if the converse of Lagrange's theorem holds for $G$.* ****Definition** 54**. *A subset $M$ of a set of positive integers is called a closed subset if, for all $k\in M$, all factors of $k$ also belong to $M$.* Let $G$ be a group and $\pi_e(G)$ be the set of element orders of $G$. It is not all closed subsets of $\pi_e(G)$ can become the set of element orders of a subgroup of $G$. Compared to $\mathop{\mathrm{CLT}}$ group, we introduce the following definition: ****Definition** 55**. *We say that a finite group $G$ is a $\mathop{\mathrm{COE}}$ group if any closed subset $M$ of $\pi_e(G)$, there exists a subgroup $H$ such that $\pi_e(H)=M$.* ****Theorem** 56** (See [@174]). *Suppose $G$ is a $\mathop{\mathrm{COE}}$ group, then $|\pi(G)|\leqslant 3$, and one of the following holds.* - *$G$ is a $p$-group;* - *$|G|=p^aq^b$ where $p\neq q$, $ab\neq 0$, and if the exponent of $G$ is $p^mq^n$ then $G$ has an $\mathop{\mathrm{EPPO}}$-subgroup with an exponent of $p^mq^n$.* - *$|G|=2^a3^b5^c$ where $abc\neq 0$ and the exponent of $G$ is $30$, $60$ or $120$. Furthermore, $G$ has a proper subgroup that is isomorphic to $A_5$, and $G$ is not an $\mathop{\mathrm{EPPO}}$-group.* ## Group and graph are closely interconnected   Gruenberg and Kegel [^5] first introduced the definition of prime graph $\Gamma(G)$ of group $G$ (See Definition [**Definition** 14](#2.11){reference-type="ref" reference="2.11"}) and its classification. Subsequently, the literature [@83; @102; @103; @201] provided explicit descriptions for the connected components of the prime graphs of the finite simple groups. Lucido defined the diameter of the prime graph as follows (See [@111]): ****Definition** 57**. *We say that $$\mathop{\mathrm{diam}}(\Gamma(G))=max\{ d(p,q)|p,q \textit{ in the same connected component of } \Gamma(G)\}$$ is the diameter of the prime graph of a group $G$, where $d(p,q)$ denotes the distance between elements $p$, $q$ if they are in the same connected component of $\Gamma(G)$.* If $G$ is a finite slvable group then $\mathop{\mathrm{diam}}(\Gamma(G))\leqslant 3$. In [@111] Lucido demonstrated that for all finite groups $G$, we have $\mathop{\mathrm{diam}}(\Gamma(G))\leqslant 5$ and characterized almost simple groups with $\mathop{\mathrm{diam}}(\Gamma(G))=5$. Recall that a group $B$ is called an almost simple group if there exists a nonabelian simple group $A$ such that $A\leqslant B\leqslant \mathop{\mathrm{Aut}}(A)$. Finite groups with $\mathop{\mathrm{diam}}(\Gamma(G))=5$ were obtained in [@63]. Note that, when $\mathop{\mathrm{diam}}(\Gamma(G))=1$, the group $G$ is an $\mathop{\mathrm{EPPO}}$-group (See [@185]. So, we pose the following question: ****Problem** 58**. *Study finite groups with $\mathop{\mathrm{diam}}(\Gamma(G))=2$, $3$, $4$.* In the case of solvable groups, the literature [@96; @152] respectively investigated finite solvable groups with $\mathop{\mathrm{diam}}(\Gamma(G))=2$ and $\mathop{\mathrm{diam}}(\Gamma(G))=3$. In [@112] they further discussed the case of a prime graph being a tree based on the work in reference [@111], i.e., a connected graph without loops. The following result was obtained: ****Theorem** 59**. *If the prime graph of a finite group $G$ is a tree, then $|\pi(G)|\leqslant 8$.* In [@112] Lucido also gave an example with $|\pi(G)|= 8$. Compared to the prime graph $\Gamma(G)$ of $G$, the definition of solvable graph was introduced in [@4] as follows: ****Definition** 60**. *Let $G$ be a finite group. We define the solvable graph $\Gamma_s(G)$ of $G$ as follows: its vertices are the primes dividing the order $|G|$ of $G$, and vertices $p$ and $q$ are joined by an edge if and only if there exists a solvable subgroup whose order is divisible by $pq$.* In [@71] the author proved that $\mathop{\mathrm{diam}}(\Gamma_s(G))\leqslant 4$. There is a lot of work in study groups using graphs, Erdös defined non-commuting graph in 1976 (See [@147]). ****Definition** 61**. *Let $G$ be a finite group and let $Z(G)$ be the center of $G$. We define the non-commuting graph $\triangledown(G)$ of $G$ as follows: Take $G\backslash Z(G)$ as the vertices of $\triangledown(G)$ and join two distinct vertices $x$ and $y$ if and only if $xy\neq yx$.* The following problem was conjectured in [@1; @140]: Let $G$ and $H$ be two nonabelian finite groups such that $\triangledown(G)=\triangledown(H)$, then $|G|=|H|$. In [@137] they gave some counterexamples to this conjecture. Moreover, in [@1], the following conjecture was posed, which is called AAM (Abdollahi-Akbari-Maimani) Conjecture: ****Conjecture** 62**. *Let $S$ be a nonabelian simple group and $G$ a group. If $\triangledown(G)=\triangledown(S)$, then $G\cong S$.* It was proved in [@199; @220] that for alternating simple group $A_{10}$ and $L_4(4)$, whose prime graphs are disconnected, Conjecture [**Conjecture** 62](#6.14){reference-type="ref" reference="6.14"} is valid. In addition, the following result was also proved in [@199]: ****Proposition** 63**. *Let $S$ be a nonabelian simple group and let $G$ be a group with $Z(G)=1$. If $\triangledown(G)=\triangledown(S)$, then $N(G)=N(S)$.* Because Thompson Conjecture has been proved (See sections 4.1 and 4.2), consequently, AAM Conjecture is correct. ****Definition** 64** (OD-characterization of simple group). *Let $G$ be a finite group and $|G|=p^aq^b\cdots r^c$, where $p<q<\cdots <r$ are primes, and $a$, $b\cdots, c$ are integers. For $s\in \pi(G)$, let $deg(s)=|\{t\in \pi(G)|s\sim t\}|$, which we call the degree of $s$. We also define $D(G)=(deg(p), deg(q),\cdots, deg(r))$, we call $D(G)$ the degree pattern of $G$. A group $M$ is called $k$-fold OD-characterizable if there exist exactly $k$ non-isomorphic groups $G$ such that $|G|=|M|$ and $D(G)=D(M)$. Moreover, a $1$-fold OD-characterizable group is simply called an OD-characterizable group.* Extensive research has been conducted by A. R. Moghaddamfar, L. C. Zhang, et al. on $OD$-characterizable ($k$-fold $OD$-characterizable) simple group (almost simple group) (See [@12; @141; @142; @143; @218; @219]). ## Adjacency criterion for the vertices of the prime graph (GK graph) of a finite simple group   Both the characterization of finite simple groups by using the set of element orders $\pi_e(G)$ and the aforementioned $OD$-characterization of finite simple groups are inseparable from the study of the prime graph of groups (GK graph) (See Definition [**Definition** 14](#2.11){reference-type="ref" reference="2.11"}). Therefore, studying the adjacency of the vertices in prime graph is a very important subject. The literature [@197; @198] gave the adjacency criterion for the vertices of prime graphs (GK graph) associated with each nonabelian simple group. Especially in the case of vertex adjacency to vertex "2\" and in the case of "independent sets\" in which vertices are not adjacent to each other. The literature [@197; @198] provided a large number of graphs and information for all simple groups in tabular form. Reference [@198] is a continuation of reference [@197], and it includes corrections to some content in reference [@197]. ## Unrecognizable finite group by prime graph   Finite simple groups $E_6(2)$, $E_6(3)$ and ${}^2E_6(3)$ were characterized by its prime graph $\Gamma(G)$ in [@69; @99]. Clearly, there are many simple groups that cannot be only characterized using prime graphs. Thus, in [@29] they studied the criteria for finite groups that cannot be characterized using prime graphs and they aimed to study the following questions: 1. Which groups $G$ are uniquely determined by their prime graph $\Gamma(G)$? 2. For which groups are there only finitely many groups with the same prime graph $\Gamma(G)$ as $G$? 3. Which groups $G$ are uniquely determined by isomorphism type of their prime graph? There are differences between having the same prime graph and having an isomorphic prime graph. For example, $\Gamma(A_{10})\neq\Gamma(\mathop{\mathrm{Aut}}(J_2))$ but $\Gamma(A_{10})$ and $\Gamma(\mathop{\mathrm{Aut}}(J_2))$ are isomorphic as abstract graphs. ****Problem** 65**. *Which simple groups can be characterized by their prime graphs and orders?* The above question is a generalization of the work that characterizes all simple groups using "two orders\". Consider the classical groups $B_n(q)$ and $C_n(q)$ (See [@176]), when $n\geqslant 3$ and $q$ is odd, their prime graphs and orders are same. Therefore, not all simple groups can be characterized by their prime graphs and orders. ## Graphs defined on groups   There are various graphs defined on a group, but the one most closely associated with quantitative characterization is the prime graph (GK graph). Cameron in [@27] introduced the following graphs: Vertex set of graphs is a group $G$ and whose edges reflect the structure of $G$ in some way, so that the automorphism group of $G$ acts as automorphisms of the graph. These include the commuting graph (two vertices $x$ and $y$ are joined if and only if $xy=yx$; this graph was first studied in 1955, see [@22]), the generating graph ($x$ and $y$ joined if and only if $\langle x, y\rangle=G$; the generating graph was first introduced in 1996, see reference [@23; @109]), the power graph (vertex $x$ is joined to vertex $y$ if and only if one of $x$ and $y$ is a power of the other; this graph was first studied in 2000, see [@93; @193]), the enhanced power graph (two vertices $x$ and $y$ are joined if and only if $\langle x, y\rangle$ is not cyclic; this graph was first studied in 2007, see [@2; @3]) and the deep commuting graph (two elements of $G$ are joined in the deep commuting graph if and only if their preimages in every central extension of $G$ (that is, every group $H$ with a central subgroup $Z$ such that $H/Z\cong G$) commute, see [@28]). This paper mainly discusses the quantitative characterization of finite simple groups, which is an essential component of quantitative properties of groups and a significant topic of group theory research. Studying finite groups using the set of element orders and characterizing finite simple groups by "two orders\" were first proposed and investigated by the author, which have a significant influence on the field of group theory at home and abroad. Since the 1980s, the author has successively worked at Southwest Normal University (including a concurrent doctoral supervisor position at Sichuan University), Soochow University, and Chongqing University of Arts and Sciences. Additionally, the author has received funding for 11 projects from the National Natural Science Foundation of China, enabling continuous progress in this work. Our intention in writing this article is not only to systematically summarize our work, distill new viewpoints, ideas, and methods to solve related problems, but also to further delve into this work and find more significant applications. # Acknowledgement {#acknowledgement .unnumbered} We sincerely thank the reviewers for their valuable suggestions for revisions. Professors Ping Jin and Guohua Qian provided many valuable insights into the writing and specific content of this paper. Drs. Nanying Yang and Rulin Shen also offered their revision suggestions for this paper. In particular, Dr. Jinbao Li dedicated a considerable amount of effort to the editing and revising of this paper. The author would like to extend his heartfelt gratitude to all of them for their contributions. 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G. dated January 4, 1988 to the author posed this conjecture, which is also introduced in Khukhro E I, Mazurov V D. *Unsolved problems in group theory.* The Kourovka notebook, No. 20. 2022, 12.38, p.59. [^3]: Yan Q F, Shen Z C, Zhang J S, et al. A new characteristic of sporadic simple groups $J_1$ and $J_4$. Submitted [^4]: Reference [@200] contains conclusions without specific proofs, and there are omissions (such as $L_3(8)$ and $U_3(7)$) and repetitions (for instance $O_7(2)$ and $S_6(2)$ are the same group) in some places, furthermore, Suzuki simple groups can be completely determined. [^5]: Gruenberg K W, Kegel O H. Unpublished manuscript, 1975

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